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

International Tables for Crystallography (2013). Vol. D, ch. 3.4, pp. 486-490

Section Principal and basic domain states

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: Principal and basic domain states

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As for all crystalline materials, domain structures can be approached in two ways: In the microscopic description, a crystal is treated as a regular arrangement of atoms. Domains differ in tiny differences of atomic positions which can be determined only indirectly, e.g. by diffraction techniques. In what follows, we shall pay main attention to the continuum description, in which a crystal is treated as an anisotropic continuum. Then the crystal properties are described by property tensors (see Section 1.1.1[link] ) and the crystal symmetry is expressed by crystallographic point groups. In this approach, domains exhibit different tensor properties that enable one to visualize domains by optical or other methods.

The domain structure observed in a microscope appears to be a patchwork of homogeneous regions – domains – that have various colours and shapes (see Fig.[link]). Indeed, the usual description considers a domain structure as a collection of domains and contact regions of domains called domain walls. Strictly speaking, by a domain [{\bf D}_i] one understands a connected part of the crystal, called the domain region, which is filled with a homogeneous low-symmetry crystal structure. Domain walls can be associated with the boundaries of domain regions. The interior homogeneous bulk structure within a domain region will be called a domain state. Equivalent terms are variant or structural variant (Van Tendeloo & Amelinckx, 1974[link]). We shall use different adjectives to specify domain states. In the microscopic description, domain states associated with the primary order parameter will be referred to as primary (microscopic, basic) domain states. Corresponding domain states in the macroscopic description will be called principal domain states, which correspond to Aizu's orientation states. (An exact definition of principal domain states is given below.)

Further useful division of domain states is possible (though not generally accepted): Domain states that are specified by a constant value of the spontaneous strain are called ferroelastic domain states; similarly, ferroelectric domain states exhibit constant spontaneous polarization etc. Domain states that differ in some tensor properties are called ferroic or tensorial domain states etc. If no specification is given, the statements will apply to any of these domain states.

A domain [{\bf D}_i] is specified by a domain state [{\bf S}_j ] and by domain region [Q_k]: [{\bf D}_i =] [{\bf D}_i({\bf S}_j,Q_k)]. Different domains may possess the same domain state but always differ in the domain region that specifies their shape and position in space.

The term `domain' has also often been used for a domain state. Clear distinction of these two notions is essential in further considerations and is illustrated in Fig.[link]. A ferroelectric domain structure (Fig.[link]) consists of six ferroelectric domains [{\bf D}_1], [{\bf D}_2], [\ldots], [{\bf D}_6] but contains only two domain states [{\bf S}_1], [{\bf S}_2] characterized by opposite directions of the spontaneous polarization depicted in Fig.[link]). Neighbouring domains have different domain states but non-neighbouring domains may possess the same domain state. Thus domains with odd serial number have the domain state [{\bf S}_1] (spontaneous polarization `down'), whereas domains with even number have domain state [{\bf S}_2] (spontaneous polarization `up').


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Hierarchy in domain-structure analysis. (a) Domain structure consisting of domains [{\bf D}_1, {\bf D}_2,\ldots, {\bf D}_6] and domain walls [{\bf W}_{12}] and [{\bf W}_{21}]; (b) domain twin and reversed twin (with reversed order of domain states); (c) domain pair consisting of two domain states [{\bf S}_1] and [{\bf S}_2]; (d) domain states [{\bf S}_1] and [{\bf S}_2 ].

A great diversity of observed domain structures are connected mainly with various dimensions and shapes of domain regions, whose shapes depend sensitively on many factors (kinetics of the phase transition, local stresses, defects etc.). It is, therefore, usually very difficult to interpret in detail a particular observed domain pattern. Domain states of domains are, on the other hand, governed by simple laws, as we shall now demonstrate.

We shall consider a ferroic phase transition with a symmetry lowering from a parent (prototypic, high-symmetry) phase with symmetry described by a point group G to a ferroic phase with the point-group symmetry [F_1 ], which is a subgroup of G. We shall denote this dissymmetrization by a group–subgroup symbol [G \supset F_1] (or [G\Downarrow F_1 ] in Section 3.1.3[link] ) and call it a symmetry descent, dissymmetrization, symmetry lowering or reduction.

As an illustrative example, we choose a phase transition with parent symmetry [G =4_z/m_zm_xm_{xy}] and ferroic symmetry [F_1=2_xm_ym_z] (see Fig.[link]). Strontium bismuth tantalate (SBT) crystals, for instance, exhibit a phase transition with this symmetry descent (Chen et al., 2000[link]). Symmetry elements in the symbols of G and [F_1] are supplied with subscripts specifying the orientation of the symmetry elements with respect to the reference coordinate system. The necessity of this extended notation is exemplified by the fact that the group [G = 4_z/m_zm_xm_{xy}] has six subgroups with the same `non-oriented' symbol [mm2]: [m_xm_y2_z], [2_xm_ym_z], [m_x2_ym_z ], [m_{x{\bar y}}m_{xy}2_z], [2_{x{\bar y}}m_{xy}m_z], [m_{x{\bar y}}2_{xy}m_z]. Lower indices thus specify these subgroups unequivocally and the example illustrates an important rule of domain-structure analysis: All symmetry operations, groups and tensor components must be related to a common reference coordinate system and their orientation in space must be clearly specified.


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Exploded view of single-domain states [{\bf S}_1], [{\bf S}_2], [{\bf S}_3] and [{\bf S}_4 ] (solid rectangles with arrows of spontaneous polarization) formed at a phase transition from a parent phase with symmetry [G=4_z/m_zm_xm_{xy} ] to a ferroic phase with symmetry [F_1=2_xm_ym_z]. The parent phase is represented by a dashed square in the centre with the symmetry elements of the parent group [G=4_z/m_zm_xm_{xy}] shown.

The physical properties of crystals in the continuum description are expressed by property tensors. As explained in Section 1.1.4[link] , the crystal symmetry reduces the number of independent components of these tensors. Consequently, for each property tensor the number of independent components in the low-symmetry ferroic phase is the same or higher than in the high-symmetry parent phase. Those tensor components or their linear combinations that are zero in the high-symmetry phase and nonzero in the low-symmetry phase are called morphic tensor components or tensor parameters and the quantities that appear only in the low-symmetry phase are called spontaneous quantities (see Section[link] ). The morphic tensor components and spontaneous quantities thus reveal the difference between the high- and low-symmetry phases. In our example, the symmetry [F_1 =2_xm_ym_z ] allows a nonzero spontaneous polarization [{\rm P}^{(1)}_0] [=(P,0,0) ], which must be zero in the high-symmetry phase with [G=4_z/m_zm_xm_{xy} ].

We shall now demonstrate in our example that the symmetry lowering at the phase transition leads to the existence of several equivalent variants (domain states) of the low-symmetry phase. In Fig.[link], the parent high-symmetry phase is represented in the middle by a dashed square that is a projection of a square prism with symmetry [4_z/m_zm_xm_{xy} ]. A possible variant of the low-symmetry phase can be represented by an oblong prism with a vector representing the spontaneous polarization. In Fig.[link], the projection of this oblong prism is drawn as a rectangle which is shifted out of the centre for better recognition. We denote by [{\bf S}_1] a homogeneous low-symmetry phase with spontaneous polarization [{\rm P}^{(1)}_0=(P,0,0)] and with symmetry F1 = [2_xm_ym_z]. Let us, mentally, increase the temperature to above the transition temperature and then apply to the high-symmetry phase an operation [2_z], which is a symmetry operation of this high-symmetry phase but not of the low-symmetry phase. Then decrease the temperature to below the transition temperature. The appearance of another variant of the low-symmetry phase [{\bf S}_2] with spontaneous polarization [{\rm P}^{(2)}_0=(-P,0,0) ] obviously has the same probability of appearing as had the variant [{\bf S}_1]. Thus the two variants of the low-symmetry phase [{\bf S}_1 ] and [{\bf S}_2] can appear with the same probability if they are related by a symmetry operation suppressed (lost) at the transition, i.e. an operation that was a symmetry operation of the high-symmetry phase but is not a symmetry operation of the low-symmetry phase [{\bf S}_1 ]. In the same way, the lost symmetry operations [4_z] and [4^3_z] generate from [{\bf S}_1] two other variants, [{\bf S}_3 ] and [{\bf S}_4], with spontaneous polarizations [(0,P,0)] and [(0,-P,0)], respectively. Variants of the low-symmetry phase that are related by an operation of the high-symmetry group G are called crystallographically equivalent (in G) variants. Thus we conclude that crystallographically equivalent (in G) variants of the low-symmetry phase have the same chance of appearing.

We shall now make similar considerations for a general ferroic phase transition with a symmetry descent [G\supset F_1]. By the state S of a crystal we shall understand, in the continuum description, the set of all its properties expressed by property (matter) tensors in the reference Cartesian crystallophysical coordinate system of the parent phase (see Example[link] in Section[link] ). A state defined in this way may change not only with temperature and external fields but also with the orientation of the crystal in space.

We denote by [{\bf S}_1] a state of a homogeneous ferroic phase. If we apply to [{\bf S}_1] a symmetry operation [g_i] of the group G, then the ferroic phase in a new orientation will have the state [{\bf S}_j], which may be identical with [{\bf S}_1] or different. Using the concept of group action (explained in detail in Section[link] ) we express this operation by a simple relation: [g_j{\bf S}_1 = {\bf S}_j, \quad g_j \in G. \eqno( ]

Let us first turn our attention to operations [f_j \in G] that do not change the state [{\bf S}_1]: [f_j{\bf S}_1 = {\bf S}_j,\quad f_j \in G. \eqno( ]The set of all operations of G that leave [{\bf S}_1] invariant form a group called a stabilizer (or isotropy group) of a state [{\bf S}_1] in the group G. This stabilizer, denoted by [I_G({\bf S}_1)], can be expressed explicitly in the following way: [I_G({\bf S}_1) \equiv \{g \in G|g{\bf S}_1 = {\bf S}_1\}, \eqno( ]where the right-hand part of the equation should be read as `a set of all operations of G that do not change the state [{\bf S}_1]' (see Section[link] ).

Here we have to explain the difference between the concept of a stabilizer of an object and the symmetry of that object. By the symmetry group F of an object one understands the set of all operations (isometries) that leave this object [{\bf S}] invariant. The symmetry group F of an object is considered to be an inherent property that does not depend on the orientation and position of the object in space. (The term eigensymmetry is used in Chapter 3.3[link] for symmetry groups defined in this way. Another expression is non-oriented symmetry.) In this case, the symmetry elements of F are `attached' to the object.

A stabilizer describes the symmetry properties of an object in another way, in which the object and the group of isometries are decoupled. One is given a group G, the symmetry elements of which have a defined orientation in a fixed reference system. The object can have any orientation in this reference system. Those operations of G that map the object in a given orientation onto itself form the stabilizer [I_G({\bf S}_1)] of [{\bf S}_i] in the group G. In this case, the stabilizer depends on the orientation of the object in space and is expressed by an `oriented' group symbol [F_1] with subscripts defining the orientation of the symmetry elements of [F_1]. Only for certain `prominent' orientations will the stabilizer acquire a symmetry group of the same crystal class (crystallographic point group) as the eigensymmetry of the object.

We shall define a single-domain orientation as a prominent orientation of the crystal in which the stabilizer [I_G({\bf S}_1)] of its state [{\bf S}_1] is equal to the symmetry group [F_1] which is, after removing subscripts specifying the orientation, identical with the eigensymmetry of the ferroic phase: [I_G({\bf S}_1) = F_1. \eqno( ]This equation thus declares that the crystal in the state [{\bf S}_1 ] has a prominent single-domain orientation.

The concept of the stabilizer allows us to identify the `eigensymmetry' of a domain state (or an object in general) [{\bf S}_i] with the crystallographic class (non-oriented point group) of the stabilizer of this state in the group of all rotations O(3), [I_{O(3)}({\bf S}_i) ].

Since we shall further deal mainly with states of the ferroic phase in single-domain orientations, we shall use the term `state' for a `state of the crystal in a single-domain orientation', unless mentioned otherwise. Then the stabilizer [I_G({\bf S}_1)] will usually be replaced by the group [F_1], although all statements have been derived and hold for stabilizers.

The difference between symmetry groups of a crystal and stabilizers will become more obvious in the treatment of secondary domain states in Section[link] and in discussing disoriented ferroelastic domain states (see Section[link]).

As we have seen in our illustrative example, the suppressed operations generate from the first state [{\bf S}_1] other states. Let [g_j ] be such a suppressed operation, i.e. [g_j \in G ] but [g_j \not\in F_1]. Since all operations that retain [{\bf S}_1] are collected in [F_1], the operation [g_j] must transform [{\bf S}_1] into another state [{\bf S}_j], [g_j{\bf S}_1 = {\bf S}_j \not= {\bf S}_1,\quad g_j\in G,\quad g_j \not\in F_1, \eqno( ]and we say that the state [{\bf S}_j] is crystallographically equivalent (in G) with the state [{\bf S}_1], [{\bf S}_j \buildrel {G} \over \sim {\bf S}_1 ].

We define principal domain states as crystallographically equivalent (in G) variants of the low-symmetry phase in single-domain orientations that can appear with the same probability in the ferroic phase. They represent possible macroscopic bulk structures of (1) ferroic single-domain crystals, (2) ferroic domains in non-ferroelastic domain structures (see Section[link]), or (3) ferroic domains in any ferroic domain structure, if all spontaneous strains are suppressed [this is the so-called parent clamping approximation (PCA), see Section[link]]. In what follows, any statement formulated for principal domain states or for single-domain states applies to any of these three situations. Principal domain states are in one-to-one correspondence with orientation states (Aizu, 1969[link]) or orientation variants (Van Tendeloo & Amelinckx, 1974[link]). The adjective `principal' distinguishes these domain states from primary (microscopic, basic – see Section[link]) domain states and from degenerate domain states, defined in Section[link], and implies that any two of these domain states differ in principal tensor parameters (these are linear combinations of morphic tensor components that transform as the primary order parameter of an equitranslational phase transition with a point-group symmetry descent [G\supset F_1], see Sections[link] and[link]). A simple criterion for a principal domain state [{\bf S}_1] is that its stabilizer in G is equal to the symmetry [F_1] of the ferroic phase [see equation ([link])].

When one applies to a principal domain state [{\bf S}_1] all operations of the group G, one gets all principal domain states that are crystallographically equivalent with [{\bf S}_1]. The set of all these states is denoted [G{\bf S}_1] and is called an G-orbit of [{\bf S}_1] (see also Section[link] ), [G{\bf S}_1 = \{{\bf S}_1, {\bf S}_2,\ldots,{\bf S}_n\}. \eqno( ]In our example, the G-orbit is [4_z/m_zm_xm_{xy}{\bf S}_1=\{{\bf S}_1,{\bf S}_2,{\bf S}_3,{\bf S}_4\} ].

Note that any operation g from the parent group G leaves the orbit [G{\bf S}_1] invariant since its action results only in a permutation of all principal domain states. This change does not alter the orbit, since the orbit is a set in which the sequence (order) of objects is irrelevant. Therefore, the orbit [G{\bf S}_1] is invariant under the action of the parent group G, [GG{\bf S}_1 = G{\bf S}_1].

A ferroic phase transition is thus a paradigmatic example of the law of symmetry compensation (see Section 3.2.2[link] ): The dissymmetrization of a high-symmetry parent phase into a low-symmetry ferroic phase produces variants of the low-symmetry ferroic phase (single-domain states). Any two single-domain states are related by some suppressed operations of the parent symmetry that are missing in the ferroic symmetry and the set of all single-domain states (G-orbit of domain states) recovers the symmetry of the parent phase. If the domain structure contains all domain states with equal partial volumes then the average symmetry of this polydomain structure is, in the first approximation, identical to the symmetry of the parent phase.

Now we find a simple formula for the number n of principal domain states in the orbit [G{\bf S}_1] and a recipe for an efficient generation of all principal domain states in this orbit.

The fact that all operations of the group [I_G({\bf S}_1)=F_1] leave [{\bf S}_1] invariant can be expressed in an abbreviated form in the following way [see equation ([link] )]: [F_1{\bf S}_1 = {\bf S}_1. \eqno(]We shall use this relation to derive all operations that transform [{\bf S}_1] into [{\bf S}_j = g_j{\bf S}_1]: [g_j{\bf S}_1 = g_j(F_1{\bf S}_1) = (g_jF_1){\bf S}_1 = {\bf S}_j, \quad g_j\in G. \eqno( ]The second part of equation ([link]) shows that all lost operations that transform [{\bf S}_1] into [{\bf S}_j] are contained in the left coset [g_jF_1] (for left cosets see Section[link] ).

It is shown in group theory that two left cosets have no operation in common. Therefore, another left coset [g_kF_1] generates another principal domain state [{\bf S}_k] that is different from principal domain states [{\bf S}_1] and [{\bf S}_j]. Equation ([link] defines, therefore, a one-to-one relation between principal domain states of the orbit [G{\bf S}_1] and left cosets of [F_1] [see equation ([link] )], [{\bf S}_j \leftrightarrow g_{j}F_1, \quad F_1= I_G({\bf S}_1), \quad j = 1,2,\ldots, n. \eqno( ]From this relation follow two conclusions:

  • (1) The number n of principal domain states equals the number of left cosets of [F_1]. All different left cosets of [F_1] constitute the decomposition of the group G into left cosets of [F_1] [see equation ([link] )], [G = g_1F_{1} \cup g_2F_{1} \cup\ldots\cup g_jF_{1}\cup\ldots \cup g_nF_{1}, \eqno( ]where the symbol [\cup] is a union of sets and the number n of left cosets is called the index of G in [F_1] and is denoted by the symbol [[G:F_1]]. Usually, one chooses for [g_1 ] the identity operation e; then the first left coset equals [F_1]. Since each left coset contains [|F_1|] operations, where [|F_1|] is number of operations of [F_1] (order of [F_1]), the number of left cosets in the decomposition ([link]) is [n=[G:F_1]=|G|:|F_1|, \eqno(]where [|G|, |F_1|] are orders of the point groups [G, F_1], respectively. The index n is a quantitative measure of the degree of dissymmetrization [G \supset F_1]. Thus the number of principal domain states in orbit [G{\bf S}_1] is equal to the index of [F_1] in G, i.e. to the number of operations of the high-symmetry group G divided by the number of operations of the low-symmetry phase [F_1]. In our illustrative example we get [n =|4_z/m_zm_xm_{xy}|:|2_xm_ym_z|= 16:4= 4].

    The basic formula ([link]) expresses a remarkable result: the number n of principal domain states is determined by how many times the number of symmetry operations increases at the transition from the low-symmetry group [F_1] to the high-symmetry group G, or, the other way around, the fraction [{{1}\over{n}}] is a quantitative measure of the symmetry decrease from G to [F_1], [|F_1| = {{1}\over{n}}|G|]. Thus it is not the concrete structural change, nor even the particular symmetries of both phases, but only the extent of dissymmetrization that determines the number of principal domain states. This conclusion illustrates the fundamental role of symmetry in domain structures.

  • (2) Relation ([link]) yields a recipe for calculating all principal domain states of the orbit [G{\bf S}_1]: One applies successively to the first principal domain states [{\bf S}_1] the representatives of all left cosets of [F_1]: [G{\bf S}_1 = \{{\bf S}_1, g_2{\bf S}_1,\ldots, g_j{\bf S}_1,\ldots, g_n{\bf S}_1\}, \eqno( ]where the operations [g_1 = e,g_2,\ldots,g_j,\ldots,g_n] are the representatives of left cosets in the decomposition ([link]) and e is an identity operation. We add that any operation of a left coset can be chosen as its representative, hence the operation [g_j] can be chosen arbitrarily from the left coset [g_jF_1], [j=1,2,\ldots,n ].

This result can be illustrated in our example. Table[link] presents in the first column the four left cosets [g_j\{2_xm_ym_z\}] of the group [F_1 = 2_xm_ym_z]. The corresponding principal domain states [{\bf S}_j], [j = 1,2,3,4,] and the values of spontaneous polarization in these principal domain states are given in the second and the third columns, respectively. It is easy to verify in Fig.[link] that all operations of each left coset transform the first principal domain state [{\bf S}_1] into one principal domain state [{\bf S}_j], [j = 2, 3, 4.]

Table| top | pdf |
Left and double cosets, principal and secondary domain states and their tensor parameters for the phase transition with [G=4_z/m_zm_xm_{xy} ] and [F_1=2_xm_ym_z]

P00) and (0±P0): polarization; (000±g00) and (0000±g0): optical activity; u1, u2: strain; s11, s22: elastic compliances (see Fig.[link]).

Left cosets [g_j{\bf S}_1]Principal domain statesSecondary domain states
[1] [2_x] [m_y] [m_z] [{\bf S}_1] [(P00) ] [(000g00) ] [{\bf R}_1] [u_{1}-u_{2}] [s_{11}-s_{22}]
[\bar{1} ] [m_x] [2_y] [2_z] [{\bf S}_2 ] [(-P00) ] [(000{-g}00) ]
[2_{xy} ] [4_z ] [\bar{4}^3_{z} ] [m_{x\bar{y}}] [{\bf S}_3 ] [(0P0) ] [(0000{-g}0) ] [{\bf R}_2 ] [u_{2}-u_{1} ] [s_{22}-s_{11} ]
[2_{x\bar{y}} ] [4^3_z] [\bar{4}_{z}] [m_{xy}] [{\bf S}_4 ] [(0{-P}0) ] [(0000g0)]

The left coset decompositions of all crystallographic point groups and their subgroup symmetry are available in the software GI[\star ]KoBo-1, path: Subgroups\View\Twinning Group.

Let us turn briefly to the symmetries of the principal domain states. From Fig.[link] we deduce that two domain states [{\bf S}_1 ] and [{\bf S}_2] in our illustrative example have the same symmetry, [F_1 =] [F_2 =2_xm_ym_z], whereas two others [{\bf S}_3] and [{\bf S}_4] have another symmetry, [F_3 =F_4 =m_x2_ym_z]. We see that symmetry does not specify the principal domain state in a unique way, although a principal domain state [{\bf S}_j] has a unique symmetry [F_i =I_G({\bf S}_j)].

It turns out that if [g_j] transforms [{\bf S}_1] into [{\bf S}_j], then the symmetry group [F_j] of [{\bf S}_j] is conjugate by [g_j] to the symmetry group [F_1] of [{\bf S}_1 ] [see Section[link] , Proposition[link] and equation ([link] )]: [\hbox {if }{\bf S}_j = g_j{\bf S}_1, \hbox { then } F_j = g_jF_1g_j^{-1}. \eqno( ]One can easily check that in our example each operation of the second left coset of [F_1=2_xm_ym_z] (second row in Table[link]) transforms [F_1=2_xm_ym_z] into itself, whereas operations from the third and fourth left cosets yield [F_3=F_4=m_x2_ym_z]. We shall return to this issue again at the end of Section[link].


Aizu, K. (1969). Possible species of `ferroelastic' crystals and of simultaneously ferroelectric and ferroelastic crystals. J. Phys. Soc. Jpn, 27, 387–396.
Chen, X. J., Liu, J. S., Zhu, J. S. & Wang, Y. N. (2000). Group theoretical analysis of the domain structure of SrBi2Ta2O9 ferroelectric ceramic. J Phys. Condens. Matter, 12, 3745–3749.
Van Tendeloo, G. & Amelinckx, S. (1974). Group-theoretical considerations concerning domain formation in ordered alloys. Acta Cryst. A30, 431–440.

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