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scattering kernel at point r’ from direction
Ω ' at energy E' to direction Ω with energy E, and
S( r ', Ω,
E)
is the neutron source at point r’ of direction Ω with energy E. In the above equation,
while the fission neutron source is not expressed explicitly, it may be included in the source term.
Here, we assume that the scattering and the source are isotropic;
1
Σ s ( r,
Ω'
→ Ω,
E'
→ E)
= Σ s ( r,
E'
→ E)
4π
,
(7.1-2a)
1
S( r,
Ω , E)
= S(
r,
E)
(7.1-2b)
4π
Integrating Eq.(7.1-1) over the whole angle of Ω , we obtain,
1
( , ) = ∞
Ω exp( − Σ )
⎡
' Σ ( ', ' → ) ( ', ')
+ ( ', )
⎤
4 4 0
⎢⎣ ∫
∞
ϕ r E ∫ d ∫ dR R dE s r E E ϕ r E S r E
π π
0
⎥⎦
where ϕ ( r,
E)
is the neutron flux at point r with E, and is defined by
,
(7.1-3)
∫
ϕ( r , E)
= dΩ
ϕ(
r,
Ω,
E)
(7.1-4)
π
4
Equation (7.1-3) can be rewritten by the relation
2
dr
' = R dRdΩ
∞
Σ(
r,
E)
ϕ(
r,
E)
= ' ( ' → , )
⎡
' Σ ( ', ' → ) ( ', ')
+ ( ', )
⎤
∫ dr
P r r E
⎢⎣ ∫ dE s r E E ϕ r E S r E
0 ⎥⎦
,
(7.1-5)
where
Σ(
r)
P(
r ' → r,
E)
= exp
⎛
⎞
⎜−
Σ ⎟
⎝ ∫ R s ( s)
ds
(7.1-6)
2
4πR
0 ⎠
By the form of Eq.(7.1-6), the reciprocity relation holds,
Σ( r',
E)
P(
r'
→ r,
E)
= Σ(
r,
E)
P(
r → r',
E)
(7.1-7)
We divide the whole system under consideration into several regions. Each region is assumed
homogeneous with respect to its nuclear properties, but different regions are not necessarily of
different materials. The region is the spatial variable in the collision probability method. We denote
space dependent cross sections with subscript i which are associated to the region i. Integrating
Eq.(7.1-5) over V j , we obtain
Σ ( E)
j
∫
V j
∞
ϕ ( r,
E)
dr
= dr
dr'
⎡
s ( r',
E'
E)
( r',
E'
) dE'
S(
r',
E)
⎤
∑∫
⋅ P(
r'
→ r,
E) .
V ∫ Σ →
+
j Vi
⎢⎣ ∫
ϕ
(7.1-8)
0
⎥⎦
i
We make the flat flux approximation so that the neutron flux ϕ ( r,
E)
is assumed constant in
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