Starting with the 1DX code (Ref. 19), various attempts have been made (Refs. 20 and 21) to correct the elastic removal based on the shape of the computed flux using an iterative scheme. However, the complicated consequences of resonances in the cross sections make this dangerous except in regions where the calculated flux varies smoothly. Therefore, TRANSX does not attempt to correct for variations that only show up in one or two groups. Instead it smoothes the flux by averaging over several groups above and below the group in question. This results in an effective slope for the flux near the bottom of the group, which can be used to correct the removal as shown below. The corrected cross-section set should give improved answers for any responses that depend on the average removal effect such as k$_{\rm eff}$, but the actual flux shape near a particular resonance may not be improved at all.
The removal from group $g$ to group $g^{\prime}$ can be written
R_{g^{\prime} \leftarrow g}^M = \int_g du\,\int_{g^{\prime}}du^{\prime} \sigma_e (u^{\prime}{\leftarrow}u)\, \phi^M (u)~~,
where $u$ is lethargy, $\sigma_e$ is the elastic scattering differential cross section, and $\phi^M$ is the model flux. If the actual flux at the bottom of the group has a different slope than the model flux,
\phi (u) = [ a-b(u_g -u) ] \phi^M (u)~~,
where $u_g$ is the lethargy at the bottom of group $g$, the removal rate becomes
R_{g^{\prime} \leftarrow g} = R_{g^{\prime} \leftarrow g} [ a-b\gamma_{g^{\prime} \leftarrow g} ]~~,
where
\gamma_{g^{\prime} \leftarrow g} = \frac{\displaystyle \int_g du \int_{g^{\prime}} du^{\prime} \,(u_g{-}u)\, \sigma_e (u^{\prime}{\leftarrow}u)\, \phi^M (u)} {\displaystyle\int_g du\int_{g^{\prime}} du^{\prime}\, \sigma_e (u^{\prime}{\leftarrow}u)\, \phi^M (u)}~~.
The quantity $\gamma_g = \sum_g \gamma_{g^{\prime} \leftarrow g}$ is available on the MATXS library, so this $\gamma$ is used for all groups and Legendre orders to obtain the corrections to the elastic scattering matrix elements.
It is then assumed that the smoothed flux ratio can be represented by
\ln (\phi/\phi^M) = x_0 + x_1 (u_g{-}u) + x_2 (u_g{-}u)^2~~,
in the group $g$. The coefficients are computed using a least squares fit to six values of the ratio assigned to the lethargy center of each of the six groups (three above $u_g$ and three below). The parameters $a$ and $b$ are determined at the bottom of the group, and the smoothed value of the ratio at the center of the group is used to find the change in cross section that will give the desired change in removal rate.
The method is very stable. It avoids the divergences in removal cross sections often seen with other methods, and it usually converges adequately in one iteration.
TRANSX HyperText Manual
TRANSX HyperText Help Package
T-2 Nuclear Information Service