Scattering amplitude

The scattering amplitude is a quantity of scattering theory, which describes the dependence of the scattering direction of wave when a plane wave is scattered at a scattering center. It has the dimension of length and connects the S- matrix with the cross section.

Definition

The scattering amplitude is defined by the S- operator:

Where and are the eigenstates of the momentum operator are. The scattering amplitude is defined only for or because is null. Further, the operator S, and thus the scattering amplitude is invariant under rotations. Therefore, the scattering amplitude can also be as a function of the energy of the incoming state and the angle between and written.

In the following, an alternative representation is presented, which is often used as a definition.

When a plane wave is assumed to be parallel to the z- axis for the incoming wave, this results in the following:

Cross-section

The differential cross-section is given by

To the total cross section, there is a connection via the optical theorem

Partialwellenentwicklung

In Partialwellenentwicklung the scattering amplitude is expressed by a sum of partial waves,

The partial scattering amplitude, the Legendre polynomial and the index for the angular momentum.

Partial scattering amplitude can be expressed by the S- matrix element and the dispersion phase:

It should be noted that partial scattering amplitude, the S- matrix element and the scattering phase functions are implicitly scattering power or the pulse.

The scattering length can be defined by means of the partial scattering amplitude:

Usually, however, only the s-wave scattering length is called the scattering length.