Rutherford Scattering (Homework 9, Physics 325)

3. Rutherford scattering. 

It can be shown that for impact parameter b the a particle of kinetic energy $E$ and charge $q$ is scattered at an angle $θ$ by a stationary particle of charge $q_0$ at the origin, where $tan(θ/2) = (qq0/8π\epsilon_0E)/b(θ)$

a) Determine the differential cross-section $D(\Omega) = dσ/d\Omega= (b/ sin θ)|db/dθ|$. Hint: differ-entiate both sides with respect to theta.

b) Show that the total cross-section $σ =\int D(\Omega)d\Omega$ for Rutherford scattering diverges dueto the singular behavior of $D$ at small angles $θ$


$\tan⁡ θ/2=(qq_0)/(8πϵ_0 E)*1/b(θ)$    or $\tan⁡ θ/2=A/b(θ)$     with $A=(2kqq_0)/E$

Find  $D(Ω)=dσ/dθ=b/\sin ⁡θ *|db/dθ|$

Differentiate red equation:

$1/(2 \cos^2⁡ θ/2)=-A/(b^2 (θ))*db/dθ$    or   $db/dθ=-b^2/(2A \cos^2⁡ θ/2)$
So that

$D(Ω)=dσ/dθ=b/\sin⁡θ *|db/dθ|=b/\sin⁡θ *b^2/(2 A*\cos^2 θ/2)=b^3/(2A*\sinθ*cos^2⁡ θ/2)$

$D(Ω)=b^3/(2A*\sin⁡ (θ/2)*\cos^3⁡ θ/2)$


$σ=∫ D(Ω)*dΩ=∫ b^3/(2A*\sinθ*\cos^2⁡ (θ/2))*(\sinθ*dθdφ)=$

$=b^3/2A ∫_0^π dθ/\cos^2⁡ (θ/2) *∫_0^{2π} dφ$

$σ=(2πb^3)/A*∫_0^{(π/2)} dx/\cos^2⁡x =(2πb^3)/A*\tan ⁡x  |_0 ^{π/2}→∞$ 

since $\tan⁡(π/2)→∞$