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G j
Pij
( lattice)
= Pij
( isolated ) + Pis
. (7.1-24)
1 − G
s
7.1.2 Collision Probabilities for Slab Lattice
In a one-dimensional slab geometry shown in Fig.7.1-1, we have
x'
−x
R = ,
cosθ
dr
= dx ,
dΩ = 2π
sinθdθ
i
j
dx’
ds
x
θ
dθ
x’
x i-1 x i x j-1 x j
Fig.7.1-1 Coordinates in slab geometry
We assume that the system is divided into an array of slabs. The slab i has its left edge at x i-1 and its
total cross section denoted by Σ i . Then we have
P
ij
= 1 xi
x j π / 2 sinθ
x
dx dx Σ x ⋅
t dt θ dθ
x x xi x j
θ ⎩ ⎨⎧ − Σ
−
∫ ∫ ∫
∫
'
' ( ')
exp ( ) / cos
2( )
x
⎭ ⎬⎫
−1
−1
0 cos
i
i−1
(7.1-25)
for the case x i < x j -1, and the optical distance which appears in the exponential term in Eq.(7.1-25) is
reduced to
x'
j 1
∫ ∑ − Σ ( t ) dt = Σ i ( xi
− x ) + Σ j ( x ' −x
j− 1 ) +
x
k = i+
where λ Σ ( x − x 1) .
k = k k k −
Then we can carry out the integration over x and x’, and we get
λ
1
k
227