Cylinder by the Bell approximation,
C=\frac{1}{1+\lambda}~~;
Cylinder in an hexagonal lattice by the Sauer approximation,
C=\frac{{\rm e}^{-\tau\lambda}}{1+(1-\tau)\lambda}~~;
where
\tau=\left( .9523\sqrt{1+\frac{V_M}{V_M+V_C}}-1\right) \frac{\sqrt{1+V_C}}{V_M}-.12\left(1+\frac{1}{2} \sqrt{\frac{V_C}{1+V_C}}\right)~~,
Cylinder in a square lattice by the Sauer approximation,
C=\frac{{\rm e}^{-\tau\lambda}}{1+(1-\tau)\lambda}~~;
where
\tau=\left( .8862\sqrt{1+\frac{V_M}{V_M+V_C}}-1\right) \frac{\sqrt{1+V_C}}{V_M}-.08\left(1+\frac{1}{2} \sqrt{\frac{V_C}{1+V_C}}\right)~~,
Slab cell by the Bell approximation,
C=\frac{0.5}{1+\lambda_R}+\frac{0.5}{1+\lambda_L}~~;
and
Slab cell by E$_3$ collision probabilities,
C={\rm E}_3(\lambda_R)+{\rm E}_3(\lambda_L)~~.
In these expressions, $\lambda$ stands for the optical path in the moderator region, $\overline{\ell}_m\sigma_m$, $\lambda_L$ and $\lambda_R$ are the optical paths to the left and right from the region containing material $i$ to the next slab containing the same material, $V_M$ and $V_C$ are the ratios of the moderator volume and clad volume to the fuel volume, and E$_3$ is the third-order elliptic integral. In all the cases, the Dancoff correction results in an escape cross section of
\sigma^i_{xg}=\frac{1}{N_i\overline{\ell}}\, \frac{b_1(1-C)}{1+(b_2-1)C}~~,
where $b_1$ and $b_2$ are called Bell corrections (appropriate values are 1.09 for slabs and 1.35 for cylinders). When $b_1{=}b_2$, the constant is usually called $A$, the Levine factor. TRANSX provides for the automatic calculation of fuel and moderator mean chords and homogenized moderator cross sections, given the volumes and compositions of the various regions of a cell. Examples are given later in this report.
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