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

International Tables for Crystallography (2013). Vol. D, ch. 1.9, pp. 231-245
doi: 10.1107/97809553602060000908

Chapter 1.9. Atomic displacement parameters

W. F. Kuhsa*

aGZG 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.

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[link]) shows that the atomic thermal DWF Tα 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 uα are the individual atomic displacement vectors and the brackets symbolize the thermodynamic (time–space) average over all contributions uα. In the harmonic (Gaussian) approximation, (1.9.1.1)[link] 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)[link] 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[link] ).

Similar formulations are found for the static atomic DWF Sα, where the average of the atomic static displacements Δuα may also be approximated [though with weaker theoretical justification, see Kuhs (1992[link])] 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)[link], 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[link]). 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 Uij and βij (Trueblood et al., 1996[link]).

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 βij, 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[link] ); 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 βij as their dimensioned representations Uij.

In the harmonic approximation, the atomic displacements are fully described by the fully symmetric second-order tensor given in (1.9.2.1)[link]. 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)[link] and (1.9.2.3)[link], 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[link]), equation (1.2.12.9[link] ) and more completely in Kuhs (1988[link], 1992[link]). 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)[link] and (1.9.2.5)[link] are useful, as they can be entered directly in a structure-factor equation (see Chapter 1.2[link] 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)[link] 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[link]) 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)[link] is called a Gram–Charlier series expansion. Discussions of its merits as compared to the Edgeworth series are given in Zucker & Schulz (1982a[link],b[link]), Kuhs (1983[link], 1988[link], 1992[link]) and Scheringer (1985[link]).

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)[link], (1.9.2.5)[link] and (1.9.2.7)[link] 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 bij is identical to βij. 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[link]), their tensor symmetry thus is [bN]. The number of independent tensor coefficients depends on the site symmetry of the atom and is tabulated in Sirotin (1960[link]) as well as in Tables 1.9.3.1[link]–1.9.3.6[link][link][link][link][link]. 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[link].

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)[link] 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 ÅN (or pmN); ai etc. are reciprocal-lattice vectors. Nowadays, the published structural results usually quote Uij 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 gij; for further details on tensor contraction, see Section 1.1.3.3.3[link] . 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 lengthN 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 Ueq is also obtained by this contraction process. Noting that Uij may be expressed in terms of bij (= βij) according to (1.9.2.9)[link] 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 Beq, which is related to Ueq through the identity [B_{\rm eq}=8\pi^2U_{\rm eq}]. The use of Beq is now discouraged (Trueblood et al., 1996[link]). 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[link]) and its units are lengthN − 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[link]) 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[link]).

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[link] 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[link] of IT C (2004[link]), 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[link]). To use these tables, first the site symmetry has to be identified. The site symmetries are given in IT A (2005[link]) 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 bijk are allowed to vary freely for all atoms (Hazell & Willis, 1978[link]). Even if this is strictly true only for the Edgeworth-series expansion, it also holds in practice for the Gram–Charlier case (Kuhs, 1992[link]).

1.9.3.1. Calculation procedures

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Levy (1956[link]) and Peterse & Palm (1966[link]) 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[link]–1.9.3.6[link][link][link][link]) needs to be looked up in Table 1.9.3.1[link]. The line entry corresponding to the cross-reference number in Tables 1.9.3.2[link]–1.9.3.6[link][link][link][link] 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[link] to 1.9.3.6[link][link][link][link] for restrictions on the symmetry of various thermal-motion tensors

Hex denotes hexagonal axes.

Point symmetry at special positionPosition [x,y,z]Cross-reference for tensor tables
Symmetry axesPoint-group generators1B1C1D1E1F
[m3m]   [4[0,0,1]] [3[1,1,1]] [\bar 1] [0,0,0] B1 C0 D1 E0 F1
[\bar 43m]   [\bar 4[0,0,1]] [3[1,1,1]]   [0,0,0] B1 C1 D1 E1 F1
[432]   [4[0,0,1]] [3[1,1,1]]   [0,0,0] B1 C0 D1 E0 F1
[m3]   [3[1,1,1]] [2[0,0,1]] [\bar 1] [0,0,0] B1 C0 D1 E0 F2
[23]   [3[1,1,1]] [2[0,0,1]]   [0,0,0] B1 C1 D1 E1 F2
[6/mmm] Hex [6[0,0,1]] [2[1,0,0]] [\bar 1] [0,0,0] B9 C0 D2 E0 F3
[\bar 6m2] Hex [\bar 6[0,0,1]] [2[1,0,0]]   [0,0,0] B9 C9 D2 E5 F3
[\bar 6m2] Hex [\bar 6[0,0,1]] [2[1,2,0]]   [0,0,0] B9 C10 D2 E6 F3
[6mm] Hex [6[0,0,1]] [\bar 2[1,0,0]]   [0,0,z] B9 C19 D2 E17 F3
[622] Hex [6[0,0,1]] [2[1,0,0]]   [0,0,0] B9 C0 D2 E0 F4
[6/m] Hex [6[0,0,1]] [\bar 1]   [0,0,0] B9 C0 D2 E0 F4
[\bar 6] Hex [\bar 6[0,0,1]]     [0,0,0] B9 C20 D2 E24 F4
[6] Hex [6[0,0,1]]     [0,0,z] B9 C19 D2 E17 F4
[4/mmm]   [4[0,0,1]] [2[1,0,0]] [\bar 1] [0,0,0] B2 C0 D3 E0 F5
[4/mmm]   [4[0,1,0]] [2[0,0,1]] [\bar 1] [0,0,0] B3 C0 D4 E0 F6
[4/mmm]   [4[1,0,0]] [2[0,1,0]] [\bar 1] [0,0,0] B4 C0 D5 E0 F7
[\bar 42m]   [\bar 4[0,0,1]] [2[1,0,0]]   [0,0,0] B2 C1 D3 E7 F5
[\bar 42m]   [\bar 4[0,0,1]] [2[1,1,0]]   [0,0,0] B2 C2 D3 E8 F5
[\bar 42m]   [\bar 4[0,1,0]] [2[0,0,1]]   [0,0,0] B3 C1 D4 E9 F6
[\bar 42m]   [\bar 4[0,1,0]] [2[1,0,1]]   [0,0,0] B3 C3 D4 E10 F6
[\bar 42m]   [\bar 4[1,0,0]] [2[0,1,0]]   [0,0,0] B4 C1 D5 E11 F7
[\bar 42m]   [\bar 4[1,0,0]] [2[0,1,1]]   [0,0,0] B4 C4 D5 E12 F7
[4mm]   [4[0,0,1]] [\bar 2[1,0,0]]   [0,0,z] B2 C13 D3 E25 F5
[4mm]   [4[0,1,0]] [\bar 2[0,0,1]]   [0,y,0] B3 C14 D4 E26 F6
[4mm]   [4[1,0,0]] [\bar 2[0,1,0]]   [x,0,0] B4 C15 D5 E27 F7
[422]   [4[0,0,1]] [2[1,0,0]]   [0,0,0] B2 C0 D3 E2 F5
[422]   [4[0,1,0]] [2[0,0,1]]   [0,0,0] B3 C0 D4 E3 F6
[422]   [4[1,0,0]] [2[0,1,0]]   [0,0,0] B4 C0 D5 E4 F7
[4/m]   [4[0,0,1]] [\bar 1]   [0,0,0] B2 C0 D12 E0 F14
[4/m]   [4[0,1,0]] [\bar 1]   [0,0,0] B3 C0 D13 E0 F15
[4/m]   [4[1,0,0]] [\bar 1]   [0,0,0] B4 C0 D14 E0 F16
[\bar 4]   [\bar 4[0,0,1]]     [0,0,0] B2 C16 D12 E28 F14
[\bar 4]   [\bar 4[0,1,0]]     [0,0,0] B3 C17 D13 E29 F15
[\bar 4]   [\bar 4[1,0,0]]     [0,0,0] B4 C18 D14 E30 F16
[4]   [4[0,0,1]]     [0,0,z] B2 C13 D12 E31 F14
[4]   [4[0,1,0]]     [0,y,0] B3 C14 D13 E32 F15
[4]   [4[1,0,0]]     [x,0,0] B4 C15 D14 E33 F16
[\bar 3m]   [3[1,1,1]] [2[1,\bar 1,0]] [\bar 1] [0,0,0] B5 C0 D6 E0 F8
[\bar 3m]   [3[1,1,\bar 1]] [2[1,\bar 1,0]] [\bar 1] [0,0,0] B6 C0 D7 E0 F9
[\bar 3m]   [3[1,\bar 1,1]] [2[1,1,0]] [\bar 1] [0,0,0] B7 C0 D8 E0 F10
[\bar 3m]   [3[\bar 1,1,1]] [2[1,1,0]] [\bar 1] [0,0,0] B8 C0 D9 E0 F11
[\bar 3m] Hex [3[0,0,1]] [2[1,0,0]] [\bar 1] [0,0,0] B9 C0 D10 E0 F12
[\bar 3m] Hex [3[0,0,1]] [2[1,2,0]] [\bar 1] [0,0,0] B9 C0 D11 E0 F13
[3m]   [3[1,1,1]] [\bar 2[1,\bar 1,0]]   [x,x,x] B5 C33 D6 E34 F8
[3m]   [3[1,1,\bar 1]] [\bar 2[1,\bar 1,0]]   [x,x,\bar x] B6 C34 D7 E35 F9
[3m]   [3[1,\bar 1,1]] [\bar 2[1,1,0]]   [x,\bar x,x] B7 C35 D8 E36 F10
[3m]   [3[\bar 1,1,1]] [\bar 2[1,1,0]]   [\bar x,x,x] B8 C36 D9 D37 F11
[3m] Hex [3[0,0,1]] [\bar 2[1,0,0]]   [0,0,z] B9 C37 D10 E38 F12
[3m] Hex [3[0,0,1]] [\bar 2[1,2,0]]   [0,0,z] B9 C38 D11 E39 F13
[32]   [3[1,1,1]] [2[1,\bar 1,0]]   [0,0,0] B5 C5 D6 E13 F8
[32]   [3[1,1,\bar 1]] [2[1,\bar 1,0]]   [0,0,0] B6 C6 D7 E14 F9
[32]   [3[1,\bar 1,1]] [2[1,1,0]]   [0,0,0] B7 C7 D8 E15 F10
[32]   [3[\bar 1,1,1]] [2[1,1,0]]   [0,0,0] B8 C8 D9 E16 F11
[32] Hex [3[0,0,1]] [2[1,0,0]]   [0,0,0] B9 C9 D10 E5 F12
[32] Hex [3[0,0,1]] [2[1,2,0]]   [0,0,0] B9 C10 D11 E6 F13
[\bar 3]   [\bar 3[1,1,1]]     [0,0,0] B5 C0 D15 E0 F17
[\bar 3]   [\bar 3[1,1,\bar 1]]     [0,0,0] B6 C0 D16 E0 F18
[\bar 3]   [\bar 3[1,\bar 1,1]]     [0,0,0] B7 C0 D17 E0 F19
[\bar 3]   [\bar 3[\bar 1,1,1]]     [0,0,0] B8 C0 D18 E0 F20
[\bar 3] Hex [\bar 3[0,0,1]]     [0,0,0] B9 C0 D19 E0 F21
[3]   [3[1,1,1]]     [x,x,x] B5 C54 D15 E58 F17
[3]   [3[1,1,\bar 1]]     [x,x,\bar x] B6 C55 D16 E59 F18
[3]   [3[1,\bar 1,1]]     [x,\bar x,x] B7 C56 D17 E60 F19
[3]   [3[\bar 1,1,1]]     [\bar x,x,x] B8 C57 D18 E61 F20
[3] Hex [3[0,0,1]]     [0,0,z] B9 C58 D19 E62 F21
[mmm]   [2[0,0,1]] [2[1,0,0]] [\bar 1] [0,0,0] B10 C0 D20 E0 F22
[mmm]   [2[0,0,1]] [2[1,1,0]] [\bar 1] [0,0,0] B11 C0 D21 E0 F23
[mmm]   [2[0,1,0]] [2[1,0,1]] [\bar 1] [0,0,0] B12 C0 D22 E0 F24
[mmm]   [2[1,0,0]] [2[0,1,1]] [\bar 1] [0,0,0] B13 C0 D23 E0 F25
[mmm] Hex [2[0,0,1]] [2[1,0,0]] [\bar 1] [0,0,0] B14 C0 D24 E0 F26
[mmm] Hex [2[0,0,1]] [2[1,1,0]] [\bar 1] [0,0,0] B11 C0 D21 E0 F23
[mmm] Hex [2[0,0,1]] [2[0,1,0]] [\bar 1] [0,0,0] B15 C0 D25 E0 F27
[mm]   [2[0,0,1]] [\bar 2[1,0,0]]   [0,0,z] B10 C21 D20 E40 F22
[mm]   [2[0,0,1]] [\bar 2[1,1,0]]   [0,0,z] B11 C22 D21 E41 F23
[mm]   [2[0,1,0]] [\bar 2[0,0,1]]   [0,y,0] B10 C23 D20 E42 F22
[mm]   [2[0,1,0]] [\bar 2[1,0,1]]   [0,y,0] B12 C24 D22 E43 F24
[mm]   [2[1,0,0]] [\bar 2[0,0,1]]   [x,0,0] B10 C25 D20 E44 F22
[mm]   [2[1,0,0]] [\bar 2[0,1,1]]   [x,0,0] B13 C26 D23 E45 F25
[mm]   [2[1,1,0]] [\bar 2[0,0,1]]   [x,x,0] B11 C27 D21 E46 F23
[mm]   [2[1,\bar 1,0]] [\bar 2[0,0,1]]   [x,\bar x,0] B11 C28 D21 E47 F23
[mm]   [2[1,0,1]] [\bar 2[0,1,0]]   [x,0,x] B12 C29 D22 E48 F24
[mm]   [2[1,0,\bar 1]] [\bar 2[0,1,0]]   [x,0,\bar x] B12 C30 D22 E49 F24
[mm]   [2[0,1,1]] [\bar 2[1,0,0]]   [0,y,y] B13 C31 D23 E50 F25
[mm]   [2[0,1,\bar 1]] [\bar 2[1,0,0]]   [0,y,\bar y] B13 C32 D23 E51 F25
[mm] Hex [2[0,0,1]] [\bar 2[1,0,0]]   [0,0,z] B14 C40 D24 E52 F26
[mm] Hex [2[0,0,1]] [\bar 2[1,1,0]]   [0,0,z] B11 C22 D21 E41 F23
[mm] Hex [2[0,0,1]] [\bar 2[0,1,0]]   [0,0,z] B15 C39 D25 E53 F27
[mm] Hex [2[1,0,0]] [\bar 2[0,0,1]]   [x,0,0] B14 C41 D24 E54 F26
[mm] Hex [2[2,1,0]] [\bar 2[0,0,1]]   [2x,x,0] B15 C42 D25 E55 F27
[mm] Hex [2[1,1,0]] [\bar 2[0,0,1]]   [x,x,0] B11 C27 D21 E46 F23
[mm] Hex [2[1,2,0]] [\bar 2[0,0,1]]   [x,2x,0] B14 C43 D24 E56 F26
[mm] Hex [2[0,1,0]] [\bar 2[0,0,1]]   [0,y,0] B15 C44 D25 E57 F27
[mm] Hex [2[1,\bar 1,0]] [\bar 2[0,0,1]]   [x,\bar x,0] B11 C28 D21 E47 F23
[222]   [2[0,0,1]] [2[1,0,0]]   [0,0,0] B10 C1 D20 E18 F22
[222]   [2[0,0,1]] [2[1,1,0]]   [0,0,0] B11 C2 D21 E19 F23
[222]   [2[0,1,0]] [2[1,0,1]]   [0,0,0] B12 C3 D22 E20 F24
[222]   [2[1,0,0]] [2[0,1,1]]   [0,0,0] B13 C4 D23 E21 F25
[222] Hex [2[0,0,1]] [2[1,0,0]]   [0,0,0] B14 C11 D24 E22 F26
[222] Hex [2[0,0,1]] [2[1,1,0]]   [0,0,0] B11 C2 D21 E19 F23
[222] Hex [2[0,0,1]] [2[0,1,0]]   [0,0,0] B15 C12 D25 E23 F27
[2/m]   [2[0,0,1]] [\bar 1]   [0,0,0] B16 C0 D26 E0 F28
[2/m]   [2[0,1,0]] [\bar 1]   [0,0,0] B17 C0 D27 E0 F29
[2/m]   [2[1,0,0]] [\bar 1]   [0,0,0] B18 C0 D28 E0 F30
[2/m]   [2[1,1,0]] [\bar 1]   [0,0,0] B19 C0 D29 E0 F31
[2/m]   [2[1,\bar 1,0]] [\bar 1]   [0,0,0] B20 C0 D30 E0 F32
[2/m]   [2[1,0,1]] [\bar 1]   [0,0,0] B21 C0 D31 E0 F33
[2/m]   [2[1,0,\bar 1]] [\bar 1]   [0,0,0] B22 C0 D32 E0 F34
[2/m]   [2[0,1,1]] [\bar 1]   [0,0,0] B23 C0 D33 E0 F35
[2/m]   [2[0,1,\bar 1]] [\bar 1]   [0,0,0] B24 C0 D34 E0 F36
[2/m] Hex [2[0,0,1]] [\bar 1]   [0,0,0] B16 C0 D26 E0 F28
[2/m] Hex [2[1,0,0]] [\bar 1]   [0,0,0] B25 C0 D35 E0 F37
[2/m] Hex [2[2,1,0]] [\bar 1]   [0,0,0] B26 C0 D36 E0 F38
[2/m] Hex [2[1,1,0]] [\bar 1]   [0,0,0] B19 C0 D29 E0 F31
[2/m] Hex [2[1,2,0]] [\bar 1]   [0,0,0] B27 C0 D37 E0 F39
[2/m] Hex [2[0,1,0]] [\bar 1]   [0,0,0] B28 C0 D38 E0 F40
[2/m] Hex [2[1,\bar 1,0]] [\bar 1]   [0,0,0] B20 C0 D30 E0 F32
m   [\bar 2[0,0,1]]     [x,y,0] B16 C63 D26 E76 F28
m   [\bar 2[0,1,0]]     [x,0,z] B17 C64 D27 E77 F29
m   [\bar 2[1,0,0]]     [0,y,z] 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]]     [x,x,z] 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]]     [x,y,x] 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]]     [x,y,y] B24 C71 D34 E84 F36
m Hex [\bar 2[0,0,1]]     [x,y,0] B16 C63 D26 E76 F28
m Hex [\bar 2[1,0,0]]     [x,2x,z] B25 C72 D35 E85 F37
m Hex [\bar 2[2,1,0]]     [0,y,z] 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]]     [x,0,z] B27 C74 D37 E87 F39
m Hex [\bar 2[0,1,0]]     [2x,x,z] B28 C75 D38 E88 F40
m Hex [\bar 2[1,\bar 1,0]]     [x,x,z] B20 C67 D30 E80 F32
[2]   [2[0,0,1]]     [0,0,z] B16 C45 D26 E63 F28
[2]   [2[0,1,0]]     [0,y,0] B17 C46 D27 E64 F29
[2]   [2[1,0,0]]     [x,0,0] B18 C47 D28 E65 F30
[2]   [2[1,1,0]]     [x,x,0] B19 C48 D29 E66 F31
[2]   [2[1,\bar 1,0]]     [x,\bar x,0] B20 C49 D30 E67 F32
[2]   [2[1,0,1]]     [x,0,x] B21 C50 D31 E68 F33
[2]   [2[1,0,\bar 1]]     [x,0,\bar x] B22 C51 D32 E69 F34
[2]   [2[0,1,1]]     [0,y,y] B23 C52 D33 E70 F35
[2]   [2[0,1,\bar 1]]     [0,y,\bar y] B24 C53 D34 E71 F36
[2] Hex [2[0,0,1]]     [0,0,z] B16 C45 D26 E63 F28
[2] Hex [2[1,0,0]]     [x,0,0] B25 C59 D35 E72 F37
[2] Hex [2[2,1,0]]     [2x,x,0] B26 C60 D36 E73 F38
[2] Hex [2[1,1,0]]     [x,x,0] B19 C48 D29 E66 F31
[2] Hex [2[1,2,0]]     [x,2x,0] B27 C61 D37 E74 F39
[2] Hex [2[0,1,0]]     [0,y,0] B28 C62 D38 E75 F40
[2] Hex [2[1,\bar 1,0]]     [x,\bar x,0] B20 C49 D30 E67 F32
[\bar 1]   [\bar 1]     [0,0,0] B29 C0 D39 E0 F41
[\bar 1] Hex [\bar 1]     [0,0,0] B29 C0 D39 E0 F41
[1]   [1]     [x,y,z] B29 C76 D39 E89 F41
[1] Hex [1]     [x,y,z] B29 C76 D39 E89 F41

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

Cross-referenceNo. of independent variablesSymbols and coefficient indices
ABCDEF
(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-referenceNo. of independent variablesSymbols and coefficient indices
ABCDEFGHIJ
(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-referenceNo. of independent variablesSymbols and coefficient indices
ABCDEFGH
(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-referenceNo. of independent variablesSymbols and coefficient indices
IJKLMNP
(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-referenceNo. of independent coefficientsSymbols and coefficient indices
ABCDEFGHIJK
12311111111
12311111111
12311111222
12311223223
12323233233
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-referenceNo. of independent coefficientsSymbols and coefficient indices
LMNPQRSTUV
1111112222
1222232223
3222332233
3223332333
3233333333
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-referenceNo. of independent parametersSymbols and coefficient indices
ABCDEFGHIJKLMN
12311111111111
12311111111111
12311111111122
12311111222322
12311223223322
12323233233323
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