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each region, for example, it is expressed by ϕ (E)
in the region i. This assumption leads the
equation,
i
Σ ( E ) V ϕ ( E)
=
⎡
)
⎤
j
j
j
∞
∑ Pi
j ( E)
Vi
⎢⎣ ∫ Σ s i ( E'
→ E)
ϕi
( E)
dE'
+ Si
( E
0
⎥⎦
i
(7.1-9)
where the collision probability P i j (E) is defined by
P
i j
Σ j ( E)
( E)
= d
4 V
∫ r
π ∫
i
V j V i
exp( −ΣR)
dr'
2
R
(7.1-10)
It is explained as the probability that a neutron emitted uniformly and isotropically in the region i has
its next collision in the region j. We divide the neutron energy range into multi-groups. The average
flux in the energy interval
simultaneous equation,
ΔE g
is denoted by
ϕ ig
. Then from Eq.(7.1-9), we obtain the
ΔE
g
Σ
jg
V ϕ
j
jg
=
∑
i
P
ijg
⎡
Vi
⎢
⎢⎣
∑
g'
ΔE
g'
Σ
s i g'
→g
ϕ
i g'
+ ΔE
S
g
ig
⎤
⎥
⎥⎦
(7.1-11)
E g
E g '
where Δ and Δ are the energy width of the group g and g’ and
cross-section in the region i from the group g’ to g, and is defined by
Σ ' is the scattering
s i g →g
Σ = dE'
dE Σ ( E'
→ E)
ϕ ( E'
) dE'
ϕ ( E'
)
(7.1-12)
s i g'
→g
∫
ΔEg'
∫
ΔEg
si
i
As seen in the above derivation, once we obtain the collision probabilities, the neutron flux can
be easily obtained by solving the simultaneous equation Eq.(7.1-11) by means of a matrix inversion or
an iterative process.
Now we focus our considerations on the collision probability. The definition of the collision
probability Eq.(7.1-10) can be expressed equivalently by
∫
ΔEg
'
i
P
ij
= 1
R j +
⎛ ⎞
∫ ∫ Ω∫
Σ ⎜−
Σ ⎟
⎝ ∫
R
dr d dR j exp ( s)
ds
(7.1-13)
4πV
Vi 4π
R j −
0 ⎠
i
where the subscript to indicate the energy group is, hereafter, dropped for simplicity, and R j- and R j+
denote the distances from point r to the inner and outer boundaries of the region j along the line
through the points r and r'.
From the form of Eq.(7.1-10), it can be seen easily that the reciprocity relation holds,
P Σ V = P Σ V
(7.1-14)
ji
j
j
ij
i
i
This relation is, as shown later, utilized to reduce the angular range of numerical integration.
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