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

dR/R

d*fi/*5

Cross sections from few groups are calculated from a multi-group cross section by using neutron flux ф8 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:

Ф8 —— ф8 + $ф8

Therefore we obtain

Here we apply the narrow resonance approximation to express the flux perturbation caused by cross-section change.

where C is a constant, Ni is number density of nuclide i, eg is microscopic total cross section of nuclide i, and o0 is background cross section. When using only the j reaction cross section of nuclide i, og, the flux perturbation is expressed by

where o0 — o0 + — і*] in the first-order approximation. Introducing the preced-

j—j

ing equation to Eq. (17.25) leads to

We change the multi-group cross sections eg at constant rate a (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

SG=£ Sg

g є G

We use this relationship to choose energy groups N (G = 1-N) such that

SG ^2 sg

g є G

As an example, we calculated keff 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.