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]

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).


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.