Scattering amplitude

The scattering amplitude describes the amplitude of an outgoing, elementary, spherical wave relative to a plane, incoming wave scattered on a point size particle. Since the spherical wave amplitude varies with $$\frac{1}{R}$$ (note, the intensity is the |square| of the amplitude), $$R$$ being the distance to the point scatterer, the unit of the scattering amplitude includes the unit of a length with respect to the units of the incoming wave. Therefore, the scattering amplitude is often also called scattering length.

Cross section
The scattering amplitude f relates to the scattering cross section $$ s = 4*\pi*f^2. $$

X-rays
The scattering length for X-rays is the Thompson scattering length or classical electron radius r0.

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

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