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. 491-492

Section 3.4.2.2.1. Ferroelastic domain state

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

3.4.2.2.1. Ferroelastic domain state

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The distinction ferroelastic–non-ferroelastic is a basic division in domain structures. Ferroelastic transitions are ferroic transitions involving a spontaneous distortion of the crystal lattice that entails a change of shape of the crystallographic or conventional unit cell (Wadhawan, 2000[link]). Such a transformation is accompanied by a change in the number of independent nonzero components of a symmetric second-rank tensor [u ] that describes spontaneous strain.

In discussing ferroelastic and non-ferroelastic domain structures, the concepts of crystal family and holohedry of a point group are useful (IT A , 2005[link]). Crystallographic point groups (and space groups as well) can be divided into seven crystal systems and six crystal families (see Table 3.4.2.2[link]). A symmetry descent within a crystal family does not entail a qualitative change of the spontaneous strain – the number of independent nonzero tensor components of the strain tensor u remains unchanged.

Table 3.4.2.2| top | pdf |
Crystal systems, holohedries, crystal families and number of spontaneous strain components

Point group MCrystal systemHolohedry HolMSpontaneous strain componentsCrystal familyFamily group FamM
IndependentNonzero
[23], [m\bar3], [432], [\bar43m], [m\bar3m] Cubic [m\bar3m] 1 3 Cubic [m\bar3m]
[6], [\bar6], [6/m], [622], [6mm], [\bar62m], [6/mmmm] Hexagonal [6/mmm] 2 3 Hexagonal 6/mmm
[3], [\bar3], [32], [3m], [\bar3m] Trigonal [\bar3m] 2 3
[4], [\bar4], [4/m], [422], [4mm], [\bar42m], [4/mmm] Tetragonal [4/mmm] 2 3 Tetragonal 4/mmm
[222], [mm2], [mmm] Orthorhombic [mmm] 3 3 Orthorhombic mmm
[2], m, [2/m] Monoclinic [2/m] 4 4 Monoclinic 2/m
[1], [\bar1] Triclinic [\bar1] 6 6 Triclinic [\bar 1]

We shall call the largest group of the crystal family to which the group M belongs the family group of M (symbol FamM). Then a simple criterion for a ferroic phase transition with symmetry descent [G \subset F] to be a non-ferroelastic phase transition is[F \subset G, \quad{\rm Fam}F ={\rm Fam}G. \eqno(3.4.2.25) ]

A necessary and sufficient condition for a ferroelastic phase transition is[F \subset G, \quad{\rm Fam}F \neq {\rm Fam}G. \eqno(3.4.2.26) ]

A ferroelastic domain state [{\bf R}_i] is defined as a state with a homogeneous spontaneous strain [u^{(i)}]. [We drop the suffix `s' or `(s)' if the serial number of the domain state is given as the superscript [(i)]. The definition of spontaneous strain is given in Section 3.4.3.6.1[link].] Different ferroelastic domain states differ in spontaneous strain. The symmetry of a ferroelastic domain state Ri is specified by the stabilizer [I_G(u^{(i)}) ] of the spontaneous strain [u^{(i)}] of the principal domain state [{\bf S}_i] [see (3.4.2.16[link])]. This stabilizer, which we shall denote by [A_i], can be expressed as an intersection of the parent group G and the family group of [F_i] (see Table 3.4.2.2[link]):[A_i \equiv I_G(u^{(i)})=G\cap {\rm Fam}F_i. \eqno(3.4.2.27) ]This equation indicates that the ferroelastic domain state Ri has a prominent single-domain orientation. Further on, the term `ferroelastic domain state' will mean a `ferroelastic domain state in single-domain orientation'.

In our illustrative example, [\eqalign{A_1 &= I_{4_z/m_zm_xm_{xy}}(u_{11}-u_{22})\cr &= {\rm Fam}(2_xm_ym_z)\cap m4_z/m_zm_xm_{xy}\cr &=m_xm_ym_z \cap 4_z/m_zm_xm_{xy}= m_xm_ym_z.\cr} ]

The number [n_a] of ferroelastic domain states is given by[n_a = [G:A_1] = |G|:|A_1|. \eqno(3.4.2.28)]In our example, [n_a=|4_z/m_zm_xm_{xy}|:|m_xm_ym_z|=16:8=2]. In Table 3.4.2.7[link], last column, the number [n_a] of ferroelastic domain states is given for all possible ferroic phase transitions.

The number [d_a] of principal domain states compatible with one ferroelastic domain state (degeneracy of ferroelastic domain states) is given by[d_a=[A_1:F_1]=|A_1|:|F_1|. \eqno(3.4.2.29) ]In our example, [d_a=|m_xm_ym_z|:|2_xm_ym_z|=8:4=2], i.e. two non-ferroelastic principal domain states are compatible with each of the two ferroelastic domain states (cf. Fig. 3.4.2.2[link]).

The product of [n_a] and [d_a] is equal to the number n of all principal domain states [see equation (3.4.2.19[link])],[n_ad_a=[G:A_1][A_1:F_1]=[G:F_1]=n. \eqno(3.4.2.30) ]The number [d_a] of principal domain states in one ferroelastic domain state can be calculated for all ferroic phase transitions from the ratio of numbers n and [n_a] that are given in Table 3.4.2.7[link].

According to Aizu (1969[link]), we can recognize three possible cases (see also Table 3.4.2.3[link]):

  • (i) Full ferroelastics: All principal domain states differ in spontaneous strain. In this case, [n_a=n], i.e. [A_1=F_1], ferroelastic domain states are identical with principal domain states.

    Table 3.4.2.3| top | pdf |
    Classification of ferroic phases according to ferroelastic and ferroelectric domain states

    ne, na and n: number of ferroelectric, ferroelastic and principal domain states, respectively. Terms in parentheses were introduced by Aizu (1970a[link]).

    FerroelasticFerroelectric
    na = n 1 < na < n na = 1 ne = n 1 < ne < n ne = 1 ne = 0
    faithful (full) degenerate (partial) non-ferroelastic (non-ferroelastic) faithful (full) degenerate (partial) trivial zero
    f or af d or ad n or an f or ef d or ed t or et z or ez
    non-ferroelectric (non-ferroelectric)
    n or en
  • (ii) Partial ferroelastics: Some but not all principal domain states differ in spontaneous strain. A necessary and sufficient condition is [1 \,\lt\, n_a \,\lt\, n], or, equivalently, [F_1 \subset A_1 \subset G]. In this case, ferroelastic domain states are degenerate secondary domain states with degeneracy [n>d_a=|A_1|:|F_1|>1 ]. In this case, the phase transition [G\supset F_1] can also be classified as an improper ferro­elastic one (see Section 3.1.3.2[link] ).

  • (iii) Non-ferroelastics: All principal domain states have the same spontaneous strain. The criterion is [n_a=1 ], i.e. [A_1=G].

Example 3.4.2.1. Domain states in leucite.  Leucite (KAlSi2O6) (see e.g. Hatch et al., 1990[link]) undergoes at about 938 K a ferro­elastic phase transition from cubic symmetry [G=m\bar3m] to tetragonal symmetry [L=4/mmm]. This phase can appear in [|m\bar3m|:|4/mmm|=3] single-domain states, which we denote [{\bf R}_1], [{\bf R}_2], [{\bf R}_3 ]. The symmetry group of the first domain state [{\bf R}_1] is [L_1=4_x/m_xm_ym_z]. This group equals the stabilizer [I_G(u^{(1)}) ] of the spontaneous strain [u^{(1)}] of [{\bf R}_1] since Fam([4_x/m_xm_ym_z)] [=4_x/m_xm_ym_z] (see Table 3.4.2.2[link]), hence this phase is a full ferroelastic one.

At about 903 K, another phase transition reduces the symmetry [4/mmm] to [F= 4/m]. Let us suppose that this transition has taken place in a domain state [{\bf R}_1] with symmetry [L_1=4_x/m_xm_ym_z]; then the room-temperature ferroic phase has symmetry [F_1=4_x/m_x]. The [4_x/m_xm_ym_z \supset 4_x/m_x] phase transition is a non-ferroelastic one [[{\rm Fam}(4_x/m_x) = 4_x/m_xm_ym_z]] with [|4_x/m_xm_ym_z|:|4_x/m_x|=8:4=2] non-ferro­elastic domain states, which we denote [{\bf S}_1] and [{\bf S}_2]. Similar considerations performed with initial domain states R2 and R3 generate another two couples of principal domain states [{\bf S}_3 ], [{\bf S}_4] and [{\bf S}_5], [{\bf S}_6], respectively. Thus the room-temperature phase is a partially ferroelastic phase with three degenerate ferroelastic domain states, each of which can contain two principal domain states. Both ferroelastic domains and non-ferroelastic domains within each ferroelastic domain have been observed [see Fig. 3.3.10.13[link] in Chapter 3.3[link] , Palmer et al. (1988[link]) and Putnis (1992)[link]].

References

International Tables for Crystallography (2005). Vol. A, Space-Group Symmetry, 5th ed., edited by Th. Hahn. Heidelberg: Springer.
Aizu, K. (1969). Possible species of `ferroelastic' crystals and of simultaneously ferroelectric and ferroelastic crystals. J. Phys. Soc. Jpn, 27, 387–396.
Hatch, D. M., Ghose, S. & Stokes, H. (1990). Phase transitions in leucite, KAl2O6. I. Symmetry analysis with order parameter treatment and the resulting microscopic distortions. Phys. Chem. Mineral. 17, 220–227.
Palmer, D. C., Putnis, A. & Salje, E. K. H. (1988). Twinning in tetragonal leucite. Phys. Chem. Mineral. 16, 298–303.
Putnis, A. (1992). Introduction to Mineral Sciences. Cambridge University Press.
Wadhawan, V. K. (2000). Introduction to Ferroic Materials. The Netherlands: Gordon and Breach.








































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