�
�
�
�
�
�
�
�
�
Taking the two-dimensional Fourier transform with respect to r
� ,
and applying the convolution theorem, we obtain
∂
1
˜
˜
F (l, z) = − F (l, z) · [ 1 − σ ˜(l) ],
(4.212)
∂z
µ
where we have made use of the radial symmetry of F and σ, as
before. This is immediately integrated to reproduce the previous
result.
To this point we have only considered the distribution with respect
to angle or slope. It is also of considerable interest to discuss the
distribution with respect to transverse coordinates. This governs
the lateral broadening of electron probes in thick films, as well as
the resolution in transmission electron microscopes for thick specimens. In this case we must include the two-dimensional transverse
position r and the two-dimensional slope vector r
� . The geometry
is shown in Figure 4.9. We define a distribution function F (r, r
� ; z)
as the probability per unit area per unit solid angle for the particle at depth z, in the presence of plural scattering. Applying the
preceding logic, we expect F to satisfy
d
1
1
F (r, r ; z) = − F (r, r ; z) +
d
2 r 0 F (r, r 0 ; z) σ(|r
� − r |).
ds
µ
µ
0
(4.213)
To solve this equation for F , we begin by considering only one
transverse coordinate x, and one transverse slope component x
� .
This is equivalent to a projection of the plural scattering problem
onto the longitudinal xz-plane. The transport equation in this special case reduces to
d
1
1 d
2
�
F (x, x ; z) = − F (x, x ; z) +
r 0 F (x, x 0 ; z) τ (x
� − x 0 ).
ds
µ
µ
(4.214)
The single scattering distribution τ (x
� ) is a projection of the twodimensional single scattering distribution σ(r
� ). For the special
case of screened Coulomb scattering, this is given by
�2
r
1
τ (x
� ) =
∞
dy
� σ(x
� , y
� ) =
W
.
(4.215)
�2 + r
�2
−∞
2 (x
W ) 3/2
4.9. Small angle plural scattering of fast electrons
287
�
�
�
�
�
�
�
�
Taking the two-dimensional Fourier transform with respect to r
� ,
and applying the convolution theorem, we obtain
∂
1
˜
˜
F (l, z) = − F (l, z) · [ 1 − σ ˜(l) ],
(4.212)
∂z
µ
where we have made use of the radial symmetry of F and σ, as
before. This is immediately integrated to reproduce the previous
result.
To this point we have only considered the distribution with respect
to angle or slope. It is also of considerable interest to discuss the
distribution with respect to transverse coordinates. This governs
the lateral broadening of electron probes in thick films, as well as
the resolution in transmission electron microscopes for thick specimens. In this case we must include the two-dimensional transverse
position r and the two-dimensional slope vector r
� . The geometry
is shown in Figure 4.9. We define a distribution function F (r, r
� ; z)
as the probability per unit area per unit solid angle for the particle at depth z, in the presence of plural scattering. Applying the
preceding logic, we expect F to satisfy
d
1
1
F (r, r ; z) = − F (r, r ; z) +
d
2 r 0 F (r, r 0 ; z) σ(|r
� − r |).
ds
µ
µ
0
(4.213)
To solve this equation for F , we begin by considering only one
transverse coordinate x, and one transverse slope component x
� .
This is equivalent to a projection of the plural scattering problem
onto the longitudinal xz-plane. The transport equation in this special case reduces to
d
1
1 d
2
�
F (x, x ; z) = − F (x, x ; z) +
r 0 F (x, x 0 ; z) τ (x
� − x 0 ).
ds
µ
µ
(4.214)
The single scattering distribution τ (x
� ) is a projection of the twodimensional single scattering distribution σ(r
� ). For the special
case of screened Coulomb scattering, this is given by
�2
r
1
τ (x
� ) =
∞
dy
� σ(x
� , y
� ) =
W
.
(4.215)
�2 + r
�2
−∞
2 (x
W ) 3/2
4.9. Small angle plural scattering of fast electrons
287