Probability table self-shielding

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Self-shielding is an essential aspect of calculating reaction rates which can have very strong effects on numerous observables from criticality to activation. There are two general types of self-shielding which are often classified as energy and spatial self shielding. Methods for correcting reaction rates for both of these self-shielding issues have been implemented within FISPACT-II.

The essential feature of energy self shielding is that cross sections often contain resonances which are defined for very specific energy regions, which often have large contributions toward the total reaction rate. The large resonance will cause a depression in the neutron population around that energy which may not be reflected in simulations where this feedback is not fully taken into account. It is particularly problematic where cross sections and particle spectra are discretised into some multi-group where the depression in the population for a specific energy cannot be resolved.

Spatial self-shielding deals with the fact that neutron flux and local spectra can be significantly changed over short distances. In cases with strong resonances, the change in the particle spectrum over short distances can be very large in the region around the resonance.


FISPACT-II employs probability tables generated by CALENDF-2010 to offer material and dilution specific reaction rates. Probability tables are provided for macro-partial cross sections including elastic scattering, absorption, fission, inelastic scattering and neutron production (excluding fission). An infinite dilution cross section is calculated based on the raw spectrum and cross sections, which may be expanded in Gauss quatratures:

[math] \sigma(x,d=\infty)=\frac{1}{E_{max}-E_{min}}\int_{E_{min}}^{E_{max}} \sigma(E)\, dE= \sum_{n=1}^N P(x,n) \sigma(x,n) [/math]

The cross sections σ and probility tables P are dependent upon the parent nuclide [math]p[/math], energy group [math]g[/math], macro-partial index [math]x[/math] and quadrature index [math]n[/math]. When a nuclide is a constituent of a homogeneous mixture, the effective cross sections in the resonance regsions are reduced and cna be parameterised using the dilution:

Dilutions are calculated based on the material composition and an iterative algorithm using the library cross section values. These dilutions are used to recalculate the cross sections for each macro-partial. Two different approaches which are available through the PROBTABLE keyword, are to scale the reaction rates base don the total cross section or macro-partials, which also initiates different dilution algorithms. For more details see the models appendix of the user manual.

The self-shielding factors are applied through the use of either SSFFUEL (which uses isotopic definitions just as the FUEL keyword) or SSFMASS (which employs natural elements by mass %). Specific dilution values may be overwritten by hand using the SSFDILUTION keyword. More details on the use of these functionalities can be found in the inputs section of the manual.

Application example