Elastic scattering

The partial wave method

Assume a central potential of finite range, i.e. .  Then stationary state solutions of the eigenvalue equation  with the asymptotic form

exist.  Here 

is the scattering amplitude, d_l is called the phase shift of the lth partial wave.

The differential scattering cross section is and the total scattering cross section is

Let R be the range of the potential. If  , then d_l=0.

Low energy scattering

Assume  for all l except l=0.  Then , and .  This is called s-wave scattering.

For a finite range potential U(r) of range R the phase shift d₀ is negative if  and it is positive if  . If  then if kR<<1.

For  we can have   .  If  .  This is called the Ramsauer-Townsend effect.

The Born approximation

The elastic scattering cross section in the Born Approximation is , and m is the reduced mass.