Scattering amplitude

In quantum physics, the scattering amplitude is the amplitude of the outgoing spherical wave relative to the incoming plane wave in the stationary-state scattering process. The latter is described by the wavefunction

$\psi(\mathbf{r}) = e^{ikz} + f(\theta)\frac{e^{ikr}}{r} \;,$

where $\mathbf{r}\equiv\{x,y,z\}$ is the coordinate vector; $r\equiv|\mathbf{r}|$; e^ikz is the incoming plane wave with the wave-vector k along the z axis; e^ikr / r is the outgoing spherical wave; θ is the scattering angle; and f(θ) is the scattering amplitude. The dimension of the scattering amplitude is length.

The differential cross-section is given as

$\frac{d\sigma}{d\Omega} = |f(\theta)|^2 \;.$

In the low-energy regime the scattering amplitude is determined by the scattering length.

Partial wave expansion

In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,

$f(\theta)=\sum_{l=0}^\infty (2l+1) f_l(k) P_l(\cos(\theta)) \;,$

where f_l(k) is the partial amplitude and P_l(cos(θ)) is the Legendre polynomial.

The partial amplitude can be expressed via the S-matrix element $S_l=e^{2i\delta_l}$ and the scattering phase δ_l as

$f_l = \frac{S_l-1}{2ik} = \frac{e^{2i\delta_l}-1}{2ik} = \frac{e^{i\delta_l} \sin\delta_l}{k} = \frac{1}{k\cot\delta_l-ik} \;.$

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.