##

Scattering from a Spherical Well *****

For the scattering problem, the energy is greater than zero.
We must choose the Bessel function in the region containing the origin.

For large
, we can have a linear combination of functions.

Matching the logarithmic derivative, we get

Recalling that for
,

and that our formula with the phase shift is

we can **identify the phase shift** easily.

We need to use the boundary condition to get this phase shift.

For
, we get

We can now get the phase shift.

With just the
term, the differential scattering cross section is.

The cross section will have zeros when

There will be many solutions to this and the cross section will look like diffraction.

Jim Branson
2013-04-22