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diff --git a/doc/IGE351/SectDmicrolib.tex b/doc/IGE351/SectDmicrolib.tex new file mode 100644 index 0000000..433c7b6 --- /dev/null +++ b/doc/IGE351/SectDmicrolib.tex @@ -0,0 +1,1476 @@ +\section{Contents of a \dir{microlib} directory}\label{sect:microlibdir} + +A \dir{microlib} directory contains the set of multigroup microscopic +cross sections associated with a set of isotopes. It also includes a \dir{macrolib} directory +where the macroscopic cross sections for the mixtures to which are associated these +isotopes are stored (see \Sect{macrolibdir}). Finally it may contains a \dir{depletion} directory +(see \Sect{microlibdirdepletion}) which is required for burnup calculation and a \dir{selfshield} +directory which is generated by the \moc{SHI:} or \moc{USS:} operator (see +\Sect{subgroupdirselfshield}). It is +therefore multi-level, namely, it contains sub-directories. Note that the contents of such a +directory will vary depending on the operator which was used to create or modify it. Here for +convenience we will define the variable +$\mathcal{M}$ to identify the creation operator: +\begin{displaymath} +\mathcal{M} = \left\{ +\begin{array}{ll} +1 & \textrm{if the microlib is created or modified by the \moc{LIB:} or \moc{EVO:} operator}\\ +2 & \textrm{if the microlib is created or modified by the \moc{EDI:} or +\moc{C2M:} operator}\\ +3 & \textrm{if the microlib is modified by the \moc{SHI:} or \moc{USS:} operator}\\ +4 & \textrm{if the microlib is part of a {\sc compo} object and is created by the \moc{COMPO:} operator} +\end{array} \right. +\end{displaymath} + +In the case where the \moc{LIB:} or \moc{C2M:} operator is used to create the microlib, it appears on the root +level of the data structure. For the other case it is embedded as a +subdirectory of a surrounding data structure. + +\subsection{State vector content for the \dir{microlib} data structure}\label{sect:microlibstate} + +The dimensioning parameters for the \dir{microlib} data structure, which are stored in +the state vector $\mathcal{S}^{m}$, represent: + +\begin{itemize} +\item The maximum number of mixtures $M_{m}=\mathcal{S}^{m}_{1}$ +\item The number of isotopes $N_{I}=\mathcal{S}^{m}_{2}$ +\item The number of groups ${G}=\mathcal{S}^{m}_{3}$ +\item The order for the scattering anisotropy $L=\mathcal{S}^{m}_{4}$ +($L=1$ is an isotropic collision; $L=2$ is a linearly anisotropic collision, +etc.) +\item The transport correction option $I_{tr}=\mathcal{S}^{m}_{5}$ +\begin{displaymath} +I_{tr} = \left\{ +\begin{array}{ll} +0 & \textrm{do not use a transport correction}\\ +1 & \textrm{use an APOLLO-type transport correction (micro-reversibility at +all energies)}\\ +2 & \textrm{recover a transport correction from the cross-section library}\\ +3 & \textrm{use a WIMS-type transport correction (micro-reversibility below +4eV;}\\ + & \textrm{1/E current spectrum elsewhere)}\\ +4 & \textrm{use a leakage correction based on {\tt NTOT1} data.} +\end{array} \right. +\end{displaymath} +\item Format of the included \dir{macrolib} $I_{p}=\mathcal{S}^{m}_{6}$ +\begin{displaymath} +I_{p} = \left\{ +\begin{array}{ll} +0 & \textrm{for the direct macroscopic cross sections} \\ +1 & \textrm{for the adjoint macroscopic cross sections} +\end{array} \right. +\end{displaymath} +\item Option for removing delayed neutron effects from the \dir{microlib} +$I_{t}=\mathcal{S}^{m}_{7}$ +\begin{displaymath} +I_{t} = \left\{ +\begin{array}{ll} +1 & \textrm{include the delayed and prompt neutron effect} \\ +2 & \textrm{consider only the prompt neutrons. This option is only available +with}\\ +& \textrm{MATXS--type libraries.} +\end{array} \right. +\end{displaymath} +\item The number of independent libraries $N_{\mathrm{lib}}=\mathcal{S}^{m}_{8}$ +\item The number of fast groups without self-shielding $N_{g,f}=\mathcal{S}^{m}_{9}$ + +Represents the number of fast energy groups to be treated without including resonance +effects. It is automatically determined from the cross-section libraries. This value, +which is only used by the self-shielding operator, can be modified using the keyword \moc{GRMAX}. + +\item The maximum index of all groups with self-shielding $N_{g,e}=\mathcal{S}^{m}_{10}$. + +In the case of a WIMS--type library, it represents the total number of energy groups above +$4.0$ eV. Otherwise, it is automatically determined from the cross-section libraries. This +value, is used by the self-shielding operator and can be modified locally in +this operator using the keyword \moc{GRMIN}. + +\item The number of depleting isotopes $N_{d}=\mathcal{S}^{m}_{11}$ +\item The number of depleting mixtures $N_{d,f}=\mathcal{S}^{m}_{12}$ +\item The number of additional $\phi$--weighted editing cross sections $N_{e}=\mathcal{S}^{m}_{13}$ +\item The number of mixtures $N_{m}=\mathcal{S}^{m}_{14}$ +\item The number of resonant mixtures $N_{r}=\mathcal{S}^{m}_{15}$ +\item The number of energy-dependent fission spectra $G_{\rm chi}=\mathcal{S}^{m}_{16}$. By default ($G_{\rm chi}=0$), +a unique fission spectrum is used. The theory of multiple fission spectra is presented in Ref.~\citen{mosca}. +\item Option for processing the cross-section libraries $I_{\rm proc}=\mathcal{S}^{m}_{17}$ +\begin{displaymath} +I_{\rm proc} = \left\{ +\begin{array}{ll} +-1 & \textrm{skip the library processing (i.e., no interpolation).} \\ +0 & \textrm{perform an interpolation in temperature and dilution.} \\ +1 & \textrm{perform an interpolation in temperature and compute probability} \\ + & \textrm{tables based on the tabulation in dilution.} \\ +2 & \textrm{perform an interpolation in temperature and build a new temperature-} \\ + & \textrm{independent cross-section library in \dir{microlib} format.} \\ +3 & \textrm{perform an interpolation in temperature and compute CALENDF--type} \\ + & \textrm{mathematical probability tables based on BIN--type cross sections. Do} \\ + & \textrm{not compute the slowing-down correlated weight matrices. Option} \\ + & \textrm{compatible with the subgroup projection method (SPM).} \\ +4 & \textrm{perform an interpolation in temperature and compute CALENDF--type} \\ + & \textrm{mathematical probability tables and slowing-down correlated weight} \\ + & \textrm{matrices based on BIN--type cross sections. Option compatible with} \\ + & \textrm{the Ribon extended method.} \\ +5 & \textrm{perform an interpolation in temperature and compute CALENDF--type} \\ + & \textrm{mathematical probability tables based on BIN--type cross sections. This} \\ + & \textrm{option is similar to the $I_{\rm proc} =3$ procedure. Here, the base points of the} \\ + & \textrm{probability tables corresponding to fission and scattering cross sections} \\ + & \textrm{and to components of the transfer scattering matrix are also obtained} \\ + & \textrm{using the CALENDF approach.} \\ +6 & \textrm{perform an interpolation in temperature and compute RSE--type proba-} \\ + & \textrm{bility tables based on BIN--type cross sections. RSE is the resonance} \\ + & \textrm{spectrum expansion method.} \\ +\end{array} \right. +\end{displaymath} +\item Option for computing the macrolib $I_{\rm mac}=\mathcal{S}^{m}_{18}$ +\begin{displaymath} +I_{\rm mac} = \left\{ +\begin{array}{ll} +0 & \textrm{do not build an embedded macrolib.} \\ +1 & \textrm{build an embedded macrolib. Mandatory if the microlib is to be used to} \\ + & \textrm{perform micro-depletion.} +\end{array} \right. +\end{displaymath} +\item The number of precursor groups producing delayed neutrons $N_{\rm del}=\mathcal{S}^{m}_{19}$. +\item The number of fissile isotopes producing fission products with {\tt PYIELD} data $N_{\rm dfi}=\mathcal{S}^{m}_{20}$ (see Table~\ref{tabl:tabiso3}) +\item Option for completing the depletion chains with the missing isotopes $I_{\rm cmp}=\mathcal{S}^{m}_{21}$ +\begin{displaymath} +I_{\rm cmp} = \left\{ +\begin{array}{ll} +0 & \textrm{complete} \\ +1 & \textrm{do not complete.} +\end{array} \right. +\end{displaymath} +\item The maximum number of isotopes per mixture $M_{\rm I}=\mathcal{S}^{m}_{22}$. +\item An integer index (1, 2, 3 or 4) used to set the accuracy of the CALENDF probability +tables. The highest the value, the more accurate are the tables. $N_{\rm +ipreci}=\mathcal{S}^{m}_{23}$. +\item Discontinuity factor flag $I_{\rm df}=\mathcal{S}^{m}_{24}$. This information is available in \dir{macrolib} directory (see \Sect{macrolibdir}) +\begin{displaymath} +I_{\rm df} = \left\{ +\begin{array}{ll} +0 & \textrm{no discontinuity factor information}\\ +1 & \textrm{multigroup boundary current information is available}\\ +2 & \textrm{boundary flux information (see \Sect{macroADF}) is available}\\ +3 & \textrm{discontinuity factor information (see \Sect{macroADF}) is available}\\ +4 & \textrm{matrix ($G \times G$) discontinuity factor information (see \Sect{macroADF}) is available.} +\end{array} \right. +\end{displaymath} +\item The maximum Legendre order of the weighting functions $I_{w}=\mathcal{S}^{m}_{25}$ +\begin{displaymath} +I_{w} = \left\{ +\begin{array}{ll} +0 & \textrm{use the flux as weighting function for all cross sections}\\ +1 & \textrm{use the fundamental current ${\cal J}$ as weighting function for +scattering cross sections}\\ +& \textrm{with order $\ge 1$ and compute both $\phi$-- and +${\cal J}$--weighted total cross sections.} +\end{array} \right. +\end{displaymath} +\item Number of companion particles in coupled sets $I_{\rm part}=\mathcal{S}^{M}_{26}$ +\begin{displaymath} +I_{\rm part} = \left\{ +\begin{array}{ll} +0 & \textrm{the microlib doesn't include coupled sets}\\ +>0 & \textrm{number of companion particles.} +\end{array} \right. +\end{displaymath} +\item Option for performing the Sternheimer density correction for charged particle cases $I_{\rm ster}=\mathcal{S}^{m}_{27}$ +\begin{displaymath} +I_{\rm ster} = \left\{ +\begin{array}{ll} +0 & \textrm{do not perform the correction.} \\ +1 & \textrm{perform Sternheimer correction applied for both restricted total stopping power}\\ +& \textrm{and heat deposition cross section.} +\end{array} \right. +\end{displaymath} +\end{itemize} + +\goodbreak +\clearpage + +\subsection{The main \dir{microlib} directory}\label{sect:microlibdirmain} + +The following records and sub-directories will be found on the first level of a \dir{microlib} +directory: +\begin{DescriptionEnregistrement}{Main records and sub-directories in \dir{microlib}}{7.0cm} +\CharEnr + {SIGNATURE\blank{3}}{$*12$} + {Signature of the \dir{microlib} data structure ($\mathsf{SIGNA}=${\tt L\_LIBRARY\blank{3}}).} +\IntEnr + {STATE-VECTOR}{$40$} + {Vector describing the various parameters associated with this data structure $\mathcal{S}^{m}_{i}$, + as defined in \Sect{microlibstate}.} +\RealEnr + {ENERGY\blank{6}}{$G+1$}{eV} + {Energy groups limits $E_{g}$} +\RealEnr + {DELTAU\blank{6}}{$G$}{} + {Lethargy width of each group $U_{g}$} +\OptRealEnr + {CHI-ENERGY\blank{2}}{$G_{\rm chi}+1$}{$G_{\rm chi}\ne 0$}{eV} + {$E_{\rm chi}(g)$: Group energy limits defining the energy-dependent fission spectra. By default, a unique fission spectra is used.} +\OptIntEnr + {CHI-LIMITS\blank{2}}{$G_{\rm chi}+1$}{$G_{\rm chi}\ne 0$} + {$N_{\rm chi}(g)$: Group limit indices defining the energy-dependent fission spectra. By default, a unique fission spectra is used.} +\DirlEnr + {ISOTOPESLIST}{$N_{I}$} + {List of {\sc isotope} directories. Each component of this list follows the \dir{isotope} specification + presented in Tables~\ref{tabl:tabiso1} to \ref{tabl:tabiso5} and is containing the cross section + information associated with a specific isotope. The name of these isotopes is specified by + $\mathsf{NALIAS}_{i}$ as given in record {\tt ISOTOPESUSED}.} +\CharEnr + {ISOTOPESUSED}{$(N_{I})*12$} + {Alias name associated with each isotope $\mathsf{NALIAS}_{i}$. The first eight characters of the name of a macroscopic residual are set to {\tt '*MAC*RES'}.} +\OptCharEnr + {ISOTOPERNAME}{$(N_{I})*12$}{$\mathcal{M}=1,3$} + {Reference name associated with each isotope $\mathsf{NISO}_{i}$} +\OptIntEnr + {ISOTOPESMIX\blank{1}}{$N_{I}$}{$\mathcal{M}\ne 4$} + {Mixture number associated with each isotope $N_{I}$} +\RealEnr + {ISOTOPESDENS}{$N_{I}$}{(cm b)$^{-1}\ $} + {Isotopic density $\rho_{i}$} +\RealEnr + {ISOTOPESTEMP}{$N_{I}$}{K} + {Isotope temperature $T_{i}$} +\OptIntEnr + {ISOTOPESTODO}{$N_{I}$}{$\mathcal{M}=1,3$} + {=0: automatic detection of depletion for isotope $i$; =1: isotope $i$ is + forced to be non depleting (keeps its capability to produce energy); =2: isotope $i$ is + forced to be depleting; =3: isotope $i$ is at saturation.} +\IntEnr + {ISOTOPESTYPE}{$N_{I}$} + {Type index associated with each isotope $\mathsf{ITYP}_{i}$. $=1$: the isotope is + not fissile and not a fission product; $=2$: fissile isotope; $=3$: fission + product.} +\OptRealEnr + {ISOTOPESVOL\blank{1}}{$N_{I}$}{$\mathcal{M}=2, 4$}{cm$^{3}$} + {Volume occupied by isotope $V_{i}$} +\OptCharEnr + {ILIBRARYTYPE}{$(N_{I})*8$}{$N_{\mathrm{lib}}\ge 1$} + {Library type associated with each isotope $\mathsf{NLTY}_{i}$} +\OptCharEnr + {ILIBRARYNAME}{$(N_{\mathrm{lib}})*64$}{$N_{\mathrm{lib}}\ge 1$} + {Name associated with each cross-section library} +\OptIntEnr + {ILIBRARYINDX}{$N_{I}$}{$N_{\mathrm{lib}}\ge 1$} + {Index of the cross-section library associated with each isotope $1 \le \mathsf{LLIB}_{i}\le N_{\mathrm{lib}}$} +\OptCharEnr + {ISOTOPESCOH\blank{1}}{$(N_{I})*8$}{$N_{\mathrm{lib}}\ge 1$} + {Name of coherent scattering type at thermal energies $\mathsf{NCOH}_{i}$} +\OptCharEnr + {ISOTOPESINC\blank{1}}{$(N_{I})*8$}{$N_{\mathrm{lib}}\ge 1$} + {Name of incoherent scattering type at thermal energies $\mathsf{NINC}_{i}$} +\OptCharEnr + {ISOTOPESRESK}{$(N_{I})*8$}{$N_{\mathrm{lib}}\ge 1$} + {Name of resonance elastic scattering kernel (RESK) type at epithermal energies $\mathsf{NRSK}_{i}$} +\OptIntEnr + {ISOTOPESNTFG}{$N_{I}$}{$N_{\mathrm{lib}}\ge 1$} + {Number of thermal groups involved in coherent or incoherent scattering $G_{s,i}$} +\OptCharEnr + {ISOTOPESHIN\blank{1}}{$(N_{I})*12$}{$N_{\mathrm{lib}}\ge 1$} + {Name of resonant isotope associated with each isotope $\mathsf{NSHI}_{i}$} +\OptIntEnr + {ISOTOPESSHI\blank{1}}{$N_{I}$}{$N_{\mathrm{lib}}\ge 1$} + {Resonant mixture associated with each isotope $I_{R,i}$} +\OptRealEnr + {ISOTOPESDSN\blank{1}}{$G \times N_{I}$}{${\displaystyle N_{\mathrm{lib}}\ge 1 \atop + \displaystyle I_{\rm proc}=0}$}{b} + {Standard dilution cross section for isotope $\sigma_{\mathrm{dil},i}$ in each energy group} +\OptRealEnr + {ISOTOPESDSB\blank{1}}{$G \times N_{I}$}{${\displaystyle N_{\mathrm{lib}}\ge 1 \atop + \displaystyle I_{\rm proc}=0}$}{b} + {Livolant-Jeanpierre dilution cross section for isotope $\sigma_{\mathrm{LJ},i}$ in each energy group} +\OptIntEnr + {ISOTOPESNIR\blank{1}}{$N_{I}$}{$N_{\mathrm{lib}}\ge 1$} + {Use Goldstein-Cohen factor $\lambda_i$ in groups with index $\ge N^{\rm ir}_i$. + Use $\lambda=1$ in other groups} +\OptRealEnr + {ISOTOPESGIR\blank{1}}{$N_{I}$}{$N_{\mathrm{lib}}\ge 1$}{1} + {Goldstein-Cohen parameter in low-energy resonant groups $\lambda_i$. Set to -998.0 if + $I_{\rm proc}=3$, to -999.0 if $I_{\rm proc}=4$, to -1000.0 if $I_{\rm proc}=5$ and to -1001.0 if $I_{\rm proc}=6$.} +\OptRealEnr + {MIXTURESVOL\blank{1}}{$N_{m}$}{$\mathcal{M}=2, 4$}{cm$^{3}$} + {Volume occupied by each mixture} +\OptRealEnr + {MIXTURESDENS}{$N_{m}$}{$\mathcal{M}=1$}{g/cm$^{3}$~~} + {Volumetric mass density of each mixture $\rho_{m}$} +\OptCharEnr + {ADDXSNAME-P0}{$(N_{e})*8$}{$N_{e}\ge 1$} + {Names of the additional $\phi$--weighted editing cross sections $\mathsf{ADDXS}_{k}$ stored on \dir{macrolib}} +\OptCharEnr + {PARTICLE\blank{4}}{$*1$}{$I_{\rm part}\ge 1$} + {Character name of the particle associated to the microlib. Usual names for + particles are {\tt N} (neutrons), {\tt G} (photons), {\tt B} (electrons), + {\tt C} (positrons) and {\tt P} (protons).} +\OptCharEnr + {PARTICLE-NAM}{($I_{\rm part}+1$)$*1$}{$I_{\rm part}\ge 1$} + {Character name associated to each particle.} +\OptIntEnr + {PARTICLE-NGR}{$I_{\rm part}+1$}{$I_{\rm part}\ge 1$} + {Number of energy groups associated to each particle.} +\OptRealEnr + {PARTICLE-MC2}{$I_{\rm part}+1$}{$I_{\rm part}\ge 1$}{eV} + {Rest energy associated to each particle.} +\OptRealVar + {\listedir{penergy}}{$G_i+1$}{$I_{\rm part}\ge 1$}{eV} + {Set of arrays containing energy groups limits for a companion particle. The character name + of each sub-directory is the concatenation of the character*1 name of the particle with ``{\tt ENERGY}''. + For example, {\tt GENERGY} contains the energy mesh of secondary photons ($G_i+1$ values).} +\OptRealEnr + {TIMESPER\blank{4}}{$2\times 3$}{$\mathcal{M}=2$}{} + {Array $T_{j,i}$ that contains $T_{j,1}=t$, $T_{j,2}=B$ and $T_{j,3}=w$, the + lower ($j=1$) and upper bounds ($j=2$) for the reference time in days, burnup + in MW day T$^{-1}$ and irradiation in Kb$^{-1}$ respectively for which the + perturbative expansion is valid} +\OptRealEnr + {K-EFFECTIVE\blank{1}}{$1$}{*}{} + {Effective multiplication constant $k_{\mathrm{eff}}$} +\OptRealEnr + {K-INFINITY\blank{2}}{$1$}{*}{} + {Infinite multiplication constant $k_{\mathrm{inf}}$} +\OptRealEnr + {B2\blank{2}B1HOM\blank{3}}{$1$}{*}{cm$^{-2}$~~} + {Homogeneous Buckling $B_{\mathrm{hom}}$} +\OptDirEnr + {MACROLIB\blank{4}}{$I_{\rm mac} = 1$} + {Sub-directory containing the \dir{macrolib} associated with this + library, following the specification presented in \Sect{macrolibdirmain}.} +\OptDirEnr + {DEPL-CHAIN\blank{2}}{$N_{d} \ge 1$} + {Sub-directory containing the \dir{depletion} associated with this library, following + the specification presented in \Sect{microlibdirdepletion}.} +\OptDirEnr + {SHIBA\blank{7}}{$\mathcal{M}=3$} + {Sub-directory containing the \dir{selfshield} associated with this + library, following the specification presented in \Sect{shibadirselfshield}. + This data is used by the \moc{SHI:} self-shielding module.} +\OptDirEnr + {SHIBA\_SG\blank{4}}{$\mathcal{M}=3$} + {Sub-directory containing the \dir{uss-selfshield} associated with this + library, following the specification presented in \Sect{subgroupdirselfshield}. + This sub-directory is present in the library builded by \moc{USS:} self-shielding module and used by \moc{USS:}.} +\IntEnr + {MIXTUREGAS\blank{2}}{$N_{m}$} + {State of each mixture (used for stopping power correction).} +\OptDirEnr + {INDEX\blank{7}}{*} + {Sub-directory containing indexing or table-of-content data for specific library + files} +\end{DescriptionEnregistrement} + +One will find in \Sect{macrolibdir} the description of a +\dir{macrolib} directory and in +\Sect{isotopedir} the contents of an \dir{isotope} directory. Note that if $N_{I}=2$ and +\begin{displaymath} +\mathsf{NALIAS}_{i} = \left\{ +\begin{array}{lll} +\texttt{U235 0001} & \textrm{for}& i=1\\ +\texttt{Pu239 0003} & \textrm{for}& i=2 +\end{array} \right. +\end{displaymath} +then \listedir{isotope} will correspond to the following two directories: + +\begin{DescriptionEnregistrement}{Examples of isotopes directory in a \dir{microlib}}{7.5cm} +\DirEnr + {U235\blank{4}0001} + {Directory where the microscopic cross sections of \Iso{U}{235} are stored. These are + self-shielded cross section already interpolated in temperature. They correspond to the + properties of mixture $1$} +\DirEnr + {Pu239\blank{3}0003} + {Directory where the microscopic cross sections of \Iso{Pu}{239} are stored. These are + self-shielded cross section already interpolated in temperature. They correspond to the + properties of mixture $3$} +\end{DescriptionEnregistrement} + +\subsection{State vector content for the depletion sub-directory}\label{sect:chainlibstate} + +The dimensioning parameters for the depletion sub-directory, which are stored in +the state vector $\mathcal{S}^{d}$, represent: + +\begin{itemize} +\item The number of depleting isotopes $N_{\mathrm{depl}}=\mathcal{S}^{d}_{1}$ +\item The number of direct fissile isotopes (i.e., producing fission products) $N_{\mathrm{dfi}}=\mathcal{S}^{d}_{2}$ +\item The number of fission fragments $N_{\mathrm{dfp}}=\mathcal{S}^{d}_{3}$. A fission fragment is produced directly by the +fission reaction. A fission product is a fission fragment or a daughter isotope +produced by decay or neutron-induced reaction. +\item The number of heavy isotopes $N_{\mathrm{H}}=\mathcal{S}^{d}_{4}$ + +This number represents the combination of fissile isotopes and the other isotopes produced from +these isotopes by reactions other than fission. + +\item The number of fission products $N_{\mathrm{fp}}=\mathcal{S}^{d}_{5}$ + +This number represents the combination of fission fragments and the other +daughter isotopes produced by any reaction (decay or neutron induced). + +\item The number of other isotopes $N_{\mathrm{O}}=\mathcal{S}^{d}_{6}$ + +This number represents the other depleting isotopes which are not produced by fission or by reaction +with fission isotopes or fission products but have a depletion chain. + +\item The number of stable isotopes $N_{\mathrm{H}}=\mathcal{S}^{d}_{7}$ + +This number represents the non-depleting isotopes producing energy (mainly +by radiative capture). An isotope is considered to be stable if: +\begin{itemize} +\item its radioactive decay constant is zero +\item the isotope has no father and no daughter +\item energy is produced by the isotope. +\end{itemize} + +\item The maximum number of depleting reactions, including radioactive decay and +neutron-induced reactions $M_{\mathrm{R}}=\mathcal{S}^{d}_{8}$ + +\item The maximum number of parent isotopes leading to the production of an isotope in the +depletion chain $M_{\mathrm{S}}=\mathcal{S}^{d}_{9}$ + +\item The number of energy-dependent fission yield matrices $N_{\mathrm{ndp}}=\mathcal{S}^{d}_{10}$ + +\end{itemize} + +\subsection{The depletion sub-directory \dir{depletion} in +\dir{microlib}}\label{sect:microlibdirdepletion} + +The following records and sub-directories will be found on the first level of a +\dir{depletion} directory: + +\begin{DescriptionEnregistrement}{Main records and sub-directories in +\dir{depletion}}{6.0cm} +\label{tabl:tabchain} +\IntEnr + {STATE-VECTOR}{$40$} + {$\mathcal{S}^{d}_{i}$ is the vector describing the various parameters associated with this data structure, + as defined in \Sect{chainlibstate}.} +\CharEnr + {ISOTOPESDEPL}{$(N_{\mathrm{depl}})*12$} + {Reference name of the isotopes $\mathsf{NISOD}_{i}$ present in the depletion chain} +\IntEnr + {CHARGEWEIGHT}{$N_{\mathrm{depl}}$} + {6-digit (integer number) nuclide identifier with atomic number $Z$ (2 + digits), mass number $A$ (3 digits) and energy state $E$ (0 for ground state, 1 + for first excited level, etc.). This identifier is not defined for pseudo + fission products.} +\CharEnr + {DEPLETE-IDEN}{$(M_{\mathrm{R}})*8$} + {Reference name of the depletion reactions} +\IntEnr + {DEPLETE-REAC}{$M_{\mathrm{R}}\times N_{\mathrm{depl}}$} + {$K_{r,i}^{\rm d}$ is the list of identifier for the depletion of an isotope.} +\RealEnr + {DEPLETE-ENER}{$M_{\mathrm{R}}\times N_{\mathrm{depl}}$}{Mev} + {$R_{r,i}^{\rm d}$ is the energy produced with each depletion reaction $r$ of the father isotope. If {\tt H-FACTOR} + information is available for an isotope $i$, $R_{r,i}^{\rm d}$ contains only decay energy contributions of lumped isotopes + produced by reaction $r$.} +\RealEnr + {DEPLETE-DECA}{$N_{\mathrm{depl}}$}{$10^{-8}$ s$^{-1}\ $} + {Radioactive decay constants.} +\IntEnr + {PRODUCE-REAC}{$M_{\mathrm{S}}\times N_{\mathrm{depl}}$} + {$K_{s,i}^{\rm p}$ is the list of identifier for the production of an isotope.} +\RealEnr + {PRODUCE-RATE}{$M_{\mathrm{S}}\times N_{\mathrm{depl}}$}{1} + {$R_{s,i}^{\rm p}$ is the branching ratio associated with each production reaction.} +\RealEnr + {FISSIONYIELD}{$N_{\mathrm{ndp}} \times N_{\mathrm{dfi}}\times N_{\mathrm{dfp}}$}{1} + {$Y_{k,{i\to j}}$ is the fission yield for each direct fissile isotope $i$ to each fission fragment $j$ in fission yield + macrogroup $k$.} +\OptRealEnr + {ENERGY-YIELD}{$N_{\mathrm{ndp}+1}$}{$N_\mathrm{ndp}\ge 2$}{eV} + {$E_{k}^{\rm fiss}$ are the energy limits of fission yield macrogroups.} +\end{DescriptionEnregistrement} + +An isotope $\mathsf{NISO}_{i}$ defined in \Sect{microlibdirmain} is considered +to be part of the depletion chain only if one can find a value of $1 \le j \le N_{\rm depl}$ +such that $\mathsf{NISO}_{i}= \mathsf{NISOD}_{j}$. +Some depleting isotopes may be automatically added to the \dir{microlib} directory. +In this case, the reference name in record {\tt ISOTOPERNAME} is taken equal +to its reference name in {\tt ISOTOPESDEPL} and the alias name in record +{\tt ISOTOPESUSED} is taken equal to the +first 8 characters of its reference name in {\tt ISOTOPESDEPL}, completed by a +4-digit mixture identifier. If the reference name contains an underscore, the +alias name is truncated at the first underscore. For example, an isotope +present in mixture 2 with a reference name equal to {\tt D2O\_3\_P5} is +translated into an alias name equal to {\tt D2O\blank{5}0002}. + +\vskip 0.2cm + +The contents of the variables $K_{r,i}^{\rm d}$ is used to identify the type of isotope under +consideration. For each isotope $i$, $r$ will take +successively the values $1$ to $M_{\mathrm{D}}$ depending on the type of +reaction $\mathsf{NREAD}_{r}$ one wishes to analyze, namely + +\vskip 0.2cm + +\begin{tabular}{|l|l|} +\hline +$\mathsf{NREAD}_{1}=${\tt DECAY\blank{3}} & isotope may undergo radioactive decay \\ +$\mathsf{NREAD}_{2}=${\tt NFTOT\blank{3}} & isotope may undergo fission or is a +fission fragment \\ +& $^{1}_{0}n + ^{A}_{Z}X \to \ ^{A+1-\nu-B}_{Z-Y}U + ^{B}_{Y}V + \nu \ ^{1}_{0}n + \gamma$ \\ +$\mathsf{NREAD}_{3}=${\tt NG\blank{6}} & isotope may undergo neutron capture (mt$=$102) \\ +& $^{1}_{0}n + ^{A}_{Z}X \to \ ^{A+1}_{Z}X + \gamma$ \\ +$\mathsf{NREAD}_{4}=${\tt N2N\blank{5}} & isotope may undergo (n,2n) reaction (mt$=$16) \\ +& $^{1}_{0}n + ^{A}_{Z}X \to \ ^{A-1}_{Z}X + 2 \ ^{1}_{0}n + \gamma$ \\ +$\mathsf{NREAD}_{5}=${\tt N3N\blank{5}} & isotope may undergo (n,3n) reaction (mt$=$17) \\ +$\mathsf{NREAD}_{6}=${\tt N4N\blank{5}} & isotope may undergo (n,4n) reaction (mt$=$37) \\ +$\mathsf{NREAD}_{7}=${\tt NA\blank{6}} & isotope may undergo (n,$\alpha$) reaction (mt$=$107) \\ +$\mathsf{NREAD}_{8}=${\tt NP\blank{6}} & isotope may undergo (n,p) reaction (mt$=$103) \\ +$\mathsf{NREAD}_{9}=${\tt N2A\blank{5}} & isotope may undergo (n,2$\alpha$) reaction (mt$=$108) \\ +$\mathsf{NREAD}_{10}=${\tt NNP\blank{5}} & isotope may undergo (n,np) reaction (mt$=$28) \\ +$\mathsf{NREAD}_{11}=${\tt ND\blank{6}} & isotope may undergo (n,d) reaction (mt$=$104)\\ +$\mathsf{NREAD}_{12}=${\tt NT\blank{6}} & isotope may undergo (n,t) reaction (mt$=$105) \\ +\hline +\end{tabular} + +\vskip 0.3cm + +\noindent where symbols n, $\alpha$, p, d and t represent neutron, alpha particle, proton, deuteron +and triton, respectively. + +\vskip 0.2cm + +The contents of the variable $K_{r,i}^{\rm d}$ is used to specify the +properties of reaction $r$ for each isotope $i$ under consideration. +Here $K_{r,i}^{\rm d}$ contains two different types of informations, namely +$d(r)$ and $i(r)$ which are defined as follows: + +\begin{equation} + d(r)=K_{r,i}^{\rm d} \bmod \ 100 \ \ \ \ {\rm and} \ \ \ \ i(r)={K_{r,i}^{\rm d} \over 100} +\end{equation} + +\noindent where + +\begin{displaymath} +d(r) = \left\{ +\begin{array}{ll} +0 & \textrm{isotope $i$ does not deplete by reaction $\mathsf{NREAD}_{r}$} \\ +1 & \textrm{isotope $i$ will deplete by reaction $\mathsf{NREAD}_{r}$} \\ +2 & \textrm{isotope $i$ does not deplete by reaction $\mathsf{NREAD}_{r}$ but yields energy production} \\ +3 & \textrm{isotope $i$ is fissile without fission yield. Valid only for $r$ such +that $\mathsf{NREAD}_{r}=${\tt NFTOT}} \\ +4 & \textrm{isotope $i$ is fissile with fission yield. Valid only for $r$ such +that $\mathsf{NREAD}_{r}=${\tt NFTOT}} \\ +5 & \textrm{isotope $i$ is a fission fragment. Valid only for $r$ such +that $\mathsf{NREAD}_{r}=${\tt NFTOT}} +\end{array} \right. +\end{displaymath} + +\noindent and $i(r)=0$ unless $4\le d(r)\le 5$. When $d(r)=4$, $i(r)$ represents the fissile +isotope index while for $d(r)=5$, $i(r)$ represents the fission fragment index. +The fractional yield for the production of the fission fragment $i(r')$ from the +fissile isotope $i(r)$ is stored in matrix $Y_{i(r)\to i(r')}$. +The contents of the vector $R_{r,i}^{\rm d}$ is the energy in MeV emitted per +decay or reaction. + +\vskip 0.2cm + +The contents of the variables $K_{s,i}^{\rm p}$ is used to identify explicitly the parent isotope +which can generate the current isotope $i$. The maximum number of parent reaction for this +depletion chain is $M_{\mathrm{S}}$. $K_{s,i}^{\rm p}$ contains two different types of information, +namely $r(s)$ and $i(s)$ which are defined as follows: + +\begin{equation} + r(s)=K_{s,i}^{\rm p}\bmod 100 \ \ \ \ {\rm and} \ \ \ \ i(s)={{K_{s,i}^{\rm p}}\over{100}} +\end{equation} + +\noindent where $r(s)=0$ indicates that the list of parent isotopes is complete while $r(s)>0$ +refers to the reaction type $\mathsf{NREAD}_{r(s)}$ and can take the following values: + +\begin{displaymath} +r(s) = \left\{ +\begin{array}{ll} +1 & \textrm{isotope $i$ produced by radioactive decay}\\ +2 & \textrm{isotope $i$ produced by fission (this contribution is kept apart from record} \\ + & \textrm{{\tt 'FISSIONYIELD'})} \\ +3 & \textrm{isotope $i$ produced by neutron capture} \\ +\ge 4 & \textrm{isotope $i$ produced by $\mathsf{NREAD}_{r(s)}$ reaction} +\end{array} \right. +\end{displaymath} + +In the case where $r(s)>0$, $i(s)$ represents the isotope index associated +with the parent isotope and $R_{s,i}^{\rm p}$ represents the branching +ratio in fraction for the production of isotope $\mathsf{NISOD}_{i}$ from a neutron +reaction with the parent isotope $\mathsf{NISOD}_{i(s)}$. + +\goodbreak + +\subsection{State vector content for the {\sc shiba} self-shielding sub-directory}\label{sect:ssshibastate} + +The dimensioning parameters for the self-shielding sub-directory, which are stored in the state vector +$\mathcal{S}^{s}$, represent: + +\begin{itemize} +\item The first group for which self-shielding takes place $G_{\mathrm{min}}=\mathcal{S}^{s}_{1}$ + By default $G_{\mathrm{min}}=N_{g,f}+1$ + +\item The last group for which self-shielding takes place $G_{\mathrm{max}}=\mathcal{S}^{s}_{2}$ + By default $G_{\mathrm{max}}=N_{g,e}$ + +\item The maximum number of iterations in the self-shielding calculation $M_{r}=\mathcal{S}^{s}_{3}$ + +\item Enabling flag for the Livolant-Jeanpierre normalization $I_{\mathrm{lj}}=\mathcal{S}^{s}_{4}$ + +\item Enabling flag for the use of Goldstein-Cohen parameters $I_{\mathrm{gc}}=\mathcal{S}^{s}_{5}$ + +\item The transport correction option used in self-shielding $I_{\mathrm{tc}}=\mathcal{S}^{s}_{6}$ +\begin{displaymath} +I_{\mathrm{tc}} = \left\{ +\begin{array}{ll} +0 & \textrm{no transport correction applied in self-shielding calculation} \\ +1 & \textrm{use transport corrected cross section in self-shielding calculation} +\end{array} \right. +\end{displaymath} + +\item Type of self-shielding model $I_{\mathrm{level}}=\mathcal{S}^{s}_{7}$ +\begin{displaymath} +I_{\mathrm{level}} = \left\{ +\begin{array}{ll} +0 & \textrm{Stamm'ler model without distributed self-shielding effects} \\ +1 & \textrm{Stamm'ler model with the Nordheim (PIC) distributed self-shielding model} \\ +2 & \textrm{Stamm'ler model with both Nordheim (PIC) distributed self-shielding model} \\ + & \textrm{and Riemann integration method.} +\end{array} \right. +\end{displaymath} + +\item The option to indicate whether a specific flux solver or collision probability matrices +are used to perform the self-shielding calculation $I_{\mathrm{flux}}=\mathcal{S}^{s}_{8}$ +(see \moc{PIJ} and \moc{ARM} keyword in \moc{SHI:} operator input option) +\begin{displaymath} +I_{\mathrm{flux}} = \left\{ +\begin{array}{rl} + 1 & \textrm{use a specific flux solver (the \moc{ARM} keyword was selected)} \\ + 2 & \textrm{use collision probability matrices (the \moc{PIJ} keyword was selected)} +\end{array} \right. +\end{displaymath} + +\end{itemize} + +\subsection{State vector content for the subgroup self-shielding sub-directory}\label{sect:sssubgroupstate} + +The dimensioning parameters for the self-shielding sub-directory, which are stored in the state vector +$\mathcal{S}^{s}$, represent: + +\begin{itemize} +\item The first group for which self-shielding takes place $G_{\mathrm{min}}=\mathcal{S}^{s}_{1}$ + By default $G_{\mathrm{min}}=N_{g,f}+1$ + +\item The last group for which self-shielding takes place $G_{\mathrm{max}}=\mathcal{S}^{s}_{2}$ + By default $G_{\mathrm{max}}=N_{g,e}$ + +\item SPH enabling flag $I_{\mathrm{sph}}=\mathcal{S}^{s}_{3}$ + +\begin{displaymath} +I_{\mathrm{sph}} = \left\{ +\begin{array}{ll} +0 & \textrm{skip the multigroup equivalence procedure} \\ +1 & \textrm{perform a multigroup equivalence procedure (SPH procedure or} \\ + & \textrm{Livolant-Jeanpierre equivalence)} +\end{array} \right. +\end{displaymath} + +\item The transport correction option used in self-shielding $I_{\mathrm{tc}}=\mathcal{S}^{s}_{4}$ +\begin{displaymath} +I_{\mathrm{tc}} = \left\{ +\begin{array}{ll} +0 & \textrm{no transport correction applied in self-shielding calculation} \\ +1 & \textrm{use transport corrected cross section in self-shielding calculation} +\end{array} \right. +\end{displaymath} + +\item The number of iterations in the self-shielding calculation $M_{r}=\mathcal{S}^{s}_{5}$ + +\item The option to indicate whether a specific flux solver or collision probability matrices +are used to perform the self-shielding calculation $I_{\mathrm{flux}}=\mathcal{S}^{s}_{6}$ +(see \moc{PIJ} and \moc{ARM} keyword in \moc{USS:} operator input option) +\begin{displaymath} +I_{\mathrm{flux}} = \left\{ +\begin{array}{rl} + 1 & \textrm{use a specific flux solver (the \moc{ARM} keyword was selected)} \\ + 2 & \textrm{use collision probability matrices (the \moc{PIJ} keyword was selected)} +\end{array} \right. +\end{displaymath} + +\item The $\gamma$ factor enabling flag $I_{\mathrm{\gamma}}=\mathcal{S}^{s}_{7}$. These factors +are used to represent the moderator absorption effect in the Sanchez--Coste self-shielding method. +\begin{displaymath} +I_{\mathrm{\gamma}} = \left\{ +\begin{array}{ll} +0 & \textrm{the $\gamma$ factors are set to 1.0} \\ +1 & \textrm{the $\gamma$ factors are computed} +\end{array} \right. +\end{displaymath} + +\item The simplified self-shielding enabling flag $I_{\mathrm{calc}}=\mathcal{S}^{s}_{8}$ +\begin{displaymath} +I_{\mathrm{calc}} = \left\{ +\begin{array}{ll} +0 & \textrm{perform a delailed self-shielding calculation} \\ +1 & \textrm{perform a simplified self-shielding calculation using data recovered from the} \\ + & {\tt -DATA-CALC-} \textrm{ directory} +\end{array} \right. +\end{displaymath} + +\item The flag for ignoring the activation of the mutual resonance shielding model $I_{\mathrm{noco}}=\mathcal{S}^{s}_{9}$ +\begin{displaymath} +I_{\mathrm{noco}} = \left\{ +\begin{array}{ll} +0 & \textrm{follow the directives set by {\tt LIB}} \\ +1 & \textrm{ignore the directives set by {\tt LIB}} +\end{array} \right. +\end{displaymath} + +\item Maximum number of fixed point iterations for the ST scattering source convergence $I_{\mathrm{max}}=\mathcal{S}^{s}_{10}$ + +\item Type of elastic slowing-down kernel in Autosecol $I_{\mathrm{ialt}}=\mathcal{S}^{s}_{11}$ +\begin{displaymath} +I_{\mathrm{ialt}} = \left\{ +\begin{array}{ll} +0 & \textrm{use exact elastic kernel} \\ +1 & \textrm{use an approximate kernel for the resonant isotopes} +\end{array} \right. +\end{displaymath} + +\item Maximum storage size for the slowing-down kernel values in Autosecol $I_{\mathrm{tra}}=\mathcal{S}^{s}_{12}$ + +\item Normalization flag for the collision probabilities $I_{\mathrm{norm}}=\mathcal{S}^{s}_{13}$ +\begin{displaymath} +I_{\mathrm{norm}} = \left\{ +\begin{array}{ll} +0 & \textrm{no normalization} \\ +1 & \textrm{remove any remaining leakage from collision probabilities} +\end{array} \right. +\end{displaymath} + +\item Seed integer used by the random number generator $I_{\mathrm{seed}}=\mathcal{S}^{s}_{14}$. + +\end{itemize} + +\clearpage + +\subsection{The {\sc shiba} self-shielding sub-directory \dir{selfshield} in +\dir{microlib}}\label{sect:shibadirselfshield} + +\begin{DescriptionEnregistrement}{Main records and sub-directories in \dir{selfshield}}{7.5cm} +\IntEnr + {STATE-VECTOR}{$40$} + {Vector describing the various parameters associated with this data structure $\mathcal{S}^{s}_{i}$, + as defined in \Sect{ssshibastate}.} +\RealEnr + {EPS-SHIBA\blank{3}}{$1$}{1} + {Value of the relative convergence criterion for the self-shielding iterations in {\tt SHI:}. } +\end{DescriptionEnregistrement} + +\subsection{The subgroup self-shielding sub-directory \dir{uss-selfshield} in +\dir{microlib}}\label{sect:subgroupdirselfshield} + +\begin{DescriptionEnregistrement}{Main records and sub-directories in \dir{uss-selfshield}}{7.5cm} +\IntEnr + {STATE-VECTOR}{$40$} + {Vector describing the various parameters associated with this data structure $\mathcal{S}^{s}_{i}$, + as defined in \Sect{sssubgroupstate}.} +\OptDirEnr + {-DATA-CALC-\blank{1}}{$I_{\mathrm{calc}} = 1$} + {Name of directory containing the data required by a simplified self-shielding + calculation. This type of calculation allows the definition of a single + self-shielded isotope in several resonant mixtures.} +\DirVar + {\listedir{isodir}} + {List of sub-directories that contain isotopic subgroup information collected by the {\tt USS:} module.} +\end{DescriptionEnregistrement} + +The list of directory \listedir{isodir} named $\mathsf{ISODIR}$ will be composed according to +\begin{quote} +\verb|WRITE(|$\mathsf{ISODIR}$,\verb|'(1HC,I5,1H/,I5)')| $iso$,$nbiso$ +\end{quote} +\noindent where $iso$ is the isotope index and $nbiso$ is the total number of isotopes. \listedir{isodir} is defined in Table~\ref{table:isodir}. + +\begin{DescriptionEnregistrement}{Main records and sub-directories in \listedir{isodir}}{7.5cm}\label{table:isodir} +\DirVar + {\listedir{cordir}} + {List of sub-directories that contain correlated isotopic subgroup information collected by the {\tt USS:} module.} +\end{DescriptionEnregistrement} + +The list of directory \listedir{cordir} named $\mathsf{CORDIR}$ will be composed according to +\begin{quote} +\verb|WRITE(|$\mathsf{CORDIR}$,\verb|'(3HCOR,I4,1H/,I4)')| $ires$,$nires$ +\end{quote} +\noindent where $ires$ is the correlated isotope index and $nires$ is the total number of correlated isotopes. \listedir{cordir} is defined +in Table~\ref{table:cordir}. + +\begin{DescriptionEnregistrement}{Main records and sub-directories in \listedir{cordir}}{7.5cm}\label{table:cordir} +\DirlEnr + {NWT0-PT\blank{5}}{$G$} + {List of real arrays. Each component of this list contains subgroup flux information in correlated fuel regions, as computed by {\tt USS:}. + Each real array has dimension $N_{\rm nbnrs}\times K_g$, where $N_{\rm nbnrs}$ is the number of correlated fuel regions and $K_g$ is the + number of base points in energy group $g$.} +\OptDirlEnr + {ASSEMB-PHYS\blank{1}}{$N_{\rm asm}$}{$I_{\mathrm{calc}} = 1$} + {List of {\sc assemb-phys} directories. Each component of this list contains subgroup assembly information for the subgroup method with + physical probability tables. The specification of this directory is given in Sect.~\ref{sect:asminfodhdirgroup} or~\ref{sect:asminfodirgroup} + depending if a double-heterogeneity is present or not. A double-heterogeneity is present if $\mathcal{S}^{t}_{40}=1$ + in the {\sc tracking} object.} +\OptDirlEnr + {ASSEMB-RIBON}{$N_{\rm asm}$}{$I_{\mathrm{calc}} = 3,4$} + {List of {\sc assemb-ribon} directories. Each component of this list contains subgroup assembly information for the subgroup projection + or Ribon extended method. The specification of this directory is given in Sect.~\ref{sect:asminfodhdirgroup} or~\ref{sect:asminfodirgroup} + depending if a double-heterogeneity is present or not. A double-heterogeneity is present if $\mathcal{S}^{t}_{40}=1$ + in the {\sc tracking} object.} +\OptDirlEnr + {ASSEMB-RSE\blank{2}}{$N_{\rm asm}$}{$I_{\mathrm{calc}} = 6$} + {List of {\sc assemb-rse} directories. Each component of this list contains subgroup assembly information for the resonance spectrum + expansion method. The specification of this directory is given in Sect.~\ref{sect:asminfodhdirgroup} or~\ref{sect:asminfodirgroup} + depending if a double-heterogeneity is present or not. A double-heterogeneity is present if $\mathcal{S}^{t}_{40}=1$ + in the {\sc tracking} object.} +\end{DescriptionEnregistrement} + +\goodbreak + +\subsection{Contents of an \dir{isotope} directory}\label{sect:isotopedir} + +Each isotope directory always contains a cross section identifier record {\tt SCAT-SAVED\blank{2}} +which must be used to verify if a given cross section type has +been saved for this isotope. + +\begin{DescriptionEnregistrement}{Isotopic cross section identifier records}{7.5cm} +\label{tabl:tabiso1} +\OptCharEnr + {ALIAS\blank{7}}{$*12$}{$\mathcal{M} \ge 0$} + {Alias character*12 name of a microlib isotope. This record is not provided in {\sc draglib} objects.} +\IntEnr + {SCAT-SAVED\blank{2}}{$L$} + {Vector $\kappa^{\rm scat}_{k}$ to identify the various type of + Legendre-dependent cross sections saved for this isotope} +\RealEnr + {AWR\blank{9}}{$1$}{nau} + {Ratio of the isotope mass divided by the neutron mass} +\OptDirEnr + {PT-TABLE\blank{4}}{$I_{\rm proc}\ge 1$} + {Sub-directory containing probability table information, following the specification given in \Sect{pt-table}. + $I_{\rm proc}$ is defined in \Sect{microlibdir}. This sub-directory is present in the microlib builded by the {\tt LIB:} module.} +\end{DescriptionEnregistrement} + +Delayed neutron data can be present for some fissile isotopes on the \dir{isotope} directory. If $N_{\rm +del}\ge 1$ precursor groups are used, the following information is available: + +\begin{DescriptionEnregistrement}{Delayed neutron reaction +records}{7.5cm} +\label{tabl:tabiso2} +\OptRealVar + {\{nusid\}}{$G$}{$N_{del}\ge 1$}{b} + {$\nu\sigma_{{\rm f},\ell}^{{\rm D},g}$: The product of $\sigma_{\rm f}^{g}$, the fission cross section with + $\nu_{\ell}^{{\rm D},g}$, the averaged number of fission--emitted delayed + neutron produced in the precursor group $\ell$.} +\OptRealVar + {\{chid\}}{$G$}{$N_{del}\ge 1$}{1} + {$\chi^{{\rm D},g}_\ell$: Delayed fission spectrum, normalized to one, for the delayed fission + neutrons in precursor group $\ell$.} +\OptRealEnr + {LAMBDA-D\blank{4}}{$N_{\rm del}$}{$N_{\rm del}\ge 1$}{s$^{-1}$} + {$\lambda^{\rm D}_\ell$: Decay constant associated with the precursor group $\ell$. We must have + $0 <\lambda^{\rm D}_\ell<\lambda^{\rm D}_{\ell+1}$.} +\end{DescriptionEnregistrement} + +The delayed component of the fission yields in each precursor group $\ell$ is given as +$\nu_\ell^{{\rm D},g}$. The quantities $\pi^{{\rm D},g}$ and $\nu_\ell^{{\rm D},g} \ \sigma_{\rm f}^g$ are defined as +$$\pi^{{\rm D},g}={\nu^{{\rm D},g} \ \sigma_{\rm f}^g \over + \left( \nu^g \sigma_{\rm f}^g \right)^{\rm ss}} \ \ .$$ + +\noindent and + +$$\nu_\ell^{{\rm D},g} \ \sigma_{\rm f}^g=\omega_\ell \ \pi^{{\rm D},g} \ + \left( \nu^g \sigma_{\rm f}^g \right)^{\rm ss}$$ + +\noindent where the superscript ${\rm ss}$ indicates steady-state values. The +delayed neutron records {\sl \{nusid\}} and {\sl \{chid\}} will be +composed, using the following FORTRAN instructions, as $\mathsf{NUSIGD}$ and $\mathsf{CHID}$: + \begin{displaymath} + \mathtt{WRITE(}\mathsf{NUSIGD}\mathtt{,'(A6,I2.2)')} \ \mathtt{'NUSIGF'},ell + \end{displaymath} + \begin{displaymath} + \mathtt{WRITE(}\mathsf{CHID}\mathtt{,'(A3,I2.2)')} \ \mathtt{'CHI'},ell + \end{displaymath} +for $1\leq ell \leq N_{\rm del}$. For example, in the case where two group cross sections are considered +($N_{\rm del}=2$), the following records would be generated: + +\begin{DescriptionEnregistrement}{Example of delayed--neutron records in +\dir{isotope}}{8.0cm} +\OptRealEnr + {NUSIGF01\blank{4}}{$G$}{$N_{\rm del}\ge 1$}{b} + {$\nu\sigma_{{\rm f},1}^{{\rm D},g}$: The product of $\sigma_{\rm f}^{g}$, the fission cross section with + $\nu_1^{{\rm D},g}$, the averaged number of fission--emitted delayed + neutron produced in the precursor group 1.} +\OptRealEnr + {NUSIGF02\blank{4}}{$G$}{$N_{\rm del}\ge 2$}{b} + {$\nu\sigma_{{\rm f},2}^{{\rm D},g}$: The product of $\sigma_{\rm f}^{g}$, the fission cross section with + $\nu_2^{{\rm D},g}$, the averaged number of fission--emitted delayed + neutron produced in the precursor group 2.} +\OptRealEnr + {CHI01\blank{7}}{$G$}{$N_{\rm del}\ge 1$}{1} + {$\chi^{{\rm D},g}_1$: Delayed fission spectrum, + normalized to one, for the delayed fission neutrons in + precursor group 1.} +\OptRealEnr + {CHI02\blank{7}}{$G$}{$N_{\rm del}\ge 2$}{1} + {$\chi^{{\rm D},g}_2$: Delayed fission spectrum, + normalized to one, for the delayed fission neutrons in + precursor group 2.} +\end{DescriptionEnregistrement} + +\vskip 0.2cm + +In cases where the /isotope/ directory is produced by the edition module, some +depletion-related information may be available in this directory, in order to facilitate +subsequent data processing. This information is described in +Table~\ref{tabl:tabiso3}. + +\begin{DescriptionEnregistrement}{Depletion-related information}{7.5cm} +\label{tabl:tabiso3} +\OptRealEnr + {MEVG\blank{8}}{$1$}{$N_d \ge 1$}{MeV} + {Energy in MeV produced by radiative capture. $N_d$ is defined in \Sect{microlibdir}.} +\OptRealEnr + {MEVF\blank{8}}{$1$}{$N_d \ge 1$}{MeV} + {Energy in MeV produced by fission.} +\OptRealEnr + {DECAY\blank{7}}{$1$}{$N_d \ge 1$}{10$^{-8}$ s$^{-1}$} + {Radioactive decay constant} +\OptRealEnr + {YIELD\blank{7}}{$G+1$}{$N_d \ge 1$}{1} + {Fission fragment yield per energy group. The first value is the average yield + over all the energy spectrum. This record is given only for fission fragments.} +\OptIntEnr + {PIFI\blank{8}}{$N_{\rm dfi}$}{$N_{\rm dfi} \ge 1$} + {Position in {\tt ISOTOPESUSED} of the mother fissile isotopes. This record is + given only for fission fragments.} +\OptRealEnr + {PYIELD\blank{6}}{$N_{\rm dfi}$}{$N_{\rm dfi} \ge 1$}{1} + {Fission product yield per fissile isotope. This record is given only for + fission fragments.} +\end{DescriptionEnregistrement} + +\vskip 0.2cm + +We will first consider the more usual case where constant vector reactions are +stored on the isotopic directory. A typical example of the microscopic cross +section directory may be: + +\begin{DescriptionEnregistrement}{Example of isotopic vector reaction records}{7.0cm} +\label{tabl:tabiso4} +\RealEnr + {NTOT0\blank{7}}{$G$}{b} + {The $\phi$--weighted multigroup total cross section $\sigma_0^{g}$} +\RealEnr + {TRANC\blank{7}}{$G$}{b} + {The multigroup transport correction $\sigma_{tc}^{g}$} +\RealEnr + {NUSIGF\blank{6}}{$G$}{b} + {The product of $\sigma_{f}^{g}$, the multigroup fission cross section with + $\nu^{g}$, the steady-state number of neutron produced per fission, + $\nu\sigma_{f}^{{\rm ss},g}$} +\RealEnr + {NFTOT\blank{7}}{$G$}{b} + {The multigroup fission cross section $\sigma_{f}^{g}$} +\OptRealEnr + {CHI\blank{9}}{$G$}{$G_{\rm chi}=0$}{} + {The multigroup energy spectrum of the neutron emitted by fission $\chi^{g}$} +\OptRealEnr + {CHI--01\blank{5}}{$G$}{$G_{\rm chi}\ge 1$}{} + {The first energy-dependent multigroup energy spectrum of the neutron emitted by fission $\chi^{g,1}$} +\OptRealEnr + {CHI--02\blank{5}}{$G$}{$G_{\rm chi}\ge 2$}{} + {The second energy-dependent multigroup energy spectrum of the neutron emitted by fission $\chi^{g,2}$} +\OptRealEnr + {CHI--03\blank{5}}{$G$}{$G_{\rm chi}\ge 3$}{} + {The third energy-dependent multigroup energy spectrum of the neutron emitted by fission $\chi^{g,3}$} +\OptRealEnr + {CHI--04\blank{5}}{$G$}{$G_{\rm chi}\ge 4$}{} + {The fourth energy-dependent multigroup energy spectrum of the neutron emitted by fission $\chi^{g,4}$} +\RealEnr + {NG\blank{10}}{$G$}{b} + {The multigroup neutron capture cross section $\sigma_{c}^{g}$} +\RealEnr + {H-FACTOR\blank{4}}{$G$}{eV b} + {Energy production coefficients $H^{g}$ (product of each microscopic cross section + times the energy emitted by this reaction).} +\OptRealEnr + {C-FACTOR\blank{4}}{$G$}{*}{electron b} + {Charge deposition coefficients $C^{g}$ (product of each microscopic cross section + times the charge deposed by this reaction). Information provided if {\tt PARTICLE}$=${\tt B}, {\tt C} or {\tt P}.} +\RealEnr + {N2N\blank{9}}{$G$}{b} + {The multigroup cross section + $\sigma_{(n,2n)}^{g}$ for the reaction + $^{A}_{Z}X+n \to ^{A-1}_{Z}X+2n$} +\RealEnr + {N3N\blank{9}}{$G$}{b} + {The multigroup cross section + $\sigma_{(n,3n)}^{g}$ for the reaction + $^{A}_{Z}X+n \to ^{A-2}_{Z}X+3n$} +\RealEnr + {N4N\blank{9}}{$G$}{b} + {The multigroup cross section + $\sigma_{(n,4n)}^{g}$ for the reaction + $^{A}_{Z}X+n \to ^{A-3}_{Z}X+4n$} +\RealEnr + {NP\blank{10}}{$G$}{b} + {The multigroup cross section + $\sigma_{(n,p)}^{g}$ for the reaction + $^{A}_{Z}X+n \to ^{A}_{Z-1}X+p$} +\RealEnr + {NA\blank{10}}{$G$}{b} + {The multigroup cross section + $\sigma_{(n,\alpha)}^{g}$ for the reaction + $^{A}_{Z}X+n \to ^{A-3}_{Z-2}X+\alpha$ } +\RealEnr + {NGOLD\blank{7}}{$G$}{} + {The multigroup Goldstein-Cohen parameters as recovered from {\tt GIR} array in main \dir{microlib} directory + $\lambda^{g}$} +\RealEnr + {NWT0\blank{8}}{$G$}{s$^{-1}$cm$^{-2}$} + {The multigroup neutron flux spectrum $\phi_{w}^{g}$} +\RealEnr + {STRD\blank{8}}{$G$}{b} + {The multigroup transport cross section + homogenized over all directions + $\sigma_{\rm strd}^{g}$} +\RealEnr + {STRD-X\blank{6}}{$G$}{b} + {The $x-$directed multigroup transport cross + section $\sigma_{{\rm strd},x}^{g}$} +\RealEnr + {STRD-Y\blank{6}}{$G$}{b} + {The $y-$directed multigroup transport cross + section $\sigma_{{\rm strd},y}^{g}$} +\RealEnr + {STRD-Z\blank{6}}{$G$}{b} + {The $z-$directed multigroup transport cross + section $\sigma_{{\rm strd},z}^{g}$} +\RealEnr + {OVERV\blank{7}}{$G$}{cm$^{-1}$s} + {The average of the inverse neutron velocity \hbox{$<1/v>_{m}^g$}} +\RealEnr + {NTOT1\blank{7}}{$G$}{b} + {The ${\cal J}$--weighted multigroup total cross section $\sigma_1^{g}$} +\RealEnr + {NWT1\blank{8}}{$G$}{s$^{-1}$cm$^{-2}$} + {The multigroup fundamental current spectrum ${\cal J}_{w}^{g}$} +\RealEnr + {NWAT0\blank{7}}{$G$}{1} + {The multigroup neutron adjoint flux spectrum $\phi_{w}^{*g}$} +\RealEnr + {NWAT1\blank{7}}{$G$}{1} + {The multigroup fundamental adjoint current spectrum ${\cal J}_{w}^{*g}$} +\end{DescriptionEnregistrement} + +\vskip 0.2cm + +We can also use this isotopic directory to store time dependent cross sections in the form of a power series expansion: +\begin{equation} + v_{k}^{g}(t)=\sum_{i=0}^{I} v_{k,i}^{g} t^{i} +\label{eq:TimeSerie} +\end{equation} +where the presence of these various terms is specified using $\kappa_{k}$. +Note that the last three characters of each of the records in Table~\ref{tabl:tabiso4} correspond to the extension $\mathsf{EXT}$=\verb*|' '| that is +associated with term $i=0$ in the power series expansion for the cross sections (see +\Eq{TimeSerie}). For $i=1, 2$, the extension takes successively the value $\mathsf{EXT}$=\verb*|'LIN'| and $\mathsf{EXT}$=\verb*|'QUA'|. +For example, if one considers the total cross section and assumes that $F_{i}(\kappa_{1})=1$ for $i=0,2$, then this implies the presence +of the following additional records in the \dir{isotope}: + +\begin{DescriptionEnregistrement}{Additional total cross section records for $I=2$}{6.0cm} +\RealEnr + {TOTAL\blank{4}LIN}{$G$}{d$^{-1}$b} + {array $v_{1,1}^{g}=\Delta\sigma^{g}$ containing the first order coefficients in the power series expansion for the multigroup total +cross section} +\RealEnr + {TOTAL\blank{4}QUA}{$G$}{d$^{-2}$b} + {array $v_{1,2}^{g}=\Delta^{2}\sigma^{g}$ containing the second order coefficients in the power series expansion for the multigroup +total cross section} +\end{DescriptionEnregistrement} + +\vskip 0.2cm + +The multigroup scattering cross section matrix, which gives the probability for a +neutron in group $h$ to appear in group $g$ after a collision with this isotope +is represented by the form: + \begin{displaymath} + \sigma_{s}^{h\to g}(\vec{\Omega}\to\vec{\Omega}') + =\sum_{\ell=0}^{L}{{2\ell+1}\over{4\pi}} P_{\ell}(\vec{\Omega}\cdot\vec{\Omega}') + \sigma_{\ell}^{h\to g} + =\sum_{\ell=0}^{L}\sum_{m=-\ell}^{\ell} + Y_{\ell}^{m}(\vec{\Omega})Y_{\ell}^{m}(\vec{\Omega}')\sigma_{\ell}^{h\to g} + \end{displaymath} +using a spherical harmonic series expansion to order $L-1$. Assuming these +spherical harmonic are orthonormalized, namely: + \begin{displaymath} + \int_{4\pi} d^{2}\Omega \ Y_{\ell}^{m}(\vec{\Omega}) Y_{l'}^{m'}(\vec{\Omega})= + \delta_{m,m'}\delta_{\ell,\ell'} + \end{displaymath} +we can define $\sigma_{\ell}^{h\to g}$ in terms of $\sigma_{s}^{h\to +g}(\vec{\Omega}\to\vec{\Omega}')$ using the following integral: + \begin{displaymath} + \sigma_{\ell}^{h\to g} + =\int_{4\pi}d^{2}\Omega \ \sigma_{s}^{h\to g}(\vec{\Omega}\to\vec{\Omega}') + P_{\ell}(\vec{\Omega}\cdot\vec{\Omega}') + \end{displaymath} +Note that this definition of $\sigma_{\ell}^{h\to g}$ is not unique and some authors +include the factor $2l+1$ directly in different angular moments of the +scattering cross section. + +\vskip 0.2cm + +Here instead of storing on these $G\times G$ +matrices $\sigma_{\ell}^{h\to g}$, a vector which contains a compress form for this +matrix will be considered. This choice is justified by the fact that the number +of energy groups which will lead to scattering in a specific group is generally +relatively small compared to the total number of groups in the library and that +these groups are clustered around the final energy group. +Here we will first define two different integer vectors $n_{\ell}^{g}$ and +$h_{\ell}^{g}$ for each order in the scattering cross section and for each final +energy group $g$ which will contain respectively the number of +successive initial energy groups for which the scattering cross section does +not vanish and the maximum energy group number for which scattering to the +final group $g$ does not vanishes. Accordingly, for a scattering cross section +of the form: + +\begin{center} +\begin{tabular}{c||cccc} +$\sigma_{0}^{h\to g}$ &$g=1$ & $g=2$ & $g=3$ & $g=4$ \\ \hline\hline +$h=1$ & $a_{1}$ & $a_{2}$ & 0 & 0 \\ +$h=2$ & 0 & $a_{3}$ & $a_{4}$ & $a_{5}$ \\ +$h=3$ & 0 & $a_{6}$ & $a_{7}$ & 0 \\ +$h=4$ & 0 & $a_{8}$ & 0 & $a_{9}$ \\ \hline\hline +$h_{0}^{g}$ & 1 & 4 & 3 & 4 \\ +$n_{0}^{g}$ & 1 & 4 & 2 & 3 \\ +\end{tabular} +\end{center} + +The compress scattering matrix will then contain the following information: + \begin{displaymath} + \sigma_{\ell,c}=\left(\sigma_{\ell}^{h^{1}\to 1},\sigma_{\ell}^{h^{1}-1\to 1}, + \ldots,\sigma_{\ell}^{h^{1}-n_{1}+1\to 1},\sigma_{\ell}^{h^{2}\to + 2},\ldots,\sigma_{\ell}^{h^{G}-n_{G}+1\to G}\right) + \end{displaymath} +which for the example above leads to + \begin{displaymath} + \sigma_{\ell,c}=\left(a_{1},a_{8},a_{6}, + a_{3},a_{2},a_{7},a_{4},a_{9},0,a_{5}\right) + \end{displaymath} +As a result $\sigma_{\ell}^{h\to g}$ can be +reconstructed using +\begin{displaymath} +\sigma_{\ell}^{h\to g} = \left\{ +\begin{array}{lll} +0 & \textrm{if} & h > h_{\ell}^{g}\\ +0 & \textrm{if} & h < h_{\ell}^{g}-n_{\ell}^{g}+1\\ +\sigma_{\ell,c}^{k} & \textrm{otherwise} & k=\sum_{h=1}^{g-1} n_{\ell}^{h} + +h_{\ell}^{g}-h+1 +\end{array} \right. +\end{displaymath} + +Finally, we will also save the total scattering cross section vector of order +$\ell$ which is defined as + \begin{displaymath} + \sigma_{\ell,s}^{h}=\sum_{g=1}^{G} \sigma_{\ell}^{h\to g} + \end{displaymath} +In the case where only the order $\ell=0$ moment of scattering cross section is non +vanishing (isotropic scattering) the following records can be found on the +isotopic directory. + +\begin{DescriptionEnregistrement}{Optional scattering records}{7.0cm} +\label{tabl:tabiso5} +\RealEnr + {SIGS00\blank{6}}{$G$}{b} + {The isotropic component ($\ell=0$) of the multigroup total scattering cross + section + $\sigma_{0,s}^{g}$} +\IntEnr + {IJJS00\blank{6}}{$G$} + {Highest energy group number for which + the isotropic component of the scattering cross section to group $g$ does not + vanish, $h_{0}^{g}$} +\IntEnr + {NJJS00\blank{6}}{$G$} + {Number of energy groups for which + the isotropic component of the scattering cross section to group $g$ does not + vanish, $n_{0}^{g}$} +\RealEnr + {SCAT00\blank{6}}{$\sum_{g=1}^{G} n_{0}^{g}$}{b} + {Compressed isotropic component of the scattering matrix + $\sigma_{0,c}^{k}$} +\OptDirVar + {\listedir{subiso}}{$I_{\rm part}\ge 1$} + {Set of sub-directories containing scattering information towards a companion particle. \listedir{subiso} + is the name of the companion particle (set to {\tt N}, {\tt G}, {\tt B}, {\tt C} or {\tt P}). This information + is used to construct coupled sets of cross sections.} +\end{DescriptionEnregistrement} + +If the scattering cross section is +expanded to order $L>1$ in Legendre polynomials, additional set of scattering +records similar to those described above will be presentin the cross section directory. +The first four characters and last 6 characters in the names of +these records will again be identical to those described above while character 5 +and 6 will differ from level to level. For example, the order +$\ell=5$ compressed scattering matrix will be identified by +\texttt{SCAT05\blank{6}} while for order +$\ell=50$ we will use \texttt{SCAT50\blank{6}}. + +\vskip 0.2cm + +The {\tt STRD} cross sections are normalized in such a way to permit the +calculation of a diffusion coefficient using the following formula: + +\begin{equation} +D^g={\displaystyle 1\over\displaystyle 3 \ \sum_i N_i \ \sigma_{{\rm strd},i}^g} +\end{equation} + +\noindent where $N_i$ is the isotopic density of isotope $i$ and $\sigma_{{\rm strd},i}^g$ +is the {\tt STRD} cross section of isotope $i$ in energy group $g$. The sum is +performed over {\sl all} isotopes present in the mixture. The {\tt STRD} cross +sections for isotope $i$ are defined as + +\begin{eqnarray} +\sigma_{{\rm strd},i}^g&=&{1\over (\mu^g)^2} \ {\left<\phi \right>_g \over 3 +\left<(\Sigma_1-\Sigma_{\rm s1}){\cal J}\right>_g} \ (\sigma_{1,i}^g-\sigma_{{\rm +s1},i}^g) \ \ +\ {\rm if \ a \ streaming \ model \ is \ used} \\ +&=&{1\over (\mu^g)^2} \ {\left<\phi \right>_g^2 \over 3 \left< D \phi \right>_g +\left<(\Sigma_0-\Sigma_{\rm s1})\phi\right>_g} \ (\sigma_{0,i}^g-\sigma_{{\rm +s1},i}^g) \ \ +\ {\rm if \ no \ streaming \ model \ used} +\end{eqnarray} +\noindent where + +\begin{description} +\item [$\phi^g$] fundamental flux +\item [${\cal J}^g$] fundamental current +\item [$\mu^g$] SPH equivalence factor +\item [$\Sigma_0^g$] $\phi$--weighted macroscopic total cross section of the +mixture +\item [$\Sigma_1^g$] ${\cal J}$--weighted macroscopic total cross section of the +mixture +\item [$\Sigma_{\rm s1}^g$] macroscopic $P_1$ scattering cross section of the +mixture (${\cal J}$--weighted +if a streaming model is used; $\phi$--weighted if no streaming model used) +\item [$D^g$] diffusion coefficient +\item [$\sigma_{0,i}^g$] $\phi$--weighted microscopic total cross section for +isotope $i$ +\item [$\sigma_{1,i}^g$] ${\cal J}$--weighted microscopic total cross section for +isotope $i$ +\item [$\sigma_{{\rm s1}.i}^g$] microscopic $P_1$ scattering cross section for +isotope $i$ (${\cal J}$--weighted +if a streaming model is used; $\phi$--weighted if no streaming model used) +\end{description} + +\vskip 0.2cm + +On the other hand the so-called directional cross +section {\tt STRD\blank{1}X}, {\tt STRD\blank{1}Y} +and {\tt STRD\blank{1}Z} are obtained in such a way that + +\begin{equation} +D_k^g={\displaystyle 1\over\displaystyle 3 \ \sum_i N_i \ \sigma_{{\rm strd},k,i}^g} +\ ; \ \ \ k=x,\ y \ {\rm or} \ z \ \ \ . +\end{equation} + +\vskip 0.2cm + +For example, for an isotope with only total and scattering cross sections, we will find the +following records on the cross section directory. + +\begin{DescriptionEnregistrement}{Example of cross section records}{7.5cm} +\RealEnr + {NTOT0\blank{7}}{$G$}{b} + {The multigroup total cross section $\sigma^{g}$} +\RealEnr + {SIGS00\blank{6}}{$G$}{b} + {The isotropic component ($\ell=1$)of the multigroup total scattering cross + section + $\sigma_{0,s}^{g}$} +\IntEnr + {IJJS00\blank{6}}{$G$} + {Highest energy group number for which + the isotropic component of the scattering cross section to group $g$ does not + vanishes, $h_{0}^{g}$} +\IntEnr + {IJJS00\blank{3}}{$G$} + {Highest energy group number for which the first order perturbation in + the isotropic component of the scattering cross section to group $g$ does not + vanishes, $h_{0,1}^{g}$} +\IntEnr + {NJJS00\blank{6}}{$G$} + {Number of energy groups for which + the isotropic component of the scattering cross section to group $g$ does not + vanishes, $n_{0}^{g}$} +\RealEnr + {SCAT00\blank{6}}{$\sum_{g=1}^{G} n_{0}^{g}$}{b} + {Compressed isotropic component of the scattering matrix + $\sigma_{0,c}^{k}$} +\RealEnr + {SIGS01\blank{6}}{$G$}{b} + {The linearly anisotropic component ($\ell=1$) + of the multigroup total scattering cross section + $\sigma_{1,s}^{g}$} +\IntEnr + {IJJS01\blank{6}}{$G$} + {Highest energy group number for which + the linearly anisotropic component of the scattering cross section + to group $g$ does not vanishes, + $h_{1}^{g}$} +\IntEnr + {NJJS01\blank{6}}{$G$} + {Number of energy groups for which + the linearly anisotropic component of the scattering cross section + to group $g$ does not vanishes, + $n_{1}^{g}$} +\RealEnr + {SCAT01\blank{6}}{$\sum_{g=1}^{G} n_{1}^{g}$}{b} + {Compressed linearly anisotropic component of the scattering matrix + $\sigma_{1,c}^{k}$} +\end{DescriptionEnregistrement} + +Note that most of these cross sections are not required to perform a cell +calculation. In fact, in a typical transport calculation, only +$\sigma^{g}$, $\sigma_{tc}^{g}$, $\nu\sigma_{f}^{g}$, $\chi^{g}$ and the +isotropic and linearly anisotropic scattering matrix are +used. For burnup calculations, depending on the depletion chain prescribed, +the following cross sections may be required: +$\sigma_{f}^{g}$, $\sigma_{c}^{g}$, $\sigma_{(n,2n)}^{g}$, $\sigma_{(n,3n)}^{g}$, +$\sigma_{(n,4n)}^{g}$, $\sigma_{(n,p)}^{g}$, $\sigma_{(n,\alpha)}^{g}$. +Finally, when editing isotopic cross sections, all the cross sections types in +the library will be processed. + +\vskip 0.15cm + +A final note on the use of the transport correction and the homogenized and +directional transport cross section. In DARGON, the transport correction cross +section is used to correct the total and isotropic scattering cross +section using the relations +\begin{eqnarray*} +\sigma_{c}^{g} &=& \sigma^{g} -\sigma_{tc}^{g}\\ +\sigma_{c,0}^{g\to g}&=& \sigma_{0}^{g\to g}-\sigma_{tc}^{g} +\end{eqnarray*} + +\goodbreak + +\subsubsection{The probability table directory {\tt PT-TABLE} in \dir{isotope}}\label{sect:pt-table} + +Physical probability tables ($I_{\rm proc}=1$) are obtained from a least-square fit of the +self-shielded cross sections against dilution. Mathematical probability tables ($I_{\rm proc}\ge 3$) are obtained from +Autolib data using the CALENDF formalism. +Resonance spectrum expansion (RSE) information ($I_{\rm proc}=6$) is obtained from Autolib data using a singular value decomposition (SVD) of the +form $\shadowA=\shadowU \shadowW \shadowV^\top$ where +\begin{description} +\item[$\shadowA$:] snapshot flux matrix of size $N_{{\rm ufg},g}\times N_{\rm dil}$ recovered from the Draglib or Apollo2 file, +\item[$\shadowU$:] first orthogonal SVD matrix of size $N_{{\rm ufg},g}\times K_g$, +\item[$\shadowW$:] singular-value diagonal matrix of size $K_g\times K_g$, +\item[$\shadowV$:] second orthogonal SVD matrix of size $N_{\rm dil}\times K_g$ +\end{description} +\noindent where $N_{{\rm ufg},g}$ is the number of ultra-fine groups in coarse group $g$, $N_{\rm dil}$ is the number of snapshot ultra-fine group +flux distributions in coarse group $g$ (corresponding to the number of dilutions) and $K_g$ is the SVD rank in coarse group $g$. + +\begin{DescriptionEnregistrement}{Probability tables or RSE tables in \dir{isotope}}{7.5cm} +\OptDirlEnr + {GROUP-PT\blank{4}}{$G$}{$I_{\rm proc}\ne 6$} + {List of energy-group sub-directories. Each component of the list is a directory containing + the probability-table information associated with a specific group. See table~\ref{table:pt}.} +\OptDirlEnr + {GROUP-RSE\blank{3}}{$G$}{$I_{\rm proc}= 6$} + {List of energy-group sub-directories. Each component of the list is a directory containing + the resonance spectrum expansion information associated with a specific coarse group. See table~\ref{table:rse}.} +\OptDirVar + {\listedir{isotope2}}{$I_{\rm proc}= 6$} + {Set of sub-directories containing subgroup projection for {\sl isotope2} cross section information with + respect of {\sl isotope} base points for the RSE method. Subgroup projection of scattering cross sections is present for all + resonant isotopes, even for {\sl isotope2} $\equiv$ {\sl isotope}. Subgroup projection of total cross sections is + present only for {\sl isotope2} $\neq$ {\sl isotope}. See table~\ref{table:spmrse}.} +\IntEnr + {NOR\blank{9}}{$G$} + {Order $K_g$ of the probability table or of the resonance spectrum expansion tables in each energy group $g$. + If $I_{\rm proc}= 6$, the RSE rank $K_g \le N_{\rm dil}$ where $N_{\rm dil}$ is the number of dilutions.} +\IntEnr + {NDEL\blank{8}}{$1$} + {Number of delayed neutron precursor groups for this resonant isotope.} +\OptRealEnr + {SVD-EPS\blank{5}}{$1$}{$I_{\rm proc}= 6$}{~} + {Rank accuracy of the SVD.} +\end{DescriptionEnregistrement} + +\vskip -0.4cm + +\begin{DescriptionEnregistrement}{Group-dependent non-RSE directories in \dir{isotope}}{7.5cm}\label{table:pt} +\RealEnr + {PROB-TABLE\blank{2}}{$12,N_{\rm part}$}{~} + {Probability tables. $N_{\rm part}$ is the total number of reactions + represented by probability tables. 12 is the maximum allowed order of a + probability table.} +\OptRealEnr + {SIGQT-SIGS\blank{2}}{$K_g$}{$I_{\rm proc}=4$}{b} + {Probability table in secondary slowing-down cross section.} +\OptRealEnr + {SIGQT-SLOW\blank{2}}{$K_g,K_g$}{$I_{\rm proc}=4$}{b} + {Slowing-down correlated weight matrix.} +\OptRealVar + {\listedir{isotope2}}{$K_g,L_g$}{*}{1} + {Set of records, each containing the correlated weights + between the current total xs and the total xs of {\sl isotope2}. $L_g$ is the + order of the probability table for {\sl isotope2}. (*) This data is optional + and is provided only if $I_{\rm proc}\ge 3$ and if the mutual self-shielding + effect is to be taken into account.} +\IntEnr + {ISM-LIMITS\blank{2}}{$2,L$} + {Minimum (index 1) and maximum (index 2) secondary group for each Legendre + order of the scattering matrices} +\end{DescriptionEnregistrement} + +\vskip -0.4cm + +\begin{DescriptionEnregistrement}{Group-dependent RSE directories in \dir{isotope}}{7.5cm}\label{table:rse} +\DbleEnr + {RSE-TABLE\blank{3}}{$N_{\rm part}, K_g$}{~} + {Resonance spectrum expansion (RSE) table $\shadowP_g$. $N_{\rm part}$ is the total number of flux and reactions + represented by RSE tables and $K_g$ is the RSE rank (equal to the number of base points). The table is obtained as + $$\shadowP_g=\shadowX_g \shadowV_g \shadowW_g^{-1} \shadowT_g$$ + \noindent where $\shadowX_g$ is a $N_{\rm part} \times N_{\rm dil}$ double precision matrix containing dilution-dependent + homogeneous flux and effective cross sections in group $g$, as recovered from the Draglib or Apollo2 file.} +\DbleEnr + {SIGT\_V\blank{6}}{$K_g$}{~} + {Double precision vector corresponding to the base points in microscopic total cross sections. These + values are the eigenvalues of the linear transformation. These base points are located in the following diagonal matrix: + $$\left[{\rm diag}(\sigma^*_{k,g})\right]=\shadowT_g^\top \shadowU_g^\top \left[{\rm diag}(\sigma^{*(m)}_g)\right] \shadowU_g \shadowT_g$$ + \noindent where $\left[{\rm diag}(\sigma^{*(m)}_g)\right]$ is a $N_{{\rm ufg},g} \times N_{{\rm ufg},g}$ diagonal matrix containing UFG microscopic total cross section values.} +\DbleEnr + {WEIGHT\_V\blank{4}}{$K_g$}{~} + {Double precision weight vector $$\bff(\omega)_g={1\over{u_g-u_{g-1}}} \bff(1)^\top \shadowU_g \shadowT_g$$ + \noindent where $\bff(1)^\top$ is a row vector of ones used to sum over indices $m$ of the UFG mesh.} +\DbleEnr + {GAMMA\_V\blank{5}}{$K_g$}{~} + {Double precision gamma vector $$\bff(\gamma)_g=\bff(1)^\top \left[{\rm diag}(\Delta u_g^{(m)})\right] \shadowU_g \shadowT_g .$$} +\IntEnr + {ISM-LIMITS\blank{2}}{$2,L$} + {Minimum (index 1) and maximum (index 2) secondary group for each Legendre + order of the scattering matrices} +\end{DescriptionEnregistrement} + +\vskip -0.4cm + +\begin{DescriptionEnregistrement}{Subgroup projection of {\sl isotope2} for the RSE method}{7.5cm}\label{table:spmrse} +\OptDirlEnr + {SIGT\_M\blank{6}}{$G$}{*} + {Set of matrices representing the {\sl subgroup projection} of {\sl isotope2} microscopic total cross sections on {\sl isotope} base + points in group $g$. Each matrix of size $K_g \times K_g$ is defined as + $$\left[\sigma^{*b/a}_{k,\ell,g}\right]=\shadowT_g^\top \shadowU_g^\top \left[{\rm diag}(\sigma^{*b(m)}_g)\right] \shadowU_g \shadowT_g$$ + \noindent where $\left[{\rm diag}(\sigma^{*b(m)}_g)\right]$ is a $N_{{\rm ufg},g} \times N_{{\rm ufg},g}$ diagonal matrix containing UFG microscopic total cross section values for {\sl isotope2}. + (*) This data is optional and is provided only if {\sl isotope2} is resonant and if {\sl isotope2} $\neq$ {\sl isotope}.} +\DirlEnr + {SCAT\_M\blank{6}}{$n_{\rm pos}$} + {Set of matrices representing the {\sl subgroup projection} of {\sl isotope2} microscopic scattering cross sections on {\sl isotope} base + points in group $g$. Each matrix of size $K_g \times K_h$ is defined as + $$\left[\sigma^{*b/a}_{k,\ell,g\leftarrow h}\right]=\shadowT_g^\top \shadowU_g^\top \left[\sigma^{*b(m\leftarrow n)}_{{\rm s},g \leftarrow h}\right] \shadowU_h \shadowT_h$$ + \noindent where $\left[\sigma^{*b(m\leftarrow n)}_{{\rm s},g \leftarrow h}\right]$ is a $N_{{\rm ufg},g} \times N_{{\rm ufg},h}$ matrix containing UFG microscopic scattering cross section values for {\sl isotope2}. Here, $n_{\rm pos}$ is the total number of scattering double precision matrices taking into account self-scattering and out-of-group scattering. Record {\sl isotope} (i. e., {\sl isotope2} $\equiv$ {\sl isotope} and $a\equiv b$) is always present.} +\IntEnr + {NJJS00\blank{6}}{$G$} + {Bandwidth $n_{{\rm njj},g}$ of records {\tt SCAT\_M} for {\sl isotope2}. $n_{\rm pos}=\sum_g n_{{\rm njj},g}$.} +\end{DescriptionEnregistrement} +\eject |