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. 512-515

Section 3.4.3.5. Non-ferroelastic domain pairs: twin laws, domain distinction and switching fields, synoptic table

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

3.4.3.5. Non-ferroelastic domain pairs: twin laws, domain distinction and switching fields, synoptic table

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Two domain states [{\bf S}_1] and [{\bf S}_j] form a non-ferroelastic domain pair [({\bf S}_1,{\bf S}_j)] if the spontaneous strain in both domain states is the same, [{\bf u}_0^{(1)}={\bf u}_0^{(j)} ]. This is so if the twinning group [K_{1j}] of the pair and the symmetry group [F_1] of domain state [{\bf S}_1] belong to the same crystal family (see Table 3.4.2.2[link]): [{\rm Fam}K_{1j}={\rm Fam}F_{1}.\eqno(3.4.3.40)]

It can be shown that all non-ferroelastic domain pairs are completely transposable domain pairs (Janovec et al., 1993[link]), i.e. [F_{1j}=F_1=F_j\eqno(3.4.3.41)]and the twinning group [K_{1j}] is equal to the symmetry group [J_{1j} ] of the unordered domain pair [see equation (3.4.3.24[link])]: [K_{1j}^{\star}=J_{1j}^{\star} = F_{1} \cup g^{\star}_{1j}F_{1}.\eqno(3.4.3.42) ](Complete transposability is only a necessary, but not a sufficient, condition of a non-ferroelastic domain pair, since there are also ferroelastic domain pairs that are completely transposable – see Table 3.4.3.6[link].)

The relation between domain states in a non-ferroelastic domain twin, in which two domain states coexist, is the same as that of a corresponding non-ferroelastic domain pair consisting of single-domain states. Transposing operations [g^{\star}_{1j}] are, therefore, also twinning operations.

Synoptic Table 3.4.3.4[link] lists representative domain pairs of all orbits of non-ferroelastic domain pairs. Each pair is specified by the first domain state [{\bf S}_1] with symmetry group [F_1] and by transposing operations [g^{\star}_{1j}] that transform [{\bf S}_1] into [{\bf S}_j], [{\bf S}_j =g^{\star}_{1j}{\bf S}_1 ]. Twin laws in dichromatic notation are presented and basic data for tensor distinction and switching of non-ferroelastic domains are given.

3.4.3.5.1. Explanation of Table 3.4.3.4[link]

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The first three columns specify domain pairs.

Table 3.4.3.4| top | pdf |
Non-ferroelastic domain pairs, domain twin laws and distinction of non-ferroelastic domains

[F_1]: symmetry of [{\bf S}_1]; [g_{1j}^{\star} ]: twinning operations of second order; [K_{1j}^{\star}]: twinning group signifying the twin law of domain pair [({\bf S}_1,g_{1j}^{\star}{\bf S}_1) ]; [J_{1j}^{\star}]: symmetry group of the pair; [\Gamma_{\alpha} ]: irreducible representation of [K_{1j}^{\star}]; [\rho], [{P}_i,\ldots], [{Q}_{{\mu\nu}} ]: components of property tensors (see Table 3.4.3.5[link]): [a|c]: number of distinct[|]equal nonzero independent tensor components of property tensors.

[F_1][g_{1j}^{\star} ][K_{1j}^{\star}=J_{1j}^{\star} ][\Gamma_{\alpha}]Diffraction intensities[\rho][{P}_i ][g_{\mu}][{d}_{i{\mu}} ][A_{i\mu}][{s}_{\mu\nu} ][{Q}_{{\mu\nu}} ]
1 [\bar1^{\star}] [\bar1^{\star}] [A_u] = [1|0] [3|0] [6|0] [18|0] [0|18] [0|21] [0|36]
[2_u] [\bar1^{\star}], [m^{\star}_u] [2_u/m_u^{\star}] [A_u] = [1|0 ] [1|0 ] [4|0 ] [8|0 ] [0|8 ] [0|13] [0|20]
[m_u] [\bar1^{\star}], [2^{\star}_u] [2_u^{\star}/m_u] [B_u] = [0|0] [2|0] [2|0] [10|0] [0|8] [0|13] [0|20]
[2_x2_y2_z] [\bar1^{\star}], [m^{\star}_x], [m^{\star}_y], [m^{\star}_z] [m_x^{\star}m_y^{\star}m_z^{\star}] [A_{u}] = [1|0 ] [0|0 ] [3|0 ] [3|0 ] [0|3 ] [0|9 ] [0|12]
[2_{x\bar{y}}2_{xy}2_z] [\bar1^{\star}], [m^{\star}_{xy} ], [m^{\star}_{x\bar{y}}], [m^{\star}_z] [m_{x\bar{y}}^{\star}m_{xy}^{\star}m_z^{\star} ] [A_{u}] [= ] [1|0 ] [0|0 ] [3|0 ] [3|0 ] [0|3 ] [0|9 ] [0|12]
[m_xm_y2_z] [\bar1^{\star}], [m^{\star}_z], [2^{\star}_x], [2^{\star}_y] [m_xm_ym_z^{\star}] [B_{1u}] [= ] [0|0 ] [1|0 ] [1|0 ] [5|0 ] [0|3 ] [0|9 ] [0|12]
[2_xm_ym_z] [\bar1^{\star}], [m^{\star}_x], [2^{\star}_y], [2^{\star}_z] [m_x^{\star}m_ym_z] [B_{1u}] = [0|0 ] [1|0 ] [1|0 ] [5|0 ] [0|3 ] [0|9 ] [0|12]
[m_x2_ym_z] [\bar1^{\star}], [m^{\star}_y], [2^{\star}_x], [2^{\star}_z] [m_xm_y^{\star}m_z] [B_{1u}] [= ] [0|0 ] [1|0 ] [1|0 ] [5|0 ] [0|3 ] [0|9 ] [0|12]
[m_{x\bar{y}}m_{xy}2_z] [\bar1^{\star}], [m^{\star}_z], [2^{\star}_{xy}], [2^{\star}_{x\bar{y}}] [m_{x\bar{y}}m_{xy}m_z^{\star}] [B_{1u}] [= ] [0|0 ] [1|0 ] [1|0 ] [5|0 ] [0|3 ] [0|9 ] [0|12]
[4_z] [\bar1^{\star}], [m^{\star}_z] [4_z/m_z^{\star}] [A_{u}] [= ] [1|0 ] [1|0 ] [2|0 ] [4|0 ] [0|4 ] [0|7 ] [0|10]
[4_z] [2_x^{\star}], [2^{\star}_y], [2^{\star}_{xy}], [2^{\star}_{x\bar y}] [4_z2_x^{\star}2_{xy}^{\star}] [A_{2}] [\not= ] [0|1 ] [1|0 ] [0|2 ] [3|1 ] [3|1 ] [1|6 ] [3|7]
[4_z] [m_x^{\star}], [m^{\star}_y], [m^{\star}_{xy}], [m^{\star}_{x\bar y}] [4_zm_x^{\star}m_{xy}^{\star}] [A_{2} ] [\not=] [1|0 ] [0|1 ] [2|0 ] [1|3 ] [3|1 ] [1|6 ] [3|7]
[\bar{4}_z ] [\bar1^{\star}], [m^{\star}_z] [4^{\star}_z/m_z^{\star}] [B_{u}] [= ] [0|0 ] [0|0 ] [2|0 ] [4|0 ] [0|4 ] [0|7 ] [0|10]
[\bar{4}_z] [m_{xy}^{\star}], [m^{\star}_{x\bar y} ], [2^{\star}_x], [2^{\star}_y] [\bar{4}_z2_x^{\star}m_{xy}^{\star}] [A_{2}] [\not= ] [0|0 ] [0|0 ] [1|1 ] [2|2 ] [3|1 ] [1|6 ] [3|7]
[\bar{4}_z] [m_x^{\star}], [m^{\star}_y], [2^{\star}_{xy}], [2^{\star}_{x\bar y}] [\bar{4}_zm_x^{\star}2_{xy}^{\star}] [A_{2}] [\not= ] [0|0 ] [0|0 ] [1|1 ] [2|2 ] [3|1 ] [1|6 ] [3|7]
[4_z/m_z] [m^{\star}_x], [m^{\star}_y], [m^{\star}_{xy}], [m^{\star}_{x\bar y}], [2_x^{\star}], [2^{\star}_y], [2^{\star}_{xy}], [ 2^{\star}_{x\bar y}] [4_z/m_zm_x^{\star}m_{xy}^{\star}] [A_{2g}] [\not=] [0|0 ] [0|0 ] [0|0 ] [0|0 ] [3|1 ] [1|6 ] [3|7]
[4_z2_x2_{xy} ] [\bar1^{\star}], [m^{\star}_z], [m_x^{\star}], [m^{\star}_y], [m^{\star}_{xy}], [m^{\star}_{x\bar y} ] [4_z/m_z^{\star}m_x^{\star}m_{xy}^{\star} ] [A_{1u}] [= ] [1|0 ] [0|0 ] [2|0 ] [1|0 ] [0|1 ] [0|6 ] [0|7]
[4_zm_xm_{xy}] [\bar1^{\star}], [m^{\star}_z], [2_x^{\star}], [2^{\star}_y], [2^{\star}_{xy}], [2^{\star}_{x\bar y} ] [4_z/m_z^{\star}m_xm_{xy} ] [A_{2u}] [= ] [0|0 ] [1|0 ] [0|0 ] [3|0 ] [0|1 ] [0|6 ] [0|7]
[\bar{4}_z2_xm_{xy}] [\bar1^{\star}], [m^{\star}_z], [m_x^{\star}], [m^{\star}_y], [2^{\star}_{xy}], [2^{\star}_{x\bar y} ] [4_z^{\star}/m_z^{\star}m_x^{\star}m_{xy} ] [B_{1u}] [= ] [0|0 ] [0|0 ] [1|0 ] [2|0 ] [0|1 ] [0|6 ] [0|7]
[\bar{4}_zm_x2_{xy}] [\bar1^{\star}], [m^{\star}_z], [m^{\star}_{xy}], [m^{\star}_{x\bar y}], [2_x^{\star} ], [2^{\star}_y] [4_z^{\star}/m_z^{\star}m_xm_{xy}^{\star} ] [B_{1u}] [= ] [0|0 ] [0|0 ] [1|0 ] [2|0 ] [0|1 ] [0|6 ] [0|7]
[3_v] [\bar{1}^{\star} ] [\bar{3}_v^{\star} ] [A_{u}] [= ] [1|0 ] [1|0 ] [2|0 ] [6|0 ] [0|6 ] [0|7 ] [0|12]
[3_z ] [2_x^{\star}], [2^{\star}_{x{^\prime}} ], [2^{\star}_{x{^\prime}{^\prime}}] [3_z2_x^{\star} ] [A_{2}] [\not=] [0|1 ] [1|0 ] [0|2 ] [4|2 ] [4|2] [1|6 ] [4|8]
[3_z] [2_y^{\star}], [2^{\star}_{y{^\prime}} ], [2^{\star}_{y{^\prime}{^\prime}}] [3_z2_y^{\star} ] [A_{2}] [\not=] [0|1 ] [1|0 ] [0|2 ] [4|2 ] [4|2 ] [1|6 ] [4|8]
[3_p] [2_{x\bar{y}}^{\star}], [2^{\star}_{y\bar{z}} ], [2^{\star}_{z\bar{x}}] [3_p2_{x\bar{y}}^{\star} ] [A_{2}] [\not=] [0|1 ] [1|0 ] [0|2 ] [4|2 ] [4|2] [1|6 ] [4|8]
[3_z ] [m_x^{\star}], [m^{\star}_{x{^\prime}} ], [m^{\star}_{x{^\prime}{^\prime}}] [3_zm_x^{\star} ] [A_{2}] [\not=] [1|0 ] [0|1 ] [2|0 ] [2|4 ] [4|2 ] [1|6 ] [4|8]
[3_z] [m_y^{\star}], [m^{\star}_{y{^\prime}} ], [m^{\star}_{y{^\prime}{^\prime}}] [3_zm_y^{\star} ] [A_{2}] [\not= ] [1|0 ] [0|1 ] [2|0 ] [2|4 ] [4|2 ] [1|6 ] [4|8]
[3_p ] [m_{x\bar{y}}^{\star}], [m^{\star}_{y\bar{z}} ], [m^{\star}_{z\bar{x}}] [3_pm_x^{\star} ] [A_{2}] [\not=] [1|0 ] [0|1 ] [2|0 ] [2|4 ] [4|2 ] [1|6 ] [4|8]
[3_z] [2_z^{\star} ] [6_z^{\star} ] B [\not=] [0|1 ] [0|1 ] [0|2 ] [2|4 ] [2|4 ] [2|5 ] [4|8]
[3_z ] [m_z^{\star} ] [\bar{6}_z^{\star} ] [A^{{^\prime}{^\prime}}] [\not= ] [1|0 ] [1|0 ] [2|0 ] [4|2 ] [2|4 ] [2|5 ] [4|8]
[\bar{3}_z] [m_x^{\star}], [m^{\star}_{x{^\prime}} ], [m^{\star}_{x{^\prime}{^\prime}}], [2_x^{\star} ], [2^{\star}_{x{^\prime}}], [2^{\star}_{x{^\prime}{^\prime}}] [\bar{3}_zm_x^{\star}] [A_{2g}] [\not=] [0|0 ] [0|0 ] [0|0 ] [0|0 ] [4|2 ] [1|6 ] [4|8]
[\bar{3}_z] [m_y^{\star}], [m^{\star}_{y{^\prime}} ], [m^{\star}_{y{^\prime}{^\prime}}], [2_y^{\star}], [2^{\star}_{y{^\prime}}], [2^{\star}_{y{^\prime}{^\prime}}] [\bar{3}_zm_y^{\star}] [A_{2g}] [\not= ] [0|0 ] [0|0 ] [0|0 ] [0|0 ] [4|2 ] [1|6 ] [4|8]
[\bar{3}_p ] [m_{x\bar{y}}^{\star}], [m^{\star}_{y\bar{z}} ], [m^{\star}_{z\bar{x}}], [2_{x\bar{y}}^{\star}], [2^{\star}_{y\bar{z}}], [2^{\star}_{z\bar{x}}] [\bar{3}_pm_x^{\star}] [A_{2g}] [\not=] [0|0 ] [0|0 ] [0|0 ] [0|0 ] [4|2 ] [1|6 ] [4|8]
[\bar{3}_z] [m^{\star}_z], [2_z^{\star}] [6_z^{\star}/m_z^{\star}] [B_{g} ] [\not= ] [0|0 ] [0|0 ] [0|0 ] [0|0 ] [2|4 ] [2|5] [4|8]
[3_z2_x ] [\bar1^{\star}], [m_x^{\star}], [m^{\star}_{x{^\prime}}], [m^{\star}_{x{^\prime}{^\prime}}] [\bar{3}_z^{\star}m_x^{\star}] [A_{1u}] [= ] [1|0 ] [0|0] [2|0 ] [2|0 ] [0|2 ] [0|6 ] [0|8]
[3_z2_y ] [\bar1^{\star}], [m_y^{\star}], [m^{\star}_{y{^\prime}}], [m^{\star}_{y{^\prime}{^\prime}}] [\bar{3}_z^{\star}m_y^{\star}] [A_{1u}] = [1|0 ] [0|0] [2|0 ] [2|0 ] [0|2 ] [0|6 ] [0|8]
[3_z2_x ] [2_z^{\star}], [2_y^{\star}], [2^{\star}_{y{^\prime}}], [2^{\star}_{y{^\prime}{^\prime}}] [6_z^{\star}2_x2_y^{\star}] [B_{1}] [\not= ] [0|1 ] [0|0 ] [0|2 ] [1|1 ] [1|1 ] [1|5] [2|6]
[3_z2_y] [2_z^{\star}], [2_x^{\star}], [2^{\star}_{x{^\prime}}], [2^{\star}_{x{^\prime}{^\prime}}] [6_z^{\star}2_x^{\star}2_y ] [B_{1}] [\not=] [0|1 ] [0|0 ] [0|2 ] [1|1 ] [1|1 ] [1|5 ] [2|6]
[3_p2_{x\bar{y}}] [\bar1^{\star}], [m_{x\bar{y}}^{\star} ], [m^{\star}_{y\bar{z}}], [m^{\star}_{z\bar{x}}] [\bar{3}_p^{\star}m_x^{\star}] [A_{1u}] [= ] [1|0] [0|0 ] [2|0 ] [2|0 ] [0|2 ] [0|6 ] [0|8]
[3_z2_x ] [m_z^{\star}], [m_y^{\star}], [m^{\star}_{y{^\prime}}], [m^{\star}_{y{^\prime}{^\prime}}] [\bar{6}_z^{\star}2_xm_y^{\star}] [A^{{^\prime}{^\prime}}_{1}] [\not= ] [1|0 ] [0|0] [2|0 ] [1|1 ] [1|1 ] [1|5 ] [2|6]
[3_z2_y ] [m_z^{\star}], [m_x^{\star}], [m^{\star}_{x{^\prime}}], [m^{\star}_{x{^\prime}{^\prime}}] [\bar{6}_z^{\star}m_x2_y^{\star}] [A^{{^\prime}{^\prime}}_{1}] [\not= ] [1|0 ] [0|0] [2|0 ] [1|1 ] [1|1 ] [1|5 ] [2|6]
[3_pm_{x\bar{y}} ] [\bar1^{\star}], [2_{x\bar{y}}^{\star} ], [2^{\star}_{y\bar{z}}], [2^{\star}_{z\bar{x}}] [\bar{3}_p^{\star}m_x ] [A_{2u}] = [0|0 ] [1|0 ] [0|0 ] [4|0 ] [0|2 ] [0|6 ] [0|8]
[3_zm_x ] [\bar1^{\star}], [2_x^{\star}], [2^{\star}_{x{^\prime}}], [2^{\star}_{x{^\prime}{^\prime}}] [\bar{3}_z^{\star}m_x ] [A_{2u}] = [0|0 ] [1|0 ] [0|0 ] [4|0 ] [0|2 ] [0|6 ] [0|8]
[3_zm_y] [\bar1^{\star}], [2_y^{\star}], [2^{\star}_{y{^\prime}}], [2^{\star}_{y{^\prime}{^\prime}}] [\bar{3}_z^{\star}m_y ] [A_{2u}] = [0|0 ] [1|0 ] [0|0 ] [4|0 ] [0|2 ] [0|6 ] [0|8]
[3_zm_x ] [2_z^{\star}], [m_y^{\star}], [m^{\star}_{y{^\prime}}], [m^{\star}_{y{^\prime}{^\prime}}] [6_z^{\star}m_xm_y^{\star} ] [B_{2}] [\not=] [0|0 ] [0|1 ] [0|0 ] [1|3 ] [1|1 ] [1|5 ] [2|6]
[3_zm_y] [m_x^{\star}], [m^{\star}_{x{^\prime}} ], [m^{\star}_{x{^\prime}{^\prime}}] [6_z^{\star}m_x^{\star}m_y ] [B_{2}] [\not=] [0|0 ] [0|1 ] [0|0 ] [1|3 ] [1|1 ] [1|5 ] [2|6]
[3_zm_x ] [m_z^{\star}], [2_y^{\star}], [2^{\star}_{y{^\prime}}], [2^{\star}_{y{^\prime}{^\prime}}] [\bar{6}_z^{\star}m_x2_y^{\star}] [A_{2}^{{^\prime}{^\prime}}] [\not=] [0|0 ] [1|0 ] [0|0 ] [3|1 ] [1|1 ] [1|5 ] [2|6]
[3_zm_y ] [m_z^{\star}], [2_x^{\star}], [2^{\star}_{x{^\prime}}], [2^{\star}_{x{^\prime}{^\prime}}] [\bar{6}_z^{\star}2_x^{\star}m_y] [A_{2}^{{^\prime}{^\prime}}] [\not=] [0|0 ] [1|0 ] [0|0 ] [3|1 ] [1|1 ] [1|5 ] [2|6]
[\bar{3}_zm_x ] [m_z^{\star}], [m_y^{\star}], [m^{\star}_{y{^\prime}}], [m^{\star}_{y{^\prime}{^\prime}}] [6_z^{\star}/m_z^{\star}m_xm_y^{\star} ] [B_{1g}] [\not=] [0|0 ] [0|0 ] [0|0 ] [0|0 ] [1|1 ] [1|5 ] [2|6]
[\bar{3}_zm_y ] [m_z^{\star}], [m_x^{\star}], [m^{\star}_{x{^\prime}}], [m^{\star}_{x{^\prime}{^\prime}}] [6_z^{\star}/m_z^{\star}m_x^{\star}m_y ] [B_{1g}] [\not= ] [0|0 ] [0|0 ] [0|0 ] [0|0 ] [1|1 ] [1|5 ] [2|6]
[6_z ] [\bar1^{\star}], [m^{\star}_z] [6_z/m_z^{\star} ] [A_{u}] [= ] [1|0 ] [1|0 ] [2|0 ] [4|0 ] [0|4 ] [0|5 ] [0|8]
[6_z ] [2_x^{\star}], [2^{\star}_{x{^\prime}} ], [2^{\star}_{x{^\prime}{^\prime}}], [2_y^{\star}], [2^{\star}_{y{^\prime}}], [2^{\star}_{y{^\prime}{^\prime}}] [6_z2_x^{\star}2_y^{\star} ] [A_{2}] [\not=] [0|1 ] [1|0 ] [0|2 ] [3|1 ] [3|1 ] [0|5 ] [2|6]
[6_z] [m_x^{\star}], [m^{\star}_{x{^\prime}} ], [m^{\star}_{x{^\prime}{^\prime}}], [m_y^{\star}], [m^{\star}_{y{^\prime}}], [m^{\star}_{y{^\prime}{^\prime}} ] [6_zm_x^{\star}m_y^{\star} ] [A_{2}] [\not=] [1|0 ] [0|1 ] [2|0 ] [1|3 ] [3|1 ] [0|5 ] [2|6]
[\bar{6}_z ] [\bar1^{\star}], [2^{\star}_z] [6_z^{\star}/m_z ] [B_{u}] [= ] [0|0 ] [0|0 ] [0|0 ] [2|0 ] [0|4 ] [0|5 ] [0|8]
[\bar{6}_z ] [m_x^{\star}], [m^{\star}_{x{^\prime}} ], [m^{\star}_{x{^\prime}{^\prime}}], [2_y^{\star}], [2^{\star}_{y{^\prime}}], [2^{\star}_{y{^\prime}{^\prime}} ] [\bar{6}_zm_x^{\star}2_y^{\star}] [A_{2}^{{^\prime}}] [\not= ] [0|0 ] [0|0 ] [0|0 ] [1|1 ] [3|1 ] [0|5 ] [2|6]
[\bar{6}_z ] [m_y^{\star}], [m^{\star}_{y{^\prime}} ], [m^{\star}_{y{^\prime}{^\prime}}], [2_x^{\star}], [2^{\star}_{x{^\prime}}], [2^{\star}_{x{^\prime}{^\prime}} ] [\bar{6}_z2_x^{\star}m_y^{\star}] [A_{2}^{{^\prime}}] [\not= ] [0|0 ] [0|0 ] [0|0 ] [1|1 ] [3|1 ] [0|5 ] [2|6]
[6_z/m_z ] [m_x^{\star}], [m^{\star}_{x{^\prime}} ], [m^{\star}_{x{^\prime}{^\prime}}], [m_y^{\star}], [m^{\star}_{y{^\prime}}], [m^{\star}_{y{^\prime}{^\prime}}], [2_x^{\star}], [2^{\star}_{x{^\prime}}], [2^{\star}_{x{^\prime}{^\prime}} ], [2_y^{\star}], [2^{\star}_{y{^\prime}}], [2^{\star}_{y{^\prime}{^\prime}} ] [6_z/m_zm_x^{\star}m_y^{\star} ] [A_{2g}] [\not=] [0|0 ] [0|0 ] [0|0 ] [0|0 ] [3|1 ] [0|5 ] [2|6]
[6_z2_x2_y ] [\bar1^{\star}], [m_z^{\star}], [m_x^{\star}], [m^{\star}_{x{^\prime}}], [m^{\star}_{x{^\prime}{^\prime}} ], [m_y^{\star}], [m^{\star}_{y{^\prime}}][, m^{\star}_{y{^\prime}{^\prime}} ] [6_z/m_z^{\star}m_x^{\star}m^{\star}_y ] [A_{1u}] [= ] [1|0 ] [0|0 ] [2|0 ] [1|0 ] [0|1 ] [0|5 ] [0|6]
[6_zm_xm_y ] [\bar1^{\star}], [m^{\star}_z], [2_x^{\star}], [2^{\star}_{x{^\prime}}], [2^{\star}_{x{^\prime}{^\prime}} ], [2_y^{\star}], [2^{\star}_{y{^\prime}}], [2^{\star}_{y{^\prime}{^\prime}} ] [6_z/m_z^{\star}m_xm_y ] [A_{2u}] = [0|0 ] [1|0 ] [0|0 ] [3|0 ] [0|1 ] [0|5 ] [0|6]
[\bar{6}_z2_xm_y ] [\bar1^{\star}], [2^{\star}_z], [m_x^{\star}], [m^{\star}_{x{^\prime}}], [m^{\star}_{x{^\prime}{^\prime}} ], [2_y^{\star}], [2^{\star}_{y{^\prime}}], [2^{\star}_{y{^\prime}{^\prime}} ] [6_z^{\star}/m_zm_x^{\star}m_y] [B_{2u}] [= ] [0|0 ] [0|0 ] [0|0 ] [1|0 ] [0|1 ] [0|5 ] [0|6]
[\bar{6}_zm_x2_y ] [\bar1^{\star}], [2^{\star}_z], [m_y^{\star}], [m^{\star}_{y{^\prime}}], [m^{\star}_{y{^\prime}{^\prime}} ], [2_x^{\star}, 2^{\star}_{x{^\prime}}], [2^{\star}_{x{^\prime}{^\prime}} ] [6_z^{\star}/m_zm_xm_y^{\star}] [B_{2u}] [= ] [0|0 ] [0|0 ] [0|0 ] [1|0 ] [0|1 ] [0|5 ] [0|6]
23 [\bar1^{\star}], [m_x^{\star}], [m^{\star}_y], [m^{\star}_z] [m^{\star}\bar{3} ] [A_{u}] = [1|0 ] [0|0 ] [1|0 ] [1|0 ] [0|1 ] [0|3 ] [0|4]
23 [2_{xy}^{\star}], [2_{yz}^{\star} ], [2_{zx}^{\star}], [2^{\star}_{x\bar y}], [2^{\star}_{y\bar z} ], [2^{\star}_{z\bar x}] [4^{\star}32^{\star}] [A_{2}] [\not= ] [0|1 ] [0|0 ] [0|1] [1|0 ] [1|0 ] [0|3 ] [1|3]
23 [m_{xy}^{\star}], [m_{yz}^{\star} ], [m_{zx}^{\star}], [m^{\star}_{x\bar y}], [m^{\star}_{y\bar z} ], [m^{\star}_{z\bar x}] [\bar{4}^{\star}3m^{\star} ] [A_{2}] [\not= ] [1|0 ] [0|0 ] [1|0 ] [0|1 ] [1|0 ] [0|3 ] [1|3]
[m\bar{3}] [m_{xy}^{\star}], [m_{yz}^{\star} ], [m_{zx}^{\star}], [m^{\star}_{x\bar y}], [m^{\star}_{y\bar z} ], [m^{\star}_{z\bar x}], [2^{\star}_{xy}], [2^{\star}_{yz} ], [2^{\star}_{zx}], [2^{\star}_{x\bar y}], [2^{\star}_{y\bar z} ], [2^{\star}_{z\bar x}] [m\bar{3}m^{\star}] [A_{2g}] [\not=] [0|0 ] [0|0 ] [0|0 ] [0|0 ] [1|0 ] [0|3 ] [1|3]
432 [\bar1^{\star}], [m_x^{\star}], [m^{\star}_y], [m^{\star}_z], [m_{xy}^{\star}], [m_{yz}^{\star} ], [m_{zx}^{\star}], [m^{\star}_{x\bar y}], [m^{\star}_{y\bar z} ], [m^{\star}_{z\bar x}] [m^{\star}\bar{3}m^{\star}] [A_{1u}] [= ] [1|0] [0|0] [1|0] [0|0] [0|0] [0|3] [0|3]
[\bar{4}3m ] [\bar1^{\star}], [m_x^{\star}], [m^{\star}_y], [m^{\star}_z], [2^{\star}_{xy}], [2^{\star}_{yz} ], [2^{\star}_{zx}], [2^{\star}_{x\bar y} ], [2^{\star}_{y\bar z} ], [2^{\star}_{z\bar x}] [m^{\star}\bar{3}m] [A_{2u}] [= ] [0|0] [0|0] [0|0] [1|0] [0|0] [0|3] [0|3]
[u = z,x(x{^\prime},x{^\prime}{^\prime}),y(y{^\prime},y{^\prime}{^\prime}),xy(x\bar{y},zx,z\bar{x},yz,y\bar{z}) ].
[v =z,p(q,r,s)].

  • [F_1]: point-group symmetry (stabilizer in [K_{1j}]) of the first domain state [{\bf S}_1)] in a single-domain orientation. There are two domain states with the same [F_1]; one has to be chosen as [{\bf S}_1]. Subscripts of generators in the group symbol specify their orientation in the Cartesian (rectangular) crystallophysical coordinate system of the group [K_{1j}] (see Tables 3.4.2.5[link], 3.4.2.6[link] and Figs. 3.4.2.3[link], 3.4.2.4[link]).

  • [g_{1j}^{\star}]: switching operations that specify domain pair [({\bf S}_1,g_{1j}^{\star}{\bf S}_1)=] [({\bf S}_1,{\bf S}_j)]. Subscripts of symmetry operations specify the orientation of the corresponding symmetry element in the Cartesian (rectangular) crystallophysical coordinate system of the group [K_{1j} ]. In hexagonal and trigonal systems, [x', y'] and [x'', y'' ] denote the Cartesian coordinate system rotated about the z axis through 120 and 240°, respectively, from the Cartesian coordinate axes x and y; diagonal directions are abbreviated: [p =[111]], [q=[\bar 1\bar 11]], [r=[1\bar 1\bar 1]], [s=[\bar 1 1 \bar 1]] (for further details see Tables 3.4.2.5[link] and 3.4.2.6[link], and Figs. 3.4.2.3[link] and 3.4.2.4[link]).

    All switching operations of the second order are given, switching operations of higher order are omitted. The star symbol signifies that the operation is both a transposing and a twinning operation.

  • [K_{1j}^{\star}=J_{1j}^{\star}]: twinning group of the domain pair [({\bf S}_1,{\bf S}_j)]. This group is equal to the symmetry group [J_{1j}^{\star}] of the completely transposable unordered domain pair [\{{\bf S}_1,{\bf S}_j\}] [see equation (3.4.3.24[link])]. The dichromatic symbol of the group [K^{\star}_{1j}=J_{1j}^{\star} ] designates the twin law of the non-ferroelastic domain pair [\{{\bf S}_1,{\bf S}_j \}] and the twin law of all non-ferroelastic twins with domains containing [{\bf S}_1] and [{\bf S}_j] (see Section 3.4.3.1[link]).

The second part of the table concerns the distinction and switching of domain states of the non-ferroelastic domain pair [({\bf S}_1,{\bf S}_j) =] [({\bf S}_1,g_{1j}^{\star}{\bf S}_1)].

  • [{\Gamma}_{\alpha} ]: irreducible representation of [K_{1j}] that defines the transformation properties of the principal tensor parameters of the symmetry descent [K_{1j}\supset F_1] and thus specifies the components of principal tensor parameters that are given explicitly in Table 3.1.3.1[link] , in the software GI[\star]KoBo-1 and in Kopský (2001[link]), where one replaces G by [K_{1j}].

  • Diffraction intensities: the entries in this column characterize the differences of diffraction intensities from two domain states of the domain pair:

    = signifies that the twinning operations belong to the Laue class of [F_1]. Then the reflection intensities per unit volume are the same for both domain states if anomalous scattering is zero, i.e. if Friedel's law is valid. For nonzero anomalous scattering, the intensities from the two domain states differ, but when the partial volumes of both states are equal the diffraction pattern is centrosymmetric;

    [\neq] signifies that the twinning operations do not belong to the Laue class of [F_1]. Then the reflection intesities per unit volume of the two domain states are different [for more details, see Chapter 3.3[link] ; Catti & Ferraris (1976[link]); Koch (2004[link])].

  • [\rho], [{P}_i], [g_{\mu},\ldots, ] [{Q}_{\mu\nu}]: components (in matrix notation) of important property tensors that are specified in Table 3.4.3.5[link]. The same symbol may represent several property tensors (given in the same row of Table 3.4.3.5[link]) of the same rank and intrinsic symmetry. Bold-face symbols signify polar tensors. For each type of property tensor two numbers [a|c] are given; number a in front of the vertical bar | is the number of independent covariant components (in most cases identical with Cartesian components) that have the same absolute value but different sign in domain states [{\bf S}_1 ] and [{\bf S}_j]. The number c after the vertical bar | gives the number of independent nonzero tensor parameters that have equal values in both domain states of the domain pair [({\bf S}_1,{\bf S}_j)]. These tensor components are already nonzero in the parent phase.

    The principal tensor parameters are one-dimensional and have the same absolute value but opposite sign in [{\bf S}_1] and [{\bf S}_j=g^{\star}_{1j}{\bf S}_1]. Principal tensor parameters for symmetry descents [K_{1j}\supset F_1] and the associated [\Gamma_{\alpha} ] of all non-ferroelastic domain pairs can be found for property tensors of lower rank in Table 3.1.3.1[link] and for all tensors appearing in Table 3.4.3.4[link] in the software GI[\star]KoBo-1 and in Kopský (2001[link]), where one replaces G by [K_{1j}].

    When [a\neq 0] for a polar tensor (in bold-face components), then switching fields exist in the combination given in the last column of Table 3.4.3.5[link]. Components of these fields can be determined from the explicit form of corresponding principal tensor parameters expressed in Cartesian components.

Table 3.4.3.5[link] lists important property tensors up to fourth rank. Property tensor components that appear in the column headings of Table 3.4.3.4[link] are given in the first column, where bold face is used for the polar tensors significant for specifying the switching fields appearing in schematic form in the last column. In the third and fourth columns, those propery tensors appear for which hold all the results presented in Table 3.4.3.4[link] for the symbols given in the first column of Table 3.4.3.5[link].

Table 3.4.3.5| top | pdf |
Property tensors and switching fields

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

Table 3.4.3.4[link]Other propertiesSwitching field
ComponentProperty tensorComponentProperty tensor
[\rho] Enantiomorphism [\rho] Optical rotatory power  
[{P}_i] Polarization [p_i] Pyroelectricity E
[\varepsilon_{ij}] Permittivity     EE
[u_{\mu}] Strain     [\boldsigma]
[\sigma_{\mu}] Mechanical stress      
[g_{\mu}] Optical activity      
[{d}_{i{\mu}}] Piezoelectricity [r_{ijk}] Electro-optics [{\bf E}\boldsigma]
[A_{i{\mu}}] Electrogyration      
[{s}_{\mu\nu}] Elastic compliance [c_{\mu\nu}] Elastic stiffness [\boldsigma\boldsigma]
[{Q}_{\mu\nu}] Electrostriction [\pi_{{\mu\nu}}] Piezo-optics [{\bf E}{\bf E}\boldsigma]
For contracted notation, see Section 1.1.4.10.5[link] .

We turn attention to Section 3.4.5[link] (Glossary), which describes the difference between the notation of tensor components in matrix notation given in Chapter 1.1[link] and those used in the software GI[\star]KoBo-1 and in Kopský (2001[link]).

The numbers a in front of the vertical bar | in Table 3.4.3.4[link] provide global information about the tensor distinction of two domain states and enables one to classify domain pairs. Thus, for example, the first number a in column [P_i] gives the number of nonzero components of the spontaneous polarization that differ in sign in both domain states; if [a\neq 0], this domain pair can be classified as a ferroelectric domain pair.

Similarly, the first number a in column [g_{\mu}] determines the number of independent components of the tensor of optical activity that have opposite sign in domain states [{\bf S}_1] and [{\bf S}_j]; if [a\neq 0], the two domain states in the pair can be distinguished by optical activity. Such a domain pair can be called a gyrotropic domain pair. As in Table 3.4.3.1[link] for the ferroelectric (ferroelastic) domain pairs, we can define a gyrotropic phase as a ferroic phase with gyrotropic domain pairs. The corresponding phase transition to a gyrotropic phase is called a gyrotropic phase transition (Koňák et al., 1978[link]; Wadhawan, 2000[link]). If it is possible to switch gyrotropic domain states by an external field, the phase is called a ferrogyrotropic phase (Wadhawan, 2000[link]). Further division into full and partial subclasses is possible.

One can also define piezoelectric (electro-optic) domain pairs, electrostrictive (elasto-optic) domain pairs and corresponding phases and transitions.

As we have already stated, domain states in a domain pair [({\bf S}_1,{\bf S}_j) ] differ in principal tensor parameters of the transition [K_{1j}\supset F_1 ]. These principal tensor parameters are Cartesian tensor components or their linear combinations that transform according to an irreducible representation [\Gamma_{\alpha}] specifying the primary order parameter of the transition [K_{1j}\supset F_1] (see Section 3.1.3[link] ). Owing to a special form of [K_{1j}] expressed by equation (3.4.3.42[link]), this representation is a real one-dimensional irreducible representation of [K_{1j}]. Such a representation associates +1 with operations of [F_1 ] and −1 with operations from the left coset [g^{\star}_{1j} ]. This means that the principal tensor parameters are one-dimensional and have the same absolute value but opposite sign in [{\bf S}_1] and [{\bf S}_j=g^{\star}_{1j}{\bf S}_1]. Principal tensor parameters for symmetry descents [K_{1j}\supset F_1] and associated [\Gamma_{\alpha}]'s of all non-ferroelastic domain pairs can be found for property tensors of lower rank in Table 3.1.3.1[link] and for all tensors appearing in Table 3.4.3.5[link] in the software GI[\star]KoBo-1 and in Kopský (2001[link]).

These specific properties of non-ferroelastic domain pairs allow one to formulate simple rules for tensor distinction that do not use principal tensor parameters and that are applicable for property tensors of lower rank.

  • (i) Symmetry descents [K_{1j}\supset F_1 ] of non-ferroelastic domain pairs for lower-rank property tensors lead only to the appearance of independent Cartesian morphic tensor components and not to the breaking of relations between these components. These morphic Cartesian tensor components can be found by comparing matrices of property tensors in the twinning group [K_{1j}] and the low-symmetry group [F_1] as those components that appear in [F_1] but are zero in [K_{1j}].

  • (ii) As follows from Table 3.4.3.4[link], one can always find a twinning operation that is either inversion, or a twofold axis or a mirror plane with a prominent crystallographic orientation. By applying the method of direct inspection (see Section 1.1.4.6.3[link] ), one can in most cases easily find morphic Cartesian components in the second domain state of the domain pair considered and prove that they differ only in sign.

Example 3.4.3.4. Tensor distinction of domains and switching in lead germanate.  Lead germanate (Pb5Ge3O11) undergoes a phase transition with symmetry descent [G=\bar 6 \supset 3=F_1] for which we find in Table 3.4.2.7[link], column [K_{1j} ], just one twinning group [K_{1j}={\bar 6}^{\star}], i.e. [K_{1j}^{\star}=G]. This means that there is only one G-orbit of domain pairs. Since Fam3 = Fam[\bar 6] [see Table 3.4.2.2[link] and equation (3.4.3.40[link])] this orbit comprises non-ferro­elastic domain pairs. In Table 3.4.3.4[link], we find for [F_1=3] and [F_{1j}^{\star}=\bar 6] that the two domain states differ in some components of all property tensors listed in this table. The first polar tensor is the spontaneous polarization (the pair is ferroelectric) with one component [(a=1)] that has opposite sign in the two domain states. In Table 3.1.3.1[link] , we find for [G (=K_{1j})=\bar6] and [F_1 = 3] that this component is [P_3=P_z ]. From Table 3.4.3.1[link], it follows that the state shift is electrically first order with switching field [{\bf E}=({0,0,E}_z)].

The first optical tensor, which could enable the visualization of the domain states, is the optical activity [g_\mu] with two independent components which have opposite sign in the two domain states. In the software GI[\star]KoBo-1, path: Subgroups\View\Domains or in Kopský (2001[link]) we find these components: [g_3, g_1+g_2]. Shur et al. (1989[link]) have visualized in this way the domain structure of lead germanate with excellent black and white contrast (see Fig. 3.4.3.3[link]). Other examples are given in Shuvalov & Ivanov (1964[link]) and especially in Koňák et al. (1978[link]).

[Figure 3.4.3.3]

Figure 3.4.3.3 | top | pdf |

Domain structure in lead germanate observed using a polarized-light microscope. Visualization based on the opposite sign of the optical activity coefficient in the two domain states. Courtesy of Vl. Shur, Ural State University, Ekaterinburg.

Table 3.4.3.4[link] can be used readily for twinning by merohedry [see Chapter 3.3[link] and e.g. Cahn (1954[link]); Koch (2004[link])], where it enables an easy determination of the tensor distinction of twin components and the specification of external fields for possible switching and detwinning.

Example 3.4.3.5. Tensor distinction and switching of Dauphiné twins in quartz.  Quartz undergoes a phase transition from [G=6_z2_x2_y] to [F_1=3_z2_x]. Using the same procedure as in the previous example, we come to following conclusions: There are only two domain states [{\bf S}_1], [{\bf S}_2] and the twinning group, expressing the twin law, is equal to the high-symmetry group [K_{12}^{\star}=6_z2_x2_z]. In Table 3.4.3.4[link], we find that these two states differ in one independent component of the piezoelectric tensor and in one elastic compliance component. Comparison of the matrices for [6_z2_x2_y] and [3_z2_x] (see Sections 1.1.4.10.3[link] and 1.1.4.10.4[link] ) yields the following morphic tensor components in the first domain state [{\bf S}_1]: [d^{(1)}_{11} =] [-d^{(1)}_{12}=] [-2d^{(1)}_{26}] and [s^{(1)}_{14}=] [-s^{(1)}_{24} =] [2s^{(1)}_{56}]. According to the rule given above, the values of morphic components in the second domain state [{\bf S}_2] are [d^{(2)}_{11}=] [-d^{(1)}_{11} =] [-d^{(2)}_{12}=] [d^{(1)}_{12}=] [-2d^{(2)}_{26}=] [2d^{(1)}_{26}] and [s^{(2)}_{14}=] [-s^{(1)}_{14}=] [-s^{(2)}_{24}=] [s^{(1)}_{24}=] [2s^{(2)}_{56}=] [-2s^{(1)}_{56}] [see Section 3.4.5[link] (Glossary)]. These results show that there is an elastic state shift of second order and an electromechanical state shift of second order. Nonzero components [d_{14}=-d_{25}] in [6_z2_x2_y] are the same in both domain states. Similarly, one can find five independent components of the tensor [s_{\mu\nu} ] that are nonzero in [6_z2_x2_y] and equal in both domain states. For the piezo-optic tensor [\pi_{\mu\nu} ], one can proceed in a similar way. Aizu (1973[link]) has used the ferrobielastic character of the domain pairs for visualizing domains and realizing switching in quartz. Other methods for switching and visualizing domains in quartz are known (see e.g. Bertagnolli et al., 1978[link], 1979[link]).

References

Aizu, K. (1973). Second-order ferroic state shifts. J. Phys. Soc. Jpn, 34, 121–128.
Bertagnolli, E., Kittinger, E. & Tichý, J. (1978). Observation of reversible elastic Dauphiné twinning in alpha-quartz. J. Phys. (Paris) Lett. 39, 295–297.
Bertagnolli, E., Kittinger, E. & Tichý, J. (1979). Ferrobielastic hysteresis in alpha-quartz. J. Appl. Phys. 50, 6267–6271.
Cahn, R. W. (1954). Twinned crystals. Adv. Phys. 3, 363–445.
Catti, M. & Ferraris, G. (1976). Twinning by merohedry and X-ray structure determination. Acta Cryst. A32, 163–165.
Janovec, V., Richterová, L. & Litvin, D. B. (1993). Non-ferroelastic twin laws and distinction of domains in non-ferroelastic phases. Ferroelectrics, 140, 95–100.
Koch, E. (2004). Twinning. In International Tables for Crystallography, Vol. C, Mathematical, Physical and Chemical Tables, 3rd ed., edited by E. Prince, ch. 1.3. Dordrecht: Kluwer Academic Publishers.
Kopský, V. (2001). Tensor parameters of ferroic phase transitions I. Theory and tables. Phase Transit. 73, 1–422.
Koňák, Č., Kopský, V. & Smutný, F. (1978). Gyrotropic phase transitions. J. Phys. Solid State Phys. 11, 2493–2518.
Shur, V. Ya., Gruverman, A. L., Letuchev, V. V., Rumyantsev, E. L. & Subbotin, A. L. (1989). Domain structure of lead germanate. Ferroelectrics, 98, 29–49.
Shuvalov, L. A. & Ivanov, N. R. (1964). Change in the optical activity of ferroelectric crystals on reversal of polarization. Sov. Phys. Crystallogr. 9, 290–299. (Kristallografiya, 9, 363–372.)
Wadhawan, V. K. (2000). Introduction to Ferroic Materials. The Netherlands: Gordon and Breach.








































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