S_g = \sum_{g^{\prime}} \sigma_{fg \leftarrow g^{\prime}}\, \phi_{0g^{\prime}}~~,
where $\sigma_{fg\leftarrow g^{\prime}}$ is the fission matrix that is available on the MATXS file. However, most existing transport codes do not use this matrix form because the upscatter is expensive to handle and a reasonably accurate alternative exists. Except for relatively high neutron energies, the spectrum of fission neutrons is only weakly dependent on initial energy. Therefore, the fission source can be written as
S_g = \chi_g \sum_{g^{\prime}} \bar{\nu}_{g^{\prime}}\, \sigma_{fg^{\prime}}\,\phi_{0g^{\prime}}~~,
where $\bar{\nu}_g$ is the fission neutron yield, $\sigma_{fg}$ is the fission cross sections, and $\chi_g$ is the average fission spectrum, which can be defined by
\chi_g = \frac{\displaystyle{\sum_{g^{\prime}}}\sigma_ {fg\leftarrow g^{\prime}}\, \phi_{0g^{\prime}}}{\displaystyle{\sum_ {g^{\prime}}}\bar{\nu}_{g^{\prime}}\, \sigma_{fg^{\prime}}\, \phi_{0g^{\prime}}}~~,
where the fission neutron production rate can also be written as
\sum_{g^{\prime}} \bar{\nu}_{g^{\prime}}\, \sigma_{fg^{\prime}}\, \phi_{0g^{\prime}} = \sum_g \sum_{g^{\prime}} \sigma_{fg \leftarrow g^{\prime}}\, \phi_{0g^{\prime}}~~.
Clearly, $\chi_g$ as given by Eq.~(\ref{eq218}) depends on the flux in the system of interest. The dependence is weak except for high incident energies, and a rough guess for $\phi_{0g}$ usually gives an accurate spectrum. When this is not the case, a sequence of calculations can be made, using the flux from each step to improve the $\chi_g$ for the next step. The matrix in a MATXS file represents the prompt part of fission only. Steady-state (SS) fission is obtained using two auxiliary pieces of data: delayed $\bar{\nu}$ and delayed $\chi$. Therefore,
\bar{\nu}^{SS}_{g^{\prime}} \sigma_{fg^{\prime}} = \sum_g \sigma_{fg\leftarrow g^{\prime}} + \bar{\nu}_{g^{\prime}}^D\, \sigma_{fg^{\prime}}~~,
and
\chi_{g}^{SS} = \frac{\displaystyle{\sum_{g^{\prime}}}\sigma_ {fg\leftarrow g^{\prime}}\, \phi_{0g^{\prime}} + \chi_g^D\displaystyle{\sum_{g^{\prime}}}\bar{\nu}_{g^{\prime}}^D\, \sigma_{fg^{\prime}}\, \phi_{0g^{\prime}}}{\displaystyle{\sum_g\sum_{g^{\prime}}} \sigma_{fg\leftarrow g^{\prime}}\, \phi_{o0^{\prime}} + \displaystyle{\sum_ {g^{\prime}}}\bar{\nu}_{g^{\prime}}^D\, \sigma_{fg^{\prime}}\, \phi_{0g}^{\prime}}~~.
For large group structures, the square fission matrices can consume a large part of the library. As already noted, the shape of the fission spectrum is not strongly dependent on initial energy. In fact, the shape is independent of energy at low energies. Therefore, the fission matrix can be split into slow and fast parts,
\sigma_{fg\leftarrow g^{\prime}} = \chi_g^S\,(\bar{\nu} \sigma_f)_{g^{\prime}}^S + \sigma_{fg\leftarrow g^{\prime}}^F~~.
Starting with NJOY 91.0, the GROUPR module automatically determines the energy range over which the fission spectrum is constant, and it generates a production cross section and normalized spectrum for this range. At higher energies, it produces a rectangular fission matrix. The new version of the MATXS format established for NJOY 91 uses a generalized format for transfer matrices that allows for this decomposition of a matrix into two vectors and a smaller matrix. TRANSX 2.0 can then reconstruct the full representation of fission using Eq.~(\ref{eq222}).
The fission representation for some materials may be divided into the separate reactions $(n,f)$, $(n,n^\prime f)$, $(n,2nf)$, and $(n,3nf)$. Because the second-, third-, and fourth-chance fission channels are only open at high energies (for example, above 6 MeV for U-235), their corresponding fission matrices are always rectangular on the MATXS library and do not cause a space problem. In any case, the prompt fission source produced by TRANSX includes all of these partial matrices. This full matrix treatment is important for 14-MeV source problems (for example, fusion-fission hybrid design).
This approach gives TRANSX a very general and efficient fission source capability. The same library can be used to compute $\chi$ for a fast or thermal system; in fact, if region fluxes are available, $\chi$ for a given material can be different in each region of a problem.
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