# Burn-up Sensitivity

To calculate reliable MA transmutation rates, it is important to evaluate the uncertainty of calculated MA transmutation rates. The uncertainty can be calculated when the sensitivity of the MA transmutation rates to the cross sections called burnup sensitivity is known. Therefore, we developed a calculation code of burn-up sensitivity based on the generalized perturbation theory [9]. The burn-up sensitivity e*S* relative to effective cross sections is calculated by

In Eq. (17.18), the term containing ∂*R* is called the direct term; the second term is the ∂σ number density term, which represeents the effect of the change of nuclide number densities caused by cross-section changes; the third term shows the effect of the change of flux from to cross-section changes; the fourth term shows the effect of the change of adjoint flux caused by cross-section changes, and the last term shows the effect of constant power production even when there are cross-section changes. The adjoint number density *N** is calculated from the end of a burn-up period to the beginning of the period, and the generalized flux and generalized adjoint flux are calculated at each burn-up step. The adjoint number density *N** is not continuous but has a discontinuity at each burn-up step. To calculate true burn-up sensitivities *S* relative to infinite-dilution cross sections, we introduce e*S* to Eq. (17.14) to obtain *S*.

# Dependence of Sensitivities on Numbers of Energy Groups

In sensitivity and uncertainty analysis, multi-group sensitivities are usually used, but there is no theoretical basis for the effect of number of energy groups to sensitivities. Here we derive a relationship between sensitivities calculated with different numbers of energy groups by considering the case where multi-groups are collapsed to a few groups. The sensitivity of core parameter *R* to the microscopic cross section of nuclide *i* and reaction *j* in group *g* in multi-groups is denoted by *S* and is defined by

The sensitivity of *R* to microscopic cross section in few groups is given by

Cross sections from few groups are calculated from a multi-group cross section by using neutron flux ϕ*g* in group *g*:

where the summation about *g* is performed over energy groups *g* included in few groups *G*. Let us consider the case where multi-group cross sections change as follows:

With the cross-section change, the neutron flux also changes:

The few group cross sections change as follows:

Therefore we obtain

where *C* is a constant, *Ni* is number density of nuclide *i*, σ*g* is microscopic total cross section of nuclide *i*, and σ*g* is background cross section. When using only the *j* reaction cross section of nuclide *i*, σ*g* , the flux perturbation is expressed by

in the first-order approximation. Introducing the preceding equation to Eq. (17.25) leads to

We change the multi-group cross sections σ*g* at constant rate α (for example, 1 %) within few groups *G*:

In this case, the few-groups cross-section change is expressed by the multi-group sensitivity as follows

Therefore, the few groups sensitivity is given by

Thus, in general,

We use this relationship to choose energy groups *N* (*G* ¼ 1–*N* ) such that

As an example, we calculated *k*eff sensitivities in 7, 33, and 70 energy groups, and compared sensitivities. In 7 groups, the sensitivities to 235U capture cross section are different from the corresponding integrated sensitivities calculated from 70 groups by 10–20 % above 100 eV. However, in 33 groups, the sensitivities are different from the 70 groups result by at most 5 %. This result convinced us that calculations of sensitivities for 33 groups or 70 groups are sufficient.