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i.e.,
X 0 j / X j = 1 ( j ∉ R)
. (7.3.3-11)
Under these assumptions, Eq.(7.3.3-8) can be rewritten as
⎛
⎞
ϕ ⎜
⎟
i
( u)
→ X −
→ ∞
0i
∑γ
ij
X
i
for X
i . (7.3.3-12)
⎝ j∉R
⎠
On the other hand, an equivalent theorem between homogeneous and heterogeneous systems
means that the flux in the absorber lump should be expressed by the spectrum in homogeneous
medium
X 0i
+ bi
ϕ i ( u)
= → ( X 0i
+ bi
) / X i for X i → ∞ . (7.3.3-13)
X + b
i
i
where the flux is normalized to be unity at off-resonance energy.
For Eqs.(7.3.3-12) and (7.3.3-13) to be held, the following identity must be satisfied:
bi
= −
∑
j∉R
γ
ij
(7.3.3-14)
For a special case where the unit cell under study consists of fuel region (f) and moderator region (m),
we can prove the conventional relation 61)
b
i
= − γ = 1 − C , (7.3.3-15)
fm
where C is the Dancoff correction factor and
− γ fm corresponds to the first-flight blackness for
neutrons leaving the fuel region 61) . Figure 7.3.3-1 shows the expression of b i for other simple
geometries.
Consequently, from the above examples, Eq.(7.3.3-14) is considered to be a generalization of
the previous works introduced in the previous section. Equation (7.3.3-14) shows that the Dancoff
correction factor for complex geometry can be obtained by calculating the generalized blackness γ fm .
Here, the blackness γ fm is calculated by using the collision probability package in the SRAC system,
that is, by using the value of P ij for a sufficiently large value of Σ ( j ∈ R)
, as Σ j =300 cm -1 , based on
Eq.(7.3.3-5).
j
256