�
�
F (x, x
� ; z) in principle. Typically, this last integral is performed
numerically, after subtracting the unscattered beam exp(−z/µ) =
exp(−n) from F ˜ .
It is instructive to investigate several limiting cases. In the limit
of zero thickness, z = 0, we find immediately that
˜
F (k, l; 0) = 1.
(4.232)
Performing the inverse transform, this leads to
F (x, x
� ; 0) = δ(x) · δ(x
� ),
(4.233)
thus recovering the incident beam, as required.
In the limit k → 0, a Taylor expansion gives us
g ˜(l + kz) = ˜
g(l) + ˜
g
� (l) kz,
(4.234)
to first order in k. In this limit, F ˜ reduces to
z
˜
F (0, l; z) = exp − [ 1 − τ ˜(l) ] ,
(4.235)
µ
which represents the projected angular distribution. This is expected, as k = 0 in Fourier space represents an integral over all x
in direct space.
In the limit l = 0, we obtain
1
˜
F (k, 0; z) = exp
[ ˜
g(0) − g ˜(kz) ] .
(4.236)
kµ
Setting l = 0 in Fourier space represents an integral over all scattering angles in direct space. The distribution F (x, 0; z) in direct
space is obtained from the inverse Fourier transform
F (x, 0; z) =
1 ∞
dk exp(−ikx) F ˜ (k, 0; z).
(4.237)
2π −∞
Physically, this represents the line spread function, corresponding
to scanning an incident probe beam along the infinite y-axis, and
291
4.9. Small angle plural scattering of fast electrons
�
F (x, x
� ; z) in principle. Typically, this last integral is performed
numerically, after subtracting the unscattered beam exp(−z/µ) =
exp(−n) from F ˜ .
It is instructive to investigate several limiting cases. In the limit
of zero thickness, z = 0, we find immediately that
˜
F (k, l; 0) = 1.
(4.232)
Performing the inverse transform, this leads to
F (x, x
� ; 0) = δ(x) · δ(x
� ),
(4.233)
thus recovering the incident beam, as required.
In the limit k → 0, a Taylor expansion gives us
g ˜(l + kz) = ˜
g(l) + ˜
g
� (l) kz,
(4.234)
to first order in k. In this limit, F ˜ reduces to
z
˜
F (0, l; z) = exp − [ 1 − τ ˜(l) ] ,
(4.235)
µ
which represents the projected angular distribution. This is expected, as k = 0 in Fourier space represents an integral over all x
in direct space.
In the limit l = 0, we obtain
1
˜
F (k, 0; z) = exp
[ ˜
g(0) − g ˜(kz) ] .
(4.236)
kµ
Setting l = 0 in Fourier space represents an integral over all scattering angles in direct space. The distribution F (x, 0; z) in direct
space is obtained from the inverse Fourier transform
F (x, 0; z) =
1 ∞
dk exp(−ikx) F ˜ (k, 0; z).
(4.237)
2π −∞
Physically, this represents the line spread function, corresponding
to scanning an incident probe beam along the infinite y-axis, and
291
4.9. Small angle plural scattering of fast electrons