\sigma_{\ell g \leftarrow g}^{SN} = \sigma_{\ell g \leftarrow g}^{PN} - ( \sigma_{\ell tg}^{PN} - \sigma_{otg}^{PN} ) - \Delta^N_g~~,
and
\sigma_g^{SN} = \sigma_{otg}^{PN} - \Delta_g^N~~.
The term in parentheses corrects for the anisotropy in the total reaction rate term of the Boltzmann equation, and $\Delta_g^N$ can be chosen to minimize the effects of truncating the Legendre expansion at $\ell{=}N$. The recipes available in TRANSX are as follows:
Consistent-P approximation:
\Delta_g^N = 0 ,
Inconsistent-P approximation:
\Delta_g^N = \sigma_{otg}^{PN} - \sigma_{N+1, tg}^{PN}~~,
Diagonal transport approximation:
\Delta_g^N = \sigma_{otg}^{PN} - \sigma_{N+1, tg}^{PN} + \sigma_{N+1, g \leftarrow g}^{PN}~~,
Bell-Hansen-Sandmeier or extended transport approximation:
\Delta_g^N = \sigma_{otg}^{PN} - \sigma_{N+1, tg}^{PN} + \sum_{g^{\prime}} \sigma_{N+1, g^{\prime} \leftarrow g}^{PN}~~,
and
Inflow transport approximation:
\Delta_g^N = \sigma_{otg}^{PN} - \sigma_{N+1, tg}^{PN} + \frac{\displaystyle{\sum_{g^{\prime}}}\,\sigma_{N+1, g \leftarrow g^ {\prime}}^{PN}\,\phi_{N+1, g^{\prime}}}{\phi_{N+1,g}}~~.
The first two approximations are most appropriate when the scattering orders above N are small. The inconsistent option removes most of the delta-function of forward scattering introduced by the correction for the anisotropy in the total scattering rate and should normally be more convergent than the consistent option. For libraries produced with an $\ell$-independent flux guess and in the absence of self-shielding, the difference between ``consistent'' and ``inconsistent'' vanishes.
The diagonal and Bell-Hansen-Sandmeier (BHS) recipes make an attempt to correct for anisotropy in the scattering matrix and are especially effective for forward-peaked scattering. The BHS form is most often used, but the diagonal option can be substituted when BHS produces negative values.
The inflow recipe makes the N+1 term of the PN expansion vanish, but it requires a good knowledge of the N+1 flux moment from some previous calculation. Inflow reduces to BHS for systems in equilibrium by detail balance (i.e., the thermal region). The diffusion approximation obtained using the inflow formula is equivalent to a P1 transport solution.
These corrections require data from the N+1-th Legendre moments of the cross sections to prepare a corrected N-table set. The user of the TRANSX program should be certain that, in fact, N+1 tables are available on the MATXS file when producing an N-table transport-corrected set. Absence of the N+1 tables is not fatal, but obviously the cross sections will not contain the desired corrections.
TRANSX HyperText Manual
TRANSX HyperText Help Package
T-2 Nuclear Information Service