From 7977024bd154462b4da26289921812c552d2d674 Mon Sep 17 00:00:00 2001 From: Batson Iii <3tv@mac107054.ornl.gov> Date: Tue, 17 Nov 2020 23:45:30 -0500 Subject: [PATCH] Adding chapter 7 --- .DS_Store | Bin 14340 -> 16388 bytes BONAMI.rst | 1393 + CAJUN.rst | 114 + CENTRM.rst | 3367 +++ CHOPS.rst | 200 + CRAWDAD.rst | 264 + MCDancoff.rst | 404 + ... and Cross Section Processing Overview.rst | 208 + PMC.rst | 1373 + PMCAppAB.rst | 30 + XSProc.rst | 2945 +++ XSProcAppA.rst | 903 + XSProcAppB.rst | 725 + XSProcAppC.rst | 971 + _build/doctrees/BONAMI.doctree | Bin 0 -> 223085 bytes _build/doctrees/CAAScapability.doctree | Bin 0 -> 129400 bytes _build/doctrees/CAJUN.doctree | Bin 0 -> 15523 bytes _build/doctrees/CENTRM.doctree | Bin 0 -> 645472 bytes _build/doctrees/CHOPS.doctree | Bin 0 -> 31758 bytes _build/doctrees/CRAWDAD.doctree | Bin 0 -> 31815 bytes _build/doctrees/CSAS5.doctree | Bin 0 -> 321982 bytes _build/doctrees/CSAS5App.doctree | Bin 0 -> 511145 bytes _build/doctrees/CSAS6.doctree | Bin 0 -> 134484 bytes _build/doctrees/CSAS6App.doctree | Bin 0 -> 41598 bytes .../Criticality Safety Overview.doctree | Bin 0 -> 43465 bytes _build/doctrees/DEVC.doctree | Bin 0 -> 137840 bytes _build/doctrees/K5C5.doctree | Bin 0 -> 30541 bytes _build/doctrees/KMART.doctree | Bin 0 -> 32538 bytes _build/doctrees/Keno.doctree | Bin 0 -> 3716198 bytes _build/doctrees/KenoA.doctree | Bin 0 -> 24475 bytes _build/doctrees/KenoB.doctree | Bin 0 -> 94026 bytes _build/doctrees/KenoC.doctree | Bin 0 -> 305716 bytes _build/doctrees/MAVRIC.doctree | Bin 0 -> 597716 bytes _build/doctrees/MCDancoff.doctree | Bin 0 -> 55346 bytes ... 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+Regardless of the method used to obtain the flux spectrum, the +parameterized shielded cross sections for absorber nuclide “r” are +computed from the expression, + +.. math:: + :label: eq7-3-14 + + \sigma _{\text{X,g}}^{\text{(r)}}(\sigma \,_{0}^{(r)}\,,T)\quad =\quad \,\frac{\int_{g}{\ \ \,\sigma _{X}^{(r)}(E,T)\ \,\Phi (E;\,\,\sigma \,_{0}^{(r)}\,,T)\ dE}}{\int_{g}{\ \,\Phi (E;\,\,\sigma \,_{0}^{(r)}\,,T)\ \,dE}}\quad , + +where :math:`\Phi (E;\,\,\sigma \,_{0}^{(r)}\,,T)` is the flux for a given value +of :math:`\sigma \,_{0}^{(r)}` and *T*. + +Rather than storing self-shielded cross sections in the master library, +AMPX converts them to Bondarenko shielding factors, also called +f-factors, defined as the ratio of the shielded cross section to the +infinitely dilute cross section. Thus the MG libraries in SCALE contain +Bondarenko data consisting of f‑factors defined as + +.. math:: + :label: eq7-3-15 + + f_{\text{X,g}}^{\text{(r)}}(\sigma \,_{0}^{{}}\,,T)\quad =\quad \,\frac{\sigma _{\text{X,g}}^{\text{(r)}}(\sigma \,_{0}^{{}},T)}{\sigma _{\text{X,g}}^{\text{(r)}}(\infty )}\quad , + +and infinitely dilute cross sections defined as, + +.. math:: + :label: eq7-3-16 + + \sigma _{\text{X,g}}^{\text{(r)}}(\infty )\quad =\quad \,\sigma _{\text{X,g}}^{\text{(r)}}(\sigma \,_{0}^{{}}=\infty ,T={{T}_{ref}}) \to \ \ \,\frac{\int_{g}{\ \sigma _{X}^{(r)}(E,{{T}_{ref}})\ C(E)\ \,dE}}{\int_{g}{\ \,C(E)\ \,dE}}\quad . + +In AMPX, the reference temperature for the infinitely dilute cross +section is normally taken to be 293 K. Bondarenko data on SCALE +libraries are provided for all energy groups and for five reaction +types: total, radiative capture, fission, within-group scattering, and +elastic scatter. Recent SCALE libraries include f-factors at ~10–30 +background cross section values (depending on nuclide) ranging from +~10\ :sup:`−3` to ~10\ :sup:`10` barns, which span the range of +self-shielding conditions. Typically the f-factor data are tabulated at +five temperature values. Background cross sections and temperatures +available for each nuclide in the SCALE MG libraries are given in the +SCALE Cross Section Libraries chapter. + +.. _7-3-2-3: + +Background Cross Section Options in BONAMI +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +To compute self-shielded cross sections for nuclide *r*, BONAMI first +computes the appropriate background cross section for the system of +interest and then interpolates the library Bondarenko data to obtain the +f-factor corresponding to this σ\ :sub:`0` and nuclide temperature. +Several options are available in BONAMI to compute the background cross +section, based on :eq:`eq7-3-10` and :eq:`eq7-3-12` in the preceding section. The options are +specified by input parameter “\ **iropt**\ ” and have the following +definitions: + +(a) iropt = 0 => NR approximation with Bondarenko iterations: + +Background cross sections for all nuclides are computed using :eq:`eq7-3-12` with +λ=1; therefore, + +.. math:: + :label: eq7-3-17 + + \sigma _{0}^{\text{(r)}}\ =\ \frac{1}{{{\text{N}}^{\text{(r)}}}}\,\,\sum\limits_{j\ne r}{\Sigma _{\text{t,g}}^{\text{(j)}}} . + +Since the background cross section for each nuclide depends on the shielded +total cross sections of all other nuclides in the mixture, +“Bondarenko iterations” are performed in BONAMI to obtain a consistent set of +shielded cross sections. Bondarenko iterations provide a crude method of +accounting for resonance interference effects that are ignored by the +approximation for :math:`\sigma \,_{0}^{(r)}` in :eq:`eq7-3-10`. The BONAMI +iterative algorithm generally converges in a few iterations. Prior to +SCALE-6.2, this option was the only one available in BONAMI, and it is still the default for XSProc. + +(b) iropt = 1 => IR approximation with no resonance interference + (potential cross sections): + +Background cross sections for all nuclides are computed using :eq:`eq7-3-10`. No +Bondarenko iterations are needed. + +(c) iropt t = 2 => IR approximation with Bondarenko iterations, but no + resonance scattering: + +Background cross sections for all nuclides are computed using :eq:`eq7-3-12` with +the scattering cross section approximated by the potential value; +therefore, + +.. math:: + :label: eq7-3-18 + + \sigma _{0}^{\text{(r)}}\ \ =\ \ \frac{1}{{{\text{N}}^{\text{(r)}}}}\,\,\sum\limits_{j\ne r}{\left( \Sigma _{\text{a,g}}^{\text{(j)}}+\lambda _{\text{g}}^{\text{(j)}}\,\Sigma _{\text{p}}^{\text{(j)}}\, \right)} + + +Since the background cross section for each resonance nuclide includes the +shielded absorption cross sections of all other nuclides, Bondarenko +interactions are performed. + +(d) iropt = 3 => IR approximation with Bondarenko iterations: + +Background cross sections for all nuclides are computed using the full +IR expression in :eq:`eq7-3-12`. Bondarenko interactions are performed. + +Computation of the background cross sections in BONAMI generally +requires group-dependent values for the IR parameter λ. These are +calculated by a module in AMPX during the library process and are stored +in the MG libraries under the reaction identifier (MT number), MT=2000. + +.. _7-3-2-4: + +Self-Shielded Cross Sections for Heterogeneous Media +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Equivalence theory can be used to obtain shielded cross sections for +heterogeneous systems containing one or more “lumps” of resonance +absorber mixtures separated by moderators, such as reactor lattices. It +can be shown that if the fuel escape probability is represented by the +Wigner rational approximation, the collision probability formulation of +the neutron transport equation for an absorber body in a heterogeneous +medium can be reduced to a form identical to :eq:`eq7-3-3`. This can be done for +an “equivalent” infinite homogeneous medium consisting of the same +absorber body mixture plus an additional NR scatterer with a constant +cross section called the “escape cross section” :cite:`lamarsh_introduction_1966`. +Equivalence +theory states that the self-shielded cross section for resonance +absorber *r* in the heterogeneous medium is equal to the self-shielded +cross section of *r* in the equivalent infinite homogeneous medium; +therefore the f-factors that were calculated for homogenous mixtures can +also be used to compute self-shielded cross sections for heterogeneous +media by simply interpolating the tabulated f-factors in the library to +the modified sigma-zero value of + +.. math:: + :label: eq7-3-19 + + \hat{\sigma }_{0}^{(r)}\quad =\quad \sigma _{0}^{(r)}\ +\ \ \,\sigma _{esc}^{(r)} + +where, + + :math:`\hat{\sigma }_{0}^{(r)}` = background cross section of r in the absorber lump of the heterogeneous system; + + :math:`\sigma \,_{0}^{(r)}` = background cross section defined in :ref:`7-3-2-1` for an infinite homogeneous medium of the + absorber body mixture; + + :math:`\sigma _{esc}^{(r)}` = microscopic escape cross section for nuclide *r*, defined as + +.. math:: + :label: eq7-3-20 + + \sigma _{esc}^{(r)}\quad =\quad \frac{{{\Sigma }_{esc}}}{{{N}^{(r)}}} + +.. + + :math:`{{\Sigma }_{esc}}` = macroscopic escape cross section for the absorber lump defined in BONAMI as + +.. math:: + :label: eq7-3-21 + + {{\Sigma }_{esc}}\quad =\quad \,\frac{(1\quad -\quad c)A}{\bar{\ell }\ \,\ \left[ 1\quad +\quad \left( A\quad -\quad 1 \right)c \right]} + +where + + :math:`\bar{\ell }` = average chord length of the absorber body = :math:`4\ \ \,\times \ \frac{volume}{surface\ \ area}`; + + A = Bell factor, used to improve the accuracy of the Wigner rational approximation; + + c = lattice Dancoff factor, which is equal to the probability that a neutron escaping from one + absorber body will reach another absorber body before colliding in the intervening moderator. + +Values for the mean chord length :math:`\bar{\ell }` are computed in BONAMI for slab, +sphere, and cylinder absorber bodies. In the most common mode of operation where +BONAMI is executed through the XSProc module in SCALE, Dancoff factors for +uniform lattices are computed automatically and provided as input to BONAMI. +For nonuniform lattices—such as those containing water holes, control rods, +etc.—it may be desirable for the user to run the SCALE module MCDancoff to +compute Dancoff factors using Monte Carlo for an arbitrary 3D configuration. +In this case the values are provided in the MORE DATA input block of XSProc. +The Bell factor “A” is a correction factor to account for errors caused by use +of the Wigner rational approximation to represent the escape probability from a +lump. Two optional Bell factor corrections are included in BONAMI. The first uses +expressions developed by Otter that essentially force the Wigner escape +probability for an isolated absorber lump to agree with the exact escape +probability for the particular geometry by determining a value of A as a function of +:math:`{{\Sigma }_{T}}\bar{\ell }` for slab, cylindrical, or spherical +geometries. Since the Otter expression was developed for isolated bodies, +it does not account for errors in the Wigner rational approximation due to +lattice effects. BONAMI also includes a Bell factor correction based on a +modified formulation developed by Leslie :cite:`leslie_improvements_1965` that is a function of the Dancoff factor. + +.. _7-3-3: + +Interpolation Scheme +-------------------- + +After the background cross section for a system has been computed, +BONAMI interpolates f-factors at the appropriate σ\ :sub:`0` and +temperature from the tabulated values in the library. :numref:`fig7-3-1` shows +a typical variation of the f-factor vs. background cross sections for +the capture cross section of :sup:`238`\ U in the SCALE 252 group +library. + +.. _fig7-3-1: +.. figure:: figs/BONAMI/fig1.png + :align: center + :width: 500 + + Plot of f-factor variation for :sup:`238`\ U capture reaction. + +Interpolation of the f-factors can be problematic, and several different +schemes have been developed for this purpose. Some of the interpolation +methods that have been used in other codes are constrained +Lagrangian, :cite:`davis_sphinx_1977` arc-tangent fitting, :cite:`kidman_improved_1974` and an approach developed by +Segev :cite:`segev_interpolation_1981`. All of these were tested and found to be inadequate for use +with the SCALE libraries, which may have multiple energy groups within a +single resonance. BONAMI uses a unique interpolation method developed by +Greene, which is described in :cite:`greene_method_1982`. Greene’s interpolation method +is essentially a polynomial approach in which the powers of the +polynomial terms can vary within a panel, as shown in :eq:`eq7-3-25`: + +.. math:: + :label: eq7-3-22 + + f\left( \sigma \right)\quad =\quad f\left( \sigma {{\,}_{1}} \right)\quad +\quad \frac{\sigma {{\,}^{q(\sigma )}}\quad -\quad \sigma \,_{1}^{q(\sigma )}}{\sigma \,_{2}^{q(\sigma )}\quad -\quad \sigma \,_{1}^{q(\sigma )}}\quad \left( f\left( {{\sigma }_{2}} \right)\quad -\quad f\left( {{\sigma }_{1}} \right) \right)\quad , + +where + +.. math:: + :label: eq7-3-23 + + q\left( \sigma \right)\quad =\quad q\left( \sigma {{\,}_{1}} \right)\quad +\quad \frac{\sigma \quad -\quad \sigma \,_{1}^{{}}}{\sigma \,_{2}^{{}}\quad -\quad \sigma \,_{1}^{{}}}\quad \left( q\left( {{\sigma }_{2}} \right)\quad -\quad q\left( {{\sigma }_{1}} \right) \right)\quad . + +:numref:`fig7-3-2` illustrates the expected behavior of :eq:`eq7-3-22` caused by varying +the powers in a panel. + +By allowing the power *q* to vary as a function of independent +variable σ, we can move between the various monotonic curves on the +graph in a monotonic fashion. Note that when *p* crosses the +*p* = 1 curve, the shape changes from concave to convex, or vice versa. +This shape change means that we can use the scheme to introduce an +inflection point, which is exactly the situation needed for +interpolating f-factors. + +.. _fig7-3-2: +.. figure:: figs/BONAMI/fig2.png + :align: center + :width: 500 + + Illustration of the effects of varying “powers” in the Greene interpolation method. + +:numref:`fig7-3-3` and :numref:`fig7-3-3` show typical “fits” of the f-factors using +the Greene interpolation scheme for two example cases. Note, in +particular, that since this scheme has guaranteed monotonicity, it +easily accommodates the end panels that have the smooth asymptotic +variation. Even considering the extra task of having to determine the +powers for temperature and σ\ :sub:`0` interpolations, the method is not +significantly more time-consuming than the alternative schemes for most +applications. + +.. _fig7-3-3: +.. figure:: figs/BONAMI/fig3.png + :align: center + :width: 500 + + Use of Greene’s method to fit the σ\ :sub:`0` variation of Bondarenko factors for case 1. + +.. _fig7-3-4: +.. figure:: figs/BONAMI/fig4.png + :align: center + :width: 500 + + Use of Greene’s method to fit the σ\ :sub:`0` variation of Bondarenko factors for case 2. + +.. _7-3-4: + +Input Instructions +------------------ + +BONAMI is most commonly used as an integral component of XSProc through +SCALE automated analysis sequences. XSProc automatically prepares all +the input data for BONAMI and links it with the other self-shielding +modules. During a SCALE sequence execution, the data are provided +directly to BONAMI in memory through XSProc. Some of the input +parameters can be modified in the MOREDATA block in XSProc. + +However, the legacy interface to execute stand-alone BONAMI calculations +has been preserved for expert users. The legacy input to BONAMI uses the +FIDO schemes described in the FIDO chapter of the SCALE manual. The +BONAMI input for standalone execution is given below, where the MOREDATA +input keywords are marked in bold. + +.. centered:: Data Block 1 + +0$ Logical Unit Assignments [4] + + 1. masterlib— input master library (Default = 23) + + 2. mwt—not used + + 3. msc—not used + + 4. newlib—output master library (Default = 22) + +1$ Case Description [6] + + 1. cellgeometry—geometry description + + 0 homogeneous + + 1 slab + + 2 cylinder + + 3 sphere + + 2. numzones—number of zones or material regions + + 3. mixlength—mixing table length. This is the total number of entries + needed to describe the concentrations of all constituents in all + mixtures in the problem. + + 4. ib—not used + + 5. **crossedt**—output edit option + + 0 no output (Default) + + 1 input echo + + 2 iteration list, timing + + 3 background cross section calculation details + + 4 shielded cross sections, Bondarenko factors + + 6. issopt—not used + + 7. **iropt—**\ resonance approximation option + + 0 NR (Default) (Bondarenko iterations) + + 1 IR with potential scattering + + 2 IR with absorption and potential scattering (Bondarenko iterations) + + 3 IR with absorption and elastic scattering (Bondarenko iterations) + + 8. **bellopt—**\ Bell factor calculation option + + 0 Otter + 1 Leslie (Default) + + 9. **escxsopt—**\ escape cross section calculation option + + 0 consistent + + 1 inconsistent (Default) + +2\* Floating-Point Constants [2] + + 1. **bonamieps**—convergence criteria for the Bondarenko iteration + (Default = 0.001) + + 2. **bellfact**—geometrical escape probability adjustment factor. See + notes below on this parameter (Default = 0.0). + +T Terminate Data Block 1. + +.. centered:: Data Block 2 + +3$ Mixture numbers in the mixing table [mixlength] +4$ Component (nuclide) identifiers in the mixing table [mixlength] +5* Concentrations (atoms/b-cm) in the mixing table [mixlength] +6$ Mixtures by zone [numzones] +7* Outer radii (cm) by zone [numzones] +8* Temperature (k) by zone [numzones] +9* Escape cross section (cm\ :sup:`-1`) by zone [numzones] +10$ Not used +11$ Not used +12* Temperature (K) of the nuclide in a one-to-one correspondence with the mixing table arrays. +13* Dancoff factors by zone [numzones] +14* Lbar (:math:`\bar{\ell }`) factors by zone [numzones] + +T Terminate Data Block 2. + +This concludes the input data required by BONAMI. + +.. _7-3-4-1: + +Notes on input +~~~~~~~~~~~~~~ + +In the 1$ array, *cellgeometry* specifies the geometry. The geometry +information is used in conjunction with the 7* array to calculate mean +chord length *Lbar* if it is not provided by the user in the 14\* array. + +*numzones*, the number of zones, may or may not model a real situation. +It may, for example, be used to specify *numzones* independent media to +perform a cell calculation in parallel with one or more infinite medium +calculations. The geometry description in 1$ array applies only to mean +chord length calculations unless it is provided in 14*. + +In the 2* array, *bonamieps* is used to specify the convergence expected +on all macroscopic total values by zone, that is, each :math:`{{\Sigma }_{t}}(g,j)` in group g and +zone j is converged so that + +.. math:: + :label: eq7-3-24 + + \frac{\left| \,\Sigma \,_{t}^{i}(g,j)\quad -\quad \Sigma \,_{t}^{i-1}(g,j)\, \right|}{\Sigma \,_{t}^{i}(g,j)}\quad \le \quad bonamieps\quad . + +The “Bell” factor in the 2* array is the parameter used to adjust the +Wigner rational approximation for the escape probability to a more +correct value. It has been suggested that if one wishes to use one +constant value, the Bell factor should be 1.0 for slabs and 1.35 +otherwise. In the ordinary case, BONAMI defaults the Bell factor to zero +and uses a prescription by Otter :cite:`otter_escape_1964` to determine a cross-section +geometry-dependent value of the Bell factor for isolated absorber +bodies. It uses a prescription by Leslie\ :sup:`6` to determine the +Dancoff factor–dependent values of the Bell factor for lattices, which +are much more accurate than the single value. The user who wishes to +determine the constant value can, however, use it by inputting a value +other than zero. + +The 3$, 4$, and 5* arrays are used to specify the concentrations of the +constituents of all mixtures in the problem as follows: + +Entry 3$ (Mixture Number) 4$ (Nuclide ID) 5\* (Concentrations) + +1 + +2 + +. + +. + +. + +. + +mixlength + +. + +Because of the manner in which BONAMI references the nuclides in a +calculation, each nuclide in the problem must have a unique entry in the +mixing table. Thus one cannot specify a mixture and subsequently load it +into more than one zone, as can be the case with many modules requiring +this type of data. + +The 12* array is used to allow varying the temperatures by nuclide +within a zone. In the event this array is omitted, the 12* array will +default by nuclide to the temperature of the zone containing the +nuclide. + +The mixture numbers in each zone are specified in the 6$ array. Mixture +numbers are arbitrary and need only match up with those used in the +3$ array. + +The radii in the 7* array are referenced to a zero value at the left +boundary of the system. + +In the event the temperatures in the 8* array are not bounded by +temperature values in the Bondarenko tables, BONAMI will extrapolate +using the three temperature points closest to the value. For example, a +request for 273 K for a nuclide with Bondarenko sets at 300, 900, and +2,100 K would use the polynomial fit from those three temperature points +to extrapolate the 273 K value. + +The escape cross sections in the 9* array allow a macro escape cross +section (:math:`\Sigma _{e}^{input}`) to be specified by zone. (This array can be ignored if +Dancoff factors are provided.) If the Dancoff factor for a zone is +specified as −1 in the input, then the user-specified escape cross +section is used in calculating the background cross sections σ\ :sub:`0` +as follows: + +.. math:: + :label: eq7-3-25 + + {{\sigma }_{0}}\quad =\quad \frac{\sum\limits_{n\ne i}{{{N}_{n}}\ \sigma \,_{t}^{n}\quad +\quad \Sigma _{e}^{input}}}{{{N}_{i}}}\quad + +.. _7-3-5: + +Sample Problem +-------------- + +In most cases, the input data to BONAMI are simple and obvious because +the complicated parameters are determined internally based on the +options selected. The user describes his geometry, the materials +contained therein, the temperatures, and a few options. + +This problem is for a system of iron-clad uranium (U\ :sup:`238` – +U\ :sup:`235` ) fuel pins arranged in a square lattice in a water pool. + +.. image:: figs/BONAMI/ex1.png + :align: center + :width: 300 + +Our number densities are + +Fuel: + + :math:`{{N}_{{}^{235}U}}` = 1.4987 × 10\ :sup:`−4` + + :math:`{{N}_{{}^{238}U}}` = 2.0664× 10\ :sup:`−-2` + +Clad: + + :math:`{{N}_{{}^{56}Fe}}` = 9.5642× 10\ :sup:`−5` + +Water: + + N\ :sub:`H` = 6.6662 × 10\ :sup:`−2` + + N\ :sub:`O` = 3.3331 × 10\ :sup:`−2` + +For the problem, we choose *iropt* = 1 (IR approximation with scattering +approximated by λΣ\ :sub:`p`) and *crossedt* = 4 for the most detailed +output edits. An 8-group test library is used for fast execution and a +short output file. + +The XSProc/CSAS1X SCALE sequence input file, the corresponding i_bonami +FIDO input file created by the sequence under the temporary working +directory, and an abbreviated copy of the output from this case follows. + +.. highlight:: scale + +:: + + =csas1x + Assembly pin + test-8grp + read comp + ' fuel + u-235 1 0 1.4987e-4 297.15 end + u-238 1 0 2.0664e-2 297.15 end + ' clad + fe-56 2 0 9.5642e-5 297.15 end + ' coolant + h 3 0 6.6662e-2 297.15 end + o 3 0 3.3331e-2 297.15 end + end comp + ' ==================================================================== + read celldata + latticecell squarepitch pitch=1.26 3 fuelr=0.405765 1 + cladr=0.47498 2 end + + moredata iropt=1 crossedt=4 end moredata + end celldata + ' ==================================================================== + end + +FIDO input i_bonami + +:: + + -1$$ a0001 + 500000 + e + 0$$ a0001 + 11 0 18 1 + e + 1$$ a0001 + 1 3 5 0 4 1010 + 1 -1 -1 + e + 2** a0001 + 1.00000E-03 0.00000E+00 + e + t + 3$$ a0001 + 1 1 2 3 3 + e + 4$$ a0001 + 92235 92238 26056 1001 8016 + e + 5** a0001 + 1.49870E-04 2.06640E-02 9.56420E-05 6.66620E-02 3.33310E-02 + e + 6$$ a0001 + 1 2 3 + e + 7** a0001 + 4.05765E-01 4.74980E-01 7.10879E-01 + e + 8** a0001 + 2.97150E+02 2.97150E+02 2.97150E+02 + e + 9** a0001 + 1.11870E+00 4.15813E+00 1.78119E-01 + e + 10$$ a0001 + 92235 92238 26056 1001 8016 + e + 11$$ a0001 + 0 0 0 + e + 13** a0001 + 2.71260E-01 5.20852E-01 9.24912E-01 + e + 14** a0001 + 8.11530E-01 1.38430E-01 4.71798E-01 + e + 15** a0001 + 0.00000E+00 0.00000E+00 0.00000E+00 0.00000E+00 + e + 16$$ a0001 + 2 2 2 + e + 17$$ a0001 + 0 0 0 0 + e + t + +:: + + + + + + program verification information + + code system: SCALE version: 6.2 + + + + + program: bonami + + creation date: unknown + + library: /home02/u2m/Workfolder/sampletmp + + + test code: bonami + + version: 6.2.0 + + jobname: u2m + + machine name: node22.ornl.gov + + date of execution: 04_dec_2013 + + time of execution: 21:43:54.38 + +:: + + 1 + BONAMI CELL PARAMETERS + --------------------------------------------- + Bonami Print Option : 4 + BellFactor : 0 + Bondarenko Iteration eps : 0.001 + Resonance Option : 1 + Bell Factor Option : LESLIE + Escape CrossSection Option : INCONSISTENT + CellGeometry : 2 + MasterLibrary : + Number oF Neutron Groups : 8 + First Thermal Neutron Group : 5 + __________________________________________ + Processing Zone : 1 + Mixture Number : 1 + Number Of Nuclides : 2 + Dancoff Factor : 0.27126 + Lbar : 0.81153 + Escape Cross Section Input : 1.1187 + Material Temeprature : 297.15 + + Processing Nuclide : 92235 Number Density : 0.00014987 + Processing Nuclide : 92238 Number Density : 0.020664 + + Bondarenko Iterations + iteration Nuclide Group MaxChange Selfsig0 Effsig0 + 1 92235 0 0 0 0 + 1 92238 0 0 0 0 + + Total number of Bondarenko Iterations : 1 + Max Change in Group : 0 + + Group Eff Macro Sig0 Escape Xsec + 1 0.2351032 0.9075513 + 2 0.2351032 0.9075513 + 3 0.2351032 0.9075513 + 4 0.2351032 0.9075513 + 5 0.2351032 0.9075513 + 6 0.2351032 0.9075513 + 7 0.2351032 0.9075513 + 8 0.2351032 0.9075513 + + --------------------------------------------------- + +:: + + Shielding Nuclide 92235 + + mt Group sig0 infDiluted Xsec f-factor shielded Xsec + 1 1 7612.71875 7.19131 0.999998 7.19129 + 1 2 7612.71875 10.2521 0.999616 10.2481 + 1 3 7612.71875 24.9361 1.00241 24.9963 + 1 4 7612.71875 75.1109 1.05902 79.5436 + 1 5 7612.71875 56.0286 1.00205 56.1434 + 1 6 7612.71875 198.645 1.0008 198.805 + 1 7 7612.71875 347.945 1.00024 348.028 + 1 8 7612.71875 761.257 1.0066 766.282 + + mt Group sig0 infDiluted Xsec f-factor shielded Xsec + 2 1 7612.71875 3.71448 0.999999 3.71448 + 2 2 7612.71875 7.63235 0.99935 7.62739 + 2 3 7612.71875 11.841 0.999444 11.8345 + 2 4 7612.71875 11.5408 1.00561 11.6055 + 2 5 7612.71875 12.5449 1.00001 12.545 + 2 6 7612.71875 14.2501 1.00007 14.2511 + 2 7 7612.71875 14.8125 1.00003 14.8128 + 2 8 7612.71875 15.1274 1.00015 15.1297 + + mt Group sig0 infDiluted Xsec f-factor shielded Xsec + 18 1 7612.71875 1.21846 0.999996 1.21846 + 18 2 7612.71875 1.40834 1.0002 1.40862 + 18 3 7612.71875 8.92885 1.00132 8.94062 + 18 4 7612.71875 39.2086 1.06274 41.6686 + 18 5 7612.71875 32.7026 1.00105 32.737 + 18 6 7612.71875 153.511 1.00089 153.647 + 18 7 7612.71875 285.775 1.00026 285.848 + 18 8 7612.71875 636.445 1.00655 640.611 + + mt Group sig0 infDiluted Xsec f-factor shielded Xsec + 102 1 7612.71875 0.060296 1 0.0602962 + 102 2 7612.71875 0.317627 1.00352 0.318746 + 102 3 7612.71875 4.16593 1.01325 4.22113 + 102 4 7612.71875 24.3615 1.07832 26.2695 + 102 5 7612.71875 10.781 1.00749 10.8618 + 102 6 7612.71875 30.8844 1.00074 30.9073 + 102 7 7612.71875 47.3579 1.0002 47.3671 + 102 8 7612.71875 109.685 1.00781 110.542 + + mt Group sig0 infDiluted Xsec f-factor shielded Xsec + 1007 1 7612.71875 0 0 0 + 1007 2 7612.71875 0 0 0 + 1007 3 7612.71875 0 0 0 + 1007 4 7612.71875 0 0 0 + 1007 5 7612.71875 12.5448 1.00001 12.5449 + 1007 6 7612.71875 14.2501 1.00007 14.2511 + 1007 7 7612.71875 14.8125 1.00003 14.8129 + 1007 8 7612.71875 15.1278 1.00015 15.13 + + --------------------------------------------------- + +:: + + Shielding Nuclide 92238 + + mt Group sig0 infDiluted Xsec f-factor shielded Xsec + 1 1 44.0034676 7.33815 0.999983 7.33803 + 1 2 44.0034676 10.3566 1.00418 10.3999 + 1 3 44.0034676 15.0517 0.976844 14.7032 + 1 4 44.0034676 15.951 0.983793 15.6925 + 1 5 44.0034676 9.43867 1.00002 9.43887 + 1 6 44.0034676 10.1008 1.00008 10.1015 + 1 7 44.0034676 10.7744 1.00004 10.7748 + 1 8 44.0034676 12.2124 1.00145 12.2301 + + mt Group sig0 infDiluted Xsec f-factor shielded Xsec + 2 1 44.0034676 4.0228 0.999974 4.0227 + 2 2 44.0034676 9.05886 1.00575 9.11093 + 2 3 44.0034676 14.0213 0.979923 13.7398 + 2 4 44.0034676 11.9032 0.98795 11.7598 + 2 5 44.0034676 8.86555 0.999984 8.86541 + 2 6 44.0034676 9.24452 1.00002 9.24471 + 2 7 44.0034676 9.2797 1.00002 9.27987 + 2 8 44.0034676 9.3077 1.00009 9.30853 + + mt Group sig0 infDiluted Xsec f-factor shielded Xsec + 18 1 44.0034676 0.376356 1.00001 0.376361 + 18 2 44.0034676 0.000528746 1.00019 0.000528845 + 18 3 44.0034676 0.000308061 0.966052 0.000297603 + 18 4 44.0034676 4.75014e-06 0.967842 4.59738e-06 + 18 5 44.0034676 2.60878e-06 1.00006 2.60893e-06 + 18 6 44.0034676 5.27139e-06 1.00071 5.27512e-06 + 18 7 44.0034676 9.3235e-06 1.00018 9.32514e-06 + 18 8 44.0034676 1.81868e-05 1.00588 1.82937e-05 + + mt Group sig0 infDiluted Xsec f-factor shielded Xsec + 102 1 44.0034676 0.0554327 1.00006 0.0554359 + 102 2 44.0034676 0.17972 0.978628 0.175879 + 102 3 44.0034676 1.03011 0.934934 0.963087 + 102 4 44.0034676 4.04777 0.971568 3.93268 + 102 5 44.0034676 0.573119 1.0006 0.573462 + 102 6 44.0034676 0.856257 1.00068 0.856839 + 102 7 44.0034676 1.49471 1.00017 1.49497 + 102 8 44.0034676 2.90465 1.00586 2.92168 + + mt Group sig0 infDiluted Xsec f-factor shielded Xsec + 1007 1 44.0034676 0 0 0 + 1007 2 44.0034676 0 0 0 + 1007 3 44.0034676 0 0 0 + 1007 4 44.0034676 0 0 0 + 1007 5 44.0034676 8.86549 0.999984 8.86535 + 1007 6 44.0034676 9.24445 1.00002 9.24463 + 1007 7 44.0034676 9.27974 1.00002 9.27992 + 1007 8 44.0034676 9.30769 1.00009 9.30852 + Zone Calculation is completed in 0 seconds + BONAMI CELL PARAMETERS + --------------------------------------------- + Bonami Print Option : 4 + BellFactor : 0 + Bondarenko Iteration eps : 0.001 + Resonance Option : 1 + Bell Factor Option : LESLIE + Escape CrossSection Option : INCONSISTENT + CellGeometry : 2 + MasterLibrary : + Number oF Neutron Groups : 8 + First Thermal Neutron Group : 5 + __________________________________________ + Processing Zone : 2 + Mixture Number : 2 + Number Of Nuclides : 1 + Dancoff Factor : 0.520852 + Lbar : 0.13843 + Escape Cross Section Input : 4.15813 + Material Temeprature : 297.15 + + Processing Nuclide : 26056 Number Density : 9.5642e-05 + + Bondarenko Iterations + iteration Nuclide Group MaxChange Selfsig0 Effsig0 + 1 26056 0 0 0 0 + + Total number of Bondarenko Iterations : 1 + Max Change in Group : 0 + + Group Eff Macro Sig0 Escape Xsec + 1 0.0003553244 3.487286 + 2 0.0003553244 3.487286 + 3 0.0003553244 3.487286 + 4 0.0003553244 3.487286 + 5 0.0003553244 3.487286 + 6 0.0003553244 3.487286 + 7 0.0003553244 3.487286 + 8 0.0003553244 3.487286 + + --------------------------------------------------- + +:: + + Shielding Nuclide 26056 + + mt Group sig0 infDiluted Xsec f-factor shielded Xsec + 1 1 36461.8672 3.07957 1.00005 3.07972 + 1 2 36461.8672 4.68958 1.00091 4.69382 + 1 3 36461.8672 7.85712 0.999843 7.85589 + 1 4 36461.8672 12.0029 1 12.0029 + 1 5 36461.8672 12.3689 1.00001 12.369 + 1 6 36461.8672 12.8598 1.00003 12.8602 + 1 7 36461.8672 13.5237 0.999906 13.5224 + 1 8 36461.8672 15.0714 0.99949 15.0637 + + mt Group sig0 infDiluted Xsec f-factor shielded Xsec + 2 1 36461.8672 2.26476 1.00047 2.26583 + 2 2 36461.8672 4.6817 1.0009 4.68592 + 2 3 36461.8672 7.81457 0.999813 7.81311 + 2 4 36461.8672 11.9143 1 11.9143 + 2 5 36461.8672 12.0468 1.00001 12.0469 + 2 6 36461.8672 12.065 1.00002 12.0653 + 2 7 36461.8672 12.0887 1.00005 12.0893 + 2 8 36461.8672 12.2042 1.00013 12.2057 + + mt Group sig0 infDiluted Xsec f-factor shielded Xsec + 102 1 36461.8672 0.00206393 1.0015 0.00206702 + 102 2 36461.8672 0.00787763 1.0035 0.00790524 + 102 3 36461.8672 0.0425504 1.00623 0.0428155 + 102 4 36461.8672 0.0885525 1 0.0885529 + 102 5 36461.8672 0.322101 1.00002 0.322109 + 102 6 36461.8672 0.794804 1.0002 0.79496 + 102 7 36461.8672 1.43496 0.998734 1.43314 + 102 8 36461.8672 2.86723 0.996792 2.85803 + + mt Group sig0 infDiluted Xsec f-factor shielded Xsec + 1007 1 36461.8672 0 0 0 + 1007 2 36461.8672 0 0 0 + 1007 3 36461.8672 0 0 0 + 1007 4 36461.8672 0 0 0 + 1007 5 36461.8672 12.0468 1.00001 12.0469 + 1007 6 36461.8672 12.065 1.00002 12.0653 + 1007 7 36461.8672 12.0887 1.00005 12.0893 + 1007 8 36461.8672 12.2042 1.00013 12.2057 + Zone Calculation is completed in 0 seconds + +:: + + BONAMI CELL PARAMETERS + --------------------------------------------- + Bonami Print Option : 4 + BellFactor : 0 + Bondarenko Iteration eps : 0.001 + Resonance Option : 1 + Bell Factor Option : LESLIE + Escape CrossSection Option : INCONSISTENT + CellGeometry : 2 + MasterLibrary : + Number oF Neutron Groups : 8 + First Thermal Neutron Group : 5 + __________________________________________ + Processing Zone : 3 + Mixture Number : 3 + Number Of Nuclides : 2 + Dancoff Factor : 0.924912 + Lbar : 0.471798 + Escape Cross Section Input : 0.178119 + Material Temeprature : 297.15 + + Processing Nuclide : 1001 Number Density : 0.066662 + Processing Nuclide : 8016 Number Density : 0.033331 + + Bondarenko Iterations + iteration Nuclide Group MaxChange Selfsig0 Effsig0 + 1 1001 0 0 0 0 + 1 8016 0 0 0 0 + + Total number of Bondarenko Iterations : 1 + Max Change in Group : 0 + + Group Eff Macro Sig0 Escape Xsec + 1 1.494705 0.1593803 + 2 1.494705 0.1593803 + 3 1.494705 0.1593803 + 4 1.494705 0.1593803 + 5 1.494705 0.1593803 + 6 1.494705 0.1593803 + 7 1.494705 0.1593803 + 8 1.494705 0.1593803 + + --------------------------------------------------- + +:: + + Shielding Nuclide 1001 + + mt Group sig0 infDiluted Xsec f-factor shielded Xsec + 1 1 4.33502197 2.98905 0.999485 2.98751 + 1 2 4.33502197 9.87269 0.999169 9.86448 + 1 3 4.33502197 19.9332 0.999972 19.9326 + 1 4 4.33502197 20.4672 0.998926 20.4453 + 1 5 4.33502197 21.1735 1.00001 21.1736 + 1 6 4.33502197 26.1886 0.99995 26.1873 + 1 7 4.33502197 35.0621 0.999821 35.0558 + 1 8 4.33502197 54.9507 0.997361 54.8057 + + mt Group sig0 infDiluted Xsec f-factor shielded Xsec + 2 1 4.33502197 2.98901 0.999485 2.98747 + 2 2 4.33502197 9.8726 0.999168 9.86439 + 2 3 4.33502197 19.9315 0.999972 19.9309 + 2 4 4.33502197 20.4556 0.998926 20.4336 + 2 5 4.33502197 21.1321 1.00001 21.1322 + 2 6 4.33502197 26.0865 0.99995 26.0852 + 2 7 4.33502197 34.8778 0.999829 34.8718 + 2 8 4.33502197 54.5786 0.997396 54.4365 + + mt Group sig0 infDiluted Xsec f-factor shielded Xsec + 102 1 4.33502197 3.56422e-05 1.00003 3.56433e-05 + 102 2 4.33502197 8.96827e-05 0.998165 8.9518e-05 + 102 3 4.33502197 0.00171679 0.999794 0.00171643 + 102 4 4.33502197 0.0116042 1.00002 0.0116044 + 102 5 4.33502197 0.0413709 1.00001 0.0413714 + 102 6 4.33502197 0.102043 1.00007 0.10205 + 102 7 4.33502197 0.184322 0.998389 0.184025 + 102 8 4.33502197 0.372079 0.992163 0.369163 + + mt Group sig0 infDiluted Xsec f-factor shielded Xsec + 1007 1 4.33502197 0 0 0 + 1007 2 4.33502197 0 0 0 + 1007 3 4.33502197 0 0 0 + 1007 4 4.33502197 0 0 0 + 1007 5 4.33502197 21.1312 1.00005 21.1322 + 1007 6 4.33502197 26.0871 0.999927 26.0852 + 1007 7 4.33502197 34.8779 0.999826 34.8718 + 1007 8 4.33502197 54.5802 0.997367 54.4365 + + --------------------------------------------------- + +:: + + Shielding Nuclide 8016 + + mt Group sig0 infDiluted Xsec f-factor shielded Xsec + 1 1 45.7377472 2.36081 0.980071 2.31376 + 1 2 45.7377472 3.95917 0.995755 3.94236 + 1 3 45.7377472 3.84394 1 3.84394 + 1 4 45.7377472 3.85289 1 3.85291 + 1 5 45.7377472 3.85531 1.00002 3.85537 + 1 6 45.7377472 3.8648 1.00006 3.86502 + 1 7 45.7377472 3.89139 1.00014 3.89194 + 1 8 45.7377472 4.01909 1.00036 4.02056 + + mt Group sig0 infDiluted Xsec f-factor shielded Xsec + 2 1 45.7377472 2.34636 0.979939 2.29929 + 2 2 45.7377472 3.95907 0.995756 3.94226 + 2 3 45.7377472 3.84393 1 3.84393 + 2 4 45.7377472 3.85288 1 3.8529 + 2 5 45.7377472 3.85529 1.00002 3.85535 + 2 6 45.7377472 3.86474 1.00006 3.86496 + 2 7 45.7377472 3.89128 1.00014 3.89183 + 2 8 45.7377472 4.01888 1.00037 4.02035 + + mt Group sig0 infDiluted Xsec f-factor shielded Xsec + 102 1 45.7377472 0.00010111 1.0002 0.00010113 + 102 2 45.7377472 0.000101635 0.991292 0.00010075 + 102 3 45.7377472 7.10826e-06 1.00036 7.11084e-06 + 102 4 45.7377472 7.2788e-06 1.00001 7.27885e-06 + 102 5 45.7377472 2.38519e-05 1.00003 2.38526e-05 + 102 6 45.7377472 5.83622e-05 1.00019 5.83734e-05 + 102 7 45.7377472 0.000105195 0.998729 0.000105061 + 102 8 45.7377472 0.000209981 0.996663 0.00020928 + + mt Group sig0 infDiluted Xsec f-factor shielded Xsec + 1007 1 45.7377472 0 0 0 + 1007 2 45.7377472 0 0 0 + 1007 3 45.7377472 0 0 0 + 1007 4 45.7377472 0 0 0 + 1007 5 45.7377472 3.85523 1.00003 3.85535 + 1007 6 45.7377472 3.86483 1.00003 3.86496 + 1007 7 45.7377472 3.89119 1.00016 3.89183 + 1007 8 45.7377472 4.01794 1.0006 4.02035 + Zone Calculation is completed in 0 seconds + module: BonamiM has terminated after a cpu usage of 0.0100 seconds + + + + + + + + +.. bibliography:: bibs/BONAMI.bib diff --git a/CAJUN.rst b/CAJUN.rst new file mode 100644 index 0000000..2eb055e --- /dev/null +++ b/CAJUN.rst @@ -0,0 +1,114 @@ +.. _7-9: + +CAJUN: Module for Combining and Manipulating CENTRM Continuous-Energy Libraries +=============================================================================== + +*L. M. Petrie and N. M. Greene* + +.. _7-9-1: + +Introduction +------------ + +CAJUN is a program used to combine continuous-energy (CE) cross section +libraries for use in the cross-section processing codes CENTRM and PMC. +It is used primarily by SCALE sequences when processing pointwise cross +section data for DOUBLEHET unit cells with the XSProc module (see +:ref:`7-1-1`), although it can also be run standalone in conjunction +with CRAWDAD. CAJUN combines multiple CE libraries into a single +library, adds selected nuclides from one library into another library, +deletes nuclides from a specified library, or renames nuclides in a +library. CAJUN performs an analogous function for CE libraries as AJAX +does for multigroup libraries. + +In order for input CE libraries to be read by CAJUN, they must be +assigned a unit number from 1 to 99. The SHELL module can be used to +link individual CE libraries to appropriate file names that are +accessible by unit number. + +.. _7-9-2: + +CAJUN Input Data +---------------- + +Input data for CAJUN is read into the program using FIDO type input. The +data is divided into three data blocks. The first data block provides +the output file number and number of libraries processed. The second +data block provides input library numbers, number of nuclides in each +library, and whether nuclides are selected by MAT or ZA number. The +third data block provides current nuclide MAT or ZA numbers and new +nuclide MAT or ZA numbers. Detailed description of the CAJUN input data +is provided below. + +.. highlight:: none + +:: + + Data Block 1 + + 0$$ unit assignments (1) + + 1. lcen – logical unit number of output CE CENTRM library (1) + + 1$$ number of files to process (2) + + 1. nfile – the number of CENTRM CE libraries to process (1) + 2. idtap – identifier for the new library (0) + + T terminate data block 1 + + *********************************************************************** + *** repeat data block(s) 2 and 3 "nfile" times to create new library + *********************************************************************** + + Data Block 2 + + 2$$ file selection and treatment option (4) + + 1. log – logical unit number of input CENTRM CE library (77) + 2. inum – number of nuclides selected from this library (0) + 0 - select all nuclides on the library as is. + n – select all nuclides on the library as is except for those indicated in the 3$$ array. + n – select the "n" nuclides on the library listed in the 3$$ array. + 3. iopt – select nuclides by 'mat' or 'nza' number (1) + 0 – mat + 1 – nza (default) + 4. nsq – sequence number of file opened on unit LOG (1) + + T terminate data block 2 + + Data Block 3 (enter if inum is non-zero) + + 3$$ nuclide selection list (inum) + enter a positive identifier to select a nuclide. + enter a negative identifier to exclude a nuclide. + + 4$$ new nuclide identifiers (inum) + enter the new nuclide identifiers in the locations corresponding + to the positive identifier entry in the 3$ array. + + 5$$ version of the data (–1,0,1,2,3,4/unk,ENDF,JEF,JENDL,BROND,CENDL) (inum) + + 6$$ 8-Character Identifier for Thermal Kinematics Data (inum) + + 7$$ ZA-override Values. Non-zero values will replace the ZA values in the + Header Record. The ZA values in the Data Directory records are not changed. (inum) + + T terminate data block 3 + +.. _7-9-3: + +CAJUN I/O Units +--------------- + +CAJUN requires the following I/O devices. + ++----------+--+---------------------------+ +| Unit No. | | Purpose | ++----------+--+---------------------------+ +| 5 | | Standard definition input | +| | | | +| 6 | | Output | +| | | | +| 18 | | Scratch file | ++----------+--+---------------------------+ diff --git a/CENTRM.rst b/CENTRM.rst new file mode 100644 index 0000000..9a97f1e --- /dev/null +++ b/CENTRM.rst @@ -0,0 +1,3367 @@ +.. _7-4: + + +CENTRM: A Neutron Transport Code for Computing Continuous-Energy Spectra in General One-Dimensional Geometries and Two-Dimensional Lattice Cells +================================================================================================================================================ + +*M. L. Williams* + +Abstract + +CENTRM computes continuous-energy neutron spectra for infinite media, +general one-dimensional (1D) systems, and two-dimensional (2D) unit +cells in a lattice, by solving the Boltzmann transport equation using a +combination of pointwise and multigroup nuclear data. Several +calculational options are available, including a slowing-down +computation for homogeneous infinite media, 1D discrete ordinates in +slab, spherical, or cylindrical geometries; a simplified two-region +solution; and 2D method of characteristics for a unit cell within a +square-pitch lattice. In SCALE, CENTRM is used mainly to calculate +problem-specific fluxes on a fine energy mesh (10,000–70,000 points), +which may be used to generate self-shielded multigroup cross sections +for subsequent radiation transport computations. + +ACKNOWLEDGMENTS + +Several current and former ORNL staff made valuable contributions to the +CENTRM development. The author acknowledges the contributions of former +ORNL staff members D. F. Hollenbach and N. M. Greene; as well as current +staff member L. M. Petrie. Special thanks go to Kang-Seog Kim who +developed the 2D method of characteristics option for CENTRM. Portions +of the original code development were performed by M. Asgari as partial +fulfillment of his PhD dissertation research at Louisiana State +University (LSU); and Riyanto Raharjo from LSU made significant +programming contributions for the inelastic scattering and thermal +calculations. + +.. _7-4-1: + +Introduction +------------ + +CENTRM (**C**\ ontinuous **EN**\ ergy **TR**\ ansport **M**\ odule) +computes “continuous-energy” neutron spectra using various deterministic +approximations to the Boltzmann transport equation. Computational +methods are available for infinite media, general one-dimensional (1-D) +geometries, and two-dimensional (2D) unit cells in a square-pitch +lattice. The purpose of the code is to provide fluxes and flux moments +for applications that require a high resolution of the fine-structure +variation in the neutron energy spectrum. The major function of CENTRM +is to determine problem-specific fluxes for processing multigroup (MG) +data with the XSProc self-shielding module (Introduction in XSProc +chapter), which is executed by all SCALE MG sequences. XSProc calls an +application program interface (API) to perform a CENTRM calculation for +a representative model (e.g., a unit cell in a lattice), and then +utilizes the spectrum as a *problem-dependent* weight function for MG +averaging. The MG data processing is done in XSProc by calling an API +for the PMC code, which uses the CENTRM continuous-energy (CE) flux +spectra and cross-section data to calculate group-averaged +cross sections over some specified energy range. The resulting +application-specific library is used for MG neutron transport +calculations within SCALE sequences. In this approach the CENTRM/PMC +cross-section processing in XSProc becomes an active component in the +overall transport analysis. CENTRM can also be executed as a standalone +code, if the user provides all required input data and nuclear data +libraries; but execution through XSProc is much simpler and less prone +to error. + +.. _7-4-1-1: + +Description of problem solved +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +CENTRM uses a combination of MG and pointwise (PW) solution techniques +to solve the neutron transport equation over the energy range ~0 to +20 MeV. The calculated CE spectrum consists of PW values for the flux +per unit lethargy defined on a discrete energy mesh, for which a linear +variation of the flux between energy points is assumed. Depending on the +specified transport approximation, the flux spectrum may vary as a +function of space and direction, in addition to energy. Spherical +harmonic moments of the angular flux, which may be useful in processing +MG matrices for higher order moments of the scattering cross section, +can also be determined as a function of space and energy mesh. + +CENTRM solves the fixed-source (inhomogeneous) form of the transport +equation, with a user-specified fixed source term. The input source may +correspond to MG histogram spectra for volumetric or surface sources or +it may be a “fission source” which has a continuous-energy +fission-spectrum distribution (computed internally) appropriate for each +fissionable mixture. Note that eigenvalue calculations are *not* +performed in CENTRM—these must be performed by downstream MG transport +codes that utilize the self-shielded data processed with the CENTRM +spectra. + +.. _7-4-1-2: + +Nuclear data required for CENTRM +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +A MG cross section library, a CE cross section library, and a CE thermal +kernel [S(α, β)] library are required for the CENTRM transport +calculation. During XSProc execution for a given unit cell in the CELL +DATA block, the MG library specified in the input is processed by BONAMI +prior to the CENTRM calculation, in order to provide self-shielded data +based on the Bondarenko approximation for the MG component of the CENTRM +solution. The shielded MG cross sections are also used in CENTRM to +correct infinitely dilute CE data in the unresolved resoance range. The +CRAWDAD module is executed by XSProc to generate the CENTRM CE cross +section and thermal kernel libraries, respectively, by concatenating +discrete PW data read from individual files for the nuclides in the unit +cell mixtures. CE resonance profiles are based strictly on +specifications in the nuclear data evaluations; e.g., Reich-Moore +formalism is specified for most materials in ENDF/B-VII. PW +data in the CENTRM library are processed such that values at any energy +can be obtained by linear interpolation within some error tolerance +specified during the library generation (usually ~0.1% or less). CRAWDAD +also interpolates the CE cross section data and the Legendre moments of +the thermal scattering kernels to the appropriate temperatures for the +unit cell mixtures. The format of the CENTRM library is described in +:ref:`7-4-6-1`. + +.. _7-4-1-3: + +Code assumptions and features +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +As shown in :numref:`fig7-4-1`, the energy range of interest is divided into +three intervals called the Upper Multigroup Range (UMR), Pointwise Range +(PW), and Lower Multigroup Range (LMR), respectively, which are defined +by input. MG fluxes are computed using standard multigroup techniques +for the UMR and LMR, and these values are then divided by the group +lethargy width to obtain the average flux per lethargy within each +group. This “pseudo-pointwise” flux is assigned to the midpoint lethargy +of the group, so that there is one energy point per group in the UMR and +LMR energy intervals. However, for each group in the PW range there are +generally several, and possibly many, energy points for which CENTRM +computes flux values. In this manner a problem-dependent spectrum is +obtained over the entire energy range. + +The default PW range goes from 0.001 eV to 20 keV, but the user can +modify the PW limits. The energy range for the PW transport calculation +is usually chosen to include the interval where the important absorber +nuclides have resolved resonances, while MG calculations are performed +where the cross sections characteristically have a smoother variation or +where shielding effects are less important. In the SCALE libraries the +thermal range is defined to be energies less than 5.0 eV. Above thermal +energies, scattering kinematics are based on the stationary nucleus +model, while molecular motion and possible chemical binding effects are +taken into account for thermal scattering, which can result in an +incease in the neutron incident energy. The CENTRM thermal calculation +uses Legendre coefficients from the CE kernel library that describes +point-to-point energy transfers for incoherent and coherent scattering, +as function of temperature, for all moderators that have thermal +scattering law data provided in ENDF/B. Thermal kernels for all other +materials are generated internally by CENTRM based on the free-gas +model. + +.. _fig7-4-1: +.. figure:: figs/CENTRM/fig1.png + :align: center + :width: 500 + + Definition of UMR, PW, and LMR energy ranges. + +Several transport computation methods are available for both MG and PW +calculations. These include a space-independent slowing down calculation +for infinite homogeneous media, 1D discrete ordinates or P1 methods for +slab, spherical, and cylindrical geometries, and a 2D method of +chracateristics (MoC) method for lattice unit cells. A simplified +two-region collision-probablity method is also available for ther +pointwise solution. In general the user may specify different transport +methods for the UMR, PW, and LMR, respectively; however, if the 2D MoC +method is specified for any range, it will be used for all. + +The CENTRM 1D discrete ordinates calculation option has many of the same +features as the SCALE MG code XSDRNPM. It represents the directional +dependence of the angular flux with an arbitrary symmetric-quadrature +order, and uses Legendre expansions up to P\ :sub:`5` to represent the +scattering source. No restrictions are placed on the material +arrangement or the number of spatial intervals in the calculation, and +general boundary conditions (vacuum, reflected, periodic, albedo) can be +applied on either boundary of the 1D geometry. Lattice cells are +represented in the CENTRM discrete ordinates option by a 1D Wigner-Sitz +cylindrical or spherical model with a white boundary condition on the +outer surface. + +Starting with SCALE-6.2, CENTRM also includes a 2D MoC solver for +lattice cell geometries consisting of a cylindrical fuel rod +(fuel/gap/clad) contained within a rectangular moderator region. The MoC +calculation is presently limited to square lattices. The 2D unit cell +uses a reflected boundary condition on the outer square surface, which +provides a more rigorous treatment than the 1D Wigner-Seitz model; +however the MoC option requires a longer execution time than the 1D +discrete ordinates method. The MoC option has been found to improve +results compared to the 1D Wigner-Seitz cell model for many cases, but +in other cases the improvement is marginal. + +A variable PW energy mesh is generated internally to accurately +represent the fine-structure flux spectrum for the system of interest. +This gives CENTRM the capability to rigorously account for resonance +interference effects in systems with multiple resonance absorbers. +Because CENTRM calculates the space-dependent PW flux spectrum, the +spatial variation of the self-shielded cross sections within an absorber +body can be obtained. A radial temperature distribution can also be +specified, so that space-dependent Doppler broadening can be treated in +the transport solution. Within the epithermal PW range, the slowing-down +source due to elastic and discrete-level inelastic reactions is computed +with the analytical scatter kernel based upon the neutron kinematic +relations for *s*-wave scattering. Continuum inelastic scatter is +approximated by an analytical evaporation spectrum, assumed isotropic in +the laboratory system. For many thermal reactor and criticality safety +problems, self-shielding of inelastic cross sections has a minor impact, +and by default these options are turned off for faster execution. As +previously discussed, the thermal scatter kernel is based on the ENDF/B +scattering law data for bound moderators, and uses the free-gas model +for other materials. + +.. _7-4-2: + +Theory and Analytical Models +---------------------------- + +This section describes the coupled MG and PW techniques used to solve +the neutron transport equation. + +.. _7-4-2-1: + +Energy/lethargy ranges for MG and PW calculations +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The combined MG/PW CENTRM calculation is performed over the energy range +spanned by the group structure in the input MG library. The energy +boundaries for the “IGM” neutron groups specified on the MG library +divide the entire energy range into energy intervals. The lowest energy +group contained in the UMR is defined to be “MGHI”; while the highest +energy group in the LMR is designated “MGLO.” The boundary between the +PW and UMR energy intervals is set by the energy value “DEMAX,” while +“DEMIN” is the boundary between the PW and LMR. The default values of +0.001 eV and 20 keV for DEMIN and DEMAX, respectively, can be modified +by user input, but the input values are altered by the code to +correspond to the closest group boundaries. Hence, DEMAX is always equal +to the lower energy boundary of group MGHI and DEMIN the upper energy +boundary of MGLO. The PW calculation is performed in terms of lethargy +(u), rather than energy (E). The origin (u=0) of the lethargy coordinate +corresponds to the energy E=DEMAX, which is the top of the PW range. See +:numref:`fig7-4-2`. + +The highest energy group of the thermal range is defined by the +parameter “IFTG,” obtained from the MG library. If DEMIN is less than +the upper energy boundary of IFTG, the PW range extends into thermal. In +this case, scattering in the PW region of the thermal range is based on +the PW scattering kernel data; and the LMR calculation uses 2D transfer +matrices for incoherent and coherent scattering on the MG library. +Coupling between the MG and PW thermal calculations is treated, and +outer iterations are required to address effects of upscattering. + +.. _fig7-4-2: +.. figure:: figs/CENTRM/fig2.png + :align: center + :width: 500 + + Definition of *High* and *Transition* regions in upper multigroup region. + +With the exception of hydrogen moderation, elastic down-scattering +coupling the UMR and PW ranges, occurs only within a limited sub-range +of the UMR called the “transition region”. The highest energy group in +the transition region is designated “MGTOP.” The precise definition of +the transition region is given in :ref:`7-4-2-6-1`. + +Energy boundaries of the group structure on the input MG library +correspond to the IGM+1 values, { G\ :sub:`1`, G\ :sub:`2,` ... +G\ :sub:`g,` G\ :sub:`g+1`, ..., G\ :sub:`IGM+1`}. It is convenient to +designate the number of groups in the UMR, PW, and LMR ranges equal to +NG\ :sub:`U`, NG\ :sub:`P`, and NG\ :sub:`L`, respectively, so that IGM += NG\ :sub:`U` + NG\ :sub:`P` + NG\ :sub:`L`; or in terms of the +parameters MGHI and MGLO introduced previously: + + NG\ :sub:`U` = MGHI; NG\ :sub:`P` = MGLO − MGHI − 1; NG\ :sub:`L` = IGM + − MGLO + 1. + +The flux per unit lethargy is calculated for a discrete energy (or +lethargy) mesh spanning the MG structure. Groups in the UMR and LMR each +contain a single energy mesh point, while groups in the PW range +generally contain several points. The number of mesh *points* in the +UMR, PW, and LMR is equal respectively to NG\ :sub:`U`, N\ :sub:`P`, and +NG\ :sub:`L`; and the total number of points in the entire energy mesh +is designated as “N\ :sub:`T`,” which is equal to NG\ :sub:`U` + +N\ :sub:`P` + NG\ :sub:`L`. Thus the lethargy (u) mesh consists of the +set of points: {u:sub:`1`,....u\ :sub:`NGU,` +u\ :sub:`NGU+1,`....u\ :sub:`NGU+NP`, +u\ :sub:`NGU+NP+1`,...u\ :sub:`NT`}. Based on the lethargy origin at +E=DEMAX, the lethargy “u\ :sub:`n`\ ” associated with any energy point +“E\ :sub:`n`\ ” is equal to, + + u\ :sub:`n` = ln(DEMAX/E\ :sub:`n`). + +Lethargy points are arranged in order of increasing value. The lethargy +origin is at point NG\ :sub:`U`\ +1, the lower energy boundary of group +MGHI; i.e., \ **u**\ :sub:`NGU+1`\ =0. Note that the entire UMR +(E>DEMAX) corresponds to negative lethargy values. Lethargy values for +the first NG\ :sub:`U` and the last NG\ :sub:`L` points in the mesh are +defined to be the midpoint lethargies of groups in the UMR and LMR +ranges, respectively. For example, for the NG\ :sub:`U` groups within +the UMR, + + u\ :sub:`1` = 0.5[ln(DEMAX/G\ :sub:`1`) + ln(DEMAX/G\ :sub:`2`)]; + + u\ :sub:`NGU` = 0.5[ln(DEMAX/G\ :sub:`MGHI`) + ln(DEMAX/G\ :sub:`MGHI + 1`)]; + +and similarly for the NG\ :sub:`L` groups in the LMR, + + u\ :sub:`NGU + NP + 1` = 0.5[ln(DEMAX/G\ :sub:`MGLO`) + ln(DEMAX/G\ :sub:`MGLO + 1`)] + + u\ :sub:`NT` = 0.5[ln(DEMAX/G\ :sub:`IGM`) + ln(DEMAX/G\ :sub:`IGM + 1`)] + +The remaining N\ :sub:`P` points in the mesh (i.e., values +u\ :sub:`NGU+1` to u\ :sub:`NGU+NP`) are contained within the +NG\ :sub:`P` groups that span the PW range. By definition the first +point in the PW range is the lower energy boundary of group MGHI. The +other mesh points are computed internally by CENTRM, based on the +behavior of the macroscopic PW total cross sections and other criteria. + +The neutron flux, as a function of space and direction, is calculated +for each energy/lethargy point in the mesh by solving the Boltzmann +transport equation. The transport equation at each lethargy point +generally includes a source term representing the production rate due to +elastic and inelastic scatter from other lethargies, which couples the +solutions at different lethargy mesh points. Except in the thermal +range, neutrons can only gain lethargy (lose energy) in a scattering +reaction; thus the PW flux is computed by solving the transport equation +at successive mesh points, sweeping from low to high lethargy values. + +.. _7-4-2-2: + +The Boltzmann equation for neutron transport +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The steady state neutron transport equation shown below represents a +particle balance-per unit phase space, at an arbitrary point ρ in phase +space, + +.. math:: + :label: eq7-4-1 + + \Omega \cdot \nabla \Psi(\rho)+\sum_{t}(\mathrm{r}, \mathrm{u}) \Psi(\rho)=\int_{0}^{\infty} \int_{0}^{4 \pi} \Sigma\left(\mathrm{u}^{\prime} \rightarrow \mathrm{u} ; \mu_{0}\right) \Psi\left(\mathrm{u}^{\prime}, \Omega^{\prime}\right) \mathrm{d} \mathrm{u}^{\prime}+\mathrm{Q}_{\mathrm{ext}}(\rho) + +where: + + ψ(p) = angular flux (per lethargy) at phase space coordinate ρ; + + ρ = (r,u,Ω) = phase space point defined by the six independent + variables; + + r = (x\ :sub:`1`,x\ :sub:`2`,x\ :sub:`3`) = space coordinates; + + u = ln(E\ :sub:`ref`/E) = lethargy at energy E, relative to an origin + (u=0) at E\ :sub:`ref`; + + Ω = (μ,ζ) = neutron direction defined by polar cosine μ and azimuthal + angle ζ; + + Σ\ :sub:`t`\ (r,u) = macroscopic total cross section; + + Σ(u′→u;μ\ :sub:`0`) = double differential scatter cross section; + + μ\ :sub:`0` = cosine of scatter angle, measured in laboratory coordinate + system; + + Q\ :sub:`ext`\ (ρ) = external source term, including fission source; + +The left and right sides of :eq:`eq7-4-1` respectively, are equal to the neutron +loss and production rates, per unit volume-direction-lethargy. In CENTRM +the spatial distribution of the fission source is input as a component +of the external source Q; hence, a fixed source rather than an +eigenvalue calculation is required for the transport solution. + +The angular dependence of the double-differential macroscopic scatter +cross section of an arbitrary nuclide “j” is represented by a finite +Legendre expansion of arbitrary order L: + +.. math:: + :label: eq7-4-2 + + \Sigma^{(\mathrm{j})}\left(\mathrm{u}^{\prime} \rightarrow \mathrm{u} ; \mu_{0}\right)=\sum_{=0}^{\mathrm{L}} \frac{2+1}{2} \Sigma^{(\mathrm{j})}\left(\mathrm{u}^{\prime} \rightarrow \mathrm{u}\right) \quad \mathrm{P}\left(\mu_{0}\right) + +where P\ :sub:`ℓ`\ (μ:sub:`0`) = Legendre polynomial evaluated at the +laboratory scattering cosine μ\ :sub:`0`; and + + :math:`\Sigma^{(j)}\left(u^{\prime} \rightarrow u\right)` = cross section moments of nuclide j, defined by the expression + +.. math:: + :label: eq7-4-3 + + \Sigma^{(j)}\left(u^{\prime} \rightarrow u\right)=\int_{-1}^{1} \Sigma^{(j)}\left(u^{\prime} \rightarrow u ; \mu_{0}\right) P\left(\mu_{0}\right) d \mu_{0} + +After substitution of the above Legendre expansions for the scattering +data of each nuclide, and applying the spherical harmonic addition +theorem in the usual manner, the scattering source on the right side of +:eq:`eq7-4-1` becomes :cite:`bell_nuclear_1970`: + +.. math:: + :label: eq7-4-4 + + \mathrm{S}(\mathrm{r}, \mathrm{u}, \Omega) \equiv \int_{0}^{\infty} \int_{0}^{4 \pi} \Sigma\left(\mathrm{u}^{\prime} \rightarrow \mathrm{u} ; \mu_{0}\right) \Psi\left(\mathrm{u}^{\prime}, \Omega^{\prime}\right) \mathrm{d} \Omega^{\prime} \mathrm{d} \mathrm{u}^{\prime}=\sum_{\mathrm{k}=1}^{\mathrm{LK}} \frac{2+1}{2} \mathrm{Y}_{\mathrm{k}}(\Omega) \mathrm{S}_{\mathrm{k}}(\mathrm{r}, \mathrm{u}) + +wherein, + + :math:`\mathrm{Y}_{\mathrm{k}}(\Omega)=\mathrm{Y}_{\mathrm{k}}(\mu, \zeta)` = the spherical harmonic function evaluated at direction Ω + + S\ :sub:`k` = spherical harmonic moments of the scatter source, per unit letharagy. + +The summation index “ℓk” indicates a double sum over ℓ and k indices; in +the most general case it is defined as: + +.. math:: + + \sum_{\mathrm{k}=1}^{\mathrm{LK}}=\sum_{=0}^{\mathrm{L}} \sum_{\mathrm{k}=0} + +where “L” is the input value for the maximum order of scatter (input +parameter “ISCT”). + +Due to symmetry conditions, some of the source moments may be zero. The +parameter LK is defined to be the total number of non-zero moments +(including scalar flux) for the particular geometry of interest, and is +equal to, + + LK = L + 1 for 1D slabs and spheres; + + LK = L*(L+4)/4+1 for 1D cylinders, and + + LK = L*(L+3)/2+1 for 2D MoC cells + +More details concerning the 1-D Boltzmann equation can be found in the +XSDRNPM chapter of the SCALE manual. + +.. _7-4-2-3: + +The S\ :sub:`ℓk` moments in :eq:`eq7-4-4`  correspond to expansion coefficients in +a spherical harmonic expansion of the scatter source. These can be +expressed in terms of the cross section and flux moments by + +.. math:: + :label: eq7-4-5 + + \mathrm{S}_{\mathrm{k}}(\mathrm{u})=\sum_{\mathrm{j}} \int_{\mathrm{u}^{\prime}} \mathrm{S}_{\mathrm{k}}^{(\mathrm{j})}\left(\mathrm{u}^{\prime} \rightarrow \mathrm{u}\right) \mathrm{d} \mathrm{u}^{\prime}=\sum_{\mathrm{j}} \int_{\mathrm{u}^{\prime}} \Sigma^{(\mathrm{j})}\left(\mathrm{u}^{\prime} \rightarrow \mathrm{u}\right) \Psi_{\mathrm{k}}\left(\mathrm{u}^{\prime}\right) \mathrm{d} \mathrm{u}^{\prime} + +where ψ\ :sub:`ℓk`\ (u) = spherical harmonic moments of the angular +flux; + +.. math:: + :label: eq7-4-6 + + = \int_{0}^{4 \pi} \mathrm{Y}_{\mathrm{k}}(\Omega) \Psi(\Omega) \mathrm{d} \Omega + +and S\ :sub:`ℓk`\ :sup:`(j)`\ (u′→u) = moments of the differential +scatter rate from lethargy u′ to u, for nuclide “j”; + +.. math:: + :label: eq7-4-7 + + = \Sigma^{(\mathrm{j})}\left(\mathrm{u}^{\prime} \rightarrow \mathrm{u}\right) \Psi_{\mathrm{k}}\left(\mathrm{u}^{\prime}\right) + +The ψ\ :sub:`ℓk` flux moments are the well known coefficients appearing +in a spherical harmonic expansion of the angular flux. These usually are +the desired output from the transport calculation. In particular, the +ℓ=0, k=0 moment corresponds to the scalar flux [indicated here as +Φ(r,u)], + +.. math:: + :label: eq7-4-8 + + \Psi_{0,0}(\mathrm{r}, \mathrm{u})=\Phi(\mathrm{r}, \mathrm{u})=\int_{0}^{4 \pi} \Psi(\mathrm{r}, \mathrm{u}, \Omega) d \Omega + + +In general the epithermal component of the scatter source in :eq:`eq7-4-4`  +contains contributions from both elastic and inelastic scatter +reactions; however, inelastic scatter is only possible above the +threshold energy corresponding to the lowest inelastic level. The +inelastic Q values for most materials are typically above 40 keV; +therefore, elastic scatter is most important for slowing down +calculations in the resolved resonance range of most absorber materials +of interest. For example, the inelastic Q values of :sup:`238`\ U, iron, +and oxygen are approximately 45 keV, 846 keV, and 6 MeV, respectively; +while the upper energy of the :sup:`238`\ U resolved resonance range is +20 keV in ENDF/B-VII. The inelastic thresholds of some fissile materials +like :sup:`235`\ U and :sup:`239`\ Pu are on the order of 10 keV; +however, with the exception of highly enriched fast systems, these +inelastic reactions usually contribute a negligible amount to the +overall scattering source. CENTRM assumes that continuum inelastic +scatter is isotropic in the laboratory system, while discrete level +inelastic scatter is isotropic in the *center of mass* (CM) coordinate +system. + +Over a broad energy range, *elastic* scatter from most moderators can +usually be assumed isotropic +(*s*-wave) in the neutron-nucleus CM coordinate system. In the case of +hydrogen, this is true up to approximately 13 MeV; for carbon up to +2 MeV; and for oxygen up to 100 keV. However, it is well known that +isotropic CM scatter does not result in isotropic scattering in the +laboratory system. For *s‑*\ wave elastic scatter the average +scatter-cosine in the laboratory system is given by: :math:`\bar{\mu}_{0}=0.667 / \mathrm{~A};^{3}` where +“A” is the mass number (in neutron mass units) of the scattering +material. This relation indicates that *s*-wave, elastic scattering +from low A materials tends to be more anisotropic in the laboratory, +and that the laboratory scattering distribution approaches isotropic +:math:`\left(\bar{\mu}_{0}=0 ; \theta_{0}=90\right)` as A becomes large. For example, the :math:`\bar{\mu}_{0}` +of hydrogen +is 0.667 (48.2°); while it is about 0.042 (87.6°) for oxygen. Because +*s*\ ‑wave scattering from heavy materials is nearly isotropic in the +laboratory system, the differential scattering cross section (and thus +the scattering source) can usually be expressed accurately by a low +order Legendre expansion. On the other hand light moderators like +hydrogen may require more terms—depending on the flux anisotropy—to +accurately represent the elastic scatter source in the laboratory +system. The default settings in CENTRM are to use P0 (isotropic lab +scatter) for mass numbers greater than A=100, and P1 for lighter +masses. + +An analytical expression can be derived for the cross-section moments in +the case of two-body reactions, such as elastic and discrete-level +inelastic scattering from “stationary” nuclei. Stationary here implies +that the effect of nuclear motion on neutron scattering kinematics is +neglected. + +.. note:: The stationary nucleus approximation for treating + scattering kinematics does not imply that the effect of nuclear motion + on Doppler broadening of resonance cross sections is ignored, since this + effect is included in the PW cross-section data. + +In CENTRM the stationary nucleus approximation is applied above the +thermal cutoff, typically around 3-5 eV, but is not valid for low energy +neutrons. CENTRM has the capability to perform a PW transport +calculation in the thermal energy range using tabulated thermal +scattering law data for bound molecules, combined with the analytical +free-gas kernel for other materials. In this case the cross-section +moments appearing in :eq:`eq7-4-3`  include upscattering effects. The expressions +used in CENTRM to compute the PW scatter source moments in the thermal +range are given in :ref:`7-4-2-6`. + +The following two sections discuss the evaluation of the scatter source +moments for epithermal elastic and inelastic reactions, respectively. + +.. _7-4-2-3-1: + +Epithermal Elastic Scatter +^^^^^^^^^^^^^^^^^^^^^^^^^^ + +Consider a neutron with energy E′, traveling in a direction Ω′, that +scatters elastically from an arbitrary material “j,” having a +mass A\ :sup:`(j)` in neutron-mass units. Conservation of kinetic energy +and momentum requires that there be a unique relation between the angle +that the neutron scatters (relative to the initial direction) and its +final energy E after the collision. If the nucleus is assumed to be +stationary in the laboratory coordinate system, then the +cosine (μ\ :sub:`0`) of the scatter angle (θ\ :sub:`0`) measured in the +laboratory system, as a function of the initial and final energies, is +found to be + +.. math:: + :label: eq7-4-9 + + \mu_{0} \equiv \Omega^{\prime} \cdot \Omega=\mathrm{G}^{(\mathrm{j})}\left(\mathrm{E}^{\prime}, \mathrm{E}\right) , + +where the kinematic correlation function G relating E′, E, and +μ\ :sub:`0` for elastic scatter is equal to + +.. math:: + :label: eq7-4-10 + + \begin{array}{l} + \mathrm{G}^{(\mathrm{j})}\left(\mathrm{E}^{\prime}, \mathrm{E}\right)=\mathrm{a}_{1}^{(\mathrm{j})} \times\left[\mathrm{E} / \mathrm{E}^{\prime}\right]^{\frac{1}{2}}-\mathrm{a}_{2}^{(\mathrm{j})} \times\left[\mathrm{E}^{\prime} / \mathrm{E}\right]^{\frac{1}{2}} \\ + \text { and } \mathrm{a}_{1}^{(\mathrm{j})}=\left(\mathrm{A}^{(\mathrm{j})}+1\right) / 2 \quad ; \quad \mathrm{a}_{2}^{(\mathrm{j})}=\left(\mathrm{A}^{(\mathrm{j})}-1\right) / 2 + \end{array} . + +The final energy E of an elastically scattered neutron is restricted to +the range, + +.. math:: + :label: eq7-4-11 + + \alpha^{(j)} E^{\prime} \leq E \leq E^{\prime} + +where α\ :sup:`(j)` =  maximum fractional energy lost by elastic scatter + +.. math:: + :label: eq7-4-12 + + = \left[\mathrm{a}_{2}^{(\mathrm{j})} / \mathrm{a}_{1}^{(\mathrm{j})}\right]^{2} + +The corresponding range of scattered neutrons in terms of lethargy is equal to + +.. math:: + :label: eq7-4-13 + + \mathrm{u}^{\prime} \leq \mathrm{u} \leq \mathrm{u}^{\prime}+\varepsilon^{(\mathrm{j})} + +where + +.. math:: + :label: eq7-4-14 + + \begin{aligned} + \mathrm{u}, \mathrm{u}^{\prime} &=\mathrm{u}(\mathrm{E}), \mathrm{u}^{\prime}\left(\mathrm{E}^{\prime}\right)=\text { lethargies corresponding to } \mathrm{E} \text { and } \mathrm{E}^{\prime}, \text { respectively; and } \\ + \varepsilon^{(\mathrm{j})} &=\text { maximum increase in lethargy, per elastic scatter }=\ln \left[1 / \alpha^{(j)}\right] + \end{aligned} + +The double-differential scatter kernel of nuclide j (per unit lethargy +and solid angle) for *s-*\ wave elastic scatter of neutrons from +stationary nuclei, is equal to + +.. math:: + :label: eq7-4-15 + + \begin{aligned} + \Sigma^{(\mathrm{j})}\left(\mathrm{u}^{\prime} \rightarrow \mathrm{u} ; \mu_{0}\right) &=\frac{\mathrm{E} \Sigma^{(\mathrm{i})}\left(\mathrm{u}^{\prime}\right)}{\mathrm{E}^{\prime}\left(1-\alpha^{(\mathrm{j})}\right)} \delta\left[\mu_{0}-\mathrm{G}^{(\mathrm{j})}\left(\mathrm{E}^{\prime}, \mathrm{E}\right)\right], \text { for } \mathrm{u}^{\prime} \leq \mathrm{u} \leq \mathrm{u}^{\prime}+\varepsilon^{(\mathrm{j})} \\ + &=0 \quad \mathrm{u}<\mathrm{u}^{\prime} \text { or } \mathrm{u}>\mathrm{u}^{\prime}+\varepsilon^{(\mathrm{j})} + \end{aligned} + +The presence of the Dirac delta function completely correlates the angle +of scatter and the values of the initial and final energies. +Substituting the double differential cross-section expression from :eq:`eq7-4-15` +into :eq:`eq7-4-3`  gives the single-differential Legendre moments of the +cross section, per final lethargy: + +.. math:: + :label: eq7-4-16 + + \begin{aligned} + \Sigma^{(\mathrm{j})}\left(\mathrm{u}^{\prime} \rightarrow \mathrm{u}\right) &=\frac{\mathrm{E} \mathrm{P}\left[\mathrm{G}^{(\mathrm{j})}\right] \Sigma^{(\mathrm{j})}\left(\mathrm{u}^{\prime}\right)}{\mathrm{E}^{\prime}\left(1-\alpha^{(\mathrm{j})}\right)}, \text { for } \mathrm{u}^{\prime} \leq \mathrm{u} \leq \mathrm{u}^{\prime}+\varepsilon^{(\mathrm{j})} \\ + &=0 \quad \mathrm{u}^{\prime} \text { or } \mathrm{u}>\mathrm{u}^{\prime}+\varepsilon^{(\mathrm{j})} + \end{aligned} + +where P\ :sub:`ℓ` = Legendre polynomial evaluated at argument +G\ :sup:`(j)` equal to the scatter cosine. + +When the above expressions are used in :eq:`eq7-4-5` , the following is obtained +for the ℓk moment of the epithermal elastic scattering source at +lethargy u: + +.. math:: + :label: eq7-4-17 + + \mathrm{S}_{\mathrm{k}}^{(\mathrm{j})}(\mathrm{u})=\sum_{\mathrm{j}} \int_{\left.\mathrm{u}-\varepsilon^{(\mathrm{i}}\right)}^{\mathrm{u}} \mathrm{S}_{\mathrm{k}}\left(\mathrm{u}^{\prime} \rightarrow \mathrm{u}\right) \mathrm{du}^{\prime}=\sum_{\mathrm{j}} \int_{\mathrm{u}-\varepsilon^{(j)}}^{\mathrm{u}} \frac{\mathrm{E} \Sigma^{(\mathrm{j})}\left(\mathrm{u}^{\prime}\right) \mathrm{P}\left[\mathrm{G}^{(\mathrm{j})}\right]}{\mathrm{E}^{\prime}\left(1-\alpha^{(\mathrm{j})}\right)} \Psi_{\mathrm{k}}\left(\mathrm{u}^{\prime}\right) \mathrm{du}^{\prime} . + +.. _7-4-2-3-2: + +Epithermal Inelastic Scatter +^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +If the input value of DEMAX is set above the inelastic threshold of some +materials in the problem, then inelastic scattering can occur in the +PW range. The pointwise transport calculation may optionally include +discrete-level and continuum inelastic reactions in computing the +PW scatter source moments. The multigroup calculations always consider +inelastic reactions. + +Discrete-level inelastic reactions are two-body interactions, so that +kinematic relations can be derived relating the initial and final +energies and the angle of scatter. It can be shown that the kinematic +correlation function for discrete-level inelastic scatter can be written +in a form identical to that for elastic scatter by redefining the +parameter a\ :sub:`1` in :eq:`eq7-4-10`  to be the energy dependent function :cite:`williams_submoment_2000`, + +.. math:: + :label: eq7-4-18 + + \mathrm{a}_{1}^{(\mathrm{~m}, \mathrm{j})} = \frac{\left(\mathrm{A}^{(\mathrm{j})}+1\right)}{2}+\frac{\left(-\mathrm{Q}^{(\mathrm{m}, \mathrm{j})}\right) \mathrm{A}^{(\mathrm{j})}}{2 \mathrm{E}} + +The parameter Q\ :sup:`(m, j)` is the Q-value for the m\ :sub:`th` level +of nuclide “j”. The Q value is negative for inelastic scattering, while +it is zero for elastic scatter. The threshold energy in the laboratory +coordinate system is proportional to the Q-value of the inelastic level, +and is given by: + +.. math:: + + \mathrm{E}_{\mathrm{T}}^{(\mathrm{m}, \mathrm{j})}=\frac{\mathrm{A}^{(\mathrm{j})}+1}{\mathrm{~A}^{(\mathrm{j})}} \times\left(-\mathrm{Q}^{(\mathrm{m}, \mathrm{j})}\right) + +The range of energies that can contribute to the scatter source at E, +due to inelastic scatter from the m\ :sub:`th` level of nuclide j is +defined to be [E:sub:`L` , E:sub:`H` ], where +E\ :sub:`H` >E:sub:`L` >E:sub:`T` . This energy range has a +corresponding lethargy range of [u:sub:`LO` , u:sub:`HI` ] which is +equal to, + +.. math:: + + \begin{array}{l} + \mathrm{u}_{\mathrm{LO}}=\mathrm{u}-\ln \left(\frac{1}{\alpha_{1}^{(\mathrm{j})}\left(\mathrm{E}_{\mathrm{H}}\right)}\right)=\mathrm{u}-\varepsilon_{1}^{(\mathrm{j})} \\ + \mathrm{u}_{\mathrm{HI}}=\mathrm{u}-\ln \left(\frac{1}{\alpha_{2}^{(\mathrm{j})}\left(\mathrm{E}_{\mathrm{L}}\right)}\right)=\mathrm{u}-\varepsilon_{2}^{(\mathrm{j})} + \end{array} + +The energy-dependent alpha parameters in the above expressions are defined as, + +.. math:: + + \begin{array}{l} + \alpha_{1}(\mathrm{E})=\left(\frac{\mathrm{A} \Delta^{(\mathrm{m}, \mathrm{j})}(\mathrm{E})-1}{\mathrm{~A}+1}\right)^{2} \\ + \alpha_{2}(\mathrm{E})=\left(\frac{\mathrm{A} \Delta^{(\mathrm{m}, \mathrm{j})}(\mathrm{E})+1}{\mathrm{~A}+1}\right)^{2} + \end{array} + +where + +.. math:: + :label: eq7-4-19 + + \Delta^{(\mathrm{m}, \mathrm{j})}(\mathrm{E})=\sqrt{1-\frac{\mathrm{E}_{\mathrm{T}}^{(\mathrm{m}, \mathrm{j})}}{\mathrm{E}}} + +Modifying the epithermal elastic scatter source in :eq:`eq7-4-17`  to include +discrete-level inelastic scatter gives the following expression + +.. math:: + :label: eq7-4-20 + + \mathrm{S}_{\mathrm{k}}(\mathrm{u})=\sum_{m, j} \int_{u_{L O}}^{u_{H I}} \frac{\mathrm{E}}{\mathrm{E}^{\prime}} \frac{\Sigma^{(\mathrm{m}, \mathrm{j})}\left(\mathrm{E}^{\prime}\right) \mathrm{P}\left[\mathrm{G}^{(\mathrm{m}, \mathrm{j})}\right]}{\left(1-\alpha^{j}\right) \Delta^{(\mathrm{m}, \mathrm{j})}\left(\mathrm{E}^{\prime}\right)} \Psi_{\mathrm{k}}\left(\mathrm{u}^{\prime}\right) \mathrm{d} u^{\prime} + +Detailed expressions for the lethargy limits are given in :cite:`williams_submoment_2000`. Since +Δ\ :sup:`(m,j)` is equal to unity for elastic scatter, the above +equation reduces to :eq:`eq7-4-14` if there is no discrete-level inelastic +contribution. + +At high energies, the inelastic levels of the nucleus become a +continuum. In this case CENTRM represents the energy distribution of the +scattered neutrons by an evaporation spectrum with an isotropic angular +distribution in the lab system; thus, only the P\ :sub:`0` moment +appears in the continuum inelastic scattering source. Including +continuum inelastic reactions in the PW calculation usually has a small +impact on the spectrum used for resonance self-shielding, and may +adversely impact the computer memory requirements and execution time. +Therefore, by default, CENTRM does not include continuum inelastic +reactions in the pointwise solution; however, it is always included in +the UMR solution. + +.. _7-4-2-3-3: + +Thermal Scatter +^^^^^^^^^^^^^^^ + +Since thermal neutrons have energies comparable to the mean kinetic +energy of molecules in thermal equilibrium, the scattering kernels must +account for molecular motion. The scatter moments include both +downscatter as well as upscatter contributions; hence, the integration +limits appearing in :eq:`eq7-4-17`  must be extended from the lowest to the highest +energy of the thermal range. Furthermore the cross-section moments +correspond to the Legendre expansion coefficients of the thermal scatter +kernel, which has a substantially different form than the epithermal +kernel discussed in the previous two sections. In general the +ℓ\ :sub:`th` Legendre moment of the thermal scattering kernel at +temperature T, describing scattering from E to E′, is given by + +.. math:: + + \sigma \quad\left(\mathrm{E}^{\prime} \rightarrow \mathrm{E} ; \mathrm{T}\right)=\frac{\sigma_{b}}{T} \sqrt{\frac{\mathrm{E}}{\mathrm{E}^{\prime}}} e^{-\frac{\beta\left(\mathrm{E}^{\prime} \rightarrow \mathrm{E}\right)}{2}} \int \mathrm{P}\left(\mu_{0}\right) \mathrm{S}[\alpha, \beta ; T] \quad d \mu_{0} + +where β(E′→E) and α(E′,E,μ\ :sub:`0`) are dimensionless variables +(functions of temperature) defining the energy and momentum exchange, +respectively, of the collision :cite:`bell_nuclear_1970`; σ\ :sub:`b` is the rigidly +bound scatter cross section, which is proportional to the free atom +cross section; and S(α, β; T) describes the temperature-dependent +thermal scattering law. + +If atomic bonding effects are neglected, the atoms of a material behave +like a gas in thermal equilibrium at the temperature of the medium. In +this case S(α, β) can be expressed by an analytical function. CENTRM +uses the free gas model for all nuclides except those materials that +have thermal scattering laws available in the ENDF/B nuclear data files. +The ENDF/B scattering law data account for the effects of molecular +bonding and possible polyatomic crystalline structure. While free-gas +kernels are computed internally in CENTRM, the kernel moments describing +bound thermal scatterers are stored in a data file that can be accessed +by CENTRM. + +.. _7-4-2-3-4: + +Bound thermal kernels +^^^^^^^^^^^^^^^^^^^^^ + +Thermal scattering from bound atoms is classified either as an +“inelastic reaction,” in which the neutron energy may change, or an +“elastic reaction,” in which the neutron changes direction, but does not +change energy. In ENDF/B the former reactions are treated as incoherent +inelastic scattering with a doubly differential kernel describing the +secondary neutron energy and angle distribution. The latter reactions +are usual treated as coherent elastic scatter characterized by +diffractive interference of the scattered deBroglie waves, although a +few materials are modeled by the incoherent elastic approximation. +Legendre moments for thermal elastic kernels describe the secondary +angular distribution with no energy exchange, at a given neutron energy. +Bound scatter kernels have been processed by the AMPX code system for +most of the ~25 compounds with thermal scatter laws in ENDF/B, and are +stored in individual kinematics files distributed with the SCALE code +system. These include materials such as: H in water, H and C in +polyethylene, H and Zr in ZrH, C in graphite, deuterium in heavy water, +Be metal, Be in BeO, etc. The CRAWDAD module processes scattering kernel +data for individual nuclides into a combined library used in CENTRM, and +also interpolates the kernels to the appropariate temperatures. + +The bound scatter kernels are tabulated at different energy points from +the flux solution mesh; therefore it is necessary to map the data onto +the desired energy mesh in the CENTRM calculation. Because thermal +elastic scattering results in no energy loss, the elastic moments only +appear in the within-point term of the scattering source in the CENTRM +thermal calculation. Thus the coherent elastic data is easily +interpolated since it only involves a single energy index and +temperature. However, the incoherent inelastic moments are 2-D arrays in +terms of the initial and final energies, so that a 2-D interpolation +must be done for each temperature. CENTRM uses a simple type of +“unit-base transform” method to interpolate incoherent inelastic kernels +onto the flux solution mesh. The method attempts to preserve the +absolute peak of the secondary energy distribution, at given initial +energy. For water-bound hydrogen and several other moderators, this is +quite adequate, since the kernel generally has only a single maximum. +However, if more than one local extrema is present, such as for +graphite, the other local peaks are not explicitly preserved in the +interpolation method. For this reason it is necessary to include a +fairly dense set of initial energies in the tabulated kernels of +graphite and similar materials, to avoid gross changes in the kernel +shape at adjacent initial energy panels. + +.. _7-4-2-3-4-1: + +Free gas thermal kernels +........................ + +CENTRM computes free-gas kernels using the approach proposed by +Robinson :cite:`robinson_notes_1981` as a modification to the original +FLANGE :cite:`honeck_flange-ii_1971` methodology. +Legendre moments of the free-gas scatter kernel per unit lethargy are +expressed as, + +.. math:: + :label: eq7-4-21 + + \Sigma^{(\mathrm{j})}\left(\mathrm{u}^{\prime} \rightarrow \mathrm{u}\right)=\mathrm{A}^{(\mathrm{j})} \Sigma_{\mathrm{free}}^{(\mathrm{j})} \frac{\mathrm{E}}{\mathrm{E}^{\prime}} \mathrm{e}^{-\beta / 2} \sum_{\mathrm{n}=0} \mathrm{W}_{\mathrm{n}} \mathrm{H}_{\mathrm{n}}\left(\mathrm{E}^{\prime}, \mathrm{E}\right) + +where W\ :sub:`ℓn` are constant coefficients associated with the +Legendre polynomial of order ℓ; Σ\ :sub:`free` is the constant free-atom +cross section for the material; and H\ :sub:`n` are the α-moments of the +free-gas scatter law, given as + +.. math:: + + \mathrm{H}_{\mathrm{n}}\left(\mathrm{E}^{\prime}, \mathrm{E}\right)=\frac{1}{\sqrt{\pi}} \int_{\alpha_{\mathrm{L}}}^{\alpha_{\mathrm{H}}} \alpha^{\mathrm{n}} \times\left(\frac{\mathrm{e}^{-\frac{\alpha^{2}+\beta^{2}}{4 \alpha}}}{2 \sqrt{\alpha}}\right) \mathrm{d} \alpha + +The limits on the above integral correspond to: + +.. math:: + + \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`: + +.. math:: + + \mathrm{H}_{\mathrm{n}}\left(\mathrm{E}^{\prime}, \mathrm{E}\right)=2(2 \mathrm{n}-1) \mathrm{H}_{\mathrm{n}-1}+\beta^{2} \mathrm{H}_{\mathrm{n}-2}-\left[\mathrm{F}_{\mathrm{n}}\left(\sqrt{\alpha_{\mathrm{H}}}, \beta\right)-\mathrm{F}_{\mathrm{n}}\left(\sqrt{\alpha_{\mathrm{L}}}, \beta\right)\right] + +where F\ :sub:`n` is the function, + +.. math:: + + \mathrm{F}_{\mathrm{n}}(\mathrm{t}, \beta)=\frac{\mathrm{t}^{2 \mathrm{n}-1} \mathrm{e}^{-\frac{1}{4}\left(\frac{\beta^{2}}{\mathrm{t}^{2}}+\mathrm{t}^{2}\right)}}{\sqrt{\pi} / 2} + +Analytical expressions for the initial two moments, +H\ :sub:`0` and H\ :sub:` −1`, are given in :cite:`robinson_notes_1981`. + +The standard free-gas kernel is based on the assumption of a constant +free atom cross section. When averaged over the molecular velocity +distribution, this gives a 1/v variation in the effective free-gas +cross section at low energies. To approximately account for nuclear +structure effects on the energy dependence of the thermal cross section +(e.g., low energy resonances), the free-gas moments are multiplied by +the ratio σ\ :sub:`s`\ (E)/σ\ :sub:`FG`\ (E), where σ\ :sub:`s` is the +Doppler broadened scatter cross section processed from ENDF/B data; and +σ\ :sub:`FG` is the effective free-gas cross section, + +.. math:: + + \sigma_{\mathrm{FG}}\left(\mathrm{E}^{\prime}\right)=\frac{\sigma_{\mathrm{free}}}{\mathrm{y}^{2}}\left[\left(\mathrm{y}^{2}+1 / 2\right) \operatorname{erf}(\mathrm{y})+\frac{\mathrm{y} \mathrm{e}^{-\mathrm{y}^{2}}}{\sqrt{\pi}}\right] + +where :math:`\mathrm{y}^{2}=\mathrm{~A} \frac{\mathrm{E}}{\mathrm{kT}}`. + +.. _7-4-2-4: + +Sub-moment expansion of the epithermal scattering source +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +One difficulty in computing the epithermal scatter source moments is +that the Legendre polynomial in the integrand of :eq:`eq7-4-17`  and :eq:`eq7-4-20` is a function +of both the initial and final lethargy (or energy) of the scattered +neutrons, due to the correlation function G\ :sup:`(j)`\ (E,E′). At each +lethargy u this requires that the u′-integral be recomputed over all +lower lethargies, for every nuclide and moment. A more efficient +algorithm would be possible if the differential scattering moments +appearing in the integrand could be factored into a product of a +function of u multiplied by a function of u′ such as + +.. math:: + :label: eq7-4-22 + + \mathrm{S}_{\mathrm{k}}^{(\mathrm{j})}\left(\mathrm{u}^{\prime} \rightarrow \mathrm{u}\right)=\mathrm{F}^{(\mathrm{j})}\left(\mathrm{u}^{\prime}\right) \quad \mathrm{H}^{(\mathrm{j})}(\mathrm{u}) + +where F and H are the two factors (to be specified later). + +If this is done, the u-function can be factored from the scatter source +integrals, leaving only integrals over the u′-function as shown below: + +.. math:: + + \mathrm{S}^{(\mathrm{j})}(\mathrm{u})=\int_{\mathrm{u}^{\prime}} \mathrm{S}_{\mathrm{k}}^{(\mathrm{j})}\left(\mathrm{u}^{\prime} \rightarrow \mathrm{u}\right) \mathrm{du}^{\prime}=\mathrm{H}^{(\mathrm{j})}(\mathrm{u}) \int_{\mathrm{u}^{\prime}} \mathrm{F}^{(\mathrm{j})}\left(\mathrm{u}^{\prime}\right) \mathrm{du}^{\prime} + +Because the factored integrand does not depend on the variable u, a +running summation over all u′ points can be accumulated and saved as the +calculation sweeps from low to high lethargy. For example, note that the +ℓ = 0 moment in :eq:`eq7-4-17` is already separable into a product of u times u′ +because P\ :sub:`0` is equal to one at all values of G. Thus the +isotropic component of the elastic differential scatter rate (per unit +lethargy) from u′ to u is proportional to E/E′, where + +.. math:: + + \mathrm{E}=\mathrm{E}(\mathrm{u})=\mathrm{E}_{\mathrm{ref}} \mathrm{e}^{-\mathrm{u}}, \quad \text { and } \quad \mathrm{E}^{\prime}=\mathrm{E}^{\prime}\left(\mathrm{u}^{\prime}\right)=\mathrm{E}_{\mathrm{ref}} \mathrm{e}^{-\mathrm{u}^{\prime}} + +Therefore, the two separable factors in the lowest moment, +S\ :sup:`(j)`\ :sub:`0.0`\ (u′→u), are identified as, + +.. math:: + + \begin{array}{l} + \mathrm{H}(\mathrm{u})=\mathrm{E} /\left(1-\alpha^{(\mathrm{j})}\right), \quad \text { and } \\ + \mathrm{F}\left(\mathrm{u}^{\prime}\right)=\Sigma^{(\mathrm{j})}\left(\mathrm{u}^{\prime}\right) \Psi_{00}\left(\mathrm{u}^{\prime}\right) / \mathrm{E}^{\prime} + \end{array} + +However, the higher order Legendre moments contain the term +P\ :sub:`ℓ`\ (G) in the integrand; and the expression for G(E′,E) is a +difference of two terms that depend on both E and E′. A new method +called a “sub‑moment expansion” has been developed for CENTRM that +allows the Legendre polynomials appearing in the differential scatter +moments to be factored into the desired separable form. Each spherical +harmonic moment of the scattering source appears expanded in a series of +factored “sub‑moments.” + +The Legendre polynomial of order ℓ is a polynomial containing terms up +to the ℓ\ :sup:`th` power. Applying the binomial expansion theorem and +some algebraic manipulation, the standard expression for P\ :sub:`ℓ` +evaluated at “G” can be expressed as + +.. math:: + :label: eq7-4-23 + + \mathrm{P}_{\ell}(\mathrm{G})=\frac{\mathrm{E}^{\prime}}{\mathrm{E}} \times \mathrm{a}_{1}^{\ell} \sum_{\mathrm{K}=-\ell}^{\ell} \tilde{\mathrm{g}}_{\ell, \mathrm{K}}(\mathrm{E}) \quad \mathrm{h}_{\mathrm{K}}(\mathrm{E}) \mathrm{h}_{\mathrm{K}}^{-1}\left(\mathrm{E}^{\prime}\right) + +where h\ :sub:`k`\ (E)=E\ :sup:`1+K/2`; and the expansion coefficients +:math:`\tilde{\mathrm{g}}_{\ell, \mathrm{k}}` are equal to, +:math:`\tilde{\mathrm{g}}_{\ell \mathrm{K}}=\frac{\mathrm{g}_{\ell \mathrm{K}}}{N_{\ell} \times \alpha_{1}^{\ell}}` +where the g\ :sub:`ℓ,K` (no tilde) +parameters were defined in :cite:`williams_computation_1995` to be: + +.. math:: + :label: eq7-4-24 + + \begin{aligned} + &\mathrm{g}_{, \mathrm{K}}=\frac{\left(1+(-1)^{+\mathrm{K}}\right)}{2} \sum_{K^{\prime}=0}^{\frac{-K}{2}}(-1)^{K^{\prime}} b_{2 K^{\prime}+K},\left(\begin{array}{r} + 2 K^{\prime}+K \\ + K^{\prime} + \end{array}\right) \quad a_{1}^{K+K^{\prime}} a_{2}^{K^{\prime}} ; \quad \text { for } \quad \mathrm{K} \geq 0\\ + &\text { and }\\ + &=\left(-a_{2} / a_{1}\right)^{|K|} \quad g_{,|K|} \quad ; \quad \text { for } \quad K<0 + \end{aligned} + +In :eq:`eq7-4-23`\ –\ :eq:`eq7-4-24`, the constants b\ :sub:`m,ℓ` and N\ :sub:`ℓ` are the standard +Legendre constants and normalization factors, respectively, which are +tabulated in :numref:`tab7-4-1` for orders through P\ :sub:`7`; and :math:`\left(\begin{array}{c} +\mathrm{m} \\ +\mathrm{i} +\end{array}\right)=` +the binomial expansion coefficients\ :sup:`(20)` :math:`= \frac{\mathrm{m} !}{(\mathrm{m}-\mathrm{i}) ! \quad \mathrm{i} !}` + +.. _tab7-4-1: +.. list-table:: Constants appearing in Legendre polynomials. + :align: center + + * - .. image:: figs/CENTRM/tab1.svg + :width: 800 + +The explicit dependence of the constants a\ :sub:`1` and a\ :sub:`2` on +the nuclide index j [see :eq:`eq7-4-10`] has been suppressed to simplify notation. +For discrete-level inelastic scatter the parameter a\ :sub:`1` is an +energy dependent function given by :eq:`eq7-4-18`, but for elastic scatter this +reduces to the constant in :eq:`eq7-4-10`. Note that the g\ :sub:`ℓ,K` value is zero unless ℓ and K are both +even or both odd, respectively, so that about half the terms appearing +in the summation of :eq:`eq7-4-23` vanish. :numref:`tab7-4-2` through :numref:`tab7-4-4` give values +for the submoment expansion coefficients for +several nuclides. + +The sub-moment expansion of the scattering source, including both +elastic and discrete-level inelastic reactions, is obtained by +substituting the expansion of the Legendre polynomial from :eq:`eq7-4-23`  into +:eq:`eq7-4-21`, giving + +.. math:: + :label: eq7-4-25 + + \mathrm{S}_{\mathrm{k}}(\mathrm{u})=\sum_{m, j} \sum_{K=-} \mathrm{Z}_{, \mathrm{K}}^{(\mathrm{m}, \mathrm{j})}(\mathrm{E}) \mathrm{h}_{\mathrm{K}}(\mathrm{E}) \int_{u_{L O}^{(m, j)}}^{u_{H}^{(m, j)}} \psi_{\mathrm{k}}\left(\mathrm{u}^{\prime}\right) \Sigma^{(\mathrm{m}, \mathrm{j})}\left(\mathrm{E}^{\prime}\right) \frac{\mathrm{h}_{\mathrm{K}}^{-1}\left(\mathrm{E}^{\prime}\right)}{\Delta^{(\mathrm{m}, \mathrm{j})}\left(\mathrm{E}^{\prime}\right)} \mathrm{du}^{\prime} + +where :math:`Z_{\ell \mathrm{K}}^{(\mathrm{m}, \mathrm{j})}(E)=a_{1}^{\ell}(E) \frac{\tilde{g}_{\ell, K}^{(m, j)}(E)}{\left(1-\alpha^{(j)}\right)}`. +For elastic scatter, the Z coefficients are independent +of energy. + +With this approach the scatter source moments in :eq:`eq7-4-26` have been further +expanded into a summation of “submoments” identified by index K +(although some of these terms are equal to zero, due to the behavior of +the g\ :sub:`ℓ,K `\ coefficients). Each term has the desired factored +form expressed in :eq:`eq7-4-22`; i.e., separable in terms of the variables u and +u′ with + +.. math:: + :label: eq7-4-26 + + \mathrm{H}_{, \mathrm{K}}^{(\mathrm{j})}(\mathrm{u})=\mathrm{Z}_{, \mathrm{K}}^{(\mathrm{j})}(\mathrm{E}) \mathrm{h}_{\mathrm{K}}(\mathrm{E}), \quad \text { and } \quad \mathrm{F}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{)})}\left(\mathrm{u}^{\prime}\right)=\frac{\mathrm{h}_{\mathrm{K}}^{-1}\left(\mathrm{E}^{\prime}\right) \Psi_{\mathrm{k}}\left(\mathrm{u}^{\prime}\right) \Sigma^{(\mathrm{j})}\left(\mathrm{u}^{\prime}\right)}{\Delta^{(\mathrm{m}, \mathrm{j})}\left(\mathrm{E}^{\prime}\right)} + +so that the lk\ :sub:`th` moment of the scatter source can be written +as + +.. math:: + :label: eq7-4-27 + + \mathrm{S}_{\mathrm{k}}(\mathrm{u})=\sum_{m, j} \sum_{K=-1} \mathrm{H}_{, \mathrm{K}}^{(\mathrm{j})}(\mathrm{u}) \int_{\left.\mathrm{u}_{\mathrm{LO}}^{(\mathrm{m}, \mathrm{j}}\right)}^{\mathrm{u}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{m}, \mathrm{j})}} \mathrm{F}_{\mathrm{k}}^{(\mathrm{j})}\left(\mathrm{u}^{\prime}\right) \mathrm{d} \mathrm{u}^{\prime} + +.. _7-4-2-4-1: + +Characteristics and Properties of the Sub-Moment Expansion +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +The expansion in :eq:`eq7-4-23` becomes numerically unstable for heavy nuclides +(large A), with high Legendre orders. Using double precision arithmetic, +it was found that the accuracy of the expansion begins to break down for +heavy nuclides (A100) if the order of scatter exceeds P\ :sub:`5`; +although the expansion for lighter nuclides (viz, moderators) is very +accurate even for scattering orders as high as P\ :sub:`7` or more. For +this reason CENTRM has an option to restrict the Legendre expansion to +lower orders for heavy masses, while using the input value of “ISCT” for +lighter nuclides. The restricted Legendre order and mass cut-off value +can be controlled by user input, but the default is P\ :sub:`0` (i.e., +isotropic lab scattering) for A>100. :numref:`tab7-4-5` shows the maximum error +observed in the series representation of Legendre polynomials up to +P\ :sub:`5`, for selected mass numbers. These values were obtained by +evaluating the series expansion for P\ *ℓ*\ (G(x)) in :eq:`eq7-4-23`, and +comparing to the exact value computed at 11 equally spaced values for +E/E′. The observed error in the P\ :sub:`5` polynomial expansion is < 1% +even for heavy materials such as :sup:`238`\ U, while nuclides whose +mass is < 100 are computed nearly exactly by the expansion. + +Although the accuracy of the submoment expansion is good through +P\ :sub:`7` scattering in moderators, Legendre expansions above P3 are +not recommended because the number of terms in the expansion increases +rapidly with increasing scattering order, especially for 2D MoC and 1D +cylindrical systems. The number of spherical harmonic moments appearing +in the scattering source depends on the order (L=ISCAT) of the Legendre +expansion used to represent the differential scatter cross section, as +well as on the type of geometry (slab, spherical, cylindrical, or 2D +MoC) used in the transport calculation. The submoment method further +expands each source moment. :numref:`tab7-4-6` shows the number of moments in +the cross-section expansion, and the number of moments and submoments in +the scatter source expansion, as a function of scatter order and +geometry type. Although the use of cumulative integrals discussed below +allows the sub-moments to be evaluated rapidly, the large number of +terms becomes prohibative for high scattering orders. Fortunately a +P\ :sub:`1` Legendre order is sufficient for most self-shielding +calculations, and orders beyond P\ :sub:`2` should seldom be required +for reactor physics and criticality applications. + +.. _tab7-4-2: +.. list-table:: Coefficients in expansion of Pℓ[G(x)]* for hydrogen (A = 1). + :align: center + + * - .. image:: figs/CENTRM/tab2.svg + :width: 600 + +.. _tab7-4-3: +.. list-table:: Coefficients in expansion on Pℓ[G(x)]* for oxygen (A = 16). + :align: center + + * - .. image:: figs/CENTRM/tab3.svg + :width: 600 + +.. _tab7-4-4: +.. list-table:: Coefficients in expansion of Pℓ[G(x)]* for U-238 (A = 236). + :align: center + + * - .. image:: figs/CENTRM/tab4.svg + :width: 600 + +.. _tab7-4-5: +.. list-table:: Fractional error :sup:`(1)` in series expansion [Eq. (F18.2.25)] of Legendre polynomials. + :align: center + + * - .. image:: figs/CENTRM/tab5.svg + :width: 600 + +.. _tab7-4-6: +.. list-table:: Number of moments and submoments as function of scattering order. + :align: center + + * - .. image:: figs/CENTRM/tab5.svg + :width: 600 + +.. _7-4-2-4-2: + +Scattering moments expressed with cumulative integral operator +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +It will be convenient to express the scatter source moments in terms of +an integral operator :math:`\mathbb{C}`, designated here as the “cumulative integral.” The +domain of this operator is the vector space of all integrable lethargy +functions. The operator is defined for an arbitrary domain element +f(u'), at an arbitrary lethargy limit U, to be: + +.. math:: + :label: eq7-4-28 + + \mathbb{C}(\mathrm{f} ; \mathrm{U})=\int_{\mathrm{u}_{0}}^{\mathrm{U}} \mathrm{f}\left(\mathrm{u}^{\prime}\right) \mathrm{d} \mathrm{u}^{\prime} + +where u\ :sub:`0` is an arbitrary reference point. In implementing this +method in CENTRM, it is convenient to set u\ :sub:`0`\ =u\ :sub:`L`; +i.e., the negative lethargy value corresponding to highest energy of the +transition range. + +The cumulative integral at some lethargy mesh point u\ :sub:`n` is +related to the value at the previous lethargy mesh point u\ :sub:`n−1` +by the expression + +.. math:: + :label: eq7-4-29 + + f\left(\text{f} ; \text{u}_{n}\right) = f\left(\text{f} ; \text{u}_{n-1}\right)+\int_{u_{n-1}}^{u_{n}} f\left(u^{\prime}\right) d u^{\prime} + +where u\ :sub:`n` > u\ :sub:`n−1`. + +Note that only *a single panel of integration* over the interval +[u\ :sub:`n−1`, u\ :sub:`n`] must be performed to update the cumulative +integrals. + +The sub-moment expansion of the scatter source in :eq:`eq7-4-25` can be expressed +in terms of the cumulative integral operator as follows: + +.. math:: + :label: eq7-4-30 + + \mathrm{S}_{\mathrm{k}}^{(\mathrm{j})}(\mathrm{u})=\sum_{j} \sum_{K=-} \mathrm{H}_{, \mathrm{K}}^{(\mathrm{j})}(\mathrm{u}) \times\left[\mathrm{f}\left(\mathrm{F}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{j})} ; u_{H I}^{(m, j)}\right)-\mathrm{f}\left(\mathrm{F}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{j})} ; u_{L O}^{(m, j)}\right)\right] + +For elastic scatter the value of :math:`\mathrm{u}_{\mathrm{LO}}^{(\mathrm{m}, \mathrm{j})}` is equal to (u−ε\ :sup:`(j)`), +and :math:`\mathrm{u}_{\mathrm{HI}}^{(\mathrm{m}, \mathrm{j})}` is equal to u. + +.. _7-4-2-5: + +Multigroup Boltzmann equation +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The MG form of the transport equation used in the UMR and LMR is derived +by integrating :eq:`eq7-4-1`  over the energy intervals defined by the group +structure in the MG library. Details concerning the MG transport +equation, including its solution using the discrete ordinates method, +can be found in the SCALE documentation of XSDRNPM. The CENTRM MG +solution is similar to the XSDRNPM method: however; the outer iteration +loop in CENTRM is limited to the thermal groups, since no eigenvalue +calculation is performed in CENTRM. The MG scatter source in the thermal +range has upscatter contributions that depend on group fluxes from lower +energy groups in the LMR, so that outer iterations are performed over +thermal groups in the LMR until the upscatter portion of the MG scatter +source converges. + +.. _7-4-2-5-1: + +Multigroup data for CENTRM calculation +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +Group cross-section data for the MG calculations are taken from the +input MG library which should include a combined 2D transfer matrix +representing all pertinent scatter reactions (viz, elastic, inelastic, +coherent and incoherent thermal reactions, n-2n, etc). MG cross sections +also should be problem-dependent values. This is done by processing the +data with BONAMI prior to the CENTRM calculation. BONAMI converts the +problem-independent cross-sections into problem-dependent values by +using the Bondarenko factors on the MG library. + +.. _7-4-2-5-2: + +Conversion of multigroup fluxes to pseudo-pointwise values +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +The MG flux solution provides the integrated flux over lethargy, for +each group interval. The average flux within a group is assumed to +approximate the value of the flux per unit lethargy at the midpoint +lethargy of the group; thus a set of “pseudo-pointwise” angular fluxes +and moments can be obtained for the NG\ :sub:`U` and NG\ :sub:`L` mesh +points in the UMR and LMR, respectively. For lethargy point u\ :sub:`n` +, corresponding to the midpoint lethargy of group g contained within the +LMR and UMR, a PW flux value is computed from the expression, + +.. math:: + :label: eq7-4-31 + + \Psi\left(\mathrm{u}_{\mathrm{n}}\right)=\Psi_{\mathrm{g}} / \Delta \mathrm{u}_{\mathrm{g}} + +where Δu\ :sub:`g` is the lethargy width of group g. :eq:`eq7-4-31` provides +PW flux values for lethargy mesh points, + +.. math:: + + \mathrm{u}_{1} \ldots \mathrm{u}_{\mathrm{NGU},} \quad \text { and } \quad \mathrm{u}_{\mathrm{NGU}+\mathrm{NP}+1} \cdots \ldots \mathrm{u}_{\mathrm{NT}} + +A linear variation of the flux per unit lethargy is assumed between +lethargy points to obtain a continuous representation in the UMR and +LMR. + +.. _7-4-2-6: + +The Boltzmann equation within the PW range +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +In contrast to the “pseudo-pointwise” fluxes obtained from the MG +transport calculation, a true PW solution is performed for the +N\ :sub:`P` lethargy points between DEMAX and DEMIN. The PW solution is +performed within a loop over energy groups: i.e., for each of the +NG\ :sub:`P` groups in the PW range there is an additional loop over all +lethargy mesh points contained inside the group. This approach +facilitates coupling of the scatter source from the UMR to the PW range +and from the PW and LMR. + +Evaluating :eq:`eq7-4-1` at each of the N\ :sub:`P` energy mesh-points in the +PW range gives a system of integro-differential equations that can be +solved to obtain the PW flux moments, per lethargy, for the +N\ :sub:`P` energy mesh points in the range DEMAX to DEMIN—which +correspond to the lethargy points, :math:`\mathrm{U}_{\mathrm{NGU}+1}, \ldots \mathrm{U}_{\mathrm{NGU}+\mathrm{NP}}`. +Again linear variation of +the flux between lethargy points is assumed, to obtain a continuous +spectrum. Substituting :eq:`eq7-4-4` into :eq:`eq7-4-1`, the PW transport equation at mesh +point n is found to be, + +.. math:: + :label: eq7-4-32 + + \Omega \bullet \nabla \Psi_{\mathrm{n}}(\mathrm{r}, \Omega)+\Sigma_{\mathrm{t}, \mathrm{n}}(\mathrm{r}) \Psi_{\mathrm{n}}(\mathrm{r}, \Omega)=\sum_{\mathrm{k}} \frac{2+1}{2} \mathrm{Y}_{\mathrm{k}}(\Omega) \quad \mathrm{S}_{\mathrm{k}, \mathrm{n}}(\mathrm{r})+\mathrm{Q}_{\mathrm{n}}(\mathrm{r}, \Omega) + +for :math:`\mathrm{n}=\left(\mathrm{NG}_{\mathrm{U}}+1\right), \ldots .,\left(\mathrm{NG}_{\mathrm{U}}+\mathrm{N}_{\mathrm{P}}\right)` + +where + +.. math:: + + \begin{aligned} + \sum_{\mathrm{t}, \mathrm{n}}(\mathbf{r}) &=\sum_{\mathrm{t}}\left(\mathbf{r}, \mathrm{u}_{\mathrm{n}}\right) \\ + \Psi_{\mathrm{n}}(\mathbf{r}, \Omega) &=\Psi\left(\mathbf{r}, \Omega, \mathrm{u}_{\mathrm{n}}\right) \\ + \mathrm{S}_{\mathrm{k}, \mathrm{n}}(\mathbf{r}) &=\mathrm{S}_{\mathrm{k}}\left(\mathbf{r}, \mathrm{u}_{\mathrm{n}}\right) + \end{aligned} + +Aside from the definition of the cross-section data, the above equation +appears identical in form to the MG transport equation, and can be +solved with virtually the same algorithm as the MG solution, once the +scatter source moments are determined. The same computer routines in +CENTRM calculate both the MG and PW fluxes. However, a major conceptual +difference between the PW and MG transport equations is that the PW +equation describes a differential neutron balance per unit lethargy *at +an energy point*, while the MG equation represents an integral balance +over an interval of lethargy points. Although this type of point +solution is not inherently conservative over the intervals defined by +the energy mesh, the particle balance for each interval has been found +to be very good. It should also be noted that exact particle +conservation is not a strict requirement for this type of application +where flux spectra rather than particle balances are primarily of +interest. + +In the PW range the scatter source is composed of (a) MG-to-PW scatter +from the UMR and possibly upscatter from the LMR if the PW range extends +into thermal, and (b) PW-to-PW scatter from points in the PW range. The +submoment expansion method described previously is used in CENTRM to +provide an efficient method of evaluating the PW-to-PW downscatter +source for the epithermal range, which includes most of the resolved +resonances. + +.. _7-4-2-6-1: + +Scattering sources for the PW range +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +In the case of elastic scatter from nuclide “j,” only the lethargy +interval below u\ :sub:`n` −ε\ :sup:`(j)` can scatter to a lethargy +point u\ :sub:`n` in the PW range. If u\ :sub:`n` −ε\ :sup:`(j)` is +negative, then some portion of the source at u\ :sub:`n` is due to +UMR-to-PW from energies above DEMAX, since zero-lethargy is equal to the +top energy of the PW range. Otherwise, the elastic source is entirely +PW-to-PW. + +For any given nuclide j, the lowest lethargy in the UMR range that +contributes to the elastic scatter source in the PW range is equal to +−ε\ :sup:`(j)`. Let “jL” represent the lightest *non-hydrogen* nuclide +(i.e., having the smallest A value greater than unity) in the system. +The associated fractional energy loss for this material is indicated as +α\ :sub:`L`, so that the highest energy neutron in the UMR range that +can scatter into the PW range from an elastic collision with any +non-hydrogenous moderator will have an energy equal to +DEMAX/α\ :sub:`L`. The corresponding lethargy is equal to be the +negative value −ε\ :sup:`(L)`, or −ln(1/α\ :sub:`L`). The value of +−ε\ :sup:`(L)` is actually adjusted in CENTRM to coincide with the +immediately preceding multigroup boundary, which has a lethargy value +designated as u\ :sub:`L`. The interval of negative lethargy in the UMR +between u\ :sub:`L` and 0 has been defined previously to be transition +range, because the elastic slowing-down source from this interval +provides a transition between the UMR and PW solutions, respectively. +The transition range always contains an integer number of groups, +corresponding to MGTOP to MGHI. The total downscatter source from the +UMR to lethargy u\ :sub:`n` is composed of elastic and inelastic +contributions from the transition range between [u\ :sub:`L`,0]; and +contributions from the “\ *high*\ ” energy range from lethargies below +u\ :sub:`L`. The high contribution comes from inelastic and hydrogen +elastic reactions in the energy interval above the transition range. + +The downscatter source at u\ :sub:`n` in the PW range can thus be +expressed as the sum of three distinct contributions — S\ :sub:`HI`, +S\ :sub:`Tr`, and S\ :sub:`PW` —, that correspond to scatter from the +high region of the UMR, the transition region of the UMR, and the PW +ranges, respectively. The source moments appearing in :eq:`eq7-4-32`  can thus be +expressed as: + +.. math:: + :label: eq7-4-33 + + \begin{aligned} + \mathrm{S}_{\mathrm{k}, \mathrm{n}}(\mathrm{r}) &=\mathrm{S}_{\mathrm{k}, \mathrm{HI}}\left(\mathrm{r}, \mathrm{u}_{\mathrm{n}}\right)+\mathrm{S}_{\mathrm{k}, \mathrm{Tr}}\left(\mathrm{r}, \mathrm{u}_{\mathrm{n}}\right)+\mathrm{S}_{\mathrm{k}, \mathrm{PR}}\left(\mathrm{r}, \mathrm{u}_{\mathrm{n}}\right) \\ + &=\int_{-\infty}^{\mathrm{u}_{\mathrm{L}}} \mathrm{S}_{\mathrm{k}}\left(\mathrm{u}^{\prime} \rightarrow \mathrm{u}\right) \mathrm{d} \mathrm{u}^{\prime}+\int_{\mathrm{u}_{\mathrm{L}}}^{0} \mathrm{S}_{\mathrm{k}}\left(\mathrm{u}^{\prime} \rightarrow \mathrm{u}\right) \mathrm{du}^{\prime}+\int_{0}^{\mathrm{u}_{\mathrm{n}}} \mathrm{S}_{\mathrm{k}}\left(\mathrm{u}^{\prime} \rightarrow \mathrm{u}\right) \mathrm{du}^{\prime} + \end{aligned} + +.. _7-4-2-6-2: + +**Downscatter source from** high **region of the UMR to the PW range (SHI)** +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +The high region of the UMR corresponds to groups 1 through MGTOP-1. The +MG-to-PW scattering source (S\ :sub:`HI`) from high energy region +originates in the energy range above DEMAX/α\ :sub:`L`; i.e., lethargies +below u\ :sub:`L` (see :numref:`fig7-4-2`). In this region, inelastic +reactions may scatter neutrons to the PW range; but due to the +definition of u\ :sub:`L`, the only elastic reactions that scatter to +the PW range are due to hydrogen. Therefore in general, the MG matrices +describing scatter from groups in high region to groups in the PW range +correspond to discrete and continuum inelastic reactions, and elastic +scatter from hydrogen. If g′ is an arbitrary group in the UMR range +above the transition interval and g is a fixed group interval in the +PW range, then the rate that neutrons scatter from all groups g′ in the +high region to all energy points in g, for a given direction Ω, is +obtained from the usual expression for MG-to-MG transfers, and is equal +to + +.. math:: + + \mathrm{S}_{\mathrm{g}}(\mathrm{r}, \Omega)=\sum_{\mathrm{k}} \frac{2+1}{2} \quad \mathrm{Y}_{\mathrm{k}}(\Omega) \mathrm{S}_{\mathrm{k}, \mathrm{g}} + +where MGLO > g > MGHI, and the MG source moments are, + +.. math:: + :label: eq7-4-34 + + \mathrm{S}_{\mathrm{k}, \mathrm{g}}=\sum_{\mathrm{g}^{\prime}=1}^{\text {MGTOP-1 }} \Sigma_{, \mathrm{g}^{\prime} \rightarrow \mathrm{g}} \Psi_{\mathrm{k}, \mathrm{g}^{\prime}} + +while :eq:`eq7-4-34` gives the moments of the overall scatter rate from all groups +in the high range into the *entire* PW group g, it is necessary to +determine how the group source should be distributed over the PW energy +mesh contained within the group; i.e., it is desired to extract the PW +source moments, from the group moments by applying some “intra-group” +distribution H\ :sub:`ℓk,g`\ (E) such that, + +.. math:: + :label: eq7-4-35 + + \mathrm{S}_{\mathrm{k}, \mathrm{HI}}(\mathrm{u})=\mathrm{S}_{\mathrm{k}, \mathrm{g}} \quad \mathrm{H}_{\mathrm{k}, \mathrm{g}}(\mathrm{E}), \quad \text { for } \mathrm{u}(\mathrm{E}) \varepsilon \operatorname{group} \mathrm{g} + +The intra-group distribution has units of “per unit lethargy,” and its +integral over the group is normalized to unity. This form of the scatter +source preserves the MG moments S\ :sub:`ℓk,g`, whenever +S\ :sub:`ℓk,HI`\ (u) is integrated over group g, insuring that the +correct number of neutrons (as determined from the UMR calculation) will +always be transferred from the high range into the PW group. Only the +distribution within the group is approximate. + +Recall that the scatter source of concern here is due only to elastic +scatter from hydrogen and inelastic scatter from all other materials. In +the case of *s*-wave elastic scatter from hydrogen, the P\ :sub:`0` and +P\ :sub:`1` moments per unit lethargy, respectively, can be rigorously +expressed in the form of :eq:`eq7-4-35` with + +.. math:: + :label: eq7-4-36 + + \begin{array}{lllll} + \mathrm{H}_{0} & \propto \mathrm{E} & , & \text { and } & \mathrm{H}_{1} & \propto \mathrm{E}^{3 / 2} + \end{array} + +These expressions can be inferred directly from the moments of the +scatter kernel in :eq:`eq7-4-16` . The higher order scatter moments for hydrogen +have a somewhat more complicated form containing sums of energy +functions; but since these moments are usually less important than the +first two moments, a less rigorous treatment of their intra-group +distribution is used. The intra-group distribution due to inelastic +scatter depends on the Q values for the individual levels, and these are +not available on the multigroup libraries. Fortunately, the scatter +source in the PW range is not very sensitive to the assumed intra-group +distribution for inelastic scatter, as long as the total inelastic +source for the group is computed correctly. As a reasonable trade-off +between rigor and complexity, the high energy component of the UMR-to-PW +scatter source is approximated using H\ :sub:`0` for the intra-group +distribution of all P\ :sub:`0` moments, and H\ :sub:`1` for all higher +order moments. This approximation produces the correct intra-group +variation for the lowest two moments of the hydrogen scatter source, but +the higher order moments of hydrogen and the inelastic scatter source +are not distributed exactly throughout the group. However, the +integrated source moments are correct in all cases. Again, it should be +emphasized that the approximations discussed here only apply to the +UMR-to-PW component designated as S\ :sub:`HI`, which comes from +reactions above the transition range (energies above +E\ :sub:`HI`/α\ :sub:`L`). This is often a small contribution to the +overall PW source term. + +.. _7-4-2-6-3: + +**Scattering sources from UMR** transition **region and epithermal PW range** +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +Most coupling between the UMR and the PW range is due usually to elastic +scatter from energies immediately above DEMAX. The contribution to the +PW source due to downscatter source from this transition range has been +designated S\ :sub:`Tr`\ (u:sub:`n`). The other component of the PW +source, S\ :sub:`PW`\ (u\ :sub:`n`), accounts for the scattering source +coming from all lethargies lower than u\ :sub:`n` in the PW range. It is +convenient to combine the two sources together as the PW epithermal +source called “S\ :sub:`Ep`,” which has an lk\ :sub:`th` moment given by +:eq:`eq7-4-22`, + +.. math:: + :label: eq7-4-37 + + \begin{aligned} + S_{k, E p} &=\int_{\mathrm{u}_{\mathrm{L}}}^{\mathrm{u}_{\mathrm{n}}} \mathrm{S}_{\mathrm{k}}\left(\mathrm{r}, \mathrm{u}^{\prime} \rightarrow \mathrm{u}\right) \mathrm{du}^{\prime} \\ + &=\sum_{\mathrm{j}} \sum_{\mathrm{K}=-} \mathrm{Z}_{, \mathrm{k}}^{(\mathrm{j})} \mathrm{h}_{\mathrm{K}}(\mathrm{E}) \int_{\mathrm{u}_{\mathrm{n}}-\varepsilon^{(\mathrm{j})}}^{\mathrm{u}_{\mathrm{k}}} \Psi_{\mathrm{k}}\left(\mathrm{u}^{\prime}\right) \Sigma^{(\mathrm{j})}\left(\mathrm{u}^{\prime}\right) \quad \mathrm{h}_{\mathrm{K}}^{-1}\left(\mathrm{E}^{\prime}\right) \mathrm{du}^{\prime} + \end{aligned} + +This is done because CENTRM uses the submoment expansion technique to +compute both the PW-to-PW epithermal source from the PW range as well as +the MG-to-PW source from the transition range of the UMR. Note that +elastic scattering from the transition range only impacts the PW scatter +source at the initial mesh points in the PW range; i.e., those contained +in the interval 0 < u\ :sub:`n` <ε:sup:`(j)`, for nuclide j. Beyond +these mesh points the elastic scatter source is due only to PW-to-PW +scatter, as illustrated in :numref:`fig7-4-2`. + +The epithermal elastic source at u\ :sub:`n`, coming from the range +u\ :sub:`L` to u\ :sub:`n`, is expressed as an integral over the +immediately preceding lethargy mesh interval from u\ :sub:`n−1` to +u\ :sub:`n` plus the integral over the remaining lethargy interval, as +illustrated in :numref:`fig7-4-3`. The former integral is designated as +I(u\ :sub:`n−1`,u\ :sub:`n`) and the latter as +I(u\ :sub:`L`,u\ :sub:`n−1`), so that + +.. math:: + + \mathrm{S}_{\mathrm{k}, \mathrm{E}}\left(\mathrm{u}_{\mathrm{n}}\right)=\mathrm{I}\left(\mathrm{u}_{\mathrm{n}-1}, \mathrm{u}_{\mathrm{n}}\right)+\mathrm{I}\left(\mathrm{u}_{\mathrm{L}}, \mathrm{u}_{\mathrm{n}-1}\right) + +.. _fig7-4-3: +.. figure:: figs/CENTRM/fig3.png + :align: center + :width: 500 + + Definition of cumulative integral elements. + +The lethargy mesh in CENTRM is constrained such that the maximum +lethargy gain in an elastic reaction (ε\ :sup:`(j)`) is always greater +than the maximum mesh interval size, which insures that +I(u\ :sub:`n−1`,u\ :sub:`n`) always includes the full panel from +u\ :sub:`n−1` to u\ :sub:`n`. In the above and subsequent equations the +explicit dependence of S\ :sub:`Ep` on independent variables other than +lethargy is not shown for notational convenience. The integral +I(u\ :sub:`n−1`,u\ :sub:`n`) is evaluated approximately by applying the +trapezoidal rule, which leads to, + +.. math:: + :label: eq7-4-38 + + \mathrm{I}\left(\mathrm{u}_{\mathrm{n}}, \mathrm{u}_{\mathrm{n}-1}\right)=\int_{\mathrm{u}_{\mathrm{n}-1}}^{\mathrm{u}_{\mathrm{n}}} \mathrm{S}_{\mathrm{k}}\left(\mathrm{u}^{\prime} \rightarrow \mathrm{u}_{\mathrm{n}}\right) \mathrm{du}^{\prime} \sim \frac{\left[\mathrm{S}_{\mathrm{k}}\left(\mathrm{u}_{\mathrm{n}} \rightarrow \mathrm{u}_{\mathrm{n}}\right)+\mathrm{S}_{\mathrm{k}}\left(\mathrm{u}_{\mathrm{n}-1} \rightarrow \mathrm{u}_{\mathrm{n}}\right)\right]}{2} \times\left(\mathrm{u}_{\mathrm{n}}-\mathrm{u}_{\mathrm{n}-1}\right) + +Using the submoment expansion from :eq:`eq7-4-25`, :eq:`eq7-4-38` can be written for elastic scatter as + +.. math:: + :label: eq7-4-39 + + \mathrm{I}\left(\mathrm{u}_{\mathrm{n}-1}, \mathrm{u}_{\mathrm{n}}\right)=\Sigma_{\mathrm{n} \rightarrow \mathrm{n}} \Psi_{\mathrm{k}, \mathrm{n}}+\sum_{\mathrm{K}} Z_{\mathrm{K}}^{(\mathrm{j})} \mathrm{h}_{\mathrm{K}}\left(\mathrm{E}_{\mathrm{n}}\right) \mathrm{h}_{\mathrm{K}}^{-1}\left(\mathrm{E}_{\mathrm{n}-1}\right) \Sigma^{(\mathrm{j})}\left(\mathrm{u}_{\mathrm{n}-1}\right) \Psi_{\mathrm{k}, \mathrm{n}-1} \frac{\Delta \mathrm{u}_{\mathrm{n}-1}}{2} . + +The first term on the right side of :eq:`eq7-4-39` corresponds to the +“within-point” component of elastic scatter from u\ :sub:`n` to +u\ :sub:`n`, which only occurs for straight ahead scatter +(μ\ :sub:`0`\ =1). The within-point cross section is defined as, + +.. math:: + :label: eq7-4-40 + + \Sigma_{\mathrm{n} \rightarrow \mathrm{n}}=\frac{\Delta \mathrm{u}_{\mathrm{n}-1}}{2} \sum_{\mathrm{j}} \frac{\Sigma_{\mathrm{n}}^{(\mathrm{j})}}{\left(1-\alpha^{(\mathrm{j})}\right)} . + +In deriving this term the following relation has been used for each nuclide: + +.. math:: + :label: eq7-4-41 + + \sum_{\mathrm{K}} \mathrm{Z}_{\mathrm{K}}=\frac{1}{1-\alpha} . + +The I(u\ :sub:`L`,u\ :sub:`n−1`) portion of the integral in :eq:`eq7-4-37` is +equal to + +.. math:: + :label: eq7-4-42 + + \mathrm{I}\left(\mathrm{u}_{\mathrm{L}}, \mathrm{u}_{\mathrm{n}-1}\right)=\sum_{j} \sum_{K=-} \mathrm{Z}_{, \mathrm{K}}^{(\mathrm{j})} \mathrm{h}_{\mathrm{K}}\left(\mathrm{E}_{\mathrm{n}}\right) \int_{\mathrm{U}_{\mathrm{n}}-\varepsilon^{(j)}}^{\mathrm{u}_{\mathrm{n}-1}} \Psi_{\mathrm{k}}\left(\mathrm{u}^{\prime}\right) \Sigma^{(\mathrm{j})}\left(\mathrm{u}^{\prime}\right) \quad \mathrm{h}_{\mathrm{K}}^{-1}\left(\mathrm{E}^{\prime}\right) \mathrm{d} \mathrm{u}^{\prime} . + +Note that the lower lethargy limit of the integral is restricted to +u\ :sub:`n` − ε\ :sup:`(j)`, since this is the maximum limit of lethargy +that can scatter to u\ :sub:`n` in an elastic reaction. In terms of the +cumulative integral operator, the integral in :eq:`eq7-4-42`  over the interval +[u\ :sub:`n` − ε\ :sup:`(j)`, u\ :sub:`n−1`] is equal to + +.. math:: + :label: eq7-4-43 + + \int_{\mathrm{u}_{\mathrm{n}}-\varepsilon(\mathrm{j})}^{\mathrm{u}_{\mathrm{n}}-1} \Psi_{\mathrm{k}}\left(\mathrm{u}^{\prime}\right) \Sigma^{(\mathrm{j})}\left(\mathrm{u}^{\prime}\right) \mathrm{h}_{\mathrm{K}}\left(\mathrm{E}^{\prime}\right)^{-1} \mathrm{du}^{\prime}=\left[\mathrm{f}\left(\mathrm{F}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{j})} ; \mathrm{u}_{\mathrm{n}-1}\right)-\mathrm{f}\left(\mathrm{F}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{j})} ; \mathrm{u}_{\mathrm{n}}-\varepsilon^{(\mathrm{j})}\right)\right] + +.. check this one. + +where F\ :sub:`ℓk,K` has been defined in :eq:`eq7-4-26`. In order to evaluate :eq:`eq7-4-43`  +it is necessary to determine the cumulative integral values at +u\ :sub:`n−1` and u\ :sub:`n` − ε\ :sup:`(j)`. The lethargy u\ :sub:`n−1` +will always correspond to a mesh point value, but in general +u\ :sub:`n` − ε\ :sup:`(j)` can fall between mesh points. Evaluation of +the cumulative integrals at an arbitrary limit such as +u\ :sub:`n` − ε\ :sup:`(j)` is performed in CENTRM by interpolation of +previously calculated values stored for all the mesh points below +u\ :sub:`n` during the transport calculation at lower lethargies. The +interpolated value of the cumulative integral at +u\ :sub:`n` − ε\ :sup:`(j)` that is subtracted in :eq:`eq7-4-43` is called the +“\ *excess integral*\ ” in CENTRM. At each lethargy point, excess +integrals must be found as a function of space, nuclide, moment, and +submoment. Also note that for some initial mesh points +(i.e., u\ :sub:`n` < ε\ :sup:`(j)`) the value u\ :sub:`n` − ε\ :sup:`(j)` can +be negative, indicating that a portion of the PW scatter source at +u\ :sub:`n` is due to elastic scattering from the negative lethargy +range above DEMAX. This means that cumulative integrals must be known +for mesh intervals in the transition as well as in the PW range. Values +of the cumulative integrals at all points within the transition range +are first computed from the results from the UMR calculation, prior to +the PW transport calculation (but after the UMR calculation). Additional +cumulative integrals are then calculated successively during the +PW transport solution at all mesh points and are stored as the +calculation proceeds from low to high lethargy. Thus in evaluating +S\ :sub:`ℓk,Ep`\ (u\ :sub:`n`), the cumulative integrals at every space +interval already will have been stored at all energy points up to (n−1), +in an array called CUM\ :sup:`(j)`\ :sub:`ℓk,K`, for each nuclide j, +moment ℓk, and submoment K: + +.. math:: + :label: eq7-4-44 + + \mathrm{CUM}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{j})}=\left\{\mathrm{f}\left(\mathrm{F}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{j})} ; \mathrm{u}_{\mathrm{n}^{\prime}}\right), \quad \mathrm{n}^{\prime}=1, \mathrm{n}-1\right\} + +so that the excess integral values can be interpolated from the above +array. The first N\ :sub:`Tr` elements of the array +CUM\ :sup:`(j)`\ :sub:`ℓk,K` correspond to lethargy points in the +transition range, and the remainder are in the PW range, where + + +-----------------------+-----------------------+-----------------------+ + | N\ :sub:`Tr` | = | G\ :sub:`U` − | + | | | g\ :sub:`Tr` + 1; | + +=======================+=======================+=======================+ + | g\ :sub:`Tr` | = | MGTOP, the highest | + | | | energy group in the | + | | | transition range; | + | | | (i.e., the group | + | | | whose high energy | + | | | boundary corresponds | + | | | to u\ :sub:`L`); | + +-----------------------+-----------------------+-----------------------+ + | G\ :sub:`U` | = | Lowest energy group | + | | | in the transition | + | | | range. | + +-----------------------+-----------------------+-----------------------+ + +Elastic cumulative integrals contained in array +CUM\ :sup:`(j)`\ :sub:`ℓk,K` are calculated at each lethargy point +u\ :sub:`n` with the expression: + +.. math:: + :label: eq7-4-45 + + \begin{array}{l} + f\left(\mathrm{F}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{j})} ; \mathrm{u}_{\mathrm{n}}\right)=\mathrm{f}\left(\mathrm{F}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{j})} ; \mathrm{u}_{\mathrm{n}-1}\right)+\int_{\mathrm{u}_{\mathrm{n}-1}}^{\mathrm{u}_{\mathrm{n}}} \mathrm{F}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{j})}\left(\mathrm{u}^{\prime}\right) \mathrm{d} u^{\prime} \\ + =\mathrm{f}\left(\mathrm{F}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{j})} ; \mathrm{u}_{\mathrm{n}-1}\right)+\int_{\mathrm{u}_{\mathrm{n}}-1}^{\mathrm{u}_{\mathrm{n}}} \Psi_{\mathrm{k}}\left(\mathrm{u}^{\prime}\right) \Sigma^{(\mathrm{j})}\left(\mathrm{u}^{\prime}\right) \quad \mathrm{h}_{\mathrm{K}}\left(\mathrm{E}^{\prime}\right)^{-1} \mathrm{du}^{\prime} + \end{array} + +After completing the calculation of PW angular fluxes and moments at +u\ :sub:`n` the integral over the most current lethargy panel +[u\ :sub:`n−1`,u\ :sub:`n`] is evaluated with the trapezoidal +approximation, resulting in an updated cumulative integral array +containing the value at lethargy u\ :sub:`n`: + +.. math:: + :label: eq7-4-46 + + \mathrm{f}\left(\mathrm{F}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{j})} ; \mathrm{u}_{\mathrm{n}}\right) ; \quad \mathrm{f}\left(\mathrm{F}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{j})} ; \mathrm{u}_{\mathrm{n}-1}\right)+\Delta \mathrm{u}_{\mathrm{n}-1} \frac{\left[\mathrm{h}_{\mathrm{K}}^{-1}\left(\mathrm{E}_{\mathrm{n}-1}\right) \Sigma_{\mathrm{n}-1}^{(\mathrm{j})} \Psi_{\mathrm{k}, \mathrm{n}-1}+\mathrm{h}_{\mathrm{K}}^{-1}\left(\mathrm{E}_{\mathrm{n}}\right) \Sigma_{\mathrm{n}}^{(\mathrm{j})} \Psi_{\mathrm{k}, \mathrm{n}}\right]}{2} , + + +where the cumulative integrals at the preceding mesh point are known +from the previous calculation, and the flux moments +Ψ\ :sub:`ℓk`\ (u\ :sub:`n`) are determined from the transport calculation +at the current lethargy point. Only a single panel of integration is +required to update the cumulative integrals, significantly reducing the +amount of computation compared to recomputing the entire summation again +at each new energy point. The integration is performed rapidly with the +trapezoidal approximation, which should be accurate since the energy +mesh is defined to reproduce the macroscopic cross sections linearly +between mesh points. In order to avoid loss of numerical significance, +the set of stored cumulative integrals is periodically “renormalized,” +by translating to a new reference lethargy point (recall that only the +*differences* of cumulative integrals is needed). + +Elastic cumulative integrals for the transition range are calculated +with a slightly different expression, using MG flux moments obtained in +the UMR calculation. Because the transition interval is part of the UMR, +it is convenient to evaluate cumulative integrals at lethargy values +corresponding to group boundaries. This requires approximating the +energy distribution of the flux spectrum within each group in the +transition range. To evaluate the cumulative interval in the transition +range of some nuclide j, the scalar flux per energy (at a given space +location) within a transition group is approximated as: +Φ(E) = M\ :sup:`(j)`/E, where M\ :sup:`(j)` is a normalization constant +defined so that the MG outscatter rate (i.e., slowing-down density) from +the group is preserved. It can be shown that this normalization +condition requires that + +.. math:: + :label: eq7-4-47 + + \mathbf{M}^{(j)}=\frac{\left[\Sigma_{t, g^{\prime}}^{(j)}-\Sigma_{a, g^{\prime}}^{(j)}-\Sigma_{g^{\prime}}^{(j)},\right] \Delta u_{g^{\prime}}}{\xi^{(j)} \Sigma_{s, g^{\prime}}^{(j)}} \times\left[\phi_{g^{\prime}} / \Delta u_{g^{\prime}}\right] + +where ξ is the average lethargy gain in an elastic reaction and +Σ\ :sub:`g′g′`, is the within-group MG scatter cross section. Thus the +scalar flux per unit lethargy used to evaluate cumulative integrals of +nuclide j is: + +.. math:: + + \phi\left(u^{\prime}\right)=M^{(j)} ; \quad \text { for } u^{\prime} \varepsilon g^{\prime}, \text { and } g^{\prime} \varepsilon \text { Transition region of } U M R + +Within-group energy spectra for the higher order flux-moments could be +approximated in similar manner by preserving the higher order Legendre +moments of the slowing-down density, but CENTRM simply uses the same +form in :eq:`eq7-4-47` for all flux moments, so that in general the within-group +energy distribution for any ℓk\ :sub:`th` moment in the transition range +is approximated as, + +.. math:: + :label: eq7-4-48 + + \Psi_{k}\left(u^{\prime}\right)=\frac{\left[\sum_{t, g^{\prime}}^{(j)}-\Sigma_{a, g^{\prime}}^{(j)}-\Sigma_{g^{\prime}}^{(j)}\right] \Delta u_{g^{\prime}}}{\xi^{(j)} \sum_{s, g^{\prime}}^{(j)}} \times\left[\Psi_{k, g^{\prime}} / \Delta u_{g^{\prime}}\right] + +for u′εg′, and g′ε transition region of UMR. Therefore the following +integrals can be evaluated: + +.. math:: + :label: eq7-4-49 + + \int_{\mathrm{ug}^{\prime}}^{\mathrm{u}_{\mathrm{g}^{\prime}+1}} \mathrm{h}_{\mathrm{K}}^{-1}\left(\mathrm{E}^{\prime}\right) \Sigma^{(\mathrm{j})}\left(\mathrm{u}^{\prime}\right) \Psi_{\mathrm{k}}\left(\mathrm{u}^{\prime}\right) \mathrm{du}^{\prime}=\frac{\Sigma_{\mathrm{r}, \mathrm{g}^{\prime}}^{(\mathrm{j})} \Delta \mathrm{u}_{\mathrm{g}^{\prime}}}{\xi^{(\mathrm{j})}} \frac{\Psi_{\mathrm{k}, \mathrm{g}^{\prime}}}{\Delta \mathrm{u}_{\mathrm{g}^{\prime}}} \int_{\mathrm{ug}^{\prime}}^{\mathrm{u}_{\mathrm{g}^{\prime}+1}} \mathrm{h}_{\mathrm{K}}^{-1}\left(\mathrm{E}^{\prime}\right) \mathrm{du}^{\prime} . + +Integration of the h\ :sub:`k`\ :sup:`−1` function is performed +analytically to give the cumulative integral at any group boundary +u\ :sub:`g` in the transition range: + +.. math:: + :label: eq7-4-50 + + \begin{array}{l} + f\left(\mathrm{F}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{j})} ; \mathrm{u}_{\mathrm{g}}\right) ; \sum_{\mathrm{g}^{\prime}=\mathrm{g}}^{\mathrm{G}_{\mathrm{U}}} \frac{\Sigma_{\mathrm{r}, \mathrm{g}^{\prime}}^{(\mathrm{j})}{\xi^{(\mathrm{j})}} \mathrm{g}_{\mathrm{g}^{\prime}}}{\frac{\Psi_{\mathrm{k}, \mathrm{g}^{\prime}}}{\Delta \mathrm{u}_{\mathrm{g}^{\prime}}} \times\left[\frac{2}{\mathrm{~K}+2}\right]}\left[\mathrm{h}_{\mathrm{K}}^{-1}\left(\mathrm{E}_{\mathrm{g}^{\prime}+1}\right)-\mathrm{h}_{\mathrm{K}}^{-1}\left(\mathrm{E}_{\mathrm{g}^{\prime}}\right)\right] \\ + \mathrm{g}=\mathrm{g}_{\mathrm{Tr}}, \quad \mathrm{g}_{\mathrm{Tr}+1}, \quad \mathrm{G}_{\mathrm{U}} + \end{array} + +:eq:`eq7-4-50` is used to obtain the initial N\ :sub:`Tr` values of the +cumulative integrals, corresponding to the transition range. If the +lower limit of the integral in :eq:`eq7-4-43` is negative, then the cumulative +integral at u\ :sub:`n` − ε\ :sup:`(j)` is interpolated from among the set +of N\ :sub:`Tr` tabulated values generated by :eq:`eq7-4-50`; otherwise it is +interpolated from the values that were computed with :eq:`eq7-4-46`. The following +algorithm is used to interpolate cumulative integrals for negative +lethargy arguments (i.e., in the transition range ): + +.. math:: + :label: eq7-4-51 + + \begin{array}{l} + f\left(\mathrm{F}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{j})} ; \mathrm{u}\right)=\mathrm{f}\left(\mathrm{F}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{j})} ; \mathrm{u}_{\mathrm{g}}\right)+\frac{\mathrm{h}_{\mathrm{K}}^{-1}(\mathrm{E})-\mathrm{h}_{\mathrm{K}}^{-1}\left(\mathrm{E}_{\mathrm{g}}\right)}{\mathrm{h}_{\mathrm{K}}^{-1}\left(\mathrm{E}_{\mathrm{g}+1}\right)-\mathrm{h}_{\mathrm{K}}^{-1}\left(\mathrm{E}_{\mathrm{g}}\right)} \\ + \times\left[\mathrm{f}\left(F_{k, K}^{(j)} ; u_{g+1}\right)-\mathrm{f}\left(F_{k, K}^{(j)} ; u_{g}\right)\right] + \end{array} + +.or u(E) ε g; and g ε transition range of UMR. + +Because the energy mesh in the PW range is very fine, simple linear +interpolation of the cumulative integrals is used for positive lethargy +arguments. + +The complete epithermal elastic scatter source S(r,Ω,u\ :sub:`n`) +appearing in :eq:`eq7-4-32` at any mesh point u\ :sub:`n` corresponds to a +spherical harmonic expansion using the previously derived moments of +S\ :sub:`HI` and S\ :sub:`Ep`. This angular scatter source is equal +to,\ + +.. math:: + :label: eq7-4-52 + + \begin{array}{l} + \mathrm{S}\left(\mathbf{r}, \Omega, \mathrm{u}_{\mathrm{n}}\right)=\Sigma_{\mathrm{n} \rightarrow \mathrm{n}} \Psi_{\mathrm{n}}(\mathbf{r}, \Omega) \\ + +\sum_{\mathrm{k}} \frac{2+1}{2} \mathrm{Y}_{\mathrm{k}}(\Omega)\left\{\mathrm{H}\left(\mathrm{E}_{\mathrm{n}}\right) \sum_{\mathrm{g}^{\prime}=1}^{\mathrm{g}_{\mathrm{Tr}}-1} \sum_{, \mathrm{g}^{\prime} \rightarrow \mathrm{g}} \psi_{\mathrm{k}, \mathrm{g}^{\prime}}\right. \\ + \left.+\sum_{j} \sum_{\mathrm{K}} Z_{\mathrm{K}}^{(\mathrm{j})} \mathrm{h}_{\mathrm{K}}\left(\mathrm{E}_{\mathrm{n}}\right)\left[0.5 \Delta \mathrm{u}_{\mathrm{n}-1} \mathrm{F}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{j})}\left(\mathrm{u}_{\mathrm{n}-1}\right)+\mathrm{f}\left(\mathrm{F}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{j})} ; \mathrm{u}_{\mathrm{n}-1}\right)-\mathrm{f}\left(\mathrm{F}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{j})} ; \mathrm{u}_{\mathrm{n}}-\varepsilon^{(\mathrm{j})}\right)\right]\right\} + \end{array} + +.. check equation + +The above expression was written explicitly for the case of elastic +scatter; however, the discrete level inelastic PW source can be +incorporated with little modification. The only changes are that +additional cumulative integral terms corresponding to each inelastic +level will appear in :eq:`eq7-4-52`; the cumulative integrals for the inelastic +levels must be computed by integrating the more general expression in +:eq:`eq7-4-26`; and the lethargy arguments for the inelastic cumulative integrals +are the generalized lethargy limits u\ :sub:`LO` and u\ :sub:`HI` +defined in :ref:`7-4-2-4` and :cite:`williams_submoment_2000`. + +Note that :eq:`eq7-4-52` contains the term Σ\ :sub:`n→n` Ψ\ :sub:`n`\ (r,Ω) which +can be subtracted from both sides of the transport equation in Eq.  to +give a slightly altered form of the PW transport equation that contains +a modified scatter source and a modified total cross section. The +modified source component is identical to the expression in :eq:`eq7-4-52` with +the within-point term Σ\ :sub:`n→n` Ψ\ :sub:`n`\ (r,Ω) removed. The +modified total cross section, represented by :math:`\Sigma_{\mathrm{t}, \mathrm{n}}` has the +appearance of a “transport‑corrected” cross section given below: + +.. math:: + :label: eq7-4-53 + + \Sigma_{\mathrm{t}, \mathrm{n}}=\Sigma_{\mathrm{t}, \mathrm{n}}-\Sigma_{\mathrm{n} \rightarrow \mathrm{n}} + +An interesting and significant consequence of this operation is that the +right side of Eq.  no longer contains the unknown flux +Ψ\ :sub:`n`\ (r,Ω) since the within-point term is eliminated. The +resulting modified transport equation has the same form as a purely +absorbing medium with a known source term; and thus can be solved +without requiring scatter-source iterations in the epithermal range. +However, iterations may still be required for cell cases with two +reflected or albedo boundary conditions. + +.. _7-4-2-6-4: + +PW thermal scatter source +^^^^^^^^^^^^^^^^^^^^^^^^^ + +There are significant differences in the CENTRM epithermal and thermal +PW transport solutions. In the epithermal range neutrons can only lose +energy in scattering reactions, so that a single sweep from high to low +energy (i.e., low to high lethargy) is required in the solution. On the +other hand, since low energy neutrons may gain as well as lose energy in +scattering reactions, outer iterations are required to converge the +thermal scattering source. Furthermore, the PW scatter kernels +Σ\ :sub:`ℓ`\ (u′→u) in the epithermal range represent two-body +interactions (such as elastic and discrete-level inelastic reactions) +between a neutron and a stationary nucleus. The simple kinematic +relations for these cases allow the efficient sub-moment expansion +method to be utilized in computing scattering source moments. Thermal +scattering reactions are not two body reactions, but rather represent an +effective average over the molecular velocity distribution; thus, there +is no simple kinematic relationship between neutron energy loss and the +angle of scatter relative to its initial direction. In solving the +transport equation for thermal neutrons, the scatter source at lethargy +u\ :sub:`n` is approximated as a summation over the “N” mesh points in +the thermal range, + +.. math:: + :label: eq7-4-54 + + \int_{\text {thermal }} \Sigma^{(\mathrm{j})}\left(\mathrm{u}^{\prime} \rightarrow \mathrm{u}_{\mathrm{n}}\right) \Psi_{\mathrm{k}}\left(\mathrm{u}^{\prime}\right) \mathrm{d} \mathrm{u}^{\prime}=\sum_{\mathrm{m}=1}^{\mathrm{N}} \mathrm{W}_{\mathrm{m}} \Sigma^{(j)}\left(\mathrm{u}_{\mathrm{m}} \rightarrow \mathrm{u}_{\mathrm{n}}\right) \Psi_{\mathrm{k}}\left(\mathrm{u}_{\mathrm{m}}\right) + +where + + m = 1 is the thermal/epithermal boundary point; + + m = N is the lowest thermal energy point; and + + W\ :sub:`m` are standard quadrature weights for trapezoidal integration + with N-1 lethargy panels: + +.. math:: + + \begin{aligned} + \mathrm{W}_{\mathrm{m}}=& 0.5 \times\left(\Delta \mathrm{u}_{\mathrm{m}}+\Delta \mathrm{u}_{\mathrm{m}+1}\right) \quad ; \quad \text { for } \mathrm{m}=2,3, \ldots \mathrm{N}-1 \\ + & 0.5 \times \Delta \mathrm{u}_{\mathrm{m}} \quad ; \quad \text { for } \mathrm{m}=1 \quad \text { or } \quad \mathrm{N} + \end{aligned} + +Point-to-point cross-section moments in the thermal range are computed +from the free-gas or bound kernels evaluated at the desired initial +(u\ :sub:`m`) and final (u\ :sub:`n`) lethargy mesh points. For a given +outer iteration, the summation in :eq:`eq7-4-54` is evaluated using the most +recently computed flux moments. In many instances the main purpose of +the CENTRM calculation will be to obtain a PW spectrum for resonance +self-shielding calculations. In these cases the thermal flux does not +have to be converged very tightly to obtain a reasonable thermal +spectrum for self-shielding low energy resonances, so that only a few +outer iterations are typically employed. + +An additional complication in the thermal calculation is that inner +iterations are necessary to converge the “within-point” (no energy loss) +contribution of the thermal scattering source, due to the presence of +PW flux moments at lethargy point m = n. No inner iterations are +required to converge the within-point elastic scatter term in the +epithermal PW calculation because there can be no change in the neutron +direction if there is no energy loss, unlike the thermal range. + +A space-dependent rebalance calculation for the entire thermal energy +band is performed between outer iterations in order to speed up +convergence of the solution. Reaction rates and leakage values appearing +in the thermal-band rebalance equation are obtained by integrating +PW values over the thermal range. Other acceleration techniques, such as +over-relaxation, extrapolation, and renormalization, are also employed. + +.. _7-4-2-6-5: + +Downscatter source from the epithermal PW range to the LMR +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +MG transport calculations performed in the energy range below DEMIN, +which includes the thermal energy range, are coupled to the epithermal +PW range transport calculations by the slowing down source. The +epithermal PW-to-LMR scatter source represents the contribution to the +multigroup source in some fixed group g contained in the LMR, from +scatter reactions in the epithermal range above DEMIN. The lethargy +value corresponding to the energy DEMIN (i.e., the bottom energy of the +PW range) will be indicated as u\ :sub:`PW`, thus +u\ :sub:`PW` = ln(DEMAX/DEMIN); while the lethargy corresponding to the +thermal energy boundary will be designated as u\ :sub:`TH`. The cut-off +lethargy for the epithermal PW range will correspond to: +u\ :sub:`cut` = min(u\ :sub:`PW`,u\ :sub:`TH`). If there is no PW thermal +calculation in CENTRM, then u\ :sub:`cut` = u\ :sub:`PW`; otherwise, +u\ :sub:`cut` = u:sub:`TH`. For a given nuclide j, the lowest lethargy +in the epithermal PW range from which a neutron can scatter elastically +into the LMR is equal to (u:sub:`cut` − ε\ :sup:`(j)`). If the value of +(u − ε\ :sup:`(j)`) is greater than u\ :sub:`cut`, then an elastic +collision with nuclide j cannot moderate an epithermal neutron from the +PW range to u. Therefore in general for a given material zone, only a +limited number of nuclides (possibly none) and a limited portion of the +epithermal PW energy range may be able to scatter neutrons elastically +to any particular group in the LMR. Utilizing the elastic scatter kernel +and applying a sub-moment expansion to the resulting expression, the +source moment describing scatter from the PW epithermal range to a +lethargy u in the LMR is found to be + +.. math:: + + \mathrm{S}_{\mathrm{k}}^{(\mathrm{j})}(\mathrm{u})=\sum_{\mathrm{K}} Z_{\mathrm{K}}^{(\mathrm{j})} \quad \mathrm{h}_{\mathrm{K}}(\mathrm{E}) \int_{\mathrm{U}-\varepsilon^{-6}}^{\mathrm{u}_{\mathrm{Cut}}} \mathrm{F}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{j})}\left(\mathrm{u}^{\prime}\right) \mathrm{d} \mathrm{u}^{\prime} + +where u in group g; and g ε LMR. + +The integral in the above expression can be evaluated from cumulative +integrals stored during the epithermal PW transport calculation. Thus +the source moment per unit lethargy at u in the LMR range, due to +epithermal scattering from nuclide j, can be written as, + +.. math:: + :label: eq7-4-55 + + \mathrm{S}_{\mathrm{k}}^{(\mathrm{j})}(\mathrm{u})=\sum_{\mathrm{K}} \mathrm{Z}_{\mathrm{K}}^{(\mathrm{j})} \quad \mathrm{h}_{\mathrm{K}}(\mathrm{E}) \quad\left[\mathrm{f}\left(\mathrm{F}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{j})} ; \mathrm{u}_{\mathrm{cut}}\right)-\mathrm{f}\left(\mathrm{F}_{\mathrm{k}, \mathrm{K}}^{(\mathrm{j})} ; \mathrm{u}-\varepsilon^{(\mathrm{j})}\right)\right] , + +for uε group g and u−ε(j) < u\ :sub:`PW`. + +The source per unit lethargy in Eq.  is integrated over the “sink group” +g in the LMR to determine the desired MG scatter source moment due to +reactions in the epithermal PW range. The actual integral over group g +is performed numerically by introducing a three-point (two panel) +integration mesh within the group, as follows: + +.. math:: + + \begin{aligned} + \mathrm{u}_{\mathrm{I}} &=\text { initial integration point in group } \mathrm{g}=\text { lethargy at top energy of group } \mathrm{g}=\mathrm{u}_{\mathrm{g}} \\ + \mathrm{u}_{\mathrm{F}}^{(\mathrm{j})} &=\text { final integration point in group } \mathrm{g} \\ + &=\operatorname{MIN}\left\{\mathrm{u}_{\mathrm{g}+1} ; \mathrm{u}_{\mathrm{cut}}+\varepsilon^{(j)}\right\}, \quad \text { where } \mathrm{u}_{\mathrm{g}+1}=\text { lethargy at bottom of energy of group } \mathrm{g} \\ + \mathrm{u}_{\mathrm{A}}^{(\mathrm{j})} &=\text { middle integration point in group } \mathrm{g}=0.5\left(\mathrm{u}_{\mathrm{F}}^{(\mathrm{j})}+\mathrm{u}_{\mathrm{I}}\right) + \end{aligned} + +Note that the final and middle points of integration +(i.e., u\ :sub:`F`\ :sup:`(j)` and u:sub:`A`\ :sup:`(j)`) may be nuclide +dependent; and if (u\ :sub:`I` − ε\ :sup:`(j)`) > u\ :sub:`cut`, then +nuclide j does not contribute to the pointwise-to-LMR scatter source in +g. Applying the two-panel Simpson’s approximation for integration over +group g results in + +.. math:: + :label: eq7-4-56 + + \mathrm{S}_{\mathrm{k}, \mathrm{g}}^{(\mathrm{)}}=\Delta^{(\mathrm{j})} / 3\left[\mathrm{S}_{\mathrm{k}}^{(\mathrm{j})}\left(\mathrm{u}_{\mathrm{I}}\right)+4 \mathrm{S}_{\mathrm{k}}^{(\mathrm{j})}\left(\mathrm{u}_{\mathrm{A}}^{(\mathrm{j})}\right)+\mathrm{S}_{\mathrm{k}}^{(\mathrm{j})}\left(\mathrm{u}_{\mathrm{F}}^{(\mathrm{j})}\right)\right] + +where Δ\ :sup:`(j)` = 0.5(u\ :sub:`F`\ :sup:`(j)` − u\ :sub:`I`). + +The values for S\ :sub:`ℓk`\ :sup:`(j)`\ (u\ :sub:`I`), +S\ :sub:`ℓk`\ :sup:`(j)`\ (u\ :sub:`A`\ :sup:`(j)`), and +S\ :sub:`ℓk`\ :sup:`(j)`\ (u\ :sub:`F`\ :sup:`(j)`) in :eq:`eq7-4-56` are obtained +by evaluating :eq:`eq7-4-55` at the lethargy values u\ :sub:`I`, +u\ :sub:`A`\ :sup:`(j)`, and u\ :sub:`F`\ :sup:`(j)`, respectively. Use +of more than two panels for the group integration was found to have an +insignificant impact. + +The complete epithermal PW-to-LMR source in group g is finally obtained +by summing :eq:`eq7-4-55` over all nuclides and then substituting the spherical +harmonic moments into the Legendre expansion of the MG scatter source, +resulting in + +.. math:: + :label: eq7-4-57 + + \mathrm{S}_{\mathrm{PW} \rightarrow \mathrm{g}}=\sum_{\mathrm{k}} \frac{2+1}{2} \mathrm{Y}_{\mathrm{k}}(\Omega) \sum_{\mathrm{j}} \Delta^{(\mathrm{j})} / 3\left[\mathrm{S}_{\mathrm{k}}^{(\mathrm{j})}\left(\mathrm{u}_{\mathrm{I}}\right)+4 \mathrm{S}_{\mathrm{k}}^{(\mathrm{j})}\left(\mathrm{u}_{\mathrm{A}}^{(\mathrm{j})}\right)+\mathrm{S}_{\mathrm{k}}^{(\mathrm{j})}\left(\mathrm{u}_{\mathrm{F}}^{(\mathrm{j})}\right)\right] + +.. _7-4-2-6-6: + +Thermal scatter sources from LMR and PW range +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +If the value of DEMIN is specified to be below the thermal energy +boundary, the portion of the PW range between DEMIN and the thermal +cutoff, as well as the entire LMR, will be contained in the thermal +range. In this case thermal neutrons will downscatter from the thermal +PW range to the LMR, and upscatter from the LMR to the thermal PW range. + +The latter thermal source (LMR-to-PW) is computed in exactly the same +manner as used to compute the UMR-to-PW source S\ :sub:`HI`, described +in :ref:`7-4-2-6-2`. On the other hand, the scatter source from the +thermal PW to the LMR is computed with a similar approach as given in +the previous section for epithermal PW-to-LMR scatter. In this case :eq:`eq7-4-57`  +is used as before, except the source moments are not obtained from the +submoment expansion in :eq:`eq7-4-55`, but rather by evaluating the PW thermal +scatter expression in :eq:`eq7-4-54`. + +In performing the transport calculation for any group g in the LMR +range, the PW-to-MG source component in :eq:`eq7-4-57` is added to the MG-to-MG +scattering into g from all groups in the UMR and LMR ranges, +respectively, to obtain the total scatter source. + +.. _7-4-2-7: + +Determination of energy mesh for PW flux calculation +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The energy mesh for the PW flux computation is determined for a specific +problem as follows: (a) for each zone-composition, microscopic +cross-section data are interpolated (if necessary) to the desired +zone-temperature, and a union energy mesh is formed from the energy +meshes of PW total cross sections of all materials in that zone, plus +the MG boundaries; (b) macroscopic total cross sections are computed for +the union meshes in each zone; (c) union meshes for each zone are +thinned (i.e., some energy points eliminated) in a manner that allows +the zone macroscopic cross section to be interpolated linearly, within +some input error tolerance; (d) a union mesh is created from the thinned +energy meshes for each zone thus producing a “global” energy mesh; +(e) the global mesh is checked to insure that it still contains group +boundaries and midpoint-energies of the input MG library, and finally, +(f) still more points may be added to constrain the maximum interval +width between successive lethargy points to be less than some fraction +of the maximum lethargy gained by elastic scatter from a fictitious +nuclide having a mass of approximately 400. The fraction used in +limiting the maximum size of any lethargy interval can be set by the +input value of “FLET,” but is defaulted to a value of 1/3. + +The mesh thinning procedure is effective in reducing the number of +energy points in the PW transport calculation, while preserving +essential features of the macroscopic cross-section data that affect the +flux spectrum; viz, the mesh is typically fine in energy regions +corresponding to important resonances, but coarser where there is little +variation in the macroscopic cross-section data. The default thinning +tolerance is 0.1%. A less stringent thinning tolerance may give a large +reduction in computation time, but also can affect the accuracy. + +.. _7-4-2-8: + +CENTRM cross sections and fixed sources +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +.. _7-4-2-8-1: + +CENTRM PW cross-section libraries +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +SCALE includes CE nuclear data for all materials and all reaction types +available in ENDF/B, processed for several different temperatures. The +CE data, spanning the energy range from 10\ :sup:`-5` eV to 20 MeV, are +stored in separate files for individual nuclides, which can be used for +CENTRM as well as CE Monte Carlo calculations. The CRAWDAD module reads +these files and merges the data to form a single CENTRM formatted +library containing only the particular materials, cross section types, +temperatures, and energy range needed for a given calculation (see +:ref:`8-1-5`). In general the CENTRM library includes CE data for the +unresolved, as well as the resolved, resonance range. Unresolved +resonance data typically have rather smooth variations, but in reality +the cross sections represent average values for very closely spaced +resonances that can not be measured individually. + +.. _7-4-2-8-2: + +Linearization of MG cross sections and fixed sources +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +Shielded group cross sections from the input MG library are always +required for the UMR and LMR portions of the CENTRM calculation. Two +approaches are available to translate multigroup cross sections into +pseudo-PW data at energy points within a group. The first is a “step” +approximation in which σ(E\ :sub:`n`) = σ\ :sub:`g`, where E\ :sub:`n` +is any energy point contained in group g. This leads to a histogram +representation of σ(E) that is discontinuous at the group boundaries. If +the multigroup data show significant variation between adjacent groups, +then the histogram approach can introduce discontinuities and +oscillations into the pointwise flux. An alternative approach is to +“linearize” the multigroup cross sections, using a linear representation +that preserves the group-average values and is continuous at the group +boundaries. Although the resulting cross section is continuous and does +not cause distortions in the flux spectrum, the data does not +necessarily represent the actual energy variation of the cross section. + +Input fixed source terms are treated in a similar manner. The multigroup +spectra that are input by the user may be converted either to a +discontinuous histogram function in lethargy; or may be linearized by +group. In the latter case the resulting groupwise-linear function is +evaluated at the energy mesh points to obtain the pointwise source term. + +.. _7-4-3: + +Available Methods for Solving Transport Equation +------------------------------------------------ + +CENTRM offers several calculation options for solving the Boltzmann +equation. Some of these are only available for either the MG or PW +calculations, respectively. In the case of the MG methods, the +calculation procedures are similar to those described in the XSDRNPM +documentation. The following sections briefly describe the PW transport +approximations available in CENTRM. + +.. _7-4-3-1: + +Discrete ordinates +~~~~~~~~~~~~~~~~~~ + +The discrete ordinates method can be used for both MG and PW solutions. +The main difference in the solution is the computation of the scattering +sources: the multigroup method uses group-to-group scatter matrices, +while the PW method uses the submoment expansion technique described +earlier. Also, as previously discussed, the pointwise discrete ordinates +equation has the same form as the transport equation for a purely +absorbing medium; so that inner iterations are not required to converge +the pointwise scattering source. The XSDRNPM documentation shows the +finite-difference form of the discrete ordinate equations, and includes +a discussion of S\ :sub:`N` quadratures, the weighted-difference model, +angular streaming coefficients, treatment of boundary conditions, and +other standard procedures used in the CENTRM 1-D discrete ordinates +solution. + +.. _7-4-3-2: + +Homogenized infinite medium +^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +A homogenized infinite medium calculation can be performed for either +the MG or the PW energy ranges. This method is essentially a +“zero-dimensional” model that has no spatial or angular variation in the +flux (only energy dependence). The materials contained in all zones are +“smeared” into a single homogenized mixture using volume weighting of +the number densities, and the effective external source is defined to be +the volume-weighted source density. The resulting homogenized +composition is then solved as an infinite medium, so that the PW scalar +flux is equal to, + +.. math:: + :label: eq7-4-58 + + \Phi\left(u_{n}\right)=\frac{\int \Sigma\left(u^{\prime} \rightarrow u_{n}\right) \Phi\left(u^{\prime}\right) d u^{\prime}+Q_{e x t}(E)}{\Sigma_{t}\left(u_{n}\right)}=\frac{S\left(u_{n}\right)+Q_{e x t}\left(u_{n}\right)}{\Sigma_{t}\left(u_{n}\right)} + +where :math:`\Sigma_{\mathrm{t}}, \mathrm{S}, \mathrm{Q}_{\mathrm{ext}},=` homogenized cross section, scatter source, and external +source, respectively. + +The total cross section in the PW calculation is reduced by the value of +the “within-point” cross section, as discussed in :ref:`7-4-2-6`. The PW +scattering source is computed from a P\ :sub:`0` submoment expansion +using cumulative integrals, as described in :ref:`7-4-2-4`. The scalar +flux at all space-intervals and the angular flux in all directions are +set to the above value, while higher order flux moments are equal to +zero. + +.. _7-4-3-3: + +Zonewise infinite medium +~~~~~~~~~~~~~~~~~~~~~~~~ + +The zonewise infinite medium solution is an option for the both PW and +MG calculations. This method is similar to the homogenized infinite +medium option, except each zone is treated independently as an infinite +medium, so that no spatial homogenization is required; i.e., Eq.  is +solved for each individual zone. The zonewise source corresponds to the +*input* external source within the respective zone. + +.. note:: *THERE IS + NO COUPLING BETWEEN THE CALCULATIONS FOR EACH ZONE.* + +Hence, any zones +that do not contain an input external source will have zero fluxes, +since there is no leakage between zones in the zonewise infinite medium +model. To avoid this problem the user must either input a volumetric +source, or specify a fission source and add a minute amount of +fissionable material to generate a fission spectrum source. + +.. _7-4-3-4: + +Collision probability method +~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +A collision probability (CP) solution of the 1-D integral transport +equation is available in CENTRM as an alternative to the pointwise +discrete ordinates approach. This option is only available for the +pointwise solution, and has the following additional restrictions +compared to the pointwise S\ :sub:`N` method: + + (a) only 1-D cylindrical or slab geometry is treated (no spherical + geometry); + + (b) periodic boundary conditions have not been implemented; + + (c) interior surface sources cannot be treated; + + (d) P\ :sub:`0` scattering (isotropic *laboratory* scatter) is + assumed; no transport correction is applied to the pointwise + cross sections in CENTRM. + + (e) due to assumption (d), computation of the angular flux is not + required in order to obtain the scalar flux; therefore, only the + P\ :sub:`0` flux moment (scalar flux) of the angular flux is + calculated. + + (f) PW thermal calculations are not supported for the CP method. + +**This method has only had a limited amount of testing, and has not been +validated as extensively as the discrete ordinates and MoC options.** + +The PW scattering source for the integral transport equation is similar +to that for the discrete ordinates method, except that only the +P\ :sub:`0` source moment is considered, and the “within point” +correction is not applied to the total cross section for the collision +probability method. Therefore only the spatial and directional treatment +is substantially different in the two transport approaches. In the +collision probability method the total interaction rate within a space +interval is expressed in terms of collision probabilities P\ :sub:`i′→i` +corresponding to the probability that neutrons born uniformly in +volume V\ :sub:`i′`, at lethargy u\ :sub:`n` will collide in interval i +with volume V\ :sub:`i`. The scalar flux in space interval “i,” at +lethargy point u\ :sub:`n` obeys the integral transport equation, + +.. math:: + :label: eq7-4-59 + + \Sigma_{\mathrm{i}}\left(\mathrm{u}_{\mathrm{n}}\right) \phi_{\mathrm{i}}\left(\mathrm{u}_{\mathrm{n}}\right) \mathrm{V}_{\mathrm{i}}=\sum_{\mathrm{i}^{\prime}} \mathrm{P}_{\mathrm{i}^{\prime} \rightarrow \mathrm{i}}\left(\mathrm{u}_{\mathrm{n}}\right)\left[\Sigma_{\mathrm{i}^{\prime} ; \mathrm{n} \rightarrow \mathrm{n}} \quad \Phi_{\mathrm{i}}\left(\mathrm{u}_{\mathrm{n}}\right)+\mathrm{Q}_{\mathrm{eff}, \mathrm{i}^{\prime}}\left(\mathrm{u}_{\mathrm{n}}\right)\right] \mathrm{V}_{\mathrm{i}^{\prime}} + +where + + :math:`\sum_{\mathrm{i}^{\prime}, \mathrm{n} \rightarrow \mathrm{n}}` is the within-point cross section at i′, defined in :eq:`eq7-4-40`; and + + :math:`\mathrm{Q}_{\text {eff }, \mathrm{i}^{\prime}}` is the external source (Q\ :sub:`n`) plus the + P\ :sub:`0` downscatter source in i′ and at u\ :sub:`n`, which is + computed in a similar manner as for the discrete ordinates solution. + +The boundary condition at the center of a cylinder is assumed to be +reflected; while the outer boundary condition can either be vacuum or +albedo. An albedo boundary condition returns a fraction of the outward +leakage, using a cosine distribution for the return-current. An albedo +of unity corresponds to a “white” boundary condition. + +Unlike the integro-differential transport equation used by the discrete +ordinates method, the integral equation at space interval i, contains +the unknown fluxes at *all other* space intervals i′. This requires that +inner iterations be performed for the PW solution using the CP option. + +Collision Probabilities (CP) in 1-D cylindrical geometry are computed +using the method developed by Carlvik :cite:`carlvik_method_1964`. The CPs for a vacuum outer +boundary condition are first computed. In this case the probability that +a neutron born in annular interval i with outer radius R\ :sub:`i`, will +have its next collision in annular interval j with outer radius +R\ :sub:`j` , is computed from the expression: + +.. math:: + :label: eq7-4-60 + + P_{i \rightarrow j}=\delta_{i j} \Sigma_{j} V_{j}+2\left(S_{i-1, j-1}-S_{i-1, j}-S_{i, j-1}-S_{i, j}\right) + + +where δ\ :sub:`ij` represents the Kronecker delta function and + +.. math:: + :label: eq7-4-61 + + S_{i j}=\int_{0}^{R_{t}}\left[K i_{3}\left(\tau_{i j}^{+}\right)-K i_{3}\left(\tau_{i j}^{-}\right)\right] d y + +In the above expression Ki\ :sub:`3` corresponds to a 3rd order Bickley +function, and :math:`\tau_{i j}^{+} \text {and } \tau_{i j}^{-}` are, + +.. math:: + + \tau_{i f}^{\pm}=\Sigma \times\left(\sqrt{R_{j}^{2}-y^{2}} \pm \sqrt{R_{i}^{2}-y^{2}}\right) + +Integration of the Bickley functions is performed numerically using +Gauss-Jacobi quadrature. + +The probability that a neutron born in interval “i” will escape +uncollided from a cell with a vacuum outer boundary is equal to one +minus the probability that it has a collision in any interval of the +cell, so that + +.. math:: + :label: eq7-4-62 + + P_{e s c, i}=1-\sum_{j} P_{i \rightarrow j} + +The “blackness” (Γ\ :sub:`i`) of interval i is defined to be the +probability that neutrons entering the cell with a cosine-current +angular distribution on the surface will collide within interval i. The +total blackness (Γ) is the probability that a neutron entering the outer +boundary will collide anywhere in the cell. The blackness is computed +from the escape probability: + +.. math:: + :label: eq7-4-63 + + \Gamma_{i}=\frac{4 V_{i} \Sigma_{i}}{S} \times P_{e s c, i} \quad ; \quad \text { and } \quad \Gamma=\sum_{i} \Gamma_{i} + +where S is the outer surface area of the cell. + +The vacuum CPs can be modified to account for the effect of an albedo +condition on the outer surface. In these cases a fraction of the +neutrons reaching the outer boundary are returned isotropically back +into the cell, and “bounce” back and forth between the two boundaries, +with each traverse reducing the number of uncollided neutrons. +Mathematically this corresponds to a converging geometric series that +accounts for the cumulative effect of the “infinite” number of traverses +through the cell. Evaluating the geometric sum, the collision +probability (P\ :sup:`A`) for an albedo of α can be expressed as + +.. math:: + :label: eq7-4-64 + + P_{i \rightarrow j}^{A}=P_{i \rightarrow j}+\frac{S}{4 V_{i} \Sigma_{i}} \times \frac{\alpha \Gamma_{i} \Gamma_{j}}{1-\alpha(1-\Gamma)} + +In the case of a slab with two vacuum boundaries, the CP’s are expressed +in terms of transmission probabilities T\ :sub:`i→j`, corresponding to +the probability that a neutron born in volume i will reach surface j +without a collision. The transmission probability is equal to, + +.. math:: + :label: eq7-4-65 + + T_{i \rightarrow j}=\frac{E_{3}\left(\tau_{i j}\right)-E_{3}\left(\tau_{i j}+\tau_{i}\right)}{2 \tau_{i}} + +where: + + τ\ :sub:`ij` is the optical distance between surface S\ :sub:`j` and the + surface volume V\ :sub:`i` that is closest to S\ :sub:`j`; + + τ\ :sub:`i` is the optical thickness of volume V\ :sub:`i`; and + + E\ :sub:`3` is the third order Exponential Function. + +In a slab with vacuum boundaries, the CP for a neutron born in +interval i to have its next collision in some interval j bounded by +surfaces S\ :sub:`j` and S\ :sub:`j+1`, is given by the expression, + +.. math:: + :label: eq7-4-66 + + P_{i \rightarrow j}=T_{i \rightarrow j}-T_{i \rightarrow j+1} + +A slab with a specular-reflected boundary condition on one side and a +vacuum on the other can be represented as an “expanded” geometry with +two vacuum boundaries, simply by adding the cell’s mirror image at the +reflected boundary. Hence, collision probabilities for these cases can +also be obtained from the expressions for two vacuum boundaries. For +slabs with an albedo (or white) boundary on one side and a reflected +boundary on the other, the same expression in :eq:`eq7-4-64` presented earlier for +cylinders can also be used to obtain albedo-corrected collision +probabilities from the vacuum values. However, unlike cylindrical +geometry, a slab may have specular-reflected boundary conditions on both +sides, corresponding to an infinite array of repeating cells. In CENTRM +the collision probabilities for a doubly-reflected slab geometry are +obtained by explicitly including two reflected cells in addition to the +“primary” cell, and then applying a white boundary condition on the +outer surface of the expanded geometry. The resulting modified geometry, +consisting of three cells (i.e., the primary plus two reflected cells) +with a reflected center and outer albedo boundary, can be treated as +described previously. + +.. _7-4-3-5: + +P\ :sub:`1` (“Diffusion Theory”) method +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The CENTRM MG calculation has an option for a P\ :sub:`1` calculation; +however, this option is not available for the PW calculation. The method +is essentially identical to that used in XSDRNPM. The P\ :sub:`1` option +is called diffusion theory in both XSDRNPM and CENTRM input +descriptions; but since the P\ :sub:`1` component of the scatter source +is explicitly treated and since Fick’s Law is not assumed, the method is +actually more closely related to the P\ :sub:`1` spherical harmonic +solution approach. The P\ :sub:`1` method is also used in CENTRM to +generate a flux guess for the thermal flux distribution in an +S\ :sub:`N` calculation. Several outer iterations over the thermal +groups are always performed with P\ :sub:`1` theory, prior to beginning +the discrete ordinates calculation for the thermal range. In the case of +PW thermal calculations, the multigroup thermal flux guess is converted +in PW flux values by multiplying by the ratio of MG to PW cross-section +values at each lethargy point. + +.. _7-4-3-6: + +Two-region (2R) method +~~~~~~~~~~~~~~~~~~~~~~ + +A two-region (2R) calculation similar to the Nordheim method is also +available as a PW option in CENTRM. In general the PW S\ :sub:`N` or MoC +solutions provide a more rigorous approach to compute self-shielded +cross sections than the 2R method; however the 2R approximation gives +accurate results for a wide range of “conventional cases.” Verification +studies have shown that the 2R option produces eigenvalues comparable to +those obtained with the S\ :sub:`N` and MoC options for numerous cases +of interest, and the execution time is often significantly faster since +only the zone-averaged PW scalar flux is computed. Thus the CENTRM 2R +option is an adequate and attractive alternative for many applications. + +The CENTRM 2R solution provides several advantages over the original +Nordheim method. CENTRM uses pre-processed PW nuclear data, rather than +the Nordheim approach of using input resonance parameters for a built-in +resonance formula (Breit-Wigner). This allows the CENTRM-2R calculation +to utilize PW cross sections processed from Reich-Moore resonance data +in ENDF/B, while the conventional Nordheim method is limited to +Breit-Wigner resonance formulae that do not treat level-level +interference. Other advantages of the CENTRM-2R calculation include a +rigorous treatment of resonance overlap from mixed absorbers, ability to +address mixtures of arbitrary moderators with energy dependent cross +sections, and capability to include inelastic and thermal scattering +effects in the flux calculation. + +The 2R approximation, which is a simplified version of the general +collision probability method, represents the system by an interior +region containing the mixture of materials to be self-shielded, and an +exterior moderator region where the asympototic flux per lethargy is +approximated by a simple analytical expression (e.g., constant for +epithermal energies). + +CENTRM performs a separate 2R calculation for each zone. For example, if +a problem consists of fuel and moderator zones, then two 2R calculations +are done: one has fuel as the interior region and moderator as the +exterior, and the other has moderator as interior and fuel as exterior. +In the case of lattices, multiple bodies composed of the same mixture +are addressed by introducing a Dancoff factor. + +The 2R equation solved by CENTRM for interior region “I” is equal to, + +.. math:: + :label: eq7-4-67 + + \Sigma_{\mathrm{I}}\left(\mathrm{u}_{\mathrm{n}}\right) \phi_{\mathrm{I}}\left(\mathrm{u}_{\mathrm{n}}\right)=\left(1-\mathrm{P}_{\mathrm{I}}^{(e s c)}\left(\mathrm{u}_{\mathrm{n}}\right)\right)\left(\mathrm{S}_{\mathrm{I}}\left(\mathrm{u}_{\mathrm{n}}\right)+\mathrm{Q}_{\mathrm{I}}^{e x t}\left(\mathrm{u}_{\mathrm{n}}\right)\right)+\Sigma_{\mathrm{I}}\left(\mathrm{u}_{\mathrm{n}}\right) \mathrm{P}_{\mathrm{I}}^{(e s c)}\left(\mathrm{u}_{\mathrm{n}}\right) \varphi_{\mathrm{asy}}\left(\mathrm{u}_{\mathrm{n}}\right) + +In the above equation, S\ :sub:`I` and Q\ :sub:`I` are the scatter and +fixed sources, respectively, for interior region “I”; and :math:`\mathrm{P}_{\mathrm{I}}^{(e s c)}` is +the probability that a neutron born in the interior region will escape +and have its next collision in the external region. For an interior +region consisting of multiple bodies of the same composition separated +by an exterior region, the escape probability is equal to + +.. math:: + + \mathrm{P}_{\mathrm{I}}^{(e s c)}\left(\mathrm{u}_{\mathrm{n}}\right)=\frac{\mathrm{P}_{0}^{(e s c)}\left(\mathrm{u}_{\mathrm{n}}\right)\left[1-C_{\mathrm{I}}\right]}{\left[1-C_{\mathrm{I}}\right]+C_{\mathrm{I}}^{-} \Sigma_{\mathrm{I}}\left(\mathrm{u}_{\mathrm{n}}\right) \mathrm{P}_{0}^{(e s c)}\left(\mathrm{u}_{\mathrm{n}}\right)} + +where :math:`\mathrm{P}_{0}^{(e s c)}` is the escape probability from a single, isolated body +in the interior region; :math:`^{-}\text { I }` is the average chord length of bodies +in the interior region; and C\ :sub:`I` is the Dancoff factor, +corresponding to the probability that a neutron escaping one interior +body will pass through the exterior region and have its next collision +in another body of the interior region. Values for :math:`:math:`\mathrm{P}_{0}^{(e s c)}` are +computed internally by the code at each energy mesh point, but Dancoff +factors must be provided by input for each zone. In the standard XSProc +computational sequence, Dancoff values are automatically computed and +provided to CENTRM. + +The asymptotic flux :math:`\varphi_{\text {asy }}` for the exterior region is represented by +analytical expressions in the fast, epithermal, and thermal energy +ranges, respectively, as summarized in :numref:`tab7-4-7`. The +constants C\ :sub:`1`, C\ :sub:`2` and C\ :sub:`3` are defined to impose +continuity at the energy boundaries, and also include the overall +normalization condition that the PW asymptotic flux at DEMAX is equal to +the MG flux for group MGHI obtained from the UMR calculation. + +.. _tab7-4-7: +.. table:: Asymptotic flux representation. + + +-----------------+-----------------+-----------------+-----------------+ + | **Upper | **Description** | **Distribution**| **Distribution | + | energy** | | | (per unit | + | | | | lethargy)** | + | | | **(per unit | | + | | | energy)** | | + +=================+=================+=================+=================+ + | 20 MeV | Fission | C\ :sub:`3`\ E\ | C\ :sub:`3`\ E\ | + | | spectrum | :sup:`1/2` | :sup:`3/2` | + | | | e\ :sup:`−E/θ` | e\ :sup:`−E/θ` | + | | | (*) | (*\**) | + +-----------------+-----------------+-----------------+-----------------+ + | 200 keV | Slowing-down | C\ :sub:`2`/E | C\ :sub:`2` | + +-----------------+-----------------+-----------------+-----------------+ + | 5kT | Maxwellian | C\ :sub:`1`\ E | C\ :sub:`1`\ E\ | + | | | e\ :sup:`−E/kT` | :sup:`2` | + | | | | e\ :sup:`−E/kT` | + +-----------------+-----------------+-----------------+-----------------+ + | (*\**) θ= | | | | + | fission | | | | + | spectrum | | | | + | “temperature” = | | | | + | 1 MeV. | | | | + +-----------------+-----------------+-----------------+-----------------+ + +Equation :eq:`eq7-4-67` is solved similarly to the zonewise infinite medium equation; +the only difference being that the interior sources are multiplied by +(1-\ :math:`\mathrm{P}_{0}^{(e s c)}`), and the presence of an inhomogeneous source term coming +from the exterior region. Like all CENTRM transport options, the +PW scattering source is computed using cumulative integrals as given in +:eq:`eq7-4-30` In the 2R option, only the P\ :sub:`0` scatter-source moment is +needed. + +.. _7-4-3-7: + +Method of characteristics for a 2D unit cell +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The 2D MoC option is available for square-lattice unit cell cases, with +both multigroup and pointwisecross sections. The geometry for the MoC +cell calculation consists of a number of concentric cylindrical zones +contained within a square outer surface with reflected boundary +conditions. The MoC calculation is not currently functional for +triangular lattices. A multiregion cell may be used in the MoC option, +but it must correspond to a similar geometry as a square lattice cell. +Because the 2D MoC solution is limited to 2D rectangular lattice cell +geometries, the code performs a number of internal checks to verify the +input geometry is permitted. The energy mesh generation, scattering +source calculation, etc. are performed in the same manner as for the +discrete ordinates option, except that only P\ :sub:`0` scatter is +treated. Unlike the CENTRM discrete ordinates method which solves the 1D +transport equation, the MoC option solves the 2D transport equation for +planar XY slice through a square-pitch lattice of cylindrical pins. +Because the the MoC calculation treats the outer rectangular boundary of +the unit cell correctly, it provides a more rigorous calculation than +using an equivalent 1D cylindrical Wigner-Seitz cell. The geometry input +for the MoC calculation is identical to the input for the 1D cylindrical +model--- the code internally converts the input value for the equivalent +cylindrical radius into the outer rectangular cell dimension (pitch) by +preserving the total cell area. Thus the length (L) of a side of the +rectangular cell in the MoC calculation is computed form the relation +:math:`L=\sqrt{\pi R_{e q}^{2}}`, where *R\ eq* is the equivalent outer radius of the +Wigner-Seitz cell, given in the CENTRM input array of radial dimensions. +Due to symmetry conditions, the MoC calculation is performed for only +1/8 of the rectangular cell (i.e., a 45 degree sector) and for the +upward directions. + +MoC uses a integrating factor to convert the divergence term in the +transport equation into a directional derivative along the direction of +neutron transport (Ω). For a given direction in the angular quadrature +set, the spatial domain is spanned by a set of parallel characteristic +rays originating at one boundary and terminating at the opposite +boundary. The default separation distance between the parallel rays is +0.02 cm. in CENTRM. The angular flux in given direction is computed by +integrating the directional derivative along the characteristic +direction. A reflected boundary condition is normally applied along the +rectangular outer surface of the unit cell. The default directional +quadrature, which defines the characterstic directions and is used for +integration, consists of 8 azimuthal angles for the 45 degree sector in +the X-Y plane and 3 postive polar angles relative to the Z-axis. + +The spatial domain of the unit cell is defined by zones of uniform +mixtures (e.g., fuel, clad, moderator) in which the cross section at a +fixed energy point or group is constant. These uniform composition +regions may be further divided into sufficiently small “flat-source” +sub-divisons in which the sources (scattering and external) can be +approximated as being constant. The MoC computes the average flux in the +flat-source region by summing over all characteristics in the volume. + +:cite:`kim_method_2012` gives more information about the CENTRM MoC solution +method. + +.. _7-4-4: + +CENTRM Input Data +----------------- + +The standard mode for executing CENTRM is through the XSProc +self-shielding module which automatically defines the CENTRM options, +mixing table, and geometry, and also prepares the necessary MG and PW +nuclear data files and passes the CENTRM PW flux results to the +downstream code PMC [see section describing XSProc]. However CENTRM can +also be executed in standalone using the FIDO input provided in this +section. When executed through XSProc, the user can set values for most +CENTRM input parameters given in this section by using keywords in the +CENTRM DATA block [see XSProc section]. Note that if CENTRM is run as a +standalone module, the input data files defined in :numref:`tab7-4-8` must be +provided, and the default values for some input parameters are +different. + +.. centered:: FIDO INPUT FOR STANDALONE CENTRM CALCULATIONS + +\****\* Title Card - (72 Characters) + + DATA BLOCK 1 : GENERAL PROBLEM DATA + +**1$$** GENERAL PROBLEM DESCRIPTION (18 entries. Defaults shown in +parenthesis) + +1. IGE = Problem Geometry: + + 0/1/2/3 = inf. hom. medium /plane/ cylinder/ sphere + +2. IZM = Number of Zones + +3. IM = Number of Spatial Intervals + +4. IBL = Left Boundary Condition: + +0/1/2/3 = vacuum/reflected/periodic/albedo + +5. IBR = Right Boundary Condition: + + 0/1/2/3 = vacuum/reflected/periodic/albedo + +6. MXX = Number of Mixtures + +7. MS = Mixing Table Length + +8. ISN = S\ :sub:`N` Quadrature Order + +9. ISCT = Order of Elastic Scattering + +10. ISRC = Problem Type: 0/1/2 = fixed source/input fission source/both +(0) + + [see Note 1] + +11. IIM = Inner Iteration Maximum (10) + +12. IUP = Upscatter Outer Iterations in Thermal Range (3) [see Note 2] + +13. NFST = Multigroup Calculation Option in Upper MG Range [E>DEMAX] (3) + + 0/1/2/3/4/5/6 = S\ :sub:`N` / diffusion / homogenized infinite medium / + + zonewise infinite medium/(option 5 deprecated)/2D MoC cell + + [see Note 3] + +14. NTHR = Multigroup Calculation Option in Lower MG Range [E 16 = allowable values (8) + +20. kern = Thermal Neutron Scattering Treatment: + + 0/1 = all free-gas kernels/use bound S(alpha, beta) data if available + (1) + +21. ISCTI = P\ :sub:`N` Order of Scattering for PW inelastic [<= ISCT] +(1) + +22. NMF6 = PW Inelastic Scatter Option (-1) + + −1/0/1 = no inelastic/ discrete level inelastic/ discrete and continuum + +**2$$** EDITING AND OTHER OPTIONS (12 entries. Defaults shown in +parenthesis) + +1. IPRT = Mixture Cross-Section Print Option: (−3) + + −3/−2/−1/N = none / write PW macro cross sections to output file, in + tab1 format/ + + print 1-D MG cross sections /print P\ :sub:`0`\ →P\ :sub:`N` MG + scatter matrices + +2. ID1 = Flux Print/Punch Options (−1) + + −1/0/1/2/ = none / print MG flux / also print M.G moments + + /save PW zone-average flux in ascii file + +3. IPBT = Balance Tables Print Option (PW thermal B.T. edit not +functional) (0) + + 0/1 = none / print balance tables + +4. IQM = Use Volumetric Sources: 0/N = No / Yes (0) + +5. IPM = Use Boundary Source: 0/N = No / Yes (0) + +6. IPN = Group Diffusion Coefficient Option (2) + + 0/1/2 = [*see XSDRNPM input*] + +7. IDFM = Use Density Factors 0/1 = No / Yes (0) + +8. IXPRT = Extra Print Option: + +0/1 = minimum print /regular print (0) + +9. MLIM = Mass Value Restriction on Order of Scattering (100) + + 0/M = no effect / restrict nuclides with mass ≥ M to have scatter + + order ≤ NLIM + +10. NLIM = Restrictive Scatter Order (0) + + + +**3*\*** CONVERGENCE CRITERIA AND OTHER CONSTANTS (9 entries. Defaults +in parenthesis) + +1. EPS = Upscatter Integral Convergence Criterion (0.001) + +2. PTC = Point Convergence Criterion (0.001) [see Note 5] + +3. XNF = Source Normalization Factor (1.0) [see Note 1] + +4. B2 = Material Buckling Value [units of cm\ :sup:`−2`] (0.0) + +5. DEMIN = Lowest Energy of Pointwise Flux Calculation, in eV (0.001) + +6. DEMAX = Highest Energy of Pointwise Flux Calculation, in eV (2.0E4) + +7. TOLE = Tolerance Used in Thinning Pointwise Cross Sections (0.001) +[see Note 7] + +8. MOCRAY = Distance (cm) Between MoC Rays; only for npxs=6 (0.02) + +9. FLET = Lethargy-Gain Fraction For Determining Energy Mesh (0.1) [see +Note 7] + +10. ALUMP = 0.0\ :math:`\rightarrow`\ 1.0, Criterion for Lumping of Materials by +Mass (0.0) [see Notes] + + T *[TERMINATE DATA BLOCK 1]* + + DATA BLOCK 2 : MIXING TABLE + +**12$$** COMPOSITION NUMBERS [AS IN MG LIBRARY] (MS entries) + +**13$$** MIXTURE NUMBERS (MS entries) + +**14$$** NUCLIDE IDENTIFIERS [AS IN MG LIBRARY] (MS entries) + + [*Note on 14$$*: Negative entry excludes material from PW treatment; + i.e., MG data used] + +**15*\*** NUCLIDE CONCENTRATIONS (MS entries) + + T *[TERMINATE DATA BLOCK 2]* + + **DATA BLOCK 3: SOURCE DATA** + +**30$$** SOURCE NO. BY INTERVAL (IM entries, if IQM or IPM > 0) + +**31*\*** VOLUMETRIC MULTIGROUP SOURCE SPECTRA (IQM*IGM entries, if IQM +>0) + +**32*\*** MULTIGROUP BOUNDARY ANGULAR SOURCE SPECTRA (IPM*IGM*MM, if IPM +>0) + +**34*\*** SPACE-DEPENDENT FISSION SOURCE (IM entries, if ISRC > 0) + + T *[ TERMINATE DATA BLOCK 3 ]* + + **DATA BLOCK 4: OTHER INPUT ARRAYS** + +**35*\*** INTERVAL BOUNDARIES (IM+1 entries) + +**36$$** ZONE NUMBER BY INTERVAL (IM entries) + +**38*\*** DENSITY FACTORS (IM entries, if IDFM>0) + +**39$$** MIXTURE NUMBER BY ZONE (IZM entries) + +**41*\*** TEMPERATURE [kelvin] BY ZONE (IZM entries: Default = F300.0) + +**47*\*** RIGHT BOUNDARY ALBEDOS (IGM entries if IBR=3: Default = F1.0) + +**48*\*** LEFT BOUNDARY ALBEDOS (IGM entries if IBL=3: Default = F1.0) + +**49*\*** DANCOFF FACTOR BY ZONE (IZM entries: Default = 0.0) + + T *[ TERMINATE DATA BLOCK 4 ]* + + \******\* END OF CENTRM INPUT DATA \******\* + +.. _7-4-4-1: + +CENTRM data notes +~~~~~~~~~~~~~~~~~ + +1. If ISRC = 0, an external volumetric or boundary source is used, as +specified in the 30$$, 31**, and 32** arrays (these are similar to same +arrays in XSDRNPM). This option is only available for standalone CENTRM +calculations; it is not an option for XSProc execution. If ISRC=1, the +magnitude of the fission source density (neutrons/cm\ :sup:`3`-s) is input +in the 34** array, and the source energy spectrum is assumed to be χ(E) +for the interval (if no fissionable material is present in an interval, +then χ=0 regardless of value in 34** array). Both external and fission +sources may be used if ISRC = 2. The source normalization parameter XNF +in the 3* array applies to the combined sources specified in data +block 3. + +2. If the value of DEMIN in the 3* array is set lower than the thermal +cutoff energy of the AMPX MG library, a PW thermal calculation is +performed over the energy range between DEMIN and the thermal cutoff. A +MG thermal calculation is performed over the remainder of the thermal +range. PW thermal calculations always start with ten outer iterations of +MG theory to obtain an initial flux guess. The value “IUP” indicates how +many *additional* outer iterations are to be performed. + +3. The solutions in the upper and lower MG ranges, as well as the +PW solution, provide several calculational options in addition to the +default method. These are described in :ref:`7-4-3`. + +4. The parameter ISVAR indicates how MG values for sources and +cross sections are mapped onto the PW energy mesh. See :ref:`7-4-2-8`. + +5. “Point Convergence” refers to the worst convergence over all space +intervals, between successive iterations. It is applied to (a) inner +iterations for MG S\ :sub:`N` solution; (b) inner iterations of +doubly-reflected boundary conditions in PW S\ :sub:`N` solution; and +(c) outer iterations of MG and PW solution in thermal range. + +6. The values for DEMAX and DEMIN determine the energy range of the +PW calculation. If the purpose of the CENTRM calculation is to obtain +PW fluxes for resonance self-shielding calculations with PMC, then the +PW energy range should at least span the resolved resonance ranges of +all materials for which self shielding is significant. + +7. The energy mesh for the PW flux calculation is based on several +factors, as described in :ref:`7-4-2-7`. + +8. If parameter alump > 0, epithermal and thermal scattering sources for +individual nuclides with similar masses are combined into macroscopic +“lumps,” based on a fractional mass deviation of “alump.” For example, +alump=0.2 means that materials are combined into one or more lumps such +that their masses are within +/- 20% of the effective mass of the lump. +The effective mass of the lump is defined to preserve the macroscopic +slowing down power. Mass lumping up to 0.2 will often reduce execution +time with little impact on the results. + +.. _7-4-4-2: + +CENTRM I/O files +~~~~~~~~~~~~~~~~ + +:numref:`tab7-4-8` shows filenames used by CENTRM. Some files are not required +for some calculations. + +.. _tab7-4-8: +.. table:: Default File Names + + +----------------------+--------------------------------------------------+ + | **File Name** | **Description** | + +======================+==================================================+ + | ft04f001 | Input MG library (only for standalone execution) | + +----------------------+--------------------------------------------------+ + | ft81f001 | Input PW cross-section library from Crawdad PW | + | | | + | lib_cen_kern | Input PW thermal scatter kernels from Crawdad | + | | | + | \_centrm.pw.flux | Output PW flux by zone (only for standalone) | + | | | + | \_centrm.pw_macro_xs | Output PW macro cross-section (optional) | + +----------------------+--------------------------------------------------+ + +.. _7-4-4-3: + +Description of the CENTRM CE cross section file +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The CENTRM CE cross section library is typically created using the +CRAWDAD module. CRAWDAD is executed automatically during XSProc cases; +but when CENTRM run standalone, the CE library must be pre-generated +prior to the CENTRM calculation (e.g., by running CRAWDAD). CRAWDAD +reads the SCALE CE data files for individual nuclides, and creates a +combined CENTRM library in the binary format shown below. + +.. list-table:: Header Records. + :align: center + + * - .. image:: figs/CENTRM/header-records.svg + :width: 700 + +.. list-table:: Nuclide Dependent Records (repeat for each nuclide) + :align: center + + * - .. image:: figs/CENTRM/nuclide-dependent-records.svg + :width: 700 + +.. _7-4-4-4: + +Description of the CENTRM output PW flux file +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +.. list-table: 3 Header Records Described Below. + :align: center + + * - .. image:: figs/CENTRM/header-records-2.svg + :width: 800 + + +*[IZM Records Containing Zone-Averaged PW Fluxes and Moments]* + ++--------------------------------+ +| DO 1 N = 1 , IZM | ++================================+ +| 1 PXJ( NTOTP , M), M=1, JT+1) | ++--------------------------------+ +| *(DP) DOUBLE PRECISION ARRAYS* | ++--------------------------------+ + +.. _7-4-5: + +Example Case +------------ + +In this section an example standalone CENTRM calculation is demonstrated +for a 1-D slab geometry model of a highly enriched uranium solution with +an iron-56 reflector. To execute CENTRM in standalone mode, nuclear data +libraries must be provided for MG cross sections (file ft04f001), PW +cross sections (file ft81f001), and PW thermal scatter kernels +(lib_cen_kernel). The shell script (=shell) in front of the CENTRM input +is used to link the necessary nuclear data libraries to the CENTRM +calculation. Here it is assumed that the Crawdad module has been +previously run to produce the PW cross section and thermal kernel +libraries, which are located in the same directory from which that the +job is submitted, while the MG library is the 8 group test library from +the SCALE data directory. + +.. _7-4-5-1: + +CENTRM input for example case +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +.. highlight:: scale + +:: + + =shell + ln -sf /scale/scale_data/test8g_v7.1 ft04f001 + ln -sf $RTNDIR/ft81f001 ft81f001 + ln -sf $RTNDIR/lib_cen_kernel lib_cen_kernel + end + + =centrm + centrm standalone example + 1$$ 1 2 15 1 0 2 5 e 2$$ -3 0 a8 1 e + 3** a5 0.0001 4000.0 e + t + 12$$ 1 1 1 1 2 13$$ 1 1 1 1 2 + 14$$ 92235 1001 8016 7014 26056 + 15** 0.003 0.06 0.04 0.018 0.08 + t + 34** f1.0 t + 35** 9i 0.0 4i 20.0 30.0 36$$ 10r1 5r2 39$$ 1 2 41** f300.0 + t + end + +.. _7-4-5-2: + +CENTRM output for example case +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +:: + + ****************************************************************** + + + module shell will be called on Tue Mar 15 16:52:40 2016. + sequence specification record:=shell + + Input Data: + ln -sf /scale/scale_dev_data/test8g_v7.1 ft04f001 + ln -sf $RTNDIR/ft81f001 ft81f001 + ln -sf $RTNDIR/lib_cen_kernel lib_cen_kernel + end + + module shell is finished. completion code 0 + + module centrm will be called on Tue Mar 15 16:52:40 2016. + sequence specification record:=centrm + + Input Data: + centrm standalone example + 1$$ 1 2 15 1 0 2 5 e 2$$ -3 0 a8 1 e + 3** a5 0.0001 4000.0 e + t + 12$$ f0 + 13$$ 1 1 1 1 2 + 14$$ 92235 1001 8016 7014 26056 + 15** 0.003 0.06 0.04 0.018 0.08 + t + 34** f1.0 t + 35** 9i 0.0 4i 20.0 30.0 + 36$$ 10r1 5r2 39$$ 1 2 41** f300.0 + t + end + +:: + + 1$ array 18 entries read + + 2$ array 12 entries read + + 3* array 9 entries read + + 0t + + 12$ array 5 entries read + + 13$ array 5 entries read + + 14$ array 5 entries read + + 15* array 5 entries read + + 0t + + 34* array 15 entries read + + 0t + + 35* array 16 entries read + + 36$ array 15 entries read + + 39$ array 2 entries read + + 41* array 2 entries read + + 0t + + CENTRM MATERIALS + + nuclides on Mixing Table + multi-grp lib. composition mixture component atom density small + 1 92235 0 1 92235 3.00000E-03 1 + 2 1001 0 1 1001 6.00000E-02 1 + 3 8016 0 1 8016 4.00000E-02 1 + 4 7014 0 1 7014 1.80000E-02 1 + 5 26056 0 2 26056 8.00000E-02 1 + + elapsed time .00 min. + =time after return from setup_centrm. + + ft81f001 = pointwise XS library from Crawdad + +:: + + MULTIGROUP STRUCTURE + + igm = 8 number of energy groups + mmt = 8 number of neutron groups + mcr = 0 number of gamma groups + iftg = 5 first thermal group number + + + GENERAL PROBLEM DATA + + ige = 1 problem geometry + 0/1/2/3 = inf. hom. medium/plane/cylinder/sphere + izm = 2 number of zones + im = 15 number of spatial intervals + ibl = 1 left boundary condition: + 0/1/2/3 = vacuum/reflected/periodic/albedo + ibr = 0 right boundary condition: + 0/1/2/3 = vacuum/reflected/periodic/albedo + mxx = 2 number of mixtures + ms = 5 mixing table length + isn = 8 SN quadrature order (not used for npxs=6) + isct = 3 order of elastic scattering + isrc = 1 type of source spectrum: + 0/1/2 = multigroup input spectrum/fission spectrum/both (1) + iim = 20 inner iterations maximum (10) + iup = 3 upscatter outer iterations in thermal range (3) + nfst = 0 multigroup calculation option in upper energy range [option nfst=5 is deprecated]: + 0/1/2/3/4/6 = sn/diffusion/homogeneous/zonewise homogeneous/BN/2D MoC cell (0) + nthr = 0 multigroup calculation option in lower energy range [option nthr=5 is deprecated]: + 0/1/2/3/4/6 = sn/diffusion/homogeneous/zonewise homogeneous/BN/2D MoC cell (0) + npxs = 1 pointwise calculation option: + <=0/1/2/3/4/5/6 = no pointwise: do multigroup as in nfst/sn/coll. prob./ + homogeneous/zonewise homogeneous/zonewise two-region/2D MoC cell (1) + isvar = 3 multigroup source & cross section linearization option : + 0/1/2/3 = none/linearize source/linearize group cross sections/both (3) + mocMesh = ***** meshing options for MoC calculation; npxs=6 only: + 0/1/2 = coarse/regular/fine mesh/input by zone (0) + mocPol = 0 number of polar angles for MoC quadrature; npxs=6 only: + 2/3/4 = allowable values (3) + mocAzi = 0 number of azimuthal angles for MoC quadrature; npxs=6 only: + 1 -> 16 = allowable values (8) + kern = 0 thermal neutron scattering treatment: + 0/1 = all free-gas thermal kernels/use bound S(alpha, beta) data if available (0) + iscti = 1 inelastic scattering order (0) + nmf6 = 0 inelastic scattering option: + -1/0/1 = no inelastic/discrete level only/discrete+continuum (0) + + +:: + + EDITTING AND OTHER OPTIONS + + iprt = -3 -3/-2/-1/n = mixture cross sections print options: + none/write P.W. energy & macro x-sections to output file/print 1-D M.G. x-sections + /print nth p (p0,p1..) M.G. x-section matrix (-3) + id1 = 0 -1/0/1/2 = flux print options: + none/print M.G flux/also print M.G moments/save PW zone flux in ascii file + ipbt = 0 0/1 = none/group summary table print (0) + iqm = 0 input multigrp volumetric sources (0/n=no/yes) + ipm = 0 input multigrp boundary angular sources (0/n=no/yes) + ipn = 2 0/1/2 diff. coef. param (2) + idfm = 0 0/1 = none/density factors (0) + ixprt = 1 0/1 = minimum print/regular print (0) + mlim = 100 0/m = no effect/restrict nuclides with mass >= m to have PW scatter order <= nlim (100) + nlim = 3 restrictive scatter order (1) + + + FLOATING POINT VALUES + + eps = 0.10000E-03 upscatter integral convergence (0.001) + ptc = 0.10000E-03 inner iteration convergence (0.001) + xnf = 0.10000E+01 source normalization factor (1.0) + b2 = 0.00000E+00 buckling value ( cm**(-2) ) + demin = 0.10000E-03 lowest energy of pointwise calculation in (eV) + demax = 0.40000E+04 highest energy of pointwise calculation in (eV) + tole = 0.10000E-02 tolerance used in thinning the pointwise cross sections (0.001) + mocRay = 0.00000E+00 distance between MoC rays; npxs=6 only. (0.02) + flet = 0.10000E+00 fractional lethargy used in construction of flux energy mesh (0.1) + alump = 0.00000E+00 mass-lumping criterion (0) + +:: + + ......................................................................................................... + + Input DEMIN and DEMAX values may be modified to lie on multigroup boundaries. + Min and Max energies (eV) defining range for pointwise flux calculation = 4.000E-02 2.000E+04 + ......................................................................................................... + 1 + POINTWISE MATERIALS USED IN THIS PROBLEM + + **** ZONE NO. 1 + Nuclide ID No. of Micro XS Energy Points in Energy Range 0.040 to 20000.000 + 1001 260 + 7014 604 + 8016 448 + 92235 118346 + No. of Macro XS Energy Points for Zone (After Thinning): 7133 + + **** ZONE NO. 2 + Nuclide ID No. of Micro XS Energy Points in Energy Range 0.040 to 20000.000 + 26056 1379 + No. of Macro XS Energy Points for Zone (After Thinning): 363 + + + NUMBER OF POINTS IN FINAL ENERGY MESH FOR FLUX CALCULATION 12687 + NUMBER OF POINTS IN PW THERMAL RANGE 458 + + + No unit supplied for PW kinematics data. + All PW thermal scatter kernels are treated as Free Gas + + +:: + + NEUTRON MULTIGROUP PARAMETERS + + + gp energy lethargy mid pt pointwise calc right left + boundaries boundaries velocities flx pts type albedo albedo + 1 2.00000E+07 -6.93147E-01 2.51321E+09 0 0 + 2 8.20000E+05 2.50104E+00 4.31276E+08 0 0 + 3 2.00000E+04 6.21461E+00 4.01867E+07 8332 1 + 4 1.05000E+02 1.14641E+01 6.03421E+06 3893 1 + 5 5.00000E+00 1.45087E+01 1.78251E+06 238 1 + 6 6.50000E-01 1.65489E+01 5.78908E+05 134 1 + 7 1.50000E-01 1.80152E+01 3.45722E+05 85 1 + 8 4.00000E-02 1.93370E+01 1.55701E+05 0 0 + 9 1.00000E-05 2.76310E+01 + + + DESCRIPTION OF ZONES + + mixture temperature Dancoff factr + by zone by zone by zone + 1 1 3.00000E+02 0 + 2 2 3.00000E+02 0 + + + SN QUADRATURE CONSTANTS + + weights directions refl direc wt x cos + 1 0 -1.00000E+00 9 0 + 2 8.69637E-02 -9.30568E-01 9 -8.09257E-02 + 3 1.63036E-01 -6.69991E-01 8 -1.09233E-01 + 4 1.63036E-01 -3.30009E-01 7 -5.38035E-02 + 5 8.69637E-02 -6.94318E-02 6 -6.03805E-03 + 6 8.69637E-02 6.94318E-02 5 6.03805E-03 + 7 1.63036E-01 3.30009E-01 4 5.38035E-02 + 8 1.63036E-01 6.69991E-01 3 1.09233E-01 + 9 8.69637E-02 9.30568E-01 2 8.09257E-02 + + CONSTANTS FOR P( 3) SCATTERING IN SN CALCULATION + + angl set 1 set 2 set 3 + 1 -1.000E+00 1.000E+00 -1.000E+00 + 2 -9.306E-01 7.989E-01 -6.187E-01 + 3 -6.700E-01 1.733E-01 2.531E-01 + 4 -3.300E-01 -3.366E-01 4.052E-01 + 5 -6.943E-02 -4.928E-01 1.033E-01 + 6 6.943E-02 -4.928E-01 -1.033E-01 + 7 3.300E-01 -3.366E-01 -4.052E-01 + 8 6.700E-01 1.733E-01 -2.531E-01 + 9 9.306E-01 7.989E-01 6.187E-01 + + elapsed time .02 min. + =time at after return from drtran... calling calc. + +:: + + GEOMETRY DESCRIPTION + + 1 int radii mid pts zone no. areas volumes dens fact + 1 0 1.00000E+00 1 1.00000E+00 2.00000E+00 + 2 2.00000E+00 3.00000E+00 1 1.00000E+00 2.00000E+00 + 3 4.00000E+00 5.00000E+00 1 1.00000E+00 2.00000E+00 + 4 6.00000E+00 7.00000E+00 1 1.00000E+00 2.00000E+00 + 5 8.00000E+00 9.00000E+00 1 1.00000E+00 2.00000E+00 + 6 1.00000E+01 1.10000E+01 1 1.00000E+00 2.00000E+00 + 7 1.20000E+01 1.30000E+01 1 1.00000E+00 2.00000E+00 + 8 1.40000E+01 1.50000E+01 1 1.00000E+00 2.00000E+00 + 9 1.60000E+01 1.70000E+01 1 1.00000E+00 2.00000E+00 + 10 1.80000E+01 1.90000E+01 1 1.00000E+00 2.00000E+00 + 11 2.00000E+01 2.10000E+01 2 1.00000E+00 2.00000E+00 + 12 2.20000E+01 2.30000E+01 2 1.00000E+00 2.00000E+00 + 13 2.40000E+01 2.50000E+01 2 1.00000E+00 2.00000E+00 + 14 2.60000E+01 2.70000E+01 2 1.00000E+00 2.00000E+00 + 15 2.80000E+01 2.90000E+01 2 1.00000E+00 2.00000E+00 + 16 3.00000E+01 1.00000E+00 + 1 + + EXTERNAL AND FISSION SOURCE DATA + + int radii mid pts fixd src spec normalized fiss src + 1 0 1.00000E+00 5.00000E-02 + 2 2.00000E+00 3.00000E+00 5.00000E-02 + 3 4.00000E+00 5.00000E+00 5.00000E-02 + 4 6.00000E+00 7.00000E+00 5.00000E-02 + 5 8.00000E+00 9.00000E+00 5.00000E-02 + 6 1.00000E+01 1.10000E+01 5.00000E-02 + 7 1.20000E+01 1.30000E+01 5.00000E-02 + 8 1.40000E+01 1.50000E+01 5.00000E-02 + 9 1.60000E+01 1.70000E+01 5.00000E-02 + 10 1.80000E+01 1.90000E+01 5.00000E-02 + 11 2.00000E+01 2.10000E+01 0 + 12 2.20000E+01 2.30000E+01 0 + 13 2.40000E+01 2.50000E+01 0 + 14 2.60000E+01 2.70000E+01 0 + 15 2.80000E+01 2.90000E+01 0 + 16 3.00000E+01 + 1 + +:: + + MULTIGROUP SCALAR FLUX (P0 MOMENT) + + int. grp. 1 grp. 2 grp. 3 grp. 4 grp. 5 grp. 6 grp. 7 grp. 8 + 1 1.49464E-01 7.16136E-02 1.97417E-01 7.40893E-02 3.79816E-02 1.46041E-02 5.93163E-03 4.53285E-03 + 2 1.49408E-01 7.16093E-02 1.97385E-01 7.40770E-02 3.79734E-02 1.46035E-02 5.98650E-03 4.79191E-03 + 3 1.49422E-01 7.15720E-02 1.97300E-01 7.40356E-02 3.79569E-02 1.45962E-02 5.95732E-03 4.61562E-03 + 4 1.49164E-01 7.15343E-02 1.97118E-01 7.39612E-02 3.79044E-02 1.45759E-02 5.95761E-03 4.72210E-03 + 5 1.49106E-01 7.13917E-02 1.96726E-01 7.37835E-02 3.78199E-02 1.45410E-02 5.94751E-03 4.64476E-03 + 6 1.48182E-01 7.11713E-02 1.95901E-01 7.33956E-02 3.75685E-02 1.44420E-02 5.89064E-03 4.64109E-03 + 7 1.47512E-01 7.05913E-02 1.93957E-01 7.24791E-02 3.71327E-02 1.42704E-02 5.84992E-03 4.59489E-03 + 8 1.43892E-01 6.93993E-02 1.89432E-01 7.09230E-02 3.61956E-02 1.39005E-02 5.64907E-03 4.43077E-03 + 9 1.38788E-01 6.61071E-02 1.82365E-01 6.80018E-02 3.47040E-02 1.33102E-02 5.46977E-03 4.33076E-03 + 10 1.14667E-01 5.81119E-02 1.75350E-01 6.17556E-02 3.03161E-02 1.13955E-02 4.57365E-03 3.58528E-03 + 11 7.10033E-02 4.72163E-02 1.70191E-01 5.14605E-02 2.15414E-02 7.71994E-03 2.57498E-03 1.07163E-03 + 12 4.41942E-02 3.58651E-02 1.47094E-01 3.99690E-02 1.38534E-02 4.46652E-03 1.23499E-03 1.02896E-04 + 13 3.07018E-02 2.56932E-02 1.17584E-01 2.91750E-02 9.12693E-03 2.42263E-03 6.13347E-04 4.02885E-05 + 14 2.00333E-02 1.74870E-02 8.62759E-02 1.88824E-02 5.53088E-03 1.18988E-03 2.76717E-04 1.54836E-05 + 15 1.25992E-02 9.76450E-03 5.01845E-02 8.56130E-03 2.43234E-03 4.58629E-04 1.02719E-04 5.89663E-06 + + elapsed time .03 min. + =time at end of centrm_Execute. + + module centrm used 2.34 seconds cpu time for the current pass. + + module centrm is finished. completion code 0. total cpu time used 0 seconds. + + SCALE is finished on Tue Mar 15 16:52:42 2016. + + -------------------------- Summary -------------------------- + shell finished. used 0.02 seconds. + centrm finished. used 2.34 seconds. + ------------------------ End Summary ------------------------ + +.. _7-4-6: + +CENTRM PW library and flux file formats +--------------------------------------- + +:ref:`7-4-6-1` below describes the format for the input CENTRM PW data +library (binary). The subsequent :ref:`7-4-6-2`, describes the format of +the output PW flux file (binary) produced by CENTRM, which is input to +the PMC code. + +.. _7-4-6-1: + +Description of the CENTRM CE cross section file +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The CENTRM CE cross section library is typically created using the +CRAWDAD module in SCALE. CRAWDAD reads the SCALE CE data files for +individual nuclides, and creates the combined CENTRM library. + +.. list-table:: 17 Header Records described below. + :align: center + + * - .. image:: figs/CENTRM/header-records-3.svg + :width: 700 + +.. list-table:: [NUCLIDE DIRECTORY : 1 RECORD/NUCLIDE]. + :align: center + + * - .. image:: figs/CENTRM/nuclide-dependent-records-2.svg + :width: 700 + +.. _7-4-6-2: + +Description of the CENTRM output PW flux file +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +.. list-table:: [3 Header Records Described Below]. + :align: center + + * - .. image:: figs/CENTRM/header-records-4.svg + :width: 700 + +*[IZM Records Containing Zone-Averaged P.W. Fluxes and Moments, +Described Below]* + ++--------------------------------+ +| DO 1 N = 1 , IZM | ++================================+ +| 1 PXJ( NTOTP , M), M=1, JT+1) | ++--------------------------------+ +| *(DP) DOUBLE PRECISION ARRAYS* | ++--------------------------------+ + +.. _7-4-7: + +CENTRM Error Messages +--------------------- + +CENTRM prints several of the same error messages as the XSDRNPM code. +Users should refer to the XSDRNPM chapter for these. CENTRM also prints +the additional error messages shown below. + ++-----------------------+-----------------------+-----------------------+ +| STATEMENT IN PRINTOUT | | USER ACTION | ++-----------------------+-----------------------+-----------------------+ +| ERROR. DEMIN exceeds | | Reduce input value of | +| maximum energy in | | DEMIN | +| working library | | | +| | | | +| STOP 400 | | | ++-----------------------+-----------------------+-----------------------+ +| ERROR. DEMAX exceeds | | Check input values | +| DEMIN | | for PW energy limits | +| | | | +| STOP 401 | | | ++-----------------------+-----------------------+-----------------------+ +| \***CALCULATION | | Check AMPX working | +| TERMINATED. | | library | +| | | | +| No neutron groups in | | | +| working library | | | ++-----------------------+-----------------------+-----------------------+ +| WARNING: Thermal | | Check input source | +| Source is 0 in ACCEL | | and thermal | +| **–OR-** | | cross-section data | +| | | (often caused by | +| WARNING: Thermal | | using Zonewise | +| Absorption + Leakage | | Infinite Medium | +| is 0 in ACCEL | | option with no source | +| | | in some zone) | ++-----------------------+-----------------------+-----------------------+ +| Requests nuclide | | Check nuclide ID’s | +| *XXX* which is not on | | and composition | +| your working library. | | numbers in mixing | +| | | table to make sure | +| | | they are on MG | +| | | library; Verify that | +| | | PW and MG | +| | | | +| | | libraries are | +| | | consistent | ++-----------------------+-----------------------+-----------------------+ + + + + + + + + + + +.. bibliography:: bibs/CENTRM.bib diff --git a/CHOPS.rst b/CHOPS.rst new file mode 100644 index 0000000..3846e0e --- /dev/null +++ b/CHOPS.rst @@ -0,0 +1,200 @@ +.. _7-6: + +CHOPS: Module to Compute Pointwise Disadvantage Factors and Produce a Cell-Homogenized CENTRM Library +===================================================================================================== + +*M. L. Williams and L. M. Petrie* + +.. _7-6-1: + +Introduction +------------ + +CHOPS (**C**\ ompute **HO**\ mogenized **P**\ ointwise **S**\ tuff) +computes pointwise (PW) disadvantage factors from the PW zone fluxes on +a CENTRM output file, and then multiples the disadvantage factors by +continuous-energy (CE) cross section data in a CENTRM library to +generate a new cell-homogenized CENTRM CE library. The PW disadvantage +factor for zone “Z”, as a function of energy E, is calculated from the +expression, + +.. math:: + :label: eq7-6-1 + + D_{Z}(E)=\frac{\Phi_{Z}(E)}{\Phi_{C}(E)} + +where :math:`\Phi_{Z}(E)` is the CENTRM PW flux spectrum averaged over the volume +of zone Z in the cell, and :math:`\Phi_{C}(E)` is the PW flux averaged over the +entire cell volume. The cell-homogenized CE cross section for a nuclide +“j” is equal to + +.. math:: + :label: eq7-6-2 + + \sigma_{C}^{(j)}(E)=\sum_{Z} F_{Z}^{(j)} D_{Z}(E) \sigma_{Z}^{(j)}(E) + +where :math:`F_{Z}^{(j)}` is the fraction of all nuclide–j atoms contained in zone +Z, and :math:`\sigma_{Z}^{(j)}(E)` is the CE cross section for nuclide-j at the +temperature of zone Z. When multiplied by the cell-homogenized number +density of nuclide-j and by the cell-average flux, the cross section +expression in :eq:`eq7-6-1` gives the correct average reaction rate at +energy E. + +.. equation in here says 8.5.1 + +CHOPS is used in the automated double heterogeneity sequence in SCALE, +in which a low-level heterogeneity, such as microspheres in a granular +fuel element, are smeared into a homogenized absorber region appearing +in the second level heterogeneity, such as fuel pellet or pebble +appearing in a lattice. The disadvantage factors provide for flux +weighting of the PW XS data so that the spatial self-shielding is +treated correctly in the homogenized geometry. A second CENTRM PW +transport calculation is performed with the cell-averaged PW library +output by CHOPS in order to account for the additional self-shielding of +the absorber pellets/pebbles in the lattice. CHOPS is called +automatically by the XSProc module for double-heterogeneous unit cells, +or it can run as a standalone code. + +.. _7-6-2: + +CHOPS Input Data +---------------- + +**DATA BLOCK 1** + +**0$$ LOGICAL UNIT ASSIGNMENTS** (10 entries. Default values given in +parentheses) + +1. lold -- logical unit number of input CENTRM XS library (1) + +2. lnew -- logical unit number of output CENTRM homogenized XS library +(2) + +3. lflx -- logical unit number of input CENTRM PW flux library (3) + +4. ldis -- logical unit number for edit of PW disadvantage factors (0) + +5. n15 -- logical unit number for scratch (15) + +6. n16 -- logical unit number for scratch (16) + +7. n17 -- logical unit number for scratch (17) + +8. n18 -- logical unit number for scratch (18) + +9. n19 -- logical unit number for scratch (19) + +10. nsq -- sequence number used in filename on unit “lnew” (1) + +[Example: if *lnew=11* and *nsq=3*: output filename of homogenized +library\ *= ft11f003]* + + +**1$$ INTEGER PARAMETERS** (5 entries ) + +1. idtap -- identifier for the new library (55555) + +[for macro library, the value of *idtap* is made negative] + +2. nprt -- output print option: 0 = > min print; 1 = > normal; 2 = > max +print (0) + +3. iden -- if=0 = > define homogenized XS id = id on CENTRM flux file +(0) + +if>0 = > define homogenized XS id to be, (*iden*\ \*10\ :sup:`6` + ZA) + +4. macr -- type of XS output: 0 = > microscopic ; 1 = > macroscopic (0) + +5. icorr -- not used (0) + + +**2*\* REAL PARAMETERS** (3 entries ) + +1. tole -- tolerance used to thin pointwise cross-sections (0.0025) + +( 0.0 means no thinning is done ) + +2. cleth -- maximum lethargy between thinned pointwise cross-sections + +points that allow a point to be discarded (0.25) + +3. vfrac -- multiplier applied to all output XS’s [eg, grain fraction] +(1.0) + +T [ TERMINATE DATA BLOCK 1 ] + +.. _7-6-3: + +CHOPS I/O units +--------------- + +:numref:`tab7-6-1` shows default logical unit numbers used by CHOPS. These +values may be changed in the 0$$ array of input. + +.. _tab7-6-1: +.. table:: Default I/O unit assignments for CHOPS. + :align: center + + +-------------+-------------------------------------------+ + | Unit number | Description | + +-------------+-------------------------------------------+ + | 1 | Input CENTRM CE data library | + | | | + | 2 | Output homogenized CENTRM CE data library | + | | | + | 3 | Input pointwise CENTRM flux file | + | | | + | 15 | Scratch file | + | | | + | 16 | Scratch file | + | | | + | 17 | Scratch file | + | | | + | 18 | Scratch file | + | | | + | 19 | Scratch file | + +-------------+-------------------------------------------+ + +.. _7-6-4: + +CHOPS Sample Input +------------------ + +The sample case in :numref:`list7-6-1` first executes a CENTRM unit cell +geometry calculation using the CSAS-MG sequence, which by default +generates the PW flux file on unit 15, as well as the CE nuclear data +library on unit 81 for input to CHOPS. The standalone CHOPS code then +computes a cell-homogenized CE library for the unit cell. The new +homogenized CENTRM CE library is output on unit 91with filename: +*ft91f001* + +.. code-block:: scale + :caption: CHOPS sample input. + :name: list7-6-1 + + =CSAS-MG parm=centrm + test case for CHOPS + v7-252n + READ COMP + ' Fuel pellet + o 1 0 4.59675e-2 900.0 end + u-235 1 0 4.88385e-4 900.0 end + u-238 1 0 2.24804e-2 900.0 end + ' Clad + zr 2 0 4.99789e-2 600.0 end + ' Coolant + h 3 0 4.76619e-2 600.0 end + o 3 0 2.38310e-2 600.0 end + END COMP + READ CELLDATA + latticecell squarepitch pitch=1.6 3 + fueld=1.262 1 cladd=1.350 2 end + END CELLDATA + END + =CHOPS + 0$$ 81 91 15 93 92 e + 1$$ a2 1 e + 2** a3 1.0 e + t + END diff --git a/CRAWDAD.rst b/CRAWDAD.rst new file mode 100644 index 0000000..a506558 --- /dev/null +++ b/CRAWDAD.rst @@ -0,0 +1,264 @@ +.. _7-7: + +CRAWDAD: Module to Produce CENTRM-Formatted Continuous-Energy Nuclear Data Libraries +==================================================================================== + +*M. L. Williams, D. Wiarda, and S. W. D. Hart* + +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: + +Introduction +------------ + +SCALE uses the code CRAWDAD (Code to Read And Write DAta for Discretized +solution) to read nuclear data from the SCALE-6 continuous (CE) library +files, and write it to an output file in the particular format needed +for the discretized energy solution in CENTRM. Prior to SCALE-6, the CE +data used by the CENTRM and PMC modules were distributed directly in the +CENTRM library format. However beginning with SCALE-6, the same CE data +are used by both CENTRM/PMC and by the CE versions of the KENO and +Monaco Monte Carlo codes. CE nuclear data for each nuclide are stored in +individual files contained in the SCALE permanent data directory. +CRAWDAD reads the files for each material appearing in a problem and +combines all data into a single problem-dependent CENTRM library file +stored in the temporary directory for execution. + +All SCALE-6 calculations that use modules CENTRM and PMC for +self-shielding multigroup (MG) cross sections must first execute the +CRAWDAD computational module. During execution of SCALE sequences, the +XSProc self-shielding module automatically executes CRAWDAD whenever the +CENTRM/PMC method is specified. CRAWDAD also can be run in stand-alone +mode to process and save a CENTRM-formatted library for subsequent +CENTRM/PMC calculations. + +PMC allows the energy range of the CE data to be selected, as well as +which reactions are placed on the output CENTRM library. The output +CENTRM library always contains the following “standard” nuclear data +types for all materials: total (1); elastic (2); complete inelastic (4); +radiative capture (102); fission (18); total/prompt/delayed nubars (452, +455, 456); and (n,α) cross section for 10B and 7Li. In this list, the +number shown in parenthesis corresponds to the ENDF/B “mt numbers.” + +CE data are obtained for arbitrary energies by linear interpolation of +discrete cross sections defined on a pointwise (PW) energy mesh. The PW +energy mesh for a given nuclide is sufficiently fine that error +introduced by linear interpolation between any two points is less than +0.1%. CRAWDAD also interpolates the CE data to the specific temperatures +needed for the problem. The default temperature interpolation method +uses square-root of temperature below 1200 Kelvin and a finite +difference procedure above this temperature :cite:`hart_creation_2016`. + +.. _7-7-2: + +CRAWDAD Input Data +------------------ + +For standalone CRAWDAD execution, the user prepares the FIDO input deck +as described below. However during a SCALE sequence computation, the +XSProc module always executes CRAWDAD for CENTRM/PMC self-shielding +calculations, and defines appropriate CRAWDAD parameter values based on +specified CENTRM and PMC options. This is the recommended mode of +operation. Some XSProc default values for CRAWDAD can be changed using +keywords in the CENTRM DATA block; e.g., see parameters *mtout=* and +*kernel=* in :ref:`7-4-4`. Several options available for stand-alone +execution cannot be controlled by keywords in the sequence runs, as +these are set automatically + +.. this reference needs to be checked. + +.. highlight:: none + +:: + + CRAWDAD STANDALONE INPUT + + ************** DATA BLOCK 1 + + 0$$ LOGICAL UNIT ASSIGNMENTS [4 entries. Default values given in parentheses] + + Entry Number Variable Name Description Default Value + 1 lcen logical unit number of output CENTRM library (81) + 2 n17 logical unit for scratch (17) + 3 n18 logical unit for scratch (18) + 4 n19 logical unit for reading CE-KENO libraries (88) + + 1$$ INTEGER PARAMETERS [10 entries ] + + 1 num_nucs number of PW nuclides to process (1) + 2 idtap identifier placed on header of output CENTRM library (66666) + 3 iprt print out option (1) + -1 no print out AT ALL + 0 hardly any print + 1 normal print + 2 debug print + 4 obsolete feature + 5 iterp temperature interpolation method for PW cross sections (0) + 0 square-root-T interpolation for T<1200 K and finite difference for T>1200 K + 1 square-root-T interpolation for all temperatures + 2 finite difference interpolation for all temperatures + 6 libth create CENTRM thermal kernel library for bound moderators (1) + 0 no + 1 yes (output kernel file is named lib_cen_kern) + 7–10 N/A extra integer parameters (not used) (0) + + + 1** REAL PARAMETERS [10 entries] + + 1 teps tolerance on temperature differences (5.0) + ( temperatures within +/- "teps" are assumed equal) + 2 tole not implemented + 3–10 N/A extra real parameters (not used) (0.0) + + T terminate data block 1 + +:: + + ************** DATA BLOCK 2 + ***** Repeat data block(s) 2 and 3, stacked "num_nucs" times to create a new + CENTRM library containing specified temperatures and reaction types + + 2$$ NUCLIDE INFORMATION [5 entries] + + Entry Number Variable Name Description Default Value + + 1 za zaid for this nuclide in PW XS library + 2 lver version number of evaluated nuclear data (e.g, 7 for ENDF/B-VII) + 3 mod desired mod number of evaluated nuclear (-1) + -1 => use latest mod + 4 inum desired number of temperatures for this nuclide (0) + 0 - put all available temperatures on output CENTRM library + n - include data at the "n" temperatures in 4** array + 5 mtout MTS to be included on output CENTRM PW library (2) + 0 - output PW data for all available MTs + 1 - output PW data only for default standard MTs: + 1, 2, 4, 102, 18, 452, 455, 456 for all materials; and 107 for 10B and 7Li + 2 - output standard MTs, plus inelastic levels and (n,2n) + 3 - standard MTs plus those listed in 5$$ array + -3 - out all MTs EXCEPT those listed in 5$$ + 6 kmod mod number for ENDF thermal scattering law data (-1) + ≥ 0 – use cross section data with this thermal mod number + -1 – use cross section data with latest thermal mod and kernel (if available) + -2 – do not include bound kernel data (i.e., free-gas scattering will be used in CENTRM) + 7 lsrc Source of nuclear data (0 only allowed at present) + 0/1/2/3/4 => ENDF/JEF/JENDL/BROND/CENDL + + 3** ENERGY LIMITS [2 entries] + + 1 pemin minimum energy for PW data (0.0001 eV) + 2 pemax maximum energy for PW data (20 MeV) + + T terminate data block 2 + +:: + + ************** DATA BLOCK 3 + ***** Only enter if inum >0, or mtout= +/- 3 ) + + 4** DESIRED TEMPERATURES for this nuclide [inum entries] + 5$$ MT VALUES (if mtout = +/-3) [always end with an "E"] + + T terminate data block 3 + + Optional 72 character title for the CENTRM library + +.. _7-7-3: + +CRAWDAD Sample Input +-------------------- + +:numref:`list7-7-1` shows an example input file for standalone execution of +CRAWDAD. The CRAWDAD output for this case is shown in :numref:`list7-7-2`. In +more typical cases where CRAWDAD is executed automatically by the XSProc +module as part of a SCALE sequence calculation, no CRAWDAD input is +needed, but similar CRAWDAD output will be printed. + +.. code-block:: scale + :name: list7-7-1 + :caption: CRAWDAD input generated by CSAS1 sample. + + =crawdad + 0$$ 81 17 18 77 e + 1$$ 5 66666 0 0 2 1 e + 1** 5.00E+00 e + t + 2$$ 8016 7 3 2 2 -1 0 + 3** 1.00-03 1.30+04 2t + 4** 6.00+02 9.00+02 3t + 2$$ 13027 7 1 1 2 -1 0 + 3** 1.00E-03 1.30E+04 2t + 4** 6.50E+02 3t + 2$$ 92235 7 7 1 2 -1 0 + 3** 1.00E-03 1.30E+04 2t + 4** 9.00E+02 3t + 2$$ 92238 7 5 1 2 -1 0 + 3** 1.00E-03 1.30E+04 2t + 4** 9.00E+02 3t + 2$$ 1001 7 5 1 2 0 0 + 3** 1.00E-03 1.30E+04 2t + 4** 6.00E+02 3t + end + ‚ move the generated PW CENTRM library to execution directory + =shell + mv ft81f001 $RTNDIR + end shell + ‘ …………………………………………………………………… + +.. code-block:: scale + :name: list7-7-2 + :caption: Sample output edit from CRAWDAD. + + A new centrm library has been written on unit number: 81 + The number of input nuclides was: 5 + Number of Nuclides on output PW library: 5 + Directory containing input PW library files: /scale/scale6.dev/data/cekenolib_7.0 + + + Description of Output CENTRM Library + + Entry ZA Data Src Vers No. Mod No. MT-Optn Thermal ID XS temperatures + ----- ----- -------- -------- ------- ------- ---------- --------------- + 1 8016 endf 7 3 2 0 600.00 + 900.00 + 2 13027 endf 7 1 2 0 650.00 + 3 92235 endf 7 7 2 0 900.00 + 4 92238 endf 7 5 2 0 900.00 + 5 1001 endf 7 5 2 7000001 600.00 + + + + Nuclides in Problem-Dependent Thermal Kernel Library + + Library Identifier: 901 + Number of kernels: 1 + Maximum Order of Scattering: 6 + Maximum Number of Temperatures: 9 + + Library Directory + Nuclide Identifier Sigfree File + ---------- ---------- ------- ------------------------------------------- + h(h2o) 7000001 20.48 endf_b/vers7/1-0 + + =============================================================================== + logical 18 (problem dependent centrm thermal kernel library) + dataset name: /usr/tmp/xmw.9890/lib_cen_kernel + volume: + =============================================================================== + + CRAWDAD has terminated normally + + + + + + + + + + + +.. bibliography:: bibs/CRAWDAD.bib diff --git a/MCDancoff.rst b/MCDancoff.rst new file mode 100644 index 0000000..d4a376f --- /dev/null +++ b/MCDancoff.rst @@ -0,0 +1,404 @@ +.. _7-8: + +MCDancoff Data Guide +==================== + +*L. M. Petrie, B. T. Rearden* + +ABSTRACT + +The MCDancoff program is used to calculate Dancoff factors in +complicated, three-dimensional (3-D) geometries using Monte Carlo +integrations. The geometries are standard SCALE geometry descriptions, +with the current restriction that Dancoff factors can only be calculated +for regions bounded by cuboids, spheres, or cylinders. Multiple Dancoff +factors can be calculated with one input file. + +ACKNOWLEDGMENT + +This work was sponsored in part by Atomic Energy of Canada, Ltd. The +contribution of S. J. Poarch in preparing this document is gratefully +acknowledged. + +.. _7-8-1: + +Introduction +------------ + +MCDancoff (Monte Carlo Dancoff) is a program that calculates Dancoff +factors for complicated, three-dimensional geometries. Its input is a +slight modification of a CSAS6 input file which uses the standard SCALE +geometry as detailed for KENO-VI. The modifications to the input involve +different input in the START data block describing which Dancoff factors +are to be calculated. The calculation involves starting histories +isotropically on the surface of the region for which the Dancoff factor +is to be calculated and following the path of each history until it has +encountered all the elements of the material in the region, or until it +has exited the system. A one group cross-section library is used to +determine the total cross sections of the mixtures in the problem. + +A current restriction of MCDancoff is that it can only calculate Dancoff +factors for regions bounded by cylinders, spheres, or cuboids. Other +simple bodies could be added in the future, but a general bounding +surface would be impractical. + +The Dancoff factors are used in SCALE to correctly self-shield +multigroup cross sections for a given problem; either as input to BONAMI +or to determine an equivalent cell for CENTRM. This is most typically +accomplished through the MORE DATA and CENTRM DATA blocks. + +The Dancoff factors are actually calculated by a modified version of the +KENO-VI code called KENO_Dancoff. All printed output from these +calculations is suppressed by default. If there is a need to see this +output (for example, to find an error message), it can be turned on by +setting an environment variable **print_dancoff=yes**. + +.. _7-8-2: + +Input data description +---------------------- + +MCDancoff input data is the same as CSAS6 input data with the following +exceptions. A special one group cross-section library will be used. It +can be specified as **xn01** in the input but will be set to this if +anything else is entered for the library. Because MCDancoff is running a +fixed source problem, and the Dancoff factor doesn’t need to be +calculated with the same accuracy as an eigenvalue, there are useful +changes that can be made to the parameters in the PARAMETER data block. +:ref:`7-8-3` discusses this in more detail. Finally, the START data +block is used to define which Dancoff factors will be calculated. This +data block is defined below. + +**READ START** Begins the data block + +1. **dancoff** + +begins defining a new Dancoff factor. Always start +relative to the global unit in the geometry. + +2. **array** + +step into an array contained in the current unit – followed +by **karray**, **nbx**, **nby**, **nbz** where **karray** is the +region containing the array in the current unit, **nbx** is the x +position in the array of the next unit, **nby** is the y position +in the array of the next unit, and **nbz** is the z position in +the array of the next unit. + +3. **hole** + +step into a hole contained in the current unit – followed by +**nhole**, the hole number relative to the current unit. + +4. **unit** + +final unit in the nesting chain – followed by **nn**, the +unit number + +5. **region** + +region to calculate the Dancoff factor for – followed by +**k**, the relative geometry word in unit **nn** defining the +outer bound of the region. + +6. **nst** + +if input, must be 0 (defaults to 0). + +Repeat 2 and 3 to get from the global unit to the final unit **nn**. + +Repeat 1–5 for each Dancoff factor to be calculated. + +**END START** Ends the data block + + +.. _7-8-3: + +Calculation and use of 3D Dancoff factors +----------------------------------------- + +1. The 3-D Dancoff factors are computed with KENO-VI geometry. If + beginning with CSAS5 model, use C5TOC6 to convert to CSAS6. + +2. Change sequence name from CSAS6 to MCDancoff and change cross-section + library to **xn01**. + +3. Input appropriate parameter data. + +.. + + Since the Dancoff calculation is fixed source integration, there is + no need to skip generations, and **nsk** should be set to 0. Since + small changes to the Dancoff have very minor effects on the cross + sections, fewer histories are probably needed for calculating the + Dancoff than for calculating *k\ eff*. Thirty thousand histories + divided as 100 generations of 300 histories per generation has + produced Dancoff factors with deviations of less than 1 percent. It + may be advantageous to turn off plots at this point. Since the same + parameters can be entered more than once, with the final entry being + the one used, adding a separate record with these values immediately + before the **end parameter** keywords would override the original + KENO-VI parameters. + + Example: + + +.. highlight:: scale + + :: + + read param + ......... + nsk=0 npg=300 gen=100 nub=no fdn=no flx=yes plt=no + end param + + +4. Identify the region for which Dancoff factors are desired in START + data. + +.. + + The start type needs to be set to **0** for the Dancoff calculation + (this is the default). All KENO-VI START data should be removed or + commented out by placing an apostrophe in column 1. Each region for + which a Dancoff calculation is desired then starts with the keyword + **dancoff**. This is followed by data that specify the relationship + of the global unit to the specific geometry description of the + region. If the region is nested inside an array, then the keyword, + **array**, is specified, followed by four integers. The first integer + is the indices of the media record specifying the array relative to + the current unit. The next 3 integers are the X, Y, and Z indexes of + the position of the next unit in the array. If the region is nested + in a hole, then the keyword, **hole**, is specified, followed by the + relative count of the correct hole in the unit. The preceding data + are repeated (in the correct nesting order starting with the global + unit) until reaching the unit where the region is located. Then the + keyword, **unit**, followed by the unit number is given, followed by + the keyword, **region**, followed by the relative index of the + geometry keyword describing the desired region with respect to that + unit. Currently, only cylinders, spheres, and cuboids are programmed + for calculating Dancoff factors. + + Examples: + + :: + + read start + nst=0 + dancoff hole 1 unit=1 reg=1 + end start + + read start + dancoff array 1 1 1 1 array 1 17 17 2 unit 10 region 1 + end start + + +5. Execute MCDANCOFF *filename.*\ input file like any other SCALE input + file. + + +6. Examine *filename*.dancoff file, which will contain Dancoff factors + for each nuclide in the specified region + +:: + + index nuclide dancoff deviation + 1 92234 3.36340E-01 1.81134E-03 + 2 92235 3.36340E-01 1.81134E-03 + 3 92236 3.36340E-01 1.81134E-03 + 4 92238 3.36340E-01 1.81134E-03 + 5 8016 1.00000E+00 0.00000E+00 + +7. Once all desired Dancoff factors are obtained, return to original model and + enter CENTRM DATA for each cell with dan2pitch(mix) specified. + +:: + + read celldata + latticecell triangpitch fuelr=0.633 1 gapr=0.637 0 cladr=0.675 10 hpitch=0.867 14 end + centrm data + dan2pitch(1)=0.336 + end centrm + +8. If executing TSUNAMI-3D, additional steps are necessary because + TSUNAMI‑3D does not treat the dan2pitch input parameter. + +.. + + Return to the original TSUNAMI-3D input file and replace the sequence + name to “CSAS-MG PARM=CHECK” and delete all data after the unit cell + data to quickly obtain revised pitch values. (Note: CSAS will not + modify cell dimensions to more than 20 cm, so a revised moderator + density may need to be entered to obtain the desired Dancoff factor.) + Search for the word “\ *desired”* in output file to find new pitch + values for each cell. + + :: + + unit cell = 1 + original pitch = 1.7340E+00 + Dancoff for orig pitch = 2.9728E-01 + desired Dancoff = 3.3600E-01 + + pitch to produce desired Dancoff= 1.6845E+00 + +9. Enter revised pitch and revised moderator density (for cell + calculation only, not for geometry model) in TSUNAMI model. + +.. _7-8-4: + +Example Case +------------ + +The following is a contrived case to illustrate an input file using both +holes, arrays, and multiple sets of Dancoff factors (although both +factors apply to the same pin, so only one set can be used). The case +represents two fuel assemblies in a cylindrical tank, each assembly +having a poisoned central pin, and four water holes. The Dancoff factors +are calculated for each central pin. The input file is listed in +:numref:`list7-8-1`. + +.. code-block:: scale + :name: list7-8-1 + :caption: Example input file (continued below). + + =mcdancoff + sample case demonstrating calculating Dancoff factors + xn01 + read composition + uo2 1 den=10.38 1 294 92234 .0303 92235 4.7378 92236 .1364 92238 95.0955 end uo2 + zirc4 2 1 294 end zirc4 + h2o 3 1 294 end h2o + uo2 4 den=10.08 1 294 92234 .0303 92235 4.7378 92236 .1364 92238 95.0955 end uo2 + gd 4 den= 0.3 1 294 end gd + end composition + read param + nsk=0 gen=100 npg=300 + end param + read geometry + unit 1 + com=!fuel pin! + cylinder 10 0.395 40.0 -40.0 + cylinder 20 0.410 40.0 -40.0 + cylinder 30 0.470 40.0 -40.0 + cuboid 40 4p0.65 2p40.0 + media 1 1 10 + media 0 1 20 -10 + media 2 1 30 -20 + media 3 1 40 -30 + boundary 40 + unit 2 + com=!water hole! + cuboid 40 4p0.65 2p40.0 + media 3 1 40 + boundary 40 + unit 3 + com=!unit containing a 2x2 array of fuel pins! + cuboid 10 4p1.30 2p40.0 + array 1 10 place 1 1 1 -0.65 -0.65 0.0 + boundary 10 + unit 4 + com=!unit containing a 1x2 array of fuel pins! + cuboid 10 2p0.65 2p1.30 2p40.0 + array 2 10 place 1 1 1 0.0 -0.65 0.0 + boundary 10 + unit 5 + com=!unit containing a 2x1 array of fuel pins! + cuboid 10 2p1.30 2p0.65 2p40.0 + array 3 10 place 1 1 1 -0.65 0.0 0.0 + boundary 10 + unit 6 + com=!unit containing a 5x5 array of fuel pins! + cuboid 10 4p3.25 2p40.0 + array 4 10 place 2 2 1 0.0 0.0 0.0 + boundary 10 + unit 7 + com=!unit containing a 5x5 array of fuel pins - water hole in the middle! + cuboid 10 4p3.25 2p40.0 + array 5 10 place 2 2 1 0.0 0.0 0.0 + boundary 10 + + +:: + + unit 8 + com=!unit containing a 5x5 array of fuel pins - poisoned pin in the middle! + cuboid 10 4p3.25 2p40.0 + array 6 10 place 2 2 1 0.0 0.0 0.0 + boundary 10 + unit 9 + com=!poisoned fuel pin! + cylinder 10 0.395 40.0 -40.0 + cylinder 20 0.410 40.0 -40.0 + cylinder 30 0.470 40.0 -40.0 + cuboid 40 4p0.65 2p40.0 + media 4 1 10 + media 0 1 20 -10 + media 2 1 30 -20 + media 3 1 40 -30 + boundary 40 + unit 10 + com=!unit containing a 15x15 fuel assembly! + cuboid 10 4p9.75 2p40.0 + array 7 10 place 2 2 1 0.0 0.0 0.0 + boundary 10 + global + unit 11 + com=!global unit with 2 fuel assemblies! + cylinder 10 25.0 60.0 -60.0 + hole 10 origin x=-10.0 + hole 10 origin x= 10.0 + media 3 1 10 + boundary 10 + end geometry + read array + ara=1 typ=square nux=2 nuy=2 nuz=1 fill f1 end fill + ara=2 typ=square nux=1 nuy=2 nuz=1 fill f1 end fill + ara=3 typ=square nux=2 nuy=1 nuz=1 fill f1 end fill + ara=4 typ=square nux=3 nuy=3 nuz=1 fill 3 4 3 5 1 5 3 4 3 end fill + ara=5 typ=square nux=3 nuy=3 nuz=1 fill 3 4 3 5 2 5 3 4 3 end fill + ara=6 typ=square nux=3 nuy=3 nuz=1 fill 3 4 3 5 9 5 3 4 3 end fill + ara=7 typ=square nux=3 nuy=3 nuz=1 fill 7 6 7 6 8 6 7 6 7 end fill + end array + read start + ' first Dancoff - calculate for the poisoned fuel pin in unit 9 for the x=-10 assembly + dancoff + ' hole 1 is unit 10 at x=-10 + hole 1 + ' array in first region of unit 10 is array 7 - 2 2 1 position is unit 8 + array 1 2 2 1 + ' array in first region of unit 8 is array 6 - 2 2 1 position is unit 9 + array 1 2 2 1 + ' cylinder labeled 10 in unit 9 is the first region + unit 9 region 1 + ' second Dancoff - calculate for the poisoned fuel pin in unit 9 for the x=+10 assembly + dancoff + ' hole 2 is unit 10 at x=+10 + hole 2 + ' array in first region of unit 10 is array 7 - 2 2 1 position is unit 8 + array 1 2 2 1 + ' array in first region of unit 8 is array 6 - 2 2 1 position is unit 9 + array 1 2 2 1 + ' cylinder labeled 10 in unit 9 is the first region + unit 9 region 1 + end start + end data + end + + +This input file creates two files of Dancoff factors. The first such file is listed in :numref:`list7-8-2`. + +.. code-block:: scale + :name: list7-8-2 + :caption: Output file of Dancoff factors. + + Unit 9 at global x -1.00000E+01 y 0.00000E+00 z 0.00000E+00 + index nuclide dancoff deviation + 1 92234 2.20873E-01 1.03436E-03 + 2 92235 2.20873E-01 1.03436E-03 + 3 92236 2.20873E-01 1.03436E-03 + 4 92238 2.20873E-01 1.03436E-03 + 5 8016 9.64748E-01 4.28121E-04 + 6 64000 2.82254E-10 3.61320E-11 + +The second file is statistically the same, as it solved for the mirror image Dancoff factor. diff --git a/Material Specification and Cross Section Processing Overview.rst b/Material Specification and Cross Section Processing Overview.rst new file mode 100644 index 0000000..2b89562 --- /dev/null +++ b/Material Specification and Cross Section Processing Overview.rst @@ -0,0 +1,208 @@ +.. _7-0: + +Material Specification and Cross Section Processing Overview +============================================================ + +*Introduction by M. L. Williams and B. T. Rearden* + +**XSProc** (Cross Section Processing) provides material input and +multigroup (MG) cross section preparation for most SCALE sequences. +XSProc allows users to specify problem materials using easily remembered +and easily recognizable keywords associated with mixtures, elements, +nuclides, and fissile solutions provided in the SCALE **Standard +Composition Library**. For MG calculations, XSProc provides cross +section temperature correction and resonance self-shielding as well as +energy group collapse and spatial homogenization for systems that can be +represented in *celldata* input as infinite media, finite 1D systems, or +repeating structures of 1D systems, such as uniform arrays of fuel +units. Improved resonance self-shielding treatment for nonuniform +lattices can be achieved through the use of the **MCDancoff** (Monte +Carlo Dancoff) code that generates Dancoff factors for generalized 3D +geometries for subsequent use in XSProc. Cross sections are generated on +a microscopic and/or macroscopic basis as needed. Although XSProc is +most often used as part of an integrated sequence, it can be run without +subsequent calculations to generate problem-dependent MG data for use in +other tools. + +This chapter provides detailed descriptions of the methods and modules +used for self-shielding. Self-shielding calculations are effectively a +problem-specific extension of the processing procedures used to create +the SCALE cross section libraries. SCALE includes continuous energy (CE) +and several MG (MG) cross section libraries described in the chapter on +SCALE Cross Section Libraries. The AMPX nuclear data processing +system :cite:`wiarda_ampx_2015` was used to convert evaluated data from ENDF/B into CE cross +sections, which were then averaged into problem-independent MG data at a +reference temperature of 300K, weighted with a generic energy spectrum +(see the SCALE Cross Section Libraries chapter). After being transformed +in probability distributions by AMPX, the CE data require no further +modifications for application to a specific problem except for possible +interpolation to the required temperatures. However, in MG calculations, +reaction rates depend strongly on the problem-specific energy +distribution of the flux, which implies that the problem-independent MG +data on the library should be modified into problem-dependent values +representative of the actual flux spectrum rather than the library +generic spectrum. The neutron energy spectrum is especially sensitive to +the concentrations and heterogeneous arrangement of resonance absorbers, +which may dramatically reduce the flux at the resonance peaks of a +nuclide, thus reducing its own reaction rate —a phenomenon known as +self-shielding. In general, the higher the concentration of a resonance +nuclide and the more the interaction between heterogeneous lumps (e.g. +fuel pins), the greater the degree of self-shielding for the nuclide. + +Reference :cite:`williams_resonance_2011` gives a general description of the SCALE self-shielding +methods. The individual computational modules perform distinct functions +within the overall all self-shielding methodology of XSProc. More +theoretical details about individual computational modules are given in +:ref:`7-2` through :ref:`7-7`. XSProc provides capabilities for two different types +of self-shielding methods, which are summarized below. + +**Bondarenko Method** + +The Bondarenko approach :cite:`ilich_bondarenko_group_1964` uses MG cross sections pre-computed over a +range of self-shielding conditions, varying from negligibly (infinitely +dilute) to highly self-shielded. Based on the following +approximations :cite:`stammler_methods_1983` it can be shown that the degree of self-shielding in +both homogeneous and heterogeneous systems depends only on a single +parameter called the background cross section, “sigma0,” and on the +Doppler broadening temperature: + +(a) neglect of resonance interference effects, + +(b) intermediate resonance approximation, and + +(c) equivalence theory. + +During the SCALE MG library processing with AMPX, self-shielded cross +sections are computed using a CE flux calculated at several background +cross section values and temperatures. These are used to calculate +ratios of the shielded to unshielded cross sections, called “Bondarenko +factors” (a.k.a. shielding factors or f-factors). As described in the +SCALE Cross Section Libraries chapter, Bondarenko factors are tabulated +on the SCALE libraries as a function of sigma0 values and Doppler +temperatures for all energy groups of each nuclide. + +Bondarenko factors are multiplicative correction factors that convert +the generic unshielded data into problem-dependent self-shielded values. +The BONAMI computational module performs self-shielding calculations +with the Bondarenko method by using the input concentrations and unit +cell geometry to calculate a sigma0 value for each nuclide and then +interpolating the appropriate MG shielding factors from the tabulated +library values. + +**CENTRM/PMC Method** + +Self-shielding calculations with BONAMI are fast and are always +performed for all SCALE MG sequences. However, due to the approximations +(a)–(c) listed in the previous section, a more rigorous method is also +provided which can replace the BONAMI results over a specified energy +range, usually encompassing the resolved resonance ranges of important +absorber nuclides. This approach is designated as the CENTRM/PMC method, +named after the two main computational modules, although several +additional modules are also used. CENTRM/PMC eliminates the main +approximations of the BONAMI approach by performing detailed neutron +transport calculations with a combination of MG and CE cross sections +for the actual problem-dependent compositions and unit cell +descriptions :cite:`williams_computation_1995`. This provides a problem-dependent pointwise (PW) flux +spectrum for averaging MG cross sections, which reflects resonance +cross-interference effects, an accurate slowing down treatment, and +geometry-specific transport calculations using several available +options. Shielded MG cross sections processed with CENTRM/PMC are +usually more accurate than BONAMI, so it is the default for most SCALE +MG sequences. However, depending on the selected transport option, +CENTRM/PMC may run considerably longer than BONAMI alone. + +The CENTRM/PMC methodology first executes BONAMI, which provides +shielded cross sections outside the specified range of the PW flux +calculation. Then the computational module CRAWDAD reads CE cross +section files and bound thermal scatter kernels and interpolates the +data to the desired temperatures for CENTRM. Using a combination of +shielded MG data from BONAMI and CE data from CRAWDAD, CENTRM calculates +PW flux spectra by solving the deterministic neutron transport equation +for all unit cells described in the input. CENTRM calculations cover the +energy interval 10\ :sup:`-5` eV to 2 × 10\ :sup:`7` eV spanned by the +SCALE MG libraries. This energy range is subdivided into three sections: +(a) upper MG range: E>\ *demax*, (b) PW range: *demin*\ = 0, the +discrete-level PW inelastic (MTs 50-90) and continuum inelastic (MT-91) +data are also included in the CENTRM PW library. + +PW data for the unresolved resonance range are infinitely dilute on the +CENTRM library; therefore PMC does not use PW cross sections to compute +self-shielded data for the unresolved range. Instead, self-shielded +cross sections in the unresolved range are calculated using the +Bondarenko method in BONAMI prior to the CENTRM and PMC calculations. +This step is automatically performed by XSProc in the SCALE calculation +sequences. + +PMC offers two methods to compute the total cross section. In the first +method the MG value for the total cross section (MT=1) is processed +directly from the PW MT-1 data on the CENTRM library. Total cross +sections are generally considered the most accurate type of evaluated +reaction data (due to measurement techniques); however if PW data for +MT-1 are processed as an independent cross section, there is no +guarantee that the sum of the partial cross sections will sum to the +total. These small imbalances in cross sections affect the neutron +balance, and may impact eigenvalue calculations. For this reason the PMC +default option does not compute the total cross section by weighting the +MT-1 PW data, but rather by summing the MG partial cross sections +(including the original MG data not re-processed in PMC). + +The 1-D cross sections can be weighted using either the P\ :sub:`0` +(scalar flux) or P\ :sub:`1` (current) PW Legendre moment. In almost all +cases flux weighting is more desirable, since resonance reaction rates +are usually the dominant factor in the PW range. However, +current-weighting may be more accurate for certain problems where +spatial transport and leakage strongly influence the spectrum in the +resonance range, such as when the leakage spectrum is greatly impacted +by cross section interference minima such as occur in iron media. The +current-weighting option has been successfully applied for criticality +calculations involving mixtures of highly-enriched uranium and iron. An +alternative approach to using the current-weighted total cross section +is to include a Legendre expansion of the angular-flux-weighted total +cross section, which modifies the diagonal elements of the 2D elastic +scattering moments.\ :sup:`7` This option is specified by setting PMC +input parameter n2d=±2, as discussed in :ref:`7-5-2-4`. + +.. _7-5-2-2: + +Spatial averaging of 1D cross sections +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +PMC computes MG microscopic cross sections for each material mixture in +a given CENTRM calculation, using the spatially averaged PW spectrum +within the mixture. In SCALE this method is called “zone-weighting”, and +it is the default for PMC. Zone-weighted cross sections are generated +for every mixture zone in the unit cell. In configurations containing +fuel/absorber mixtures (e.g., lattices) in multiple unit cells, +CENTRM/PMC calculations may be performed for each mixture, resulting in +multiple mixture-weighted cross sections for the same nuclide ID. For +this reason, both the nuclide ID and a mixture number are generally +required to uniquely identify any specific cross section data generated +by PMC. + +PMC also has an option to calculate “cell-weighted” (i.e., homogenized) +MG data, which applies disadvantage factors to preserve the +cell-averaged reaction rates for the entire unit cell. This is not +typically done, except for treating doubly-heterogeneous cells with +SCALE. In this case the PMC cell-weighting option is performed to +produce homogenized MG cross sections for the low level heterogeneity +(e.g., fuel grain in a fuel pebble). The XSProc control module +automatically sets the correct PMC weighing flag based on the type of +unit cell. + +.. _7-5-2-3: + +Energy ranges for multigroup weighting +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The energy range of the MG and CE libraries in SCALE typically spans +10\ :sup:`−5` to 2*10\ :sup:`7` eV. In general this encompasses the (a) +thermal region where upscatter is treated, (b) resolved and unresolved +resonance ranges, and (c) high energy region above the resonance ranges. +The thermal range for the current SCALE libraries is defined to be below +5 eV. Energy limits for the resolved and unresolved resonance ranges are +defined by the individual ENDF/B evaluations for each nuclide, and these +limits are included in the CENTRM PW library. + +As discussed in section 8.3, the CENTRM PW flux file contains values of +the zone-flux (and moments) per unit lethargy, calculated over the +entire energy range 10\ :sup:`−5` eV to 20 MeV; however, only the fluxes +in the energy range from DEMAX to DEMIN are computed from the PW +transport solution and exhibit the spectral fine-structure due to +resonance reactions. The flux outside interval [DEMAX, DEMIN] is +represented by the smoother “pseudo-pointwise” values obtained from +CENTRM’s MG solution. PMC provides two options to define the +nuclide-specific energy range for computing problem-dependent MG data: + +Option (1). Compute MG cross sections of a given nuclide only over the +resolved resonance range of the nuclide. If the CENTRM PW calculation +does not encompass the entire resolved resonance range for the nuclide, +pseudo-point fluxes are be used in the self-shielding calculations for +some groups in the resolved regions. The pseudo-point fluxes are +generally a good representation for the gross spectrum shape, but do not +reflect fine-structure effects caused by resonance absorption; therefore +with this option, the user should take care that the CENTRM PW limits +are appropriate for the resonance nuclides of interest. + +Option (2). Compute MG cross sections for a given nuclide over the +entire energy range for which PW flux values are calculated in the +CENTRM. In this case PMC computes MG cross sections only over the +portion of the PW data that is contained within the PW flux range; i.e., +the pseudo PW spectrum is not used to process any data. Shielded cross +sections for groups not included in the PW calculation are based on the +BONAMI self-shielding method. + +Option (2) above is default in PMC. SCALE-6.2 has DEMIN and DEMAX +default values of 0.001 eV and 20 keV. This is sufficient for resonance +self-shielding of essentially all actinide and important fission product +nuclides; but some structural materials such as iron have resonances +above 20 keV which would be shielded by BONAMI (:ref:`7-3`). + +.. _7-5-2-4: + +Options for treatment of 2-D cross sections +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The input parameter N2D defines five PMC options for processing +problem-dependent, 2-D elastic scattering matrices. The first approach, +N2D=0, simply multiplies the elastic scattering matrices by the ratio of +the new to old 1-D elastic cross sections for the specified reaction +process, where the “old” data are the 1-D values in the original MG +library, and the “new” data are the problem-dependent MG cross sections +processed using the PW flux as described above. The P\ :sub:`ℓ` Legendre +moments as well as the P\ :sub:`0` matrix are scaled by the same ratio +for a given group. This method is also always used for discrete-level +and continuum inelastic cross sections, as well as any other 2-D data +other than elastic. The basic assumption is that the relative +group-to-group scattering distribution does not change from the +distribution in the original MG library, which is processed with an +infinitely dilute spectrum— i.e., self-shielding only affects the total +scatter rate. This approach gives good results for many applications, +and is very efficient computationally. However, for intermediate and +high mass materials, the elastic removal rate from a group may be +sensitive to the problem-dependent CE spectrum. In these cases the +scaling approximation may not give the correct elastic removal rate from +the group, because the within-group elastic cross section will be in +error. In these cases the alternate approaches described below can be +used. + +The option N2D= −1 corrects for the impact of resonance self-shielding +on the elastic removal from an energy group. This option recomputes a +new value for the within-group cross section by applying a correction +factor based on the ratio of shielded versus unshielded removal +probabilities for *s*-wave scatter (isotropic center-of-mass scatter). +The P\ :sub:`0` out-scattering cross sections are then renormalized to +give the correct 1D shielded cross section for the group. This approach +provides a reasonable and computationally efficient approximation to +process 2D elastic matrices in the resolved resonance range of actinide +nuclides. However the assumption of s-wave scatter may not be valid in +the resolved resonance range of a structural material such as iron; +therefore users should beware when applying the approximation if the PW +range is extended above 50 keV, for systems with large sensitivity to +structural materials. + +Option N2D=1 uses the CENTRM PW flux to recompute the entire set of +group-to-group scatter data (including Legendre moments) assuming +*s*-wave kinematics. Since the CENTRM PW flux is used as the weighting +function, this approach is sometimes more accurate for groups with large +spectral gradients as discussed above. As with the N2D=-1 option, the +main limitation is the *s*-wave scattering approximation for the +secondary energy distribution. This option requires more computation +time than the N2D methods discussed previously, and usually gives +similar results as N2D=-1. + +A rigorous derivation of the MG transport equation from the CE equation +results in a directionally dependent total cross section. PMC option +N2D=2 uses the method in :cite:`bell_nuclear_1970` to address this effect by modifying +the Legendre moments of the 2D elastic matrix. For cross section moment +“n”, the diagonal term (i.e., within-group scatter) is modified by +adding a term equal to the difference in the MG total cross section +weighted with the PW scalar flux and the MG total cross section weighted +with the n\ :sub:`th` Legendre moment of the PW flux. + +Option N2D=-2 is essentially a combination of options N2D=2 and N2D=-1. +This option applies the elastic removal correction to the diagonal term +of the P\ :sub:`0` moment of the elastic 2D matrix, and applies the PL +correction described above to the diagonal term of the PL Legendre +moment of the elastic matrix. + +The thermal energy range presents a particularly difficult challenge for +processing problem-dependent 2‑D scattering data, due to the complicated +kinematics associated with molecular motion, chemical binding, and +coherent scattering effects. PMC currently the scaling approximation +(N2D=0 option) for the thermal energy range, regardless of the input +value of N2D. + +.. _7-5-3: + +Calculation of Problem-Dependent Multigroup Cross Sections +---------------------------------------------------------- + +.. _7-5-3-1: + +1-D cross sections +~~~~~~~~~~~~~~~~~~ + +.. math:: + :label: eq7-5-1 + + \sigma_{z, r, g}^{j}=\frac{\int_{\Delta E_{g}} \sigma_{z, r}^{j}(E) \Phi_{z}(E) d E}{\int_{\Delta E_{g}} \Phi_{z}(E) d E}=\frac{\int_{\Delta E_{g}} \sigma_{z, r}^{j}(E) \Phi_{z}(E) d E}{\Phi_{z, g}} + +where + + Φ\ :sub:`z,g` is the multigroup zone flux, + + σ\ :sup:`j`\ :sub:`z,r,g` is the zone-average, group cross section, and + + ∆E\ :sub:`g` is the energy interval of group g. + +The integration in :eq:`eq7-5-1` is performed by summing over a discrete energy +mesh within the group boundaries. Since the CE cross section and the PW +flux generally have different energy grids, the integration mesh for the +numerator is formed by taking the union of the two. The CE +cross sections and the PW flux are mapped onto the union mesh, and the +integral is evaluated using the trapezoidal method. :eq:`eq7-5-1` is used to +compute weighted group data for all MT’s for which CE data are available +on the CENTRM library, except in the case of the fission neutron yield +ν. Instead of using the PW scalar flux as the weighting function, the MG +value for ν is weighted by the product of the PW flux and the PW fission +cross section for the material. + +.. _7-5-3-2: + +2-D scattering cross sections +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The 2-D MG cross section moments are defined as the weighted +group-average of terms appearing in a Legendre (PL) expansion of the CE +double-differential scatter cross section, which describes the transfer +of neutrons from one energy to another, for a given angle of scatter. +The PL Legendre moments on the original MG library are fully consistent +with the ENDF/B kinematic specifications. Thus the specified anisotropy +in elastic or inelastic data in the center-of-mass (CM) system is +reflected in the PL scattering matrices; however the library MG data are +processed with an infinitely dilute flux spectrum. PMC provides several +options for modifying these data to correct for problem-specific +spectral effects, such as self-shielding. First, consider the scaling +method (N2D=0) in which all the elements of the original scatter matrix +(i.e., on the input Master library) for a given initial group are +multiplied by the ratio of 1-D scatter cross sections. This has the +effect of normalizing the original scatter matrix to the +problem-dependent value calculated for the 1-D scatter data. In this +case the l\ :sub:`th` Legendre moment of the 2-D multigroup +cross section for reaction type “s” of nuclide “j” in zone “z” (at a +specified temperature), for scatter from initial group g′ to final +group g, is computed by: + +.. math:: + :label: eq7-5-2 + + \sigma_{l, z, s, g^{\prime} \rightarrow g}^{j}=\frac{\left(\sigma_{z, s, g^{\prime}}^{j}\right)_{n e w}}{\left(\sigma_{s, g^{\prime}}^{j}\right)_{o r i g}} \times\left(\sigma_{l, s, g^{\prime} \rightarrow g}^{j}\right)_{o r i g} + +where the subscripts “\ *orig*\ ” and “\ *new,*\ ” respectively, refer +to the original MG data on the Master library, and the new +problem-dependent data computed by PMC. The types of reactions for which +problem-dependent 2-D cross sections may be processed using the scaling +method are elastic (MT=2), discrete-level inelastic (MT’s 50–89), +continuum inelastic (MT=90), and (n,2n) (MT=16). This approach is also +applied to obtain problem-dependent thermal scatter matrices, which +contain upscatter as well as down-scatter reactions. The CENTRM nuclear +data libraries include PW cross sections for incoherent (MT=1007) and +coherent (MT=1008, if available) thermal scattering reactions, which can +be processed into 1-D MG data by PMC in the same manner as other +reaction types. The 1-D weighted thermal scattering data are then used +to normalize the 2-D thermal matrices on the input Master library. For +materials with both coherent and incoherent thermal scatter data, each +matrix is scaled by the corresponding type of 1-D data. The coherent +scattering matrix only contains within-group terms. + +The option N2D= −1 recomputes the P\ :sub:`0` within-group elastic +cross section based on the assumption of s-wave scatter kinematics, and +scales the other terms of the original P0 elastic matrix by the modified +removal rate. This procedure approximately corrects for effects of +resonance self-shielding on the group removal probability, without +having to recompute the entire matrix assuming *s*-wave scatter, as done +for N2D=1. Suppressing the zone index for simplicity, the P\ :sub:`0` +within-group XS is defined as: + +.. math:: + :label: eq7-5-3 + + \sigma_{\mathrm{g}, \mathrm{g}} \equiv \frac{\int_{\mathrm{g}} \sigma_{\mathrm{s}}(\mathrm{E})\left[1-\mathrm{p}_{\mathrm{r}}(\mathrm{E})\right] \Phi(\mathrm{E}) \mathrm{d} \mathrm{E}}{\int_{\mathrm{g}} \Phi(\mathrm{E}) \mathrm{d} \mathrm{E}} + +where p\ :sub:`r`\ (E) is the probability that a neutron at energy E, +within group g, will scatter to an energy below the lower boundary of +the group. For *s*-wave scattering this equation becomes, + +.. math:: + :label: eq7-5-4 + + \sigma_{\text{g,g}} = \frac{\int^{\text{min}\left(\text{E}_{\text{Hi}}, \frac{\text{E}_{\text{Lo}}}{\alpha}\right)}_{\text{E}_{\text{Lo}}} \sigma_{\text{s}}(\text{E})\left[\frac{\text{E}-\text{E}_{\text{L}}}{\text{E}(1-\alpha)}\right] \Phi(\text{E})\text{dE}}{\int_{\text{g}}\Phi(\text{E})\text{dE}} + + +The N2D= −1 option recomputes a modified P\ :sub:`0` within-group +cross section from the expression, + +.. math:: + :label: eq7-5-5 + + \left(\sigma_{\mathrm{g}, \mathrm{g}}\right)_{\text {new}}=\frac{\widetilde{\sigma}_{\mathrm{g}, \mathrm{g}}^{(\varphi)}}{\widetilde{\sigma}_{\mathrm{g}, \mathrm{g}}^{\infty}}\left(\sigma_{\mathrm{g}, \mathrm{g}}\right)_{\text {orig}} + +where + + (σ\ :sub:`g,g`)\ :sub:`orig` is the original within-group + cross section on the MG library, based on actual kinematics and weighted + with an infinitely dilute spectrum; + + :math:`\widetilde{\sigma}_{\mathrm{g}, \mathrm{g}}^{(\infty)}` is the infinitely dilute within-group cross section based on + s-wave kinematics, which is computed from :eq:`eq7-5-4`  using an infinitely + dilute spectrum + + :math:`\widetilde{\sigma}_{\mathrm{g}, \mathrm{g}}^{(\varphi)}` is the self-shielded within-group based on s-wave kinematics, + computed from :eq:`eq7-5-4` using Φ(E) →CENTRM PW flux. + +If the effects of resonance self-shielding are small, then there will be +little change in the original within-group value, since in this case +:math:`\widetilde{\sigma}_{\mathrm{g}, \mathrm{g}}^{(\varphi)} \sim \widetilde{\mathrm{O}}_{\mathrm{g}, \mathrm{g}}^{(\infty)}`. + +The P\ :sub:`0` group-to-group out-scatter terms for N2D=-1 are scaled +as follows: + +.. math:: + :label: eq7-5-6 + + \sigma_{g \rightarrow g^{\prime}}=\frac{\left(\sigma_{\mathrm{s}, \mathrm{g}}\right)_{\mathrm{new}}-\widetilde{\sigma}_{\mathrm{g}, \mathrm{g}}^{(\varphi)}}{\left(\sigma_{\mathrm{s}, \mathrm{g}}^{\infty}\right)_{\mathrm{new}}-\widetilde{\sigma}_{\mathrm{g}, \mathrm{g}}^{\infty}} \times\left(\sigma_{\mathrm{g} \rightarrow \mathrm{g}^{\prime}}\right)_{\mathrm{orig}} + +Again if there is little self-shielding, the change in off-diagonal +matrix elements is small, so that the original secondary energy +distribution is preserved. Finally the entire modified P\ :sub:`0` +scatter matrix is renormalized to correspond to the self-shielded 1-D +scatter cross section. + +For the option N2D=1, an entirely new PL elastic scattering matrix is +computed. The l\ :sub:`th` Legendre moment of the 2-D MG elastic +cross section of nuclide “j” in zone “z” (at a specified temperature), +for scattering from initial group g′ to final group g is rigorously +defined as, :cite:`bell_nuclear_1970` + +.. math:: + :label: eq7-5-7 + + \sigma_{l, g^{\prime} \rightarrow g}^{j}=\frac{\int_{\Delta E_{g}} \int_{\Delta E_{g^{\prime}}} \sigma_{l}^{j}\left(E^{\prime} \rightarrow E\right) \Phi_{l, z}\left(E^{\prime}\right) d E^{\prime} d E}{\int_{\Delta E_{g^{\prime}}} \Phi_{l, z}\left(E^{\prime}\right) d E^{\prime}}=\frac{\int_{\Delta E_{g}} \int_{\Delta E_{g^{\prime}}} \sigma^{j}\left(E^{\prime}\right) f_{l}^{j}\left(E^{\prime} \rightarrow E\right) \Phi_{l, z}\left(E^{\prime}\right) d E^{\prime} d E}{\int_{\Delta E_{g^{\prime}}} \Phi_{l, z}\left(E^{\prime}\right) d E^{\prime}} + +where σ\ :sub:`z`\ (E) is the CE elastic cross-section data from the +CENTRM nuclear data file, evaluated at the appropriate temperature for +zone z;\ :math:`f_{l}^{j}` (E′→E) is the secondary neutron energy distribution +from elastic scattering; and Φ\ :sub:`l,z`\ (E) is the lth PW flux +moment averaged over zone Z. PMC assumes *s*-wave scattering from +stationary nuclei to evaluate the scattering distribution, and uses the +P\ :sub:`0` flux moment (i.e., scalar flux) as for the weighting +function for all PL matrices; therefore the expression evaluated by PMC +for N2D=1 is: + +.. math:: + :label: eq7-5-8 + + \sigma_{l, z, g^{\prime} \rightarrow g}^{j}=\frac{\int_{g^{\prime}} \int_{g} \frac{\sigma_{z}^{j}(\mathrm{E}) \Phi_{z}\left(E^{\prime}\right) P_{l}\left(G^{j}\right)}{\left(1-\alpha^{j}\right) E^{\prime}} d E^{\prime} d E}{\int_{g} \Phi_{z}\left(E^{\prime}\right) d E^{\prime}} + +here P\ *l* is the *l*\ :sub:`th` order Legendre polynomial; and +G\ :sup:`j` is the kinematics relation expressing the cosine of the +scattering angle as a function of E and E’, for elastic scattering from +nuclear mass A\ :sup:`j`. The kinematics function for nuclide j is +defined as, + +.. math:: + :label: eq7-5-9 + + \mathrm{G}^{\mathrm{j}}\left(\mathrm{E}^{\prime}, \mathrm{E}\right)=\frac{\mathrm{A}^{\mathrm{j}}+1}{2} \sqrt{\frac{\mathrm{E}}{\mathrm{E}^{\prime}}}-\frac{\mathrm{A}^{\mathrm{j}}-1}{2} \sqrt{\frac{\mathrm{E}^{\prime}}{\mathrm{E}}} , + +where G\ :sup:`j`\ (E′,E) is equal to the cosine of the angle of scatter +between the initial and final directions. The integral over the final +group (g) is evaluated analytically using routines developed by +J. A. Bucholz :cite:`bucholz_method_1978`. Integration over the initial group (g′) is then +performed numerically using the same method as for evaluating the +problem-dependent 1-D cross sections. + +Option N2D=2 adds the following term to the diagonal of the *l*\ :sub:`th` +moment of the PL elastic scatter matrix, + +.. math:: + :label: eq7-5-10 + + \left(\sigma_{l ; g, g}^{j}\right)_{n e w}=\left(\sigma_{l ; g, g}^{j}\right)_{o r i g}+\sigma_{t ; g}^{j}-\sigma_{t, l ; g}^{j} ; \quad 0 +XS_dilute. The expression used in PMC to compute the background cross +section :math:`\sigma_{0}^{(\mathrm{j})}` is given in the BONAMI section. + +.. _7-5-4: + +PMC Input Data +-------------- + +The Fido input blocks shown in this section are only required when +executing PMC as a standalone module. In the more typical case where PMC +is executed through the XSProc module during a SCALE sequence +calculation, the default parameter values are automatically defined +within XSProc. Default values for XSProc execution can be overridden +using keyword input in the CENTRM DATA block (see :ref:`7-4-4`). The +keyword input names correspond to the variable names given in this +section. + +.. centered:: **DATA BLOCK 1** + +**0$$ LOGICAL UNIT ASSIGNMENTS** (8 entries. Default values given in +parenthesis)\* + +1. LIBM = Input AMPX Master nuclear data library (22) + +2. LIBX = Input CENTRM pointwise nuclear data library (90) + +3. LIBF = Pointwise flux file produced by CENTRM (91) + +4. LIBNM = Output problem-dependent Master library created by PMC (92) + +5. LIBSC = Scratch unit (18) + +6. LIBSX = Scratch unit (24) + +*(*) Parameters in the 0$$ array cannot be modified for XSProc +execution.* + + +**1$$ INTEGER PARAMETERS** (10 entries ) + +1. MRANGE + + = 0, obsolete option + + = 1, Compute new group cross sections over resolved resonance range of + pointwise nuclides [from EUPR to ELOR given in CENTRM data library] + + = 2, Compute new group cross sections over pointwise flux range [from + DEMAX to DEMIN in CENTRM flux calculation] (2). + +2. N2D + + = -2, Apply removal correction to P0 elastic scatter matrix AND + apply consistent PN correction to higher order Legendre components; + normalize to 1D. + + −1, Apply elastic removal correction to P0 elastic scatter matrix; + normalize to 1D. + + = 0, Normalize P\ :sub:`N` components of original elastic scattering + matrix to new 1-D elastic value. + + = 1, Compute new P\ :sub:`N` components of elastic matrix, using scalar + flux as weighting function. + + = 2, Modify diagonal elements of the PN moments of the elastic matrix + using the consistent PN method (-1). + +3. NTHRM + + = 0 Treatment of thermal scatter kernels [not functional] (0) + +4. NPRT + + = −1, Minimum printed output; + + = 0, Standard print out; + + = 1, Also print new weighted cross sections for MT’s 1, 2, 18, and 102. + + = 2, Maximum amount of printed output includes 2D matrices (−1). + +5. NWT + + = 0, Generate zone-weighted multigroup data; + + = 1, Generate cell-weighted multigroup data (0). + +6. MTT + + = 0, Process all MT’s included in LIBX. [**NOTE:** With this + option, total cross section may not equal to sum of partials]; + + = 1, Process all MT’s except 1, 27, 101; then compute: + + MT 101 = sum of MT’s 102-114, + + MT 27 = sum of MT’s 18 and 101, + + MT 1 = sum of MT’s 2, 4, 16, 17, and 27 (1). + +7. PMC_OMIT + + = 0, Process all pointwise nuclides used in CENTRM + calculation; + + = 1, Process only nuclides in fuel zones. + + > 1, Process all materials except those in 2$$ array + +8. IXTR2 + + = 0, PMC run in CSAS standard sequence; + + = 1, PMC run in stand-alone mode (1); + + = 2 PMC run in CSAS double-heterogeneous cell sequence + +9. IXTR3 + + = −1, Process new data for all Legendre components on the input + AMPX master library up to P\ :sub:`7`. + + = N, Process new data through P\ :sub:`N` moments. [N=Scattering + Order+1] (−1). + +10. N1D + + = 0 Use CENTRM scalar flux for weighting function; + + = 1, Use the absolute value of CENTRM current for weighting function + (0). + +*1*\* REAL PARAMETERS** (10 entries) + +1. XS_DILUTE = background cross section (barns) considered to be +infinitely dilute (10\ :sup:`10`) + +2-10. Fill with 0.0 + + **T [ TERMINATE DATA BLOCK 1 ]** + +.. centered:: DATA BLOCK 2 : INDIVIDUAL NUCLIDES OMITTED FROM PROCESSING + +.. note:: This data cannot be entered for XSProc execution. + +**2$$ ISOTOPE IDENTIFIERS** (PMC_OMIT entries). Only enter PMC_OMIT > 1 + +[IDs of nuclides to be omitted from pointwise processing] + + **T [TERMINATE DATA BLOCK** + + + **END OF PMC INPUT DATA** + +.. _7-5-4-1: + +Notes for PMC users +~~~~~~~~~~~~~~~~~~~ + +1. N2D specifies the method used to process the P\ :sub:`N` components +of the 2-D elastic scattering matrices. In the option N2D=0, the +P\ :sub:`N` components of the original elastic scattering matrix are +simply re‑normalized using the new, problem-dependent 1-D elastic +values. This simple scaling approach often works well, but it does not +account for the impact of resonance self-shielding on the group removal +probability. The default option N2D= −1 approximately corrects the P0 +elastic matrix for removal self-shielding effects on and is usually +preferred to N2D=0, except for fast systems. Option N2D=1 re-computes +all the P\ :sub:`N` components of 2-D elastic cross sections using the +scalar flux as a weighting function, along with the assumption of +*s*-wave scattering within the PW energy range. This approach takes +significantly more execution time than N2D=-1, and usually is not +necessary. Option N2D=2 corrects the diagonal terms of the Legendre +moments, using the consistent PN expression. Option N2D=-2 is similar to +N2D=2, except the elastic removal correction is applied to the P0 moment +(Like for N2D=-1). Option N2D=-2 has been found to improve results for +many infinite lattice cases. + +2. NWT specifies whether the new multigroup cross sections are +zone-weighted or cell-weighted. When PMC is executed through XSProc, +nuclides are always zone-weighted unless the double-heterogeneous option +is specified in the CELLDATA block of the sequence input. Except for +double-heterogeneous cells, cell-weighting of the MG cross sections +should be done by the multigroup XSDRNPM calculation. + +3. PMC_OMIT is used to indicate which pointwise nuclides are processed +when computing new group cross sections. If PMC_OMIT=1, only nuclides in +fuel mixtures are processed. Fuel mixtures are defined as having at +least one material with Z ≥ 90. Option PMC_OMIT>1 only works for PMC +standalone runs, since there is no mechanism for inputting the 2$$ array +in sequences. + +4. IXTR3 is used to indicate through what Legendre order the scattering +matrices are to be processed. By default, in stand-alone mode all +P\ :sub:`N` moments on the Master library are processed, where as in a +SCALE sequence only through order N=5 are processed. With few +exceptions, the SCALE multigroup libraries contain scattering data +through P\ :sub:`5`. + +5. If input parameter XS_DILUTE > 0.0, PMC computes background cross +sections (σ\ :sub:`0`) for each material, and bypasses processing +materials with σ\ :sub:`0` > XS_DILUTE. The default of XS_DILUTE +=10\ :sup:`10` barns causes essentially all materials to be processed +regardless of dilution. Smaller XS_DILUTE values may reduce the number +of materials being processed, and hence reduce the execution time; +however, XS_DILUTE should not be so low that important absorbers are not +shielded. + +.. _7-5-5: + +Example Case +------------ + +Usually PMC is executed through one of the automated SCALE sequences +such as CSAS or TRITON where it is called by XSProc in conjunction with +other SCALE modules, such as CRAWDAD which provides the pointwise +nuclear data library and CENTRM which provides pointwise fluxes. In such +cases the user does not have to prepare input directly for PMC. + +.. _7-5-5-1: + +PMC input for example case +~~~~~~~~~~~~~~~~~~~~~~~~~~ + +An example of PMC stand-alone execution is given below, but it should be +noted that this PMC case cannot be executed unless it is linked to the +output data files produced by other modules. The example problem given +in the CENTRM chapter shows the coupled execution of several stand-alone +modules, including PMC, which mimics the function of XSProc. + +.. highlight:: scale + +:: + + =pmc + 0$$ -42 81 15 -42 18 19 17 + 1$$ 2 -1 0 0 0 1 0 0 5 0 + 1t + end + +.. _7-5-5-2: + +PMC output for example case +~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Only the printed output produced by PMC for the example problem is shown +here. In this case the “standard” PMC editing option (NPRT=0) was +specified. The XSProc default of “minimum” print in the SCALE sequences +produces considerably less output. + +:: + + program verification information + + code system: scale version: 6.0 + + + + program: pmc + + creation date: 18_nov_2008 + + library: /scale/scale6/Linux_x86_64/bin + + production code: pmc + + version: 6.0.9 + + jobname: xmw + + machine name: node12.ornl.gov + + date of execution: 05_dec_2008 + + time of execution: 13:22:19.23 + +:: + + 1 + + 0$ array 7 entries read + + 1$ array 10 entries read + + 1t + + **** LOGICAL UNITS **** + + nin = 5 Card Image Input Unit + nout = 6 Print Output Unit + libm = -42 Input Master Library + libx = 81 Input Pointwise XS Library + libf = 15 Input Pointwise Flux File + libnm = -42 Output Master Library + libsc = 18 Scratch Unit 1 + libsx = 19 Scratch Unit 2 + libsm = 17 scratch unit (master library) + + + + **** INPUT PARAMETERS **** + + mrange = 2 Option for choosing energy range 0 Averaging over pointwise xs limits + 1 Averaging over resolved resonance range + 2 Averaging over pointwise flux limits + + n2d = -1 Option for 2-D scat. calculation -1 Recompute self-scatter, then normalize 2-D elastic + data to shielded 1-D value + 0 Normalize 2-D elastic data to shielded 1-D value + 1 Recompute 2-D elastic using flux and s-wave kernel + 2 Recompute 2-D moments with flux-moments weighting + + nthrm = 0 Option for thermal scatter kernal + (NOT FUNCTIONAL) + + nprt = 0 Option for PMC print output -1 Minimum data printed + 0 Standard printed output + 1 Print 1-D XSs + 2 Print both 1-D and 2-D XSs + + nwt = 0 Option for XS averaging 0 Zone average + 1 Cell average + + mtt = 1 Option for total XS calculation 0 Average independently + 1 As sum of partial XS + + ixtr(1)= 0 Option for Processing PW Materials 0 Process all Pointwise Materials Used in CENTRM + N Omit N Materials + + ixtr(2)= 0 Option for calculation sequence 0 CSAS Standard Sequence + 1 Independant (stand-alone) Execution + 2 CSAS Doubly-Heterogeneous Cell Sequence + + ixtr(3)= 5 Legendre expansion order -1 Process all Legendre expansion moments found on AMPX LIB. + =0,...N Process only up through PN moments + + n1d = 0 Option for 1-D cross-sections 0 Weight using using scalar flux + 1 Weight using using abs value of current (1st moment) + +:: + + **** POINTWISE CROSS SECTION LIBRARY **** + + tape identifier 66666 + No. of nuclides 9 + Max no. of temperatures 2 + Max no. of processes 9 + Max no. of energy points 174194 + + + + **** POINTWISE FLUX FILE **** + + No. of nuclides 10 + No. flux moments 1 + No. of zones 3 + No. of energy points 48313 + Upper energy limit,demax 0.25000E+05 + Lower energy limit,demin 0.10000E-02 + + + + **** AMPX INPUT MASTER LIBRARY **** + + ID of the tape 238000 + No. of nuclides 10 + No. of neutron groups 238 + No. of gamma groups 0 + + + **** POINTWISE CROSS SECTION DIRECTORY **** + + ZA Pointwise Pointwise Unresolved Resolved Resolved + EMAX EMIN EMAX EMAX EMIN + 8016 0.2500E+05 0.1000E-02 0.0000E+00 0.0000E+00 0.0000E+00 + 40090 0.2500E+05 0.1000E-02 0.4000E+06 0.6000E+05 0.0000E+00 + 40091 0.2500E+05 0.1000E-02 0.1000E+06 0.2000E+05 0.0000E+00 + 40092 0.2500E+05 0.1000E-02 0.1000E+06 0.7100E+05 0.0000E+00 + 40094 0.2500E+05 0.1000E-02 0.1000E+06 0.9000E+05 0.0000E+00 + 40096 0.2500E+05 0.1000E-02 0.1000E+06 0.1000E+06 0.0000E+00 + 92235 0.2500E+05 0.1000E-02 0.2500E+05 0.2250E+04 0.0000E+00 + 92238 0.2500E+05 0.1000E-02 0.1490E+06 0.2000E+05 0.0000E+00 + 1001 0.2500E+05 0.1000E-02 0.0000E+00 0.0000E+00 0.0000E+00 + +:: + + **** NUCLIDES IN POINTWISE FLUX CALCULATION **** + Zone IR(# of nuclides) Temperature + 1 3 900.0 + 2 5 600.0 + 3 2 600.0 + + -- Nuclide by Zone -- + 0 -- no; 1 -- yes + + ID:: 1008016 3008016 2040090 2040091 2040092 2040094 + ZONE:: + 1 1 0 0 0 0 0 + 2 0 0 1 1 1 1 + 3 0 1 0 0 0 0 + ID:: 2040096 1092235 1092238 3001001 + ZONE:: + 1 0 1 1 0 + 2 1 0 0 0 + 3 0 0 0 1 + + -- Atom Density by Zone -- + ID:: 1008016 3008016 2040090 2040091 2040092 2040094 + ZONE:: + 1 4.5968E-02 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 + 2 0.0000E+00 0.0000E+00 2.5714E-02 5.6076E-03 8.5714E-03 8.6863E-03 + 3 0.0000E+00 2.3831E-02 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 + + -- Averaged Cell Atom Density -- + 2.2461E-02 1.0506E-02 1.8133E-03 3.9544E-04 6.0445E-04 6.1255E-04 + + + ID:: 2040096 1092235 1092238 3001001 + ZONE:: + 1 0.0000E+00 4.8838E-04 2.2480E-02 0.0000E+00 + 2 1.3994E-03 0.0000E+00 0.0000E+00 0.0000E+00 + 3 0.0000E+00 0.0000E+00 0.0000E+00 4.7662E-02 + + -- Averaged Cell Atom Density -- + 9.8685E-05 2.3863E-04 1.0984E-02 2.1012E-02 + + + + **** INPUT MASTER LIB. DIRECTORY **** + + nmt: No. of 1-D Neutron Processes + nbond: No. of Sets of Bondarenko Data + nrec: No. of Records for this Nuclide + + id za nmt nbond nrec + + 1008016 8016.0 49 0 3 + 3008016 8016.0 49 0 3 + 2040090 40090.0 86 0 3 + 2040091 40091.0 45 0 3 + 2040092 40092.0 47 0 3 + 2040094 40094.0 39 0 3 + 2040096 40096.0 32 0 3 + 1092235 92235.0 77 0 3 + 1092238 92238.0 77 0 3 + 3001001 1001.0 10 0 3 + +:: + + P r o c e s s i n g N u c l i d e 1008016 + + Energy Range for Multigroup Averaging of this Data + EH = 2.50000E+04 EL = 1.00000E-03 + + INFORMATION ON CENTRM POINTWISE XS LIB: + + MT ENERGY POINTS TEMPERATURE (K) + 1 459 600.0 900.0 + 2 459 600.0 900.0 + 102 459 600.0 900.0 + + <<<<< ZONE: 1 >>>>> + + PROCESSING MT = 1 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 2 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 102 + Generating new multigroup data from group 55 through group 234 + + ====>> Done Processing Shielded Zone-Averaged Cross Section 1008016 + + + P r o c e s s i n g N u c l i d e 3008016 + + Energy Range for Multigroup Averaging of this Data + EH = 2.50000E+04 EL = 1.00000E-03 + + INFORMATION ON CENTRM POINTWISE XS LIB: + + MT ENERGY POINTS TEMPERATURE (K) + 1 459 600.0 900.0 + 2 459 600.0 900.0 + 102 459 600.0 900.0 + + <<<<< ZONE: 3 >>>>> + + PROCESSING MT = 1 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 2 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 102 + Generating new multigroup data from group 55 through group 234 + + ====>> Done Processing Shielded Zone-Averaged Cross Section 3008016 + +:: + + P r o c e s s i n g N u c l i d e 2040090 + + Energy Range for Multigroup Averaging of this Data + EH = 2.50000E+04 EL = 1.00000E-03 + + INFORMATION ON CENTRM POINTWISE XS LIB: + + MT ENERGY POINTS TEMPERATURE (K) + 1 4488 600.0 + 2 4488 600.0 + 102 4488 600.0 + + <<<<< ZONE: 2 >>>>> + + PROCESSING MT = 1 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 2 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 102 + Generating new multigroup data from group 55 through group 234 + + ====>> Done Processing Shielded Zone-Averaged Cross Section 2040090 + + + P r o c e s s i n g N u c l i d e 2040091 + + Energy Range for Multigroup Averaging of this Data + EH = 2.50000E+04 EL = 1.00000E-03 + + INFORMATION ON CENTRM POINTWISE XS LIB: + + MT ENERGY POINTS TEMPERATURE (K) + 1 24295 600.0 + 2 24295 600.0 + 102 24295 600.0 + + <<<<< ZONE: 2 >>>>> + + PROCESSING MT = 1 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 2 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 102 + Generating new multigroup data from group 55 through group 234 + + ====>> Done Processing Shielded Zone-Averaged Cross Section 2040091 + +:: + + + P r o c e s s i n g N u c l i d e 2040092 + + Energy Range for Multigroup Averaging of this Data + EH = 2.50000E+04 EL = 1.00000E-03 + + INFORMATION ON CENTRM POINTWISE XS LIB: + + MT ENERGY POINTS TEMPERATURE (K) + 1 8142 600.0 + 2 8142 600.0 + 102 8142 600.0 + + <<<<< ZONE: 2 >>>>> + + PROCESSING MT = 1 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 2 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 102 + Generating new multigroup data from group 55 through group 234 + + ====>> Done Processing Shielded Zone-Averaged Cross Section 2040092 + + + P r o c e s s i n g N u c l i d e 2040094 + + Energy Range for Multigroup Averaging of this Data + EH = 2.50000E+04 EL = 1.00000E-03 + + INFORMATION ON CENTRM POINTWISE XS LIB: + + MT ENERGY POINTS TEMPERATURE (K) + 1 8068 600.0 + 2 8068 600.0 + 102 8068 600.0 + + <<<<< ZONE: 2 >>>>> + + PROCESSING MT = 1 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 2 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 102 + Generating new multigroup data from group 55 through group 234 + + ====>> Done Processing Shielded Zone-Averaged Cross Section 2040094 + + + P r o c e s s i n g N u c l i d e 2040096 + + Energy Range for Multigroup Averaging of this Data + EH = 2.50000E+04 EL = 1.00000E-03 + + INFORMATION ON CENTRM POINTWISE XS LIB: + + MT ENERGY POINTS TEMPERATURE (K) + 1 4944 600.0 + 2 4944 600.0 + 102 4944 600.0 + + <<<<< ZONE: 2 >>>>> + + PROCESSING MT = 1 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 2 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 102 + Generating new multigroup data from group 55 through group 234 + + ====>> Done Processing Shielded Zone-Averaged Cross Section 2040096 + +:: + + P r o c e s s i n g N u c l i d e 1092235 + + Energy Range for Multigroup Averaging of this Data + EH = 2.50000E+04 EL = 1.00000E-03 + + INFORMATION ON CENTRM POINTWISE XS LIB: + + MT ENERGY POINTS TEMPERATURE (K) + 1 59851 900.0 + 2 59851 900.0 + 18 59851 900.0 + 102 59851 900.0 + 51 94 0.0 + 52 74 0.0 + 452 48 0.0 + 455 6 0.0 + 456 48 0.0 + + <<<<< ZONE: 1 >>>>> + + PROCESSING MT = 1 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 2 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 18 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 102 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 51 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 52 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 452 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 455 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 456 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 1018 + Collapsing 2D chi to effective 1D + + ====>> Done Processing Shielded Zone-Averaged Cross Section 1092235 + +:: + + P r o c e s s i n g N u c l i d e 1092238 + + Energy Range for Multigroup Averaging of this Data + EH = 2.50000E+04 EL = 1.00000E-03 + + INFORMATION ON CENTRM POINTWISE XS LIB: + + MT ENERGY POINTS TEMPERATURE (K) + 1 174194 900.0 + 2 174194 900.0 + 18 174194 900.0 + 102 174194 900.0 + 452 10 0.0 + 455 4 0.0 + 456 10 0.0 + + <<<<< ZONE: 1 >>>>> + + PROCESSING MT = 1 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 2 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 18 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 102 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 452 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 455 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 456 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 1018 + Collapsing 2D chi to effective 1D + + ====>> Done Processing Shielded Zone-Averaged Cross Section 1092238 + + + P r o c e s s i n g N u c l i d e 3001001 + + Energy Range for Multigroup Averaging of this Data + EH = 2.50000E+04 EL = 1.00000E-03 + + INFORMATION ON CENTRM POINTWISE XS LIB: + + MT ENERGY POINTS TEMPERATURE (K) + 1 324 600.0 + 2 324 600.0 + 102 324 600.0 + + <<<<< ZONE: 3 >>>>> + + PROCESSING MT = 1 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 2 + Generating new multigroup data from group 55 through group 234 + PROCESSING MT = 102 + Generating new multigroup data from group 55 through group 234 + + ====>> Done Processing Shielded Zone-Averaged Cross Section 3001001 + + + elapsed time 0.01 min. + +:: + + Number of nuclides in new master library 10 + + The Output AMPX Master Library Produced by PMC + + Logical Unit No. -42 + Tape ID No. 238000 + No. of Weighted Cross Section Sets 10 + No. of Neutron Groups 238 + No. of Gamma Groups 0 + First Thermal Neutron Group 149 + + Contents of Output Master Library + + o16 825 endfb7 rel8 rev7 mod3 08/13/08 ID 1008016 + o16 825 endfb7 rel8 rev7 mod3 08/13/08 ID 3008016 + zr90 4025 endfb7 rel0 rev7 mod1 08/13/08 ID 2040090 + zr91 4028 endfb7 rel0 rev7 mod1 08/13/08 ID 2040091 + zr92 4031 endfb7 rel3 rev7 mod4 08/13/08 ID 2040092 + zr94 4037 endfb7 rel3 rev7 mod1 08/13/08 ID 2040094 + zr96 4043 endfb7 rel0 rev7 mod1 08/13/08 ID 2040096 + u235 9228 endfb7 rel0 rev7 mod7 08/13/08 ID 1092235 + u238 9237 endfb7 rel6 rev7 mod5 08/13/08 ID 1092238 + h_h2o 1 endfbv7 rel0 rev7 mod0 09/29/08 ID 3001001 + + elapsed time 0.02 min. + + **** PMC CALCULATION COMPLETED **** + +.. _7-5-6: + +Formats of Data Files +--------------------- + +The CENTRM chapter of the SCALE manual describes the format for the CENTRM +PW nuclear data library and the format of the output PW flux file produced by +CENTRM, which is input to the PMC code. + + +.. bibliography:: bibs/PMC.bib diff --git a/PMCAppAB.rst b/PMCAppAB.rst new file mode 100644 index 0000000..2c92a70 --- /dev/null +++ b/PMCAppAB.rst @@ -0,0 +1,30 @@ +.. _7-5a: + +PMC Appendices A and B +====================== + +.. _7-5a-1: + +Alphabetical Index of Subroutines +--------------------------------- +This section provides a convenient alphabetical index of the subroutines used +in PMC, the subroutines that call them and the subroutines they call. + +.. list-table:: + :align: center + + * - .. image:: figs/PMCAppAB/tab1.svg + :width: 600 + +.. _7-5b-1: + +Alphabetical Index of Modules +----------------------------- + +This section provides a list of the modules used in PMC and the subroutines that reference them. + +.. list-table:: + :align: center + + * - .. image:: figs/PMCAppAB/tab2.svg + :width: 500 diff --git a/XSProc.rst b/XSProc.rst new file mode 100644 index 0000000..9e8f6df --- /dev/null +++ b/XSProc.rst @@ -0,0 +1,2945 @@ +.. _7-1: + +XSPROC: The Material and Cross Section Processing Module for SCALE +================================================================== + +*M. L. Williams, L. M. Petrie, R. A. Lefebvre, K. T. Clarno, J. P. +Lefebvre, U. Merturyek, D. Wiarda, and B. T. Rearden* + +ABSTRACT + +The modern material and cross section processing module of SCALE +(XSProc) was developed for the 6.2 release to prepare data for +continuous-energy and multigroup calculations. XSProc expands material +input from Standard Composition Library definitions into atom number +densities and, for multigroup calculations, performs cross section +resonance self-shielding, energy group collapse, and spatial +homogenization. XSProc implements capabilities for problem-dependent +temperature interpolation, calculation of Dancoff factors, resonance +self-shielding using Bondarenko factors with full-range intermediate +resonance treatment, as well as use of continuous energy resonance +self-shielding in the resolved resonance region. XSProc integrates and +enhances the capabilities previously implemented independently in +BONAMI, CENTRM, PMC, WORKER, ICE, and XSDRNPM, along with some +additional capabilities that were provided by MIPLIB and SCALELIB. The +use of the modern XSProc sequence instead of the legacy codes of +previous versions of SCALE generally results in the preparation of cross +sections in less time, with substantial speedups for more I/O bound +problems. Additionally, the memory requirements of XSProc are improved +by generating only the data needed for a particular calculation instead +of generating a general-purpose library that contains substantial +amounts of data that are not needed for a particular calculation. + +ACKNOWLEDGMENTS + +XSProc has evolved from the concept of a Material Information Processor +library (MIPLIB) that used alphanumeric material specifications, which +was initially proposed and developed by R. M. Westfall. J. R. Knight and +J. A. Bucholz expanded and refined MIPLIB in early SCALE releases. +Through SCALE 6.1, many enhancements were made by S. Goluoglu, D. F. +Hollenbach, N. F. Landers, J. A. Bucholz, C. F. Weber, and C. M. Hopper, +with L. M. Petrie taking the lead responsibility. With the SCALE +modernization initiative beginning in SCALE 6.2, MIPLIB is no longer +part of the XSProc analysis, but the original concepts and input +formatting were preserved in the new implementation. The authors wish to +thank Dan Ilas for helping convert the original MIPLIB documentation and +Sheila Walker for editing and formatting this document. Special thanks +to Don Mueller for his detailed review and checking of the document. + +.. _7-1-1: + +Introduction +------------ + +Self-shielding of multigroup cross sections is required in SCALE +sequences for criticality safety, reactor physics, radiation shielding, +and sensitivity analysis. In all previous versions of SCALE, resonance +self-shielding calculations were done by executing a series of +stand-alone executable codes, each dedicated to a specific aspect of the +self-shielding operations. Each sequence had its own unique internal +coding to launch the executable codes. Multigroup (MG) and +continuous-energy (CE) cross sections and other data were passed between +the individual executable codes by external I/O, which could require a +substantial amount of clock time. In the modern version of SCALE, all +self-shielding operations are consolidated into a single driver module +named XSProc, and the stand-alone executable codes have been transformed +into callable “computational modules" :cite:`rearden_modernization_2015`. The +functions of XSProc are to (a) read input data, (b) generate in-memory +data structures (objects) containing problem-definition information +(compositions, cell geometries, computation options), as well as +self-shielding information (MG and CE cross sections and fluxes), and +(c) execute appropriate computational modules for the requested +self-shielding option. Calculated results produced by one module may be +stored in the internal data objects and passed to other modules through +application program interfaces (APIs). At the completion of XSProc the +self-shielded MG cross sections on the data objects can be passed along +to transport solvers for continued execution of the control sequence or +can be written to an external AMPX library file. + +In the future, XSProc will be extended to parallel computations in which +self-shielding calculations are done simultaneously for multiple types +of unit cells. At the present time, however, XSProc is limited to serial +computations; but even in serial mode it typically requires less time +than older versions of SCALE to process shielded cross sections, and +significant speedups have been observed for heavily I/O bound problems. +Integrating the self-shielding capabilities into a single module has a +number of additional benefits as well, including maintainability, +extensibility, and the ability to easily replace an entire computational +module with a future implementation containing new features. +Additionally, the size of the problem-dependent MG library generated by +XSProc may be greatly reduced compared to previous versions of SCALE +because macroscopic cross sections are stored rather than a +general-purpose library of microscopic data. + +.. _7-1-2: + +Techniques +---------- + +XSProc integrates and enhances the capabilities previously implemented +independently in BONAMI, CENTRM, PMC, WORKER, ICE, and XSDRNPM, as well +as other capabilities formerly provided by MIPLIB and SCALELIB. It +provides capabilities for problem-dependent temperature interpolation of +both CE and MG nuclear data, calculation of Dancoff factors, and +resonance self-shielding of MG cross sections using several available +options. XSProc produces shielded microscopic data for each nuclide or +macroscopic data for each material. Additionally, a flux-weighting +spectrum can be applied to collapse cross sections to a coarser group +structure and/or to integrate over volumes for homogenized cross +sections. The flux-weighting spectrum can be input by the user or +calculated using one-dimensional (1-D) coupled neutron/gamma transport +model. These operations are performed by the sequences CSAS-MG, CSAS1, +CSASI, and T-XSEC described in :ref:`7-1-3-2`. + +.. _7-1-2-1: + +Overview of XSProc procedures +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +XSProc reads the COMPOSITION and CELL DATA blocks of the SCALE input, +which are described in the following sections. After reading the user +input data, XSProc loads the specified MG library to be self-shielded +and, depending on the selected self-shielding method, additional CE data +files for nuclides appearing in the problem specification. Finally +XSProc performs MG self-shielding calculations for all compositions by +calling APIs to computational modules such as BONAMI (**BON**\ darenko +**AM**\ PX **I**\ nterpolator), CRAWDAD (**C**\ ode to **R**\ ead +**A**\ nd **W**\ rite **DA**\ ta for **D**\ iscretized solution), CENTRM +(**C**\ ontinuous **EN**\ ergy **TR**\ ansport **M**\ odule), PMC +(**P**\ roduce **M**\ ultigroup **C**\ ross sections), CHOPS +(**C**\ ompute **HO**\ mogenized **P**\ ointwise **S**\ tuff), CAJUN +(**C**\ E **AJ**\ AX **UN**\ iter), WAX (**W**\ orking +**A**\ JA\ **X**), XSDRNPM (**X**\ ‑\ **S**\ ection **D**\ evelopment +for **R**\ eactor **N**\ ucleonics with **P**\ etrie +**M**\ odifications), and/or MIXMACRO to provide a problem-dependent +cross section library. Many computational modules have been modernized +compared to earlier executable codes distributed in previous versions of +SCALE. + +Like earlier versions of SCALE, XSProc provides several options for +self-shielding an input MG library :cite:`williams_resonance_2011`. The first, based on the +Bondarenko method :cite:`ilich_bondarenko_group_1964`, uses the computational module BONAMI. BONAMI is +always used to compute self-shielded cross sections for all energy +groups. If *parm=bonami* is specified, the shielded cross sections +provided by BONAMI are the final values output from XSProc. However the +Bondarenko method has several limitations, especially in the resolved +resonance range. Therefore XSProc provides another self-shielding +method, with several computation options, which often produces more +accurate MG data in the resolved resonance and thermal energy ranges. If +*parm=centrm* or *parm=2region* is specified on the sequence line, +XSProc calls APIs for the modules CRAWDAD, CENTRM, and PMC to compute CE +flux spectra for processing problem-specific, self-shielded cross +sections “on the fly :cite:`williams_computation_1995`. CENTRM performs MG transport calculations in +the fast and lower energy ranges, coupled to pointwise (PW) transport +calculations that use CE cross sections in the resonance range. PMC uses +the PW flux spectra from CENTRM to compute MG values, which replace the +previous values obtained from BONAMI over the specified range of the CE +calculation. The original BONAMI shielded cross sections are retained +for all other groups. + +The CENTRM/PMC approach is the default for criticality and lattice +physics calculations, while the BONAMI-only method is default for +radiation shielding calculations. The end results of an XSProc +calculation are self-shielded macroscopic and/or microscopic MG cross +sections stored in memory for subsequent transport calculations; or +alternatively a shielded MG AMPX library can be written to an external +file and saved for future use. + +.. _7-1-2-2: + +Standard composition material processing +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +A primary function of the XSProc module is to expand user input in the +COMPOSITION block into nuclear number densities (nuclei/b-cm) for every +nuclide in each defined mixture. Mixtures can be specified through the +direct use of materials presented in the Standard Composition Library, +which includes individual nuclides, elements with natural abundances, +numerous compounds, alloys and mixtures found in engineering practice, +as well several variations of fissile solutions. Additionally, users may +define their own materials as atom percent or weight percent +combinations. Nuclear masses and theoretical densities are provided in +the Standard Composition Library, and methods are available to determine +equilibrium states for fissile solutions. Input options for composition +data are described in :ref:`7-1-3-3` with several examples provided in +Appendix A. + +.. _7-1-2-3: + +Unit cells for MG resonance self-shielding +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +XSProc utilizes a unit cell description to provide information for +resonance self-shielding calculations of the input mixtures. As many +unit cells as needed to describe the problem may be specified; however, +each mixture (other than 0 for a void mixture) can appear only in one +unit cell in the CELLDATA block. If a nuclide appears in more than one +mixture, multiple sets of self-shielded cross sections are calculated +for the nuclide—one for each mixture in each unit cell. Four types of +cells are available for self-shielding calculations: **INFHOMMEDIUM**, +**LATTICECELL**, **MULTIREGION**, and **DOUBLEHET**. The default +calculation type is CENTRM/PMC for CSAS (see :ref:`CSAS5`), TRITON, (see +:ref:`3-0`) and TSUNAMI (see :ref:`6-0`) sequences and BONAMI for MAVRIC. +All materials not specified in a unit cell are treated as infinite +homogeneous media and shielded with BONAMI only, unless the mixture +contains a fissionable nuclide, in which case an infinite medium +CENTRM/PMC model is used. Note that previous versions of SCALE used +infinite medium CENTRM/PMC calculations for all unassigned mixtures. The +default type of self-shielding calculation can be overridden, as +described in :ref:`7-1-3-2`. The following is a brief description of +the types of unit cells that can be input in CELLDATA and the +computation procedures used. + +.. _7-1-2-3-1: + +INFHOMMEDIUM (infinite homogeneous medium) Treatment +^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ + +The **INFHOMMEDIUM** treatment is best suited for large masses of +materials where the size of each material is large compared with the +average mean-free path of the material or where the fraction of the +material that is a mean-free path from the surface of the material is +very small. When **INFHOMMEDIUM** cell is specified, the material in the +unit cell is treated as an infinite homogeneous lump. Systems composed +of small fuel lumps or resonance nuclides sandwiched between moderating +regions should not be treated as infinite homogeneous media. In these +cases a MULTIREGION or LATTICECELL geometry should be used. + +.. _7-1-2-3-2: + +LATTICECELL Treatment +^^^^^^^^^^^^^^^^^^^^^^ + +The **LATTICECELL** model is appropriate for arrays of resonance +absorber mixtures—with or without clad—arranged in a square or a +triangular pitch configuration within a moderator. Annular fuel (e.g., +with an internal moderator in the center) can also be addressed. Input +data for the **LATTICECELL** treatment are described in :ref:`7-1-3-5`. +Self-shielded cross sections are generated for each material zone in a +unit cell of the lattice. If a nuclide appears in more than one zone, +self-shielded cross sections are produced for each zone where the +nuclide is present. Limitations of the **LATTICECELL** treatment are +listed below. + +1. The cell description is limited to unit cells for arrays of + spherical, plate (slab), or cylindrical fuel bodies. In the case + of cylindrical pins in a square-pitch lattice, the default + (*parm=centrm*) self-shielding calculation uses the CENTRM method + of characteristics (MoC) option to represent the 2D rectangular + unit cell with reflected boundary conditions. By default, + self-shielding for all other arrays uses a CENTRM 1D S\ :sub:`N` + calculation for the unit cell (spherical and cylindrical + geometries use Wigner-Seitz cells). If *parm=bonami* is specified, + heterogeneous self-shielding effects are treated by equivalence + theory :cite:`williams_resonance_2011` The computation option *parm=2region*, described + in :ref:`7-1-3-1`, can also be used for self-shielding lattice + cells. + +2. Only predefined choices of cell configurations are available. The + available options are described in detail in :ref:`7-1-3-5`. + +3. The basic treatment for **LATTICECELL** assumes an infinite, uniform + array of unit cells. This assumption is a good approximation for + interior fuel regions within a large, uniform array. The + approximation becomes less rigorous for fuel regions on the + periphery of the array or adjacent to a nonuniformity (e.g., + control rod, water hole, etc.) in the lattice. For some cases it + may be desirable to address this issue by specifying a different + lattice cell for this type of fuel pin and using a modified + procedure to define an effective unit cell, as described below. + +*\****\* LATTICECELL treatment for nonuniform arrays*. + +Nonuniform lattice effects may be treated in CENTRM calculation by +specifying the keyword **DAN2PITCH=**\ *dancoff* in the optional CENTRM +DATA (see :ref:`7-1-3-9`). In this approach, the SCALE standalone code +MCDancoff must be run prior to the self-shielding calculation in order +to compute Dancoff factors for the fuel regions of interest in the +nonuniform lattice configuration. MCDancoff performs a simplified +one-group Monte Carlo calculation to compute Dancoff factors for complex +geometries (see :ref:`7-8`). The Dancoff value for the fuel region of +interest is assigned to the DAN2PITCH keyword in the input for the +corresponding cell. Using an iterative procedure, CENTRM computes the +pitch of a uniform lattice that has the same Dancoff value as the +nonuniform lattice. + +.. _7-1-2-3-3: + +MULTIREGION Treatment +^^^^^^^^^^^^^^^^^^^^^ + +The **MULTIREGION** treatment is appropriate for 1-D geometric regions +where the geometry effects may be important, but the limited number of +zones and boundary conditions in the **LATTICECELL** treatment are not +applicable. The **MULTIREGION** unit cell allows more flexibility in the +placement of the mixtures but requires all regions of the cell to have +the same geometric shape (i.e., slab, cylinder, sphere, buckled slab, or +buckled cylinder). Lattice arrangements can be approximated by +specifying a non-vacuum boundary condition on the outer boundary. See +:ref:`7-1-3-6` for more details. Limitations of the **MULTIREGION** +cell treatment are listed below. + +1. A **MULTIREGION** cell is limited to a 1-D approximation of the + system being represented. An exact 1D model can be defined for the + following multizone geometries with vacuum boundary conditions: + spheres, infinitely long cylinders, and slabs; and for an infinite + array of slabs with reflected or periodic boundaries. + +2. The shape of the outer boundary of the **MULTIREGION** cell is the + same as the shape of the inner regions. Cells with curved outer + surfaces cannot be stacked physically to create arrays; however, + arrays can be approximated by a Wigner-Sietz cell with a white + outer boundary condition, where the outer radius is defined to + preserve the area of the true rectangular or hexagonal unit cell. + +3. Boundary conditions available in a **MULTIREGION** problem include + vacuum (eliminated at the boundary), reflected (reflected about + the normal to the surface at the point of impact), periodic + (a particle exiting the surface effectively enters an identical + cell having the same orientation and continues traveling in the + same direction), and white (isotropic return about the point of + impact). Reflected and periodic boundary conditions on a slab can + represent real physical situations but are not valid on a curved + outer surface. A single, non-interacting cell has a vacuum outer + boundary condition. If the cell outer boundary condition is not a + vacuum boundary, the unit cell approximates some type of array. + +4. When using the CENTRM/PMC self-shielding method, the MULTIREGION cell + model must include fissionable material. This can be accomplished + by adding a trace amount of a fissionable material to one or more + mixtures, or by modeling a region of homogenized fuel and water, + or by adding a thin (e.g., 1e-6 cm-thick) layer containing at + least a trace of a fissionable nuclide on the periphery of the + problem. + +.. _7-1-2-3-4: + +DOUBLEHET Treatment +^^^^^^^^^^^^^^^^^^^ + +**DOUBLEHET** cells use a specialized CENTRM/PMC calculational approach +to treat resonance self-shielding in “doubly heterogeneous” systems. The +fuel for these systems typically consists of small, heterogeneous, +spherical fuel particles (grains) embedded in a moderator matrix to form +the fuel compact. The fuel-grain/matrix compact constitutes the first +level of heterogeneity. Cylindrical(rod). spherical (pebble), or slab +fuel elements composed of the compact material are arranged in a +moderating medium to form a regular or irregular lattice, producing the +second level of heterogeneity. The fuel elements are also referred to as +“macro cells.” Advanced reactor fuel designs that use TRISO +(tri-material, isotopic) or fully ceramic microencapsulated (FCM) fuel +require the **DOUBLEHET** treatment to account for both levels of +heterogeneities in the self-shielding calculations. Simply ignoring the +double-heterogeneity by volume-weighting the fuel grains and matrix +material into a homogenized compact mixture can result in a large +reactivity bias. + +In the **DOUBLEHET** cell input, keywords and the geometry description +for grains are similar to those of the **MULTIREGION** treatment, while +keywords and the geometry for the fuel element (macro-cell) are similar +to those of the **LATTICECELL** treatment. The following rules apply to +the **DOUBLEHET** cell treatment and must be followed. Violation of any +rules may cause a fatal error. + +1. As many grain types as needed may be specified for each unique fuel + element. Note that grain type is different from the number of grains + of a certain type. For example, a fuel element that contains both + UO\ :sub:`2` and PuO\ :sub:`2` grains has two grain types. The same + fuel element may contain 10000 UO\ :sub:`2` grains and + 5000 PuO\ :sub:`2` grains. In this case, the number of grains of type + UO\ :sub:`2` is 10000, and the number of grains of type PuO\ :sub:`2` + is 5000. + +2. As many fuel elements as needed may be specified, each requiring its + own **DOUBLEHET** cell. This may be the case for systems with many + fuel elements at different fuel enrichments, burnable poisons, etc. + Each fuel element may have one or more grain types. + +3. Since the grains are homogenized into a new mixture to be used in the + fuel element (macro-cell) cell calculation, a unique fuel mixture + number must be entered. XSProc creates a new material with the new + mixture number designated by the keyword f\ *uelmix=*, containing all + the nuclides that are homogenized. The user must assign the new + mixture number in the transport solver geometry (e.g., KENO) input + unless a cell-weighted mixture is created. + +4. The type of lattice or array configuration for the fuel-element may + be spheres on a triangular pitch (**SPHTRIANGP**), spheres on a + square pitch (**SPHSQUAREP**), annular spheres on a triangular pitch + (**ASPHTRIANGP)**, annular spheres on a square pitch + (**ASPHSQUAREP)**, cylindrical rods on a triangular pitch + (**TRIANGPITCH**), cylindrical rods on a square pitch + (**SQUAREPITCH**),annular cylinderical rods on a triangular pitch + (**ATRIANGPITCH)**, annular cylindrical rods on a square pitch + (**ASQUAREPITCH)**, a symmetric slab (**SYMMSLABCELL)**, or an + asymmetric slab (**ASYMSLABCELL)**. + +5. If there is only one grain type for a fuel element, the user must + enter either the pitch, the aggregate number of particles in the + element, or the volume fraction for the grains. The code needs the + pitch and will directly use it if entered. If pitch is not given, + then the volume fraction (if given) is used to calculate the pitch. + If neither the pitch nor the volume fraction is given, then the + number of particles is used to calculate the pitch and the volume + fraction. The user should only enter one of these items. + +.. + + If the fuel matrix contains more than one grain type, all types are + homogenized into a single mixture for the compact. As for the one + grain type case, the pitch is needed for the spherical cell + calculations. However, the pitch by itself is not sufficient to + perform the homogenization. Since each grain’s volume is known (grain + dimensions must always be entered), entering the number of particles + for each grain type essentially provides the total volume of each + grain type and therefore enables the calculation of the volume + fraction and the pitch. Likewise, entering the volume fraction for + each grain type essentially provides the total volume of each grain + type and therefore enables the calculation of the number of particles + and the pitch. Therefore, one of these two quantities must be entered + for multiple grain types. In these cases, since pitch is not given, + the available matrix material is distributed around the grains of + each grain type proportional to the grain volume and is used to + calculate the corresponding pitch. Over-specification is allowed as + long as the values are not inconsistent to greater than 0.01%. + +6. For cylindrical rods and for slabs, fuel height must also be + specified. For slabs the slab width must also be specified. + +7. The CENTRM calculation option must be S\ :sub:`n`. + +.. _7-1-2-4: + +Cell weighting of MG cross sections +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Cell-weighted self-shielded cross sections are created when +**CELLMIX**\ = is specified in a **LATTICECELL** or **MULTIREGION** cell +input. In this case, after finishing the self-shielding calculations for +all mixtures in the cell, XSProc calls the computational module XSDRNPM, +which solves the 1-D MG transport equation to obtain k\ :sub:`∞` and +space-dependent MG fluxes for the cell. The resultant fluxes are used to +compute MG flux disadvantage factors for processing cell-weighted +cross sections of all nuclides in the cell. When the cell-weighted +cross sections are used with *homogenized* number densities of the cell +nuclides, the reaction rates of the homogenized mixture preserve the +spatially averaged reactions rates of the heterogeneous configuration. +The user must input a new mixture ID to identify the homogenized mixture +associated with the cell-weighted cross sections. **This homogenized +mixture should not be used in the heterogeneous geometry data for other +transport codes such as KENO, NEWT, etc.** Instead, the cell-homogenized +mixture that is created should be used at the location of the original +cell. Also, cell weighted homogenized cross sections should not be used +in MG sensitivity data calculations performed using the TSUNAMI +sequences. + +.. _7-1-3: + +XSPROC Input Data Guide +----------------------- + +XSProc input data are entered in free form, allowing alphanumeric data, +floating-point data, and integer data to be entered in an unstructured +manner. Up to 252 characters per line are allowed. Data can usually +start or end in any column. Each data entry must be followed by one or +more blanks to terminate the data entry. For numeric data, either a +comma or a blank can be used to terminate each data entry. Integers may +be entered for floating values. For example, 10 will be interpreted as +10.0 if a floating point value is required. Imbedded blanks are not +allowed within a data entry unless an E precedes a single blank as in an +unsigned exponent in a floating-point number. For example, 1.0E 4 would +be correctly interpreted as 1.0 × 10\ :sup:`4`. A number with a negative +exponent must include an “E”. For example 1.0-4 cannot be used for +1.0E-4. + +The word “END” is a special data item. An END may have a name or label +associated with it. The name or label associated with an END is +separated from the END by a single blank and is a maximum of +12 characters long. *At least two blanks or a new line MUST follow every +labeled and unlabeled END. WARNING: It is the user’s responsibility to +ensure compliance with this restriction. Failure to observe this +restriction can result in the use of incorrect or incomplete data +without the benefit of warning or error messages.* + +Multiple entries of the same data value can be achieved by specifying +the number of times the data value is to be entered, followed by either +R, \*, or $, followed by the data value to be repeated. Imbedded blanks +are not allowed between the number of repeats and the repeat flag. For +example, 5R12, 5*12, 5$12, or 5R 12, etc., will enter five successive +12’s in the input data. Multiple zeros can be specified as nZ where n is +the number of zeroes to be entered. + +.. _7-1-3-1: + +XSProc data checking and resonance processing options +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +.. highlight:: scale + +To check the XSProc input data, run CSAS-MG and specify PARM=CHECK or +PARM=CHK after the sequence specification as shown below. + +:: + + =CSAS-MG PARM=CHK + +In this case the actual XSProc cross section processing calculations are +not performed. The input data are checked, the problem description is +printed, appropriate error and warning messages are printed, and a table +of additional data is printed. + +Resonance processing will automatically be performed by the default +method for the sequence selected. The default methods are CENTRM/PMC for +CSAS, TRITON, and TSUNAMI sequences and BONAMI for the MAVRIC sequences. +Alternatively, a resonance processing procedure may be chosen by +entering PARM=\ *option*, where *option* CENTRM selects the recommended +CENTRM/PMC transport method for each cell type, *option* 2REGION selects +the CENTRM/PMC two-region calculation, and *option* BONAMI applies full +range Bondarenko factors to all energy groups without utilizing +CENTRM/PMC. For example, to run CSAS1X sequence using only BONAMI for +self-shielding, rather than the default CENTRM/PMC method, enter the +computational sequence specification shown below. + +:: + + =CSAS1X PARM=BONAMI + +Multiple PARM options are specified by enclosing parameters in +parenthesis, such as + +:: + + =CSAS1X PARM=(CHK, BONAMI) + +XSProc resonance self-shielding options are summarized below. + +PARM=BONAMI. + + This is the fastest MG processing method. It performs + resonance self-shielding for all energy groups using the Bondarenko + method. BONAMI computes the appropriate background cross section of a + given unit cell and then interpolates the corresponding shielding factor + from Bondarenko factors on the MG library. Dancoff factors needed to + evaluate the background cross section for lattices are computed + internally, but these can be overridden by input values in the MORE DATA + block. More details on this method are given in the BONAMI section of + the manual. + +PARM=CENTRM. + + This executes the CENTRM/PMC modules to process shielded MG + cross sections using CE flux spectra calculated with the recommended + type of CE transport solver for the designated type of cell. The + CENTRM-recommended CE transport solvers are (a) infinite homogeneous + medium calculation for INFHOMMEDIUM cells; (b) 2D MoC transport + calculation for a LATTICECELL consisting of cylindrical fuel pins in a + square lattice; and (c) 1-D discrete S\ :sub:`n` transport for all other + LATTICECELLs and for all MULTIREGION cells. The recommended type of + transport solver can be overridden for individual cells, as well as for + selected energy ranges, by using the CENTRM DATA block described in + :ref:`7-1-3-9`. + +PARM=2REGION. + + The CENTRM two-region (2R) option computes the PW flux using a + simplified collision probability method for an absorber (e.g., fuel) + region surrounded by an external moderator region which has an + asymptotic energy spectrum. To account for the heterogeneous effects of + a lattice, a correction known as the Dancoff factor is applied to the + escape probabilities in the 2R calculation (see the CENTRM chapter of + the SCALE manual). These Dancoff factors are calculated internally by + XSProc for a uniform array of mixtures in slab, spherical, or + cylindrical geometries. These mixture-dependent Dancoff factors can be + modified by user input using the DAN parameters contained in the MORE + DATA block, as defined in :ref:`7-1-3-8`. + +*Note on CENTRM/PMC self-shielding options:* + +The energy range of the CENTRM flux calculation is subdivided into three +sections: fast, PW, and low energy. PMC only computes self-shielded +cross sections for groups within the PW range defined by parameters +*demax* and *demin*, which, respectively, define the upper and lower +energies of the CENTRM PW flux calculation. Problem-dependent cross +sections for groups in the fast and low energy ranges are obtained with +the more approximate BONAMI method. Default values for parameters +*demax* and *demin* are defined appropriately for self-shielding of +important resonance materials in thermal reactor systems. The PW +self-shielding range can be extended or decreased for individual cells +by modifying these parameters using CENTRM DATA. + +.. _7-1-3-2: + +XSProc input data +~~~~~~~~~~~~~~~~~ + +The types of input data required for XSProc are given in :numref:`tab7-1-1`, +and individual entries are explained in the text following the table. +The title, cross section library name (either CE or MG), and standard +composition specification data (**READ COMP** input block) are required +for all sequences that use XSProc. The name of the cross section library +is used to determine if the transport solver is executed using CE or MG +data (e.g., CE or MG KENO calculations). The unit cell descriptions +(**READ CELL** input block) are only used for MG self-shielding +calculations. If the specified sequence executes in CE mode, the cell +data input can be omitted, or it will be skipped if present. If the cell +data information is omitted for MG calculations, all mixtures are +self-shielded using the infinite medium approximation. + +There are seven standard SCALE sequences that run just XSProc, and +produce a MG cross section library or libraries. + +**=XSPROC** produces three libraries with an optional fourth library. + +- **sysin.microLib** is a self-shielded library of the individual + nuclides in the problem for use in a later transport calculation, + +- **sysin.macroLib** is a self-shielded library of the mixture cross + sections in the problem for use in a later transport calculation, + +- **sysin.smallMicroLib** is a self-shielded library of specific + reaction rate cross sections and the elastic and total inelastic + scattering transfer matrices for later use in calculating reaction + rates and sensitivity values, and + +- **sysin.xsdrnWeightedLib** is an optional library produced if the + input specifies having **XSDRN** do a weighting calculation. This can + be a cell weighted and/or a group collapse calculation. The library + can be either individual nuclides or mixtures, depending on input. + +**=CSAS-MG** produces an **ft04f001** library that is equivalent to the +**sysin.microLib**. With appropriate input it can also produce an +**ft03f001** which is equivalent to **sysin.xsdrnWeightedLib** above. + +**=CSASI** or **=CSASIX** produce an **ft04f001** library that is +equivalent to **sysin.microLib**, and an **ft02f001** library that is +equivalent to **sysin.macroLib**. CSASIX will run an **XSDRN** on the +first cell without any MOREDATA input. With appropriate input they both +can produce an **ft03f001** that is the equivalent of +**sysin.xsdrnWeightedLib**. + +**=CSAS1** or **=CSAS1X** produce an **ft04f001** library that is the +equivalent of **sysin.microLib**. Both sequences will run an **XSDRN** +on the first cell. With appropriate input, they both can produce an +**ft03f001** that is the equivalent of **sysin.xsdrnWeightLib**. + +**=T-XSEC** produces an **ft04f001** library that is equivalent to +**sysin.macroLib**. and an **ft44f001** library that is equivalent to +sysin.microLib. + +The reactions (MT numbers) written to each library are listed in the +``SequenceNeutronMT.txt`` file located in the etc directory installed with +SCALE. + +.. _tab7-1-1: + +.. table:: Outline of XSProc input data + :align: center + + +-----------------+-----------------+-----------------+-----------------+ + | Data Position | Type of Data | Data Entry | Comments | + +=================+=================+=================+=================+ + | 1 | Title | Enter title | Limit to 80 | + | | | | characters | + +-----------------+-----------------+-----------------+-----------------+ + | 2 | Cross section | | The currently | + | | library name | | available | + | | | | libraries are | + | | | | listed in the | + | | | | table *Standard | + | | | | SCALE | + | | | | cross-section | + | | | | libraries* of | + | | | | the XSLib | + | | | | chapter. | + +-----------------+-----------------+-----------------+-----------------+ + | 3 | Standard | Enter the | | Begin this | + | | composition | appropriate | data block | + | | | data | with | + | | specification | | | **READ COMP** | + | | data | | | + | | | | | and terminate | + | | | | with | + | | | | | **END COMP**. | + | | | | | + | | | | See Section | + | | | | :ref:`7-1-3-3`. | + +-----------------+-----------------+-----------------+-----------------+ + | 4 | Unit cell(s) | | Begin this data | + | | description | | block with | + | | | | READ CELL (or | + | | for MG | | CELLDATA) | + | | calculations | | | + | | | | | + | | only | | | + +-----------------+-----------------+-----------------+-----------------+ + | | a. Type of self | **INFHOMMEDIUM**| These are the | + | | shielding | | available | + | | calculation | | options. | + | | | **LATTICECELL** | | + | | | | See the | + | | | **MULTIREGION** | explanation in | + | | | | Section | + | | | DOUBLEHET | :ref:`7-1-3-2`. | + +-----------------+-----------------+-----------------+-----------------+ + | | b. Unit cell | Enter the | See | + | | geometry | appropriate | :ref:`7-1-3-4` | + | | specification | data | **INFHOMMEDIUM**| + | | | | | + | | | | | + | | | | See Section | + | | | | :ref:`7-1-3-5` | + | | | | **LATTICECELL** | + | | | | . | + | | | | | + | | | | See Section | + | | | | :ref:`7-1-3-6` | + | | | | **MULTIREGION** | + | | | | . | + | | | | | + | | | | See Section | + | | | | :ref:`7-1-3-7` | + | | | | DOUBLEHET. | + +-----------------+-----------------+-----------------+-----------------+ + | | c. Optional | Enter the | | Begin this | + | | MORE parameter | desired data | data block | + | | data | | with | + | | | | | **MORE DATA** | + | | | | (or | + | | | | **MOREDATA**) | + | | | | | + | | | | | and terminate | + | | | | with | + | | | | | **END MORE** | + | | | | (or END | + | | | | **MOREDATA**) | + | | | | . | + | | | | | + | | | | Use only if | + | | | | MORE parameter | + | | | | data are to be | + | | | | entered; | + | | | | otherwise, omit | + | | | | these data | + | | | | entirely. See | + | | | | :ref:`7-1-3-8` | + | | | | | + +-----------------+-----------------+-----------------+-----------------+ + | | d. Optional | Enter the | Begin this data | + | | CENTRM | desired data | block with | + | | parameter data | | | + | | | | **CENTRM DATA** | + | | | | (or | + | | | | **CENTRMDATA**) | + | | | | | + | | | | and terminate | + | | | | with | + | | | | | + | | | | **END CENTRM** | + | | | | (or END | + | | | | **CENTRMDATA**) | + | | | | . | + | | | | | + | | | | Use only if | + | | | | CENTRM | + | | | | parameter data | + | | | | are to be | + | | | | entered; | + | | | | otherwise, omit | + | | | | these data | + | | | | entirely. | + +-----------------+-----------------+-----------------+-----------------+ + | | e. End of unit | | Terminate with | + | | cell data | | END CELL (or | + | | | | END CELLDATA) | + +-----------------+-----------------+-----------------+-----------------+ + | Repeat | | | | + | positions 4a–4d | | | | + | as needed to | | | | + | specify all | | | | + | unit cells. | | | | + | Position 4 data | | | | + | are applicable | | | | + | to the MG | | | | + | calculations | | | | + | only. | | | | + +-----------------+-----------------+-----------------+-----------------+ + +1. TITLE. An 80-character maximum title is required. The title is the + first 80 characters of the XSPROC data. + +2. CROSS SECTION LIBRARY NAME. This item specifies the cross section + library that is to be used in the calculation. See Table *Standard + SCALE cross-section libraries* in the XSLIB chapter of the SCALE + manual for a discussion of the available libraries. + +3. The keywords **READ COMP** followed by the standard compositions + specifications. These data are used to define mixtures used in the + problem. See :ref:`7-1-3-3` and :numref:`tab7-1-2` for a description of + the standard composition specification data. These data are + required for every problem. After all mixtures have been entered, + the keywords **END COMP** must be entered. + +4. The keywords **READ CELLDATA** followed by the input describing each + unit cell as defined below. After all unit cells are described, + the keywords **END CELLDATA** terminate this input block. + + a. TYPE OF CALCULATION. The options are **INFHOMMEDIUM**, + **LATTICECELL**, **MULTIREGION**, **DOUBLEHET**, or nothing. A + description of these cell types and the associated + computational methods are provided in :ref:`7-1-2-3`. If all + input mixtures are to be treated as infinite homogeneous media, + the **CELLDATA** block can be omitted. In this case the + self-shielding calculations will not account for any + geometrical effects, so users should be careful in applying + this approach. Similarly, mixtures not explicitly assigned to a + cell are treated as infinite homogeneous media in the manner + discussed in :ref:`7-1-2-3`. + + b. CELL GEOMETRY SPECIFICATION. See :ref:`7-1-3-4` and :numref:`tab7-1-3` + for an explanation of the optional unit cell data associated + with an **INFHOMMEDIUM** problem. See :ref:`7-1-3-5` for an + explanation of the data associated with **LATTICECELL** + problems. :ref:`7-1-3-6` explains the data required for a + **MULTIREGION** problem. :ref:`7-1-3-7` explains the required + data for a **DOUBLEHET** problem. The **DOUBLEHET** input may + be thought of as a combination of **MULTIREGION** input for the + fuel grains and **LATTICECELL** input for the fuel element. + + c. OPTIONAL MORE PARAMETER DATA. This option allows certain defaulted + parameters to be re-specified by the user. This block begins + with **MORE DATA** and is used by XSDRN. These data apply only + to the unit cell immediately preceding them. Data placed prior + to all unit cell data apply to all materials not listed in any + unit cell and are treated as infinite homogeneous media. Omit + these data unless they are needed. This block ends with **END + MORE**. See :ref:`7-1-3-8`. + + d. OPTIONAL CENTRM PARAMETER DATA. This optional data block begins + with **CENTRM DATA** and ends with **END CENTRM**. These data + allow the user to override default data for CENTRM and PMC. + These data apply only to the unit cell immediately preceding + them. Data placed prior to all unit cell data apply to all + materials not listed in any unit cell and are treated as + infinite homogeneous media. + +.. _7-1-3-3: + +Standard composition specification data +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +Mixtures utilized in the problem are defined using standard composition +specification data. The standard composition input begins with the +keywords **READ COMP**, followed by standard composition specifications +for all mixtures in the problem. When all mixtures have been described, +enter the words **END COMP** to signal the completion of this block of +data. XSProc computes macroscopic cross sections for all mixtures +defined in the **COMP** block. + +The required input for the standard composition specification data +varies, depending on the type of standard composition material. However, +every standard composition specification must include the following: + +1. a standard composition material name. + +2. a mixture number (MX) that contains this material, and + +3. a terminator for the standard composition specification data (enter + the word END). + +The types of standard compositions in SCALE are (a) basic mixtures, (b) +fissile solutions, (c) chemical compounds, and (d) alloys. The four +general options for inputting these types of data are shown in :numref:`tab7-1-2`. +For some cases, more than one option could possibly be used to +specify the mixture. The user may select whichever options are most +convenient to define a particular mixture, and these may be entered in +any order. + +.. _tab7-1-2: + +.. table:: Outline of standard composition specification options (Mixtures can be defined using one or more of these options in any order). + :align: center + + +-----------------------------------+-----------------------------------+ + | Input data | Comments | + | | | + | name | | + +-----------------------------------+-----------------------------------+ + | **READ COMP** | Enter once for a problem. | + | | Enter the words **READ COMP** | + | | prior to entering any standard | + | | composition data. | + +-----------------------------------+-----------------------------------+ + | **sc** | This option is used for | + | | defining basic mixtures. Enter | + | | one of the alphanumeric | + | | identifiers, symbols, or names | + | | from Standard Composition | + | | Library tables *Isotopes in | + | | standard composition library*, | + | | *Isotopes and their natural | + | | abundances*, *Elements and | + | | special nuclide symbols*, | + | | *Compounds*, or *Alloys and | + | | mixtures* in place of SC. This | + | | indicates the isotope, | + | | nuclide, compound, or alloy | + | | that will make up this | + | | standard composition. | + | | See :numref:`tab7-1-2a` for | + | | additional required and | + | | optional data for each | + | | standard composition. | + +-----------------------------------+-----------------------------------+ + | **SOLUTION** | This option is used to specify | + | | a fissile solution mixture. | + | | See :numref:`tab7-1-2b` for | + | | additional required and | + | | optional data for each | + | | solution. End the data with an | + | | **END**. | + +-----------------------------------+-----------------------------------+ + | **ATOM** | This option creates a chemical | + | | compound mixture composed of | + | | the specified nuclide in | + | | the compound. Each nuclide is | + | | entered followed by the | + | | relative number of atoms of | + | | the nuclide in the compound. | + | | All compounds must begin with | + | | the four letters \ **ATOM** | + | | followed by up to eight | + | | additional alphanumeric | + | | characters. :numref:`tab7-1-2c`| + | | shows additional required and | + | | optional data for each | + | | compound. | + +-----------------------------------+-----------------------------------+ + | **WTPT** | This option creates a | + | | mixture/alloy composed of the | + | | specified nuclides in | + | | the mixture/alloy. Each | + | | nuclide is entered followed by | + | | the weight percent of the | + | | nuclide in the mixture/alloy. | + | | All mixture/alloys must begin | + | | with the four letters **WTPT** | + | | followed by up to eight | + | | additional alphanumeric | + | | characters. :numref:`tab7-1-2d`| + | | shows additional required and | + | | optional data for each | + | | arbitrary physical mixture or | + | | alloy. | + +-----------------------------------+-----------------------------------+ + | **END COMP** | Enter once for a problem. | + | | Enter the exact words **END | + | | COMP** when all the standard | + | | composition components have | + | | been described. At least two | + | | blanks or a new line must | + | | follow the words **END COMP** | + | | prior to continuing data | + | | entry. | + +-----------------------------------+-----------------------------------+ + +Names of the standard composition materials (the alphanumeric +identifiers) appearing in the **COMP** block input must be selected from +the tables of elements, compounds, solutions, and alloys given in the +SCALE manual section describing the Standard Composition Library. An +error message will be printed if the user enters an invalid standard +composition material name or if any isotopes in the compound do not +exist in the specified library + +Input data to define each of the standard composition types in +:numref:`tab7-1-2` are summarized in :numref:`tab7-1-2a` through :numref:`tab7-1-2d`. Optional input is +indicated by curly brackets { }. *Since some of the input is not keyword +based, the order of entries is important in the standard composition +specification*. The temperature specification is used for Doppler +broadening and/or determination of the proper thermal scattering data. +Input material densities are not modified for temperature effects. +Additional description of the standard composition input for each type +of material is given following all the tables. As in the tables, input +parameters enclosed by curly brackets { } indicate that these are +optional. + +.. _tab7-1-2a: +.. list-table:: Standard composition specification for basic mixtures. + :align: center + + + * - .. image:: figs/XSProc/tab2a.svg + :align: center + :width: 800 + +.. _tab7-1-2b: +.. list-table:: Standard composition specification for solutions. + :align: center + + * - .. image:: figs/XSProc/tab2b.svg + :width: 800 + +.. _tab7-1-2c: +.. list-table:: Standard composition specification for chemical compounds. + :align: center + + * - .. image:: figs/XSProc/tab2c.svg + :width: 800 + +.. _tab7-1-2d: +.. list-table:: Input specification for user-specified mixture/alloy data. + :align: center + + * - .. image:: figs/XSProc/tab2d.svg + :width: 800 + +STANDARD COMPOSITION INPUT FOR BASIC MIXTURES (see :numref:`tab7-1-2a`). + +Two input syntaxes are available for standard composition specifications +of basic mixtures in the **COMP** block. The first uses information +(e.g., densities, atomic weights, physical constants, etc.) contained in +the Standard Composition Library, along with user specified input, to +automatically compute the number densities for mixture components. In +the second option, the user computes the nuclide number densities, and +inputs these directly for each component of the mixture. XSProc +recognizes syntax 2 if the third entry of the composition specification +is zero, as shown below. It is allowable to use syntax 1 for some +standard composition specifications and syntax 2 for others. The two +syntaxes to define basic mixtures with the standard composition +specifications are shown below. + +**syntax 1: Standard Composition Library data used to compute number +densities.** + + **sc** *mx* **DEN**\ =\ *roth* {**VF**\ =}\ *vf* *temp* *iza*\ :sub:`1` *wtp*\ :sub:`1` … + *iza*\ :sub:`N` *wtp*\ :sub:`N` **END** + +**syntax 2: User input number densities** + + **sc** *mx* 0 *aden* *temp* **END** + +The definitions for these input parameters are given below. + +A1. **sc** + + STANDARD COMPOSITION MATERIAL NAME. This corresponds to one + of the material names given in the Standard Composition Library for + isotopes, elements, thermal moderators and activity materials, chemical + compounds, and alloys/mixtures. Some types of these materials require + entering certain data such as the volume fraction or theoretical density + and other engineering-type data. For standard compositions containing + more than one isotope of an element (such as UO\ :sub:`2`), the user is + free to specify the weight percent for each isotope, such that they + total 100%. See the Basic standard composition specifications section + for examples of basic standard compositions. + +A2. *mx* + + MIXTURE ID NUMBER. An arbitrary mixture number is required on + every standard composition specification for both syntaxes. It defines + the mixture that contains the material defined by the standard + composition specification data. The mixture numbers are utilized in the + CELLDATA block Cell Block for INFHOMMEDIUM, LATTICECELL, MULTIREGION, or + DOUBLEHET problems and the geometry data. + +A3. **DEN**\ =\ *roth* + + MIXTURE DENSITY. The keyword **DEN** is assigned + a value of *roth,* where *roth* is the specified density of the mixture + component in g/cm\ :sup:`3`. It should always be entered for materials + that contain enriched multi-isotopic nuclides. The effective density of + the material component is equal to the product of *roth* and *vf.* An + example of this is demonstrated in Appendix A. + +A4. {**VF**\ =}\ *vf* + + VOLUME FRACTION. The keyword **VF** is assigned a + value of *vf*. It is also allowable to omit the keyword **VF=** and just + enter the value *vf* . The default value of the volume fraction is 1.0. + The volume fraction can be interpreted as + + a. the volume fraction of this standard composition component in the + mixture, + + b. the density of the standard composition component in this + application divided by the theoretical or default density given in + the Standard Composition Library, or + + c. the product of (a) and (b). + + Appendix A discusses the interaction between *roth* and *vf*. For + example, assume a homogenized mixture representing the water + moderator and Zircaloy cladding around a fuel pin is to be described. + If the volume of the clad is 5.32 cc and the volume of the water + moderator is 44.68 cc, the mixture can be described using + H\ :sub:`2`\ O with a volume fraction of 0.8936 + [i.e., 44.68/(44.68+5.32)] and ZIRCALOY with a volume fraction of + 0.1064 [i.e., 5.32/(44.68+5.32)]. + +A5. *aden* + + NUMBER DENSITY (not used for syntax 1, but required for 2). + The number density is entered ONLY if 3\ :sup:`rd` entry on the standard + composition specification is entered as zero. The number density is + entered in units of atoms per barn-cm. + +A6. *temp* + + TEMPERATURE. The default value of the temperature is 300 K. + The temperature can be omitted if entries A7 and A8 are also omitted. + +A7. *iza* + + ISOTOPE ZA NUMBER. Enter a value for each isotope in the + standard composition component, entry 1. Do not enter a value if the + volume fraction, **VF**, is zero (A4 above). + + The ZA number of the isotope is entered if the user wishes to specify + the isotopic distribution. This is done by entering *iza* and *wtp* + for each isotope until all the desired isotopes have been described. + In most cases the “ZA” ID number is (A+1000*Z), where A is the atomic + mass or weight of the isotope, and Z is the atomic number. For + example, the ZA number for :sup:`235`\ U is 92235. + + Entries A7 and A8 can be skipped if the default values listed in + :numref:`tab7-1-2` are acceptable. + +A8. *wtp* + + WEIGHT PERCENT OF THE ISOTOPE. If entry A7 is entered, a value + must also be entered for A8. The weight percent of the isotope is the + percent of this isotope in the element. The weight percent of all + specified isotopes of the element must sum to 100 (± 0.01). + +A9. **END** + + The word **END** is entered to terminate the input data for + a standard composition component. This **END** can have a label + associated with it that can be as long as 12 characters. The label is + optional, and if entered must be preceded from the **END** by a single + blank. At least two blanks or a new line must separate this item from + the next data entry. + +STANDARD COMPOSITION INPUT FOR FISSILE SOLUTIONS (see :numref:`tab7-1-2b`). + +Syntax: + +:: + + SOLUTION {MIX=}mx RHO[fuelsalt]=fd (izai wtpi) MOLAR[acid]=aml + MASSFRAC[name]=mfrac MOLEFRAC[name] =molfrac + MOLALITY[name]=molal DENSITY=roth + TEMPERATURE=temp VOL_FRAC=vf + END SOLUTION + +where + + *mx* is the mixture number, + + *fuelsalt* is the Standard Composition Library component name of one + of the fissile compounds + + *fd* is the fuel density in grams of uranium or plutonium per liter + of solution + + *acid* is one of the Standard Composition Library acid compounds + (e.g., HNO3 or HFACID) + + *name* is one of the Standard Composition Library solution components + + *aml* is the acid molarity of the *acid* component (moles of *acid/* + liter of solution) + + *mfrac* is the mass fraction of *name* in the solution (grams of + metal in *name*/gram solution) + + *molfrac* is the mole fraction of *name* in the solution (moles of + *name*/mole solution) + + *molal* is the mass fraction of *name* in the solution (moles of + *name*/kg water) + + *roth* is the density of the solution, + + *vf* is the density multiplier (ratio of actual to theoretical + density of the solution), + + *temp* is the temperature in Kelvin, + + *iza* is the isotope ID number from table *Available fissile solution + components*, and + + *wtp* is the weight percent of the isotope in the material. + + +Below are the input data for fissile solutions. + +1. **SOLUTION** + + Keyword starting a solution specification. + Solutions require the specification of the mixture and at least one component. + Current possible components are given in the Standard Composition Library table, + *Available fissile solution components*. Only the mixture number and one component are required. + Appendix A contains examples of the input data for solutions. + +2. *mx* + + MIXTURE ID NUMBER. A mixture number is required on every standard composition specification. + It defines the mixture that contains the material defined by the standard composition + specification data. The mixture numbers are utilized in the Unit Cell Specification for + INFHOMMEDIUM, LATTICECELL, or MULTIREGION. + + + +| 3. **RHO**\ [*fuelsalt* ]=\ *fd* +| **MOLAR**\ [*acid*]=\ *aml* +| **MASSFRAC**\ [*name*]=\ *mfrac* +| **MOLEFRAC**\ [*name*]=\ *molfrac* +| **MOLALITY**\ [*name*]=\ *molal* + + KEYWORD PARAMETERS TO DEFINE CONCENTRATIONS OF SOLUTION COMPONENTS. + Each keyword specifies the unit, the component name from the Standard + Composition Library and the component value, as shown :numref:`tab7-1-2b`. Up + to three components can be specified for a solution if one is an acid. After + the value, the isotopic enrichments of the nuclides can be given as pairs of + isotope IDs and weight percent. **NOTE: the square brackets [ ] containing the + component name are required.** + +4. **DENSITY=**\ *roth* + + Keyword specifying the overall solution density as grams per cubic centimeter + or as a “?”, meaning it is to be solved for. Solving for the density is the + default behavior, but the density can be given, and a component value can be + solved for instead. + +5. **TEMPERATURE=**\ *temp* + + Keyword defining temperature of the solution. The default value is 300 K. + +6. **VOLFRAC=**\ *vf* + + Keyword defining volume fraction — the default volume fraction is 1.0. This value must be greater than 0.0. The volume fraction can be interpreted as: + a. the volume fraction of this solution specification in the mixture, + b. the density of the solution in this application divided by the calculated density of the solution, or + c. the product of (a) and (b). + +7. **END SOLUTION** + +STANDARD COMPOSITION INPUT FOR CHEMICAL COMPOUNDS (see :numref:`tab7-1-2c`) + +**Syntax:** + +**ATOM\ nn** *mx* *roth* *nel* *ncza*\ :sub:`1` *atpm*\ :sub:`1` … *ncza*\ :sub:`nel` *atpm*\ :sub:`nel` +{*vf* {*temp* {*iza*\ :sub:`1` *wtp*\ :sub:`1` …} } } **END** + +Below are the data for user-defined chemical compounds. + + +C1. **ATOM\ nn** + + COMPOUND NAME. User-specified compounds (also defined + as “arbitrary” in older versions of SCALE) require the user to provide + all the information normally found in the Standard Composition Library. + This option allows specifying a compound not available in the Standard + Composition Library by utilizing nuclides and elements available in the + library. An user-specified compound name must start with the four + characters “\ **ATOM**.” A maximum of twelve characters is allowed for + the compound name, and imbedded blanks are not allowed. + +C2. *mx* + + MIXTURE ID NUMBER. A mixture number is required on every + standard composition specification. It defines the mixture that contains + the material defined by the compound specification data. The mixture + numbers are utilized in the Unit Cell Specification for + **INFHOMMEDIUM**, **LATTICECELL**, or **MULTIREGION** problems and the + KENO V.a or KENO-VI geometry data. + +C3. *roth* + + MIXTURE DENSITY. The density of the arbitrary material is + entered in units of g/cm\ :sup:`3`. *roth* and *vf* interact to produce + the density of the mixture used in the problem. Note that this is a + required entry and does not use “\ **DEN**\ =” keyword. + +C4. *nel* + + NUMBER OF ELEMENTS IN THE MATERIAL. Enter the number of + components from the Standard Composition Library that are to be used to + define this material. + +C5. *ncza* + + ID NUMBER. This is the “ZA” ID number for the element or + isotope. Usually, *ncza*\ =A+1000*Z, where A is the atomic mass or + weight of the nuclide, and Z is the atomic number. + +C6. *atpm* + + ATOMIC or ELEMENT ABUNDANCE. Enter the number of atoms of + this element per molecule of compound. Repeat the sequence *ncza* and + *atpm* (C5 and C6) for every element in the compound before going to + entry C7. + +C7. *vf* + + VOLUME FRACTION. The default value of the volume fraction is + 1.0. This value must be greater than 0.0. The volume fraction can be + interpreted as + + a. the volume fraction of this compound in the mixture, + + b. the density of the compound in this application divided by the input + density of the compound, or + + c. the product of (a) and (b). + +C8. *temp* + + TEMPERATURE. The default value of the temperature is 300 K. + The temperature can be omitted if entries C9 and C10 are also omitted. + +C9. *iza* + + ISOTOPE ZA NUMBER. Enter a value for each isotope in the + element in the compound. The ZA number of the isotope is entered if the + user wishes to specify the isotopic distribution. This is done by + entering *iza* and *wtp* for each isotope until all the desired isotopes + have been described. In most cases the “ZA” ID number is (A+1000*Z), + where A is the atomic mass or weight of the isotope, and Z is the atomic + number. + + Entries C9 and C10 can be skipped if the default values listed in + :numref:`tab7-1-2` of :ref:`7-1` are acceptable. + +C10. *wtp* WEIGHT PERCENT OF THE ISOTOPE. If entry C9 is entered, a +value must also be entered for C10. The weight percent of the isotope is +the percent of this isotope in the element. The weight percents of all +specified isotopes of the element must sum to 100 (± 0.01). + + Repeat the sequence *iza* *wtp* until the sum of the *wtp*\ s sum to + 100. The sequence *iza* *wtp* is repeated until all of the desired + isotopes have been specified. + +C11. **END** + + The word **END** is entered to terminate the input data for + compound. This **END** can have a label associated with it that can be + as long as 12 characters. The label is optional, and if entered must be + preceded from the **END** by a single blank. At least two blanks or a + new line must separate item C11 from the next data entry. + +STANDARD COMPOSITION INPUT FOR MIXTURES AND ALLOYS (see :numref:`tab7-1-2d`) + +**Syntax:** + +**WTPTnn** *mx* *roth* *nel* *ncza*\ :sub:`1` *wpct*\ :sub:`1` … *ncza*\ :sub:`nel` *wpct*\ :sub:`nel` +{*vf* {*temp* {*iza*\ :sub:`1` *wtp*\ :sub:`1` …} }} **END** + +Below are the input data for arbitrary (i.e., user-defined) physical +mixture or alloy. + +D1. **WTPTnn** + + ARBITRARY MIXTURE/ALLOY NAME. The arbitrary + user-specified mixture/alloy option allows specifying a mixture or an + alloy not available in the Standard Composition Library by utilizing the + nuclides and elements available in the library. An arbitrary + mixture/alloy name must start with the four characters “\ **WTPT**.” A + maximum of 12 characters is allowed for the arbitrary mixture/alloy + name. Imbedded blanks are not allowed in an arbitrary mixture/alloy + name. Appendix A contains input data for arbitrary mixture/alloys. + +D2. *mx* + + MIXTURE ID NUMBER. A mixture number is required on every + standard composition specification. It defines the mixture that contains + the material defined by the arbitrary compound specification data. The + mixture numbers are utilized in the Unit Cell Specification for + **INFHOMMEDIUM**, **LATTICECELL**, **MULTIREGION**, or **DOUBLEHET** + problems and the KENO V.a or KENO-VI geometry data. + +D3. *roth* + + MIXTURE DENSITY. The density of the arbitrary material is + entered in units of g/cm\ :sup:`3`. *roth* and *vf* interact to produce + the density of the mixture used in the problem. Note that this is a + required entry and does not use “\ **DEN**\ =” keyword. + +D4. *nel* + + NUMBER OF ELEMENTS IN THE MATERIAL. Enter the number of + components from the Standard Composition Library that are to be used to + define this arbitrary material. + +D5. *ncza* + + ID NUMBER. This is the “ZA” ID number for the element or + isotope. Usually, *ncza*\ =A+1000*Z, where A is the atomic mass or + weight of the nuclide, and Z is the atomic number. + +D6. *wpct* + + ATOMIC or ELEMENT ABUNDANCE. Enter the weight percent of this + element in the arbitrary alloy. The sum of all the weight percents for + each specified element in the arbitrary alloy MUST be 100.0. Repeat the + sequence *ncza* and *wpct* (D5 and D6) for every element in the + arbitrary mixture/alloy before going to entry D7. + +D7. *vf* + + VOLUME FRACTION. The default value of the volume fraction is + 1.0. This value must be greater than 0.0. The volume fraction can be + interpreted as: + + a. the volume fraction of this mixture or alloy in the mixture, + + b. the density of the mixture or alloy in this application divided by + the input density (*roth*) of the mixture or alloy, or + + c. the product of (a) and (b). + +D8. *temp* + + TEMPERATURE. The default value of the temperature is 300 K. + The temperature can be omitted if entries D9 and D10 are also omitted. + +D9. *iza* + + ISOTOPE ZA NUMBER. Enter a value for each isotope in the + element in the arbitrary alloy. The ZA number of the isotope is entered + if the user wishes to specify the isotopic distribution. This is done by + entering *iza* and *wtp* for each isotope until all the desired isotopes + have been described. In most cases the “ZA” ID number is (A+1000*Z), + where A is the atomic mass or weight of the isotope, and Z is the atomic + number. + + Entries D9 and D10 can be skipped if the default values listed in + :numref:`tab7-1-2` are acceptable. + +D10. *wtp* + + WEIGHT PERCENT OF THE ISOTOPE. If entry D9 is entered, a + value must also be entered for D10. The weight percent of the isotope is + the percent of this isotope in the element. Weight percents of all + specified isotopes of the element must sum to 100 (±0.01). + +D11. **END** + + The word **END** is entered to terminate the input data for + an arbitrary compound. This **END** can have a label associated with it + that can be as long as 12 characters. The label is optional and if + entered must be preceded from the **END** by a single blank. At least + two blanks or a new line must separate this item from the next data + entry. + + +.. _7-1-3-4: + +Unit cell specification for infinite homogeneous problems +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +This section describes the unit cell data that can be entered for an +**INFHOMMEDIUM** problem. Additional information is available in +Appendix B. + +Syntax: + +**INFHOMMEDIUM** *mx* {**CELLMIX**\ {=}*mix*} **END** + +The data required to specify the unit cell for an **INFHOMMEDIUM** unit +cell are given in :numref:`tab7-1-3`. The individual entries are explained in +the following text. + +1. **celltype** + + **INFHOMMEDIUM**. The keyword **INFHOMMEDIUM** is + entered to indicate this unit cell contains one mixture with no geometry + corrections. This data must be entered. The keyword may be truncated to + any number of characters as long as the characters present are identical + from the beginning of the keyword (i.e., INF is acceptable). All + mixtures not in a defined unit cell are by default processed as + infhommedium. + +2. *mx* + + MIXTURE NUMBER. The mixture number defines the mixture to be + used in the cell. This data must be entered. Be sure the mixture number + entered is defined in the standard composition data. + +3. **CELLMIX**\ =\ *mix* + + CELL-WEIGHTED MIXTURE NUMBER. (the = sign can + be replaced by a space if desired). Enter ONLY if a cell-weighted + mixture is to be generated. Enter a unique mixture number to be used by + XSDRN to create the cell-weighted mixture (:ref:`7-1-2-4`). For + **INFHOMMEDIUM** cells, cross sections for the cell mixture are equal to + the shielded values of the original mixture. + +4. **END** + + The word **END** is entered to terminate the **INFHOMMEDIUM** + data. An optional label can be associated with this **END**. The label + can be as many as 12 characters long and is separated from the **END** + by a single blank. At least two blanks must follow this entry. + +.. _tab7-1-3: +.. table:: Unit cell specifications for INFHOMMEDIUM problems. + :align: center + + +-----------------+-----------------+-----------------+-----------------+ + | Entry | Input | Data | Comments | + | | | | | + | no. | data | type | | + +-----------------+-----------------+-----------------+-----------------+ + | 1 | **INFHOMMEDIUM**| Keyword | Keyword to | + | | | | begin | + | | | | infhommedium | + | | | | unit cell. | + | | | | Enter the | + | | | | keyword | + | | | | INFHOMMEDIUM. | + | | | | This word may | + | | | | be truncated to | + | | | | any number of | + | | | | letters as long | + | | | | as they exactly | + | | | | replicate the | + | | | | beginning part | + | | | | of the keyword | + | | | | (e.g., INF is | + | | | | acceptable). | + +-----------------+-----------------+-----------------+-----------------+ + | 2 | *mx* | Cell mixture | Specifies the | + | | | number | mixture number | + | | | | to be used in | + | | | | the cell. | + +-----------------+-----------------+-----------------+-----------------+ + | 3 | **CELLMIX**\ = | Keyword + | Enter the | + | | *mix* | | keyword | + | | | new mixture | **CELLMIX**\ = | + | | | number | followed | + | | | | immediately by | + | | | | a unique | + | | | | positive | + | | | | integer | + | | | | (*mix*). The | + | | | | integer will be | + | | | | a new mixture | + | | | | number that has | + | | | | the neutronic | + | | | | properties of | + | | | | the | + | | | | self-shielded | + | | | | unit cell.\ | + | | | | :sup:`a` | + +-----------------+-----------------+-----------------+-----------------+ + | 4 | **END** | | Terminate | + | | | | **INFHOMMEDIUM**| + | | | | data | + | | | | | + +-----------------+-----------------+-----------------+-----------------+ + | :sup:`a`\ Note: If CELLMIX is entered for a **INFHOMMEDIUM** cell, | + | XSDRNPM is executed to compute k\ :sub:`∞`, but cross sections for the| + | “homogenized” mixture are identical to the shielded values for the | + | original cell. | + +-----------------------------------------------------------------------+ + +.. _7-1-3-5: + + +Unit cell specification for LATTICECELL problems +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +This section describes the unit cell input data for a **LATTICECELL** +problem. The **LATTICECELL** description is especially suited to +self-shield arrays of repeated cells such as a fuel assembly lattice. +The unit cell specification plays a major role in providing accurate +problem-dependent cross sections using the computational procedures +described in :ref:`7-1-2-3`. Unit cells are limited to (a) infinitely +long cylindrical rods in a square or triangular lattice, (b) spheres in +a cubic or triangular lattice, (c) a symmetric array of slabs, or (d) an +asymmetric array of slabs. Both “regular” and “annular” fuel geometries +can be used in **LATTICECELL** problems. “Regular” cells allow a +concentric spherical, cylindrical, or symmetric slab configuration, +where the central region is fuel, surrounded by an optional gap, an +optional clad, and an external moderator. “Annular” cells also allow +concentric spherical, cylindrical, or asymmetric slab configurations, +but the central region corresponds to an inner moderator region which is +surrounded by a fuel region having an optional gap and optional clad on +each side of the fuel. An inner gap may be specified inside the fuel +region, and an outer gap may be specified outside the fuel region. +Similarly an inner clad may be specified inside the fuel region, and an +outer clad may be specified outside the fuel region. For both regular +and annular fuel cells, the outer boundary of the unit cell is +determined from the square or triangular pitch of the array. + +Regular cells are **SQUAREPITCH**, **TRIANGPITCH**, **SPHSQUAREP**, +**SPHTRIANGP**, and **SYMMSLABCELL**. + +Annular cells are **ASQUAREPITCH** (or **ASQP**), **ATRIANGPITCH** (or +**ATRP**), **ASPHSQUAREP** (or **ASSP**), **ASPHTRIANGP** (or **ASTP**), +and **ASYMSLABCELL** + +Syntax: + + **celltype** **ctp PITCH** (or **HPITCH**) *pitch* *mm* **FUELD (or + FUELR**) *fuel mf* + + **GAPD (or GAPR**) *gap mg* **CLADD (or CLADR**) *clad mc* + + **IMODD (or IMODR**) *imod mim* **IGAPD** (**or IGAPR**) *igap mig* + + **ICLADD** (or **ICLADR**) *iclad mic* {**CELLMIX**\ =\ *mix*} + **END** + +The unit cell geometry data required to specify a LATTICECELL problem +are given in :numref:`tab7-1-4`. The individual entries are explained in the +text below. + +1. **celltype** + + **LATTICECELL.** The keyword **LATTICECELL** is entered + to indicate this unit cell contains mixtures that are positioned in a + regular array. This data must be entered. The keyword may be truncated + to any number of characters as long as the characters present are + identical from the beginning of the keyword (e.g., **LAT** is + acceptable). This unit cell is normally used for regular arrays of + materials such as fuel pins in an assembly. + +2. **ctp** + + TYPE OF LATTICE. This defines the type of lattice or array + configuration. Any one of the following alphanumeric descriptions may be + used. Note that the alphanumeric description must be separated from + subsequent data entries by one or more blanks. :numref:`fig7-1-1` Mixture and + position data are entered using keywords. Mixture number 0 may be + entered for void and may be used multiple times in each and all unit + cells. For regular cells, the minimum requirement is that a fuel region + and a moderator region are specified and no other inner components are + specified. For annular cells, the minimum requirement is the fuel and + outer moderator and inner moderator regions are specified. Regular and + annular cell configurations are specified as shown below. + + **Regular Cells** + + **SQUAREPITCH** is used for an array of cylinders arranged in a + square lattice, as shown in :numref:`fig7-1-1`. The clad and/or gap can be + omitted. + + **TRIANGPITCH** is used for an array of cylinders arranged in a + triangular-pitch lattice as shown in :numref:`fig7-1-2`. The clad and/or + gap can be omitted. + + **SPHSQUAREP** is used for an array of spheres arranged in a + square-pitch lattice. A cross section view through a cell is + represented by :numref:`fig7-1-1`. The clad and/or gap can be omitted. + + **SPHTRIANGP** is used for an array of spheres arranged in a + triangular-pitch (dodecahedral) lattice. A cross section view through + a cell is represented by :numref:`fig7-1-2`. The clad and/or gap can be + omitted. + + **SYMMSLABCELL** is used for an infinite array of symmetric slab + cells, as shown in :numref:`fig7-1-3`. The clad and/or gap can be omitted. + + **Annular Cells** + + **ASQUAREPITCH** or **ASQP** is used for annular cylindrical rods in + a square-pitch lattice as shown in :numref:`fig7-1-4`. The inner and outer + clad and gap are independently entered so they must be different + materials and dimensions. Note that each mixture in the problem can + be used only once and in only one zone of a cell. + + **ATRIANGPITCH** or **ATRP** is used for annular cylindrical rods in + a triangular-pitch lattice as shown in :numref:`fig7-1-5`. The inner and + outer clad and gap are independently entered, so they must be + different materials and dimensions. + + **ASPHSQUAREP** or **ASSP** is used for spherical shells in a + square-pitch lattice as shown in :numref:`fig7-1-4`. The inner and outer + clad and gap are independently entered, so they must be different + materials and dimensions. + + **ASPHTRIANGP** or **ASTP** is used for spherical shells in a + triangular-pitch (dodecahedral) lattice as shown in :numref:`fig7-1-5`. The + inner and outer clad and gap are independently entered, so they must + be different materials and dimensions. + + **ASYMSLABCELL** is used for a periodic, but asymmetric, array of + slabs as shown in :numref:`fig7-1-6`. The inner and outer clad and gap are + independently entered, so they may be different materials and + dimensions. + +3. **PITCH** or **HPITCH** + + ARRAY PITCH. This is the center-to-center spacing + or half-spacing between the fuel lumps (rods, pellets, or slabs), *pitch*, in + cm followed by the outer moderator material number, mm, as shown in + :numref:`fig7-1-1` through :numref:`fig7-1-6`. + +4. **FUELD** or **FUELR** + + OUTSIDE DIMENSION OF FUEL. This is the outside diameter or radius of the + fuel, fuel, in cm followed by the fuel mixture number, *mf*, as shown in + :numref:`fig7-1-1` through :numref:`fig7-1-6`. + +5. **GAPD** or **GAPR** + + OUTSIDE DIMENSION OF OUTER GAP. Enter only if outer gap is present. + This is the outside diameter or radius of the outer gap, *gap*, in cm + followed by the gap mixture number, mg, as shown in :numref:`fig7-1-1` through + :numref:`fig7-1-6`. + +6. **CLADD** or **CLADR** + + OUTSIDE DIMENSION OF OUTER CLAD. Enter ONLY if a clad is present. + This is the outside diameter or radius of the outer clad, *clad*, in cm + followed by the clad mixture number, mc, as shown in :numref:`fig7-1-1` through + :numref:`fig7-1-6`. + +7. **IMODD** or **IMODR** + + DIMENSION OF INNER MODERATOR. Enter ONLY if an annular cell is specified. + This is the outside diameter or radius of the inner moderator, *imod*, in cm + followed by the inner moderator mixture number, *mim*, as shown + in :numref:`fig7-1-4` through :numref:`fig7-1-6`. + +8. **IGAPD** or **IGAPR** + + OUTSIDE DIMENSION OF INNER GAP. Enter ONLY if an annular cell is specified and + inner gap is present. This is the outside diame*ter or radius of the inner gap, + *igap*, in cm followed by the inner gap mixture number, *mig*, as shown in + :numref:`fig7-1-4` through :numref:`fig7-1-6`. + +9. **ICLADD** or **ICLADR** + + OUTSIDE DIMENSION OF INNER CLAD. Enter ONLY if an annular cell is + specified and inner clad is present. This is the outside diameter or + radius of the inner clad, *iclad*, in cm followed by the inner clad mixture + number, *mic*, as shown in :numref:`fig7-1-4` through :numref:`fig7-1-6`. + +10. {**CELLMIX\ =}\ mix** + + CELL-WEIGHTED MIXTURE NUMBER. [the = sign can be replaced by a space if desired). + Enter ONLY if a cell-weighted mixture is to be generated. Enter a unique mixture + number to be used by XSDRN to create the cell-weighted mixture (:ref:`7-1-2-4`). + +11. END + + The word **END** is entered to terminate the **LATTICECELL** data. + An optional label can be associated with this **END**. The label can be as many + as 12 characters long and is separated from the **END** by a single blank. At least + two blanks must follow this entry. Must not start in column 1. + +.. _tab7-1-4: +.. list-table:: Unit cell specification for LATTICECELL problems. + :align: center + + * - .. image:: figs/XSProc/tab4.svg + :width: 1000 + +.. _fig7-1-1: +.. figure:: figs/XSProc/fig1.png + :align: center + :width: 400 + + Arrangement of materials in a SQUAREPITCH and SPHSQUAREP unit cell. + +.. _fig7-1-2: +.. figure:: figs/XSProc/fig2.png + :align: center + :width: 400 + + Arrangement of materials in a TRIANGPITCH and SPHTRIANGP unit cell. + +.. _fig7-1-3: +.. figure:: figs/XSProc/fig3.png + :align: center + :width: 600 + + Arrangement of materials in a SYMMSLABCELL unit cell having reflected + left and right boundary conditions. + +.. _fig7-1-4: +.. figure:: figs/XSProc/fig4.png + :align: center + :width: 400 + + Arrangement of materials in an ASQUAREPITCH and ASPHSQUAREP unit cell. + +.. _fig7-1-5: +.. figure:: figs/XSProc/fig5.png + :align: center + :width: 400 + + Arrangement of materials in an ATRIANGPITCH and ASPHTRIANGP unit cell. + +.. _fig7-1-6: +.. figure:: figs/XSProc/fig6.png + :align: center + :width: 600 + + Arrangement of materials in an ASYMSLABCELL unit cell having reflected + left and right boundary conditions. + +.. _7-1-3-6: + +Unit cell specification for MULTIREGION cells +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +A **MULTIREGION** cell can be used to define a 1-D geometric arrangement +that is more general than allowed by a **LATTICECELL**. It can also be +used for large geometric regions where the geometry effects for the +cross sections are small. For CENTRM/PMC self-shielding, lattice effects +can be approximated by applying reflected, periodic, or white external +boundary conditions to a MULTIREGION cell. HOWEVER, MULTIREGION CELLS +SHOULD NOT BE USED FOR BONAMI-ONLY SELF-SHIELDING OF AN ARRAY UNIT CELL. +In this case a LATTICECELL should always be used for BONAMI +self-shielding in order to incorporate the proper Dancoff effects. + +The data required for a MULTIREGION cell are given in :numref:`tab7-1-5` and +explained in the following text. + +1. **celltype** + + **MULTIREGION**. The keyword **MULTIREGION** is used to + represent arbitrary 1‑D geometries, with no restrictions the on number + or placement of mixtures in the cell. The keyword may be truncated to + any number of characters as long as the characters presented are + identical from the beginning of the keyword (i.e., M is acceptable). + +2. **cs** + + TYPE OF GEOMETRY. The type of geometry must always be + specified for a **MULTIREGION** cell. The available geometry options are + listed below. + + **SLAB**. This is used to describe a slab geometry. + + **CYLINDRICAL**. This is used to describe cylindrical geometry. + + **SPHERICAL**. This is used to describe spherical geometry. + + **BUCKLEDSLAB**. This is used for slab geometry with a buckling + correction for the two transverse directions. + + **BUCKLEDCYL**. This is used for cylindrical geometry with a buckling + correction in the axial direction. + +3. **RIGHT_BDY** + + RIGHT BOUNDARY CONDITION. This is defaulted to + **VACUUM**. The available options and their qualifications are listed + below. + + **VACUUM**. This imposes a vacuum at the boundary of the system. + + **REFLECTED**. This imposes mirror image reflection at the boundary. + Do not use for **CYLINDRICAL** or **SPHERICAL**. + + **PERIODIC**. This imposes periodic reflection at the boundary. Do + not use for **CYLINDRICAL** or **SPHERICAL**. + + **WHITE**. This imposes isotropic return at the boundary. + +4. **LEFT_BDY** + + LEFT BOUNDARY CONDITION. This is defaulted to + **REFLECTED**. The available options and their qualifications are listed + below. + + **VACUUM**. This imposes a vacuum at the boundary of the system. + + **REFLECTED**. This imposes mirror image reflection at the boundary. + For **CYLINDRICAL** or **SPHERICAL**, this is the only valid boundary + condition because the left boundary corresponds to the centerline of + the cylinder or the center of the sphere. + + **PERIODIC**. This imposes periodic reflection at the boundary. + + **WHITE**. This imposes isotropic return at the boundary. + +5. **ORIGIN** + + LOCATION OF LEFT BOUNDARY ON THE ORIGIN. The default value + of **ORIGIN** is 0.0. This is the only value allowed for **CYLINDRICAL** + or **SPHERICAL** geometry. For **SLAB**\ s, enter the location of the + left boundary on the X-axis perpendicular to the slab (in cm). + +6. **DY** + + BUCKLING HEIGHT. This is the buckling height in cm. It + corresponds to one of the transverse dimensions of an actual 3-D slab + assembly or the length of a finite cylinder. + +7. **DZ** + + BUCKLING DEPTH. This is the buckling width in cm. + It corresponds to the second transverse dimension of an actual 3-D slab + assembly. + +8. **CELLMIX** + + CELL-WEIGHTED MIXTURE NUMBER. Enter ONLY if a + cell-weighted mixture is required. Enter a unique mixture number used to + create a cell-weighted homogeneous mixture (:ref:`7-1-2-4`). + +9. **END** + + The word **END** is entered to terminate these data before + entering the zone description data. It must not be entered in columns 1 + through 3, and at least two blanks must separate it from the zone + description. A label can be associated with this **END**. The label can + be a maximum of 12 characters and is separated from the **END** by a + single blank. At least two blanks must follow this entry. + + The zone description data are entered at this point. Entries 10 and + 11 are entered for each zone, and the sequence is repeated until all + the desired zones have been described. To terminate the data, enter + the words END ZONE. Zone dimensions must be in increasing order. + +10. **mxz** + + MIXTURE NUMBER IN THE ZONE. Enter the mixture number of the + material that is present in the zone. Enter a zero for a void. Repeat + the sequence of entries 10 and 11 for each zone. Mixtures other than + zero must not be used more than once in a cell and may be used in no + more than one cell. + +11. **rz** + + OUTSIDE RADIUS OF THE ZONE. Enter the outside dimension of + the zone in cm. + + In **SLAB** geometry, **rz** is the location of the zone’s right + boundary on the X-axis. Repeat the sequence of entries 10 and 11 for + each zone. + +12. **END ZONE** + + Is used to terminate the **MULTIREGION** zone data. + Enter the words **END** **ZONE** when all the zones have been described. + Note that **ZONE** is a label associated with this **END**. This label + can be as long as 12 characters, but the first four characters must be + **ZONE**. At least two blanks must follow this entry. + +.. _tab7-1-5: +.. list-table:: Unit cell specification for MULTIREGION problems. + :align: center + + * - .. image:: figs/XSProc/tab5.svg + :width: 1000 + +.. _7-1-3-7: + +Unit cell specification for doubly heterogeneous (DOUBLEHET) cells +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The data required for a **DOUBLEHET** cell are given in :numref:`tab7-1-6` and +explained in the following text. + +Details about the computation procedures for **DOUBLEHET** cells can be +found in :ref:`7-1-2-3`. + +“Grain” refers to a spherical fuel particle surrounded by one or more +coating zones and does not include the matrix material the grains are +in. “Grain type” refers to a grain that has specified dimensions and +mixtures such as a 0.025-cm-radius UO\ :sub:`2` fuel kernel with a +0.01-cm-thick carbon coating. Another grain type could be a +0.012-cm-radius PuO\ :sub:`2` fuel kernel with a 0.01-cm-thick carbon +coating. The user must first define all grain types in a fuel element. +Next, all fuel element–related data must be entered. + +Since all grains and the matrix material are homogenized into a single +uniform mixture for the fuel element, there are restrictions on how each +grain type must be defined so that the volume fraction of each grain +type within the homogenized fuel mixture can be determined. Related +entries are **PITCH**, **NUMPAR** (number of particles), and **VF** +(volume fraction). If there is only one grain type for a fuel element, +the code needs the pitch and will directly use the input value if +entered. If **PITCH** is not given, then the **VF** (if given) is used +to calculate the pitch. If neither **PITCH** nor **VF** is given, then +**NUMPAR** is used to calculate the pitch and the volume fraction. The +user should only enter one of these items. + +If more than one grain type is present, additional information is needed +since all grain types are homogenized into a single mixture. Similar to +the one grain type case, the pitch is needed to perform the CENTRM +spherical cell calculations. However, the pitch by itself is not +sufficient to perform the homogenization. Therefore, the user needs to +input **VF** or **NUMPAR** for each grain type. Since each grain’s +volume is known (grain dimensions must always be entered), entering +**NUMPAR** or **VF** for each grain type essentially provides the total +volume of each grain type and therefore enables the calculation of the +other unknowns (**VF** or **NUMPAR**, and **PITCH**). In this case, +since pitch is not given, the available matrix material is distributed +around the grains of each grain type proportional to the grain volume to +calculate the corresponding pitch. + +Syntax: + +**DOUBLEHET** *fuelmix* **END** + +**GF**\ (**D**\ \|\ **R**)=\ *fuel mg* +(**COAT**\ (**D**\ \|\ **R**)=\ *coat mc*)|(\ **COATT**\ =\ *coat mc*) +{**H**}\ **PITCH**\ =\ *mod* **MATRIX**\ =\ *mm* **NUMPAR**\ =\ *npar* +**VF**\ =\ *vf* **END GRAIN** + +**mct** **ctp** **FUEL**\ (**D**\ \|\ **R**)=\ *mfuel* +{**FUELH**\ =\ *hfuel*} {**FUELW**\ =wfuel} +{**GAP**\ (**D**\ \|\ **R**)=\ *mgap mmg*} +{**CLAD**\ (**D**\ \|\ **R**)=\ *mclad* *mmc*} +{**H**}\ **PITCH**\ =\ *mpitch* *mmm* {**CELLMIX**\ =\ *mcmx*} **END** + + +1. **celltype** + + DOUBLEHET. The keyword DOUBLEHET is used to represent a doubly heterogeneous + problem such as fuel units that are composed of grains of fuel. + +2. *fuelmix* + + HOMOGENIZED MIXTURE NUMBER. Enter a unique mixture number to be + used for the homogenized grains and matrix material. + +3. **END** + + The word **END** is entered to terminate these data before entering the grain and + fuel element description data. It must not be entered in columns 1 through 3, + and at least two blanks must separate it from the zone description. + A label can be associated with this **END**. The label can be a maximum of + 12 characters and is separated from the **END** by a single blank. At least + two blanks must follow this entry. + + +The grain description data are entered at this point. Entries 5 through +12 are entered for each grain, and the sequence is repeated until all +the grains have been described. To terminate the data, enter the words +**END GRAIN**. Data may be entered in any order. + +4. **PITCH** or **HPITCH** + + EQUIVALENT CELL DIMENSION. This is the equivalent spherical diameter (or radius), + in cm, of the "average" unit cell for this grain, as shown in :numref:`fig7-1-7`. + Physically, the volume of the average unit cell is equal to the volume of the + fuel element divided by the total number of all grain types. + +5. **GFD** or **GFR** + + OUTSIDE DIMENSION OF FUEL. This is the outside diameter or radius of the fuel + zone in a grain, *fuel*, in cm followed by the fuel mixture number, *mg*, as shown in + :numref:`fig7-1-7`. + +6. **COATD** or **COATR** + + OUTSIDE DIMENSION OF COATING. This is the outside diameter or radius of a + coating zone, *coat*, in cm followed by the coating mixture number, *mc*, as shown in + :numref:`fig7-1-7`. As many coating-mixture pairs as desired may be entered. + If the coating dimensions are entered using COATD or COATR, then the COATT keyword should not be used. + +7. **COATT** + + THICKNESS OF COATING. This is the thickness of a coating zone, *coat*, + in cm followed by the coating mixture number, *mc*, as shown in :numref`fig7-1-7`. + As many coating-mixture pairs as desired may be entered. If the coating + dimensions are entered using COATT, then the COATD or COATR keyword should not be used. + +8. **MATRIX** + + MIXTURE NUMBER OF THE MATRIX MATERIAL. This is the mixture number, *mm*, of the matrix material that encloses the grains. + +9. **NUMPAR** + + NUMBER OF PARTICLES. This is the number of grains, *npar*, of this type in each fuel element. + +10. **VF** + + VOLUME FRACTION. This is the volume fraction, *vf*, of grains of this type in each + fuel element’s fuel zone. A fuel element’s fuel zone is entered using the entry + number 16—\ **FUELD** (or **FUELR**). + +11. **END GRAIN** + + This is used to terminate the grain zone data for this grain type. At least two blanks must follow this entry. + +REPEAT ENTRIES 4-11 FOR EACH GRAIN TYPE IN A FUEL ELEMENT. + +12. **mct** + + TYPE OF FUEL ELEMENT (macro cell type). One of the keywords **PEBBLE** or **ROD** + or SLAB is entered to indicate the type of the fuel element, i.e., the second + layer of heterogeneity. This data must be entered. The keyword may NOT be + truncated. **PEBBLE** is used for spherical fuel elements; **ROD** is used for + cylindrical fuel elements; and SLAB for plate fuel elements. + +13. **ctp** + + TYPE OF LATTICE. This defines the type of lattice or array configuration. + Any one of the following alphanumeric descriptions may be used. Note that the + alphanumeric description must be separated from subsequent data entries by one + or more blanks. :numref:`fig7-1-1` Mixture and position data are entered using keywords. + Mixture number 0 may be entered for void and may be used multiple times + in each and all unit cells. For regular cells, the minimum requirement is that + a fuel region and a moderator region are specified and no inner components are + specified. For annular cells, the minimum requirement is the fuel and outer + moderator and inner moderator regions are specified. Regular and annular cell + configurations are specified as shown below. + +**Regular Cells** + + **SQUAREPITCH** is used for an array of cylinders arranged in a + square lattice, as shown in :numref:`fig7-1-1`. The clad and/or gap can be + omitted. + + **TRIANGPITCH** is used for an array of cylinders arranged in a + triangular-pitch lattice as shown in :numref:`fig7-1-2`. The clad and/or + gap can be omitted. + + **SPHSQUAREP** is used for an array of spheres arranged in a + square-pitch lattice. A cross section view through a cell is + represented by :numref:`fig7-1-1`. The clad and/or gap can be omitted. + + **SPHTRIANGP** is used for an array of spheres arranged in a + triangular-pitch (dodecahedral) lattice. A cross section view through + a cell is represented by :numref:`fig7-1-2`. The clad and/or gap can be + omitted. + + **SYMMSLABCELL** is used for an infinite array of symmetric slab + cells, as shown in :numref:`fig7-1-3`. The clad and/or gap can be omitted. + + **Annular Cells** + + **ASQUAREPITCH** or **ASQP** is used for annular cylindrical rods in + a square-pitch lattice as shown in :numref:`fig7-1-4`. The inner and outer + clad and gap are independently entered so they may be different + materials and dimensions. + + **ATRIANGPITCH** or **ATRP** is used for annular cylindrical rods in + a triangular-pitch lattice as shown in :numref:`fig7-1-5`. The inner and + outer clad and gap are independently entered, so they may be + different materials and dimensions. + + **ASPHSQUAREP** or **ASSP** is used for spherical shells in a + square-pitch lattice as shown in :numref:`fig7-1-4`. The inner and outer + clad and gap are independently entered, so they may be different + materials and dimensions. + + **ASPHTRIANGP** or **ASTP** is used for spherical shells in a + triangular-pitch (dodecahedral) lattice as shown in :numref:`fig7-1-5`. The + inner and outer clad and gap are independently entered, so they may + be different materials and dimensions. + + **ASYMSLABCELL** is used for a periodic, but asymmetric, array of + slabs as shown in :numref:`fig7-1-6`. The inner and outer clad and gap are + independently entered, so they may be different materials and + dimensions. + +14. **PITCH** or **HPITCH** + + ARRAY PITCH. This is the center-to-center spacing or half-spacing between the + fuel lumps (pebbles or rods or slabs), *mpitch*, in cm followed by the outer + moderator material number, *mmm*, as shown in :numref:`fig7-1-1` and :numref:`fig7-1-2`. + +15. **FUELD** or **FUELR** + + OUTSIDE DIMENSION OF FUEL. This is the outside dimension (diameter or radius + for sphere/cylinder or x-thickness for slab) of the fuel region, *mfuel*, in cm, + as shown in :numref:`fig7-1-1` and :numref:`fig7-1-2`. + +16. **FUELH** + + HEIGHT OF FUEL ROD OR SLAB. This is the height (z-dimension) of the fuel plate, + *hfuel*, in cm. (only used to compute volume of fuel plate). + +17. **FUELW** + + WIDTH OF FUEL ROD or slab. This is the width/depth (y-dimension) of the fuel + plate, *wfuel*, in cm. (only used to compute volume of fuel plate). + +18. **GAPD** or **GAPR** + + OUTSIDE DIMENSION OF GAP. Enter only if outer gap is present. This is the + outside diameter or radius of the outer gap, *mgap*, in cm followed by the gap + mixture number, *mmg*, as shown in :numref:`fig7-1-1` and :numref:`fig7-1-2`. + +19. **CLADD** or **CLADR** + + OUTSIDE DIMENSION OF CLAD. Enter ONLY if a clad is present. This is the + outside diameter or radius of the outer clad, *mclad*, in cm followed by the + clad mixture number, *mmc*, as shown in :numref:`fig7-1-1` and :numref:`fig7-1-2`. + +20. **CELLMIX** + + CELL-WEIGHTED MIXTURE NUMBER. Enter ONLY if cell-weighted mixture, *mcmx*, is to be created. + +21. **IMODD** or **IMODR** + + DIMENSION OF INNER MODERATOR. Enter ONLY if an annular cell is specified. + This is the outside diameter or radius of the inner moderator, *imod*, in + cm followed by the inner moderator mixture number, *mim*, as shown in + :numref:`fig7-1-4` through :numref:`fig7-1-6`. + +22. **IGAPD** or **IGAPR** + + OUTSIDE DIMENSION OF INNER GAP. Enter ONLY if an annular cell is specified and + inner gap is present. This is the outside diameter or radius of the inner gap, + *igap*, in cm followed by the inner gap mixture number, *mig*, as shown in + :numref:`fig7-1-4` through :numref:`fig7-1-6`. + +23. **ICLADD** or **ICLADR** + + OUTSIDE DIMENSION OF INNER CLAD. Enter ONLY if an annular cell is specified + and inner clad is present. This is the outside diameter or radius of the inner + clad, *iclad*, in cm followed by the inner clad mixture number, *mic*, as shown + in :numref:`fig7-1-4` through :numref:`fig7-1-6`. + +24. **END** + + The word **END** is entered to terminate the **DOUBLEHET** data. + An optional label can be associated with this **END**. The label can be + as many as 12 characters long and is separated from the **END** by a single + blank. At least two blanks must follow this entry. + +.. _fig7-1-7: +.. figure:: figs/XSProc/fig7.png + :align: center + :width: 500 + + Arrangement of materials in a grain (first level cell) in a DOUBLEHET unit cell. + +.. _tab7-1-6: +.. list-table:: Unit cell specification for DOUBLEHET problems + :align: center + + * - .. image:: figs/XSProc/tab6.svg + :width: 1000 + +.. list-table:: Unit cell specification for DOUBLEHET problems (continued). + :align: center + + * - .. image:: figs/XSProc/tab6cont.svg + :width: 1000 + +.. list-table:: Unit cell specification for DOUBLEHET problems (continued). + :align: center + + * - .. image:: figs/XSProc/tab6cont2.svg + :width: 1000 + +.. _7-1-3-8: + +Optional MORE DATA parameter data +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +**MORE DATA** is an optional sub-block of the **READ CELL** block. +**MORE DATA** parameters allow certain default options in BONAMI and +XSDRNPM to be modified for individual cell calculations. Each **MORE +DATA** sub-block applies only to the unit cell immediately preceding it. +However a **MORE DATA** sub-block placed prior to all unit cell +definitions applies to all mixtures not assigned to a unit cell, which +are treated as infinite homogeneous media. If the default parameters are +acceptable, this section of input data should be omitted in its +entirety. Non-default values for one or more of the parameters can be +specified by entering the words **MORE DATA** followed by the desired +keyword parameters and their associated values. One or more of the +parameters can be entered in any order. Default values are used for +parameters that are not entered. Each parameter is entered by spelling +its name, followed immediately by an equal sign and the value to be +entered. There should not be a blank between the parameter name and the +equal sign. Each parameter specification must be separated from the rest +by at least one blank. For example, if an XSDRNPM calculation is +performed for particular unit cell (e.g., *cellmix=* is specified), + +**MORE DATA ISN**\ =16 **EPS**\ =0.00008 **END MORE** + +would result in using an S16 angular quadrature set and tightening the +convergence criteria to 0.00008 in the XSDRNPM calculation. + +A description of each entry is given. (Also see sections on BONAMI and XSDRNPM input description.) + +1. **MORE DATA** These words, followed by one or more blanks, are + entered ONLY if optional parameter data are to be entered. Entries 2 + through 42 can be entered in any order. + +2. **NSENSX** This is the XSDRNPM sensitivity output file for TSUNAMI + sequences. + +3. **CROSSEDT** BONAMI CROSS SECTION EDIT. Cross section print option + for BONAMI 0/1 –no/yes (default is 0). + +4. **FFACTEDT** BONDARENKO FACTOR EDIT. Bondarenko factor (f-factor) + print option 0/1 –no/yes (default is 0). + +5. **ISSOPT** BONAMI BACKGROUND XSEC OPTIONS. BONAMI background cross + section selection option if > 1000 potential; otherwise, total cross + section is used (default is –1). + +6. **IROPT** BONAMI IR/NR CALCULATION OPTION. BONAMI uses intermediate + resonance (IR) if iropt=1 and narrow resonance (NR) approximation + for iropt=0 (default is 0). + +7. **BELLOPT** BELL FACTOR OPTION. Optional user-defined bell factor + calculation option (default is -1). + +8. **BELLFACT** BELL FACTOR. Optional user-defined bell factor for + BONAMI (default is 0.0). + +9. **ESCXSOPT** ESCAPE CROSS SECTION CALC OPTION. Escape cross section + calculation for IR calculations. 0/1 =consistent/inconsis tent + (default is 0). + +10. **BONAMIEPS** BONAMI CONVERGENCE CRITERIA. BONAMI Bondarenko + iteration convergence criteria (default is 0.001). + +11. **LBARIN** INPUT MEAN CORD LENGTH. Mean cord length for each zone + (default is 0.00). + +12. **ADJTHERM** ADJUST 1D THERMAL CROSS SECTIONS TO MATCH SUM OF 2D + CROSS SECTIONS. Flag determining whether 1-D cross sections are + scaled to match the 2-D cross sections or the 2-D cross sections are + scaled to match the 1-D cross sections. + +13. **EXSIG** ESCAPE CROSS SECTION. External escape cross section for + BONAMI (default is 0.00). + +14. **IEVT** XSDRNPM CALCULATION TYPE. The type of calculation to be + performed— fixed source, eigenvalue, alpha, zone width search, outer + radius search, buckling search, direct buckling search (default is + 1). + +15. **ICLC** THEORY OPTION. Number of outer iterations to use an + alternative theory (diffusion, infinite medium, or B\ :sub:`N`) + before using discrete ordinates. Negative values indicate + alternative theory (default is 0). + +16. **IPVT** PARAMETRIC EIGENVALUE SEARCH. 0 – none; 1 – search for + eigenvalue equal PV; 2 – alpha search (default is 0). + +17. **IPP** WEIGHTED CROSS SECTION PRINT. 2 -> No print; -1 -> 1-D edit; + 0-N – edit through PN cross section arrays (default is 2). + +18. **IFLU** GENERALIZED ADJOINT CALCULATION. 0 is a standard + calculation; 1 is a generalized adjoint calculation (default is 0). + +19. **IFSN** FISSION SOURCE SUPPRESSION. Non-zero suppresses the fission + source in a fixed source calculation (default is 0). + +20. **IQM** VOLUMETRIC FIXED SOURCES. The number of volumetric sources + in a fixed source problem (default is 0). + +21. **IPM** BOUNDARY FIXED SOURCES. The number of boundary sources in a + fixed source problem (default is 0). + +22. **XNF** SOURCE NORMALIZATION FACTOR. The value used to normalize the + problem source (default is 1.0). + +23. **VSC** VOID STREAMING CORRECTION. The height of a void streaming + path in a cylinder or slab in centimeters (default is 0.0). + +24. **EV** EIGENVALUE GUESS. Starting eigenvalue guess for a search + calculation (default is 0.0). + +25. **EQL** INITIAL SEARCH CONVERGENCE. Initial eigenvalue search + convergence (default is 0.0001). + +26. **XNPM** DAMPING FACTOR. Damping factor used in search calculations + (default is 0.75). + +27. **ISN** ORDER OF ANGULAR QUADRATURE FOR XSDRNPM. Quadrature sets are + geometry-dependent quantities that are defaulted to order 8 by the + XSProc for **LATTICECELL** and cylindrical **MULTIREGION**. The + default is 32 for **MULTIREGION** slabs and spheres. See the + automatic quadrature generator and Appendix B for a more detailed + explanation. + +28. **SZF** SPATIAL MESH SIZE FACTOR FOR XSDRNPM. The size of the mesh + intervals can be adjusted by entering a value for **SZF**, which is + a multiplier of the mesh size. The default value is 1.0. A value + between zero and 1.0 yields a finer mesh; a value greater than 1.0 + yields a coarser mesh. If **SZF** ≤ 0, the user specifies the number + of mesh intervals in each zone immediately following the **MORE + DATA** block. If **SZF** = 0, the interval spacing is automatically + generated, while if **SZF** < 0 the intervals are equally spaced + intervals in each zone. + +29. **IIM** MAXIMUM NUMBER OF INNER ITERATIONS FOR XSDRNPM. This is the + maximum number of inner iterations to be used in the XSDRNPM + calculation. The default value is 20. See Appendix B for a more + detailed explanation. + +30. **ICM** MAXIMUM NUMBER OF OUTER ITERATIONS FOR XSDRNPM. This is the + maximum number of outer iterations to be used in the XSDRNPM + calculation. The default value is 25. If the calculation reaches the + outer iteration limit, a larger value should be used. See Appendix B + for a more detailed explanation. + +31. **EPS** OVERALL CONVERGENCE CRITERIA FOR XSDRNPM. This is used by + XSDRNPM after each outer iteration to determine if the problem has + converged. The default value of **EPS** is 0.00001. A value less + than 0.00001 tightens the convergence criteria; a larger value + loosens the convergence criteria. + +32. **PTC** POINTWISE CONVERGENCE CRITERIA FOR XSDRNPM. This is the + point flux convergence criteria used by XSDRNPM to determine if + convergence has been achieved after an inner iteration. The default + value for PTC is 0.000001. A smaller value tightens convergence; a + larger value loosens it. + +33. **BKL** BUCKLING FACTOR FOR XSDRNPM. A buckling factor should be + used ONLY for a **MULTIREGION** **BUCKLEDSLAB** or **BUCKLEDCYL** + problem. Because cylinders are assumed to be infinitely long and + slabs are assumed to be infinite in both transverse directions, the + analytic sequence may tend to overestimate the total flux for a + finite system. A buckling correction can be used to approximate the + leakage from the system in the transverse direction(s). The + extrapolation distance factor, **BKL**, is defaulted to 1.420892. + +34. **IUS** UPSCATTER SCALING FLAG for XSDRNPM. This option allows the + use of upscatter scaling to accelerate the solution or force + convergence. The default value is zero, in which case upscatter + scaling is not used. **IUS**\ =1 facilitates upscatter scaling. + Guidelines are not available to indicate when upscatter scaling is + needed. Some problems will not converge with it, and some will not + converge without it. See Appendix B for a more detailed explanation. + +35. **DAN**\ (mm) DANCOFF FACTOR for the specified mixtures used in + BONAMI and in the CENTRM 2REGION option. This value overrides the + internally computed Dancoff factor used in the resonance correction + for the specified mixture *mm*. The Dancoff data are entered in the + form **DAN**\ (mm) = Dancoff factor. Note that the parentheses must + be entered as part of the data, and the mixture number, mm, must be + enclosed in the parentheses. See Appendix B for additional details. + (Note: this is not to be confused with the DAN2PITCH parameter in + CENTRMDATA) + +36. **BAL** BALANCE TABLE PRINT FLAG for XSDRNPM. This allows control of + the balance table print from **XSDRNPM**. The default value is + **FINE**. **BAL**\ =\ **NONE** suppresses the balance table print. + **BAL**\ =\ **ALL** prints all of the balance tables. + **BAL**\ =\ **FINE** prints only the fine-group balance tables. See + Appendix B for additional details. + +37. **DY** FIRST TRANSVERSE DIMENSION for XSDRNPM. This is the first + transverse dimension, in cm, used in a buckling correction to + calculate the leakage normal to the principal calculation direction + (the height of a slab or cylinder). It should only be entered if + XSDRNPM is to create cell-weighted cross sections and/or calculate + the eigenvalue of a cylinder or slab system of finite height for a + **LATTICECELL** problem. **DY**\ = is defaulted to an infinite + height, or is set to **DY** for a buckled **MULTIREGION** cell + description. A value entered here overrides any buckling height + value entered in the **MULTIREGION** data. + +38. **DZ** SECOND TRANSVERSE DIMENSION for XSDRNPM. This is the second + transverse dimension, in cm, used for a buckling correction for a + slab of finite width. It should only be entered if XSDRNPM is to + create cell-weighted cross sections and/or calculate the eigenvalue + of a **LATTICECELL** slab of finite width. **DZ**\ = is defaulted to + an infinite width, or is set to **DZ** for a buckled **MULTIREGION** + slab cell of finite width. A value entered here overrides any + buckling depth value entered in the **MULTIREGION** data. + +39. **COF** DIFFUSION COEFFICIENT FOR TRANSVERSE LEAKAGE CORRECTIONS IN + XSDRNPM. The default value is 3. The available options are as + follows. + + **COF**\ =0 sets a transport-corrected cross section for each zone + + **COF**\ =1 use a spatially averaged diffusion coefficient for each + zone + + **COF**\ =2 use a diffusion coefficient for all zones that is + one-third of the diffusion coefficient determined from the spatially + averaged transport cross section for all zones + + **COF**\ =3 use a flux and volume weighting across all zones + + See Appendix B or XSDRNPM Input/Output Assignments in the XSDRNPM + chapter, 3$ array, variable **IPN** for more details. + +40. **NT3** UNIT WHERE XSDRNPM WRITES THE WEIGHTED LIBRARY. If XSDRN + does a weighting calculation, this is the unit number it uses to + write the weighted library on (default is 3). + +41. **NT4** UNIT WHERE XSDRNPM WRITES THE ANGULAR FLUXES. XSDRN writes + the angular fluxes on this unit if it is non-zero (default is 16). + +42. **ADJ** Adjoint mode flag for XSDRNPM. Set to 1 to cause XSDRNPM to + solve the adjoint problem (default is 0). + +43. **NTA** UNIT WHERE XSDRNPM WRITES THE ACTIVITIES. XSDRN writes the + calculated activities on this unit if it is non-zero (default is + 75). + +44. **NBU** UNIT WHERE XSDRNPM WRITES BALANCE TABLES. If the balance + tables file is to be saved, enter the unit number where it is to be + written (default is 76). + +45. **NTC** UNIT WHERE XSDRNPM WRITES THE DERIVED DATA. XSDRN writes the + derived input data on this unit if it is non-zero (default is 73). + +46. **NTD** UNIT WHERE XSDRNPM WRITES THE DATA FOR A SENSITIVITY + ANALYSIS. XSDRN writes the data for a sensitivity analysis on this + unit if it is non-zero (default is 0). + +47. **FRD** UNIT WHERE XSDRNPM READS INPUT FLUX GUESS. If greater than + 0, a flux guess will be read from this unit. + +48. **FWR** UNIT WHERE XSDRNPM WRITES OUPUT FLUX. If greater than 0, the + space-dependent multigroup scalar flux is written in binary format + to this unit. + +49. **WGT** CROSS SECTION WEIGHTING FLAG for XSDRNPM. The default is 0, + not to perform cross section weighting. To turn on cross section + weighting, a positive value should be entered. A value of 1 will + weight the cross sections by nuclide; 2 will weight by mixture. + +50. **ZMD**\ (iz) ZONE WIDTH MODIFIERs for an XSDRNPM search problem. + This allows entering a zone width modifier for zone iz in the + XSDRNPM problem description. The zone width data are entered in the + following form: + + **ZMD(iz)=modifier** + + Note that the parentheses must be entered as part of the keyword. + The zone number iz, to which the modifier is applied, must be + enclosed in the parentheses. The modifier is entered after the equal + sign. See the “Dimension Search Calculations” description in the + XSDRNPM chapter for more information. + +51. **INT**\ (iz) NUMBER OF MESH INTERVALS FOR ZONE IZ in XSDRNPM. The + default is 0, which causes the number to be calculated. The data are + entered in the following form: + +.. + + **INT(iz)=number** + + Note that the parentheses must be entered as part of the keyword. The + zone number iz, for which the number of intervals is specified, must + be enclosed in the parentheses. The number of intervals is entered + after the equal sign. + +52. **KEF** DESIRED VALUE OF *k*\ :sub:`EFF` for an XSDRNPM zone width search. + The default value is 1.0. If it is desired to search for some other + value, such as 0.9, then input it here. + +53. **KFM** The first eigenvalue modifier used in an XSDRNPM search. + This value is used to make the first change in the XSDRNPM search. + The default value is −0.1. A user may sometimes need to change this + to make the search converge. + +54. **ID1** SCALAR FLUX PRINT CONTROL. The default value is −1, which + suppresses printing the scalar fluxes in XSDRNPM. See the XSDRNPM + Input/Output Assignments section in the XSDRNPM chapter, 2$ array, + variable **ID1** for allowed values and corresponding actions. + +55. **ISCT** ORDER OF SCATTERING for XSDRMPM. The default is 5 for all + libraries. + +56. **ICON** TYPE OF WEIGHTING (see Cross-Section Weighting section in + the XSDRNPM chapter). + +.. + + **INNERCELL** − followed by integer N (zones in the cell). Cell + weighting is performed over the N innermost regions in the problem. + Nuclides outside these regions are not weighted. + + **CELL** − cell weighting + + **ZONE** − zone weighting + + **REGION** − region weighting + +58. **ITP** COLLAPSED OUTPUT FORMAT. The default is 0. + +.. + + 0−19 − cross sections are written only in the AMPX weighted library + formats on logical 3. A weighted library is always written when IFG= + 1. + + The various values of ITP (modulo 10) are used to select the + different transport cross section weighting options mentioned + earlier. The options are as follows: + + ITP = 0, 10, ... :math:`\sqrt{(\psi^{g}_{1} + (DG\psi))^{2}}` + + ITP = 1 ,11, ... absolute value of current + + ITP = 2, 12, ... :math:`DB^{2}\psi_{g}` + outside leakage + + ITP = 3, 13, ... :math:`\frac{\psi}{\Sigma^{g}_{t}}` + + ITP = 4, 14, ... :math:`DB\psi_{g}` + + ITP = Other values are reserved for future development and should not be used. + +59. **GAMMA_MT_LIST** LIST OF GAMMA 1D REACTIONS ASSOCIATED WITH INPUT. + A list of 1-D gamma reactions to be included on a condensed library + for later use gamma_mt_list= numberEntries mt1 mt2 ... + mt_numberEntries. + +60. **NEUTRON_MT_LIST** LIST OF NEUTRON 1D REACTIONS ASSOCIATED WITH + INPUT. A list of 1-D neutron reactions to be included on a condensed + library for later use neutron_mt_list= numberEntries mt1 mt2 ... + mt_numberEntries. + +61. **NEUTRON_2D_LIST** LIST OF NEUTRON 2D ARRAYS FOR THE MICRO LIBRARY. + This list flags the finalizer to place 2-D arrays (currently MT 2, + 4, 16) on the micro library for use in SAMS. + +62. **ACTIVITY** Enter: + + IAZ (number of activities) + + IAI (calculate activities by zone or interval) + + 0 – zone + + 1 – interval + + LACFX (unit number to which activities are written) + + LAZ (IAZ sets of numbers consisting of the nuclide and process + numbers for each activity) + +63. **BAND** NUMBER OF REBALANCE BANDS for XSDRNPM (default is 1). + +64. **IPRT** CROSS SECTION PRINT CONTROL. The default value is −2, which + suppresses printing the cross sections in XSDRNPM. See XSDRNPM + chapter, 2$ array, variable IPRT for allowed value, and + corresponding actions. + +65. **GRAIN_K** Flag to control execution of XSDRNPM after each grain + calculation for a **DOUBLEHET** cell. + +66. **SOURCE**\ (iz) ZONE SOURCE for an XSDRNPM fixed source problem. + This allows entering a source spectrum for zone iz in the XSDRNPM + problem description. The source spectrum data are entered in the + following form: + + **SOURCE(iz)= numEntries spectrum_grp_1 … spectrum_grp_numEntries** + + Note that the parentheses must be entered as part of the keyword. The + zone number, iz, to which the spectrum is applied, must be enclosed by + the parentheses. The numEntries follows the equal sign and must be less + than or equal to the number of energy groups for the problem. It is + followed by numEntries numbers defining the spectrum for the first + numEntries groups for zone iz. Groups not defined are set to zero. The + spectrum applies uniformly to zone iz. A different spectrum may be + entered for different zones. + + +67. **END MORE** Terminate the optional parameter data. + +.. _7-1-3-9: + +Optional CENTRM DATA parameter data +~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + +The CENTRM DATA block defines input parameter values for the CENTRM, PMC +and CRAWDAD modules. XSProc defines default values for these parameters +which are adequate for most applications. If all default values are +acceptable, this section of input data can be omitted. The CENTRM DATA +block applies only to the unit cell immediately preceding it. CENTRM +DATA placed prior to all unit cell data applies to all materials not +listed in any unit cell. Parameter values are assigned by entering the +words **CENTRM DATA** followed by the desired keyword parameters and +their associated values. One or more parameters can be entered in any +order. There should not be a blank between the parameter name and the +equal sign. Each parameter specification must be separated from the rest +by at least one blank. For example, + +:: + + **CENTRM DATA ISN**\ =16 **PTC**\ =0.0008 **N1D**\ =1 **END CENTRM + DATA** + +A description of CENTRM DATA parameters is given below. + +1. **CENTRM DATA** These words, followed by one or more blanks, are + entered ONLY if optional parameter data are to be entered. They must + precede all other optional parameter data. Entries 2 through 42 can + be entered in any order. + +2. **ISN** ORDER OF SN ANGULAR QUADRATURE FOR CENTRM. SN Quadrature + sets are geometry-dependent quantities. Default value for **ISN** is + 6 (only used for **NFST** and **NTHR**\ =0; and **NPXS**\ =1). + +3. **ISCT** LEGENDRE POLYNOMIAL P\ :sub:`N` ORDER OF SCATTERING. These + are used to determine the number of moments calculated for the + scattering cross sections. Default value is 0 for 2-D MoC option and + 1 for 1-D S\ :sub:`n`, which have been found adequate for nearly all + cases. + +4. **IIM** MAXIMUM NUMBER OF INNER ITERATIONS. This is the maximum + number of inner iterations for Sn transport calculations in CENTRM. + Default value is 10. + +5. **IUP** MAXIMUM NUMBER OF OUTER ITERATIONS IN THERMAL RANGE. This is + the maximum number of outer iterations used to converge PW flux + changes caused by upscattering in the thermal range. Default value + is 3. More iterations (~ 15) may be required for higher accuracy in + some cases. + +6. **NFST** FAST RANGE MULTIGROUP CALCULATION OPTION, E > \ **DEMAX**. + This determines what type of calculation is done above **DEMAX**. + The options are (0) S:sub:`N`, (1) diffusion theory, (2) homogenized + infinite medium, (3) zonewise infinite medium, or (6) 2D MoC lattice + cell [NOTE: NFST=4,5 are deprecated]. Default value is 0 + (S\ :sub:`N`). + +7. **NTHR** THERMAL RANGE MULTIGROUP CALCULATION OPTION, + E < \ **DEMIN**. This determines what type of calculation is done + below **DEMIN**. The options include (0) S:sub:`N`, (1) diffusion, + (2) homogenized infinite medium, (3) zonewise infinite medium, or + (6) 2-D MoC lattice cell [NOTE: NTHR=4,5 are deprecated]. Default + value is 0 (S\ :sub:`N` ). + +8. **NPXS** POINTWISE RANGE MULTIGROUP CALCULATION OPTION, + **DEMIN** < E < **DEMAX**. This determines what type of calculation + is done between **DEMIN** and **DEMAX**. The options include (0) MG + calculation, (1) 1-D S\ :sub:`N`, (2) collision probability, + (3) homogenized infinite medium, (4) zonewise infinite medium, + (5) two-region, or (6) 2-D MoC lattice cell. Default value + is 1 (S\ :sub:`N`), except for square-pitch LATTICECELL where the + default is 6 (2D MoC). + +9. **ISVAR** LINEARIZATION OPTION. This determines if the MG source + and/or the cross sections are linearized in CENTRM calculations. + Options for linearizing are (0) neither, (1) source, (2) cross + section, or (3) both. Default value is 3. + +10. **ISCTI** LEGENDRE POLYNOMIAL P\ :sub:`N` ORDER OF SCATTERING IN THE + INELASTIC RANGE. These are used to determine the number of moments + calculated for the inelastic scattering cross sections. Default + value is 0, isotropic. + +11. **NMF6** INELASTIC FLAG. This determines if inelastic data are used. + The options are to include (−1) no inelastic data, (0) discrete + inelastic data, and (1) discrete inelastic and continuum. Default + value is −1. Use of **NMF6**\ =1 is not recommended due to long + running times. + +12. **IPRT** MIXTURE CROSS-SECTION OUTPUT OPTION. This determines the + output of cross section. The options include (−3) none, (−2) output + macro PW cross sections to file “_­centrm.pw.macroxs”, (−1) 1-D MG + cross sections, (N) P\ :sub:`0` to P\ :sub:`N` MG 2-D matrices. + Default value is −3, none. + +13. **ID1** FLUX EDIT OPTION. This option determines the output of flux + energy spectra. The options are (−1) none, (0) print MG fluxes, + (1) also print MG flux moments, (2) save CE fluxes on output file, + “_centrm.pw_flux”. Default value is −1. + +14. **KERNEL** BOUND KERNELS. This indicates use of CENTRM PW thermal + kernel data [S(α,β)] for bound nuclides if **KERNEL**\ =1. If + **KERNEL**\ =0, all thermal kernels are treated as free gas; Default + is 1, use bound scattering kernels if available. + +15. **IPBT** PRINT GROUP SUMMARY TABLES. Group summary tables for each + zone are printed in CENTRM if greater than 0. Default is 0. Balance + ratios are not computed in thermal groups or for MoC option. + +16. **IPN** GROUP DIFFUSION COEFFICENT. Used for DB\ :sup:`2` loss term. + See XSDRNPM chapter for more information. Default is 2. + +17. **IXPRT** PRINT OPTION FOR CENTRM. This value is >0 if more + information is printed to output. Default value is 0, minimum + output. + +18. **MLIM** MASS VALUE RESTRICTION ON ORDER OF SCATTERING. Nuclides + with mass ratios greater than **MLIM** are limited to a **NLIM** + order of scattering. Default value is 100. + +19. **NLIM** ORDER OF SCATTERING RESTRICTION. This is the limiting order + of scattering for all nuclides with mass ratios greater than + **MLIM**. Default value is 0. + +20. **EPS** INTEGRAL CONVERGENCE CRITERIA. This is used by CENTRM after + each outer iteration to determine if the problem has converged. + Default value is 0.001. A value less than 0.0001 tightens the + convergence criteria; a larger value loosens the convergence + criteria. + +21. **PTC** POINTWISE CONVERGENCE CRITERIA. This is the point flux + convergence criteria used by CENTRM to determine if convergence has + been achieved after an inner iteration. Default value is 0.0001. A + smaller value tightens convergence; a larger value loosens it. + +22. **B2** MATERIAL BUCKLING FACTOR (cm\ :sup:`-2`). This is used with a + buckled system. If a buckled system is specified for a unit cell, + the code will use this value. Default value is 0.0. + +23. **DEMIN** LOWEST ENERGY OF POINTWISE FLUX CALCULATION. This value is + the lowest energy (eV) for which CENTRM calculates PW fluxes. + Default is 0.001 eV. + +24. **DEMAX** HIGHEST ENERGY OF POINTWISE FLUX CALCULATION. This value + is the highest energy (eV) for which CENTRM calculates PW fluxes. + Default is 20,000.0 eV, which encompasses the resolved resonance + range of all actinides. It is recommended that DEMAX be <500 keV. + +25. **TOLE** CENTRM PW THINNING TOLERANCE. This is the tolerance used to + thin the PW material cross sections after they are mixed. Default + value is 0.001. + +26. **FLET** FRACTIONAL LETHARGY CONSTRAINT. This is the maximum + lethargy difference between points in the flux solution energy mesh. + Smaller values increase the number of energy points. Default value + is 0.1. + +27. **DAN2PITCH** CENTRM DANCOFF FACTOR SEARCH. Fuel Dancoff factor to + search for a Dancoff-equivalent pitch used in the CENTRM cell + calculation. Only applicable in LATTICECELL cases with fuel in + center region, with SN or MoC transport solvers. Default is 0, which + indicates no pitch modification. NOTE! This option should not be + used to enter Dancoff factors for the CENTRM *2REGION* transport + option—use EDAN(m) array in **MOREDATA** for these values. + +28. **MRANGE** PMC GROUP CROSS-SECTION PROCESSING RANGE. This option + determines the range over which the group cross sections will be + processed. The options are (0) compute new group cross section over + the PW range, (1) over the resolved resonance range of each nuclide, + or (2) over the PW flux range (**DEMAX** to **DEMIN**). Default + value is 2. + +29. **N2D** PMC ELASTIC MATRIX PROCESSING FLAG. This option determines + how MG P\ :sub:`N` elastic scattering matrices are obtained. Options + are (-2) perform operations in both (-1) and (2); (−1) compute P0 + self-scatter, then renormalize matrix to shielded 1-D elastic + values; (0) normalize original scatter matrix to shielded 1-D + elastic values; (1) compute new P\ :sub:`N` moments of elastic + matrix using scalar flux and S‑wave kinematics; or (2)  use + flux-moments to compute “consistent PN” correction for diagonal + elements of elastic P\ :sub:`N` components. Default value is −1. For + unit cell calculations in reactor lattices, option -2 may improve + results. NOTE: option 0 is always used in thermal range + +30. **IXTR3** PMC P\ :sub:`N` ORDER FLAG. This option determines the + maximum order of Legendre moments to be retained on output MG + library. The default is 5; i.e., retain scattering moments up to + P\ :sub:`5` if available on the input MG library. If (−1) is + entered, all elastic moments on the MG library are included. + +31. **NPRT** PMC PRINT FLAG. This option determines what is printed to + output. The options include (−1) minimum output, (0) standard + output, (1) print 1-D cross sections, (2) print both 1-D and 2-D + cross sections. Default value is −1, minimum output. + +32. **NWT** PMC MULTIGROUP SPATIAL-WEIGHTING FLAG. This option + determines if the MG data are (0) zone-weighted or + (1) cell-weighted. Default value is 0. + +33. **MTT** PMC MT PROCESSING FLAG. This option determines if reaction + MTs are processed individually or treat dependencies explicitly. If + **MTT**\ =0 all MTs are processed independently; if **MTT**\ =1, all + MTs are processed except 1, 27, and 101. These are then computed as + follows: MT101 = sum MT102 − 114, MT27 = MT18 + MT101, MT1 = MT2 + + MT4 + MT16 + MT17 + MT27. Default value is 1. + +34. **N1D** PMC WEIGHTING FUNCTION FLAG. This is used to determine if + (0) flux weighting or (1) current weighting is used to collapse the + cross sections. Default value is 0, flux weighting. + +35. **PMC_DILUTE** PMC INFINITELY DILUTE BACKGROUND. The background + cross section :math:`\sigma_{0}` value above which materials are considered + to be infinitely dilute in PMC. No resonance shielding corrections + are performed for materials with background cross sections greater + than *pmc_dilute*. Higher values of *pmc_dilute* result in more + nuclides being processed. The default value is 1.0E10. + +36. **MTOUT** PW REACTION TYPES. Reactions included by CRAWDAD on PW + library for MG processing in PMC: (0) all; (1) output only MTs 1, 2, + 4, 102, 18, 452, 455, 456 (and 107 for :sup:`10`\ B or + :sup:`7`\ Li); (2) all from option (1) and all inelastic MTs and 16. + Default is **MTOUT**\ =1 for **NMF6**\ =-1 and **MTOUT**\ =2 for + **NMF6**>-1. + +37. **IBR** CENTRM RIGHT BOUNDARY TYPE. Type of boundary condition on + right boundary of unit cell for CENTRM **LATTICECELL** calculations. + See allowable IBR values in CENTRM. Default is white (**IBR** = 3) + for 1D SN; 2D MoC transport option always uses reflected. + +38. **IBL** CENTRM LEFT BOUNDARY TYPE. Same as **IBR**, but for left + boundary. Default is reflected (**IBL** = 1). + +39. **ALUMP** MASS LUMPING FRACTION. A value in range [0.0, 1.0] + indicates fractional mass lumping criterion for CENTRM. Value of 0 + indicates no lumping applied. For example, **ALUMP**\ =0.3 means + that materials are combined into one or more lumps such that their + masses are within +/‑30% of the effective lump mass, while + preserving the slowing-down power. This approximation reduces + execution time. Default value is 0.2. + +40. **PMC_OMIT** PMC NUCLIDES SKIPPED. PMC normally processes + problem-dependent (e.g., self-shielded) MG cross sections for all + materials. If **PMC_OMIT**\ =1, processing is only performed for + materials contained in fuel mixtures. Default value is 0 (all + materials processed). + +41. **PXSMEM** CENTRM PW DATA STORAGE. Option to store PW data in memory + or in external file during centrm execution. If **PXSMEM**\ =1, PW + cross section data are stored by group in external scratch file + during CENTRM calculation; if **PXSMEM**\ =0 (default value), all PW + cross sections are kept in memory. + +42. **MOCMESH** CENTRM MOC MESH OPTION. Pre-defined space mesh intervals + for CENTRM MoC calculation: 0=>coarse mesh (1 interval per zone); + 1=>regular mesh (4 intervals in fuel, 2 in moderator, 1 in others); + 2=> fine mesh (8 in fuel, 4 in moderator, 1 in others). Default=0. + +43. **MOCRAY** CENTRM MOC RAY SPACING. Distance between characteristic + rays in CENTRM MoC calculation. Default=0.02. + +44. **MOCPOL** CENTRM NUMBER OF MOC POLAR ANGLES. Allowable values are + 2, 3, 4. Default=3 (only used for **NPXS**\ =6). + +45. **MOCAZI** CENTRM NUMBER OF MOC AZIMUTHAL ANGLES. Allowable values + are 2–16. Default=8 (only used for **NPXS**\ =6). + +46. **MOCZONE_INT** CENTRM MOC MESH BY ZONE. User-defined mesh intervals + by zone; e.g., moczone_int(1)=5 defines five intervals for zone 1; + zero value means not used. This overrides the predefined meshs + described by **MOCMESH** + +47. **ISRC** CENTRM SOURCE TYPE. CENTRM can use a fission-spectrum + source (isrc=1), an input source spectrum (isrc=0), or a + combination(isrc=3) for transport. Default=1. + +48. **XNF** CENTRM SOURCE NORMALIZATION. The integrated source + (fission-spectrum and/or fixed source spectrum) is normalized to + this value. Default=1.0. + +49. **ITERP** CRAWDAD TEMPERATURE INTERPOLATION METHOD. Method to use + for CE cross section interpolation 0=>combination of square-root(T) + and finite-difference; 1=>only square-root(T); 2=> only + finite-difference. Default is 0. + +50. **END CENTRM** The word **END** is entered to terminate the optional + parameter data. A label can be associated with this **END**. The + label can be as long as 12 characters but must be preceded by a + single blank. If this **END** is entered without a label, it must + not begin in column 1. At least two blanks must follow this entry. diff --git a/XSProcAppA.rst b/XSProcAppA.rst new file mode 100644 index 0000000..adcc2e2 --- /dev/null +++ b/XSProcAppA.rst @@ -0,0 +1,903 @@ +.. _7-1a: + +XSProc: Standard Composition Examples +===================================== + +.. _7-1a-1: + +Standard composition fundamentals +--------------------------------- + +The standard composition specification data are used to define mixtures +using standardized engineering data entered in a free-form format. The +XSProc uses the standard composition specification data and information +from the Standard Composition Library to provide number densities for +each nuclide of every defined mixture according to :eq:`eq7-1a-1`. + +.. math:: + :label: eq7-1a-1 + + NO = \frac{RHO \times AVN \times C}{AWT} , + +where + + NO is the number density of the nuclide in atoms/b-cm, + + RHO is the actual density of the nuclide in g/cm\ :sup:`3`, + + AVN is Avogadro’s number, 6.02214199 × 10\ :sup:`23`, in atoms/mol, + + C is a constant, 10\ :sup:`−24` cm\ :sup:`2`/b, + + AWT is the atomic or molecular weight of the nuclide in g/mol. + + +The actual density, RHO, is defined by + +.. math:: + :label: eq7-1a-2 + + RHO = ROTH \times VF \times WGTF , + +where + + RHO is the actual density of the standard composition in + g/cm\ :sup:`3`, + + ROTH is either the specified density of the standard composition or + the theoretical density of the standard composition in + g/cm\ :sup:`3`, + + VF is a density multiplier compatible with ROTH as defined by Eq. , + + WGTF is the weight fraction of the nuclide in the standard + composition. This value is automatically obtained by the code from + the Standard Composition Library. WGTF is 1.0 for a single nuclide + standard composition. + +.. math:: + :label: eq7-1a-3 + + VF = DFRAC \times VFRAC , + +where + + VF is the density multiplier, + + DFRAC is the density fraction, + + VFRAC is the volume fraction. + +To illustrate the interaction between ROTH and VF, consider an Inconel +having a density of 8.5 g/cm\ :sup:`3`. It is 7.0% by weight iron, 15.5% +chromium, and 77.5% nickel. The Inconel occupies a volume of +4 cm\ :sup:`3`. + +**Method 1**: + + +To describe the iron, enter 8.5 for ROTH and 0.07 for VF. + +To describe the chromium, enter 8.5 for ROTH and 0.155 for VF. + +To describe the nickel, enter 8.5 for ROTH and 0.775 for VF. + +**Method 2**: + + +Do not enter the density, and by default the theoretical density of each +component will be used for ROTH. DFRAC will be the ratio of the +specified density to the theoretical density. The specified density of +each component is the density of the Inconel × the weight fraction of +that component. + +Thus, the density of the iron is 8.5 × 0.07 = 0.595 g/cm\ :sup:`3` + + chromium is 8.5 × 0.155 = 1.318 g/cm\ :sup:`3` + + nickel is 8.5 × 0.775 = 6.588 g/cm\ :sup:`3`. + +To calculate DFRAC, the theoretical density of each material must be +obtained from the table *Elements and special nuclide symbols* in the +STDCMP chapter. These values are + +7.86 g/cm\ :sup:`3` for iron + +8.90 g/cm\ :sup:`3` for nickel + +7.20 g/cm\ :sup:`3` for chromium + +The DFRAC entered for the iron is 0.595/7.86 = 0.0757 + + for the nickel is 1.318/8.90 = 0.1481 + + for the chromium is 6.588/7.20 = 0.9163. + +Since there are no volumetric corrections, VFRAC is 1.0 and the values of DFRAC are entered for VF. + +**Method 3**: + + +Assume the Inconel, which occupies 4 cm\ :sup:`3`, is to be spread over +a volume of 5 cm\ :sup:`3`. Then the volume fraction, VFRAC, is +4 cm\ :sup:`3`/5 cm\ :sup:`3` = 0.8 and can be combined with the density +fraction, DFRAC, to obtain the density multiplier, VF. + +To describe the iron, enter 8.5 for ROTH and 0.07 × 0.8 = 0.056 for VF + + or chromium, enter 8.5 for ROTH and 0.155 × 0.8 = 0.124 for VF + + for nickel, enter 8.5 for ROTH and 0.775 × 0.8 = 0.620 for VF. + + +Alternatively, the volume fraction can be applied to the density before +it is entered. Then the ROTH can be entered as 8.5 g/cm\ :sup:`3` × 0.8 += 6.8 g/cm\ :sup:`3`, and DFRAC is entered for the density multiplier, +VF. + +To describe the iron, enter 6.8 for ROTH and 0.07 for VF + + for chromium, enter 6.8 for ROTH and 0.155 for VF + + for nickel, enter 6.8 for ROTH and 0.775 for VF. + +**Method 4**: + + +Assume the Inconel, which occupies 4 cm\ :sup:`3`, is to be spread over +a volume of 5 cm\ :sup:`3`. Then the volume fraction, VFRAC, is +4 cm\ :sup:`3`/5 cm\ :sup:`3` = 0.8. Do not enter the density, and by +default the theoretical density of each component will be used for ROTH. + +VF is then entered as the product of VFRAC and DFRAC according to Eq. . +The specified density of each component is the density of the Inconel × +the weight fraction of that component. + +Thus, the density of the iron is 8.5 × 0.07 = 0.595 g/cm\ :sup:`3` + + chromium is 8.5 × 0.155 = 1.318 g/cm\ :sup:`3` + + nickel is 8.5 × 0.775 = 6.588 g/cm\ :sup:`3`. + +To calculate DFRAC, the theoretical density of each material must be obtained from :numref:`tab7-2-3`. These values are + + 7.86 g/cm\ :sup:`3` for iron + 8.90 g/cm\ :sup:`3` for nickel + 7.20 g/cm\ :sup:`3` for chromium. + +Then DFRAC for the iron is 0.595/7.86 = 0.0756 + + for nickel is 1.318/8.90 = 0.1481 + + for chromium is 6.588/7.20 = 0.9150. + + +Then VF is DFRAC × VFRAC + +VF for the iron is 0.0757 × 0.8 = 0.0606 + + for nickel is 0.1481 × 0.8 = 0.1185 + + for chromium is 0.9150 × .8 = 0.7320. + + +.. _7-1a-2: + +Basic standard composition specifications +----------------------------------------- + +EXAMPLE 1. Material name is given. Create a mixture 3 that is Plexiglas. + + Since no other information is given, the information on the Standard + Composition Library can be assumed to be adequate. Therefore, the + only data to be entered are the standard composition name and the + mixture number + +.. highlight:: scale + +:: + + PLEXIGLAS 3 END + +EXAMPLE 2. Material name and density (g/cm\ :sup:`3`) are given. + + Create a mixture 3 that is Plexiglas at a density of + 1.15 g/cm\ :sup:`3`. Since no other data are specified, the defaults + from the Standard Composition Library will be used. Therefore, the + only data to be entered are the standard composition name, the + mixture number, and the density. + +:: + + PLEXIGLAS 3 DEN=1.15 END + +EXAMPLE 3. Material name and number density (atoms/b-cm) are given. Create a mixture 2 that is aluminum having a number density of 0.060244. + + :: + + AL 2 0 .060244 END + +EXAMPLE 4. Material name, density (g/cm\ :sup:`3`) and isotopic abundance are given. + + Create a mixture 1 that is uranium metal at 18.76 g/cm\ :sup:`3` whose + isotopic composition is 93.2 wt % :sup:`235`\ U, 5.6 wt % :sup:`238`\ U, + and 1.0 wt % :sup:`234`\ U, and 0.2 wt % :sup:`236`\ U. This example + uses the DEN= keyword to enter the density and define the standard + composition. Example 5 demonstrates another method of defining the + standard composition. + +:: + + URANIUM 1 DEN=18.76 1 300 92235 93.2 92238 5.6 92234 1.0 92236 0.2 END + +EXAMPLE 5. Material name, density (g/cm\ :sup:`3`) and isotopic abundance are given. + + Create a mixture 7 defining B\ :sub:`4`\ C with a density of + 2.45 g/cm\ :sup:`3`. The boron is 40 wt % :sup:`10`\ B and 60 wt % + :sup:`11`\ B. This example utilizes the **DEN**\ = keyword. Example 6 + illustrates an alternative description. + +:: + + B4C 7 DEN=2.45 1.0 300 5010 40.0 5011 60.0 END + +EXAMPLE 6. Material name, density (g/cm\ :sup:`3`) and isotopic abundance are given. + + Create a mixture 7 defining B\ :sub:`4`\ C with a density of + 2.45 g/cm\ :sup:`3`. The boron is 40 wt % :sup:`10`\ B and 60 wt % + :sup:`11`\ B. This example incorporates the known density into the + density multiplier, *vf*, rather than using the **DEN**\ = keyword. + The default density for B\ :sub:`4`\ C given in the COMPOUNDS table + in the SCL section 7.2 is equal to 2.52 g/cm\ :sup:`3`. + +:: + + B4C 7 0.9722 300 5010 40.0 5011 60.0 END + +.. note:: In the above examples, the actual density is input for + materials containing enriched multi-isotope nuclides (uranium in + Examples 4 and 5 and boron in Examples 6 and 7). The default density + should never be used for enriched materials, especially low atomic mass + neutron absorbers such as boron and lithium. The default density is a + fixed value for nominal conditions and naturally occurring distributions + of isotopes. Use of the default density for enriched materials will + likely result in incorrect number densities + +.. _7-1a-3: + +User-defined (arbitrary) chemical compound specifications +--------------------------------------------------------- + +The user-defined compound option allows the user to specify materials +that are not found in the Standard Composition Library and can be +specified by the number of atoms of each element or isotope that are +contained in the molecule. To define a user-defined compound, the first +four characters of the standard composition component name must be +**ATOM**. The remaining characters of the standard composition component +name are chosen by the user. The maximum length of the standard +composition name is 16 characters. All the information that would +normally be found in the Standard Composition Library must be entered in +the user-defined compound specification. :ref:`7-1-3-3` contains data +input details for arbitrary compounds. + +EXAMPLE 1. Density and chemical equation are given. + + Create a mixture 3 that is a hydraulic fluid, + C\ :sub:`2`\ H\ :sub:`6`\ SiO, with a density of 0.97 g/cm\ :sup:`3`. + The input data for this user-defined compound are given below: + +:: + + ATOM 3 0.97 4 6000 2 1001 6 14000 1 8000 1 END + +EXAMPLE 2. Density and chemical equation are given. Create a mixture 7, +TBP, also known as phosphoric acid tributyl ester or tributylphosphate, +(C\ :sub:`4`\ H\ :sub:`9`\ O)\ :sub:`3`\ PO, having a density of 0.973 +g/cm\ :sup:`3`. + +:: + + ATOMtbp 7 0.973 4 1001 27 6000 12 8016 4 15031 1 end + +.. _7-1a-4: + +User-defined (arbitrary) mixture/alloy specifications +----------------------------------------------------- + +The user-defined compound or alloy option allows the user to specify +materials that are not found in the Standard Composition Library and are +defined by specifying the weight percent of each element or isotope +contained in the material. To define a user-defined weight percent +mixture, the first four characters of the standard composition component +name must be *wtpt*. The remaining characters of the standard +composition component name are chosen by the user. The maximum length of +the standard composition name is 16 characters. All the information that +would normally be found in the Standard Composition Library must be +entered in the arbitrary mixture/alloy specification. :ref:`7-1-3-3` +contains data input details for user-defined compounds. + +EXAMPLE 1. Density and weight percents are given. + + Create a mixture 5 that defines a borated aluminum that is 2.5 wt % + natural boron. The density of the borated aluminum is + 2.65 g/cm\ :sup:`3`. + +:: + + SOLUTION MIX=2 RHO[UO2(NO3)2]=415 92235 92.6 92238 5.9 92234 1 92236 + 0.5 MASSFRAC[HNO3]=6.339-6 TEMPERATURE=293 END SOLUTION + +EXAMPLE 2. Density, weight percents, and isotopic abundance are given. + + Create a mixture 5 that defines a borated aluminum that is 2.5 wt % + boron. The boron is 90 wt % :sup:`10`\ B and 10 wt % :sup:`11`\ B. + The density of the borated aluminum is 2.65 g/cm\ :sup:`3`. The + minimum generic input specification for this arbitrary material is + +:: + + WTPTBAL 5 2.65 2 5000 2.5 13027 97.5 1 293 5010 90. 5011 10. END + +.. _7-1a-5: + +Fissile solution specifications +------------------------------- + +Solutions of fissile materials are available in the XSProc. A list of +the available solution salts and acids is given in the table *Available +fissile solution components* in :ref:`7-2-3`. When the XSProc processes +a solution, it breaks the solution into its component parts (basic +standard composition specifications) and uses the solution density to +calculate the volume fractions. + +EXAMPLE 1. Fuel density, excess acid and isotopic abundance are given. + + Create a mixture 2 that is a highly enriched uranyl nitrate solution + with 415 g/L and 0.39 mg of excess nitrate per gram of solution. The + uranium isotopic content is 92.6 wt % :sup:`235`\ U, 5.9 wt % + :sup:`238`\ U, 1.0 wt % :sup:`234`\ U, and 0.5 wt % :sup:`236`\ U. + The temperature is 293 Kelvin. + +:: + + SOLUTION MIX=2 RHO[UO2(NO3)2]=415 92235 92.6 92238 5.9 92234 1 92236 + 0.5 MASSFRAC[HNO3]=6.339-6 TEMPERATURE=293 END SOLUTION + +where + + The molecular weight of NO\ :sub:`3` is 62.0049 g/mole, of H is + 1.0078 g/mole, so the grams of excess H per gram of solution is + 1.0078 / 62.0049 × (0.39 mg/g) × (1 g/1000 mg) = 6.339 × + 10\ :sup:`-6`. + +.. _7-1a-6: + +Combinations of standard composition materials to define a mixture +------------------------------------------------------------------ + +Frequently more than one standard composition is required to define a +mixture. This section contains such examples. + +EXAMPLE 1. Boral from B\ :sub:`4`\ C and Aluminum. + + Create a mixture 6 that is Boral, 15 wt % B\ :sub:`4`\ C and 85 wt % + Al, having a density of 2.64 g/cm\ :sup:`3`. Natural boron is used in + the B\ :sub:`4`\ C. Note that Example 2 demonstrates the use of the + keyword **DEN**\ = to enter the density of the mixture and avoid + having to look up the theoretical density from the table *Isotopes in standard + composition library,* in the section 7.2.2, and calculate the density + multiplier (VF) + +:: + + B4C 6 0.1571 END +  AL 6 0.8305 END + +EXAMPLE 2. Boral from B\ :sub:`4`\ C and Aluminum. + + This is the same problem as Example 1 using a different method of + specifying the input data. Create a mixture 6 that is Boral, 15 wt % + B\ :sub:`4`\ C and 85 wt % Al, having a density of + 2.64 g/cm\ :sup:`3`. Natural boron is used in the B\ :sub:`4`\ C. + +:: + + B4C 6 DEN=2.64 0.15 END +  AL 6 DEN=2.64 0.85 END + +EXAMPLE 3. Boral from Boron, Carbon, and Aluminum. + + If neither Boral nor B\ :sub:`4`\ C were available in the Standard + Composition Library, Boral could be described as follows: + + Create a mixture 2 that is Boral composed of 35 wt % B\ :sub:`4`\ C and + 65 wt % aluminum with an overall density of 2.64 g/cm\ :sup:`3`. The + boron is natural boron. + + *vf* is the density multiplier. (The density multiplier is the ratio + of actual to theoretical density.) From the Standard Composition + Library chapter, table *Isotopes in standard composition library*, + the theoretical density of aluminum is 2.702 g/cm\ :sup:`3`; boron is + 2.37 g/cm\ :sup:`3`; and carbon is 2.1 g/cm\ :sup:`3`. The density + multiplier, *vf*, for Al is (0.65)(2.64)/2.702 = 0.63509. The + isotopic abundances in natural boron are known to have some + variability. Here it is assumed that natural boron is 18.4309 wt % + :sup:`10`\ B at 10.0129 amu and 81.5691 wt % :sup:`11`\ B at + 11.0096 amu. C is 12.000 amu. + + Convert the weight percents to atom percents for the natural boron where + *w* denotes weight fraction, *a* denotes atom fraction, and *M* denotes + atomic mass: + +.. math:: + + w_{B10} = 0.184309 \equiv \frac{a_{B10}M_{B10}}{a_{B10}M_{B10} + a_{B11}M_{B11}} = \frac{a_{B10}(10.0129)}{a_{B10}(10.0129) + (1-a_{B10}))(11.0093)} + +Solving for :math:`a_{B10}` gives: + +.. math:: + + [{{\text{a}}_{\text{B10}}}\text{=0.184309}\ \ \text{=}\ \ \frac{\text{(0.184309)}\ \text{(11.0093)}}{\ \text{(0.184309)}\ \text{(11.0093)-(0.184309)}\ \text{(10.0129)+(10.0129)}}\quad \text{=}\ \ \text{19.900} + +Therefore the atom percent of :sup:`11`\ B is, *a\ B*\ :sub:`11` = 80.1 +a%. + +Similarly, the mass of the B\ :sub:`4`\ C molecule is + + [(0.199 × 4 × 10.0129) + (0.801 × 4 × 11.0093) + (12.000)] = + 55.24407 amu. + + +The mass of the boron is (55.24407 − 12.000) = 43.24407 amu. + +The *vf* of boron would be :math:`\left( \frac{43.24407}{55.24407} \right)\left( \frac{(0.35)(2.64)}{2.37} \right)` = 0.30519 + +The *vf* of C would be + +.. math:: + + \left( \frac{12.0000}{55.24407} \right)\left( \frac{(0.35)(2.64)}{2.1} \right) = 0.09558 + + +.. math:: + + \left(\frac{12.000}{55.25045}\right)\left[\frac{(0.35)(2.64)}{2.30}\right] = 0.08725 + +The standard composition input data for the Boral follows: + +:: + + AL 2 0.63509 END + BORON 2 0.30519 END + C 2 0.09558 END + +EXAMPLE 4. Boral from :sup:`10`\ B, :sup:`11`\ B, Carbon, and Aluminum. + + Create a mixture 2 that is Boral composed of 35 wt % B\ :sub:`4`\ C + and 65 wt % aluminum. The Boral density is 2.64 g/cm\ :sup:`3`. The + boron is natural boron. + + *vf* is the density multiplier. Use 0.63581 for AL and 0.08725 for C + as explained in Example 3 above. From the Standard Composition + Library chapter, *Isotopes in standard composition library* table, + the theoretical density of :sup:`10`\ B is 1.00 g/cm\ :sup:`3` and + :sup:`11`\ B is 1.00 g/cm\ :sup:`3`. As computed in Example 3, the + mass of the B\ :sub:`4`\ C molecule is 55.25045 amu, and the boron is + 19.764 atom % :sup:`10`\ B and 80.236 atom % :sup:`11`\ B. The mass + of :sup:`10`\ B is 10.0129 amu and the :sup:`11`\ B is 11.0096. Thus, + the *vf* of :sup:`10`\ B is + + .. math:: + + \left( \frac{(4)(0.199)(10.0129)}{55.24407} \right)\left( \frac{(0.35)(2.64)}{1.0} \right)\ \ =\ \ 0.13331\ . + + The *vf* of :sup:`11`\ B is + + .. math:: + + \left( \frac{(4)(0.801)(11.0093)}{55.24407} \right)\left( \frac{(0.35)(2.64)}{1.0} \right)\ \ =\ \ 0.58998\ . + +The standard composition input data for the Boral are given as + +:: + + AL 2 0.63509 END + B-10 2 0.13331 END + B-11 2 0.58998 END + C 2 0.09558 END + + +EXAMPLE 5. Specify all of the number densities in a mixture. + + Create a mixture 1 that is vermiculite, defined as + + hydrogen at a number density of 6.8614−4 atoms/b-cm + + oxygen at a number density of 2.0566−3 atoms/b-cm + + magnesium at a number density of 3.5780−4 atoms/b-cm + + aluminum at a number density of 1.9816−4 atoms/b-cm + + silicon at a number density of 4.4580−4 atoms/b-cm + + potassium at a number density of 1.0207−4 atoms/b-cm + + iron at a number density of 7.7416−5 atoms/b-cm + + In this example we use the 2\ :sup:`nd` syntax option described in + :ref:`7-1-3-3`, in which the 3rd entry must be 0. The standard + composition input data for the vermiculite are given below: + + :: + + H 1 0 6.8614-4 END + O 1 0 2.0566-3 END + MG 1 0 3.5780-4 END + AL 1 0 1.9816-4 END + SI 1 0 4.4580-4 END + K 1 0 1.0207-4 END + FE 1 0 7.7416-5 END + +.. _7-1a-7: + +Combinations of user-defined compound and user-defined mixture/alloy to define a mixture +---------------------------------------------------------------------------------------- + +Mixtures can usually be created using only basic standard composition +specifications. Occasionally, it is convenient to create two or more +user-defined materials for a given mixture. This procedure is +demonstrated in the following example. + +EXAMPLE 1. Specify Boral using a user-defined compound and user-defined mixture/alloy. + + Create a mixture 6 that is Boral, 15 wt % B\ :sub:`4`\ C and 85 wt % + Al, having a density of 2.64 g/cm\ :sup:`3`. Natural boron is used in + the B\ :sub:`4`\ C. Boral can be described in several ways. + For demonstration purposes, it will be described as a combination of + a user-defined compound and user-defined mixture/alloy. This is not + necessary, because both B\ :sub:`4`\ C and Al are available as + standard compositions. A method of describing the Boral without using + user-defined compounds or user-defined mixtures/alloys is given in + Examples 1 and 2 of :ref:`7-1a-6`. The minimum generic input + specifications for this user-defined compound and alloy are + + :: + + ATOM-B4C 6 2.64 2 5000 4 6012 1 0.15 END + WTPT-AL 6 2.64 1 13027 100.0 0.85 END + +.. _7-1a-8: + +Combinations of solutions to define a mixture +--------------------------------------------- + +This section demonstrates the use of more than one solution definition +to describe a single mixture. The assumptions used in processing the +cross sections are likely to be inadequate for solutions of mixed oxides +of uranium and plutonium. Therefore, this section is given purely for +demonstration purposes. + +EXAMPLE 1. Solution of uranyl nitrate and plutonium nitrate. + + Note that the assumptions used in processing the cross sections are + likely to only be adequate for CENTRM/PMC calculations of mixed-oxide + solutions. This example is given purely for demonstration purposes. + Create a mixture 1 consisting of a mixture of plutonium nitrate + solution and uranyl nitrate solution. The specific gravity of the + mixed solution is 1.4828. The solution contains 325.89 g (U + Pu)/L + soln. The acid molarity of the solution is 0.53. In this solution + 77.22 wt % of the U+Pu is uranium. The isotopic abundance of the + uranium is 0.008% :sup:`234`\ U, 0.7% :sup:`235`\ U, 0.052% + :sup:`236`\ U, and 99.24% :sup:`238`\ U. The isotopic abundance of + the plutonium is 0.028% :sup:`238`\ Pu, 91.114% :sup:`239`\ Pu, 8.34% + :sup:`240`\ Pu, 0.426% :sup:`241`\ Pu, and 0.092% :sup:`242`\ Pu. + Note that a single quote in the first column indicates a comment line + in SCALE input. + + :: + + ' Uranium density of 77.22% of 325.89 g/L + SOLUTION MIX=1 RHO[UO2(NO3)2]=251.65 92234 .008 92235 .700 92236 .052 + 92238 99.240 + ' Plutonium density if 22.78% of 325.89 g/L + RHO[PU(NO3)4]=74.24 94238 .028 94239 91.114 94240 8.34 + 94241 .426 94242 .092 + ' Acid molarity is 0.53 M + MOLAR[HNO3]=0.53 + ' Specifying the density over specifies the problem, which means the solution may + ' not be in thermodynamic equilibrium. The specification below adds about 0.3% + ' extra hydrogen to the problem + DENSITY=1.4828 + END SOLUTION + +.. _7-1a-9: + +Combinations of basic and user-defined standard compositions to define a mixture +-------------------------------------------------------------------------------- + +EXAMPLE 1. Burnable poison from B\ :sub:`4`\ C and Al\ :sub:`2`\ O\ :sub:`3`. + + Create a mixture 6 that is a burnable poison with a density of + 3.7 g/cm\ :sup:`3` and composed of Al\ :sub:`2`\ O\ :sub:`3` and + B\ :sub:`4`\ C. The material is 1.395 wt % B\ :sub:`4`\ C. The boron + is natural boron. This material can be easily specified using a + combination of user-defined material to describe the + Al\ :sub:`2`\ O\ :sub:`3` and a simple standard composition to define + the B\ :sub:`4`\ C. The minimum generic input specification for this + user-defined material and the standard composition are + + The density multiplier of the B\ :sub:`4`\ C is the density of the + material times the weight percent, divided by the theoretical density + of B\ :sub:`4`\ C [(3.7 × 0.01395)/2.52] or 0.02048; the density + multiplier of the Al\ :sub:`2`\ O\ :sub:`3` is 1.0 – 0.01395 or + 0.98605 (the theoretical density of B\ :sub:`4`\ C was obtained from + *Isotopes in standard composition library* table in the STDCMP + chapter). + + The input data for the burnable poison are given below: + + :: + + ATOM-AL2O3 6 3.70 2 13027 2 8016 3 0.98605 END + B4C 6 2.048-2 END + + The B\ :sub:`4`\ C input can be specified using the **DEN**\ = parameter + as shown below: + + :: + + ATOM-AL2O3 6 3.70 2 13027 2 8016 3 0.98605 END + B4C 6 DEN=3.7 0.01395 END + + The fraction of B\ :sub:`4`\ C in the mixture is ((3.7 × 0.01395)/2.52) + = 0.02048. The fraction of Al\ :sub:`2`\ O\ :sub:`3` in the mixture is + 1.0 – 0.02048 = 0.979518. The density of the Al\ :sub:`2`\ O\ :sub:`3` + can be calculated as shown below. + + .. image:: figs/XSProcAppA/math1.png + :align: center + :width: 500 + + Input data using the density of Al\ :sub:`2`\ O\ :sub:`3` are given + below: + + :: + + ATOM-AL2O3 6 3.72467 2 13027 2 8016 3 END + B4C 6 2.048-2 END + +EXAMPLE 2. Borated water from H\ :sub:`3`\ BO\ :sub:`3` and water. + + Create a mixture 2 that is borated water at 4350 parts per million + (ppm) by weight, resulting from the addition of boric acid, + H\ :sub:`3`\ BO\ :sub:`3` to water. The density of the borated water + is 1.0078 g/cm\ :sup:`3` (see “Specific Gravity of Boric Acid Solutions,” Handbook of Chemistry, 1162, Compiled and Edited by Norbert A. Lange, Ph.D, 1956.). The solution temperature + is 15ºC and the boron is natural boron. + +An easy way to describe this mixture is to use a combination of a +user-defined compound to describe the boric acid, and a basic +composition to describe the water. + +STEP 1. INPUT DATA TO DESCRIBE THE USER-DEFINED COMPOUNDThe generic input data +for the boric acid are given below. The actual input data are derived in steps 2 through 5. + +:: + + ATOMH3BO3 2 0.025066 3 5000 1 1001 3 8016 3 1.0 288.15 END + +STEP 2. AUXILIARY CALCULATIONS FOR THE USER-DEFINED COMPOUND INPUT DATA + +In calculating the molecular weights, use the atomic weights from SCALE, +which are available in the table *Isotopes in standard composition +library* in :ref:`7-2-2` of the SCALE manual. The atomic weights used +in SCALE may differ from some periodic tables. The SCALE atomic weights +used in this problem are listed below: + + H (1001) 1.0078 + + O (8016) 15.9949 + + :sup:`10`\ B 10.0129 + + :sup:`11`\ B 11.0093 + +The natural boron abundance, in weight percent, is defined to be: + + :sup:`10`\ B 18.4309 + + :sup:`11`\ B 81.5691 + +The molecular weight of natural boron is given by + + DEN nat B/AWT nat B = DEN :sup:`10`\ B/AWT :sup:`10`\ B + DEN :sup:`11`\ B/AWT :sup:`11`\ B + + DEN :sup:`10`\ B = WTF :sup:`10`\ B × DEN nat B + + DEN :sup:`11`\ B = WTF :sup:`11`\ B × DEN nat B + +where: + + DEN is density in g/cm\ :sup:`3`, + + AWT is the atomic weight in g/mol, + + WTF is the weight fraction of the isotope. + +Substituting, + + DEN nat B/AWT nat B = DEN nat B × ((WTF :sup:`10`\ B/AWT + :sup:`10`\ B) + (WTF :sup:`11`\ B/AWT :sup:`11`\ B)) + +Solving for AWT nat B yields: + + AWT nat B = 1/((WTF :sup:`10`\ B/AWT :sup:`10`\ B) + (WTF + :sup:`11`\ B/AWT :sup:`11`\ B)) + +The atomic weight of natural boron is thus + + 1.0/((0.184309 g :sup:`10`\ B/g nat B/10.0129 g :sup:`10`\ B/mol + :sup:`10`\ B) + + (0.815691 g :sup:`11`\ B/g nat B/11.0093 g /mol :sup:`11`\ B)) = + 10.81103 g nat B/mol nat B + +The molecular weight of the boric acid, H\ :sub:`3`\ BO\ :sub:`3` is +given by: + + (3 × 1.0078) + 10.81103 + (3 × 15.9949) = 61.8191 + +Calculate the grams of boric acid in a gram of solution: + + Boric acid, H\ :sub:`3`\ BO\ :sub:`3` is 61.8222 g/mol + + Natural boron is 10.81261 g/mol + + (4350 × 10\ :sup:`–6` g B/g soln) × (1 mol/10.81261 g B) × (61.8191 g + boric acid/mol) = + + 0.024874 g boric acid/g soln (2.4874 wt %) + +Interpolating from the referenced page from Lange's Handbook of Chemistry, the specific gravity of the boric acid +solution at 2.4872 weight percent is 1.0087. This value is based on +water at 15ºC. The density of pure air free water at 15°C is +0.99913 g/cm\ :sup:`3`. Therefore, the density of the boric acid +solution is 1.0087 × 0.99913 g/cm\ :sup:`3` = 1.0078 g +soln/cm\ :sup:`3`. + +Calculate ROTH, the theoretical density of the boric acid. + + 1.0078 g soln/cm\ :sup:`3` × 0.024874 g boric acid/g soln = + 0.025068 g boric acid/cm\ :sup:`3` + +STEP 3. DESCRIBE THE BASIC STANDARD COMPOSITION INPUT DATA + +:: + + H2O 2 0.984507 288.15 END + +where the volume fraction =0.984506 (see step 4 auxiliary calculations below) + +STEP 4. AUXILIARY CALCULATIONS FOR THE BASIC STANDARD COMPOSITION INPUT +DATA + +Calculate the volume fraction of the water in the solution, assuming +0.9982 is the theoretical density of water from :numref:`tab7-2-4`. Each gram +of solution contains 0.024872 g of boric acid, so there is 0.975128 g of +water in each gram of solution. The volume fraction of water is then +given by: + + (1.0078 g soln/cm\ :sup:`3` × 0.975128 g water/g soln)/0.9982 g + water/cm\ :sup:`3` = 0.984506 + +STEP 5. CREATE THE MIXTURE FOR BORATED WATER + +:: + + ATOMH3BO3 2 0.025068 3 5000 1 1001 3 8016 3 1.0 288.15 END + H2O 2 0.984506 288.15 END + +.. _7-1a-10: + +Combinations of basic and solution standard compositions to define a mixture +---------------------------------------------------------------------------- + +The solution specification is the easiest way of specifying the +solutions listed in the *Available fissile solution components* table in +:ref:`7-2-3`. A combination of solution and basic standard compositions +can be used to describe a mixture that contains more than just a +solution as demonstrated in the following example. + +EXAMPLE 1. Uranyl nitrate solution containing gadolinium. + + Create a 4.306% enriched uranyl nitrate solution containing 0.184 g + gadolinium per liter. The uranium in the nitrate is 95.65% + :sup:`238`\ U, 0.022% :sup:`236`\ U, 4.306% :sup:`235`\ U, and 0.022% + :sup:`234`\ U. The uranium concentration is 195.8 g U/L and the + specific gravity of the uranyl nitrate is 1.254. There is no excess + acid in the solution. The presence of the gadolinium is assumed to + produce no significant change in the solution density. The solution + is defined to be mixture 3. + +:: + + SOLUTION MIX=3 + RHO[UO2(NO3)2]=195.8 92238 95.65 92236 0.022 92235 4.306 92234 0.022 + VOL_FRAC=0.99985 + DENSITY=1.254 + END SOLUTION + GD 3 0.000184 293 END + +.. _7-1a-11: + +Combinations of user-defined compound and solution to define a mixture +---------------------------------------------------------------------- + +The solution specification is the easiest way of specifying the +solutions listed in the *Available fissile solution components* table in +:ref:`7-2-3` of the SCALE manual. A solution specification and +user-defined compound specification can be used to describe a mixture +that contains more than just a solution as demonstrated in the following +example. + +EXAMPLE 1. Uranyl nitrate solution with gadolinium nitrate. + + Create a 4.306% enriched uranyl nitrate solution containing + gadolinium in the form of Gd(NO\ :sub:`3`)\ :sub:`3`. The uranium in + the nitrate is 95.65% :sup:`238`\ U, 0.022% :sup:`236`\ U, 4.306% + :sup:`235`\ U, and 0.022% :sup:`234`\ U. The uranium concentration is + 195.8 g U/L and the density of the uranyl nitrate is 1.254. There is + no excess acid in the solution. The concentration of the gadolinium + is 0.184 g/L. The volume fraction of the mixture that is uranyl + nitrate (0.99985 = 1.254/ (1.254 + 0.000184)). The solution is + defined to be mixture 3. + +:: + + SOLUTION MIX=3 + RHO[UO2(NO3)2]=195.8 92238 95.65 92236 0.022 92235 4.306 92234 0.022 + VOL_FRAC=0.99985 + DENSITY=1.254 + END SOLUTION + +The density of the gadolinium is given as 0.184 g/L. To describe the +user-defined compound, the density of the Gd(NO\ :sub:`3`)\ :sub:`3` is +needed. The atomic weights from the Standard Composition Library are: + + Gd 157.25 + + N 14.0067 + + O 15.999 + +Therefore, the density of the Gd(NO\ :sub:`3`)\ :sub:`3` = 0.000184 g +Gd/cm\ :sup:`3` × (157.25 + 3(14.0067 + 3(15.999))/157.25) = +0.0004017 g/cm\ :sup:`3`. + +The input data for this user-defined compound are given below: + +:: + + ATOMGD(NO3)3 3 .0004017 3 64000 1 7014 3 8016 9 1.0 300 END + +The complete input data for the mixture of uranyl nitrate and gadolinium nitrate are given as: + +:: + + SOLUTION MIX=3 + RHO[UO2(NO3)2]=195.8 92238 95.65 92236 0.022 92235 4.306 92234 0.022 + VOL_FRAC=0.99985 + DENSITY=1.254 + END SOLUTION + ATOMGD(NO3)3 3 .0004017 3 64000 1 7014 3 8016 9 1.0 300 END + +.. note:: Since the default temperature (300 K) is to be used, it can be + omitted from the user-defined compound standard composition. The + temperature must be entered if the standard composition contains a + multiple-isotope nuclide whose isotopic abundance is to be specified. + + + + + + +.. diff --git a/XSProcAppB.rst b/XSProcAppB.rst new file mode 100644 index 0000000..37071ff --- /dev/null +++ b/XSProcAppB.rst @@ -0,0 +1,725 @@ +.. _7-1b: + +XSProc Standard Composition Examples +==================================== + +.. _7-1b-1: + +Infinite homogeneous medium unit cell data +------------------------------------------ + +EXAMPLE 1. A single mixture 1. + + Consider a single cylindrical configuration of mixture 1, composed of + 10% enriched UO\ :sub:`2` having a radius of 35 cm and a height of + 20 cm. This fuel region is sufficient large to model as an infinite + medium. Mixture 100 may be used in subsequent multigroup neutron + transport calculations. + +.. highlight:: scale + +:: + + INFHOMMEDIUM 1 CELLMIX=10 END + +XSDRNPM will calculate the eigenvalue of an infinite mass of 10% +enriched UO\ :sub:`2`. + +.. _7-1b-2: + +LATTICECELL unit cell data +-------------------------- + +Examples of “regular” **LATTICECELL** unit cells are given in +Examples 1–5, and examples of “annular” **LATTICECELL** unit cells are +given in Examples 6–10 below. + +EXAMPLE 1. SQUAREPITCH (infinitely long cylindrical pins in a square-pitched array). + + Consider a large array of UO\ :sub:`2` fuel pins having a fuel O.D. + of 0.79 cm, a 0.015-cm gap, and a 0.06-cm-thick aluminum clad. The + array is a square-pitched array with a center-to-center spacing of + 1.60 cm and is completely flooded with water. In the standard + composition data, UO\ :sub:`2` is defined to be mixture 1, the + aluminum clad is defined to be mixture 2, and the water moderator is + defined to be mixture 3. + +:: + + LATTICECELL SQUAREPITCH PITCH=1.60 3 FUELD=0.79 1 CLADD=0.94 2 GAPD=0.82 0 END + +EXAMPLE 2. TRIANGPITCH (infinitely long cylinders in a triangular-pitched array). + + Consider an array of UO\ :sub:`2` pins with a diameter of 0.635 m. + The outside diameter of the clad is 0.78 cm. There is no gap between + the fuel and clad. The array is a triangular-pitched array with a + center-to-center spacing of 5.0 cm and is flooded with water. In the + standard composition data, the UO\ :sub:`2` is defined to be + mixture 1, the aluminum is defined to be mixture 2, and the water + moderator is defined to be mixture 3. + +:: + + LATTICECELL TRIANGPITCH PITCH=5.0 3 FUELD=.635 1 CLAD=.78 2 END + +EXAMPLE 3. SPHSQUAREP (spheres in a square-pitched array). + + Consider a large array of U\ :sub:`3`\ O\ :sub:`8` spheres having a + fuel O.D. of 18.6 cm, with an aluminum clad that is 0.18 cm thick. + The array is a triangular-pitched array with a center-to-center + spacing of 19.0 cm and is unmoderated. In the standard composition + data, the aluminum is defined to be mixture 1 and the + U\ :sub:`3`\ O\ :sub:`8` is defined to be mixture 2. There is no + moderator material, so 0 will be used to represent a void. Also, have + XSDRNPM make a cell weighted material 20 from this unit cell. + +:: + + LATTICECELL SPHSQUAREP PITCH=19.0 0 FUELD=18.6 2 CLADD=18.96 1 CELLMIX=20 END + +EXAMPLE 4. SPHTRIANGP (spheres in a triangular-pitched array). + + Consider a large array of U\ :sub:`3`\ O\ :sub:`8` spheres having a + fuel O.D. of 18.6 cm, with an aluminum clad that is 0.18 cm thick. + The array is a triangular-pitched array with a center-to-center + spacing of 19.0 cm and is flooded with water. In the standard + composition data, the aluminum is defined to be mixture 1, the + U\ :sub:`3`\ O\ :sub:`8` is defined to be mixture 2, and the water + moderator is defined to be mixture 3. + +:: + + LATTICECELL SPHTRIANGP PITCH=19.0 3 FUELD=18.6 2 CLADD=18.96 1 END + +EXAMPLE 5. SYMMSLABCELL (slabs repeated in a symmetric fashion). + + Consider a system of alternating slabs of U\ :sub:`3`\ O\ :sub:`8` + and low-density water. Each U\ :sub:`3`\ O\ :sub:`8` region is + 1.27 cm thick, and each water region is 15.0 cm thick. In the + standard composition data, the U\ :sub:`3`\ O\ :sub:`8` is defined to + be mixture 1, and the low-density water is defined to be mixture 2. + +:: + + LATTICECELL SYMMSLABCELL PITCH=16.27 2 FUELD=1.27 1 END + +EXAMPLE 5a. SYMMSLABCELL (slabs repeated in a symmetric fashion). + + Consider a system of alternating slabs of U\ :sub:`3`\ O\ :sub:`8` + and low-density water. Each U\ :sub:`3`\ O\ :sub:`8` region is + 1.27 cm thick, and each water region is 15.0 cm thick. The + U\ :sub:`3`\ O\ :sub:`8` regions have a 0.01-cm gap and 0.24-cm-thick + aluminum clad on each face. In the standard composition data, the + U\ :sub:`3`\ O\ :sub:`8` is defined to be mixture 1, the aluminum is + defined to be mixture 2, and the low-density water is defined to be + mixture 3. Also, have XSDRNPM make a cell-weighted material 100 from + this unit cell. + +:: + + LATTICECELL SYMMSLABCELL PITCH=16.77 3 FUELD=1.27 1 + CLADD=1.77 2 GAPD=1.29 0 CELLMIX=100 END + + +EXAMPLE 6. ASQUAREPITCH (infinitely long annular cylindrical rods in a square-pitched array). + + Consider an array of uranium metal pipes having an inside diameter of + 5.0 cm and an outer diameter of 6.75 cm. A gap of 0.025 cm and a clad + of 0.25 cm exist on both the inner and outer surfaces of the fuel. + The fuel rods are arranged in a square-pitched array. + The center-to-center spacing is 8.0 cm. The array is completely + flooded with water. In the standard composition data, the uranium + metal is defined to be mixture 1, the outer clad is mixture 2, the + inner clad is mixture 7, the inner moderator is Plexiglas and is + mixture 3, the gap is a void, and the external moderator is water, + defined to be mixture 4. + +:: + + LATTICECELL ASQUAREPITCH PITCH=8.0 4 FUELD=6.75 1 GAPD=6.8 0 + CLADD=7.3 2 IMODD=4.45 3 ICLADD=4.95 7 IGAPD=5.0 0 END + +EXAMPLE 6a. ASQUAREPITCH (infinitely long annular cylindrical rods in a square-pitched array). + + Consider an array of uranium metal pipes having an inside diameter of + 5.0 cm and an outer diameter of 6.75 cm arranged in a square-pitched + array. The center-to-center spacing is 8.0 cm. The array is + completely flooded with water. In the standard composition data, the + uranium metal is defined to be mixture 1, the water moderator is + defined to be mixture 2, and the inside water moderator is defined as + mixture 3. + +:: + + LATTICECELL ASQUAREPITCH PITCH=8.0 2 FUELD=6.75 1 IMODD=5.0 3 END + +.. note:: This problem defines two water mixtures. If mixture 2 were + entered twice, i.e., for both the inner and outer moderator, an error + message would be printed and the calculation terminated. + +EXAMPLE 7. ATRIANGPITCH (infinitely long annular cylindrical rods in a triangular-pitched array). + + Consider an array of uranium metal pipes having an inside diameter of + 8.0 cm and a wall thickness of 0.75 cm arranged in a square-pitched + array. The center-to-center spacing is 9.75 cm. The array is + completely flooded with water. A Plexiglas rod fills the center of + the uranium pipe. In the standard compositions data, the uranium + metal is defined to be mixture 1, the Plexiglas is defined to be + mixture 2, and the external water moderator is mixture 3. + +:: + + LATTICECELL ATRIANGPITCH PITCH=9.75 3 FUELD=9.5 1 IMODD=8.0 2 END + +EXAMPLE 8. ASPHSQUAREP (spherical annuli in a square-pitched array). + + Consider a large array of hollow U\ :sub:`3`\ O\ :sub:`8` spheres + having a fuel I.D. of 8.0 cm and O.D. of 18.6 cm. The centers of the + spheres are empty. The external moderator is water. The spheres are + stacked in a square-pitched array with a center-to-center spacing of + 19.0 cm. In the standard composition data, the + U\ :sub:`3`\ O\ :sub:`8` is defined to be mixture 1, and the water is + defined to be mixture 2. The centers of the spheres are defined to be + void, mixture 0. + +:: + + LATTICECELL ASPHSQUAREP HPITCH=9.5 2 FUELR=9.3 1 IMODR=4.0 0 END + +EXAMPLE 9. ASPHTRIANGP (spheres in a triangular-pitched array). + + Consider a large array of hollow U\ :sub:`3`\ O\ :sub:`8` spheres + having a fuel I.D. of 8.0 cm and a fuel O.D. of 18.6 cm. A + 0.18-cm-thick aluminum clad exists outside the fuel. The interior of + each sphere is void. The array is a triangular-pitched array with a + center-to-center spacing of 19.0 cm and is flooded with water. In the + standard composition data, the aluminum is defined to be mixture 1, + the U\ :sub:`3`\ O\ :sub:`8` is defined to be mixture 2, and the + water moderator is defined to be mixture 3. The void in the center of + each sphere is entered as mixture 0. + +:: + + LATTICECELL ASPHTRIANGP HPITCH=9.5 3 FUELR=9.3 2 IMODR=4.0 0 CLADR=9.48 1 END + +EXAMPLE 10. ASYMSLABCELL (repeated slabs having different moderator conditions on the left and right boundaries). + + Consider an array of U\ :sub:`3`\ O\ :sub:`8` slabs with an inner + moderator region composed of full-density water with a half thickness + of 8.0 cm, and a low-density water outer moderator with a 16 cm half + thickness of 16 cm half thickness. Each U\ :sub:`3`\ O\ :sub:`8` slab + is 10.54 cm thick. In the standard composition data, the + U\ :sub:`3`\ O\ :sub:`8` is defined to be mixture 1, the full density + water is defined to be mixture 2, and the low-density water is + mixture 3. Also, have XSDRNPM create a cell weighted mixture 100 from + this unit cell. + +:: + + LATTICECELL ASYMSLABCELL CELLMIX=100 IMODR=8.0 2 FUELR=18.54 1 HPITCH=34.54 3 END + +EXAMPLE 10a. ASYMSLABCELL (repeated slabs having different moderator conditions on the left and right boundaries). + + Consider an array of U\ :sub:`3`\ O\ :sub:`8` fuel plates with an + inner moderator region of full-density water with a half-thickness of + 8.0 cm, and with a 16 cm thick low-density outer moderator. Each fuel + plate includes a 10.54 cm thick U\ :sub:`3`\ O\ :sub:`8` slab with a + 0.01 cm gap and 0.24-cm-thick aluminum clad on each face. In the + standard composition data, the U\ :sub:`3`\ O\ :sub:`8` is defined to + be mixture 1, the full density water is defined to be mixture 2, and + the low-density water is mixture 3, the inner aluminum is mixture 4, + the outer aluminum clad is mixture 5, and both gaps are voids. + +:: + + + LATTICECELL ASYMSLABCELL IMODR=8.0 2 ICLADR=8.24 5 IGAPR=8.25 0 FUELR=18.79 1 + GAPR 18.80 0 CLADR 19.04 4 HPITCH=27.04 3 END + +.. _7-1b-3: + +MULTIREGION unit cell data +-------------------------- + +Examples of **MULTIREGION** unit cells follow: + +EXAMPLE 1. SLAB. + + Consider a 5-cm-thick slab of fuel (mixture 1) with 0.5 cm of + aluminum (mixture 3) and 15 cm of water (mixture 2) on each face. The + unit cell data for this problem could be entered as follows: + +:: + + MULTIREGION SLAB LEFT_BDY=REFLECTED RIGHT_BDY=VACUUM ORIGIN=0 END + 1 2.5 3 3.0 2 18.0 END ZONE + +EXAMPLE 2. CYLINDRICAL. + + Consider a large array of fuel pins. Each pin is UO\ :sub:`2` + (mixture 1) with a radius of 0.465 cm, a 0.009-cm gap (mixture 0), + and a Zircaloy clad (mixture 9) 0.062 cm thick, centered in a water + (mixture 8) region surrounded by a flooded support structure + represented by homogenized water and Zircaloy (mixture 10). The outer + radius of the water-Zircaloy region is 0.844 cm and it is 0.037 cm + thick. This problem cannot be described as a **LATTICECELL** problem + because the **LATTICECELL** configuration is limited to + fuel-gap-clad-cell boundary and this problem is + fuel-gap-clad-moderator-outer region. When **MULTIREGION** is used, + lattice effects are accounted for by specifying a **WHITE**, + **PERIODIC**, or **REFLECTED** right boundary condition, as long as + the CENTRM/PMC self-shielding method is used. **MULTIREGION** cells + should not be used for arrays if BONAMI-only method is specified + + +:: + + MULTIREGION CYLINDRICAL RIGHT_BDY=WHITE END + 1 0.465 0 0.474 9 0.536 8 0.807 10 0.844 END ZONE + +EXAMPLE 3. SPHERICAL. + + Describe a bare sphere of uranium metal 8.72 cm in radius. The + uranium metal is defined to be mixture 1. Also, have XSDRNPM create a + cell weighted mixture 100 and calculate and eigenvalue. The unit cell + data for this problem could be entered as follows: + +:: + + MULTIREGION SPHERICAL CELLMIX=100 END 1 8.72 END ZONE + +EXAMPLE 4. BUCKLEDSLAB. + + Consider a plate of fuel 4 cm thick, reflected by 3 cm of water on + both faces. The plate is 32 cm tall and 16 cm deep. The fuel is + mixture 1 and the water is mixture 2. Also, have XSDRNPM create a + cell weighted mixture 100 and calculate and eigenvalue. + +:: + + MULTIREGION BUCKLEDSLAB CELLMIX=100 LEFT_BDY=REFLECTED RIGHT_BDY=VACUUM + DY=32 DZ=16.0 END 1 2.0 2 5.0 END ZONE + +EXAMPLE 5. BUCKLEDCYL. + + Consider a solution of uranyl nitrate contained in a cylindrical + stainless-steel container reflected by 33 cm of water. The inside + dimensions of the steel container are 7.62 cm in radius and 130.0 cm + tall. The steel is 0.15 cm thick. The uranyl nitrate is defined to be + mixture 1, the steel is defined to be mixture 2, and the water is + defined to be mixture 3. + +:: + + MULTIREGION BUCKLEDCYL DY=130 END + 1 7.62 2 7.77 3 40.77 END ZONE + +.. _7-1b-4: + +DOUBLEHET unit cell data +------------------------ + +Unit cell data are always required for **DOUBLEHET** calculations. As +many unit cells as needed may be defined in the problem. If +**CELLMIX**\ =\ *mx* is entered after the fuel element (macro cell) +description, XSProc calls XSDRNPM to calculate the eigenvalue of the +cell and to create a homogenized cell-weighted cross section having the +characteristics of the doubly-heterogeneous cell configuration. + +EXAMPLE 1: A doubly-heterogeneous spherical fuel element with 15,000 UO\ :sub:`2` particles in a graphite matrix. + + Grain fuel radius is 0.025 cm. Grain contains one coating layer that + is 0.009-cm-thick. Pebbles are in a triangular pitch on a + 6.4-cm-pitch. Fuel pebble fuel zone is 2.5‑cm in radius and contains + a 0.5-cm-thick graphite clad that contains small amounts of + :sup:`10`\ B. Pebbles are surrounded by :sup:`4`\ He. Assume the + composition block is below: + +:: + + ' UO2 FUEL KERNEL + U-235 1 0 1.92585E-3 293.6 END + O 1 0 4.64272E-2 293.6 END + ' FIRST COATING + C 2 0 5.26449E-2 293.6 END + ' GRAPHITE MATRIX + C 6 0 8.77414E-2 293.6 END + ' CARBON PEBBLE OUTER COATING + C 7 0 8.77414E-2 293.6 END + B-10 7 0 9.64977E-9 293.6 END + HE-4 8 0 2.65156E-5 293.6 END + +The cell data for the **DOUBLEHET** cell follows: + +:: + + DOUBLEHET FUELMIX=10 END + GFR=0.025 1 COATT=0.009 2 MATRIX=6 NUMPAR=15000 END GRAIN + PEBBLE SPHTRIANGP RIGHT_BDY=WHITE HPITCH=3.2 8 FUELR=2.5 CLADR=3.0 7 END + +In this case we designated the homogenized mixture as mixture 10. If we +have a KENO V.a or KENO-VI input section, we would use mixture 10 in +that section. Note that the keyword “\ **FUELR**\ =” is followed by the +fuel dimension only, i.e., no mixture number. That is because the fuel +mixture number is specified with “\ **FUELMIX**\ =” and therefore need +not be repeated. + +EXAMPLE 2: Same as Example 1, except volume fraction of the grain type is known and is 0.037732. + +:: + + DOUBLEHET RIGHT_BDY=WHITE FUELMIX=10 END + GFR=0.025 1 COATT=0.009 2 MATRIX=6 VF=0.037732 END GRAIN + PEBBLE SPHTRIANGP RIGHT_BDY=WHITE HPITCH=3.2 8 FUELR=2.5 CLADR=3.0 7 END + +EXAMPLE 3: Same as Example 1, except halfpitch of the grain type is known and is 0.10137 cm. + +:: + + DOUBLEHET FUELMIX=10 END + GFR=0.025 1 COATT=0.009 2 HPITCH=0.10137 MATRIX=6 END GRAIN + PEBBLE SPHTRIANGP RIGHT_BDY=WHITE HPITCH=3.2 8 FUELR=2.5 CLADR=3.0 7 END + +EXAMPLE 4: A doubly-heterogeneous spherical fuel element with 10,000 UO2 particles +and 5,000 PuO2 particles in a graphite matrix. + + Grain fuel radii for UO\ :sub:`2` and PuO\ :sub:`2` particles are + 0.025 cm and 0.012 cm, respectively. UO\ :sub:`2` grains contain one + coating layer that is 0.009‑cm-thick. PuO\ :sub:`2` grains contain one + coating layer that is 0.0095-cm-thick. Pebbles are in a triangular pitch + on a 6.4-cm-pitch. Fuel pebble fuel zone is 2.5-cm in radius and + contains a 0.5-cm-thick graphite clad that contains small amounts of + :sup:`10`\ B. Pebbles are surrounded by :sup:`4`\ He. Assume the + composition block is given below: + +:: + + ' UO2 FUEL KERNEL + U-235 1 0 1.92585E-3 293.6 END + O 1 0 4.64272E-2 293.6 END + ' FIRST COATING + C 2 0 5.26449E-2 293.6 END + ' GRAPHITE MATRIX + C 6 0 8.77414E-2 293.6 END + ' CARBON PEBBLE OUTER COATING + C 7 0 8.77414E-2 293.6 END + B-10 7 0 9.64977E-9 293.6 END + HE-4 8 0 2.65156E-5 293.6 END + ' PUO2 FUEL KERNEL + PU-239 11 0 1.24470E-02 293.6 END + O 11 0 4.60983E-02 293.6 END + ' FIRST COATING + C 12 0 5.26449E-2 293.6 END + ' GRAPHITE MATRIX + C 16 0 8.77414E-2 293.6 END + +The cell data for the **DOUBLEHET** cell follows: + +:: + + DOUBLEHET FUELMIX=10 END + GFR=0.025 1 COATT=0.009 2 MATRIX=6 NUMPAR=10000 END GRAIN + GFR=0.012 11 COATT=0.0095 12 MATRIX=16 NUMPAR=5000 END GRAIN + PEBBLE SPHTRIANGP RIGHT_BDY=WHITE HPITCH=3.2 8 FUELR=2.5 CLADR=3.0 7 END + +Since number of particles is entered, the total volume fraction and the pitch can be calculated by the code. + +EXAMPLE 5: Same as Example 4 above except the volume fractions of UO\ :sub:`2` +and PuO\ :sub:`2` grains are 0.02511 and 0.00318, respectively. + +:: + + DOUBLEHET RIGHT_BDY=WHITE FUELMIX=10 END + GFR=0.025 1 COATT=0.009 2 MATRIX=6 VF=0.02511 END GRAIN + GFR=0.012 11 COATT=0.0095 12 MATRIX=16 VF=0.00318 END GRAIN + PEBBLE SPHTRIANGP RIGHT_BDY=WHITE HPITCH=3.2 8 FUELR=2.5 CLADR=3.0 7 END + +EXAMPLE 6: Same as Example 4 above except pitch is also known. + + UO\ :sub:`2` grains have a pitch of 0.25 cm. PuO\ :sub:`2` grains + have a pitch of 0.20 cm. + +:: + + DOUBLEHET FUELMIX=10 END + GFR=0.025 1 COATT=0.009 2 + MATRIX=6 NUMPAR=10000 PITCH=0.25 END GRAIN + GFR=0.012 11 COATT=0.0095 12 + MATRIX=16 NUMPAR=5000 PITCH=0.20 END GRAIN + PEBBLE SPHTRIANGP RIGHT_BDY=WHITE HPITCH=3.2 8 FUELR=2.5 CLADR=3.0 7 END + +Since number of particles is sufficient to perform the homogenization, +it is used. However, instead of calculating the pitch for the 1-D cell +calculation for each grain type, the user input pitch is used. Hence, +the calculated *k*\ :sub:`eff` of Example 6 will be different from those of +Examples 4 and 5. + +**EXAMPLE 7: Same as Example 6 except the doubly-heterogeneous cell will +be cell-weighted.** + + The final cell-weighted mixture number is 17. + +:: + + + DOUBLEHET FUELMIX=10 END + GFR=0.025 1 COATT=0.009 2 + NUMPAR=10000 PITCH=0.25 MATRIX=6 END GRAIN + GFR=0.012 11 COATT=0.0095 12 + NUMPAR=5000 PITCH=0.20 MATRIX=16 END GRAIN + PEBBLE SPHTRIANGP RIGHT_BDY=WHITE HPITCH=3.2 8 FUELR=2.5 CLADR=3.0 7 CELLMIX=17 END + +EXAMPLE 8: A doubly-heterogeneous spherical fuel element with 15,000 UO\ :sub:`2` particles in a graphite matrix. + + Grain fuel radius is 0.012 cm. Grain contains four coating layers + that are 0.0095, 0.004, 0.0035, and 0.004-cm-thick. Pebbles are in a + square pitch on a 6.0‑cm-pitch. Fuel pebble fuel zone is 2.5-cm in + radius and contains a 0.5-cm-thick graphite clad that contains small + amounts of :sup:`10`\ B. Pebbles are surrounded by :sup:`4`\ He. + Assume the composition block is given below: + +:: + + + ' UO2 FUEL KERNEL + U-235 1 0 1.92585E-3 293.6 END + O 1 0 4.64272E-2 293.6 END + ' FIRST COATING + C 2 0 5.26449E-2 293.6 END + ' INNER PYRO CARBON + C 3 0 9.52621E-2 293.6 END + ' SILICON CARBIDE + C 4 0 4.77240E-2 293.6 END + SI 4 0 4.77240E-2 293.6 END + ' OUTER PYRO CARBON + C 5 0 9.52621E-2 293.6 END + ' GRAPHITE MATRIX + GRAPHITE 6 0 8.77414E-2 293.6 END + ' CARBON PEBBLE OUTER COATING + C 7 0 8.77414E-2 293.6 END + B-10 7 0 9.64977E-9 293.6 END + HE-4 8 0 2.65156E-5 293.6 END + +The cell data for the **DOUBLEHET** cell follows: + +:: + + DOUBLEHET FUELMIX=10 END + GFR=0.012 1 COATT=0.0095 2 COATT=0.004 3 COATT=0.0035 4 COATT=0.004 5 MATRIX=6 NUMPAR=15000 VF=0.0245 END GRAIN + PEBBLE SPHSQUAREP RIGHT_BDY=WHITE HPITCH=3.0 8 FUELR=2.5 CLADR=3.0 7 END + +Note that the grains are overspecified and the numbers are inconsistent. +A **VF** value of 0.0245 results in a total number of particles of +10652.32 which is considerably less than 15,000. In this case, the code +will issue a warning to this effect and will use **VF** value in the +calculations (i.e., ignore **NUMPAR**\ =15000 entry). + +EXAMPLE 9: Similar to Example 8 except radii for grain regions are entered. + +:: + + DOUBLEHET FUELMIX=10 END + GFR=0.012 1 COATR=0.0215 2 COATR=0.0255 3 COATR=0.029 4 COATR=0.033 5 MATRIX=6 NUMPAR=15000 VF=0.0245 END GRAIN + PEBBLE SPHSQUAREP RIGHT_BDY=WHITE HPITCH=3.0 8 FUELR=2.5 CLADR=3.0 7 END + +EXAMPLE 10: A doubly-heterogeneous spherical fuel element with two UO\ :sub:`2` grain types. + + First grain type has a fuel radius of 0.025 cm. Second grain type + fuel radius is 0.004 cm. First grain type has one coating that is + 0.009-cm-thick. Second grain type has two coatings each + 0.004-cm-thick. Each grain type has a volume fraction of 0.45. + Pebbles are in a triangular pitch on a 7.0-cm-pitch. Fuel pebble fuel + zone is 2.5-cm in radius and contains a 0.5-cm-thick graphite clad + that contains small amounts of :sup:`10`\ B and :sup:`11`\ B. Pebbles + are surrounded by :sup:`4`\ He. Assume the composition block is given + below: + +:: + + ' FUEL KERNEL + U-238 1 0 2.12877E-2 END + U-235 1 0 1.92585E-3 END + O 1 0 4.64272E-2 END + B-10 1 0 1.14694E-7 END + B-11 1 0 4.64570E-7 END + ' FIRST COATING + C 2 0 5.26449E-2 END + ' INNER PYRO CARBON + C 3 0 9.52621E-2 END + ' SILICON CARBIDE + C 4 0 4.77240E-2 END + SI 4 0 4.77240E-2 END + ' FUEL KERNEL + U-238 5 0 2.12877E-2 END + U-235 5 0 1.92585E-3 END + O 5 0 4.64272E-2 END + B-10 5 0 1.14694E-7 END + B-11 5 0 4.64570E-7 END + ' GRAPHITE MATRIX + C 6 0 8.77414E-2 END + B-10 6 0 9.64977E-9 END + B-11 6 0 3.90864E-8 END + ' CARBON PEBBLE OUTER COATING + C 7 0 8.77414E-2 END + B-10 7 0 9.64977E-9 END + B-11 7 0 3.90864E-8 END + ' HELIUM + HE 8 0.000164 END + ' GRAPHITE MATRIX + C 9 0 8.77414E-2 END + B-10 9 0 9.64977E-9 END + B-11 9 0 3.90864E-8 END + +The cell data for the **DOUBLEHET** cell follows: + +:: + + DOUBLEHET FUELMIX=10 END + GFR=0.025 1 COATR=0.034 2 MATRIX=6 VF=0.45 END GRAIN + COATT=0.004 3 GFR=0.4 5 COATT=0.004 4 MATRIX=9 VF=0.45 END GRAIN + PEBBLE SPHTRIANGP RIGHT_BDY=WHITE HPITCH=3.5 8 FUELD=5.0 + CLADD=6.0 7 END + +EXAMPLE 11: A doubly-heterogeneous hexagonal block type fuel element +with UO\ :sub:`2` grains in a cylindrical fuel region. + + Grain fuel radius is 0.025 cm. Grain coating is 0.009-cm-thick. + Grains have a volume fraction of 0.45. Hexagonal rods are in a 7-cm + triangular pitch. Fuel rod fuel zone is 2.5-cm in radius, 10-cm-high + and contains a 0.5-cm-thick graphite clad that contains small amounts + of :sup:`10`\ B. Assume the composition block is below: + +:: + + ' FUEL KERNEL + U-238 1 0 2.12877E-2 END + U-235 1 0 1.92585E-3 END + O 1 0 4.64272E-2 END + B-10 1 0 1.14694E-7 END + ' FIRST COATING + C 2 0 5.26449E-2 END + ' GRAPHITE MATRIX + C 6 0 8.77414E-2 END + B-10 6 0 9.64977E-9 END + ' CARBON PEBBLE OUTER COATING + C 7 0 8.77414E-2 END + B-10 7 0 9.64977E-9 END + ' IRON CLADDING + FE 8 END + +The cell data for the **DOUBLEHET** cell follows: + +:: + + DOUBLEHET FUELMIX=10 END + GFR=0.025 1 COATR=0.034 2 MATRIX=6 VF=0.45 END GRAIN + ROD TRIANGP RIGHT_BDY=WHITE HPITCH=3.5 7 FUELD=5.0 + FUELH=10 END + +EXAMPLE 12: This is the same as Example 11 except the fuel elements (cylindrical rods) have 0.05‑cm-thick iron cladding. + +The cell data for the **DOUBLEHET** cell follows: + +:: + + DOUBLEHET FUELMIX=10 END + GFR=0.025 1 COATR=0.034 2 MATRIX=6 VF=0.45 END GRAIN + ROD TRIANGP RIGHT_BDY=WHITE HPITCH=3.5 7 FUELR=2.5 + CLADD=5.1 8 FUELH=10 END + +.. _7-1b-5: + +Optional parameter data +----------------------- + +The optional parameter data provide a means of providing additional +information or alternative data to the cross-section processing codes. +There are two types of optional parameter data. The first type of data +is used by XSDRNPM and BONAMI for cross-section processing and +cell-weighting cross sections. This type of data is initiated using the +keywords **MORE DATA** and ends with the keywords **END MORE**. This +input is described in :ref:`7-1-3-8`. The second type of optional +parameter data is used by CENTRM and PMC for cross-section processing. +This type of data is initiated using the keywords **CENTRM DATA** and +ends with the keywords **END CENTRM**. This input is described in +:ref:`7-1-3-9`. It is possible to input both types of data for a unit +cell. The optional parameter data specified apply only to the unit cell +that immediately precedes it. + + +MORE DATA examples + +Consider a problem in which it is desirable to increase the number of +inner iterations in XSDRNPM to 30 and to tighten the overall convergence +criteria to a value of 0.000075. This could be accomplished by entering +the data as follows: + +:: + + MORE DATA IIM=30 EPS=0.000075 END + +The order of the data entry is not important, and it can be continued +across several lines. However, a keyword and its value cannot be +separated across lines. The terminator for the optional parameter data, +END, must not begin in column 1 unless you assign a name to it. An +alternative method of entering the above data is given below. + +:: + + MORE DATA + IIM=30 EPS=0.000075 + END MORE + +or, + +:: + + MORE DATA IIM=30 EPS=0.000075 END MORE DATA + +.. _7-1b-6: + +CENTRM DATA examples +-------------------- + +Consider a problem in which it is desirable to increase the upper energy +of the CENTRM CE transport calculation from the default of 20000 eV to a +value 50000 eV, and to extend the default lower energy from 0.001 eV to +0.0001. This is accomplished by entering the data as follows: + +:: + + CENTRM DATA DEMAX=50000 DEMIN=0.0001 END CENTRM + +As with the **MORE DATA** block, an alternative method of entering the +above data is given below. + +:: + + CENTRMDATA + DEMAX=50000 DEMIN=0.0001 + END CENTRMDATA + +**CENTRM and PMC** computation options can also be controlled with +**CENTRM DATA.** A complete description of the CENTRM/PMC computational +methods and options can be found the corresponding sections of the SCALE +manual. The following example specifies that: + +(a) discrete-level inelastic scattering will be used in CENTRM and +processed in PMC [nmf6]; + +(b) the CENTRM 1D discrete S\ :sub:`N` transport solver will be used in +the upper MG energy range [nfst] and the CE energy range [npxs], while +the infinitie medium model will be used for the thermal energy range +[nthr]; + +(c) a P3 scattering order [isct] will be used in the transport +calculations; + +(d) PMC will perform “consistent PN” corrections on Legendre moments of +the 2D elastic matrices [n2d]; (e) additional output information will be +provided by CENTRM [ixprt] and by PMC [nprt]. + +:: + + CENTRM DATA NMF6=0 NFST=0 NTHR=2 ISCT=3 + N2D=-2 IXPRT=1 NPRT=1 END CENTRM DATA diff --git a/XSProcAppC.rst b/XSProcAppC.rst new file mode 100644 index 0000000..04716e4 --- /dev/null +++ b/XSProcAppC.rst @@ -0,0 +1,971 @@ +.. _7-1c: + +Examples of Complete XSProc Input Data +====================================== + +.. _7-1c-1: + +Infinite homogeneous medium input data +-------------------------------------- + +Examples of XSProc input data for infinite homogeneous media problems +are given below. In these cases the cross section library name “fine_n” +indicates that the latest recommended fine-group SCALE library will used +in the calculations. + +EXAMPLE 1. Default cell definition. + + + Consider a cylindrical billet of 20 wt % enriched UO\ :sub:`2`, + having a density of 10.85 g/cm\ :sup:`3` that is 26 cm in diameter + and 26 cm tall. + +The average mean-free path in the uranium dioxide is on the order of +2.5 cm. Because only a small fraction of the billet is within a +mean-free path of the surface, the material can be treated as an +infinite homogeneous medium; therefore the CELL DATA block can be +omitted. The XSProc data follows: + +.. highlight:: scale + +:: + + 20% ENRICHED UO2 BILLET + fine_n + READ COMP + UO2 1 0.99 293 92235 20 92238 80 END + END COMP + +The volume fraction used for the UO\ :sub:`2`, 0.99, is calculated by +dividing the actual density by the theoretical density obtained from the +*Isotopes in standard composition library* table in the STDCMP chapter, +(10.85/10.96). Since the enrichment was specified as 20%, it is assumed +that the remainder is :sup:`238`\ U. + +An alternative input data description follows: + +:: + + 20% ENRICHED UO2 BILLET + fine_n + READ COMP + UO2 1 DEN=10.85 1 293 92235 20 92238 80 END + END COMP + +EXAMPLE 2. Specify the cell definition. + + + Consider a 5-liter Plexiglas bottle with an inner radius of 9.525 cm + and inner height of 17.78 cm that is filled with highly enriched + uranyl nitrate solution at 415 g/L and 0.39 mg of excess nitrate per + gram of solution. The uranium isotopic content of the nitrate + solution is 92.6 wt % :sup:`235`\ U, 5.9 wt % :sup:`238`\ U, 1.0 wt % + :sup:`234`\ U, and 0.5 wt % :sup:`236`\ U. Solution density will be + calculated from the given data. + +The size of the nitrate solution is on the order of 16 to 20 cm in +diameter and height. The average mean-free path in the nitrate solution +is on the order of 0.5 cm. Therefore, infinite homogeneous medium is an +appropriate choice for this problem. By default BONAMI is used for +self-shielding the infinite medium of Plexiglas, while CENTRM is used to +shield the infinite medium fissile solution. + +:: + + SET UP 5 LITER URANYL NITRATE SOLUTION IN A PLEXIGLAS CONTAINER + fine_n + READ COMP + PLEXIGLAS 1 END + SOLUTION MIX=2 RHO[UO2(NO3)2]=415 + 92235 92.6 92238 5.9 92236 0.5 + END SOLUTION + END COMP + READ CELLDATA + INFHOMMEDIUM 2 END + END CELLDATA + +.. _7-1c-2: + +LATTICECELL input data +---------------------- + +Examples of XSProc input data for **LATTICECELL** problems are given +below. + +EXAMPLE 1. SQUAREPITCH ARRAY. + + + Consider an infinite planar array (infinite in X and Y and one layer + in Z) of 20 wt % enriched U metal rods with a 1-cm pitch. Each fuel + rod is bare uranium metal, 0.75 cm OD × 30.0 cm long. The rods are + submerged in water. + +Because the diameter of the fuel rod, 0.75 cm, is only slightly larger +than the average mean-free path in the uranium metal, approximately 0.5, +and because the configuration is a regular array, **LATTICECELL** is the +appropriate choice for proper cross-section processing. The *parm* field +is not provided, so the default CENTRM/PMC self-shielding method is +used. XSProc data follows: + +:: + + INFINITE PLANAR ARRAY OF 20% U METAL RODS + fine_n + READ COMP + URANIUM 1 1 293 92235 20 92238 80 END + H2O 2 END + END COMP + READ CELLDATA + LATTICECELL SQUAREPITCH PITCH=1.0 2 FUELD=0.75 1 END + END CELLDATA + +Since the MORE DATA and CENTRM DATA blocks were omitted, default options +will be used in the self-shielding calculations. The default CENTRM/PMC +computation options for a square pitch lattice cell are the +method-of-characteristics (MoC) method with P0 scatter in CENTRM +calculations. + +EXAMPLE 2. SQUAREPITCH PWR LATTICE. + + + Consider an infinite, uniform planar array (infinite in X and Y and + one layer in Z) of PWR-like fuel pins of 2.35% enriched UO\ :sub:`2` + clad with zirconium. The density of the UO\ :sub:`2` is + 9.21 g/cm\ :sup:`3`. The fuel in each pin is 0.823 cm in diameter, + the clad is 0.9627 cm in diameter, and the length of each pin is + 366 cm. The fuel pins are separated by 0.3124 cm of water in the + horizontal plane. + +**LATTICECELL** is the appropriate choice for cross-section processing. +Assume that all defaults are appropriate; thus the CENTRM/PMC +methodology is used, and the MORE DATA and CELL DATA blocks are not +entered. The input cross section library named “broad_n” indicates that +the recommended broad group SCALE library will be used. In this case +CENTRM uses the 2D MoC transport solver. The XSProc data follows: + +:: + + PWR-LIKE FUEL BUNDLE; uniform infinite array model. + broad_n + READ COMP + UO2 1 .84 293. 92235 2.35 92238 97.65 END + ZR 2 1 END + H2O 3 1 END + END COMP + READ CELLDATA + LATTICECELL SQUAREPITCH PITCH=1.2751 3 FUELD=0.823 1 CLADD=0.9627 2 END + END CELLDATA + +EXAMPLE 3. SQUAREPITCH PWR LATTICE, with non-uniform Dancoff. + + +This example is a single PWR assembly of fuel pins of the type described +above, contained in a water pool. The interior pins in the assembly can +be self-shielded using the same uniform, infinite lattice model in +previous example. However self-shielding of the outer boundary-edge pins +will be modified to account for being adjacent to a water reflector, +rather than surrounded on all sides by similar pins. This requires that +the MCDancoff module be executed previously to obtain non-uniform +Dancoff factors for the edge pins. The average edge-pin value of 0.61 is +used to represent Dancoff factors of all boundary pins. The default +CENTRM MoC transport solver is used for both cells, but the original +pitch of 1.2751 cm for the second cell (i.e., boundary pin) is modified +to a new pitch corresponding to a Dancoff value of 0.61. + +:: + + PWR-LIKE FUEL BUNDLE, with boundary-pin corrections + broad_n + READ COMP + ' mixtures for interior pins + UO2 1 .84 293. 92235 2.35 92238 97.65 END + ZR 2 1 END + H2O 3 1 END + ' mixtures for boundary pins + UO2 4 .84 293. 92235 2.35 92238 97.65 END + ZR 5 1 END + H2O 6 1 END + END COMP + READ CELLDATA + LATTICECELL SQUAREPITCH PITCH=1.2751 3 FUELD=0.823 1 CLADD=0.9627 2 END + LATTICECELL SQUAREPITCH PITCH=1.2751 6 FUELD=0.823 4 CLADD=0.9627 5 END + CENTRM DATA DAN2PITCH=0.61 END CENTRM + END CELLDATA + +EXAMPLE 6. SPHTRIANGP ARRAY. + + + Consider an infinite array of spherical pellets of 2.67% enriched + UO\ :sub:`2` with a density of 10.3 g/cm\ :sup:`3` and a diameter of + 1.0724 cm arranged in a “triangular” pitch, flooded with borated + water at 4350 ppm. The boron is natural boron; the borated water is + created by adding boric acid, H\ :sub:`3`\ BO\ :sub:`3`, and has a + density of 1.0078 g/cm\ :sup:`3`. The temperature is 15ºC and the + pitch is 1.1440 cm. The standard composition data for the borated + water are given in Example 2 of :ref:`7-1a-9`. + +Because the diameter of the fuel pellet, 1.0724 cm, is smaller than the +average mean-free path in the UO\ :sub:`2`, approximately 1.5 cm, and +because the configuration is a regular array, **LATTICECELL** is the +appropriate choice for proper cross-section processing. + +The density fraction for the UO\ :sub:`2` is the ratio of actual to +theoretical density (10.3/10.96 = 0.9398). Assume that the U is all +:sup:`235`\ U and :sup:`238`\ U. See :ref:`7-1a-9` for how to define +borated water. + +The XSProc data follows: + +:: + + SPHERICAL PELLETS IN BORATED WATER + fine_n + READ COMP + UO2 1 .9398 288 92235 2.67 92238 97.33 END + ATOMH3BO3 2 0.025066 3 5000 1 1001 3 8016 3 + 1.0 288 END + H2O 2 0.984507 288 END + END COMP + READ CELLDATA + LATTICECELL SPHTRIANGP PITCH=1.1440 2 FUELD=1.0724 1 END + END CELLDATA + +.. _7-1c-3: + +MULTIREGION input data +---------------------- + +Examples of XSProc input data for **MULTIREGION** problems are given +below. + +EXAMPLE 1. SPHERICAL. + + + Consider a small highly enriched uranium sphere supported by a + Plexiglas collar in a tank of water. The uranium metal sphere has a + diameter of 13.1075 cm, is 97.67% enriched, and has a density of + 18.794 g/cm\ :sup:`3`. The cylindrical Plexiglas collar has a + 4.1275-cm-radius central hole, extends to a radius of 12.7 cm and is + 2.54 cm thick. The water filled tank is 60 cm in diameter. + +The density fraction of the uranium metal is the ratio of actual to +theoretical density, where the theoretical density is obtained from the +*Isotopes in standard composition library* table in section 7.2.1. Thus, +the density multiplier is 18.794/19.05 = 0.9866. The abundance of +uranium is not stated beyond 97.67% enriched, so it is reasonable to +assume the remainder is :sup:`238`\ U. The Plexiglas collar is not +significantly different from water and does not surround the fuel, so it +can be ignored. If it is ignored, the problem becomes a 1-D geometry +that can be defined using the **MULTIREGION** type of calculation, and +the eigenvalue of the system can be obtained without additional data by +executing CSAS1. However, the Plexiglas has been included in this data +so it can be passed to a code such as KENO V.a which can describe the +geometry rigorously. The XSProc data follow: + +:: + + SMALL WATER REFLECTED SPHERE ON PLEXIGLAS COLLAR + fine_n + READ COMP + URANIUM 1 .9866 293. 92235 97.67 92238 2.33 END + PLEXIGLAS 2 END + H2O 3 END + END COMP + READ CELLDATA + MULTIREGION SPHERICAL RIGHT_BDY=VACUUM END 1 6.55375 3 30.0 END ZONE + END CELLDATA + +EXAMPLE 2. BUCKLEDSLAB. + + + This example features a 93.2% enriched uranyl-fluoride solution + inside a rectangular Plexiglas container immersed in water. The + fissile solution contains 578.7 g of UO\ :sub:`2`\ F\ :sub:`2` per + liter and has no excess acid. The critical thickness of the fuel is + 5.384 cm. The finite height of the fuel slab is 147.32 cm, and the + depth is 71.58 cm. The Plexiglas container is 1.905 cm thick and is + reflected by 20.32 cm of water. + +The half thickness of the fuel (2.692) will be used with a reflected +left boundary and a vacuum right boundary (default). The XSProc data +follow: + +:: + + CRITICAL SLAB EXPERIMENT USING URANYL-FLUORIDE SOLUTION + fine_n + READ COMP + SOLUTION MIX=1 RHO[UO2F2]=578.7 + 92235 93.2 92238 6.8 TEMP=300 + END SOLUTION + PLEXIGLAS 2 END + H2O 3 END + END COMP + READ CELLDATA + MULTIREGION BUCKLEDSLAB LEFT_BDY=REFLECTED + DY=71.58 DZ=147.32 END 1 2.692 2 4.597 3 24.917 END ZONE + END CELLDATA + +.. _7-1c-4: + +DOUBLEHET input data +-------------------- + +EXAMPLE 1: A doubly-heterogeneous spherical fuel element with 15,000 UO\ :sub:`2` particles in a graphite matrix. + + + Grain fuel radius is 0.025 cm. Grain contains one coating layer that + is 0.009-cm-thick. Pebbles are in a triangular pitch on a + 6.4-cm-pitch. Fuel pebble fuel zone is 2.5‑cm in radius and contains + a 0.5-cm-thick graphite clad that contains small amounts of + :sup:`10`\ B. Pebbles are surrounded by :sup:`4`\ He. In this case we + designated the homogenized mixture as mixture 10. If we have a + KENO V.a or KENO-VI input section, we would use mixture 10 in that + section. Note that the keyword “FUELR=” is followed by the fuel + dimension only, i.e., no mixture number. That is because the fuel + mixture number is specified with “FUELMIX=” and therefore need not be + repeated. + +:: + + INFINITE ARRAY OF UO2-FUELLED PEBBLES + fine_n + READ COMP + ' UO2 FUEL KERNEL + U-235 1 0 1.92585E-3 293.6 END + O 1 0 4.64272E-2 293.6 END + ' FIRST COATING + C 2 0 5.26449E-2 293.6 END + ' GRAPHITE MATRIX + C 6 0 8.77414E-2 293.6 END + ' CARBON PEBBLE OUTER COATING + C 7 0 8.77414E-2 293.6 END + B-10 7 0 9.64977E-9 293.6 END + HE-4 8 0 2.65156E-5 293.6 END + END COMP + READ CELLDATA + DOUBLEHET RIGHT_BDY=WHITE FUELMIX=10 END + GFR=0.025 1 COATT=0.009 2 MATRIX=6 NUMPAR=15000 END GRAIN + PEBBLE SPHTRIANGP RIGHT_BDY=WHITE HPITCH=3.2 8 FUELR=2.5 CLADR=3.0 7 END + END CELLDATA + +EXAMPLE 2: A doubly-heterogeneous spherical fuel element with 10,000 UO\ :sub:`2` particles and 5,000 PuO\ :sub:`2` particles in a graphite matrix. + + + Grain fuel radii for UO\ :sub:`2` and PuO\ :sub:`2` particles are + 0.025 cm and 0.012 cm, respectively. UO\ :sub:`2` grains contain one + coating layer that is 0.009‑cm-thick. PuO\ :sub:`2` grains contain + one coating layer that is 0.0095-cm-thick. Pebbles are in a + triangular pitch on a 6.4-cm-pitch. Fuel pebble fuel zone is 2.5-cm + in radius and contains a 0.5-cm-thick graphite clad that contains + small amounts of :sup:`10`\ B. Pebbles are surrounded by + :sup:`4`\ He. Since number of particles is entered, the total volume + fraction and the pitch can be calculated by the code. + +:: + + INFINITE ARRAY OF UO2- AND PUO2-FUELLED PEBBLES + fine_n + READ COMP + ' UO2 FUEL KERNEL + U-235 1 0 1.92585E-3 293.6 END + O 1 0 4.64272E-2 293.6 END + ' FIRST COATING + C 2 0 5.26449E-2 293.6 END + ' GRAPHITE MATRIX + C 6 0 8.77414E-2 293.6 END + ' CARBON PEBBLE OUTER COATING + C 7 0 8.77414E-2 293.6 END + B-10 7 0 9.64977E-9 293.6 END + HE-4 8 0 2.65156E-5 293.6 END + ' PUO2 FUEL KERNEL + PU-239 11 0 1.24470E-02 293.6 END + O 11 0 4.60983E-02 293.6 END + ' FIRST COATING + C 12 0 5.26449E-2 293.6 END + ' GRAPHITE MATRIX + C 16 0 8.77414E-2 293.6 END + END COMP + READ CELLDATA + DOUBLEHET RIGHT_BDY=WHITE FUELMIX=10 END + GFR=0.025 1 COATT=0.009 2 MATRIX=6 NUMPAR=10000 END GRAIN + GFR=0.012 11 COATT=0.0095 12 MATRIX=16 NUMPAR=5000 END GRAIN + PEBBLE SPHTRIANGP RIGHT_BDY=WHITE HPITCH=3.2 8 FUELR=2.5 CLADR=3.0 7 END + END CELLDATA + +EXAMPLE 3: A doubly-heterogeneous slab fuel element with flibe salt coolant + + + Grain fuel radii for UO\ :sub:`2` particles are 0.025 cm. The + UO\ :sub:`2` grains contain four coating layers with thicknesses of + 0.01, 0.0035, 0.003, and 0.004 cm, respectively. The fuel grains are + embedded in a carbon matrix material to form the fuel compact. The + x-dimension of fuel plate consists of a 0.5 cm (half-thickness) fuel + compact region, a carbon clad with outer dimension of 1.27, followed + by the flibe coolant with an outer reflected dimension of 1.62 cm. + The width (y-dimension) of the slab plate is 22.5 cm and the height + (z-dimension) is 500 cm. The y and z dimensions are only used to + define volumes for the fuel plate. + +:: + + slab doublehet sample problem: double-het for slab + v7.1-252n + read comp + ' fuel kernel + u-238 1 0 2.12877e-2 293.6 end + u-235 1 0 1.92585e-3 293.6 end + o 1 0 4.64272e-2 293.6 end + b-10 1 0 1.14694e-7 293.6 end + b-11 1 0 4.64570e-7 293.6 end + ' first coating + c 2 0 5.26449e-2 293.6 end + ' inner pyro carbon + c 3 0 9.52621e-2 293.6 end + ' silicon carbide + c 4 0 4.77240e-2 293.6 end + si 4 0 4.77240e-2 293.6 end + ' outer pyro carbon + c 5 0 9.52621e-2 293.6 end + ' graphite matrix + c 6 0 8.77414e-2 293.6 end + b-10 6 0 9.64977e-9 293.6 end + b-11 6 0 3.90864e-8 293.6 end + ' carbon slab outer coating + c 7 0 8.77414e-2 293.6 end + b-10 7 0 9.64977e-9 293.6 end + b-11 7 0 3.90864e-8 293.6 end + Li-6 8 0 1.38344E-06 948.15 end + Li-7 8 0 2.37205E-02 948.15 end + Be 8 0 1.18609E-02 948.15 end + F 8 0 4.74437E-02 948.15 end + end comp + read celldata + doublehet fuelmix=10 end + gfr=0.02135 1 + coatt=0.01 2 + coatt=0.0035 3 + coatt=0.003 4 + coatt=0.004 5 + vf=0.4 + matrix=6 + end grain + slab symmslabcell + hpitch=1.62 8 + cladr=1.27 7 + fuelr=0.5 + fuelh=500 + fuelw=22.500 + end + centrm data ixprt=1 isn=8 end centrm + end celldata + +.. _7-1c-5: + +Two methods of specifying a fissile solution +-------------------------------------------- + +The standard composition specification data offer flexibility in the +choice of input data. This section illustrates two methods of specifying +the same fissile solution. + +Create a mixture 3 that is aqueous uranyl nitrate solution: + + UO\ :sub:`2`\ (NO\ :sub:`3`)\ :sub:`2`, solution density = 1.555 g + cm\ :sup:`3`/ + + 0.2669 g U/g-soln., 0.415 g U/ cm\ :sup:`3`; excess nitrate = + 0.39 mg/g-soln + + Uranium isotopic content: 92.6 wt % U-235 5.9 wt % U-238 + + 1.0 wt % U-234 and 0.5 wt % U-236 + +The SCALE atomic weights used in this problem are listed as follows: + + H 1.0078 + + O 15.999 + + N 14.0067 + + U-234 234.041 + + U-235 235.0439 + + U-236 236.0456 + + U-238 238.0508 + +Two methods of describing the uranyl nitrate solution will be demonstrated. +Method 1 is more rigorous, and method 2 is easier and as accurate. + +.. centered:: METHOD 1: + + +This method involves breaking the solution into its component parts +[(HNO\ :sub:`3`, UO\ :sub:`2`\ (NO\ :sub:`3`)\ :sub:`2`, and +H\ :sub:`2`\ O)] and entering the basic standard composition +specifications for each. + +1. Calculate the density of the HNO\ :sub:`3` 0.39 × 10\ :sup:`−3` g + NO\ :sub:`3`/g soln × [(62.997 g HNO\ :sub:`3`/mole + HNO\ :sub:`3`)/(61.990 g NO\ :sub:`3`/mole NO\ :sub:`3`)] × 1.555 g + soln/ cm\ :sup:`3`\ soln = 6.16 × 10\ :sup:`−4` g HNO\ :sub:`3`/cc + soln. + +2. Calculate the density fraction of HNO\ :sub:`3` (actual + density/theoretical density). In the Standard Composition Library the + theoretical density of HNO\ :sub:`3` is 1.0. 6.16 × 10\ :sup:`−4`/1.0 + = 6.16 × 10\ :sup:`−4`. + +3. Calculate the molecular weight of the uranium + +.. + + The number of atoms in a mole of uranium is the sum of the number of + atoms of each isotope in the mole of uranium. + + Let AU = the average molecular weight of uranium, g U/mole U + + GU = the density of uranium in g/cm\ :sup:`3`. + + Then the number of atoms in a mol of uranium = + + (6.023 × 10\ :sup:`+23` \* 10\ :sup:`−24` \* GU)/AU + + or 0.6023 \* GU/AU. + + The weight fraction of each isotope is the weight % \* 100. + + Therefore, F235 = 0.926, the weight fraction of U-235 in the U + + F238 = 0.059, the weight fraction of U-238 in the U + + F236 = 0.005, the weight fraction of U-236 in the U + + F234 = 0.010, the weight fraction of U-234 in the U + + A235 = 235.0442, the molecular weight of U-235 + + A238 = 238.0510, the molecular weight of U-238 + + A236 = 236.0458, the molecular weight of U-236 + + A234 = 234.0406, the molecular weight of U-234. + + Then the number of atoms of isotopes in a mol of uranium = + + 6.023 × 10\ :sup:`+23` \* 10\ :sup:`−24` \* ( (GU*F235/A235) + + (GU*F238/A238) + + + GU*F236/A236) + (GU*F234/A234) ) + + or + + 0.6023*GU \* ( 0.926/235.0442 + 0.059/238.0510 + + + 0.005/236.0458 + 0.010/234.0406 ). + + Because the number of atoms of uranium equals the sum of the atoms of + isotopes, + + 0.6023 \* GU/AU = 0.6023 \* GU \*( 0.926/235.0442 + 0.059/238.0510 + + + 0.005/236.0458 + 0.010/234.0406 ) + + 1/AU = 0.926/235.0442 + 0.059/238.0510 + 0.005/236.0458 + + 0.010/234.0406 + + AU = 235.2144. + +4. Calculate the molecular weight of the + UO\ :sub:`2`\ (NO\ :sub:`3`)\ :sub:`2`. + +.. + + 235.2144 + (8 × 15.9954) + (2 × 14.0033) = 391.184 g + UO\ :sub:`2`\ (NO\ :sub:`3`)\ :sub:`2`/mole + +5. Calculate the density of UO\ :sub:`2`\ (NO\ :sub:`3`)\ :sub:`2` + +.. + + 0.415 g U/cc × [(391.184 g + UO\ :sub:`2`\ (NO\ :sub:`3`)\ :sub:`2`/mol)/(235.2144 g U/mole)] = + + 0.69018 g UO\ :sub:`2`\ (NO\ :sub:`3`)\ :sub:`2`/ cm\ :sup:`3`.soln. + +Calculate the density fraction (actual density/theoretical density) of +UO\ :sub:`2`\ (NO\ :sub:`3`)\ :sub:`2`. + + [In the Standard Composition Library the theoretical density of + UO\ :sub:`2`\ (NO\ :sub:`3`)\ :sub:`2` is given as + 2.2030 g/cm\ :sup:`3`.] + + The density fraction is 0.69018/2.2030 = 0.31329. + +6. Calculate the amount of water in the solution + +.. + + 1.555 g soln/ cm\ :sup:`3`. soln − 6.16 × 10\ :sup:`−4` g + HNO\ :sub:`3`/cm\ :sup:`3` soln − 0.69018 g + UO\ :sub:`2`\ (NO\ :sub:`3`)\ :sub:`2`\ LL/ cm\ :sup:`3`. soln = + 0.8642 g H\ :sub:`2`\ O/cc soln. + +7. Calculate the density fraction (actual density/theoretical density) + of water. + +:: + + HNO3 3 6.16-4 293 END + UO2(NO3)2 3 .31329 293 92235 92.6 92238 5.9 92234 1.0 + 92236 0.5 END + H2O 3 .86575 293 END + +.. centered:: METHOD 2: + +This method utilizes the solution option available in the standard +composition specification data. Because the density is specified in the +input data, this method should yield correct number densities that +should agree with method 1 except for calculational round-off. + +1. Calculate the fuel density + +.. + + 0.415 g U/cc is 415 g U/L. + +2. The molecular weight of nitrate NO\ :sub:`3` is 61.9895. + +3. Calculate the molarity of the solution. + +.. + + 0.39 mg nitrate/g soln × 1000 cm\ :sup:`3`\ soln/L soln × 1 g/1000 mg + × 1.555 g soln/ cm\ :sup:`3`\ soln = 0.60645 g excess nitrate/L soln. + + A 1-molar solution is 1 mole of acid/L of solution: + + (For nitric acid 1 molar is 1 normal because there is only one atom + of hydrogen per molecule of acid in HNO\ :sub:`3`.) + + (0.60645 g nitrate/L soln)/(61.9895 g NO\ :sub:`3`/mole NO\ :sub:`3`) + = 9.783 × 10\ :sup:`−3` mole nitrate/L is identical to mole of + acid/L, which is identical to molarity. + +4. The density fraction of the solution is 1.0. Do not try to use the + density of the solution divided by the theoretical density of + UO\ :sub:`2`\ (NO\ :sub:`3`)\ :sub:`2` from the Standard Composition + Library for your density multiplier. The + UO\ :sub:`2`\ (NO\ :sub:`3`)\ :sub:`2` listed there is the solid, not + the solution. + +.. + + The solution specification data follow: + +:: + + SOLUTION MIX=1 RHO[UO2(NO3)2] = 415 92235 92.6 92238 5.9 + 92234 1.0 92236 0.5 + MOLAR [HNO3] = 9.783-3 + TEMP = 293 DENSITY = 1.555 END SOLUTION + +.. centered:: Comparison of number densities from the two methods + +The number densities of methods 1 and 2 should agree within the limits +of the input data. The density multipliers in method 1 are 5 digits and +the density multipliers in method 2 are 4 digits. Therefore, the number +densities calculated by the two methods should agree to 4 or 5 digits. + ++----------------+--------------+--------------+ +| | Method 1 | Method 2 | ++----------------+--------------+--------------+ +| Nuclide number | Atom density | Atom density | ++----------------+--------------+--------------+ +| 92235 | 9.84603E−04 | 9.84603E−04 | ++----------------+--------------+--------------+ +| 92238 | 6.19415E−05 | 6.19415E−05 | ++----------------+--------------+--------------+ +| 92234 | 1.06784E−05 | 1.06784E−05 | ++----------------+--------------+--------------+ +| 92236 | 5.29387E−06 | 5.29387E−06 | ++----------------+--------------+--------------+ +| 07014 | 2.13092E−03 | 2.13092E−03 | ++----------------+--------------+--------------+ +| 08016 | 3.74135E−02 | 3.7410E−02 | ++----------------+--------------+--------------+ +| 01001 | 5.77973E−02 | 5.77983E−02 | ++----------------+--------------+--------------+ + +.. _7-1c-6: + +Multiple unit cells in a single problem +--------------------------------------- + +Consider a problem that involves three different UO\ :sub:`2` fuel +assemblies: a 1.98%-enriched assembly, a 2.64%-enriched assembly, and a +2.96%-enriched assembly. All fuel rods are UO\ :sub:`2` at +10.138 g/cm\ :sup:`3` and are 0.94 cm in diameter. The Zircaloy-4 clad +has an inside radius of 0.4875 cm and an outside radius of 0.545 cm. The +rod pitch is 1.44 cm. Each fuel assembly is a 15 × 15 array of fuel pins +with water holes, instrumentation holes, and burnable poison rods. For +cross-section processing, the presence of the water holes, +instrumentation holes, and burnable poison rods in the assemblies are +ignored. + +The following XSProc input use the CENTRM/PMC method for self-shielding +three latticecells with different fuel enrichments. The remaining +mixture (SS-304), not specified in a unit cell, is processed as an +infinite homogeneous medium using the BONAMI method. Each mixture can +appear only in a single zone of one unit cell. For square pitch +latticecells the default CENTRM transport solver is MoC with P0 scatter; +however in this input, the solver for the 3\ :sup:`rd` cell is modified +through CENTRM DATA to use the two-region approximation for the CE +calculation [npxs=5], and discrete S\ :sub:`N` transport calculation +with P1 anisotropic scatteringfor the MG solutions in the fast and +thermal energy ranges [nfst=0, nthr=0]. + +:: + + DEMONSTRATION PROBLEM WITH MULTIPLE RESONANCE CORRECTIONS REQUIRED + broad_n + READ COMP + UO2 1 .925 300 92235 1.98 92238 98.02 END + UO2 2 .925 300 92235 2.64 92238 97.36 END + UO2 3 .925 300 92235 2.96 92238 97.04 END + ZIRC4 4 1.0 300 END + H2O 5 1.0 300 END + ZIRC4 6 1.0 300 END + H2O 7 1.0 300 END + ZIRC4 8 1.0 300 END + H2O 9 1.0 300 END + SS304 10 1.0 300 END + END COMP + READ CELLDATA + LATTICECELL SQUAREPITCH PITCH=1.44 5 FUELD=0.94 1 CLADD=1.09 4 GAPD=0.975 0 END + LATTICECELL SQUAREPITCH PITCH=1.44 7 FUELD=0.94 2 CLADD=1.09 6 GAPD=0.975 0 END + LATTICECELL SQUAREPITCH PITCH=1.44 9 FUELD=0.94 3 CLADD=1.09 8 GAPD=0.975 0 END + CENTRM DATA npxs=5 nthr=0 nfst=0 isct=1 END CENTRM DATA + END CELLDATA + +.. _7-1c-7: + +Multiple fissile mixtures in a single unit cell +----------------------------------------------- + +The following problem involves large units having the bulk of their +fissile material more than one mean-free path away from the surface of +the unit. The interaction between the units that occurs in the resonance +range is a very small fraction of the total interaction because an +overwhelming percentage of the interaction occurs deep within each unit. +Therefore, the resonance range interaction between the units can be +ignored, and the default infinite homogeneous medium cross-section +processing in the resonance range can be considered adequate for this +particular application. + +Consider a problem that consists of four 20.96-kg 93.2%-enriched uranium +metal cylinders, density 18.76 g/cm\ :sup:`3`, and four 5-liters +Plexiglas bottles filled with highly enriched uranyl nitrate solution at +415 g/L, a specific gravity of 1.555, and 0.39 mg of excess nitrate per +gram of solution. The isotopic content of the uranium metal is 93.2 wt % +:sup:`235`\ U, 5.6 wt % :sup:`238`\ U, 1.0 wt % :sup:`234`\ U, and +0.2 wt % :sup:`236`\ U. The uranium isotopic content of the nitrate +solution is 92.6 wt % :sup:`235`\ U, 5.9 wt % :sup:`238`\ U, 1.0 wt % +:sup:`234`\ U and 0.5 wt % :sup:`236`\ U. The size of the metal +cylinders is between 10 and 12 cm in diameter and height, and the size +of the nitrate solution is on the order of 16 and 20 cm in diameter and +height. The average mean-free path in the uranium metal is on the order +of 1.5 cm, and the average mean free path in the nitrate solution is on +the order of 0.5 cm. Therefore, infinite homogeneous medium is an +appropriate choice for this problem and the use of CENTRM/PMC is valid. + +See Examples 1–4 of :ref:`7-1a-2` for data input details for the +Plexiglas and uranium metal. See Example 1 of :ref:`7-1a-5` for data +input details for the uranyl nitrate solution. The XSProc data for this +problem follow: + +:: + + SET UP 4 AQUEOUS 4 METAL + fine_n + READ COMP + URANIUM 1 0.985 293 92235 93.2 92238 5.6 92234 1.0 92236 0.2 END + SOLUTION 2 RHO[UO2(NO3)2]=415 92235 92.6 92238 5.9 92234 1.0 92236 0.5 + MOLAR[HNO3]=9.783-3 DENSITY=1.555 TEMPERATURE=293 END SOLUTION + PLEXIGLAS 3 END + END COMP + +Consider the same materials above except rearrange them so that a 10 cm +diameter uranium metal sphere sits inside a 50 cm diameter spherical +tank of uranyl nitrate solution having a 1-cm thick Plexiglas wall. This +problem can be modeled in SCALE but only CENTRM/PMC will treat the +resonance processing correctly. This problem is modeled below. + +:: + + SET UP 4 AQUEOUS 4 METAL + fine_n + READ COMP + URANIUM 1 0.985 293 92235 93.2 92238 5.6 92234 1.0 92236 0.2 END + SOLUTION 2 RHO[UO2(NO3)2]=415 92235 92.6 92238 5.9 92234 1.0 92236 0.5 + MOLAR[HNO3]=9.783-3 DENSITY=1.555 TEMPERATURE=293 END SOLUTION + PLEXIGLAS 3 END + END COMP + READ CELLDATA + MULTIREGION SPHERICAL END 1 5.0 2 25.0 3 26.0 END ZONE + END CELLDATA + +.. _7-1c-8: + +Cell weighting an infinite homogeneous problem +---------------------------------------------- + +Cell weighting an infinite homogeneous medium has no effect on the +cross sections because there is only one zone and one set of +cross sections. However, a cell-weighted mixture number can still be +supplied using the keyword **CELLMIX**\ = followed by an unique mixture +number. This cell-weighted mixture number can be used in subsequent +codes and will produce results similar to the cross sections of the +original mixture. + +EXAMPLE 1 + +This problem would probably be run with CSAS1 to provide the k-infinity +of 20%-enriched UO\ :sub:`2`. + +:: + + 20% ENRICHED UO2 BILLET + fine_n + READ COMP + UO2 1 0.99 293 92235 20 92238 80 END + END COMP + READ CELLDATA + INFHOMMEDIUM 1 CELLMIX=100 END + END CELLDATA + +.. _7-1c-9: + +Cell weighting a LATTICECELL problem +------------------------------------ + +Cell weighting used with a **LATTICECELL** problem creates cell-weighted +homogeneous cross sections that represent the characteristics of the +heterogeneous unit cell. This cell-weighted mixture can then be used in +a subsequent code for the overall volume where the cells are located +without having to mock up the actual 3-D heterogeneous array of cells. +This cell-weighted homogeneous mixture is designated by the user with +the keyword **CELLMIX**\ = immediately followed by an unused mixture +number. This needs to follow immediately after the cell description. +Note that the mixtures used in the unit cell data cannot be used in a +subsequent code because they have been flux weighted to create the user +specified mixture. Therefore, if a mixture used in the unit cell +description is also to be used in a subsequent code, another mixture +must be created that is identical except for the mixture number. Every +mixture that is to be used in a subsequent code except zero (i.e., void) +must be defined in the standard composition data. + +A byproduct of the cell-weighting calculation is the eigenvalue +(k-effective) of an infinite array of the cell described as the unit +cell. + +EXAMPLE 1 + +Consider a cylindrical stainless steel tank filled with spherical +pellets of 2.67%-enriched UO\ :sub:`2` arranged in a close-packed +“triangular” pitch, flooded with borated water at 4350 ppm. The +cylindrical stainless tank is sitting in a larger tank filled with +borated water at 4350 ppm. + +The data for the UO\ :sub:`2` and borated water were developed in detail +in Example 3 of :ref:`7-1c-2`. The stainless steel must be defined, and +mixture 3 was chosen because mixture 1 was the UO\ :sub:`2` and +mixture 2 was the borated water. Because the borated water will be used +as a reflector for the stainless steel tank and has been used in the +unit cell data, it must be repeated with a different mixture number (in +this case, as mixture 4). + +In the subsequent calculation, user specified cell mixture 100 will be +used to represent the UO\ :sub:`2` pellets in the borated water, +mixture 3 will represent the stainless steel tank, and mixture 4 will +represent the borated water reflector around the stainless-steel tank. + +The XSProc data for creating the cell-weighted cross sections on +mixture 100 follow: + +:: + + SPHERICAL PELLETS IN BORATED WATER + fine_n + READ COMP + UO2 1 .9398 293. 92235 2.67 92238 97.33 END + ATOMH3BO3 2 0.025066 3 5000 1 1001 3 8016 3 1.0 293 END + H2O 2 0.984507 293 END + SS304 3 1.0 293 END + ATOMH3BO3 4 0.025066 3 5000 1 1001 3 8016 3 1.0 293 END + H2O 4 0.984507 293 END + END COMP + READ CELLDATA + LATTICECELL SPHTRIANGP PITCH 1.0724 2 FUELD 1.0724 1 CELLMIX=100 END + END CELLDATA + +.. _7-1c-10: + +Cell weighting a MULTIREGION problem +------------------------------------ + +A **MULTIREGION** problem is cell weighted primarily to obtain a +cell-weighted homogeneous cross section that represents the +characteristics of the heterogeneous unit cell. The eigenvalue obtained +for a **MULTIREGION** problem with cylindrical or spherical geometry +having a white boundary condition specified on the right boundary +approximates an infinite array of the cells. A vacuum boundary condition +would represent a single cell. A slab with reflected boundary conditions +for both boundaries represents an infinite array of slab cells. The +cell-weighted cross sections for spherical or cylindrical geometries +with a white right boundary condition do not use a Dancoff correction +and thus may not be accurate for representing a large array of the +specified units. + + +EXAMPLE 1 + + +Consider a small, highly enriched uranium sphere supported by a +Plexiglas collar in a tank of water. The uranium metal sphere has a +diameter of 13.1075 cm, is 97.67% enriched, and has a density of +18.794 g/cm\ :sup:`3`. The cylindrical Plexiglas collar has a 4.1275-cm +radius central hole, extends to a radius of 12.7 cm and is 2.54 cm +thick. The water-filled tank is 60 cm in diameter. + +The Plexiglas collar is not significantly different from water and does +not surround the fuel, so it will be ignored. Because this makes the +problem a 1-D geometry, it can be defined using the **MULTIREGION** type +of calculation and the eigenvalue of the system can be obtained without +additional data by executing CSAS1 with CENTRM/PMC, if PARM=CENTRM is +specified on the command line. The abundance of uranium is not stated +beyond 97.67% enriched, so assume the remainder is :sup:`238`\ U. The +XSProc data follow: + +:: + + =CSAS5 + SMALL WATER REFLECTED SPHERE ON PLEXIGLAS COLLAR + fine_n + READ COMP + URANIUM 1 DEN=18.794 1 293. 92235 97.67 92238 2.33 END + H2O 2 END + END COMP + READ CELLDATA + MULTI SPHERICAL CELLMIX=100 END 1 6.5537 2 30.0 END ZONE + END CELLDATA + • + • + • + KENO DATA THAT USES MIX=100 FOR A HOMOGENEOUS SPHERE OF 30-CM RADIUS GOES HERE. + • + • + END diff --git a/_build/doctrees/BONAMI.doctree b/_build/doctrees/BONAMI.doctree new file mode 100644 index 0000000000000000000000000000000000000000..39991f6191fba56dcb7b169b3acf174728bc6ff6 GIT binary patch literal 223085 zcmeFa3!EJ1RVON0mMoWL$FJCqW8!jR%*=RZG!IF(-CaF1vShmhp3!tyeUI~iv`{4bDN3gKQZtN@Ceb)m=4^AB! ztw)Wu=3MFG_@c-!G=f@H1E;jTw1YM^fG@Wv-FD603QEV@p$Q#r5o?NkYR8! zKb8aNz(cR@_R#*)O}sxEk@t&z-YJluu=n6y$ES}Syn9+~ zV7K*wSNV)eX5<&$M$KLH>+`jG#j&ue;FSx_vPZZZx%kqpYmAHtBO`^8qpqB}HrMciYUC`| zYjb74G7?ma{-Tc`)rMQB*P_Ukw15@ju&ufDSnj-6Xw>SMuwJV;*hjP62o~zK=At_= 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