From 387985b0a7c15bec3112679f532efab10c0aff80 Mon Sep 17 00:00:00 2001 From: Batson Iii <3tv@mac107054.ornl.gov> Date: Tue, 19 Jan 2021 14:14:38 -0500 Subject: [PATCH] Docs minus Keno --- .DS_Store | Bin 16388 -> 22532 bytes AMPXUtil.rst | 132 + CENTRM.rst | 2 +- COMPOZ.rst | 251 + COVLIB.rst | 3258 +++ COVLIBAppA.rst | 99 + CRAWDAD.rst | 2 +- ..., and Spent Fuel Source Terms Overview.rst | 106 + Deterministic Transport Intro.rst | 116 + FIDO.rst | 344 + ICE.rst | 295 + MALOCS2.rst | 210 + MCDancoff.rst | 5 +- NEWT.rst | 5307 +++++ Nuclear Data Libraries Overview.rst | 40 + ORIGAMI.rst | 3072 +++ ORIGEN-Data.rst | 3468 +++ ORIGEN.rst | 5418 +++++ Origenutil.rst | 568 + PMC.rst | 37 +- Polaris.rst | 3360 ++- PolarisA.rst | 1448 +- SLIG.rst | 664 + ...vity and Uncertainty Analysis Overview.rst | 278 + TRITON.rst | 5697 +++++ TRITONAppABC.rst | 1711 ++ UtilOverview.rst | 24 + WORKER.rst | 424 + XSD.rst | 3563 +++ XSDAppAB.rst | 421 + XSLib.rst | 2795 +++ XSLibAppA.rst | 1032 + XSProc.rst 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Wiarda, L. M. Petrie* + +Abstract + +The purpose of this section is to document selected AMPX modules that +can benefit the analyst interested in editing, converting, or combining +cross-section libraries normally used by the SCALE system modules. The +input description for these codes is provided in the documentation of +the AMPX nuclear data processing code system that is distributed with +SCALE package. + +.. _11-1-1: + +Introduction +------------ + +AMPX is a modular system [1]_ that generates continuous energy (CE) and +multigroup (MG) cross section data from evaluated nuclear data files +such as ENDF/B. All the nuclear data libraries distributed with SCALE +have been processed using AMPX. In addition to data processing modules, +AMPX also includes a number of useful utility modules for checking, +manipulating, and editing the libraries in SCALE. This section lists and +briefly describes some of the AMPX utility codes that may be useful to +SCALE users. Input instructions for these codes can be found the AMPX +code documentation, which is distributed with the SCALE code package. +Additional AMPX modules of interest may also found in the documentation. + +.. _11-1-2: + +AJAX: MODULE TO MERGE, COLLECT, ASSEMBLE, REORDER, JOIN, AND/OR COPY SELECTED DATA FROM AMPX MASTER LIBRARIES +------------------------------------------------------------------------------------------------------------- + +AJAX (**A**\ utomatic **J**\ oining of **A**\ MPX **X**-Sections) is a +module to combine data from different AMPX libraries. Options are +provided to allow merging from any number of files. + +.. _11-1-3: + +ALPO: MODULE TO CONVERT AMPX LIBRARIES INTO ANISN FORMAT +-------------------------------------------------------- + +ALPO (**A**\ NISN **L**\ IBRARY **P**\ RODUCTION **O**\ PTION) is a +module for converting AMPX working libraries into the library format +used by the legacy discrete ordinates transport codes ANISN and +DORT/TORT contained in the DOORS package. [2]_ + +.. _11-1-4: + +CADILLAC: MODULE TO MERGE MULTIPLE COVARIANCE DATA FILES +-------------------------------------------------------- + +CADILLAC (**C**\ ombine **A**\ ll **D**\ ata **I**\ dentifiers +**L**\ isted in **L**\ ogical **A**\ MPX **C**\ overx-format) is a +module that can be used to combine multiple covariance data files in +COVERX format into a single covariance data file. The material IDs can +be changed as needed by the user. + +.. _11-1-5: + +COGNAC: MODULE TO CONVERT COVARIANCE DATA FILES IN COVERX FORMAT +---------------------------------------------------------------- + +COGNAC (**C**\ onversion **O**\ perations for **G**\ roup-dependent +**N**\ uclides in **A**\ MPX **C**\ overx-format) is a module that can +be used to convert a single COVERX-formatted data file from bcd format +to binary. Also, COGNAC can be used to convert from binary to bcd, +binary to binary, and bcd to bcd. + +.. _11-1-6: + +LAVA: MODULE TO MAKE AN AMPX WORKING LIBRARY FROM AN ANISN LIBRARY +------------------------------------------------------------------ + +LAVA (**L**\ et **A**\ NISN **V**\ isit **A**\ MPX) is a module that can +convert an ANISN formatted library (neutron, gamma, or coupled +neutron-gamma) to an AMPX working library that can be used in XSDRNPM. + +.. _11-1-7: + +MALOCS: MODULE TO COLLAPSE AMPX MASTER CROSS-SECTION LIBRARIES +-------------------------------------------------------------- + +MALOCS (**M**\ iniature **A**\ MPX **L**\ ibrary **O**\ f **C**\ ross +**S**\ ections) is a module to collapse AMPX master cross-section +libraries. The module can be used to collapse neutron, gamma-ray, or +coupled neutron-gamma master libraries. + +.. _11-1-8: + +PALEALE: MODULE TO LIST INFORMATION FROM AMPX LIBRARIES +-------------------------------------------------------- + +PALEALE lists selected data by nuclide, reaction, data-type from AMPX +master and working libraries. + +.. _11-1-9: + +RADE: MODULE TO CHECK AMPX CROSS-SECTION LIBRARIES +-------------------------------------------------- + +RADE (**R**\ ancid **A**\ MPX **D**\ ata **E**\ xposer) is provided to +check AMPX- and ANISN-formatted multigroup libraries. It will check +neutron, gamma, or coupled neutron-gamma libraries. + +.. _11-1-10: + +TOC: MODULE TO PRINT AN AMPX LIBRARY TABLE OF CONTENTS +------------------------------------------------------ + +Program TOC is a utility program to print a sorted table of contents of +an AMPX cross section library. It is designed to be run interactively, +with the cross section library specified as the argument. + +References +~~~~~~~~~~ + +.. [1] + D. Wiarda, M. L. Williams, C. Celik, and M. E. Dunn, “AMPX: A + Modern Cross Section Processing System For Generating Nuclear Data + Libraries,” *Proceedings of International Conference on Nuclear + Criticality Safety,* Charlotte, NC, Sept 13-17 2015. + +.. [2] + **“**\ \ DOORS3.2a:   One, Two- and Three-Dimensional Discrete + Ordinates Neutron/Photon Transport Code System”, Radiation Shielding + Information Center package CCC-650, Oak Ridge National Laboratory + (2003). diff --git a/CENTRM.rst b/CENTRM.rst index 9a97f1e..7b33d84 100644 --- a/CENTRM.rst +++ b/CENTRM.rst @@ -848,7 +848,7 @@ The limits on the above integral correspond to: \alpha_{\mathrm{L}}\left(\mathrm{E}^{\prime}, \mathrm{E}\right)=\alpha\left(\mathrm{E}^{\prime}, \mathrm{E}, \mu_{0}=-1\right) \quad \text { and } \quad \alpha_{\mathrm{H}}\left(\mathrm{E}^{\prime}, \mathrm{E}\right)=\alpha\left(\mathrm{E}^{\prime}, \mathrm{E}, \mu_{0}=1\right) . The alpha moments for n > 0 can be evaluated very efficiently using a -recursive relation :cite:`illiams_submoment_2000`: +recursive relation :cite:`williams_submoment_2000`: .. math:: diff --git a/COMPOZ.rst b/COMPOZ.rst new file mode 100644 index 0000000..afdc3a5 --- /dev/null +++ b/COMPOZ.rst @@ -0,0 +1,251 @@ +.. _11-3: + +COMPOZ Data Guide +================= + +*J. R. Knight* [1]_ *and L. M. Petrie*  + +ABSTRACT + +The COMPOZ program used to create the Standard Composition Library is +described. Of particular importance is documentation of the COMPOZ input +data file structure. Knowledge of the file structure allows users to +edit the data file and subsequently create their own site-specific +composition library. + + +ACKNOWLEDGMENT + +This work was originally funded by the Office of Nuclear Material Safety +and Safeguards of the U.S. Nuclear Regulatory Commission. + +.. _11-3-1: + +Introduction +------------ + +COMPOZ is the program that creates (writes) the SCALE Standard +Composition Library. Data are input in free form. A text data file +containing the input to COMPOZ (and the Standard Composition Library) is +available with the SCALE system. Execution of COMPOZ using this data +file creates the Standard Composition Library currently available with +the SCALE package. This section provides documentation of the data file +structure. Knowledge of the data file structure allows users to edit the +data file and subsequently create their own site-specific or +user-specific composition library. + +COMPOZ is intended to create or make *permanent* *changes* to and/or to +print the composition library and should not be used for any other +purpose. To avoid confusion with the Standard Composition Library +provided with SCALE, it is strongly recommended that only *new* keywords +and compositions be used in any site-specific or user-specific library. + +.. _11-3-2: + +Input Data Description +---------------------- + +COMPOZ input data are entered in free form. All data must be followed by +at least one blank. The COMPOZ input data file contains *five* data +records or blocks: + +1. COMPOZ mode flag selects whether a new standard composition library + will be created, or an old standard composition library will be + listed. Only if a new library is being created are the following data + records entered. A new library is created with a filename of + “xfile089”. If an old library is being dumped as an ASCII file, it + will be named “_sclN…N” where N…N is an 18 digit sequence number that + is incremented starting from 0 for each library dumped in the same + directory. + +2. The header record contains the library identification, a set of + parameters describing the size of the library, and library title with + 80 characters per line. + +3. The standard composition table contains the name, theoretical + density, number of elements, and other information about each + standard composition. Individual nuclides, mixtures, and compounds + are all included in the table. + +4. The nuclide information table contains the nuclide identification + number, atomic mass, and resonance energy cross sections. + +5. The isotopic distribution table contains the nuclide identification + number and the atom percent of each isotope used in specifying the + default enrichment. + +.. note:: For executing COMPOZ via SCALE, an =COMPOZ is required in the + first eight columns of a record preceding the mode flag, and an END is + required in the first three columns of a record inserted after the last + data record. If debug output is desired, then use =COMPOZ PRINTDEBUG to + execute compoz. + +.. _11-3-2-1: + +COMPOZ mode selector +~~~~~~~~~~~~~~~~~~~~ + +1. LGEN = + + 0 – create a new library and list it + + 1 – list an existing library + + >1 – list an existing library and write an ASCII input file. + +If LGEN is 0, then input the following data to create a new standard +composition library. + +.. _11-3-2-2: + +Library heading information +~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +1. IDT – library identification number + +2. TITLE – 1 line of 80 characters used to identify the library + +.. _11-3-2-3: + +Standard composition table +~~~~~~~~~~~~~~~~~~~~~~~~~~ + +1. SCID – Composition name, maximum of 12 characters. + +2. ROTH – Theoretical density, gm/cm\ :sup:`3`. + +3. ICP – + + 0 for a mixture, + + 1 for a compound. + +4. NCZA – Element or nuclide ID + +5. ATPM – Weight percent if ICP = 0. Number of atoms per molecule if ICP = 1. + +6. END – Keyword END to terminate this standard composition. + +For each composition, items 4 and 5 are repeated until all components of +the composition are described. Items 1 through 6 are entered in a +similar fashion for all compositions. After all the standard +compositions are read, terminate the table with an END [label], where +[label] represents an optional label. + +.. _11-3-2-4: + +Nuclide information table +~~~~~~~~~~~~~~~~~~~~~~~~~ + +1. NZA – Nuclide ID. This should be the mass number + 1000 \* the atomic +number. + +2. AM – Atomic mass, C-12 scale. + +3. SIGS – Resonance energy scattering cross section, barns. + +4. SIGT – Resonance energy total cross section, barns. + +5. NU*SIGF – Resonance energy nu*sigf cross section, barns. + +The resonance energy cross sections are averaged over the appropriate +energy range for the nuclide. Items 1–5 are repeated for all nuclides. +After all nuclides are entered, terminate the nuclide table with an END +[label]. + +.. _11-3-2-5: + +Isotopic distribution table +~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +1. NZN – 1000 \* atomic number of variable isotope elements. + +2. ISZA – Isotope ID. + +3. ABWP – Default abundance, atom percent. + +4. END – Keyword END to terminate this isotopic specification. + +The default abundance is generally the naturally occurring abundance. +For each element, items 2 and 3 are repeated until 100% total abundance +is described, making a set for this element. The next element is +described in the same fashion in the next set, etc. After all isotopic +distributions are entered, terminate the isotopic distribution table +with an END [label]. + +.. _11-3-3: + +Sample Problem +-------------- + +The following sample problem first lists the SCALE standard composition +library, then creates a new, short standard composition library, then +lists and outputs an ASCII copy of this new library, and finally copies +this new copy back to the output directory. + +.. code:: scale + :class: long + + =compoz + ' print the current standard composition library + 1 + end + =compoz + ' create a new standard composition library + 0 + ' library identification number + 101 + ' library title + scale-X standard composition library + ' standard composition table + ' all nuclide IDs here must be in the nuclide table + h 1.0000 0 1001 100.0000 end + o 1.0000 0 8016 100.0000 end + u 19.0500 0 92000 100.0000 end + h2o 0.9982 1 1001 2 + 8016 1 end + uo2 10.9600 1 92000 1 + 8016 2 end + ' end of standard composition table + end stdcmp + ' nuclide table + ' ID AWR SigmaS SigmaT nuSigmaF + 1001 1.00783 20.38087 20.38782 0.00000 + 1002 2.01410 3.39486 3.39487 0.00000 + 8016 15.99491 3.88696 3.88696 0.00000 + 8017 16.99913 3.74000 3.74501 0.00000 + 8018 17.99916 3.79000 3.79000 0.00000 + 92233 233.03964 12.46693 37.62292 100.78482 + 92234 234.04095 12.18716 16.09542 2.66969 + 92235 235.04393 11.90249 35.22383 90.23152 + 92236 236.04556 12.27302 14.93351 1.33334 + 92237 237.04874 14.24581 24.68619 1.93695 + 92238 238.05080 12.32636 14.62708 0.65970 + ' end of nuclide table + end nuclides + ' isotope distribution table + ' all nuclide IDs here must be in the nuclide table + 1000 1001 99.9885 + 1002 0.0115 end + 8000 8016 99.7570 + 8017 0.0380 + 8018 0.2050 end + 92000 92234 0.0054 + 92235 0.7204 + 92238 99.2742 end + ' end of isotope distribution table + end isotopes + ' end of compoz input + end + =compoz + ' print and create an ASCII copy of the current standard composition library + ' (created in the previous step) + 2 + end + =shell + # copy the ASCII copy back to the output directory + copy_file _scl000000000000000000 ${OUTBASE}.stdcmplib + end + +.. [1] + Formerly with Oak Ridge National Laboratory. diff --git a/COVLIB.rst b/COVLIB.rst new file mode 100644 index 0000000..dff6830 --- /dev/null +++ b/COVLIB.rst @@ -0,0 +1,3258 @@ +.. _10-2: + +SCALE Nuclear Data Covariance Library +===================================== + +*M. L. Williams, D. Wiarda, G. Arbanas, and B. L. Broadhead* + +ABSTRACT + +An updated cross section covariance library has been created for use +with the sensitivity and uncertainty modules in SCALE 6.2. The data has +been assembled from a variety sources, including high-fidelity +covariance evaluations from ENDF/B-VII.1 as well as approximate +uncertainties obtained from a collaborative project performed by +Brookhaven National Laboratory, Los Alamos National Laboratory, and Oak +Ridge National Laboratory. This document describes the assumptions in +generating the data, the library contents, and processing procedure for +the SCALE 56-group and 252-group covariance libraries. The SCALE +44-group covariance library distributed with SCALE 6.0 and SCALE 6.1 is +retained for backwards compatibility. + +ACKNOWLEDGMENT + +We gratefully acknowledge the sponsorship of the US Department of Energy +Nuclear Criticality Safety Program in the development of the SCALE 6.2 +covariance libraries. + +.. _10-2-1: + +Introduction +------------ + +The SCALE 6.2 covariance library is based on available ENDF/B-VII.1 :cite:`chadwick_endfb-vii_2011` +data for 187 nuclides, combined with the previous SCALE 6.1 covariance +data are retained for the ~215 nuclides not available in ENDF/B‑VII.1. +The ENDF/B-VII.1 uncertainties were modified for a few nuclides, as +described in :ref:`10-2-2-3`. In addition, the covariance library now has +a 56-group structure for broad group analysis, as well as the 252-group +structure for fine-group analysis. These covariance libraries were +generated for compatibility with the ENDF/B-VII.1 cross section +libraries distributed with SCALE 6.2, and they may also be applied for +the 238-group ENDF/B-VII.0 library. The previous SCALE 6.0 and SCALE 6.1 +44‑group library (44groupcov) was based on older covariance data and is +retained in SCALE 6.2 for backwards compatibility. However, the 56- and +252-group covariance libraries (56groupcov7.1 and 252groupcov7.1) are +now recommended for all applications. The 56-group library—which is +default for SCALE uncertainty analysis—and the 252 fine-group library +generally produce similar results, except for some threshold reactions +such as (n,2n). The 252-group library may be used to improve uncertainty +estimates from these types of data, but it typically takes more +execution time than the default 56-group library. Because the 56- and +252-group covariance data in many cases are based on newer uncertainty +evaluations than the previous 44-group library, some differences will +occur between these sets of results. + +The covariance data correspond to relative uncertainties assembled from +a variety of sources, including evaluations from ENDF/B-VII.1, +ENDF/B-VI, and approximated uncertainties from a collaborative project +performed by Brookhaven National Laboratory (BNL), Los Alamos National +Laboratory (LANL), and Oak Ridge National Laboratory (ORNL). Because +SCALE uncertainty data come from several different sources, the +application of a single generic covariance library to all multigroup +cross section libraries raises questions about consistency with any +given data evaluation. In reality, much of the approximate uncertainty +data in the library is based on simplifying approximations that do not +depend on specific ENDF evaluations and thus can be applied to all cross +section libraries within the limitations of the assumed methodology. In +other cases in which a covariance evaluation has been taken from a +specific nuclear data file (e.g., ENDF/B-VII.1, ENDF/B-VI, or JENDL), it +is assumed that the same *relative* (rather than *absolute*) +uncertainties can be applied to all cross section libraries, even if +these are not strictly consistent with the nuclear data evaluations. The +assumption is partially justified by the fact that different evaluations +often use many of the same experimental measurements since there is a +limited amount of this information available. In some cases, older data +evaluations have been carried over into the newer ENDF versions. Also, +because many important nuclear data are now known rather well, newer +evaluations in many instances correspond to rather modest variations +from previous ones and are expected to lie within the earlier +uncertainties. As shown by plots in :ref:`11-3a`, the nuclear data +evaluations from ENDF/B-VII, ENDF/B-VI, JEF-3.1, and JENDL-3.3 tend to +agree well for many types of cross sections, so it is reasonable to +assume that the uncertainties in these data are similar. + +No inherently “true” uncertainty can be defined for nuclear data. For +example, in theory, two independent evaluations could produce similar +nuclear data with very different uncertainties. Differences in nuclear +data evaluations directly impact calculations that can be affirmed by +comparisons with benchmark experiments; but there is no established +procedure to quantify the reliability of uncertainty estimates. In +general, the SCALE covariance library should be viewed as a +best-estimate assessment of data uncertainties based upon the specific +methodology described in the following section. While this methodology +is not unique and other approaches could have been used, the SCALE +covariance library is a reasonable representation of the nuclear data +uncertainties for most applications given the current lack of +information. Furthermore, it is the only available comprehensive library +that has been created in a well-defined, systematic manner. + +.. _10-2-2: + +Covariance Data Description +--------------------------- + +.. _10-2-2-1: + +Evaluated covariances from nuclear data files +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +A rigorous, modern evaluation of nuclear data typically uses a +regression algorithm that adjusts parameters in a nuclear physics model +(e.g., Reich-Moore resonance formula, optical model, etc.), to fit a set +of differential experimental measurements that have various sources of +statistical and systematic uncertainties :cite:`larson_systematic_2006`. Information from the +regression analysis of the model parameters can be propagated to +uncertainties and correlations in the evaluated differential data. In +this manner, the differential nuclear data and covariances are +consistent and are coupled together by evaluation processes. +Unfortunately, only a limited number of cross section evaluations have +produced high-fidelity covariances in this rigorous manner. All other +nuclear data uncertainties must be estimated from approximations in +which the uncertainty assessment is decoupled from the original +evaluation procedure. + +The SCALE covariance library is based on several different uncertainty +approximations with varying degrees of fidelity relative to the actual +nuclear data evaluation. The library includes high-fidelity evaluated +covariances obtained from ENDF/B-VII.1, and ENDF/B-VI whenever +available. As discussed in :ref:`10-2-1`, it is assumed that covariances +taken from one data evaluation, such as ENDF/B-VI, can also be applied +to other evaluations of the same data, such as ENDF/B-VII.1. If this is +done judiciously for cases in which the nuclear data evaluations are +similar, then the covariances taken from one source should be a +reasonable representation of uncertainties for the other evaluations. + +.. _10-2-2-2: + +Approximate covariance data +~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +At the other end of the spectrum from high fidelity data, low-fidelity +(lo-fi) covariances are defined to be those estimated independently of a +specific data evaluation. The approximate covariance data in SCALE are +based on results from a collaborative project funded by the US +Department of Energy Nuclear Criticality Safety Program to generate +lo-fi covariances over the energy range from 10\ :sup:`-5` eV to 20 MeV +for materials without covariances in ENDF/B-VII.1. Nuclear data experts +at BNL, LANL, and ORNL devised simple procedures to estimate data +uncertainties in the absence of high fidelity covariance evaluations. +The result of this project is a set of covariance data in ENDF/B file 33 +format that can be processed into multigroup covariances :cite:`little_low-fidelity_2008`. Some of +these data were later revised and included in ENDF/B‑VII.1, while others +were carried over from SCALE 6.1 to the SCALE 6.2 library. In this +documentation, these data are known as BLO (BNL-LANL-ORNL) uncertainty +data, which were generated as described below. + +ORNL used uncertainties in integral experiment measurements of thermal +cross sections, resonance integrals, and potential cross sections to +approximate the standard deviations of capture, fission, and elastic +scattering reactions for the thermal (<0.5 eV) and resonance ranges (0.5 +eV- 5 keV). Full energy correlation was assumed for the covariances +within each of these respective ranges :cite:`williams_approximate_2007,williams_scale-6_2008` This +procedure was originally introduced for the approximate uncertainty data +in SCALE 5.1. However, the current version includes updated integral +measurement uncertainties, using the more recent values tabulated by +Mughabghab in the *Atlas of Neutron Resonances* :cite:`mughabghab_atlas_2006`. The lo-fi relative +uncertainty is computed as the absolute uncertainty in the integral +parameter (i.e., thermal cross section or resonance integral) taken from +the *Atlas*, divided by the average of the measured parameter and the +calculated value computed from ENDF/B-VII differential data: + +.. math:: + :label: eq10-2-1 + + \mathrm{U}=\frac{\Delta_{\mathrm{I}}}{0.5 \times\left(\mathrm{X}_{\mathrm{I}}+\mathrm{X}_{\mathrm{D}}\right)} , + +where: + + U is the relative lo-fi uncertainty included in SCALE, + + Δ\ :sub:`I` is the absolute uncertainty in the integral measurement + (obtained from Mughabghab), and + + X\ :sub:`I` and X\ :sub:`D` are the measured and computed (from + ENDF/B differential data) integral parameter values, respectively. + +In some cases the integral measurement value from the Mughabghab +*Atlas*\ :sup:`6` and the corresponding value computed from the +ENDF/B-VII differential evaluation are inconsistent—defined here as +having a difference greater than two standard deviations in the measured +and computed integral parameters. In these cases, the lo-fi relative +standard deviation is defined as half the difference relative to the +average of the measured and calculated values: + +.. math:: + :label: eq10-2-2 + + \mathrm{U}=\frac{\left|\mathrm{X}_{\mathrm{I}}-\mathrm{X}_{\mathrm{D}}\right|}{\mathrm{X}_{\mathrm{I}}+\mathrm{X}_{\mathrm{D}}} ; \text { for }\left|\mathrm{X}_{\mathrm{I}}-\mathrm{X}_{\mathrm{D}}\right|>2 \Delta_{\mathrm{I}} . + +In some instances this expression may exceed 100%. For these cases, a +100% uncertainty was assigned. Also, the *Atlas* does not include +uncertainties in integral measurements for several isotopes, which +typically are not of great interest for most applications. In this case +the integral uncertainty was defined as a +/-5 in the least significant +digit for these materials; e.g., 1.23 is assign an uncertainty of +/- +5E-3. + +BNL and LANL provided estimates in the fast energy range from 5 keV to +20 MeV for covariances of capture, fission, elastic, inelastic, (n,2n) +cross sections, and prompt nubar. BNL used optical model calculations +with estimated uncertainties in model parameters to compute covariances +in the fast range for about 300 structural isotopes, fission products, +and non-fissionable heavy nuclei. Estimated uncertainties in model +parameters were based on previous work and expert judgment :cite:`pigni_extensive_2009`. +Covariances for 14 actinide isotopes were obtained from earlier work +performed by BNL for Subgroup-26 (SG-26) :cite:`rochman_preliminary_2007`. The SG-26 actinide +covariances cover the full energy range, including thermal, resonance, +and fast regions. If the thermal data uncertainties estimated by the +SG-26 approach exceed the thermal uncertainty given in reference 6, the +thermal data covariances are represented by ORNL’s integral uncertainty +technique. + +LANL produced covariances in the fast range for an additional 47 +actinide materials. The LANL actinide covariances were based on +empirical estimates of nuclear reaction models :cite:`kawano_evaluation_2008`. Full energy range +covariances were also produced by LANL for 16 light isotopes ranging +from hydrogen to fluorine :cite:`hale_covariances_2008`. These included high fidelity +covariances from R-matrix analyses for :sup:`1`\ H, :sup:`6`\ Li, and +:sup:`10`\ B, along with lo-fi uncertainties for the other materials, +based on approximations such as least-squares fitting to experimental +data, statistical model calculations at higher energies, or sometimes +simply best-judgment estimation :cite:`little_low-fidelity_2008`. + +.. _10-2-2-3: + +Modifications to covariance data +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +In generating earlier covariance libraries, some omissions or +inconsistencies were identified and corrected in the current covariance +library: + +- If the absolute correlation is larger than 1, it is set to 1. + +- If a relative uncertainty is larger than 1, it is set to 1. + +- If cross section data exist but covariance data do not span the + entire range, then the diagonal element for the higher energy groups + is repeated for the lower energy groups. + +- If total inelastic scattering covariance is not supplied, it is + calculated from the uncertainties in the discrete level inelastic + data. + +- If total nubar covariance is not supplied, it is calculated from the + the prompt and delayed nubar uncertainties + +A few inconsistencies were found in the ENDF/B-VII.1 uncertainty data, +and these were modified for the SCALE 6.2 covariance library :cite:`williams_applications_2014`. The +corrections were also conveyed to the National Nuclear Data Center, +where they were added to the ENDF/A file for possible inclusion in the +future release of ENDF/B-VII.2. These modifications are summarized +below: + +(a) :sup:`235`\ U thermal nubar: standard deviation was decreased from + 0.7% to 0.3% in energy range from 0.0 to 0.5 eV, consistent with + JENDL-3.3. + +(b) :sup:`239`\ Pu thermal nubar: standard deviation was increased from + 0.01% to 0.15% in energy range from 0.0 to 0.01 eV, consistent with + ENDF/B-VII.1 uncertainty at 0.01 eV. + +(c) H thermal capture: standard deviation reduced from 2.5% to 0.2%, + consistent with Williams and Rearden 2008 :cite:`williams_scale-6_2008`, + +(d) :sup:`103`\ Rh thermal capture: reduced from ~4% to 1.04%, +consistent with Williams and Rearden 2008 :cite:`williams_scale-6_2008`. + +(e) :sup:`151`\ Sm thermal capture: modified to ~1.8%, consistent with +Williams and Rearden 2008 :cite:`williams_scale-6_2008`. + +(f) :sup:`147`\ Pm: standard deviation was reduced from 24% to 5% in the +energy range 0.5–5000 eV, consistent with the quoted resonance integral +uncertainty in Williams and Rearden 2008 :cite:`williams_scale-6_2008`. + +Several modifications were also made to the uncertainties obtained from +the original BLO data used in SCALE 6.1. The energy boundary between the +thermal and resonance covariance blocks was modified from 0.5 to 0.625 +eV in order to coincide with a 56-group boundary. The BLO lo-fi data do +not include thermal or resonance range uncertainties for isotope +reactions that do not have integral uncertainties given in the +Mughabghab text :cite:`mughabghab_atlas_2006`. These occur mainly for relatively unimportant +data such as elastic cross sections of several fission products. +Therefore in these cases the uncertainties were estimated using +different approaches. For example, the thermal data uncertainty was +sometimes used to represent the epithermal uncertainty if it was not +available in the Mughabghab tabulation, and sometimes the high-energy +uncertainty was extended to lower energies. The uncertainty in the +:sup:`149`\ Sm resonance capture integral is not provided in the 2006 +edition of Mughabghab’s text, so it was set to the value of 5.7%, which +was obtained from an earlier tabulation by Mughabghab :cite:`mughabghab_thermal_2003`. + +.. _10-2-2-4: + +Covariance data for fission spectra +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +As of ENDF/B-VII.1, covariance matrices are now provided for the fission +exit energy distribution. The data are given as a function of incident +energy. The incident energy grid is very broad, and the exit energy +distribution is constant over a given incident energy group. Since the +COVERX library file only allows one multigroup fission spectrum (χ) +covariance matrix per nuclide, the exit energy spectrum is used for the +average energy of fission. If ν is nubar, *f* is fission, and *w* is the +appropriate flux, then the average energy of fission is calculated as: + +.. math:: + :label: eq10-2-3 + + 10^{7}exp\left( - \frac{\sum_{}^{}{\text{vfw}\frac{1}{2}\left( \log\left( \frac{10^{7}}{E_{g1}} \right) + log\left( \frac{10^{7}}{E_{g2}} \right) \right)}}{\sum_{}^{}\text{νfw}} \right) , + +where the sum is over all groups and E\ :sub:`g1` and E\ :sub:`g2` are +the group boundaries for group g. ENDF/B-VII.1 provides covariance data +for exit energy distributions for 64 nuclides. This includes all +nuclides for which fission spectrum (χ) covariance matrices where +provided in the previous covariance library. Some additional +χ-covariance matrices were taken from JENDL-4.0. The new 56-group and +252-group fission spectrum covariances are more complete and +significantly improved compared to the earlier 44-group chi uncertainty +data, which were based on the Watt fission spectrum in ENDF/B-V. (see +:ref:`10-2-5`). + +.. _10-2-3: + +Multigroup Covariance Processing +-------------------------------- + +Covariance data were processed with the AMPX code PUFF-IV. PUFF-IV has +major improvements in the treatment of the resolved and unresolved +resonance parameter uncertainties over previous code versions :cite:`wiarda_recent_2008`. All +nuclides with resonance parameter uncertainty files were processed with +the full sensitivity option in PUFF-IV. + +.. _10-2-4: + +Contents of the SCALE 6.2 Covariance Library +-------------------------------------------- + +The SCALE covariance library provides uncertainty data in 56- and +252-group formats for a total of 456 materials, including some +duplication for materials with multiple thermal scattering kernels. +:numref:`tab10-2-1` describes the contents of the library using the following +nomenclature: + +1. ENDF/B-VII.1: evaluated covariance data released with ENDF/B-VII.1 + +2. ENDF/B-VII.2-prelim: recently evaluated data proposed for future + release of ENDF/B-VII.2 + +3. ENDF/B-VI: evaluated covariance data released with ENDF/B-VI + +4. BLO approximate data: lo-fi covariances from BLO project + +5. SG-26: approximate covariances from WPEC Subgroup-26 + +6. JENDL-4.0: evaluated covariance data released with JENDL-4.0 + +Several covariance evaluations include cross correlations between +reactions. These are summarized in :numref:`tab10-2-2`. + +.. tabularcolumns:: |m{2cm}|m{2cm}|m{3cm}|m{7cm}| + +.. _tab10-2-1: +.. table:: Contents of SCALE 6.2 covariance libraries. + :align: center + :class: longtable + + +-----------------+-----------------+-----------------+-----------------+ + | **SCALE name** | **SCALE ID** | **Data source** | **Comment** | + +=================+=================+=================+=================+ + | ac-225 | 89225 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | ac-226 | 89226 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | ac-227 | 89227 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | ag-107 | 47107 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ag-109 | 47109 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | ag-110m | 1047110 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ag-111 | 47111 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | al-27 | 13027 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | albound | 1013027 | ENDF/B-VII.1 | Duplicate of | + | | | | al-27 | + +-----------------+-----------------+-----------------+-----------------+ + | am-240 | 95240 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | am-241 | 95241 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | | | χ covariance | | + | | | JENDL-4.0 | | + +-----------------+-----------------+-----------------+-----------------+ + | am-242 | 95242 | SG-26 | Thermal | + | | | | uncertainty | + | | | | replaced by | + | | | | Mughabghab | + | | | | value | + +-----------------+-----------------+-----------------+-----------------+ + | | | χ covariance | | + | | | JENDL-4.0 | | + +-----------------+-----------------+-----------------+-----------------+ + | am-242m | 1095242 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | am-243 | 95243 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | | | χ covariance | | + | | | JENDL-4.0 | | + +-----------------+-----------------+-----------------+-----------------+ + | am-244 | 95244 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | | | χ covariance | | + | | | JENDL-4.0 | | + +-----------------+-----------------+-----------------+-----------------+ + | am-244m | 1095244 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ar-36 | 18036 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ar-38 | 18038 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ar-40 | 18040 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | as-74 | 33074 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | as-75 | 33075 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | au-197 | 79197 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | b-10 | 5010 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | b-11 | 5011 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | ba-130 | 56130 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ba-132 | 56132 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ba-133 | 56133 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ba-134 | 56134 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ba-135 | 56135 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ba-136 | 56136 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ba-137 | 56137 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ba-138 | 56138 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ba-140 | 56140 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | be-7 | 4007 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | be-9 | 4009 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | be-beo | 5004009 | ENDF/B-VII.1 | Duplicate of | + | | | | be-9 | + +-----------------+-----------------+-----------------+-----------------+ + | bebound | 3004009 | ENDF/B-VII.1 | Duplicate of | + | | | | be-9 | + +-----------------+-----------------+-----------------+-----------------+ + | bi-209 | 83209 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | bk-245 | 97245 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | bk-246 | 97246 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | bk-247 | 97247 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | bk-248 | 97248 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | bk-249 | 97249 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | bk-250 | 97250 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | br-79 | 35079 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | br-81 | 35081 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | c | 6000 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | ca | 20000 | BLO | | + | | | approximation | | + | | | dataca | | + +-----------------+-----------------+-----------------+-----------------+ + | ca-40 | 20040 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ca-42 | 20042 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ca-43 | 20043 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ca-44 | 20044 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ca-46 | 20046 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ca-48 | 20048 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | cd | 48000 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | cd-106 | 48106 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | cd-108 | 48108 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | cd-110 | 48110 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | cd-111 | 48111 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | cd-112 | 48112 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | cd-113 | 48113 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | cd-114 | 48114 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | cd-115m | 1048115 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | cd-116 | 48116 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ce-136 | 58136 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ce-138 | 58138 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ce-139 | 58139 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ce-140 | 58140 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ce-141 | 58141 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | ce-142 | 58142 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ce-143 | 58143 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ce-144 | 58144 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | cf-246 | 98246 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cf-248 | 98248 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cf-249 | 98249 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cf-250 | 98250 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cf-251 | 98251 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cf-252 | 98252 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cf-253 | 98253 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cf-254 | 98254 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cl | 17000 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | cl-35 | 17035 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cl-37 | 17037 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cm-240 | 96240 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cm-241 | 96241 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cm-242 | 96242 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cm-243 | 96243 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cm-244 | 96244 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cm-245 | 96245 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cm-246 | 96246 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cm-247 | 96247 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cm-248 | 96248 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cm-249 | 96249 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cm-250 | 96250 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | co-58 | 27058 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | co-58m | 1027058 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | co-59 | 27059 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cr-50 | 24050 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cr-52 | 24052 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cr-53 | 24053 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cr-54 | 24054 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cs-133 | 55133 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cs-134 | 55134 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | cs-135 | 55135 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | cs-136 | 55136 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | cs-137 | 55137 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | cu-63 | 29063 | ENDF/B-VI | | + +-----------------+-----------------+-----------------+-----------------+ + | cu-65 | 29065 | ENDF/B-VI | | + +-----------------+-----------------+-----------------+-----------------+ + | d | 1002 | ENDF/B-VII.1 | Duplicate of | + | | | | h-2 | + +-----------------+-----------------+-----------------+-----------------+ + | d-cryo_ortho | 4001002 | ENDF/B-VII.1 | Duplicate of | + | | | | h-2 | + +-----------------+-----------------+-----------------+-----------------+ + | d-cryo_para | 5001002 | ENDF/B-VII.1 | Duplicate of | + | | | | h-2 | + +-----------------+-----------------+-----------------+-----------------+ + | dfreegas | 8001002 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | dy-156 | 66156 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | dy-158 | 66158 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | dy-160 | 66160 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | dy-161 | 66161 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | dy-162 | 66162 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | dy-163 | 66163 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | dy-164 | 66164 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | er-162 | 68162 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | er-164 | 68164 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | er-166 | 68166 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | er-167 | 68167 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | er-168 | 68168 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | er-170 | 68170 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | es-251 | 99251 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | es-252 | 99252 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | es-253 | 99253 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | es-254 | 99254 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | es-254m | 1099254 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | es-255 | 99255 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | eu-151 | 63151 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | eu-152 | 63152 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | eu-153 | 63153 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | eu-154 | 63154 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | eu-155 | 63155 | ENDF/B-VII.1 | Uses | + | | | | ENDF/B-VII.1 | + | | | | data | + | | | | uncertainty in | + | | | | the thermal | + | | | | range for | + | | | | MT=102 | + +-----------------+-----------------+-----------------+-----------------+ + | eu-156 | 63156 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | eu-157 | 63157 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | f-19 | 9019 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | fe-54 | 26054 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | fe-56 | 26056 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | fe-57 | 26057 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | fe-58 | 26058 | ENDF/B-VI | | + +-----------------+-----------------+-----------------+-----------------+ + | febound | 1026000 | ENDF/B-VII.1 | Duplicate of | + | | | | fe-56 | + +-----------------+-----------------+-----------------+-----------------+ + | fm-255 | 100255 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | ga | 31000 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ga-69 | 31069 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ga-71 | 31071 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | gd-152 | 64152 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | gd-153 | 64153 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | gd-154 | 64154 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | gd-155 | 64155 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | gd-156 | 64156 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | gd-157 | 64157 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | gd-158 | 64158 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | gd-160 | 64160 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | ge-70 | 32070 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ge-72 | 32072 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ge-73 | 32073 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ge-74 | 32074 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ge-76 | 32076 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | graphite | 3006000 | ENDF/B-VII.1 | Duplicate of c | + +-----------------+-----------------+-----------------+-----------------+ + | h | 1001 | ENDF/B-VII.2 | Duplicate of h1 | + | | | prelim | | + +-----------------+-----------------+-----------------+-----------------+ + | h-3 | 1003 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | h-benzene | 6001001 | ENDF/B-VII.2 | Duplicate of | + | | | prelim | h-1 | + +-----------------+-----------------+-----------------+-----------------+ + | h-benzene | 5006000 | ENDF/B-VII.1 | Duplicate of c | + +-----------------+-----------------+-----------------+-----------------+ + | h-cryo_ortho | 4001001 | ENDF/B-VII.2 | Duplicate of | + | | | prelim | h-1 | + +-----------------+-----------------+-----------------+-----------------+ + | h-cryo_para | 5001001 | ENDF/B-VII.2 | Duplicate of | + | | | prelim | h-1 | + +-----------------+-----------------+-----------------+-----------------+ + | h-liquid_ch4 | 1001001 | ENDF/B-VII.2 | Duplicate of | + | | | prelim | h-1 | + +-----------------+-----------------+-----------------+-----------------+ + | h-poly | 9001001 | ENDF/B-VII.2 | Duplicate of | + | | | prelim | h-1 | + +-----------------+-----------------+-----------------+-----------------+ + | h-solid_ch4 | 2001001 | ENDF/B-VII.2 | Duplicate of | + | | | prelim | h-1 | + +-----------------+-----------------+-----------------+-----------------+ + | h-zrh2 | 7001001 | ENDF/B-VII.2 | Duplicate of | + | | | prelim | h-1 | + +-----------------+-----------------+-----------------+-----------------+ + | he-3 | 2003 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | he-4 | 2004 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | hf | 72000 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | hf-174 | 72174 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | hf-176 | 72176 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | hf-177 | 72177 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | hf-178 | 72178 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | hf-179 | 72179 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | hf-180 | 72180 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | hfreegas | 8001001 | ENDF/B-VII.2 | | + | | | prelim | | + +-----------------+-----------------+-----------------+-----------------+ + | hg-196 | 80196 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | hg-198 | 80198 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | hg-199 | 80199 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | hg-200 | 80200 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | hg-201 | 80201 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | hg-202 | 80202 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | hg-204 | 80204 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ho-165 | 67165 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ho-166m | 1067166 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | i-127 | 53127 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | i-129 | 53129 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | i-130 | 53130 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | i-131 | 53131 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | i-135 | 53135 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | in | 49000 | ENDF/B-VI | | + +-----------------+-----------------+-----------------+-----------------+ + | in-113 | 49113 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | in-115 | 49115 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ir-191 | 77191 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | ir-193 | 77193 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | k | 19000 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | k-39 | 19039 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | k-40 | 19040 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | k-41 | 19041 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | kr-78 | 36078 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | kr-80 | 36080 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | kr-82 | 36082 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | kr-83 | 36083 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | kr-84 | 36084 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | kr-85 | 36085 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | kr-86 | 36086 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | la-138 | 57138 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | la-139 | 57139 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | la-140 | 57140 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | li-6 | 3006 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | li-7 | 3007 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | lu-175 | 71175 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | lu-176 | 71176 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | mg | 12000 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | mg-24 | 12024 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | mg-25 | 12025 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | mg-26 | 12026 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | mn-55 | 25055 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | mo | 42000 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | mo-100 | 42100 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | mo-92 | 42092 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | mo-94 | 42094 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | mo-95 | 42095 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | mo-96 | 42096 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | mo-97 | 42097 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | mo-98 | 42098 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | mo-99 | 42099 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | n-14 | 7014 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | n-15 | 7015 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | na-23 | 11023 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | nb-93 | 41093 | ENDF/B-VI | | + +-----------------+-----------------+-----------------+-----------------+ + | nb-94 | 41094 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | nb-95 | 41095 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | nd-142 | 60142 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | nd-143 | 60143 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | nd-144 | 60144 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | nd-145 | 60145 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | nd-146 | 60146 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | nd-147 | 60147 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | nd-148 | 60148 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | nd-148 | 60148 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | nd-150 | 60150 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ni-58 | 28058 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | ni-59 | 28059 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ni-60 | 28060 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | ni-61 | 28061 | ENDF/B-VI | | + +-----------------+-----------------+-----------------+-----------------+ + | ni-62 | 28062 | ENDF/B-VI | | + +-----------------+-----------------+-----------------+-----------------+ + | ni-64 | 28064 | ENDF/B-VI | | + +-----------------+-----------------+-----------------+-----------------+ + | np-234 | 93234 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | np-235 | 93235 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | np-236 | 93236 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | np-237 | 93237 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | | | χ covariance | | + | | | JENDL-4.0 | | + +-----------------+-----------------+-----------------+-----------------+ + | np-238 | 93238 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | np-239 | 93239 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | o-16 | 8016 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | o-17 | 8017 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | o-beo | 5008016 | ENDF/B-VII.1 | Duplicate of | + | | | | o-16 | + +-----------------+-----------------+-----------------+-----------------+ + | o-uo2 | 1008016 | ENDF/B-VII.1 | Duplicate of | + | | | | o-16 | + +-----------------+-----------------+-----------------+-----------------+ + | p-31 | 15031 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | pa-229 | 91229 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | pa-230 | 91230 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | pa-231 | 91231 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | | | χ covariance | | + | | | JENDL-4.0 | | + +-----------------+-----------------+-----------------+-----------------+ + | pa-232 | 91232 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | pa-233 | 91233 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | | | χ covariance | | + | | | JENDL-4.0 | | + +-----------------+-----------------+-----------------+-----------------+ + | pb-204 | 82204 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | pb-206 | 82206 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | pb-207 | 82207 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | pb-208 | 82208 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | pd-102 | 46102 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | pd-104 | 46104 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | pd-105 | 46105 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | pd-106 | 46106 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | pd-107 | 46107 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | pd-108 | 46108 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | pd-110 | 46110 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | pm-147 | 61147 | ENDF/B-VII.1 | Thermal and | + | | | | resonance range | + | | | | uncertainty | + | | | | values from | + | | | | Mughabghab | + | | | | | + +-----------------+-----------------+-----------------+-----------------+ + | pm-148 | 61148 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | pm-148m | 1061148 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | pm-149 | 61149 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | pm-151 | 61151 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | pr-141 | 59141 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | pr-142 | 59142 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | pr-143 | 59143 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | pu-236 | 94236 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | pu-237 | 94237 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | pu-238 | 94238 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | pu-239 | 94239 | ENDF/B-VII.2 | | + | | | prelim | | + +-----------------+-----------------+-----------------+-----------------+ + | pu-240 | 94240 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | pu-241 | 94241 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | | | χ covariance | | + | | | JENDL-4.0 | | + +-----------------+-----------------+-----------------+-----------------+ + | pu-242 | 94242 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | pu-243 | 94243 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | pu-244 | 94244 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | pu-246 | 94246 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | rb-85 | 37085 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | rb-86 | 37086 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | rb-87 | 37087 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | re-185 | 75185 | ENDF/B-VI | | + +-----------------+-----------------+-----------------+-----------------+ + | re-187 | 75187 | ENDF/B-VI | | + +-----------------+-----------------+-----------------+-----------------+ + | rh-103 | 45103 | ENDF/B-VII.1 | Uses | + | | | | ENDF/B-VII.1 | + | | | | data | + | | | | uncertainty in | + | | | | the thermal | + | | | | range for | + | | | | MT=102 | + +-----------------+-----------------+-----------------+-----------------+ + | rh-105 | 45105 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ru-100 | 44100 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ru-101 | 44101 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | ru-102 | 44102 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | ru-103 | 44103 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | ru-104 | 44104 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | ru-105 | 44105 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ru-106 | 44106 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | ru-96 | 44096 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ru-98 | 44098 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ru-99 | 44099 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | s | 16000 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | s-32 | 16032 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | s-33 | 16033 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | s-34 | 16034 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | s-36 | 16036 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sb-121 | 51121 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sb-123 | 51123 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sb-124 | 51124 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sb-125 | 51125 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sb-126 | 51126 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sc-45 | 21045 | ENDF/B-VI | | + +-----------------+-----------------+-----------------+-----------------+ + | se-74 | 34074 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | se-76 | 34076 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | se-77 | 34077 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | se-78 | 34078 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | se-79 | 34079 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | se-80 | 34080 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | se-82 | 34082 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | si | 14000 | ENDF/B-VI | | + +-----------------+-----------------+-----------------+-----------------+ + | si-28 | 14028 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | si-29 | 14029 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | si-30 | 14030 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | si-28 in | 14728 | ENDF/B-VII.1 | Duplicate of | + | SiO\ :sub:`2` | | | si-28 | + +-----------------+-----------------+-----------------+-----------------+ + | si-29 in | 14729 | ENDF/B-0VII.1 | Duplicate of | + | SiO\ :sub:`2` | | | si-29 | + +-----------------+-----------------+-----------------+-----------------+ + | si-30 in | 14730 | ENDF/B-VII.1 | Duplicate of | + | SiO\ :sub:`2` | | | si-30 | + +-----------------+-----------------+-----------------+-----------------+ + | sm-144 | 62144 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sm-147 | 62147 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sm-148 | 62148 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sm-149 | 62149 | ENDF/B-VII.1 | Uses | + | | | | ENDF/B-VII.1 | + | | | | data | + | | | | uncertainty in | + | | | | the thermal | + | | | | range for | + | | | | MT=102 | + +-----------------+-----------------+-----------------+-----------------+ + | sm-149 | 62149 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sm-150 | 62150 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sm-151 | 62151 | ENDF/B-VII.1 | Uses | + | | | | ENDF/B-VII.1 | + | | | | data | + | | | | uncertainty in | + | | | | the thermal | + | | | | range for | + | | | | MT=102 | + +-----------------+-----------------+-----------------+-----------------+ + | sm-152 | 62152 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | sm-153 | 62153 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sm-154 | 62154 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sn-112 | 50112 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sn-113 | 50113 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sn-114 | 50114 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sn-115 | 50115 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sn-116 | 50116 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sn-117 | 50117 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sn-118 | 50118 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sn-119 | 50119 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sn-120 | 50120 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sn-122 | 50122 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sn-123 | 50123 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sn-124 | 50124 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sn-125 | 50125 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sn-126 | 50126 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sr-84 | 38084 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sr-86 | 38086 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sr-87 | 38087 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sr-88 | 38088 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sr-89 | 38089 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | sr-90 | 38090 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ta-181 | 73181 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ta-182 | 73182 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | tb-159 | 65159 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | tb-160 | 65160 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | tc-99 | 43099 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | te-120 | 52120 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | te-122 | 52122 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | te-123 | 52123 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | te-124 | 52124 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | te-125 | 52125 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | te-126 | 52126 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | te-127m | 1052127 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | te-128 | 52128 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | te-129m | 1052129 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | te-130 | 52130 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | te-132 | 52132 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | th-227 | 90227 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | th-228 | 90228 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | th-229 | 90229 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | th-230 | 90230 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | th-231 | 90231 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | th-232 | 90232 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | | | χ covariance | | + | | | JENDL-4.0 | | + +-----------------+-----------------+-----------------+-----------------+ + | th-233 | 90233 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | th-234 | 90234 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | ti | 22000 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | ti-46 | 22046 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | ti-47 | 22047 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | ti-48 | 22048 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | ti-49 | 22049 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | ti-50 | 22050 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | tl-203 | 81203 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | tl-205 | 81205 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | tm-169 | 69169 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | tm-170 | 69170 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | u-230 | 92230 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | u-231 | 92231 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | u-232 | 92232 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | u-233 | 92233 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | | | χ covariance | | + | | | JENDL-4.0 | | + +-----------------+-----------------+-----------------+-----------------+ + | u-234 | 92234 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | u-235 | 92235 | ENDF/B-VII.2 | | + | | | prelim | | + +-----------------+-----------------+-----------------+-----------------+ + | u-236 | 92236 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | u-237 | 92237 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | u-238 | 92238 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | u-239 | 92239 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | u-240 | 92240 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | u-241 | 92241 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | u-uo2 | 1092235 | ENDF/B-VII.1 | Duplicate of | + | | | | u-235 | + +-----------------+-----------------+-----------------+-----------------+ + | v | 23000 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | w | 74000 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | w-180 | 74180 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | w-182 | 74182 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | w-183 | 74183 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | w-184 | 74184 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | w-186 | 74186 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | xe-123 | 54123 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | xe-124 | 54124 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | xe-126 | 54126 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | xe-128 | 54128 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | xe-129 | 54129 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | xe-130 | 54130 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | xe-131 | 54131 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | xe-132 | 54132 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | xe-133 | 54133 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | xe-134 | 54134 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | xe-135 | 54135 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | xe-136 | 54136 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | y-89 | 39089 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | y-90 | 39090 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | y-91 | 39091 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | zr | 40000 | BLO | | + | | | approximation | | + | | | data | | + +-----------------+-----------------+-----------------+-----------------+ + | zr-90 | 40090 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | zr-91 | 40091 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | zr-92 | 40092 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | zr-93 | 40093 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | zr-94 | 40094 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | zr-95 | 40095 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | zr-96 | 40096 | ENDF/B-VII.1 | | + +-----------------+-----------------+-----------------+-----------------+ + | zr-90-zr5h8 | 1040090 | ENDF/B-VII.1 | Duplicate of | + | | | | zr-90 | + +-----------------+-----------------+-----------------+-----------------+ + | zr-91-zr5h8 | 1040091 | ENDF/B-VII.1 | Duplicate of | + | | | | zr-91 | + +-----------------+-----------------+-----------------+-----------------+ + | zr-92-zr5h8 | 1040092 | ENDF/B-VII.1 | Duplicate of | + | | | | zr-92 | + +-----------------+-----------------+-----------------+-----------------+ + | zr-93-zr5h8 | 1040093 | ENDF/B-VII.1 | Duplicate of | + | | | | zr-93 | + +-----------------+-----------------+-----------------+-----------------+ + | zr-94-zr5h8 | 1040094 | ENDF/B-VII.1 | Duplicate of | + | | | | zr-94 | + +-----------------+-----------------+-----------------+-----------------+ + | zr-95-zr5h8 | 1040095 | ENDF/B-VII.1 | Duplicate of | + | | | | zr-95 | + +-----------------+-----------------+-----------------+-----------------+ + | zr-96-zr5h8 | 1040096 | ENDF/B-VII.1 | Duplicate of | + | | | | zr-96 | + +-----------------+-----------------+-----------------+-----------------+ + +.. _tab10-2-2: +.. table:: Covariance data with cross-correlations between nuclide reactions. + :align: center + + + +----------------+------------+----------------+-------------------+ + | Nuclide 1 | Reaction 1 | Nuclide 2 | Reaction 2 | + +================+============+================+===================+ + | :sup:`239`\ Pu | Fission | :sup:`6`\ Li | Triton production | + +----------------+------------+----------------+-------------------+ + | :sup:`239`\ Pu | Fission | :sup:`197`\ Au | Capture | + +----------------+------------+----------------+-------------------+ + | :sup:`239`\ Pu | Fission | :sup:`235`\ U | Fission | + +----------------+------------+----------------+-------------------+ + | :sup:`239`\ Pu | Fission | :sup:`238`\ U | Fission | + +----------------+------------+----------------+-------------------+ + | :sup:`235`\ U | Fission | :sup:`197`\ Au | Capture | + +----------------+------------+----------------+-------------------+ + | :sup:`235`\ U | Fission | :sup:`6`\ Li | Triton production | + +----------------+------------+----------------+-------------------+ + | :sup:`238`\ U | Capture | :sup:`197`\ Au | Capture | + +----------------+------------+----------------+-------------------+ + | :sup:`238`\ U | Capture | :sup:`235`\ U | Fission | + +----------------+------------+----------------+-------------------+ + +.. _10-2-5: + +SCALE 6.1 44-group covariance library +------------------------------------- + +The older 44-group covariance library distributed with SCALE 6.0 and +SCALE 6.1 is included with this distribution for backwards +compatibility. The 44-group covariance library provides uncertainty data +for a total of 401 materials, including some duplication for materials +with multiple thermal scattering kernels. However, the 44-group library +was created prior to the official release of ENDF/B-VII.1. Therefore, it +is recommended that the 56- or 252-group covariances be used rather than +the 44-group. As discussed in :ref:`10-2-1`, it is assumed that +covariances taken from one data evaluation such as ENDF/B-VI or +JENDL-3.3 can also be applied to other evaluations of the same data, +such as ENDF/B-VII. If this is done judiciously for cases in which the +nuclear data evaluations are similar, then the covariances taken from +one source should be a reasonable representation of uncertainties for +the other evaluations. Among the materials in the SCALE 44-group library +with covariances taken from high-fidelity nuclear data evaluations are +the following: + +a) ENDF/B-VII evaluations *(includes both VII.0 and pre-release +covariances proposed for VII.1, but no official ENDF/B-VII.1)*: + + Au, :sup:`209`\ Bi, :sup:`59`\ Co, :sup:`152,154,155,156`\ Gd, + :sup:`191,193`\ I, :sup:`7`\ Li, :sup:`23`\ Na, :sup:`93`\ Nb, + :sup:`58`\ Ni, :sup:`99`\ Tc,\ :sup:`232`\ Th, :sup:`48`\ Ti, + :sup:`239`\ Pu, :sup:`233,235,238`\ U,V + +(b) ENDF/B-VI evaluations: + + Al, :sup:`241`\ Am, :sup:`10`\ B, :sup:`12`\ C, + :sup:`50,52,53,54`\ Cr, :sup:`63,65`\ Cu, :sup:`54,56,57`\ Fe, In, + :sup:`55`\ Mn, :sup:`60,61,62,64`\ Ni, :sup:`206,207,208`\ Pb, + :sup:`242`\ Pu, :sup:`28,29`\ Si + +(c) JENDL-3.3 evaluations: + + :sup:`11`\ B, :sup:`1`\ H, :sup:`16`\ O, :sup:`240,241`\ Pu + +Two modifications were also made to the ENDF/B-VII evaluated nubar +covariances. These nubar uncertainties are believed to be more +realistic. The ENDF/B-VII.0 :sup:`235`\ U thermal nubar uncertainty of +0.71% was revised to the JENDL-3.3 value of 0.31%. In addition, the +thermal nubar certainty in the pre-released ENDF/B-VII.1 :sup:`233`\ U +evaluation was modified to the value in a recent ORNL data +evaluation :cite:`leal_233_2008`. This ORNL :sup:`233`\ U cross section evaluation also +provided the thermal and resonance cross sections for the prereleased +ENDF/B‑VII.1 data. The ENDF/B-VII.1 pre-release nubar data for +:sup:`239`\ Pu was incomplete when the 44-group covariance library was +generated, so :sup:`239`\ Pu nubar data are included from ENDF/B-V, the +most current data available at that time. This value is much higher than +the current estimated uncertainty in :sup:`239`\ Pu nubar. The basic +ENDF/B uncertainty files that were changed are described in +:numref:`tab10-2-3`. + +Several modifications were also made to the uncertainties obtained from +the BLO data. The BLO thermal uncertainties for :sup:`1`\ H capture and +elastic and for :sup:`16`\ O elastic were modified to the JENDL-3.3 +values of 0.5% and 0.1%, respectively. Similarly, the uncertainty in the +:sup:`10`\ B (n,alpha) thermal cross section was modified to the +ENDF/B-VI value of about 0.2%, since this is more consistent with the +Mughabghab integral uncertainty. The uncertainty in the :sup:`149`\ Sm +resonance capture integral is not provided in the 2006 edition of +Mughabghab’s text; therefore it was set to the value of 5.7% which was +obtained from an earlier tabulation by Mughabghab :cite:`mughabghab_thermal_2003`. + +.. _tab10-2-3: +.. table:: Summary of changes made to covariance evaluations for the 44-group library. + :align: center + + +-----------------------------------+-----------------------------------+ + | ENDF/B-VII.1 pre-release | Data were incomplete at time of | + | | library generation, so ENDF/B-V | + | :sup:`239`\ Pu | data were used for nubar. | + +===================================+===================================+ + | ENDF/B-VII | Thermal nubar modified to | + | | JENDL-3.3 value | + | :sup:`235`\ U | | + +-----------------------------------+-----------------------------------+ + | ENDF/B-VII | Thermal nubar modified to value | + | | from ORNL internal evaluation | + | :sup:`233`\ U | | + +-----------------------------------+-----------------------------------+ + | ENDF/B-VI | Thermal uncertainties were added | + | | to total cross section (set equal | + | :sup:`241`\ Am | to capture uncertainties) | + +-----------------------------------+-----------------------------------+ + | ENDF/B-VI | In elastic scatter uncertainty, | + | | corrected cross reference to | + | :sup:`28`\ Si, :sup:`29`\ Si, | MT=102 from original value of | + | :sup:`30`\ Si, :sup:`206`\ Pb, | MT=1.02 | + | :sup:`57`\ Fe | | + +-----------------------------------+-----------------------------------+ + | ENDF/B-VI | Removed MT=3 due to inconsistency | + | | with other MT values, resulting | + | :sup:`208`\ Pb, :sup:`207`\ Pb | in very large uncertainty | + | | predictions | + +-----------------------------------+-----------------------------------+ + +At the time of the preparation of the 44-group covariance library, +ENDF/B did not provide fission spectra uncertainty estimates. The +methodology used to construct these data for the 44-group covariance +library is described in Broadhead and Wagschal :cite:`broadhead_fission_2004`. In this approach, +the fission spectrum is represented as either a Watt or Maxwellian +distribution. These energy distributions are widely used to represent +fission spectra and have been commonly employed in many ENDF/B +evaluations. For example, Watt and Maxwellian expressions were used +almost exclusively to describe fission spectra in ENDF/B-V and also for +many ENDF/B-VI evaluations. More recent evaluations for some important +fissionable nuclides have replaced the simple Watt and Maxwellian +analytical expressions by distributions such as the Madland-Nix spectrum +obtained from more phenomenological nuclear fission models. However, it +is assumed here that uncertainties based on an appropriate Watt or +Maxwellian representation of the fission spectrum can be transferred to +the actual fission spectra contained in the different multigroup cross +section libraries. + +The methodology in Broadhead and Wagschal :cite:`broadhead_fission_2004` determines +energy-dependent covariances from uncertainties and correlations in the +*a* and *b* parameters for the Watt spectrum or the *T* parameter for a +Maxwellian spectrum, appearing the analytical expressions given below: + +Watt Spectrum: :math:`\chi(\mathrm{E})=\frac{\mathrm{e}^{-\mathrm{E} / \mathrm{a}}}{\mathrm{I}} \sinh (\sqrt{\mathrm{bE}})` + +Maxwellian Spectrum: :math:`\chi(\mathrm{E})=\frac{\sqrt{\mathrm{E}} \mathrm{e}^{-\mathrm{E} / \mathrm{T}}}{\mathrm{I}}` + +In these expressions, the parameter “I” is the normalization factor +required to normalize the integrated spectrum to unity. The value of “I” +is fixed by the values of the other parameters. Due to the normalization +constraint, the fission spectrum covariance includes anti-correlations. +The assumed fission spectra parameters and uncertainties are given in +Maerker, Marable, and Wagschal 1980 :cite:`maerker_estimation_1980` and in Howerton and Doyas +1971 :cite:`howerton_fission_1971`. + +:numref:`tab10-2-4` shows that fission spectra covariances are not provided for +all fissionable materials in the SCALE multigroup cross sections. +:numref:`tab10-2-5` lists the fissionable nuclides without fission spectra +covariances on the 44-group covariance library. + +.. |t| replace:: :sup:`16` +.. |n| replace:: :sup:`17` +.. _tab10-2-4: +.. table:: Source of fission spectrum parameters and uncertainties + :align: center + + +---------+---------+---------+---------+---------+---------+---------+ + | Watt | *a* or | *b* | Source | ∂\ *a* | ∂\ *b* | Source | + | spectru\| *T* | | of | or | | of | + | m | | | paramet\| ∂\ *T* | (%) | uncerta\| + | | | | ers | (%) | | inty | + +=========+=========+=========+=========+=========+=========+=========+ + | :sup:`2\| 0.988 | 2.249 | ENDF/B-\| 1.2 | 5.9 |TANS\ |t|| + | 35`\ U | | | V | | | | + | | | | | | | | + +---------+---------+---------+---------+---------+---------+---------+ + | :sup:`2\| 0.881 | 3.401 | ENDF/B-\| 1.2 | 5.9 |TANS\ |t|| + | 38`\ U | | | V | | | | + | | | | | | | | + +---------+---------+---------+---------+---------+---------+---------+ + | :sup:`2\| 0.977 | 2.546 | ENDF/B-\| 1.2 | 5.9 |TANS\ |t|| + | 33`\ U | | | V | | | | + | | | | | | | | + +---------+---------+---------+---------+---------+---------+---------+ + | :sup:`2\| 0.966 | 2.842 | ENDF/B-\| 1.2 | 5.9 |TANS\ |t|| + | 39`\ Pu | | | V | | | | + | | | | | | | | + +---------+---------+---------+---------+---------+---------+---------+ + | :sup:`2\| 1.0888 | 1.6871 | ENDF/B-\| 1.2 | 5.9 |TANS\ |t|| + | 32`\ Th | | | V | | | | + | | | | | | | | + +---------+---------+---------+---------+---------+---------+---------+ + | :sup:`2\| 1.025 | 2.926 | ENDF/B-\| 1.2 | 5.9 |TANS\ |t|| + | 52`\ Cf | | | V | | | | + | | | | | | | | + +---------+---------+---------+---------+---------+---------+---------+ + | Maxwell\| | | | | | | + | ian | | | | | | | + | Spectru\| | | | | | | + | m | | | | | | | + +---------+---------+---------+---------+---------+---------+---------+ + | :sup:`2\| 1.330 | — | ENDF/B-\| 3.01 | — |NSE\ |n| | + | 38`\ Pu | | | V | | | | + +---------+---------+---------+---------+---------+---------+---------+ + | :sup:`2\| 1.346 | — | ENDF/B-\| 2.97 | — |NSE\ |n| | + | 40`\ Pu | | | V | | | | + +---------+---------+---------+---------+---------+---------+---------+ + | :sup:`2\| 1.3597 | — | ENDF/B-\| 2.50 | — |NSE\ |n| | + | 41`\ Pu | | | V | | | | + +---------+---------+---------+---------+---------+---------+---------+ + | :sup:`2\| 1.337 | — | ENDF/B-\| 5.24 | — |NSE\ |n| | + | 42`\ Pu | | | V | | | | + +---------+---------+---------+---------+---------+---------+---------+ + +.. _tab10-2-5: +.. table:: Fissionable nuclides with missing fission spectrum uncertainty data in covariance library. + :align: center + + +----------------+----------------+----------------+ + | :sup:`241`\ Am | :sup:`244`\ Cm | :sup:`238`\ Pu | + +================+================+================+ + | :sup:`242`\ Am | :sup:`245`\ Cm | :sup:`243`\ Pu | + +----------------+----------------+----------------+ + | :sup:`243`\ Am | :sup:`246`\ Cm | :sup:`244`\ Pu | + +----------------+----------------+----------------+ + | :sup:`249`\ Bk | :sup:`247`\ Cm | :sup:`230`\ Th | + +----------------+----------------+----------------+ + | :sup:`249`\ Cf | :sup:`248`\ Cm | :sup:`232`\ U | + +----------------+----------------+----------------+ + | :sup:`250`\ Cf | :sup:`237`\ Np | :sup:`234`\ U | + +----------------+----------------+----------------+ + | :sup:`251`\ Cf | :sup:`238`\ Np | :sup:`236`\ U | + +----------------+----------------+----------------+ + | :sup:`253`\ Cf | :sup:`239`\ Np | :sup:`237`\ U | + +----------------+----------------+----------------+ + | :sup:`242`\ Cm | :sup:`231`\ Pa | | + +----------------+----------------+----------------+ + | :sup:`243`\ Cm | :sup:`233`\ Pa | | + +----------------+----------------+----------------+ + +*Table 10.2.6 describes the contents of the library using the following +nomenclature:* + +1. ENDF/B-VII.0: evaluated covariance data released with ENDF/B-VII.0 + +2. ENDF/B-VII-p: recently evaluated data proposed for future release of + ENDF/B-VII.1 + +3. ENDF/B-VI: evaluated covariance data released with ENDF/B-VI + +4. JENDL-3.3: evaluated covariance data in JENDL-3.3 + +5. BLO approximate data: lo-fi covariances from BLO project + +6. BLO LANL evaluation: LANL R-matrix evaluation from BLO project + +7. SG-26: approximate covariances from WPEC Subgroup-26 + +.. tabularcolumns:: |m{3cm}|m{5em}|m{7cm}| + +.. _tab10-2-6: +.. table:: Contents of SCALE 6.1 44-group covariance library. + :align: center + :class: longtable + + +-----------------------+-----------------------+-----------------------+ + | **SCALE name** | **Data source** | **Comments** | + +-----------------------+-----------------------+-----------------------+ + | ac-225 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ac-226 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ac-227 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ag-107 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ag-109 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ag-110m | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ag-111 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | al-27 | ENDF/B-VI | | + +-----------------------+-----------------------+-----------------------+ + | am-241 | ENDF/B-VI | MT=452 added | + | | | corrections for total | + | | | and elastic) | + +-----------------------+-----------------------+-----------------------+ + | am-242 | SG-26 | Thermal uncertainty | + | | | replaced by | + | | | Mughabghab value | + +-----------------------+-----------------------+-----------------------+ + | am-242m | SG-26 | Thermal uncertainty | + | | | replaced by | + | | | Mughabghab value | + +-----------------------+-----------------------+-----------------------+ + | am-243 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | am-244 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | am-244m | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ar-36 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ar-38 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ar-40 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | as-74 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | as-75 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | au-197 | ENDF/B-VII-p | Pre-released | + | | | evaluation proposed | + | | | for ENDF/B-VII.1 | + +-----------------------+-----------------------+-----------------------+ + | b-10 | BLO LANL evaluation | LANL high-fidelity | + | | +ENDF/B-VI | covariance, with | + | | | ENDF/B-VI for thermal | + +-----------------------+-----------------------+-----------------------+ + | b-11 | JENDL 3.3 | | + +-----------------------+-----------------------+-----------------------+ + | ba-130 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ba-132 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ba-133 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ba-135 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ba-136 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ba-137 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ba-138 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ba-140 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | be-7 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | be-9 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | Bebound | BLO approximate data | Duplicate of | + | | | :sup:`9`\ Be | + +-----------------------+-----------------------+-----------------------+ + | bi-209 | ENDF/B-VII-p | Pre-released | + | | | evaluation proposed | + | | | for ENDF/B-VII.1 | + +-----------------------+-----------------------+-----------------------+ + | bk-249 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | bk-250 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | br-79 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | br-81 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | C | ENDF/B-VI | | + +-----------------------+-----------------------+-----------------------+ + | C-graphite | ENDF/B-VI | Duplicate of carbon | + +-----------------------+-----------------------+-----------------------+ + | Ca | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ca-40 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ca-42 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ca-43 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ca-44 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ca-46 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ca-48 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | Cd | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cd-106 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cd-108 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cd-110 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cd-111 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cd-112 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cd-113 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cd-114 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cd-115m | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cd-116 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cd-136 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cd-138 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cd-139 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cd-140 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cd-141 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cd-142 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ce-143 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ce-144 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cf-249 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cf-250 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cf-251 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cf-252 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cf-253 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cf-254 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | Cl | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cl-35 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cl-37 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cm-241 | BLO approximate data | Thermal uncertainty | + +-----------------------+-----------------------+-----------------------+ + | cm-242 | SG-26 | Mughabghab value | + +-----------------------+-----------------------+-----------------------+ + | cm-243 | SG-26 | Thermal uncertainty | + | | | replaced by | + | | | Mughabghab value | + +-----------------------+-----------------------+-----------------------+ + | cm-244 | SG-26 | Thermal uncertainty | + | | | replaced by | + | | | Mughabghab value | + +-----------------------+-----------------------+-----------------------+ + | cm-245 | SG-26 | Thermal uncertainty | + | | | replaced by | + | | | Mughabghab value | + +-----------------------+-----------------------+-----------------------+ + | cm-246 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cm-247 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cm-248 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cm-249 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cm-250 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | co-58 | BLO approximate data | Pre-released | + | | | evaluation proposed | + | co-58m | BLO approximate data | for ENDF/B-VII.1 | + | | | | + | co-59 | ENDF/B-VII-p | | + +-----------------------+-----------------------+-----------------------+ + | cr-50 | ENDF/B-VI | LB=8 representation | + | | | caused problematic | + | | | representation of | + | | | cross section | + | | | uncertainty due to | + | | | use of fine energy | + | | | group structure. | + | | | Tests were performed | + | | | to determine how to | + | | | handle this problem. | + | | | LB=8 data were | + | | | removed in the final | + | | | results. | + +-----------------------+-----------------------+-----------------------+ + | cr-52 | ENDF/B-VI | LB=8 representation | + | | | caused problematic | + | | | representation of | + | | | cross section | + | | | uncertainty due to | + | | | use of fine energy | + | | | group structure. | + | | | Tests were performed | + | | | to determine how to | + | | | handle this problem. | + | | | LB=8 data were | + | | | removed in the final | + | | | results. | + +-----------------------+-----------------------+-----------------------+ + | cr-53 | ENDF/B-VI | LB=8 representation | + | | | caused problematic | + | | | representation of | + | | | cross section | + | | | uncertainty due to | + | | | use of fine energy | + | | | group structure. | + | | | Tests were performed | + | | | to determine how to | + | | | handle this problem. | + | | | LB=8 data were | + | | | removed in the final | + | | | results. | + +-----------------------+-----------------------+-----------------------+ + | cr-54 | ENDF/B-VI | LB=8 representation | + | | | caused problematic | + | | | representation of | + | | | cross section | + | | | uncertainty due to | + | | | use of fine energy | + | | | group structure. | + | | | Tests were performed | + | | | to determine how to | + | | | handle this problem. | + | | | LB=8 data were | + | | | removed in the final | + | | | results. | + +-----------------------+-----------------------+-----------------------+ + | cs-133 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cs-134 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cs-135 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cs-136 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cs-137 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | cu-63 | ENDF/B-VI | | + +-----------------------+-----------------------+-----------------------+ + | cu-65 | ENDF/B-VI | | + +-----------------------+-----------------------+-----------------------+ + | dy-156 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | dy-158 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | dy-160 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | dy-161 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | dy-162 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | dy-163 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | dy-164 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | er-162 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | er-164 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | er-166 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | er-167 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | er-168 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | er-170 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | es-253 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | es-254 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | es-255 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | eu-151 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | eu-152 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | eu-153 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | eu-154 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | eu-155 | BLO approximate data | | + | | BLO approximate data | | + | eu-156 | BLO approximate data | | + | | | | + | eu-157 | | | + +-----------------------+-----------------------+-----------------------+ + | f-19 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | fe-54 | ENDF/B-VI | LB=8 representation | + | | | caused problematic | + | | | representation of | + | | | cross section | + | | | uncertainty due to | + | | | use of fine energy | + | | | group structure. | + | | | Tests were performed | + | | | to determine how to | + | | | handle this problem. | + | | | LB=8 data were | + | | | removed in the final | + | | | results. | + +-----------------------+-----------------------+-----------------------+ + | fe-56 | ENDF/B-VI | LB=8 representation | + | | | caused problematic | + | | | representation of | + | | | cross section | + | | | uncertainty due to | + | | | use of fine energy | + | | | group structure. | + | | | Tests were performed | + | | | to determine how to | + | | | handle this problem. | + | | | LB=8 data were | + | | | removed in the final | + | | | results. | + +-----------------------+-----------------------+-----------------------+ + | fe-57 | ENDF/B-VI | Error in file | + | | | corrected | + | | | LB=8 representation | + | | | caused problematic | + | | | representation of | + | | | cross section | + | | | uncertainty due to | + | | | use of fine energy | + | | | group structure. | + | | | Tests were performed | + | | | to determine how to | + | | | handle this problem. | + | | | LB=8 data were | + | | | removed in the final | + | | | results. | + +-----------------------+-----------------------+-----------------------+ + | fe-58 | ENDF/B-VI | LB=8 representation | + | | | caused problematic | + | fm-255 | BLO approximate data | representation of | + | | | cross section | + | | | uncertainty due to | + | | | use of fine energy | + | | | group structure. | + | | | Tests were performed | + | | | to determine how to | + | | | handle this problem. | + | | | LB=8 data were | + | | | removed in the final | + | | | results. New material | + | | | not in previous | + | | | | + | | | SCALE 5.1 covariance | + | | | libraries. | + +-----------------------+-----------------------+-----------------------+ + | Ga | BLO approximate data | | + | | | | + | ga-69 | BLO approximate data | | + | | | | + | ga-71 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | gd-152 | ENDF/B-VII.0 | | + | | | | + | gd-153 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | gd-154 | ENDF/B-VII.0 | | + +-----------------------+-----------------------+-----------------------+ + | gd-155 | ENDF/B-VII.0 | | + +-----------------------+-----------------------+-----------------------+ + | gd-156 | ENDF/B-VII.0 | | + +-----------------------+-----------------------+-----------------------+ + | gd-157 | ENDF/B-VII.0 | | + +-----------------------+-----------------------+-----------------------+ + | gd-158 | ENDF/B-VII.0 | | + +-----------------------+-----------------------+-----------------------+ + | gd-160 | ENDF/B-VII.0 | | + +-----------------------+-----------------------+-----------------------+ + | ge-70 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ge-72 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ge-73 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ge-74 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ge-76 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | h-1 | BLO LANL evaluation | LANL covariance above | + | | +JENDL 3.3 | 5 keV; | + | | | JENDL values below 5 | + | | | keV | + +-----------------------+-----------------------+-----------------------+ + | h-ZrH | BLO LANL evaluation | Duplicate of | + | | +JENDL 3. | :sup:`1`\ H | + +-----------------------+-----------------------+-----------------------+ + | h-poly | BLO LANL evaluation | Duplicate of | + | | +JENDL 3. | :sup:`1`\ H | + +-----------------------+-----------------------+-----------------------+ + | Hfreegas | BLO LANL evaluation | Duplicate of | + | | +JENDL 3. | :sup:`1`\ H | + +-----------------------+-----------------------+-----------------------+ + | h-2 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | Dfreegas | BLO approximate data | Duplicate of | + | | | :sup:`2`\ H | + +-----------------------+-----------------------+-----------------------+ + | h-3 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | he-3 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | he-4 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | Hf | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | hf-174 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | hf-176 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | fh-177 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | hf-178 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | hf-179 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | hf-180 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | hg-196 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | hg-198 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | hg-199 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | hg-200 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | hg-201 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | hg-202 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | hg-204 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ho-165 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | i-127 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | i-129 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | i-130 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | i-131 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | i-135 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | In | ENDF/B-VI | | + +-----------------------+-----------------------+-----------------------+ + | in-113 | BLO approximate data | | + | | | | + | in-115 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ir-191 | ENDF/B-VII.0 | | + +-----------------------+-----------------------+-----------------------+ + | ir-193 | ENDF/B-VII.0 | | + +-----------------------+-----------------------+-----------------------+ + | K | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | k-39 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | k-40 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | k-41 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | kr-78 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | kr-80 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | kr-82 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | kr-83 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | kr-84 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | kr-85 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | kr-86 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | la-138 | BLO approximate data | | + | | BLO approximate data | | + | la-139 | | | + +-----------------------+-----------------------+-----------------------+ + | la-140 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | li-6 | BLO-LANL evaluation | | + +-----------------------+-----------------------+-----------------------+ + | li-7 | ENDF/B-VII.0 | | + +-----------------------+-----------------------+-----------------------+ + | lu-175 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | lu-176 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | Mg | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | mg-24 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | mg-25 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | mg-26 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | mn-55 | ENDF/B-VI | LB=8 representation | + | | | caused problematic | + | | | representation of | + | | | cross section | + | | | uncertainty due to | + | | | use of fine energy | + | | | group structure. | + | | | Tests were performed | + | | | to determine how to | + | | | handle this problem. | + | | | LB=8 data were | + | | | removed in the final | + | | | results. | + +-----------------------+-----------------------+-----------------------+ + | Mo | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | mo-92 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | mo-94 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | mo-95 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | mo-96 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | mo-97 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | mo-98 | BLO approximate data | | + | | BLO approximate data | | + | mo-99 | | | + | | BLO approximate data | | + | mo-100 | | | + +-----------------------+-----------------------+-----------------------+ + | n-14 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | n-15 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | na-23 | ENDF/B-VII-p | Pre-released | + | | | evaluation proposed | + | | | for ENDF/B-VII.1 | + +-----------------------+-----------------------+-----------------------+ + | nb-93 | ENDF/B-VII-p | Pre-released | + | | | evaluation proposed | + | | | for ENDF/B-VII.1 | + +-----------------------+-----------------------+-----------------------+ + | nb-94 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | nb-95 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | nd-142 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | nd-143 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | nd-144 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | nd-145 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | nd-146 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | nd-147 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | nd-148 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | nd-150 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ni-58 | ENDF/B-VII-p | Pre-released | + | | | evaluation proposed | + | ni-59 | BLO approximate data | for ENDF/B-VII.1 | + +-----------------------+-----------------------+-----------------------+ + | ni-60 | ENDF/B-VI | LB=8 representation | + | | | caused problematic | + | | | representation of | + | | | cross section | + | | | uncertainty due to | + | | | use of fine energy | + | | | group structure. | + | | | Tests were performed | + | | | to determine how to | + | | | handle this problem. | + | | | LB=8 data were | + | | | removed in the final | + | | | results. | + +-----------------------+-----------------------+-----------------------+ + | ni-61 | ENDF/B-VI | LB=8 representation | + | | | caused problematic | + | | | representation of | + | | | cross section | + | | | uncertainty due to | + | | | use of fine energy | + | | | group structure. | + | | | Tests were performed | + | | | to determine how to | + | | | handle this problem. | + | | | LB=8 data were | + | | | removed in the final | + | | | results. | + +-----------------------+-----------------------+-----------------------+ + | ni-62 | ENDF/B-VI | LB=8 representation | + | | | caused problematic | + | | | representation of | + | | | cross section | + | | | uncertainty due to | + | | | use of fine energy | + | | | group structure. | + | | | Tests were performed | + | | | to determine how to | + | | | handle this problem. | + | | | LB=8 data were | + | | | removed in the final | + | | | results. | + +-----------------------+-----------------------+-----------------------+ + | ni-64 | ENDF/B-VI | LB=8 representation | + | | | caused problematic | + | | | representation of | + | | | cross section | + | | | uncertainty due to | + | | | use of fine energy | + | | | group structure. | + | | | Tests were performed | + | | | to determine how to | + | | | handle this problem. | + | | | LB=8 data were | + | | | removed in the final | + | | | results. | + +-----------------------+-----------------------+-----------------------+ + | np-235 | BLO approximate data | Thermal uncertainty | + | | | replaced by | + | np-236 | BLO approximate data | Mughabghab value | + | | | | + | np-237 | SG-26 | | + +-----------------------+-----------------------+-----------------------+ + | np-238 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | np-239 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | o-16 | JENDL 3.3+BLO | BLO covariances from | + | | | LANL used above 5 keV | + +-----------------------+-----------------------+-----------------------+ + | o-17 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | p-31 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | pa-231 | BLO approximate data | | + | | | | + | pa-232 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | pa-233 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | pb-204 | BLO approximate data | Error in file | + | | ENDF/B-VI | corrected | + | pb-206 | | | + +-----------------------+-----------------------+-----------------------+ + | pb-207 | ENDF/B-VI | MT=3 removed, Error | + | | | in file corrected | + +-----------------------+-----------------------+-----------------------+ + | bp-208 | ENDF/B-VI | MT=3 removed, Error | + | | | in file corrected | + +-----------------------+-----------------------+-----------------------+ + | pd-102 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | pd-104 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | pd-105 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | pd-106 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | pd-107 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | pd-108 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | pd-110 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | pm-147 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | pm-148 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | pm-148m | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | pm-149 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | pm-151 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | pr-141 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | pr-142 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | pr-143 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | pu-236 | BLO approximate data | Thermal uncertainty | + | | | replaced by | + | pu-237 | BLO approximate data | Mughabghab value | + | | | | + | pu-238 | SG-26 | | + +-----------------------+-----------------------+-----------------------+ + | pu-239 | ENDF/B-VII-p | Pre-released | + | | | evaluation proposed | + | | | for ENDF/B-VII.1; | + | | | nubar data from | + | | | ENDF/B-V | + | | | | + | | | | Cross | + | | | nuclide-to-nuclide | + | | | matrices present; | + | | | covariances due to | + | | | fission cross | + | | | sections / nubar | + | | | for each nuclide | + | | | | :numref:`tab10-2-2` | + +-----------------------+-----------------------+-----------------------+ + | pu-240 | JENDL 3.3 | Cross | + | | | nuclide-to-nuclide | + | | | matrices present; | + | | | covariances due to | + | | | fission cross | + | | | sections / nubar for | + | | | each nuclide (\ | + | | | :numref:`tab10-2-2`). | + +-----------------------+-----------------------+-----------------------+ + | pu-241 | JENDL 3.3 | Cross | + | | | nuclide-to-nuclide | + | | | matrices present; | + | | | covariances due to | + | | | fission cross | + | | | sections / nubar for | + | | | each nuclide (\ | + | | | :numref:`tab10-2-2`). | + +-----------------------+-----------------------+-----------------------+ + | pu-242 | ENDF/B-VI | | + +-----------------------+-----------------------+-----------------------+ + | pu-243 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | pu-244 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | pu-246 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | rb-85 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | rb-86 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | rb-87 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | re-185 | ENDF/B-VI | MT=2 added from | + | | | Mughabghab. LB=8 | + | | | representation caused | + | | | problematic | + | | | representation of | + | | | cross section | + | | | uncertainty due to | + | | | use of fine energy | + | | | group structure. | + | | | Tests were performed | + | | | to determine how to | + | | | handle this problem. | + | | | LB=8 data were | + | | | removed in the final | + | | | results. | + +-----------------------+-----------------------+-----------------------+ + | re-187 | ENDF/B-VI | MT=2 added from | + | | | Mughabghab. LB=8 | + | | | representation caused | + | | | problematic | + | | | representation of | + | | | cross section | + | | | uncertainty due to | + | | | use of fine energy | + | | | group structure. | + | | | Tests were performed | + | | | to determine how to | + | | | handle this problem. | + | | | LB=8 data were | + | | | removed in the final | + | | | results. | + +-----------------------+-----------------------+-----------------------+ + | rh-103 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | rh-105 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ru-96 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ru-98 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ru-103 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ru-99 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ru-100 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ru-101 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ru-102 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ru-104 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ru-105 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ru-106 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | S | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | s-32 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | s-33 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | s-34 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | s-36 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sb-123 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sb-124 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sb-125 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sb-126 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sc-45 | ENDF/B-VI | | + +-----------------------+-----------------------+-----------------------+ + | se-74 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | se-76 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | se-77 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | se-78 | BLO approximate data | | + | | BLO approximate data | | + | se-79 | | | + +-----------------------+-----------------------+-----------------------+ + | se-80 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | se-82 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | Si | ENDF/B-VI | | + +-----------------------+-----------------------+-----------------------+ + | si-28 | ENDF/B-VI | Error in file | + | | | corrected LB=8 | + | | | representation caused | + | | | problematic | + | | | representation of | + | | | cross section | + | | | uncertainty due to | + | | | use of fine energy | + | | | group structure. | + | | | Tests were performed | + | | | to determine how to | + | | | handle this problem. | + | | | LB=8 data were | + | | | removed in the final | + | | | results. | + +-----------------------+-----------------------+-----------------------+ + | si-29 | ENDF/B-VI | Error in file | + | | | corrected LB=8 | + | | | representation caused | + | | | problematic | + | | | representation of | + | | | cross section | + | | | uncertainty due to | + | | | use of fine energy | + | | | group structure. | + | | | Tests were performed | + | | | to determine how to | + | | | handle this problem. | + | | | LB=8 data were | + | | | removed in the final | + | | | results. | + +-----------------------+-----------------------+-----------------------+ + | si-30 | ENDF/B-VI | Error in file | + | | | corrected LB=8 | + | | | representation caused | + | | | problematic | + | | | representation of | + | | | cross section | + | | | uncertainty due to | + | | | use of fine energy | + | | | group structure. | + | | | Tests were performed | + | | | to determine how to | + | | | handle this problem. | + | | | LB=8 data were | + | | | removed in the final | + | | | results. | + +-----------------------+-----------------------+-----------------------+ + | sm-144 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sm-147 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sm-148 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sm-149 | BLO approximate data | Resonance range | + | | | uncertainty from | + | | | Kawano 2008 | + +-----------------------+-----------------------+-----------------------+ + | sm-150 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sm-151 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sm-152 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sm-153 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sm-154 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sn-112 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sn-113 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sn-114 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sn-115 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sn-116 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sn-117 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sn-118 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sn-119 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sn-120 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sn-122 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sn-123 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sn-124 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sn-125 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sr-84 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sr-86 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sr-87 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sr-88 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sr-89 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | sr-90 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ta-181 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ta-182 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | tb-159 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | tb-160 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | tc-99 | ENDF/B-VII.0 | | + +-----------------------+-----------------------+-----------------------+ + | te-120 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | te-122 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | te-123 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | te-124 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | te-125 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | te-126 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | te-127m | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | te-128 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | te-129m | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | te-130 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | th-227 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | th-228 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | th-229 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | th-230 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | th-232 | ENDF/B-VII.0 | | Cross | + | | | nuclide-to-nuclide | + | th-233 | BLO approximate data | matrices present; | + | | | covariances due to | + | th-234 | BLO approximate data | fission cross | + | | | sections / nubar | + | Ti | BLO approximate data | for each nuclide (\ | + | | | | :numref:`tab10-2-2`)| + | ti-46 | BLO approximate data | | + | | | Pre-released | + | ti-47 | BLO approximate data | evaluation proposed | + | | | for ENDF/B-VII.1 | + | ti-48 | ENDF/B-VII-p | | + +-----------------------+-----------------------+-----------------------+ + | ti-49 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | ti-50 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | u-232 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | u-233 | ENDF/B-VII-p | Pre-released | + | | | evaluation proposed | + | | | for ENDF/B-VII.1; | + | | | nubar uncertainty | + | | | from Ref. 14Cross | + | | | nuclide-to-nuclide | + | | | matrices present; | + | | | covariances due to | + | | | fission cross | + | | | sections / nubar for | + | | | each nuclide (\ | + | | | | + | | | :numref:`tab10-2-2`). | + +-----------------------+-----------------------+-----------------------+ + | u-234 | SG-26 | Thermal uncertainty | + | | | replaced by | + | | | Mughabghab value | + +-----------------------+-----------------------+-----------------------+ + | u-235 | ENDF/B-VII-p | Pre-released | + | | | evaluation proposed | + | | | for ENDF/B-VII.1; | + | | | nubar uncertainty | + | | | from JENDL-3.1 Cross | + | | | nuclide-to-nuclide | + | | | matrices present; | + | | | covariances due to | + | | | fission cross | + | | | sections / nubar for | + | | | each nuclide (\ | + | | | :numref:`tab10-2-2`). | + +-----------------------+-----------------------+-----------------------+ + | u-236 | SG-26 | Thermal uncertainty | + | | | replaced by | + | | | Mughabghab value | + +-----------------------+-----------------------+-----------------------+ + | u-237 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | u-238 | ENDF/B-VII-p | Pre-released | + | | | evaluation proposed | + | u-239 | BLO approximate data | for ENDF/B-VII.1 | + | | | Cross | + | u-240 | BLO approximate data | nuclide-to-nuclide | + | | | matrices present; | + | u-241 | BLO approximate data | covariances due to | + | | | fission cross | + | | | sections / nubar for | + | | | each nuclide (\ | + | | | :numref:`tab10-2-2`). | + +-----------------------+-----------------------+-----------------------+ + | V | ENDF/B-VII-p | Pre-released | + | | | evaluation proposed | + | | | for ENDF/B-VII.1 | + +-----------------------+-----------------------+-----------------------+ + | W | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | w-182 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | w-183 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | w-184 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | w-186 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | xe-123 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | xe-124 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | xe-126 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | xe-128 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | xe-129 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | xe-130 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | xe-131 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | xe-132 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | xe-134 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | xe-135 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | xe-136 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | y-89 | ENDF/B-VI | | + +-----------------------+-----------------------+-----------------------+ + | y-90 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | y-91 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | Zr | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | zr-90 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | zr-91 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | zr-92 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | zr-93 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | zr-94 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | zr-95 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + | zr-96 | BLO approximate data | | + +-----------------------+-----------------------+-----------------------+ + +.. _tab10-2-7: +.. table:: Covariance data with cross correlations between nuclide reactions. + :align: center + + +----------------+------------+----------------+------------+ + | Nuclide 1 | Reaction 1 | Nuclide 2 | Reaction 2 | + +================+============+================+============+ + | :sup:`240`\ Pu | Fission | :sup:`239`\ Pu | Fission | + +----------------+------------+----------------+------------+ + | :sup:`240`\ Pu | Fission | :sup:`233`\ U | Fission | + +----------------+------------+----------------+------------+ + | :sup:`240`\ Pu | Fission | :sup:`238`\ U | Fission | + +----------------+------------+----------------+------------+ + | :sup:`241`\ Pu | Fission | :sup:`239`\ Pu | Fission | + +----------------+------------+----------------+------------+ + | :sup:`241`\ Pu | Fission | :sup:`240`\ Pu | Fission | + +----------------+------------+----------------+------------+ + | :sup:`241`\ Pu | Fission | :sup:`233`\ U | Fission | + +----------------+------------+----------------+------------+ + | :sup:`241`\ Pu | Fission | :sup:`235`\ U | Fission | + +----------------+------------+----------------+------------+ + | :sup:`241`\ Pu | Fission | :sup:`238`\ U | Fission | + +----------------+------------+----------------+------------+ + | :sup:`235`\ U | Fission | :sup:`240`\ Pu | Fission | + +----------------+------------+----------------+------------+ + +.. bibliography:: bibs/COVLIB.bib diff --git a/COVLIBAppA.rst b/COVLIBAppA.rst new file mode 100644 index 0000000..8e290bd --- /dev/null +++ b/COVLIBAppA.rst @@ -0,0 +1,99 @@ +.. _10-2a: + +COVLIB Appendix A: Cross section plots for U, Pu, TH, B, H, He, and Gd Nuclides +=============================================================================== + +Plots of cross section differences between various evaluations are shown +below. The legend below applies to all plots shown in this appendix. + +.. image:: figs/COVLIBAppA/img1.png + :align: center + :width: 500 + +.. _fig10-2a-1: +.. figure:: figs/COVLIBAppA/fig1.png + :align: center + :width: 600 + + :sup:`239`\ Pu fission and capture comparison between ENDF/B-VI, + JENDL 3.3, and JEF 3.1. + +.. _fig10-2a-2: +.. figure:: figs/COVLIBAppA/fig2.png + :align: center + :width: 600 + + :sup:`240`\ Pu fission and capture comparison between ENDF/B-VI, + JENDL 3.3 and JEF 3.1. + +.. _fig10-2a-3: +.. figure:: figs/COVLIBAppA/fig3.png + :align: center + :width: 600 + + :sup:`241` fission and capture comparison between ENDF/B-VI, + JENDL 3.3 and JEF 3.1. + +.. _fig10-2a-4: +.. figure:: figs/COVLIBAppA/fig4.png + :align: center + :width: 600 + + :sup:`233` fission and capture comparison between ENDF/B-VI, + JENDL 3.3 and JEF 3.1. + +.. _fig10-2a-5: +.. figure:: figs/COVLIBAppA/fig5.png + :align: center + :width: 600 + + :sup:`235` fission and capture comparison between ENDF/B-VI, + JENDL 3.3 and JEF 3.1. + +.. _fig10-2a-6: +.. figure:: figs/COVLIBAppA/fig6.png + :align: center + :width: 600 + + :sup:`238`\ U capture comparison between ENDF/B-VI, JENDL 3.3 + and JEF 3.1. + +.. _fig10-2a-7: +.. figure:: figs/COVLIBAppA/fig7.png + :align: center + :width: 600 + + :sup:`232`\ Th capture comparison between ENDF/B-VII + (beta2), ENDF/B‑VI, JENDL 3.3 and JEF 3.1. + +.. _fig10-2a-8: +.. figure:: figs/COVLIBAppA/fig8.png + :align: center + :width: 600 + + :sup:`10`\ B capture and :sup:`3`\ He elastic comparison + between ENDF/B-VII (beta2), ENDF/B-VI, JENDL 3.3 and JEF 3.1. + +.. _fig10-2a-9: +.. figure:: figs/COVLIBAppA/fig9.png + :align: center + :width: 600 + + :sup:`1`\ H and :sup:`2`\ H elastic comparison between + ENDF/B-VII (beta2), ENDF/B-VI, JENDL 3.3 and JEF 3.1. + +.. _fig10-2a-10: +.. figure:: figs/COVLIBAppA/fig10.png + :align: center + :width: 600 + + :sup:`152`\ Gd and :sup:`154`\ Gd capture comparison + between ENDF/B-VII (beta2), ENDF/B-VI, JENDL 3.3 and JEF 3.1. + +.. _fig10-2a-11: +.. figure:: figs/COVLIBAppA/fig11.png + :align: center + :width: 500 + + :sup:`155`\ Gd capture comparison between ENDF/B-VII + (beta2), ENDF/B-VI, JENDL 3.3 and JEF 3.1. diff --git a/CRAWDAD.rst b/CRAWDAD.rst index a506558..7a5d51f 100644 --- a/CRAWDAD.rst +++ b/CRAWDAD.rst @@ -10,7 +10,7 @@ ACKNOWLEDGMENTS The authors would like to acknowledge the important contributions to CRAWDAD made by former ORNL staff N. M. Greene and D. F. Hollenbach. -.. _7-1: +.. _7-7-1: Introduction ------------ diff --git a/Depletion, Activation, and Spent Fuel Source Terms Overview.rst b/Depletion, Activation, and Spent Fuel Source Terms Overview.rst new file mode 100644 index 0000000..50d2925 --- /dev/null +++ b/Depletion, Activation, and Spent Fuel Source Terms Overview.rst @@ -0,0 +1,106 @@ +.. _5-0: + +Depletion, Activation, and Spent Fuel Source Terms Overview +=========================================================== + +*Introduction by W. A. Wieselquist* + +SCALE’s general depletion, activation, and spent fuel source terms analysis +capabilities are enabled through a family of modules related to the main ORIGEN +depletion/irradiation/decay solver. The nuclide tracking in ORIGEN is based on +the principle of explicitly modeling all available nuclides and transitions in +the current fundamental nuclear data for decay and neutron-induced transmutation +and relies on fundamental cross section and decay data in ENDF/B VII. Cross +section data for materials and reaction processes not available in ENDF/B-VII +are obtained from the JEFF-3.0/A special purpose European activation library +containing 774 materials and 23 reaction channels with 12,617 neutron-induced +reactions below 20 MeV. Resonance cross section corrections in the resolved and +unresolved range are performed using a continuous-energy treatment by data +modules in SCALE. All nuclear decay data, fission product yields, and gamma-ray +emission data are developed from ENDF/B-VII.1 evaluations. Decay data include +all ground and metastable state nuclides with half-lives greater than 1 +millisecond. Using these data sources, ORIGEN currently tracks 174 actinides, +1149 fission products, and 974 activation products. The purpose of this chapter +is to describe the stand-alone capabilities and underlying methodology of +ORIGEN—as opposed to the integrated depletion capability it provides in all +coupled neutron transport/depletion sequences in SCALE, as described in other +chapters. Through the stand-alone capabilities, there is generality to handle +arbitrary systems (e.g., fast reactor fuel depletion or structural activation) +by providing arbitrary flux spectra and arbitrary one-group cross sections to +the module COUPLE, which in turn creates ORIGEN library (.f33) files containing +the problem-dependent, one-group reaction coefficients required to solve the +actual equations governing depletion/decay. These libraries are required input +for the ORIGEN module, along with the initial isotopics and irradiation history, +in terms of either a time-dependent power or flux level. Two high-performance +equation solvers are available: the hybrid linear chains and matrix exponential +method and a new Chebyshev Rational Approximation Method (CRAM). Typical +execution times are on the order of a few seconds for a multi-step solution, +with each individual solution (step) taking approximately 10 milliseconds. +ORIGEN also includes capabilities for continuous feed and removal by element. +Output capabilities include isotopics (moles or grams), source spectra (alpha, +beta, gamma, and neutron), activity (becquerels or curies), decay heat (total +watts or gamma only), and radiological hazard factors (maximum permissible +concentrations in air or water). These results can be displayed in the output +file (.out extension) and/or archived in an ORIGEN binary results file (.f71 +extension). The use of current, fundamental data resources is a key feature of +ORIGEN, including nuclear decay data, multigroup neutron reaction cross +sections, neutron-induced fission product yields, and decay emission data for +photons, neutrons, alpha particles, and beta particles. The nuclear decay data +are based primarily on ENDF/B-VII.1 evaluations. The multigroup nuclear reaction +cross section libraries now include evaluations from the JEFF 3.0/A neutron +activation file containing data for 774 target nuclides, more than 12,000 +neutron-induced reactions, and more than 20 different reaction types below 20 +MeV, provided in various energy group structures. Energy-dependent +ENDF/B-VII.0-based fission product yields are available for 30 fissionable +actinides. Gamma-ray and x-ray emission data libraries are based on +ENDF/B-VII.1. The photon libraries contain discrete photon line energy and +intensity data for decay gamma-ray and x-rays emission for 1,132 radionuclides, +prompt and delayed continuum spectra for spontaneous fission, (α,n) reactions in +oxide fuel, and bremsstrahlung from decay beta (electron and positron) particles +slowing down in either a UO2 fuel or water matrix. Methods and data libraries +used to calculate the neutron yields and energy spectra for spontaneous fission, +(α,n) reactions, and delayed neutron emission are adopted from the SOURCES4C +code. Capabilities to calculate the beta and alpha particle emission source and +spectra have also been added. + +The ORIGEN reactor libraries distributed with SCALE include a set of +pre-calculated ORIGEN libraries (with TRITON) for a variety of fuel assembly +designs: + + - BWR 7×7, 8×8-1, 8×8-2, 9×9-2, 9×9-9, 10×10-9, 10×10-8, SVEA-64, + SVEA-96, and SVEA-100; + + - PWR 14×14, 15×15, 16×16, 17×17, 18×18; + + - CANDU reactor (19-, 28-, and 37-element bundle designs); + + - Magnox graphite reactor; + + - Advanced Gas-Cooled Reactor (AGR); + + - VVER 440 and VVER 1000; + + - RBMK; + + - IRT; + + - MOX BWR 7×7, 8×8-1, 8×8-2, 9×9-2, 9×9-9, 10×10-9, 10×10-8, SVEA-64, SVEA-96, and + SVEA-100; + + - MOX PWR 14×14, 15×15, 16×16, 17×17, 18×18. + +These libraries may be +used to rapidly assess spent fuel isotopics and source terms in these systems +for arbitrary burnups and decay times. For UO2-based assembly isotopics, the new +ORIGAMI sequence provides a very convenient, easy-to-use interface. The most +general capability, and requiring more user input, is available using the ARP +interpolator module in conjunction with the ORIGEN solver module. Finally, with +regards to user interfaces, ORIGEN has a new keyword-based input in SCALE 6.2 +but also maintains the ability to read SCALE 6.1 input. Both ORIGEN and ORIGAMI +are tightly integrated with the SCALE graphical user interface, Fulcrum, which +includes syntax highlighting, input checking with immediate feedback, and (.f71) +output viewing. Additionally, Fulcrum provides an ORIGAMI Automator project +interface to characterize the fuel inventory for an entire reactor site and +generate data needed for severe accident analysis. ORIGAMI Automator is not +documented in this chapter, but a primer is available with step by step +instructions on its use. diff --git a/Deterministic Transport Intro.rst b/Deterministic Transport Intro.rst new file mode 100644 index 0000000..e753fe8 --- /dev/null +++ b/Deterministic Transport Intro.rst @@ -0,0 +1,116 @@ +Deterministic Transport Overview +================================ + +*Introduction by S. M. Bowman* + +SCALE deterministic transport capabilities enable criticality safety, +depletion, sensitivity, and uncertainty analysis, as well as hybrid +approaches to Monte Carlo analysis. SCALE provides a one-dimensional +(1D) transport solver for eigenvalue neutronics and fixed source +neutron-gamma analysis with XSDRN, two-dimensional (2D) eigenvalue +neutronics with NEWT, and a three-dimensional (3D) transport solver for +hybrid acceleration of Monte Carlo fixed source and eigenvalue +calculations with Denovo. Generally, the use of these transport solvers +in SCALE is best accessed through the capability specific sequences: +CSAS and Sourcerer for criticality safety, TRITON for 1D and 2D +depletion, TSUNAMI‑1D and TSUNAMI-2D for sensitivity and uncertainty +analysis, and MAVRIC for 3D fixed source hybrid Monte Carlo analysis. + +XSDRN +----- + +XSDRN is a multigroup discrete-ordinates code that solves the 1D +Boltzmann equation in slab, cylindrical, or spherical coordinates. +Alternatively, the user can select P1 diffusion theory, infinite medium +theory, or Bn theory. A variety of calculational types is available, +including fixed source, eigenvalue, or search calculations. In SCALE, +XSDRN is used for several purposes: eigenvalue (*k*\ :sub:`eff`) determination; +cross section collapsing; and computation of fundamental-mode or +generalized adjoint functions for sensitivity analysis. + +NEWT +---- + +NEWT (New ESC-based Weighting Transport code) is a multigroup +discrete-ordinates radiation transport computer code with flexible +meshing capabilities that allow 2D neutron transport calculations using +complex geometric models. The differencing scheme employed by NEWT—the +Extended Step Characteristic approach—allows a computational mesh based +on arbitrary polygons. Such a mesh can be used to closely approximate +curved or irregular surfaces to provide the capability to model problems +that were formerly difficult or impractical to model directly with +discrete-ordinates methods. Automated grid generation capabilities +provide a simplified user input specification in which elementary bodies +can be defined and placed within a problem domain. NEWT can be used for +eigenvalue, critical-buckling correction, and source calculations, and +it can be used to prepare collapsed weighted cross sections in AMPX +working library format. + +Like other SCALE modules, NEWT can be run as a standalone module or as +part of a SCALE sequence. NEWT has been incorporated into SCALE TRITON +control module sequences. TRITON can be used simply to prepare +cross sections for a NEWT transport calculation and then automatically +execute NEWT. TRITON also provides the capability to perform 2D +depletion calculations in which the transport capabilities of NEWT are +combined with multiple ORIGEN depletion calculations to perform 2D +depletion of complex geometries. In the TRITON depletion sequence, NEWT +can also be used to generate lattice-physics parameters and +cross sections for use in subsequent nodal core simulator calculations. +In addition, the SCALE TSUNAMI-2D sequence can be used to perform +sensitivity and uncertainty analysis of 2D geometries in which NEWT is +used to compute the adjoint flux solution to generate sensitivity +coefficients for *k\ eff* and other responses of interest with respect +to the cross sections used in the NEWT model. + +DENOVO +------ + +Denovo [1]_ is a parallel 3D discrete-ordinates code available in SCALE +as part of two control module sequences for different applications, as +described below. Because Denovo can only be run in SCALE via the Monaco +with Automated Variance Reduction using Importance Calculations (MAVRIC) +or Denovo Eigenvalue Calculation (DEVC) as developed for use with +Sourcerer, it is not documented separately in the section entitled +“Deterministic Transport” in this manual. + +The MAVRIC hybrid Monte Carlo radiation shielding sequence employs the +Consistent Adjoint Driven Importance Sampling (CADIS) and +Forward-Weighted CADIS (FW-CADIS) methodologies. Denovo is used to +generate adjoint (and, for FW-CADIS, forward) scalar fluxes for the +CADIS methods in MAVRIC. This adjoint flux information is then used by +MAVRIC to construct a space- and energy-dependent importance map (i.e., +weight windows) to be used for biasing during Monte Carlo particle +transport and as a mesh-based biased source distribution. For use in +MAVRIC/CADIS, it is highly desirable that the S\ :sub:`N` code be fast, +positive, and robust. The phase-space shape of the forward and adjoint +fluxes, as opposed to a highly accurate solution, is the most important +quality for Monte Carlo weight-window generation. Accordingly, Denovo +provides a step-characteristics spatial differencing option that +produces positive scalar fluxes as long as the source (volume plus +in-scatter) is positive. Denovo uses an orthogonal, nonuniform mesh that +is ideal for CADIS applications because of the speed and robustness of +calculations on this mesh type. Denovo can be run stand-alone in MAVRIC +to perform fixed source calculations using the *PARM=forward* (for +forward Denovo) or *PARM=adjoint* (for adjoint Denovo). See the MAVRIC +chapter for details. + +The other sequence that uses Denovo is the DEVC sequence. DEVC generates +a reasonably accurate starting source through a Denovo eigenvalue +calculation so that Sourcerer can improve the KENO/CSAS Monte Carlo +calculation by (1) reducing the number of skipped generations required +to converge the fission source distribution in the KENO solution, and +(2) increasing the reliability of the final eigenvalue +(:math:`k_{\mathrm{\text{eff}}}`) for problems with loosely coupled +fissionable areas. Denovo can be run stand-alone in DEVC for calculating +criticality eigenvalue problems. This sequence reads an input file very +similar to a CSAS6 input file that contains an extra block of input for +describing the Denovo mesh grid and calculational parameters. See the +Sourcerer chapter for details. + +Reference +--------- + +.. [1] + T. M. Evans, A. S. Stafford, R. N. Slaybaugh, and K. T. Clarno, + “Denovo: A New Three-Dimensional Parallel Discrete Ordinates Code in + SCALE,” *Nuclear Technology* **171**, 171–200 (2010). diff --git a/FIDO.rst b/FIDO.rst new file mode 100644 index 0000000..4176566 --- /dev/null +++ b/FIDO.rst @@ -0,0 +1,344 @@ +.. _11-5: + +FIDO Input System +================= + +*L. M. Petrie* + +ABSTRACT + +This document provides a description of the FIDO input system being used +in conjunction with several SCALE functional modules. The FIDO system is +a widely used method of entering or modifying large data arrays with +minimum effort. Special advantage is taken of patterns of repetition or +symmetry whenever possible. + + +ACKNOWLEDGMENTS + +This document was funded by the Office of Nuclear Material Safety and +Safeguards, U.S. Nuclear Regulatory Commission, for inclusion within the +SCALE manual. + +.. _11-5-1: + +Introduction +------------ + +The FIDO input method is specially devised to allow entering or +modifying large data arrays with minimum effort. Advantage is taken of +patterns of repetition or symmetry wherever possible. The FIDO system +was patterned after the input method used with the FLOCO coding system +at Los Alamos and was first applied to the DTF-II code. Since that time, +numerous features requested by users have been added, a free-field +option has been developed, and the application of FIDO has spread to +innumerable codes. Starting with SCALE 5, the FIDO routines have been +converted to Fortran 90, and the requirement that arrays be held in a +large container array has been removed. + +The data are entered in units called “arrays.” An array comprises a +group of contiguous storage locations that are to be filled with data at +the same time. These arrays usually correspond on a one-to-one basis +with Fortran arrays used in the program. A group of one or more arrays +read with a single call to the FIDO package forms a “block,” and a +special delimiter is required to signify the end of each block. Arrays +within a block may be read in any order with respect to each other, but +an array belonging to one block must not be shifted to another. The same +array can be entered repeatedly within the same block. For example, an +array could be filled with “0” using a special option, and then a few +scattered locations could be changed by reading in a new set of data for +that array. If no entries to the arrays in a block are required, the +delimiter alone satisfies the input requirement. + +Three major types of input are available: fixed-field input, free-field +input, and user-field input. + +.. _11-5-2: + +Fixed-Field Input +----------------- + +The fixed-field input option is documented here for completeness. + +.. note:: The use of fixed-field input is NOT recommended. Use the free-field input + option documented in :ref:`11-5-3`. + +Each record is divided into six 12-column data fields, each of which is +divided into three subfields. The following sketch illustrates a typical +data field. The three subfields always comprise 2, 1, and 9 columns, +respectively. + +.. image:: figs/FIDO/img1.png + :align: center + :width: 500 + +To begin the first array of a block, an array originator field is placed +in any field on a record: + +Subfield 1: An integer array identifier < 100 specifying the data array to be read in. + +Subfield 2: An array-type indicator: + + “$” if the array is integer data + + “*” if the array is real data + + “#” if the array is double-precision data + +Subfield 3: Blank + +Data are then placed in successive fields until the required number of +entries has been accounted for. + +In entering data, it is convenient to think of an “index” or “pointer” +as a designator that is under the control of the user and which +specifies the position in the array into which the next data entry is to +go. The pointer is always positioned at array location #1 by entering +the array originator field. The pointer subsequently moves according to +the data operator chosen. Blank fields are a special case in that they +do not cause any data modification and do not move the pointer. + +A data field has the following form: + +Subfield 1: The data numerator, an integer <100. We refer to this entry +as N\ :sub:`1` in the following discussion. + +Subfield 2: One of the special data operators listed below. + +Subfield 3: A nine-character data entry, to be read in F9.0 format. It +will be converted to an integer if the array is a “$” array or if a +special array operator such as Q is being used. Note that an exponent is +permissible but not required. Likewise, a decimal is permissible but not +required. If no decimal is supplied, it is assumed to be immediately to +the left of the exponent, if any; and otherwise to the right of the last +column. This entry is referred to as N3 in the following discussion. + +A list of data operators and their effect on the array being input +follows: + +“Blank” + +indicates a single entry of data. The data entry in the third +subfield is entered in the location indicated by the pointer, and the +pointer is advanced by one. However, an entirely blank field is ignored. + +“+” or “–” + +indicates exponentiation. The data entry in the third field +is entered and multiplied by :math:`10^{\pm N_{1}}` where N\ :sub:`1` is the data numerator in +the first subfield, given the sign indicated by the data operator +itself. The pointer advances by one. In cases where an exponent is +needed, this option allows the entering of more significant figures than +the blank option. + +“&” + +has the same effect as “+.” + +“R” + +indicates that the data entry is to be repeated N\ :sub:`1` times. +The pointer advances by N\ :sub:`1`. The entry 5R1 is equivalent to 1 1 +1 1 1. + +“I” + +indicates linear interpolation. The data numerator, N\ :sub:`1`, +indicates the number of interpolated points to be supplied. The data +entry in the third subfield N\ :sub:`3` is entered, followed by Nj +interpolated entries equally spaced between that value and the data +entry found in the third subfield of the next nonblank field. The +pointer is advanced by N\ :sub:`1` + 1. The field following an “I” field +is than processed normally, according to its own data operator. The “I” +entry is especially valuable for specifying a spatial mesh. For example, +the entry 3I 10 50 is equivalent to 10 20 30 40 50. In “$” arrays, +interpolated values will be rounded to the nearest integer. + +“L” + +indicates logarithmic interpolation. The effect is the same as that +of “I” except that the resulting data are evenly separated in log-space. +This feature is especially convenient for specifying an energy mesh. For +example, the entry 3L 1 1+4 is equivalent to 1 10 100 1000 10000. + +“Q” + +is used to repeat sequences of numbers. The length of the sequence +is given by the third subfield, N\ :sub:`3`. The sequence of N\ :sub:`3` +entries is to be repeated N\ :sub:`1` times. The pointer advances by +N\ :sub:`1`\ \*N\ :sub:`3`. If either N\ :sub:`1` or N\ :sub:`3` is 0, +then a sequence of N\ :sub:`1` + N\ :sub:`3` is repeated one time only, +and the pointer advances by N\ :sub:`1` + N\ :sub:`3`. This feature is +especially valuable for geometry specification. + +The “N” option + +has the same effect as “Q,” except that the order of the +sequence is reversed each time it is entered. This feature is valuable +for the type of symmetry possessed by S\ :sub:`n` quadrature +coefficients. + +“M” + +has the same effect as “N,” except that the sign of each entry in +the sequence is reversed each time the sequence is entered. For example, +the entries + + 1 2 3 2M2 + + would be equivalent to + + 1 2 3 –3 –2 2 3. + +This option is also useful in entering discrete ordinates angular +quadrature coefficients. + +“Z” + +causes N\ :sub:`1` + N\ :sub:`3` locations to be set at 0. The +pointer is advanced by N\ :sub:`1` + N\ :sub:`3`. + +“C” + +causes the position of the last array entered to be printed. This is +the position of the pointer, less 1. The pointer is not moved. + +“O” + +causes the print trigger to be changed. The trigger is originally +off. Successive “O” fields turn it on and off alternately. When the +trigger is on, each record is listed as it is read. + +“S” + +indicates that the pointer is to skip N\ :sub:`1` positions leaving +those array positions unchanged. If the third subfield is blank, the +pointer is advanced by N\ :sub:`1`. If the third subfield is nonblank, +that data entry is entered following the skip, and the pointer is +advanced by N\ :sub:`1` + 1. + +“A” + +moves the pointer to the position, N\ :sub:`3` specified in the +third subfield. + +"F" + +fills the remainder of the array with the datum entered in the third +subfield. For example, F9 will fill all positions of the array with a +value of 9. + +“E” + +skips over the remainder of the array. The array length criterion is +always satisfied by an E, no matter how many entries have been +specified. No more entries to an array may be given following an “E,” +except that data entry may be restarted with an “A.” + +The reading of data to an array is terminated when a new array origin +field is supplied, or when the block is terminated. If an incorrect +number of positions has been filled, an error edit is given; and a flag +is set which will later abort execution of the problem. FIDO then +continues with the next array if an array origin was read. Otherwise, +control is returned to the calling program. + +A block termination consists of a field having “T” in the second +subfield. Entries following “T” on a record are ignored, and control is +returned from FIDO to the calling program. + +Comment records can be entered within a block by placing an apostrophe +(') in column 1. Then columns 2–80 will be listed, with column 2 being +used for printer carriage control. Such records have no effect on the +data array or pointer. + +.. _11-5-3: + +Free-Field Input +---------------- + +With free-field input, data are written without fixed restrictions as to +field and subfield size and positioning on the record. The options used +with fixed-field input are available, although some are slightly +restricted in form. In general, fewer data records are required for a +problem, the interpreting print is easier to read, a record listing is +more intelligible, the records are easier to enter, and certain common +data entry errors are tolerated without affecting the problem. Data +arrays using fixed- and free-field input can be intermingled at will +within a given block. + +The concept of three subfields per field is still applicable to +free-field input; but if no entry for a field is required, no space for +it need be left. Only columns 1–72 may be used, as with fixed-field +input. A field may not be split across records. + +The array originator field can begin in any position. The array +identifiers and type indicators are used as in fixed-field input. The +type indicator is entered twice to designate free-field input (i.e., +“$$,” “\*\*,” or “##”). The blank third subfield required in fixed-field +input is not required. For example, + + 31*\* + +indicates that array 31, a real-data array, will follow in free-field +format. + +Data fields may follow the array origin field immediately. The data +field entries are identical to the fixed-field entries with the +following restrictions: + +1. Any number of blanks may separate fields, but at least one blank must +follow a third subfield entry if one is used. + +2. If both first- and second-subfield entries are used, no blanks may +separate them (i.e., 24S, but not 24 S). + +3. Numbers written with exponents must not have imbedded blanks (i.e., +1.0E+4, 1.0−E4, 1.0+4, or even 1+4, but *not* 1.0 E4). A zero should +never be entered with an exponent. For example, 0.00 − 5 or 0.00E − 5 +will be interpreted as − 5 × 10\ :sup:`–2`. + +4. In third-subfield data entries only 9 digits, including the decimal +but not including the exponent field, can be used (i.e., +123456.89E07, but *not* 123456.789E07). + +5. The Z entry must be of the form: 738Z, *not* Z738 or 738 Z. + +6. The + or − data operators are not needed and are not available. + +7. The Q, N, and M entries are restricted: 3Q4, 1N4, M4, but *not* 4Q, +4N, or 4M. + +.. _11-5-4: + +User-Field Input +---------------- + +If the user follows the array identifier in the array originator field +with the character “U” or “V,” the input format is to be specified by +the user. If “U” is specified, the FORTRAN FORMAT to be used must be +supplied in columns 1–72 of the next record. The format must be enclosed +by the usual parentheses. Then the data for the entire array must follow +on successive records. The rules of ordinary FORTRAN input as to +exponents, blanks, etc., apply. If the array data do not fill the last +record, the remainder must be left blank. + +“V” has the same effect as “U,” except that the format read in the last +preceding “U” array is used. + +.. _11-5-5: + +Character Input +--------------- + +If the user wishes to enter character data into an array, at least three +options are available. The user may specify an arbitrary format using a +“U” and reading in the format. The user may follow the array identifier +by a “/.” The next two entries into subfield 3 specify the beginning and +ending indices in the array into which data will be read. The character +data are then read starting with the next data record in an 18A4 format +if going to a real or integer array, and 9AB if going to a double +precision array. + +Finally, the user may specify the array as a free-form “*” array and +then specify the data entries as “nH” character data where n specifies +how many characters follow H. diff --git a/ICE.rst b/ICE.rst new file mode 100644 index 0000000..7ae9676 --- /dev/null +++ b/ICE.rst @@ -0,0 +1,295 @@ +.. _11-4: + +ICE: Module to Mix Multigroup Cross Sections +============================================ + +*N. M. Greene,*\ :sup:`\*` *L. M. Petrie, S. K. Fraley*\ :sup:`\*` [1]_ + +ABSTRACT + +ICE is a legacy SCALE utility program that reads microscopic cross +sections from an AMPX working library and uses input mixture number +densities to produce macroscopic cross sections, which are written to an +output file in the AMPX working library format. User input is entered +with the FIDO procedures. + +.. _11-4-1: + +Introduction +------------ + +ICE (**I**\ ntermixed **C**\ ross Sections **E**\ ffortlessly) is a +legacy SCALE utility program that reads microscopic cross sections from +an AMPX working library and uses input mixture number densities to +produce macroscopic cross sections, which are written to an AMPX working +library output file. The code was originally developed to allow +efficient cross section mixing with minimum user effort and with reduced +core storage requirements. The SCALE version of ICE is the latest in a +series [2]_ of versions of the program. + +In previous versions of SCALE several sequences employed the ICE module +as a component in the self-shielding procedure; however in modern +sequences the functionality of ICE has been replaced by new routines in +the XSProc module. The ICE module is retained in the modern version of +SCALE mainly for use as a standalone executable module to compute +macroscopic cross sections and to provide backward compatibility with +legacy inputs. + +.. _11-4-2: + +Cross Section Mixing Expressions +-------------------------------- + +The mixing operations in ICE use the simple expressions presented below. + +.. _11-4-2-1: + +Cross-section mixing for AMPX libraries +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +For the options that produce AMPX working libraries, the mixing of +cross sections involves a very simple summing of constituent values +times a number density for the constituent, that is, Σ, a macroscopic +value, is determined by + +.. math:: + + \Sigma=\sum_{j} N_{j} \sigma_{j} + + +where the j are the individual nuclides in the mixture whose number +density and microscopic cross sections are N\ :sub:`j` and σ\ :sub:`j`, +respectively. + +The only exceptions to the above rule are for fissionable mixtures where +the number of neutrons per fission, υ\ :sub:`g`, or a fission spectrum, +χ\ :sub:`g`, is required: + +.. math:: + + v_{\mathrm{g}}=\frac{\sum_{j} N_{j} v_{g j} \sigma_{g j}}{\sum_{j} N_{j} \sigma_{f g j}} + +χ\ :sub:`g` is defined as the fraction of the fission neutrons produced +by the mixture which fall in group g. By definition, + +.. math:: + + \sum_{\mathrm{g}} \chi_{\mathrm{g}}=1.0 + +ICE uses the following scheme to determine χ. First, terms F\ :sub:`g` +are determined by + +.. math:: + :label: eq11-4-1 + + \mathrm{F}_{\mathrm{g}}=\sum_{\mathrm{j}} \mathrm{N}_{\mathrm{j}} \chi_{\mathrm{g}, \mathrm{j}} \sum_{\mathrm{g}^{\prime}} \overline{\mathrm{v} \sigma}_{\mathrm{fg}^{\prime}, \mathrm{j}} \hat{\phi}_{\mathrm{g}^{\prime}} , + + +where :math:`\text { VO } \mathrm{fg}, \mathrm{j}` is the average of the product of υ times σ\ :sub:`f` for the +nuclide, χ\ :sub:`g,j` is the nuclide fission spectrum, and :math:`\hat{\phi}_{\mathrm{g}}` is an +estimate for the integrated flux in group g´. Once the F\ :sub:`g` are +determined, χ\ :sub:`g` is determined by normalizing the sum of +F\ :sub:`g` to unity. + +In many AMPX libraries, the integrals of the spectrum used to determine +the multigroup values are carried on the library for each nuclide. ICE +uses this nuclide-dependent spectrum to determine χ\ :sub:`g`. This +option should be exercised with caution, however, for no attempt is made +to ensure that the individual spectra are consistently normalized. + +.. _11-4-3: + +Input Instructions +------------------ + +The input to ICE uses the FIDO schemes described in the FIDO chapter. In +the descriptions, the number of entries expected in an array is given in +square brackets. + +\*******************************************************************************\* + +Card A (20A4) + +Title card + +Data Block 1 + +–1$ Direct-Access Specifications [4] + + 1. NB8 No longer used. + + 2. NL8 No longer used. + + 3. NB9 No longer used. + + 4. NL9 No longer used. + +0$ Logical Unit Specifications for Various Cross-Section Libraries [5] + + 1. INTAPE Input AMPX working library unit; default 4. + + 2. IOT1 Output AMPX working library unit; default 3. + + 3. IOT2 No longer used. + + 4. IOT3 No longer used. + + 5. IOT4 No longer used. + +1$ Problem Size and Major Options [7] + + 1. MIX Number of cross-section mixtures to be made. + + 2. NMIX Number of mixing operations (elements times density operations) to be performed. + + 3. IFLAG(1) Set greater than ten if AMPX working library output desired + + 4. IFLAG(2) No longer used.. + + 5. IFLAG(3) No longer used.. + + 6. IFLAG(4) No longer used. + + 7. KOPT No longer used. + +T - Terminate Block 1 + +Data Block 2 + +2$ [NMIX] + +1. (KM(I),I=1,NMIX) Mixture numbers in the mixture specification table – +values range from 1 to MIX. + +3$ [NMIX] + +1. (KE(I),I=1,NMIX) Element identifiers for the mixture specification +table. + +4\* [NMIX] + +1. (RHO(I),I=1,NMIX) Atom densities for the mixture specification table. + +5$ [MIX] + +1. (NCOEF(I),I=1,MIX) Number of Legendre coefficients, including +P\ :sub:`o`, to be mixed for each mixture. + +6\* [NG+4] + +No longer used. + +12$ [NMIX] + +\`1.(NUCMX(I),I=1,NMIX) Element mixture identifiers for the mixture +specification table. + +7$ No longer used. + +T - Terminate Data Block 2 + +Data Block 3 + +8$ [MIX] Required only if IFLAG(1) > 0 + +1. (MID(I),I=1,MIX) Mixture ID numbers for AMPX working library; +default (MID(I)=I,I=1,MIX) + +9$ [N] No longer used. + +. + +. + +10 No longer used. + +11 No longer used. + +T - Terminate Data Block 3 + +.. _11-4-4: + +Sample Problem +-------------- + +A simple case has been selected to demonstrate the use of ICE. In this +case, it is desired to produce mixture cross sections for UO\ :sub:`2` +and H\ :sub:`2`\ O using basic data from ENDF version 7 238 group SCALE +library. Information pertinent to the basic data is given in the +following table: + ++---------------+--+------------+--+---------------------+ +| Nuclide | | Identifier | | Order of Scattering | ++===============+==+============+==+=====================+ +| :sup:`235`\ U | | 92235 | | 5 | ++---------------+--+------------+--+---------------------+ +| :sup:`238`\ U | | 92238 | | 5 | ++---------------+--+------------+--+---------------------+ +| O | | 8016 | | 5 | ++---------------+--+------------+--+---------------------+ +| H | | 1001 | | 5 | ++---------------+--+------------+--+---------------------+ + +The atom densities to be used are: + +UO\ :sub:`2` + + N(\ :sup:`235`\ U) = 0.01 atoms/(barn-cm) + + N(\ :sup:`238`\ U) = 0.04 atoms/(barn-cm) + + N(O) = 0.08 atoms/(barn-cm) + +Water + + N(H) = 0.06 atoms/(barn-cm) + + N(O) = 0.03 atoms/(barn-cm) + +In the sample case, we have elected to make an AMPX working library on +logical 61, + +We have selected further to identify UO\ :sub:`2` with a 111 on the AMPX +working library. + +CSAS-MG PARM=CHECK is run to set up the master library, then WORKER is +run to produce a working library for ICE. + +A listing of the input follows: + +.. highlight:: scale + +:: + + =csas-mg parm=(check) + cross sections for ice sample problem + v7-238 + read composition + atom 1 1 4 1001 1 8016 1 92235 1 92238 1 end atom + end composition + end + =ice + sample ice problem + 0$$ 4 61 62 63 64 + 1$$ 2 5 13 13 13 13 2 1t + 2$$ 3r1 2r2 + 3$$ 92235 92238 8016 1001 8016 + 4** 0.01 0.04 0.08 0.06 0.03 + 5$$ 1 2 + 12$$ f1 + 2t + 8$$ 111 222 9$$ 1 2 3 11$$ 100 1111 2222 3t + End + + + +Reference +~~~~~~~~~ + +.. [1] + :sup:`∗` Formerly with Oak Ridge National Laboratory. + +.. [2] + S. K. Fraley, *User’s Guide for ICE,* ORNL/CSD/TM-9, Union Carbide + Corporation (Nuclear Division), Oak Ridge National Laboratory, July + 1976. diff --git a/MALOCS2.rst b/MALOCS2.rst new file mode 100644 index 0000000..7c22048 --- /dev/null +++ b/MALOCS2.rst @@ -0,0 +1,210 @@ +.. _11-6: + +MALOCS2: Module To Collapse AMPX Master Cross Section libraries +=============================================================== + +*L.M. Petrie* + +.. _11-6-1: + +Introduction +------------ + +MALOCS2 (**M**\ iniature **A**\ MPX **L**\ ibrary **O**\ f **C**\ ross +**S**\ ections) is a module to collapse AMPX master cross-section +libraries. The SCALE MALOCS2 module is an extension of the AMPX module +MALOCS. MALOCS2 provides capability to read the collapsing spectrum from +the output flux file produced by XSDRNPM, and also has extended options +for collapsing Legendre moments of the 2D elastic scattering matrix. The +module can be used to collapse neutron, gamma-ray, or coupled +neutron-gamma master libraries. + +.. _11-6-2: + +MALOCS Input Data +----------------- + +.. describe:: broadfilename + + filename of the collapsed library [no default] + +.. describe:: crosssectionprint + + cross section printing option [none] + + none - don't print any cross sections + + onedxsecs - print the 1D cross sections + + twodxsecs N - print the 2D cross sections through Legendre order N + +.. describe:: epsilon + + epsilon for when to print invalid moment messages[0.05] + +.. describe:: finefilename + + filename of the input library [no default] + +.. describe:: fluxfilename + + filename of an xsdrn flux file to be used in the collapse + [no default] + +.. describe:: numgammagroups + + the number of fine gamma groups [no default] + +.. describe:: gammacollapse + + the broad group by fine group collapse structure for the + gammas + + must come after "numgammagroups" + +.. describe:: latticezones + + identifies the zones to be used as fuel, gap, clad, and + moderator [1,2,3,4] + +.. describe:: max2dweightorder + + maximum Legendre order to be collapsed [max Legendre + order of the nuclide] + +.. describe:: numneutrongroups + + the number of fine neutron groups [no default] + +.. describe:: neutroncollapse + + the broad group by fine group collapse structure for the + neutrons + + must come after "numneutrongroups" + +.. describe:: printepsilon + + not used [2.0D-6] + +.. describe:: problemfilename + + filename of the xsdrn data file that corresponds to the + flux file [no default] + +.. describe:: sigmatotalpl + + flag to turn on doing a within group correction using the + Pl weighted sigma total + + 'y' or 'yes' is true, anything else is false [true] + +.. describe:: updatechi + + flag to turn on updating the total chi + + 'y' or 'yes' is true, anything else is false [true] + +.. describe:: validate2ds + + flag to validate the Legendre moments of the collapsed 2D + cross sections + + 'y' or 'yes' is true, anything else is false [true] + +.. describe:: weighttype + + type of weighting to be done + + innercell - cell weight over a subset of the zones + + innercell is followed by the largest zone number in the innercell + + cell - cell weight over the whole cell + + zone - weight each zone independently + + region - cell weight each nuclide over only the zones it is in + + default is region + +.. describe:: wgtsource + + source of the weighting flux + + nuclideflux - use the flux from the nuclide on the fine group library + + inputflux - read a flux from input + + [default is to use an xsdrn flux] + +.. describe:: end + + terminates input stream + +.. _11-6-3: + +MALOCS Example Problem +---------------------- + +The following problem shortens the 56 group library to just the nuclides +that will be used to run a fixed source, 1-D discrete ordinates +calculation of a void sphere with a neutron source in it, surrounded by +a sphere of water, and then surrounded by an iron sphere. The flux from +the discrete ordinates problem is then used to collapse the short +library to 14 groups using a zone collapse method. Finally, the +collapsed library is listed showing the nuclides on it, and copied back +to the input directory. + +.. code:: scale + :class: long + + =shell + ln -s ${DATA}/scale.rev04.xn56v7.1 ft51f001 + end + =ajax + 0$$ 52 e + 1$$ 1 1t + 2$$ 51 8 2t + 3$$ 1001 1002 8016 8017 26054 26056 26057 26058 3t + end + =csas1 parm=bonami + generate a flx file to be used to collapse a library + v7-56n + read composition + iron 1 1.0 293.0 end iron + water 2 1.0 293.0 end water + end composition + read celldata + multiregion spherical end + 0 1.0 2 10.0 1 15.0 end zone + moredata + ievt=0 iqm=1 ntd=61 fwr=62 source(1)=15 + 0.2 0.2 0.2 0.5 0.5 0.5 0.5 0.5 0.5 0.2 0.2 0.2 0.05 0.05 0.05 + end moredata + end celldata + end + =malocs2 + ' the input fine group cross section library to be collapsed + finefilename=ft52f001 + ' the output collapsed cross section library + broadfilename=ft53f001 + ' the file with the fluxes from xsdrn to be used to collapse the XSs + fluxfilename=ft62f001 + ' the file containing the description of the xsdrn problem + problemfilename=ft61f001 + ' number of fine neutron groups + numneutrongroups=56 + ' fine group to broad group correspondence array + neutroncollapse + 4r1 4r2 4r3 4r4 4r5 4r6 4r7 4r8 4r9 4r10 4r11 4r12 4r13 4r14 + ' type of weighting to be used in doing the collapse + weighttype=zone + end + end + =paleale + 0$$ 53 e 1$$ 0 1t + end + =shell + cp ft53f001 ${OUTDIR} + end diff --git a/MCDancoff.rst b/MCDancoff.rst index d4a376f..f478d1e 100644 --- a/MCDancoff.rst +++ b/MCDancoff.rst @@ -145,7 +145,10 @@ Calculation and use of 3D Dancoff factors .. highlight:: scale - :: +.. + + +:: read param ......... diff --git a/NEWT.rst b/NEWT.rst new file mode 100644 index 0000000..2848677 --- /dev/null +++ b/NEWT.rst @@ -0,0 +1,5307 @@ +.. _9-2: + +NEWT: A New Transport Algorithm for Two-Dimensional Discrete-Ordinates Analysis in Non-Orthogonal Geometries +============================================================================================================= + +.. |Om| replace:: :math:`\Omega` + +M. A. Jessee, M. D. DeHart [1]_ + +ABSTRACT + +NEWT (**N**\ ew **E**\ SC-based **W**\ eighting **T**\ ransport code) is +a multigroup discrete-ordinates radiation transport computer code with +flexible meshing capabilities that allow two-dimensional (2-D) neutron +transport calculations using complex geometric models. The differencing +scheme employed by NEWT, the Extended Step Characteristic approach, +allows a computational mesh based on arbitrary polygons. Such a mesh can +be used to closely approximate curved or irregular surfaces to provide +the capability to model problems that were formerly difficult or +impractical to model directly with discrete-ordinates methods. Automated +grid generation capabilities provide a simplified user input +specification in which elementary bodies can be defined and placed +within a problem domain. NEWT can be used for eigenvalue, +critical-buckling correction, and source calculations and it can be used +to prepare collapsed weighted cross sections in AMPX working library +format. + +Like other SCALE modules, NEWT can be run as a standalone module or as +part of a SCALE sequence. NEWT has been incorporated into the SCALE +TRITON control module sequences. TRITON can be used simply to prepare +cross sections for a NEWT transport calculation and then automatically +execute NEWT. TRITON also provides the capability to perform 2-D +depletion calculations, in which the transport capabilities of NEWT are +combined with multiple ORIGEN depletion calculations to perform 2-D +depletion of complex geometries. In the TRITON depletion sequence, NEWT +can also be used to generate lattice-physics parameters and +cross sections for use in subsequent nodal core simulator calculations. +In addition, the SCALE TSUNAMI-2D sequence can be used to perform +sensitivity and uncertainty analysis of 2-D geometries, where NEWT is +used to compute the adjoint flux solution to generate sensitivity +coefficients for *k*\ :sub:`eff` and other responses of interest, with respect +to the cross sections used in the NEWT model. + +ACKNOWLEDGMENTS + +The author expresses gratitude to B. T. Rearden and S. M. Bowman for +their supervision of the SCALE project and review of the manuscript. The +author acknowledges R. Y. Lee of the U.S. Nuclear Regulatory Commission +(NRC) and A. P. Ulses (formerly NRC) for their support of the +development of NEWT. Appreciation is extended to G. Ilas, B. L. +Broadhead, Deokjung Lee (formerly ORNL), and B. J. Ade for their review +of this or previous versions of the manuscript. The efforts of Z. Zhong +(Argonne National Laboratory), A. P. Ulses (formerly NRC), K. S. Kim, C. +F. Weber, G. Ilas, and K. T. Clarno (Oak Ridge National Laboratory) in +methods development and testing of the code have been invaluable in the +continued evolution and improvement of the code. + +.. _9-2-1: + +Introduction +------------ + +NEWT (**N**\ ew **E**\ SC-based **W**\ eighting **T**\ ransport code) is +a two-dimensional (2-D) discrete-ordinates transport code developed +based on the Extended Step Characteristic (ESC) approach :cite:`dehart_discrete_1992` for +spatial discretization on an arbitrary mesh structure. This +discretization scheme makes NEWT an extremely powerful and versatile +tool for deterministic calculations in real-world non-orthogonal problem +domains. The NEWT computer code evolved from the earlier +proof-of-principle CENTAUR code :cite:`dehart_discrete_1992` and has been developed to run +within SCALE. Thus, NEWT uses AMPX-formatted cross sections processed by +other SCALE modules. If cross sections are properly prepared, NEWT can +be run in stand-alone mode. NEWT can also be used within the TRITON +control module for transport analysis, depletion analysis, and +sensitivity and uncertainty analysis. + +.. _9-2-1-1: + +How to use this manual +~~~~~~~~~~~~~~~~~~~~~~ + +This users’ manual is intended to assist both the novice and the expert +in the application of NEWT for transport analysis. As such, the document +is divided into subsections, each with a specific purpose. Not all +sections will be of value to all users. It is not intended that the user +of this manual read through the manual from start to end. Rather, the +manual is designed to serve as a reference, with each section meeting +different needs. This introductory section has been written to provide a +general overview of the background, nature, functionality, and +applications of NEWT; it should prove of interest to users at all +levels. :ref:`9-2-2` provides detail on the theory of NEWT in terms of +derivations, equations, and relationships used in the NEWT solution. +This information will be of interest to those with a background in +transport methods desiring a comprehensive understanding of the NEWT +solution scheme. However, this information may provide too much detail +or simply not be relevant for the beginning user or someone desiring to +improve or expand an existing model. These users will find :ref:`9-2-3` +to be of more value, where input data requirements and formats are +described in detail, along with examples of each data type. This +information is supplemented by :ref:`9-2-4`, in which complete sample +inputs with descriptions of the features of each model are provided. +:ref:`9-2-5` describes the components of an output listing obtained +from a successful NEWT calculation. + +.. _9-2-1-2: + +Background +~~~~~~~~~~ + +The radiation transport equation, a linearized derivative of the +Boltzmann equation, provides an exact description of a neutral-particle +radiation field in terms of the position, direction of travel, and +energy of every particle in the field. Both stochastic (Monte Carlo +simulation) and deterministic (direct numerical solution) forms of the +transport equation have been developed and are used extensively in +nuclear applications. Each approach has its strengths and weaknesses. +Stochastic approaches are extremely effective for problems with complex +geometries where the calculations of integral quantities, such as +radiation dose and neutron multiplication factors, are desired. However, +calculations to obtain accurate differential information, such as the +neutron flux as a function of space and energy, can be difficult and +inefficient at best and prone to inaccuracies (even if the integral +quantity is correct). Deterministic techniques, such as integral +transport, collision probability, diffusion theory, and +discrete-ordinates methods, are better suited for problems where +differential quantities, such as the neutron flux as a function of +energy or space, are desired. However, integral transport, collision +probability, and diffusion approximations are based on simplifying +assumptions, which can limit their applicability. The discrete-ordinates +approach is a more rigorous approximation to the transport equation but +is typically very limited in its flexibility to describe complex +geometric systems. + +Discrete-ordinates approaches are derived from the integro-differential +form of the Boltzmann transport equation, where space, time, and energy +dependencies are normally treated by the use of a finite‑difference +grid, while angular behavior is treated by considering a number of +discrete directions in space. The angular solution is coupled to a +scalar spatial solution via some form of numerical integration. Because +of the direct angular treatment of the discrete-ordinates approach, +angularly dependent distribution functions can be computed; thus, this +approach is the preferred method of solution in many specific +applications where angular anisotropy is important. However, as +indicated earlier, it is often limited in applicability because of the +geometric constraints of the orthogonal grid system associated with the +finite-difference numerical approximation. + +.. _9-2-1-3: + +Discrete-ordinates solution on an arbitrary grid +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The ESC approach was developed to obtain a discrete-ordinates solution +in complicated geometries to handle the needs of irregular +configurations. Deterministic solutions to the transport equation +generally calculate a solution in terms of the particle flux; the flux +is the product of particle density and speed and is a useful quantity in +the determination of reaction rates that characterize nuclear systems. +General 2-D *xy* discrete-ordinates methods perform calculations that +provide four side-averaged fluxes and a cell-averaged flux for each cell +in a rectangular problem grid; iteration is performed to obtain a +converged distribution. This approach is usually termed the +diamond-difference approach. Using the ESC approach, a more flexible and +completely arbitrary problem grid may be defined in terms of completely +arbitrary polygons. Side-averaged fluxes for each polygon in the problem +domain are computed and are used to calculate a cell-averaged flux. This +process is repeated for each cell in the problem domain, and as with the +traditional approach, iteration is performed for convergence. This +geometric flexibility is a significant enhancement to existing +technology, as it provides the capability to model problems that are +currently difficult or impractical to model directly. + +.. _9-2-1-4: + +Functions performed +~~~~~~~~~~~~~~~~~~~ + +NEWT provides multiple capabilities that can potentially be used in a +wide variety of application areas. These include 2-D eigenvalue +calculations, forward and adjoint flux solutions, multigroup flux +spectrum calculations, and cross section collapse calculations. NEWT +provides significant functionality to support lattice-physics +calculations, including assembly cross section homogenization and +collapse, calculation of assembly discontinuity factors (for internal +and reflected assemblies), diffusion coefficients, pin powers, and group +form factors. Used as part of the TRITON depletion sequence, NEWT +provides spatial fluxes, weighted multigroup cross sections, and power +distributions used for multi-material depletion calculations and coupled +depletion and branch calculations needed for lattice-physics analysis. + +.. _9-2-2: + +Theory and Procedures +--------------------- + +This section provides the theoretical basis for the ESC discretization +technique, the NEWT solution algorithm, and cross section processing +procedures used by NEWT. Although this information is not necessary to +be able to use NEWT for transport calculations, it provides a deeper +understanding of the basic operations performed within NEWT. + +.. _9-2-2-1: + +Boltzmann transport equation +~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The neutron transport equation may be presented in various forms, and +simplifications are often applied to tailor the equation to the +requirements of a specific application. In nuclear engineering +applications, the transport equation is often written in terms of the +angular neutron flux as the dependent variable. The angular neutron flux +is defined as the product of the angular neutron density and the neutron +velocity. The time-independent form of the linear transport equation is +then expressed as :cite:`duderstadt_nuclear_nodate` + +.. math:: + :label: eq9-2-1 + + \Omega \cdot \nabla \psi(\mathbf{r}, \Omega, E)+\sigma_{t}(\mathbf{r}, E) \psi(\mathbf{r}, \Omega, E)=Q(\mathbf{r}, \Omega, E) , + +where + + :math:`\psi(\mathbf{r}, \Omega, E)` ≡ angular flux at position per unit volume, in direction :math:`\Omega` per unit solid + angle and at energy E per unit energy; + + :math:`\sigma_{t}(\mathbf{r}, E)` ≡ total macroscopic cross section at position **r** and energy E; and + + Q ≡ source at position **r** per unit volume, in direction :math:`\Omega` per unit solid + angle and at energy E per unit energy. + +The source Q is generally composed of three terms: + +1. a scattering source, + +.. math:: + :label: eq9-2-2 + + S(\mathbf{r}, \Omega, E)=\int_{4 \pi} d \Omega^{\prime} \int_{0}^{\infty} d E^{\prime} \sigma_{s}\left(\mathbf{r}, \Omega^{\prime} \rightarrow \Omega, E^{\prime} \rightarrow E\right) \psi\left(\mathbf{r}, \Omega^{\prime}, E^{\prime}\right) , + +where + + :math:`\sigma_{s}\left(\mathbf{r}, \Omega^{\prime} \rightarrow \Omega, E^{\prime} \rightarrow E\right)`≡ macroscopic scattering cross section at position **r** from initial energy + E′ and direction :math:`\Omega`′ to final energy E and direction :math:`\Omega`, + +2. a fission source, + +.. math:: + :label: eq9-2-3 + + F(\mathbf{r}, \Omega, E)=\chi(\mathbf{r}, E) \int_{0}^{\infty} d E^{\prime} v\left(\mathbf{r}, E^{\prime}\right) \sigma_{f}\left(\mathbf{r}, E^{\prime}\right) \psi\left(\mathbf{r}, \Omega, E^{\prime}\right) , + +where + + :math:`\sigma_{f}\left(\mathbf{r}, E^{\prime}\right)` ≡ macroscopic fission cross section at position **r** and energy E′ (assumed + to be isotropic), + + :math:`v\left(\mathbf{r}, E^{\prime}\right)` ≡ number of neutrons released per fission event at position **r** and + energy E′, + + :math:`\chi(\mathbf{r}, E)` ≡ fraction of neutrons that are born at **r** and at energy E, and + +3. an external or fixed source, S(**r** ,E). + +In general, the transport equation can be difficult to apply and can be +solved analytically only for highly idealized cases. Hence, +simplifications and numerical approximations are often necessary to +apply the equation in engineering applications. Traditional +discrete-ordinates methods are based on a finite-difference +approximation to solve the flux streaming (leakage) term. Such +differencing schemes are intimately tied to the coordinate system in +which the differencing equations are developed, and it becomes difficult +to represent non-orthogonal volumes within that coordinate system. For +example, it is not possible to exactly represent a cylinder in a 2-D +Cartesian coordinate system; one must approximate the cylinder with a +number of rectangular cells. A close approximation can require a large +number of computational cells. However, the ESC approach for +discretizing computational cells allows the use of non-orthogonal +computational cells composed of arbitrary polygons. Using this method, +practically any shape can be represented within a Cartesian grid to a +very close approximation. The ESC approach is discussed in the following +sections. + +.. _9-2-2-2: + +The step characteristic approximation +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Efficient application of discrete-ordinates methods is difficult when +dealing with complicated non-orthogonal geometries because of the nature +of finite difference approximations for spatial derivatives. An +alternative to the discrete representation of the spatial variable is +achieved in the method of characteristics, in which the transport +equation is solved analytically along characteristic directions within a +computational cell. The angular flux  is solved along the *s*-axis, +where this axis is oriented along the characteristic direction :math:`\Omega`. Since +only the angular flux in direction :math:`\Omega` is of concern, then the streaming +term can be rewritten as + +.. math:: + :label: eq9-2-4 + + \Omega \cdot \nabla \psi(\mathbf{r}, \Omega, E)=\frac{d \psi(s, E)}{d s} . + +Hence :eq:`eq9-2-1` can be written in the characteristic form (omitting *E* for +clarity) as + +.. math:: + :label: eq9-2-5 + + \frac{d \psi(s)}{d s}+\sigma_{t}(s) \psi(\mathrm{s})=Q(s) , + +which has a solution of the form :cite:`hildebrand_advanced_1976` + +.. math:: + :label: eq9-2-6 + + \psi(s)=\psi_{0} e^{-\sigma_{t} s}+e^{-\sigma_{t} s} \int_{0}^{s} Q e^{\sigma_{t} s^{\prime}} d s^{\prime} , + +where s is the distance along the characteristic direction :math:`\Omega`, and +ψ\ :sub:`0` is the known angular flux at *s*\ =0. The value for +ψ\ :sub:`0` is given from boundary conditions for known cell sides, and +angular fluxes on unknown sides are computed using Eq. (9.2.6). Methods +for the determination of an appropriate value for ψ\ :sub:`0` and for +evaluation of the integral term vary in different solution +techniques.\ :sup:`4–9`\ :cite:`lewis_j_nodate,hildebrand_advanced_1976,alcouffe_review_1981,lathrop_spatial_1969,alcouffe_computational_1979,larsen_linear_1981,lathrop_spatial_1968`. +One of the simplest schemes employing the Method of Characteristics is +the Step Characteristic (SC) method developed by Lathrop :cite:`alcouffe_review_1981`. In +this approach, the source Q and macroscopic total cross section σt are +assumed to be constant within a computational cell and the angular flux +is assumed constant on the cell boundaries of incoming direction. +Integration of Eq. :eq:`eq9-2-6` can be performed to obtain + +.. math:: + :label: eq9-2-7 + + \psi(s)=\psi_{0} e^{-\sigma_{t} s}+\frac{Q}{\sigma_{t}}\left(1-e^{-\sigma_{t} s}\right) . + +:numref:`fig9-2-1` shows a sample computational cell in which the SC method +can be applied. For a given characteristic direction :math:`\Omega`, the angular flux +on any unknown side may be expressed in terms of a suitable average of +fluxes from known sides, which contribute to the unknown side. For the +characteristic direction :math:`\Omega` shown in :numref:`fig9-2-1`, the unknown “top” flux +ψ\ :sub:`T` may be computed as a linearly weighted average of +contributions from known sides ψ\ :sub:`B` and ψ\ :sub:`L`. The fluxes +on each of the two known sides are taken to be constant along the length +of each side, representing the average angular flux in direction :math:`\Omega` and +must be specified from external boundary conditions or from a completed +calculation in an adjacent cell. + +The set of characteristic directions is chosen from a quadrature set, so +that the resulting angular fluxes may be numerically integrated to +obtain a scalar flux. Knowing the lengths of the sides of a rectangular +cell (∆x and ∆y) and the direction cosines of :math:`\Omega` in the *x-y* plane +(μ and η), a function for the length *s* can easily be determined. The +solution for from Eq. :eq:`eq9-2-7` can then be integrated along the length of +each unknown side to determine the average angular flux of the unknown +side. Once the angular flux is known on all four sides, a neutron +balance on the cell can be used to determine the cell’s average angular +flux. + +Although the SC method described above is based on rectangular cells, +the derivation of Eq. :eq:`eq9-2-7` makes no assumptions about the shape of +the cell. It merely requires knowledge of the relationship between cell +edges along the direction of the characteristic. Hence, the method is +not restricted to any particular geometry. Because it is an extension of +the SC approach into generalized cells, the method developed here for +generalized geometries is referred to as the Extended Step +Characteristic (ESC) method. + +.. _fig9-2-1: +.. figure:: figs/NEWT/fig1.png + :align: center + :width: 400 + + Typical rectangular cell used in the step characteristic approach. + +.. _9-2-2-3: + +The Extended Step Characteristic approach +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The theory of the ESC approach is developed and explained in detail in +:cite:`dehart_discrete_1992`. However, the work has evolved significantly from that time, most +notably in the elimination of a requirement for non-reentrant polygons +(convex). The following subsections describe the primary equations +applied in the ESC approach as currently applied in NEWT. + +.. _9-2-2-3-1: + +Cell properties and geometries +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +The two primary assumptions of the ESC method are that (1) within each +computational cell all properties (i.e., σt and Q) are uniform and +(2) cell boundaries are defined by straight lines. The restriction of a +computational cell to boundaries consisting of a set of straight lines +results in computational cells that are limited to polygons. However, as +will be seen later, no restrictions are placed on the shape of the +polygon or on the number of sides in the polygon. However, the size of +the polygon will be limited. In practical applications, properties are +unlikely to remain constant over significant volumes. Thus this +approach, like many other differencing schemes, is a poor approximation +when cell volumes become too large. Although σt is a material property +and may remain spatially constant, the source term Q, which depends on +the neutron flux, will vary with position. However, since the solution +would become exact in an infinitesimally small cell, it is expected that +the approximation will be reasonable for computational cells in which +the change in the flux (and therefore the source) is small over the +domain of the cell. + +As a result of this geometric configuration, each side of a cell can +have one of three possible attributes relative to particle flow in a +given characteristic direction, as illustrated in :numref:`fig9-2-2`: (1) flow +can enter the cell when crossing a side (as shown by sides E and F in +the figure); (2) flow can exit the cell when crossing a side (sides B +and C); or (3) in a special case, flow may be parallel to the +orientation of a given side (sides A and D). Expressed mathematically, +these relationships become + +.. math:: + :label: eq9-2-8 + + \text { Category } 1: \Omega_{k} \cdot \hat{n}_{i}<0 + +.. math:: + :label: eq9-2-9 + + \text { Category } 2: \Omega_{k} \cdot \hat{n}_{i}>0 + +.. math:: + :label: eq9-2-10 + + \text { Category } 1: \Omega_{k} \cdot \hat{n}_{i}=0 + +where :math:`\hat{n}_{i}` is a unit vector in the cell-outward direction normal to +side \ *i*, and :math:`\Omega_{k}` is the *k*\ :sup:`th` discrete element of a set of +characteristic directions. A category 1 side will be termed an +“incoming” side with respect to the direction :math:`\Omega_{k}`, and a category 2 side +will be referred to as an “outgoing” side. For simplicity, the +definition of Eq. :eq:`eq9-2-10` will be included as a special case of +Eq. :eq:`eq9-2-8` for an incoming side. Thus, Eq. :eq:`eq9-2-8` can be rewritten as + +.. math:: + :label: eq9-2-11 + + \text { Side } i \text { is incoming with respect to } \Omega_{k}: \Omega_{k} \cdot \hat{n}_{i} \leq 0 + +.. math:: + :label: eq9-2-12 + + \text { Side } i \text { is outgoing with respect to } \Omega_{k}: \Omega_{k} \cdot \hat{n}_{i}>0 + +To solve for fluxes (flow) on outgoing sides of a cell, one must know fluxes on +all incoming sides. Each incoming side of each cell will be given from a +boundary condition or will be the outgoing side of an adjacent cell. + +.. _fig9-2-2: +.. figure:: figs/NEWT/fig2.png + :align: center + :width: 500 + + Orientation of the sides of a cell with respect to a given direction vector. + +.. _9-2-2-3-2: + +Relationships between cells +^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +In the ESC method, the shape of the computational cell and the form of +the neutron balance differ from that used in traditional +discrete-ordinates methods. Nevertheless, the relationships between +cells are treated essentially as they would be in traditional +approaches. The entire problem domain is mapped in terms of a set of +finite cells. Each side of each cell is adjacent to either an external +boundary condition or another cell. For each discrete direction, cells +are swept in a predetermined order beginning at a known boundary (from a +specified external boundary condition) moving in the given direction. +The precise order of sweep is such that as the solution for one cell is +obtained, the cell provides sufficient boundary conditions for the +solution of an adjacent cell. Hence, cells sharing a given side share +the value of the angular flux on that side. Knowledge of the flux on all +incoming sides of a cell is sufficient to solve for all outgoing sides. +Once the angular flux has been determined for all sides of the cell for +the given direction, it is possible to use a neutron balance to compute +the average value of the angular flux within the cell. + +The sweeping of cells continues for a given direction until all cell +fluxes have been calculated. The procedure is then repeated for the next +direction until all directions have been computed. At this point, the +cell average angular fluxes are known for each cell for each direction +used. Numerical quadrature can then be used to determine the average +scalar flux in each cell in the problem domain. The scalar fluxes are +used to determine fission and scattering reaction rates in each cell and +to update the value of the cell average source, Q. The process is +repeated, and the iteration continues until all scalar fluxes converge +to within a specified tolerance. + +This approach can be performed assuming a single energy group or any +number of discretized energy groups. The multigroup approach used in the +ESC method is the standard approach used in most multigroup methods and +is independent of the shape of each computational cell. Hence, the +details of the multigroup formalism will be omitted from this +discussion. + +.. _9-2-2-3-3: + +The set of characteristic directions +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ +.. |Omk| replace:: :math:`\Omega_{k}` + +The characteristic solution to the transport equation gives only the +angular flux in the direction of the characteristic direction vector |Omk|. +To compute interaction rates within a cell, one must compute scalar +fluxes. In computing the scalar flux from the set of angular fluxes, it +is convenient to choose the set of characteristic directions from an +appropriate quadrature set. Then the set of computed angular fluxes can +be combined with appropriate directional weights and summed to obtain a +scalar flux solution within a cell. Therefore, it is most appropriate to +choose characteristic directions from an established set of base points +and weights. Such quadrature sets that have been developed and used in +numerous earlier discrete- ordinates approaches are used in NEWT. No +restriction is placed on the nature or order of the quadrature set, as +long as it is sufficient to adequately represent the scalar flux from +computed angular fluxes. + +.. _9-2-2-3-4: + +Angular flux at a cell boundary +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +As in the development of the SC method, as well as most +finite-difference methods, the ESC approach does not explicitly +determine the flux distribution as a function of position along the +sides of a computational cell. Instead, the angular flux on each cell +side is represented in terms of the average angular flux along the +length of the side. This is sufficient to determine the net leakage +across each cell side, which is needed in order to maintain a cell +balance. An average value of the flux for an incoming side must be +specified from a boundary condition or from the prior solution of an +adjacent cell. The average flux along a given outgoing side can be +computed by integrating the flux along the side and dividing by the +length of the side. However, the form of the distribution of the angular +flux on the side must be known to perform this integration. This +distribution can be determined from the properties of the cell and from +the average flux on each of the known incoming sides. + +Because the characteristic solution [Eq. :eq:`eq9-2-6`] allows calculation of +the angular flux at any point *s* in a single cell given an initial +condition, the exact value of the flux can be computed at any point on +any outgoing side if the flux along each incoming side is known. As an +initial condition, it is assumed that the angular flux in some +characteristic direction is known at some starting point, *s* = 0 +[i.e., ψ(0) = ψ\ :sub:`0`], on an incoming side. To determine the flux at +some point on an outgoing side, one need know only the distance *s* +measured along a characteristic direction to the appropriate incoming +side. This method can then be expanded to determine a functional form of +the flux for every point on the outgoing side, which can be integrated +to produce the average outgoing flux on the side. + +To develop a mathematical relationship between two arbitrary sides of a +cell, one should first consider two arbitrary coplanar line segments in +space whose endpoints each lie on a pair of parallel lines laid in the +direction |Omk|, as shown in :numref:`fig9-2-3`. Points B\ :sub:`1` and B\ :sub:`2` +can be considered to be the “projections” of A\ :sub:`1` and +A\ :sub:`2`, respectively, relative to |Omk|. Because *s* is the distance +between a point on segment A and its projection on segment B, it can be +seen that *s* varies linearly in moving from the “beginning” to the +“end” of the pair of segments. + +.. _fig9-2-3: +.. figure:: figs/NEWT/fig3.png + :align: center + :width: 400 + + Line endpoints for computation of average fluxes. + +If α is the distance along segment B measured from endpoint B\ :sub:`1` +and B has a total length L, then the distance *s* between A and B along +direction |Omk| can be written as a linear function in terms of the +position α: + +.. math:: + :label: eq9-2-13 + + s(\alpha)=s_{1}+\left(\frac{s_{2}-s_{1}}{L}\right) \alpha , + +where *s*\ :sub:`1` and *s*\ :sub:`2` are related to the distances along +the characteristic direction between A\ :sub:`1`, B\ :sub:`1` and +A\ :sub:`2`, B\ :sub:`2`, respectively. (It is important to note that +the length *s* is the same as the distance between the endpoints only +when the characteristic vector lies in the plane of the computational +cell. This is not necessarily the case, depending on the choice of +quadrature directions. This situation is discussed in more detail +later.) + +If ψ(α) is the angular flux on side B at a distance α from B\ :sub:`1`, +then :math:`\bar{\psi}_{\mathrm{B}}`, the average value of ψ on B, is given by + +.. math:: + :label: eq9-2-14 + + \bar{\psi}_{B}=\frac{\int_{0}^{L} \psi(s(\alpha)) d \alpha}{\int_{0}^{L} d \alpha} . + +Equation :eq:`eq9-2-6`, the solution to the characteristic equation in the +step approximation, can be rewritten in terms of the average known +angular flux on side A + +.. math:: + :label: eq9-2-15 + + \psi_{B}(s)=\left(\bar{\psi}_{A}-Q / \sigma_{t}\right) e^{-\sigma_{t} s}+Q / \sigma_{t} . + +Inserting Eqs. :eq:`eq9-2-13` and :eq:`eq9-2-15` into Eq. :eq:`eq9-2-14` and simplifying +yields + +.. math:: + :label: eq9-2-16 + + \bar{\psi}_{B}=\frac{1}{L} \int_{0}^{L}\left[\left(\bar{\psi}_{A}-Q / \sigma_{t}\right) \exp \left(-\sigma_{t}\left(s_{1}+\left(\frac{s_{2}-s_{1}}{L}\right) \alpha\right)\right)+Q / \sigma_{t}\right] d \alpha . + +For the special case in which A and B are parallel, +*s*\ :sub:`1` = *s*\ :sub:`2` and the second term in the exponential +drops out. Equation :eq:`eq9-2-16` can easily be integrated to obtain + +.. math:: + :label: eq9-2-17 + + \bar{\psi}_{B}=\left(\bar{\psi}_{A}-Q / \sigma_{t}\right) e^{-\sigma_{t} s_{1}}+Q / \sigma_{t} . + +In the more general case, *s*\ :sub:`1` ≠ *s*\ :sub:`2`, the result is +slightly more complicated: + +.. math:: + :label: eq9-2-18 + + \bar{\psi}_{B}=\frac{\left(\bar{\psi}_{A}-Q / \sigma_{t}\right)}{\sigma_{t}\left(s_{2}-s_{1}\right)}\left[e^{-\sigma_{t} s_{1}}-e^{-\sigma_{t} s_{2}}\right]+Q / \sigma_{t} . + +Equations :eq:`eq9-2-17` and :eq:`eq9-2-18` can also be written in a simplified form: + +.. math:: + :label: eq9-2-19 + + \bar{\psi}_{B}=\beta_{A B} \bar{\psi}_{A}+\left(1-\beta_{A B}\right) Q / \sigma_{t} + +where + + .. math:: + + \beta_{A B}=\left\{\begin{array}{cc} + \frac{e^{-\sigma_{t} s_{1}}-e^{-\sigma_{t} s_{2}}}{\sigma_{t}\left(s_{2}-s_{1}\right)} & s_{1} \neq s_{2} \\ + e^{-\sigma_{t} s_{1}} & s_{1}=s_{2} + \end{array}\right. + +Thus far, this development has considered only the special case where +contributions to side B are the result only of the cell internal source +and a single incoming side (i.e., side A). For an arbitrarily shaped +cell and discrete direction |Omk|, it is likely that the outgoing side would +receive contributions from two or more incoming sides, as illustrated in +:numref:`fig9-2-4`, for a cell with three incoming sides (X, Y, and Z) +contributing to the flux on a single outgoing side (B). In such a +situation, the outgoing side can be subdivided into multiple components. +Side B of :numref:`fig9-2-4` can be represented by three components, +B\ :sub:`X`, B\ :sub:`Y`, and B\ :sub:`Z`, representing contributions +from line segments X, Y, and Z, respectively. The average angular flux +:math:`\bar{\psi}` can be computed for each component of side B using +Eq. :eq:`eq9-2-19`; then :math:`\bar{\psi}_{B}`, the average flux for the entire length of +B, can be calculated by the length-weighted average of each component. +In general, for a given side B composed of *n* components, the average +flux of the side is given by + +.. math:: + :label: eq9-2-20 + + \bar{\psi}_{B}=\sum_{i=1}^{n} \frac{\bar{\psi}_{i} i}{L_{B}} + +where + + :math:`\ell_{i}` is the length of the projection of the ith side onto B, and + + :math:`\bar{\psi}_{i}` is the average flux computed for segment B\ :sub:`i` due to the + flux on side *i* + +Using Eqs. :eq:`eq9-2-19` and :eq:`eq9-2-20`, one can compute the average flux on +each of the outgoing sides for a given cell, once the angular flux on +each incoming side is known. At this point, only distances *s*\ :sub:`1` +and *s*\ :sub:`2` and the lengths :math:`\ell_{i}` and L need be determined to estimate +fluxes in an iterative process. These can be computed from the geometry +of the cell and the direction |Omk|. + +.. _fig9-2-4: +.. figure:: figs/NEWT/fig4.png + :align: center + :width: 500 + + Contributions of multiple incoming sides to an outgoing side. + +.. _9-2-2-3-5: + +Mapping a characteristic vector into the two-dimensional problem domain +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +Even in a 2-D x-y system in which the scalar flux is constant with +respect to the z axis, the angular flux has components in the +z direction. Thus, to obtain the scalar flux at a point on the +x-y plane, one must integrate over the unit sphere in all 4π directions +of |Om|. Recall that the choices of characteristic directions for this model +were selected to be the same as the set of directions composing a +conventional quadrature set. Quadrature sets specified in the +literature :cite:`carlson_transport_1970,carlson_discrete_1965,lee_discrete_1962` and used in other +discrete-ordinates codes :cite:`lathrop_twotran-ii_1973,engle_jr_users_1967` are based on a unit +sphere and are usually specified in terms of μ\ :sub:`k` and η\ :sub:`k`, +the respective x and y components of |Omk|, where is one of a set of discrete +directions composing the quadrature set. Because |Omk| is a unit vector, :math:`\xi_{k}`, the +z component of the direction, is implicit: :math:`\xi_{k}=\sqrt{1-\mu_{k}^{2}-\eta_{k}^{2}}`. +However, because of the 2‑D +nature of the problem, the z component is never explicitly used. It is +therefore sufficient to evaluate the angular flux at a finite number of +points in 4π of |Om| -space in terms of just the μ\ :sub:`k` and η\ :sub:`k` +components of the discrete directions |Omk|. One must recognize, however, +that the length of the path traveled by particles moving in a direction +out of the x-y plane is always longer than the x-y projection of the +path, by a factor of (μ\ :sup:`2` + η\ :sup:`2`)\ :sup:`–1/2`. Thus, for any +path length *s*' measured in the x‑y plane for a given direction |Omk|, the +true path length traveled is *s*, where + +.. math:: + :label: eq9-2-21 + + s=\frac{s^{\prime}}{\sqrt{\mu^{2}+\eta^{2}}} . + +This is illustrated in :numref:`fig9-2-5`. + +.. _fig9-2-5: +.. figure:: figs/NEWT/fig5.png + :align: center + :width: 500 + + Relationship between *s*\ :sub:`1` and *s*\ :sub:`2` and + their projections in the x-y plane. + +.. _9-2-2-3-6: + +Neutron balance within a computational cell +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +Once angular fluxes have been computed for all sides of a cell, it is +necessary to compute the cell-averaged angular flux. To enforce +conservation, a balance condition is applied to the cell. This provides +the equation necessary to determine the average flux in the cell. The +neutron balance for an arbitrary cell in steady state may be expressed +as + +.. math:: + :label: eq9-2-22 + + \left[\begin{array}{c} + \text { net number of } \\ + \text { neutrons moving in } \\ + \text { direction } \hat{\Omega} \text { escaping } \\ + \text { from the cell } + \end{array}\right]+\left[\begin{array}{c} + \text { number of neutrons } \\ + \text { removed from the cell } \\ + \text { or from direction } \hat{\Omega} \\ + \text { by interactions } + \end{array}\right]=\left[\begin{array}{c} + \text { number of } \\ + \text { neutrons produced } \\ + \text { in the cell moving } \\ + \text { in direction } \hat{\Omega} + \end{array}\right] + +or, expressed mathematically, + +.. math:: + :label: eq9-2-23 + + \oint_{s} n \cdot \hat{\Omega}_{k} \psi d S+\sigma_{t} \bar{\psi} V=Q V, + +where :math:`n` is the outward normal direction at each side of the cell and V is +the 2-D volume of the cell. Note that in this context, *S* represents +the surface area or perimeter of the cell. Hence, for a cell with *m* +sides, each of the sides having a constant angular flux :math:`\bar{\psi}_{i}` and an outward +normal direction :math:`\mathrm{n}_{i}`, + +.. math:: + :label: eq9-2-24 + + \bar{\psi}_{c e l l}=\frac{Q}{\sigma_{t}}-\frac{1}{\sigma_{t} V} \sum_{i=1}^{m} \bar{\psi}_{i} \int_{S_{i}} \mathrm{n}_{i} \cdot \hat{\Omega}_{k} d S_{i} . + +Because each cell is restricted to be a polygon, each side in the cell +will be a straight line and :math:`\mathrm{n}_{i} \cdot \hat{\Omega}_{k}` will be constant along the length of the +side. Equation :eq:`eq9-2-24` can then be simplified to obtain + +.. math:: + :label: eq9-2-25 + + \bar{\psi}_{c e l l}=\frac{Q}{\sigma_{t}}-\frac{1}{\sigma_{t} V} \sum_{i=1}^{m} \bar{\psi}_{i}\left(\mathrm{n}_{i} \cdot \hat{\Omega}_{k} \mathrm{~L}_{i}\right) , + +where L\ :sub:`i` is the length of the *i*\ th side and the term in +parentheses represents a leakage coefficient for the side. + +.. _9-2-2-4: + +Coarse-mesh finite-difference acceleration +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Beyond cell discretization and solution described above for the ESC +approach, the NEWT iterative approach is similar to that used in other +discrete-ordinates methods. Inner iterations are used to solve spatial +fluxes in each energy group to generate updated source terms; outer +iterations use these source terms to converge all energy groups. This +source-iteration approach can be somewhat slow to converge, especially +when significant scattering is present. Hence, it is desirable to apply +some form of acceleration to the iterative solution used by NEWT. To +this end, a coarse-mesh finite-difference acceleration (CMFD) approach +has been added to NEWT. The CMFD formulation uses a simplified +representation of a complex problem, in which selected rectangular +regions are derived from the global NEWT Cartesian grid and homogenized. +The CMFD formulation utilizes coupling correction factors for each +homogenized cell to dynamically homogenize the constituent ESC-based +polygonal cells during the iterative solution process such that the +heterogeneous transport solution can be preserved. Dynamic-group +collapse is also possible with a two-level CMFD formulation in which +alternating multigroup and two-group calculations are performed. By +extending the concept of the equivalence theory to energy and angle, it +is possible to apply a consistent lower-order formulation in the form of +a homogenized pin-cell, few-group, diffusion-like finite-difference +scheme. This simplified lower-order formulation is much less expensive +to solve, and its solution can be used to accelerate the original +higher-order transport solution in NEWT, resulting in much faster +convergence of the fission and scattering source distributions. This +work is described in detail in :cite:`zhong_implementation_2008` and in previous versions of +the NEWT manual. + +Although the original implementation of the CMFD acceleration method is +extremely efficient and actively maintained, its use is limited to +rectangular-domain configurations (e.g., square-pitched fuel lattices). +An alternative CMFD acceleration method has been developed to support +triangular- and hexagonal-domain configurations (e.g., +triangular-pitched fuel lattices such as the VVER or prismatic graphite +models). The new CMFD acceleration method does not require the +coarse-mesh cells to be rectangles but rather arbitrary polygons. +However in the current implementation, the “unstructured” coarse-mesh +cells are still constructed from the global NEWT Cartesian grid. +Therefore, for a hexagonal configuration, interior coarse-mesh cells +will be rectangular shape whereas cells near the boundary will be +triangular or trapezoidal shapes. + +The new unstructured CMFD iterative solution scheme is essentially +identical to the original solution scheme; the two methods differ only +in how the lower-order system is solved. Additionally the two-group +acceleration is not employed in the unstructured CMFD method. Input +options for both CMFD methods are described in :ref:`9-2-3-2`. + +.. _9-2-2-5: + +Assembly discontinuity factors +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + + +In nodal multi-assembly or core calculations, lattice transport +solutions are used to generate few-group homogenized cross sections. +These cross sections are in general obtained from single-assembly +transport calculations with zero-current boundary conditions. Generation +of few-group homogenized cross sections for nodal calculations typically +includes the generation of discontinuity factors (i.e., additional +parameters used to preserve both reaction rates and the interface +currents in the homogenization process). The discontinuity of the flux +at an assembly interface that can arise by the use of homogenized +cross sections is illustrated in :numref:`fig9-2-6`. The so-called +“homogeneous” flux, computed in the nodal calculation, is discontinuous +at the assembly interface, as opposed to the exact “heterogeneous” flux, +computed in the transport calculation, which is continuous at the +assembly interface. The interface condition employed in nodal +calculations between two assemblies (nodes) *i* and *i*\ +1 is given as + +.. math:: + :label: eq9-2-26 + + \phi_{i, \text { homogeneous }}^{+} \cdot F_{i}^{+}=\phi_{i+1, \text { homogeneous }}^{-} \cdot F_{i+1}^{-} , + +where :math:`F_{i}^{+}` and :math:`F_{i+1}^{-}` are assembly discontinuity factors (ADFs) on each side of the +interface between assemblies *i* and *i*\ +1. + +The ADF on the assembly interface is defined as the ratio of the +heterogeneous flux :math:`\phi_{\text {heterogeneous }}` at that assembly interface to the homogeneous flux +evaluated at the interface, denoted :math:`\phi_{i, \text { homogeneous }}^{+}` (or :math:`\phi_{i+1, \text { homogeneous }}^{-}`): + +.. math:: + :label: eq9-2-27 + + F_{i}^{+}=\frac{\phi_{\text {heterogeneous }}}{\phi_{i, \text { homogeneous }}^{+}}, F_{i+1}^{-}=\frac{\phi_{\text {heterogeneous }}}{\phi_{i+1, \text { homogeneous }}^{-}} . + +Fluxes, and therefore ADFs, vary with energy; therefore, few-group +homogenized cross sections are always accompanied by corresponding +few-group ADFs. + +In a single-assembly calculation with zero-current boundary conditions, +the heterogeneous flux at each boundary is easily calculated as the +surface-averaged scalar flux on the boundary, whereas the homogenous +flux at each boundary is simply the assembly-averaged flux. Hence, for +each energy group, the ADF is calculated for each boundary as the ratio +of the average flux on that boundary to the average flux across the +assembly. + +In other configurations, such as a multi-assembly calculation or an +assembly located on the edge of a core next to the core baffle and +reflector, the ADF calculation requires more effort. For reflector +situations, NEWT applies a simple one-dimensional (1-D) multigroup +diffusion approximation to determine the ADF at the assembly boundary. +In this approximation, it is assumed that the reflector is infinite and +that the scalar flux goes to zero at infinity. The reflector ADF can be +determined analytically using this boundary condition along with the +known surface-averaged current and scalar flux evaluated at the +assembly/reflector interface. + +The reflector ADFs computed by NEWT may potentially be different from +the ADFs calculated using the diffusion approximations employed by the +nodal code. Moreover, ADFs computed for multi-assembly or +hexagonal-domain configurations will depend on the nodal method +employed. For these reasons, NEWT supports the option to edit +surface-averaged scalar flux and current values along user-defined line +segments so that appropriate ADFs can be computed directly by the nodal +code. The input options for the single-assembly ADF, reflector ADF, and +arbitrary line-segment edit are discussed in :ref:`9-2-3-11`. + +.. _fig9-2-6: +.. figure:: figs/NEWT/fig6.png + :align: center + :width: 500 + + Heterogeneous vs homogeneous fluxes in a multi-assembly solution. + +.. _9-2-3: + +Input Formats +------------- + +NEWT input is free form and keyword based, similar in form to the input for many +other modules in the SCALE code package. Input may start with a title card +record, but this line may be omitted if desired; remaining data are supplied in +data blocks. The order of the data blocks is arbitrary (with two exceptions), +and many blocks are optional. Only one instance of a data block is allowed. + +.. _9-2-3-1: + +Overview of newt data blocks +~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The NEWT input deck data blocks are defined by keyword delimiters in the +following form: + +.. highlight:: none + +:: + + read keyword [data] end keyword + +Read routines are terminated by the “end *keyword*\ ” label, and any +intervening carriage returns or line feeds are ignored. Thus, data can +also be entered in this format: + +:: + + read keyword + [data] + [data] + end keyword + +Within each block, specific control or specification parameters are +input. Each block contains a fixed set of input parameters (also defined +by keyword). + +As with other keyword-driven modules within SCALE, lines beginning with +a single quote (') in the first column are treated as comments and +ignored. + +The keyword name and general contents of each data block are as follows: + +.. table:: + :align: center + :class: longtable + + +-----------------------+-----------------------+-----------------------+ + | **Block type** | **Recognized | **Description** | + | | keywords** | | + +-----------------------+-----------------------+-----------------------+ + | Problem control | parameter, | General problem | + | parameters | parameters, param, | parameters—must | + | | parm, para | follow title card, if | + | :ref:`9-2-3-2` | | used (optional) | + +-----------------------+-----------------------+-----------------------+ + | Material properties | material, materials, | Assigns | + | | matl | characteristics | + | :ref:`9-2-3-3` | | (e.g., P\ :sub:`n` | + | | | scattering order and | + | | | material description) | + | | | for each material | + | | | ID—must follow | + | | | problem control block | + | | | or must follow title | + | | | card if control block | + | | | is omitted (required) | + +-----------------------+-----------------------+-----------------------+ + | Broad group collapse | collapse, coll | Defines broad group | + | | | energy ranges to be | + | :ref:`9-2-3-5` | | created from the | + | | | original fine group | + | | | library when cross | + | | | section collapse is | + | | | desired (optional) | + +-----------------------+-----------------------+-----------------------+ + | Simple-body geometry | geometry, geom | Defines basic grid | + | | | structure and all | + | :ref:`9-2-3-6` | | bodies to be placed | + | | | within this structure | + | | | (required unless | + | | | geometry restart file | + | | | is available) | + +-----------------------+-----------------------+-----------------------+ + | Boundary conditions | bounds, bnds | Defines boundary | + | | | conditions to be | + | :ref:`9-2-3-7` | | applied on outer | + | | | boundaries of global | + | | | unit (optional, | + | | | default is reflective | + | | | on all sides) | + +-----------------------+-----------------------+-----------------------+ + | Array specifications | array | Defines composition | + | | | of all arrays (unit | + | :ref:`9-2-3-9` | | placement within each | + | | | array). Each array | + | | | placed within the | + | | | geometry block must | + | | | be defined in the | + | | | array block | + +-----------------------+-----------------------+-----------------------+ + | Homogenization | homog, hmog, homo | Defines mixtures to | + | instructions | | be flux weighted and | + | | | homogenized in the | + | :ref:`9-2-3-10` | | preparation of a | + | | | homogenized cross | + | | | section library | + | | | (optional) | + +-----------------------+-----------------------+-----------------------+ + | Assembly | adf | Assigns type and | + | discontinuity factors | | location of planes at | + | | | which assembly | + | :ref:`9-2-3-11` | | discontinuity factors | + | | | (ADFs) are calculated | + | | | (optional) | + +-----------------------+-----------------------+-----------------------+ + | Flux plane | flux | Allows definition of | + | | | an x- or y-axis line | + | :ref:`9-2-3-12` | | (plane) for which | + | | | average fluxes are | + | | | computed and printed | + | | | (optional) | + +-----------------------+-----------------------+-----------------------+ + | Mixing table | mixtable, mixt | Mixing table | + | | | specification | + | :ref:`9-2-3-13` | | (optional) | + +-----------------------+-----------------------+-----------------------+ + | Source definition | src, source | Defines particle | + | | | source strength for | + | :ref:`9-2-3-4` | | use in source | + | | | calculations | + +-----------------------+-----------------------+-----------------------+ + +Each of the following subsections describes the parameters associated +with a specific data block, lists default values (if available), and +describes meaning of the parameter and its effect on a NEWT calculation. + +.. _9-2-3-2: + +Parameter block +~~~~~~~~~~~~~~~ + +**Parameter Block keyword = param, parm, para, parameter, or +parameters** + +The Parameter block contains problem control parameters and must come +immediately after the title card if one is used. Valid parameter +specifications are described below. For each keyword, allowable values +are listed in parentheses, and the default (if any) is listed in +brackets. Input that can take an arbitrary integer value is indicated by +an *IN*; similarly, any parameter that can take an arbitrary +real/floating point value is indicated by *RN* as the allowable value. +However, note that SCALE read routines do allow input of integers for +real numbers, and vice versa; the number will be converted accordingly. +The order of the parameters within the block is arbitrary, and may be +skipped if a default value is desired for that parameter. Control +parameters are set in the order in which they are input; this means that +the same parameter may be listed multiple times, but only the final +value is used. + +.. _9-2-3-2-1: + +Convergence and acceleration parameters +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +**epseigen=**\ (*RN*) — Convergence criterion for *k*\ :sub:`eff`. [0.0001] + +**epsinner=**\ (*RN*) — Spatial convergence criterion for inner +iterations. [0.0001] + +**epsouter=**\ (*RN*) — Spatial convergence criterion for outer +iterations. [0.0001] + +**epsthrm=**\ (*RN*) — Spatial convergence criterion for +thermal-upscattering iterations, if enabled. [same value as +**epsouter**] + +**epsilon=**\ (*RN*) — Simultaneously sets all (spatial and eigenvalue) +convergence criteria to the same value. [uses individual defaults] + +**converg**\ =(\ *cell/mix*) — Sets the region within which convergence +testing is applied. Use of *cell* will force converged scalar fluxes in +every computation cell, while *mix* will relax convergence such that +averaged scalar fluxes within a mixture are converged. The latter is +useful for mixtures in which fluxes become very small—large reflectors +or near a vacuum BC. [cell] + +**therm**\ =(\ *yes/no*) — Enables/disables thermal-upscattering +iterations. [yes] + +**inners=**\ (*IN*) — Maximum number of inner iterations in an energy +group. [5] + +**therms=**\ (*IN*) — Maximum number of thermal-upscattering iterations, +if enabled. [2] + +**outers=**\ (*IN*) — Maximum number of outer iterations. NEWT will stop +with an error code if more than *outers* outer iterations are required +for convergence. [250] + +**inrcvrg**\ =(\ *yes/no*) — If inrcvrg=yes, NEWT will continue outer +iterations until all convergence criteria are met. If inrcvrg=no, NEWT +will stop whenever outer iteration and *k*\ :sub:`eff` convergence criterion +are met, regardless of the convergence of inner or thermal-upscattering +iterations. [no] + +**cmfd=**\ (*no/rect/yes/part*) — CMFD acceleration option. If cmfd=no, +CMFD acceleration is not employed. If cmfd=rect, the CMFD method is +employed. The original NEWT CMFD method can be applied only to +rectangular-domain configurations. If cmfd=yes, the unstructured CMFD +method is employed. The new unstructured CMFD method can be applied to +rectangular-, triangular-, and hexagonal- domain configurations. If +cmfd=part, an alternative version of the unstructured CMFD method is +employed and uses a “partial-current” acceleration scheme. +Alternatively, users can use cmfd=0/1/2/3 for no, rect, yes, and part, +respectively. [no] + +**cmfd2g=**\ (*yes/no*) — Enables/disables the second-level two-group +CMFD accelerator within the CMFD solver. This parameter has an effect +only when cmfd=rect is set. [yes] + +**accel=**\ (*yes/no*) — Enables/disables source (*k*\ :sub:`eff`) +acceleration. This parameter is automatically disabled if unstructured +CMFD is employed (cmfd=yes or cmfd=part). [yes] + +**xcmfd**\ =(\ *IN),* **ycmfd**\ =(\ *IN),* **xycmfd**\ =(\ *IN)* — +These inputs specify the number of fine-mesh cells in the global NEWT +grid per coarse-mesh cell. These options are used only when CMFD +acceleration is enabled. The parameter *xcmfd* specifies the number +fine-mesh cells per coarse-mesh cell in the x‑direction. Likewise, +*ycmfd* specifies the number of fine-mesh cells per coarse-mesh cell in +the y‑direction. The parameter *xycmfd* simultaneously sets *xcmfd* and +*ycmfd* to the same value. In a special case for rectangular-domain +configurations in which the entire domain is completely filled by a +square-type array (see :ref:`9-2-3-9`), *xycmfd=0* sets the coarse mesh +based on the size of the array elements. [1] + +.. important:: Default convergence parameters are recommended for general analysis. + Larger convergence criteria are useful for debugging if shorter run time + is desired over solution accuracy. Smaller convergence criteria are + recommended for generating reference solutions or benchmark + calculations. + + CMFD acceleration should be applied whenever possible. The CMFD method + with second-level 2-group acceleration should be applied for + rectangular-domain configurations [e.g., light water reactor (LWR) + assembly models (*\ **cmfd=rect**\ *), by default*\ **cmfd2g=yes**\ *]. + The unstructured CMFD method should be applied for triangular- or + hexagonal-domain configurations (*\ **cmfd=yes**\ *). If NEWT detects an + unstable CMFD condition, a warning message is printed and NEWT continues + with CMFD disabled. NEWT may also provide a terminating error message if + improper selection of the coarse mesh is detected. Internal + investigation has shown that the coarse mesh should be approximately the + same size as the unit cell used in the model. For LWR assembly models, a + fine mesh of 4 x 4 is recommended for the square-pitched unit cell, + implying that*\ **xycmfd**\ *should be 4 only if the global unit has a + mesh. If individual meshes are used in each unit definition, then the + global unit coarse-mesh cells should be sized based on the unit cell + size and, therefore, xycmfd=1 should be used. The values + of*\ **xcmfd**\ *and*\ **ycmfd**\ *do not have to be a common factor of + the number of fine-mesh cells in a given direction (NEWT will make the + last coarse-mesh cell smaller than the other coarse-mesh cells), but it + is highly recommended. + + Users can gauge solution convergence by the outer iteration edit as it + is printed to the terminal window (*\ **echo=yes**\ *, see below). One + can terminate a calculation prematurely (via the Control-C option on + most platforms) if convergence or iteration parameters need to be + modified. + + The TRITON control module supports a sensitivity and uncertainty + analysis sequence TSUNAMI-2D (See TRITON chapter, section S/U Analysis + Sequences (TSUNAMI-2D, TSUNAMI-2DC)). TSUNAMI-2D calculations require + NEWT to be run in both forward mode and adjoint mode. In adjoint mode, + CMFD acceleration is not currently supported and NEWT automatically + disables its use if*\ **cmfd=yes**\ *,*\ **=rect**\ *, + or*\ **=part**\ *. In adjoint mode with defined fixed source [i.e., + generalized perturbation theory (GPT) analysis], it is observed that + tighter convergence and iteration parameters are needed to properly + remove fundamental mode contamination. (For more details, see SAMS + chapter: Generalized Perturbation Theory.) To facilitate the CMFD + options and larger convergence criteria for the forward calculations as + well as smaller convergence criteria for GPT adjoint calculations, the + following parameters are also available. + +**gptepsinner=**\ (*RN*) — Spatial convergence criterion for inner +iterations in GPT analysis. [0.0001] + +**gptepsouter=**\ (*RN*) — Spatial convergence criterion for outer +iterations in GPT analysis. [0.001] + +**gptepsthrm=**\ (*RN*) — Spatial convergence criterion for +thermal-upscattering iterations, if enabled, in GPT analysis. [same +value as **gptepsouter**] + +**gptsepsilon=**\ (*RN*) — Simultaneously sets all spatial convergence +criteria to the same value in GPT analysis. [uses individual defaults] + +**gpttherm**\ =(\ *yes/no*) — Enables/disables thermal-upscattering +iterations in GPT analysis. [yes] + +**gptinners=**\ (*IN*) — Maximum number of inner iterations in an energy +group in GPT analysis. [500] + +**gpttherms=**\ (*IN*) — Maximum number of thermal-upscattering +iterations, if enabled, in GPT analysis. [10] + +**gptouters=**\ (*IN*) — Maximum number of outer iterations in GPT +analysis. NEWT will stop with an error code if more than *outers* outer +iterations are required for convergence. [2000] + +.. important:: Default values for GPT convergence may change with future releases, as + more experience is gained and user feedback is received. If the GPT + calculation is not converging because of fundamental mode contamination, + it is recommended that convergence criteria be decreased and/or inner + and thermal-upscattering iteration limits be increased. If the solution + convergence is slow,*\ **gptinners**\ *can potentially be decreased. + Again, it is highly recommended that*\ **echo=yes**\ *be used to monitor + speed of convergence. + +.. _9-2-3-2-2: + +Output editing +^^^^^^^^^^^^^^ + +**drawit=**\ (*yes/no*) — Create a PostScript file showing the grid +structure determined from input. Two files are created—the first showing +the grid structure and the second showing the material placement. +(Features and use of this simple graphics capability are described +further in :ref:`9-2-5-14`) [no] + +**echo=**\ (*yes/no*) — During the iteration phase of execution, output +is generated at the beginning of each outer iteration. This same +information can be printed to SCALE message file (.msg) during iteration +by setting echo=yes. [no] + +**prtbalnc**\ =\ *(yes/no)* — Flag indicating whether or not balance +tables for fine-group mixtures should be printed. [no] + +**prtbroad=**\ (*yes/no/1d*) — Flag indicating whether or not broad +group cross sections should be printed in problem output. The *1d* +option indicates that 2-D scattering tables are not to be printed. This +flag has no effect if collapse=no is specified. [no] + +**prthmmix=**\ (*yes/no*) — Flag indicating whether or not homogenized +mixture macroscopic cross sections should be printed in problem output. +Homogenized cross sections are printed only if Homogenization Block is +provided (:ref:`9-2-3-10`). [yes] + +**prtflux**\ *=(yes/no)* — Create a PostScript plot file showing flux +distribution for each energy group in problem. If an energy collapse is +performed, a second plot file is generated for the fluxes of the +collapsed group structures. [no] + +**prtmxsec=**\ (*yes/no/1d*) — Flag indicating whether or not mixture +macroscopic cross sections should be printed in problem output. The *1d* +option indicates that 2-D scattering tables are not to be printed. [no] + +**prtmxtab=**\ (*yes/no*) — Flag indicating whether or not the input +mixing table should be printed in problem output. [no] + +**prtxsec=**\ (*yes/no/1d*) — Flag indicating whether or not input +microscopic cross sections should be printed in problem output. The *1d* +option indicates that 2-D scattering tables are not to be printed. [no] + +**timed=**\ (yes/no) — Turns on printing of iteration timing and CPU use +data. [no] + +**det=**\ (*IN*) — Specifies the mixture used to represent a local power +range monitor (LPRM) and/or Traversing In-core Probe (TIP) detector +located within a fuel lattice. The mixture must also be included in a +homogenization block in order to obtain detector cross sections. [has no +default] + +.. important:: With the exception of*\ **prthmmix**\ *, all output edit options are + disabled unless requested by the user. The output edits are disabled by + default to minimize the size of the output. The*\ **drawit**\ *option is + recommended to generate PostScript plots of the model grid structure and + material placement. As previously mentioned, + the*\ **echo**\ *and*\ **timed**\ *options are recommended to monitor + solution convergence. If the timed option is enabled, each line in the + outer iteration edit will be longer than 80 characters. Therefore, it is + recommended that Windows users should increase the Command Window size + from 80 characters to 132* characters. + +.. _9-2-3-2-3: + +Angular quadrature +^^^^^^^^^^^^^^^^^^ + +**sn=**\ (*2/4/6/8/10/12/14/16*) — Order of Sn level symmetric +quadrature set. [6] + +**nazim**\ =(\ *IN*) — Number of equally spaced azimuthal directions in +a product quadrature set. Used in tandem with *npolar* keyword (both +must be specified). Total number of angles in the product quadrature set +is the product of *nazim* and *npolar*. [No default. If not specified, +level symmetric quadrature default is used.] + +**npolar**\ =(\ *IN*) — Number of polar angles in a product quadrature +set (determined using a Gauss-Legendre polynomial). Used in tandem with +*nazim* keyword (both must be specified). Total number of angles in the +product quadrature set is the product of *nazim* and *npolar*. [No +default. If not specified, level symmetric quadrature default is used.] + +**dgauss=**\ (yes/no) — Enables/disables use of double Gauss-Legendre +product quadrature set. If disabled, single Gauss-Legendre product +quadrature sets are used. [no] + +.. important:: If both level symmetric quadrature sets and product quadrature sets are + requested, the level symmetric quadrature set is to be used. Level + symmetric quadrature sets are recommended for general analysis. If + reflective boundary conditions are desired for hexagonal-domain + configurations, product quadrature sets must be used and nazim must be a + multiple of 3. If reflective boundary conditions are desired for + triangular-domain configurations, product quadrature sets must be used + and*\ **nazim**\ *must be an odd number. + +.. _9-2-3-2-4: + +Control options +^^^^^^^^^^^^^^^ + +**adjoint=**\ (*yes/no*) — This keyword specifies either a forward +(adjoint=no) or adjoint (adjoint=yes) calculation. [no] + +**forward=**\ (*yes/no*) — This keyword specifies either a forward +(forward=yes) or adjoint (forward=no) calculation. If adjoint and +forward are both specified, NEWT uses the last specification. [yes] + +**gpt=**\ (*yes/no*) — This keyword specifies whether this is a GPT +adjoint calculation. The *gpt* keyword is active only for adjoint +calculations. [no] + +.. important:: The TRITON control module automatically sets the values for forward, + adjoint, and gpt keywords; therefore, they can typically be omitted from + the Parameter Block. Default values are recommended unless running + stand-alone NEWT adjoint calculations. + +**run=**\ (*yes/no*) — A run=no calculation will perform all setup +calculations normally performed before beginning iterations and then +will stop. It is useful for debugging input and obtaining plots of the +input geometry. Run=yes will perform a complete calculation. [yes] + +**premix=**\ (*yes/no*) — This flag indicates whether the cross section +library contains microscopic (premix=no) or macroscopic (premix=yes) +cross sections. In essence, it creates a mixing table with a mixture +fraction of 1.0 for each mixture on the library. Other mixing tables are +ignored. The premixed cross section option is active only for +stand-alone NEWT calculations. [no] + +**kguess=**\ (*RN*) — Initial guess at eigenvalue for an eigenvalue +calculation. This parameter may be entered but is not used if a source +calculation is performed or a restart file is used to determine the +initial guess. [1.0] + +**restart=**\ (*yes/no*) — If restart=yes is specified, NEWT will open +file *restart_newt* and read scalar fluxes and fission rates, enabling a +restart from the point at which a previous calculation ended. The file +*restart_newt* is always written by NEWT at the end of every successful +calculation. The code assumes that all geometry is unchanged from the +previous calculation but does allow restart with a different angular +quadrature set and P\ :sub:`n` scattering coefficients. A low-order +solution can be used to accelerate a higher-order solution by restarting +using the converged flux of the lower-order solution. [no] + +**savrest=**\ (*yes/no*) — Determines whether or not a geometry restart +file *worf* is written at the end of a calculation. If written, it will +overwrite any existing geometry restart file. [yes] + +.. important:: The default values of savrest and kguess are recommended. The TRITON + control module automates generation and reuse of the geometry restart + file, as well as the initial guess of the eigenvalue. Keywords run, + premix, and restart can generally be omitted unless the following + conditions are applicable: + + - TRITON T-NEWT sequence calculation or stand-alone NEWT calculation + with user-supplied restart file, restart=yes. + + - Stand-alone NEWT calculation with user-supplied premixed cross + section file, premix=yes. + + - Interested only in performing setup calculations to debug input and + generate geometry plots, run=no, and/or PARM=CHECK in the TRITON + sequence input. + +**solntype**\ =(keff/b1/src) — Specifies solution mode type: keff is +eigenvalue, b1 is eigenvalue mode followed by a buckling correction, and +src is fixed source (no eigenvalue calculation). Fixed source +calculations require additional data for the source specification (see +Materials and Source data blocks in :ref:`9-2-3-3` and :ref:`9-2-3-4`). [keff] + +**collapse=**\ (*yes/no*) — If collapse=yes is specified, a +flux-weighted collapse is performed by material number; cross sections +for each nuclide in each material in the problem are collapsed to a +specified (or default) group structure based on the average flux in that +material. If collapse=yes, NEWT will look for the *collapse* parameter +block; if not found, NEWT will generate cross sections based on the +original group structure. If a Homogenization block is present, then +collapse is always set to yes. [no] + +**saveangflx=**\ (*yes/no*) — Option to save angular flux solution. The +angular flux is saved to a binary file used in the TSUNAMI-2D sequence +of the TRITON control module. Because the angular flux can require +significant file storage, it is not saved by default. The angular flux +solution can and should be saved for TSUNAMI-2D calculations to generate +more accurate sensitivity coefficients. [no] + +.. important:: Keyword threads should be omitted in favor of the SCALE command line–I + option. Keywords solntype, collapse, and saveangflx should be omitted + unless the following conditions are applicable. + + - For homogenized few-group cross section generation for nodal + calculations, solntype* *should be b1. This option will perform a + critical spectrum calculation, which will be folded into cross + section homogenization calculation. The critical spectrum is also + folded into the generation of ADFs and reaction rates for + depletion calculations. + + - Generation of a new collapsed cross section library, collapse=yes. + + - For TSUNAMI-2D calculations, saveangflx=yes. + +.. _9-2-3-2-5: + +Geometry processing options +''''''''''''''''''''''''''' + +**combine**\ =(\ *yes/no*) — Automatic grid generation can result in +very small grid cells in some locations. Setting parameter combine to +*yes* performs automatic combination of smaller grid cells into adjacent +neighbor of same material, if possible. Combine is automatically set to +*no* if CMFD is enabled; this setting cannot be overridden. [no] + +**clearint**\ =(\ *yes/no*) — Grid generation option that removes the +global NEWT grid if a local unit grid is supplied. (For meshing options, +see the *boundary* keyword in the Geometry block description in +:ref:`9-2-3-6`) By default, clearint is set to yes, which means the +global grid is removed if local grids are provided. If CMFD acceleration +is enabled, clearint is set to no, which means both the global grid and +optional local grids are used. [yes] + +**grid_tol=**\ (*RN*) — Tolerance used in determining if polygon +vertices are numerically identical during NEWT grid generation. +[0.000001] + +**cell_tol=**\ (*RN*) — Tolerance used in determining if polygon +vertices are numerically identical during NEWT cell generation. +[0.000001] + +**line_tol=**\ (*RN*) — Tolerance used in determining if polygon +vertices are numerically identical during NEWT line generation. +[1.0e-10] + +.. important:: The default values for all geometry-processing keywords are recommended + and can be omitted. For problems with very fine mesh, tighter grid and + cell tolerances should be applied. For problems that terminate with a + ray-tracing error (i.e., tracer error), tighter grid and cell tolerances + should be applied. + +.. _9-2-3-2-6: + +Critical spectrum options +^^^^^^^^^^^^^^^^^^^^^^^^^ + +**useb1**\ =(\ *yes/no*) —Turns on/off the use of the B1 approximation +to determine the critical spectrum. If useb1 is set to no, the P1 +approximation is used. [yes] + +**b2=**\ (*RN*) — Material buckling factor, in units of 1/cm\ :sup:`2`. +[0.0] + +**height=**\ (*RN*) — Height (transverse dimension) in centimeters. Used +in a geometric buckling correction to calculate leakage normal to the +plane of the input 2-D model. Keywords **dz=** and **deltaz=** are +equivalent. When set to zero (default), no buckling correction is +performed. [0.0] + +**bf=**\ (*RN*) — Twice the extrapolation distance multiplier used to +determine the geometric buckling correction. [1.420892] + +.. important:: If critical spectrum corrections are to be applied, the default values + listed above are recommended along with **solntype=b1**\ . In this + option, NEWT will search for the material buckling value such that the + homogenized infinite-medium system is critical. NEWT currently uses the + B1 approximation as the default. If the P1 approximation is preferred, + useb1should be set to no. The infinite-medium B1 (or P1) buckling search + is performed in the energy group structure as the original model. + + Alternatively, the user can supply the material buckling value using + the b2 keyword, and specifying the B1 (default) or P1 + approximation ( **useb1=no** ). In this + case, **solntype** should be set to **keff**\ . + + Alternatively, if the user knows the transverse dimension, a geometry + buckling factor can be applied, derived from the + user-defined **height** and extrapolation distance + term **bf** as the following: + + :math:`B_{g}^{2}=\left(\frac{\pi}{H+z / \sigma_{t r}}\right)^{2}` + + In this formula, H is keyword height, z is keyword bf, and :math:`\sigma_{t r}` is the + collapsed, homogenized macroscopic transport cross section. + +.. _9-2-3-2-7: + +File unit options +^^^^^^^^^^^^^^^^^ + +.. important:: It is highly recommended that the file unit options below be omitted or + that default values be used. Alternate file unit values are acceptable + for stand-alone NEWT calculations, but changing their values may + adversely impact other SCALE modules if NEWT is invoked through a SCALE + sequence. + +**hmoglib=**\ (*IN, 00) is retained because +x mode is specified. +Since the cylinder is centered at (0, 0), this chord cuts the cylinder +in half and retains the right half of the cylinder. The unit in +:numref:`fig9-2-16` uses the same cylinder but with a chord cutting at the +plane located at y=0. The bottom half (y < 0) is kept because –y is +specified. + + +:numref:`fig9-2-17` is somewhat more complicated but represents perhaps the +most common use of chords in lattice models. In this case, it is +desired to create a one-quarter cylinder located in the bottom right +quadrant of a cuboid. A 1 by 1 cm square cuboid is centered at (0, 0), +and a cylinder is placed at +(0.5, –0.5), which is the lower right-hand corner of the cuboid. Since +we are interested only in the portion of the cylinder within the +cuboid, we choose to keep the top (+x) and left (–y) portions of the +cylinder. This requires two separate chord modifiers. (Each chord +specifies only one cutting plane.) Additionally, because the cylinder +was relocated to a new origin, the chords are specified such that the +cuts go through the new origin. + +Note that there is no requirement that a chord cut through the origin of +a body. :numref:`fig9-2-18` illustrates the use of four chords to set four +cutting planes. A 0.5 cm cylinder is specified centered within the unit +cuboid. All four of the four permitted cutting planes are specified. We +have effectively created a cuboid by retaining the portion of the +cylinder above (+y) the xz plane located at y= –0.25, below (–y) the +plane at y=+0.25, to the right (+x) of the yz plane at x= –0.25, and to +the left (–x) of the plane located at x=+0.25. There is, of course, a +much more direct means to create a cuboid—this example is provided only +for illustrative purposes. + +For guidance on how to cut a cylinder at an oblique angle, refer to +:ref:`9-2-3-6-1-14`. + +.. _9-2-3-6-1-12: + +Com +''' + +The *com* modifier is a means to label specific bodies. It is provided +primarily for consistency with KENO‑VI. At this time, NEWT simply reads +and then ignores *com* data. It can, however, be used as a means to help +annotate an input listing. The format for the *com* modifier is as +follows: + +:: + + com=”comment string” + +where *“comment string”* is any text description of up to 132 +characters, delimited by single (′) or double quotes (″). For example, +the input description of :numref:`fig9-2-18` is shown here with comments added +via *com* modifiers. + +:: + + unit 10 + cuboid 1 0.5 -0.5 0.5 -0.5 com=”unit cuboid centered at (0,0)” + cylinder 2 0.5 com=”cylinder with four chords” + chord -x=0.25 chord +x=-0.25 + chord -y=0.25 chord +y=-0.25 + +.. _9-2-3-6-1-13: + +Sides +''''' + +The *sides* modifier applies only to cylinders and is unique to NEWT +(i.e., it is not used in KENO-VI). Because NEWT’s solution grid is based +on arbitrary polygons, all cells must be straight sided. Hence, the +curved surfaces of a cylinder are approximated as an N-sided regular +polygon. By default, N=12. The *sides* operator allows the user to +override the default. The format is very simple: + +:: + + sides=N + +where N is the number of sides desired for the full cylinder. In +general, a 12-sided polygon provides an adequate approximation of a +cylinder. Use of additional sides will create a cylinder that has a +smoother appearance and increase the computational effort required to +solve the cells associated with the cylinder. + +:numref:`fig9-2-19` shows a model built with three nested cylinders inside a +unit cuboid. Cylinder 10 is the innermost cylinder, with no *sides* +modifier; hence, it uses the default 12-sided approximation. The second +cylinder is specified with sides=16; the refinement in this +approximation is seen in the figure. Finally, cylinder 30 is specified +with 40 sides—this is visually a very close approximation to a cylinder. + +.. _fig9-2-19: +.. figure:: figs/NEWT/fig19.png + :align: center + :width: 500 + + Use of the sides modifier for cylinders. + +.. _9-2-3-6-1-14: + +Holes +''''' + +The next level of complexity within a *unit* is provided through the use +of a *hole* specification. The *hole* specification is simply a means by +which one unit may be placed within another unit. In some instances, a +well-defined set of structures, assembled as a *unit*, needs to be +placed within a larger unit. NEWT provides two methods to do this—holes +and arrays. Arrays are used to place a unit (or a number of similar +units) in a regular repeating pattern within an enclosing unit. A +*hole*, on the other hand, is a means to place a single unit. This is +often used when units being placed do not have a regular repeating +pattern. + +The format for a *hole* specification within a unit is as follows: + +:: + + hole unit_id [modifier_list] + +where *unit_id* is the identification number for the unit that is being +placed within the current unit. (A unit cannot be placed within itself.) +Unlike the shapes described earlier, holes do not have a distinct +identification number of their own—they are simply a mechanism to place +a unit defined elsewhere. + +By default, the *hole* operator places the origin of the new unit at the +origin (0,0) of the current unit. The *origin* modifier may also be used +with a *hole* specification to position the placed unit at a location +other than (0,0) of the current unit. However, placement of the body is +**always** relative to the origin of the original unit, which can be +defined in a number of different ways. + +Holes are also associated with a particular shape. Hole specifications +must immediately follow the shape into which they are being placed. +Holes redefine the boundaries of a shape by figuratively cutting holes +in that shape into which units are placed. When mixtures are defined for +a given shape (through media specifications, described below), the +mixture is placed throughout the region, except in the space excluded by +the hole placements. + +The *rotate* modifier can also be applied to a hole, as can the *com* +modifier. However, *chord* specifications cannot be used to remove a +portion of a hole. To construct a cylinder that is cut at an oblique +angle, users should construct a cylinder that is cut by a chord and then +use the *hole* operator combined with the *origin* and *rotate* +modifiers to place and rotate the unit to the desired position and +orientation. This can be particularly useful in hexagonal or triangular +geometries. + +As an example, consider a unit, unit_id=1, consisting of two concentric +cylinders, and a second unit, consisting of two concentric cuboids. +Descriptions for these two units are given below. Note that these are +incomplete unit specifications; other components of the unit +specification have not yet been introduced. However, for the purposes of +this example, incomplete unit specification will suffice. + +:: + + unit 1 + cylinder 12 0.8 + cylinder 13 0.6 + + unit 2 + cuboid 12 0.8 -0.8 0.8 -0.8 + cuboid 13 0.6 -0.6 0.6 -0.6 + +Now suppose that we wished to place two of the unit 1 cells and one of +the unit 2 cells inside unit 3, with unit 2 rotated by 45°. We can +define a cuboid as unit 3 and place the units 1 and 2 inside the cuboid +using *hole* specifications: + +:: + + unit 3 + cuboid 10 4.5 0.0 4.5 0.0 + hole 1 origin x=1.3 y=1.3 + hole 1 origin x=1.3 y=3.3 + hole 2 origin x=3.1 y=2.3 rotate a1=45 + +In this example, a square cuboid is defined such that its lower-left +corner is situated at (0,0). Three hole operations are used: the first +to place unit 1 at (1.3, 1.3), the second to place another instance of +unit 1 2 cm above the first, at (1.3, 3.3). Lastly, unit 2 is placed +inside unit 3 at (3.1, 2.3) and then rotated 45°. :numref:`fig9-2-20` +illustrates how such a unit would appear. + +.. _fig9-2-20: +.. figure:: figs/NEWT/fig20.png + :align: center + :width: 500 + + Unit placement within a unit using holes. + +.. _9-2-3-6-1-15: + +Array placement +''''''''''''''' + +As indicated in the previous section, arrays are a method for arranging +one or more units within another unit. Arrays specifications are +typically used when units are placed in a repeating pattern. While the +*hole* specification is used to place different units within a given +unit, the array placement specification is used to place an *array* +within a unit\ *.* + +All arrays are specified (declaration of size, type, and fill) in the +array data block, as described in :ref:`9-2-3-9`. The array placement +operator is used to locate an array within a unit. The format for the +array placement operator is as follows: + +:: + + array arrayid body_id place i j xij yij + +where *arrayid* is the identification number assigned to the array in +the array data block and *body_id* is the identification number of the +*shape* into which the array is placed. The remainder of the array +placement operator is used to fix the position of the array within the +body, identified by *body_id*. A general discussion of this concept +follows, after which the actual placement of the array is described. + +Arrays are defined by two dimensioning parameters—the number of rows and +the number of columns. Each element of an array is filled by a unit; +each unit has its own local coordinate system. In other words, one unit +may have the origin (0,0) in its local coordinate system defined as the +lower-left corner while another unit may have its origin defined at its +geometric center. The array itself has no coordinate system; it is +simply a list of relative positions of units, defined by their +row/column position. The *place* directive of the array placement +operator is used to locate the array within the body into which it is +being placed. + +In the *place* directive, *i* represents the column (counting from left +to right) and *j* represents the row (counting from bottom to top) of a +specific element of the array. The coordinate system of that specific +unit is used to set the position of the entire array. The coordinates +*x\ ij* and *y\ ij* represent the location in the current unit where the +array is to be placed. Placement occurs by situating the origin of the +unit in column i, row j at coordinate (x:sub:`ij`,y\ :sub:`ij`). +Placement of arrays within a unit is best understood through examples. +Consider three (partial) unit specifications, as defined and illustrated +in :numref:`fig9-2-21`. + +.. _fig9-2-21: +.. figure:: figs/NEWT/fig21.png + :align: center + :width: 600 + + Example of units to be placed in an array. + +Unit 1 is a simple 1  by 1 cm cuboid with its origin located at its +bottom-left corner; unit 2 is a similar-sized cuboid but with its origin +located at its geometric center and with a cylinder centered in it; and +unit 3 is identical to unit 1 but with a cylinder centered in it. +Units 2 and 3 are identical in structure but use a different local +coordinate system. + +Now assume an array has been defined in the array data block and +assigned *arrayid*\ =50. The relative positions of the units are shown +in :numref:`fig9-2-22`; unit 3 is located in row 1, column 1. + +We desire to place this array within a unit 4, a 2 by 2 cm cuboid with +its lower-left corner located at (0,0). Because there are four different +array positions, this array has four possible placements: + +.. _fig9-2-22: +.. figure:: figs/NEWT/fig22.png + :align: center + :width: 600 + + Layout of units in array 50. + +For the first example, the unit located in row 1, column 1 (i.e., +unit 3) is placed such that its origin (its lower-left corner) is +located at (0,0), which is the origin of unit 4. For the second example, +the unit located in column 2, row 1, is placed such that its local +origin, which is in the center of the unit, is located at x=1.5, y=0.5 +in the coordinate system of unit 4. + +.. _9-2-3-6-2: + +Media specifications +^^^^^^^^^^^^^^^^^^^^ + +A unit is only partially specified by its constituent bodies. At this +point, no composition has been associated with the various regions of +the problem nor has the outer boundary of the unit been defined. This +section provides information on the use of *media* specifications to +define the contents of each shape that has been defined. + +Each shape statement defines a basic shape, with optional modifiers, +which represents a spatial region within the unit. Assignment of +compositions to regions is performed via *media* specifications. + +As discussed earlier in the introduction to shapes, input processing in +the SGGP is combinatorial. This permits intersection of shapes, and +different compositions (or media) may be assigned to different portions +of intersecting bodies. + +The format of a media specification is as follows: + +:: + + media materialid bias_placeholder reg_def_vector + +where *materialid* is the composition number being placed in this entry, +*bias_placeholder* is a simple placeholder that is required but not +used, and *reg_def_vector* is the region definition vector used to +define the shape or shapes to which the mixture is assigned. + +The *bias_placeholder* is used to be as consistent as possible with +KENO-VI input. KENO-VI allows the user to assign biases within the media +assignments to improve the Monte Carlo solution performance. Biases have +no meaning in NEWT, so the field has no meaning. In KENO-VI, if no +special biasing is desired, a value of 1 is assigned. If it is desired +to move models between NEWT and KENO-VI format, a placeholder value of 1 +is recommended. However, the value itself has no meaning within NEWT; it +is simply read and ignored. (This may change in a future release.) + +The region definition vector is used to describe the location of the +composition within the current unit. This is done by providing a list of +shapes for which the media is either “inside” or “outside.” The sense of +the media with respect to a shape is specified by listing the shape +number with a negative sign if “outside” and with a positive (or no) +sign when the media is placed “inside” the shape. + +Consider a simple cylinder inside a cuboid. Assume composition 1 is to +be placed inside the cylinder and composition 2 outside the cylinder but +inside the cuboid. The shape and media specifications could have the +following format: + +:: + + unit 1 + cylinder 10 0.5 + cuboid 20 1.0 -1.0 1.0 -1.0 + media 1 1 10 + media 2 1 20 -10 + +In the above example, the first media record places mixture 1 inside all +of shape 10 (the cylinder). The second media entry places mixture 2 in +all regions that are outside shape 10 but inside shape 20. Note also +that a bias placeholder value of 1 is used in each media statement. + +It is necessary to specify media for all regions of the unit. If any +regions remain unassigned, NEWT will stop with an error message. If the +second record had been omitted, regions outside the cylinder would be +unspecified and the code would stop. Note also that if the second media +statement had read only + +:: + + media 2 1 20 + +then composition 2 would have been placed inside all of cuboid 20, +including inside the cylinder 10. The fact that the contents of 10 have +already been specified is ignored. The above statement directs the code +to put mixture 2 everywhere inside the boundaries of the cuboid. + +Each region definition vector combines all specifications with a logical +AND. In other words, the second media record in “``media 2 1 20``” places +composition 2 in all regions that are simultaneously outside shape 10 +**and** inside shape 20. Separate media specifications are required to +place a composition in two independent shapes. The following represents +a cuboid with two nonintersecting cylinders. + +:: + + unit 1 + cylinder 10 0.4 origin x=0.5 + cylinder 20 0.4 origin x=1.5 + cuboid 30 2.0 0.0 1.0 -1.0 + media 1 1 10 + media 1 1 20 + media 2 1 30 -10 -20 + +A media statement is necessary to place composition 1 inside shape 10; a +similar statement is necessary to place composition 1 inside shape 20. +Finally, all space inside cuboid 30 but outside both 10 and 20 is filled +with composition 2. If one attempted to fill both 10 and 20 with +composition 1 in a single media record, for example, + +:: + + media 1 1 10 20 + +then an error would occur. The code would attempt to place composition 1 +in all space that is simultaneously within shape 10 **and** within shape +20—and no such space exists. + +A more common example is the configuration of a fuel pin +(composition 1), gas gap (composition 2), clad (composition 3), and +moderator (composition 4) in a lattice. Consider a pin in a hexagonal +lattice: + +:: + + unit 1 + cylinder 10 0.4 + cylinder 20 0.41 + cylinder 30 0.5 + hexprism 40 0.8 + media 1 1 10 + media 2 1 20 -10 + media 3 1 30 -20 + media 4 1 40 -30 + +In this example, for the hexagonal moderator region outside the clad, it +is sufficient to specify that mixture 4 is inside 40 **and** outside 30. +Although it is true that the moderator is also outside shapes 10 and 20, +it is not necessary to specify this. Logically, since 10 and 20 are +inside 30, then everything outside 30 must be outside 10 and outside 20. +The use of a hexprism in this example is irrelevant. If the outer body +had been a cuboid, the result would have been the same. + +As a final example, consider a unit with intersecting bodies. It becomes +possible to assign a unique composition to each intersection of shapes +(:numref:`fig9-2-23`). + +.. _fig9-2-23: +.. figure:: figs/NEWT/fig23.png + :align: center + :width: 500 + + Media assignments in overlapping regions. + +In this model, cylinder 10 is on the right, 20 is the lower left, and 30 +is the upper left. Media 1, placed inside cylinder 10 but outside 20 and +30, is represented by the partially filled right-hand side of the right +cylinder. The central region of the unit is filled with composition 4 +and represents all areas that are simultaneously within shapes 10, 20, +and 30. The outermost region is everything that is inside 40, but +outside 10, 20, and 30. + +Media statements apply only to shapes, and only to those shapes within +the unit. Holes and array specifications are used to define placement of +one or more units in which media have already been specified in the +corresponding unit definitions. Like shape statements, media statements +may occur in any order. However, if one region is erroneously assigned +two different compositions in two different media statements, the code +will allow this and will proceed with the calculation. The last +specification for a shape will always take precedence. Thus, it is +important that newly developed models be visually inspected using +mixture plots (files named “*.newtmatl.ps”) created using *drawit=yes* +in the parameter block. + +.. _9-2-3-6-3: + +Unit boundary +''''''''''''' + +The final section of a unit description is the *boundary* specification. +This input record serves two purposes: to specify the shape that defines +the outer bounds of the unit, and hence the shape of the unit, and +(optionally) to specify the underlying grid associated with the unit. +The format of the boundary specification is as follows: + +:: + + boundary body_id [x-discretization y-discretization] + +where *body_id* is the identification number for the body that is to +serve as the unit boundary. The *x‑discretization* and +*y-discretization* terms are integers (≥2) that specify the number of +rectangular cells to be placed in the unit in the x-direction and +y-direction, respectively. A grid specification is **required** for the +global unit but is optional for other units. If a grid is specified for +a grid other than the global unit, that grid replaces the base grid. (An +exception to this principle is discussed later.) + +In general, grid refinement should be such that cell sizes are on the +order of or smaller than a mean free path for a neutron. Grid spacing +can be easily varied in order to converge on the parameter of interest. +Global factors, such as a system eigenvalue, can tolerate a relatively +coarse grid. However, if fluxes are known to vary rapidly in space, then +a more refined grid may be necessary. NEWT does provide the ability to +locally refine a grid structure so that detail can be modeled where +needed, without having to pay the computational penalty of refining the +grid everywhere. NEWT does place one limit on grid refinement: every +shape, hole, or array placed within a unit must be intersected by at +least one gridline. The grid may be locally defined or part of the +global grid, but it must intersect each body at least once. Thus, if +small geometric shapes are modeled, a detailed grid structure is +generally necessary. + +Examples of boundary specifications follow, as parts of partial unit +specifications. Media descriptions are omitted for simplicity. +Accompanying figures illustrate the grid structure(s) associated with +each specification. :numref:`fig9-2-24` shows a single (global) unit with a 2 +by 2 base grid. Cuboid 10 serves as the boundary for the unit. This +represents the minimum grid structure that can be specified for a unit. +:numref:`fig9-2-25` shows a more complex configuration in which a unit defined +with a 5 by 5 grid is placed in the center of a larger enclosing unit, +specified to have a 3 by 3 grid. Note that because the first unit has +its own (local) grid, the underlying grid structure is removed in favor +of the local grid structure. The grid is applied to the boundary shape +of the unit, which is cuboid 10. + +:numref:`fig9-2-26` shows a similar structure; however, the cuboid was removed +from unit 1 and the outer hexprism was defined as the unit boundary. +Note that the grid structure applied to the nonrectangular body is the +same as the one that would be assigned for a cuboid with the same minima +and maxima in x and y directions. :numref:`fig9-2-27` illustrates the grid +structure that would be applied to the same model as was used in the +previous figure but with CMFD acceleration enabled. Because CMFD is +normally applied to a coarse mesh defined by the base global grid +(unless xycmfd=0), the global grid is always retained when CMFD +acceleration is used. Finally, :numref:`fig9-2-28` illustrates the use of a +base grid only. In this case, no grid structure is assigned for unit 1; +the bodies are inlaid but are adapted to the base global grid structure. + +.. _fig9-2-24: +.. figure:: figs/NEWT/fig24.png + :align: center + :width: 500 + + Unit with 2 by 2 grid in a simple pin-cell model. + +.. _fig9-2-25: +.. figure:: figs/NEWT/fig26.png + :align: center + :width: 500 + + Unit with 5 by 5 grid inset into unit with 3 by 3 grid. + +.. _fig9-2-26: +.. figure:: figs/NEWT/fig26.png + :align: center + :width: 500 + + Effect of boundary grid specification on noncuboidal unit placed as a hole within a larger unit. + +.. _fig9-2-27: +.. figure:: figs/NEWT/fig27.png + :align: center + :width: 500 + + Effect of coarse-mesh finite-difference acceleration on grid structure. + + +.. _fig9-2-28: +.. figure:: figs/NEWT/fig28.png + :align: center + :width: 500 + + Use of base grid without localized grid specification. + +The following section provides examples of complete geometry specifications for +various models, including bodies, media, and boundary specifications. Each will +include one or more uses or boundary specifications for units. + +.. _9-2-3-6-4: + +Geometry block examples +^^^^^^^^^^^^^^^^^^^^^^^ + +The following three subsections present geometry block examples to show +how various models may be assembled. Each listing is described briefly +and is followed by a figure showing the NEWT grid structure generated +for each set of geometry instructions. + +.. _9-2-3-6-4-1: + +Simple pin cell +''''''''''''''' + +The following geometry block (:numref:`fig9-2-29`) shows the specifications +necessary to define a single pin-cell. The model is reduced to a ¼ cell +to take advantage of symmetry. Mixture 1 is fuel, mixture 2 is fill gas, +mixture 3 is clad, and mixture 4 is moderator. Features of this model +include the use of chords to obtain ¼ cylinders and the specification of +20 sides for each cylinder (five sides for a ¼ cylinder). + +.. _fig9-2-29: +.. figure:: figs/NEWT/fig29.png + :align: center + :width: 500 + + Geometry model for infinite-lattice pin cell with fuel, gap, clad, and moderator. + +.. _9-2-3-6-4-2: + +Hexagonal assembly +'''''''''''''''''' + +The geometry block below (:numref:`fig9-2-30`) is used to describe a hexagonal +fuel assembly within a hexagonal shroud. Each cylinder is placed +individually, followed by a series of media statements that fill each +cylinder. The hexagonal moderator area is surrounded by a hexagonal +shroud of cladding material. Note that NEWT allows only cuboid and +hexprisms as outer boundaries for the global unit. This model could also +have been assembled with a unit definition for a set of cylinders, which +could then be placed in the global unit using holes or by defining a +single pin cell and placing it using an array. + + +.. _fig9-2-30: +.. figure:: figs/NEWT/fig30.png + :align: center + :width: 500 + + Geometry model of hexagonal fuel assembly. + +.. _9-2-3-6-4-3: + +Advanced CANDU reactor ACR-700 assembly +''''''''''''''''''''''''''''''''''''''' + +This example of a geometry block (:numref:`fig9-2-31`) is included to illustrate the complexity of design that +is possible through the use of simple bodies, units, and holes. The +ACR-700 design cannot be modeled using an array because pins are not +placed in a repeating lattice pattern. Features of this example include +use of holes; use of noncuboidal units placed in holes; and localized +pin-cell grid refinement by (1) decreased mesh size in fuel elements +(three outer rings) and (2) increased radial discretization (central +pin). + +.. _fig9-2-31: +.. figure:: figs/NEWT/fig31.png + :align: center + :width: 500 + + ACR-700 fuel assembly model. + + (continued from :numref:`fig9-2-31`) + + :: + + hole 1 + hole 2 origin x=-1.56318 y= 0.75279 + hole 2 origin x=-0.38607 y= 1.69150 + hole 2 origin x= 1.08176 y= 1.35648 + hole 2 origin x= 1.73500 + hole 2 origin x= 1.08176 y=-1.35648 + hole 2 origin x=-0.38607 y=-1.69150 + hole 2 origin x=-1.56318 y=-0.75279 + hole 3 origin x=-2.99790 y= 0.68425 + hole 3 origin x=-2.40413 y= 1.91723 + hole 3 origin x=-1.33419 y= 2.77048 + hole 3 origin y= 3.07500 + hole 3 origin x= 1.33419 y= 2.77048 + hole 3 origin x= 2.40413 y= 1.91723 + hole 3 origin x= 2.99790 y= 0.68425 + hole 3 origin x= 2.99790 y=-0.68425 + hole 3 origin x= 2.40413 y=-1.91723 + hole 3 origin x= 1.33419 y=-2.77048 + hole 3 origin x= 0.00000 y=-3.07500 + hole 3 origin x=-1.33419 y=-2.77048 + hole 3 origin x=-2.40413 y=-1.91723 + hole 3 origin x=-2.99790 y=-0.68425 + hole 4 origin x=-4.33602 y= 0.65355 + hole 4 origin x=-3.95075 y= 1.90258 + hole 4 origin x=-3.21443 y= 2.98256 + hole 4 origin x=-2.19250 y= 3.79752 + hole 4 origin x=-0.97575 y= 4.27506 + hole 4 origin x= 0.32769 y= 4.37274 + hole 4 origin x= 1.60202 y= 4.08188 + hole 4 origin x= 2.73400 y= 3.42833 + hole 4 origin x= 3.62306 y= 2.47016 + hole 4 origin x= 4.19019 y= 1.29250 + hole 4 origin x= 4.38500 + hole 4 origin x= 4.19019 y=-1.29250 + hole 4 origin x= 3.62306 y=-2.47016 + hole 4 origin x= 2.73400 y=-3.42833 + hole 4 origin x= 1.60202 y=-4.08188 + hole 4 origin x= 0.32769 y=-4.37274 + hole 4 origin x=-0.97575 y=-4.27506 + hole 4 origin x=-2.19250 y=-3.79752 + hole 4 origin x=-3.21443 y=-2.98256 + hole 4 origin x=-3.95075 y=-1.90258 + + hole 4 origin x=-4.33602 y=-0.65355 + cylinder 501 5.8169 sides=20 + cylinder 502 7.55 sides=24 + cylinder 503 7.8 sides=24 + media 13 1 500 + media 5 1 501 -500 + media 6 1 502 -501 + media 7 1 503 -502 + media 8 1 100 -503 + boundary 100 50 50 + end geom + +.. _9-2-3-6-5: + +Summary of geometry specifications +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +This section is provided as a quick reference for geometry statements. +Details and examples of the usage of each geometry specification are +provided in previous subsections of this manual. For each definition, +the lists of permitted modifiers are listed. + +.. _9-2-3-6-5-1: + +Unit definition statements +'''''''''''''''''''''''''' +:: + + [global] unit unit_id + +The *unit* statement is used to initiate the definition of each unit +used. *unit_id* is an integer identification label for the unit and must +be unique. One (and only one) global unit is required in each geometry +model. Modifiers: none. + +:: + + cuboid body_id xmax xmin ymax ymin [modifier_list] + +The *cuboid* statement is used to define a rectangular shape. *body_id* +is the integer identification label for the cuboid and must be unique +within the unit it is used. The rectangle is defined such that the +coordinates (x\ :sub:`min`, y\ :sub:`min`) and (x\ :sub:`max`, +y\ :sub:`max`) represent the lower-left and upper-right vertices of the +cuboid. Modifiers: *rotate, com*. + +:: + + cylinder body_id radius [modifier_list] + +The *cylinder* statement defines a circle with integer label *body_id* +and radius *radius*, placed with its center at the origin (0,0) of the +unit. Modifiers: *origin, rotate, chord, sides, com*. + +:: + + hexprism body_id radius [modifier_list] + +The *hexprism* statement defines a standard hexagon with integer label +*body_id* and inscribed radius *radius*, placed with its center at the +origin (0,0) of the unit. A standard hexagon has vertices located on +north (top) and south (bottom) regions of the shape. Modifiers: *origin, +rotate, chord, sides, com.* + +:: + + rhexprism body_id radius [modifier_list] + +The *rhexprism* statement defines a rotated hexagon with integer label +*body_id* and inscribed radius *radius*, placed with its center at the +origin (0,0) of the unit. A rotated hexagon has vertices located on east +(right) and west (left) regions of the shape. Modifiers: *origin, +rotate, chord, sides, com.* + +:: + + wedge body_id xbase xpt ypt [modifier_list] + +The *wedge* statement defines a triangle with integer label *body_id* +placed with a vertex at (0,0), a vertex at (x\ :sub:`base`,0), and a +vertex at (x\ :sub:`pt`,y\ :sub:`pt`). Modifiers: *origin, rotate, com.* + +:: + + array arrayid body_id place i j xij yij [modifier_list] + +The *array* placement statement specifies the placement of an array with +identification number *arrayid* (defined in the array data block), +within shape *body_id*. If the *place* statement is used, the array +element located in row \ *i* (counted from the bottom) and column \ *j* +(counted from the left) is placed such that its origin is located at +spatial coordinate (*x\ ij*, *y\ ij*) of the unit in which it is placed. +Modifiers: *com*. + +:: + + hole unit_id [modifier_list] + +The *hole* statement is used to place a different unit identified by +label *unit_id* within the current unit. The origin of the unit being +placed will be located at the origin of the current unit but can be +repositioned using the *origin* modifier. Modifiers: *origin, rotate, +com*. + +:: + + media materialid bias_placeholder reg_def_vector + +The *media* statement assigns material properties associated with +mixture index *materialid* to a shape region defined within a unit. The +*bias_placeholder* term is not currently used but is retained for +consistency with KENO-VI; typically it is assigned a value of 1. +*reg_def_vector* is the region definition vector and assigns the mixture +placement relative to shapes within the unit. If the index is positive, +the shape region is included in the material assignment; if negative, it +is excluded. Modifiers: none. + +:: + + boundary body_id [x-discretization y-discretization] + +The *boundary* statement is used to define the outer boundary of the +unit, corresponding to the outer bounds of the shape *body_id* within +the unit. This shape must exist and must contain all other bodies +associated with the unit. The *x-discretization* and *y-discretization* +terms are integers (≥2) that specify the number of cells to be placed in +the unit in the x-direction and y-direction, respectively. A grid +specification is required for the global unit but is optional for other +units. Modifiers: none. + +.. _9-2-3-6-5-2: + +Geometry modifiers +'''''''''''''''''' + +:: + + origin x=xnew y=ynew + +Used to relocate the origin of cylinders, hexprisms, rhexprisms, and +holes to new co-ordinate (x\ :sub:`new`, y\ :sub:`new`). The default (if +not specified) is to place the origin of the body at (0, 0). + +:: + + rotate a1=A + +Causes a body to be rotated by an angle of *A* degrees +(counterclockwise) around its origin. It can be applied to holes and all +basic shapes; the default is 0 degrees. + +:: + + chord ±x=xplane + chord ±y=yplane + +Chords are used to truncate a shape at the line x=\ *chord* (or +y=\ *chord*). Multiple chord commands are allowed, but only one line +(either in x- or y-direction) is specified for each. If the negative +keyword ‘–x’ (or ‘–y’) appears, then the part of the shape to the left +of (below) the chord cut is retained. Similarly, if the positive keyword +(‘+x’ or ‘+y’) is used, then the portion of the shape to the right of or +above the chord is retained. Chords may be applied to cylinders, +hexprisms, and rhexprisms only. + +:: + + sides=N + +For use with the cylinder statement, the *sides* modifier specifies the +number of sides on the regular polygon used to approximate the cylinder. +Its default is N = 12. The radius of the polygon is adjusted such that +the area of the polygon matches the area of the cylinder it is +approximating. The *sides* modifier is unique to NEWT and is not used by +KENO-VI. + +.. _9-2-3-7: + +Boundary conditions +~~~~~~~~~~~~~~~~~~~ + +The *geometry data* block is generally followed by the *bounds* data +block, in which boundary conditions for the sides of the bounding shape +in the global array are specified. NEWT supports the use of cuboid, +hexprism, and wedge shapes to define outer boundaries. This results in +the need to specify up to 6 surface boundaries, on up to 8 spatial +orientations. In other words, a cuboid will have boundaries on +x, ‑x, ++y, and –y faces. A regular hexprism will have boundaries on +x and -x +faces; it will also have boundaries on the four sloped sides of the hex. +In order to identify the sense of sides for specification of boundary +conditions, NEWT applies an eight-point compass nomenclature. The four +permitted rectangular boundary surfaces are identified as +x, -x, +y, +and –y, corresponding to east (E), west (W), north (N), and south (S) +faces, as illustrated in :numref:`fig9-2-32` Sloped (non-rectangular) surfaces +are identified as +x+y, +x‑y, -x-y, and –x+y, for northeast (NE), +southeast (SE), southwest (SW), and northwest (NW) surfaces. No +assumptions are made on the slope of the various non-rectangular +surfaces; for the bodies available within NEWT, it is not possible to +have more than one surface in each octant. + +.. _fig9-2-32: +.. figure:: figs/NEWT/fig32.png + :align: center + :width: 400 + + Two-dimensional boundary condition surface orientations. + +Currently, full specification of boundary conditions is permitted only +when the boundary body for the global unit is a cuboid. Only white and +vacuum boundary conditions are permitted on non-rectangular surfaces. + +Boundary conditions therefore may only be specified for the ±x and +±y faces of a boundary cuboid. Four boundary conditions are currently +supported: + + 1. reflective (default), + + 2. white, + + 3. vacuum, and + + 4. periodic. + +Albedo boundary conditions are not yet supported but will be available +in a future release. Additional information on the meaning of each +boundary condition type is provided in the following section. + +.. _9-2-3-7-1: + +Boundary conditions descriptions +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +Boundary conditions are mathematical approximations used to describe the +behavior of neutrons when they cross a problem boundary. Typically, +transport methods provide for reflective (or mirror), white, vacuum, or +periodic boundary conditions. The following subsections describe and +illustrate these four types of boundary conditions. + +.. _9-2-3-7-1-1: + +Reflective boundary condition +''''''''''''''''''''''''''''' + +For the reflective boundary condition, the incoming angular flux is set +equal to the outgoing angular flux in the direction corresponding to +mirror or specular reflection. As shown in :numref:`fig9-2-33`, a given +quantity of neutrons leaving a boundary (dotted line) in a particular +direction will be returned (solid line of same color) to the system with +the same quantity but at a mirrored angle to the initial leakage +direction. + +.. note:: In the following figures, a dashed arrow indicates neutrons + leaving the system while a solid arrow represents those returning to the + system. The length of the arrow is proportional to the number of + neutrons; therefore, longer arrows represent more neutrons than do + shorter arrows. + +.. _fig9-2-33: +.. figure:: figs/NEWT/fig33.png + :align: center + :width: 500 + + Reflective boundary condition. + +.. _9-2-3-7-1-2: + +White boundary condition +'''''''''''''''''''''''' + +For the white boundary condition, the incoming angular fluxes are each +set equal to a single value chosen such that the net flow across the +boundary is zero. The white boundary provides isotropic return (solid +lines) at a boundary (:numref:`fig9-2-34`). + +.. _fig9-2-34: +.. figure:: figs/NEWT/fig34.png + :align: center + :width: 500 + + White boundary condition. + +.. _9-2-3-7-1-3: + +Vacuum boundary condition +''''''''''''''''''''''''' + +A vacuum boundary condition means that no neutrons will reenter the +boundary. Thus, any neutron exiting the system through a boundary with a +vacuum boundary condition is permanently lost to the system. This +condition is illustrated in :numref:`fig9-2-35`. + +.. _fig9-2-35: +.. figure:: figs/NEWT/fig35.png + :align: center + :width: 500 + + Vacuum boundary condition. + +.. _9-2-3-7-1-4: + +Periodic boundary condition +''''''''''''''''''''''''''' + +For the periodic boundary condition, the incoming angular flux on a +boundary is set equal to the outgoing angular flux on the opposite +boundary. :numref:`fig9-2-36` shows the leakage leaving each boundary (dotted +lines) being returned at the same quantity and angle on the opposite +boundary (solid line of same color). When the periodic boundary +condition is used, it must be applied to both opposing boundaries. + +.. _fig9-2-36: +.. figure:: figs/NEWT/fig36.png + :align: center + :width: 500 + + Periodic boundary condition. + +.. _9-2-3-7-2: + +Boundary condition specification +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +The standard format for boundary condition specifications is as follows: + +:: + + read bounds + -x=west_BC +x=east_BC -y=south_BC +y=north_BC + +x+y=northeast_BC +x-y=southeast_BC + -x+y=northwest_BC -x-y=southwest_BC + end bounds + +where *west_BC, east_BC, south_BC, north_BC, northeast_BC, southeast_BC, +northwest_BC, southwest_BC* are each one of the eight possible boundary +condition options. The name of the boundary condition requires only the +number of leading characters required to make the name unique. For this +set, the first letter is sufficient; that is, +x=v, +x=vac +x=vacu, and ++x=vacuum are all equivalent and specify a vacuum (zero return) boundary +condition on the right side of the global cuboid. + +In keeping with KENO-VI, multiple shortcuts exist to simplify the +specification. For example, a single boundary condition can be assigned +to all four sides simultaneously with the *all=* specifier: + +:: + + read bounds + all=refl + end bounds + +All KENO-VI boundary face keywords that do not reference the z-dimension +are allowed and are listed in :numref:`tab9-2-1`. + +.. _tab9-2-1: +.. table:: Boundary condition specifiers accepted by NEWT. + :align: center + + +------------------+---------------------------------+ + | **Keyword** | **Boundary edge** | + +==================+=================================+ + | +x, +xb | East (right) | + +------------------+---------------------------------+ + | -x, -xb | West (left) | + +------------------+---------------------------------+ + | +y, +yb | North (top) | + +------------------+---------------------------------+ + | -y, -yb | South (bottom) | + +------------------+---------------------------------+ + | all, xyf, yxf | All boundaries | + +------------------+---------------------------------+ + | +xy, +yx, +fc | East (right) + north (top) | + +------------------+---------------------------------+ + | -xy, -yx, -fc | West (left) + south (bottom) | + +------------------+---------------------------------+ + | xfc | West (left) + east (right) | + +------------------+---------------------------------+ + | yfc | North (top) + south (bottom) | + +------------------+---------------------------------+ + | +x+y | Northeast (top right) | + +------------------+---------------------------------+ + | -x+y | Southeast (bottom right) | + +------------------+---------------------------------+ + | -x-y | Southwest (bottom left) | + +------------------+---------------------------------+ + | -x+y | Northwest (top left) | + +------------------+---------------------------------+ + +.. _9-2-3-8: + +General cross section weighting +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +NEWT performs cross section weighting by mixture and optionally by +homogenization zone (:ref:`9-2-3-10`). Weighting is always used in +conjunction with an energy collapse. Cross section weighting is +performed over a spatial and energy domain; the resulting average +(weighted) cross sections will preserve all reaction rates in the +collapsed cross section set; that is, + +.. math:: + :label: eq9-2-28 + + \sigma_{G}^{i}=\frac{\int_{r} N^{i}(r) d r \int_{G} \sigma^{i}(E, r) W(E, r) d E}{\int_{r} N^{i}(r) d r \int_{G} W(E, r) d E} , + +where + + :math:`\sigma_{G}^{i}` ≡ average (weighted) cross section in energy group \ *G* for + nuclide \ *I*, + + :math:`N^{i}(r)` ≡ number density of nuclide *i* in region *r*, + + :math:`W(E, r)` ≡ the weighting function within region *r,* + + :math:`\sigma^{i}(E, r)` ≡ the cross section from the input library for nuclide \ *i* in + region \ *r*. + +Within NEWT, each collapsing region is the spatial region or regions in +which a given mixture is placed. Hence, for most of the cross section +types, an average cross section for the mixture associated with the +spatial domain *r* (which may include one or more defined regions +occupied by that mixture) is calculated by weighting the original +problem-specific cross section data for that mixture using as a +weighting function the neutron spectrum calculated within spatial domain +*r*. + +In practice, the integration of :eq:`eq9-2-28` is performed as a simple +summation over all cells *j* within region \ *r*: + +.. math:: + :label: eq9-2-29 + + \sigma_{i, G}=\frac{N_{r}^{i} \sum_{j \in r} \sum_{g \in G} \sigma_{g, j}^{i} W_{g, j}}{N_{r}^{i} \sum_{j \in r} \sum_{g \in G} W_{g, j}} . + +Because any region *r* is defined as the sum of all spatial regions +containing a given mixture, is constant everywhere within *r*. + +All multigroup cross sections and related data present on the AMPX +library are weighted using appropriate weighting functions. For most +basic cross sections, the multigroup flux obtained from the transport +solution is the appropriate weighting function and :math:`W_{g, r}` in :eq:`eq9-2-29` +becomes :math:`\phi_{g, r}`. However, special cross sections and data need special +treatment, as described in the following sections. + +.. _9-2-3-8-1: + +Scattering cross section transfer matrix weighting +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +In weighting scattering cross sections, the form of the weighting is +slightly more complex: + +.. math:: + :label: eq9-2-30 + + \sigma_{s, G \rightarrow G^{\prime}}^{i}=\frac{\int_{r} N^{i}(r) d r \int_{G} W(E, r) d E \int_{G^{\prime}} \sigma^{i}\left(E \rightarrow E^{\prime}, r\right) d E^{\prime}}{\int_{r} N^{i}(r) d r \int_{G} W(E, r) d E} , + +or, in multigroup format, + +.. math:: + :label: eq9-2-31 + + \sigma_{s, G \rightarrow G^{\prime}}^{i}=\frac{N_{r}^{i} \sum_{j \in r} \sum_{g \in G} W_{g, r} \sum_{g^{\prime} \in G^{\prime}} \sigma^{i}\left(g \rightarrow g^{\prime}\right)}{N_{r}^{i} \sum_{j \in r} \sum_{g \in G} W_{g, r}} . + +In general, the scalar flux, *ϕ*\ :sub:`g,r` is the appropriate weighting +function for scattering cross sections: + +.. math:: + :label: eq9-2-32 + + \sigma_{G}^{i}=\frac{N_{r}^{i} \sum_{j \in r} \sum_{g \in G} \sigma_{g, r}^{i} \phi_{g, r}}{N_{r}^{i} \sum_{j \in r} \sum_{g \in G} \phi_{g, r}} . + +This is an approximation for the higher order moments (*l* > 0) of the +scattering cross sections, which should be more properly weighted using +the *l*\ th moment of the flux instead of the 0th moment (scalar) flux +as used in Eq. :eq:`eq9-2-32`. However, because flux moments are generally +both positive and negative, NEWT performs higher-order-moment scattering +cross section weighting using the positive scalar flux. + +.. _9-2-3-8-2: + +Weighting of the collapsed fission spectrum, :math:`\chi` +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +Weighting is not required for collapsing a fission spectrum vector; the +format for a collapsed fission spectrum (χ\ :sub:`G`) is a very +straightforward summation of the fission spectra in energy groups \ *g* +spanning the energy domain of the collapsed energy group \ *G*: + +.. math:: + :label: eq9-2-33 + + \chi_{G}=\sum_{g \in G} \chi_{g} . + +.. _9-2-3-8-3: + +Weighting of the number of neutrons per fission, :math:`\boldsymbol{V}` +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +Accurate weighting of :math:`v` in an energy and space domain requires weighting +by the fission rate; that is, + +.. math:: + :label: eq9-2-34 + + W(E, r)=\sigma_{\text {fission }}^{i}(E, r) \phi(E, r) . + +Hence, for :math:`v`, Eq. :eq:`eq9-2-32` has the form + +.. math:: + :label: eq9-2-35 + + v_{G}^{i}=\frac{N_{r}^{i} \sum_{j \in r} \sum_{g \in G} v_{g}^{i} \sigma_{f i s s i o n, g, r}^{i} \phi_{g, r}}{N_{r}^{i} \sum_{j \in r} \sum_{g \in G} \sigma_{f i s s i o n, g, r}^{i} \phi_{g, r}} . + +.. _9-2-3-8-4: + +Weighting of (n,2n), (n,3n), and (n,4n) cross sections +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +In creating AMPX weighted libraries with NEWT, all data on the original +library are collapsed and written to a collapsed working-format library, +for each reaction type and for each nuclide. Therefore, each of the +(n,*X*\ n) libraries is collapsed independently using Eq. :eq:`eq9-2-29` and +is written on the weighted library. + +However, during NEWT transport calculations, NEWT computes and stores a +single effective (n,2n) reaction rate, determined as the weighted sum of +the individual reactions: + +.. math:: + :label: eq9-2-36 + + \sigma_{n, 2 n}^{\text {effective }}=\sigma_{n, 2 n}+2 \sigma_{n, 3 n}+3 \sigma_{n, 4 n} . + +The (n,2n) reaction rates reported in NEWT output are those computed for +the effective cross section. The effective (n,2n) cross section is not +saved to the weighted library. + +.. _9-2-3-8-5: + +Calculation and weighting of transport cross sections +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +Transport cross sections are processed in a different manner from other +cross sections. The transport cross section does not represent a purely +measurable quantity. Introduced within the P\ :sub:`1` (diffusion) +approximation to the neutron transport equation, it attempts to preserve +a flux gradient in addition to reaction rate information. Hence, the +magnitude of a microscopic transport cross section is affected by both +the physics properties of the nuclide in question and the geometrical +attributes of the spatial domain where the nuclide resides and the other +nuclides present in the same vicinity. + +Consistent with XSDRNPM, NEWT provides two options to generate a +microscopic transport cross section—based on the “consistent” and +“inconsistent” methods for solving the P\ :sub:`l` transport equations. +These approximations are referred to as the “outscatter” and “inscatter” +approximations because of the nature of the equations used. + +.. _9-2-3-8-5-1: + +Outscatter approximation (inconsistent method) +'''''''''''''''''''''''''''''''''''''''''''''' + +In the outscatter approximation, the following assumption is made for +the transport cross section in group \ *g*: + +.. math:: + :label: eq9-2-37 + + \sigma_{t r}^{g}=\sigma_{t}^{g}-\bar{\mu}^{g} \sigma_{s}^{g} , + +where σ\ :sub:`t`\ :sup:`g` and σ\ :sub:`s`\ :sup:`g` are the total and +scattering cross section in group \ *g*, + +.. math:: + :label: eq9-2-38 + + \bar{\mu}^{g}=\frac{\sigma_{s, 1}^{g}}{3 \sigma_{s, 0}^{g}} + +and + +.. math:: + :label: eq9-2-39 + + \sigma_{s, N}^{g}=\sum_{g^{\prime}} \sigma_{s, N}\left(g \rightarrow g^{\prime}\right) . + +Note that the :math:`\sigma_{s, N}\left(g \rightarrow g^{\prime}\right)` terms are the P\ :sub:`N` coefficients of the scattering +matrix, hence the origin of the term “outscatter.” + +.. _9-2-3-8-5-2: + +Inscatter approximation (consistent method) +''''''''''''''''''''''''''''''''''''''''''' + +In the “consistent” P\ :sub:`1` approximation of the transport equation, +the transport cross section is defined as + +.. math:: + :label: eq9-2-40 + + \sigma_{t r}(E)=\sigma_{t}(E)-\frac{1}{3 J(E)} \int_{0}^{\infty} \sigma_{s, 1}\left(E^{\prime} \rightarrow E\right) J\left(E^{\prime}\right) d E^{\prime} + +where *J* is the neutron current and :math:`\sigma_{s, 1}` is the first moment (P\ :sub:`1` +coefficient) of the scattering transfer matrix. + +If one multiplies Eq. :eq:`eq9-2-40` by J(E), integrates over group \ *g*, and +converts to a group-averaged form by dividing by :math:`\int_{g} J(E) d E`, the following +expression is derived: + +.. math:: + :label: eq9-2-41 + + \sigma_{t r}^{g}=\sigma_{t}^{g}-\frac{1}{3 J_{g}} \sum_{g^{\prime}} \sigma_{s, 1}\left(g^{\prime} \rightarrow g\right) J_{g^{\prime}} . + +This is the “inscatter” approximation. It is consistent because the +transport values are explicitly derived from the P\ :sub:`0` and +P\ :sub:`1` equations. + +Weighting function for transport cross section +'''''''''''''''''''''''''''''''''''''''''''''' + +Internal investigations have shown that transport cross sections +computed using the “outscatter” approximation are more robust in +subsequent nodal core calculations as compared with transport cross +sections computed using the “inscatter approximation.” NEWT computes +transport cross sections using the outscatter approximation and +collapses the cross section with the scalar flux. + +.. _9-2-3-9: + +Array definition +~~~~~~~~~~~~~~~~ + +Any arrays specified in unit definitions within the *read geom* block +are defined in terms of form and content in the *read array* data block. +The block has the form shown below: + +:: + + read array + ara=arrayid nux=nx nuy=ny typ=aratype [pinpow=yes/no] + fill i1 i2 i3 i4 … iN end fill + … + end array + +where *arrayid* is a unique integer identifier for the array, *nx* is +the number of array elements moving left to right (i.e., columns), and +*ny* is the number of array elements moving from bottom to top +(i.e., rows). The type of array is indicated by *aratype* (e.g., square, +hexagonal). The optional parameter *pinpow* may be specified as either +*yes* or *no* (default is no) and is used to enable/disable pin power +edits for units within the array. The *fill*\ …\ *end fill* specifier +set is used to input the elements of the array. A total of *N *\ entries +are required, where *N* = nx*ny. + +Each of these portions of the array definition statement is described in +more detail below. + +.. _9-2-3-9-1: + +Array types +^^^^^^^^^^^ + +NEWT supports arrays of cuboids, hexprisms, and rotated hexprism +elements. Four different array types may be selected. :numref:`tab9-2-2` lists +the supported array types and corresponding element types. The array +type given in the first column lists the keyword associated with each +type, as used in the typ= specifier; in some cases, multiple keywords +are associated with a specific array type. The element type in the +second column provides the boundary shape that can be used in the given +array type. For example, a cuboidal (square) array may only be filled +with cuboidal units. + +.. _tab9-2-2: +.. table:: NEWT array types with corresponding element types. + :align: center + + +-----------------------+------------------+ + | **Array type** | **Element type** | + +=======================+==================+ + | Cuboidal, square | Cuboid | + +-----------------------+------------------+ + | Hexagonal, triangular | Hexprism | + +-----------------------+------------------+ + | Shexagonal | Hexprism | + +-----------------------+------------------+ + | Rhexagonal | Rhexprism | + +-----------------------+------------------+ + +All arrays are filled in a 2-D *i, j* pattern, with *i* varying from 1 +to *nux* and *j* varying from 1 to *nuy*. All \ *nux*nuy* elements of +each array must be filled. :numref:`fig9-2-37` illustrates the layout of a +conceptual 4 by 4 cuboidal array, showing the row/column index for each +element of the array. :numref:`fig9-2-38` shows the row/column designation for +a 4 by 4 hexagonal array. Because of the shape of a hexprism, the array +itself is staggered. However, the row/column numbering is simple to +understand. + +The stacked hexagon (shexagon) layout, as shown in :numref:`fig9-2-39`, was +developed to simply allow an alternate placement algorithm for +hexprisms. Any model that can be specified with *typ=hexagonal* can also +be specified with *type=shexagonal*; the choice of which form to use is +simply one of user preference. It is important to note that beginning +with row 3, units will be placed in a physical location different from +that of the hexagonal layout when the shexagonal layout is used. + +Finally, the rotated hexprism (rhexprism) array is pictured in +:numref:`fig9-2-40`. This array is intended to facilitate placement of +rhexprisms. The numbering of cells is somewhat irregular because of the +staggered rows, but it is easy to follow if one is aware of the fill +pattern as illustrated in the figure. Note that the layout of a +rhexagonal array can be replicated exactly using a hexagonal or +rhexagonal array, placed in a hole, and rotated 90°. Thus, the type of +hexprism-based array used can always be tailored to the preferences of +the user and all can be used to create the same model. + +It is often the case, especially for hexagonal-type arrays, that one +does not need to fill all array positions. While the array fill +procedure does require that all positions be filled, NEWT provides a +mechanism to fill a position with a null unit, effectively skipping that +position. This is discussed further under Fill Operations. + +.. _fig9-2-37: +.. figure:: figs/NEWT/fig37.png + :align: center + :width: 400 + + Layout of a 4 by 4 cuboidal array. + +.. _fig9-2-38: +.. figure:: figs/NEWT/fig38.png + :align: center + :width: 400 + + Layout of a 4 by 4 hexagonal array. + +.. _fig9-2-39: +.. figure:: figs/NEWT/fig39.png + :align: center + :width: 400 + + Layout of a 4 by 4 stacked hexagonal (shexagonal) array. + +.. _fig9-2-40: +.. figure:: figs/NEWT/fig40.png + :align: center + :width: 400 + + Layout of a 4 by 4 rotated hexagonal (rhexagonal) array. + +Although all elements of cuboidal arrays **must** be cuboids, they need +not be the same size. Elements of each row must have the same height but +may have varying widths. Similarly, elements of each column must be of a +single common width but may vary in height. Less flexibility is +available in hex-based arrays, because of their very nature. Hexagonal +and stacked hexagonal arrays may contain only hexprisms, and all must be +of the same outer size (although unit contents may vary as needed). +Rotated hexagonal arrays likewise are limited to rhexprisms with a +single boundary size. + +.. _9-2-3-9-2: + +Pin-power edits +^^^^^^^^^^^^^^^ + +In lattice-physics calculations, it is often necessary to obtain a +pin-power edit showing the power produced in each fuel pin cell. NEWT +uses the array functionality to define pin cells. When *pinpow=yes* is +specified, an extra edit is produced that gives the normalized pin power +in each pin cell. A pin cell is defined as any element within the array +that contains a fissionable nuclide. Pin powers are normalized such that +the average of all fuel-bearing array elements is 1.0. Array elements +such as burnable poison rods or water holes, which produce no *fission* +power, are not included in the power normalization process. The *pinpow* +functionality is not available for hexagonal, shexagonal, or rhexagonal +lattices. + +Output provides an edit of each assembly for which *pinpow=yes* is +specified. In addition, a final edit is provided for the entire system, +normalized to all fuel cells in all arrays for which *pinpow=ye*\ s is +specified. + +Pin-power edits are shown in the description of output in :ref:`9-2-5`. + +.. _9-2-3-9-3: + +Fill operations +^^^^^^^^^^^^^^^ + +The final section of an array specification is the *fill* list. +Delimited by *fill* and *end fill* keywords, NEWT expects a list of +N=nux*nuy unit numbers, specifying the unit to be placed at each array +position. Arrays are filled left to right, starting at the bottom +left-hand corner and moving up a row after all columns in the current +row are filled. In other words, any of the 4 × 4 arrays in the figures +above would be filled in the following order: (1,1), (2,1), (3,1), +(4,1), (1,2), (2,2), (3,2), (4,2), (1,3), (2,3), (3,3), (4,3), (1,4), +(2,4), (3,4), (4,4). + +The list of elements used to fill an array consists of unit numbers. +Each unit used in the fill list must be defined in the geometry block +and must be of the shape and size required for the array type and +position. However, NEWT provides the ability to fill an array location +with a null unit, which in essence skips the current array location. +This is accomplished simply by entering unit number “0” (a “null” unit) +at the location to be skipped. + +.. _9-2-3-9-4: + +Examples of array definitions +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +Consider a simple 3 × 3 square array with array ID 10, with unit 1 to be +placed in the center of the array, surrounded by unit 2 cells. Such a +specification would take the following form: + +:: + + ara=10 nux=3 nuy=3 type=cuboidal fill 2 2 2 2 1 2 2 2 2 end fill + +This may also be written spanning several lines to help visualize the layout: + +:: + + + ara=10 nux=3 nuy=3 type=cuboidal fill + 2 2 2 + 2 1 2 + 2 2 2 end fill + +Note that the array is being filled from the bottom so that the actual unit +layout is inverted relative to the fill description. A fill specification that +places unit 1 at the top center position in this array would be input as +follows, where the second-to-last entry in the list is placed in the horizontal +center of the top row. + +:: + + ara=10 nux=3 nuy=3 type=cuboidal fill + 2 2 2 + 2 1 2 + 2 1 2 end fill + +Arrays may be nested within arrays. Each array must be placed in a unit, but a +unit containing an array may be placed within another array. The following +example demonstrates the use of nested arrays, along with the use of a null +unit. + +In this example, we define two units containing cells, a unit containing a +smaller array and a global unit containing a larger array: + + +.. code:: none + :class: long + + unit 1 + '0.5/0.6 cm radius pin + cylinder 30 0.5 sides=20 + cylinder 20 0.6 sides=20 + cuboid 10 4p0.75 + media 1 1 30 + media 2 1 20 -30 + media 3 1 10 -20 + boundary 10 3 3 + + unit 2 + '0.25/0.3 cm radius pin + cylinder 30 0.25 sides=8 + cylinder 20 0.3 sides=8 + cuboid 10 4p0.375 + media 1 1 30 + media 2 1 20 -30 + media 3 1 10 -20 + boundary 10 3 3 + + 'small array + unit 3 + cuboid 10 1.5 0 1.5 0 + array 1 10 place 1 1 0.375 0.375 + media 3 1 10 + boundary 10 5 5 'large array + global unit 4 + cuboid 10 3.0 0.0 3.0 0.0 + array 2 10 place 1 1 0.75 0.75 + media 3 1 10 + boundary 10 5 5 + +Now, in a *read array* block, we define array 1, a 2 by 2 array filled +with unit 2 cells, and array 2, filled with two unit 1 cells, one unit 3 +cell (containing array 1), and one null unit: + +:: + + read array + ara=1 nux=2 nuy=2 typ=cuboidal + fill 2 2 + 2 2 end fill + ' + ara=2 nux=2 nuy=2 typ=cuboidal + fill 1 3 + 0 1 end fill + end array + +When assembled, the model would appear as shown in :numref:`fig9-2-41`. Note +that local grids override the global grid in each array location but +that the global grid is seen where the null unit is placed. + +.. _fig9-2-41: +.. figure:: figs/NEWT/fig41.png + :align: center + :width: 500 + + Example of nested arrays and a null unit specification. + +.. _9-2-3-10: + +Homogenization block +~~~~~~~~~~~~~~~~~~~~ + +**Homogenization block keyword = homog, hmog, homo** + +As discussed earlier, NEWT can be used to collapse cross sections to a +reduced broad-group format. The cross sections produced from this +operation are written as microscopic cross sections for each nuclide in +each mixture. NEWT also provides the ability to produce macroscopic +weighted cross sections homogenized over one or more mixtures. +Homogenized cross sections are created using the collapsing energy +structure defined in the collapse data block or the original library’s +group structure if no collapsing instructions are provided. +Flux-weighted collapsed cross sections are combined with number +densities and added such that reaction rates in homogenized materials +are conserved. + +Within the homogenization block, multiple homogenization records are +permitted, and the same mixtures may be repeated in different records. +Each record provides a recipe defining the mixtures to be homogenized. +Homogenization records have the following form: + +:: + + Homogenized Mixture List of end + Mixture ID Description Mixtures + +The homogenized mixture ID is the “nuclide” number under which the +mixture is saved on the homogenized cross section library. The value is +arbitrary and serves only as a means to identify the cross section set +on the library, although each ID must be unique. The mixture description +is an alphanumeric label of up to 12 characters that is associated with +the mixture; this label provides a little more descriptive ability than +the ID itself. The label may not contain blanks. Finally, the label is +followed by the list of mixtures to be homogenized, terminated by the +end keyword. The list may contain up to 1000 unique mixtures. + +A sample homogenization block is shown below. In this illustration, two +homogenized mixtures are created. This first consists of five different +fuel mixtures (201–205); this could be used to obtain the average fuel +cross section for an assembly containing five different fuel types. The +homogenized cross sections will be written to the homogenized library as +nuclide 500, with label “fuel.” The second instruction homogenizes +mixtures 201, 210, and 220 from the original problem; this could be +used, for example, to homogenize the fuel, clad, and moderator of a fuel +pin cell. This cross section set would be written on the same library as +nuclide 501, with the label “fuel_cell201.” + +:: + + read hmog + 500 fuel 201 202 203 204 205 end + 501 fuel_cell201 201 210 220 end + end hmog + +Homogenized (macroscopic) cross sections are saved in an AMPX +working-format library at the unit specified by the *hmoglib=* parameter +(default=13) [ft13f001]. + +.. _9-2-3-11: + +Assembly discontinuity factors +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +**Assembly discontinuity factor (adf) block keyword = adf** + +Because discontinuity factors have meaning only with respect to +homogenized cross sections, ADFs are calculated only if homogenized +cross sections are also specified via the Homogenization data block +(see :ref:`9-2-3-10`). Calculation of ADFs is specified in the *read adf* +data block. The three supported formats of this data block are as +follows. For a single-assembly model, the following format is used: + +:: + + read adf + 1 homg_assm_id n=Y1 s=Y2 e=X1 w=X2 + end adf + +For a reflected assembly model, the following format is used: + +:: + + read adf + 2 homg_assm_id homog_refl_id w=Xi + end adf + +:: + + read adf + 3 homg_assm_id line1 line2 line3 ... + end adf + +where + + ``hmog_assm_id`` is the identifying label assigned to the homogenized assembly, + + ``hmog_refl_id`` is the label assigned to the homogenized reflector region, + + ``Y1`` is the y-ordinate of the north boundary of the assembly + (typically this is y\ :sub:`max` for the global unit), + + ``Y2`` is the y-ordinate of the south boundary of the assembly (typically this is y\ :sub:`min` for the global unit), + + ``X1`` is the x-ordinate of the east boundary of the assembly + (typically this is x\ :sub:`max` for the global unit), + + ``X2`` is the x-ordinate of the east boundary of the assembly + (typically this is x\ :sub:`max` for the global unit), + + ``Xi`` is the x-ordinate for the fuel/reflector interface, + + ``linei`` is a series of two ordered pairs (``X1``,``Y2``), (``X1``,``Y2``)that define a line segment in the NEWT grid. + +In a single-assembly calculation, only a single homogenized mixture is +specified. Leading index 1 indicates that ADFs for the fuel assembly are +being calculated; ADFs may be calculated on any or all faces of a +rectangular assembly. In a reflected assembly, the leading index is 2, +followed by the homogenized mixture ID for the fuel assembly first, then +the homogenized mixture ID for the reflector region. An ADF may be +requested only for one location, at the fuel/reflector interface. In any +configuration, ADFs may be requested for a set of arbitrary line +segments defined in the NEWT geometry. In this case, the leading index +is 3, followed by the homogenized mixture ID, followed by up to 12 line +segments, which are defined by their beginning and ending points. ADFs +along these lines are defined as the surface-averaged flux divided by +the average flux defined for the associated homogenized mixture. +Surface-averaged currents are also edited for each arbitrary line +segment; both full and partial currents in both the x- and y- directions +are provided. The net current is also provided in the few-group cross +section database file *xfile016*. The orientation of the net current +across the line segment is further discussed in Appendix A of TRITON +chapter. + +Although any homogenized set of mixtures can be specified for each +homogenized region, the ADF will have physical meaning only if the +homogenized set represents all mixtures in the assembly. Similarly, if a +reflector calculation is performed, the *hmog_refl_id* should represent +the set of homogenized reflector mixtures. The average collapsed flux in +the homogenized mixtures is used to calculate the homogeneous flux for a +single-assembly ADF. In a reflector calculation, the homogenized +cross sections for the reflector are used to solve the multigroup +diffusion approximation (:ref:`9-2-2-5`). + +Examples illustrate the use of the ADF input specification. Consider a +17 × 17 pressurized-water-reactor (PWR) design. Because of symmetry, it +can be modeled a ¼ assembly; therefore, we choose to model the upper +right quadrant, as shown in :numref:`fig9-2-42`. Mixtures 1, 2, 3, and 4 +represent the fuel, clad, moderator, and guide tube materials, +respectively. The west and south sides of the model are the assembly +midplanes, so ADFs calculated on these boundaries would have no physical +meaning. (These are not real assembly boundaries.) However, because of +the symmetry of the assembly, fluxes would be identical on all +boundaries. Therefore, selection of either the north or east boundary +will yield a valid ADF for all boundaries. We choose to request an ADF +calculation for the east (right) side of the assembly. + +.. _fig9-2-42: +.. figure:: figs/NEWT/fig42.png + :align: center + :width: 400 + + Upper-right quadrant of an unreflected PWR assembly. + +Assuming we are collapsing 44 energy groups to 2 energy groups, we would +specify the following *collapse*, *homog*, and *adf* blocks to obtain +2-group ADFs representative of this assembly: + +:: + + read collapse + 22r1 22r2 + end collapse + read hmog + 500 assm 1 2 3 4 end + end hmog + read adf + 1 500 e=10.752 + end adf + +Again, recall that for a single assembly, the ADF in each energy group +is simply the average flux on the specified boundary divided by the +average flux for the entire assembly, which in this case is the flux in +homogenized mixture 500. + +ADF can also be calculated using the arbitrary line-segment ADF type. +Using this new ADF type, the ADF input would be the following: + +:: + + read collapse + 22r1 22r2 + end collapse + read hmog + 500 assm 1 2 3 4 end + end hmog + read adf + 3 500 10.752 0.0 10.752 10.752 + end adf + +In this example, the east-side ADF will be calculated along the line +segment starting at (10.752,0) and ending at (10.752,10.752). The values +of the line segments depend on a coordinate system of the global unit. + +For a reflected model, consider the same type of assembly but +representing an assembly placed on the core periphery. It is bounded on +one side by a 2 cm stainless steel baffle and 10 cm of water, beyond +which is treated as vacuum. Because the reflector calculation is a 1-D +solution, it is not possible to directly model a fuel assembly with two +reflector boundaries. Typically the assumption is made that the same ADF +may be applied in any assembly/reflector boundary and that a 1-D +reflector model is all that is necessary. This model is pictured in +:numref:`fig9-2-43`. Notice that different mixtures are used in the reflector +model. Fluxes used in homogenized mixtures and for generating +homogenized cross sections are spatially averaged; thus, it is important +to separate the moderator in the reflector from that in the fuel +assembly such that average fluxes in each region properly characterize +that region. For example, the flux in the reflector will be +significantly different (far more thermal) from the flux within the +assembly. + +.. _fig9-2-43: +.. figure:: figs/NEWT/fig43.png + :align: center + :width: 500 + + Upper-right quadrant of a PWR assembly with baffle and reflector. + +Assuming again a collapse from 44 energy groups to 2, the following +*collapse*, *homog*, and *adf* blocks would be specified to obtain +2-group ADFs representative of the assembly and the reflector; 2-group +cross sections for each homogenized region would also be prepared from +the collapse and homogenization instructions: + +:: + + read collapse + 22r1 22r2 + end collapse + read hmog + 500 assm 1 2 3 4 end + 501 reflector 5 6 end + end hmog + read adf + 2 500 501 w=10.752 + end adf + +.. _9-2-3-12: + +Flux planes +~~~~~~~~~~~ + +**Fluxplane block keyword = fluxplane, fluxplan, flux, fluxplanes** + +The fluxplane block is a special output edit that lets one obtain the +average scalar flux and currents along any line segment or any +continuous set of collinear line segments. One must simply specify the +start and end points of a line segment for which a linearly averaged +flux is desired. This line segment must correspond to one or more line +segments in the model’s grid structure, which requires some knowledge of +where grid lines exist in the model. + +The format of a flux plane specification is as follows: + +:: + + read fluxplane + text_label homg_assm_id xstart ystart xend yend + … + end fluxplane + +where *text_label* is an alphanumeric description used to label the +selected plane in the output, *homg_assm_id* is the identifier in the +*homog* block, and *(xstart, y\ start)* and *(xend, y\ end)* are the +start and end points, respectively, for the line segment for which an +average flux is desired. The *text_label* string must not contain white +space and may be up to 16 characters in length. + +As an example, we consider a simple model consisting of two dissimilar +pin cells (1/4 cells): + +:: + + global unit 1 + cuboid 1 1.26 0.0 0.63 0.0 + cylinder 2 .4750 chord +x=0 chord +y=0 sides=20 + cylinder 3 .4095 chord +x=0 chord +y=0 sides=20 + cylinder 4 .6030 chord +y=0 chord -x=1.26 origin x=1.26 sides=20 + cylinder 5 .5630 chord +y=0 chord -x=1.26 origin x=1.26 sides=20 + media 1 1 3 + media 2 1 2 -3 + media 10 1 5 + media 2 1 4 -5 + media 3 1 1 -2 -4 + boundary 1 4 2 + +.. image:: figs/NEWT/img1.png + :align: center + :width: 400 + +We know a line segment (actually, two) exists at *x* = 0, *x* = 0.63, +and *x* = 1.26, between *y* = 0 and *y* = 0.63. Thus, a legitimate set +of flux plane specifications would be the following: + +:: + + read fluxplane + cell_interface 0.63 0.0 0.63 0.63 + midplane_cell1 0.0 0.0 0.0 0.63 + midplane_cell2 1.26 0.0 1.26 0.63 + end fluxplane + + +This will provide a summary of fluxes and currents at each line segment in +fine-group structure, and if a collapse is performed, in broad-group structure. +Results from a calculation with a two-group collapse appear as follows: + + +:: + + Broad Group Fluxes: + Group cell_interface midplane_cell1 midplane_cell2 + 1 5.906586D+01 5.903113D+01 5.930420D+01 + 2 7.645792D+00 7.641424D+00 6.705809D+00 + + Broad Group Currents (x): + Group cell_interface midplane_cell1 midplane_cell2 + 1 -3.481505D-01 0.000000D+00 0.000000D+00 + 2 3.004759D-01 0.000000D+00 0.000000D+00 + + Broad Group Currents (y): + Group cell_interface midplane_cell1 midplane_cell2 + 1 1.081450D-01 1.652161D-01 1.120160D-01 + 2 -1.031163D-01 -1.647396D-01 -9.993214D-02 + + +Output also includes +x, –x, +y, and –y components of currents. + +.. _9-2-3-13: + +Mixing table block +~~~~~~~~~~~~~~~~~~ + +**Mixing table block keyword = mixt, mixtable** + +Generally, NEWT calculations are performed using a cross section library +and mixing table prepared in advance by other SCALE modules. However, +NEWT allows the user the ability to manually specify the isotopic +composition of each mixture used in a NEWT model. If a mixing table +block is read, any existing mixing table data file is ignored. +Therefore, all mixtures specified in the material block must be mixed in +the mixing table block. + +The format of the mixing table is simple and straightforward. For each +nuclide used, three parameters must be supplied: (1) *mixid*, the +mixture ID number into which the nuclide is to be mixed; +(2) *nuclideid*, the SCALE ID number for the nuclide (which must exist +on the cross section library being referenced); and (3) *concentration*, +the number density (atoms/b-cm) of the nuclide in this mixture. The same +nuclide may appear in multiple mixtures or more than once in a single +mixture if desired. Macroscopic cross sections are determined for each +mixture by the following formula: + +.. math:: + + \Sigma^{R}=\sum_{i} \sigma_{i}^{R} N_{i} + +where + + :math:`\Sigma^{R}` is the mixed macroscopic cross section for reaction *R* in the mixture, + + N\ :sub:`i` is the number density of nuclide *i*, + + :math:`\sigma_{i}^{R}` is the microscopic cross section for reaction *R* in nuclide *i*. + +The form of the mixing table block is as follows: + +:: + + read mixt + mixid1 nuclideid1 concentration1 + mixid2 nuclideid2 concentration2 + mixid3 nuclideid3 concentration3 + … + mixidN nuclideidN concentrationN + end mixt + +This concludes this list of input blocks available within NEWT. The +following section provides a list of sample inputs used to represent a +variety of configurations and use of codes. These examples are intended +to provide a broader illustration of the use of NEWT in a range of +potential applications. + +.. _9-2-4: + +Examples of Inputs +------------------ + +This section provides annotated sample input listings for three +different model types, showing the use of a number of different options +and approaches in model development for a variety of applications. These +samples use the TRITON T-XSEC sequence to prepare cross sections for +stand-alone NEWT calculations. In general, this is more easily +accomplished as a TRITON T-NEWT calculation in which cross section +processing and a NEWT transport solution are combined into a single +calculation. However, the user is directed to the TRITON user’s manual +(Chapter `T1.4 `__ of +the SCALE manual) for details on T-XSEC and T‑NEWT calculations. The +examples are intended simply to demonstrate the use of the NEWT code. +The T-XSEC data are included to allow a user to observe the mixture +definitions used in the NEWT input in its calculation. These problems +are also available as sample problems in the SCALE distribution. + +.. _9-2-4-1: + +Sample 1 +~~~~~~~~ + +Sample 1 illustrates the use of a series of three consecutive +stand-alone NEWT calculations. Annotated input for this problem is given +in :numref:`fig9-2-44`. The calculation begins with SCALE standard composition +specifications used to prepare a problem-specific weighted cross section +library and mixing table for use by NEWT. In this case the T-XSEC +sequence of the TRITON control module is used. This input is described +in the TRITON chapter and is not described further here. + +The first NEWT case uses no parameter block; thus, all default +parameters are applied. The default is an eigenvalue calculation, with +cross sections read from ft04f001 (xnlib=4) and collapsed cross sections +written to ft30f001 (wtdlib=30). The 238-group cross section library is +collapsed to a 44‑group library using mixture-weighted fluxes. The model +calculates the eigenvalue for a simple 1/4 pin cell. The center of the +pin is placed at the origin, the lower-left corner of the global unit +boundary, inlaid into a 2 by 2 base grid. The grid structure is +illustrated in :numref:`fig9-2-45`. + +The second case performs the same calculation using the collapsed cross +section library created by the first case. Parameter *restart*\ =no is +set to prevent the code from attempting a restart from the existing +library. Because the first case saved 238-group fluxes and the second +case uses 44 energy groups from the collapsed set, a restart is not +possible. + +The third NEWT case is a calculation identical to the second case, +although the input is different. In this case, the flux restart file +from the previous calculation is used as a first guess for fluxes. This +is permitted since both cases used the same cross section library and +therefore have the same energy boundaries. For this case, the “read +geom” data block is omitted, telling NEWT to use the geometry restart +file from the previous case. This allows a rapid restart, since no +geometric data need to be recomputed. Because no other parameters are +changed, this case will converge after one outer iteration to the same +eigenvalue as in the first case. + +.. _fig9-2-44: +.. figure:: figs/NEWT/fig44.svg + :align: center + :width: 1000 + + Sample 1 input listing (annotated). + +.. _fig9-2-45: +.. figure:: figs/NEWT/fig45.png + :align: center + :width: 500 + + Grid structure for 1/4 pin cell of Sample 1. + +.. _9-2-4-2: + +Sample 2 +~~~~~~~~ + +Sample 2 (shown in Figure 9.2.46 and Figure 9.2.47) illustrates the use +of multiple bodies within a single unit. It highlights the use of media +definitions to include and exclude regions when various bodies are used. +Although an array can be used to place bodies, this example illustrates +a method suitable for use in developing a model for a configuration with +an irregular non-array-type structure. This sample problem also +highlights the use of partial-current unstructured-mesh CMFD +acceleration, which reduces the number of outer iterations from 35 to 21 +and the CPU run time by ~25%. + +.. _fig9-2-46: +.. figure:: figs/NEWT/fig46.svg + :align: center + :width: 1000 + + Sample 2 input listing (annotated). + +.. _fig9-2-47: +.. figure:: figs/NEWT/fig47.png + :align: center + :width: 500 + + Mixture placement and grid structure for model described in Sample 2. + +.. _9-2-4-3: + +Sample 3 +^^^^^^^^ + +Sample 3 demonstrates the development of a VVER-440 hexagonal fuel +assembly. Annotated input for this problem is given in Figure 9.2.48. +The output plot for this model is shown in Figure 9.2.49. The key +attributes of this model are as follows: + +1. the use of hexagonal (hexprism) units in a stacked hexagonal array, + +2. the use of null units as placeholders in the array, + +3. a full model within a rhexagonal outer boundary, + +4. the use of white boundary conditions, + +5. the use of the new partial-current-based unstructured CMFD + acceleration for hexagonal-domain configurations, and + +6. new type-3 ADF inputs. + +Using CMFD acceleration, the number of outer iterations needed for +convergence decreased from 21 to 8 with a run-time speedup of ~2.58. + +.. _fig9-2-48: +.. figure:: figs/NEWT/fig48.svg + :align: center + :width: 1000 + :class: long + + Sample 3 input listing (annotated). + +.. _fig9-2-49: +.. figure:: figs/NEWT/fig49.png + :align: center + :width: 600 + + Grid structure and material placement for VVER-440 model. + +.. _9-2-4-4: + +Sample 4 +~~~~~~~~ + +Sample 4 demonstrates the use of NEWT in modeling a larger, more complex +configuration. Annotated input for this problem is given in +:numref:`fig9-2-50`. The calculation begins with the use of SCALE standard +composition specifications to prepare a problem-specific weighted cross +section library and mixing table for use by NEWT. In this case the +T-XSEC sequence of the TRITON control module is used. + +This NEWT case is used to calculate the eigenvalue of an infinite +lattice of fuel assemblies. Symmetry at the assembly center is used to +reduce a 15 by 15 assembly lattice to a smaller one-quarter assembly. +The grid structure is illustrated in :numref:`fig9-2-51`. A similar +illustration showing media placements by color is given in +:numref:`fig9-2-52`. + +This input illustrates several features of NEWT modeling capabilities. +Some important features of this model are as follows. + +- In this sample problem, S-6 quadrature, P-1 scattering (P-2 in the + moderator), spatial convergence criteria of 0.005, and an eigenvalue + convergence criteria of 0.001 are used. These are an order of a + magnitude larger than the values typically used for LWR lattice + calculations. + +- Two sets of UO\ :sub:`2` cross sections are prepared in the T-XSEC + calculation. These cross sections are identical with the exception of + the mixture number. Since NEWT reports fluxes, reaction rates, etc., + by mixture, the placement of a unique mixture at a specific location + in a model allows one to determine, for example, the reaction rates + at that model location. Mixture 7, placed in unit 9 in this model, + occurs in only one pin location in the model. Mixture 1, placed in + all other fuel rod locations, will yield reaction rates close to the + average of those for all fuel in the assembly. If the flux or + reaction rate was needed in each unique fuel location, a unique + mixture would be needed for each location. + +- The use of chords for cutting cylinders allows inclusion of one-half + and one-quarter fuel cells in the quarter-assembly model. Because the + fuel assembly has an odd number of rods in each dimension, use of + symmetry at the assembly midplanes requires the rods to be bisected. + +- In this model, local grid spacing was selected such common grid + spacings occur in all cells. However, this is not a requirement. For + example, a much more refined local grid could have been specified for + unit 9. There is no requirement that grid lines match between + different elements of an array. + +- Unstructured coarse-mesh finite-difference acceleration (cmfd=2 or + cmfd=yes) was employed to accelerate the convergence of the solution. + For this case, 14 outer iterations were required for full spatial + convergence as compared with 30 outer iterations when CMFD is + disabled. The CMFD-accelerated case ran 2.5 times faster than its + unaccelerated counterpart. In this sample problem, xycmfd=2 was used + to define the coarse-mesh grid to have two fine-mesh cells per + coarse-mesh cell. + +- Two-group homogenized cross sections were generated along with + single-assembly (i.e., type 1) ADFs derived from the Collapse block, + ADF block, and the Homogenization block. In addition, a B1 critical + spectrum search is computed after the transport calculation, which is + folded into the transport solution to generated homogenized + constants. + +.. _fig9-2-50: +.. figure:: figs/NEWT/fig50.svg + :align: center + :width: 1000 + :class: long + + Sample 4 input listing (annotated). + +.. _fig9-2-51: +.. figure:: figs/NEWT/fig51.png + :align: center + :width: 500 + + Grid structure for one-quarter assembly of Sample 4. + +.. _fig9-2-52: +.. figure:: figs/NEWT/fig52.png + :align: center + :width: 500 + + Mixture placement for quarter-assembly model of Sample 4. + +.. _9-2-4-5: + +Sample 5 +~~~~~~~~ + +Sample 5 (:numref:`fig9-2-53`) illustrates a calculation for a fuel assembly +with a large water boundary and a vacuum boundary condition. The +calculation begins with the use of SCALE standard composition +specifications to prepare a problem-specific weighted cross section +library and mixing table for use by NEWT. + +In this model, seven UO\ :sub:`2` pins are adjacent to eight MOX pins, +which, in turn, are adjacent to a large reflector region. The outer +boundary of the reflector is vacuum. Reflection on the top and bottom +boundaries makes the problem infinite in the y direction. The grid +structure for this problem is illustrated in :numref:`fig9-2-54`. This problem +illustrates the use of the original CMFD acceleration scheme in NEWT +(cmfd=1 or cmfd=rect). Because of the large degree of scattering within +the reflector region, the problem can be relatively slow to converge. +Without CMFD acceleration, 40 outer iterations are required for spatial +convergence as compared with 12 when CMFD is enabled. A total run-time +speedup of ~1.4 is achieved with the CMFD acceleration scheme. + +In addition to the application of CMFD, Sample 5 also illustrates the +use of NEWT’s reflector ADF capability. Reflector ADFs are computed +along the fuel/reflector interface. + +.. _fig9-2-53: +.. figure:: figs/NEWT/fig53.svg + :align: center + :width: 1000 + :class: long + + Sample 5 input listing (annotated). + +.. _fig9-2-54: +.. figure:: figs/NEWT/fig54.png + :align: center + :width: 500 + + Grid structure for 15-pin row of Sample 5. + +.. _9-2-5: + +Description of Output +^^^^^^^^^^^^^^^^^^^^^ + +This section contains a brief description and explanation of NEWT +output. Portions of the output will not be printed for every problem. +Some output is optional, depending on user input specifications and is +so noted in the description. As with any SCALE module, output begins +with an input echo, module execution records with times and completion +codes, and the program verification information banner page. These +outputs are common to all SCALE modules and are not described here. + +.. _9-2-5-1: + +NEWT banner +~~~~~~~~~~~ + +Following the SCALE program verification information, the first section +unique to NEWT output is the NEWT banner, which appears as shown in +:numref:`fig9-2-55`. The bottom of the banner gives the title of the case as +given in input. The NEWT banner is printed only if the command line +option –p is used to run SCALE. + +.. _fig9-2-55: +.. figure:: figs/NEWT/fig55.png + :align: center + :width: 500 + + NEWT copyright banner page and case title. + +.. _9-2-5-2: + +Input summary +~~~~~~~~~~~~~ + +The next several pages of output provide a summary of input parameters. +As described in :ref:`9-2-5`, default parameters are used when no user +specification is supplied. The input summary lists all parameters and +states used in the calculation, whether user supplied or default. The +following subsections describe the various blocks of output information +provided in the input summary. + +.. _9-2-5-2-1: + +Control options +^^^^^^^^^^^^^^^ + +The Control Options block lists global control parameters that determine +the type of analysis being performed. A sample Control Options page is +shown in :numref:`fig9-2-56`. Parameters are self-explanatory. More +information is available in the description of the keywords in +:ref:`9-2-5-2`. + +.. _fig9-2-56: +.. figure:: figs/NEWT/fig56.svg + :align: center + :width: 600 + + Control Options page. + +.. _9-2-5-2-2: + +Output options +^^^^^^^^^^^^^^ + +The Output Options block (:numref:`fig9-2-57`) lists selections made for +output. Portions of the output listing will be printed only if the +appropriate printing option was selected. + +.. _fig9-2-57: +.. figure:: figs/NEWT/fig57.svg + :align: center + :width: 600 + + Output Options page. + +.. _9-2-5-2-3: + +Input/output unit assignments +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +The Input/Output (I/O) Unit Assignments block (:numref:`fig9-2-58`) simply +lists the unit numbers selected for reading or writing various data +files, as appropriate for the calculation. + +.. _fig9-2-58: +.. figure:: figs/NEWT/fig58.svg + :align: center + :width: 600 + + Input/Output Unit Assignments page. + +.. _9-2-5-2-4: + +Convergence control parameters +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +The Convergence Control block (:numref:`fig9-2-59`) summarizes all parameters +used to control spatial, angular, and eigenvalue convergence for the +iterative phases of the solution process. + +.. _fig9-2-59: +.. figure:: figs/NEWT/fig59.svg + :align: center + :width: 600 + + Convergence Control Parameters page. + +.. _9-2-5-2-5: + +Pin-power edit requests +^^^^^^^^^^^^^^^^^^^^^^^ + +If pin-power edits are requested for one or more arrays, a listing is +provided of the arrays for which this request was made (:numref:`fig9-2-60`). + +.. _fig9-2-60: +.. figure:: figs/NEWT/fig60.svg + :align: center + :width: 600 + + Pin-power edit request summary. + +.. _9-2-5-2-6: + +Geometry specifications +^^^^^^^^^^^^^^^^^^^^^^^ + +The Geometry Specifications block (:numref:`fig9-2-61`) lists parameters +associated with the geometric model specified by the user. The first +section lists the characteristics of the global unit. This is followed +by a listing of the four boundary conditions. Finally, the last section +in this block lists all bodies specified for the model. The appearance +and contents of this section of input depend on the nature of the input +model. + +.. _fig9-2-61: +.. figure:: figs/NEWT/fig61.svg + :align: center + :width: 600 + + Geometry Specifications page. + +.. _9-2-5-2-7: + +Homogenization region specifications +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +The Homogenization Region Specifications block (:numref:`fig9-2-62`) +summarizes all sets of homogenized cross sections requested in user +input. + +.. _fig9-2-62: +.. figure:: figs/NEWT/fig62.svg + :align: center + :width: 600 + + Homogenization Region Specifications page. + +.. _9-2-5-2-8: + +Material specifications +^^^^^^^^^^^^^^^^^^^^^^^ + +The Material Specification block (:numref:`fig9-2-63`) lists the NEWT material +number, counting in the order read in; the SCALE mixture number; and the +P\ :sub:`n` order assigned for that mixture. + +.. _fig9-2-63: +.. figure:: figs/NEWT/fig63.svg + :align: center + :width: 600 + + Material Specifications page. + +.. _9-2-5-2-9: + +Derived parameters +^^^^^^^^^^^^^^^^^^ + +The Derived Parameters block (:numref:`fig9-2-64`) lists values not +specifically input but derived from other sources of input. Some of this +information comes from the cross section library, some from the model +geometry, and some from the S\ :sub:`n` and P\ :sub:`n` values +specified. + +.. _fig9-2-64: +.. figure:: figs/NEWT/fig64.svg + :align: center + :width: 600 + + Derived Parameters page. + +.. _9-2-5-2-10: + +Energy group structure listing +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +The Energy Group Structures block (:numref:`fig9-2-65`) lists the energy and +lethargy boundaries found in the cross section library. If a broad-group +collapse was requested, the boundaries of the broad-group library that +will be produced are also identified. This example shows the structure +of the SCALE 44GROUPNDF5 library and 2-group fast/thermal collapse +structure. The final entry (group 45, broad group 3) indicates the lower +bound of the previous energy group. + +.. _fig9-2-65: +.. figure:: figs/NEWT/fig65.svg + :align: center + :width: 600 + + Energy Group Structure Listing page. + +.. _9-2-5-2-11: + +Quadrature parameters +^^^^^^^^^^^^^^^^^^^^^ + +The Quadrature Parameters block (:numref:`fig9-2-66`) lists the first-quadrant +angles and weights used for the specified order of quadrature. The same +angles and weights are applied in the other three quadrants; however, +the signs of the angles vary with the quadrant. Also listed are the +P\ :sub:`n` moments associated with the maximum P\ :sub:`n` scattering +order requested in all materials. Of course, only a subset of these +moments applies to the lower-order P\ :sub:`n` assignments. + +.. _fig9-2-66: +.. figure:: figs/NEWT/fig66.svg + :align: center + :width: 600 + + Quadrature Parameters page. + +.. _9-2-5-2-12: + +Mixture volumes listing +^^^^^^^^^^^^^^^^^^^^^^^ + +The Mixture Volumes block (:numref:`fig9-2-67`) provides a summary of the +volume and volume fraction of each mixture in the problem, together with +the total volume. This block can be used as a simple check of the input +model by ensuring that the calculated volumes of mixtures used for a +given problem match the expected volumes or volume fractions. + +.. _fig9-2-67: +.. figure:: figs/NEWT/fig67.svg + :align: center + :width: 500 + + Mixture Volumes page. + +.. _9-2-5-2-13: + +Mixing table listing +^^^^^^^^^^^^^^^^^^^^ + +The Mixing Table block summarizes the input mixing table, whether user +supplied or read from a SCALE-generated file. Number densities are in +units of atoms per barn-centimeter. Although optional, the mixing table +is printed by default. This default setting can be disabled by +specifying *prtmxtab*\ =no in the Parameter block. A sample mixing table +is shown in :numref:`fig9-2-68`. + +.. _fig9-2-68: +.. figure:: figs/NEWT/fig68.svg + :align: center + :width: 500 + + Mixing Table Listing page. + +.. _9-2-5-2-14: + +Nuclide cross sections +^^^^^^^^^^^^^^^^^^^^^^ + +The Nuclide Cross Section block is optional and is printed only when +*prtxsec*\ =yes is specified in the Parameter Block. The volume of +output generated is quite extensive, especially when a very fine group +library is used and/or a large number of nuclides are included in the +mixing table. The nuclide data are taken directly from the working +library used for the calculation. A sample showing a partial listing for +a single nuclide is shown in :numref:`fig9-2-69`. + +Following the block header, nuclide data are listed for all nuclides. +For each record, the same format is used. Nuclide data begin with a +listing of nuclide header information. This is followed by a listing of +the 1-D cross sections that are important in NEWT calculations. The +sample below shows a partial listing of the 1-D cross sections. +Following the 1-D cross section listing is the scattering matrix for the +nuclide. This abbreviated listing shows a portion of the +P\ :sub:`0` matrix for this nuclide; however, in a full listing, all +higher-order elements are printed as well. + +As was indicated in the input description, specification of +*prtxsec*\ =1d can be used to obtain header and 1‑D cross section data +only, skipping the printing of scattering matrices. + + +.. _fig9-2-69: +.. figure:: figs/NEWT/fig69.svg + :align: center + :width: 1000 + + Partial listing of Nuclide Cross section data pages. + +.. _9-2-5-2-15: + +Mixture cross sections +^^^^^^^^^^^^^^^^^^^^^^ + +The Mixture Cross Section block provides mixed macroscopic +cross sections for each mixture provided in the input mixing table. The +block is also optional and is printed only when *prtmxsec*\ =yes is +specified in the Parameter Block. Although the volume of output +generated is not as extensive as that of the nuclide listings, the +mixture cross section print can be voluminous, especially when a very +fine group library is used and/or a large number of mixtures are +included in the mixing table. A sample showing a partial listing for a +single mixture is shown in :numref:`fig9-2-70`. + +Following the block header, information is provided for each mixture +using the same format. Mixture data begin with a listing of general +mixture information, including the mixing table for that mixture. This +is followed by a listing of 1-D cross sections important in NEWT +calculations. The sample below shows a partial listing of the 1-D mixed +macroscopic cross sections. Following the 1-D cross section listing is +the scattering matrix for the nuclide for all moments requested for the +mixture. This abbreviated listing shows a portion of the +P\ :sub:`0` matrix for this nuclide; however, in a full listing, all +higher-order elements are printed as well if greater than +P\ :sub:`0` scattering was requested. + +As was indicated in the input description, specification of +*prtmxsec*\ =1d can be used to skip the printing of scattering matrices. + +.. _fig9-2-70: +.. figure:: figs/NEWT/fig70.svg + :align: center + :width: 1000 + + Partial listing of Mixture Cross section data pages. + +.. _9-2-5-3: + +Iteration history +~~~~~~~~~~~~~~~~~ + +The next portion of NEWT output lists the iteration convergence +history for the iterative solution +(:numref:`fig9-2-71`). This information can be used to track and understand +the performance of the outer loop of the iterative solution. The first +column provides the outer iteration count. The second column lists the +system eigenvalue after each outer iteration. The third column lists +the change in the eigenvalue from the last outer iteration; this is +one of the parameters tested for convergence. The fourth column, “Max +Flux Delta,” gives the maximum change in cell flux for all cells and +all energy groups; this is also used as a convergence test. The next +column lists the cell number and energy group corresponding to the +maximum flux change in this iteration. The next two columns list the +same flux information for mixtures with fissionable nuclides. This can +be used to track spatial convergence in fuel when convergence is +slowed by significant scattering outside the fuel. Finally, the last +column provides information on the convergence of inners in each outer +iteration. Inner iterations do not need to converge within early outer +iterations, but final convergence will not be achieved until all +inners are converged. The maximum number of inner iterations per +energy group is set by the *inners=* parameter in the parameter input +block. After convergence is achieved, the table is terminated by +printing the final version of *k*\ :sub:`eff`. + +If the parameter keyword *timed=* is set to *yes*, four additional +columns are introduced that give timing information on the solution +process, listing real (“wall clock”) time, elapsed CPU time since +beginning the iteration process, elapsed CPU time per outer iteration, +and an estimate of the fractional CPU usage during each outer. +:numref:`fig9-2-72` illustrates the form of output produced when *timed=yes* +is input. Additionally, a supplementary edit follows the iteration edit +when *timed=yes*, giving information on average time per transport sweep +(outer iteration) within different components of the solution. This edit +is especially useful when coarse-mesh finite-difference acceleration is +used, to assess the overhead of the CMFD accelerator. + + +.. _fig9-2-71: +.. figure:: figs/NEWT/fig71.svg + :align: center + :width: 600 + + Nominal iteration history output. + + +.. _fig9-2-72: +.. figure:: figs/NEWT/fig72.svg + :align: center + :width: 600 + + Timed iteration history output. + +.. _9-2-5-4: + +Four-factor formula +~~~~~~~~~~~~~~~~~~~ + +Following the iteration history listing, NEWT output provides edit +listing the four traditional components of the four-factor formula. This +is followed by an alternate three-group formulation that separates out +resonance and fast escape probabilities (:numref:`fig9-2-73`). + +.. _fig9-2-73: +.. figure:: figs/NEWT/fig73.svg + :align: center + :width: 500 + + Four-factor formula with alternate three-group formulation. + +.. _9-2-5-5: + +Fine-group balance tables +~~~~~~~~~~~~~~~~~~~~~~~~~ + +Following the iteration history and flux convergence, a fine-group +balance table is provided for each mixture used in the calculation. Fine +group refers to the group structure of the library used for the +calculation. Broad-group data, discussed later, refer to a group +structure collapsed from the original fine-group structure. After tables +for all mixtures are printed, a last table provides a fine-group summary +for the entire problem (i.e., the volume-weighted average for all +mixtures). Balance tables are printed by default but may be disabled by +setting *prtbalnc=no* in the Parameter block. + +:numref:`fig9-2-74` shows a clipped excerpt from the fine-group summary of an +output listing. Similar tables are produced for each mixture in the +problem for all energy groups in the problem. The header lists the NEWT +mixture number; the mixture ID (i.e., the SCALE mixture number); and the +mixture description, if provided in the original input specification. +The header also gives the number of computational cells in which the +mixture was present and the volume of the mixture in the problem. + +For each mixture, two tables are printed. The first table provides a +balance of all sources and loss terms: the fixed source, the fission +source, in-scatter, out-scatter, absorption, leakage, n-2n production, +and the net balance of all terms for each energy group. The final row +lists the mixture total for all groups. The fixed source lists the +user-supplied source for fixed-source problems. This field is disabled +(set to zero) for eigenvalue calculations. The fission source is the +number of neutrons born into each energy group in the mixture. In this +example, the mixture is water, which is not fissile; hence, no fission +source is present. In‑scatter represents the number of neutrons +scattered into each group from all other groups; conversely, out‑scatter +is the loss from each energy group by scattering. Absorption is the +number of neutrons absorbed in reactions that do not emit a neutron +(e.g., n-γ). Leakage is the net loss of neutrons from the mixture to +another mixture or a nonreflective boundary, and n-2n is the effective +n-2n production rate calculated from a weighted sum of all n-\ *x*\ n +reactions. The balance table is the ratio of production to loss in each +energy group. + +The second fine-group balance table, also shown in Figure 9.2.74, lists +other reactions rates of interest. The first two columns after the group +number list in-scatter broken into its upscatter and downscatter +components. The subsequent two columns provide a similar breakdown for +out-scatter from the energy group. Self-scatter is the amount of +within-group scattering occurring within each energy group. The fission +rate is the number of (n-fission) reactions occurring in each group. The +next column provides the transverse leakage (i.e., the product of the +flux and the DB\ :sup:`2` term). This column will provide only nonzero +values when a nonzero buckling height is specified in input. The final +column lists the total (scalar) flux for each energy group. + + +.. _fig9-2-74: +.. figure:: figs/NEWT/fig74.svg + :align: center + :width: 1000 + + Partial mixture fine-group balance table output. + +.. _9-2-5-6: + +Planar fluxes and currents +~~~~~~~~~~~~~~~~~~~~~~~~~~ + +If planar fluxes are requested, an edit is printed to provide fluxes and +currents on each line segment specified, identified by label +(:numref:`fig9-2-75`). Fine-group fluxes are listed for each energy group, +followed by x and y net currents and partial currents (+x, –x, +y, and +–y). Fluxes and currents are printed for each group in the input group +structure. The example below shows only partial listings of each for +simplicity. If a broad-group collapse is requested, the fine-group +output is followed by the set of fluxes and currents for each broad +energy group. + +Note that discontinuity factors make internal use of planar fluxes to +determine the flux and current on each boundary. Hence, planar flux +edits will be present any time an ADF calculation is performed. + +.. _fig9-2-75: +.. figure:: figs/NEWT/fig75.svg + :align: center + :width: 600 + + Example of planar flux and current output (continued below). + +.. image:: figs/NEWT/fig75-2.svg + :align: center + :width: 600 + +.. _9-2-5-7: + +Pin-power edits +~~~~~~~~~~~~~~~ + +The next section of the NEWT output listing is the pin-power edit +(:numref:`fig9-2-76`). This information is printed only if *pinpow=yes* is set +in one or more arrays. Two maps are provided. The first is the power of +each fuel location relative to all other fuel pins in all other arrays +for which *pinpow=yes*. This is useful if multiple fuel assemblies are +present or if more than one array is used to describe a fuel assembly. +The second power map shows the pin edit normalized to the set of pins +within the single array. Both maps are identical in a relative sense; +different normalization factors are applied. If *pinpow=yes* is +specified for only one array, then the two edits will have the same +normalization factor and will be identical. The location of each pin is +identified by the (x,y) coordinate of the center of each element of the +array, in centimeters. Please note that the pin-power option is +available only for square (cuboidal) arrays. + +Following the two maps is a one-line edit identifying the location and +magnitude of the maximum pin power. + +.. _fig9-2-76: +.. figure:: figs/NEWT/fig76.svg + :align: center + :width: 800 + + Example of pin-power edit. + +.. _9-2-5-8: + +Broad-group collapse +~~~~~~~~~~~~~~~~~~~~ + +.. _9-2-5-8-1: + +Broad-group summary data +^^^^^^^^^^^^^^^^^^^^^^^^ + +The next section of the NEWT output listing is the broad-group summary +listing (:numref:`fig9-2-77`). This is printed only if a broad-group collapse +is performed. This section lists broad-group data calculated based on +the collapsing scheme applied. First, the energy group structure is +printed, followed by cell-averaged fluxes in each mixture, for all +collapsed groups. This is followed by flux disadvantage factors for each +mixture and each broad group. + +Note that when NEWT is used as the transport solver within TRITON +depletion calculations, a three-group collapse is always done +automatically. If a second user-specified collapse is requested, +broad-group summary data will be provided for both collapsing +structures. + +.. _fig9-2-77: +.. figure:: figs/NEWT/fig77.svg + :align: center + :width: 800 + + Broad-group summary output. + +.. _9-2-5-8-2: + +Broad-group cross section data +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +The next section of data in a NEWT output listing is the broad-group +cross section output +(:numref:`fig9-2-78`). This is printed only if a broad-group collapse is +performed and if *prtbroad=yes* is specified in Parameter block input. +This block lists the collapsed cross section data for key reactions +for each nuclide used in the calculation. This is a summary form of +the data that are written to the collapsed cross section library. It +does not list all reactions. Such data may be read directly from the +working-format library by other SCALE utilities if needed. The listing +below shows the data printed for a single nuclide. Data are written in +the same format for all nuclides used in the analysis. + +.. _fig9-2-78: +.. figure:: figs/NEWT/fig78.svg + :align: center + :width: 800 + + Partial broad-group cross section listing. + +.. _9-2-5-9: + +Critical spectrum edit +~~~~~~~~~~~~~~~~~~~~~~ + +When a critical buckling correction is requested (e.g., *solntype=b1* is +set in the NEWT parameter block or user-defined material buckling or +transverse height), the critical spectrum is computed using either the +B1 approximation or the P1 approximation (:numref:`fig9-2-79`). The output +lists the buckling in 1/cm\ :sup:`2`, the method (B1 or P1), and the +computed critical spectrum as a function of energy. Note that the +spectrum is normalized “per unit lethargy” to be equal to 1.0. In +addition to the critical spectrum, the critical adjoint spectrum and the +zero-buckling spectra (forward and adjoint) are also edited. + +.. _fig9-2-79: +.. figure:: figs/NEWT/fig79.svg + :align: center + :width: 800 + + Partial collapsing spectra listing for a case with critical buckling correction. + +.. _9-2-5-10: + +Assembly discontinuity factors +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +When calculation of assembly discontinuity factors (ADFs) is requested, +a broad-group edit is provided for each face for which an ADF was +selected (:numref:`fig9-2-80`). Up to four ADFs may be printed for the fuel +region. If the model contains a reflector region in addition to the +required fuel region, then ADFs are printed for a single face, typically +the fuel/moderator interface. Discontinuity factors for user-input +surfaces may also be edited. + + +.. _fig9-2-80: +.. figure:: figs/NEWT/fig80.svg + :align: center + :width: 400 + + Output of assembly discontinuity factors. + +.. _9-2-5-11: + +Groupwise form factors +~~~~~~~~~~~~~~~~~~~~~~ + +Whenever homogenization is performed and pin-power edits are requested, +NEWT will automatically calculate groupwise form factors (GFFs). GFFs +are used in pin-power reconstruction calculations for homogenized +assemblies used in nodal diffusion methods (:numref:`fig9-2-81`). + +.. _fig9-2-81: +.. figure:: figs/NEWT/fig81.svg + :align: center + :width: 800 + + Partial collapsing spectra listing for a case with no critical buckling correction. + +.. _9-2-5-12: + +Homogenized cross sections +^^^^^^^^^^^^^^^^^^^^^^^^^^ + +When homogenization is performed and parameter *prthmmix=yes* is set, +the final output section of a NEWT calculation is the homogenized cross +section edit, as shown in :numref:`fig9-2-82`. This information is generally +passed to nodal analysis codes and hence is presented in a slightly +different format from other cross sections. Output includes a +region-averaged k-infinity value, transport-corrected cross section, and +two interpretations of absorption. The first is the directly collapsed +absorption cross section, while the second (Total-Scatter) is a more +consistent definition of absorption as applied in nodal calculations. +The difference between the two definitions is the effective (n-2n) +cross section. Both cross sections exclude contributions from +:sup:`135`\ Xe and :sup:`149`\ Sm; microscopic cross sections and number +densities for these two nuclides are printed explicitly elsewhere in the +table. Nu*fission is the product of the fission cross section and the +number of neutrons produced per fission, while Kappa*fission is the +product of the fission cross section and the energy release per fission +(J). Inverse velocity is the inverse (1/x) of the group neutron speed. + +The table also lists the two-group isotropic scattering matrix and the +prompt fission fraction distribution. Finally, NEWT lists approximate +six-group decay constants (lambdas) and group fractions (betas) for each +group. + +.. _fig9-2-82: +.. figure:: figs/NEWT/fig82.svg + :align: center + :width: 1000 + + Homogenized cross section edit for nodal diffusion applications. + +.. _9-2-5-13: + +End-of-calculation banner +~~~~~~~~~~~~~~~~~~~~~~~~~ + +NEWT output listings are terminated with an end-of-calculation banner +(shown in :numref:`fig9-2-83`) upon successful completion of a calculation. If +this banner is not present, then the calculation ended abnormally, and +the output listing must be reviewed to determine the cause of the error. +In general, the final lines of an output file describe the error +condition that caused the calculation to stop. + +.. _fig9-2-83: +.. figure:: figs/NEWT/fig83.svg + :align: center + :width: 800 + + End-of-calculation banner listing. + +.. _9-2-5-14: + +Postscript graphics files +^^^^^^^^^^^^^^^^^^^^^^^^^ + +Two user-selectable options within NEWT provide the ability to generate +PostScript-based graphics files for visualization of both input +specifications and output results. By specification of *drawit=yes* in +the NEWT parameter block, NEWT will generate two PostScript-based plot +files: :file:`newtgrid.ps` and :file:`newtmatl.ps`. The former, a grayscale plot of the +line segments generated by NEWT based on the input specification, will +be generated if all body placement input is valid. If input contains +errors such that the code stops before grid generation routines are +completed, no :file:`newtgrid.ps` output is created. + +The :file:`newtmatl.ps` plot illustrates the same grid structure but with +material placement indicated by color. At this time, no user control is +provided for color assignment or plot control. This plot also requires +complete grid generation; additionally, it requires completion of all +media placement routines before the plot will be produced. + +Figures used throughout this manual were generated from newtgrid and +newtmatl PostScript plot files. Files :file:`newtgrid.ps` and :file:`newtmatl.ps` are +automatically copied back from SCALE’s temporary directory to the +original location of the input case, with the names +*casename*.newtgrid.ps and *casename*.newtmatl.ps. + +When *prtflux=yes* is input, NEWT will generate a set of flux plots +showing relative neutron number densities in each energy group. A plot +file will be generated with the name fluxplot\_\ *N*\ g.ps, where *N* is +the number of energy groups in the problem. If an energy collapse is +performed, an additional file named fluxplot\_\ *M*\ g.ps is created, +where M is the number of energy groups in the collapsed set. +:numref:`fig9-2-84` is an example of a flux plot output for the fast group of +a two-group flux collapse. + +.. _fig9-2-84: +.. figure:: figs/NEWT/fig84.png + :align: center + :width: 600 + + Example of a flux plot image created with prtflux=yes. + +.. _9-2-5-15: + +Media zone edits +~~~~~~~~~~~~~~~~ + +NEWT automatically determines “zones” representing spatially independent +regions of the same media. For example, in a fuel pin cell, the fuel, +clad, and moderator are all considered separate zones. In an array of +such pin cells, each unique location is a unique zone. Zone numbers and +the geometric location of each zone are listed in the *Geometry +Specification*\ ” in :ref:`9-2-5-2-6`. + +Upon completion of a calculation, NEWT provides an output edit of each +zone by number, giving the mixture number, average flux, fission power, +and volume, as shown in :numref:`fig9-2-85`. + +.. _fig9-2-85: +.. figure:: figs/NEWT/fig85.svg + :align: center + :width: 500 + + Media zone output edit. + +Notes +~~~~~~~~~~ + +.. [1] + Formerly with Oak Ridge National Laboratory. + +.. bibliography:: bibs/NEWT.bib + + + + + + + + + + + + + + +.. diff --git a/Nuclear Data Libraries Overview.rst b/Nuclear Data Libraries Overview.rst new file mode 100644 index 0000000..632d38a --- /dev/null +++ b/Nuclear Data Libraries Overview.rst @@ -0,0 +1,40 @@ +.. _10-0: + +SCALE Nuclear Data Libraries +============================ + +*Introduction by M. L. Williams and D. Wiarda* + +Chapter 10 describes the SCALE cross section data libraries for use with +deterministic and Monte Carlo radiation transport modules. All cross section +libraries were processed from ENDF/B-VII.0 or -VII.1 evaluated data files using +the AMPX code system. [1]_ SCALE includes continuous-energy libraries, as well as +multigroup libraries with several group structures. Libraries are available for +neutron, gamma, and coupled neutron-gamma transport calculations. The fine and +broad multigroup libraries provided for reactor physics and criticality safety +applications in SCALE 6.2 include intermediate resonance parameters (lambdas) +and improved Bondarenko data for self-shielding calculations using the +Bondarenko method, or the traditional CENTRM-based procedures in SCALE can be +used for self-shielding. Section 10.1 in this chapter describes the available +cross section libraries. + +Fine and broad group covariance libraries containing cross section uncertainties +and correlations are also distributed with SCALE for sensitivity/uncertainty +analysis with the Sampler and TSUNAMI modules. The covariance libraries include +a comprehensive collection of data for all nuclides included in the SCALE cross +section libraries. New 252-group and 56-group covariances based on ENDF/B-VII.1 +and other data sources are available, along with the older 44-group covariance +library distributed with earlier releases of SCALE. The Covariance Libraries +chapter describes the contents of the SCALE 6.2 covariance libraries and +explains how they were processed. + +Additional libraries used for transmutation calculations with ORIGEN are +described in the ORIGEN Data Resources section of the ORIGEN chapter. These +libraries include fission product yields, decay data, decay gamma spectra, etc., +as well as supplemental cross section data not available in ENDF/B. + +Reference +--------- + +.. [1] + D. Wiarda, M. L. Williams, C. Celik, and M. E. Dunn, “AMPX: A Modern Cross Section Processing System for Generating Nuclear Data Libraries,” Proceedings of International Conference on Nuclear Criticality Safety, Charlotte, NC, Sept. 13–17, 2015. diff --git a/ORIGAMI.rst b/ORIGAMI.rst new file mode 100644 index 0000000..50e442f --- /dev/null +++ b/ORIGAMI.rst @@ -0,0 +1,3072 @@ +.. _5-4: + +ORIGAMI: A Code for Computing Assembly Isotopics with ORIGEN +============================================================ + +.. |rarr| replace:: :math:`\rightarrow` + +.. codeauthor:: M. L. Williams, S. E. Skutnik [#utk]_, I. C. Gauld, + W. A. Wieselquist, and R. A. LeFebvre + +.. [#utk] University of Tennessee + +.. include:: +.. program:: origami + +ABSTRACT + +ORIGAMI computes detailed isotopic compositions for light water reactor +assemblies containing UO\ :sub:`2` fuel by using the ORIGEN +transmutation code with pre-generated ORIGEN libraries, for a specified +assembly power distribution. The fuel may be modeled using either lumped +or pinwise representations with the option of including axial zones. In +either case, ORIGAMI performs ORIGEN burnup calculations for each of the +specified power regions to obtain the spatial distribution of isotopes +in the burned fuel. Multiple cycles with varying burn-times and +downtimes may be used. ORIGAMI produces several types of output files, +including one containing stacked ORIGEN binary output data ("ft71 file") +for each depletion zone; files with nuclide concentrations at the last +time-step for each axial depletion region, in the format of SCALE +standard composition input data or as MCNP material cards; a file +containing the axial decay heat at the final time-step; and gamma and +neutron radiation source spectra. + +ACKNOWLEDGMENTS + +ORIGAMI is based on the PinDeplete code developed by Steve Skutnik of +the University of Tennessee, and it also includes techniques taken from +the Orella code written by Ian Gauld. Support for development of ORIGAMI +was provided by the U.S. Department of Energy, Office of Nuclear Energy, +Nuclear Fuels Storage and Transportation Planning Project. + +.. _5-4-1: + +Introduction +------------ + +ORIGAMI (**ORIG**\ EN **A**\ sse\ **m**\ bly **I**\ sotopics) provides +the capability to perform isotopic depletion and decay calculations for +a light water reactor fuel assembly model using one or more ORIGEN +calculations. The assembly may be modeled using either lumped or pinwise +representations with the option of including axial zones. ORIGAMI +automates the performance of ORIGEN depletion calculations for each +region and thus simulates zero-, one-, two-, and three -dimensional (0D, +1D, 2D, and 3D) modeling of a fuel assembly. Multiple cycles with +different specific powers and exposure and decay times may be treated, +and the power distribution is described in terms of fractional pin +powers in the XY plane and axial distributions along the Z axis, which +define the burnup regions for the ORIGEN computations. ORIGAMI allows +for easy and flexible material composition specification through the +standard SCALE mixture processor for composition input, the same as in +TRITON (see XSPROC chapter). While ORIGAMI cannot presently treat +axially non-uniform lattice features (e.g. axially varying enrichment or +the partial-length rods found in many boiling water reactor designs) +within a single input, these problems can still be easily modeled by +splitting the problem across sequential ORIGAMI input cases residing on +the same file. + +The ORIGEN calculations performed by ORIGAMI use the methodology +originally established for the SCALE sequence ORIGEN-ARP (see ARP in +:ref:`5-1`). This approach provides an efficient mechanism to +perform stand-alone reactor depletion calculations using pre-generated +ORIGEN libraries which contain self-shielded, collapsed one-group cross +sections as a function of selected independent variables, such as burnup, +enrichment, and moderator density, for different reactor systems. Typically the +data in these libraries are obtained from 2D, multigroup lattice transport +calculations (e.g., TRITON) coupled with depletion calculations for burnup. +The library cross sections may be flux-weighted over the lattice to obtain data +representative of the entire homogenized assembly for lumped depletion; or +alternatively, it is also possible to generate multiple ORIGEN libraries +corresponding to individual or groups of pins within the lattice for multi-pin +depletion. The burnup-dependent ORIGEN libraries are analogous to the +parameterized cross section data produced by lattice physics codes for reactor +core simulators, except that data for many more nuclides and reactions are +included to allow ORIGEN to compute detailed isotopics for more than 2200 +nuclides. + +ORIGAMI extends the capabilities previously provided by ORIGEN-ARP to +perform a suite of ORIGEN calculations in order to represent the +isotopic distribution of fuel within an assembly in more detail. The +pre-generated ORIGEN libraries provided with SCALE tabulate the +assembly-average one-group cross sections, in order to accurately +reproduce assembly-average isotopics. When performing pin-by-pin +calculations in ORIGAMI, users can increase the fidelity with respect to +proximity to features such as assembly edges, water holes, burnable +poison rods, etc. by creating and employing zone-specific libraries for +different pins. By specifying the individual library assignments for +each pin, users can capture these local spectrum changes in the ORIGAMI +calculation through the use of one-group libraries based on these local +conditions. Currently, the specification of individual libraries is +limited to pin-level specification only (i.e., the same library is used +for all axial zones corresponding to a pin for 3D cases) with an allowed +axial moderator density distribution and radial and axial power +distributions. + +ORIGAMI can produce the following output files in addition to the +standard ORIGEN output for each depletion zone: + + * isotopics in ORIGEN binary concentration (ft71) files + + * in each depletion zone at times specified by the :command:`options` block, + :option:`ft71` key + + * in each axial zone (summed over all pins at a particular axial + level) at the final time; + + * nuclide concentrations by axial zone, written as a SCALE "standard + composition block" that can be used directly as input for SCALE + transport codes such as the KENO Monte Carlo criticality code; + + * axially-dependent decay heat source for input to a thermal analysis + code such as COBRA, so that the temperature distribution within a + storage cask can be computed; + + * nuclide concentrations for each axial zone, given in the format of + MCNP material cards; + + * space-dependent radiation source energy spectra and magnitudes in a + simple text file. + + +ORIGAMI is tightly integrated with the SCALE Graphical User Interface, +Fulcrum. Using Fulcrum and the "UO2 express form (configurable)", one +can create a simple UO\ :sub:`2` assembly depletion case in seconds (see +:numref:`fig-origami-uox-express`). Finally, ORIGAMI has the ability to +perform the depletion/decay calculations for each zone in parallel using the +MPI (Message Passing Interface), however this requires a special SCALE +installation built with MPI in order to do so :cite:`SHLG2013`. + +.. _fig-origami-uox-express: +.. figure:: figs/ORIGAMI/fig1.png + :align: center + :width: 500 + + Fulcrum UO\ :sub:`2` express form for creating ORIGAMI input. + +.. _5-4-2: + +Computational Methods +--------------------- + +.. _5-4-2-1: + +ORIGAMI assembly model +~~~~~~~~~~~~~~~~~~~~~~ + +The basic model for ORIGAMI is a fuel assembly, which may be modeled in +several ways with varying degrees of complexity. The most primitive +model represents the assembly materials as a single mass lump that is +depleted using the value of the specific power input in the +power-history block. In this case, a single ORIGEN calculation is +performed to obtain isotopics representing the entire assembly. This 0D +model is equivalent to the current ORIGEN-ARP procedure. A more detailed +model applies an input axial power profile to the (radially) lumped +assembly materials. This lumped axial depletion model produces a 1D +axially varying burnup distribution, but no allowance is made for +variations in the relative pin powers within the assembly. Thus, if the +axial power distribution is defined by N\ :sub:`Z` axial zones, ORIGEN +calculations are performed for N\ :sub:`Z` different depletion regions. +The 1D axial depletion model has been found to be adequate for most +criticality and decay heat analysis of spent fuel +assemblies :cite:`RGIW2012`. Note that both the 0D and 1D modes are fully +consistent with the 2D TRITON calculations used to generate ORIGEN +reactor libraries distributed with SCALE, in that these modes employ +spatially-homogenized cross-sections to represent assembly-averaged flux +and cross-sections. For 2D and 3D depletion models (wherein individual +pin-specific libraries may optionally be specified), the user is advised +that the ORIGEN reactor data libraries distributed with SCALE are +representative of an assembly axial plane as a whole; in as much, the +user is advised to generate their own zone-specific libraries (i.e., +based on individual material zones) within TRITON if they wish to +capture regional neutronic effects within the assembly (such as +proximity to water holes, burnable absorbers, etc.) + +By specifying a radial pin-power map, a 2D or 3D calculation may be +performed. Currently the axial and radial power shapes are fixed for the +entire calculation but do still result in a fully 3D isotopic +distribution :cite:`SHLG2013,SGRT2012`. If there are N\ :sub:`P` pins in the +assembly and each has N\ :sub:`Z` axial zones, ORIGAMI will perform ORIGEN +calculations for N\ :sub:`P` × N\ :sub:`Z` depletion regions. For +example, an assembly with a 17×17 array with 264 fuel pins and +N\ :sub:`Z` = 24 axial zones requires 6336 independent ORIGEN +calculations. For these types of simulations, the parallel mode with MPI +is highly recommended. + +.. _5-4-2-2: + +Definition of initial composition +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The initial mass in metric tons of heavy metal is M\ :sub:`mtu`, set by the +input parameter :option:`mtu`. The default value of :option:`mtu` is equal to 1.0, so +that by default the ORIGEN calculations are performed on the basis of +"per metric ton of heavy metal". Given that the sum over all zones must +have the total heavy metal content (M\ :sub:`mtu`), one arrives at zone-wise +heavy metal masses of: + +.. math:: + M_{xy,z} = \dfrac{M_{\text{mtu}}} + { \sum \limits_{z = 1}^{N_{z}}{ f_{z} } \sum \limits_{xy = 1}^{N_{P}}{ m_{xy}} } \cdot f_{z} \cdot m_{xy} + :label: eq-origami-mass-norm + +where the relative amount of heavy metal in each radial position, +:math:`m_{xy}`, is calculated from the mixture specification; +fractional axial height, :math:`f_{z}`, from the zone specification; +and :math:`N_P: and :math:`N_Z` are the total number of fuel pins and axial zones, +respectively. Note that some pin locations in the assembly may not +contain fuel, and these are not included in the value of :math:`N_P`. +The fractional axial height is given + + :math:`f_{z} = \frac{\Delta Z }{Z_{\text{tot}}}` is the fraction of + the active fuel height occupied by axial zone Z + :math:`\Delta Z` is the length of axial zone + + Z\ :sub:`tot` is the total length of the active fuel. + + +Whenever an axial zone mesh is input (with array :option:`meshz`), the value of +f\ :sub:`Z` is computed from the values of the zone boundaries (see input +description in :ref:`5-4-3-6`). If an axial mesh array +is not input, the axial zones are assumed to be uniformly distributed. In this +case, the axial zones all have the same height, so that :math:`f=\frac{1}{N_Z}`, +where N\ :sub:`Z` is the number of uniform axial zones in the assembly. + +The uranium mass in a single axial zone for all N\ :sub:`P` fuel pins in +the assembly (M\ :sub:`z`) is thus: + +.. math:: + M_Z = N_P \times M_{XY, Z} = M_{\text{mtu}} \times f_Z + :label: eq-origami-mass-ax-norm + + +In addition to the fuel mixture in an assembly, non-fuel materials +(e.g., structural materials) may also be present. These materials +contribute to the overall power production due to the energy produced by +neutron capture reactions.. For a given value of the total assembly +power, this reduces the power from the fuel mass and thus may slightly +alter the fuel burnup and isotopics. In addition, activation of non-fuel +materials produces additional radiation source terms in the spent fuel, +which contribute to the decay heat and activity. Therefore ORIGAMI +provides an option for including the non-fuel elements in the input +array, :option:`nonfuel`. The units of the non-fuel element masses are kg per +MTU, and the materials are distributed uniformly within all fuel +depletion zones. Note that the input non-fuel materials should not +include oxygen in UO\ :sub:`2` if UO\ :sub:`2` is specified as the fuel +material, as oxygen is already included in proportion to the uranium +mass basis. Finally, because ORIGAMI accesses the StdComp library, any +SCALE StdComp composition, e.g. "zirc4" for reactor cladding material +Zircaloy-4, may be used in either structural or fuel materials. + +.. _5-4-2-3: + +Restart cases +------------- + +ORIGAMI also allows the initial nuclide concentrations to be obtained +from a previously produced ORIGEN binary output file. A restart case is +indicated by setting :command:`restart=yes` in the parameter array. The restart +file has the name :file:`assembly_restart.f71` and must be copied (or linked) +to the SCALE temporary directory used for calculations. The restart file +is normally obtained from an earlier ORIGAMI calculation, which always +produces an ORIGEN restart file named :file:`${OUTBASENAME}.assm.f71`, where +:envvar:`${OUTBASENAME}` is an output prefix defined by the name of the input file +and any user-specified prefix with the :command:`prefix` key. Generally the +restart file from ORIGAMI contains stacked concentrations, corresponding +to each axial zone and then a final entry for the lumped assembly +concentrations; hence, the initial composition for a restart case varies +with axial zone, unlike the case for fresh fuel. ORIGAMI does not +currently allow pin-dependent restart calculations. A restart case may +be useful for performing decay-only calculations of spent fuel +inventory, using the burned fuel composition previously computed for the +assembly exposure during reactor operation. For decay-only cases, a +value for the input parameter :option:`nz` must be input in order to indicate +the number of axial depletion regions in the previous burnup +calculation. + +.. _5-4-2-4: + +Definition of power distribution +-------------------------------- + +The radial power distribution is defined by the XY fractional pin powers in the +input array :option:`pxy`, and the axial fractional powers in the input array :option:`pz`. +The input values in arrays :option:`pxy` and :option:`pz` are normalized to unity by the code. +The fractional power for a fuel pin "XY" is designated here to be +r\ :sub:`XY`, with the normalization :math:`\sum_{XY = 1}^{N_{P}}r_{XY}`. +Similarly the fractional axial power for an axial zone Z is a\ :sub:`Z`, +which is normalized to :math:`\sum_{Z = 1}^{N_{Z}}a_{Z}`. The shapes of +both the radial XY and axial Z distributions must be obtained prior to the +ORIGAMI calculation, either from neutron transport calculations or experimental +measurements. The input distributions remain constant during the ORIGEN burn +calculations for all cycles; but in reality, the power distributions may vary +with time—for example, the initial axial power distribution tends to flatten +after a period of burnup since the higher power zones deplete the fuel faster. +For this reason it is strongly recommended to use the relative burnup +distribution (at final discharge) rather than the relative power density +distribution for the input values. The burnup shape corresponds to the +shape of the time-averaged flux distribution during the exposure period. This +ensures that the final burnup distribution matches the desired shape. + +For a given cycle, the assembly-specific power P\ :sup:`(SP)` is equal +to the value of input variable :option:`power`, read in the power-history block +(see :ref:`5-4-3-3`). The assembly-specific power has units +of megawatts per MTU (MW/MTU). Therefore, the total power produced by the fuel +assembly is: + +.. math:: + P_{\text{tot}} = P_{A}^{\left( \text{SP} \right) } \cdot M_{\text{mtu}} + :label: eq-origami-tot-pow + +where P\ :sub:`tot` is the assembly total power, and +:math:`P_{A}^{\left( \text{SP} \right)}` is the specific power for the +assembly, read from input. + +The absolute power (MW) in fuel pin "XY" is: + +.. math:: + P_{P} = P_{\text{tot}} \cdot r_{xy} = P_{A}^{\left( \text{SP} \right)} + \cdot M_{\text{mtu}} \cdot r_{xy} + :label: eq-origami-pin-pow + + +and the power produced in axial zone Z of this fuel pin XY is: + +.. math:: + P_{XY,Z} = P_{\text{tot}} \cdot r_{xy} \cdot a_{Z} = P_{A}^{\left( \text{SP} + \right)} \cdot M_{\text{mtu}} \cdot r_{xy} \cdot a_{Z} + :label: eq-origami-node-pow + +The absolute power produced in a single axial zone Z for all pins is: + +.. math:: + P_{Z} = \sum_{XY = 1}^{N_{P}}{P_{XY,Z} = P_{\text{tot}} + \times a_{Z} = P_{A}^{\left( \text{SP} \right)} \times M_{\text{mtu}} + \times a_{Z}} + :label: eq-origami-ax-zone-pow + +The ORIGEN depletion calculations are performed with the absolute powers +defined in :eq:`eq-origami-pin-pow` and :eq:`eq-origami-ax-zone-pow` +for each depletion region in the 2D/3D pin-wise or 0D/1D axial depletion models, +respectively. However, cross sections in the ORIGEN libraries are parameterized +as a function of burnup, which depends on the specific power rather than absolute +power for a given depletion region. The specific power (MW/MTU) in axial zone Z +of pin XY is equal to: + +.. math:: + P_{XY,Z}^{(SP)} = \frac{P_{XY,Z}}{M_{XY,Z}} + :label: eq-origami-pin-sp-pow + +Substituting :eq:`eq-origami-mass-norm` and :eq:`eq-origami-node-pow` into +:eq:`eq-origami-pin-sp-pow` gives: + +.. math:: + P_{XY,Z}^{(SP)} = \frac{P_{A}^{(SP)} \cdot r_{\text{xy}} \cdot a_{Z} + \cdot N_{P}}{f_{Z}} + + +In a similar manner, it can be shown that the specific power for all +fuel pins in axial plane Z is: + +.. math:: + P_{Z}^{(SP)} = \frac{P_{A}^{(SP)} \cdot a_{Z}}{f_{Z}} + :label: eq-origami-ax-zone-sp-pow + +ORIGAMI permits two modes for user-specified power distributions along the +axial and radial meshes: *absolute* fractions (i.e., where powers along the +axial mesh points are expressed as fractions of the total assembly power in +MW) and *relative* normalization (i.e., in which *specific powers*\ –- +in MW/MTU-–of axial zones are expressed as a relative modifiers of the +assembly specific powers input in the power history block). Relative power +shape modifiers assume that the specific powers expressed in the power history +block represent the *average* assembly specific power(s) thus, ORIGAMI will +convert these factors into axial & pin power *fractions* – i.e., the factors +r\ :sub:`xy` and a\ :sub:`z` found in :eq:`eq-origami-pin-pow` and +:eq:`eq-origami-ax-zone-pow` used to calculate the absolute pin power +and axial zone power, respectively. The conversion from *relative* specific +power modifiers to *absolute* power fractions is accomplished through the +following normalization procedure :eq:`eq-origami-rel-pow-norm`: + + +.. math:: + \left( a_{Z} \right)_{i} = + \frac{ \left( R_{Z} \right)_{i} \cdot M_{\text{MTU}} \cdot \left( \frac{\Delta Z}{Z_{\text{tot}}} \right)_{i}} + {\sum{\left( R_{Z} \right)_{i} \cdot M_{\text{MTU}} \cdot \left( \frac{\Delta Z}{Z_{\text{tot}}} \right)_{i}}} + = \frac{\left( R_{Z} \right)_{i} \cdot \left( \frac{\Delta Z}{Z_{\text{tot}}} \right)_{i}}{\sum{\left( R_{Z} \right)_{i} + \cdot \left( \frac{\Delta Z}{Z_{\text{tot}}} \right)_{i}}} + :label: eq-origami-rel-pow-norm + + +where :math:`\left( a_{Z} \right)_{i}` is the axial power fraction for axial +zone *i* and :math:`\left( R_{Z} \right)_{i}` is the relative axial zone +specific power modifier for axial zone *i.* Obviously, for a uniformly-spaced +axial mesh, the conversion from relative specific powers (using relative power +modifiers) is precisely the same as that for absolute fractional axial zone +powers; i.e., the relative power modifiers simply become axial power fractions +by virtue of the fact that the term +:math:`\left( \frac{\Delta Z}{Z_{\text{tot}}} \right)_{i}` becomes a constant, +thereby reducing :eq:`eq-origami-rel-pow-norm` back to a direct +calculation of the fractional axial power based on a relative power modifier +following normalization. + +Because it is assumed that the assembly mass is uniformly distributed across +the pins, it can similarly be shown that the use of relative power modifiers +for the XY pin map :math:`\left( r_{XY} \right)_{i}` will always produce the +same result as using pre-normalized absolute fractional powers in the pin map, +i.e. :eq:`eq-origami-pin-pow-norm`: + + .. math:: + \left( r_{XY} \right)_{i} = \frac{\left( R_{XY} \right)_{i} \cdot + \frac{M_{\text{MTU}}}{N_{P}}}{\sum_{}^{}{\left( R_{XY} \right)_{i} + \cdot \frac{M_{\text{MTU}}}{N_{P}}}} = + \frac{\left( R_{XY} \right)_{i}}{\sum_{}^{}\left( R_{XY} \right)_{i}} + :label: eq-origami-pin-pow-norm + + +This option is provided as the :option:`relnorm` option in the parameters block +(discussed further in :ref:`5-4-3-2`). The motivation +for providing an alternative normalization for axial power shape factors is +twofold. First, it is generally assumed that information on the axial power +shape is obtained from axial measurements relative to an assembly-average +value (i.e., axial gamma scans to determine the burnup profile based +upon the gross gamma intensity or isotopic ratios of burnup indicators +such as :sup:`134`\ Cs / :sup:`137`\ Cs, etc.). Therefore, by using the +relative normalization option (i.e., treating axial power shape factors +as *relative* modifiers of the assembly specific power), users can +directly input shape factors obtained from techniques such as +non-destructive analysis (NDA) fuel measurements into ORIGAMI to model +assembly isotopic distributions. + +The second motivation for the relative normalization option comes from +potential problems that can arise if treating axial power shape factors +as absolute fractional powers (:command:`relnorm=no`) in conjunction with +non-uniform axial mesh spacing defined by the user in the :command:`z` array (see +:ref:`5-4-3-7` for details). + +.. Important:: + If using the :command:`relnorm=no` option, the fractional axial powers **must** be + consistent with the axial mesh sizes defined or else **incorrect** + zone-specific powers will result from :eq:`eq-origami-ax-zone-sp-pow`, + therefore leading to incorrect results and likely causing the ARP sequence + to fail (and therefore the ORIGAMI calculation to halt) due to calculated + burnup values for the depletion zone being out of the library range. + + Users are thus **strongly cautioned** when using absolute fractional + axial powers (:command:`relnorm=no`) to ensure proper consistency between the axial + power fractions and the axial mesh sizes. + + For this reason, relative power shape factor normalization is + **turned on** by default (:command:`relnorm=yes`). + +.. _5-4-2-5: + +Computation of neutron and gamma energy spectra +----------------------------------------------- + +ORIGAMI includes an option to generate multi-group neutron and gamma +source spectra due to radioactive decay, for each depletion zone. +Multi-group values are calculated by binning the discrete line and +continuum spectra produced by radioactive decay and nuclear reactions +into arbitrary energy group structures defined by user input. Whenever +neutron energy group boundaries are input in array :option:`ngrp`, neutron +source spectra due to spontaneous fission, delayed neutron emission, and +:math:`\left( \alpha, n\right)` reactions are calculated. + +Similarly, gamma source spectra are computed if gamma energy group bounds are +input in array :option:`ggrp`. The gamma source includes photons produced by all types +of radioactive decays, and also may include bremsstrahlung radiation produced +by beta interactions. Input options can specify the type of nuclides included in +the source term (i.e., light elements, actinides, fission products, or all +nuclides), and the materials used for :math:`\left( \alpha,n \right)` +reactions and bremsstrahlung production. If source spectra are calculated, the +values are always included in the ORIGEN output ft71 binary file; and +optionally the source spectra may also be output in a text file. The source +text file only includes the average over all pins for each axial zone, while +the ft71 file includes sources for all pins and axial zones. + +The source spectra output by ORIGAMI are calculated in ORIGEN using the +expression outlined in :eq:`eq-origami-spectra`: + + +.. math:: + S_{\text{Z.g}}^{(p)} = \sum_{i = 1}^{\text{itot}}{Y_{i,g}^{(p)}\lambda_{i} + \frac{M_{Z}^{(i)}}{A^{(i)}} \cdot N_{A}} + :label: eq-origami-spectra + +where + + :math:`S_{\text{Z.g}}^{(p)}` = source spectrum (p/s) in energy group *g* + for particles of type *p* and axial zone *Z*; + + :math:`Y_{i,g}^{(p)}` = number of particles of type *p* emitted per + decay of nuclide *i*; with energy in group *g*; + + :math:`M_{Z}^{(i)}` = mass (g) of nuclide *i* in axial zone *Z*, + obtained from ORIGEN calculation; + + N\ :sub:`A` = Avogadro’s number (number atoms of nuclide *i* per mole); + + A\ :sup:`(i)` = mass (g) of 1 mole of nuclide *i*; + + :math:`\lambda_i` = decay constant (s\ :sup:`-1`) for nuclide *i*, + + itot = total number of nuclides in burned fuel. + + +More details on the ORIGEN calculation of the source spectra can be +found in the ORIGEN section (:ref:`5-1`) of +the SCALE documentation. + +.. _5-4-3: + +ORIGAMI Input Description +------------------------- + +ORIGAMI uses free-form, keyword-driven input with the SCALE Object +Notation (SON) syntax also used for ORIGEN input, and is described in +more detail there. The general outline of ORIGAMI input is as follows. + + (a) Case Identifier + + (b) Options + + (c) Fuel Composition + + (d) Power-History + + (e) Source-Options + + (f) Output-Print Options + + (g) Input Data + +The above input data may be entered in any order. Data blocks and +parameters which are not needed, or for which default values are +desired, can be omitted. :numref:`ex-origami-input` provides a template +containing all of the ORIGAMI input data blocks and arrays, with example +values assigned. Note that much of the information shown in the template is +optional, and typically is not needed for many cases. The following +subsections provide a more detailed description of the input. + +.. code-block:: scale + :caption: Template for ORIGAMI input data + :name: ex-origami-input + + =origami + % Case identifier information + title= 'input template example' + prefix= example + asmid=1 + % Parameter options + options{ + pitch= 19.718 + mtu= 0.4 + decayheat=yes + fracnf=0.08 + nburn=15 + ndecay=12 + temper=300.0 + stdcomp=yes + restart=no + interp=spline + output=cycle + ft71=all + } + % Array containing ORIGEN library names + libs=[ ce14x14 ce16x16 ] + % Fuel Composition + fuelcomp{ + uox(fuel1){ enrich=3.21 } + uox(fuel2){ enrich=3.50 } + uox(fuel3){ enrich=2.80 } + mix(1){ comps[ fuel1=98.2 Gd2O3=1.8 ] } + mix(2){ comps[ fuel2=100 ] } + mix(3){ comps[ fuel2=97.5 Gd2O3=2.5 ] } + mix(4){ comps[ fuel3=96.9 Gd2O3=3.1 ] } + } + % Map ORIGEN library names to XY pin layout + libmap=[ 1 2 + 2 1 ] + % Map individual compositions XY pin layout + compmap=[ 1 2 + 3 4 ] + % XY relative power distribution (code renormalizes to unity) + pxy=[ 0.2 0.3 + 0.4 0.5 ] + % Z-axial relative power distribution (code renormalizes to unity) + pz=[ 0.6 0.4 ] + % Axial interval boundaries (for MTU mass distribution & plotting) + meshz=[ 0.0 15.0 30.0 ] + % Non-fuel nuclides distributed within fuel material + nonfuel=[ cr=3.366 mn=0.1525 fe=6.309 co=0.0302 + ni=2.366 zr=516.3 sn=8.412 gd=2.860 ] + % Axial variation of moderator density fraction + modz=[ 0.73 0.715 ] + % Irradiation/decay information + hist[ + cycle{ power=35.0 burn=200.0 nlib=7 down=50.0 } + ] + % Optional neutron/gamma source information + ggrp=[ 10.0e6 2.0e6 1.0e6 0.5e6 0.01 ] + ngrp=[ 20.0e6 1.0e6 1.0e5 1.0e4 1.0e3 10.0 0.01 ] + srcopt{ sublib=ac brem_medium=uo2 alphan_medium=case print=yes } + % Output edit options + print{ + nuc{sublibs=[lt ac] total=no units=[grams] } + } + % Nuclides included in comp file (OPTIONAL: overrides default) + nuccomp=[ + 92232 92233 92234 92235 92236 92237 92238 92239 92240 + 92241 93235 93236 93237 93238 93239 94236 94237 94238 + 94239 94240 94241 94242 94243 94244 94246 95241 95242 + 95243 95244 95246 96241 96242 96243 96244 96245 96246 + 96247 96248 96249 96250 97249 97250 98249 98250 98251 + 98252 98253 98254 99253 99254 99255 + ] + end + +.. _5-4-3-1: + +Case and identifier information +------------------------------- + +ORIGAMI has three optional identifiers for the case. The :option:`title` is +included as a descriptor in the printed output file. The character string +:option:`prefix` is added to the front of the output file names described in +ref:`5-4-4` and in :numref:`tab-origami-io-files`. Finally, +the integer variable :option:`asmid` is an arbitrary assembly identifier used +in defining mixture numbers in the SCALE standard composition output file. +:eq:`eq-origami-mix-num` in :ref:`5-4-4-1` describes how +the mixture ID is determined. + +.. option:: title= + + Title (up to 50 characters) describing the case. Enclosed in quotes if + using embedded blanks. + + (**Default:** none) + +.. option:: prefix= + + Prefix (up to 16 characters) to append to output file names. + + (**Default:** none) + +.. option:: asmid= + + Integer used to identify mixture ID in generated SCALE standard composition + block [see :eq:`eq-origami-mix-num`]. + + (**Default:** 1) + + +.. table:: Keywords for case identifier + :name: tab-origami-id-kw + :widths: 13 65 12 + + +-------------+----------------------------------------+-------------+ + | **Keyword** | **Description** | **Default** | + +=============+========================================+=============+ + | title= | up to 50 characters describing the | blank | + | | case title, quoted if embedded blanks | | + +-------------+----------------------------------------+-------------+ + | prefix= | up to 16 characters (no embedded | blank | + | | blanks) appended to output file names | | + +-------------+----------------------------------------+-------------+ + | asmid= | integer used to identify mixture ID in | 1 | + | | generated SCALE standard composition | | + | | block [see :eq:`eq-origami-mix-num`] | | + +-------------+----------------------------------------+-------------+ + +.. _5-4-3-2: + +Options block +~~~~~~~~~~~~~ + +The ``options`` block has the following form: + +.. option:: options {… keyword blocks …} + + The :option:`options` block allows the user to control problem features such + as the total mass basis (:option:`mtu`), non-fuel mass (:option:`fracnf`), + axial power normalization (:option:`relnorm`), exercise fine-grained control + over depletion calculations (:option:`solver`, :option:`interp`, + option:`nburn`, :option:`ndecay`), perform restart calculations from a prior + ORIGAMI run (:option:`restart`), specify the number of axial zones + (:option:`nz`), specify optional parameters used for visualization and + post-processing (:option:`pitch`, :option:`temper`, :option:`fdens`), + and control which outputs to generate (:option:`small`, :option:`mcnp`, + :option:`stdcomp`, :option:`decayheat`). + +.. :numref:`tab-origami-options-kw` shows the keywords for allowable parameters. + +Each of the allowable parameter keywords is explained below. An example parameter block would be: :: + + options{ stdcomp=yes decayheat=yes } + + +.. option:: mtu= + + Metric tons of heavy metal in the assembly. + + (**Default:** `1.0`) + + +.. option:: fracnf= + + Total non-fuel mass in the assembly, given as a fraction of the heavy metal + mass defined in :option:`mtu`. + + .. seealso::`nonfuel` + + (**Default:** `none`) + + +.. option:: nz= + + Number of axial intervals. If not input, :option:`nz` is equal to the number of + entries in the input axial power array :option:`pz`. + + **Required** for decay-only restarts. + + (**Default:** Determined by code via :option:`pz`) + + +.. option:: nburn= + + Number of substeps used in ORIGEN burn calculations + + (**Default:** 10) + + +.. option:: ndecay= + + Number of substeps used in ORIGEN decay calculations + + (**Default:** 10) + +.. option:: pitch= + + Assembly pitch (cm), if > 0.0. Only used to define XY mesh in viewing + results. If this parameter is input, array :option:`pxy` must also be + entered. + + (**Default:** 0.0) + + +.. option:: temper= + + Temperature (in degrees Kelvin) for mixtures output to the SCALE Standard + Composition output (:file:`compBlock`). + + (**Default:** 293.0) + + +.. option:: fdens= + + Fuel density in g/cm\ :sup:`3`. + + (**Default:** 10.4) + + +.. option:: offsetz= + + Axial numbering offset; used for sequential ORIGAMI cases to uniquely + identify axial zones (i.e., such as when using sequential cases to modify + changing axial geometry). + + (**Default:** 0) + +.. option:: relnorm= + + Normalization of axial power shaping factors (:option:`pz`) to be used + + **no** |rarr| axial power shape factors treated as absolute fractions (does + not normalize all axial burnups to 1.0) + + **yes** |rarr| axial power shape factors treated as relative modifiers of + assembly specific power (i.e., power= entries in the power history block) + + (**Default:** `yes`) + + +.. option:: mcnp= + + Generate MCNP input stubs containing data on material concentrations and/or + gamma and neutron emissions for each depletion node in the problem. + + (**Default:** `yes`) + + +.. option:: stdcomp= + + Generate a text-based standard composition file containing burnup-credit + nuclide number densities for each axial zone. + + (**Default:** `no`) + + +.. option:: decayheat= + + Produce a decay heat file containing decay powers (in W) for each axial + zone. + + (**Default:** `no`) + + +.. option:: restart= + + Perform a restart calculation using initial compositions from a + previously-generated ORIGEN ft71 file. + + (**Default:** `no`) + + +.. option:: solver= + + Use the standard ("MATREX") solver or the Chebyshev Rational Approximation + Method (CRAM) solver. + + (**Default:** `matrex`) + + +.. option:: small= + + keep .out file small by suppressing all spectra and concentrations output + except for lumped, assembly-averaged concentrations and spectra + + .. note:: + + Full results are still written to other relevant files + + (**Default:** `no`) + + +.. option:: interp= + + Method for interpolating cross sections in ARP; Lagrangian polynomial (`lagrange` or monotonic cubic spline `spline`) + + (**Default:** `lagrange`) + + +.. .. option:: output= + + Time steps for printed output + (**Default:** `last`) + + +.. .. option:: ft71= + + Time steps included in output ft71 file + (**Default:** `last`) + +.. option:: ft71=, output= + + Controls output of saved / printed output concentrations. + + ``last`` saves / prints results only for the substeps in last step of the + last cycle (**default**) + + ``cycle`` saves results for substeps in the last irradiation and decay + steps in every cycle + + ``all`` saves results for all substeps of all irradiation and decay steps + in every cycle + + (**Default:** ``last``) + + +.. table:: Keywords in ORIGAMI options + :name: tab-origami-options-kw + :widths: 13 72 15 + + +-------------+---------------------------------+--------------------+ + | **Keyword** | **Description** | **Default** | + +=============+=================================+====================+ + | mtu= | Metric tons of heavy metal in | 1.0 | + | | the assembly | | + +-------------+---------------------------------+--------------------+ + | fracnf= | Total non-fuel mass in | none | + | | assembly, given as fraction of | | + | | heavy metal mass defined by | | + | | input *mtu=* . See description | | + | | of input array | | + | | :option:`nonfuel` | | + +-------------+---------------------------------+--------------------+ + | nz= | Number of axial intervals. If | Determined by code | + | | not input, *nz* is equal to the | | + | | number of entries in the input | | + | | axial power array :option:`pz`. | | + | | Required for decay-only | | + | | restarts. | | + +-------------+---------------------------------+--------------------+ + | nburn= | Number of substeps used in | 10 | + | | ORIGEN burn calculations | | + +-------------+---------------------------------+--------------------+ + | ndecay= | Number of substeps used in | 10 | + | | ORIGEN decay calculations | | + +-------------+---------------------------------+--------------------+ + | pitch= | Assembly pitch (cm), if > 0.0. | 0.0 | + | | Only used to define XY mesh in | | + | | viewing results. If this | | + | | parameter is input, array | | + | | :option:`pxy` must also be | | + | | entered. | | + +-------------+---------------------------------+--------------------+ + | temper= | Temperature (Kelvin) assigned | 293.0 | + | | to materials in standard | | + | | composition file | | + +-------------+---------------------------------+--------------------+ + | offsetz= | Axial numbering offset; used | 0 | + | | for sequential ORIGAMI cases to | | + | | uniquely identify axial zones | | + | | (i.e., such as when using | | + | | sequential cases to modify | | + | | changing axial geometry). | | + | | [integer] | | + +-------------+---------------------------------+--------------------+ + | relnorm= | Normalization of axial power | Yes | + | | shaping factors (:option:`pz`) | | + | | to be used | | + | | | | + | | **no:** axial power shape | | + | | factors treated as absolute | | + | | fractions (does not normalize | | + | | all axial burnups to 1.00) | | + | | | | + | | **yes:** axial power shape | | + | | factors treated as **relative** | | + | | modifiers of assembly specific | | + | | power (i.e., *power=* entries | | + | | in the power history block) | | + | | [yes/no] | | + +-------------+---------------------------------+--------------------+ + | mcnp= | no/yes |rarr| do not / do | Yes | + | | generate an MCNP material and | | + | | gamma/neutron file | | + +-------------+---------------------------------+--------------------+ + | stdcomp= | no/yes |rarr| do not / do | No | + | | generate a standard composition | | + | | file containing burnup-credit | | + | | nuclide number densities for | | + | | each axial zone. | | + +-------------+---------------------------------+--------------------+ + | decayheat= | no/yes |rarr| do not / do | No | + | | produce a decay heat file | | + | | containing decay powers (in W) | | + | | for each axial zone. | | + +-------------+---------------------------------+--------------------+ + | restart= | no/yes |rarr| do not / do | No | + | | restart using initial | | + | | compositions from a | | + | | previously-generated ORIGEN | | + | | ft71 file. | | + +-------------+---------------------------------+--------------------+ + | solver= | matrex/cram |rarr| use the | Matrex | + | | standard ("MATREX") solver or | | + | | the Chebyshev Rational | | + | | Approximation Method (CRAM) | | + | | solver. | | + +-------------+---------------------------------+--------------------+ + | small= | no/yes |rarr| keep .out file | No | + | | small by suppressing all | | + | | spectra and concentrations | | + | | output except for lumped, | | + | | assembly-averaged | | + | | concentrations and spectra | | + | | (**Note:** all results are | | + | | still written to other relevant | | + | | files). | | + +-------------+---------------------------------+--------------------+ + | interp= | lagrange/spline |rarr| method | Lagrange | + | | for interpolating cross | | + | | sections in ARP | | + +-------------+---------------------------------+--------------------+ + | output= | last/cycle/all |rarr| time | Last | + | | steps for output print edits | | + +-------------+---------------------------------+--------------------+ + | ft71= | last/cycle/all |rarr| time | Last | + | | steps included in output ft71 | | + | | file | | + +-------------+---------------------------------+--------------------+ + + +Additional notes on input parameters: + + (a) :option:`pitch` is only used for visualization of the results, and may be + omitted if this is not of interest; + + (b) :option:`mtu` is discussed in :ref:`5-4-2-2` + + (c) :option:`nz` is not required except decay-only restart cases; it must equal + the number of entries in the array :option:`pz`; + + (d) :option:`nburn` and :option:`ndecay` are discussed in :ref:`5-4-3-3`; + + (e) :option:`fracnf` is discussed in :ref:`5-4-3-7`, where the + input array of non-fuel materials is described; + + (f) :option:`relnorm` is discussed in :ref:`5-4-2-4`, in the + definition of the assembly power distribution; + + (g) :option:`stdcomp`, :option:`fdens`, and :option:`temper` are discussed + in :ref:`5-4-4`; + + (h) :option:`offsetz` is an optional feature designed to allow for ORIGAMI + cases to be split across multiple inputs to capture axially-dependent + features (such as partial-length rods); its use is discussed in further + detail in the context of output generation in + :ref:`5-4-4`; + + (i) :option:`decayheat` is discussed in :ref:`5-4-4-3`; + + (j) :option:`restart` is discussed in :ref:`5-4-2-3`. + + (k) :option:`output`, :option:`ft71`, are discussed in + :ref:`5-4-3-6`. + +.. _5-4-3-3: + +Fuel composition block +~~~~~~~~~~~~~~~~~~~~~~ + +The purpose of the :option:`fuelcomp` block is to create a set of mixtures (via +the :command:`mix` blocks inside) to specify the pin-wise distribution of initial +isotopics. The example below, defines three mixtures (with IDs 1, 2, and +3); these are referenced in the :option:`compmap` array for this 2x2 array of +fuel pins. + +.. option:: fuelcomp= { mixture blocks } + + Specifies fuel mixtures to be used by ORIGAMI in the :option:`compmap` + array. *Numbered* :option:`mix` blocks are used by :option:`compmap`, + which can be composed of other named mixtures. + + +.. option:: mix= { SCALE standard composition } + + Mixture blocks identify specific pin-wise composition to be used by ORIGAMI, + using the standard SCALE mixture composition syntax. Mixtures must be given + an integer identifier (e.g., ``mix(1)``, ``mix(2)``, etc.) + +.. option:: compmap= [ mixture IDs ] + + Specifies the distribution of fuel compositions / mixtures for each pin for + 2-D and 3-D depletion cases. Mixture ID numbers correspond to those in the + :option:`fuelcomp` block. + + **Required** if :option:`libmap` is explicitly specified beyond one element. + + (**Default:** ``[1]``) + +.. code-block:: scale + :name: ex-origami-fuel-comps + :caption: Example specification of uranium oxide-based fuel mixtures in + ORIGAMI, including 1) Mixed urania-gadolina fuel, 2) 4\% enriched + UO\ :sub:`2` fuel, and 3) 2\% enriched UO\ :sub:2 fuel. + + fuelcomp{ + uox(fuel_3pct){ enrich=3.20 dens=10.42 } + uox(fuel_4pct){ enrich=4.00 dens=10.45 } + uox(fuel_2pct){ enrich=2.10 dens=10.43 } + mix(1){ comps[ fuel_3pct=99.0 Gd2O3=1.0 ] } + mix(2){ comps[ fuel_4pct=100] } + mix(3){ comps[ fuel_2pct=100] } + } + compmap=[ 1 2 + 2 3 ] + + + +The *mix* block defines an array of compositions by their weight %. +For example, in the case of mix 2 and 3, it is 100% the "fuel_4pct" and +"fuel_2pct" compositions defined on the *uox* blocks above. In the case +of mix 1, it is 99% by weight `fuel_3pct` and 1% by weight the SCALE +StdComp `Gd2O3` (gadolinia). Each mixture number (defined by numbered +*mix* objects) is then referenced in the *compmap* array to define an +individual pin composition. For UO\ :sub:`x`-based fuels, ORIGAMI +automatically calculates the pin enrichment for cross-section library +interpolation via ARP. (Interpolation for MOX-based fuels is not +supported by ORIGAMI at this time.) + +The *uox* keyword is an ORIGAM-specific shortcut to allow for easy +specification of UO\ :sub:`2`-based fuels along with their enrichment; +ORIGAMI automatically expands the *uox* keyword into a SCALE StdComp +block with a UO\ :sub:`2` base and explicitly-calculated uranium +isotopics per :numref:`tab-origami-uox-formula`. For example, the *uox* block +"fuel_3pct"expands to the following (:numref:`ex-origami-stdcmp-uox`): + +.. code-block:: scale + :name: ex-origami-stdcmp-uox + :caption: Equivalent explicit expansion of the "fuel_3pct" block + + stdcomp(fuel_3pct){ + base=uo2 + iso[92234=0.02848 92235=3.2 92236=0.01472 92238=96.7568] + } + + +For *uox*-based entries, the uranium isotopic distribution is calculated +from the user-specified enrichment per the formula outlined in +:numref:`tab-origami-uox-formula` :cite:`HPR1994,RGI2014`: + +.. table:: Uraniumm isotope dependent on X wt% :sup:`235`\ U + :name: tab-origami-uox-formula + :align: center + + ============= =============== + **Isotope** **Isotope wt%** + ============= =============== + :sup:`234`\ U 0.0089 X + :sup:`235`\ U 1.0000 X + :sup:`236`\ U 0.0046 X + :sup:`238`\ U 100 – 1.0135 X + ============= =============== + + +Users may also specify materials directly using SCALE mixture processor +conventions; for example, the user could simply enter fuel mixture 2 +directly as a StdComp as shown in :numref:`ex-origami-mix-direct` and +:numref:`ex-origami-mix-indirect`: + +.. code-block:: none + :caption: Direct specification of materials in ORIGAMI (i.e., within the mixture block) + :name: ex-origami-mix-direct + + mix(2){ + stdcomp(fuel_4pct){ + base=uo2 + iso[92234=XXX 92235=XXX 92236=XXX 92238=XXX] + } + } + +Or similarly, one can refer to a composition by its alias: + +.. code-block:: none + :caption: Indirect specification of fuel material mixtures (outside the mixture block) + :name: ex-origami-mix-indirect + + stdcomp(fuel_4pct){ + base=uo2 + iso[92234=XXX 92235=XXX 92236=XXX 92238=XXX] + } + mix(2){ comps[ fuel_4pct=100.0 ] } + + +The *uox* keyword is thus useful when a user wishes to quickly specify a +UO\ :sub:`2`-based fuel; however, in cases where the user wishes to +specify the isotopic fractions of each uranium isotope, the use of a +StdComp object is recommended. + + +.. Caution:: + The mixture composition system in ORIGAMI is very flexible but the user is + cautioned that ORIGAMI does not rigorously check that the specified + composition is neutronically similar to that used to generate the ORIGEN + library used in the calculation. + + For example, use of gadolinia burnable absorbers in the ORIGAMI input will + yield incorrect results if the ORIGEN library was generated without + gadolinia, due to the extreme thermal flux depression that gadolinia + creates. It is therefore **up to the user** to verify that the libraries + specified for the depletion zone are matched neutronically to the + compositions specified. + +.. _5-4-3-4: + +Power history block +------------------- + +The data contained in the power history block is the same as in the +*BURNDATA* block of the TRITON lattice physics depletion sequence in +SCALE (see TRITON chapter, section *BURNDATA* block). The power-history +block describes the burnup and decay of the assembly and has the +following general form: + +.. code-block:: scale + :caption: Origami power history block + :name: ex-origami-history-kw + + hist[ + cycle{ keywords for cycle-1 } + cycle{ keywords for cycle-2 } + … *(repeat for total number of cycles) …* + ] + + +Because the cycles must be processed in order, the array syntax with +"[]" is used for the "hist" block. (The block syntax "{}" implies no +order for its contents.) The "hist" array consists of one or more +"cycle" blocks, each describing the assembly irradiation and/or decay +for some period of time. Each cycle is defined by (a) the assembly total +specific power; (b) number of exposure days at this power; (c) the +number of ORIGEN library burnup interpolations during the exposure +period; and (d) number of days of decay following the exposure period. + +The keywords defining this information are: + +.. given in :numref:`tab-origami-hist-kw`. + + +.. option:: power= + + Assembly specific power (MW/MTU) for the cycle + + (**Default:** 0.0) + + +.. option:: burn= + + Length of cycle exposure period in days + + (**Default:** 0.0) + + +.. option:: down= + + Downtime (decay) in days following exposure + + (**Default:** 0.0) + +.. option:: nlib= + + Number of ORIGEN library burnup-interpolations during the cycle + + (**Default:** 1) + + +.. .. table:: Keywords in the power history (hist) block hist-repeat_ + :name: tab-origami-hist-kw + :widths: 13 65 10 + :align: center + + +-------------+----------------------------------------+-------------+ + | **Keyword** | **Description** | **Default** | + +=============+========================================+=============+ + | power= | assembly specific power (MW/MTU) for | 0.0 | + | | the cycle | | + +-------------+----------------------------------------+-------------+ + | burn= | length of the cycle exposure period in | 0.0 | + | | days | | + +-------------+----------------------------------------+-------------+ + | nlib= | number of ORIGEN library | 1 | + | | burnup-interpolations during the cycle | | + +-------------+----------------------------------------+-------------+ + | down= | downtime in days following the | 0.0 | + | | exposure | | + +-------------+----------------------------------------+-------------+ +.. [#hist-repeat] Keywords are repeated for each cycle. + + +:numref:`ex-origami-history` demonstrates the use of the power-history block for four cycles: + +.. code-block:: scale + :name: ex-origami-history + :caption: Example of the ORIGAMI "hist" block for irradiation cycle history + + hist[ + cycle{ power=35.6 burn=400 nlib=6 down=30 } + cycle{ power=38.2 burn=350 nlib=6 down=30 } + cycle{ power=30.0 burn=200 nlib=4 down=30 } + cycle{ down=10000 } + ] + + +ORIGAMI discretizes time intervals first by *cycles* (composed of a +fixed power over a set burn time interval and / or decay time), where +each *cycle* is composed of a number of *substeps*. The power-history +block, along with values of :option:`nburn` and :option:`ndecay` from the input +parameter block, define various types of nested time intervals +(substeps) for the ORIGEN calculations. The entire time period for an +ORIGAMI case is first of all divided into the cycles defined within the +power-history block. Each cycle is divided into an exposure interval +(:option:`burn`) and a decay (:option:`down`) interval. The exposure interval has a +constant specific power, but it is further subdivided into a number of +equally spaced burnup steps defined by :option:`nlib` in the power-history +block. This parameter specifies the number of burnup-dependent ORIGEN +libraries to use during the exposure interval. Cross section values for +each burnup step are interpolated using the burnup at the midpoint of +the step and remain constant throughout the burnup step. The burnup +period associated with a single ORIGEN library, or a decay period, is +called a time "step." Finally, each burnup step, as well as the entire +decay step, is divided into a number of computational "substeps"—the +actual time steps used in the ORIGEN solver kernel. The number of +substeps in each burnup step is given by the value of :option:`nburn`, while +the number of decay substeps is equal to the value :option:`ndecay`. The +default number of substeps for both burnup and decay is equal to 10. The +substeps for irradiation are equally spaced but for decay follow the +rule of threes, i.e. each substep increases in duration by a factor of +three over the previous substep. + +For the example given above, there are four cycles. The first three +cycles include both exposure and decay intervals, while the last cycle +is decay only. In the first cycle, the assembly-specific power is +35.6 MW/MTU, which remains constant over the 400-day exposure interval; +therefore, the total burnup for the exposure period is 400*35.6 = 14240 +MWD/MTU. This exposure period is divided into six burnup steps of 66.67 +days, each with a cross-section library based on the midpoint burnup of +that step. Thus, ORIGEN libraries are interpolated at 1186.7, 3560.0, +5933.3, 8306.7, 10680.0, and 13053.3 MWD/MTU. Each of the six burnup +steps is further subdivided into 10 computational substeps. Likewise, +the decay interval of 30 days is divided into 10 computational substeps. + +.. _5-4-3-5: + +Source options block +~~~~~~~~~~~~~~~~~~~~ + +This block defines options used in computing neutron and gamma sources. +The block is only used if the input energy group boundary arrays :option:`ggrp` +or :option:`ngrp` is given, which indicates that radiation decay source spectra +are to be computed. The general form of this block is: + +.. option:: srcopt { … keyword-value pairs … } + + Where the following blocks are permitted: + + * :option:`sublibs` + * :option:`brem_medium` + * :option:`alphan_medium` + * :option:`print` + + +The following (:numref:`ex-origami-srcopt`) is an example of the +:option:`srcopt` input block: + +.. code-block:: scale + :name: ex-origami-srcopt + :caption: Template of the ORIGAMI "srcopt" block options + + srcopt{ + sublib= … + brem_medium= … + alphan_medium= … + print= … + + } + +If `print=yes`, then text files with axial neutron and gamma sources are +created. + +.. option:: sublib= [ lt / fp / ac / all ] + + Gamma sources from light elements / fission products / actinides / all + nuclides. + + (**Default:** all) + + +.. option:: brem_medium= [ H2O / UO2 / none ] + + Medium for Bremsstrahlung production based on water (H2O), uranium oxide + (UO2), or no Bremsstrahlung calculation (none) + + (**Default:** UO2) + + +.. option:: alphan_medium= [ UO2 / borosilicate / case ] + + Target medium used for :math:`\left(\alpha,n\right)` source caclulation; + UO\ :sub:`2`, borosilicate glass, or case-specific mixture. + + (**Default:** case) + + +.. option:: print= [ yes / no ] + + Write text-based output file containing source information / only write + radiation source terms to binary ft71 file. + + (**Default:** no) + +.. .. table:: Keywords in the ORIGAMI source options (srcopt) block + :name: tab-origami-srcopt-kw + :widths: 20 60 10 + :align: center + + +----------------+-------------------------------------+-------------+ + | **Keyword** | **Description** | **Default** | + +================+=====================================+=============+ + | sublib= | *lt / fp / ac / all* |rarr| gamma | all | + | | sources from: | | + | | | | + | | light elements / fission products / | | + | | actinides / all nuclides | | + +----------------+-------------------------------------+-------------+ + | brem_medium= | *none* / *H2O* / *UO2* / |rarr| | uo2 | + | | bremsstrahlung production based on: | | + | | | | + | | no bremsstrahlung / water / | | + | | UO\ :sub:`2` | | + +----------------+-------------------------------------+-------------+ + | alphan_medium= | *UO2* / *borosilicate*/ *case* | case | + | | |rarr| (alpha,n) source computed | | + | | for: | | + | | | | + | | UO­\ :sub:`2`/ borosilicate glass / | | + | | case-specific mixture | | + +----------------+-------------------------------------+-------------+ + | print= | *yes* / *no* |rarr| write output | no | + | | text file containing sources / only | | + | | write sources in binary output ft71 | | + | | file | | + +----------------+-------------------------------------+-------------+ + +.. _5-4-3-6: + +Output print-options block +~~~~~~~~~~~~~~~~~~~~~~~~~~ + +This block defines the desired ORIGEN output response edits to be +printed by ORIGAMI. + +The following is an example input which edits response values for the +mass in grams, activities in Curies, and concentrations in +atoms/barn-cm, for all nuclides (isotopes) broken down by actinides or +fission products as well as curies by element, totaled over all nuclide +sub-libraries (sublibs). + +.. code-block:: scale + :caption: Example of Origami's "print" block for specifying output print + options + + print{ + nuc{ units=[grams curies atoms-per-barn-cm] sublibs=[fp ac] } + ele{ units=[curies] total=yes } + } + + +.. option:: nuc= { }, ele={ } + + Block to specify print options for output by individual nuclides / elements + + +.. option:: + units= [ moles / gram-atoms / grams / curies / becquerels / watts + / g-watts / m3_air / m3_water / weight_ppm / atoms_ppm / atoms-per-barn-cm ] + + Output concentrations in units of gram-atoms (moles), grams, curies, + becquerels, total thermal power (alpha, beta, and gamma), thermal + power from gammas only, radiotoxicity / dilution factors for air and water, + mass fraction (in ppm), atom fraction (in ppm), atoms / barn-cm [#bncm]_\ , + respectively. + + One or more output units may be specified, separated by commas. + + (**Default:** gram-atoms) + +.. [#bncm] Requires volume input + + +.. option:: sublibs= [ le / fp / ac / all ] + + Output concentration units for light element sublibrary, fission product + sublibrary, actinide sublibrary, or all nuclides. + + (**Default:** all) + +.. option:: total= [ no / yes ] + + Print out total concentration for nuclides / elements for each selected unit + type. + + (**Default:** yes) + +.. table:: Keywords in ORIGAMI "print" block + :name: tab-origami-print-kw + :widths: 13 77 10 + :align: center + + +-------------+------------------------------------------+-------------+ + | **Keyword** | **Description** | **Default** | + +=============+==========================================+=============+ + | nuc / ele | Specify print options for output by | N/A | + | | individual nuclides / elements | | + +-------------+------------------------------------------+-------------+ + | units= | *moles / gram-atoms / grams / curies /* | all | + | | *becquerels / watts / g-watts / m3_air* | | + | | */ m3_water / weight_ppm / atoms_ppm /* | | + | | *atoms-per-barn-cm* | | + | | | | + | | Output concentrations in units of | | + | | gram-atoms (moles), grams, curies, | | + | | becquerels, total thermal power | | + | | (alpha, beta, and gamma), thermal | | + | | power from gammas only, radiotoxicity | | + | | / dilution factors for air and water, | | + | | mass fraction (in ppm), atom fraction | | + | | (in ppm), atoms / barn-cm, | | + | | respectively. | | + +-------------+------------------------------------------+-------------+ + | sublibs= | *le / fp / ac / all* |rarr| output | all | + | | concentration units for light element | | + | | / fission product / actinide | | + | | sub-libraries | | + +-------------+------------------------------------------+-------------+ + | total= | *yes / no* |rarr| print out total | yes | + | | concentration for nuclides / elements | | + | | for each output unit type | | + +-------------+------------------------------------------+-------------+ + +.. _5-4-3-7: + +Input data arrays +----------------- + +.. highlight:: scale + +.. :numref:`tab-origami-input-arrays` shows the remaining input arrays for ORIGAMI. + +For all other input arrays in ORIGAMI, the input values are entered in either +of the general forms (with or without ``=``) :: + + array[ … values … ] + + array=[ … values … ] + +The array :option:`libs`, which defines the ORIGEN library files, is the only +one that is strictly required for all cases. Cases that simulate 0D or +1D lumped-assembly models typically only require one entry for a single +ORIGEN library (assuming uniform axial enrichment), while the simulated +3D depletion model may utilize multiple libraries if specific ORIGEN +libraries are pre-generated for different pin locations (e.g., adjacent +to a water hole, Gd rods, etc.). If multiple libraries are used, the +array :option:`libmap` is required to identify the pin locations associated +with the input libraries. The numbering of these libraries in the libmap +array corresponds to the ordering of libraries in the :option:`libs` array; +i.e., a "1" corresponds to the first library specified, "2" to the +second, and so on. A zero-value entry in the array indicates that the +location is not to be depleted (i.e., a non-fuel region, such as a water +hole or guide tube). + +For **single** array values, the array bracket syntax is not required. +For example, each of the following is equivalent: :: + + compmap=[1] + + compmap[1] + + compmap= + +Note that the assignment operator (``=``) is likewise optional for arrays +when using the square-bracket syntax. + +Unless the 0D lumped-assembly model (i.e., lumped mass with no axial +power variation) is used, at least one of the arrays (:option:`pz`, +:option:`pxy`) describing the power variations must also be entered. The 1D +axial depletion model requires that the :option:`pz` array be entered, while +the pin-wise depletion model additionally requires the array :option:`pxy` . +The data in arrays :option:`pxy` and :option:`pz` correspond to the variables +r\ :sub:`xy` and a\ :sub:`z`, respectively, described in +:ref:`5-4-2-3`. The axial and XY power distributions are +normalized to unity inside the code, so that only the ratios of the input +array values are significant. As discussed in :ref:`5-4-2-3`, +it is generally recommended to use the final burnup distributions rather than +the relative power distributions for the values in the :option:`pxy` and +:option:`pz` arrays. + +The array :option:`nuccomp` defines the nuclides to be included in the output +compBlock file, described in more detail in :ref:`5-4-4`. +The nuclides in the array are identified by their seven digit IZZZAAA +identifier defined as ID = I \* 1000000 + Z \* 1000 + A, where Z is the atomic +number; A is the mass number, and I is the isomeric state (I=0 for ground; I=1 +for first metastable; etc.). For example, identifiers for :sup:`16`\ O and +:sup:`242m`\ Am are 8016 and 1095242, respectively. If this array is +omitted, the nuclides in :numref:`tab-origami-stdcmp-default` are used. +This is described in more detail in :ref:`5-4-4-1`. + +The optional array describing the non-fuel elements in the assembly +contains pairs of values (element, mass), where "element" is the +chemical symbol for a particular element, and "mass" is the mass of the +element in kilograms per MTU. For example, the + +.. highlight:: none + +:: + + nonfuel=[ zr=520.3 sn=8.4 ] + +indicates that the assembly contains 520.3 kilograms of zirconium and +8.4 kilograms of tin for each metric ton of uranium (MTU) in the +assembly. Note that elemental masses are specified — the isotopic masses +are computed internally by the code using natural abundances in the data +library. It is also possible to normalize the total mass of non-fuel +elements to a specified fraction of the MTU mass using the parameter +:option:`fracnf` in the parameter block. In this case, only the relative +amounts of each non-fuel element are needed for the :option:`nonfuel` array. +Non-fuel masses are distributed uniformly among all the fuel depletion +regions. + +.. option:: libs= [ ... ] + + List of ORIGEN one or more library file names for fuel in assembly + + **Required** + +.. option:: libmap= [ integer(s) ] + + XY map of library identifiers associated with each pin in assembly. Library + identifiers correspond to the order of the ORIGEN libraries entered in the + :option:`libs` array (i.e., index positions) + + (**Default:** ``[1]``) + + .. seealso:: :option:`libs` + + +.. option:: commap= [ integer(s) ] + + XY map of mixture identifiers that correspond to the mixture ID in the fuelcomp block. + + (**Default:** ``[1]``) + + .. seealso:: :option:`fuelcomp` + + +.. option:: pxy= [ real number(s) ] + + XY map of pin power shaping factors / fractional powers. Must be a square + array (e.g., 15×15). Defaults to lumped assembly model (no individual pins). + + (**Default:** ``[1.0]``) + + +.. option:: pz= [ real number(s) ] + + Axial (Z) power shaping factors / fractional power distribution for the + assembly. + + (**Default:** ``[1.0]``) + + +.. option:: meshz= [ real number(s) ] + + Axial mesh boundaries (cm) for the axial relative power zones. Only + required to define axial mesh for viewing results; but if entered, it must + be consistent with axial power shape. The number of entries should be one + greater than number of entries in :option:`pz` array. + + (**Default:** `none`) + + .. seealso:: :option:`pz` + + +.. option:: modz= [ real number(s) ] + + Axial variation in water density (g/cc) corresponding to the axial power zones. + + (**Default:** ``[0.723]``) + + +.. option:: nonfuel= [ key-value pairs ] + + Non-fuel materials contained in assembly. Values are entered in pairs of + ``element-symbol=mass`` (kg per mtu of HM ). If parameter :option:`fracnf` + is input, mass of non-fuel materials is normalized to this fraction of fuel + :option:`mtu`. + + .. note:: + + Oxygen mass in UO\ :sub:`2` should not be entered here (i.e., this is + pre-supplied by ORIGAMI). + + (**Default:** `None`) + + +.. option:: ggrp= [ real numbers ] + + Energy boundaries (eV) for defining decay gamma source spectrum, in monotonically + increasing order. + + (**Default:** `None`) + + +.. option:: ngrp= [ real numbers ] + + Energy boundaries (eV) for defining :math:`\left(\alpha,n\right)` and + fission neutron source spectrum. + + (**Default:** Nuclides in :numref:`tab-origami-stdcmp-default`) + + +.. option:: nuccomp= [ IZZZAAA values ] + + User-specified list of nuclides (in IZZZAAA format) to be included in the + :file:`compBlock` file. + + (**Default:** Nuclides specified in :numref:`tab-origami-stdcmp-default`). + + +.. table:: Description of ORIGAMI input arrays + :name: tab-origami-input-arrays + :widths: 15 65 20 + :align: center + + +----------------+------------------------------------+---------------------------------------+ + | **Array Name** | **Description** | **Default** | + +================+====================================+=======================================+ + | **libs** [#rq]_| List of ORIGEN library file names | None | + | | for fuel in assembly. [characters] | | + +----------------+------------------------------------+---------------------------------------+ + | libmap | XY map of library identifiers | 1 | + | | associated with each pin in | | + | | assembly. Library identifiers | | + | | correspond to the order of the | | + | | ORIGEN libraries entered in the | | + | | *libs* block. [integers] | | + +----------------+------------------------------------+---------------------------------------+ + | compmap | XY map of mixture identifiers that | 1 | + | | correspond to the mixture ID in | | + | | the :option:`fuelcomp` block. | | + | | [integers] | | + +----------------+------------------------------------+---------------------------------------+ + | pxy | XY map of pin power shaping | 1.0 | + | | factors / fractional powers. Must | | + | | be a square array (e.g., 15×15). | | + | | Defaults to lumped assembly model | | + | | (no individual pins). [real | | + | | numbers] | | + +----------------+------------------------------------+---------------------------------------+ + | pz | Axial (Z) power shaping factors / | 1.0 | + | | fractional power distribution for | | + | | the assembly. [real numbers] | | + +----------------+------------------------------------+---------------------------------------+ + | meshz | Axial mesh boundaries (cm) for the | None | + | | axial relative power zones. Only | | + | | required to define axial mesh for | | + | | viewing results; but if entered, | | + | | it must be consistent with axial | | + | | power shape. The number of entries | | + | | should be one greater than number | | + | | of entries in :option:`pz` array. | | + | | [real numbers] | | + +----------------+------------------------------------+---------------------------------------+ + | modz | Axial variation in water density | 0.723 | + | | (g/cc) corresponding to the axial | | + | | power zones. [real numbers] | | + +----------------+------------------------------------+---------------------------------------+ + | nonfuel | Non-fuel materials contained in | None | + | | assembly. Values are entered in | | + | | pairs of (element-symbol=mass(kg) | | + | | per mtu of HM ). If parameter | | + | | *fracnf* is input, mass of | | + | | non-fuel materials is normalized | | + | | to this fraction of fuel mtu. | | + | | **NOTE:** Oxygen mass in | | + | | UO\ :sub:`2` should **not** be | | + | | entered here (i.e., this is | | + | | pre-supplied by ORIGAMI). | | + | | [character / real number pairs] | | + +----------------+------------------------------------+---------------------------------------+ + | ggrp | Energy boundaries (eV) for | None | + | | defining decay gamma source | | + | | spectrum. [real numbers] | | + +----------------+------------------------------------+---------------------------------------+ + | ngrp | Energy boundaries (eV) for | None | + | | defining | | + | | :math:`\left(\alpha,n\right)` and | | + | | fission neutron source spectrum. | | + | | | | + | | [real numbers] | | + +----------------+------------------------------------+---------------------------------------+ + | nuccomp | List of nuclide IZZZAAAs to be | :numref:`tab-origami-stdcmp-default` | + | | included in output *compBlock* | | + | | file. | Nuclides | + +----------------+------------------------------------+---------------------------------------+ +.. [#rq] indicates required + +.. _5-4-4: + +ORIGAMI Input/Output Files +-------------------------- + +:numref:`tab-origami-io-files` gives the input and output files for ORIGAMI. +ORIGAMI produces printed output results as well as several optional output +files described in this section. In order to reduce the potentially voluminous +amount of printout, by default ORIGAMI only prints the concentrations in +grams for selected actinides in each axial zone of every pin, and only +for the last time step (e.g., decay step) of the last cycle in the +power-history block. Time-dependent results are given for all substeps +in the last step (i.e, there are :option:`nburn` and :option:`ndecay` substeps within a +burn step or decay step, respectively) In addition, the blended actinide +concentrations over all pins are printed for each axial zone, and for +the entire lumped assembly. Additional types of printed output can be +specified in the :option:`print` block. The concentrations, as well as optional +neutron and gamma source spectra information, for all nuclides, in all +pins and axial zones are also stored in the ORIGEN binary output file, +often called an "ft71" file. The contents and format of the binary file +are described in the ORIGEN documentation of the SCALE manual. The +binary file information can be edited by the OPUS module in SCALE. Like +the printed output, the ft71 file is written by default only for the +last step of the last cycle. However, both the printed output and binary +file results can be obtained at additional time steps by specifying the +input variables output and ft71, respectively, in the OPTIONS input +block. These input parameters can have the keywords: + +The output files are written in the user output directory for the +calculation (i.e., the same directory where the printed output file is +written — the default is the directory from where the case was +submitted). File names are prefixed by an extension consisting of the +input file base-name appended to an optional character string given by +the input keyword :option:`prefix` . For example, if the ORIGAMI input file is +named file:`ORIGAMICase.inp`, the base-name is :file:`ORIGAMICase`. Thus, if the +keyword :option:`prefix` is not included in the input, the file containing the +axial decay heat results is named file:`ORIGAMICase_AxialDecayHeat`. On the +other hand, if the input contains the keyword :command:`prefix=CE16X16`, the file +is named :file:`ORIGAMICase_CE16X16_AxialDecayHeat`. + +In order to capture axially dependent features of an assembly (such as +partial-length rods), users may elect to construct sequential ORIGAMI +cases that modify the XY pin map features (e.g., library and enrichment +maps) between cases. In order to allow for these types of "continuation" +cases (in which the sequential case represents an adjacent axial span of +the assembly), the :option:`offsetz` feature is provided, which adjusts the +axial numbering for ORIGAMI outputs (such as for MCNP materials & +spectra cards, axial decay heat, etc.). The :option:`offsetz` parameter offsets +the axial numbering for these output files, where the (integer) value +provided corresponds to the *last* axial zone number calculated by +ORIGAMI (default: 0). For more details on the syntax of the :command:`options` +block, see :ref:`5-4-3-2`. + +.. table:: ORIGAMI input/output files + :name: tab-origami-io-files + :widths: 20 60 10 10 + :class: longtable + :align: center + + +--------------------------------+---------------------+----------+------------+ + | **File Name** [#prefix]_ | **Description** | **Type** | **Format** | + +================================+=====================+==========+============+ + | :file:`compBlock` | Mixture | out | text | + | | compositions in | | | + | | standard | | | + | | composition format | | | + | | for input to SCALE | | | + | | codes such as KENO | | | + +--------------------------------+---------------------+----------+------------+ + | :file:`MCNP_matls.inp` | Nuclide identifiers | out | text | + | | and weight | | | + | | fractions in format | | | + | | for MCNP material | | | + | | cards | | | + +--------------------------------+---------------------+----------+------------+ + | :file:`MCNP_gamma.inp` | Total gamma source | out | text | + | | intensity in MCNP | | | + | | source format. Only | | | + | | output if gamma | | | + | | energy group | | | + | | boundaries are | | | + | | entered in input | | | + | | array | | | + | | :option:`ggrp` | | | + +--------------------------------+---------------------+----------+------------+ + | :file:`MCNP_neutron.inp` | Total neutron | out | text | + | | source intensity in | | | + | | MCNP source format. | | | + | | Only output if | | | + | | neutron energy | | | + | | group boundaries | | | + | | are entered in | | | + | | input array | | | + | | :option:`ngrp` | | | + +--------------------------------+---------------------+----------+------------+ + | :file:`AxialGammaSpec` | Gamma spectrum | out | text | + | | (photons/sec) by | | | + | | axial zone, enabled | | | + | | by "srcopt{ | | | + | | print=yes }". | | | + +--------------------------------+---------------------+----------+------------+ + | :file:`AxialNeutSpec` | Neutron spectrum | out | text | + | | (neutron/sec) by | | | + | | axial zone, enabled | | | + | | by "srcopt{ | | | + | | print=yes }". | | | + +--------------------------------+---------------------+----------+------------+ + | :file:`AxialDecayHeat` | Decay heat source | out | text | + | | (watts) by axial | | | + | | zone, enabled by | | | + | | "options{ | | | + | | decayheat=yes }" | | | + +--------------------------------+---------------------+----------+------------+ + | :file:`assm.f71` | Output stacked | out | binary | + | | ORIGEN ft71 files | | | + | | for each axial zone | | | + +--------------------------------+---------------------+----------+------------+ + | :file:`assembly_restart.f71` | Input stacked | in | binary | + | | ORIGEN ft71 files | | | + | | for each axial zone | | | + +--------------------------------+---------------------+----------+------------+ + | :file:`.f71` | Output of stacked | out | binary | + | | ORIGEN ft71 files | | | + | | for each pin and | | | + | | axial zone | | | + +--------------------------------+---------------------+----------+------------+ + | :file:`actinideMesh.3dmap` | Binary MeshView | out | binary | + | | file of selected | | | + | | actinide masses by | | | + | | depletion cell | | | + +--------------------------------+---------------------+----------+------------+ + | :file:`actinideMesh.ASCII.txt` | Plaintext MeshView | out | text | + | | file of selected | | | + | | actinide masses by | | | + | | depletion cell | | | + +--------------------------------+---------------------+----------+------------+ + | :file:`fpMesh.3dmap` | Binary MeshView | out | binary | + | | file of selected | | | + | | fission product | | | + | | masses by depletion | | | + | | cell | | | + +--------------------------------+---------------------+----------+------------+ + | :file:`fpMesh.ASCII.txt` | Plaintext MeshView | out | text | + | | file of selected | | | + | | fission product | | | + | | masses by depletion | | | + | | cell | | | + +--------------------------------+---------------------+----------+------------+ + | :file:`burnupMesh.3dmap` | Binary MeshView | out | binary | + | | file of depletion | | | + | | node burnups | | | + +--------------------------------+---------------------+----------+------------+ + +.. [#prefix] Note that all file names are prefixed by an identifier + :envvar:`${OUTBASENAME}`, where :envvar:`${OUTBASENAME}` is a prefix + constructed from the input file base name followed by the character string + given by input keyword ``prefix= *.*`` For example, the input file + named "my.inp" with :command:`prefix=sample` would give an output prefix + ``my_sample``; e.g., :file:`my_sample.f71`, :file:`my_sample.assm.f71`, + :file:`my_sample_MCNP_matls.inp`, etc. + +.. _5-4-4-1: + +Generation of SCALE standard composition data file +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +If input parameter ``stdcomp=yes`` is specified, ORIGAMI produces a text +file containing a SCALE standard composition description for each axial +interval. The file is written in the form of a ``stdcomp`` block that can +be directly used as input to any SCALE module that requires a +composition block. If a 1D axial depletion model is used for the +assembly, the composition for each axial zone is given a unique mixture +number defined for an axial node "\ *Z*\ " as: + +.. math:: + \begin{align} + \text{(1D\ axial\ model)}\ \ &\ \text{mix} = 1000 + (\textit{asmid} -1) \times N_Z + Z + \end{align} + +where N\ :sub:`Z` is the number of axial zones and :option:`asmid` is the input +identifier. For example, if there are 12 axial zones and the input for +:option:`asmid` is 20, then the mixture number associated with axial zone number +1 is mix = 1229, and the mixture for zone 12 is mix= 1240. If an +assembly is represented by a 3D multiple-pin model, the mixture number +is defined, + +.. math:: + \begin{align} + \text{(3D\ model)}\ \ &\ \text{mix} = 1000 + (\textit{asmid} -1) \times N_Z + + Z + X \times 100000 + Y \times 10000000 + \end{align} + :label: eq-origami-mix-num + + +where X and Y correspond to the row and column numbers of the pin. + +The nuclides components of the mixtures may be specified in the input +array :option:`nuccomp`, or by default the mixture may consist of the nuclides +given in :numref:`tab-origami-stdcmp-default`, which are the nuclides +recommended in :cite:`RGIW2012` for burnup credit analysis, plus :sup:`16`\ O. + +The temperatures of the mixtures are set by the value of parameter +:option:`temper`, which defaults to a value of 293 Kelvin. The number densities +of the nuclides in the mixtures are calculated using the following expressions: + +.. math:: + N_{Z}^{ \left( i \right) } & = + \rho \frac{M_{Z}^{(i)}}{M_{Z} \cdot 10^6} \cdot \frac{N_{A}}{A^{\left( i \right)}} \cdot 0.8814 \cdot 10^{-24} \\ + & = \rho \frac{ M_{Z}^{(i)} }{ M_{Z} \cdot A^{\left( i \right)} } \cdot 5.309 \cdot 10^{-7} + :label: eq-origami-num-dens + +Where: + + :math:`N_{Z}^{\left( i \right)}` = number density of nuclide "i" in zone + Z, in units of atoms of "i" per barn-cm of UO\ :sub:`2`; + + :math:`\rho` = density of UO\ :sub:`2` (g/cc), defined by the input parameter + :option:`fdens` (default is 10.4 g/cc); + + :math:`M_{Z}^{(i)}` = mass (g) of nuclide i in axial zone Z , obtained + from ORIGEN calculation; + + :math:`M_{Z} \cdot 10^{6}` = mass (g) of uranium in axial zone Z, where + M\ :sub:`Z` is given by :eq:`eq-origami-mass-ax-norm`; + + A\ :sup:`(i)` = mass (g) of 1 mole of nuclide i; + + 0.8814 = weight fraction of uranium in UO\ :sub:`2`; + + 10\ :sup:`-24` = cm\ :sup:`3` per barn-cm. + + +The definitions of other parameters appearing in this equation are given +in :ref:`5-4-2-5`. An example of the standard composition +file produced by ORIGAMI is given in :ref:`5-4-6`, +:numref:`ex-origami-prob2-stdcmp` (illustrated in sample problem 2, +:numref:`ex-origami-sample2`). + +.. table:: Default burnup credit nuclides in Standard Composition output + :name: tab-origami-stdcmp-default + :align: center + + ============== ======== ================ + **Nuclide** **ZAID** **Nuclide type** + ============== ======== ================ + :sup:`16`\ O 8016 light element + :sup:`234`\ U 92234 actinide + :sup:`235`\ U 92235 actinide + :sup:`236`\ U 92236 actinide + :sup:`238`\ U 92238 actinide + :sup:`237`\ Np 93237 actinide + :sup:`238`\ Pu 94238 actinide + :sup:`239`\ Pu 94239 actinide + :sup:`240`\ Pu 94240 actinide + :sup:`241`\ Pu 94241 actinide + :sup:`242`\ Pu 94242 actinide + :sup:`241`\ Am 95241 actinide + :sup:`243`\ Am 95243 actinide + :sup:`95`\ Mo 42095 fission product + :sup:`99`\ Tc 43099 fission product + :sup:`101`\ Ru 44101 fission product + :sup:`103`\ Rh 45103 fission product + :sup:`109`\ Ag 47109 fission product + :sup:`133`\ Cs 55133 fission product + :sup:`143`\ Nd 60143 fission product + :sup:`145`\ Nd 60145 fission product + :sup:`147`\ Sm 62147 fission product + :sup:`149`\ Sm 62149 fission product + :sup:`150`\ Sm 62150 fission product + :sup:`151`\ Sm 62151 fission product + :sup:`152`\ Sm 62152 fission product + :sup:`151`\ Eu 63151 fission product + :sup:`153`\ Eu 63153 fission product + :sup:`155`\ Gd 64155 fission product + ============== ======== ================ + +.. _5-4-4-2: + +MCNP data files +~~~~~~~~~~~~~~~ + +If the input parameter ``mcnp=yes`` is set in the ``options`` block, the +computed weight fractions for the materials in each axial zone also are +output in a file in the format of MCNP material cards. These material +cards are designed to be coupled to a corresponding MCNP assembly +geometry using the same numbering convention for the depletion zones. +:ref:`5-4-6` shows an example of the MCNP material +information produced by ORIGAMI. The numbering convention of the MCNP +materials cards works by combining the axial and pin numbers into a material +card, where pins are counted sequentially by row, starting with the bottom-left +row of input, counting from left to right across each row to the top-right pin +(i.e., the bottom-left pin is pin #1, etc.). The pin numbers reset with +each axial zone, starting from the bottom zone, counting up from 1. The +naming convention for materials cards is thus the pin number (1-999) +followed by the zone number (1-99); for example, pin #15 of axial zone +#12 would be **m1512**. Accompanying each material card is a list of +ZAID numbers and final concentrations (following depletion/decay) for the +cell expressed in weight fractions. The weight fractions are given as negative +values in accordance with MCNP convention. The fuel density, which may be used +in the MCNP cell card, is equal to the value of the input parameter +:option:`fdens`. + +When parameter ``mcnp=yes`` is set, ORIGAMI also produces output files +containing the fuel assembly radiation source magnitude by depletion +zone, to support modeling with MCNP. The gamma/neutron source term cards +correspond to the total gamma or neutron intensity (particles/s) from +each respective depletion region, using the same numbering convention as +that for the MCNP material cards. The source magnitude is computed by +summing over the MG source spectra defined in :eq:`eq-origami-spectra`. + + .. math:: + S_{Z}^{\left( p \right)} = \sum_{g}^{}{S_{Z,g}^{\left( p \right)}\ } + :label: eq-origami-ax-rad-source + + +Where: + + :math:`S_{Z}^{\left( p \right)}` = total source magnitude (p/s) for + particles of type *p*; + + :math:`S_{Z,g}^{\left( p \right)}` = multigroup source magnitude (p/s) + for energy group *g*, and particles of type *p* + +More details on the ORIGEN calculation of the source terms can be found +in the ORIGEN section of SCALE documentation. + +.. _5-4-4-3: + +Decay heat calculation +~~~~~~~~~~~~~~~~~~~~~~ + +When input parameter ``decayheat=yes`` is specified in the input, a text +file containing the decay heat source by axial zone, summed over all +pins, is generated as output. The decay heat in zone *Z* is given in +watts and is computed from the + + + .. math:: + H_{Z} = \sum_{i = 1}^{\text{itot}}{Q_{i}\lambda_{i}\frac{M_{Z}^{(i)}}{A^{(i)}} + \cdot 1.602 \cdot 10^{-13} \cdot N_{A}} = 9.65 \cdot 10^{10} + \sum_{i = 1}^{\text{itot}}{Q_{i}\lambda_{i}\frac{M_{Z}^{(i)}}{A^{(i)}}} + :label: eq-origami-ax-dh + +where: + + Q\ :sub:`i` = decay energy in MeV for nuclide *i*; + + :math:`\lambda_i` = decay constant (s\ :sup:`-1`) for nuclide *i*; + + :math:`M_{Z}^{(i)}` = mass (g) of nuclide *i* in axial zone *Z*, + obtained from ORIGEN calculation; + + A\ :sup:`(i)` = mass (g) of 1 mole of nuclide *i*; + + itot = total number of nuclides in burned fuel, + + 1.602×10\ :sup:`-19` = number of joules per MeV. + +An example output decay heat file produced by ORIGAMI is shown in +:ref:`5-4-6`, :numref:`ex-origami-prob2-dh-file` (from +sample problem 2). + +.. _5-4-4-4: + +ORIGEN results files +~~~~~~~~~~~~~~~~~~~~ + +The ORIGEN computation for each depletion region produces an ORIGEN +binary concentrations output file, historically called an "ft71" because +it was written on "Fortran tape" number 71. The file named +:file:`${OUTBASENAME}.f71` contains the concentrations for all depletion +regions, stacked within a single binary file, where :envvar:`${OUTBASENAME}` is +the base of the output file name, e.g. the "my" in :file:`my.out`. + +The order of stored cases on the f71 file corresponds to the order in +which ORGAMI processes individual depletion cases, starting with the +bottom-left row in the user-supplied power map (pin #1) and looping left +to right, progressively up through the series of rows. This process +repeats for each axial zone, starting from the bottom of the assembly +and working upward (i.e., starting with pin #1, axial zone #1, looping +through each pin on axial zone #1, and then proceeding to pin #1 on +axial zone #2, etc.). This convention is the same as that used for +TRITON arrays. + +In addition, the compositions are blended over all pins for each axial +zone to obtain the axially-dependent compositions for the lumped +assembly, stored in a file named :file:`${OUTBASENAME}.assm.f71`. If saved, +this file may be input as a restart file, as discussed in +:ref:`5-4-2-3`. + +.. _5-4-4-5: + +Plotting features +----------------- + +ORIGAMI creates three separate mesh summaries of material inventories +for individual depletion regions, useful for 3D visualization and +inspection. These include maps of (1) depletion region burnups, (2) +selected actinide concentrations (including isotopes of U, Pu, A m, and +Cm), and (3) selected fission products typically used for burnup +evaluation, including isotopes of Cs, Y, Ag, Rh, Ru, Eu, Sm, Nd, Gd, and +others). Additionally, ORIGAMI outputs a separate mesh tally of +individual node burnups. These outputs are described in +:numref:`tab-origami-io-files`. + +.. note:: + + The mesh files are only created if the user specifies the (optional) input + arguments for assembly pitch (:option:`pitch`) and axial zone locations + (:option:`meshz`). + +These output mesh-dependent maps can be visualized using the Java-based +Mesh File Viewer program included with SCALE. An example MeshView +visualization of one of these outputs is shown in :numref:`fig-origami-mv-xz` +and :numref:`fig-origami-mv-xy`. MeshView is installed in :file:`${SCALE}/Meshview`, +where :envvar:`${SCALE}` is the installation directory. A script to run MeshView is +located at :file:`${SCALE}/cmds/meshview`. + + +.. _fig-origami-mv-xz: +.. figure:: figs/ORIGAMI/fig2xz.png + :width: 50% + :align: center + + MeshView plot of total plutonium content in the 3D depletion regions (XZ plane) + +.. _fig-origami-mv-xy: +.. figure:: figs/ORIGAMI/fig2xy.png + :width: 50% + :align: center + + MeshView plot of total plutonium content in the 3D depletion regions (XY plane). + +.. _5-4-5: + +Parallel Execution on Linux Clusters +------------------------------------ + +For large 3D depletion problems it is advantageous to execute the ORIGEN +calculations for different depletion regions in parallel. This can be +done on Linux clusters using MPI. When parallel execution mode is +enabled, ORIGAMI distributes the individual depletion cases across the +pin rows, columns, and axial zones across several processors; the +depletion calculation is thus split across several processors. ORIGAMI +then collects the inventories from each calculation node and +concatenates the output. + +To execute ORIGAMI in parallel mode, a parallel-enabled MPI build of +SCALE must be used and ORIGAMI should be invoked with the percent (%) +prefix: :: + + =% + + + +Additionally, for parallel jobs spanning multiple computational nodes +(as opposed to those just using multiple processors on the same node, it +is recommended to use the :command:`–T` option to specify a common temporary +directory (such as a network-mounted directory accessible to all nodes). +This is due to the way ORIGAMI divides the problem space in parallel +mode; each computational node stores its respective binary dump file of +the individual pin/zone concentrations. Upon completion of execution, +the master node must be able to locate these individual problem +node-generated binary dump files; thus, by using a common temporary +directory, ORIGAMI can correctly re-assembly the individual pinwise +dumpfiles into a single consolidated "master" dump file. + +The following is a typical execution command line to execute ORIGAMI in +parallel. + +:command:`scalerte –N [number of nodes] -M [machine file] –T [tmpdir] [input_file.inp]` + +For more information on executing SCALE in parallel, see the SCALE +Readme file. + +.. _5-4-6: + +Sample Problems +--------------- + +This section shows sample problems for each of the three types of +simulated assembly models: 0D fully lumped, 2D lumped axial depletion, +and 3D pinwise depletion, and also demonstrates a restart case. + +.. _5-4-6-1: + +Sample problem 1: fully lumped assembly model +--------------------------------------------- + +The first example, :numref:`ex-origami-prob1`, corresponds to a fully-lumped +assembly model in which the materials are depleted with a space-independent +(i.e., assembly average) flux distribution. The assembly contains 0.38 MTU, +and the fuel is 2.8 wt% enriched. The assembly also includes several non-fuel +materials corresponding to cladding and other structural materials. Note +that the non-fuel concentrations are specified in kg/MTU, and thus are +not the actual total non-fuel masses in the 0.38 MTU assembly. The +assembly is depleted for three cycles with specific powers of 40.0, +38.6, and 25.2 MW/MTU, respectively. The ORIGEN library data are +interpolated for eight different burnup steps during the irradiation +periods of the first two cycles, and for six burnup steps in the last +cycle. + +:numref:`tab-origami-prob1-results` gives the calculated actinide +concentrations at the end of the third cycle. + +.. code-block:: scale + :caption: Input for ORIGAMI sample problem 1 + :name: ex-origami-prob1 + + =origami + title='fully lumped assembly model' + libs=[ ce14x14 ] + fuelcomp{ + stdcomp(fuel){ + base=uo2 iso[92234=0.02848 92235=3.2 92236=0.01472 92238=96.7568] } + mix(1){ comps[fuel=100] } + } + options{ mtu=0.38 ft71=all } + nonfuel=[ cr=3.366 mn=0.1525 fe=6.309 co=0.0302 + ni=2.366 zr=516.3 sn=8.412 gd=2.860 ] + hist[ + cycle{ power=40 burn=284 nlib=4 down=54 } + cycle{ power=38.6 burn=300 nlib=4 down=28 } + cycle{ power=25.2 burn=250 nlib=3 down=30 } + ] + print{ + nuc { + sublibs=[ac fp] + units=[grams moles] + total=no } + } + end + + +.. table:: Calculated Actinide inventories for sample problem 1 + :name: tab-origami-prob1-results + :align: center + + =============== ========= + Nuclide [#act]_ Mass (g) + =============== ========= + :sup:`234`\ U 6.820E+01 + :sup:`235`\ U 3.621E+03 + :sup:`236`\ U 1.487E+03 + :sup:`238`\ U 3.598E+05 + :sup:`237`\ Np 1.348E+02 + :sup:`238`\ Pu 3.862E+01 + :sup:`239`\ Pu 1.919E+03 + :sup:`240`\ Pu 7.820E+02 + :sup:`241`\ Pu 3.960E+02 + :sup:`242`\ Pu 1.394E+02 + :sup:`241`\ Am 1.474E+01 + :sup:`243`\ Am 2.491E+01 + :sup:`242`\ Cm 2.663E+00 + :sup:`244`\ Cm 5.698E+00 + TOTAL 6.820E+01 + =============== ========= + +.. [#act] Actinides with concentrations less than 0.0001 are not shown. + +.. _5-4-6-2: + +Sample problem 2: lumped axial depletion assembly model +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The second example has the same lumped assembly and power history as +sample problem 1, except in this case an axial power distribution is +provided for eight zones, so that the fuel burnup will vary axially; +the ORIGAMI input for this case is provided as :numref:`ex-origami-sample2`. +Also, the options to generate standard composition and decay output +files are requested. + +:numref:`tab-origami-prob2-results` gives the computed actinide +concentrations in grams for the first four of the eight axial zones. +Since the input axial power distribution is symmetrical about the assembly +midplane, the last four zones have identical concentrations as the first four. +The last column in the table shows actinide masses for the entire assembly. + +:numref:`ex-origami-prob2-stdcmp` is a listing of the contents of the +compBlock file, which contains standard composition input for the eight +axial zones in the assembly at the end of cycle 3. A complete description of +the SCALE standard composition input format is given in the MIPLIB chapter. The +first entry on each line in :numref:`ex-origami-prob2-stdcmp` corresponds to +the SCALE nuclide identifier. Only the default burnup credit analysis are +included. The second entry is the mixture number associated with a +particular axial zone. The mixture number for an axial zone is obtained +using :eq:`eq-origami-mix-num`. The third entry is always zero in this +file, and the fourth entry corresponds to the number density in atoms per +barn-cm. The next entry on the line is the temperature, which has the default +value of 293.0 since the input parameter :option:`temper` was not specified. The +final entry is an "end" statement. The information in this file can be +used as the ``read comp`` input block for any SCALE module. + +:numref:`ex-origami-prob2-dh-file` shows a listing of the file AxialDecayHeat, +which contains the heat source at the end of the third cycle. The entries in +the file correspond to the decay power in watts for the eight axial zones, +which are computed using :eq:`eq-origami-ax-dh`. + +.. code-block:: scale + :caption: Input for ORIGAMI sample problem 2 + :name: ex-origami-sample2 + + =origami + title= 'lumped axial-deplete assembly model' + libs=[ ce14x14 ] + fuelcomp{ + uox(fuel2){ enrich=3.2 } + mix(1){ comps[fuel2=100] } + } + options{ + mtu=0.38 stdcomp=yes decayheat=yes + } + pz=[ 1.0 2.0 3.0 4.0 4.0 3.0 2.0 1.0 ] + nonfuel=[ cr=3.366 mn=0.1525 fe=6.309 co=0.0302 + ni=2.366 zr=516.3 sn=8.412 gd=2.860 ] + hist[ + cycle{ power=40 burn=284 nlib=4 down=54 } + cycle{ power=38.6 burn=300 nlib=4 down=28 } + cycle{ power=25.2 burn=250 nlib=3 down=30 } + ] + end + + +.. table:: Calculated actinide inventories by axial zone for sample problem 2 + :name: tab-origami-prob2-results + :widths: 15 17 17 17 17 17 + :class: longtable + + +----------------+--------------+--------------+--------------+--------------+------------+ + | Nuclide | Axial Zone 1 | Axial Zone 2 | Axial Zone 3 | Axial Zone 4 | TOTAL | + | | | | | | | + | | Mass (g) | Mass (g) | Mass (g) | Mass (g) | Mass (g) | + +================+==============+==============+==============+==============+============+ + | :sup:`234`\ U | 1.1384E+01 | 9.4326E+00 | 7.6666E+00 | 6.11E+00 | 6.92E+01 | + +----------------+--------------+--------------+--------------+--------------+------------+ + | :sup:`235`\ U | 9.7324E+02 | 5.9453E+02 | 3.3795E+02 | 1.78E+02 | 4.17E+03 | + +----------------+--------------+--------------+--------------+--------------+------------+ + | :sup:`236`\ U | 1.0392E+02 | 1.6544E+02 | 2.0053E+02 | 2.15E+02 | 1.37E+03 | + +----------------+--------------+--------------+--------------+--------------+------------+ + | :sup:`238`\ U | 4.5604E+04 | 4.5202E+04 | 4.4752E+04 | 4.43E+04 | 3.60E+05 | + +----------------+--------------+--------------+--------------+--------------+------------+ + | :sup:`237`\ Np | 4.7874E+00 | 1.2622E+01 | 2.0984E+01 | 2.83E+01 | 1.33E+02 | + +----------------+--------------+--------------+--------------+--------------+------------+ + | :sup:`238`\ Pu | 5.5445E-01 | 2.9024E+00 | 7.1722E+00 | 1.26E+01 | 4.65E+01 | + +----------------+--------------+--------------+--------------+--------------+------------+ + | :sup:`239`\ Pu | 1.7378E+02 | 2.2968E+02 | 2.4438E+02 | 2.46E+02 | 1.79E+03 | + +----------------+--------------+--------------+--------------+--------------+------------+ + | :sup:`240`\ Pu | 3.3124E+01 | 7.7894E+01 | 1.1457E+02 | 1.40E+02 | 7.30E+02 | + +----------------+--------------+--------------+--------------+--------------+------------+ + | :sup:`241`\ Pu | 1.2974E+01 | 3.8401E+01 | 5.8714E+01 | 7.09E+01 | 3.62E+02 | + +----------------+--------------+--------------+--------------+--------------+------------+ + | :sup:`242`\ Pu | 1.4543E+00 | 1.0115E+01 | 2.6393E+01 | 4.75E+01 | 1.71E+02 | + +----------------+--------------+--------------+--------------+--------------+------------+ + | :sup:`241`\ Am | 5.3938E-01 | 1.5121E+00 | 2.0433E+00 | 2.12E+00 | 1.24E+01 | + +----------------+--------------+--------------+--------------+--------------+------------+ + | :sup:`243`\ Am | 9.5029E-02 | 1.4341E+00 | 5.6093E+00 | 1.29E+01 | 4.00E+01 | + +----------------+--------------+--------------+--------------+--------------+------------+ + | :sup:`242`\ Cm | [#epsilon]_ | 2.0213E-01 | 4.7744E-01 | 7.56E-01 | 2.93E+00 | + +----------------+--------------+--------------+--------------+--------------+------------+ + | :sup:`244`\ Cm | [#epsilon]_ | 2.4685E-01 | 1.6315E+00 | 5.54E+00 | 1.49E+01 | + +----------------+--------------+--------------+--------------+--------------+------------+ + | :sup:`245`\ Cm | [#epsilon]_ | [#epsilon]_ | 5.8836E-02 | 2.40E-01 | 6.11E-01 | + +----------------+--------------+--------------+--------------+--------------+------------+ + | TOTAL | 4.6920E+04 | 4.6346E+04 | 4.5780E+04 | 4.5221E+04 | 3.6854E+05 | + +----------------+--------------+--------------+--------------+--------------+------------+ +.. [#epsilon] Values < 0.0001 are not shown + + +.. code-block:: none + :caption: Sample Problem 2: Standard composition file + (default burnup credit nuclides) + :name: ex-origami-prob2-stdcmp + + o-16 1001 0 4.6395E-02 293.0 end + u-234 1001 0 5.6529E-06 293.0 end + u-235 1001 0 4.8120E-04 293.0 end + u-236 1001 0 5.1165E-05 293.0 end + u-238 1001 0 2.2263E-02 293.0 end + np-237 1001 0 2.3470E-06 293.0 end + pu-238 1001 0 2.7067E-07 293.0 end + pu-239 1001 0 8.4481E-05 293.0 end + pu-240 1001 0 1.6036E-05 293.0 end + pu-241 1001 0 6.2547E-06 293.0 end + pu-242 1001 0 6.9823E-07 293.0 end + am-241 1001 0 2.6003E-07 293.0 end + am-243 1001 0 4.5435E-08 293.0 end + mo-95 1001 0 1.5700E-05 293.0 end + tc-99 1001 0 1.7475E-05 293.0 end + ru-101 1001 0 1.5272E-05 293.0 end + rh-103 1001 0 9.4877E-06 293.0 end + ag-109 1001 0 7.6473E-07 293.0 end + cs-133 1001 0 1.8563E-05 293.0 end + nd-143 1001 0 1.4504E-05 293.0 end + nd-145 1001 0 1.0445E-05 293.0 end + sm-147 1001 0 1.5451E-06 293.0 end + sm-149 1001 0 8.0469E-08 293.0 end + sm-150 1001 0 3.3359E-06 293.0 end + sm-151 1001 0 3.5230E-07 293.0 end + sm-152 1001 0 1.7469E-06 293.0 end + eu-151 1001 0 1.5582E-09 293.0 end + eu-153 1001 0 9.1077E-07 293.0 end + gd-155 1001 0 1.3966E-09 293.0 end + o-16 1002 0 4.6394E-02 293.0 end + u-234 1002 0 4.6837E-06 293.0 end + u-235 1002 0 2.9395E-04 293.0 end + u-236 1002 0 8.1452E-05 293.0 end + u-238 1002 0 2.2067E-02 293.0 end + np-237 1002 0 6.1878E-06 293.0 end + pu-238 1002 0 1.4169E-06 293.0 end + pu-239 1002 0 1.1165E-04 293.0 end + pu-240 1002 0 3.7710E-05 293.0 end + pu-241 1002 0 1.8513E-05 293.0 end + pu-242 1002 0 4.8561E-06 293.0 end + am-241 1002 0 7.2896E-07 293.0 end + am-243 1002 0 6.8566E-07 293.0 end + mo-95 1002 0 2.9706E-05 293.0 end + tc-99 1002 0 3.3424E-05 293.0 end + ru-101 1002 0 3.0467E-05 293.0 end + rh-103 1002 0 1.8486E-05 293.0 end + ag-109 1002 0 2.2784E-06 293.0 end + cs-133 1002 0 3.5311E-05 293.0 end + nd-143 1002 0 2.4869E-05 293.0 end + nd-145 1002 0 1.9393E-05 293.0 end + sm-147 1002 0 2.4415E-06 293.0 end + sm-149 1002 0 1.0154E-07 293.0 end + sm-150 1002 0 7.3182E-06 293.0 end + sm-151 1002 0 4.4543E-07 293.0 end + sm-152 1002 0 3.4915E-06 293.0 end + eu-151 1002 0 1.1064E-09 293.0 end + eu-153 1002 0 2.5199E-06 293.0 end + gd-155 1002 0 2.5157E-09 293.0 end + o-16 1003 0 4.6392E-02 293.0 end + u-234 1003 0 3.8068E-06 293.0 end + u-235 1003 0 1.6709E-04 293.0 end + u-236 1003 0 9.8728E-05 293.0 end + u-238 1003 0 2.1847E-02 293.0 end + np-237 1003 0 1.0287E-05 293.0 end + pu-238 1003 0 3.5014E-06 293.0 end + pu-239 1003 0 1.1880E-04 293.0 end + pu-240 1003 0 5.5465E-05 293.0 end + pu-241 1003 0 2.8306E-05 293.0 end + pu-242 1003 0 1.2671E-05 293.0 end + am-241 1003 0 9.8507E-07 293.0 end + am-243 1003 0 2.6819E-06 293.0 end + mo-95 1003 0 4.2205E-05 293.0 end + tc-99 1003 0 4.7742E-05 293.0 end + ru-101 1003 0 4.5361E-05 293.0 end + rh-103 1003 0 2.6134E-05 293.0 end + ag-109 1003 0 4.1770E-06 293.0 end + cs-133 1003 0 5.0069E-05 293.0 end + nd-143 1003 0 3.1519E-05 293.0 end + nd-145 1003 0 2.6993E-05 293.0 end + sm-147 1003 0 2.8511E-06 293.0 end + sm-149 1003 0 1.2277E-07 293.0 end + sm-150 1003 0 1.1691E-05 293.0 end + sm-151 1003 0 5.2472E-07 293.0 end + sm-152 1003 0 5.0112E-06 293.0 end + eu-151 1003 0 9.0563E-10 293.0 end + eu-153 1003 0 4.4446E-06 293.0 end + gd-155 1003 0 4.0054E-09 293.0 end + o-16 1004 0 4.6390E-02 293.0 end + u-234 1004 0 3.0343E-06 293.0 end + u-235 1004 0 8.7951E-05 293.0 end + u-236 1004 0 1.0588E-04 293.0 end + u-238 1004 0 2.1605E-02 293.0 end + np-237 1004 0 1.3864E-05 293.0 end + pu-238 1004 0 6.1605E-06 293.0 end + pu-239 1004 0 1.1946E-04 293.0 end + pu-240 1004 0 6.7542E-05 293.0 end + pu-241 1004 0 3.4172E-05 293.0 end + pu-242 1004 0 2.2786E-05 293.0 end + am-241 1004 0 1.0210E-06 293.0 end + am-243 1004 0 6.1463E-06 293.0 end + mo-95 1004 0 5.3319E-05 293.0 end + tc-99 1004 0 6.0395E-05 293.0 end + ru-101 1004 0 5.9831E-05 293.0 end + rh-103 1004 0 3.2151E-05 293.0 end + ag-109 1004 0 6.2338E-06 293.0 end + cs-133 1004 0 6.2779E-05 293.0 end + nd-143 1004 0 3.4995E-05 293.0 end + nd-145 1004 0 3.3350E-05 293.0 end + sm-147 1004 0 2.9202E-06 293.0 end + sm-149 1004 0 1.4469E-07 293.0 end + sm-150 1004 0 1.6101E-05 293.0 end + sm-151 1004 0 6.0099E-07 293.0 end + sm-152 1004 0 6.3684E-06 293.0 end + eu-151 1004 0 8.1899E-10 293.0 end + eu-153 1004 0 6.4006E-06 293.0 end + gd-155 1004 0 5.5315E-09 293.0 end + o-16 1005 0 4.6390E-02 293.0 end + u-234 1005 0 3.0343E-06 293.0 end + u-235 1005 0 8.7951E-05 293.0 end + u-236 1005 0 1.0588E-04 293.0 end + u-238 1005 0 2.1605E-02 293.0 end + np-237 1005 0 1.3864E-05 293.0 end + pu-238 1005 0 6.1605E-06 293.0 end + pu-239 1005 0 1.1946E-04 293.0 end + pu-240 1005 0 6.7542E-05 293.0 end + pu-241 1005 0 3.4172E-05 293.0 end + pu-242 1005 0 2.2786E-05 293.0 end + am-241 1005 0 1.0210E-06 293.0 end + am-243 1005 0 6.1463E-06 293.0 end + mo-95 1005 0 5.3319E-05 293.0 end + tc-99 1005 0 6.0395E-05 293.0 end + ru-101 1005 0 5.9831E-05 293.0 end + rh-103 1005 0 3.2151E-05 293.0 end + ag-109 1005 0 6.2338E-06 293.0 end + cs-133 1005 0 6.2779E-05 293.0 end + nd-143 1005 0 3.4995E-05 293.0 end + nd-145 1005 0 3.3350E-05 293.0 end + sm-147 1005 0 2.9202E-06 293.0 end + sm-149 1005 0 1.4469E-07 293.0 end + sm-150 1005 0 1.6101E-05 293.0 end + sm-151 1005 0 6.0099E-07 293.0 end + sm-152 1005 0 6.3684E-06 293.0 end + eu-151 1005 0 8.1899E-10 293.0 end + eu-153 1005 0 6.4006E-06 293.0 end + gd-155 1005 0 5.5315E-09 293.0 end + o-16 1006 0 4.6392E-02 293.0 end + u-234 1006 0 3.8068E-06 293.0 end + u-235 1006 0 1.6709E-04 293.0 end + u-236 1006 0 9.8728E-05 293.0 end + u-238 1006 0 2.1847E-02 293.0 end + np-237 1006 0 1.0287E-05 293.0 end + pu-238 1006 0 3.5014E-06 293.0 end + pu-239 1006 0 1.1880E-04 293.0 end + pu-240 1006 0 5.5465E-05 293.0 end + pu-241 1006 0 2.8306E-05 293.0 end + pu-242 1006 0 1.2671E-05 293.0 end + am-241 1006 0 9.8507E-07 293.0 end + am-243 1006 0 2.6819E-06 293.0 end + mo-95 1006 0 4.2205E-05 293.0 end + tc-99 1006 0 4.7742E-05 293.0 end + ru-101 1006 0 4.5361E-05 293.0 end + rh-103 1006 0 2.6134E-05 293.0 end + ag-109 1006 0 4.1770E-06 293.0 end + cs-133 1006 0 5.0069E-05 293.0 end + nd-143 1006 0 3.1519E-05 293.0 end + nd-145 1006 0 2.6993E-05 293.0 end + sm-147 1006 0 2.8511E-06 293.0 end + sm-149 1006 0 1.2277E-07 293.0 end + sm-150 1006 0 1.1691E-05 293.0 end + sm-151 1006 0 5.2472E-07 293.0 end + sm-152 1006 0 5.0112E-06 293.0 end + eu-151 1006 0 9.0563E-10 293.0 end + eu-153 1006 0 4.4446E-06 293.0 end + gd-155 1006 0 4.0054E-09 293.0 end + o-16 1007 0 4.6394E-02 293.0 end + u-234 1007 0 4.6837E-06 293.0 end + u-235 1007 0 2.9395E-04 293.0 end + u-236 1007 0 8.1452E-05 293.0 end + u-238 1007 0 2.2067E-02 293.0 end + np-237 1007 0 6.1878E-06 293.0 end + pu-238 1007 0 1.4169E-06 293.0 end + pu-239 1007 0 1.1165E-04 293.0 end + pu-240 1007 0 3.7710E-05 293.0 end + pu-241 1007 0 1.8513E-05 293.0 end + pu-242 1007 0 4.8561E-06 293.0 end + am-241 1007 0 7.2896E-07 293.0 end + am-243 1007 0 6.8566E-07 293.0 end + mo-95 1007 0 2.9706E-05 293.0 end + tc-99 1007 0 3.3424E-05 293.0 end + ru-101 1007 0 3.0467E-05 293.0 end + rh-103 1007 0 1.8486E-05 293.0 end + ag-109 1007 0 2.2784E-06 293.0 end + cs-133 1007 0 3.5311E-05 293.0 end + nd-143 1007 0 2.4869E-05 293.0 end + nd-145 1007 0 1.9393E-05 293.0 end + sm-147 1007 0 2.4415E-06 293.0 end + sm-149 1007 0 1.0154E-07 293.0 end + sm-150 1007 0 7.3182E-06 293.0 end + sm-151 1007 0 4.4543E-07 293.0 end + sm-152 1007 0 3.4915E-06 293.0 end + eu-151 1007 0 1.1064E-09 293.0 end + eu-153 1007 0 2.5199E-06 293.0 end + gd-155 1007 0 2.5157E-09 293.0 end + o-16 1008 0 4.6395E-02 293.0 end + u-234 1008 0 5.6529E-06 293.0 end + u-235 1008 0 4.8120E-04 293.0 end + u-236 1008 0 5.1165E-05 293.0 end + u-238 1008 0 2.2263E-02 293.0 end + np-237 1008 0 2.3470E-06 293.0 end + pu-238 1008 0 2.7067E-07 293.0 end + pu-239 1008 0 8.4481E-05 293.0 end + pu-240 1008 0 1.6036E-05 293.0 end + pu-241 1008 0 6.2547E-06 293.0 end + pu-242 1008 0 6.9823E-07 293.0 end + am-241 1008 0 2.6003E-07 293.0 end + am-243 1008 0 4.5435E-08 293.0 end + mo-95 1008 0 1.5700E-05 293.0 end + tc-99 1008 0 1.7475E-05 293.0 end + ru-101 1008 0 1.5272E-05 293.0 end + rh-103 1008 0 9.4877E-06 293.0 end + ag-109 1008 0 7.6473E-07 293.0 end + cs-133 1008 0 1.8563E-05 293.0 end + nd-143 1008 0 1.4504E-05 293.0 end + nd-145 1008 0 1.0445E-05 293.0 end + sm-147 1008 0 1.5451E-06 293.0 end + sm-149 1008 0 8.0469E-08 293.0 end + sm-150 1008 0 3.3359E-06 293.0 end + sm-151 1008 0 3.5230E-07 293.0 end + sm-152 1008 0 1.7469E-06 293.0 end + eu-151 1008 0 1.5582E-09 293.0 end + eu-153 1008 0 9.1077E-07 293.0 end + gd-155 1008 0 1.3966E-09 293.0 end + + +.. code-block:: none + :caption: Sample problem 2: Decay heat file with axial decay heat by zone + (Watts) + :name: ex-origami-prob2-dh-file + + 7.11397E+02 + 1.43786E+03 + 2.18631E+03 + 2.95428E+03 + 2.95428E+03 + 2.18631E+03 + 1.43786E+03 + 7.11397E+02 + +.. _5-4-6-3: + +Sample problem 3: restart decay calculation for lumped axial depletion assembly model +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The third example shows a restart decay-only calculation, using the +ORIGEN ft71 binary file obtained from sample problem 2. This case +calculates the composition of the burned fuel produced in sample problem +2 after 100,000 additional days of decay. The input this problem is +given in :numref:`ex-origami-prob4`. Because the input parameter +:command:`restart=yes` is specified, the initial composition of the assembly is +obtained from a file named :file:`assembly_restart.f71`. The shell input that +precedes the ORIGAMI input in :numref:`ex-origami-prob4` copies the output ft71 +file produced in sample problem 2, which was named :file:`assembly_dump.f71`, +into a file named :file:`assembly_restart.f71` in the temporary directory for +SCALE calculations. The restart file contains the complete inventory of +nuclide compositions for eight axial zones. Because this restart case is +decay only (i.e., :option:`power` value is not given in the power-history +block), it is necessary to provide the input parameter :command:`nz=8` because +this value is used to determine how many axial zones were used in the previous +burnup calculations. + +:numref:`tab-origami-prob3-act-results` shows the actinide composition of the +first four axial of the (symmetrical) eight zones after 100,000 days of decay. +The initial masses of these nuclides before decay are the values given in +`numref:`ex-origami-prob2-stdcmp`. The last column in +:numref:`tab-origami-prob3-act-results` shows actinide masses for the entire +assembly after the decay period. + +.. code-block:: scale + :name: ex-origami-prob3-input + :caption: Sample problem 3: restart decay for a lumped axial depletion model + + =origami + title= 'lumped axial-deplete assembly model' + libs=[ ce14x14 ] + fuelcomp{ + %3.2 w/o + uox(fuel){ enrich=3.2 } + mix(1){ comps[fuel=100] } + } + options[ mtu=0.38 relnorm=no ] + pz=[ 1.0 2.0 3.0 4.0 4.0 3.0 2.0 1.0 ] + nonfuel=[ cr=3.366 mn=0.1525 fe=6.309 co=0.0302 + ni=2.366 zr=516.3 sn=8.412 gd=2.860 ] + hist[ + cycle{ power=40 burn=284 nlib=8 down=54 } + cycle{ power=38.6 burn=300 nlib=8 down=28 } + cycle{ power=25.2 burn=250 nlib=6 down=30 } + ] + end + =shell + mv \*.assm.f71 assembly_restart.f71 + end + =origami + title= 'restart decay' + asmid= 22 + libs=[ ce14x14 ] + prefix=origam3 + options{ + stdcomp=yes decayheat=yes relnorm=no restart=yes nz=8 + } + pz=[ 1.0 2.0 3.0 4.0 4.0 3.0 2.0 1.0 ] + hist[ + cycle{ down=100000 } + ] + end + =shell + rm assembly_restart.f71 + rm ${OUTDIR}/\*origam3\* + end + + +.. table:: Calculated actinide inventories by axial zone for sample problem 3 + :name: tab-origami-prob3-act-results + :widths: 15 17 17 17 17 17 + :align: center + :class: longtable + + +----------------+--------------+--------------+--------------+--------------+--------------+ + | Nuclide | Axial Zone 1 | Axial Zone 2 | Axial Zone 3 | Axial Zone 4 | TOTAL | + | | | | | | | + | | Mass (g) | Mass (g) | Mass (g) | Mass (g) | Mass (g) | + +================+==============+==============+==============+==============+==============+ + | :sup:`234`\ U | 1.1889E+01 | 1.2133E+01 | 1.4319E+01 | 1.7738E+01 | 1.1216E+02 | + +----------------+--------------+--------------+--------------+--------------+--------------+ + | :sup:`235`\ U | 9.7454E+02 | 5.9616E+02 | 3.3963E+02 | 1.7955E+02 | 4.1798E+03 | + +----------------+--------------+--------------+--------------+--------------+--------------+ + | :sup:`236`\ U | 1.0486E+02 | 1.6764E+02 | 2.0378E+02 | 2.1907E+02 | 1.3907E+03 | + +----------------+--------------+--------------+--------------+--------------+--------------+ + | :sup:`238`\ U | 4.5604E+04 | 4.5202E+04 | 4.4752E+04 | 4.4256E+04 | 3.5963E+05 | + +----------------+--------------+--------------+--------------+--------------+--------------+ + | :sup:`237`\ Np | 9.2242E+00 | 2.5723E+01 | 4.0929E+01 | 5.2246E+01 | 2.5625E+02 | + +----------------+--------------+--------------+--------------+--------------+--------------+ + | :sup:`238`\ Pu | 6.8535E-02 | 3.6104E-01 | 8.8478E-01 | 1.5422E+00 | 5.7132E+00 | + +----------------+--------------+--------------+--------------+--------------+--------------+ + | :sup:`239`\ Pu | 1.7243E+02 | 2.2794E+02 | 2.4262E+02 | 2.4413E+02 | 1.7742E+03 | + +----------------+--------------+--------------+--------------+--------------+--------------+ + | :sup:`240`\ Pu | 3.2192E+01 | 7.5957E+01 | 1.1297E+02 | 1.4095E+02 | 7.2412E+02 | + +----------------+--------------+--------------+--------------+--------------+--------------+ + | :sup:`241`\ Pu | [#epsilon2]_ | [#epsilon2]_ | [#epsilon2]_ | [#epsilon2]_ | [#epsilon2]_ | + +----------------+--------------+--------------+--------------+--------------+--------------+ + | :sup:`242`\ Pu | 1.4538E+00 | 1.0107E+01 | 2.6379E+01 | 4.7456E+01 | 1.7079E+02 | + +----------------+--------------+--------------+--------------+--------------+--------------+ + | :sup:`241`\ Am | 8.9945E+00 | 2.6564E+01 | 4.0451E+01 | 4.8623E+01 | 2.4927E+02 | + +----------------+--------------+--------------+--------------+--------------+--------------+ + | :sup:`243`\ Am | 9.2577E-02 | 1.3965E+00 | 5.4616E+00 | 1.2516E+01 | 3.8934E+01 | + +----------------+--------------+--------------+--------------+--------------+--------------+ + | :sup:`242`\ Cm | [#epsilon2]_ | [#epsilon2]_ | [#epsilon2]_ | [#epsilon2]_ | [#epsilon2]_ | + +----------------+--------------+--------------+--------------+--------------+--------------+ + | :sup:`244`\ Cm | [#epsilon2]_ | [#epsilon2]_ | [#epsilon2]_ | [#epsilon2]_ | [#epsilon2]_ | + +----------------+--------------+--------------+--------------+--------------+--------------+ + | :sup:`245`\ Cm | [#epsilon2]_ | [#epsilon2]_ | 5.7288E-02 | 2.3382E-01 | 5.9479E-01 | + +----------------+--------------+--------------+--------------+--------------+--------------+ + | TOTAL | 4.6920E+04 | 4.6346E+04 | 4.5780E+04 | 4.5220E+04 | 3.6853E+05 | + +----------------+--------------+--------------+--------------+--------------+--------------+ + +.. [#epsilon2] Values < 0.0001 are not shown + +.. _5-4-6-4: + +Sample problem 4: Simplified 3D multi-pin model +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The fourth example is a simulation of a simplified 3D depletion model. +The ORIGAMI 3D model normally includes all fuel pins within the +assembly, such as a 14×14 array. However to keep this example case +simple and the execution time low, only a 2×2 array of four individual +pins is considered for illustrative purposes. An axial fractional power +distribution is also specified for two axial zones. Therefore the total +number of depletion regions will be eight – two axial for each of the +four pins. Two different ORIGEN libraries are used to obtain cross +sections for the four pins. This is done whenever fuel pins in different +locations in the assembly have significantly different neutron spectra, +such as if some pins are adjacent to a control rod. In this sample +problem, ORIGEN libraries for two different types of assembly designs CE +(Combustion Engineering) 14×14 and 16×16 assembly designs, respectively +are used to demonstrate the use of pin-dependent libraries, although in +reality the ORIGEN libraries normally would be pre-generated for +different pin locations within a single type of assembly configuration. +The values specified in :option:`libmap` indicate which library is to be used +for each pin location. :numref:`ex-origami-prob4` shows the input for this +sample problem. + +.. code-block:: scale + :caption: Input for ORIGAMI Sample Problem 4: a simplified multi-pin, multi-axial model + :name: ex-origami-prob4 + + =origami + title= 'multi-pin; multi-library pin-deplete model' + prefix= origam4 + libs=[ ce14x14 ce16x16 ] + + fuelcomp{ + %3.2 w/o + stdcomp(fuel){ base=uo2 iso[92234=0.028569 92235=3.21 92236=0.014766 + 92238=96.746665] } + mix(1){ comps[fuel=100] } + } + + options{ mtu=0.4 decayheat=yes } + libmap=[ 1 1 + 2 2 ] + pxy=[ 0.284 0.283 + 0.218 0.215 ] + pz=[ 0.55 0.45 ] + hist[ + cycle{ power=39.78 burn=284.0 nlib=2 down=54.0 } + ] + end + + +:numref:`tab-origami-prob4-results-pins` shows selected actinide compositions +for the first row of two pins, that is, locations (1,1) and (1,2), for each of the two axial zones. The blended compositions over all fuel pins, for the two +axial zones, are given in :numref:`tab-origami-prob4-axblend`. The output +decay heat file for the assembly is shown in :numref:`tab-origami-prob4-dh`, +as a function of axial zone, summed over all pins. Note that this file has the +prefix "sample4\_" appended to the standard file name, since *prefix=sample4* +is specified in the input. + +.. table:: Actinide inventories by axial zone for pins (1,1) and (1,2) + in sample problem 4 + :name: tab-origami-prob4-results-pins + :widths: 14 22 22 22 22 + :align: center + + ============== ============ ============ ============= ============ + Nuclide pin (1,1) pin (1,1) pin (1,2) pin (1,2) + Axial Zone 1 Axial Zone 2 Axial Zone 1 Axial Zone 2 + Mass (g) Mass (g) Mass (g) Mass (g) + ============== ============ ============ ============= ============ + :sup:`234`\ U 1.2115E+01 1.2493E+01 1.2143E+01 1.2516E+01 + :sup:`235`\ U 1.0701E+03 1.1528E+03 1.0762E+03 1.1581E+03 + :sup:`236`\ U 1.0383E+02 8.9286E+01 1.0277E+02 8.8348E+01 + :sup:`237`\ U 1.0664E-03 [#epsilon3]_ 1.0428E-03 [#epsilon3]_ + :sup:`238`\ U 4.8003E+04 4.8074E+04 4.8009E+04 4.8078E+04 + :sup:`237`\ Np 4.8000E+00 3.6009E+00 4.7055E+00 3.5303E+00 + :sup:`239`\ Np 5.8742E-07 4.4485E-07 5.7555E-07 4.3692E-07 + :sup:`238`\ Pu 4.7198E-01 2.9453E-01 4.5684E-01 2.8512E-01 + :sup:`239`\ Pu 1.8855E+02 1.6706E+02 1.8705E+02 1.6560E+02 + :sup:`240`\ Pu 3.3235E+01 2.5225E+01 3.2625E+01 2.4736E+01 + :sup:`241`\ Pu 1.3906E+01 9.2953E+00 1.3537E+01 9.0304E+00 + :sup:`242`\ Pu 1.3733E+00 7.3260E-01 1.3161E+00 7.0081E-01 + :sup:`241`\ Am 2.3603E-01 1.5773E-01 2.2978E-01 1.5322E-01 + :sup:`243`\ Am 8.4143E-02 [#epsilon3]_ 7.9365E-02 [#epsilon3]_ + TOTAL 4.9432E+04 4.9535E+04 4.9440E+04 4.9541E+04 + ============== ============ ============ ============= ============ + +.. [#epsilon3] Values < 0.0001 are not shown + + +.. table:: Blended actinide inventories by axial zone (all pins) + for sample problem 4 + :name: tab-origami-prob4-axblend + :align: center + + + ============== ============ ============ ========== + Nuclide Axial Zone 1 Axial Zone 2 TOTAL + Mass (g) Mass (g) Mass (g) + ============== ============ ============ ========== + :sup:`234`\ U 4.7388E+01 4.9082E+01 9.6469E+01 + :sup:`235`\ U 4.0115E+03 4.3778E+03 8.3894E+03 + :sup:`236`\ U 4.5851E+02 3.9537E+02 8.5387E+02 + :sup:`238`\ U 1.9184E+05 1.9216E+05 3.8399E+05 + :sup:`237`\ Np 2.2821E+01 1.7132E+01 3.9953E+01 + :sup:`238`\ Pu 2.5739E+00 1.6049E+00 4.1789E+00 + :sup:`239`\ Pu 7.8246E+02 6.9963E+02 1.4821E+03 + :sup:`240`\ Pu 1.5436E+02 1.1801E+02 2.7238E+02 + :sup:`241`\ Pu 6.7918E+01 4.6455E+01 1.1437E+02 + :sup:`242`\ Pu 8.1693E+00 4.4411E+00 1.2610E+01 + :sup:`241`\ Am 1.1476E+00 7.8696E-01 1.9346E+00 + :sup:`243`\ Am 6.0150E-01 2.5995E-01 8.6145E-01 + TOTAL 1.9740E+05 1.9787E+05 3.9526E+05 + ============== ============ ============ ========== + +.. table:: Sample problem 4: Decay heat by axial zone (Watts) + :name: tab-origami-prob4-dh + :align: center + + +-------------+ + | 6.96472E+03 | + | | + | 5.72539E+03 | + +-------------+ + +.. _5-4-6-5: + +Sample problem 5: PWR 3D assembly model +--------------------------------------- + +This sample problem shows the input for a simulated full 3D pressurized +water reactor (PWR) assembly-depletion model, which corresponds to a +16×16 lattice with 26 axial zones. :numref:`ex-origami-sample-prob4` shows the +ORIGAMI input for this case. The arrays :option:`pxy` and :option:`pz` define the 3D XY-Z +fractional power distribution. Four different pre-processed ORIGEN libraries +are used to describe the pin-averaged ORIGEN cross-sections for the assembly. +The array :option:`libmap` assigns these libraries to the appropriate pin locations. +It can be seen that the :option:`libmap` array contains values of zero at 21 locations. +These correspond to non-depleting (i.e., zero power) locations. The input +includes the information (parameter :option:`pitch`, and array *z=* ) necessary to +generate 3D mesh summary maps for subsequent visualization. In this case, the +axial mesh is not uniform. Since a total of 6110 ORIGEN cases are executed for +the depletable pins, this ORIGAMI calculation was performed in parallel using +MPI. :numref:`fig5-4-3` shows a plot of the axial burnup distribution, summed over +all pins. + +.. code-block:: scale + :name: ex-origami-sample-prob4 + :caption: Input for ORIGAMI Sample Problem 4 + + =%origami + title= 'PWR 3D deplete model' + prefix= pwr + options{ pitch= 19.816 } + fuelcomp{ + uox(fuel){ enrich=3.5 } + mix(1){ comps[fuel=100] } + } + libs=[ lib1 lib2 lib3 lib4] + libmap=[ + 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 + 2 1 1 1 1 4 1 1 1 1 4 1 1 1 1 2 + 2 1 1 4 4 0 4 4 1 4 0 4 4 1 1 2 + 2 1 4 0 4 4 4 0 4 1 4 4 0 4 1 2 + 2 1 4 4 1 4 1 4 1 1 4 1 4 4 1 2 + 2 4 0 4 4 0 4 1 1 4 0 4 4 0 4 2 + 2 1 4 4 1 4 1 1 1 1 4 1 1 4 1 2 + 2 1 4 0 4 1 1 4 1 1 1 1 4 1 1 2 + 2 1 1 4 1 1 4 0 4 1 1 4 0 4 1 2 + 2 1 4 1 1 4 1 4 1 1 4 1 4 4 1 2 + 2 4 0 4 4 0 4 1 1 4 0 4 4 0 4 2 + 2 1 4 4 1 4 1 4 1 1 4 1 4 4 1 2 + 2 1 4 0 4 4 4 0 4 1 4 4 0 4 1 2 + 2 1 1 4 4 0 4 4 1 4 0 4 4 1 1 2 + 2 1 1 1 1 4 1 1 1 1 4 1 1 1 1 2 + 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 ] + pxy=[ + 0.99 0.98 0.98 0.99 0.99 0.99 0.99 0.99 + 0.99 0.99 0.99 0.99 0.98 0.98 0.97 0.98 + 0.99 0.99 0.99 1.00 1.01 1.02 1.00 1.00 + 1.00 1.01 1.02 1.00 0.99 0.98 0.98 0.98 + 1.00 1.00 1.01 1.03 1.03 0.00 1.03 1.01 + 1.03 1.04 0.00 1.03 1.02 1.00 0.99 0.98 + 1.01 1.01 1.03 0.00 1.04 1.04 1.02 0.00 + 1.03 1.04 1.04 1.04 0.00 1.02 1.00 0.99 + 1.01 1.02 1.02 1.05 0.73 1.04 1.02 1.02 + 1.03 1.03 1.04 0.72 1.04 1.04 1.01 1.00 + 1.02 1.04 0.00 1.05 1.04 0.00 1.03 1.01 + 1.01 1.03 0.00 1.03 1.04 0.00 1.02 1.00 + 1.02 1.03 1.02 1.05 1.04 1.04 1.02 1.01 + 1.01 1.02 1.03 1.02 1.02 1.03 1.01 1.00 + 1.01 1.02 1.04 0.00 1.04 1.02 1.02 1.03 + 1.01 1.01 1.01 1.02 1.03 1.02 1.00 1.00 + 1.00 1.01 1.02 1.03 1.02 1.02 1.03 0.00 + 1.02 1.01 1.01 1.03 0.00 1.02 0.99 0.98 + 1.00 1.01 1.03 1.02 1.02 1.03 1.03 1.03 + 1.01 1.01 1.03 1.02 1.03 1.03 1.00 0.99 + 1.01 1.02 0.00 1.04 1.03 0.00 1.03 1.01 + 1.01 1.02 0.00 1.03 1.03 0.00 1.01 0.98 + 1.00 1.01 1.04 1.04 0.72 1.04 1.03 1.03 + 1.01 1.01 1.02 0.71 1.03 1.02 0.99 0.98 + 1.00 1.00 1.02 0.00 1.04 1.04 1.04 0.00 + 1.02 1.01 1.03 1.03 0.00 1.01 0.98 0.97 + 0.99 0.99 1.01 1.03 1.04 0.00 1.04 1.03 + 1.01 1.02 0.00 1.02 1.01 0.99 0.97 0.97 + 0.99 0.99 0.99 1.00 1.01 1.03 1.01 1.00 + 1.00 1.00 1.01 1.00 0.99 0.97 0.97 0.97 + 1.00 0.99 1.00 1.00 1.01 1.01 1.01 1.00 + 0.99 0.99 0.99 0.99 0.98 0.97 0.97 0.97 ] + + pz=[0.486645842 + 0.510544887 + 0.641121243 + 0.798557507 + 0.931372279 + 1.063949280 + 1.173174524 + 1.178015382 + 1.241701554 + 1.247451593 + 1.203231683 + 1.228462686 + 1.237668911 + 1.221002529 + 1.191997899 + 1.231513011 + 1.222065701 + 1.172711869 + 1.200902470 + 1.164812132 + 1.083204453 + 0.931028309 + 0.810656652 + 0.700324838 + 0.611466339 + 0.516416427 ] + + meshz=[ 0.0 2.0 6.0 10.0 16.5 23.0 37.0 57.0 77.0 97.0 16.0 + 136.0 156.0 176.0 196.0 216.0 236.0 256.0 276.0 296.0 + 316.0 328.5 344.0 352.0 355.5 359.0 366.0 ] + hist[ + cycle{ power=49.395 burn=385 nlib=3 down=52 } + cycle{ power=43.772 burn=360 nlib=2 down=7673 } + ] + end + +.. _fig5-4-3: +.. figure:: figs/ORIGAMI/fig3.png + :align: center + :width: 500 + + MeshView Plot of Axial Burnup Map (MWd/MTU) for Sample Problem 5. + +.. bibliography:: bibs/origami-references.bib diff --git a/ORIGEN-Data.rst b/ORIGEN-Data.rst new file mode 100644 index 0000000..fbfa175 --- /dev/null +++ b/ORIGEN-Data.rst @@ -0,0 +1,3468 @@ +.. _5-2: + +Origen Data Resources +===================== + +*I. C. Gauld, D. Wiarda, M. Pigni, W. Wieslequist* + +ABSTRACT + +.. |nbsp| unicode:: 0xA0 + :trim: + +ORIGEN data resources include nuclear decay data, multigroup neutron +reaction cross sections, neutron-induced fission product yields, and +decay emission data for photons, neutrons, alpha particles and beta +particles. The nuclear decay data are based primarily on ENDF/B-VII.1 +evaluations. The multigroup nuclear reaction cross section libraries now +include evaluations from the JEFF‑3.0/A neutron activation file +containing data for 774 target nuclides, more than +12,000 neutron-induced reactions, and more than 20 different reaction +types below 20 MeV, provided in various energy group structures. +Energy-dependent ENDF/B-VII.0-based fission product yields are available +for 30 fissionable actinides. Gamma-ray and x-ray emission data +libraries are based on ENDF/B-VII.1. The photon libraries contain +discrete photon line energy and intensity data for decay gamma and +x-rays emission for 1,132 radionuclides, prompt and delayed continuum +spectra for spontaneous fission, :math:`\left( \alpha,n \right)` reactions in oxide fuel, +and bremsstrahlung from decay beta (electron and positron) particles slowing +down in either a UO\ :sub:`2` fuel or water matrix. Methods and data +libraries used to calculate the neutron yields and energy spectra for +spontaneous fission, :math:`\left( \alpha,n \right)` reactions, and delayed neutron +emission are adopted from the SOURCES4C code. Capabilities to calculate +the beta and alpha particle emission source and spectra have also been added. + +ACKNOWLEDGMENTS + +Development and testing of ORIGEN data resources, libraries, and methods +have been sponsored by many organizations including the US Nuclear +Regulatory Commission (NRC), the US Department of Energy (DOE), and +nuclear power and research institutions. + +VERSION INFORMATION + +Following is a description of the data resources available for use with +ORIGEN in different SCALE versions. Methodologies and algorithms used in +applying the data are described in the ORIGEN chapter. + +.. centered:: Version 6.2 (2016) + +*Data lead(s):* I. C. Gauld, D. Wiarda, M. Pigni, and W. Wieselquist + +Nuclear data in ORIGEN are unchanged from SCALE 6.1.3 except for the +modification of independent fission yields for thermal fission of +:sup:`235`\ U and :sup:`241`\ Pu and fast fission of :sup:`238`\ U to +provide greater compatibility between the direct and cumulative fission +yields when using the updated decay data from ENDF/B-VII.1. +Additionally, ORIGEN no longer has its own independent source of nuclide +mass and abundance data and now relies on the SCALE Standard Composition +library so that there is consistency in this data across SCALE. D. +Mueller and W. Wieselquist are acknowledged for testing of the new yield +data. W. Wieselquist and S. Hart are acknowledged for the revision of +this chapter. + +.. centered:: Version 6.1.3 (2011) + +*Data lead(s):* I. C. Gauld and D. Wiarda + +SCALE 6.1 represented a complete revision and update of the nuclear data +available in ORIGEN. The following is a summary from the SCALE 6.1 +manual. + + The ORIGEN data libraries include nuclear decay data, neutron + reaction cross sections, neutron induced fission product yields, + delayed gamma-ray emission data, and neutron emission data. The + nuclear decay data libraries have been updated based on ENDF/B-VII + evaluations and expanded to include 903 activation products and + structural materials, 174 actinides, and 1149 fission products. The + cross section libraries have been revised using evaluations from the + JEFF-3.0/A neutron activation file, containing data for 774 target + nuclides, more than 12,000 neutron-induced reactions, and more than + 20 different reaction types below 20 MeV. The JEFF-3.0/A activation + file is processed into several multigroup cross section libraries, + from 44 groups to 238 groups, that can be used to determine the + neutron reaction transition rates in ORIGEN. Energy-dependent + ENDF/B-VII fission product yields are provided for 30 fissionable + actinides. Photon yield data libraries have been updated based on the + most recent ENSDF nuclear structure evaluations processed using the + NuDat program. The photon libraries contain discrete photon line + energy and intensity data for decay gamma and x-rays emission for 982 + radionuclides, prompt and equilibrium continuum fission product + spectra from spontaneous fission, :math:`\left( \alpha,n \right)` reactions in oxide fuel, and + bremsstrahlung from decay beta (negatron and positron) particles + slowing down in either UO\ :sub:`2` fuel or water matrix. Methods and data + libraries used to calculate the neutron yields and energy spectra for + spontaneous fission, :math:`\left( \alpha,n \right)` reactions in any matrix, and delayed + neutron emission are adopted from the SOURCES code. The libraries + used by ORIGEN can be coupled directly with detailed and + problem-dependent physics calculations to obtain self-shielded + problem-dependent cross sections based on the most recent evaluations + of ENDF/B-VII. In addition, the library formats allow multiple sets + of cross section data to be stored on a library to represent the + changes in cross sections during irradiation. + +.. _5-2-1: + +Introduction +------------ + +ORIGEN data resources include nuclear decay data, multigroup neutron +reaction cross sections, neutron-induced fission product yields, and +decay emission data for photons, neutrons, alpha particles and beta +particles. The available resources are summarized in +:numref:`tab-origen-resources` and described in greater detail in the +subsequent sections. The "Unit" column shows the corresponding unit number +for use with FIDO input systems (e.g. with COUPLE). + + +.. table:: Available resources in ORIGEN + :name: tab-origen-resources + :align: center + :class: longtable + + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + | **Description** | **Alias** | **Unit** | **Category** | **Location in SCALE data directory** | + +================================+===========+==========+==============+===========================================+ + | ENDF/B-VII.1 decay data | decay | 27 | Decay | origen_data/origen.rev03.decay.data | + | | | | | | + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + | ENDF/B-VII.0-based | yields | 17 | Yield | origen_data/origen.rev05.yields.data | + | fission yield data | | | | | + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + | JEFF-3.0/A – 44g | n44 | 79 | Reaction | origen.rev03.jeff44g | + | | | | | ev03.jeff44g | + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + | JEFF-3.0/A – 47g | n47 | 22 | Reaction | origen.rev003.jeff47g | + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + | JEFF-3.0/A – 49g | n49 | 77 | Reaction | origen.rev03.jeff49g | + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + | JEFF-3.0/A – 56g | n56 | 75 | Reaction | origen.rev01.jeff56g | + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + | JEFF-3.0/A – 200g | n200 | 78 | Reaction | origen.rev03.jeff200g | + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + | JEFF-3.0/A – 238g | n238 | 80 | Reaction | origen.rev03.jeff238g | + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + | JEFF-3.0/A – 252g | n252 | 74 | Reaction | origen.rev01.jeff252g | + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + | JEFF-3.0/A – 999g | n999 | 76 | Reaction | origen.rev01.jeff999g | + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + | Energy per fission and capture | | | Energy | n/a | + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + | Master photon (x-ray and | | | Emission | origen_data/origen.rev04.mpkkxgam.data | + | gamma) emission data | | | | | + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + | Spontaneous fission and | | | Emission | origen_data/origen.rev00.mpsfangm.data | + | :math:`\left(\alpha,n \right)` | | | | | + | reaction gamma rays | | | | | + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + | Bremsstrahlung from beta | | | Emission | origen_data/origen.rev00.mpbrh2om.data | + | particles slowing down in | | | | | + | water | | | | | + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + | Bremsstrahlung from positrons | | | Emission | origen_data/origen.rev00.mpbrh2op.data | + | slowing down in water | | | | | + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + | Bremsstrahlung from beta | | | Emission | origen_data/origen.rev00.mpbruo2m.data | + | particles slowing down in | | | | igen_data/or | + | UO\ :sub:`2` | | | | | + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + | Bremsstrahlung from positrons | | | Emission | origen_data/origen.rev00.mpbruo2p.data | + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + | Neutron source emission and | | | Emission | origen_data/origen.rev01.alphdec.data | + | alpha decay data | | | | rigen_data/o | + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + | Alpha particle stopping cross | | | Emission | origen_data/origen.rev00.stcoeff.data | + | section expansion coeffcients | | | | | + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + | Target | | | Emission | origen_data/origen.rev00.alphyld.data | + | :math:`\left( \alpha,n\right)` | | | | | + | product excited level | | | | | + | branching data | | | | | + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + | Target | | | Emission | origen_data/origen.rev00.alphaxs.data | + | :math:`\left(\alpha,n \right)` | | | | | + | cross section data | | | | | + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + | Beta source emission data | | | Emission | origen_data/origen.rev00.ensdf95beta.data | + +--------------------------------+-----------+----------+--------------+-------------------------------------------+ + +.. _5-2-2: + +Decay Resource +-------------- + +The nuclear data stored on the decay resource is based on ENDF/B-VII.1 +evaluations :cite:`ENDF-VII.1`, including half-lives, decay modes and branching +fractions, and recoverable energy per disintegration. Decay modes +include beta (:math:`\beta^-`), positron (:math:`\beta^+`) and electron capture (EC), +isomeric transition (IT), alpha (:math:`\alpha`), spontaneous fission (SF), delayed +neutron (:math:`\beta^-\,n`) emission, neutron emission (n), double beta decay +:math:`\left( \beta^- \beta^- \right)`, and decay by beta and alpha emission (:math:`\beta^- \alpha`). +The decay resource also includes radiotoxicity factors based on the radioactivity concentration +guides (RCGs) for air and water as defined in Part 10, Title 20, of the +Code of Federal Regulations (10CFR20) :cite:`10CFR20`. RCGs specify the maximum +permissible concentrations of an isotope in soluble and insoluble forms +for both ingestion and inhalation and for occupational and unrestricted +exposure. The radiotoxicity is calculated as the dilution volume of a +nuclide for cases of direct ingestion or inhalation. The values are +defined to be the smaller (i.e., more toxic) of the values for soluble +and insoluble forms of the isotope. The maximum permissible RCGs for air +and water are the public exposure limits for adult ingestion and +inhalation dose coefficients of ICRP Publication 72 :cite:`ICRP72`. External +exposure dose coefficients for noble gases were obtained from the +Environmental Protection Agency (EPA) Federal Guidance Report 12 :cite:`EPA1993`. +Recoverable energy includes the delayed energy from all electron-related +radiations (e.g., :math:`\beta^-`, :math:`\beta^+`, Auger electrons), all gamma rays, x-rays, +annihilation radiations, and the average energy of all heavy charged +particles and delayed neutrons. The average alpha energy includes the +energy of the recoil nucleus. A part of the recoverable energy per decay +not included in the ENDF/B-VII.1 values is the additional contribution +from spontaneous fission. This energy was calculated as the product of +the spontaneous fission branching fraction and recoverable energy per +fission using a value of 200 MeV per fission and then added to the +ENDF/B-VII.1 recoverable Q energy. A value of 12.56 MeV gamma energy per +fission was used in computing the fraction of recoverable spontaneous +fission energy from gamma rays. External Bremsstrahlung radiation is +**not** included in the Q-value since the Bremsstrahlung spectrum +depends on electron interactions with the medium that contains the decay +nuclide. The energy from capture gamma rays accompanying :math:`\left( \alpha,n \right)` reactions +is not included either since it also depends on the medium. + +Appendix A describes the decay resource file format. It is important to +note that the decay resource not only defines fundamental decay data, +but also the complete ORIGEN nuclide set, including the "duplicates" of +nuclides across sublibraries. For example, a version of :sup:`155`\ Gd +is contained in both the light nuclide/activation product and fission +product sublibraries. Appendix D includes the full list of the nuclides +on the ORIGEN decay library "end7dec" created by COUPLE based on the +current decay resource, including duplicates. Appendix E contains a list +of the fundamental decay data only, without duplicates. To consider a +different set of nuclides in an ORIGEN calculation, the current process +is to alter the decay resource and then regenerate the "end7dec" decay +library with COUPLE. By default, all subsequent libraries created from +COUPLE using problem-dependent reaction transitions are based on the +"end7dec" decay library and will therefore include the modified nuclide +set. + +.. _5-2-3: + +Neutron Reaction Resource +------------------------- + +The neutron cross sections defining the nuclear reaction transmutation +rates use a comprehensive collection of nuclear data evaluations +compiled from the JEFF-3.0/A neutron activation files :cite:`JEFFDOC-982`. +The JEFF-3.0/A files contain continuous energy neutron data for 774 target +nuclei, including ground and metastable excited states, and +12,617 neutron-induced reactions below 20 MeV. The JEFF-3.0/A cross +section data are developed directly from the European Activation File +(EAF‑2003) :cite:`FUS-486` formatted as standard ENDF-6 format data. JEFF-3.0/A +cross sections are stored using File 3, multiplicities on File 10, and +isomeric branching to different metastable levels using File 9. +The evaluations include many reactions that may be important for +modeling fast fission and other high-energy systems. Neutron reactions +are available for 23 reaction types, including +:math:`\left(n,n' \right)`, :math:`\left(n,2n \right)`, +:math:`\left(n,3n \right)`, :math:`\left(n,f \right)`, +:math:`\left(n,n' \alpha \right)`, :math:`\left(n,2n\alpha\right)`, +:math:`\left(n,3n\alpha \right)`, :math:`\left(n,n'p \right)`, +:math:`\left(n,n2\alpha \right)`, :math:`\left(n,n'd \right)`, +:math:`\left(n,n't \right)`, :math:`\left(n,n'{}^3 He \right)`, +:math:`\left(n,4n \right)`, :math:`\left(n,2np \right)`, +:math:`\left(n,\gamma \right)`, :math:`\left(n,p \right)`, +:math:`\left(n,d \right)`, :math:`\left(n,t \right)`, +:math:`\left(n,{}^3\ He \right)`, :math:`\left(n,\alpha \right)`, +:math:`\left(n,2\alpha \right)`, :math:`\left(n,2p \right)`, +and :math:`\left(n,p\alpha \right)`. + +The JEFF-3.0/A evaluations also include extensive compilations of +energy-dependent branching fractions that define neutron reaction +transitions to ground and metastable energy states. Energy-dependent +branching is fully implemented in the ORIGEN cross section libraries. +Implementation of the JEFF-3.0/A cross sections as ORIGEN multigroup +data was accomplished by processing and collapsing the JEFF-3.0/A +pointwise cross sections into a standard multigroup AMPX format using +ENDF data-processing modules of the AMPX :cite:`DuGr2002` cross section +processing code system. The collapse is performed using a thermal +Maxwellian–1/E–fission–1/E weighting spectrum (see +:numref:`fig-origen-collapse-flux`) to provide infinite dilution multigroup +cross sections. + +.. _fig-origen-collapse-flux: +.. figure:: figs/ORIGENdata/fig1.png + :align: center + + Pointwise flux spectrum used to generate collapsed cross section libraries. + + +Neutron reactions with transitions to multiple states of the daughter +product are represented using separate cross sections to the ground and +metastable states. A special reaction identifier (MT') is defined for +this implementation of metastable transitions as + +.. math:: + \text{MT}' = \text{MT}*10000 + 100*\text{LP} + \text{LT} + :label: eq-MT-special + +where MT is the reaction identifier, LP is the product metastable state, +and LT is the target metastable state. Using the +:sup:`187`\ W(n,3n)\ :sup:`185`\ W cross section (MT=17) as an example, +the reaction identifier 170000 defines the partial cross section to the +ground state of :sup:`185`\ W, and 170100 defines the cross section to +metastable :sup:`185m`\ W. + +Cross section data from the JEFF-3.0/A neutron activation file are first +converted to point-wise cross section data, are Doppler broadened to +900K, and then they are collapsed to different group structures. The +following group strucures are available in SCALE: + + * 238-group neutron (thermal applications), + * 252-group neutron (thermal applications), + * 56-group neutron (thermal applications), + * 200-group neutron (fast applications and shielding), + * 47-group neutron (applications using the BUGLE shielding transport + library), + * 49-group neutron (collapsed version of 238 groups), + * 44-group neutron (collapsed version of 238 groups), and + * 999-group neutron (multipurpose). + +Several minor modifications were made to the JEFF-3.0/A data: + + * The :sup:`239`\ Np radiative neutron capture cross section was + replaced with data from ENDF/B-VII.0. Neutron capture using + JEFF-3.0/A cross sections was significantly larger than ENDF/B-VII.0 + due to differences in the resonance cross section region. Although + experimental resonance parameters are not available for + :sup:`239`\ Np, comparisons of :sup:`240`\ Pu production during + irradiation :cite:`Gump1954` obtained using the two evaluations showed that + better agreement with the experiment was obtained using the + ENDF/B-VII.0 evaluation. + + * The :math:`{}^{241} Am (n,\gamma)` branching fraction to the :sup:`242`\ Am + ground and metastable states was replaced by the evaluation from + ENDF/B-VII.0 to yield better agreement with the results of + destructive radiochemical assay measurements of irradiated fuels. The + branching fraction of :sup:`241`\ Am to :sup:`242m`\ Am for thermal + neutron capture changed from 8.2% in JEFF-3.0/A to 10.0% in + ENDF/B-VII.0. + +The cross section library header record information and a complete list +of nuclides in JEFF-3.0/A libraries developed for ORIGEN are provided in +Appendix E. + +Because JEFF-3.0/A-based libraries are formatted as standard AMPX +working libraries, they can be accessed and/or manipulated using +standard AMPX utility modules in SCALE. For example, multigroup cross +sections may be listed using the PALEALE module. Additionally, the data +may be visualized using the Fulcrum user interface. Cross section plots +of the 238-group JEFF‑3.0/A library are illustrated in +:numref:`fig-origen-jeff-w187` for :math:`(n,\gamma)`, :math:`(n,\alpha)`, +(n,2n), and (n,3n) cross sections to the ground and metastable states. + +Before the cross sections in ORIGEN can be used, they must be collapsed +with a user-defined multigroup flux to a one-group cross section and +added to the ORIGEN binary library (see the COUPLE input description). + +.. _fig-origen-jeff-w187: +.. figure:: figs/ORIGENdata/fig2.png + :align: center + + 238-group JEFF-3.0/A cross sections for :sup:187:\ W. + +.. _5-2-4: + +Fission Yield Resource +---------------------- + +The fission-yield resource contains the energy-dependent direct yields +of each fission product for 30 fissionable actinides, including +:sup:`227,228,232`\ Th, :sup:`231`\ Pa, :sup:`232--238`\ U, +:sup:`238-242`\ Pu, :sup:`241,242m,243`\ Am, :sup:`237,238`\ Np, +:sup:`242-246,248`\ Cm, :sup:`249,252`\ Cf, and :sup:`254`\ Es. +Independent (direct) fission product yields are stored as atom percent +per fission, and except for :sup:`235`\ U(thermal), :sup:`238`\ U(fast), +and :sup:`241`\ Pu(thermal), they are obtained from ENDF/B‑VII.0 :cite:`ENDF-VII.0` +File 8 and MT=454. + +Revised independent fission product yields for :sup:`235`\ U(thermal), +:sup:`238`\ U(fast), and :sup:`241`\ Pu(thermal) were adopted to address +inconsistencies between the direct and cumulative fission yields in +ENDF/B-VII.0 caused by the use of updated nuclear decay schemes in the +decay sublibrary :cite:`PiFrGa2015,FrWePi2015`. Namely, recent changes in the +decay data, particularly the delayed neutron branching fractions, result +in calculated fission product concentrations that do not agree with the +cumulative fission yields in the ENDF/B-VII.0 library. These issues were +particularly evident for the three cited isotopes because their +fissioning systems result in a preferential formation of fragments that +are sensitive to the changes in the decay data. For example, a study on +:sup:`239`\ Pu(thermal) showed negligible differences between cumulative +yields calculated (using the recent decay data sublibrary) and the +cumulative yields in ENDF/B-VII.0. Energy-dependent product yields are +available for thermal, fast, and high-energy incident neutron energies. +For fast fission, the value of the energy of incident neutron was +modified from the value of 500 keV tabulated in ENDF/B-VII.0 to more +accurately represent the relationship between the energy distribution of +the neutrons causing fission and the and the fission neutron spectrum +energy. For this implementation of the yield data, the effective +incident neutron energy for fast fission was adjusted from 500 keV to +2.0 MeV to better reflect the average fission energy of most nuclides. +The neutron energies for thermal fission (0.0253 eV) and high energy +fission (14 MeV) are unchanged. + +The fission product yields also include cumulative ternary yields from +the JEF-2.2 fission yield library :cite:`JEF-2.2` for :sup:`3`\ H and +:sup:`4`\ He. The nuclide :sup:`3`\ He was also added to the fission +product library since it is a decay product of tritium. + +Note that inclusion of fission yields for each actinide in an ORIGEN +library can be controlled by the user through COUPLE. Actinides not +assigned with explicit yields do not produce fission products during +fission. + +.. table:: Fissionable isotopes having explicit fission yields + :name: tab-origen-fiss-isotopes + :align: center + :class: longtable + + +----------------------+-------------------------------------------+ + | **Nuclide** | **Neutron-induced fission energies** | + | | [#fiss-energy]_ | + +======================+======================+======+=============+ + | :sup:`227`\ Th | Thermal | | | + +----------------------+----------------------+------+-------------+ + | :sup:`229`\ Th | Thermal | | | + +----------------------+----------------------+------+-------------+ + | :sup:`232`\ Th | | Fast | High energy | + +----------------------+----------------------+------+-------------+ + | :sup:`231`\ Pa | | Fast | | + +----------------------+----------------------+------+-------------+ + | :sup:`232`\ U | Thermal | | | + +----------------------+----------------------+------+-------------+ + | :sup:`233`\ U | Thermal | | | + +----------------------+----------------------+------+-------------+ + | :sup:`234`\ U | | Fast | High energy | + +----------------------+----------------------+------+-------------+ + | :sup:`235`\ U | Thermal | Fast | High energy | + +----------------------+----------------------+------+-------------+ + | :sup:`236`\ U | | Fast | High energy | + +----------------------+----------------------+------+-------------+ + | :sup:`237`\ U | | Fast | | + +----------------------+----------------------+------+-------------+ + | :sup:`238`\ U | | Fast | High energy | + +----------------------+----------------------+------+-------------+ + | :sup:`237`\ Np | Thermal | Fast | High energy | + +----------------------+----------------------+------+-------------+ + | :sup:`238`\ Np | | Fast | | + +----------------------+----------------------+------+-------------+ + | :sup:`238`\ Pu | | Fast | | + +----------------------+----------------------+------+-------------+ + | :sup:`239`\ Pu | Thermal | Fast | High energy | + +----------------------+----------------------+------+-------------+ + | :sup:`240`\ Pu | Thermal | Fast | High energy | + +----------------------+----------------------+------+-------------+ + | :sup:`241`\ Pu | Thermal | Fast | | + +----------------------+----------------------+------+-------------+ + | :sup:`242`\ Pu | Thermal | Fast | High energy | + +----------------------+----------------------+------+-------------+ + | :sup:`241`\ Am | Thermal | Fast | High energy | + +----------------------+----------------------+------+-------------+ + | :sup:`242\ m`\ Am | Thermal | | | + +----------------------+----------------------+------+-------------+ + | :sup:`243`\ Am | | Fast | | + +----------------------+----------------------+------+-------------+ + | :sup:`242`\ Cm | | Fast | | + +----------------------+----------------------+------+-------------+ + | :sup:`243`\ Cm | Thermal | Fast | | + +----------------------+----------------------+------+-------------+ + | :sup:`244`\ Cm | | Fast | | + +----------------------+----------------------+------+-------------+ + | :sup:`245`\ Cm | Thermal | | | + +----------------------+----------------------+------+-------------+ + | :sup:`246`\ Cm | | Fast | | + +----------------------+----------------------+------+-------------+ + | :sup:`248`\ Cm | | Fast | | + +----------------------+----------------------+------+-------------+ + | :sup:`249`\ Cf | Thermal | | | + +----------------------+----------------------+------+-------------+ + | :sup:`251`\ Cf | Thermal | | | + +----------------------+----------------------+------+-------------+ + | :sup:`254`\ Es | Thermal | | | + +----------------------+----------------------+------+-------------+ + +.. [#fiss-energy] Neutron energy causing fission + +.. _5-2-5: + +Energy Resource +=============== + +The energy resource is a set of data defined internally to ORIGEN to +compute the total power during irradiation if the flux is known, or the +total flux if the power is known. The data include the energy +contributed by fission and capture. The recoverable energy values taken +primarily from ENDF/B evaluations are listed in +:numref:`tab-origen-recoverable-en-fc` and :numref:`tab-origen-recoverable-en-cap`. +The recoverable energy for fission and neutron capture for nuclides not listed in +the tables are assumed to be 200 MeV and 5.0 MeV, respectively. + +.. table:: Recoverable energy (MeV) values for actinides + :name: tab-origen-recoverable-en-fc + :align: center + + =============== =========== =========== + **Nuclide** **Fission** **Capture** + :sup:`230`\ Th 190.00 5.010 + :sup:`232`\ Th 189.21 4.786 + :sup:`233`\ Th 190.00 6.080 + :sup:`231`\ Pa 190.00 5.660 + :sup:`233`\ Pa 189.10 5.197 + :sup:`232`\ U 200.00 5.930 + :sup:`233`\ U 191.29 6.841 + :sup:`234`\ U 190.30 5.297 + :sup:`235`\ U 194.02 6.545 + :sup:`236`\ U 192.80 5.124 + :sup:`238`\ U 198.12 4.804 + :sup:`237`\ Np 195.10 5.490 + :sup:`239`\ Np 200.00 4.970 + :sup:`238`\ Pu 197.80 5.550 + :sup:`239`\ Pu 200.05 6.533 + :sup:`240`\ Pu 199.79 5.241 + :sup:`241`\ Pu 202.22 6.301 + :sup:`242`\ Pu 200.62 5.071 + :sup:`243`\ Pu 200.00 6.020 + :sup:`241`\ Am 202.30 5.529 + :sup:`242m`\ Am 202.29 6.426 + :sup:`243`\ Am 202.10 5.363 + :sup:`244`\ Cm 200.00 6.451 + :sup:`245`\ Cm 200.00 6.110 + =============== =========== =========== + + +.. table:: Recoverable energy (MeV) values for activation and fission products + :name: tab-origen-recoverable-en-cap + :align: center + + =============== =========== + **Nuclide** **Capture** + :sup:`1`\ H 2.225 + :sup:`10`\ B 2.790 + :sup:`16`\ O 4.143 + :sup:`56`\ Fe 7.600 + :sup:`58`\ Ni 9.020 + :sup:`90`\ Zr 7.203 + :sup:`91`\ Zr 8.635 + :sup:`92`\ Zr 6.758 + :sup:`96`\ Zr 5.571 + :sup:`95`\ Mo 9.154 + :sup:`95`\ Tc 7.710 + :sup:`101`\ Ru 9.216 + :sup:`103`\ Rh 6.999 + :sup:`105`\ Rh 7.094 + :sup:`109`\ Ag 6.825 + :sup:`131`\ Xe 8.936 + :sup:`135`\ Xe 7.880 + :sup:`133`\ Cs 6.704 + :sup:`134`\ Cs 6.550 + :sup:`143`\ Nd 7.817 + :sup:`145`\ Nd 7.565 + :sup:`147`\ Pm 5.900 + :sup:`148`\ Pm 7.266 + :sup:`148m`\ Pm 7.266 + :sup:`147`\ Sm 8.140 + :sup:`149`\ Sm 7.982 + :sup:`150`\ Sm 5.596 + :sup:`151`\ Sm 8.258 + :sup:`152`\ Sm 5.867 + :sup:`153`\ Eu 6.444 + :sup:`154`\ Eu 8.167 + :sup:`155`\ Eu 6.490 + =============== =========== + +.. _5-2-6: + +Emission Resources +------------------ + +The two main groups for emission resources are the photon (gamma) +resource, which includes beta particle emission data, and the neutron +resource, which includes alpha emission data. + +.. _5-2-6-1: + +Gamma Emission +~~~~~~~~~~~~~~ + +The resources for gamma emission are stored as separate files (see +:numref:`tab-origen-photon-files`) containing the photon data associated with +different modes of decay or photon production. The photon data sets include decay +gamma and x-ray line-energy data, gamma rays accompanying spontaneous fission, +gamma rays accompanying :math:`\left( \alpha,n \right)` reactions in oxide fuels, and +Bremsstrahlung spectra from decay electrons/positrons slowing down in +UO\ :sub:`2` and water. The photon energy spectra can be generated in +any energy group structure for all activation products, actinides, and +fission product nuclides with photon yield data. + +.. table:: Photon data files + :name: tab-origen-photon-files + :align: center + + ============== ================================================================== + **File name** **Description** + ============== ================================================================== + MPDKXGAM x-ray and gamma emissions line data + MPSFANGM spontaneous fission and :math:`\left( \alpha,n \right)` reactions + MPBRH2OM bremsstrahlung from beta particles slowing down in water + MPBRH2OP bremsstrahlung from positrons slowing down in water + MPBRUO2M bremsstrahlung from beta particles slowing down in UO\ :sub:`2` + MPBRUO2P bremsstrahlung from positrons slowing down in UO\ :sub:`2` + ============== ================================================================== + + +All photon data sets are constructed with the same format (see Appendix +C). The majority of the photon emissions are discrete energy lines. +Photon continuum data, used to represent Bremstrahlung and some other +gamma-ray emission spectra, are stored at discrete energies and +approximately expanded to a continuum, as needed. +Gamma and x-ray yields are directly from ENDF/B-VII.1 decay files +containing spectral data for decay transitions of 1,132 nuclides. A +separate file contains emission spectra for gamma rays accompanying +spontaneous fission and for gamma rays accompanying :math:`\left( \alpha, n \right)` +reactions in oxide fuels :cite:`CrHaGr1979`. The spontaneous fission spectra +combine prompt and equilibrium fission product gamma-ray components. The prompt +spectrum is similar to that of :sup:`235`\ U, and the delayed fission product gamma +intensity at equilibrium is about 0.75 of that from the prompt fission +gamma rays. Based on measured prompt fission gamma spectra from +:sup:`235`\ U, spontaneous fission spectra are computed from the +following approximation: + +.. math:: + N\left(E \right) \cong + \begin{cases} + 11.5 & 0.1 \leq\ E \leq 0.6\ MeV \\ + 35.4 e^{-1.78 E} & 0.6 \leq E < 1.5\ MeV \\ + 12.6 e^{-1.09 E} & 1.5 \leq E \leq 10.5\ MeV \\ + 0 & \text{otherwise.} + \end{cases} + :label: eq-origen-sfn-spec + + +where + + N(E) = number of photons per unit energy per fission (photons/MeV per + fission) at energy E, where E is the photon energy (MeV). + + +For medical and industrial spontaneous fission source applications, a +more accurate simulation of the source may be desirable. Work has been +performed on :sup:`252`\ Cf source modeling to explicity represent the +fission product generation from fission and the delayed gamma emission. +In this application, the equilibrium spontaneous fission gamma spectrum +was replaced with an evaluation of the :sup:`252`\ Cf prompt gamma +spectrum, and the delayed fission product gamma rays was modeled +explicitly in ORIGEN by generating the time-dependent fission products +using :sup:`252`\ Cf spontaneous fission product yields from +ENDF/B-VII.0 :cite:`FoGaWa2011`. This was performed by adding decay transitions +to the ORIGEN library from the actinides to the fission products. +The spectrum of gamma rays accompanying :math:`\left( \alpha,n \right)` +reactions is based on reaction data for alpha interactions on :sup:`18`\ O +and from studies for :sup:`238`\ PuO\ :sub:`2` systems. The spectrum is +computed from the approximation: + +.. math:: + :label: eq5-2-2 + + N\left( E \right) \cong 2.13 \cdot 10^{-8}\ e^{- 1.38E} + +where + + N(E) = number of photons per unit energy per alpha decay (photons/MeV + per disintegration) at energy E (MeV). + +The photon yields in this data set are continuum spectra represented by +discrete lines with an energy width of 500 keV and range from 250 keV to +10.25 MeV. + +Two photon data sets contain bremsstrahlung spectra from decay electrons +and positrons slowing down in a UO\ :sub:`2` fuel matrix. The yields are +in the form of continuum spectra represented in the data sets as +discrete lines using up to 70 quasi-logarithmic spaced energy points +over the energy range between 0 and 13.5 MeV. Two libraries contain +bremsstrahlung spectra from decay electrons and positrons slowing down +in water. Bremsstrahlung spectra were calculated using a computer +program developed by Dillman *et al.* :cite:`DiSnFo1973` using beta spectra +derived from **Evaluated Nuclear Structure Data Files (**\ ENSDF) decay data +with a computer program written by Gove and Martin :cite:`GoMa1971`. + +.. _5-2-6-2: + +Neutron Emission +~~~~~~~~~~~~~~~~ + +There are four neutron emission resources used by ORIGEN to calculate +the neutron intensities and spectra: (1) neutron decay data, (2) an +\alpha-particle stopping power, (3) a target :math:`\left( \alpha,n \right)` +cross section, and (4) a target :math:`\left( \alpha,n \right)` product level +branching. All of the neutron data are stored in a text format with names and +descriptions given in :numref:`tab-origen-neutron-libs`. The neutron decay data +contain basic decay information for decay processes that lead to direct and +indirect emission of neutrons, including spontaneous fission branching +fractions, alpha decay branching fractions, delayed neutron branching fractions, +alpha-particle decay energies, Watt fission spectrum parameters, and delayed +neutron spectra. The stopping cross sections, :math:`\left( \alpha,n \right)` +target cross sections, and product-level branching data are used in calculating +the neutron yield and spectra from :math:`\left( \alpha,n \right)` reactions. + +The neutron data were obtained directly from the updated SOURCES-4B code +package. The sources of the neutron data are described by Shores :cite:`Shores2000`. +An update was made to correct an error in the :sup:`250`\ Cf spontaneous fission +neutron branching fraction in the neutron source decay data distributed with the +SOURCES code. The :sup:`250`\ Cf branching fraction was incorrectly assigned the +value from :sup:`252`\ Cf of :math:`3.092 \cdot 10^{-2}`. A corrected value of +:math:`7.700 \cdot 10^{-4}` from ENDF/B‑VII.1 is used. + +.. table:: Neutron source data libraries + :name: tab-origen-neutron-libs + :align: center + + ============= ==================================================================== + **File name** **Description** + ALPHDEC Neutron source decay data + STCOEFF Stopping cross section expansion coefficients + ALPHYLD Target :math:`\left( \alpha,n \right)` product level branching + ALPHAXS Target :math:`\left( \alpha,n \right)` cross section + ============= ==================================================================== + +The neutron source decay contains spontaneous fission data for the +49 actinides listed in :numref:`origen-nucl-sfn`. These data include the +spontaneous fission branching fraction, the number of neutrons per fission +(:math:`nu`), and the watt spectrum parameters for spontaneous fission. The +spontaneous fission neutron energy spectrum is approximated using spectral +parameters A and B, such that + +.. math:: + N\left( E \right) \cong \text{C\ }e^{ - \frac{E}{A}}\sinh \sqrt{\text{BE}} + :label: eq-watt-spec + + +where *E* is the neutron energy and *C* is a normalization constant. + + +.. table:: Nuclides with spontaneous fission data and spectral parameters + :name: origen-nucl-sfn + :align: center + + ================ ================ ================ ================ ================ + :sup:`230`\ Th :sup:`239`\ U :sup:`240`\ Pu :sup:`244`\ Am :sup:`250`\ Cm + :sup:`232`\ Th :sup:`236`\ Np :sup:`241`\ Pu :sup:`244m`\ Am :sup:`249`\ Bk + :sup:`231`\ Pa :sup:`236m`\ Np :sup:`242`\ Pu :sup:`240`\ Cm :sup:`248`\ Cf + :sup:`232`\ U :sup:`237`\ Np :sup:`243`\ Pu :sup:`241`\ Cm :sup:`250`\ Cf + :sup:`233`\ U :sup:`238`\ Np :sup:`244`\ Pu :sup:`242`\ Cm :sup:`252`\ Cf + :sup:`234`\ U :sup:`239`\ Np :sup:`240`\ Am :sup:`243`\ Cm :sup:`254`\ Cf + :sup:`235`\ U :sup:`236`\ Pu :sup:`241`\ Am :sup:`244`\ Cm :sup:`253`\ Es + :sup:`236`\ U :sup:`237`\ Pu :sup:`242`\ Am :sup:`245`\ Cm :sup:`254m`\ Es + :sup:`237`\ U :sup:`238`\ Pu :sup:`242m`\ Am :sup:`246`\ Cm :sup:`255`\ Es + :sup:`238`\ U :sup:`239`\ Pu :sup:`243`\ Am :sup:`248`\ Cm + ================ ================ ================ ================ ================ + + +Delayed neutron branching fractions and neutron spectra for 105 fission +products are listed in :numref:`tab-origen-nucl-dn`. The delayed neutron +spectra are tabulated in discrete 10 keV bins from 50 keV to about 2 MeV. + + +.. table:: Nuclides with delayed neutron emission spectral data + :name: tab-origen-nucl-dn + :align: center + + ============== ============== =============== =============== =============== + :sup:`79`\ Zn :sup:`89`\ Br :sup:`97`\ Y :sup:`128`\ In :sup:`41`\ I + :sup:`79`\ Ga :sup:`90`\ Br :sup:`97m`\ Y :sup:`129`\ In :sup:`42`\ I + :sup:`80`\ Ga :sup:`91`\ Br :sup:`98`\ Y :sup:`129m`\ In :sup:`43`\ I + :sup:`81`\ Ga :sup:`92`\ Br :sup:`98m`\ Y :sup:`130`\ In :sup:`141`\ Xe + :sup:`82`\ Ga :sup:`93`\ Br :sup:`99`\ Y :sup:`131`\ In :sup:`142`\ Xe + :sup:`83`\ Ga :sup:`92`\ Kr :sup:`100`\ Y :sup:`132`\ In :sup:`143`\ Xe + :sup:`83`\ Ge :sup:`93`\ Kr :sup:`104`\ Zr :sup:`133`\ Sn :sup:`144`\ Xe + :sup:`84`\ Ge :sup:`94`\ Kr :sup:`105`\ Zr :sup:`134`\ Sn :sup:`141`\ Cs + :sup:`85`\ Ge :sup:`95`\ Kr :sup:`103`\ Nb :sup:`135`\ Sn :sup:`142`\ Cs + :sup:`86`\ Ge :sup:`92`\ Rb :sup:`104`\ Nb :sup:`134m`\ Sb :sup:`143`\ Cs + :sup:`84`\ As :sup:`93`\ Rb :sup:`105`\ Nb :sup:`135`\ Sb :sup:`144`\ Cs + :sup:`85`\ As :sup:`94`\ Rb :sup:`106`\ Nb :sup:`136`\ Sb :sup:`145`\ Cs + :sup:`86`\ As :sup:`95`\ Rb :sup:`109`\ Mo :sup:`137`\ Sb :sup:`146`\ Cs + :sup:`87`\ As :sup:`96`\ Rb :sup:`110`\ Mo :sup:`136`\ Te :sup:`147`\ Cs + :sup:`87`\ Se :sup:`97`\ Rb :sup:`109`\ Tc :sup:`137`\ Te :sup:`147`\ Ba + :sup:`88`\ Se :sup:`98`\ Rb :sup:`110`\ Tc :sup:`138`\ Te :sup:`148`\ Ba + :sup:`89`\ Se :sup:`99`\ Rb :sup:`122`\ Ag :sup:`139`\ Te :sup:`149`\ Ba + :sup:`90`\ Se :sup:`97`\ Sr :sup:`123`\ Ag :sup:`137`\ I :sup:`150`\ Ba + :sup:`91`\ Se :sup:`98`\ Sr :sup:`128`\ Cd :sup:`138`\ I :sup:`147`\ La + :sup:`87`\ Br :sup:`99`\ Sr :sup:`127`\ In :sup:`139`\ I :sup:`149`\ La + :sup:`88`\ Br :sup:`100`\ Sr :sup:`127m`\ In :sup:`140`\ I :sup:`150`\ La + ============== ============== =============== =============== =============== + + +Neutron yields from \alpha-particle interaction are available for 19 :math:`\left( \alpha,n \right)` +target nuclides: :sup:`7`\ Li, :sup:`9`\ Be, :sup:`10`\ B, :sup:`11`\ B, +:sup:`13`\ C, :sup:`14`\ N, :sup:`17`\ O, :sup:`18`\ O, :sup:`19`\ F, +:sup:`21`\ Ne, :sup:`22`\ Ne, :sup:`23`\ Na, :sup:`25`\ Mg, +:sup:`26`\ Mg, :sup:`27`\ Al, :sup:`29`\ Si, :sup:`30`\ Si, +:sup:`31`\ P, and :sup:`37`\ Cl. The neutron decay data contain discrete +alpha-particle energies and branching fractions for 89 actinides and +7 fission products listed in :numref:`tab-origen-nucl-alpha`. The sources of +the level branching fraction data and the :math:`\left( \alpha,n \right)` +cross section data are listed in :numref:`tab-origen-an-cxs`. The stopping +cross sections and :math:`\left( \alpha,n \right)` target cross section +and product level branching libraries are used in calculating the neutron yield +and spectra from Ziegler :cite:`Zieg1977` for all elements with Z < 93, and +from Wilson :cite:`WPS1983` for all elements \geq 93. + + +.. table:: Nuclides with :math:`\alpha`-particle emission data for neutron yield calculations + :name: tab-origen-nucl-alpha + :align: center + + ================ ================ ================ ================= ================= + :sup:`142`\ Ce :sup:`216`\ Po :sup:`226`\ Ac :sup:`237`\ Np :sup:`245`\ Cm + :sup:`144`\ Nd :sup:`218`\ Po :sup:`227`\ Ac :sup:`235`\ Pu :sup:`246`\ Cm + :sup:`146`\ Sm :sup:`215`\ At :sup:`226`\ Th :sup:`236`\ Pu :sup:`247`\ Cm + :sup:`147`\ Sm :sup:`217`\ At :sup:`227`\ Th :sup:`237`\ Pu :sup:`248`\ Cm + :sup:`148`\ Sm :sup:`218`\ At :sup:`228`\ Th :sup:`238`\ Pu :sup:`249`\ Bk + :sup:`149`\ Sm :sup:`219`\ At :sup:`229`\ Th :sup:`239`\ Pu :sup:`248`\ Cf + :sup:`152`\ Gd :sup:`217`\ Rn :sup:`230`\ Th :sup:`240`\ Pu :sup:`249`\ Cf + :sup:`210`\ Pb :sup:`218`\ Rn :sup:`232`\ Th :sup:`241`\ Pu :sup:`250`\ Cf + :sup:`210`\ Bi :sup:`219`\ Rn :sup:`230`\ Pa :sup:`242`\ Pu :sup:`251`\ Cf + :sup:`211`\ Bi :sup:`220`\ Rn :sup:`231`\ Pa :sup:`244`\ Pu :sup:`252`\ Cf + :sup:`212`\ Bi :sup:`222`\ Rn :sup:`230`\ U :sup:`240`\ Am :sup:`253`\ Cf + :sup:`213`\ Bi :sup:`221`\ Fr :sup:`231`\ U :sup:`241`\ Am :sup:`254`\ Cf + :sup:`214`\ Bi :sup:`222`\ Fr :sup:`232`\ U :sup:`242m`\ Am :sup:`253`\ Es + :sup:`210`\ Po :sup:`223`\ Fr :sup:`233`\ U :sup:`243`\ Am :sup:`254`\ Es + :sup:`211`\ Po :sup:`222`\ Ra :sup:`234`\ U :sup:`240`\ Cm :sup:`254m`\ Es + :sup:`212`\ Po :sup:`223`\ Ra :sup:`235`\ U :sup:`241`\ Cm :sup:`255`\ Es + :sup:`213`\ Po :sup:`224`\ Ra :sup:`236`\ U :sup:`242`\ Cm :sup:`254`\ Fm + :sup:`214`\ Po :sup:`226`\ Ra :sup:`238`\ U :sup:`243`\ Cm :sup:`255`\ Fm + :sup:`215`\ Po :sup:`225`\ Ac :sup:`235`\ Np :sup:`244`\ Cm :sup:`256`\ Fm + |nbsp| |nbsp| |nbsp| |nbsp| :sup:`257`\ Fm + ================ ================ ================ ================= ================= + + +.. table:: Target :math:`(\alpha,n)` cross section and branching level isotopes and sources + :name: tab-origen-an-cxs + :align: center + :class: longtable + + +------------------+----------+--------------------------+------------------------+ + | **Isotope** | **ZAID** | **Level branching** | **Cross section data** | + | | | **fraction source data** | | + +==================+==========+==========================+========================+ + | :sup:`7`\ Li | 30070 | GNASH | Gibbons and Macklin | + | | | | :cite:`GiMa1959` | + +------------------+----------+--------------------------+------------------------+ + | :sup:`9`\ Be | 40090 | Geiger and Van der | Geiger and Van der | + | | | Zwain :cite:`GeZw1975` | Zwain :cite:`GeZw1975` | + +------------------+----------+--------------------------+------------------------+ + | :sup:`10`\ B | 50010 | GNASH | Bair *et al.* | + | | | | :cite:`BaCa1979` | + +------------------+----------+--------------------------+------------------------+ + | :sup:`11`\ B | 50110 | GNASH | Bair *et al.* | + | | | | :cite:`BaCa1979` | + +------------------+----------+--------------------------+------------------------+ + | :sup:`13`\ C | 60130 | GNASH\ *a* | Bair and Haas | + | | | | :cite:`BaHa1973` | + +------------------+----------+--------------------------+------------------------+ + | :sup:`14`\ N | 70140 | N/A | GNASH | + +------------------+----------+--------------------------+------------------------+ + | :sup:`17`\ O | 80170 | Lessor and Schenter | Perry and Wilson | + | | | :cite:`LeSc1971` | :cite:`PeWi1981` | + +------------------+----------+--------------------------+------------------------+ + | :sup:`18`\ O | 80180 | Lesser and Schenter | Perry and Wilson | + | | | :cite:`LeSc1971` | :cite:`PeWi1981` | + +------------------+----------+--------------------------+------------------------+ + | :sup:`19`\ F | 90190 | Lesser and Schenter | Balakrishnan *et al.* | + | | | :cite:`LeSc1971` | :cite:`BaKaMe1978` | + +------------------+----------+--------------------------+------------------------+ + | :sup:`21`\ Ne | 100210 | N/A | GNASH | + +------------------+----------+--------------------------+------------------------+ + | :sup:`22`\ Ne | 100220 | N/A | GNASH | + +------------------+----------+--------------------------+------------------------+ + | :sup:`23`\ Na | 110230 | GNASH | GNASH\ *a* | + +------------------+----------+--------------------------+------------------------+ + | :sup:`25`\ Mg | 120250 | GNASH | GNASH | + +------------------+----------+--------------------------+------------------------+ + | :sup:`26`\ Mg | 120260 | GNASH | GNASH | + +------------------+----------+--------------------------+------------------------+ + | :sup:`27`\ Al | 130270 | GNASH | GNASH\ *a* | + +------------------+----------+--------------------------+------------------------+ + | :sup:`29`\ Si | 140290 | GNASH | GNASH\ *a* | + +------------------+----------+--------------------------+------------------------+ + | :sup:`30`\ Si | 140300 | GNASH | GNASH\ *a* | + +------------------+----------+--------------------------+------------------------+ + | :sup:`31`\ P | 150310 | GNASH | GNASH | + +------------------+----------+--------------------------+------------------------+ + | :sup:`37`\ Cl | 170370 | GNASH | Woosley et. al. | + | | | | :cite:`WoFoHoZi1975` | + +------------------+----------+--------------------------+------------------------+ + +.. [#gnash] GNASH-calculated data and measured data are available for these + nuclides in the library. By default, the GNASH values are + used. To use the measured data, the user must reverse the + order of th GNASH and measured data in the library since the code + uses the first set encountered in the library (GNASH set). + +.. _5-2-6-3: + +Beta Emission +~~~~~~~~~~~~~ + +Beta emission rates and energy spectra are calculated using an +analytic expression for the kinetic energy of the emitted +:math:`\beta^-` particles :cite:`GoMa1971`: + + .. math:: + N\left( Z,W \right) = \frac{g^{2}}{{2\pi}^{3}} + F\left( Z,W \right) \rho W \left( W_{0}- W \right)^{2}S_{n}\left( W \right) dW + +where + + :math:`Z =` atomic number of the daugher nucleus + + :math:`g =` weak interaction coupling constant + + :math:`W =` kinetic energy of beta particle (in :math:`m_{e}c^{2}` units) + + :math:`F\left( Z,W \right) =` Fermi function + + :math:`W_{0} =` endpoint beta energy + + :math:`\rho = \sqrt{W^{2} - 1}` = electron momentum + + :math:`S_{n}\left( W \right) =` spectral shape factor based on transition type + + :math:`n =` classification of the transition type + + +**Internal conversion electron emission is not considered.** + +The calculation requires nuclear data on the fraction of the beta +transition to each exicited state of the daughter nucleus, the maximum +endpoint energy of the transition (W\ 0), and a classification of the +beta transition (n) defined by the spin and parity change of the +transition which defines the spectral shape factor. The transition +classification uses n\ =0 for allowed and forbidden non-unique +transitions, n\ =1 for first forbidden unique transitions, n\ =2 for +second forbidden unique transitions, and n\ =3 for third forbidden +unique transitions. These data are not stored in the decay data resource +but are included in a separate beta decay resource used only for the +beta calculation. + +The beta decay data are stored in the formatted file +origen.rev00.ensdf95beta.data. The data are derived from ENSDF as +compiled in 1995. The file includes beta decay information for 715 beta +decay nuclides and has 8486 beta transition branches. + +.. _5-2-6-4: + +Alpha Emission +-------------- + +Calculation of the alpha emission intensity and spectrum requires +detailed information that is not available on the decay resource. The +calculation requires the alpha particle energy and branching fraction +for each transition branch. Unlike the beta spectrum, the alpha +particles are emitted with discrete energies, and the source spectrum +may be generated by straightforward binning into the user-defined group +structure. Alpha particle emission data are also used in the +:math:`\left( \alpha,n \right)` neutron source calculation. +Therefore, the alpha emission spectra are calculated using the same +alpha decay library in the neutron emission resource: ``origen.rev01.alphdec.data``. + +.. _tab5-2-9: +.. list-table:: Nuclides with :math:`\alpha`-particle emission data for neutron yield calculations. + :align: center + + * - .. image:: figs/ORIGENdata/tab9.svg + :width: 800 + +.. _tab5-2-10: +.. list-table:: Target (:math:`\alpha`,n) cross section and branching level isotopes and sources. + :align: center + + * - .. image:: figs/ORIGENdata/tab10.svg + :width: 800 + +.. _5-2a: + +Decay Resource Format +===================== + +The decay resource is a simple text format file that can be processed by +COUPLE to create a binary decay-only library that can be used directly +by ORIGEN. In general, this is not necessary, as the decay resource +distributed with SCALE has already been processed with COUPLE to produce +the end7dec ORIGEN decay-only binary library file. Modifying the decay +data or the set of nuclides ORIGEN tracks requires modification of the +decay resource file. The format is described in +:numref:`tab-origen-decay-resource`. Note that as of the SCALE 6.2 release, +ORIGEN now uses the SCALE Standard Composition resource for abundance data +and the "ABUND" field shown below is ignored by COUPLE when reading the decay +resource. + +.. table:: Definitions of data in the decay resource + :name: tab-origen-decay-resource + :align: center + + +---------------+-----------------------------------------------------+ + | **Data name** | **Definition** | + +---------------+-----------------------------------------------------+ + | LIB | Nuclide sublib (used by COUPLE) | + +---------------+-----------------------------------------------------+ + | NUC1 | Nuclide identifier | + +---------------+-----------------------------------------------------+ + | IU | Units for the half life value | + | | (see numref:`tab-origen-hl-units`) | + +---------------+-----------------------------------------------------+ + | HALFL | Value of the half life in IU units | + +---------------+-----------------------------------------------------+ + | FB1 | Beta decay transition leading to a daughter in the | + | | metastable state | + +---------------+-----------------------------------------------------+ + | FP | Positron emission decay fraction or orbital | + | | electron capture to the ground state | + +---------------+-----------------------------------------------------+ + | FP1 | Positron emission decay fraction or orbital | + | | electron capture to a metastable state | + +---------------+-----------------------------------------------------+ + | FA | Alpha particle emission decay fraction | + +---------------+-----------------------------------------------------+ + | FT | Isomeric transition decay fraction | + +---------------+-----------------------------------------------------+ + | LIB1 | Nuclide type in the library | + +---------------+-----------------------------------------------------+ + | FSF | Spontaneous fission decay fraction | + +---------------+-----------------------------------------------------+ + | FBN | Delayed neutron decay (beta particle and a neutron) | + | | fraction | + +---------------+-----------------------------------------------------+ + | Q | Recoverable energy per decay (MeV) | + +---------------+-----------------------------------------------------+ + | ABUND | Natural atom isotopic abundance in percent | + | | (**no longer used**) | + +---------------+-----------------------------------------------------+ + | AMPC | Maximum permissible concentration in air | + +---------------+-----------------------------------------------------+ + | WMPC | Maximum permissible concentration in water | + +---------------+-----------------------------------------------------+ + | LIB1 | Nuclide type in the library (used by COUPLE) | + +---------------+-----------------------------------------------------+ + | FG | Fraction of recoverable decay energy Q associated | + | | with gamma rays | + +---------------+-----------------------------------------------------+ + | FB | Beta decay transition leading to a daughter in the | + | | ground state | + +---------------+-----------------------------------------------------+ + | FBB | Double beta decay fraction | + +---------------+-----------------------------------------------------+ + | FN | Neutron decay fraction | + +---------------+-----------------------------------------------------+ + | FBA | Beta decay plus an alpha particle emission decay | + | | fraction | + +---------------+-----------------------------------------------------+ + +The variable ``LIB`` (and ``LIB1``)defines the nuclide sublibrary +(1/2/3=activation product/actinide/fission product). Variable ``LIB1`` is +included for formatting purposes only. + +The nuclide identifier is read in variable ``NUC1`` and is subsequently +stored in array ``NUCL``. The nuclide identifier is given by + +.. math:: + \text{NUCL} = \text{Z} * 10000 + \text{A} * 10 + \text{I} + :label: eq-origen-nuc-id + + +where Z is the atomic number, A is the atomic mass number, and I is the +isomeric state, where :math:`I=0` designates a ground state, and :math:`I=1` +is the first metastable state. + +The variable ``HALFL`` is the physical half-life in units designated by the +variable IU, as shown in :numref:`tab-origen-hl-units`. The definitions of 11 +variables representing the different decay mode branching fractions are given in +:numref:`tab-origen-decay-resource`. The decay branching fractions are used in +constructing the transition matrix. + + +.. table:: Units of half-life indicated by the variable IU + :name: tab-origen-hl-units + :align: center + + ====== ====================== + **IU** **Units of half-life** + 1 seconds + 2 minutes + 3 hours + 4 days + 5 years + 6 stable + 7 10\ :sup:`3` years + 8 10\ :sup:`6` years + 9 10\ :sup:`9` years + ====== ====================== + +The variable Q is the total amount of recoverable energy (MeV) per +disintegration released by radioactive decay used for decay heat +calculations. It does not include the energy of neutrinos emitted during +beta decay transitions. The variable FG is the fraction of recoverable +energy per disintegration that comes from gamma rays and x-rays. +The value of Q is obtained directly from ENDF/B-VII.1 as the sum of the +average beta, gamma, and alpha decay energy values. The quantity +includes the energy from all electron- related radiations such as :math:`\beta^-`, +:math:`\beta^+`, Auger electrons, etc., all gamma rays, x-rays, and annihilation +radiations, and the average energy of all heavy charged particles and +delayed neutrons. The contribution from alpha decay energy includes the +energy of the recoil nucleus. A part of the recoverable energy per decay +not included in the ENDF/B-VII.1 values is the additional contribution +from spontaneous fission. This energy was calculated as the product of +the spontaneous fission branching fraction and recoverable energy per +fission using a value of 200 MeV per fission and added to the +ENDF/B-VII.1 recoverable Q energy. A value of 12.56 MeV gamma energy per +fission was used in computing the fraction of recoverable spontaneous +fission energy from gamma rays. + +External bremsstrahlung radiation is **not** included in the values of +FG since the bremsstrahlung spectrum depends on electron interactions +with the medium that contains the decay nuclide. The energy from capture +gamma rays accompanying :math:`\left( \alpha,n \right)` reactions is also not included since it +also depends on the medium. The variable ABUND is the atom percent +abundance of naturally occurring isotopes. + +An example of the decay resource content for selected fission products is +presented as :numref:`ex-dec-res-fp`. + +.. literalinclude:: figs/ORIGENdata/dec-fp.txt + :caption: Example of the ENDF/B-VII.1 decay data resource entries for selected fission products. + :name: ex-dec-res-fp + :language: none + +.. _5-2b: + +Fission Yield Resource Format +============================= + +The independent fission product yields are stored as a formatted text +file. The header record for each set of fission product yields includes +the fissionable nuclide ID and an unused entry (0.0), followed by the +number of incident neutron energies included for this nuclide. The +fission yields for each energy are preceeded by a single record +containing the incident neutron energy (eV), an unused entry (0.0), an +index for the incident energy, the number of data entries per fission +product, the total number of entries for each incident energy, and the +number of fission products. The fission product yields for each +fissionable nuclide and incident neutron energy are then listed as pairs +of entries for the fission product nuclide ID and the independent +(direct) fission yield as atom percent per fission. An example of the +format is shown below in :numref:`ex-origen-fy-th227` for :sup:`227`\ Th. + +The number and order of the fission product yields must be the same for +all fissionable nuclides and must correspond to the fission products in +the nuclear decay data. The fission product yields for each fissionable +nuclide, excluding the yields for the terniary fission products +:sup:`3`\ H, :sup:`3`\ He, and :sup:`4`\ He, sum to 200. + +The fissionable nuclides and the tabulated incident neutron energies for +which yields are available are listed in :numref:`tab-origen-fiss-isotopes`. + + +.. literalinclude:: figs/ORIGENdata/fy-th227.txt + :caption: Fission yield format example showing a portion of :sup:`227`\ Th. + :name: ex-origen-fy-th227 + :language: none + +.. _5-2c: + +Gamma Resource Format +===================== + +An example of the photon data entries for the emissions from +:sup:`140`\ La decay is shown below in :numref:`ex-origen-dec-la140`. The header +record for each nuclide contains the nuclide ID, the total number of emission +lines in the evaluation, as well as the number of discrete x-ray lines, +discrete gamma lines, and number of pseudo lines used to represent +continuum data if present in an evaluation used to reconstruct +continuous energy emission spectra from the discrete representation. The +last entries in the header record include the total gamma energy (MeV), +and the character nuclide name. The emission spectrum is listed using +pairs of entries for the photon energy (MeV) and photon emission +(photons per disintegration). + +.. literalinclude:: figs/ORIGENdata/gam-la140.txt + :caption: Gamma resource format example showing :sup:`140`\ La decay photon + emission. + :name: ex-origen-dec-la140 + :language: none + +.. _5-2d: + +ORIGEN "end7dec" Nuclide Set +============================ + +:numref:`tab-origen-end7dec-nucs` shows a list of the 2,237 nuclides on the +origen.rev04.end7dec ORIGEN binary decay-only library, and because this +library provides the basis for all other libraries, effectively the set +of nuclides tracked by ORIGEN in any decay or irradiation calculation. +The "index" column is the index of that nuclide in the set (internally +every ORIGEN isotopics vector has this order), the "sublib" column is +the sublibrary (LT=light nuclide, AC=actinide, FP=fission product) in +which the nuclide resides, the "nuclide" column is the nuclide +identifier, the "mass" column is the mass of the nuclide in grams per +mole, the "abundance" column is the natural abundance in atom percent +for the nuclide (note only light nuclides have abundances), and the +"decay" column is the decay constant. Note that the mass and abundance +data are embedded on the library with the values from the current SCALE +Standard Composition Library. + +.. table:: Nuclide listing for "end7dec" ORIGEN library + :name: tab-origen-end7dec-nucs + :class: longtable + + ========= ========== =========== ========= =========== ========= + **index** **sublib** **nuclide** **mass (\ **abundance **decay** + g/mol)** (atom%)** + **(1/s)** + ========= ========== =========== ========= =========== ========= + 1 LT 1-H-1 1.0078 1.00E+02 0.00E+00 + 2 LT 1-H-2 2.0141 1.15E-02 0.00E+00 + 3 LT 1-H-3 3.0161 0.00E+00 1.78E-09 + 4 LT 2-He-3 3.0160 1.00E-04 0.00E+00 + 5 LT 2-He-4 4.0026 1.00E+02 0.00E+00 + 6 LT 2-He-5 5.0122 0.00E+00 6.93E+02 + 7 LT 2-He-6 6.0189 0.00E+00 8.59E-01 + 8 LT 3-Li-6 6.0151 7.59E+00 0.00E+00 + 9 LT 3-Li-7 7.0160 9.24E+01 0.00E+00 + 10 LT 3-Li-8 8.0225 0.00E+00 8.27E-01 + 11 LT 4-Be-7 7.0169 0.00E+00 1.51E-07 + 12 LT 4-Be-8 8.0053 0.00E+00 6.93E+02 + 13 LT 4-Be-9 9.0122 1.00E+02 0.00E+00 + 14 LT 4-Be-10 10.0135 0.00E+00 1.45E-14 + 15 LT 4-Be-11 11.0217 0.00E+00 5.02E-02 + 16 LT 5-B-10 10.0129 1.99E+01 0.00E+00 + 17 LT 5-B-11 11.0093 8.01E+01 0.00E+00 + 18 LT 5-B-12 12.0143 0.00E+00 3.43E+01 + 19 LT 6-C-12 12.0000 9.89E+01 0.00E+00 + 20 LT 6-C-13 13.0034 1.07E+00 0.00E+00 + 21 LT 6-C-14 14.0032 0.00E+00 3.85E-12 + 22 LT 6-C-15 15.0106 0.00E+00 2.83E-01 + 23 LT 7-N-13 13.0057 0.00E+00 1.16E-03 + 24 LT 7-N-14 14.0031 9.96E+01 0.00E+00 + 25 LT 7-N-15 15.0001 3.64E-01 0.00E+00 + 26 LT 7-N-16 16.0061 0.00E+00 9.72E-02 + 27 LT 8-O-16 15.9949 9.98E+01 0.00E+00 + 28 LT 8-O-17 16.9991 3.80E-02 0.00E+00 + 29 LT 8-O-18 17.9992 2.05E-01 0.00E+00 + 30 LT 8-O-19 19.0036 0.00E+00 2.58E-02 + 31 LT 9-F-19 18.9984 1.00E+02 0.00E+00 + 32 LT 9-F-20 20.0000 0.00E+00 6.21E-02 + 33 LT 10-Ne-20 19.9924 9.05E+01 0.00E+00 + 34 LT 10-Ne-21 20.9939 2.70E-01 0.00E+00 + 35 LT 10-Ne-22 21.9914 9.25E+00 0.00E+00 + 36 LT 10-Ne-23 22.9945 0.00E+00 1.86E-02 + 37 LT 11-Na-22 21.9944 0.00E+00 8.44E-09 + 38 LT 11-Na-23 22.9898 1.00E+02 0.00E+00 + 39 LT 11-Na-24 23.9910 0.00E+00 1.28E-05 + 40 LT 11-Na-24m 23.9910 0.00E+00 3.43E+01 + 41 LT 11-Na-25 24.9900 0.00E+00 1.17E-02 + 42 LT 12-Mg-24 23.9850 7.90E+01 0.00E+00 + 43 LT 12-Mg-25 24.9858 1.00E+01 0.00E+00 + 44 LT 12-Mg-26 25.9826 1.10E+01 0.00E+00 + 45 LT 12-Mg-27 26.9843 0.00E+00 1.22E-03 + 46 LT 12-Mg-28 27.9839 0.00E+00 9.21E-06 + 47 LT 13-Al-26 25.9869 0.00E+00 3.06E-14 + 48 LT 13-Al-27 26.9815 1.00E+02 0.00E+00 + 49 LT 13-Al-28 27.9819 0.00E+00 5.15E-03 + 50 LT 13-Al-29 28.9804 0.00E+00 1.76E-03 + 51 LT 13-Al-30 29.9830 0.00E+00 1.91E-01 + 52 LT 14-Si-28 27.9769 9.22E+01 0.00E+00 + 53 LT 14-Si-29 28.9765 4.69E+00 0.00E+00 + 54 LT 14-Si-30 29.9738 3.09E+00 0.00E+00 + 55 LT 14-Si-31 30.9754 0.00E+00 7.34E-05 + 56 LT 14-Si-32 31.9741 0.00E+00 1.44E-10 + 57 LT 15-P-31 30.9738 1.00E+02 0.00E+00 + 58 LT 15-P-32 31.9739 0.00E+00 5.62E-07 + 59 LT 15-P-33 32.9717 0.00E+00 3.17E-07 + 60 LT 15-P-34 33.9736 0.00E+00 5.58E-02 + 61 LT 16-S-32 31.9721 9.50E+01 0.00E+00 + 62 LT 16-S-33 32.9715 7.50E-01 0.00E+00 + 63 LT 16-S-34 33.9679 4.25E+00 0.00E+00 + 64 LT 16-S-35 34.9690 0.00E+00 9.17E-08 + 65 LT 16-S-36 35.9671 1.00E-02 0.00E+00 + 66 LT 16-S-37 36.9711 0.00E+00 2.29E-03 + 67 LT 17-Cl-35 34.9688 7.58E+01 0.00E+00 + 68 LT 17-Cl-36 35.9683 0.00E+00 7.30E-14 + 69 LT 17-Cl-37 36.9659 2.42E+01 0.00E+00 + 70 LT 17-Cl-38 37.9680 0.00E+00 3.10E-04 + 71 LT 17-Cl-38m 37.9680 0.00E+00 9.69E-01 + 72 LT 18-Ar-36 35.9675 3.37E-01 0.00E+00 + 73 LT 18-Ar-37 36.9668 0.00E+00 2.29E-07 + 74 LT 18-Ar-38 37.9627 6.32E-02 0.00E+00 + 75 LT 18-Ar-39 38.9643 0.00E+00 8.17E-11 + 76 LT 18-Ar-40 39.9624 9.96E+01 0.00E+00 + 77 LT 18-Ar-41 40.9645 0.00E+00 1.05E-04 + 78 LT 18-Ar-42 41.9631 0.00E+00 6.68E-10 + 79 LT 19-K-39 38.9637 9.33E+01 0.00E+00 + 80 LT 19-K-40 39.9640 1.17E-02 1.76E-17 + 81 LT 19-K-41 40.9618 6.73E+00 0.00E+00 + 82 LT 19-K-42 41.9624 0.00E+00 1.56E-05 + 83 LT 19-K-43 42.9607 0.00E+00 8.63E-06 + 84 LT 19-K-44 43.9616 0.00E+00 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5.00E-08 + 745 LT 72-Hf-170 169.9396 0.00E+00 1.20E-05 + 746 LT 72-Hf-171 170.9405 0.00E+00 1.59E-05 + 747 LT 72-Hf-172 171.9395 0.00E+00 1.17E-08 + 748 LT 72-Hf-173 172.9405 0.00E+00 8.16E-06 + 749 LT 72-Hf-174 173.9400 1.60E-01 1.10E-23 + 750 LT 72-Hf-175 174.9415 0.00E+00 1.15E-07 + 751 LT 72-Hf-176 175.9414 5.26E+00 0.00E+00 + 752 LT 72-Hf-177 176.9432 1.86E+01 0.00E+00 + 753 LT 72-Hf-177m 176.9432 0.00E+00 6.36E-01 + 754 LT 72-Hf-178 177.9437 2.73E+01 0.00E+00 + 755 LT 72-Hf-178m 177.9437 0.00E+00 1.73E-01 + 756 LT 72-Hf-179 178.9458 1.36E+01 0.00E+00 + 757 LT 72-Hf-179m 178.9458 0.00E+00 3.71E-02 + 758 LT 72-Hf-180 179.9465 3.51E+01 0.00E+00 + 759 LT 72-Hf-180m 179.9465 0.00E+00 3.50E-05 + 760 LT 72-Hf-181 180.9491 0.00E+00 1.89E-07 + 761 LT 72-Hf-182 181.9505 0.00E+00 2.47E-15 + 762 LT 73-Ta-177 176.9445 0.00E+00 3.40E-06 + 763 LT 73-Ta-178 177.9458 0.00E+00 1.24E-03 + 764 LT 73-Ta-179 178.9459 0.00E+00 1.21E-08 + 765 LT 73-Ta-180m 179.9475 1.20E-02 0.00E+00 + 766 LT 73-Ta-180 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75-Re-184 183.9525 0.00E+00 2.27E-07 + 789 LT 75-Re-184m 183.9525 0.00E+00 4.75E-08 + 790 LT 75-Re-185 184.9530 3.74E+01 0.00E+00 + 791 LT 75-Re-186 185.9550 0.00E+00 2.16E-06 + 792 LT 75-Re-186m 185.9550 0.00E+00 1.10E-13 + 793 LT 75-Re-187 186.9557 6.26E+01 5.07E-19 + 794 LT 75-Re-188 187.9581 0.00E+00 1.13E-05 + 795 LT 75-Re-188m 187.9581 0.00E+00 6.21E-04 + 796 LT 75-Re-189 188.9592 0.00E+00 7.92E-06 + 797 LT 76-Os-182 181.9521 0.00E+00 8.82E-06 + 798 LT 76-Os-183 182.9531 0.00E+00 1.48E-05 + 799 LT 76-Os-184 183.9525 2.00E-02 0.00E+00 + 800 LT 76-Os-185 184.9540 0.00E+00 8.57E-08 + 801 LT 76-Os-186 185.9538 1.59E+00 1.10E-23 + 802 LT 76-Os-187 186.9557 1.96E+00 0.00E+00 + 803 LT 76-Os-188 187.9558 1.32E+01 0.00E+00 + 804 LT 76-Os-189 188.9581 1.62E+01 0.00E+00 + 805 LT 76-Os-189m 188.9581 0.00E+00 3.31E-05 + 806 LT 76-Os-190 189.9585 2.63E+01 0.00E+00 + 807 LT 76-Os-190m 189.9585 0.00E+00 1.17E-03 + 808 LT 76-Os-191 190.9609 0.00E+00 5.21E-07 + 809 LT 76-Os-191m 190.9609 0.00E+00 1.47E-05 + 810 LT 76-Os-192 191.9615 4.08E+01 0.00E+00 + 811 LT 76-Os-193 192.9642 0.00E+00 6.39E-06 + 812 LT 76-Os-194 193.9652 0.00E+00 3.66E-09 + 813 LT 77-Ir-185 184.9567 0.00E+00 1.34E-05 + 814 LT 77-Ir-186 185.9579 0.00E+00 1.16E-05 + 815 LT 77-Ir-188 187.9588 0.00E+00 4.64E-06 + 816 LT 77-Ir-189 188.9587 0.00E+00 6.08E-07 + 817 LT 77-Ir-189m 188.9587 0.00E+00 5.21E+01 + 818 LT 77-Ir-190 189.9606 0.00E+00 6.81E-07 + 819 LT 77-Ir-191 190.9606 3.73E+01 0.00E+00 + 820 LT 77-Ir-191m 190.9606 0.00E+00 1.41E-01 + 821 LT 77-Ir-192 191.9626 0.00E+00 1.09E-07 + 822 LT 77-Ir-192m 191.9626 0.00E+00 7.97E-03 + 823 LT 77-Ir-193 192.9629 6.27E+01 0.00E+00 + 824 LT 77-Ir-193m 192.9629 0.00E+00 7.62E-07 + 825 LT 77-Ir-194 193.9651 0.00E+00 9.99E-06 + 826 LT 77-Ir-194m 193.9651 0.00E+00 2.18E+01 + 827 LT 77-Ir-196 195.9684 0.00E+00 1.33E-02 + 828 LT 77-Ir-196m 195.9684 0.00E+00 1.38E-04 + 829 LT 78-Pt-188 187.9594 0.00E+00 7.87E-07 + 830 LT 78-Pt-189 188.9608 0.00E+00 1.77E-05 + 831 LT 78-Pt-190 189.9599 1.40E-02 3.38E-20 + 832 LT 78-Pt-191 190.9617 0.00E+00 2.86E-06 + 833 LT 78-Pt-192 191.9610 7.82E-01 0.00E+00 + 834 LT 78-Pt-193 192.9630 0.00E+00 4.39E-10 + 835 LT 78-Pt-193m 192.9630 0.00E+00 1.85E-06 + 836 LT 78-Pt-194 193.9627 3.30E+01 0.00E+00 + 837 LT 78-Pt-195 194.9648 3.38E+01 0.00E+00 + 838 LT 78-Pt-195m 194.9648 0.00E+00 2.00E-06 + 839 LT 78-Pt-196 195.9650 2.52E+01 0.00E+00 + 840 LT 78-Pt-197 196.9673 0.00E+00 9.68E-06 + 841 LT 78-Pt-197m 196.9673 0.00E+00 1.21E-04 + 842 LT 78-Pt-198 197.9679 7.16E+00 0.00E+00 + 843 LT 78-Pt-199 198.9706 0.00E+00 3.75E-04 + 844 LT 78-Pt-199m 198.9706 0.00E+00 5.10E-02 + 845 LT 78-Pt-200 199.9714 0.00E+00 1.53E-05 + 846 LT 79-Au-193 192.9642 0.00E+00 1.09E-05 + 847 LT 79-Au-194 193.9654 0.00E+00 5.06E-06 + 848 LT 79-Au-195 194.9650 0.00E+00 4.31E-08 + 849 LT 79-Au-195m 194.9650 0.00E+00 2.27E-02 + 850 LT 79-Au-196 195.9666 0.00E+00 1.30E-06 + 851 LT 79-Au-197 196.9666 1.00E+02 0.00E+00 + 852 LT 79-Au-197m 196.9666 0.00E+00 8.97E-02 + 853 LT 79-Au-198 197.9682 0.00E+00 2.98E-06 + 854 LT 79-Au-198m 197.9682 0.00E+00 3.53E-06 + 855 LT 79-Au-199 198.9688 0.00E+00 2.56E-06 + 856 LT 79-Au-200 199.9707 0.00E+00 2.39E-04 + 857 LT 79-Au-200m 199.9707 0.00E+00 1.03E-05 + 858 LT 80-Hg-193 192.9667 0.00E+00 5.07E-05 + 859 LT 80-Hg-193m 192.9667 0.00E+00 1.63E-05 + 860 LT 80-Hg-194 193.9654 0.00E+00 4.95E-11 + 861 LT 80-Hg-195 194.9667 0.00E+00 1.83E-05 + 862 LT 80-Hg-195m 194.9667 0.00E+00 4.63E-06 + 863 LT 80-Hg-196 195.9658 1.50E-01 0.00E+00 + 864 LT 80-Hg-197 196.9672 0.00E+00 3.00E-06 + 865 LT 80-Hg-197m 196.9672 0.00E+00 8.09E-06 + 866 LT 80-Hg-198 197.9668 9.97E+00 0.00E+00 + 867 LT 80-Hg-199 198.9683 1.69E+01 0.00E+00 + 868 LT 80-Hg-199m 198.9683 0.00E+00 2.71E-04 + 869 LT 80-Hg-200 199.9683 2.31E+01 0.00E+00 + 870 LT 80-Hg-201 200.9703 1.32E+01 0.00E+00 + 871 LT 80-Hg-202 201.9706 2.99E+01 0.00E+00 + 872 LT 80-Hg-203 202.9729 0.00E+00 1.72E-07 + 873 LT 80-Hg-204 203.9735 6.87E+00 0.00E+00 + 874 LT 80-Hg-205 204.9761 0.00E+00 2.25E-03 + 875 LT 80-Hg-206 205.9775 0.00E+00 1.39E-03 + 876 LT 81-Tl-200 199.9710 0.00E+00 7.38E-06 + 877 LT 81-Tl-201 200.9708 0.00E+00 2.64E-06 + 878 LT 81-Tl-202 201.9721 0.00E+00 6.52E-07 + 879 LT 81-Tl-203 202.9723 2.95E+01 0.00E+00 + 880 LT 81-Tl-204 203.9739 0.00E+00 5.81E-09 + 881 LT 81-Tl-205 204.9744 7.05E+01 0.00E+00 + 882 LT 81-Tl-206 205.9761 0.00E+00 2.75E-03 + 883 LT 81-Tl-207 206.9774 0.00E+00 2.42E-03 + 884 LT 82-Pb-200 199.9718 0.00E+00 8.96E-06 + 885 LT 82-Pb-202 201.9722 0.00E+00 4.18E-13 + 886 LT 82-Pb-203 202.9734 0.00E+00 3.71E-06 + 887 LT 82-Pb-204 203.9730 1.40E+00 1.58E-25 + 888 LT 82-Pb-205 204.9745 0.00E+00 1.27E-15 + 889 LT 82-Pb-205m 204.9745 0.00E+00 1.25E+02 + 890 LT 82-Pb-206 205.9745 2.41E+01 0.00E+00 + 891 LT 82-Pb-207 206.9759 2.21E+01 0.00E+00 + 892 LT 82-Pb-207m 206.9759 0.00E+00 8.60E-01 + 893 LT 82-Pb-208 207.9767 5.24E+01 0.00E+00 + 894 LT 82-Pb-209 208.9811 0.00E+00 5.92E-05 + 895 LT 82-Pb-210 209.9842 0.00E+00 9.89E-10 + 896 LT 83-Bi-205 204.9774 0.00E+00 5.24E-07 + 897 LT 83-Bi-206 205.9785 0.00E+00 1.29E-06 + 898 LT 83-Bi-207 206.9785 0.00E+00 6.96E-10 + 899 LT 83-Bi-208 207.9797 0.00E+00 5.97E-14 + 900 LT 83-Bi-209 208.9804 1.00E+02 1.16E-27 + 901 LT 83-Bi-210 209.9841 0.00E+00 1.60E-06 + 902 LT 83-Bi-210m 209.9841 0.00E+00 7.23E-15 + 903 LT 83-Bi-211 210.9873 0.00E+00 5.40E-03 + 904 LT 84-Po-206 205.9805 0.00E+00 9.12E-07 + 905 LT 84-Po-207 206.9816 0.00E+00 3.32E-05 + 906 LT 84-Po-208 207.9812 0.00E+00 7.58E-09 + 907 LT 84-Po-209 208.9824 0.00E+00 2.15E-10 + 908 LT 84-Po-210 209.9829 0.00E+00 5.80E-08 + 909 LT 84-Po-211 210.9866 0.00E+00 1.34E+00 + 910 LT 84-Po-211m 210.9866 0.00E+00 2.75E-02 + 911 AC 2-He-3 3.0160 0.00E+00 0.00E+00 + 912 AC 2-He-4 4.0026 0.00E+00 0.00E+00 + 913 AC 3-Li-6 6.0151 0.00E+00 0.00E+00 + 914 AC 3-Li-7 7.0160 0.00E+00 0.00E+00 + 915 AC 4-Be-7 7.0169 0.00E+00 1.51E-07 + 916 AC 6-C-12 12.0000 0.00E+00 0.00E+00 + 917 AC 80-Hg-206 205.9775 0.00E+00 1.39E-03 + 918 AC 81-Tl-203 202.9723 0.00E+00 0.00E+00 + 919 AC 81-Tl-205 204.9744 0.00E+00 0.00E+00 + 920 AC 81-Tl-206 205.9761 0.00E+00 2.75E-03 + 921 AC 81-Tl-207 206.9774 0.00E+00 2.42E-03 + 922 AC 81-Tl-208 207.9820 0.00E+00 3.78E-03 + 923 AC 81-Tl-209 208.9854 0.00E+00 5.25E-03 + 924 AC 81-Tl-210 209.9901 0.00E+00 8.89E-03 + 925 AC 82-Pb-203 202.9734 0.00E+00 3.71E-06 + 926 AC 82-Pb-204 203.9730 0.00E+00 1.58E-25 + 927 AC 82-Pb-205 204.9745 0.00E+00 1.27E-15 + 928 AC 82-Pb-206 205.9745 0.00E+00 0.00E+00 + 929 AC 82-Pb-207 206.9759 0.00E+00 0.00E+00 + 930 AC 82-Pb-207m 206.9759 0.00E+00 8.60E-01 + 931 AC 82-Pb-208 207.9767 0.00E+00 0.00E+00 + 932 AC 82-Pb-209 208.9811 0.00E+00 5.92E-05 + 933 AC 82-Pb-210 209.9842 0.00E+00 9.89E-10 + 934 AC 82-Pb-211 210.9887 0.00E+00 3.20E-04 + 935 AC 82-Pb-212 211.9919 0.00E+00 1.81E-05 + 936 AC 82-Pb-214 213.9998 0.00E+00 4.31E-04 + 937 AC 83-Bi-206 205.9785 0.00E+00 1.29E-06 + 938 AC 83-Bi-207 206.9785 0.00E+00 6.96E-10 + 939 AC 83-Bi-208 207.9797 0.00E+00 5.97E-14 + 940 AC 83-Bi-209 208.9804 0.00E+00 1.16E-27 + 941 AC 83-Bi-210m 209.9841 0.00E+00 7.23E-15 + 942 AC 83-Bi-210 209.9841 0.00E+00 1.60E-06 + 943 AC 83-Bi-211 210.9873 0.00E+00 5.40E-03 + 944 AC 83-Bi-212 211.9913 0.00E+00 1.91E-04 + 945 AC 83-Bi-212m 211.9913 0.00E+00 4.62E-04 + 946 AC 83-Bi-213 212.9944 0.00E+00 2.53E-04 + 947 AC 83-Bi-214 213.9987 0.00E+00 5.81E-04 + 948 AC 84-Po-207 206.9816 0.00E+00 3.32E-05 + 949 AC 84-Po-208 207.9812 0.00E+00 7.58E-09 + 950 AC 84-Po-209 208.9824 0.00E+00 2.15E-10 + 951 AC 84-Po-210 209.9829 0.00E+00 5.80E-08 + 952 AC 84-Po-211m 210.9866 0.00E+00 2.75E-02 + 953 AC 84-Po-211 210.9866 0.00E+00 1.34E+00 + 954 AC 84-Po-212 211.9889 0.00E+00 6.93E+02 + 955 AC 84-Po-213 212.9929 0.00E+00 6.93E+02 + 956 AC 84-Po-214 213.9952 0.00E+00 6.93E+02 + 957 AC 84-Po-215 214.9994 0.00E+00 3.89E+02 + 958 AC 84-Po-216 216.0019 0.00E+00 4.78E+00 + 959 AC 84-Po-218 218.0090 0.00E+00 3.73E-03 + 960 AC 85-At-216 216.0024 0.00E+00 6.93E+02 + 961 AC 85-At-217 217.0047 0.00E+00 2.15E+01 + 962 AC 85-At-218 218.0087 0.00E+00 4.62E-01 + 963 AC 86-Rn-216 216.0003 0.00E+00 6.93E+02 + 964 AC 86-Rn-217 217.0039 0.00E+00 6.93E+02 + 965 AC 86-Rn-218 218.0056 0.00E+00 1.98E+01 + 966 AC 86-Rn-219 219.0095 0.00E+00 1.75E-01 + 967 AC 86-Rn-220 220.0114 0.00E+00 1.25E-02 + 968 AC 86-Rn-222 222.0176 0.00E+00 2.10E-06 + 969 AC 87-Fr-220 220.0123 0.00E+00 2.53E-02 + 970 AC 87-Fr-221 221.0143 0.00E+00 2.36E-03 + 971 AC 87-Fr-222 222.0175 0.00E+00 8.14E-04 + 972 AC 87-Fr-223 223.0197 0.00E+00 5.25E-04 + 973 AC 88-Ra-220 220.0110 0.00E+00 3.85E+01 + 974 AC 88-Ra-222 222.0154 0.00E+00 1.92E-02 + 975 AC 88-Ra-223 223.0185 0.00E+00 7.02E-07 + 976 AC 88-Ra-224 224.0202 0.00E+00 2.19E-06 + 977 AC 88-Ra-225 225.0236 0.00E+00 5.38E-07 + 978 AC 88-Ra-226 226.0254 0.00E+00 1.37E-11 + 979 AC 88-Ra-227 227.0292 0.00E+00 2.74E-04 + 980 AC 88-Ra-228 228.0311 0.00E+00 3.82E-09 + 981 AC 89-Ac-224 224.0217 0.00E+00 6.93E-05 + 982 AC 89-Ac-225 225.0232 0.00E+00 8.02E-07 + 983 AC 89-Ac-226 226.0261 0.00E+00 6.56E-06 + 984 AC 89-Ac-227 227.0278 0.00E+00 1.01E-09 + 985 AC 89-Ac-228 228.0310 0.00E+00 3.13E-05 + 986 AC 90-Th-226 226.0249 0.00E+00 3.78E-04 + 987 AC 90-Th-227 227.0277 0.00E+00 4.29E-07 + 988 AC 90-Th-228 228.0287 0.00E+00 1.15E-08 + 989 AC 90-Th-229 229.0318 0.00E+00 2.99E-12 + 990 AC 90-Th-230 230.0331 0.00E+00 2.91E-13 + 991 AC 90-Th-231 231.0363 0.00E+00 7.55E-06 + 992 AC 90-Th-232 232.0381 0.00E+00 1.56E-18 + 993 AC 90-Th-233 233.0416 0.00E+00 5.18E-04 + 994 AC 90-Th-234 234.0436 0.00E+00 3.33E-07 + 995 AC 91-Pa-228 228.0311 0.00E+00 8.75E-06 + 996 AC 91-Pa-229 229.0321 0.00E+00 5.35E-06 + 997 AC 91-Pa-230 230.0345 0.00E+00 4.61E-07 + 998 AC 91-Pa-231 231.0359 0.00E+00 6.70E-13 + 999 AC 91-Pa-232 232.0386 0.00E+00 6.08E-06 + 1000 AC 91-Pa-233 233.0403 0.00E+00 2.97E-07 + 1001 AC 91-Pa-234m 234.0433 0.00E+00 9.97E-03 + 1002 AC 91-Pa-234 234.0433 0.00E+00 2.87E-05 + 1003 AC 91-Pa-235 235.0454 0.00E+00 4.73E-04 + 1004 AC 92-U-230 230.0339 0.00E+00 3.86E-07 + 1005 AC 92-U-231 231.0363 0.00E+00 1.91E-06 + 1006 AC 92-U-232 232.0372 0.00E+00 3.19E-10 + 1007 AC 92-U-233 233.0396 0.00E+00 1.38E-13 + 1008 AC 92-U-234 234.0410 0.00E+00 8.95E-14 + 1009 AC 92-U-235 235.0439 0.00E+00 3.12E-17 + 1010 AC 92-U-235m 235.0439 0.00E+00 4.44E-04 + 1011 AC 92-U-236 236.0456 0.00E+00 9.38E-16 + 1012 AC 92-U-237 237.0487 0.00E+00 1.19E-06 + 1013 AC 92-U-238 238.0508 0.00E+00 4.92E-18 + 1014 AC 92-U-239 239.0543 0.00E+00 4.93E-04 + 1015 AC 92-U-240 240.0566 0.00E+00 1.37E-05 + 1016 AC 92-U-241 241.0603 0.00E+00 2.31E-03 + 1017 AC 93-Np-234 234.0429 0.00E+00 1.82E-06 + 1018 AC 93-Np-235 235.0441 0.00E+00 2.03E-08 + 1019 AC 93-Np-236m 236.0466 0.00E+00 8.56E-06 + 1020 AC 93-Np-236 236.0466 0.00E+00 1.44E-13 + 1021 AC 93-Np-237 237.0482 0.00E+00 1.02E-14 + 1022 AC 93-Np-238 238.0509 0.00E+00 3.79E-06 + 1023 AC 93-Np-239 239.0529 0.00E+00 3.41E-06 + 1024 AC 93-Np-240m 240.0562 0.00E+00 1.60E-03 + 1025 AC 93-Np-240 240.0562 0.00E+00 1.87E-04 + 1026 AC 93-Np-241 241.0582 0.00E+00 8.31E-04 + 1027 AC 94-Pu-236 236.0461 0.00E+00 7.69E-09 + 1028 AC 94-Pu-237m 237.0484 0.00E+00 3.85E+00 + 1029 AC 94-Pu-237 237.0484 0.00E+00 1.76E-07 + 1030 AC 94-Pu-238 238.0496 0.00E+00 2.50E-10 + 1031 AC 94-Pu-239 239.0522 0.00E+00 9.11E-13 + 1032 AC 94-Pu-240 240.0538 0.00E+00 3.35E-12 + 1033 AC 94-Pu-241 241.0569 0.00E+00 1.54E-09 + 1034 AC 94-Pu-242 242.0587 0.00E+00 5.88E-14 + 1035 AC 94-Pu-243 243.0620 0.00E+00 3.89E-05 + 1036 AC 94-Pu-244 244.0642 0.00E+00 2.71E-16 + 1037 AC 94-Pu-245 245.0677 0.00E+00 1.83E-05 + 1038 AC 94-Pu-246 246.0702 0.00E+00 7.40E-07 + 1039 AC 94-Pu-247 247.0741 0.00E+00 3.53E-06 + 1040 AC 95-Am-239 239.0530 0.00E+00 1.62E-05 + 1041 AC 95-Am-240 240.0553 0.00E+00 3.79E-06 + 1042 AC 95-Am-241 241.0568 0.00E+00 5.08E-11 + 1043 AC 95-Am-242m 242.0595 0.00E+00 1.56E-10 + 1044 AC 95-Am-242 242.0596 0.00E+00 1.20E-05 + 1045 AC 95-Am-243 243.0614 0.00E+00 2.98E-12 + 1046 AC 95-Am-244m 244.0646 0.00E+00 4.44E-04 + 1047 AC 95-Am-244 244.0643 0.00E+00 1.91E-05 + 1048 AC 95-Am-245 245.0665 0.00E+00 9.39E-05 + 1049 AC 95-Am-246 246.0698 0.00E+00 2.96E-04 + 1050 AC 95-Am-246m 246.0698 0.00E+00 4.62E-04 + 1051 AC 95-Am-247 247.0721 0.00E+00 5.02E-04 + 1052 AC 96-Cm-240 240.0555 0.00E+00 2.97E-07 + 1053 AC 96-Cm-241 241.0576 0.00E+00 2.45E-07 + 1054 AC 96-Cm-242 242.0588 0.00E+00 4.92E-08 + 1055 AC 96-Cm-243 243.0614 0.00E+00 7.55E-10 + 1056 AC 96-Cm-244 244.0627 0.00E+00 1.21E-09 + 1057 AC 96-Cm-245 245.0655 0.00E+00 2.58E-12 + 1058 AC 96-Cm-246 246.0672 0.00E+00 4.61E-12 + 1059 AC 96-Cm-247 247.0703 0.00E+00 1.41E-15 + 1060 AC 96-Cm-248 248.0724 0.00E+00 6.31E-14 + 1061 AC 96-Cm-249 249.0759 0.00E+00 1.80E-04 + 1062 AC 96-Cm-250 250.0784 0.00E+00 2.65E-12 + 1063 AC 96-Cm-251 251.0823 0.00E+00 6.88E-04 + 1064 AC 97-Bk-245 245.0664 0.00E+00 1.62E-06 + 1065 AC 97-Bk-246 246.0687 0.00E+00 4.46E-06 + 1066 AC 97-Bk-247 247.0703 0.00E+00 1.59E-11 + 1067 AC 97-Bk-248 248.0731 0.00E+00 2.44E-09 + 1068 AC 97-Bk-248m 248.0731 0.00E+00 8.12E-06 + 1069 AC 97-Bk-249 249.0750 0.00E+00 2.51E-08 + 1070 AC 97-Bk-250 250.0783 0.00E+00 5.99E-05 + 1071 AC 97-Bk-251 251.0808 0.00E+00 2.08E-04 + 1072 AC 98-Cf-246 246.0688 0.00E+00 5.39E-06 + 1073 AC 98-Cf-248 248.0722 0.00E+00 2.41E-08 + 1074 AC 98-Cf-249 249.0748 0.00E+00 6.26E-11 + 1075 AC 98-Cf-250 250.0764 0.00E+00 1.68E-09 + 1076 AC 98-Cf-251 251.0796 0.00E+00 2.45E-11 + 1077 AC 98-Cf-252 252.0816 0.00E+00 8.30E-09 + 1078 AC 98-Cf-253 253.0851 0.00E+00 4.50E-07 + 1079 AC 98-Cf-254 254.0873 0.00E+00 1.33E-07 + 1080 AC 98-Cf-255 255.0910 0.00E+00 1.36E-04 + 1081 AC 99-Es-251 251.0800 0.00E+00 5.83E-06 + 1082 AC 99-Es-252 252.0830 0.00E+00 1.70E-08 + 1083 AC 99-Es-253 253.0848 0.00E+00 3.92E-07 + 1084 AC 99-Es-254m 254.0880 0.00E+00 4.90E-06 + 1085 AC 99-Es-254 254.0880 0.00E+00 2.91E-08 + 1086 AC 99-Es-255 255.0903 0.00E+00 2.02E-07 + 1087 FP 1-H-3 3.0161 0.00E+00 1.78E-09 + 1088 FP 2-He-3 3.0160 0.00E+00 0.00E+00 + 1089 FP 2-He-4 4.0026 0.00E+00 0.00E+00 + 1090 FP 26-Fe-65 64.9454 0.00E+00 8.56E-01 + 1091 FP 27-Co-65 64.9365 0.00E+00 5.98E-01 + 1092 FP 28-Ni-65 64.9301 0.00E+00 7.65E-05 + 1093 FP 29-Cu-65 64.9278 0.00E+00 0.00E+00 + 1094 FP 24-Cr-66 65.9734 0.00E+00 6.93E+01 + 1095 FP 25-Mn-66 65.9611 0.00E+00 1.08E+01 + 1096 FP 26-Fe-66 65.9468 0.00E+00 1.58E+00 + 1097 FP 27-Co-66 65.9398 0.00E+00 3.47E+00 + 1098 FP 28-Ni-66 65.9291 0.00E+00 3.53E-06 + 1099 FP 29-Cu-66 65.9289 0.00E+00 2.26E-03 + 1100 FP 30-Zn-66 65.9260 0.00E+00 0.00E+00 + 1101 FP 31-Ga-66 65.9316 0.00E+00 2.03E-05 + 1102 FP 32-Ge-66 65.9338 0.00E+00 8.52E-05 + 1103 FP 24-Cr-67 66.9796 0.00E+00 1.39E+01 + 1104 FP 25-Mn-67 66.9641 0.00E+00 1.47E+01 + 1105 FP 26-Fe-67 66.9510 0.00E+00 1.67E+00 + 1106 FP 27-Co-67 66.9409 0.00E+00 1.63E+00 + 1107 FP 28-Ni-67 66.9316 0.00E+00 3.30E-02 + 1108 FP 29-Cu-67 66.9277 0.00E+00 3.11E-06 + 1109 FP 30-Zn-67 66.9271 0.00E+00 0.00E+00 + 1110 FP 31-Ga-67 66.9282 0.00E+00 2.46E-06 + 1111 FP 32-Ge-67 66.9327 0.00E+00 6.11E-04 + 1112 FP 25-Mn-68 67.9693 0.00E+00 2.48E+01 + 1113 FP 26-Fe-68 67.9537 0.00E+00 3.71E+00 + 1114 FP 27-Co-68 67.9449 0.00E+00 3.48E+00 + 1115 FP 28-Ni-68 67.9319 0.00E+00 2.39E-02 + 1116 FP 29-Cu-68 67.9296 0.00E+00 2.23E-02 + 1117 FP 29-Cu-68m 67.9296 0.00E+00 3.08E-03 + 1118 FP 30-Zn-68 67.9248 0.00E+00 0.00E+00 + 1119 FP 31-Ga-68 67.9280 0.00E+00 1.71E-04 + 1120 FP 32-Ge-68 67.9281 0.00E+00 2.96E-08 + 1121 FP 25-Mn-69 68.9728 0.00E+00 4.95E+01 + 1122 FP 26-Fe-69 68.9588 0.00E+00 6.36E+00 + 1123 FP 27-Co-69 68.9463 0.00E+00 3.15E+00 + 1124 FP 28-Ni-69 68.9356 0.00E+00 6.08E-02 + 1125 FP 29-Cu-69 68.9294 0.00E+00 4.05E-03 + 1126 FP 30-Zn-69 68.9266 0.00E+00 2.05E-04 + 1127 FP 30-Zn-69m 68.9266 0.00E+00 1.40E-05 + 1128 FP 31-Ga-69 68.9256 0.00E+00 0.00E+00 + 1129 FP 32-Ge-69 68.9280 0.00E+00 4.93E-06 + 1130 FP 33-As-69 68.9323 0.00E+00 7.59E-04 + 1131 FP 26-Fe-70 69.9615 0.00E+00 7.37E+00 + 1132 FP 27-Co-70 69.9510 0.00E+00 5.82E+00 + 1133 FP 28-Ni-70 69.9365 0.00E+00 1.16E-01 + 1134 FP 29-Cu-70 69.9324 0.00E+00 1.56E-02 + 1135 FP 29-Cu-70m 69.9324 0.00E+00 2.10E-02 + 1136 FP 30-Zn-70 69.9253 0.00E+00 0.00E+00 + 1137 FP 31-Ga-70 69.9260 0.00E+00 5.46E-04 + 1138 FP 32-Ge-70 69.9242 0.00E+00 0.00E+00 + 1139 FP 26-Fe-71 70.9667 0.00E+00 2.48E+01 + 1140 FP 27-Co-71 70.9529 0.00E+00 8.77E+00 + 1141 FP 28-Ni-71 70.9407 0.00E+00 2.71E-01 + 1142 FP 29-Cu-71 70.9327 0.00E+00 3.55E-02 + 1143 FP 30-Zn-71 70.9277 0.00E+00 4.72E-03 + 1144 FP 30-Zn-71m 70.9277 0.00E+00 4.86E-05 + 1145 FP 31-Ga-71 70.9247 0.00E+00 0.00E+00 + 1146 FP 32-Ge-71 70.9249 0.00E+00 7.02E-07 + 1147 FP 32-Ge-71m 70.9249 0.00E+00 3.40E+01 + 1148 FP 33-As-71 70.9271 0.00E+00 2.95E-06 + 1149 FP 26-Fe-72 71.9696 0.00E+00 6.93E+02 + 1150 FP 27-Co-72 71.9578 0.00E+00 1.16E+01 + 1151 FP 28-Ni-72 71.9421 0.00E+00 4.42E-01 + 1152 FP 29-Cu-72 71.9358 0.00E+00 1.05E-01 + 1153 FP 30-Zn-72 71.9269 0.00E+00 4.14E-06 + 1154 FP 31-Ga-72 71.9264 0.00E+00 1.37E-05 + 1155 FP 31-Ga-72m 71.9264 0.00E+00 1.75E+01 + 1156 FP 32-Ge-72 71.9221 0.00E+00 0.00E+00 + 1157 FP 33-As-72 71.9268 0.00E+00 7.41E-06 + 1158 FP 34-Se-72 71.9271 0.00E+00 9.55E-07 + 1159 FP 27-Co-73 72.9602 0.00E+00 1.69E+01 + 1160 FP 28-Ni-73 72.9465 0.00E+00 8.25E-01 + 1161 FP 29-Cu-73 72.9367 0.00E+00 1.65E-01 + 1162 FP 30-Zn-73 72.9298 0.00E+00 2.95E-02 + 1163 FP 31-Ga-73 72.9252 0.00E+00 3.96E-05 + 1164 FP 32-Ge-73 72.9235 0.00E+00 0.00E+00 + 1165 FP 32-Ge-73m 72.9235 0.00E+00 1.39E+00 + 1166 FP 33-As-73 72.9238 0.00E+00 9.99E-08 + 1167 FP 34-Se-73 72.9268 0.00E+00 2.69E-05 + 1168 FP 34-Se-73m 72.9268 0.00E+00 2.90E-04 + 1169 FP 27-Co-74 73.9654 0.00E+00 2.31E+01 + 1170 FP 28-Ni-74 73.9481 0.00E+00 1.02E+00 + 1171 FP 29-Cu-74 73.9399 0.00E+00 3.96E-01 + 1172 FP 30-Zn-74 73.9295 0.00E+00 7.25E-03 + 1173 FP 31-Ga-74 73.9269 0.00E+00 1.42E-03 + 1174 FP 31-Ga-74m 73.9269 0.00E+00 7.30E-02 + 1175 FP 32-Ge-74 73.9212 0.00E+00 0.00E+00 + 1176 FP 33-As-74 73.9239 0.00E+00 4.51E-07 + 1177 FP 34-Se-74 73.9225 0.00E+00 0.00E+00 + 1178 FP 27-Co-75 74.9683 0.00E+00 2.04E+01 + 1179 FP 28-Ni-75 74.9529 0.00E+00 1.16E+00 + 1180 FP 29-Cu-75 74.9419 0.00E+00 5.66E-01 + 1181 FP 30-Zn-75 74.9329 0.00E+00 6.80E-02 + 1182 FP 31-Ga-75 74.9265 0.00E+00 5.50E-03 + 1183 FP 32-Ge-75 74.9229 0.00E+00 1.40E-04 + 1184 FP 32-Ge-75m 74.9229 0.00E+00 1.45E-02 + 1185 FP 33-As-75 74.9216 0.00E+00 0.00E+00 + 1186 FP 33-As-75m 74.9216 0.00E+00 3.93E+01 + 1187 FP 34-Se-75 74.9225 0.00E+00 6.70E-08 + 1188 FP 35-Br-75 74.9258 0.00E+00 1.19E-04 + 1189 FP 28-Ni-76 75.9553 0.00E+00 2.91E+00 + 1190 FP 29-Cu-76 75.9453 0.00E+00 1.06E+00 + 1191 FP 30-Zn-76 75.9333 0.00E+00 1.22E-01 + 1192 FP 31-Ga-76 75.9288 0.00E+00 2.13E-02 + 1193 FP 32-Ge-76 75.9214 0.00E+00 0.00E+00 + 1194 FP 33-As-76 75.9224 0.00E+00 7.34E-06 + 1195 FP 34-Se-76 75.9192 0.00E+00 0.00E+00 + 1196 FP 28-Ni-77 76.9605 0.00E+00 1.14E+01 + 1197 FP 29-Cu-77 76.9479 0.00E+00 1.48E+00 + 1198 FP 30-Zn-77 76.9370 0.00E+00 3.33E-01 + 1199 FP 31-Ga-77 76.9292 0.00E+00 5.25E-02 + 1200 FP 32-Ge-77 76.9236 0.00E+00 1.70E-05 + 1201 FP 32-Ge-77m 76.9236 0.00E+00 1.31E-02 + 1202 FP 33-As-77 76.9206 0.00E+00 4.96E-06 + 1203 FP 34-Se-77 76.9199 0.00E+00 0.00E+00 + 1204 FP 34-Se-77m 76.9199 0.00E+00 3.99E-02 + 1205 FP 35-Br-77 76.9214 0.00E+00 3.38E-06 + 1206 FP 35-Br-77m 76.9214 0.00E+00 2.70E-03 + 1207 FP 36-Kr-77 76.9247 0.00E+00 1.55E-04 + 1208 FP 28-Ni-78 77.9632 0.00E+00 6.30E+00 + 1209 FP 29-Cu-78 77.9520 0.00E+00 2.07E+00 + 1210 FP 30-Zn-78 77.9384 0.00E+00 4.72E-01 + 1211 FP 31-Ga-78 77.9316 0.00E+00 1.36E-01 + 1212 FP 32-Ge-78 77.9229 0.00E+00 1.31E-04 + 1213 FP 33-As-78 77.9218 0.00E+00 1.27E-04 + 1214 FP 34-Se-78 77.9173 0.00E+00 0.00E+00 + 1215 FP 35-Br-78 77.9212 0.00E+00 1.79E-03 + 1216 FP 36-Kr-78 77.9204 0.00E+00 0.00E+00 + 1217 FP 29-Cu-79 78.9546 0.00E+00 3.69E+00 + 1218 FP 30-Zn-79 78.9426 0.00E+00 6.97E-01 + 1219 FP 31-Ga-79 78.9329 0.00E+00 2.43E-01 + 1220 FP 32-Ge-79 78.9254 0.00E+00 3.65E-02 + 1221 FP 32-Ge-79m 78.9254 0.00E+00 1.78E-02 + 1222 FP 33-As-79 78.9210 0.00E+00 1.28E-03 + 1223 FP 34-Se-79 78.9185 0.00E+00 7.45E-14 + 1224 FP 34-Se-79m 78.9185 0.00E+00 2.95E-03 + 1225 FP 35-Br-79 78.9183 0.00E+00 0.00E+00 + 1226 FP 35-Br-79m 78.9183 0.00E+00 1.43E-01 + 1227 FP 36-Kr-79 78.9201 0.00E+00 5.49E-06 + 1228 FP 36-Kr-79m 78.9201 0.00E+00 1.39E-02 + 1229 FP 37-Rb-79 78.9240 0.00E+00 5.04E-04 + 1230 FP 29-Cu-80 79.9609 0.00E+00 4.08E+00 + 1231 FP 30-Zn-80 79.9443 0.00E+00 1.28E+00 + 1232 FP 31-Ga-80 79.9365 0.00E+00 4.14E-01 + 1233 FP 32-Ge-80 79.9254 0.00E+00 2.35E-02 + 1234 FP 33-As-80 79.9225 0.00E+00 4.56E-02 + 1235 FP 34-Se-80 79.9165 0.00E+00 0.00E+00 + 1236 FP 35-Br-80 79.9185 0.00E+00 6.53E-04 + 1237 FP 35-Br-80m 79.9185 0.00E+00 4.36E-05 + 1238 FP 36-Kr-80 79.9164 0.00E+00 0.00E+00 + 1239 FP 30-Zn-81 80.9505 0.00E+00 2.17E+00 + 1240 FP 31-Ga-81 80.9378 0.00E+00 5.70E-01 + 1241 FP 32-Ge-81 80.9288 0.00E+00 9.12E-02 + 1242 FP 32-Ge-81m 80.9288 0.00E+00 9.12E-02 + 1243 FP 33-As-81 80.9221 0.00E+00 2.08E-02 + 1244 FP 34-Se-81 80.9180 0.00E+00 6.26E-04 + 1245 FP 34-Se-81m 80.9180 0.00E+00 2.02E-04 + 1246 FP 35-Br-81 80.9163 0.00E+00 0.00E+00 + 1247 FP 36-Kr-81 80.9166 0.00E+00 9.59E-14 + 1248 FP 36-Kr-81m 80.9166 0.00E+00 5.29E-02 + 1249 FP 37-Rb-81 80.9190 0.00E+00 4.21E-05 + 1250 FP 30-Zn-82 81.9544 0.00E+00 1.33E+01 + 1251 FP 31-Ga-82 81.9430 0.00E+00 1.16E+00 + 1252 FP 32-Ge-82 81.9296 0.00E+00 1.52E-01 + 1253 FP 33-As-82 81.9245 0.00E+00 3.63E-02 + 1254 FP 33-As-82m 81.9245 0.00E+00 5.10E-02 + 1255 FP 34-Se-82 81.9167 0.00E+00 0.00E+00 + 1256 FP 35-Br-82 81.9168 0.00E+00 5.46E-06 + 1257 FP 35-Br-82m 81.9168 0.00E+00 1.88E-03 + 1258 FP 36-Kr-82 81.9135 0.00E+00 0.00E+00 + 1259 FP 30-Zn-83 82.9610 0.00E+00 1.61E+01 + 1260 FP 31-Ga-83 82.9470 0.00E+00 2.25E+00 + 1261 FP 32-Ge-83 82.9346 0.00E+00 3.75E-01 + 1262 FP 33-As-83 82.9250 0.00E+00 5.17E-02 + 1263 FP 34-Se-83 82.9191 0.00E+00 5.18E-04 + 1264 FP 34-Se-83m 82.9191 0.00E+00 9.89E-03 + 1265 FP 35-Br-83 82.9152 0.00E+00 8.02E-05 + 1266 FP 36-Kr-83 82.9141 0.00E+00 0.00E+00 + 1267 FP 36-Kr-83m 82.9141 0.00E+00 1.05E-04 + 1268 FP 37-Rb-83 82.9151 0.00E+00 9.31E-08 + 1269 FP 38-Sr-83 82.9176 0.00E+00 5.94E-06 + 1270 FP 31-Ga-84 83.9527 0.00E+00 8.15E+00 + 1271 FP 32-Ge-84 83.9375 0.00E+00 7.27E-01 + 1272 FP 33-As-84 83.9291 0.00E+00 1.65E-01 + 1273 FP 34-Se-84 83.9185 0.00E+00 3.54E-03 + 1274 FP 35-Br-84 83.9165 0.00E+00 3.64E-04 + 1275 FP 35-Br-84m 83.9165 0.00E+00 1.93E-03 + 1276 FP 36-Kr-84 83.9115 0.00E+00 0.00E+00 + 1277 FP 37-Rb-84 83.9144 0.00E+00 2.44E-07 + 1278 FP 38-Sr-84 83.9134 0.00E+00 0.00E+00 + 1279 FP 31-Ga-85 84.9570 0.00E+00 1.44E+01 + 1280 FP 32-Ge-85 84.9430 0.00E+00 1.30E+00 + 1281 FP 33-As-85 84.9320 0.00E+00 3.43E-01 + 1282 FP 34-Se-85 84.9222 0.00E+00 2.19E-02 + 1283 FP 35-Br-85 84.9156 0.00E+00 3.98E-03 + 1284 FP 36-Kr-85 84.9125 0.00E+00 2.04E-09 + 1285 FP 36-Kr-85m 84.9125 0.00E+00 4.30E-05 + 1286 FP 37-Rb-85 84.9118 0.00E+00 0.00E+00 + 1287 FP 38-Sr-85 84.9129 0.00E+00 1.24E-07 + 1288 FP 38-Sr-85m 84.9129 0.00E+00 1.71E-04 + 1289 FP 39-Y-85 84.9164 0.00E+00 7.18E-05 + 1290 FP 31-Ga-86 85.9631 0.00E+00 2.39E+01 + 1291 FP 32-Ge-86 85.9465 0.00E+00 7.30E+00 + 1292 FP 33-As-86 85.9365 0.00E+00 7.33E-01 + 1293 FP 34-Se-86 85.9243 0.00E+00 4.85E-02 + 1294 FP 35-Br-86 85.9188 0.00E+00 1.26E-02 + 1295 FP 36-Kr-86 85.9106 0.00E+00 0.00E+00 + 1296 FP 37-Rb-86 85.9112 0.00E+00 4.31E-07 + 1297 FP 37-Rb-86m 85.9112 0.00E+00 1.14E-02 + 1298 FP 38-Sr-86 85.9093 0.00E+00 0.00E+00 + 1299 FP 32-Ge-87 86.9525 0.00E+00 4.95E+00 + 1300 FP 33-As-87 86.9399 0.00E+00 1.24E+00 + 1301 FP 34-Se-87 86.9285 0.00E+00 1.26E-01 + 1302 FP 35-Br-87 86.9207 0.00E+00 1.25E-02 + 1303 FP 36-Kr-87 86.9134 0.00E+00 1.51E-04 + 1304 FP 37-Rb-87 86.9092 0.00E+00 4.57E-19 + 1305 FP 38-Sr-87 86.9089 0.00E+00 0.00E+00 + 1306 FP 38-Sr-87m 86.9089 0.00E+00 6.84E-05 + 1307 FP 39-Y-87 86.9109 0.00E+00 2.41E-06 + 1308 FP 39-Y-87m 86.9109 0.00E+00 1.44E-05 + 1309 FP 40-Zr-87 86.9148 0.00E+00 1.15E-04 + 1310 FP 32-Ge-88 87.9569 0.00E+00 1.05E+01 + 1311 FP 33-As-88 87.9449 0.00E+00 6.19E+00 + 1312 FP 34-Se-88 87.9314 0.00E+00 4.53E-01 + 1313 FP 35-Br-88 87.9241 0.00E+00 4.26E-02 + 1314 FP 36-Kr-88 87.9145 0.00E+00 6.78E-05 + 1315 FP 37-Rb-88 87.9113 0.00E+00 6.50E-04 + 1316 FP 38-Sr-88 87.9056 0.00E+00 0.00E+00 + 1317 FP 39-Y-88 87.9095 0.00E+00 7.52E-08 + 1318 FP 40-Zr-88 87.9102 0.00E+00 9.62E-08 + 1319 FP 32-Ge-89 88.9638 0.00E+00 1.78E+01 + 1320 FP 33-As-89 88.9494 0.00E+00 1.17E+01 + 1321 FP 34-Se-89 88.9364 0.00E+00 1.69E+00 + 1322 FP 35-Br-89 88.9264 0.00E+00 1.58E-01 + 1323 FP 36-Kr-89 88.9176 0.00E+00 3.67E-03 + 1324 FP 37-Rb-89 88.9123 0.00E+00 7.63E-04 + 1325 FP 38-Sr-89 88.9074 0.00E+00 1.59E-07 + 1326 FP 39-Y-89 88.9059 0.00E+00 0.00E+00 + 1327 FP 39-Y-89m 88.9059 0.00E+00 4.43E-02 + 1328 FP 40-Zr-89 88.9089 0.00E+00 2.46E-06 + 1329 FP 40-Zr-89m 88.9089 0.00E+00 2.78E-03 + 1330 FP 41-Nb-89 88.9134 0.00E+00 9.48E-05 + 1331 FP 33-As-90 89.9555 0.00E+00 1.61E+01 + 1332 FP 34-Se-90 89.9400 0.00E+00 4.31E+00 + 1333 FP 35-Br-90 89.9306 0.00E+00 3.61E-01 + 1334 FP 36-Kr-90 89.9195 0.00E+00 2.14E-02 + 1335 FP 37-Rb-90 89.9148 0.00E+00 4.39E-03 + 1336 FP 37-Rb-90m 89.9148 0.00E+00 2.69E-03 + 1337 FP 38-Sr-90 89.9077 0.00E+00 7.63E-10 + 1338 FP 39-Y-90 89.9072 0.00E+00 3.01E-06 + 1339 FP 39-Y-90m 89.9072 0.00E+00 6.04E-05 + 1340 FP 40-Zr-90 89.9047 0.00E+00 0.00E+00 + 1341 FP 40-Zr-90m 89.9047 0.00E+00 8.57E-01 + 1342 FP 41-Nb-90 89.9113 0.00E+00 1.32E-05 + 1343 FP 41-Nb-90m 89.9113 0.00E+00 3.69E-02 + 1344 FP 42-Mo-90 89.9139 0.00E+00 3.40E-05 + 1345 FP 33-As-91 90.9604 0.00E+00 1.58E+01 + 1346 FP 34-Se-91 90.9460 0.00E+00 2.57E+00 + 1347 FP 35-Br-91 90.9340 0.00E+00 1.28E+00 + 1348 FP 36-Kr-91 90.9234 0.00E+00 8.09E-02 + 1349 FP 37-Rb-91 90.9165 0.00E+00 1.19E-02 + 1350 FP 38-Sr-91 90.9102 0.00E+00 2.00E-05 + 1351 FP 39-Y-91 90.9073 0.00E+00 1.37E-07 + 1352 FP 39-Y-91m 90.9073 0.00E+00 2.32E-04 + 1353 FP 40-Zr-91 90.9056 0.00E+00 0.00E+00 + 1354 FP 41-Nb-91 90.9070 0.00E+00 3.23E-11 + 1355 FP 41-Nb-91m 90.9070 0.00E+00 1.32E-07 + 1356 FP 42-Mo-91 90.9118 0.00E+00 7.46E-04 + 1357 FP 33-As-92 91.9668 0.00E+00 2.57E+01 + 1358 FP 34-Se-92 91.9499 0.00E+00 7.45E+00 + 1359 FP 35-Br-92 91.9393 0.00E+00 2.02E+00 + 1360 FP 36-Kr-92 91.9262 0.00E+00 3.77E-01 + 1361 FP 37-Rb-92 91.9197 0.00E+00 1.54E-01 + 1362 FP 38-Sr-92 91.9110 0.00E+00 7.10E-05 + 1363 FP 39-Y-92 91.9090 0.00E+00 5.44E-05 + 1364 FP 40-Zr-92 91.9050 0.00E+00 0.00E+00 + 1365 FP 41-Nb-92 91.9072 0.00E+00 6.33E-16 + 1366 FP 41-Nb-92m 91.9072 0.00E+00 7.90E-07 + 1367 FP 42-Mo-92 91.9068 0.00E+00 0.00E+00 + 1368 FP 34-Se-93 92.9563 0.00E+00 1.12E+01 + 1369 FP 35-Br-93 92.9430 0.00E+00 6.80E+00 + 1370 FP 36-Kr-93 92.9313 0.00E+00 5.39E-01 + 1371 FP 37-Rb-93 92.9220 0.00E+00 1.19E-01 + 1372 FP 38-Sr-93 92.9140 0.00E+00 1.56E-03 + 1373 FP 39-Y-93 92.9096 0.00E+00 1.89E-05 + 1374 FP 39-Y-93m 92.9096 0.00E+00 8.45E-01 + 1375 FP 40-Zr-93 92.9065 0.00E+00 1.44E-14 + 1376 FP 41-Nb-93 92.9064 0.00E+00 0.00E+00 + 1377 FP 41-Nb-93m 92.9064 0.00E+00 1.36E-09 + 1378 FP 42-Mo-93 92.9068 0.00E+00 5.49E-12 + 1379 FP 42-Mo-93m 92.9068 0.00E+00 2.81E-05 + 1380 FP 43-Tc-93 92.9102 0.00E+00 7.00E-05 + 1381 FP 34-Se-94 93.9605 0.00E+00 1.17E+01 + 1382 FP 35-Br-94 93.9487 0.00E+00 9.90E+00 + 1383 FP 36-Kr-94 93.9344 0.00E+00 3.27E+00 + 1384 FP 37-Rb-94 93.9264 0.00E+00 2.57E-01 + 1385 FP 38-Sr-94 93.9154 0.00E+00 9.20E-03 + 1386 FP 39-Y-94 93.9116 0.00E+00 6.18E-04 + 1387 FP 40-Zr-94 93.9063 0.00E+00 0.00E+00 + 1388 FP 41-Nb-94 93.9073 0.00E+00 1.08E-12 + 1389 FP 41-Nb-94m 93.9073 0.00E+00 1.84E-03 + 1390 FP 42-Mo-94 93.9051 0.00E+00 0.00E+00 + 1391 FP 35-Br-95 94.9529 0.00E+00 1.05E+01 + 1392 FP 36-Kr-95 94.9398 0.00E+00 6.08E+00 + 1393 FP 37-Rb-95 94.9293 0.00E+00 1.84E+00 + 1394 FP 38-Sr-95 94.9194 0.00E+00 2.90E-02 + 1395 FP 39-Y-95 94.9128 0.00E+00 1.12E-03 + 1396 FP 40-Zr-95 94.9080 0.00E+00 1.25E-07 + 1397 FP 41-Nb-95 94.9068 0.00E+00 2.29E-07 + 1398 FP 41-Nb-95m 94.9068 0.00E+00 2.22E-06 + 1399 FP 42-Mo-95 94.9058 0.00E+00 0.00E+00 + 1400 FP 43-Tc-95 94.9077 0.00E+00 9.63E-06 + 1401 FP 43-Tc-95m 94.9077 0.00E+00 1.32E-07 + 1402 FP 44-Ru-95 94.9104 0.00E+00 1.17E-04 + 1403 FP 35-Br-96 95.9585 0.00E+00 1.65E+01 + 1404 FP 36-Kr-96 95.9431 0.00E+00 8.66E+00 + 1405 FP 37-Rb-96 95.9343 0.00E+00 3.41E+00 + 1406 FP 38-Sr-96 95.9217 0.00E+00 6.48E-01 + 1407 FP 39-Y-96 95.9159 0.00E+00 1.30E-01 + 1408 FP 39-Y-96m 95.9159 0.00E+00 7.22E-02 + 1409 FP 40-Zr-96 95.9083 0.00E+00 1.10E-27 + 1410 FP 41-Nb-96 95.9081 0.00E+00 8.25E-06 + 1411 FP 42-Mo-96 95.9047 0.00E+00 0.00E+00 + 1412 FP 43-Tc-96 95.9079 0.00E+00 1.87E-06 + 1413 FP 44-Ru-96 95.9076 0.00E+00 0.00E+00 + 1414 FP 35-Br-97 96.9628 0.00E+00 1.73E+01 + 1415 FP 36-Kr-97 96.9486 0.00E+00 1.10E+01 + 1416 FP 37-Rb-97 96.9373 0.00E+00 4.10E+00 + 1417 FP 38-Sr-97 96.9261 0.00E+00 1.62E+00 + 1418 FP 39-Y-97 96.9181 0.00E+00 1.85E-01 + 1419 FP 39-Y-97m 96.9181 0.00E+00 5.92E-01 + 1420 FP 40-Zr-97 96.9109 0.00E+00 1.15E-05 + 1421 FP 41-Nb-97 96.9081 0.00E+00 1.60E-04 + 1422 FP 41-Nb-97m 96.9081 0.00E+00 1.18E-02 + 1423 FP 42-Mo-97 96.9060 0.00E+00 0.00E+00 + 1424 FP 43-Tc-97 96.9064 0.00E+00 5.22E-15 + 1425 FP 43-Tc-97m 96.9064 0.00E+00 8.82E-08 + 1426 FP 44-Ru-97 96.9076 0.00E+00 2.83E-06 + 1427 FP 36-Kr-98 97.9519 0.00E+00 1.51E+01 + 1428 FP 37-Rb-98 97.9418 0.00E+00 6.08E+00 + 1429 FP 38-Sr-98 97.9285 0.00E+00 1.06E+00 + 1430 FP 39-Y-98 97.9222 0.00E+00 1.26E+00 + 1431 FP 39-Y-98m 97.9222 0.00E+00 3.47E-01 + 1432 FP 40-Zr-98 97.9127 0.00E+00 2.26E-02 + 1433 FP 41-Nb-98 97.9103 0.00E+00 2.42E-01 + 1434 FP 41-Nb-98m 97.9103 0.00E+00 2.25E-04 + 1435 FP 42-Mo-98 97.9054 0.00E+00 0.00E+00 + 1436 FP 43-Tc-98 97.9072 0.00E+00 5.23E-15 + 1437 FP 44-Ru-98 97.9053 0.00E+00 0.00E+00 + 1438 FP 36-Kr-99 98.9576 0.00E+00 2.57E+01 + 1439 FP 37-Rb-99 98.9454 0.00E+00 1.28E+01 + 1440 FP 38-Sr-99 98.9332 0.00E+00 2.57E+00 + 1441 FP 39-Y-99 98.9246 0.00E+00 4.72E-01 + 1442 FP 40-Zr-99 98.9165 0.00E+00 3.30E-01 + 1443 FP 41-Nb-99 98.9116 0.00E+00 4.62E-02 + 1444 FP 41-Nb-99m 98.9116 0.00E+00 4.62E-03 + 1445 FP 42-Mo-99 98.9077 0.00E+00 2.92E-06 + 1446 FP 43-Tc-99 98.9062 0.00E+00 1.04E-13 + 1447 FP 43-Tc-99m 98.9062 0.00E+00 3.21E-05 + 1448 FP 44-Ru-99 98.9059 0.00E+00 0.00E+00 + 1449 FP 45-Rh-99 98.9081 0.00E+00 4.98E-07 + 1450 FP 45-Rh-99m 98.9081 0.00E+00 4.10E-05 + 1451 FP 46-Pd-99 98.9118 0.00E+00 5.40E-04 + 1452 FP 36-Kr-100 99.9611 0.00E+00 9.90E+01 + 1453 FP 37-Rb-100 99.9499 0.00E+00 1.36E+01 + 1454 FP 38-Sr-100 99.9353 0.00E+00 3.43E+00 + 1455 FP 39-Y-100 99.9278 0.00E+00 9.43E-01 + 1456 FP 40-Zr-100 99.9178 0.00E+00 9.76E-02 + 1457 FP 41-Nb-100 99.9142 0.00E+00 4.62E-01 + 1458 FP 41-Nb-100m 99.9142 0.00E+00 2.32E-01 + 1459 FP 42-Mo-100 99.9075 0.00E+00 3.01E-27 + 1460 FP 43-Tc-100 99.9077 0.00E+00 4.48E-02 + 1461 FP 44-Ru-100 99.9042 0.00E+00 0.00E+00 + 1462 FP 37-Rb-101 100.9532 0.00E+00 2.17E+01 + 1463 FP 38-Sr-101 100.9405 0.00E+00 5.87E+00 + 1464 FP 39-Y-101 100.9303 0.00E+00 1.54E+00 + 1465 FP 40-Zr-101 100.9211 0.00E+00 3.01E-01 + 1466 FP 41-Nb-101 100.9153 0.00E+00 9.76E-02 + 1467 FP 42-Mo-101 100.9103 0.00E+00 7.91E-04 + 1468 FP 43-Tc-101 100.9073 0.00E+00 8.14E-04 + 1469 FP 44-Ru-101 100.9056 0.00E+00 0.00E+00 + 1470 FP 45-Rh-101 100.9062 0.00E+00 6.66E-09 + 1471 FP 45-Rh-101m 100.9062 0.00E+00 1.85E-06 + 1472 FP 46-Pd-101 100.9083 0.00E+00 2.27E-05 + 1473 FP 37-Rb-102 101.9589 0.00E+00 1.87E+01 + 1474 FP 38-Sr-102 101.9430 0.00E+00 1.00E+01 + 1475 FP 39-Y-102 101.9336 0.00E+00 1.93E+00 + 1476 FP 40-Zr-102 101.9230 0.00E+00 2.39E-01 + 1477 FP 41-Nb-102 101.9180 0.00E+00 1.61E-01 + 1478 FP 41-Nb-102m 101.9180 0.00E+00 5.33E-01 + 1479 FP 42-Mo-102 101.9103 0.00E+00 1.02E-03 + 1480 FP 43-Tc-102 101.9092 0.00E+00 1.31E-01 + 1481 FP 43-Tc-102m 101.9092 0.00E+00 2.66E-03 + 1482 FP 44-Ru-102 101.9044 0.00E+00 0.00E+00 + 1483 FP 45-Rh-102 101.9068 0.00E+00 3.87E-08 + 1484 FP 45-Rh-102m 101.9068 0.00E+00 5.87E-09 + 1485 FP 46-Pd-102 101.9056 0.00E+00 0.00E+00 + 1486 FP 38-Sr-103 102.9490 0.00E+00 1.02E+01 + 1487 FP 39-Y-103 102.9367 0.00E+00 3.01E+00 + 1488 FP 40-Zr-103 102.9266 0.00E+00 5.33E-01 + 1489 FP 41-Nb-103 102.9191 0.00E+00 4.62E-01 + 1490 FP 42-Mo-103 102.9132 0.00E+00 1.03E-02 + 1491 FP 43-Tc-103 102.9092 0.00E+00 1.28E-02 + 1492 FP 44-Ru-103 102.9063 0.00E+00 2.04E-07 + 1493 FP 45-Rh-103 102.9055 0.00E+00 0.00E+00 + 1494 FP 45-Rh-103m 102.9055 0.00E+00 2.06E-04 + 1495 FP 46-Pd-103 102.9061 0.00E+00 4.72E-07 + 1496 FP 47-Ag-103 102.9090 0.00E+00 1.76E-04 + 1497 FP 38-Sr-104 103.9523 0.00E+00 1.61E+01 + 1498 FP 39-Y-104 103.9410 0.00E+00 3.85E+00 + 1499 FP 40-Zr-104 103.9288 0.00E+00 5.78E-01 + 1500 FP 41-Nb-104 103.9225 0.00E+00 1.41E-01 + 1501 FP 41-Nb-104m 103.9225 0.00E+00 7.37E-01 + 1502 FP 42-Mo-104 103.9138 0.00E+00 1.16E-02 + 1503 FP 43-Tc-104 103.9115 0.00E+00 6.31E-04 + 1504 FP 44-Ru-104 103.9054 0.00E+00 0.00E+00 + 1505 FP 45-Rh-104 103.9067 0.00E+00 1.64E-02 + 1506 FP 45-Rh-104m 103.9067 0.00E+00 2.66E-03 + 1507 FP 46-Pd-104 103.9040 0.00E+00 0.00E+00 + 1508 FP 38-Sr-105 104.9586 0.00E+00 1.25E+01 + 1509 FP 39-Y-105 104.9449 0.00E+00 7.88E+00 + 1510 FP 40-Zr-105 104.9331 0.00E+00 1.16E+00 + 1511 FP 41-Nb-105 104.9239 0.00E+00 2.35E-01 + 1512 FP 42-Mo-105 104.9170 0.00E+00 1.95E-02 + 1513 FP 43-Tc-105 104.9117 0.00E+00 1.52E-03 + 1514 FP 44-Ru-105 104.9078 0.00E+00 4.34E-05 + 1515 FP 45-Rh-105 104.9057 0.00E+00 5.45E-06 + 1516 FP 45-Rh-105m 104.9057 0.00E+00 1.73E-02 + 1517 FP 46-Pd-105 104.9051 0.00E+00 0.00E+00 + 1518 FP 47-Ag-105 104.9065 0.00E+00 1.94E-07 + 1519 FP 47-Ag-105m 104.9065 0.00E+00 1.60E-03 + 1520 FP 48-Cd-105 104.9095 0.00E+00 2.08E-04 + 1521 FP 39-Y-106 105.9498 0.00E+00 1.05E+01 + 1522 FP 40-Zr-106 105.9359 0.00E+00 2.57E+00 + 1523 FP 41-Nb-106 105.9280 0.00E+00 7.45E-01 + 1524 FP 42-Mo-106 105.9181 0.00E+00 7.94E-02 + 1525 FP 43-Tc-106 105.9144 0.00E+00 1.95E-02 + 1526 FP 44-Ru-106 105.9073 0.00E+00 2.16E-08 + 1527 FP 45-Rh-106 105.9073 0.00E+00 2.31E-02 + 1528 FP 45-Rh-106m 105.9073 0.00E+00 8.82E-05 + 1529 FP 46-Pd-106 105.9035 0.00E+00 0.00E+00 + 1530 FP 47-Ag-106 105.9067 0.00E+00 4.82E-04 + 1531 FP 47-Ag-106m 105.9067 0.00E+00 9.69E-07 + 1532 FP 48-Cd-106 105.9065 0.00E+00 0.00E+00 + 1533 FP 39-Y-107 106.9541 0.00E+00 2.31E+01 + 1534 FP 40-Zr-107 106.9408 0.00E+00 4.62E+00 + 1535 FP 41-Nb-107 106.9303 0.00E+00 2.31E+00 + 1536 FP 42-Mo-107 106.9217 0.00E+00 1.98E-01 + 1537 FP 43-Tc-107 106.9151 0.00E+00 3.27E-02 + 1538 FP 44-Ru-107 106.9099 0.00E+00 3.08E-03 + 1539 FP 45-Rh-107 106.9068 0.00E+00 5.32E-04 + 1540 FP 46-Pd-107 106.9051 0.00E+00 3.38E-15 + 1541 FP 46-Pd-107m 106.9051 0.00E+00 3.25E-02 + 1542 FP 47-Ag-107 106.9051 0.00E+00 0.00E+00 + 1543 FP 47-Ag-107m 106.9051 0.00E+00 1.56E-02 + 1544 FP 48-Cd-107 106.9066 0.00E+00 2.96E-05 + 1545 FP 49-In-107 106.9103 0.00E+00 3.57E-04 + 1546 FP 39-Y-108 107.9595 0.00E+00 1.44E+01 + 1547 FP 40-Zr-108 107.9440 0.00E+00 8.66E+00 + 1548 FP 41-Nb-108 107.9348 0.00E+00 3.59E+00 + 1549 FP 42-Mo-108 107.9234 0.00E+00 6.36E-01 + 1550 FP 43-Tc-108 107.9185 0.00E+00 1.34E-01 + 1551 FP 44-Ru-108 107.9102 0.00E+00 2.54E-03 + 1552 FP 45-Rh-108 107.9087 0.00E+00 4.13E-02 + 1553 FP 45-Rh-108m 107.9087 0.00E+00 1.93E-03 + 1554 FP 46-Pd-108 107.9039 0.00E+00 0.00E+00 + 1555 FP 47-Ag-108 107.9060 0.00E+00 4.85E-03 + 1556 FP 47-Ag-108m 107.9060 0.00E+00 5.01E-11 + 1557 FP 48-Cd-108 107.9042 0.00E+00 0.00E+00 + 1558 FP 40-Zr-109 108.9492 0.00E+00 5.92E+00 + 1559 FP 41-Nb-109 108.9376 0.00E+00 3.65E+00 + 1560 FP 42-Mo-109 108.9278 0.00E+00 1.31E+00 + 1561 FP 43-Tc-109 108.9200 0.00E+00 8.06E-01 + 1562 FP 44-Ru-109 108.9132 0.00E+00 2.01E-02 + 1563 FP 45-Rh-109 108.9087 0.00E+00 8.66E-03 + 1564 FP 46-Pd-109 108.9060 0.00E+00 1.41E-05 + 1565 FP 46-Pd-109m 108.9060 0.00E+00 2.46E-03 + 1566 FP 47-Ag-109 108.9047 0.00E+00 0.00E+00 + 1567 FP 47-Ag-109m 108.9047 0.00E+00 1.75E-02 + 1568 FP 48-Cd-109 108.9050 0.00E+00 1.74E-08 + 1569 FP 49-In-109 108.9072 0.00E+00 4.62E-05 + 1570 FP 40-Zr-110 109.9529 0.00E+00 7.07E+00 + 1571 FP 41-Nb-110 109.9424 0.00E+00 4.08E+00 + 1572 FP 42-Mo-110 109.9297 0.00E+00 2.31E+00 + 1573 FP 43-Tc-110 109.9238 0.00E+00 7.53E-01 + 1574 FP 44-Ru-110 109.9141 0.00E+00 5.98E-02 + 1575 FP 45-Rh-110 109.9111 0.00E+00 2.17E-01 + 1576 FP 45-Rh-110m 109.9111 0.00E+00 2.43E-02 + 1577 FP 46-Pd-110 109.9052 0.00E+00 0.00E+00 + 1578 FP 47-Ag-110 109.9061 0.00E+00 2.82E-02 + 1579 FP 47-Ag-110m 109.9062 0.00E+00 3.21E-08 + 1580 FP 48-Cd-110 109.9030 0.00E+00 0.00E+00 + 1581 FP 41-Nb-111 110.9456 0.00E+00 8.66E+00 + 1582 FP 42-Mo-111 110.9344 0.00E+00 3.47E+00 + 1583 FP 43-Tc-111 110.9257 0.00E+00 2.39E+00 + 1584 FP 44-Ru-111 110.9177 0.00E+00 3.27E-01 + 1585 FP 45-Rh-111 110.9116 0.00E+00 6.30E-02 + 1586 FP 46-Pd-111 110.9077 0.00E+00 4.94E-04 + 1587 FP 46-Pd-111m 110.9077 0.00E+00 3.50E-05 + 1588 FP 47-Ag-111 110.9053 0.00E+00 1.08E-06 + 1589 FP 47-Ag-111m 110.9053 0.00E+00 1.07E-02 + 1590 FP 48-Cd-111 110.9042 0.00E+00 0.00E+00 + 1591 FP 48-Cd-111m 110.9042 0.00E+00 2.38E-04 + 1592 FP 49-In-111 110.9051 0.00E+00 2.86E-06 + 1593 FP 49-In-111m 110.9051 0.00E+00 1.50E-03 + 1594 FP 50-Sn-111 110.9077 0.00E+00 3.27E-04 + 1595 FP 41-Nb-112 111.9508 0.00E+00 1.00E+01 + 1596 FP 42-Mo-112 111.9368 0.00E+00 2.42E+00 + 1597 FP 43-Tc-112 111.9292 0.00E+00 2.48E+00 + 1598 FP 44-Ru-112 111.9190 0.00E+00 3.96E-01 + 1599 FP 45-Rh-112 111.9144 0.00E+00 3.30E-01 + 1600 FP 46-Pd-112 111.9073 0.00E+00 9.16E-06 + 1601 FP 47-Ag-112 111.9070 0.00E+00 6.15E-05 + 1602 FP 48-Cd-112 111.9028 0.00E+00 0.00E+00 + 1603 FP 49-In-112 111.9055 0.00E+00 7.72E-04 + 1604 FP 49-In-112m 111.9055 0.00E+00 5.62E-04 + 1605 FP 50-Sn-112 111.9048 0.00E+00 0.00E+00 + 1606 FP 41-Nb-113 112.9547 0.00E+00 2.31E+01 + 1607 FP 42-Mo-113 112.9419 0.00E+00 6.93E+00 + 1608 FP 43-Tc-113 112.9316 0.00E+00 4.33E+00 + 1609 FP 44-Ru-113 112.9225 0.00E+00 8.66E-01 + 1610 FP 45-Rh-113 112.9155 0.00E+00 2.48E-01 + 1611 FP 46-Pd-113 112.9101 0.00E+00 7.45E-03 + 1612 FP 47-Ag-113 112.9066 0.00E+00 3.59E-05 + 1613 FP 47-Ag-113m 112.9066 0.00E+00 1.01E-02 + 1614 FP 48-Cd-113 112.9044 0.00E+00 2.73E-24 + 1615 FP 48-Cd-113m 112.9044 0.00E+00 1.56E-09 + 1616 FP 49-In-113 112.9041 0.00E+00 0.00E+00 + 1617 FP 49-In-113m 112.9041 0.00E+00 1.16E-04 + 1618 FP 50-Sn-113 112.9052 0.00E+00 6.97E-08 + 1619 FP 50-Sn-113m 112.9052 0.00E+00 5.40E-04 + 1620 FP 51-Sb-113 112.9094 0.00E+00 1.73E-03 + 1621 FP 42-Mo-114 113.9449 0.00E+00 8.66E+00 + 1622 FP 43-Tc-114 113.9359 0.00E+00 4.62E+00 + 1623 FP 44-Ru-114 113.9243 0.00E+00 1.33E+00 + 1624 FP 45-Rh-114 113.9188 0.00E+00 3.75E-01 + 1625 FP 46-Pd-114 113.9104 0.00E+00 4.77E-03 + 1626 FP 47-Ag-114 113.9088 0.00E+00 1.51E-01 + 1627 FP 48-Cd-114 113.9034 0.00E+00 0.00E+00 + 1628 FP 49-In-114 113.9049 0.00E+00 9.64E-03 + 1629 FP 49-In-114m 113.9049 0.00E+00 1.62E-07 + 1630 FP 50-Sn-114 113.9028 0.00E+00 0.00E+00 + 1631 FP 42-Mo-115 114.9503 0.00E+00 7.53E+00 + 1632 FP 43-Tc-115 114.9387 0.00E+00 9.50E+00 + 1633 FP 44-Ru-115 114.9287 0.00E+00 9.37E-01 + 1634 FP 45-Rh-115 114.9203 0.00E+00 7.00E-01 + 1635 FP 46-Pd-115 114.9137 0.00E+00 2.77E-02 + 1636 FP 47-Ag-115 114.9088 0.00E+00 5.78E-04 + 1637 FP 47-Ag-115m 114.9088 0.00E+00 3.85E-02 + 1638 FP 48-Cd-115 114.9054 0.00E+00 3.60E-06 + 1639 FP 48-Cd-115m 114.9051 0.00E+00 1.80E-07 + 1640 FP 49-In-115 114.9039 0.00E+00 4.98E-23 + 1641 FP 49-In-115m 114.9039 0.00E+00 4.29E-05 + 1642 FP 50-Sn-115 114.9033 0.00E+00 0.00E+00 + 1643 FP 51-Sb-115 114.9066 0.00E+00 3.60E-04 + 1644 FP 52-Te-115 114.9119 0.00E+00 1.99E-03 + 1645 FP 43-Tc-116 115.9434 0.00E+00 7.70E+00 + 1646 FP 44-Ru-116 115.9308 0.00E+00 3.40E+00 + 1647 FP 45-Rh-116 115.9241 0.00E+00 1.02E+00 + 1648 FP 46-Pd-116 115.9142 0.00E+00 5.87E-02 + 1649 FP 47-Ag-116 115.9114 0.00E+00 2.92E-03 + 1650 FP 47-Ag-116m 115.9114 0.00E+00 3.47E-02 + 1651 FP 48-Cd-116 115.9048 0.00E+00 7.09E-28 + 1652 FP 49-In-116 115.9053 0.00E+00 4.92E-02 + 1653 FP 49-In-116m 115.9053 0.00E+00 2.13E-04 + 1654 FP 50-Sn-116 115.9017 0.00E+00 0.00E+00 + 1655 FP 43-Tc-117 116.9465 0.00E+00 1.73E+01 + 1656 FP 44-Ru-117 116.9356 0.00E+00 4.88E+00 + 1657 FP 45-Rh-117 116.9260 0.00E+00 1.58E+00 + 1658 FP 46-Pd-117 116.9178 0.00E+00 1.61E-01 + 1659 FP 47-Ag-117 116.9117 0.00E+00 9.52E-03 + 1660 FP 47-Ag-117m 116.9117 0.00E+00 1.30E-01 + 1661 FP 48-Cd-117 116.9072 0.00E+00 7.73E-05 + 1662 FP 48-Cd-117m 116.9072 0.00E+00 5.73E-05 + 1663 FP 49-In-117 116.9045 0.00E+00 2.67E-04 + 1664 FP 49-In-117m 116.9045 0.00E+00 9.94E-05 + 1665 FP 50-Sn-117 116.9029 0.00E+00 0.00E+00 + 1666 FP 50-Sn-117m 116.9029 0.00E+00 5.90E-07 + 1667 FP 51-Sb-117 116.9048 0.00E+00 6.88E-05 + 1668 FP 52-Te-117 116.9087 0.00E+00 1.86E-04 + 1669 FP 43-Tc-118 117.9515 0.00E+00 1.05E+01 + 1670 FP 44-Ru-118 117.9378 0.00E+00 5.64E+00 + 1671 FP 45-Rh-118 117.9301 0.00E+00 2.61E+00 + 1672 FP 46-Pd-118 117.9190 0.00E+00 3.65E-01 + 1673 FP 47-Ag-118 117.9146 0.00E+00 1.84E-01 + 1674 FP 47-Ag-118m 117.9146 0.00E+00 3.47E-01 + 1675 FP 48-Cd-118 117.9069 0.00E+00 2.30E-04 + 1676 FP 49-In-118 117.9063 0.00E+00 1.39E-01 + 1677 FP 49-In-118m 117.9063 0.00E+00 2.60E-03 + 1678 FP 50-Sn-118 117.9016 0.00E+00 0.00E+00 + 1679 FP 51-Sb-118 117.9055 0.00E+00 3.21E-03 + 1680 FP 51-Sb-118m 117.9055 0.00E+00 3.85E-05 + 1681 FP 52-Te-118 117.9058 0.00E+00 1.34E-06 + 1682 FP 44-Ru-119 118.9428 0.00E+00 4.28E+00 + 1683 FP 45-Rh-119 118.9321 0.00E+00 4.05E+00 + 1684 FP 46-Pd-119 118.9231 0.00E+00 7.53E-01 + 1685 FP 47-Ag-119 118.9157 0.00E+00 3.30E-01 + 1686 FP 48-Cd-119 118.9099 0.00E+00 4.29E-03 + 1687 FP 48-Cd-119m 118.9099 0.00E+00 5.25E-03 + 1688 FP 49-In-119 118.9059 0.00E+00 4.81E-03 + 1689 FP 49-In-119m 118.9059 0.00E+00 6.42E-04 + 1690 FP 50-Sn-119 118.9033 0.00E+00 0.00E+00 + 1691 FP 50-Sn-119m 118.9033 0.00E+00 2.74E-08 + 1692 FP 51-Sb-119 118.9039 0.00E+00 5.04E-06 + 1693 FP 52-Te-119 118.9064 0.00E+00 1.20E-05 + 1694 FP 44-Ru-120 119.9453 0.00E+00 4.65E+00 + 1695 FP 45-Rh-120 119.9364 0.00E+00 5.10E+00 + 1696 FP 46-Pd-120 119.9247 0.00E+00 1.39E+00 + 1697 FP 47-Ag-120 119.9188 0.00E+00 5.64E-01 + 1698 FP 47-Ag-120m 119.9188 0.00E+00 2.17E+00 + 1699 FP 48-Cd-120 119.9099 0.00E+00 1.36E-02 + 1700 FP 49-In-120 119.9080 0.00E+00 2.25E-01 + 1701 FP 49-In-120m 119.9080 0.00E+00 1.50E-02 + 1702 FP 50-Sn-120 119.9022 0.00E+00 0.00E+00 + 1703 FP 51-Sb-120 119.9051 0.00E+00 7.27E-04 + 1704 FP 51-Sb-120m 119.9051 0.00E+00 1.39E-06 + 1705 FP 52-Te-120 119.9040 0.00E+00 0.00E+00 + 1706 FP 45-Rh-121 120.9387 0.00E+00 4.59E+00 + 1707 FP 46-Pd-121 120.9289 0.00E+00 2.43E+00 + 1708 FP 47-Ag-121 120.9199 0.00E+00 8.89E-01 + 1709 FP 48-Cd-121 120.9130 0.00E+00 5.13E-02 + 1710 FP 48-Cd-121m 120.9130 0.00E+00 8.35E-02 + 1711 FP 49-In-121 120.9079 0.00E+00 3.00E-02 + 1712 FP 49-In-121m 120.9079 0.00E+00 2.98E-03 + 1713 FP 50-Sn-121 120.9042 0.00E+00 7.12E-06 + 1714 FP 50-Sn-121m 120.9042 0.00E+00 5.00E-10 + 1715 FP 51-Sb-121 120.9038 0.00E+00 0.00E+00 + 1716 FP 52-Te-121 120.9049 0.00E+00 4.19E-07 + 1717 FP 52-Te-121m 120.9049 0.00E+00 4.89E-08 + 1718 FP 53-I-121 120.9074 0.00E+00 9.08E-05 + 1719 FP 45-Rh-122 121.9432 0.00E+00 6.42E+00 + 1720 FP 46-Pd-122 121.9305 0.00E+00 3.96E+00 + 1721 FP 47-Ag-122 121.9235 0.00E+00 1.31E+00 + 1722 FP 47-Ag-122m 121.9235 0.00E+00 3.47E+00 + 1723 FP 48-Cd-122 121.9133 0.00E+00 1.32E-01 + 1724 FP 49-In-122 121.9103 0.00E+00 4.62E-01 + 1725 FP 49-In-122m 121.9103 0.00E+00 6.73E-02 + 1726 FP 50-Sn-122 121.9034 0.00E+00 0.00E+00 + 1727 FP 51-Sb-122 121.9052 0.00E+00 2.95E-06 + 1728 FP 51-Sb-122m 121.9052 0.00E+00 2.76E-03 + 1729 FP 52-Te-122 121.9030 0.00E+00 0.00E+00 + 1730 FP 46-Pd-123 122.9349 0.00E+00 2.84E+00 + 1731 FP 47-Ag-123 122.9249 0.00E+00 2.31E+00 + 1732 FP 48-Cd-123 122.9170 0.00E+00 3.30E-01 + 1733 FP 48-Cd-123m 122.9170 0.00E+00 3.81E-01 + 1734 FP 49-In-123 122.9104 0.00E+00 1.12E-01 + 1735 FP 49-In-123m 122.9104 0.00E+00 1.46E-02 + 1736 FP 50-Sn-123 122.9057 0.00E+00 6.21E-08 + 1737 FP 50-Sn-123m 122.9057 0.00E+00 2.88E-04 + 1738 FP 51-Sb-123 122.9042 0.00E+00 0.00E+00 + 1739 FP 52-Te-123 122.9043 0.00E+00 0.00E+00 + 1740 FP 52-Te-123m 122.9043 0.00E+00 6.73E-08 + 1741 FP 53-I-123 122.9056 0.00E+00 1.46E-05 + 1742 FP 46-Pd-124 123.9369 0.00E+00 1.82E+01 + 1743 FP 47-Ag-124 123.9286 0.00E+00 4.03E+00 + 1744 FP 48-Cd-124 123.9176 0.00E+00 5.55E-01 + 1745 FP 49-In-124 123.9132 0.00E+00 2.22E-01 + 1746 FP 49-In-124m 123.9132 0.00E+00 1.87E-01 + 1747 FP 50-Sn-124 123.9053 0.00E+00 0.00E+00 + 1748 FP 51-Sb-124 123.9059 0.00E+00 1.33E-07 + 1749 FP 51-Sb-124m 123.9059 0.00E+00 7.45E-03 + 1750 FP 52-Te-124 123.9028 0.00E+00 0.00E+00 + 1751 FP 53-I-124 123.9062 0.00E+00 1.92E-06 + 1752 FP 54-Xe-124 123.9059 0.00E+00 0.00E+00 + 1753 FP 47-Ag-125 124.9304 0.00E+00 4.18E+00 + 1754 FP 48-Cd-125 124.9212 0.00E+00 1.02E+00 + 1755 FP 49-In-125 124.9136 0.00E+00 2.94E-01 + 1756 FP 49-In-125m 124.9136 0.00E+00 5.68E-02 + 1757 FP 50-Sn-125 124.9078 0.00E+00 8.32E-07 + 1758 FP 50-Sn-125m 124.9078 0.00E+00 1.21E-03 + 1759 FP 51-Sb-125 124.9053 0.00E+00 7.96E-09 + 1760 FP 52-Te-125 124.9044 0.00E+00 0.00E+00 + 1761 FP 52-Te-125m 124.9044 0.00E+00 1.40E-07 + 1762 FP 53-I-125 124.9046 0.00E+00 1.35E-07 + 1763 FP 54-Xe-125 124.9064 0.00E+00 1.14E-05 + 1764 FP 54-Xe-125m 124.9064 0.00E+00 1.22E-02 + 1765 FP 47-Ag-126 125.9345 0.00E+00 6.48E+00 + 1766 FP 48-Cd-126 125.9223 0.00E+00 1.35E+00 + 1767 FP 49-In-126 125.9165 0.00E+00 4.53E-01 + 1768 FP 49-In-126m 125.9165 0.00E+00 4.23E-01 + 1769 FP 50-Sn-126 125.9077 0.00E+00 9.55E-14 + 1770 FP 51-Sb-126 125.9072 0.00E+00 6.50E-07 + 1771 FP 51-Sb-126m 125.9072 0.00E+00 6.03E-04 + 1772 FP 52-Te-126 125.9033 0.00E+00 0.00E+00 + 1773 FP 53-I-126 125.9056 0.00E+00 6.20E-07 + 1774 FP 54-Xe-126 125.9043 0.00E+00 0.00E+00 + 1775 FP 47-Ag-127 126.9368 0.00E+00 6.36E+00 + 1776 FP 48-Cd-127 126.9264 0.00E+00 1.87E+00 + 1777 FP 49-In-127 126.9174 0.00E+00 6.36E-01 + 1778 FP 49-In-127m 126.9174 0.00E+00 1.89E-01 + 1779 FP 50-Sn-127 126.9104 0.00E+00 9.17E-05 + 1780 FP 50-Sn-127m 126.9104 0.00E+00 2.80E-03 + 1781 FP 51-Sb-127 126.9069 0.00E+00 2.08E-06 + 1782 FP 52-Te-127 126.9052 0.00E+00 2.06E-05 + 1783 FP 52-Te-127m 126.9052 0.00E+00 7.36E-08 + 1784 FP 53-I-127 126.9045 0.00E+00 0.00E+00 + 1785 FP 54-Xe-127 126.9052 0.00E+00 2.20E-07 + 1786 FP 54-Xe-127m 126.9052 0.00E+00 1.00E-02 + 1787 FP 55-Cs-127 126.9074 0.00E+00 3.08E-05 + 1788 FP 47-Ag-128 127.9412 0.00E+00 1.20E+01 + 1789 FP 48-Cd-128 127.9278 0.00E+00 2.48E+00 + 1790 FP 49-In-128 127.9202 0.00E+00 8.25E-01 + 1791 FP 49-In-128m 127.9202 0.00E+00 9.63E-01 + 1792 FP 50-Sn-128 127.9105 0.00E+00 1.96E-04 + 1793 FP 50-Sn-128m 127.9105 0.00E+00 1.07E-01 + 1794 FP 51-Sb-128 127.9092 0.00E+00 2.14E-05 + 1795 FP 51-Sb-128m 127.9092 0.00E+00 1.11E-03 + 1796 FP 52-Te-128 127.9045 0.00E+00 2.50E-27 + 1797 FP 53-I-128 127.9058 0.00E+00 4.62E-04 + 1798 FP 54-Xe-128 127.9035 0.00E+00 0.00E+00 + 1799 FP 47-Ag-129 128.9437 0.00E+00 1.51E+01 + 1800 FP 48-Cd-129 128.9321 0.00E+00 2.57E+00 + 1801 FP 49-In-129 128.9217 0.00E+00 1.14E+00 + 1802 FP 49-In-129m 128.9217 0.00E+00 5.64E-01 + 1803 FP 50-Sn-129 128.9135 0.00E+00 5.18E-03 + 1804 FP 50-Sn-129m 128.9135 0.00E+00 1.67E-03 + 1805 FP 51-Sb-129 128.9091 0.00E+00 4.38E-05 + 1806 FP 51-Sb-129m 128.9091 0.00E+00 6.53E-04 + 1807 FP 52-Te-129 128.9066 0.00E+00 1.66E-04 + 1808 FP 52-Te-129m 128.9074 0.00E+00 2.39E-07 + 1809 FP 53-I-129 128.9050 0.00E+00 1.40E-15 + 1810 FP 54-Xe-129 128.9048 0.00E+00 0.00E+00 + 1811 FP 54-Xe-129m 128.9048 0.00E+00 9.03E-07 + 1812 FP 55-Cs-129 128.9061 0.00E+00 6.01E-06 + 1813 FP 56-Ba-129 128.9087 0.00E+00 8.63E-05 + 1814 FP 47-Ag-130 129.9505 0.00E+00 1.39E+01 + 1815 FP 48-Cd-130 129.9339 0.00E+00 4.28E+00 + 1816 FP 49-In-130 129.9250 0.00E+00 2.39E+00 + 1817 FP 49-In-130m 129.9250 0.00E+00 1.28E+00 + 1818 FP 50-Sn-130 129.9140 0.00E+00 3.11E-03 + 1819 FP 50-Sn-130m 129.9140 0.00E+00 6.80E-03 + 1820 FP 51-Sb-130 129.9117 0.00E+00 2.92E-04 + 1821 FP 51-Sb-130m 129.9117 0.00E+00 1.83E-03 + 1822 FP 52-Te-130 129.9062 0.00E+00 0.00E+00 + 1823 FP 53-I-130 129.9067 0.00E+00 1.56E-05 + 1824 FP 53-I-130m 129.9067 0.00E+00 1.31E-03 + 1825 FP 54-Xe-130 129.9035 0.00E+00 0.00E+00 + 1826 FP 48-Cd-131 130.9407 0.00E+00 1.02E+01 + 1827 FP 49-In-131 130.9268 0.00E+00 2.48E+00 + 1828 FP 49-In-131m 130.9268 0.00E+00 1.98E+00 + 1829 FP 50-Sn-131 130.9170 0.00E+00 1.24E-02 + 1830 FP 50-Sn-131m 130.9170 0.00E+00 1.19E-02 + 1831 FP 51-Sb-131 130.9120 0.00E+00 5.02E-04 + 1832 FP 52-Te-131 130.9085 0.00E+00 4.62E-04 + 1833 FP 52-Te-131m 130.9085 0.00E+00 5.79E-06 + 1834 FP 53-I-131 130.9061 0.00E+00 1.00E-06 + 1835 FP 54-Xe-131 130.9051 0.00E+00 0.00E+00 + 1836 FP 54-Xe-131m 130.9051 0.00E+00 6.78E-07 + 1837 FP 55-Cs-131 130.9055 0.00E+00 8.28E-07 + 1838 FP 56-Ba-131 130.9069 0.00E+00 6.98E-07 + 1839 FP 48-Cd-132 131.9456 0.00E+00 7.15E+00 + 1840 FP 49-In-132 131.9330 0.00E+00 3.35E+00 + 1841 FP 50-Sn-132 131.9178 0.00E+00 1.75E-02 + 1842 FP 51-Sb-132 131.9145 0.00E+00 4.14E-03 + 1843 FP 51-Sb-132m 131.9145 0.00E+00 2.82E-03 + 1844 FP 52-Te-132 131.9086 0.00E+00 2.50E-06 + 1845 FP 53-I-132 131.9080 0.00E+00 8.39E-05 + 1846 FP 53-I-132m 131.9080 0.00E+00 1.39E-04 + 1847 FP 54-Xe-132 131.9041 0.00E+00 0.00E+00 + 1848 FP 55-Cs-132 131.9064 0.00E+00 1.24E-06 + 1849 FP 56-Ba-132 131.9051 0.00E+00 0.00E+00 + 1850 FP 49-In-133 132.9378 0.00E+00 4.20E+00 + 1851 FP 50-Sn-133 132.9238 0.00E+00 4.75E-01 + 1852 FP 51-Sb-133 132.9153 0.00E+00 4.62E-03 + 1853 FP 52-Te-133 132.9110 0.00E+00 9.24E-04 + 1854 FP 52-Te-133m 132.9110 0.00E+00 2.09E-04 + 1855 FP 53-I-133 132.9078 0.00E+00 9.26E-06 + 1856 FP 53-I-133m 132.9078 0.00E+00 7.70E-02 + 1857 FP 54-Xe-133 132.9059 0.00E+00 1.53E-06 + 1858 FP 54-Xe-133m 132.9059 0.00E+00 3.66E-06 + 1859 FP 55-Cs-133 132.9055 0.00E+00 0.00E+00 + 1860 FP 56-Ba-133 132.9060 0.00E+00 2.09E-09 + 1861 FP 57-La-133 132.9082 0.00E+00 4.92E-05 + 1862 FP 49-In-134 133.9442 0.00E+00 4.95E+00 + 1863 FP 50-Sn-134 133.9283 0.00E+00 6.60E-01 + 1864 FP 51-Sb-134 133.9204 0.00E+00 8.89E-01 + 1865 FP 51-Sb-134m 133.9204 0.00E+00 6.88E-02 + 1866 FP 52-Te-134 133.9114 0.00E+00 2.76E-04 + 1867 FP 53-I-134 133.9097 0.00E+00 2.20E-04 + 1868 FP 53-I-134m 133.9097 0.00E+00 3.28E-03 + 1869 FP 54-Xe-134 133.9054 0.00E+00 0.00E+00 + 1870 FP 54-Xe-134m 133.9054 0.00E+00 2.39E+00 + 1871 FP 55-Cs-134 133.9067 0.00E+00 1.06E-08 + 1872 FP 55-Cs-134m 133.9067 0.00E+00 6.61E-05 + 1873 FP 56-Ba-134 133.9045 0.00E+00 0.00E+00 + 1874 FP 49-In-135 134.9493 0.00E+00 7.53E+00 + 1875 FP 50-Sn-135 134.9347 0.00E+00 1.31E+00 + 1876 FP 51-Sb-135 134.9252 0.00E+00 4.13E-01 + 1877 FP 52-Te-135 134.9164 0.00E+00 3.65E-02 + 1878 FP 53-I-135 134.9100 0.00E+00 2.93E-05 + 1879 FP 54-Xe-135 134.9072 0.00E+00 2.11E-05 + 1880 FP 54-Xe-135m 134.9072 0.00E+00 7.56E-04 + 1881 FP 55-Cs-135 134.9060 0.00E+00 9.55E-15 + 1882 FP 55-Cs-135m 134.9060 0.00E+00 2.18E-04 + 1883 FP 56-Ba-135 134.9057 0.00E+00 0.00E+00 + 1884 FP 56-Ba-135m 134.9057 0.00E+00 6.71E-06 + 1885 FP 57-La-135 134.9070 0.00E+00 9.87E-06 + 1886 FP 58-Ce-135 134.9091 0.00E+00 1.09E-05 + 1887 FP 50-Sn-136 135.9393 0.00E+00 2.77E+00 + 1888 FP 51-Sb-136 135.9303 0.00E+00 7.51E-01 + 1889 FP 52-Te-136 135.9201 0.00E+00 3.96E-02 + 1890 FP 53-I-136 135.9147 0.00E+00 8.31E-03 + 1891 FP 53-I-136m 135.9147 0.00E+00 1.48E-02 + 1892 FP 54-Xe-136 135.9072 0.00E+00 0.00E+00 + 1893 FP 55-Cs-136 135.9073 0.00E+00 6.10E-07 + 1894 FP 55-Cs-136m 135.9073 0.00E+00 3.65E-02 + 1895 FP 56-Ba-136 135.9046 0.00E+00 0.00E+00 + 1896 FP 56-Ba-136m 135.9046 0.00E+00 2.25E+00 + 1897 FP 50-Sn-137 136.9460 0.00E+00 3.65E+00 + 1898 FP 51-Sb-137 136.9353 0.00E+00 1.54E+00 + 1899 FP 52-Te-137 136.9253 0.00E+00 2.78E-01 + 1900 FP 53-I-137 136.9179 0.00E+00 2.83E-02 + 1901 FP 54-Xe-137 136.9116 0.00E+00 3.03E-03 + 1902 FP 55-Cs-137 136.9071 0.00E+00 7.30E-10 + 1903 FP 56-Ba-137 136.9058 0.00E+00 0.00E+00 + 1904 FP 56-Ba-137m 136.9058 0.00E+00 4.53E-03 + 1905 FP 57-La-137 136.9065 0.00E+00 3.66E-13 + 1906 FP 58-Ce-137 136.9078 0.00E+00 2.14E-05 + 1907 FP 51-Sb-138 137.9408 0.00E+00 4.13E+00 + 1908 FP 52-Te-138 137.9292 0.00E+00 4.95E-01 + 1909 FP 53-I-138 137.9223 0.00E+00 1.11E-01 + 1910 FP 54-Xe-138 137.9140 0.00E+00 8.20E-04 + 1911 FP 55-Cs-138 137.9110 0.00E+00 3.46E-04 + 1912 FP 55-Cs-138m 137.9110 0.00E+00 3.97E-03 + 1913 FP 56-Ba-138 137.9052 0.00E+00 0.00E+00 + 1914 FP 57-La-138 137.9071 0.00E+00 2.15E-19 + 1915 FP 58-Ce-138 137.9060 0.00E+00 0.00E+00 + 1916 FP 51-Sb-139 138.9460 0.00E+00 5.46E+00 + 1917 FP 52-Te-139 138.9347 0.00E+00 2.00E+00 + 1918 FP 53-I-139 138.9261 0.00E+00 3.04E-01 + 1919 FP 54-Xe-139 138.9188 0.00E+00 1.75E-02 + 1920 FP 55-Cs-139 138.9134 0.00E+00 1.25E-03 + 1921 FP 56-Ba-139 138.9088 0.00E+00 1.39E-04 + 1922 FP 57-La-139 138.9064 0.00E+00 0.00E+00 + 1923 FP 58-Ce-139 138.9066 0.00E+00 5.83E-08 + 1924 FP 58-Ce-139m 138.9066 0.00E+00 1.26E-02 + 1925 FP 59-Pr-139 138.9089 0.00E+00 4.37E-05 + 1926 FP 52-Te-140 139.9388 0.00E+00 2.28E+00 + 1927 FP 53-I-140 139.9310 0.00E+00 8.06E-01 + 1928 FP 54-Xe-140 139.9216 0.00E+00 5.10E-02 + 1929 FP 55-Cs-140 139.9173 0.00E+00 1.09E-02 + 1930 FP 56-Ba-140 139.9106 0.00E+00 6.29E-07 + 1931 FP 57-La-140 139.9095 0.00E+00 4.78E-06 + 1932 FP 58-Ce-140 139.9054 0.00E+00 0.00E+00 + 1933 FP 59-Pr-140 139.9091 0.00E+00 3.41E-03 + 1934 FP 60-Nd-140 139.9095 0.00E+00 2.38E-06 + 1935 FP 52-Te-141 140.9447 0.00E+00 3.25E+00 + 1936 FP 53-I-141 140.9350 0.00E+00 1.61E+00 + 1937 FP 54-Xe-141 140.9267 0.00E+00 4.01E-01 + 1938 FP 55-Cs-141 140.9200 0.00E+00 2.79E-02 + 1939 FP 56-Ba-141 140.9144 0.00E+00 6.32E-04 + 1940 FP 57-La-141 140.9110 0.00E+00 4.91E-05 + 1941 FP 58-Ce-141 140.9083 0.00E+00 2.47E-07 + 1942 FP 59-Pr-141 140.9077 0.00E+00 0.00E+00 + 1943 FP 60-Nd-141 140.9096 0.00E+00 7.73E-05 + 1944 FP 60-Nd-141m 140.9096 0.00E+00 1.12E-02 + 1945 FP 61-Pm-141 140.9136 0.00E+00 5.53E-04 + 1946 FP 52-Te-142 141.9491 0.00E+00 3.47E+00 + 1947 FP 53-I-142 141.9402 0.00E+00 3.12E+00 + 1948 FP 54-Xe-142 141.9297 0.00E+00 5.64E-01 + 1949 FP 55-Cs-142 141.9243 0.00E+00 4.12E-01 + 1950 FP 56-Ba-142 141.9164 0.00E+00 1.09E-03 + 1951 FP 57-La-142 141.9141 0.00E+00 1.27E-04 + 1952 FP 58-Ce-142 141.9092 0.00E+00 0.00E+00 + 1953 FP 59-Pr-142 141.9100 0.00E+00 1.01E-05 + 1954 FP 59-Pr-142m 141.9100 0.00E+00 7.91E-04 + 1955 FP 60-Nd-142 141.9077 0.00E+00 0.00E+00 + 1956 FP 53-I-143 142.9446 0.00E+00 2.34E+00 + 1957 FP 54-Xe-143 142.9351 0.00E+00 2.31E+00 + 1958 FP 55-Cs-143 142.9274 0.00E+00 3.87E-01 + 1959 FP 56-Ba-143 142.9206 0.00E+00 4.78E-02 + 1960 FP 57-La-143 142.9161 0.00E+00 8.14E-04 + 1961 FP 58-Ce-143 142.9124 0.00E+00 5.83E-06 + 1962 FP 59-Pr-143 142.9108 0.00E+00 5.91E-07 + 1963 FP 60-Nd-143 142.9098 0.00E+00 0.00E+00 + 1964 FP 61-Pm-143 142.9109 0.00E+00 3.03E-08 + 1965 FP 62-Sm-143 142.9146 0.00E+00 1.32E-03 + 1966 FP 62-Sm-143m 142.9146 0.00E+00 1.05E-02 + 1967 FP 53-I-144 143.9500 0.00E+00 3.57E+00 + 1968 FP 54-Xe-144 143.9385 0.00E+00 6.03E-01 + 1969 FP 55-Cs-144 143.9321 0.00E+00 6.97E-01 + 1970 FP 56-Ba-144 143.9229 0.00E+00 6.03E-02 + 1971 FP 57-La-144 143.9196 0.00E+00 1.70E-02 + 1972 FP 58-Ce-144 143.9137 0.00E+00 2.82E-08 + 1973 FP 59-Pr-144 143.9133 0.00E+00 6.69E-04 + 1974 FP 59-Pr-144m 143.9133 0.00E+00 1.60E-03 + 1975 FP 60-Nd-144 143.9101 0.00E+00 9.59E-24 + 1976 FP 61-Pm-144 143.9126 0.00E+00 2.21E-08 + 1977 FP 62-Sm-144 143.9120 0.00E+00 0.00E+00 + 1978 FP 54-Xe-145 144.9441 0.00E+00 3.69E+00 + 1979 FP 55-Cs-145 144.9355 0.00E+00 1.18E+00 + 1980 FP 56-Ba-145 144.9276 0.00E+00 1.61E-01 + 1981 FP 57-La-145 144.9216 0.00E+00 2.80E-02 + 1982 FP 58-Ce-145 144.9172 0.00E+00 3.84E-03 + 1983 FP 59-Pr-145 144.9145 0.00E+00 3.22E-05 + 1984 FP 60-Nd-145 144.9126 0.00E+00 0.00E+00 + 1985 FP 61-Pm-145 144.9128 0.00E+00 1.24E-09 + 1986 FP 62-Sm-145 144.9134 0.00E+00 2.36E-08 + 1987 FP 54-Xe-146 145.9478 0.00E+00 1.88E+00 + 1988 FP 55-Cs-146 145.9403 0.00E+00 2.16E+00 + 1989 FP 56-Ba-146 145.9302 0.00E+00 3.12E-01 + 1990 FP 57-La-146 145.9258 0.00E+00 1.11E-01 + 1991 FP 57-La-146m 145.9258 0.00E+00 6.93E-02 + 1992 FP 58-Ce-146 145.9188 0.00E+00 8.54E-04 + 1993 FP 59-Pr-146 145.9176 0.00E+00 4.78E-04 + 1994 FP 60-Nd-146 145.9131 0.00E+00 0.00E+00 + 1995 FP 61-Pm-146 145.9147 0.00E+00 3.97E-09 + 1996 FP 62-Sm-146 145.9130 0.00E+00 2.13E-16 + 1997 FP 54-Xe-147 146.9536 0.00E+00 6.93E+00 + 1998 FP 55-Cs-147 146.9442 0.00E+00 3.01E+00 + 1999 FP 56-Ba-147 146.9350 0.00E+00 7.75E-01 + 2000 FP 57-La-147 146.9282 0.00E+00 1.71E-01 + 2001 FP 58-Ce-147 146.9227 0.00E+00 1.23E-02 + 2002 FP 59-Pr-147 146.9190 0.00E+00 8.62E-04 + 2003 FP 60-Nd-147 146.9161 0.00E+00 7.31E-07 + 2004 FP 61-Pm-147 146.9151 0.00E+00 8.37E-09 + 2005 FP 62-Sm-147 146.9149 0.00E+00 2.07E-19 + 2006 FP 63-Eu-147 146.9167 0.00E+00 3.33E-07 + 2007 FP 64-Gd-147 146.9191 0.00E+00 5.06E-06 + 2008 FP 55-Cs-148 147.9492 0.00E+00 4.75E+00 + 2009 FP 56-Ba-148 147.9377 0.00E+00 1.13E+00 + 2010 FP 57-La-148 147.9322 0.00E+00 5.50E-01 + 2011 FP 58-Ce-148 147.9244 0.00E+00 1.24E-02 + 2012 FP 59-Pr-148 147.9221 0.00E+00 5.04E-03 + 2013 FP 59-Pr-148m 147.9221 0.00E+00 5.75E-03 + 2014 FP 60-Nd-148 147.9169 0.00E+00 0.00E+00 + 2015 FP 61-Pm-148 147.9175 0.00E+00 1.49E-06 + 2016 FP 61-Pm-148m 147.9207 0.00E+00 1.94E-07 + 2017 FP 62-Sm-148 147.9148 0.00E+00 3.14E-24 + 2018 FP 55-Cs-149 148.9529 0.00E+00 1.39E+01 + 2019 FP 56-Ba-149 148.9426 0.00E+00 2.02E+00 + 2020 FP 57-La-149 148.9347 0.00E+00 6.60E-01 + 2021 FP 58-Ce-149 148.9284 0.00E+00 1.31E-01 + 2022 FP 59-Pr-149 148.9237 0.00E+00 5.11E-03 + 2023 FP 60-Nd-149 148.9202 0.00E+00 1.11E-04 + 2024 FP 61-Pm-149 148.9183 0.00E+00 3.63E-06 + 2025 FP 62-Sm-149 148.9172 0.00E+00 0.00E+00 + 2026 FP 63-Eu-149 148.9179 0.00E+00 8.62E-08 + 2027 FP 64-Gd-149 148.9193 0.00E+00 8.65E-07 + 2028 FP 55-Cs-150 149.9582 0.00E+00 1.39E+01 + 2029 FP 56-Ba-150 149.9457 0.00E+00 2.31E+00 + 2030 FP 57-La-150 149.9388 0.00E+00 8.06E-01 + 2031 FP 58-Ce-150 149.9304 0.00E+00 1.73E-01 + 2032 FP 59-Pr-150 149.9267 0.00E+00 1.12E-01 + 2033 FP 60-Nd-150 149.9209 0.00E+00 2.78E-27 + 2034 FP 61-Pm-150 149.9210 0.00E+00 7.18E-05 + 2035 FP 62-Sm-150 149.9173 0.00E+00 0.00E+00 + 2036 FP 55-Cs-151 150.9622 0.00E+00 1.39E+01 + 2037 FP 56-Ba-151 150.9508 0.00E+00 2.68E+00 + 2038 FP 57-La-151 150.9417 0.00E+00 8.91E-01 + 2039 FP 58-Ce-151 150.9340 0.00E+00 3.94E-01 + 2040 FP 59-Pr-151 150.9283 0.00E+00 3.67E-02 + 2041 FP 60-Nd-151 150.9238 0.00E+00 9.29E-04 + 2042 FP 61-Pm-151 150.9212 0.00E+00 6.78E-06 + 2043 FP 62-Sm-151 150.9199 0.00E+00 2.44E-10 + 2044 FP 63-Eu-151 150.9198 0.00E+00 0.00E+00 + 2045 FP 64-Gd-151 150.9203 0.00E+00 6.47E-08 + 2046 FP 65-Tb-151 150.9231 0.00E+00 1.09E-05 + 2047 FP 56-Ba-152 151.9543 0.00E+00 3.04E+00 + 2048 FP 57-La-152 151.9462 0.00E+00 1.54E+00 + 2049 FP 58-Ce-152 151.9365 0.00E+00 4.95E-01 + 2050 FP 59-Pr-152 151.9315 0.00E+00 1.91E-01 + 2051 FP 60-Nd-152 151.9247 0.00E+00 1.01E-03 + 2052 FP 61-Pm-152 151.9235 0.00E+00 2.80E-03 + 2053 FP 61-Pm-152m 151.9235 0.00E+00 1.54E-03 + 2054 FP 62-Sm-152 151.9197 0.00E+00 0.00E+00 + 2055 FP 63-Eu-152 151.9217 0.00E+00 1.62E-09 + 2056 FP 63-Eu-152m 151.9217 0.00E+00 2.07E-05 + 2057 FP 64-Gd-152 151.9198 0.00E+00 2.03E-22 + 2058 FP 56-Ba-153 152.9596 0.00E+00 4.39E+00 + 2059 FP 57-La-153 152.9496 0.00E+00 2.03E+00 + 2060 FP 58-Ce-153 152.9406 0.00E+00 7.08E-01 + 2061 FP 59-Pr-153 152.9338 0.00E+00 1.62E-01 + 2062 FP 60-Nd-153 152.9277 0.00E+00 2.19E-02 + 2063 FP 61-Pm-153 152.9241 0.00E+00 2.20E-03 + 2064 FP 62-Sm-153 152.9221 0.00E+00 4.14E-06 + 2065 FP 63-Eu-153 152.9212 0.00E+00 0.00E+00 + 2066 FP 64-Gd-153 152.9218 0.00E+00 3.34E-08 + 2067 FP 65-Tb-153 152.9234 0.00E+00 3.43E-06 + 2068 FP 57-La-154 153.9545 0.00E+00 3.04E+00 + 2069 FP 58-Ce-154 153.9434 0.00E+00 8.94E-01 + 2070 FP 59-Pr-154 153.9375 0.00E+00 3.01E-01 + 2071 FP 60-Nd-154 153.9295 0.00E+00 2.68E-02 + 2072 FP 61-Pm-154 153.9265 0.00E+00 6.68E-03 + 2073 FP 61-Pm-154m 153.9265 0.00E+00 4.31E-03 + 2074 FP 62-Sm-154 153.9222 0.00E+00 0.00E+00 + 2075 FP 63-Eu-154 153.9230 0.00E+00 2.55E-09 + 2076 FP 63-Eu-154m 153.9230 0.00E+00 2.51E-04 + 2077 FP 64-Gd-154 153.9209 0.00E+00 0.00E+00 + 2078 FP 57-La-155 154.9583 0.00E+00 3.77E+00 + 2079 FP 58-Ce-155 154.9480 0.00E+00 1.47E+00 + 2080 FP 59-Pr-155 154.9401 0.00E+00 8.14E-01 + 2081 FP 60-Nd-155 154.9329 0.00E+00 7.79E-02 + 2082 FP 61-Pm-155 154.9281 0.00E+00 1.67E-02 + 2083 FP 62-Sm-155 154.9246 0.00E+00 5.18E-04 + 2084 FP 63-Eu-155 154.9229 0.00E+00 4.62E-09 + 2085 FP 64-Gd-155 154.9226 0.00E+00 0.00E+00 + 2086 FP 64-Gd-155m 154.9226 0.00E+00 2.17E+01 + 2087 FP 65-Tb-155 154.9235 0.00E+00 1.51E-06 + 2088 FP 66-Dy-155 154.9258 0.00E+00 1.94E-05 + 2089 FP 58-Ce-156 155.9513 0.00E+00 1.88E+00 + 2090 FP 59-Pr-156 155.9443 0.00E+00 9.46E-01 + 2091 FP 60-Nd-156 155.9350 0.00E+00 1.26E-01 + 2092 FP 61-Pm-156 155.9311 0.00E+00 2.60E-02 + 2093 FP 62-Sm-156 155.9255 0.00E+00 2.05E-05 + 2094 FP 63-Eu-156 155.9247 0.00E+00 5.28E-07 + 2095 FP 64-Gd-156 155.9221 0.00E+00 0.00E+00 + 2096 FP 65-Tb-156 155.9247 0.00E+00 1.50E-06 + 2097 FP 65-Tb-156m 155.9247 0.00E+00 7.89E-06 + 2098 FP 66-Dy-156 155.9243 0.00E+00 0.00E+00 + 2099 FP 58-Ce-157 156.9563 0.00E+00 2.85E+00 + 2100 FP 59-Pr-157 156.9474 0.00E+00 1.16E+00 + 2101 FP 60-Nd-157 156.9390 0.00E+00 3.64E-01 + 2102 FP 61-Pm-157 156.9330 0.00E+00 6.56E-02 + 2103 FP 62-Sm-157 156.9284 0.00E+00 1.44E-03 + 2104 FP 63-Eu-157 156.9254 0.00E+00 1.27E-05 + 2105 FP 64-Gd-157 156.9240 0.00E+00 0.00E+00 + 2106 FP 65-Tb-157 156.9240 0.00E+00 3.09E-10 + 2107 FP 66-Dy-157 156.9255 0.00E+00 2.37E-05 + 2108 FP 59-Pr-158 157.9520 0.00E+00 5.17E+00 + 2109 FP 60-Nd-158 157.9416 0.00E+00 5.21E-01 + 2110 FP 61-Pm-158 157.9366 0.00E+00 1.44E-01 + 2111 FP 62-Sm-158 157.9300 0.00E+00 2.18E-03 + 2112 FP 63-Eu-158 157.9279 0.00E+00 2.52E-04 + 2113 FP 64-Gd-158 157.9241 0.00E+00 0.00E+00 + 2114 FP 65-Tb-158 157.9254 0.00E+00 1.22E-10 + 2115 FP 65-Tb-158m 157.9254 0.00E+00 6.48E-02 + 2116 FP 66-Dy-158 157.9244 0.00E+00 0.00E+00 + 2117 FP 59-Pr-159 158.9555 0.00E+00 6.57E+00 + 2118 FP 60-Nd-159 158.9461 0.00E+00 8.97E-01 + 2119 FP 61-Pm-159 158.9390 0.00E+00 4.72E-01 + 2120 FP 62-Sm-159 158.9332 0.00E+00 6.10E-02 + 2121 FP 63-Eu-159 158.9291 0.00E+00 6.38E-04 + 2122 FP 64-Gd-159 158.9264 0.00E+00 1.04E-05 + 2123 FP 65-Tb-159 158.9254 0.00E+00 0.00E+00 + 2124 FP 66-Dy-159 158.9257 0.00E+00 5.56E-08 + 2125 FP 67-Ho-159 158.9277 0.00E+00 3.50E-04 + 2126 FP 67-Ho-159m 158.9277 0.00E+00 8.35E-02 + 2127 FP 60-Nd-160 159.9491 0.00E+00 1.18E+00 + 2128 FP 61-Pm-160 159.9430 0.00E+00 4.44E-01 + 2129 FP 62-Sm-160 159.9351 0.00E+00 7.22E-02 + 2130 FP 63-Eu-160 159.9320 0.00E+00 1.82E-02 + 2131 FP 64-Gd-160 159.9270 0.00E+00 0.00E+00 + 2132 FP 65-Tb-160 159.9272 0.00E+00 1.11E-07 + 2133 FP 66-Dy-160 159.9252 0.00E+00 0.00E+00 + 2134 FP 60-Nd-161 160.9539 0.00E+00 1.42E+00 + 2135 FP 61-Pm-161 160.9459 0.00E+00 6.51E-01 + 2136 FP 62-Sm-161 160.9388 0.00E+00 1.44E-01 + 2137 FP 63-Eu-161 160.9337 0.00E+00 2.67E-02 + 2138 FP 64-Gd-161 160.9297 0.00E+00 3.16E-03 + 2139 FP 65-Tb-161 160.9276 0.00E+00 1.16E-06 + 2140 FP 66-Dy-161 160.9269 0.00E+00 0.00E+00 + 2141 FP 67-Ho-161 160.9279 0.00E+00 7.76E-05 + 2142 FP 67-Ho-161m 160.9279 0.00E+00 1.03E-01 + 2143 FP 68-Er-161 160.9300 0.00E+00 6.00E-05 + 2144 FP 61-Pm-162 161.9503 0.00E+00 2.59E+00 + 2145 FP 62-Sm-162 161.9412 0.00E+00 2.89E-01 + 2146 FP 63-Eu-162 161.9370 0.00E+00 6.54E-02 + 2147 FP 64-Gd-162 161.9310 0.00E+00 1.38E-03 + 2148 FP 65-Tb-162 161.9295 0.00E+00 1.52E-03 + 2149 FP 66-Dy-162 161.9268 0.00E+00 0.00E+00 + 2150 FP 67-Ho-162 161.9291 0.00E+00 7.70E-04 + 2151 FP 67-Ho-162m 161.9291 0.00E+00 1.72E-04 + 2152 FP 68-Er-162 161.9288 0.00E+00 0.00E+00 + 2153 FP 61-Pm-163 162.9537 0.00E+00 3.47E+00 + 2154 FP 62-Sm-163 162.9454 0.00E+00 3.97E-01 + 2155 FP 63-Eu-163 162.9392 0.00E+00 9.00E-02 + 2156 FP 64-Gd-163 162.9340 0.00E+00 1.02E-02 + 2157 FP 65-Tb-163 162.9306 0.00E+00 5.92E-04 + 2158 FP 66-Dy-163 162.9287 0.00E+00 0.00E+00 + 2159 FP 67-Ho-163 162.9287 0.00E+00 4.81E-12 + 2160 FP 67-Ho-163m 162.9287 0.00E+00 6.36E-01 + 2161 FP 68-Er-163 162.9300 0.00E+00 1.54E-04 + 2162 FP 62-Sm-164 163.9483 0.00E+00 5.65E-01 + 2163 FP 63-Eu-164 163.9430 0.00E+00 2.44E-01 + 2164 FP 64-Gd-164 163.9359 0.00E+00 1.54E-02 + 2165 FP 65-Tb-164 163.9333 0.00E+00 3.85E-03 + 2166 FP 66-Dy-164 163.9292 0.00E+00 0.00E+00 + 2167 FP 67-Ho-164 163.9302 0.00E+00 3.98E-04 + 2168 FP 67-Ho-164m 163.9302 0.00E+00 3.08E-04 + 2169 FP 68-Er-164 163.9292 0.00E+00 0.00E+00 + 2170 FP 62-Sm-165 164.9530 0.00E+00 9.07E-01 + 2171 FP 63-Eu-165 164.9457 0.00E+00 3.01E-01 + 2172 FP 64-Gd-165 164.9394 0.00E+00 6.73E-02 + 2173 FP 65-Tb-165 164.9349 0.00E+00 5.47E-03 + 2174 FP 66-Dy-165 164.9317 0.00E+00 8.25E-05 + 2175 FP 66-Dy-165m 164.9317 0.00E+00 9.19E-03 + 2176 FP 67-Ho-165 164.9303 0.00E+00 0.00E+00 + 2177 FP 68-Er-165 164.9307 0.00E+00 1.86E-05 + 2178 FP 69-Tm-165 164.9324 0.00E+00 6.41E-06 + 2179 FP 63-Eu-166 165.9500 0.00E+00 1.73E+00 + 2180 FP 64-Gd-166 165.9416 0.00E+00 1.44E-01 + 2181 FP 65-Tb-166 165.9380 0.00E+00 2.76E-02 + 2182 FP 66-Dy-166 165.9328 0.00E+00 2.36E-06 + 2183 FP 67-Ho-166 165.9323 0.00E+00 7.18E-06 + 2184 FP 67-Ho-166m 165.9324 0.00E+00 1.83E-11 + 2185 FP 68-Er-166 165.9303 0.00E+00 0.00E+00 + 2186 FP 69-Tm-166 165.9335 0.00E+00 2.50E-05 + 2187 FP 70-Yb-166 165.9339 0.00E+00 3.40E-06 + 2188 FP 63-Eu-167 166.9532 0.00E+00 3.47E+00 + 2189 FP 64-Gd-167 166.9456 0.00E+00 2.31E-01 + 2190 FP 65-Tb-167 166.9400 0.00E+00 3.57E-02 + 2191 FP 66-Dy-167 166.9357 0.00E+00 1.86E-03 + 2192 FP 67-Ho-167 166.9331 0.00E+00 6.21E-05 + 2193 FP 68-Er-167 166.9321 0.00E+00 0.00E+00 + 2194 FP 68-Er-167m 166.9321 0.00E+00 3.05E-01 + 2195 FP 69-Tm-167 166.9328 0.00E+00 8.67E-07 + 2196 FP 70-Yb-167 166.9350 0.00E+00 6.60E-04 + 2197 FP 64-Gd-168 167.9484 0.00E+00 2.31E+00 + 2198 FP 65-Tb-168 167.9436 0.00E+00 8.45E-02 + 2199 FP 66-Dy-168 167.9371 0.00E+00 1.33E-03 + 2200 FP 67-Ho-168 167.9355 0.00E+00 3.86E-03 + 2201 FP 68-Er-168 167.9324 0.00E+00 0.00E+00 + 2202 FP 69-Tm-168 167.9342 0.00E+00 8.62E-08 + 2203 FP 70-Yb-168 167.9339 0.00E+00 0.00E+00 + 2204 FP 64-Gd-169 168.9529 0.00E+00 6.93E-01 + 2205 FP 65-Tb-169 168.9462 0.00E+00 3.47E-01 + 2206 FP 66-Dy-169 168.9403 0.00E+00 1.78E-02 + 2207 FP 67-Ho-169 168.9369 0.00E+00 2.45E-03 + 2208 FP 68-Er-169 168.9346 0.00E+00 8.54E-07 + 2209 FP 69-Tm-169 168.9342 0.00E+00 0.00E+00 + 2210 FP 70-Yb-169 168.9352 0.00E+00 2.51E-07 + 2211 FP 70-Yb-169m 168.9352 0.00E+00 1.51E-02 + 2212 FP 71-Lu-169 168.9377 0.00E+00 5.65E-06 + 2213 FP 71-Lu-169m 168.9377 0.00E+00 4.33E-03 + 2214 FP 65-Tb-170 169.9503 0.00E+00 2.31E-01 + 2215 FP 66-Dy-170 169.9424 0.00E+00 2.31E-02 + 2216 FP 67-Ho-170 169.9396 0.00E+00 4.19E-03 + 2217 FP 67-Ho-170m 169.9396 0.00E+00 1.61E-02 + 2218 FP 68-Er-170 169.9355 0.00E+00 0.00E+00 + 2219 FP 69-Tm-170 169.9358 0.00E+00 6.24E-08 + 2220 FP 70-Yb-170 169.9348 0.00E+00 0.00E+00 + 2221 FP 65-Tb-171 170.9533 0.00E+00 1.39E+00 + 2222 FP 66-Dy-171 170.9462 0.00E+00 1.16E-01 + 2223 FP 67-Ho-171 170.9415 0.00E+00 1.31E-02 + 2224 FP 68-Er-171 170.9380 0.00E+00 2.56E-05 + 2225 FP 69-Tm-171 170.9364 0.00E+00 1.14E-08 + 2226 FP 70-Yb-171 170.9363 0.00E+00 0.00E+00 + 2227 FP 71-Lu-171 170.9379 0.00E+00 9.74E-07 + 2228 FP 71-Lu-171m 170.9379 0.00E+00 8.77E-03 + 2229 FP 72-Hf-171 170.9405 0.00E+00 1.59E-05 + 2230 FP 66-Dy-172 171.9488 0.00E+00 2.31E-01 + 2231 FP 67-Ho-172 171.9448 0.00E+00 2.77E-02 + 2232 FP 68-Er-172 171.9394 0.00E+00 3.91E-06 + 2233 FP 69-Tm-172 171.9384 0.00E+00 3.03E-06 + 2234 FP 70-Yb-172 171.9364 0.00E+00 0.00E+00 + 2235 FP 71-Lu-172 171.9391 0.00E+00 1.20E-06 + 2236 FP 71-Lu-172m 171.9391 0.00E+00 3.12E-03 + 2237 FP 72-Hf-172 171.9395 0.00E+00 1.17E-08 + ========= ========== =========== ========= =========== ========= + +.. bibliography:: bibs/data-resource-refs.bib diff --git a/ORIGEN.rst b/ORIGEN.rst new file mode 100644 index 0000000..f7b2fce --- /dev/null +++ b/ORIGEN.rst @@ -0,0 +1,5418 @@ +.. _5-1: + +Origen: Neutron Activation, Actinide Transmutation, Fission Product Generation, and Radiation Source Term Calculation +===================================================================================================================== + +.. |rarr| replace:: :math:`\rightarrow` + +W. Wieselquist, S. Hart, A. Isotalo [#f1]_, F. Havlůj [#f2]_, +S. Skutnik [#f3]_, R. Lefebvre, I. Gauld, D. Wiarda, J. Lefebvre, +G. Hu [#f4]_, N. Sly [#f3]_, and D. Lago [#f5]_ + +ABSTRACT + +ORIGEN (Oak Ridge Isotope G\ eneration code) calculates time-dependent +concentrations, activities, and radiation source terms for a large +number of isotopes simultaneously generated or depleted by neutron +transmutation, fission, and radioactive decay. ORIGEN is used internally +within SCALE’s TRITON and Polaris sequences to perform depletion and +decay. As a stand-alone SCALE module, ORIGEN provides additional unique +capabilities to (1) simulate continuous nuclide feed and chemical +removal, which can be used to model reprocessing or liquid fuel systems, +and (2) generate alpha, beta, neutron and gamma decay emission spectra. +A standard decay library is provided to perform decay calculations. For +neutron activation and fuel depletion problems, neutron +spectrum-dependent ORIGEN libraries are required and may be created from +(1) user-defined spectrum and self-shielded cross sections using the +COUPLE module or (2) interpolation of existing ORIGEN reactor libraries +(precalculated by TRITON) using the Automated Rapid Processing (ARP) +module. Post-processing using the OPUS module allows calculated +isotopics and spectra to be sorted, ranked, and converted to other +units. + +ACKNOWLEDGEMENTS + +Development and maintenance of ORIGEN and related codes and methods have +been sponsored by many organizations including the US Nuclear Regulatory +Commission (NRC), the US Department of Energy (DOE), and nuclear power +and research institutions. + +Version Information + +.. centered:: Version 6.2 (2016) + +**Code Responsible(s):** W. A. Wieselquist + +The ORIGEN (Oak Ridge Isotope G\ eneration) code :cite:`bell_origen_1973` was developed +at Oak Ridge National Laboratory (ORNL) to calculate nuclide compositions +and radioactivity of fission products, activation products, and products +of heavy metal transmutation. Since 1991, ORIGEN has been developed as +the depletion/decay module in SCALE with support from the NRC. ORIGEN in +SCALE is the only version supported at ORNL, and it supersedes all +earlier versions. The following is a brief description of the major +enhancements in each version. Data are described in the ORIGEN Data +Resources chapter. + +A major modernization effort for ORIGEN was initiated by I. Gauld in +2011 and has resulted in approximately 5 person-years of effort +refactoring the ORIGEN and related codes to be more efficient and easily +testable. The major enhancements and responsible parties are listed +below. + + - Extensive refactor and modernization of Fortran 77 to Fortran 90+ + performed by F. Havlůj, including substantial extension of the output + capability + + - Implementation of an alpha and beta spectrum calculation by F. Havlůj + and I. Gauld + + - Introduction of C++ core data structures with Fortran bindings, + implemented by S. Skutnik using R. Lefebvre’s C++/C/Fortran binding + generator created for this purpose + + - Testing suite developed by S. Skutnik, W. Wieselquist, D. Lago, and + N. Sly + + - Standardization of codebase while developing application programming + interface (API) for high-performance depletion in the Consortium for + Advanced Simulation of Light Water Reactors (CASL) and Nuclear Energy + Advanced Modeling and Simulation (NEAMS) projects performed by W. + Wieselquist + + - Unification of readers/writers for ORIGEN data files developed by W. + Wieselquist + + - Improvement of binary formats for the ORIGEN library (f33) and ORIGEN + concentration file (f71) by J. Lefebvre, R. Lefebvre, and W. + Wieselquist + + - Implementation of Chebyshev Rational Approximation Method (CRAM) + solver by A. Isotalo + + - Development of new input format (ORIGEN sequence only) by S. Hart and + W. Wieselquist using the SCALE Object Notation (SON) syntax developed + by R. Lefebvre + + - Improvement of cubic spline interpolation scheme for ARP by S. + Skutnik and W. Wieselquist with monotonicity fix-up determined by G. + Hu + + - Major revision of manuals by W. Wieselquist, combining ORIGEN, ARP, + COUPLE, and OPUS into a single manual + +Additional guidance provided by D. Wiarda and I. Gauld with testing by +J. W. Hu. + +.. centered:: Version 6.1 (2011) + +The following section acknowledgements appeared in the SCALE 6.1 manual. + +ORIGEN + + +**Code Responsible(s):** I. C. Gauld + +The ORIGEN code was first developed by M. J. Bell with contributions +from J. P. Nichols and other members of the Chemical Technology Division +at ORNL. Development of the ORIGEN code as a depletion module of the +SCALE code system was performed by O. W. Hermann with contributions from +R. M. Westfall, supported by the NRC. + +COUPLE + + +**Code Responsible(s):** D. Wiarda and I. C. Gauld + + +The COUPLE code was originally developed by O. W. Hermann with guidance +from staff members including L. M. Petrie, N. M. Greene, W. E. Ford III, +and R. M. Westfall, who contributed greatly to formulation of the +methods, design of the data library interface with other modules, and +testing. Many valuable suggestions concerning code applications were +received from J. C. Ryman, J. R. Knight, and E. J. Allen. + +ARP + + +**Code Responsible(s):** I. C. Gauld, S. M. Bowman, and J. E. Horwedel + +The authors thank S. B. Ludwig for his support in earlier stages of this +work. The authors are grateful for the technical advice received from B. +L. Broadhead, M. D. DeHart, N. M. Greene, O. W. Hermann, C. V. Parks, +L. M. Petrie, and J. C. Ryman. The authors thank Germina Ilas and +Georgeta Radulescu for reviewing the manual and Willena Carter for +preparation of the manuscript. + +OPUS + + +**Code Responsible(s):** I. C. Gauld and J. E. Horwedel + +The work of O. W. Hermann in developing the PLORIGEN program, from which +OPUS was later developed, and the work of D. L. Barnett in developing +the original version of PlotOPUS, are acknowledged. Appreciation is +extended to J. C. Ryman for his review and testing of the program. +Finally the authors thank S. J. Poarch for formatting the manuscript. + + +.. [#f1] Aalto University, Finland + +.. [#f2] ÚJV Řež, a. s., Czech Republic + +.. [#f3] University of Tennessee, Knoxville + +.. [#f4] University of Illinois, Urbana-Champaign + +.. [#f5] Georgia Tech + +.. _5-1-1: + +Introduction +------------ + +ORIGEN solves the system of ordinary differential equations (ODEs) that +describe nuclide generation, depletion, and decay, + +.. math:: + \frac{dN_{i}}{\text{dt}} = \sum_{j \neq i}{(l_{\text{ij}} + \lambda_{j} + f_{\text{ij}}\sigma_{j}\Phi})N_{j}\left( t \right) - + \left( \lambda_{i} + \sigma_{i}\Phi \right) N_{i}(t) + S_{i}(t) + :label: eq-origen-odes + +where + + - :math:`N_{i}` = amount of nuclide *i* (atoms)\ *,* + + - :math:`\lambda_{i}` = decay constant of nuclide *i* (1/s)\ *,* + + - :math:`l_{\text{ij}}` = fractional yield of nuclide *i* from decay of + nuclide *j,* + + - :math:`\sigma_{i}` = spectrum-averaged removal cross section for + nuclide *i* (barn)\ *,* + + - :math:`f_{\text{ij}}` = fractional yield of nuclide *i* from + neutron-induced removal of nuclide *j*, + + - :math:`\Phi` = angle- and energy-integrated time-dependent neutron + flux (neutrons/cm\ :sup:`2`-s), and + + - :math:`S_{i}` = time-dependent source/feed term (atoms/s). + + +Note that :eq:`eq-origen-odes` has no spatial dependence and can be interpreted as either +a solution at a point in space or the spatial average over some volume. +The latter interpretation is preferred here, such that :math:`\Phi` is +the spatially averaged neutron flux magnitude, and all energy-dependence +is embedded in the one-group flux-weighted average cross sections +:math:`\sigma_{i}` and reaction yields :math:`f_{\text{ij}}`. :eq:`eq-origen-odes` is +conveniently written in matrix form as + +.. math:: + \frac{d\overrightarrow{N}}{dt} = \mathbf{A} + \overrightarrow{N}\left( t \right) + \overrightarrow{S}(t) + :label: eq-origen-tr-matrix + +with a :math:`\mathbf{A}` commonly referred to as the "transition +matrix." The representation of the transition matrix as +:math:`\mathbf{A = A}_{\sigma}\Phi\mathbf{+}\mathbf{A}_{\lambda}`, where +:math:`\mathbf{A}_{\sigma}` is the part of the transition matrix +containing reaction terms and :math:`\mathbf{A}_{\lambda}` is the part +containing decay terms, is convenient, as the numerical solution of this +system of ODEs holds the reaction, flux, and feed terms constant over +step :math:`n`, + +.. math:: + \frac{d\overrightarrow{N}}{\text{dt}} = \left( \mathbf + {A}_{\sigma,n}\Phi_{n}\mathbf{+}\mathbf{A}_{\lambda} \right) + \overrightarrow{N}\left( t \right) + {\overrightarrow{S}}_{n} + :label: eq-origen-tr-matrix-soln + +over time step :math:`t_{n - 1} \leq t \leq t_{n}.` + +Adding a continuous removal process described with rate constant +:math:`\lambda_{i,rem}` simply modifies the decay constant, +:math:`\lambda_{i} \rightarrow \lambda_{i} + \lambda_{i,rem}`, whereas a +continuous feed process defines a nonzero component of the +:math:`{\overrightarrow{S}}` vector. + +ORIGEN can also compute the alpha, beta, neutron, and gamma emission +spectra during decay. For the "stand-alone" ORIGEN calculations +described here, the transition matrix is loaded from an ORIGEN binary +library file (f33), which uses sparse-matrix storage to store one or +more transition matrices. The f33s may be created using COUPLE, saved +from TRITON depletion calculations, or interpolated using ARP from a set +of precompiled f33s distributed with SCALE. + +Results from ORIGEN calculations may be stored on a binary concentration +file (f71), which facilitates transfer of isotopics to other codes in +SCALE. The f71 file can also store calculated emission spectra. Within +ORIGEN, the f71 can be used to restart calculations from an existing set +of compositions. + +.. _5-1-2: + +Methodology +----------- + +This section describes the methodology used in performing the following +main functions: + + - generation of problem-dependent transition matrices, + + - solution of the system of depletion/decay equations, + + - conversion from power to flux (important for reactor applications), + + - calculation of emission spectra, and + + - interpolation of pregenerated sets of transition matrices. + +.. _5-1-2-1: + +Generation of Problem-dependent Transition Matrix +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +In the transition matrix :math:`\mathbf{A}` from :eq:`eq-origen-tr-matrix`, +each matrix element :math:`a_{\text{ij}}` is the first-order rate constant for the +formation of nuclide *i* from nuclide *j* given below. + +.. math:: + a_{ij} = + \begin{cases} + l_{ij}\lambda_{j} + f_{ij}\sigma_{j}\Phi & i \neq j \\ + \lambda_{i} - \sigma_{i} \phi & \text{otherwise} + \end{cases} + :label: eq-origen-trm-terms + +The transition matrix coefficients for decay and reaction transitions +are stored separately and reaction transitions are always stored with +:math:`\Phi = 1` and later during solution of the system, depending on +the step-average flux level the actual transition matrix +:math:`{\mathbf{A}_{n}\mathbf{= A}}_{\sigma,n}\Phi_{n}\mathbf{+}\mathbf{A}_{\lambda}` +is reconstructed using step-average flux, :math:`\Phi_{n}`. + +The decay coefficients :math:`l_{\text{ij}}\lambda_{j}` and +:math:`\lambda_{i}` are generated directly from ORIGEN decay resource +data. The reaction coefficients :math:`f_{\text{ij}}\sigma_{j}` and +:math:`\sigma_{i}` are generated using the following two-stage +procedure. + + 1. Calculate all removal cross sections :math:`\sigma_{i}` and yields + :math:`f_{\text{ij}}`, including isomeric branching ratios and + fission yields, by folding provided flux spectrum :math:`\phi^{g}` + with multigroup cross sections from the ORIGEN reaction resource + and energy-dependent fission yield data from the ORIGEN yield + resource. + + 2. Overwrite specific removal cross sections and yields based on a + provided multigroup cross section library **[SCALE Cross Section + Libraries chapter]** and/or user-provided one-group cross sections + and yields. + + +The second stage is optional, but it is important for cases which there +is significant self-shielding because ORIGEN's reaction resource assumes +infinite dilution for its multigroup data. The decay, reaction, and yeld +resources mentioned here are described in the ORIGEN Data Resources +chapter. The collapse to a one-group cross section in either stage is +given by + +.. math:: + \sigma_{\text{ri}} = \frac{\sum_{g}{\sigma_{\text{ri}}^{g}\phi^{g}}}{\sum_{g}\phi^{g}} + :label: eq-origen-collapse + +for reaction type :math:`r`, nuclide :math:`i`, and provided multigroup +flux :math:`\phi^{g}`. Different reaction types are recognized by their +ENDF MT numbers [SCALE Cross Section Libraries chapter] on the +appropriate data resource For example, MT=16 is +:math:`\left( n,2n \right),` and MT=107 is +:math:`\left( n,\alpha \right)`. The removal cross section +:math:`\sigma_{i}` is simply calculated as the sum over all relevant +reactions for a particular nuclide, +:math:`\sigma_{i} = \ \sum_{r}\sigma_{\text{ri}}`. This type of +reaction-dependent multigroup data may be contained in either the data +sources available in stage 1 or 2 above. However, only two types of data +are expected to be available in stage 1 reaction resource data: (1) +isomeric branching and (2) fission yields. + +The energy-dependent isomeric branching that describes the yield of each +excited level (metastable state) of a daughter nucleus is calculated in +a similar way, + +.. math:: + f_{\text{rim}} = \frac{\sum_{g}{f_{\text{rim}}^{g} {\sigma_{\text{ri}}^{g}\phi}^{g}}} + {\sum_{g}{\sigma_{\text{ri}}^{g}\phi^{g}}} + :label: eq-origen-meta-branching + +where :math:`m` indicates the possible metastable states and the +fractions always satisfy :math:`\sum_{m}f_{\text{rim}}^{\ } = 1`. + +Fission product yields are typically tabulated at discrete neutron +energies such as thermal (0.0253 eV), fission (500 keV), and high energy +(14 MeV). The yield for each fissionable nuclide is calculated in stage +1 by linearly interpolating the tabulated data using the computed +average energy of fission, + +.. math:: + {\overline{E}}_{\text{fi}} = \frac{\sum_{g}{{\overline{E}}^{g}{ + \sigma_{\text{fi}}^{g}\phi}^{g}}}{\sum_{g}{\sigma_{\text{fi}}^{g}\phi^{g}}} + :label: eq-origen-fpy + +where :math:`\sigma_{\text{fi}}^{g}` is the multigroup fission cross +section, and :math:`{\overline{E}}^{g}` is the average energy in +the group (simple midpoint energy used). In addition to generating +transition data for daughter/residual nuclides, the coefficients for +byproducts such as He-4/:math:`\alpha` byproducts from +:math:`\left( n,\alpha \right)` reactions are also retained in the +transition matrix and associated to an appropriate nuclide in the +system: hydrogen, deuterium, tritium, :sup:`3`\ He, or :sup:`4`\ He. + +.. _5-1-2-2: + +Solution of the Depletion/Decay Equations +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +ORIGEN includes two solver kernels that can solve the depletion/decay +equations of :eq:`eq-origen-tr-matrix-soln`: + + 1. a hybrid matrix exponential/linear chains method (MATREX) and + + 2. a Chebyshev Rational Approximation Method (CRAM). + +They are described in the following sections. + +.. _5-1-2-2-1: + +MATREX +^^^^^^ + +Referring to the system of ODEs shown in :eq:`eq-origen-tr-matrix` and setting +the external feed/source :math:`S\left( t \right) = 0`, there is a formal +solution by matrix exponential (an analog to the solution of a single ODE of +this type by exponential), + +.. math:: + \overrightarrow{N}\left(t \right) = \exp\left(\mathbf{A}t \right) \overrightarrow{N}\left( 0 \right) + :label: eq-origen-homo-soln + + +where :math:`\overrightarrow{N}\left( 0 \right)` is a vector of initial +nuclide concentrations, by defining the series expansion of +:math:`exp(\mathbf{A}t\mathbf{)}` to be + +.. math:: + \exp\left( \mathbf{A}t \right) = \mathbf{I + A}t + + \frac{\left( \mathbf{A}t \right)^{2}}{2} + \ldots = + \sum_{k = 0}^{\infty}\frac{\left( \mathbf{A}t \right)^{k}}{k!} + :label: eq-origen-series-exp + + +with :math:`\mathbf{I}` the identity matrix. :eq:`eq-origen-homo-soln` and +:eq:`eq-origen-series-exp` describe the matrix exponential method, which +yields a complete solution to the problem. However, in certain instances +related to limitation in computer precision, difficulties occur in generating +accurate values of the matrix exponential function. Under these circumstances, +alternative procedures using either the generalized Bateman equations +:cite:`bateman_solution_1910` or Gauss-Seidel iterative techniques are applied. These +alternative procedures will be discussed in further sections. + +A straightforward solution of :eq:`eq-origen-homo-soln` and +:eq:`eq-origen-series-exp` would require storage of the complete +transition matrix. To avoid excessive memory requirements, a recursion +relation has been developed. Substituting :eq:`eq-origen-series-exp` into +:eq:`eq-origen-homo-soln`, + +.. math:: + \overrightarrow{N}\left( t \right)\mathbf{=}\left + \lbrack \mathbf{I + A}t + \frac{\left( \mathbf{A}t \right)^{2}}{2} + + \ldots \right\rbrack\overrightarrow{N}\left( 0 \right) + :label: eq-origen-recursion + +one may recognize a recursion relation for a particular nuclide, +:math:`N_{i}(t)`. + +.. math:: + N_{i}\left( t \right) = N_{i}\left( 0 \right) + t\sum_{j}{a_{ij}N_{j} + \left( 0 \right)} + \frac{t}{2}\sum_{k}\left\lbrack a_{ik} + t\sum_{j}{a_{kj} N_{j}\left( 0 \right)} \right\rbrack \\ + + \frac{t}{3}\sum_{m}\left\{ a_{im} \frac{t}{2}\sum_{k}\left\lbrack a_{mk}t + \sum_{j}{a_{kj} N_{j}\left( 0 \right)} \right\rbrack \right\} + \ldots + :label: eq-origen-nuclide-recursion + + +where the range of indices, *j*, *k*, *m*, is 1 to *M* for matrix +:math:`\mathbf{A}` of size :math:`M \times M`. The result is a series of terms +that arise from the successive post-multiplication of the transition +matrix by the vector of nuclide concentration increments produced from +the computation of the previous terms. Within the accuracy of the series +expansion approximation, physical values of the nuclide concentrations +are obtained by summing a converged series of these vector terms. By +defining the terms :math:`C_{i}^{n}\left( t \right)` as + +.. math:: + C_{i}^{0} = N_{i}\left( 0 \right) \\ + C_{i}^{n + 1} = \frac{t}{n + 1}\sum_{j}{a_{\text{ij}}C_{j}^{n}} + :label: eq-origen-conc-soln-1 + + +the solution for :math:`N_{i}\left( t \right)` is given as + +.. math:: N_{i}\left( t \right) = \sum_{n = 0}^{\infty}C_{i}^{n} + :label: eq-origen-conc-soln-2 + +The use of :eq:`eq-origen-conc-soln-1` and :eq:`eq-origen-conc-soln-2` requires +storage of only two vectors--:math:`{\overrightarrow{C}}^{n}` and +:math:`{\overrightarrow{C}}^{n + 1}` ----in addition to the current +value of the solution. However, the series summation solution in +:eq:`eq-origen-conc-soln-2` is not valid until a finite limit is +identified which can achieve a reasonable accuracy, i.e., + +.. math:: + N_{i}\left( t \right) = \sum_{n = 0}^{n_{\text{term}}}C_{i}^{n} + \epsilon_{\text{trunc}} + :label: eq-origen-series-trunc + +where :math:`n_{\text{term}}` is the number of terms and +:math:`\epsilon_{\text{trunc}}` is the truncation error. The key is to +split the nuclides into two sets: those that are long-lived and permit a +rapid, accurate solution via :eq:`eq-origen-series-trunc`, and those that are +short-lived and require an alternate solution. + +.. _5-1-2-2-1-1: + +Solution for Long-Lived Nuclides +"""""""""""""""""""""""""""""""" + +This section describes the various tests used to ensure that the +summations indicated in :eq:`eq-origen-series-trunc` do not lose accuracy +due to large changes in magnitudes or small differences between positive and +negative rate constants. Nuclides with large rate constants (short-lived) are +removed from the transition matrix and treated separately. For example, +in the decay chain :math:`\mathrm{A\rightarrow B \rightarrow C}`, if the +decay constant for B is large, a new rate constant is inserted in the matrix for +:math:`\mathrm{A \rightarrow C}`. This technique was originally employed by Ball +and Adams :cite:`ball_matexpgeneral_1967`. The key to determining which transitions should be +removed involves calculation of the matrix norm. The norm of matrix +:math:`\mathbf{A}` is defined by Lapidus and Luus :cite:`lapidus_optimal_1967` as being the +smaller of the maximum-row absolute sum and the maximum-column absolute sum, + +.. math:: + \lbrack\mathbf{A}\rbrack = \min\left\{ {\max_{j}{\sum_{i}\left| + a_{\text{ij}} \right|}}{,\max_{i}{\sum_{j}\left| a_{\text{ij}} \right|}\ } \right\} + :label: eq-origen-tr-removal + +To maintain precision in performing the summations of :eq:`eq-origen-series-trunc`, +the matrix norm is used to balance the user-specified time step, *t*, with +the precision associated with the word len>h employed in the machine +calculation. The constraint on the matrix norm has been chosen as + +.. math:: + \left\lbrack \mathbf{A} \right\rbrack t \leq \ - 2\ \ln(0.001) = 13.8155 + :label: eq-origen-norm-constraint + +The remainder of this section shows that this constraint serves two +purposes. + + - It allows reasonable accuracy for a reasonable number (20--60) of + matrix exponential terms. + + - It defines what "short-lived" means over a particular time step, + dictating which concentrations must be solved by alternate means. + + +A relationship between *m* digits of machine precision and *p* +significant digits required in all results can be stated by the +following inequality: + +.. math:: + \text{(Largest term in series)} \times 10^{-m} + \leq \text{(Series result)} \times 10^{-p} + :label: eq-origen-precision-1 + +In this particular series, the relationship may be represented as + +.. math:: + \max_{n}\frac{\left| \left\lbrack \mathbf{A} \right\rbrack t \right|^{n}}{n!}10^{- m} + \leq \ e^{- \left\lbrack \mathbf{A} \right\rbrack t}10^{- p}`, + :label: eq-origen-precision-2 + +or alternatively, + +.. math:: + \max_{n}\frac{\left| \left\lbrack \mathbf{A} \right\rbrack t \right|^{n}}{n!} + e^{\left\lbrack \mathbf{A} \right\rbrack t} \leq \ 10^{m - p}. + :label: eq-origen-precision-3 + +Lapidus and Luus have shown that the maximum term in the summation for +any element in the matrix exponential function cannot exceed +:math:`\frac{\left( \left\lbrack \mathbf{A} \right\rbrack t \right)^{n}}{n!},\ `\ where +:math:`n` is the largest integer not larger than +:math:`\left\lbrack \mathbf{A} \right\rbrack t`. For the constraint in +:eq:`eq-origen-norm-constraint`, this yields *n*\ =13 and yields limit +:math:`\frac{\left( \left\lbrack \mathbf{A} \right\rbrack t \right)^{n}}{n!} \approx 10^{5}`. +With :math:`e^{\left\lbrack \mathbf{A} \right\rbrack t} \approx 10^{6}` +and standard double precision with *m=16*, :eq:`eq-origen-precision-3` evaluates +to :math:`10^{11} \leq 10^{16 - j}`, which indicates that five significant +figures will be maintained in values as small as 10\ :sup:`--6`. The +number of terms required to converge the matrix exponential series can +be investigated by a plot of the +:math:`\frac{\left| \left\lbrack \mathbf{A} \right\rbrack t \right|^{n}}{n!}e^{\left\lbrack \mathbf{A} \right\rbrack t}` +as a function of term index *n* in :eq:`eq-origen-precision-3`, as shown in :numref:`fig-series-conv` + +.. _fig-series-conv: +.. figure:: figs/ORIGEN/fig1.png + :align: center + + Values of terms in series for various values of the matrix norm. + + +The intersection between the black line in :numref:`fig-series-conv` and the +various curves indicates the number of terms needed to achieve +:math:`\epsilon_{\text{trunc}} \leq 0.1\%`. For example, with +:math:`\left\lbrack \mathbf{A} \right\rbrack t = 13.8155`, +:math:`n_{\text{term}} = 54` is required, and with +:math:`\left\lbrack \mathbf{A} \right\rbrack t = 13.8155/2`, +approximately :math:`n_{\text{term}} = 29` is required. This behavior +has been used to develop the heuristic + +.. math:: + n_{\text{term}} = 7\ \lbrack\mathbf{A}\rbrack t/2 + 6. + :label: eq-origen-solver-nterms + +Thus it has been shown that the limit imposed in :eq:`eq-origen-norm-constraint` leads to a +maximum of :math:`n_{\text{term}} = 54` terms with +:math:`\epsilon_{\text{trunc}} \leq 0.1\%`. + +It remains to be shown that any arbitrary system can be modified so that +it does not violate :eq:`eq-origen-norm-constraint`. Because the time step :math:`t` is +provided and fixed, +:math:`\left\lbrack \mathbf{A} \right\rbrack t \leq \ - 2\ \ln(0.001)` +cannot be satisfied unless the system is modified. The physical nature +of the system leads to +:math:`\max_{j}{\sum_{i}\left| a_{\text{ij}} \right|} \leq \max_{j}{2|a_{\text{jj}}|}` +based on production rates equal to loss rates when both parent and +daughter nuclide are included in the system. The maximum column sum in +:eq:`eq-origen-tr-removal` can then be bounded by twice the maximum diagonal term, +:math:`\max_{j}{2|a_{\text{jj}}|}`. Using this upper limit as the matrix +norm and substituting into :eq:`eq-origen-norm-constraint` yields + +.. math:: + \left\lbrack \mathbf{A} \right\rbrack t \leq 2\max_{j} + \left| a_{\text{jj}} \right| \leq - 2\ln\left( 0.001 \right) + :label: eq-origen-sln-ineq-1 + + +Rearranging :eq:`eq-origen-sln-ineq-1` leads to the condition + +.. math:: + e^{-\|a_{jj}\|t} < 0.001 + :label: eq-origen-sln-ineq-2 + + +which is used to mark nuclide *j* as a short-lived nuclide for this time +step, to be solved with linear chains instead of the series-based matrix +exponential. An alternative interpretation of the short-lived condition +can be made by rewriting :eq:`eq-origen-sln-ineq-2` in terms of an effective +half-life, :math:`t_{1/2} = \frac{\ln\left( 2 \right)}{|a_{\text{jj}}|}`, which +results in +:math:`t_{1/2} < \frac{{- ln}\left( 2 \right)}{\ln\left( 0.001 \right)}t \approx 0.1t`. +In other words, when a nuclide's effective half-life (including +destruction by both decay and reaction mechanisms) is less than 10% of +the time step, it can be considered short-lived. + +Finally, as a note for applications where the nuclides of interest are +in long transmutation chains, it has been found that the above algorithm +may not yield accurate concentrations for those nuclides near the end of +the chain that are significantly affected by those near the beginning of +the chain. In these applications, specifying the minimum +:math:`n_{\text{term}}` as + +.. math:: + n_{\text{term}} \geq \left| \Delta Z \right| + \left| \Delta A \right| + 5 + :label: eq-origen-sln-truncation + + +where :math:`\Delta Z` is the atomic number difference and +:math:`\Delta A` is the mass number difference, has been found to +ameliorate the issue. + +.. _5-1-2-2-1-2: + +Solution for Short-Lived Nuclides +""""""""""""""""""""""""""""""""" + +The condition in :eq:`eq-origen-sln-ineq-2` forms the basis for declaring a +nuclide short-lived, and its solution is found via solution of the nuclide +chain equations. In conjunction with maintaining the transition matrix norm +below the prescribed level, a queue is formed of the short-lived +precursors of each long-lived isotope. These queues extend back up the +several chains to the last preceding long-lived precursor. According to +:eq:`eq-origen-sln-ineq-2`, the queues will include all nuclides whose effective +half-lives are less than 10% of the time interval. A generalized form of +the Bateman equations developed by Vondy :cite:`vondy_development_1963` is used to solve for +the concentrations of the short-lived nuclides at the end of the time step. +For an arbitrary forward-branching chain, Vondy's form of the Bateman +solution is given by, + + +.. math:: + N_{i}\left( t \right) = N_{i}\left( 0 \right)e^{- d_{i}t} + \sum_{k = 1}^{i - 1} + {N_{k}(0)\left\lbrack \sum_{j = k}^{i - 1}\frac{e^{- d_{j}t} + - e^{- d_{i}t}}{d_{i} - d_{j}}a_{j + 1,j}\prod_{\begin{matrix} + n = k \\ + n \neq j \\ + \end{matrix}}^{i - 1}\frac{a_{n + 1,n}}{d_{n} - d_{j}} \right\rbrack} + :label: eq-origen-vondy-soln + + +where :math:`N_{1}\left( 0 \right)` is the initial concentration of the +first precursor, :math:`N_{2}\left( 0 \right)` is that of the second +precursor, etc. + +As in :eq:`eq-origen-trm-terms`, :math:`a_{\text{ij}}` is the first-order rate +constant, and :math:`d_{i} = {- a}_{\text{ii}}` which is the magnitude +of the diagonal element. Bell recast Vondy's form of the solution +through multiplication and division by +:math:`\prod_{n = k}^{i - 1}d_{n}` and rearranged to obtain + +.. math:: + N_{i}\left( t \right) = N_{i}\left( 0 \right)e^{- d_{i}t} + + \sum_{k = 1}^{i - 1}{N_{k}(0)\prod_{n = k}^{i - 1}\frac{a_{n + 1,n}}{d_{n}} + \left\lbrack \sum_{j = k}^{i - 1}{d_{j}\frac{e^{- d_{j}t} + - e^{- d_{i}t}}{d_{i} - d_{j}}}\prod_{ + \begin{matrix} + n = k \\ + n \neq j \\ + \end{matrix}}^{i - 1}\frac{d_{n}}{d_{n} - d_{j}} \right\rbrack} + :label: eq-origen-bell-soln + + +The first product over isotopes *n* is the fraction of atoms that +remains after the *k\ th* particular sequence of decays and captures. If +this product becomes less than 10\ :sup:`-6`, the contribution of this +sequence to the concentration of nuclide *i* is neglected. Indeterminate +forms that arise when *d\ i\ =d\ j* or *d\ n\ =d\ j* are evaluated using +L'Hôpital's rule. These forms occur when two isotopes in a chain have +the same diagonal element. + +:eq:`eq-origen-bell-soln` is applied to calculate all contributions to the "queue +end-of-interval concentrations" of each short-lived nuclide from the +initial concentrations of all others in the queue described above. It is +also applied to calculate contributions from the initial concentrations +of all short-lived nuclides in the queue to the long-lived nuclide that +follows the queue, in addition to the total contribution to its daughter +products. These values are appropriately applied either before or after +the matrix expansion calculation is performed to correctly compute +concentrations of long-lived nuclides and the long-lived or short-lived +daughters. :eq:`eq-origen-bell-soln` is also used to adjust to certain elements +of the final transition matrix, which now excludes the short-lived +nuclides. The value of the element must be determined for the new +transition between the long-lived precursor and the long-lived daughter +of a short-lived queue. The element is adjusted so that the +end-of-interval concentration of the long-lived daughter calculated from +the single link between the two long-lived nuclides (using the new +element) is the same as what would be determined from the chain +including all short-lived nuclides. The method assumes zero +concentrations for precursors to the long-lived precursor. The computed +values asymptotically approach the correct value with successive steps +through time. For this reason, at least five to ten time intervals +during the decay of discharged fuel is reasonable, because long-lived +nuclides have built up by that time. + +If a short-lived nuclide has a long-lived precursor, an additional +solution is required. First, the amount of short-lived nuclide *i* due +to the decay of the initial concentration of long-lived precursor *j* is +calculated as + +.. math:: + N_{j \rightarrow i}\left( t \right) = N_{j} + \left( 0 \right)a_{ij} \frac{e^{- d_{j}t}}{d_{i} - d_{j}} + :label: eq-origen-lln-sln + +from :eq:`eq-origen-vondy-soln`, assuming :math:`e^{- d_{i}t} \ll \ e^{- d_{j}t}`. +However, the total amount of nuclide *i* produced depends on the +contribution from the precursors of precursor *j*, in addition to that +given by :eq:`eq-origen-lln-sln`. The quantity of nuclide *j* not accounted for in +:eq:`eq-origen-lln-sln` is denoted by :math:`N_{j}'\left( t \right)`, the +end-of-interval concentration, minus the amount that would have remained +had there been no precursors to nuclide *j*: + +.. math:: + N_{j}^{'\left( t \right)} = N_{j}\left( t \right) - N_{j}(0)e^{- d_{j}t} + :label: eq-origen-lln-nef + +Then the short-lived daughter and subsequent short-lived progeny are +assumed to be in secular equilibrium with their parents, which implies +that the time derivative is zero, + + +.. math:: + \frac{ dN_{i}}{\text{dt}} = \sum_{j}{a_{\text{ij}}N_{j}(t)} = 0. + :label: eq-origen-sln-se + +The queue end-of-interval concentrations of all the short-lived nuclides +following the long-lived precursor are augmented by amounts calculated +with :eq:`eq-origen-bell-soln`. The concentration of the long-lived precursor +used in :eq:`eq-origen-lln-nef` is that given by :eq:`eq-origen-lln-sln`. +The set of linear algebraic equations given by :eq:`eq-origen-sln-se` +is solved by the Gauss-Seidel iterative technique. This algorithm involves +an inversion of the diagonal terms and an iterated improvement of an estimate +for :math:`N_{i}(t)` through the expression + +.. math:: + + N_{i}^{k + 1} = - \frac{1}{a_{\text{ii}}}\sum_{j}{a_{\text{ij}}N_{j}^{k}} + :label: eq-origen-sln-soln + + +Since short-lived isotopes are usually not their own precursors, this +iteration often reduces to a direct solution. + +.. _5-1-2-2-1-3: + +Solution of the Nonhomogeneous Equation +""""""""""""""""""""""""""""""""""""""" + +The previous sections have presented the solution of the homogeneous +equation in :eq:`eq-origen-homo-soln`, applicable to fuel burnup, activation, and +decay calculations. However, the solution of a nonhomogeneous equation +is required to simulate reprocessing or other systems that require an +external feed term, :math:`S\left( t \right) \neq 0`. The nonhomogeneous +equation is given in matrix form (assumed constant over a step *n*) as + + +.. math:: + \frac{d\overrightarrow{N}}{\text{dt}} = \mathbf{A} + \overrightarrow{N}\left( t \right) + {\overrightarrow{S}} + :label: eq-origen-nhe-matrix + + +for a fixed feed or removal rate, :math:`{\overrightarrow{S}}`. A +particular solution of :eq:`eq-origen-nhe-matrix` will be determined and added +to the solution of the homogeneous equation given by :eq:`eq-origen-recursion`. +As before, the matrix exponential method is used for the long-lived nuclides, +and solution by linear chains is used for the short-lived nuclides. Assume +:math:`\overrightarrow{C}` an arbitrary vector with which to test a +particular solution of the form + +.. math:: + \overrightarrow{N}\left( t \right) = \sum_{k = 0}^{\infty} + \frac{\left( \mathbf{A}t \right)^{k}}{\left( k + 1 \right)!}\overrightarrow{C}t + :label: eq-origen-nhe-matrix-soln-1 + + +Substituting :eq:`eq-origen-nhe-matrix-soln-1` into :eq:`eq-origen-nhe-matrix` +yields + +.. math:: + \sum_{k = 0}^{\infty}\frac{A^{k}t^{k}}{k!}\overrightarrow{C} = \sum_{k = 0}^{\infty} + \frac{A^{k + 1}t^{k + 1}}{(k + 1)!}\overrightarrow{C} + \overrightarrow{S} + :label: eq-origen-nhe-matrix-soln-2 + + +in which the *k=0* term may be extracted from the LHS, + +.. math:: + \overrightarrow{C} + \sum_{k = 1}^{\infty} + \frac{A^{k}t^{k}}{k!}\overrightarrow{C} = \sum_{k = 0}^{\infty} + \frac{A^{k + 1}t^{k + 1}}{\left( k+ 1 \right)!}\overrightarrow{C} + \overrightarrow{S} + :label: eq-origen-nhe-matrix-soln-3 + + +which allows the summations on the left and right to be easily +shown equal. This proves the particular solution is indeed +valid if the arbitrary vector is in fact the feed term +:math:`\overrightarrow{C} = \overrightarrow{S}`. The solution +to the nonhomogeneous problem is therefore (as a series), + +.. math:: + \overrightarrow{N}\left( t \right) = \sum_{k= 0}^{\infty} + \frac{\left( \mathbf{A}t \right)^{k}}{k!}\overrightarrow{N} + \left( 0 \right) + \sum_{k = 0}^{\infty}\frac{\left( \mathbf{A}t \right)^{k}}{ + \left( k +1 \right)!}\overrightarrow{S}t + :label: eq-origen-nhe-matrix-soln-4 + + +For the second term in :eq:`eq-origen-nhe-matrix-soln-4`, a new recursion +relation is developed for the particular solution in the same manner as +was done for the homogeneous solution, + +.. math:: + N_{i}^{P}\left( t \right) = \sum_{n = 1}^{\infty}D_{i}^{n} + :label: eq-origen-nhe-particular-soln-1 + + +where + +.. math:: + D_{i}^{1} = S_{i}t;\ D_{i}^{n+ 1} = \frac{1}{n + 1}\sum_{j}^{\ }{a_{ij}D}_{j}^{n} + :label: eq-origen-nhe-particular-soln-2 + + +For the short-lived nuclides, the secular equilibrium equations are +modified to become + + +.. math:: + \frac{dN_{i}}{\text{dt}} = + \sum_{j}{a_{\text{ij}}N_{j}\left( t \right) + S_{i}} = 0. + :label: eq-origen-sln-se-eqns` + + +The Gauss-Seidel iterative method is applied to determine the solution. +The complete solution to the nonhomogeneous equation in +:eq:`eq-origen-nhe-matrix-soln-1` is given by the sum of the homogeneous +solutions described in previous sections and the particular solutions described +here. + +.. _5-1-2-2-2: + +CRAM +^^^^ + +The solver kernel based on the Chebyshev Rational Approximation Method +(CRAM) is described in detail in references +:cite:`pusa_computing_2010,pusa_rational_2011,isotalo_comparison_2011,pusa_numerical_2013`. Compared to the MATREX solver, +CRAM generally has similar runtimes but is more accurate and robust on a +larger range of problems. CRAM relies on the lower upper (LU) decomposition, +so the SuperLU library has been used. The accuracy of CRAM is related to the +order, with an order 16 solution having a truncation error less than 0.01% +for all nuclides in most problems. + +Unlike many methods for solving this type of system of ODEs, the len>h +of a step does not significantly affect the accuracy of CRAM. However, +any significant errors from CRAM will shrink rapidly over multiple steps +as long as there are no large changes in reaction rates. The CRAM solver +has an efficient internal substepping algorithm that can perform +multiple same-sized substeps (with the same transition matrix) very +efficiently by reusing the LU decomposition. When using internal +substepping, 2--4 substeps are typical, with a large gain in accuracy for +marginal increase in runtime. + +.. _5-1-2-3: + +Power Calculation +~~~~~~~~~~~~~~~~~ + +The following is formula is used to calculate power during irradiation +(:math:`\Phi > 0`), + +.. math:: + P\left( t \right) = \sum_{i} + {\left( \kappa_{fi} \sigma_{fi} + \kappa_{ci} + \sigma_{ci} \right) \phi N_{i}\left( t \right)} + :label: eq-origen-power + + +where :math:`\kappa_{\text{fi}}` and :math:`\kappa_{\text{ci}}` are +nuclide-dependent energy released per fission and "capture," with +*capture* defined as removal minus fission: +:math:`\sigma_{\text{ci}} = \sigma_{i} - \sigma_{\text{fi}}`. The +:math:`\sigma_{\text{fi}}` and :math:`\sigma_{\text{ci}}` terms are +extracted from the transition matrix itself, whereas the +:math:`\kappa_{\text{fi}}` and :math:`\kappa_{\text{ci}}` are available +from a separate ORIGEN energy resource (see ORIGEN Data Resources +chapter). If the flux :math:`\phi` is specified, then the power +can be calculated at any time according to :eq:`eq-origen-power`. +However in reactor fuel systems, it is convenient to be able to specify the power +produced by the system and internally to the depletion code, to convert +the power to an equivalent flux. Solving :eq:`eq-origen-power` +for the flux, however, + +.. math:: + \Phi(t) = \frac{P}{\sum_{i}{{\left( \kappa_{\text{fi}}\sigma_{\text{fi}} + + \kappa_{\text{ci}}\sigma_{\text{ci}} \right)N}_{i}\left( t \right)}} + :label: eq-origen-flux + + +it is apparent that a fixed power over a time step :math:`n` does not +lead to a fixed flux, due to changing isotopics that produce different +amounts of power per fission and capture. ORIGEN performs a +flux-correction calculation to obtain an estimate of the average flux +over the step. The beginning-of-step flux is first calculated for the +initial compositions: :eq:`eq-origen-flux` is evaluated as +:math:`\Phi(t_{n - 1})`, and then :eq:`eq-origen-power` is solved +with that flux. The flux is then recalculated at the end of step :math:`\Phi(t_{n})` using +the estimated end-of-step isotopics, and the step-average flux +:math:`\Phi_{n}` is estimated as the simple average of the beginning and +end-of-step fluxes, i.e. :eq:`eq-origen-pc-flux`, + +.. math:: + \Phi_{n} = 0.5\lbrack\Phi\left( t_{n} \right) + + \Phi^{\text{pred}}\left( t_{n + 1} \right)\rbrack + :label: eq-origen-pc-flux + + +noting that the "predicted" flux at end-of-step +:math:`\Phi^{\text{pred}}\left( t_{n + 1} \right)` is based on +"predicted" end-of-step isotopics, based on a beginning-of-step flux +level. + +.. _5-1-2-4: + +Decay Emission Sources Calculation +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +ORIGEN can calculate the emission sources (and spectra) during decay for +alpha, beta, neutron, and photon particles according to + +.. math:: + R_{x}^{g}\left( t \right) = \sum_{i}{{\lambda_{i}N}_{i}\left( t + \right)\int_{E^{g}}^{E^{g - 1}}{w_{i,x}\left( E \right)\text{dE}}} + :label: eq-origen-decay + + +where :math:`w_{i,x}\left( E \right)` is the number of particles of type +:math::`x` emitted per disintegration of nuclide :math:`\text{i\ }`\ at +energy :math:`E`, using provided energy bins defined by energy bounds +:math:`E^{g}` to :math:`E^{g - 1},` where :math:`g` is an energy index. +The fundamental data resources for performing emission source +calculations are described in the ORIGEN Data Resources chapter. + +.. _5-1-2-4-1: + +Neutron Sources +^^^^^^^^^^^^^^^ + +Computed neutron sources include neutrons spontaneous fission, :math:`\left(\alpha,n \right)` +reactions, and delayed :math:`\left( \beta^-,n \right)` neutron emission, + +.. math:: + w_{i,n}\left( E \right) = w_{i,SFn}\left( E \right) + + w_{i,\left( \alpha,n \right)}(E) + w_{i,Dn}\left( E \right) + :label: eq-origen-neutron + +with components that will be described below. The method of computing +the spontaneous fission and delayed neutron source is independent of the +medium containing the fuel. However, :math:`\left(\alpha,n \right)` production varies +significantly with the composition of the medium. The homogeneous medium +:math:`\left(\alpha,n \right)` calculation methodology has been adopted from the Los Alamos code +SOURCES 4B :cite:`shores_data_2001,wilson_sources_2005`. + +The total yield of spontaneous fission neutrons from decay of nuclide +*i* is + +.. math:: + Y_{i,SFn} = \frac{\lambda_{i,SFn}}{\lambda_{i}}` + :label: eq-origen-sfn-yield + + +where :math:`\frac{\lambda_{i,SFn}}{\lambda_{i}}` is the fraction of +decays which undergo spontaneous fission. The distribution of +spontaneous fission neutrons, :math:`w_{i,SFn}\left( E \right)` is given +by a Watt fission spectrum, + +.. math:: + w_{i,SFn}\left( E \right) = Y_{i,SFn}\ C_{i}\ e^{- E/A_{i}}\sinh\sqrt{B_{i}E} + :label: eq-origen-sfn-watt + + +where *A\ i*, *B\ i,* and *C\ i* are model parameters. + +The :math:`\left(\alpha,n \right)` neutron source is strongly dependent on the low-Z content of +the medium containing the alpha-emitting nuclides and requires modeling +the slowing down of the alpha particles and the probability of neutron +production as the :math:`\alpha` particle slows down. The calculation assumes (1) a +homogeneous mixture in which the alpha-emitting nuclides are uniformly +intermixed with the target nuclides and (2) that the dimensions of the +target are much larger than the range of the alpha particles. Thus, all +alpha particles are stopped within the mixture. The yield of a +particular :math:`\alpha` is given by + +.. math:: + Y^\ell_{i,\alpha} = f^\ell_{i,\ \alpha}\frac{\lambda_{i,\alpha}}{\lambda_{i}} + :label: eq-origen-an + +where :math:`\frac{\lambda_{i,\alpha}}{\lambda_{i}}` is the +relative probability of :math:`\alpha`--decay, and :math:`f^\ell_{i,\ \alpha}` +is the fraction of those :math:`\alpha`--decays producing an :math:`\alpha` particle with initial +energy :math:`E^\ell_{i,\alpha}`, and is considered fundamental +data. The *total* neutron yield from an alpha particle +:math:`\ell` emitted by nuclide *i* and interacting with target +*k* is given by the following, + +.. math:: + Y^\ell_{i,k,\left( \alpha,n \right)} = Y^\ell_{i,\alpha} + \frac{N_{k}}{N}\int_{0}^{E^\ell_{i,\alpha}}{\frac{\sigma_{k, + \left(\alpha,n \right)}(E_{\alpha})}{S(E_{\alpha})}dE_{\alpha}\ } + :label: eq-origen-an-alpha-per-target + +where :math:`S\left( E_{\alpha} \right)` is the total stopping power of +the medium, :math:`\sigma_{k,\left( \alpha,n \right)}(E_{\alpha})` is +the :math:`\left(\alpha,n \right)` reaction cross section for target nuclide +*k*, and :math:`\frac{N_{k}}{N}` is the fraction of atoms in the medium +composed of nuclide *k*. This expression is used to calculate the neutron yield +for each target nuclide and from each discrete-energy alpha particle +emitted by all alpha-emitting nuclides in the material. The stopping +power for compounds, rather than pure elements, is approximated using +the Bragg-Kleeman additivity rule. The energy-dependent elemental +stopping cross sections are determined as parametric fits to evaluated +data. :eq:`eq-origen-an-alpha-per-target` is solved for the total +neutron yields from the alpha particle, as it slows down in the medium by +subdividing the maximum energy :math:`E^\ell_{i,\alpha}` into a number of +discrete energy bins and evaluating stopping power and +:math:`\left(\alpha,n \right)` reaction cross section at the midpoint energy of +the bin. The distribution of :math:`\left(\alpha,n \right)` neutrons as required +by :eq:`eq-origen-decay` is + +.. math:: + w_{i,(\alpha,n)}\left( E \right) = \sum_{k}{\sum_{\ell \in i} + Y^{\ell}_{i,k}\left( \alpha,n\right) X^{\ell}_{i,k} + \left( \alpha,n \right)}\left( E \right) + :label: eq:origen-an-dist + +with the distribution of :math:`\left(\alpha,n \right)` neutrons in energy, +:math:`X^\ell_{i,k,\left( \alpha,n \right)}\left( E \right)`, +calculated using nuclear reaction kinematics, assuming that the :math:`\left(\alpha,n \right)` +reaction emits neutrons with an isotropic angular distribution in the +center-of-mass system. The maximum and minimum permissible energies of +the emitted neutron are determined by applying mass, momentum, and +energy balance for each product's nuclide energy level. The product +nuclide levels, the product level branching data, the :math:`\left(\alpha,n \right)` reaction +Q values, the excitation energy of each product nuclide level, and the +branching fraction of :math:`\left(\alpha,n \right)` reactions result in the production of +product levels. A more detailed discussion of the theory and derivation +of the kinematic equations can be found in :cite:`shores_data_2001`. + +Delayed neutrons are emitted by decay of short-lived fission products. +The observed delay is due to the decay of the precursor nuclide. The +total yield of delayed neutrons from decay of nuclide *i* is + +.. math:: + Y_{i,Dn} = \frac{\lambda_{i,Dn}}{\lambda_{i}} + :label: eq-origen-dn-yield + +where :math:`\frac{\lambda_{i,Dn}}{\lambda_{i}}` is the fraction of +decays which emit delayed neutrons. The delayed neutrons emitted per +decay of nuclide *i* at energy *E* is given by + +.. math:: + w_{i,Dn}\left( E \right) = Y_{i,Dn} X_{i,Dn} \left( E \right) + :label: eq-origen-dn-rate + +where the spectrum :math:`X_{i,Dn}\left( E \right)` is fundamental library data. +Delayed neutrons are not important in typical spent fuel applications +due to the very short half-lives of the parent nuclides, dropping off +significantly after ~10 seconds, but they may be of value in specialized +applications where calculating time-dependent delayed neutron source +spectra is important. + +.. _5-1-2-4-2: + +Alpha Sources +^^^^^^^^^^^^^ + +An :math:`\alpha` slowing down calculation is performed as part of the :math:`\left(\alpha,n \right)` neutron +calculation. However, the alpha source (i.e. without considering slowing +down in the media) is also available, simply as the sum of delta +functions at the discrete initial alpha particle energies +:math:`w_{i,\alpha}\left( E \right) = \sum_{\ell \in i} {Y^{\ell}_{i,\alpha} \delta \left(E - E^{\ell}_{i,\alpha} \right)}` +with yields :math:`Y^{\ell}_{i,\alpha}`, as required by :eq:`eq-origen-decay`. + +.. _5-1-2-4-3: + +Beta Sources +^^^^^^^^^^^^ + +The beta source (i.e. without considering slowing down in the media) is +available as the sum of the continuous emission spectra for each +:math:`\beta^{-}` decay in nuclide *i*. The total yield of beta +particles from decay of nuclide *i* is + +.. math:: + Y_{i,\beta} = \frac{\lambda_{i,\beta}}{\lambda_{i}} + :label: eq-origen-beta-yield + +where :math:`\frac{\lambda_{i,\beta}}{\lambda_{i}}` is the fraction of +decays which emit betas. The betas emitted by nuclide *i* at energy *E* +is given by + +.. math:: + w_{i,\beta}\left( E \right) = Y_{i,\beta} X_{i,\beta}\left( E \right) + :label: eq-origen-beta-rate + +where the spectrum :math:`X_{i,\beta}(E)` is fundamental data, +independent of the media. The spectrum includes betas from allowed +transitions and first, second, and third forbidden transitions. + +.. _5-1-2-4-4: + +Photon Sources +^^^^^^^^^^^^^^ + +The total yield of photons from decay of nuclide *i* is + +.. math:: + Y_{i,\gamma} = \frac{\lambda_{i,\gamma}}{\lambda_{i}} + :label: eq-origen-gamma-yield + +where :math:`\frac{\lambda_{i,\gamma}}{\lambda_{i}}` is the fraction of +decays which emit photons. The photons emitted by nuclide *i* at energy +*E* is given by + +.. math:: + w_{i,\gamma}\left( E \right) = Y_{i,\gamma} X_{i,\gamma}\left( E \right) + :label: eq-origen-gamma-rate + +where the spectrum :math:`X_{i,\gamma}(E)` is fundamental data and +includes both line data from x-rays, gamma-rays and continuum data from +Bremsstrahlung, spontaneous fission gamma rays, and gamma rays +accompanying :math:`\left(\alpha,n \right)` reactions. The Bremsstrahlung component of the photon +source has been tabulated for various media and no on-the-fly slowing +down calculation is performed. + +.. _5-1-2-5: + +Library Interpolation +~~~~~~~~~~~~~~~~~~~~~ + +Accurate solution of fuel depletion with :eq:`eq-origen-odes` requires +coupling to self-shielding and neutron transport to accurately capture the +time-dependent change in space and energy flux distribution and 1‑group +cross sections with isotopic change. This is in generally a fairly +computationally intensive problem compared to stand-alone depletion. In +typical assembly design and analysis, the same basic assembly problem +must be solved repeatedly with variations in power history, different +periods of decay/burnup, different moderator density, etc. A question +naturally arises: could the isotopics from numerous well-constructed +cases be saved and interpolated to the actual system? Interpolating the +isotopics themselves is fraught with difficulty. For example, consider +two cases with the same burnup but different periods of decay between +cycles. A better approach---the ORIGEN Automated Rapid Processing +(ORIGEN-ARP)---was developed with the key realization that one can +reconstruct very accurate isotopics from stand-alone depletion +calculations by interpolating *transition matrices* rather than +*isotopics*. + +The accuracy of the interpolation methodology compared to the coupled +transport/depletion solution (e.g., with TRITON) depends on the +suitability of the interpolation parameters and the deviation of the +desired system from the systems used to create the library. For example, +for thermal systems with uranium-based fuels, it was found that +enrichment, water density, and burnup were the dominant independent +variables and thus were best suited for interpolation. An example of the +variation of removal cross sections for key actinides is shown in +:numref:`fig-PWR-cx` for a Westinghouse 17 × 17 pressurized water reactor +(PWR) assembly type with 5% initial enrichment in :sup:`235`\ U. Each cross +section has been divided by its initial value at zero burnup to show the +variation more clearly. :sup:`240`\ Pu has been observed to have the +most variation with spectral changes, with ~60% reduction in cross +section from beginning to end of life. The variations in :sup:`240`\ Pu +with respect to enrichment and moderator density are shown in +:numref:`fig-BWR-CX-BU`, :numref:`fig-BWR-CX-mod`, :numref:`fig-BWR-CX-enr`, +and :numref:`fig-BWR-CX-BU-enr`. + + +.. _fig-PWR-CX: +.. figure:: figs/ORIGEN/fig2.png + + Relative removal cross section as a function of burnup for + key actinides (Westinghouse 17 × 17 assembly with 5\\% enrichment). + + +.. _fig-BWR-CX-BU: +.. figure:: figs/ORIGEN/fig3.png + + :sup:`240`\ Pu-240 removal cross section as a function of + burnup for various enrichments (GE 10 × 10 assembly). + +.. _fig-BWR-CX-mod: +.. figure:: figs/ORIGEN/fig4.png + + :sup:`240`\ Pu removal cross section as a function of burnup + for various moderator densities (GE 10 × 10 assembly). + + +.. _fig-BWR-CX-enr: +.. figure:: figs/ORIGEN/fig5.png + + :sup:`240`\ Pu removal cross section as a function of + initial enrichment for various burnups (GE 10 × 10 assembly). + + +.. _fig-BWR-CX-BU-enr: +.. figure:: figs/ORIGEN/fig6.png + + :sup:`240`\ Pu removal cross section as a function of + moderator density for various burnups (GE 10 × 10 assembly). + +Currently there are two interpolation methods: a Lagrangian based on +low-order polynomials and a cubic spline with an optional monotonicity +fix-up. + +.. _5-1-2-5-1: + +Lagrangian Interpolation +^^^^^^^^^^^^^^^^^^^^^^^^ + +Lagrangian interpolation :cite:`funderlic_programmers_1968` seeks the unique *n-1* order polynomial +that will pass through *n*-points of the function and then interpolating +to the desired point by evaluating the polynomial, + +.. math:: + y\left( x \right) = \prod_{i = 1}^{n}{\left( x - x_{i} \right)\sum_{k = 1} + ^{n}\frac{y_{k}}{\left( x - x_{k} \right)\prod_{\begin{matrix} + i = 1 \\ + i \neq k \\ + \end{matrix}}^{n}{(x_{k} - x_{i})}}} + :label: eq-origen-lag-interp + +where :math:`x_{i}` and :math:`y_{i}` are the known x- and y-values in +the neighborhood of the desired x-value *x*, with *n* the number of data +points/order of Lagrangian interpolation. Note that Lagrangian +interpolation is by definition *local*, involving only points in the +neighborhood of the desired value. Global alternatives such as Hermite +cubic splines use the entire data set to construct the interpolants. +Common interpolation methods based on polynomials can have difficulty +with data that vary quickly and have uneven *x‑*\ spacing, as is +expected with transition data. Polynomials tend to produce unphysical +oscillations in these cases. In cases with very small *y-*\ values +(~10\ :sup:`-10`), oscillations of the interpolant can produce negative +interpolated values. + +.. _5-1-2-5-2: + +Cubic Spline Interpolation +^^^^^^^^^^^^^^^^^^^^^^^^^^ + +Cubic spline interpolation has been observed to produce fewer, lower +frequency oscillations. Oscillations can be effectively eliminated by +enforcing monotonicity on the interpolation: that is, additional max +maxima or minima are not introduced by the interpolant between known +values of the function. Monotonic cubic splines :cite:`wolberg_energy-minimization_2002` have shown +particularly stable behavior and have been implemented as an +interpolation option. + +.. _5-1-3: + +ORIGEN Family of Modules +------------------------ + +This section describes how to perform the calculations and evaluations +described in :ref:`5-2` using the ORIGEN family of modules in SCALE. +These modules are summarized briefly below. + +1. The COUPLE module is used to create ORIGEN libraries. The ORIGEN + library contains transition matrices **A** and other relevant data in + order to solve the depletion/decay equation of :eq:`eq-origen-odes`. + COUPLE requires an input flux spectrum in order to perform the multigroup + cross section collapse. Optionally, one-group self-shielded cross sections + can be provided. Generally, in order to solve a non-decay problem, a + library must be created with either COUPLE or ARP. + +2. The ARP module is also used to create an ORIGEN library, but it is + created by way of a special interpolation scheme on a set of existing + libraries rather than by specifying a flux spectrum and one-group + cross sections as in COUPLE. A set of ORIGEN libraries is distributed + with SCALE for use with ARP and is described in ORIGEN Reactor + Libraries chapter. + +3. The ORIGEN module is used to solve depletion, decay, activation, and + feed problems described by :eq:`eq-origen-odes`, as well as the decay + emission calculations described by :eq:`eq-origen-decay`. For spent fuel + calculations using the ARP interpolation methodology, it may be more + convenient to use ORIGAMI, as described in ORIGAMI chapter. + +4. The OPUS module is used to perform post processing and analysis on + ORIGEN results contained in ORIGEN concentrations files, including + sorting, ranking, and unit conversion. + +Two types of files are an integral part of the ORIGEN family of modules: +the library file and the concentrations files. + +- The library file is a binary file, usually either with the complete + filename “ft33f001” or with extension “.f33,” and it contains a + collection of transition matrices **A**\ *,* usually corresponding to + different burnups. It is typically called an “ORIGEN library,” + “ft33,” or “f33” file. + +- The concentrations file is also a binary file, usually either with + the complete filename “ft71f001” or with extension “.f71.” The f71 is + a solution archive containing isotopics vectors + :math:`{\overrightarrow{N}}_{\ }` corresponding to different + materials or different points in time. + +.. _5-1-3-1: + +COUPLE Module +~~~~~~~~~~~~~ + +COUPLE is a coupling code that prepares the transition matrix **A** from +:eq:`eq-origen-tr-matrix`, which contains the decay and cross section +transition rate constants according to the procedures defined in +:ref:`5-1-2-1`. The transition matrix and other important data +are stored on an ORIGEN library (f33) file for use by other modules. COUPLE has +two distinct modes of operation: + + 1. to create a new decay-only ORIGEN library from an ORIGEN decay + resource, and + + 2. to add new or to update existing reaction transitions yield resource, + reaction resource, and optionally an AMPX working library containing + multigroup cross sections. + +Details on the decay, yield, and reaction resources may be found in the +ORIGEN Data Resources chapter. + +.. _5-1-3-1-1: + +Key Features +^^^^^^^^^^^^ + +This section briefly highlights some key features in COUPLE and +describes how they are used. + +.. _5-1-3-1-1-1: + +AMPX multi-group libraries +"""""""""""""""""""""""""" + +AMPX multigroup libraries contain multigroup cross sections by nuclide +and material-zone identifiers. If the working library is the result of a +multiregion transport calculation, then it is important to specify the +correct zone identifier, e.g. corresponding to the fuel in a problem +with moderator, clad, and fuel zones. The neutron flux is also stored on +the AMPX library associated with a nuclide and a zone as are the cross +sections. An AMPX library flux can be used to perform the cross section +collapse as an alternative to providing a flux spectrum in the COUPLE +input. New transitions may be added to the ORIGEN binary library for all +reactions for which there are data in the weighted AMPX library if both +the target and product nuclides are present in the ORIGEN library. + +.. _5-1-3-1-1-2: + +Nuclide Specification +""""""""""""""""""""" + +In COUPLE, the following nuclide identifier is used: + +.. code-block:: scale + + Nuclide identifier = Z \* 10000 + A \* 10 + I + +where + + Z = atomic number, + + A = mass number, + + I = metastable/isomeric state (0 is ground/1 is first metastable) + +Examples include 922350 for :sup:`235`\ U and 952421 for +:sup:`242m`\ Am. Note that this varies from the identifiers used in +other ORIGEN-related modules in which the isomeric state *I* comes +first, as in 1095242 for :sup:`242m`\ Am. + +.. _5-1-3-1-1-3: + +Adding new transitions and user-defined transitions +""""""""""""""""""""""""""""""""""""""""""""""""""" + +The use of a transition matrix in ORIGEN allows any nuclide to +transition to any other nuclide. By default, when the reaction data on +the library is updated, then the transition matrix’s sparse storage is +expanded to include the new reaction transition if both the target and +the reaction product nuclide are in the library. The user may request +that the code does not add new transitions by setting Block1 ``1$$ JADD=0``. +This option ensures that the matrix structure on the input library is +identical to that of the output library. The user may explicitly set +one-group transition coefficients by setting Block1 ``1$$ LBUP=1`` and +entering Block6 and Block8 data. + +.. _5-1-3-1-1-4: + +Unit numbers and Aliases +"""""""""""""""""""""""" + +In COUPLE, a unit number is used instead of a full file name to specify +files, where unit number XY links to the data file “ftXYf001” in the +working directory. For example, unit number 33 means file ft33f001. +There are several predefined unit numbers that are controlled by a +special “origen_filenames” file, which creates an alias for the local +file “ftXYf001” to a file in the data directory. :numref:`table-couple-units` +shows the basic COUPLE unit numbers, their aliases, and a description of the +file. + +.. _table-couple-units: +.. table:: Basic COUPLE unit numbers + :widths: 8 15 40 + :align: center + + +----------+-----------+-------------------------------------------+ + | **Unit** | **Alias** | **Description** | + +==========+===========+===========================================+ + | 17 | YIELDS | ORIGEN Yield Resource | + +----------+-----------+-------------------------------------------+ + | 21 | END7DEC | ORIGEN library | + | | | | + | | | *ENDF/B-VII-based decay transitions only* | + +----------+-----------+-------------------------------------------+ + | 27 | DECAY | ORIGEN Decay Resource | + +----------+-----------+-------------------------------------------+ + | 80 | JEFF252G | ORIGEN Reaction Resource (252 groups) | + +----------+-----------+-------------------------------------------+ + +.. _5-1-3-1-2: + +Input Description +^^^^^^^^^^^^^^^^^ + +COUPLE uses the FIDO input system, except for title entries. The input +is arranged in blocks, with each block containing one or more arrays, +followed by the FIDO block terminator “t.” Each input parameter is named +and defined below in the order in which it appears, with the index of +the parameter in the array. Some options have been deprecated over time +and thus the first available entry may not correspond to index “1” and +some indices may be skipped. Default values are given in parentheses. In +the SCALE code system, COUPLE input appears between “=couple” and “end.” + +.. _5-1-3-1-2-1: + +Block1: titles, unit numbers, and case controls. +"""""""""""""""""""""""""""""""""""""""""""""""" + +TITLE – Title lines + + Title lines can provide information about the ORIGEN library cre