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

1
i
∑ − 1
k = j+
1
λ ij = λ k
for r i >r j (7.1-48)
We have not yet considered the case where the shell i coincides with the shell j. In this case the
optical distances are reduced to
x'
⎧Σ
i ( x'
−x)
for x'
> x ,
∫ Σ(
t)
dt = ⎨
x
⎩Σ
i ( x − x'
) for x > x'
,
∫
x'
−x
Σ(
t)
dt
=
Σ ( x'
−x
) + Σ ( x − x
i
i
i
i−1
i− 1)
+ 2∑
k = 1
λ
k
.
In the integration over x’ for the first term on R.H.S. of Eq.(7.1-44), we must divide the range
into (x i-1 , x) and (x, x i ) and then we have
P
ii
2 ri
−1
π / 2
∫ ∫
⎢ ⎢ ⎡
⎧ ⎛ ⎞⎫
−
2
λ
⎨ − ⎜ −
i
= dρ
dθ
2λi
sinθ
2sin θ 1 exp ⎟⎬
ΣiV
0 0
i
⎣
⎩ ⎝ sinθ
⎠⎭
2
i−1
⎧ ⎛ λ ⎞⎫
⎛ ⎞⎤
+ ⎨ − ⎜ −
i
2
λ
⎟⎬
⎜ −
k
1 exp sin θ exp 2∑
⎟⎥
⎩ ⎝ sinθ
⎠⎭
⎝ k = 1 sinθ
⎠⎥
⎦
2
+
Σ V
i
i
∫
ri
ri
−1
dρ
∫
π / 2
0
⎡
dθ
⎢2λi
sinθ
− sin
⎢⎣
2
⎧ ⎛ λ ⎞⎫⎤
⎨ − ⎜ −
i
θ 1 exp 2 ⎟⎬⎥
⎩ ⎝ sinθ
⎠⎭⎥⎦
Using the K in function we get the final form of P ii as follows:
P
ii
2
=
Σ V
i
2
+
Σ V
i
i
i
∫
ri
−1
0
∫
ri
ri
−1
dρ
dρ
{ 2λ
− 2K
(0) + 2K
( λ ) + K ( λ ) − 2K
( λ + λ ) + K ( λ + 2λ
)}
i
i3
{ 2λ
− K (0) + K (2λ
)}
i
i3
i3
i3
i
i
i3
ii
i3
ii
i
i3
ii
i
(7.1-49)
where
i
∑ − 1
2
k = 1
λ ii = λ k
(7.1-50)
If the
λ i
’s are so small that the differences in the brackets of Eq.(7.1-46) and (7.1-49) can not
be obtained accurately in numerical calculation, we should use instead of Eqs.(7.1-46) and (7.1-49),
the following differential forms:
P
ij
2
=
Σ V
i
i
r i
{
1
2
∫ dρλ iλ
j K i1 ( λij
) + K i1(
λij
)} (7.1-51)
0
P
ii
=
Σ V
1 ri
i
i
1
{
∫ − 2
2
dρ λi
K i1(
λi
/ 2) + λi
K i1(
λii
)} (7.1-52)
0
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