PMC.rst.txt 58.5 KB
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L. Williams, D. F. Hollenbach, U. Merteryuk* ABSTRACT PMC generates problem-dependent multigroup cross sections from an existing multigroup cross-section library, a pointwise nuclear data library, and a pointwise neutron flux file produced by the CENTRM continuous-energy transport code. In the SCALE sequences, PMC is a computational module called from XSProc to produce self-shielded multigroup (MG) cross-sections over a specified energy range (e.g., resolved resonance range) of individual nuclides in the system of interest. The self-shielded cross sections are obtained by integrating the pointwise (PW) nuclear data using the CENTRM problem-specific, continuous-energy flux as a weight function for each spatial mixture in the system. Several options are available in PMC to specify various types of weighting methods for the one-dimensional and two-dimensional MG data. PMC outputs problem-dependent self-shielded cross sections that can be used in XSDRNPM, KENO, NEWT or other MG transport codes. ACKNOWLEDGMENTS The authors acknowledge the suggestions and direct contributions of L. M. Petrie of Oak Ridge National Laboratory, and former ORNL staff N. M. Greene, and R. M. Westfall. .. _7-5-1: Introduction ------------ PMC is a computational module used for the CENTRM/PMC self-shielding method performed by the XSProc driver module [see :ref:7-1]. It can also be run in standalone mode. PMC (**P**\ roduce **M**\ ultigroup **C**\ ross sections) computes multigroup (MG) cross sections by utilizing the pointwise (PW) neutron spectra calculated in CENTRM :cite:williams_computation_1995 to weight cross sections in a continuous-energy (CE) library file. This provides problem-dependent, self-shielded MG data representative of the fine-structure variation in the neutron energy spectrum for the system of interest. PMC only computes shielded cross sections within the energy interval of the CENTRM PW flux calculation, defined by the energy limits DEMIN and DEMAX. By default the lower limit is DEMIN=0.001 eV, and the upper energy is DEMAX=20,000 eV; however these parameters can be modified by the user in the CENTRM DATA input block. Outside of this energy interval, the shielded cross sections previously computed with the Bondarenko method in BONAMI are retained. PMC is automatically called from the XSProc driver module during execution of a SCALE sequence, and the resulting zone-averaged, problem-dependent cross sections can be passed to MG transport solvers (e.g., KENO, NEWT, XSDRNPM, etc.) called by the sequence. .. _7-5-1-1: Description of PMC input nuclear data ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The nuclear data input to PMC consists of both MG and CE cross sections. Input MG data are obtained from the MG library specified in the sequence input. During SCALE execution, CE data used by both CENTRM and PMC are prepared by the code CRAWDAD (:ref:7-7), which reads CE files for individual nuclides, interpolates the data to the appropriate temperatures for the specified mixtures, and concatenates the data into a one problem-specific, multiple-nuclide CENTRM PW library. In general each nuclide has its own unique energy mesh defined such that the cross section at any energy value can be interpolated linearly from the library point data to accuracy better than 0.1%. Although cross sections in the original CE data files include values over the full energy range of 0-20 MeV, CRAWDAD reduces the energy range to interval of the CENTRM PW calculation (i.e., DEMIN→DEMAX). It is this combined PW library that is accessed by PMC. The format of the CENTRM PW library is described in :ref:7-4. .. _7-5-1-2: Description of PMC input pointwise flux data ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In addition to the input nuclear data, PMC also requires PW flux values calculated in CENTRM to be provided. Depending on the CENTRM transport approximation, the flux data includes the PW scalar flux spectrum as a function of energy and spatial-zone, and also may include PW spherical harmonic moments of the angular flux (e.g., the current), which can be used in processing MG scattering matrices for higher-order Legendre moments. The non-uniform energy-mesh of the PW flux is determined during the CENTRM calculation in order to represent the spectrum variation with a minimum number of energy points. Like the CE cross section data, the flux spectrum at any energy value can be obtained within a specified tolerance by linear interpolation of the PW flux values. .. _7-5-2: Code Features ------------- Two types of MG data are processed by PMC: 1-D cross sections and 2-D scatter matrices. The 1-D cross sections are weighted-average values over each energy group, by nuclide and reaction type. If there are “G” energy groups on the input library, then the 1-D cross section for each reaction type can be viewed as a 1-D vector with G values (of course some may be zero). Depending on the options and PW energy range specified, PMC will generally only re-compute and replace some of the G-group data. The 2‑D cross sections correspond to group-to-group transfers (and corresponding Legendre moments) associated with various types of scatter reactions. These data can be arranged into a 2-D G by G matrix. For most materials this matrix is quite sparse. The 2-D data depend not only on the cross-section data, but also on the energy/angular distributions of the secondary neutrons, which are represented by Legendre moments. PMC always re-normalizes the 2-D elastic and inelastic scattering matrices (including moments) to be consistent with the respective self-shielded 1-D data. In the case of elastic scattering, PMC also has rigorous options that can be used to modify the secondary energy distribution to account for self-shielding effects, such as by correcting the group removal cross section. .. _7-5-2-1: Options for treatment of 1-D cross sections ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ PMC computes new MG data for each reaction type (MT) and each nuclide on the input MG library, which also has CE data on the CENTRM PW library. Cross sections for reactions on the input MG library which do not have corresponding PW reaction data are not replaced; i.e., the original MG values are retained. SCALE CE library files for individual nuclides contain all reaction types included in the ENDF/B data; however the CRAWDAD module, executed prior to PMC, only includes certain ones when it produces the problem-specific CENTRM library. By default the CENTRM PW nuclear data library always includes cross sections for the total (MT-1), radiative capture (MT-102), and elastic scattering reactions (MT-2) of all nuclides; as well as fission (MT-18), and prompt, delayed, and total-nubar values (MTs-456, 455, 452, respectively) for fissionable nuclides. The (n,alpha) cross sections (MT-107) for B-10 and Li-6 are also always included if these nuclides are present in a mixture. If the CENTRM PW transport calculation includes the inelastic scattering option, indicated by CENTRM input parameter nmf6 >= 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