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

International Tables for Crystallography (2013). Vol. D, ch. 3.4, pp. 508-510

Section 3.4.3.2. Twinning group, distinction of two domain states

V. Janoveca* and J. Přívratskáb

aInstitute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-18221 Prague 8, Czech Republic, and bDepartment of Mathematics and Didactics of Mathematics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic
Correspondence e-mail:  janovec@fzu.cz

3.4.3.2. Twinning group, distinction of two domain states

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We have seen that for transposable domain pairs the symmetry group [J_{1j}] of a domain pair [({\bf S}_1,{\bf S}_j)] specifies transposing operations [g^{\star}_{1j}F_1] that transform [{\bf S}_1] into [{\bf S}_j]. This does not apply to non-transposable domain pairs, where the symmetry group [J_{1j}=F_{1j}] does not contain any switching operation. Another group exists, called the twinning group, which is associated with a domain pair and which does not have this drawback. The twinning group determines the distinction of two domain states, specifies the external fields needed to switch one domain state into another one and enables one to treat domain pairs independently of the transition [G\supset F_1]. This facilitates the tabulation of the properties of non-equivalent domain pairs that appear in all possible ferroic phases.

The twinning group [K_{1j}] of a domain pair [({\bf S}_1,{\bf S}_j)] is defined as the minimal subgroup of G that contains both [F_1] and a switching operation [g_{1j} ] of the domain pair [({\bf S}_1,{\bf S}_j)], [{\bf S}_j=g_{1j}{\bf S}_1 ] (Fuksa & Janovec, 1995[link]; Fuksa, 1997[link]), [F_1\subset K_{1j}\subseteq G, \quad g_{1j}\in K_{1j},\eqno(3.4.3.22)]where no group [K'_{1j}] exists such that [F_1\subset K'_{1j}\subset K_{1j}, \quad g_{1j}\in K'_{1j}.\eqno(3.4.3.23) ]

The twinning group [K_{1j}] is identical to the embracing (fundamental) group used in bicrystallography (see Section 3.2.2[link] ). In Section 3.3.4[link] it is called a composite symmetry of a twin.

Since [K_{1j}] is a group, it must contain all products of [g_{1j}] with operations of [F_1], i.e. the whole left coset [g_{1j}F_1]. For completely transposable domain pairs, the union of [F_1] and [g_{1j}^{\star}F_1] forms a group that is identical with the symmetry group [J_{1j}^{\star}] of the unordered domain pair [\{{\bf S}_1,{\bf S}_j\}]: [K_{1j}^{\star}=J_{1j}^{\star} = F_1 \cup g^{\star}_{1j}F_1, \quad g_{1j}^{\star}\in K_{1j}, \quad F_1=F_j.\eqno(3.4.3.24) ]

In a general case, the twinning group [K_{1j}], being a supergroup of [F_1], can always be expressed as a decomposition of the left cosets of [F_1], [K_{1j} = F_1 \cup g_{1j}F_1 \cup g_{1k}F_1 \cup\ldots\cup g_{1c}\in G. \eqno(3.4.3.25) ]

We can associate with the twinning group a set of c domain states, the [K_{1j}]-orbit of [{\bf S}_1], which can be generated by applying to [{\bf S}_1] the representatives of the left cosets in decomposition (3.4.3.25[link]), [K_{1j}{\bf S}_1=\{{\bf S}_1,{\bf S}_j,\ldots, {\bf S}_c\}.\eqno(3.4.3.26) ]This orbit is called the generic orbit of domain pair [({\bf S}_1,{\bf S}_j) ].

Since the generic orbit (3.4.3.26[link]) contains both domain states of the domain pair [({\bf S}_1,{\bf S}_j)], one can find different and equal nonzero tensor components in two domain states [{\bf S}_1 ] and [{\bf S}_j] by a similar procedure to that used in Section 3.4.2.3[link] for ascribing principal and secondary tensor parameters to principal and secondary domain states. All we have to do is just replace the group G of the parent phase by the twinning group [K_{1j}]. There are, therefore, three kinds of nonzero tensor components in [{\bf S}_1] and [{\bf S}_j]:

  • (1) Domain states [{\bf S}_1] and [{\bf S}_j ] differ in the principal tensor parameters [\kappa_a] of the `virtual' phase transition with symmetry descent [K_{1j}\supset F_1], [\kappa_a^{(1)}\neq \kappa_a^{(j)}, \quad a=1,2,\ldots,\eqno(3.4.3.27) ]where [\kappa_a^{(1)}] and [\kappa_a^{(j)}] are the principal tensor parameters in domain states [{\bf S}_1] and [{\bf S}_j]; in the symbol of the principal tensor parameter [\kappa_{a}] we explicitly write only the lower index a, which numbers different principal tensor parameters, but omit the upper index labelling the representation of [K_{1j}], according to which [\kappa_a] transforms, and the second lower index denoting the components of the principal tensor parameter (see Section 3.4.2.3[link] and the manual of the software GI[\star]KoBo-1, path: Subgroups\View\Domains and Kopský, 2001[link]).

    The principal tensor parameters [\kappa_a^{(1)}] of lower rank in domain state [{\bf S}_1] can be found for [G=K_{1j}] in Table 3.1.3.1[link] of Section 3.1.3.3[link] , where we replace G by [K_{1j}], and for all important property tensors in the software GI[\star]KoBo-1, path: Subgroups\View\Domains and in Kopský (2001[link]), where we again replace G by [K_{1j}]. Tensor parameters in domain state [{\bf S}_j ] can be obtained by applying to the principal tensor parameters in [{\bf S}_1] the operation [g_{1j}].

  • (2) If there exists an intermediate group [L_{1j} ] in between [F_1] and [K_{1j}] that does not – contrary to [K_{1j}] – contain the switching operation [g_{1j}] of the domain pair [({\bf S}_1,{\bf S}_j)], [F_1\subset L_{1j}\subseteq K_{1j}, \quad g_{1j}\in L_{1j}, \eqno(3.4.3.28) ][cf. relation (3.4.3.23[link])] then domain states [{\bf S}_1] and [{\bf S}_j] differ not only in the principal tensor parameters [\kappa_a], but also in the secondary tensor parameters [\lambda_{b}]: [\lambda_{b}^{(1)}\neq \lambda_{b}^{(j)}, \quad I_{K_{1j}}(\lambda_{b}^{(1)})=L_{1j}, \quad b=1,\ldots, \eqno(3.4.3.29) ]where [\lambda_{b}^{(1)}] and [\lambda_{b}^{(j)}] are the secondary tensor parameters in domain states [{\bf S}_1] and [{\bf S}_j ]; the last equation, in which [I_{K_{1j}}(\lambda_{b}^{(1)})] is the stabilizer of [\lambda_{b}^{(1)}] in [K_{1j}], expresses the condition that [\lambda_{b}] is the principal tensor parameter of the transition [K_{1j}\supset L_{1j}] [see equation (3.4.2.40[link])].

    The secondary tensor parameters [\lambda_{b}^{(1)} ] of lower rank in domain state [{\bf S}_1] can be found for [G=K_{1j}] in Table 3.1.3.1[link] of Section 3.1.3.3[link] , and for all important property tensors in the soft­ware GI[\star]KoBo-1, path: Subgroups\View\Domains and in Kopský (2001[link]). Tensor parameters [\lambda_{b}^{(j)}] in domain state [{\bf S}_j] can be obtained by applying to the secondary tensor parameters [\lambda_{b}^{(1)}] in [{\bf S}_1] the operation [g_{1j}].

  • (3) All nonzero tensor components that are the same in domain states [{\bf S}_1] and [{\bf S}_j] are identical with nonzero tensor components of the group [K_{1j}]. These components are readily available for all important material tensors in Section 1.1.4[link] , in the software GI[\star]KoBo-1, path: Subgroups\View\Domains and in Kopský (2001[link]).

Cartesian tensor components corresponding to the tensor parameters can be calculated by means of conversion equations [for details see the manual of the software GI[\star]KoBo-1, path: Subgroups\View\Domains and Kopský (2001[link])].

Let us now illustrate the above recipe for finding tensor distinctions by two simple examples.

Example 3.4.3.2.  The domain pair [({\bf S}_1,{\bf S}_2)] in Fig. 3.4.3.1[link](a) is a completely transposable pair, therefore, according to equations (3.4.3.24[link]) and (3.4.3.18[link]), [K_{12}^{\star}=J_{12}^{\star}=2_xm_ym_z \cup m^{\star}_x\{2_xm_ym_z\} = m_x^{\star}m_ym_z.\eqno(3.4.3.30) ]

In Table 3.1.3.1[link] , we find that the first principal tensor parameter [\kappa^{(1)}] of the transition G = [K_{1j}=] [m_xm_ym_z\supset 2_xm_ym_z =] [F_1] is the x-component [P_1] of the spontaneous polarization, [\kappa_1^{(1)}=P_1]. Since the switching operation [g_{12}^{\star}] is for example the inversion [\bar 1], the tensor parameter [\kappa_1^{(2)}] in the second domain state [{\bf S}_2] is [\kappa_1^{(2)}=-P_1].

Other principal tensor parameters can be found in the software GI[\star]KoBo-1 or in Kopský (2001[link]), p. 185. They are: [\kappa_2^{(1)}=d_{12},] [ \kappa_3^{(1)}=d_{13},] [\kappa_4^{(1)}=d_{26},] [\kappa_5^{(1)}=d_{35}] (the physical meaning of the components is explained in Table 3.4.3.5[link]). In the second domain state [{\bf S}_2], these components have the opposite sign. No other tensor components exist that would be different in [{\bf S}_1 ] and [{\bf S}_2], since there is no intermediate group [L_{1j} ] in between [F_1] and [K_{1j}].

Nonzero components that are the same in both domain states are nonzero components of property tensors in the group [mmm] and are listed in Section 1.1.4.7[link] or in the software GI[\star]KoBo-1 or in Kopský (2001[link]).

The numbers of independent tensor components that are different and those that are the same in two domain states are readily available for all non-ferroelastic domain pairs and important property tensors in Table 3.4.3.4[link].

Example 3.4.3.3.  The twinning group of the partially transposable domain pair [({\bf S}_1,{\bf S}_3) ] in Fig. 3.4.3.1[link](b) with [{\bf S}_3=2_{xy}{\bf S}_1 ] has the twinning group [\eqalignno{K_{13} &= 2_xm_ym_z \cup 2_{xy}\{2_xm_ym_z\} \cup 2_z\{2_xm_ym_z\} \cup 2_{x\bar y}\{2_xm_ym_z\} &\cr&= 4_z/m_zm_xm_{xy}. &(3.4.3.31)} ]Domain states [{\bf S}_1] and [{\bf S}_3] differ in the principal tensor parameter of the transition [4_z/m_zm_xm_{xy} \subset 2_xm_ym_z ], which is two-dimensional and which we found in Example 3.4.2.4[link]: [\kappa_1^{(1)}=(P,0)]. Then in the domain state [{\bf S}_3] it is [\kappa_1^{(3)}=D(2_{xy})(P,0)=(0,P)]. Other principal tensors are: [\kappa_2^{(1)}=(g_4,0),] [\kappa_3^{(1)}=(d_{11},0), ] [\kappa_4^{(1)}=(d_{12},0),] [\kappa_5^{(1)}=(d_{13},0),] [\kappa_6^{(1)}=(d_{26},0),] [\kappa_7^{(1)}=(d_{35},0)] (the physical meaning of the components is explained in Table 3.4.3.5[link]). In the domain state [{\bf S}_3] they keep their absolute value but appear as the second nonzero components, as with spontaneous polarization.

There is an intermediate group [L_{13}=m_xm_ym_z] between [F_1=2_xm_ym_z ] and [K_{13}=4_z/m_zm_xm_{xy}], since [L_{13}=m_xm_ym_z] does not contain [g_{13}=2_{xy}]. The one-dimensional secondary tensor parameters for the symmetry descent [K_{13}=] [4_z/m_zm_xm_{xy}\supset L_{13}=] [m_xm_ym_z] was also found in Example 3.4.2.4[link]: [\lambda_{1}^{(1)}=u_1-u_2; ] [\lambda_{2}^{(1)}=A_{14}+A_{25},A_{36};] [\lambda_{3}^{(1)}=s_{11}-s_{22}, ] [s_{13}-s_{23},] [s_{44}-s_{55};] [\lambda_{4}^{(1)}=Q_{11}-Q_{22}, ] [Q_{12}-Q_{21},] [Q_{13}-Q_{23},] [Q_{31}-Q_{32},] [Q_{44}-Q_{55}]. All these parameters have the opposite sign in [{\bf S}_3 ].

The tensor distinction of two domain states [{\bf S}_1] and [{\bf S}_j] in a domain pair [({\bf S}_1,{\bf S}_j)] provides a useful classification of domain pairs given in the second and the third columns of Table 3.4.3.1[link]. This classification can be extended to ferroic phases which are named according to domain pairs that exist in this phase. Thus, for example, if a ferroic phase contains ferroelectric (ferroelastic) domain pair(s), then this phase is a ferroelectric (ferroelastic) phase. Finer division into full and partial ferroelectric (ferro­elastic) phases specifies whether all or only some of the possible domain pairs in this phase are ferroelectric (ferroelastic) ones. Another approach to this classification uses the notions of principal and secondary tensor parameters, and was explained in Section 3.4.2.2[link].

Table 3.4.3.1| top | pdf |
Classification of domain pairs, ferroic phases and of switching (state shifts)

[P_{0i}^{(k)}] and [u_{0\mu}^{(k)}] are components of the spontaneous polarization and spontaneous strain in the domain state [{\bf S}_k], where [k=1] or [k=j]; similarly, [d_{i\mu}^{(k)} ] are components of the piezoelectric tensor, [{\varepsilon}_0{\kappa}_{ij}^{(k)} ] are components of electric susceptibility, [s_{\mu\nu}^{(k)}] are compliance components and [Q_{ij\mu}^{(k)}] are components of electrostriction (components with Greek indices are expressed in matrix notation) [see Chapter 1.1[link] or e.g. Nye (1985[link]) and Sirotin & Shaskolskaya (1982[link])]. Text in italics concerns the classification of ferroic phases. [{\bf E}] is the electric field and [{\sigma} ] is the mechanical stress.

Ferroic classDomain pair – at least in one pairDomain pair – phaseSwitching (state shift)Switching field
Primary At least one [P_{0i}^{(j)}-P_{0i}^{(1)}\neq 0 ] Ferroelectric Electrically first order E
At least one [u_{0\mu}^{(j)}-u_{0\mu}^{(1)}\neq 0 ] Ferroelastic Mechanically first order [{\sigma} ]
Secondary At least one [P_{0i}^{(j)}-P_{0i}^{(1)}\neq 0 ] and at least one [u_{0\mu}^{(j)}-u_{0\mu}^{(1)}\neq 0] Ferroelastoelectric Electromechanically first order E[\sigma]
  All [P_{0i}^{(j)}-P_{0i}^{(1)}= 0] and at least one [{\varepsilon}_0({\kappa}_{ik}^{(j)}-{\kappa}_{ik}^{(1)})\neq 0 ] Ferrobielectric Electrically second order EE
  All [u_{0\mu}^{(j)}-u_{0\mu}^{(1)}= 0 ] and at least one [s_{\mu\nu}^{(j)}-s_{\mu\nu}^{(1)}\neq 0] Ferrobielastic Mechanically second order [{\sigma\sigma}]
[\ldots] [\ldots] [\ldots] [\ldots] [\ldots]

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

A discussion of and many examples of secondary ferroic phases are available in papers by Newnham & Cross (1974a[link],b[link]) and Newnham & Skinner (1976[link]), and tertiary ferroic phases are discussed by Amin & Newnham (1980[link]).

We shall now show that the tensor distinction of domain states is closely related to the switching of domain states by external fields.

References

Amin, A. & Newnham, R. E. (1980). Tertiary ferroics. Phys. Status Solidi A, 61, 215–219.
Fuksa, J. (1997). The role of the twinning group of a domain pair in tensor distinction of domain states. Ferroelectrics, 204, 135–155.
Fuksa, J. & Janovec, V. (1995). Permutation classification of domain pairs. Ferroelectrics, 172, 343–350.
Kopský, V. (2001). Tensor parameters of ferroic phase transitions I. Theory and tables. Phase Transit. 73, 1–422.
Newnham, R. E. & Cross, L. E. (1974a). Symmetry of secondary ferroics I. Mater. Res. Bull. 9, 927–934.
Newnham, R. E. & Cross, L. E. (1974b). Symmetry of secondary ferroics II. Mater. Res. Bull. 9, 1021–1032.
Newnham, R. E. & Skinner, D. P. Jr (1976). Polycrystalline secondary ferroics. Mater. Res. Bull. 11, 1273–1284.








































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