In a typical scattering experiment a target is struck by a beam of
mono-energetic particles.

Let F_{i} be the incident flux, i.e. the number of particles per
unit area per unit time.

F_{i} = n_{p}v,
where n_{p} is the number of particles per unit volume.

Typically n_{p}
is very small and we can neglect any interaction between different incident
particles.

We measure the number ΔN_{p} of
particles scattered per unit time into a solid angle ΔΩ about the
direction defined by the spherical coordinates θ and φ.

We
expect ΔN_{p} ∝ F_{i},
and ΔN_{p} ∝ ΔΩ .

We define ΔN_{p }= σ_{t}(θ,φ)F_{i}ΔΩ.
Here σ_{t}(θ,φ) is
the **differential scattering cross section of the target**. It has the
units of an area. Commonly used units are cm^{2} and barn = 10^{-24 }cm^{2}.

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 HΦ_{k}(**r**)
= E_{k}Φ_{k}(**r**)
with the **asymptotic form
**Φ

exist. Here

f

is the

The

σ

and the

σ

Let R be the range of the potential. If kR << (l(l + 1))

Assume kR << (l(l + 1))^{½} for all l except l = 0. Then

f_{k}(θ) = (1/k)exp(iδ_{0})sinδ_{0},
σ_{k}(θ)
= (1/k^{2})sin^{2}δ_{0},
and σ_{k} = (4п/k^{2})sin^{2}δ_{0}.

This is called **s-wave scattering**.

[To derive the expression f_{k}(θ)
= (1/k)∑_{l=0}^{∞}(2l+1)exp(iδ_{l})sinδ_{l}P_{l}(cosθ)
we expand Φ_{k}(r)
in terms of the common eigenfunctions of H, L^{2} and L_{z}, |klm>
= (u(r)/r)Y_{lm}(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 b_{l}(k) = (1/k)(l(l + 1))^{½}.

b_{l}(k) is the radius where the angular momentum barrier potential, U_{eff}(r) = l(l+1)ħ^{2}/(2mr^{2}) is
equal to the total energy of the free particle.

l(l+1)ħ^{2}/(2m(b_{l}(k))^{2})
= ħ^{2}k^{2}/(2m), b_{l}(k)
= (1/k)(l(l + 1))^{½}.

In Quantum Mechanics, for r < b_{l}(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
δ_{l} is approximately zero.]

For a finite range potential U(r) of range R the phase shift δ_{0} is negative if U(r)_{r<R} > 0,
and it is positive if U(r)_{r<R} < 0. If U(r)_{r<R} =
∞ then δ_{0} = kR, σ_{k}
= 4пR^{2}, if kR << 1.

For U(r)_{r<R} < 0 we can have δ_{0}
= п/2 while kR << 1, then σ_{k}
= 4п/k^{2} --> ∞ as
E --> 0. If δ_{0} =
п, then σ_{k} =
0. This is called the **Ramsauer-Townsend effect**.

The elastic scattering cross section in the Born Approximation is

σ_{k}^{B}(θ,φ)
= σ_{k}^{B}(**k**,**k**') = [μ^{2}/(4п^{2}ħ^{4})]|∫d^{3}r'
exp(-i**q∙r**')U(**r**')|^{2},

where **q** = **k'** - **k**,
**k** =
μ**v**_{0}/ħ,
**k**' = μv_{0}/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).