International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2013 
International Tables for Crystallography (2013). Vol. D, ch. 3.4, pp. 528529
Section 3.4.4.1. Formal description of simple domain twins and planar domain walls of zero thickness^{a}Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ18221 Prague 8, Czech Republic, and ^{b}Department of Mathematics and Didactics of Mathematics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, Czech Republic 
In this section, we examine crystallographic properties of planar compatible domain walls and simple domain twins. The symmetry of these objects is described by layer groups. Since this concept is not yet common in crystallography, we briefly explain its meaning in Section 3.4.4.2. The exposition is performed in the continuum description, but most of the results apply with slight generalizations to the microscopic treatment that is illustrated with an example in Section 3.4.4.7.
We shall consider a simple domain twin that consists of two domains and which meet along a planar domain wall of zero thickness. Let us denote by p a plane of the domain wall, in brief wall plane of . This plane is specified by Miller indices , or by a normal to the plane which also defines the sidedness (plus and minus side) of the plane p. By orientation of the plane p we shall understand a specification which can, but may not, include the sidedness of p. If both the orientation and the sidedness are given, then the plane p divides the space into two halfspaces. Using the bra–ket symbols, mentioned in Section 3.4.3.6, we shall denote by the halfspace on the negative side of p and by the halfspace on the positive side of p.
A simple twin consists of two (theoretically semiinfinite) domains and with domain states and , respectively, that join along a planar domain wall the orientation of which is specified by the wall plane p with normal n. A symbol specifies the domain twin unequivocally: domain , with domain region filled with domain state , is on the negative side of p and domain is on the positive side of p (see Fig. 3.4.4.1a).

Symbols of a simple twin. (a) Two different symbols with antiparallel normal . (b) Symbols of the reversed twin. 
If we were to choose the normal of opposite direction, i.e. , the same twin would have the symbol (see Fig. 3.4.4.1a). Since these two symbols signify the same twin, we have the identity Thus, if we invert the normal n and simultaneously exchange domain states and in the twin symbol, we obtain an identical twin (see Fig. 3.4.4.1a). This identity expresses the fact that the specification of the twin by the symbol introduced above does not depend on the chosen direction of the wall normal .
The full symbol of the twin can be replaced by a shorter symbol if we accept a simple convention that the first lower index signifies the domain state that occupies the half space on the negative side of . Then the identity (3.4.4.1) in short symbols is
If the orientation and sidedness of the plane p of a wall is known from the context or if it is not relevant, the specification of n in the symbol of the domain twin and domain wall can be omitted.
A twin , or , can be formed by sectioning the ordered domain pair by a plane p with normal and removing the domain state on the negative side and domain state on the positive side of the normal . This is the same procedure that is used in bicrystallography when an ideal bicrystal is derived from a dichromatic complex (see Section 3.2.2 ).
A twin with reversed order of domain states is called a reversed twin. The symbol of the twin reversed to the initial twin is or A reversed twin is depicted in Fig. 3.4.4.1(b).
A planar domain wall is the interface between the domains and of the associated simple twin. Even a domain wall of zero thickness is specified not only by its orientation in space but also by the domain states that adhere to the minus and plus sides of the wall plane p. The symbol for the wall is, therefore, analogous to that of the twin, only in the explicit symbol the brackets ( ) are replaced by square brackets [ ] and T in the short symbol is replaced by W: or by a shorter equivalent symbol