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. 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 can be determined by the scattering length.

Partial wave expansion

Main article: Partial wave analysis

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

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

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

${\frac {d\sigma }{d\Omega }}=|f(\theta )|^{2}={\frac {1}{k^{2}}}\left|\sum _{\ell =0}^{\infty }(2\ell +1)e^{i\delta _{\ell }}\sin \delta _{\ell }P_{\ell }(\cos \theta )\right|^{2}$,

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.