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. 485-486

## Section 3.4.1.2. Scope of this chapter

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

#### 3.4.1.2. Scope of this chapter

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This chapter is devoted to the crystallographic aspects of static domain structures, especially to the symmetry analysis of these structures. The main aim is to explain basic concepts, derive relations that govern the formation of domain structures and provide tables with useful ready-to-use data on domain structures of ferroic phases. The exposition uses algebraic tools that are explained in Section 3.2.3 , but the important points are illustrated with simple examples comprehensible even without mathematical details. The synoptic tables in Sections 3.4.2 and 3.4.3 present the main results of the analysis for all possible ferroic domain structures. More detailed information on certain points can be found in the software GIKoBo-1.

All these results are definite – their validity does not depend on any particular model or approximation – and form thus a firm basis for further more detailed quantitative treatments. `For the most part, the only exact statements which can be made about a solid state system are those which arise as a direct consequence of symmetry alone.' (Knox & Gold, 1967.)

The exposition starts with domain states, continues with pairs of domain states and domain distinction, and terminates with domain twins and walls. This is also the sequence of steps in domain-structure analysis, which proceeds from the simplest to more complicated objects.

In Section 3.4.2, we explain the concept of domain states (also called variants or orientational states), define different types of domain states (principal, ferroelastic, ferroelectric, basic), find simple formulae for their number, and disclose their hierarchy and relation with symmetry lowering and with order parameters of the transition. Particular results for all possible ferroic phase transitions can be found in synoptic Table 3.4.2.7, which lists all possible crystallographically non-equivalent point-group symmetry descents that may appear at a ferroic phase transition. For each descent, all independent twinning groups (characterizing the relation between two domain states) are given together with the number of principal, ferroelastic and ferroelectric domain states and other data needed in further analysis.

Section 3.4.3 deals with pairs of domain states and with the relationship between two domain states in a pair. This relationship, in mineralogy called a `twin law', determines the distinction between domain states, specifies switching processes between two domain states and forms a starting point for discussing domain walls and twins. We show different ways of expressing the relation between two domain states of a domain pair, derive a classification of domain pairs, find non-equivalent domain pairs and determine which tensor properties are different and which are the same in two domain states of a domain pair.

The presentation of non-equivalent domain pairs is divided into two parts. Synoptic Table 3.4.3.4 lists all representative non-equivalent non-ferroelastic domain pairs, and for each pair gives the twinning groups, and the number of tensor components that are different and that are the same in two domain states. These numbers are given for all important property tensors up to rank four. We also show how these data can be used to determine switching forces between two non-ferroelastic domain states.

Then we explain specific features of ferroelastic domain pairs: compatible (permissible) domain walls and disorientation of domain states in ferroelastic domain twins. A list of all non-equivalent ferroelastic domain pairs is presented in two tables. Synoptic Table 3.4.3.6 contains all non-equivalent ferroelastic domain pairs with compatible (coherent) domain walls. This table gives the orientation of compatible walls and their symmetry properties. Table 3.4.3.7 lists all non-equivalent ferroelastic domain pairs with no compatible ferroelastic domain walls.

Column K1j in Table 3.4.2.7 specifies all representative non-equivalent domain pairs that can appear in each particular phase transition; in combination with Tables 3.4.3.4 and 3.4.3.6, it allows one to determine the main features of any ferroic domain structure.

Section 3.4.4 is devoted to domain twins and domain walls. We demonstrate that the symmetry of domain twins and domain walls is described by layer groups, give a classification of domain twins and walls based on their symmetry, and present possible layer groups of non-ferroelastic and ferroelastic domain twins and walls. Then we discuss the properties of finite-thickness domain walls. In an example, we illustrate the symmetry analysis of microscopic domain walls and present conclusions that can be drawn from this analysis about the microscopic structure of domain walls.

The exposition is given in the continuum description with crystallographic point groups and property tensors. In this approach, all possible cases are often treatable and where possible are covered in synoptic tables or – in a more detailed form – in the software GIKoBo-1. Although the group-theoretical tools are almost readily transferable to the microscopic description (using the space groups and atomic positions), the treatment of an inexhaustible variety of microscopic situations can only be illustrated by particular examples.

Our attempt to work with well defined notions calls for introducing several new, and generalizing some accepted, concepts. Also an extended notation for the symmetry operations and groups has turned out to be indispensable. Since there is no generally accepted terminology on domain structures yet, we often have to choose a term from several existing more-or-less equivalent variants.

The specialized scope of this chapter does not cover many other important aspects of domain-structure studies. More information can be found in the following references. The only comprehensive monograph, by Tagancev et al. (2010), provides a thorough treatise on major topics of domain structure in non-magnetic ferroic crystals and thin films. There are two specialized monographs: the booklet by Fesenko et al. (1990) deals with experimental aspects of domain structure in multi-axial ferroelectrics and Sidorkin (2002, 2006) concentrates on the theoretical treatment of phenomenological models of various problems of domain-structure studies. The main concepts of domain structures of ferroic materials are explained in the book by Wadhawan (2000) and in a review by Schranz (1995). Ferroelastic domain structures are reviewed in Boulesteix (1984) and Wadhawan (1991), and are treated in detail by Salje (1990, 1991, 2000a,b). The main properties of ferroelectric domain structures are explained in older books or reviews on ferroelectric crystals: Känzig (1957), Jona & Shirane (1962), Fatuzzo & Merz (1967), Mitsui et al. (1976), Lines & Glass (1977), Smolenskii et al. (1984), Zheludev (1988) and Strukov & Levanyuk (1998). Recent books and reviews on ferroic materials rarely contain new, informed and unifying views on admirable experimental achievements in this field. Applications of ferroelectrics are also described in now older books by Xu (1991) and Uchino (2000). Principles and technical aspects of ferroelectric memories were reviewed by Scott (1998, 2000).

### References

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