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. 3.3, pp. 414418
Section 3.3.2. Basic concepts and definitions of twinning^{a}Institut für Kristallographie, Rheinisch–Westfälische Technische Hochschule, D52056 Aachen, Germany, and ^{b}MineralogischPetrologisches Institut, Universität Bonn, D53113 Bonn, Germany 
Because twinning is a rather complex and widespread phenomenon, several definitions have been presented in the literature. Two of them are quoted here because of the particular engagement of their authors in this topic.
George Friedel (1904; 1926, p. 421): A twin is a complex crystalline edifice built up of two or more homogeneous portions of the same crystal species in contact (juxtaposition) and oriented with respect to each other according to welldefined laws.
These laws, as formulated by Friedel, are specified in his book (Friedel, 1926). His `lattice theory of twinning' is discussed in Sections 3.3.8 and 3.3.9 of the present chapter.
Paul Niggli (1919, 1920/1924/1941): If several crystal individuals of the same species are intergrown in such a way that all analogous faces and edges are parallel, then one speaks of parallel intergrowth. If for two crystal individuals not all but only some of the (morphological) elements (edges or faces), at least two independent ones, are parallel or antiparallel, and if such an intergrowth due to its frequent occurrence is not `accidental', then one speaks of twins or twin formation. The individual partners of typical twins are either mirror images with respect to a common plane (`twinplane law'), or they appear rotated by 180° around a (common) direction (`zoneaxis law', `hemitropic twins'), or both features occur together. These planes or axes, or both, for all frequently occurring twins turn out to be elements with relatively simple indices (referred to the growth morphology). (Niggli, 1941, p. 137.)
Both definitions are geometric. They agree in the essential fact that the `well defined' laws, i.e. the orientation relations between two twin partners, refer to rational planes and directions. Morphologically, these relations find their expression in the parallelism of some crystal edges and crystal faces. In these and other classical definitions of twins, the structure and energy of twin boundaries were not included. This aspect was first introduced by Buerger in 1945.
In a more extended fashion we define twinning as follows:
An intergrowth of two or more macroscopic, congruent or enantiomorphic, individuals of the same crystal species is called a twin, if the orientation relations between the individuals occur frequently and are `crystallographic'. The individuals are called twin components, twin partners or twin domains. A twin is characterized by the twin law, i.e. by the orientation and chirality relation of two twin partners, as well as by their contact relation (twin interface, composition plane, domain boundary).
In the following, the crystallographic orientation and chirality relations of two or more twin components, only briefly mentioned in the definition, are explained in detail. Two categories of orientation relations have to be distinguished: those arising from binary twin operations (binary twin elements), i.e. operations of order 2, and those arising from pseudo nfold twin rotations (nfold twin axes), i.e. operations of order .
The (crystallographic) orientation relation of two twin partners can be expressed either by a twin operation or by its corresponding twin element. Binary twin elements can be either twin mirror planes or twofold twin axes or twin inversion centres. The former two twin elements must be parallel or normal to (possible) crystal faces and edges (macroscopic description) or, equivalently, parallel or normal to lattice planes and lattice rows (microscopic lattice description). Twin elements may be either rational (integer indices) or irrational (irrational indices which, however, can always be approximated by sufficiently large integer indices). Twin reflection planes and twin axes parallel to lattice planes or lattice rows are always rational. Twin axes and twin mirror planes normal to lattice planes or lattice rows are either rational or irrational. In addition to planes and axes, points can also occur as twin elements: twin inversion centres.
There exist seven kinds of binary twin elements that define the seven general twin laws possible for noncentrosymmetric triclinic crystals (crystal class 1):

All these binary twin elements – no matter whether rational or irrational – lead to crystallographic orientation relations, as defined in Section 3.3.2.2, because the following lattice items belong to both twin partners:

In this context one realizes which wide range of twinning is covered by the requirement of a crystallographic orientation relation: the `minimal' condition is provided by the complex twins (v) and (vi): only a onedimensional lattice row is `common', two lattice planes are `parallel' and all twin elements are irrational (Fig. 3.3.2.3). The `maximal' condition, a `common' threedimensional lattice, occurs for inversion twins (`merohedral' or `parallellattice twins'), case (vii); for displacement vector t = 0, the threedimensional lattice is even `coincident'.
In noncentrosymmetric triclinic crystals, the above twin elements define seven different twin laws, but for centrosymmetric crystals only three of them represent different orientation relations, because both in lattices and in centrosymmetric crystals a twin mirror plane defines the same orientation relation as the twofold twin axis normal to it, and vice versa. Consequently, the twin elements of the three pairs (i) + (ii), (iii) + (iv) and (v) + (vi) represent the same orientation relation. Case (vii) does not apply to centrosymmetric crystals, since here the inversion centre already belongs to the symmetry of the crystal.
For symmetries higher than triclinic, even more twin elements may define the same orientation relation, i.e. form the same twin law. Example: the dovetail twin of gypsum (point group ) with twin mirror plane (100) can be described by the four alternative twin elements (i), (ii), (iii), (iv) (cf. Section 3.3.4, Fig. 3.3.4.1). Furthermore, with increasing symmetry, the twin elements (i) and (iii) may become even more special, and the nature of the twin type may change as follows:
In both cases, the threedimensional lattice (or a sublattice of it) is now common to both twin partners, i.e. a `merohedral' twin results (cf. Section 3.3.9).
There is one more binary twin type which seems to reduce even further the abovementioned `minimal' condition for a crystallographic orientation relation, the socalled `median law' (German: Mediangesetz) of Brögger (1890), described by Tschermak & Becke (1915, p. 99). So far, it has been found in one mineral only: hydrargillite (modern name gibbsite), Al(OH)_{3}. The acceptability of this orientation relation as a twin law is questionable; see Section 3.3.6.11.
There is a longlasting controversy in the literature, e.g. Hartman (1956, 1960), Buerger (1960b), Curien (1960), about the acceptance of three, four and sixfold rotation axes as twin elements, for the following reason:
Twin operations of order two (reflection, twofold rotation, inversion) are `exact', i.e. in a component pair they transform the orientation state of one component exactly into that of the other and vice versa. There occur, in addition, many cases of multiple twins, which can be described by three, four and sixfold twin axes. These axes, however, are pseudo axes because their rotation angles are close to but not exactly equal to 120, 90 or 60°, due to metrical deviations (no matter how small) from a highersymmetry lattice. A well known example is the triple twin (German: Drilling) of orthorhombic aragonite, where the rotation angle (which transforms the orientation state of one component exactly into that of the other) deviates significantly from the 120° angle of a proper threefold rotation (Fig. 3.3.2.4). Another case of n = 3 with a very small metrical deviation is provided by ammonium lithium sulfate (γ = 119.6°).

(a) Triple growth twin of orthorhombic aragonite, CaCO_{3}, with pseudothreefold twin axis. The gap angle is 11.4° (= 360° − 3 × 116.2°). The exact description of the twin aggregate by means of two symmetryequivalent twin mirror planes (110) and () is indicated. In actual crystals, the gap is usually closed as shown in (b). 
All these (pseudo) nfold rotation twins, however, can also be described by (exact) binary twin elements, viz by a cyclic sequence of twin mirror planes or twofold twin axes. This is also illustrated and explained in Fig. 3.3.2.4. This possibility of describing cyclic twins by `exact' binary twin operations is the reason why Hartman (1956, 1960) and Curien (1960) do not consider `nonexact' three, four and sixfold rotations as proper twin operations.
The crystals forming twins with pseudo nfold rotation axes always exhibit metrical pseudosymmetries. In the case of transformation twins and domain structures, the metrical pseudosymmetries of the lowsymmetry (deformed) phase result from the true structural symmetry of the parent phase (cf. Section 3.3.7.2). This aspect caused several authors [e.g. Friedel, 1926, pp. 435 and 464; Donnay (cf. Hurst et al., 1956); Buerger, 1960b] to accept these pseudo axes for the treatment of twinning. The present authors also recommend including three, four and sixfold rotations as permissible twin operations. The consequences for the definition of the twin law will be discussed in Section 3.3.4 and in Section 3.4.3 . For a further extension of this concept to fivefold and tenfold multiple growth twins, see Note (6) below and Example 3.3.6.9.

References
Bloss, F. D. (1971). Crystallography and Crystal Chemistry, pp. 324–338. New York: Holt, Rinehart & Winston.Brögger, W. C. (1890). Hydrargillit. Z. Kristallogr. 16, second part, pp. 16–43, especially pp. 24–43 and Plate 1.
Buerger, M. J. (1945). The genesis of twin crystals. Am. Mineral. 30, 469–482.
Buerger, M. J. (1960b). Introductory remarks. Twinning with special regard to coherence. In Symposium on twinning. Cursillos y Conferencias, Fasc. VII, pp. 3 and 5–7. Madrid: CSIC.
Curien, H. (1960). Sur les axes de macle d'ordre supérieur à deux. In Symposium on twinning. Cursillos y Conferencias, Fasc. VII, pp. 9–11. Madrid: CSIC.
Ellner, M. (1995). Polymorphic phase transformation of Fe_{4}Al_{13} causing multiple twinning with decagonal pseudosymmetry. Acta Cryst. B51, 31–36.
Ellner, M. & Burkhardt, U. (1993). Zur Bildung von Drehmehrlingen mit pentagonaler Pseudosymmetrie beim Erstarrungsvorgang des Fe_{4}Al_{13}. J. Alloy. Compd. 198, 91–100.
Friedel, G. (1904). Etude sur les groupements cristallins. Extrait du Bulletin de la Société d' Industrie Minérale, Quatrième Série, Tomes III et IV. Saint Etienne: Imprimerie Théolier J. et Cie.
Friedel, G. (1926). Lecons de cristallographie, ch. 15. Nancy, Paris, Strasbourg: BergerLevrault. [Reprinted (1964). Paris: Blanchard].
Hartman, P. (1956). On the morphology of growth twins. Z. Kristallogr. 107, 225–237.
Hartman, P. (1960). Epitaxial aspects of the atacamite twin. In Symposium on twinning. Cursillos y Conferencias, Fasc. VII, pp. 15–18. Madrid: CSIC.
Hurst, V. J., Donnay, J. D. H. & Donnay, G. (1956). Staurolite twinning. Mineral. Mag. 31, 145–163.
Klapper, H., Hahn, Th. & Chung, S. J. (1987). Optical, pyroelectric and Xray topographic studies of twin domains and twin boundaries in KLiSO_{4}. Acta Cryst. B43, 147–159. Erratum: Acta Cryst. B43, >406.
Menzer, G. (1955). Über Kristallzwillingsgesetze. Z. Kristallogr. 106, 193–198.
Mügge, O. (1911). Über die Zwillingsbildung der Kristalle. Fortschr. Mineral. Kristallogr. Petrogr. 1, 18–47.
Nespolo, M., Ferraris, G. & Takeda, H. (2000). Twins and allotwins of basic mica polytypes: theoretical derivation and identification in the reciprocal space. Acta Cryst. A56, 132–148.
Nespolo, M., Ferraris, G., Takeda, H. & Takeuchi, Y. (1999). Plesiotwinning: oriented crystal associations based on a large coincidencesite lattice. Z. Kristallogr. 214, 378–382.
Nespolo, M., Kogure, T. & Ferraris, G. (1999). Allotwinning: oriented crystal association of polytypes – some warnings on consequences. Z. Kristallogr. 214, 5–8.
Niggli, P. (1919). Geometrische Kristallographie des Diskontinuums, especially pp. 551–560. Leipzig: Gebrüder Borntraeger. [Reprinted (1973). Wiesbaden: Sändig].
Niggli, P. (1920/1924/1941). Lehrbuch der Mineralogie und Kristallchemie, 1st ed. 1920, 2nd ed. 1924, 3rd ed., Part I, 1941, especially pp. 136–153, 401–415. Berlin: Gebrüder Borntraeger.
Penn, R. L. & Banfield, J. F. (1998). Oriented attachment and growth, twinning, polytypism, and formation of metastable phases: insights from nanocrystalline TiO_{2}. Am. Mineral. 83, 1077–1082.
Phillips, F. C. (1971). An Introduction to Crystallography, 4th ed. London: Longman.
Sunagawa, I. & Tomura, S. (1976). Twinning in phlogopite. Am. Mineral. 61, 939–943.
Takeuchi, Y. (1997). Tropochemical CellTwinning. Tokyo: Terra Scientific Publishing Company.
Tertsch, H. (1936). Bemerkungen zur Frage der Verbreitung und zur Geometrie der Zwillingsbildungen. Z. Kristallogr. 94, 461–490.
Tschermak, G. (1884, 1905). Lehrbuch der Mineralogie, 1st ed. 1884, 6th ed. 1905. Wien: Alfred Hölder.
Tschermak, G. (1904). Einheitliche Ableitung der Kristallisations und Zwillingsgesetze. Z. Kristallogr. 39, 433–462, especially 456–462.
Tschermak, G. & Becke, F. (1915). Lehrbuch der Mineralogie, 7th ed., pp. 93–114. Wien: Alfred Hölder.
Wadhawan, V. K. (1997). A tensor classification of twinning in crystals. Acta Cryst. A53, 546–555.
Wadhawan, V. K. (2000). Introduction to Ferroic Materials, ch. 7. Amsterdam: Gordon and Breach.