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7.4 Solution of Linear Equation
It is one of the features of the SRAC code system to execute the cell calculations by the
collision probability method to cover the whole neutron energy range. In this section we shall describe
how to solve the linear equation introduced in Sect.7.1.
Because of the difference of physical characteristics, the different specialized equation is
formulated separately by neutron energy range. Although the concatenation of the equations into a set
of equations so as to describe the quantities in the whole neutron energy is available, the cell
calculation is usually achieved separately by neutron energy range. Partly because there occurs no
up-scattering in the epi-thermal and fast neutron energy range, but does in the thermal neutron range
where any iterative process among the energy group variables is required. Partly because the neutron
flux distribution in the fast and epi-thermal neutron range is relatively flat, which allows coarse spatial
division of the cell model or the overall flat flux assumption coupled with some suitable resonance
shielding treatments. However, the distribution in the thermal neutron range shows sharp spatial
change due to small flight path. In this energy range needs fine spatial division. It is to be noted that
although there occurs the sharp flux depression due to the strong resonance structure of the fertile
nuclides near the resonance energy, fine spatial division is not necessary to evaluate the overall
resonance absorption. Because neutrons scarcely come out from the place where the depression occurs,
the shape of the depression does not affect the absorption rate.
7.4.1 General Form of Linear Equation
When the system under consideration is divided into N regions and the neutron energy range
is divided into G groups, Eq.(7.1-9) is rewritten as
Σ
jg
N
⎪⎧
G
⎪⎫
ϕ
jg = ∑P
ijg ⎨∑Σ
sig'
→gϕig
' + Sig
⎬
(7.4.1-1)
i= 1 ⎪⎩ g'
= 1
⎪ ⎭
The physical quantities are redefined as follow s:
• the volume of the region i;
V
i
= ∫ dV
V i
• the integral flux over the region i for the energy group g;
∫
ϕ ig = ∫ dV dE ϕ(r, E)
Δ
V i E g
• the fixed source;
∫
S ig = ∫ dV dE S(r
, E)
Δ
V i E g
• the collision probability from the region i to j for the group g;
P ijg
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