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7.4.3 Solution by Matrix Inversion in Fast Neutron Range
For the fixed source problem in the fast neutron range, the emission rate can be rewritten by
H ig = S ig
+
g
G
∑Σ
smg' →gϕig
' + χ mg ∑νΣ
fmg'
ϕig'
.
g' = 1
g'
= 1
(7.4.3-1)
The fixed source term S ig usually consists of the thermal fission neutron. The scattering term is
determined by the fluxes of the upper energy groups. Given the fast fission term, the fluxes can be
successively solved starting at the highest energy group. The number of the unknowns to be
simultaneously solved is the total number of regions, N.
We have a choice for the solution whether by an iterative method or by a matrix inversion. As
far as N is less than about 40, the round-off error due to the limited computer precision encountered in
the matrix inversion can be negligible. The computer time required for the matrix inversion
(proportional to N 3 ) does not so much exceed that for the iterative method (proportional to the iteration
count*N 2 ). In the SRAC code system, the matrix inversion is applied preferring its definitive solution.
After finding the flux distribution for all the energy groups, we have to modify the assumed fast
fission source distribution by an iterative process. This power iteration converges rapidly for the case
of a thermal reactor because the ratio of fast fission to thermal fission is small. Contrary, the procedure
of this section can not be applied for the fast reactor where the matrix may have an eigenvalue greater
than unity.
The procedure described in this section is summarized as follows;
Step 1. Normalize the fixed source S ig .
Step 2. Set the initial guess for the fast fission distribution.
Step 3. Starting at the highest energy group g=1, calculate emission rates of a group H ig by
Eq.(7.4.3-1).
Calculate fluxes of a group by a matrix inversion.
Repeat Step 3 for all groups.
Step 4. Calculate fast fission distributions and modify it by an SOR.
Repeat Step 3 and 4, until the fast fission distribution converges.
7.4.4 Iterative Procedure for Eigenvalue Problem in Whole Energy Range
Considering the fact that while the iteration count in the thermal energy range amounts to
several tens, in the fast energy range any iterative process is not needed, the treatment for the thermal
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