↑ arvau

Wikipedia / 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 is also called 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 reflection 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 denotes either thermal or time averaging.

Assuming harmonicity of the scattering centers in the material under study, the Boltzmann distribution is normally distributed with zero mean. 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 is the mean squared displacement, then

In the context of protein structures, the term B-factor is used. The B-factor is defined as

It is measured in units of Å2. 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.

Derivation

Introduction

Scattering experiments are a common method for learning about crystals. 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. Mathematical expressions relating the scattering pattern, properties of the probe, properties of the experimental apparatus, and properties of the crystal then allow one to derive desired features of the crystalline sample.

The following derivation is based on chapter 14 of Simon's The Oxford Solid State Basics and on the report Atomic Displacement Parameter Nomenclature by Trueblood et al. (available under #External links). It is recommended to consult these sources for a more explicit discussion. Background on the quantum mechanics involved may be found in Sakurai and Napolitano's Modern Quantum Mechanics.

Scattering experiments often consist of a particle with initial crystal momentum. This situation is described by Fermi's golden rule, to the energy eigenstate.

  1. . (1)

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.

  1. . (2)

.

  1. . (3)

Laue equation

According to the Poisson summation formula:

  1. . (4)

is a reciprocal lattice is the volume of its unit cell. By comparison of (3) and (4), we find that the Laue equation must be satisfied for scattering to occur:

  1. . (5)

(5) is a statement of the conservation of crystal momentum. Particles scattered in a crystal experience a change in wave vector equal to a reciprocal lattice vector of the crystal. When they do, the contribution to the matrix element is simply a finite constant. Thus, we find an important link between scattered particles and the scattering crystal. The Laue condition, which states that crystal momentum must be conserved, is equivalent to the Bragg condition, which demands constructive interference for scattered particles. Now that we see how the first factor of (3) determines whether or not incident particles are scattered, we consider how the second factor influences scattering.

Structure factor

The second term on the right hand side of (3) is the structure factor.

  1. (6)

For a given reciprocal lattice vector (corresponding to a family of lattice planes labeled by Miller indices), the intensity of scattered particles is proportional to the square of the structure factor.

  1. . (7)

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

Debye–Waller factor

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. Taking the scattering potential to be proportional to the density of scattering matter, we rewrite the structure factor.

  1. . (8)

is the density of scattering matter. The angle brackets indicate a temporal average of each unit cell followed by a spatial average over every unit cell. We further assume that each atom is displaced independently of the other atoms.

  1. . (9)

-functions weighted by a scattering length for the respective nucleus (see Fermi pseudopotential). Note that in the above discussion, we assumed the atoms were not deformable. With this in mind, (9) may be plugged into expression (8) for the structure factor.

  1. . (10)

. Substitute into (10).

  1. . (11)

. But, according to the convolution theorem.

  1. . (12)

Substitute (12) into (10).

  1. . (13)

That is:

  1. . (14)

is the atomic form factor in the article's opening is different because of the decision to take the lattice or time average.

Anisotropic displacement parameter, U

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).

  1. . (15)

(see to Trueblood et al. for a more complete discussion). Finally, we can find the relationship between the Debye–Waller factor and the anisotropic displacement parameter.

  1. . (16)

.

Applications

is some constant. Such "vibration ellipsoids" have been used to illustrate crystal structures.. See the external links "Gallery of ray-traced ORTEP's", "2005 paper by Rowsell et al.", and "2009 paper by Korostelev and Noller" for more images. Anisotropic displacement parameters are also refined in programs (e.g. GSAS-II) to resolve scattering spectra during Rietveld refinement.

The B-factor can also be used to analyze radiation damage caused by the X-ray crystallography process. Localized radiation damage tends to show up as an increase in B-factor of particular "damaged" atoms as the atoms are allowed to move around due to radiation causing chemical changes. By comparing the B-factors of damage-prone atoms (aspartate/glutamate side-chain oxygen atoms) and non-damage-prone atoms, an estimate can be made about the degree of radiation damage present.

References