JAEA-Data/Code 2007-004 - Welcome to Research Group for ...

ϕ
i ( 0
u ) = ∑ Pij
( u)
W j ( u)
X j / X j ( u)
j
, (7.3.3-1)
W j ( u)
= S j ( u)
/ Σ 0 j
, (7.3.3-2)
X
0 j ( 0 j
j j u
u)
= l jΣ
, X = l jΣ
( ) , (7.3.3-3)
where the subscript j denotes a spatial region j, S j (u) the slowing-down source,
length, Σ 0j , the non-resonance part of Σ j (u) and the other notation is conventional.
l j , the mean chord
Now, let us consider the limit at which the resonance cross-section of one resonant isotope, say
σ t , tends to be infinite. This black limit corresponds to a physical situation encountered near a
resonance energy. Then, all the macroscopic cross-sections of the region with the resonance isotope
under consideration will also tend to be infinite. We denote these regions by the symbol R.
64), 65) show
General arguments on asymptotic behaviors of the collision probability, P ij , at the black limit 27),
P ( σ ) ≡ P ( u)
→ δ − γ / X ( i ∈ R)
for X → ∞
(7.3.3-4)
ij
t
ij
ij
ij
i
i
with γ ij ≡ lim { δ ij − P ij ( σ t )} X i
σ
t →∞
. (7.3.3-5)
From the conservation law
∑
j
P ≡1
ij
(7.3.3-6)
we have
∑
j
γ
ij
≡ 0
or
γ
ii
= −
∑
j≠i
γ
ij
. (7.3.3-7)
Therefore, we have for the flux ϕ ( u)
( i ∈ R)
i
−2
( X ) for X → ∞
⎛
⎞
⎜ ∞
∞
ϕ ⎟
i
( u)
→ W −
+
i
X
0i
∑W
j
X
0 jγ
ij
/ X
j
X
i
O
i
i
(7.3.3-8)
⎝
j∉R
⎠
∞ ≡
σ →∞
W j lim W j
t
. (7.3.3-9)
Here, since we try to treat the higher energy region, say E ≥ several hundred eV, the NRA is a
reasonable approximation, i.e.,
W j
( u)
= const.
≡ 1
. (7.3.3-10)
Moreover, we assume that accidental overlapping between different resonance sequences is negligible,
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