17.4.3 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
S ¼
dR=R
dσ
g
i, j =σ
g
i, j
ð17:19Þ
The sensitivity of R to microscopic cross section in few groups is given by
S ¼
dR=R
dσ
G
i, j =σ
G
i, j
ð17:20Þ
Cross sections from few groups are calculated from a multi-group cross section by
using neutron flux ϕ
g in group g:
σ
G
i, j ¼
X
g ∈ G
σ
g
i, j ϕ
g
=
X
g ∈ G
ϕ
g
ð17:21Þ
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:
σ
g
i, j ! σ
g
i, j þ δσ
g
i, j
ð17:22Þ
With the cross-section change, the neutron flux also changes:
ϕ
g
! ϕ
g
þ δϕ
g
ð17:23Þ
The few group cross sections change as follows:
σ
G
i, j ! σ
G
i, j þ δσ
G
i, j ¼ σ
G
i, j 1 þ
X
g ∈ G
δϕ
g
σ
g
i, j þ δσ
g
i, j ϕ
g
X
g ∈ G
ϕ
g
σ
g
i, j
À
X
g ∈ G
δϕ
g
X
g ∈ G
ϕ
g
8
> > <
> > :
9
> > =
> > ;
ð17:24Þ
Therefore we obtain
190
T. Takeda et al.