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 3.4.3.5.1. Explanation of Table 3.4.3.4[link]

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

3.4.3.5.1. Explanation of Table 3.4.3.4[link]

| top | pdf |

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.

References

Catti, M. & Ferraris, G. (1976). Twinning by merohedry and X-ray structure determination. Acta Cryst. A32, 163–165.
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.








































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