�
�
and l φ is regarded as constant under the integral.
Investigating the limiting cases, we see that, for z = 0,
˜
F (k, l r , l φ ; 0) = 1.
(4.250)
Performing the inverse transform,
F (r, r
� , φ
� ; 0) = δ(r) δ(r
� ),
(4.251)
thus recovering the incident beam, as required.
In the limit k → 0, we can write the Taylor expansion for
g ˜(l r + kz, l φ ) to first order as
∂
g ˜(l r + kz, l φ ) = ˜
g(l r , l φ ) + kz ·
g ˜(l r , l φ ),
(4.252)
∂l r
in which case,
z
F ˜ (k, l r , l φ ; z) = F ˜ (0, l r , l φ ; z) = exp − [ 1 − σ ˜(l) ] , (4.253)
µ
remembering that l = |l| = l r
2 + l φ
2 . This is immediately recognizable as the angular distribution. This is expected, as k = 0 in
Fourier space represents an integral over the entire range of radial
coordinate, 0 ≤ r < ∞ in direct space. This result is superfluous,
as it was derived previously by simpler methods.
Finally, setting l = 0, we find
F ˜ (k, l r , l φ ; z) = F ˜ (k, 0, 0; z) = exp
1 [ ˜
g(0, 0) − g ˜(kz, 0) ] ,
kµ
(4.254)
where this represents the integral over all slopes |l| in direct space.
The distribution in the transverse radial coordinate r is found by
performing the inverse Bessel transform,
F (r; z) =
∞
dk k J 0 (kr) F ˜ (k, 0, 0; z).
(4.255)
0
294
Chapter 4. Particle scattering
�
and l φ is regarded as constant under the integral.
Investigating the limiting cases, we see that, for z = 0,
˜
F (k, l r , l φ ; 0) = 1.
(4.250)
Performing the inverse transform,
F (r, r
� , φ
� ; 0) = δ(r) δ(r
� ),
(4.251)
thus recovering the incident beam, as required.
In the limit k → 0, we can write the Taylor expansion for
g ˜(l r + kz, l φ ) to first order as
∂
g ˜(l r + kz, l φ ) = ˜
g(l r , l φ ) + kz ·
g ˜(l r , l φ ),
(4.252)
∂l r
in which case,
z
F ˜ (k, l r , l φ ; z) = F ˜ (0, l r , l φ ; z) = exp − [ 1 − σ ˜(l) ] , (4.253)
µ
remembering that l = |l| = l r
2 + l φ
2 . This is immediately recognizable as the angular distribution. This is expected, as k = 0 in
Fourier space represents an integral over the entire range of radial
coordinate, 0 ≤ r < ∞ in direct space. This result is superfluous,
as it was derived previously by simpler methods.
Finally, setting l = 0, we find
F ˜ (k, l r , l φ ; z) = F ˜ (k, 0, 0; z) = exp
1 [ ˜
g(0, 0) − g ˜(kz, 0) ] ,
kµ
(4.254)
where this represents the integral over all slopes |l| in direct space.
The distribution in the transverse radial coordinate r is found by
performing the inverse Bessel transform,
F (r; z) =
∞
dk k J 0 (kr) F ˜ (k, 0, 0; z).
(4.255)
0
294
Chapter 4. Particle scattering