Scattering amplitude


In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process.
The latter is described by the wavefunction
where is the position vector; ; is the incoming plane wave with the wavenumber along the axis; is the outgoing spherical wave; is the scattering angle; and is the scattering amplitude. The dimension of the scattering amplitude is length.
The scattering amplitude is a probability amplitude; the differential cross-section as a function of scattering angle is given as its modulus squared,

Partial wave expansion

In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,
where is the partial scattering amplitude and are the Legendre polynomials.
The partial amplitude can be expressed via the partial wave S-matrix element and the scattering phase shift as
Then the differential cross section is given by
and the total elastic cross section becomes
where is the imaginary part of.

X-rays

The scattering length for X-rays is the Thomson scattering length or classical electron radius, 0.

Neutrons

The nuclear neutron scattering process involves the coherent neutron scattering length, often described by.

Quantum mechanical formalism

A quantum mechanical approach is given by the S matrix formalism.

Measurement

The scattering amplitude can be determined by the scattering length in the low-energy regime.