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 the stationary-state scattering process.[1] The latter is described by the wavefunction
where
is the coordinate vector;
; eikz is the incoming plane wave with the wave-vector k along the z axis; eikr / r is the outgoing spherical wave; θ is the scattering angle; and f(θ) is the scattering amplitude. The dimension of the scattering amplitude is length.
The differential cross-section is given as
In the low-energy regime the scattering amplitude is determined by the scattering length.
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[edit] Partial wave expansion
In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,[2]
where fl(k) is the partial amplitude and Pl(cos(θ)) is the Legendre polynomial.
The partial amplitude can be expressed via the S-matrix element
and the scattering phase δl as
[edit] X-rays
The scattering length for X-rays is the Thompson scattering length or classical electron radius, r0.
[edit] Neutrons
The nuclear neutron scattering process involves the coherent neutron scattering length, often described by b.
[edit] Quantum mechanical formalism
A quantum mechanical approach is given by the S matrix formalism.



