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P
θ
Q
R
t’
P
Q
r
M
t
C
ρ
C
a
b
Fig.7.1-3 Spherical coordinates
To perform analytically the integration as far as possible, the coordinates shown in Fig.7.1-3a
are transformed into the new coordinates shown in Fig.7.1-3b where the perpendicular length CM is ρ.
The positions of points P and Q are defined by the distances t and t’, respectively, from the point M.
The following relations among variables are found:
r
2
= t
2
+ ρ
2
r sinθ
= ρ ,
R = t'
−t
.
,
(7.1-58)
The Jacobian is then obtained as follows:
∂(
r,
θ,
R)
1
= −
∂(
ρ,
t,
t'
) r
. (7.1-59)
The probability is rewritten using the new variables by
P
ij
πΣ
j rN
=
⎧
∫ ρdρ∫ dt ∫ dt'
exp⎨−
V 0 t⊂V
t'
⊂V
⎩ ∫
i
j
2 t−t'
i
0
Σ(
s)
ds
⎫
⎬
⎭
. (7.1-60)
As the nuclear cross-section in each shell is uniform, we can integrate Eq.(7.1-60) over t and t’.
Finally the shell-to-shell collision probabilities are given in the form of single integral (See Fig.7.1-4);
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