�
�
�
�
�
�
�
�
We will not assume this special dependency in the following analysis, but rather a general form for small-angle scattering τ (x
� ).
Applying the chain rule, and expanding the total derivative as
before, we obtain
d
ds
F (x, x
� ; z) =
dx
ds
∂
∂x
+
dx
�
ds
∂
∂x � +
dz
ds
∂
∂z
F (x, x
� ; z).
(4.216)
Again dx
� /ds = 0. For small angles, dx/ds ≈ x
� and dz/ds ≈ 1,
leading to
∂
∂
1
1
x
+
F (x, x ; z) = − F (x, x ; z) + F (x, x ; z) ∗ τ (x
� ).
∂x ∂z
µ
µ
(4.217)
288
Chapter 4. Particle scattering
Figure 4.9: Geometry for plural scattering at depth z.
�
�
�
�
�
�
�
We will not assume this special dependency in the following analysis, but rather a general form for small-angle scattering τ (x
� ).
Applying the chain rule, and expanding the total derivative as
before, we obtain
d
ds
F (x, x
� ; z) =
dx
ds
∂
∂x
+
dx
�
ds
∂
∂x � +
dz
ds
∂
∂z
F (x, x
� ; z).
(4.216)
Again dx
� /ds = 0. For small angles, dx/ds ≈ x
� and dz/ds ≈ 1,
leading to
∂
∂
1
1
x
+
F (x, x ; z) = − F (x, x ; z) + F (x, x ; z) ∗ τ (x
� ).
∂x ∂z
µ
µ
(4.217)
288
Chapter 4. Particle scattering
Figure 4.9: Geometry for plural scattering at depth z.