General remarks

A twin consists of two or more single crystals of the same species but in different orientation, its twin componentsTwin(s)components. They are intergrown in such a way that at least some of their lattice directions are parallel. The twin lawTwin(s)law describes the geometrical relation between the twin components. It specifies a symmetry operation, the twin operationTwin(s)operation, that brings one of the twin components into parallel orientation with the other. The corresponding symmetry element is called the twin elementTwin(s)element.

There are several kinds of twin lawsTwin(s)law:

In many cases, there does not exist a unique twin lawTwin(s)law, but a twin may be described equally well by more than one twin law. (a) If the crystal structure of the twin componentsTwin(s)components contains an evenfold rotation or screw-rotation axis, an inversion twin cannot be distinguished from a reflection twinReflection twinsTwin(s)reflection with twin planeTwin(s)plane perpendicular to that axis. (b) If the crystal structure contains a mirror or a glide plane, an inversion twinTwin(s)inversion cannot be distinguished from a rotation twinRotation twinsTwin(s)rotation with a twofold twin axis perpendicular to that plane. (c) If for a centrosymmetrical crystal structure the normal of a twin plane runs parallel to a lattice vector or a twin axisTwin(s)axis runs perpendicular to a net plane, the twin may be described equally well as a reflection twin or as a rotation twin.

The twin componentsTwin(s)components are grown together in a surface called composition surfaceComposition surface, twin interfaceTwin(s)interface or twin boundaryTwin(s)boundary. In most cases, the composition surfacesComposition surface are low-energy surfaces with good structural fit. For a reflection twinReflection twinsTwin(s)reflection, it is usually a plane parallel to the twin plane. The composition surface of a rotation twin may either be a plane parallel to the twin axisTwin(s)axis or be a non-planar surface with irregular shape.

If more than two componentsTwin(s)components are twinned according to the same law, the twin is called a repeated twinRepeated twinsTwin(s)repeated or a multiple twinMultiple twinsTwin(s)multiple. If all the twin boundaries are parallel planes, it is a polysynthetic twinPolysynthetic twinsTwin(s)polysynthetic, otherwise it is called a cyclic twinCyclic twinsTwin(s)cyclic. If the twin components are related to each other by more than one twin lawTwin(s)law, the shape and the mutual arrangement of the twin domains may be very irregular.

With respect to the formation process, one may distinguish between growth twinsGrowth twinsTwin(s)growth, transformation twinsTransformation(s)twinsTwin(s)transformation, and mechanical (deformation, glideMechanical (deformation, glide) twinsTwin(s)mechanical (deformation, glide)) twinsMechanical twins. Transformation twinsTransformation(s)twinsTwin(s)transformation result from phase transitions, e.g. of ferroelectric or ferromagnetic crystals. The corresponding twin domains are usually small and the number of such domains is high. Mechanical twinning is due to mechanical stress and may often be described in terms of shear of the crystal structure. This includes ferroelasticity.

Twins are observable by, for example, macroscopic or microscopic observation of re-entrant angles between crystal faces, by etching, by means of different extinction positions for the twin componentsTwin(s)components between cross polarizers of a polarization microscope, by different rotation angles of the plane of polarization of a beam of plane-polarized light passing through the components of a twin showing optical activity, by a splitting of part of the X-ray diffraction spots (except for twins by merohedry), by means of domain contrast or boundary contrast in an X-ray topogram, or by investigation with a transmission electron microscope.

The phenomenon of twinning has frequently been described and discussed in the literature and it is impossible, therefore, to give a complete list of references. Further details may be learned, e.g. from a review article by Cahn (1954) or from appropriate textbooks. A comprehensive survey of X-ray topography of twinned crystals is given by Klapper (1987). The following papers are related to twinning by merohedryTwinningby merohedry or pseudo-merohedry: Catti & Ferraris (1976), Grimmer (1984, 1989a, b), Grimmer & Warrington (1985), Donnay & Donnay (1974), Le Page, Donnay & Donnay (1984), Hahn (1981, 1984), Klapper, Hahn & Chung (1987), Flack (1987).

Twin latticesLattice(s)twinTwin(s)lattices

For reflectionReflection twinsTwin(s)reflection and rotation twinsRotation twinsTwin(s)rotation described in the last section, a special situation arises whenever there exists a lattice vector perpendicular to the twin planeTwin(s)plane or a lattice plane perpendicular to a rational twofold twin axisRational twin axisTwin(s)axis, rational. Such a situation occurs systematically for all reflection and rotation twins with cubic symmetry and for certain twins with non-cubic symmetry (cf. Table 1.3.2.1). In addition, such a perpendicularity may occur occasionally if equation (Summary of general formulaeE. Koch1.1.2.12 ) is satisfied.

Lattice planes and rows that are perpendicular to each other independently of the metrical parameters Lattice planes and rows that are perpendicular to each other independently of the metrical parameters

In the case of a noncentrosymmetric crystal structure, different twins result from a twin axisTwin(s)axis [uvw] with a perpendicular lattice plane (hkl), or from a twin planeTwin(s)plane (hkl) with a perpendicular lattice row [uvw]: the reflection twinReflection twins consists of two enantiomorphous twin componentsTwin(s)components whereas the rotation twin is built up from two crystals with the same handedness (cf., for example, Brazil twinsBrazil twinsTwin(s)Brazil and Dauphiné twinsDauphiné twinsTwin(s)Dauphiné of quartzQuartz twinsTwin(s)quartz). With respect to the first twin component, the lattice of the second component has the same orientation in both cases. For a centrosymmetrical crystal structure, both twin lawsTwin(s)law give rise to the same twin.

Whenever a twin planeTwin(s)plane or twin axisTwin(s)axis is perpendicular to a lattice vector or a net plane, respectively, the vector lattices of the twin componentsTwin(s)components have a three-dimensional subset in common. This sublattice [derivative latticeDerivative latticeLattice(s)derivative, cf. IT A (2005, Chapter Derivative latticesY. BillietE. F. Bertaut13.2 )] is called the twin lattice. It corresponds uniquely to the intersection group of the two translation groups referring to the twin components. The respective subgroup index i is called the twin indexTwin(s)index. It is equal to the ratio of the volumes of the primitive unit cells for the twin lattice and the crystal structure. If one subdivides the crystal lattice into nets parallel to the twin plane or perpendicular to the twin axis, each ith of these nets belongs to the common twin lattice of the two twin components (cf. Fig. 1.3.2.1

Figure 1.3.2.1 | top | pdf | (a) Projection of the lattices of the twin componentsTwin(s)components of a monoclinic twinned crystal (unique axis c, γ = 93°) with twin indexTwin(s)index 3. The twin may be interpreted either as a rotation twinRotation twins with twin axisTwin(s)axis [210] or as a reflection twinReflection twins with twin planeTwin(s)plane (110). (b) Projection of the corresponding reciprocal lattices.

(a) Projection of the lattices of the twin componentsTwin(s)components of a monoclinic twinned crystal (unique axis c, γ = 93°) with twin indexTwin(s)index 3. The twin may be interpreted either as a rotation twinRotation twins with twin axisTwin(s)axis [210] or as a reflection twinReflection twins with twin planeTwin(s)plane (110). (b) Projection of the corresponding reciprocal lattices.

(a) Projection of the lattices of the twin componentsTwin(s)components of a monoclinic twinned crystal (unique axis c, γ = 93°) with twin indexTwin(s)index 3

). Important examples are cubic twins with [111] as twofold twin axisTwin(s)axis or (111) as twin plane and rhombohedral twins with [001] as twin axes or (001) as twin plane (hexagonal description). In all these cases, the twin indexTwin(s)index i equals 3.

For every twin lattice, its twin indexTwin(s)index i can be calculated from the Miller indicesMiller indices of the net plane (hkl) and the coprime coefficients u, v, w of the lattice vector t perpendicular to (hkl). Referred to a primitive latticeTwin(s)primitive lattice basis, i is simply related to the modulus of the scalar product j of the two vectors and : The same procedure – but with modified coefficients – may be applied to a centred latticeTwin(s)centred lattice described with respect to a conventionally chosen basis: The coprime Miller indicesMiller indices h, k, l that characterize the net plane have to be replaced by larger non-coprime indices h′, k′, l′, if h, k, l do not refer to a (non-extinct) point of the reciprocal lattice. The integer coefficients u, v, w specifying the lattice vector perpendicular to (hkl) have to be replaced by smaller non-integer coefficients u′, v′, w′, if the centred lattice contains such a vector in the direction [uvw].

1.3.2.1. Examples

| top | pdf |

Examples

Projection of the lattices of the twin componentsTwin(s)components of an orthorhombic twinned crystal (oC, b = 2a) with twin indexTwin(s)index 4

(4) Rhombohedral lattice in hexagonal description with : is perpendicular to .

Because of the R centring, has to be replaced by .

As refers to an `extinct reflection' of an R lattice, the triplet has to be replaced by .

  even

Projection of the lattices of the twin componentsTwin(s)components of an orthorhombic twinned crystal (oC, b = 2a) with twin indexTwin(s)index 4. The twin may be interpreted either as a rotation twinRotation twinsTwin(s)rotation with twin axisTwin(s)axis [210] or as a reflection twinReflection twins with twin planeTwin(s)plane (120).

(1) Cubic P lattice: [111] is perpendicular to (111).  odd . (2) Orthorhombic lattice with : [310] is perpendicular to (110). Projection of the lattices of the twin componentsTwin(s)components of an orthorhombic twinned crystal (oP, b = a) with twin indexTwin(s)index 2 (i) P lattice (cf. Fig. 1.3.2.2):  even (ii) C lattice (cf. also Fig. 1.3.2.2): Because of the C centring, [310] has to be replaced by .  even Figure 1.3.2.2 | top | pdf | Projection of the lattices of the twin componentsTwin(s)components of an orthorhombic twinned crystal (oP, b = a) with twin indexTwin(s)index 2. The twin may be interpreted either as a rotation twinRotation twins with twin axisTwin(s)axis [310] or as a reflection twinReflection twins with twin planeTwin(s)plane (110). The figure shows, in addition, that twin index 1 results if the oP lattice is replaced by an oC lattice in this example (twinning by pseudomerohedryTwinningby pseudomerohedry). Projection of the lattices of the twin componentsTwin(s)components of an orthorhombic twinned crystal (oP, b = a) with twin indexTwin(s)index 2. The twin may be interpreted either as a rotation twinRotation twins with twin axisTwin(s)axis [310] or as a reflection twinReflection twins with twin planeTwin(s)plane (110). The figure shows, in addition, that twin index 1 results if the oP lattice is replaced by an oC lattice in this example (twinning by pseudomerohedryTwinningby pseudomerohedry). (3) Orthorhombic C lattice with b = 2a: [210] is perpendicular to (120) (cf. Fig. 1.3.2.3). As (120) refers to an `extinct reflection' of a C lattice, the triplet 240 has to be used in the calculation. even . Figure 1.3.2.3 | top | pdf | Projection of the lattices of the twin componentsTwin(s)components of an orthorhombic twinned crystal (oC, b = 2a) with twin indexTwin(s)index 4. The twin may be interpreted either as a rotation twinRotation twinsTwin(s)rotation with twin axisTwin(s)axis [210] or as a reflection twinReflection twins with twin planeTwin(s)plane (120).

Implication of twinning in reciprocal spaceTwinningreciprocal-space implications

As shown above, the direct lattices of the components of any twin coincide in at least one row. The same is true for the corresponding reciprocal lattices. They coincide in all rows perpendicular to parallel net planes of the direct lattices.

For a reflection twinReflection twinsTwin(s)reflection with twin planeTwin(s)plane (hkl), the reciprocal lattices of the twin components have only the lattice points with coefficients nh, nk, nl in common.

For a rotation twinRotation twinsTwin(s)rotation with twofold twin axisTwin(s)axis [uvw], the reciprocal lattices of the twin components coincide in all points of the plane perpendicular to [uvw], i.e. in all points with coefficients h, k, l that fulfil the condition .

For a rotation twin with irrational twin axis parallel to a net plane (hkl), only reciprocal-lattice points with coefficients nh, nk, nl are common to both twin components.

As the entire direct lattices of the two twin components coincide for an inversion twinInversion twinsTwin(s)inversion, the same must be true for their reciprocal lattices.

For a reflection or rotation twin with a twin lattice of index i, the corresponding reciprocal lattices, too, have a sublattice with index i in common (cf. Fig. 1.3.2.1b). In analogy to direct space, the twin lattice in reciprocal space consists of each ith lattice plane parallel to the twin planeTwin(s)plane or perpendicular to the twin axisTwin(s)axis. If the twin indexTwin(s)index equals 1, the entire reciprocal lattices of the twin components coincide.

If for a reflection twinReflection twinsTwin(s)reflection there exists only a lattice row [uvw] that is almost (but not exactly) perpendicular to the twin plane (hkl), then the lattices of the two twin components nearly coincide in a three-dimensional subset of lattice points. The corresponding misfit is described by the quantity , the twin obliquity. It is the angle between the lattice row [uvw] and the direction perpendicular to the twin plane (hkl). In an analogous way, the twin obliquity is defined for a rotation twinRotation twinsTwin(s)rotation. If (hkl) is a net plane almost (but not exactly) perpendicular to the twin axisTwin(s)axis [uvw], then is the angle between [uvw] and the direction perpendicular to (hkl).

Twinning by merohedryTwinningby merohedry

A twin is called a twin by merohedryTwinningby merohedry if its twin operation belongs to the point group of its vector lattice, i.e. to the corresponding holohedryHolohedry. As each lattice is centrosymmetric, an inversion twinInversion twinsTwin(s)inversion is necessarily a twin by merohedry. Only crystals from merohedral (i.e. non-holohedral) point groups may form twins by merohedry; 159 out of the 230 types of space groups belong to merohedral point groupsMerohedral point groupsPoint group(s)merohedral.

For a twin by merohedry, the vector lattices of all twin components coincide in direct and in reciprocal space. The twin indexTwin(s)index is 1. The maximal number of differently oriented twin components equals the subgroup index m of the point group of the crystal with respect to its holohedry.

Table 1.3.4.1 displays all possibilities for twinning by merohedryTwinningby merohedry. For each holohedral point group (column 1), the types of Bravais lattices (column 2) and the corresponding merohedral point groups (column 3) are listed. Column 4 gives the subgroup index m of a merohedral point group in its holohedry. Column 5 shows m − 1 possible twin operations referring to the different twin components. These twin operations are not uniquely defined (except for point group 1), but may be chosen arbitrarily from the corresponding right coset of the crystal point group in its holohedry. It is always possible, however, to choose an inversion, a reflection, or a twofold rotation as twin operation.

Possible twin operations for twins by merohedry Possible twin operations for twins by merohedry

A twin that is not a twin by merohedry as defined above but, because of metrical specialization, has a twin lattice with twin index 1 is called a twin by pseudo-merohedry.

Two kinds of twins by merohedry may be distinguished.

Type 1: The twin can be described as an inversion twinInversion twinsTwin(s)inversion. Then, only two twin components exist and the twin operation belongs to the Laue classLaue class of the crystal. As a consequence, the reciprocal lattices of the twin components are superimposed so that coinciding lattice points refer to Bragg reflections with the same values as long as Friedel's law is valid. In that case, no differences with respect to symmetry, or to reflection conditions, or to relative intensities occur between two sets of Bragg intensities measured from a single crystal on the one hand and from a twin on the other hand (whether or not the twin components differ in their volumes). If anomalous scattering is observed and the twin components differ in size, the intensities of Bragg reflections are changed in comparison with the untwinned crystal but the symmetry of the diffraction pattern is unchanged. For equal volumes of the twin components, however, the diffraction pattern is centrosymmetric again. The occurrence of anomalous scattering does not produce additional difficulties for space-group determination. The change of the Bragg intensities in comparison with the untwinned crystals, however, makes a structure determination more difficult.

Type 2: The twin operation does not belong to the Laue class of the crystal. Such twins can occur only in point groups marked by an asterisk in Table 1.3.4.1, i.e. in 55 out of the 159 types of space groups mentioned above. If the different twin components occur with equal volumes, the corresponding diffraction pattern shows enhanced symmetryEnhanced symmetrySymmetryenhanced. On the contrary, the reflection conditions are unchanged in comparison to those for a single crystal, except for . As a consequence, for 51 out of the 55 space-group types, the derivation of `possible space groups', as described in IT A (2005, Part Space-group determination and diffraction symbolsA. Looijenga-VosM. J. Buerger3 ), gives incorrect results. For , and , the combination of the simulated Laue class of the twinTwin(s)simulated Laue class of and the (unchanged) extinction symbolExtinctionsymbol does not occur for single crystals. Therefore, the symmetry of these twins can be determined uniquely. In the case of , the reflection conditions differ for the two twin componentsReflectionconditions, for a twinned crystalTwinned crystalreflection conditions. [This is because the holohedryHolohedry of is whereas the Laue class of the Euclidean normalizer of is ; cf. IT A (2005, Part Introduction and definitionsE. KochW. FischerU. Müller15 ).] As a consequence, the reflection conditions for such a twinned crystalTwinned crystalreflection conditions differ from all conditions that may be observed for single crystals (hkl cyclically permutable: 0kl only with k = 2n or l = 2n; 00l only with l = 2n) and, therefore, the true symmetry can be identified without uncertainty.

In Table 1.3.4.2, all simulated Laue classesLaue classTwin(s)simulated Laue class of (column 1) are listed that may be observed for twins by merohedry of type 2. Column 2 shows the corresponding extinction symbolsExtinctionsymbol. The symbols of the simulated `possible space groups' that follow from IT A (2005, Part Space-group determination and diffraction symbolsA. Looijenga-VosM. J. Buerger3 ) are gathered in column 3. The last column displays the symbols of those space groups which may be the true symmetry groups for twins by merohedry showing such diffraction patterns.

Table 1.3.4.2| top | pdf | Simulated Laue classesLaue classTwin(s)simulated Laue class of, extinction symbolsExtinctionsymbol, simulated `possible space groups', and possible true space groups for crystals twinned by merohedryTwinningby merohedry (type 2) Twinned crystal Single crystal Simulated Laue classTwin(s)simulated Laue class of Twin extinction symbolExtinctionsymbol Simulated `possible space groups' Possible true space groups 4/mmm P - - - , - - - - - - - - – I - - - - - - - – P - - - , P3, P\bar3 - - P - - - , - - R - - P - - - - - P - - - , , , - - - - , , - - P - - c P - c - P - - - - - - - I - - - Ia - - – F - - - Fd - - - - –

Simulated Laue classesLaue classTwin(s)simulated Laue class of, extinction symbolsExtinctionsymbol, simulated `possible space groups', and possible true space groups for crystals twinned by merohedryTwinningby merohedry (type 2) Simulated Laue classesLaue classTwin(s)simulated Laue class of, extinction symbolsExtinctionsymbol, simulated `possible space groups', and possible true space groups for crystals twinned by merohedryTwinningby merohedry (type 2)

Calculation of the twin elementTwin(s)elementTwin(s)elementCalculation of the twin element

If the twin elementTwin(s)element cannot be recognized by direct macroscopic or microscopic inspection, it may be calculated as described below. Given are two analogous bases a, b, c and a′, b′, c′ referring to the two twin components. If possible, both basis systems should be chosen with the same handedness. If no such bases exist, the twin is a reflection twin and one of the bases has to be replaced by its centrosymmetrical one, e.g. a′, b′, c′ by −a′, −b′, −c′. The relation between the two bases is described by The coefficients e_ij have to be obtained by measurement.

Basis a, b, c may be mapped onto a′, b′, c′ by a pure rotation that brings a to a′, b to b′, and c to c′. To derive the direction of the rotation axis, calculate the three vectors a₁, b₁, c₁ bisect the angles , , and , respectively. Calculate three further vectors of arbitrary length a₂, b₂, c₂ which are perpendicular to the planes defined by a and a′, b and b′, and c and c′, respectively, from the scalar products The plane defined by a₁ and a₂ is perpendicular to the plane defined by a and a′ and bisects the angle . Analogous planes refer to b₁ and b₂, and c₁ and c₂. Vectors , , and lying within one of these planes may be described as linear combinations of a₁ and a₂, b₁ and b₂, or c₁ and c₂, respectively: The common intersection line of these three planes is parallel to the twin axis. It may be calculated by solving any of the three equations : choose arbitrarily equal to 1. Solve the inhomogeneous system of three equations that corresponds to this vector equation for the three variables , , and . Calculate the vector . Its components with respect to a, b, c describe the direction of the twin axis.

The angle τ of the twin rotation may then be calculated by with .

If the basis a, b, c is orthogonal, τ may be obtained from If the coefficients of r are rational and τ equals 180°, then r describes the direction either of the twofold twin axis or of the normal of the twin plane. If r is rational and τ equals 60, 90 or 120°, r is parallel to the twin axis. If r is irrational, but τ equals 180° and there exists, in addition, a net plane perpendicular to r, this net plane describes the twin plane.

If none of these conditions is fulfilled, one has to repeat the calculations with a differently chosen basis system for one of the twin components. The number of possibilities for this choice depends on the lattice symmetry. The following list gives all equivalent basis systems for all descriptions of Bravais lattices used in IT A (2005):