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. 521-527

Section 3.4.3.6.4. Ferroelastic domain pairs with compatible domain walls, 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

3.4.3.6.4. Ferroelastic domain pairs with compatible domain walls, synoptic table

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As we have seen, for each ferroelastic domain pair for which condition (3.4.3.54[link]) for the existence of coherent domain walls is fulfilled, there exist two perpendicular equally deformed planes. On each of these planes two ferroelastic twins can be formed; these two twins are in a simple relation (one is a reversed twin of the other), have the same symmetry, and can therefore be represented by one of these twins. Then we can say that from one ferroelastic domain pair two different twins can be formed. Each of these twins represents a different `twin law' that has arisen from the initial domain pair. All four ferroelastic twins can be described in terms of mechanical twinning with the same value of the shear angle [\omega].

3.4.3.6.4.1. Explanation of Table 3.4.3.6[link]

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Table 3.4.3.6[link] presents representative domain pairs of all classes of ferroelastic domain pairs for which compatible domain walls exist. The first five columns concern the domain pair. In subsequent columns, each row splits into two rows describing the orientation of two associated perpendicular equally deformed planes and the symmetry properties of the four domain twins that can be formed from the given domain pair. We explain the meaning of each column in detail.

Table 3.4.3.6| top | pdf |
Ferroelastic domain pairs and twins with compatible domain walls

[F_1]: symmetry of domain state [{\bf S}_1]; [g_{1j}]: switching operation, [g_{1j}{\bf S}_1={\bf S}_j]; [K(F_1,g_{1j})]: twinning group, group extension of [F_1] by [g_{1j}]; Axis h: intersection of compatible walls; Equation: component B expressed as a function of strain components or lattice parameters (see end of table); Wall normals: coordinates of normals n1 and n2 of two perpendicular compatible walls, subscript e: wall is charged (see Explanation[link]); [\omega]: obliquity, for numbers (n) see end of table; [{\sf \overline J}_{1j} ]: extended layer-group symmetry of the twin and the wall; [{\underline t}_{1j}^{\star}]: non-trivial twinning operation of the twin; [{\sf T}_{1j}]: layer-group symmetry of the twin and the wall, twin law of the ferroelastic twin; Classification: classification of the twin and the wall (see Table 3.4.4.3[link]).

[F_1][g_{1j}][K(F_1, g_{1j} )]Axis hEquationWall normals n[\omega][{\sf \overline J}_{1j}][\underline t^{\star}_{1j}][{\sf T}_{1j}]Classification
[1] [2^{\star}_z] [2^{\star}_z] [[B\bar{1}0]] (a)   [[001]] (1) [2^{\star}_z]   1 [{\rm AR}^{\star}]
[[1B0]_e] [\underline2^{\star}_z] [\underline 2^{\star}_z] [\underline 2^{\star}_z] SI
[1] [m^{\star}_z] [m^{\star}_z] [[B\bar{1}0]] (a)   [[001]_e] (1) [{\underline m}^{\star}_z] [{\underline m}^{\star}_z] [{\underline m}^{\star}_z] SI
[[1B0]] [m^{\star}_z]   1 [{\rm AR}^{\star}]
[\bar{1}] [m^{\star}_z, 2^{\star}_z] [2^{\star}_z/m^{\star}_z] [[B\bar{1}0]] (a)   [[001]] (1) [2_z^{\star}/\underline{m}^{\star}_z] [\underline{m}^{\star}_z] [\underline{m}^{\star}_z] SR
[[1B0]] [\underline2^{\star}_z/m^{\star}_z] [\underline2^{\star}_z] [\underline2^{\star}_z] SR
[2_z] [2^{\star}_x,2^{\star}_y] [2^{\star}_x2^{\star}_y2_z] [[001]]     [[100]] (2) [2^{\star}_x\underline2^{\star}_y\underline2_z] [\underline2^{\star}_y] [\underline2^{\star}_y] SR
[[010]] [\underline2^{\star}_x2^{\star}_y\underline2_z] [\underline2^{\star}_x] [\underline2^{\star}_x] SR
[2_z] [m^{\star}_x,m^{\star}_y] [m^{\star}_xm^{\star}_y2_z] [[001]]     [[100]] (2) [\underline{m}^{\star}_xm^{\star}_y\underline2_z] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x] SR
[[010]] [m^{\star}_x\underline{m}^{\star}_y\underline2_z] [\underline{m}^{\star}_y] [\underline{m}^{\star}_y] SR
[2_z] [4^{\star}_z,4_z^{3 \star}] [4^{\star}_z] [[001]] (b) [\Bigl[] [[1B0]] (3) [\underline2_z]   1 AR
[[B\bar{1}0]] [\underline2_z] 1 AR
[2_z] [\bar4^{\star}_z,\bar4_z^{*3}] [\bar{4}^{\star}_z] [[001]] (b) [\Bigl[] [[1B0]] (3) [\underline2_z]   1 AR
[[B\bar{1}0]] [\underline2_z] 1 AR
[2_z] [3_z,6_z^5] [6_{z}] [[001]] (c)   [[1B0]] (4) [\underline2_z]   1 AR
[[B\bar{1}0]] [\underline2_z] 1 AR
[3_z^2,6_z] [6_z] [[001]] (c) [[1B0]] (4) [\underline2_z] 1 AR
[[B\bar{1}0]] [\underline2_z] 1 AR
[2_z] [\bar3_z^5,\bar6_z] [6_{z}/m_z] [[001]] (c)   [[1B0]] (4) [\underline2_z]   1 AR
[[B\bar{1}0]] [\underline2_z] 1 AR
[\bar3_z,\bar6_z^5] [6_z/m_z] [[001]] (c) [[1B0]] (4) [\underline2_z] 1 AR
[[B\bar{1}0]] [\underline2_z] 1 AR
[2_x] [2^{\star}_{xy},4_z] [4_z2_x2_{xy}] [[\bar{B}B2]] (d)   [[110]] (5) [2^{\star}_{xy}]   1 [{\rm AR}^{\star}]
[[1\bar{1}B]_e] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] SI
[2_x] [m^{\star}_{xy},\bar4_z] [\bar4_z2_xm_{xy}] [[\bar{B}B2]] (d)   [[110]_e] (5) [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] SI
[[1\bar{1}B]] [m^{\star}_{xy}]   1 [{\rm AR}^{\star}]
[2_x] [2^{\star}_{x'},3_z^2] [3_z2_x] [[\sqrt{3}B,B,\bar4]] (e)   [[\bar1\sqrt{3}0]] (6) [2^{\star}_{x'}]   1 [{\rm AR}^{\star}]
[[\sqrt{3}1B]_e] [\underline2^{\star}_{x'}] [\underline2^{\star}_{x'}] [\underline2^{\star}_{x'}] SI
[2_x] [m^{\star}_{x'},\bar3_z^5] [\bar{3}_zm_x] [[\sqrt{3}B,B,\bar4]] (e)   [[\bar1\sqrt{3}0]_e] (6) [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] SI
[[\sqrt{3}1B]] [m^{\star}_{x'}]   1 [{\rm AR}^{\star}]
[2_x] [2^{\star}_{y'},6_z] [6_z2_x2_y] [[\bar{B},\sqrt{3}B,\bar4]] (f)   [[\sqrt{3}10]] (7) [2^{\star}_{y'}]   1 [{\rm AR}^{\star}]
[[\bar1\sqrt{3}B]_e] [\underline2^{\star}_{y'}] [\underline2^{\star}_{y'}] [\underline2^{\star}_{y'}] SI
[2_x] [m^{\star}_{y'},\bar6_z] [\bar{6}_z2_xm_y] [[\bar{B},\sqrt{3}B,\bar4]] (f)   [[\sqrt{3}10]_e] (7) [\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] SI
[[\bar1\sqrt{3}B]] [m^{\star}_{y'}]   1 [{\rm AR}^{\star}]
[2_{xy}] [m^{\star}_{x},\bar4_z^{3}] [\bar4_zm_x2_{xy}] [[0B\bar{1}]] (g)   [[100]_e] (8) [\underline{m}^{\star}_{x}] [\underline{m}^{\star}_{x}] [\underline{m}^{\star}_{x}] SI
[[01B]] [m^{\star}_{x}]   1 [{\rm AR}^{\star}]
[m_z] [m^{\star}_x,2^{\star}_y] [m^{\star}_x2^{\star}_ym_z] [[001]]     [[100]_e] (2) [\underline{m}^{\star}_x\underline2^{\star}_ym_z] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x\underline2^{\star}_ym_z] SI
[[010]] [m_x^{\star}2_y^{\star}m_z]   [m_z] [{\rm AR}^{\star}]
[m_{z}] [4_z,\bar4_z^3] [4_z/m_{z}] [[001]] (b)   [[1B0]_{e0}] (3) [m_z]   [m_z] AI
[[B\bar{1}0]_{0e}] [m_z]   [m_z] AI
[4_z^3,\bar4_z] [4_z/m_z] [[001]] (b) [[1B0]_{e0}] (3) [m_z]   [m_z] AI
[[B\bar{1}0]_{0e}] [m_z]   [m_z] AI
[m_z] [3_z,\bar6^5_z] [\bar6_{z}] [[001]] (c)   [[1B0]_{e0}] (4) [m_z]   [m_z] AI
[[B\bar{1}0]_{0e}] [m_z]   [m_z] AI
[3^2_z,\bar6_z] [\bar6_z] [[001]] (c) [[1B0]_{e0}] (4) [m_z]   [m_z] AI
[[B\bar{1}0]_{0e}] [m_z]   [m_z] AI
[m_z] [\bar3_z,6_z^5] [6_{z}/m_z] [[001]] (c)   [[1B0]_{e0}] (4) [m_z]   [m_z] AI
[[B\bar{1}0]_{0e}] [m_z]   [m_z] AI
[\bar3_z^5,6_z] [6_z/m_z] [[001]] (c) [[1B0]_{e0}] (4) [m_z]   [m_z] AI
[[B\bar{1}0]_{0e}] [m_z]   [m_z] AI
[m_x] [m^{\star}_{xy},4_z] [4_zm_xm_{xy}] [[\bar{B}B2]] (d)   [[110]_e] (5) [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] SI
[[1\bar{1}B]] [m^{\star}_{xy}]   1 [{\rm AR}^{\star}]
[m_x] [2^{\star}_{xy},\bar4_z] [\bar{4}_zm_x2_{xy}] [[\bar{B}B2]] (d)   [[110]] (5) [2^{\star}_{xy}]   1 [{\rm AR}^{\star}]
[[1\bar{1}B]_e] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] SI
[m_x] [m^{\star}_{x'},3_z^2] [3_zm_x] [[\sqrt{3}B,B,\bar4]] (e)   [[\bar1\sqrt{3}0]_e] (6) [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] SI
[[\sqrt{3}1B]] [m^{\star}_{x'}]   1 [{\rm AR}^{\star}]
[m_x] [2^{\star}_{x'},\bar3_z^5] [\bar3_zm_x] [[\sqrt{3}B,B,\bar4]] (e)   [[\bar1\sqrt{3}0]] (6) [2^{\star}_{x'}]   1 [{\rm AR}^{\star}]
[[\sqrt{3}1B]_e] [\underline2^{\star}_{x'}] [\underline2^{\star}_{x'}] [\underline2^{\star}_{x'}] SI
[m_x] [m^{\star}_{y'},6_z] [6_zm_xm_y] [[\bar{B},\sqrt{3}B,\bar4]] (f)   [[\sqrt{3}10]_e] (6) [\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] SI
[[\bar1\sqrt{3}B]] [m^{\star}_{y'}]   1 [{\rm AR}^{\star}]
[m_x] [2^{\star}_{y'},\bar6_z] [\bar{6}_zm_x2_y] [[\bar{B},\sqrt{3}B,\bar4]] (f)   [[\sqrt{3}10]] (6) [2^{\star}_{y'}]   1 [{\rm AR}^{\star}]
[[\bar1\sqrt{3}B]_e] [\underline2^{\star}_{y'}] [\underline2^{\star}_{y'}] [\underline2^{\star}_{y'}] SI
[m_{xy}] [2^{\star}_{x},\bar4^3_z] [\bar{4}_z2_xm_{xy}] [[0B\bar{1}]] (h)   [[100]] (9) [2^{\star}_{x}]   1 [{\rm AR}^{\star}]
[[01B]_e] [\underline2^{\star}_{x}] [\underline2^{\star}_{x}] [\underline2^{\star}_{x}] SI
[2_z/m_z] [m^{\star}_x,m^{\star}_y] [m^{\star}_xm^{\star}_ym_z] [[001]]     [[100]] (2) [\underline{m}^{\star}_xm^{\star}_ym_z] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x\underline2^{\star}_ym_z] SR
[[010]] [m^{\star}_x\underline{m}^{\star}_ym_z] [\underline{m}^{\star}_y] [\underline2^{\star}_x\underline{m}^{\star}_ym_z] SR
[2_z/m_z] [4^{\star}_z,4^{3 \star}_z] [4^{\star}_z/m_z] [[001]] (b) [\Bigl[] [[1B0]] (3) [\underline2_z/m_z]   [m_z] AR
[[B\bar{1}0]] [\underline2_z/m_z]   [m_z] AR
[2_z/m_z] [3_z,6_z^5] [6_{z}/m_z] [[001]] (c)   [[1B0]] (4) [\underline2_z/m_z]   [m_z] AR
[[B\bar{1}0]] [\underline2_z/m_z]   [m_z] AR
[3_z^2,6_z] [6_z/m_z] [[001]] (c) [[1B0]] (4) [\underline2_z/m_z]   [m_z] AR
[[B\bar{1}0]] [\underline2_z/m_z]   [m_z] AR
[2_x/m_x] [m^{\star}_{xy},4_z] [4_z/m_zm_xm_{xy}] [[\bar{B}B2]] (d)   [[110]] (5) [2^{\star}_{xy}/\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] SR
[[1\bar{1}B]] [\underline2^{\star}_{xy}/m^{\star}_{xy}] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] SR
[2_x/m_x] [m^{\star}_{x'},3_z^2] [\bar{3}_zm_x] [[\sqrt{3}B,B,\bar4]] (e)   [[\bar1\sqrt{3}0]] (6) [2^{\star}_{x'}/\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] SR
[[\sqrt{3}1B]] [\underline2^{\star}_{x'}/m^{\star}_{x'}] [\underline2^{\star}_{x'}] [\underline2^{\star}_{x'}] SR
[2_x/m_x] [m^{\star}_{y'},6_z] [6_z/m_zm_xm_y] [[\bar{B},\sqrt{3}B,\bar4]] (f)   [[\sqrt{3}10]] (6) [2^{\star}_{y'}/\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] SR
[[\bar1\sqrt{3}B]] [\underline2^{\star}_{y'}/m^{\star}_{y'}] [\underline2^{\star}_{y'}] [\underline2^{\star}_{y'}] SR
[2_x2_y2_z] [2^{\star}_{x\bar{y}}, 2^{\star}_{xy}] [4^{\star}_z2_x2^{\star}_{xy}] [[001]]   [\Bigl[] [[110]] (11) [2^{\star}_{xy}\underline2^{\star}_{x\bar{y}}\underline2_z] [\underline2^{\star}_{x\bar{y}}] [\underline2^{\star}_{x\bar{y}}] SR
[[1\bar10]] [\underline2^{\star}_{xy}2^{\star}_{x\bar{y}}\underline2_z] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] SR
[2_x2_y2_z] [m^{\star}_{x\bar{y}},m^{\star}_{xy}] [\bar{4}^{\star}_z2_xm^{\star}_{xy}] [[001]]   [\Bigl[] [[110]] (11) [\underline{m}^{\star}_{xy}m^{\star}_{x\bar{y}}\underline2_z] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] SR
[[1\bar10]] [m^{\star}_{xy}\underline{m}^{\star}_{x\bar{y}}\underline2_z] [\underline{m}^{\star}_{x\bar{y}}] [\underline{m}^{\star}_{x\bar{y}}] SR
[2_x2_y2_z] [2^{\star}_{x'},2^{\star}_{y'}] [6_z2_x2_y] [[001]]     [[\bar1\sqrt{3}0]] (10) [2^{\star}_{x'}\underline2^{\star}_{y'}\underline2_z] [\underline2^{\star}_{y'}] [\underline2^{\star}_{y'}] SR
[[\sqrt{3}10]] [\underline2^{\star}_{x'}2^{\star}_{y'}\underline2_z] [\underline2^{\star}_{x'}] [\underline2^{\star}_{x'}] SR
[2_x2_y2_z] [m^{\star}_{x'},m^{\star}_{y'}] [6_z/m_zm_xm_y] [[001]]     [[\bar1\sqrt{3}0]] (10) [\underline{m}^{\star}_{x'}m^{\star}_{y'}\underline2_z] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] SR
[[\sqrt{3}10]] [{m}^{\star}_{x'}\underline{m}^{\star}_{y'}\underline2_z] [\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] SR
[2_{x\bar{y}}2_{xy}2_z] [m^{\star}_{x},m^{\star}_{y}] [\bar{4}^{\star}_zm_x2^{\star}_{xy}] [[001]]   [\Bigl[] [[100]] (13) [\underline{m}^{\star}_{x}m^{\star}_{y}\underline2_z] [\underline{m}^{\star}_{x}] [\underline{m}^{\star}_{x}] SR
[[010]] [m^{\star}_{x}\underline{m}^{\star}_{y}\underline2_z] [\underline{m}^{\star}_{y}] [\underline{m}^{\star}_{y}] SR
[2_{x\bar{y}}2_{xy}2_z] [2^{\star}_{xz},4_y] [4_z3_{p}2_{xy}] [[B2\bar{B}]] (k)   [[101]] (12) [2^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[[\bar{1}B1]] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] SI
[2_{x\bar{y}}2_{xy}2_z] [m_{xz}^{\star},\bar4_y] [m_z\bar{3}_{p}m_{xy}] [[B2\bar{B}]] (k)   [[101]] (12) [\underline{m}_{xz}^{\star}] [\underline{m}_{xz}^{\star}] [\underline{m}_{xz}^{\star}] SI
[[\bar{1}B1]] [m_{xz}^{\star}]   1 [{\rm AR}^{\star}]
[m_xm_y2_z] [m^{\star}_{x\bar{y}},m^{\star}_{xy}] [4^{\star}_zm_xm^{\star}_{xy}] [[001]]   [\Bigl[] [[110]] (11) [m^{\star}_{x\bar{y}}\underline{m}^{\star}_{xy}\underline2_z] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] SR
[[1\bar10]] [\underline{m}^{\star}_{x\bar{y}}{m}^{\star}_{xy}\underline2_z] [\underline{m}^{\star}_{x\bar{y}}] [\underline{m}^{\star}_{x\bar{y}}] SR
[m_xm_y2_z] [2^{\star}_{x\bar{y}},2^{\star}_{xy}] [\bar{4}^{\star}_zm_x2^{\star}_{xy}] [[001]]   [\Bigl[] [[110]] (11) [2^{\star}_{xy}\underline2^{\star}_{x\bar{y}}\underline2_z] [\underline2^{\star}_{x\bar{y}}] [\underline2^{\star}_{x\bar{y}}] SR
[[1\bar10]] [\underline2^{\star}_{xy}2^{\star}_{x\bar{y}}\underline2_z] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] SR
[m_xm_y2_z] [m^{\star}_{x'},m^{\star}_{y'}] [6_zm_xm_y] [[001]]     [[\bar1\sqrt{3}0]] (10) [\underline{m}^{\star}_{x'}m^{\star}_{y'}\underline2_z] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}] SR
[[\sqrt{3}10]] [{m}^{\star}_{x'}\underline{m}^{\star}_{y'}\underline2_z] [\underline{m}^{\star}_{y'}] [\underline{m}^{\star}_{y'}] SR
[m_xm_y2_z] [2^{\star}_{x'},2^{\star}_{y'}] [6_z/m_zm_xm_y] [[001]]     [[\bar1\sqrt{3}0]] (10) [2^{\star}_{x'}\underline2^{\star}_{y'}\underline2_z] [\underline2^{\star}_{y'}] [\underline2^{\star}_{y'}] SR
[[\sqrt{3}10]] [\underline2^{\star}_{x'}2^{\star}_{y'}\underline2_z] [\underline2^{\star}_{x'}] [\underline2^{\star}_{x'}] SR
[m_x2_ym_z] [m^{\star}_{x'},2^{\star}_{y'}] [\bar{6}_zm_x2_y] [[001]]     [[\bar1\sqrt{3}0]_e] (10) [\underline{m}^{\star}_{x'}\underline2^{\star}_{y'}m_z] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}\underline2^{\star}_{y'}m_z] SI
[[\sqrt{3}10]] [{m}^{\star}_{x'}2^{\star}_{y'}m_z]   [m_z] [{\rm AR}^{\star}]
[2_xm_ym_z] [m^{\star}_{x\bar{y}},2^{\star}_{xy}] [4_z/m_zm_xm_{xy}] [[001]]     [[110]] (11) [2^{\star}_{xy}m^{\star}_{x\bar{y}}m_z]   [m_z] [{\rm AR}^{\star}]
[[1\bar10]_e] [\underline2^{\star}_{xy}\underline{m}^{\star}_{x\bar{y}}m_z] [\underline{m}^{\star}_{x\bar{y}}] [\underline2^{\star}_{xy}\underline{m}^{\star}_{x\bar{y}}m_z] SI
[2_xm_ym_z] [m^{\star}_{y'},2^{\star}_{x'}] [\bar{6}_z2_xm_y] [[001]]     [[\bar1\sqrt{3}0]] (10) [2^{\star}_{x'}m^{\star}_{y'}m_z]   [m_z] [{\rm AR}^{\star}]
[[\sqrt{3}10]_e] [\underline{2}^{\star}_{x'}\underline{m}^{\star}_{y'}m_z] [\underline{m}_{y'}] [\underline{2}^{\star}_{x'}\underline{m}^{\star}_{y'}m_z] SI
[2_xm_ym_z] [m^{\star}_{x'},2^{\star}_{y'}] [6_z/m_zm_xm_y] [[001]]     [[\bar1\sqrt{3}0]_e] (10) [\underline{m}^{\star}_{x'}\underline2^{\star}_{y'}m_z] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}\underline2^{\star}_{y'}m_z] SI
[[\sqrt{3}10]] [m^{\star}_{x'}2^{\star}_{y'}m_z]   [m_z] [{\rm AR}^{\star}]
[m_{x\bar{y}}m_{xy}2_z] [2^{\star}_{x},2^{\star}_{y}] [\bar{4}^{\star}_z2^{\star}_xm_{xy}] [[001]]   [\Bigl[] [[100]] (13) [2^{\star}_{x}\underline2^{\star}_{y}\underline2_z] [\underline2^{\star}_{y}] [\underline2^{\star}_{y}] SR
[[010]] [\underline2^{\star}_{x}2^{\star}_{y}\underline2_z] [\underline2^{\star}_{x}] [\underline2^{\star}_{x}] SR
[m_{x\bar{y}}m_{xy}2_z] [m^{\star}_{xz},\bar4_y] [\bar{4}_z3_{p}m_{xy}] [[B2\bar{B}]] (k)   [[101]_e] (12) [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SI
[[\bar{1}B1]] [m^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[m_{x\bar{y}}m_{xy}2_z] [2^{\star}_{xz},4_y] [m_z\bar{3}_{p}m_{xy}] [[B2\bar{B}]] (k)   [[101]] (12) [2^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[[\bar{1}B1]_e] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] SI
[m_{x\bar{y}}2_{xy}m_z] [m^{\star}_{xz},4_y] [m_z\bar{3}_{p}m_{xy}(m^{\star}_{xz})] [[B2\bar{B}]] (k)   [[101]_e] (12) [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SI
[[\bar{1}B1]] [m^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[m_{x\bar{y}}2_{xy}m_z] [2^{\star}_{xz},\bar4_y] [m_z\bar{3}_{p}m_{xy}(2^{\star}_{xz})] [[B2\bar{B}]] (k)   [[101]] (12) [2^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[[\bar{1}B1]_e] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] SI
[m_xm_ym_z] [m^{\star}_{xy},m^{\star}_{x\bar{y}}] [4^{\star}_z/m_zm_xm^{\star}_{xy}] [[001]]   [\Bigl[] [[110]] (10) [m^{\star}_{x\bar{y}}\underline{m}^{\star}_{xy}m_z] [\underline{m}^{\star}_{xy}] [\underline2^{\star}_{x\bar{y}}\underline{m}^{\star}_{xy}m_z] SR
[[1\bar10]] [\underline{m}^{\star}_{x\bar{y}}{m}^{\star}_{xy}m_z] [\underline{m}^{\star}_{x\bar{y}}] [\underline{m}^{\star}_{x\bar{y}}\underline{2}^{\star}_{xy}m_z] SR
[m_xm_ym_z] [m^{\star}_{x'},m^{\star}_{y'}] [6_z/m_zm_xm_y] [[001]]     [[\bar1\sqrt{3}0]] (10) [\underline{m}^{\star}_{x'}m^{\star}_{y'}m_z] [\underline{m}^{\star}_{x'}] [\underline{m}^{\star}_{x'}\underline2^{\star}_{y'}m_z] SR
[[\sqrt{3}10]] [m^{\star}_{x'}\underline{m}^{\star}_{y'}m_z] [\underline{m}^{\star}_{y'}] [\underline{2}^{\star}_{x'}\underline{m}^{\star}_{y'}m_z] SR
[m_{xy}m_{\bar{x}y}m_z] [m^{\star}_{xz},4_y] [m_z\bar{3}_{p}m_{xy}] [[B2\bar{B}]] (k)   [[101]] (12) [2^{\star}_{xz}/\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SR
[[\bar{1}B1]] [\underline2^{\star}_{xz}/m^{\star}_{xz}] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] SR
[4_z] [2^{\star}_{xz},4_y] [4_z3_{p}2_{xy}] [[010]]     [[101]] (14) [2^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[[\bar101]_e] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] SI
[4_z] [m^{\star}_{xz},\bar{4}_y] [m_z\bar{3}_{p}m_{xy}] [[010]]     [[101]_e] (14) [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SI
[[\bar101]] [m^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[\bar{4}_z] [m^{\star}_{xz},\bar4_y] [\bar{4}_z3_{p}m_{xy}] [[010]]     [[101]] (14) [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SI
[[\bar101]] [m^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[\bar{4}_z] [2^{\star}_{xz},4_y] [m_z\bar{3}_{p}m_{xy}] [[010]]     [[101]] (14) [2^{\star}_{xz}]   1 [{\rm AR}^{\star}]
[[\bar101]_e] [\underline2_{xz}^{\star}] [\underline2_{xz}^{\star}] [\underline2_{xz}^{\star}] SI
[4_z/m_z] [m^{\star}_{xz},4_y] [m_z\bar{3}_{p}m_{xy}] [[010]]     [[101]] (14) [2^{\star}_{xz}/\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SR
[[\bar101]] [\underline2^{\star}_{xz}/m^{\star}_{xz}] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] SR
[4_z2_x2_{xy}] [2^{\star}_{xz},2^{\star}_{x\bar{z}}] [4_z3_{p}2_{xy}] [[010]]   [\Bigl[] [[101]] (14) [2^{\star}_{xz}\underline2^{\star}_{x\bar{z}}\underline2_{y}] [\underline2^{\star}_{\bar{x}z}] [\underline2^{\star}_{\bar{x}z}] SR
[[\bar101]] [\underline2^{\star}_{xz}2^{\star}_{x\bar{z}}\underline2_y] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}] SR
[4_z2_x2_{xy}] [m^{\star}_{xz},m^{\star}_{x\bar{z}}] [m_z\bar{3}_{p}m_{xy}] [[010]]   [\Bigl[] [[101]] (14) [\underline{m}^{\star}_{xz}m^{\star}_{x\bar{z}}\underline2_y] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SR
[[\bar101]] [m^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}\underline2_y] [\underline{m}^{\star}_{x\bar{z}}] [\underline{m}^{\star}_{x\bar{z}}] SR
[4_zm_xm_{xy}] [m^{\star}_{x\bar{z}},2^{\star}_{xz}] [m_z\bar{3}_{p}m_{xy}] [[010]]     [[101]] (14) [2^{\star}_{xz}m^{\star}_{x\bar{z}}m_y]   [m_y] [{\rm AR}^{\star}]
[[\bar101]_e] [\underline2^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}m_y] [\underline2^{\star}_{xz}] [\underline2^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}m_y] SI
[\bar{4}_z2_xm_{xy}] [m^{\star}_{xz},m^{\star}_{x\bar{z}}] [\bar{4}_z3_{p}m_{xy}] [[010]]   [\Bigl[] [[101]] (14) [\underline{m}^{\star}_{xz}m^{\star}_{x\bar{z}}\underline2_y] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}] SR
[[\bar101]] [m^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}\underline2_y] [\underline{m}^{\star}_{x\bar{z}}] [\underline{m}^{\star}_{x\bar{z}}] SR
[\bar{4}_zm_x2_{xy}] [m_{x\bar{z}}^{\star},2_{xz}^{\star}] [m_z\bar{3}_{p}m_{xy}] [[010]]     [[101]] (14) [2^{\star}_{xz}m^{\star}_{x\bar{z}}m_y]   [m_y] [{\rm AR}^{\star}]
[[\bar101]] [\underline2^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}m_y] [\underline{m}^{\star}_{x\bar{z}}] [\underline2^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}m_y] SR
[\bar{4}_z2_xm_{xy}] [2^{\star}_{xz},2^{\star}_{x\bar{z}}] [m_z\bar{3}_{p}m_{xy}] [[010]]     [[101]] (14) [2^{\star}_{xz}2^{\star}_{x\bar{z}}2_y] [\underline2^{\star}_{x\bar{z}}] [\underline2^{\star}_{x\bar{z}}] SR
[[\bar101]] [\underline2^{\star}_{xz}\underline{2}^{\star}_{x\bar{z}}2_y] [\underline{2}^{\star}_{x\bar{z}}] [\underline2^{\star}_{xz}\underline{2}^{\star}_{x\bar{z}}2_y] SI
[4_z/m_zm_xm_{xy}] [m^{\star}_{xz},m^{\star}_{\bar{x}z}] [m_z\bar{3}_{p}m_{xy}] [[010]]   [\Bigl[] [[101]] (14) [\underline{m}^{\star}_{xz}m^{\star}_{\bar{x}z}m_y] [\underline{m}^{\star}_{xz}] [\underline{m}^{\star}_{xz}\underline{2}^{\star}_{\bar{x}z}m_y] SR
[[\bar101]] [m^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}m_y] [\underline{m}^{\star}_{x\bar{z}}] [\underline2^{\star}_{xz}\underline{m}^{\star}_{x\bar{z}}m_y] SR
[3_{p}] [2^{\star}_{x},3_r] [2_z 3_{p}] [[01\bar1]]     [[100]] (15) [2^{\star}_x]   1 [{\rm AR}^{\star}]
[[011]_e] [\underline2^{\star}_x] [\underline2^{\star}_x] [\underline2^{\star}_x] SI
[3_{p}] [m^{\star}_{x},\bar3_r] [m_z \bar3_{p}] [[01\bar1]]     [[100]_e] (15) [\underline{m}^{\star}_x] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x] SI
[[011]] [m^{\star}_x]   1 [{\rm AR}^{\star}]
[3_{p}] [2^{\star}_{xy},4_y] [4_z 3_{p} 2_{xy}] [[1\bar10]]     [[001]_e] (15) [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] SI
[[110]] [2^{\star}_{xy}]   1 [{\rm AR}^{\star}]
[3_{p}] [m^{\star}_{xy},\bar4_y] [\bar4_z3_{p}m_{xy}] [[1\bar10]]     [[001]] (15) [m^{\star}_{xy}]   1 [{\rm AR}^{\star}]
[[110]_e] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] SI
[\bar3_{p}] [m^{\star}_{x},3_r] [m_z\bar3_{p}] [[01\bar1]]     [[100]] (15) [2^{\star}_x/\underline{m}^{\star}_x] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x] SR
[[011]] [\underline2^{\star}_x/m^{\star}_x] [\underline2^{\star}_x] [\underline2^{\star}_x] SR
[\bar3_{p}] [m^{\star}_{xy},4_y] [m_z\bar3_{p}m_{xy}] [[1\bar10]]     [[001]] (15) [\underline2^{\star}_{xy}/m^{\star}_{xy}] [\underline2^{\star}_{xy}] [\underline2^{\star}_{xy}] SR
[[110]] [2^{\star}_{xy}/\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] [\underline{m}^{\star}_{xy}] SR
[3_{p} 2_{x\bar{y}}] [2^{\star}_{x},2^{\star}_{yz}] [4_z3_{p} 2_{xy}] [[01\bar1]]     [[100]] (15) [2^{\star}_x\underline2^{\star}_{yz}\underline2_{y\bar{z}}] [\underline2^{\star}_{yz}] [\underline2^{\star}_{yz}] SR
[[011]] [\underline2^{\star}_x2^{\star}_{yz}\underline2_{y\bar{z}}] [\underline2^{\star}_x] [\underline2^{\star}_x] SR
[3_{p} 2_{x\bar{y}}] [m^{\star}_{x},m^{\star}_{yz}] [m_z\bar3_{p}m_{xy}] [[01\bar1]]     [[100]] (15) [\underline{m}^{\star}_xm_{yz}^{\star}\underline2_{y\bar{z}}] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x] SR
[[011]] [m^{\star}_x\underline{m}^{\star}_{yz}\underline2_{y\bar{z}}] [\underline{m}^{\star}_{yz}] [\underline{m}^{\star}_{yz}] SR
[3_{p} m_{x\bar{y}}] [2^{\star}_{x},m^{\star}_{yz}] [\bar4_z3_{p}m_{xy}] [[01\bar1]]     [[100]] (15) [m^{\star}_{yz}m_{y\bar{z}}2^{\star}_x]   [m_{y\bar{z}}] [{\rm AR}^{\star}]
[[011]_e] [\underline{m}^{\star}_{yz}m_{y\bar{z}}\underline2^{\star}_x] [\underline{m}^{\star}_{yz}] [\underline{m}^{\star}_{yz}m_{y\bar{z}}\underline2^{\star}_x] SI
[3_{p} m_{x\bar{y}}] [m^{\star}_x,2^{\star}_{yz}] [m_z\bar3_{p}m_{xy}] [[01\bar1]]     [[100]_e] (15) [\underline{m}^{\star}_x\underline2^{\star}_{yz}m_{y\bar{z}}] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x\underline2^{\star}_{yz}m_{y\bar{z}}] SI
[[011]] [m^{\star}_x2^{\star}_{yz}m_{y\bar{z}}]   [m_{y\bar{z}}] [{\rm AR}^{\star}]
[\bar3_{p}m_{x\bar{y}}] [m^{\star}_x,m^{\star}_{yz}] [m_z \bar3_{p}m_{xy}] [[01\bar1]]     [[100]] (15) [\underline{m}^{\star}_xm^{\star}_{yz}m_{y\bar{z}}] [\underline{m}^{\star}_x] [\underline{m}^{\star}_x\underline2^{\star}_{yz}m_{y\bar{z}}] SR
[[011]] [m^{\star}_x\underline{m}^{\star}_{yz}m_{y\bar{z}}] [\underline{m}^{\star}_{yz}] [\underline2^{\star}_x\underline{m}^{\star}_{yz}m_{y\bar{z}}] SR

Expressions for obliquity [\omega] as a function of spontaneous strain components and lattice parameters

 [\omega] as a function of spontaneous strain components [\omega] as a function of lattice parameters a, b, c; [\alpha = \angle (b;c)], [\beta = \angle (a;c)], [\gamma = \angle (a;b)]
Expression[\pmatrix{q&v&u\cr v&r& t\cr u&t&s\cr}] [Scheme scheme1]
(1) [\omega= 2 \sqrt{t^2+u^2}] [\omega=\left|\arccos {{\displaystyle \sqrt{1-\cos^2\alpha -\cos^2\beta -\cos^2\gamma +2\cos\alpha \cos\beta \cos\gamma }}\over{\displaystyle \sin\gamma }} \right|]
(2) [\omega= 2|v|] [\omega= |\pi /2 -\gamma |]
(3) [\omega=\sqrt{(q-r)^2+4v^2}] [\omega=\left |\arcsin {{\displaystyle \sqrt{(2ab\cos\gamma)^2+(b^2-a^2)}}\over{\displaystyle b^2-a^2}}\right|]
(4) [\omega={{\sqrt3}\over{2}}\sqrt{(q-r)^2+4v^2}] [\omega= \left|\pi/2-\psi_1- \psi_2\right|]
    [\psi _1={\rm arccotan}{{\displaystyle c^2(a^2+b^2–2d^2)-(a^2-b^2)^2+D(a^2-b^2-d^2)}\over {\displaystyle (D-b^2+d^2)\sqrt{4a^2d^2-(a^2-b^2-d^2)^2}}}]
    [\psi_2={\rm arccotan}{{\displaystyle b^2(a^2-b^2)+d^2(a^2-b^2)-D(a^2-b^2+d^2)}\over {\displaystyle (D-b^2+d^2)\sqrt{4a^2d^2-(a^2-b^2-d^2)^2}}}]
    [D=\sqrt{(a^2-d^2)^2-(a^2-b^2)(b^2-d^2)}]
(5) [\omega=\sqrt{(q-r)^2+2t^2}] [\omega=\left|\arcsin {{\displaystyle c^2(a^2+b^2)\sin^2 \alpha -b^2(a+c\cos\alpha)(2Da+c\cos\alpha)}\over {\displaystyle \sqrt{c^2(a^2+b^2) \sin^2\alpha +b^2(a+c\cos\alpha )^2} \sqrt{ c^2(a^2+b^2)\sin^2\alpha+b^2(2Da+c\cos\alpha)^2}}}\right|]
    [D={{\displaystyle ac\cos\alpha }\over {\displaystyle b^2-a^2}}]
(6) [\omega={{\sqrt3}\over{2}}\sqrt{(q-r)^2+4t^2}] [\omega=\left|\arcsin {{\displaystyle (4a^2-b^2)\left[\left(1-{ {c\cos\beta }\over{a+b}}\right)b\cos\beta -{{c}\over{2}}\right]+{{3cb^2\sin^2\beta }\over{2}}}\over {\displaystyle \sqrt{4a^2-b^2(1–9\sin^2\beta )} \sqrt {(ac\sin\beta)^2+(4a^2-b^2)\left[1-{{c\cos\beta }\over{a+b}})b-{{c\cos\beta }\over{2}}\right]^2}}}\right| \quad(^*)]
(7) [\omega=2|t|] [\omega=|\pi /2 -\alpha |]
(8) [\omega={{\sqrt3}\over{2}}\sqrt{(q-r)^2+4t^2}] [\omega=\left|\arcsin {{\displaystyle 3b^2c-c(4a^2-b^2)\sin^2\alpha +2 b^2D\sqrt{4a^2-b^2}\cos\alpha }\over {\displaystyle \sqrt{b^2+(4a^2-b^2)\sin^2\alpha }\sqrt{ 9b^2c^2+(4a^2-b^2)(c^2\sin^2\alpha +4b^2D^2)+12b^2Dc\sqrt{4a^2-b^2}}}} \right|\quad(^*)]
    [D={{\displaystyle 2ac\cos\alpha }\over {\displaystyle b^2-a^2}} ]
(9) [\omega=2\sqrt{q^2+f^2}] [\omega=\left|\arccos {{\displaystyle \sqrt{1-\cos^2\alpha -\cos^2\beta -\cos^2\gamma +2\cos\alpha \cos\beta \cos\gamma }}\over{\displaystyle \sin\alpha }}\right|]
(10) [\omega={{\sqrt3}\over{2}} |q-r| ] [\omega=\left|\arcsin{{\displaystyle b^2-a^2}\over{\displaystyle a\sqrt{2b^2+a^2}}}\right|]
(11) [\omega=|q-r| ] [\omega=\left|\arcsin {{\displaystyle a^2-b^2}\over{\displaystyle b^2+a^2}}\right|]
(12) [\omega=\sqrt{(q-s)^2+2v^2}] [\omega=\left|\arcsin {{\displaystyle c^2(D\cos\gamma -1)-a^2\sin^2\gamma }\over {\displaystyle \sqrt{c^2+a^2 \sin^2\gamma} \sqrt{ 4c^2(D^2-D\cos\gamma )+c^2+a^2\sin^2\gamma }}}\right|]
    [D={{\displaystyle 2a^2\cos\gamma }\over {\displaystyle c^2-a^2}} ]
(13) [\omega=2|v| ] [\omega= |\pi /2 -\gamma |]
(14) [\omega=|q-s|] [\omega=\left|\arcsin {{\displaystyle a^2-c^2}\over{\displaystyle c^2+a^2}}\right|]
(15) [\omega=2\sqrt2 |v|] [\omega=\left|\arcsin {{\displaystyle \sqrt 2\cos\alpha }\over {\displaystyle 1+\cos\alpha}} \right|]

Expressions for component B of wall normal as a function of spontaneous strain components and lattice parameters

EquationB as a function of spontaneous strain components [\pmatrix{q&v&u\cr v&r& t\cr u&t&s\cr}]B as a function of lattice parameters a, b, c; [\alpha = \angle (b;c)], [\beta = \angle (a;c)], [\gamma = \angle (a;b)]
(a) [B={{\displaystyle t}\over{\displaystyle u}}]  
(b) [B={{\displaystyle 2v+\sqrt{(q-r)^2+4v^2}}\over{\displaystyle q-r}}] [B={{\displaystyle -2ab\cos\gamma +\sqrt{(2ab\cos\gamma)^2+(b^2-a^2)}}\over{\displaystyle b^2-a^2}}]
(c) [B={{\displaystyle (q-r)+2\sqrt{3}v+4\sqrt{(q-r)^2+4v^2}}\over{\displaystyle \sqrt3(r-q)+2v}}] [B=2{{\displaystyle a^2-c^2-\sqrt{(a^2-c^2)^2-(a^2-b^2)(b^2-c^2)}}\over{\displaystyle a^2-b^2}}-1]
(d) [B={{\displaystyle 2t}\over{\displaystyle q-r}}]  
(e) [B={{\displaystyle 4t}\over{\displaystyle r-q}}]  
(f) [B={{\displaystyle 4t}\over{\displaystyle q-r}}]  
(g) [B={{\displaystyle 4t}\over{\displaystyle r-q}}]  
(h) [B={{\displaystyle -u}\over {\displaystyle v}}]  
(k) [B={{\displaystyle 2v}\over{\displaystyle s-v}}] [B={{\displaystyle 2a^2\cos\gamma }\over {\displaystyle c^2-a^2}}]

The first three columns specify domain pairs.

  • [F_1]: point-group symmetry (stabilizer in [K_{1j}]) of the first domain state [{\bf S}_1] in a single-domain orientation.

  • [g_{1j}]: switching operations (if available) that specify the domain pair [({\bf S}_1,{\bf S}_j =g_{1j}{\bf S}_1)]. Subscripts [x,y,z] specify the orientation of the symmetry operations in the Cartesian coordinate system of [K_{1j}]. Subscripts [x^\prime, y^\prime ] and [x^{\prime\prime}, y^{\prime\prime}] denote a 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]]. Where possible, mirror planes and 180° rotations are chosen such that the two perpendicular permissible walls have crystallographic orientations.

  • [K_{1j}]: twinning group [K(F_1,g_{1j})] of the domain pair [({\bf S}_1,{\bf S}_j)]. For the pair with [F_1=m_{x\bar y}2_{xy}m_{z} ] and [K=m\bar3 m], where the twinning group does not specify the domain pair unambiguously, we add after [K_{1j}] in parentheses a switching operation [2^{\star}_{xz}] or [m^{\star}_{xz}] that defines the domain pair.

  • Axis: axis of ferroelastic domain pair around which single-domain states must be rotated to establish a contact along a compatible domain wall. This axis is parallel to the intersection of the two compatible domain walls given in the column Wall normals and its direction [{\bf h}] is defined by a vector product [{\bf h}={\bf n}_1 \times {\bf n}_2 ] of normal vectors [{\bf n}_1] and [{\bf n}_2] of these walls. The letter B denotes components of h which depend on spontaneous strain.

  • Equation: a reference to an expression, given at the end of the table, for the direction [{\bf h}] of the axis, where the component B in the column Axis is expressed as functions of spontaneous strain components, and the matrices above these expressions give the form of the `absolute' spontaneous strain.

  • Wall normals: orientation of equally deformed planes. As explained above, each plane represents two mutually reversed compatible domain walls. Numbers or parameters B, C given in parentheses can be interpreted either as components of normal vectors to compatible walls or as intercepts analogous to Miller indices: Planes of compatible domain walls [Ax_1+Bx_2+Cx_3=0] and [A{^\prime}x_1+B{^\prime}x_2+C{^\prime}x_3=0 ] [see equations (3.4.3.55[link])] pass through the origin of the Cartesian coordinate system of [K_{1j}] and have normal vectors [{\bf n}_1=[ABC]] and [{\bf n}_2=[A{^\prime}B{^\prime}C{^\prime}]]. It is possible to find a plane with the same normal vector [[ABC]] but not passing through the origin, e.g. [Ax_1+Bx_2+Cx_3=1]. Then parameters A, B and C can be interpreted as the reciprocal values of the oriented intercepts on the coordinate axes cut by this plane, [[x_1/(1/A)] + [x_2/(1/B)]+[x_3/(1/C)] =1]. In analogy with Miller indices, the symbol [(ABC)] is used for expressing the orientation of a wall. However, parameters A, B and C are not Miller indices, since they are expressed in an orthonormal and not a crystallographic coordinate system. A left square bracket [ in front of two equally deformed planes signifies that the two domain walls (domain twins) associated with one equally deformed plane are crystallographically equivalent (in [K_{1j}]) with two domain walls (twins) associated with the perpendicular equally deformed plane, i.e. all four compatible domain walls (domain twins) that can be formed from domain pair [({\bf S}_1,{\bf S}_j) ] are crystallographically equivalent in [K_{1j}] (see Fig. 3.4.3.8[link]).

    The subscript e indicates that the wall carries a nonzero polarization charge, [\hbox{Div }{\bf P}\neq0]. This can happen in ferroelectric domain pairs with spontaneous polarization not parallel to the axis of the pair. If one domain wall is charged then the perpendicular wall is not charged. In a few cases, polarization and/or orientation of the domain wall is not determined by symmetry; then it is not possible to specify which of the two walls is charged. In such cases, a subscript [e0] or [0e] indicates that one of the two walls is charged and the other is not.

  • [\omega]: reference to an expression, given at the end of the table, in which the shear angle [\omega] (in radians) is given as a function of the `absolute' spontaneous strain components, defined in a matrix given above the equations.

  • [{\sf \overline J}_{1j}]: symmetry of the `twin pair'. The meaning of this group and its symbol is explained in the next section. This group specifies the symmetry properties of a ferroelastic domain twin and the reversed twin with compatible walls of a given orientation and with domain states [{\bf S}_1^+], [{\bf S}_j^-] and [{\bf S}_1^+], [{\bf S}_j^-]. This group can be used for designating a twin law of the ferroelastic domain twin.

  • [{\underline t}_{1j}^{\star}]: one non-trivial twinning operation of the twin [{\bf S}_1[ABC]{\bf S}_j] and the wall. An underlined symbol with a star symbol signifies an operation that inverts the wall normal and exchanges the domain states (see the next section).

  • [{\sf T}_{1j}]: layer-group symmetry of the ferroelastic domain twin and the reversed twin with compatible walls of a given orientation. Contains all trivial and non-trivial symmetry operations of the domain twin (see the next section).

  • Classification: symbol that specifies the type of domain twin and the wall. Five types of twins and domain walls are given in Table 3.4.4.3[link]. The letter S denotes a symmetric domain twin (wall) in which the structures in two half-spaces are related by a symmetry operation of the twin, A denotes an asymmetric twin where there is no such relation. The letters R (reversible) and I (irreversible) signify whether a twin and reversed twin are, or are not, crystallographically equivalent in [K_{1j}].

Example 3.4.3.7. The rhombohedral phase of perovskite crystals.  Examples include PZN-PT and PMN-PT solid solutions (see e.g. Erhart & Cao, 2001[link]) and BaTiO3 below 183 K. The phase transition has symmetry descent [m\bar 3m\supset 3m].

In Table 3.4.2.7[link] we find that there are eight domain states and eight ferroelectric domain states. In this fully ferroelectric phase, domain states can be specified by unit vectors representing the direction of spontaneous polarization. We choose [{\bf S}_1\equiv [111]] with corresponding symmetry group [F_1=3_pm_{z\bar y}].

From eight domain states one can form [7\times8=56] domain pairs. These pairs can be divided into classes of equivalent pairs which are specified by different twinning groups. In column [K_{1j}] of Table 3.4.2.7[link] we find three twinning groups:

  • (i) The first twin law [\bar3_p^{\star}m_{x\bar y} ] characterizes a non-ferroelastic pair (Fam[\bar3_p^{\star}m_{x\bar y} =] Fam[3_p^{\star}m_{x\bar y}]) with inversion [\bar 1] as a twinning operation of this pair. A representative domain pair is [({\bf S}_1,g_{12}{\bf S}_1=] [{\bf S}_2) =] [([111],[\bar1\bar1\bar1])], domain pairs consist of two domain states with antiparallel spontaneous polarization (`180° pairs'). Domain walls of low energy are not charged, i.e. they are parallel with the spontaneous polarization.

  • (ii) The second twinning group [K_{13}=\bar4 3m ] characterizes a ferro­elastic domain pair (Fam[\bar4 3m=] [m\bar 3m] [\neq] Fam[F_1=] [\bar3_pm_{z\bar y}]). In Table 3.4.3.6[link], we find [g_{13}^{\star}=2^{\star}_x], which defines the representative pair [([111],[1\bar 1\bar 1])] (`109° pairs'). Orientations of compatible domain walls of this domain pair are [(100)] and [(011)_e] (this wall is charged). All equivalent orientations of these compatible walls will appear if all crystallographically equivalent pairs are considered.

  • (iii) The third twinning group [K_{14}=m\bar 3m ] also represents ferroelastic domain pairs with representative pair [([111],m^{\star}_x[111])=] [([111],[\bar 111]) ] (`71° pairs') and compatible wall orientations [(100)_e ] and (011). We see that for a given crystallographic orientation both charged and non-charged domain walls exist; for a given orientation the charge specifies to which class the domain wall belongs.

These conclusions are useful in deciphering the `domain-engineered structures' of these crystals (Yin & Cao, 2000[link]).

References

Erhart, J. & Cao, W. (2001). Effective symmetry and physical properties of twinned perovskite ferroelectric single crystals. J. Mater. Res. 16, 570–578.
Yin, J. & Cao, W. (2000). Domain configurations in domain engineered 0.995Pb(Zn1/3Nb2/3)O30.045PbTiO3 single crystals. J. Appl. Phys. 87, 7438–7441.








































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