�
�
�
�
�
�
�
�
�
observing in the xz-plane.
With these mathematical methods established, we are now in a
position to solve for the full three-dimensional distribution function F (r, r
� ; z) as a function of transverse coordinate r and slope
r
� at depth z. The rate of change of F with path length in polar
coordinates (r, φ) is given by the chain rule as
d
dr ∂
dφ ∂
dz ∂
F (r, φ, r
� , φ
� ; z) =
+
+
F (r, φ, r
� , φ
� ; z),
ds
ds ∂r ds ∂φ ds ∂z
(4.238)
where dr
� /ds = 0, and dφ
� /ds = 0, because the trajectories form
straight lines between scattering events. Making use of the axial
symmetry, F is independent of azimuth φ, in which case the second term on the right vanishes. We note that F depends on the
azimuthal slope component φ
� , as the scattering angle r
� has a
skew component in general for two or more scattering events. For
small angle scattering, dz/ds ≈ 1, in which case we can substitute
d
∂
∂
F (r, r
� , φ
� ; z) = r
�
+
F (r, r
� , φ
� ; z).
(4.239)
ds
∂r ∂z
Applying the logic of the preceding section, the transport equation
is
∂
∂
1
r
�
+
F (r, r
� , φ
� ; z) = − F (r, r
� , φ
� ; z)
∂r ∂z
µ
1 d
2
� − r
+
r 0 F (r, r 0 , φ
�
0 ; z) σ(|r
|),
0
µ
(4.240)
where σ(|r
� |) is the differential cross section for elastic scattering,
normalized to unity. As before, the first term on the right represents absorption into all scattering angles from the angle of interest, and the second term on the right represents emission from
all scattering angles into the angle of interest. This equation is
formally similar to the preceding case. Consequently, the preceding analysis can be adopted, being mindful of the various vector
292
Chapter 4. Particle scattering
�
�
�
�
�
�
�
�
observing in the xz-plane.
With these mathematical methods established, we are now in a
position to solve for the full three-dimensional distribution function F (r, r
� ; z) as a function of transverse coordinate r and slope
r
� at depth z. The rate of change of F with path length in polar
coordinates (r, φ) is given by the chain rule as
d
dr ∂
dφ ∂
dz ∂
F (r, φ, r
� , φ
� ; z) =
+
+
F (r, φ, r
� , φ
� ; z),
ds
ds ∂r ds ∂φ ds ∂z
(4.238)
where dr
� /ds = 0, and dφ
� /ds = 0, because the trajectories form
straight lines between scattering events. Making use of the axial
symmetry, F is independent of azimuth φ, in which case the second term on the right vanishes. We note that F depends on the
azimuthal slope component φ
� , as the scattering angle r
� has a
skew component in general for two or more scattering events. For
small angle scattering, dz/ds ≈ 1, in which case we can substitute
d
∂
∂
F (r, r
� , φ
� ; z) = r
�
+
F (r, r
� , φ
� ; z).
(4.239)
ds
∂r ∂z
Applying the logic of the preceding section, the transport equation
is
∂
∂
1
r
�
+
F (r, r
� , φ
� ; z) = − F (r, r
� , φ
� ; z)
∂r ∂z
µ
1 d
2
� − r
+
r 0 F (r, r 0 , φ
�
0 ; z) σ(|r
|),
0
µ
(4.240)
where σ(|r
� |) is the differential cross section for elastic scattering,
normalized to unity. As before, the first term on the right represents absorption into all scattering angles from the angle of interest, and the second term on the right represents emission from
all scattering angles into the angle of interest. This equation is
formally similar to the preceding case. Consequently, the preceding analysis can be adopted, being mindful of the various vector
292
Chapter 4. Particle scattering