(International Tables for Crystallography) Domain structures

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

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International Tables for Crystallography (2013). Vol. D, ch. 3.4, p. 539

Section 3.4.5. Glossary

V. Janovec^a ^* and J. Přívratská^b

^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

3.4.5. Glossary

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Note: the correspondence between contracted Greek indices and the Cartesian vector components used in Sections 3.1.3 , in the present chapter and in the software GI $[\star]$ KoBo-1, is defined in the following way: $[\matrix{\hbox{Cartesian components}\hfill & 11 & 22 & 33 & 23,32& 31,13 & 12,21\cr \hbox{Contracted notation}\hfill & 1 & 2 & 3 & 4 & 5 & 6} ]$

In this designation, coefficients with contracted indices 4, 5, 6 appear two times, e.g. index 4 replaces yz in one coefficient and zy in the other coefficient. With this convention, the coefficients transform in tensor space as vector components, but some coefficients differ from the usual matrix notation (Voigt matrices) by numerical factors [see Section 1.1.4.10 ; Nye (1985); Sirotin & Shaskolskaya, Appendix E (1982)].

(a) Objects

	domain region
d	scanning vector (basis vector of a scanning group)
$[{\bf D}_i({\bf S}_k]$ ,	the ith domain, with domain state $[{\bf S}_k]$ in the mth domain region
$[G{\bf S}_1]$	G-orbit of principal single-domain states
$[G({\bf S}_{1},{\bf S}_{j})]$	G-orbit of domain pairs
$[G({\bf S}_{1}\|{\bf n}\|{\bf S}_{j})]$	G-orbit of simple domain twins
n	normal to a plane p
	equally deformed plane (EDP), plane of a domain wall, domain wall plane
$[{\bf R}_{1},]$ $[{\bf R}_{2},]$ $[\ldots,]$ $[{\bf R}_{i}, ]$ $[\ldots]$	secondary ferroic single-domain states
$[{\bf R}_{1}^+,]$ $[{\bf R}_{1}^-,]$ $[{\bf R}_{2}^+,]$ $[{\bf R}_{2}^-,]$ $[\ldots]$	disoriented secondary ferroic domain states
$[s{\bf d}]$ $[(0\leq s \,\lt\, 1)]$	location of a plane in crystal lattice
$[{\bf S}_{1},]$ $[{\bf S}_{2},]$ $[\ldots,]$ $[{\bf S}_{i}, ]$ $[\ldots]$	principal single-domain states (orientation states, variants)
$[{\bf S}_{1}^+,]$ $[{\bf S}_{1}^-,]$ $[{\bf S}_{2}^+,]$ $[{\bf S}_{2}^-, ]$ $[\ldots]$	disoriented domain states
$[{\sf S}_1,]$ $[{\sf S}_2,]$ $[\ldots,]$ $[{\sf S}_i,]$ $[\ldots]$	basic (microscopic) single-domain states (structural variants)
$[({\bf S}_{i},{\bf S}_{k})]$	ordered domain pair = ordered pair of domain states $[{\bf S}_{i}]$ and $[{\bf S}_{k} ]$
$[\{{\bf S}_i,{\bf S}_k\}]$	unordered domain pair = unordered pair of domain states $[{\bf S}_{i}]$ and $[{\bf S}_{k}]$
$[({\bf S}_{i}\|{\bf n}\|{\bf S}_{k})]$	simple domain twin formed from single-domain states
$[({\bf S}_{i}^{+}\|{\bf n}\|{\bf S}_{k}^{-})]$	simple ferroelastic domain twin with a compatible domain wall
$[[{\bf S}_{i}\|{\bf n}\|{\bf S}_{k}]]$	domain wall in the simple twin $[({\bf S}_{i}\|{\bf n}\|{\bf S}_{k})]$
$[{\bf T}_{ik}({\bf n})]$ or $[{\bf T}_{ik}]$	simple domain twin – short symbol
$[{\bf W}_{ik}({\bf n})]$ or $[{\bf W}_{ik}]$	domain wall – short symbol
$[\omega]$	shear angle, obliquity
$[\pm{{1}\over{2}}\omega]$	disorientation angle of a domain state

(b) Symmetry groups – point groups in a continuum description and space groups in a microscopic description

F	point-group symmetry of the ferroic phase (domain state not specified)
$[{\cal F}]$	space-group symmetry of the ferroic phase (domain state not specified)
	point-group symmetry of a principal domain state $[{\bf S}_i]$
$[{\cal F}_i]$	space-group symmetry of a basic (microscopic) domain state $[{\sf S}_i]$
$[F_{ik}]$	point-group symmetry (stabilizer in G) of the ordered domain pair $[({\bf S}_{i},{\bf S}_{k})]$
$[{\cal F}_{ik}]$	space-group symmetry (stabilizer in $[{\cal G}]$ ) of the ordered domain pair $[({\sf S}_i,{\sf S}_k)]$
$[\overline {\sf F}_{ik}]$	sectional layer group of $[F_{ik}]$
$[\widehat {\sf F}_{ik} ]$	face group, trivial layer group, scanning group of $[F_{ik}]$
Fam $[\,G]$	crystal family of the group G
G	point-group symmetry of the parent phase
$[{\cal G}]$	space-group symmetry of the parent phase
g	point-group symmetry operation of the group $[G({\cal G})]$
$[{\sf g}]$	space-group symmetry operation of the group $[{\cal G}]$
$[g_{ik}]$	switching operation in domain pair $[({\bf S}_i,{\bf S}_k)]$ , transforms $[{\bf S}_i]$ into $[{\bf S}_k]$
$[g_{ik}^{\star}]$	transposing operation in domain pair $[({\bf S}_i,{\bf S}_k)]$ , exchanges $[{\bf S}_i]$ and $[{\bf S}_k]$ , twinning operation of a non-ferroelastic domain pair $[({\bf S}_i, {\bf S}_k)]$
$[I_{G}({\bf S}_i)]$	stabilizer (isotropy group) of $[{\bf S}_i]$ in G
$[{\cal I}_{\cal G}({\sf S}_{i})]$	stabilizer (isotropy group) of $[{\sf S}_i]$ in $[{\cal G}]$
$[J_{ik}]$	point-group symmetry (stabilizer in G) of the unordered domain pair $[\{{\bf S}_i, {\bf S}_k\}]$
$[J_{ik}^{\star}]$	point-group symmetry (stabilizer in G) of a completely transposable domain pair $[\{{\bf S}_i,{\bf S}_k\} ]$
$[{\cal J}_{ik}]$	space-group symmetry (stabilizer in $[{\cal G}]$ ) of the unordered domain pair $[\{{\sf S}_i,{\sf S}_k\}]$
$[K_{ik}]$	twinning group of the domain pair $[({\bf S}_i,{\bf S}_k)]$
$[K_{ik}^{\star}]$	twinning group of a completely transposable domain pair $[({\bf S}_i,{\bf S}_k)]$
	intermediate group, $[F_i \subset L_i \subset G]$
$[{\overline {\sf J}}_{ik}]$	sectional layer group of $[J_{ik}]$
$[{\widehat {\sf J}}_{ik} ]$	face group, floating subgroup of sectional group of $[J_{ik}]$
$[r^{\star}_{ik}]$	symmetry operation that exchanges $[{\bf S}_{i}]$ and $[{\bf S}_{k} ]$
$[{\underline s}_{ik}]$	symmetry operation that inverts n into −n
$[\underline t^{\star}_{ik}]$	symmetry operation that exchanges $[{\bf S}_{i}]$ and $[{\bf S}_{k} ]$ and inverts n into −n
$[{\sf T}_{ik}({\bf n})]$	symmetry group of the twin $[{\bf T}_{ik}({\bf n})]$
$[{\sf W}_{ik}({\bf n})]$	symmetry group of the domain wall $[{\bf W}_{ik}({\bf n})]$
$[{\cal T}_i]$	translational subgroup of $[{\cal F}_i]$
$[{\cal T}_{ik}]$	translational subgroup of $[{\cal F}_{ik}]$

$[\rho]$	enantiomorphism
	polarization
$[\varepsilon_{ij}]$	dielectric permittivity
$[u_{\mu}]$	strain
$[\sigma_{\mu}]$	mechanical stress
$[g_{\mu}]$	optical activity
$[d_{i\mu}]$	piezoelectricity
$[r_{i\mu}]$	electro-optics
$[A_{i\mu}]$	electrogyration
$[s_{\mu\nu}]$	linear elasticity
$[Q_{\mu\nu}]$	electrostriction (contracted notation, see Section 1.1.4.10.5 )
$[\pi_{\mu\nu}]$	piezo-optics (contracted notation, see Section 1.1.4.10.5 )

$[i = 1, 2, 3; \mu,\nu = 1, 2, \ldots, 6.]$

References

International Tables for Crystallography (2005). Vol. A, Space-Group Symmetry, 5th ed., edited by Th. Hahn. Heidelberg: Springer.
Aizu, K. (1970a). Possible species of ferromagnetic, ferroelectric and ferroelastic crystals. Phys. Rev. B, 2, 754–772.
Altmann, S. L. & Herzig, P. (1994). Point-Group Theory Tables. Oxford: Clarendon Press.
Bradley, C. J. & Cracknell, A. P. (1972). The Mathematical Theory of Symmetry in Solids. Oxford: Clarendon Press.
Jona, F. & Shirane, G. (1962). Ferroelectric Crystals. Oxford: Pergamon Press.
Nye, J. F. (1985). Physical Properties of Crystals. Oxford: Clarendon Press.
Rosová, A. (1999). Real domain structure origination in (110) mechanical twinning in YBa₂Cu₃O_7−y. In Studies of High Temperature Superconductors, Vol. 28, edited by A. Narlikar, pp. 125–148. New York: Nova Science Publishers.
Salje, E. K. H. (1990). Phase Transitions in Ferroelastic and Co-elastic Crystals, 1st ed. Cambridge University Press.
Shur, V. Ya., Rumyantsev, E. L., Nikolaeva, E. V., Shishkin, E. I., Batchko, R. G., Fejer, M. M. & Byer, R. L. (2001). Recent achievements in domain engineering in lithium niobate and lithium tantalate. Ferroelectrics, 257, 191–202.
Sirotin, Yu. I. & Shaskolskaya, M. P. (1982). Fundamentals of Crystal Physics. Moscow: Mir.

International Tables for Crystallography (2013). Vol. D, ch. 3.4, p. 539