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E. Peplow and C. Celik* Monaco is a general-purpose, fixed-source, Monte Carlo shielding code for the SCALE package. It is a functional module that uses either AMPX cross sections or continuous energy libraries to calculate neutron and photon fluxes and responses to specific geometry regions, to point detectors and to mesh tallies. Basic multigroup transport methods are inherited from Monaco’s predecessor, MORSE. Continuous energy physics has been incorporated into the code with a new physics package that uses the same CE data as CE-KENO-VI, with extensions for simulating photons. Variance reduction capabilities include source biasing and weight windows, either by geometry region or by using a mesh-based importance map. User input includes the cross section file unit number; the geometry description using the SCALE General Geometry Package; source description as a function of position, energy, and direction; tally descriptions (fluxes in which regions, at what point detectors, or over what mesh grids); and response functions (functions of energy). Output consists of tables detailing the region and point detector fluxes (and their responses), as well as files for mesh tallies. Introduction ------------ Monaco is a neutron/photon, fixed-source Monte Carlo shielding code for the SCALE code package. Monaco uses the SCALE General Geometry Package (SGGP)---the same geometry description as KENO-VI. Monaco has many options available to the user for specifying source distributions, many tally options, and many variance reduction capabilities. Monaco was originally based on the MORSE Monte Carlo code but has been extensively modified to modernize the coding, increase the number of capabilities in terms of sources and tallies, and allow for either multigroup or continuous energy (CE) transport through the use of the new SCALE CE Modular Physics Package (SCEMPP). Monaco was developed to address a number of long-term goals for the Monte Carlo shielding capabilities in SCALE. The principal goals for this project included (1) unification of geometric descriptions between the SCALE shielding and criticality Monte Carlo codes, (2) implementation of a mesh-based importance map and mesh-based biased source distribution so that automated variance reduction could be used, and (3) establishment of a code using modern programming practices from which to continue future development. The addition of a continuous-energy transport capability is a significant change as well. Monaco is the key component of the MAVRIC sequence, which also uses Denovo to create the mesh-based importance map and mesh-based biased source distribution for general 3-D automated variance reduction. See the MAVRIC chapter for more information. Monaco Capabilities ------------------- Monaco has a wide range of source descriptions and tallies for performing general radiation transport calculations. Note that Monaco can work with either the AMPX-based multigroup libraries or the newer AMPX-based CE libraries. Note that for CE calculations, tallies still employ a multigroup energy structure to store and report results. Source Descriptions ~~~~~~~~~~~~~~~~~~~ Multiple sources can be defined for a Monaco calculation. Sampling of the different sources can be biased by the user. Each source is specified by its spatial distribution, its energy distribution, its directional distribution, and its strength. Distributions defined by the user can also be biased and can be used multiple times by different sources. The Monaco tallies assume that the sources all have units of particles/second. If the source strengths are given in other units, the user will have to incorporate the proper conversion to the tally results and remember to interpret the results accordingly. Distributions ^^^^^^^^^^^^^ Two types of basic distributions are used by Monaco – binned histograms and a set of value/function pairs. The binned histogram type is defined by :math:n + 1 bin boundaries and *n* values, representing the integrated amount in each bin. For the true distribution\ :math:f(x), the bin boundaries :math:\left\lbrack x_{0},\ x_{1},\ \ldots,\ x_{n} \right\rbrack and the integrated amounts :math:F_{i} = \ \int_{x_{i - 1}}^{x_{i}}{f\left( x \right)\text{dx}} are given. The distribution will be normalized by Monaco after reading. The user can optionally bias a binned histogram distribution by supplying one of the following: the biased sampling distribution amounts, :math:G_{i} = \ \int_{x_{i - 1}}^{x_{i}}{g\left( x \right)\text{dx}}; the importance of each bin, :math:I_{i}; or the suggested weight for each bin, :math:w_{i}. Based on what type of input is given, Monaco will compute a properly normalized probability distribution function for sampling. If the importances are given, the sampling distribution is computed as .. math:: :label: Monaco-1 G_{i} = \frac{I_{i}F_{i}}{\sum_{i}^{}{I_{i}F_{i}}} If suggested weights are given, then the sampling distribution is computed as .. math:: :label: Monaco-2 G_{i} = \frac{\frac{F_{i}}{w_{i}}}{\sum_{i}^{}\frac{F_{i}}{w_{i}}} for bins with non-zero weight. The sampling distribution for bins with a suggested weight of zero are set to :math:G_{i} = \ 0. When sampled, particles are assigned a weight of :math:\frac{F_{i}}{G_{i}}. The second type of distribution that a user can define is for a series of point values of a function. For a set of :math:n + 1 point pairs, :math:\left( x_{i},\ f_{i} \right) for :math:i \in \left\lbrack 0\ldots n \right\rbrack, defining :math:n intervals, a distribution can be made by linearly interpolating between adjacent point pairs. This type of distribution can also be biased by supplying one of the following: the biased sampling distribution function value :math:g_{i} at each point, the importance of each point, :math:I_{i}; or the suggested weight for each point, :math:w_{i}. Similar to above, if importances or weights are given, Monaco computes the biased distribution for sampling. For the value/function point pairs type of distribution, the weight assigned to the sampled particle is a continuous function. Some commonly used distributions are built into Monaco and can be used by simple keywords. Monaco can produce a graph of any distribution so that the user can verify that the input was entered correctly. Spatial energy and directional attributes ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Each Monaco source is described by three separable components: spatial, energy and directional. The spatial component of a source in Monaco is simple but very flexible. First, the general shape of the source region is defined in global coordinates. The basic solid shapes and their allowed degenerate cases are listed in :numref:Monaco-tab1. The user can reference any of the defined distributions to describe the source distribution in any coordinate (*x*, *y*, and *z* for cuboids, *r* and *z* for cylinders and *r* for spheres) to use for sampling or leave the source distribution as uniform over each dimension for the solid shape. The source region can be limited by the underlying SGGP geometry variables of unit, media, and mixture. This way, source volumes (or planes, lines, or points) can be defined that are independent or dependent on the model geometry. A cylinder or cylindrical shell region can be oriented with its axis in any direction. .. _Monaco-tab1: .. table:: Available source shapes and their allowed degenerate cases +-----------------------------------+-----------------------------------+ | **Shape** | **Allowable degenerate cases** | +===================================+===================================+ | cuboid | rectangular plane, line, point | +-----------------------------------+-----------------------------------+ | cylinder | circular plane, line, point | +-----------------------------------+-----------------------------------+ | cylindrical shell | cylinder, planar annulus, | | | circular plane, cylindrical | | | surface, line, ring, point | +-----------------------------------+-----------------------------------+ | sphere | point | +-----------------------------------+-----------------------------------+ | spherical shell | sphere, spherical surface, point | +-----------------------------------+-----------------------------------+ Monaco samples the source position using either the given distributions or uniformly over the basic solid shape and then uses rejection if any of the optional SGGP geometry limiters have been specified. For sources that are confined to a particular unit, media, or mixture, users should make sure the basic solid shape tightly bounds the desired region for efficient sampling. For the energy component of each source, either type of distribution described above can be used. Biasing can be used in the energy component of the source as well. The Watt spectrum is a built-in distribution which uses the Froehner and Spencer :cite:froehner_method_1981 method for sampling. If the defined energy distribution has point(s) that are out of the problem’s energy range for a CE problem, these points will be rejected in the source energy sampling and an error message will be generated. The warnings will be suppressed if the number of rejected source points exceeds a pre-defined threshold (1000). Distributions can be used to define the directional component of the source. A function of the cosine of the polar angle, with respect to some reference direction in global coordinates, can be used by Monaco. If no directional distribution is specified, the default is an isotropic distribution (one directional bin from *µ*\ = −1 to *µ*\ =1). The default reference direction is the positive *z*-axis (<0,0,1>). Monaco mesh source map files ^^^^^^^^^^^^^^^^^^^^^^^^^^^^ An alternative to specifying the separate spatial and energy distributions, a Monaco mesh source file can be used. A mesh source consists of a 3D Cartesian mesh that overlay the geometry. Each mesh cell has some probability of emitting a source particle, and within each mesh cell, a different energy distribution can be sampled. Position within each mesh cell is sampled uniformly, and the emission direction is sampled from the standard directional distribution. Monaco mesh source files are typically produced by the MAVRIC sequence or by other Monaco calculations (see the mesh source saver option in the source input). For a source constructed from the separable spatial and energy distributions, Monaco can create a mesh source file which can then be visualized using the Mesh File Viewer. This is a convenient way to ensure that the source being used is what was intended. Tallies ~~~~~~~ Monaco offers three tally types: point detectors, region tallies, and mesh tallies. Each is useful in determining quantities of interest in the simulation. Any number of each can be used, up to the limit of machine memory. The tallies will compute flux for each group, the total neutron and total photon fluxes, and any number of dose-like responses. A typical dose-like response, *R*, is the integral over energy of the product of a response function, :math:f\left( E \right), and the flux, :math:\phi\left( E \right). .. math:: :label: Monaco-3 R = \int_{}^{}{f\left(E \right)\phi \left( E \right)\ } dE In multigroup calculations, the total response would be expressed as the sum over all groups :math:R = \sum_{}^{}{f_{g}\phi_{g}}. For CE calculations, tallies can be segmented into energy and time bins which can be thought of as “groups”. All three of the tally types can be scaled with a constant – for example, to account for units conversions. Tally statistics ^^^^^^^^^^^^^^^^ The three Monaco tallies are really just collections of simple and extended tallies for each group, each total, and each group contribution to a response or total response. The simple tally works in the following way: a history score :math:h_{i} is zeroed out at the start of history :math:i. During the course of the history, when an event occurs during substep :math:j, a score consisting of some contribution :math:c_{\text{ij}} weighted by the current particle weight :math:w_{\text{ij}} is calculated and added to :math:h_{i}. At the end of the history, the history score is the total weighted score for each substep :math:j in the history. .. math:: :label: Monaco-4 h_{i} = \sum_{j}^{}w_{\text{ij}}c_{\text{ij}} Note that the values for the contribution :math:c_{\text{ij}} and when it is added to the accumulator are determined by the tally type. At the end of the each history, the history score is added to two accumulators (power sums) - the first accumulator is for finding the tally average, :math:S_{1}, and the second accumulator is for finding the uncertainty in the tally average, :math:S_{2}. .. math:: :label: Monaco-5 S_{1} = \ \sum_{i}^{}h_{i}^{\ } .. math:: :label: Monaco-6 S_{2} = \ \sum_{i}^{}h_{i}^{2} At the end of all :math:N histories, the second sample central moment is found from the power sums .. math:: :label: Monaco-7 m_{2} = \frac{S_{2}}{N} - \ \frac{S_{1}^{2}}{N^{2}} and then the tally average is computed as :math:\overline{x} = \frac{S_{1}}{N} and the uncertainty in the tally average is :math:u = \sqrt{\frac{m_{2}}{N}}. The extended tally uses four accumulators – the first and second are the same as the simple tally – with the third and fourth accumulators used for finding the variance of the variance (VOV). These extra accumulators, :math:S_{3} and :math:S_{4}, are calculated as .. math:: :label: Monaco-8 S_{3} = \ \sum_{i}^{}h_{i}^{3} .. math:: :label: Monaco-9 S_{4} = \ \sum_{i}^{}h_{i}^{4} At the end of all :math:N histories, the tally average :math:\overline{x}\ \ and uncertainty in the tally average :math:u are found in the same way as a simple tally. For the VOV calculation, the third and fourth sample central moments are found as At the end of all :math:N histories, the tally average :math:\overline{x}\ \ and uncertainty in the tally average :math:u are found in the same way as a simple tally. For the VOV calculation, the third and fourth sample central moments are found as .. math:: :label: Monaco-10 m_{3} = \frac{S_{3}}{N} - \frac{3S_{1}S_{2}}{N^{2}} + \frac{2S_{1}^{3}}{N^{3}} .. math:: :label: Monaco-11 m_{4} = \frac{S_{4}}{N} - \frac{4S_{1}S_{3}}{N^{2}} + \ \frac{6S_{1}^{2}S_{2}}{N^{3}} - \ \frac{3S_{1}^{4}}{N^{4}} and then the VOV :cite:pederson_confidence_1997 and figure-of-merit (FOM) are found using .. math:: :label: Monaco-12 \mathrm{\text{VOV}} = \frac{m_{4} - \ m_{2}^{2}}{Nm_{2}^{2}} .. math:: :label: Monaco-13 \mathrm{\text{FOM}} = \ \frac{1}{\left( \frac{u}{\overline{x}} \right)^{2} \ T} where *T* is the calculation time (in minutes). Extended tallies are used for the total neutron flux, total photon flux and any responses for the Monaco tallies. Simple tallies are used for each group’s flux and each group’s contribution to a response. Detailed, group-wise results for each tally are saved to separate files at the end of each batch of particles. Users can view these files (in the SCALE temporary directory) as the Monaco simulation progresses. Summaries of the extended tallies appear in the final Monaco output file. Statistical tests ^^^^^^^^^^^^^^^^^ Statistical tests are performed on the extended tallies at the end of each batch. Results for each batch are stored in files and the results for the final batch are shown in the main output tally summary. The six tests are: +-------------+-------------+-------------+-------------+------------------+ | | **Quantity**| **Test** | **Goal** | **Within** | | | | | | | +=============+=============+=============+=============+==================+ | 1. | mean | relative | = 0.00 | ±0.10 | | | | slope of | | | | | | linear fit | | | +-------------+-------------+-------------+-------------+------------------+ | 2. | standard | exponent of | = -0.50 | .. math:: | | | deviation | power fit | | R^{2} > 0.99 | +-------------+-------------+-------------+-------------+------------------+ | 3. | relative | final value | < 0.05 | | | | uncertainty | | | | +-------------+-------------+-------------+-------------+------------------+ | 4. | relative | exponent of | = -1.00 | .. math:: | | | VOV | power fit | | R^{2} > 0.95 | +-------------+-------------+-------------+-------------+------------------+ | 5. | relative | final value | < 0.10 | | | | VOV | | | | +-------------+-------------+-------------+-------------+------------------+ | 6. | figure-of-m | relative | = 0.00 | ±0.10 | | | erit | slope of | | | | | | linear fit | | | +-------------+-------------+-------------+-------------+------------------+ For the tests that are fit to a function with respect to batch (1, 2, 4, and 6), only the last half of the simulation is used. The basis for these tests is that in a well-behaved Monte Carlo, the mean should not increase or decrease as a function of the number of histories (:math:N), the standard deviation should decrease with :math:\frac{1}{\sqrt{N}}, the variance of the variance should decrease with :math:\frac{1}{N} and the figure-of-merit should neither increase or decrease as a function of the number of histories (proportional to time). For tests 2 and 4, the coefficient of determination, :math:R^{2}, from a forced fit to a function with the right exponent is used as the tally test. Point detector tallies ^^^^^^^^^^^^^^^^^^^^^^ Point detectors are a form of variance reduction in computing the flux or response at a specific point. At the source emission site and at every interaction in the particle’s history, an estimate is made of the probability of the particle striking the position of the point detector. For each point detector, Monaco tallies the uncollided and total flux for each energy group, the total for all neutron groups, and the total for all photon groups. Any number of optional dose-like responses can be calculated as well. Multigroup .......... After a source particle of group *g* is started, the distance *R* between the source position and the detector position is calculated. Along the line connecting the source and detector positions, the sum of the distance *s\ j* through each region *j* multiplied by the total cross section :math:\Sigma_{j}^{g}\ for that region is also calculated. The contribution *c\ g* to the uncollided flux estimator is then made to the tally for group *g*. .. math:: :label: Monaco-14 c_{g} = \frac{1}{4\pi R^{2}}\mathrm{\exp}\left( - \sum_{j}^{}{s_{j}\Sigma_{j}^{g}} \right) Continuous Energy ................. After a source particle with energy *E* is started, the distance *R* between the source position and the detector position is calculated. For each bin :math:g of the tally energy structure, a specific energy :math:E_{g} is sampled uniformly within the bin. Along the line connecting the source and detector positions, the sum of the distance *s\ j* through each region *j* multiplied by the total cross section :math:\Sigma_{j}\left( E_{g} \right) for that region. The contribution *c\ g* to the uncollided flux estimator is then made to the tally for group *g*. total cross section :math:\Sigma_{j}\left( E \right) : .. math:: :label: Monaco-15 c_{g} = \frac{1}{4\pi R^{2}}\mathrm{\exp}\left( - \sum_{j}^{}{s_{j}\Sigma_{j}\left( E \right)} \right) Only source particles contribute to the uncollided flux tally. At each interaction point during the life of the particle, similar contributions are made to each of the tallies. For each group *g′* that the particle could scatter into and reach the detector location, a contribution is made that also includes the probability to scatter from the current direction towards the detector and having the energy change from group *g* to group *g′.* This type of tally is costly, since ray-tracing through the geometry from the current particle position to the detector location is required many times over the particle history. Point detectors should be located in regions made of void material, so that contributions from interactions arbitrarily close to the point detector cannot overwhelm the total estimated flux (as :math:\frac{1}{4\pi R^{2} \rightarrow \infty}). Care must be taken in using point detectors in deep penetration problems to ensure that the entire phase space that could contribute has been well sampled—so that the point detector is not underestimating the flux by leaving out areas far from the source but close to the point detector position. One way to check this is by examining how the tally average and uncertainty change with each batch of particles used in the simulation. Large fluctuations in either quantity could indicate that the phase space is not being sampled well. Region tallies ^^^^^^^^^^^^^^ Region tallies are used for calculating the flux and/or responses over one of the regions listed in the SGGP geometry. Both the track-length estimate of the flux and the collision density estimate of the flux are calculated—and for each, the region tally contains simple tallies for finding flux in each group, the total neutron flux, and the total photon flux. For each of the optional response functions, the region tally also contains simple tallies for each group and the total response. For the track-length estimate of flux, each time a particle of energy :math:E moves through the region of interest, a contribution of :math:l (the length of the step in the region) is made to the history score for the simple tally for flux for tally group \ *g*. The same contribution is made for the history score for the simple tally for total particle flux, neutron or photon, depending on the particle type. If any optional response functions were requested with the tally, then the contribution of :math:\text{lf}\left( E \right)\ is made for the response group, where :math:f\left( E \right) is the response function value for energy :math:E. The history score for the total response function is also incremented using :math:\text{lf}\left( E \right). At the end of all of the histories, the averages and uncertainties of all of the simple tallies for fluxes are found for every group and each total. These results then represent the average track-length over the region. To determine flux, these results are divided by the volume of the region. If the volume :math:V of the region was not given in the geometry input nor calculated by Monaco, then the tally results will be just the average track lengths and their uncertainties. A reminder message is written to the tally detail file if the volume of the region was not set. For the collision density estimate of the flux, each time a particle of energy :math:E has a collision in the region of interest, a contribution of :math:\frac{1}{\Sigma} (the reciprocal of the total macroscopic cross section) is made to the history scores for the simple tally for flux for tally energy group *g* and for the total particle flux. At the end of the simulation, the averages and uncertainties of all of the simple tallies for every group flux and total flux are found and then divided by the region volume, if available. Similar to the point detector tallies, region tallies produce a file listing the tally average and uncertainty at the end of each batch of source particles (a \*.chart file). This file can be plotted using the simple 2-D plotter (ChartPlot) to observe the tally convergence behavior. Mesh tallies ^^^^^^^^^^^^ For a D Cartesian mesh or a cylindrical mesh (independent of the SGGP geometry), Monaco can calculate the track-length estimate of the flux. Since the number of cells (voxels) in a mesh can become quite large, the mesh tallies are not updated at the end of each history but are instead updated at the end of each batch of particles. This prevents the mesh tally accumulation from taking too much time but means that the estimate of the statistical uncertainty is slightly low. Like the other tallies, mesh tallies can calculate optional response functions. Since a mesh tally consists of many actual tallies, the statistical tests are a bit more complex than for the region and point detector tallies. Several statistical quantities and tests are used in Monaco similar to those in several recent studies :cite:kiedrowski_statistical_2011,kiedrowski_evaluating_2011 which look at a distribution of relative variances over the mesh tally. In Monaco, the basis of the statistical tests center on the distribution of relative uncertainties and its mean, :math:\overline{r}, of the voxels (:math:V) with score. .. math:: :label: Monaco-16 \overline{r} = \frac{1}{V}\sum_{}^{}R_{i} where :math:R_{i} is the relative uncertainty of the flux or dose in voxel :math:i. If every voxel has been sampled well and its relative uncertainty :math:R_{i} \propto \frac{1}{\sqrt{N}}, then the mean relative uncertainty of the voxels should also behave as :math:\frac{1}{\sqrt{N}}. The variance of the mean relative uncertainty can be calculated and a figure of merit (FOM) for the mesh tally can be constructed using .. math:: :label: Monaco-17 FOM = \frac{1}{{\overline{r}}^{2}T} with the time\ :math:\text{\ T} in minutes. The four tests measure over the simulation: 1) if :math:\zeta, the fraction of voxels with non-zero score, is constant; 2) if the mean relative uncertainty is decreasing as :math:\frac{1}{\sqrt{N}} (as measured by the coefficient of determination, :math:R^{2}, of a fit to a curve with power of -0.5); 3) if the variance of the mean relative uncertainty is decreasing with :math:\frac{1}{N}; and 4) if the FOM is constant. +----+------------------------------------------------------------+------------------------------+----------+---------------------------+ | | Quantity | Test | Goal | Within | +====+============================================================+==============================+==========+===========================+ | 1. | :math:\zeta, fraction with score | relative slope of linear fit | = 0.00 | ±0.10 | +----+------------------------------------------------------------+------------------------------+----------+---------------------------+ | 2. | :math:\overline{r}, mean relative uncertainty | exponent of power fit | = -0.50 | .. math:: R^{2} > 0.99 | +----+------------------------------------------------------------+------------------------------+----------+---------------------------+ | 3. | variance of :math:\overline{r} | exponent of power fit | = -1.00 | .. math:: R^{2} > 0.95 | +----+------------------------------------------------------------+------------------------------+----------+---------------------------+ | 4. | figure-of-merit | exponent of power fit | = 0.00 | ±0.10 | +----+------------------------------------------------------------+------------------------------+----------+---------------------------+ For non-uniform meshes (especially cylindrical), these tests may not be the best measure of performance since different size voxels will have a wider variety of relative uncertainties. The user is also cautioned that if there are individual voxels within the mesh tally that have relative uncertainties that are not decreasing as :math:\frac{1}{\sqrt{N}}, then the mesh tally statistical tests will not be meaningful. It is ultimately up to the user to decide if the mesh tally is performing well (is the goal of the mesh tally just to calculate dose, not flux?; are all spatial areas of the mesh tally equally important?; are all magnitudes of the flux or response values equally important?; etc.) Mesh tallies can be viewed with the Mesh File Viewer, a Java utility that can be run from GeeWiz (on PC systems) or can be run separately (on any system). The Mesh File Viewer will show the flux for each group, the total flux for each type of particle and the optional responses. Uncertainties and relative uncertainties can also be shown for mesh tallies using the Mesh File Viewer. For more information on the Mesh File Viewer, see its on-line documentation. Continuous Energy Transport ~~~~~~~~~~~~~~~~~~~~~~~~~~~ Using multigroup data in Monte Carlo transport calculations is generally sufficient for most problems (both shielding and criticality). Many of the reaction cross sections vary slowly with energy, so energy “groups” can be made with one set of properties for the group. Multigroup treatments can further simplify radiation transport by combining the different types of reactions that can occur into a simple scattering matrix – particles then have certain probabilities to scatter from their current energy group to another energy group. If the user is not interested in knowing which specific type of interaction happened at each collision, this simplification can increase calculation efficiency. One major drawback of the multigroup approach is in representing discrete gammas, such as the decay radiation from common isotopic sources. Consider a simple shielding simulation using cobalt-60. This isotope gives off two high-energy gamma rays when it decays (1173230 eV with intensity 99.85% and 1332490 eV with intensity 99.9826%). In the SCALE multigroup calculations, a cobalt-60 source spectrum is represented by a broad pdf, controlled by the group structure. This is shown in :numref:fig8-1. for the fine 47-group structure and the broad 19-group structure. .. _fig8-1: .. figure:: figs/Monaco/Picture1.png :width: 400 :align: center The multigroup representation of a cobalt-60 source. Note that in both group structures, 1.33 MeV is a group boundary, so the 1332490 eV line is represented by group that covers higher energies. The cross section for that group is lower than the cross section for the specific line, so multigroup transport calculations will tend to overestimate the number of photons penetrating a shield, which will overestimate dose rates. Using CE and the two multigroup libraries, the total cross sections for the cobalt lines are listed in :numref:tab8-2. :numref:fig8-2. shows the total cross section of photons in tungsten, in both CE and the two SCALE multigroup structures. On the whole, the multigroup data represents the CE data well. :numref:fig8-3. shows the same cross section information near the two cobalt lines, which shows how the multigroup cross sections average over quite large energy ranges. .. _tab8-2: .. table:: Total macroscopic cross section in tungsten (/cm). :align: center +----------+------------+------------+ | | 1173230 eV | 1332490 eV | +==========+============+============+ | SCALE CE | 1.03353 | 0.94864 | +----------+------------+------------+ | SCALE 47 | 1.09066 | 0.92743 | +----------+------------+------------+ | SCALE 19 | 1.05167 | 0.89289 | +----------+------------+------------+ The small differences in cross section can make large differences in the transport. Consider just 5 cm of tungsten. Using the cross sections in :numref:tab8-2, the attenuation (:math:e^{- \mu x}) of either line can vary by 30%. In addition to source representation problems, multigroup transport is not adequate for applications where line spectra are measured. Because of the group structure, tally results will be averaged out within a group. With the fixed boundaries, specific lines in the tallies will not be able to be seen. For examples, in the 19-group library, there is no group around the 511 keV annihilation gammas – they are averaged in with other photons from 400 to 600 keV. No multigroup structure could contain thin groups around every line of interest. .. _fig8-2: .. figure:: figs/Monaco/8-2.png :width: 500 :align: center Photon total cross section in tungsten. The energies of the cobalt-60 are displayed as lines at 1173230 and 1332490 eV. .. _fig8-3: .. figure:: figs/Monaco/8-3.png :width: 500 :align: center Photon total cross section in tungsten, near the cobalt lines. The energies of the cobalt-60 are displayed as lines at 1173230 and 1332490 eV. A sample problem involving a cobalt source and a slab of tungsten will compare the use of continuous-energy transport to multigroup transport, to demonstrate the large difference in results for single-line sources. For distributions, differences between multigroup and continuous-energy may not be very significant. Monaco Input Files ------------------ The input file for Monaco consists of two lines of text (“=monaco” command line and one for the problem title) and then several blocks, with each block starting with “read xxxx” and ending with “end xxxx”. There are three blocks that are required and seven blocks that are optional. The cross section and geometry blocks must be listed first and in the specified order. Other blocks may be listed in any order. Blocks (must be in this order): - Cross Sections – (required) lists the cross-section file and the mixing table information - Geometry – (required) SCALE general geometry description - Array – optional addition to the above geometry description - Volume – optional calculation or listing of region volumes - Plot – create 2D slices of the SGGP geometry Other Blocks (any order, following the blocks listed above): - Definitions – defines locations, response functions, grid geometries, cylindrical mesh geometries, energy bin boundaries, time bin boundaries and various distributions used by other blocks - Source – (required) description of multiple sources; with the spatial, energy, and directional distributions and particle type for each - Tallies – description of what to calculate: point detector tallies, region tallies, or mesh tallies - Parameters – how to perform the simulation (random number seed, how many histories, etc.) - Biasing – data for reducing the variance of the simulation The physical model blocks (Geometry, Array, Volume and Plot) follow the standard SCALE format. See the other SCALE references as noted in the following sections for details. For the other six blocks, scalar variables are set by “keyword=value”, fixed length arrays are set with “keyword value\ :sub:1 ... value\ :sub:N\ ”, variable length arrays are set with “keyword value\ :sub:1 ... value\ :sub:N end”, and some text and filenames are read in as quoted strings. Single keywords to set options are also used in some instances. The indention, comment lines, and upper/lower case shown in this document are not required—they are used in the examples only for clarity. Except for strings in quotes (like filenames), SCALE is not case sensitive. After all of the blocks are listed, a single line with “end data” should be listed. A final “end” should also be listed, to signify the end of all Monaco input. See :numref:tab8-3 for an overview of the Monaco input file structure. Cross sections block ~~~~~~~~~~~~~~~~~~~~ Monaco does its own mixing, so it needs a mixing table. For each element of each mixture, an identifier and a number density must be supplied. These can be found in the output of whatever sequence was used to make the cross-section file, such as CSAS-MG. Two coupled neutron/photon multigroup libraries were created specifically for shielding problems from ENDF/B-VII.0 data—the v7-200n47g fine-group and the v7-27n19g coarse-group libraries. CE libraries made from ENDF/BVII.0 are also available in SCALE. .. _tab8-3: .. list-table:: Overall input format for Monaco :align: center * - Input File - Comment * - .. code:: scale =monaco Some title for this problem read crossSections ... end crossSections read geometry ... end geometry read array ... end array read volume ... end volume read plot ... end plot read definitions ... end definitions read sources ... end sources read tallies ... end tallies read parameters ... end parameters read biasing ... end biasing end data end - .. code:: scale name of sequence title List of isotopes/mixtures [required block] SCALE SGGP geometry [required block] SCALE SGGP arrays [optional block] SCALE SGGP volume calc [optional block] SGGP Plots [optional block] Definitions [possibly required] Sources definition [required block] Tally specifications [optional block] Monte Carlo parameters [optional block] Biasing information [optional block] end of all blocks end of Monaco input For example, if CSAS-MG was used to produce an AMPX file using the following input, .. code:: scale =csas-mg Demonstration problem, three mixtures v7-200n47g read composition uo2 1 0.2 293.0 92234 0.0055 92235 3.5 92238 96.4945 end ss304 2 1.0 293.0 end h2o 4 1.0 293.0 end end composition end in addition to creating an AMPX file, the output would include a tables similar to .. code:: scale m i x i n g t a b l e (THREAD = 00 ) entry mixture isotope number density new identifier explicit temperature 1 1 92234 2.73451E-07 92234 293.0 2 1 92235 1.73272E-04 92235 293.0 3 1 92238 4.71674E-03 92238 293.0 4 1 8016 9.78057E-03 8016 293.0 m i x i n g t a b l e (THREAD = 00 ) entry mixture isotope number density new identifier explicit temperature 1 2 6000 3.18488E-04 6000 293.0 2 2 14028 1.57010E-03 14028 293.0 3 2 14029 7.97625E-05 14029 293.0 4 2 14030 5.26416E-05 14030 293.0 5 2 15031 6.94688E-05 15031 293.0 6 2 24050 7.59178E-04 24050 293.0 7 2 24052 1.46400E-02 24052 293.0 8 2 24053 1.66006E-03 24053 293.0 9 2 24054 4.13224E-04 24054 293.0 10 2 25055 1.74072E-03 25055 293.0 11 2 26054 3.42190E-03 26054 293.0 12 2 26056 5.37166E-02 26056 293.0 13 2 26057 1.24055E-03 26057 293.0 14 2 26058 1.65094E-04 26058 293.0 15 2 28058 5.26873E-03 28058 293.0 16 2 28060 2.02951E-03 28060 293.0 17 2 28061 8.82212E-05 28061 293.0 18 2 28062 2.81288E-04 28062 293.0 19 2 28064 7.16357E-05 28064 293.0 m i x i n g t a b l e (THREAD = 00 ) entry mixture isotope number density new identifier explicit temperature 1 4 1001 6.67531E-02 1001 293.0 2 4 8016 3.33765E-02 8016 293.0 which can be used to construct the Monaco cross-section block mixing table. .. highlight:: scale :: read crossSections ampxFileUnit=4 mixture 1 element 92234 2.73451E-07 element 92235 1.73272E-04 element 92238 4.71674E-03 element 8016 9.78057E-03 end mixture mixture 2 element 6000 3.18488E-04 element 14028 1.57010E-03 element 14029 7.97625E-05 element 14030 5.26416E-05 element 15031 6.94688E-05 element 24050 7.59178E-04 element 24052 1.46400E-02 element 24053 1.66006E-03 element 24054 4.13224E-04 element 25055 1.74072E-03 element 26054 3.42190E-03 element 26056 5.37166E-02 element 26057 1.24055E-03 element 26058 1.65094E-04 element 28058 5.26873E-03 element 28060 2.02951E-03 element 28061 8.82212E-05 element 28062 2.81288E-04 element 28064 7.16357E-05 end mixture mixture 4 element 1001 6.67531E-02 element 8016 3.33765E-02 end mixture end crossSections For a CE calculation, instead of the keyword “ampxFileUnit=” (which refers to a given AMPX library), the keyword “ceLibrary=” should be used with a CE library name, enclosed in quotes. Also for CE, a default temperature can be set before any mixtures are defined using the “ceTempDefault=” temperature (in Kelvins). With each mixture, a specific temperature can be set using “temperature.” Other keywords that can be used in the cross-section block for multigroup problems include flags to turn on printing of different aspects of the cross-section mixing process (“printTotals”, “printScatters”, “printAngleProb”, “printFissionChi”, “printExtra”, and “printLegendre”). The keyword “fullyCoupled” can be used to specify all groups to be treated as primary groups. These keywords do not work in CE problems since the point wise data contain an enormous number of points. Users are encouraged to use Monaco by running the MAVRIC sequence, which creates the cross-section mixing table automatically, for both multigroup and CE calculations. Geometry block ~~~~~~~~~~~~~~ The geometry input uses the standard SGGP, similar to KENO-VI. Input instructions can be found in *Geometry Data* in the KENO-VI chapter of the SCALE manual. Shielding calculations (Monaco, MAVRIC, SAS4) differ from their criticality cousins (KENO V.a, KENO-VI) in a very special way—sources and detectors can be located outside of the materials where the transport takes place. To accommodate this fact in Monaco and MAVRIC, make sure that a void region (a media record using mixture 0) surrounds the source area and any point detectors, if they are not located in a region of the actual geometry. For example, if the objective is to calculate the effectiveness of a simple slab shield, the model geometry would consist of just one slab of material. The source would be on one side of the slab, and a detector would be on the other side of the slab. In Monaco (and the MAVRIC sequence), the input should list at least two regions: (1) the slab itself and (2) a void region outside of the slab containing both the source and detector positions. Monaco tracks particles through the SGGP geometry as well as other geometries used for mesh tallies or mesh importance maps. Because Monaco must track through all of these geometries at the same time, users should not use the reflective boundary capability in the SGGP geometry. The graphical user interfaces GeeWiz and Keno3D can be used on Windows platforms to develop and view the geometry. Array, volume, and plot blocks ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Geometry array input uses the standard SGGP, similar to KENO-VI. Input instructions can be found in KENO-VI chapter on *Array Data* of the SCALE manual. Volumes of various geometry regions are used to calculate fluxes for those regions. Volumes can be input as part of the geometry input block above, or calculated by the SGGP using one of two different methods. See 
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