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• the reduced collision probability which has finite value even if the collision region j is
vacuum;
P
#
ijg P ijg
= / Σ
jg
• the emission rate of the region i for the group g;
H ig
• Nuclear constants of the material m;
Σ mg
= total cross-section,
ν Σ fmg
= production cross-section,
Σ amg
= absorption cross-section,
Σ smg ' →g = scattering cross-section from the group g’ to the group g
Σ
rmg
=
∑
g'
∉G
Σ
∑
smg→g
' + χ mg ' νΣ
fmg ,
g ' ∉G
scattering-out cross-section which is used to evaluate the neutron balance of the
system. It has non-zero value when a fixed source problem of a limited energy
range is considered. The quantity χ mg stands for the fission yield to the group g.
Using the above definitions, the equation to be solved is written by
ϕ
ig
=
∑
j
P # jig
H
jg
(7.4.1-2)
The emission rate for the fixed source problem is written by
H ig = S ig
+
G
∑
Σ
ϕ
∑
smg ' →g
ig '
mg
g ' = 1
g ' = 1
+
χ
G
νΣ
fmg '
ϕ
ig '
(7.4.1-3a)
where m denotes the material assigned to the region i.
For the eigenvalue problem,
H ig
=
G
∑
g'
= 1
Σ
smg'
→g
χ G
mg
ϕ
ig '
+ ∑νΣ
fmg'
ϕig
'
(7.4.1-3b)
λ
g'
= 1
Equation (7.4.1-2) coupled with Eq.(7.4.1-3a) forms inhomogeneous equations and that coupled with
the Eq.(7.4.1-3b) forms homogeneous equations. In both problems, the number of unknown is N*G.
The general matrix of the same rank consists of (N*G) 2 elements, however, the computer storage
required for the above equations is at most (N 2 G+MG 2 < NG(N+G): (N 2 G) for the collision probability
and (MG 2 ) for scattering matrix, where M is the number of materials. The size (G 2 ) for the scattering
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