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Equation(7.3.2-13) can be written in the standard form of Eq.(7.3.2-1):
λσ p + σ
(1)
b
ϕ
f
( u)
=
σ + λσ + σ
a
s
b
(7.3.2-16)
where
*
* κΣ
am μ g(
C)(1
− C)
σ
b
= κσ
am
+ μ s = +
(7.3.2-17)
n n l
f
f
f
Particularly for the NRA (κ=1, μ ∗ =1), the above equation can generally be written as
1 g(
C)(1
− C)
σ n jσ
j +
(7.3.2-18)
n l
b = σ am + s = ∑
n f j≠
f
f
f
The simple result of Eqs.(7.3.2-17) and (7.3.2-18) again reveals the following equivalence
relation with practical usefulness: A heterogeneous system with σ b has the same effective resonance
cross-section as a homogeneous system of Eq.(7.3.1-1) with the same σ b . In other words, the effective
cross-sections in a heterogeneous system can be estimated by using a cross-section set of the
Bondarenko type 51) which is calculated based on Eq.(7.3.1-1).
Several groups of the multigroup structure around 100 eV include more than one resonance
levels of 238 U. Since the background cross-section σ b somewhat fluctuates among the resonance levels,
the arithmetic average value of σ b ’s in each group is in practice taken as the background cross-section
for the table-look-up. In the SRAC system, the IR method is applied only to 238 U, 232 Th, and 240 Pu.
7.3.3 Generalized Dancoff Correction Factor
The effective resonance cross-sections in heterogeneous systems have been calculated by
combined use of an equivalence theorem 61) and a cross-section set of the Bondarenko type 51) . The
equivalence theorem between homogeneous and heterogeneous systems is usually established through
a Dancoff correction factor. This factor has been derived under the assumption that the system under
study consists of an infinite array of two-region cells. Meneghetti 62) derived a generalized Dancoff
correction factor for a more general situation of infinite array of multi-region plate cell. In a complex
situation as encountered in the lattice of the PROTEUS-LWHCR (Light Water High Converter
Reactor) cores 63) where MOX pin and depleted U pin are alternatively placed in a hexagonal array,
however, it is generally difficult to see how one can construct a unit cell for which the Dancoff
correction factor can be calculated. A generalized Dancoff correction factor was derived for infinite
array of multi-region cells including several absorber lumps with different absorber densities 19) . The
equivalence theorem based on this Dancoff correction factor is incorporated in the SRAC system.
The neutron slowing-down equation in a cell may be written by using the collision probabilities
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