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

where
λ
λ
1
is
2
is
=
=
N
∑
k
k = i+
1
i−1
∑
λ
λ
+
∑
k
k = 1 k = 1
,
N
λ
k
.
(7.1-55)
7.1.4 Collision Probabilities for Spherical System
A spherical system is divided into N spherical shells. We define the shell i that is bounded by
two spherical surfaces of radii r i-1 and r i . The shells are numbered by increasing order of r i . In general,
a probability P ij that a neutron emitted in the region i has its first collision in the region j is defined as
P
ij
1
=
4πV
i
⎧
⎩
∫ dV∫
dΩ∫
dRΣ
j exp⎨−
⊂ ∫ Σ(
s)
ds
⎭ ⎬⎫
V
4π
R
i V j
0
R
(7.1-56)
The integrand on R.H.S. of Eq.(7.1-56) is interpreted as follows by referring Fig.7.1-3a. A neutron
emitted at a point P in the region i moves toward a point Q which is in distance R from the point P, has
the exponential decay by the optical length
R
∫ Σ ( s)
ds
0
and suffers its collision at the layer of
thickness dR in region j of the cross-section Σ j . In the spherically symmetric system the position of the
point P is defined only by the distance r from the center of the system, C. The position of the point Q
is defined by the distance R from the point P, and the angle θ made by the lines PQ and PC (See
Fig.7.1-3a). In this coordinate system,
dΩ = 2π
sinθdθ
2
dV = 4πr
dr
0 < r < r
N
0 ≤ θ ≤ π ,
,
and Eq.(7.1-56) is rewritten by a triple integral form:
4πΣ
j rN
2
π
R
P
⎧
ij = ∫ r dr∫ sinθdθ∫ dR exp⎨−
⊂ ⎩ ∫ Σ(
s)
ds
⎭ ⎬⎫ . (7.1-57)
V 0 0 R V j
0
i
235