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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 : \text = \left\langle \exp\left(i \mathbf\cdot \mathbf\right) \right\rangle^2 where u is the displacement of a scattering center, and \langle \ldots \rangle 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 \mathbf\cdot \mathbf 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 : \text = \exp\left( -\langle mathbf\cdot \mathbf2 \rangle \right) 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 : \text = \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, and \langle u^2 \rangle 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 U are often given where U = \langle u^2 \rangle. Note that if the incident wave has wavelength \lambda, and it is elastically scattered by an angle of 2\theta, then :q = \frac 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 :B = \frac \langle u^2 \rangle 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 ...
\vec incident on a solid. The particle passes through a potential distributed in space, V(\vec), and exits with crystal momentum \vec'. 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, \Gamma(\vec',\vec), to the energy eigenstate E_ from the energy eigenstate E_ due to the weak perturbation caused by our potential V(\vec). :\Gamma(\vec',\vec)=\frac \left \vert \langle \vec', V , \vec \rangle \right \vert ^2 \delta(E_-E_). (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. :\langle \vec', V , \vec \rangle = \frac\int d^3\vec V(\vec) e^ . (2) Above, the length of the sample is denoted by L. We now assume that our solid is a periodic crystal with each unit cell labeled by a lattice position vector \vec . Position within a unit cell is given by a vector \vec such that the overall position in the crystal may be expressed as \vec = \vec + \vec. Because of the translational invariance of our unit cells, the potential distribution of every cell is identical and V(\vec) = V(\vec + \vec). :\langle \vec', V , \vec \rangle = \left \frac \sum_ e^ \right \left \int_ d^3\vec V(\vec) e^ \right /math> . (3)


Laue equation

According to the
Poisson summation formula In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function (mathematics), function to values of the function's continuous Fourier transform. Consequently, the ...
: :\sum_ e^ = \frac \sum_ \delta (\vec - \vec) . (4) \vec is a
reciprocal lattice In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial fu ...
vector of the periodic potential and v is the volume of its
unit cell In geometry, biology, mineralogy and solid state physics, a unit cell is a repeating unit formed by the vectors spanning the points of a lattice. Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessari ...
. By comparison of (3) and (4), we find that the Laue equation must be satisfied for scattering to occur: :\vec'-\vec = \vec. (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 m \lambda = 2 d \sin \theta, 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. :F(\vec) = \int_ d^3\vec V(\vec) e^ . (6) For a given reciprocal lattice vector (corresponding to a family of lattice planes labeled by
Miller indices Miller indices form a notation system in crystallography for lattice planes in crystal (Bravais) lattices. In particular, a family of lattice planes of a given (direct) Bravais lattice is determined by three integers ''h'', ''k'', and ''� ...
(hkl)), the intensity of scattered particles is proportional to the square of the structure factor. :I_ \propto , F_, ^2 . (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. :F(\vec) = \int d^3\vec \langle \rho(\vec) \rangle e^ . (8) The integral from here onwards is understood to be taken over the unit cell. \rho(\vec) 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. :\langle \rho(\vec) \rangle \simeq \sum_^N n_k \int d^3\vec_k \rho_k(\vec-\vec_k) p_k(\vec_k-\vec_) . (9) The number of atoms in the unit cell is N and the occupancy factor for atom k is n_k. \vec represents the point in the unit cell for which we would like to know the density of scattering matter. \rho_k(\vec-\vec_k) is the density of scattering matter from atom k at a position separated from the nuclear position \vec_k by a vector \vec-\vec_k. p_k(\vec_k-\vec_) is the probability density function for displacement. \vec_ is the reference lattice site from which atom k may be displaced to a new position \vec_. If \rho_k is symmetrical enough (e.g. spherically symmetrical), \vec_ is simply the mean nuclear position. When considering X-ray scattering, the scattering matter density consists of electron density around the nucleus. For neutron scattering, we have \delta-functions weighted by a scattering length b_k 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. :F(\vec) \simeq \sum_^N n_k F_k(\vec); F_k(\vec) = \int d^3\vec \left \int d^3\vec_k \rho_k(\vec-\vec_k) p_k(\vec_k-\vec_) \right e^ . (10) Now we see the overall structure factor may be represented as a weighted sum of structure factors F_k(\vec) corresponding to each atom. 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 \vec = \vec - \vec_. Do the same for the displacement between the displaced and reference nuclear positions \vec = \vec_k - \vec_. Substitute into (10). :F_k(\vec) = \left \ e^ . (11) Within the square brackets of (11), we convolve the density of scattering matter of atom k with the probability density function for some nuclear displacement. Then, in the curly brackets, we Fourier transform the resulting convolution. The final step is to multiply by a phase depending on the reference (e.g. mean) position of atom k. But, according to the
convolution theorem In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e. ...
, Fourier transforming a convolution is the same as multiplying the two Fourier transformed functions. 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 \vec = \vec - \vec_ = \vec - \vec. :F_k(\vec) = \left \int d^3\vec \rho_k(\vec) e^ \right \left \int d^3\vec p_k(\vec) e^ \right e^ . (12) Substitute (12) into (10). :F(\vec) = \sum_^N n_k \left \int d^3\vec \rho_k(\vec) e^ \right \left \int d^3\vec p_k(\vec) e^ \right e^ . (13) That is: :F(\vec) = \sum_^N n_k f_k(\vec) T_k(\vec) e^; f_k(\vec) = \int d^3 \vec \rho_k(\vec) e^ , T_k(\vec) = \int d^3 \vec p_k(\vec) e^ . (14) f_k(\vec) is the
atomic form factor In physics, the atomic form factor, or atomic scattering factor, is a measure of the scattering amplitude of a wave by an isolated atom. The atomic form factor depends on the type of scattering, which in turn depends on the nature of the incident ...
of the atom k; it determines how the distribution of scattering matter about the nuclear position influences scattering. T_k(\vec) is the atomic Debye–Waller factor; it determines how the propensity for nuclear displacement from the reference lattice position influences scattering. The expression given for \text 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).


Anisotropic displacement parameter, U

A common simplification to (14) is the harmonic approximation, in which the probability density function is modeled as a
Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymo ...
. 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). :p(\vec) \equiv \sqrt e^; \vec \equiv \sum_^3 \Delta \xi^j a^j \vec_j; \mathsf^ \equiv \langle \Delta \xi^j \Delta \xi^l \rangle. (15) We've dropped the atomic index. \vec_j belongs to the direct lattice while \vec^j would belong to the reciprocal lattice. By choosing the convenient dimensionless basis a^j \vec_j, we guarantee that \Delta \xi^j will have units of length and describe the displacement. The tensor \mathsf in (15) is the anisotropic displacement parameter. With dimension (length)^2, it is associated with mean square displacements. For the mean square displacement along unit vector \hat, simply take \hat^\mathsf \mathsf \hat. Related schemes use the parameters \beta or B rather than \mathsf (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. :T(\vec) = \langle e^ \rangle = e^ = e^ = e^. (16) From equations (7) and (14), the Debye–Waller factor T(\vec) contributes to the observed intensity of a diffraction experiment. And based on (16), we see that our anisotropic displacement factor \mathsf is responsible for determining T(\vec). Additionally, (15) shows that \mathsf may be directly related to the probability density function p for a nuclear displacement \vec from the mean position. As a result, it's possible to conduct a scattering experiment on a crystal, fit the resulting spectrum for the various atomic \mathsf values, and derive each atom's tendency for nuclear displacement from p.


Applications

Anisotropic displacement parameters are often useful for visualizing matter. From (15), we may define ellipsoids of constant probability for which \gamma = \vec^\mathsf \mathsf \vec, where \gamma is some constant. Such " vibration ellipsoids" have been used to illustrate crystal structures. Alternatively, mean square displacement surfaces along \hat may be defined by \langle \vec^2 \rangle_ = \hat^\mathsf \mathsf \hat. 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.


References


External links

* 2019 paper by Cristiano Malica and Dal Corso. Introduction to Debye–Waller factor and applications within Density Functional Theory
Temperature-dependent atomic B factor: an ab initio calculation
* Gallery of ray-traced ORTEP's

* 2005 paper by Rowsell ''et al''. depicting metal-organic framework thermal ellipsoids

* 2009 paper by Korostelev and Noller depicting tRNA thermal ellipsoids
Analysis of Structural Dynamics in the Ribosome by TLS Crystallographic Refinement
* Cruickshank's 1956 ''Acta Crystallogr.'' paper
The analysis of the anisotropic thermal motion of molecules in crystals
* 1996 report by Trueblood ''et al. -'

{{DEFAULTSORT:Debye-Waller factor Crystallography Scattering Condensed matter physics Peter Debye