�
�
�
components. The transformed equation is
−k
∂
∂l r
+
∂
∂z
˜
F (k, l r , l φ ; z) = −
1
µ
˜
F (k, l r , l φ ; z) [ 1 − ˜
σ(l) ],
(4.241)
where (k, l r , l φ ) are the Fourier transform variables corresponding
to (r, r
� , φ
� ), respectively, and
l = |l| = l 2 + l
2
(4.242)
r
φ .
Applying the method of characteristics, we define the variables
ξ = l r + kz,
η = l r − kz,
(4.243)
in which case, the transformed equation reduces to
∂
1
˜
˜
−2k
F = − F [ 1 − σ ˜(l) ].
(4.244)
∂η
µ
Integrating this with respect to η,
1
˜
ln F =
dη [ 1 − σ ˜(l) ].
(4.245)
2kµ ξ
Noting that
2 l r = ξ + η,
2 dl r = dξ + dη
(4.246)
with ξ = const, and hence dξ = 0 for the integration. This gives
lr
1
ln F ˜ =
dl r 1 − σ ˜
l r
2 + l φ
2
.
(4.247)
µk lr+kz
This is immediately integrated to give
1
˜
F (k, l; z) = exp
[ ˜
g(l r , l φ ) − g ˜(l r + kz, l φ ) ] ,
(4.248)
kµ
where we have defined ˜
g(l r , l φ ) by the indefinite integral
g ˜(l r , l φ ) = dl r 1 − σ ˜
l r
2 + l φ
2
,
(4.249)
293
4.9. Small angle plural scattering of fast electrons