International
Tables for Crystallography Volume E Subperiodic groups Edited by V. Kopský and D. B. Litvin © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. E, ch. 5.2, pp. 414419
Section 5.2.5. Applications^{a}Freelance research scientist, Bajkalská 1170/28, 100 00 Prague 10, Czech Republic, and ^{b}Department of Physics, The Eberly College of Science, Penn State – Berks Campus, The Pennsylvania State University, PO Box 7009, Reading, PA 19610–6009, USA 
The following two examples show the use of layer symmetries in the description of crystal structures.
Example 1
Fig. 5.2.5.1 shows the crystal structure of cadmium iodide, CdI_{2}. The space group of this crystal is , (No. 164). The anions form a hexagonal close packing of spheres and the cations occupy half of the octahedral holes, filling one of the alternate layers. In closepacking notation, the CdI_{2} structure is:
From the scanning tables, we obtain for planes with the (0001) orientation and at heights 0c or a sectional layer symmetry (L72), and for planes of this orientation at any other height a sectional layer symmetry (L69).
The plane at height 0c contains cadmium ions. This plane defines the orbit of planes of orientation (0001) located at points , where (Z is the set of all integers). All these planes contain cadmium ions in the same arrangement (C layer filled with Cd).
The plane at height defines the orbit of planes of orientation (0001) located at points , where . All these planes lie midway between A and B layers of iodine ions with the B layer below, the A layer above the plane. They contain only voids.
The planes at levels and contain B and A layers of iodine ions, respectively. These planes and all planes produced by translations from them belong to the same orbit because the operations exchange the A and B layers.
Example 2
The space group of cadmium chloride, CdCl_{2}, is , (No. 166). Fig. 5.2.5.2 shows the structure of CdCl_{2} in its triple hexagonal cell. The anions form a cubic close packing of spheres and the cations occupy half of the octahedral holes of each alternate layer. In closepacking notation, the CdCl_{2} structure is:
We choose the origin at a cadmium ion and the hexagonal basis vectors a, b as shown in Fig. 5.2.5.2. This corresponds to the obverse setting for which the scanning table is given in Part 6. The planes with the (0001) orientation at the heights 0c, , , , and have a sectional layer group of the type (L72) and at any other height have a sectional layer group of the type (L69).
The scanning table also specifies the location of the sectional layer groups. The position along the c axis, where the basis vector specifies the scanning direction, is given by fractions of d or by in the case of a general position. At the heights 0c and , the sectional layer group is the group (L72), while at the heights and it is the group (L72), and at the heights and it is the group , (L72), where the vectors in brackets mean the shift of the group in space. The planes at the heights 0d, and belong to one translation orbit and the layers contain cadmium ions which are shifted relative to each other by the vectors and . The planes at the heights , and contain the voids and are located midway between layers of chlorine ions; they belong to another linear orbit and again are shifted relative to each other by the vectors and .
The scanning for the sectional layer groups is a procedure which finds applications in the theory of bicrystals and their interfaces. The first of these two terms was introduced in the study of grain boundaries (Pond & Bollmann, 1979; Pond & Vlachavas, 1983; Vlachavas, 1985; Kalonji, 1985). An ideal bicrystal is understood to be an aggregate of two semiinfinite crystals of identical structure, meeting at a common planar boundary called the interface, where one of the structures, occupying halfspace on one side of the interface, is misoriented and/or displaced relative to the other structure occupying the other halfspace. The word interface is a synonym for a boundary and interfaces considered here are homophase interfaces, in contrast with heterophase interfaces, where the two structures are different (Sutton & Balluffi, 1995).
An independent study of domain and twin boundaries (Janovec, 1981; Zikmund, 1984) resulted in a terminology parallel to that of the bicrystallography. The basic concept here is the domain twin, which is technically a particular case of a bicrystal. In this section, we use the terminology of bicrystals, giving the terminology of domain twins, used in the next section, parenthetically.
In both cases, the aim of the analysis is to determine the symmetry group of a bicrystal (domain twin), corresponding to a certain orientation and location of the interface (domain wall or twin boundary), which is a certain layer group.
In the first step of the analysis, one constructs a dichromatic complex or pattern [(unordered) domain pair].
It is an abstract construction, a superposition of two infinite crystals which have the same structure, orientation and/or location as the two semiinfinite crystals of the bicrystal (domain twin) when extended to infinity. The two components are referred to as black and white crystals or variants (single domain states).
The symmetry group of the dichromatic complex (domain pair) is the group of those Euclidean motions which either leave both black and white crystals (domain states) invariant or which exchange them. Planes of various orientations and locations, representing the interface, are then considered as transecting the dichromatic complex (domain pair). To each such plane there corresponds a sectional layer group , the elements of which leave invariant the dichromatic pattern (domain pair) and the plane. A bicrystal (domain twin) is obtained by deleting from one side of the plane the atoms of one of the components of the dichromatic pattern (single domain states) and the atoms of the second component (single domain state) from the other side of the plane. The symmetry of the bicrystal (domain twin) is a layer group which contains those elements of the sectional layer group of the dichromatic pattern (domain pair) that satisfy one of the following two conditions:
Example
Consider the bicrystal consisting of two facecentred cubic crystals misoriented by a rotation of 36.9° about the [001] direction. The corresponding dichromatic complex is shown in Fig. 5.2.5.3. The symmetry group of the complex is the space group , (No. 139).
Vlachavas (1985) has tabulated the symmetries of bicrystals arising when the above dichromatic complex is transected with planes of various orientations and locations. For planes of the orientation (001), given with reference to the tetragonal coordinate system shown in Fig. 5.2.5.3, Vlachavas lists
The position of the plane is given in terms of a fraction of the basis vector of the tetragonal c axis. The `p' in the symbol of the symmetry groups of the bicrystal denotes all translations in the (001) plane.
From the subtable for the space group , (No. 139) in the scanning tables, Part 6, one finds
The symmetry group of the bicrystal is that subgroup of the corresponding sectional layer group consisting of all elements that satisfy one of the two conditions given above. For example, for the plane at position `0', the sectional layer group is (L61). None of the mirror planes satisfies either of the conditions. The mirror plane perpendicular to [001] inverts the normal to the plane but leaves invariant both black and white crystals. The mirror planes perpendicular to [001] and [010] leave the normal to the plane invariant, but exchange the black and white crystals. The fourfold rotation satisfies condition (i), and the twofold rotations about auxiliary axes satisfy condition (ii). Consequently, from the sectional layer groups (L61), (L64) and (L55) one obtains the respective symmetries of the bicrystal with different locations of interfaces: (L53), (L54) and (L49), as listed by Vlachavas.
The symmetry of domain twins with planar coherent domain walls and the symmetry of domain walls themselves are also described by layer groups (see e.g. Janovec et al., 1989), from which conclusions about the structure and tensorial properties of the domain walls can be deduced. The derivation of the layer symmetries of twins and domain walls is again facilitated by the scanning tables. As shown below, the symmetry of a twin is in general expressed through four sectional layer groups, where the central plane of the interface is considered as the section plane of an ordered and unordered domain pair. The relations between the symmetries and possible conclusions about the structure of the wall will be illustrated by an analysis of a domain twin in univalent mercurous halide (calomel) crystals.
A twin is a particular case of a bicrystal in which the relative orientation and/or displacement of the two components is not arbitrary; it is required that the operation that sends one of the components to the other is crystallographic. A domain twin is a special case where the structures and of the two components (domains) are distortions of a certain parent structure , the symmetry of which is a certain space group , called the parent group. The parent structure is either a real structure, the distortions of which are due to a structural phase transition, or it is a hypothetical structure. If the symmetry of one of the distorted structures is , then, from the coset decompositionwe obtain equivalent distorted structures , , with symmetries which form a set of conjugate subgroups of .
Hence, a domain twin is always connected with a certain symmetry descent from a space group to a set of conjugate subgroups . The distorted structures are called the single domain states. A domain twin consists of two semiinfinite regions (halfspaces), called domains, separated by a planar interface called the central plane. The structures at infinite distance from this plane coincide with the domain states. The structure in the vicinity of the central plane is called the domain wall. The aim of the symmetry analysis is to determine the possible structure of the domain wall.
Basic theory: We consider a domain twin in which the domains are occupied by single domain states and . To define the twin uniquely, we first observe that Miller indices or corresponding normal to the interface (central plane of the domain wall) define not only the orientation of the wall but also its sidedness, so that one can distinguish between the two halfspaces. The normal points from one of the halfspaces to the other while points in the opposite direction. The twin is then defined uniquely by the symbol = , which means that the domains are separated by the plane of orientation and location , where is the scanning vector. The symbol also specifies that the normal points from the halfspace occupied by domain state to the halfspace occupied by domain state .
Now we consider the changes of the twin under the action of those isometries which leave the plane invariant. The action of such an isometry g on the twin is expressed by , where is the linear constituent of the isometry g and . Among these isometries, there are in general two kinds which define the symmetry of the twin and two which reverse the twin. The symbols for these four kinds of operations, their action on the initial twin , their graphical representation and the names of the resulting twins are as shown in Fig. 5.2.5.4.
An auxiliary notation has been introduced in which an asterisk labels operations that exchange the domain states and an underline labels operations that change the normal to the plane of the wall. To avoid misinterpretation (the symbolism is similar to that of the symmetry–antisymmetry groups), let us emphasize that neither the asterisk nor the underline have any meaning of an operation; they are just suitable labels which can be omitted without changing the meaning of general or specific symbols of the isometries. Operations with these labels mean the same as if the labels are dropped.
The operations leave invariant the normal as well as the states and . Such operations are called the trivial symmetry operations of a domain twin and they constitute a certain layer group . The exchange the halfspaces because they invert the normal and at the same time they exchange the domain states and . As a result they leave the twin invariant, changing only the direction of the normal. These operations are called the nontrivial symmetry operations of a domain twin. If is one such operation, then all these operations are contained in a coset . Operations , called the sidereversing operations, exchange the halfspaces, leaving the domain states and invariant, and operations , called the statereversing operations, exchange the domain states and , leaving the halfspaces invariant.
The symmetry group , or in short , of the twin can therefore be generally be expressed aswhere is a group of all trivial symmetry operations and is the coset of all nontrivial symmetry operations of the twin.
The group is a layer group which can be deduced from four sectional layer groups of two space groups which describe the symmetry of two kinds of domain pairs formed from the domain states and (Janovec, 1972):
An ordered domain pair is an analogue of the dichromatic complex in which we keep track of the two components. The symmetry group of this pair must therefore leave invariant both domain states and is expressed as the intersectionof symmetry groups and of the respective single domain states and , where is an operation transforming into : .
The sectional layer group of the central plane with normal n and at a position under the action of the space group is generally expressed aswhere the halving subgroup is the floating sectional layer group at a general position . The operation inverts the normal and thus exchanges halfspaces on the left and right sides of the wall, where the left side is occupied by the state and the right side by the state in the initial twin. These operations appear only at special positions of the central plane. Since the halfspaces are occupied by domain states and , their exchange is accompanied by an exchange of domain states on both sides of the wall. The operation changes neither nor and hence it results in a reversed domain twin which has domain state on the left side and the domain state on the right side of the wall.
The unordered domain pair has the symmetry described by the group where is an operation that exchanges and , , . Since for an unordered domain pair , the symmetry operations of the left coset are also symmetry operations of the unordered domain pair . If such an operation and hence the whole coset of statereversing operations exists, then the domain pair is called transposable. Otherwise and the domain pair is called nontransposable.
The sectional layer group of the space group can therefore be generally written in the formIn the general case, the group contains three halving subgroups which intersect at the subgroup of index four: the subgroup is the floating subgroup of ; the coset is present if and only if the domain pair is transposable. The group is the sectional layer group for the ordered domain pair defined above. Finally, the group is the symmetry group of the twin [see (5.2.5.2)]. Notice that it is itself not a sectional layer group of the space groups and involved unless , which occurs in the case of a nontransposable domain pair and of a general position of the central plane.
Since the cosets can be settheoretically expressed as differences of groups: and , while , we receive a compact settheoretical expression for the symmetry group of the twin in terms of four sectional layer groups:Thus the symmetry group of the twin can be expressed in terms of two sectional layer groups , and their floating subgroups , , respectively. These four sectional layer groups can be found in the scanning tables.
As an illustrative example, we consider below a domain twin with a ferroelastic wall in the orthorhombic ferroelastic phase of the calomel crystal Hg_{2}Cl_{2}. Original analysis which includes the domain twin with antiphase boundary is given by Janovec & Zikmund (1993). Another analysis performed prior to the scanning tables is that of the domain twin in the KSCN crystal (Janovec et al., 1989). Various cases of domain twins in fullerene C_{60} have also been analysed with the use of scanning tables (Janovec & Kopský, 1997; SaintGrégoire et al., 1997).
Example
The parent phase of calomel has a tetragonal bodycentred structure of spacegroup symmetry (), where lattice points are occupied by calomel molecules which have the form of Cl–Hg–Hg–Cl chains along the c axis. The crystallographic coordinate system is defined by vectors of the conventional tetragonal basis , , with reference to the Cartesian basis and the origin P is chosen at the centre of gravity of one of the calomel molecules. The parent structure projected onto the plane is depicted in the middle of Fig. 5.2.5.5, where full and empty circles denote the centres of gravity at the levels and , respectively.
The ferroic phase is orthorhombic with a spacegroup symmetry of the type (), the conventional orthorhombic cell is based on vectors , , and contains two original cells. The conventional cell of the original tetragonal structure and the cells of the four single domain states , , and are shaded in Fig. 5.2.5.5. The arrows represent spontaneous shifts , , and of gravity centres of molecules. The two single domain states and have the symmetry ( or ); the other two single domain states and have the symmetry ( or ), where the Hermann–Mauguin symbols refer to the orthorhombic basis. There are two classes of domain pairs, represented by the pairs and , which result in domain walls referred to as a ferroelastic domain wall and an antiphase boundary, respectively. We shall consider the first of these cases.
The two single domain states , and the unordered pair are represented in Fig. 5.2.5.6. The symmetries of the single domain states and of both the ordered and unordered pair are given in Table 5.2.5.1, where subscripts indicate the orientation of symmetry elements with reference to the Cartesian basis and an asterisk denotes operations that exchange the single domain states.

We consider the domain walls of the orientation (100) with reference to the original tetragonal basis . This is the orientation with the Miller indices (110) with reference to the orthorhombic basis . Consulting the scanning table No. 136 for the group (), we find the scanning group () with reference to its conventional basis , where , , . Applying the results of the scanning table with the shift by , we obtain the sectional layer groups and and their floating subgroup (for ). Analogously, for the space group , we obtain the sectional layer groups and and their floating subgroup (for ). All these groups are collected in the Table 5.2.5.2 in two notations. In this table, with a specified basis, each standard symbol contains the same information as the optional symbol. Optional symbols contain subscripts which explicitly specify the orientations of symmetry elements with reference to the Cartesian coordinate system , asterisks and underlines have the meaning specified above. The lattice symbol p means the common lattice of all sectional layer groups and twin symmetries. The Hermann–Mauguin symbols are written with reference to the coordinate systems .

The twin symmetry is determined by the relation (5.2.5.7). This means, in practice, that we have to find the groups , , and from which we obtain the group . If tables of subgroups of layer groups were available, it would be sufficient to look up the subgroups which lie between and and recognize the three groups , and .
Optional symbols facilitate this determination considerably. To get the twin symmetry , we look up the optional symbol for the group and eliminate elements that are either only underlined or that are only labelled by an asterisk. Or, vice versa, we leave only those elements that are not labelled at all or that are at the same time underlined and labelled by an asterisk. The resulting twin symmetries are given in the lower part of Table 5.2.5.2.
The implications of this symmetry analysis on the structure of domain walls at and are illustrated in Fig. 5.2.5.7. Shaded areas represent the domain states at infinity. The lefthand part of the figure corresponds to the location of the central plane at , the righthand part to the location at . The twin symmetries and determine the relationship between the structures in the two halfspaces. The trivial symmetry operations form the layer group in both cases and leave invariant the structures in both halfspaces. The nontrivial symmetry operations map the structure in one of the halfspaces onto the structure in the other halfspace and back. The symmetry of the central plane is given by the groups and because the states and meet at this plane.

The structures and symmetries of domain twins in calomel corresponding to two different special positions of the wall. 
The arrows that represent the shift of calomel molecules in the plane may rotate and change their amplitude as we approach the central plane because the symmetry requirements are relaxed to those imposed by the layer group consisting of trivial symmetry operations of the twin. The nontrivial twin symmetries determine the relationship between the structures in the two halfspaces, so that the rotation and change of amplitude in these two halfspaces are correlated. The symmetry of the central plane requires, in the lefthand part of the figure, that the arrows at black circles are aligned along the plane and that they are of the same lengths and alternating direction. The arrows at the empty circles in the righthand part of the figure are nearly perpendicular to the plane, of the same lengths and of alternating direction in accordance with the centralplane symmetry. They are shown in the figure as strictly perpendicular to the plane; however, slight shifts of the atoms parallel to the plane can be expected because the arrows mean that the atoms are actually already out of the central plane.
Summary: In the analysis of domain twins, we know the structures of the two domain states, in our case the orientation of arrows, at infinity. In the example above, we considered two cases in both of which the layer group contains all four types of the twin operations – two types of symmetry operations and two types of twinreversing operations. In this case, we summarize the results of the symmetry analysis as follows. (i) The floating layer group determines the allowed changes of the structures on the path from infinity (physically this means the domain bulk) towards the central plane. (ii) Operations of the coset correlate the changes in the two halfspaces. (iii) The group as the symmetry of the central plane where the two halfspaces meet contains the twin symmetry as its halving subgroup and therefore imposes additional conditions on the structure of the central plane in comparison with the conditions in its vicinity.
As always, the symmetry determines only the character of possible changes but neither their magnitude nor their dependence on the distance from the central plane. Thus, in the example considered, the symmetry arguments cannot predict the detailed dependence of the angle of rotation on the distance from the wall and they cannot predict whether and how the lengths of these arrows change.
References
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