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.[1][2] 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[3].) 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).

The basic expression for the DWF is given by

\( \text{DWF} = \left\langle \exp\left(i \mathbf{q}\cdot \mathbf{u}\right) \right\rangle^2 \)

where u is the displacement of a scattering center, and <...> denotes either thermal or time averaging.

Assuming harmonicity of the scattering centers in the material under study, the Boltzmann distribution implies that \( \mathbf{q}\cdot \mathbf{u} \) is normally distributed with zero mean. Then, using for example the expression of the corresponding characteristic function, the DWF takes the form

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

Note that although the above reasoning is classical, the same holds also quantum mechanically.

Assuming also isotropy of the harmonic potential, one may write

\( \text{DWF} = \exp\left( -q^2 \langle u^2 \rangle / 3 \right) \)

where q, u are the magnitudes (or absolute values) of the vectors q, u respectively. Note that if the incident wave has wavelength λ, and it is elastically scattered by an angle of 2θ, then q = 4π sin(θ) / λ.

In the context of protein structures, the B-factors can be taken as indicating the relative vibrational motion of different parts of the structure. Atoms with low B-factors belong to a part of the structure that is well-ordered. Atoms with large B-factors generally belong to part of the structure that is very flexible. Each ATOM record (PDB file format) of a crystal structure deposited with the Protein Data Bank contains a B-factor for that atom.

References

^ Debye, Peter (1913). "Interferenz von Röntgenstrahlen und Wärmebewegung" (in German). Ann. d. Phys. 348 (1): 49–92. Bibcode 1913AnP...348...49D. doi:10.1002/andp.19133480105.
^ Waller, Ivar (1923). "Zur Frage der Einwirkung der Wärmebewegung auf die Interferenz von Röntgenstrahlen" (in German). Zeitschrift für Physik A (Berlin / Heidelberg: Springer) 17: 398–408. Bibcode 1923ZPhy...17..398W. doi:10.1007/BF01328696.
^ Lipkin, Harry (2004). "Physics of Debye-Waller Factors". arXiv:cond-mat/0405023v1.

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