The magnitude of this self-shielding effect is in general a complicated function of the geometry and composition of the system. However, it has been found that a simple model called the Bondarenko Method (Ref. 16) or Background Cross Section Method (Ref. 17) does a surprisingly good job of representing the effects for many applications. The flux is assumed to vary inversly as the total macroscopic cross section. The model flux for the group averages for isotope $i$ is written as
\phi^i_\ell (E) = \frac{C(E)}{[ \sigma_0^i + \sigma_t^i(E) ]^{l+1}}~~,
where $C(E)$ is the smooth part of the shape of the flux, and $\sigma_0^i$ is called the background cross section (it represents the effects of all the other isotopes). The effect of the total cross section in the denominator is to put a dip in the flux for each peak in the cross section, and $\sigma_0^i$ controls the relative size of the dip ($\sigma_0^i{=}\infty$ gives the infinitely dilute case discussed above). The $\ell$ dependence shown here is appropriate for a large system with nearly isotropic scattering (i.e., the B0 approximation), and it was used when the MATXS files were generated. The result in Eq.~(\ref{eq225}) is based on the narrow resonance approximation, and it will be less accurate for some of the wider low-energy resonances important in thermal systems. The multigroup form of this model flux is
\phi^i_{\ell g} = \frac{C_g}{[ \sigma_{0g}^i + \sigma_{tg}^i ]^{l+1}}~~.
For an infinite homogeneous mixture, the appropriate background cross section is
\sigma^i_{0g}=\frac{1}{N_i}\,\sum_{j\ne i} N_j\,\sigma^j_{tg}(\sigma^j_{0g})~~,
where $N_i$ is the number density for the isotope. Because $\sigma_t$ depends on $\sigma_0$, finding $\sigma_0$ is clearly an iterative process.
For a lump of resonance material embedded in a large moderating region, escapes from the lump also increase the background cross section. This additional escape cross section is given by
\sigma^i_{xg}=\frac{1}{N_i\overline{\ell}}~~,
where $\overline{\ell}$ is the mean chord length of the lump given by
\overline{\ell}=\frac{4V}{S}~~,
and where $V$ and $S$ are the volume and surface area of the lump (e.g., for a cylinder, $\overline{\ell}{=}2r$). The mean chord $\overline{\ell}$ can be adjusted away from the geometric value of Eq.~(\ref{eq282}) to compensate for the presence of other lumps (Dancoff corrections) or for shortcoming in the escape probability model used to obtain Eq.~(\ref{eq281}) (Bell factors); the main point is that TRANSX option provides an additional component for the background cross section that is parameterized by $\overline{\ell}$.
Eqs.~(\ref{eq280}) and (\ref{eq281}) show that there is some kind of equivalence between the self-shielding effects of an admixed moderator and the self-shielding effects of heterogeneity. Physically, in the narrow-resonance regime, if a neutron is produced at the energy of an absorption resonance of material $i$, either losing energy by scattering from one of the competing materials $j$ or escaping from the lump will assure that it will not be absorbed by that resonance. In fact, the effects are additive. As an example, consider a 1-cm rod of uranium oxide surrounded by a large region of moderator:
\sigma_0 & = & \frac{.046\,\hbox{at/b-cm} \times 3.76\,\hbox{b/at}} {.023\,\hbox{at/b-cm}}+\frac{1\,\hbox{cm}}{.023\,\hbox{at/b-cm}} \\ & = & 7.52\,\hbox{b/at} + 43.48\,\hbox{b/at} \\ & = & 51.0\,\hbox{b/at}
where .023 and .046 are the atomic densities of the $^{238}$U and oxygen, respectively, and 3.76 barns is the oxygen cross section.
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