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5.2
Discretize directions (“discrete ordinates”) and locations
( )
( )
(
)
( )
( )
( )
(
)
( )
( )
att
prim ph,
scat ph,
phot
prim ph,
scat ph,
0
0
ph ph,
prim ph,
N l
N l
m
m
m
m
m
m
l
i
l
i
l
l
i
l
i
l
l
m l
l
m l
l
i
l
E
E Y
E
E Y
E
Ω
φ
φ
Ω σ
φ
φ
Ω
σ
φ
= =−
= =−
→
⋅ ∇
+
+
⋅
+
=
′
=
∑ ∑
∑ ∑
uur
( )
( )
(
)
( )
scat ph,
0
....and similar for the electron equation
i
N l
m
m
m
i
l
i
l
i
E l
m l
E
E Y
E
∆
φ
Ω ∆
′ = =−
′
′
′
′
+
∑∑ ∑
Setting up equations:
The fluence moments contains an integral of the unknown fluence –
...,
m
φ
( )
( )
...,
...,
...,
d
n
m
m
m
l
l
n l
n
n
Y
w Y
φ
′
′
Ω
Ω
Ω
′
′
′
= Ω ⋅ Φ Ω ≈ ⋅
Ω ⋅ Φ
∑
∫
approximate it by a quadrature summation over a set of angles :
l
n
Ω
Finally, discretize also the locations
(Finite Elements), to yield a large,
sparse system of linear equations which
can be solved iteratively.
Gifford etal, PMB 51, 2253-65
10