Born approximation: Yukawa → Rutherford

physics

Problem

A particle of mass

m

and energy

E = \hbar^2 k^2/2m

scatters off the Yukawa potential

V(r) = V_0\, e^{-\mu r}/(\mu r)

. (a) In the first Born approximation, find the scattering amplitude

f(\theta)

and the differential cross section

d\sigma/d\Omega

. (b) Take the limit

\mu\to 0

with

V_0/\mu \to Z_1 Z_2 e^2/(4\pi\varepsilon_0)

held fixed, and recover the Rutherford cross section.