International
Tables for Crystallography Volume D Physical properties of crystals Edited by A. Authier © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. D. ch. 3.3, p. 426
Section 3.3.10.1. Contact relations in twinning
a
Institut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D-52056 Aachen, Germany, and bMineralogisch-Petrologisches Institut, Universität Bonn, D-53113 Bonn, Germany |
So far, twinning has been discussed only in terms of symmetry and orientation relations of the (bulk) twin components. In this chapter, the very important aspect of contact relations is discussed. This topic concerns the orientation and the structure of the twin boundary, which is also called twin interface, composition plane, contact plane, domain boundary or domain wall. It is the twin boundary and its structure and energy which determine the occurrence or non-occurrence of twinning. In principle, for each crystal species an infinite number of orientation relations obey the requirements for twinning, as set out in Section 3.3.2, because any rational lattice plane (hkl), as well as any rational lattice row [uvw], common to both partners would lead to a legitimate reflection or rotation twin. Nevertheless, only a relatively small number of crystal species exhibit twinning at all, and, if so, with only a few twin laws. This wide discrepancy between theory and reality shows that a permissible crystallographic orientation relation (twin law) is a necessary, but not at all a sufficient, condition for twinning. In other words, the contact relations play the decisive role: a permissible orientation relation can only lead to actual twinning if a twin interface of good structural fit and low energy is available.
In principle, a twin boundary is a special kind of grain boundary connecting two `homophase' component crystals which exhibit a crystallographic orientation relation, as defined in Section 3.3.2. For a given orientation relation of the twin partners, crystallographic or general, the interface energy depends on the orientation of their boundary. It is intuitively clear that crystallographic orientation relations lead to energetically more favourable boundaries than noncrystallographic ones. As a rule, twin boundaries are planar (at least in segments), but for certain types of twins curved and irregular interfaces have been observed. This is discussed later in this section.
In order to determine theoretically for a given twin law the optimal interface, the interface energy has to be calculated or at least estimated for various boundary orientations. This problem has not been solved for the general case so far. The special situation of reflection twins with coinciding twin mirror and composition planes has recently been treated by Fleming et al. (1997). These authors calculated the interface energies for three possible reflection twin laws in each of aragonite, gibbsite, corundum, rutile and sodium oxalate, and they compared the results with the observed twinning. In all cases, the twin law with lowest boundary energy corresponds to the twin law actually observed. Another calculation of the twin interface energy has been performed by Lieberman et al. (1998) for the reflection twins of monoclinic saccharin crystals. In this study, the boundary energy was calculated for different shifts of the two twin components with respect to each other. It was shown that a minimum of the boundary energy is achieved for a particular `twin displacement vector' (cf. Section 3.3.10.4.1).
Calculations of interface energies, as performed by Fleming et al. (1997) and Lieberman et al. (1998), however, require knowledge of the atomic potentials and their parameters for each pair of bonded atoms. They are, therefore, restricted to specific crystals for which these parameters are known. Similarly, high-resolution electron microscopy (HRTEM) images of twin boundaries have been obtained so far for only a small number of crystals.
It is possible, however, to predict for a given twin law low-energy twin boundaries on the basis of symmetry considerations, even without knowledge of the crystal structure, as discussed in the following section. This prediction has been carried out by Sapriel (1975) for ferroelastic crystals. His treatment assumes a phase transition from a real or hypothetical parent phase (supergroup ) to a `distorted' (daughter) phase of lower eigensymmetry (subgroup ), leading to two (or more) domain states of equal but opposite shear strain. The subgroup must belong to a lower-symmetry crystal system than the supergroup , as explained in Section 3.3.7.3(ii). Similar criteria, but restricted to ferroelectric materials, had previously been devised in 1969 by Fousek & Janovec (1969). A review of ferroelastic domains and domain walls is provided by Boulesteix (1984) and an extension of the Sapriel procedure to phase boundaries between a ferroelastic and its `prototypic' (parent) phase is given by Boulesteix et al. (1986).
References
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