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.

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 reflection peaks.

Assuming harmonicity of the scattering centers in the material under study, the Boltzmann distribution implies that

Then, using for example the expression of the corresponding characteristic function, the DWF takes the form Note that although the above reasoning is classical, the same holds in quantum mechanics.

Assuming also isotropy of the harmonic potential, one may write where q, u are the magnitudes (or absolute values) of the vectors q, u respectively, and

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.

Such experiments typically involve a probe (e.g. X-rays or neutrons) and a crystalline solid.

A well-characterized probe propagating towards the crystal may interact and scatter away in a particular manner.

The following derivation is based on chapter 14 of Simon's The Oxford Solid State Basics[5] and on the report Atomic Displacement Parameter Nomenclature by Trueblood et al.[6] (available under #External links).

[7] Scattering experiments often consist of a particle with initial crystal momentum

This situation is described by Fermi's golden rule, which gives the probability of transition per unit time,

By inserting a complete set of position states, then utilizing the plane-wave expression relating position and momentum, we find that the matrix element is simply a Fourier transform of the potential.

We now assume that our solid is a periodic crystal with each unit cell labeled by a lattice position vector

By comparison of (3) and (4), we find that the Laue equation must be satisfied for scattering to occur: (5) is a statement of the conservation of crystal momentum.

), the intensity of scattered particles is proportional to the square of the structure factor.

Buried in (6) are detailed aspects of the crystal structure that are worth distinguishing and discussing.

Consideration of the structure factor (and our assumption about translational invariance) is complicated by the fact that atoms in the crystal may be displaced from their respective lattice sites.

represents the point in the unit cell for which we would like to know the density of scattering matter.

Set the displacement between the location in space for which we would like to know the scattering density and the reference position for the nucleus equal to a new variable

Within the square brackets of (11), we convolve the density of scattering matter of atom

The final step is to multiply by a phase depending on the reference (e.g. mean) position of atom

Set the displacement between the location in space for which we would like to know the scattering density and the position for the nucleus equal to a new variable

is the atomic Debye–Waller factor; it determines how the propensity for nuclear displacement from the reference lattice position influences scattering.

in the article's opening is different because of 1) the decision to take the thermal or time average, 2) the arbitrary choice of negative sign in the exponential, and 3) the decision to square the factor (which more directly connects it to the observed intensity).

A common simplification to (14) is the harmonic approximation, in which the probability density function is modeled as a Gaussian.

Under this approximation, static displacive disorder is ignored and it is assumed that atomic displacements are determined entirely by motion (alternative models in which the Gaussian approximation is invalid have been considered elsewhere[8]).

Finally, we can find the relationship between the Debye–Waller factor and the anisotropic displacement parameter.

A 50% probability thermal ellipsoid model of H 8 Si 8 O 12 constructed with ORTEP-3 [ 9 ] from a .cif file on ICSD. [ 10 ] Analysis following a diffraction experiment consists of fitting to the observed spectrum of scattered particles. U may be refined for each distinct atom during the process. For the above 50% probability model, in equation (15). This defines a surface of nuclear displacements for each U. Therefore, we expect each ellipsoid to vary depending on the type and environment of its atom. Note that surfaces represent nuclear displacements; thermal ellipsoid models should not be confused with other models (e.g. electron density, Van der Waals radii). Fewer than 28 atoms are displayed due to redundancy from symmetry considerations.