The latter is described by the wavefunction
where is the position vector; ; is the incoming plane wave with the wavenumber k along the z axis; is the outgoing spherical wave; θ is the scattering angle; and is the scattering amplitude. The dimension of the scattering amplitude is length.
Partial wave expansionEdit
In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,
where fℓ is the partial scattering amplitude and Pℓ are the Legendre polynomials.
The partial amplitude can be expressed via the partial wave S-matrix element Sℓ ( ) and the scattering phase shift δℓ as
Then the differential cross section is given by
and the total elastic cross section becomes
where Im f(0) is the imaginary part of f(0).
The nuclear neutron scattering process involves the coherent neutron scattering length, often described by b.
Quantum mechanical formalismEdit
A quantum mechanical approach is given by the S matrix formalism.
The scattering amplitude can be determined by the scattering length in the low-energy regime.
- Quantum Mechanics: Concepts and Applications Archived 2010-11-10 at the Wayback Machine. By Nouredine Zettili, 2nd edition, page 623. ISBN 978-0-470-02679-3 Paperback 688 pages January 2009
- Michael Fowler/ 1/17/08 Plane Waves and Partial Waves
- Schiff, Leonard I. (1968). Quantum Mechanics. New York: McGraw Hill. pp. 119–120.