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hyper-fine groups to give their representative values for the medium. That is, the homogenized
cross-section σ h is obtained by the flux-volume weighted average of the resonance cross-section σ f ,
i.e.,
σ h ( u)
=
V
σ ( u)
V
f
f
ϕ ( u)
+ V
f
f
ϕ ( u)
f
ϕ ( u)
m
m
.
If we define the self-shielding factor f
f
( V
f
+ Vm
) ϕ
f
( u)
= (7.3.5-12)
V ϕ ( u)
+ V ϕ ( u)
f
f
m
m
we can obtain σ h by a simple volume-weighted average of the effective cross-section, fσ f as
σ ( u)
=
h
f σ ( u)
V
V
f
f
+ V
m
f
.
Thus, the effective microscopic cross-section fσ f can be treated as if it is the resonance cross-section of
one constituent in a homogeneous medium. The resulted variation of the cross-section with neutron
energy is much smooth, compared with the original cross-section, due to the spatial shielding. The
shielded cross-sections are prepared for all the reactions of the resonant nuclides under consideration.
(2) Collision Rate Method
As noted before, the fraction of collision rate σl was introduced as to be independent of the
origin of neutron. We can suppose several origins of neutron, for example, a neutron just having
escaped from the absorber grain, a neutron emitted from the diluent, a neutron impinging on the outer
surface of the imaginary cell, or a neutron beam injected into the medium of grain structure, etc.
Now we introduce a quasi-analytic expression of the collision probability for the imaginary cell
to give a physical explanation. The following approximation was derived by Nordheim 72) :
⎡ 1 − c ⎤
Q(
f , m)
= Pe ⎢
⎥ ,
(7.3.5-13)
⎢⎣
1 − (1 − l f Σ f Pe
) c ⎥⎦
where P e (=1-P c ) is the escape probability 27) from an individual grain, c, the Dancoff correction, and Σ f
is the macroscopic cross-section of the absorber grain and its mean chord length
l f
is given by
4V
f
l
f
= ,
(7.3.5-14)
S
f
where S f is the surface area of the absorber grain. The conservation and reciprocity relations give other
elements of Q(l, k).
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