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/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.h](h/Option N2D=1 uses the CENTRM PW flux to recompute the entire set of
group-to-group scatter data (including Legendre moments) assuming
}(hOption N2D=1 uses the CENTRM PW flux to recompute the entire set of
group-to-group scatter data (including Legendre moments) assuming
h jhhh!NhNubhA)}(h*s*h]h/s}(hhh jubah}(h]h]h]h]h]uhh@h jubh/-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 }(h-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 h jhhh!NhNubhA)}(h*s*h]h/s}(hhh jubah}(h]h]h]h]h]uhh@h jubh/-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.}(h-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.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!h"hM+h j
hhubh;)}(hX5A 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.h](h/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 }(hA 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 h jhhh!NhNubh)}(hbell_nuclear_1970h]h)}(hjh]h/[bell_nuclear_1970]}(hhh jubah}(h]h]h]h]h]uhhh jubah}(h]id11ah]j.ah]h]h] refdomainj3reftypej5 reftargetjrefwarnsupport_smartquotesuhhh!h"hM5h jhhubh/XM 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 }(hXM 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\ h jhhh!NhNubj)}(h :sub:`th`h]h/th}(hhh jubah}(h]h]h]h]h]uhjh jubh/ Legendre moment of the PW flux.}(h Legendre moment of the PW flux.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!h"hM5h j
hhubh;)}(hX7Option 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.h](h/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 }(hOption 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\ h jhhh!NhNubj)}(h:sub:`0`h]h/0}(hhh j
ubah}(h]h]h]h]h]uhjh jubh/ 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.}(h 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.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!h"hM>h j
hhubh;)}(hXnThe 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.h]h/XnThe 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.}(hj%h j#hhh!NhNubah}(h]h]h]h]h]uhh:h!h"hMDh j
hhubh)}(h
.. _7-5-3:h]h}(h]h]h]h]h]hid12uhh
hMKh j
hhh!h"ubeh}(h](+options-for-treatment-of-2-d-cross-sectionsjeh]h](+options for treatment of 2-d cross sections7-5-2-4eh]h]uhh#h jhhh!h"hMj}jBjsj}jjsubeh}(h](
code-featuresjeh]h](
code features7-5-2eh]h]uhh#h h%hhh!h"hKbj}jMjsj}jjsubh$)}(hhh](h))}(h:Calculation of Problem-Dependent Multigroup Cross Sectionsh]h/:Calculation of Problem-Dependent Multigroup Cross Sections}(hjWh jUhhh!NhNubah}(h]h]h]h]h]uhh(h jRhhh!h"hMNubh)}(h.. _7-5-3-1:h]h}(h]h]h]h]h]hid13uhh
hMPh jRhhh!h"ubh$)}(hhh](h))}(h1-D cross sectionsh]h/1-D cross sections}(hjsh jqhhh!NhNubah}(h]h]h]h]h]uhh(h jnhhh!h"hMSubh)}(hhh]h}(h]h]h]h]h]hequation-eq7-5-1uhh
h jnhhh!h"hNubh
math_block)}(h\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}}h]h/\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}}}(hhh jubah}(h]jah]h]h]h]docnamehьnumberKlabeleq7-5-1nowrap xml:spacepreserveuhjh!h"hMUh jnhhj}j}jjsubh;)}(hwhereh]h/where}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!h"hMZh jnhhubh block_quote)}(hhh](h;)}(h+Φ\ :sub:`z,g` is the multigroup zone flux,h](h/Φ }(hΦ\ h jubj)}(h
:sub:`z,g`h]h/z,g}(hhh jubah}(h]h]h]h]h]uhjh jubh/ is the multigroup zone flux,}(h is the multigroup zone flux,h jubeh}(h]h]h]h]h]uhh:h!h"hM\h jubh;)}(hIσ\ :sup:`j`\ :sub:`z,r,g` is the zone-average, group cross section, andh](h/σ }(hσ\ h jubj)}(h:sup:`j`h]h/j}(hhh jubah}(h]h]h]h]h]uhjh jubh/ }(h\ h jubj)}(h:sub:`z,r,g`h]h/z,r,g}(hhh jubah}(h]h]h]h]h]uhjh jubh// is the zone-average, group cross section, and}(h/ is the zone-average, group cross section, andh jubeh}(h]h]h]h]h]uhh:h!h"hM^h jubh;)}(h2∆E\ :sub:`g` is the energy interval of group g.h](h/∆E }(h∆E\ h jubj)}(h:sub:`g`h]h/g}(hhh jubah}(h]h]h]h]h]uhjh jubh/$ is the energy interval of group g.}(h$ is the energy interval of group g.h jubeh}(h]h]h]h]h]uhh:h!h"hM`h jubeh}(h]h]h]h]h]uhjh jnhhh!h"hNubh;)}(hXThe 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.h](h/The integration in }(hThe integration in h j4hhh!NhNubh)}(h
:eq:`eq7-5-1`h]h literal)}(hj?h]h/eq7-5-1}(hhh jCubah}(h]h](hČeqeh]h]h]uhjAh j=ubah}(h]h]h]h]h]refdochь refdomainmathreftypejMrefexplicitrefwarnheq7-5-1uhhh!h"hMbh j4ubh/Xy 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. }(hXy 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. h j4hhh!NhNubh)}(h
:eq:`eq7-5-1`h]jB)}(hjeh]h/eq7-5-1}(hhh jgubah}(h]h](hČeqeh]h]h]uhjAh jcubah}(h]h]h]h]h]refdochь refdomainjYreftypejqrefexplicitrefwarnheq7-5-1uhhh!h"hMbh j4ubh/XR 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.}(hXR 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.h j4hhh!NhNubeh}(h]h]h]h]h]uhh:h!h"hMbh jnhhubh)}(h.. _7-5-3-2:h]h}(h]h]h]h]h]hid14uhh
hMnh jnhhh!h"ubeh}(h](d-cross-sectionsjmeh]h](1-d cross sections7-5-3-1eh]h]uhh#h jRhhh!h"hMSj}jjcsj}jmjcsubh$)}(hhh](h))}(h2-D scattering cross sectionsh]h/2-D scattering cross sections}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!h"hMqubh;)}(hX: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:h](h/XXThe 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 }(hXXThe 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\ h jhhh!NhNubj)}(h :sub:`th`h]h/th}(hhh jubah}(h]h]h]h]h]uhjh jubh/ 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:}(h 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:h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!h"hMsh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq7-5-2uhh
h jhhh!h"hNubj)}(h\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}h]h/\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}}(hhh jubah}(h]jah]h]h]h]docnamehьnumberKlabeleq7-5-2nowrapjjuhjh!h"hMh jhhj}j}jjsubh;)}(hXSwhere 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.<h](h/where the subscripts “ }(hwhere the subscripts “\ h jhhh!NhNubhA)}(h*orig*h]h/orig}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ ” and “ }(h\ ” and “\ h jhhh!NhNubhA)}(h*new,*h]h/new,}(hhh jubah}(h]h]h]h]h]uhh@h jubh/X ” 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.}(hX\ ” 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.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!h"hMh jhhubh;)}(hXThe 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:h](h/(The option N2D= −1 recomputes the P }(h(The option N2D= −1 recomputes the P\ h j)hhh!NhNubj)}(h:sub:`0`h]h/0}(hhh j2ubah}(h]h]h]h]h]uhjh j)ubh/X] 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 }(hX] 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 h j)hhh!NhNubhA)}(h*s*h]h/s}(hhh jEubah}(h]h]h]h]h]uhh@h j)ubh/T-wave scatter, as done
for N2D=1. Suppressing the zone index for simplicity, the P }(hT-wave scatter, as done
for N2D=1. Suppressing the zone index for simplicity, the P\ h j)hhh!NhNubj)}(h:sub:`0`h]h/0}(hhh jXubah}(h]h]h]h]h]uhjh j)ubh/
within-group XS is defined as:}(h
within-group XS is defined as:h j)hhh!NhNubeh}(h]h]h]h]h]uhh:h!h"hMh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq7-5-3uhh
h jhhh!h"hNubj)}(h\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}}h]h/\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}}}(hhh j{ubah}(h]jzah]h]h]h]docnamehьnumberKlabeleq7-5-3nowrapjjuhjh!h"hMh jhhj}j}jzjqsubh;)}(hwhere 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,h](h/ where p }(h where p\ h jhhh!NhNubj)}(h:sub:`r`h]h/r}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (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 }(h\ (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 h jhhh!NhNubhA)}(h*s*h]h/s}(hhh jubah}(h]h]h]h]h]uhh@h jubh/'-wave scattering this equation becomes,}(h'-wave scattering this equation becomes,h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!h"hMh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq7-5-4uhh
h jhhh!h"hNubj)}(hX+\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}}h]h/X+\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}}}(hhh jubah}(h]jah]h]h]h]docnamehьnumberKlabeleq7-5-4nowrapjjuhjh!h"hMh jhhj}j}jjsubh;)}(hhThe N2D= −1 option recomputes a modified P\ :sub:`0` within-group
cross section from the expression,h](h//The N2D= −1 option recomputes a modified P }(h/The N2D= −1 option recomputes a modified P\ h jhhh!NhNubj)}(h:sub:`0`h]h/0}(hhh jubah}(h]h]h]h]h]uhjh jubh/1 within-group
cross section from the expression,}(h1 within-group
cross section from the expression,h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!h"hMh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq7-5-5uhh
h jhhh!h"hNubj)}(h\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}}h]h/\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}}}(hhh jubah}(h]jah]h]h]h]docnamehьnumberKlabeleq7-5-5nowrapjjuhjh!h"hMh jhhj}j}jjsubh;)}(hwhereh]h/where}(hj'h j%hhh!NhNubah}(h]h]h]h]h]uhh:h!h"hMh jhhubj)}(hhh](h;)}(h(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;h](h/(σ }(h(σ\ h j6ubj)}(h
:sub:`g,g`h]h/g,g}(hhh j?ubah}(h]h]h]h]h]uhjh j6ubh/) }(h)\ h j6ubj)}(h:sub:`orig`h]h/orig}(hhh jRubah}(h]h]h]h]h]uhjh j6ubh/ is the original within-group
cross section on the MG library, based on actual kinematics and weighted
with an infinitely dilute spectrum;}(h is the original within-group
cross section on the MG library, based on actual kinematics and weighted
with an infinitely dilute spectrum;h j6ubeh}(h]h]h]h]h]uhh:h!h"hMh j3ubh;)}(h: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 spectrumh](h jY)}(h>:math:`\widetilde{\sigma}_{\mathrm{g}, \mathrm{g}}^{(\infty)}`h]h/6\widetilde{\sigma}_{\mathrm{g}, \mathrm{g}}^{(\infty)}}(hhh jpubah}(h]h]h]h]h]uhjYh jkubh/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 jkubh)}(h
:eq:`eq7-5-4`h]jB)}(hjh]h/eq7-5-4}(hhh jubah}(h]h](hČeqeh]h]h]uhjAh jubah}(h]h]h]h]h]refdochь refdomainjYreftypejrefexplicitrefwarnheq7-5-4uhhh!h"hMh jkubh/& using an infinitely
dilute spectrum}(h& using an infinitely
dilute spectrumh jkubeh}(h]h]h]h]h]uhh:h!h"hMh j3ubh;)}(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](jo)}(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]uhjYh 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]jB)}(hjh]h/eq7-5-4}(hhh jubah}(h]h](hČeqeh]h]h]uhjAh jubah}(h]h]h]h]h]refdochь refdomainjYreftypejrefexplicitrefwarnheq7-5-4uhhh!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 j3ubeh}(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 jhhh!NhNubjo)}(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 jubah}(h]h]h]h]h]uhjYh jubh/.}(hjh jhhh!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 j? ubah}(h]j> ah]h]h]h]docnamehьnumberKlabeleq7-5-6nowrapjjuhjh!h"hMh jhhj}j}j> j5 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 jT hhh!NhNubj)}(h:sub:`0`h]h/0}(hhh j] ubah}(h]h]h]h]h]uhjh jT 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 jT 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 jv hhh!NhNubj)}(h :sub:`th`h]h/th}(hhh j ubah}(h]h]h]h]h]uhjh jv 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 jv 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 jv 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!NhNubjo)}(h:math:`f_{l}^{j}`h]h/ f_{l}^{j}}(hhh j ubah}(h]h]h]h]h]uhjYh 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 jL
ubah}(h]jK
ah]h]h]h]docnamehьnumberKlabeleq7-5-8nowrapjjuhjh!h"hMh jhhj}j}jK
jB
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 ja
hhh!NhNubhA)}(h*l*h]h/l}(hhh jj
ubah}(h]h]h]h]h]uhh@h ja
ubh/ is the }(h is the h ja
hhh!NhNubhA)}(h*l*h]h/l}(hhh j}
ubah}(h]h]h]h]h]uhh@h ja
ubh/ }(h\ h ja
hhh!NhNubj)}(h :sub:`th`h]h/th}(hhh j
ubah}(h]h]h]h]h]uhjh ja
ubh/# order Legendre polynomial; and
G }(h# order Legendre polynomial; and
G\ h ja
hhh!NhNubj)}(h:sup:`j`h]h/j}(hhh j
ubah}(h]h]h]h]h]uhjh ja
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 ja
hhh!NhNubj)}(h:sup:`j`h]h/j}(hhh j
ubah}(h]h]h]h]h]uhjh ja
ubh/7. The kinematics function for nuclide j is
defined as,}(h7. The kinematics function for nuclide j is
defined as,h ja
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 j
hhh!NhNubj)}(h:sup:`j`h]h/j}(hhh j
ubah}(h]h]h]h]h]uhjh j
ubh/ (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 j
hhh!NhNubh)}(hbucholz_method_1978h]h)}(hjh]h/[bucholz_method_1978]}(hhh jubah}(h]h]h]h]h]uhhh j
ubah}(h]id16ah]j.ah]h]h] refdomainj3reftypej5 reftargetjrefwarnsupport_smartquotesuhhh!h"hMh j
hhubh/. 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 j
hhh!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/