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In Fig.7.1-5, the line PQ’ defined by ρ and ϕ is the projection of the neutron line PQ on the
horizontal plane. The points P and Q are, respectively, the source and collision positions. The point A
is the origin of measures of t, t’ and s. The points B, C, D and E are the points of intersection of the
line PQ’ with the region boundaries. A restriction on the moving direction of a neutron is imposed so
that a neutron moves only to the positive direction of t along the line PQ’. If the line PQ’ enters the
region j more than once, a sum of Eq.(7.1-66) is required.
The self collision probability, P ii is expressed by the following Eq.(7.1-67), where the point Q is
in the region i.
P
ii
⎡
= ⎢
⎢⎣
+∞
∫ ρdρ∫ dϕ∫ sinθdθ∫
−∞
⎡
⎢⎣
+∞
∫ ρdρ∫ dϕ∫ sinθdθ∫
−∞
2π
0
2π
0
π / 2
0
π / 2
0
AC
AB
dt
AC
AB
∫
t
AC
dt
⎤
⎥⎦
Σ ⎪⎧
Σ t − t'
⎪⎫
⎤
i
i
dt'
exp⎨−
⎬⎥
sinθ
⎪⎩ sinθ
⎪⎭ ⎥⎦
(7.1-67)
If the line PQ’ reenters the region i, a sum of such a term as Eq.(7.1-66) is also required for obtaining
P ii .
The six-fold integrals of Eqs.(7.1-66) and (7.1-67) are reduced to the double integrals as
follows:
P
ij
i
i
{ K ( λ ) − K ( λ + λ ) − K ( λ + λ ) + K ( λ + λ + λ }
1 +∞ 2π
=
Σ
∫ dρ
−∞ ∫ dϕ
i3
0 i3
0 i i3
0 j i3
0 i j ) , (7.1-68)
2π
V
0
P
ii
1 +∞ 2π
=
Σ
∫ dρ
−∞ ∫ dϕ
i i3
i3
i )
2π
V
0
i
i
{ λ − K (0) + K ( λ }
, (7.1-69)
where λ i and λ j denote the optical path lengths (the physical path multiplied by the macroscopic total
cross section), λ i =BC*Σ i , λ j =DE*Σ j and λ 0 stands for the sum of optical path lengths between C and
D; and K i3 is the third order Bickley-Naylor function.
The escape probability P is defined as a neutron emitted in the region i escapes from the surface
without suffering collision, is expressed as
P
is
1 +∞ 2π
=
Σ ∫ dρ
−∞ ∫ dϕ
i3
is i3
is i )
2π
V
0
i
i
{ K ( λ ) − K ( λ + λ }
, (7.1-70)
where λ is is the optical path length along the line from the edge of the region i to the surface of the
system.
As for the directional probabilities, similarly to the case of the one-dimensional cylinder, it is
not necessary to write the whole components and hence a few samples are shown here:
P
ijr
{ K ( λ ) − K ( λ + λ ) − K ( λ + λ ) + K ( λ + λ + λ }
3 +∞ 2π
=
Σ
∫ dρ
−∞ ∫ dϕ
i5
0 i5
0 i i5
0 j i5
0 i j )
4π
V
0
i
i
(7.1-71)
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