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7.3.2 Table-look-up Method of f-tables Based on IR Approximation
We start with the IRA of resonance absorption in homogeneous systems, to give an insight into
the relationship with the table-look-up method.
From the two extreme cases representing the limits of NR and wide resonance (WR) for the
53), 54)
slowing down kernel, the first-order solution for ϕ(u) of Eq.(7.3.1-1) can be written as
λσ p + σ b
ϕ( u)
≅
(7.3.2-1)
σ ( u)
+ λσ ( u)
+ σ
a
s
b
where λ is the IRA parameter for the absorber. The value of λ can be determined by solving a
transcendental equation for λ 53, 54) .
For a homogeneous system including many moderator nuclides described by Eq.(7.3.1-4), the
corresponding first-order solution can be given by
λσ p + σ ' b
ϕ(
u)
≅ (7.3.2-2)
σ ( u)
+ λσ ( u)
+ σ '
a
s
b
with
1
σ ' b = ∑ λ j ( n jσ
j )
(7.3.2-3)
n
f
j
where λ j is the IR parameter for moderator j and can be again determined by solving a coupled set of
transcendental equations 54) .
Here, it should be noted that both the fluxes obtained from a numerical integration of
Eq.(7.3.1-1) and given by Eq.(7.3.2-1) or (7.3.2-2) are the weighting functions for cross-section
averaging. Hence, they can be assumed to give the same value for the effective cross-section in the
extent of the accuracy of the IRA. Consequently, a homogeneous system with σ b ’ has the same
effective cross-section as the homogeneous system with the same σ b of Eq.(7.3.1-1). That is, the
effective cross-sections can be calculated by determining the IR parameters and σ b ’ as to be used for
the table-look-up of the resonance shielding tables. The case of letting all the λ j s equal to unity
corresponds just to the NRA applied to the slowing down of moderators σ b ’ = σ m , as discussed in the
previous subsection.
The IRA method described above can be applied only to a zero temperature system. For nonzero
temperature, the IR parameter λ for absorber depends on temperature when the interference between
potential and resonance scattering is taken into consideration 55) . A simple way to take account of this
dependence is to multiply the interference scattering term by a factor with temperature dependence 55) .
Next consider the IR treatment of resonance absorption in heterogeneous systems. Assuming a
flat flux in each spatial region, the slowing down equation in the two-region system, consisting of an
absorbing lump (f) and a non-absorbing moderator (m), can be written as
{ σ
amK
am
( ϕ
f
) + K
f
( σ
sϕ
f
)} + (1 − p
ff
) σ
f ∑{ Rk
K
k
( ϕm
}
σ ϕ = p )
(7.3.2-4)
f
f
ff
k
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