gsphinx.addnodesdocument)}( rawsourcechildren](docutils.nodestarget)}(h.. _9-0:h]
attributes}(ids]classes]names]dupnames]backrefs]refidid1utagnameh
lineKparenthhhsourceG/Users/john/Documents/SCALE-test/docs/Deterministic Transport Intro.rstubh section)}(hhh](h title)}(h Deterministic Transport Overviewh]h Text Deterministic Transport Overview}(hh,h h*hhh!NhNubah}(h]h]h]h]h]uhh(h h%hhh!h"hKubh paragraph)}(h*Introduction by S. M. Bowman*h]h emphasis)}(hh>h]h/Introduction by S. M. Bowman}(hhh hBubah}(h]h]h]h]h]uhh@h hubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh j:ubah}(h]h]h]h]h]refdocj refdomainjHreftypenumrefrefexplicitrefwarnjtab9-1-1uhjh!jhMph jubh/M gives
leakage terms expressed in conservation form for the three geometries.}(hM gives
leakage terms expressed in conservation form for the three geometries.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMph j hhubh)}(h
.. _tab9-1-1:h]h}(h]h]h]h]h]htab9-1-1uhh
hMh j hhh!jubh table)}(hhh](h))}(hOne-dimensional leakage terms.h]h/One-dimensional leakage terms.}(hjwh juubah}(h]h]h]h]h]uhh(h!jhMyh jrubh tgroup)}(hhh](h colspec)}(hhh]h}(h]h]h]h]h]colwidthK2uhjh jubj)}(hhh]h}(h]h]h]h]h]jK2uhjh jubh tbody)}(hhh](h row)}(hhh](h entry)}(hhh]h;)}(hGeometryh]h/Geometry}(hjh jubah}(h]h]h]h]h]uhh:h!jhM|h jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(h=:math:`\overset{\rightharpoonup}{\Omega} \bullet \nabla \psi`h]jr)}(h=:math:`\overset{\rightharpoonup}{\Omega} \bullet \nabla \psi`h]h/5\overset{\rightharpoonup}{\Omega} \bullet \nabla \psi}(hhh jubah}(h]h]h]h]h]uhjqh jubah}(h]h]h]h]h]uhh:h!jhM}h jubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(hSlabh]h/Slab}(hjh jubah}(h]h]h]h]h]uhh:h!jhM~h jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(h,:math:`\mu \frac{\partial \psi}{\partial x}`h]jr)}(h,:math:`\mu \frac{\partial \psi}{\partial x}`h]h/$\mu \frac{\partial \psi}{\partial x}}(hhh jubah}(h]h]h]h]h]uhjqh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(hCylinderh]h/Cylinder}(hj0h j.ubah}(h]h]h]h]h]uhh:h!jhMh j+ubah}(h]h]h]h]h]uhjh j(ubj)}(hhh]h;)}(ho:math:`\frac{\mu}{r} \frac{\partial(r \psi)}{\partial r}-\frac{1}{r} \frac{\partial(\eta \psi)}{\partial \phi}`h]jr)}(ho:math:`\frac{\mu}{r} \frac{\partial(r \psi)}{\partial r}-\frac{1}{r} \frac{\partial(\eta \psi)}{\partial \phi}`h]h/g\frac{\mu}{r} \frac{\partial(r \psi)}{\partial r}-\frac{1}{r} \frac{\partial(\eta \psi)}{\partial \phi}}(hhh jIubah}(h]h]h]h]h]uhjqh jEubah}(h]h]h]h]h]uhh:h!jhMh jBubah}(h]h]h]h]h]uhjh j(ubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(hSphereh]h/Sphere}(hjqh joubah}(h]h]h]h]h]uhh:h!jhMh jlubah}(h]h]h]h]h]uhjh jiubj)}(hhh]h;)}(h:math:`\frac{\mu}{r^{2}} \frac{\partial\left(r^{2} \psi\right)}{\partial r}+\frac{1}{r} \frac{\partial\left[\left(1-\mu^{2}\right) \psi\right]}{\partial \mu}`h]jr)}(h:math:`\frac{\mu}{r^{2}} \frac{\partial\left(r^{2} \psi\right)}{\partial r}+\frac{1}{r} \frac{\partial\left[\left(1-\mu^{2}\right) \psi\right]}{\partial \mu}`h]h/\frac{\mu}{r^{2}} \frac{\partial\left(r^{2} \psi\right)}{\partial r}+\frac{1}{r} \frac{\partial\left[\left(1-\mu^{2}\right) \psi\right]}{\partial \mu}}(hhh jubah}(h]h]h]h]h]uhjqh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jiubeh}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]colsKuhjh jrubeh}(h](id258joeh]h]tab9-1-1ah]h]aligncenteruhjph j hhh!NhNjf}jjesjh}jojesubh)}(h
.. _fig9-1-1:h]h}(h]h]h]h]h]hfig9-1-1uhh
hMh j hhh!jubh figure)}(hhh](h image)}(he.. figure:: figs/XSDRNPM/fig1.png
:align: center
:width: 600
Three common coordinate systems.
h]h}(h]h]h]h]h]width600urifigs/XSDRNPM/fig1.png
candidates}*jsuhjh jh!jhMubh caption)}(h Three common coordinate systems.h]h/ Three common coordinate systems.}(hjh jubah}(h]h]h]h]h]uhjh!jhMh jubeh}(h](id259jeh]h]fig9-1-1ah]h]jcenteruhjhMh j hhh!jjf}jjsjh}jjsubh)}(h
.. _fig9-1-2:h]h}(h]h]h]h]h]hfig9-1-2uhh
hMh j hhh!jubj)}(hhh](j)}(hb.. figure:: figs/XSDRNPM/fig2.png
:align: center
:width: 600
Three 1-D coordinate systems.
h]h}(h]h]h]h]h]width600urifigs/XSDRNPM/fig2.pngj}jjsuhjh jh!jhMubj)}(hThree 1-D coordinate systems.h]h/Three 1-D coordinate systems.}(hj h jubah}(h]h]h]h]h]uhjh!jhMh jubeh}(h](id260jeh]h]fig9-1-2ah]h]jcenteruhjhMh j hhh!jjf}j1jsjh}jjsubh)}(h.. _9-1-2-2:h]h}(h]h]h]h]h]hid22uhh
hMh j hhh!jubeh}(h]()one-dimensional-discrete-ordinates-theoryj eh]h]()one-dimensional discrete-ordinates theory9-1-2-1eh]h]uhh#h j hhh!jhM+jf}jHj sjh}j j subh$)}(hhh](h))}(h-Multigroup one-dimensional Boltzmann equationh]h/-Multigroup one-dimensional Boltzmann equation}(hjRh jPhhh!NhNubah}(h]h]h]h]h]uhh(h jMhhh!jhMubh;)}(hXIn multigroup schemes, the continuous-energy (CE) balance equations are
converted to multigroup form by first selecting an energy structure and
then writing a multigroup equivalent of the point equation which
requires multigroup constants that tend to preserve the reaction rates
that would arise from integrating the CE equations by group. First we
define the following multigroup values for g,h]h/XIn multigroup schemes, the continuous-energy (CE) balance equations are
converted to multigroup form by first selecting an energy structure and
then writing a multigroup equivalent of the point equation which
requires multigroup constants that tend to preserve the reaction rates
that would arise from integrating the CE equations by group. First we
define the following multigroup values for g,}(hj`h j^hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jMhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-4uhh
h jMhhh!jhNubj )}(h-\psi_{g}(x, \mu)=\int_{g} d E \psi(x, E, \mu)h]h/-\psi_{g}(x, \mu)=\int_{g} d E \psi(x, E, \mu)}(hhh jvubah}(h]juah]h]h]h]docnamejnumberKlabeleq9-1-4nowrapjyjzuhj h!jhMh jMhhjf}jh}jujlsubh;)}(handh]h/and}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jMhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-5uhh
h jMhhh!jhNubj )}(h0\psi_{g}(x)=\int_{-1}^{1} d \mu \psi_{g}(x, \mu)h]h/0\psi_{g}(x)=\int_{-1}^{1} d \mu \psi_{g}(x, \mu)}(hhh jubah}(h]jah]h]h]h]docnamejnumberKlabeleq9-1-5nowrapjyjzuhj h!jhMh jMhhjf}jh}jjsubh;)}(handh]h/and}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jMhhubj )}(hV\Sigma_{t g}(x)=\frac{\int_{g} d E \Sigma_{t g}(x, E) W(x, E)}{\int_{g} d E W(x, E)} ,h]h/V\Sigma_{t g}(x)=\frac{\int_{g} d E \Sigma_{t g}(x, E) W(x, E)}{\int_{g} d E W(x, E)} ,}(hhh jubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh jMhhubh;)}(hX+where *W(x.E)* is the weighting function used to compute the multigroup
cross sections at a particular location. To rigorously conserve reaction
rates, the weight function should be angle-dependent, but this causes
the multigroup cross section to vary with direction; therefore the usual
approach is to represent the weight function by an approximation to the
scalar flux spectrum. In energy ranges where the CE cross sections have
fine-structure due to resonances, the multigroup data must be
self-shielded prior to the multigroup transport calculations.h](h/where }(hwhere h jhhh!NhNubhA)}(h*W(x.E)*h]h/W(x.E)}(hhh jubah}(h]h]h]h]h]uhh@h jubh/X is the weighting function used to compute the multigroup
cross sections at a particular location. To rigorously conserve reaction
rates, the weight function should be angle-dependent, but this causes
the multigroup cross section to vary with direction; therefore the usual
approach is to represent the weight function by an approximation to the
scalar flux spectrum. In energy ranges where the CE cross sections have
fine-structure due to resonances, the multigroup data must be
self-shielded prior to the multigroup transport calculations.}(hX is the weighting function used to compute the multigroup
cross sections at a particular location. To rigorously conserve reaction
rates, the weight function should be angle-dependent, but this causes
the multigroup cross section to vary with direction; therefore the usual
approach is to represent the weight function by an approximation to the
scalar flux spectrum. In energy ranges where the CE cross sections have
fine-structure due to resonances, the multigroup data must be
self-shielded prior to the multigroup transport calculations.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jMhhubh;)}(hPThe following multi-group form of 1-D equation can be derived for the slab case:h]h/PThe following multi-group form of 1-D equation can be derived for the slab case:}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jMhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-6uhh
h jMhhh!jhNubj )}(h{\mu \frac{\partial \psi_{g}(x, \mu)}{\partial x}+\sum_{t g}(x) \psi_{g}(x, \mu)=S_{g}(x, \mu)+F_{g}(x, \mu)+Q_{g}(x, \mu) .h]h/{\mu \frac{\partial \psi_{g}(x, \mu)}{\partial x}+\sum_{t g}(x) \psi_{g}(x, \mu)=S_{g}(x, \mu)+F_{g}(x, \mu)+Q_{g}(x, \mu) .}(hhh jubah}(h]jah]h]h]h]docnamejnumberKlabeleq9-1-6nowrapjyjzuhj h!jhMh jMhhjf}jh}jjsubh;)}(hThe equations for the cylinder and sphere are essentially the same, in this
notation, except for the differences in the leakage terms from :numref:`tab9-1-1`.h](h/The equations for the cylinder and sphere are essentially the same, in this
notation, except for the differences in the leakage terms from }(hThe equations for the cylinder and sphere are essentially the same, in this
notation, except for the differences in the leakage terms from h j'hhh!NhNubj)}(h:numref:`tab9-1-1`h]jc)}(hj2h]h/tab9-1-1}(hhh j4ubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh j0ubah}(h]h]h]h]h]refdocj refdomainj>reftypenumrefrefexplicitrefwarnjtab9-1-1uhjh!jhMh j'ubh/.}(hjWh j'hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jMhhubh;)}(hIn Eq. :eq:`eq9-1-6` , *S*\ :sub:`g`, *F*\ :sub:`g`, and *Q*\ :sub:`g` are the scattering, fission, and
fixed sources, respectively. The scattering term is discussed in
:ref:`9-1-2-3`. The multigroup form of the fission source ish](h/In Eq. }(hIn Eq. h jZhhh!NhNubj)}(h
:eq:`eq9-1-6`h]jc)}(hjeh]h/eq9-1-6}(hhh jgubah}(h]h](jneqeh]h]h]uhjbh jcubah}(h]h]h]h]h]refdocj refdomainjqreftypejqrefexplicitrefwarnjeq9-1-6uhjh!jhMh jZubh/ , }(h , h jZhhh!NhNubhA)}(h*S*h]h/S}(hhh jubah}(h]h]h]h]h]uhh@h jZubh/ }(h\ h jZhhh!NhNubh)}(h:sub:`g`h]h/g}(hhh jubah}(h]h]h]h]h]uhhh jZubh/, }(h, h jZhhh!NhNubhA)}(h*F*h]h/F}(hhh jubah}(h]h]h]h]h]uhh@h jZubh/ }(h\ h jZubh)}(h:sub:`g`h]h/g}(hhh jubah}(h]h]h]h]h]uhhh jZubh/, and }(h, and h jZhhh!NhNubhA)}(h*Q*h]h/Q}(hhh jubah}(h]h]h]h]h]uhh@h jZubh/ }(hjh jZubh)}(h:sub:`g`h]h/g}(hhh jubah}(h]h]h]h]h]uhhh jZubh/c are the scattering, fission, and
fixed sources, respectively. The scattering term is discussed in
}(hc are the scattering, fission, and
fixed sources, respectively. The scattering term is discussed in
h jZhhh!NhNubj)}(h:ref:`9-1-2-3`h]j#)}(hjh]h/9-1-2-3}(hhh jubah}(h]h](jnstdstd-refeh]h]h]uhj"h jubah}(h]h]h]h]h]refdocj refdomainjreftyperefrefexplicitrefwarnj9-1-2-3uhjh!jhMh jZubh/.. The multigroup form of the fission source is}(h.. The multigroup form of the fission source ish jZhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jMhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-7uhh
h jMhhh!jhNubj )}(hsF_{g}(x, \mu)=\frac{\chi_{g}}{2 \pi k} \sum_{g^{\prime}} \overline{v \sum_{f g^{\prime}}}(x) \psi_{g^{\prime}}(x) ,h]h/sF_{g}(x, \mu)=\frac{\chi_{g}}{2 \pi k} \sum_{g^{\prime}} \overline{v \sum_{f g^{\prime}}}(x) \psi_{g^{\prime}}(x) ,}(hhh j,ubah}(h]j+ah]h]h]h]docnamejnumberKlabeleq9-1-7nowrapjyjzuhj h!jhMh jMhhjf}jh}j+j"subh;)}(hwhere *χ*\ :sub:`g` is the fraction of the fission neutrons that are produced
in group g, and is the average of the product of *υ*, the average number
of neutrons produced per fission and Σ\ :sub:`f`, the fission cross section.h](h/where }(hwhere h jAhhh!NhNubhA)}(h*χ*h]h/χ}(hhh jJubah}(h]h]h]h]h]uhh@h jAubh/ }(h\ h jAhhh!NhNubh)}(h:sub:`g`h]h/g}(hhh j]ubah}(h]h]h]h]h]uhhh jAubh/m is the fraction of the fission neutrons that are produced
in group g, and is the average of the product of }(hm is the fraction of the fission neutrons that are produced
in group g, and is the average of the product of h jAhhh!NhNubhA)}(h*υ*h]h/υ}(hhh jpubah}(h]h]h]h]h]uhh@h jAubh/>, the average number
of neutrons produced per fission and Σ }(h>, the average number
of neutrons produced per fission and Σ\ h jAhhh!NhNubh)}(h:sub:`f`h]h/f}(hhh jubah}(h]h]h]h]h]uhhh jAubh/, the fission cross section.}(h, the fission cross section.h jAhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jMhhubh)}(h.. _9-1-2-3:h]h}(h]h]h]h]h]hid23uhh
hMLh jMhhh!jubeh}(h](-multigroup-one-dimensional-boltzmann-equationjAeh]h](-multigroup one-dimensional boltzmann equation9-1-2-2eh]h]uhh#h j hhh!jhMjf}jj7sjh}jAj7subh$)}(hhh](h))}(hScattering source termh]h/Scattering source term}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMubh;)}(hIn discrete-ordinates theory, one typically calculates the Legendre
moments of the flux, *ψ*\ :math:`_{g,l}`, defined for slab and spherical geometries
byh](h/YIn discrete-ordinates theory, one typically calculates the Legendre
moments of the flux, }(hYIn discrete-ordinates theory, one typically calculates the Legendre
moments of the flux, h jhhh!NhNubhA)}(h*ψ*h]h/ψ}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ }(h\ h jhhh!NhNubjr)}(h:math:`_{g,l}`h]h/_{g,l}}(hhh jubah}(h]h]h]h]h]uhjqh jubh/., defined for slab and spherical geometries
by}(h., defined for slab and spherical geometries
byh jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-8uhh
h jhhh!jhNubj )}(hF\psi_{g, l}=\frac{1}{2} \int_{-1}^{1} d \mu \psi_{g}(\mu) P_{1}(\mu) .h]h/F\psi_{g, l}=\frac{1}{2} \int_{-1}^{1} d \mu \psi_{g}(\mu) P_{1}(\mu) .}(hhh jubah}(h]jah]h]h]h]docnamejnumberKlabeleq9-1-8nowrapjyjzuhj h!jhMh jhhjf}jh}jjsubh;)}(hCylindrical geometry has a similar expression containing spherical
harmonic functions rather than Legendre polynomials, shown in the next
section.h]h/Cylindrical geometry has a similar expression containing spherical
harmonic functions rather than Legendre polynomials, shown in the next
section.}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(hdThe group-to-group scattering coefficients are, themselves, fit with
Legendre polynomials, such thath]h/dThe group-to-group scattering coefficients are, themselves, fit with
Legendre polynomials, such that}(hj'h j%hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-9uhh
h jhhh!jhNubj )}(h\sigma\left(g^{\prime} \rightarrow g, \mu\right)=\sum_{l=0}^{I S C T} \frac{2 l+1}{2} \sigma_{l}\left(g^{\prime} \rightarrow g\right) P_{l}(\mu) .h]h/\sigma\left(g^{\prime} \rightarrow g, \mu\right)=\sum_{l=0}^{I S C T} \frac{2 l+1}{2} \sigma_{l}\left(g^{\prime} \rightarrow g\right) P_{l}(\mu) .}(hhh j=ubah}(h]j<ah]h]h]h]docnamejnumberK labeleq9-1-9nowrapjyjzuhj h!jhMh jhhjf}jh}j<j3subh;)}(h-In this example, we have a fit of order ISCT.h]h/-In this example, we have a fit of order ISCT.}(hjTh jRhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jhhubh note)}(hAMPX
cross-section libraries contain the 2\ *l* + 1 factor in the
:math:`\sigma_{t}\left(g^{\prime} \rightarrow g\right)` matrix.h]h;)}(hAMPX
cross-section libraries contain the 2\ *l* + 1 factor in the
:math:`\sigma_{t}\left(g^{\prime} \rightarrow g\right)` matrix.h](h/,AMPX
cross-section libraries contain the 2 }(h,AMPX
cross-section libraries contain the 2\ h jfubhA)}(h*l*h]h/l}(hhh joubah}(h]h]h]h]h]uhh@h jfubh/ + 1 factor in the
}(h + 1 factor in the
h jfubjr)}(h7:math:`\sigma_{t}\left(g^{\prime} \rightarrow g\right)`h]h//\sigma_{t}\left(g^{\prime} \rightarrow g\right)}(hhh jubah}(h]h]h]h]h]uhjqh jfubh/ matrix.}(h matrix.h jfubeh}(h]h]h]h]h]uhh:h!jhMh jbubah}(h]h]h]h]h]uhj`h jhhh!jhNubh)}(h.. _9-1-2-3-1:h]h}(h]h]h]h]h]hid24uhh
hMlh jhhh!jubh$)}(hhh](h))}(hSlab and Spherical Geometriesh]h/Slab and Spherical Geometries}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMubh;)}(hXzBecause of the symmetries in 1-D slabs and spheres, only one angle is needed to
describe a “direction.” In the case of the slab, the angle is taken with
reference to the x-axis, while for the sphere; it is with reference to a radius
vector between the point and the center of the sphere. This means that the flux
can be expanded in ordinary Legendre polynomials, such thath]h/XzBecause of the symmetries in 1-D slabs and spheres, only one angle is needed to
describe a “direction.” In the case of the slab, the angle is taken with
reference to the x-axis, while for the sphere; it is with reference to a radius
vector between the point and the center of the sphere. This means that the flux
can be expanded in ordinary Legendre polynomials, such that}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-10uhh
h jhhh!jhNubj )}(h\begin{array}{l}
\psi(r, E, \mu)=\sum_{l=0}^{\infty} \psi_{l}(r, E) P_{l}(\mu) \\
\psi_{l}(r, E)=\int_{-1}^{1} \frac{d \mu}{2} P_{l}(\mu) \psi(r, E, \mu)
\end{array} .h]h/\begin{array}{l}
\psi(r, E, \mu)=\sum_{l=0}^{\infty} \psi_{l}(r, E) P_{l}(\mu) \\
\psi_{l}(r, E)=\int_{-1}^{1} \frac{d \mu}{2} P_{l}(\mu) \psi(r, E, \mu)
\end{array} .}(hhh jubah}(h]jah]h]h]h]docnamejnumberK
labeleq9-1-10nowrapjyjzuhj h!jhMh jhhjf}jh}jjsubh;)}(hWhen Eq. :eq:`eq9-1-10` and Eq. :eq:`eq9-1-9` are introduced into Eq. :eq:`eq9-1-2`, the following
expression is derived for the scattering source:h](h/ When Eq. }(h When Eq. h jhhh!NhNubj)}(h:eq:`eq9-1-10`h]jc)}(hjh]h/eq9-1-10}(hhh jubah}(h]h](jneqeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjqreftypejrefexplicitrefwarnjeq9-1-10uhjh!jhMh jubh/ and Eq. }(h and Eq. h jhhh!NhNubj)}(h
:eq:`eq9-1-9`h]jc)}(hjh]h/eq9-1-9}(hhh jubah}(h]h](jneqeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjqreftypej$refexplicitrefwarnjeq9-1-9uhjh!jhMh jubh/ are introduced into Eq. }(h are introduced into Eq. h jhhh!NhNubj)}(h
:eq:`eq9-1-2`h]jc)}(hj;h]h/eq9-1-2}(hhh j=ubah}(h]h](jneqeh]h]h]uhjbh j9ubah}(h]h]h]h]h]refdocj refdomainjqreftypejGrefexplicitrefwarnjeq9-1-2uhjh!jhMh jubh/A, the following
expression is derived for the scattering source:}(hA, the following
expression is derived for the scattering source:h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-11uhh
h jhhh!jhNubj )}(hS(r, E, \mu)=2 \pi P_{l}(\mu) \int_{0}^{\infty} d E^{\prime} \int_{-1}^{1} d \mu^{\prime} \sum_{l=0}^{I S C T} \frac{2 l+1}{2} \sum_{s_{l}}\left(r, E^{\prime} \rightarrow E\right) P_{l}\left(\mu^{\prime}\right) \psi_{l}\left(r, E^{\prime}\right)h]h/S(r, E, \mu)=2 \pi P_{l}(\mu) \int_{0}^{\infty} d E^{\prime} \int_{-1}^{1} d \mu^{\prime} \sum_{l=0}^{I S C T} \frac{2 l+1}{2} \sum_{s_{l}}\left(r, E^{\prime} \rightarrow E\right) P_{l}\left(\mu^{\prime}\right) \psi_{l}\left(r, E^{\prime}\right)}(hhh jlubah}(h]jkah]h]h]h]docnamejnumberKlabeleq9-1-11nowrapjyjzuhj h!jhMh jhhjf}jh}jkjbsubh;)}(hBwhere *ISCT* is the order of fit to the fluxes and cross sections.h](h/where }(hwhere h jhhh!NhNubhA)}(h*ISCT*h]h/ISCT}(hhh jubah}(h]h]h]h]h]uhh@h jubh/6 is the order of fit to the fluxes and cross sections.}(h6 is the order of fit to the fluxes and cross sections.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(h.. _9-1-2-3-2:h]h}(h]h]h]h]h]hid25uhh
hMh jhhh!jubeh}(h](slab-and-spherical-geometriesjeh]h](slab and spherical geometries 9-1-2-3-1eh]h]uhh#h jhhh!jhMjf}jjsjh}jjsubh$)}(hhh](h))}(hCylindrical Geometryh]h/Cylindrical Geometry}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMubh;)}(hThe situation is more complicated in the case of the 1-D cylinder where the flux
(and cross section) must be given as a function of two angles. Consider
:numref:`fig9-1-3`.h](h/The situation is more complicated in the case of the 1-D cylinder where the flux
(and cross section) must be given as a function of two angles. Consider
}(hThe situation is more complicated in the case of the 1-D cylinder where the flux
(and cross section) must be given as a function of two angles. Consider
h jhhh!NhNubj)}(h:numref:`fig9-1-3`h]jc)}(hjh]h/fig9-1-3}(hhh jubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnjfig9-1-3uhjh!jhMh jubh/.}(hjWh jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(h
.. _fig9-1-3:h]h}(h]h]h]h]h]hfig9-1-3uhh
hMh jhhh!jubj)}(hhh](j)}(hy.. figure:: figs/XSDRNPM/fig3.png
:align: center
:width: 400
One-dimensional cylindrical scattering coordinates.
h]h}(h]h]h]h]h]width400urifigs/XSDRNPM/fig3.pngj}jjsuhjh jh!jhMubj)}(h3One-dimensional cylindrical scattering coordinates.h]h/3One-dimensional cylindrical scattering coordinates.}(hjh jubah}(h]h]h]h]h]uhjh!jhMh jubeh}(h](id261jeh]h]fig9-1-3ah]h]jcenteruhjhMh jhhh!jjf}j-jsjh}jjsubh;)}(hThe addition theorem for associated Legendre polynomials can be used to
transform from scattering angle coordinates to the real coordinates required in
the cylindrical case:h]h/The addition theorem for associated Legendre polynomials can be used to
transform from scattering angle coordinates to the real coordinates required in
the cylindrical case:}(hj5h j3hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-12uhh
h jhhh!jhNubj )}(hP_{l}\left(\mu_{0}\right)=\sum_{n=-1}^{1} \frac{(l-n) !}{(l+n) !} P_{l}^{n}(\mu) P_{l}^{n}\left(\mu^{\prime}\right) e^{i n\left(\zeta-\zeta^{\prime}\right)} ,h]h/P_{l}\left(\mu_{0}\right)=\sum_{n=-1}^{1} \frac{(l-n) !}{(l+n) !} P_{l}^{n}(\mu) P_{l}^{n}\left(\mu^{\prime}\right) e^{i n\left(\zeta-\zeta^{\prime}\right)} ,}(hhh jKubah}(h]jJah]h]h]h]docnamejnumberKlabeleq9-1-12nowrapjyjzuhj h!jhM#h jhhjf}jh}jJjAsubh;)}(h\where *μ*\ :sub:`0` = cos\ *θ\ 0* ; *μ *\ = cos \ *θ* and
*μ′* = cos *θ′*.h](h/where }(hwhere h j`hhh!NhNubhA)}(h*μ*h]h/μ}(hhh jiubah}(h]h]h]h]h]uhh@h j`ubh/ }(h\ h j`hhh!NhNubh)}(h:sub:`0`h]h/0}(hhh j|ubah}(h]h]h]h]h]uhhh j`ubh/ = cos }(h = cos\ h j`hhh!NhNubhA)}(h*θ\ 0*h]h/θ 0}(hhh jubah}(h]h]h]h]h]uhh@h j`ubh/ ; }(h ; h j`hhh!NhNubhA)}(h*μ *\ = cos \ *θ*h]h/μ * = cos *θ}(hhh jubah}(h]h]h]h]h]uhh@h j`ubh/ and
}(h and
h j`hhh!NhNubhA)}(h*μ′*h]h/μ′}(hhh jubah}(h]h]h]h]h]uhh@h j`ubh/
= cos }(h
= cos h j`hhh!NhNubhA)}(h*θ′*h]h/θ′}(hhh jubah}(h]h]h]h]h]uhh@h j`ubh/.}(hjWh j`hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM(h jhhubh;)}(hIf we note thath]h/If we note that}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM+h jhhubj )}(hX.\begin{aligned}
\sigma_{s}\left(r, E^{\prime} \rightarrow E, \Omega^{\prime} \rightarrow \Omega\right) &=\sigma_{s}\left(r, E^{\prime} \rightarrow E\right), P_{l}\left(\Omega^{\prime} \bullet \Omega\right) \\
&=\sigma_{s}\left(r, E^{\prime} \rightarrow E\right), P_{l}\left(\mu_{0}\right)
\end{aligned}h]h/X.\begin{aligned}
\sigma_{s}\left(r, E^{\prime} \rightarrow E, \Omega^{\prime} \rightarrow \Omega\right) &=\sigma_{s}\left(r, E^{\prime} \rightarrow E\right), P_{l}\left(\Omega^{\prime} \bullet \Omega\right) \\
&=\sigma_{s}\left(r, E^{\prime} \rightarrow E\right), P_{l}\left(\mu_{0}\right)
\end{aligned}}(hhh jubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhM-h jhhubh;)}(hDEq. :eq:`eq9-1-12` can be introduced into Eq. :eq:`eq9-1-2` to yieldh](h/Eq. }(hEq. h jhhh!NhNubj)}(h:eq:`eq9-1-12`h]jc)}(hjh]h/eq9-1-12}(hhh j
ubah}(h]h](jneqeh]h]h]uhjbh j ubah}(h]h]h]h]h]refdocj refdomainjqreftypejrefexplicitrefwarnjeq9-1-12uhjh!jhM4h jubh/ can be introduced into Eq. }(h can be introduced into Eq. h jhhh!NhNubj)}(h
:eq:`eq9-1-2`h]jc)}(hj.h]h/eq9-1-2}(hhh j0ubah}(h]h](jneqeh]h]h]uhjbh j,ubah}(h]h]h]h]h]refdocj refdomainjqreftypej:refexplicitrefwarnjeq9-1-2uhjh!jhM4h jubh/ to yield}(h to yieldh jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM4h jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-13uhh
h jhhh!jhNubj )}(hX\begin{array}{c}
S(r, E, \mu)=\int_{0}^{\infty} d E^{\prime} \int_{-1}^{1} d \mu^{\prime} \int_{0}^{2 x} d \zeta \psi\left(r, E^{\prime}, \mu^{\prime}, \zeta^{\prime}\right) \sum_{l=0}^{I S C T} \frac{2 l+1}{2} \sigma_{s_{l}}\left(r, E^{\prime} \rightarrow E\right) \\
\times \sum_{n=-1}^{l} \frac{(l-n) !}{(l+n) !} P_{l}^{n}(\mu) P_{l}^{n}\left(\mu^{\prime}\right) e^{i n\left(\zeta-\zeta^{\prime}\right)}
\end{array} .h]h/X\begin{array}{c}
S(r, E, \mu)=\int_{0}^{\infty} d E^{\prime} \int_{-1}^{1} d \mu^{\prime} \int_{0}^{2 x} d \zeta \psi\left(r, E^{\prime}, \mu^{\prime}, \zeta^{\prime}\right) \sum_{l=0}^{I S C T} \frac{2 l+1}{2} \sigma_{s_{l}}\left(r, E^{\prime} \rightarrow E\right) \\
\times \sum_{n=-1}^{l} \frac{(l-n) !}{(l+n) !} P_{l}^{n}(\mu) P_{l}^{n}\left(\mu^{\prime}\right) e^{i n\left(\zeta-\zeta^{\prime}\right)}
\end{array} .}(hhh j_ubah}(h]j^ah]h]h]h]docnamejnumberK
labeleq9-1-13nowrapjyjzuhj h!jhM6h jhhjf}jh}j^jUsubh;)}(h#Now it is convenient to recall thath]h/#Now it is convenient to recall that}(hjvh jthhh!NhNubah}(h]h]h]h]h]uhh:h!jhM>h jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-14uhh
h jhhh!jhNubj )}(h$\cos x=\frac{e^{+i x}+e^{-i x}}{2} ,h]h/$\cos x=\frac{e^{+i x}+e^{-i x}}{2} ,}(hhh jubah}(h]jah]h]h]h]docnamejnumberKlabeleq9-1-14nowrapjyjzuhj h!jhM@h jhhjf}jh}jjsubh;)}(hFwhich can be introduced into Eq. :eq:`eq9-1-13` and rearranged to giveh](h/!which can be introduced into Eq. }(h!which can be introduced into Eq. h jhhh!NhNubj)}(h:eq:`eq9-1-13`h]jc)}(hjh]h/eq9-1-13}(hhh jubah}(h]h](jneqeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjqreftypejrefexplicitrefwarnjeq9-1-13uhjh!jhMEh jubh/ and rearranged to give}(h and rearranged to giveh jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMEh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-15uhh
h jhhh!jhNubj )}(hXf\begin{aligned}
S(r, E, \mu)=& \sum_{l=0}^{I S C T} \frac{2 l+1}{2} \int_{0}^{\infty} d E^{\prime} \sigma_{s_{l}}\left(r, E^{\prime} \rightarrow E\right)\left[P_{l}(\mu) \int_{-1}^{1} d \mu^{\prime} \int_{0}^{2 \pi} d \zeta \psi\left(r, E^{\prime}, \mu^{\prime}, \zeta^{\prime}\right) P_{l}\left(\mu^{\prime}\right)\right] \\
&+\sum_{n=1}^{l} 2 \frac{(l-n) !}{(l+n) !} P_{l}^{n}(\mu)\left[\int_{-1}^{1} d \mu^{\prime} \int_{0}^{2 \pi} d \zeta \psi\left(r, E^{\prime}, \mu^{\prime}, \zeta^{\prime}\right) P_{l}^{n}\left(\mu^{\prime}\right) \cos \left[n\left(\zeta-\zeta^{\prime}\right)\right]\right]
\end{aligned} .h]h/Xf\begin{aligned}
S(r, E, \mu)=& \sum_{l=0}^{I S C T} \frac{2 l+1}{2} \int_{0}^{\infty} d E^{\prime} \sigma_{s_{l}}\left(r, E^{\prime} \rightarrow E\right)\left[P_{l}(\mu) \int_{-1}^{1} d \mu^{\prime} \int_{0}^{2 \pi} d \zeta \psi\left(r, E^{\prime}, \mu^{\prime}, \zeta^{\prime}\right) P_{l}\left(\mu^{\prime}\right)\right] \\
&+\sum_{n=1}^{l} 2 \frac{(l-n) !}{(l+n) !} P_{l}^{n}(\mu)\left[\int_{-1}^{1} d \mu^{\prime} \int_{0}^{2 \pi} d \zeta \psi\left(r, E^{\prime}, \mu^{\prime}, \zeta^{\prime}\right) P_{l}^{n}\left(\mu^{\prime}\right) \cos \left[n\left(\zeta-\zeta^{\prime}\right)\right]\right]
\end{aligned} .}(hhh jubah}(h]jah]h]h]h]docnamejnumberKlabeleq9-1-15nowrapjyjzuhj h!jhMGh jhhjf}jh}jjsubh;)}(h-We now define moments of the flux, *ψ\ l* byh](h/#We now define moments of the flux, }(h#We now define moments of the flux, h jhhh!NhNubhA)}(h*ψ\ l*h]h/ψ l}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ by}(h byh jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMOh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-16uhh
h jhhh!jhNubj )}(h\phi_{l}(r, E)=\int_{-1}^{1} d \mu^{\prime} \int_{0}^{2 \pi} d \zeta \psi\left(r, E, \mu^{\prime}, \zeta^{\prime}\right) P_{1}\left(\mu^{\prime}\right)h]h/\phi_{l}(r, E)=\int_{-1}^{1} d \mu^{\prime} \int_{0}^{2 \pi} d \zeta \psi\left(r, E, \mu^{\prime}, \zeta^{\prime}\right) P_{1}\left(\mu^{\prime}\right)}(hhh jubah}(h]jah]h]h]h]docnamejnumberKlabeleq9-1-16nowrapjyjzuhj h!jhMQh jhhjf}jh}jjsubh;)}(hCIt is also convenient to make use of the trigonometric relationshiph]h/CIt is also convenient to make use of the trigonometric relationship}(hj5h j3hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMVh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-17uhh
h jhhh!jhNubj )}(h|\cos \left[n\left(\zeta-\zeta^{\prime}\right)\right]=\cos n \zeta \cos n \zeta^{\prime}+\sin n \zeta \sin n \zeta^{\prime} ,h]h/|\cos \left[n\left(\zeta-\zeta^{\prime}\right)\right]=\cos n \zeta \cos n \zeta^{\prime}+\sin n \zeta \sin n \zeta^{\prime} ,}(hhh jKubah}(h]jJah]h]h]h]docnamejnumberKlabeleq9-1-17nowrapjyjzuhj h!jhMXh jhhjf}jh}jJjAsubh;)}(handh]h/and}(hjbh j`hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM]h jhhubj )}(h\psi_{l}^{n}(r, E)=\sqrt{2 \frac{(l-n) !}{(l+n) !} \int_{-1}^{1} d \mu^{\prime}} \int_{0}^{2 \pi} d \zeta \psi\left(r, E, \mu^{\prime}, \zeta^{\prime}\right) P_{l}^{n}\left(\mu^{\prime}\right) \sin n \zeta^{\prime}h]h/\psi_{l}^{n}(r, E)=\sqrt{2 \frac{(l-n) !}{(l+n) !} \int_{-1}^{1} d \mu^{\prime}} \int_{0}^{2 \pi} d \zeta \psi\left(r, E, \mu^{\prime}, \zeta^{\prime}\right) P_{l}^{n}\left(\mu^{\prime}\right) \sin n \zeta^{\prime}}(hhh jnubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhM_h jhhubh comment)}(h)this problem had a corrupt label in word.h]h/)this problem had a corrupt label in word.}(hhh jubah}(h]h]h]h]h]jyjzuhjh jhhh!jhMdubh;)}(hWith a 1-D cylinder, the flux is symmetric in *ζ*; therefore, it is an
even function, and the terms involving sin n \ *ζ* will vanish. This
fact yields the following expression for Eq. :eq:`eq9-1-15` :h](h/.With a 1-D cylinder, the flux is symmetric in }(h.With a 1-D cylinder, the flux is symmetric in h jhhh!NhNubhA)}(h*ζ*h]h/ζ}(hhh jubah}(h]h]h]h]h]uhh@h jubh/G; therefore, it is an
even function, and the terms involving sin n }(hG; therefore, it is an
even function, and the terms involving sin n \ h jhhh!NhNubhA)}(h*ζ*h]h/ζ}(hhh jubah}(h]h]h]h]h]uhh@h jubh/@ will vanish. This
fact yields the following expression for Eq. }(h@ will vanish. This
fact yields the following expression for Eq. h jhhh!NhNubj)}(h:eq:`eq9-1-15`h]jc)}(hjh]h/eq9-1-15}(hhh jubah}(h]h](jneqeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjqreftypejrefexplicitrefwarnjeq9-1-15uhjh!jhMeh jubh/ :}(h :h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMeh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-18uhh
h jhhh!jhNubj )}(hXA\begin{array}{c}
S(r, E, \mu)=\sum_{l=0}^{I S C T} \frac{2 l+1}{2} \int_{0}^{\infty} d E^{\prime} \sigma_{s_{l}}\left(r, E^{\prime} \rightarrow E\right) \\
\left.P_{l}(\mu) \psi_{l}\left(r, E^{\prime}\right)+\sum_{n=l}^{l} \sqrt{2 \frac{(l-n) !}{(l+n) !}} P_{n}^{l}(\mu) \cos n \zeta \psi_{l}^{n}(r, E)\right]
\end{array}h]h/XA\begin{array}{c}
S(r, E, \mu)=\sum_{l=0}^{I S C T} \frac{2 l+1}{2} \int_{0}^{\infty} d E^{\prime} \sigma_{s_{l}}\left(r, E^{\prime} \rightarrow E\right) \\
\left.P_{l}(\mu) \psi_{l}\left(r, E^{\prime}\right)+\sum_{n=l}^{l} \sqrt{2 \frac{(l-n) !}{(l+n) !}} P_{n}^{l}(\mu) \cos n \zeta \psi_{l}^{n}(r, E)\right]
\end{array}}(hhh jubah}(h]jah]h]h]h]docnamejnumberKlabeleq9-1-18nowrapjyjzuhj h!jhMih jhhjf}jh}jjsubh;)}(hWe observe further that for an even function in *ζ*, the odd *l* and odd
(*l*-n) moments must all vanish, such that the following moments are
nonzero for various orders of scattering:h](h/0We observe further that for an even function in }(h0We observe further that for an even function in h jhhh!NhNubhA)}(h*ζ*h]h/ζ}(hhh jubah}(h]h]h]h]h]uhh@h jubh/
, the odd }(h
, the odd h jhhh!NhNubhA)}(h*l*h]h/l}(hhh j#ubah}(h]h]h]h]h]uhh@h jubh/
and odd
(}(h
and odd
(h jhhh!NhNubhA)}(h*l*h]h/l}(hhh j6ubah}(h]h]h]h]h]uhh@h jubh/j-n) moments must all vanish, such that the following moments are
nonzero for various orders of scattering:}(hj-n) moments must all vanish, such that the following moments are
nonzero for various orders of scattering:h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMqh jhhubjq)}(hhh]j)}(hhh](j)}(hhh]h}(h]h]h]h]h]jK2uhjh jRubj)}(hhh]h}(h]h]h]h]h]jK2uhjh jRubh thead)}(hhh]j)}(hhh](j)}(hhh]h;)}(hISCTh]h/ISCT}(hjth jrubah}(h]h]h]h]h]uhh:h!jhMyh joubah}(h]h]h]h]h]uhjh jlubj)}(hhh]h;)}(hNonzero flux momentsh]h/Nonzero flux moments}(hjh jubah}(h]h]h]h]h]uhh:h!jhMzh jubah}(h]h]h]h]h]uhjh jlubeh}(h]h]h]h]h]uhjh jiubah}(h]h]h]h]h]uhjgh jRubj)}(hhh](j)}(hhh](j)}(hhh]h;)}(h0h]h/0}(hjh jubah}(h]h]h]h]h]uhh:h!jhM{h jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(h:math:`\psi_{0}`h]jr)}(h:math:`\psi_{0}`h]h/\psi_{0}}(hhh jubah}(h]h]h]h]h]uhjqh jubah}(h]h]h]h]h]uhh:h!jhM|h jubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(h1h]h/1}(hjh jubah}(h]h]h]h]h]uhh:h!jhM}h jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(h&:math:`\psi_{0}`, :math:`\psi_{1}^{1}`h](jr)}(h:math:`\psi_{0}`h]h/\psi_{0}}(hhh jubah}(h]h]h]h]h]uhjqh j
ubh/, }(h, h j
ubjr)}(h:math:`\psi_{1}^{1}`h]h/\psi_{1}^{1}}(hhh j!ubah}(h]h]h]h]h]uhjqh j
ubeh}(h]h]h]h]h]uhh:h!jhM~h jubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(h2h]h/2}(hjIh jGubah}(h]h]h]h]h]uhh:h!jhMh jDubah}(h]h]h]h]h]uhjh jAubj)}(hhh]h;)}(h6:math:`\psi_{0}, \psi_{1}^{1}, \psi_{2}, \psi_{2}^{2}`h]jr)}(h6:math:`\psi_{0}, \psi_{1}^{1}, \psi_{2}, \psi_{2}^{2}`h]h/.\psi_{0}, \psi_{1}^{1}, \psi_{2}, \psi_{2}^{2}}(hhh jbubah}(h]h]h]h]h]uhjqh j^ubah}(h]h]h]h]h]uhh:h!jhMh j[ubah}(h]h]h]h]h]uhjh jAubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(h3h]h/3}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(hR:math:`\psi_{0}, \psi_{1}^{1}, \psi_{2}, \psi_{2}^{2}, \psi_{3}^{1}, \psi_{3}^{3}`h]jr)}(hR:math:`\psi_{0}, \psi_{1}^{1}, \psi_{2}, \psi_{2}^{2}, \psi_{3}^{1}, \psi_{3}^{3}`h]h/J\psi_{0}, \psi_{1}^{1}, \psi_{2}, \psi_{2}^{2}, \psi_{3}^{1}, \psi_{3}^{3}}(hhh jubah}(h]h]h]h]h]uhjqh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(h4h]h/4}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(hx:math:`\psi_{0}, \psi_{1}^{1}, \psi_{2}, \psi_{2}^{2}, \psi_{3}^{1}, \psi_{3}^{3}, \psi_{4}, \psi_{4}^{2}, \psi_{4}^{4}`h]jr)}(hx:math:`\psi_{0}, \psi_{1}^{1}, \psi_{2}, \psi_{2}^{2}, \psi_{3}^{1}, \psi_{3}^{3}, \psi_{4}, \psi_{4}^{2}, \psi_{4}^{4}`h]h/p\psi_{0}, \psi_{1}^{1}, \psi_{2}, \psi_{2}^{2}, \psi_{3}^{1}, \psi_{3}^{3}, \psi_{4}, \psi_{4}^{2}, \psi_{4}^{4}}(hhh jubah}(h]h]h]h]h]uhjqh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jRubeh}(h]h]h]h]h]colsKuhjh jOubah}(h]h]h]h]h]jcenteruhjph jhhh!NhNubh;)}(h=In general, [ISCT(ISCT + 4)/4] + 1 flux moments are required.h]h/=In general, [ISCT(ISCT + 4)/4] + 1 flux moments are required.}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(h.. _9-1-2-4:h]h}(h]h]h]h]h]hid26uhh
hMh jhhh!jubeh}(h](cylindrical-geometryjeh]h](cylindrical geometry 9-1-2-3-2eh]h]uhh#h jhhh!jhMjf}j7jsjh}jjsubeh}(h](scattering-source-termjeh]h](scattering source term9-1-2-3eh]h]uhh#h j hhh!jhMjf}jBjsjh}jjsubh$)}(hhh](h))}(h'Discrete-ordinates difference equationsh]h/'Discrete-ordinates difference equations}(hjLh jJhhh!NhNubah}(h]h]h]h]h]uhh(h jGhhh!jhMubh$)}(hhh](h))}(h'Discrete-ordinates difference equationsh]h/'Discrete-ordinates difference equations}(hj]h j[hhh!NhNubah}(h]h]h]h]h]uhh(h jXhhh!jhMubh;)}(hIn formulating the |Sn| equations, several symbols are defined which
relate to a flux in an energy group g, in a spatial interval i, and in
an angle m.h](h/In formulating the }(hIn formulating the h jihhh!NhNubjr)}(hjh]h/S_n}(hhh jrhhh!NhNubah}(h]h]h]h]h]uhjqh!NhNh jihhubh/ equations, several symbols are defined which
relate to a flux in an energy group g, in a spatial interval i, and in
an angle m.}(h equations, several symbols are defined which
relate to a flux in an energy group g, in a spatial interval i, and in
an angle m.h jihhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jXhhubh;)}(hTypically, the flux is quoted as an integral of the flux in an energy
group g, whose upper and lower bounds are :math:`E_{g}^{U}` and :math:`E_{g}^{L}` respectively.h](h/qTypically, the flux is quoted as an integral of the flux in an energy
group g, whose upper and lower bounds are }(hqTypically, the flux is quoted as an integral of the flux in an energy
group g, whose upper and lower bounds are h jhhh!NhNubjr)}(h:math:`E_{g}^{U}`h]h/ E_{g}^{U}}(hhh jubah}(h]h]h]h]h]uhjqh jubh/ and }(h and h jhhh!NhNubjr)}(h:math:`E_{g}^{L}`h]h/ E_{g}^{L}}(hhh jubah}(h]h]h]h]h]uhjqh jubh/ respectively.}(h respectively.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jXhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-19uhh
h jXhhh!jhNubj )}(h3\psi_{g}=\int_{E_{g}^{L}}^{E_{g}^{U}} d E \psi(E) .h]h/3\psi_{g}=\int_{E_{g}^{L}}^{E_{g}^{U}} d E \psi(E) .}(hhh jubah}(h]jah]h]h]h]docnamejnumberKlabeleq9-1-19nowrapjyjzuhj h!jhMh jXhhjf}jh}jjsubh;)}(hXA mechanical quadrature is taken in space, typically IM intervals with
IM + 1 boundaries. Likewise, an angular quadrature is picked compatible
with the particular 1-D geometry, typically MM angles with associated
directional coordinates and integration weights.h]h/XA mechanical quadrature is taken in space, typically IM intervals with
IM + 1 boundaries. Likewise, an angular quadrature is picked compatible
with the particular 1-D geometry, typically MM angles with associated
directional coordinates and integration weights.}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jXhhubh;)}(hXThe different equations are formulated in a manner which involves
calculating so-called angular fluxes, *ψ*\ :sub:`g,i,m` at each of the
spatial interval boundaries, and also cell-centered fluxes,
:math:`\psi_{g, i+1 / 2, m}` at the centers of the spatial intervals. The
centered fluxes are related to the angular boundary fluxes by “weighted
diamond difference” assumptions as will be described below.h](h/hThe different equations are formulated in a manner which involves
calculating so-called angular fluxes, }(hhThe different equations are formulated in a manner which involves
calculating so-called angular fluxes, h jhhh!NhNubhA)}(h*ψ*h]h/ψ}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ }(h\ h jhhh!NhNubh)}(h:sub:`g,i,m`h]h/g,i,m}(hhh jubah}(h]h]h]h]h]uhhh jubh/M at each of the
spatial interval boundaries, and also cell-centered fluxes,
}(hM at each of the
spatial interval boundaries, and also cell-centered fluxes,
h jhhh!NhNubjr)}(h:math:`\psi_{g, i+1 / 2, m}`h]h/\psi_{g, i+1 / 2, m}}(hhh jubah}(h]h]h]h]h]uhjqh jubh/ at the centers of the spatial intervals. The
centered fluxes are related to the angular boundary fluxes by “weighted
diamond difference” assumptions as will be described below.}(h at the centers of the spatial intervals. The
centered fluxes are related to the angular boundary fluxes by “weighted
diamond difference” assumptions as will be described below.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jXhhubh;)}(hUnits on angular fluxes are per unit solid angle *w*\ :sub:`m` and per unit
area. Units on the centered fluxes are track length per unit volume of
the interval. In both cases the fluxes are integrated in energy over the
group g.h](h/1Units on angular fluxes are per unit solid angle }(h1Units on angular fluxes are per unit solid angle h j4hhh!NhNubhA)}(h*w*h]h/w}(hhh j=ubah}(h]h]h]h]h]uhh@h j4ubh/ }(h\ h j4hhh!NhNubh)}(h:sub:`m`h]h/m}(hhh jPubah}(h]h]h]h]h]uhhh j4ubh/ and per unit
area. Units on the centered fluxes are track length per unit volume of
the interval. In both cases the fluxes are integrated in energy over the
group g.}(h and per unit
area. Units on the centered fluxes are track length per unit volume of
the interval. In both cases the fluxes are integrated in energy over the
group g.h j4hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jXhhubh;)}(hOThe areas and volumes for the three geometries are listed in :numref:`tab9-1-2`h](h/=The areas and volumes for the three geometries are listed in }(h=The areas and volumes for the three geometries are listed in h jihhh!NhNubj)}(h:numref:`tab9-1-2`h]jc)}(hjth]h/tab9-1-2}(hhh jvubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jrubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnjtab9-1-2uhjh!jhMh jiubeh}(h]h]h]h]h]uhh:h!jhMh jXhhubh)}(h
.. _tab9-1-2:h]h}(h]h]h]h]h]htab9-1-2uhh
hM*h jXhhh!jubjq)}(hhh](h))}(h"One-dimensional areas and volumes.h]h/"One-dimensional areas and volumes.}(hjh jubah}(h]h]h]h]h]uhh(h!jhMh jubj)}(hhh](j)}(hhh]h}(h]h]h]h]h]jK!uhjh jubj)}(hhh]h}(h]h]h]h]h]jK!uhjh jubj)}(hhh]h}(h]h]h]h]h]jK!uhjh jubjh)}(hhh]j)}(hhh](j)}(hhh]h;)}(hGeometryh]h/Geometry}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(hAreah]h/Area}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(hVolumeh]h/Volume}(hjh j ubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubah}(h]h]h]h]h]uhjgh jubj)}(hhh](j)}(hhh](j)}(hhh]h;)}(hSlabh]h/Slab}(hj4h j2ubah}(h]h]h]h]h]uhh:h!jhMh j/ubah}(h]h]h]h]h]uhjh j,ubj)}(hhh]h;)}(h1.0h]h/1.0}(hjKh jIubah}(h]h]h]h]h]uhh:h!jhMh jFubah}(h]h]h]h]h]uhjh j,ubj)}(hhh]h;)}(h:math:`x_{i+1}-x_{i}`h]jr)}(hjbh]h/
x_{i+1}-x_{i}}(hhh jdubah}(h]h]h]h]h]uhjqh j`ubah}(h]h]h]h]h]uhh:h!jhMh j]ubah}(h]h]h]h]h]uhjh j,ubeh}(h]h]h]h]h]uhjh j)ubj)}(hhh](j)}(hhh]h;)}(hCylinderh]h/Cylinder}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(h:math:`2 \pi r_{i}`h]jr)}(h:math:`2 \pi r_{i}`h]h/2 \pi r_{i}}(hhh jubah}(h]h]h]h]h]uhjqh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(h.:math:`\pi \left(r_{i+1}^{2}-r_{i}^{2}\right)`h]jr)}(h.:math:`\pi \left(r_{i+1}^{2}-r_{i}^{2}\right)`h]h/&\pi \left(r_{i+1}^{2}-r_{i}^{2}\right)}(hhh jubah}(h]h]h]h]h]uhjqh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh j)ubj)}(hhh](j)}(hhh]h;)}(hSphereh]h/Sphere}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(h:math:`4 \pi r^{2}`h]jr)}(h:math:`4 \pi r^{2}`h]h/4 \pi r^{2}}(hhh jubah}(h]h]h]h]h]uhjqh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(h2:math:`4/3 \pi \left(r_{i+1}^{3}-r_{i}^{3}\right)`h]jr)}(h2:math:`4/3 \pi \left(r_{i+1}^{3}-r_{i}^{3}\right)`h]h/*4/3 \pi \left(r_{i+1}^{3}-r_{i}^{3}\right)}(hhh j'ubah}(h]h]h]h]h]uhjqh j#ubah}(h]h]h]h]h]uhh:h!jhMh j ubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh j)ubeh}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]colsKuhjh jubeh}(h](id262jeh]h]tab9-1-2ah]h]jcenteruhjph jXhhh!NhNjf}jYjsjh}jjsubh)}(h.. _9-1-2-4-1:h]h}(h]h]h]h]h]hid28uhh
hM<h jXhhh!jubeh}(h]id27ah]h]h]'discrete-ordinates difference equationsah]uhh#h jGhhh!jhMjKubh$)}(hhh](h))}(h&Discrete-ordinates equation for a slabh]h/&Discrete-ordinates equation for a slab}(hjwh juhhh!NhNubah}(h]h]h]h]h]uhh(h jrhhh!jhMubh;)}(hXConsider a spatial cell bounded by *(xi,x\ i+1)* and write the loss term
for flow through the cell in direction *μ*\ :sub:`m`. The net flow in the
x-direction out the right side is the product of the angular flux times
the area times the solid angle times the cosine of the angle:h](h/#Consider a spatial cell bounded by }(h#Consider a spatial cell bounded by h jhhh!NhNubhA)}(h
*(xi,x\ i+1)*h]h/(xi,x i+1)}(hhh jubah}(h]h]h]h]h]uhh@h jubh/@ and write the loss term
for flow through the cell in direction }(h@ and write the loss term
for flow through the cell in direction h jhhh!NhNubhA)}(h*μ*h]h/μ}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ }(h\ h jhhh!NhNubh)}(h:sub:`m`h]h/m}(hhh jubah}(h]h]h]h]h]uhhh jubh/. The net flow in the
x-direction out the right side is the product of the angular flux times
the area times the solid angle times the cosine of the angle:}(h. The net flow in the
x-direction out the right side is the product of the angular flux times
the area times the solid angle times the cosine of the angle:h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jrhhubj )}(h(w_{m} \mu_{m} A_{i+1} \psi_{g, i+1, m} .h]h/(w_{m} \mu_{m} A_{i+1} \psi_{g, i+1, m} .}(hhh jubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh jrhhubh;)}(hSThe net loss from the cell is the difference between the flow over both boundaries:h]h/SThe net loss from the cell is the difference between the flow over both boundaries:}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jrhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-20uhh
h jrhhh!jhNubj )}(hIw_{m} \mu_{m}\left(A_{i+1} \psi_{g, i+1, m}-A_{i} \psi_{g, i, m}\right) .h]h/Iw_{m} \mu_{m}\left(A_{i+1} \psi_{g, i+1, m}-A_{i} \psi_{g, i, m}\right) .}(hhh jubah}(h]jah]h]h]h]docnamejnumberKlabeleq9-1-20nowrapjyjzuhj h!jhMh jrhhjf}jh}jjsubh;)}(hThe loss in the spatial cell due to collisions is given by the product of the
centered angular flux (in per unit volume units) times the total macroscopic
cross section times the solid angle times the volume:h]h/The loss in the spatial cell due to collisions is given by the product of the
centered angular flux (in per unit volume units) times the total macroscopic
cross section times the solid angle times the volume:}(hjh j
hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jrhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-21uhh
h jrhhh!jhNubj )}(h6w_{m} \sigma_{g, i+1 / 2} V_{i} \psi_{g, i+1 / 2}, m .h]h/6w_{m} \sigma_{g, i+1 / 2} V_{i} \psi_{g, i+1 / 2}, m .}(hhh j"ubah}(h]j!ah]h]h]h]docnamejnumberKlabeleq9-1-21nowrapjyjzuhj h!jhMh jrhhjf}jh}j!jsubh;)}(hThe sources in direction *μ*\ :sub:`m` are given by the product of the solid
angle times the interval volume times the volume-averaged source (sum of
fixed, fission, and scattering) in the direction m:h](h/The sources in direction }(hThe sources in direction h j7hhh!NhNubhA)}(h*μ*h]h/μ}(hhh j@ubah}(h]h]h]h]h]uhh@h j7ubh/ }(h\ h j7hhh!NhNubh)}(h:sub:`m`h]h/m}(hhh jSubah}(h]h]h]h]h]uhhh j7ubh/ are given by the product of the solid
angle times the interval volume times the volume-averaged source (sum of
fixed, fission, and scattering) in the direction m:}(h are given by the product of the solid
angle times the interval volume times the volume-averaged source (sum of
fixed, fission, and scattering) in the direction m:h j7hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jrhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-22uhh
h jrhhh!jhNubj )}(hw_{m} V_{i} S_{g, i+1 / 2}, m .h]h/w_{m} V_{i} S_{g, i+1 / 2}, m .}(hhh jvubah}(h]juah]h]h]h]docnamejnumberKlabeleq9-1-22nowrapjyjzuhj h!jhMh jrhhjf}jh}jujlsubh;)}(hThe slab equation is obtained by using Eqs. :eq:`eq9-1-20`, :eq:`eq9-1-21`, and :eq:`eq9-1-22` and
substituting proper values for area and volume:h](h/,The slab equation is obtained by using Eqs. }(h,The slab equation is obtained by using Eqs. h jhhh!NhNubj)}(h:eq:`eq9-1-20`h]jc)}(hjh]h/eq9-1-20}(hhh jubah}(h]h](jneqeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjqreftypejrefexplicitrefwarnjeq9-1-20uhjh!jhMh jubh/, }(h, h jhhh!NhNubj)}(h:eq:`eq9-1-21`h]jc)}(hjh]h/eq9-1-21}(hhh jubah}(h]h](jneqeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjqreftypejrefexplicitrefwarnjeq9-1-21uhjh!jhMh jubh/, and }(h, and h jhhh!NhNubj)}(h:eq:`eq9-1-22`h]jc)}(hjh]h/eq9-1-22}(hhh jubah}(h]h](jneqeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjqreftypejrefexplicitrefwarnjeq9-1-22uhjh!jhMh jubh/4 and
substituting proper values for area and volume:}(h4 and
substituting proper values for area and volume:h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jrhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-23uhh
h jrhhh!jhNubj )}(hw_{m} \mu_{m}\left(\psi_{g, i+l, m}-\psi_{g, i, m}\right)+w_{m} \sigma_{g, i+1 / 2} \psi_{g, i+1 / 2, m}\left(x_{i+l}-x_{i}\right)=w_{m} S_{g, i+1 / 2^{\prime} m}\left(x_{i+l}-x_{i}\right) .h]h/w_{m} \mu_{m}\left(\psi_{g, i+l, m}-\psi_{g, i, m}\right)+w_{m} \sigma_{g, i+1 / 2} \psi_{g, i+1 / 2, m}\left(x_{i+l}-x_{i}\right)=w_{m} S_{g, i+1 / 2^{\prime} m}\left(x_{i+l}-x_{i}\right) .}(hhh j
ubah}(h]jah]h]h]h]docnamejnumberKlabeleq9-1-23nowrapjyjzuhj h!jhMh jrhhjf}jh}jjsubh;)}(hIn an MM angle quadrature set, there are MM of these equations and they are
coupled through the assumption on how the cell-centered flux relates to the
boundary angular fluxes, the sources, and the boundary conditions, as will be
discussed later.h]h/In an MM angle quadrature set, there are MM of these equations and they are
coupled through the assumption on how the cell-centered flux relates to the
boundary angular fluxes, the sources, and the boundary conditions, as will be
discussed later.}(hj$h j"hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jrhhubh)}(h.. _9-1-2-4-2:h]h}(h]h]h]h]h]hid29uhh
hMph jrhhh!jubeh}(h](&discrete-ordinates-equation-for-a-slabjieh]h](&discrete-ordinates equation for a slab 9-1-2-4-1eh]h]uhh#h jGhhh!jhMjf}jAj_sjh}jij_subh$)}(hhh](h))}(h4Discrete-ordinates equations for sphere and cylinderh]h/4Discrete-ordinates equations for sphere and cylinder}(hjKh jIhhh!NhNubah}(h]h]h]h]h]uhh(h jFhhh!jhMubh;)}(hThe development of the equations for these geometries is analogous to that for
the slab except that the leakage terms are more complicated. Consider
:numref:`fig9-1-4`.h](h/The development of the equations for these geometries is analogous to that for
the slab except that the leakage terms are more complicated. Consider
}(hThe development of the equations for these geometries is analogous to that for
the slab except that the leakage terms are more complicated. Consider
h jWhhh!NhNubj)}(h:numref:`fig9-1-4`h]jc)}(hjbh]h/fig9-1-4}(hhh jdubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh j`ubah}(h]h]h]h]h]refdocj refdomainjnreftypenumrefrefexplicitrefwarnjfig9-1-4uhjh!jhMh jWubh/.}(hjWh jWhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jFhhubh)}(h
.. _fig9-1-4:h]h}(h]h]h]h]h]hfig9-1-4uhh
hMyh jFhhh!jubj)}(hhh](j)}(hr.. figure:: figs/XSDRNPM/fig4.png
:align: center
:width: 400
Angular redistribution in spherical geometry.
h]h}(h]h]h]h]h]width400urifigs/XSDRNPM/fig4.pngj}jjsuhjh jh!jhMubj)}(h-Angular redistribution in spherical geometry.h]h/-Angular redistribution in spherical geometry.}(hjh jubah}(h]h]h]h]h]uhjh!jhMh jubeh}(h](id263jeh]h]fig9-1-4ah]h]jcenteruhjhMh jFhhh!jjf}jjsjh}jjsubh;)}(hXRecall that the directions are taken with reference to the radius vector
for a sphere. A particle traveling in direction *μ*\ :sub:`m` at *r*\ :sub:`i`s will
intersect the radius vector to the next point *r*\ :sub:`i`\ :sub:`+1` at a
different angle :math:`\mu_{m}^{*}`. The same effect also exists for the cylinder, though in
this case the direction coordinates are more complicated. Because of the
effect, a loss term is included for the “angular redistribution.” It is
defined in a manner analogous to Eq. :eq:`eq9-1-20` ash](h/yRecall that the directions are taken with reference to the radius vector
for a sphere. A particle traveling in direction }(hyRecall that the directions are taken with reference to the radius vector
for a sphere. A particle traveling in direction h jhhh!NhNubhA)}(h*μ*h]h/μ}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ }(h\ h jhhh!NhNubh)}(h:sub:`m`h]h/m}(hhh jubah}(h]h]h]h]h]uhhh jubh/ at }(h at h jhhh!NhNubhA)}(h*r*h]h/r}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ }(h\ h jubh)}(hJ:sub:`i`s will
intersect the radius vector to the next point *r*\ :sub:`i`h]h/Ci`s will
intersect the radius vector to the next point *r* :sub:`i}(hhh j ubah}(h]h]h]h]h]uhhh jubh/ }(hjh jubh)}(h :sub:`+1`h]h/+1}(hhh j ubah}(h]h]h]h]h]uhhh jubh/ at a
different angle }(h at a
different angle h jhhh!NhNubjr)}(h:math:`\mu_{m}^{*}`h]h/\mu_{m}^{*}}(hhh j' ubah}(h]h]h]h]h]uhjqh jubh/. The same effect also exists for the cylinder, though in
this case the direction coordinates are more complicated. Because of the
effect, a loss term is included for the “angular redistribution.” It is
defined in a manner analogous to Eq. }(h. The same effect also exists for the cylinder, though in
this case the direction coordinates are more complicated. Because of the
effect, a loss term is included for the “angular redistribution.” It is
defined in a manner analogous to Eq. h jhhh!NhNubj)}(h:eq:`eq9-1-20`h]jc)}(hj< h]h/eq9-1-20}(hhh j> ubah}(h]h](jneqeh]h]h]uhjbh j: ubah}(h]h]h]h]h]refdocj refdomainjqreftypejH refexplicitrefwarnjeq9-1-20uhjh!jhMh jubh/ as}(h ash jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jFhhubh definition_list)}(hhh]h definition_list_item)}(h,.. math::
:label: eq9-1-24
\alpha_{i+1 / 2}, m+1 / 2 \quad \psi_{g, i+1 / 2, m+1 / 2}-\alpha_{i+1 / 2, m-1 / 2} \psi_{g, i+1 / 2, m-1 / 2}
h](h term)}(h
,.. math::h]h/
,.. math::}(hjr h jp ubah}(h]h]h]h]h]uhjn h!jhMh jj ubh
definition)}(hhh](h
field_list)}(hhh]h field)}(hhh](h
field_name)}(hlabelh]h/label}(hj h j ubah}(h]h]h]h]h]uhj h j h!jhKubh
field_body)}(h eq9-1-24
h]h;)}(heq9-1-24h]h/eq9-1-24}(hj h j ubah}(h]h]h]h]h]uhh:h!jhMh j ubah}(h]h]h]h]h]uhj h j ubeh}(h]h]h]h]h]uhj h!jhMh j ubah}(h]h]h]h]h]uhj h j ubh;)}(ho\alpha_{i+1 / 2}, m+1 / 2 \quad \psi_{g, i+1 / 2, m+1 / 2}-\alpha_{i+1 / 2, m-1 / 2} \psi_{g, i+1 / 2, m-1 / 2}h]h/oalpha_{i+1 / 2}, m+1 / 2 quad psi_{g, i+1 / 2, m+1 / 2}-alpha_{i+1 / 2, m-1 / 2} psi_{g, i+1 / 2, m-1 / 2}}(ho\alpha_{i+1 / 2}, m+1 / 2 \quad \psi_{g, i+1 / 2, m+1 / 2}-\alpha_{i+1 / 2, m-1 / 2} \psi_{g, i+1 / 2, m-1 / 2}h j ubah}(h]h]h]h]h]uhh:h!jhMh j ubeh}(h]h]h]h]h]uhj~ h jj ubeh}(h]h]h]h]h]uhjh h!jhMh je ubah}(h]h]h]h]h]uhjc h jFhhh!jhNubh;)}(hXwhere the *α* coefficients are to be defined in such a manner as to
preserve particle balance. In this case one speaks of m+1 and m−½ as the
corresponding angles to *μ*\ :sub:`m` on the I + lth and ith boundaries,
respectively. (See :numref:`fig9-1-5`) Here we are interested in an angle
*μ\ m* at the center of interval i which redistributes to *μ*\ :sub:`m−½` at
boundary i and to *μ*\ :sub:`m+½` at boundary I + 1.h](h/
where the }(h
where the h j hhh!NhNubhA)}(h*α*h]h/α}(hhh j ubah}(h]h]h]h]h]uhh@h j ubh/ coefficients are to be defined in such a manner as to
preserve particle balance. In this case one speaks of m+1 and m−½ as the
corresponding angles to }(h coefficients are to be defined in such a manner as to
preserve particle balance. In this case one speaks of m+1 and m−½ as the
corresponding angles to h j hhh!NhNubhA)}(h*μ*h]h/μ}(hhh j!ubah}(h]h]h]h]h]uhh@h j ubh/ }(h\ h j hhh!NhNubh)}(h:sub:`m`h]h/m}(hhh j!ubah}(h]h]h]h]h]uhhh j ubh/9 on the I + lth and ith boundaries,
respectively. (See }(h9 on the I + lth and ith boundaries,
respectively. (See h j hhh!NhNubj)}(h:numref:`fig9-1-5`h]jc)}(hj(!h]h/fig9-1-5}(hhh j*!ubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh j&!ubah}(h]h]h]h]h]refdocj refdomainj4!reftypenumrefrefexplicitrefwarnjfig9-1-5uhjh!jhMh j ubh/%) Here we are interested in an angle
}(h%) Here we are interested in an angle
h j hhh!NhNubhA)}(h*μ\ m*h]h/μ m}(hhh jK!ubah}(h]h]h]h]h]uhh@h j ubh/5 at the center of interval i which redistributes to }(h5 at the center of interval i which redistributes to h j hhh!NhNubhA)}(h*μ*h]h/μ}(hhh j^!ubah}(h]h]h]h]h]uhh@h j ubh/ }(h\ h j ubh)}(h
:sub:`m−½`h]h/m−½}(hhh jq!ubah}(h]h]h]h]h]uhhh j ubh/ at
boundary i and to }(h at
boundary i and to h j hhh!NhNubhA)}(h*μ*h]h/μ}(hhh j!ubah}(h]h]h]h]h]uhh@h j ubh/ }(hj!h j ubh)}(h:sub:`m+½`h]h/m+½}(hhh j!ubah}(h]h]h]h]h]uhhh j ubh/ at boundary I + 1.}(h at boundary I + 1.h j hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jFhhubh)}(h
.. _fig9-1-5:h]h}(h]h]h]h]h]hfig9-1-5uhh
hMh jFhhh!jubj)}(hhh](j)}(h\.. figure:: figs/XSDRNPM/fig5.png
:align: center
:width: 500
Angular redistribution.
h]h}(h]h]h]h]h]width500urifigs/XSDRNPM/fig5.pngj}jj!suhjh j!h!jhMubj)}(hAngular redistribution.h]h/Angular redistribution.}(hj!h j!ubah}(h]h]h]h]h]uhjh!jhMh j!ubeh}(h](id264j!eh]h]fig9-1-5ah]h]jcenteruhjhMh jFhhh!jjf}j!j!sjh}j!j!subh;)}(hObviously, it is necessary that the net effect of all redistributing be zero, in
order to maintain particle balance. This condition is met ifh]h/Obviously, it is necessary that the net effect of all redistributing be zero, in
order to maintain particle balance. This condition is met if}(hj!h j!hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM h jFhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-25uhh
h jFhhh!jhNubj )}(h\sum_{m=1}^{M M} \alpha_{m-1 / 2} \psi_{m-1 / 2}+\alpha_{m+1 / 2} \psi_{m+1 / 2}=\alpha_{1 / 2} \psi_{1 / 2}+\alpha_{M M+1 / 2} \psi_{M M+1 / 2}=0 ,h]h/\sum_{m=1}^{M M} \alpha_{m-1 / 2} \psi_{m-1 / 2}+\alpha_{m+1 / 2} \psi_{m+1 / 2}=\alpha_{1 / 2} \psi_{1 / 2}+\alpha_{M M+1 / 2} \psi_{M M+1 / 2}=0 ,}(hhh j!ubah}(h]j!ah]h]h]h]docnamejnumberKlabeleq9-1-25nowrapjyjzuhj h!jhM#h jFhhjf}jh}j!j!subh;)}(h5where we have dropped the group and interval indexes.h]h/5where we have dropped the group and interval indexes.}(hj"h j"hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM(h jFhhubh;)}(hIn order to develop an expression for determining the α’s consider an infinite
medium with a constant isotropic flux. In this case, there is no leakage and
the transport equation reduces toh]h/In order to develop an expression for determining the α’s consider an infinite
medium with a constant isotropic flux. In this case, there is no leakage and
the transport equation reduces to}(hj""h j "hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM*h jFhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-26uhh
h jFhhh!jhNubj )}(h\sum_{t} \phi=S .h]h/\sum_{t} \phi=S .}(hhh j8"ubah}(h]j7"ah]h]h]h]docnamejnumberKlabeleq9-1-26nowrapjyjzuhj h!jhM.h jFhhjf}jh}j7"j."subh;)}(hThis condition requires thath]h/This condition requires that}(hjO"h jM"hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM3h jFhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-27uhh
h jFhhh!jhNubj )}(h\mu_{m} w_{m}\left(A_{i+1} \psi_{g, i+1, m}-A_{i} \psi_{g, i, m}\right)+\alpha_{i+1 / 2}, m+1 / 2 \quad \psi_{g, i+1 / 2}, m+1 / 2-\alpha_{i+1 / 2, m-1 / 2} \psi_{g, i+1 / 2, m-1 / 2}=0 ,h]h/\mu_{m} w_{m}\left(A_{i+1} \psi_{g, i+1, m}-A_{i} \psi_{g, i, m}\right)+\alpha_{i+1 / 2}, m+1 / 2 \quad \psi_{g, i+1 / 2}, m+1 / 2-\alpha_{i+1 / 2, m-1 / 2} \psi_{g, i+1 / 2, m-1 / 2}=0 ,}(hhh je"ubah}(h]jd"ah]h]h]h]docnamejnumberKlabeleq9-1-27nowrapjyjzuhj h!jhM5h jFhhjf}jh}jd"j["subh;)}(hXwhich when we note that all the *ψ* terms in the infinite medium case are equal becomesh](h/ which when we note that all the }(h which when we note that all the h jz"hhh!NhNubhA)}(h*ψ*h]h/ψ}(hhh j"ubah}(h]h]h]h]h]uhh@h jz"ubh/4 terms in the infinite medium case are equal becomes}(h4 terms in the infinite medium case are equal becomesh jz"hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM:h jFhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-28uhh
h jFhhh!jhNubj )}(hL\mu_{m} w_{m}\left(A_{i+1}-A_{i}\right)=-\alpha_{m+1 / 2}+\alpha_{m-1 / 2} ,h]h/L\mu_{m} w_{m}\left(A_{i+1}-A_{i}\right)=-\alpha_{m+1 / 2}+\alpha_{m-1 / 2} ,}(hhh j"ubah}(h]j"ah]h]h]h]docnamejnumberKlabeleq9-1-28nowrapjyjzuhj h!jhM<h jFhhjf}jh}j"j"subh;)}(h)which is a recursion relationship for α.h]h/)which is a recursion relationship for α.}(hj"h j"hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMAh jFhhubh;)}(hNFrom Eq. :eq:`eq9-1-25` we see that the conservation requirement can be met ifh](h/ From Eq. }(h From Eq. h j"hhh!NhNubj)}(h:eq:`eq9-1-25`h]jc)}(hj"h]h/eq9-1-25}(hhh j"ubah}(h]h](jneqeh]h]h]uhjbh j"ubah}(h]h]h]h]h]refdocj refdomainjqreftypej"refexplicitrefwarnjeq9-1-25uhjh!jhMCh j"ubh/7 we see that the conservation requirement can be met if}(h7 we see that the conservation requirement can be met ifh j"hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMCh jFhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-29uhh
h jFhhh!jhNubj )}(h#\alpha_{1 / 2}=\alpha_{M M+1 / 2}=0h]h/#\alpha_{1 / 2}=\alpha_{M M+1 / 2}=0}(hhh j#ubah}(h]j#ah]h]h]h]docnamejnumberKlabeleq9-1-29nowrapjyjzuhj h!jhMEh jFhhjf}jh}j#j"subh;)}(hX)for any values of flux, and is, therefore, used to evaluate the α’s along with
Eq. eq:`eq9-1-27` or eq:`eq9-1-28`. (Note that had we included the redistribution term in
the slab equation, Eq. eq:`eq9-1-28` would have given zeroes for the terms, which is
as one would expect for this geometry.)h](h/Yfor any values of flux, and is, therefore, used to evaluate the α’s along with
Eq. eq:}(hYfor any values of flux, and is, therefore, used to evaluate the α’s along with
Eq. eq:h j#hhh!NhNubh title_reference)}(h
`eq9-1-27`h]h/eq9-1-27}(hhh j%#ubah}(h]h]h]h]h]uhj##h j#ubh/ or eq:}(h or eq:h j#hhh!NhNubj$#)}(h
`eq9-1-28`h]h/eq9-1-28}(hhh j8#ubah}(h]h]h]h]h]uhj##h j#ubh/S. (Note that had we included the redistribution term in
the slab equation, Eq. eq:}(hS. (Note that had we included the redistribution term in
the slab equation, Eq. eq:h j#hhh!NhNubj$#)}(h
`eq9-1-28`h]h/eq9-1-28}(hhh jK#ubah}(h]h]h]h]h]uhj##h j#ubh/X would have given zeroes for the terms, which is
as one would expect for this geometry.)}(hX would have given zeroes for the terms, which is
as one would expect for this geometry.)h j#hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMJh jFhhubh;)}(hThe final discrete-ordinates expression for spheres and cylinders is then
derived by summing expressions Eqs. eq:`eq9-1-20`, eq:`eq9-1-24`, eq:`eq9-1-21` and setting it
equal to expression Eq. eq:`eq9-1-22`.h](h/qThe final discrete-ordinates expression for spheres and cylinders is then
derived by summing expressions Eqs. eq:}(hqThe final discrete-ordinates expression for spheres and cylinders is then
derived by summing expressions Eqs. eq:h jd#hhh!NhNubj$#)}(h
`eq9-1-20`h]h/eq9-1-20}(hhh jm#ubah}(h]h]h]h]h]uhj##h jd#ubh/, eq:}(h, eq:h jd#hhh!NhNubj$#)}(h
`eq9-1-24`h]h/eq9-1-24}(hhh j#ubah}(h]h]h]h]h]uhj##h jd#ubh/, eq:}(hj#h jd#ubj$#)}(h
`eq9-1-21`h]h/eq9-1-21}(hhh j#ubah}(h]h]h]h]h]uhj##h jd#ubh/+ and setting it
equal to expression Eq. eq:}(h+ and setting it
equal to expression Eq. eq:h jd#hhh!NhNubj$#)}(h
`eq9-1-22`h]h/eq9-1-22}(hhh j#ubah}(h]h]h]h]h]uhj##h jd#ubh/.}(hjWh jd#hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMOh jFhhubh)}(h.. _9-1-2-4-3:h]h}(h]h]h]h]h]hid30uhh
hMh jFhhh!jubeh}(h](4discrete-ordinates-equations-for-sphere-and-cylinderj:eh]h](4discrete-ordinates equations for sphere and cylinder 9-1-2-4-2eh]h]uhh#h jGhhh!jhMjf}j#j0sjh}j:j0subh$)}(hhh](h))}(h|Sn| quadratures for slabsh](jr)}(hjh]h/S_n}(hhh j#hhh!NhNubah}(h]h]h]h]h]uhjqh!NhNh j#hhubh/ quadratures for slabs}(h quadratures for slabsh j#hhh!NhNubeh}(h]h]h]h]h]uhh(h j#hhh!jhMVubh;)}(hXSDRNPM will automatically calculate quadrature sets for each of the 1-D
geometries, or a user can, if he wants, input a quadrature.h]h/XSDRNPM will automatically calculate quadrature sets for each of the 1-D
geometries, or a user can, if he wants, input a quadrature.}(hj#h j#hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMXh j#hhubh;)}(hIn the case of the 1-D slab, the quadrature is a double Gauss-Legendre set based
on recommendations from :cite:`carlson_discrete_1965`.h](h/iIn the case of the 1-D slab, the quadrature is a double Gauss-Legendre set based
on recommendations from }(hiIn the case of the 1-D slab, the quadrature is a double Gauss-Legendre set based
on recommendations from h j$hhh!NhNubj)}(hcarlson_discrete_1965h]j#)}(hj$h]h/[carlson_discrete_1965]}(hhh j
$ubah}(h]h]h]h]h]uhj"h j $ubah}(h]id31ah]j5ah]h]h] refdomainj:reftypej< reftargetj$refwarnsupport_smartquotesuhjh!jhM[h j$hhubh/.}(hjWh j$hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM[h j#hhubh;)}(hIThe ordering of the directions for a slab is shown in :numref:`fig9-1-6`.h](h/6The ordering of the directions for a slab is shown in }(h6The ordering of the directions for a slab is shown in h j0$hhh!NhNubj)}(h:numref:`fig9-1-6`h]jc)}(hj;$h]h/fig9-1-6}(hhh j=$ubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh j9$ubah}(h]h]h]h]h]refdocj refdomainjG$reftypenumrefrefexplicitrefwarnjfig9-1-6uhjh!jhM^h j0$ubh/.}(hjWh j0$hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM^h j#hhubh)}(h
.. _fig9-1-6:h]h}(h]h]h]h]h]hfig9-1-6uhh
hMh j#hhh!jubj)}(hhh](j)}(hw.. figure:: figs/XSDRNPM/fig6.png
:align: center
:width: 500
Ordering of |Sn| directions for slabs and spheres.
h]h}(h]h]h]h]h]width500urifigs/XSDRNPM/fig6.pngj}jj~$suhjh jn$h!jhMeubj)}(h2Ordering of |Sn| directions for slabs and spheres.h](h/Ordering of }(hOrdering of h j$ubjr)}(hjh]h/S_n}(hhh j$ubah}(h]h]h]h]h]uhjqh!NhNh j$ubh/" directions for slabs and spheres.}(h" directions for slabs and spheres.h j$ubeh}(h]h]h]h]h]uhjh!jhMeh jn$ubeh}(h](id265jm$eh]h]fig9-1-6ah]h]jcenteruhjhMeh j#hhh!jjf}j$jc$sjh}jm$jc$subh;)}(hXdNote that in referring to the quadratures for any of the geometries, we do not
attempt to define an explicit area on a unit sphere, but rather speak of
characteristic directions with associated weights. In the case of the slab, it
is convenient to think of “directions” which are shaped like cones, because of
the azimuthal symmetry around the x-axis.h]h/XdNote that in referring to the quadratures for any of the geometries, we do not
attempt to define an explicit area on a unit sphere, but rather speak of
characteristic directions with associated weights. In the case of the slab, it
is convenient to think of “directions” which are shaped like cones, because of
the azimuthal symmetry around the x-axis.}(hj$h j$hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMgh j#hhubh;)}(hXSIn an nth order quadrature, there are n +1 angles with the first angle being
taken at μ = -1.0. This first angle is not required for the slab, but is needed
for the curvilinear geometries because of the angular redistribution terms, as
will be noted later. It is included in the slab case for reasons of uniformity
of programming, etc.h]h/XSIn an nth order quadrature, there are n +1 angles with the first angle being
taken at μ = -1.0. This first angle is not required for the slab, but is needed
for the curvilinear geometries because of the angular redistribution terms, as
will be noted later. It is included in the slab case for reasons of uniformity
of programming, etc.}(hj$h j$hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMmh j#hhubh;)}(hUSeveral requirements are made regarding the angles and weights in the quadrature set.h]h/USeveral requirements are made regarding the angles and weights in the quadrature set.}(hj$h j$hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMsh j#hhubh;)}(hMThe arguments relating to angular redistribution can be expected to show thath]h/MThe arguments relating to angular redistribution can be expected to show that}(hj$h j$hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMuh j#hhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-30uhh
h j#hhh!jhNubj )}(h$\sum_{m=1}^{M M} \mu_{m} w_{m}=0.0 .h]h/$\sum_{m=1}^{M M} \mu_{m} w_{m}=0.0 .}(hhh j$ubah}(h]j$ah]h]h]h]docnamejnumberKlabeleq9-1-30nowrapjyjzuhj h!jhMwh j#hhjf}jh}j$j$subh;)}(hThis situation is ensured if the weight of the *μ* = −1.0 direction is
zero and the other directions and weights are symmetric about *μ* = 0.
(The *μ* = 0 direction is never included in the quadrature set because
of its singularity.)h](h//This situation is ensured if the weight of the }(h/This situation is ensured if the weight of the h j%hhh!NhNubhA)}(h*μ*h]h/μ}(hhh j%ubah}(h]h]h]h]h]uhh@h j%ubh/W = −1.0 direction is
zero and the other directions and weights are symmetric about }(hW = −1.0 direction is
zero and the other directions and weights are symmetric about h j%hhh!NhNubhA)}(h*μ*h]h/μ}(hhh j%ubah}(h]h]h]h]h]uhh@h j%ubh/
= 0.
(The }(h
= 0.
(The h j%hhh!NhNubhA)}(h*μ*h]h/μ}(hhh j2%ubah}(h]h]h]h]h]uhh@h j%ubh/U = 0 direction is never included in the quadrature set because
of its singularity.)}(hU = 0 direction is never included in the quadrature set because
of its singularity.)h j%hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM|h j#hhubh;)}(hFurther, it is required thath]h/Further, it is required that}(hjM%h jK%hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j#hhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-31uhh
h j#hhh!jhNubj )}(h\sum_{m=1}^{M M} w_{m}=1.0 .h]h/\sum_{m=1}^{M M} w_{m}=1.0 .}(hhh jc%ubah}(h]jb%ah]h]h]h]docnamejnumberKlabeleq9-1-31nowrapjyjzuhj h!jhMh j#hhjf}jh}jb%jY%subh;)}(hDue to the above normalization of the quadrature weights, the discrete
ordinates angular flux is not “per steradian” but rather “per
direction-weight”. The calculated angular flux can be converted to
steradians by dividing by 4π.h]h/Due to the above normalization of the quadrature weights, the discrete
ordinates angular flux is not “per steradian” but rather “per
direction-weight”. The calculated angular flux can be converted to
steradians by dividing by 4π.}(hjz%h jx%hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j#hhubh)}(h.. _9-1-2-4-4:h]h}(h]h]h]h]h]hid32uhh
hMh j#hhh!jubeh}(h](sn-quadratures-for-slabsj#eh]h](sn quadratures for slabs 9-1-2-4-3eh]h]uhh#h jGhhh!jhMVjf}j%j#sjh}j#j#subh$)}(hhh](h))}(h|Sn| quadratures for spheresh](jr)}(hjh]h/S_n}(hhh j%hhh!NhNubah}(h]h]h]h]h]uhjqh!NhNh j%hhubh/ quadratures for spheres}(h quadratures for spheresh j%hhh!NhNubeh}(h]h]h]h]h]uhh(h j%hhh!jhMubh;)}(hqThe quadratures generated for spheres are Gauss-Legendre coefficients as
recommended by :cite:`emett_morse_1975`.h](h/XThe quadratures generated for spheres are Gauss-Legendre coefficients as
recommended by }(hXThe quadratures generated for spheres are Gauss-Legendre coefficients as
recommended by h j%hhh!NhNubj)}(hemett_morse_1975h]j#)}(hj%h]h/[emett_morse_1975]}(hhh j%ubah}(h]h]h]h]h]uhj"h j%ubah}(h]id33ah]j5ah]h]h] refdomainj:reftypej< reftargetj%refwarnsupport_smartquotesuhjh!jhMh j%hhubh/.}(hjWh j%hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j%hhubh;)}(hMThe ordering and symmetry requirements for spheres are the same as for
slabs.h]h/MThe ordering and symmetry requirements for spheres are the same as for
slabs.}(hj%h j%hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j%hhubh;)}(hXFIn the case of the sphere, the initial (*μ* = −1.0) direction is
required, because the difference equations involve three unknown values
for each direction, *μ*\ :sub:`m`:*ψ*\ :sub:`m` and the fluxes at the two
“redistributed” angles *ψ*\ :sub:`m`\ :sub:`−½` and *ψ*\ :sub:`m`\ :sub:`+½`. It is
obvious that an angle along the radius will not involve the
redistribution; hence, the expression for this direction involves only
*ψ*\ (*μ* = −1.0) as unknowns. Angle 2 proceeds by assuming
*ψ*\ :sub:`2−½` is given by *ψ*\ :sub:`1` and also uses a weighted
diamond difference model to relate *ψ*\ :sub:`m`,\ *ψ* :sub:`m`\ :sub:`−½` and
*ψ*\ :sub:`m`\ :sub:`+½`, as will be described below. Subsequent angles will
then have values for *ψ*\ :sub:`m`\ :sub:`−½` calculated by the previous angle
equations.h](h/(In the case of the sphere, the initial (}(h(In the case of the sphere, the initial (h j%hhh!NhNubhA)}(h*μ*h]h/μ}(hhh j&ubah}(h]h]h]h]h]uhh@h j%ubh/v = −1.0) direction is
required, because the difference equations involve three unknown values
for each direction, }(hv = −1.0) direction is
required, because the difference equations involve three unknown values
for each direction, h j%hhh!NhNubhA)}(h*μ*h]h/μ}(hhh j&ubah}(h]h]h]h]h]uhh@h j%ubh/ }(h\ h j%hhh!NhNubh)}(h:sub:`m`h]h/m}(hhh j(&ubah}(h]h]h]h]h]uhhh j%ubh/:}(h:h j%hhh!NhNubhA)}(h*ψ*h]h/ψ}(hhh j;&ubah}(h]h]h]h]h]uhh@h j%ubh/ }(h\ h j%ubh)}(h:sub:`m`h]h/m}(hhh jN&ubah}(h]h]h]h]h]uhhh j%ubh/6 and the fluxes at the two
“redistributed” angles }(h6 and the fluxes at the two
“redistributed” angles h j%hhh!NhNubhA)}(h*ψ*h]h/ψ}(hhh ja&ubah}(h]h]h]h]h]uhh@h j%ubh/ }(h\ h j%ubh)}(h:sub:`m`h]h/m}(hhh jt&ubah}(h]h]h]h]h]uhhh j%ubh/ }(h\ h j%ubh)}(h:sub:`−½`h]h/−½}(hhh j&ubah}(h]h]h]h]h]uhhh j%ubh/ and }(h and h j%hhh!NhNubhA)}(h*ψ*h]h/ψ}(hhh j&ubah}(h]h]h]h]h]uhh@h j%ubh/ }(h\ h j%ubh)}(h:sub:`m`h]h/m}(hhh j&ubah}(h]h]h]h]h]uhhh j%ubh/ }(h\ h j%ubh)}(h
:sub:`+½`h]h/+½}(hhh j&ubah}(h]h]h]h]h]uhhh j%ubh/. It is
obvious that an angle along the radius will not involve the
redistribution; hence, the expression for this direction involves only
}(h. It is
obvious that an angle along the radius will not involve the
redistribution; hence, the expression for this direction involves only
h j%hhh!NhNubhA)}(h*ψ*h]h/ψ}(hhh j&ubah}(h]h]h]h]h]uhh@h j%ubh/ (}(h\ (h j%hhh!NhNubhA)}(h*μ*h]h/μ}(hhh j&ubah}(h]h]h]h]h]uhh@h j%ubh/8 = −1.0) as unknowns. Angle 2 proceeds by assuming
}(h8 = −1.0) as unknowns. Angle 2 proceeds by assuming
h j%hhh!NhNubhA)}(h*ψ*h]h/ψ}(hhh j&ubah}(h]h]h]h]h]uhh@h j%ubh/ }(h\ h j%ubh)}(h
:sub:`2−½`h]h/2−½}(hhh j'ubah}(h]h]h]h]h]uhhh j%ubh/
is given by }(h
is given by h j%hhh!NhNubhA)}(h*ψ*h]h/ψ}(hhh j'ubah}(h]h]h]h]h]uhh@h j%ubh/ }(h\ h j%ubh)}(h:sub:`1`h]h/1}(hhh j2'ubah}(h]h]h]h]h]uhhh j%ubh/= and also uses a weighted
diamond difference model to relate }(h= and also uses a weighted
diamond difference model to relate h j%hhh!NhNubhA)}(h*ψ*h]h/ψ}(hhh jE'ubah}(h]h]h]h]h]uhh@h j%ubh/ }(h\ h j%ubh)}(h:sub:`m`h]h/m}(hhh jX'ubah}(h]h]h]h]h]uhhh j%ubh/, }(h,\ h j%hhh!NhNubhA)}(h*ψ*h]h/ψ}(hhh jk'ubah}(h]h]h]h]h]uhh@h j%ubh/ }(hjh j%hhh!NhNubh)}(h:sub:`m`h]h/m}(hhh j}'ubah}(h]h]h]h]h]uhhh j%ubh/ }(h\ h j%ubh)}(h:sub:`−½`h]h/−½}(hhh j'ubah}(h]h]h]h]h]uhhh j%ubh/ and
}(h and
h j%hhh!NhNubhA)}(h*ψ*h]h/ψ}(hhh j'ubah}(h]h]h]h]h]uhh@h j%ubh/ }(h\ h j%ubh)}(h:sub:`m`h]h/m}(hhh j'ubah}(h]h]h]h]h]uhhh j%ubh/ }(h\ h j%ubh)}(h
:sub:`+½`h]h/+½}(hhh j'ubah}(h]h]h]h]h]uhhh j%ubh/J, as will be described below. Subsequent angles will
then have values for }(hJ, as will be described below. Subsequent angles will
then have values for h j%hhh!NhNubhA)}(h*ψ*h]h/ψ}(hhh j'ubah}(h]h]h]h]h]uhh@h j%ubh/ }(h\ h j%ubh)}(h:sub:`m`h]h/m}(hhh j'ubah}(h]h]h]h]h]uhhh j%ubh/ }(hj'&h j%ubh)}(h:sub:`−½`h]h/−½}(hhh j(ubah}(h]h]h]h]h]uhhh j%ubh/, calculated by the previous angle
equations.}(h, calculated by the previous angle
equations.h j%hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j%hhubh)}(h.. _9-1-2-4-5:h]h}(h]h]h]h]h]hid34uhh
hM h j%hhh!jubeh}(h](sn-quadratures-for-spheresj%eh]h](sn quadratures for spheres 9-1-2-4-4eh]h]uhh#h jGhhh!jhMjf}j+(j%sjh}j%j%subh$)}(hhh](h))}(h|Sn| quadratures for cylindersh](jr)}(hjh]h/S_n}(hhh j7(hhh!NhNubah}(h]h]h]h]h]uhjqh!NhNh j3(hhubh/ quadratures for cylinders}(h quadratures for cylindersh j3(hhh!NhNubeh}(h]h]h]h]h]uhh(h j0(hhh!jhMubh;)}(hThe quadrature sets for cylinders are more complicated (see
:numref:`fig9-1-2`) because the directions must be specified with two angles,
*ζ* and *η* where *α* ≡ sin *η* cos *ζ* and *β* ≡ cos *η*.h](h/Ordering of the directions for an S\ :sub:`6` cylindrical set.h](h/%Ordering of the directions for an S }(h%Ordering of the directions for an S\ h j)ubh)}(h:sub:`6`h]h/6}(hhh j)ubah}(h]h]h]h]h]uhhh j)ubh/ cylindrical set.}(h cylindrical set.h j)ubeh}(h]h]h]h]h]uhjh!jhMh j)ubeh}(h](id266j)eh]h]fig9-1-7ah]h]jcenteruhjhMh j0(hhh!jjf}j)j)sjh}j)j)subh;)}(hXCIn general, an n\ *th* order quadrature will contain n(n + 4)/4 angles.
The cosines, *μ*, and the weights are stored in two arrays internally in
the code; and, since the weights for the 1\ *st*, 4\ *th*, and
9\ *th* angles are zero, the cosines for the corresponding levels are
placed in these locations in the arrays.h](h/In general, an n }(hIn general, an n\ h j)hhh!NhNubhA)}(h*th*h]h/th}(hhh j)ubah}(h]h]h]h]h]uhh@h j)ubh/B order quadrature will contain n(n + 4)/4 angles.
The cosines, }(hB order quadrature will contain n(n + 4)/4 angles.
The cosines, h j)hhh!NhNubhA)}(h*μ*h]h/μ}(hhh j*ubah}(h]h]h]h]h]uhh@h j)ubh/e, and the weights are stored in two arrays internally in
the code; and, since the weights for the 1 }(he, and the weights are stored in two arrays internally in
the code; and, since the weights for the 1\ h j)hhh!NhNubhA)}(h*st*h]h/st}(hhh j%*ubah}(h]h]h]h]h]uhh@h j)ubh/, 4 }(h, 4\ h j)hhh!NhNubhA)}(h*th*h]h/th}(hhh j8*ubah}(h]h]h]h]h]uhh@h j)ubh/ , and
9 }(h , and
9\ h j)hhh!NhNubhA)}(h*th*h]h/th}(hhh jK*ubah}(h]h]h]h]h]uhh@h j)ubh/h angles are zero, the cosines for the corresponding levels are
placed in these locations in the arrays.}(hh angles are zero, the cosines for the corresponding levels are
placed in these locations in the arrays.h j)hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j0(hhubh;)}(hThe cylindrical sets are based on Gauss-Tschebyscheff schemes as
recommended by :cite:`emett_morse_1975` with Gaussian quadratures in *β* and Tschebyscheff
quadratures in α.h](h/PThe cylindrical sets are based on Gauss-Tschebyscheff schemes as
recommended by }(hPThe cylindrical sets are based on Gauss-Tschebyscheff schemes as
recommended by h jd*hhh!NhNubj)}(hemett_morse_1975h]j#)}(hjo*h]h/[emett_morse_1975]}(hhh jq*ubah}(h]h]h]h]h]uhj"h jm*ubah}(h]id35ah]j5ah]h]h] refdomainj:reftypej< reftargetjo*refwarnsupport_smartquotesuhjh!jhMh jd*hhubh/ with Gaussian quadratures in }(h with Gaussian quadratures in h jd*hhh!NhNubhA)}(h*β*h]h/β}(hhh j*ubah}(h]h]h]h]h]uhh@h jd*ubh/% and Tschebyscheff
quadratures in α.}(h% and Tschebyscheff
quadratures in α.h jd*hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j0(hhubh)}(h.. _9-1-2-5:h]h}(h]h]h]h]h]hid36uhh
hMDh j0(hhh!jubeh}(h](sn-quadratures-for-cylindersj$(eh]h](sn quadratures for cylinders 9-1-2-4-5eh]h]uhh#h jGhhh!jhMjf}j*j(sjh}j$(j(subeh}(h]('discrete-ordinates-difference-equationsj0eh]h]9-1-2-4ah]jpah]uhh#h j hhh!jhMjKjf}j*j&sjh}j0j&subh$)}(hhh](h))}(h@Weighted-difference formulation for discrete-ordinates equationsh]h/@Weighted-difference formulation for discrete-ordinates equations}(hj*h j*hhh!NhNubah}(h]h]h]h]h]uhh(h j*hhh!jhMubh;)}(hIn order to solve the discrete-ordinates equations, an assumption is required concerning the
relationship of the various flux terms: :math:`\psi_{i, m}, \psi_{i+1, m}, \psi_{i+1 / 2, m}, \psi_{i, m-1 / 2}, \psi_{i+1, m-1 / 2}`.h](h/In order to solve the discrete-ordinates equations, an assumption is required concerning the
relationship of the various flux terms: }(hIn order to solve the discrete-ordinates equations, an assumption is required concerning the
relationship of the various flux terms: h j*hhh!NhNubjr)}(h]:math:`\psi_{i, m}, \psi_{i+1, m}, \psi_{i+1 / 2, m}, \psi_{i, m-1 / 2}, \psi_{i+1, m-1 / 2}`h]h/U\psi_{i, m}, \psi_{i+1, m}, \psi_{i+1 / 2, m}, \psi_{i, m-1 / 2}, \psi_{i+1, m-1 / 2}}(hhh j*ubah}(h]h]h]h]h]uhjqh j*ubh/.}(hjWh j*hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j*hhubh;)}(hXJThe solution of the equations involves three major loops: an outer loop over
energy groups, a loop over angles, and a loop over the spatial mesh. The
spatial loop is made either from the origin to the outside boundary or from the
outside to the origin, depending on whether the angle is directed outward or
inward, respectively.h]h/XJThe solution of the equations involves three major loops: an outer loop over
energy groups, a loop over angles, and a loop over the spatial mesh. The
spatial loop is made either from the origin to the outside boundary or from the
outside to the origin, depending on whether the angle is directed outward or
inward, respectively.}(hj*h j*hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j*hhubh;)}(h~Two models are widely used for expressing flux relationships: (1) the step
model and (2) the diamond-difference linear model.h]h/~Two models are widely used for expressing flux relationships: (1) the step
model and (2) the diamond-difference linear model.}(hj
+h j+hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j*hhubh;)}(hThe “step model” is a histogram model whereby one sets the centered flux value
to the appropriate boundary value, depending on which way the mesh sweep is
going. If, for example, the sweep is to the right in space, thenh]h/The “step model” is a histogram model whereby one sets the centered flux value
to the appropriate boundary value, depending on which way the mesh sweep is
going. If, for example, the sweep is to the right in space, then}(hj+h j+hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j*hhubj )}(h\psi_{i+1 / 2, m}=\psi_{i, m}h]h/\psi_{i+1 / 2, m}=\psi_{i, m}}(hhh j$+ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh j*hhubh;)}(hor if to the left,h]h/or if to the left,}(hj8+h j6+hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j*hhubj )}(h!\psi_{i+1 / 2}, m=\psi_{i+1, m} .h]h/!\psi_{i+1 / 2}, m=\psi_{i+1, m} .}(hhh jD+ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh j*hhubh;)}(hLikewise, in angle:h]h/Likewise, in angle:}(hjX+h jV+hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j*hhubj )}(h0\psi_{\mathrm{i}+1 / 2}, m=\psi_{i+1, m-1 / 2} .h]h/0\psi_{\mathrm{i}+1 / 2}, m=\psi_{i+1, m-1 / 2} .}(hhh jd+ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh j*hhubh;)}(hThe step model involves a very crude approximation, but has the marked advantage
of helping to ensure positivity of flux values as long as scattering sources are
positive.h]h/The step model involves a very crude approximation, but has the marked advantage
of helping to ensure positivity of flux values as long as scattering sources are
positive.}(hjx+h jv+hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j*hhubh;)}(hcIn the “diamond-difference” model, the centered fluxes are assumed linear with
the edge values:h]h/cIn the “diamond-difference” model, the centered fluxes are assumed linear with
the edge values:}(hj+h j+hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j*hhubj )}(h\begin{array}{c}
\psi_{i+1 / 2}=0.5\left(\psi_{i}+\psi_{i+1}\right) \\
\psi_{m}=0.5\left(\psi_{m-1 / 2}+\psi_{m+1 / 2}\right)
\end{array}h]h/\begin{array}{c}
\psi_{i+1 / 2}=0.5\left(\psi_{i}+\psi_{i+1}\right) \\
\psi_{m}=0.5\left(\psi_{m-1 / 2}+\psi_{m+1 / 2}\right)
\end{array}}(hhh j+ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh j*hhubh;)}(hXUnfortunately, though the linear model is clearly a better model than
the step model, care must be taken by selecting a fine spatial mesh, or
the linear extrapolation can lead to negative flux values. In some
cases, the situation is so severe that it is impractical to take enough
mesh points to eliminate the problems. Because of these difficulties
XSDRNPM uses a different approach, as described below.h]h/XUnfortunately, though the linear model is clearly a better model than
the step model, care must be taken by selecting a fine spatial mesh, or
the linear extrapolation can lead to negative flux values. In some
cases, the situation is so severe that it is impractical to take enough
mesh points to eliminate the problems. Because of these difficulties
XSDRNPM uses a different approach, as described below.}(hj+h j+hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j*hhubh;)}(hThe weighted diamond difference model :cite:`rhoades_new_1977` was developed in an attempt
to take advantage of the “correctness” of the linear model, while
retaining the positive flux advantages of the step model.h](h/&The weighted diamond difference model }(h&The weighted diamond difference model h j+hhh!NhNubj)}(hrhoades_new_1977h]j#!)}(hj+h]h/[rhoades_new_1977]}(hhh j+ubah}(h]h]h]h]h]uhj"h j+ubah}(h]id37ah]j5ah]h]h] refdomainj:reftypej< reftargetj+refwarnsupport_smartquotesuhjh!jhMh j+hhubh/ was developed in an attempt
to take advantage of the “correctness” of the linear model, while
retaining the positive flux advantages of the step model.}(h was developed in an attempt
to take advantage of the “correctness” of the linear model, while
retaining the positive flux advantages of the step model.h j+hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j*hhubh;)}(hA solution in some |Sn| codes is to use the linear model in all cases
where positive fluxes are obtained and to revert to step model
otherwise. Unfortunately, this method leads to artificial distortions in
the fluxes.h](h/A solution in some }(hA solution in some h j+hhh!NhNubjr)}(hjh]h/S_n}(hhh j+hhh!NhNubah}(h]h]h]h]h]uhjqh!NhNh j+hhubh/ codes is to use the linear model in all cases
where positive fluxes are obtained and to revert to step model
otherwise. Unfortunately, this method leads to artificial distortions in
the fluxes.}(h codes is to use the linear model in all cases
where positive fluxes are obtained and to revert to step model
otherwise. Unfortunately, this method leads to artificial distortions in
the fluxes.h j+hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j*hhubh;)}(hNote that if one writesh]h/Note that if one writes}(hj,h j,hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j*hhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-32uhh
h j*hhh!jhNubj )}(h*\psi_{i+1 / 2}=a \psi_{i}+(1-a) \psi_{i+1}h]h/*\psi_{i+1 / 2}=a \psi_{i}+(1-a) \psi_{i+1}}(hhh j,ubah}(h]j,ah]h]h]h]docnamejnumberKlabeleq9-1-32nowrapjyjzuhj h!jhMh j*hhjf}jh}j,j,subh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-33uhh
h j*hhh!jhNubj )}(h.\psi_{m}=b \psi_{m-1 / 2}+(1-b) \psi_{m+1 / 2}h]h/.\psi_{m}=b \psi_{m-1 / 2}+(1-b) \psi_{m+1 / 2}}(hhh j;,ubah}(h]j:,ah]h]h]h]docnamejnumberK labeleq9-1-33nowrapjyjzuhj h!jhMh j*hhjf}jh}j:,j1,subh;)}(hthat the same expression can be used to express linear or step model
(e.g., *a = b =* ½ is equivalent to linear, while a = b = 1.0 can be
used for the step model).h](h/Mthat the same expression can be used to express linear or step model
(e.g., }(hMthat the same expression can be used to express linear or step model
(e.g., h jP,hhh!NhNubhA)}(h*a = b =*h]h/
a = b =}(hhh jY,ubah}(h]h]h]h]h]uhh@h jP,ubh/T ½ is equivalent to linear, while a = b = 1.0 can be
used for the step model).}(hT ½ is equivalent to linear, while a = b = 1.0 can be
used for the step model).h jP,hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j*hhubh;)}(hIn the weighted model, the intention is to use the linear model when
fluxes are positive but to select values for a and b in the rangeh]h/In the weighted model, the intention is to use the linear model when
fluxes are positive but to select values for a and b in the range}(hjt,h jr,hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j*hhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-34uhh
h j*hhh!jhNubj )}(h$1 / 2 \leq a \text { or } b \leq 1.0h]h/$1 / 2 \leq a \text { or } b \leq 1.0}(hhh j,ubah}(h]j,ah]h]h]h]docnamejnumberK!labeleq9-1-34nowrapjyjzuhj h!jhMh j*hhjf}jh}j,j,subh;)}(h2that ensure positivity, if the source is positive.h]h/2that ensure positivity, if the source is positive.}(hj,h j,hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM$h j*hhubh;)}(hAt this point, it is convenient to rewrite the discrete-ordinates
expression in a simplified notation, without the obvious subscripts on
energy group, angle, etc.h]h/At this point, it is convenient to rewrite the discrete-ordinates
expression in a simplified notation, without the obvious subscripts on
energy group, angle, etc.}(hj,h j,hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM&h j*hhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-35uhh
h j*hhh!jhNubj )}(hw \mu\left(A_{i+1} \psi_{i+1}-A_{i} \psi_{i}\right)+\alpha_{m+1 / 2} \psi_{m+1 / 2}-\alpha_{m-1 / 2} \psi_{m-1 / 2}+w \sigma V \psi=w V S .h]h/w \mu\left(A_{i+1} \psi_{i+1}-A_{i} \psi_{i}\right)+\alpha_{m+1 / 2} \psi_{m+1 / 2}-\alpha_{m-1 / 2} \psi_{m-1 / 2}+w \sigma V \psi=w V S .}(hhh j,ubah}(h]j,ah]h]h]h]docnamejnumberK"labeleq9-1-35nowrapjyjzuhj h!jhM*h j*hhjf}jh}j,j,subh;)}(hCombining Eqs. :eq:`eq9-1-35` and :eq:`eq9-1-32` or :eq:`eq9-1-33` yields the following expressions for
*ψ*\ :sub:`l+1` and *ψ\ m*\ :sub:`+½`:h](h/Combining Eqs. }(hCombining Eqs. h j,hhh!NhNubj)}(h:eq:`eq9-1-35`h]jc)}(hj,h]h/eq9-1-35}(hhh j,ubah}(h]h](jneqeh]h]h]uhjbh j,ubah}(h]h]h]h]h]refdocj refdomainjqreftypej,refexplicitrefwarnjeq9-1-35uhjh!jhM/h j,ubh/ and }(h and h j,hhh!NhNubj)}(h:eq:`eq9-1-32`h]jc)}(hj-h]h/eq9-1-32}(hhh j
-ubah}(h]h](jneqeh]h]h]uhjbh j-ubah}(h]h]h]h]h]refdocj refdomainjqreftypej-refexplicitrefwarnjeq9-1-32uhjh!jhM/h j,ubh/ or }(h or h j,hhh!NhNubj)}(h:eq:`eq9-1-33`h]jc)}(hj+-h]h/eq9-1-33}(hhh j--ubah}(h]h](jneqeh]h]h]uhjbh j)-ubah}(h]h]h]h]h]refdocj refdomainjqreftypej7-refexplicitrefwarnjeq9-1-33uhjh!jhM/h j,ubh/' yields the following expressions for
}(h' yields the following expressions for
h j,hhh!NhNubhA)}(h*ψ*h]h/ψ}(hhh jL-ubah}(h]h]h]h]h]uhh@h j,ubh/ }(h\ h j,hhh!NhNubh)}(h
:sub:`l+1`h]h/l+1}(hhh j_-ubah}(h]h]h]h]h]uhhh j,ubh/ and }(hj-h j,ubhA)}(h*ψ\ m*h]h/ψ m}(hhh jq-ubah}(h]h]h]h]h]uhh@h j,ubh/ }(hj^-h j,ubh)}(h
:sub:`+½`h]h/+½}(hhh j-ubah}(h]h]h]h]h]uhhh j,ubh/:}(hj:&h j,hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM/h j*hhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-36uhh
h j*hhh!jhNubj )}(h[\psi_{i+1}=\frac{S V+C_{2} \psi_{m-1 / 2}+\left[\mu A_{i}-(1-a) D_{1}\right] \psi_{i}}{a D}h]h/[\psi_{i+1}=\frac{S V+C_{2} \psi_{m-1 / 2}+\left[\mu A_{i}-(1-a) D_{1}\right] \psi_{i}}{a D}}(hhh j-ubah}(h]j-ah]h]h]h]docnamejnumberK#labeleq9-1-36nowrapjyjzuhj h!jhM2h j*hhjf}jh}j-j-subh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-37uhh
h j*hhh!jhNubj )}(hr\psi_{m+1 / 2}=\frac{S V+C_{1} \psi_{i}+\left[\frac{\alpha_{m-1 / 2}}{w}-(1-b) D_{2}\right] \psi_{m-1 / 2}}{b D} ,h]h/r\psi_{m+1 / 2}=\frac{S V+C_{1} \psi_{i}+\left[\frac{\alpha_{m-1 / 2}}{w}-(1-b) D_{2}\right] \psi_{m-1 / 2}}{b D} ,}(hhh j-ubah}(h]j-ah]h]h]h]docnamejnumberK$labeleq9-1-37nowrapjyjzuhj h!jhM7h j*hhjf}jh}j-j-subh;)}(hwhereh]h/where}(hj-h j-hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM<h j*hhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-38uhh
h j*hhh!jhNubj )}(hBC_{1}=\mu\left[A_{i}+A_{i+1}\left(\frac{1}{\alpha}-1\right)\right]h]h/BC_{1}=\mu\left[A_{i}+A_{i+1}\left(\frac{1}{\alpha}-1\right)\right]}(hhh j-ubah}(h]j-ah]h]h]h]docnamejnumberK%labeleq9-1-38nowrapjyjzuhj h!jhM>h j*hhjf}jh}j-j-subh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-39uhh
h j*hhh!jhNubj )}(hUC_{2}=\frac{\alpha_{m-1 / 2}}{w}+\frac{\alpha_{m+1 / 2}}{w}\left(\frac{1}{b}-1\right)h]h/UC_{2}=\frac{\alpha_{m-1 / 2}}{w}+\frac{\alpha_{m+1 / 2}}{w}\left(\frac{1}{b}-1\right)}(hhh j.ubah}(h]j.ah]h]h]h]docnamejnumberK&labeleq9-1-39nowrapjyjzuhj h!jhMCh j*hhjf}jh}j.j.subh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-40uhh
h j*hhh!jhNubj )}(hAD=\Sigma V_{i}+\frac{\mu A_{i+1}}{a}+\frac{\alpha_{m+1 / 2}}{w b}h]h/AD=\Sigma V_{i}+\frac{\mu A_{i+1}}{a}+\frac{\alpha_{m+1 / 2}}{w b}}(hhh j/.ubah}(h]j..ah]h]h]h]docnamejnumberK'labeleq9-1-40nowrapjyjzuhj h!jhMHh j*hhjf}jh}j..j%.subh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-41uhh
h j*hhh!jhNubj )}(hD_{1}=D-\frac{\mu A_{i+1}}{a}h]h/D_{1}=D-\frac{\mu A_{i+1}}{a}}(hhh jN.ubah}(h]jM.ah]h]h]h]docnamejnumberK(labeleq9-1-41nowrapjyjzuhj h!jhMMh j*hhjf}jh}jM.jD.subh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-42uhh
h j*hhh!jhNubj )}(h&D_{2}=D-\frac{\alpha_{m+1 / 2}}{w b} .h]h/&D_{2}=D-\frac{\alpha_{m+1 / 2}}{w b} .}(hhh jm.ubah}(h]jl.ah]h]h]h]docnamejnumberK)labeleq9-1-42nowrapjyjzuhj h!jhMRh j*hhjf}jh}jl.jc.subh;)}(hXIn determining a and b, the “theta-weighted” model uses arbitrary
multipliers *θ*\ :sub:`s` on SV and *θ*\ :sub:`n` on the
*C*\ :sub:`2`\ ψ\ m\ :sub:`−½` or *C*\ :sub:`1`\ *ψ*\ :sub:`i` terms in Eqs. :eq:`eq9-1-36` and :eq:`eq9-1-37`.
(In :cite:`tomlinson_flux_1980`, a thorough discussion is given on the history of using
different choices of *θ\ s* and *θ\ n* and the advantages and
disadvantages of each method.) In XSDRNPM, a value of 0.9 is used for
*θ\ s* and *θ\ n* following the practice in the DOT-IV code.7h](h/RIn determining a and b, the “theta-weighted” model uses arbitrary
multipliers }(hRIn determining a and b, the “theta-weighted” model uses arbitrary
multipliers h j.hhh!NhNubhA)}(h*θ*h]h/θ}(hhh j.ubah}(h]h]h]h]h]uhh@h j.ubh/ }(h\ h j.hhh!NhNubh)}(h:sub:`s`h]h/s}(hhh j.ubah}(h]h]h]h]h]uhhh j.ubh/ on SV and }(h on SV and h j.hhh!NhNubhA)}(h*θ*h]h/θ}(hhh j.ubah}(h]h]h]h]h]uhh@h j.ubh/ }(h\ h j.ubh)}(h:sub:`n`h]h/n}(hhh j.ubah}(h]h]h]h]h]uhhh j.ubh/ on the
}(h on the
h j.hhh!NhNubhA)}(h*C*h]h/C}(hhh j.ubah}(h]h]h]h]h]uhh@h j.ubh/ }(h\ h j.ubh)}(h:sub:`2`h]h/2}(hhh j.ubah}(h]h]h]h]h]uhhh j.ubh/ ψ m }(h \ ψ\ m\ h j.hhh!NhNubh)}(h:sub:`−½`h]h/−½}(hhh j.ubah}(h]h]h]h]h]uhhh j.ubh/ or }(h or h j.hhh!NhNubhA)}(h*C*h]h/C}(hhh j/ubah}(h]h]h]h]h]uhh@h j.ubh/ }(h\ h j.ubh)}(h:sub:`1`h]h/1}(hhh j#/ubah}(h]h]h]h]h]uhhh j.ubh/ }(h\ h j.ubhA)}(h*ψ*h]h/ψ}(hhh j6/ubah}(h]h]h]h]h]uhh@h j.ubh/ }(hj.h j.ubh)}(h:sub:`i`h]h/i}(hhh jH/ubah}(h]h]h]h]h]uhhh j.ubh/ terms in Eqs. }(h terms in Eqs. h j.hhh!NhNubj)}(h:eq:`eq9-1-36`h]jc)}(hj]/h]h/eq9-1-36}(hhh j_/ubah}(h]h](jneqeh]h]h]uhjbh j[/ubah}(h]h]h]h]h]refdocj refdomainjqreftypeji/refexplicitrefwarnjeq9-1-36uhjh!jhMWh j.ubh/ and }(h and h j.hhh!NhNubj)}(h:eq:`eq9-1-37`h]jc)}(hj/h]h/eq9-1-37}(hhh j/ubah}(h]h](jneqeh]h]h]uhjbh j~/ubah}(h]h]h]h]h]refdocj refdomainjqreftypej/refexplicitrefwarnjeq9-1-37uhjh!jhMWh j.ubh/.
(In }(h.
(In h j.hhh!NhNubj)}(htomlinson_flux_1980h]j#)}(hj/h]h/[tomlinson_flux_1980]}(hhh j/ubah}(h]h]h]h]h]uhj"h j/ubah}(h]id38ah]j5ah]h]h] refdomainj:reftypej< reftargetj/refwarnsupport_smartquotesuhjh!jhMWh j.hhubh/N, a thorough discussion is given on the history of using
different choices of }(hN, a thorough discussion is given on the history of using
different choices of h j.hhh!NhNubhA)}(h*θ\ s*h]h/θ s}(hhh j/ubah}(h]h]h]h]h]uhh@h j.ubh/ and }(h and h j.hhh!NhNubhA)}(h*θ\ n*h]h/θ n}(hhh j/ubah}(h]h]h]h]h]uhh@h j.ubh/_ and the advantages and
disadvantages of each method.) In XSDRNPM, a value of 0.9 is used for
}(h_ and the advantages and
disadvantages of each method.) In XSDRNPM, a value of 0.9 is used for
h j.hhh!NhNubhA)}(h*θ\ s*h]h/θ s}(hhh j/ubah}(h]h]h]h]h]uhh@h j.ubh/ and }(hj/h j.ubhA)}(h*θ\ n*h]h/θ n}(hhh j/ubah}(h]h]h]h]h]uhh@h j.ubh/, following the practice in the DOT-IV code.7}(h, following the practice in the DOT-IV code.7h j.hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMWh j*hhubh;)}(hxFor *ψ*\ :sub:`i`\ :sub:`+1` in Eq. :eq:`eq9-1-36` to be positive, the numerator should be
positive, thereby requiringh](h/For }(hFor h j0hhh!NhNubhA)}(h*ψ*h]h/ψ}(hhh j0ubah}(h]h]h]h]h]uhh@h j0ubh/ }(h\ h j0hhh!NhNubh)}(h:sub:`i`h]h/i}(hhh j00ubah}(h]h]h]h]h]uhhh j0ubh/ }(hj/0h j0ubh)}(h :sub:`+1`h]h/+1}(hhh jB0ubah}(h]h]h]h]h]uhhh j0ubh/ in Eq. }(h in Eq. h j0hhh!NhNubj)}(h:eq:`eq9-1-36`h]jc)}(hjW0h]h/eq9-1-36}(hhh jY0ubah}(h]h](jneqeh]h]h]uhjbh jU0ubah}(h]h]h]h]h]refdocj refdomainjqreftypejc0refexplicitrefwarnjeq9-1-36uhjh!jhM_h j0ubh/D to be positive, the numerator should be
positive, thereby requiring}(hD to be positive, the numerator should be
positive, thereby requiringh j0hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM_h j*hhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-43uhh
h j*hhh!jhNubj )}(hF\left[\mu A_{i}-(1-a) D_{1}\right] \psi_{i}~~ 0) = 0, etc.).h](h/1. Vacuum boundary ─ all angular fluxes that are directed inward at the
boundary are set to zero (e.g., at the left-hand boundary of slab,
}(h1. Vacuum boundary ─ all angular fluxes that are directed inward at the
boundary are set to zero (e.g., at the left-hand boundary of slab,
h jR1hhh!NhNubhA)}(h*ψ*h]h/ψ}(hhh j[1ubah}(h]h]h]h]h]uhh@h jR1ubh/ (μ > 0) = 0, etc.).}(h\ (μ > 0) = 0, etc.).h jR1hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM|h j31hhubh;)}(h2. Reflected boundary ─ the incoming angular flux at a boundary is set
equal to the outgoing angular flux in the reflected direction
(e.g., at the left-hand boundary of a slab),h]h/2. Reflected boundary ─ the incoming angular flux at a boundary is set
equal to the outgoing angular flux in the reflected direction
(e.g., at the left-hand boundary of a slab),}(hjv1h jt1hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j31hhubj )}(h(\psi_{i n}(\mu)=\psi_{\text {out}}(-\mu)h]h/(\psi_{i n}(\mu)=\psi_{\text {out}}(-\mu)}(hhh j1ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh j31hhubh;)}(h3. Periodic boundary ─ the incoming angular flux at a boundary is set
equal to the outgoing angular flux in the same angle at the opposite
boundary.h]h/3. Periodic boundary ─ the incoming angular flux at a boundary is set
equal to the outgoing angular flux in the same angle at the opposite
boundary.}(hj1h j1hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j31hhubh;)}(h4. White boundary ─ the angular fluxes of all incoming angles on a
boundary are set equal to a constant value such that the net flow across
the boundary is zero, that is,h]h/4. White boundary ─ the angular fluxes of all incoming angles on a
boundary are set equal to a constant value such that the net flow across
the boundary is zero, that is,}(hj1h j1hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j31hhubj )}(hc\psi_{i n}=\frac{\sum_{m}^{o u t} w_{m} \mu_{m} \psi_{m}}{\sum_{m}^{i n} w_{m}\left|\mu_{m}\right|}h]h/c\psi_{i n}=\frac{\sum_{m}^{o u t} w_{m} \mu_{m} \psi_{m}}{\sum_{m}^{i n} w_{m}\left|\mu_{m}\right|}}(hhh j1ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh j31hhubj)}(hhh]h}(h]h]h]h]h]jyjzuhjh j31hhh!jhMubh block_quote)}(hhh]h;)}(hThis boundary condition is generally used as an outer-boundary
condition for cell calculation of cylinders and spheres that occur in
lattice geometries.h]h/This boundary condition is generally used as an outer-boundary
condition for cell calculation of cylinders and spheres that occur in
lattice geometries.}(hj1h j1ubah}(h]h]h]h]h]uhh:h!jhMh j1ubah}(h]h]h]h]h]uhj1h j31hhh!jhNubh;)}(h5. Albedo boundary ─ this option is for the white boundary condition
except that a user-supplied group-dependent albedo multiplies the
incoming angular fluxes. This option is rarely used, as it is difficult
to relate to most practical situations.h]h/5. Albedo boundary ─ this option is for the white boundary condition
except that a user-supplied group-dependent albedo multiplies the
incoming angular fluxes. This option is rarely used, as it is difficult
to relate to most practical situations.}(hj1h j1hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j31hhubh)}(h.. _9-1-2-7:h]h}(h]h]h]h]h]hid40uhh
hMh j31hhh!jubeh}(h](boundary-conditionsj'1eh]h]9-1-2-6ah]boundary conditionsah]uhh#h j hhh!jhMwjKjf}j2j1sjh}j'1j1subh$)}(hhh](h))}(h
Fixed sourcesh]h/
Fixed sources}(hj
2h j2hhh!NhNubah}(h]h]h]h]h]uhh(h j2hhh!jhMubh;)}(hHTwo types of inhomogeneous or fixed sources can be specified in XSDRNPM.h]h/HTwo types of inhomogeneous or fixed sources can be specified in XSDRNPM.}(hj2h j2hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j2hhubh;)}(hIn the first case, an isotropic group-dependent volumetric source can be
specified for any or all spatial intervals in a system.h]h/In the first case, an isotropic group-dependent volumetric source can be
specified for any or all spatial intervals in a system.}(hj)2h j'2hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j2hhubh;)}(hXIn the second case, an angle- and group-dependent boundary source can be
specified for any or all boundaries between spatial intervals in a
system, excepting the left-most boundary. In this case, one specifies
not a source but a flux condition on the boundary. If one uses the
“track length” definition for flux, it is easy to show that the flux
condition is related to a source condition byh]h/XIn the second case, an angle- and group-dependent boundary source can be
specified for any or all boundaries between spatial intervals in a
system, excepting the left-most boundary. In this case, one specifies
not a source but a flux condition on the boundary. If one uses the
“track length” definition for flux, it is easy to show that the flux
condition is related to a source condition by}(hj72h j52hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j2hhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-45uhh
h j2hhh!jhNubj )}(h$\psi_{m}^{s}=\frac{S_{m}}{\mu_{m}} .h]h/$\psi_{m}^{s}=\frac{S_{m}}{\mu_{m}} .}(hhh jM2ubah}(h]jL2ah]h]h]h]docnamejnumberK,labeleq9-1-45nowrapjyjzuhj h!jhMh j2hhjf}jh}jL2jC2subh;)}(h(This equation says that an isotropic source on a boundary would be input as a constant divided by the cosine
of the direction.)h]h/(This equation says that an isotropic source on a boundary would be input as a constant divided by the cosine
of the direction.)}(hjd2h jb2hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j2hhubh;)}(hIn conventional fixed-source calculations, the total fixed source in the system
can be normalized to an input parameter, XNF. In the volumetric source case,
the source values will be normalized such thath]h/In conventional fixed-source calculations, the total fixed source in the system
can be normalized to an input parameter, XNF. In the volumetric source case,
the source values will be normalized such that}(hjr2h jp2hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j2hhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-46uhh
h j2hhh!jhNubj )}(h:X N F=\sum_{g=1}^{I G M} \sum_{i=1}^{I M} Q_{g, i} V_{i} ,h]h/:X N F=\sum_{g=1}^{I G M} \sum_{i=1}^{I M} Q_{g, i} V_{i} ,}(hhh j2ubah}(h]j2ah]h]h]h]docnamejnumberK-labeleq9-1-46nowrapjyjzuhj h!jhMh j2hhjf}jh}j2j~2subh;)}(h and in the boundary source case,h]h/ and in the boundary source case,}(hj2h j2hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j2hhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-47uhh
h j2hhh!jhNubj )}(heX N F=\sum_{g=1}^{I G M} \sum_{i=1}^{I M} A_{i+1} \sum_{m=1}^{M M} \mu_{m} \psi_{m}^{s}(g, i) w_{m} .h]h/eX N F=\sum_{g=1}^{I G M} \sum_{i=1}^{I M} A_{i+1} \sum_{m=1}^{M M} \mu_{m} \psi_{m}^{s}(g, i) w_{m} .}(hhh j2ubah}(h]j2ah]h]h]h]docnamejnumberK.labeleq9-1-47nowrapjyjzuhj h!jhMh j2hhjf}jh}j2j2subh;)}(hiIn the case where both volumetric and boundary sources are specified, the two
sums are normalized to XNF.h]h/iIn the case where both volumetric and boundary sources are specified, the two
sums are normalized to XNF.}(hj2h j2hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j2hhubh;)}(hXThe fixed source for a generalized adjoint calculation corresponds to a
particular response ratio of interest. The generalized adjoint equation only
has a solution for responses that are ratios of linear functionals of the flux,
and in this case the source will contain both positive and negative components.
These types of sources are described in more detail in ref:`9-1-2-15-1` and in
the SAMS chapter, in *Generalized Perturbation Theory*.h](h/XqThe fixed source for a generalized adjoint calculation corresponds to a
particular response ratio of interest. The generalized adjoint equation only
has a solution for responses that are ratios of linear functionals of the flux,
and in this case the source will contain both positive and negative components.
These types of sources are described in more detail in ref:}(hXqThe fixed source for a generalized adjoint calculation corresponds to a
particular response ratio of interest. The generalized adjoint equation only
has a solution for responses that are ratios of linear functionals of the flux,
and in this case the source will contain both positive and negative components.
These types of sources are described in more detail in ref:h j2hhh!NhNubj$#)}(h`9-1-2-15-1`h]h/
9-1-2-15-1}(hhh j2ubah}(h]h]h]h]h]uhj##h j2ubh/ and in
the SAMS chapter, in }(h and in
the SAMS chapter, in h j2hhh!NhNubhA)}(h!*Generalized Perturbation Theory*h]h/Generalized Perturbation Theory}(hhh j2ubah}(h]h]h]h]h]uhh@h j2ubh/.}(hjWh j2hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j2hhubh)}(h.. _9-1-2-8:h]h}(h]h]h]h]h]hid41uhh
hMMh j2hhh!jubeh}(h](
fixed-sourcesj1eh]h](
fixed sources9-1-2-7eh]h]uhh#h j hhh!jhMjf}j3j1sjh}j1j1subh$)}(hhh](h))}(hDimension search calculationsh]h/Dimension search calculations}(hj'3h j%3hhh!NhNubah}(h]h]h]h]h]uhh(h j"3hhh!jhMubh$)}(hhh](h))}(hDimension search calculationsh]h/Dimension search calculations}(hj83h j63hhh!NhNubah}(h]h]h]h]h]uhh(h j33hhh!jhMubh;)}(hXSDRNPM has three options for searching for dimensions such that the
system will produce a specified effective multiplication factor,
*k*\ :sub:`eff`. The options are selected by a parameter IEVT in the 1$ array
and are as follows:h](h/XSDRNPM has three options for searching for dimensions such that the
system will produce a specified effective multiplication factor,
}(hXSDRNPM has three options for searching for dimensions such that the
system will produce a specified effective multiplication factor,
h jD3hhh!NhNubhA)}(h*k*h]h/k}(hhh jM3ubah}(h]h]h]h]h]uhh@h jD3ubh/ }(h\ h jD3hhh!NhNubh)}(h
:sub:`eff`h]h/eff}(hhh j`3ubah}(h]h]h]h]h]uhhh jD3ubh/S. The options are selected by a parameter IEVT in the 1$ array
and are as follows:}(hS. The options are selected by a parameter IEVT in the 1$ array
and are as follows:h jD3hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j33hhubj)}(hhh](j)}(h zone width search (IEVT = 4),
h]h;)}(hzone width search (IEVT = 4),h]h/zone width search (IEVT = 4),}(hj3h j3ubah}(h]h]h]h]h]uhh:h!jhMh j|3ubah}(h]h]h]h]h]uhjh jy3hhh!jhNubj)}(h"outer radius search (IEVT = 5),
h]h;)}(h!outer radius search (IEVT = 5),h]h/!outer radius search (IEVT = 5),}(hj3h j3ubah}(h]h]h]h]h]uhh:h!jhMh j3ubah}(h]h]h]h]h]uhjh jy3hhh!jhNubj)}(hbuckling search (IEVT = 6).
h]h;)}(hbuckling search (IEVT = 6).h]h/buckling search (IEVT = 6).}(hj3h j3ubah}(h]h]h]h]h]uhh:h!jhMh j3ubah}(h]h]h]h]h]uhjh jy3hhh!jhNubeh}(h]h]h]h]h]jSjCjUhjVjWuhjh j33hhh!jhMubh;)}(hBy default, the search is made to produce a *k*\ :sub:`eff` value of unity. For
*k*\ :sub:`eff`\ ’s other than unity, IPVT (3$ array) is set to unity and the
desired *k*\ :sub:`eff` is input as PV (5* array).h](h/,By default, the search is made to produce a }(h,By default, the search is made to produce a h j3hhh!NhNubhA)}(h*k*h]h/k}(hhh j3ubah}(h]h]h]h]h]uhh@h j3ubh/ }(h\ h j3hhh!NhNubh)}(h
:sub:`eff`h]h/eff}(hhh j3ubah}(h]h]h]h]h]uhhh j3ubh/ value of unity. For
}(h value of unity. For
h j3hhh!NhNubhA)}(h*k*h]h/k}(hhh j3ubah}(h]h]h]h]h]uhh@h j3ubh/ }(h\ h j3ubh)}(h
:sub:`eff`h]h/eff}(hhh j4ubah}(h]h]h]h]h]uhhh j3ubh/J ’s other than unity, IPVT (3$ array) is set to unity and the
desired }(hJ\ ’s other than unity, IPVT (3$ array) is set to unity and the
desired h j3hhh!NhNubhA)}(h*k*h]h/k}(hhh j4ubah}(h]h]h]h]h]uhh@h j3ubh/ }(hj3h j3ubh)}(h
:sub:`eff`h]h/eff}(hhh j14ubah}(h]h]h]h]h]uhhh j3ubh/ is input as PV (5* array).}(h is input as PV (5* array).h j3hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j33hhubh;)}(hX$Other input parameters which apply specifically to all search
calculations are in the 5* array and are EV, the starting eigenvalue
guess, EVM, the eigenvalue modifier, EQL, the eigenvalue convergence,
and XNPM, the new parameter modifier. These parameters are discussed in
more detail below.h]h/X$Other input parameters which apply specifically to all search
calculations are in the 5* array and are EV, the starting eigenvalue
guess, EVM, the eigenvalue modifier, EQL, the eigenvalue convergence,
and XNPM, the new parameter modifier. These parameters are discussed in
more detail below.}(hjL4h jJ4hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j33hhubh)}(h.. _9-1-2-8-1:h]h}(h]h]h]h]h]hid43uhh
hMjh j33hhh!jubeh}(h]id42ah]h]h]dimension search calculationsah]uhh#h j"3hhh!jhMjKubh$)}(hhh](h))}(hZone-width search (IEVT = 4)h]h/Zone-width search (IEVT = 4)}(hjp4h jn4hhh!NhNubah}(h]h]h]h]h]uhh(h jk4hhh!jhMubh;)}(hWith this option, one can vary the width of any or all zones in a case. Note
that it is also possible to change zone widths at different rates.h]h/With this option, one can vary the width of any or all zones in a case. Note
that it is also possible to change zone widths at different rates.}(hj~4h j|4hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jk4hhubh;)}(hThis option requires the inputting of a zone width modifier array (41*) which is
used to specify the relative movements of the zones according to the following
expression:h]h/This option requires the inputting of a zone width modifier array (41*) which is
used to specify the relative movements of the zones according to the following
expression:}(hj4h j4hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jk4hhubj )}(h?\Delta Z_{j}^{f}=\Delta Z_{j}^{i}\left(1+E V^{*} Z M_{j}\right)h]h/?\Delta Z_{j}^{f}=\Delta Z_{j}^{i}\left(1+E V^{*} Z M_{j}\right)}(hhh j4ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh jk4hhubh;)}(hXYwhere :math:`\Delta Z_{j}^{i}, \Delta Z_{j}^{f}` are the initial and final widths of zone j, respectively, *ZM*\ :sub:`j`
is the zone width modifier for the zone (as input in the 41* array), and
EV is the final “eigenvalue” for the problem. Note that a zero value for
ZM will specify a fixed zone width. Negative values for ZM are allowed.h](h/where }(hwhere h j4hhh!NhNubjr)}(h*:math:`\Delta Z_{j}^{i}, \Delta Z_{j}^{f}`h]h/"\Delta Z_{j}^{i}, \Delta Z_{j}^{f}}(hhh j4ubah}(h]h]h]h]h]uhjqh j4ubh/< are the initial and final widths of zone j, respectively, }(h< are the initial and final widths of zone j, respectively, h j4hhh!NhNubhA)}(h*ZM*h]h/ZM}(hhh j4ubah}(h]h]h]h]h]uhh@h j4ubh/ }(h\ h j4hhh!NhNubh)}(h:sub:`j`h]h/j}(hhh j4ubah}(h]h]h]h]h]uhhh j4ubh/
is the zone width modifier for the zone (as input in the 41* array), and
EV is the final “eigenvalue” for the problem. Note that a zero value for
ZM will specify a fixed zone width. Negative values for ZM are allowed.}(h
is the zone width modifier for the zone (as input in the 41* array), and
EV is the final “eigenvalue” for the problem. Note that a zero value for
ZM will specify a fixed zone width. Negative values for ZM are allowed.h j4hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jk4hhubh)}(h.. _9-1-2-8-2:h]h}(h]h]h]h]h]hid44uhh
hMh jk4hhh!jubeh}(h](zone-width-search-ievt-4jb4eh]h](zone-width search (ievt = 4) 9-1-2-8-1eh]h]uhh#h j"3hhh!jhMjf}j5jX4sjh}jb4jX4subh$)}(hhh](h))}(hOuter radius search (IEVT = 5)h]h/Outer radius search (IEVT = 5)}(hj
5h j5hhh!NhNubah}(h]h]h]h]h]uhh(h j5hhh!jhMubh;)}(hWith this option, all zones are scaled uniformly in order to make the
system attain the specified *k*\ :sub:`eff`. The final zone widths are found by
multiplying the initial values by the final “eigenvalue:”h](h/bWith this option, all zones are scaled uniformly in order to make the
system attain the specified }(hbWith this option, all zones are scaled uniformly in order to make the
system attain the specified h j5hhh!NhNubhA)}(h*k*h]h/k}(hhh j"5ubah}(h]h]h]h]h]uhh@h j5ubh/ }(h\ h j5hhh!NhNubh)}(h
:sub:`eff`h]h/eff}(hhh j55ubah}(h]h]h]h]h]uhhh j5ubh/b. The final zone widths are found by
multiplying the initial values by the final “eigenvalue:”}(hb. The final zone widths are found by
multiplying the initial values by the final “eigenvalue:”h j5hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM h j5hhubj )}(h1Z_{j}^{f}=E V\left(Z_{j}^{i} / Z_{p}^{i}\right) .h]h/1Z_{j}^{f}=E V\left(Z_{j}^{i} / Z_{p}^{i}\right) .}(hhh jN5ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhM
h j5hhubh)}(h.. _9-1-2-8-3:h]h}(h]h]h]h]h]hid45uhh
hMh j5hhh!jubeh}(h](outer-radius-search-ievt-5j4eh]h](outer radius search (ievt = 5) 9-1-2-8-2eh]h]uhh#h j"3hhh!jhMjf}jq5j4sjh}j4j4subh$)}(hhh](h))}(hBuckling search (IEVT = 6)h]h/Buckling search (IEVT = 6)}(hj{5h jy5hhh!NhNubah}(h]h]h]h]h]uhh(h jv5hhh!jhMubh;)}(hThis option is used to search for “transverse” dimensions that will
yield a specified *k*\ :sub:`eff` for a system. This means that the search is
for the height for a 1-D cylinder or the y- and/or z-dimensions in a
1-D slab.h](h/ZThis option is used to search for “transverse” dimensions that will
yield a specified }(hZThis option is used to search for “transverse” dimensions that will
yield a specified h j5hhh!NhNubhA)}(h*k*h]h/k}(hhh j5ubah}(h]h]h]h]h]uhh@h j5ubh/ }(h\ h j5hhh!NhNubh)}(h
:sub:`eff`h]h/eff}(hhh j5ubah}(h]h]h]h]h]uhhh j5ubh/} for a system. This means that the search is
for the height for a 1-D cylinder or the y- and/or z-dimensions in a
1-D slab.}(h} for a system. This means that the search is
for the height for a 1-D cylinder or the y- and/or z-dimensions in a
1-D slab.h j5hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jv5hhubh;)}(h2For this option, the final dimensions are given byh]h/2For this option, the final dimensions are given by}(hj5h j5hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jv5hhubj1)}(hhh]h;)}(hDY = *DY*\ :sub:`0` × EV,h](h/DY = }(hDY = h j5ubhA)}(h*DY*h]h/DY}(hhh j5ubah}(h]h]h]h]h]uhh@h j5ubh/ }(h\ h j5ubh)}(h:sub:`0`h]h/0}(hhh j5ubah}(h]h]h]h]h]uhhh j5ubh/ × EV,}(h × EV,h j5ubeh}(h]h]h]h]h]uhh:h!jhMh j5ubah}(h]h]h]h]h]uhj1h jv5hhh!jhNubh;)}(handh]h/and}(hj
6h j6hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jv5hhubj1)}(hhh]h;)}(hDZ = *DZ*\ :sub:`0` × EV,h](h/DZ = }(hDZ = h j6ubhA)}(h*DZ*h]h/DZ}(hhh j"6ubah}(h]h]h]h]h]uhh@h j6ubh/ }(h\ h j6ubh)}(h:sub:`0`h]h/0}(hhh j56ubah}(h]h]h]h]h]uhhh j6ubh/ × EV,}(h × EV,h j6ubeh}(h]h]h]h]h]uhh:h!jhM!h j6ubah}(h]h]h]h]h]uhj1h jv5hhh!jhNubh;)}(hXwhere *DY*\ :sub:`0` , *DZ*\ :sub:`0` are the initial dimensions input in the 5* array.h](h/where }(hwhere h jT6hhh!NhNubhA)}(h*DY*h]h/DY}(hhh j]6ubah}(h]h]h]h]h]uhh@h jT6ubh/ }(h\ h jT6hhh!NhNubh)}(h:sub:`0`h]h/0}(hhh jp6ubah}(h]h]h]h]h]uhhh jT6ubh/ , }(h , h jT6hhh!NhNubhA)}(h*DZ*h]h/DZ}(hhh j6ubah}(h]h]h]h]h]uhh@h jT6ubh/ }(hjo6h jT6ubh)}(h:sub:`0`h]h/0}(hhh j6ubah}(h]h]h]h]h]uhhh jT6ubh/3 are the initial dimensions input in the 5* array.}(h3 are the initial dimensions input in the 5* array.h jT6hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM#h jv5hhubh)}(h.. _9-1-2-8-4:h]h}(h]h]h]h]h]hid46uhh
hMh jv5hhh!jubeh}(h](buckling-search-ievt-6jj5eh]h](buckling search (ievt = 6) 9-1-2-8-3eh]h]uhh#h j"3hhh!jhMjf}j6j`5sjh}jj5j`5subh$)}(hhh](h))}(hSearch calculation strategyh]h/Search calculation strategy}(hj6h j6hhh!NhNubah}(h]h]h]h]h]uhh(h j6hhh!jhM(ubh;)}(hX]All the “dimension searches” use the same simple strategy. The
calculations start by using the input eigenvalue (EV from the 5* array)
to determine initial dimensions for the system. These dimensions allow
the code to calculate a *k*\ :sub:`eff` . The eigenvalue modifier (EVM in the
5* array) is then used to change the dimensions as follows:h](h/All the “dimension searches” use the same simple strategy. The
calculations start by using the input eigenvalue (EV from the 5* array)
to determine initial dimensions for the system. These dimensions allow
the code to calculate a }(hAll the “dimension searches” use the same simple strategy. The
calculations start by using the input eigenvalue (EV from the 5* array)
to determine initial dimensions for the system. These dimensions allow
the code to calculate a h j6hhh!NhNubhA)}(h*k*h]h/k}(hhh j6ubah}(h]h]h]h]h]uhh@h j6ubh/ }(h\ h j6hhh!NhNubh)}(h
:sub:`eff`h]h/eff}(hhh j6ubah}(h]h]h]h]h]uhhh j6ubh/c . The eigenvalue modifier (EVM in the
5* array) is then used to change the dimensions as follows:}(hc . The eigenvalue modifier (EVM in the
5* array) is then used to change the dimensions as follows:h j6hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM*h j6hhubh;)}(hIOPT = 4 (Zone width search)h]h/IOPT = 4 (Zone width search)}(hj7h j
7hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM0h j6hhubj )}(hG\Delta Z_{j}^{f}=\Delta Z_{j}^{i}\left[1+(E V M+E V)^{*} Z M_{j}\right]h]h/G\Delta Z_{j}^{f}=\Delta Z_{j}^{i}\left[1+(E V M+E V)^{*} Z M_{j}\right]}(hhh j7ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhM2h j6hhubh;)}(hIOPT = 5 (Outer radius search)h]h/IOPT = 5 (Outer radius search)}(hj,7h j*7hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM6h j6hhubj )}(h9\Delta Z_{j}^{f}=(E V M+E V)\left(\Delta Z_{j}^{i}\right)h]h/9\Delta Z_{j}^{f}=(E V M+E V)\left(\Delta Z_{j}^{i}\right)}(hhh j87ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhM8h j6hhubh;)}(hIOPT = 6 (Buckling search)h]h/IOPT = 6 (Buckling search)}(hjL7h jJ7hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM<h j6hhubj1)}(hhh](h;)}(hDY = *DY*\ :sub:`0` (EV + EVM)h](h/DY = }(hDY = h j[7ubhA)}(h*DY*h]h/DY}(hhh jd7ubah}(h]h]h]h]h]uhh@h j[7ubh/ }(h\ h j[7ubh)}(h:sub:`0`h]h/0}(hhh jw7ubah}(h]h]h]h]h]uhhh j[7ubh/ (EV + EVM)}(h (EV + EVM)h j[7ubeh}(h]h]h]h]h]uhh:h!jhM>h jX7ubh;)}(hDZ = *DZ*\ :sub:`0` (EV + EVM).h](h/DZ = }(hDZ = h j7ubhA)}(h*DZ*h]h/DZ}(hhh j7ubah}(h]h]h]h]h]uhh@h j7ubh/ }(h\ h j7ubh)}(h:sub:`0`h]h/0}(hhh j7ubah}(h]h]h]h]h]uhhh j7ubh/ (EV + EVM).}(h (EV + EVM).h j7ubeh}(h]h]h]h]h]uhh:h!jhM@h jX7ubeh}(h]h]h]h]h]uhj1h j6hhh!jhNubh;)}(hfThe new dimensions are then used in a new calculation which determines a
second *k*\ :sub:`eff` value.h](h/PThe new dimensions are then used in a new calculation which determines a
second }(hPThe new dimensions are then used in a new calculation which determines a
second h j7hhh!NhNubhA)}(h*k*h]h/k}(hhh j7ubah}(h]h]h]h]h]uhh@h j7ubh/ }(h\ h j7hhh!NhNubh)}(h
:sub:`eff`h]h/eff}(hhh j7ubah}(h]h]h]h]h]uhhh j7ubh/ value.}(h value.h j7hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMBh j6hhubh;)}(hXSDRNPM searches for a unity value of *k*\ :sub:`eff` by default; however, when
IPVT = 1 (3$ array), a nonunity value can be specified in PV (5* array)
and the search will be made on this value.h](h/&XSDRNPM searches for a unity value of }(h&XSDRNPM searches for a unity value of h j8hhh!NhNubhA)}(h*k*h]h/k}(hhh j 8ubah}(h]h]h]h]h]uhh@h j8ubh/ }(h\ h j8hhh!NhNubh)}(h
:sub:`eff`h]h/eff}(hhh j8ubah}(h]h]h]h]h]uhhh j8ubh/ by default; however, when
IPVT = 1 (3$ array), a nonunity value can be specified in PV (5* array)
and the search will be made on this value.}(h by default; however, when
IPVT = 1 (3$ array), a nonunity value can be specified in PV (5* array)
and the search will be made on this value.h j8hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMEh j6hhubh;)}(hOnce the two *k*\ :sub:`eff`\ ’s are known, which are based on eigenvalues of
EV and EV + EVM, respectively, a linear fit is used to project to the
next value for EV. This yields an expression of the formh](h/
Once the two }(h
Once the two h j58hhh!NhNubhA)}(h*k*h]h/k}(hhh j>8ubah}(h]h]h]h]h]uhh@h j58ubh/ }(h\ h j58hhh!NhNubh)}(h
:sub:`eff`h]h/eff}(hhh jQ8ubah}(h]h]h]h]h]uhhh j58ubh/ ’s are known, which are based on eigenvalues of
EV and EV + EVM, respectively, a linear fit is used to project to the
next value for EV. This yields an expression of the form}(h\ ’s are known, which are based on eigenvalues of
EV and EV + EVM, respectively, a linear fit is used to project to the
next value for EV. This yields an expression of the formh j58hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMIh j6hhubj )}(hVE V_{\text {next}}=E V M \frac{\left(P V-k_{1}\right)}{\left(k_{2}-k_{1}\right)}+E V ,h]h/VE V_{\text {next}}=E V M \frac{\left(P V-k_{1}\right)}{\left(k_{2}-k_{1}\right)}+E V ,}(hhh jj8ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMMh j6hhubh;)}(hwhere *k*\ :sub:`1` and *k*\ :sub:`2` are the first and second value of *k\ eff ,*
respectively. After this iteration, the procedure is to fit a quadratic
to the three most recent *k*\ :sub:`eff` values in order to obtain an estimate
for the next EV.h](h/where }(hwhere h j|8hhh!NhNubhA)}(h*k*h]h/k}(hhh j8ubah}(h]h]h]h]h]uhh@h j|8ubh/ }(h\ h j|8hhh!NhNubh)}(h:sub:`1`h]h/1}(hhh j8ubah}(h]h]h]h]h]uhhh j|8ubh/ and }(h and h j|8hhh!NhNubhA)}(h*k*h]h/k}(hhh j8ubah}(h]h]h]h]h]uhh@h j|8ubh/ }(h\ h j|8ubh)}(h:sub:`2`h]h/2}(hhh j8ubah}(h]h]h]h]h]uhhh j|8ubh/# are the first and second value of }(h# are the first and second value of h j|8hhh!NhNubhA)}(h
*k\ eff ,*h]h/k eff ,}(hhh j8ubah}(h]h]h]h]h]uhh@h j|8ubh/b
respectively. After this iteration, the procedure is to fit a quadratic
to the three most recent }(hb
respectively. After this iteration, the procedure is to fit a quadratic
to the three most recent h j|8hhh!NhNubhA)}(h*k*h]h/k}(hhh j8ubah}(h]h]h]h]h]uhh@h j|8ubh/ }(hj8h j|8ubh)}(h
:sub:`eff`h]h/eff}(hhh j8ubah}(h]h]h]h]h]uhhh j|8ubh/7 values in order to obtain an estimate
for the next EV.}(h7 values in order to obtain an estimate
for the next EV.h j|8hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMQh j6hhubh;)}(hdThe procedure continues until a relative convergence of EQL (5* array)
or better is obtained on EV.h]h/dThe procedure continues until a relative convergence of EQL (5* array)
or better is obtained on EV.}(hj9h j9hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMVh j6hhubh;)}(hyTo prevent oscillations in the search, extrapolations are limited by
XNPM, the new parameter modifier from the 5* array.h]h/yTo prevent oscillations in the search, extrapolations are limited by
XNPM, the new parameter modifier from the 5* array.}(hj9h j9hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMYh j6hhubh)}(h.. _9-1-2-9:h]h}(h]h]h]h]h]hid47uhh
hMh j6hhh!jubeh}(h](search-calculation-strategyj6eh]h](search calculation strategy 9-1-2-8-4eh]h]uhh#h j"3hhh!jhM(jf}j<9j6sjh}j6j6subeh}(h](dimension-search-calculationsj3eh]h]9-1-2-8ah]ji4ah]uhh#h j hhh!jhMjKjf}jF9j3sjh}j3j3subh$)}(hhh](h))}(hAlpa Searchh]h/Alpa Search}(hjP9h jN9hhh!NhNubah}(h]h]h]h]h]uhh(h jK9hhh!jhM_ubh;)}(hX
It is possible to make some of the searches described in :ref:`9-1-2-8` in a more
“direct” fashion than the strategy described in :ref:`9-1-2-8-4`. XSDRNPM has
two such options: (1) the alpha search and (2) a direct buckling search. These
are described below.h](h/9It is possible to make some of the searches described in }(h9It is possible to make some of the searches described in h j\9hhh!NhNubj)}(h:ref:`9-1-2-8`h]j#)}(hjg9h]h/9-1-2-8}(hhh ji9ubah}(h]h](jnstdstd-refeh]h]h]uhj"h je9ubah}(h]h]h]h]h]refdocj refdomainjs9reftyperefrefexplicitrefwarnj9-1-2-8uhjh!jhMah j\9ubh/? in a more
“direct” fashion than the strategy described in }(h? in a more
“direct” fashion than the strategy described in h j\9hhh!NhNubj)}(h:ref:`9-1-2-8-4`h]j#)}(hj9h]h/ 9-1-2-8-4}(hhh j9ubah}(h]h](jnstdstd-refeh]h]h]uhj"h j9ubah}(h]h]h]h]h]refdocj refdomainj9reftyperefrefexplicitrefwarnj 9-1-2-8-4uhjh!jhMah j\9ubh/t. XSDRNPM has
two such options: (1) the alpha search and (2) a direct buckling search. These
are described below.}(ht. XSDRNPM has
two such options: (1) the alpha search and (2) a direct buckling search. These
are described below.h j\9hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMah jK9hhubh)}(h.. _9-1-2-9-1:h]h}(h]h]h]h]h]hid48uhh
hMh jK9hhh!jubh$)}(hhh](h))}(hAlpha searchh]h/Alpha search}(hj9h j9hhh!NhNubah}(h]h]h]h]h]uhh(h j9hhh!jhMiubh;)}(hThe time-dependent form of the Boltzmann equation is identical with Eq. :eq:`eq9-1-1`,
except for the inclusion of a time-gradient term on the left-hand side:h](h/HThe time-dependent form of the Boltzmann equation is identical with Eq. }(hHThe time-dependent form of the Boltzmann equation is identical with Eq. h j9hhh!NhNubj)}(h
:eq:`eq9-1-1`h]jc)}(hj9h]h/eq9-1-1}(hhh j9ubah}(h]h](jneqeh]h]h]uhjbh j9ubah}(h]h]h]h]h]refdocj refdomainjqreftypej9refexplicitrefwarnjeq9-1-1uhjh!jhMkh j9ubh/I,
except for the inclusion of a time-gradient term on the left-hand side:}(hI,
except for the inclusion of a time-gradient term on the left-hand side:h j9hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMkh j9hhubj )}(hH\frac{1}{\mathrm{v}} \frac{\partial \psi(r, E, \Omega, t)}{\partial t} .h]h/H\frac{1}{\mathrm{v}} \frac{\partial \psi(r, E, \Omega, t)}{\partial t} .}(hhh j:ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMnh j9hhubh;)}(hPAll other flux terms in the expression also would include the time (t) argument.h]h/PAll other flux terms in the expression also would include the time (t) argument.}(hj:h j:hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMrh j9hhubh;)}(hhIn some analyses it is reasonable to assume that the time variation of the flux
is exponential, that is,h]h/hIn some analyses it is reasonable to assume that the time variation of the flux
is exponential, that is,}(hj%:h j#:hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMth j9hhubj )}(h2\psi(r, E, \Omega, t)=\psi(r, E, \Omega) e^{a t} .h]h/2\psi(r, E, \Omega, t)=\psi(r, E, \Omega) e^{a t} .}(hhh j1:ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMwh j9hhubh;)}(hWhen this variation is introduced into the expanded form of Eq. :eq:`eq9-1-1`, the
exponential terms all cancel leaving a leading term:h](h/@When this variation is introduced into the expanded form of Eq. }(h@When this variation is introduced into the expanded form of Eq. h jC:hhh!NhNubj)}(h
:eq:`eq9-1-1`h]jc)}(hjN:h]h/eq9-1-1}(hhh jP:ubah}(h]h](jneqeh]h]h]uhjbh jL:ubah}(h]h]h]h]h]refdocj refdomainjqreftypejZ:refexplicitrefwarnjeq9-1-1uhjh!jhM{h jC:ubh/:, the
exponential terms all cancel leaving a leading term:}(h:, the
exponential terms all cancel leaving a leading term:h jC:hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM{h j9hhubj )}(h,\frac{\alpha}{\mathrm{v}} \psi(r, E, \Omega)h]h/,\frac{\alpha}{\mathrm{v}} \psi(r, E, \Omega)}(hhh ju:ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhM~h j9hhubh;)}(h;which is in the same form as the *Σ*\ :sub:`t`\ *ψ* term.h](h/!which is in the same form as the }(h!which is in the same form as the h j:hhh!NhNubhA)}(h*Σ*h]h/Σ}(hhh j:ubah}(h]h]h]h]h]uhh@h j:ubh/ }(h\ h j:hhh!NhNubh)}(h:sub:`t`h]h/t}(hhh j:ubah}(h]h]h]h]h]uhhh j:ubh/ }(hj:h j:ubhA)}(h*ψ*h]h/ψ}(hhh j:ubah}(h]h]h]h]h]uhh@h j:ubh/ term.}(h term.h j:hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j9hhubh;)}(hdIf one considers integrating over energy, angle, and space, the following expression can be derived:h]h/dIf one considers integrating over energy, angle, and space, the following expression can be derived:}(hj:h j:hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j9hhubj )}(hP-A-L-\alpha V=0h]h/P-A-L-\alpha V=0}(hhh j:ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh j9hhubh;)}(hwhereh]h/where}(hj:h j:hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j9hhubh;)}(hwhereh]h/where}(hj:h j:hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j9hhubj1)}(hhh](h;)}(hP ≡ production in the system,h]h/P ≡ production in the system,}(hj;h j
;ubah}(h]h]h]h]h]uhh:h!jhMh j
;ubh;)}(h A ≡ absorptions in the system,h]h/ A ≡ absorptions in the system,}(hj;h j;ubah}(h]h]h]h]h]uhh:h!jhMh j
;ubh;)}(hL ≡ leakage from the system,h]h/L ≡ leakage from the system,}(hj+;h j);ubah}(h]h]h]h]h]uhh:h!jhMh j
;ubh;)}(hV ≡ :math:`\int_{0}^{\infty} d E \int_{0}^{4 \pi} d \overset{\rightharpoonup}{\Omega} \int_{s y s t e m} d \overset{\rightharpoonup}{r} \frac{\psi(\overset{\rightharpoonup}{r}, E, \overset{\rightharpoonup}{\Omega})}{\mathrm{v}}`h](h/V ≡ }(hV ≡ h j7;ubjr)}(h:math:`\int_{0}^{\infty} d E \int_{0}^{4 \pi} d \overset{\rightharpoonup}{\Omega} \int_{s y s t e m} d \overset{\rightharpoonup}{r} \frac{\psi(\overset{\rightharpoonup}{r}, E, \overset{\rightharpoonup}{\Omega})}{\mathrm{v}}`h]h/\int_{0}^{\infty} d E \int_{0}^{4 \pi} d \overset{\rightharpoonup}{\Omega} \int_{s y s t e m} d \overset{\rightharpoonup}{r} \frac{\psi(\overset{\rightharpoonup}{r}, E, \overset{\rightharpoonup}{\Omega})}{\mathrm{v}}}(hhh j@;ubah}(h]h]h]h]h]uhjqh j7;ubeh}(h]h]h]h]h]uhh:h!jhMh j
;ubeh}(h]h]h]h]h]uhj1h j9hhh!jhNubh;)}(hXRSince all terms other than α can be determined from a calculation, it is
possible to determine α directly, thereby avoiding a scheme like that
used for dimension searches. In the balance expression, the fission
component of the production term is adjusted for the case of a non-unity
*k*\ :sub:`eff` value (IPVT = 1 in the 3$ array).h](h/XSince all terms other than α can be determined from a calculation, it is
possible to determine α directly, thereby avoiding a scheme like that
used for dimension searches. In the balance expression, the fission
component of the production term is adjusted for the case of a non-unity
}(hXSince all terms other than α can be determined from a calculation, it is
possible to determine α directly, thereby avoiding a scheme like that
used for dimension searches. In the balance expression, the fission
component of the production term is adjusted for the case of a non-unity
h jZ;hhh!NhNubhA)}(h*k*h]h/k}(hhh jc;ubah}(h]h]h]h]h]uhh@h jZ;ubh/ }(h\ h jZ;hhh!NhNubh)}(h
:sub:`eff`h]h/eff}(hhh jv;ubah}(h]h]h]h]h]uhhh jZ;ubh/% value (IPVT = 1 in the 3$ array).}(h% value (IPVT = 1 in the 3$ array).h jZ;hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j9hhubh;)}(hXAn α-search has several practical applications. If, for example, a
subcritical assembly is pulsed by a source, the time-dependence of the
flux is expected to die off exponentially. Another way to interpret the
α-search is as that amount of 1/v absorber which could be added or taken
away from a system in order to achieve criticality. This number could be
of interest when certain control materials are used, such as
:sup:`10`\ *B*\ :sub:`5` , which is a “1/v” material.h](h/XAn α-search has several practical applications. If, for example, a
subcritical assembly is pulsed by a source, the time-dependence of the
flux is expected to die off exponentially. Another way to interpret the
α-search is as that amount of 1/v absorber which could be added or taken
away from a system in order to achieve criticality. This number could be
of interest when certain control materials are used, such as
}(hXAn α-search has several practical applications. If, for example, a
subcritical assembly is pulsed by a source, the time-dependence of the
flux is expected to die off exponentially. Another way to interpret the
α-search is as that amount of 1/v absorber which could be added or taken
away from a system in order to achieve criticality. This number could be
of interest when certain control materials are used, such as
h j;hhh!NhNubh superscript)}(h :sup:`10`h]h/10}(hhh j;ubah}(h]h]h]h]h]uhj;h j;ubh/ }(h\ h j;hhh!NhNubhA)}(h*B*h]h/B}(hhh j;ubah}(h]h]h]h]h]uhh@h j;ubh/ }(hj;h j;ubh)}(h:sub:`5`h]h/5}(hhh j;ubah}(h]h]h]h]h]uhhh j;ubh/" , which is a “1/v” material.}(h" , which is a “1/v” material.h j;hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j9hhubh)}(h.. _9-1-2-9-2:h]h}(h]h]h]h]h]hid49uhh
hMh j9hhh!jubeh}(h](alpha-searchj9eh]h](alpha search 9-1-2-9-1eh]h]uhh#h jK9hhh!jhMijf}j;j9sjh}j9j9subh$)}(hhh](h))}(hDirect-buckling searchh]h/Direct-buckling search}(hj;h j;hhh!NhNubah}(h]h]h]h]h]uhh(h j;hhh!jhMubh;)}(hX0A “direct”-buckling search can be made using a procedure analogous to that
described in :ref:`9-1-2-9-1`. Recall that the buckling is introduced in order
to represent a transverse leakage through the use of a *DB*\ :sup:`2`\ *ψ* term.
This suggests that the foregoing balance
expression be written:h](h/\A “direct”-buckling search can be made using a procedure analogous to that
described in }(h\A “direct”-buckling search can be made using a procedure analogous to that
described in h j;hhh!NhNubj)}(h:ref:`9-1-2-9-1`h]j#)}(hj
<h]h/ 9-1-2-9-1}(hhh j<ubah}(h]h](jnstdstd-refeh]h]h]uhj"h j<ubah}(h]h]h]h]h]refdocj refdomainj<reftyperefrefexplicitrefwarnj 9-1-2-9-1uhjh!jhMh j;ubh/j. Recall that the buckling is introduced in order
to represent a transverse leakage through the use of a }(hj. Recall that the buckling is introduced in order
to represent a transverse leakage through the use of a h j;hhh!NhNubhA)}(h*DB*h]h/DB}(hhh j-<ubah}(h]h]h]h]h]uhh@h j;ubh/ }(h\ h j;hhh!NhNubj;)}(h:sup:`2`h]h/2}(hhh j@<ubah}(h]h]h]h]h]uhj;h j;ubh/ }(hj?<h j;ubhA)}(h*ψ*h]h/ψ}(hhh jR<ubah}(h]h]h]h]h]uhh@h j;ubh/F term.
This suggests that the foregoing balance
expression be written:}(hF term.
This suggests that the foregoing balance
expression be written:h j;hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j;hhubj )}(hP-A-L-\alpha D B^{2} X=0 ,h]h/P-A-L-\alpha D B^{2} X=0 ,}(hhh jk<ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh j;hhubh;)}(hwhereh]h/where}(hj<h j}<hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j;hhubj )}(hhX \equiv \int_{0}^{\infty} d E \int_{0}^{4 \pi} d \Omega \int_{\text {system }} d r \psi(r, E, \Omega) .h]h/hX \equiv \int_{0}^{\infty} d E \int_{0}^{4 \pi} d \Omega \int_{\text {system }} d r \psi(r, E, \Omega) .}(hhh j<ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh j;hhubh;)}(hLIn this case, the diffusion coefficients, *D*\ :sub:`g`, are determined fromh](h/*In this case, the diffusion coefficients, }(h*In this case, the diffusion coefficients, h j<hhh!NhNubhA)}(h*D*h]h/D}(hhh j<ubah}(h]h]h]h]h]uhh@h j<ubh/ }(h\ h j<hhh!NhNubh)}(h:sub:`g`h]h/g}(hhh j<ubah}(h]h]h]h]h]uhhh j<ubh/, are determined from}(h, are determined fromh j<hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j;hhubj )}(h$D_{g}=\frac{1}{3 \Sigma_{t r_{g}}} ,h]h/$D_{g}=\frac{1}{3 \Sigma_{t r_{g}}} ,}(hhh j<ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh j;hhubh;)}(hwhereh]h/where}(hj<h j<hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j;hhubj )}(h0\Sigma_{t r_{g}}=\Sigma_{t_{g}}-\Sigma_{1_{g}} ,h]h/0\Sigma_{t r_{g}}=\Sigma_{t_{g}}-\Sigma_{1_{g}} ,}(hhh j<ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh j;hhubh;)}(hXand *Σ*\ :sub:`1` is the within-group term from the
*P*\ :sub:`1` scattering matrixes:h](h/and }(hand h j=hhh!NhNubhA)}(h*Σ*h]h/Σ}(hhh j
=ubah}(h]h]h]h]h]uhh@h j=ubh/ }(h\ h j=hhh!NhNubh)}(h:sub:`1`h]h/1}(hhh j =ubah}(h]h]h]h]h]uhhh j=ubh/# is the within-group term from the
}(h# is the within-group term from the
h j=hhh!NhNubhA)}(h*P*h]h/P}(hhh j3=ubah}(h]h]h]h]h]uhh@h j=ubh/ }(hj=h j=ubh)}(h:sub:`1`h]h/1}(hhh jE=ubah}(h]h]h]h]h]uhhh j=ubh/ scattering matrixes:}(h scattering matrixes:h j=hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j;hhubj )}(hZ\Sigma_{1_{\mathrm{g}}}=\sum_{t}\left(\mathrm{~g} \rightarrow \mathrm{g}^{\prime}\right) .h]h/Z\Sigma_{1_{\mathrm{g}}}=\sum_{t}\left(\mathrm{~g} \rightarrow \mathrm{g}^{\prime}\right) .}(hhh j^=ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh j;hhubh;)}(hXThe original *B*\ :sup:`2` value is determined as specified in
:ref:`9-1-2-12`, and the α is the square of the search parameter, that
one multiplies by the original *B*\ :sup:`2` value in order to determine
the final buckling and, hence, the dimensions of the system.h](h/
The original }(h
The original h jp=hhh!NhNubhA)}(h*B*h]h/B}(hhh jy=ubah}(h]h]h]h]h]uhh@h jp=ubh/ }(h\ h jp=hhh!NhNubj;)}(h:sup:`2`h]h/2}(hhh j=ubah}(h]h]h]h]h]uhj;h jp=ubh/& value is determined as specified in
}(h& value is determined as specified in
h jp=hhh!NhNubj)}(h:ref:`9-1-2-12`h]j#)}(hj=h]h/9-1-2-12}(hhh j=ubah}(h]h](jnstdstd-refeh]h]h]uhj"h j=ubah}(h]h]h]h]h]refdocj refdomainj=reftyperefrefexplicitrefwarnj9-1-2-12uhjh!jhMh jp=ubh/X, and the α is the square of the search parameter, that
one multiplies by the original }(hX, and the α is the square of the search parameter, that
one multiplies by the original h jp=hhh!NhNubhA)}(h*B*h]h/B}(hhh j=ubah}(h]h]h]h]h]uhh@h jp=ubh/ }(hj=h jp=ubj;)}(h:sup:`2`h]h/2}(hhh j=ubah}(h]h]h]h]h]uhj;h jp=ubh/Z value in order to determine
the final buckling and, hence, the dimensions of the system.}(hZ value in order to determine
the final buckling and, hence, the dimensions of the system.h jp=hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j;hhubh)}(h
.. _9-1-2-10:h]h}(h]h]h]h]h]hid50uhh
hMLh j;hhh!jubeh}(h](direct-buckling-searchj;eh]h](direct-buckling search 9-1-2-9-2eh]h]uhh#h jK9hhh!jhMjf}j>j;sjh}j;j;subeh}(h](alpa-searchj59eh]h](alpa search9-1-2-9eh]h]uhh#h j hhh!jhM_jf}j>j+9sjh}j59j+9subh$)}(hhh](h))}(hIteration and convergence testsh]h/Iteration and convergence tests}(hj>h j>hhh!NhNubah}(h]h]h]h]h]uhh(h j>hhh!jhMubh;)}(hX$Two parameters are used to specify the required levels of convergence on
an XSDRNPM calculation. These are EPS and PTC, both given in the
5* array. The flux calculations proceed through a series of iterations
until either convergence is achieved or the specified iteration limit is
exceeded.h]h/X$Two parameters are used to specify the required levels of convergence on
an XSDRNPM calculation. These are EPS and PTC, both given in the
5* array. The flux calculations proceed through a series of iterations
until either convergence is achieved or the specified iteration limit is
exceeded.}(hj#>h j!>hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j>hhubh;)}(hXThe basic iteration strategy in XSDRNPM is now described. The
discrete-ordinates difference equation is solved for the first angle and
the first energy group. This sweep generally is made from the last
interval boundary to the center of the system, and it uses the flux
guess supplied as part of the input along with the boundary conditions.
The second angle is then calculated, etc., until all angles in the
quadrature are treated. At the end of this sweep, new scalar fluxes for
the midpoints of all intervals have been determined. The angular sweep
continues until either the point scalar fluxes are converged to within
PTC or until the code makes IIM inner iterations. An exception to this
“inner iteration” pattern occurs on the first outer (defined below)
iteration whenever a fission density guess is used, instead of the flux
guess. In this case, the program uses 1-D diffusion theory to determine
a scalar flux value for all intervals and the angular sweeps are not
made until the second outer iteration. After the first group is
completed, the calculation goes to the second group and repeats the
above procedure. This continues until all groups have been treated.h]h/XThe basic iteration strategy in XSDRNPM is now described. The
discrete-ordinates difference equation is solved for the first angle and
the first energy group. This sweep generally is made from the last
interval boundary to the center of the system, and it uses the flux
guess supplied as part of the input along with the boundary conditions.
The second angle is then calculated, etc., until all angles in the
quadrature are treated. At the end of this sweep, new scalar fluxes for
the midpoints of all intervals have been determined. The angular sweep
continues until either the point scalar fluxes are converged to within
PTC or until the code makes IIM inner iterations. An exception to this
“inner iteration” pattern occurs on the first outer (defined below)
iteration whenever a fission density guess is used, instead of the flux
guess. In this case, the program uses 1-D diffusion theory to determine
a scalar flux value for all intervals and the angular sweeps are not
made until the second outer iteration. After the first group is
completed, the calculation goes to the second group and repeats the
above procedure. This continues until all groups have been treated.}(hj1>h j/>hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j>hhubh;)}(hXThe pass through all groups, angles, and intervals is called an outer
iteration. Most of the convergence checks on the outer iteration have to
do with reaction rates involving all energy groups and are made against
the EPS parameter mentioned above. For a coupled neutron-gamma problem,
outer iterations are only performed for the neutron groups until
convergence is achieved, then the final converged pass is made over all
groups. In discussing these checks, it is convenient to define several
terms:h]h/XThe pass through all groups, angles, and intervals is called an outer
iteration. Most of the convergence checks on the outer iteration have to
do with reaction rates involving all energy groups and are made against
the EPS parameter mentioned above. For a coupled neutron-gamma problem,
outer iterations are only performed for the neutron groups until
convergence is achieved, then the final converged pass is made over all
groups. In discussing these checks, it is convenient to define several
terms:}(hj?>h j=>hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j>hhubh;)}(h&Q ≡ total fixed source in the systemh]h/&Q ≡ total fixed source in the system}(hjM>h jK>hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j>hhubh;)}(h(F ≡ total fission source in the systemh]h/(F ≡ total fission source in the system}(hj[>h jY>hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j>hhubh;)}(h)D ≡ total outscatter rate in the systemh]h/)D ≡ total outscatter rate in the system}(hji>h jg>hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j>hhubh;)}(hD ≡ :math:`\sum_{i}^{intervals} \sum_{g}^{groups } \sum^{groups}_{\mathrm{g}^{\prime} \neq \mathrm{g}} \psi_{i, g} \sigma_{g \rightarrow g^{\prime}} \mathrm{V}_{\mathrm{i}}`h](h/D ≡ }(hD ≡ h ju>hhh!NhNubjr)}(h:math:`\sum_{i}^{intervals} \sum_{g}^{groups } \sum^{groups}_{\mathrm{g}^{\prime} \neq \mathrm{g}} \psi_{i, g} \sigma_{g \rightarrow g^{\prime}} \mathrm{V}_{\mathrm{i}}`h]h/\sum_{i}^{intervals} \sum_{g}^{groups } \sum^{groups}_{\mathrm{g}^{\prime} \neq \mathrm{g}} \psi_{i, g} \sigma_{g \rightarrow g^{\prime}} \mathrm{V}_{\mathrm{i}}}(hhh j~>ubah}(h]h]h]h]h]uhjqh ju>ubeh}(h]h]h]h]h]uhh:h!jhMh j>hhubh;)}(h@:math:`\psi_{i, g}` ≡ scalar flux in intervals i and group gh](jr)}(h:math:`\psi_{i, g}`h]h/\psi_{i, g}}(hhh j>ubah}(h]h]h]h]h]uhjqh j>ubh/- ≡ scalar flux in intervals i and group g}(h- ≡ scalar flux in intervals i and group gh j>hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j>hhubh;)}(hj:math:`\sigma_{g \rightarrow g^{\prime}}` ≡ macroscopic scattering cross section from group g to grouph](jr)}(h):math:`\sigma_{g \rightarrow g^{\prime}}`h]h/!\sigma_{g \rightarrow g^{\prime}}}(hhh j>ubah}(h]h]h]h]h]uhjqh j>ubh/A ≡ macroscopic scattering cross section from group g to group}(hA ≡ macroscopic scattering cross section from group g to grouph j>hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j>hhubh;)}(h9:math:`\mathrm{V}_{\mathrm{i}}` ≡ volume of interval ih](jr)}(h:math:`\mathrm{V}_{\mathrm{i}}`h]h/\mathrm{V}_{\mathrm{i}}}(hhh j>ubah}(h]h]h]h]h]uhjqh j>ubh/ ≡ volume of interval i}(h ≡ volume of interval ih j>hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j>hhubh;)}(hk ≡ outer iteration numberh]h/k ≡ outer iteration number}(hj>h j>hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j>hhubh;)}(h%IGM ≡ total number of energy groupsh]h/%IGM ≡ total number of energy groups}(hj>h j>hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM h j>hhubh;)}(h9:math:`\lambda_{k}` ≡ :math:`\frac{Q+F_{k}}{Q+F_{k-1}}`h](jr)}(h:math:`\lambda_{k}`h]h/\lambda_{k}}(hhh j ?ubah}(h]h]h]h]h]uhjqh j?ubh/ ≡ }(h ≡ h j?hhh!NhNubjr)}(h!:math:`\frac{Q+F_{k}}{Q+F_{k-1}}`h]h/\frac{Q+F_{k}}{Q+F_{k-1}}}(hhh j?ubah}(h]h]h]h]h]uhjqh j?ubeh}(h]h]h]h]h]uhh:h!jhMh j>hhubh;)}(h/:math:`G_{k}` ≡ :math:`\frac{D_{k}}{Q+F_{k}}`h](jr)}(h
:math:`G_{k}`h]h/G_{k}}(hhh j4?ubah}(h]h]h]h]h]uhjqh j0?ubh/ ≡ }(h ≡ h j0?hhh!NhNubjr)}(h:math:`\frac{D_{k}}{Q+F_{k}}`h]h/\frac{D_{k}}{Q+F_{k}}}(hhh jG?ubah}(h]h]h]h]h]uhjqh j0?ubeh}(h]h]h]h]h]uhh:h!jhM
h j>hhubh;)}(h>:math:`\lambda_{k}^{\prime}` ≡ :math:`\frac{G_{k-1}}{G_{k}}`h](jr)}(h:math:`\lambda_{k}^{\prime}`h]h/\lambda_{k}^{\prime}}(hhh j_?ubah}(h]h]h]h]h]uhjqh j[?ubh/ ≡ }(h ≡ h j[?hhh!NhNubjr)}(h:math:`\frac{G_{k-1}}{G_{k}}`h]h/\frac{G_{k-1}}{G_{k}}}(hhh jr?ubah}(h]h]h]h]h]uhjqh j[?ubeh}(h]h]h]h]h]uhh:h!jhMh j>hhubh;)}(h:math:`U_{k}` ≡ total upscatter rate = :math:`\sum_{i} \sum_{g} \sum_{g^{\prime}hhubh;)}(h`:math:`\lambda_{k}^{\prime \prime}` ≡ :math:`U_{K} / U_{k-1}, U_{k-1} \neq 0_{j}=1, U_{k-1}=0`h](jr)}(h#:math:`\lambda_{k}^{\prime \prime}`h]h/\lambda_{k}^{\prime \prime}}(hhh j?ubah}(h]h]h]h]h]uhjqh j?ubh/ ≡ }(h ≡ h j?hhh!NhNubjr)}(h8:math:`U_{K} / U_{k-1}, U_{k-1} \neq 0_{j}=1, U_{k-1}=0`h]h/0U_{K} / U_{k-1}, U_{k-1} \neq 0_{j}=1, U_{k-1}=0}(hhh j?ubah}(h]h]h]h]h]uhjqh j?ubeh}(h]h]h]h]h]uhh:h!jhMh j>hhubh;)}(hXoAn inner iteration in XSDRNPM consists of sweeping one time through the
entire spatial mesh for all the *S*\ :sub:`n` angles for one energy group. When
the fluxes for a particular group are being calculated, inner iterations
(j) will continue until (a) the number of inner iterations for this
outer iteration exceeds IIM (the inner iteration maximum) or (b) untilh](h/hAn inner iteration in XSDRNPM consists of sweeping one time through the
entire spatial mesh for all the }(hhAn inner iteration in XSDRNPM consists of sweeping one time through the
entire spatial mesh for all the h j?hhh!NhNubhA)}(h*S*h]h/S}(hhh j?ubah}(h]h]h]h]h]uhh@h j?ubh/ }(h\ h j?hhh!NhNubh)}(h:sub:`n`h]h/n}(hhh j?ubah}(h]h]h]h]h]uhhh j?ubh/ angles for one energy group. When
the fluxes for a particular group are being calculated, inner iterations
(j) will continue until (a) the number of inner iterations for this
outer iteration exceeds IIM (the inner iteration maximum) or (b) until}(h angles for one energy group. When
the fluxes for a particular group are being calculated, inner iterations
(j) will continue until (a) the number of inner iterations for this
outer iteration exceeds IIM (the inner iteration maximum) or (b) untilh j?hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j>hhubj)}(hhh]j)}(hh.. math::
\max _{i}\left|\frac{\psi_{i, g}^{j}-\psi_{i, g}^{j-1}}{\psi_{i, g}^{j}}\right| \leq P T C
h]j )}(hZ\max _{i}\left|\frac{\psi_{i, g}^{j}-\psi_{i, g}^{j-1}}{\psi_{i, g}^{j}}\right| \leq P T Ch]h/Z\max _{i}\left|\frac{\psi_{i, g}^{j}-\psi_{i, g}^{j-1}}{\psi_{i, g}^{j}}\right| \leq P T C}(hhh j@ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh j@ubah}(h]h]h]h]h]uhjh j@hhh!NhNubah}(h]h]h]h]h]jSjCjU(jV)uhjh j>hhh!jhMubh;)}(h@At the end of an outer iteration, the following checks are made:h]h/@At the end of an outer iteration, the following checks are made:}(hj:@h j8@hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j>hhubj)}(hhh](j)}(h5.. math::
\left|1.0-\lambda_{k}\right| \leq E P S
h]j )}(h'\left|1.0-\lambda_{k}\right| \leq E P Sh]h/'\left|1.0-\lambda_{k}\right| \leq E P S}(hhh jM@ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhM!h jI@ubah}(h]h]h]h]h]uhjh jF@hhh!NhNubj)}(h?.. math::
R\left|1.0-\lambda_{k}^{\prime}\right| \leq E P S
h]j )}(h1R\left|1.0-\lambda_{k}^{\prime}\right| \leq E P Sh]h/1R\left|1.0-\lambda_{k}^{\prime}\right| \leq E P S}(hhh ji@ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhM%h je@ubah}(h]h]h]h]h]uhjh jF@hhh!NhNubj)}(hF.. math::
R\left|1.0-\lambda_{k}^{\prime \prime}\right| \leq E P S
h]j )}(h8R\left|1.0-\lambda_{k}^{\prime \prime}\right| \leq E P Sh]h/8R\left|1.0-\lambda_{k}^{\prime \prime}\right| \leq E P S}(hhh j@ubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhM)h j@ubah}(h]h]h]h]h]uhjh jF@hhh!NhNubeh}(h]h]h]h]h]jSjCjUj6@jVj7@jXKuhjh j>hhh!jhM!ubh;)}(hX=R is a convergence relaxation factor and is set internally to 0.5 in XSDRNPM.
If all convergence criteria are met, if ICM (the outer iteration maximum) is
reached, or if ITMX (the maximum execution time) is exceeded, the problem will
be terminated with full output; otherwise, another outer iteration will be
started.h]h/X=R is a convergence relaxation factor and is set internally to 0.5 in XSDRNPM.
If all convergence criteria are met, if ICM (the outer iteration maximum) is
reached, or if ITMX (the maximum execution time) is exceeded, the problem will
be terminated with full output; otherwise, another outer iteration will be
started.}(hj@h j@hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM-h j>hhubh)}(h
.. _9-1-2-11:h]h}(h]h]h]h]h]hid51uhh
hMh j>hhh!jubeh}(h](iteration-and-convergence-testsj=eh]h](iteration and convergence tests9-1-2-10eh]h]uhh#h j hhh!jhMjf}j@j=sjh}j=j=subh$)}(hhh](h))}(h!Group banding (scaling rebalance)h]h/!Group banding (scaling rebalance)}(hj@h j@hhh!NhNubah}(h]h]h]h]h]uhh(h j@hhh!jhM6ubh;)}(hXAs described above, the normal mode of operation in XSDRNPM is to do inner
iterations on a group until it converges, then go to the next group. For groups
where there is no upscatter, the scattering source to a group depends only on
higher energy groups for which the fluxes have already been calculated. A fixed
source problem with no fission and no upscattering can, therefore, be converged
in one outer iteration. Since fission sources and upscattering sources are
calculated with fluxes from the previous outer iteration, multiple outer
iterations must be done to converge problems involving these kinds of sources.
For problems involving many fine thermal groups (groups with both upscatter and
downscatter), a special convergence problem arises. Because the groups are
fine, within-group scattering is small and the flux calculation is dominated by
scattering sources from other groups. This situation leads to a very slow
reduction in scattering source errors from one outer iteration to the next.
XSDRNPM has a special “group banding” option for treating this problem. It
involves collecting several groups together into a band and doing one inner for
each group in the band while collecting particle balance information. This
balance information is then used to solve for one set of flux rebalance factors
to apply to each group in the band. Because the band is much wider than an
individual group, the scattering that remains within the band is a much larger
fraction of the total scattering source for the band. This condition leads to
considerably faster convergence from one outer iteration to the next. The group
banding option in XSDRNPM is triggered by the seventh entry in the 2$ array.
The absolute value of this entry indicates the number of bands to be used. If
the number is negative, these bands are only for the thermal groups. Normally
there is no need to band together groups other than the thermal groups. An
entry of 1 indicates that all the thermal groups will be treated as one band.
This mode is one that is used successfully for many problems, but occasionally
will cause a problem to not converge. For these problems using two or three
bands for the thermal groups has been successful.h]h/XAs described above, the normal mode of operation in XSDRNPM is to do inner
iterations on a group until it converges, then go to the next group. For groups
where there is no upscatter, the scattering source to a group depends only on
higher energy groups for which the fluxes have already been calculated. A fixed
source problem with no fission and no upscattering can, therefore, be converged
in one outer iteration. Since fission sources and upscattering sources are
calculated with fluxes from the previous outer iteration, multiple outer
iterations must be done to converge problems involving these kinds of sources.
For problems involving many fine thermal groups (groups with both upscatter and
downscatter), a special convergence problem arises. Because the groups are
fine, within-group scattering is small and the flux calculation is dominated by
scattering sources from other groups. This situation leads to a very slow
reduction in scattering source errors from one outer iteration to the next.
XSDRNPM has a special “group banding” option for treating this problem. It
involves collecting several groups together into a band and doing one inner for
each group in the band while collecting particle balance information. This
balance information is then used to solve for one set of flux rebalance factors
to apply to each group in the band. Because the band is much wider than an
individual group, the scattering that remains within the band is a much larger
fraction of the total scattering source for the band. This condition leads to
considerably faster convergence from one outer iteration to the next. The group
banding option in XSDRNPM is triggered by the seventh entry in the 2$ array.
The absolute value of this entry indicates the number of bands to be used. If
the number is negative, these bands are only for the thermal groups. Normally
there is no need to band together groups other than the thermal groups. An
entry of 1 indicates that all the thermal groups will be treated as one band.
This mode is one that is used successfully for many problems, but occasionally
will cause a problem to not converge. For these problems using two or three
bands for the thermal groups has been successful.}(hj@h j@hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM8h j@hhubh;)}(hnThe code generates a default banding structure, but this structure can be
overridden by inputting a 52$ array.h]h/nThe code generates a default banding structure, but this structure can be
overridden by inputting a 52$ array.}(hj@h j@hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMVh j@hhubh)}(h
.. _9-1-2-12:h]h}(h]h]h]h]h]hid52uhh
hMh j@hhh!jubeh}(h](group-banding-scaling-rebalancej@eh]h](!group banding (scaling rebalance)9-1-2-11eh]h]uhh#h j hhh!jhM6jf}jAj@sjh}j@j@subh$)}(hhh](h))}(hBuckling correctionh]h/Buckling correction}(hjAh j
Ahhh!NhNubah}(h]h]h]h]h]uhh(h j
Ahhh!jhM\ubh;)}(hXSDRNPM allows “buckling” corrections to be made for the transverse
(non-calculated) dimensions in its 1-D slab and cylindrical geometries. Three
input parameters-DY, DZ, and BF (5* array)-may be involved.h]h/XSDRNPM allows “buckling” corrections to be made for the transverse
(non-calculated) dimensions in its 1-D slab and cylindrical geometries. Three
input parameters-DY, DZ, and BF (5* array)-may be involved.}(hjAh jAhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM^h j
Ahhubh;)}(hXIn the case of the 1-D slab, the height DY and the width DZ can be input. The
buckling correction uses an expression based on asymptotic diffusion theory to
account for leakage in the transverse direction and is treated analogous to an
absorption cross section, that is,h]h/XIn the case of the 1-D slab, the height DY and the width DZ can be input. The
buckling correction uses an expression based on asymptotic diffusion theory to
account for leakage in the transverse direction and is treated analogous to an
absorption cross section, that is,}(hj+Ah j)Ahhh!NhNubah}(h]h]h]h]h]uhh:h!jhMbh j
Ahhubj1)}(hhh]h;)}(h(Transverse Leakage :math:`=D B^{2} \psi`h](h/Transverse Leakage }(hTransverse Leakage h j:Aubjr)}(h:math:`=D B^{2} \psi`h]h/
=D B^{2} \psi}(hhh jCAubah}(h]h]h]h]h]uhjqh j:Aubeh}(h]h]h]h]h]uhh:h!jhMgh j7Aubah}(h]h]h]h]h]uhj1h j
Ahhh!jhNubh;)}(h1where B is the geometric buckling and is given byh]h/1where B is the geometric buckling and is given by}(hj_Ah j]Ahhh!NhNubah}(h]h]h]h]h]uhh:h!jhMih j
Ahhubj )}(hCB^{2}=\left(\frac{\pi}{Y}\right)^{2}+\left(\frac{\pi}{Z}\right)^{2}h]h/CB^{2}=\left(\frac{\pi}{Y}\right)^{2}+\left(\frac{\pi}{Z}\right)^{2}}(hhh jkAubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMkh j
Ahhubh;)}(hdand Y and Z are the height and width of the slab, respectively, and include extrapolation distances.h]h/dand Y and Z are the height and width of the slab, respectively, and include extrapolation distances.}(hjAh j}Ahhh!NhNubah}(h]h]h]h]h]uhh:h!jhMoh j
Ahhubh;)}(hXRecall that the “extrapolation distance” is defined as the linear
extrapolation distance such that if one extrapolated to a zero flux
value at this distance from the boundary, the interior flux shape in the
body would be correctly represented. The distance can be shown to occur
at 0.71 \ *λ*\ :sub:`tr`, where *λ*\ :sub:`tr` is the transport mean free path given
by 1/Σ\ :sub:`tr`. Note that for a slab, there are two extrapolation
distances to include (one on either side) for the height and width, such
thath](h/X&Recall that the “extrapolation distance” is defined as the linear
extrapolation distance such that if one extrapolated to a zero flux
value at this distance from the boundary, the interior flux shape in the
body would be correctly represented. The distance can be shown to occur
at 0.71 }(hX&Recall that the “extrapolation distance” is defined as the linear
extrapolation distance such that if one extrapolated to a zero flux
value at this distance from the boundary, the interior flux shape in the
body would be correctly represented. The distance can be shown to occur
at 0.71 \ h jAhhh!NhNubhA)}(h*λ*h]h/λ}(hhh jAubah}(h]h]h]h]h]uhh@h jAubh/ }(h\ h jAhhh!NhNubh)}(h :sub:`tr`h]h/tr}(hhh jAubah}(h]h]h]h]h]uhhh jAubh/, where }(h, where h jAhhh!NhNubhA)}(h*λ*h]h/λ}(hhh jAubah}(h]h]h]h]h]uhh@h jAubh/ }(hjAh jAubh)}(h :sub:`tr`h]h/tr}(hhh jAubah}(h]h]h]h]h]uhhh jAubh/0 is the transport mean free path given
by 1/Σ }(h0 is the transport mean free path given
by 1/Σ\ h jAhhh!NhNubh)}(h :sub:`tr`h]h/tr}(hhh jAubah}(h]h]h]h]h]uhhh jAubh/. Note that for a slab, there are two extrapolation
distances to include (one on either side) for the height and width, such
that}(h. Note that for a slab, there are two extrapolation
distances to include (one on either side) for the height and width, such
thath jAhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMqh j
Ahhubj )}(hY=D Y+1.42 \lambda_{t r} ,h]h/Y=D Y+1.42 \lambda_{t r} ,}(hhh jAubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMzh j
Ahhubh;)}(handh]h/and}(hjBh j
Bhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM~h j
Ahhubj )}(h+\mathrm{Z}=\mathrm{DZ}+1.42 \lambda_{t r} .h]h/+\mathrm{Z}=\mathrm{DZ}+1.42 \lambda_{t r} .}(hhh jBubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh j
Ahhubh;)}(hQThe 1.42 factor (= 2 *X* 0.71) is input in the BF parameter of the
5\ \* array.h](h/The 1.42 factor (= 2 }(hThe 1.42 factor (= 2 h j*Bhhh!NhNubhA)}(h*X*h]h/X}(hhh j3Bubah}(h]h]h]h]h]uhh@h j*Bubh/8 0.71) is input in the BF parameter of the
5 * array.}(h8 0.71) is input in the BF parameter of the
5\ \* array.h j*Bhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j
Ahhubh;)}(h_In calculating *λ*\ :sub:`tr`, a transport cross section, *Σ*\ :sub:`tr`, is
determined fromh](h/In calculating }(hIn calculating h jLBhhh!NhNubhA)}(h*λ*h]h/λ}(hhh jUBubah}(h]h]h]h]h]uhh@h jLBubh/ }(h\ h jLBhhh!NhNubh)}(h :sub:`tr`h]h/tr}(hhh jhBubah}(h]h]h]h]h]uhhh jLBubh/, a transport cross section, }(h, a transport cross section, h jLBhhh!NhNubhA)}(h*Σ*h]h/Σ}(hhh j{Bubah}(h]h]h]h]h]uhh@h jLBubh/ }(hjgBh jLBubh)}(h :sub:`tr`h]h/tr}(hhh jBubah}(h]h]h]h]h]uhhh jLBubh/, is
determined from}(h, is
determined fromh jLBhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j
Ahhubj )}(h$\Sigma_{t r}=\Sigma_{t}-\Sigma_{s 1}h]h/$\Sigma_{t r}=\Sigma_{t}-\Sigma_{s 1}}(hhh jBubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh j
Ahhubh;)}(hwhich varies as a function of energy group and zone. The
Σ\ :sub:`s\ 1` term is the within-group term from the
*P*\ :sub:`1` scattering matrix.h](h/=which varies as a function of energy group and zone. The
Σ }(h=which varies as a function of energy group and zone. The
Σ\ h jBhhh!NhNubh)}(h:sub:`s\ 1`h]h/s 1}(hhh jBubah}(h]h]h]h]h]uhhh jBubh/) term is the within-group term from the
}(h) term is the within-group term from the
h jBhhh!NhNubhA)}(h*P*h]h/P}(hhh jBubah}(h]h]h]h]h]uhh@h jBubh/ }(h\ h jBhhh!NhNubh)}(h:sub:`1`h]h/1}(hhh jBubah}(h]h]h]h]h]uhhh jBubh/ scattering matrix.}(h scattering matrix.h jBhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j
Ahhubh;)}(hvIn the case of the 1-D cylinder, the procedure is the same as for the slab
except that the buckling is determined fromh]h/vIn the case of the 1-D cylinder, the procedure is the same as for the slab
except that the buckling is determined from}(hjCh jChhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j
Ahhubj )}(h&B^{2}=\left(\frac{\pi}{Y}\right)^{2} ,h]h/&B^{2}=\left(\frac{\pi}{Y}\right)^{2} ,}(hhh jCubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh j
Ahhubh;)}(h.since only one transverse dimension is needed.h]h/.since only one transverse dimension is needed.}(hj"Ch j Chhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j
Ahhubh;)}(h@The diffusion coefficient in the leakage term is determined fromh]h/@The diffusion coefficient in the leakage term is determined from}(hj0Ch j.Chhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j
Ahhubj )}(hD=\frac{1}{3 \Sigma_{t r}}h]h/D=\frac{1}{3 \Sigma_{t r}}}(hhh jIubah}(h]h]h]h]h]uhh:h!jhMgh j:Iubah}(h]h]h]h]h]uhjh j7Ihhh!jhNubah}(h]h]h]h]h]jSjCjUhjVjWjXKuhjh jHhhh!jhMgubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-56uhh
h jHhhh!jhNubj )}(h{\overline{\sigma_{G}^{j}} \equiv \frac{\sum_{g \varepsilon G} \sigma_{g}^{j} W_{g}^{j}}{\sum_{g \varepsilon G} W_{G}^{j}} .h]h/{\overline{\sigma_{G}^{j}} \equiv \frac{\sum_{g \varepsilon G} \sigma_{g}^{j} W_{g}^{j}}{\sum_{g \varepsilon G} W_{G}^{j}} .}(hhh jbIubah}(h]jaIah]h]h]h]docnamejnumberK7labeleq9-1-56nowrapjyjzuhj h!jhMih jHhhjf}jh}jaIjXIsubj)}(hhh]j)}(hRegion weighting
h]h;)}(hRegion weightingh]h/Region weighting}(hjIh j~Iubah}(h]h]h]h]h]uhh:h!jhMnh jzIubah}(h]h]h]h]h]uhjh jwIhhh!jhNubah}(h]h]h]h]h]jSjCjUhjVjWjXKuhjh jHhhh!jhMnubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-57uhh
h jHhhh!jhNubj )}(h\bar{\sigma}_{G} \equiv \frac{\sum_{j} N^{j} \sum_{g \varepsilon G} \sigma_{g}^{j} W_{g}^{j}}{\sum_{j} N^{j} \sum_{g \varepsilon G} W_{g}^{j}} .h]h/\bar{\sigma}_{G} \equiv \frac{\sum_{j} N^{j} \sum_{g \varepsilon G} \sigma_{g}^{j} W_{g}^{j}}{\sum_{j} N^{j} \sum_{g \varepsilon G} W_{g}^{j}} .}(hhh jIubah}(h]jIah]h]h]h]docnamejnumberK8labeleq9-1-57nowrapjyjzuhj h!jhMph jHhhjf}jh}jIjIsubh)}(h.. _9-1-2-14-6:h]h}(h]h]h]h]h]hid61uhh
hMh jHhhh!jubeh}(h](multigroup-weighting-equationsjHeh]h](multigroup weighting equations
9-1-2-14-5eh]h]uhh#h jMDhhh!jhMRjf}jIjHsjh}jHjHsubh$)}(hhh](h))}(hTransfer matricesh]h/Transfer matrices}(hjIh jIhhh!NhNubah}(h]h]h]h]h]uhh(h jIhhh!jhMxubh;)}(hCollapsing transfer matrices is not quite so simple as collapsing cross sections
with a single value per group. A group-to-group term in the broad group sense
conserves the scattering rate from one group to the other, that is,h]h/Collapsing transfer matrices is not quite so simple as collapsing cross sections
with a single value per group. A group-to-group term in the broad group sense
conserves the scattering rate from one group to the other, that is,}(hjIh jIhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMzh jIhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-58uhh
h jIhhh!jhNubj )}(h\bar{N}^{*} \bar{\sigma}\left(G \rightarrow G^{\prime}\right) \psi_{G} \equiv \int_{\text {space}} d r N(r) \int_{g} d E \psi(E, r) \int_{g^{\prime}} d E^{\prime} \sigma\left(E \rightarrow E^{\prime}\right)h]h/\bar{N}^{*} \bar{\sigma}\left(G \rightarrow G^{\prime}\right) \psi_{G} \equiv \int_{\text {space}} d r N(r) \int_{g} d E \psi(E, r) \int_{g^{\prime}} d E^{\prime} \sigma\left(E \rightarrow E^{\prime}\right)}(hhh jIubah}(h]jIah]h]h]h]docnamejnumberK9labeleq9-1-58nowrapjyjzuhj h!jhM~h jIhhjf}jh}jIjIsubh;)}(hwhere the asterisk (*) denotes that the number density on the left side
of the equation is consistent with the weighting desired. Therefore, the
multigroup forms of the weighting equations for components of the
transfer matrices are as follows:h]h/where the asterisk (*) denotes that the number density on the left side
of the equation is consistent with the weighting desired. Therefore, the
multigroup forms of the weighting equations for components of the
transfer matrices are as follows:}(hwhere the asterisk (*) denotes that the number density on the left side
of the equation is consistent with the weighting desired. Therefore, the
multigroup forms of the weighting equations for components of the
transfer matrices are as follows:h jJhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jIhhubj)}(hhh]j)}(hCell weighting
h]h;)}(hCell weightingh]h/Cell weighting}(hj#Jh j!Jubah}(h]h]h]h]h]uhh:h!jhMh jJubah}(h]h]h]h]h]uhjh jJhhh!jhNubah}(h]h]h]h]h]jSjCjUhjVjWuhjh jIhhh!jhMubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-59uhh
h jIhhh!jhNubj )}(h\bar{\sigma}_{G \rightarrow G^{\prime}} \equiv \frac{\sum_{j}^{I Z M} N^{j} \sum_{g \varepsilon G} W_{g}^{j} \sum_{g^{\prime} \varepsilon G^{\prime}} \sigma^{j}\left(g \rightarrow g^{\prime}\right)}{\bar{N} \sum_{j}^{IZM} \sum_{g \varepsilon G} W_{g}^{j}}h]h/\bar{\sigma}_{G \rightarrow G^{\prime}} \equiv \frac{\sum_{j}^{I Z M} N^{j} \sum_{g \varepsilon G} W_{g}^{j} \sum_{g^{\prime} \varepsilon G^{\prime}} \sigma^{j}\left(g \rightarrow g^{\prime}\right)}{\bar{N} \sum_{j}^{IZM} \sum_{g \varepsilon G} W_{g}^{j}}}(hhh jEJubah}(h]jDJah]h]h]h]docnamejnumberK:labeleq9-1-59nowrapjyjzuhj h!jhMh jIhhjf}jh}jDJj;Jsubj)}(hhh]j)}(hZone weighting
h]h;)}(hZone weightingh]h/Zone weighting}(hjcJh jaJubah}(h]h]h]h]h]uhh:h!jhMh j]Jubah}(h]h]h]h]h]uhjh jZJhhh!jhNubah}(h]h]h]h]h]jSjCjUhjVjWjXKuhjh jIhhh!jhMubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-60uhh
h jIhhh!jhNubj )}(h\bar{\sigma}_{G \rightarrow G^{\prime}} \equiv \frac{\sum_{g \varepsilon G} W_{g}^{j} \sum_{g^{\prime} \varepsilon G^{\prime}} \sigma^{j}\left(g \rightarrow g^{\prime}\right)}{\sum_{g \varepsilon G} W_{g}^{j}} .h]h/\bar{\sigma}_{G \rightarrow G^{\prime}} \equiv \frac{\sum_{g \varepsilon G} W_{g}^{j} \sum_{g^{\prime} \varepsilon G^{\prime}} \sigma^{j}\left(g \rightarrow g^{\prime}\right)}{\sum_{g \varepsilon G} W_{g}^{j}} .}(hhh jJubah}(h]jJah]h]h]h]docnamejnumberK;labeleq9-1-60nowrapjyjzuhj h!jhMh jIhhjf}jh}jJj{Jsubj)}(hhh]j)}(hRegion weighting
h]h;)}(hRegion weightingh]h/Region weighting}(hjJh jJubah}(h]h]h]h]h]uhh:h!jhMh jJubah}(h]h]h]h]h]uhjh jJhhh!jhNubah}(h]h]h]h]h]jSjCjUhjVjWjXKuhjh jIhhh!jhMubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-61uhh
h jIhhh!jhNubj )}(h\bar{\sigma}_{G \rightarrow G^{\prime}} \equiv \frac{\sum_{j} N^{j} \sum_{g \varepsilon G} W_{g}^{j} \sum_{g^{\prime} \varepsilon G^{\prime}} \sigma^{j}\left(g \rightarrow g^{\prime}\right)}{\sum_{j} N^{j} \sum_{g \varepsilon G} W_{g}^{j}} .h]h/\bar{\sigma}_{G \rightarrow G^{\prime}} \equiv \frac{\sum_{j} N^{j} \sum_{g \varepsilon G} W_{g}^{j} \sum_{g^{\prime} \varepsilon G^{\prime}} \sigma^{j}\left(g \rightarrow g^{\prime}\right)}{\sum_{j} N^{j} \sum_{g \varepsilon G} W_{g}^{j}} .}(hhh jJubah}(h]jJah]h]h]h]docnamejnumberK<labeleq9-1-61nowrapjyjzuhj h!jhMh jIhhjf}jh}jJjJsubh;)}(hXTheoretically, the higher-than-zero order :math:`\sigma_{l}\left(g \rightarrow g^{\prime}\right)` should be weighted over
*ψ*\ :sub:`l`. Since these functions are generally positive-negative, *ψ*\ :sub:`l`
weighting does not always work in practice, and XSDRNPM weights the :math:`\sigma_{l}\left(g \rightarrow g^{\prime}\right), \quad>0`,
by the scalar flux, which is positive. This procedure gives usable values
for most cases.h](h/*Theoretically, the higher-than-zero order }(h*Theoretically, the higher-than-zero order h jJhhh!NhNubjr)}(h7:math:`\sigma_{l}\left(g \rightarrow g^{\prime}\right)`h]h//\sigma_{l}\left(g \rightarrow g^{\prime}\right)}(hhh jJubah}(h]h]h]h]h]uhjqh jJubh/ should be weighted over
}(h should be weighted over
h jJhhh!NhNubhA)}(h*ψ*h]h/ψ}(hhh jJubah}(h]h]h]h]h]uhh@h jJubh/ }(h\ h jJhhh!NhNubh)}(h:sub:`l`h]h/l}(hhh j Kubah}(h]h]h]h]h]uhhh jJubh/9. Since these functions are generally positive-negative, }(h9. Since these functions are generally positive-negative, h jJhhh!NhNubhA)}(h*ψ*h]h/ψ}(hhh jKubah}(h]h]h]h]h]uhh@h jJubh/ }(hjKh jJubh)}(h:sub:`l`h]h/l}(hhh j.Kubah}(h]h]h]h]h]uhhh jJubh/E
weighting does not always work in practice, and XSDRNPM weights the }(hE
weighting does not always work in practice, and XSDRNPM weights the h jJhhh!NhNubjr)}(h@:math:`\sigma_{l}\left(g \rightarrow g^{\prime}\right), \quad>0`h]h/8\sigma_{l}\left(g \rightarrow g^{\prime}\right), \quad>0}(hhh jAKubah}(h]h]h]h]h]uhjqh jJubh/[,
by the scalar flux, which is positive. This procedure gives usable values
for most cases.}(h[,
by the scalar flux, which is positive. This procedure gives usable values
for most cases.h jJhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jIhhubh)}(h.. _9-1-2-14-7:h]h}(h]h]h]h]h]hid62uhh
hMh jIhhh!jubeh}(h](transfer-matricesjIeh]h](transfer matrices
9-1-2-14-6eh]h]uhh#h jMDhhh!jhMxjf}jkKjIsjh}jIjIsubh$)}(hhh](h))}(hWeighting of :math:`\bar{v}`h](h/
Weighting of }(h
Weighting of h jsKhhh!NhNubjr)}(h:math:`\bar{v}`h]h/\bar{v}}(hhh j|Kubah}(h]h]h]h]h]uhjqh jsKubeh}(h]h]h]h]h]uhh(h jpKhhh!jhMubh;)}(hXIn weighting parameters such as :math:`\bar{v}`, the average number of neutrons produced
per fission, one is interested in preserving the fission source;
therefore, the weighting is over *σ*\ :sub:`f`\ ψ* instead of just *ψ*. The
weighting procedure in XSDRNPM is to calculate :math:`\left(\overline{v \sigma_{f}}\right)_{G}` and (*σ*\ :sub:`f`)\ *G* using
the appropriate choice from Eqs. :eq:`eq9-1-59`, :eq:`eq9-1-60`, or :eq:`eq9-1-61`. Thenh](h/ In weighting parameters such as }(h In weighting parameters such as h jKhhh!NhNubjr)}(h:math:`\bar{v}`h]h/\bar{v}}(hhh jKubah}(h]h]h]h]h]uhjqh jKubh/, the average number of neutrons produced
per fission, one is interested in preserving the fission source;
therefore, the weighting is over }(h, the average number of neutrons produced
per fission, one is interested in preserving the fission source;
therefore, the weighting is over h jKhhh!NhNubhA)}(h*σ*h]h/σ}(hhh jKubah}(h]h]h]h]h]uhh@h jKubh/ }(h\ h jKhhh!NhNubh)}(h:sub:`f`h]h/f}(hhh jKubah}(h]h]h]h]h]uhhh jKubh/ ψ* instead of just }(h\ ψ* instead of just h jKhhh!NhNubhA)}(h*ψ*h]h/ψ}(hhh jKubah}(h]h]h]h]h]uhh@h jKubh/5. The
weighting procedure in XSDRNPM is to calculate }(h5. The
weighting procedure in XSDRNPM is to calculate h jKhhh!NhNubjr)}(h0:math:`\left(\overline{v \sigma_{f}}\right)_{G}`h]h/(\left(\overline{v \sigma_{f}}\right)_{G}}(hhh jKubah}(h]h]h]h]h]uhjqh jKubh/ and (}(h and (h jKhhh!NhNubhA)}(h*σ*h]h/σ}(hhh jKubah}(h]h]h]h]h]uhh@h jKubh/ }(hjKh jKubh)}(h:sub:`f`h]h/f}(hhh j
Lubah}(h]h]h]h]h]uhhh jKubh/) }(h)\ h jKhhh!NhNubhA)}(h*G*h]h/G}(hhh jLubah}(h]h]h]h]h]uhh@h jKubh/( using
the appropriate choice from Eqs. }(h( using
the appropriate choice from Eqs. h jKhhh!NhNubj)}(h:eq:`eq9-1-59`h]jc)}(hj2Lh]h/eq9-1-59}(hhh j4Lubah}(h]h](jneqeh]h]h]uhjbh j0Lubah}(h]h]h]h]h]refdocj refdomainjqreftypej>Lrefexplicitrefwarnjeq9-1-59uhjh!jhMh jKubh/, }(h, h jKhhh!NhNubj)}(h:eq:`eq9-1-60`h]jc)}(hjULh]h/eq9-1-60}(hhh jWLubah}(h]h](jneqeh]h]h]uhjbh jSLubah}(h]h]h]h]h]refdocj refdomainjqreftypejaLrefexplicitrefwarnjeq9-1-60uhjh!jhMh jKubh/, or }(h, or h jKhhh!NhNubj)}(h:eq:`eq9-1-61`h]jc)}(hjxLh]h/eq9-1-61}(hhh jzLubah}(h]h](jneqeh]h]h]uhjbh jvLubah}(h]h]h]h]h]refdocj refdomainjqreftypejLrefexplicitrefwarnjeq9-1-61uhjh!jhMh jKubh/. Then}(h. Thenh jKhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jpKhhubj )}(hZ\bar{v}_{G}=\frac{\left(\overline{v \sigma_{F}}\right)_{G}}{\left(\sigma_{f}\right)_{G}} .h]h/Z\bar{v}_{G}=\frac{\left(\overline{v \sigma_{F}}\right)_{G}}{\left(\sigma_{f}\right)_{G}} .}(hhh jLubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh jpKhhubh)}(h.. _9-1-2-14-8:h]h}(h]h]h]h]h]hid63uhh
hM-h jpKhhh!jubeh}(h](weighting-of-bar-vjdKeh]h](weighting of \bar{v}
9-1-2-14-7eh]h]uhh#h jMDhhh!jhMjf}jLjZKsjh}jdKjZKsubh$)}(hhh](h))}(hTransport cross sectionsh]h/Transport cross sections}(hjLh jLhhh!NhNubah}(h]h]h]h]h]uhh(h jLhhh!jhMubh;)}(hXTransport cross sections are not as directly related to the physical
properties of a material as much as other group-averaged values. Instead
of a reaction rate, these numbers must attempt to preserve a “flux
gradient,” which not only depends on the cross sections of the material,
but is also very strongly influenced by the geometry and the other
nuclides in the vicinity of a material.h]h/XTransport cross sections are not as directly related to the physical
properties of a material as much as other group-averaged values. Instead
of a reaction rate, these numbers must attempt to preserve a “flux
gradient,” which not only depends on the cross sections of the material,
but is also very strongly influenced by the geometry and the other
nuclides in the vicinity of a material.}(hjLh jLhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jLhhubh;)}(hXJTwo options are provided in XSDRNPM to generate transport
cross sections—options based on the “consistent” and “inconsistent”
methods for solving the *P\ l* transport equations. These approximations
are referred to as the “outscatter” and “inscatter” approximations
because of the nature of the equations used.h](h/Two options are provided in XSDRNPM to generate transport
cross sections—options based on the “consistent” and “inconsistent”
methods for solving the }(hTwo options are provided in XSDRNPM to generate transport
cross sections—options based on the “consistent” and “inconsistent”
methods for solving the h jLhhh!NhNubhA)}(h*P\ l*h]h/P l}(hhh jLubah}(h]h]h]h]h]uhh@h jLubh/ transport equations. These approximations
are referred to as the “outscatter” and “inscatter” approximations
because of the nature of the equations used.}(h transport equations. These approximations
are referred to as the “outscatter” and “inscatter” approximations
because of the nature of the equations used.h jLhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jLhhubh)}(h.. _9-1-2-14-8-1:h]h}(h]h]h]h]h]hid64uhh
hM?h jLhhh!jubh$)}(hhh](h))}(h.Outscatter approximation (inconsistent method)h]h/.Outscatter approximation (inconsistent method)}(hjMh jMhhh!NhNubah}(h]h]h]h]h]uhh(h jMhhh!jhMubh;)}(h\sigma_{t r}^{g}=\sigma_{t}^{g}-\bar{\mu}^{g} \sigma_{s}^{g} .h]h/>\sigma_{t r}^{g}=\sigma_{t}^{g}-\bar{\mu}^{g} \sigma_{s}^{g} .}(hhh jlabeleq9-1-63nowrapjyjzuhj h!jhMh jMhhjf}jh}jhMj_Msubh;)}(hand thath]h/and that}(hjMh j~Mhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jMhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-64uhh
h jMhhh!jhNubj )}(hR\sigma_{1}^{g}=\sum_{g^{\prime}} \sigma_{1}\left(g \rightarrow g^{\prime}\right) ,h]h/R\sigma_{1}^{g}=\sum_{g^{\prime}} \sigma_{1}\left(g \rightarrow g^{\prime}\right) ,}(hhh jMubah}(h]jMah]h]h]h]docnamejnumberK?labeleq9-1-64nowrapjyjzuhj h!jhMh jMhhjf}jh}jMjMsubh;)}(hwhere the :math:`\sigma_{l}\left(g \rightarrow g^{\prime}\right)` terms are the *P*\ :sub:`1` coefficients of the scattering
matrix, the origin of the term “outscatter” to designate the
approximation is evident.h](h/
where the }(h
where the h jMhhh!NhNubjr)}(h7:math:`\sigma_{l}\left(g \rightarrow g^{\prime}\right)`h]h//\sigma_{l}\left(g \rightarrow g^{\prime}\right)}(hhh jMubah}(h]h]h]h]h]uhjqh jMubh/ terms are the }(h terms are the h jMhhh!NhNubhA)}(h*P*h]h/P}(hhh jMubah}(h]h]h]h]h]uhh@h jMubh/ }(h\ h jMhhh!NhNubh)}(h:sub:`1`h]h/1}(hhh jMubah}(h]h]h]h]h]uhhh jMubh/{ coefficients of the scattering
matrix, the origin of the term “outscatter” to designate the
approximation is evident.}(h{ coefficients of the scattering
matrix, the origin of the term “outscatter” to designate the
approximation is evident.h jMhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jMhhubh)}(h.. _9-1-2-14-8-2:h]h}(h]h]h]h]h]hid65uhh
hM]h jMhhh!jubeh}(h](,outscatter-approximation-inconsistent-methodjMeh]h]9-1-2-14-8-1ah].outscatter approximation (inconsistent method)ah]uhh#h jLhhh!jhMjKjf}jNjMsjh}jMjMsubh$)}(hhh](h))}(h+Inscatter approximation (consistent method)h]h/+Inscatter approximation (consistent method)}(hjNh jNhhh!NhNubah}(h]h]h]h]h]uhh(h j Nhhh!jhMubh;)}(hfIn the “consistent” solution of the *P*\ :sub:`1` point transport
equations, it can be shown thath](h/(In the “consistent” solution of the }(h(In the “consistent” solution of the h jNhhh!NhNubhA)}(h*P*h]h/P}(hhh j#Nubah}(h]h]h]h]h]uhh@h jNubh/ }(h\ h jNhhh!NhNubh)}(h:sub:`1`h]h/1}(hhh j6Nubah}(h]h]h]h]h]uhhh jNubh/1 point transport
equations, it can be shown that}(h1 point transport
equations, it can be shown thath jNhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j Nhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-65uhh
h j Nhhh!jhNubj )}(h\sigma_{t r}(E)=\sigma_{t}(E)-\frac{1}{3 J(E)} \int_{0}^{\infty} d E^{\prime} \sigma_{1}\left(E^{\prime} \rightarrow E\right) J\left(E^{\prime}\right) ,h]h/\sigma_{t r}(E)=\sigma_{t}(E)-\frac{1}{3 J(E)} \int_{0}^{\infty} d E^{\prime} \sigma_{1}\left(E^{\prime} \rightarrow E\right) J\left(E^{\prime}\right) ,}(hhh jYNubah}(h]jXNah]h]h]h]docnamejnumberK@labeleq9-1-65nowrapjyjzuhj h!jhMh j Nhhjf}jh}jXNjONsubh;)}(h%where *J*\ (*E*\ ′) is the current.h](h/where }(hwhere h jnNhhh!NhNubhA)}(h*J*h]h/J}(hhh jwNubah}(h]h]h]h]h]uhh@h jnNubh/ (}(h\ (h jnNhhh!NhNubhA)}(h*E*h]h/E}(hhh jNubah}(h]h]h]h]h]uhh@h jnNubh/ ′) is the current.}(h\ ′) is the current.h jnNhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j Nhhubh;)}(hIf one multiplies the equation by J(E), integrates over group g, and
converts to group-averaged form by dividing by :math:`\int_{g} J(E) d E` the following expression
is derived:h](h/uIf one multiplies the equation by J(E), integrates over group g, and
converts to group-averaged form by dividing by }(huIf one multiplies the equation by J(E), integrates over group g, and
converts to group-averaged form by dividing by h jNhhh!NhNubjr)}(h:math:`\int_{g} J(E) d E`h]h/\int_{g} J(E) d E}(hhh jNubah}(h]h]h]h]h]uhjqh jNubh/% the following expression
is derived:}(h% the following expression
is derived:h jNhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j Nhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-66uhh
h j Nhhh!jhNubj )}(h\sigma_{t r}^{g}=\sigma_{t}^{g}-\frac{1}{3 J_{g}} \sum_{g^{\prime}} \sigma_{1}\left(g^{\prime} \rightarrow g\right) J_{g^{\prime}} .h]h/\sigma_{t r}^{g}=\sigma_{t}^{g}-\frac{1}{3 J_{g}} \sum_{g^{\prime}} \sigma_{1}\left(g^{\prime} \rightarrow g\right) J_{g^{\prime}} .}(hhh jNubah}(h]jNah]h]h]h]docnamejnumberKAlabeleq9-1-66nowrapjyjzuhj h!jhMh j Nhhjf}jh}jNjNsubh;)}(hXThis is the “inscatter” approximation. It is consistent because the
transport values are explicitly derived from the *P*\ :sub:`0` and
*P*\ :sub:`1` equations. As a general rule, the transport values from
this treatment are “better” than those from the “inconsistent”
treatment. However, in some cases (notably hydrogen at lower energies),
negative numbers may be calculated which are unusable and the more
approximate approach must be used.h](h/yThis is the “inscatter” approximation. It is consistent because the
transport values are explicitly derived from the }(hyThis is the “inscatter” approximation. It is consistent because the
transport values are explicitly derived from the h jNhhh!NhNubhA)}(h*P*h]h/P}(hhh jNubah}(h]h]h]h]h]uhh@h jNubh/ }(h\ h jNhhh!NhNubh)}(h:sub:`0`h]h/0}(hhh jOubah}(h]h]h]h]h]uhhh jNubh/ and
}(h and
h jNhhh!NhNubhA)}(h*P*h]h/P}(hhh jOubah}(h]h]h]h]h]uhh@h jNubh/ }(hjNh jNubh)}(h:sub:`1`h]h/1}(hhh j%Oubah}(h]h]h]h]h]uhhh jNubh/X. equations. As a general rule, the transport values from
this treatment are “better” than those from the “inconsistent”
treatment. However, in some cases (notably hydrogen at lower energies),
negative numbers may be calculated which are unusable and the more
approximate approach must be used.}(hX. equations. As a general rule, the transport values from
this treatment are “better” than those from the “inconsistent”
treatment. However, in some cases (notably hydrogen at lower energies),
negative numbers may be calculated which are unusable and the more
approximate approach must be used.h jNhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j Nhhubh)}(h.. _9-1-2-14-8-3:h]h}(h]h]h]h]h]hid66uhh
hM}h j Nhhh!jubeh}(h]()inscatter-approximation-consistent-methodjMeh]h]9-1-2-14-8-2ah]+inscatter approximation (consistent method)ah]uhh#h jLhhh!jhMjKjf}jNOjMsjh}jMjMsubh$)}(hhh](h))}(h.Weighting function for transport cross sectionh]h/.Weighting function for transport cross section}(hjYOh jWOhhh!NhNubah}(h]h]h]h]h]uhh(h jTOhhh!jhMubh;)}(hXUnfortunately, the matter of choosing a current to use in the
“transport” weighting is not simple. In real problems, currents are
positive-negative as a function of energy and space. When cross sections
are averaged over positive-negative functions, the “law-of-the-mean” no
longer holds and the average value can be anything. This unbounded
nature leads to real problems in diffusion calculations.h]h/XUnfortunately, the matter of choosing a current to use in the
“transport” weighting is not simple. In real problems, currents are
positive-negative as a function of energy and space. When cross sections
are averaged over positive-negative functions, the “law-of-the-mean” no
longer holds and the average value can be anything. This unbounded
nature leads to real problems in diffusion calculations.}(hjgOh jeOhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jTOhhubh;)}(hApproximations that inherently guarantee positive currents are generally
used in other codes that circumvent the positive-negative problem. For
example, in *B\ n* theory the current is given byh](h/Approximations that inherently guarantee positive currents are generally
used in other codes that circumvent the positive-negative problem. For
example, in }(hApproximations that inherently guarantee positive currents are generally
used in other codes that circumvent the positive-negative problem. For
example, in h jsOhhh!NhNubhA)}(h*B\ n*h]h/B n}(hhh j|Oubah}(h]h]h]h]h]uhh@h jsOubh/ theory the current is given by}(h theory the current is given byh jsOhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jTOhhubj )}(h
j \sim B \psih]h/
j \sim B \psi}(hhh jOubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh jTOhhubh;)}(h#where B and *ψ* are both positive.h](h/where B and }(hwhere B and h jOhhh!NhNubhA)}(h*ψ*h]h/ψ}(hhh jOubah}(h]h]h]h]h]uhh@h jOubh/ are both positive.}(h are both positive.h jOhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jTOhhubh;)}(hIn XSDRNPM, more direct routes that ensure positivity are taken
(e.g., one might set :math:`\mathrm{W}_{\mathrm{g}} \equiv\left|\mathrm{W}_{\mathrm{g}}\right|`).
This is crudely supported by the following
argument:h](h/WIn XSDRNPM, more direct routes that ensure positivity are taken
(e.g., one might set }(hWIn XSDRNPM, more direct routes that ensure positivity are taken
(e.g., one might set h jOhhh!NhNubjr)}(hJ:math:`\mathrm{W}_{\mathrm{g}} \equiv\left|\mathrm{W}_{\mathrm{g}}\right|`h]h/B\mathrm{W}_{\mathrm{g}} \equiv\left|\mathrm{W}_{\mathrm{g}}\right|}(hhh jOubah}(h]h]h]h]h]uhjqh jOubh/7).
This is crudely supported by the following
argument:}(h7).
This is crudely supported by the following
argument:h jOhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jTOhhubh;)}(hiConsider a 1-D cylindrical calculation. In two dimensions, the current is a vector combination, that is,h]h/iConsider a 1-D cylindrical calculation. In two dimensions, the current is a vector combination, that is,}(hjOh jOhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jTOhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-67uhh
h jTOhhh!jhNubj )}(hJ=J_{r}+J_{z} .h]h/J=J_{r}+J_{z} .}(hhh jPubah}(h]jPah]h]h]h]docnamejnumberKBlabeleq9-1-67nowrapjyjzuhj h!jhMh jTOhhjf}jh}jPjOsubh;)}(hRIn XSDRNPM, the z direction is treated by using a buckling approximation, that is,h]h/RIn XSDRNPM, the z direction is treated by using a buckling approximation, that is,}(hjPh jPhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM$h jTOhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-68uhh
h jTOhhh!jhNubj )}(hJ_{z}=B \psi .h]h/J_{z}=B \psi .}(hhh j0Pubah}(h]j/Pah]h]h]h]docnamejnumberKClabeleq9-1-68nowrapjyjzuhj h!jhM&h jTOhhjf}jh}j/Pj&Psubh;)}(hIn the weighting calculation, we want to weight over the magnitude of the
current. In XSDRNPM, the z current is imaginary, since we are not calculating a
z-direction:h]h/In the weighting calculation, we want to weight over the magnitude of the
current. In XSDRNPM, the z current is imaginary, since we are not calculating a
z-direction:}(hjGPh jEPhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM+h jTOhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-69uhh
h jTOhhh!jhNubj )}(hJ=J_{r}+i B \psi .h]h/J=J_{r}+i B \psi .}(hhh j]Pubah}(h]j\Pah]h]h]h]docnamejnumberKDlabeleq9-1-69nowrapjyjzuhj h!jhM/h jTOhhjf}jh}j\PjSPsubh;)}(h&The magnitude of a complex quantity ish]h/&The magnitude of a complex quantity is}(hjtPh jrPhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM4h jTOhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-70uhh
h jTOhhh!jhNubj )}(h{J=\frac{\left(J_{r}+i B \psi\right)\left(J_{r}-i B \psi\right)}{\sqrt{\left(\text {Value}_{r}\right)^{2}+B^{2} \psi^{2}}} ,h]h/{J=\frac{\left(J_{r}+i B \psi\right)\left(J_{r}-i B \psi\right)}{\sqrt{\left(\text {Value}_{r}\right)^{2}+B^{2} \psi^{2}}} ,}(hhh jPubah}(h]jPah]h]h]h]docnamejnumberKElabeleq9-1-70nowrapjyjzuhj h!jhM6h jTOhhjf}jh}jPjPsubh;)}(hwhich is always positive.h]h/which is always positive.}(hjPh jPhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM;h jTOhhubh;)}(hfIn a discrete-ordinates calculation, the current is easily obtained since it is the first flux moment.h]h/fIn a discrete-ordinates calculation, the current is easily obtained since it is the first flux moment.}(hjPh jPhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM=h jTOhhubh;)}(h>XSDRNPM has the following options for calculating the current:h]h/>XSDRNPM has the following options for calculating the current:}(hjPh jPhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM?h jTOhhubj)}(hhh]j)}(hhh]h}(h]h]h]h]h]uhjh jPhhh!jhMBubah}(h]h]h]h]h]jSjCjUhjVjWuhjh jTOhhh!jhMAubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-71uhh
h jTOhhh!jhNubj )}(hHJ_{g}=\sqrt{\left(\psi_{1}^{g}\right)^{2}+\left(D B \psi_{g}\right)^{2}}h]h/HJ_{g}=\sqrt{\left(\psi_{1}^{g}\right)^{2}+\left(D B \psi_{g}\right)^{2}}}(hhh jPubah}(h]jPah]h]h]h]docnamejnumberKFlabeleq9-1-71nowrapjyjzuhj h!jhMCh jTOhhjf}jh}jPjPsubj)}(hhh]j)}(hhh]h}(h]h]h]h]h]uhjh jPhhh!jhMIubah}(h]h]h]h]h]jSjCjUhjVjWjXKuhjh jTOhhh!jhMHubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-72uhh
h jTOhhh!jhNubj )}(hJ_{g}=\left|\psi_{1}^{g}\right|h]h/J_{g}=\left|\psi_{1}^{g}\right|}(hhh jQubah}(h]jQah]h]h]h]docnamejnumberKGlabeleq9-1-72nowrapjyjzuhj h!jhMJh jTOhhjf}jh}jQjQsubj)}(hhh]j)}(hhh]h}(h]h]h]h]h]uhjh j+Qhhh!jhMPubah}(h]h]h]h]h]jSjCjUhjVjWjXKuhjh jTOhhh!jhMOubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-73uhh
h jTOhhh!jhNubj )}(hZJ_{g}=D B^{2} \psi_{g}+\int_{0}^{1} d \mu \mu \psi\left(g, r_{\text {outside}}, \mu\right)h]h/ZJ_{g}=D B^{2} \psi_{g}+\int_{0}^{1} d \mu \mu \psi\left(g, r_{\text {outside}}, \mu\right)}(hhh jGQubah}(h]jFQah]h]h]h]docnamejnumberKHlabeleq9-1-73nowrapjyjzuhj h!jhMQh jTOhhjf}jh}jFQj=Qsubj)}(hhh]j)}(hhh]h}(h]h]h]h]h]uhjh j\Qhhh!jhMWubah}(h]h]h]h]h]jSjCjUhjVjWjXKuhjh jTOhhh!jhMVubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-74uhh
h jTOhhh!jhNubj )}(h'J_{g}=\frac{\psi_{0}^{g}}{\sum_{t}^{g}}h]h/'J_{g}=\frac{\psi_{0}^{g}}{\sum_{t}^{g}}}(hhh jxQubah}(h]jwQah]h]h]h]docnamejnumberKIlabeleq9-1-74nowrapjyjzuhj h!jhMXh jTOhhjf}jh}jwQjnQsubj)}(hhh]j)}(hhh]h}(h]h]h]h]h]uhjh jQhhh!jhM^ubah}(h]h]h]h]h]jSjCjUhjVjWjXKuhjh jTOhhh!jhM]ubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-75uhh
h jTOhhh!jhNubj )}(hJ_{g}=D B \psi_{g}h]h/J_{g}=D B \psi_{g}}(hhh jQubah}(h]jQah]h]h]h]docnamejnumberKJlabeleq9-1-75nowrapjyjzuhj h!jhM_h jTOhhjf}jh}jQjQsubh;)}(hX\The first option is the recommended option; option 2 treats only the
current in the primary direction; option 3 will always be positive and
is a weighting over the total leakage from the system. Option 4 is
sometimes referred to as a “bootstrap” approximation; option 5 is
equivalent to that used in codes that employ *B*\ :sub:`n` theory.h](h/XFThe first option is the recommended option; option 2 treats only the
current in the primary direction; option 3 will always be positive and
is a weighting over the total leakage from the system. Option 4 is
sometimes referred to as a “bootstrap” approximation; option 5 is
equivalent to that used in codes that employ }(hXFThe first option is the recommended option; option 2 treats only the
current in the primary direction; option 3 will always be positive and
is a weighting over the total leakage from the system. Option 4 is
sometimes referred to as a “bootstrap” approximation; option 5 is
equivalent to that used in codes that employ h jQhhh!NhNubhA)}(h*B*h]h/B}(hhh jQubah}(h]h]h]h]h]uhh@h jQubh/ }(h\ h jQhhh!NhNubh)}(h:sub:`n`h]h/n}(hhh jQubah}(h]h]h]h]h]uhhh jQubh/ theory.}(h theory.h jQhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMdh jTOhhubh;)}(hOnce the currents are determined, the transport values are determined as
set forth in the equations discussed above. For example, consider cell
weighting and the “inscatter” approximation,h]h/Once the currents are determined, the transport values are determined as
set forth in the equations discussed above. For example, consider cell
weighting and the “inscatter” approximation,}(hjQh jQhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMjh jTOhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-76uhh
h jTOhhh!jhNubj )}(h\sigma_{t r}^{G}=\frac{\sum_{j} N^{j} \sum_{g \varepsilon G}\left\{J_{g} \sigma_{t}^{g}-\frac{1}{3} \sum_{g^{\prime}} \sigma_{1}\left(g^{\prime} \rightarrow g\right) J_{g}^{\prime}\right\}}{\bar{N} \sum^{IZM}_{j} \sum_{g \varepsilon G} J_{g}}h]h/\sigma_{t r}^{G}=\frac{\sum_{j} N^{j} \sum_{g \varepsilon G}\left\{J_{g} \sigma_{t}^{g}-\frac{1}{3} \sum_{g^{\prime}} \sigma_{1}\left(g^{\prime} \rightarrow g\right) J_{g}^{\prime}\right\}}{\bar{N} \sum^{IZM}_{j} \sum_{g \varepsilon G} J_{g}}}(hhh jRubah}(h]j
Rah]h]h]h]docnamejnumberKKlabeleq9-1-76nowrapjyjzuhj h!jhMnh jTOhhjf}jh}j
RjRsubh;)}(h:For cell weighting and the “outscatter” approximation,h]h/:For cell weighting and the “outscatter” approximation,}(hj"Rh j Rhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMsh jTOhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-77uhh
h jTOhhh!jhNubj )}(h\sigma_{t r}^{G}=\frac{\sum_{j} N^{j} \sum_{g \varepsilon G} J_{g}\left\{\sigma_{t}^{g}-\frac{1}{3} \sum_{g^{\prime}} \sigma_{1}\left(g^{\prime} \rightarrow g\right)\right\}}{\bar{N} \sum_{j}^{IZM} \sum_{g \varepsilon G} J_{g}}h]h/\sigma_{t r}^{G}=\frac{\sum_{j} N^{j} \sum_{g \varepsilon G} J_{g}\left\{\sigma_{t}^{g}-\frac{1}{3} \sum_{g^{\prime}} \sigma_{1}\left(g^{\prime} \rightarrow g\right)\right\}}{\bar{N} \sum_{j}^{IZM} \sum_{g \varepsilon G} J_{g}}}(hhh j8Rubah}(h]j7Rah]h]h]h]docnamejnumberKLlabeleq9-1-77nowrapjyjzuhj h!jhMuh jTOhhjf}jh}j7Rj.Rsubh)}(h
.. _9-1-2-15:h]h}(h]h]h]h]h]hid67uhh
hMh jTOhhh!jubeh}(h](.weighting-function-for-transport-cross-sectionjHOeh]h]9-1-2-14-8-3ah].weighting function for transport cross sectionah]uhh#h jLhhh!jhMjKjf}j]Rj>Osjh}jHOj>Osubeh}(h](transport-cross-sectionsjLeh]h](transport cross sections
9-1-2-14-8eh]h]uhh#h jMDhhh!jhMjf}jiRjLsjh}jLjLsubeh}(h](cross-section-weightingjADeh]h](cross-section weighting9-1-2-14eh]h]uhh#h j hhh!jhMjf}jtRj7Dsjh}jADj7Dsubh$)}(hhh](h))}(hAdjoint calculationsh]h/Adjoint calculations}(hj~Rh j|Rhhh!NhNubah}(h]h]h]h]h]uhh(h jyRhhh!jhM}ubh;)}(hQXSDRNPM will, upon option, solve the adjoint forms of the 1-D transport
equation.h]h/QXSDRNPM will, upon option, solve the adjoint forms of the 1-D transport
equation.}(hjRh jRhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jyRhhubh;)}(h=Several special procedures apply for the adjoint calculation:h]h/=Several special procedures apply for the adjoint calculation:}(hjRh jRhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jyRhhubj)}(hhh](j)}(hThe iteration pattern discussed in :ref:`9-1-2-10` is reversed in
energy. The scheme starts with the last (lowest energy) group and
proceeds to the first group.
h]h;)}(hThe iteration pattern discussed in :ref:`9-1-2-10` is reversed in
energy. The scheme starts with the last (lowest energy) group and
proceeds to the first group.h](h/#The iteration pattern discussed in }(h#The iteration pattern discussed in h jRubj)}(h:ref:`9-1-2-10`h]j#)}(hjRh]h/9-1-2-10}(hhh jRubah}(h]h](jnstdstd-refeh]h]h]uhj"h jRubah}(h]h]h]h]h]refdocj refdomainjRreftyperefrefexplicitrefwarnj9-1-2-10uhjh!jhMh jRubh/n is reversed in
energy. The scheme starts with the last (lowest energy) group and
proceeds to the first group.}(hn is reversed in
energy. The scheme starts with the last (lowest energy) group and
proceeds to the first group.h jRubeh}(h]h]h]h]h]uhh:h!jhMh jRubah}(h]h]h]h]h]uhjh jRhhh!jhNubj)}(hThe angular quadrature is treated as if it has the reverse directions
associated with the angle (e.g., many quadratures start with
*μ*\ :sub:`1` = −1.0). In the adjoint case, this direction is for
*μ*\ :sub:`1` = +1.0.
h]h;)}(hThe angular quadrature is treated as if it has the reverse directions
associated with the angle (e.g., many quadratures start with
*μ*\ :sub:`1` = −1.0). In the adjoint case, this direction is for
*μ*\ :sub:`1` = +1.0.h](h/The angular quadrature is treated as if it has the reverse directions
associated with the angle (e.g., many quadratures start with
}(hThe angular quadrature is treated as if it has the reverse directions
associated with the angle (e.g., many quadratures start with
h jRubhA)}(h*μ*h]h/μ}(hhh jRubah}(h]h]h]h]h]uhh@h jRubh/ }(h\ h jRubh)}(h:sub:`1`h]h/1}(hhh jSubah}(h]h]h]h]h]uhhh jRubh/9 = −1.0). In the adjoint case, this direction is for
}(h9 = −1.0). In the adjoint case, this direction is for
h jRubhA)}(h*μ*h]h/μ}(hhh jSubah}(h]h]h]h]h]uhh@h jRubh/ }(hjSh jRubh)}(h:sub:`1`h]h/1}(hhh j,Subah}(h]h]h]h]h]uhhh jRubh/
= +1.0.}(h
= +1.0.h jRubeh}(h]h]h]h]h]uhh:h!jhMh jRubah}(h]h]h]h]h]uhjh jRhhh!jhNubj)}(hAll edits of input fluxes and collapsed cross sections are given in
their normal ordering, as opposed to many codes which require their
reversal.
h]h;)}(hAll edits of input fluxes and collapsed cross sections are given in
their normal ordering, as opposed to many codes which require their
reversal.h]h/All edits of input fluxes and collapsed cross sections are given in
their normal ordering, as opposed to many codes which require their
reversal.}(hjQSh jOSubah}(h]h]h]h]h]uhh:h!jhMh jKSubah}(h]h]h]h]h]uhjh jRhhh!jhNubeh}(h]h]h]h]h]jSjCjUhjVjWuhjh jyRhhh!jhMubh;)}(hAdjoint calculations have many uses and advantages. As opposed to the
forward calculation which yields particle density values, the adjoint
fluxes are more abstract and can be thought of as particle importance.h]h/Adjoint calculations have many uses and advantages. As opposed to the
forward calculation which yields particle density values, the adjoint
fluxes are more abstract and can be thought of as particle importance.}(hjkSh jiShhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jyRhhubh;)}(hXConsider, for example, the problem of determining the response of a
detector to particles as a function of their energy and direction.
Assume the detector is a cylindrical fission chamber that utilizes a
foil of :sup:`235`\ U. The most obvious way to attack this problem is to
mock up the detector and make a series of runs that contain sources of
identical strength in different angles and energy groups. If an
*S*\ :sub:`8` (24 angles) quadrature were used with 50 energy groups,
the 12 × 50 or 600 independent calculations could be used to completely
determine the responses. (Here we have taken note that half of the
angles will point away from a detector and, hence, produce no response.)
The adjoint calculation produces all 600 responses in one run that is no
more difficult and time consuming than the typical forward case. In the
adjoint case, the detector response (i.e., the fission cross section of
:sup:`235`\ U would be specified as a source in the foil region and the
adjoint fluxes given as a function of energy and angle would be
interpreted as the source of neutrons necessary to produce a response of
the magnitude to which one required the response to be normalized.h](h/Consider, for example, the problem of determining the response of a
detector to particles as a function of their energy and direction.
Assume the detector is a cylindrical fission chamber that utilizes a
foil of }(hConsider, for example, the problem of determining the response of a
detector to particles as a function of their energy and direction.
Assume the detector is a cylindrical fission chamber that utilizes a
foil of h jwShhh!NhNubj;)}(h
:sup:`235`h]h/235}(hhh jSubah}(h]h]h]h]h]uhj;h jwSubh/ U. The most obvious way to attack this problem is to
mock up the detector and make a series of runs that contain sources of
identical strength in different angles and energy groups. If an
}(h\ U. The most obvious way to attack this problem is to
mock up the detector and make a series of runs that contain sources of
identical strength in different angles and energy groups. If an
h jwShhh!NhNubhA)}(h*S*h]h/S}(hhh jSubah}(h]h]h]h]h]uhh@h jwSubh/ }(h\ h jwShhh!NhNubh)}(h:sub:`8`h]h/8}(hhh jSubah}(h]h]h]h]h]uhhh jwSubh/X (24 angles) quadrature were used with 50 energy groups,
the 12 × 50 or 600 independent calculations could be used to completely
determine the responses. (Here we have taken note that half of the
angles will point away from a detector and, hence, produce no response.)
The adjoint calculation produces all 600 responses in one run that is no
more difficult and time consuming than the typical forward case. In the
adjoint case, the detector response (i.e., the fission cross section of
}(hX (24 angles) quadrature were used with 50 energy groups,
the 12 × 50 or 600 independent calculations could be used to completely
determine the responses. (Here we have taken note that half of the
angles will point away from a detector and, hence, produce no response.)
The adjoint calculation produces all 600 responses in one run that is no
more difficult and time consuming than the typical forward case. In the
adjoint case, the detector response (i.e., the fission cross section of
h jwShhh!NhNubj;)}(h
:sup:`235`h]h/235}(hhh jSubah}(h]h]h]h]h]uhj;h jwSubh/X U would be specified as a source in the foil region and the
adjoint fluxes given as a function of energy and angle would be
interpreted as the source of neutrons necessary to produce a response of
the magnitude to which one required the response to be normalized.}(hX \ U would be specified as a source in the foil region and the
adjoint fluxes given as a function of energy and angle would be
interpreted as the source of neutrons necessary to produce a response of
the magnitude to which one required the response to be normalized.h jwShhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jyRhhubh;)}(hXoA second important use of adjoint calculations is to establish good
biasing factors for Monte Carlo codes. Two recent
reports :cite:`hoffman_xsdrnpm-s_1982` :sup:`,`\ :cite:`hoffman_optimization_1982` discuss the time and accuracy advantages
of this approach for shielding and criticality applications and give
some real examples as to how to make the calculations.h](h/A second important use of adjoint calculations is to establish good
biasing factors for Monte Carlo codes. Two recent
reports }(hA second important use of adjoint calculations is to establish good
biasing factors for Monte Carlo codes. Two recent
reports h jShhh!NhNubj)}(hhoffman_xsdrnpm-s_1982h]j#)}(hjSh]h/[hoffman_xsdrnpm-s_1982]}(hhh jSubah}(h]h]h]h]h]uhj"h jSubah}(h]id68ah]j5ah]h]h] refdomainj:reftypej< reftargetjSrefwarnsupport_smartquotesuhjh!jhMh jShhubh/ }(hjh jShhh!NhNubj;)}(h:sup:`,`h]h/,}(hhh jSubah}(h]h]h]h]h]uhj;h jSubh/ }(h\ h jShhh!NhNubj)}(hhoffman_optimization_1982h]j#)}(hjTh]h/[hoffman_optimization_1982]}(hhh jTubah}(h]h]h]h]h]uhj"h jTubah}(h]id69ah]j5ah]h]h] refdomainj:reftypej< reftargetjTrefwarnsupport_smartquotesuhjh!jhMh jShhubh/ discuss the time and accuracy advantages
of this approach for shielding and criticality applications and give
some real examples as to how to make the calculations.}(h discuss the time and accuracy advantages
of this approach for shielding and criticality applications and give
some real examples as to how to make the calculations.h jShhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jyRhhubh;)}(hXTPerturbation theory uses adjoint and forward fluxes in combination in a
manner that determines changes in responses that would arise from
changing parameters used in a calculation. One :cite:`weisbin_review_1979` interesting
application is to determine the sensitivity of a calculation to changes
in one or more cross-section value changes.h](h/Perturbation theory uses adjoint and forward fluxes in combination in a
manner that determines changes in responses that would arise from
changing parameters used in a calculation. One }(hPerturbation theory uses adjoint and forward fluxes in combination in a
manner that determines changes in responses that would arise from
changing parameters used in a calculation. One h j7Thhh!NhNubj)}(hweisbin_review_1979h]j#)}(hjBTh]h/[weisbin_review_1979]}(hhh jDTubah}(h]h]h]h]h]uhj"h j@Tubah}(h]id70ah]j5ah]h]h] refdomainj:reftypej< reftargetjBTrefwarnsupport_smartquotesuhjh!jhMh j7Thhubh/ interesting
application is to determine the sensitivity of a calculation to changes
in one or more cross-section value changes.}(h interesting
application is to determine the sensitivity of a calculation to changes
in one or more cross-section value changes.h j7Thhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jyRhhubh)}(h.. _9-1-2-15-1:h]h}(h]h]h]h]h]hid71uhh
hM. h jyRhhh!jubh$)}(hhh](h))}(h Generalized adjoint calculationsh]h/ Generalized adjoint calculations}(hjxTh jvThhh!NhNubah}(h]h]h]h]h]uhh(h jsThhh!jhMubh;)}(hXjGeneralized adjoint solutions are needed for generalized perturbation theory
(GPT) applications such as sensitivity and uncertainty analysis. The
generalized adjoint solution differs from both a conventional external source
case and a fundamental mode eigenvalue calculation: It has the transport
operator for an adjoint eigenvalue equation, but contains a fixed source term as
well. The eigenvalue transport operator is singular, which forces certain
restrictions on the allowable sources. The generalized adjoint source term is
associated with a particular response ratio of interest in a critical system,
such ash]h/XjGeneralized adjoint solutions are needed for generalized perturbation theory
(GPT) applications such as sensitivity and uncertainty analysis. The
generalized adjoint solution differs from both a conventional external source
case and a fundamental mode eigenvalue calculation: It has the transport
operator for an adjoint eigenvalue equation, but contains a fixed source term as
well. The eigenvalue transport operator is singular, which forces certain
restrictions on the allowable sources. The generalized adjoint source term is
associated with a particular response ratio of interest in a critical system,
such as}(hjTh jThhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jsThhubj )}(hR=\frac{\sum_{g=1}^{I G M} \sum_{i=1}^{I M} H_{N}(g, i) \psi_{i, g} V_{i}}{\sum_{g=1}^{I G M} \sum_{i=1}^{I M} H_{D}(g, i) \psi_{i, g} V_{i}}h]h/R=\frac{\sum_{g=1}^{I G M} \sum_{i=1}^{I M} H_{N}(g, i) \psi_{i, g} V_{i}}{\sum_{g=1}^{I G M} \sum_{i=1}^{I M} H_{D}(g, i) \psi_{i, g} V_{i}}}(hhh jTubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh jsThhubh;)}(hXwhere H\ :sub:`N` and H\ :sub:`D` are response functions defining the
response of interest and :math:`\psi_{i, g}` is the scalar flux from a prior forward
eigenvalue solution of the same problem. The generalized adjoint source
for this response is defined ash](h/ where H }(h where H\ h jThhh!NhNubh)}(h:sub:`N`h]h/N}(hhh jTubah}(h]h]h]h]h]uhhh jTubh/ and H }(h and H\ h jThhh!NhNubh)}(h:sub:`D`h]h/D}(hhh jTubah}(h]h]h]h]h]uhhh jTubh/> are response functions defining the
response of interest and }(h> are response functions defining the
response of interest and h jThhh!NhNubjr)}(h:math:`\psi_{i, g}`h]h/\psi_{i, g}}(hhh jTubah}(h]h]h]h]h]uhjqh jTubh/ is the scalar flux from a prior forward
eigenvalue solution of the same problem. The generalized adjoint source
for this response is defined as}(h is the scalar flux from a prior forward
eigenvalue solution of the same problem. The generalized adjoint source
for this response is defined ash jThhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jsThhubj )}(hQ^{*}(g, i) \equiv \frac{1}{R} \frac{\partial R}{\partial \psi_{i, g}}=\frac{H_{N}(g, i)}{\sum_{g=1}^{IGM} \sum_{i=1}^{I M} H_{N}(g, i) \psi_{i, g} V_{i}}-\frac{H_{D}(g, i)}{\sum_{g=1}^{I G M} \sum_{i=1}^{I M} H_{D}(g, i) \psi_{i, g} V_{i}}h]h/Q^{*}(g, i) \equiv \frac{1}{R} \frac{\partial R}{\partial \psi_{i, g}}=\frac{H_{N}(g, i)}{\sum_{g=1}^{IGM} \sum_{i=1}^{I M} H_{N}(g, i) \psi_{i, g} V_{i}}-\frac{H_{D}(g, i)}{\sum_{g=1}^{I G M} \sum_{i=1}^{I M} H_{D}(g, i) \psi_{i, g} V_{i}}}(hhh jTubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh jsThhubh;)}(hThe above source expression is computed automatically whenever XSDRN is
executed in the TSUNAMI-1D sequence, but must it be computed and input
by the user if XSDRN is run standalone for a generalized adjoint case.h]h/The above source expression is computed automatically whenever XSDRN is
executed in the TSUNAMI-1D sequence, but must it be computed and input
by the user if XSDRN is run standalone for a generalized adjoint case.}(hjUh jThhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jsThhubh;)}(hXlIn order to obtain a unique solution and avoid numerical problems, the
generalized adjoint solution is “normalized” to contain no fundamental
harmonic of the adjoint eigenvalue calculation. This is done by sweeping
out the adjoint fundamental mode “contamination” from the fission source
after each outer iteration, as described in the SAMS chapter, in
*Generalized Perturbation Theory*. This operation requires both forward
and adjoint eigenvalue solutions from prior XSDRN calculations. External
files containing the fundamental mode forward and adjoint fluxes are
input to the generalized adjoint calculation.h](h/XiIn order to obtain a unique solution and avoid numerical problems, the
generalized adjoint solution is “normalized” to contain no fundamental
harmonic of the adjoint eigenvalue calculation. This is done by sweeping
out the adjoint fundamental mode “contamination” from the fission source
after each outer iteration, as described in the SAMS chapter, in
}(hXiIn order to obtain a unique solution and avoid numerical problems, the
generalized adjoint solution is “normalized” to contain no fundamental
harmonic of the adjoint eigenvalue calculation. This is done by sweeping
out the adjoint fundamental mode “contamination” from the fission source
after each outer iteration, as described in the SAMS chapter, in
h jUhhh!NhNubhA)}(h!*Generalized Perturbation Theory*h]h/Generalized Perturbation Theory}(hhh jUubah}(h]h]h]h]h]uhh@h jUubh/. This operation requires both forward
and adjoint eigenvalue solutions from prior XSDRN calculations. External
files containing the fundamental mode forward and adjoint fluxes are
input to the generalized adjoint calculation.}(h. This operation requires both forward
and adjoint eigenvalue solutions from prior XSDRN calculations. External
files containing the fundamental mode forward and adjoint fluxes are
input to the generalized adjoint calculation.h jUhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jsThhubh;)}(hX
Unlike conventional fixed source and eigenvalue calculations, the
generalized adjoint flux has both negative and positive components. This
causes some XSDRN acceleration features such as space-dependent
rebalance and group-banding to not function properly; and thus these are
turned off internally. Typically the outer iterations for generalized
adjoint solution converge much slower than an eigenvalue calculation.
More background on GPT and generalized adjoint properties can be found
in :cite:`williams_perturbation_1986`.h](h/XUnlike conventional fixed source and eigenvalue calculations, the
generalized adjoint flux has both negative and positive components. This
causes some XSDRN acceleration features such as space-dependent
rebalance and group-banding to not function properly; and thus these are
turned off internally. Typically the outer iterations for generalized
adjoint solution converge much slower than an eigenvalue calculation.
More background on GPT and generalized adjoint properties can be found
in }(hXUnlike conventional fixed source and eigenvalue calculations, the
generalized adjoint flux has both negative and positive components. This
causes some XSDRN acceleration features such as space-dependent
rebalance and group-banding to not function properly; and thus these are
turned off internally. Typically the outer iterations for generalized
adjoint solution converge much slower than an eigenvalue calculation.
More background on GPT and generalized adjoint properties can be found
in h j.Uhhh!NhNubj)}(hwilliams_perturbation_1986h]j#)}(hj9Uh]h/[williams_perturbation_1986]}(hhh j;Uubah}(h]h]h]h]h]uhj"h j7Uubah}(h]id72ah]j5ah]h]h] refdomainj:reftypej< reftargetj9Urefwarnsupport_smartquotesuhjh!jhMh j.Uhhubh/.}(hjWh j.Uhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jsThhubh)}(h
.. _9-1-2-16:h]h}(h]h]h]h]h]hid73uhh
hMa h jsThhh!jubeh}(h]( generalized-adjoint-calculationsjrTeh]h]( generalized adjoint calculations
9-1-2-15-1eh]h]uhh#h jyRhhh!jhMjf}joUjhTsjh}jrTjhTsubeh}(h](adjoint-calculationsjWReh]h](adjoint calculations9-1-2-15eh]h]uhh#h j hhh!jhM}jf}jzUjMRsjh}jWRjMRsubh$)}(hhh](h))}(h#Coupled neutron-photon calculationsh]h/#Coupled neutron-photon calculations}(hjUh jUhhh!NhNubah}(h]h]h]h]h]uhh(h jUhhh!jhMubh;)}(hXIn XSDRNPM, it is possible to do a neutron or a photon calculation,
depending only on whether the input libraries are for neutrons or gamma
rays. It is also possible to do a “coupled neutron-photon” calculation
which automatically determines the gamma-ray sources arising from
neutron induced interactions in its photon calculation. This
calculation, of course, requires an input cross-section library
containing three classes of data:h]h/XIn XSDRNPM, it is possible to do a neutron or a photon calculation,
depending only on whether the input libraries are for neutrons or gamma
rays. It is also possible to do a “coupled neutron-photon” calculation
which automatically determines the gamma-ray sources arising from
neutron induced interactions in its photon calculation. This
calculation, of course, requires an input cross-section library
containing three classes of data:}(hjUh jUhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jUhhubj)}(hhh](j)}(hIneutron cross sections, including neutron-to-neutron transfer
matrices,
h]h;)}(hHneutron cross sections, including neutron-to-neutron transfer
matrices,h]h/Hneutron cross sections, including neutron-to-neutron transfer
matrices,}(hjUh jUubah}(h]h]h]h]h]uhh:h!jhMh jUubah}(h]h]h]h]h]uhjh jUhhh!jhNubj)}(hSphoton production cross sections (i.e., neutron-to-gamma transfer
matrices), and
h]h;)}(hRphoton production cross sections (i.e., neutron-to-gamma transfer
matrices), andh]h/Rphoton production cross sections (i.e., neutron-to-gamma transfer
matrices), and}(hjUh jUubah}(h]h]h]h]h]uhh:h!jhMh jUubah}(h]h]h]h]h]uhjh jUhhh!jhNubj)}(hOgamma-ray cross sections, including gamma-ray-to-gamma-ray transfer
matrices.
h]h;)}(hNgamma-ray cross sections, including gamma-ray-to-gamma-ray transfer
matrices.h]h/Ngamma-ray cross sections, including gamma-ray-to-gamma-ray transfer
matrices.}(hjUh jUubah}(h]h]h]h]h]uhh:h!jhMh jUubah}(h]h]h]h]h]uhjh jUhhh!jhNubeh}(h]h]h]h]h]jSjCjUhjVjWuhjh jUhhh!jhMubh;)}(hXBAt present there are no provisions for treating neutrons produced from
gamma interactions other than having the user introduce these sources by
hand in a sort of iterative procedure, though this reaction is certainly
not unknown (cf., deuterium, beryllium-9, and carbon-13). There are
several cases where the (*γ*,\ *n*) interaction can be important. If,
for example, one looks at neutrons in a water-moderated pool reactor or
in a water spent fuel storage tank at large distances from the fuel, the
dominant source is from the neutrons produced by the deuterium in the
water.h](h/X7At present there are no provisions for treating neutrons produced from
gamma interactions other than having the user introduce these sources by
hand in a sort of iterative procedure, though this reaction is certainly
not unknown (cf., deuterium, beryllium-9, and carbon-13). There are
several cases where the (}(hX7At present there are no provisions for treating neutrons produced from
gamma interactions other than having the user introduce these sources by
hand in a sort of iterative procedure, though this reaction is certainly
not unknown (cf., deuterium, beryllium-9, and carbon-13). There are
several cases where the (h jUhhh!NhNubhA)}(h*γ*h]h/γ}(hhh jUubah}(h]h]h]h]h]uhh@h jUubh/, }(h,\ h jUhhh!NhNubhA)}(h*n*h]h/n}(hhh jVubah}(h]h]h]h]h]uhh@h jUubh/X) interaction can be important. If,
for example, one looks at neutrons in a water-moderated pool reactor or
in a water spent fuel storage tank at large distances from the fuel, the
dominant source is from the neutrons produced by the deuterium in the
water.}(hX) interaction can be important. If,
for example, one looks at neutrons in a water-moderated pool reactor or
in a water spent fuel storage tank at large distances from the fuel, the
dominant source is from the neutrons produced by the deuterium in the
water.h jUhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jUhhubh;)}(hXNormally the neutron-photon calculation requires no more input than a
single particle run, except in the case where extraneous neutron and/or
gamma-ray sources need to be specified. Most output edits will be split
into a neutron and a gamma-ray part and will be labeled as suchh]h/XNormally the neutron-photon calculation requires no more input than a
single particle run, except in the case where extraneous neutron and/or
gamma-ray sources need to be specified. Most output edits will be split
into a neutron and a gamma-ray part and will be labeled as such}(hj&Vh j$Vhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM h jUhhubh)}(h
.. _9-1-2-17:h]h}(h]h]h]h]h]hid74uhh
hM h jUhhh!jubeh}(h](#coupled-neutron-photon-calculationsjhUeh]h](#coupled neutron-photon calculations9-1-2-16eh]h]uhh#h j hhh!jhMjf}jCVj^Usjh}jhUj^Usubh$)}(hhh](h))}(hDiffusion theory optionh]h/Diffusion theory option}(hjMVh jKVhhh!NhNubah}(h]h]h]h]h]uhh(h jHVhhh!jhM ubh;)}(hXSDRNPM can make a 1-D diffusion theory calculation in user-specified
energy groups (enter 1’s for the appropriate groups of the 46$ array).
In this case, the *P*\ :sub:`1` diffusion equations :cite:`alder_methods_1963` are solved:h](h/XSDRNPM can make a 1-D diffusion theory calculation in user-specified
energy groups (enter 1’s for the appropriate groups of the 46$ array).
In this case, the }(hXSDRNPM can make a 1-D diffusion theory calculation in user-specified
energy groups (enter 1’s for the appropriate groups of the 46$ array).
In this case, the h jYVhhh!NhNubhA)}(h*P*h]h/P}(hhh jbVubah}(h]h]h]h]h]uhh@h jYVubh/ }(h\ h jYVhhh!NhNubh)}(h:sub:`1`h]h/1}(hhh juVubah}(h]h]h]h]h]uhhh jYVubh/ diffusion equations }(h diffusion equations h jYVhhh!NhNubj)}(halder_methods_1963h]j#)}(hjVh]h/[alder_methods_1963]}(hhh jVubah}(h]h]h]h]h]uhj"h jVubah}(h]id75ah]j5ah]h]h] refdomainj:reftypej< reftargetjVrefwarnsupport_smartquotesuhjh!jhM h jYVhhubh/ are solved:}(h are solved:h jYVhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM h jHVhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-78uhh
h jHVhhh!jhNubj )}(h`A_{I+1} \psi_{1, I+1}-A_{I} \psi_{1, I}+\sigma_{0}\left(\psi_{0, I}+\psi_{0, I}\right)=S_{0}^{*}h]h/`A_{I+1} \psi_{1, I+1}-A_{I} \psi_{1, I}+\sigma_{0}\left(\psi_{0, I}+\psi_{0, I}\right)=S_{0}^{*}}(hhh jVubah}(h]jVah]h]h]h]docnamejnumberKMlabeleq9-1-78nowrapjyjzuhj h!jhM h jHVhhjf}jh}jVjVsubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-79uhh
h jHVhhh!jhNubj )}(hn\bar{A}_{I}\left(\psi_{0, I+1}-\psi_{0, I}\right)+\sigma_{1}\left(\psi_{1, I+1}+\psi_{1, I}\right)=S_{1}^{*} ,h]h/n\bar{A}_{I}\left(\psi_{0, I+1}-\psi_{0, I}\right)+\sigma_{1}\left(\psi_{1, I+1}+\psi_{1, I}\right)=S_{1}^{*} ,}(hhh jVubah}(h]jVah]h]h]h]docnamejnumberKNlabeleq9-1-79nowrapjyjzuhj h!jhM h jHVhhjf}jh}jVjVsubh;)}(hwhereh]h/where}(hjVh jVhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM h jHVhhubj )}(h'\psi_{1} \equiv P_{1} \text { current }h]h/'\psi_{1} \equiv P_{1} \text { current }}(hhh jVubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhM h jHVhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-80uhh
h jHVhhh!jhNubj )}(hP\sigma_{0}=\left[\Sigma_{t}-\Sigma_{0}(g \rightarrow g)\right] \frac{V_{I}}{2.0}h]h/P\sigma_{0}=\left[\Sigma_{t}-\Sigma_{0}(g \rightarrow g)\right] \frac{V_{I}}{2.0}}(hhh jWubah}(h]jWah]h]h]h]docnamejnumberKOlabeleq9-1-80nowrapjyjzuhj h!jhM$ h jHVhhjf}jh}jWjWsubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-81uhh
h jHVhhh!jhNubj )}(hT\sigma_{1}=\left[3.0 \Sigma_{t}-\Sigma_{1}(g \rightarrow g)\right] \frac{V_{I}}{2.0}h]h/T\sigma_{1}=\left[3.0 \Sigma_{t}-\Sigma_{1}(g \rightarrow g)\right] \frac{V_{I}}{2.0}}(hhh j7Wubah}(h]j6Wah]h]h]h]docnamejnumberKPlabeleq9-1-81nowrapjyjzuhj h!jhM) h jHVhhjf}jh}j6Wj-Wsubj )}(h h jWubh/ for }(h for h jWhhh!NhNubjr)}(h:math:`\psi_{0, I+1}`h]h/
\psi_{0, I+1}}(hhh jWubah}(h]h]h]h]h]uhjqh jWubh/ and substituting into Eq. }(h and substituting into Eq. h jWhhh!NhNubj)}(h:eq:`eq9-1-79`h]jc)}(hjWh]h/eq9-1-79}(hhh jWubah}(h]h](jneqeh]h]h]uhjbh jWubah}(h]h]h]h]h]refdocj refdomainjqreftypejWrefexplicitrefwarnjeq9-1-79uhjh!jhM> h jWubh/, one can write}(h, one can writeh jWhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM> h jHVhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-82uhh
h jHVhhh!jhNubj )}(h\psi_{1, I+1}=\frac{\bar{A}_{I} S_{0}^{*}-2 \sigma_{0} \bar{A}_{I} \psi_{0, I}-\sigma_{0} S_{1}^{*}+\psi_{1, I}\left(\sigma_{0} \sigma_{1}+\bar{A}_{I} A_{I}\right)}{\bar{A}_{I} A_{I+1}-\sigma_{0} \sigma_{1}} .h]h/\psi_{1, I+1}=\frac{\bar{A}_{I} S_{0}^{*}-2 \sigma_{0} \bar{A}_{I} \psi_{0, I}-\sigma_{0} S_{1}^{*}+\psi_{1, I}\left(\sigma_{0} \sigma_{1}+\bar{A}_{I} A_{I}\right)}{\bar{A}_{I} A_{I+1}-\sigma_{0} \sigma_{1}} .}(hhh jXubah}(h]jXah]h]h]h]docnamejnumberKQlabeleq9-1-82nowrapjyjzuhj h!jhM@ h jHVhhjf}jh}jXjWsubh;)}(hlSolving Eq. :eq:`eq9-1-79` for :math:`\psi_{1, I}` and substituting into Eq. :eq:`eq9-1-78` , one can writeh](h/
Solving Eq. }(h
Solving Eq. h jXhhh!NhNubj)}(h:eq:`eq9-1-79`h]jc)}(hj&Xh]h/eq9-1-79}(hhh j(Xubah}(h]h](jneqeh]h]h]uhjbh j$Xubah}(h]h]h]h]h]refdocj refdomainjqreftypej2Xrefexplicitrefwarnjeq9-1-79uhjh!jhMF h jXubh/ for }(h for h jXhhh!NhNubjr)}(h:math:`\psi_{1, I}`h]h/\psi_{1, I}}(hhh jGXubah}(h]h]h]h]h]uhjqh jXubh/ and substituting into Eq. }(h and substituting into Eq. h jXhhh!NhNubj)}(h:eq:`eq9-1-78`h]jc)}(hj\Xh]h/eq9-1-78}(hhh j^Xubah}(h]h](jneqeh]h]h]uhjbh jZXubah}(h]h]h]h]h]refdocj refdomainjqreftypejhXrefexplicitrefwarnjeq9-1-78uhjh!jhMF h jXubh/ , one can write}(h , one can writeh jXhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMF h jHVhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-83uhh
h jHVhhh!jhNubj )}(h\psi_{0, I+1}=\frac{A_{I+1} S_{1}^{*}-2 \sigma_{1} \bar{A}_{I} \psi_{1, I}-\sigma_{1} S_{0}^{*}+\psi_{0, I}\left(\sigma_{0} \sigma_{1}+\bar{A}_{I} A_{I+1}\right)}{\bar{A}_{I} A_{I+1}-\sigma_{0} \sigma_{1}} .h]h/\psi_{0, I+1}=\frac{A_{I+1} S_{1}^{*}-2 \sigma_{1} \bar{A}_{I} \psi_{1, I}-\sigma_{1} S_{0}^{*}+\psi_{0, I}\left(\sigma_{0} \sigma_{1}+\bar{A}_{I} A_{I+1}\right)}{\bar{A}_{I} A_{I+1}-\sigma_{0} \sigma_{1}} .}(hhh jXubah}(h]jXah]h]h]h]docnamejnumberKRlabeleq9-1-83nowrapjyjzuhj h!jhMH h jHVhhjf}jh}jXjXsubh;)}(hIf one assumesh]h/If one assumes}(hjXh jXhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMM h jHVhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-84uhh
h jHVhhh!jhNubj )}(h+\psi_{1, I+1}=P_{I+1} \psi_{0, I+1}-q_{I+1}h]h/+\psi_{1, I+1}=P_{I+1} \psi_{0, I+1}-q_{I+1}}(hhh jXubah}(h]jXah]h]h]h]docnamejnumberKSlabeleq9-1-84nowrapjyjzuhj h!jhMO h jHVhhjf}jh}jXjXsubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-85uhh
h jHVhhh!jhNubj )}(h#\psi_{1, I}=P_{I} \psi_{0, I}-q_{I}h]h/#\psi_{1, I}=P_{I} \psi_{0, I}-q_{I}}(hhh jXubah}(h]jXah]h]h]h]docnamejnumberKTlabeleq9-1-85nowrapjyjzuhj h!jhMT h jHVhhjf}jh}jXjXsubh;)}(haand plugs Eqs. :eq:`eq9-1-82` and :eq:`eq9-1-83` into , solving for :math:`\psi_{1, I}` yields:h](h/and plugs Eqs. }(hand plugs Eqs. h jXhhh!NhNubj)}(h:eq:`eq9-1-82`h]jc)}(hjXh]h/eq9-1-82}(hhh jXubah}(h]h](jneqeh]h]h]uhjbh jXubah}(h]h]h]h]h]refdocj refdomainjqreftypejYrefexplicitrefwarnjeq9-1-82uhjh!jhMY h jXubh/ and }(h and h jXhhh!NhNubj)}(h:eq:`eq9-1-83`h]jc)}(hjYh]h/eq9-1-83}(hhh jYubah}(h]h](jneqeh]h]h]uhjbh jYubah}(h]h]h]h]h]refdocj refdomainjqreftypej(Yrefexplicitrefwarnjeq9-1-83uhjh!jhMY h jXubh/ into , solving for }(h into , solving for h jXhhh!NhNubjr)}(h:math:`\psi_{1, I}`h]h/\psi_{1, I}}(hhh j=Yubah}(h]h]h]h]h]uhjqh jXubh/ yields:}(h yields:h jXhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMY h jHVhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-86uhh
h jHVhhh!jhNubj )}(hX\begin{array}{l}
\psi_{1, I}=\frac{P_{I+1}\left(\sigma_{0} \sigma_{1}+\bar{A}_{I} A_{I+1}\right)+2 \sigma_{0} \bar{A}_{I}}{\sigma_{0} \sigma_{1}+\bar{A}_{I} A_{I}+P_{I+1} 2 \sigma_{1} \bar{A}_{I}} \psi_{0, I} \\
-\frac{S_{0}^{*}\left(\bar{A}_{I}+\sigma_{1} P_{I+1}\right)-S_{1}^{*}\left(\sigma_{0}+A_{I+1} P_{I+1}\right)+q_{I+1}\left(\bar{A}_{I} A_{I+1}-\sigma_{0} \sigma_{1}\right)}{\sigma_{0} \sigma_{1}+\bar{A}_{I} A_{I}+P_{I+1} 2 \sigma_{1} \bar{A}_{I}}
\end{array}h]h/X\begin{array}{l}
\psi_{1, I}=\frac{P_{I+1}\left(\sigma_{0} \sigma_{1}+\bar{A}_{I} A_{I+1}\right)+2 \sigma_{0} \bar{A}_{I}}{\sigma_{0} \sigma_{1}+\bar{A}_{I} A_{I}+P_{I+1} 2 \sigma_{1} \bar{A}_{I}} \psi_{0, I} \\
-\frac{S_{0}^{*}\left(\bar{A}_{I}+\sigma_{1} P_{I+1}\right)-S_{1}^{*}\left(\sigma_{0}+A_{I+1} P_{I+1}\right)+q_{I+1}\left(\bar{A}_{I} A_{I+1}-\sigma_{0} \sigma_{1}\right)}{\sigma_{0} \sigma_{1}+\bar{A}_{I} A_{I}+P_{I+1} 2 \sigma_{1} \bar{A}_{I}}
\end{array}}(hhh j`Yubah}(h]j_Yah]h]h]h]docnamejnumberKUlabeleq9-1-86nowrapjyjzuhj h!jhM[ h jHVhhjf}jh}j_YjVYsubh;)}(hrwhich by inspection and comparison with Eq. :eq:`eq9-1-85` gives expressions for
*P*\ :sub:`I` and *q*\ :sub:`I`.h](h/-which by inspection and comparison with Eq. }(h-which by inspection and comparison with Eq. h juYhhh!NhNubj)}(h:eq:`eq9-1-85`h]jc)}(hjYh]h/eq9-1-85}(hhh jYubah}(h]h](jneqeh]h]h]uhjbh j~Yubah}(h]h]h]h]h]refdocj refdomainjqreftypejYrefexplicitrefwarnjeq9-1-85uhjh!jhMc h juYubh/ gives expressions for
}(h gives expressions for
h juYhhh!NhNubhA)}(h*P*h]h/P}(hhh jYubah}(h]h]h]h]h]uhh@h juYubh/ }(h\ h juYhhh!NhNubh)}(h:sub:`I`h]h/I}(hhh jYubah}(h]h]h]h]h]uhhh juYubh/ and }(h and h juYhhh!NhNubhA)}(h*q*h]h/q}(hhh jYubah}(h]h]h]h]h]uhh@h juYubh/ }(hjYh juYubh)}(h:sub:`I`h]h/I}(hhh jYubah}(h]h]h]h]h]uhhh juYubh/.}(hjWh juYhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMc h jHVhhubh;)}(h~Equations :eq:`eq9-1-84` and :eq:`eq9-1-85` can be substituted into Eq. :eq:`eq9-1-78` and solved for :math:`\psi_{0, I+1}`:h](h/Equations }(hEquations h jYhhh!NhNubj)}(h:eq:`eq9-1-84`h]jc)}(hjYh]h/eq9-1-84}(hhh jYubah}(h]h](jneqeh]h]h]uhjbh jYubah}(h]h]h]h]h]refdocj refdomainjqreftypejZrefexplicitrefwarnjeq9-1-84uhjh!jhMf h jYubh/ and }(h and h jYhhh!NhNubj)}(h:eq:`eq9-1-85`h]jc)}(hjZh]h/eq9-1-85}(hhh j!Zubah}(h]h](jneqeh]h]h]uhjbh jZubah}(h]h]h]h]h]refdocj refdomainjqreftypej+Zrefexplicitrefwarnjeq9-1-85uhjh!jhMf h jYubh/ can be substituted into Eq. }(h can be substituted into Eq. h jYhhh!NhNubj)}(h:eq:`eq9-1-78`h]jc)}(hjBZh]h/eq9-1-78}(hhh jDZubah}(h]h](jneqeh]h]h]uhjbh j@Zubah}(h]h]h]h]h]refdocj refdomainjqreftypejNZrefexplicitrefwarnjeq9-1-78uhjh!jhMf h jYubh/ and solved for }(h and solved for h jYhhh!NhNubjr)}(h:math:`\psi_{0, I+1}`h]h/
\psi_{0, I+1}}(hhh jcZubah}(h]h]h]h]h]uhjqh jYubh/:}(hj:&h jYhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMf h jHVhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-87uhh
h jHVhhh!jhNubj )}(h\psi_{0, I+1}=\frac{\psi_{0, I}\left(A_{I} P_{I}-\sigma_{0}\right)+A_{I+1} q_{I+1}-A_{I} q_{I}+S_{0}^{*}}{A_{I+1} P_{I+1}+\sigma_{0}} ,h]h/\psi_{0, I+1}=\frac{\psi_{0, I}\left(A_{I} P_{I}-\sigma_{0}\right)+A_{I+1} q_{I+1}-A_{I} q_{I}+S_{0}^{*}}{A_{I+1} P_{I+1}+\sigma_{0}} ,}(hhh jZubah}(h]jZah]h]h]h]docnamejnumberKVlabeleq9-1-87nowrapjyjzuhj h!jhMh h jHVhhjf}jh}jZj{Zsubh;)}(hwhich is the expression used in XSDRNPM. The procedure solves for arrays
of *P*\ :sub:`I` and *q*\ :sub:`I` which are plugged back into the above expression to
yield the fluxes.h](h/Lwhich is the expression used in XSDRNPM. The procedure solves for arrays
of }(hLwhich is the expression used in XSDRNPM. The procedure solves for arrays
of h jZhhh!NhNubhA)}(h*P*h]h/P}(hhh jZubah}(h]h]h]h]h]uhh@h jZubh/ }(h\ h jZhhh!NhNubh)}(h:sub:`I`h]h/I}(hhh jZubah}(h]h]h]h]h]uhhh jZubh/ and }(h and h jZhhh!NhNubhA)}(h*q*h]h/q}(hhh jZubah}(h]h]h]h]h]uhh@h jZubh/ }(hjZh jZubh)}(h:sub:`I`h]h/I}(hhh jZubah}(h]h]h]h]h]uhhh jZubh/F which are plugged back into the above expression to
yield the fluxes.}(hF which are plugged back into the above expression to
yield the fluxes.h jZhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMm h jHVhhubh)}(h
.. _9-1-2-18:h]h}(h]h]h]h]h]hid76uhh
hM h jHVhhh!jubeh}(h](diffusion-theory-optionj, cross sections in a
multiregion system are not homogenized.}(h>, cross sections in a
multiregion system are not homogenized.h j[hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM h j[hhubh;)}(hDThe *B*\ :sub:`N` equations :cite:`henryson_mc_1976` can be writtenh](h/The }(hThe h jX\hhh!NhNubhA)}(h*B*h]h/B}(hhh ja\ubah}(h]h]h]h]h]uhh@h jX\ubh/ }(h\ h jX\hhh!NhNubh)}(h:sub:`N`h]h/N}(hhh jt\ubah}(h]h]h]h]h]uhhh jX\ubh/ equations }(h equations h jX\hhh!NhNubj)}(hhenryson_mc_1976h]j#)}(hj\h]h/[henryson_mc_1976]}(hhh j\ubah}(h]h]h]h]h]uhj"h j\ubah}(h]id78ah]j5ah]h]h] refdomainj:reftypej< reftargetj\refwarnsupport_smartquotesuhjh!jhM h jX\hhubh/ can be written}(h can be writtenh jX\hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM h j[hhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-89uhh
h j[hhh!jhNubj )}(hX
\begin{aligned}
\frac{l+1}{2 l+1} i B \psi_{l+1} &+\frac{l}{2 l+1} i B \psi_{l-1}+\Sigma_{t} \psi_{l}=S(u) \delta_{l}^{0} \\
&+\int d u^{\prime} \Sigma_{s}^{l}\left(u^{\prime} \rightarrow u\right) \psi_{l}\left(u^{\prime}\right) \quad l=0,1, \quad, N-1
\end{aligned}h]h/X
\begin{aligned}
\frac{l+1}{2 l+1} i B \psi_{l+1} &+\frac{l}{2 l+1} i B \psi_{l-1}+\Sigma_{t} \psi_{l}=S(u) \delta_{l}^{0} \\
&+\int d u^{\prime} \Sigma_{s}^{l}\left(u^{\prime} \rightarrow u\right) \psi_{l}\left(u^{\prime}\right) \quad l=0,1, \quad, N-1
\end{aligned}}(hhh j\ubah}(h]j\ah]h]h]h]docnamejnumberKXlabeleq9-1-89nowrapjyjzuhj h!jhM h j[hhjf}jh}j\j\subh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-90uhh
h j[hhh!jhNubj )}(h\frac{N}{2 N+1} i B \psi_{N-1}+\gamma \Sigma_{t} \psi_{N}=\int d u^{\prime} \Sigma_{g}^{N}\left(u^{\prime} \rightarrow u\right) \psi_{N}\left(u^{\prime}\right)h]h/\frac{N}{2 N+1} i B \psi_{N-1}+\gamma \Sigma_{t} \psi_{N}=\int d u^{\prime} \Sigma_{g}^{N}\left(u^{\prime} \rightarrow u\right) \psi_{N}\left(u^{\prime}\right)}(hhh j\ubah}(h]j\ah]h]h]h]docnamejnumberKYlabeleq9-1-90nowrapjyjzuhj h!jhM h j[hhjf}jh}j\j\subh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-91uhh
h j[hhh!jhNubj )}(h\psi_{-1}=0h]h/\psi_{-1}=0}(hhh j\ubah}(h]j\ah]h]h]h]docnamejnumberKZlabeleq9-1-91nowrapjyjzuhj h!jhM h j[hhjf}jh}j\j\subh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-92uhh
h j[hhh!jhNubj )}(h\gamma=1+\frac{N+1}{2 N+1} \frac{i B}{\Sigma_{T}} \frac{Q_{N+1}\left(-\Sigma_{t} / i B\right)}{Q_{N}\left(-\Sigma_{t} / i B\right)}h]h/\gamma=1+\frac{N+1}{2 N+1} \frac{i B}{\Sigma_{T}} \frac{Q_{N+1}\left(-\Sigma_{t} / i B\right)}{Q_{N}\left(-\Sigma_{t} / i B\right)}}(hhh j]ubah}(h]j]ah]h]h]h]docnamejnumberK[labeleq9-1-92nowrapjyjzuhj h!jhM h j[hhjf}jh}j]j]subh;)}(hwhere :math:`\delta_{l}^{0}` is the Kronecker delta function and *Q*\ :sub:`N` is a Legendre
polynomial of the second kind. In mutigroup form, the above expressions
become:h](h/where }(hwhere h j+]hhh!NhNubjr)}(h:math:`\delta_{l}^{0}`h]h/\delta_{l}^{0}}(hhh j4]ubah}(h]h]h]h]h]uhjqh j+]ubh/% is the Kronecker delta function and }(h% is the Kronecker delta function and h j+]hhh!NhNubhA)}(h*Q*h]h/Q}(hhh jG]ubah}(h]h]h]h]h]uhh@h j+]ubh/ }(h\ h j+]hhh!NhNubh)}(h:sub:`N`h]h/N}(hhh jZ]ubah}(h]h]h]h]h]uhhh j+]ubh/^ is a Legendre
polynomial of the second kind. In mutigroup form, the above expressions
become:}(h^ is a Legendre
polynomial of the second kind. In mutigroup form, the above expressions
become:h j+]hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM h j[hhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-93uhh
h j[hhh!jhNubj )}(hX\begin{aligned}
\frac{l+1}{2 l+1} i B \psi_{l+1}^{g} &+\frac{l}{2 l+1} i B \psi_{l-1}^{g}+\Sigma_{t}^{g} \psi_{l}^{g}=S_{g} \delta_{l}^{0} \\
&+\sum_{g^{\prime}} \Sigma_{l}\left(g^{\prime} \rightarrow g\right) \psi_{l}^{g} \quad l=0,1, \quad N-1
\end{aligned}h]h/X\begin{aligned}
\frac{l+1}{2 l+1} i B \psi_{l+1}^{g} &+\frac{l}{2 l+1} i B \psi_{l-1}^{g}+\Sigma_{t}^{g} \psi_{l}^{g}=S_{g} \delta_{l}^{0} \\
&+\sum_{g^{\prime}} \Sigma_{l}\left(g^{\prime} \rightarrow g\right) \psi_{l}^{g} \quad l=0,1, \quad N-1
\end{aligned}}(hhh j}]ubah}(h]j|]ah]h]h]h]docnamejnumberK\labeleq9-1-93nowrapjyjzuhj h!jhM h j[hhjf}jh}j|]js]subh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-94uhh
h j[hhh!jhNubj )}(h\frac{N}{2 N+1} i B \psi_{N-1}^{g}+\gamma \Sigma_{t}^{g} \psi_{N}^{g}=\sum_{g^{\prime}} \Sigma_{N}\left(g^{\prime} \rightarrow g\right) \psi_{N}^{g}h]h/\frac{N}{2 N+1} i B \psi_{N-1}^{g}+\gamma \Sigma_{t}^{g} \psi_{N}^{g}=\sum_{g^{\prime}} \Sigma_{N}\left(g^{\prime} \rightarrow g\right) \psi_{N}^{g}}(hhh j]ubah}(h]j]ah]h]h]h]docnamejnumberK]labeleq9-1-94nowrapjyjzuhj h!jhM h j[hhjf}jh}j]j]subh)}(hhh]h}(h]h]h]h]h]hequation-eq9-1-95uhh
h j[hhh!jhNubj )}(h\psi_{-1}^{g}=0 .h]h/\psi_{-1}^{g}=0 .}(hhh j]ubah}(h]j]ah]h]h]h]docnamejnumberK^labeleq9-1-95nowrapjyjzuhj h!jhM h j[hhjf}jh}j]j]subh;)}(hpIn Eqs. :eq:`eq9-1-89` and :eq:`eq9-1-93`, the S term includes fission, fixed, and scattering source components.h](h/In Eqs. }(hIn Eqs. h j]hhh!NhNubj)}(h:eq:`eq9-1-89`h]jc)}(hj]h]h/eq9-1-89}(hhh j]ubah}(h]h](jneqeh]h]h]uhjbh j]ubah}(h]h]h]h]h]refdocj refdomainjqreftypej]refexplicitrefwarnjeq9-1-89uhjh!jhM h j]ubh/ and }(h and h j]hhh!NhNubj)}(h:eq:`eq9-1-93`h]jc)}(hj]h]h/eq9-1-93}(hhh j^ubah}(h]h](jneqeh]h]h]uhjbh j]ubah}(h]h]h]h]h]refdocj refdomainjqreftypej
^refexplicitrefwarnjeq9-1-93uhjh!jhM h j]ubh/G, the S term includes fission, fixed, and scattering source components.}(hG, the S term includes fission, fixed, and scattering source components.h j]hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM h j[hhubh)}(h
.. _9-1-3:h]h}(h]h]h]h]h]hid79uhh
hM?
h j[hhh!jubeh}(h](bn-theory-optionj[eh]h](bn theory option9-1-2-19eh]h]uhh#h j hhh!jhM jf}j6^j[sjh}j[j[subeh}(h](theory-and-proceduresj eh]h]9-1-2ah]theory and proceduresah]uhh#h jhhh!jhM#jKjf}j@^j sjh}j j subh$)}(hhh](h))}(hXSDRNPM Input Datah]h/XSDRNPM Input Data}(hjK^h jI^hhh!NhNubah}(h]h]h]h]h]uhh(h jF^hhh!jhM ubh;)}(hThe input data to XSDRNPM consist of a title card and up to five data blocks,
depending on the particular problem. All data in these blocks are entered using
the FIDO formats discussed in the chapter on FIDO.h]h/The input data to XSDRNPM consist of a title card and up to five data blocks,
depending on the particular problem. All data in these blocks are entered using
the FIDO formats discussed in the chapter on FIDO.}(hjY^h jW^hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM h jF^hhubh;)}(hXIIn the description that follows, the quantity in square brackets is the number
of items in an array. The quantity in braces is the condition which requires
the array to be input. If no condition is specified, an array must be input.
Default parameters that are used if an array is not input are shown in
parentheses if nonzero.h]h/XIIn the description that follows, the quantity in square brackets is the number
of items in an array. The quantity in braces is the condition which requires
the array to be input. If no condition is specified, an array must be input.
Default parameters that are used if an array is not input are shown in
parentheses if nonzero.}(hjg^h je^hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM h jF^hhubj1)}(hhh]h;)}(hE\******************************************************************\*h]h/E*******************************************************************}(hE\******************************************************************\*h jv^ubah}(h]h]h]h]h]uhh:h!jhM h js^ubah}(h]h]h]h]h]uhj1h jF^hhh!jhNubh;)}(hTitle Card - Format (20A4)h]h/Title Card - Format (20A4)}(hj^h j^hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM h jF^hhubh;)}(hUThis is the title card for the problem. It will be used to label the problem output.h]h/UThis is the title card for the problem. It will be used to label the problem output.}(hj^h j^hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM h jF^hhubhd)}(hData Block 1h]h/Data Block 1}(hData Block 1h j^hhh!NhNubah}(h]h]h]h]h]uhhch jF^hhh!jhM ubh;)}(hThis block contains information to set up various array dimensions and most
calculational and editing options. Various convergence criteria and special
constants can be input.h]h/This block contains information to set up various array dimensions and most
calculational and editing options. Various convergence criteria and special
constants can be input.}(hj^h j^hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM h jF^hhubh;)}(h0$$ Logical Assignments [17]h]h/0$$ Logical Assignments [17]}(hj^h j^hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM h jF^hhubj1)}(hhh]j)}(hhh](j)}(h6LPUN – Logical number for punched card output (7).
h]h;)}(h5LPUN – Logical number for punched card output (7).h]h/5LPUN – Logical number for punched card output (7).}(hj^h j^ubah}(h]h]h]h]h]uhh:h!jhM h j^ubah}(h]h]h]h]h]uhjh j^ubj)}(h1LRSF – Random-access scratch for fluxes (10).
h]h;)}(h0LRSF – Random-access scratch for fluxes (10).h]h/0LRSF – Random-access scratch for fluxes (10).}(hj^h j^ubah}(h]h]h]h]h]uhh:h!jhM h j^ubah}(h]h]h]h]h]uhjh j^ubj)}(h*LAWL – Input AMPX working library (4).
h]h;)}(h)LAWL – Input AMPX working library (4).h]h/)LAWL – Input AMPX working library (4).}(hj_h j_ubah}(h]h]h]h]h]uhh:h!jhM h j_ubah}(h]h]h]h]h]uhjh j^ubj)}(h4LANC – ANISN binary or CCCC ISOTXS library (20).
h]h;)}(h3LANC – ANISN binary or CCCC ISOTXS library (20).h]h/3LANC – ANISN binary or CCCC ISOTXS library (20).}(hj&_h j$_ubah}(h]h]h]h]h]uhh:h!jhM h j _ubah}(h]h]h]h]h]uhjh j^ubj)}(h'LOWL – Output weighted library (3).
h]h;)}(h&LOWL – Output weighted library (3).h]h/&LOWL – Output weighted library (3).}(hj>_h j<_ubah}(h]h]h]h]h]uhh:h!jhM h j8_ubah}(h]h]h]h]h]uhjh j^ubj)}(h*LANG – Angular flux scratch file (16).
h]h;)}(h)LANG – Angular flux scratch file (16).h]h/)LANG – Angular flux scratch file (16).}(hjV_h jT_ubah}(h]h]h]h]h]uhh:h!jhM h jP_ubah}(h]h]h]h]h]uhjh j^ubj)}(h(LSFF – Scalar flux output file (17).
h]h;)}(h'LSFF – Scalar flux output file (17).h]h/'LSFF – Scalar flux output file (17).}(hjn_h jl_ubah}(h]h]h]h]h]uhh:h!jhM h jh_ubah}(h]h]h]h]h]uhjh j^ubj)}(h)LSF2 – Sequential scratch space (18).
h]h;)}(h(LSF2 – Sequential scratch space (18).h]h/(LSF2 – Sequential scratch space (18).}(hj_h j_ubah}(h]h]h]h]h]uhh:h!jhM h j_ubah}(h]h]h]h]h]uhjh j^ubj)}(h)LSF3 – Sequential scratch space (19).
h]h;)}(h(LSF3 – Sequential scratch space (19).h]h/(LSF3 – Sequential scratch space (19).}(hj_h j_ubah}(h]h]h]h]h]uhh:h!jhM h j_ubah}(h]h]h]h]h]uhjh j^ubj)}(hDLRSM – Random-access scratch for macroscopic cross sections (8).
h]h;)}(hCLRSM – Random-access scratch for macroscopic cross sections (8).h]h/CLRSM – Random-access scratch for macroscopic cross sections (8).}(hj_h j_ubah}(h]h]h]h]h]uhh:h!jhM h j_ubah}(h]h]h]h]h]uhjh j^ubj)}(hDLRSX – Random-access scratch for macroscopic cross sections (9).
h]h;)}(hCLRSX – Random-access scratch for macroscopic cross sections (9).h]h/CLRSX – Random-access scratch for macroscopic cross sections (9).}(hj_h j_ubah}(h]h]h]h]h]uhh:h!jhM h j_ubah}(h]h]h]h]h]uhjh j^ubj)}(h'LACF – Activities output file (75).
h]h;)}(h&LACF – Activities output file (75).h]h/&LACF – Activities output file (75).}(hj_h j_ubah}(h]h]h]h]h]uhh:h!jhM h j_ubah}(h]h]h]h]h]uhjh j^ubj)}(h*LBTF – Balance table output file (76).
h]h;)}(h)LBTF – Balance table output file (76).h]h/)LBTF – Balance table output file (76).}(hj_h j_ubah}(h]h]h]h]h]uhh:h!jhM h j_ubah}(h]h]h]h]h]uhjh j^ubj)}(h LIDF – Input dump file (73).
h]h;)}(hLIDF – Input dump file (73).h]h/LIDF – Input dump file (73).}(hj`h j`ubah}(h]h]h]h]h]uhh:h!jhM h j`ubah}(h]h]h]h]h]uhjh j^ubj)}(h'LSEN – Sensitivity output file (6).
h]h;)}(h&LSEN – Sensitivity output file (6).h]h/&LSEN – Sensitivity output file (6).}(hj.`h j,`ubah}(h]h]h]h]h]uhh:h!jhM h j(`ubah}(h]h]h]h]h]uhjh j^ubj)}(hLEXT – Not used (0).
h]h;)}(hLEXT – Not used (0).h]h/LEXT – Not used (0).}(hjF`h jD`ubah}(h]h]h]h]h]uhh:h!jhM h j@`ubah}(h]h]h]h]h]uhjh j^ubj)}(h,LISF – Scalar flux input guess file (0).
h]h;)}(h+LISF – Scalar flux input guess file (0).h]h/+LISF – Scalar flux input guess file (0).}(hj^`h j\`ubah}(h]h]h]h]h]uhh:h!jhM
h jX`ubah}(h]h]h]h]h]uhjh j^ubeh}(h]h]h]h]h]jSjCjUhjVjWuhjh j^ubah}(h]h]h]h]h]uhj1h jF^hhh!NhNubh;)}(h$1$ General Problem Description [15]h]h/$1$ General Problem Description [15]}(hj~`h j|`hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM
h jF^hhubj1)}(hhh](j)}(hhh]j)}(hIGE – problem geometry (1)
h]h;)}(hIGE – problem geometry (1)h]h/IGE – problem geometry (1)}(hj`h j`ubah}(h]h]h]h]h]uhh:h!jhM
h j`ubah}(h]h]h]h]h]uhjh j`ubah}(h]h]h]h]h]jSjCjUhjVjWuhjh j`ubj1)}(hhh]h;)}(h0 – homogeneous (This causes a BN calculation to be made for all zones – :ref:`9-1-2-19`)
1 – slab
2 – cylinder
3 – sphereh](h/N0 – homogeneous (This causes a BN calculation to be made for all zones – }(hN0 – homogeneous (This causes a BN calculation to be made for all zones – h j`ubj)}(h:ref:`9-1-2-19`h]j#)}(hj`h]h/9-1-2-19}(hhh j`ubah}(h]h](jnstdstd-refeh]h]h]uhj"h j`ubah}(h]h]h]h]h]refdocj refdomainj`reftyperefrefexplicitrefwarnj9-1-2-19uhjh!jhM
h j`ubh/()
1 – slab
2 – cylinder
3 – sphere}(h()
1 – slab
2 – cylinder
3 – sphereh j`ubeh}(h]h]h]h]h]uhh:h!jhM
h j`ubah}(h]h]h]h]h]uhj1h j`ubj)}(hhh](j)}(h 0, XSDRNPM will calculate an angular quadrature for the appropriate geometry. If ISN < 0, the calculation is bypassed, and the user must supply a set in the 42# and 43# arrays.
h]h;)}(hISN – the order of angular quadrature to be used. If ISN > 0, XSDRNPM will calculate an angular quadrature for the appropriate geometry. If ISN < 0, the calculation is bypassed, and the user must supply a set in the 42# and 43# arrays.h]h/ISN – the order of angular quadrature to be used. If ISN > 0, XSDRNPM will calculate an angular quadrature for the appropriate geometry. If ISN < 0, the calculation is bypassed, and the user must supply a set in the 42# and 43# arrays.}(hjbh jaubah}(h]h]h]h]h]uhh:h!jhM$
h jaubah}(h]h]h]h]h]uhjh jaubj)}(hXISCT – the order of scattering. Flux moments will be calculated through this order.
h]h;)}(hWISCT – the order of scattering. Flux moments will be calculated through this order.h]h/WISCT – the order of scattering. Flux moments will be calculated through this order.}(hjbh jbubah}(h]h]h]h]h]uhh:h!jhM&
h jbubah}(h]h]h]h]h]uhjh jaubj)}(h'IEVT – the type of calculation. (1)
h]h;)}(h&IEVT – the type of calculation. (1)h]h/&IEVT – the type of calculation. (1)}(hj0bh j.bubah}(h]h]h]h]h]uhh:h!jhM(
h j*bubah}(h]h]h]h]h]uhjh jaubeh}(h]h]h]h]h]jSjCjUhjVjWjXKuhjh jaubj1)}(hhh]h;)}(h0 – fixed source
1 – k calculation
2 – α calculation (flux is assumed to have an et time variation)
3 – inoperable in present version
4 – zone width search
5 – outer radius search
6 – buckling search
7 – direct buckling searchh]h/0 – fixed source
1 – k calculation
2 – α calculation (flux is assumed to have an et time variation)
3 – inoperable in present version
4 – zone width search
5 – outer radius search
6 – buckling search
7 – direct buckling search}(hjMbh jKbubah}(h]h]h]h]h]uhh:h!jhM*
h jHbubah}(h]h]h]h]h]uhj1h jaubj)}(hhh](j)}(hGIIM – the inner iteration maximum used in an Sn calculation. (10)
h]h;)}(hFIIM – the inner iteration maximum used in an Sn calculation. (10)h]h/FIIM – the inner iteration maximum used in an Sn calculation. (10)}(hjhbh jfbubah}(h]h]h]h]h]uhh:h!jhM3
h jbbubah}(h]h]h]h]h]uhjh j_bubj)}(h,ICM – the outer iteration maximum. (10)
h]h;)}(h+ICM – the outer iteration maximum. (10)h]h/+ICM – the outer iteration maximum. (10)}(hjbh j~bubah}(h]h]h]h]h]uhh:h!jhM5
h jzbubah}(h]h]h]h]h]uhjh j_bubeh}(h]h]h]h]h]jSjCjUhjVjWjXKuhjh jaubj1)}(hhh]h;)}(hAfter ICM outer iterations, the problem will be forced into the termination
phase and the program will continue as if full convergence was attained. A
message to this effect is printed.h]h/After ICM outer iterations, the problem will be forced into the termination
phase and the program will continue as if full convergence was attained. A
message to this effect is printed.}(hjbh jbubah}(h]h]h]h]h]uhh:h!jhM7
h jbubah}(h]h]h]h]h]uhj1h jaubj)}(hhh]j)}(hICLC – theory option. (0)
h]h;)}(hICLC – theory option. (0)h]h/ICLC – theory option. (0)}(hjbh jbubah}(h]h]h]h]h]uhh:h!jhM;
h jbubah}(h]h]h]h]h]uhjh jbubah}(h]h]h]h]h]jSjCjUhjVjWjXK
uhjh jaubj1)}(hhh](j1)}(hhh]h;)}(h0 – use |Sn| theory always
N – use alternative theory (diffusion, infinite medium, or Bn) for N outer iterations, after which revert back to Sn theory.h](h/0 – use }(h0 – use h jbubjr)}(hjh]h/S_n}(hhh jbubah}(h]h]h]h]h]uhjqh!NhNh jbubh/ theory always
N – use alternative theory (diffusion, infinite medium, or Bn) for N outer iterations, after which revert back to Sn theory.}(h theory always
N – use alternative theory (diffusion, infinite medium, or Bn) for N outer iterations, after which revert back to Sn theory.h jbubeh}(h]h]h]h]h]uhh:h!jhM=
h jbubah}(h]h]h]h]h]uhj1h jbubh;)}(h)-N – always use alternative theoryh]h/)-N – always use alternative theory}(hjbh jbubah}(h]h]h]h]h]uhh:h!jhM?
h jbubeh}(h]h]h]h]h]uhj1h jaubeh}(h]h]h]h]h]uhj1h jF^hhh!jhNubj)}(hhh]h}(h]h]h]h]h]jyjzuhjh jF^hhh!jhMA
ubj1)}(hhh](j)}(hhh]j)}(h(ITH – forward/adjoint selector. (0)
h]h;)}(h'ITH – forward/adjoint selector. (0)h]h/'ITH – forward/adjoint selector. (0)}(hj,ch j*cubah}(h]h]h]h]h]uhh:h!jhMC
h j&cubah}(h]h]h]h]h]uhjh j#cubah}(h]h]h]h]h]jSjCjUhjVjWjXKuhjh j cubj1)}(hhh]h;)}(hc0 – solve the forward Boltzmann equation.
1 – solve the adjoint Boltzmann equation.h]h/c0 – solve the forward Boltzmann equation.
1 – solve the adjoint Boltzmann equation.}(hjIch jGcubah}(h]h]h]h]h]uhh:h!jhME
h jDcubah}(h]h]h]h]h]uhj1h j cubj)}(hhh]j)}(hIFLU – Generalized adjoint calculation flag. (0)
0 – standard calculation 1 – Generalized adjoint calculation. Requires
input forward and adjoint fundamental mode fluxes on units 31 and 32,
respectively
h](h;)}(h3IFLU – Generalized adjoint calculation flag. (0)h]h/3IFLU – Generalized adjoint calculation flag. (0)}(hjdch jbcubah}(h]h]h]h]h]uhh:h!jhMH
h j^cubh;)}(h0 – standard calculation 1 – Generalized adjoint calculation. Requires
input forward and adjoint fundamental mode fluxes on units 31 and 32,
respectivelyh]h/0 – standard calculation 1 – Generalized adjoint calculation. Requires
input forward and adjoint fundamental mode fluxes on units 31 and 32,
respectively}(hjrch jpcubah}(h]h]h]h]h]uhh:h!jhMJ
h j^cubeh}(h]h]h]h]h]uhjh j[cubah}(h]h]h]h]h]jSjCjUhjVjWjXKuhjh j cubeh}(h]h]h]h]h]uhj1h jF^hhh!NhNubh;)}(h$2$ Editing and Special Options [10]h]h/$2$ Editing and Special Options [10]}(hjch jchhh!NhNubah}(h]h]h]h]h]uhh:h!jhMN
h jF^hhubj)}(hhh]h}(h]h]h]h]h]jyjzuhjh jF^hhh!jhMP
ubj1)}(hhh](j)}(hhh]j)}(h7IPRT– fine-group mixture cross-section edits. (-1)
h]h;)}(h6IPRT– fine-group mixture cross-section edits. (-1)h]h/6IPRT– fine-group mixture cross-section edits. (-1)}(hjch jcubah}(h]h]h]h]h]uhh:h!jhMR
h jcubah}(h]h]h]h]h]uhjh jcubah}(h]h]h]h]h]jSjCjUhjVjWuhjh jcubj1)}(hhh]h;)}(h-2 – no edits
-1 – edit 1-D cross sections
0–N – edit through PN cross sections.
1-D edits are made, also.h]h/-2 – no edits
-1 – edit 1-D cross sections
0–N – edit through PN cross sections.
1-D edits are made, also.}(hjch jcubah}(h]h]h]h]h]uhh:h!jhMT
h jcubah}(h]h]h]h]h]uhj1h jcubeh}(h]h]h]h]h]uhj1h jF^hhh!jhNubj)}(hhh]h}(h]h]h]h]h]jyjzuhjh jF^hhh!jhMY
ubj1)}(hhh](j)}(hhh]j)}(h$ID1 – flux editing options. (0)
h]h;)}(h#ID1 – flux editing options. (0)h]h/#ID1 – flux editing options. (0)}(hjch jcubah}(h]h]h]h]h]uhh:h!jhM[
h jcubah}(h]h]h]h]h]uhjh jcubah}(h]h]h]h]h]jSjCjUhjVjWjXKuhjh jcubj1)}(hhh]jq)}(hhh]j)}(hhh](j)}(hhh]h}(h]h]h]h]h]colwidthKuhjh jdubj)}(hhh]h}(h]h]h]h]h]colwidthKuhjh jdubj)}(hhh]h}(h]h]h]h]h]colwidthKuhjh jdubj)}(hhh]h}(h]h]h]h]h]colwidthKuhjh jdubj)}(hhh](j)}(hhh](j)}(hhh]h;)}(hID1h]h/ID1}(hjQdh jOdubah}(h]h]h]h]h]uhh:h!jhM^
h jLdubah}(h]h]h]h]h]uhjh jIdubj)}(hhh](h;)}(hPrinth]h/Print}(hjhdh jfdubah}(h]h]h]h]h]uhh:h!jhM^
h jcdubh;)}(hAngular fluxesh]h/Angular fluxes}(hjvdh jtdubah}(h]h]h]h]h]uhh:h!jhM`
h jcdubeh}(h]h]h]h]h]uhjh jIdubj)}(hhh](h;)}(hPrinth]h/Print}(hjdh jdubah}(h]h]h]h]h]uhh:h!jhM^
h jdubh;)}(h
Scalar fluxesh]h/
Scalar fluxes}(hjdh jdubah}(h]h]h]h]h]uhh:h!jhM`
h jdubeh}(h]h]h]h]h]uhjh jIdubj)}(hhh](h;)}(h
Punch\ *a*h](h/Punch }(hPunch\ h jdubhA)}(h*a*h]h/a}(hhh jdubah}(h]h]h]h]h]uhh@h jdubeh}(h]h]h]h]h]uhh:h!jhM^
h jdubh;)}(h
Scalar fluxesh]h/
Scalar fluxes}(hjdh jdubah}(h]h]h]h]h]uhh:h!jhM`
h jdubeh}(h]h]h]h]h]uhjh jIdubeh}(h]h]h]h]h]uhjh jFdubj)}(hhh](j)}(hhh]h;)}(h−1h]h/−1}(hjdh jdubah}(h]h]h]h]h]uhh:h!jhMb
h jdubah}(h]h]h]h]h]uhjh jdubj)}(hhh]h;)}(hNoh]h/No}(hjeh jeubah}(h]h]h]h]h]uhh:h!jhMb
h jeubah}(h]h]h]h]h]uhjh jdubj)}(hhh]h;)}(hNoh]h/No}(hjeh jeubah}(h]h]h]h]h]uhh:h!jhMb
h jeubah}(h]h]h]h]h]uhjh jdubj)}(hhh]h;)}(hNoh]h/No}(hj4eh j2eubah}(h]h]h]h]h]uhh:h!jhMb
h j/eubah}(h]h]h]h]h]uhjh jdubeh}(h]h]h]h]h]uhjh jFdubj)}(hhh](j)}(hhh]h;)}(hjh]h/0}(hjh jReubah}(h]h]h]h]h]uhh:h!jhMd
h jOeubah}(h]h]h]h]h]uhjh jLeubj)}(hhh]h;)}(hNoh]h/No}(hjjeh jheubah}(h]h]h]h]h]uhh:h!jhMd
h jeeubah}(h]h]h]h]h]uhjh jLeubj)}(hhh]h;)}(hYesh]h/Yes}(hjeh jeubah}(h]h]h]h]h]uhh:h!jhMd
h j|eubah}(h]h]h]h]h]uhjh jLeubj)}(hhh]h;)}(hNoh]h/No}(hjeh jeubah}(h]h]h]h]h]uhh:h!jhMd
h jeubah}(h]h]h]h]h]uhjh jLeubeh}(h]h]h]h]h]uhjh jFdubj)}(hhh](j)}(hhh]h;)}(hjh]h/1}(hjh jeubah}(h]h]h]h]h]uhh:h!jhMf
h jeubah}(h]h]h]h]h]uhjh jeubj)}(hhh]h;)}(hYesh]h/Yes}(hjeh jeubah}(h]h]h]h]h]uhh:h!jhMf
h jeubah}(h]h]h]h]h]uhjh jeubj)}(hhh]h;)}(hYesh]h/Yes}(hjeh jeubah}(h]h]h]h]h]uhh:h!jhMf
h jeubah}(h]h]h]h]h]uhjh jeubj)}(hhh]h;)}(hNoh]h/No}(hjeh jeubah}(h]h]h]h]h]uhh:h!jhMf
h jeubah}(h]h]h]h]h]uhjh jeubeh}(h]h]h]h]h]uhjh jFdubj)}(hhh](j)}(hhh]h;)}(hjIh]h/2}(hjIh jfubah}(h]h]h]h]h]uhh:h!jhMh
h jfubah}(h]h]h]h]h]uhjh jfubj)}(hhh]h;)}(hNoh]h/No}(hj2fh j0fubah}(h]h]h]h]h]uhh:h!jhMh
h j-fubah}(h]h]h]h]h]uhjh jfubj)}(hhh]h;)}(hYesh]h/Yes}(hjIfh jGfubah}(h]h]h]h]h]uhh:h!jhMh
h jDfubah}(h]h]h]h]h]uhjh jfubj)}(hhh]h;)}(hYesh]h/Yes}(hj`fh j^fubah}(h]h]h]h]h]uhh:h!jhMh
h j[fubah}(h]h]h]h]h]uhjh jfubeh}(h]h]h]h]h]uhjh jFdubj)}(hhh](j)}(hhh]h;)}(hjh]h/3}(hjh j~fubah}(h]h]h]h]h]uhh:h!jhMj
h j{fubah}(h]h]h]h]h]uhjh jxfubj)}(hhh]h;)}(hYesh]h/Yes}(hjfh jfubah}(h]h]h]h]h]uhh:h!jhMj
h jfubah}(h]h]h]h]h]uhjh jxfubj)}(hhh]h;)}(hYesh]h/Yes}(hjfh jfubah}(h]h]h]h]h]uhh:h!jhMj
h jfubah}(h]h]h]h]h]uhjh jxfubj)}(hhh]h;)}(hYesh]h/Yes}(hjfh jfubah}(h]h]h]h]h]uhh:h!jhMj
h jfubah}(h]h]h]h]h]uhjh jxfubeh}(h]h]h]h]h]uhjh jFdubj)}(hhh](j)}(hhh]h;)}(hZ*a* The
fluxes will
be punched
in a format
suitable for
restarting
an XSDRNPM
calculation.h](hA)}(h*a*h]h/a}(hhh jfubah}(h]h]h]h]h]uhh@h jfubh/W The
fluxes will
be punched
in a format
suitable for
restarting
an XSDRNPM
calculation.}(hW The
fluxes will
be punched
in a format
suitable for
restarting
an XSDRNPM
calculation.h jfubeh}(h]h]h]h]h]uhh:h!jhMl
h jfubah}(h]h]h]h]h]uhjh jfubj)}(hhh]h}(h]h]h]h]h]uhjh jfubj)}(hhh]h}(h]h]h]h]h]uhjh jfubj)}(hhh]h}(h]h]h]h]h]uhjh jfubeh}(h]h]h]h]h]uhjh jFdubeh}(h]h]h]h]h]uhjh jdubeh}(h]h]h]h]h]colsKuhjh jdubah}(h]h]h]h]h]jdefaultuhjph jdubah}(h]h]h]h]h]uhj1h jcubj)}(hhh]j)}(h#IPBT – balance table edits. (0)
h]h;)}(h"IPBT – balance table edits. (0)h]h/"IPBT – balance table edits. (0)}(hjIgh jGgubah}(h]h]h]h]h]uhh:h!jhMv
h jCgubah}(h]h]h]h]h]uhjh j@gubah}(h]h]h]h]h]jSjCjUhjVjWjXKuhjh jcubj1)}(hhh]jd)}(hhh]j)}(hr1 – none
0 – make fine-group balance tables
1 – make fine- and broad-group balance tables
h]h;)}(hq1 – none
0 – make fine-group balance tables
1 – make fine- and broad-group balance tablesh]h/q1 – none
0 – make fine-group balance tables
1 – make fine- and broad-group balance tables}(hjmgh jkgubah}(h]h]h]h]h]uhh:h!jhMx
h jggubah}(h]h]h]h]h]uhjh jdgubah}(h]h]h]h]h]jBjCuhjch!jhMx
h jagubah}(h]h]h]h]h]uhj1h jcubj)}(hhh](j)}(hOISX – broad-group flux edit as a function of interval. (0) (0/1 = no/yes)
h]h;)}(hNISX – broad-group flux edit as a function of interval. (0) (0/1 = no/yes)h]h/NISX – broad-group flux edit as a function of interval. (0) (0/1 = no/yes)}(hjgh jgubah}(h]h]h]h]h]uhh:h!jhM|
h jgubah}(h]h]h]h]h]uhjh jgubj)}(h 0 means no fission source if IEVT=0. (0)
h]h;)}(h8IFSN – If > 0 means no fission source if IEVT=0. (0)h]h/8IFSN – If > 0 means no fission source if IEVT=0. (0)}(hjhh jhubah}(h]h]h]h]h]uhh:h!jhM
h j
hubah}(h]h]h]h]h]uhjh j
hubj)}(h :math:`\sqrt{\left(\psi_{1}^{g}+\left(D G \psi_{g}\right)\right)^{2}}`h](h/ITP = 0 => }(hITP = 0 => h jKlubjr)}(hF:math:`\sqrt{\left(\psi_{1}^{g}+\left(D G \psi_{g}\right)\right)^{2}}`h]h/>\sqrt{\left(\psi_{1}^{g}+\left(D G \psi_{g}\right)\right)^{2}}}(hhh jTlubah}(h]h]h]h]h]uhjqh jKlubeh}(h]h]h]h]h]uhh:h!jhM
h j:lubh;)}(h'ITP = 1 => absolute value of currenth]h/'ITP = 1 => absolute value of current}(hjjlh jhlubah}(h]h]h]h]h]uhh:h!jhMh j:lubh;)}(hIITP = 2 => :math:`\mathrm{DB}^{2} \psi_{\mathrm{g}}+` outside leakageh](h/ITP = 2 => }(hITP = 2 => h jvlubjr)}(h*:math:`\mathrm{DB}^{2} \psi_{\mathrm{g}}+`h]h/"\mathrm{DB}^{2} \psi_{\mathrm{g}}+}(hhh jlubah}(h]h]h]h]h]uhjqh jvlubh/ outside leakage}(h outside leakageh jvlubeh}(h]h]h]h]h]uhh:h!jhMh j:lubh;)}(h;ITP = 3 => :math:`\psi / \sum_{\mathrm{t}}^{\mathrm{g}}`h](h/ITP = 3 => }(hITP = 3 => h jlubjr)}(h-:math:`\psi / \sum_{\mathrm{t}}^{\mathrm{g}}`h]h/%\psi / \sum_{\mathrm{t}}^{\mathrm{g}}}(hhh jlubah}(h]h]h]h]h]uhjqh jlubeh}(h]h]h]h]h]uhh:h!jhMh j:lubh;)}(h3ITP = 4 => :math:`\mathrm{DB} \psi_{\mathrm{g}}`h](h/ITP = 4 => }(hITP = 4 => h jlubjr)}(h%:math:`\mathrm{DB} \psi_{\mathrm{g}}`h]h/\mathrm{DB} \psi_{\mathrm{g}}}(hhh jlubah}(h]h]h]h]h]uhjqh jlubeh}(h]h]h]h]h]uhh:h!jhMh j:lubh;)}(hNITP = other values are reserved for future development and should not be used.h]h/NITP = other values are reserved for future development and should not be used.}(hjlh jlubah}(h]h]h]h]h]uhh:h!jhM h j:lubeh}(h]h]h]h]h]uhj1h jxkubj)}(hhh](j)}(hIPP – weighted cross-section edit option -1.
−2 – none
−1 – edit 1-D data
0−N – edit through *P\ N* cross-section arrays. 1-D edits are given.
h](h;)}(h0IPP – weighted cross-section edit option -1.h]h/0IPP – weighted cross-section edit option -1.}(hjlh jlubah}(h]h]h]h]h]uhh:h!jhMh jlubj1)}(hhh](h;)}(h
−2 – noneh]h/
−2 – none}(hjmh jlubah}(h]h]h]h]h]uhh:h!jhM
h jlubh;)}(h−1 – edit 1-D datah]h/−1 – edit 1-D data}(hjmh jmubah}(h]h]h]h]h]uhh:h!jhMh jlubh;)}(hH0−N – edit through *P\ N* cross-section arrays. 1-D edits are given.h](h/0−N – edit through }(h0−N – edit through h jmubhA)}(h*P\ N*h]h/P N}(hhh j#mubah}(h]h]h]h]h]uhh@h jmubh/+ cross-section arrays. 1-D edits are given.}(h+ cross-section arrays. 1-D edits are given.h jmubeh}(h]h]h]h]h]uhh:h!jhMh jlubeh}(h]h]h]h]h]uhj1h jlubeh}(h]h]h]h]h]uhjh jlubj)}(h IHTF – (deprecated feature)
h]h;)}(hIHTF – (deprecated feature)h]h/IHTF – (deprecated feature)}(hjNmh jLmubah}(h]h]h]h]h]uhh:h!jhMh jHmubah}(h]h]h]h]h]uhjh jlubj)}(h NDSF – (deprecated feature)
h]h;)}(hNDSF – (deprecated feature)h]h/NDSF – (deprecated feature)}(hjfmh jdmubah}(h]h]h]h]h]uhh:h!jhMh j`mubah}(h]h]h]h]h]uhjh jlubj)}(hNUSF – (deprecated feature)
h]h;)}(hNUSF – (deprecated feature)h]h/NUSF – (deprecated feature)}(hj~mh j|mubah}(h]h]h]h]h]uhh:h!jhMh jxmubah}(h]h]h]h]h]uhjh jlubj)}(hIAP – (deprecated feature)
h]h;)}(hIAP – (deprecated feature)h]h/IAP – (deprecated feature)}(hjmh jmubah}(h]h]h]h]h]uhh:h!jhMh jmubah}(h]h]h]h]h]uhjh jlubj)}(h(deprecated feature)
h]h;)}(h(deprecated feature)h]h/(deprecated feature)}(hjmh jmubah}(h]h]h]h]h]uhh:h!jhMh jmubah}(h]h]h]h]h]uhjh jlubeh}(h]h]h]h]h]jSjCjUhjVjWjXKuhjh jxkubh;)}(h45* Convergence Criteria and Assorted Constants [12]h]h/45* Convergence Criteria and Assorted Constants [12]}(hjmh jmubah}(h]h]h]h]h]uhh:h!jhMh jxkubj)}(hhh](j)}(h5EPS – overall problem convergence. (10:sup:`−4`)
h]h;)}(h4EPS – overall problem convergence. (10:sup:`−4`)h](h/-EPS – overall problem convergence. (10:sup:}(h-EPS – overall problem convergence. (10:sup:h jmubj$#)}(h`−4`h]h/−4}(hhh jmubah}(h]h]h]h]h]uhj##h jmubh/)}(hj7@h jmubeh}(h]h]h]h]h]uhh:h!jhMh jmubah}(h]h]h]h]h]uhjh jmubj)}(h1PTC – scalar flux convergence. (10:sup:`−5`)
h]h;)}(h0PTC – scalar flux convergence. (10:sup:`−5`)h](h/)PTC – scalar flux convergence. (10:sup:}(h)PTC – scalar flux convergence. (10:sup:h jnubj$#)}(h`−5`h]h/−5}(hhh jnubah}(h]h]h]h]h]uhj##h jnubh/)}(hj7@h jnubeh}(h]h]h]h]h]uhh:h!jhM!h jnubah}(h]h]h]h]h]uhjh jmubj)}(hXXNF – normalization factor. (1.0)
When IEVT = 0, the fixed sources are normalized to XNF.
For IEVT > 0, the fission source is normalized to XNF.
When XNF = 0.0, no normalization is made.
XNF should only be specified as 0 for a fixed source problem (IEVT = 0).
h](h;)}(h#XNF – normalization factor. (1.0)h]h/#XNF – normalization factor. (1.0)}(hj3nh j1nubah}(h]h]h]h]h]uhh:h!jhM#h j-nubh;)}(h9When IEVT = 0, the fixed sources are normalized to XNF.h]h/9When IEVT = 0, the fixed sources are normalized to XNF.}(hjAnh j?nubah}(h]h]h]h]h]uhh:h!jhM%h j-nubh;)}(h8For IEVT > 0, the fission source is normalized to XNF.h]h/8For IEVT > 0, the fission source is normalized to XNF.}(hjOnh jMnubah}(h]h]h]h]h]uhh:h!jhM'h j-nubh;)}(h+When XNF = 0.0, no normalization is made.h]h/+When XNF = 0.0, no normalization is made.}(hj]nh j[nubah}(h]h]h]h]h]uhh:h!jhM)h j-nubh;)}(hJXNF should only be specified as 0 for a fixed source problem (IEVT = 0).h]h/JXNF should only be specified as 0 for a fixed source problem (IEVT = 0).}(hjknh jinubah}(h]h]h]h]h]uhh:h!jhM+h j-nubeh}(h]h]h]h]h]uhjh jmubj)}(h:EV – starting eigenvalue guess for search calculations.
h]h;)}(h9EV – starting eigenvalue guess for search calculations.h]h/9EV – starting eigenvalue guess for search calculations.}(hjnh jnubah}(h]h]h]h]h]uhh:h!jhM-h j}nubah}(h]h]h]h]h]uhjh jmubj)}(hXEVM – eigenvalue modifier used in a search calculation. The following
is a tabulation of recommended values for EV and EVM.
+------+------------------------+-----------------------+---------+
| IEVT | Calculation type | EV | EVM |
+------+------------------------+-----------------------+---------+
| 0 | Fixed source | 0 | 0 |
+------+------------------------+-----------------------+---------+
| 1 | k-calculation | 0 | 0 |
+------+------------------------+-----------------------+---------+
| 2 | Direct α-search | 0 | 0 |
+------+------------------------+-----------------------+---------+
| 3 | -- | -- | -- |
+------+------------------------+-----------------------+---------+
| 4 | Zone width search | 0 | −0.1 |
+------+------------------------+-----------------------+---------+
| 5 | Outer radius search | Starting outer radius | −0.1∗EV |
+------+------------------------+-----------------------+---------+
| 6 | Buckling search | 1.0 | −0.1 |
+------+------------------------+-----------------------+---------+
| 7 | Direct buckling search | 0.0 | 0.0 |
+------+------------------------+-----------------------+---------+
h](h;)}(h}EVM – eigenvalue modifier used in a search calculation. The following
is a tabulation of recommended values for EV and EVM.h]h/}EVM – eigenvalue modifier used in a search calculation. The following
is a tabulation of recommended values for EV and EVM.}(hjnh jnubah}(h]h]h]h]h]uhh:h!jhM/h jnubjq)}(hhh]j)}(hhh](j)}(hhh]h}(h]h]h]h]h]colwidthKuhjh jnubj)}(hhh]h}(h]h]h]h]h]colwidthKuhjh jnubj)}(hhh]h}(h]h]h]h]h]colwidthKuhjh jnubj)}(hhh]h}(h]h]h]h]h]colwidthK uhjh jnubj)}(hhh](j)}(hhh](j)}(hhh]h;)}(hIEVTh]h/IEVT}(hjnh jnubah}(h]h]h]h]h]uhh:h!jhM3h jnubah}(h]h]h]h]h]uhjh jnubj)}(hhh]h;)}(hCalculation typeh]h/Calculation type}(hjnh jnubah}(h]h]h]h]h]uhh:h!jhM3h jnubah}(h]h]h]h]h]uhjh jnubj)}(hhh]h;)}(hEVh]h/EV}(hjoh joubah}(h]h]h]h]h]uhh:h!jhM3h j oubah}(h]h]h]h]h]uhjh jnubj)}(hhh]h;)}(hEVMh]h/EVM}(hj%oh j#oubah}(h]h]h]h]h]uhh:h!jhM3h j oubah}(h]h]h]h]h]uhjh jnubeh}(h]h]h]h]h]uhjh jnubj)}(hhh](j)}(hhh]h;)}(hjh]h/0}(hjh jCoubah}(h]h]h]h]h]uhh:h!jhM5h j@oubah}(h]h]h]h]h]uhjh j=oubj)}(hhh]h;)}(hFixed sourceh]h/Fixed source}(hj[oh jYoubah}(h]h]h]h]h]uhh:h!jhM5h jVoubah}(h]h]h]h]h]uhjh j=oubj)}(hhh]h;)}(hjh]h/0}(hjh jpoubah}(h]h]h]h]h]uhh:h!jhM5h jmoubah}(h]h]h]h]h]uhjh j=oubj)}(hhh]h;)}(hjh]h/0}(hjh joubah}(h]h]h]h]h]uhh:h!jhM5h joubah}(h]h]h]h]h]uhjh j=oubeh}(h]h]h]h]h]uhjh jnubj)}(hhh](j)}(hhh]h;)}(hjh]h/1}(hjh joubah}(h]h]h]h]h]uhh:h!jhM7h joubah}(h]h]h]h]h]uhjh joubj)}(hhh]h;)}(h
k-calculationh]h/
k-calculation}(hjoh joubah}(h]h]h]h]h]uhh:h!jhM7h joubah}(h]h]h]h]h]uhjh joubj)}(hhh]h;)}(hjh]h/0}(hjh joubah}(h]h]h]h]h]uhh:h!jhM7h joubah}(h]h]h]h]h]uhjh joubj)}(hhh]h;)}(hjh]h/0}(hjh joubah}(h]h]h]h]h]uhh:h!jhM7h joubah}(h]h]h]h]h]uhjh joubeh}(h]h]h]h]h]uhjh jnubj)}(hhh](j)}(hhh]h;)}(hjIh]h/2}(hjIh jpubah}(h]h]h]h]h]uhh:h!jhM9h jpubah}(h]h]h]h]h]uhjh jpubj)}(hhh]h;)}(hDirect α-searchh]h/Direct α-search}(hjph jpubah}(h]h]h]h]h]uhh:h!jhM9h jpubah}(h]h]h]h]h]uhjh jpubj)}(hhh]h;)}(hjh]h/0}(hjh j4pubah}(h]h]h]h]h]uhh:h!jhM9h j1pubah}(h]h]h]h]h]uhjh jpubj)}(hhh]h;)}(hjh]h/0}(hjh jJpubah}(h]h]h]h]h]uhh:h!jhM9h jGpubah}(h]h]h]h]h]uhjh jpubeh}(h]h]h]h]h]uhjh jnubj)}(hhh](j)}(hhh]h;)}(hjh]h/3}(hjh jipubah}(h]h]h]h]h]uhh:h!jhM;h jfpubah}(h]h]h]h]h]uhjh jcpubj)}(hhh]h;)}(h--h]h/–}(hjph jpubah}(h]h]h]h]h]uhh:h!jhM;h j|pubah}(h]h]h]h]h]uhjh jcpubj)}(hhh]h;)}(h--h]h/–}(hjph jpubah}(h]h]h]h]h]uhh:h!jhM;h jpubah}(h]h]h]h]h]uhjh jcpubj)}(hhh]h;)}(h--h]h/–}(hjph jpubah}(h]h]h]h]h]uhh:h!jhM;h jpubah}(h]h]h]h]h]uhjh jcpubeh}(h]h]h]h]h]uhjh jnubj)}(hhh](j)}(hhh]h;)}(hjh]h/4}(hjh jpubah}(h]h]h]h]h]uhh:h!jhM=h jpubah}(h]h]h]h]h]uhjh jpubj)}(hhh]h;)}(hZone width searchh]h/Zone width search}(hjph jpubah}(h]h]h]h]h]uhh:h!jhM=h jpubah}(h]h]h]h]h]uhjh jpubj)}(hhh]h;)}(hjh]h/0}(hjh jpubah}(h]h]h]h]h]uhh:h!jhM=h jpubah}(h]h]h]h]h]uhjh jpubj)}(hhh]h;)}(h−0.1h]h/−0.1}(hjqh jqubah}(h]h]h]h]h]uhh:h!jhM=h j
qubah}(h]h]h]h]h]uhjh jpubeh}(h]h]h]h]h]uhjh jnubj)}(hhh](j)}(hhh]h;)}(h5h]h/5}(hj2qh j0qubah}(h]h]h]h]h]uhh:h!jhM?h j-qubah}(h]h]h]h]h]uhjh j*qubj)}(hhh]h;)}(hOuter radius searchh]h/Outer radius search}(hjIqh jGqubah}(h]h]h]h]h]uhh:h!jhM?h jDqubah}(h]h]h]h]h]uhjh j*qubj)}(hhh]h;)}(hStarting outer radiush]h/Starting outer radius}(hj`qh j^qubah}(h]h]h]h]h]uhh:h!jhM?h j[qubah}(h]h]h]h]h]uhjh j*qubj)}(hhh]h;)}(h−0.1∗EVh]h/−0.1∗EV}(hjwqh juqubah}(h]h]h]h]h]uhh:h!jhM?h jrqubah}(h]h]h]h]h]uhjh j*qubeh}(h]h]h]h]h]uhjh jnubj)}(hhh](j)}(hhh]h;)}(h6h]h/6}(hjqh jqubah}(h]h]h]h]h]uhh:h!jhMAh jqubah}(h]h]h]h]h]uhjh jqubj)}(hhh]h;)}(hBuckling searchh]h/Buckling search}(hjqh jqubah}(h]h]h]h]h]uhh:h!jhMAh jqubah}(h]h]h]h]h]uhjh jqubj)}(hhh]h;)}(h1.0h]h/1.0}(hjqh jqubah}(h]h]h]h]h]uhh:h!jhMAh jqubah}(h]h]h]h]h]uhjh jqubj)}(hhh]h;)}(h−0.1h]h/−0.1}(hjqh jqubah}(h]h]h]h]h]uhh:h!jhMAh jqubah}(h]h]h]h]h]uhjh jqubeh}(h]h]h]h]h]uhjh jnubj)}(hhh](j)}(hhh]h;)}(h7h]h/7}(hjqh jqubah}(h]h]h]h]h]uhh:h!jhMCh jqubah}(h]h]h]h]h]uhjh jqubj)}(hhh]h;)}(hDirect buckling searchh]h/Direct buckling search}(hjrh jrubah}(h]h]h]h]h]uhh:h!jhMCh jrubah}(h]h]h]h]h]uhjh jqubj)}(hhh]h;)}(h0.0h]h/0.0}(hj*rh j(rubah}(h]h]h]h]h]uhh:h!jhMCh j%rubah}(h]h]h]h]h]uhjh jqubj)}(hhh]h;)}(h0.0h]h/0.0}(hjArh j?rubah}(h]h]h]h]h]uhh:h!jhMCh j 0. When
IPVT = 1 and IEVT > 1, this is the value of k-effective on which the
search calculation is to be made (0.0)h]h/10. PV – parametric eigenvalue or value for α used when IPVT > 0. When
IPVT = 1 and IEVT > 1, this is the value of k-effective on which the
search calculation is to be made (0.0)}(hjrh jrubah}(h]h]h]h]h]uhh:h!jhMVh jxkubj)}(hhh](j)}(h=EQL – eigenvalue convergence for a search. (10:sup:`−3`)
h]h;)}(h0), this
data block is omitted. Both arrays in this block are double-precision
arrays, which will require the use of the “#” array designator;
otherwise the number of entries read into the arrays will be incorrect
or may contain nonsensical values for the starting guess.h]h/XThis data block contains starting guesses for fluxes and fission
densities. If fluxes are read from an external device (LISF>0), this
data block is omitted. Both arrays in this block are double-precision
arrays, which will require the use of the “#” array designator;
otherwise the number of entries read into the arrays will be incorrect
or may contain nonsensical values for the starting guess.}(hjuh j uhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(h33# Flux Guess [IM*IGM] {IFN=0}h]h/33# Flux Guess [IM*IGM] {IFN=0}}(hjuh juhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(hXA guess for the scalar flux is specified in the order:
((FLUX(I,J),I=1,IM),J=1,IGM), where IM is the number of spatial
intervals and IGM is the total number of energy groups. For fixed-source
problems, without better information, use zeroes. For eigenvalue
problems, a nonzero flux guess must be used. The fluxes punched by using
the ID1 parameter in the 2$ array can be used here in restart
calculations.h]h/XA guess for the scalar flux is specified in the order:
((FLUX(I,J),I=1,IM),J=1,IGM), where IM is the number of spatial
intervals and IGM is the total number of energy groups. For fixed-source
problems, without better information, use zeroes. For eigenvalue
problems, a nonzero flux guess must be used. The fluxes punched by using
the ID1 parameter in the 2$ array can be used here in restart
calculations.}(hj'uh j%uhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(h&34# Fission Density Guess [IM] {IFN=1}h]h/&34# Fission Density Guess [IM] {IFN=1}}(hj5uh j3uhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(hThis is a guess at the number of fission neutrons produced in an
interval. When IFN = 1, XSDRNPM uses diffusion theory for the first
outer iteration, after which it reverts to the normal mode.h]h/This is a guess at the number of fission neutrons produced in an
interval. When IFN = 1, XSDRNPM uses diffusion theory for the first
outer iteration, after which it reverts to the normal mode.}(hjCuh jAuhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(hT Terminate this data block.h]h/T Terminate this data block.}(hjQuh jOuhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubhd)}(hData Block 5h]h/Data Block 5}(hData Block 5h j]uhhh!NhNubah}(h]h]h]h]h]uhhch jF^hhh!jhMubh;)}(hIThis block contains the remaining data needed for an XSDRNPM
calculation.h]h/IThis block contains the remaining data needed for an XSDRNPM
calculation.}(hjnuh jluhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(h$35\* Interval Boundaries [IM+1] (cm)h]h/$35* Interval Boundaries [IM+1] (cm)}(h$35\* Interval Boundaries [IM+1] (cm)h jzuhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(hXThis array describes the spatial quadrature into which the problem model
is divided. The boundaries are nonnegative for curvilinear geometries
and in increasing order. Usually they will start with a zero value,
though this is not necessary. The origin for slab geometry may be
negative.h]h/XThis array describes the spatial quadrature into which the problem model
is divided. The boundaries are nonnegative for curvilinear geometries
and in increasing order. Usually they will start with a zero value,
though this is not necessary. The origin for slab geometry may be
negative.}(hjuh juhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(h.36$ Zone Number for Each Spatial Interval [IM]h]h/.36$ Zone Number for Each Spatial Interval [IM]}(hjuh juhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(h#Spatial zones should be contiguous.h]h/#Spatial zones should be contiguous.}(hjuh juhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(h438\* Density Factors by Interval [IM] {IDFM>0} (1.0)h]h/438* Density Factors by Interval [IM] {IDFM>0} (1.0)}(h438\* Density Factors by Interval [IM] {IDFM>0} (1.0)h juhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(hThese factors are used to effect a density variation in a mixture as a
function of spatial interval. Zero for a density factor affords a
convenient way for modeling a void region.h]h/These factors are used to effect a density variation in a mixture as a
function of spatial interval. Zero for a density factor affords a
convenient way for modeling a void region.}(hjuh juhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(h!39$ Mixture Numbers by Zone [IZM]h]h/!39$ Mixture Numbers by Zone [IZM]}(hjuh juhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(h0The mixture that is in a zone is specified here.h]h/0The mixture that is in a zone is specified here.}(hjuh juhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(h,40$ Order of Scattering by Zone [IZM] (ISCT)h]h/,40$ Order of Scattering by Zone [IZM] (ISCT)}(hjuh juhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(hThis is the order, *l*, of the \*S\ n Pl \*\ calculation which is desired
in a zone. This number should be no larger than ISCT.h](h/This is the order, }(hThis is the order, h juhhh!NhNubhA)}(h*l*h]h/l}(hhh jvubah}(h]h]h]h]h]uhh@h juubh/k, of the *S n Pl * calculation which is desired
in a zone. This number should be no larger than ISCT.}(hk, of the \*S\ n Pl \*\ calculation which is desired
in a zone. This number should be no larger than ISCT.h juhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(h,41\* Radius Modifiers by Zone [IZM] {IEVT=4}h]h/,41* Radius Modifiers by Zone [IZM] {IEVT=4}}(h,41\* Radius Modifiers by Zone [IZM] {IEVT=4}h jvhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(hThese parameters specify the relative movement of the width of a zone in
a zone width search. A zero indicates that a zone’s width is fixed. (See
:ref:`9-1-2-8`)h](h/These parameters specify the relative movement of the width of a zone in
a zone width search. A zero indicates that a zone’s width is fixed. (See
}(hThese parameters specify the relative movement of the width of a zone in
a zone width search. A zero indicates that a zone’s width is fixed. (See
h j+vhhh!NhNubj)}(h:ref:`9-1-2-8`h]j#)}(hj6vh]h/9-1-2-8}(hhh j8vubah}(h]h](jnstdstd-refeh]h]h]uhj"h j4vubah}(h]h]h]h]h]refdocj refdomainjBvreftyperefrefexplicitrefwarnj9-1-2-8uhjh!jhMh j+vubh/)}(hj7@h j+vhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(hH42# Weights of the Angles in the Discrete-Ordinates Quadrature
[MM [2]_]h](h/C42# Weights of the Angles in the Discrete-Ordinates Quadrature
[MM }(hC42# Weights of the Angles in the Discrete-Ordinates Quadrature
[MM h j^vhhh!NhNubh footnote_reference)}(h[2]_h]h/2}(hhh jivubah}(h]id81ah]h]h]h]hid91jjuhjgvh j^vresolvedKubh/]}(h]h j^vhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(h~Input this set if you wish to override those provided by XSDRNPM. See
:ref:`9-1-2-4-3`, :ref:`9-1-2-4-4`, or :ref:`9-1-2-4-5`.h](h/FInput this set if you wish to override those provided by XSDRNPM. See
}(hFInput this set if you wish to override those provided by XSDRNPM. See
h jvhhh!NhNubj)}(h:ref:`9-1-2-4-3`h]j#)}(hjvh]h/ 9-1-2-4-3}(hhh jvubah}(h]h](jnstdstd-refeh]h]h]uhj"h jvubah}(h]h]h]h]h]refdocj refdomainjvreftyperefrefexplicitrefwarnj 9-1-2-4-3uhjh!jhMh jvubh/, }(h, h jvhhh!NhNubj)}(h:ref:`9-1-2-4-4`h]j#)}(hjvh]h/ 9-1-2-4-4}(hhh jvubah}(h]h](jnstdstd-refeh]h]h]uhj"h jvubah}(h]h]h]h]h]refdocj refdomainjvreftyperefrefexplicitrefwarnj 9-1-2-4-4uhjh!jhMh jvubh/, or }(h, or h jvhhh!NhNubj)}(h:ref:`9-1-2-4-5`h]j#)}(hjvh]h/ 9-1-2-4-5}(hhh jvubah}(h]h](jnstdstd-refeh]h]h]uhj"h jvubah}(h]h]h]h]h]refdocj refdomainjvreftyperefrefexplicitrefwarnj 9-1-2-4-5uhjh!jhMh jvubh/.}(hjWh jvhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(hC43# Cosines of the Angles in the Discrete-Ordinates Quadrature [MM]h]h/C43# Cosines of the Angles in the Discrete-Ordinates Quadrature [MM]}(hjwh jwhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(h~Input this set if you wish to override those provided by XSDRNPM. See
:ref:`9-1-2-4-3`, :ref:`9-1-2-4-4`, or :ref:`9-1-2-4-5`.h](h/FInput this set if you wish to override those provided by XSDRNPM. See
}(hFInput this set if you wish to override those provided by XSDRNPM. See
h jwhhh!NhNubj)}(h:ref:`9-1-2-4-3`h]j#)}(hjwh]h/ 9-1-2-4-3}(hhh jwubah}(h]h](jnstdstd-refeh]h]h]uhj"h jwubah}(h]h]h]h]h]refdocj refdomainj'wreftyperefrefexplicitrefwarnj 9-1-2-4-3uhjh!jhMh jwubh/, }(h, h jwhhh!NhNubj)}(h:ref:`9-1-2-4-4`h]j#)}(hj@wh]h/ 9-1-2-4-4}(hhh jBwubah}(h]h](jnstdstd-refeh]h]h]uhj"h j>wubah}(h]h]h]h]h]refdocj refdomainjLwreftyperefrefexplicitrefwarnj 9-1-2-4-4uhjh!jhMh jwubh/, or }(h, or h jwhhh!NhNubj)}(h:ref:`9-1-2-4-5`h]j#)}(hjewh]h/ 9-1-2-4-5}(hhh jgwubah}(h]h](jnstdstd-refeh]h]h]uhj"h jcwubah}(h]h]h]h]h]refdocj refdomainjqwreftyperefrefexplicitrefwarnj 9-1-2-4-5uhjh!jhMh jwubh/.}(hjWh jwhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(h046$ Calculational Option by Group [IGM] {ICLC>0}h]h/046$ Calculational Option by Group [IGM] {ICLC>0}}(hjwh jwhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubj1)}(hhh](j1)}(hhh]h;)}(h:0 - perform discrete-ordinates calculation for this group.h]h/:0 - perform discrete-ordinates calculation for this group.}(hjwh jwubah}(h]h]h]h]h]uhh:h!jhMh jwubah}(h]h]h]h]h]uhj1h jwubh;)}(hw1 - perform a diffusion calculation for this group for ICLC outer
iterations; use discrete-ordinates theory after this.h]h/w1 - perform a diffusion calculation for this group for ICLC outer
iterations; use discrete-ordinates theory after this.}(hjwh jwubah}(h]h]h]h]h]uhh:h!jhMh jwubj1)}(hhh]h;)}(h~2 - perform a homogeneous calculation for this group for ICLC outer
iterations; then revert back to discrete-ordinates theory.h]h/~2 - perform a homogeneous calculation for this group for ICLC outer
iterations; then revert back to discrete-ordinates theory.}(hjwh jwubah}(h]h]h]h]h]uhh:h!jhMh jwubah}(h]h]h]h]h]uhj1h jwubh;)}(hJ3 - perform a homogeneous calculation using *B\ n* theory for this
group.h](h/,3 - perform a homogeneous calculation using }(h,3 - perform a homogeneous calculation using h jwubhA)}(h*B\ n*h]h/B n}(hhh jwubah}(h]h]h]h]h]uhh@h jwubh/ theory for this
group.}(h theory for this
group.h jwubeh}(h]h]h]h]h]uhh:h!jhMh jwubeh}(h]h]h]h]h]uhj1h jF^hhh!jhNubh;)}(h847\* Right-Boundary Albedos by Group [IGM] {IBR=3} (1.0)h]h/847* Right-Boundary Albedos by Group [IGM] {IBR=3} (1.0)}(h847\* Right-Boundary Albedos by Group [IGM] {IBR=3} (1.0)h jxhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(hsA right-boundary albedo is specified for each fine group. The return
current is distributed isotropically in angle.h]h/sA right-boundary albedo is specified for each fine group. The return
current is distributed isotropically in angle.}(hjxh jxhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(h748\* Left-Boundary Albedos by Group [IGM] {IBL=3} (1.0)h]h/748* Left-Boundary Albedos by Group [IGM] {IBL=3} (1.0)}(h748\* Left-Boundary Albedos by Group [IGM] {IBL=3} (1.0)h jxhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM
h jF^hhubh;)}(hAs for the 47\* array but for the left boundary. Note that if IBR or IBL
is 3 and the corresponding 47\* or 48* array is omitted, XSDRNPM fills
the array with 1.0’s effecting a boundary with zero net current and with
isotropic neutron return.h]h/As for the 47* array but for the left boundary. Note that if IBR or IBL
is 3 and the corresponding 47* or 48* array is omitted, XSDRNPM fills
the array with 1.0’s effecting a boundary with zero net current and with
isotropic neutron return.}(hAs for the 47\* array but for the left boundary. Note that if IBR or IBL
is 3 and the corresponding 47\* or 48* array is omitted, XSDRNPM fills
the array with 1.0’s effecting a boundary with zero net current and with
isotropic neutron return.h j.xhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(h049$ Material Number for Activities [IAZ] {IAZ>0}h]h/049$ Material Number for Activities [IAZ] {IAZ>0}}(hj?xh j=xhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(h/50$ Process Number for Activities [IAZ] {IAZ>0}h]h//50$ Process Number for Activities [IAZ] {IAZ>0}}(hjMxh jKxhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubh;)}(hThe 49$ and 50$ arrays provide a means of obtaining the activity
(reaction rate) for any process for which cross sections are available
in the XSDRNPM calculation. A representative activity table entry is
shown below:h]h/The 49$ and 50$ arrays provide a means of obtaining the activity
(reaction rate) for any process for which cross sections are available
in the XSDRNPM calculation. A representative activity table entry is
shown below:}(hj[xh jYxhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jF^hhubjq)}(hhh]j)}(hhh](j)}(hhh]h}(h]h]h]h]h]colwidthKuhjh jjxubj)}(hhh]h}(h]h]h]h]h]colwidthKuhjh jjxubj)}(hhh](j)}(hhh](j)}(hhh]h;)}(hACTIVITY TABLE ENTRYh]h/ACTIVITY TABLE ENTRY}(hjxh jxubah}(h]h]h]h]h]uhh:h!jhMh jxubah}(h]h]h]h]h]uhjh jxubj)}(hhh]h}(h]h]h]h]h]uhjh jxubeh}(h]h]h]h]h]uhjh jxubj)}(hhh](j)}(hhh]h;)}(h49$h]h/49$}(hjxh jxubah}(h]h]h]h]h]uhh:h!jhMh jxubah}(h]h]h]h]h]uhjh jxubj)}(hhh]h;)}(h50$h]h/50$}(hjxh jxubah}(h]h]h]h]h]uhh:h!jhMh jxubah}(h]h]h]h]h]uhjh jxubeh}(h]h]h]h]h]uhjh jxubj)}(hhh](j)}(hhh]h;)}(hMh]h/M}(hjxh jxubah}(h]h]h]h]h]uhh:h!jhMh jxubah}(h]h]h]h]h]uhjh jxubj)}(hhh]h;)}(hNh]h/N}(hjyh jyubah}(h]h]h]h]h]uhh:h!jhMh jxubah}(h]h]h]h]h]uhjh jxubeh}(h]h]h]h]h]uhjh jxubeh}(h]h]h]h]h]uhjh jjxubeh}(h]h]h]h]h]colsKuhjh jgxubah}(h]h]h]h]h]jj9guhjph jF^hhh!jhNubh;)}(h~This entry specifies that the activity N for material M be calculated
for all parts of the system which contain that material.h]h/~This entry specifies that the activity N for material M be calculated
for all parts of the system which contain that material.}(hj0yh j.yhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM"h jF^hhubh;)}(hIf N is < 0, a density of 1.0 is used to calculate activities instead of
densities in the mixing table. Allowable process identifiers are given
in Appendix M4.B.h]h/If N is < 0, a density of 1.0 is used to calculate activities instead of
densities in the mixing table. Allowable process identifiers are given
in Appendix M4.B.}(hj>yh j0}h]h/%51$ Broad-Group Numbers [IGM] {IFG>0}}(hjyh jyhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM2h jF^hhubh;)}(hX=This array contains the broad-group numbers into which the fine groups
are collapsed in a flux-weighting calculation. For example, if the first
five fine groups are to be collapsed to the first broad group, the first
five entries in the 51$ array are 1, etc. A zero value can be used to
ignore (or truncate) a group.h]h/X=This array contains the broad-group numbers into which the fine groups
are collapsed in a flux-weighting calculation. For example, if the first
five fine groups are to be collapsed to the first broad group, the first
five entries in the 51$ array are 1, etc. A zero value can be used to
ignore (or truncate) a group.}(hjyh jyhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM4h jF^hhubh;)}(h%52$ Lower Band Group Numbers [NBANDS]h]h/%52$ Lower Band Group Numbers [NBANDS]}(hjyh jyhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM:h jF^hhubh;)}(hlGroup numbers giving the last group in a flux rebalance band. Overrides
the default set supplied by XSDRNPM.h]h/lGroup numbers giving the last group in a flux rebalance band. Overrides
the default set supplied by XSDRNPM.}(hjyh jyhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM<h jF^hhubh)}(h.. _9-1-3-1:h]h}(h]h]h]h]h]hid82uhh
hMh jF^hhh!jubh$)}(hhh](h))}(h%Abbreviated XSDRNPM input descriptionh]h/%Abbreviated XSDRNPM input description}(hjyh jyhhh!NhNubah}(h]h]h]h]h]uhh(h jyhhh!jhMBubjq)}(hhh]j)}(hhh](j)}(hhh]h}(h]h]h]h]h]colwidthK$uhjh jyubj)}(hhh]h}(h]h]h]h]h]colwidthK#uhjh jyubj)}(hhh](j)}(hhh](j)}(hhh]h;)}(hTitle Card - (18A4)h]h/Title Card - (18A4)}(hjzh jzubah}(h]h]h]h]h]uhh:h!jhMEh jzubah}(h]h]h]h]h]uhjh jzubj)}(hhh]h}(h]h]h]h]h]uhjh jzubeh}(h]h]h]h]h]uhjh j
zubj)}(hhh](j)}(hhh]h;)}(hDATA BLOCK 1h]h/DATA BLOCK 1}(hjAzh j?zubah}(h]h]h]h]h]uhh:h!jhMGh j{h j<{ubah}(h]h]h]h]h]uhh:h!jhM[h j8{ubah}(h]h]h]h]h]uhjh jzubj)}(h
Scratch (18)
h]h;)}(hScratch (18)h]h/Scratch (18)}(hjV{h jT{ubah}(h]h]h]h]h]uhh:h!jhM]h jP{ubah}(h]h]h]h]h]uhjh jzubj)}(h
Scratch (19)
h]h;)}(hScratch (19)h]h/Scratch (19)}(hjn{h jl{ubah}(h]h]h]h]h]uhh:h!jhM_h jh{ubah}(h]h]h]h]h]uhjh jzubj)}(hDirect Access (8)
h]h;)}(hDirect Access (8)h]h/Direct Access (8)}(hj{h j{ubah}(h]h]h]h]h]uhh:h!jhMah j{ubah}(h]h]h]h]h]uhjh jzubj)}(hDirect Access (9)
h]h;)}(hDirect Access (9)h]h/Direct Access (9)}(hj{h j{ubah}(h]h]h]h]h]uhh:h!jhMch j{ubah}(h]h]h]h]h]uhjh jzubj)}(hActivities (75)
h]h;)}(hActivities (75)h]h/Activities (75)}(hj{h j{ubah}(h]h]h]h]h]uhh:h!jhMeh j{ubah}(h]h]h]h]h]uhjh jzubj)}(hBalances (76)
h]h;)}(h
Balances (76)h]h/
Balances (76)}(hj{h j{ubah}(h]h]h]h]h]uhh:h!jhMgh j{ubah}(h]h]h]h]h]uhjh jzubj)}(hInput Dump (73)
h]h;)}(hInput Dump (73)h]h/Input Dump (73)}(hj{h j{ubah}(h]h]h]h]h]uhh:h!jhMih j{ubah}(h]h]h]h]h]uhjh jzubj)}(hSensitivity Data (0)
h]h;)}(hSensitivity Data (0)h]h/Sensitivity Data (0)}(hj{h j{ubah}(h]h]h]h]h]uhh:h!jhMkh j{ubah}(h]h]h]h]h]uhjh jzubj)}(h
Not Used (0)
h]h;)}(hNot Used (0)h]h/Not Used (0)}(hj|h j|ubah}(h]h]h]h]h]uhh:h!jhMmh j|ubah}(h]h]h]h]h]uhjh jzubj)}(hFlux Guess (0)h]h;)}(hj*|h]h/Flux Guess (0)}(hj*|h j,|ubah}(h]h]h]h]h]uhh:h!jhMoh j(|ubah}(h]h]h]h]h]uhjh jzubeh}(h]h]h]h]h]jSjCjUhjVjWuhjh jezubeh}(h]h]h]h]h]uhjh jbzubj)}(hhh](h;)}(h1$ - General Description [15]h]h/1$ - General Description [15]}(hjP|h jN|ubah}(h]h]h]h]h]uhh:h!jhMIh jK|ubj)}(hhh](j)}(hIGE - geometry
h]h;)}(hIGE - geometryh]h/IGE - geometry}(hje|h jc|ubah}(h]h]h]h]h]uhh:h!jhMKh j_|ubah}(h]h]h]h]h]uhjh j\|ubj)}(hIZM - number of zones
h]h;)}(hIZM - number of zonesh]h/IZM - number of zones}(hj}|h j{|ubah}(h]h]h]h]h]uhh:h!jhMMh jw|ubah}(h]h]h]h]h]uhjh j\|ubj)}(hIM - number of intervals
h]h;)}(hIM - number of intervalsh]h/IM - number of intervals}(hj|h j|ubah}(h]h]h]h]h]uhh:h!jhMOh j|ubah}(h]h]h]h]h]uhjh j\|ubj)}(hIBL - left boundary condition
h]h;)}(hIBL - left boundary conditionh]h/IBL - left boundary condition}(hj|h j|ubah}(h]h]h]h]h]uhh:h!jhMQh j|ubah}(h]h]h]h]h]uhjh j\|ubj)}(hIBR - right boundary condition
h]h;)}(hIBR - right boundary conditionh]h/IBR - right boundary condition}(hj|h j|ubah}(h]h]h]h]h]uhh:h!jhMSh j|ubah}(h]h]h]h]h]uhjh j\|ubj)}(hMXX - number of mixtures
h]h;)}(hMXX - number of mixturesh]h/MXX - number of mixtures}(hj|h j|ubah}(h]h]h]h]h]uhh:h!jhMUh j|ubah}(h]h]h]h]h]uhjh j\|ubj)}(hMS - mixing table length
h]h;)}(hMS - mixing table lengthh]h/MS - mixing table length}(hj|h j|ubah}(h]h]h]h]h]uhh:h!jhMWh j|ubah}(h]h]h]h]h]uhjh j\|ubj)}(hISN - angular quadrature
h]h;)}(hISN - angular quadratureh]h/ISN - angular quadrature}(hj
}h j}ubah}(h]h]h]h]h]uhh:h!jhMYh j}ubah}(h]h]h]h]h]uhjh j\|ubj)}(hISCT - order of scattering
h]h;)}(hISCT - order of scatteringh]h/ISCT - order of scattering}(hj%}h j#}ubah}(h]h]h]h]h]uhh:h!jhM[h j}ubah}(h]h]h]h]h]uhjh j\|ubj)}(hIEVT - problem type
h]h;)}(hIEVT - problem typeh]h/IEVT - problem type}(hj=}h j;}ubah}(h]h]h]h]h]uhh:h!jhM]h j7}ubah}(h]h]h]h]h]uhjh j\|ubj)}(hIIM - inner iteration maximum
h]h;)}(hIIM - inner iteration maximumh]h/IIM - inner iteration maximum}(hjU}h jS}ubah}(h]h]h]h]h]uhh:h!jhM_h jO}ubah}(h]h]h]h]h]uhjh j\|ubj)}(hICM - outer iteration maximum
h]h;)}(hICM - outer iteration maximumh]h/ICM - outer iteration maximum}(hjm}h jk}ubah}(h]h]h]h]h]uhh:h!jhMah jg}ubah}(h]h]h]h]h]uhjh j\|ubj)}(hICLC - optional theory
h]h;)}(hICLC - optional theoryh]h/ICLC - optional theory}(hj}h j}ubah}(h]h]h]h]h]uhh:h!jhMch j}ubah}(h]h]h]h]h]uhjh j\|ubj)}(hITH - forward or adjoint
h]h;)}(hITH - forward or adjointh]h/ITH - forward or adjoint}(hj}h j}ubah}(h]h]h]h]h]uhh:h!jhMeh j}ubah}(h]h]h]h]h]uhjh j\|ubj)}(h%IFLU – GPT calculation flag
h]h;)}(hIFLU – GPT calculation flagh]h/IFLU – GPT calculation flag}(hj}h j}ubah}(h]h]h]h]h]uhh:h!jhMgh j}ubah}(h]h]h]h]h]uhjh j\|ubeh}(h]h]h]h]h]jSjCjUhjVjWuhjh jK|ubeh}(h]h]h]h]h]uhjh jbzubeh}(h]h]h]h]h]uhjh j
zubeh}(h]h]h]h]h]uhjh jyubeh}(h]h]h]h]h]colsKuhjh jyubah}(h]h]h]h]h]jj9guhjph jyhhh!NhNubjq)}(hhh]j)}(hhh](j)}(hhh]h}(h]h]h]h]h]colwidthK(u!hjh j}ubj)}(hhh]h}(h]h]h]h]h]colwidthK#uhjh j}ubj)}(hhh](j)}(hhh](j)}(hhh]h;)}(hDATA BLOCK 1 (continued)h]h/DATA BLOCK 1 (continued)}(hj~h j~ubah}(h]h]h]h]h]uhh:h!jhMsh j~ubah}(h]h]h]h]h]uhjh j ~ubj)}(hhh]h}(h]h]h]h]h]uhjh j ~ubeh}(h]h]h]h]h]uhjh j~ubj)}(hhh](j)}(hhh](h;)}(h%2$ - Editing and Control Options [10]h]h/%2$ - Editing and Control Options [10]}(hj:~h j8~ubah}(h]h]h]h]h]uhh:h!jhMuh j5~ubj)}(hhh](j)}(hFine-Group-Mixture Edit
h]h;)}(hFine-Group-Mixture Edith]h/Fine-Group-Mixture Edit}(hjO~h jM~ubah}(h]h]h]h]h]uhh:h!jhMwh jI~ubah}(h]h]h]h]h]uhjh jF~ubj)}(hFine-Group-Flux Edit
h]h;)}(hFine-Group-Flux Edith]h/Fine-Group-Flux Edit}(hjg~h je~ubah}(h]h]h]h]h]uhh:h!jhMyh ja~ubah}(h]h]h]h]h]uhjh jF~ubj)}(hBalance Table Edit
h]h;)}(hBalance Table Edith]h/Balance Table Edit}(hj~h j}~ubah}(h]h]h]h]h]uhh:h!jhM{h jy~ubah}(h]h]h]h]h]uhjh jF~ubj)}(hBroad-Group-Flux Edit
h]h;)}(hBroad-Group-Flux Edith]h/Broad-Group-Flux Edit}(hj~h j~ubah}(h]h]h]h]h]uhh:h!jhM}h j~ubah}(h]h]h]h]h]uhjh jF~ubj)}(h Not Used
h]h;)}(hNot Usedh]h/Not Used}(hj~h j~ubah}(h]h]h]h]h]uhh:h!jhMh j~ubah}(h]h]h]h]h]uhjh jF~ubj)}(hOuters-Group-Limit-Option
h]h;)}(hOuters-Group-Limit-Optionh]h/Outers-Group-Limit-Option}(hj~h j~ubah}(h]h]h]h]h]uhh:h!jhMh j~ubah}(h]h]h]h]h]uhjh jF~ubj)}(hNumber of Bands
h]h;)}(hNumber of Bandsh]h/Number of Bands}(hj~h j~ubah}(h]h]h]h]h]uhh:h!jhMh j~ubah}(h]h]h]h]h]uhjh jF~ubj)}(hSuppress Fixed-Source Fission
h]h;)}(hSuppress Fixed-Source Fissionh]h/Suppress Fixed-Source Fission}(hj~h j~ubah}(h]h]h]h]h]uhh:h!jhMh j~ubah}(h]h]h]h]h]uhjh jF~ubj)}(h Not Used
h]h;)}(hNot Usedh]h/Not Used}(hjh j
ubah}(h]h]h]h]h]uhh:h!jhMh j ubah}(h]h]h]h]h]uhjh jF~ubj)}(hNot Used
h]h;)}(hNot Usedh]h/Not Used}(hj'h j%ubah}(h]h]h]h]h]uhh:h!jhMh j!ubah}(h]h]h]h]h]uhjh jF~ubeh}(h]h]h]h]h]jSjCjUhjVjWuhjh j5~ubeh}(h]h]h]h]h]uhjh j2~ubj)}(hhh](h;)}(h3$ - Other Options [12]h]h/3$ - Other Options [12]}(hjJh jHubah}(h]h]h]h]h]uhh:h!jhMuh jEubj)}(hhh](j)}(hIFG - Weighting Option
h]h;)}(hIFG - Weighting Optionh]h/IFG - Weighting Option}(hj_h j]ubah}(h]h]h]h]h]uhh:h!jhMwh jYubah}(h]h]h]h]h]uhjh jVubj)}(hIQM - Volumetric Sources
h]h;)}(hIQM - Volumetric Sourcesh]h/IQM - Volumetric Sources}(hjwh juubah}(h]h]h]h]h]uhh:h!jhMyh jqubah}(h]h]h]h]h]uhjh jVubj)}(hIPM - Boundary Sources
h]h;)}(hIPM - Boundary Sourcesh]h/IPM - Boundary Sources}(hjh jubah}(h]h]h]h]h]uhh:h!jhM{h jubah}(h]h]h]h]h]uhjh jVubj)}(hIFN - Starting Guess
h]h;)}(hIFN - Starting Guessh]h/IFN - Starting Guess}(hjh jubah}(h]h]h]h]h]uhh:h!jhM}h jubah}(h]h]h]h]h]uhjh jVubj)}(hITMX - Time Shut-off
h]h;)}(hITMX - Time Shut-offh]h/ITMX - Time Shut-off}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jVubj)}(hIDAT1 - Storage Scheme
h]h;)}(hIDAT1 - Storage Schemeh]h/IDAT1 - Storage Scheme}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jVubj)}(hIPN - Diff. Coeff. Option
h]h;)}(hIPN - Diff. Coeff. Optionh]h/IPN - Diff. Coeff. Option}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jVubj)}(hIDFM - Density Factors
h]h;)}(hIDFM - Density Factorsh]h/IDFM - Density Factors}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jVubj)}(hIAZ - Activities by Zone
h]h;)}(hIAZ - Activities by Zoneh]h/IAZ - Activities by Zone}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jVubj)}(hIAI - Activities by Interval
h]h;)}(hIAI - Activities by Intervalh]h/IAI - Activities by Interval}(hj7h j5ubah}(h]h]h]h]h]uhh:h!jhMh j1ubah}(h]h]h]h]h]uhjh jVubj)}(hIFCT - Thermal Scaling
h]h;)}(hIFCT - Thermal Scalingh]h/IFCT - Thermal Scaling}(hjOh jMubah}(h]h]h]h]h]uhh:h!jhMh jIubah}(h]h]h]h]h]uhjh jVubj)}(hIPVT - Search on k≠1h]h;)}(hjch]h/IPVT - Search on k≠1}(hjch jeubah}(h]h]h]h]h]uhh:h!jhMh jaubah}(h]h]h]h]h]uhjh jVubeh}(h]h]h]h]h]jSjCjUhjVjWuhjh jEubeh}(h]h]h]h]h]uhjh j2~ubeh}(h]h]h]h]h]uhjh j~ubj)}(hhh](j)}(hhh](h;)}(h4$ - Weighting Options [9]h]h/4$ - Weighting Options [9]}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubj)}(hhh](j)}(hType of weighting
h]h;)}(hType of weightingh]h/Type of weighting}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(hNumber of broad groups
h]h;)}(hNumber of broad groupsh]h/Number of broad groups}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(hOutput format
h]h;)}(h
Output formath]h/
Output format}(hj׀h jՀubah}(h]h]h]h]h]uhh:h!jhMh jрubah}(h]h]h]h]h]uhjh jubj)}(hEdit option
h]h;)}(hEdit optionh]h/Edit option}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(h%σ\ :sub:`T` position or number CCCC
h]h;)}(h$σ\ :sub:`T` position or number CCCCh](h/σ }(hσ\ h jubh)}(h:sub:`T`h]h/T}(hhh jubah}(h]h]h]h]h]uhhh jubh/ position or number CCCC}(h position or number CCCCh jubeh}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(hσ\ :sub:`gg` position
h]h;)}(hσ\ :sub:`gg` positionh](h/σ }(hσ\ h j1ubh)}(h :sub:`gg`h]h/gg}(hhh j:ubah}(h]h]h]h]h]uhhh j1ubh/ position}(h positionh j1ubeh}(h]h]h]h]h]uhh:h!jhMh j-ubah}(h]h]h]h]h]uhjh jubj)}(h
Table length
h]h;)}(hTable lengthh]h/Table length}(hj_h j]ubah}(h]h]h]h]h]uhh:h!jhMh jYubah}(h]h]h]h]h]uhjh jubj)}(hANISN edit option
h]h;)}(hANISN edit optionh]h/ANISN edit option}(hjwh juubah}(h]h]h]h]h]uhh:h!jhMh jqubah}(h]h]h]h]h]uhjh jubj)}(hExtra cross sections
h]h;)}(hExtra cross sectionsh]h/Extra cross sections}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]jSjCjUhjVjWuhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](h;)}(h 5\* - Floating Point Values [12]h]h/ 5* - Floating Point Values [12]}(h 5\* - Floating Point Values [12]h jubah}(h]h]h]h]h]uhh:h!jhMh jubj)}(hhh](j)}(hEPS - overall convergence
h]h;)}(hEPS - overall convergenceh]h/EPS - overall convergence}(hjȁh jƁubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(hPTC - Point flux convergence
h]h;)}(hPTC - Point flux convergenceh]h/PTC - Point flux convergence}(hjh jށubah}(h]h]h]h]h]uhh:h!jhMh jځubah}(h]h]h]h]h]uhjh jubj)}(hXNF - normalization
h]h;)}(hXNF - normalizationh]h/XNF - normalization}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(hEV - starting guess for search
h]h;)}(hEV - starting guess for searchh]h/EV - starting guess for search}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh j
ubah}(h]h]h]h]h]uhjh jubj)}(hEVM - modifier for search
h]h;)}(hEVM - modifier for searchh]h/EVM - modifier for search}(hj(h j&ubah}(h]h]h]h]h]uhh:h!jhMh j"ubah}(h]h]h]h]h]uhjh jubj)}(hBF - buckling factor
h]h;)}(hBF - buckling factorh]h/BF - buckling factor}(hj@h j>ubah}(h]h]h]h]h]uhh:h!jhMh j:ubah}(h]h]h]h]h]uhjh jubj)}(hDY - height
h]h;)}(hDY - heighth]h/DY - height}(hjXh jVubah}(h]h]h]h]h]uhh:h!jhMh jRubah}(h]h]h]h]h]uhjh jubj)}(hDZ - width
h]h;)}(h
DZ - widthh]h/
DZ - width}(hjph jnubah}(h]h]h]h]h]uhh:h!jhMh jjubah}(h]h]h]h]h]uhjh jubj)}(hVSC - void streaming height
h]h;)}(hVSC - void streaming heighth]h/VSC - void streaming height}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(hPV - k for search
h]h;)}(hPV - k for searchh]h/PV - k for search}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(hEQL - search convergence
h]h;)}(hEQL - search convergenceh]h/EQL - search convergence}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(hXNPM - search modifier
h]h;)}(hXNPM - search modifierh]h/XNPM - search modifier}(hjЂh jubah}(h]h]h]h]h]uhh:h!jhMh jʂubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]jSjCjUhjVjWuhjh jubh;)}(hT Terminate Data Block 1h]h/T Terminate Data Block 1}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubeh}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh j~ubeh}(h]h]h]h]h]uhjh j}ubeh}(h]h]h]h]h]colsKuhjh j}ubah}(h]h]h]h]h]jj9guhjph jyhhh!jhNubjq)}(hhh]j)}(hhh](j)}(hhh]h}(h]h]h]h]h]colwidthKGuhjh jubj)}(hhh](j)}(hhh]j)}(hhh]h;)}(hTitle Card – (18A4)h]h/Title Card – (18A4)}(hj0h j.ubah}(h]h]h]h]h]uhh:h!jhMh j+ubah}(h]h]h]h]h]uhjh j(ubah}(h]h]h]h]h]uhjh j%ubj)}(hhh]j)}(hhh]h;)}(hDATA BLOCK 2h]h/DATA BLOCK 2}(hjPh jNubah}(h]h]h]h]h]uhh:h!jhMh jKubah}(h]h]h]h]h]uhjh jHubah}(h]h]h]h]h]uhjh j%ubj)}(hhh]j)}(hhh](h;)}(h110$ CCCC Transport Cross Section Selector [IHTF]h]h/110$ CCCC Transport Cross Section Selector [IHTF]}(hjph jnubah}(h]h]h]h]h]uhh:h!jhMh jkubh;)}(h11$ Composition numbersh]h/11$ Composition numbers}(hj~h j|ubah}(h]h]h]h]h]uhh:h!jhMh jkubh;)}(h:12$ Additional Processes to be put on ANISN Library [MSCM]h]h/:12$ Additional Processes to be put on ANISN Library [MSCM]}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jkubh;)}(h13$ Mixture Numbers [MS]h]h/13$ Mixture Numbers [MS]}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jkubh;)}(h14$ Isotope Identifiers [MS]h]h/14$ Isotope Identifiers [MS]}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jkubh;)}(h 15\* Isotope Concentrations [MS]h]h/ 15* Isotope Concentrations [MS]}(h 15\* Isotope Concentrations [MS]h jubah}(h]h]h]h]h]uhh:h!jhMh jkubh;)}(h016$ CCCC Identifiers from Working Library [IHTF]h]h/016$ CCCC Identifiers from Working Library [IHTF]}(hjŃh jÃubah}(h]h]h]h]h]uhh:h!jhMh jkubh;)}(h,18U or 18# CCCC Identifiers on ISOTXS [IHTF]h]h/,18U or 18# CCCC Identifiers on ISOTXS [IHTF]}(hjӃh jуubah}(h]h]h]h]h]uhh:h!jhMh jkubh;)}(h"T Terminate the second Data Block.h]h/"T Terminate the second Data Block.}(hjh j߃ubah}(h]h]h]h]h]uhh:h!jhMh jkubeh}(h]h]h]h]h]uhjh jhubah}(h]h]h]h]h]uhjh j%ubj)}(hhh]j)}(hhh]h;)}(hDATA BLOCK 3h]h/DATA BLOCK 3}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubah}(h]h]h]h]h]uhjh j%ubj)}(hhh]j)}(hhh](h;)}(h+{Required only when IQM or IPM is nonzero.}h]h/+{Required only when IQM or IPM is nonzero.}}(hj!h jubah}(h]h]h]h]h]uhh:h!jhMh jubh;)}(h*30$ Spectrum Number by Interval (IQM>0) orh]h/*30$ Spectrum Number by Interval (IQM>0) or}(hj/h j-ubah}(h]h]h]h]h]uhh:h!jhMh jubh;)}(h)Right-Hand Interval Boundary (IPM>0) [IM]h]h/)Right-Hand Interval Boundary (IPM>0) [IM]}(hj=h j;ubah}(h]h]h]h]h]uhh:h!jhMh jubh;)}(h!31\* Volumetric Sources [IQM*IGM]h]h/!31* Volumetric Sources [IQM*IGM]}(h!31\* Volumetric Sources [IQM*IGM]h jIubah}(h]h]h]h]h]uhh:h!jhMh jubh;)}(h"32\* Boundary Sources [IPM*IGM*MM]h]h/"32* Boundary Sources [IPM*IGM*MM]}(h"32\* Boundary Sources [IPM*IGM*MM]h jXubah}(h]h]h]h]h]uhh:h!jhMh jubh;)}(h!T Terminate the third Data Block.h]h/!T Terminate the third Data Block.}(hjih jgubah}(h]h]h]h]h]uhh:h!jhMh jubeh}(h]h]h]h]h]uhjh jubah}(h]h]h]h]h]uhjh j%ubj)}(hhh]j)}(hhh]h;)}(hDATA BLOCK 4h]h/DATA BLOCK 4}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubah}(h]h]h]h]h]uhjh j%ubj)}(hhh]j)}(hhh](h;)}(hK{When fluxes are read from an external device-IFN>3-this block is
omitted.}h]h/K{When fluxes are read from an external device-IFN>3-this block is
omitted.}}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubh;)}(h33# Flux Guess [IM*IGM] {IFN=0}h]h/33# Flux Guess [IM*IGM] {IFN=0}}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubh;)}(h&34# Fission Density Guess [IM] {IFN=1}h]h/&34# Fission Density Guess [IM] {IFN=1}}(hjńh jÄubah}(h]h]h]h]h]uhh:h!jhMh jubh;)}(h"T Terminate the fourth Data Block.h]h/"T Terminate the fourth Data Block.}(hjӄh jфubah}(h]h]h]h]h]uhh:h!jhMh jubeh}(h]h]h]h]h]uhjh jubah}(h]h]h]h]h]uhjh j%ubeh}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]colsKuhjh jubah}(h]h]h]h]h]jj9guhjph jyhhh!jhNubjq)}(hhh]j)}(hhh](j)}(hhh]h}(h]h]h]h]h]colwidthK5uhjh jubj)}(hhh](j)}(hhh]j)}(hhh]h;)}(hTitle Card - (18A4)h]h/Title Card - (18A4)}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubah}(h]h]h]h]h]uhjh jubj)}(hhh]j)}(hhh]h;)}(hDATA BLOCK 5h]h/DATA BLOCK 5}(hj9h j7ubah}(h]h]h]h]h]uhh:h!jhMh j4ubah}(h]h]h]h]h]uhjh j1ubah}(h]h]h]h]h]uhjh jubj)}(hhh]j)}(hhh](h;)}(h!35\* Interval Boundaries [IM + 1]h]h/!35* Interval Boundaries [IM + 1]}(h!35\* Interval Boundaries [IM + 1]h jWubah}(h]h]h]h]h]uhh:h!jhMh jTubh;)}(h!36$ Zone Numbers by Interval [IM]h]h/!36$ Zone Numbers by Interval [IM]}(hjhh jfubah}(h]h]h]h]h]uhh:h!jhMh jTubh;)}(h38\* Density Factors [IM]h]h/38* Density Factors [IM]}(h38\* Density Factors [IM]h jtubah}(h]h]h]h]h]uhh:h!jhMh jTubh;)}(h 39$ Mixture Number by Zone [IZM]h]h/ 39$ Mixture Number by Zone [IZM]}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jTubh;)}(h%40$ Order of Scattering by Zone [IZM]h]h/%40$ Order of Scattering by Zone [IZM]}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jTubh;)}(h+41\* Radius Modifier by Zone [IZM] {IEVT=4}h]h/+41* Radius Modifier by Zone [IZM] {IEVT=4}}(h+41\* Radius Modifier by Zone [IZM] {IEVT=4}h jubah}(h]h]h]h]h]uhh:h!jhMh jTubh;)}(h#42# Discrete-Ordinates Cosines [MM]h]h/#42# Discrete-Ordinates Cosines [MM]}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jTubh;)}(h#43# Discrete-Ordinates Weights [MM]h]h/#43# Discrete-Ordinates Weights [MM]}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jTubh;)}(h/46$ Alternative Theory Selection [IGM] {ICLC>0}h]h//46$ Alternative Theory Selection [IGM] {ICLC>0}}(hj̅h jʅubah}(h]h]h]h]h]uhh:h!jhMh jTubh;)}(h!47\* Right-Boundary Albedos [IGM]h]h/!47* Right-Boundary Albedos [IGM]}(h!47\* Right-Boundary Albedos [IGM]h jubah}(h]h]h]h]h]uhh:h!jhMh jTubh;)}(h 48\* Left-Boundary Albedos [IGM]h]h/ 48* Left-Boundary Albedos [IGM]}(h 48\* Left-Boundary Albedos [IGM]h jubah}(h]h]h]h]h]uhh:h!jhMh jTubh;)}(h.49$ Activity Material or Nuclide Numbers [IAZ]h]h/.49$ Activity Material or Nuclide Numbers [IAZ]}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jTubh;)}(h350$ Activity Process Numbers [IAZ] (See Appendix B)h]h/350$ Activity Process Numbers [IAZ] (See Appendix B)}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jTubh;)}(h%51$ Broad-Group Numbers [IGM] {IFG>0}h]h/%51$ Broad-Group Numbers [IGM] {IFG>0}}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jTubh;)}(h%52$ Lower-Band-Group Numbers [NBANDS]h]h/%52$ Lower-Band-Group Numbers [NBANDS]}(hj"h j ubah}(h]h]h]h]h]uhh:h!jhMh jTubh;)}(h!T Terminate the fifth Data Block.h]h/!T Terminate the fifth Data Block.}(hj0h j.ubah}(h]h]h]h]h]uhh:h!jhM
h jTubeh}(h]h]h]h]h]uhjh jQubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]colsKuhjh jubah}(h]h]h]h]h]jj9guhjph jyhhh!jhNubh)}(h
.. _9-1-4:h]h}(h]h]h]h]h]hid83uhh
hM~
h jyhhh!jubeh}(h](%abbreviated-xsdrnpm-input-descriptionjyeh]h](%abbreviated xsdrnpm input description9-1-3-1eh]h]uhh#h jF^hhh!jhMBjf}jljysjh}jyjysubeh}(h](xsdrnpm-input-dataj/^eh]h](xsdrnpm input data9-1-3eh]h]uhh#h jhhh!jhM jf}jwj%^sjh}j/^j%^subh$)}(hhh](h))}(h XSDRNPM Input/Output Assignmentsh]h/ XSDRNPM Input/Output Assignments}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h j|hhh!jhM
ubh;)}(hFThe following logical units can be required in an XSDRNPM calculation.h]h/FThe following logical units can be required in an XSDRNPM calculation.}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM
h j|hhubjq)}(hhh]j)}(hhh](j)}(hhh]h}(h]h]h]h]h]colwidthK#uhjh jubj)}(hhh]h}(h]h]h]h]h]colwidthK#uhjh jubjh)}(hhh]j)}(hhh](j)}(hhh](h;)}(hDefaulth]h/Default}(hjh jubah}(h]h]h]h]h]uhh:h!jhM
h jubh;)}(hLogical Unith]h/Logical Unit}(hjΆh j̆ubah}(h]h]h]h]h]uhh:h!jhM
h jubeh}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(hPurposeh]h/Purpose}(hjh jubah}(h]h]h]h]h]uhh:h!jhM
h jubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubah}(h]h]h]h]h]uhjgh jubj)}(hhh](j)}(hhh](j)}(hhh]h;)}(hjh]h/3}(hjh jubah}(h]h]h]h]h]uhh:h!jhM
h j ubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(h&Weighted Library (Produced by
XSDRNPM)h]h/&Weighted Library (Produced by
XSDRNPM)}(hj$h j"ubah}(h]h]h]h]h]uhh:h!jhM
h jubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(hjh]h/4}(hjh jBubah}(h]h]h]h]h]uhh:h!jhM
h j?ubah}(h]h]h]h]h]uhjh j<ubj)}(hhh]h;)}(hWorking Library (Input)h]h/Working Library (Input)}(hjZh jXubah}(h]h]h]h]h]uhh:h!jhM
h jUubah}(h]h]h]h]h]uhjh j<ubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(hj2qh]h/5}(hj2qh jxubah}(h]h]h]h]h]uhh:h!jhM
h juubah}(h]h]h]h]h]uhjh jrubj)}(hhh]h;)}(h
Card Inputh]h/
Card Input}(hjh jubah}(h]h]h]h]h]uhh:h!jhM
h jubah}(h]h]h]h]h]uhjh jrubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(hjqh]h/6}(hjqh jubah}(h]h]h]h]h]uhh:h!jhM
h jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(hStandard Outputh]h/Standard Output}(hjƇh jćubah}(h]h]h]h]h]uhh:h!jhM
h jubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(hjqh]h/7}(hjqh jubah}(h]h]h]h]h]uhh:h!jhM
h jubah}(h]h]h]h]h]uhjh jއubj)}(hhh]h;)}(hPunch Fluxes or ANISN Librariesh]h/Punch Fluxes or ANISN Libraries}(hjh jubah}(h]h]h]h]h]uhh:h!jhM
h jubah}(h]h]h]h]h]uhjh jއubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(h8h]h/8}(hjh jubah}(h]h]h]h]h]uhh:h!jhM
h jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(h?Scratch Direct-Access Device for
External Cross-Section Storageh]h/?Scratch Direct-Access Device for
External Cross-Section Storage}(hj3h j1ubah}(h]h]h]h]h]uhh:h!jhM
h j.ubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(h9h]h/9}(hjSh jQubah}(h]h]h]h]h]uhh:h!jhM
h jNubah}(h]h]h]h]h]uhjh jKubj)}(hhh]h;)}(h@Scratch Direct-Access Device for
Mixing and Weighting Operationsh]h/@Scratch Direct-Access Device for
Mixing and Weighting Operations}(hjjh jhubah}(h]h]h]h]h]uhh:h!jhM
h jeubah}(h]h]h]h]h]uhjh jKubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(h10h]h/10}(hjh jubah}(h]h]h]h]h]uhh:h!jhM#
h jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(h=Scratch Direct-Access Device for
External Flux Moment Storageh]h/=Scratch Direct-Access Device for
External Flux Moment Storage}(hjh jubah}(h]h]h]h]h]uhh:h!jhM#
h jubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(h16h]h/16}(hjh jubah}(h]h]h]h]h]uhh:h!jhM&
h jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(hAngular Fluxesh]h/Angular Fluxes}(hj؈h jֈubah}(h]h]h]h]h]uhh:h!jhM&
h jӈubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(h17h]h/17}(hjh jubah}(h]h]h]h]h]uhh:h!jhM(
h jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(h
Scalar Fluxesh]h/
Scalar Fluxes}(hjh j
ubah}(h]h]h]h]h]uhh:h!jhM(
h j
ubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(h18h]h/18}(hj/h j-ubah}(h]h]h]h]h]uhh:h!jhM*
h j*ubah}(h]h]h]h]h]uhjh j'ubj)}(hhh]h;)}(hScratch Deviceh]h/Scratch Device}(hjFh jDubah}(h]h]h]h]h]uhh:h!jhM*
h jAubah}(h]h]h]h]h]uhjh j'ubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(h19h]h/19}(hjfh jdubah}(h]h]h]h]h]uhh:h!jhM,
h jaubah}(h]h]h]h]h]uhjh j^ubj)}(hhh]h;)}(hScratch Deviceh]h/Scratch Device}(hj}h j{ubah}(h]h]h]h]h]uhh:h!jhM,
h jxubah}(h]h]h]h]h]uhjh j^ubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(h20h]h/20}(hjh jubah}(h]h]h]h]h]uhh:h!jhM.
h jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(h/ANISN Binary Libraries or CCCC
ISOTXS Interfaceh]h//ANISN Binary Libraries or CCCC
ISOTXS Interface}(hjh jubah}(h]h]h]h]h]uhh:h!jhM.
h jubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(h31h]h/31}(hjԉh j҉ubah}(h]h]h]h]h]uhh:h!jhM1
h jωubah}(h]h]h]h]h]uhjh j̉ubj)}(hhh]h;)}(h]Fundamental mode forward angular
flux input unit for generalized
adjoint calculation (iflu>0)h]h/]Fundamental mode forward angular
flux input unit for generalized
adjoint calculation (iflu>0)}(hjh jubah}(h]h]h]h]h]uhh:h!jhM1
h jubah}(h]h]h]h]h]uhjh j̉ubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(h32h]h/32}(hjh j ubah}(h]h]h]h]h]uhh:h!jhM5
h jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(h]Fundamental mode adjoint angular
flux input unit for generalized
adjoint calculation (iflu>0)h]h/]Fundamental mode adjoint angular
flux input unit for generalized
adjoint calculation (iflu>0)}(hj"h j ubah}(h]h]h]h]h]uhh:h!jhM5
h jubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(h73h]h/73}(hjBh j@ubah}(h]h]h]h]h]uhh:h!jhM9
h j=ubah}(h]h]h]h]h]uhjh j:ubj)}(hhh]h;)}(hDump of Input and Derived Datah]h/Dump of Input and Derived Data}(hjYh jWubah}(h]h]h]h]h]uhh:h!jhM9
h jTubah}(h]h]h]h]h]uhjh j:ubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(h75h]h/75}(hjyh jwubah}(h]h]h]h]h]uhh:h!jhM;
h jtubah}(h]h]h]h]h]uhjh jqubj)}(hhh]h;)}(h
Activitiesh]h/
Activities}(hjh jubah}(h]h]h]h]h]uhh:h!jhM;
h jubah}(h]h]h]h]h]uhjh jqubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(h76h]h/76}(hjh jubah}(h]h]h]h]h]uhh:h!jhM=
h jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(hBalance Tablesh]h/Balance Tables}(hjǊh jŊubah}(h]h]h]h]h]uhh:h!jhM=
h jubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]colsKuhjh jubah}(h]h] longtableah]h]h]jj9guhjph j|hhh!jhNubh)}(h
.. _9-1-5:h]h}(h]h]h]h]h]hid84uhh
hM
h j|hhh!jubeh}(h]( xsdrnpm-input-output-assignmentsjeeh]h]( xsdrnpm input/output assignments9-1-4eh]h]uhh#h jhhh!jhM
jf}jj[sjh}jej[subh$)}(hhh](h))}(hXSDRN Sample Problemh]h/XSDRN Sample Problem}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h j hhh!jhMC
ubh;)}(hIn this section, the input and output for a sample case involving a
bare, homogeneous 16-cm sphere of a 93% enriched *UO\ 2-F2* solution is
presented. The input AMPX working format cross-section library will be
read from logical unit 4.h](h/uIn this section, the input and output for a sample case involving a
bare, homogeneous 16-cm sphere of a 93% enriched }(huIn this section, the input and output for a sample case involving a
bare, homogeneous 16-cm sphere of a 93% enriched h jhhh!NhNubhA)}(h
*UO\ 2-F2*h]h/UO 2-F2}(hhh j#ubah}(h]h]h]h]h]uhh@h jubh/n solution is
presented. The input AMPX working format cross-section library will be
read from logical unit 4.}(hn solution is
presented. The input AMPX working format cross-section library will be
read from logical unit 4.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhME
h j hhubh;)}(hpThe input working library is a temporary 238-group library created by
CSAS-mg containing the following nuclides:h]h/pThe input working library is a temporary 238-group library created by
CSAS-mg containing the following nuclides:}(hj>h j<hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMJ
h j hhubjq)}(hhh]j)}(hhh](j)}(hhh]h}(h]h]h]h]h]colwidthK#uhjh jMubj)}(hhh]h}(h]h]h]h]h]colwidthK#uhjh jMubjh)}(hhh]j)}(hhh](j)}(hhh]h;)}(hNuclideh]h/Nuclide}(hjoh jmubah}(h]h]h]h]h]uhh:h!jhMN
h jjubah}(h]h]h]h]h]uhjh jgubj)}(hhh]h;)}(hIdentifier\ :sup:`a`h](h/Identifier }(hIdentifier\ h jubj;)}(h:sup:`a`h]h/a}(hhh jubah}(h]h]h]h]h]uhj;h jubeh}(h]h]h]h]h]uhh:h!jhMN
h jubah}(h]h]h]h]h]uhjh jgubeh}(h]h]h]h]h]uhjh jdubah}(h]h]h]h]h]uhjgh jMubj)}(hhh](j)}(hhh](j)}(hhh]h;)}(h:sup:`235`\ U\ :sub:`92`h](j;)}(h
:sup:`235`h]h/235}(hhh jubah}(h]h]h]h]h]uhj;h jubh/ U }(h\ U\ h jubh)}(h :sub:`92`h]h/92}(hhh jӋubah}(h]h]h]h]h]uhhh jubeh}(h]h]h]h]h]uhh:h!jhMP
h jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(h92235h]h/92235}(hjh jubah}(h]h]h]h]h]uhh:h!jhMP
h jubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(h:sup:`238`\ U\ :sub:`92`h](j;)}(h
:sup:`238`h]h/238}(hhh jubah}(h]h]h]h]h]uhj;h jubh/ U }(h\ U\ h jubh)}(h :sub:`92`h]h/92}(hhh j'ubah}(h]h]h]h]h]uhhh jubeh}(h]h]h]h]h]uhh:h!jhMR
h j
ubah}(h]h]h]h]h]uhjh j
ubj)}(hhh]h;)}(h92238h]h/92238}(hjFh jDubah}(h]h]h]h]h]uhh:h!jhMR
h jAubah}(h]h]h]h]h]uhjh j
ubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(h:sup:`1`\ Hh](j;)}(h:sup:`1`h]h/1}(hhh jhubah}(h]h]h]h]h]uhj;h jdubh/ H}(h\ Hh jdubeh}(h]h]h]h]h]uhh:h!jhMT
h jaubah}(h]h]h]h]h]uhjh j^ubj)}(hhh]h;)}(h
1001\ :sup:`bh](h/1001 :sup:}(h1001\ :sup:h jubj)}(h`h]h/`}(hhh jubah}(h]id86ah]h]h]h]refidid85uhjh jubh/b}(hbh jubeh}(h]h]h]h]h]uhh:h!jhMT
h jubah}(h]h]h]h]h]uhjh j^ubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(h:sup:`16`\ O\ :sub:`8`h](j;)}(h :sup:`16`h]h/16}(hhh jŌubah}(h]h]h]h]h]uhj;h jubh/ O }(h\ O\ h jubh)}(h:sub:`8`h]h/8}(hhh j،ubah}(h]h]h]h]h]uhhh jubeh}(h]h]h]h]h]uhh:h!jhMV
h jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(h8016h]h/8016}(hjh jubah}(h]h]h]h]h]uhh:h!jhMV
h jubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh]h;)}(h:sup:`19`\ F\ :sub:`9`h](j;)}(h :sup:`19`h]h/19}(hhh jubah}(h]h]h]h]h]uhj;h jubh/ F }(h\ F\ h jubh)}(h:sub:`9`h]h/9}(hhh j,ubah}(h]h]h]h]h]uhhh jubeh}(h]h]h]h]h]uhh:h!jhMX
h jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(h9019h]h/9019}(hjKh jIubah}(h]h]h]h]h]uhh:h!jhMX
h jFubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh](h;)}(hY*a* These are the identifiers
of the sets of data on the
library created for the
problem.h](hA)}(h*a*h]h/a}(hhh jmubah}(h]h]h]h]h]uhh@h jiubh/V These are the identifiers
of the sets of data on the
library created for the
problem.}(hV These are the identifiers
of the sets of data on the
library created for the
problem.h jiubeh}(h]h]h]h]h]uhh:h!jhMZ
h jfubh;)}(h*b* Water-bound kernel.h](hA)}(h*b*h]h/b}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ Water-bound kernel.}(h Water-bound kernel.h jubeh}(h]h]h]h]h]uhh:h!jhM_
h jfubeh}(h]h]h]h]h]uhjh jcubj)}(hhh]h}(h]h]h]h]h]uhjh jcubeh}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jMubeh}(h]h]h]h]h]colsKuhjh jJubah}(h]h]h]h]h]jj9guhjph j hhh!jhNubh;)}(hAn S\ :sub:`16` quadrature is selected with 32 spatial intervals.
Activities are requested for :sup:`235`\ U\ :sub:`92` absorption and
fission and :sup:`238`\ U\ :sub:`92` absorption.h](h/An S }(hAn S\ h jˍhhh!NhNubh)}(h :sub:`16`h]h/16}(hhh jԍubah}(h]h]h]h]h]uhhh jˍubh/P quadrature is selected with 32 spatial intervals.
Activities are requested for }(hP quadrature is selected with 32 spatial intervals.
Activities are requested for h jˍhhh!NhNubj;)}(h
:sup:`235`h]h/235}(hhh jubah}(h]h]h]h]h]uhj;h jˍubh/ U }(h\ U\ h jˍhhh!NhNubh)}(h :sub:`92`h]h/92}(hhh jubah}(h]h]h]h]h]uhhh jˍubh/ absorption and
fission and }(h absorption and
fission and h jˍhhh!NhNubj;)}(h
:sup:`238`h]h/238}(hhh j
ubah}(h]h]h]h]h]uhj;h jˍubh/ U }(hjh jˍubh)}(h :sub:`92`h]h/92}(hhh jubah}(h]h]h]h]h]uhhh jˍubh/ absorption.}(h absorption.h jˍhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMb
h j hhubh
literal_block)}(hX=xsdrn
93% uo2f2 solution sphere
1$$ 3 1 32 1 0 1 5 16 1 1 30 20 0 0 0
2$$ a7 -1 e
3$$ 1 a9 3 1 e
4$$ 0 4 0 -1 5 e
5** 1.-7 1.-8 e 1t
13$$ 1 1 1 1 1
14$$ 1001 8016 9019 92235 92238
15** 6.548-2 3.342-2 6.809-4 3.169-4 2.355-5
16$$ 1001 8016 9019 92235 92238
18## 6hh-1 6ho-16 6hf-19 6hu-235 6hu-238 2t
33## f1 4t
35** 31i0.0 16.0
36$$ f1
49$$ 92235 92235 92238
50$$ 18 27 27
51$$ 74r1 74r2 45r3 45r4 5t
endh]h/X=xsdrn
93% uo2f2 solution sphere
1$$ 3 1 32 1 0 1 5 16 1 1 30 20 0 0 0
2$$ a7 -1 e
3$$ 1 a9 3 1 e
4$$ 0 4 0 -1 5 e
5** 1.-7 1.-8 e 1t
13$$ 1 1 1 1 1
14$$ 1001 8016 9019 92235 92238
15** 6.548-2 3.342-2 6.809-4 3.169-4 2.355-5
16$$ 1001 8016 9019 92235 92238
18## 6hh-1 6ho-16 6hf-19 6hu-235 6hu-238 2t
33## f1 4t
35** 31i0.0 16.0
36$$ f1
49$$ 92235 92235 92238
50$$ 18 27 27
51$$ 74r1 74r2 45r3 45r4 5t
end}(hhh j:ubah}(h]h]h]h]h]forcehighlight_args}jyjzlanguagescaleuhj8h!jhMf
h j hhubh)}(h
.. _9-1-6:h]h}(h]h]h]h]h]hid87uhh
hM
h j hhh!jubeh}(h](xsdrn-sample-problemjeh]h](xsdrn sample problem9-1-5eh]h]uhh#h jhhh!jhMC
jf}j^jsjh}jjsubh$)}(hhh](h))}(hOutput Cross Sectionsh]h/Output Cross Sections}(hjhh jfhhh!NhNubah}(h]h]h]h]h]uhh(h jchhh!jhM
ubh;)}(hOne of the most common uses of XSDRNPM is to collapse cross sections and
write them onto a file for input into another computer code. At present,
two options are allowed:h]h/One of the most common uses of XSDRNPM is to collapse cross sections and
write them onto a file for input into another computer code. At present,
two options are allowed:}(hjvh jthhh!NhNubah}(h]h]h]h]h]uhh:h!jhM
h jchhubj)}(hhh](j)}(hrOutput library in AMPX Working Library format. This library is always
written when cross sections are collapsed.
h]h;)}(hqOutput library in AMPX Working Library format. This library is always
written when cross sections are collapsed.h]h/qOutput library in AMPX Working Library format. This library is always
written when cross sections are collapsed.}(hjh jubah}(h]h]h]h]h]uhh:h!jhM
h jubah}(h]h]h]h]h]uhjh jhhh!jhNubj)}(hX"Output library in ANISN binary or BCD *format*.\ :cite:`joanou_gam-ii_1963` Engle The binary
library is written on logical 20 by default; the BCD library is
produced on logical 7. The identifiers on this library range from 1
to the total number of blocks required to accommodate the data.
h]h;)}(hX!Output library in ANISN binary or BCD *format*.\ :cite:`joanou_gam-ii_1963` Engle The binary
library is written on logical 20 by default; the BCD library is
produced on logical 7. The identifiers on this library range from 1
to the total number of blocks required to accommodate the data.h](h/&Output library in ANISN binary or BCD }(h&Output library in ANISN binary or BCD h jubhA)}(h*format*h]h/format}(hhh jubah}(h]h]h]h]h]uhh@h jubh/. }(h.\ h jubj)}(hjoanou_gam-ii_1963h]j#)}(hjh]h/[joanou_gam-ii_1963]}(hhh jubah}(h]h]h]h]h]uhj"h jubah}(h]id88ah]j5ah]h]h] refdomainj:reftypej< reftargetjrefwarnsupport_smartquotesuhjh!jhM
h jubh/ Engle The binary
library is written on logical 20 by default; the BCD library is
produced on logical 7. The identifiers on this library range from 1
to the total number of blocks required to accommodate the data.}(h Engle The binary
library is written on logical 20 by default; the BCD library is
produced on logical 7. The identifiers on this library range from 1
to the total number of blocks required to accommodate the data.h jubeh}(h]h]h]h]h]uhh:h!jhM
h jubah}(h]h]h]h]h]uhjh jhhh!jhNubeh}(h]h]h]h]h]jSjCjUhjVjWuhjh jchhh!jhM
ubh)}(h
.. _9-1-7:h]h}(h]h]h]h]h]hid89uhh
hMh jchhh!jubeh}(h](output-cross-sectionsjWeh]h](output cross sections9-1-6eh]h]uhh#h jhhh!jhM
jf}jjMsjh}jWjMsubh$)}(hhh](h))}(hError messagesh]h/Error messages}(hjh j
hhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhM
ubh;)}(hX'During the course of a problem, XSDRNPM makes many checks to determine
if input data are in the required form. If inconsistencies are spotted,
a message is printed, and the problem may be terminated. Some of these
messages are listed below along with a brief description of their
possible cause.h]h/X'During the course of a problem, XSDRNPM makes many checks to determine
if input data are in the required form. If inconsistencies are spotted,
a message is printed, and the problem may be terminated. Some of these
messages are listed below along with a brief description of their
possible cause.}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM
h jhhubh;)}(hDATA N N arrays have been input with incorrect length. See the messages
produced as arrays are read to determine specific arrays.h]h/DATA N N arrays have been input with incorrect length. See the messages
produced as arrays are read to determine specific arrays.}(hj(h j&hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM
h jhhubh;)}(hMSN-1 N The Nth entry in the |Sn| quadrature directions is zero.
(43# array)h](h/SN-1 N The Nth entry in the }(hSN-1 N The Nth entry in the h j4hhh!NhNubjr)}(hjh]h/S_n}(hhh j=hhh!NhNubah}(h]h]h]h]h]uhjqh!NhNh j4hhubh/- quadrature directions is zero.
(43# array)}(h- quadrature directions is zero.
(43# array)h j4hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM
h jhhubh;)}(h8SN-2 0 The |Sn| weights do not sum to 1.0. (42# array)h](h/SN-2 0 The }(hSN-2 0 The h jUhhh!NhNubjr)}(hjh]h/S_n}(hhh j^hhh!NhNubah}(h]h]h]h]h]uhjqh!NhNh jUhhubh/) weights do not sum to 1.0. (42# array)}(h) weights do not sum to 1.0. (42# array)h jUhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM
h jhhubh;)}(hSN-3 0 The sum of the products of |Sn| weights and directions is not
0.0, that is, the directions are not symmetric. (42# and 43# arrays)h](h/"SN-3 0 The sum of the products of }(h"SN-3 0 The sum of the products of h jvhhh!NhNubjr)}(hjh]h/S_n}(hhh jhhh!NhNubah}(h]h]h]h]h]uhjqh!NhNh jvhhubh/e weights and directions is not
0.0, that is, the directions are not symmetric. (42# and 43# arrays)}(he weights and directions is not
0.0, that is, the directions are not symmetric. (42# and 43# arrays)h jvhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM
h jhhubh;)}(hTFIXS 0 Fixed source calculation requested (IEVT=0) and total fixed
sources are zero.h]h/TFIXS 0 Fixed source calculation requested (IEVT=0) and total fixed
sources are zero.}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM
h jhhubh;)}(h^Q-HI N A volumetric source spectrum numbered N has been requested where
N is greater than IQM.h]h/^Q-HI N A volumetric source spectrum numbered N has been requested where
N is greater than IQM.}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM
h jhhubh;)}(h\B-HI N A boundary source spectrum numbered N has been requested where N
is greater than IPM.h]h/\B-HI N A boundary source spectrum numbered N has been requested where N
is greater than IPM.}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM
h jhhubh;)}(h5FISS 0 IEVT≥1 and the total fission source is zero.h]h/5FISS 0 IEVT≥1 and the total fission source is zero.}(hjÏh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM
h jhhubh;)}(h'8101 N The N\ *th* radius is negative.h](h/8101 N The N }(h8101 N The N\ h jϏhhh!NhNubhA)}(h*th*h]h/th}(hhh j؏ubah}(h]h]h]h]h]uhh@h jϏubh/ radius is negative.}(h radius is negative.h jϏhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM
h jhhubh;)}(h=8102 N The N\ *th* radius is less than the (N-1)*th* radius.h](h/8102 N The N }(h8102 N The N\ h jhhh!NhNubhA)}(h*th*h]h/th}(hhh jubah}(h]h]h]h]h]uhh@h jubh/+ radius is less than the (N-1)*th* radius.}(h+ radius is less than the (N-1)*th* radius.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM
h jhhubh;)}(hE8103 N Zone N dimensions have become negative in a zone width search.h]h/E8103 N Zone N dimensions have become negative in a zone width search.}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM
h jhhubh;)}(hMIX N A request has been made to use the N\ *th* component from the
mixing table, but this nuclide has not been requested from a library.h](h/,MIX N A request has been made to use the N }(h,MIX N A request has been made to use the N\ h j!hhh!NhNubhA)}(h*th*h]h/th}(hhh j*ubah}(h]h]h]h]h]uhh@h j!ubh/Z component from the
mixing table, but this nuclide has not been requested from a library.}(hZ component from the
mixing table, but this nuclide has not been requested from a library.h j!hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM
h jhhubh;)}(htSeveral messages may be encountered during an XSDRNPM run which indicate
problems with either the code or the setup:h]h/tSeveral messages may be encountered during an XSDRNPM run which indicate
problems with either the code or the setup:}(hjEh jChhh!NhNubah}(h]h]h]h]h]uhh:h!jhM
h jhhubh;)}(hOROOT L The polynomials from which the default angles are derived are
incorrect.h]h/OROOT L The polynomials from which the default angles are derived are
incorrect.}(hjSh jQhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM
h jhhubh;)}(hJBAND N The number of bands specified is greater than the number of
groups.h]h/JBAND N The number of bands specified is greater than the number of
groups.}(hjah j_hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM
h jhhubh;)}(hCWAT1 N The number of sets of weighted cross sections is incorrect.h]h/CWAT1 N The number of sets of weighted cross sections is incorrect.}(hjoh jmhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM
h jhhubh;)}(hCWAT2 N The number of sets of weighted cross sections is incorrect.h]h/CWAT2 N The number of sets of weighted cross sections is incorrect.}(hj}h j{hhh!NhNubah}(h]h]h]h]h]uhh:h!jhM
h jhhubj9)}(hXg*******************************************************INFACE************************************
*** WARNING YOU REQUESTED nn SETS OF CROSS SECTIONS, BUT ONLY mm SETS WERE FOUND *************
*******************************************************MIXEM*************************************
MAGIC WORD ERROR DETECTED IN MIXEM, MW= xx
*******************************************************SPOUT*************************************
NO PROGRAMMING PROVIDED FOR ITP= nn
*******************************************************FIDAS*************************************
****** ERROR nn ENTRIES REQUIRED IN xx? ARRAY
DATA EDIT CONTINUES
*******************************************************FIDAS*************************************
****** FILL OPTION IGNORED IN xx? ARRAY
*******************************************************FIDAS*************************************
****** WARNING ADDRESS aa IS BEYOND LIMITS OF xx? ARRAY
*******************************************************STORXS************************************
MAGIC WORD ERROR IN STORXS - GROUP gg MIXTURE mm L ln
MAGIC WORD mw IGI ig MXI mx MNI mn LLL 11
*******************************************************STORXS************************************
ERROR #1 IN STORXS.$
*******************************************************STORXS************************************
ERROR #2 IN STORXS.$h]h/Xg*******************************************************INFACE************************************
*** WARNING YOU REQUESTED nn SETS OF CROSS SECTIONS, BUT ONLY mm SETS WERE FOUND *************
*******************************************************MIXEM*************************************
MAGIC WORD ERROR DETECTED IN MIXEM, MW= xx
*******************************************************SPOUT*************************************
NO PROGRAMMING PROVIDED FOR ITP= nn
*******************************************************FIDAS*************************************
****** ERROR nn ENTRIES REQUIRED IN xx? ARRAY
DATA EDIT CONTINUES
*******************************************************FIDAS*************************************
****** FILL OPTION IGNORED IN xx? ARRAY
*******************************************************FIDAS*************************************
****** WARNING ADDRESS aa IS BEYOND LIMITS OF xx? ARRAY
*******************************************************STORXS************************************
MAGIC WORD ERROR IN STORXS - GROUP gg MIXTURE mm L ln
MAGIC WORD mw IGI ig MXI mx MNI mn LLL 11
*******************************************************STORXS************************************
ERROR #1 IN STORXS.$
*******************************************************STORXS************************************
ERROR #2 IN STORXS.$}(hhh jubah}(h]h]h]h]h]forcehighlight_args}jyjzjKnoneuhj8h!jhM
h jhhubh;)}(hlFor the cryptic messages above (e.g., the last two), contact the code developers as to their possible cause.h]h/lFor the cryptic messages above (e.g., the last two), contact the code developers as to their possible cause.}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM
h jhhubh;)}(h Footnotesh]h/ Footnotes}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM
h jhhubj)}(h-Formerly with Oak Ridge National Laboratory.
h](j)}(h1h]h/1}(hhh jubah}(h]h]h]h]h]uhjh jubh;)}(h,Formerly with Oak Ridge National Laboratory.h]h/,Formerly with Oak Ridge National Laboratory.}(hjːh jɐubah}(h]h]h]h]h]uhh:h!jhM
h jubeh}(h]id90ah]h]h]jah]jjuhjh!jhM
jKh jhhubj)}(hS:sup:`∗` MM = ISN + 1 for slabs and spheres,
= ISN*(ISN + 4)/4 for a cylinder.
h](j)}(h2h]h/2}(hhh jubah}(h]h]h]h]h]uhjh jސubh;)}(h.:sup:`∗` MM = ISN + 1 for slabs and spheres,h](j;)}(h
:sup:`∗`h]h/∗}(hhh jubah}(h]h]h]h]h]uhj;h jubh/$ MM = ISN + 1 for slabs and spheres,}(h$ MM = ISN + 1 for slabs and spheres,h jubeh}(h]h]h]h]h]uhh:h!jhM
h jސubh;)}(h!= ISN*(ISN + 4)/4 for a cylinder.h]h/!= ISN*(ISN + 4)/4 for a cylinder.}(hjh j
ubah}(h]h]h]h]h]uhh:h!jhM
h jސubeh}(h]jxvah]h]2ah]h]jsvajjuhjh!jhM
h jhhjyvKubh;)}(hhh](h j:)}(hhh](j)}(hhh]h/alder_methods_1963}(hhh j)ubah}(h]h]h]h]h]support_smartquotesuhjh j&ubh;)}(hhh](h/Berni Alder.}(hBerni Alder.h j7ubh/ }(hjh h;)}(hhh](h/Zhaopeng Zhong, Thomas}(hZhaopeng Zhong, Thomash jCubh/ }(h h jCubh/J. Downar, Yunlin Xu, Mark}(hJ. Downar, Yunlin Xu, Markh jCubh/ }(hjOh jCubh/D. DeHart, and Kevin}(hD. DeHart, and Kevinh jCubjWh/
T. Clarno.}(h
T. Clarno.h jCubh/ }(hjh jCubh/Implementation of two-level coarse-mesh finite difference acceleration in an arbitrary geometry, two-dimensional discrete ordinates transport method.}(hImplementation of two-level coarse-mesh finite difference acceleration in an arbitrary geometry, two-dimensional discrete ordinates transport method.h jCubjAhA)}(hhh]h/Nuclear science and engineering}(hNuclear science and engineeringh jlubah}(h]h]h]h]h]uhh@h jCubh/, 158(3):289–298, 2008.}(h, 158(3):289–298, 2008.h jCubjAh/Publisher: Taylor & Francis.}(hPublisher: Taylor & Francis.h jCubeh}(h]h]h]h]h]uhh:h j%)}(hhh](j)}(hhh]h/zhong_implementation_2008}(hhh jubah}(h]h]h]h]h]j6uhjh jubjCeh}(h]zhong-implementation-2008ah]j5ah]zhong_implementation_2008ah]h]id131ajjuhj:h h;)}(hhh](j%)}(hhh](j)}(hhh]h/alcouffe_review_1981}(hhh jubah}(h]h]h]h]h]j6uhjh jubh;)}(hhh](h/R.}(hR.h jubjWh/E. Alcouffe and E.}(hE. Alcouffe and E.h jubjWh/
W. Larsen.}(h
W. Larsen.h jubjAh/MReview of characteristic methods used to solve the linear transport equation.}(hMReview of characteristic methods used to solve the linear transport equation.h jubjAh/=Technical Report, Los Alamos Scientific Lab., NM (USA), 1981.}(h=Technical Report, Los Alamos Scientific Lab., NM (USA), 1981.h jubeh}(h]h]h]h]h]uhh:h jubeh}(h]alcouffe-review-1981ah]j5ah]alcouffe_review_1981ah]h](id111id116ejjuhj:h jjyvKubj%)}(hhh](j)}(hhh]h/alcouffe_computational_1979}(hhh jubah}(h]h]h]h]h]j6uhjh jubh;)}(hhh](h/R.}(hR.h jubjWh/E. Alcouffe, E.}(hE. Alcouffe, E.h jubjWh/
W. Larsen, W.}(h
W. Larsen, W.h jubjWh/ F. Miller}(h F. Millerh jubjWh/
Jr, and B.}(h
Jr, and B.h jubjWh/
R. Wienke.}(h
R. Wienke.h jubjAh/Computational efficiency of numerical methods for the multigroup, discrete-ordinates neutron transport equations: the slab geometry case.}(hComputational efficiency of numerical methods for the multigroup, discrete-ordinates neutron transport equations: the slab geometry case.h jubjAhA)}(hhh]h/Nuclear Science and Engineering}(hNuclear Science and Engineeringh jubah}(h]h]h]h]h]uhh@h jubh/, 71(2):111–127, 1979.}(h, 71(2):111–127, 1979.h jubjAh/Publisher: Taylor & Francis.}(hPublisher: Taylor & Francis.h jubeh}(h]h]h]h]h]uhh:h jubeh}(h]alcouffe-computational-1979ah]j5ah]alcouffe_computational_1979ah]h]id113ajjuhj:h jjyvKubj%)}(hhh](j)}(hhh]h/carlson_discrete_1965}(hhh jBubah}(h]h]h]h]h]j6uhjh j?ubh;)}(hhh](h/B.}(hB.h jOubjWh/G. Carlson and K.}(hG. Carlson and K.h jOubjWh/D. Lathrop.}(hD. Lathrop.h jOubjAh/HDiscrete ordinates angular quadrature of the neutron transport equation.}(hHDiscrete ordinates angular quadrature of the neutron transport equation.h jOubjAhA)}(hhh]h/'LA-3186, Los Alamos National Laboratory}(h'LA-3186, Los Alamos National Laboratoryh jfubah}(h]h]h]h]h]uhh@h jOubh/, 1965.}(h, 1965.h jOubeh}(h]h]h]h]h]uhh:h j?ubeh}(h]id257ah]j5ah]h]carlson_discrete_1965ah]jjuhj:jKh jubj%)}(hhh](j)}(hhh]h/carlson_transport_1970}(hhh jubah}(h]h]h]h]h]j6uhjh jubh;)}(hhh](h/Bengt}(hBength jubjWh/G. Carlson.}(hG. Carlson.h jubh/ }(h h jubh/ TRANSPORT}(h TRANSPORTh jubh/ }(h h jubh/THEORY}(hTHEORYh jubh/: }(h: h jubh/DISCRETE}(hDISCRETEh jubh/ }(h h jubh/ ORDINATES}(h ORDINATESh jubh/ }(h h jubh/
QUADRATURE}(h
QUADRATUREh jubh/ }(h h jubh/OVER}(hOVERh jubh/ }(h h jubh/THE}(hTHEh jubh/ }(h h jubh/UNIT}(hUNITh jubh/ }(hjh jubh/SPHERE}(hSPHEREh jubh/.}(h.h jubjAh/`__ of
the SCALE manual) for details on T-XSEC and T‑NEWT calculations. The
examples are intended simply to demonstrate the use of the NEWT code.
The T-XSEC data are included to allow a user to observe the mixture
definitions used in the NEWT input in its calculation. These problems
are also available as sample problems in the SCALE distribution.h](h/X/This section provides annotated sample input listings for three
different model types, showing the use of a number of different options
and approaches in model development for a variety of applications. These
samples use the TRITON T-XSEC sequence to prepare cross sections for
stand-alone NEWT calculations. In general, this is more easily
accomplished as a TRITON T-NEWT calculation in which cross section
processing and a NEWT transport solution are combined into a single
calculation. However, the user is directed to the TRITON user’s manual
(Chapter }(hX/This section provides annotated sample input listings for three
different model types, showing the use of a number of different options
and approaches in model development for a variety of applications. These
samples use the TRITON T-XSEC sequence to prepare cross sections for
stand-alone NEWT calculations. In general, this is more easily
accomplished as a TRITON T-NEWT calculation in which cross section
processing and a NEWT transport solution are combined into a single
calculation. However, the user is directed to the TRITON user’s manual
(Chapter h j@hhh!NhNubh reference)}(h9`T1.4 `__h]h/T1.4}(hT1.4h jKubah}(h]h]h]h]h]namejSrefuri)file:///\nstdsrvusersm8jmrrsT01triton.pdfuhjIh j@ubh/X[ of
the SCALE manual) for details on T-XSEC and T‑NEWT calculations. The
examples are intended simply to demonstrate the use of the NEWT code.
The T-XSEC data are included to allow a user to observe the mixture
definitions used in the NEWT input in its calculation. These problems
are also available as sample problems in the SCALE distribution.}(hX[ of
the SCALE manual) for details on T-XSEC and T‑NEWT calculations. The
examples are intended simply to demonstrate the use of the NEWT code.
The T-XSEC data are included to allow a user to observe the mixture
definitions used in the NEWT input in its calculation. These problems
are also available as sample problems in the SCALE distribution.h j@hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j/hhubh)}(h.. _9-2-4-1:h]h}(h]h]h]h]h]hid215uhh
hM h j/hhh!jubh$)}(hhh](h))}(hSample 1h]h/Sample 1}(hjxh jvhhh!NhNubah}(h]h]h]h]h]uhh(h jshhh!jhMubh;)}(hXSample 1 illustrates the use of a series of three consecutive
stand-alone NEWT calculations. Annotated input for this problem is given
in :numref:`fig9-2-44`. The calculation begins with SCALE standard composition
specifications used to prepare a problem-specific weighted cross section
library and mixing table for use by NEWT. In this case the T-XSEC
sequence of the TRITON control module is used. This input is described
in the TRITON chapter and is not described further here.h](h/Sample 1 illustrates the use of a series of three consecutive
stand-alone NEWT calculations. Annotated input for this problem is given
in }(hSample 1 illustrates the use of a series of three consecutive
stand-alone NEWT calculations. Annotated input for this problem is given
in h jhhh!NhNubj)}(h:numref:`fig9-2-44`h]jc)}(hjh]h/ fig9-2-44}(hhh jubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnj fig9-2-44uhjh!jhMh jubh/XC. The calculation begins with SCALE standard composition
specifications used to prepare a problem-specific weighted cross section
library and mixing table for use by NEWT. In this case the T-XSEC
sequence of the TRITON control module is used. This input is described
in the TRITON chapter and is not described further here.}(hXC. The calculation begins with SCALE standard composition
specifications used to prepare a problem-specific weighted cross section
library and mixing table for use by NEWT. In this case the T-XSEC
sequence of the TRITON control module is used. This input is described
in the TRITON chapter and is not described further here.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jshhubh;)}(hXVThe first NEWT case uses no parameter block; thus, all default
parameters are applied. The default is an eigenvalue calculation, with
cross sections read from ft04f001 (xnlib=4) and collapsed cross sections
written to ft30f001 (wtdlib=30). The 238-group cross section library is
collapsed to a 44‑group library using mixture-weighted fluxes. The model
calculates the eigenvalue for a simple 1/4 pin cell. The center of the
pin is placed at the origin, the lower-left corner of the global unit
boundary, inlaid into a 2 by 2 base grid. The grid structure is
illustrated in :numref:`fig9-2-45`.h](h/XBThe first NEWT case uses no parameter block; thus, all default
parameters are applied. The default is an eigenvalue calculation, with
cross sections read from ft04f001 (xnlib=4) and collapsed cross sections
written to ft30f001 (wtdlib=30). The 238-group cross section library is
collapsed to a 44‑group library using mixture-weighted fluxes. The model
calculates the eigenvalue for a simple 1/4 pin cell. The center of the
pin is placed at the origin, the lower-left corner of the global unit
boundary, inlaid into a 2 by 2 base grid. The grid structure is
illustrated in }(hXBThe first NEWT case uses no parameter block; thus, all default
parameters are applied. The default is an eigenvalue calculation, with
cross sections read from ft04f001 (xnlib=4) and collapsed cross sections
written to ft30f001 (wtdlib=30). The 238-group cross section library is
collapsed to a 44‑group library using mixture-weighted fluxes. The model
calculates the eigenvalue for a simple 1/4 pin cell. The center of the
pin is placed at the origin, the lower-left corner of the global unit
boundary, inlaid into a 2 by 2 base grid. The grid structure is
illustrated in h jhhh!NhNubj)}(h:numref:`fig9-2-45`h]jc)}(hj×h]h/ fig9-2-45}(hhh jŗubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjϗreftypenumrefrefexplicitrefwarnj fig9-2-45uhjh!jhMh jubh/.}(hjWh jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jshhubh;)}(hXfThe second case performs the same calculation using the collapsed cross
section library created by the first case. Parameter *restart*\ =no is
set to prevent the code from attempting a restart from the existing
library. Because the first case saved 238-group fluxes and the second
case uses 44 energy groups from the collapsed set, a restart is not
possible.h](h/}The second case performs the same calculation using the collapsed cross
section library created by the first case. Parameter }(h}The second case performs the same calculation using the collapsed cross
section library created by the first case. Parameter h jhhh!NhNubhA)}(h *restart*h]h/restart}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ =no is
set to prevent the code from attempting a restart from the existing
library. Because the first case saved 238-group fluxes and the second
case uses 44 energy groups from the collapsed set, a restart is not
possible.}(h\ =no is
set to prevent the code from attempting a restart from the existing
library. Because the first case saved 238-group fluxes and the second
case uses 44 energy groups from the collapsed set, a restart is not
possible.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jshhubh;)}(hXThe third NEWT case is a calculation identical to the second case,
although the input is different. In this case, the flux restart file
from the previous calculation is used as a first guess for fluxes. This
is permitted since both cases used the same cross section library and
therefore have the same energy boundaries. For this case, the “read
geom” data block is omitted, telling NEWT to use the geometry restart
file from the previous case. This allows a rapid restart, since no
geometric data need to be recomputed. Because no other parameters are
changed, this case will converge after one outer iteration to the same
eigenvalue as in the first case.h]h/XThe third NEWT case is a calculation identical to the second case,
although the input is different. In this case, the flux restart file
from the previous calculation is used as a first guess for fluxes. This
is permitted since both cases used the same cross section library and
therefore have the same energy boundaries. For this case, the “read
geom” data block is omitted, telling NEWT to use the geometry restart
file from the previous case. This allows a rapid restart, since no
geometric data need to be recomputed. Because no other parameters are
changed, this case will converge after one outer iteration to the same
eigenvalue as in the first case.}(hjh j
hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jshhubh)}(h.. _fig9-2-44:h]h}(h]h]h]h]h]h fig9-2-44uhh
hM h jshhh!jubj)}(hhh](j)}(hg.. figure:: figs/NEWT/fig44.svg
:align: center
:width: 1000
Sample 1 input listing (annotated).
h]h}(h]h]h]h]h]width1000urifigs/NEWT/fig44.svgj}jj6suhjh j&h!jhMubj)}(h#Sample 1 input listing (annotated).h]h/#Sample 1 input listing (annotated).}(hj:h j8ubah}(h]h]h]h]h]uhjh!jhMh j&ubeh}(h](id315j%eh]h] fig9-2-44ah]h]jcenteruhjhMh jshhh!jjf}jKjsjh}j%jsubh)}(h.. _fig9-2-45:h]h}(h]h]h]h]h]h fig9-2-45uhh
hM h jshhh!jubj)}(hhh](j)}(ho.. figure:: figs/NEWT/fig45.png
:align: center
:width: 500
Grid structure for 1/4 pin cell of Sample 1.
h]h}(h]h]h]h]h]width500urifigs/NEWT/fig45.pngj}jjlsuhjh j\h!jhMubj)}(h,Grid structure for 1/4 pin cell of Sample 1.h]h/,Grid structure for 1/4 pin cell of Sample 1.}(hjph jnubah}(h]h]h]h]h]uhjh!jhMh j\ubeh}(h](id316j[eh]h] fig9-2-45ah]h]jcenteruhjhMh jshhh!jjf}jjQsjh}j[jQsubh)}(h.. _9-2-4-2:h]h}(h]h]h]h]h]hid216uhh
hM h jshhh!jubeh}(h](sample-1jreh]h](sample 19-2-4-1eh]h]uhh#h j/hhh!jhMjf}jjhsjh}jrjhsubh$)}(hhh](h))}(hSample 2h]h/Sample 2}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMubh;)}(hXOSample 2 (shown in Figure 9.2.46 and Figure 9.2.47) illustrates the use
of multiple bodies within a single unit. It highlights the use of media
definitions to include and exclude regions when various bodies are used.
Although an array can be used to place bodies, this example illustrates
a method suitable for use in developing a model for a configuration with
an irregular non-array-type structure. This sample problem also
highlights the use of partial-current unstructured-mesh CMFD
acceleration, which reduces the number of outer iterations from 35 to 21
and the CPU run time by ~25%.h]h/XOSample 2 (shown in Figure 9.2.46 and Figure 9.2.47) illustrates the use
of multiple bodies within a single unit. It highlights the use of media
definitions to include and exclude regions when various bodies are used.
Although an array can be used to place bodies, this example illustrates
a method suitable for use in developing a model for a configuration with
an irregular non-array-type structure. This sample problem also
highlights the use of partial-current unstructured-mesh CMFD
acceleration, which reduces the number of outer iterations from 35 to 21
and the CPU run time by ~25%.}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(h.. _fig9-2-46:h]h}(h]h]h]h]h]h fig9-2-46uhh
hM h jhhh!jubj)}(hhh](j)}(hg.. figure:: figs/NEWT/fig46.svg
:align: center
:width: 1000
Sample 2 input listing (annotated).
h]h}(h]h]h]h]h]width1000urifigs/NEWT/fig46.svgj}jjטsuhjh jǘh!jhMubj)}(h#Sample 2 input listing (annotated).h]h/#Sample 2 input listing (annotated).}(hjۘh j٘ubah}(h]h]h]h]h]uhjh!jhMh jǘubeh}(h](id317jƘeh]h] fig9-2-46ah]h]jcenteruhjhMh jhhh!jjf}jjsjh}jƘjsubh)}(h.. _fig9-2-47:h]h}(h]h]h]h]h]h fig9-2-47uhh
hM!h jhhh!jubj)}(hhh](j)}(h.. figure:: figs/NEWT/fig47.png
:align: center
:width: 500
Mixture placement and grid structure for model described in Sample 2.
h]h}(h]h]h]h]h]width500urifigs/NEWT/fig47.pngj}jj
suhjh jh!jhMubj)}(hEMixture placement and grid structure for model described in Sample 2.h]h/EMixture placement and grid structure for model described in Sample 2.}(hjh jubah}(h]h]h]h]h]uhjh!jhMh jubeh}(h](id318jeh]h] fig9-2-47ah]h]jcenteruhjhMh jhhh!jjf}j"jsjh}jjsubh)}(h.. _9-2-4-3:h]h}(h]h]h]h]h]hid217uhh
hM!h jhhh!jubh$)}(hhh](h))}(hSample 3h]h/Sample 3}(hj8h j6hhh!NhNubah}(h]h]h]h]h]uhh(h j3hhh!jhMubh;)}(hSample 3 demonstrates the development of a VVER-440 hexagonal fuel
assembly. Annotated input for this problem is given in Figure 9.2.48.
The output plot for this model is shown in Figure 9.2.49. The key
attributes of this model are as follows:h]h/Sample 3 demonstrates the development of a VVER-440 hexagonal fuel
assembly. Annotated input for this problem is given in Figure 9.2.48.
The output plot for this model is shown in Figure 9.2.49. The key
attributes of this model are as follows:}(hjFh jDhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j3hhubj)}(hhh](j)}(hDthe use of hexagonal (hexprism) units in a stacked hexagonal array,
h]h;)}(hCthe use of hexagonal (hexprism) units in a stacked hexagonal array,h]h/Cthe use of hexagonal (hexprism) units in a stacked hexagonal array,}(hj[h jYubah}(h]h]h]h]h]uhh:h!jhMh jUubah}(h]h]h]h]h]uhjh jRhhh!jhNubj)}(h4the use of null units as placeholders in the array,
h]h;)}(h3the use of null units as placeholders in the array,h]h/3the use of null units as placeholders in the array,}(hjsh jqubah}(h]h]h]h]h]uhh:h!jhMh jmubah}(h]h]h]h]h]uhjh jRhhh!jhNubj)}(h1a full model within a rhexagonal outer boundary,
h]h;)}(h0a full model within a rhexagonal outer boundary,h]h/0a full model within a rhexagonal outer boundary,}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jRhhh!jhNubj)}(h&the use of white boundary conditions,
h]h;)}(h%the use of white boundary conditions,h]h/%the use of white boundary conditions,}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jRhhh!jhNubj)}(htthe use of the new partial-current-based unstructured CMFD
acceleration for hexagonal-domain configurations, and
h]jd )}(hhh]ji )}(hqthe use of the new partial-current-based unstructured CMFD
acceleration for hexagonal-domain configurations, and
h](jo )}(h:the use of the new partial-current-based unstructured CMFDh]h/:the use of the new partial-current-based unstructured CMFD}(hjh jubah}(h]h]h]h]h]uhjn h!jhMh jubj )}(hhh]h;)}(h5acceleration for hexagonal-domain configurations, andh]h/5acceleration for hexagonal-domain configurations, and}(hjәh jљubah}(h]h]h]h]h]uhh:h!jhMh jΙubah}(h]h]h]h]h]uhj~ h jubeh}(h]h]h]h]h]uhjh h!jhMh jubah}(h]h]h]h]h]uhjc h jubah}(h]h]h]h]h]uhjh jRhhh!NhNubj)}(hnew type-3 ADF inputs.
h]h;)}(hnew type-3 ADF inputs.h]h/new type-3 ADF inputs.}(hjh jubah}(h]h]h]h]h]uhh:h!jhM
h jubah}(h]h]h]h]h]uhjh jRhhh!jhNubeh}(h]h]h]h]h]jSjCjUhjVjWuhjh j3hhh!jhMubh;)}(hUsing CMFD acceleration, the number of outer iterations needed for
convergence decreased from 21 to 8 with a run-time speedup of ~2.58.h]h/Using CMFD acceleration, the number of outer iterations needed for
convergence decreased from 21 to 8 with a run-time speedup of ~2.58.}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j3hhubh)}(h.. _fig9-2-48:h]h}(h]h]h]h]h]h fig9-2-48uhh
hM"!h j3hhh!jubj)}(hhh](j)}(hv.. figure:: figs/NEWT/fig48.svg
:align: center
:width: 1000
:class: long
Sample 3 input listing (annotated).
h]h}(h]h]longah]h]h]width1000urifigs/NEWT/fig48.svgj}jj?suhjh j.h!jhMubj)}(h#Sample 3 input listing (annotated).h]h/#Sample 3 input listing (annotated).}(hjCh jAubah}(h]h]h]h]h]uhjh!jhMh j.ubeh}(h](id319j-eh]h] fig9-2-48ah]h]jcenteruhjhMh j3hhh!jjf}jTj#sjh}j-j#subh)}(h.. _fig9-2-49:h]h}(h]h]h]h]h]h fig9-2-49uhh
hM*!h j3hhh!jubj)}(hhh](j)}(h|.. figure:: figs/NEWT/fig49.png
:align: center
:width: 600
Grid structure and material placement for VVER-440 model.
h]h}(h]h]h]h]h]width600urifigs/NEWT/fig49.pngj}jjusuhjh jeh!jhMubj)}(h9Grid structure and material placement for VVER-440 model.h]h/9Grid structure and material placement for VVER-440 model.}(hjyh jwubah}(h]h]h]h]h]uhjh!jhMh jeubeh}(h](id320jdeh]h] fig9-2-49ah]h]jcenteruhjhMh j3hhh!jjf}jjZsjh}jdjZsubh)}(h.. _9-2-4-4:h]h}(h]h]h]h]h]hid218uhh
hM1!h j3hhh!jubeh}(h](sample-3j2eh]h](sample 39-2-4-3eh]h]uhh#h jhhh!jhMjf}jj(sjh}j2j(subeh}(h](sample-2jeh]h](sample 29-2-4-2eh]h]uhh#h j/hhh!jhMjf}jjsjh}jjsubh$)}(hhh](h))}(hSample 4h]h/Sample 4}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhM!ubh;)}(hXSample 4 demonstrates the use of NEWT in modeling a larger, more complex
configuration. Annotated input for this problem is given in
:numref:`fig9-2-50`. The calculation begins with the use of SCALE standard
composition specifications to prepare a problem-specific weighted cross
section library and mixing table for use by NEWT. In this case the
T-XSEC sequence of the TRITON control module is used.h](h/Sample 4 demonstrates the use of NEWT in modeling a larger, more complex
configuration. Annotated input for this problem is given in
}(hSample 4 demonstrates the use of NEWT in modeling a larger, more complex
configuration. Annotated input for this problem is given in
h jhhh!NhNubj)}(h:numref:`fig9-2-50`h]jc)}(hj͚h]h/ fig9-2-50}(hhh jϚubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh j˚ubah}(h]h]h]h]h]refdocj refdomainjٚreftypenumrefrefexplicitrefwarnj fig9-2-50uhjh!jhM#h jubh/. The calculation begins with the use of SCALE standard
composition specifications to prepare a problem-specific weighted cross
section library and mixing table for use by NEWT. In this case the
T-XSEC sequence of the TRITON control module is used.}(h. The calculation begins with the use of SCALE standard
composition specifications to prepare a problem-specific weighted cross
section library and mixing table for use by NEWT. In this case the
T-XSEC sequence of the TRITON control module is used.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM#h jhhubh;)}(hXdThis NEWT case is used to calculate the eigenvalue of an infinite
lattice of fuel assemblies. Symmetry at the assembly center is used to
reduce a 15 by 15 assembly lattice to a smaller one-quarter assembly.
The grid structure is illustrated in :numref:`fig9-2-51`. A similar
illustration showing media placements by color is given in
:numref:`fig9-2-52`.h](h/This NEWT case is used to calculate the eigenvalue of an infinite
lattice of fuel assemblies. Symmetry at the assembly center is used to
reduce a 15 by 15 assembly lattice to a smaller one-quarter assembly.
The grid structure is illustrated in }(hThis NEWT case is used to calculate the eigenvalue of an infinite
lattice of fuel assemblies. Symmetry at the assembly center is used to
reduce a 15 by 15 assembly lattice to a smaller one-quarter assembly.
The grid structure is illustrated in h jhhh!NhNubj)}(h:numref:`fig9-2-51`h]jc)}(hjh]h/ fig9-2-51}(hhh jubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainj
reftypenumrefrefexplicitrefwarnj fig9-2-51uhjh!jhM*h jubh/G. A similar
illustration showing media placements by color is given in
}(hG. A similar
illustration showing media placements by color is given in
h jhhh!NhNubj)}(h:numref:`fig9-2-52`h]jc)}(hj&h]h/ fig9-2-52}(hhh j(ubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh j$ubah}(h]h]h]h]h]refdocj refdomainj2reftypenumrefrefexplicitrefwarnj fig9-2-52uhjh!jhM*h jubh/.}(hjWh jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM*h jhhubh;)}(h|This input illustrates several features of NEWT modeling capabilities.
Some important features of this model are as follows.h]h/|This input illustrates several features of NEWT modeling capabilities.
Some important features of this model are as follows.}(hjPh jNhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM1h jhhubjd)}(hhh](j)}(hXIn this sample problem, S-6 quadrature, P-1 scattering (P-2 in the
moderator), spatial convergence criteria of 0.005, and an eigenvalue
convergence criteria of 0.001 are used. These are an order of a
magnitude larger than the values typically used for LWR lattice
calculations.
h]h;)}(hXIn this sample problem, S-6 quadrature, P-1 scattering (P-2 in the
moderator), spatial convergence criteria of 0.005, and an eigenvalue
convergence criteria of 0.001 are used. These are an order of a
magnitude larger than the values typically used for LWR lattice
calculations.h]h/XIn this sample problem, S-6 quadrature, P-1 scattering (P-2 in the
moderator), spatial convergence criteria of 0.005, and an eigenvalue
convergence criteria of 0.001 are used. These are an order of a
magnitude larger than the values typically used for LWR lattice
calculations.}(hjeh jcubah}(h]h]h]h]h]uhh:h!jhM4h j_ubah}(h]h]h]h]h]uhjh j\hhh!jhNubj)}(hXTwo sets of UO\ :sub:`2` cross sections are prepared in the T-XSEC
calculation. These cross sections are identical with the exception of
the mixture number. Since NEWT reports fluxes, reaction rates, etc.,
by mixture, the placement of a unique mixture at a specific location
in a model allows one to determine, for example, the reaction rates
at that model location. Mixture 7, placed in unit 9 in this model,
occurs in only one pin location in the model. Mixture 1, placed in
all other fuel rod locations, will yield reaction rates close to the
average of those for all fuel in the assembly. If the flux or
reaction rate was needed in each unique fuel location, a unique
mixture would be needed for each location.
h]h;)}(hXTwo sets of UO\ :sub:`2` cross sections are prepared in the T-XSEC
calculation. These cross sections are identical with the exception of
the mixture number. Since NEWT reports fluxes, reaction rates, etc.,
by mixture, the placement of a unique mixture at a specific location
in a model allows one to determine, for example, the reaction rates
at that model location. Mixture 7, placed in unit 9 in this model,
occurs in only one pin location in the model. Mixture 1, placed in
all other fuel rod locations, will yield reaction rates close to the
average of those for all fuel in the assembly. If the flux or
reaction rate was needed in each unique fuel location, a unique
mixture would be needed for each location.h](h/Two sets of UO }(hTwo sets of UO\ h j{ubh)}(h:sub:`2`h]h/2}(hhh jubah}(h]h]h]h]h]uhhh j{ubh/X cross sections are prepared in the T-XSEC
calculation. These cross sections are identical with the exception of
the mixture number. Since NEWT reports fluxes, reaction rates, etc.,
by mixture, the placement of a unique mixture at a specific location
in a model allows one to determine, for example, the reaction rates
at that model location. Mixture 7, placed in unit 9 in this model,
occurs in only one pin location in the model. Mixture 1, placed in
all other fuel rod locations, will yield reaction rates close to the
average of those for all fuel in the assembly. If the flux or
reaction rate was needed in each unique fuel location, a unique
mixture would be needed for each location.}(hX cross sections are prepared in the T-XSEC
calculation. These cross sections are identical with the exception of
the mixture number. Since NEWT reports fluxes, reaction rates, etc.,
by mixture, the placement of a unique mixture at a specific location
in a model allows one to determine, for example, the reaction rates
at that model location. Mixture 7, placed in unit 9 in this model,
occurs in only one pin location in the model. Mixture 1, placed in
all other fuel rod locations, will yield reaction rates close to the
average of those for all fuel in the assembly. If the flux or
reaction rate was needed in each unique fuel location, a unique
mixture would be needed for each location.h j{ubeh}(h]h]h]h]h]uhh:h!jhM:h jwubah}(h]h]h]h]h]uhjh j\hhh!jhNubj)}(hXThe use of chords for cutting cylinders allows inclusion of one-half
and one-quarter fuel cells in the quarter-assembly model. Because the
fuel assembly has an odd number of rods in each dimension, use of
symmetry at the assembly midplanes requires the rods to be bisected.
h]h;)}(hXThe use of chords for cutting cylinders allows inclusion of one-half
and one-quarter fuel cells in the quarter-assembly model. Because the
fuel assembly has an odd number of rods in each dimension, use of
symmetry at the assembly midplanes requires the rods to be bisected.h]h/XThe use of chords for cutting cylinders allows inclusion of one-half
and one-quarter fuel cells in the quarter-assembly model. Because the
fuel assembly has an odd number of rods in each dimension, use of
symmetry at the assembly midplanes requires the rods to be bisected.}(hjh jubah}(h]h]h]h]h]uhh:h!jhMFh jubah}(h]h]h]h]h]uhjh j\hhh!jhNubj)}(hX*In this model, local grid spacing was selected such common grid
spacings occur in all cells. However, this is not a requirement. For
example, a much more refined local grid could have been specified for
unit 9. There is no requirement that grid lines match between
different elements of an array.
h]h;)}(hX)In this model, local grid spacing was selected such common grid
spacings occur in all cells. However, this is not a requirement. For
example, a much more refined local grid could have been specified for
unit 9. There is no requirement that grid lines match between
different elements of an array.h]h/X)In this model, local grid spacing was selected such common grid
spacings occur in all cells. However, this is not a requirement. For
example, a much more refined local grid could have been specified for
unit 9. There is no requirement that grid lines match between
different elements of an array.}(hjh jubah}(h]h]h]h]h]uhh:h!jhMKh jubah}(h]h]h]h]h]uhjh j\hhh!jhNubj)}(hXUnstructured coarse-mesh finite-difference acceleration (cmfd=2 or
cmfd=yes) was employed to accelerate the convergence of the solution.
For this case, 14 outer iterations were required for full spatial
convergence as compared with 30 outer iterations when CMFD is
disabled. The CMFD-accelerated case ran 2.5 times faster than its
unaccelerated counterpart. In this sample problem, xycmfd=2 was used
to define the coarse-mesh grid to have two fine-mesh cells per
coarse-mesh cell.
h]h;)}(hXUnstructured coarse-mesh finite-difference acceleration (cmfd=2 or
cmfd=yes) was employed to accelerate the convergence of the solution.
For this case, 14 outer iterations were required for full spatial
convergence as compared with 30 outer iterations when CMFD is
disabled. The CMFD-accelerated case ran 2.5 times faster than its
unaccelerated counterpart. In this sample problem, xycmfd=2 was used
to define the coarse-mesh grid to have two fine-mesh cells per
coarse-mesh cell.h]h/XUnstructured coarse-mesh finite-difference acceleration (cmfd=2 or
cmfd=yes) was employed to accelerate the convergence of the solution.
For this case, 14 outer iterations were required for full spatial
convergence as compared with 30 outer iterations when CMFD is
disabled. The CMFD-accelerated case ran 2.5 times faster than its
unaccelerated counterpart. In this sample problem, xycmfd=2 was used
to define the coarse-mesh grid to have two fine-mesh cells per
coarse-mesh cell.}(hjٛh jכubah}(h]h]h]h]h]uhh:h!jhMQh jӛubah}(h]h]h]h]h]uhjh j\hhh!jhNubj)}(hXUTwo-group homogenized cross sections were generated along with
single-assembly (i.e., type 1) ADFs derived from the Collapse block,
ADF block, and the Homogenization block. In addition, a B1 critical
spectrum search is computed after the transport calculation, which is
folded into the transport solution to generated homogenized
constants.
h]h;)}(hXTTwo-group homogenized cross sections were generated along with
single-assembly (i.e., type 1) ADFs derived from the Collapse block,
ADF block, and the Homogenization block. In addition, a B1 critical
spectrum search is computed after the transport calculation, which is
folded into the transport solution to generated homogenized
constants.h]h/XTTwo-group homogenized cross sections were generated along with
single-assembly (i.e., type 1) ADFs derived from the Collapse block,
ADF block, and the Homogenization block. In addition, a B1 critical
spectrum search is computed after the transport calculation, which is
folded into the transport solution to generated homogenized
constants.}(hjh jubah}(h]h]h]h]h]uhh:h!jhMZh jubah}(h]h]h]h]h]uhjh j\hhh!jhNubeh}(h]h]h]h]h]jBjCuhjch!jhM4h jhhubh)}(h.. _fig9-2-50:h]h}(h]h]h]h]h]h fig9-2-50uhh
hMt!h jhhh!jubj)}(hhh](j)}(hv.. figure:: figs/NEWT/fig50.svg
:align: center
:width: 1000
:class: long
Sample 4 input listing (annotated).
h]h}(h]h]longah]h]h]width1000urifigs/NEWT/fig50.svgj}jj%suhjh jh!jhMgubj)}(h#Sample 4 input listing (annotated).h]h/#Sample 4 input listing (annotated).}(hj)h j'ubah}(h]h]h]h]h]uhjh!jhMgh jubeh}(h](id321jeh]h] fig9-2-50ah]h]jcenteruhjhMgh jhhh!jjf}j:j sjh}jj subh)}(h.. _fig9-2-51:h]h}(h]h]h]h]h]h fig9-2-51uhh
hM|!h jhhh!jubj)}(hhh](j)}(hw.. figure:: figs/NEWT/fig51.png
:align: center
:width: 500
Grid structure for one-quarter assembly of Sample 4.
h]h}(h]h]h]h]h]width500urifigs/NEWT/fig51.pngj}jj[suhjh jKh!jhMnubj)}(h4Grid structure for one-quarter assembly of Sample 4.h]h/4Grid structure for one-quarter assembly of Sample 4.}(hj_h j]ubah}(h]h]h]h]h]uhjh!jhMnh jKubeh}(h](id322jJeh]h] fig9-2-51ah]h]jcenteruhjhMnh jhhh!jjf}jpj@sjh}jJj@subh)}(h.. _fig9-2-52:h]h}(h]h]h]h]h]h fig9-2-52uhh
hM!h jhhh!jubj)}(hhh](j)}(h|.. figure:: figs/NEWT/fig52.png
:align: center
:width: 500
Mixture placement for quarter-assembly model of Sample 4.
h]h}(h]h]h]h]h]width500urifigs/NEWT/fig52.pngj}jjsuhjh jh!jhMuubj)}(h9Mixture placement for quarter-assembly model of Sample 4.h]h/9Mixture placement for quarter-assembly model of Sample 4.}(hjh jubah}(h]h]h]h]h]uhjh!jhMuh jubeh}(h](id323jeh]h] fig9-2-52ah]h]jcenteruhjhMuh jhhh!jjf}jjvsjh}jjvsubh)}(h.. _9-2-4-5:h]h}(h]h]h]h]h]hid219uhh
hM!h jhhh!jubeh}(h](sample-4jeh]h](sample 49-2-4-4eh]h]uhh#h j/hhh!jhM!jf}jjsjh}jjsubh$)}(hhh](h))}(hSample 5h]h/Sample 5}(hjǜh jŜhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMzubh;)}(hX:Sample 5 (:numref:`fig9-2-53`) illustrates a calculation for a fuel assembly
with a large water boundary and a vacuum boundary condition. The
calculation begins with the use of SCALE standard composition
specifications to prepare a problem-specific weighted cross section
library and mixing table for use by NEWT.h](h/Sample 5 (}(hSample 5 (h jӜhhh!NhNubj)}(h:numref:`fig9-2-53`h]jc)}(hjޜh]h/ fig9-2-53}(hhh jubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jܜubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnj fig9-2-53uhjh!jhM|h jӜubh/X) illustrates a calculation for a fuel assembly
with a large water boundary and a vacuum boundary condition. The
calculation begins with the use of SCALE standard composition
specifications to prepare a problem-specific weighted cross section
library and mixing table for use by NEWT.}(hX) illustrates a calculation for a fuel assembly
with a large water boundary and a vacuum boundary condition. The
calculation begins with the use of SCALE standard composition
specifications to prepare a problem-specific weighted cross section
library and mixing table for use by NEWT.h jӜhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM|h jhhubh;)}(hXIn this model, seven UO\ :sub:`2` pins are adjacent to eight MOX pins,
which, in turn, are adjacent to a large reflector region. The outer
boundary of the reflector is vacuum. Reflection on the top and bottom
boundaries makes the problem infinite in the y direction. The grid
structure for this problem is illustrated in :numref:`fig9-2-54`. This problem
illustrates the use of the original CMFD acceleration scheme in NEWT
(cmfd=1 or cmfd=rect). Because of the large degree of scattering within
the reflector region, the problem can be relatively slow to converge.
Without CMFD acceleration, 40 outer iterations are required for spatial
convergence as compared with 12 when CMFD is enabled. A total run-time
speedup of ~1.4 is achieved with the CMFD acceleration scheme.h](h/In this model, seven UO }(hIn this model, seven UO\ h jhhh!NhNubh)}(h:sub:`2`h]h/2}(hhh jubah}(h]h]h]h]h]uhhh jubh/X! pins are adjacent to eight MOX pins,
which, in turn, are adjacent to a large reflector region. The outer
boundary of the reflector is vacuum. Reflection on the top and bottom
boundaries makes the problem infinite in the y direction. The grid
structure for this problem is illustrated in }(hX! pins are adjacent to eight MOX pins,
which, in turn, are adjacent to a large reflector region. The outer
boundary of the reflector is vacuum. Reflection on the top and bottom
boundaries makes the problem infinite in the y direction. The grid
structure for this problem is illustrated in h jhhh!NhNubj)}(h:numref:`fig9-2-54`h]jc)}(hj%h]h/ fig9-2-54}(hhh j'ubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh j#ubah}(h]h]h]h]h]refdocj refdomainj1reftypenumrefrefexplicitrefwarnj fig9-2-54uhjh!jhMh jubh/X. This problem
illustrates the use of the original CMFD acceleration scheme in NEWT
(cmfd=1 or cmfd=rect). Because of the large degree of scattering within
the reflector region, the problem can be relatively slow to converge.
Without CMFD acceleration, 40 outer iterations are required for spatial
convergence as compared with 12 when CMFD is enabled. A total run-time
speedup of ~1.4 is achieved with the CMFD acceleration scheme.}(hX. This problem
illustrates the use of the original CMFD acceleration scheme in NEWT
(cmfd=1 or cmfd=rect). Because of the large degree of scattering within
the reflector region, the problem can be relatively slow to converge.
Without CMFD acceleration, 40 outer iterations are required for spatial
convergence as compared with 12 when CMFD is enabled. A total run-time
speedup of ~1.4 is achieved with the CMFD acceleration scheme.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(hIn addition to the application of CMFD, Sample 5 also illustrates the
use of NEWT’s reflector ADF capability. Reflector ADFs are computed
along the fuel/reflector interface.h]h/In addition to the application of CMFD, Sample 5 also illustrates the
use of NEWT’s reflector ADF capability. Reflector ADFs are computed
along the fuel/reflector interface.}(hjPh jNhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(h.. _fig9-2-53:h]h}(h]h]h]h]h]h fig9-2-53uhh
hM!h jhhh!jubj)}(hhh](j)}(hv.. figure:: figs/NEWT/fig53.svg
:align: center
:width: 1000
:class: long
Sample 5 input listing (annotated).
h]h}(h]h]longah]h]h]width1000urifigs/NEWT/fig53.svgj}jjxsuhjh jgh!jhMubj)}(h#Sample 5 input listing (annotated).h]h/#Sample 5 input listing (annotated).}(hj|h jzubah}(h]h]h]h]h]uhjh!jhMh jgubeh}(h](id324jfeh]h] fig9-2-53ah]h]jcenteruhjhMh jhhh!jjf}jj\sjh}jfj\subh)}(h.. _fig9-2-54:h]h}(h]h]h]h]h]h fig9-2-54uhh
hM!h jhhh!jubj)}(hhh](j)}(hm.. figure:: figs/NEWT/fig54.png
:align: center
:width: 500
Grid structure for 15-pin row of Sample 5.
h]h}(h]h]h]h]h]width500urifigs/NEWT/fig54.pngj}jjsuhjh jh!jhMubj)}(h*Grid structure for 15-pin row of Sample 5.h]h/*Grid structure for 15-pin row of Sample 5.}(hjh jubah}(h]h]h]h]h]uhjh!jhMh jubeh}(h](id325jeh]h] fig9-2-54ah]h]jcenteruhjhMh jhhh!jjf}jÝjsjh}jjsubh)}(h
.. _9-2-5:h]h}(h]h]h]h]h]hid220uhh
hM!h jhhh!jubh$)}(hhh](h))}(hDescription of Outputh]h/Description of Output}(hjٝh jםhhh!NhNubah}(h]h]h]h]h]uhh(h jԝhhh!jhMubh;)}(hXThis section contains a brief description and explanation of NEWT
output. Portions of the output will not be printed for every problem.
Some output is optional, depending on user input specifications and is
so noted in the description. As with any SCALE module, output begins
with an input echo, module execution records with times and completion
codes, and the program verification information banner page. These
outputs are common to all SCALE modules and are not described here.h]h/XThis section contains a brief description and explanation of NEWT
output. Portions of the output will not be printed for every problem.
Some output is optional, depending on user input specifications and is
so noted in the description. As with any SCALE module, output begins
with an input echo, module execution records with times and completion
codes, and the program verification information banner page. These
outputs are common to all SCALE modules and are not described here.}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jԝhhubh)}(h.. _9-2-5-1:h]h}(h]h]h]h]h]hid221uhh
hM!h jԝhhh!jubeh}(h](description-of-outputjӝeh]h](description of output9-2-5eh]h]uhh#h jhhh!jhMjf}jjɝsjh}jӝjɝsubeh}(h](sample-5jeh]h](sample 59-2-4-5eh]h]uhh#h j/hhh!jhMzjf}jjsjh}jjsubh$)}(hhh](h))}(hNEWT bannerh]h/NEWT banner}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMubh;)}(hX?Following the SCALE program verification information, the first section
unique to NEWT output is the NEWT banner, which appears as shown in
:numref:`fig9-2-55`. The bottom of the banner gives the title of the case as
given in input. The NEWT banner is printed only if the command line
option –p is used to run SCALE.h](h/Following the SCALE program verification information, the first section
unique to NEWT output is the NEWT banner, which appears as shown in
}(hFollowing the SCALE program verification information, the first section
unique to NEWT output is the NEWT banner, which appears as shown in
h j%hhh!NhNubj)}(h:numref:`fig9-2-55`h]jc)}(hj0h]h/ fig9-2-55}(hhh j2ubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh j.ubah}(h]h]h]h]h]refdocj refdomainj<reftypenumrefrefexplicitrefwarnj fig9-2-55uhjh!jhMh j%ubh/. The bottom of the banner gives the title of the case as
given in input. The NEWT banner is printed only if the command line
option –p is used to run SCALE.}(h. The bottom of the banner gives the title of the case as
given in input. The NEWT banner is printed only if the command line
option –p is used to run SCALE.h j%hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(h.. _fig9-2-55:h]h}(h]h]h]h]h]h fig9-2-55uhh
hM!h jhhh!jubj)}(hhh](j)}(hm.. figure:: figs/NEWT/fig55.png
:align: center
:width: 500
NEWT copyright banner page and case title.
h]h}(h]h]h]h]h]width500urifigs/NEWT/fig55.pngj}jjtsuhjh jdh!jhMubj)}(h*NEWT copyright banner page and case title.h]h/*NEWT copyright banner page and case title.}(hjxh jvubah}(h]h]h]h]h]uhjh!jhMh jdubeh}(h](id326jceh]h] fig9-2-55ah]h]jcenteruhjhMh jhhh!jjf}jjYsjh}jcjYsubh)}(h.. _9-2-5-2:h]h}(h]h]h]h]h]hid222uhh
hM!h jhhh!jubeh}(h](newt-bannerjeh]h](newt banner9-2-5-1eh]h]uhh#h j/hhh!jhMjf}jjsjh}jjsubh$)}(hhh](h))}(h
Input summaryh]h/
Input summary}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMubh;)}(hXThe next several pages of output provide a summary of input parameters.
As described in :ref:`9-2-5`, default parameters are used when no user
specification is supplied. The input summary lists all parameters and
states used in the calculation, whether user supplied or default. The
following subsections describe the various blocks of output information
provided in the input summary.h](h/XThe next several pages of output provide a summary of input parameters.
As described in }(hXThe next several pages of output provide a summary of input parameters.
As described in h jhhh!NhNubj)}(h:ref:`9-2-5`h]j#)}(hjh]h/9-2-5}(hhh jÞubah}(h]h](jnstdstd-refeh]h]h]uhj"h jubah}(h]h]h]h]h]refdocj refdomainj͞reftyperefrefexplicitrefwarnj9-2-5uhjh!jhMh jubh/X, default parameters are used when no user
specification is supplied. The input summary lists all parameters and
states used in the calculation, whether user supplied or default. The
following subsections describe the various blocks of output information
provided in the input summary.}(hX, default parameters are used when no user
specification is supplied. The input summary lists all parameters and
states used in the calculation, whether user supplied or default. The
following subsections describe the various blocks of output information
provided in the input summary.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(h.. _9-2-5-2-1:h]h}(h]h]h]h]h]hid223uhh
hM!h jhhh!jubh$)}(hhh](h))}(hControl optionsh]h/Control options}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMubh;)}(hX"The Control Options block lists global control parameters that determine
the type of analysis being performed. A sample Control Options page is
shown in :numref:`fig9-2-56`. Parameters are self-explanatory. More
information is available in the description of the keywords in
:ref:`9-2-5-2`.h](h/The Control Options block lists global control parameters that determine
the type of analysis being performed. A sample Control Options page is
shown in }(hThe Control Options block lists global control parameters that determine
the type of analysis being performed. A sample Control Options page is
shown in h jhhh!NhNubj)}(h:numref:`fig9-2-56`h]jc)}(hjh]h/ fig9-2-56}(hhh jubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnj fig9-2-56uhjh!jhMh jubh/g. Parameters are self-explanatory. More
information is available in the description of the keywords in
}(hg. Parameters are self-explanatory. More
information is available in the description of the keywords in
h jhhh!NhNubj)}(h:ref:`9-2-5-2`h]j#)}(hj6h]h/9-2-5-2}(hhh j8ubah}(h]h](jnstdstd-refeh]h]h]uhj"h j4ubah}(h]h]h]h]h]refdocj refdomainjBreftyperefrefexplicitrefwarnj9-2-5-2uhjh!jhMh jubh/.}(hjWh jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(h.. _fig9-2-56:h]h}(h]h]h]h]h]h fig9-2-56uhh
hM!h jhhh!jubj)}(hhh](j)}(hX.. figure:: figs/NEWT/fig56.svg
:align: center
:width: 600
Control Options page.
h]h}(h]h]h]h]h]width600urifigs/NEWT/fig56.svgj}jjysuhjh jih!jhMubj)}(hControl Options page.h]h/Control Options page.}(hj}h j{ubah}(h]h]h]h]h]uhjh!jhMh jiubeh}(h](id327jheh]h] fig9-2-56ah]h]jcenteruhjhMh jhhh!jjf}jj^sjh}jhj^subh)}(h.. _9-2-5-2-2:h]h}(h]h]h]h]h]hid225uhh
hM!h jhhh!jubeh}(h](jid224eh]h] 9-2-5-2-1ah]control optionsah]uhh#h jhhh!jhMjKjf}jjsjh}jjsubh$)}(hhh](h))}(hOutput optionsh]h/Output options}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMubh;)}(hThe Output Options block (:numref:`fig9-2-57`) lists selections made for
output. Portions of the output listing will be printed only if the
appropriate printing option was selected.h](h/The Output Options block (}(hThe Output Options block (h jhhh!NhNubj)}(h:numref:`fig9-2-57`h]jc)}(hjƟh]h/ fig9-2-57}(hhh jȟubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jğubah}(h]h]h]h]h]refdocj refdomainjҟreftypenumrefrefexplicitrefwarnj fig9-2-57uhjh!jhMh jubh/) lists selections made for
output. Portions of the output listing will be printed only if the
appropriate printing option was selected.}(h) lists selections made for
output. Portions of the output listing will be printed only if the
appropriate printing option was selected.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(h.. _fig9-2-57:h]h}(h]h]h]h]h]h fig9-2-57uhh
hM!h jhhh!jubj)}(hhh](j)}(hW.. figure:: figs/NEWT/fig57.svg
:align: center
:width: 600
Output Options page.
h]h}(h]h]h]h]h]width600urifigs/NEWT/fig57.svgj}jj
suhjh jh!jhMubj)}(hOutput Options page.h]h/Output Options page.}(hjh jubah}(h]h]h]h]h]uhjh!jhMh jubeh}(h](id328jeh]h] fig9-2-57ah]h]jcenteruhjhMh jhhh!jjf}jjsjh}jjsubh)}(h.. _9-2-5-2-3:h]h}(h]h]h]h]h]hid226uhh
hM"h jhhh!jubeh}(h](output-optionsjeh]h](output options 9-2-5-2-2eh]h]uhh#h jhhh!jhMjf}j6jsjh}jjsubh$)}(hhh](h))}(hInput/output unit assignmentsh]h/Input/output unit assignments}(hj@h j>hhh!NhNubah}(h]h]h]h]h]uhh(h j;hhh!jhMubh;)}(hThe Input/Output (I/O) Unit Assignments block (:numref:`fig9-2-58`) simply
lists the unit numbers selected for reading or writing various data
files, as appropriate for the calculation.h](h//The Input/Output (I/O) Unit Assignments block (}(h/The Input/Output (I/O) Unit Assignments block (h jLhhh!NhNubj)}(h:numref:`fig9-2-58`h]jc)}(hjWh]h/ fig9-2-58}(hhh jYubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jUubah}(h]h]h]h]h]refdocj refdomainjcreftypenumrefrefexplicitrefwarnj fig9-2-58uhjh!jhMh jLubh/w) simply
lists the unit numbers selected for reading or writing various data
files, as appropriate for the calculation.}(hw) simply
lists the unit numbers selected for reading or writing various data
files, as appropriate for the calculation.h jLhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j;hhubh)}(h.. _fig9-2-58:h]h}(h]h]h]h]h]h fig9-2-58uhh
hM
"h j;hhh!jubj)}(hhh](j)}(hf.. figure:: figs/NEWT/fig58.svg
:align: center
:width: 600
Input/Output Unit Assignments page.
h]h}(h]h]h]h]h]width600urifigs/NEWT/fig58.svgj}jjsuhjh jh!jhMubj)}(h#Input/Output Unit Assignments page.h]h/#Input/Output Unit Assignments page.}(hjh jubah}(h]h]h]h]h]uhjh!jhMh jubeh}(h](id329jeh]h] fig9-2-58ah]h]jcenteruhjhMh j;hhh!jjf}jjsjh}jjsubh)}(h.. _9-2-5-2-4:h]h}(h]h]h]h]h]hid227uhh
hM"h j;hhh!jubeh}(h](input-output-unit-assignmentsj/eh]h](input/output unit assignments 9-2-5-2-3eh]h]uhh#h jhhh!jhMjf}jǠj%sjh}j/j%subh$)}(hhh](h))}(hConvergence control parametersh]h/Convergence control parameters}(hjѠh jϠhhh!NhNubah}(h]h]h]h]h]uhh(h j̠hhh!jhMubh;)}(hThe Convergence Control block (:numref:`fig9-2-59`) summarizes all parameters
used to control spatial, angular, and eigenvalue convergence for the
iterative phases of the solution process.h](h/The Convergence Control block (}(hThe Convergence Control block (h jݠhhh!NhNubj)}(h:numref:`fig9-2-59`h]jc)}(hjh]h/ fig9-2-59}(hhh jubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnj fig9-2-59uhjh!jhMh jݠubh/) summarizes all parameters
used to control spatial, angular, and eigenvalue convergence for the
iterative phases of the solution process.}(h) summarizes all parameters
used to control spatial, angular, and eigenvalue convergence for the
iterative phases of the solution process.h jݠhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j̠hhubh)}(h.. _fig9-2-59:h]h}(h]h]h]h]h]h fig9-2-59uhh
hM"h j̠hhh!jubj)}(hhh](j)}(hg.. figure:: figs/NEWT/fig59.svg
:align: center
:width: 600
Convergence Control Parameters page.
h]h}(h]h]h]h]h]width600urifigs/NEWT/fig59.svgj}jj,suhjh jh!jhMubj)}(h$Convergence Control Parameters page.h]h/$Convergence Control Parameters page.}(hj0h j.ubah}(h]h]h]h]h]uhjh!jhMh jubeh}(h](id330jeh]h] fig9-2-59ah]h]jcenteruhjhMh j̠hhh!jjf}jAjsjh}jjsubh)}(h.. _9-2-5-2-5:h]h}(h]h]h]h]h]hid228uhh
hM!"h j̠hhh!jubeh}(h](convergence-control-parametersjeh]h](convergence control parameters 9-2-5-2-4eh]h]uhh#h jhhh!jhMjf}jXjsjh}jjsubh$)}(hhh](h))}(hPin-power edit requestsh]h/Pin-power edit requests}(hjbh j`hhh!NhNubah}(h]h]h]h]h]uhh(h j]hhh!jhMubh;)}(hIf pin-power edits are requested for one or more arrays, a listing is
provided of the arrays for which this request was made (:numref:`fig9-2-60`).h](h/~If pin-power edits are requested for one or more arrays, a listing is
provided of the arrays for which this request was made (}(h~If pin-power edits are requested for one or more arrays, a listing is
provided of the arrays for which this request was made (h jnhhh!NhNubj)}(h:numref:`fig9-2-60`h]jc)}(hjyh]h/ fig9-2-60}(hhh j{ubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jwubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnj fig9-2-60uhjh!jhMh jnubh/).}(h).h jnhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j]hhubh)}(h.. _fig9-2-60:h]h}(h]h]h]h]h]h fig9-2-60uhh
hM)"h j]hhh!jubj)}(hhh](j)}(hb.. figure:: figs/NEWT/fig60.svg
:align: center
:width: 600
Pin-power edit request summary.
h]h}(h]h]h]h]h]width600urifigs/NEWT/fig60.svgj}jjsuhjh jh!jhMubj)}(hPin-power edit request summary.h]h/Pin-power edit request summary.}(hjh jubah}(h]h]h]h]h]uhjh!jhMh jubeh}(h](id331jeh]h] fig9-2-60ah]h]jcenteruhjhMh j]hhh!jjf}jҡjsjh}jjsubh)}(h.. _9-2-5-2-6:h]h}(h]h]h]h]h]hid229uhh
hM0"h j]hhh!jubeh}(h](pin-power-edit-requestsjQeh]h](pin-power edit requests 9-2-5-2-5eh]h]uhh#h jhhh!jhMjf}jjGsjh}jQjGsubh$)}(hhh](h))}(hGeometry specificationsh]h/Geometry specifications}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhM ubh;)}(hXThe Geometry Specifications block (:numref:`fig9-2-61`) lists parameters
associated with the geometric model specified by the user. The first
section lists the characteristics of the global unit. This is followed
by a listing of the four boundary conditions. Finally, the last section
in this block lists all bodies specified for the model. The appearance
and contents of this section of input depend on the nature of the input
model.h](h/#The Geometry Specifications block (}(h#The Geometry Specifications block (h jhhh!NhNubj)}(h:numref:`fig9-2-61`h]jc)}(hj
h]h/ fig9-2-61}(hhh jubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnj fig9-2-61uhjh!jhM"h jubh/X|) lists parameters
associated with the geometric model specified by the user. The first
section lists the characteristics of the global unit. This is followed
by a listing of the four boundary conditions. Finally, the last section
in this block lists all bodies specified for the model. The appearance
and contents of this section of input depend on the nature of the input
model.}(hX|) lists parameters
associated with the geometric model specified by the user. The first
section lists the characteristics of the global unit. This is followed
by a listing of the four boundary conditions. Finally, the last section
in this block lists all bodies specified for the model. The appearance
and contents of this section of input depend on the nature of the input
model.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM"h jhhubh)}(h.. _fig9-2-61:h]h}(h]h]h]h]h]h fig9-2-61uhh
hM="h jhhh!jubj)}(hhh](j)}(h`.. figure:: figs/NEWT/fig61.svg
:align: center
:width: 600
Geometry Specifications page.
h]h}(h]h]h]h]h]width600urifigs/NEWT/fig61.svgj}jjNsuhjh j>h!jhM/ubj)}(hGeometry Specifications page.h]h/Geometry Specifications page.}(hjRh jPubah}(h]h]h]h]h]uhjh!jhM/h j>ubeh}(h](id332j=eh]h] fig9-2-61ah]h]jcenteruhjhM/h jhhh!jjf}jcj3sjh}j=j3subh)}(h.. _9-2-5-2-7:h]h}(h]h]h]h]h]hid230uhh
hMD"h jhhh!jubeh}(h](geometry-specificationsjeh]h](geometry specifications 9-2-5-2-6eh]h]uhh#h jhhh!jhM jf}jzjءsjh}jjءsubh$)}(hhh](h))}(h$Homogenization region specificationsh]h/$Homogenization region specifications}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhM4ubh;)}(hThe Homogenization Region Specifications block (:numref:`fig9-2-62`)
summarizes all sets of homogenized cross sections requested in user
input.h](h/0The Homogenization Region Specifications block (}(h0The Homogenization Region Specifications block (h jhhh!NhNubj)}(h:numref:`fig9-2-62`h]jc)}(hjh]h/ fig9-2-62}(hhh jubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnj fig9-2-62uhjh!jhM6h jubh/M)
summarizes all sets of homogenized cross sections requested in user
input.}(hM)
summarizes all sets of homogenized cross sections requested in user
input.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM6h jhhubh)}(h.. _fig9-2-62:h]h}(h]h]h]h]h]h fig9-2-62uhh
hMM"h jhhh!jubj)}(hhh](j)}(hm.. figure:: figs/NEWT/fig62.svg
:align: center
:width: 600
Homogenization Region Specifications page.
h]h}(h]h]h]h]h]width600urifigs/NEWT/fig62.svgj}jjߢsuhjh jϢh!jhM?ubj)}(h*Homogenization Region Specifications page.h]h/*Homogenization Region Specifications page.}(hjh jubah}(h]h]h]h]h]uhjh!jhM?h jϢubeh}(h](id333jeh]h] fig9-2-62ah]h]jcenteruhjhM?h jhhh!jjf}jjĢsjh}jjĢsubh)}(h.. _9-2-5-2-8:h]h}(h]h]h]h]h]hid231uhh
hMT"h jhhh!jubeh}(h]($homogenization-region-specificationsjseh]h]($homogenization region specifications 9-2-5-2-7eh]h]uhh#h jhhh!jhM4jf}jjisjh}jsjisubh$)}(hhh](h))}(hMaterial specificationsh]h/Material specifications}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMDubh;)}(hThe Material Specification block (:numref:`fig9-2-63`) lists the NEWT material
number, counting in the order read in; the SCALE mixture number; and the
P\ :sub:`n` order assigned for that mixture.h](h/"The Material Specification block (}(h"The Material Specification block (h j!hhh!NhNubj)}(h:numref:`fig9-2-63`h]jc)}(hj,h]h/ fig9-2-63}(hhh j.ubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh j*ubah}(h]h]h]h]h]refdocj refdomainj8reftypenumrefrefexplicitrefwarnj fig9-2-63uhjh!jhMFh j!ubh/f) lists the NEWT material
number, counting in the order read in; the SCALE mixture number; and the
P }(hf) lists the NEWT material
number, counting in the order read in; the SCALE mixture number; and the
P\ h j!hhh!NhNubh)}(h:sub:`n`h]h/n}(hhh jOubah}(h]h]h]h]h]uhhh j!ubh/! order assigned for that mixture.}(h! order assigned for that mixture.h j!hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMFh jhhubh)}(h.. _fig9-2-63:h]h}(h]h]h]h]h]h fig9-2-63uhh
hM]"h jhhh!jubj)}(hhh](j)}(h`.. figure:: figs/NEWT/fig63.svg
:align: center
:width: 600
Material Specifications page.
h]h}(h]h]h]h]h]width600urifigs/NEWT/fig63.svgj}jjsuhjh jsh!jhMOubj)}(hMaterial Specifications page.h]h/Material Specifications page.}(hjh jubah}(h]h]h]h]h]uhjh!jhMOh jsubeh}(h](id334jreh]h] fig9-2-63ah]h]jcenteruhjhMOh jhhh!jjf}jjhsjh}jrjhsubh)}(h.. _9-2-5-2-9:h]h}(h]h]h]h]h]hid232uhh
hMd"h jhhh!jubeh}(h](material-specificationsjeh]h](material specifications 9-2-5-2-8eh]h]uhh#h jhhh!jhMDjf}jjsjh}jjsubh$)}(hhh](h))}(hDerived parametersh]h/Derived parameters}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMTubh;)}(hXThe Derived Parameters block (:numref:`fig9-2-64`) lists values not
specifically input but derived from other sources of input. Some of this
information comes from the cross section library, some from the model
geometry, and some from the S\ :sub:`n` and P\ :sub:`n` values
specified.h](h/The Derived Parameters block (}(hThe Derived Parameters block (h jţhhh!NhNubj)}(h:numref:`fig9-2-64`h]jc)}(hjУh]h/ fig9-2-64}(hhh jңubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jΣubah}(h]h]h]h]h]refdocj refdomainjܣreftypenumrefrefexplicitrefwarnj fig9-2-64uhjh!jhMVh jţubh/) lists values not
specifically input but derived from other sources of input. Some of this
information comes from the cross section library, some from the model
geometry, and some from the S }(h) lists values not
specifically input but derived from other sources of input. Some of this
information comes from the cross section library, some from the model
geometry, and some from the S\ h jţhhh!NhNubh)}(h:sub:`n`h]h/n}(hhh jubah}(h]h]h]h]h]uhhh jţubh/ and P }(h and P\ h jţhhh!NhNubh)}(h:sub:`n`h]h/n}(hhh jubah}(h]h]h]h]h]uhhh jţubh/ values
specified.}(h values
specified.h jţhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMVh jhhubh)}(h.. _fig9-2-64:h]h}(h]h]h]h]h]h fig9-2-64uhh
hMo"h jhhh!jubj)}(hhh](j)}(h[.. figure:: figs/NEWT/fig64.svg
:align: center
:width: 600
Derived Parameters page.
h]h}(h]h]h]h]h]width600urifigs/NEWT/fig64.svgj}jj:suhjh j*h!jhMaubj)}(hDerived Parameters page.h]h/Derived Parameters page.}(hj>h j<ubah}(h]h]h]h]h]uhjh!jhMah j*ubeh}(h](id335j)eh]h] fig9-2-64ah]h]jcenteruhjhMah jhhh!jjf}jOjsjh}j)jsubh)}(h.. _9-2-5-2-10:h]h}(h]h]h]h]h]hid233uhh
hMv"h jhhh!jubeh}(h](derived-parametersjeh]h](derived parameters 9-2-5-2-9eh]h]uhh#h jhhh!jhMTjf}jfjsjh}jjsubh$)}(hhh](h))}(hEnergy group structure listingh]h/Energy group structure listing}(hjph jnhhh!NhNubah}(h]h]h]h]h]uhh(h jkhhh!jhMfubh;)}(hXThe Energy Group Structures block (:numref:`fig9-2-65`) lists the energy and
lethargy boundaries found in the cross section library. If a broad-group
collapse was requested, the boundaries of the broad-group library that
will be produced are also identified. This example shows the structure
of the SCALE 44GROUPNDF5 library and 2-group fast/thermal collapse
structure. The final entry (group 45, broad group 3) indicates the lower
bound of the previous energy group.h](h/#The Energy Group Structures block (}(h#The Energy Group Structures block (h j|hhh!NhNubj)}(h:numref:`fig9-2-65`h]jc)}(hjh]h/ fig9-2-65}(hhh jubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnj fig9-2-65uhjh!jhMhh j|ubh/X) lists the energy and
lethargy boundaries found in the cross section library. If a broad-group
collapse was requested, the boundaries of the broad-group library that
will be produced are also identified. This example shows the structure
of the SCALE 44GROUPNDF5 library and 2-group fast/thermal collapse
structure. The final entry (group 45, broad group 3) indicates the lower
bound of the previous energy group.}(hX) lists the energy and
lethargy boundaries found in the cross section library. If a broad-group
collapse was requested, the boundaries of the broad-group library that
will be produced are also identified. This example shows the structure
of the SCALE 44GROUPNDF5 library and 2-group fast/thermal collapse
structure. The final entry (group 45, broad group 3) indicates the lower
bound of the previous energy group.h j|hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMhh jkhhubh)}(h.. _fig9-2-65:h]h}(h]h]h]h]h]h fig9-2-65uhh
hM"h jkhhh!jubj)}(hhh](j)}(hg.. figure:: figs/NEWT/fig65.svg
:align: center
:width: 600
Energy Group Structure Listing page.
h]h}(h]h]h]h]h]width600urifigs/NEWT/fig65.svgj}jjˤsuhjh jh!jhMuubj)}(h$Energy Group Structure Listing page.h]h/$Energy Group Structure Listing page.}(hjϤh jͤubah}(h]h]h]h]h]uhjh!jhMuh jubeh}(h](id336jeh]h] fig9-2-65ah]h]jcenteruhjhMuh jkhhh!jjf}jjsjh}jjsubh)}(h.. _9-2-5-2-11:h]h}(h]h]h]h]h]hid234uhh
hM"h jkhhh!jubeh}(h](energy-group-structure-listingj_eh]h](energy group structure listing
9-2-5-2-10eh]h]uhh#h jhhh!jhMfjf}jjUsjh}j_jUsubh$)}(hhh](h))}(hQuadrature parametersh]h/Quadrature parameters}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMzubh;)}(hXThe Quadrature Parameters block (:numref:`fig9-2-66`) lists the first-quadrant
angles and weights used for the specified order of quadrature. The same
angles and weights are applied in the other three quadrants; however,
the signs of the angles vary with the quadrant. Also listed are the
P\ :sub:`n` moments associated with the maximum P\ :sub:`n` scattering
order requested in all materials. Of course, only a subset of these
moments applies to the lower-order P\ :sub:`n` assignments.h](h/!The Quadrature Parameters block (}(h!The Quadrature Parameters block (h j
hhh!NhNubj)}(h:numref:`fig9-2-66`h]jc)}(hjh]h/ fig9-2-66}(hhh jubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainj$reftypenumrefrefexplicitrefwarnj fig9-2-66uhjh!jhM|h j
ubh/) lists the first-quadrant
angles and weights used for the specified order of quadrature. The same
angles and weights are applied in the other three quadrants; however,
the signs of the angles vary with the quadrant. Also listed are the
P }(h) lists the first-quadrant
angles and weights used for the specified order of quadrature. The same
angles and weights are applied in the other three quadrants; however,
the signs of the angles vary with the quadrant. Also listed are the
P\ h j
hhh!NhNubh)}(h:sub:`n`h]h/n}(hhh j;ubah}(h]h]h]h]h]uhhh j
ubh/) moments associated with the maximum P }(h) moments associated with the maximum P\ h j
hhh!NhNubh)}(h:sub:`n`h]h/n}(hhh jNubah}(h]h]h]h]h]uhhh j
ubh/w scattering
order requested in all materials. Of course, only a subset of these
moments applies to the lower-order P }(hw scattering
order requested in all materials. Of course, only a subset of these
moments applies to the lower-order P\ h j
hhh!NhNubh)}(h:sub:`n`h]h/n}(hhh jaubah}(h]h]h]h]h]uhhh j
ubh/ assignments.}(h assignments.h j
hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM|h jhhubh)}(h.. _fig9-2-66:h]h}(h]h]h]h]h]h fig9-2-66uhh
hM"h jhhh!jubj)}(hhh](j)}(h^.. figure:: figs/NEWT/fig66.svg
:align: center
:width: 600
Quadrature Parameters page.
h]h}(h]h]h]h]h]width600urifigs/NEWT/fig66.svgj}jjsuhjh jh!jhMubj)}(hQuadrature Parameters page.h]h/Quadrature Parameters page.}(hjh jubah}(h]h]h]h]h]uhjh!jhMh jubeh}(h](id337jeh]h] fig9-2-66ah]h]jcenteruhjhMh jhhh!jjf}jjzsjh}jjzsubh)}(h.. _9-2-5-2-12:h]h}(h]h]h]h]h]hid235uhh
hM"h jhhh!jubeh}(h](quadrature-parametersjeh]h](quadrature parameters
9-2-5-2-11eh]h]uhh#h jhhh!jhMzjf}jjsjh}jjsubh$)}(hhh](h))}(hMixture volumes listingh]h/Mixture volumes listing}(hj˥h jɥhhh!NhNubah}(h]h]h]h]h]uhh(h jƥhhh!jhMubh;)}(hX]The Mixture Volumes block (:numref:`fig9-2-67`) provides a summary of the
volume and volume fraction of each mixture in the problem, together with
the total volume. This block can be used as a simple check of the input
model by ensuring that the calculated volumes of mixtures used for a
given problem match the expected volumes or volume fractions.h](h/The Mixture Volumes block (}(hThe Mixture Volumes block (h jץhhh!NhNubj)}(h:numref:`fig9-2-67`h]jc)}(hjh]h/ fig9-2-67}(hhh jubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnj fig9-2-67uhjh!jhMh jץubh/X/) provides a summary of the
volume and volume fraction of each mixture in the problem, together with
the total volume. This block can be used as a simple check of the input
model by ensuring that the calculated volumes of mixtures used for a
given problem match the expected volumes or volume fractions.}(hX/) provides a summary of the
volume and volume fraction of each mixture in the problem, together with
the total volume. This block can be used as a simple check of the input
model by ensuring that the calculated volumes of mixtures used for a
given problem match the expected volumes or volume fractions.h jץhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jƥhhubh)}(h.. _fig9-2-67:h]h}(h]h]h]h]h]h fig9-2-67uhh
hM"h jƥhhh!jubj)}(hhh](j)}(hX.. figure:: figs/NEWT/fig67.svg
:align: center
:width: 500
Mixture Volumes page.
h]h}(h]h]h]h]h]width500urifigs/NEWT/fig67.svgj}jj&suhjh jh!jhMubj)}(hMixture Volumes page.h]h/Mixture Volumes page.}(hj*h j(ubah}(h]h]h]h]h]uhjh!jhMh jubeh}(h](id338jeh]h] fig9-2-67ah]h]jcenteruhjhMh jƥhhh!jjf}j;jsjh}jjsubh)}(h.. _9-2-5-2-13:h]h}(h]h]h]h]h]hid236uhh
hM"h jƥhhh!jubeh}(h](mixture-volumes-listingjeh]h](mixture volumes listing
9-2-5-2-12eh]h]uhh#h jhhh!jhMjf}jRjsjh}jjsubh$)}(hhh](h))}(hMixing table listingh]h/Mixing table listing}(hj\h jZhhh!NhNubah}(h]h]h]h]h]uhh(h jWhhh!jhMubh;)}(hX}The Mixing Table block summarizes the input mixing table, whether user
supplied or read from a SCALE-generated file. Number densities are in
units of atoms per barn-centimeter. Although optional, the mixing table
is printed by default. This default setting can be disabled by
specifying *prtmxtab*\ =no in the Parameter block. A sample mixing table
is shown in :numref:`fig9-2-68`.h](h/XThe Mixing Table block summarizes the input mixing table, whether user
supplied or read from a SCALE-generated file. Number densities are in
units of atoms per barn-centimeter. Although optional, the mixing table
is printed by default. This default setting can be disabled by
specifying }(hXThe Mixing Table block summarizes the input mixing table, whether user
supplied or read from a SCALE-generated file. Number densities are in
units of atoms per barn-centimeter. Although optional, the mixing table
is printed by default. This default setting can be disabled by
specifying h jhhhh!NhNubhA)}(h
*prtmxtab*h]h/prtmxtab}(hhh jqubah}(h]h]h]h]h]uhh@h jhubh/@ =no in the Parameter block. A sample mixing table
is shown in }(h@\ =no in the Parameter block. A sample mixing table
is shown in h jhhhh!NhNubj)}(h:numref:`fig9-2-68`h]jc)}(hjh]h/ fig9-2-68}(hhh jubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnj fig9-2-68uhjh!jhMh jhubh/.}(hjWh jhhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jWhhubh)}(h.. _fig9-2-68:h]h}(h]h]h]h]h]h fig9-2-68uhh
hM"h jWhhh!jubj)}(hhh](j)}(h].. figure:: figs/NEWT/fig68.svg
:align: center
:width: 500
Mixing Table Listing page.
h]h}(h]h]h]h]h]width500urifigs/NEWT/fig68.svgj}jjɦsuhjh jh!jhMubj)}(hMixing Table Listing page.h]h/Mixing Table Listing page.}(hjͦh j˦ubah}(h]h]h]h]h]uhjh!jhMh jubeh}(h](id339jeh]h] fig9-2-68ah]h]jcenteruhjhMh jWhhh!jjf}jަjsjh}jjsubh)}(h.. _9-2-5-2-14:h]h}(h]h]h]h]h]hid237uhh
hM"h jWhhh!jubeh}(h](mixing-table-listingjKeh]h](mixing table listing
9-2-5-2-13eh]h]uhh#h jhhh!jhMjf}jjAsjh}jKjAsubh$)}(hhh](h))}(hNuclide cross sectionsh]h/Nuclide cross sections}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMubh;)}(hXThe Nuclide Cross Section block is optional and is printed only when
*prtxsec*\ =yes is specified in the Parameter Block. The volume of
output generated is quite extensive, especially when a very fine group
library is used and/or a large number of nuclides are included in the
mixing table. The nuclide data are taken directly from the working
library used for the calculation. A sample showing a partial listing for
a single nuclide is shown in :numref:`fig9-2-69`.h](h/EThe Nuclide Cross Section block is optional and is printed only when
}(hEThe Nuclide Cross Section block is optional and is printed only when
h jhhh!NhNubhA)}(h *prtxsec*h]h/prtxsec}(hhh jubah}(h]h]h]h]h]uhh@h jubh/Xp =yes is specified in the Parameter Block. The volume of
output generated is quite extensive, especially when a very fine group
library is used and/or a large number of nuclides are included in the
mixing table. The nuclide data are taken directly from the working
library used for the calculation. A sample showing a partial listing for
a single nuclide is shown in }(hXp\ =yes is specified in the Parameter Block. The volume of
output generated is quite extensive, especially when a very fine group
library is used and/or a large number of nuclides are included in the
mixing table. The nuclide data are taken directly from the working
library used for the calculation. A sample showing a partial listing for
a single nuclide is shown in h jhhh!NhNubj)}(h:numref:`fig9-2-69`h]jc)}(hj)h]h/ fig9-2-69}(hhh j+ubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh j'ubah}(h]h]h]h]h]refdocj refdomainj5reftypenumrefrefexplicitrefwarnj fig9-2-69uhjh!jhMh jubh/.}(hjWh jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(hXJFollowing the block header, nuclide data are listed for all nuclides.
For each record, the same format is used. Nuclide data begin with a
listing of nuclide header information. This is followed by a listing of
the 1-D cross sections that are important in NEWT calculations. The
sample below shows a partial listing of the 1-D cross sections.
Following the 1-D cross section listing is the scattering matrix for the
nuclide. This abbreviated listing shows a portion of the
P\ :sub:`0` matrix for this nuclide; however, in a full listing, all
higher-order elements are printed as well.h](h/XFollowing the block header, nuclide data are listed for all nuclides.
For each record, the same format is used. Nuclide data begin with a
listing of nuclide header information. This is followed by a listing of
the 1-D cross sections that are important in NEWT calculations. The
sample below shows a partial listing of the 1-D cross sections.
Following the 1-D cross section listing is the scattering matrix for the
nuclide. This abbreviated listing shows a portion of the
P }(hXFollowing the block header, nuclide data are listed for all nuclides.
For each record, the same format is used. Nuclide data begin with a
listing of nuclide header information. This is followed by a listing of
the 1-D cross sections that are important in NEWT calculations. The
sample below shows a partial listing of the 1-D cross sections.
Following the 1-D cross section listing is the scattering matrix for the
nuclide. This abbreviated listing shows a portion of the
P\ h jQhhh!NhNubh)}(h:sub:`0`h]h/0}(hhh jZubah}(h]h]h]h]h]uhhh jQubh/e matrix for this nuclide; however, in a full listing, all
higher-order elements are printed as well.}(he matrix for this nuclide; however, in a full listing, all
higher-order elements are printed as well.h jQhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(hAs was indicated in the input description, specification of
*prtxsec*\ =1d can be used to obtain header and 1‑D cross section data
only, skipping the printing of scattering matrices.h](h/ =1d can be used to skip the printing of scattering matrices.}(h>\ =1d can be used to skip the printing of scattering matrices.h jmhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(h.. _fig9-2-70:h]h}(h]h]h]h]h]h fig9-2-70uhh
hM#h jhhh!jubj)}(hhh](j)}(hx.. figure:: figs/NEWT/fig70.svg
:align: center
:width: 1000
Partial listing of Mixture Cross section data pages.
h]h}(h]h]h]h]h]width1000urifigs/NEWT/fig70.svgj}jjsuhjh jh!jhMubj)}(h4Partial listing of Mixture Cross section data pages.h]h/4Partial listing of Mixture Cross section data pages.}(hjh jubah}(h]h]h]h]h]uhjh!jhMh jubeh}(h](id341jeh]h] fig9-2-70ah]h]jcenteruhjhMh jhhh!jjf}jjsjh}jjsubh)}(h.. _9-2-5-3:h]h}(h]h]h]h]h]hid239uhh
hM#h jhhh!jubeh}(h](mixture-cross-sectionsjէeh]h](mixture cross sections
9-2-5-2-15eh]h]uhh#h jhhh!jhMjf}j֨j˧sjh}jէj˧subeh}(h](
input-summaryjeh]h](
input summary9-2-5-2eh]h]uhh#h j/hhh!jhMjf}jjsjh}jjsubh$)}(hhh](h))}(hIteration historyh]h/Iteration history}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMubh;)}(hXThe next portion of NEWT output lists the iteration convergence
history for the iterative solution
(:numref:`fig9-2-71`). This information can be used to track and understand
the performance of the outer loop of the iterative solution. The first
column provides the outer iteration count. The second column lists the
system eigenvalue after each outer iteration. The third column lists
the change in the eigenvalue from the last outer iteration; this is
one of the parameters tested for convergence. The fourth column, “Max
Flux Delta,” gives the maximum change in cell flux for all cells and
all energy groups; this is also used as a convergence test. The next
column lists the cell number and energy group corresponding to the
maximum flux change in this iteration. The next two columns list the
same flux information for mixtures with fissionable nuclides. This can
be used to track spatial convergence in fuel when convergence is
slowed by significant scattering outside the fuel. Finally, the last
column provides information on the convergence of inners in each outer
iteration. Inner iterations do not need to converge within early outer
iterations, but final convergence will not be achieved until all
inners are converged. The maximum number of inner iterations per
energy group is set by the *inners=* parameter in the parameter input
block. After convergence is achieved, the table is terminated by
printing the final version of *k*\ :sub:`eff`.h](h/dThe next portion of NEWT output lists the iteration convergence
history for the iterative solution
(}(hdThe next portion of NEWT output lists the iteration convergence
history for the iterative solution
(h jhhh!NhNubj)}(h:numref:`fig9-2-71`h]jc)}(hjh]h/ fig9-2-71}(hhh jubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnj fig9-2-71uhjh!jhMh jubh/X). This information can be used to track and understand
the performance of the outer loop of the iterative solution. The first
column provides the outer iteration count. The second column lists the
system eigenvalue after each outer iteration. The third column lists
the change in the eigenvalue from the last outer iteration; this is
one of the parameters tested for convergence. The fourth column, “Max
Flux Delta,” gives the maximum change in cell flux for all cells and
all energy groups; this is also used as a convergence test. The next
column lists the cell number and energy group corresponding to the
maximum flux change in this iteration. The next two columns list the
same flux information for mixtures with fissionable nuclides. This can
be used to track spatial convergence in fuel when convergence is
slowed by significant scattering outside the fuel. Finally, the last
column provides information on the convergence of inners in each outer
iteration. Inner iterations do not need to converge within early outer
iterations, but final convergence will not be achieved until all
inners are converged. The maximum number of inner iterations per
energy group is set by the }(hX). This information can be used to track and understand
the performance of the outer loop of the iterative solution. The first
column provides the outer iteration count. The second column lists the
system eigenvalue after each outer iteration. The third column lists
the change in the eigenvalue from the last outer iteration; this is
one of the parameters tested for convergence. The fourth column, “Max
Flux Delta,” gives the maximum change in cell flux for all cells and
all energy groups; this is also used as a convergence test. The next
column lists the cell number and energy group corresponding to the
maximum flux change in this iteration. The next two columns list the
same flux information for mixtures with fissionable nuclides. This can
be used to track spatial convergence in fuel when convergence is
slowed by significant scattering outside the fuel. Finally, the last
column provides information on the convergence of inners in each outer
iteration. Inner iterations do not need to converge within early outer
iterations, but final convergence will not be achieved until all
inners are converged. The maximum number of inner iterations per
energy group is set by the h jhhh!NhNubhA)}(h *inners=*h]h/inners=}(hhh j%ubah}(h]h]h]h]h]uhh@h jubh/ parameter in the parameter input
block. After convergence is achieved, the table is terminated by
printing the final version of }(h parameter in the parameter input
block. After convergence is achieved, the table is terminated by
printing the final version of h jhhh!NhNubhA)}(h*k*h]h/k}(hhh j8ubah}(h]h]h]h]h]uhh@h jubh/ }(h\ h jhhh!NhNubh)}(h
:sub:`eff`h]h/eff}(hhh jKubah}(h]h]h]h]h]uhhh jubh/.}(hjWh jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(hXIf the parameter keyword *timed=* is set to *yes*, four additional
columns are introduced that give timing information on the solution
process, listing real (“wall clock”) time, elapsed CPU time since
beginning the iteration process, elapsed CPU time per outer iteration,
and an estimate of the fractional CPU usage during each outer.
:numref:`fig9-2-72` illustrates the form of output produced when *timed=yes*
is input. Additionally, a supplementary edit follows the iteration edit
when *timed=yes*, giving information on average time per transport sweep
(outer iteration) within different components of the solution. This edit
is especially useful when coarse-mesh finite-difference acceleration is
used, to assess the overhead of the CMFD accelerator.h](h/If the parameter keyword }(hIf the parameter keyword h jchhh!NhNubhA)}(h*timed=*h]h/timed=}(hhh jlubah}(h]h]h]h]h]uhh@h jcubh/ is set to }(h is set to h jchhh!NhNubhA)}(h*yes*h]h/yes}(hhh jubah}(h]h]h]h]h]uhh@h jcubh/X", four additional
columns are introduced that give timing information on the solution
process, listing real (“wall clock”) time, elapsed CPU time since
beginning the iteration process, elapsed CPU time per outer iteration,
and an estimate of the fractional CPU usage during each outer.
}(hX", four additional
columns are introduced that give timing information on the solution
process, listing real (“wall clock”) time, elapsed CPU time since
beginning the iteration process, elapsed CPU time per outer iteration,
and an estimate of the fractional CPU usage during each outer.
h jchhh!NhNubj)}(h:numref:`fig9-2-72`h]jc)}(hjh]h/ fig9-2-72}(hhh jubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnj fig9-2-72uhjh!jhMh jcubh/. illustrates the form of output produced when }(h. illustrates the form of output produced when h jchhh!NhNubhA)}(h*timed=yes*h]h/ timed=yes}(hhh jubah}(h]h]h]h]h]uhh@h jcubh/N
is input. Additionally, a supplementary edit follows the iteration edit
when }(hN
is input. Additionally, a supplementary edit follows the iteration edit
when h jchhh!NhNubhA)}(h*timed=yes*h]h/ timed=yes}(hhh jʩubah}(h]h]h]h]h]uhh@h jcubh/, giving information on average time per transport sweep
(outer iteration) within different components of the solution. This edit
is especially useful when coarse-mesh finite-difference acceleration is
used, to assess the overhead of the CMFD accelerator.}(h, giving information on average time per transport sweep
(outer iteration) within different components of the solution. This edit
is especially useful when coarse-mesh finite-difference acceleration is
used, to assess the overhead of the CMFD accelerator.h jchhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(h.. _fig9-2-71:h]h}(h]h]h]h]h]h fig9-2-71uhh
hM4#h jhhh!jubj)}(hhh](j)}(he.. figure:: figs/NEWT/fig71.svg
:align: center
:width: 600
Nominal iteration history output.
h]h}(h]h]h]h]h]width600urifigs/NEWT/fig71.svgj}jjsuhjh jh!jhM&ubj)}(h!Nominal iteration history output.h]h/!Nominal iteration history output.}(hjh jubah}(h]h]h]h]h]uhjh!jhM&h jubeh}(h](id342jeh]h] fig9-2-71ah]h]jcenteruhjhM&h jhhh!jjf}jjsjh}jjsubh)}(h.. _fig9-2-72:h]h}(h]h]h]h]h]h fig9-2-72uhh
hM<#h jhhh!jubj)}(hhh](j)}(hb.. figure:: figs/NEWT/fig72.svg
:align: center
:width: 600
Timed iteration history output.
h]h}(h]h]h]h]h]width600urifigs/NEWT/fig72.svgj}jj4suhjh j$h!jhM.ubj)}(hTimed iteration history output.h]h/Timed iteration history output.}(hj8h j6ubah}(h]h]h]h]h]uhjh!jhM.h j$ubeh}(h](id343j#eh]h] fig9-2-72ah]h]jcenteruhjhM.h jhhh!jjf}jIjsjh}j#jsubh)}(h.. _9-2-5-4:h]h}(h]h]h]h]h]hid240uhh
hMC#h jhhh!jubeh}(h](iteration-historyjϨeh]h](iteration history9-2-5-3eh]h]uhh#h j/hhh!jhMjf}j`jŨsjh}jϨjŨsubh$)}(hhh](h))}(hFour-factor formulah]h/Four-factor formula}(hjjh jhhhh!NhNubah}(h]h]h]h]h]uhh(h jehhh!jhM3ubh;)}(hXFollowing the iteration history listing, NEWT output provides edit
listing the four traditional components of the four-factor formula. This
is followed by an alternate three-group formulation that separates out
resonance and fast escape probabilities (:numref:`fig9-2-73`).h](h/Following the iteration history listing, NEWT output provides edit
listing the four traditional components of the four-factor formula. This
is followed by an alternate three-group formulation that separates out
resonance and fast escape probabilities (}(hFollowing the iteration history listing, NEWT output provides edit
listing the four traditional components of the four-factor formula. This
is followed by an alternate three-group formulation that separates out
resonance and fast escape probabilities (h jvhhh!NhNubj)}(h:numref:`fig9-2-73`h]jc)}(hjh]h/ fig9-2-73}(hhh jubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnj fig9-2-73uhjh!jhM5h jvubh/).}(h).h jvhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM5h jehhubh)}(h.. _fig9-2-73:h]h}(h]h]h]h]h]h fig9-2-73uhh
hMM#h jehhh!jubj)}(hhh](j)}(h~.. figure:: figs/NEWT/fig73.svg
:align: center
:width: 500
Four-factor formula with alternate three-group formulation.
h]h}(h]h]h]h]h]width500urifigs/NEWT/fig73.svgj}jjŪsuhjh jh!jhM?ubj)}(h;Four-factor formula with alternate three-group formulation.h]h/;Four-factor formula with alternate three-group formulation.}(hjɪh jǪubah}(h]h]h]h]h]uhjh!jhM?h jubeh}(h](id344jeh]h] fig9-2-73ah]h]jcenteruhjhM?h jehhh!jjf}jڪjsjh}jjsubh)}(h.. _9-2-5-5:h]h}(h]h]h]h]h]hid241uhh
hMT#h jehhh!jubeh}(h](four-factor-formulajYeh]h](four-factor formula9-2-5-4eh]h]uhh#h j/hhh!jhM3jf}jjOsjh}jYjOsubh$)}(hhh](h))}(hFine-group balance tablesh]h/Fine-group balance tables}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMDubh;)}(hXWFollowing the iteration history and flux convergence, a fine-group
balance table is provided for each mixture used in the calculation. Fine
group refers to the group structure of the library used for the
calculation. Broad-group data, discussed later, refer to a group
structure collapsed from the original fine-group structure. After tables
for all mixtures are printed, a last table provides a fine-group summary
for the entire problem (i.e., the volume-weighted average for all
mixtures). Balance tables are printed by default but may be disabled by
setting *prtbalnc=no* in the Parameter block.h](h/X2Following the iteration history and flux convergence, a fine-group
balance table is provided for each mixture used in the calculation. Fine
group refers to the group structure of the library used for the
calculation. Broad-group data, discussed later, refer to a group
structure collapsed from the original fine-group structure. After tables
for all mixtures are printed, a last table provides a fine-group summary
for the entire problem (i.e., the volume-weighted average for all
mixtures). Balance tables are printed by default but may be disabled by
setting }(hX2Following the iteration history and flux convergence, a fine-group
balance table is provided for each mixture used in the calculation. Fine
group refers to the group structure of the library used for the
calculation. Broad-group data, discussed later, refer to a group
structure collapsed from the original fine-group structure. After tables
for all mixtures are printed, a last table provides a fine-group summary
for the entire problem (i.e., the volume-weighted average for all
mixtures). Balance tables are printed by default but may be disabled by
setting h jhhh!NhNubhA)}(h
*prtbalnc=no*h]h/prtbalnc=no}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ in the Parameter block.}(h in the Parameter block.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMFh jhhubh;)}(hX:numref:`fig9-2-74` shows a clipped excerpt from the fine-group summary of an
output listing. Similar tables are produced for each mixture in the
problem for all energy groups in the problem. The header lists the NEWT
mixture number; the mixture ID (i.e., the SCALE mixture number); and the
mixture description, if provided in the original input specification.
The header also gives the number of computational cells in which the
mixture was present and the volume of the mixture in the problem.h](j)}(h:numref:`fig9-2-74`h]jc)}(hj/h]h/ fig9-2-74}(hhh j1ubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh j-ubah}(h]h]h]h]h]refdocj refdomainj;reftypenumrefrefexplicitrefwarnj fig9-2-74uhjh!jhMPh j)ubh/X shows a clipped excerpt from the fine-group summary of an
output listing. Similar tables are produced for each mixture in the
problem for all energy groups in the problem. The header lists the NEWT
mixture number; the mixture ID (i.e., the SCALE mixture number); and the
mixture description, if provided in the original input specification.
The header also gives the number of computational cells in which the
mixture was present and the volume of the mixture in the problem.}(hX shows a clipped excerpt from the fine-group summary of an
output listing. Similar tables are produced for each mixture in the
problem for all energy groups in the problem. The header lists the NEWT
mixture number; the mixture ID (i.e., the SCALE mixture number); and the
mixture description, if provided in the original input specification.
The header also gives the number of computational cells in which the
mixture was present and the volume of the mixture in the problem.h j)hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMPh jhhubh;)}(hXFor each mixture, two tables are printed. The first table provides a
balance of all sources and loss terms: the fixed source, the fission
source, in-scatter, out-scatter, absorption, leakage, n-2n production,
and the net balance of all terms for each energy group. The final row
lists the mixture total for all groups. The fixed source lists the
user-supplied source for fixed-source problems. This field is disabled
(set to zero) for eigenvalue calculations. The fission source is the
number of neutrons born into each energy group in the mixture. In this
example, the mixture is water, which is not fissile; hence, no fission
source is present. In‑scatter represents the number of neutrons
scattered into each group from all other groups; conversely, out‑scatter
is the loss from each energy group by scattering. Absorption is the
number of neutrons absorbed in reactions that do not emit a neutron
(e.g., n-γ). Leakage is the net loss of neutrons from the mixture to
another mixture or a nonreflective boundary, and n-2n is the effective
n-2n production rate calculated from a weighted sum of all n-\ *x*\ n
reactions. The balance table is the ratio of production to loss in each
energy group.h](h/XVFor each mixture, two tables are printed. The first table provides a
balance of all sources and loss terms: the fixed source, the fission
source, in-scatter, out-scatter, absorption, leakage, n-2n production,
and the net balance of all terms for each energy group. The final row
lists the mixture total for all groups. The fixed source lists the
user-supplied source for fixed-source problems. This field is disabled
(set to zero) for eigenvalue calculations. The fission source is the
number of neutrons born into each energy group in the mixture. In this
example, the mixture is water, which is not fissile; hence, no fission
source is present. In‑scatter represents the number of neutrons
scattered into each group from all other groups; conversely, out‑scatter
is the loss from each energy group by scattering. Absorption is the
number of neutrons absorbed in reactions that do not emit a neutron
(e.g., n-γ). Leakage is the net loss of neutrons from the mixture to
another mixture or a nonreflective boundary, and n-2n is the effective
n-2n production rate calculated from a weighted sum of all n- }(hXVFor each mixture, two tables are printed. The first table provides a
balance of all sources and loss terms: the fixed source, the fission
source, in-scatter, out-scatter, absorption, leakage, n-2n production,
and the net balance of all terms for each energy group. The final row
lists the mixture total for all groups. The fixed source lists the
user-supplied source for fixed-source problems. This field is disabled
(set to zero) for eigenvalue calculations. The fission source is the
number of neutrons born into each energy group in the mixture. In this
example, the mixture is water, which is not fissile; hence, no fission
source is present. In‑scatter represents the number of neutrons
scattered into each group from all other groups; conversely, out‑scatter
is the loss from each energy group by scattering. Absorption is the
number of neutrons absorbed in reactions that do not emit a neutron
(e.g., n-γ). Leakage is the net loss of neutrons from the mixture to
another mixture or a nonreflective boundary, and n-2n is the effective
n-2n production rate calculated from a weighted sum of all n-\ h jXhhh!NhNubhA)}(h*x*h]h/x}(hhh jaubah}(h]h]h]h]h]uhh@h jXubh/Y n
reactions. The balance table is the ratio of production to loss in each
energy group.}(hY\ n
reactions. The balance table is the ratio of production to loss in each
energy group.h jXhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMXh jhhubh;)}(hXThe second fine-group balance table, also shown in Figure 9.2.74, lists
other reactions rates of interest. The first two columns after the group
number list in-scatter broken into its upscatter and downscatter
components. The subsequent two columns provide a similar breakdown for
out-scatter from the energy group. Self-scatter is the amount of
within-group scattering occurring within each energy group. The fission
rate is the number of (n-fission) reactions occurring in each group. The
next column provides the transverse leakage (i.e., the product of the
flux and the DB\ :sup:`2` term). This column will provide only nonzero
values when a nonzero buckling height is specified in input. The final
column lists the total (scalar) flux for each energy group.h](h/XDThe second fine-group balance table, also shown in Figure 9.2.74, lists
other reactions rates of interest. The first two columns after the group
number list in-scatter broken into its upscatter and downscatter
components. The subsequent two columns provide a similar breakdown for
out-scatter from the energy group. Self-scatter is the amount of
within-group scattering occurring within each energy group. The fission
rate is the number of (n-fission) reactions occurring in each group. The
next column provides the transverse leakage (i.e., the product of the
flux and the DB }(hXDThe second fine-group balance table, also shown in Figure 9.2.74, lists
other reactions rates of interest. The first two columns after the group
number list in-scatter broken into its upscatter and downscatter
components. The subsequent two columns provide a similar breakdown for
out-scatter from the energy group. Self-scatter is the amount of
within-group scattering occurring within each energy group. The fission
rate is the number of (n-fission) reactions occurring in each group. The
next column provides the transverse leakage (i.e., the product of the
flux and the DB\ h jzhhh!NhNubj;)}(h:sup:`2`h]h/2}(hhh jubah}(h]h]h]h]h]uhj;h jzubh/ term). This column will provide only nonzero
values when a nonzero buckling height is specified in input. The final
column lists the total (scalar) flux for each energy group.}(h term). This column will provide only nonzero
values when a nonzero buckling height is specified in input. The final
column lists the total (scalar) flux for each energy group.h jzhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMkh jhhubh)}(h.. _fig9-2-74:h]h}(h]h]h]h]h]h fig9-2-74uhh
hM#h jhhh!jubj)}(hhh](j)}(ht.. figure:: figs/NEWT/fig74.svg
:align: center
:width: 1000
Partial mixture fine-group balance table output.
h]h}(h]h]h]h]h]width1000urifigs/NEWT/fig74.svgj}jjsuhjh jh!jhM}ubj)}(h0Partial mixture fine-group balance table output.h]h/0Partial mixture fine-group balance table output.}(hjh jubah}(h]h]h]h]h]uhjh!jhM}h jubeh}(h](id345jeh]h] fig9-2-74ah]h]jcenteruhjhM}h jhhh!jjf}j̫jsjh}jjsubh)}(h.. _9-2-5-6:h]h}(h]h]h]h]h]hid242uhh
hM#h jhhh!jubeh}(h](fine-group-balance-tablesjeh]h](fine-group balance tables9-2-5-5eh]h]uhh#h j/hhh!jhMDjf}jjsjh}jjsubh$)}(hhh](h))}(hPlanar fluxes and currentsh]h/Planar fluxes and currents}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMubh;)}(hX<If planar fluxes are requested, an edit is printed to provide fluxes and
currents on each line segment specified, identified by label
(:numref:`fig9-2-75`). Fine-group fluxes are listed for each energy group,
followed by x and y net currents and partial currents (+x, –x, +y, and
–y). Fluxes and currents are printed for each group in the input group
structure. The example below shows only partial listings of each for
simplicity. If a broad-group collapse is requested, the fine-group
output is followed by the set of fluxes and currents for each broad
energy group.h](h/If planar fluxes are requested, an edit is printed to provide fluxes and
currents on each line segment specified, identified by label
(}(hIf planar fluxes are requested, an edit is printed to provide fluxes and
currents on each line segment specified, identified by label
(h jhhh!NhNubj)}(h:numref:`fig9-2-75`h]jc)}(hjh]h/ fig9-2-75}(hhh jubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnj fig9-2-75uhjh!jhMh jubh/X). Fine-group fluxes are listed for each energy group,
followed by x and y net currents and partial currents (+x, –x, +y, and
–y). Fluxes and currents are printed for each group in the input group
structure. The example below shows only partial listings of each for
simplicity. If a broad-group collapse is requested, the fine-group
output is followed by the set of fluxes and currents for each broad
energy group.}(hX). Fine-group fluxes are listed for each energy group,
followed by x and y net currents and partial currents (+x, –x, +y, and
–y). Fluxes and currents are printed for each group in the input group
structure. The example below shows only partial listings of each for
simplicity. If a broad-group collapse is requested, the fine-group
output is followed by the set of fluxes and currents for each broad
energy group.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(hNote that discontinuity factors make internal use of planar fluxes to
determine the flux and current on each boundary. Hence, planar flux
edits will be present any time an ADF calculation is performed.h]h/Note that discontinuity factors make internal use of planar fluxes to
determine the flux and current on each boundary. Hence, planar flux
edits will be present any time an ADF calculation is performed.}(hj/h j-hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(h.. _fig9-2-75:h]h}(h]h]h]h]h]h fig9-2-75uhh
hM#h jhhh!jubj)}(hhh](j)}(h.. figure:: figs/NEWT/fig75.svg
:align: center
:width: 600
Example of planar flux and current output (continued below).
h]h}(h]h]h]h]h]width600urifigs/NEWT/fig75.svgj}jjVsuhjh jFh!jhMubj)}(hh]h/ fig9-2-81}(hhh j@ubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh j<ubah}(h]h]h]h]h]refdocj refdomainjJreftypenumrefrefexplicitrefwarnj fig9-2-81uhjh!jhM"h j3ubh/).}(h).h j3hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM"h j"hhubh)}(h.. _fig9-2-81:h]h}(h]h]h]h]h]h fig9-2-81uhh
hM:$h j"hhh!jubj)}(hhh](j)}(h.. figure:: figs/NEWT/fig81.svg
:align: center
:width: 800
Partial collapsing spectra listing for a case with no critical buckling correction.
h]h}(h]h]h]h]h]width800urifigs/NEWT/fig81.svgj}jjsuhjh jrh!jhM,ubj)}(hSPartial collapsing spectra listing for a case with no critical buckling correction.h]h/SPartial collapsing spectra listing for a case with no critical buckling correction.}(hjh jubah}(h]h]h]h]h]uhjh!jhM,h jrubeh}(h](id352jqeh]h] fig9-2-81ah]h]jcenteruhjhM,h j"hhh!jjf}jjgsjh}jqjgsubh)}(h
.. _9-2-5-12:h]h}(h]h]h]h]h]hid252uhh
hMA$h j"hhh!jubh$)}(hhh](h))}(hHomogenized cross sectionsh]h/Homogenized cross sections}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhM1ubh;)}(hXWhen homogenization is performed and parameter *prthmmix=yes* is set,
the final output section of a NEWT calculation is the homogenized cross
section edit, as shown in :numref:`fig9-2-82`. This information is generally
passed to nodal analysis codes and hence is presented in a slightly
different format from other cross sections. Output includes a
region-averaged k-infinity value, transport-corrected cross section, and
two interpretations of absorption. The first is the directly collapsed
absorption cross section, while the second (Total-Scatter) is a more
consistent definition of absorption as applied in nodal calculations.
The difference between the two definitions is the effective (n-2n)
cross section. Both cross sections exclude contributions from
:sup:`135`\ Xe and :sup:`149`\ Sm; microscopic cross sections and number
densities for these two nuclides are printed explicitly elsewhere in the
table. Nu*fission is the product of the fission cross section and the
number of neutrons produced per fission, while Kappa*fission is the
product of the fission cross section and the energy release per fission
(J). Inverse velocity is the inverse (1/x) of the group neutron speed.h](h//When homogenization is performed and parameter }(h/When homogenization is performed and parameter h jhhh!NhNubhA)}(h*prthmmix=yes*h]h/prthmmix=yes}(hhh j°ubah}(h]h]h]h]h]uhh@h jubh/k is set,
the final output section of a NEWT calculation is the homogenized cross
section edit, as shown in }(hk is set,
the final output section of a NEWT calculation is the homogenized cross
section edit, as shown in h jhhh!NhNubj)}(h:numref:`fig9-2-82`h]jc)}(hjװh]h/ fig9-2-82}(hhh jٰubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jհubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnj fig9-2-82uhjh!jhM3h jubh/XC. This information is generally
passed to nodal analysis codes and hence is presented in a slightly
different format from other cross sections. Output includes a
region-averaged k-infinity value, transport-corrected cross section, and
two interpretations of absorption. The first is the directly collapsed
absorption cross section, while the second (Total-Scatter) is a more
consistent definition of absorption as applied in nodal calculations.
The difference between the two definitions is the effective (n-2n)
cross section. Both cross sections exclude contributions from
}(hXC. This information is generally
passed to nodal analysis codes and hence is presented in a slightly
different format from other cross sections. Output includes a
region-averaged k-infinity value, transport-corrected cross section, and
two interpretations of absorption. The first is the directly collapsed
absorption cross section, while the second (Total-Scatter) is a more
consistent definition of absorption as applied in nodal calculations.
The difference between the two definitions is the effective (n-2n)
cross section. Both cross sections exclude contributions from
h jhhh!NhNubj;)}(h
:sup:`135`h]h/135}(hhh jubah}(h]h]h]h]h]uhj;h jubh/ Xe and }(h \ Xe and h jhhh!NhNubj;)}(h
:sup:`149`h]h/149}(hhh j
ubah}(h]h]h]h]h]uhj;h jubh/X Sm; microscopic cross sections and number
densities for these two nuclides are printed explicitly elsewhere in the
table. Nu*fission is the product of the fission cross section and the
number of neutrons produced per fission, while Kappa*fission is the
product of the fission cross section and the energy release per fission
(J). Inverse velocity is the inverse (1/x) of the group neutron speed.}(hX\ Sm; microscopic cross sections and number
densities for these two nuclides are printed explicitly elsewhere in the
table. Nu*fission is the product of the fission cross section and the
number of neutrons produced per fission, while Kappa*fission is the
product of the fission cross section and the energy release per fission
(J). Inverse velocity is the inverse (1/x) of the group neutron speed.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM3h jhhubh;)}(hThe table also lists the two-group isotropic scattering matrix and the
prompt fission fraction distribution. Finally, NEWT lists approximate
six-group decay constants (lambdas) and group fractions (betas) for each
group.h]h/The table also lists the two-group isotropic scattering matrix and the
prompt fission fraction distribution. Finally, NEWT lists approximate
six-group decay constants (lambdas) and group fractions (betas) for each
group.}(hj(h j&hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMEh jhhubh)}(h.. _fig9-2-82:h]h}(h]h]h]h]h]h fig9-2-82uhh
hM]$h jhhh!jubj)}(hhh](j)}(h.. figure:: figs/NEWT/fig82.svg
:align: center
:width: 1000
Homogenized cross section edit for nodal diffusion applications.
h]h}(h]h]h]h]h]width1000urifigs/NEWT/fig82.svgj}jjOsuhjh j?h!jhMOubj)}(h@Homogenized cross section edit for nodal diffusion applications.h]h/@Homogenized cross section edit for nodal diffusion applications.}(hjSh jQubah}(h]h]h]h]h]uhjh!jhMOh j?ubeh}(h](id353j>eh]h] fig9-2-82ah]h]jcenteruhjhMOh jhhh!jjf}jdj4sjh}j>j4subh)}(h
.. _9-2-5-13:h]h}(h]h]h]h]h]hid253uhh
hMd$h jhhh!jubeh}(h](homogenized-cross-sectionsjeh]h](homogenized cross sections9-2-5-12eh]h]uhh#h j"hhh!jhM1jf}j{jsjh}jjsubeh}(h](groupwise-form-factorsjeh]h](groupwise form factors9-2-5-11eh]h]uhh#h j/hhh!jhM jf}jjsjh}jjsubh$)}(hhh](h))}(hEnd-of-calculation bannerh]h/End-of-calculation banner}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMTubh;)}(hXNEWT output listings are terminated with an end-of-calculation banner
(shown in :numref:`fig9-2-83`) upon successful completion of a calculation. If
this banner is not present, then the calculation ended abnormally, and
the output listing must be reviewed to determine the cause of the error.
In general, the final lines of an output file describe the error
condition that caused the calculation to stop.h](h/PNEWT output listings are terminated with an end-of-calculation banner
(shown in }(hPNEWT output listings are terminated with an end-of-calculation banner
(shown in h jhhh!NhNubj)}(h:numref:`fig9-2-83`h]jc)}(hjh]h/ fig9-2-83}(hhh jubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnj fig9-2-83uhjh!jhMVh jubh/X1) upon successful completion of a calculation. If
this banner is not present, then the calculation ended abnormally, and
the output listing must be reviewed to determine the cause of the error.
In general, the final lines of an output file describe the error
condition that caused the calculation to stop.}(hX1) upon successful completion of a calculation. If
this banner is not present, then the calculation ended abnormally, and
the output listing must be reviewed to determine the cause of the error.
In general, the final lines of an output file describe the error
condition that caused the calculation to stop.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMVh jhhubh)}(h.. _fig9-2-83:h]h}(h]h]h]h]h]h fig9-2-83uhh
hMp$h jhhh!jubj)}(hhh](j)}(he.. figure:: figs/NEWT/fig83.svg
:align: center
:width: 800
End-of-calculation banner listing.
h]h}(h]h]h]h]h]width800urifigs/NEWT/fig83.svgj}jjsuhjh j۱h!jhMbubj)}(h"End-of-calculation banner listing.h]h/"End-of-calculation banner listing.}(hjh jubah}(h]h]h]h]h]uhjh!jhMbh j۱ubeh}(h](id354jڱeh]h] fig9-2-83ah]h]jcenteruhjhMbh jhhh!jjf}jjбsjh}jڱjбsubh)}(h
.. _9-2-5-14:h]h}(h]h]h]h]h]hid254uhh
hMw$h jhhh!jubh$)}(hhh](h))}(hPostscript graphics filesh]h/Postscript graphics files}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMgubh;)}(hXrTwo user-selectable options within NEWT provide the ability to generate
PostScript-based graphics files for visualization of both input
specifications and output results. By specification of *drawit=yes* in
the NEWT parameter block, NEWT will generate two PostScript-based plot
files: :file:`newtgrid.ps` and :file:`newtmatl.ps`. The former, a grayscale plot of the
line segments generated by NEWT based on the input specification, will
be generated if all body placement input is valid. If input contains
errors such that the code stops before grid generation routines are
completed, no :file:`newtgrid.ps` output is created.h](h/Two user-selectable options within NEWT provide the ability to generate
PostScript-based graphics files for visualization of both input
specifications and output results. By specification of }(hTwo user-selectable options within NEWT provide the ability to generate
PostScript-based graphics files for visualization of both input
specifications and output results. By specification of h j"hhh!NhNubhA)}(h*drawit=yes*h]h/
drawit=yes}(hhh j+ubah}(h]h]h]h]h]uhh@h j"ubh/R in
the NEWT parameter block, NEWT will generate two PostScript-based plot
files: }(hR in
the NEWT parameter block, NEWT will generate two PostScript-based plot
files: h j"hhh!NhNubjc)}(h:file:`newtgrid.ps`h]h/newtgrid.ps}(hnewtgrid.psh j>ubah}(h]h]fileah]h]h]rolefileuhjbh j"ubh/ and }(h and h j"hhh!NhNubjc)}(h:file:`newtmatl.ps`h]h/newtmatl.ps}(hnewtmatl.psh jUubah}(h]h]fileah]h]h]rolefileuhjbh j"ubh/X. The former, a grayscale plot of the
line segments generated by NEWT based on the input specification, will
be generated if all body placement input is valid. If input contains
errors such that the code stops before grid generation routines are
completed, no }(hX. The former, a grayscale plot of the
line segments generated by NEWT based on the input specification, will
be generated if all body placement input is valid. If input contains
errors such that the code stops before grid generation routines are
completed, no h j"hhh!NhNubjc)}(h:file:`newtgrid.ps`h]h/newtgrid.ps}(hnewtgrid.psh jlubah}(h]h]fileah]h]h]rolefileuhjbh j"ubh/ output is created.}(h output is created.h j"hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMih jhhubh;)}(hXYThe :file:`newtmatl.ps` plot illustrates the same grid structure but with
material placement indicated by color. At this time, no user control is
provided for color assignment or plot control. This plot also requires
complete grid generation; additionally, it requires completion of all
media placement routines before the plot will be produced.h](h/The }(hThe h jhhh!NhNubjc)}(h:file:`newtmatl.ps`h]h/newtmatl.ps}(hnewtmatl.psh jubah}(h]h]fileah]h]h]rolefileuhjbh jubh/XB plot illustrates the same grid structure but with
material placement indicated by color. At this time, no user control is
provided for color assignment or plot control. This plot also requires
complete grid generation; additionally, it requires completion of all
media placement routines before the plot will be produced.}(hXB plot illustrates the same grid structure but with
material placement indicated by color. At this time, no user control is
provided for color assignment or plot control. This plot also requires
complete grid generation; additionally, it requires completion of all
media placement routines before the plot will be produced.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMsh jhhubh;)}(hXEFigures used throughout this manual were generated from newtgrid and
newtmatl PostScript plot files. Files :file:`newtgrid.ps` and :file:`newtmatl.ps` are
automatically copied back from SCALE’s temporary directory to the
original location of the input case, with the names
*casename*.newtgrid.ps and *casename*.newtmatl.ps.h](h/kFigures used throughout this manual were generated from newtgrid and
newtmatl PostScript plot files. Files }(hkFigures used throughout this manual were generated from newtgrid and
newtmatl PostScript plot files. Files h jhhh!NhNubjc)}(h:file:`newtgrid.ps`h]h/newtgrid.ps}(hnewtgrid.psh jubah}(h]h]fileah]h]h]rolefileuhjbh jubh/ and }(h and h jhhh!NhNubjc)}(h:file:`newtmatl.ps`h]h/newtmatl.ps}(hnewtmatl.psh jϲubah}(h]h]fileah]h]h]rolefileuhjbh jubh/} are
automatically copied back from SCALE’s temporary directory to the
original location of the input case, with the names
}(h} are
automatically copied back from SCALE’s temporary directory to the
original location of the input case, with the names
h jhhh!NhNubhA)}(h
*casename*h]h/casename}(hhh jubah}(h]h]h]h]h]uhh@h jubh/.newtgrid.ps and }(h.newtgrid.ps and h jhhh!NhNubhA)}(h
*casename*h]h/casename}(hhh jubah}(h]h]h]h]h]uhh@h jubh/
.newtmatl.ps.}(h
.newtmatl.ps.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMyh jhhubh;)}(hXWhen *prtflux=yes* is input, NEWT will generate a set of flux plots
showing relative neutron number densities in each energy group. A plot
file will be generated with the name fluxplot\_\ *N*\ g.ps, where *N* is
the number of energy groups in the problem. If an energy collapse is
performed, an additional file named fluxplot\_\ *M*\ g.ps is created,
where M is the number of energy groups in the collapsed set.
:numref:`fig9-2-84` is an example of a flux plot output for the fast group of
a two-group flux collapse.h](h/When }(hWhen h jhhh!NhNubhA)}(h
*prtflux=yes*h]h/prtflux=yes}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ is input, NEWT will generate a set of flux plots
showing relative neutron number densities in each energy group. A plot
file will be generated with the name fluxplot_ }(h is input, NEWT will generate a set of flux plots
showing relative neutron number densities in each energy group. A plot
file will be generated with the name fluxplot\_\ h jhhh!NhNubhA)}(h*N*h]h/N}(hhh j.ubah}(h]h]h]h]h]uhh@h jubh/ g.ps, where }(h\ g.ps, where h jhhh!NhNubhA)}(h*N*h]h/N}(hhh jAubah}(h]h]h]h]h]uhh@h jubh/y is
the number of energy groups in the problem. If an energy collapse is
performed, an additional file named fluxplot_ }(hy is
the number of energy groups in the problem. If an energy collapse is
performed, an additional file named fluxplot\_\ h jhhh!NhNubhA)}(h*M*h]h/M}(hhh jTubah}(h]h]h]h]h]uhh@h jubh/P g.ps is created,
where M is the number of energy groups in the collapsed set.
}(hP\ g.ps is created,
where M is the number of energy groups in the collapsed set.
h jhhh!NhNubj)}(h:numref:`fig9-2-84`h]jc)}(hjih]h/ fig9-2-84}(hhh jkubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jgubah}(h]h]h]h]h]refdocj refdomainjureftypenumrefrefexplicitrefwarnj fig9-2-84uhjh!jhMh jubh/U is an example of a flux plot output for the fast group of
a two-group flux collapse.}(hU is an example of a flux plot output for the fast group of
a two-group flux collapse.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(h.. _fig9-2-84:h]h}(h]h]h]h]h]h fig9-2-84uhh
hM$h jhhh!jubj)}(hhh](j)}(hy.. figure:: figs/NEWT/fig84.png
:align: center
:width: 600
Example of a flux plot image created with prtflux=yes.
h]h}(h]h]h]h]h]width600urifigs/NEWT/fig84.pngj}jjsuhjh jh!jhMubj)}(h6Example of a flux plot image created with prtflux=yes.h]h/6Example of a flux plot image created with prtflux=yes.}(hjh jubah}(h]h]h]h]h]uhjh!jhMh jubeh}(h](id355jeh]h] fig9-2-84ah]h]jcenteruhjhMh jhhh!jjf}j³jsjh}jjsubh)}(h
.. _9-2-5-15:h]h}(h]h]h]h]h]hid255uhh
hM$h jhhh!jubeh}(h](postscript-graphics-filesjeh]h](postscript graphics files9-2-5-14eh]h]uhh#h jhhh!jhMgjf}jٳjsjh}jjsubeh}(h](end-of-calculation-bannerjteh]h](end-of-calculation banner9-2-5-13eh]h]uhh#h j/hhh!jhMTjf}jjjsjh}jtjjsubh$)}(hhh](h))}(hMedia zone editsh]h/Media zone edits}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMubh;)}(hXNEWT automatically determines “zones” representing spatially independent
regions of the same media. For example, in a fuel pin cell, the fuel,
clad, and moderator are all considered separate zones. In an array of
such pin cells, each unique location is a unique zone. Zone numbers and
the geometric location of each zone are listed in the *Geometry
Specification*\ ” in :ref:`9-2-5-2-6`.h](h/XWNEWT automatically determines “zones” representing spatially independent
regions of the same media. For example, in a fuel pin cell, the fuel,
clad, and moderator are all considered separate zones. In an array of
such pin cells, each unique location is a unique zone. Zone numbers and
the geometric location of each zone are listed in the }(hXWNEWT automatically determines “zones” representing spatially independent
regions of the same media. For example, in a fuel pin cell, the fuel,
clad, and moderator are all considered separate zones. In an array of
such pin cells, each unique location is a unique zone. Zone numbers and
the geometric location of each zone are listed in the h jhhh!NhNubhA)}(h*Geometry
Specification*h]h/Geometry
Specification}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ ” in }(h \ ” in h jhhh!NhNubj)}(h:ref:`9-2-5-2-6`h]j#)}(hjh]h/ 9-2-5-2-6}(hhh jubah}(h]h](jnstdstd-refeh]h]h]uhj"h jubah}(h]h]h]h]h]refdocj refdomainj$reftyperefrefexplicitrefwarnj 9-2-5-2-6uhjh!jhMh jubh/.}(hjWh jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(hUpon completion of a calculation, NEWT provides an output edit of each
zone by number, giving the mixture number, average flux, fission power,
and volume, as shown in :numref:`fig9-2-85`.h](h/Upon completion of a calculation, NEWT provides an output edit of each
zone by number, giving the mixture number, average flux, fission power,
and volume, as shown in }(hUpon completion of a calculation, NEWT provides an output edit of each
zone by number, giving the mixture number, average flux, fission power,
and volume, as shown in h j@hhh!NhNubj)}(h:numref:`fig9-2-85`h]jc)}(hjKh]h/ fig9-2-85}(hhh jMubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jIubah}(h]h]h]h]h]refdocj refdomainjWreftypenumrefrefexplicitrefwarnj fig9-2-85uhjh!jhMh j@ubh/.}(hjWh j@hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(h.. _fig9-2-85:h]h}(h]h]h]h]h]h fig9-2-85uhh
hM$h jhhh!jubj)}(hhh](j)}(hZ.. figure:: figs/NEWT/fig85.svg
:align: center
:width: 500
Media zone output edit.
h]h}(h]h]h]h]h]width500urifigs/NEWT/fig85.svgj}jjsuhjh j~h!jhMubj)}(hMedia zone output edit.h]h/Media zone output edit.}(hjh jubah}(h]h]h]h]h]uhjh!jhMh j~ubeh}(h](id356j}eh]h] fig9-2-85ah]h]jcenteruhjhMh jhhh!jjf}jjssjh}j}jssubeh}(h](media-zone-editsjҳeh]h](media zone edits9-2-5-15eh]h]uhh#h j/hhh!jhMjf}jjȳsjh}jҳjȳsubjeh}(h](examples-of-inputsid214eh]h](examples of inputs9-2-4eh]h]uhh#h h$)}(hhh](h))}(hmNEWT: A New Transport Algorithm for Two-Dimensional Discrete-Ordinates Analysis in Non-Orthogonal Geometriesh]h/mNEWT: A New Transport Algorithm for Two-Dimensional Discrete-Ordinates Analysis in Non-Orthogonal Geometries}(hjôh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhKubj)}(hhh]j)}(hA. Jessee, M. D. DeHart [1]_
h]j)}(hhh]j)}(hJessee, M. D. DeHart [1]_
h]h;)}(hJessee, M. D. DeHart [1]_h](h/Jessee, M. D. DeHart }(hJessee, M. D. DeHart h jݴubj)}(h[1]_h]h/[1]_}(hhh jubah}(h]id93ah]h]h]h]refidid359uhjh jݴubeh}(h]h]h]h]h]uhh:h!jhKh jٴubah}(h]h]h]h]h]uhjh jִubah}(h]h]h]h]h]jSjTjUhjVjWuhjh jҴubah}(h]h]h]h]h]uhjh jϴhhh!NhNubah}(h]h]h]h]h]jSjTjUhjVjWjXK
uhjh jhhh!jhKubh;)}(hABSTRACTh]h/ABSTRACT}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhK
h jhhubh;)}(hXNEWT (**N**\ ew **E**\ SC-based **W**\ eighting **T**\ ransport code) is
a multigroup discrete-ordinates radiation transport computer code with
flexible meshing capabilities that allow two-dimensional (2-D) neutron
transport calculations using complex geometric models. The differencing
scheme employed by NEWT, the Extended Step Characteristic approach,
allows a computational mesh based on arbitrary polygons. Such a mesh can
be used to closely approximate curved or irregular surfaces to provide
the capability to model problems that were formerly difficult or
impractical to model directly with discrete-ordinates methods. Automated
grid generation capabilities provide a simplified user input
specification in which elementary bodies can be defined and placed
within a problem domain. NEWT can be used for eigenvalue,
critical-buckling correction, and source calculations and it can be used
to prepare collapsed weighted cross sections in AMPX working library
format.h](h/NEWT (}(hNEWT (h j#hhh!NhNubj)}(h**N**h]h/N}(hhh j,ubah}(h]h]h]h]h]uhjh j#ubh/ ew }(h\ ew h j#hhh!NhNubj)}(h**E**h]h/E}(hhh j?ubah}(h]h]h]h]h]uhjh j#ubh/ SC-based }(h\ SC-based h j#hhh!NhNubj)}(h**W**h]h/W}(hhh jRubah}(h]h]h]h]h]uhjh j#ubh/ eighting }(h\ eighting h j#hhh!NhNubj)}(h**T**h]h/T}(hhh jeubah}(h]h]h]h]h]uhjh j#ubh/X ransport code) is
a multigroup discrete-ordinates radiation transport computer code with
flexible meshing capabilities that allow two-dimensional (2-D) neutron
transport calculations using complex geometric models. The differencing
scheme employed by NEWT, the Extended Step Characteristic approach,
allows a computational mesh based on arbitrary polygons. Such a mesh can
be used to closely approximate curved or irregular surfaces to provide
the capability to model problems that were formerly difficult or
impractical to model directly with discrete-ordinates methods. Automated
grid generation capabilities provide a simplified user input
specification in which elementary bodies can be defined and placed
within a problem domain. NEWT can be used for eigenvalue,
critical-buckling correction, and source calculations and it can be used
to prepare collapsed weighted cross sections in AMPX working library
format.}(hX\ ransport code) is
a multigroup discrete-ordinates radiation transport computer code with
flexible meshing capabilities that allow two-dimensional (2-D) neutron
transport calculations using complex geometric models. The differencing
scheme employed by NEWT, the Extended Step Characteristic approach,
allows a computational mesh based on arbitrary polygons. Such a mesh can
be used to closely approximate curved or irregular surfaces to provide
the capability to model problems that were formerly difficult or
impractical to model directly with discrete-ordinates methods. Automated
grid generation capabilities provide a simplified user input
specification in which elementary bodies can be defined and placed
within a problem domain. NEWT can be used for eigenvalue,
critical-buckling correction, and source calculations and it can be used
to prepare collapsed weighted cross sections in AMPX working library
format.h j#hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhKh jhhubh;)}(hXLike other SCALE modules, NEWT can be run as a standalone module or as
part of a SCALE sequence. NEWT has been incorporated into the SCALE
TRITON control module sequences. TRITON can be used simply to prepare
cross sections for a NEWT transport calculation and then automatically
execute NEWT. TRITON also provides the capability to perform 2-D
depletion calculations, in which the transport capabilities of NEWT are
combined with multiple ORIGEN depletion calculations to perform 2-D
depletion of complex geometries. In the TRITON depletion sequence, NEWT
can also be used to generate lattice-physics parameters and
cross sections for use in subsequent nodal core simulator calculations.
In addition, the SCALE TSUNAMI-2D sequence can be used to perform
sensitivity and uncertainty analysis of 2-D geometries, where NEWT is
used to compute the adjoint flux solution to generate sensitivity
coefficients for *k*\ :sub:`eff` and other responses of interest, with respect
to the cross sections used in the NEWT model.h](h/XLike other SCALE modules, NEWT can be run as a standalone module or as
part of a SCALE sequence. NEWT has been incorporated into the SCALE
TRITON control module sequences. TRITON can be used simply to prepare
cross sections for a NEWT transport calculation and then automatically
execute NEWT. TRITON also provides the capability to perform 2-D
depletion calculations, in which the transport capabilities of NEWT are
combined with multiple ORIGEN depletion calculations to perform 2-D
depletion of complex geometries. In the TRITON depletion sequence, NEWT
can also be used to generate lattice-physics parameters and
cross sections for use in subsequent nodal core simulator calculations.
In addition, the SCALE TSUNAMI-2D sequence can be used to perform
sensitivity and uncertainty analysis of 2-D geometries, where NEWT is
used to compute the adjoint flux solution to generate sensitivity
coefficients for }(hXLike other SCALE modules, NEWT can be run as a standalone module or as
part of a SCALE sequence. NEWT has been incorporated into the SCALE
TRITON control module sequences. TRITON can be used simply to prepare
cross sections for a NEWT transport calculation and then automatically
execute NEWT. TRITON also provides the capability to perform 2-D
depletion calculations, in which the transport capabilities of NEWT are
combined with multiple ORIGEN depletion calculations to perform 2-D
depletion of complex geometries. In the TRITON depletion sequence, NEWT
can also be used to generate lattice-physics parameters and
cross sections for use in subsequent nodal core simulator calculations.
In addition, the SCALE TSUNAMI-2D sequence can be used to perform
sensitivity and uncertainty analysis of 2-D geometries, where NEWT is
used to compute the adjoint flux solution to generate sensitivity
coefficients for h j~hhh!NhNubhA)}(h*k*h]h/k}(hhh jubah}(h]h]h]h]h]uhh@h j~ubh/ }(h\ h j~hhh!NhNubh)}(h
:sub:`eff`h]h/eff}(hhh jubah}(h]h]h]h]h]uhhh j~ubh/\ and other responses of interest, with respect
to the cross sections used in the NEWT model.}(h\ and other responses of interest, with respect
to the cross sections used in the NEWT model.h j~hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhKh jhhubh;)}(hACKNOWLEDGMENTSh]h/ACKNOWLEDGMENTS}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhK,h jhhubh;)}(hXThe author expresses gratitude to B. T. Rearden and S. M. Bowman for
their supervision of the SCALE project and review of the manuscript. The
author acknowledges R. Y. Lee of the U.S. Nuclear Regulatory Commission
(NRC) and A. P. Ulses (formerly NRC) for their support of the
development of NEWT. Appreciation is extended to G. Ilas, B. L.
Broadhead, Deokjung Lee (formerly ORNL), and B. J. Ade for their review
of this or previous versions of the manuscript. The efforts of Z. Zhong
(Argonne National Laboratory), A. P. Ulses (formerly NRC), K. S. Kim, C.
F. Weber, G. Ilas, and K. T. Clarno (Oak Ridge National Laboratory) in
methods development and testing of the code have been invaluable in the
continued evolution and improvement of the code.h]h/XThe author expresses gratitude to B. T. Rearden and S. M. Bowman for
their supervision of the SCALE project and review of the manuscript. The
author acknowledges R. Y. Lee of the U.S. Nuclear Regulatory Commission
(NRC) and A. P. Ulses (formerly NRC) for their support of the
development of NEWT. Appreciation is extended to G. Ilas, B. L.
Broadhead, Deokjung Lee (formerly ORNL), and B. J. Ade for their review
of this or previous versions of the manuscript. The efforts of Z. Zhong
(Argonne National Laboratory), A. P. Ulses (formerly NRC), K. S. Kim, C.
F. Weber, G. Ilas, and K. T. Clarno (Oak Ridge National Laboratory) in
methods development and testing of the code have been invaluable in the
continued evolution and improvement of the code.}(hjõh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhK.h jhhubh)}(h
.. _9-2-1:h]h}(h]h]h]h]h]hid94uhh
hMMh jhhh!jubh$)}(hhh](h))}(hIntroductionh]h/Introduction}(hjߵh jݵhhh!NhNubah}(h]h]h]h]h]uhh(h jڵhhh!jhK=ubh;)}(hXNEWT (**N**\ ew **E**\ SC-based **W**\ eighting **T**\ ransport code) is
a two-dimensional (2-D) discrete-ordinates transport code developed
based on the Extended Step Characteristic (ESC) approach :cite:`dehart_discrete_1992` for
spatial discretization on an arbitrary mesh structure. This
discretization scheme makes NEWT an extremely powerful and versatile
tool for deterministic calculations in real-world non-orthogonal problem
domains. The NEWT computer code evolved from the earlier
proof-of-principle CENTAUR code :cite:`dehart_discrete_1992` and has been developed to run
within SCALE. Thus, NEWT uses AMPX-formatted cross sections processed by
other SCALE modules. If cross sections are properly prepared, NEWT can
be run in stand-alone mode. NEWT can also be used within the TRITON
control module for transport analysis, depletion analysis, and
sensitivity and uncertainty analysis.h](h/NEWT (}(hNEWT (h jhhh!NhNubj)}(h**N**h]h/N}(hhh jubah}(h]h]h]h]h]uhjh jubh/ ew }(h\ ew h jhhh!NhNubj)}(h**E**h]h/E}(hhh jubah}(h]h]h]h]h]uhjh jubh/ SC-based }(h\ SC-based h jhhh!NhNubj)}(h**W**h]h/W}(hhh jubah}(h]h]h]h]h]uhjh jubh/ eighting }(h\ eighting h jhhh!NhNubj)}(h**T**h]h/T}(hhh j-ubah}(h]h]h]h]h]uhjh jubh/ ransport code) is
a two-dimensional (2-D) discrete-ordinates transport code developed
based on the Extended Step Characteristic (ESC) approach }(h\ ransport code) is
a two-dimensional (2-D) discrete-ordinates transport code developed
based on the Extended Step Characteristic (ESC) approach h jhhh!NhNubj)}(hdehart_discrete_1992h]j#)}(hjBh]h/[dehart_discrete_1992]}(hhh jDubah}(h]h]h]h]h]uhj"h j@ubah}(h]jKah]j5ah]h]h] refdomainj:reftypej< reftargetjBrefwarnsupport_smartquotesuhjh!jhK?h jhhubh/X( for
spatial discretization on an arbitrary mesh structure. This
discretization scheme makes NEWT an extremely powerful and versatile
tool for deterministic calculations in real-world non-orthogonal problem
domains. The NEWT computer code evolved from the earlier
proof-of-principle CENTAUR code }(hX( for
spatial discretization on an arbitrary mesh structure. This
discretization scheme makes NEWT an extremely powerful and versatile
tool for deterministic calculations in real-world non-orthogonal problem
domains. The NEWT computer code evolved from the earlier
proof-of-principle CENTAUR code h jhhh!NhNubj)}(hdehart_discrete_1992h]j#)}(hjch]h/[dehart_discrete_1992]}(hhh jeubah}(h]h]h]h]h]uhj"h jaubah}(h]jLah]j5ah]h]h] refdomainj:reftypej< reftargetjcrefwarnsupport_smartquotesuhjh!jhK?h jhhubh/XY and has been developed to run
within SCALE. Thus, NEWT uses AMPX-formatted cross sections processed by
other SCALE modules. If cross sections are properly prepared, NEWT can
be run in stand-alone mode. NEWT can also be used within the TRITON
control module for transport analysis, depletion analysis, and
sensitivity and uncertainty analysis.}(hXY and has been developed to run
within SCALE. Thus, NEWT uses AMPX-formatted cross sections processed by
other SCALE modules. If cross sections are properly prepared, NEWT can
be run in stand-alone mode. NEWT can also be used within the TRITON
control module for transport analysis, depletion analysis, and
sensitivity and uncertainty analysis.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhK?h jڵhhubh)}(h.. _9-2-1-1:h]h}(h]h]h]h]h]hid98uhh
hM`h jڵhhh!jubh$)}(hhh](h))}(hHow to use this manualh]h/How to use this manual}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhKPubh;)}(hXThis users’ manual is intended to assist both the novice and the expert
in the application of NEWT for transport analysis. As such, the document
is divided into subsections, each with a specific purpose. Not all
sections will be of value to all users. It is not intended that the user
of this manual read through the manual from start to end. Rather, the
manual is designed to serve as a reference, with each section meeting
different needs. This introductory section has been written to provide a
general overview of the background, nature, functionality, and
applications of NEWT; it should prove of interest to users at all
levels. :ref:`9-2-2` provides detail on the theory of NEWT in terms of
derivations, equations, and relationships used in the NEWT solution.
This information will be of interest to those with a background in
transport methods desiring a comprehensive understanding of the NEWT
solution scheme. However, this information may provide too much detail
or simply not be relevant for the beginning user or someone desiring to
improve or expand an existing model. These users will find :ref:`9-2-3`
to be of more value, where input data requirements and formats are
described in detail, along with examples of each data type. This
information is supplemented by :ref:`9-2-4`, in which complete sample
inputs with descriptions of the features of each model are provided.
:ref:`9-2-5` describes the components of an output listing obtained
from a successful NEWT calculation.h](h/X}This users’ manual is intended to assist both the novice and the expert
in the application of NEWT for transport analysis. As such, the document
is divided into subsections, each with a specific purpose. Not all
sections will be of value to all users. It is not intended that the user
of this manual read through the manual from start to end. Rather, the
manual is designed to serve as a reference, with each section meeting
different needs. This introductory section has been written to provide a
general overview of the background, nature, functionality, and
applications of NEWT; it should prove of interest to users at all
levels. }(hX}This users’ manual is intended to assist both the novice and the expert
in the application of NEWT for transport analysis. As such, the document
is divided into subsections, each with a specific purpose. Not all
sections will be of value to all users. It is not intended that the user
of this manual read through the manual from start to end. Rather, the
manual is designed to serve as a reference, with each section meeting
different needs. This introductory section has been written to provide a
general overview of the background, nature, functionality, and
applications of NEWT; it should prove of interest to users at all
levels. h jhhh!NhNubj)}(h:ref:`9-2-2`h]j#)}(hjh]h/9-2-2}(hhh jubah}(h]h](jnstdstd-refeh]h]h]uhj"h jubah}(h]h]h]h]h]refdocj refdomainjreftyperefrefexplicitrefwarnj9-2-2uhjh!jhKRh jubh/X provides detail on the theory of NEWT in terms of
derivations, equations, and relationships used in the NEWT solution.
This information will be of interest to those with a background in
transport methods desiring a comprehensive understanding of the NEWT
solution scheme. However, this information may provide too much detail
or simply not be relevant for the beginning user or someone desiring to
improve or expand an existing model. These users will find }(hX provides detail on the theory of NEWT in terms of
derivations, equations, and relationships used in the NEWT solution.
This information will be of interest to those with a background in
transport methods desiring a comprehensive understanding of the NEWT
solution scheme. However, this information may provide too much detail
or simply not be relevant for the beginning user or someone desiring to
improve or expand an existing model. These users will find h jhhh!NhNubj)}(h:ref:`9-2-3`h]j#)}(hjԶh]h/9-2-3}(hhh jֶubah}(h]h](jnstdstd-refeh]h]h]uhj"h jҶubah}(h]h]h]h]h]refdocj refdomainjreftyperefrefexplicitrefwarnj9-2-3uhjh!jhKRh jubh/
to be of more value, where input data requirements and formats are
described in detail, along with examples of each data type. This
information is supplemented by }(h
to be of more value, where input data requirements and formats are
described in detail, along with examples of each data type. This
information is supplemented by h jhhh!NhNubj)}(h:ref:`9-2-4`h]j#)}(hjh]h/9-2-4}(hhh jubah}(h]h](jnstdstd-refeh]h]h]uhj"h jubah}(h]h]h]h]h]refdocj refdomainjreftyperefrefexplicitrefwarnj9-2-4uhjh!jhKRh jubh/`, in which complete sample
inputs with descriptions of the features of each model are provided.
}(h`, in which complete sample
inputs with descriptions of the features of each model are provided.
h jhhh!NhNubj)}(h:ref:`9-2-5`h]j#)}(hjh]h/9-2-5}(hhh j ubah}(h]h](jnstdstd-refeh]h]h]uhj"h jubah}(h]h]h]h]h]refdocj refdomainj*reftyperefrefexplicitrefwarnj9-2-5uhjh!jhKRh jubh/[ describes the components of an output listing obtained
from a successful NEWT calculation.}(h[ describes the components of an output listing obtained
from a successful NEWT calculation.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhKRh jhhubh)}(h.. _9-2-1-2:h]h}(h]h]h]h]h]hid99uhh
hM|h jhhh!jubeh}(h](how-to-use-this-manualjeh]h](how to use this manual9-2-1-1eh]h]uhh#h jڵhhh!jhKPjf}jXjsjh}jjsubh$)}(hhh](h))}(h
Backgroundh]h/
Background}(hjbh j`hhh!NhNubah}(h]h]h]h]h]uhh(h j]hhh!jhKlubh;)}(hXThe radiation transport equation, a linearized derivative of the
Boltzmann equation, provides an exact description of a neutral-particle
radiation field in terms of the position, direction of travel, and
energy of every particle in the field. Both stochastic (Monte Carlo
simulation) and deterministic (direct numerical solution) forms of the
transport equation have been developed and are used extensively in
nuclear applications. Each approach has its strengths and weaknesses.
Stochastic approaches are extremely effective for problems with complex
geometries where the calculations of integral quantities, such as
radiation dose and neutron multiplication factors, are desired. However,
calculations to obtain accurate differential information, such as the
neutron flux as a function of space and energy, can be difficult and
inefficient at best and prone to inaccuracies (even if the integral
quantity is correct). Deterministic techniques, such as integral
transport, collision probability, diffusion theory, and
discrete-ordinates methods, are better suited for problems where
differential quantities, such as the neutron flux as a function of
energy or space, are desired. However, integral transport, collision
probability, and diffusion approximations are based on simplifying
assumptions, which can limit their applicability. The discrete-ordinates
approach is a more rigorous approximation to the transport equation but
is typically very limited in its flexibility to describe complex
geometric systems.h]h/XThe radiation transport equation, a linearized derivative of the
Boltzmann equation, provides an exact description of a neutral-particle
radiation field in terms of the position, direction of travel, and
energy of every particle in the field. Both stochastic (Monte Carlo
simulation) and deterministic (direct numerical solution) forms of the
transport equation have been developed and are used extensively in
nuclear applications. Each approach has its strengths and weaknesses.
Stochastic approaches are extremely effective for problems with complex
geometries where the calculations of integral quantities, such as
radiation dose and neutron multiplication factors, are desired. However,
calculations to obtain accurate differential information, such as the
neutron flux as a function of space and energy, can be difficult and
inefficient at best and prone to inaccuracies (even if the integral
quantity is correct). Deterministic techniques, such as integral
transport, collision probability, diffusion theory, and
discrete-ordinates methods, are better suited for problems where
differential quantities, such as the neutron flux as a function of
energy or space, are desired. However, integral transport, collision
probability, and diffusion approximations are based on simplifying
assumptions, which can limit their applicability. The discrete-ordinates
approach is a more rigorous approximation to the transport equation but
is typically very limited in its flexibility to describe complex
geometric systems.}(hjph jnhhh!NhNubah}(h]h]h]h]h]uhh:h!jhKnh j]hhubh;)}(hXfDiscrete-ordinates approaches are derived from the integro-differential
form of the Boltzmann transport equation, where space, time, and energy
dependencies are normally treated by the use of a finite‑difference
grid, while angular behavior is treated by considering a number of
discrete directions in space. The angular solution is coupled to a
scalar spatial solution via some form of numerical integration. Because
of the direct angular treatment of the discrete-ordinates approach,
angularly dependent distribution functions can be computed; thus, this
approach is the preferred method of solution in many specific
applications where angular anisotropy is important. However, as
indicated earlier, it is often limited in applicability because of the
geometric constraints of the orthogonal grid system associated with the
finite-difference numerical approximation.h]h/XfDiscrete-ordinates approaches are derived from the integro-differential
form of the Boltzmann transport equation, where space, time, and energy
dependencies are normally treated by the use of a finite‑difference
grid, while angular behavior is treated by considering a number of
discrete directions in space. The angular solution is coupled to a
scalar spatial solution via some form of numerical integration. Because
of the direct angular treatment of the discrete-ordinates approach,
angularly dependent distribution functions can be computed; thus, this
approach is the preferred method of solution in many specific
applications where angular anisotropy is important. However, as
indicated earlier, it is often limited in applicability because of the
geometric constraints of the orthogonal grid system associated with the
finite-difference numerical approximation.}(hj~h j|hhh!NhNubah}(h]h]h]h]h]uhh:h!jhKh j]hhubh)}(h.. _9-2-1-3:h]h}(h]h]h]h]h]hid100uhh
hMh j]hhh!jubeh}(h](
backgroundjQeh]h](
background9-2-1-2eh]h]uhh#h jڵhhh!jhKljf}jjGsjh}jQjGsubh$)}(hhh](h))}(h0Discrete-ordinates solution on an arbitrary gridh]h/0Discrete-ordinates solution on an arbitrary grid}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhKubh;)}(hXThe ESC approach was developed to obtain a discrete-ordinates solution
in complicated geometries to handle the needs of irregular
configurations. Deterministic solutions to the transport equation
generally calculate a solution in terms of the particle flux; the flux
is the product of particle density and speed and is a useful quantity in
the determination of reaction rates that characterize nuclear systems.
General 2-D *xy* discrete-ordinates methods perform calculations that
provide four side-averaged fluxes and a cell-averaged flux for each cell
in a rectangular problem grid; iteration is performed to obtain a
converged distribution. This approach is usually termed the
diamond-difference approach. Using the ESC approach, a more flexible and
completely arbitrary problem grid may be defined in terms of completely
arbitrary polygons. Side-averaged fluxes for each polygon in the problem
domain are computed and are used to calculate a cell-averaged flux. This
process is repeated for each cell in the problem domain, and as with the
traditional approach, iteration is performed for convergence. This
geometric flexibility is a significant enhancement to existing
technology, as it provides the capability to model problems that are
currently difficult or impractical to model directly.h](h/XThe ESC approach was developed to obtain a discrete-ordinates solution
in complicated geometries to handle the needs of irregular
configurations. Deterministic solutions to the transport equation
generally calculate a solution in terms of the particle flux; the flux
is the product of particle density and speed and is a useful quantity in
the determination of reaction rates that characterize nuclear systems.
General 2-D }(hXThe ESC approach was developed to obtain a discrete-ordinates solution
in complicated geometries to handle the needs of irregular
configurations. Deterministic solutions to the transport equation
generally calculate a solution in terms of the particle flux; the flux
is the product of particle density and speed and is a useful quantity in
the determination of reaction rates that characterize nuclear systems.
General 2-D h jhhh!NhNubhA)}(h*xy*h]h/xy}(hhh jubah}(h]h]h]h]h]uhh@h jubh/Xe discrete-ordinates methods perform calculations that
provide four side-averaged fluxes and a cell-averaged flux for each cell
in a rectangular problem grid; iteration is performed to obtain a
converged distribution. This approach is usually termed the
diamond-difference approach. Using the ESC approach, a more flexible and
completely arbitrary problem grid may be defined in terms of completely
arbitrary polygons. Side-averaged fluxes for each polygon in the problem
domain are computed and are used to calculate a cell-averaged flux. This
process is repeated for each cell in the problem domain, and as with the
traditional approach, iteration is performed for convergence. This
geometric flexibility is a significant enhancement to existing
technology, as it provides the capability to model problems that are
currently difficult or impractical to model directly.}(hXe discrete-ordinates methods perform calculations that
provide four side-averaged fluxes and a cell-averaged flux for each cell
in a rectangular problem grid; iteration is performed to obtain a
converged distribution. This approach is usually termed the
diamond-difference approach. Using the ESC approach, a more flexible and
completely arbitrary problem grid may be defined in terms of completely
arbitrary polygons. Side-averaged fluxes for each polygon in the problem
domain are computed and are used to calculate a cell-averaged flux. This
process is repeated for each cell in the problem domain, and as with the
traditional approach, iteration is performed for convergence. This
geometric flexibility is a significant enhancement to existing
technology, as it provides the capability to model problems that are
currently difficult or impractical to model directly.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhKh jhhubh)}(h.. _9-2-1-4:h]h}(h]h]h]h]h]hid101uhh
hMh jhhh!jubeh}(h](0discrete-ordinates-solution-on-an-arbitrary-gridjeh]h](0discrete-ordinates solution on an arbitrary grid9-2-1-3eh]h]uhh#h jڵhhh!jhKjf}jjsjh}jjsubh$)}(hhh](h))}(hFunctions performedh]h/Functions performed}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhKubh;)}(hX5NEWT provides multiple capabilities that can potentially be used in a
wide variety of application areas. These include 2-D eigenvalue
calculations, forward and adjoint flux solutions, multigroup flux
spectrum calculations, and cross section collapse calculations. NEWT
provides significant functionality to support lattice-physics
calculations, including assembly cross section homogenization and
collapse, calculation of assembly discontinuity factors (for internal
and reflected assemblies), diffusion coefficients, pin powers, and group
form factors. Used as part of the TRITON depletion sequence, NEWT
provides spatial fluxes, weighted multigroup cross sections, and power
distributions used for multi-material depletion calculations and coupled
depletion and branch calculations needed for lattice-physics analysis.h]h/X5NEWT provides multiple capabilities that can potentially be used in a
wide variety of application areas. These include 2-D eigenvalue
calculations, forward and adjoint flux solutions, multigroup flux
spectrum calculations, and cross section collapse calculations. NEWT
provides significant functionality to support lattice-physics
calculations, including assembly cross section homogenization and
collapse, calculation of assembly discontinuity factors (for internal
and reflected assemblies), diffusion coefficients, pin powers, and group
form factors. Used as part of the TRITON depletion sequence, NEWT
provides spatial fluxes, weighted multigroup cross sections, and power
distributions used for multi-material depletion calculations and coupled
depletion and branch calculations needed for lattice-physics analysis.}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhKh jhhubh)}(h
.. _9-2-2:h]h}(h]h]h]h]h]hid103uhh
hMh jhhh!jubeh}(h](jݷid102eh]h]9-2-1-4ah]jqah]uhh#h jڵhhh!jhKjKjf}jjӷsjh}jݷjӷsubeh}(h](jٵid95eh]h]9-2-1ah]j ah]uhh#h jhhh!jhK=jKjf}j"jϵsjh}jٵjϵsubh$)}(hhh](h))}(hTheory and Proceduresh]h/Theory and Procedures}(hj,h j*hhh!NhNubah}(h]h]h]h]h]uhh(h j'hhh!jhKubh;)}(hXTThis section provides the theoretical basis for the ESC discretization
technique, the NEWT solution algorithm, and cross section processing
procedures used by NEWT. Although this information is not necessary to
be able to use NEWT for transport calculations, it provides a deeper
understanding of the basic operations performed within NEWT.h]h/XTThis section provides the theoretical basis for the ESC discretization
technique, the NEWT solution algorithm, and cross section processing
procedures used by NEWT. Although this information is not necessary to
be able to use NEWT for transport calculations, it provides a deeper
understanding of the basic operations performed within NEWT.}(hj:h j8hhh!NhNubah}(h]h]h]h]h]uhh:h!jhKh j'hhubh)}(h.. _9-2-2-1:h]h}(h]h]h]h]h]hid105uhh
hMh j'hhh!jubh$)}(hhh](h))}(hBoltzmann transport equationh]h/Boltzmann transport equation}(hjVh jThhh!NhNubah}(h]h]h]h]h]uhh(h jQhhh!jhKubh;)}(hXThe neutron transport equation may be presented in various forms, and
simplifications are often applied to tailor the equation to the
requirements of a specific application. In nuclear engineering
applications, the transport equation is often written in terms of the
angular neutron flux as the dependent variable. The angular neutron flux
is defined as the product of the angular neutron density and the neutron
velocity. The time-independent form of the linear transport equation is
then expressed as :cite:`duderstadt_nuclear_nodate`h](h/XThe neutron transport equation may be presented in various forms, and
simplifications are often applied to tailor the equation to the
requirements of a specific application. In nuclear engineering
applications, the transport equation is often written in terms of the
angular neutron flux as the dependent variable. The angular neutron flux
is defined as the product of the angular neutron density and the neutron
velocity. The time-independent form of the linear transport equation is
then expressed as }(hXThe neutron transport equation may be presented in various forms, and
simplifications are often applied to tailor the equation to the
requirements of a specific application. In nuclear engineering
applications, the transport equation is often written in terms of the
angular neutron flux as the dependent variable. The angular neutron flux
is defined as the product of the angular neutron density and the neutron
velocity. The time-independent form of the linear transport equation is
then expressed as h jbhhh!NhNubj)}(hduderstadt_nuclear_nodateh]j#)}(hjmh]h/[duderstadt_nuclear_nodate]}(hhh joubah}(h]h]h]h]h]uhj"h jkubah}(h]jah]j5ah]h]h] refdomainj:reftypej< reftargetjmrefwarnsupport_smartquotesuhjh!jhKh jbhhubeh}(h]h]h]h]h]uhh:h!jhKh jQhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-1uhh
h jQhhh!jhNubj )}(h\Omega \cdot \nabla \psi(\mathbf{r}, \Omega, E)+\sigma_{t}(\mathbf{r}, E) \psi(\mathbf{r}, \Omega, E)=Q(\mathbf{r}, \Omega, E) ,h]h/\Omega \cdot \nabla \psi(\mathbf{r}, \Omega, E)+\sigma_{t}(\mathbf{r}, E) \psi(\mathbf{r}, \Omega, E)=Q(\mathbf{r}, \Omega, E) ,}(hhh jubah}(h]jah]h]h]h]docnamejnumberK_labeleq9-2-1nowrapjyjzuhj h!jhKh jQhhjf}jh}jjsubh;)}(hwhereh]h/where}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhKh jQhhubj1)}(hhh](h;)}(h:math:`\psi(\mathbf{r}, \Omega, E)` ≡ angular flux at position per unit volume, in direction :math:`\Omega` per unit solid
angle and at energy E per unit energy;h](jr)}(h#:math:`\psi(\mathbf{r}, \Omega, E)`h]h/\psi(\mathbf{r}, \Omega, E)}(hhh jubah}(h]h]h]h]h]uhjqh jubh/< ≡ angular flux at position per unit volume, in direction }(h< ≡ angular flux at position per unit volume, in direction h jubjr)}(h:math:`\Omega`h]h/\Omega}(hhh jԸubah}(h]h]h]h]h]uhjqh jubh/7 per unit solid
angle and at energy E per unit energy;}(h7 per unit solid
angle and at energy E per unit energy;h jubeh}(h]h]h]h]h]uhh:h!jhKh jubh;)}(hl:math:`\sigma_{t}(\mathbf{r}, E)` ≡ total macroscopic cross section at position **r** and energy E; andh](jr)}(h!:math:`\sigma_{t}(\mathbf{r}, E)`h]h/\sigma_{t}(\mathbf{r}, E)}(hhh jubah}(h]h]h]h]h]uhjqh jubh/3 ≡ total macroscopic cross section at position }(h3 ≡ total macroscopic cross section at position h jubj)}(h**r**h]h/r}(hhh jubah}(h]h]h]h]h]uhjh jubh/ and energy E; and}(h and energy E; andh jubeh}(h]h]h]h]h]uhh:h!jhKh jubh;)}(hQ ≡ source at position **r** per unit volume, in direction :math:`\Omega` per unit solid
angle and at energy E per unit energy.h](h/Q ≡ source at position }(hQ ≡ source at position h jubj)}(h**r**h]h/r}(hhh j&ubah}(h]h]h]h]h]uhjh jubh/ per unit volume, in direction }(h per unit volume, in direction h jubjr)}(h:math:`\Omega`h]h/\Omega}(hhh j9ubah}(h]h]h]h]h]uhjqh jubh/7 per unit solid
angle and at energy E per unit energy.}(h7 per unit solid
angle and at energy E per unit energy.h jubeh}(h]h]h]h]h]uhh:h!jhKh jubeh}(h]h]h]h]h]uhj1h jQhhh!jhNubh;)}(h2The source Q is generally composed of three terms:h]h/2The source Q is generally composed of three terms:}(hjZh jXhhh!NhNubah}(h]h]h]h]h]uhh:h!jhKh jQhhubj)}(hhh]j)}(ha scattering source,
h]h;)}(ha scattering source,h]h/a scattering source,}(hjoh jmubah}(h]h]h]h]h]uhh:h!jhKh jiubah}(h]h]h]h]h]uhjh jfhhh!jhNubah}(h]h]h]h]h]jSjCjUhjVjWuhjh jQhhh!jhKubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-2uhh
h jQhhh!jhNubj )}(hS(\mathbf{r}, \Omega, E)=\int_{4 \pi} d \Omega^{\prime} \int_{0}^{\infty} d E^{\prime} \sigma_{s}\left(\mathbf{r}, \Omega^{\prime} \rightarrow \Omega, E^{\prime} \rightarrow E\right) \psi\left(\mathbf{r}, \Omega^{\prime}, E^{\prime}\right) ,h]h/S(\mathbf{r}, \Omega, E)=\int_{4 \pi} d \Omega^{\prime} \int_{0}^{\infty} d E^{\prime} \sigma_{s}\left(\mathbf{r}, \Omega^{\prime} \rightarrow \Omega, E^{\prime} \rightarrow E\right) \psi\left(\mathbf{r}, \Omega^{\prime}, E^{\prime}\right) ,}(hhh jubah}(h]jah]h]h]h]docnamejnumberK`labeleq9-2-2nowrapjyjzuhj h!jhKh jQhhjf}jh}jjsubh;)}(hwhereh]h/where}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhKh jQhhubj1)}(hhh]h;)}(hX:math:`\sigma_{s}\left(\mathbf{r}, \Omega^{\prime} \rightarrow \Omega, E^{\prime} \rightarrow E\right)`≡ macroscopic scattering cross section at position **r** from initial energy
E′ and direction :math:`\Omega`′ to final energy E and direction :math:`\Omega`,h](jr)}(h:math:`\sigma_{s}\left(\mathbf{r}, \Omega^{\prime} \rightarrow \Omega, E^{\prime} \rightarrow E\right)`≡ macroscopic scattering cross section at position **r** from initial energy
E′ and direction :math:`h]h/_\sigma_{s}\left(\mathbf{r}, \Omega^{\prime} \rightarrow \Omega, E^{\prime} \rightarrow E\right)}(hhh jubah}(h]h]h]h]h]uhjqh jubh/+Omega`′ to final energy E and direction }(h+\Omega`′ to final energy E and direction h jubjr)}(h:math:`\Omega`h]h/\Omega}(hhh jιubah}(h]h]h]h]h]uhjqh jubh/,}(hj
h jubeh}(h]h]h]h]h]uhh:h!jhKh jubah}(h]h]h]h]h]uhj1h jQhhh!jhNubj)}(hhh]j)}(ha fission source,
h]h;)}(ha fission source,h]h/a fission source,}(hjh jubah}(h]h]h]h]h]uhh:h!jhKh jubah}(h]h]h]h]h]uhjh jhhh!jhNubah}(h]h]h]h]h]jSjCjUhjVjWjXKuhjh jQhhh!jhKubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-3uhh
h jQhhh!jhNubj )}(hF(\mathbf{r}, \Omega, E)=\chi(\mathbf{r}, E) \int_{0}^{\infty} d E^{\prime} v\left(\mathbf{r}, E^{\prime}\right) \sigma_{f}\left(\mathbf{r}, E^{\prime}\right) \psi\left(\mathbf{r}, \Omega, E^{\prime}\right) ,h]h/F(\mathbf{r}, \Omega, E)=\chi(\mathbf{r}, E) \int_{0}^{\infty} d E^{\prime} v\left(\mathbf{r}, E^{\prime}\right) \sigma_{f}\left(\mathbf{r}, E^{\prime}\right) \psi\left(\mathbf{r}, \Omega, E^{\prime}\right) ,}(hhh jubah}(h]jah]h]h]h]docnamejnumberKalabeleq9-2-3nowrapjyjzuhj h!jhKh jQhhjf}jh}jj
subh;)}(hwhereh]h/where}(hj.h j,hhh!NhNubah}(h]h]h]h]h]uhh:h!jhKh jQhhubj1)}(hhh](h;)}(h:math:`\sigma_{f}\left(\mathbf{r}, E^{\prime}\right)` ≡ macroscopic fission cross section at position **r** and energy E′ (assumed
to be isotropic),h](jr)}(h5:math:`\sigma_{f}\left(\mathbf{r}, E^{\prime}\right)`h]h/-\sigma_{f}\left(\mathbf{r}, E^{\prime}\right)}(hhh jAubah}(h]h]h]h]h]uhjqh j=ubh/4 ≡ macroscopic fission cross section at position }(h4 ≡ macroscopic fission cross section at position h j=ubj)}(h**r**h]h/r}(hhh jTubah}(h]h]h]h]h]uhjh j=ubh/+ and energy E′ (assumed
to be isotropic),}(h+ and energy E′ (assumed
to be isotropic),h j=ubeh}(h]h]h]h]h]uhh:h!jhKh j:ubh;)}(h:math:`v\left(\mathbf{r}, E^{\prime}\right)` ≡ number of neutrons released per fission event at position **r** and
energy E′,h](jr)}(h,:math:`v\left(\mathbf{r}, E^{\prime}\right)`h]h/$v\left(\mathbf{r}, E^{\prime}\right)}(hhh jqubah}(h]h]h]h]h]uhjqh jmubh/? ≡ number of neutrons released per fission event at position }(h? ≡ number of neutrons released per fission event at position h jmubj)}(h**r**h]h/r}(hhh jubah}(h]h]h]h]h]uhjh jmubh/ and
energy E′,}(h and
energy E′,h jmubeh}(h]h]h]h]h]uhh:h!jhMh j:ubh;)}(h`:math:`\chi(\mathbf{r}, E)` ≡ fraction of neutrons that are born at **r** and at energy E, andh](jr)}(h:math:`\chi(\mathbf{r}, E)`h]h/\chi(\mathbf{r}, E)}(hhh jubah}(h]h]h]h]h]uhjqh jubh/+ ≡ fraction of neutrons that are born at }(h+ ≡ fraction of neutrons that are born at h jubj)}(h**r**h]h/r}(hhh jubah}(h]h]h]h]h]uhjh jubh/ and at energy E, and}(h and at energy E, andh jubeh}(h]h]h]h]h]uhh:h!jhMh j:ubeh}(h]h]h]h]h]uhj1h jQhhh!jhNubj)}(hhh]j)}(h*an external or fixed source, S(**r** ,E).
h]h;)}(h)an external or fixed source, S(**r** ,E).h](h/an external or fixed source, S(}(han external or fixed source, S(h jںubj)}(h**r**h]h/r}(hhh jubah}(h]h]h]h]h]uhjh jںubh/ ,E).}(h ,E).h jںubeh}(h]h]h]h]h]uhh:h!jhMh jֺubah}(h]h]h]h]h]uhjh jӺhhh!jhNubah}(h]h]h]h]h]jSjCjUhjVjWjXKuhjh jQhhh!jhMubh;)}(hXIn general, the transport equation can be difficult to apply and can be
solved analytically only for highly idealized cases. Hence,
simplifications and numerical approximations are often necessary to
apply the equation in engineering applications. Traditional
discrete-ordinates methods are based on a finite-difference
approximation to solve the flux streaming (leakage) term. Such
differencing schemes are intimately tied to the coordinate system in
which the differencing equations are developed, and it becomes difficult
to represent non-orthogonal volumes within that coordinate system. For
example, it is not possible to exactly represent a cylinder in a 2-D
Cartesian coordinate system; one must approximate the cylinder with a
number of rectangular cells. A close approximation can require a large
number of computational cells. However, the ESC approach for
discretizing computational cells allows the use of non-orthogonal
computational cells composed of arbitrary polygons. Using this method,
practically any shape can be represented within a Cartesian grid to a
very close approximation. The ESC approach is discussed in the following
sections.h]h/XIn general, the transport equation can be difficult to apply and can be
solved analytically only for highly idealized cases. Hence,
simplifications and numerical approximations are often necessary to
apply the equation in engineering applications. Traditional
discrete-ordinates methods are based on a finite-difference
approximation to solve the flux streaming (leakage) term. Such
differencing schemes are intimately tied to the coordinate system in
which the differencing equations are developed, and it becomes difficult
to represent non-orthogonal volumes within that coordinate system. For
example, it is not possible to exactly represent a cylinder in a 2-D
Cartesian coordinate system; one must approximate the cylinder with a
number of rectangular cells. A close approximation can require a large
number of computational cells. However, the ESC approach for
discretizing computational cells allows the use of non-orthogonal
computational cells composed of arbitrary polygons. Using this method,
practically any shape can be represented within a Cartesian grid to a
very close approximation. The ESC approach is discussed in the following
sections.}(hj
h jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jQhhubh)}(h.. _9-2-2-2:h]h}(h]h]h]h]h]hid107uhh
hM.h jQhhh!jubeh}(h](boltzmann-transport-equationjPeh]h](boltzmann transport equation9-2-2-1eh]h]uhh#h j'hhh!jhKjf}j'jFsjh}jPjFsubh$)}(hhh](h))}(h%The step characteristic approximationh]h/%The step characteristic approximation}(hj1h j/hhh!NhNubah}(h]h]h]h]h]uhh(h j,hhh!jhMubh;)}(hXEfficient application of discrete-ordinates methods is difficult when
dealing with complicated non-orthogonal geometries because of the nature
of finite difference approximations for spatial derivatives. An
alternative to the discrete representation of the spatial variable is
achieved in the method of characteristics, in which the transport
equation is solved analytically along characteristic directions within a
computational cell. The angular flux is solved along the *s*-axis,
where this axis is oriented along the characteristic direction :math:`\Omega`. Since
only the angular flux in direction :math:`\Omega` is of concern, then the streaming
term can be rewritten ash](h/XEfficient application of discrete-ordinates methods is difficult when
dealing with complicated non-orthogonal geometries because of the nature
of finite difference approximations for spatial derivatives. An
alternative to the discrete representation of the spatial variable is
achieved in the method of characteristics, in which the transport
equation is solved analytically along characteristic directions within a
computational cell. The angular flux is solved along the }(hXEfficient application of discrete-ordinates methods is difficult when
dealing with complicated non-orthogonal geometries because of the nature
of finite difference approximations for spatial derivatives. An
alternative to the discrete representation of the spatial variable is
achieved in the method of characteristics, in which the transport
equation is solved analytically along characteristic directions within a
computational cell. The angular flux is solved along the h j=hhh!NhNubhA)}(h*s*h]h/s}(hhh jFubah}(h]h]h]h]h]uhh@h j=ubh/F-axis,
where this axis is oriented along the characteristic direction }(hF-axis,
where this axis is oriented along the characteristic direction h j=hhh!NhNubjr)}(h:math:`\Omega`h]h/\Omega}(hhh jYubah}(h]h]h]h]h]uhjqh j=ubh/,. Since
only the angular flux in direction }(h,. Since
only the angular flux in direction h j=hhh!NhNubjr)}(h:math:`\Omega`h]h/\Omega}(hhh jlubah}(h]h]h]h]h]uhjqh j=ubh/; is of concern, then the streaming
term can be rewritten as}(h; is of concern, then the streaming
term can be rewritten ash j=hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM h j,hhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-4uhh
h j,hhh!jhNubj )}(hJ\Omega \cdot \nabla \psi(\mathbf{r}, \Omega, E)=\frac{d \psi(s, E)}{d s} .h]h/J\Omega \cdot \nabla \psi(\mathbf{r}, \Omega, E)=\frac{d \psi(s, E)}{d s} .}(hhh jubah}(h]jah]h]h]h]docnamejnumberKblabeleq9-2-4nowrapjyjzuhj h!jhM+h j,hhjf}jh}jjsubh;)}(h\Hence :eq:`eq9-2-1` can be written in the characteristic form (omitting *E* for
clarity) ash](h/Hence }(hHence h jhhh!NhNubj)}(h
:eq:`eq9-2-1`h]jc)}(hjh]h/eq9-2-1}(hhh jubah}(h]h](jneqeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjqreftypejrefexplicitrefwarnjeq9-2-1uhjh!jhM0h jubh/6 can be written in the characteristic form (omitting }(h6 can be written in the characteristic form (omitting h jhhh!NhNubhA)}(h*E*h]h/E}(hhh jлubah}(h]h]h]h]h]uhh@h jubh/ for
clarity) as}(h for
clarity) ash jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM0h j,hhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-5uhh
h j,hhh!jhNubj )}(h;\frac{d \psi(s)}{d s}+\sigma_{t}(s) \psi(\mathrm{s})=Q(s) ,h]h/;\frac{d \psi(s)}{d s}+\sigma_{t}(s) \psi(\mathrm{s})=Q(s) ,}(hhh jubah}(h]jah]h]h]h]docnamejnumberKclabeleq9-2-5nowrapjyjzuhj h!jhM3h j,hhjf}jh}jjsubh;)}(hAwhich has a solution of the form :cite:`hildebrand_advanced_1976`h](h/!which has a solution of the form }(h!which has a solution of the form h jhhh!NhNubj)}(hhildebrand_advanced_1976h]j#)}(hjh]h/[hildebrand_advanced_1976]}(hhh jubah}(h]h]h]h]h]uhj"h jubah}(h]jah]j5ah]h]h] refdomainj:reftypej< reftargetjrefwarnsupport_smartquotesuhjh!jhM8h jhhubeh}(h]h]h]h]h]uhh:h!jhM8h j,hhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-6uhh
h j,hhh!jhNubj )}(hl\psi(s)=\psi_{0} e^{-\sigma_{t} s}+e^{-\sigma_{t} s} \int_{0}^{s} Q e^{\sigma_{t} s^{\prime}} d s^{\prime} ,h]h/l\psi(s)=\psi_{0} e^{-\sigma_{t} s}+e^{-\sigma_{t} s} \int_{0}^{s} Q e^{\sigma_{t} s^{\prime}} d s^{\prime} ,}(hhh j=ubah}(h]j<ah]h]h]h]docnamejnumberKdlabeleq9-2-6nowrapjyjzuhj h!jhM:h j,hhjf}jh}j<j3subh;)}(hXwhere s is the distance along the characteristic direction :math:`\Omega`, and
ψ\ :sub:`0` is the known angular flux at *s*\ =0. The value for
ψ\ :sub:`0` is given from boundary conditions for known cell sides, and
angular fluxes on unknown sides are computed using Eq. (9.2.6). Methods
for the determination of an appropriate value for ψ\ :sub:`0` and for
evaluation of the integral term vary in different solution
techniques.\ :sup:`4–9`\ :cite:`lewis_j_nodate,hildebrand_advanced_1976,alcouffe_review_1981,lathrop_spatial_1969,alcouffe_computational_1979,larsen_linear_1981,lathrop_spatial_1968`.
One of the simplest schemes employing the Method of Characteristics is
the Step Characteristic (SC) method developed by Lathrop :cite:`alcouffe_review_1981`. In
this approach, the source Q and macroscopic total cross section σt are
assumed to be constant within a computational cell and the angular flux
is assumed constant on the cell boundaries of incoming direction.
Integration of Eq. :eq:`eq9-2-6` can be performed to obtainh](h/;where s is the distance along the characteristic direction }(h;where s is the distance along the characteristic direction h jRhhh!NhNubjr)}(h:math:`\Omega`h]h/\Omega}(hhh j[ubah}(h]h]h]h]h]uhjqh jRubh/
, and
ψ }(h
, and
ψ\ h jRhhh!NhNubh)}(h:sub:`0`h]h/0}(hhh jnubah}(h]h]h]h]h]uhhh jRubh/ is the known angular flux at }(h is the known angular flux at h jRhhh!NhNubhA)}(h*s*h]h/s}(hhh jubah}(h]h]h]h]h]uhh@h jRubh/ =0. The value for
ψ }(h\ =0. The value for
ψ\ h jRhhh!NhNubh)}(h:sub:`0`h]h/0}(hhh jubah}(h]h]h]h]h]uhhh jRubh/ is given from boundary conditions for known cell sides, and
angular fluxes on unknown sides are computed using Eq. (9.2.6). Methods
for the determination of an appropriate value for ψ }(h is given from boundary conditions for known cell sides, and
angular fluxes on unknown sides are computed using Eq. (9.2.6). Methods
for the determination of an appropriate value for ψ\ h jRhhh!NhNubh)}(h:sub:`0`h]h/0}(hhh jubah}(h]h]h]h]h]uhhh jRubh/Q and for
evaluation of the integral term vary in different solution
techniques. }(hQ and for
evaluation of the integral term vary in different solution
techniques.\ h jRhhh!NhNubj;)}(h:sup:`4–9`h]h/4–9}(hhh jubah}(h]h]h]h]h]uhj;h jRubh/ }(h\ h jRhhh!NhNubj)}(hlewis_j_nodateh]j#)}(hjϼh]h/[lewis_j_nodate]}(hhh jѼubah}(h]h]h]h]h]uhj"h jͼubah}(h]jܖah]j5ah]h]h] refdomainj:reftypej< reftargetjϼrefwarnsupport_smartquotesuhjh!jhM?h jRhhubj)}(hhildebrand_advanced_1976h]j#)}(hjh]h/[hildebrand_advanced_1976]}(hhh jubah}(h]h]h]h]h]uhj"h jubah}(h]jah]j5ah]h]h] refdomainj:reftypej< reftargetjrefwarnsupport_smartquotesuhjh!jhM?h jRhhubj)}(halcouffe_review_1981h]j#)}(hjh]h/[alcouffe_review_1981]}(hhh j ubah}(h]h]h]h]h]uhj"h jubah}(h]jah]j5ah]h]h] refdomainj:reftypej< reftargetjrefwarnsupport_smartquotesuhjh!jhM?h jRhhubj)}(hlathrop_spatial_1969h]j#)}(hj#h]h/[lathrop_spatial_1969]}(hhh j%ubah}(h]h]h]h]h]uhj"h j!ubah}(h]jah]j5ah]h]h] refdomainj:reftypej< reftargetj#refwarnsupport_smartquotesuhjh!jhM?h jRhhubj)}(halcouffe_computational_1979h]j#)}(hj?h]h/[alcouffe_computational_1979]}(hhh jAubah}(h]h]h]h]h]uhj"h j=ubah}(h]j>ah]j5ah]h]h] refdomainj:reftypej< reftargetj?refwarnsupport_smartquotesuhjh!jhM?h jRhhubj)}(hlarsen_linear_1981h]j#)}(hj[h]h/[larsen_linear_1981]}(hhh j]ubah}(h]h]h]h]h]uhj"h jYubah}(h]j2ah]j5ah]h]h] refdomainj:reftypej< reftargetj[refwarnsupport_smartquotesuhjh!jhM?h jRhhubj)}(hlathrop_spatial_1968h]j#)}(hjwh]h/[lathrop_spatial_1968]}(hhh jyubah}(h]h]h]h]h]uhj"h juubah}(h]jvah]j5ah]h]h] refdomainj:reftypej< reftargetjwrefwarnsupport_smartquotesuhjh!jhM?h jRhhubh/.
One of the simplest schemes employing the Method of Characteristics is
the Step Characteristic (SC) method developed by Lathrop }(h.
One of the simplest schemes employing the Method of Characteristics is
the Step Characteristic (SC) method developed by Lathrop h jRhhh!NhNubj)}(halcouffe_review_1981h]j#)}(hjh]h/[alcouffe_review_1981]}(hhh jubah}(h]h]h]h]h]uhj"h jubah}(h]jah]j5ah]h]h] refdomainj:reftypej< reftargetjrefwarnsupport_smartquotesuhjh!jhM?h jRhhubh/. In
this approach, the source Q and macroscopic total cross section σt are
assumed to be constant within a computational cell and the angular flux
is assumed constant on the cell boundaries of incoming direction.
Integration of Eq. }(h. In
this approach, the source Q and macroscopic total cross section σt are
assumed to be constant within a computational cell and the angular flux
is assumed constant on the cell boundaries of incoming direction.
Integration of Eq. h jRhhh!NhNubj)}(h
:eq:`eq9-2-6`h]jc)}(hjh]h/eq9-2-6}(hhh jubah}(h]h](jneqeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjqreftypejŽrefexplicitrefwarnjeq9-2-6uhjh!jhM?h jRubh/ can be performed to obtain}(h can be performed to obtainh jRhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM?h j,hhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-7uhh
h j,hhh!jhNubj )}(hY\psi(s)=\psi_{0} e^{-\sigma_{t} s}+\frac{Q}{\sigma_{t}}\left(1-e^{-\sigma_{t} s}\right) .h]h/Y\psi(s)=\psi_{0} e^{-\sigma_{t} s}+\frac{Q}{\sigma_{t}}\left(1-e^{-\sigma_{t} s}\right) .}(hhh jubah}(h]jah]h]h]h]docnamejnumberKelabeleq9-2-7nowrapjyjzuhj h!jhMMh j,hhjf}jh}jjsubh;)}(hX!:numref:`fig9-2-1` shows a sample computational cell in which the SC method
can be applied. For a given characteristic direction :math:`\Omega`, the angular flux
on any unknown side may be expressed in terms of a suitable average of
fluxes from known sides, which contribute to the unknown side. For the
characteristic direction :math:`\Omega` shown in :numref:`fig9-2-1`, the unknown “top” flux
ψ\ :sub:`T` may be computed as a linearly weighted average of
contributions from known sides ψ\ :sub:`B` and ψ\ :sub:`L`. The fluxes
on each of the two known sides are taken to be constant along the length
of each side, representing the average angular flux in direction :math:`\Omega` and
must be specified from external boundary conditions or from a completed
calculation in an adjacent cell.h](j)}(h:numref:`fig9-2-1`h]jc)}(hjh]h/fig9-2-1}(hhh jubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnjfig9-2-1uhjh!jhMRh jubh/p shows a sample computational cell in which the SC method
can be applied. For a given characteristic direction }(hp shows a sample computational cell in which the SC method
can be applied. For a given characteristic direction h jhhh!NhNubjr)}(h:math:`\Omega`h]h/\Omega}(hhh j(ubah}(h]h]h]h]h]uhjqh jubh/, the angular flux
on any unknown side may be expressed in terms of a suitable average of
fluxes from known sides, which contribute to the unknown side. For the
characteristic direction }(h, the angular flux
on any unknown side may be expressed in terms of a suitable average of
fluxes from known sides, which contribute to the unknown side. For the
characteristic direction h jhhh!NhNubjr)}(h:math:`\Omega`h]h/\Omega}(hhh j;ubah}(h]h]h]h]h]uhjqh jubh/
shown in }(h
shown in h jhhh!NhNubj)}(h:numref:`fig9-2-1`h]jc)}(hjPh]h/fig9-2-1}(hhh jRubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jNubah}(h]h]h]h]h]refdocj refdomainj\reftypenumrefrefexplicitrefwarnjfig9-2-1uhjh!jhMRh jubh/!, the unknown “top” flux
ψ }(h!, the unknown “top” flux
ψ\ h jhhh!NhNubh)}(h:sub:`T`h]h/T}(hhh jsubah}(h]h]h]h]h]uhhh jubh/W may be computed as a linearly weighted average of
contributions from known sides ψ }(hW may be computed as a linearly weighted average of
contributions from known sides ψ\ h jhhh!NhNubh)}(h:sub:`B`h]h/B}(hhh jubah}(h]h]h]h]h]uhhh jubh/ and ψ }(h and ψ\ h jhhh!NhNubh)}(h:sub:`L`h]h/L}(hhh jubah}(h]h]h]h]h]uhhh jubh/. The fluxes
on each of the two known sides are taken to be constant along the length
of each side, representing the average angular flux in direction }(h. The fluxes
on each of the two known sides are taken to be constant along the length
of each side, representing the average angular flux in direction h jhhh!NhNubjr)}(h:math:`\Omega`h]h/\Omega}(hhh jubah}(h]h]h]h]h]uhjqh jubh/n and
must be specified from external boundary conditions or from a completed
calculation in an adjacent cell.}(hn and
must be specified from external boundary conditions or from a completed
calculation in an adjacent cell.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMRh j,hhubh;)}(hXThe set of characteristic directions is chosen from a quadrature set, so
that the resulting angular fluxes may be numerically integrated to
obtain a scalar flux. Knowing the lengths of the sides of a rectangular
cell (∆x and ∆y) and the direction cosines of :math:`\Omega` in the *x-y* plane
(μ and η), a function for the length *s* can easily be determined. The
solution for from Eq. :eq:`eq9-2-7` can then be integrated along the length of
each unknown side to determine the average angular flux of the unknown
side. Once the angular flux is known on all four sides, a neutron
balance on the cell can be used to determine the cell’s average angular
flux.h](h/XThe set of characteristic directions is chosen from a quadrature set, so
that the resulting angular fluxes may be numerically integrated to
obtain a scalar flux. Knowing the lengths of the sides of a rectangular
cell (∆x and ∆y) and the direction cosines of }(hXThe set of characteristic directions is chosen from a quadrature set, so
that the resulting angular fluxes may be numerically integrated to
obtain a scalar flux. Knowing the lengths of the sides of a rectangular
cell (∆x and ∆y) and the direction cosines of h jžhhh!NhNubjr)}(h:math:`\Omega`h]h/\Omega}(hhh jξubah}(h]h]h]h]h]uhjqh jžubh/ in the }(h in the h jžhhh!NhNubhA)}(h*x-y*h]h/x-y}(hhh jubah}(h]h]h]h]h]uhh@h jžubh/0 plane
(μ and η), a function for the length }(h0 plane
(μ and η), a function for the length h jžhhh!NhNubhA)}(h*s*h]h/s}(hhh jubah}(h]h]h]h]h]uhh@h jžubh/6 can easily be determined. The
solution for from Eq. }(h6 can easily be determined. The
solution for from Eq. h jžhhh!NhNubj)}(h
:eq:`eq9-2-7`h]jc)}(hj h]h/eq9-2-7}(hhh jubah}(h]h](jneqeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjqreftypejrefexplicitrefwarnjeq9-2-7uhjh!jhM^h jžubh/X can then be integrated along the length of
each unknown side to determine the average angular flux of the unknown
side. Once the angular flux is known on all four sides, a neutron
balance on the cell can be used to determine the cell’s average angular
flux.}(hX can then be integrated along the length of
each unknown side to determine the average angular flux of the unknown
side. Once the angular flux is known on all four sides, a neutron
balance on the cell can be used to determine the cell’s average angular
flux.h jžhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM^h j,hhubh;)}(hX Although the SC method described above is based on rectangular cells,
the derivation of Eq. :eq:`eq9-2-7` makes no assumptions about the shape of
the cell. It merely requires knowledge of the relationship between cell
edges along the direction of the characteristic. Hence, the method is
not restricted to any particular geometry. Because it is an extension of
the SC approach into generalized cells, the method developed here for
generalized geometries is referred to as the Extended Step
Characteristic (ESC) method.h](h/^Although the SC method described above is based on rectangular cells,
the derivation of Eq. }(h^Although the SC method described above is based on rectangular cells,
the derivation of Eq. h j0hhh!NhNubj)}(h
:eq:`eq9-2-7`h]jc)}(hj;h]h/eq9-2-7}(hhh j=ubah}(h]h](jneqeh]h]h]uhjbh j9ubah}(h]h]h]h]h]refdocj refdomainjqreftypejGrefexplicitrefwarnjeq9-2-7uhjh!jhMih j0ubh/X makes no assumptions about the shape of
the cell. It merely requires knowledge of the relationship between cell
edges along the direction of the characteristic. Hence, the method is
not restricted to any particular geometry. Because it is an extension of
the SC approach into generalized cells, the method developed here for
generalized geometries is referred to as the Extended Step
Characteristic (ESC) method.}(hX makes no assumptions about the shape of
the cell. It merely requires knowledge of the relationship between cell
edges along the direction of the characteristic. Hence, the method is
not restricted to any particular geometry. Because it is an extension of
the SC approach into generalized cells, the method developed here for
generalized geometries is referred to as the Extended Step
Characteristic (ESC) method.h j0hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMih j,hhubh)}(h
.. _fig9-2-1:h]h}(h]h]h]h]h]hfig9-2-1uhh
hMh j,hhh!jubj)}(hhh](j)}(h.. figure:: figs/NEWT/fig1.png
:align: center
:width: 400
Typical rectangular cell used in the step characteristic approach.
h]h}(h]h]h]h]h]width400urifigs/NEWT/fig1.pngj}jj}suhjh jmh!jhMwubj)}(hBTypical rectangular cell used in the step characteristic approach.h]h/BTypical rectangular cell used in the step characteristic approach.}(hjh jubah}(h]h]h]h]h]uhjh!jhMwh jmubeh}(h](id270jleh]h]fig9-2-1ah]h]jcenteruhjhMwh j,hhh!jjf}jjbsjh}jljbsubh)}(h.. _9-2-2-3:h]h}(h]h]h]h]h]hid117uhh
hMh j,hhh!jubeh}(h](%the-step-characteristic-approximationj eh]h](%the step characteristic approximation9-2-2-2eh]h]uhh#h j'hhh!jhMjf}jjsjh}j jsubh$)}(hhh](h))}(h)The Extended Step Characteristic approachh]h/)The Extended Step Characteristic approach}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhM|ubh;)}(hXiThe theory of the ESC approach is developed and explained in detail in
:cite:`dehart_discrete_1992`. However, the work has evolved significantly from that time, most
notably in the elimination of a requirement for non-reentrant polygons
(convex). The following subsections describe the primary equations
applied in the ESC approach as currently applied in NEWT.h](h/GThe theory of the ESC approach is developed and explained in detail in
}(hGThe theory of the ESC approach is developed and explained in detail in
h jhhh!NhNubj)}(hdehart_discrete_1992h]j#)}(hjʿh]h/[dehart_discrete_1992]}(hhh j̿ubah}(h]h]h]h]h]uhj"h jȿubah}(h]jMah]j5ah]h]h] refdomainj:reftypej< reftargetjʿrefwarnsupport_smartquotesuhjh!jhM~h jhhubh/X. However, the work has evolved significantly from that time, most
notably in the elimination of a requirement for non-reentrant polygons
(convex). The following subsections describe the primary equations
applied in the ESC approach as currently applied in NEWT.}(hX. However, the work has evolved significantly from that time, most
notably in the elimination of a requirement for non-reentrant polygons
(convex). The following subsections describe the primary equations
applied in the ESC approach as currently applied in NEWT.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM~h jhhubh)}(h.. _9-2-2-3-1:h]h}(h]h]h]h]h]hid119uhh
hMh jhhh!jubh$)}(hhh](h))}(hCell properties and geometriesh]h/Cell properties and geometries}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMubh;)}(hX{The two primary assumptions of the ESC method are that (1) within each
computational cell all properties (i.e., σt and Q) are uniform and
(2) cell boundaries are defined by straight lines. The restriction of a
computational cell to boundaries consisting of a set of straight lines
results in computational cells that are limited to polygons. However, as
will be seen later, no restrictions are placed on the shape of the
polygon or on the number of sides in the polygon. However, the size of
the polygon will be limited. In practical applications, properties are
unlikely to remain constant over significant volumes. Thus this
approach, like many other differencing schemes, is a poor approximation
when cell volumes become too large. Although σt is a material property
and may remain spatially constant, the source term Q, which depends on
the neutron flux, will vary with position. However, since the solution
would become exact in an infinitesimally small cell, it is expected that
the approximation will be reasonable for computational cells in which
the change in the flux (and therefore the source) is small over the
domain of the cell.h]h/X{The two primary assumptions of the ESC method are that (1) within each
computational cell all properties (i.e., σt and Q) are uniform and
(2) cell boundaries are defined by straight lines. The restriction of a
computational cell to boundaries consisting of a set of straight lines
results in computational cells that are limited to polygons. However, as
will be seen later, no restrictions are placed on the shape of the
polygon or on the number of sides in the polygon. However, the size of
the polygon will be limited. In practical applications, properties are
unlikely to remain constant over significant volumes. Thus this
approach, like many other differencing schemes, is a poor approximation
when cell volumes become too large. Although σt is a material property
and may remain spatially constant, the source term Q, which depends on
the neutron flux, will vary with position. However, since the solution
would become exact in an infinitesimally small cell, it is expected that
the approximation will be reasonable for computational cells in which
the change in the flux (and therefore the source) is small over the
domain of the cell.}(hj
h jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(hX
As a result of this geometric configuration, each side of a cell can
have one of three possible attributes relative to particle flow in a
given characteristic direction, as illustrated in :numref:`fig9-2-2`: (1) flow
can enter the cell when crossing a side (as shown by sides E and F in
the figure); (2) flow can exit the cell when crossing a side (sides B
and C); or (3) in a special case, flow may be parallel to the
orientation of a given side (sides A and D). Expressed mathematically,
these relationships becomeh](h/As a result of this geometric configuration, each side of a cell can
have one of three possible attributes relative to particle flow in a
given characteristic direction, as illustrated in }(hAs a result of this geometric configuration, each side of a cell can
have one of three possible attributes relative to particle flow in a
given characteristic direction, as illustrated in h jhhh!NhNubj)}(h:numref:`fig9-2-2`h]jc)}(hj$h]h/fig9-2-2}(hhh j&ubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh j"ubah}(h]h]h]h]h]refdocj refdomainj0reftypenumrefrefexplicitrefwarnjfig9-2-2uhjh!jhMh jubh/X<: (1) flow
can enter the cell when crossing a side (as shown by sides E and F in
the figure); (2) flow can exit the cell when crossing a side (sides B
and C); or (3) in a special case, flow may be parallel to the
orientation of a given side (sides A and D). Expressed mathematically,
these relationships become}(hX<: (1) flow
can enter the cell when crossing a side (as shown by sides E and F in
the figure); (2) flow can exit the cell when crossing a side (sides B
and C); or (3) in a special case, flow may be parallel to the
orientation of a given side (sides A and D). Expressed mathematically,
these relationships becomeh jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-8uhh
h jhhh!jhNubj )}(h4\text { Category } 1: \Omega_{k} \cdot \hat{n}_{i}<0h]h/4\text { Category } 1: \Omega_{k} \cdot \hat{n}_{i}<0}(hhh jWubah}(h]jVah]h]h]h]docnamejnumberKflabeleq9-2-8nowrapjyjzuhj h!jhMh jhhjf}jh}jVjMsubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-9uhh
h jhhh!jhNubj )}(h4\text { Category } 2: \Omega_{k} \cdot \hat{n}_{i}>0h]h/4\text { Category } 2: \Omega_{k} \cdot \hat{n}_{i}>0}(hhh jvubah}(h]juah]h]h]h]docnamejnumberKglabeleq9-2-9nowrapjyjzuhj h!jhMh jhhjf}jh}jujlsubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-10uhh
h jhhh!jhNubj )}(h4\text { Category } 1: \Omega_{k} \cdot \hat{n}_{i}=0h]h/4\text { Category } 1: \Omega_{k} \cdot \hat{n}_{i}=0}(hhh jubah}(h]jah]h]h]h]docnamejnumberKhlabeleq9-2-10nowrapjyjzuhj h!jhMh jhhjf}jh}jjsubh;)}(hX'where :math:`\hat{n}_{i}` is a unit vector in the cell-outward direction normal to
side \ *i*, and :math:`\Omega_{k}` is the *k*\ :sup:`th` discrete element of a set of
characteristic directions. A category 1 side will be termed an
“incoming” side with respect to the direction :math:`\Omega_{k}`, and a category 2 side
will be referred to as an “outgoing” side. For simplicity, the
definition of Eq. :eq:`eq9-2-10` will be included as a special case of
Eq. :eq:`eq9-2-8` for an incoming side. Thus, Eq. :eq:`eq9-2-8` can be rewritten ash](h/where }(hwhere h jhhh!NhNubjr)}(h:math:`\hat{n}_{i}`h]h/\hat{n}_{i}}(hhh jubah}(h]h]h]h]h]uhjqh jubh/B is a unit vector in the cell-outward direction normal to
side }(hB is a unit vector in the cell-outward direction normal to
side \ h jhhh!NhNubhA)}(h*i*h]h/i}(hhh jubah}(h]h]h]h]h]uhh@h jubh/, and }(h, and h jhhh!NhNubjr)}(h:math:`\Omega_{k}`h]h/
\Omega_{k}}(hhh jubah}(h]h]h]h]h]uhjqh jubh/ is the }(h is the h jhhh!NhNubhA)}(h*k*h]h/k}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ }(h\ h jhhh!NhNubj;)}(h :sup:`th`h]h/th}(hhh jubah}(h]h]h]h]h]uhj;h jubh/ discrete element of a set of
characteristic directions. A category 1 side will be termed an
“incoming” side with respect to the direction }(h discrete element of a set of
characteristic directions. A category 1 side will be termed an
“incoming” side with respect to the direction h jhhh!NhNubjr)}(h:math:`\Omega_{k}`h]h/
\Omega_{k}}(hhh jubah}(h]h]h]h]h]uhjqh jubh/o, and a category 2 side
will be referred to as an “outgoing” side. For simplicity, the
definition of Eq. }(ho, and a category 2 side
will be referred to as an “outgoing” side. For simplicity, the
definition of Eq. h jhhh!NhNubj)}(h:eq:`eq9-2-10`h]jc)}(hj'h]h/eq9-2-10}(hhh j)ubah}(h]h](jneqeh]h]h]uhjbh j%ubah}(h]h]h]h]h]refdocj refdomainjqreftypej3refexplicitrefwarnjeq9-2-10uhjh!jhMh jubh/, will be included as a special case of
Eq. }(h, will be included as a special case of
Eq. h jhhh!NhNubj)}(h
:eq:`eq9-2-8`h]jc)}(hjJh]h/eq9-2-8}(hhh jLubah}(h]h](jneqeh]h]h]uhjbh jHubah}(h]h]h]h]h]refdocj refdomainjqreftypejVrefexplicitrefwarnjeq9-2-8uhjh!jhMh jubh/" for an incoming side. Thus, Eq. }(h" for an incoming side. Thus, Eq. h jhhh!NhNubj)}(h
:eq:`eq9-2-8`h]jc)}(hjmh]h/eq9-2-8}(hhh joubah}(h]h](jneqeh]h]h]uhjbh jkubah}(h]h]h]h]h]refdocj refdomainjqreftypejyrefexplicitrefwarnjeq9-2-8uhjh!jhMh jubh/ can be rewritten as}(h can be rewritten ash jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-11uhh
h jhhh!jhNubj )}(hf\text { Side } i \text { is incoming with respect to } \Omega_{k}: \Omega_{k} \cdot \hat{n}_{i} \leq 0h]h/f\text { Side } i \text { is incoming with respect to } \Omega_{k}: \Omega_{k} \cdot \hat{n}_{i} \leq 0}(hhh jubah}(h]jah]h]h]h]docnamejnumberKilabeleq9-2-11nowrapjyjzuhj h!jhMh jhhjf}jh}jjsubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-12uhh
h jhhh!jhNubj )}(ha\text { Side } i \text { is outgoing with respect to } \Omega_{k}: \Omega_{k} \cdot \hat{n}_{i}>0h]h/a\text { Side } i \text { is outgoing with respect to } \Omega_{k}: \Omega_{k} \cdot \hat{n}_{i}>0}(hhh jubah}(h]jah]h]h]h]docnamejnumberKjlabeleq9-2-12nowrapjyjzuhj h!jhMh jhhjf}jh}jjsubh;)}(hTo solve for fluxes (flow) on outgoing sides of a cell, one must know fluxes on
all incoming sides. Each incoming side of each cell will be given from a
boundary condition or will be the outgoing side of an adjacent cell.h]h/To solve for fluxes (flow) on outgoing sides of a cell, one must know fluxes on
all incoming sides. Each incoming side of each cell will be given from a
boundary condition or will be the outgoing side of an adjacent cell.}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(h
.. _fig9-2-2:h]h}(h]h]h]h]h]hfig9-2-2uhh
hMh jhhh!jubj)}(hhh](j)}(h.. figure:: figs/NEWT/fig2.png
:align: center
:width: 500
Orientation of the sides of a cell with respect to a given direction vector.
h]h}(h]h]h]h]h]width500urifigs/NEWT/fig2.pngj}jjsuhjh jh!jhMubj)}(hLOrientation of the sides of a cell with respect to a given direction vector.h]h/LOrientation of the sides of a cell with respect to a given direction vector.}(hjh jubah}(h]h]h]h]h]uhjh!jhMh jubeh}(h](id271jeh]h]fig9-2-2ah]h]jcenteruhjhMh jhhh!jjf}jjsjh}jjsubh)}(h.. _9-2-2-3-2:h]h}(h]h]h]h]h]hid120uhh
hMh jhhh!jubeh}(h](cell-properties-and-geometriesjeh]h](cell properties and geometries 9-2-2-3-1eh]h]uhh#h jhhh!jhMjf}j'jsjh}jjsubh$)}(hhh](h))}(hRelationships between cellsh]h/Relationships between cells}(hj1h j/hhh!NhNubah}(h]h]h]h]h]uhh(h j,hhh!jhMubh;)}(hXIn the ESC method, the shape of the computational cell and the form of
the neutron balance differ from that used in traditional
discrete-ordinates methods. Nevertheless, the relationships between
cells are treated essentially as they would be in traditional
approaches. The entire problem domain is mapped in terms of a set of
finite cells. Each side of each cell is adjacent to either an external
boundary condition or another cell. For each discrete direction, cells
are swept in a predetermined order beginning at a known boundary (from a
specified external boundary condition) moving in the given direction.
The precise order of sweep is such that as the solution for one cell is
obtained, the cell provides sufficient boundary conditions for the
solution of an adjacent cell. Hence, cells sharing a given side share
the value of the angular flux on that side. Knowledge of the flux on all
incoming sides of a cell is sufficient to solve for all outgoing sides.
Once the angular flux has been determined for all sides of the cell for
the given direction, it is possible to use a neutron balance to compute
the average value of the angular flux within the cell.h]h/XIn the ESC method, the shape of the computational cell and the form of
the neutron balance differ from that used in traditional
discrete-ordinates methods. Nevertheless, the relationships between
cells are treated essentially as they would be in traditional
approaches. The entire problem domain is mapped in terms of a set of
finite cells. Each side of each cell is adjacent to either an external
boundary condition or another cell. For each discrete direction, cells
are swept in a predetermined order beginning at a known boundary (from a
specified external boundary condition) moving in the given direction.
The precise order of sweep is such that as the solution for one cell is
obtained, the cell provides sufficient boundary conditions for the
solution of an adjacent cell. Hence, cells sharing a given side share
the value of the angular flux on that side. Knowledge of the flux on all
incoming sides of a cell is sufficient to solve for all outgoing sides.
Once the angular flux has been determined for all sides of the cell for
the given direction, it is possible to use a neutron balance to compute
the average value of the angular flux within the cell.}(hj?h j=hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j,hhubh;)}(hXThe sweeping of cells continues for a given direction until all cell
fluxes have been calculated. The procedure is then repeated for the next
direction until all directions have been computed. At this point, the
cell average angular fluxes are known for each cell for each direction
used. Numerical quadrature can then be used to determine the average
scalar flux in each cell in the problem domain. The scalar fluxes are
used to determine fission and scattering reaction rates in each cell and
to update the value of the cell average source, Q. The process is
repeated, and the iteration continues until all scalar fluxes converge
to within a specified tolerance.h]h/XThe sweeping of cells continues for a given direction until all cell
fluxes have been calculated. The procedure is then repeated for the next
direction until all directions have been computed. At this point, the
cell average angular fluxes are known for each cell for each direction
used. Numerical quadrature can then be used to determine the average
scalar flux in each cell in the problem domain. The scalar fluxes are
used to determine fission and scattering reaction rates in each cell and
to update the value of the cell average source, Q. The process is
repeated, and the iteration continues until all scalar fluxes converge
to within a specified tolerance.}(hjMh jKhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j,hhubh;)}(hXbThis approach can be performed assuming a single energy group or any
number of discretized energy groups. The multigroup approach used in the
ESC method is the standard approach used in most multigroup methods and
is independent of the shape of each computational cell. Hence, the
details of the multigroup formalism will be omitted from this
discussion.h]h/XbThis approach can be performed assuming a single energy group or any
number of discretized energy groups. The multigroup approach used in the
ESC method is the standard approach used in most multigroup methods and
is independent of the shape of each computational cell. Hence, the
details of the multigroup formalism will be omitted from this
discussion.}(hj[h jYhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j,hhubh)}(h.. _9-2-2-3-3:h]h}(h]h]h]h]h]hid121uhh
hMh j,hhh!jubeh}(h](relationships-between-cellsj eh]h](relationships between cells 9-2-2-3-2eh]h]uhh#h jhhh!jhMjf}jxjsjh}j jsubh$)}(hhh](h))}(h$The set of characteristic directionsh]h/$The set of characteristic directions}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h j}hhh!jhMubh;)}(hXThe characteristic solution to the transport equation gives only the
angular flux in the direction of the characteristic direction vector |Omk|.
To compute interaction rates within a cell, one must compute scalar
fluxes. In computing the scalar flux from the set of angular fluxes, it
is convenient to choose the set of characteristic directions from an
appropriate quadrature set. Then the set of computed angular fluxes can
be combined with appropriate directional weights and summed to obtain a
scalar flux solution within a cell. Therefore, it is most appropriate to
choose characteristic directions from an established set of base points
and weights. Such quadrature sets that have been developed and used in
numerous earlier discrete- ordinates approaches are used in NEWT. No
restriction is placed on the nature or order of the quadrature set, as
long as it is sufficient to adequately represent the scalar flux from
computed angular fluxes.h](h/The characteristic solution to the transport equation gives only the
angular flux in the direction of the characteristic direction vector }(hThe characteristic solution to the transport equation gives only the
angular flux in the direction of the characteristic direction vector h jhhh!NhNubjr)}(h:math:`\Omega_{k}`h]h/
\Omega_{k}}(hhh jhhh!NhNubah}(h]h]h]h]h]uhjqh!NhNh jhhubh/X%.
To compute interaction rates within a cell, one must compute scalar
fluxes. In computing the scalar flux from the set of angular fluxes, it
is convenient to choose the set of characteristic directions from an
appropriate quadrature set. Then the set of computed angular fluxes can
be combined with appropriate directional weights and summed to obtain a
scalar flux solution within a cell. Therefore, it is most appropriate to
choose characteristic directions from an established set of base points
and weights. Such quadrature sets that have been developed and used in
numerous earlier discrete- ordinates approaches are used in NEWT. No
restriction is placed on the nature or order of the quadrature set, as
long as it is sufficient to adequately represent the scalar flux from
computed angular fluxes.}(hX%.
To compute interaction rates within a cell, one must compute scalar
fluxes. In computing the scalar flux from the set of angular fluxes, it
is convenient to choose the set of characteristic directions from an
appropriate quadrature set. Then the set of computed angular fluxes can
be combined with appropriate directional weights and summed to obtain a
scalar flux solution within a cell. Therefore, it is most appropriate to
choose characteristic directions from an established set of base points
and weights. Such quadrature sets that have been developed and used in
numerous earlier discrete- ordinates approaches are used in NEWT. No
restriction is placed on the nature or order of the quadrature set, as
long as it is sufficient to adequately represent the scalar flux from
computed angular fluxes.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j}hhubh)}(h.. _9-2-2-3-4:h]h}(h]h]h]h]h]hid122uhh
hM!h j}hhh!jubeh}(h]($the-set-of-characteristic-directionsjqeh]h]($the set of characteristic directions 9-2-2-3-3eh]h]uhh#h jhhh!jhMjf}jjgsjh}jqjgsubh$)}(hhh](h))}(hAngular flux at a cell boundaryh]h/Angular flux at a cell boundary}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMubh;)}(hXAs in the development of the SC method, as well as most
finite-difference methods, the ESC approach does not explicitly
determine the flux distribution as a function of position along the
sides of a computational cell. Instead, the angular flux on each cell
side is represented in terms of the average angular flux along the
length of the side. This is sufficient to determine the net leakage
across each cell side, which is needed in order to maintain a cell
balance. An average value of the flux for an incoming side must be
specified from a boundary condition or from the prior solution of an
adjacent cell. The average flux along a given outgoing side can be
computed by integrating the flux along the side and dividing by the
length of the side. However, the form of the distribution of the angular
flux on the side must be known to perform this integration. This
distribution can be determined from the properties of the cell and from
the average flux on each of the known incoming sides.h]h/XAs in the development of the SC method, as well as most
finite-difference methods, the ESC approach does not explicitly
determine the flux distribution as a function of position along the
sides of a computational cell. Instead, the angular flux on each cell
side is represented in terms of the average angular flux along the
length of the side. This is sufficient to determine the net leakage
across each cell side, which is needed in order to maintain a cell
balance. An average value of the flux for an incoming side must be
specified from a boundary condition or from the prior solution of an
adjacent cell. The average flux along a given outgoing side can be
computed by integrating the flux along the side and dividing by the
length of the side. However, the form of the distribution of the angular
flux on the side must be known to perform this integration. This
distribution can be determined from the properties of the cell and from
the average flux on each of the known incoming sides.d}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(hX>Because the characteristic solution [Eq. :eq:`eq9-2-6`] allows calculation of
the angular flux at any point *s* in a single cell given an initial
condition, the exact value of the flux can be computed at any point on
any outgoing side if the flux along each incoming side is known. As an
initial condition, it is assumed that the angular flux in some
characteristic direction is known at some starting point, *s* = 0
[i.e., ψ(0) = ψ\ :sub:`0`], on an incoming side. To determine the flux at
some point on an outgoing side, one need know only the distance *s*
measured along a characteristic direction to the appropriate incoming
side. This method can then be expanded to determine a functional form of
the flux for every point on the outgoing side, which can be integrated
to produce the average outgoing flux on the side.h](h/*Because the characteristic solution [Eq. }(h*Because the characteristic solution [Eq. h jhhh!NhNubj)}(h
:eq:`eq9-2-6`h]jc)}(hjh]h/eq9-2-6}(hhh jubah}(h]h](jneqeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjqreftypejrefexplicitrefwarnjeq9-2-6uhjh!jhM#h jubh/6] allows calculation of
the angular flux at any point }(h6] allows calculation of
the angular flux at any point h jhhh!NhNubhA)}(h*s*h]h/s}(hhh jubah}(h]h]h]h]h]uhh@h jubh/X* in a single cell given an initial
condition, the exact value of the flux can be computed at any point on
any outgoing side if the flux along each incoming side is known. As an
initial condition, it is assumed that the angular flux in some
characteristic direction is known at some starting point, }(hX* in a single cell given an initial
condition, the exact value of the flux can be computed at any point on
any outgoing side if the flux along each incoming side is known. As an
initial condition, it is assumed that the angular flux in some
characteristic direction is known at some starting point, h jhhh!NhNubhA)}(h*s*h]h/s}(hhh j$ubah}(h]h]h]h]h]uhh@h jubh/ = 0
[i.e., ψ(0) = ψ }(h = 0
[i.e., ψ(0) = ψ\ h jhhh!NhNubh)}(h:sub:`0`h]h/0}(hhh j7ubah}(h]h]h]h]h]uhhh jubh/q], on an incoming side. To determine the flux at
some point on an outgoing side, one need know only the distance }(hq], on an incoming side. To determine the flux at
some point on an outgoing side, one need know only the distance h jhhh!NhNubhA)}(h*s*h]h/s}(hhh jJubah}(h]h]h]h]h]uhh@h jubh/X
measured along a characteristic direction to the appropriate incoming
side. This method can then be expanded to determine a functional form of
the flux for every point on the outgoing side, which can be integrated
to produce the average outgoing flux on the side.}(hX
measured along a characteristic direction to the appropriate incoming
side. This method can then be expanded to determine a functional form of
the flux for every point on the outgoing side, which can be integrated
to produce the average outgoing flux on the side.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM#h jhhubh;)}(hXkTo develop a mathematical relationship between two arbitrary sides of a
cell, one should first consider two arbitrary coplanar line segments in
space whose endpoints each lie on a pair of parallel lines laid in the
direction |Omk|, as shown in :numref:`fig9-2-3`. Points B\ :sub:`1` and B\ :sub:`2`
can be considered to be the “projections” of A\ :sub:`1` and
A\ :sub:`2`, respectively, relative to |Omk|. Because *s* is the distance
between a point on segment A and its projection on segment B, it can be
seen that *s* varies linearly in moving from the “beginning” to the
“end” of the pair of segments.h](h/To develop a mathematical relationship between two arbitrary sides of a
cell, one should first consider two arbitrary coplanar line segments in
space whose endpoints each lie on a pair of parallel lines laid in the
direction }(hTo develop a mathematical relationship between two arbitrary sides of a
cell, one should first consider two arbitrary coplanar line segments in
space whose endpoints each lie on a pair of parallel lines laid in the
direction h jchhh!NhNubjr)}(hjh]h/
\Omega_{k}}(hhh jlhhh!NhNubah}(h]h]h]h]h]uhjqh!NhNh jchhubh/, as shown in }(h, as shown in h jchhh!NhNubj)}(h:numref:`fig9-2-3`h]jc)}(hjh]h/fig9-2-3}(hhh jubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh j~ubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnjfig9-2-3uhjh!jhM0h jcubh/
. Points B }(h
. Points B\ h jchhh!NhNubh)}(h:sub:`1`h]h/1}(hhh jubah}(h]h]h]h]h]uhhh jcubh/ and B }(h and B\ h jchhh!NhNubh)}(h:sub:`2`h]h/2}(hhh jubah}(h]h]h]h]h]uhhh jcubh/5
can be considered to be the “projections” of A }(h5
can be considered to be the “projections” of A\ h jchhh!NhNubh)}(h:sub:`1`h]h/1}(hhh jubah}(h]h]h]h]h]uhhh jcubh/ and
A }(h and
A\ h jchhh!NhNubh)}(h:sub:`2`h]h/2}(hhh jubah}(h]h]h]h]h]uhhh jcubh/, respectively, relative to }(h, respectively, relative to h jchhh!NhNubjr)}(hjh]h/
\Omega_{k}}(hhh jhhh!NhNubah}(h]h]h]h]h]uhjqh!NhNh jchhubh/
. Because }(h
. Because h jchhh!NhNubhA)}(h*s*h]h/s}(hhh jubah}(h]h]h]h]h]uhh@h jcubh/e is the distance
between a point on segment A and its projection on segment B, it can be
seen that }(he is the distance
between a point on segment A and its projection on segment B, it can be
seen that h jchhh!NhNubhA)}(h*s*h]h/s}(hhh jubah}(h]h]h]h]h]uhh@h jcubh/] varies linearly in moving from the “beginning” to the
“end” of the pair of segments.}(h] varies linearly in moving from the “beginning” to the
“end” of the pair of segments.h jchhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM0h jhhubh)}(h
.. _fig9-2-3:h]h}(h]h]h]h]h]hfig9-2-3uhh
hMMh jhhh!jubj)}(hhh](j)}(hs.. figure:: figs/NEWT/fig3.png
:align: center
:width: 400
Line endpoints for computation of average fluxes.
h]h}(h]h]h]h]h]width400urifigs/NEWT/fig3.pngj}jjHsuhjh j8h!jhM?ubj)}(h1Line endpoints for computation of average fluxes.h]h/1Line endpoints for computation of average fluxes.}(hjLh jJubah}(h]h]h]h]h]uhjh!jhM?h j8ubeh}(h](id272j7eh]h]fig9-2-3ah]h]jcenteruhjhM?h jhhh!jjf}j]j-sjh}j7j-subh;)}(hIf α is the distance along segment B measured from endpoint B\ :sub:`1`
and B has a total length L, then the distance *s* between A and B along
direction |Omk| can be written as a linear function in terms of the
position α:h](h/BIf α is the distance along segment B measured from endpoint B }(hBIf α is the distance along segment B measured from endpoint B\ h jchhh!NhNubh)}(h:sub:`1`h]h/1}(hhh jlubah}(h]h]h]h]h]uhhh jcubh/0
and B has a total length L, then the distance }(h0
and B has a total length L, then the distance h jchhh!NhNubhA)}(h*s*h]h/s}(hhh jubah}(h]h]h]h]h]uhh@h jcubh/! between A and B along
direction }(h! between A and B along
direction h jchhh!NhNubjr)}(hjh]h/
\Omega_{k}}(hhh jhhh!NhNubah}(h]h]h]h]h]uhjqh!NhNh jchhubh/B can be written as a linear function in terms of the
position α:}(hB can be written as a linear function in terms of the
position α:h jchhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMAh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-13uhh
h jhhh!jhNubj )}(h;s(\alpha)=s_{1}+\left(\frac{s_{2}-s_{1}}{L}\right) \alpha ,h]h/;s(\alpha)=s_{1}+\left(\frac{s_{2}-s_{1}}{L}\right) \alpha ,}(hhh jubah}(h]jah]h]h]h]docnamejnumberKklabeleq9-2-13nowrapjyjzuhj h!jhMFh jhhjf}jh}jjsubh;)}(hXwhere *s*\ :sub:`1` and *s*\ :sub:`2` are related to the distances along
the characteristic direction between A\ :sub:`1`, B\ :sub:`1` and
A\ :sub:`2`, B\ :sub:`2`, respectively. (It is important to note that
the length *s* is the same as the distance between the endpoints only
when the characteristic vector lies in the plane of the computational
cell. This is not necessarily the case, depending on the choice of
quadrature directions. This situation is discussed in more detail
later.)h](h/where }(hwhere h jhhh!NhNubhA)}(h*s*h]h/s}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ }(h\ h jhhh!NhNubh)}(h:sub:`1`h]h/1}(hhh jubah}(h]h]h]h]h]uhhh jubh/ and }(h and h jhhh!NhNubhA)}(h*s*h]h/s}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ }(hjh jubh)}(h:sub:`2`h]h/2}(hhh j
ubah}(h]h]h]h]h]uhhh jubh/L are related to the distances along
the characteristic direction between A }(hL are related to the distances along
the characteristic direction between A\ h jhhh!NhNubh)}(h:sub:`1`h]h/1}(hhh jubah}(h]h]h]h]h]uhhh jubh/, B }(h, B\ h jhhh!NhNubh)}(h:sub:`1`h]h/1}(hhh j0ubah}(h]h]h]h]h]uhhh jubh/ and
A }(h and
A\ h jhhh!NhNubh)}(h:sub:`2`h]h/2}(hhh jCubah}(h]h]h]h]h]uhhh jubh/, B }(hj/h jubh)}(h:sub:`2`h]h/2}(hhh jUubah}(h]h]h]h]h]uhhh jubh/9, respectively. (It is important to note that
the length }(h9, respectively. (It is important to note that
the length h jhhh!NhNubhA)}(h*s*h]h/s}(hhh jhubah}(h]h]h]h]h]uhh@h jubh/X
is the same as the distance between the endpoints only
when the characteristic vector lies in the plane of the computational
cell. This is not necessarily the case, depending on the choice of
quadrature directions. This situation is discussed in more detail
later.)}(hX
is the same as the distance between the endpoints only
when the characteristic vector lies in the plane of the computational
cell. This is not necessarily the case, depending on the choice of
quadrature directions. This situation is discussed in more detail
later.)h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMKh jhhubh;)}(hIf ψ(α) is the angular flux on side B at a distance α from B\ :sub:`1`,
then :math:`\bar{\psi}_{\mathrm{B}}`, the average value of ψ on B, is given byh](h/CIf ψ(α) is the angular flux on side B at a distance α from B }(hCIf ψ(α) is the angular flux on side B at a distance α from B\ h jhhh!NhNubh)}(h:sub:`1`h]h/1}(hhh jubah}(h]h]h]h]h]uhhh jubh/,
then }(h,
then h jhhh!NhNubjr)}(h:math:`\bar{\psi}_{\mathrm{B}}`h]h/\bar{\psi}_{\mathrm{B}}}(hhh jubah}(h]h]h]h]h]uhjqh jubh/+, the average value of ψ on B, is given by}(h+, the average value of ψ on B, is given byh jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMTh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-14uhh
h jhhh!jhNubj )}(hT\bar{\psi}_{B}=\frac{\int_{0}^{L} \psi(s(\alpha)) d \alpha}{\int_{0}^{L} d \alpha} .h]h/T\bar{\psi}_{B}=\frac{\int_{0}^{L} \psi(s(\alpha)) d \alpha}{\int_{0}^{L} d \alpha} .}(hhh jubah}(h]jah]h]h]h]docnamejnumberKllabeleq9-2-14nowrapjyjzuhj h!jhMWh jhhjf}jh}jjsubh;)}(hEquation :eq:`eq9-2-6`, the solution to the characteristic equation in the
step approximation, can be rewritten in terms of the average known
angular flux on side Ah](h/
Equation }(h
Equation h jhhh!NhNubj)}(h
:eq:`eq9-2-6`h]jc)}(hjh]h/eq9-2-6}(hhh jubah}(h]h](jneqeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjqreftypejrefexplicitrefwarnjeq9-2-6uhjh!jhM\h jubh/, the solution to the characteristic equation in the
step approximation, can be rewritten in terms of the average known
angular flux on side A}(h, the solution to the characteristic equation in the
step approximation, can be rewritten in terms of the average known
angular flux on side Ah jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM\h jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-15uhh
h jhhh!jhNubj )}(hY\psi_{B}(s)=\left(\bar{\psi}_{A}-Q / \sigma_{t}\right) e^{-\sigma_{t} s}+Q / \sigma_{t} .h]h/Y\psi_{B}(s)=\left(\bar{\psi}_{A}-Q / \sigma_{t}\right) e^{-\sigma_{t} s}+Q / \sigma_{t} .}(hhh jubah}(h]jah]h]h]h]docnamejnumberKmlabeleq9-2-15nowrapjyjzuhj h!jhM`h jhhjf}jh}jjsubh;)}(haInserting Eqs. :eq:`eq9-2-13` and :eq:`eq9-2-15` into Eq. :eq:`eq9-2-14` and simplifying
yieldsh](h/Inserting Eqs. }(hInserting Eqs. h j&hhh!NhNubj)}(h:eq:`eq9-2-13`h]jc)}(hj1h]h/eq9-2-13}(hhh j3ubah}(h]h](jneqeh]h]h]uhjbh j/ubah}(h]h]h]h]h]refdocj refdomainjqreftypej=refexplicitrefwarnjeq9-2-13uhjh!jhMeh j&ubh/ and }(h and h j&hhh!NhNubj)}(h:eq:`eq9-2-15`h]jc)}(hjTh]h/eq9-2-15}(hhh jVubah}(h]h](jneqeh]h]h]uhjbh jRubah}(h]h]h]h]h]refdocj refdomainjqreftypej`refexplicitrefwarnjeq9-2-15uhjh!jhMeh j&ubh/ into Eq. }(h into Eq. h j&hhh!NhNubj)}(h:eq:`eq9-2-14`h]jc)}(hjwh]h/eq9-2-14}(hhh jyubah}(h]h](jneqeh]h]h]uhjbh juubah}(h]h]h]h]h]refdocj refdomainjqreftypejrefexplicitrefwarnjeq9-2-14uhjh!jhMeh j&ubh/ and simplifying
yields}(h and simplifying
yieldsh j&hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMeh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-16uhh
h jhhh!jhNubj )}(h\bar{\psi}_{B}=\frac{1}{L} \int_{0}^{L}\left[\left(\bar{\psi}_{A}-Q / \sigma_{t}\right) \exp \left(-\sigma_{t}\left(s_{1}+\left(\frac{s_{2}-s_{1}}{L}\right) \alpha\right)\right)+Q / \sigma_{t}\right] d \alpha .h]h/\bar{\psi}_{B}=\frac{1}{L} \int_{0}^{L}\left[\left(\bar{\psi}_{A}-Q / \sigma_{t}\right) \exp \left(-\sigma_{t}\left(s_{1}+\left(\frac{s_{2}-s_{1}}{L}\right) \alpha\right)\right)+Q / \sigma_{t}\right] d \alpha .}(hhh jubah}(h]jah]h]h]h]docnamejnumberKnlabeleq9-2-16nowrapjyjzuhj h!jhMhh jhhjf}jh}jjsubh;)}(hFor the special case in which A and B are parallel,
*s*\ :sub:`1` = *s*\ :sub:`2` and the second term in the exponential
drops out. Equation :eq:`eq9-2-16` can easily be integrated to obtainh](h/4For the special case in which A and B are parallel,
}(h4For the special case in which A and B are parallel,
h jhhh!NhNubhA)}(h*s*h]h/s}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ }(h\ h jhhh!NhNubh)}(h:sub:`1`h]h/1}(hhh jubah}(h]h]h]h]h]uhhh jubh/ = }(h = h jhhh!NhNubhA)}(h*s*h]h/s}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ }(hjh jubh)}(h:sub:`2`h]h/2}(hhh jubah}(h]h]h]h]h]uhhh jubh/= and the second term in the exponential
drops out. Equation }(h= and the second term in the exponential
drops out. Equation h jhhh!NhNubj)}(h:eq:`eq9-2-16`h]jc)}(hjh]h/eq9-2-16}(hhh jubah}(h]h](jneqeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjqreftypejrefexplicitrefwarnjeq9-2-16uhjh!jhMmh jubh/# can easily be integrated to obtain}(h# can easily be integrated to obtainh jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMmh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-17uhh
h jhhh!jhNubj )}(h`\bar{\psi}_{B}=\left(\bar{\psi}_{A}-Q / \sigma_{t}\right) e^{-\sigma_{t} s_{1}}+Q / \sigma_{t} .h]h/`\bar{\psi}_{B}=\left(\bar{\psi}_{A}-Q / \sigma_{t}\right) e^{-\sigma_{t} s_{1}}+Q / \sigma_{t} .}(hhh jDubah}(h]jCah]h]h]h]docnamejnumberKolabeleq9-2-17nowrapjyjzuhj h!jhMqh jhhjf}jh}jCj:subh;)}(heIn the more general case, *s*\ :sub:`1` ≠ *s*\ :sub:`2`, the result is
slightly more complicated:h](h/In the more general case, }(hIn the more general case, h jYhhh!NhNubhA)}(h*s*h]h/s}(hhh jbubah}(h]h]h]h]h]uhh@h jYubh/ }(h\ h jYhhh!NhNubh)}(h:sub:`1`h]h/1}(hhh juubah}(h]h]h]h]h]uhhh jYubh/ ≠ }(h ≠ h jYhhh!NhNubhA)}(h*s*h]h/s}(hhh jubah}(h]h]h]h]h]uhh@h jYubh/ }(hjth jYubh)}(h:sub:`2`h]h/2}(hhh jubah}(h]h]h]h]h]uhhh jYubh/*, the result is
slightly more complicated:}(h*, the result is
slightly more complicated:h jYhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMvh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-18uhh
h jhhh!jhNubj )}(h\bar{\psi}_{B}=\frac{\left(\bar{\psi}_{A}-Q / \sigma_{t}\right)}{\sigma_{t}\left(s_{2}-s_{1}\right)}\left[e^{-\sigma_{t} s_{1}}-e^{-\sigma_{t} s_{2}}\right]+Q / \sigma_{t} .h]h/\bar{\psi}_{B}=\frac{\left(\bar{\psi}_{A}-Q / \sigma_{t}\right)}{\sigma_{t}\left(s_{2}-s_{1}\right)}\left[e^{-\sigma_{t} s_{1}}-e^{-\sigma_{t} s_{2}}\right]+Q / \sigma_{t} .}(hhh jubah}(h]jah]h]h]h]docnamejnumberKplabeleq9-2-18nowrapjyjzuhj h!jhMyh jhhjf}jh}jjsubh;)}(hUEquations :eq:`eq9-2-17` and :eq:`eq9-2-18` can also be written in a simplified form:h](h/
Equations }(h
Equations h jhhh!NhNubj)}(h:eq:`eq9-2-17`h]jc)}(hjh]h/eq9-2-17}(hhh jubah}(h]h](jneqeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjqreftypejrefexplicitrefwarnjeq9-2-17uhjh!jhM~h jubh/ and }(h and h jhhh!NhNubj)}(h:eq:`eq9-2-18`h]jc)}(hjh]h/eq9-2-18}(hhh jubah}(h]h](jneqeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjqreftypejrefexplicitrefwarnjeq9-2-18uhjh!jhM~h jubh/* can also be written in a simplified form:}(h* can also be written in a simplified form:h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM~h jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-19uhh
h jhhh!jhNubj )}(hS\bar{\psi}_{B}=\beta_{A B} \bar{\psi}_{A}+\left(1-\beta_{A B}\right) Q / \sigma_{t}h]h/S\bar{\psi}_{B}=\beta_{A B} \bar{\psi}_{A}+\left(1-\beta_{A B}\right) Q / \sigma_{t}}(hhh j1ubah}(h]j0ah]h]h]h]docnamejnumberKqlabeleq9-2-19nowrapjyjzuhj h!jhMh jhhjf}jh}j0j'subh;)}(hwhereh]h/where}(hjHh jFhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jhhubj1)}(hhh]j )}(h\beta_{A B}=\left\{\begin{array}{cc}
\frac{e^{-\sigma_{t} s_{1}}-e^{-\sigma_{t} s_{2}}}{\sigma_{t}\left(s_{2}-s_{1}\right)} & s_{1} \neq s_{2} \\
e^{-\sigma_{t} s_{1}} & s_{1}=s_{2}
\end{array}\right.h]h/\beta_{A B}=\left\{\begin{array}{cc}
\frac{e^{-\sigma_{t} s_{1}}-e^{-\sigma_{t} s_{2}}}{\sigma_{t}\left(s_{2}-s_{1}\right)} & s_{1} \neq s_{2} \\
e^{-\sigma_{t} s_{1}} & s_{1}=s_{2}
\end{array}\right.}(hhh jWubah}(h]h]h]h]h]docnamejnumberNlabelNnowrapjyjzuhj h!jhMh jTubah}(h]h]h]h]h]uhj1h jhhh!NhNubh;)}(hXaThus far, this development has considered only the special case where
contributions to side B are the result only of the cell internal source
and a single incoming side (i.e., side A). For an arbitrarily shaped
cell and discrete direction |Omk|, it is likely that the outgoing side would
receive contributions from two or more incoming sides, as illustrated in
:numref:`fig9-2-4`, for a cell with three incoming sides (X, Y, and Z)
contributing to the flux on a single outgoing side (B). In such a
situation, the outgoing side can be subdivided into multiple components.
Side B of :numref:`fig9-2-4` can be represented by three components,
B\ :sub:`X`, B\ :sub:`Y`, and B\ :sub:`Z`, representing contributions
from line segments X, Y, and Z, respectively. The average angular flux
:math:`\bar{\psi}` can be computed for each component of side B using
Eq. :eq:`eq9-2-19`; then :math:`\bar{\psi}_{B}`, the average flux for the entire length of
B, can be calculated by the length-weighted average of each component.
In general, for a given side B composed of *n* components, the average
flux of the side is given byh](h/Thus far, this development has considered only the special case where
contributions to side B are the result only of the cell internal source
and a single incoming side (i.e., side A). For an arbitrarily shaped
cell and discrete direction }(hThus far, this development has considered only the special case where
contributions to side B are the result only of the cell internal source
and a single incoming side (i.e., side A). For an arbitrarily shaped
cell and discrete direction h johhh!NhNubjr)}(hjh]h/
\Omega_{k}}(hhh jxhhh!NhNubah}(h]h]h]h]h]uhjqh!NhNh johhubh/u, it is likely that the outgoing side would
receive contributions from two or more incoming sides, as illustrated in
}(hu, it is likely that the outgoing side would
receive contributions from two or more incoming sides, as illustrated in
h johhh!NhNubj)}(h:numref:`fig9-2-4`h]jc)}(hjh]h/fig9-2-4}(hhh jubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnjfig9-2-4uhjh!jhMh joubh/, for a cell with three incoming sides (X, Y, and Z)
contributing to the flux on a single outgoing side (B). In such a
situation, the outgoing side can be subdivided into multiple components.
Side B of }(h, for a cell with three incoming sides (X, Y, and Z)
contributing to the flux on a single outgoing side (B). In such a
situation, the outgoing side can be subdivided into multiple components.
Side B of h johhh!NhNubj)}(h:numref:`fig9-2-4`h]jc)}(hjh]h/fig9-2-4}(hhh jubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjreftypenumrefrefexplicitrefwarnjfig9-2-4uhjh!jhMh joubh/, can be represented by three components,
B }(h, can be represented by three components,
B\ h johhh!NhNubh)}(h:sub:`X`h]h/X}(hhh jubah}(h]h]h]h]h]uhhh joubh/, B }(h, B\ h johhh!NhNubh)}(h:sub:`Y`h]h/Y}(hhh jubah}(h]h]h]h]h]uhhh joubh/ , and B }(h , and B\ h johhh!NhNubh)}(h:sub:`Z`h]h/Z}(hhh jubah}(h]h]h]h]h]uhhh joubh/d, representing contributions
from line segments X, Y, and Z, respectively. The average angular flux
}(hd, representing contributions
from line segments X, Y, and Z, respectively. The average angular flux
h johhh!NhNubjr)}(h:math:`\bar{\psi}`h]h/
\bar{\psi}}(hhh j
ubah}(h]h]h]h]h]uhjqh joubh/: can be computed for each component of side B using
Eq. }(h: can be computed for each component of side B using
Eq. h johhh!NhNubj)}(h:eq:`eq9-2-19`h]jc)}(hj"h]h/eq9-2-19}(hhh j$ubah}(h]h](jneqeh]h]h]uhjbh j ubah}(h]h]h]h]h]refdocj refdomainjqreftypej.refexplicitrefwarnjeq9-2-19uhjh!jhMh joubh/; then }(h; then h johhh!NhNubjr)}(h:math:`\bar{\psi}_{B}`h]h/\bar{\psi}_{B}}(hhh jCubah}(h]h]h]h]h]uhjqh joubh/, the average flux for the entire length of
B, can be calculated by the length-weighted average of each component.
In general, for a given side B composed of }(h, the average flux for the entire length of
B, can be calculated by the length-weighted average of each component.
In general, for a given side B composed of h johhh!NhNubhA)}(h*n*h]h/n}(hhh jVubah}(h]h]h]h]h]uhh@h joubh/5 components, the average
flux of the side is given by}(h5 components, the average
flux of the side is given byh johhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-20uhh
h jhhh!jhNubj )}(h<\bar{\psi}_{B}=\sum_{i=1}^{n} \frac{\bar{\psi}_{i} i}{L_{B}}h]h/<\bar{\psi}_{B}=\sum_{i=1}^{n} \frac{\bar{\psi}_{i} i}{L_{B}}}(hhh jyubah}(h]jxah]h]h]h]docnamejnumberKrlabeleq9-2-20nowrapjyjzuhj h!jhMh jhhjf}jh}jxjosubh;)}(hwhereh]h/where}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jhhubj1)}(hhh](h;)}(hL:math:`\ell_{i}` is the length of the projection of the ith side onto B, andh](jr)}(h:math:`\ell_{i}`h]h/\ell_{i}}(hhh jubah}(h]h]h]h]h]uhjqh jubh/< is the length of the projection of the ith side onto B, and}(h< is the length of the projection of the ith side onto B, andh jubeh}(h]h]h]h]h]uhh:h!jhMh jubh;)}(hh:math:`\bar{\psi}_{i}` is the average flux computed for segment B\ :sub:`i` due to the
flux on side *i*h](jr)}(h:math:`\bar{\psi}_{i}`h]h/\bar{\psi}_{i}}(hhh jubah}(h]h]h]h]h]uhjqh jubh/. is the average flux computed for segment B }(h. is the average flux computed for segment B\ h jubh)}(h:sub:`i`h]h/i}(hhh jubah}(h]h]h]h]h]uhhh jubh/ due to the
flux on side }(h due to the
flux on side h jubhA)}(h*i*h]h/i}(hhh jubah}(h]h]h]h]h]uhh@h jubeh}(h]h]h]h]h]uhh:h!jhMh jubeh}(h]h]h]h]h]uhj1h jhhh!jhNubh;)}(hXUsing Eqs. :eq:`eq9-2-19` and :eq:`eq9-2-20`, one can compute the average flux on
each of the outgoing sides for a given cell, once the angular flux on
each incoming side is known. At this point, only distances *s*\ :sub:`1`
and *s*\ :sub:`2` and the lengths :math:`\ell_{i}` and L need be determined to estimate
fluxes in an iterative process. These can be computed from the geometry
of the cell and the direction |Omk|.h](h/Using Eqs. }(hUsing Eqs. h jhhh!NhNubj)}(h:eq:`eq9-2-19`h]jc)}(hjh]h/eq9-2-19}(hhh j
ubah}(h]h](jneqeh]h]h]uhjbh j ubah}(h]h]h]h]h]refdocj refdomainjqreftypejrefexplicitrefwarnjeq9-2-19uhjh!jhMh jubh/ and }(h and h jhhh!NhNubj)}(h:eq:`eq9-2-20`h]jc)}(hj.h]h/eq9-2-20}(hhh j0ubah}(h]h](jneqeh]h]h]uhjbh j,ubah}(h]h]h]h]h]refdocj refdomainjqreftypej:refexplicitrefwarnjeq9-2-20uhjh!jhMh jubh/, one can compute the average flux on
each of the outgoing sides for a given cell, once the angular flux on
each incoming side is known. At this point, only distances }(h, one can compute the average flux on
each of the outgoing sides for a given cell, once the angular flux on
each incoming side is known. At this point, only distances h jhhh!NhNubhA)}(h*s*h]h/s}(hhh jOubah}(h]h]h]h]h]uhh@h jubh/ }(h\ h jhhh!NhNubh)}(h:sub:`1`h]h/1}(hhh jbubah}(h]h]h]h]h]uhhh jubh/
and }(h
and h jhhh!NhNubhA)}(h*s*h]h/s}(hhh juubah}(h]h]h]h]h]uhh@h jubh/ }(hjah jubh)}(h:sub:`2`h]h/2}(hhh jubah}(h]h]h]h]h]uhhh jubh/ and the lengths }(h and the lengths h jhhh!NhNubjr)}(h:math:`\ell_{i}`h]h/\ell_{i}}(hhh jubah}(h]h]h]h]h]uhjqh jubh/ and L need be determined to estimate
fluxes in an iterative process. These can be computed from the geometry
of the cell and the direction }(h and L need be determined to estimate
fluxes in an iterative process. These can be computed from the geometry
of the cell and the direction h jhhh!NhNubjr)}(hjh]h/
\Omega_{k}}(hhh jhhh!NhNubah}(h]h]h]h]h]uhjqh!NhNh jhhubh/.}(hjWh jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(h
.. _fig9-2-4:h]h}(h]h]h]h]h]hfig9-2-4uhh
hMh jhhh!jubj)}(hhh](j)}(h.. figure:: figs/NEWT/fig4.png
:align: center
:width: 500
Contributions of multiple incoming sides to an outgoing side.
h]h}(h]h]h]h]h]width500urifigs/NEWT/fig4.pngj}jjsuhjh jh!jhMubj)}(h=Contributions of multiple incoming sides to an outgoing side.h]h/=Contributions of multiple incoming sides to an outgoing side.}(hjh jubah}(h]h]h]h]h]uhjh!jhMh jubeh}(h](id273jeh]h]fig9-2-4ah]h]jcenteruhjhMh jhhh!jjf}jjsjh}jjsubh)}(h.. _9-2-2-3-5:h]h}(h]h]h]h]h]hid123uhh
hMh jhhh!jubeh}(h](angular-flux-at-a-cell-boundaryjeh]h](angular flux at a cell boundary 9-2-2-3-4eh]h]uhh#h jhhh!jhMjf}jjsjh}jjsubh$)}(hhh](h))}(hGMapping a characteristic vector into the two-dimensional problem domainh]h/GMapping a characteristic vector into the two-dimensional problem domain}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMubh;)}(hXEven in a 2-D x-y system in which the scalar flux is constant with
respect to the z axis, the angular flux has components in the
z direction. Thus, to obtain the scalar flux at a point on the
x-y plane, one must integrate over the unit sphere in all 4π directions
of |Om|. Recall that the choices of characteristic directions for this model
were selected to be the same as the set of directions composing a
conventional quadrature set. Quadrature sets specified in the
literature :cite:`carlson_transport_1970,carlson_discrete_1965,lee_discrete_1962` and used in other
discrete-ordinates codes :cite:`lathrop_twotran-ii_1973,engle_jr_users_1967` are based on a unit
sphere and are usually specified in terms of μ\ :sub:`k` and η\ :sub:`k`,
the respective x and y components of |Omk|, where is one of a set of discrete
directions composing the quadrature set. Because |Omk| is a unit vector, :math:`\xi_{k}`, the
z component of the direction, is implicit: :math:`\xi_{k}=\sqrt{1-\mu_{k}^{2}-\eta_{k}^{2}}`.
However, because of the 2‑D
nature of the problem, the z component is never explicitly used. It is
therefore sufficient to evaluate the angular flux at a finite number of
points in 4π of |Om| -space in terms of just the μ\ :sub:`k` and η\ :sub:`k`
components of the discrete directions |Omk|. One must recognize, however,
that the length of the path traveled by particles moving in a direction
out of the x-y plane is always longer than the x-y projection of the
path, by a factor of (μ\ :sup:`2` + η\ :sup:`2`)\ :sup:`–1/2`. Thus, for any
path length *s*' measured in the x‑y plane for a given direction |Omk|, the
true path length traveled is *s*, whereh](h/XEven in a 2-D x-y system in which the scalar flux is constant with
respect to the z axis, the angular flux has components in the
z direction. Thus, to obtain the scalar flux at a point on the
x-y plane, one must integrate over the unit sphere in all 4π directions
of }(hXEven in a 2-D x-y system in which the scalar flux is constant with
respect to the z axis, the angular flux has components in the
z direction. Thus, to obtain the scalar flux at a point on the
x-y plane, one must integrate over the unit sphere in all 4π directions
of h j!hhh!NhNubjr)}(h:math:`\Omega`h]h/\Omega}(hhh j*hhh!NhNubah}(h]h]h]h]h]uhjqh!NhNh j!hhubh/. Recall that the choices of characteristic directions for this model
were selected to be the same as the set of directions composing a
conventional quadrature set. Quadrature sets specified in the
literature }(h. Recall that the choices of characteristic directions for this model
were selected to be the same as the set of directions composing a
conventional quadrature set. Quadrature sets specified in the
literature h j!hhh!NhNubj)}(hcarlson_transport_1970h]j#)}(hj?h]h/[carlson_transport_1970]}(hhh jAubah}(h]h]h]h]h]uhj"h j=ubah}(h]jah]j5ah]h]h] refdomainj:reftypej< reftargetj?refwarnsupport_smartquotesuhjh!jhMh j!hhubj)}(hcarlson_discrete_1965h]j#)}(hj[h]h/[carlson_discrete_1965]}(hhh j]ubah}(h]h]h]h]h]uhj"h jYubah}(h]id125ah]j5ah]h]h] refdomainj:reftypej< reftargetj[refwarnsupport_smartquotesuhjh!jhMh j!hhubj)}(hlee_discrete_1962h]j#)}(hjxh]h/[lee_discrete_1962]}(hhh jzubah}(h]h]h]h]h]uhj"h jvubah}(h]jnah]j5ah]h]h] refdomainj:reftypej< reftargetjxrefwarnsupport_smartquotesuhjh!jhMh j!hhubh/, and used in other
discrete-ordinates codes }(h, and used in other
discrete-ordinates codes h j!hhh!NhNubj)}(hlathrop_twotran-ii_1973h]j#)}(hjh]h/[lathrop_twotran-ii_1973]}(hhh jubah}(h]h]h]h]h]uhj"h jubah}(h]jah]j5ah]h]h] refdomainj:reftypej< reftargetjrefwarnsupport_smartquotesuhjh!jhMh j!hhubj)}(hengle_jr_users_1967h]j#)}(hjh]h/[engle_jr_users_1967]}(hhh jubah}(h]h]h]h]h]uhj"h jubah}(h]jah]j5ah]h]h] refdomainj:reftypej< reftargetjrefwarnsupport_smartquotesuhjh!jhMh j!hhubh/F are based on a unit
sphere and are usually specified in terms of μ }(hF are based on a unit
sphere and are usually specified in terms of μ\ h j!hhh!NhNubh)}(h:sub:`k`h]h/k}(hhh jubah}(h]h]h]h]h]uhhh j!ubh/ and η }(h and η\ h j!hhh!NhNubh)}(h:sub:`k`h]h/k}(hhh jubah}(h]h]h]h]h]uhhh j!ubh/(,
the respective x and y components of }(h(,
the respective x and y components of h j!hhh!NhNubjr)}(hjh]h/
\Omega_{k}}(hhh jhhh!NhNubah}(h]h]h]h]h]uhjqh!NhNh j!hhubh/U, where is one of a set of discrete
directions composing the quadrature set. Because }(hU, where is one of a set of discrete
directions composing the quadrature set. Because h j!hhh!NhNubjr)}(hjh]h/
\Omega_{k}}(hhh jhhh!NhNubah}(h]h]h]h]h]uhjqh!NhNh j!hhubh/ is a unit vector, }(h is a unit vector, h j!hhh!NhNubjr)}(h:math:`\xi_{k}`h]h/\xi_{k}}(hhh jubah}(h]h]h]h]h]uhjqh j!ubh/2, the
z component of the direction, is implicit: }(h2, the
z component of the direction, is implicit: h j!hhh!NhNubjr)}(h1:math:`\xi_{k}=\sqrt{1-\mu_{k}^{2}-\eta_{k}^{2}}`h]h/)\xi_{k}=\sqrt{1-\mu_{k}^{2}-\eta_{k}^{2}}}(hhh j1ubah}(h]h]h]h]h]uhjqh j!ubh/.
However, because of the 2‑D
nature of the problem, the z component is never explicitly used. It is
therefore sufficient to evaluate the angular flux at a finite number of
points in 4π of }(h.
However, because of the 2‑D
nature of the problem, the z component is never explicitly used. It is
therefore sufficient to evaluate the angular flux at a finite number of
points in 4π of h j!hhh!NhNubjr)}(hj,h]h/\Omega}(hhh jDhhh!NhNubah}(h]h]h]h]h]uhjqh!NhNh j!hhubh/! -space in terms of just the μ }(h! -space in terms of just the μ\ h j!hhh!NhNubh)}(h:sub:`k`h]h/k}(hhh jVubah}(h]h]h]h]h]uhhh j!ubh/ and η }(h and η\ h j!hhh!NhNubh)}(h:sub:`k`h]h/k}(hhh jiubah}(h]h]h]h]h]uhhh j!ubh/(
components of the discrete directions }(h(
components of the discrete directions h j!hhh!NhNubjr)}(hjh]h/
\Omega_{k}}(hhh j|hhh!NhNubah}(h]h]h]h]h]uhjqh!NhNh j!hhubh/. One must recognize, however,
that the length of the path traveled by particles moving in a direction
out of the x-y plane is always longer than the x-y projection of the
path, by a factor of (μ }(h. One must recognize, however,
that the length of the path traveled by particles moving in a direction
out of the x-y plane is always longer than the x-y projection of the
path, by a factor of (μ\ h j!hhh!NhNubj;)}(h:sup:`2`h]h/2}(hhh jubah}(h]h]h]h]h]uhj;h j!ubh/ + η }(h + η\ h j!hhh!NhNubj;)}(h:sup:`2`h]h/2}(hhh jubah}(h]h]h]h]h]uhj;h j!ubh/) }(h)\ h j!hhh!NhNubj;)}(h
:sup:`–1/2`h]h/–1/2}(hhh jubah}(h]h]h]h]h]uhj;h j!ubh/. Thus, for any
path length }(h. Thus, for any
path length h j!hhh!NhNubhA)}(h*s*h]h/s}(hhh jubah}(h]h]h]h]h]uhh@h j!ubh/7’ measured in the x‑y plane for a given direction }(h5' measured in the x‑y plane for a given direction h j!hhh!NhNubjr)}(hjh]h/
\Omega_{k}}(hhh jhhh!NhNubah}(h]h]h]h]h]uhjqh!NhNh j!hhubh/#, the
true path length traveled is }(h#, the
true path length traveled is h j!hhh!NhNubhA)}(h*s*h]h/s}(hhh jubah}(h]h]h]h]h]uhh@h j!ubh/, where}(h, whereh j!hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-21uhh
h jhhh!jhNubj )}(h.s=\frac{s^{\prime}}{\sqrt{\mu^{2}+\eta^{2}}} .h]h/.s=\frac{s^{\prime}}{\sqrt{\mu^{2}+\eta^{2}}} .}(hhh jubah}(h]jah]h]h]h]docnamejnumberKslabeleq9-2-21nowrapjyjzuhj h!jhMh jhhjf}jh}jjsubh;)}(h*This is illustrated in :numref:`fig9-2-5`.h](h/This is illustrated in }(hThis is illustrated in h j$hhh!NhNubj)}(h:numref:`fig9-2-5`h]jc)}(hj/h]h/fig9-2-5}(hhh j1ubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh j-ubah}(h]h]h]h]h]refdocj refdomainj;reftypenumrefrefexplicitrefwarnjfig9-2-5uhjh!jhMh j$ubh/.}(hjWh j$hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(h
.. _fig9-2-5:h]h}(h]h]h]h]h]hfig9-2-5uhh
hMh jhhh!jubj)}(hhh](j)}(h.. figure:: figs/NEWT/fig5.png
:align: center
:width: 500
Relationship between *s*\ :sub:`1` and *s*\ :sub:`2` and
their projections in the x-y plane.
h]h}(h]h]h]h]h]width500urifigs/NEWT/fig5.pngj}jjrsuhjh jbh!jhMubj)}(h\Relationship between *s*\ :sub:`1` and *s*\ :sub:`2` and
their projections in the x-y plane.h](h/Relationship between }(hRelationship between h jtubhA)}(h*s*h]h/s}(hhh j}ubah}(h]h]h]h]h]uhh@h jtubh/ }(h\ h jtubh)}(h:sub:`1`h]h/1}(hhh jubah}(h]h]h]h]h]uhhh jtubh/ and }(h and h jtubhA)}(h*s*h]h/s}(hhh jubah}(h]h]h]h]h]uhh@h jtubh/ }(hjh jtubh)}(h:sub:`2`h]h/2}(hhh jubah}(h]h]h]h]h]uhhh jtubh/( and
their projections in the x-y plane.}(h( and
their projections in the x-y plane.h jtubeh}(h]h]h]h]h]uhjh!jhMh jbubeh}(h](id274jaeh]h]fig9-2-5ah]h]jcenteruhjhMh jhhh!jjf}jjWsjh}jajWsubh)}(h.. _9-2-2-3-6:h]h}(h]h]h]h]h]hid129uhh
hMh jhhh!jubeh}(h](Gmapping-a-characteristic-vector-into-the-two-dimensional-problem-domainjeh]h](Gmapping a characteristic vector into the two-dimensional problem domain 9-2-2-3-5eh]h]uhh#h jhhh!jhMjf}jjsjh}jjsubh$)}(hhh](h))}(h+Neutron balance within a computational cellh]h/+Neutron balance within a computational cell}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMubh;)}(hX]Once angular fluxes have been computed for all sides of a cell, it is
necessary to compute the cell-averaged angular flux. To enforce
conservation, a balance condition is applied to the cell. This provides
the equation necessary to determine the average flux in the cell. The
neutron balance for an arbitrary cell in steady state may be expressed
ash]h/X]Once angular fluxes have been computed for all sides of a cell, it is
necessary to compute the cell-averaged angular flux. To enforce
conservation, a balance condition is applied to the cell. This provides
the equation necessary to determine the average flux in the cell. The
neutron balance for an arbitrary cell in steady state may be expressed
as}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-22uhh
h jhhh!jhNubj )}(hX
\left[\begin{array}{c}
\text { net number of } \\
\text { neutrons moving in } \\
\text { direction } \hat{\Omega} \text { escaping } \\
\text { from the cell }
\end{array}\right]+\left[\begin{array}{c}
\text { number of neutrons } \\
\text { removed from the cell } \\
\text { or from direction } \hat{\Omega} \\
\text { by interactions }
\end{array}\right]=\left[\begin{array}{c}
\text { number of } \\
\text { neutrons produced } \\
\text { in the cell moving } \\
\text { in direction } \hat{\Omega}
\end{array}\right]h]h/X
\left[\begin{array}{c}
\text { net number of } \\
\text { neutrons moving in } \\
\text { direction } \hat{\Omega} \text { escaping } \\
\text { from the cell }
\end{array}\right]+\left[\begin{array}{c}
\text { number of neutrons } \\
\text { removed from the cell } \\
\text { or from direction } \hat{\Omega} \\
\text { by interactions }
\end{array}\right]=\left[\begin{array}{c}
\text { number of } \\
\text { neutrons produced } \\
\text { in the cell moving } \\
\text { in direction } \hat{\Omega}
\end{array}\right]}(hhh jubah}(h]jah]h]h]h]docnamejnumberKtlabeleq9-2-22nowrapjyjzuhj h!jhMh jhhjf}jh}jjsubh;)}(hor, expressed mathematically,h]h/or, expressed mathematically,}(hj/h j-hhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-23uhh
h jhhh!jhNubj )}(hH\oint_{s} n \cdot \hat{\Omega}_{k} \psi d S+\sigma_{t} \bar{\psi} V=Q V,h]h/H\oint_{s} n \cdot \hat{\Omega}_{k} \psi d S+\sigma_{t} \bar{\psi} V=Q V,}(hhh jEubah}(h]jDah]h]h]h]docnamejnumberKulabeleq9-2-23nowrapjyjzuhj h!jhMh jhhjf}jh}jDj;subh;)}(hXewhere :math:`n` is the outward normal direction at each side of the cell and V is
the 2-D volume of the cell. Note that in this context, *S* represents
the surface area or perimeter of the cell. Hence, for a cell with *m*
sides, each of the sides having a constant angular flux :math:`\bar{\psi}_{i}` and an outward
normal direction :math:`\mathrm{n}_{i}`,h](h/where }(hwhere h jZhhh!NhNubjr)}(h :math:`n`h]h/n}(hhh jcubah}(h]h]h]h]h]uhjqh jZubh/z is the outward normal direction at each side of the cell and V is
the 2-D volume of the cell. Note that in this context, }(hz is the outward normal direction at each side of the cell and V is
the 2-D volume of the cell. Note that in this context, h jZhhh!NhNubhA)}(h*S*h]h/S}(hhh jvubah}(h]h]h]h]h]uhh@h jZubh/N represents
the surface area or perimeter of the cell. Hence, for a cell with }(hN represents
the surface area or perimeter of the cell. Hence, for a cell with h jZhhh!NhNubhA)}(h*m*h]h/m}(hhh jubah}(h]h]h]h]h]uhh@h jZubh/9
sides, each of the sides having a constant angular flux }(h9
sides, each of the sides having a constant angular flux h jZhhh!NhNubjr)}(h:math:`\bar{\psi}_{i}`h]h/\bar{\psi}_{i}}(hhh jubah}(h]h]h]h]h]uhjqh jZubh/" and an outward
normal direction }(h" and an outward
normal direction h jZhhh!NhNubjr)}(h:math:`\mathrm{n}_{i}`h]h/\mathrm{n}_{i}}(hhh jubah}(h]h]h]h]h]uhjqh jZubh/,}(hj
h jZhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-24uhh
h jhhh!jhNubj )}(h\bar{\psi}_{c e l l}=\frac{Q}{\sigma_{t}}-\frac{1}{\sigma_{t} V} \sum_{i=1}^{m} \bar{\psi}_{i} \int_{S_{i}} \mathrm{n}_{i} \cdot \hat{\Omega}_{k} d S_{i} .h]h/\bar{\psi}_{c e l l}=\frac{Q}{\sigma_{t}}-\frac{1}{\sigma_{t} V} \sum_{i=1}^{m} \bar{\psi}_{i} \int_{S_{i}} \mathrm{n}_{i} \cdot \hat{\Omega}_{k} d S_{i} .}(hhh jubah}(h]jah]h]h]h]docnamejnumberKvlabeleq9-2-24nowrapjyjzuhj h!jhMh jhhjf}jh}jjsubh;)}(hBecause each cell is restricted to be a polygon, each side in the cell
will be a straight line and :math:`\mathrm{n}_{i} \cdot \hat{\Omega}_{k}` will be constant along the length of the
side. Equation :eq:`eq9-2-24` can then be simplified to obtainh](h/cBecause each cell is restricted to be a polygon, each side in the cell
will be a straight line and }(hcBecause each cell is restricted to be a polygon, each side in the cell
will be a straight line and h jhhh!NhNubjr)}(h-:math:`\mathrm{n}_{i} \cdot \hat{\Omega}_{k}`h]h/%\mathrm{n}_{i} \cdot \hat{\Omega}_{k}}(hhh jubah}(h]h]h]h]h]uhjqh jubh/: will be constant along the length of the
side. Equation }(h: will be constant along the length of the
side. Equation h jhhh!NhNubj)}(h:eq:`eq9-2-24`h]jc)}(hjh]h/eq9-2-24}(hhh jubah}(h]h](jneqeh]h]h]uhjbh jubah}(h]h]h]h]h]refdocj refdomainjqreftypejrefexplicitrefwarnjeq9-2-24uhjh!jhMh jubh/! can then be simplified to obtain}(h! can then be simplified to obtainh jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-25uhh
h jhhh!jhNubj )}(h\bar{\psi}_{c e l l}=\frac{Q}{\sigma_{t}}-\frac{1}{\sigma_{t} V} \sum_{i=1}^{m} \bar{\psi}_{i}\left(\mathrm{n}_{i} \cdot \hat{\Omega}_{k} \mathrm{~L}_{i}\right) ,h]h/\bar{\psi}_{c e l l}=\frac{Q}{\sigma_{t}}-\frac{1}{\sigma_{t} V} \sum_{i=1}^{m} \bar{\psi}_{i}\left(\mathrm{n}_{i} \cdot \hat{\Omega}_{k} \mathrm{~L}_{i}\right) ,}(hhh j5ubah}(h]j4ah]h]h]h]docnamejnumberKwlabeleq9-2-25nowrapjyjzuhj h!jhMh jhhjf}jh}j4j+subh;)}(hwhere L\ :sub:`i` is the length of the *i*\ th side and the term in
parentheses represents a leakage coefficient for the side.h](h/ where L }(h where L\ h jJhhh!NhNubh)}(h:sub:`i`h]h/i}(hhh jSubah}(h]h]h]h]h]uhhh jJubh/ is the length of the }(h is the length of the h jJhhh!NhNubhA)}(h*i*h]h/i}(hhh jfubah}(h]h]h]h]h]uhh@h jJubh/U th side and the term in
parentheses represents a leakage coefficient for the side.}(hU\ th side and the term in
parentheses represents a leakage coefficient for the side.h jJhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM h jhhubh)}(h.. _9-2-2-4:h]h}(h]h]h]h]h]hid130uhh
hM6h jhhh!jubeh}(h](+neutron-balance-within-a-computational-celljeh]h](+neutron balance within a computational cell 9-2-2-3-6eh]h]uhh#h jhhh!jhMjf}jjsjh}jjsubeh}(h]()the-extended-step-characteristic-approachjeh]h]()the extended step characteristic approach9-2-2-3eh]h]uhh#h j'hhh!jhM|jf}jjsjh}jjsubh$)}(hhh](h))}(h*Coarse-mesh finite-difference accelerationh]h/*Coarse-mesh finite-difference acceleration}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhM&ubh;)}(hX*Beyond cell discretization and solution described above for the ESC
approach, the NEWT iterative approach is similar to that used in other
discrete-ordinates methods. Inner iterations are used to solve spatial
fluxes in each energy group to generate updated source terms; outer
iterations use these source terms to converge all energy groups. This
source-iteration approach can be somewhat slow to converge, especially
when significant scattering is present. Hence, it is desirable to apply
some form of acceleration to the iterative solution used by NEWT. To
this end, a coarse-mesh finite-difference acceleration (CMFD) approach
has been added to NEWT. The CMFD formulation uses a simplified
representation of a complex problem, in which selected rectangular
regions are derived from the global NEWT Cartesian grid and homogenized.
The CMFD formulation utilizes coupling correction factors for each
homogenized cell to dynamically homogenize the constituent ESC-based
polygonal cells during the iterative solution process such that the
heterogeneous transport solution can be preserved. Dynamic-group
collapse is also possible with a two-level CMFD formulation in which
alternating multigroup and two-group calculations are performed. By
extending the concept of the equivalence theory to energy and angle, it
is possible to apply a consistent lower-order formulation in the form of
a homogenized pin-cell, few-group, diffusion-like finite-difference
scheme. This simplified lower-order formulation is much less expensive
to solve, and its solution can be used to accelerate the original
higher-order transport solution in NEWT, resulting in much faster
convergence of the fission and scattering source distributions. This
work is described in detail in :cite:`zhong_implementation_2008` and in previous versions of
the NEWT manual.h](h/XBeyond cell discretization and solution described above for the ESC
approach, the NEWT iterative approach is similar to that used in other
discrete-ordinates methods. Inner iterations are used to solve spatial
fluxes in each energy group to generate updated source terms; outer
iterations use these source terms to converge all energy groups. This
source-iteration approach can be somewhat slow to converge, especially
when significant scattering is present. Hence, it is desirable to apply
some form of acceleration to the iterative solution used by NEWT. To
this end, a coarse-mesh finite-difference acceleration (CMFD) approach
has been added to NEWT. The CMFD formulation uses a simplified
representation of a complex problem, in which selected rectangular
regions are derived from the global NEWT Cartesian grid and homogenized.
The CMFD formulation utilizes coupling correction factors for each
homogenized cell to dynamically homogenize the constituent ESC-based
polygonal cells during the iterative solution process such that the
heterogeneous transport solution can be preserved. Dynamic-group
collapse is also possible with a two-level CMFD formulation in which
alternating multigroup and two-group calculations are performed. By
extending the concept of the equivalence theory to energy and angle, it
is possible to apply a consistent lower-order formulation in the form of
a homogenized pin-cell, few-group, diffusion-like finite-difference
scheme. This simplified lower-order formulation is much less expensive
to solve, and its solution can be used to accelerate the original
higher-order transport solution in NEWT, resulting in much faster
convergence of the fission and scattering source distributions. This
work is described in detail in }(hXBeyond cell discretization and solution described above for the ESC
approach, the NEWT iterative approach is similar to that used in other
discrete-ordinates methods. Inner iterations are used to solve spatial
fluxes in each energy group to generate updated source terms; outer
iterations use these source terms to converge all energy groups. This
source-iteration approach can be somewhat slow to converge, especially
when significant scattering is present. Hence, it is desirable to apply
some form of acceleration to the iterative solution used by NEWT. To
this end, a coarse-mesh finite-difference acceleration (CMFD) approach
has been added to NEWT. The CMFD formulation uses a simplified
representation of a complex problem, in which selected rectangular
regions are derived from the global NEWT Cartesian grid and homogenized.
The CMFD formulation utilizes coupling correction factors for each
homogenized cell to dynamically homogenize the constituent ESC-based
polygonal cells during the iterative solution process such that the
heterogeneous transport solution can be preserved. Dynamic-group
collapse is also possible with a two-level CMFD formulation in which
alternating multigroup and two-group calculations are performed. By
extending the concept of the equivalence theory to energy and angle, it
is possible to apply a consistent lower-order formulation in the form of
a homogenized pin-cell, few-group, diffusion-like finite-difference
scheme. This simplified lower-order formulation is much less expensive
to solve, and its solution can be used to accelerate the original
higher-order transport solution in NEWT, resulting in much faster
convergence of the fission and scattering source distributions. This
work is described in detail in h jhhh!NhNubj)}(hzhong_implementation_2008h]j#)}(hjh]h/[zhong_implementation_2008]}(hhh jubah}(h]h]h]h]h]uhj"h jubah}(h]jah]j5ah]h]h] refdomainj:reftypej< reftargetjrefwarnsupport_smartquotesuhjh!jhM(h jhhubh/- and in previous versions of
the NEWT manual.}(h- and in previous versions of
the NEWT manual.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM(h jhhubh;)}(hXLAlthough the original implementation of the CMFD acceleration method is
extremely efficient and actively maintained, its use is limited to
rectangular-domain configurations (e.g., square-pitched fuel lattices).
An alternative CMFD acceleration method has been developed to support
triangular- and hexagonal-domain configurations (e.g.,
triangular-pitched fuel lattices such as the VVER or prismatic graphite
models). The new CMFD acceleration method does not require the
coarse-mesh cells to be rectangles but rather arbitrary polygons.
However in the current implementation, the “unstructured” coarse-mesh
cells are still constructed from the global NEWT Cartesian grid.
Therefore, for a hexagonal configuration, interior coarse-mesh cells
will be rectangular shape whereas cells near the boundary will be
triangular or trapezoidal shapes.h]h/XLAlthough the original implementation of the CMFD acceleration method is
extremely efficient and actively maintained, its use is limited to
rectangular-domain configurations (e.g., square-pitched fuel lattices).
An alternative CMFD acceleration method has been developed to support
triangular- and hexagonal-domain configurations (e.g.,
triangular-pitched fuel lattices such as the VVER or prismatic graphite
models). The new CMFD acceleration method does not require the
coarse-mesh cells to be rectangles but rather arbitrary polygons.
However in the current implementation, the “unstructured” coarse-mesh
cells are still constructed from the global NEWT Cartesian grid.
Therefore, for a hexagonal configuration, interior coarse-mesh cells
will be rectangular shape whereas cells near the boundary will be
triangular or trapezoidal shapes.}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMDh jhhubh;)}(hXPThe new unstructured CMFD iterative solution scheme is essentially
identical to the original solution scheme; the two methods differ only
in how the lower-order system is solved. Additionally the two-group
acceleration is not employed in the unstructured CMFD method. Input
options for both CMFD methods are described in :ref:`9-2-3-2`.h](h/XAThe new unstructured CMFD iterative solution scheme is essentially
identical to the original solution scheme; the two methods differ only
in how the lower-order system is solved. Additionally the two-group
acceleration is not employed in the unstructured CMFD method. Input
options for both CMFD methods are described in }(hXAThe new unstructured CMFD iterative solution scheme is essentially
identical to the original solution scheme; the two methods differ only
in how the lower-order system is solved. Additionally the two-group
acceleration is not employed in the unstructured CMFD method. Input
options for both CMFD methods are described in h jhhh!NhNubj)}(h:ref:`9-2-3-2`h]j#)}(hjh]h/9-2-3-2}(hhh jubah}(h]h](jnstdstd-refeh]h]h]uhj"h jubah}(h]h]h]h]h]refdocj refdomainjreftyperefrefexplicitrefwarnj9-2-3-2uhjh!jhMRh jubh/.}(hjWh jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMRh jhhubh)}(h.. _9-2-2-5:h]h}(h]h]h]h]h]hid132uhh
hMkh jhhh!jubeh}(h](*coarse-mesh-finite-difference-accelerationjeh]h](*coarse-mesh finite-difference acceleration9-2-2-4eh]h]uhh#h j'hhh!jhM&jf}j3jsjh}jjsubh$)}(hhh](h))}(hAssembly discontinuity factorsh]h/Assembly discontinuity factors}(hj=h j;hhh!NhNubah}(h]h]h]h]h]uhh(h j8hhh!jhM[ubh;)}(hX
In nodal multi-assembly or core calculations, lattice transport
solutions are used to generate few-group homogenized cross sections.
These cross sections are in general obtained from single-assembly
transport calculations with zero-current boundary conditions. Generation
of few-group homogenized cross sections for nodal calculations typically
includes the generation of discontinuity factors (i.e., additional
parameters used to preserve both reaction rates and the interface
currents in the homogenization process). The discontinuity of the flux
at an assembly interface that can arise by the use of homogenized
cross sections is illustrated in :numref:`fig9-2-6`. The so-called
“homogeneous” flux, computed in the nodal calculation, is discontinuous
at the assembly interface, as opposed to the exact “heterogeneous” flux,
computed in the transport calculation, which is continuous at the
assembly interface. The interface condition employed in nodal
calculations between two assemblies (nodes) *i* and *i*\ +1 is given ash](h/XIn nodal multi-assembly or core calculations, lattice transport
solutions are used to generate few-group homogenized cross sections.
These cross sections are in general obtained from single-assembly
transport calculations with zero-current boundary conditions. Generation
of few-group homogenized cross sections for nodal calculations typically
includes the generation of discontinuity factors (i.e., additional
parameters used to preserve both reaction rates and the interface
currents in the homogenization process). The discontinuity of the flux
at an assembly interface that can arise by the use of homogenized
cross sections is illustrated in }(hXIn nodal multi-assembly or core calculations, lattice transport
solutions are used to generate few-group homogenized cross sections.
These cross sections are in general obtained from single-assembly
transport calculations with zero-current boundary conditions. Generation
of few-group homogenized cross sections for nodal calculations typically
includes the generation of discontinuity factors (i.e., additional
parameters used to preserve both reaction rates and the interface
currents in the homogenization process). The discontinuity of the flux
at an assembly interface that can arise by the use of homogenized
cross sections is illustrated in h jIhhh!NhNubj)}(h:numref:`fig9-2-6`h]jc)}(hjTh]h/fig9-2-6}(hhh jVubah}(h]h](jnstd
std-numrefeh]h]h]uhjbh jRubah}(h]h]h]h]h]refdocj refdomainj`reftypenumrefrefexplicitrefwarnjfig9-2-6uhjh!jhM^h jIubh/XU. The so-called
“homogeneous” flux, computed in the nodal calculation, is discontinuous
at the assembly interface, as opposed to the exact “heterogeneous” flux,
computed in the transport calculation, which is continuous at the
assembly interface. The interface condition employed in nodal
calculations between two assemblies (nodes) }(hXU. The so-called
“homogeneous” flux, computed in the nodal calculation, is discontinuous
at the assembly interface, as opposed to the exact “heterogeneous” flux,
computed in the transport calculation, which is continuous at the
assembly interface. The interface condition employed in nodal
calculations between two assemblies (nodes)
h jIhhh!NhNubhA)}(h*i*h]h/i}(hhh jwubah}(h]h]h]h]h]uhh@h jIubh/ and }(h and h jIhhh!NhNubhA)}(h*i*h]h/i}(hhh jubah}(h]h]h]h]h]uhh@h jIubh/ +1 is given as}(h\ +1 is given ash jIhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM^h j8hhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-26uhh
h j8hhh!jhNubj )}(hm\phi_{i, \text { homogeneous }}^{+} \cdot F_{i}^{+}=\phi_{i+1, \text { homogeneous }}^{-} \cdot F_{i+1}^{-} ,h]h/m\phi_{i, \text { homogeneous }}^{+} \cdot F_{i}^{+}=\phi_{i+1, \text { homogeneous }}^{-} \cdot F_{i+1}^{-} ,}(hhh jubah}(h]jah]h]h]h]docnamejnumberKxlabeleq9-2-26nowrapjyjzuhj h!jhMnh j8hhjf}jh}jjsubh;)}(hwhere :math:`F_{i}^{+}` and :math:`F_{i+1}^{-}` are assembly discontinuity factors (ADFs) on each side of the
interface between assemblies *i* and *i*\ +1.h](h/where }(hwhere h jhhh!NhNubjr)}(h:math:`F_{i}^{+}`h]h/ F_{i}^{+}}(hhh jubah}(h]h]h]h]h]uhjqh jubh/ and }(h and h jhhh!NhNubjr)}(h:math:`F_{i+1}^{-}`h]h/F_{i+1}^{-}}(hhh jubah}(h]h]h]h]h]uhjqh jubh/\ are assembly discontinuity factors (ADFs) on each side of the
interface between assemblies }(h\ are assembly discontinuity factors (ADFs) on each side of the
interface between assemblies h jhhh!NhNubhA)}(h*i*h]h/i}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ and }(hjh jubhA)}(h*i*h]h/i}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ +1.}(h\ +1.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMsh j8hhubh;)}(hX1The ADF on the assembly interface is defined as the ratio of the
heterogeneous flux :math:`\phi_{\text {heterogeneous }}` at that assembly interface to the homogeneous flux
evaluated at the interface, denoted :math:`\phi_{i, \text { homogeneous }}^{+}` (or :math:`\phi_{i+1, \text { homogeneous }}^{-}`):h](h/TThe ADF on the assembly interface is defined as the ratio of the
heterogeneous flux }(hTThe ADF on the assembly interface is defined as the ratio of the
heterogeneous flux h jhhh!NhNubjr)}(h%:math:`\phi_{\text {heterogeneous }}`h]h/\phi_{\text {heterogeneous }}}(hhh j%ubah}(h]h]h]h]h]uhjqh jubh/X at that assembly interface to the homogeneous flux
evaluated at the interface, denoted }(hX at that assembly interface to the homogeneous flux
evaluated at the interface, denoted h jhhh!NhNubjr)}(h+:math:`\phi_{i, \text { homogeneous }}^{+}`h]h/#\phi_{i, \text { homogeneous }}^{+}}(hhh j8ubah}(h]h]h]h]h]uhjqh jubh/ (or }(h (or h jhhh!NhNubjr)}(h-:math:`\phi_{i+1, \text { homogeneous }}^{-}`h]h/%\phi_{i+1, \text { homogeneous }}^{-}}(hhh jKubah}(h]h]h]h]h]uhjqh jubh/):}(h):h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMvh j8hhubh)}(hhh]h}(h]h]h]h]h]hequation-eq9-2-27uhh
h j8hhh!jhNubj )}(hF_{i}^{+}=\frac{\phi_{\text {heterogeneous }}}{\phi_{i, \text { homogeneous }}^{+}}, F_{i+1}^{-}=\frac{\phi_{\text {heterogeneous }}}{\phi_{i+1, \text { homogeneous }}^{-}} .h]h/F_{i}^{+}=\frac{\phi_{\text {heterogeneous }}}{\phi_{i, \text { homogeneous }}^{+}}, F_{i+1}^{-}=\frac{\phi_{\text {heterogeneous }}}{\phi_{i+1, \text { homogeneous }}^{-}} .}(hhh jnubah}(h]jmah]h]h]h]docnamejnumberKylabeleq9-2-27nowrapjyjzuhj h!jhMzh j8hhjf}jh}jmjdsubh;)}(hFluxes, and therefore ADFs, vary with energy; therefore, few-group
homogenized cross sections are always accompanied by corresponding
few-group ADFs.h]h/Fluxes, and therefore ADFs, vary with energy; therefore, few-group
homogenized cross sections are always accompanied by corresponding
few-group ADFs.}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j8hhubh;)}(hXIn a single-assembly calculation with zero-current boundary conditions,
the heterogeneous flux at each boundary is easily calculated as the
surface-averaged scalar flux on the boundary, whereas the homogenous
flux at each boundary is simply the assembly-averaged flux. Hence, for
each energy group, the ADF is calculated for each boundary as the ratio
of the average flux on that boundary to the average flux across the
assembly.h]h/XIn a single-assembly calculation with zero-current boundary conditions,
the heterogeneous flux at each boundary is easily calculated as the
surface-averaged scalar flux on the boundary, whereas the homogenous
flux at each boundary is simply the assembly-averaged flux. Hence, for
each energy group, the ADF is calculated for each boundary as the ratio
of the average flux on that boundary to the average flux across the
assembly.}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j8hhubh;)}(hXIn other configurations, such as a multi-assembly calculation or an
assembly located on the edge of a core next to the core baffle and
reflector, the ADF calculation requires more effort. For reflector
situations, NEWT applies a simple one-dimensional (1-D) multigroup
diffusion approximation to determine the ADF at the assembly boundary.
In this approximation, it is assumed that the reflector is infinite and
that the scalar flux goes to zero at infinity. The reflector ADF can be
determined analytically using this boundary condition along with the
known surface-averaged current and scalar flux evaluated at the
assembly/reflector interface.h]h/XIn other configurations, such as a multi-assembly calculation or an
assembly located on the edge of a core next to the core baffle and
reflector, the ADF calculation requires more effort. For reflector
situations, NEWT applies a simple one-dimensional (1-D) multigroup
diffusion approximation to determine the ADF at the assembly boundary.
In this approximation, it is assumed that the reflector is infinite and
that the scalar flux goes to zero at infinity. The reflector ADF can be
determined analytically using this boundary condition along with the
known surface-averaged current and scalar flux evaluated at the
assembly/reflector interface.}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j8hhubh;)}(hXZThe reflector ADFs computed by NEWT may potentially be different from
the ADFs calculated using the diffusion approximations employed by the
nodal code. Moreover, ADFs computed for multi-assembly or
hexagonal-domain configurations will depend on the nodal method
employed. For these reasons, NEWT supports the option to edit
surface-averaged scalar flux and current values along user-defined line
segments so that appropriate ADFs can be computed directly by the nodal
code. The input options for the single-assembly ADF, reflector ADF, and
arbitrary line-segment edit are discussed in :ref:`9-2-3-11`.h](h/XJThe reflector ADFs computed by NEWT may potentially be different from
the ADFs calculated using the diffusion approximations employed by the
nodal code. Moreover, ADFs computed for multi-assembly or
hexagonal-domain configurations will depend on the nodal method
employed. For these reasons, NEWT supports the option to edit
surface-averaged scalar flux and current values along user-defined line
segments so that appropriate ADFs can be computed directly by the nodal
code. The input options for the single-assembly ADF, reflector ADF, and
arbitrary line-segment edit are discussed in }(hXJThe reflector ADFs computed by NEWT may potentially be different from
the ADFs calculated using the diffusion approximations employed by the
nodal code. Moreover, ADFs computed for multi-assembly or
hexagonal-domain configurations will depend on the nodal method
employed. For these reasons, NEWT supports the option to edit
surface-averaged scalar flux and current values along user-defined line
segments so that appropriate ADFs can be computed directly by the nodal
code. The input options for the single-assembly ADF, reflector ADF, and
arbitrary line-segment edit are discussed in h jhhh!NhNubj)}(h:ref:`9-2-3-11`h]j#)}(hjh]h/9-2-3-11}(hhh jubah}(h]h](jnstdstd-refeh]h]h]uhj"h jubah}(h]h]h]h]h]refdocj refdomainjreftyperefrefexplicitrefwarnj9-2-3-11uhjh!jhMh jubh/.}(hjWh jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j8hhubh)}(h
.. _fig9-2-6:h]h}(h]h]h]h]h]hfig9-2-6uhh
hMh j8hhh!jubj)}(hhh](j)}(h.. figure:: figs/NEWT/fig6.png
:align: center
:width: 500
Heterogeneous vs homogeneous fluxes in a multi-assembly solution.
h]h}(h]h]h]h]h]width500urifigs/NEWT/fig6.pngj}jjsuhjh jh!jhMubj)}(hAHeterogeneous vs homogeneous fluxes in a multi-assembly solution.h]h/AHeterogeneous vs homogeneous fluxes in a multi-assembly solution.}(hjh jubah}(h]h]h]h]h]uhjh!jhMh jubeh}(h](id275jeh]h]fig9-2-6ah]h]jcenteruhjhMh j8hhh!jjf}jjsjh}jjsubh)}(h
.. _9-2-3:h]h}(h]h]h]h]h]hid133uhh
hMh j8hhh!jubeh}(h](assembly-discontinuity-factorsj,eh]h]9-2-2-5ah]assembly discontinuity factorsah]uhh#h j'hhh!jhM[jKjf}j&j"sjh}j,j"subeh}(h](jid104eh]h]9-2-2ah]jB^ah]uhh#h jhhh!jhKjKjf}j1jsjh}jjsubh$)}(hhh](h))}(h
Input Formatsh]h/
Input Formats}(hj;h j9hhh!NhNubah}(h]h]h]h]h]uhh(h j6hhh!jhMubh;)}(hXNEWT input is free form and keyword based, similar in form to the input for many
other modules in the SCALE code package. Input may start with a title card
record, but this line may be omitted if desired; remaining data are supplied in
data blocks. The order of the data blocks is arbitrary (with two exceptions),
and many blocks are optional. Only one instance of a data block is allowed.h]h/XNEWT input is free form and keyword based, similar in form to the input for many
other modules in the SCALE code package. Input may start with a title card
record, but this line may be omitted if desired; remaining data are supplied in
data blocks. The order of the data blocks is arbitrary (with two exceptions),
and many blocks are optional. Only one instance of a data block is allowed.}(hjIh jGhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j6hhubh)}(h.. _9-2-3-1:h]h}(h]h]h]h]h]hid134uhh
hMh j6hhh!jubh$)}(hhh](h))}(hOverview of newt data blocksh]h/Overview of newt data blocks}(hjeh jchhh!NhNubah}(h]h]h]h]h]uhh(h j`hhh!jhMubh;)}(hXThe NEWT input deck data blocks are defined by keyword delimiters in the
following form:h]h/XThe NEWT input deck data blocks are defined by keyword delimiters in the
following form:}(hjsh jqhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j`hhubh
highlightlang)}(hhh]h}(h]h]h]h]h]langnoneforcelinenothresholduhjh j`hhh!jhMubj9)}(h+read keyword [data] end keywordh]h/+read keyword [data] end keyword}(hhh jubah}(h]h]h]h]h]jyjzuhj8h!jhMh j`hhubh;)}(hRead routines are terminated by the “end *keyword*\ ” label, and any
intervening carriage returns or line feeds are ignored. Thus, data can
also be entered in this format:h](h/+Read routines are terminated by the “end }(h+Read routines are terminated by the “end h jhhh!NhNubhA)}(h *keyword*h]h/keyword}(hhh jubah}(h]h]h]h]h]uhh@h jubh/{ ” label, and any
intervening carriage returns or line feeds are ignored. Thus, data can
also be entered in this format:}(h{\ ” label, and any
intervening carriage returns or line feeds are ignored. Thus, data can
also be entered in this format:h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh j`hhubj9)}(h&read keyword
[data]
[data]
end keywordh]h/&read keyword
[data]
[data]
end keyword}(hhh jubah}(h]h]h]h]h]jyjzuhj8h!jhMh j`hhubh;)}(hWithin each block, specific control or specification parameters are
input. Each block contains a fixed set of input parameters (also defined
by keyword).h]h/Within each block, specific control or specification parameters are
input. Each block contains a fixed set of input parameters (also defined
by keyword).}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j`hhubh;)}(hAs with other keyword-driven modules within SCALE, lines beginning with
a single quote (') in the first column are treated as comments and
ignored.h]h/As with other keyword-driven modules within SCALE, lines beginning with
a single quote (‘) in the first column are treated as comments and
ignored.}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j`hhubh;)}(hHThe keyword name and general contents of each data block are as follows:h]h/HThe keyword name and general contents of each data block are as follows:}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhMh j`hhubjq)}(hhh]j)}(hhh](j)}(hhh]h}(h]h]h]h]h]colwidthKuhjh jubj)}(hhh]h}(h]h]h]h]h]colwidthKuhjh jubj)}(hhh]h}(h]h]h]h]h]colwidthKuhjh jubj)}(hhh](j)}(hhh](j)}(hhh]h;)}(h**Block type**h]j)}(hj%h]h/
Block type}(hhh j'ubah}(h]h]h]h]h]uhjh j#ubah}(h]h]h]h]h]uhh:h!jhMh j ubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(h**Recognized
keywords**h]j)}(hjEh]h/Recognized
keywords}(hhh jGubah}(h]h]h]h]h]uhjh jCubah}(h]h]h]h]h]uhh:h!jhMh j@ubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(h**Description**h]j)}(hjeh]h/Description}(hhh jgubah}(h]h]h]h]h]uhjh jcubah}(h]h]h]h]h]uhh:h!jhMh j`ubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh](h;)}(hProblem control
parametersh]h/Problem control
parameters}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubh;)}(h:ref:`9-2-3-2`h]j)}(hjh]j#)}(hjh]h/9-2-3-2}(hhh jubah}(h]h](jnstdstd-refeh]h]h]uhj"h jubah}(h]h]h]h]h]refdocj refdomainjreftyperefrefexplicitrefwarnj9-2-3-2uhjh!jhMh jubah}(h]h]h]h]h]uhh:h!jhMh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(h(parameter,
parameters, param,
parm, parah]h/(parameter,
parameters, param,
parm, para}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(hGGeneral problem
parameters—must
follow title card, if
used (optional)h]h/GGeneral problem
parameters—must
follow title card, if
used (optional)}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh](h;)}(hMaterial propertiesh]h/Material properties}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubh;)}(h:ref:`9-2-3-3`h]j)}(hjh]j#)}(hjh]h/9-2-3-3}(hhh jubah}(h]h](jnstdstd-refeh]h]h]uhj"h jubah}(h]h]h]h]h]refdocj refdomainj"reftyperefrefexplicitrefwarnj9-2-3-3uhjh!jhMh jubah}(h]h]h]h]h]uhh:h!jhMh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(hmaterial, materials,
matlh]h/material, materials,
matl}(hjEh jCubah}(h]h]h]h]h]uhh:h!jhMh j@ubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(hAssigns
characteristics
(e.g., P\ :sub:`n`
scattering order and
material description)
for each material
ID—must follow
problem control block
or must follow title
card if control block
is omitted (required)h](h/"Assigns
characteristics
(e.g., P }(h"Assigns
characteristics
(e.g., P\ h jZubh)}(h:sub:`n`h]h/n}(hhh jcubah}(h]h]h]h]h]uhhh jZubh/
scattering order and
material description)
for each material
ID—must follow
problem control block
or must follow title
card if control block
is omitted (required)}(h
scattering order and
material description)
for each material
ID—must follow
problem control block
or must follow title
card if control block
is omitted (required)h jZubeh}(h]h]h]h]h]uhh:h!jhMh jWubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh](h;)}(hBroad group collapseh]h/Broad group collapse}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubh;)}(h:ref:`9-2-3-5`h]j)}(hjh]j#)}(hjh]h/9-2-3-5}(hhh jubah}(h]h](jnstdstd-refeh]h]h]uhj"h jubah}(h]h]h]h]h]refdocj refdomainjreftyperefrefexplicitrefwarnj9-2-3-5uhjh!jhMh jubah}(h]h]h]h]h]uhh:h!jhMh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(hcollapse, collh]h/collapse, coll}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(hDefines broad group
energy ranges to be
created from the
original fine group
library when cross
section collapse is
desired (optional)h]h/Defines broad group
energy ranges to be
created from the
original fine group
library when cross
section collapse is
desired (optional)}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh](h;)}(hSimple-body geometryh]h/Simple-body geometry}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubh;)}(h:ref:`9-2-3-6`h]j)}(hjh]j#)}(hjh]h/9-2-3-6}(hhh jubah}(h]h](jnstdstd-refeh]h]h]uhj"h jubah}(h]h]h]h]h]refdocj refdomainj$reftyperefrefexplicitrefwarnj9-2-3-6uhjh!jhMh jubah}(h]h]h]h]h]uhh:h!jhMh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(hgeometry, geomh]h/geometry, geom}(hjGh jEubah}(h]h]h]h]h]uhh:h!jhMh jBubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(hDefines basic grid
structure and all
bodies to be placed
within this structure
(required unless
geometry restart file
is available)h]h/Defines basic grid
structure and all
bodies to be placed
within this structure
(required unless
geometry restart file
is available)}(hj^h j\ubah}(h]h]h]h]h]uhh:h!jhMh jYubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh](h;)}(hBoundary conditionsh]h/Boundary conditions}(hj~h j|ubah}(h]h]h]h]h]uhh:h!jhMh jyubh;)}(h:ref:`9-2-3-7`h]j)}(hjh]j#)}(hjh]h/9-2-3-7}(hhh jubah}(h]h](jnstdstd-refeh]h]h]uhj"h jubah}(h]h]h]h]h]refdocj refdomainjreftyperefrefexplicitrefwarnj9-2-3-7uhjh!jhMh jubah}(h]h]h]h]h]uhh:h!jhMh jyubeh}(h]h]h]h]h]uhjh jvubj)}(hhh]h;)}(hbounds, bndsh]h/bounds, bnds}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jvubj)}(hhh]h;)}(h{Defines boundary
conditions to be
applied on outer
boundaries of global
unit (optional,
default is reflective
on all sides)h]h/{Defines boundary
conditions to be
applied on outer
boundaries of global
unit (optional,
default is reflective
on all sides)}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jvubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh](h;)}(hArray specificationsh]h/Array specifications}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubh;)}(h:ref:`9-2-3-9`h]j)}(hjh]j#)}(hjh]h/9-2-3-9}(hhh jubah}(h]h](jnstdstd-refeh]h]h]uhj"h jubah}(h]h]h]h]h]refdocj refdomainjreftyperefrefexplicitrefwarnj9-2-3-9uhjh!jhM h jubah}(h]h]h]h]h]uhh:h!jhMh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(harrayh]h/array}(hj5h j3ubah}(h]h]h]h]h]uhh:h!jhMh j0ubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(hDefines composition
of all arrays (unit
placement within each
array). Each array
placed within the
geometry block must
be defined in the
array blockh]h/Defines composition
of all arrays (unit
placement within each
array). Each array
placed within the
geometry block must
be defined in the
array block}(hjLh jJubah}(h]h]h]h]h]uhh:h!jhMh jGubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh](h;)}(hHomogenization
instructionsh]h/Homogenization
instructions}(hjlh jjubah}(h]h]h]h]h]uhh:h!jhMh jgubh;)}(h:ref:`9-2-3-10`h]j)}(hjzh]j#)}(hjzh]h/9-2-3-10}(hhh jubah}(h]h](jnstdstd-refeh]h]h]uhj"h j|ubah}(h]h]h]h]h]refdocj refdomainjreftyperefrefexplicitrefwarnj9-2-3-10uhjh!jhMh jxubah}(h]h]h]h]h]uhh:h!jhMh jgubeh}(h]h]h]h]h]uhjh jdubj)}(hhh]h;)}(hhomog, hmog, homoh]h/homog, hmog, homo}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jdubj)}(hhh]h;)}(hyDefines mixtures to
be flux weighted and
homogenized in the
preparation of a
homogenized cross
section library
(optional)h]h/yDefines mixtures to
be flux weighted and
homogenized in the
preparation of a
homogenized cross
section library
(optional)}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jdubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh](h;)}(hAssembly
discontinuity factorsh]h/Assembly
discontinuity factors}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubh;)}(h:ref:`9-2-3-11`h]j)}(hjh]j#)}(hjh]h/9-2-3-11}(hhh jubah}(h]h](jnstdstd-refeh]h]h]uhj"h jubah}(h]h]h]h]h]refdocj refdomainjreftyperefrefexplicitrefwarnj9-2-3-11uhjh!jhMh jubah}(h]h]h]h]h]uhh:h!jhMh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(hadfh]h/adf}(hj#h j!ubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(hlAssigns type and
location of planes at
which assembly
discontinuity factors
(ADFs) are calculated
(optional)h]h/lAssigns type and
location of planes at
which assembly
discontinuity factors
(ADFs) are calculated
(optional)}(hj:h j8ubah}(h]h]h]h]h]uhh:h!jhMh j5ubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh](h;)}(h
Flux planeh]h/
Flux plane}(hjZh jXubah}(h]h]h]h]h]uhh:h!jhMh jUubh;)}(h:ref:`9-2-3-12`h]j)}(hjhh]j#)}(hjhh]h/9-2-3-12}(hhh jmubah}(h]h](jnstdstd-refeh]h]h]uhj"h jjubah}(h]h]h]h]h]refdocj refdomainjwreftyperefrefexplicitrefwarnj9-2-3-12uhjh!jhM!h jfubah}(h]h]h]h]h]uhh:h!jhM h jUubeh}(h]h]h]h]h]uhjh jRubj)}(hhh]h;)}(hfluxh]h/flux}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jRubj)}(hhh]h;)}(hnAllows definition of
an x- or y-axis line
(plane) for which
average fluxes are
computed and printed
(optional)h]h/nAllows definition of
an x- or y-axis line
(plane) for which
average fluxes are
computed and printed
(optional)}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jRubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh](h;)}(hMixing tableh]h/Mixing table}(hjh jubah}(h]h]h]h]h]uhh:h!jhM%h jubh;)}(h:ref:`9-2-3-13`h]j)}(hjh]j#)}(hjh]h/9-2-3-13}(hhh jubah}(h]h](jnstdstd-refeh]h]h]uhj"h jubah}(h]h]h]h]h]refdocj refdomainjreftyperefrefexplicitrefwarnj9-2-3-13uhjh!jhM(h jubah}(h]h]h]h]h]uhh:h!jhM'h jubeh}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(hmixtable, mixth]h/mixtable, mixt}(hjh jubah}(h]h]h]h]h]uhh:h!jhM%h jubah}(h]h]h]h]h]uhjh jubj)}(hhh]h;)}(h%Mixing table
specification
(optional)h]h/%Mixing table
specification
(optional)}(hj(h j&ubah}(h]h]h]h]h]uhh:h!jhM%h j#ubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubj)}(hhh](j)}(hhh](h;)}(hSource definitionh]h/Source definition}(hjHh jFubah}(h]h]h]h]h]uhh:h!jhM)h jCubh;)}(h:ref:`9-2-3-4`h]j)}(hjVh]j#)}(hjVh]h/9-2-3-4}(hhh j[ubah}(h]h](jnstdstd-refeh]h]h]uhj"h jXubah}(h]h]h]h]h]refdocj refdomainjereftyperefrefexplicitrefwarnj9-2-3-4uhjh!jhM,h jTubah}(h]h]h]h]h]uhh:h!jhM+h jCubeh}(h]h]h]h]h]uhjh j@ubj)}(hhh]h;)}(hsrc, sourceh]h/src, source}(hjh jubah}(h]h]h]h]h]uhh:h!jhM)h jubah}(h]h]h]h]h]uhjh j@ubj)}(hhh]h;)}(h?Defines particle
source strength for
use in source
calculationsh]h/?Defines particle
source strength for
use in source
calculations}(hjh jubah}(h]h]h]h]h]uhh:h!jhM)h jubah}(h]h]h]h]h]uhjh j@ubeh}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]colsKuhjh jubah}(h]h] longtableah]h]h]jcenteruhjph j`hhh!jhNubh;)}(hEach of the following subsections describes the parameters associated
with a specific data block, lists default values (if available), and
describes meaning of the parameter and its effect on a NEWT calculation.h]h/Each of the following subsections describes the parameters associated
with a specific data block, lists default values (if available), and
describes meaning of the parameter and its effect on a NEWT calculation.}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh:h!jhM/h j`hhubh)}(h.. _9-2-3-2:h]h}(h]h]h]h]h]hid135uhh
hMFh j`hhh!jubeh}(h](overview-of-newt-data-blocksj_eh]h](overview of newt data blocks9-2-3-1eh]h]uhh#h j6hhh!jhMjf}jjUsjh}j_jUsubh$)}(hhh](h))}(hParameter blockh]h/Parameter block}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhM6ubh;)}(hI**Parameter Block keyword = param, parm, para, parameter, or
parameters**h]j)}(hjh]h/EParameter Block keyword = param, parm, para, parameter, or
parameters}(hhh jubah}(h]h]h]h]h]uhjh jubah}(h]h]h]h]h]uhh:h!jhM8h jhhubh;)}(hXThe Parameter block contains problem control parameters and must come
immediately after the title card if one is used. Valid parameter
specifications are described below. For each keyword, allowable values
are listed in parentheses, and the default (if any) is listed in
brackets. Input that can take an arbitrary integer value is indicated by
an *IN*; similarly, any parameter that can take an arbitrary
real/floating point value is indicated by *RN* as the allowable value.
However, note that SCALE read routines do allow input of integers for
real numbers, and vice versa; the number will be converted accordingly.
The order of the parameters within the block is arbitrary, and may be
skipped if a default value is desired for that parameter. Control
parameters are set in the order in which they are input; this means that
the same parameter may be listed multiple times, but only the final
value is used.h](h/X[The Parameter block contains problem control parameters and must come
immediately after the title card if one is used. Valid parameter
specifications are described below. For each keyword, allowable values
are listed in parentheses, and the default (if any) is listed in
brackets. Input that can take an arbitrary integer value is indicated by
an }(hX[The Parameter block contains problem control parameters and must come
immediately after the title card if one is used. Valid parameter
specifications are described below. For each keyword, allowable values
are listed in parentheses, and the default (if any) is listed in
brackets. Input that can take an arbitrary integer value is indicated by
an h jhhh!NhNubhA)}(h*IN*h]h/IN}(hhh j!ubah}(h]h]h]h]h]uhh@h jubh/`; similarly, any parameter that can take an arbitrary
real/floating point value is indicated by }(h`; similarly, any parameter that can take an arbitrary
real/floating point value is indicated by h jhhh!NhNubhA)}(h*RN*h]h/RN}(hhh j4ubah}(h]h]h]h]h]uhh@h jubh/X as the allowable value.
However, note that SCALE read routines do allow input of integers for
real numbers, and vice versa; the number will be converted accordingly.
The order of the parameters within the block is arbitrary, and may be
skipped if a default value is desired for that parameter. Control
parameters are set in the order in which they are input; this means that
the same parameter may be listed multiple times, but only the final
value is used.}(hX as the allowable value.
However, note that SCALE read routines do allow input of integers for
real numbers, and vice versa; the number will be converted accordingly.
The order of the parameters within the block is arbitrary, and may be
skipped if a default value is desired for that parameter. Control
parameters are set in the order in which they are input; this means that
the same parameter may be listed multiple times, but only the final
value is used.h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM;h jhhubh)}(h.. _9-2-3-2-1:h]h}(h]h]h]h]h]hid136uhh
hM]h jhhh!jubh$)}(hhh](h))}(h'Convergence and acceleration parametersh]h/'Convergence and acceleration parameters}(hj]h j[hhh!NhNubah}(h]h]h]h]h]uhh(h jXhhh!jhMMubh;)}(hM**epseigen=**\ (*RN*) — Convergence criterion for *k*\ :sub:`eff`. [0.0001]h](j)}(h
**epseigen=**h]h/ epseigen=}(hhh jmubah}(h]h]h]h]h]uhjh jiubh/ (}(h\ (h jihhh!NhNubhA)}(h*RN*h]h/RN}(hhh jubah}(h]h]h]h]h]uhh@h jiubh/ ) — Convergence criterion for }(h ) — Convergence criterion for h jihhh!NhNubhA)}(h*k*h]h/k}(hhh jubah}(h]h]h]h]h]uhh@h jiubh/ }(h\ h jihhh!NhNubh)}(h
:sub:`eff`h]h/eff}(hhh jubah}(h]h]h]h]h]uhhh jiubh/
. [0.0001]}(h
. [0.0001]h jihhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMOh jXhhubh;)}(hV**epsinner=**\ (*RN*) — Spatial convergence criterion for inner
iterations. [0.0001]h](j)}(h
**epsinner=**h]h/ epsinner=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*RN*h]h/RN}(hhh jubah}(h]h]h]h]h]uhh@h jubh/B) — Spatial convergence criterion for inner
iterations. [0.0001]}(hB) — Spatial convergence criterion for inner
iterations. [0.0001]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMQh jXhhubh;)}(hV**epsouter=**\ (*RN*) — Spatial convergence criterion for outer
iterations. [0.0001]h](j)}(h
**epsouter=**h]h/ epsouter=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*RN*h]h/RN}(hhh jubah}(h]h]h]h]h]uhh@h jubh/B) — Spatial convergence criterion for outer
iterations. [0.0001]}(hB) — Spatial convergence criterion for outer
iterations. [0.0001]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMTh jXhhubh;)}(h**epsthrm=**\ (*RN*) — Spatial convergence criterion for
thermal-upscattering iterations, if enabled. [same value as
**epsouter**]h](j)}(h**epsthrm=**h]h/epsthrm=}(hhh j#ubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*RN*h]h/RN}(hhh j6ubah}(h]h]h]h]h]uhh@h jubh/d) — Spatial convergence criterion for
thermal-upscattering iterations, if enabled. [same value as
}(hd) — Spatial convergence criterion for
thermal-upscattering iterations, if enabled. [same value as
h jhhh!NhNubj)}(h**epsouter**h]h/epsouter}(hhh jIubah}(h]h]h]h]h]uhjh jubh/]}(hj~vh jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMWh jXhhubh;)}(h**epsilon=**\ (*RN*) — Simultaneously sets all (spatial and eigenvalue)
convergence criteria to the same value. [uses individual defaults]h](j)}(h**epsilon=**h]h/epsilon=}(hhh jeubah}(h]h]h]h]h]uhjh jaubh/ (}(h\ (h jahhh!NhNubhA)}(h*RN*h]h/RN}(hhh jxubah}(h]h]h]h]h]uhh@h jaubh/y) — Simultaneously sets all (spatial and eigenvalue)
convergence criteria to the same value. [uses individual defaults]}(hy) — Simultaneously sets all (spatial and eigenvalue)
convergence criteria to the same value. [uses individual defaults]h jahhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM[h jXhhubh;)}(hX**converg**\ =(\ *cell/mix*) — Sets the region within which convergence
testing is applied. Use of *cell* will force converged scalar fluxes in
every computation cell, while *mix* will relax convergence such that
averaged scalar fluxes within a mixture are converged. The latter is
useful for mixtures in which fluxes become very small—large reflectors
or near a vacuum BC. [cell]h](j)}(h**converg**h]h/converg}(hhh jubah}(h]h]h]h]h]uhjh jubh/ =( }(h\ =(\ h jhhh!NhNubhA)}(h
*cell/mix*h]h/cell/mix}(hhh jubah}(h]h]h]h]h]uhh@h jubh/J) — Sets the region within which convergence
testing is applied. Use of }(hJ) — Sets the region within which convergence
testing is applied. Use of h jhhh!NhNubhA)}(h*cell*h]h/cell}(hhh jubah}(h]h]h]h]h]uhh@h jubh/E will force converged scalar fluxes in
every computation cell, while }(hE will force converged scalar fluxes in
every computation cell, while h jhhh!NhNubhA)}(h*mix*h]h/mix}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ will relax convergence such that
averaged scalar fluxes within a mixture are converged. The latter is
useful for mixtures in which fluxes become very small—large reflectors
or near a vacuum BC. [cell]}(h will relax convergence such that
averaged scalar fluxes within a mixture are converged. The latter is
useful for mixtures in which fluxes become very small—large reflectors
or near a vacuum BC. [cell]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM^h jXhhubh;)}(hT**therm**\ =(\ *yes/no*) — Enables/disables thermal-upscattering
iterations. [yes]h](j)}(h **therm**h]h/therm}(hhh jubah}(h]h]h]h]h]uhjh jubh/ =( }(h\ =(\ h jhhh!NhNubhA)}(h*yes/no*h]h/yes/no}(hhh jubah}(h]h]h]h]h]uhh@h jubh/=) — Enables/disables thermal-upscattering
iterations. [yes]}(h=) — Enables/disables thermal-upscattering
iterations. [yes]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMeh jXhhubh;)}(hR**inners=**\ (*IN*) — Maximum number of inner iterations in an energy
group. [5]h](j)}(h**inners=**h]h/inners=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*IN*h]h/IN}(hhh j.ubah}(h]h]h]h]h]uhh@h jubh/@) — Maximum number of inner iterations in an energy
group. [5]}(h@) — Maximum number of inner iterations in an energy
group. [5]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMhh jXhhubh;)}(hZ**therms=**\ (*IN*) — Maximum number of thermal-upscattering iterations,
if enabled. [2]h](j)}(h**therms=**h]h/therms=}(hhh jKubah}(h]h]h]h]h]uhjh jGubh/ (}(h\ (h jGhhh!NhNubhA)}(h*IN*h]h/IN}(hhh j^ubah}(h]h]h]h]h]uhh@h jGubh/H) — Maximum number of thermal-upscattering iterations,
if enabled. [2]}(hH) — Maximum number of thermal-upscattering iterations,
if enabled. [2]h jGhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMkh jXhhubh;)}(h**outers=**\ (*IN*) — Maximum number of outer iterations. NEWT will stop
with an error code if more than *outers* outer iterations are required
for convergence. [250]h](j)}(h**outers=**h]h/outers=}(hhh j{ubah}(h]h]h]h]h]uhjh jwubh/ (}(h\ (h jwhhh!NhNubhA)}(h*IN*h]h/IN}(hhh jubah}(h]h]h]h]h]uhh@h jwubh/Y) — Maximum number of outer iterations. NEWT will stop
with an error code if more than }(hY) — Maximum number of outer iterations. NEWT will stop
with an error code if more than h jwhhh!NhNubhA)}(h*outers*h]h/outers}(hhh jubah}(h]h]h]h]h]uhh@h jwubh/5 outer iterations are required
for convergence. [250]}(h5 outer iterations are required
for convergence. [250]h jwhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMnh jXhhubh;)}(hX4**inrcvrg**\ =(\ *yes/no*) — If inrcvrg=yes, NEWT will continue outer
iterations until all convergence criteria are met. If inrcvrg=no, NEWT
will stop whenever outer iteration and *k*\ :sub:`eff` convergence criterion
are met, regardless of the convergence of inner or thermal-upscattering
iterations. [no]h](j)}(h**inrcvrg**h]h/inrcvrg}(hhh jubah}(h]h]h]h]h]uhjh jubh/ =( }(h\ =(\ h jhhh!NhNubhA)}(h*yes/no*h]h/yes/no}(hhh jubah}(h]h]h]h]h]uhh@h jubh/) — If inrcvrg=yes, NEWT will continue outer
iterations until all convergence criteria are met. If inrcvrg=no, NEWT
will stop whenever outer iteration and }(h) — If inrcvrg=yes, NEWT will continue outer
iterations until all convergence criteria are met. If inrcvrg=no, NEWT
will stop whenever outer iteration and h jhhh!NhNubhA)}(h*k*h]h/k}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ }(h\ h jhhh!NhNubh)}(h
:sub:`eff`h]h/eff}(hhh jubah}(h]h]h]h]h]uhhh jubh/o convergence criterion
are met, regardless of the convergence of inner or thermal-upscattering
iterations. [no]}(ho convergence criterion
are met, regardless of the convergence of inner or thermal-upscattering
iterations. [no]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMrh jXhhubh;)}(hX{**cmfd=**\ (*no/rect/yes/part*) — CMFD acceleration option. If cmfd=no,
CMFD acceleration is not employed. If cmfd=rect, the CMFD method is
employed. The original NEWT CMFD method can be applied only to
rectangular-domain configurations. If cmfd=yes, the unstructured CMFD
method is employed. The new unstructured CMFD method can be applied to
rectangular-, triangular-, and hexagonal- domain configurations. If
cmfd=part, an alternative version of the unstructured CMFD method is
employed and uses a “partial-current” acceleration scheme.
Alternatively, users can use cmfd=0/1/2/3 for no, rect, yes, and part,
respectively. [no]h](j)}(h **cmfd=**h]h/cmfd=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*no/rect/yes/part*h]h/no/rect/yes/part}(hhh j'ubah}(h]h]h]h]h]uhh@h jubh/X]) — CMFD acceleration option. If cmfd=no,
CMFD acceleration is not employed. If cmfd=rect, the CMFD method is
employed. The original NEWT CMFD method can be applied only to
rectangular-domain configurations. If cmfd=yes, the unstructured CMFD
method is employed. The new unstructured CMFD method can be applied to
rectangular-, triangular-, and hexagonal- domain configurations. If
cmfd=part, an alternative version of the unstructured CMFD method is
employed and uses a “partial-current” acceleration scheme.
Alternatively, users can use cmfd=0/1/2/3 for no, rect, yes, and part,
respectively. [no]}(hX]) — CMFD acceleration option. If cmfd=no,
CMFD acceleration is not employed. If cmfd=rect, the CMFD method is
employed. The original NEWT CMFD method can be applied only to
rectangular-domain configurations. If cmfd=yes, the unstructured CMFD
method is employed. The new unstructured CMFD method can be applied to
rectangular-, triangular-, and hexagonal- domain configurations. If
cmfd=part, an alternative version of the unstructured CMFD method is
employed and uses a “partial-current” acceleration scheme.
Alternatively, users can use cmfd=0/1/2/3 for no, rect, yes, and part,
respectively. [no]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMxh jXhhubh;)}(h**cmfd2g=**\ (*yes/no*) — Enables/disables the second-level two-group
CMFD accelerator within the CMFD solver. This parameter has an effect
only when cmfd=rect is set. [yes]h](j)}(h**cmfd2g=**h]h/cmfd2g=}(hhh jDubah}(h]h]h]h]h]uhjh j@ubh/ (}(h\ (h j@hhh!NhNubhA)}(h*yes/no*h]h/yes/no}(hhh jWubah}(h]h]h]h]h]uhh@h j@ubh/) — Enables/disables the second-level two-group
CMFD accelerator within the CMFD solver. This parameter has an effect
only when cmfd=rect is set. [yes]}(h) — Enables/disables the second-level two-group
CMFD accelerator within the CMFD solver. This parameter has an effect
only when cmfd=rect is set. [yes]h j@hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jXhhubh;)}(h**accel=**\ (*yes/no*) — Enables/disables source (*k*\ :sub:`eff`)
acceleration. This parameter is automatically disabled if unstructured
CMFD is employed (cmfd=yes or cmfd=part). [yes]h](j)}(h
**accel=**h]h/accel=}(hhh jtubah}(h]h]h]h]h]uhjh jpubh/ (}(h\ (h jphhh!NhNubhA)}(h*yes/no*h]h/yes/no}(hhh jubah}(h]h]h]h]h]uhh@h jpubh/) — Enables/disables source (}(h) — Enables/disables source (h jphhh!NhNubhA)}(h*k*h]h/k}(hhh jubah}(h]h]h]h]h]uhh@h jpubh/ }(h\ h jphhh!NhNubh)}(h
:sub:`eff`h]h/eff}(hhh jubah}(h]h]h]h]h]uhhh jpubh/x)
acceleration. This parameter is automatically disabled if unstructured
CMFD is employed (cmfd=yes or cmfd=part). [yes]}(hx)
acceleration. This parameter is automatically disabled if unstructured
CMFD is employed (cmfd=yes or cmfd=part). [yes]h jphhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jXhhubh;)}(hX**xcmfd**\ =(\ *IN),* **ycmfd**\ =(\ *IN),* **xycmfd**\ =(\ *IN)* —
These inputs specify the number of fine-mesh cells in the global NEWT
grid per coarse-mesh cell. These options are used only when CMFD
acceleration is enabled. The parameter *xcmfd* specifies the number
fine-mesh cells per coarse-mesh cell in the x‑direction. Likewise,
*ycmfd* specifies the number of fine-mesh cells per coarse-mesh cell in
the y‑direction. The parameter *xycmfd* simultaneously sets *xcmfd* and
*ycmfd* to the same value. In a special case for rectangular-domain
configurations in which the entire domain is completely filled by a
square-type array (see :ref:`9-2-3-9`), *xycmfd=0* sets the coarse mesh
based on the size of the array elements. [1]h](j)}(h **xcmfd**h]h/xcmfd}(hhh jubah}(h]h]h]h]h]uhjh jubh/ =( }(h\ =(\ h jhhh!NhNubhA)}(h*IN),*h]h/IN),}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ }(hjh jhhh!NhNubj)}(h **ycmfd**h]h/ycmfd}(hhh jubah}(h]h]h]h]h]uhjh jubh/ =( }(h\ =(\ h jubhA)}(h*IN),*h]h/IN),}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ }(hjh jubj)}(h
**xycmfd**h]h/xycmfd}(hhh jubah}(h]h]h]h]h]uhjh jubh/ =( }(hjh jubhA)}(h*IN)*h]h/IN)}(hhh j&ubah}(h]h]h]h]h]uhh@h jubh/ —
These inputs specify the number of fine-mesh cells in the global NEWT
grid per coarse-mesh cell. These options are used only when CMFD
acceleration is enabled. The parameter }(h —
These inputs specify the number of fine-mesh cells in the global NEWT
grid per coarse-mesh cell. These options are used only when CMFD
acceleration is enabled. The parameter h jhhh!NhNubhA)}(h*xcmfd*h]h/xcmfd}(hhh j9ubah}(h]h]h]h]h]uhh@h jubh/[ specifies the number
fine-mesh cells per coarse-mesh cell in the x‑direction. Likewise,
}(h[ specifies the number
fine-mesh cells per coarse-mesh cell in the x‑direction. Likewise,
h jhhh!NhNubhA)}(h*ycmfd*h]h/ycmfd}(hhh jLubah}(h]h]h]h]h]uhh@h jubh/b specifies the number of fine-mesh cells per coarse-mesh cell in
the y‑direction. The parameter }(hb specifies the number of fine-mesh cells per coarse-mesh cell in
the y‑direction. The parameter h jhhh!NhNubhA)}(h*xycmfd*h]h/xycmfd}(hhh j_ubah}(h]h]h]h]h]uhh@h jubh/ simultaneously sets }(h simultaneously sets h jhhh!NhNubhA)}(h*xcmfd*h]h/xcmfd}(hhh jrubah}(h]h]h]h]h]uhh@h jubh/ and
}(h and
h jhhh!NhNubhA)}(h*ycmfd*h]h/ycmfd}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ to the same value. In a special case for rectangular-domain
configurations in which the entire domain is completely filled by a
square-type array (see }(h to the same value. In a special case for rectangular-domain
configurations in which the entire domain is completely filled by a
square-type array (see h jhhh!NhNubj)}(h:ref:`9-2-3-9`h]j#)}(hjh]h/9-2-3-9}(hhh jubah}(h]h](jnstdstd-refeh]h]h]uhj"h jubah}(h]h]h]h]h]refdocj refdomainjreftyperefrefexplicitrefwarnj9-2-3-9uhjh!jhMh jubh/), }(h), h jhhh!NhNubhA)}(h
*xycmfd=0*h]h/xycmfd=0}(hhh jubah}(h]h]h]h]h]uhh@h jubh/B sets the coarse mesh
based on the size of the array elements. [1]}(hB sets the coarse mesh
based on the size of the array elements. [1]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jXhhubh important)}(hXDefault convergence parameters are recommended for general analysis.
Larger convergence criteria are useful for debugging if shorter run time
is desired over solution accuracy. Smaller convergence criteria are
recommended for generating reference solutions or benchmark
calculations.
CMFD acceleration should be applied whenever possible. The CMFD method
with second-level 2-group acceleration should be applied for
rectangular-domain configurations [e.g., light water reactor (LWR)
assembly models (*\ **cmfd=rect**\ *), by default*\ **cmfd2g=yes**\ *].
The unstructured CMFD method should be applied for triangular- or
hexagonal-domain configurations (*\ **cmfd=yes**\ *). If NEWT detects an
unstable CMFD condition, a warning message is printed and NEWT continues
with CMFD disabled. NEWT may also provide a terminating error message if
improper selection of the coarse mesh is detected. Internal
investigation has shown that the coarse mesh should be approximately the
same size as the unit cell used in the model. For LWR assembly models, a
fine mesh of 4 x 4 is recommended for the square-pitched unit cell,
implying that*\ **xycmfd**\ *should be 4 only if the global unit has a
mesh. If individual meshes are used in each unit definition, then the
global unit coarse-mesh cells should be sized based on the unit cell
size and, therefore, xycmfd=1 should be used. The values
of*\ **xcmfd**\ *and*\ **ycmfd**\ *do not have to be a common factor of
the number of fine-mesh cells in a given direction (NEWT will make the
last coarse-mesh cell smaller than the other coarse-mesh cells), but it
is highly recommended.
Users can gauge solution convergence by the outer iteration edit as it
is printed to the terminal window (*\ **echo=yes**\ *, see below). One
can terminate a calculation prematurely (via the Control-C option on
most platforms) if convergence or iteration parameters need to be
modified.
The TRITON control module supports a sensitivity and uncertainty
analysis sequence TSUNAMI-2D (See TRITON chapter, section S/U Analysis
Sequences (TSUNAMI-2D, TSUNAMI-2DC)). TSUNAMI-2D calculations require
NEWT to be run in both forward mode and adjoint mode. In adjoint mode,
CMFD acceleration is not currently supported and NEWT automatically
disables its use if*\ **cmfd=yes**\ *,*\ **=rect**\ *,
or*\ **=part**\ *. In adjoint mode with defined fixed source [i.e.,
generalized perturbation theory (GPT) analysis], it is observed that
tighter convergence and iteration parameters are needed to properly
remove fundamental mode contamination. (For more details, see SAMS
chapter: Generalized Perturbation Theory.) To facilitate the CMFD
options and larger convergence criteria for the forward calculations as
well as smaller convergence criteria for GPT adjoint calculations, the
following parameters are also available.h](h;)}(hXDefault convergence parameters are recommended for general analysis.
Larger convergence criteria are useful for debugging if shorter run time
is desired over solution accuracy. Smaller convergence criteria are
recommended for generating reference solutions or benchmark
calculations.h]h/XDefault convergence parameters are recommended for general analysis.
Larger convergence criteria are useful for debugging if shorter run time
is desired over solution accuracy. Smaller convergence criteria are
recommended for generating reference solutions or benchmark
calculations.}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubh;)}(hX6CMFD acceleration should be applied whenever possible. The CMFD method
with second-level 2-group acceleration should be applied for
rectangular-domain configurations [e.g., light water reactor (LWR)
assembly models (*\ **cmfd=rect**\ *), by default*\ **cmfd2g=yes**\ *].
The unstructured CMFD method should be applied for triangular- or
hexagonal-domain configurations (*\ **cmfd=yes**\ *). If NEWT detects an
unstable CMFD condition, a warning message is printed and NEWT continues
with CMFD disabled. NEWT may also provide a terminating error message if
improper selection of the coarse mesh is detected. Internal
investigation has shown that the coarse mesh should be approximately the
same size as the unit cell used in the model. For LWR assembly models, a
fine mesh of 4 x 4 is recommended for the square-pitched unit cell,
implying that*\ **xycmfd**\ *should be 4 only if the global unit has a
mesh. If individual meshes are used in each unit definition, then the
global unit coarse-mesh cells should be sized based on the unit cell
size and, therefore, xycmfd=1 should be used. The values
of*\ **xcmfd**\ *and*\ **ycmfd**\ *do not have to be a common factor of
the number of fine-mesh cells in a given direction (NEWT will make the
last coarse-mesh cell smaller than the other coarse-mesh cells), but it
is highly recommended.h](h/CMFD acceleration should be applied whenever possible. The CMFD method
with second-level 2-group acceleration should be applied for
rectangular-domain configurations [e.g., light water reactor (LWR)
assembly models (}(hCMFD acceleration should be applied whenever possible. The CMFD method
with second-level 2-group acceleration should be applied for
rectangular-domain configurations [e.g., light water reactor (LWR)
assembly models (h jubhA)}(h*\ **cmfd=rect**h]h/ **cmfd=rect*}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ }(h\ h jubhA)}(h*), by default*h]h/
), by default}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ }(h\ h jubj)}(h**cmfd2g=yes**h]h/
cmfd2g=yes}(hhh jubah}(h]h]h]h]h]uhjh jubh/ }(h\ h jubhA)}(hh*].
The unstructured CMFD method should be applied for triangular- or
hexagonal-domain configurations (*h]h/f].
The unstructured CMFD method should be applied for triangular- or
hexagonal-domain configurations (}(hhh j,ubah}(h]h]h]h]h]uhh@h jubh/ }(h\ h jubj)}(h**cmfd=yes**h]h/cmfd=yes}(hhh j?ubah}(h]h]h]h]h]uhjh jubh/ }(h\ h jubhA)}(hX*). If NEWT detects an
unstable CMFD condition, a warning message is printed and NEWT continues
with CMFD disabled. NEWT may also provide a terminating error message if
improper selection of the coarse mesh is detected. Internal
investigation has shown that the coarse mesh should be approximately the
same size as the unit cell used in the model. For LWR assembly models, a
fine mesh of 4 x 4 is recommended for the square-pitched unit cell,
implying that*h]h/X). If NEWT detects an
unstable CMFD condition, a warning message is printed and NEWT continues
with CMFD disabled. NEWT may also provide a terminating error message if
improper selection of the coarse mesh is detected. Internal
investigation has shown that the coarse mesh should be approximately the
same size as the unit cell used in the model. For LWR assembly models, a
fine mesh of 4 x 4 is recommended for the square-pitched unit cell,
implying that}(hhh jRubah}(h]h]h]h]h]uhh@h jubh/ }(h\ h jubj)}(h
**xycmfd**h]h/xycmfd}(hhh jeubah}(h]h]h]h]h]uhjh jubh/ }(h\ h jubhA)}(h*should be 4 only if the global unit has a
mesh. If individual meshes are used in each unit definition, then the
global unit coarse-mesh cells should be sized based on the unit cell
size and, therefore, xycmfd=1 should be used. The values
of*h]h/should be 4 only if the global unit has a
mesh. If individual meshes are used in each unit definition, then the
global unit coarse-mesh cells should be sized based on the unit cell
size and, therefore, xycmfd=1 should be used. The values
of}(hhh jxubah}(h]h]h]h]h]uhh@h jubh/ }(h\ h jubj)}(h **xcmfd**h]h/xcmfd}(hhh jubah}(h]h]h]h]h]uhjh jubh/ }(h\ h jubhA)}(h*and*h]h/and}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ }(h\ h jubj)}(h **ycmfd**h]h/ycmfd}(hhh jubah}(h]h]h]h]h]uhjh jubh/ }(hjh jubj)}(hjh]h/*}(hhh jubah}(h]id138ah]h]h]h]refidid137uhjh jubh/do not have to be a common factor of
the number of fine-mesh cells in a given direction (NEWT will make the
last coarse-mesh cell smaller than the other coarse-mesh cells), but it
is highly recommended.}(hdo not have to be a common factor of
the number of fine-mesh cells in a given direction (NEWT will make the
last coarse-mesh cell smaller than the other coarse-mesh cells), but it
is highly recommended.h jubeh}(h]h]h]h]h]uhh:h!jhMh jubh;)}(hXUsers can gauge solution convergence by the outer iteration edit as it
is printed to the terminal window (*\ **echo=yes**\ *, see below). One
can terminate a calculation prematurely (via the Control-C option on
most platforms) if convergence or iteration parameters need to be
modified.h](h/jUsers can gauge solution convergence by the outer iteration edit as it
is printed to the terminal window (}(hjUsers can gauge solution convergence by the outer iteration edit as it
is printed to the terminal window (h jubhA)}(h*\ **echo=yes**h]h/
**echo=yes*}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ }(h\ h jubj)}(hjh]h/*}(hhh jubah}(h]id140ah]h]h]h]refidid139uhjh jubh/, see below). One
can terminate a calculation prematurely (via the Control-C option on
most platforms) if convergence or iteration parameters need to be
modified.}(h, see below). One
can terminate a calculation prematurely (via the Control-C option on
most platforms) if convergence or iteration parameters need to be
modified.h jubeh}(h]h]h]h]h]uhh:h!jhMh jubh;)}(hXThe TRITON control module supports a sensitivity and uncertainty
analysis sequence TSUNAMI-2D (See TRITON chapter, section S/U Analysis
Sequences (TSUNAMI-2D, TSUNAMI-2DC)). TSUNAMI-2D calculations require
NEWT to be run in both forward mode and adjoint mode. In adjoint mode,
CMFD acceleration is not currently supported and NEWT automatically
disables its use if*\ **cmfd=yes**\ *,*\ **=rect**\ *,
or*\ **=part**\ *. In adjoint mode with defined fixed source [i.e.,
generalized perturbation theory (GPT) analysis], it is observed that
tighter convergence and iteration parameters are needed to properly
remove fundamental mode contamination. (For more details, see SAMS
chapter: Generalized Perturbation Theory.) To facilitate the CMFD
options and larger convergence criteria for the forward calculations as
well as smaller convergence criteria for GPT adjoint calculations, the
following parameters are also available.h](h/XoThe TRITON control module supports a sensitivity and uncertainty
analysis sequence TSUNAMI-2D (See TRITON chapter, section S/U Analysis
Sequences (TSUNAMI-2D, TSUNAMI-2DC)). TSUNAMI-2D calculations require
NEWT to be run in both forward mode and adjoint mode. In adjoint mode,
CMFD acceleration is not currently supported and NEWT automatically
disables its use if* }(hXoThe TRITON control module supports a sensitivity and uncertainty
analysis sequence TSUNAMI-2D (See TRITON chapter, section S/U Analysis
Sequences (TSUNAMI-2D, TSUNAMI-2DC)). TSUNAMI-2D calculations require
NEWT to be run in both forward mode and adjoint mode. In adjoint mode,
CMFD acceleration is not currently supported and NEWT automatically
disables its use if*\ h jubj)}(h**cmfd=yes**h]h/cmfd=yes}(hhh jubah}(h]h]h]h]h]uhjh jubh/ }(h\ h jubhA)}(h*,*h]h/,}(hhh j1ubah}(h]h]h]h]h]uhh@h jubh/ }(h\ h jubj)}(h **=rect**h]h/=rect}(hhh jDubah}(h]h]h]h]h]uhjh jubh/ }(h\ h jubhA)}(h*,
or*h]h/,
or}(hhh jWubah}(h]h]h]h]h]uhh@h jubh/ }(h\ h jubj)}(h **=part**h]h/=part}(hhh jjubah}(h]h]h]h]h]uhjh jubh/ }(hj0h jubj)}(hjh]h/*}(hhh j|ubah}(h]id142ah]h]h]h]refidid141uhjh jubh/X. In adjoint mode with defined fixed source [i.e.,
generalized perturbation theory (GPT) analysis], it is observed that
tighter convergence and iteration parameters are needed to properly
remove fundamental mode contamination. (For more details, see SAMS
chapter: Generalized Perturbation Theory.) To facilitate the CMFD
options and larger convergence criteria for the forward calculations as
well as smaller convergence criteria for GPT adjoint calculations, the
following parameters are also available.}(hX. In adjoint mode with defined fixed source [i.e.,
generalized perturbation theory (GPT) analysis], it is observed that
tighter convergence and iteration parameters are needed to properly
remove fundamental mode contamination. (For more details, see SAMS
chapter: Generalized Perturbation Theory.) To facilitate the CMFD
options and larger convergence criteria for the forward calculations as
well as smaller convergence criteria for GPT adjoint calculations, the
following parameters are also available.h jubeh}(h]h]h]h]h]uhh:h!jhMh jubeh}(h]h]h]h]h]uhjh jXhhh!jhNubh;)}(hi**gptepsinner=**\ (*RN*) — Spatial convergence criterion for inner
iterations in GPT analysis. [0.0001]h](j)}(h**gptepsinner=**h]h/gptepsinner=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*RN*h]h/RN}(hhh jubah}(h]h]h]h]h]uhh@h jubh/R) — Spatial convergence criterion for inner
iterations in GPT analysis. [0.0001]}(hR) — Spatial convergence criterion for inner
iterations in GPT analysis. [0.0001]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jXhhubh;)}(hh**gptepsouter=**\ (*RN*) — Spatial convergence criterion for outer
iterations in GPT analysis. [0.001]h](j)}(h**gptepsouter=**h]h/gptepsouter=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*RN*h]h/RN}(hhh jubah}(h]h]h]h]h]uhh@h jubh/Q) — Spatial convergence criterion for outer
iterations in GPT analysis. [0.001]}(hQ) — Spatial convergence criterion for outer
iterations in GPT analysis. [0.001]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jXhhubh;)}(h**gptepsthrm=**\ (*RN*) — Spatial convergence criterion for
thermal-upscattering iterations, if enabled, in GPT analysis. [same
value as **gptepsouter**]h](j)}(h**gptepsthrm=**h]h/gptepsthrm=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*RN*h]h/RN}(hhh jubah}(h]h]h]h]h]uhh@h jubh/u) — Spatial convergence criterion for
thermal-upscattering iterations, if enabled, in GPT analysis. [same
value as }(hu) — Spatial convergence criterion for
thermal-upscattering iterations, if enabled, in GPT analysis. [same
value as h jhhh!NhNubj)}(h**gptepsouter**h]h/gptepsouter}(hhh j'ubah}(h]h]h]h]h]uhjh jubh/]}(hj~vh jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jXhhubh;)}(h**gptsepsilon=**\ (*RN*) — Simultaneously sets all spatial convergence
criteria to the same value in GPT analysis. [uses individual defaults]h](j)}(h**gptsepsilon=**h]h/gptsepsilon=}(hhh jCubah}(h]h]h]h]h]uhjh j?ubh/ (}(h\ (h j?hhh!NhNubhA)}(h*RN*h]h/RN}(hhh jVubah}(h]h]h]h]h]uhh@h j?ubh/x) — Simultaneously sets all spatial convergence
criteria to the same value in GPT analysis. [uses individual defaults]}(hx) — Simultaneously sets all spatial convergence
criteria to the same value in GPT analysis. [uses individual defaults]h j?hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jXhhubh;)}(hg**gpttherm**\ =(\ *yes/no*) — Enables/disables thermal-upscattering
iterations in GPT analysis. [yes]h](j)}(h**gpttherm**h]h/gpttherm}(hhh jsubah}(h]h]h]h]h]uhjh joubh/ =( }(h\ =(\ h johhh!NhNubhA)}(h*yes/no*h]h/yes/no}(hhh jubah}(h]h]h]h]h]uhh@h joubh/M) — Enables/disables thermal-upscattering
iterations in GPT analysis. [yes]}(hM) — Enables/disables thermal-upscattering
iterations in GPT analysis. [yes]h johhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jXhhubh;)}(hg**gptinners=**\ (*IN*) — Maximum number of inner iterations in an energy
group in GPT analysis. [500]h](j)}(h**gptinners=**h]h/
gptinners=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*IN*h]h/IN}(hhh jubah}(h]h]h]h]h]uhh@h jubh/R) — Maximum number of inner iterations in an energy
group in GPT analysis. [500]}(hR) — Maximum number of inner iterations in an energy
group in GPT analysis. [500]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jXhhubh;)}(ho**gpttherms=**\ (*IN*) — Maximum number of thermal-upscattering
iterations, if enabled, in GPT analysis. [10]h](j)}(h**gpttherms=**h]h/
gpttherms=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*IN*h]h/IN}(hhh jubah}(h]h]h]h]h]uhh@h jubh/Z) — Maximum number of thermal-upscattering
iterations, if enabled, in GPT analysis. [10]}(hZ) — Maximum number of thermal-upscattering
iterations, if enabled, in GPT analysis. [10]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jXhhubh;)}(h**gptouters=**\ (*IN*) — Maximum number of outer iterations in GPT
analysis. NEWT will stop with an error code if more than *outers* outer
iterations are required for convergence. [2000]h](j)}(h**gptouters=**h]h/
gptouters=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*IN*h]h/IN}(hhh jubah}(h]h]h]h]h]uhh@h jubh/i) — Maximum number of outer iterations in GPT
analysis. NEWT will stop with an error code if more than }(hi) — Maximum number of outer iterations in GPT
analysis. NEWT will stop with an error code if more than h jhhh!NhNubhA)}(h*outers*h]h/outers}(hhh j)ubah}(h]h]h]h]h]uhh@h jubh/6 outer
iterations are required for convergence. [2000]}(h6 outer
iterations are required for convergence. [2000]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jXhhubj)}(hXDefault values for GPT convergence may change with future releases, as
more experience is gained and user feedback is received. If the GPT
calculation is not converging because of fundamental mode contamination,
it is recommended that convergence criteria be decreased and/or inner
and thermal-upscattering iteration limits be increased. If the solution
convergence is slow,*\ **gptinners**\ *can potentially be decreased.
Again, it is highly recommended that*\ **echo=yes**\ *be used to monitor
speed of convergence.h]h;)}(hXDefault values for GPT convergence may change with future releases, as
more experience is gained and user feedback is received. If the GPT
calculation is not converging because of fundamental mode contamination,
it is recommended that convergence criteria be decreased and/or inner
and thermal-upscattering iteration limits be increased. If the solution
convergence is slow,*\ **gptinners**\ *can potentially be decreased.
Again, it is highly recommended that*\ **echo=yes**\ *be used to monitor
speed of convergence.h](h/XyDefault values for GPT convergence may change with future releases, as
more experience is gained and user feedback is received. If the GPT
calculation is not converging because of fundamental mode contamination,
it is recommended that convergence criteria be decreased and/or inner
and thermal-upscattering iteration limits be increased. If the solution
convergence is slow,* }(hXyDefault values for GPT convergence may change with future releases, as
more experience is gained and user feedback is received. If the GPT
calculation is not converging because of fundamental mode contamination,
it is recommended that convergence criteria be decreased and/or inner
and thermal-upscattering iteration limits be increased. If the solution
convergence is slow,*\ h jFubj)}(h
**gptinners**h]h/ gptinners}(hhh jOubah}(h]h]h]h]h]uhjh jFubh/ }(h\ h jFubhA)}(hD*can potentially be decreased.
Again, it is highly recommended that*h]h/Bcan potentially be decreased.
Again, it is highly recommended that}(hhh jbubah}(h]h]h]h]h]uhh@h jFubh/ }(h\ h jFubj)}(h**echo=yes**h]h/echo=yes}(hhh juubah}(h]h]h]h]h]uhjh jFubh/ }(hjah jFubj)}(hjh]h/*}(hhh jubah}(h]id144ah]h]h]h]refidid143uhjh jFubh/(be used to monitor
speed of convergence.}(h(be used to monitor
speed of convergence.h jFubeh}(h]h]h]h]h]uhh:h!jhMh jBubah}(h]h]h]h]h]uhjh jXhhh!jhNubh)}(h.. _9-2-3-2-2:h]h}(h]h]h]h]h]hid145uhh
hMh jXhhh!jubeh}(h]('convergence-and-acceleration-parametersjWeh]h]('convergence and acceleration parameters 9-2-3-2-1eh]h]uhh#h jhhh!jhMMjf}jjMsjh}jWjMsubh$)}(hhh](h))}(hOutput editingh]h/Output editing}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMubh;)}(hX6**drawit=**\ (*yes/no*) — Create a PostScript file showing the grid
structure determined from input. Two files are created—the first showing
the grid structure and the second showing the material placement.
(Features and use of this simple graphics capability are described
further in :ref:`9-2-5-14`) [no]h](j)}(h**drawit=**h]h/drawit=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*yes/no*h]h/yes/no}(hhh jubah}(h]h]h]h]h]uhh@h jubh/X) — Create a PostScript file showing the grid
structure determined from input. Two files are created—the first showing
the grid structure and the second showing the material placement.
(Features and use of this simple graphics capability are described
further in }(hX) — Create a PostScript file showing the grid
structure determined from input. Two files are created—the first showing
the grid structure and the second showing the material placement.
(Features and use of this simple graphics capability are described
further in h jhhh!NhNubj)}(h:ref:`9-2-5-14`h]j#)}(hjh]h/9-2-5-14}(hhh jubah}(h]h](jnstdstd-refeh]h]h]uhj"h jubah}(h]h]h]h]h]refdocj refdomainjreftyperefrefexplicitrefwarnj9-2-5-14uhjh!jhMh jubh/) [no]}(h) [no]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(h**echo=**\ (*yes/no*) — During the iteration phase of execution, output
is generated at the beginning of each outer iteration. This same
information can be printed to SCALE message file (.msg) during iteration
by setting echo=yes. [no]h](j)}(h **echo=**h]h/echo=}(hhh j(ubah}(h]h]h]h]h]uhjh j$ubh/ (}(h\ (h j$hhh!NhNubhA)}(h*yes/no*h]h/yes/no}(hhh j;ubah}(h]h]h]h]h]uhh@h j$ubh/) — During the iteration phase of execution, output
is generated at the beginning of each outer iteration. This same
information can be printed to SCALE message file (.msg) during iteration
by setting echo=yes. [no]}(h) — During the iteration phase of execution, output
is generated at the beginning of each outer iteration. This same
information can be printed to SCALE message file (.msg) during iteration
by setting echo=yes. [no]h j$hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(h}**prtbalnc**\ =\ *(yes/no)* — Flag indicating whether or not balance
tables for fine-group mixtures should be printed. [no]h](j)}(h**prtbalnc**h]h/prtbalnc}(hhh jXubah}(h]h]h]h]h]uhjh jTubh/ = }(h\ =\ h jThhh!NhNubhA)}(h
*(yes/no)*h]h/(yes/no)}(hhh jkubah}(h]h]h]h]h]uhh@h jTubh/b — Flag indicating whether or not balance
tables for fine-group mixtures should be printed. [no]}(hb — Flag indicating whether or not balance
tables for fine-group mixtures should be printed. [no]h jThhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(hX**prtbroad=**\ (*yes/no/1d*) — Flag indicating whether or not broad
group cross sections should be printed in problem output. The *1d*
option indicates that 2-D scattering tables are not to be printed. This
flag has no effect if collapse=no is specified. [no]h](j)}(h
**prtbroad=**h]h/ prtbroad=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*yes/no/1d*h]h/ yes/no/1d}(hhh jubah}(h]h]h]h]h]uhh@h jubh/j) — Flag indicating whether or not broad
group cross sections should be printed in problem output. The }(hj) — Flag indicating whether or not broad
group cross sections should be printed in problem output. The h jhhh!NhNubhA)}(h*1d*h]h/1d}(hhh jubah}(h]h]h]h]h]uhh@h jubh/}
option indicates that 2-D scattering tables are not to be printed. This
flag has no effect if collapse=no is specified. [no]}(h}
option indicates that 2-D scattering tables are not to be printed. This
flag has no effect if collapse=no is specified. [no]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(h**prthmmix=**\ (*yes/no*) — Flag indicating whether or not homogenized
mixture macroscopic cross sections should be printed in problem output.
Homogenized cross sections are printed only if Homogenization Block is
provided (:ref:`9-2-3-10`). [yes]h](j)}(h
**prthmmix=**h]h/ prthmmix=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*yes/no*h]h/yes/no}(hhh jubah}(h]h]h]h]h]uhh@h jubh/) — Flag indicating whether or not homogenized
mixture macroscopic cross sections should be printed in problem output.
Homogenized cross sections are printed only if Homogenization Block is
provided (}(h) — Flag indicating whether or not homogenized
mixture macroscopic cross sections should be printed in problem output.
Homogenized cross sections are printed only if Homogenization Block is
provided (h jhhh!NhNubj)}(h:ref:`9-2-3-10`h]j#)}(hjh]h/9-2-3-10}(hhh jubah}(h]h](jnstdstd-refeh]h]h]uhj"h jubah}(h]h]h]h]h]refdocj refdomainjreftyperefrefexplicitrefwarnj9-2-3-10uhjh!jhMh jubh/). [yes]}(h). [yes]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(h**prtflux**\ *=(yes/no)* — Create a PostScript plot file showing flux
distribution for each energy group in problem. If an energy collapse is
performed, a second plot file is generated for the fluxes of the
collapsed group structures. [no]h](j)}(h**prtflux**h]h/prtflux}(hhh j ubah}(h]h]h]h]h]uhjh jubh/ }(h\ h jhhh!NhNubhA)}(h*=(yes/no)*h]h/ =(yes/no)}(hhh j3ubah}(h]h]h]h]h]uhh@h jubh/ — Create a PostScript plot file showing flux
distribution for each energy group in problem. If an energy collapse is
performed, a second plot file is generated for the fluxes of the
collapsed group structures. [no]}(h — Create a PostScript plot file showing flux
distribution for each energy group in problem. If an energy collapse is
performed, a second plot file is generated for the fluxes of the
collapsed group structures. [no]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(h**prtmxsec=**\ (*yes/no/1d*) — Flag indicating whether or not mixture
macroscopic cross sections should be printed in problem output. The *1d*
option indicates that 2-D scattering tables are not to be printed. [no]h](j)}(h
**prtmxsec=**h]h/ prtmxsec=}(hhh jPubah}(h]h]h]h]h]uhjh jLubh/ (}(h\ (h jLhhh!NhNubhA)}(h*yes/no/1d*h]h/ yes/no/1d}(hhh jcubah}(h]h]h]h]h]uhh@h jLubh/r) — Flag indicating whether or not mixture
macroscopic cross sections should be printed in problem output. The }(hr) — Flag indicating whether or not mixture
macroscopic cross sections should be printed in problem output. The h jLhhh!NhNubhA)}(h*1d*h]h/1d}(hhh jvubah}(h]h]h]h]h]uhh@h jLubh/H
option indicates that 2-D scattering tables are not to be printed. [no]}(hH
option indicates that 2-D scattering tables are not to be printed. [no]h jLhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(h}**prtmxtab=**\ (*yes/no*) — Flag indicating whether or not the input
mixing table should be printed in problem output. [no]h](j)}(h
**prtmxtab=**h]h/ prtmxtab=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*yes/no*h]h/yes/no}(hhh jubah}(h]h]h]h]h]uhh@h jubh/e) — Flag indicating whether or not the input
mixing table should be printed in problem output. [no]}(he) — Flag indicating whether or not the input
mixing table should be printed in problem output. [no]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(h**prtxsec=**\ (*yes/no/1d*) — Flag indicating whether or not input
microscopic cross sections should be printed in problem output. The *1d*
option indicates that 2-D scattering tables are not to be printed. [no]h](j)}(h**prtxsec=**h]h/prtxsec=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*yes/no/1d*h]h/ yes/no/1d}(hhh jubah}(h]h]h]h]h]uhh@h jubh/p) — Flag indicating whether or not input
microscopic cross sections should be printed in problem output. The }(hp) — Flag indicating whether or not input
microscopic cross sections should be printed in problem output. The h jhhh!NhNubhA)}(h*1d*h]h/1d}(hhh jubah}(h]h]h]h]h]uhh@h jubh/H
option indicates that 2-D scattering tables are not to be printed. [no]}(hH
option indicates that 2-D scattering tables are not to be printed. [no]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(hU**timed=**\ (yes/no) — Turns on printing of iteration timing and CPU use
data. [no]h](j)}(h
**timed=**h]h/timed=}(hhh jubah}(h]h]h]h]h]uhjh jubh/K (yes/no) — Turns on printing of iteration timing and CPU use
data. [no]}(hK\ (yes/no) — Turns on printing of iteration timing and CPU use
data. [no]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(hX'**det=**\ (*IN*) — Specifies the mixture used to represent a local power
range monitor (LPRM) and/or Traversing In-core Probe (TIP) detector
located within a fuel lattice. The mixture must also be included in a
homogenization block in order to obtain detector cross sections. [has no
default]h](j)}(h**det=**h]h/det=}(hhh j#ubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*IN*h]h/IN}(hhh j6ubah}(h]h]h]h]h]uhh@h jubh/X) — Specifies the mixture used to represent a local power
range monitor (LPRM) and/or Traversing In-core Probe (TIP) detector
located within a fuel lattice. The mixture must also be included in a
homogenization block in order to obtain detector cross sections. [has no
default]}(hX) — Specifies the mixture used to represent a local power
range monitor (LPRM) and/or Traversing In-core Probe (TIP) detector
located within a fuel lattice. The mixture must also be included in a
homogenization block in order to obtain detector cross sections. [has no
default]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubj)}(hXWith the exception of*\ **prthmmix**\ *, all output edit options are
disabled unless requested by the user. The output edits are disabled by
default to minimize the size of the output. The*\ **drawit**\ *option is
recommended to generate PostScript plots of the model grid structure and
material placement. As previously mentioned,
the*\ **echo**\ *and*\ **timed**\ *options are recommended to monitor
solution convergence. If the timed option is enabled, each line in the
outer iteration edit will be longer than 80 characters. Therefore, it is
recommended that Windows users should increase the Command Window size
from 80 characters to 132* characters.h]h;)}(hXWith the exception of*\ **prthmmix**\ *, all output edit options are
disabled unless requested by the user. The output edits are disabled by
default to minimize the size of the output. The*\ **drawit**\ *option is
recommended to generate PostScript plots of the model grid structure and
material placement. As previously mentioned,
the*\ **echo**\ *and*\ **timed**\ *options are recommended to monitor
solution convergence. If the timed option is enabled, each line in the
outer iteration edit will be longer than 80 characters. Therefore, it is
recommended that Windows users should increase the Command Window size
from 80 characters to 132* characters.h](h/With the exception of* }(hWith the exception of*\ h jSubj)}(h**prthmmix**h]h/prthmmix}(hhh j\ubah}(h]h]h]h]h]uhjh jSubh/ }(h\ h jSubhA)}(h*, all output edit options are
disabled unless requested by the user. The output edits are disabled by
default to minimize the size of the output. The*h]h/, all output edit options are
disabled unless requested by the user. The output edits are disabled by
default to minimize the size of the output. The}(hhh joubah}(h]h]h]h]h]uhh@h jSubh/ }(h\ h jSubj)}(h
**drawit**h]h/drawit}(hhh jubah}(h]h]h]h]h]uhjh jSubh/ }(h\ h jSubhA)}(h*option is
recommended to generate PostScript plots of the model grid structure and
material placement. As previously mentioned,
the*h]h/option is
recommended to generate PostScript plots of the model grid structure and
material placement. As previously mentioned,
the}(hhh jubah}(h]h]h]h]h]uhh@h jSubh/ }(h\ h jSubj)}(h**echo**h]h/echo}(hhh jubah}(h]h]h]h]h]uhjh jSubh/ }(h\ h jSubhA)}(h*and*h]h/and}(hhh jubah}(h]h]h]h]h]uhh@h jSubh/ }(h\ h jSubj)}(h **timed**h]h/timed}(hhh jubah}(h]h]h]h]h]uhjh jSubh/ }(hjnh jSubhA)}(hX*options are recommended to monitor
solution convergence. If the timed option is enabled, each line in the
outer iteration edit will be longer than 80 characters. Therefore, it is
recommended that Windows users should increase the Command Window size
from 80 characters to 132*h]h/Xoptions are recommended to monitor
solution convergence. If the timed option is enabled, each line in the
outer iteration edit will be longer than 80 characters. Therefore, it is
recommended that Windows users should increase the Command Window size
from 80 characters to 132}(hhh jubah}(h]h]h]h]h]uhh@h jSubh/
characters.}(h
characters.h jSubeh}(h]h]h]h]h]uhh:h!jhM h jOubah}(h]h]h]h]h]uhjh jhhh!jhNubh)}(h.. _9-2-3-2-3:h]h}(h]h]h]h]h]hid146uhh
hM>h jhhh!jubeh}(h](output-editingjeh]h](output editing 9-2-3-2-2eh]h]uhh#h jhhh!jhMjf}jjsjh}jjsubh$)}(hhh](h))}(hAngular quadratureh]h/Angular quadrature}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhM.ubh;)}(hT**sn=**\ (*2/4/6/8/10/12/14/16*) — Order of Sn level symmetric
quadrature set. [6]h](j)}(h**sn=**h]h/sn=}(hhh j*ubah}(h]h]h]h]h]uhjh j&ubh/ (}(h\ (h j&hhh!NhNubhA)}(h*2/4/6/8/10/12/14/16*h]h/2/4/6/8/10/12/14/16}(hhh j=ubah}(h]h]h]h]h]uhh@h j&ubh/5) — Order of Sn level symmetric
quadrature set. [6]}(h5) — Order of Sn level symmetric
quadrature set. [6]h j&hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM0h jhhubh;)}(hXK**nazim**\ =(\ *IN*) — Number of equally spaced azimuthal directions in
a product quadrature set. Used in tandem with *npolar* keyword (both
must be specified). Total number of angles in the product quadrature set
is the product of *nazim* and *npolar*. [No default. If not specified,
level symmetric quadrature default is used.]h](j)}(h **nazim**h]h/nazim}(hhh jZubah}(h]h]h]h]h]uhjh jVubh/ =( }(h\ =(\ h jVhhh!NhNubhA)}(h*IN*h]h/IN}(hhh jmubah}(h]h]h]h]h]uhh@h jVubh/e) — Number of equally spaced azimuthal directions in
a product quadrature set. Used in tandem with }(he) — Number of equally spaced azimuthal directions in
a product quadrature set. Used in tandem with h jVhhh!NhNubhA)}(h*npolar*h]h/npolar}(hhh jubah}(h]h]h]h]h]uhh@h jVubh/j keyword (both
must be specified). Total number of angles in the product quadrature set
is the product of }(hj keyword (both
must be specified). Total number of angles in the product quadrature set
is the product of h jVhhh!NhNubhA)}(h*nazim*h]h/nazim}(hhh jubah}(h]h]h]h]h]uhh@h jVubh/ and }(h and h jVhhh!NhNubhA)}(h*npolar*h]h/npolar}(hhh jubah}(h]h]h]h]h]uhh@h jVubh/M. [No default. If not specified,
level symmetric quadrature default is used.]}(hM. [No default. If not specified,
level symmetric quadrature default is used.]h jVhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM3h jhhubh;)}(hXc**npolar**\ =(\ *IN*) — Number of polar angles in a product quadrature
set (determined using a Gauss-Legendre polynomial). Used in tandem with
*nazim* keyword (both must be specified). Total number of angles in the
product quadrature set is the product of *nazim* and *npolar*. [No
default. If not specified, level symmetric quadrature default is used.]h](j)}(h
**npolar**h]h/npolar}(hhh jubah}(h]h]h]h]h]uhjh jubh/ =( }(h\ =(\ h jhhh!NhNubhA)}(h*IN*h]h/IN}(hhh jubah}(h]h]h]h]h]uhh@h jubh/}) — Number of polar angles in a product quadrature
set (determined using a Gauss-Legendre polynomial). Used in tandem with
}(h}) — Number of polar angles in a product quadrature
set (determined using a Gauss-Legendre polynomial). Used in tandem with
h jhhh!NhNubhA)}(h*nazim*h]h/nazim}(hhh jubah}(h]h]h]h]h]uhh@h jubh/j keyword (both must be specified). Total number of angles in the
product quadrature set is the product of }(hj keyword (both must be specified). Total number of angles in the
product quadrature set is the product of h jhhh!NhNubhA)}(h*nazim*h]h/nazim}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ and }(h and h jhhh!NhNubhA)}(h*npolar*h]h/npolar}(hhh jubah}(h]h]h]h]h]uhh@h jubh/M. [No
default. If not specified, level symmetric quadrature default is used.]}(hM. [No
default. If not specified, level symmetric quadrature default is used.]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM9h jhhubh;)}(h**dgauss=**\ (yes/no) — Enables/disables use of double Gauss-Legendre
product quadrature set. If disabled, single Gauss-Legendre product
quadrature sets are used. [no]h](j)}(h**dgauss=**h]h/dgauss=}(hhh j,ubah}(h]h]h]h]h]uhjh j(ubh/ (yes/no) — Enables/disables use of double Gauss-Legendre
product quadrature set. If disabled, single Gauss-Legendre product
quadrature sets are used. [no]}(h\ (yes/no) — Enables/disables use of double Gauss-Legendre
product quadrature set. If disabled, single Gauss-Legendre product
quadrature sets are used. [no]h j(hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM?h jhhubj)}(hXIf both level symmetric quadrature sets and product quadrature sets are
requested, the level symmetric quadrature set is to be used. Level
symmetric quadrature sets are recommended for general analysis. If
reflective boundary conditions are desired for hexagonal-domain
configurations, product quadrature sets must be used and nazim must be a
multiple of 3. If reflective boundary conditions are desired for
triangular-domain configurations, product quadrature sets must be used
and*\ **nazim**\ *must be an odd number.h]h;)}(hXIf both level symmetric quadrature sets and product quadrature sets are
requested, the level symmetric quadrature set is to be used. Level
symmetric quadrature sets are recommended for general analysis. If
reflective boundary conditions are desired for hexagonal-domain
configurations, product quadrature sets must be used and nazim must be a
multiple of 3. If reflective boundary conditions are desired for
triangular-domain configurations, product quadrature sets must be used
and*\ **nazim**\ *must be an odd number.h](h/XIf both level symmetric quadrature sets and product quadrature sets are
requested, the level symmetric quadrature set is to be used. Level
symmetric quadrature sets are recommended for general analysis. If
reflective boundary conditions are desired for hexagonal-domain
configurations, product quadrature sets must be used and nazim must be a
multiple of 3. If reflective boundary conditions are desired for
triangular-domain configurations, product quadrature sets must be used
and* }(hXIf both level symmetric quadrature sets and product quadrature sets are
requested, the level symmetric quadrature set is to be used. Level
symmetric quadrature sets are recommended for general analysis. If
reflective boundary conditions are desired for hexagonal-domain
configurations, product quadrature sets must be used and nazim must be a
multiple of 3. If reflective boundary conditions are desired for
triangular-domain configurations, product quadrature sets must be used
and*\ h jIubj)}(h **nazim**h]h/nazim}(hhh jRubah}(h]h]h]h]h]uhjh jIubh/ }(h\ h jIubj)}(hjh]h/*}(hhh jeubah}(h]id148ah]h]h]h]refidid147uhjh jIubh/must be an odd number.}(hmust be an odd number.h jIubeh}(h]h]h]h]h]uhh:h!jhMCh jEubah}(h]h]h]h]h]uhjh jhhh!jhNubh)}(h.. _9-2-3-2-4:h]h}(h]h]h]h]h]hid149uhh
hM_h jhhh!jubeh}(h](angular-quadraturej eh]h](angular quadrature 9-2-3-2-3eh]h]uhh#h jhhh!jhM.jf}jjsjh}j jsubh$)}(hhh](h))}(hControl optionsh]h/Control options}(hjh jhhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMOubh;)}(h|**adjoint=**\ (*yes/no*) — This keyword specifies either a forward
(adjoint=no) or adjoint (adjoint=yes) calculation. [no]h](j)}(h**adjoint=**h]h/adjoint=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*yes/no*h]h/yes/no}(hhh jubah}(h]h]h]h]h]uhh@h jubh/e) — This keyword specifies either a forward
(adjoint=no) or adjoint (adjoint=yes) calculation. [no]}(he) — This keyword specifies either a forward
(adjoint=no) or adjoint (adjoint=yes) calculation. [no]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMQh jhhubh;)}(h**forward=**\ (*yes/no*) — This keyword specifies either a forward
(forward=yes) or adjoint (forward=no) calculation. If adjoint and
forward are both specified, NEWT uses the last specification. [yes]h](j)}(h**forward=**h]h/forward=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*yes/no*h]h/yes/no}(hhh jubah}(h]h]h]h]h]uhh@h jubh/) — This keyword specifies either a forward
(forward=yes) or adjoint (forward=no) calculation. If adjoint and
forward are both specified, NEWT uses the last specification. [yes]}(h) — This keyword specifies either a forward
(forward=yes) or adjoint (forward=no) calculation. If adjoint and
forward are both specified, NEWT uses the last specification. [yes]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMTh jhhubh;)}(h**gpt=**\ (*yes/no*) — This keyword specifies whether this is a GPT
adjoint calculation. The *gpt* keyword is active only for adjoint
calculations. [no]h](j)}(h**gpt=**h]h/gpt=}(hhh jubah}(h]h]h]h]h]uhjh j
ubh/ (}(h\ (h j
hhh!NhNubhA)}(h*yes/no*h]h/yes/no}(hhh j$ubah}(h]h]h]h]h]uhh@h j
ubh/L) — This keyword specifies whether this is a GPT
adjoint calculation. The }(hL) — This keyword specifies whether this is a GPT
adjoint calculation. The h j
hhh!NhNubhA)}(h*gpt*h]h/gpt}(hhh j7ubah}(h]h]h]h]h]uhh@h j
ubh/6 keyword is active only for adjoint
calculations. [no]}(h6 keyword is active only for adjoint
calculations. [no]h j
hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMXh jhhubj)}(hThe TRITON control module automatically sets the values for forward,
adjoint, and gpt keywords; therefore, they can typically be omitted from
the Parameter Block. Default values are recommended unless running
stand-alone NEWT adjoint calculations.h]h;)}(hThe TRITON control module automatically sets the values for forward,
adjoint, and gpt keywords; therefore, they can typically be omitted from
the Parameter Block. Default values are recommended unless running
stand-alone NEWT adjoint calculations.h]h/The TRITON control module automatically sets the values for forward,
adjoint, and gpt keywords; therefore, they can typically be omitted from
the Parameter Block. Default values are recommended unless running
stand-alone NEWT adjoint calculations.}(hjVh jTubah}(h]h]h]h]h]uhh:h!jhM\h jPubah}(h]h]h]h]h]uhjh jhhh!jhNubh;)}(hX**run=**\ (*yes/no*) — A run=no calculation will perform all setup
calculations normally performed before beginning iterations and then
will stop. It is useful for debugging input and obtaining plots of the
input geometry. Run=yes will perform a complete calculation. [yes]h](j)}(h**run=**h]h/run=}(hhh jlubah}(h]h]h]h]h]uhjh jhubh/ (}(h\ (h jhhhh!NhNubhA)}(h*yes/no*h]h/yes/no}(hhh jubah}(h]h]h]h]h]uhh@h jhubh/X) — A run=no calculation will perform all setup
calculations normally performed before beginning iterations and then
will stop. It is useful for debugging input and obtaining plots of the
input geometry. Run=yes will perform a complete calculation. [yes]}(hX) — A run=no calculation will perform all setup
calculations normally performed before beginning iterations and then
will stop. It is useful for debugging input and obtaining plots of the
input geometry. Run=yes will perform a complete calculation. [yes]h jhhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMah jhhubh;)}(hX**premix=**\ (*yes/no*) — This flag indicates whether the cross section
library contains microscopic (premix=no) or macroscopic (premix=yes)
cross sections. In essence, it creates a mixing table with a mixture
fraction of 1.0 for each mixture on the library. Other mixing tables are
ignored. The premixed cross section option is active only for
stand-alone NEWT calculations. [no]h](j)}(h**premix=**h]h/premix=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*yes/no*h]h/yes/no}(hhh jubah}(h]h]h]h]h]uhh@h jubh/Xi) — This flag indicates whether the cross section
library contains microscopic (premix=no) or macroscopic (premix=yes)
cross sections. In essence, it creates a mixing table with a mixture
fraction of 1.0 for each mixture on the library. Other mixing tables are
ignored. The premixed cross section option is active only for
stand-alone NEWT calculations. [no]}(hXi) — This flag indicates whether the cross section
library contains microscopic (premix=no) or macroscopic (premix=yes)
cross sections. In essence, it creates a mixing table with a mixture
fraction of 1.0 for each mixture on the library. Other mixing tables are
ignored. The premixed cross section option is active only for
stand-alone NEWT calculations. [no]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMfh jhhubh;)}(h**kguess=**\ (*RN*) — Initial guess at eigenvalue for an eigenvalue
calculation. This parameter may be entered but is not used if a source
calculation is performed or a restart file is used to determine the
initial guess. [1.0]h](j)}(h**kguess=**h]h/kguess=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*RN*h]h/RN}(hhh jubah}(h]h]h]h]h]uhh@h jubh/) — Initial guess at eigenvalue for an eigenvalue
calculation. This parameter may be entered but is not used if a source
calculation is performed or a restart file is used to determine the
initial guess. [1.0]}(h) — Initial guess at eigenvalue for an eigenvalue
calculation. This parameter may be entered but is not used if a source
calculation is performed or a restart file is used to determine the
initial guess. [1.0]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMmh jhhubh;)}(hXs**restart=**\ (*yes/no*) — If restart=yes is specified, NEWT will open
file *restart_newt* and read scalar fluxes and fission rates, enabling a
restart from the point at which a previous calculation ended. The file
*restart_newt* is always written by NEWT at the end of every successful
calculation. The code assumes that all geometry is unchanged from the
previous calculation but does allow restart with a different angular
quadrature set and P\ :sub:`n` scattering coefficients. A low-order
solution can be used to accelerate a higher-order solution by restarting
using the converged flux of the lower-order solution. [no]h](j)}(h**restart=**h]h/restart=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*yes/no*h]h/yes/no}(hhh jubah}(h]h]h]h]h]uhh@h jubh/7) — If restart=yes is specified, NEWT will open
file }(h7) — If restart=yes is specified, NEWT will open
file h jhhh!NhNubhA)}(h*restart_newt*h]h/restart_newt}(hhh j"ubah}(h]h]h]h]h]uhh@h jubh/} and read scalar fluxes and fission rates, enabling a
restart from the point at which a previous calculation ended. The file
}(h} and read scalar fluxes and fission rates, enabling a
restart from the point at which a previous calculation ended. The file
h jhhh!NhNubhA)}(h*restart_newt*h]h/restart_newt}(hhh j5ubah}(h]h]h]h]h]uhh@h jubh/ is always written by NEWT at the end of every successful
calculation. The code assumes that all geometry is unchanged from the
previous calculation but does allow restart with a different angular
quadrature set and P }(h is always written by NEWT at the end of every successful
calculation. The code assumes that all geometry is unchanged from the
previous calculation but does allow restart with a different angular
quadrature set and P\ h jhhh!NhNubh)}(h:sub:`n`h]h/n}(hhh jHubah}(h]h]h]h]h]uhhh jubh/ scattering coefficients. A low-order
solution can be used to accelerate a higher-order solution by restarting
using the converged flux of the lower-order solution. [no]}(h scattering coefficients. A low-order
solution can be used to accelerate a higher-order solution by restarting
using the converged flux of the lower-order solution. [no]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMrh jhhubh;)}(h**savrest=**\ (*yes/no*) — Determines whether or not a geometry restart
file *worf* is written at the end of a calculation. If written, it will
overwrite any existing geometry restart file. [yes]h](j)}(h**savrest=**h]h/savrest=}(hhh jeubah}(h]h]h]h]h]uhjh jaubh/ (}(h\ (h jahhh!NhNubhA)}(h*yes/no*h]h/yes/no}(hhh jxubah}(h]h]h]h]h]uhh@h jaubh/8) — Determines whether or not a geometry restart
file }(h8) — Determines whether or not a geometry restart
file h jahhh!NhNubhA)}(h*worf*h]h/worf}(hhh jubah}(h]h]h]h]h]uhh@h jaubh/q is written at the end of a calculation. If written, it will
overwrite any existing geometry restart file. [yes]}(hq is written at the end of a calculation. If written, it will
overwrite any existing geometry restart file. [yes]h jahhh!NhNubeh}(h]h]h]h]h]uhh:h!jhM|h jhhubj)}(hXThe default values of savrest and kguess are recommended. The TRITON
control module automates generation and reuse of the geometry restart
file, as well as the initial guess of the eigenvalue. Keywords run,
premix, and restart can generally be omitted unless the following
conditions are applicable:
- TRITON T-NEWT sequence calculation or stand-alone NEWT calculation
with user-supplied restart file, restart=yes.
- Stand-alone NEWT calculation with user-supplied premixed cross
section file, premix=yes.
- Interested only in performing setup calculations to debug input and
generate geometry plots, run=no, and/or PARM=CHECK in the TRITON
sequence input.h](h;)}(hX+The default values of savrest and kguess are recommended. The TRITON
control module automates generation and reuse of the geometry restart
file, as well as the initial guess of the eigenvalue. Keywords run,
premix, and restart can generally be omitted unless the following
conditions are applicable:h]h/X+The default values of savrest and kguess are recommended. The TRITON
control module automates generation and reuse of the geometry restart
file, as well as the initial guess of the eigenvalue. Keywords run,
premix, and restart can generally be omitted unless the following
conditions are applicable:}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubjd)}(hhh](j)}(hqTRITON T-NEWT sequence calculation or stand-alone NEWT calculation
with user-supplied restart file, restart=yes.
h]h;)}(hpTRITON T-NEWT sequence calculation or stand-alone NEWT calculation
with user-supplied restart file, restart=yes.h]h/pTRITON T-NEWT sequence calculation or stand-alone NEWT calculation
with user-supplied restart file, restart=yes.}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(hYStand-alone NEWT calculation with user-supplied premixed cross
section file, premix=yes.
h]h;)}(hXStand-alone NEWT calculation with user-supplied premixed cross
section file, premix=yes.h]h/XStand-alone NEWT calculation with user-supplied premixed cross
section file, premix=yes.}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(hInterested only in performing setup calculations to debug input and
generate geometry plots, run=no, and/or PARM=CHECK in the TRITON
sequence input.h]h;)}(hInterested only in performing setup calculations to debug input and
generate geometry plots, run=no, and/or PARM=CHECK in the TRITON
sequence input.h]h/Interested only in performing setup calculations to debug input and
generate geometry plots, run=no, and/or PARM=CHECK in the TRITON
sequence input.}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]jBjCuhjch!jhMh jubeh}(h]h]h]h]h]uhjh jhhh!NhNubh;)}(hXc**solntype**\ =(keff/b1/src) — Specifies solution mode type: keff is
eigenvalue, b1 is eigenvalue mode followed by a buckling correction, and
src is fixed source (no eigenvalue calculation). Fixed source
calculations require additional data for the source specification (see
Materials and Source data blocks in :ref:`9-2-3-3` and :ref:`9-2-3-4`). [keff]h](j)}(h**solntype**h]h/solntype}(hhh jubah}(h]h]h]h]h]uhjh j
ubh/X- =(keff/b1/src) — Specifies solution mode type: keff is
eigenvalue, b1 is eigenvalue mode followed by a buckling correction, and
src is fixed source (no eigenvalue calculation). Fixed source
calculations require additional data for the source specification (see
Materials and Source data blocks in }(hX-\ =(keff/b1/src) — Specifies solution mode type: keff is
eigenvalue, b1 is eigenvalue mode followed by a buckling correction, and
src is fixed source (no eigenvalue calculation). Fixed source
calculations require additional data for the source specification (see
Materials and Source data blocks in h j
hhh!NhNubj)}(h:ref:`9-2-3-3`h]j#)}(hj&h]h/9-2-3-3}(hhh j(ubah}(h]h](jnstdstd-refeh]h]h]uhj"h j$ubah}(h]h]h]h]h]refdocj refdomainj2reftyperefrefexplicitrefwarnj9-2-3-3uhjh!jhMh j
ubh/ and }(h and h j
hhh!NhNubj)}(h:ref:`9-2-3-4`h]j#)}(hjKh]h/9-2-3-4}(hhh jMubah}(h]h](jnstdstd-refeh]h]h]uhj"h jIubah}(h]h]h]h]h]refdocj refdomainjWreftyperefrefexplicitrefwarnj9-2-3-4uhjh!jhMh j
ubh/ ). [keff]}(h ). [keff]h j
hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(hX**collapse=**\ (*yes/no*) — If collapse=yes is specified, a
flux-weighted collapse is performed by material number; cross sections
for each nuclide in each material in the problem are collapsed to a
specified (or default) group structure based on the average flux in that
material. If collapse=yes, NEWT will look for the *collapse* parameter
block; if not found, NEWT will generate cross sections based on the
original group structure. If a Homogenization block is present, then
collapse is always set to yes. [no]h](j)}(h
**collapse=**h]h/ collapse=}(hhh jxubah}(h]h]h]h]h]uhjh jtubh/ (}(h\ (h jthhh!NhNubhA)}(h*yes/no*h]h/yes/no}(hhh jubah}(h]h]h]h]h]uhh@h jtubh/X.) — If collapse=yes is specified, a
flux-weighted collapse is performed by material number; cross sections
for each nuclide in each material in the problem are collapsed to a
specified (or default) group structure based on the average flux in that
material. If collapse=yes, NEWT will look for the }(hX.) — If collapse=yes is specified, a
flux-weighted collapse is performed by material number; cross sections
for each nuclide in each material in the problem are collapsed to a
specified (or default) group structure based on the average flux in that
material. If collapse=yes, NEWT will look for the h jthhh!NhNubhA)}(h
*collapse*h]h/collapse}(hhh jubah}(h]h]h]h]h]uhh@h jtubh/ parameter
block; if not found, NEWT will generate cross sections based on the
original group structure. If a Homogenization block is present, then
collapse is always set to yes. [no]}(h parameter
block; if not found, NEWT will generate cross sections based on the
original group structure. If a Homogenization block is present, then
collapse is always set to yes. [no]h jthhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(hX**saveangflx=**\ (*yes/no*) — Option to save angular flux solution. The
angular flux is saved to a binary file used in the TSUNAMI-2D sequence
of the TRITON control module. Because the angular flux can require
significant file storage, it is not saved by default. The angular flux
solution can and should be saved for TSUNAMI-2D calculations to generate
more accurate sensitivity coefficients. [no]h](j)}(h**saveangflx=**h]h/saveangflx=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*yes/no*h]h/yes/no}(hhh jubah}(h]h]h]h]h]uhh@h jubh/Xv) — Option to save angular flux solution. The
angular flux is saved to a binary file used in the TSUNAMI-2D sequence
of the TRITON control module. Because the angular flux can require
significant file storage, it is not saved by default. The angular flux
solution can and should be saved for TSUNAMI-2D calculations to generate
more accurate sensitivity coefficients. [no]}(hXv) — Option to save angular flux solution. The
angular flux is saved to a binary file used in the TSUNAMI-2D sequence
of the TRITON control module. Because the angular flux can require
significant file storage, it is not saved by default. The angular flux
solution can and should be saved for TSUNAMI-2D calculations to generate
more accurate sensitivity coefficients. [no]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubj)}(hXKeyword threads should be omitted in favor of the SCALE command line–I
option. Keywords solntype, collapse, and saveangflx should be omitted
unless the following conditions are applicable.
- For homogenized few-group cross section generation for nodal
calculations, solntype* *should be b1. This option will perform a
critical spectrum calculation, which will be folded into cross
section homogenization calculation. The critical spectrum is also
folded into the generation of ADFs and reaction rates for
depletion calculations.
- Generation of a new collapsed cross section library, collapse=yes.
- For TSUNAMI-2D calculations, saveangflx=yes.h](h;)}(hKeyword threads should be omitted in favor of the SCALE command line–I
option. Keywords solntype, collapse, and saveangflx should be omitted
unless the following conditions are applicable.h]h/Keyword threads should be omitted in favor of the SCALE command line–I
option. Keywords solntype, collapse, and saveangflx should be omitted
unless the following conditions are applicable.}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubjd)}(hhh](j)}(hXRFor homogenized few-group cross section generation for nodal
calculations, solntype* *should be b1. This option will perform a
critical spectrum calculation, which will be folded into cross
section homogenization calculation. The critical spectrum is also
folded into the generation of ADFs and reaction rates for
depletion calculations.
h]h;)}(hXQFor homogenized few-group cross section generation for nodal
calculations, solntype* *should be b1. This option will perform a
critical spectrum calculation, which will be folded into cross
section homogenization calculation. The critical spectrum is also
folded into the generation of ADFs and reaction rates for
depletion calculations.h](h/UFor homogenized few-group cross section generation for nodal
calculations, solntype* }(hUFor homogenized few-group cross section generation for nodal
calculations, solntype* h jubj)}(hjh]h/*}(hhh j ubah}(h]id151ah]h]h]h]refidid150uhjh jubh/should be b1. This option will perform a
critical spectrum calculation, which will be folded into cross
section homogenization calculation. The critical spectrum is also
folded into the generation of ADFs and reaction rates for
depletion calculations.}(hshould be b1. This option will perform a
critical spectrum calculation, which will be folded into cross
section homogenization calculation. The critical spectrum is also
folded into the generation of ADFs and reaction rates for
depletion calculations.h jubeh}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jubj)}(hCGeneration of a new collapsed cross section library, collapse=yes.
h]h;)}(hBGeneration of a new collapsed cross section library, collapse=yes.h]h/BGeneration of a new collapsed cross section library, collapse=yes.}(hj0h j.ubah}(h]h]h]h]h]uhh:h!jhMh j*ubah}(h]h]h]h]h]uhjh jubj)}(h,For TSUNAMI-2D calculations, saveangflx=yes.h]h;)}(hjDh]h/,For TSUNAMI-2D calculations, saveangflx=yes.}(hjDh jFubah}(h]h]h]h]h]uhh:h!jhMh jBubah}(h]h]h]h]h]uhjh jubeh}(h]h]h]h]h]jBjCuhjch!jhMh jubeh}(h]h]h]h]h]uhjh jhhh!NhNubh)}(h.. _9-2-3-2-5:h]h}(h]h]h]h]h]hid152uhh
hMh jhhh!jubh$)}(hhh](h))}(hGeometry processing optionsh]h/Geometry processing options}(hjuh jshhh!NhNubah}(h]h]h]h]h]uhh(h jphhh!jhMubh;)}(hX^**combine**\ =(\ *yes/no*) — Automatic grid generation can result in
very small grid cells in some locations. Setting parameter combine to
*yes* performs automatic combination of smaller grid cells into adjacent
neighbor of same material, if possible. Combine is automatically set to
*no* if CMFD is enabled; this setting cannot be overridden. [no]h](j)}(h**combine**h]h/combine}(hhh jubah}(h]h]h]h]h]uhjh jubh/ =( }(h\ =(\ h jhhh!NhNubhA)}(h*yes/no*h]h/yes/no}(hhh jubah}(h]h]h]h]h]uhh@h jubh/t) — Automatic grid generation can result in
very small grid cells in some locations. Setting parameter combine to
}(ht) — Automatic grid generation can result in
very small grid cells in some locations. Setting parameter combine to
h jhhh!NhNubhA)}(h*yes*h]h/yes}(hhh jubah}(h]h]h]h]h]uhh@h jubh/ performs automatic combination of smaller grid cells into adjacent
neighbor of same material, if possible. Combine is automatically set to
}(h performs automatic combination of smaller grid cells into adjacent
neighbor of same material, if possible. Combine is automatically set to
h jhhh!NhNubhA)}(h*no*h]h/no}(hhh jubah}(h]h]h]h]h]uhh@h jubh/< if CMFD is enabled; this setting cannot be overridden. [no]}(h< if CMFD is enabled; this setting cannot be overridden. [no]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jphhubh;)}(hX**clearint**\ =(\ *yes/no*) — Grid generation option that removes the
global NEWT grid if a local unit grid is supplied. (For meshing options,
see the *boundary* keyword in the Geometry block description in
:ref:`9-2-3-6`) By default, clearint is set to yes, which means the
global grid is removed if local grids are provided. If CMFD acceleration
is enabled, clearint is set to no, which means both the global grid and
optional local grids are used. [yes]h](j)}(h**clearint**h]h/clearint}(hhh jubah}(h]h]h]h]h]uhjh jubh/ =( }(h\ =(\ h jhhh!NhNubhA)}(h*yes/no*h]h/yes/no}(hhh jubah}(h]h]h]h]h]uhh@h jubh/) — Grid generation option that removes the
global NEWT grid if a local unit grid is supplied. (For meshing options,
see the }(h) — Grid generation option that removes the
global NEWT grid if a local unit grid is supplied. (For meshing options,
see the h jhhh!NhNubhA)}(h
*boundary*h]h/boundary}(hhh jubah}(h]h]h]h]h]uhh@h jubh/. keyword in the Geometry block description in
}(h. keyword in the Geometry block description in
h jhhh!NhNubj)}(h:ref:`9-2-3-6`h]j#)}(hjh]h/9-2-3-6}(hhh jubah}(h]h](jnstdstd-refeh]h]h]uhj"h jubah}(h]h]h]h]h]refdocj refdomainj"reftyperefrefexplicitrefwarnj9-2-3-6uhjh!jhMh jubh/) By default, clearint is set to yes, which means the
global grid is removed if local grids are provided. If CMFD acceleration
is enabled, clearint is set to no, which means both the global grid and
optional local grids are used. [yes]}(h) By default, clearint is set to yes, which means the
global grid is removed if local grids are provided. If CMFD acceleration
is enabled, clearint is set to no, which means both the global grid and
optional local grids are used. [yes]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jphhubh;)}(h**grid_tol=**\ (*RN*) — Tolerance used in determining if polygon
vertices are numerically identical during NEWT grid generation.
[0.000001]h](j)}(h
**grid_tol=**h]h/ grid_tol=}(hhh jCubah}(h]h]h]h]h]uhjh j?ubh/ (}(h\ (h j?hhh!NhNubhA)}(h*RN*h]h/RN}(hhh jVubah}(h]h]h]h]h]uhh@h j?ubh/y) — Tolerance used in determining if polygon
vertices are numerically identical during NEWT grid generation.
[0.000001]}(hy) — Tolerance used in determining if polygon
vertices are numerically identical during NEWT grid generation.
[0.000001]h j?hhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jphhubh;)}(h**cell_tol=**\ (*RN*) — Tolerance used in determining if polygon
vertices are numerically identical during NEWT cell generation.
[0.000001]h](j)}(h
**cell_tol=**h]h/ cell_tol=}(hhh jsubah}(h]h]h]h]h]uhjh joubh/ (}(h\ (h johhh!NhNubhA)}(h*RN*h]h/RN}(hhh jubah}(h]h]h]h]h]uhh@h joubh/y) — Tolerance used in determining if polygon
vertices are numerically identical during NEWT cell generation.
[0.000001]}(hy) — Tolerance used in determining if polygon
vertices are numerically identical during NEWT cell generation.
[0.000001]h johhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jphhubh;)}(h**line_tol=**\ (*RN*) — Tolerance used in determining if polygon
vertices are numerically identical during NEWT line generation.
[1.0e-10]h](j)}(h
**line_tol=**h]h/ line_tol=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*RN*h]h/RN}(hhh jubah}(h]h]h]h]h]uhh@h jubh/x) — Tolerance used in determining if polygon
vertices are numerically identical during NEWT line generation.
[1.0e-10]}(hx) — Tolerance used in determining if polygon
vertices are numerically identical during NEWT line generation.
[1.0e-10]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jphhubj)}(hX0The default values for all geometry-processing keywords are recommended
and can be omitted. For problems with very fine mesh, tighter grid and
cell tolerances should be applied. For problems that terminate with a
ray-tracing error (i.e., tracer error), tighter grid and cell tolerances
should be applied.h]h;)}(hX0The default values for all geometry-processing keywords are recommended
and can be omitted. For problems with very fine mesh, tighter grid and
cell tolerances should be applied. For problems that terminate with a
ray-tracing error (i.e., tracer error), tighter grid and cell tolerances
should be applied.h]h/X0The default values for all geometry-processing keywords are recommended
and can be omitted. For problems with very fine mesh, tighter grid and
cell tolerances should be applied. For problems that terminate with a
ray-tracing error (i.e., tracer error), tighter grid and cell tolerances
should be applied.}(hjh jubah}(h]h]h]h]h]uhh:h!jhMh jubah}(h]h]h]h]h]uhjh jphhh!jhNubh)}(h.. _9-2-3-2-6:h]h}(h]h]h]h]h]hid153uhh
hMh jphhh!jubeh}(h](geometry-processing-optionsjoeh]h](geometry processing options 9-2-3-2-5eh]h]uhh#h jhhh!jhMjf}jjesjh}jojesubeh}(h](control-optionsjeh]h] 9-2-3-2-4ah]jah]uhh#h jhhh!jhMOjKjf}jjsjh}jjsubh$)}(hhh](h))}(hCritical spectrum optionsh]h/Critical spectrum options}(hjh j
hhh!NhNubah}(h]h]h]h]h]uhh(h jhhh!jhMubh;)}(h**useb1**\ =(\ *yes/no*) —Turns on/off the use of the B1 approximation
to determine the critical spectrum. If useb1 is set to no, the P1
approximation is used. [yes]h](j)}(h **useb1**h]h/useb1}(hhh jubah}(h]h]h]h]h]uhjh jubh/ =( }(h\ =(\ h jhhh!NhNubhA)}(h*yes/no*h]h/yes/no}(hhh j/ubah}(h]h]h]h]h]uhh@h jubh/) —Turns on/off the use of the B1 approximation
to determine the critical spectrum. If useb1 is set to no, the P1
approximation is used. [yes]}(h) —Turns on/off the use of the B1 approximation
to determine the critical spectrum. If useb1 is set to no, the P1
approximation is used. [yes]h jhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(hO**b2=**\ (*RN*) — Material buckling factor, in units of 1/cm\ :sup:`2`.
[0.0]h](j)}(h**b2=**h]h/b2=}(hhh jLubah}(h]h]h]h]h]uhjh jHubh/ (}(h\ (h jHhhh!NhNubhA)}(h*RN*h]h/RN}(hhh j_ubah}(h]h]h]h]h]uhh@h jHubh/2) — Material buckling factor, in units of 1/cm }(h2) — Material buckling factor, in units of 1/cm\ h jHhhh!NhNubj;)}(h:sup:`2`h]h/2}(hhh jrubah}(h]h]h]h]h]uhj;h jHubh/.
[0.0]}(h.
[0.0]h jHhhh!NhNubeh}(h]h]h]h]h]uhh:h!jhMh jhhubh;)}(hX&**height=**\ (*RN*) — Height (transverse dimension) in centimeters. Used
in a geometric buckling correction to calculate leakage normal to the
plane of the input 2-D model. Keywords **dz=** and **deltaz=** are
equivalent. When set to zero (default), no buckling correction is
performed. [0.0]h](j)}(h**height=**h]h/height=}(hhh jubah}(h]h]h]h]h]uhjh jubh/ (}(h\ (h jhhh!NhNubhA)}(h*RN*h]h/RN}(hhh jubah}(h]h]h]h]h]uhh@h jubh/) — Height (transverse dimension) in centimeters. Used
in a geometric buckling correction ~~