Explanation of Table 3.1.3.1

Janovec, V.; Kopský, V.

doi:10.1107/97809553602060000642

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. 3.1, pp. 358-360

Section 3.1.3.3.1. Explanation of Table 3.1.3.1

V. Janovec^b ^* and V. Kopský^e

3.1.3.3.1. Explanation of Table 3.1.3.1

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Parent symmetry G: the short international (Hermann–Mauguin) and the Schoenflies symbol of the point group G of the parent phase are given. Subscripts specify the orientation of symmetry elements (generators) in the Cartesian crystallophysical coordinate system of the group G (see Figs. 3.4.2.3 and 3.4.2.4 , and Tables 3.4.2.5 and 3.4.2.6 ).
R -irep $[{\Gamma}_{\eta}]$ : physically irreducible representation $[{\Gamma}_{\eta}]$ of the group G in the spectroscopic notation. This representation defines transformation properties of the primary order parameter $[\eta]$ and of the principal tensor parameters . Each complex irreducible representation is combined with its complex conjugate and thus a real physically irreducible representation R-irep is formed. Matrices $[D^{(\alpha)}]$ of R-ireps are given explicitly in the the software GI $[\star]$ KoBo-1.
(La) below the symbol of the irreducible representation $[\Gamma_{\eta}]$ indicates that the Landau condition is violated, hence the transition cannot be continuous (second order). The Landau condition requires the absence of the third-degree invariant polynomial of the order-parameter components (the symmetrized triple product $[[\Gamma_{\eta}]^3]$ must not contain the identity representation of G). For more details see Lyubarskii (1960), Kociński (1983, 1990), Tolédano & Tolédano (1987), Izyumov & Syromiatnikov (1990) and Tolédano & Dmitriev (1996).
(Li) below the symbol of the irreducible representation $[\Gamma_{\eta}]$ means that the Lifshitz condition is violated, hence the transition to a homogeneous ferroic phase is not continuous. The Lifshitz condition demands the absence of invariant terms that couple bilinearly the order-parameter components with their spatial derivatives that are not exact differentials (the antisymmetric square $[{\{\Gamma}_{\eta}\}^2]$ has no representation in common with the vector representation of G). For more details see Lyubarskii (1960), Kociński (1983, 1990), Tolédano & Tolédano (1987), Izyumov & Syromiatnikov (1990) and Tolédano & Dmitriev (1996).

If there is no symbol (La) and/or (Li) below the symbol of the R-irep $[\Gamma_{\eta}]$ (i.e. if both Landau and Lifshitz conditions are fulfilled), then the R-irep is called an active representation. In the opposite case, the R-irep is a passive representation (Lyubarskii, 1960; Kociński, 1983, 1990).
Standard variables : components of the order parameter in the carrier space of the irreducible representation $[\Gamma_{\eta}]$ expressed in so-called standard variables (see the manual of the software GI $[\star]$ KoBo-1). Upper and lower indices and the typeface of standard variables allow one to identify to which irreducible representation $[\Gamma_{\eta}]$ they belong. Standard variables of one-dimensional representations are denoted by $[\sf x]$ (Sans Serif typeface), two- or three-dimensional R-ireps by or , respectively. Upper indices ⁺ and ⁻ correspond to the lower indices g (gerade) and u (ungerade) of spectroscopic notation, respectively. The lower index specifies to which irreducible representation the variable belongs.

For multidimensional representations, a general vector of the carrier space $[V_{\eta}]$ is given in the last row; this vector is invariant under the kernel of $[\Gamma_{\eta}]$ that appears as a low-symmetry group in column . The other rows contain special vectors defined by equal or zero values of some standard variables; these vectors are invariant under epikernels of $[\Gamma_{\eta}]$ given in column .
: short international (Hermann–Mauguin) and Schoenflies symbol of the point group which describes the symmetry of the first single domain state of the ferroic (low-symmetry) phase. The subscripts define the orientation of symmetry elements (generators) of in the Cartesian crystallophysical coordinate system of the group G (see Figs. 3.4.2.3 and 3.4.2.4 , and Tables 3.4.2.5 and 3.4.2.6 ). This specifies the orientation of the group , which is a prerequisite for domain structure analysis (see Chapter 3.4 ).
$[n_{F}]$ : number of subgroups conjugate to under G. If $[n_{F}=1]$ , the group is a normal subgroup of G (see Section 3.2.3 ).

Principal tensor parameters : covariant tensor components, i.e. linear combinations of Cartesian tensor components that transform according to the same matrix R-irep $[D^{(\eta)}]$ as the primary order parameter $[\eta]$ . Principal tensor parameters are given in this form in the software GI $[\star]$ KoBo-1 and in Kopský (2001).

This presentation is in certain situations not practical, since property tensors are usually described by numerical values of their Cartesian components. Then it is important to know morphic Cartesian tensor components and symmetry-breaking increments of nonzero Cartesian components that appear spontaneously in the ferroic phase. The bridge between these two presentations is provided by the conversion equations that express Cartesian tensor components as linear combinations of principal and secondary covariant components (for more details on tensorial covariants and conversion equations see Appendix E of the manual for GI $[\star]$ KoBo-1 and Kopský, 2001).

We illustrate the situation on a transition with symmetry descent $[4_z2_x2_{xy} \Downarrow 2_x2_y2_z]$ . In Table 3.1.3.1, we find that the principal tensor parameter transforms according to irreducible representation [B_1] with standard variable $[{\sf x}_3]$ . The corresponding covariant $[{\sf u}_{3}=u_{1}-u_{2}]$ can be found in Appendix E of the manual of GI $[\star]$ KoBo-1 (or in Kopský, 2001), where one also finds an invariant containing [u_1] and [u_2] : $[{\sf u}_{1,1}= u_{1}+u_{2}]$ . The corresponding conversion equations are: $[u_{1}={{1}\over{2}}({\sf u}_{1,1}+{\sf u}_{3})]$ , $[u_{2}={{1}\over{2}}({\sf u}_{1,1}-{\sf u}_{3})]$ . In the parent phase $[{\sf u}_{3}=u_{1}^{(p)}-u_{2}^{(p)}=0]$ , hence $[u_{1}^{(p)}=u_{2}^{(p)}={{1}\over{2}}{\sf u}_{1,1}]$ , whereas in the ferroic phase $[u_{1}^{(f)}=]$ $[{{1}\over{2}}({\sf u}_{1,1}+{\sf u}_{3})=]$ $[u_{1}^{(p)}+{{1}\over{2}}{\sf u}_{1,1}=]$ $[ u_{1}^{(p)}+{\delta}u_1]$ , $[u_{2}^{(f)}=]$ $[u_{2}^{(p)}-{{1}\over{2}}{\sf u}_{1,1}=]$ $[u_{2}^{(p)}+{\delta}u_2=]$ $[u_{1}^{(p)}-{\delta}u_1]$ . The symmetry-breaking increments $[{\delta}u_1=-{\delta}u_2]$ describe thus the changes of the Cartesian components that correspond to the nonzero principal tensor component $[u_{1}-u_{2}]$ .

An analogous situation occurs frequently in trigonal and hexagonal parent groups, where $[u_{1}-u_{2}]$ (or $[g_{1}-g_{2}]$ ) transforms like the first or second component of the principal tensor parameter . In these cases, the corresponding symmetry-breaking increments of Cartesian components are again related: $[\delta u_{1}=-\delta u_{2}]$ (or $[\delta g_{1}=-\delta g_{2}]$ ).

We note that relations like $[A_{11}=-A_{12}=-A_{26}]$ do not imply that these components transform as the standard variable. Though these components are proportional to the principal tensor parameter in the first domain state , they cannot be transformed to corresponding components in other domain states as easily as covariant tensor components of the principal tensor parameter .

In general, it is useful to consider a tensor parameter as a vector in the carrier space of the respective representation. Then the Cartesian components are projections of this vector on the Cartesian basis of the tensor space.

The presentation of the principal tensor parameters in the column Principal tensor parameters of this table is a compromise: whenever conversion equations lead to simple relations between morphic Cartesian components and/or symmetry-breaking increments, we present these relations, in some cases together with corresponding covariants. In the more complicated cases, only the covariants are given. The corresponding conversion equations and labelling of covariants are given at the beginning of that part of the table which covers hexagonal and cubic parent groups G. In the main tables of the software GI $[\star]$ KoBo-1, the principal tensor parameters and the secondary tensor parameters up to rank 4 are given consistently in covariant form. Labelling of covariant components and conversion equations are given in Appendix E of the manual.

The principal tensor parameters presented in Table 3.1.3.1 represent a particular choice of property tensors for standard variables given in the second column. To save space, property tensors are selected in the following way: polarization P and strain u are always listed; if none of their components transform according to $[D^{(\eta)}]$ , then components of one axial and one polar tensor (if available) appearing in Table 3.1.3.3 are given. Principal parameters of two different property tensors are separated by a semicolon. If two different components of the same property tensor transform in the same way, they are separated by a comma.

Table 3.1.3.3 | top | pdf |
Important property tensors

[i=1,2,3] ; $[\mu, \nu=1,2,\ldots,6]$ .

Tensor components	Property	Tensor components	Property
$[\varepsilon]$	enantiomorphism		chirality
	polarization		pyroelectricity
$[u_{\mu}]$	strain	$[\varepsilon_{ij}]$	dielectric permittivity
$[ g_{\mu}]$	optical activity
$[d_{i{\mu}}]$	piezoelectricity	$[r_{i{\mu}}]$	electro-optics
$[A_{i{\mu}}]$	electrogyration
$[\pi_{{\mu}{\nu}}]$	piezo-optics	$[Q_{{\mu}{\nu}}]$	electrostriction

As tensor indices we use integers [1,2,3] instead of vector components [x,y,z] and contracted indices [1,2,3,4,5,6] in matrix notation for pairs $[xx, yy, zz, yz \approx zy, zx \approx xz, xy \approx yx]$ , respectively

Important note : To make Table 3.1.3.1 compatible with the software GI $[\star]$ KoBo-1 and with Kopský (2001), coefficients of property tensors in matrix notation with contracted indices 4, 5, 6 do not contain the numerical factors 2 and 4 which are usually introduced to preserve a compact form (without these factors) of linear constitutive relations [see Chapter 1.1 , Nye (1985) and especially Appendices E and F of Sirotin & Shaskolskaya (1982)]. This explains the differences in matrix coefficients appearing in Table 3.1.3.1 and those presented in Chapter 1.1 or in Nye (1985) and in Sirotin & Shaskolskaya (1982). Thus e.g. for the symmetry descent $[6_z2_x2_y\Downarrow 3_z2_x]$ , we find in Table 3.1.3.1 the principal tensor parameters $[d_{11}=-d_{12}=-d_{26}]$ , whereas according to Chapter 1.1 or e.g. to Nye (1985) or Sirotin & Shaskolskaya (1982) these coefficients for [F_1=3_z2_x] are related by equations $[d_{11}=-d_{12}=-2d_{26}]$ .

Property tensors and symbols of their components that can be found in Table 3.1.3.1 are given in the left-hand half of Table 3.1.3.3. The right-hand half presents other tensors that transform in the same way as those on the left and form, therefore, covariant tensor components of the same form as those given in the column Principal tensor parameters. Principal and secondary tensor parameters for all property tensors that appear in Table 3.1.3.3 are available in the software GI $[\star]$ KoBo-1.

: number of ferroic single domain states that differ in the primary order parameter $[\eta]$ and in the principal tensor parameters .
$[n_{a}]$ : number of ferroelastic single domain states. If , $[n_a \,\lt\, n_f]$ or , the ferroic phase is, respectively, a full, partial or non-ferroelastic one.
$[n_{e}]$ : number of ferroelectric single domain states. If , $[n_e \,\lt\, n_f]$ or , the ferroic phase is, repectively, a full, partial or non-ferroelectric one ( or correspond to a non-polar or to a polar parent phase, respectively) (see Section 3.4.2 ).

References

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International Tables for Crystallography (2006). Vol. D. ch. 3.1, pp. 358-360