Debye-Waller factor

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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 neutron scattering caused by thermal motion or quenched disorder. It has also been called the B factor or the temperature factor.

The DWF depends on q, the absolute value of the scattering vector q. For a given q, DWF(q) gives the fraction of elastic scattering; 1-DWF(q) correspondingly the fraction of inelastic scattering. 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 (inelastic neutron scattering).

Assuming harmonicity of thermal motion in the material under study, the DWF takes the form

DWF=\exp\left( -\langle [\mathbf{q}\mathbf{u}(0)]^2 \rangle \right) = \exp\left( -q^2 \langle [\mathbf{u}(0)]^2 \rangle / 3 \right)

where <...> denotes thermal averaging, and u(t) is the displacement of a scattering center as function of time t.

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