Contributed by Massimo Nespolo
Last update: 3 February 2009

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The study of twinned crystals can be traced back to the early morphological observations ( Steno, 1669). Nowadays, twinned crystals are often considered an obstacle to automatic solution and refinement of crystal structures. As a matter of fact, the study of twins in itself developed, thanks to several prominent crystallographers especially between the XIX and the XX centuries, almost as a specialistic branch inside crystallography. For this reason, J.D.H. Donnay introduced the term geminography to specifically indicate this branch of crystallography (historical note).

Definition

Twinning is the oriented association of two or more individuals of the same crystalline phase, in which pairs of individuals are related by a geometrical operation termed twin operation. The twin operation is a symmetry operation for the twinned edifice but not for the individuals: it relates different individuals in the twin and belongs to a crystallographic point group. It cannot belong to the symmetry of the crystal, otherwise it would produce a parallel growth instead of a twin ( Friedel, 1904).

The term "individual" is used to indicate one crystal of a twin, and the term "single crystal" to mean an untwinned crystal. "Domain" is often used with special reference to domain structures where the domains are in twin relation. "Component" is also used instead of "individual" (e.g. Hahn, Janovec and Klapper, 1999).

Hereafter we concentrate specifically on twins. Wider classes like bicrystals, translation domains, growth sectors etc. share some features with twins and are described in Hahn, Janovec and Klapper, (1999) and Hahn and Klapper (2003).

Classification

Several definitions of twinning exist in the literature. Particularly striking is the difference between the "crystallographic-mineralogical" definition, given in terms of lattice, and the "physical" definition, which often is based on phenomenological concepts most typical of transformation twins. As a consequence, we are faced with different classifications that overlap only partially. A general agreement, which encompasses all the definitions, seems still far from being reachable. Even in the "crystallographic-mineralogical" tradition there are some discordances, which more or less correspond to the definitions introduced by the so-called "German school" and those that come from the so-called "French school". While these two approaches in most cases lead to an equivalent description, there are a few cases in which some differences persist. In those cases, one could summarize the different approaches saying that the German school privileges the morphological twin elements, whereas the French school gives priority to the reticular twin elements. Here we follow mainly the definitions and classifications introduced by the French school, and summarized in the two books by Georges Friedel (1904, 1926).

Basic definitions and classifications. The notion of twin lattice

Several criteria are used to classify twins. Here we briefly review the most used ones.

Formation mechanism: depending on the formation mechanism, twins are classified in:
Growth twins, which form during crystal growth, either at the nuclation stage or by oriented attachment (for a review, see Nespolo and Ferraris, 2004b)
Transformation twins, which form during a phase transition leading to a loss of point symmetry
Mechanical twins, which form as the result of a mechanical action (typically, an oriented pressure) on the crystal
Morphology, which divides twins in:
Contact twins, separated by a surface
Penetration twins, sharing a volume
Simple twins, where the individuals are nor repeated
Polysynthetic twins, where the individuals are repeated to in a more or less linear arrangement
Cyclic twins, where the individuals are repeated to form a closed edifice where the N-th individual is in contact with the first
Dimensionality of the twin lattice, i.e. the number of dimensions along which a common (reasonable) periodity can be found (Friedel, 1933):
Triperiodic twins, having a common lattice in three dimensions (by far the most common type of twins)
Diperiodic twins, for which a common lattice exists in only two dimensions
Monoperiodic twins, sharing only a lattice row
Diperiodic and monoperiodic twins can be collectively named subperiodic twins.
Lattice restoration, which applies to triperiodic twins and gives rise to the well-known four categories originally introduced by the "French school"; they are reviewed below.

A more systematic approach, from the viewpoint of the symmetry, than usually done is necessary to separate some borderline cases. First of all, we recall the basic definitions; a more detailed analysis being presented in the next section. We assume the reader be familiar with concepts like point group and translation group.

For a classification of twins, it is useful to recall the four types of point symmetry that must be considered in analyzing the symmetry of a crystal in the three-dimensional Euclidean space:


1. Twinning by merohedry

First of all, let us consider a merohedral crystal. One or more independent symmetry elements for the lattice does not belong to the point group of the crystal pattern. This element can thus act as twin element, producing one or more (depending on the order of the element) individuals with different orientation, but whose lattice is entirely and exactly coincident with that of the original individual. This is the simplest case of twinning, termed twinning by merohedry or merohedric twinning (note). In this case, the twin lattice and the lattice of the individual coincide.

merohedral pattern

Projection along [010] of a monoclinic Pm structure (lattice point symmetry 2/m) with an object with point symmetry 2mm at the origin. The symmetry of the crystal pattern is only m
twinning by merohedry

Effect of the axis 2[010] as twin element: the crystal pattern is repeated in the same positions but with a different, non-equivalent orientation (two rotated copies of the same crystal pattern are shown superposed in the projection).

2. Twinning by pseudo-merohedry: the law of Mallard

A second case occurs when the lattice of the individual has a metric close to that of a higher holohedry. For example, a monoclinic lattice with a β angle close to 90º is pseudo-orthorhombic. The law of Mallard ( Friedel, 1926, page 436; Donnay, 1940) states that twin elements are always reticular: therefore, a twin plane is a lattice plane, and a twin axis is a lattice row. These twin elements are pseudosymmetry elements for the lattice of the individual. The twin operations produce now sligthly different orientations of the lattice of the individual, which are only quasi-equivalent, and no longer equivalent, as in the case of twinning by merohedry. The twin lattice suffers a slight deviation at the composition surface in this case. This type of twinning is called twinning by pseudo-merohedry and is characterized by a specific parameter, the twin obliquity, which is normally indicated by the letter ω.

Because a lattice is always centrosymmetric, each symmetry mirror plane is perpendicular to a lattice row, and each symmetry axis is perpendicular to a lattice plane. In twinning by pseudo-merohedry, the twin plane is only a pseudo-symmetry plane for the lattice of the individual. As a consequence, it is also only quasi-perpendicular to a lattice row. Similarly. a twin axis is only pseudo-symmetry axis for the lattice of the individual. As a consequence, it also is only quasi-perpendicular to a lattice plane.

In case of twinning by merohedry, when the twin element is twofold and the crystal is centrosymmetric, the twin operation can be described either as a rotation or as a reflection (the two operations being equivalent under the action of the center). In case of twinning by pseudo-merohedry, instead, the two twin operations are no longer equivalent even in centrosymmetric crystals but produce different twins, which are called reciprocal twins ( Mügge, 1898) or corresponding twins ( Friedel, 1904, 1926). One of the most classical examples is that of albite (010) vs. pericline [010] twins in feldspars. Pairs of corresponding twins normally do not have the same frequency of occurrence, as one would be tempted to assume from the identical value of their obliquity.

For reflection twins, ω is the angle between the normal to the lattice plane which is a pseudo-symmetry plane for the lattice (and thus operates as twin plane), and a rational direction with low indices close to it. For rotation twins, ω is the angle between the non-rational plane normal to the lattice direction, which is a pseudo-symmetry axis for the lattice (and thus operates as twin axis), and a lattice plane with low indices close to it (note).

As for the twin index, also a large obliquity is unfavorable to the occurrence of a twin: the smaller is the obliquity, the higher is the probability of occurrence of that twin. Friedel (1926) gave 6º as empirical upper value for the obliquity.

twinning by pseudo-merohedry

Twinning by pseudo-merohedry: reflection twin. The lattice of the individual in its original orientation (black) is quasi-symmetric with respect to the plane perpendicular to the drawing (the intersection of these two planes in shwon in green). If this plane acts as twin plane, it transforms the black lattice into the red one.

In the left part of the drawing, only the original orientation is shown. In the rigth part, both orientations - original (black dotted - nodes omitted to improve readability of the figure) and twinned (red solid) are shown. The angular separation (exaggerated in the figure) of the two orientations of the lattice is evident. The linear separation increases with the distance from the twin plane (green). The blue solid line shows a lattice row quasi-perpendicular to the twin plane, and the dotted blue line shows the perpendicular direction, which is not a lattice row (it is not a rational direction on the lattice). The angle between these two directions is the obliquity ω.

3. Twinning by reticular merohedry

If a crystal is holohedral and its lattice does not have a specialized metric, then the point groups of the lattice and of the crystal pattern coincide: twinning by merohedry is not possible. However, among the infinitely many possible supercells there may be one (or more) of relatively low multiplicity, with a symmetry different than the conventional cell. In this case it is useful to consider a sublattice based on the nodes belonging to this supercell. The point group of this sublattice does not coincide with the point group of the original lattice, and its translation group is a subgroup of the translation group of the original lattice. The elements which are symmetry elements for the sublattice but not for the original lattice can act as twin elements. The lattices of the individuals have different orientations, related by the twin operations about the twin elements. The sublattice defined in this way is the twin lattice. This type of twinning is called twinning by reticular merohedry. It is characterized by a specific parameter, the twin index: this is the ratio of the number of lattice nodes of the individual to the number of nodes restored by the twin operation. In general, the smaller the twin index, the higher is the probability of occurence of the corresponding twin. No uniquely defined limit value can be given, twin index 6 or 7 is considered rather high ( Friedel, 1926). Nevertheless, twins with exceptionally high index are known too ( Hahn and Klapper, 2003)

twinning by reticular merohedry.

Twinning by reticular merohedry. A portion of the lattice of the individual is shown in two non-equivalent orientations (black and red), related by a reflection through the mirror plane normal to the drawing, which is the twin operation in this example. The unit cell is hatched in black. The supercell built on the nodes common to the two orientations (blue nodes) is hatched in blue and contains three nodes for each individual. For centrosymmetric crystals, the twin operation can equivalently be described as a rotation about a two-fold axis normal to the twin plane (axis contained in a plane parallel to the drawing). The twin lattice corresponds to the sublattice built on the common (blue) nodes only. One node out of three is restored by the twin operation: the twin index is thus 3.

4. Twinning by reticular pseudo-merohedry

The fourth case is easily understood as having the common features of 2. and 3. From the conventional cell of the lattice of the individual one chooses a supercell whose nodes define a sublattice. This sublattice has a pseudo-symmetry: by idealizing the sublattice, the pseudo-symmetry becomes a true symmetry and the sublattice becomes the twin lattice.

Twinning by reticular pseudo-merohedry is characterized by both a twin index > 1 and an obliquity ω > 0 and the probability of occurrence of a given twin drops when the twin index and the obliquity increase. However, the latter plays a secondary role with respect to the twin index ( Friedel, 1926).

twinning by reticular pseudo-merohedry

Twinning by reticular pseudo-merohedry. The lattice of the individual in its original orientation (black lines) and after reflection through a mirror plane perpendicular to the drawing and passing through the blue nodes (red lines). Only a part of the nodes (one out of three) is quasi restored by the twin operation. The unit cell of the lattice of the individual is hatched in black (original orientation) and in red (after reflection). The supercell built on the quasi-restored (one out of three) is hatched in blue: it has a higher pseudo-symmetry with respect to the cell of the individual and a slightly different orientation in the two individuals. The twin lattice, which is based on this supercell, extends through the twinned edifice, with a slight deviation at the composition surface.

TLS and TLQS

A coarser classification was introduced by Donnay and Donnay (1974), who distinguished between zero-obliquity twins (Twin Lattice Symmetry: TLS) and non-zero obliquity twins (Twin Lattice Quasi Symmetry: TLQS). This classification puts more emphasis on the role of the obliquity, which is responsible for the splitting of the reflections in the diffraction pattern of a TLQS twin.

The conditions of plane/direction perpendicularity for each lattice system were given by Donnay and Donnay (1959) (see also Koch, 2004) and are summarized in the following table.

Conditions of lattice plane/direction mutual perpendicularity in the seven lattice systems of the three-dimensional space
lattice systemlattice planelattice direction
triclinic------
monoclinic (b-unique)(010)[010]
orthorhombic(100)
(010)
(001)		[100]
[010]
[001]
tetragonal(001)
(hk0)		[001]
[hk0]
rhombohedral and hexagonal
(hexagonal axes)		(0001)
(hki0)		[001]
[2h+k,h+2k,0]
cubic(hkl)[hkl]

When the cell of the twin lattice is defined by a pair of lattice elements of this type, the obliquity is necessarily zero: this type of twinning is called intrinsic-TLS, (i-TLS). In all other cases, the conditions of perpendicularity, and thus the obliquity, depend on one or more metric parameters. A pair of elements (hkl)/[uvw] may thus correspond to TLS or to TLQS depending on the experimental conditions determining the metric of the lattice of the individual. Twinning in which the zero obliquity is not a consequence of the symmetry of the lattice of the individual but comes instead from its particular metric is called extrinsic-TLS (e-TLS) ( Nespolo and Ferraris, 2006).

The point symmetry of twins

If H is the point group of the individual and the twin contains N individuals, each of them has point group Hi, i = 1,N, all of the same type (same Hermann-Mauguin symbol) but differently oriented in space. Let H* be the intersection group of them, i.e. H* = ∩iHi. In H* only the symmetry elements which are parallel (or antiparallel) in all Hi are retained. A twin element added to H* gives a point group which is a supergroup of H* but in general does not have to have a group-subgroup relation with H. The twin operation is of a different nature from the operations of the point group of the individual: it is not a symmetry operation for the individual, but instead it maps the orientation of an individual on the orientation of another individual. The situation is the same as when an object contains several parts of different colors; the operations permuting the different parts of the object are divided into those which keep the colors (achromatic) and those which exchange the colors (chromatic). The Hermann-Mauguin symbol of the group is modified by adding a prime to the symbols of the operation exchanging the colors.

chromatic symmetry

An elementary example of chromatic symmetry. The square on the left is composed by four smaller squares of alternating colours. If one ignores the colours, the symmetry of this plane figure is 4mm. When the colours are taken into account, however, rotation that are odd multiples of π/4 exchange the blue and yellow colours, which those that are even multiples keep the colours. Similarly, mirrors perpendicular to 〈10〉 exchange the colours, while those perpendicular to 〈11〉 do not. The elements exchaging the colours are shown in red; in black those keeping the colours. The rotation point shows a black oval within a red square, to indicate that 2nπ/4 keeps the colours while (2n+1)π/4 exchanges them. The modified Hermann-Mauguin symbols is 4'm'm and is a dichromatic symbol.

If we associate to each individual of a twin a colour, the symmetry operations of the individual are achromatic whereas those mapping an individual on another are chromatic. The point group of the twin K is therefore the chromatic extension of the intersection group H* of the individuals in their respective orientations. For two-individual twins, the twin group K is dichromatic and the twin elements are primed (see Curien and Le Corre, 1958). For twins composed by more than two individuals, the twin point group K is polychromatic and the modified Hermann-Mauguin symbol may become more complex (see Nespolo, 2004).

Example: some twins of quartz

Quartz, a polymorph of SiO2, exists in two polymorphs, α (H = 321) at low temperature and β (H = 622) at high temperature. The lattice is hexagonal. Both polymorphs give a large number of twins: here we briefly analyse the three most common twins (Swiss, Brazil and Japan twins) to illustrate how the point symmetry of the twin can be obtained if the twin operation is known.

The Swiss twin is obtained by applying rotation by a π about the [001] direction. The point groups of the two individuals differ by their orientation in the (0001) plane, whle the [001] direction is parallel. In particular, the three axes A1, A2 and A3 normally used to index a crystal with hexagonal lattice are antiparallel in the two individuals. Along these axes, three two-fold axes for the individual exist, which are equivalent under the action of the 3[001] axis. In the intersection group, all these symmetry elements are retained because of their parallelism, so that H* = H = 321. The twin point group is obtained by adding a 2'[001] rotation, which transforms the three-fold axis for the individual into a six-fold axis for the twin. The combiation of the twin operation with the symmetry operations of the individuals introduces further twin operations, namely two-fold rotations about the 〈210〉 directions. The poing group of the twin is therefore K = 6'22'. It is isomorphic with the point group of the β polymorph, in which this twin cannot exist. In fact, it is produced by phase transition on cooling from the β to the α polymorph, when the symmetry operations that are lost at the transition remain as twin operations for the two individuals generated by the phase transition. Twinning is by merohedry.

Swiss twin in α-quartz

The Brazil twin is obtained by applying an inversion trhough the center. The point groups of the two individuals differ by their orientation, because all the lattice direction are antiparallel in the two individuals. As in the case of the Swiss twin, in the intersection group all the symmetry elements are retained because of their parallelism, so that H* = H = 321. The twin point group is obtained by adding a ¯1', which transforms the three-fold axis for the individual into a inverse three-fold axis for the twin. The combiation of the twin operation with the symmetry operations of the individuals introduces further twin operations, namely mirror reflections perpendicular to the 〈210〉 directions. The poing group of the twin is therefore K = ¯3'2/m'. Twinning is by merohedry.

Brazil twin in α-quartz

The Brazil twin can exist also in the β polymorph, the twin point group being then K = 6/m'2/m'2/m'. Twinning is by merohedry.

Brazil twin in β:-quartz

The third most common twin in quartz is the Japan twin, obtained by reflection across a {11¯22} mirror. This is a diagonal plane and the point groups of the two individuals have now no parallel symmetry elements, so that the intersection group is H* = 1 and the twin point group is simply K = m'. This same twin exists also in the β polymorph, for which again H* = 1 and K = m': it takes the name of Verespatak twin. Twinning is by reticular pseudo-merohedry with twin index n = 2 and obliquity ω = 5º27'.

The translational symmetry of twins

For a comprehensive classification of twinning, the twin lattice and the individual lattice must be compared in terms of both their (vector) point group and their translation group. Let us introduce the following definitions:

The classification introduced by the "French school" and summarized above corresponds to the following cases:

The "pseudo" cases arise when the symmetry of one or both lattices is closer to a higher holohedry: for example, a monoclinic lattice with β angle close to 90º or an orthorhombic lattice with a close to b.

There is however a further case left out in the above classification: the case in which TT ≠ TI and D(L)T = D(L)I, namely when the two lattices have the same point symmetry but differ in their translational symmetry and, consequently, in their orientation. This case does not rigorously enter in Friedel's classification, because there is no relation of merohedry between the point groups of the two lattices. For this reason, the term reticular polyholohedry has been introduced to indicate the presence of the same holohedry in the lattice of the individual and in the differently oriented sublattice(s) ( Nespolo and Ferraris, 2004a). Of course, pseudo-polyholohedry is possible as well.

Twinning by merohedry has been further subdivided on the basis of the point groups of the lattice, of the point group of the crystal, and of the corresponding holohedral point group. Let us call:

Merohedry can be classified as follows

  1. The lattice does not have an accidentally specialized higher metric and the twin operation belongs to D, the holohedral point symmetry of the crystal. This is the most common case of twinning by merohedry, which is now termed twinning by syngonic merohedry. Syngonic merohedry is subdivided, on the basis of the ratio between the order of the lattice point group and the order of the individual point group, into hemihedry (order 2), tetartohedry (order 4) and ogdohedry (order 8, possible only for the point group 3).
  2. The lattice has an accidentally specialized higher metric and the point group of the crystal is holohedral (D = H). H is however merohedral with respect to the point group of the lattice [D(L) ⊃ D = H]. The twin operation belongs to the point symmetry of the lattice, D(L), but not to the holohedral point symmetry of the crystal. This type of merohedry, and the corresponding twinning, is now termed metric merohedry.
  3. The lattice has an accidentally specialized higher metric but the point group of the crystal is not holohedral (D ⊃ H). The crystal is thus doubly merohedral: with respect to D (syngonic merohedry) and with respect to D(L) (metric merohedry). Depending on which twin operation is active, twinning is by metric or by syngonic merohedry. The two types of twinning can also coexist.

Example. A monoclinic holoaxial crystal (H = 2) with β = 90º is syngonically merohedral with respect to the monoclinic holohedry (D = 2/m) but metrically merohedral with respect to its lattice [D(L) = mmm]. The individual may undergo twinning by syngonic merohedry with respect to D, the twin point group being 2/m'It may however undergo twinning by metric merohedry with respect to D(L), the twin point group being either 2'22' or m'm2'. In this case, the monoclinic hemihedral individual is twinned with respect to the orthorhombic hemihedry but not to the monoclinic holohedry (point groups of the same order). It may finally undergo twinning with respect to the orthorhombic holohedry: in this case, both types of twinning are present simultaneously, because twinning by syngonic merohedry with respect to the monoclinic holohedry is part of the total twinning. As a result, two independent twin elements are active, which originate a four-individual twin, the twin point group being 2"/m"2/m'2"/m".

From the above analysis we can get a general conclusion. Let be KA the achromatic point group isomorphic with K. Then we can state that:

For twins having non-zero obliquity, the approximate symmetry should be taken into account rather than the exact symmetry. The acceptable degree of approximation coincides with the obliquity itself.

Twin laws

From the knowledge of K (the chromatic point group of the twin) and of H* (the intersection point group of the individuals in their respective orientation), the twin laws can be easily obtained by coset decomposition. We use KA to indicate the achromatic point group isomorphic to K, to simplify the symbols in the to coset decomposition.

Let us suppose that the twin point group corresponds to the tetragonal holohedry, KA = 4/mmm, and that H* is 2/m. The index of 2/m in 4/mmm is and corresponds to the ratio of the order of the two poing groups (16/4 = 4). This means that when KA is decomposed in terms of H* one obtains three cosets, besides the subgroup. Each coset has the same order as the subgroup, 4 in this case. There are therefore 3 twin laws, each corresponding to 4 possible twin operations, which are all equivalent under the operations of H*.

KA = {1, 2[010], ¯1, m[010]} ∪ {2[001], 2[100], m{001], m[100]}∪
{4-[001], ¯4+[001], 2[1¯10],m[1¯10]}∪ {4+[001], ¯4-[001], 2[110],m[110]}

Any of the four twin operations in the same coset (same twin law) can map the two individuals. The experiment will tell which twin operation actually acted.

To be noted that in case of (reticular) pseudo-meorhedry/polyholohedry, KA is only an approximant of the real symmetry of the twin, which is actually lower. The twin operations in a same coset are thus no longer exactly equivalent.

Effects of twinning by merohedry on the diffraction pattern

When the Laue symmetry of the individual is the same as the twin symmetry, the corresponding twins are said to belong to class I ( Catti and Ferraris, 1976). The diffraction pattern does not differ from that of a single crystal, unless anomalous scattering is substantial. The inversion centre can always be chosen as a twin operation and the set of intensities collected from a twin is indistinguishable from that collected from a single crystal. Instead, when the Laue symmetry of the crystal is lower than the symmetry, the twins belong to class II ( Catti and Ferraris, 1976) and are then subdivided into class IIA (syngonic merohedry) and class IIB (metric merohedry) ( Nespolo and Ferraris, 2000). The twin operations relate non-equivalent reflections, and the presence of twinning may hinder a correct derivation of the symmetry from the diffraction pattern. In particular, when the number of individuals coincides with the order of the twin operation and the volumes of the individuals are identical, the symmetry of the diffraction pattern is higher than the Laue symmetry of the individual. An incorrectly chosen space-group type may thus be assumed in the initial stage of the structure refinement.

Hybrid twins

The occurrence frequency of twins is roughly inversely proportional to the twin index and to the obliquity: the lower are the values of these two parameters, the better is the lattice restoration and the higher is the probability that the twin actually occurs. Friedel gave as empirical limits n = 6 and ω = 6º: twins having twin index and twin obliquity within these empirical limits are today known as Friedelian twins, the other as non-Friedelian twins.

Non-Friedelian twins are definitely less frequent than Friedelian twins; exceptions however exist. Besides, the degree of lattice restoration does not always fit the observed frequency occurrence: twins with worse lattice restoration sometimes occur more frequently than others with better lattice restoration. The reason behind this imperfect correlation is that the reticular theory considers only the lattice matching, while it is the structural match, especially close to the interface, that actually governs the formation of a twin. Nevertheless, the reticular approach has the advantage of generality, while a complete structural analysis of twins would require a case-by-case study. A generalisation of the reticular theory has however been introduced that allows a better description of twins than the classical approach, while keeping its generality.

The cell of the twin lattice is based on the twin element (twin plane or twin axis), and the lattice element (axis or plane) that is (quasi)-perpendicular to it. The former is identified experimentally by diffraction or by a morphological analysis; the other is then computed from the lattice parameters of the individual. For TLS twinning (both i-TLS and e-TLS), the choice is unique, because there exist one lattice element that is exactly perpendicular (ω = 0º) to the twin element. For TLQS twinning, instead, the choice of the "quasi-perpendicular element" is in general not unique and is often the result of a compromise between the lattice element corresponding to the minimal twin index and the one corresponding to the minimal obliquity. When these two coincide, the description of the twin is unique and unambiguous; otherwise, one or the other can be equally chosen to define the cell of LT: the description of the twin is therefore not unique because more than one sublattice based on the same twin element exists within the accepted limit on the obliquity. This type of twins are called hybrid twins (Nespolo and Ferraris, 2005).

In the hybrid description of the twin, the lattice element corresponding to the minimal obliquity is used to define the cell of LT. The other lattice elements corresponding to higher obliquities but lower indices also contribute to the overall restoration of the lattice nodes. The whole set of concurrent sublattices determine to the overall degree of lattice restoration, expressed by the effective twin index nE (Nespolo and Ferraris, 2006) This index is calculated as the inverse of the fraction of lattice nodes within the cell of LT that belong to any of the N concurrent sublattices; these are in fact the lattice nodes that are restored, within the accepted obliquity, by the twin operation, whereas the classical twin index considers only the lattice nodes restored with an approximation corresponding to the obliquity of the cell chosen to define LT. The effective twin index, coming from different, concurrent sublattices, is not limited to integer values.

The existence of hybrid twins is not limited to TLQS twinning. In fact, TLS twins characterised by a relatively high twin index may also show concurrent sublattices corresponding to non-zero obliquities and an effective twin index lower than the classical twin index: the interpretation of these twins as hybrid twins better explains their occurrence in the framework of the reticular theory of twinning.

From the reticular viewpoint, twins can finally be classified as follows, where the limit ω = 6º corresponds to the empirical limit introduced by Friedel:

Zero-obliquity TLQS

When the twin operation is of order higher than two, the twin lattice may have a pseudo-symmetry despite a zero obliquity. For example, if an orthorhombic twin lattice has a almost equal to b, the c axis acts a pseudo-tetragonal twin axis but [001] is perpendicular to (001). In a case like this, a linear, instead of angular, measure of the pseudo-symmetry is necessary. The twin misfit has been proposed, defined as the distance between the first nodes along the two shortest directions in the plane of LT perpendicular to the twin axis that are quasi-restored by the twin operation (Nespolo and Ferraris, 2007).

Selective Merohedry

In the case of OD structures, class II twins are further subdivided. The family structure may correspond to a lattice system different from both the crystal lattice and the twin lattice. When the point group of the family structure is a subgroup of the point group of the twin lattice and twinning is by class II merohedry (either IIA or IIB), one or more of the twin laws do not belong to the point group of the family structure. This kind of twin law corresponds to merohedry for the OD structure, but to reticular merohedry for the family structure. These twin operations produce incomplete overlap of the family reciprocal sublattice; in particular, in terms of the lattice of the OD structure, they overlap some of the nodes with zero weight of an individual to nodes with non-zero weight of another individual, and vice versa. Therefore, peculiar violations of the non-space-group absences along rows appear in the diffraction pattern, where indexed in terms of the actual structure. This modifies the diffraction pattern, whose geometry no longer corresponds to that of the single crystal. This kind of merohedry, which restores only a part of the family sublattice of OD structures, is termed selective merohedry, whereas twinning by merohedry of OD structures in which the twin operation belongs to the point group of the family structure and restores the whole family reciprocal sublattice is termed complete merohedry ( Nespolo, Ferraris and Ďurovič, 1999)

Beyond twins?

Allotwinning

The oriented association of two or more crystals differing only in their polytypic character is termed allotwinning, from the Greek αλλος, "different", with reference to the individuals ( Nespolo. Kogure and Ferraris, 1999). Allotwinning differs from twinning in that the individuals are not identical but have a different stacking sequence. Allotwinning differs also from oriented overgrowth (epitaxy: Royers, 1928; 1954) and oriented intergrowth (syntaxy: Ungemach, 1935) because the chemical composition is (ideally) identical and, because the building layer(s) are the same, at least two of the three parameters - those in the plane of the layer - are identical also. A cell common to the two individuals can always be found, which in general is a multiple cell for both crystals: the parameter not in the plane of the layer is the shortest one common to the cells of both individuals. As in case of triperiodic epitaxy, a three-dimensional common lattice exists (allotwin lattice): it may coincide with the lattice of one or more individuals or be a sublattice of it. Whereas a triperiodic epitaxy in general may or may not occur, depending on the degree of misfit of the lattice parameters of the individuals, there is no similar condition in allotwinning, because the individuals have a common mesh in the plane of the layer(s) even in polytypes with a different space-group type.

The allotwin operation is a symmetry operation for the allotwin lattice, which may belong to the point group of one or more individuals also. The allotwin of N individuals is characterized by N allotwin indices: the allotwin index of the j-th individual is the order of the subgroup of translation in direct space defining the allotwin lattice with respect to the lattice of the j-th individual.

Plesiotwinning

Still another kind of oriented crystal association occurs, although less frequently, whose lattice can be described in terms of a Concidence-Site Lattice (CSL: a lattice defined by the nodes common to two individual lattices, one of which is rotated with respect to the other by an angle not belonging to the holohedry), and has been termed plesiotwinning, from the Greek πλεσιος, "close to" ( Nespolo, Ferraris, Takeda, and Takéuchi, 1999). Plesiotwins are characterized by the following features:

  1. the lattice common to the individuals (plesiotwin lattice )is always a sublattice for any of the individuals; the order of the subgroup of translation (plesiotwin index) is usually higher than in twins;
  2. the operation relating the individuals corresponds to a symmetry or pseudo-symmetry element of the plesiotwin lattice but not of the individuals, and that element has high indices in the setting of the individuals;
  3. pairs of individuals are rotated about the normal to the composition plane by a non- crystallographic angle, even neglecting the obliquity.

Plesiotwinning is a macroscopic phenomenon that differs from twinning not only in a geometrical definition but also from a physical viewpoint. Whereas for twins the twin index and the twin obliquity directly influence the probability of twin occurrences, for plesiotwins a similar lattice control is not recognized. In fact, the lowest possible plesiotwin index is 5 (individuals with an qp two-dimensional lattice), but plesiotwins often show higher indices. The degree of restoration of lattice nodes is too small for a lattice control to be active. The plesiotwin formation is thus structurally controlled. Twins are usually believed to form in the early stages of crystal growth (Buerger, 1945), but the formation of twins from macroscopic crystals is also known ( Nespolo and Ferraris, 2004b). When two or more nanocrystals interact, they can adjust their relative orientation until they reach a minimum energy configuration, corresponding either to a parallel growth or to a twin. When two macrocrystals interact, the energy barrier to the mutual adjustment is higher, especially at low temperature. If two macrocrystals coalesce or exsolve taking at first a relative orientation corresponding to an unstable atomic configuration at the interface, they tend to rotate until they reach a lower energy configuration. Parallel growth and twinning correspond to minimal interface energy, whereas plesiotwinning corresponds to a less-deep minimum. However, twin orientations are less numerous and are separated by larger angles, whereas plesiotwin orientations are more numerous and separated by smaller angles. Only limited adjustments may be necessary to reach plesiotwin orientations, which may thus represent a kind of compromise between the original unstable configuration and the too distant, although more stable, configuration of twins.

Modular structures

The study of twinned crystals dates back to the very beginning of crystallography as a science. Partly for that reason, the idea of a twinned crystal is often associated with that of macroscopic edifice. Nowadays, the study at nanoscale has shown that the formation of twins may occur at a very early stage of crystal growth. This imposes to rethink the boundary separating twins from related categories. In particular, a comparison has to be made between twins and one of the subcategories of modular structures, i.e. namely those modular structures which are built by the repetition of the same module, pairs of modules being related by a symmetry operation.

When the modules have a different chemistry, a twin cannot arise. However, modular structures are known in which modules have the same composition: for example, isochemical cell-twins (cell-twins in which the building modules have the same chemistry: Takéuchi, 1997; Ferraris, Makovicky and Merlino, 2008; Nespolo, Ferraris, Ďurovič, and Takéuchi, 2004).

If we take a twin and imagine to reduce the size of the individuals down to the order of the unit cell, we obtain two main consequences:

  1. The modules grow in number, and are repeated polysynthetically
  2. The edifice is now homogeneous, and can be described by a space group

The boundary between the two extreme cases cannot be unequivocally stated in terms of the size of the individuals. We can however distinguish a twin from an isochemical cell-twin on the basis of the following criteria:



General bibliography on twinning

Bravais, M. (1851). Études cristallographiques. Troisième partie. Des macles et des hémitropies. J. Ecole Polytechn., XX (XXXIV), 248-276.

Buerger, M.J. (1945). The Genesis of Twin Crystals. Am. Mineral., 30, 469-482.

Buerger, M.J. (1954). The Diffraction Symmetry of Twins. Anaias Acad. Bras. Ciéncias, 26, 111-121 [later republished in Buerger, M.J. (1960). Crystal-structure analysis. John Wiley & Sons, New York, 53-67].

Cahn, R.W. (1954). Twinned Crystals. Adv. Phys., 3, 363-445.

Catti, M. and Ferraris, G. (1976). Twinning by Merohedry and X-ray Crystal Structure Determination. Acta Crystallogr. A32, 163-165.

Curien, H. and Donnay, J.D.H. (1959). The Symmetry of the Complete Twin. Am. Mineral., 44, 1067-1070.

Curien, H. and Le Corre, Y. (1958). Notations des macles à l'aide du symbolisme des groupes de couleurs de Choubnikov. Bull. Soc. franç Minér. Crist., 81, 126-132.

Donnay, G. (1940). Width of albite-twinning lamellae. Am. Mineral., 25, 578-586.

Donnay, J.D.H. and Donnay, G. (1959). Twinning, section 3.1.9 in International Tables for X-Ray Crystallography, Vol. III. Birmingham: Kynoch Press (1959).

Donnay, G. and Donnay, J.D.H. (1974). Classification of Triperiodic Twins. Can. Mineral., 12, 422-425.

Ferraris G., Makovicky E., Merlino S. (2008). Crystallography of modular materials. Oxford University Press

Friedel, G. (1904). Étude sur les groupements cristallins. Extrait du Bullettin de la Société de l'Industrie minérale, Quatrième série, Tomes III e IV. Saint-Étienne, Société de l'imprimerie Thèolier J. Thomas et C., 485 pp.

Friedel, G. (1920). Contribution à l'étude géométrique des macles. Bull Soc fr Minér, 43, 246-295.

Friedel, G. (1926). Leçons de Cristallographie. Berger-Levrault, Nancy, Paris, Strasbourg, XIX+602 pp.

Friedel, G. (1933). Sur un nouveau type de macles. Bull. Soc. franç Minér. 56, 262-274.

Hahn T., Janovec V. and Klapper H. (1999). Bicrystals, twins and domain structures - A comparison. Ferroelectrics 222, 11-21.

Hahn, Th. and Klapper, H. (2003). Twinning of crystals. Section 3.3 in A. Authier, Ed., International Tables for Crystallography Volume D: Physical Properties of Crystals. International Union of Crystallography / Kluwer Academic Publishers.

Hartman, P. (1956). On the Morphology of Growth Twins. Z. Kristallogr., 107, 225-237.

Holser, W.T. (1958). Relation of Symmetry to Structure in Twinning. Z. Kristallogr., 110 249-265.

Koch, E. (2004). Twinning, Sect. 1.3 in International Tables for Crystallography, Vol. C, ed. by A.C.J. Wilson, International Union of Crystallography / Kluwer Academic Publishers.

Mallard, E (1879). Traité de Cristallographie géométrique et physique, Vol. I. Paris: Dunod. 372pp.

Mallard, E (1884). Les Groupements Cristallins,. Conférence faite à l société chimique de Paris. Extrait de la revue scientifique des 30 juillet et 6 août 1887. Paris: Administration des deux revues. 51pp.

Mallard, E (1884). Traité de Cristallographie géométrique et physique, Vol. II. Paris: Dunod. 600pp.

Mallard, E (1885). Sur la théorie des macles. Bull. Soc. fr. minéral.. 8, 452-469.

Mügge, O. (1898). Über Translationen und verwandte Erscheinungen in Krystallen. Neues Jahrb. Mineral. Geol. Paläontol., 1, 71-158.

Nespolo, M. (2004). Twin point groups and the polychromatic symmetry of twins. Z. Kristallogr. 219, 57-71.

Nespolo, M. and Ferraris, G. (2000). Twinning by syngonic and metric merohedry. Analysis, classification and effects on the diffraction pattern. Z. Kristallogr. 215, 77-81.

Nespolo, M. and Ferraris, G. (2003). Geminography - The science of twinning applied to the early-stage derivation of non-merohedric twin laws. Z. Kristallogr. 218, 178-181.

Nespolo, M. and Ferraris, G. (2004a). Applied geminography - Symmetry analysis of twinned crystals and definition of twinning by reticular polyholohedry. Acta Crystallogr. A60, 89-95.

Nespolo, M. and Ferraris, G. (2004b). The oriented attachment mechanism in the formation of twins - a survey. Eur. J. Mineral. 16, 401-406.

Nespolo, M. and, Ferraris, G. (2005). Hybrid twinning - A cooperative type of oriented crystal association. Z. Kristallogr. 220, 317-323.

Nespolo, M. and Ferraris, G. (2006). The derivation of twin laws in non-merohedric twins - Application to the analysis of hybrid twins. Acta Crystallogr. A62, 336-349.

Nespolo, M. and Ferraris, G. (2007). Overlooked problems in manifold twins: twin misfit in zero-obliquity TLQS twinning and twin index calculation. Acta Crystallogr. A63, 278-286.

Nespolo, M., Ferraris, G. and Ďurovič, S. (1999). OD character and twinning - Selective merohedry in class II merohedric twins of OD polytypes. Z. Kristallogr. 214, 776-779.

Nespolo, M., Ferraris, G., Ďurovič, S. and Takéuchi, Y. (2004). Twins vs. modular crystal structures. Z. Kristallogr. 219, 773-778.

Nespolo, M., Ferraris, G., Takeda, H. and Takéuchi, Y. (1999). Plesiotwinning: oriented crystal associations based on a large coincidence-site lattice. Z. Kristallogr. 214, 378-382.

Nespolo. M., Kogure, T. and Ferraris, G. (1999). Allotwinning: oriented crystal association of polytypes - Some warnings on consequences. Z. Kristallogr. 214, 5-8.

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Acknowledgements

Critical remarks by Prof. Hans Grimmer are gratefully acknowledged.