Elastic scattering

In a typical scattering experiment a target is struck by a beam of mono-energetic particles.
Let Fi be the incident flux, i.e. the number of particles per unit area per unit time. 
Fi = npv, where np is the number of particles per unit volume. 
Typically np is very small and we can neglect any interaction between different incident particles.
We measure the number ΔNp of particles scattered per unit time into a solid angle ΔΩ about the direction defined by the spherical coordinates θ and φ. 
We expect ΔNp ∝ Fi, and ΔNp ∝ ΔΩ .
We define ΔNp = σt(θ,φ)FiΔΩ.  Here σt(θ,φ) is the differential scattering cross section of the target.  It has the units of an area.  Commonly used units are cm2 and barn = 10-24 cm2.

The partial wave method

Assume a central potential energy function of finite range, i.e. U(r) --> 0 as r --> ∞ faster than 1/r.
Then stationary state solutions of the eigenvalue equation  HΦk(r) = EkΦk(r) with the asymptotic form
Φk(r) = eikz + fk(θ) eikr/r = ∑l=0cl Pl(cosθ) sin(kr - lп/2 + δl)/(kr)
exist.  Here
fk(θ) = (1/k)∑l=0(2l+1)exp(iδl)sinδlPl(cosθ)
is the scattering amplitude, δl is called the phase shift of the lth partial wave.
The differential scattering cross section is
σk(θ) = dσk/dΩ = |fk(θ)|2 = (1/k2)|∑l=0(2l+1)exp(iδl)sinδlPl(cosθ)|2,
and the total scattering cross section is
σk = ∫∫σk(θ)sinθdθdφ = (4п/k2)∑l=0(2l+1)sin2δl.
Let R be the range of the potential.  If kR << (l(l + 1))½, then δl = 0.

Low energy scattering

Assume kR << (l(l + 1))½ for all l except l = 0.  Then
fk(θ) = (1/k)exp(iδ0)sinδ0,  σk(θ) = (1/k2)sin2δ0,  and  σk = (4п/k2)sin2δ0
This is called s-wave scattering.

[To derive the expression fk(θ) = (1/k)∑l=0(2l+1)exp(iδl)sinδlPl(cosθ) we expand Φk(r) in terms of the common eigenfunctions of H, L2 and Lz, |klm> = (u(r)/r)Ylm(θ,φ).  In classical mechanics, a free particle with angular momentum L about the origin and momentum p moves in a straight line with closest distance to the origin b = L/p;  b is called the impact parameter relative to the origin. 
If L = ħ(l(l + 1))½ and p = ħk, then bl(k) = (1/k)(l(l + 1))½.
bl(k) is the radius where the angular momentum barrier potential, Ueff(r) = l(l+1)ħ2/(2mr2) is equal to the total energy of the free particle.
l(l+1)ħ2/(2m(bl(k))2) = ħ2k2/(2m),  bl(k) = (1/k)(l(l + 1))½.
In Quantum Mechanics, for r < bl(k), in the classically forbidden region, the eigenfunctions |klm> can only have an approximately exponentially decaying tail.
In a finite range potential, if the partial wave with angular momentum quantum number l only has an approximately exponentially decaying tail in the region where the potential is non-zero, then it is minimally affected by the potential and the phase shift δl is approximately zero.]

For a finite range potential U(r) of range R the phase shift δ0 is negative if  U(r)r<R > 0, and it is positive if  U(r)r<R < 0.  If  U(r)r<R = ∞ then δ0 = kR, σk = 4пR2, if kR << 1.
For  U(r)r<R < 0 we can have δ0 = п/2 while kR << 1, then σk = 4п/k2 --> ∞ as E --> 0.  If  δ0 = п, then σk = 0.  This is called the Ramsauer-Townsend effect.

The Born approximation

The elastic scattering cross section in the Born Approximation is
σkB(θ,φ) = σkB(k,k') = [μ2/(4п2ħ4)]|∫d3r' exp(-iq∙r')U(r')|2,
where q = k' - k, k = μv0/ħ, k' = μv0/ħ (k'/k'), and μ is the reduced mass.

The differential scattering cross section is proportional to the square of the Fourier transform of the potential.
We often want to know the scattering cross section as a function of the scattering angle and not as a function of the momentum transfer.  Let θ be the angle between k and k'.  Then q = 2ksin(θ/2).