\mu\frac{\partial}{\partial x}\phi_g (\mu,x) + \sigma^{SN}_g(x)\, \phi_g(\mu,x) = \sum^N _{\ell=0} P_{\ell}(\mu) \sum_{g\prime} \sigma^{SN}_{\ell g \leftarrow g^{\prime}}(x)\,\phi_{lg^{\prime}} + S_g(\mu,x)~~,
where one-dimensional plane geometry has been used for simplicity, $\mu$ is the scattering cosine, $x$ is position, $\phi (\mu,x)$ is the angular flux for group $g$, $\phi_{lg}$ is the Legendre flux for group $g$, $P_{\ell}(\mu)$ is a Legendre polynomial, and $S_g(\mu,x)$ is the external and fission source into group $g$. The cross sections in SN must be defined to make $\phi_g$ as close as possible to the solution of the Boltzmann equation. As shown in the reference ( Ref. 11), the multigroup Boltzmann equation can be written in the PN form:
\mu\frac{\partial}{\partial x}\psi(\mu,x) + \sum_{\ell=0}^N P_{\ell} (\mu)\,\sigma^{PN}_{\ell tg}(x)\,\psi_{\ell g}= \sum^N_{\ell=0} P_{\ell}(\mu) \sum_{g\prime}\sigma^{PN}_{\ell g\leftarrow g^{\prime}} (x)\,\psi_{\ell g^{\prime}} + S_g (\mu,x)~~,
where the PN cross sections are given by the following group averages:
\sigma^{PN}_{\ell t g}= \frac{\displaystyle\int_ g\sigma_t(E)\,W_{\ell}(E)\,dE} {\displaystyle\int_g W_{\ell}(E)\,dE}~~,
and
\sigma ^{PN}_{\ell g \leftarrow g^{\prime}}= \frac{\displaystyle\int_{g^{\prime}}dE^{\prime}\, \int _{g} dE\,\sigma _{\ell} (E^{\prime}{\rightarrow}E)\, W_{\ell}(E^{\prime})} {\displaystyle\int_{g^{\prime}}dE^{\prime}\,W_{\ell}(E^{\prime})}~~.
In these formulas, $\sigma_t(E)$ and $\sigma_{\ell}(E^{\prime}{\rightarrow} E)$ are the basic energy-dependent total and scattering cross sections, and W$_{\ell}$(E) is a weighting flux that should be chosen to be as similar to $\psi$ as possible. As will be seen later, these P$_{\rm N}$ cross sections are available on the MATXS libraries produced by NJOY (Refs. 12, 13, 14,and 15). When the SN equation is compared with the PN equation, it is evident that the SN equations require
\sigma^{SN}_{\ell g \leftarrow g\prime} = \sigma^{PN}_{\ell g\leftarrow g\prime}~~~~{\rm for}~g^\prime \neq g~~,
and
\sigma_{\ell g \leftarrow g}^{SN} = \sigma_{\ell g \leftarrow g}^{PN} - \sigma_{\ell tg}^{PN} + \sigma_{g}^{SN}~~,
where $\sigma_g^{SN}$ is not determined. The choice of $\sigma_g^{SN}$ gives rise to a ``transport approximation'' and various recipes are in use.
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