Chapter 8 Scattering Theory - Particle Physics Group

CHAPTER 8. SCATTERING THEORY 154
the first Born approximation of the scattering amplitude can be written as the Fourier transform
f B = − 1 ∫
e i⃗q·⃗x U(⃗x)d⃗x (8.134)
4π
of the potential. For elastic scattering with k = k ′ and ⃗ k · ⃗k ′ = k 2 cos θ we find
q = 2k sin θ 2 , (8.135)
with θ being the scattering angle. For a central potential it is now useful to introduce polar
coordinates with angles (α,β) such that ⃗q is the polar axis. We thus find that
f B (q) = − 1
4π
= − 1 2
= − 1 q
∫ ∞
0
∫ ∞
0
∫ ∞
0
drr 2 U(r)
drr 2 U(r)
∫ π
0
∫ +1
−1
dα sin α
∫ 2π
0
iqr cos α
d(cos α)e
iqr cos α
dβe
r sin(qr)U(r)dr (8.136)
only depends on q(k,θ). The total cross-section in the first Born approximation hence becomes
∫ π
σtot(k) B = 2π |f B (q)| 2 sin θdθ = 2π ∫ 2k
|f B (q)| 2 qdq (8.137)
0
k 2 0
where we used the differential dq = k cos θ 2 dθ of (8.135) and sinθ dθ = 2 sin θ 2 cos θ 2 dθ = q k
dq
k .
8.4.1 Application: Coulomb scattering and the Yukawa potential
Since the Coulomb potential has infinite range we apply the Born approximation to the Yukawa
potential
U(r) = C e−αr
= C e−r/a
with a = α −1 , (8.138)
r r
which can be regarded as a screened Colomb potential. At the end of the calculation we can
then try to send the screening length a → ∞. For the Born approximation (8.136) we obtain
f B = − 1 q
∫ ∞
0
r sin(qr) C r e−αr dr = − C ∫ ∞
q Im e iqr−αr dr = − C q Im 1
α − iq = − C (8.139)
α 2 + q 2
and the corresponding differential cross section
of the Yukawa potential.
0
dσ B
dΩ = C 2
(α 2 + q 2 ) 2 (8.140)
The Coulomb potential. The electrostatic force between charges Q A and Q B corresponds
to the potential
V Coulomb (r) = Q AQ B 1
4πε 0 r
(8.141)