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∑
{ R K ( }
σ ϕ + σ ϕ = σ K ( ϕ ) + K ( σ ϕ ) + σ
ϕ )
(7.3.2-5)
f
f
m
m
am
am
f
f
s
f
m
k
k
k
m
where
ϕ f , ϕ m = flux per unit lethargy in the lump and moderator region, respectively,
σ ( u)
= σ ( u)
+ σ ( u)
+ σ = microscopic total cross-section of the lump,
σ
σ
f
am
= Σ
am
a
/ n
f
s
am
= scattering cross-section of admixed moderator per absorber atom,
= Σ V /( n V ), R = Σ / Σ , Σ = ∑ Σ ,
m
m
f
f
k
k
v ,v m = volumes of the lump and the moderator regions, respectively,
f
n f
m
= atomic number density of the resonance absorber in the lump,
K = slowing down operator,
p ff = collision probability in the fuel lump.
The other notation is conventional.
Wigner:
m
m
We make use of the simple interpolation formula for the collision probability as proposed by
k
k
p
ff
=
X
X σ f
=
+ g(
C)(1
− C)
σ + s
f
(7.3.2-6)
with
X
= l n σ = l Σ
(7.3.2-7)
f
f
f
f
f
a
s = g(
C)(1
− C) /( l f n f ) and g(
C)
=
(7.3.2-8)
1 + ( a −1)
C
where
l f
quantity.
: the lump mean chord length, C: the Dancoff correction factor and a is a purely geometrical
Generalized collision probability theory shows
p ff
1−
C
≅1 − for X → ∞
(7.3.2-9)
X
where the Dancoff correction factor C is zero for isolated lumps. Being based on Eq.(7.3.2-9), the
Dancoff factor C is calculated by using the value of p ff for a sufficiently large value of Σ f as Σ f = 300
cm -1 − C = { 1 − p ( Σ )} X Σ →∞
1 (7.3.2-10)
ff f |
f
A rational interpolation of p ff leads to g(C) = 1 in Eq.(7.3.2-6) when use is made only of the
behavior of p ff at
Σ f → ∞ given by Eq.(7.3.2-9). Since the bulk of resonance absorption occurs at
finite values of Σ f , we need some corrections for the rational approximation. It is this quantity a that
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