Section 3.3.10.5. Examples of structural models of twin boundaries

The earliest structural model of a twin boundary was derived for aragonite by Bragg (1924), reviewed in Bragg (1937, pp. 119–121) and Bragg & Claringbull (1965, pp. 131–133). Aragonite is orthorhombic with space group Pmcn. It exhibits a pronounced hexagonal pseudosymmetry, corresponding to a (hypothetical) parent phase of symmetry , in which the Ca ions form a hexagonal close-packed structure with the CO₃ groups filling the octahedral voids along the axes. By eliminating the threefold axis and the C-centring translation of the orthohexagonal unit cell, the above orthorhombic space group results, where the lost centring translation now appears as the glide component n. Of the three mirror planes parallel to and the three c-glide planes parallel to , one of each set is retained in the orthorhombic structure, whereas the other two appear as possible twin mirror planes and . It is noted that predominantly planes of type are observed as twin boundaries, but less frequently those of type .

From this structural pseudosymmetry the atomic structure of the twin interface was easily derived by Bragg. It is shown in Fig. 3.3.10.6. In reality, small relaxations at the twin boundary have to be assumed. It is clearly evident from the figure that the twin operation is a glide reflection with glide component (= twin displacement vector t).

Figure 3.3.10.6 | top | pdf | Structural model of the (110) twin boundary of aragonite (after Bragg, 1924), projected along the pseudo-hexagonal c axis. The ortho­rhombic unit cells of the two domains with eigensymmetry Pmcn, as well as their glide/reflection planes m and c, are indicated. The slab centred on the (110) interface between the thin lines is common to both partners. The interface coincides with a twin glide plane c and is shown as a dotted line (twin displacement vector ). The model is based on a hexagonal cell with , the true angle is . The origin of the orthorhombic cell is chosen at the inversion centre halfway between two CO3 groups along c.

For this merohedral twin (eigensymmetry 32) a real parent phase, hexagonal -quartz (622), exists. The structural relation between the two Dauphiné twin partners of -quartz is best seen in projection along [001], as shown in Fig. 3.3.10.7 and in Figure 3 of McLaren & Phakey (1966), assuming a fault vector in both cases. These figures reveal that only small deformations occur upon passing from one twin domain to the other, irrespective of the orientation of the boundary. This is in agreement with the general observation that Dauphiné boundaries are usually irregular and curved and can adopt any orientation. The electron microscopy study of Dauphiné boundaries by McLaren & Phakey confirms the fault vector . It is noteworthy that the two models of the boundary structure by Klassen-Neklyudova (1964) and McLaren & Phakey (1966) imply a slab with the -quartz structure in the centre of the transition zone (Fig. 3.3.10.7b). This is in agreement with the assumption voiced by several authors, first by Aminoff & Broomé (1931), that the central zone of a twin interface often exhibits the structure of a different (real or hypothetical) polymorph of the crystal.

Figure 3.3.10.7 | top | pdf | Simplified structural model of a Dauphiné twin boundary in quartz (after Klassen-Neklyudova, 1964). Only Si atoms are shown. (a) Arrangement of Si atoms in the low-temperature structure of quartz viewed along the trigonal axis [001]. (b) Model of the Dauphiné twin boundary C–D. Note the opposite orientation of the three electrical axes shown in the upper left and lower right corner of part (b). In this model, the structural slab centred along the twin boundary has the structure of the hexagonal high-temperature phase of quartz which is shown in (c).

There are, however, X-ray topographic studies by Lang (1967a,b) and Lang & Miuskov (1969) which show that curved Dauphiné boundaries may be fine-stepped on a scale of a few tens of microns and exhibit a pronounced change of the X-ray topographic contrast of one and the same boundary from strong to zero (invisibility), depending on the boundary orientation. This observation indicates a change of the fault vector with the boundary orientation. It is in contradiction to the electron microscopy results of McLaren & Phakey (1966) and requires further experimental elucidation.

The room-temperature phase of KLiSO₄ is hexagonal with space group . It forms a `stuffed' tridymite structure, consisting of a framework of alternating SO<sub>4</sub> and LiO<sub>4</sub> tetrahedra with the K ions `stuffed' into the framework cavities. Crystals grown from aqueous solutions exhibit merohedral growth twins with twin reflection planes (alternatively ) with extended and sharply defined (0001) twin boundaries. The twins consist of left- and right-handed partners with the same polarity. The left- and right-handed structures, projected along the polar hexagonal c axis, are shown in Figs. 3.3.10.8(a) and (b) (Klapper et al., 1987). The tetrahedra of the two tetrahedral layers within one translation period c are in a staggered orientation. A model of the twin boundary is shown in Fig. 3.3.10.8(c): the tetrahedra on both sides of the twin interface (0001), parallel to the plane of the figure, now adopt an eclipsed position, leading to an uninterrupted framework and a conformation change in second coordination across the interface. It is immediately obvious that this (0001) interface permits an excellent low-energy fit of the two partner structures. Note that all six (alternative) twin reflection planes and are normal to the twin boundary. It is not possible to establish a similar low-energy structural model of a boundary which is parallel to one of these twin mirror planes (Klapper et al., 1987).

Figure 3.3.10.8 | top | pdf | KLiSO4: Bulk tetrahedral framework structures and models of (0001) twin boundary structures of phases III and IV. Small tetrahedra: SO4; large tetrahedra: LiO4; black spheres: K. All three figures play a double role, both as bulk structure and as (0001) twin-boundary structures. (a) and (b) Left- and right-handed bulk structures of phase III (), as well as possible structures of the (0001) twin boundary in phase IV. (c) Bulk structure of phase IV (), as well as possible structure of the (0001) twin boundary in phase III. The SO4 tetrahedra covered by the LiO4 tetrahedra are shown by thin lines. Dotted line: c-glide plane. In all cases, the (0001) twin boundary is located between the two tetrahedral layers parallel to the plane of the figure.

Inspection of the boundary structure in Fig. 3.3.10.8(c) shows that the tetrahedra related by the twin reflection plane (one representative plane is indicated by the dotted line) are shifted with respect to each other by a twin displacement vector . Thus, on an atomic scale, these twin reflection planes are in reality twin c-glide planes, bringing the right- and left-hand partner structures into coincidence.

Interestingly, upon cooling below 233 K, KLiSO₄ undergoes a (very) sluggish phase transition from the phase III into the trigonal phase IV with space group by suppression of the twofold axis parallel [001] and by addition of a c-glide plane. Structure determinations show that the bulk structure of IV is exactly the atomic arrangement of the grown-in twin boundary of phase III, as presented in Fig. 3.3.10.8(c). Moreover, X-ray topography reveals transformation twins III IV, exhibiting extended and sharply defined polysynthetic (0001) twin lamellae in IV. From the X-ray topographic domain contrast, it is proven that the twin element is the twofold rotation axis parallel to [001]. The structural model of the (0001) twin interfaces is given in Figs. 3.3.10.8(a) and (b). They show that across the (0001) twin boundary the tetrahedra are staggered, in contrast to the bulk structure of IV where they are in an eclipsed orientation (Fig. 3.3.10.8c). It is immediately recognized that the two tetrahedral layers, one above and one below the (0001) twin boundary in Fig. 3.3.10.8(a) or (b), are related by screw axes.

Thus, the (idealized) (0001) twin boundary of the transformation twins of phase IV is represented by the bulk structure of the hexagonal room-temperature phase III, whereas the twin boundary of the growth twins of the hexagonal phase III is represented by the bulk structure of the trigonal low-temperature phase IV. Upon cooling from (phase III) to (phase IV), the axes are suppressed as symmetry elements, but they now act as twin elements. In the model they are located as in space group , one type being contained in the axes, the other type halfway in between. Upon heating, the re-transformation IV III restores the reflection twins with the same large (0001) boundaries in the same geometry as existed before the transition cycle, but now as result of a phase transition, not of crystal growth (strong memory effect).

Thus, KLiSO₄ is another particularly striking example of the phenomenon, mentioned above for the Dauphiné twins of quartz, that the twin-interface structure of one polymorph may resemble the bulk structure of another polymorph.

The structural models of both kinds of twin boundaries do not exhibit a fault vector . This may be explained by the compensation of the glide component of the c-glide plane in phase IV by the screw component of the screw axis in phase III and vice versa.

An explanation for the occurrence of twinning based on the `conflict' between the energetically most favourable (hence stable) crystal structure and the arrangement with the highest possible symmetry was proposed by Krafczyk et al. (1997 and references therein) for some molecular crystals. According to this theory, pseudosymmetrical structures exhibit `structural instabilities', i.e. symmetrically favourable structures occur, whereas the energetically more stable structures are not realized, but were theoretically derived by lattice-energy calculations. The differences between the two structures provide the explanation for the occurence of twins. The twin models contain characteristic `shift vectors' (twin displacement vectors). The theory was successfully applied to pentaerythrite, 1,2,4,5-tetrabromo­benzene, maleic acid and 3,5-dimethylbenzoic acid.