The Debye–Waller factor (DWF), named after
Peter Debye
Peter Joseph William Debye (; ; March 24, 1884 – November 2, 1966) was a Dutch-American physicist and physical chemistry, physical chemist, and List of Nobel laureates in Chemistry, Nobel laureate in Chemistry.
Biography
Early life
Born Petr ...
and
Ivar Waller
Ivar Waller (11 June 1898 – 12 April 1991) was a Swedish professor of theoretical physics at Uppsala University. He developed the theory of X-ray scattering by lattice vibrations of a crystal, building upon the prior work of Peter Debye. The De ...
, is used in
condensed matter physics
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases which arise from electromagnetic forces between atoms. More generally, the s ...
to describe the attenuation of
x-ray scattering
X-ray scattering techniques are a family of non-destructive analytical techniques which reveal information about the crystal structure, chemical composition, and physical properties of materials and thin films. These techniques are based on observi ...
or coherent
neutron scattering
Neutron scattering, the irregular dispersal of free neutrons by matter, can refer to either the naturally occurring physical process itself or to the man-made experimental techniques that use the natural process for investigating materials. Th ...
caused by thermal motion.
It is also called the B factor, atomic B factor, or temperature factor. Often, "Debye–Waller factor" is used as a generic term that comprises the
Lamb–Mössbauer factor In physics, the Lamb–Mössbauer factor (LMF, after Willis Lamb and Rudolf Mössbauer) or elastic incoherent structure factor (EISF) is the ratio of elastic to total incoherent neutron scattering, or the ratio of recoil-free to total nuclear resona ...
of incoherent neutron scattering and
Mössbauer spectroscopy
Mössbauer spectroscopy is a spectroscopic technique based on the Mössbauer effect. This effect, discovered by Rudolf Mössbauer (sometimes written "Moessbauer", German: "Mößbauer") in 1958, consists of the nearly recoil-free emission and abs ...
.
The DWF depends on the scattering vector q. For a given q, DWF(q) gives the fraction of
elastic scattering
Elastic scattering is a form of particle scattering in scattering theory, nuclear physics and particle physics. In this process, the kinetic energy of a particle is conserved in the center-of-mass frame, but its direction of propagation is modif ...
; 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
In physics and chemistry , Bragg's law, Wulff–Bragg's condition or Laue–Bragg interference, a special case of Laue diffraction, gives the angles for coherent scattering of waves from a crystal lattice. It encompasses the superposition of wave ...
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
Neutron scattering, the irregular dispersal of free neutrons by matter, can refer to either the naturally occurring physical process itself or to the man-made experimental techniques that use the natural process for investigating materials. Th ...
or
electron energy loss spectroscopy
In electron energy loss spectroscopy (EELS) a material is exposed to a beam of electrons with a known, narrow range of kinetic energies. Some of the electrons will undergo inelastic scattering, which means that they lose energy and have their pa ...
).
The basic expression for the DWF is given by
:
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
In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability ...
implies that
is
normally distributed
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu is ...
with zero mean. Then, using for example the expression of the corresponding
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at point ...
, the DWF takes the form
:
Note that although the above reasoning is classical, the same holds in quantum mechanics.
Assuming also
isotropy
Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
of the harmonic potential, one may write
:
where ''q'', ''u'' are the magnitudes (or absolute values) of the vectors q, u respectively, and
is the
mean squared displacement
In statistical mechanics, the mean squared displacement (MSD, also mean square displacement, average squared displacement, or mean square fluctuation) is a measure of the deviation of the position of a particle with respect to a reference positi ...
. In crystallographic publications, values of
are often given where
. Note that if the incident wave has wavelength
, and it is elastically scattered by an angle of
, then
:
In the context of
protein structure
Protein structure is the molecular geometry, three-dimensional arrangement of atoms in an amino acid-chain molecule. Proteins are polymers specifically polypeptides formed from sequences of amino acids, the monomers of the polymer. A single ami ...
s, 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
The Protein Data Bank (PDB) is a database for the three-dimensional structural data of large biological molecules, such as proteins and nucleic acids. The data, typically obtained by X-ray crystallography, NMR spectroscopy, or, increasingly, c ...
contains a B-factor for that atom.
Derivation
Introduction
Scattering experiments are a common method for learning about
crystal
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macr ...
s. Such experiments typically involve a probe (e.g.
X-ray
X-rays (or rarely, ''X-radiation'') are a form of high-energy electromagnetic radiation. In many languages, it is referred to as Röntgen radiation, after the German scientist Wilhelm Conrad Röntgen, who discovered it in 1895 and named it ' ...
s or
neutron
The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons behav ...
s) 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
''Modern Quantum Mechanics'', often called ''Sakurai'' or ''Sakurai and Napolitano'', is a standard graduate-level quantum mechanics textbook written originally by J. J. Sakurai and edited by San Fu Tuan in 1985, with later editions coauthored ...
''.
Scattering experiments often consist of a particle with initial
crystal momentum
In solid-state physics crystal momentum or quasimomentum is a momentum-like vector associated with electrons in a crystal lattice. It is defined by the associated wave vectors \mathbf of this lattice, according to
:_ \equiv \hbar
(where \hbar ...
incident on a solid. The particle passes through a potential distributed in space,
, and exits with crystal momentum
. This situation is described by
Fermi's golden rule
In quantum physics, Fermi's golden rule is a formula that describes the transition rate (the probability of a transition per unit time) from one energy eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of a ...
, which gives the probability of transition per unit time,
, to the
energy eigenstate from the energy eigenstate
due to the weak perturbation caused by our potential
.
:
. (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.
:
. (2)
Above, the length of the sample is denoted by
. We now assume that our solid is a periodic crystal with each unit cell labeled by a lattice position vector
. Position within a unit cell is given by a vector
such that the overall position in the crystal may be expressed as
. Because of the translational invariance of our unit cells, the potential distribution of every cell is identical and
.
: