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]

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

where f_ℓ is the partial scattering amplitude and P_ℓ are the Legendre polynomials.

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

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.

References

1. Quantum Mechanics: Concepts and Applications By Nouredine Zettili, 2nd edition, page 623. ISBN 978-0-470-02679-3 Paperback 688 pages January 2009, ©2008
2. Michael Fowler/ 1/17/08 Plane Waves and Partial Waves
3. Schiff, Leonard I. (1968). Quantum Mechanics. New York: McGraw Hill. pp. 119–120.