sphinx.addnodesdocument)}( rawsourcechildren](docutils.nodestarget)}(h.. _7-5:h]
attributes}(ids]classes]names]dupnames]backrefs]refidid1utagnameh
lineKparenthhhsource-/Users/john/Documents/SCALE-test/docs/PMC.rstubh section)}(hhh](h title)}(h^PMC: A Program to Produce Multigroup Cross Sections Using Pointwise Energy Spectra from CENTRMh]h Text^PMC: A Program to Produce Multigroup Cross Sections Using Pointwise Energy Spectra from CENTRM}(hh,h h*hhh!NhNubah}(h]h]h]h]h]uhh(h h%hhh!h"hKubh paragraph)}(h0*M. L. Williams, D. F. Hollenbach, U. Merteryuk*h]h emphasis)}(hh>h]h/.M. L. Williams, D. F. Hollenbach, U. Merteryuk}(hhh hBubah}(h]h]h]h]h]uhh@h hh j`ubh/X), 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
}(hX), 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
h j`hhh!NhNubh)}(h
:ref:`7-4`h]h)}(hjh]h/7-4}(hhh jubah}(h]h](hČstdstd-refeh]h]h]uhhh jubah}(h]h]h]h]h]refdochь refdomainjreftyperefrefexplicitrefwarnh7-4uhhh!h"hK>h j`ubh/.}(h.h j`hhh!NhNubeh}(h]h]h]h]h]uhh:h!h"hK>h jOhhubh)}(h.. _7-5-1-2:h]h}(h]h]h]h]h]hid5uhh
hKNh jOhhh!h"ubeh}(h](%description-of-pmc-input-nuclear-datajNeh]h](%description of pmc input nuclear data7-5-1-1eh]h]uhh#h hhhh!h"hK<expect_referenced_by_name}jjDsexpect_referenced_by_id}jNjDsubh$)}(hhh](h))}(h,Description of PMC input pointwise flux datah]h/,Description of PMC input pointwise flux data}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!h"hKQubh;)}(hXIn 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.h]h/XIn 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.}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!h"hKSh jhhubh)}(h
.. _7-5-2:h]h}(h]h]h]h]h]hid6uhh
hK_h jhhh!h"ubeh}(h](,description-of-pmc-input-pointwise-flux-datajeh]h](,description of pmc input pointwise flux data7-5-1-2eh]h]uhh#h hhhh!h"hKQj}jjsj}jjsubeh}(h](introductionheh]h](introduction7-5-1eh]h]uhh#h h%hhh!h"hK"j}jhsj}hhsubh$)}(hhh](h))}(h
Code Featuresh]h/
Code Features}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!h"hKbubh;)}(hXTwo 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.h]h/XTwo 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.}(hj$h j"hhh!NhNubah}(h]h]h]h]h]uhh:h!h"hKdh jhhubh)}(h.. _7-5-2-1:h]h}(h]h]h]h]h]hid7uhh
hKxh jhhh!h"ubh$)}(hhh](h))}(h+Options for treatment of 1-D cross sectionsh]h/+Options for treatment of 1-D cross sections}(hj@h j>hhh!NhNubah}(h]h]h]h]h]uhh(h j;hhh!h"hK{ubh;)}(hXPMC 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.h]h/XPMC 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.}(hjNh jLhhh!NhNubah}(h]h]h]h]h]uhh:h!h"hK}h j;hhubh;)}(hXPW 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.h]h/XPW 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.}(hj\h jZhhh!NhNubah}(h]h]h]h]h]uhh:h!h"hKh j;hhubh;)}(hX3PMC 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).h]h/X3PMC 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).}(hjjh jhhhh!NhNubah}(h]h]h]h]h]uhh:h!h"hKh j;hhubh;)}(hXThe 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`.h](h/:math:`\widetilde{\sigma}_{\mathrm{g}, \mathrm{g}}^{(\infty)}`h]h/6\widetilde{\sigma}_{\mathrm{g}, \mathrm{g}}^{(\infty)}}(hhh jubah}(h]h]h]h]h]uhjlh j~ubh/i is the infinitely dilute within-group cross section based on
s-wave kinematics, which is computed from }(hi is the infinitely dilute within-group cross section based on
s-wave kinematics, which is computed from h j~ubh)}(h
:eq:`eq7-5-4`h]jU)}(hjh]h/eq7-5-4}(hhh jubah}(h]h](hČeqeh]h]h]uhjTh jubah}(h]h]h]h]h]refdochь refdomainjlreftypejrefexplicitrefwarnheq7-5-4uhhh!h"hMh j~ubh/& using an infinitely
dilute spectrum}(h& using an infinitely
dilute spectrumh j~ubeh}(h]h]h]h]h]uhh:h!h"hMh jFubh;)}(h: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.h](j)}(h?:math:`\widetilde{\sigma}_{\mathrm{g}, \mathrm{g}}^{(\varphi)}`h]h/7\widetilde{\sigma}_{\mathrm{g}, \mathrm{g}}^{(\varphi)}}(hhh jubah}(h]h]h]h]h]uhjlh jubh/M is the self-shielded within-group based on s-wave kinematics,
computed from }(hM is the self-shielded within-group based on s-wave kinematics,
computed from h jubh)}(h
:eq:`eq7-5-4`h]jU)}(hjh]h/eq7-5-4}(hhh jubah}(h]h](hČeqeh]h]h]uhjTh jubah}(h]h]h]h]h]refdochь refdomainjlreftypejrefexplicitrefwarnheq7-5-4uhhh!h"hMh jubh/ using Φ(E) →CENTRM PW flux.}(h using Φ(E) →CENTRM PW flux.h jubeh}(h]h]h]h]h]uhh:h!h"hMh jFubeh}(h]h]h]h]h]uhjh jhhh!h"hNubh;)}(hXIf 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)}`.h](h/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
}(hIf the effects of resonance self-shielding are small, then there will be
little change in the original within-group value, since in this case
h j hhh!NhNubj)}(h:math:`\widetilde{\sigma}_{\mathrm{g}, \mathrm{g}}^{(\varphi)} \sim \widetilde{\mathrm{O}}_{\mathrm{g}, \mathrm{g}}^{(\infty)}`h]h/w\widetilde{\sigma}_{\mathrm{g}, \mathrm{g}}^{(\varphi)} \sim \widetilde{\mathrm{O}}_{\mathrm{g}, \mathrm{g}}^{(\infty)}}(hhh j ubah}(h]h]h]h]h]uhjlh j ubh/.}(hjh j hhh!NhNubeh}(h]h]h]h]h]uhh:h!h"hMh jhhubh;)}(hRThe P\ :sub:`0` group-to-group out-scatter terms for N2D=-1 are scaled
as follows:h](h/The P }(hThe P\ h j& hhh!NhNubj)}(h:sub:`0`h]h/0}(hhh j/ ubah}(h]h]h]h]h]uhjh j& ubh/C group-to-group out-scatter terms for N2D=-1 are scaled
as follows:}(hC group-to-group out-scatter terms for N2D=-1 are scaled
as follows:h j& hhh!NhNubeh}(h]h]h]h]h]uhh:h!h"hMh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq7-5-6uhh
h jhhh!h"hNubj)}(hXn\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}}h]h/Xn\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}}}(hhh jR ubah}(h]jQ ah]h]h]h]docnamehьnumberKlabeleq7-5-6nowrapjjuhjh!h"hMh jhhj}j}jQ jH subh;)}(hX$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.h](h/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 }(hAgain 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\ h jg hhh!NhNubj)}(h:sub:`0`h]h/0}(hhh jp ubah}(h]h]h]h]h]uhjh jg ubh/^
scatter matrix is renormalized to correspond to the self-shielded 1-D
scatter cross section.}(h^
scatter matrix is renormalized to correspond to the self-shielded 1-D
scatter cross section.h jg hhh!NhNubeh}(h]h]h]h]h]uhh:h!h"hMh jhhubh;)}(hXEFor 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`h](h/WFor the option N2D=1, an entirely new PL elastic scattering matrix is
computed. The l }(hWFor the option N2D=1, an entirely new PL elastic scattering matrix is
computed. The l\ h j hhh!NhNubj)}(h :sub:`th`h]h/th}(hhh j ubah}(h]h]h]h]h]uhjh j ubh/ 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, }(h 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, h j hhh!NhNubh)}(hbell_nuclear_1970h]h)}(hj h]h/[bell_nuclear_1970]}(hhh j ubah}(h]h]h]h]h]uhhh j ubah}(h]id15ah]j.ah]h]h] refdomainj3reftypej5 reftargetj refwarnsupport_smartquotesuhhh!h"hMh j hhubeh}(h]h]h]h]h]uhh:h!h"hMh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq7-5-7uhh
h jhhh!h"hNubj)}(hX!\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}}h]h/X!\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}}}(hhh j ubah}(h]j ah]h]h]h]docnamehьnumberKlabeleq7-5-7nowrapjjuhjh!h"hMh jhhj}j}j j subh;)}(hXEwhere 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:h](h/
where σ }(h
where σ\ h j hhh!NhNubj)}(h:sub:`z`h]h/z}(hhh j ubah}(h]h]h]h]h]uhjh j ubh/ (E) is the CE elastic cross-section data from the
CENTRM nuclear data file, evaluated at the appropriate temperature for
zone z; }(h\ (E) is the CE elastic cross-section data from the
CENTRM nuclear data file, evaluated at the appropriate temperature for
zone z;\ h j hhh!NhNubj)}(h:math:`f_{l}^{j}`h]h/ f_{l}^{j}}(hhh j
ubah}(h]h]h]h]h]uhjlh j ubh/Z (E′→E) is the secondary neutron energy distribution
from elastic scattering; and Φ }(hZ (E′→E) is the secondary neutron energy distribution
from elastic scattering; and Φ\ h j hhh!NhNubj)}(h
:sub:`l,z`h]h/l,z}(hhh j
ubah}(h]h]h]h]h]uhjh j ubh/B (E) is the lth PW flux
moment averaged over zone Z. PMC assumes }(hB\ (E) is the lth PW flux
moment averaged over zone Z. PMC assumes h j hhh!NhNubhA)}(h*s*h]h/s}(hhh j)
ubah}(h]h]h]h]h]uhh@h j ubh/a-wave scattering from
stationary nuclei to evaluate the scattering distribution, and uses the
P }(ha-wave scattering from
stationary nuclei to evaluate the scattering distribution, and uses the
P\ h j hhh!NhNubj)}(h:sub:`0`h]h/0}(hhh j<
ubah}(h]h]h]h]h]uhjh j ubh/ 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:}(h 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:h j hhh!NhNubeh}(h]h]h]h]h]uhh:h!h"hMh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq7-5-8uhh
h jhhh!h"hNubj)}(hX\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}}h]h/X\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}}}(hhh j_
ubah}(h]j^
ah]h]h]h]docnamehьnumberKlabeleq7-5-8nowrapjjuhjh!h"hMh jhhj}j}j^
jU
subh;)}(hXhere 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,h](h/here P }(hhere P\ h jt
hhh!NhNubhA)}(h*l*h]h/l}(hhh j}
ubah}(h]h]h]h]h]uhh@h jt
ubh/ is the }(h is the h jt
hhh!NhNubhA)}(h*l*h]h/l}(hhh j
ubah}(h]h]h]h]h]uhh@h jt
ubh/ }(h\ h jt
hhh!NhNubj)}(h :sub:`th`h]h/th}(hhh j
ubah}(h]h]h]h]h]uhjh jt
ubh/# order Legendre polynomial; and
G }(h# order Legendre polynomial; and
G\ h jt
hhh!NhNubj)}(h:sup:`j`h]h/j}(hhh j
ubah}(h]h]h]h]h]uhjh jt
ubh/ 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 }(h 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\ h jt
hhh!NhNubj)}(h:sup:`j`h]h/j}(hhh j
ubah}(h]h]h]h]h]uhjh jt
ubh/7. The kinematics function for nuclide j is
defined as,}(h7. The kinematics function for nuclide j is
defined as,h jt
hhh!NhNubeh}(h]h]h]h]h]uhh:h!h"hMh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq7-5-9uhh
h jhhh!h"hNubj)}(h\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}}} ,h]h/\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}}} ,}(hhh j
ubah}(h]j
ah]h]h]h]docnamehьnumberK labeleq7-5-9nowrapjjuhjh!h"hMh jhhj}j}j
j
subh;)}(hXwhere 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.h](h/ where G }(h where G\ h jhhh!NhNubj)}(h:sup:`j`h]h/j}(hhh j
ubah}(h]h]h]h]h]uhjh jubh/ (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 }(h\ (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 h jhhh!NhNubh)}(hbucholz_method_1978h]h)}(hjh]h/[bucholz_method_1978]}(hhh j!ubah}(h]h]h]h]h]uhhh jubah}(h]id16ah]j.ah]h]h] refdomainj3reftypej5 reftargetjrefwarnsupport_smartquotesuhhh!h"hMh jhhubh/. Integration over the initial group (g′) is then
performed numerically using the same method as for evaluating the
problem-dependent 1-D cross sections.}(h. Integration over the initial group (g′) is then
performed numerically using the same method as for evaluating the
problem-dependent 1-D cross sections.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!h"hMh jhhubh;)}(hsOption N2D=2 adds the following term to the diagonal of the *l*\ :sub:`th`
moment of the PL elastic scatter matrix,h](h/