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

International Tables for Crystallography (2013). Vol. D, ch. 3.4, p. 535

Section Ferroelastic domain twins and walls. Ferroelastic twin laws

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: Ferroelastic domain twins and walls. Ferroelastic twin laws

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As explained in Section[link], from a domain pair [({\bf S}_1,{\bf S}_j)] of ferroelastic single-domain states with two perpendicular equally deformed planes p and [p^\prime] one can form four different ferroelastic twins (see Fig.[link]). Two mutually reversed twins [({\bf S}_1|{\bf n}|{\bf S}_j)] and [({\bf S}_j|{\bf n}|{\bf S}_1) ] have the same twin symmetry [{\sf T}_{1j}(p)] and the same symmetry [\overline{\sf J}_{1j}(p)] of the twin pair [({\bf S}_1,{\bf S}_j|{\bf n}|{\bf S}_j,{\bf S}_1) ]. The ferroelastic twin laws can be expressed by the layer group [\overline{\sf J}_{1j}(p)] or, in a less complete way (without specification of reversibility), by the twin symmetry [{\sf T}_{1j}(p)]. The same holds for two mutually reversed twins [({\bf S}_1|{\bf n^\prime}|{\bf S}_j) ] and [({\bf S}_j|{\bf n^\prime}|{\bf S}_1)] with a twin plane [p^\prime] perpendicular to p.

Table[link] summarizes possible symmetries [{\sf T}_{1j}] of ferro­elastic domain twins and corresponding ferroelastic twin laws [\overline{\sf J}_{1j}]. Letters V and W signify strain-dependent and strain-independent (with a fixed orientation) domain walls, respectively. The classification of domain walls and twins is defined in Table[link]. The last column contains twinning groups [K_{1j}(F_1)] of ordered domain pairs [({\bf S}_1,{\bf S}_j)] from which these twins can be formed. The symbol of [K_{1j}] is followed by a symbol of the group [F_1] given in square brackets. The twinning group [K_{1j}(F_1)] specifies, up to two cases, a class of equivalent domain pairs [orbit [G({\bf S}_1,{\bf S}_j)]] (see Section[link]). More details on particular cases (orientation of domain walls, disorientation angle, twin axis) can be found in synoptic Table[link]. From this table follow two general conclusions:

  • (1) All layer groups describing the symmetry of compatible ferroelastic domain walls are polar groups, therefore all compatible ferroelastic domain walls in dielectric crystals can be spontaneously polarized. The direction of the spontaneous polarization is parallel to the intersection of the wall plane p and the plane of shear (i.e. a plane perpendicular to the axis of the ferroelastic domain pair, see Fig.[link]b and Section[link]).

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    Symmetry properties of ferroelastic domain twins and compatible domain walls

    [{\sf T}_{1j}][{\sf \overline{J}}_{1j} ]Classification[K_{1j}[F_1] ]
    [1] [\underline2 ] V [{\rm A}\underline{\rm R}]   [4^{\star}[2]], [\bar4^{\star}[2]], [6[2]], [6/m[2]]
    [1] [\underline2] V [{\rm A}\underline{\rm R}]  
    [1] [2^{\star}] W [{\rm AR}^{\star}] [\Bigl\{] [2^{\star}[1]], [422[2]], [\bar42m[m]], [32[2]], [\bar3m[m]], [622[2]], [\bar6m2[m],] [432[222]], [m\bar3m[mm2]], [m\bar3m[2^{\star}_{xy}][mm2]]
    [\underline2^{\star}] [\underline2^{\star}] V SI
    [1] [2^{\star}] W [{\rm AR}^{\star}]   [23[3]], [432[4]], [432[3]], [m\bar3m[\bar4]]
    [\underline2^{\star}] [\underline2^{\star}] W SI  
    [1] [m^{\star}] V [{\rm AR}^{\star}] [\Bigl\{] [m^{\star}[1]], [4mm[m]], [\bar42m[2]], [3m[m]], [\bar3m[2]], [6mm[m]], [\bar6m2[2]], [\bar43m[mm2]], [m\bar3m[222]], [m\bar3m[m^{\star}_{xy}][m2m]]
    [{\underline m}^{\star} ] [\underline m^{\star}] W SI
    [1] [m^{\star}] W [{\rm AR}^{\star}]   [m\bar3[3]], [\bar43m[\bar4]], [\bar43m[3]], [m\bar3m[4] ]
    [{\underline m}^{\star} ] [\underline m^{\star}] W SI  
    [\underline2^{\star}] [2^{\star}\underline2^{\star}\underline2 ] W SR [\Bigl\{] [2^{\star}2^{\star}2[2]], [4^{\star}22^{\star}[222]], [\bar4^{\star}2^{\star}m[mm2] ], [622[222]], [6/mmm[mm2]], [432[422]], [432[32] ], [m\bar3m[\bar42m]]
    [\underline2^{\star}] [\underline2^{\star}2^{\star}\underline2 ] W SR
    [\underline2^{\star}] [\underline2^{\star}/m^{\star} ] V SR   [2^{\star}/m^{\star}[\bar1]], [4/mmm[2/m]], [\bar3m[2/m]], [6/mmm[2/m]], [m\bar3m[mmm]]
    [\underline{m}^{\star} ] [2^{\star}/\underline{m}^{\star} ] W SR  
    [\underline2^{\star}] [\underline2^{\star}/m^{\star} ] W SR   [m\bar3[\bar3]], [m\bar3m[4/m]], [m\bar3m[\bar3]]
    [\underline{m}^{\star} ] [2^{\star}/\underline{m}^{\star} ] W SR  
    [m ] m V AI   [4/m[m]], [\bar6[m]], [6/m[m]]
    [m ] m V AI  
    m [\underline{2}/m ] V [{\rm A}\underline{\rm R} ]   [4^{\star}/m[2/m]], [6/m[2/m] ]
    [m ] [\underline{2}/m ] V [{\rm A}\underline{\rm R} ]  
    [\underline{m}^{\star} ] [{m}^{\star}\underline{m}^{\star}\underline2 ] W SR [\Bigl\{] [m^{\star}m^{\star}2[2]], [4^{\star}mm^{\star}[mm2]], [\bar4^{\star}2m^{\star}[222] ], [6mm[mm2]], [6/mmm[222]], [\bar43m[\bar42m]], [m\bar3m[422]], [m\bar3m[32]]
    [\underline{m}^{\star} ] [\underline{m}^{\star}m^{\star}\underline2 ] W SR
    [m ] [m^{\star}2^{\star}m ] W [{\rm AR}^{\star}] [\Bigl\{] [m^{\star}2^{\star}m[m]], [4/mmm[2mm]], [\bar6m2[m2m]], [6/mmm [m2m]], [\bar43m[3m]], [m\bar3m[4mm]], [m\bar3m[\bar42m] ], [m\bar3m[3m]]
    [\underline{m}^{\star}\underline2^{\star}m ] [\underline{m}^{\star}\underline2^{\star}m ] W SI
    [\underline{m}^{\star}\underline2^{\star}m ] [\underline{m}^{\star}m^{\star}m ] W SR [\Bigl\{] [m^{\star}m^{\star}m[2/m] ], [4^{\star}/mmm^{\star}[mmm]], [6/mmm[mmm]], [m\bar3m[4/mmm] ], [m\bar3m[\bar3m]]
    [\underline{2}^{\star}\underline{m}^{\star}m ] [m^{\star}\underline{m}^{\star}m ] W SR
  • (2) Domain twin [({\bf S}_1|{\bf n}|{\bf S}_j) ] formed in the parent clamping approximation from a single-domain pair [({\bf S}_1,{\bf S}_j)] and the relaxed domain twin [({{\bf S}_1^+}|{\bf n}|{{\bf S}_j^-})] with disoriented domain states have the same symmetry groups [{\sf T}_{1j}] and [{\sf \overline{J}}_{1j} ].

This follows from simple reasoning: all twin symmetries [{\sf {T}}_{1j} ] in Table[link] have been derived in the parent clamping approximation and are expressed by the orthorhombic group [mm2] or by some of its subgroups. As shown in Section[link], the maximal symmetry of a mechanically twinned crystal is also [mm2]. An additional simple shear accompanying the lifting of the parent clamping approximation cannot, therefore, decrease the sym­metry [{{\sf T}_{1j}}(p) ] derived in the parent clamping approximation. In a similar way, one can prove the statement for the group [{\sf\overline{J}}_{1j}(p)] of the twin pairs [({{\bf S}_1},{\bf S}_j|{\bf n}|{\bf S}_j,{{\bf S}_1})] and [({{\bf S}_1^+},] [{\bf S}_j^{-}|{\bf n}|{\bf S}_j^{-},] [{{\bf S}_1^+})].

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