We will talk about the basics theory of resonance scattering or s-wave in this post.

The asymptotic behaviour of the scattered wave is (this post or this post), ignore th noramlization factor.

For the s-wave,

The logarithmic derivative at the boundary is

This must be matched with the logarithmic derivative of the inner wave,

Solve for

For elastics scattering, is real, but in general, it can be a complex when a complex potential is used, that corresponding to an absorption, that other reaction channels are populated. Suppose

Recall that the scattering amplitude

Assume only s-wave exist,

where

We can see that the phase shift of is . It is like a scattering of a hard sphere.

The term is respond to resonance when . We can imagine that is variate with the beam energy . Express in Taylor series, the 1st order around the resonance energy .

And define,

where is scattering width, is absorption width, and is total width.

The cross section for s-wave

Near the resonance energy, ,

At resonance energy

For real potential, and .