International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 
International Tables for Crystallography (2006). Vol. D, ch. 1.9, pp. 228242
doi: 10.1107/97809553602060000636 Chapter 1.9. Atomic displacement parameters^{a}GZG Abt. Kristallographie, Goldschmidtstrasse 1, 37077 Göttingen, Germany The theory of lattice dynamics shows that the atomic thermal Debye–Waller factor is related to the atomic displacements. In the harmonic approximation, these are fully described by a fully symmetric secondorder tensor. Anharmonicity and disorder, however, cause deviations from a Gaussian distribution of the atomic displacements around the atomic position. A generalized description of atomic displacements therefore also involves first, third, fourth and even higherorder displacement terms.The description of the properties of these tensors is the purpose of this chapter. The number of independent tensor coefficients depends on the site symmetry of the atom and are given in tables. The symmetry restrictions according to the site symmetry are tabulated for second to sixthrank thermal motion tensors. A selection of representation surfaces of higherrank tensors showing the distribution of anharmonic deformation densities is given at the end of the chapter. 
Atomic thermal motion and positional disorder is at the origin of a systematic intensity reduction of Bragg reflections as a function of scattering vector Q. The intensity reduction is given as the well known Debye–Waller factor (DWF); the DWF may be of purely thermal origin (thermal DWF or temperature factor) or it may contain contributions of static atomic disorder (static DWF). As atoms of chemically or isotopically different elements behave differently, the individual atomic contributions to the global DWF (describing the weakening of Bragg intensities) vary. Formally, one may split the global DWF into the individual atomic contributions. Crystallographic experiments usually measure the global weakening of Bragg intensities and the individual contributions have to be assessed by adjusting individual atomic parameters in a leastsquares refinement.
The theory of lattice dynamics (see e.g. Willis & Pryor, 1975) shows that the atomic thermal DWF T_{α} is given by an exponential of the formwhere u_{α} are the individual atomic displacement vectors and the brackets symbolize the thermodynamic (time–space) average over all contributions u_{α}. In the harmonic (Gaussian) approximation, (1.9.1.1) reduces to
The thermodynamically averaged atomic meansquare displacements (of thermal origin) are given as , i.e. they are the thermodynamic average of the product of the displacements along the i and j coordinate directions. Thus (1.9.1.2) may be expressed with in a form more familiar to the crystallographer aswhere are the covariant Miller indices, are the reciprocalcell basis vectors and . Here and in the following, tensor notation is employed; implicit summation over repeated indices is assumed unless stated otherwise. For computational convenience one often writeswith (no summation). Both h and β are dimensionless tensorial quantities; h transforms as a covariant tensor of rank 1, β as a contravariant tensor of rank 2 (for details of the mathematical notion of a tensor, see Chapter 1.1 ).
Similar formulations are found for the static atomic DWF S_{α}, where the average of the atomic static displacements Δu_{α} may also be approximated [though with weaker theoretical justification, see Kuhs (1992)] by a Gaussian distribution:
As in equation (1.9.1.3), the static atomic DWF may be formulated with the meansquare disorder displacements as
It is usually difficult to separate thermal and static contributions, and it is often wise to use the sum of both and call them simply (meansquare) atomic displacements. A separation may however be achieved by a temperaturedependent study of atomic displacements. A harmonic diagonal tensor component of purely thermal origin extrapolates linearly to zero at 0 K; zeropoint motion causes a deviation from this linear behaviour at low temperatures, but an extrapolation from higher temperatures (where the contribution from zeropoint motion becomes negligibly small) still yields a zero intercept. Any positive intercept in such extrapolations is then due to a (temperatureindependent) static contribution to the total atomic displacements. Care has to be taken in such extrapolations, as pronounced anharmonicity (frequently encountered at temperatures higher than the Debye temperature) will change the slope, thus invalidating the linear extrapolation (see e.g. Willis & Pryor, 1975). Owing to the difficulty in separating thermal and static displacements in a standard crystallographic structure analysis, a subcommittee of the IUCr Commission on Crystallographic Nomenclature has recommended the use of the term atomic displacement parameters (ADPs) for U^{ij} and β^{ij} (Trueblood et al., 1996).
One notes that in the Gaussian approximation, the meansquare atomic displacements (composed of thermal and static contributions) are fully described by six coefficients β^{ij}, which transform on a change of the directlattice base (according to ) as
This is the transformation law of a tensor (see Section 1.1.3.2 ); the meansquare atomic displacements are thus tensorial properties of an atom α. As the tensor is contravariant and in general is described in a (nonCartesian) crystallographic basis system, its indices are written as superscripts. It is convenient for comparison purposes to quote the dimensionless coefficients β^{ij} as their dimensioned representations U^{ij}.
In the harmonic approximation, the atomic displacements are fully described by the fully symmetric secondorder tensor given in (1.9.2.1). Anharmonicity and disorder, however, cause deviations from a Gaussian distribution of the atomic displacements around the atomic position. In fact, anharmonicity in the thermal motion also provokes a shift of the atomic position as a function of temperature. A generalized description of atomic displacements therefore also involves first, third, fourth and even higherorder displacement terms. These terms are defined by a momentgenerating function M(Q) which expresses in terms of an infinite number of moments; for a Gaussian distribution of displacement vectors, all moments of order are identically equal to zero. Thus
The moments of order N may be expressed in terms of cumulants by the identity
Separating the powers of Q and u in (1.9.2.2) and (1.9.2.3), one may obtain expressions involving moments μ and cumulants k explicitly asand the cumulantgenerating function K(Q) asThe indices run in threedimensional space from 1 to 3 and refer to the crystallographic basis system. Moments may be expressed in terms of cumulants (and vice versa); the transformation laws are given in IT B (2001), equation (1.2.12.9 ) and more completely in Kuhs (1988, 1992). The moment and cumulantgenerating functions are two ways of expressing the Fourier transform of the atomic probability density function (p.d.f.). If all terms up to infinity are taken into account, M(Q) and K(Q) are [by virtue of the identity ] identical. For a finite series, however, the cumulants of order N carry implicit information on contributions of order N^{2}, N^{3} etc. in contrast to the moments. Equations (1.9.2.4) and (1.9.2.5) are useful, as they can be entered directly in a structurefactor equation (see Chapter 1.2 in IT B); however, the moments (and thus the cumulants) may also be calculated directly from the atomic p.d.f. as
The realspace expression of the p.d.f. obtained from a Fourier transform of (1.9.2.5) is called an Edgeworth series expansion. If one assumes that the underlying atomic p.d.f. is close to a Gaussian distribution, one may separate out the Gaussian contributions to the momentgenerating function as suggested by Kuznetsov et al. (1960) and formulate a generating function for quasimoments asThese quasimoments are especially useful in crystallographic structurefactor equations, as they just modify the harmonic case. The realspace expression of the p.d.f. obtained from a Fourier transformation of (1.9.2.7) is called a Gram–Charlier series expansion. Discussions of its merits as compared to the Edgeworth series are given in Zucker & Schulz (1982a,b), Kuhs (1983, 1988, 1992) and Scheringer (1985).
By separating the powers of Q and u, one obtains in equations (1.9.2.4), (1.9.2.5) and (1.9.2.7) the higherorder displacement tensors in the form of moments, cumulants or quasimoments, which we shall denote in a general way as ; note that b^{ij} is identical to β^{ij}. They transform on a change of the directlattice base according to
The higherorder displacement tensors are fully symmetric with respect to the interchange of any of their indices; in the nomenclature of Jahn (1949), their tensor symmetry thus is [b^{N}]. The number of independent tensor coefficients depends on the site symmetry of the atom and is tabulated in Sirotin (1960) as well as in Tables 1.9.3.1–1.9.3.6. For triclinic site symmetry, the numbers of independent tensor coefficients are 1, 3, 6, 10, 15, 21 and 28 for the zeroth to sixth order. Symmetry may further reduce the number of independent coefficients, as discussed in Section 1.9.3.
In many leastsquares programs for structure refinement, the atomic displacement parameters are used in a dimensionless form [as given in (1.9.1.4) for the harmonic case]. These dimensionless quantities may be transformed according to(no summation) into quantities of units Å^{N} (or pm^{N}); a^{i} etc. are reciprocallattice vectors. Nowadays, the published structural results usually quote U^{ij} for the secondorder terms; it would be good practice to publish only dimensioned atomic displacements for the higherorder terms as well.
Anisotropic or higherorder atomic displacement tensors may contain a wealth of information. However, this information content is not always worth publishing in full, either because the physical meaning is not of importance or the significance is only marginal. Quantities of higher significance or better clarity are obtained by an operation known as tensor contraction. Likewise, lowerorder terms may be expanded to higher order to impose certain (chemically implied) symmetries on the displacement tensors or to provide initial parameters for leastsquares refinements. A contraction is obtained by multiplying the contravariant tensor components (referring to the realspace basis vectors) with the covariant components of the realspace metric tensor g_{ij}; for further details on tensor contraction, see Section 1.1.3.3.3 . In the general case of atomic displacement tensors of (even) rank N, one obtains is called the trace of a tensor of rank N and is a scalar invariant; it is given in units of length^{N} and provides an easily interpretable quantity: In the case of , a positive sign indicates that the corresponding (realspace) p.d.f. is peaked, a negative sign indicates flatness of the p.d.f. The larger , the stronger the deviation from a Gaussian p.d.f. provoked by the atomic displacements of order N. The frequently quoted isotropic equivalent U value U_{eq} is also obtained by this contraction process. Noting that U^{ij} may be expressed in terms of b^{ij} (= β^{ij}) according to (1.9.2.9) and that the trace of the matrix U is given as , one obtainsNote that in all nonorthogonal bases, . In older literature, the isotropic equivalent displacement parameter is often quoted as B_{eq}, which is related to U_{eq} through the identity . The use of B_{eq} is now discouraged (Trueblood et al., 1996). Higher atomic displacement tensors of odd rank N may be reduced to simple vectors v by the following contraction:where v^{1} is the 23 trace etc. ^{N}v^{i} is sometimes called a vector invariant, as it can be uniquely assigned to the tensor in question (Pach & Frey, 1964) and its units are length^{N − 1}. The vector v is oriented along the line of maximum projected asymmetry for a given atom and vanishes for atoms with positional parameters fixed by symmetry; Johnson (1970) has named a vector closely related to ^{3}v the vector of skew divergence. The calculation of v is useful as it gives the direction of the largest antisymmetric displacements contained in oddrank higherorder thermalmotion tensors.
Atomic displacement tensors may also be partially contracted or expanded; rules for these operations are found in Kuhs (1992).
Atoms (or molecules) situated on special positions of a space group exhibit (time–space averaged) probability distributions with a symmetry corresponding to the site symmetry. The p.d.f.'s describing these distributions contain the atomic displacement tensors. The displacement tensors enter into the structurefactor equation, which is the Fourier transform of the scattering density of the unit cell, via the atomic Debye–Waller factor, which is the Fourier transform of the atomic p.d.f. (see Chapter 1.2 of IT B). As discussed above, the tensor is fully symmetric with respect to the interchange of indices (inner symmetry). The sitesymmetry restrictions (outer symmetry) of atomic displacement tensors of rank 2 are given in Chapter 8.3 of IT C (2004), where the tabulation of the constraints on the tensor coefficients are quoted for every Wyckoff position in each space group. Here the constraints for atomic displacement tensors of ranks 2, 3, 4, 5 and 6 for any crystallographic site symmetry are tabulated; some restrictions for tensors of rank 7 and 8 can be found in Kuhs (1984). To use these tables, first the site symmetry has to be identified. The site symmetries are given in IT A (2005) for the first equipoint of every Wyckoff position in each space group. The tabulated constraints may be introduced in leastsquares refinements (some programs have the constraints of secondorder displacement tensor components already imbedded). It should also be remembered that, due to arbitrary phase shifts in the structurefactor equation in a leastsquares refinement of a noncentrosymmetric structure, for all oddorder tensors one coefficient corresponding to a nonzero entry for the corresponding acentric space group has to be kept fixed (in very much the same way as for positional parameters); e.g. the term b^{123} has to be kept fixed for one atom for all refinements in all space groups belonging to the point groups or 23, while all other terms b^{ijk} are allowed to vary freely for all atoms (Hazell & Willis, 1978). Even if this is strictly true only for the Edgeworthseries expansion, it also holds in practice for the Gram–Charlier case (Kuhs, 1992).
Levy (1956) and Peterse & Palm (1966) have given algorithms for determining the constraints on anisotropic displacement tensor coefficients, which are also applicable to higherorder tensors. The basic idea is that a tensor transformation according to the symmetry operation of the site symmetry under consideration (represented by the pointgroup generators) should leave the tensor unchanged. For symmetries higher than the identity 1, this only holds true if some of the tensor coefficients are either zero or interrelated. The constraints may be obtained explicitly from solving the homogeneous system of equations of tensor transformations (with one equation for each coefficient).
After identification of the site symmetry of the atomic site under consideration, the entry point (crossreference) for the tabulation of the displacement tensors of a given rank (Tables 1.9.3.2–1.9.3.6) needs to be looked up in Table 1.9.3.1. The line entry corresponding to the crossreference number in Tables 1.9.3.2–1.9.3.6 holds the information on the constraints imposed by the outer symmetry on the tensor coefficients. The order of assignment of independency of the coefficients is as for increasing indices of the coefficients (first 1, then 2, then 3, where 1, 2 and 3 refer to the three crystallographic axes), except for the unmixed coefficients, which have highest priority in every case; this order of priority is the same as the order in the tables reading from left to right. For better readability, each coefficent is assigned a letter (or 0 if the component is equal to zero by symmetry). Constraints thus read as algebraic relations between letter variables. Some more complicated constraint relations are quoted as footnotes to the tables.




^{†}(1) −2A/5 + D; (2) −3A/5 + B/10 + 3D/2; (3) −3A/5 + D; (4) −D + 2F; (5) −A/4 + 3F/2; (6) −2A/5 + B/5 + D; (7) −A + D; (8) −A/5 + 2B/5 + F; (9) −D + 2I; (10) −2G + 3J; (11) −E/4 + 3J/2; (12) −2H + 3K; (13) −H + K; (14) −G + 2N; (15) −4G + 6J; (16) −H/4 + 3P/2; (17) −4H + 6K.

