Ferroelastic domain state

Janovec, V.; Přívratská

doi:10.1107/97809553602060000645

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

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International Tables for Crystallography (2006). Vol. D. ch. 3.4, pp. 455-457

Section 3.4.2.2.1. Ferroelastic domain state

V. Janovec^a ^* and J. Přívratská^b

^a Department of Physics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, 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). 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). 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). 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 M	Crystal system	Holohedry HolM	Spontaneous strain components		Crystal family FamM
Point group M	Crystal system	Holohedry HolM	Independent	Nonzero	Crystal family FamM
, $[m\bar3]$ , , $[\bar43m]$ , $[m\bar3m]$	Cubic	$[m\bar3m]$	1	3	Cubic
, $[\bar6]$ , , , , $[\bar62m]$ ,	Hexagonal		2	3	Hexagonal
, $[\bar3]$ , , , $[\bar3m]$	Trigonal	$[\bar3m]$	2	3	Hexagonal
, $[\bar4]$ , , , , $[\bar42m]$ ,	Tetragonal		2	3	Tetragonal
, ,	Orthorhombic		3	3	Orthorhombic
, m,	Monoclinic		4	4	Monoclinic
, $[\bar1]$	Triclinic	$[\bar1]$	6	6	Triclinic

We shall denote the crystal family of a group M by the 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.] Different ferroelastic domain states differ in spontaneous strain. The symmetry of a ferroelastic domain state R_i 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)]. This stabilizer, which we shall denote by [A_i] , can be expressed as an intersection of the parent group G and the holohedry of group [F_i] , which we shall denote Hol [F_i] (see Table 3.4.2.2): $[A_i \equiv I_G(u^{(i)})=G\cap {\rm Hol}F_i. \eqno(3.4.2.27) ]$ This equation indicates that the ferroelastic domain state R_i 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, $[\eqalign{A_1 &= I_{4_z/m_zm_xm_{xy}}(u_{11}-u_{22})\cr &= {\rm Hol}(2_xm_ym_z)\cap m4_z/m_zm_xm_{xy}\cr &=m_xm_ym_z \cap 4_z/m_zm_xm_{xy}= m_xm_ym_z.\cr} ]$

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, 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).

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)], $[n_ad_a=[G:A_1][A_1:F_1]=[G:F_1]=n. \eqno(3.4.2.30) ]$ 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.

According to Aizu (1969), we can recognize three possible cases:

(i) Full ferroelastics: All principal domain states differ in spontaneous strain. In this case, , i.e. , ferroelastic domain states are identical with principal domain states.
(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 . 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 ).
(iii) Non-ferroelastics: All principal domain states have the same spontaneous strain. The criterion is , i.e. .

A similar classification for ferroelectric domain states is given below. Both classifications are summarized in Table 3.4.2.3.

Table 3.4.2.3 | top | pdf |
Aizu's classification of ferroic phases

[n_a] is the number of ferroelastic domain states, [n_e] is the number of ferroelectric domain states and [n_f] is the number of ferroic domain states.

Ferroelastic			Ferroelectric
Fully	Partially	Non-ferroelastic	Fully	Partially	Non-ferroelectric
	$[1 \,\lt\, n_a \,\lt\, n]$			$[1 \,\lt\, n_e \,\lt\, n]$	, 1

Example 3.4.2.1. Domain states in leucite. Leucite (KAlSi₂O₆) (see e.g. Hatch et al., 1990) 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 $[|G=m\bar3m|:|4/mmm|=3]$ 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 Hol( [4_x/m_xm_ym_z)] [=4_x/m_xm_ym_z] (see Table 3.4.2.2), 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 [ $[{\rm Hol}(4_x/m_x) =]$ $[{\rm Hol}(4_x/m_xm_ym_z) =]$ [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 in Chapter 3.3 , Palmer et al. (1988) and Putnis (1992)].

References

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Aizu, K. (1969). Possible species of `ferroelastic' crystals and of simultaneously ferroelectric and ferroelastic crystals. J. Phys. Soc. Jpn, 27, 387–396.Google Scholar

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.Google Scholar

Palmer, D. C., Putnis, A. & Salje, E. K. H. (1988). Twinning in tetragonal leucite. Phys. Chem. Mineral. 16, 298–303. Google Scholar

Putnis, A. (1992). Introduction to mineral sciences. Cambridge University Press.Google Scholar

Wadhawan, V. K. (2000). Introduction to ferroic materials. The Netherlands: Gordon and Breach.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 3.4, pp. 455-457