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The integration by R between R j- and R j+ in Eq.(7.1-13) can be performed analytically in the
homogeneous region j,
∫
R
R
j +
j −
dRΣ
j
exp( −ΣR)
= exp( −ΣR
j−
= exp( −ΣR)
) ×
R=
R
{
j −
1 − exp
−
( − Σ ( R − R ))}
exp( −ΣR)
The summation over j along the direction Ω leaves only the first term of exp( Σ R ) 0 = 1,
then the
conservation law is easily shown as,
∑
j
j
j+
R=
R
j +
j−
− R=
1
P = Ω = 1
4
∫ ∫
ij d d
πV
r (7.1-15)
V i 4π
i
Similarly, the directional probabilities P ijk defined by Benoist 25) which is used to provide the
Behrens term of the anisotropic diffusion coefficients is expressed by
P
ijk
= 1
R
2 j +
Ω3Ω
Σ exp
⎛
( )
⎞
⎜−
Σ ⎟ ,
4
∫ ∫4
∫
⎝ ∫
R
dr d k dR j
s ds
(7.1-16)
πV
Vi
π
R j −
0 ⎠
i
where k stands for direction, for example the parallel or perpendicular to the boundary plane in the
case of plane lattice, and Ω k denotes the directional cosine of Ω in the direction k. The following
relation holds:
2
∑ Ω =
k
k 1
, (7.1-17)
The extension to include surfaces given by Beardwood 43) is as follows: If S is any surface (not
necessarily closed) such that no line drawn outwards from a surface point S crosses S more once,
P
is
( − S )
= 1 r ⎛
∫ dr∫
d S exp⎜
−
V
⎝ ∫
V
3
4π
i R
i
s
0
R
s
Σ(
s)
ds
⎞
⎟
(7.1-18)
⎠
is the probability that a neutron emitted from the region i crosses the outer boundary S, or an
alternative expression,
P
is
= 1
⎛
⎞
∫ ∫ Ω ⎜−
Σ ⎟
⎝ ∫
Rs
dr d exp ( s)
ds
(7.1-19)
4πV
Vi
4π
0 ⎠
i
is given, where R s is a distance from the emitting point r to the surface point S .
The isotropic reflective boundary condition is frequently used for the lattice cell calculation not
only in the collision probability method but also in the Sn calculation. We shall describe its physical
meaning and the application in the collision probability method using the explanation given by
Bonalumi 48) .
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