Atomic displacement parameters

Kuhs, W. F.

doi:10.1107/97809553602060000636

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.9, pp. 228-242
https://doi.org/10.1107/97809553602060000636

Chapter 1.9. Atomic displacement parameters

W. F. Kuhs^a ^*

^a GZG Abt. Kristallographie, Goldschmidtstrasse 1, 37077 Göttingen, Germany
Correspondence e-mail: wkuhs1@gwdg.de

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 second-order 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 higher-order 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 sixth-rank thermal motion tensors. A selection of representation surfaces of higher-rank tensors showing the distribution of anharmonic deformation densities is given at the end of the chapter.

Keywords: Gram–Charlier series; atomic displacement; cumulants; invariants; quasimoments; representation surface; site symmetry; site-symmetry restrictions; tensor contraction; tensor expansion.

1.9.1. Introduction

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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 least-squares 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 form $[T_{\alpha}({\bf Q})=\langle\exp(i{\bf Qu}_{\alpha})\rangle,\eqno(1.9.1.1)]$ where 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 $[T_{\alpha}({\bf Q})=\exp[(-1/2)\langle({\bf Qu}_{\alpha})^2\rangle].\eqno(1.9.1.2)]$

The thermodynamically averaged atomic mean-square displacements (of thermal origin) are given as $[U^{ij}=\langle u^i u^j\rangle]$ , 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 $[{\bf Q}=4\pi{\bf h}|{\bf a}|]$ in a form more familiar to the crystallographer as $[T_{\alpha}({\bf h})=\exp(-2\pi^2 h_i |{\bf a}^i|h_j|{\bf a}^j|U^{ij}_{\alpha}),\eqno(1.9.1.3)]$ where [h_i] are the covariant Miller indices, $[{\bf a}^i]$ are the reciprocal-cell basis vectors and $[1\leq\iota,\varphi\leq3]$ . Here and in the following, tensor notation is employed; implicit summation over repeated indices is assumed unless stated otherwise. For computational convenience one often writes $[T_{\alpha}({\bf h})=\exp(-h_i h_j \beta^{ij}_{\alpha})\eqno(1.9.1.4)]$ with $[\beta^{ij}_{\alpha}=2\pi^2|{\bf a}^i||{\bf a}^j|U^{ij}_{\alpha}]$ (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: $[S_{\alpha}({\bf Q})=\exp[(-1/2)\langle ({\bf Q}\Delta{\bf u}_{\alpha})^2\rangle].\eqno(1.9.1.5)]$

As in equation (1.9.1.3), the static atomic DWF may be formulated with the mean-square disorder displacements $[\Delta U^{ij}=\langle\Delta u^i\Delta u^j\rangle]$ as $[S_{\alpha}({\bf h})=\exp(-2\pi^2 h_i|{\bf a}^i|h_j|{\bf a}^j|\Delta U^{ij}_{\alpha}).\eqno(1.9.1.6)]$

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 (mean-square) atomic displacements. A separation may however be achieved by a temperature-dependent study of atomic displacements. A harmonic diagonal tensor component of purely thermal origin extrapolates linearly to zero at 0 K; zero-point motion causes a deviation from this linear behaviour at low temperatures, but an extrapolation from higher temperatures (where the contribution from zero-point motion becomes negligibly small) still yields a zero intercept. Any positive intercept in such extrapolations is then due to a (temperature-independent) 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 ).

1.9.2. The atomic displacement parameters (ADPs)

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One notes that in the Gaussian approximation, the mean-square atomic displacements (composed of thermal and static contributions) are fully described by six coefficients β^ij, which transform on a change of the direct-lattice base (according to $[{\bf a}_k =A_{ki}{\bf a}_i]$ ) as $[\beta^{kl}=A_{ki}A_{lj}\beta^{ij}.\eqno(1.9.2.1)]$

This is the transformation law of a tensor (see Section 1.1.3.2 ); the mean-square atomic displacements are thus tensorial properties of an atom α. As the tensor is contravariant and in general is described in a (non-Cartesian) 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 second-order 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 higher-order displacement terms. These terms are defined by a moment-generating function M(Q) which expresses $[\langle(\exp(i{\bf Qu}_{\alpha})\rangle]$ in terms of an infinite number of moments; for a Gaussian distribution of displacement vectors, all moments of order $[\gt\, 2]$ are identically equal to zero. Thus $[M({\bf Q})=\langle\exp(i{\bf Qu}_{\alpha})\rangle =\textstyle\sum\limits^{\infty}_{N=0}(i^N/N!)\langle ({\bf Qu}_{\alpha})^N\rangle.\eqno(1.9.2.2)]$

The moments $[\langle({\bf Qu}_{\alpha})^N\rangle]$ of order N may be expressed in terms of cumulants $[\langle({\bf Qu}_{\alpha})^N\rangle_{\rm cum}]$ by the identity $[\textstyle \sum \limits_{N=0}^{\infty}(1/N!)\langle({\bf Qu}_{\alpha})^N\rangle\equiv\exp\textstyle \sum \limits_{N=1}^{\infty}(1/N!)\langle({\bf Qu}_{\alpha})^N\rangle_{\rm cum}.\eqno(1.9.2.3)]$

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 as $[M({\bf Q})=\textstyle \sum \limits_{N=0}^{\infty}(i^N/N!)Q_i Q_j Q_k \ldots Q_n\mu^{ijk\ldots n}\eqno(1.9.2.4)]$ and the cumulant-generating function K(Q) as $[K({\bf Q})=\exp[M({\bf Q})]=\textstyle \sum \limits_{N=1}^{\infty}(I^N/N!)Q_i Q_j Q_k\ldots Q_n k^{ijk\ldots n}.\eqno(1.9.2.5)]$ The indices $[i, j, k,\ldots, n]$ run in three-dimensional 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 cumulant-generating 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 $[\exp(i{\bf Q})=\textstyle\sum(i{\bf Q})^N/N!]$ ] identical. For a finite series, however, the cumulants of order N carry implicit information on contributions of order N², N³ 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 structure-factor 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 $[\mu^{ijk\ldots n}=\textstyle \int u^i u^j u^k\ldots u^n\, {\rm p.d.f.}({\bf u})\;\rm{d}{\bf u}.\eqno(1.9.2.6)]$

The real-space 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 moment-generating function as suggested by Kuznetsov et al. (1960 ) and formulate a generating function for quasimoments as $[\tilde{M}({\bf Q})=\exp[(1/2)\langle({\bf Qu})^2\rangle]\textstyle \sum \limits_{N=3}^{\infty}(i^N/N!)Q_iQ_jQ_k\ldots Q_n\tilde{\mu}^{ijk\ldots n}.\eqno(1.9.2.7)]$ These quasimoments are especially useful in crystallographic structure-factor equations, as they just modify the harmonic case. The real-space 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 ).

1.9.2.1. Tensorial properties of (quasi)moments and cumulants

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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 higher-order displacement tensors in the form of moments, cumulants or quasimoments, which we shall denote in a general way as $[b^{ijk\ldots}]$ ; note that b^ij is identical to β^ij. They transform on a change of the direct-lattice base according to $[b^{pqr\ldots}=A_{pi}A_{qj}A_{rk}\ldots b^{ijk\ldots}.\eqno(1.9.2.8)]$

The higher-order 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 least-squares 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 $[U^{ijk\ldots n}=[N!/(2\pi)^N]b^{ijk\ldots n}|{\bf a}^i||{\bf a}^j||{\bf a}^k|\ldots|{\bf a}^n|\eqno(1.9.2.9)]$ (no summation) into quantities of units Å^N (or pm^N); aⁱ etc. are reciprocal-lattice vectors. Nowadays, the published structural results usually quote U^ij for the second-order terms; it would be good practice to publish only dimensioned atomic displacements for the higher-order terms as well.

1.9.2.2. Contraction, expansion and invariants of atomic displacement tensors

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Anisotropic or higher-order 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, lower-order terms may be expanded to higher order to impose certain (chemically implied) symmetries on the displacement tensors or to provide initial parameters for least-squares refinements. A contraction is obtained by multiplying the contravariant tensor components (referring to the real-space basis vectors) with the covariant components of the real-space 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 $[^NI_0=g_{ij}g_{kl}\ldots g_{mn}b^{ijkl\ldots mn}.\eqno(1.9.2.10)]$ [^NI_0] 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 [^4I_0] , a positive sign indicates that the corresponding (real-space) p.d.f. is peaked, a negative sign indicates flatness of the p.d.f. The larger [^NI_0] , 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 $[{\rm Tr}({\bf U})=(2\pi^2)^{-1.2}I_0]$ , one obtains $[U_{\rm eq}=(1/3)(2\pi^2)^{-1}g_{ij}b^{ij}.\eqno(1.9.2.11)]$ Note that in all non-orthogonal bases, $[{\rm Tr}({\bf U})\neq U^{11}+U^{22}+U^{33}]$ . In older literature, the isotropic equivalent displacement parameter is often quoted as B_eq, which is related to U_eq through the identity $[B_{\rm eq}=8\pi^2U_{\rm eq}]$ . 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: $[^{N}v^i=g_{jk}g_{lm}\ldots g_{np}b^{ijklm\ldots np}.\eqno(1.9.2.12)]$ where v¹ is the 23 trace etc. ^Nvⁱ 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 ³v the vector of skew divergence. The calculation of v is useful as it gives the direction of the largest antisymmetric displacements contained in odd-rank higher-order thermal-motion tensors.

Atomic displacement tensors may also be partially contracted or expanded; rules for these operations are found in Kuhs (1992 ).

1.9.3. Site-symmetry restrictions

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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 structure-factor 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 site-symmetry 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 least-squares refinements (some programs have the constraints of second-order displacement tensor components already imbedded). It should also be remembered that, due to arbitrary phase shifts in the structure-factor equation in a least-squares refinement of a noncentrosymmetric structure, for all odd-order 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 b¹²³ has to be kept fixed for one atom for all refinements in all space groups belonging to the point groups $[\bar{4}3m]$ 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 Edgeworth-series expansion, it also holds in practice for the Gram–Charlier case (Kuhs, 1992 ).

1.9.3.1. Calculation procedures

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Levy (1956 ) and Peterse & Palm (1966 ) have given algorithms for determining the constraints on anisotropic displacement tensor coefficients, which are also applicable to higher-order tensors. The basic idea is that a tensor transformation according to the symmetry operation of the site symmetry under consideration (represented by the point-group 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).

1.9.3.2. Key to tables

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After identification of the site symmetry of the atomic site under consideration, the entry point (cross-reference) 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 cross-reference 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.

Table 1.9.3.1 | top | pdf |
Site-symmetry table giving key to Tables 1.9.3.2 to 1.9.3.6 for restrictions on the symmetry of various thermal-motion tensors

Hex denotes hexagonal axes.

Point symmetry at special position					Position	Cross-reference for tensor tables
Symmetry axes		Point-group generators			Position	1B	1C	1D	1E	1F
				$[\bar 1]$		B1	C0	D1	E0	F1
$[\bar 43m]$		$[\bar 4[0,0,1]]$				B1	C1	D1	E1	F1
						B1	C0	D1	E0	F1
				$[\bar 1]$		B1	C0	D1	E0	F2
						B1	C1	D1	E1	F2
	Hex			$[\bar 1]$		B9	C0	D2	E0	F3
$[\bar 6m2]$	Hex	$[\bar 6[0,0,1]]$				B9	C9	D2	E5	F3
$[\bar 6m2]$	Hex	$[\bar 6[0,0,1]]$				B9	C10	D2	E6	F3
	Hex		$[\bar 2[1,0,0]]$			B9	C19	D2	E17	F3
	Hex					B9	C0	D2	E0	F4
	Hex		$[\bar 1]$			B9	C0	D2	E0	F4
$[\bar 6]$	Hex	$[\bar 6[0,0,1]]$				B9	C20	D2	E24	F4
	Hex					B9	C19	D2	E17	F4
				$[\bar 1]$		B2	C0	D3	E0	F5
				$[\bar 1]$		B3	C0	D4	E0	F6
				$[\bar 1]$		B4	C0	D5	E0	F7
$[\bar 42m]$		$[\bar 4[0,0,1]]$				B2	C1	D3	E7	F5
$[\bar 42m]$		$[\bar 4[0,0,1]]$				B2	C2	D3	E8	F5
$[\bar 42m]$		$[\bar 4[0,1,0]]$				B3	C1	D4	E9	F6
$[\bar 42m]$		$[\bar 4[0,1,0]]$				B3	C3	D4	E10	F6
$[\bar 42m]$		$[\bar 4[1,0,0]]$				B4	C1	D5	E11	F7
$[\bar 42m]$		$[\bar 4[1,0,0]]$				B4	C4	D5	E12	F7
			$[\bar 2[1,0,0]]$			B2	C13	D3	E25	F5
			$[\bar 2[0,0,1]]$			B3	C14	D4	E26	F6
			$[\bar 2[0,1,0]]$			B4	C15	D5	E27	F7
						B2	C0	D3	E2	F5
						B3	C0	D4	E3	F6
						B4	C0	D5	E4	F7
			$[\bar 1]$			B2	C0	D12	E0	F14
			$[\bar 1]$			B3	C0	D13	E0	F15
			$[\bar 1]$			B4	C0	D14	E0	F16
$[\bar 4]$		$[\bar 4[0,0,1]]$				B2	C16	D12	E28	F14
$[\bar 4]$		$[\bar 4[0,1,0]]$				B3	C17	D13	E29	F15
$[\bar 4]$		$[\bar 4[1,0,0]]$				B4	C18	D14	E30	F16
						B2	C13	D12	E31	F14
						B3	C14	D13	E32	F15
						B4	C15	D14	E33	F16
$[\bar 3m]$			$[2[1,\bar 1,0]]$	$[\bar 1]$		B5	C0	D6	E0	F8
$[\bar 3m]$		$[3[1,1,\bar 1]]$	$[2[1,\bar 1,0]]$	$[\bar 1]$		B6	C0	D7	E0	F9
$[\bar 3m]$		$[3[1,\bar 1,1]]$		$[\bar 1]$		B7	C0	D8	E0	F10
$[\bar 3m]$		$[3[\bar 1,1,1]]$		$[\bar 1]$		B8	C0	D9	E0	F11
$[\bar 3m]$	Hex			$[\bar 1]$		B9	C0	D10	E0	F12
$[\bar 3m]$	Hex			$[\bar 1]$		B9	C0	D11	E0	F13
			$[\bar 2[1,\bar 1,0]]$			B5	C33	D6	E34	F8
		$[3[1,1,\bar 1]]$	$[\bar 2[1,\bar 1,0]]$		$[x,x,\bar x]$	B6	C34	D7	E35	F9
		$[3[1,\bar 1,1]]$	$[\bar 2[1,1,0]]$		$[x,\bar x,x]$	B7	C35	D8	E36	F10
		$[3[\bar 1,1,1]]$	$[\bar 2[1,1,0]]$		$[\bar x,x,x]$	B8	C36	D9	D37	F11
	Hex		$[\bar 2[1,0,0]]$			B9	C37	D10	E38	F12
	Hex		$[\bar 2[1,2,0]]$			B9	C38	D11	E39	F13
			$[2[1,\bar 1,0]]$			B5	C5	D6	E13	F8
		$[3[1,1,\bar 1]]$	$[2[1,\bar 1,0]]$			B6	C6	D7	E14	F9
		$[3[1,\bar 1,1]]$				B7	C7	D8	E15	F10
		$[3[\bar 1,1,1]]$				B8	C8	D9	E16	F11
	Hex					B9	C9	D10	E5	F12
	Hex					B9	C10	D11	E6	F13
$[\bar 3]$		$[\bar 3[1,1,1]]$				B5	C0	D15	E0	F17
$[\bar 3]$		$[\bar 3[1,1,\bar 1]]$				B6	C0	D16	E0	F18
$[\bar 3]$		$[\bar 3[1,\bar 1,1]]$				B7	C0	D17	E0	F19
$[\bar 3]$		$[\bar 3[\bar 1,1,1]]$				B8	C0	D18	E0	F20
$[\bar 3]$	Hex	$[\bar 3[0,0,1]]$				B9	C0	D19	E0	F21
						B5	C54	D15	E58	F17
		$[3[1,1,\bar 1]]$			$[x,x,\bar x]$	B6	C55	D16	E59	F18
		$[3[1,\bar 1,1]]$			$[x,\bar x,x]$	B7	C56	D17	E60	F19
	Hex	$[3[\bar 1,1,1]]$			$[\bar x,x,x]$	B8	C57	D18	E61	F20
						B9	C58	D19	E62	F21
				$[\bar 1]$		B10	C0	D20	E0	F22
				$[\bar 1]$		B11	C0	D21	E0	F23
				$[\bar 1]$		B12	C0	D22	E0	F24
				$[\bar 1]$		B13	C0	D23	E0	F25
	Hex			$[\bar 1]$		B14	C0	D24	E0	F26
	Hex			$[\bar 1]$		B11	C0	D21	E0	F23
	Hex			$[\bar 1]$		B15	C0	D25	E0	F27
			$[\bar 2[1,0,0]]$			B10	C21	D20	E40	F22
			$[\bar 2[1,1,0]]$			B11	C22	D21	E41	F23
			$[\bar 2[0,0,1]]$			B10	C23	D20	E42	F22
			$[\bar 2[1,0,1]]$			B12	C24	D22	E43	F24
			$[\bar 2[0,0,1]]$			B10	C25	D20	E44	F22
			$[\bar 2[0,1,1]]$			B13	C26	D23	E45	F25
			$[\bar 2[0,0,1]]$			B11	C27	D21	E46	F23
		$[2[1,\bar 1,0]]$	$[\bar 2[0,0,1]]$		$[x,\bar x,0]$	B11	C28	D21	E47	F23
			$[\bar 2[0,1,0]]$			B12	C29	D22	E48	F24
		$[2[1,0,\bar 1]]$	$[\bar 2[0,1,0]]$		$[x,0,\bar x]$	B12	C30	D22	E49	F24
			$[\bar 2[1,0,0]]$			B13	C31	D23	E50	F25
		$[2[0,1,\bar 1]]$	$[\bar 2[1,0,0]]$		$[0,y,\bar y]$	B13	C32	D23	E51	F25
	Hex		$[\bar 2[1,0,0]]$			B14	C40	D24	E52	F26
	Hex		$[\bar 2[1,1,0]]$			B11	C22	D21	E41	F23
	Hex		$[\bar 2[0,1,0]]$			B15	C39	D25	E53	F27
	Hex		$[\bar 2[0,0,1]]$			B14	C41	D24	E54	F26
	Hex		$[\bar 2[0,0,1]]$			B15	C42	D25	E55	F27
	Hex		$[\bar 2[0,0,1]]$			B11	C27	D21	E46	F23
	Hex		$[\bar 2[0,0,1]]$			B14	C43	D24	E56	F26
	Hex		$[\bar 2[0,0,1]]$			B15	C44	D25	E57	F27
	Hex	$[2[1,\bar 1,0]]$	$[\bar 2[0,0,1]]$		$[x,\bar x,0]$	B11	C28	D21	E47	F23
						B10	C1	D20	E18	F22
						B11	C2	D21	E19	F23
						B12	C3	D22	E20	F24
						B13	C4	D23	E21	F25
	Hex					B14	C11	D24	E22	F26
	Hex					B11	C2	D21	E19	F23
	Hex					B15	C12	D25	E23	F27
			$[\bar 1]$			B16	C0	D26	E0	F28
			$[\bar 1]$			B17	C0	D27	E0	F29
			$[\bar 1]$			B18	C0	D28	E0	F30
			$[\bar 1]$			B19	C0	D29	E0	F31
		$[2[1,\bar 1,0]]$	$[\bar 1]$			B20	C0	D30	E0	F32
			$[\bar 1]$			B21	C0	D31	E0	F33
		$[2[1,0,\bar 1]]$	$[\bar 1]$			B22	C0	D32	E0	F34
			$[\bar 1]$			B23	C0	D33	E0	F35
		$[2[0,1,\bar 1]]$	$[\bar 1]$			B24	C0	D34	E0	F36
	Hex		$[\bar 1]$			B16	C0	D26	E0	F28
	Hex		$[\bar 1]$			B25	C0	D35	E0	F37
	Hex		$[\bar 1]$			B26	C0	D36	E0	F38
	Hex		$[\bar 1]$			B19	C0	D29	E0	F31
	Hex		$[\bar 1]$			B27	C0	D37	E0	F39
	Hex		$[\bar 1]$			B28	C0	D38	E0	F40
	Hex	$[2[1,\bar 1,0]]$	$[\bar 1]$			B20	C0	D30	E0	F32
m		$[\bar 2[0,1,0]]$				B17	C64	D27	E77	F29
m		$[\bar 2[1,0,0]]$				B18	C65	D28	E78	F30
m		$[\bar 2[1,1,0]]$			$[x,\bar x,z]$	B19	C66	D29	E79	F31
m		$[\bar 2[1,\bar 1,0]]$				B20	C67	D30	E80	F32
m		$[\bar 2[1,0,1]]$			$[x,y,\bar x]$	B21	C68	D31	E81	F33
m		$[\bar 2[1,0,\bar 1]]$				B22	C69	D32	E82	F34
m		$[\bar 2[0,1,1]]$			$[x,y,\bar y]$	B23	C70	D33	E83	F35
m		$[\bar 2[0,1,\bar 1]]$				B24	C71	D34	E84	F36
m	Hex	$[\bar 2[0,0,1]]$				B16	C63	D26	E76	F28
m	Hex	$[\bar 2[1,0,0]]$				B25	C72	D35	E85	F37
m	Hex	$[\bar 2[2,1,0]]$				B26	C73	D36	E86	F38
m	Hex	$[\bar 2[1,1,0]]$			$[x,\bar x,z]$	B19	C66	D29	E79	F31
m	Hex	$[\bar 2[1,2,0]]$				B27	C74	D37	E87	F39
m	Hex	$[\bar 2[0,1,0]]$				B28	C75	D38	E88	F40
m	Hex	$[\bar 2[1,\bar 1,0]]$				B20	C67	D30	E80	F32
						B16	C45	D26	E63	F28
						B17	C46	D27	E64	F29
						B18	C47	D28	E65	F30
						B19	C48	D29	E66	F31
		$[2[1,\bar 1,0]]$			$[x,\bar x,0]$	B20	C49	D30	E67	F32
						B21	C50	D31	E68	F33
		$[2[1,0,\bar 1]]$			$[x,0,\bar x]$	B22	C51	D32	E69	F34
						B23	C52	D33	E70	F35
		$[2[0,1,\bar 1]]$			$[0,y,\bar y]$	B24	C53	D34	E71	F36
	Hex					B16	C45	D26	E63	F28
	Hex					B25	C59	D35	E72	F37
	Hex					B26	C60	D36	E73	F38
	Hex					B19	C48	D29	E66	F31
	Hex					B27	C61	D37	E74	F39
	Hex					B28	C62	D38	E75	F40
	Hex	$[2[1,\bar 1,0]]$			$[x,\bar x,0]$	B20	C49	D30	E67	F32
$[\bar 1]$		$[\bar 1]$				B29	C0	D39	E0	F41
$[\bar 1]$	Hex	$[\bar 1]$				B29	C0	D39	E0	F41
						B29	C76	D39	E89	F41
	Hex					B29	C76	D39	E89	F41

Table 1.9.3.2 | top | pdf |
Symmetry restrictions on coefficients in second-order tensors

Cross-reference	No. of independent variables	Symbols and coefficient indices
		A	B	C	D	E	F
		(1)	(2)	(3)	(1)	(1)	(2)
		(1)	(2)	(3)	(2)	(3)	(3)
B1	1	A	A	A	0	0	0
B2	2	A	A	C	0	0	0
B3	2	A	B	A	0	0	0
B4	2	A	B	B	0	0	0
B5	2	A	A	A	D	D	D
B6	2	A	A	A	D	−D	−D
B7	2	A	A	A	D	−D	D
B8	2	A	A	A	D	D	−D
B9	2	A	A	C	A/2	0	0
B10	3	A	B	C	0	0	0
B11	3	A	A	C	D	0	0
B12	3	A	B	A	0	E	0
B13	3	A	B	B	0	0	F
B14	3	A	B	C	B/2	0	0
B15	3	A	B	C	A/2	0	0
B16	4	A	B	C	D	0	0
B17	4	A	B	C	0	E	0
B18	4	A	B	C	0	0	F
B19	4	A	A	C	D	E	−E
B20	4	A	A	C	D	E	E
B21	4	A	B	A	D	E	−D
B22	4	A	B	A	D	E	D
B23	4	A	B	B	D	−D	F
B24	4	A	B	B	D	D	F
B25	4	A	B	C	B/2	E	2E
B26	4	A	B	C	A/2	0	F
B27	4	A	B	C	B/2	E	0
B28	4	A	B	C	A/2	E	E/2
B29	6	A	B	C	D	E	F

Table 1.9.3.3 | top | pdf |
Symmetry restrictions on coefficients in third-rank symmetric polar tensors

Cross-reference	No. of independent variables	Symbols and coefficient indices
		A	B	C	D	E	F	G	H	I	J
		(1)	(2)	(3)	(1)	(1)	(1)	(1)	(2)	(2)	(1)
		(1)	(2)	(3)	(1)	(2)	(1)	(3)	(2)	(3)	(2)
		(1)	(2)	(3)	(2)	(2)	(3)	(3)	(3)	(3)	(3)
C0	0	0	0	0	0	0	0	0	0	0	0
C1	1	0	0	0	0	0	0	0	0	0	J
C2	1	0	0	0	0	0	F	0	−F	0	0
C3	1	0	0	0	D	0	0	0	0	−D	0
C4	1	0	0	0	0	E	0	−E	0	0	0
C5	1	0	0	0	D	−D	−D	D	D	−D	0
C6	1	0	0	0	D	−D	D	D	−D	−D	0
C7	1	0	0	0	D	D	D	−D	−D	−D	0
C8	1	0	0	0	D	D	−D	−D	D	−D	0
C9	1	0	0	0	D	D	0	0	0	0	0
C10	1	A	−A	0	A/2	−A/2	0	0	0	0	0
C11	1	0	0	0	0	0	F	0	0	0	F
C12	1	0	0	0	0	0	0	0	H	0	H
C13	2	0	0	C	0	0	F	0	F	0	0
C14	2	0	B	0	D	0	0	0	0	D	0
C15	2	A	0	0	0	E	0	E	0	0	0
C16	2	0	0	0	0	0	F	0	−F	0	J
C17	2	0	0	0	D	0	0	0	0	−D	J
C18	2	0	0	0	0	E	0	−E	0	0	J
C19	2	0	0	C	0	0	F	0	F	0	F/2
C20	2	A	−A	0	D	D − A	0	0	0	0	0
C21	3	0	0	C	0	0	F	0	H	0	0
C22	3	0	0	C	0	0	F	0	F	0	J
C23	3	0	B	0	D	0	0	0	0	I	0
C24	3	0	B	0	D	0	0	0	0	D	J
C25	3	A	0	0	0	E	0	G	0	0	0
C26	3	A	0	0	0	E	0	E	0	0	J
C27	3	A	A	0	D	D	0	G	0	G	0
C28	3	A	−A	0	D	−D	0	G	0	−G	0
C29	3	A	0	A	0	E	F	F	E	0	0
C30	3	A	0	−A	0	E	F	−F	−E	0	0
C31	3	0	B	B	D	0	D	0	H	H	0
C32	3	0	B	−B	D	0	−D	0	H	−H	0
C33	3	A	A	A	D	D	D	D	D	D	J
C34	3	A	A	−A	D	D	−D	D	−D	D	J
C35	3	A	−A	A	D	−D	−D	−D	−D	D	J
C36	3	A	−A	−A	D	−D	D	−D	D	D	J
C37	3	A	−A	C	A/2	−A/2	F	0	F	0	F/2
C38	3	0	0	C	D	D	F	0	F	0	F/2
C39	3	0	0	C	0	0	F	0	H	0	F/2
C40	3	0	0	C	0	0	F	0	H	0	H/2
C41	3	A	0	0	D	D	0	G	0	0	0
C42	3	A	B	0	A/2	A/6 + 2B/3	0	G	0	G/2	0
C43	3	A	B	0	B/6 + 2A/3	B/2	0	G	0	2G	0
C44	3	0	B	0	D	D	0	0	0	I	0
C45	4	0	0	C	0	0	F	0	H	0	J
C46	4	0	B	0	D	0	0	0	0	I	J
C47	4	A	0	0	0	E	0	G	0	0	J
C48	4	A	A	0	D	D	F	G	−F	G	0
C49	4	A	−A	0	D	−D	F	G	−F	−G	0
C50	4	A	0	A	D	E	F	F	E	−D	0
C51	4	A	0	−A	D	E	F	−F	−E	−D	0
C52	4	0	B	B	D	E	D	−E	H	H	0
C53	4	0	B	−B	D	E	−D	−E	H	−H	0
C54	4	A	A	A	D	E	E	D	D	E	J
C55	4	A	A	−A	D	E	−E	D	−D	E	J
C56	4	A	−A	A	D	E	E	−D	−D	−E	J
C57	4	A	−A	−A	D	E	−E	−D	D	−E	J
C58	4	A	−A	C	D	D − A	F	0	F	0	F/2
C59	4	A	0	0	D	D	F	G	0	0	F
C60	4	A	B	0	A/2	A/6 + 2B/3	0	G	H	G/2	H
C61	4	A	B	0	B/6 + 2A/3	B/2	F	G	0	2G	F
C62	4	0	B	0	D	D	0	0	H	I	H
C63	6	A	B	0	D	E	0	G	0	I	0
C64	6	A	0	C	0	E	F	G	H	0	0
C65	6	0	B	C	D	0	F	0	H	I	0
C66	6	A	−A	C	D	−D	F	G	F	−G	J
C67	6	A	A	C	D	D	F	G	F	G	J
C68	6	A	B	−A	D	E	F	−F	−E	D	J
C69	6	A	B	A	D	E	F	F	E	D	J
C70	6	A	B	−B	D	E	−D	E	H	−H	J
C71	6	A	B	B	D	E	D	E	H	H	J
C72	6	A	B	C	B/6 + 2A/3	B/2	F	G	H	2G	H/2
C73	6	0	B	C	D	D	F	0	H	I	F/2
C74	6	A	0	C	D	D	F	G	H	0	H/2
C75	6	A	B	C	A/2	A/6 + 2B/3	F	G	H	G/2	F/2
C76	10	A	B	C	D	E	F	G	H	I	J

Table 1.9.3.4 | top | pdf |
Symmetry restrictions on coefficients in fourth-rank symmetric polar tensors

(a) A–H.

Cross-reference	No. of independent variables	Symbols and coefficient indices
		A	B	C	D	E	F	G	H
		(1)	(2)	(3)	(1)	(1)	(1)	(1)	(1)
		(1)	(2)	(3)	(1)	(1)	(1)	(1)	(1)
		(1)	(2)	(3)	(1)	(1)	(2)	(2)	(3)
		(1)	(2)	(3)	(2)	(3)	(2)	(3)	(3)
D1	2	A	A	A	0	0	F	0	F
D2	3	A	A	C	A/2	0	A/2	0	H
D3	4	A	A	C	0	0	F	0	H
D4	4	A	B	A	0	0	F	0	H
D5	4	A	B	B	0	0	F	0	F
D6	4	A	A	A	D	D	F	G	F
D7	4	A	A	A	D	−D	F	G	F
D8	4	A	A	A	D	−D	F	G	F
D9	4	A	A	A	D	D	F	G	F
D10	4	A	A	C	A/2	E	A/2	E/2	H
D11	4	A	A	C	A/2	0	A/2	G	H
D12	5	A	A	C	D	0	F	0	H
D13	5	A	B	A	0	E	F	0	H
D14	5	A	B	B	0	0	F	0	F
D15	5	A	A	A	D	E	F	G	F
D16	5	A	A	A	D	E	F	G	F
D17	5	A	A	A	D	E	F	G	F
D18	5	A	A	A	D	E	F	G	F
D19	5	A	A	C	A/2	E	A/2	G	H
D20	6	A	B	C	0	0	F	0	H
D21	6	A	A	C	D	0	F	0	H
D22	6	A	B	A	0	E	F	0	H
D23	6	A	B	B	0	0	F	G	F
D24	6	A	B	C	D	0	B/6 + 2D/3	0	H
D25	6	A	B	C	A/2	0	F	0	H
D26	9	A	B	C	D	0	F	0	H
D27	9	A	B	C	0	E	F	0	H
D28	9	A	B	C	0	0	F	G	H
D29	9	A	A	C	D	E	F	G	H
D30	9	A	A	C	D	E	F	G	H
D31	9	A	B	A	D	E	F	G	H
D32	9	A	B	A	D	E	F	G	H
D33	9	A	B	B	D	−D	F	G	F
D34	9	A	B	B	D	D	F	G	F
D35	9	A	B	C	D	E	B/6 + 2D/3	G	H
D36	9	A	B	C	A/2	0	F	G	H
D37	9	A	B	C	D	E	B/6 + 2D/3	G	H
D38	9	A	B	C	A/2	E	F	E/2	H
D39	15	A	B	C	D	E	F	G	H

(b) I–P.

Cross-reference	No. of independent variables	Symbols and coefficient indices
		I	J	K	L	M	N	P
		(1)	(1)	(1)	(1)	(2)	(2)	(2)
		(2)	(2)	(2)	(3)	(2)	(2)	(3)
		(2)	(2)	(3)	(3)	(2)	(3)	(3)
		(2)	(3)	(3)	(3)	(3)	(3)	(3)
D1	2	0	0	0	0	0	F	0
D2	3	A/2	0	H/2	0	0	H	0
D3	4	0	0	0	0	0	H	0
D4	4	0	0	0	0	0	F	0
D5	4	0	0	0	0	0	N	0
D6	4	D	G	G	D	D	F	D
D7	4	D	G	−G	−D	−D	F	−D
D8	4	D	−G	G	−D	D	F	D
D9	4	D	−G	−G	D	−D	F	−D
D10	4	A/2	−E/2	H/2	0	−E	H	0
D11	4	A/2	G	H/2	0	0	H	0
D12	5	−D	0	0	0	0	H	0
D13	5	0	0	0	−E	0	F	0
D14	5	0	0	0	0	M	N	−M
D15	5	E	G	G	D	D	F	E
D16	5	−E	G	−G	−D	−D	F	−E
D17	5	−E	−G	G	−D	D	F	E
D18	5	E	−G	−G	D	−D	F	−E
D19	5	A/2	G − E	H/2	0	−E	H	0
D20	6	0	0	0	0	0	N	0
D21	6	D	0	K	0	0	H	0
D22	6	0	J	0	E	0	F	0
D23	6	0	0	0	0	M	N	M
D24	6	B/2	0	K	0	0	2K	0
D25	6	−A/4 + 3F/2	0	H/2	0	0	N	0
D26	9	I	0	K	0	0	N	0
D27	9	0	J	0	L	0	N	0
D28	9	0	0	0	0	M	N	P
D29	9	D	−G	K	L	−E	H	−L
D30	9	D	G	K	L	E	H	L
D31	9	I	J	−G	E	−I	F	−D
D32	9	I	J	G	E	I	F	D
D33	9	I	J	−J	−I	M	N	M
D34	9	I	J	J	I	M	N	M
D35	9	B/2	−E/8 + 3G	K	L	−E/4 + 6G	2K	2L
D36	9	−A/4 + 3F/2	G	H/2	0	M	N	P
D37	9	B/2	G	K	L	0	2K	0
D38	9	−A/4 + 3F/2	J	H/2	L	−E/4 + 3J/2	N	L/2
D39	15	I	J	K	L	M	N	P

Table 1.9.3.5 | top | pdf |
Symmetry restrictions on coefficients in fifth-rank symmetric polar tensors

(a) A–K.

Cross-reference	No. of independent coefficients	Symbols and coefficient indices
		A	B	C	D	E	F	G	H	I	J	K
		1	2	3	1	1	1	1	1	1	1	1
		1	2	3	1	1	1	1	1	1	1	1
		1	2	3	1	1	1	1	1	2	2	2
		1	2	3	1	1	2	2	3	2	2	3
		1	2	3	2	3	2	3	3	2	3	3
E0	0	0	0	0	0	0	0	0	0	0	0	0
E1	1	0	0	0	0	0	0	G	0	0	0	0
E2	1	0	0	0	0	0	0	G	0	0	0	0
E3	1	0	0	0	0	0	0	G	0	0	0	0
E4	1	0	0	0	0	0	0	0	0	0	0	0
E5	2	0	0	0	D	0	D	0	0	D	0	K
E6	2	A	−A	0	A/2	0	A/10	0	H	−A/10	0	H/2
E7	2	0	0	0	0	0	0	G	0	0	0	0
E8	2	0	0	0	0	E	0	0	0	0	0	0
E9	2	0	0	0	0	0	0	G	0	0	0	0
E10	2	0	0	0	D	0	0	0	0	I	0	0
E11	2	0	0	0	0	0	0	G	0	0	0	0
E12	2	0	0	0	0	0	F	0	−F	0	0	0
E13	2	0	0	0	D	−D	F	0	−F	−F	0	0
E14	2	0	0	0	D	D	F	0	−F	−F	0	0
E15	2	0	0	0	D	D	F	0	−F	F	0	0
E16	2	0	0	0	D	−D	F	0	−F	F	0	0
E17	3	0	0	C	0	E	0	E/2	0	0	E/2	0
E18	3	0	0	0	0	0	0	G	0	0	0	0
E19	3	0	0	0	0	E	0	G	0	0	0	0
E20	3	0	0	0	D	0	0	G	0	I	0	0
E21	3	0	0	0	0	0	F	0	−F	0	0	0
E22	3	0	0	0	0	0	0	G	0	0	2G	0
E23	3	0	0	0	0	0	0	G	0	0	G	0
E24	4	A	−A	0	D	0	(1)^†	0	H	(3)^†	0	K
E25	4	0	0	C	0	E	0	0	0	0	J	0
E26	4	0	B	0	D	0	0	0	0	I	0	K
E27	4	A	0	0	0	0	F	0	F	0	0	0
E28	4	0	0	0	0	E	0	G	0	0	0	0
E29	4	0	0	0	D	0	0	G	0	I	0	0
E30	4	0	0	0	0	0	F	G	−F	0	0	0
E31	5	0	0	C	0	E	0	G	0	0	J	0
E32	5	0	B	0	D	0	0	G	0	I	0	K
E33	5	A	0	0	0	0	F	0	F	0	0	0
E34	5	A	A	A	D	D	F	G	F	F	J	J
E35	5	A	A	−A	D	−D	F	G	F	F	J	−J
E36	5	A	−A	A	D	−D	F	G	F	−F	J	−J
E37	5	A	−A	−A	D	D	F	G	F	−F	J	J
E38	5	A	−A	C	A/2	E	A/10	E/2	H	−A/10	E/2	H/2
E39	5	0	0	C	D	E	D	E/2	0	D	E/2	K
E40	6	0	0	C	0	E	0	0	0	0	J	0
E41	6	0	0	C	0	E	0	G	0	0	J	0
E42	6	0	B	0	D	0	0	0	0	I	0	K
E43	6	0	B	0	D	0	0	G	0	I	0	K
E44	6	A	0	0	0	0	F	0	H	0	0	0
E45	6	A	0	0	0	0	F	G	H	0	0	0
E46	6	A	−A	0	D	0	F	0	H	F	0	K
E47	6	A	−A	0	D	0	F	0	H	−F	0	K
E48	6	A	0	A	0	E	F	0	H	0	J	0
E49	6	A	0	−A	0	E	F	0	H	0	G	0
E50	6	0	B	B	D	D	0	0	0	I	J	J
E51	6	0	B	−B	D	−D	0	0	0	I	J	−J
E52	6	0	0	C	0	E	0	G	0	0	J	0
E53	6	0	0	C	0	E	0	E/2	0	0	J	0
E54	6	A	0	0	D	0	F	0	H	(4)^†	0	K
E55	6	A	B	0	A/2	0	F	0	H	(5)^†	0	H/2
E56	6	A	B	0	D	0	(2)^†	0	H	(6)^†	0	K
E57	6	0	B	0	D	0	D	0	0	I	0	K
E58	7	A	A	A	D	E	F	G	H	H	J	J
E59	7	A	A	−A	D	E	F	G	H	H	J	−J
E60	7	A	−A	A	D	E	F	G	H	−H	J	−J
E61	7	A	−A	−A	D	E	F	G	H	−H	J	J
E62	7	A	−A	C	D	E	(1)^†	E/2	H	(3)^†	E/2	K
E63	9	0	0	C	0	E	0	G	0	0	J	0
E64	9	0	B	0	D	0	0	G	0	I	0	K
E65	9	A	0	0	0	0	F	G	H	0	0	0
E66	9	A	A	0	D	E	F	G	H	F	0	K
E67	9	A	−A	0	D	E	F	G	H	−F	0	K
E68	9	A	0	A	D	E	F	G	H	I	J	0
E69	9	A	0	−A	D	E	F	G	H	I	J	0
E70	9	0	B	B	D	D	F	0	−F	I	J	J
E71	9	0	B	−B	D	−D	F	0	−F	I	J	−J
E72	9	A	0	0	D	E	F	G	H	(4)^†	2G	K
E73	9	A	B	0	A/2	0	F	G	H	(5)^†	G	H/2
E74	9	A	B	0	D	E	(2)^†	G	H	(6)^†	2G	K
E75	9	0	B	0	D	0	D	G	0	I	G	K
E76	12	A	B	0	D	0	F	0	H	I	0	K
E77	12	A	0	C	0	E	F	0	H	0	J	0
E78	12	0	B	C	D	E	0	0	0	I	J	K
E79	12	A	−A	C	D	E	F	G	H	−F	J	K
E80	12	A	A	C	D	E	F	G	H	F	J	K
E81	12	A	B	−A	D	E	F	G	H	I	J	K
E82	12	A	B	A	D	E	F	G	H	I	J	K
E83	12	A	B	−B	D	−D	F	G	F	I	J	−J
E84	12	A	B	B	D	D	F	G	F	I	J	J
E85	12	A	B	C	D	E	(2)^†	G	H	(6)^†	J	K
E86	12	0	B	C	D	E	D	E/2	0	I	J	K
E87	12	A	0	C	D	E	F	G	H	(4)^†	J	K
E88	12	A	B	C	A/2	E	F	E/2	H	(5)^†	J	H/2
E89	21	A	B	C	D	E	F	G	H	I	J	K

(b) L–V.

Cross-reference	No. of independent coefficients	Symbols and coefficient indices
		L	M	N	P	Q	R	S	T	U	V
		1	1	1	1	1	1	2	2	2	2
		1	2	2	2	2	3	2	2	2	3
		3	2	2	2	3	3	2	2	3	3
		3	2	2	3	3	3	2	3	3	3
		3	2	3	3	3	3	3	3	3	3
E0	0	0	0	0	0	0	0	0	0	0	0
E1	1	0	0	G	0	G	0	0	0	0	0
E2	1	0	0	−G	0	0	0	0	0	0	0
E3	1	0	0	0	0	−G	0	0	0	0	0
E4	1	0	0	N	0	−N	0	0	0	0	0
E5	2	0	D	0	K	0	0	0	0	0	0
E6	2	0	−A/2	0	−H/2	0	0	0	0	−H	0
E7	2	0	0	G	0	Q	0	0	0	0	0
E8	2	L	0	0	0	0	0	−E	0	−L	0
E9	2	0	0	N	0	G	0	0	0	0	0
E10	2	0	0	0	0	0	0	0	−I	0	−D
E11	2	0	0	N	0	N	0	0	0	0	0
E12	2	0	M	0	0	0	−M	0	0	0	0
E13	2	F	−D	0	0	0	D	D	F	−F	−D
E14	2	−F	−D	0	0	0	D	−D	F	F	−D
E15	2	F	D	0	0	0	−D	−D	−F	−F	−D
E16	2	−F	D	0	0	0	−D	D	−F	F	−D
E17	3	L	0	E/2	0	L/2	0	E	0	L	0
E18	3	0	0	N	0	Q	0	0	0	0	0
E19	3	L	0	−G	0	0	0	−E	0	−L	0
E20	3	0	0	0	0	−G	0	0	−I	0	−D
E21	3	0	M	N	0	−N	−M	0	0	0	0
E22	3	L	0	2G	0	L	0	S	0	0	0
E23	3	0	0	N	0	Q	0	(14)^†	0	Q	0
E24	4	0	(7)^†	0	(13)^†	0	0	0	−H	0	0
E25	4	L	0	0	0	0	0	E	0	L	0
E26	4	0	0	0	0	0	0	0	I	0	D
E27	4	0	M	0	P	0	M	0	0	0	0
E28	4	L	0	G	0	Q	0	−E	0	−L	0
E29	4	0	0	N	0	G	0	0	−I	0	−D
E30	4	0	M	N	0	N	−M	0	0	0	0
E31	5	L	0	−G	0	0	0	E	0	L	0
E32	5	0	0	0	0	−G	0	0	I	0	D
E33	5	0	M	N	P	−N	M	0	0	0	0
E34	5	F	D	G	J	G	D	D	F	F	D
E35	5	−F	D	G	−J	G	D	−D	F	−F	D
E36	5	F	−D	G	J	G	−D	−D	−−F	F	D
E37	5	−F	−D	G	−J	G	−D	D	−F	−F	D
E38	5	L	−A/2	E/2	−H/2	L/2	0	E	−H	L	0
E39	5	L	D	E/2	K	L/2	0	E	0	L	0
E40	6	L	0	0	0	0	0	S	0	U	0
E41	6	L	0	G	0	Q	0	E	0	L	0
E42	6	0	0	0	0	0	0	0	T	0	V
E43	6	0	0	N	0	G	0	0	I	0	D
E44	6	0	M	0	P	0	R	0	0	0	0
E45	6	0	M	N	P	N	M	0	0	0	0
E46	6	0	D	0	K	0	R	0	H	0	R
E47	6	0	−D	0	−K	0	R	0	−H	0	−R
E48	6	H	M	0	J	0	E	M	0	F	0
E49	6	−H	M	0	−J	0	−E	−M	0	−F	0
E50	6	I	0	0	0	0	0	S	T	T	S
E51	6	−I	0	0	0	0	0	S	T	−T	−S
E52	6	L	0	(10)^†	0	Q	0	(15)^†	0	2Q	0
E53	6	L	0	(11)^†	0	L/2	0	S	0	U	0
E54	6	0	(4)^†	0	K	0	R	0	0	0	0
E55	6	0	(8)^†	0	P	0	R	0	(16)^†	0	R/2
E56	6	0	B/2	0	(12)^†	0	R	0	(17)^†	0	2R
E57	6	0	(9)^†	0	K	0	0	0	T	0	V
E58	7	F	E	G	J	G	D	D	F	H	E
E59	7	−F	−E	G	−J	G	D	−D	F	−H	−E
E60	7	F	E	G	J	G	−D	−D	−F	H	−E
E61	7	−F	−E	G	−J	G	−D	D	−F	−H	E
E62	7	L	(7)^†	E/2	(13)^†	L/2	0	E	−H	L	0
E63	9	L	0	N	0	Q	0	S	0	U	0
E64	9	0	0	N	0	Q	0	0	T	0	V
E65	9	0	M	N	P	Q	R	0	0	0	0
E66	9	L	D	−G	K	0	R	−E	H	−L	R
E67	9	L	−D	−G	−K	0	R	−E	−H	−L	−R
E68	9	H	M	0	J	−G	E	M	−I	F	−D
E69	9	−H	M	0	−J	−G	−E	−M	−I	−F	−D
E70	9	I	M	N	0	−N	−M	S	T	T	S
E71	9	−I	M	N	0	−N	−M	S	T	−T	−S
E72	9	L	(4)^†	2G	K	L	R	0	0	0	0
E73	9	0	(8)^†	N	P	Q	R	(14)^†	(16)^†	Q	R/2
E74	9	L	B/2	2G	(12)^†	L	R	0	(17)^†	0	2R
E75	9	0	(9)^†	N	K	Q	0	(14)^†	T	Q	V
E76	12	0	M	0	P	0	R	0	T	0	V
E77	12	L	M	0	P	0	R	S	0	U	0
E78	12	L	0	0	0	0	0	S	T	U	V
E79	12	L	−D	G	−K	Q	R	E	−H	L	−R
E80	12	L	D	G	J	Q	R	E	H	L	R
E81	12	−H	M	N	−J	G	−E	−M	I	−F	D
E82	12	H	M	N	J	G	E	M	I	F	D
E83	12	−I	M	N	P	N	M	S	T	−T	−S
E84	12	I	M	N	P	N	M	S	T	T	S
E85	12	L	B/2	(10)^†	(12)^†	N	M	(15)^†	(17)^†	2N	2M
E86	12	L	2I	(11)^†	K	L/2	0	S	T	U	V
E87	12	L	(4)^†	(10)^†	K	N	M	(15)^†	0	2N	0
E88	12	L	(8)^†	(11)^†	P	L/2	R	S	(16)^†	U	U/2
E89	21	L	M	N	P	Q	R	S	T	U	V

^† (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.

Table 1.9.3.6 | top | pdf |
Symmetry restrictions on coefficients in sixth-rank symmetric polar tensors

(a) A–N.

Cross-reference	No. of independent parameters	Symbols and coefficient indices
		A	B	C	D	E	F	G	H	I	J	K	L	M	N
		1	2	3	1	1	1	1	1	1	1	1	1	1	1
		1	2	3	1	1	1	1	1	1	1	1	1	1	1
		1	2	3	1	1	1	1	1	1	1	1	1	2	2
		1	2	3	1	1	1	1	1	2	2	2	3	2	2
		1	2	3	1	1	2	2	3	2	2	3	3	2	2
		1	2	3	2	3	2	3	3	2	3	3	3	2	3
F1	3	A	A	A	0	0	F	0	F	0	0	0	0	F	0
F2	4	A	A	A	0	0	F	0	H	0	0	0	0	H	0
F3	5	A	A	C	A/2	0	F	0	H	(1)^†	0	H/2	0	F	0
F4	6	A	A	C	D	0	F	0	H	(2)^†	0	H/2	0	(5)^†	0
F5	6	A	A	C	0	0	F	0	H	0	0	0	0	F	0
F6	6	A	B	A	0	0	F	0	H	0	0	0	0	M	0
F7	6	A	B	B	0	0	F	0	F	0	0	0	0	M	0
F8	7	A	A	A	D	D	F	G	F	I	J	J	I	F	J
F9	7	A	A	A	D	−D	F	G	F	I	J	−J	−I	F	J
F10	7	A	A	A	D	−D	F	G	F	I	J	−J	−I	F	−J
F11	7	A	A	A	D	D	F	G	F	I	J	J	I	F	−J
F12	7	A	A	C	A/2	E	F	E/2	H	(1)^†	E/10	H/2	I	F	−E/10
F13	7	A	A	C	A/2	0	F	G	H	(1)^†	G	H/2	0	F	G
F14	8	A	A	C	D	0	F	0	H	0	0	K	0	F	0
F15	8	A	B	A	0	E	F	0	H	0	J	0	0	M	0
F16	8	A	B	B	0	0	F	0	H	0	0	0	0	M	N
F17	10	A	A	A	D	E	F	G	H	I	J	K	I	H	K
F18	10	A	A	A	D	E	F	G	H	I	J	K	−I	H	−K
F19	10	A	A	A	D	E	F	G	H	I	J	K	−I	H	K
F20	10	A	A	A	D	E	F	G	H	I	J	K	I	H	−K
F21	10	A	A	C	D	E	F	G	H	(2)^†	(4)^†	H/2	L	(5)^†	(7)^†
F22	10	A	B	C	0	0	F	0	H	0	0	0	0	M	0
F23	10	A	A	C	D	0	F	0	H	I	0	K	0	F	0
F24	10	A	B	A	0	E	F	0	H	0	J	0	L	M	0
F25	10	A	B	B	0	0	F	G	F	0	0	0	0	M	N
F26	10	A	B	C	D	0	F	0	H	(3)^†	0	K	0	(6)^†	0
F27	10	A	B	C	A/2	0	F	0	H	(1)^†	0	H/2	0	M	0
F28	16	A	B	C	D	0	F	0	H	I	0	K	0	M	0
F29	16	A	B	C	0	E	F	0	H	0	J	0	L	M	0
F30	16	A	B	C	0	0	F	G	H	0	0	0	0	M	N
F31	16	A	A	C	D	E	F	G	H	I	J	K	L	F	−J
F32	16	A	A	C	D	E	F	G	H	I	J	K	L	F	J
F33	16	A	B	A	D	E	F	G	H	I	J	K	L	M	N
F34	16	A	B	A	D	E	F	G	H	I	J	K	L	M	N
F35	16	A	B	B	D	−D	F	G	F	I	J	−J	−I	M	N
F36	16	A	B	B	D	D	F	G	F	I	J	J	I	M	N
F37	16	A	B	C	D	E	F	G	H	(3)^†	J	K	L	(6)^†	(8)^†
F38	16	A	B	C	A/2	0	F	G	H	(1)^†	G	H/2	0	M	N
F39	16	A	B	C	D	E	F	G	H	(3)^†	J	K	L	(6)^†	(9)^†
F40	16	A	B	C	A/2	E	F	E/2	H	(1)^†	J	H/2	L	M	(10)^†
F41	28	A	B	C	D	E	F	G	H	I	J	K	L	M	N

(b) P–c.

Cross-reference	No. of independent parameters	Symbols and coefficient indices
		P	Q	R	S	T	U	V	W	X	Y	Z	a	b	c
		1	1	1	1	1	1	1	1	1	2	2	2	2	2
		1	1	1	2	2	2	2	2	3	2	2	2	2	3
		2	2	3	2	2	2	2	3	3	2	2	2	3	3
		2	3	3	2	2	2	3	3	3	2	2	3	3	3
		3	3	3	2	2	3	3	3	3	2	3	3	3	3
		3	3	3	2	3	3	3	3	3	3	3	3	3	3
F1	3	P	0	F	0	0	0	0	0	0	0	F	0	F	0
F2	4	P	0	F	0	0	0	0	0	0	0	F	0	H	0
F3	5	H/2	0	R	A/2	0	H/2	0	R/2	0	0	H	0	R	0
F4	6	H/2	0	R	(11)^†	0	H/2	0	R/2	0	0	H	0	R	0
F5	6	P	0	R	0	0	0	0	0	0	0	H	0	R	0
F6	6	P	0	H	0	0	0	0	0	0	0	M	0	F	0
F7	6	P	0	M	0	0	0	0	0	0	0	Z	0	Z	0
F8	7	P	J	F	D	G	J	J	G	D	D	F	I	F	D
F9	7	P	J	F	D	G	−J	J	−G	−D	−D	F	−I	F	−D
F10	7	P	−J	F	D	−G	−J	−J	G	−D	D	F	I	F	D
F11	7	P	−J	F	D	−G	J	−J	−G	D	−D	F	−I	F	−D
F12	7	H/2	I/2	R	A/2	−E/2	H/2	−I/2	R/2	0	−E	H	−I	R	0
F13	7	H/2	Q	R	A/2	G	H/2	Q	R/2	0	0	H	0	R	0
F14	8	P	0	R	−D	0	−K	0	0	0	0	H	0	R	0
F15	8	P	0	H	0	0	0	−J	0	−E	0	M	0	F	0
F16	8	P	−N	M	0	0	0	0	0	0	Y	Z	0	Z	−Y
F17	10	P	J	F	E	G	J	K	G	D	D	F	I	H	E
F18	10	P	J	F	−E	G	−J	−K	−G	−D	−D	F	−I	H	E
F19	10	P	−J	F	−E	−G	−J	−K	G	−D	D	F	I	H	−E
F20	10	P	−J	F	E	−G	J	K	−G	D	−D	F	−I	H	−E
F21	10	H/2	Q	R	(11)^†	(13)^†	H/2	(18)^†	R/2	0	−E	H	−L	R	0
F22	10	P	0	R	0	0	0	0	0	0	0	Z	0	b	0
F23	10	P	0	R	D	0	K	0	W	0	0	H	0	R	0
F24	10	P	0	H	0	T	0	J	0	E	0	M	0	F	0
F25	10	P	N	M	0	0	0	0	0	0	Y	Z	a	Z	Y
F26	10	P	0	R	B/2	0	(16)^†	0	W	0	0	(22)^†	0	2W	0
F27	10	P	0	R	(12)^†	0	(17)^†	0	R/2	0	0	Z	0	b	0
F28	16	P	0	R	S	0	U	0	W	0	0	Z	0	b	0
F29	16	P	0	R	0	T	0	V	0	X	0	Z	0	b	0
F30	16	P	Q	R	0	0	0	0	0	0	Y	Z	a	b	c
F31	16	P	Q	R	D	−G	K	−Q	W	X	−E	H	−L	R	−X
F32	16	P	Q	R	D	G	K	Q	W	X	E	H	L	R	X
F33	16	P	−K	H	S	T	−N	J	−G	E	−S	M	−I	F	−D
F34	16	P	K	H	S	T	N	J	G	E	S	M	I	F	D
F35	16	P	N	M	S	T	U	−U	−T	−S	Y	Z	a	Z	Y
F36	16	P	N	M	S	T	U	U	T	S	Y	Z	a	Z	Y
F37	16	P	Q	R	B/2	(14)^†	(16)^†	(19)^†	W	X	(20)^†	(22)^†	(23)^†	2W	2X
F38	16	P	Q	R	(12)^†	(15)^†	(17)^†	Q	R/2	0	Y	Z	a	b	c
F39	16	P	Q	R	B/2	(9)^†	(16)^†	Q	W	X	0	(22)^†	0	2W	0
F40	16	P	L/2	R	(12)^†	T	(17)^†	V	R/2	X	(21)^†	Z	(24)^†	b	X/2
F41	28	P	Q	R	S	T	U	V	W	X	Y	Z	a	b	c

^† (1) −A/4 + F/2; (2) A/2 − 3D/2 + 3F/2; (3) B/20 − 3D/5 + 3F/2; (4) −2E/5 + G; (5) A − 2D + F; (6) B/5 − 2D/5 + F; (7) −3E/5 + G; (8) 2E − 5G + 4J; (9) −G + 2J; (10) −E/4 + 3J/2; (11) A − D; (12) A/2 − 5F/2 + 5M/2; (13) −E + G; (14) 6E − 15G + 10J; (15) −G + 2N; (16) −2K + 3P; (17) −H/4 + 3P/2; (18) −L + Q; (19) −2L + 3Q; (20) 12E − 30G + 20J; (21) E/2 − 5J/2 + 5T/2; (22) −4K + 6P; (23) −4L + 6Q; (24) −L/4 + 3V/2.

1.9.4. Graphical representation

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Atomic displacement tensors (ADTs) described by their tensor coefficients may be represented graphically to clarify their physical meaning. Different graphical representations exist and will be discussed separately for second- and higher-order tensors in the following.

1.9.4.1. Representation surfaces of second-order ADTs

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Numerous examples of graphical representations of thermal-motion tensors (or, more generally speaking, atomic displacement tensors ) have appeared in the literature since the early days of the computer program ORTEP written by C. K. Johnson (1965 ), yet the equal-probability surface usually displayed is only one of the possible representations of a second-order atomic displacement tensor. Representation surfaces are usually calculated in a Cartesian coordinate system. Accordingly, one has to transform the second-order ADT b into U_C described in a Cartesian frame: $[{\bf U}_{C}=(2\pi^2)^{-1}{\bf F}^T{\bf b}{\bf F}.\eqno(1.9.4.1)]$ The transformation matrix depends on the choice of Cartesian axes e_i with respect to the reciprocal-cell axes aⁱ (or equally well with respect to the direct axes a_i). Choosing e₁ along a¹, e₂ in the a¹a² plane and e₃ completing the right-handed set, one obtains for the transformation matrix F (see also Willis & Pryor, 1975 ) $[{\bf F}=\pmatrix{1/a^1 & a^2\cos \gamma^* & a^3\cos\beta^*\cr 0 & a^2\sin\gamma^* & -a^3\sin\beta^*\cos\alpha\cr0 & 0 & 1/a_3}.\eqno(1.9.4.2)]$

Clearly, there is an infinite number of possible choices for relating a Cartesian frame to a crystallographic coordinate system with correspondingly different transformation matrices F (see e.g. Chapter 1.1 of IT B). The most useful representation surface of the second-order atomic displacement tensor U_C is the representation quadric defined by the tensor invariant $[^2I_0={\bf u}^T{\bf U}^{-1}_C{\bf u}\eqno(1.9.4.3)]$ where u is a displacement vector; U⁻¹ is often called the variance–covariance matrix and has (in a general axes frame) covariant components. Under the conditions of positive definiteness, $[\left.\matrix{\hfill{\rm Det}({\bf U}_C)\cr \hfill{\bf U}^{ii}_{C}\cr{\bf U}^{ii}_{C}{\bf U}^{jj}_{C}-{\bf U}^{ij}_{C}{\bf U}^{ij}_{C}\;({\rm no} \; {\rm summation})}\right\}{\rm \;all\; positive},\eqno(1.9.4.4)]$ the surface of the representation quadric is an ellipsoid whose semi-major axes (for [^2I_0=0] ) are of lengths equal to the root-mean-square displacements (r.m.s.d.'s) along the axes directions. The thermal vibration ellipsoids calculated in ORTEP are related to this surface; considering the discussion in Section 1.9.1 , they should more appropriately be called atomic displacement ellipsoids or simply ORTEP ellipsoids. One notes that the Fourier transform of the atomic DWF, the atomic probability density function P(u), is given in the case of a second-order tensor as a trivariate Gaussian distribution, $[P({\bf u})={[{\rm Det}({\bf U}_C^{-1})]^{1/2}\over(2\pi)^{3/2}}\exp\left\{-{\textstyle{1 \over 2}}{\bf u}^T{\bf U}_C^{-1}{\bf u}\right\}.\eqno(1.9.4.5)]$

On comparing (1.9.4.3) and (1.9.4.5), it is evident that (1.9.4.3) defines a surface of constant probability of finding a (displaced) atom. The integral of (1.9.4.5) over the volume inside the ellipsoid is a constant. For [^2I_0=C^2] with the integration limit [C = 1.5382] (2.5003), the integral is equal to one half (nine tenths), and the ellipsoid is then called a 50 (90) per cent probability ellipsoid.

Other representation surfaces can be defined and are useful for special considerations. The quantities of interest are either the r.m.s.d.'s or the mean-square displacements (m.s.d.'s) defined in direct space. Here a distinction has to be made between the averaged squared displacement along a certain direction and the average for all squared displacements of an atom projected onto a given direction. Representation surfaces may also be calculated in reciprocal space, related to surfaces in direct space by Fourier transformation. For further details, see Nelmes (1969 ) and Hummel et al. (1990 ).

1.9.4.2. Higher-order representations

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Representation surfaces of higher-order tensors may be calculated from their invariants. While for second-order tensors surfaces can be found that fully describe the directional aspects of the tensor involved, higher-order tensors need several different surfaces for a full description (see e.g. Wondratschek, 1958 ; Sirotin, 1961 ). This makes the graphical representation of the displacements somewhat cumbersome and it is therefore rarely used. Instead, the probability density functions [given in equations (6.1.1.46 ), (6.1.1.48 ) or (6.1.1.49 ) of IT C] are calculated from the tensor coefficients and displayed in sections or as three-dimensional surfaces. If the higher-order terms are small, it is more appropriate to display only the difference between the total p.d.f. and the related Gaussian p.d.f., which may be calculated from the second-order displacement tensor using equation (1.9.4.5). Here, the second-order terms that were refined together with the higher-order terms are usually used (not the best-fitting second-order terms of a fit in the harmonic approximation): $[P_{\rm deformation}({\bf u})=P_{\rm general}({\bf u})-P_{\rm Gaussian}({\bf u}).\eqno(1.9.4.6)]$ The resulting anharmonic deformation densities (or disorder deformation densities in the case of static disorder) P_deformation(u) may be displayed in a similar way to the total p.d.f.'s P_general(u). The graphical representations appropriate for displaying those densities are similar to those used for electronic deformation densities (see e.g. Smith et al., 1977 ). A number of examples of displacement deformation densities of high symmetry are shown in Fig. 1.9.4.1 as three-dimensional contour maps.

Figure 1.9.4.1 | top | pdf |

A selection of graphical representations of density modulations due to higher-order terms in the Gram–Charlier series expansion of a Gaussian atomic probability density function. All figures are drawn on a common scale and have a common orientation. All terms within any given order of expansion are numerically identical and refer to the same underlying isotropic second-order term; the higher-order terms of different order of expansion differ by one order of magnitude, but refer again to the same underlying isotropic second-order term. The orthonormal crystallographic axes are oriented as follows: x oblique out of the plane of the paper towards the observer, y in the plane of the paper and to the right, and z in the plane of the paper and upwards. All surfaces are scaled to 1% of the absolute value of the maximum modulation within each density distribution. Positive modulations (i.e. an increase of density) are shown in red, negative modulations are shown in blue. The source of illumination is located approximately on the [111] axis. The following graphs are shown (with typical point groups for specific cases given in parentheses). Third-order terms: (a) b²²²; (b) b²²³; (c) b¹¹³ = −b²²³ (point group $[\bar{4}]$ ); (d) b¹²³ (point group $[\bar{4}3m]$ ). Fourth-order terms: (e) b²²²²; (f) b¹¹¹¹ = b²²²²; (g) b¹¹¹¹ = b²²²² = b³³³³ (point group $[m\bar{3}m]$ ); (h) b¹²²²; (i) b¹¹¹² = b¹²²²; (j) b¹¹²²; (k) b¹¹³³ = b²²³³; (l) b¹¹²² = b¹¹³³ = b²²³³ (point group $[m\bar{3}m]$ ). Fifth-order terms: (m) b²²²²²; (n) b¹²²²³; (o) b¹¹¹²³ = b¹²²²³; (p) b¹¹¹²³ = b¹²²²³ = b¹²³³³ (point group $[\bar{4}3m]$ ). Sixth-order terms: (q) b²²²²²²; (r) b¹¹¹¹¹¹ = b²²²²²²; (s) b¹¹¹¹¹¹ = b²²²²²² = b³³³³³³ (point group $[m\bar{3}m]$ ); (t) b¹¹²²²²; (u) b¹¹¹¹³³ = b²²²²³³; (v) b¹¹³³³³ = b²²³³³³; (w) b¹¹¹¹²² = b¹¹²²²² = b¹¹¹¹³³ = b¹¹³³³³ = b²²²²³³ = b²²³³³³ (point group $[m\bar{3}m]$ ); (x) b¹¹²²³³ (point group $[m\bar{3}m]$ ).

1.9.5. Glossary

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$[b^{ijk\ldots}]$	atomic displacement tensor
$[\beta^{ij}]$ , $[U^{ijk\ldots}]$	atomic displacement parameter
g_ij	metric tensor
S _α	atomic static Debye–Waller factor
T _α	atomic thermal Debye–Waller factor
Q	scattering vector
uⁱ	atomic displacement

References

Hazell, R. G. & Willis, B. T. M. (1978). Correlations between third cumulants in the refinement of noncentrosymmetric structures. Acta Cryst. A34, 809–811.Google Scholar

Hummel, W., Raselli, A. & Bürgi, H.-B. (1990). Analysis of atomic displacement parameters and molecular motion in crystals. Acta Cryst. B46, 683–692.Google Scholar

International Tables for Crystallography (2001). Vol. B. Reciprocal space, edited by U. Shmueli. Dordrecht: Kluwer Academic Publishers.Google Scholar

International Tables for Crystallography (2004). Vol. C. Mathematical, physical and chemical tables, edited by E. Prince. Dordrecht: Kluwer Academic Publishers.Google Scholar

International Tables for Crystallography (2005). Vol. A. Space-group symmetry, edited by Th. Hahn. Heidelberg: Springer.Google Scholar

Jahn, H. A. (1949). Note on the Bhagavantam–Suryanarayana method of enumerating the physical constants of crystals. Acta Cryst. 2, 30–33.Google Scholar

Johnson, C. K. (1965). ORTEP – a FORTRAN thermal ellipsoid plot program. Report ORNL-3794. Oak Ridge National Laboratory, Tennessee, USA. Google Scholar

Johnson, C. K. (1970). Generalized treatments for thermal motion. In Thermal neutron diffraction, edited by B. T. M. Willis, pp. 132–160. Oxford University Press.Google Scholar

Kuhs, W. F. (1983). Statistical description of multimodal atomic probability densities. Acta Cryst. A39, 148–158.Google Scholar

Kuhs, W. F. (1984). Site-symmetry restrictions on thermal-motion-tensor coefficients up to rank 8. Acta Cryst. A40, 133–137.Google Scholar

Kuhs, W. F. (1988). The anharmonic temperature factor in crystallographic structure analysis. Aust. J. Phys. 41, 369–382.Google Scholar

Kuhs, W. F. (1992) Generalized atomic displacements in crystallographic structure analysis. Acta Cryst. A48, 80–98.Google Scholar

Kuznetsov, P. I., Stratonovich, R. L. & Tikhonov, V. I. (1960). Quasi-moment functions in the theory of random processes. Theory Probab. Its Appl. (USSR), 5, 80–97.Google Scholar

Levy, H. A. (1956). Symmetry relations among coefficients of the anisotropic temperature factor. Acta Cryst. 9, 679.Google Scholar

Nelmes, R. J. (1969). Representational surfaces for thermal motion. Acta Cryst. A25, 523–526.Google Scholar

Pach, K. & Frey, T. (1964). Vector and tensor analysis. Budapest: Terra.Google Scholar

Peterse, W. J. A. M. & Palm, J. H. (1966). The anisotropic temperature factor of atoms in special positions. Acta Cryst. 20, 147–150.Google Scholar

Scheringer, C. (1985). A deficiency of the cumulant expansion of the anharmonic temperature factor. Acta Cryst. A41, 79–81.Google Scholar

Sirotin, Yu. I. (1960). Group tensor spaces. Sov. Phys. Crystallogr. 5, 157–165.Google Scholar

Sirotin, Yu. I. (1961). Plotting tensors of a given symmetry. Sov. Phys. Crystallogr. 6, 263–271.Google Scholar

Smith, V. H. Jr, Price, P. F. & Absar, I. (1977). Representations of electron density and its topographical features. Isr. J. Chem. 16, 187–197.Google Scholar

Trueblood, K. N., Bürgi, H.-B., Burzlaff, H., Dunitz, J. D., Gramaccioli, C. M., Schulz, H. H., Shmueli, U. & Abrahams, S. C. (1996). Atomic displacement parameter nomenclature. Report of a subcommittee on atomic displacement parameter nomenclature. Acta Cryst. A52, 770–781.Google Scholar

Willis, B. T. M. & Pryor, A. W. (1975). Thermal vibrations in crystallography. Cambridge University Press. Google Scholar

Wondratschek, H. (1958). Über die Möglichkeit der Beschreibung kristallphysikalischer Eigenschaften durch Flächen. Z. Kristallogr. 110, 127–135. Google Scholar

Zucker, U. H. & Schulz, H. (1982a). Statistical approaches for the treatment of anharmonic thermal motion in crystals. I. A comparison of the most frequently used formalisms of anharmonic thermal vibrations. Acta Cryst. A38, 563–568.Google Scholar

Zucker, U. H. & Schulz, H. (1982b). Statistical approaches for the treatment of anharmonic thermal motion in crystals. II. Anharmonic thermal vibrations and effective atomic potentials in the fast ionic conductor lithium nitride Li₃N. Acta Cryst. A38, 568–576.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 1.9, pp. 228-242
https://doi.org/10.1107/97809553602060000636