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diff --git a/doc/IGE351/SectDmacrolib.tex b/doc/IGE351/SectDmacrolib.tex new file mode 100644 index 0000000..f00fb19 --- /dev/null +++ b/doc/IGE351/SectDmacrolib.tex @@ -0,0 +1,980 @@ +\section{Contents of a \dir{macrolib} directory}\label{sect:macrolibdir} + +A \dir{macrolib} directory always contains the set of macroscopic multigroup cross +sections associated with a set of mixtures. The structure of this directory, +is quite different to that associated with an \dir{isotope} directory (see +\Sect{isotopedir}). First, it is multi-level, namely, it contains sub-directories. +Moreover instead of having one directory per mixture which contains the +associated multigroup cross section, one will have one directory component per group containing +multi-mixture information. Finally its contents will vary depending on the operator +which was used to create it. Here for convenience we will define the variable $\mathcal{M}$ to +identify the creation operator: +\begin{displaymath} +\mathcal{M} = \left\{ +\begin{array}{ll} +0 & \textrm{if the directory is created by the \moc{MAC:} operator}\\ +1 & \textrm{if the directory is created by the \moc{LIB:} or \moc{EVO:} operator}\\ +2 & \textrm{if the directory is created by the \moc{EDI:} operator}\\ +3 & \textrm{if the directory is created by the \moc{OUT:} operator or by an interpolation operator} +\end{array} \right. +\end{displaymath} + +In the case where the \moc{LIB:} or \moc{EDI:} operator is used to create this directory, +it is embedded as a subdirectory in a \dir{microlib} or an \dir{edition} directory. +For the other cases, it appears on the root level of the \dds{macrolib} data structure. + +\subsection{State vector content for the \dir{macrolib} data structure}\label{sect:macrolibstate} + +The dimensioning parameters for the \dir{macrolib} data structure, which are stored in +the state vector $\mathcal{S}^{M}$, represent: + +\begin{itemize} +\item The number of energy groups ${G}=\mathcal{S}^{M}_{1}$ +\item The number of mixtures $N_{m}=\mathcal{S}^{M}_{2}$ +\item The order for the scattering anisotropy $L=\mathcal{S}^{M}_{3}$ +($L=1$ is an isotropic collision; $L=2$ is a linearly anisotropic collision, +etc.) +\item The maximum number of fissile isotopes in a mixture $N_{f}=\mathcal{S}^{M}_{4}$ +\item The number of additional $\phi$--weighted editing cross sections $N_{e}=\mathcal{S}^{M}_{5}$ +\item The transport correction option $I_{tr}=\mathcal{S}^{M}_{6}$ +\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}\\ +4 & \textrm{use a leakage correction based on {\tt NTOT1} data.} +\end{array} \right. +\end{displaymath} +\item The number of precursor groups for delayed neutron $N_{d}=\mathcal{S}^{M}_{7}$ +\item The number of physical albedo $N_{A}=\mathcal{S}^{M}_{8}$ +\item The type of leakage $I_{l}=\mathcal{S}^{M}_{9}$ +\begin{displaymath} +I_{l} = \left\{ +\begin{array}{ll} +0 & \textrm{no diffusion/leakage coefficient available}\\ +1 & \textrm{isotropic diffusion/leakage coefficient available}\\ +2 & \textrm{anisotropic diffusion/leakage coefficient available.} +\end{array} \right. +\end{displaymath} +\item The maximum Legendre order of the weighting functions $I_{w}=\mathcal{S}^{M}_{10}$ +\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 with}\\ +& \textrm{order $\ge 1$ and compute both $\phi$-- and +${\cal J}$--weighted total cross sections.} +\end{array} \right. +\end{displaymath} +\item The number of delta cross section sets $I_{\rm step}=\mathcal{S}^{M}_{11}$ used +for generalized perturbation theory (GPT) or kinetics calculations: +\begin{displaymath} +I_{\rm step} = \left\{ +\begin{array}{ll} +0 & \textrm{no delta cross section sets}\\ +>0 & \textrm{number of delta cross section sets.} +\end{array} \right. +\end{displaymath} +\item Discontinuity factor flag $I_{\rm df}=\mathcal{S}^{M}_{12}$: +\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 Adjoint macrolib flag $I_{\rm adj}=\mathcal{S}^{M}_{13}$: +\begin{displaymath} +I_{\rm adj} = \left\{ +\begin{array}{ll} +0 & \textrm{direct macrolib}\\ +1 & \textrm{adjoint macrolib.} +\end{array} \right. +\end{displaymath} +\item SPH-information $I_{\rm sph}=\mathcal{S}^{M}_{14}$: +\begin{displaymath} +I_{\rm sph} = \left\{ +\begin{array}{ll} +0 & \textrm{no SPH information available}\\ +1 & \textrm{SPH information is available.} +\end{array} \right. +\end{displaymath} +\item Type of weighting in {\tt EDI:} module $I_{\rm pro}=\mathcal{S}^{M}_{15}$: +\begin{displaymath} +I_{\rm pro} = \left\{ +\begin{array}{ll} +0 & \textrm{use a flux weighting}\\ +1 & \textrm{use an adjoint--direct (a.k.a., product) flux weighting. Only available if $\mathcal{M}\ge 2$} +\end{array} \right. +\end{displaymath} +\item Group form factor index $I_{\rm gff}=\mathcal{S}^{M}_{16}$: +\begin{displaymath} +I_{\rm gff} = \left\{ +\begin{array}{ll} +0 & \textrm{no group form factor information}\\ +>0 & \textrm{number of form factors per mixture and per energy group (see \Sect{macroGFF}).} +\end{array} \right. +\end{displaymath} +\item Number of companion particles in coupled sets $I_{\rm part}=\mathcal{S}^{M}_{17}$: +\begin{displaymath} +I_{\rm part} = \left\{ +\begin{array}{ll} +0 & \textrm{the macrolib doesn't include coupled sets}\\ +>0 & \textrm{number of companion particles.} +\end{array} \right. +\end{displaymath} +\end{itemize} + +\subsection{The main \dir{macrolib} directory}\label{sect:macrolibdirmain} + +The following records and sub-directories will be found on the first level of a \dir{macrolib} +directory: + +\begin{DescriptionEnregistrement}{Main records and sub-directories in \dir{macrolib}}{8.0cm} +\CharEnr + {SIGNATURE\blank{3}}{$*12$} + {Signature of the \dir{macrolib} data structure ($\mathsf{SIGNA}=${\tt L\_MACROLIB\blank{2}}).} +\IntEnr + {STATE-VECTOR}{$40$} + {Vector describing the various parameters associated with this data structure + $\mathcal{S}^{M}_{i}$, as defined in \Sect{macrolibstate}.} +\OptCharEnr + {ADDXSNAME-P0}{$(N_{e})*8$}{$N_{e} \ge 1$} + {Names of the additional $\phi$--weighted editing cross sections ($\mathsf{ADDXS}_k$). + These names should not appear in Tables~\ref{tabl:tabnonlegendre} and \ref{tabl:tablegendre}.} +\OptIntEnr + {FISSIONINDEX}{$N_{m},N_{f}$}{$N_{f} \ge 1,\mathcal{M}=1$} + {For each mixture $i$ contains the index of each fissile isotope $j$. The index is + pointing to a component of record \moc{ISOTOPESUSED} or \moc{ISOTOPERNAME} + of /microlib/.} +\OptRealEnr + {ENERGY\blank{6}}{$G+1$}{$\mathcal{M}\ge 1$}{eV} + {Energy group limits $E_{g}$} +\OptRealEnr + {DELTAU\blank{6}}{$G$}{$\mathcal{M}\ge 1$}{} + {Lethargy width of each group $U_{g}$} +\OptRealEnr + {ALBEDO\blank{6}}{$N_{A}, G$}{$N_{A}> 0$}{} + {Multigroup and surface ordered physical albedos. The dimension is R$(N_{A},G,G)$ in case where matrix albedos are used.} +\OptRealEnr + {VOLUME\blank{6}}{$N_{m}$}{$\mathcal{M}\ge 2$}{cm$^{3}$~~} + {Volume of region containing each mixture $V_{m}$} +\OptRealEnr + {MIXTURESDENS}{$N_{m}$}{$\mathcal{M}=1$}{g/cm$^{3}$~~} + {Volumetric mass density of each mixture $\rho_{m}$} +\OptRealEnr + {FLUXDISAFACT}{$G$}{$\mathcal{M}=2$}{} + {Ratio of the flux in the fuel to the flux in the cell $F_{g}$ after homogenization} +\OptRealEnr + {LAMBDA-D\blank{4}}{$N_{d},N_{f}$}{$N_{d}\ge 1$}{s$^{-1}$} + {Radioactive decay constants of each delayed neutron precursor group, for each + fissile isotope.} +\OptRealEnr + {BETA-D\blank{6}}{$N_{d},N_{f}$}{$N_{d}\ge 1$}{} + {Delayed-neutron fraction for each delayed neutron precursor group, for each + fissile isotope.} +\OptRealEnr + {K-EFFECTIVE\blank{1}}{$1$}{$N_{f} \ge 1$}{} + {Effective multiplication constant $k_{\mathrm{eff}}$} +\OptRealEnr + {K-INFINITY\blank{2}}{$1$}{$N_{f} \ge 1$}{} + {Infinite multiplication constant $k_{\infty}$} +\OptRealEnr + {B2\blank{2}B1HOM\blank{3}}{$1$}{$I_{l} \ge 1$}{cm$^{-2}$~~} + {Homogeneous Buckling $B_{\mathrm{hom}}$} +\OptRealEnr + {B2\blank{2}HETE\blank{4}}{$3$}{$I_{l}=2$}{cm$^{-2}$} + {Directional Buckling $B_{j}$} +\OptRealEnr + {TIMESTAMP\blank{3}}{$3$}{$\mathcal{M}=1$}{} + {A vector $T_{j}$ containing three elements. The first element $T_{1}=t$ is the time in days, the + second element $T_{2}=B$ is the burnup in MW day T$^{-1}$ and the third element $T_{3}=w$ is the + irradiation in Kb$^{-1}$} +\DirlEnr + {GROUP\blank{7}}{$G$} + {List of energy-group sub-directories. Each component of the list is a directory containing + the reference macroscopic cross-section information associated with a specific secondary group.} +\OptCharEnr + {PARTICLE\blank{4}}{$*1$}{$I_{\rm part}\ge 1$} + {Character name of the particle associated to the macrolib. 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).} +\OptDirlVar + {\listedir{grpdir}}{$G$}{$I_{\rm part}\ge 1$} + {List of energy-group sub-directories. Each component of the list is a directory containing + scattering transition cross-section information associated with a specific secondary group. + The directory \listedir{grpdir} name is the concatenation of {\tt GROUP-} with the character*6 + name of the companion particle responsible for scattering transitions.} +\OptDirlEnr + {STEP\blank{8}}{$I_{\rm step}$}{$I_{\rm step}\ge 1$} + {List of GPT or kinetics perturbation sub-directories. Each component of + this list contains a single + list of energy-group sub-directories following the \moc{GROUP} specification. + This \moc{GROUP} list contains variations or derivatives of the reference cross-section set.} +\OptDirEnr + {ADF\blank{9}}{$I_{\rm df} \ge 1$} + {ADF--related information as presented in \Sect{macroADF}.} +\OptDirEnr + {GFF\blank{9}}{$I_{\rm gff} \ge 1$} + {Group form factor information as presented in \Sect{macroGFF}.} +\OptDirEnr + {SPH\blank{9}}{$I_{\rm sph} = 1$} + {SPH--related input data as presented in \Sect{macroSPH}.} +\end{DescriptionEnregistrement} + +\subsection{The group sub-directory \moc{GROUP} in \dir{macrolib}}\label{sect:macrolibdirgroup} + +Each component of the list \moc{GROUP} is a directory containing cross-section information +corresponding to a single energy group. Inside each groupwise directory, the following +records associated with vectorial cross sections will be found: + +\begin{DescriptionEnregistrement}{Vectorial cross section records and directories in +\moc{GROUP}}{7.0cm} +\label{tabl:tabnonlegendre} +\RealEnr + {NTOT0\blank{7}}{$N_{m}$}{cm$^{-1}$} + {The $\phi$--weighted total cross section $\Sigma_{0,m}^{g}$} +\OptRealEnr + {NTOT1\blank{7}}{$N_{m}$}{$\mathcal{M}=2; \ I_{w}\ge 1$}{cm$^{-1}$} + {The ${\cal J}$--weighted total cross section $\Sigma_{1,m}^{g}$} +\OptRealEnr + {TRANC\blank{7}}{$N_{m}$}{$I_{tr}=2$}{cm$^{-1}$} + {The transport correction $\Sigma_{tc,m}^{g}$} +\RealEnr + {FIXE\blank{8}}{$N_{m}$}{cm$^{-3}$s$^{-1}$} + {Fixed sources $S_{m}^{g}$.} +\OptRealEnr + {NUSIGF\blank{6}}{$N_{m},N_{f}$}{$N_{f}\ge 1$}{cm$^{-1}$} + {The product of $\Sigma_{f,m}^{g}$, the fission cross section with + $\nu_{m}^{{\rm ss},g}$, the steady-state number of neutron produced per fission, + $\nu\Sigma_{f,m}^{g}$} +\OptRealEnr + {CHI\blank{9}}{$N_{m},N_{f}$}{$N_{f}\ge 1$}{} + {The steady-state energy spectrum of the neutron emitted by fission $\chi_{m}^{{\rm ss},g}$} +\OptRealVar + {\{nusid\}}{$N_{m},N_{f}$}{$N_{d}\ge 1$}{cm$^{-1}$} + {The product of $\Sigma_{f,m}^{g}$, the fission cross section with + $\nu_{m,\ell}^{{\rm D},g}$, the averaged number of fission--emitted delayed + neutron produced in the precursor group $\ell$, + $\nu\Sigma_{f,m,\ell}^{{\rm D},g}$} +\OptRealVar + {\{chid\}}{$N_{m},N_{f}$}{$N_{d}\ge 1$}{} + {The energy spectrum of the fission--emitted delayed neutron + in the precursor group $\ell$, $\chi_{m,\ell}^{{\rm D},g}$} +\OptRealEnr + {FLUX-INTG\blank{3}}{$N_{m}$}{$\mathcal{M}\ge 2$}{cm s$^{-1}$} + {The volume-integrated flux $\Phi_{m}^{g}$} +\OptRealEnr + {FLUX-INTG-P1}{$N_{m}$}{$\mathcal{M}\ge 2; \ I_{w}\ge 1$}{cm s$^{-1}$} + {The volume-integrated fundamental current ${\cal J}_{m}^{g}$} +\OptRealEnr + {COURX-INTG\blank{2}}{$N_{m}$}{$\mathcal{M}\ge 2; \ I_{\rm intcur}=1$}{cm s$^{-1}$} + {The volume-integrated net current along the $X$-axis $J_{{\rm X},m}^{g}$. Only provided + with SN and MOC discretizations.} +\OptRealEnr + {COURY-INTG\blank{2}}{$N_{m}$}{$\mathcal{M}\ge 2; \ I_{\rm intcur}=1$}{cm s$^{-1}$} + {The volume-integrated net current along the $Y$-axis $J_{{\rm Y},m}^{g}$. Only provided + with SN and MOC 2D and 3D discretizations.} +\OptRealEnr + {COURZ-INTG\blank{2}}{$N_{m}$}{$\mathcal{M}\ge 2; \ I_{\rm intcur}=1$}{cm s$^{-1}$} + {The volume-integrated net current along the $Z$-axis $J_{{\rm Z},m}^{g}$ Only provided + with SN and MOC 3D discretizations.} +\OptRealEnr + {NWAT0\blank{7}}{$N_{m}$}{$I_{\rm pro}=1$}{1} + {The multigroup neutron adjoint flux spectrum $\phi_{m}^{*g}$} +\OptRealEnr + {NWAT1\blank{7}}{$N_{m}$}{$I_{w}\ge 1; \ I_{\rm pro}=1$}{1} + {The multigroup fundamental adjoint current spectrum ${\cal J}_{m}^{*g}$} +\RealEnr + {OVERV\blank{7}}{$N_{m}$}{cm$^{-1}$s} + {The average of the inverse neutron velocity \hbox{$<1/v>_{m}^g$}} +\OptRealEnr + {DIFF\blank{8}}{$N_{m}$}{$I_{l}=1$}{cm} + {The isotropic diffusion coefficient + $D_{m}^{g}$} +\OptRealEnr + {DIFFX\blank{7}}{$N_{m}$}{$I_{l}=2$}{cm} + {The $x$-directed diffusion coefficient + $D_{x,m}^{g}$} +\OptRealEnr + {DIFFY\blank{7}}{$N_{m}$}{$I_{l}=2$}{cm} + {The $y$-directed diffusion coefficient + $D_{y,m}^{g}$} +\OptRealEnr + {DIFFZ\blank{7}}{$N_{m}$}{$I_{l}=2$}{cm} + {The $z$-directed diffusion coefficient + $D_{z,m}^{g}$} +\OptRealEnr + {NSPH\blank{8}}{$N_{m}$}{$\mathcal{M}=2$}{1} + {SPH equivalence factors $\mu_{m}^{g}$. By default, these factors are set equal to 1.0. + Otherwise, all the cross sections, diffusion coefficients and integrated fluxes stored on the {\sc + macrolib} are SPH--corrected.} +\OptRealEnr + {H-FACTOR\blank{4}}{$N_{m}$}{$\mathcal{M}=2$}{eV cm$^{-1}$} + {Energy production coefficients $H_{m}^{g}$ (product of each macroscopic cross section + times the energy emitted by this reaction).} +\OptRealEnr + {ESTOPW\blank{6}}{$N_{m},2$}{*}{MeV cm$^{-1}$} + {Initial and final stopping power. Information provided if {\tt PARTICLE}$=${\tt B}, {\tt C} or {\tt P}.} +\OptRealEnr + {EMOMTR\blank{6}}{$N_{m}$}{*}{cm$^{-1}$} + {Restricted momentum transfer cross section. Information provided only if {\tt PARTICLE}$=${\tt B}, {\tt C} or {\tt P}.} +\OptRealEnr + {C-FACTOR\blank{4}}{$N_{m}$}{*}{electron cm$^{-1}$} + {Charge deposition cross section. Information provided if {\tt PARTICLE}$=${\tt B}, {\tt C} or {\tt P}.} +\OptRealVar + {\listedir{xsname}}{$N_{m}$}{$N_{e}\ge 1$}{cm$^{-1}$} + {Set of cross section records specified by $\mathsf{ADDXS}_{k}$} +\end{DescriptionEnregistrement} + +The set of delayed neutron records {\sl \{nusid\}} and {\sl \{chid\}} will be +composed, using the following FORTRAN instructions, as $\mathsf{NUSID}$ and $\mathsf{CHID}$, +respectively + \begin{displaymath} + \mathtt{WRITE(}\mathsf{NUSID}\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 $1\leq ell \leq N_d$. For example, in the case where two group cross sections are considered +($N_d=2$), the following records would be generated: + +\begin{DescriptionEnregistrement}{Example of delayed--neutron records in +\moc{GROUP}}{8.0cm} +\OptRealEnr + {NUSIGF01\blank{4}}{$N_{m},N_{f}$}{$N_{d}\ge 1$}{cm$^{-1}$} + {The product of $\Sigma_{f,m}^{g}$, the fission cross section with + $\nu_{m,1}^{{\rm D},g}$, the averaged number of fission--emitted delayed + neutron produced in the precursor group $\ell=1$, + $\nu\Sigma_{f,m,1}^{{\rm D},g}$} +\OptRealEnr + {CHI01\blank{7}}{$N_{m},N_{f}$}{$N_{d}\ge 1$}{} + {The energy spectrum of the fission--emitted delayed neutron + in the precursor group $\ell=1$, $\chi_{m,1}^{{\rm D},g}$} +\OptRealEnr + {NUSIGF02\blank{4}}{$N_{m},N_{f}$}{$N_{d}\ge 2$}{cm$^{-1}$~~} + {The product of $\Sigma_{f,m}^{g}$, the fission cross section with + $\nu_{m,2}^{{\rm D},g}$, the averaged number of fission--emitted delayed + neutron produced in the precursor group $\ell=2$, + $\nu\Sigma_{f,m,2}^{{\rm D},g}$} +\OptRealEnr + {CHI02\blank{7}}{$N_{m},N_{f}$}{$N_{d}\ge 2$}{} + {The energy spectrum of the fission--emitted delayed neutron + in the precursor group $\ell=2$, $\chi_{m,2}^{{\rm D},g}$} +\end{DescriptionEnregistrement} + +\vskip 0.2cm + +In the case where $N_{e}=3$ and +\begin{displaymath} +\mathsf{ADDXS}_{k} = \left\{ +\begin{array}{lll} +\mathtt{NG} & \textrm{for} & k=1\\ +\mathtt{N2N}& \textrm{for} & k=2\\ +\mathtt{NFTOT}& \textrm{for} & k=3 +\end{array} \right. +\end{displaymath} +the following reactions will be available in the data structure described +in Table~\ref{tabl:tabnonlegendre}: + +\begin{DescriptionEnregistrement}{Additional cross section records}{7.0cm} +\RealEnr + {NG\blank{10}}{$N_{m}$}{cm$^{-1}$} + {The neutron capture cross section $\Sigma_{{\rm c},m}^{g}$} +\RealEnr + {N2N\blank{9}}{$N_{m}$}{cm$^{-1}$} + {The cross section + $\Sigma_{{\rm (n,2n)},m}^{g}$ for the reaction + $^{A}_{Z}X+n \to ^{A-1}_{Z}X+2n$} +\RealEnr + {NFTOT\blank{7}}{$N_{m}$}{cm$^{-1}$} + {The neutron fission cross section $\Sigma_{{\rm f},m}^{g}$} +\end{DescriptionEnregistrement} + +The information associated with the multigroup scattering matrix, which gives the probability for a +neutron in group $h$ to appear in group $g$ after a collision with an isotope in mixture $m$ +is represented by the form: + \begin{displaymath} + \Sigma_{s,m}^{h\to g}(\vec{\Omega}\to\vec{\Omega}') + =\sum_{l=0}^{L}{{2l+1}\over{4\pi}} P_{l}(\vec{\Omega}\cdot\vec{\Omega}') + \Sigma_{l,m}^{h\to g} + =\sum_{l=0}^{L}\sum_{m=-l}^{l} + Y_{l}^{m}(\vec{\Omega})Y_{l}^{m}(\vec{\Omega}')\Sigma_{l,m}^{h\to g} + \end{displaymath} +using a series expansion to order $L$ in spherical harmonic. Assuming that the +spherical harmonic are orthonormalized, +we can define $\Sigma_{l,m}^{h\to g}$ in terms of $\Sigma_{s,m}^{h\to +g}(\vec{\Omega}\to\vec{\Omega}')$ using the following integral: + \begin{displaymath} + \Sigma_{l,m}^{h\to g} + =\int_{4\pi}d^{2}\Omega \ \Sigma_{s,m}^{h\to g}(\vec{\Omega}\to\vec{\Omega}') + P_{l}(\vec{\Omega}\cdot\vec{\Omega}') + \end{displaymath} +Note that this definition of $\Sigma_{l,m}^{h\to g}$ is not unique and some authors +include the factor $2l+1$ directly in the different angular moments of the +scattering cross section. + +\vskip 0.2cm + +Here instead of storing the $G\times M$ +matrix $\Sigma_{l,m}^{h\to g}$ associated with each final energy group $g$, a vector which +contains a compress form of the scattering matrix will be considered. +We will first define three integer vectors $n_{l,m}^{g}$, +$h_{l,m}^{g}$ and $p_{l,m}^{g}$ for order $l$ in the scattering cross section, +final energy group $g$ and mixture $m$. They will contain respectively the number of +initial energy groups $h$ for which the scattering cross section to group $g$ does not vanish, the +maximum energy group index for which scattering to the final group $g$ does not vanishes and the +position in the compressed scattering vector where the data associated with mixture $m$ for each +energy group $g$ can be found. Here $p_{l,m}^{g}$ is directly related to $n_{l,m}^{g}$ by + \begin{displaymath} + p_{l,m}^{g}=1+\sum_{k=1}^{m-1} n_{l,k}^{g} + \end{displaymath} + +\begin{figure}[htbp] +\begin{center} +\epsfxsize=8cm +\centerline{ \epsffile{scat.eps}} +\parbox{14cm}{\caption{Numbering of scattering elements in {\tt 'SCAT'} matrices.}\label{fig:scat}} +\end{center} +\end{figure} + +Now consider the following 4 groups isotropic scattering cross +section matrix associated with mixture 1 and 2 ($N_{m}=2$) respectively: + +\begin{center} +\begin{tabular}{c||cccc|cccc} + &\multicolumn{4}{l|}{Mixture $m=1$} & + \multicolumn{4}{l}{Mixture $m=2$} \\ +$\sigma_{0,m}^{h\to g}$ &$g=1$ & $g=2$ & $g=3$ & $g=4$ & + $g=1$ & $g=2$ & $g=3$ & $g=4$ \\ \hline\hline +$h=1$ & $a_{1}$ & $a_{2}$ & 0 & 0 & + $b_{1}$ & $b_{2}$ & 0 & 0 \\ +$h=2$ & 0 & $a_{3}$ & $a_{4}$ & $a_{5}$ & + $b_{3}$ & $b_{4}$ & $b_{5}$ & 0 \\ +$h=3$ & 0 & $a_{6}$ & $a_{7}$ & 0 & + 0 & $b_{6}$ & $b_{7}$ & 0 \\ +$h=4$ & 0 & $a_{8}$ & 0 & $a_{9}$ & + 0 & 0 & $b_{8}$ & $b_{9}$ \\ \hline\hline +$h_{0,m}^{g}$ & 1 & 4 & 3 & 4 & + 2 & 3 & 4 & 4 \\ +$n_{0,m}^{g}$ & 1 & 4 & 2 & 3 & + 2 & 3 & 3 & 1 \\ +$p_{0,m}^{g}$ & 1 & 1 & 1 & 1 & + 2 & 5 & 3 & 4 \\ +\end{tabular} +\end{center} + +\noindent +The compressed scattering matrix will then take the following form for each final group $g$: + +\begin{eqnarray*} +\Sigma_{0,k,c}^{1}&=&\left(a_{1},b_{3},b_{1}\right) \\ +\Sigma_{0,k,c}^{2}&=&\left(a_{8},a_{6},a_{3},a_{2},b_{6},b_{4},b_{2}\right) \\ +\Sigma_{0,k,c}^{3}&=&\left(a_{7},a_{4},b_{8},b_{7},b_{5}\right) \\ +\Sigma_{0,k,c}^{4}&=&\left(a_{9},0,a_{5},b_{9}\right) +\end{eqnarray*} +Finally, we will also save the total scattering cross section vector of order +$l$ which is defined as + \begin{displaymath} + \Sigma_{l,m,s}^{g}=\sum_{h=1}^{G} \Sigma_{l,m}^{g\to h} + \end{displaymath} +and the diagonal element of the scattering matrix: + \begin{displaymath} + \Sigma_{l,m,w}^{g}=\Sigma_{l,m}^{g\to g} + \end{displaymath} +In the case where only the order $l=0$ and $l=1$ moment of scattering cross section are non +vanishing (isotropic and linearly anisotropic scattering) the following records can be found on the +group directory. + +\begin{DescriptionEnregistrement}{Scattering cross section records in \moc{GROUP}}{7.0cm} +\label{tabl:tablegendre} +\RealEnr + {SIGS00\blank{6}}{$N_{m}$}{cm$^{-1}$} + {The isotropic component ($l=0$) of the total scattering cross + section + $\Sigma_{0,m,s}^{g}$} +\RealEnr + {SIGW00\blank{6}}{$N_{m}$}{cm$^{-1}$} + {The isotropic component ($l=0$) of the within group scattering cross + section + $\Sigma_{0,m,w}^{g}$} +\IntEnr + {IJJS00\blank{6}}{$N_{m}$} + {Highest energy group number for which + the isotropic component of the scattering cross section to group $g$ does not + vanish, $h_{0,m}^{g}$} +\IntEnr + {NJJS00\blank{6}}{$N_{m}$} + {Number of energy groups for which + the isotropic component of the scattering cross section to group $g$ does not + vanish, $n_{0,m}^{g}$} +\IntEnr + {IPOS00\blank{6}}{$N_{m}$} + {Location in the isotropic compressed scattering matrix where information associated with mixture + $m$ begins $p_{0,m}^{g}$} +\RealEnr + {SCAT00\blank{6}}{$\sum_{m=1}^{N_{m}} n_{0,m}^{g}$}{cm$^{-1}$} + {Compressed isotropic component of the scattering matrix + $\Sigma_{0,k,c}^{g}$} +\OptRealEnr + {SIGS01\blank{6}}{$N_{m}$}{$L\ge 1$}{cm$^{-1}$} + {The linearly anisotropic component of the total scattering cross + section + $\Sigma_{1,m,s}^{g}$} +\OptRealEnr + {SIGW01\blank{6}}{$N_{m}$}{$L\ge 1$}{cm$^{-1}$} + {The linearly anisotropic component of the within group scattering cross + section + $\Sigma_{1,m,w}^{g}$} +\OptIntEnr + {IJJS01\blank{6}}{$N_{m}$}{$L\ge 1$} + {Highest energy group number for which + the linearly anisotropic component of the scattering cross section to group $g$ does not + vanish, $h_{1,m}^{g}$} +\OptIntEnr + {NJJS01\blank{6}}{$N_{m}$}{$L\ge 1$} + {Number of energy groups for which + the linearly anisotropic component of the scattering cross section to group $g$ does not + vanish, $n_{1,m}^{g}$} +\OptIntEnr + {IPOS01\blank{6}}{$N_{m}$}{$L\ge 1$} + {Location in the linearly anisotropic compressed scattering matrix where information + associated with mixture $m$ begins $p_{1,m}^{g}$} +\OptRealEnr + {SCAT01\blank{6}}{$\sum_{m=1}^{N_{m}} n_{1,m}^{g}$}{$L\ge 1$}{cm$^{-1}$} + {Compressed linearly anisotropic component of the scattering matrix + $\Sigma_{1,k,c}^{g}$} +\end{DescriptionEnregistrement} + +\subsection{The \moc{/ADF/} sub-directory in \dir{macrolib}}\label{sect:macroADF} + +Sub-directory containing boundary-related edition information. This information can be boundary fluxes, discontinuity factors or +assembly discontinuity factors (ADF). Boundary fluxes can be used to compute discontinuity factors or to perform Selengut-type +normalization with the {\sl superhomog\'en\'eisation} (SPH) method. + +\begin{DescriptionEnregistrement}{Records in the \moc{/ADF/} sub-directory}{7.5cm} +\OptIntEnr + {NTYPE\blank{7}}{$1$}{$I_{\rm df} \ge 2$} + {Number of ADF-type boundary edits.} +\OptCharEnr + {HADF\blank{8}}{({\tt NTYPE})$*8$}{$I_{\rm df} \ge 2$} + {Name of each ADF-type boundary flux or discontinuity factor edit. Any name can be used, but some + names are standard. Standard names are: $=$ \moc{FD\_C}: + corner flux edition; $=$ \moc{FD\_B}: surface (assembly gap) flux edition; $=$ \moc{FD\_H}: + row flux edition (these are the first row of surrounding cells in the assembly).} +\OptRealEnr + {ALBS00\blank{6}}{$G,2$}{$I_{\rm df} = 1$}{} + {Multigroup boundary currents $J^{g}_{\rm out}$ and $J^{g}_{\rm in}$. These values correspond to surfaces where + a \moc{VOID} or \moc{ALBE} boundary condition is set in DRAGON.} +\OptRealEnr + {AVG\_FLUX\blank{5}}{$N_{m},G$}{$I_{\rm df} = 2$}{} + {Averaged fluxes in the complete assembly. Used as denominator to compute the ADF in an homogeneous assembly.} +\OptRealVar + {\listedir{type}}{$N_{m},G$}{$I_{\rm df} = 2,\, 3$}{} + {Averaged surfacic fluxes ($I_{\rm df} = 2$) or discontinuity factors ($I_{\rm df} = 3$) in a material mixture. Name {\sl type} is a component of + {\tt HADF} array.} +\OptRealVar + {\listedir{type}}{$N_{m},G,G$}{$I_{\rm df} = 4$}{} + {Matrix discontinuity factors in a material mixture. Name {\sl type} is a component of {\tt HADF} array.} +\end{DescriptionEnregistrement} + +\subsection{The \moc{/GFF/} sub-directory in \dir{macrolib}}\label{sect:macroGFF} + +Sub-directory containing group form factor information. This information can be used to perform +{\sl fine power reconstruction} over a fuel assembly. + +\begin{DescriptionEnregistrement}{Records in the \moc{/GFF/} sub-directory}{7.5cm} +\DirEnr + {GFF-GEOM\blank{4}} + {Macro--geometry directory. This geometry corresponds to an unfolded fuel assembly and is compatible + for a discretization with TRIVAC. This directory follows the specification presented in \Sect{geometrydirmain}.} +\RealEnr + {VOLUME\blank{6}}{$N_{m},I_{\rm gff}$}{cm$^{3}$} + {Volumes of homogenized cells $V_{m}$} +\RealEnr + {NWT0\blank{8}}{$N_{m},I_{\rm gff},G$}{s$^{-1}$cm$^{-2}$} + {The multigroup neutron flux spectrum $\phi_{w}^{g}$} +\RealEnr + {H-FACTOR\blank{4}}{$N_{m},I_{\rm gff},G$}{eV cm$^{-1}$} + {Energy production coefficients $H_{m}^{g}$ (product of each macroscopic cross section + times the energy emitted by this reaction).} +\RealEnr + {NFTOT\blank{7}}{$N_{m},I_{\rm gff},G$}{cm$^{-1}$} + {The neutron fission cross section $\Sigma_{{\rm f},m}^{g}$} +\IntEnr + {FINF\_NUMBER\blank{1}}{$N_{\rm ifx}$} + {Array containing the $N_{\rm ifx}$ $ifx$ indices used by the user every time the multicompo were ``enriched" + with different options.} +\RealEnr + {\listedir{FINF}}{$N_{m},I_{\rm gff},G$}{s$^{-1}$cm$^{-2}$} + {The diffusion multigroup neutron flux spectrum in an infinite domain $\psi_{m,p}^{d,\infty}$. See + \moc{NAP:} module description in IGE344 user guide for details.} +\end{DescriptionEnregistrement} + +The set of diffusion multigroup neutron flux spectrum records \listedir{FINF} will be +composed, using the following FORTRAN instructions as $\mathsf{HVECT}$, + \begin{displaymath} + \mathtt{WRITE(}\mathsf{HVECT}\mathtt{,'(5HFINF\_,I3.3)')} \ \mathtt{'ifx'} + \end{displaymath} +where {\tt ifx} is a value chosen by the user (default value is 0). A different value can be chosen every time the multicompo +are ``enriched" with different options (homogeneous/heterogeneous, tracking options, etc.). + +\clearpage + +\subsection{The \moc{/SPH/} sub-directory in \dir{macrolib}}\label{sect:macroSPH} + +The first level of the macrolib directory may contains a {\sl superhomog\'en\'eisation} (SPH) sub-directory \moc{/SPH/} +containing input data: + +\begin{DescriptionEnregistrement}{Records in the \moc{/SPH/} sub-directory}{7.5cm} +\IntEnr + {STATE-VECTOR}{$40$} + {Vector describing the various parameters associated with this data structure $\mathcal{S}^{\rm sph}_{i}$.} +\OptCharEnr + {SPH\$TRK\blank{5}}{$*12$}{$\mathcal{S}^{\rm sph}_{1}\ge 2$} + {Name of the flux solution door.} +\OptRealEnr + {SPH-EPSILON\blank{1}}{$1$}{$\mathcal{S}^{\rm sph}_{1}\ge 2$}{1} + {Convergence criterion for stopping the SPH iterations.} +\end{DescriptionEnregistrement} + +The dimensioning parameters for this data structure, which are stored in the state vector +$\mathcal{S}^{\rm sph}$, represent values related to the last editing step: + +\begin{itemize} + +\item Type of SPH equivalence factors: + $I_{\rm type}=\mathcal{S}^{\rm sph}_{1}$ +\begin{displaymath} +I_{\rm type} = \left\{ +\begin{array}{ll} +0 & \textrm{no SPH correction;} \\ +1 & \textrm{the SPH factors are read from LCM;} \\ +2 & \textrm{homogeneous macro-calculation (non-iterative procedure or H\'ebert-Benoist} \\ + & \textrm{SPH-5 procedure);} \\ +3 & \textrm{any type of $P_{ij}$ macro-calculation;} \\ +4 & \textrm{any type of diffusion, $S_n$, $P_n$ or $SP_n$ macro-calculation.} +\end{array} \right. +\end{displaymath} + +\item Type of SPH equivalence normalization $I_{\rm norm}=\mathcal{S}^{\rm sph}_{2}$ +\begin{displaymath} +I_{\rm norm} = \left\{ +\begin{array}{ll} +<0 & \textrm{asymptotic normalization with respect to homoheneous mixture} -I_{\rm norm}; \\ +1 & \textrm{average flux normalization;} \\ +2 & \textrm{Selengut normalization using {\tt ALBS00} information;} \\ +3 & \textrm{Selengut normalization using {\tt FD\_B} boundary fluxes;} \\ +4 & \textrm{Generalized Selengut normalization (EDF-type);} \\ +5 & \textrm{Selengut normalization with surface leakage;} \\ +6 & \textrm{Selengut normalization with water gap normalization;} \\ +7 & \textrm{average flux normalization in fissile zones.} +\end{array} \right. +\end{displaymath} + +\item The maximum number of SPH iterations $\mathcal{S}^{\rm sph}_{3}$ + +\item The acceptable number of SPH iterations with an increase in convergence error before aborting $\mathcal{S}^{\rm sph}_{4}$ + +\item Flag for forcing the production of a macrolib or microlib at LHS $I_{\rm lhs} = \mathcal{S}^{\rm sph}_{5}$ +\begin{displaymath} +I_{\rm lhs} = \left\{ +\begin{array}{ll} +0 & \textrm{produce an object of the type of the RHS;} \\ +1 & \textrm{produce an edition object;} \\ +2 & \textrm{produce a microlib;} \\ +3 & \textrm{produce a macrolib.} +\end{array} \right. +\end{displaymath} + +\item Type of SPH factors $I_{\rm imc} = \mathcal{S}^{\rm sph}_{6}$ +\begin{displaymath} +I_{\rm imc} = \left\{ +\begin{array}{ll} +1 & \textrm{factors compatible with diffusion theory, $P_n$ and $SP_n$ equations} \\ +2 & \textrm{factors compatible with other types of transport-theory macro-calculations} \\ +3 & \textrm{factors compatible with $P_{ij}$ macro-calculations and Bell acceleration.} \\ +\end{array} \right. +\end{displaymath} + +\item The first group index where the equivalence process is applied $\mathcal{S}^{\rm sph}_{7}$ + +\item The maximum group index where the equivalence process is applied $\mathcal{S}^{\rm sph}_{8}$ + +\end{itemize} + +\subsection{Delayed neutron information} + +We will present space-time kinetics equations in the context of the diffusion +approximation (i.e. using the Fick law) and equations used in a lattice code +to produce condensed and homogenized information. These equations will be useful to understand the +information written in the {\sc macrolib} specification. Similar expressions can +be obtained in transport theory. Note that delayed neutron information +$\beta_\ell$ and $\Lambda$ can also be computed at the scale of the complete reactor +provided that bilinear direct--adjoint condensation and homogenization relations +are used. + +\vskip 0.2cm + +The continuous-energy space-time diffusion equation is written: + +\begin{eqnarray} +\nonumber {\partial\over \partial t}\left[ {1 \over v(E)} \ \phi(\vec r,E,t)\right] &=& +\sum_j \chi_j^{\rm pr}(E)\int_0^\infty dE' \ \nu_j^{\rm pr}(\vec r,E',t)\Sigma_{{\rm f},j}(\vec r,E',t) +\phi(\vec r,E',t)\\ +\nonumber &+&\sum_j\sum_\ell\chi_{\ell,j}^{\rm D}(E)\lambda_\ell c_{\ell,j}(\vec r,t) + \nabla \cdot D(\vec r,E,t) \nabla\phi(\vec r,E,t)\\ +&-& \Sigma(\vec r,E,t) \phi(\vec r,E,t) + \int_0^\infty dE' \ \Sigma_{\rm s0}(\vec r,E \leftarrow E',t) +\phi(\vec r,E',t) +\label{eq:eq1} +\end{eqnarray} + +\noindent together with the set of $N_d$ precursor equations: + +\begin{equation} +{\partial c_{\ell,j}(\vec r,t) \over \partial t}=\int_0^\infty dE \ \nu_{\ell,j}^{\rm D}(\vec r,E,t) +\Sigma_{{\rm f},j}(\vec r,E,t) \phi(\vec r,E,t)-\lambda_\ell c_{\ell,j}(\vec r,t) \ \ ; \ \ \ +\ell=1,N_d +\label{eq:eq2} +\end{equation} + +\noindent where +\begin{description} +\item [$\phi(\vec r,E,t)$=] neutron flux +\item [$\chi_j^{\rm pr}(E)$=] prompt neutron spectrum for a fission of isotope $j$ +\item [$\nu_j^{\rm pr}(\vec r,E,t)$=] number of prompt neutrons for a fission of isotope $j$ +\item [$\Sigma_{{\rm f},j}(\vec r,E,t)$=] macroscopic fission cross section for isotope $j$ +\item [$\chi_{\ell,j}^{\rm D}(E)$=] neutron spectra for delayed neutrons emitted by precursor group $\ell$ +due to a fission of isotope $j$ +\item [$\lambda_\ell$=] radioactive decay constant for precursor group $\ell$. This +constant is assumed to be independent of the fissionable isotope $j$. +\item [$c_{\ell,j}(\vec r,t)$=] concentration of the $\ell$--th precursor for a fission of isotope $j$ +\item [$D(\vec r,E,t)$=] diffusion coefficient +\item [$\Sigma(\vec r,E,t)$=] macroscopic total cross section +\item [$\Sigma_{\rm s0}(\vec r,E \leftarrow E',t)$=] macroscopic scattering cross section +\item [$\nu_{\ell,j}^{\rm D}(\vec r,E,t)$=] number of delayed neutrons in precursor group $\ell$ for a fission of isotope $j$. +\end{description} + +\vskip 0.2cm + +The neutron spectrum are normalized so that +\begin{equation} +\int_0^\infty dE \ \chi_j^{\rm ss}(E)=1 +\end{equation} + +\noindent and + +\begin{equation} +\int_0^\infty dE \ \chi_\ell^{\rm D}(E)=1 \ \ ; \ \ \ell=1,N_d \ \ \ . +\end{equation} + +\vskip 0.2cm + +After condensation over energy, Eqs.~(\ref{eq:eq1}) and~(\ref{eq:eq2}) are +written + +\begin{eqnarray} +\nonumber <1/v>^g{\partial\over \partial t}\phi^g(\vec r,t) &=& \sum_j +\chi_j^{{\rm pr},g} +\left[1-\sum_\ell\beta_{\ell,j}\right]\sum_h \nu\Sigma_{{\rm f},j}^h(\vec r,t) \phi^h(\vec r,t)\\ +\nonumber &+&\sum_j \sum_\ell\chi_{\ell,j}^{{\rm D},g}\lambda_\ell c_{\ell,j}(\vec r,t) + \nabla \cdot D^g(\vec r,t) +\nabla\phi^g(\vec r,t)\\ +&-& \Sigma^g(\vec r,t) \phi^g(\vec r,t) + +\sum_h \Sigma_{\rm s0}^{g \leftarrow h}(\vec r,t) +\phi^h(\vec r,t) +\label{eq:eq7} +\end{eqnarray} + +\noindent together with the set of $N_d$ precursor equations: + +\begin{equation} +{\partial c_{\ell,j}(\vec r,t) \over \partial t}=\beta_{\ell,j} \sum_h +\nu\Sigma_{{\rm f},j}^h(\vec r,t) \phi^h(\vec r,t)-\lambda_\ell c_{\ell,j}(\vec r,t) \ \ ; \ \ \ +\ell=1,N_d +\label{eq:eq8} +\end{equation} + +\noindent where +\begin{description} +\item [$\nu\Sigma_{{\rm f},j}^h(\vec r,t)$=] product of the number $\nu_j^{\rm ss}(\vec r,E)$ of secondary neutrons +(both prompt and delayed) for a fission of isotope $j$ times the macroscopic fission cross +section for a fission of isotope $j$. +\item [$\beta_{\ell,j}$=] delayed neutron fraction in precursor group $\ell$. +\end{description} + +\vskip 0.2cm + +The following condensation formulas have been used: + +\begin{equation} +\nu_j^{\rm ss}(\vec r,E)=\nu_j^{\rm pr}(\vec r,E)+\sum_\ell \nu_{\ell,j}^{\rm D}(\vec r,E) +\end{equation} + +\begin{equation} +\beta_{\ell,j}={\int\limits_0^\infty dE \ \nu_{\ell,j}^{\rm D}(\vec r,E)\Sigma_{{\rm f},j}(\vec r,E) +\phi(\vec r,E) \over \int\limits_0^\infty dE \ \nu_j^{\rm ss}(\vec r,E)\Sigma_{{\rm f},j}(\vec r,E) +\phi(\vec r,E)} = {\sum\limits_g \nu\Sigma_{{\rm f},\ell,j}^{{\rm D},g}(\vec r) \phi^g(\vec r) \over +\sum\limits_g \nu\Sigma_{{\rm f},j}^g(\vec r) \phi^g(\vec r)} +\end{equation} + +\begin{equation} +\left[1-\sum_\ell\beta_{\ell,j}\right]={\int\limits_0^\infty dE \ \nu_j^{\rm pr}(\vec r,E)\Sigma_{{\rm f},j}(\vec r,E) +\phi(\vec r,E) \over \int\limits_0^\infty dE \ \nu_j^{\rm ss}(\vec r,E)\Sigma_{{\rm f},j}(\vec r,E) +\phi(\vec r,E)} = {\sum\limits_g \nu\Sigma_{{\rm f},j}^{{\rm pr},g}(\vec r) +\phi^g(\vec r) \over \sum\limits_g \nu\Sigma_{{\rm f},j}^g(\vec r) \phi^g(\vec r)} +\end{equation} + +\begin{equation} +\phi^g(\vec r)=\int_{E_g}^{E_{g-1}} dE \ \phi(\vec r,E) +\end{equation} + +\begin{equation} +\chi_j^{{\rm pr},g}=\int_{E_g}^{E_{g-1}} dE \ \chi_j^{\rm pr}(E) +\end{equation} + +\begin{equation} +\chi_{\ell,j}^{{\rm D},g}=\int_{E_g}^{E_{g-1}} dE \ \chi_{\ell,j}^{\rm D}(E) \ \ ; \ \ \ +\ell=1,N_d +\end{equation} + +\begin{equation} +<1/v>^g={1 \over \phi^g(\vec r)} \int_{E_g}^{E_{g-1}} dE \ {\displaystyle 1 \over \displaystyle v(E)} \ \phi(\vec r,E) +\end{equation} + +\begin{equation} +\Sigma^g(\vec r)={1 \over \phi^g(\vec r)} \int_{E_g}^{E_{g-1}} dE \ \Sigma(\vec r,E) \ \phi(\vec r,E) +\end{equation} + +\begin{equation} +\Sigma_{\rm s0}^{g \leftarrow h}(\vec r)={1 \over \phi^h(\vec r)} \int_{E_g}^{E_{g-1}} dE \int_{E_h}^{E_{h-1}} dE' \ \Sigma_{\rm s0}(\vec r,E \leftarrow E') \ \phi(\vec r,E') +\end{equation} + +\begin{equation} +\nu\Sigma_{{\rm f},j}^g(\vec r)={1 \over \phi^g(\vec r)} \int_{E_g}^{E_{g-1}} dE \ \nu_j^{\rm ss}(\vec r,E) \ \Sigma_{{\rm f},j}(\vec r,E) \ \phi(\vec r,E) \ \ \ . +\end{equation} + +\noindent where the variable $t$ has been omitted in order to simplify +the notation. + +\vskip 0.2cm + +A steady-state fission spectrum (taking into account both prompt and delayed neutrons), for a fission of isotope $j$, is also required for solving the static neutron diffusion equation: + +\begin{equation} +\chi_j^{\rm ss}(E)=\left[1-\sum_\ell\beta_{\ell,j}\right] \chi_j^{\rm pr}(E)+\sum_\ell \beta_{\ell,j} \ \chi_{\ell,j}^{\rm D}(E) \ \ \ . +\end{equation} + +\vskip 0.2cm + +The group-integrated steady-state fission spectrum is therefore given as +\begin{equation} +\chi_j^{{\rm ss},g} = \left[1-\sum_\ell\beta_{\ell,j}\right] \chi_j^{{\rm pr},g}+\sum_\ell \beta_{\ell,j} \ \chi_{\ell,j}^{{\rm D},g} \ \ \ . +\end{equation} + +\vskip 0.2cm + +The space-time diffusion equation is generally solved by assuming a {\sl unique} averaged fissionable isotope. +In this case, the variable $N_f$ is set to 1 in the {\sc macrolib} specification +and the summations over $j$ disapears in Eqs.~(\ref{eq:eq7}) and~(\ref{eq:eq8}): + +\begin{eqnarray} +\nonumber <1/v>^g {\partial\over \partial t}\phi^g(\vec r,t) &=& \chi^{{\rm pr},g} +\left[1-\sum_\ell\beta_\ell\right]\sum_h \nu\Sigma_{\rm f}^h(\vec r,t) \phi^h(\vec r,t)\\ +\nonumber &+&\sum_\ell\chi_\ell^{{\rm D},g}\lambda_\ell c_\ell(\vec r,t) + \nabla \cdot D^g(\vec r,t) +\nabla\phi^g(\vec r,t)\\ +&-& \Sigma^g(\vec r,t) \phi^g(\vec r,t) + +\sum_h \Sigma_{\rm s0}^{g \leftarrow h}(\vec r,t) +\phi^h(\vec r,t) +\label{eq:eq9} +\end{eqnarray} + +\noindent together with the set of $n_d$ precursor equations: + +\begin{equation} +{\partial c_\ell(\vec r,t) \over \partial t}=\beta_\ell \sum_g +\nu\Sigma_{\rm f}^g(\vec r,t) \phi^g(\vec r,t)-\lambda_\ell c_\ell(\vec r,t) \ \ ; \ \ \ +\ell=1,N_d +\label{eq:eq10} +\end{equation} + +\vskip 0.2cm + +Using additional approximations, the new condensation relations are rewritten as + +\begin{equation} +\nu\Sigma_{\rm f}(\vec r,E)=\sum_j \nu\Sigma_{{\rm f},j}(\vec r,E)=\sum_j \nu_j^{\rm ss}(\vec r,E) \ \Sigma_{{\rm f},j}(\vec r,E) +\end{equation} + +\begin{equation} +\beta_\ell={\sum\limits_j{\beta_{\ell,j}\int\limits_0^\infty dE \ \nu_j^{\rm ss}(\vec r,E) \ \Sigma_{{\rm f},j}(\vec r,E) \ +\phi(\vec r,E)} \over \sum\limits_j{\int\limits_0^\infty dE \ \nu_j^{\rm ss}(\vec r,E) \ \Sigma_{{\rm f},j}(\vec r,E) \ +\phi(\vec r,E)} } = {\sum\limits_j{\beta_{\ell,j}\sum\limits_g \nu\Sigma_{{\rm +f},j}^g(\vec r) \ \phi^g(\vec r)} \over \sum\limits_j{\sum\limits_g +\nu\Sigma_{{\rm f},j}^g(\vec r) \ \phi^g(\vec r)} } \ \ \ , +\end{equation} + +\vskip 0.2cm + +\begin{eqnarray} +\nonumber \chi^{{\rm pr},g}&=&{\sum\limits_j\left[1-\sum\limits_\ell\beta_{\ell,j}\right]{\int\limits_{E_g}^{E_{g-1}} +dE \ \chi_j^{\rm pr}(E) \int\limits_0^\infty dE' \ \nu_j^{\rm ss}(\vec r,E') \ \Sigma_{{\rm f},j}(\vec r,E') +\ \phi(\vec r,E')} \over \left[1-\sum\limits_\ell\beta_\ell\right] \sum\limits_j{\int\limits_0^\infty dE +\ \nu_j^{\rm ss}(\vec r,E) \ \Sigma_{{\rm f},j}(\vec r,E) \ \phi(\vec r,E)}} \\ + &=& {\sum\limits_j\left[1-\sum\limits_\ell\beta_{\ell,j}\right]{ +\chi_j^{{\rm pr},g} \sum\limits_h \nu\Sigma_{{\rm f},j}^h(\vec r) +\ \phi^h(\vec r)} \over \left[1-\sum\limits_\ell\beta_\ell\right] \sum\limits_j{ +\sum\limits_h \nu\Sigma_{{\rm f},j}^h(\vec r) \ \phi^h(\vec r)}} +\end{eqnarray} + +\noindent and + +\begin{eqnarray} +\nonumber \chi_\ell^{{\rm D},g}&=&{\sum\limits_j \beta_{\ell,j}{\int\limits_{E_g}^{E_{g-1}} dE \ \chi_{\ell,j}^{\rm D}(E) +\int\limits_0^\infty dE' \ \nu_j^{\rm ss}(\vec r,E') \ \Sigma_{{\rm f},j}(\vec r,E') +\ \phi(\vec r,E')} \over \beta_\ell \sum\limits_j{\int\limits_0^\infty dE \ \nu_j^{\rm ss}(\vec r,E) +\ \Sigma_{{\rm f},j}(\vec r,E) \ \phi(\vec r,E)}} \ \ ; \ \ \ \ell=1,N_d \\ +&=&{\sum\limits_j \beta_{\ell,j} \ {\chi_{\ell,j}^{{\rm D},g} +\sum\limits_h \nu\Sigma_{{\rm f},j}^h(\vec r) +\ \phi^h(\vec r)} \over \beta_\ell \sum\limits_j{\sum\limits_h \nu\Sigma_{{\rm f},j}^h(\vec r) +\ \phi^h(\vec r)}} \ \ ; \ \ \ \ell=1,N_d \ \ \ . +\end{eqnarray} + +\vskip 0.2cm + +The above definitions ensure that the group-integrated steady-state fission spectrum is given as + +\begin{equation} +\chi^{{\rm ss},g} = \left[1-\sum_\ell\beta_\ell\right] \chi^{{\rm pr},g}+\sum_\ell \beta_\ell \ \chi_\ell^{{\rm D},g} \ \ \ . +\end{equation} + +\vskip 0.2cm + +A mean neutron generation time can also be written as + +\begin{equation} +\Lambda={\int\limits_0^\infty dE \ {\displaystyle 1 \over \displaystyle v(E)} \ \phi(\vec r,E) \over +\sum\limits_j{\int\limits_0^\infty dE \ +\nu_j^{\rm ss}(\vec r,E)\ \Sigma_{{\rm f},j}(\vec r,E) \ \phi(\vec r,E)}}={\sum\limits_g <1/v>^g \ \phi^g(\vec r) \over +\sum\limits_j{\sum\limits_g \nu\Sigma_{{\rm f},j}^g(\vec r) \ \phi^g(\vec r)}} \ \ \ . +\end{equation} + +\eject |