JAEA-Data/Code 2007-004 - Welcome to Research Group for ...

We shall discuss the following two models of neutron origin for calculating the collision rates.
1) Neutron emitted from absorber grain
Segev 73) proposed an expression for the self-shielding factor which was obtained by equating
the term in the brackets of Eq.(7.3.5-13) to 1/(1 + fV Σ / V Σ )] which is the probability that a
[ f f m m
neutron escaping from a lump will collide in the diluent of the homogenized medium. The resultant
expression is
f
c V Σ l f
= Pe
(7.3.5-15)
1 − c
m m
V f
Inserting this expression into Eqs.(7.3.5-5), (7.3.5-6) and (7.3.5-8), we obtain
l
f
Σ
f
Pe
c
α f ( f ) =
,
(7.3.5-16)
1 − (1 − l Σ P ) c
f
f
e
1 − c
α m ( f ) =
,
(7.3.5-17)
1 − (1 − l P ) c
f Σ f
e
where the index (f) of α f (f) denotes that the origin of neutron is fuel grain. It can be explained that the
fraction of collision rate is measured for a neutron just having escaped from the absorber grain. That is,
the probability that the neutron has the next collision with other grains is given from the expression of
Eq.(7.3.5-13)
Q(
f , f ) − P
c
= 1 − Q(
f , m)
− (1 − P )
l f Σ f Pe
=
1 − (1 − l Σ
f
2
f
c
P ) c
e
e
.
(7.3.5-18)
On the other hand, the collision with the diluent is given by Q(f, m), which can be calculated by the
reciprocity relation. Dividing these quantities by the escape probability P e , we have the fractions of
collision rates; α f and αm just defined by Eqs.(7.3.5-16) and (7.3.5-17). Inserting Eqs.(7.3.5-16) and
(7.3.5-17) into Eqs.(7.3.5-5) and (7.3.5-6), we obtain the same expression of f as Eq.(7.3.5-15).
2) Neutron emitted from the diluent
We can define the collision rate for a neutron emitted from the diluent. It is given directly by
either Q(m, f) or Q(m, m). A little algebra using the reciprocity relation gives
V f Σ f Pe
(1 − c)
α
f
( m)
=
,
(7.3.5-19)
V Σ 1 − (1 − l Σ P ) c
m
m
where the index m of α f (m) denotes that the origin of neutron is the diluent.
f
f
e
269