Coherent and incoherent twin interfaces

Hahn, Th.; Klapper, H.

doi:10.1107/97809553602060000644

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

pdf | chapter contents | chapter index | related articles

International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 443-444

Section 3.3.10.9. Coherent and incoherent twin interfaces

Th. Hahn^a ^* and H. Klapper^b

^a Institut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D-52056 Aachen, Germany, and ^bMineralogisch-Petrologisches Institut, Universität Bonn, D-53113 Bonn, Germany
Correspondence e-mail: hahn@xtal.rwth-aachen.de

3.3.10.9. Coherent and incoherent twin interfaces

| top | pdf |

At the start of Section 3.3.10, the terms compatible and incompatible twin boundaries were introduced and clearly defined. There exists another pair of terms, coherent and incoherent interfaces, which are predominantly used in bicrystallography and metallurgy for the characterization of grain boundaries, but less frequently in mineralogy and crystallography for twin boundaries. These terms, however, are defined in different and often rather diffuse ways, as the following examples show.

(1) Cahn (1954, p. 390), in his extensive review on twinning, defines coherence in metal twins as follows: `An interface parallel to a twin (reflection) plane is called a coherent interface, while any other interface is termed non-coherent'. The same definition is given by Porter & Easterling (1992, p. 122), who consider twin boundaries as `special high-angle boundaries'. This definition is widely used, especially in metallurgy, as evidenced by the following textbooks: Cottrell (1955, p. 212); Chalmers (1959, p. 125); Van Bueren (1961, pp. 251 and 450), Friedel (1964, p. 173); Klassen-Neklyudova (1964, p. 156); Kelly & Groves (1970, p. 308).

(2) Christian (1965, p. 332) distinguishes three levels of coherence of grain boundaries :

(a) Incoherent interfaces correspond to high-angle grain boundaries without any `continuity conditions for lattice vectors or lattice planes across the interface'.
(b) Semi-coherent interfaces are low-angle boundaries formed by a regular network of dislocations. `Such an interface consists of regions in which the two structures may be regarded as being in forced elastic coherence, separated by regions of misfit', i.e. there is partial local register across the boundary.
(c) Fully coherent interfaces correspond to the joining of two twin components along their rational or irrational composition plane in such a way that the lattices match exactly at the interface.

Very similar definitions are also used by Barrett & Massalski (1966, p. 493) and Sutton & Balluffi (1995, Glossary) for bicrystal boundaries. The third term, `fully coherent', corresponds to the coherence definition of Cahn. It is noted that the terms `fully coherent', `semi-coherent' and `incoherent' are also applied to the interfaces of grains of different phases (`interphase interfaces'), as well as to boundaries of second-phase precipitates, by Porter & Easterling (1992, Chapter 3.4).

(3) Putnis (1992, pp. 225 and 335) considers twin interfaces, as well as boundaries between matrix and precipitates, in minerals by their `degree of lattice matching'. He uses the term coherent twin boundaries for `perfect lattice plane matching across the interface, the strains being taken up by elastic distortions' (i.e. without the presence of dislocations). Dislocations along the twin boundary lead to a `loss of coherence'. Interfaces containing dislocations are called semi-coherent (p. 336, Fig. 11.4), which is similar to the definition by Christian quoted above.
(4) Shektman (1993, p. 24) defines the term coherence only for ferroelastics (especially YBaCu) with different systems of lamellar twin domains: boundaries between (parallel) twin lamellae are defined as coherent interfaces, whereas boundaries between different lamellae systems are called incoherent.

The above definitions have one feature in common: coherent twin boundaries are planar interfaces, which are either rational or irrational, as stated explicitely by Christian (1965, p. 332). Beyond this, the various definitions are rather vague. In particular, they do not distinguish between ferroelastic and non-ferroelastic (strictly merohedral) twins and do not consider the twin displacement vector discussed in Section 3.3.10.4.

As an attempt to fill this gap, the following elucidations of the term `coherence' are suggested here. These proposals are based on the definitions summarized above, as well as on the concepts of compatibility of interfaces (Sections 3.3.10.1 and 3.3.10.3) and on the notion of twin displacement vector (Section 3.3.10.4).

(i) For twin interfaces, only the terms coherent and incoherent are used. In view of the fact that twin interfaces do not require regular (perfect) dislocations (but may contain twinning dislocations as described above in Section 3.3.10.8), the term `semi-coherent' is reserved for grain boundaries and heterophase interfaces.
(ii) Twin boundaries are called coherent only if they are (mechanically) compatible. This holds for both rational and irrational twin boundaries, as suggested by Christian (1965, p. 332). Both cases may be distinguished by using qualifying adjectives such as `rationally coherent' and `irrationally coherent'. Note that irrational (compatible) twin boundaries are usually less perfect and of higher energy than rational ones.
(iii) For strict merohedral (non-ferroelastic ) twins (lattice index ) any twin boundary, even a curved one, is compatible and, hence, is designated here as coherent, even if the contact plane does not coincide with the twin mirror plane.
(iv) For non-merohedral (ferroelastic) twins , a pair of (rational or irrational) perpendicular compatible interfaces occurs (Section 3.3.10.2.1). The same holds for merohedral twins of lattice index (Section 3.3.10.2.4). All these compatible boundaries are considered here as coherent.
(v) In lattice and structural terms, a twin boundary is coherent if it exhibits a well defined matching of the two lattices along the entire boundary, i.e. continuity with respect to their lattice vectors and lattice planes. We want to stress that we consider the coherence of a twin boundary not as being destroyed by the presence of a nonzero twin displacement or fault vector as long as there is an optimal low-energy fit of the two partner structures. The twin displacement (fault) vector represents a `phase shift' between the two structures with the same two-dimensional periodicity along their contact plane and thus defines the continuity relation across the boundary. This statement agrees with the general opinion that stacking faults, antiphase boundaries and many merohedral twin boundaries, all possessing nonzero fault vectors, are coherent. Well known examples are stacking faults in f.c.c and h.c.p metals and Brazil twin boundaries in quartz.

It is apparent from these discussions of coherent twin interfaces that several features have to be taken into account, some readily available by experiments and observations, whereas others require geometric models (lattice matching) or even physical models (structure matching), including determination of twin displacement vectors t.

The definitions of coherence, as treated here, often do not satisfactorily agree with reality. Two examples are given:

(a) Japanese twins of quartz with twin mirror plane $[(11{\bar 2}2)]$ or twofold twin axis normal to $[(11{\bar 2}2)]$ . According to the definitions given above, the observed $[(11{\bar 2}2)]$ contact plane is coherent. Nevertheless, these $[(11{\bar 2}2)]$ boundaries are always strongly disturbed and accompanied by extended lattice distortions. Thus, in reality they must be considered as not coherent.
(b) Sodium lithium sulfate, NaLiSO₄, with polar point group and a hexagonal lattice forms merohedral growth twins with twin mirror plane (0001) normal to the polar axis. The composition plane coincides with the twin plane and has head-to-head or tail-to-tail character. According to definition (iii) above, any twin boundary of this merohedral twin is coherent. The observed (0001) contact plane, however, despite coincidence with the twin mirror plane, is always strongly disturbed and cannot be considered as coherent. In this case, the observed incoherence is obviously due to the head-to-head orientation of the boundary, which is `electrically forbidden'.

These examples demonstrate that the above formal definitions of coherence, based on geometrical viewpoints alone, are not always satisfactory and require consideration of individual cases.

With these discussions of rather subtle features of twin interfaces, this chapter on twinning is concluded. It was our aim to present this rather ancient topic in a way that progresses from classical concepts to modern considerations, from three dimensions to two and from macroscopic geometrical arguments to microscopic atomistic reasoning. Macroscopic derivations of orientation and contact relations of the twin partners (twin laws, as well as twin morphologies and twin genesis) were followed by lattice considerations and structural implications of twinning. Finally, the physical background of twinning was explored by means of the analysis of twin interfaces, their structural and energetic features. It is this latter aspect which in the future is most likely to bring the greatest progress toward the two main goals, an atomistic understanding of the phenomenon `twinning' and the ability to predict correctly its occurrence and non-occurrence.

All considerations in this chapter refer to analysis of twinning in direct space. The complementary aspect, the effect of twinning in reciprocal space, lies beyond the scope of the present treatment and, hence, had to be omitted. This concerns in particular the recognition and characterization of twinning in diffraction experiments, especially by X-rays, as well as the consideration of the problems that twinning, especially merohedral twinning, may pose in single-crystal structure determination (cf. Buerger, 1960a). Several powerful computer programs for the solution of these problems exist. For a case study, see Herbst-Irmer & Sheldrick (1998).

References

Barrett, C. S. & Massalski, T. B. (1966). Structure of metals, 3rd edition, especially pp. 406–414. New York: McGraw-Hill.Google Scholar

Buerger, M. J. (1960a). Crystal-structure analyses, especially ch. 3. New York: Wiley.Google Scholar

Cahn, R. W. (1954). Twinned crystals. Adv. Phys. 3, 202–445.Google Scholar

Chalmers, B. (1959). Physical metallurgy, especially ch. 4.4. New York: Wiley.Google Scholar

Christian, J. W. (1965). The theory of transformations in metals and alloys, especially chs. 8 and 20. Oxford: Pergamon.Google Scholar

Cottrell, A. H. (1955). Theoretical structural metallurgy, 2nd edition, especially ch. 14.5. London: Edward Arnold.Google Scholar

Friedel, J. (1964). Dislocations, especially ch. 6. Oxford: Pergamon.Google Scholar

Herbst-Irmer, R. & Sheldrick, G. M. (1998). Refinement of twinned structures with SHELXL97. Acta Cryst. B54, 443–449.Google Scholar

Kelly, A. & Groves, G. W. (1970). Crystallography and crystal defects, especially chs. 10 and 12.5. London: Longman.Google Scholar

Klassen-Neklyudova, M. V. (1964). Mechanical twinning of crystals. New York: Consultants Bureau.Google Scholar

Porter, D. A. & Easterling, K. E. (1992). Phase transformations in metals and alloys, 2nd edition, especially ch. 3. London: Chapman & Hall.Google Scholar

Putnis, A. (1992). Introduction to mineral sciences, especially chs. 7.3 and 12.4. Cambridge University Press.Google Scholar

Shektman, V. Sh. (1993). Editor. The real structure of high-T_c superconductors, especially ch. 3, Twins and structure of twin boundaries, by I. M. Shmyt'ko & V. Sh. Shektman. Berlin: Springer.Google Scholar

Sutton, A. P. & Balluffi, R. W. (1995). Interfaces in crystalline materials, Section 1.5, pp. 25–41. Oxford: Clarendon Press.Google Scholar

Van Bueren, H. G. (1961). Imperfections in crystals, especially chs. 13.4 and 19. Amsterdam: North-Holland.Google Scholar

International Tables for Crystallography (2006). Vol. D. ch. 3.3, pp. 443-444