# Sensitivity Calculation Method

## Sensitivity to Infinite-Dilution Cross Section

Core characteristics such as *k*eff, power distribution, and control rod worth are calculated by using effective cross section in deterministic methods. Sensitivities are usually calculated by using sensitivity calculation codes such as SAGEP [9], SAGEP-T [10], and SAINT [11]. However, the sensitivities are for relative changes of effective cross sections. Here we derive a calculation method of sensitivities relative to infinite-dilution cross sections for fast reactor analysis. Usually the effective cross sections are calculated by using the Bondarenko self-shielding factor method, the subgroup method. In that method the effective cross sections are expressed by the infinite-dilution cross section and the self-shielding factors *f*

The self-shielding factors depend on the background cross section and temperature. The background cross section for nuclide *i*0 in a homogeneous medium is calculated by the formula

where *Nk* is the atomic number density of light nuclide *k* and σ*k* is the microscopic total cross section. The sensitivity coefficient is defined by the relative changes of the core characteristics caused by the relative changes of the cross sections. Here we onsider the following two sensitivities, the sensitivity *S*, which results from the relative change of the infinite-dilution cross sections, and the approximate sensitivity e*S*, which is the result of the relative change of the effective cross sections. From Eq. (17.10), the change of the effective cross section can be expressed by

Therefore, the improved sensitivity is expressed by using the approximate sensitivity as follows:

Sensitivities and cross sections are dependent on nuclides, reaction types (such as fission, capture, and scattering), and energy groups. Here we consider the case where there is a perturbation in σ of nuclide *i*, reaction type *j*, in energy group *g*. This perturbation causes a change in the self-shielding factor *f* of nuclide *i*0, reaction *j*0, in energy group *g*0. The second term of the right-hand side of Eq. (17.14) has to cover the contributions for all nuclides *i*0, reaction types *j*0, in energy groups *g*0; therefore, we have to take the summation over *i*0, *j*0, and *g*0. The sensitivity for the nuclide *i*, reaction type *j*, in energy group *g* is given by

The first term is the direct contribution to *S*; it can be calculated using the conventional tools evaluating sensitivity coefficients such as SAGEP, SAGEP-T, and SAINT. The second term represents the indirect contribution through the change of self-shielding factor. These coefficients can be calculated as follows: here we apply the resonance approximation for heavy nuclides, which is suitable for treating fast reactors with hard neutron spectra rather than light water reactors. The self-shielding effect depends on the neutron spectrum; where the neutron spectrum for the heavy nuclide *i*0 is written as

Equation (17.15) indicates that when σ*i* 0 (*E*) and σ*i* change by the same factor, the *t b* neutron spectrum remains the same; this shows that the ratio *h* has an effect on the neutron spectrum, and also on the self-shielding factor. Following a similar method [12], the coefficient in the second term of the right-hand side of Eq. (17.14) is called TERM and can be written as