Scattering amplitude

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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[edit]

Main article: Partial wave analysis

In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,[2]

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

where f is the partial scattering amplitude and P are the Legendre polynomials.

The partial amplitude can be expressed via the partial wave S-matrix element S (=e^{2i\delta_\ell}) and the scattering phase shift δ 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]

\frac{d\sigma}{d\Omega} = |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 \frac{d\sigma}{d\Omega} \sin \theta \, d \theta = \frac{4 \pi}{k} \operatorname{Im} f(0),

where Im f(0) is the imaginary part of f(0).


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


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

Quantum mechanical formalism[edit]

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