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The LBTE is linear so assuming:
For any direction, we can split our system of equations in two:
1. I need to be efficient, angular discretization
How can I relax the demand on S
N
without ray effects?
)ˆ,(
)
ˆ,
(
1
,
)ˆ,(
+
=
=
r coll
g
r
P
p
unc
gp
rg
)1(
)
(
1 4
,
,
ˆ
pr r
P
p
gpq
unc
g gt
unc
g
−
=
=
+
)2(
1
,
,
,
,
ˆ
=
+
=
+
P
p
unc
scat
gpq
coll
scat
g
q
coll
g gt
coll
g
Equation (1) for the primary (uncolided) part of the fluence can be analytically solved!!!
I can ray-trace the solution for the spectrum of primary photons through the geometry and
arrive at very quick and accurate: initial guess
and q
scat,1,n
0 )
( ,1
=
iter
rn
Then I proceed to solve the system of equations (2) for the collided fluence to refine my
solution with the higher orders of scatter
This is known as the
1
st
scatter source method