3. Multigroup Cross-Section Data
- Total cross section:
The total cross section, \(\sigma_t^g\), includes all interaction processes in group \(g\). It is given by
\[\sigma_t^g = \sigma_{s,0}^g + \sigma_{\text{n,c}}^g + \sigma_{f}^g + \sum_{\text{x}=2,3,\ldots}\sigma_{\text{n,xn}}^g \,,\]where \(\sigma_{s,0}^g\) is the (elastic + inelastic) isotropic scattering cross section, \(\sigma_{\text{n,c}}^g\) includes all capture processes (e.g., (n,\(\gamma\)), (n,p), (n,\(\alpha\)), (n,d), (n,t), …), \(\sigma_{f}^g\) is the fission cross section, and \(\sigma_{\text{n,xn}}^g\) denotes the (n,2n), (n,3n) …processes (without multiplicity). The multigroup total cross section is a vector of length \(G\) and is a mandatory input of OpenSn.
- Transfer matrix:
- The transfer matrix contains transfers from (elastic + inelastic) scattering, as well as transfers with multiplicity such as from (n,xn) reactions (with x\(=2,3,4,\ldots\))\[\vartheta\sigma_{\ell}^{g'\to g} = \sigma_{s,\ell}^{g'\to g} + \sum_{\text{x}=2,3,\ldots} f_\text{x}\sigma_{\text{(n,xn)},\ell}^{g'\to g} = \sum_{\text{x}=1,2,3,\ldots} f_\text{x}\sigma_{\text{(n,xn)},\ell}^{g'\to g} \ ,\]
where \(f_\text{x} = \text{x}\) with x\(=1,2,3,4,\ldots\) hence denoting scattering by (n,1n); \(\vartheta^{g'\to g}\) denotes the effective multiplicity and \(\ell\) is the anisotropy expansion order. The effective multiplicity is
\[\vartheta^{g'\to g} = \frac{ \sum_{\text{x}=1,2,3,\ldots} f_\text{x}\sigma_{\text{(n,xn)},0}^{g'\to g}}{ \sum_{\text{x}=1,2,3,\ldots} \sigma_{\text{(n,xn)},0}^{g'\to g}}\]Of course, when (n,xn) reactions are ignored (x\(\, \ge 2\)), we have \(\vartheta^{g'\to g}=1\) and \(\sigma_{\ell}^{g'\to g} = \sigma_{s,\ell}^{g'\to g}\).
The multigroup transfer matrix is a mandatory input of OpenSn. The transfer matrix a sparse array of dimension \(G \times G \times L_{\text{max}}\), whose sparsity pattern for each anisotropy order is dictated by the physics and the chosen energy group structure. - Reduced absorption cross section:
This cross section is computed using the total cross section and the transfer matrix. It is not user-supplied but plays a role in the conservation statement. In order to account for multiplicity in the transfer matrix, the standard definition of the absorption cross section is amended as
\[\sigma_{\text{red.}a}^g = \sigma_{t}^g - \sum_{g'} \vartheta^{g\to g'}\sigma_{s,0}^{g\to g'}\]The total cross section is the sum of absorptive processes (denoted by (n,0n)), and transfers to all groups from scattering (n,1n) reactions and other reactions with multiplicity greater than one, such as (n,2n), (n,3n) reactions, and so forth:
\[\sigma_{t}^g = \sigma_{\text{(n,0n)}}^{g} + \sum_{\text{x}=1,2,3,\ldots} \sum_{g'} \sigma_{\text{(n,xn)},0}^{g\to g'}\]If one wants to define the total cross section using the transfer matrix with multiplicity, we take the above formula and add and subtract the transfer matrix with multiplicity:
\[\sigma_{t}^g = \sigma_{\text{(n,0n)}}^{g} + \sum_{\text{x}=1,2,3,\ldots} \sum_{g'} \sigma_{\text{(n,xn)},0}^{g\to g'} - \sum_{\text{x}=1,2,3,\ldots} f_\text{x} \sum_{g'} \sigma_{\text{(n,xn)},0}^{g\to g'} + \sum_{\text{x}=1,2,3,\ldots} f_\text{x} \sum_{g'} \sigma_{\text{(n,xn)},0}^{g\to g'}\]After some algebra, we obtain
\[\sigma_{t}^g = \underbrace{\sigma_{\text{(n,0n)}}^{g} + \sum_{\text{x}=2,3,\ldots} \sum_{g'} (1- f_\text{x}) \sigma_{\text{(n,xn)},0}^{g\to g'} }_{ \sigma_{\text{red.}a}^g } + \sum_{g'} \underbrace{ \sum_{\text{x}=1,2,3,\ldots} f_\text{x} \sigma_{\text{(n,xn)},0}^{g\to g'} }_{\vartheta^{g\to g'}\sigma_{s,0}^{g\to g'} }\]which leads to the definition of the reduced absorption cross section, given previously. Note that this cross section is not mandatory in OpenSn. It is computed using the total cross section and the transfer matrix and is used in conservation (balance) statements.
- Fission production:
- Production by fission requires the given of the fission spectrum \(\chi^g\) and the production by fission cross section \(\nu\sigma_f^{g'}\). The production by fission term is then\[S_f^g = \frac{\chi^g}{4\pi} \sum_{g'} \nu\sigma_f^{g'} \phi_{0,0}^{g'}\]
The multigroup fission spectrum and production by fission cross section cross section are vectors of length \(G\) and are a mandatory input of OpenSn when fissionable materials are present.
Note that, for relatively high neutron energies, the spectrum of fission neutrons is dependent on initial energy. In such a situation, a fission production matrix should be employed\[S_f^g = \frac{1}{4\pi} \sum_{g'} \nu\sigma_f^{g' \to g} \phi_{0,0}^{g'}\]However, this is not yet employed in the current version of OpenSn
- Other cross sections:
Other multigroup cross sections can be supplied in order to compute post-processing values. Examples include \(\kappa \sigma_f^{g}\) to compute fission power and heating cross sections (for dose calculations).
In summary, the mandatory cross-section inputs are:
total cross section, \(\sigma_t^g\), and
transfer matrix (possibly with multiplicity) \(\vartheta\sigma^{g \to g'}_\ell\),
when fissionable materials are present:
fission spectrum \(\chi^g\), and
production by fission cross section \(\nu\sigma_f^{g}\).