



Energy-dependent self shielding 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 quadratures: $$\sigma(x,d=\infty) = \frac{1}{E_{max}-E_{min}}\int_{E_{min}}^{E_{max}} \sigma(E)\, dE = \sum_{n=1}^N P(x,n) \sigma(x,n). $$ The cross sections $\sigma$ and probability tables $P$ are dependent upon the parent nuclide $p$, energy group $g$, macro-partial index $x$ and quadrature index $n$. When a nuclide is a constituent of a homogeneous mixture, the effective cross sections in the resonance regions are reduced and can be parameterised using the dilution: $$\sigma(x,d) =\frac{\sum_{n=1}^N P(x,n) \sigma(x,n)/(\sigma_{t}(n)+d)} {\sum_{n=1}^N P(x,n) /(\sigma_{t}(n)+d)} . $$ 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 Appendix A.4.3 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. The self-shielding factors can be recalculated for each stage of the evolution of the nuclide inventory, shifting for example as the fuel composition of a fission reactor evolves over dozens of GWd/THM with plutonium, minor actinide and fission product build-up.
Spatial/geometry self shielding 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 is a point solution code which does not directly consider the geometry of a simulated scenario, but several robust systematics exist for the treatment of geometry effects such as point-source doses and spatial self shielding. The keyword SSFGEOMETRY allows the user to employ the universal sigmoid curve model which has been demonstrated for foils, wires, spheres and cylinders to give good agreement for spatial self shielding effects. These consider one resonance in a pure target, using a universal dimensionless parameter, $$ z = \Sigma_{tot}(E_{res}) L_{eff} \sqrt{\frac{\Gamma_\gamma}{\Gamma}}, $$

Universal curve of the resonance self shielding factors against $z$ showing values for wires, foils and sphere geometries. Taken from E. Martinho et al.