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For the parallel direction we can easily obtain the explicit form, but it is not necessary because the
following relation holds:
1 2
P ij = Pij⊥ + Pij
//
(7.1-38)
3 3
The relation is derived from
∑
1 = Ω k = Ω ⊥
k
2
2
+ 2Ω
2
//
So we can obtain
P ij //
by subtraction.
At the end of this section we show the expression for the escape probability P is :
P
is
1
1
2
{ E ( λ ) − E ( λ + λ ) + E ( λ ) − E ( λ + λ )}
1 2
= i3 is i3
is i i3
is i3
is i
(7.1-39)
2λ
i
1
is
i−1
∑
∑
where λ = λ , λ = λ .
k = 1
k
2
is
N
k
k = i+
1
7.1.3 Collision Probabilities for One-dimensional Cylindrical Lattice
We consider the infinitely long cylinder which is divided into several annular shells. The outer
radius of the shell i is r i . We suppose that a neutron emitted at the point P in the shell i has its first
collision at the point Q in the shell j. The position of P is defined by only the distance from the
cylindrical axis; r. The line PQ makes an angle θ with the vertical line. We define the point Q’ as the
projection of the point Q on the horizontal cross-section so that the line PQ’ makes an angle θ with the
line PO.
The distance between P and Q’ is R. In the cylindrical coordinate system as shown in Fig.7.1-2a,
we have the collision probability P ij as,
P
ij
2 ri
π π / 2
R j +
Σ j ⎧ R Σ(
s)
⎫
= ∫ rdr∫ dβ∫ sinθ
dθ∫
dR exp⎨−
∫ ds⎬
V ri
−1 0 0 R j −
0
i
sinθ
⎩ sinθ
⎭
, (7.1-40)
where
V
2 2
i = π ( ri
− ri
−1
)
(7.1-41)
Then we transform the variables r, β and R into new ones ρ, x and x’ as illustrated in Fig.7.1-2b. We
define the perpendicular distance OM from O to the line PQ’ by ρ, the distance between P and M by x,
and the distance between Q’ and M by x’. There are three relations among variables:
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