# Scattering amplitude

In quantum physics, the scattering amplitude is the amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process.[1] 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 position vector; $r\equiv|\mathbf{r}|$; $e^{ikz}$ is the incoming plane wave with the wavenumber $k$ along the $z$ axis; $e^{ikr}/r$ is the outgoing spherical wave; $\theta$ is the scattering angle; and $f(\theta)$ is the scattering amplitude. The dimension of the scattering amplitude is length.

The scattering amplitude is a probability amplitude and the differential cross-section as a function of scattering angle is given as its modulus squared

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

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,[2]

$f(\theta)=\sum_{\ell=0}^\infty (2\ell+1) f_\ell(k) P_\ell(\cos(\theta)) \;,$

where $f_\ell(k)$ is the partial amplitude and $P_\ell(\cos(\theta))$ is the Legendre polynomial.

The partial amplitude can be expressed via the S-matrix element $S_\ell=e^{2i\delta_\ell}$ and the scattering phase shift $\delta_\ell$ as

$f_\ell = \frac{S_\ell-1}{2ik} = \frac{e^{2i\delta_\ell}-1}{2ik} = \frac{e^{i\delta_\ell} \sin\delta_\ell}{k} = \frac{1}{k\cot\delta_\ell-ik} \;.$

Then the differential cross section is given by[3]

$\sigma(\theta) = |f(\theta)|^2 = \frac{1}{k^2} \left | \sum_{\ell=0}^\infty (2\ell+1) e^{i\delta_\ell} \sin(\delta_\ell) P_\ell(\cos(\theta)) \right |^2 \;,$

and the total elastic cross section becomes

$\sigma = 2\pi \int_0^\pi \sigma(\theta) \sin(\theta) \; d\theta = \frac{4\pi}{k} \text{Im} \left(f(0)\right) \;,$

where $\text{Im}\left(f(0)\right)$ is the imaginary part of $f(0)$.

## X-rays

The scattering length for X-rays is the Thomson scattering length or classical electron radius, $r_0$.

## 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.