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

We assume the system with neither source nor absorption where the neutron flux distribution is
uniform and isotropic everywhere. Then we suppose a cell in the entire space surrounded by a surface
S is divided into N regions. We consider the balance of the collision rate in the region i
N
ϕ
Σiϕ Vi
= SGi
+ ∑ PjiΣiϕV
j
, (7.1-20)
4
j = 1
where the subscript i denotes that the quantity is assigned to the region i; and
Σ i ; total macroscopic cross section,
ϕ ; uniform scalar flux,
V i ; volume,
S ; area of the surface,
G i ; probability that a neutron impinging on the surface has its first collision in the region i,
P ji ; probability that a neutron emitted in the region j has its first collision in the region i.
The term on the left hand side (L.H.S.) denotes the collision rate in the region i. The first term on the
right hand side (R.H.S.) denotes the contribution from the outside of the surface and the second term
the contribution of the emission in each region inside of the surface. Using the reciprocity theorem
Eq.(7.1-14) and the conservation theorem;
N
∑
j = 1
P ij + P
is
=
1
(7.1-21)
where P is is the probability that a neutron emitted in the region i escapes from the outer surface S
without suffering any collision, we have
G
i
V
S
P
i
= 4 Σi
is
(7.1-22)
Then we define G s as the probability that a neutron impinging from the outer surface into the inside of
the surface escapes from the surface without suffering any collision:
N
∑
G s = 1−
G i
i=
1
(7.1-23)
When the cells are set in an array, we get the collision probabilities for the lattice cell by using
the quantities for the isolated cell as follows:
which can be rewritten as
Pij
( lattice)
= Pij
( isolated)
+ PisG
j + PisGsG
j + PisGs
G j
2
+ ...... ,
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