401
H.2. Expression for the scattering amplitude: Born approximation
Note that in the ‘forward direction’ (i.e. within a region close to the z-axis, as
determined by the collimation), the incident and scattered waves will interfere. Careful analysis reveals a depletion of the incident beam in the forward
direction (the ‘shadow’ of the scattering centre), which corresponds exactly
to the total flux scattered into all angles (Gottfried 1966, section 12.3). This
is expressed in the optical theorem:
k
Im f (0) =
σ.
(H.10)
4π
H.2 Expression for the scattering amplitude: Born
approximation
We begin by rewriting (H.2) as
(∇
2 + k
2 )φ(r) =
2m V (r)φ(r).
(H.11)
ħ 2
This equation is of exactly the form discussed in appendix G, e.g. equa∇
2 + k
2
tion (G.18) with Ω r =
. Further, we know that the Green function
for this Ω r , corresponding to the desired outgoing wave solution, is given by
(G.14). Using then (G.19) and (G.14), we can immediately write the ‘formal
solution’ of (H.11) as
∫
ik|r−r
' |
2m
1 e
ik·r
′
φ(r) = e
+
−
V (r
′ )φ(r
′ ) d
3
r
(H.12)
ħ 2
4π |r − r ′ |
where we have chosen ‘u(r)’ in (G.19) to be the incident plane wave φ inc , and
have used k · r = kz. We say ‘formal’ because of course the unknown φ(r
′ )
still appears on the right-hand side of (H.12).
It may therefore seem that we have made no progress – but in fact (H.12)
leads to a very useful expression for f (θ, φ), which is the quantity we need to
calculate. This can be found by considering the asymptotic (r → ∞) limit of
the integral term in (H.12). We have
2
′ 2
′ )
1/2
|r − r
′
| = (r + r − 2r · r
( )
∼ r − r · r
′ /r + O
1 terms.
(H.13)
r
Thus in the exponent we may write
ik|r−r
'
ik(r−r·r
' /r)
ikr −ik
' ·r
'
e
|
≈ e
= e e
where k
′ = kr ˆ is the outgoing wavevector, pointing along the direction of the
H.2. Expression for the scattering amplitude: Born approximation
Note that in the ‘forward direction’ (i.e. within a region close to the z-axis, as
determined by the collimation), the incident and scattered waves will interfere. Careful analysis reveals a depletion of the incident beam in the forward
direction (the ‘shadow’ of the scattering centre), which corresponds exactly
to the total flux scattered into all angles (Gottfried 1966, section 12.3). This
is expressed in the optical theorem:
k
Im f (0) =
σ.
(H.10)
4π
H.2 Expression for the scattering amplitude: Born
approximation
We begin by rewriting (H.2) as
(∇
2 + k
2 )φ(r) =
2m V (r)φ(r).
(H.11)
ħ 2
This equation is of exactly the form discussed in appendix G, e.g. equa∇
2 + k
2
tion (G.18) with Ω r =
. Further, we know that the Green function
for this Ω r , corresponding to the desired outgoing wave solution, is given by
(G.14). Using then (G.19) and (G.14), we can immediately write the ‘formal
solution’ of (H.11) as
∫
ik|r−r
' |
2m
1 e
ik·r
′
φ(r) = e
+
−
V (r
′ )φ(r
′ ) d
3
r
(H.12)
ħ 2
4π |r − r ′ |
where we have chosen ‘u(r)’ in (G.19) to be the incident plane wave φ inc , and
have used k · r = kz. We say ‘formal’ because of course the unknown φ(r
′ )
still appears on the right-hand side of (H.12).
It may therefore seem that we have made no progress – but in fact (H.12)
leads to a very useful expression for f (θ, φ), which is the quantity we need to
calculate. This can be found by considering the asymptotic (r → ∞) limit of
the integral term in (H.12). We have
2
′ 2
′ )
1/2
|r − r
′
| = (r + r − 2r · r
( )
∼ r − r · r
′ /r + O
1 terms.
(H.13)
r
Thus in the exponent we may write
ik|r−r
'
ik(r−r·r
' /r)
ikr −ik
' ·r
'
e
|
≈ e
= e e
where k
′ = kr ˆ is the outgoing wavevector, pointing along the direction of the