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P
is
2π
=
Σ V
i
2π
+
Σ V
i
i
i
∫
∫
ri
−1
0
ri
ri
−1
ρdρ
ρdρ
{ 1 − exp( − λ )}
i
⎛ ⋅ exp
⎜ −
⎝
⎛ −
∑
k = i+
∑
{ 1 − exp( − 2λ
) } ⎜
⎟ i exp λk
⎝ k = i+
1 ⎠
N
N
⎞ ⎪⎧
⎛
λk
⎟ ⋅ ⎨1
+ exp
⎜−
2
1 ⎠ ⎪⎩ ⎝
⎞
i−1
∑
k = 1
⎞⎪⎫
λk
− λi
⎟⎬
⎠⎪⎭
(7.1-65)
In the SRAC code, the integrands in Eqs.(7.1-61) and (7.1-62) to possible pairs of (i, j) are, first
calculated for a fixed ρ. Then, the integration over ρ is accomplished by changing the value of ρ .
7.1.5 Collision Probabilities for Two-dimensional Cylindrical Lattice
In the cylindrical system with general shape of its cross section and of infinite height, the
collision probability from a region i to another region j, P ij , is expressed by the following Eq.(7.1-66)
in the coordinate system of Fig.7.1-5, assuming flat flux in each region and isotropic emission in the
laboratory system;
z
Q
φ
ρ
y
θ
A B P
C
i
j
E
D
Q’
λ i λ 0 λ j
t
x
Fig.7.1-5 Cylindrical coordinate system
P
ij
⎡
= ⎢
⎢⎣
+∞
∫ ρdρ∫ dϕ∫ sinθdθ∫
−∞
⎪⎧
Σ
× exp⎨−
⎪⎩
⎡
⎢⎣
+∞
t'
−AD
⎪⎫
⎧
⎬exp⎨−
⎪⎭ ⎩
⎪⎧
Σ
dt exp⎨−
⎪⎩
Σ(
s)
ds
∫ ρdρ∫ dϕ∫ sinθdθ∫
−∞
j
2π
0
sinθ
2π
0
π / 2
0
π / 2
0
∫
AD
AC
AC
AB
AC
AB
dt
⎤
⎥⎦
i
⎫
⎤
sinθ
⎬ ⎥
⎭ ⎥⎦
AC − t
sinθ
⎪⎫
⎬
⎪⎭
∫
AE
AD
Σ j
dt'
sinθ
(7.1-66)
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