Debye–Waller factor

The Debye–Waller factor (DWF), named after Peter Debye and Ivar Waller, is used in condensed matter physics to describe the attenuation of x-ray scattering or coherent neutron scattering caused by thermal motion. It has also been called the B factor or the temperature factor. Often, "Debye-Waller factor" is used as a generic term that comprises the Lamb-Mössbauer factor of incoherent neutron scattering and Mössbauer spectroscopy.

The DWF depends on the scattering vector q. For a given q, DWF(q) gives the fraction of elastic scattering; 1 - DWF(q) correspondingly gives the fraction of inelastic scattering. (Strictly speaking, this probability interpretation is not true in general.) In diffraction studies, only the elastic scattering is useful; in crystals, it gives rise to distinct Bragg peaks. Inelastic scattering events are undesirable as they cause a diffuse background — unless the energies of scattered particles are analysed, in which case they carry valuable information (for instance in inelastic neutron scattering or electron energy loss spectroscopy).

The basic expression for the DWF is given by

where u is the displacement of a scattering center, and ${\displaystyle \langle \ldots \rangle }$ denotes either thermal or time averaging.

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