Formal description of simple domain twins and planar domain walls of zero thickness

Janovec, V.; Přívratská

doi:10.1107/97809553602060000645

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

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International Tables for Crystallography (2006). Vol. D. ch. 3.4, pp. 491-492

Section 3.4.4.1. Formal description of simple domain twins and planar domain walls of zero thickness

V. Janovec^a ^* and J. Přívratská^b

^a Department of Physics, Technical University of Liberec, Hálkova 6, 461 17 Liberec 1, 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

3.4.4.1. Formal description of simple domain twins and planar domain walls of zero thickness

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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 $[{\bf T}_{12}]$ that consists of two domains $[{\bf D}_1]$ and $[{\bf D}_2]$ which meet along a planar domain wall $[{\bf W}_{12}]$ of zero thickness. Let us denote by p a plane of the domain wall, in brief wall plane of $[{\bf W}_{12}]$ . This plane is specified by Miller indices [(hkl)] , or by a normal $[{\bf n}]$ 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 half-spaces. Using the bra–ket symbols, mentioned in Section 3.4.3.6, we shall denote by $[(\, |]$ the half-space on the negative side of p and by $[| \,)]$ the half-space on the positive side of p.

A simple twin consists of two (theoretically semi-infinite) domains $[{\bf D}_1]$ and $[{\bf D}_2]$ with domain states $[{\bf S}_1]$ and $[{\bf S}_2]$ , respectively, that join along a planar domain wall the orientation of which is specified by the wall plane p with normal n. A symbol $[({\bf S}_1|{\bf n}|{\bf S}_2)]$ specifies the domain twin unequivocally: domain $[({\bf S}_1|]$ , with domain region $[(\, | ]$ filled with domain state $[{\bf S}_1]$ , is on the negative side of p and domain $[|\,{\bf S}_2)]$ is on the positive side of p (see Fig. 3.4.4.1a).

Figure 3.4.4.1 | top | pdf |

Symbols of a simple twin. (a) Two different symbols with antiparallel normal $[\bf n]$ . (b) Symbols of the reversed twin.

If we were to choose the normal of opposite direction, i.e. $[-{\bf n}]$ , the same twin would have the symbol $[({\bf S}_2|-{\bf n}|{\bf S}_1) ]$ (see Fig. 3.4.4.1a). Since these two symbols signify the same twin, we have the identity $[({\bf S}_{1}|{\bf n}|{\bf S}_{2}) \equiv ({\bf S}_{2}|-{\bf n}|{\bf S}_{1}). \eqno(3.4.4.1) ]$ Thus, if we invert the normal n and simultaneously exchange domain states $[{\bf S}_{1}]$ and $[{\bf S}_{2}]$ 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 $[{\bf n}]$ .

The full symbol of the twin can be replaced by a shorter symbol $[{\bf T}_{12}({\bf n}) ]$ 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 $[{\bf n}]$ . Then the identity (3.4.4.1) in short symbols is $[{\bf T}_{12}({\bf n}) \equiv {\bf T}_{21}(-{\bf n}). \eqno(3.4.4.2) ]$

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 $[({\bf S}_{1}|{\bf n}|{\bf S}_2)]$ , or $[{\bf T}_{12}({\bf n}) ]$ , can be formed by sectioning the ordered domain pair $[({\bf S}_1,{\bf S}_2) ]$ by a plane p with normal $[{\bf n}]$ and removing the domain state $[{\bf S}_2]$ on the negative side and domain state $[{\bf S}_2]$ on the positive side of the normal $[{\bf n}]$ . 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 $[({\bf S}_{1}|{\bf n}|{\bf S}_{2}) ]$ is $[({\bf S}_{2}|{\bf n}|{\bf S}_{1}) \equiv ({\bf S}_{1}|-{\bf n}|{\bf S}_{2}) \, \eqno(3.4.4.3) ]$ or $[{\bf T}_{21}({\bf n}) \equiv {\bf T}_{12}(-{\bf n}). \eqno(3.4.4.4) ]$ A reversed twin $[({\bf S}_{2}|{\bf n}|{\bf S}_{1}) \equiv ({\bf S}_{1}|{\bf -n}|{\bf S}_{2}) ]$ is depicted in Fig. 3.4.4.1(b).

A planar domain wall is the interface between the domains $[{\bf D}_1]$ and $[{\bf D}_2]$ 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: $[[{\bf S}_{1}|{\bf n}|{\bf S}_{2}] \equiv [{\bf S}_{2}|-{\bf n}|{\bf S}_{1}] \eqno(3.4.4.5) ]$ or by a shorter equivalent symbol $[{\bf W}_{12}({\bf n}) \equiv {\bf W}_{21}(-{\bf n}). \eqno(3.4.4.6) ]$

References

International Tables for Crystallography (2006). Vol. D. ch. 3.4, pp. 491-492