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

International Tables for Crystallography (2006). Vol. D, ch. 3.3, pp. 426-444

Section 3.3.10. Twin boundaries

Th. Hahna* and H. Klapperb

aInstitut 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. Twin boundaries

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3.3.10.1. Contact relations in twinning

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So far, twinning has been discussed only in terms of symmetry and orientation relations of the (bulk) twin components. In this chapter, the very important aspect of contact relations is discussed. This topic concerns the orientation and the structure of the twin boundary, which is also called twin interface, composition plane, contact plane, domain boundary or domain wall. It is the twin boundary and its structure and energy which determine the occurrence or non-occurrence of twinning. In principle, for each crystal species an infinite number of orientation relations obey the requirements for twinning, as set out in Section 3.3.2[link], because any rational lattice plane (hkl), as well as any rational lattice row [uvw], common to both partners would lead to a legitimate reflection or rotation twin. Nevertheless, only a relatively small number of crystal species exhibit twinning at all, and, if so, with only a few twin laws. This wide discrepancy between theory and reality shows that a permissible crystallographic orientation relation (twin law) is a necessary, but not at all a sufficient, condition for twinning. In other words, the contact relations play the decisive role: a permissible orientation relation can only lead to actual twinning if a twin interface of good structural fit and low energy is available.

In principle, a twin boundary is a special kind of grain boundary connecting two `homophase' component crystals which exhibit a crystallographic orientation relation, as defined in Section 3.3.2.[link] For a given orientation relation of the twin partners, crystallographic or general, the interface energy depends on the orientation of their boundary. It is intuitively clear that crystallographic orientation relations lead to energetically more favourable boundaries than noncrystallographic ones. As a rule, twin boundaries are planar (at least in segments), but for certain types of twins curved and irregular interfaces have been observed. This is discussed later in this section.

In order to determine theoretically for a given twin law the optimal interface, the interface energy has to be calculated or at least estimated for various boundary orientations. This problem has not been solved for the general case so far. The special situation of reflection twins with coinciding twin mirror and composition planes has recently been treated by Fleming et al. (1997[link]). These authors calculated the interface energies for three possible reflection twin laws in each of aragonite, gibbsite, corundum, rutile and sodium oxalate, and they compared the results with the observed twinning. In all cases, the twin law with lowest boundary energy corresponds to the twin law actually observed. Another calculation of the twin interface energy has been performed by Lieberman et al. (1998[link]) for the [(10{\bar 2})] reflection twins of monoclinic saccharin crystals. In this study, the [(10{\bar 2})] boundary energy was calculated for different shifts of the two twin components with respect to each other. It was shown that a minimum of the boundary energy is achieved for a particular `twin displacement vector' (cf. Section 3.3.10.4.1[link]).

Calculations of interface energies, as performed by Fleming et al. (1997[link]) and Lieberman et al. (1998[link]), however, require knowledge of the atomic potentials and their parameters for each pair of bonded atoms. They are, therefore, restricted to specific crystals for which these parameters are known. Similarly, high-resolution electron microscopy (HRTEM) images of twin boundaries have been obtained so far for only a small number of crystals.

It is possible, however, to predict for a given twin law low-energy twin boundaries on the basis of symmetry considerations, even without knowledge of the crystal structure, as discussed in the following section. This prediction has been carried out by Sapriel (1975[link]) for ferroelastic crystals. His treatment assumes a phase transition from a real or hypothetical parent phase (supergroup [{\cal G}]) to a `distorted' (daughter) phase of lower eigensymmetry (subgroup [{\cal H}]), leading to two (or more) domain states of equal but opposite shear strain. The subgroup [{\cal H}] must belong to a lower-symmetry crystal system than the supergroup [{\cal G}], as explained in Section 3.3.7.3[link](ii). Similar criteria, but restricted to ferroelectric materials, had previously been devised in 1969 by Fousek & Janovec (1969[link]). A review of ferroelastic domains and domain walls is provided by Boulesteix (1984[link]) and an extension of the Sapriel procedure to phase boundaries between a ferro­elastic and its `prototypic' (parent) phase is given by Boulesteix et al. (1986[link]).

3.3.10.2. Strain compatibility of interfaces

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For a simple derivation of stress-free contact planes, we go back to the classical description of mechanical twinning by a homogeneous shear, which is illustrated by a deformation ellipsoid as shown in Fig. 3.3.10.1[link](a) (cf. Liebisch, 1891[link]; Niggli, 1941[link]; Klassen-Neklyudova, 1964[link]). In a modification of this approach, we consider two parts of a homogeneous, crystalline or noncrystalline, solid body, which are subjected to equal but opposite shear deformations [-\varepsilon] and [+\varepsilon]. The undeformed state of the body and the deformed states of its two parts are represented by a sphere ([\varepsilon = 0]) and by two ellipsoids [-\varepsilon] and [+\varepsilon], as shown in Fig. 3.3.10.1[link](b). We now look for stress-free contact planes between the two deformed parts, i.e. planes for which line segments of any direction parallel to the planes experience the same length change in both parts during the shear. This criterion is obeyed by those planes that exhibit identical cross sections through both ellipsoids. Mathematically, this is expressed by the equation (Sapriel, 1975[link])[(\varepsilon^{\rm I}_{ij}-\varepsilon^{\rm II}_{ij})x_ix_j=2\varepsilon x_1x_2=0]([\varepsilon^{\rm I}_{ij}=\varepsilon^{\rm I}_{12}=+\varepsilon, \varepsilon^{\rm II}_{ij}=\varepsilon^{\rm II}_{12}=-\varepsilon]; [x_1,x_2,x_3] are Cartesian coordinates) which has as solutions the two planes [x_1=0] (plane BB in Fig. 3.3.10.1[link]b) and [x_2=0] (plane AA). These planes are called `planes of strain compatibility' or `permissible' planes. From the solutions of the above equation and Fig. 3.3.10.1[link](b) it is apparent that two such planes, AA and BB, normal to each other exist. The intersection line of the two compatible planes is called the shear axis of the shear deformation.

[Figure 3.3.10.1]

Figure 3.3.10.1 | top | pdf |

(a) Classical description of mechanical twinning by homogeneous shear deformation (Liebisch, 1891[link], pp. 104–118; Niggli, 1941[link], pp. 145–149; Klassen-Neklyudova, 1964[link], pp. 4–10). The shear deforms a sphere into an ellipsoid of equal volume by translations (arrows) parallel to the twin (glide) plane AA. The translations are proportional to the distance from the plane AA. Shear angle [2\varepsilon]. (Only the translations in the upper half of the diagram are shown, in the lower half they are oppositely directed.) (b) Ellipsoids representing the (spontaneous) shear deformations [-\varepsilon] and [+\varepsilon] of two orientation states, referred to the (real or hypothetic) intermediate (prototypic) state with [\varepsilon = 0] (sphere). The switching of orientation state [-\varepsilon] into state [+\varepsilon] through the shear angle [2\varepsilon] is, analogous to (a), indicated by arrows. The shear ellipsoids [-\varepsilon] and [+\varepsilon] have common cross sections along the perpendicular planes AA and BB which are both, therefore, mechanically compatible contact planes of the [+\varepsilon] and [-\varepsilon] twin domains.

It is noted that during a shear deformation induced by the (horizontal) translations shown in Fig. 3.3.10.1[link](b), only the plane AA, parallel to the arrows, can be generated as a contact plane between the two domains. A contact plane BB, normal to the arrows, cannot be formed by this process, because this would lead to a gap on one side and a penetration of the material on the other side. Plane BB, however, could be formed during a (virtual) switching between [+\varepsilon] and [-\varepsilon] with `vertical' translations, parallel to BB, which would formally result in the same mutual arrangement of the ellipsoids. The compatibility criterion, as expressed by the equation above (which applies to elastic continua), does not distinguish between these two cases. Note that the planes AA and BB are mirror planes relating the deformations [+\varepsilon] and [-\varepsilon]. Both contact planes often occur simultaneously in growth twins, see for example the dovetail and the Montmartre twins of gypsum (Fig. 3.3.6.1[link]). In general, each interface coinciding with a twin mirror plane or a plane normal to a twin axis is a (mechanically) compatible contact plane.

It should be emphasized that the criterion `strain compatibility' is a purely mechanical one for which only stress and strain are considered. It leads to `mechanical' low-energy boundaries. Other physical properties, such as electrical polarization, may reduce the number of mechanically permissible boundaries, e.g. due to energetically unfavourable head-to-head or tail-to-tail orientation of the axis of spontaneous polarization in polar crystals. The mechanical compatibility criterion is, however, always applicable to centrosymmetric materials.

3.3.10.2.1. Sapriel approach to permissible (compatible) boundaries in ferroelastic (non-merohedral) transformation twins

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The general approach to strain compatibility, as given above, can be employed to derive the permissible composition planes for twins with inclined axes (non-merohedral twins; for merohedral twins see Section 3.3.10.2.3[link] below). This concept was applied by Sapriel (1975[link]) to the 94 Aizu species of ferroelastic transformation twins. According to Aizu (1969[link], 1970[link][link]a,b), each species is represented by a pair of symmetry groups, separated by the letter F (= ferroic) in the form [{\cal K}F{\cal H}], e.g. [2/m\,2/m\,2/mF 12/m1] or [m{\bar 3}mF {\bar 3}m]. The parent phase with symmetry [{\cal G}] represents the undeformed (zero-strain) reference state (the sphere in Fig. 3.3.10.1[link]b), whereas the spontaneous strain of the two orientation states of phase [{\cal H}] is represented by the two ellipsoids. Details of the calculation of the permissible domain boundaries for all ferroelastic transformation twins are given in the paper by Sapriel (1975[link]).

Two kinds of permissible boundaries are distinguished by Sapriel:

  • (a) W boundaries. These interfaces are parallel to symmetry planes of the parent phase (supergroup [{\cal K}]), which are lost in the transition and have become twin reflection planes (F operations) of the deformed phase (subgroup [{\cal H}]), i.e. they are `crystallographically prominent planes of fixed indices' (Sapriel, 1975[link], p. 5129), which are fixed by the symmetry of the parent phase. A rational lattice plane perpendicular to a lost twofold symmetry axis of the parent phase is also a W boundary. W boundaries are crystallographically invariant with respect to temperature and pressure.

  • (b) [W'] boundaries. In contrast to W boundaries, [W'] interfaces are not fixed by the symmetry of the parent phase, i.e. they do not correspond to lost symmetry elements. Their orientation depends on the direction of the spontaneous shear strain and thus changes with temperature and pressure. In general, [W'] boundaries are irrational planes.

Example. The distinction between these two types of boundaries is illustrated by the example of the (triclinic) Aizu species [2/mF{\bar 1}]. Here, the lost mirror plane of the monoclinic parent phase (F operation) yields the permissible prominent W twin boundary (010). The second permissible boundary, perpendicular to the first, is an irrational [W'] composition plane in the zone of the direction normal to triclinic (010), i.e. of the triclinic reciprocal [{\bf b}^\ast] axis. The azimuthal orientation of this boundary around the zone axis is not determined by symmetry but depends on the direction of the spontaneous shear strain of the deformed triclinic phase.

Sapriel (1975[link]) has shown that for ferroelastic crystals the pair of perpendicular permissible domain boundaries can consist either of two W planes, or of one W and one [W'] plane, or of two [W'] planes. Examples are the Aizu species [2/m\,2/m\,2/mF12/m1], [2/mF{\bar 1}] and [4F2], respectively. There are even four Aizu cases without any permissible boundaries: [3F1], [{\bar 3}F{\bar 1}], [23F222], [m{\bar 3}Fmmm]. An example is langbeinite ([23F222]), which was discussed at the end of Section 3.3.4.4[link].

Note. The two members of a pair of permissible twin boundaries are always exactly perpendicular to each other. Frequently observed slight deviations from the strict [90^\circ] orientation have been interpreted as relaxation of the perpendicularity condition in the deformed phase, resulting from the ferroelastic phase transition (cf. Sapriel, 1975[link], p. 5138). This, however, is not the reason for the deviation from [90^\circ], but rather a splitting of the two (exactly) perpendicular symmetry planes in the parent phase [{\cal G}] into two pairs of compatible twin boundaries (i.e. two independent twin laws) in the deformed phase [{\cal H}], whereby the pairs are nearly perpendicular to each other. From each pair only one interface (usually the rational one) is realized in the twin, whereas the other compatible twin boundary (usually the irrational one) is suppressed because of its unfavourable energetic situation.

Examples

  • (1) Phase transition orthorhombic ([{\cal G} = 2/m\,2/m\,2/m]) [\Rightarrow] monoclinic ([{\cal H} = 1\,2/m\,1], [\beta \approx 90^\circ]), in Aizu notation [2/m2/m2/mF12/m1]. Whereas in [{\cal G}] the mirror planes [m_x] and [m_z] are exactly perpendicular, these planes deviate slightly (by [\beta - 90^\circ]) from perpendicularity in [{\cal H}] and split into two different twin laws (cosets), the first one containing twin plane [m_x], the second one [m_z]. Each twin law contributes its rational twin boundary, [m_x] or [m_z], to the observed twin aggregate, whereas in each pair the perpendicular irrational plane is suppressed.

  • (2) In a tetragonal–orthorhombic phase transition, the two exactly perpendicular mirror planes [(110)] and [({\bar 1}10)] of the tetragonal prototype phase [{\cal G}] split into two independent (rational) twin boundaries in the deformed orthorhombic phase [{\cal H}], which are now nearly perpendicular to each other.

    The most famous example of this type of twinning is the 1023 K ferroelastic phase transition of the high-Tc superconductor YBa2Cu3O7−δ. The twinning on {110} in this compound was first extensively studied by Roth et al. (1987[link]), both in direct space (TEM) and in reciprocal space (electron and X-ray diffraction), and by Schmid et al. (1988[link]); see also Shektman (1993[link]).

3.3.10.2.2. Extension to non-merohedral growth and mechanical twins

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The treatment by Sapriel (1975[link]) was directed to (switchable) ferroelastics with a real structural phase transition from a parent phase [{\cal G}] to a deformed daughter phase [{\cal H}]. This procedure can be extended to those non-merohedral twins that lack a (real or hypothetical) parent phase, in particular to growth twins as well as to mechanical twins in the traditional sense [cf. Section 3.3.7.3[link](i)]. Here, the missing supergroup [\cal G] formally has to be replaced by the `full' or `reduced' composite symmetry [{\cal K}] or [{\cal K}^\ast] of the twin, as defined in Section 3.3.4.[link] Furthermore, we replace the spontaneous shear strain by one half of the imaginary shear deformation which would be required to transform the first orientation state into the second via a hypothetical intermediate (zero-strain) reference state. Note that this is a formal procedure only and does not occur in reality, except in mechanical twinning (cf. Section 3.3.7.3[link]). With respect to this intermediate reference state, the two twin orientations possess equal but opposite `spontaneous' strain. With these definitions, the Sapriel treatment can be applied to non-merohedral twins in general. This extension even permits the generalization of the Aizu notation of ferroelastic species to [{\cal K}F{\cal H}] and [{\cal K}^\ast F{\cal H}^\ast] (e.g. [mmmF 2/m]), whereby now [{\cal H}] and [{\cal H}^\ast] represent the [eigensymmetry] and the intersection symmetry, and [{\cal K}] and [{\cal K}^\ast] the (possibly reduced) composite symmetry of the domain pair. With these modifications, the tables of Sapriel (1975[link]) can be used to derive the permissible boundaries W and [W'] for general non-merohedral twins.

It should be emphasized that this extension of the Sapriel treatment requires a modification of the definition of the W boundary as given above in Section 3.3.10.2.1[link]: The (rational) symmetry operations of the parent phase, becoming F operations in the phase transformation, have to be replaced by the (growth) twin operations contained in the coset of the twin law. These twin operations now correspond to either rational or irrational twin elements. Consequently, the W boundaries defined by these twin elements can be either rational or irrational, whereas by Sapriel they are defined as rational. The Sapriel definition of the [W'] boundaries, on the other hand, is not modified: [W'] boundaries depend on the direction of the spontaneous shear strain and are always irrational. They cannot be derived from the twin operations in the coset and, hence, do not appear as primed twin elements in the black–white symmetry symbol of the composite symmetry [{\cal K}] or [{\cal K}^\ast].

In many cases, the derivation of the permissible twin boundaries W can be simplified by application of the following rules:

  • (i) any twin mirror plane, rational or irrational, is a permissible composition plane W;

  • (ii) the plane perpendicular to any twofold twin axis, rational or irrational, is a permissible composition plane W;

  • (iii) all these twin mirror planes and twofold twin axes can be identified in the coset of any twin law, for example by the primed twin elements in the black–white symmetry symbol of the composite symmetry [{\cal K}] (cf. Section 3.3.5[link]).

In conclusion, the following differences in philosophy between the Sapriel approach in Section 3.3.10.2.1[link] and its extension in the present section are noted: Sapriel starts from the supergroup [{\cal G}] of the parent phase and determines all permissible domain walls at once by means of the symmetry reduction [{\cal G} \rightarrow {\cal H}] during the phase transition. This includes group–subgroup relations of index [[i]> 2]. The present extension to general twins takes the opposite direction: starting from the eigensymmetry [{\cal H}] of a twin component and the twin law [k \times {\cal H}], a symmetry increase to the composite symmetry [{\cal K}] of a twin domain pair is obtained. From this composite symmetry, which is always a supergroup of [{\cal H}] of index [[i]=2], the two permissible boundaries between the two twin domains are derived. Repetition of this process, using further twin laws one by one, determines the permissible boundaries in multiple twins of index [[i]\,\gt \,2].

Examples

  • (1) Gypsum dovetail twin, eigensymmetry [{\cal H} = 2/m], twin element: reflection plane (100) (cf. Example 3.3.6.2[link], Fig. 3.3.6.1[link]).

    Intersection symmetry of two twin domains: [{\cal H}^\ast = 1\,2/m\,1] (= eigensymmetry [{\cal H}]); composite symmetry: [{\cal K} = 2'/m'\,2/m\,2'/m'] (referred to orthorhombic axes); corresponding Aizu notation: [ 2/m\,2/m\,2/m\,F 12/m1]; alternative twin elements in the coset: rational twin reflection plane [m'\parallel(100)] and irrational plane [m'] normal to [001], as well as the twofold axes normal to these planes, all referred to monoclinic axes. The two alternative twin reflection planes are at the same time permissible W boundaries.

    In most dovetail and Montmartre twins of gypsum only the rational (100) or (001) twin boundary is observed. In some cases, however, both permissible W boundaries occur, whereby the irrational interface is usually distinctly smaller and less perfect than the rational one, cf. Fig. 3.3.6.1[link].

  • (2) Multiple twins with orthorhombic eigensymmetry [ {\cal H} = 2/m\,2/m\,2/m] and equivalent twin mirror planes (110) and [(1{\bar 1}0)].

    Intersection symmetry of two or more domain states: [{\cal H}^\ast =] [112/m]; reduced composite symmetry: [{\cal K}^\ast =] [2'/m'\,2'/m'\,2/m].

    Reference is made to Fig. 3.3.4.2[link] in Section 3.3.4.2[link], where the complete cosets for both twin laws (110) and [(1{\bar 1}0)] are shown. For each twin law, the two perpendicular twin mirror planes are at the same time the two permissible W twin boundaries. The (110) boundary is rational, the second permissible boundary, perpendicular to (110), is irrational; similarly for [(1{\bar 1}0)]. The rational boundary is always observed. This rule remains valid for multiple twins, in particular for the spectacular cyclic twins with pseudo n-fold twin axes: [\arctan b/a \approx 60^\circ] (aragonite), [72^\circ] (AlMn alloy), [90^\circ] (staurolite [90^\circ] cross), [\ldots\,, 360^\circ/n]. A pentagonal twin is shown in Fig. 3.3.6.8[link].

  • (3) Twins of triclinic feldspars, eigensymmetry [{\cal H} = {\bar 1}] (cf. Example 3.3.6.11[link], Figs. 3.3.6.11[link] and 3.3.6.12[link]).

    • (a) Albite law: twin reflection plane (010) (referred to triclinic, pseudo-monoclinic axes), Fig. 3.3.6.11[link].

      Intersection symmetry: [{\cal H}^\ast = {\cal H} = {\bar 1}]; composite symmetry: [{\cal K} = 2'/m'({\bar 1})].

      The two permissible twin boundaries are the W twin plane (010) (fixed and rational) and a [W'] plane perpendicular to the first in the zone of the reciprocal axis [{\bf b^*} = [010]^*], but `floating' with respect to its azimuth. The rational W plane (010) is always observed in the form of large-area, polysynthetic twin aggregates.

    • (b) Pericline law: twofold twin rotation axis [010] (referred to triclinic, pseudo-monoclinic axes), Fig. 3.3.6.12[link].

      Intersection symmetry: [{\cal H}^\ast = {\cal H} = {\bar 1}]; composite symmetry: [{\cal K} = 2'/m'({\bar 1})].

      The two permissible contact planes are:

      • (i) the irrational W plane normal to the twin axis [010] [parallel to the reciprocal [(010)^\ast] plane];

      • (ii) the irrational [W'] plane, normal to the first W plane, in the zone of the [010] twin axis, but `floating' with respect to its azimuth.

      The latter composition plane is the famous `rhombic section' which is always observed. The azimuthal angle of the rhombic section around [010] depends on the Na/Ca ratio of the plagioclase crystal and is used for the determination of its chemical composition.

    Remark. Both twin laws (albite and pericline) occur simultaneously in microcline, KAlSi3O8 (`transformation microcline') as the result of a slow Si/Al order–disorder phase transition from monoclinic sanidine to triclinic microcline, forming crosshatched lamellae of albite and pericline twins (Aizu species [2/m\,F{\bar 1}]).

  • (4) Carlsbad twins of monoclinic orthoclase KAlSi3O8 (cf. Fig. 3.3.7.1[link]).

    Eigensymmetry: [{\cal H} = 12/m1]; twin element: twofold axis [001] (referred to monoclinic axes); intersection symmetry: [{\cal H}^\ast = {\cal H} = 12/m1]; composite symmetry: [{\cal K} = 2'/m'\,2/m\,2'/m'] (referred to orthorhombic axes).

    Permissible W twin boundaries (referred to monoclinic axes):

    • (i) [ m \perp [001]] (irrational),

    • (ii) [m \parallel (100)] (rational).

    Carlsbad twins are penetration twins. The twin boundaries are more or less irregular, as is indicated by the re-entrant edges on the surface of the crystals. From some of these edges, it can be concluded that boundary segments parallel to the permissible (100) planes as well as parallel to the non-permissible (010) planes (which are symmetry planes of the crystal) occur. This is possibly due to complications arising from the penetration morphology.

  • (5) Calcite deformation twins (e-twins) [cf. Section 3.3.7.3[link](i)[link] and Fig. 3.3.7.3[link]].

    The deformation twinning in calcite has been extensively studied by Barber & Wenk (1979[link]). Recently, these twins were discussed by Bueble & Schmahl (1999[link]) from the viewpoint of Sapriel's strain compatibility theory of domain walls.

    For calcite (space group [R{\bar 3}c]) three unit cells are in use:

    • (i) Structural triple hexagonal R-centred cell (`X-ray cell'): ahex = 4.99, chex = 17.06 Å. This cell is used by both Barber & Wenk and Bueble & Schmahl.

    • (ii) Morphological cell: amorph = ahex = 4.99, cmorph = 1/4chex = 4.26 Å. This cell is used in many mineralogical textbooks for the description of the calcite morphology and twinning.

    • (iii) Rhombohedral (pseudo-cubic) cell, F-centred, corresponding to the cleavage rhombohedron and the cell of the cubic NaCl structure: apc = 3.21 Å, αpc = 101.90°.

    Eigensymmetry: [{\cal H} = {\bar 3}2/m1]; twin reflection and interface plane: [(01{\bar1}8)_{\rm hex} = (01{\bar 1}2)_{\rm morph} = (110)_{\rm pc}] (similar for the two other equivalent planes); intersection symmetry: [{\cal H}^\ast = 2/m] along [[010]_{\rm hex} = [10{\bar 1}]_{\rm pc}]; reduced composite symmetry: [{\cal K}^\ast = 2/m\,2'/m_1'\,2'/m_2'], with [m_1] [ =] [(01{\bar 1}8)_{\rm hex} =] [(01{\bar 1}2)_{\rm morph} =] [(110)_{\rm pc}] rational and [m_2] an irrational plane normal to the edge of the cleavage rhombohedron (cf. Fig. 3.3.7.3[link]). Planes [m_1] and [m_2] are compatible W twin boundaries, of which the rational plane [m_1] is the only one observed.

    Bueble & Schmahl (1999[link]) treated the mechanical twinning of calcite by using the Sapriel formalism for ferroelastic crystals. The authors devised a virtual prototypic phase of cubic [m{\bar 3}m] symmetry with the NaCl unit cell (iii)[link] mentioned above. From a virtual ferroelastic phase transition [m{\bar 3}m \Rightarrow {\bar 3}2/m], they derived four orientation states (corresponding to compression axes along the four cube diagonals). The W boundaries are of type [\{110\}_{\rm pc} =] [\{01{\bar 1}8\}_{\rm hex} =] [\{01{\bar 1}2\}_{\rm morph}] and [\{001\}_{\rm pc} =] [\{0{\bar 1}14\}_{\rm hex} =] [\{0{\bar 1}11\}_{\rm morph}] (cleavage faces). These boundaries are observed. The e-twins (primary twins) with [\{110\}_{\rm pc}] W walls, however, dominate in calcite (primary deformation twin lamellae), whereas the secondary r-twins with [\{001\}_{\rm pc}] W boundaries are relatively rare.

    A comparison with the compatible twin boundaries [m_1] and [m_2] derived from the reduced composite symmetry [{\cal K}^\ast] shows that the [m_1 = \{110\}_{\rm pc} = \{01{\bar 1}8\}_{\rm hex}] boundary is predicted by both approaches, whereas [m_2] and [\{001\}_{\rm pc} = \{0{\bar 1}14\}_{\rm hex}] differ by an angle of [26.2^\circ]. The twin reflection planes [\{110\}_{\rm pc}] (e-twin) and [\{001\}_{\rm pc}] (r-twin) represent different twin laws and are not alternative twin elements of the same twin law, as are [m_1 = \{110\}_{\rm pc}] and [m_2]. They would be alternative elements if the rhombohedron (pseudo-cube), keeping its structural [{\bar 3}2/m] symmetry, were re-distorted into an exact cube.

    This situation explains the rather complicated deformation twin texture of calcite. Whereas two e-twin components can be stress-free attached to each other along a boundary consisting of compatible [m_1 = \{110\}_{\rm pc}] and [m_2] segments, a boundary of [\{110\}_{\rm pc}] and (incompatible) [\{001\}_{\rm pc}] segments would generate stress, which is extraordinarily high due to the extreme shear angle of [26.2^\circ]. The irrational [m_2] boundary, though mechanically compatible, is not observed in calcite and is obviously suppressed due to bad structural fit. As a consequence, the stress in the boundary regions between the mutually incompatible [{\bf e}\{110\}_{\rm pc}] and [{\bf r}\{001\}_{\rm pc}] twin systems is often buffered by the formation of needle twin lamellae (Salje & Ishibashi, 1996[link]) or structural channels along crystallographic directions (`Rose channels'; Rose, 1868[link]). Twinning dislocations and cracks (Barber & Wenk, 1979[link]) also relax high stress. In `real' ferroelastic crystals with their small shear (usually below 1°) these stress-relaxing phenomena usually do not occur.

3.3.10.2.3. Permissible boundaries in merohedral twins (lattice index [[j] = 1])

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In merohedral twins (lattice index [[j] = 1]), the twin elements map the entire lattice exactly upon itself. Hence there is no spontaneous strain, in which the twin domains would differ. The mechanical compatibility criterion means in this case that any orientation of a twin boundary is permissible, because interfaces of any orientation obey the mechanical compatibility criterion, no matter whether the planes are rational, irrational or even curved interfaces. This variety of interfaces is brought out by many actual cases, as shown by the following examples:

  • (1) Quartz.

    The boundaries of Dauphiné transformation twins ([622 \Rightarrow 32]) are usually irregular, curved interfaces without macroscopically flat parts (segments) (Frondel, 1962[link]). The boundaries of Dauphiné growth twins are usually curved too, but sometimes they exhibit large segments roughly parallel to the rhombo­hedron faces [r\{10{\bar 1}1\}] and [z\{{\bar 1}011\}]. Inserts of growth-twin domains often have the shape of rounded cones with apices located at the nucleating perturbation (usually an inclusion). X-ray topography has shown that Dauphiné boundaries are sometimes stepped on a fine scale just above the topographic resolution of a few µm (Lang, 1967[link][link]a,b).

    In contrast to Dauphiné boundaries, the contact faces of Brazil twins (which are always growth twins) strictly adopt low-index lattice planes, preferentially those of the major rhombohedron [r\{10{\bar 1}1\}], less frequently of the minor rhombohedron [z\{{\bar 1}011\}]. More rarely observed are boundary segments parallel to [\{10{\bar 1}0\}], [\{11{\bar 2}1\}] and [\{0001\}] (Frondel, 1962[link]). A special case are the differently dyed [r\{10{\bar 1}1\}] Brazil-twin lamellae of amethyst (Frondel, 1962[link]).

  • (2) Triglycine sulfate (TGS).

    This crystal exhibits a ferroelectric transition [2/m \Leftrightarrow 2] with antipolar (merohedral) inversion twins ([180^\circ] domains). The domains usually have the shape of irregular cylinders parallel to the polar axis. All boundaries parallel to the polar axis are observed, whereas boundaries inclined or normal to the polar axis are `electrically forbidden' (they would be head-to-head and tail-to-tail boundaries, cf. Section 3.3.10.3[link]), even though they are mechanically permissible.

  • (3) Lithium formate monohydrate (polar point group [mm2]).

    The crystals grown from aqueous solutions exhibit inversion twins in the shape of sharply defined (010) lamellae (non-switchable [180^\circ] domains). Other boundaries of these growth twins have not been observed (Klapper, 1973[link]).

  • (4) KLiSO4 (polar point group 6).

    Among the three different merohedral twin laws, one type, with twin reflection plane m parallel to the hexagonal axis [001], stands out. The numerous grown-in twin lamellae are bounded by large planar (0001) interfaces (normal to the polar axis). It should be noted that these very prominent twin boundaries are perpendicular to the twin reflection plane. This is a rather rare case of boundary orientation. These planes are oriented normal to the polar axis but are not electrically forbidden, due to the twin law [m \parallel [001]] which preserves the polar direction (Klapper et al., 1987[link]).

These examples demonstrate that in many merohedral twins ony a small number of rational, well defined boundaries occur, even though any boundary is permitted by the mechanical compatibility criterion. This shows that the latter criterion is a necessary, but not a sufficient, condition and that further influences, in particular electrical or structural ones, are effective.

3.3.10.2.4. Permissible twin boundaries in twins with lattice index [[j]\, \gt\, 1]

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In contrast to the mechanical compatibility of any composition plane in merohedral twins (lattice index [[j] = 1]), twins of higher lattice index [[j]> 1] are more restricted with respect to the orientation of permissible twin boundaries. In fact, these special twins can be treated in the same way as the general non-merohedral twins described in Section 3.3.10.2.2[link] above. Again, we attribute equal but opposite spontaneous shear strain to the two twin domains 1 and 2. This `spontaneous' shear strain (referred to an intermediate state of zero strain) is half the shear deformation necessary to transform the orientation of domain 1 into that of domain 2. This also means that the lattice of domain 1 is transformed into the lattice of domain 2. The essential difference to the case in Section 3.3.10.2.2[link] is the fact that by this deformation only a subset of lattice points is `restored'. This subset forms the sublattice of index [[j] \ge 2] common to both domains (coincidence-site sublattice, twin lattice). With this analogy, the Sapriel formalism can be applied to the derivation of the mechanically compatible (permissible) twin boundaries. Again, the easiest way to find the permissible planes is the construction of the black–white symmetry symbol of the twin law, in which planes parallel to primed mirror planes or normal to primed twofold axes constitute the permissible W interfaces.

It is emphasized that the concept of a deformation from domain state 1 to domain state 2 is not always a mere mental construction, as it is for growth twins. It is physical reality in some deformation twins, for example in the famous [\Sigma 3] deformation twins (spinel law) of cubic metals which are essential elements of the plasticity of these metals. During the [\Sigma 3] deformation, the {100} cube (a [90^\circ] rhombohedron) is switched from its `reverse' into its `obverse' orientation and vice versa, whereby the hexagonal P sublattice of index [[j] = 3] is restored and, thus, is common to both twin domains.

Exact lattice coincidences of twin domains result from special symmetry relations of the lattice. Such relations are systematically provided by n-fold symmetry axes of order [n> 2], i.e. by three-, four- and sixfold axes. In other words: twins of lattice index [[j]> 2] occur systematically in trigonal, hexagonal, tetragonal and cubic crystals. This may lead to trigonal, tetragonal and hexagonal intersection symmetries [{\cal H}^\ast] (reduced eigensym­metries) of domain pairs. Consequently, if there exists one pair of permissible composition planes, all pairs of planes equivalent to the first one with respect to the intersection symmetry are permissible twin boundaries as well. This is illustrated by three examples in Table 3.3.10.1[link].

Table 3.3.10.1| top | pdf |
Examples of permissible twin boundaries for higher-order merohedral twins ([j]> 1)

 [\Sigma 3] growth and deformation twins of cubic crystals, twin mirror plane (111) (spinel law)[\Sigma 5] growth twins of tetragonal rare-earth sulfides (SmS1.9), twin mirror plane (120)[\Sigma 33] deformation twins of cubic galena (PbS), twin mirror plane (441)
Eigensymmetry [{\cal H}] [4/m{\bar 3}2/m] [4/m\,2/m\,2/m] [4/m{\bar 3}2/m]
Intersection symmetry [{\cal H}^\ast] [{\bar 3}2/m] parallel to [111] [4/m] parallel to [001] [2/m] parallel to [[1{\bar 1}0]]
Reduced composite symmetry [{\cal K}^\ast] [6'/m'_1({\bar 3})2/m\,2'/m'_3] [4/m\,2'/m'_1\,2'/m'_2] [2'/m'_1\,2/m\,2'/m'_2]
Permissible twin boundaries Three pairs of perpendicular planes Two pairs of perpendicular planes One pair of permissible planes
[m_1 = (111)] & [m_3 = (11{\bar 2})] [m_1 = (120)] & [({\bar 2}10)] [m_1 \,= \,(441)] & [m_2 \,= \,(11{\bar 8})]
[m_1 = (111)] & [m_3 = ({\bar 2}11)] [m_2 = (310)] & [({\bar 1}30)]  
[m_1 = (111)] & [m_3 = (1{\bar 2}1)]    
Reference system Cubic axes Tetragonal axes Cubic axes
The existence of this deformation twin is still in doubt (cf. Seifert, 1928[link]).
The intersection symmetry [{\cal H}^\ast] and the permissible boundaries are referred to the coordinate system of the eigensymmetry; the reduced composite symmetries [{\cal K}^\ast] are based on their own conventional coordinate system derived from the intersection symmetry [{\cal H}^\ast] plus the twin law (cf. Section 3.3.4[link]).

For the cubic and rhombohedral [\Sigma 3] twins (spinel law), due to the threefold axis of the intersection symmetry, three pairs of permissible planes occur. The plane (111), normal to this threefold axis, is common to the three pairs of boundaries (threefold degeneracy), i.e. in total four different permissible W twin boundaries occur. These composition planes (111), [(11{\bar 2})], [({\bar 2}11)], [(1{\bar 2}1)] are indeed observed in the [\Sigma 3] spinel-type penetration twins, recognizable by their re-entrant edges (Fig. 3.3.6.6[link]). They also occur as twin glide planes of cubic metals. For the tetragonal [\Sigma 5] twin, two pairs of perpendicular permissible W composition planes result, (120) & ([{\bar 2}10)] and (310) & ([{\bar 1}30]), one pair bisecting the other pair under 45°. For the cubic [\Sigma 33] twin [galena PbS, cf. Section 3.3.8.3[link], example (4)], due to the low intersection symmetry, only one pair of permissible W boundaries results.

3.3.10.3. Electrical constraints of twin interfaces

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As mentioned before, the mechanical compatibility of twin boundaries is a necessary but not a sufficient criterion for the occurrence of stress-free low-energy twin interfaces. An additional restriction occurs in materials with a permanent (spontaneous) electrical polarization, i.e. in crystals belonging to one of the ten pyroelectric crystal classes which include all ferroelectric materials. In these crystals, domains with different directions of the spontaneous polarization may occur and lead to `electrically charged boundaries'.

3.3.10.3.1. Merohedral twins

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Of particular significance are merohedral twins with polar domains of antiparallel spontaneous polarization [\pm {\bf P}] (180° domains). The charge density [\rho] at a boundary between two twin domains is given by [\rho = \pm 2 P_n,]where [P_n] is the component of the polarization normal to the boundary. The interfaces with positive charge are called `head-to-head' boundaries, those with negative charge `tail-to-tail' boundaries. Interfaces parallel to the polarization direction are uncharged ([P_n = 0]) (Fig. 3.3.10.2[link]).

[Figure 3.3.10.2]

Figure 3.3.10.2 | top | pdf |

Boundaries B–B between 180° domains (merohedral twins) of pyroelectric crystals. (a) Tail-to-tail boundary. (b) Head-to-head boundary. (c) Uncharged boundary ([P_n = 0]). (d) Charged zigzag boundary, with average orientation normal to the polar axis. The charge density is significantly reduced. Note that the charges at the boundaries are usually compensated by stray charges of opposite sign.

The electrical charges on a twin boundary constitute an additional (now electrostatic) energy of the twin boundary and are `electrically forbidden'. Only boundaries parallel to the polar axes are `permitted'. This is in fact mostly observed: practically all 180° domains originating during a phase transition from a paraelectric parent phase to the polar (usually ferroelectric) daughter phase exhibit uncharged boundaries parallel to the spontaneous polarization. Uncharged boundaries have also been found in inversion growth twins obtained from aqueous solutions, such as lithium formate monohydrate and ammonium lithium sulfate. Both crystals possess the polar eigensymmetry [mm2] and contain grown-in inversion twin lamellae (180° domains) parallel to their polar axis.

`Charged' boundaries, however, may occur in crystals that are electrical conductors. In such cases, the polarization charges accumulating along head-to-head or tail-to-tail boundaries are compensated by opposite charges obtained through the electrical conductivity. This compensation may lead to a considerable reduction of the interface energy. Note that the term `charged' is often used for boundaries of head-to-head and tail-to-tail character, even if they are uncharged due to charge compensation.

Examples

  • (1) Lithium niobate, LiNbO3, exhibits a phase transition from [{\bar 3}2/m] to [3m] between 1323 and 1473 K (depending on the Li/Nb stoichiometry). Crystals are grown from the melt ([T_m = 1538] K) by Czochralski pulling along the trigonal axis [001] in the paraelectric phase. They transform into the ferroelectric polar phase when cooled below the Curie temperature. The crystals are electrically conductive at high temperatures and can be poled by an electric field parallel to the polar axis. By applying an alternating rectangular voltage between seed crystal and melt, a sequence of 180° domains is formed during the subsequent transition. The domain boundaries follow the curved growth front (crystal–melt interface) and have alternating head-to-head and tail-to-tail character (Räuber, 1978[link]).

  • (2) Orthorhombic polar potassium titanyl phosphate, KTiOPO4 (KTP), exhibits a para- to ferroelectric phase transition ([mmm \Longleftrightarrow mm2]) and a considerable conductivity of potassium ions. In this material, head-to-head and tail-to-tail boundaries are common. Sometimes strongly folded, charged zigzag boundaries occur, which contain large segments of faces nearly parallel to the spontaneous polarization (Scherf et al., 1999[link]). The average orientation of these boundaries is roughly normal to the polar axis (Fig. 3.3.10.2[link]d), but their charge density is considerably reduced by the zigzag geometry.

  • (3) Head-to-head and tail-to-tail twin boundaries are also found in crystals grown from aqueous solutions. In such cases, the polarization charges are compensated by the opposite charges present in the electrolytic solution. An interesting example is hexagonal potassium lithium sulfate KLiSO4 (point group 6) which exhibits, among other types of twins, anti-polar domains of inversion twins. The twin boundaries often have head-to-head or tail-to-tail character and frequently coincide with the growth-sector boundaries (Klapper et al., 1987[link]).

3.3.10.3.2. Non-merohedral twins

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Charged and uncharged boundaries may also occur in non-merohedral twins of pyroelectric crystals. In this case, the polar axes of the two twin domains 1 and 2 are not parallel. The charge density [\rho] of the boundary is given by [\rho = P_n(2) - P_n(1),]with [P_n(1)] and [P_n(2)] the components of the spontaneous polarization normal to the boundary. An example of both charged and uncharged boundaries is provided by the growth twins of ammonium lithium sulfate with eigensymmetry [m2m]. These crystals exhibit, besides the inversion twinning mentioned above, growth-sector twins with twin laws `reflection plane (110)' and `twofold twin axis normal to (110)'. (Both twin elements would constitute the same twin law if the crystal were centrosymmetric.) The observed and permissible composition plane for both laws is (110) itself. As is shown in Fig. 3.3.10.3[link], the (110) boundary is charged for the reflection twin and uncharged for the rotation twin. Both cases are realized for ammonium lithium sulfate. The charges of the reflection-twin boundary are compensated by the charges contained in the electrolytic aqueous solution from which the crystal is grown. On heating (cooling), however, positive (negative) charges appear along the twin boundary.

[Figure 3.3.10.3]

Figure 3.3.10.3 | top | pdf |

Charged and uncharged boundaries B–B of non-merohedral twins of pseudo-hexagonal NH4LiSO4. Point group [m2m], spontaneous polarization P along twofold axis [010]. (a) Twin element mirror plane (110): electrically charged boundary of head-to-head character. (b) Twin element twofold twin axis normal to plane (110): uncharged twin boundary (`head-to-tail' boundary).

3.3.10.3.3. Non-pyroelectric acentric crystals

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Finally, it is pointed out that electrical constraints of twin boundaries do not occur for non-pyroelectric acentric crystals. This is due to the absence of spontaneous polarization and, consequently, of electrical boundary charges. This fact is apparent for the Dauphiné and Brazil twins of quartz: they exhibit boundaries normal to the polar twofold axes which are reversed by the twin operations.

Nevertheless, it seems that among possible twin laws those leading to opposite directions of the polar axes are avoided. This can be explained for spinel twins of cubic crystals with the sphalerite structure and eigensymmetry [{\bar 4}3m]. Two twin laws, different due to the lack of the symmetry centre, are possible:

  • (i) twofold twin rotation around [111],

  • (ii) twin reflection across the plane (111).

In the first case, the sense of the polar axis [111] is not reversed, in the second case it is reversed. All publications on this kind of twinning, common in III–V and II–VI compound semiconductors (GaAs, InP, ZnS, CdTe etc.), report the twofold axis along [111] as the true twin element, not the mirror plane (111); this was discussed very early on in a significant paper by Aminoff & Broomé (1931[link]).

3.3.10.4. Displacement and fault vectors of twin boundaries

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The statements of the preceding sections on the permissibility of twin boundaries are very general and derived without any regard to the crystal structure. For example, any arbitrary reflection plane relating two partners of a crystal aggregate or even of an anisotropic continuous elastic medium represents a mechanically permissible boundary. For twin boundaries in crystals, however, additional aspects have to be taken into account, viz the atomic structure of the twin interface, i.e. the geometrical configuration of atoms, ions and molecules and their crystal-chemical interactions (bonding topology) in the transition region between the two twin partners. Only if the configurations and interactions of the atoms lead to boundaries of good structural fit and, consequently, of low energy, will the interfaces occur with the reproducibility and frequency that are prerequisites for a twin. In this respect, the mechanical and electrical permissibility conditions given in the preceding sections are necessary but not sufficient conditions for the occurrence of a twin boundary and – in the end – of the twin itself. In the following considerations, all twin boundaries are assumed to be permissible in the sense discussed above.

3.3.10.4.1. Twin displacement vector [{\bf t}]

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As a first step of the structural elucidation of a reflection-twin boundary, the mutual relation of the two lattices of the twin partners 1 and 2 at the boundary is considered. It is assumed that the unit cells of both lattices have the same origin with respect to their crystal structure, i.e. that the lattice points are located in the same structural sites of both partners. Three cases of lattice relations across the rational composition plane (hkl) (assumed to be parallel to the twin reflection plane) are considered, as outlined in Fig. 3.3.10.4[link] [see also Section 3.3.2.4[link], Note (8)].

  • (a) The composition lattice planes [(hkl)_1] and [(hkl)_2] of domains 1 and 2 coincide pointwise (Fig. 3.3.10.4[link]a), i.e. the lattices of the two twin partners coincide in the twin boundary.

    [Figure 3.3.10.4]

    Figure 3.3.10.4 | top | pdf |

    Lattice representation of twin displacement vectors. (a) Lattice representation of a `pure' twin reflection plane (t = 0). (b) Twin reflection plane with parallel displacement vector t (generalized twin glide plane). (c) Twin reflection plane with a general twin displacement vector with parallel and normal components: [{\bf t} = {\bf t}_n + {\bf t}_p]. By choosing a suitable new lattice point x (origin shift), the normal component [{\bf t}_n] disappears, preserving the parallel component [{\bf t}_p] as the true twin displacement vector and leading to case (b), as shown in (d).

  • (b) The composition lattice planes [(hkl)_1] and [(hkl)_2] are common but not pointwise coincident, i.e. the two lattices are displaced by a non-integer vector t (twin displacement vector) parallel to the composition plane (Fig. 3.3.10.4[link]b).

  • (c) The composition lattice planes [(hkl)_1] and [(hkl)_2] are displaced – in addition to the parallel component – by a component normal to the composition plane (Fig. 3.3.10.4[link]c). By an appropriate choice of the lattice points with respect to the structure, this normal component vanishes and, hence, this general case reduces to case (b)[link] (Fig. 3.3.10.4[link]d).

This shows that for the characterization of a twin with coinciding twin reflection and contact plane only the component of a twin displacement vector parallel to the twin boundary is significant. Thus, on an atomic scale, not only twin reflection planes ([{\bf t} \approx {\bf 0}]) but also `twin glide planes' ([{\bf t} \approx 1/2 {\bf v}_L], where [{\bf v}_L] is a lattice translation vector), as well as all intermediate cases, have to be considered. In principle, these considerations also apply to irrational twin reflection and composition planes. Moreover, twin displacement vectors also have to be admitted for the other types of twins, viz rotation and inversion twins. Examples are given below.

So far, the considerations about twin boundaries are based on the idealized concept that the bulk structure extends without any deformation up to the twin boundary. In reality, however, near the interface the structures are more or less deformed (relaxed), and so are their lattices. This transition region may even contain a central slab exhibiting a different structure, which is often close to a real or hypothetical polymorph or to the parent structure. [Examples: the Dauphiné twin boundary of α-quartz resembles the structure of β-quartz; the iron-cross (110) twin interface of pyrite, FeS2, resembles the structure of marcasite, another polymorph of FeS2.]

Whereas the twin displacement vector keeps its significance for small distortions of the boundary region, it loses its usefulness for large structural deformations. It should be noted that rational twin interfaces are usually observed as `good', whereas irrational twin boundaries, despite mechanical compatibility, usually exhibit irregular features and macroscopically visible deformations.

Twin displacement vectors are a consequence of the minimization of the boundary energy. This has been proven by a theoretical study of the boundary energy of reflection twins of monoclinic saccharine crystals with [(10{\bar 2})] as twin reflection and composition plane (Lieberman et al., 1998[link]). The authors calculated the boundary energy as a function of the lattice displacement vector t, which was varied within the mesh of the [(10{\bar 2})] composition plane, admitting also a component normal to the composition plane. The calculations were based on a combination of Lennard–Jones and Coulomb potentials and result in a flat energy minimum for a displacement vector t = [0.05/0.71/0.5] (referred to the monoclinic axes). The calculations were carried out for the undistorted bulk structure. The actual deformation of the structure near the twin boundary is not known and, hence, cannot be taken into account. Nevertheless, this model calculation shows that in general twin displacement vectors [{\bf t} \ne {\bf 0}] are required for the minimization of the boundary energy.

Twin displacement vectors have been considered as long as structural models of boundaries have been derived. One of the oldest examples is the model of the (110) growth-twin boundary of aragonite, suggested by Bragg (1924[link]) (cf. Section 3.3.10.5[link] below). An even more instructive model is presented by Bragg (1937[link], pp. 246–248) and Bragg & Claringbull (1965[link], pp. 302–303) for the Baveno (021) twin reflection and interface plane of feldspars. It shows that the tetrahedral framework can be continued without interruption across the twin boundary only if the twin reflection plane is a glide plane parallel to (021). A model of a twin boundary requiring a displacement vector [{\bf t} \ne {\bf 0}] was reported by Black (1955[link]) for the (110) twin reflection boundary of the alloy Fe4Al13.

In their interesting theoretical study of the morphology and twinning of gypsum, Bartels & Follner (1989[link], especially Fig. 4) conclude that the (100) twin interface of Montmartre twins is a pure twin reflection plane without displacement vector, whereas the dovetail twins exhibit a `twin glide component' [{\bf a}/2 + {\bf b}/2 + {\bf c}/2] parallel to the twin reflection plane [({\bar 1}01)]. [Note that in the present chapter, due to a different choice of coordinate system, the Montmartre twins are given as (001) and the dovetail twins as (100), cf. Example 3.3.6.2[link].]

The occurrence of twin displacement vectors can be visualized by high-resolution transmission electron microscopy (HRTEM) studies of twin boundaries. Fig. 3.3.10.5[link] shows an HRTEM micrograph of a (112) twin reflection boundary of anatase TiO2, viewed edge on (arrows) along [[1{\bar 3}1]] (Penn & Banfield, 1998[link]). The offset of the lattices along the twin boundary is clearly visible. This result is confirmed by the structural model presented by the authors, which indicates a parallel displacement vector [{\bf t} \approx 1/2 {\bf v}_L]. Twin displacement vectors have also been observed on HRTEM micrographs of sputtered Fe4Al13 alloys by Tsuchimori et al. (1992[link]).

[Figure 3.3.10.5]

Figure 3.3.10.5 | top | pdf |

HRTEM micrograph of anatase, TiO2, with a (112) reflection twin boundary (arrows), viewed edge-on along [[1{\bar 3}1]]. The twin displacement vector t = 1/2 of the boundary translation period is clearly visible. Courtesy of R. L. Penn, Madison, Wisconsin; cf. Penn & Banfield, 1998[link].

3.3.10.4.2. Fault vectors of twin boundaries in merohedral twins

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Twin displacement vectors can occur in twin boundaries of both non-merohedral (see above) and merohedral twins. For merohedral twins, the displacement vector is usually called the `fault vector', because of the close similarity of these twin boundaries with antiphase boundaries and stacking faults (cf. Section 3.3.2.4[link], Note 7[link]). In contrast to non-merohedral twins, for merohedral twins these displacement vectors can be determined by imaging the twin boundaries by means of electron or X-ray diffraction methods. The essential reason for this possibility is the exact parallelism of the lattices of the two twin partners 1 and 2, so that for any reflection hkl the electron and X-ray diffraction conditions are always simultaneously fulfilled for both partners. Thus, in transmission electron microscopy and X-ray topography, both domains 1 and 2 are simultaneously imaged under the same excitation conditions. By a proper choice of imaging reflections, both twin domains exhibit the same diffracted intensity (no `domain contrast'), and the twin boundary is imaged by fringe contrast analogously to the imaging of stacking faults and antiphase boundaries (`stacking fault contrast').

This contrast results from the `phase jump' of the structure factor upon crossing the boundary. For stacking faults and antiphase boundaries this phase jump is [2\pi {\bf f} \cdot {\bf g}_{hkl}], with f the fault vector of the boundary and [{\bf g}_{hkl}] the diffraction vector of the reflection used for imaging. For (merohedral) twin boundaries the total phase jump [\Psi_{hkl}] is composed of two parts, [\Psi_{hkl} = \phi_{hkl} + 2\pi {\bf f} \cdot {\bf g}_{hkl},]with [\phi_{hkl}] the phase change due to the twin operation and [2\pi {\bf f }\cdot {\bf g}_{hkl}] the phase change resulting from the lattice displacement vector f. The boundary contrast is strongest if the phase jump [\Psi_{hkl}] is an odd integer multiple of [\pi], and it is zero if [\Psi_{hkl}] is an integer multiple of [2\pi]. By imaging the boundary in various reflections hkl and analysing the boundary contrast, taking into account the known phase change [\phi_{hkl}] (calculated from the structure-factor phases of the reflections [hkl_1] and [hkl_2] related by the twin operation), the fault vector f can be determined (see the examples below). This procedure has been introduced into transmission electron microscopy by McLaren & Phakey (1966[link], 1969[link]) and into X-ray topography by Lang (1967[link][link]a,b) and McLaren & Phakey (1969[link]).

In the equation given above, for each reflection hkl the total phase jump [\Psi_{hkl}] is independent of the origin of the unit cell. The individual quantities [\phi_{hkl}] and [2\pi {\bf f} \cdot {\bf g}_{hkl}], however, vary with the choice of the origin but are coupled in such a way that [\Psi_{hkl}] (which alone has a physical meaning) remains constant. This is illustrated by the following simple example of an inversion twin.

The twin operation relates reflections [hkl_1] and [hkl_2 =] [{\bar h}{\bar k}{\bar l}_1]. Their structure factors are (assuming Friedel's rule to be valid)[F_1 = \vert F\vert \exp(-{i}\varphi)\ \ {\rm and}\ \ F_2 = \vert F \vert \exp(+ {i}\varphi).]

The phase difference of the two structure factors is [\phi_{hkl} = 2 \varphi] and depends on the choice of the origin. If the origin is chosen at the twin inversion centre (superscript o), the phase jump [\Psi_{hkl}] at the boundary is given by[ \Psi_{hkl} = \phi^o_{hkl} = 2\varphi^o.]

This is the total phase jump occurring for reflection pairs [hkl_1/hkl_2 = hkl/{\bar h}{\bar k}{\bar l}] at the twin boundary.

If the origin is not located at the twin inversion centre but is displaced from it by a vector [{\textstyle{1\over 2}} {\bf f}], the phases of the structure factors of reflections [hkl_1] and [hkl_2] are [\eqalign{\varphi_1 &= \varphi^o - 2\pi (\textstyle{1\over 2}{\bf f}) \cdot{\bf g}_{hkl} \ \ {\rm and}\cr \varphi_2 &= -\varphi_1 = - [\varphi^o - 2\pi (\textstyle{1\over 2}{\bf f}) \cdot {\bf g}_{hkl}].}]From these equations the phase difference of the structure factors is calculated as [\phi_{hkl} = \varphi_1 - \varphi_2 = 2\varphi^o - 2\pi {\bf f} \cdot {\bf g}_{hkl},]and the total phase jump at the boundary is [\Psi_{hkl} = 2\varphi^o = \phi_{hkl} + 2\pi {\bf f} \cdot {\bf g}_{hkl}.]

This shows that here the fault vector f has no physical meaning. It merely compensates for the phase contributions that result from an `improper' choice of the origin. If, by the procedures outlined above, a fault vector f is determined, the true twin inversion centre is located at the endpoint of the vector [{\textstyle{1\over 2}}{\bf f}] attached to the chosen origin.

Similar considerations apply to reflection and twofold rotation twins. In these cases, the components of the fault vectors normal to the twin plane or to the twin axis can also be eliminated by a proper choice of the origin. The parallel components, however, cannot be modified by changes of the origin and have a real physical significance for the structure of the boundary.

Particularly characteristic fault vectors occur in (merohedral) `antiphase domains' (APD). Often the fault vector is the lattice-translation vector lost in a phase transition.

3.3.10.4.3. Examples of fault-vector determinations

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  • (1) Inversion boundaries (180° domain walls) of ferroelectric lithium ammonium sulfate (LAS), LiNH4SO4 (Klapper, 1987[link]), cf. Example 3.3.6.1[link].

    LAS is ferroelectric at room temperature (point group [m2m]) and transforms into the paraelectric state (point group mmm) at 459 K. Crystals grown from aqueous solution at about 313 K contain grown-in inversion twins with boundaries exactly parallel to (001), appearing on X-ray topographs by stacking-fault fringe contrast. It was found that the boundaries are invisible in reflections of type [h0l] (zone of the polar axis [010]), but show contrast in reflections with [k \ne 0] with some exceptions (e.g. no contrast for reflection 040). The `zero-contrast' reflections are particularly helpful for the determination of the fault vector. Applying the procedure described above, a fault vector [{\bf f} = 1/2[010]] (parallel to the polar axis) was derived for the chosen origin, which is located on the polar twofold symmetry axis. Thus, the true twin inversion centre is located at the endpoint of the vector [{\textstyle{1\over 2}} {\bf f} = 1/4[010]]. An inspection of the LAS structure shows that this point is the location of the inversion centre of the paraelectric parent phase above 459 K. Thus, during the transition from the para- to the ferroelectric phase the structural inversion centres vanish in the bulk of the domains, but are preserved in the domain boundaries as twin inversion centres.

    For this (001) twin interface, a reasonable structural model without any breaking of the framework of SO4 and LiO4 tetrahedra could be derived easily. The tetrahedra adopt a staggered orientation across the boundary, compared with a nearly eclipsed orientation in the bulk structure.

  • (2) Brazil twin boundaries of quartz.

    Brazil twins are commonly classified as [\{11{\bar 2}0\}] reflection twins but can alternatively be considered as inversion twins, as explained in Section 3.3.6.3.2[link]. The twin boundaries are usually strictly planar and mainly parallel to one of the major rhombohedron faces [\{10{\bar 1}1\}] but, less frequently, to one of the minor rhombohedron faces [\{{\bar 1}011\}] or prism faces [\{10{\bar 1}0\}]. From electron microscopy studies of polysynthetic (lamellar) Brazil twins in amethyst (McLaren & Phakey, 1966[link]), fault vectors of type [{\bf f} = 1/2 \langle 010\rangle], i.e. one half of one of the three translations along the twofold axes, were obtained for twin boundaries parallel to [\{10{\bar 1}1\}], where f is parallel to the boundary. The same but slightly shorter fault vector [{\bf f} = 0.4 \langle010\rangle] for [\{10{\bar 1}1\}] Brazil boundaries was determined in X-ray topographic studies by Lang (1967[link][link]a,b) and Lang & Miuskov (1969[link]). Another detailed X-ray topographic investigation was carried out by Phakey (1969[link]). He confirmed the existence of the fault vector [{\bf f} = 1/2\langle010\rangle] but proved also the occurrence of further fault vectors of type [{\bf f} = \langle0, \textstyle{1\over 2}, {1\over 3}\rangle] for [\{10{\bar 1}1\}] twin boundaries. Based on these fault vectors, the structures of the Brazil twin boundaries could be derived: it was shown that no Si—O bonds are broken and that the left- and right-handed partner structures join each other with only small distortions of the tetrahedral framework. It is worth mentioning that the structural channels along the threefold axes do not continue smoothly across the boundary but are mutually displaced by the fault-vector component parallel to the basal plane (0001). McLaren, Phakey and Lang, however, did not consider the location of the twin elements (twin inversion centre or twin reflection plane) in the structure, which can be determined from the fault vectors.

    These structural studies of the Brazil twin boundaries have shown that the fault vectors f are different for different orientations of the interface. As a consequence, a `stair-rod' dislocation must occur along the bend of the twin interface from one orientation to the other. Stair-rod dislocations have been observed and characterized in the X-ray topographic study of Brazil boundaries by Phakey (1969[link]).

3.3.10.5. Examples of structural models of twin boundaries

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Until the rather recent advent of high-resolution transmission electron microscopy (HRTEM), no experimental method for the direct elucidation of the atomic structures of twin interfaces existed. Thus, many authors have devised structural models of twin interfaces based upon the (undeformed) bulk structure of the crystals and the experimentally determined orientation and contact relations. The criterion of good structural fit and low energy of a boundary was usually applied in a rather intuitive manner to the specific case in question. The first and classic example is the model of the aragonite (110) boundary by Bragg (1924[link]).

Some examples of twin-boundary models from the literature are given below. They are intended to show the wide variety of substances and kinds of models. Examples for the direct observation of twin-interface structures by HRTEM follow in Section 3.3.10.6.[link]

3.3.10.5.1. Aragonite, CaCO3

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The earliest structural model of a twin boundary was derived for aragonite by Bragg (1924[link]), reviewed in Bragg (1937[link], pp. 119–121) and Bragg & Claringbull (1965[link], pp. 131–133). Aragonite is orthorhombic with space group Pmcn. It exhibits a pronounced hexagonal pseudosymmetry, corresponding to a (hypothetical) parent phase of symmetry [P6_3/mmc], in which the Ca ions form a hexagonal close-packed structure with the CO3 groups filling the octahedral voids along the [6_3] axes. By eliminating the threefold axis and the C-centring translation of the orthohexagonal unit cell, the above orthorhombic space group results, where the lost centring translation now appears as the glide component n. Of the three mirror planes parallel to [\{11{\bar 2}0\}_{\rm hex}] and the three c-glide planes parallel to [\{10{\bar 1}0\}_{\rm hex}], one of each set is retained in the orthorhombic structure, whereas the other two appear as possible twin mirror planes [\{110\}_{\rm orth}] and [\{130\}_{\rm orth}]. It is noted that predominantly planes of type [\{110\}_{\rm orth}] are observed as twin boundaries, but less frequently those of type [\{130\}_{\rm orth}].

From this structural pseudosymmetry the atomic structure of the twin interface was easily derived by Bragg. It is shown in Fig. 3.3.10.6[link]. In reality, small relaxations at the twin boundary have to be assumed. It is clearly evident from the figure that the twin operation is a glide reflection with glide component [\textstyle{1\over2}{\bf c}] (= twin displacement vector t).

[Figure 3.3.10.6]

Figure 3.3.10.6 | top | pdf |

Structural model of the (110) twin boundary of aragonite (after Bragg, 1924[link]), projected along the pseudo-hexagonal c axis. The ortho­rhombic unit cells of the two domains with eigensymmetry Pmcn, as well as their glide/reflection planes m and c, are indicated. The slab centred on the (110) interface between the thin lines is common to both partners. The interface coincides with a twin glide plane c and is shown as a dotted line (twin displacement vector [{\bf t} = 1/2 {\bf c}]). The model is based on a hexagonal cell with [\gamma = 120^\circ], the true angle is [\gamma = 116.2^\circ]. The origin of the orthorhombic cell is chosen at the inversion centre halfway between two CO3 groups along c.

3.3.10.5.2. Dauphiné twins of [\alpha]-quartz

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For this merohedral twin (eigensymmetry 32) a real parent phase, hexagonal [\beta]-quartz (622), exists. The structural relation between the two Dauphiné twin partners of [\alpha]-quartz is best seen in projection along [001], as shown in Fig. 3.3.10.7[link] and in Figure 3 of McLaren & Phakey (1966[link]), assuming a fault vector [{\bf f} = {\bf 0}] in both cases. These figures reveal that only small deformations occur upon passing from one twin domain to the other, irrespective of the orientation of the boundary. This is in agreement with the general observation that Dauphiné boundaries are usually irregular and curved and can adopt any orientation. The electron microscopy study of Dauphiné boundaries by McLaren & Phakey confirms the fault vector [{\bf f} = {\bf 0}]. It is noteworthy that the two models of the boundary structure by Klassen-Neklyudova (1964[link]) and McLaren & Phakey (1966[link]) imply a slab with the [\beta]-quartz structure in the centre of the transition zone (Fig. 3.3.10.7[link]b). This is in agreement with the assumption voiced by several authors, first by Aminoff & Broomé (1931[link]), that the central zone of a twin interface often exhibits the structure of a different (real or hypothetical) polymorph of the crystal.

[Figure 3.3.10.7]

Figure 3.3.10.7 | top | pdf |

Simplified structural model of a [\{10{\bar 1}0\}] Dauphiné twin boundary in quartz (after Klassen-Neklyudova, 1964[link]). Only Si atoms are shown. (a) Arrangement of Si atoms in the low-temperature structure of quartz viewed along the trigonal axis [001]. (b) Model of the Dauphiné twin boundary C–D. Note the opposite orientation of the three electrical axes shown in the upper left and lower right corner of part (b). In this model, the structural slab centred along the twin boundary has the structure of the hexagonal high-temperature phase of quartz which is shown in (c).

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

3.3.10.5.3. Potassium lithium sulfate, KLiSO4

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The room-temperature phase of KLiSO4 is hexagonal with space group [P6_3]. It forms a `stuffed' tridymite structure, consisting of a framework of alternating SO4 and LiO4 tetrahedra with the K ions `stuffed' into the framework cavities. Crystals grown from aqueous solutions exhibit merohedral growth twins with twin reflection planes [\{10{\bar 1}0\}] (alternatively [\{11{\bar 2}0\}]) with extended and sharply defined (0001) twin boundaries. The twins consist of left- and right-handed partners with the same polarity. The left- and right-handed structures, projected along the polar hexagonal c axis, are shown in Figs. 3.3.10.8[link](a) and (b) (Klapper et al., 1987[link]). The tetrahedra of the two tetrahedral layers within one translation period c are in a staggered orientation. A model of the twin boundary is shown in Fig. 3.3.10.8[link](c): the tetrahedra on both sides of the twin interface (0001), parallel to the plane of the figure, now adopt an eclipsed position, leading to an uninterrupted framework and a conformation change in second coordination across the interface. It is immediately obvious that this (0001) interface permits an excellent low-energy fit of the two partner structures. Note that all six (alternative) twin reflection planes [\{10{\bar 1}0\}] and [\{11{\bar 2}0\}] are normal to the twin boundary. It is not possible to establish a similar low-energy structural model of a boundary which is parallel to one of these twin mirror planes (Klapper et al., 1987[link]).

[Figure 3.3.10.8]

Figure 3.3.10.8 | top | pdf |

KLiSO4: Bulk tetrahedral framework structures and models of (0001) twin boundary structures of phases III and IV. Small tetrahedra: SO4; large tetrahedra: LiO4; black spheres: K. All three figures play a double role, both as bulk structure and as (0001) twin-boundary structures. (a) and (b) Left- and right-handed bulk structures of phase III ([P6_3]), as well as possible structures of the (0001) twin boundary in phase IV. (c) Bulk structure of phase IV ([P31c]), as well as possible structure of the (0001) twin boundary in phase III. The SO4 tetrahedra covered by the LiO4 tetrahedra are shown by thin lines. Dotted line: [\{10{\bar 1}0\}] c-glide plane. In all cases, the (0001) twin boundary is located between the two tetrahedral layers parallel to the plane of the figure.

Inspection of the boundary structure in Fig. 3.3.10.8[link](c) shows that the tetrahedra related by the twin reflection plane [\{10{\bar 1}0\}] (one representative plane is indicated by the dotted line) are shifted with respect to each other by a twin displacement vector [{\bf t} = 1/2 [001]]. Thus, on an atomic scale, these twin reflection planes are in reality twin c-glide planes, bringing the right- and left-hand partner structures into coincidence.

Interestingly, upon cooling below 233 K, KLiSO4 undergoes a (very) sluggish phase transition from the [P6_3] phase III into the trigonal phase IV with space group [P31c] by suppression of the twofold axis parallel [001] and by addition of a c-glide plane. Structure determinations show that the bulk structure of IV is exactly the atomic arrangement of the grown-in twin boundary of phase III, as presented in Fig. 3.3.10.8[link](c). Moreover, X-ray topography reveals transformation twins III [\rightarrow] IV, exhibiting extended and sharply defined polysynthetic (0001) twin lamellae in IV. From the X-ray topographic domain contrast, it is proven that the twin element is the twofold rotation axis parallel to [001]. The structural model of the (0001) twin interfaces is given in Figs. 3.3.10.8(a) and (b). They show that across the (0001) twin boundary the tetrahedra are staggered, in contrast to the bulk structure of IV where they are in an eclipsed orientation (Fig. 3.3.10.8[link]c). It is immediately recognized that the two tetrahedral layers, one above and one below the (0001) twin boundary in Fig. 3.3.10.8[link](a) or (b), are related by [2_1] screw axes.

Thus, the (idealized) (0001) twin boundary of the transformation twins of phase IV is represented by the bulk structure of the hexagonal room-temperature phase III, whereas the twin boundary of the growth twins of the hexagonal phase III is represented by the bulk structure of the trigonal low-temperature phase IV. Upon cooling from [P6_3] (phase III) to [P31c] (phase IV), the [2_1] axes are suppressed as symmetry elements, but they now act as twin elements. In the model they are located as in space group [P6_3], one type being contained in the [6_3] axes, the other type halfway in between. Upon heating, the re-transformation IV [\rightarrow] III restores the [\{10{\bar 1}0\}/\{11{\bar 2}0\}] reflection twins with the same large (0001) boundaries in the same geometry as existed before the transition cycle, but now as result of a phase transition, not of crystal growth (strong memory effect).

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

The structural models of both kinds of twin boundaries do not exhibit a fault vector [{\bf f} \ne {\bf 0}]. This may be explained by the compensation of the glide component [\textstyle{1\over2}c] of the c-glide plane in phase IV by the screw component [\textstyle{1\over2}c] of the [2_1] screw axis in phase III and vice versa.

3.3.10.5.4. Twin models of molecular crystals

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

3.3.10.6. Observations of twin boundaries by transmission electron microscopy

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In the previous sections of this chapter, twin boundaries have been discussed from two points of view: theoretically in terms of `compatibility relations', i.e. of mechanically and electrically `permissible' interfaces (Sections 3.3.10.2 and 3.3.10.3), followed by structural aspects, viz by displacement and fault vectors (Section 3.3.10.4[link]), as well as atomistic models of twin boundaries (Section 3.3.10.5[link]), in each case accompanied by actual examples.

In the present section, a recent and very powerful method of direct experimental elucidation of the atomistic structure of twin interfaces is summarized, transmission electron microscopy (TEM), in particular high-resolution transmission electron microscopy (HRTEM). This method enjoys wider and wider application because it can provide in principle – if applied with proper caution and criticism – direct evidence for the problems discussed in earlier sections: `good structural fit', `twin displacement vector', `relaxation of the structure' across the boundary etc.

The present chapter is not a suitable place to introduce and explain the methods of TEM and HRTEM and the interpretation of the images obtained. Instead, the following books, containing treatments of the method in connection with materials science, are recommended: Wenk (1976[link]), especially Sections 2.3 and 5; Amelinckx et al. (1978[link]), especially pp. 107–151 and 217–314; McLaren (1991[link]); Buseck et al. (1992[link]), especially Chapter 11; and Putnis (1992[link]), especially pp. 67–80.

The results of HRTEM investigations of twin interfaces are not yet numerous and representative enough to provide a complete and coherent account of this topic. Instead, a selection of typical examples is provided below, from which an impression of the method and its usefulness for twinning can be gained.

3.3.10.6.1. Anatase, TiO2 (Penn & Banfield, 1998[link], 1999[link])

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This investigation has been presented already in Section 3.3.10.4.1[link] and Fig. 3.3.10.5[link] as an example of the occurrence of a twin displacement vector, leading to [{\bf t}\approx 1/2 {\bf v}_L], where [{\bf v}_L] is a lattice translation vector parallel to the (112) twin reflection plane of anatase. Another interesting result of this HRTEM study by Penn & Banfield is the formation of anatase–brookite intergrowths during the hydrothermal coarsening of TiO2 nanoparticles. The preferred contact plane is (112) of anatase and (100) of brookite, with [131] of anatase parallel to [011] of brookite in the intergrowth plane. Moreover, it is proposed that brookite may nucleate at (112) twin boundaries of anatase and develop into (100) brookite slabs sandwiched between the anatase twin components. Similarly, after hydrothermal treatment at 523 K, nuclei of rutile at the anatase (112) twin boundary were also observed by HRTEM (Penn & Banfield, 1999[link]). A detailed structural model for this anatase-to-rutile phase transition is proposed by the authors, from which a sluggish nucleation of rutile followed by rapid growth of this phase was concluded.

3.3.10.6.2. SnO2 (rutile structure)

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Twin interfaces (011) of the closely related tetragonal SnO2 (cassiterite) were investigated by Smith et al. (1983[link]). A very close agreement between HRTEM images and corresponding computer simulations was obtained for [{\bf t} = 1/2 [1{\bar 1}1](011)]. This twin is termed `glide twin' by the authors, because the twin operation is a reflection across (011) followed by a displacement vector [{\bf t} = 1/2 [1{\bar 1}1](011)] parallel to the twin plane (011).

3.3.10.6.3. [\Sigma 3] (111) twin interface in BaTiO3 [cf. Section 3.3.8.3[link](iii)[link]]

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In cubic crystals, twins of the [\Sigma 3] (111) spinel type are by far the most common. A technologically very important phase, BaTiO3 perovskite, was investigated by Rečnik et al. (1994[link]) employing HRTEM, computer simulations and EELS (spatially resolved electron-energy-loss spectroscopy). The samples were prepared by sintering at 1523 K, i.e. in the cubic phase, whereby [\Sigma 3] (111) growth twins were formed. These twins are preserved during the transition into the tetragonal phase upon cooling below Tc = 398 K. Note that these (now tetragonal) (111) twins are not transformation twins, as are the (110) ferroelastic twins.

Fig. 3.3.10.9[link](a) shows an HRTEM micrograph and Fig. 3.3.10.9[link](b) the structural model of the (111) twin boundary, both projected along [[1{\bar 1}0]]. The main results of this study can be summarized as follows.

  • (1) The twin boundary coincides exactly with the twin reflection plane (111). It is very sharp and consists of one atomic layer only, common to both twin components. It is fully `coherent' (cf. Section 3.3.10.9[link]).

    [Figure 3.3.10.9]

    Figure 3.3.10.9 | top | pdf |

    (a) HRTEM micrograph of a coherent (111) twin boundary in BaTiO3, projected along [[1{\bar 1}0]]. The intense white spots represent the [[1{\bar 1}0]] Ba–O columns, the small weak spots in between represent the Ti columns. Thickness of specimen 4 nm. (b) Structural model of the (111) twin boundary (arrow), as derived from the micrograph (a) and confirmed by computer simulation. Note that the pure oxygen columns (open circles) are not visible in the micrograph (a), due to the low scattering power of oxygen. Some slight structural deformations along the twin boundary are discussed in the text. Courtesy of W. Mader, Bonn; cf. Rečnik et al. (1994)[link].

  • (2) The twin boundary is formed by a close-packed BaO3 layer, and the TiO6 octahedra on both sides share faces to form Ti2O9 groups. These groups occur also in the hexagonal high-temperature modification of BaTiO3, i.e. this is a further example of a twin interface having the structure of another polymorph of the same compound.

  • (3) The EELS results suggest a reduction in the valence state of the Ti4+ ions in the interface towards Ti3+ which is compensated by some oxygen vacancies, leading to the composition of the interface layer BaO3−x(VO)x instead of BaO3. This result indicates that the stoichiometry of a boundary, even of a coherent one, may differ from that of the bulk.

A very interesting structural feature of the BaTiO3 (111) twin interface was discovered by Jia & Thust (1999[link]), applying sophisticated HRTEM methods to thin films of nanometre thickness (grown by the pulsed-laser deposition technique). The distance of the nearest Ti plane on either side of the (111) twin reflection plane (which is formed by a BaO3 layer, see above) from this twin plane is increased by 0.19 Å, i.e. the distance between the Ti atoms in the Ti2O9 groups, mentioned above under (2)[link], is increased from the hypothetical value of 2.32 Å for Ti in the ideal octahedral centres to 2.70 Å in the actual interface structure. This expansion is due to the strong repulsion between the two neighbouring Ti ions in the Ti2O9 groups. A similar expansion of the Ti–Ti distance in the Ti2O9 groups (from 2.34 to 2.67 Å), again due to the strong repulsion between the Ti atoms, has been observed in the bulk crystal structure of the hexagonal modification of BaTiO3.

In addition, a decrease by 0.16 Å of the distance between the two nearest BaO planes across the twin interface was found, which corresponds to a contraction of this pair of BaO planes from 2.32 Å in the bulk to 2.16 Å at the twin interface (corresponding closely to the value of 2.14 Å in the hexagonal phase).

It is remarkable that no significant (i.e. > 0.05 Å) displacements were found for second and higher pairs of both Ti–Ti and BaO–BaO layers. Moreover, no significant lateral shifts, i.e. no twin displacements vectors [{\bf t}\neq 0] parallel to the (111) twin interface, were observed.

Note that BaTiO3 is treated again in Section 3.3.10.7.5[link] below, with respect to its twin texture in polycrystalline aggregates.

3.3.10.6.4. [\Sigma = 3] bicrystal boundaries in Cu and Ag

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Differently oriented interfaces in [\Sigma 3] bicrystals of Cu and Ag were elucidated by Hoffmann & Ernst (1994[link]) and Ernst et al. (1996[link]). They prepared bicrystals of fixed [\Sigma 3] orientation relationship [corresponding to the (111) spinel twin law] but with different contact planes. The inclinations of these contact planes vary by rotations around the two directions [[{\bar 1}10]] and [[11{\bar 2}]] [both parallel to the (111) twin reflection plane] in the range [\Phi =] 0–90°, where [\Phi = 0^\circ] corresponds to the (111) `coherent twin plane', as illustrated in Figs. 3.3.10.10[link](a) and (b).

[Figure 3.3.10.10]

Figure 3.3.10.10 | top | pdf |

(a) Schematic block diagram of a [\Sigma = 3] bicrystal (spinel twin) for [\varphi[{\bar 1}10] = 0^\circ], i.e. for coinciding (111) twin reflection and composition plane. (b) Schematic block diagram of the [\Sigma = 3] bicrystal for [\varphi[{\bar 1}10] = 82^\circ]. (c) HRTEM micrograph of the [\Sigma = 3] boundary of Cu for [\varphi[{\bar 1}10] = 82^\circ], projected along [[{\bar 1}10]]. The black spots coincide with the [[{\bar 1}10]] Cu-atom columns. The micrograph reveals a thin ([\approx] 10 Å) interface slab of a rhombohedral 9R structure, which can be derived from the bulk cubic 3C structure by introducing a stacking fault SF on every third (111) plane. The (111) planes are horizontal, the interface is roughly parallel to (4.4.11) and (223), respectively. Courtesy of F. Ernst, Stuttgart; cf. Ernst et al. (1996)[link].

The boundary energies were determined from the surface tension derived from the characteristic angles of surface grooves formed along the boundaries by thermal etching. The theoretical energy values were obtained by molecular statics calculations. The measured and calculated energy curves show a deep and sharp minimum at [\Phi = 0^\circ] for rotations around both [[{\bar 1}10]] and [[11{\bar 2}]]. This corresponds to the coherent (111) [\Sigma 3] twin boundary and is to be expected. It is surprising, however, that a second, very shallow energy minimum occurs in both cases at high [\Phi] angles: [\Phi_{[{\bar 1}10]} \approx 82^\circ] and [\Phi_{[11{\bar 2}]} \approx 84^\circ], rather than at the compatible (112) contact plane for [\Phi_{[{\bar 1}10]} = 90^\circ] [the contact plane [(1{\bar 1}0)] for [\Phi_{[11{\bar 2}]} = 90^\circ] is not compatible]. For these two angular inclinations, the boundaries, as determined by HRTEM and computer modelling, exhibit complex three-dimensional boundary structures with thin slabs of unusual Cu arrangements: the [\Phi_{[{\bar 1}10]} \approx 82^\circ] slab has a rhombohedral structure of nine close-packed layers (9R) with a thickness of about 10 Å (in contrast to the f.c.c bulk structure, which is 3C). This is shown and explained in Fig. 3.3.10.10[link](c). Similarly, for the [\Phi_{[11{\bar 2}]} \approx 84^\circ] slab a b.c.c. structure (as for [\alpha]-Fe) was found, again with a thickness of [\approx 10] Å.

3.3.10.6.5. Fivefold cyclic twins in nanocrystalline materials

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Multiply twinned particles occur frequently in nanocrystalline (sphere-like or rod-shaped) particles and amorphous thin films (deposited on crystalline substrates) of cubic face-centred metals, diamond-type semiconductors (C, Si, Ge) and alloys. Hofmeister & Junghans (1993[link]) and Hofmeister (1998[link]) have carried out extensive HRTEM investigations of nanocrystalline Ge particles in amorphous Ge films. The particles reveal, among others, fivefold cyclic twins with coinciding (111) twin reflection planes and twin boundaries (spinel type). A typical example of a fivefold twin is presented in Fig. 3.3.10.11[link]: The five different {111} twin boundaries are perpendicular to the image plane ([1{\bar 1}0]) and should theoretically form dihedral angles of 70.5° (supplement to the tetrahedral angle 109.5°), which would lead to an angular gap of about 7.5°. In reality, the five twin sectors are more or less distorted with angles ranging up to 76°. The stress due to the angular mismatch is often relaxed by defects such as stacking faults (marked by arrows in Fig. 3.3.10.11[link]). The [[1{\bar 1}0]] junction line of the five sectors can be considered as a pseudo-fivefold twin axis (similar to the pseudo-trigonal twin axis of aragonite, cf. Fig. 3.3.2.4[link]; see also the fivefold twins in the alloy FeAl4, described in Example 3.3.6.8[link] and Fig. 3.3.6.8[link]).

[Figure 3.3.10.11]

Figure 3.3.10.11 | top | pdf |

HRTEM micrograph of a fivefold-twinned Ge nanocrystal (right) in an amorphous Ge film formed by vapour deposition on an NaCl cleavage plane. Projection along a [[1{\bar 1}0]] lattice row that is the junction of the five twin sectors; plane of the image: [(1{\bar 1}0)]. The coinciding {111} twin reflection and composition planes (spinel law) are clearly visible. In one twin sector, two pairs of stacking faults (indicated by arrows) occur. They reduce the stress introduced by the angular misfit of the twin sectors. The atomic model (left) shows the structural details of the bulk and of one pair of stacking faults. Courtesy of H. Hofmeister, Halle; cf. Hofmeister & Junghans (1993)[link]; Hofmeister (1998)[link].

For the formation of fivefold twins, different mechanisms have been suggested by Hofmeister (1998[link]): nucleation of noncrystallographic clusters, which during subsequent growth collapse into cyclic twins; successive growth twinning on alternate cozonal (111) twin planes; and deformation twinning (cf. Section 3.3.7[link]).

The fivefold multiple twins provide an instructive example of a twin texture, a subject which is treated in the following section.

3.3.10.7. Twin textures

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So far in Section 3.3.10, `free' twin interfaces have been considered with respect to their mechanical and electrical compatibility, their twin displacement vectors and their structural features, experimentally and by modelling. In the present section, the `textures' of twin domains, both in a `single' twin crystal and in a polycrystalline material or ceramic, are considered. With the term `twin texture', often also called `twin pattern', `domain pattern' or `twin microstructre', the size, shape and spatial distribution of the twin domains in a twinned crystal aggregate is expressed. In a (polycrystalline) ceramic, the interaction of the twin interfaces in each grain with the grain boundary is a further important aspect. Basic factors are the `form changes' and the resulting space-filling problems of the twin domains compared to the untwinned crystal. These interactions can occur during crystal growth, phase transitions or mechanical deformations.

From the point of view of form changes, two categories of twins, described in Sections 3.3.10.7.1[link] and 3.3.10.7.2[link] below, have to be distinguished. Discussions of the most important twin cases follow in Sections 3.3.10.7.3[link] to 3.3.10.7.5[link].

3.3.10.7.1. Merohedral (non-ferroelastic) twins (see Sections 3.3.9[link] and 3.3.10.2.3[link])

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In these twins, the lattices of all domains are exactly parallel (`parallel-lattice twins'). Hence, no lattice deformations (spontaneous strain) occur and the development of the domain pattern of the twins is not infringed by spatial constraints. As a result, the twin textures can develop freely, without external restraints [cf. Section 3.3.10.7.5[link] below].

It should be noted that these features apply to all merohedral twins, irrespective of origin, i.e. to growth and transformation twins and, among mechanical twins, to ferrobielastic twins [for the latter see Section 3.3.7.3[link](iii)[link]].

3.3.10.7.2. Non-merohedral (ferroelastic) twins

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Here, the lattices of the twin domains are not completely parallel (`twins with inclined axes'). As a result, severe space problems may arise during domain formation. Several different cases have to be considered:

  • (1) Only one twin law, i.e. only two domain states occur which can form two-component twins (e.g. dovetail twins of gypsum, Carlsbad twins of orthoclase) or multi-component twins (e.g. lamellar, polysynthetic twins of albite). For these twins, no spatial constraints are imposed and, hence, the twin crystal can develop freely, without external restraints. Again, this applies to both growth and transformation twins.

  • (2) Two or more twin laws, i.e. three or more domain states coexist. Here, the free development of a twin domain is impeded by the space requirements of its neighbours. For growth twins, typical cases are sector and cyclic twins (e.g. K2SO4 and aragonite). More complicated examples are the famous harmotome and phillipsite growth twins, where the combined action of several twin laws leads to a pseudo-cubic twin texture and twin morphology.

Transformation and deformation twins are extensively treated in the following Section 3.3.10.7.3[link].

3.3.10.7.3. Fitting problems of ferroelastic twins

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The real problem of space-constrained twin textures, however, is provided by non-merohedral (ferroelastic) transformation and deformation twins (including the cubic deformation twins of the spinel law). This is schematically illustrated in Fig. 3.3.10.12[link] for the very common case of orthorhombic[\longrightarrow]monoclinic transformation twins ([\beta = 90^\circ + \varepsilon)].

[Figure 3.3.10.12]

Figure 3.3.10.12 | top | pdf |

Illustration of space-filling problems of domains for a (ferroelastic) orthorhombic [\rightarrow] monoclinic phase transition with an angle [\varepsilon] (exaggerated) of spontaneous shear. (a) Orthorhombic parent crystal with symmetry [2/m\,2/m\,2/m]. (b) Domain pairs [1+2], [1+3] and [2+4] of the monoclinic daughter phase ([\beta = 90^\circ + \varepsilon]) with independent twin reflection planes (100) and (001). (c) The combination of domain pairs [1+2] and [1+3] leads to a gap with angle [90^\circ - 3\varepsilon], whereas the combination of the three domain pairs [1+2], [1+3] and [2+4] generates a wedge-shaped overlap (hatched) of domains 3 and 4 with angle [4\varepsilon]. (d) Twin lamellae systems of domain pairs [1+2] (left) and [1+3] (or [2+4]) (right) with low-energy contact planes (100) and (001). Depending on the value of [\varepsilon], adaptation problems with more or less strong lattice distortions arise in the boundary region A–A between the two lamellae systems. (e) Stress relaxation and reduction of strain energy in the region A–A by the tapering of domains 2 (`needle domains') on approaching the (nearly perpendicular) boundary of domains [3+1]. The tips of the needle lamellae may impinge on the boundary or may be somewhat withdrawn from it, as indicated in the figure. The angle between the two lamellae systems is [90^\circ - \varepsilon].

Figs. 3.3.10.12[link](a) and (b) show the `splitting' of two mirror planes (100) and (001) of parent symmetry mmm, as a result of a phase transition [mmmF12/m1], into the two independent and symmetrically non-equivalent twin reflection planes (100) and (001), each one representing a different (monoclinic) twin law. The two orientation states of each domain pair differ by the splitting angle [2\varepsilon]. Note that in transformation twins the angle [\varepsilon] (spontaneous shear strain) is small, at most one or two degrees, due to the pseudosymmetry [\cal H] of the daughter phase with respect to the parent symmetry [\cal G]. It can be large, however, for deformation twins, e.g. calcite. The resulting fitting problems in ferroelastic textures are illustrated in Fig. 3.3.10.12[link](c). Owing to the splitting angle [2\varepsilon], twin domains would form gaps or overlaps, compared to a texture with [\varepsilon = 0], where all domains fit precisely. In reality, the misfit due to [\varepsilon \ne 0] leads to local stresses and associated elastic strains around the meeting points of three domains related by two twin laws [triple junctions, cf. Palmer et al. (1988[link]), Figs. 3–6)].

For the orthorhombic[\longrightarrow]monoclinic transition considered here, the two different twin laws often lead to two and (for small [\varepsilon]) nearly perpendicular sets of polysynthetic twin lamellae. This is illustrated in Fig. 3.3.10.12[link](d). The boundaries in one set are formed by (100) planes, those in the other set by (001) planes, both of low energy. The misfit problems are located exclusively in the region AA where the two systems of lamellae meet. Here wedge-like domains (the so-called `needle domains', see below) are formed, as shown in Fig. 3.3.10.12[link](e), i.e. the twin lamellae of one system taper on approaching the perpendicular twin system (right-angled twins), forming rounded or sharp needle tips. The tips of the needle lamellae may be in contact with the perpendicular lamella or may be somewhat withdrawn from it. These effects are the consequence of strain-energy minimization in the transition region of domain systems, as compared to the large-area contacts between parallel twin lamellae.

The formation of two lamellae systems with wedge-like domains was demonstrated very early on by the polarization-optical study of the orthorhombic[\longrightarrow]monoclinic (222[\longrightarrow]2) transformation of Rochelle salt at 297 K by Chernysheva (1950[link], 1955[link]; quoted after Klassen-Neklyudova, 1964[link], pp. 27–30 and 76–77, Figs. 35, 38 and 100; see also Zheludev, 1971[link], pp. 180–185). The term `needle domains' was coined by Salje et al. (1985[link]) in their study of the monoclinic[\longrightarrow]triclinic [(2/m\longrightarrow{\bar 1}]) transition of Na-feldspar. Another detailed description of needle domains is provided by Palmer et al. (1988[link]) for the cubic[\longrightarrow]tetragonal ([4/m\,{\bar 3}\,2/m \longrightarrow 4/m\,2/m\,2/m]) transformation of leucite at 878 K. The typical domain structure resulting from this transition is shown in Fig. 3.3.10.13[link].

[Figure 3.3.10.13]

Figure 3.3.10.13 | top | pdf |

Thin section of tetragonal leucite, K(AlSi2O6), between crossed polarizers. The two nearly perpendicular systems of (101) twin lamellae result from the cubic-to-tetragonal phase transition at about 878 K. Width of twin lamellae 20–40 µm. Courtesy of M. Raith, Bonn.

A similar example is provided by the extensively investigated tetragonal[\longrightarrow]orthorhombic transformation twinning of the high-Tc superconductor YBa2Cu3O7−δ (YBaCu) below about 973 K (Roth et al., 1987[link]; Schmid et al., 1988[link]; Keester et al., 1988[link], especially Fig. 6). Here two symmetrically equivalent systems of lamellae with twin laws [m(110)] and [m({\bar 1}10)] meet at nearly right angles ([\varepsilon \approx 1^\circ]). Interesting TEM observations of tapering, impinging and intersecting twin lamellae are presented by Müller et al. (1989[link]). An extensive review on twinning of YBaCu, with emphasis on X-ray diffraction studies (including diffuse scattering), was published by Shektman (1993[link]).

A particularly remarkable case occurs for hexagonal[\longrightarrow]orthorhombic ferroelastic transformation twins. Well known examples are the pseudo-hexagonal K2SO4-type crystals (cf. Example 3.3.6.7[link]). Three (cyclic) sets of orthorhombic twin lamellae with interfaces parallel to [\{10{\bar 1}0\}_{\rm hex}] or [\{110\}_{\rm orth}] are generated by the transformation. More detailed observations on hexagonal–orthorhombic twins are available for the III[\longrightarrow]II (heating) and I[\longrightarrow]II (cooling) transformations of KLiSO4 at about 712 and 938 K (Jennissen, 1990[link]; Scherf et al., 1997[link]). The development of the three systems of twin lamellae of the orthorhombic phase II is shown by two polarization micrographs in Fig. 3.3.10.14[link]. A further example, the cubic[\longrightarrow]rhombohedral phase transition of the perovskite LaAlO3, was studied by Bueble et al. (1998[link]).

[Figure 3.3.10.14]

Figure 3.3.10.14 | top | pdf |

Twin textures generated by the two different hexagonal-to-orthorhombic phase transitions of KLiSO4. The figures show parts of [(0001)_{\rm hex}] plates (viewed along [001]) between crossed polarizers. (a) Phase boundary III[\longrightarrow]II with circular 712 K transition isotherm during heating. Transition from the inner (cooler) room-temperature phase III (hexagonal, dark) to the (warmer) high-temperature phase II (orthorhombic, birefringent). Owing to the loss of the threefold axis, lamellar [\{10{\bar 1}0\}_{\rm hex} = \{110\}_{\rm orth}] cyclic twin domains of three orientation states appear. (b) Sketch of the orientations states 1, 2, 3 and the optical extinction directions of the twin lamellae. Note the tendency of the lamellae to orient their interfaces normal to the circular phase boundary. Arrows indicate the direction of motion of the transition isotherm during heating. (c) Phase boundary I[\longrightarrow]II with 938 K transition isotherm during cooling. The dark upper region is still in the hexagonal phase I, the lower region has already transformed into the orthorhombic phase II (below 938 K). Note the much finer and more irregular domain structure compared with the III[\longrightarrow]II transition in (a). Courtesy of Ch. Scherf, PhD thesis, RWTH Aachen, 1999; cf. Scherf et al. (1997)[link].

Another surprising feature is the penetration of two or more differently oriented nano-sized twin lamellae, which is often encountered in electron micrographs (cf. Müller et al., 1989[link], Fig. 2b). In several cases, the penetration region is interpreted as a metastable area of the higher-symmetrical para-elastic parent phase.

In addition to the fitting problems discussed above, the resulting final twin texture is determined by several further effects, such as:

  • (a) the nucleation of the (twinned) daughter phase in one or several places in the crystal;

  • (b) the propagation of the phase boundary (transformation front, cf. Fig. 3.3.10.14[link]);

  • (c) the tendency of the twinned crystal to minimize the overall elastic strain energy induced by the fitting problems of different twin lamellae systems.

Systematic treatments of ferroelastic twin textures were first published by Boulesteix (1984[link], especially Section 3.3 and references cited therein) and by Shuvalov et al. (1985[link]). This topic is extensively treated in Section 3.4.4[link] of the present volume. A detailed theoretical explanation and computational simulation of these twin textures, with numerous examples, was recently presented by Salje & Ishibashi (1996[link]) and Salje et al. (1998[link]). Textbook versions of these problems are available by Zheludev (1971[link]) and Putnis (1992[link]).

3.3.10.7.4. Tweed microstructures

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The textures of ferroelastic twins and their fitting problems, discussed above, are `time-independent' for both growth and deformation twins, i.e. after twin nucleation and growth or after the mechanical deformation there occurs in general no `ripening process' with time before the final twin structure is produced. This is characteristically different for some transformation twins, both of the (slow) order–disorder and of the (fast) displacive type and for both metals and non-metals. Here, with time and/or with decreasing temperature, a characteristic microstructure is formed in between the high- and the low-temperature polymorph. This `precursor texture' was first recognized and illustrated by Putnis in the investigation of cordierite transformation twinning and called `tweed microstructure' (Putnis et al., 1987[link]; Putnis, 1992[link]). In addition to the hexagonal–orthorhombic cordierite transformation, tweed structures have been investigated in particular in the K-feldspar orthoclase (monoclinic–triclinic transformation), in both cases involving (slow) Si–Al ordering processes. Examples of tweed structures occurring in (fast) displacive trans­form­ations are provided by tetragonal–orthorhombic Co-doped YBaCu3O7−d (Schmahl et al., 1989[link]) and rhombohedral–monoclinic (Pb,Sr)3(PO4)2 and (Pb,Ba)3(PO4)2 (Bismayer et al., 1995[link]).

Tweed microstructures are precursor twin textures, intermediate between those of the high- and the low-temperature modifications, with the following characteristic features:

  • (a) With respect to long-range order, the tweed structure belongs to the (disordered) high-temperature form, as shown by synchroton radiation powder diffraction; for instance, orthoclase has macroscopic monoclinic symmetry, and the tweed structure of cordierite is strictly hexagonal on a macroscopic scale.

  • (b) Experiments that reveal short-range order, especially TEM micrographs, infrared and Raman spectra and NMR spectra, show features of the ordered low-temperature modification; in orthoclase, very fine (nanometre-size) superposed triclinic albite and pericline microdomains occur, which may even fluctuate with time; similarly for cordierite.

With annealing or cooling time these tweed structures exhibit continuous `coarsening' of their microdomains and `sharpening' of their boundaries. The tweed microstructure of orthoclase develops into the well known crosshatched `transformation microcline' texture discussed above in Section 3.3.10.2.2[link], example (3), and the cordierite tweed structure gives way to coarse twin lamellae in two orthogonal orientations of well ordered ortho­rhombic cordierite. Recently, interesting computer simulations of the time evolution of twin domains have been performed for alkali-metal feldspars by Tsatskis & Salje (1996[link]) and for cordierite by Blackburn & Salje (1999[link]). These papers also contain references to earlier work on tweed structures.

Textbook descriptions of tweed structures can be found in the following works: Putnis, 1992[link], Sections 7.3.5, 7.3.6 and 12.4.1; Salje, 1993[link], Sections 7.3 and pp. 116–201; Putnis & Salje, 1994[link].

3.3.10.7.5. Twin textures in polycrystalline aggregates

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So far, twin textures have been treated independently of their occurrence in `single crystals' or in polycrystalline aggregates. In the present section, the specific situation in polycrystalline materials such as ceramics, metals and rocks is discussed. This treatment is concerned with the extra effects that occur in addition to those discussed in Sections 3.3.10.7.1[link][link][link] to 3.3.10.7.4[link] above. These additional effects result from the fact that in a polycrystalline material a given crystal grain is surrounded by other grains and thus `clamped' with respect to form and orientation changes arising from mechanical stress, electrical polarization or magnetization. Here, this effect is called `neighbour clamping'.

Two cases of neighbour clamping occur, three-dimensional clamping of grains in the bulk of a sample and two-dimensional clamping at the surface of a sample. In addition, two-dimensional clamping can occur in thin films, either free or epitaxial. The result of this clamping is high elastic stress which is relaxed (`stress relief'; Arlt, 1990[link]) by twinning, in particular by the formation of `shape-preserving' twin textures.

Twinning in a ceramic is of great technical importance for the preparation and optimization of devices such as capacitors, piezoelectric elements and magnets. They often contain ferroelectric or ferromagnetic polycrystalline materials which undergo domain switching in an electric or magnetic field and, hence, can be poled.

In the following, we restrict our considerations to non-metallic ceramics where twinning is generated by a ferroelastic phase transition (e.g. perovskites). It is assumed that the ceramic is formed at temperatures far above the phase transition, which is accompanied on cooling by a considerable spontaneous lattice strain in the low-temperature phase, leading to the formation of non-merohedral twins. Without any formation of twins a considerable change of the grain shapes would occur and cause high inter-grain stress. The main mechanism of stress relaxation (`stress relief') is the formation of a ferroelastic twin texture which preserves the shape of the original (high-temperature phase) grain as far as possible. Note that the twin texture resulting from this `neighbour clamping' is quite different from the twin texture of a free, unclamped grain. In the free grain, only few twin lamellae with usually coherent boundaries are formed, whereas in the clamped grain several twin bands with narrow-spaced twin lamellae of different twin types occur.

In the clamped case, the significant effect of ferroelastic twin formation is the reduction of the elastic energy resulting from the clamping. On the other hand, the formation of new twin interfaces increases the twin-boundary energy. The competition of these effects leads to an energetic balance with a (relative) minimum of the overall energy of the sample. The process of twin formation does not occur sharply at the transition temperature Tc but continues over a considerable temperature range below Tc. The `ideal' state of lowest energy is hardly ever reached due to the rigidity of the original grain structure (which remains rather unchanged) and to the existence of kinetic (coercive) barriers.

The group of materials for which these effects are most typical are the ferroelectric and ferroelastic perovskites, in particular BaTiO3. A detailed study is provided by Arlt (1990[link]), who also presents extensive model calculations of relevant energy terms, as well as of average domain sizes and widths of twin bands.

Twinning phenomena in polycrystalline metals are treated by Christian (1965[link], Chapter 8).

Note. As mentioned above in Section 3.3.10.7.1[link], non-ferroelastic transitions phase transitions cause no spontaneous lattice strain and, hence, the associated merohedral twins cannot act as `stress relief' for a `clamped' twin texture.

3.3.10.8. Twinning dislocations

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In contrast to (low-angle) grain boundaries, twin boundaries do not require the existence of boundary dislocations as necessary constituents. Nevertheless, a special kind of dislocation, called `twinning dislocation', has been introduced in materials science for twin boundaries of deformation twins, i.e. for twins with a large shear angle [2\varepsilon] and with a twin boundary parallel to a rational plane (hkl) which is simultaneously the twin reflection plane (Read, 1953[link], p. 109; Friedel, 1964[link], p. 173). Geometrically, a twinning dislocation is a step in the twin boundary (Fig. 3.3.10.15[link]), i.e. a line along which the twin interface `jumps' from one lattice plane to the next. As shown in Fig. 3.3.10.15[link], this `dislocation line', which is located at the twin interface, is surrounded by lattice distortions, similar to the deformations around regular dislocations in an untwinned crystal.

[Figure 3.3.10.15]

Figure 3.3.10.15 | top | pdf |

Definition of the Burgers vector b of a twinning dislocation TD (i.e. step of twin boundary TB). (a) Closed Burgers circuit ([{\rm A} = {\rm A}']) encircling the twinning dislocation TD. (b) Analogous circuit in a reference crystal without dislocation (after Friedel, 1964[link], p. 140). The Burgers vector b is defined as the closure error [{\rm A}'\longrightarrow {\rm A}] of the reference circuit.

Using the concept of a Burgers circuit for regular (perfect) dislocations, Burgers vectors [{\bf b}_t] of twinning dislocations can also be defined. Such a Burgers vector is parallel to the (rational) direction of shear (i.e. parallel to the intersection of the shear plane and the twin plane, as shown in Fig. 3.3.10.15[link]). Its modulus is proportional to [\tan \varepsilon] and has, in general, non-integer values. For small or zero values of the shear angle [2\varepsilon] (pseudo-merohedral and merohedral twins) the Burgers vectors are small or zero, and the related `twinning dislocations' are not physically meaningful. Note that in this approach steps in twin interfaces of (strictly) merohedral twins are not dislocations at all, because [\varepsilon = 0] and [{\bf b}_t = 0].

For classical deformation twins, the shear angles [2\varepsilon] are large, and `twinning dislocations' are well defined and have a significant influence on the deformation behaviour and on the shape of twin domains. Twin interfaces exactly parallel to the twin reflection plane are dislocation-free, whereas interfaces inclined to the reflection plane consist of segments parallel to the reflection plane separated by steps (i.e. twinning dislocations). For small inclinations, the twinning dislocations are widely spaced, whereas for curved interfaces their spacing varies. This feature plays an important role for lenticular domains (needle domains) of deformation twins. Twinning dislocations are also essential for the `coherence' and `incoherence' of twin boundaries as used in materials science. This aspect will be discussed in Section 3.3.10.9.[link]

An interesting study of twinning dislocations in deformation twins of calcite by means of X-ray topography has been carried out by Sauvage & Authier (1965[link]), Authier & Sauvage (1966[link]) and Sauvage (1968[link]).

Twinning dislocations can interact with regular dislocations. Example: a twinning dislocation (i.e. the step between neighbouring interface planes) ending in an untwinned crystal. The end point must be a `triple node' of three dislocations, viz of the twinning dislocation with Burgers vector [{\bf b}_t], of a regular dislocation in twin partner 1 with Burgers vector [{\bf b}_1] and of a regular dislocation in twin partner 2 with Burgers vector [{\bf b}_2]. The three vectors obey Frank's conservation law of Burgers vectors at dislocation nodes: [{\bf b}_t + {\bf b}_1 + {\bf b}_2 = {\bf 0}.]

This reaction of dislocations is of great importance for the plastic deformation of crystals by twinning.

For more detailed information on twinning dislocations, reference is made to Read (1953[link], p. 109), Chalmers (1959[link], pp. 125 and 158), Friedel (1964[link], p. 173), Weertman & Weertman (1964[link], pp. 141–144), and the literature quoted therein.

In conclusion, it is pointed out that twinning dislocations may also occur in non-merohedral growth and transformation twins, but very little has been published on this topic so far. For transformation twins, however, twinning dislocations (in contrast to regular dislocations) as a rule are not physically meaningful because of the usually strong pseudosymmetry (i.e. small values of the shear angle [2\varepsilon]) of these twins. Nevertheless, twinning dislocations allow the formation of the tapering twin walls of needle twins as described in Section 3.3.10.7.3[link].

3.3.10.9. Coherent and incoherent twin interfaces

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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[link], 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[link], 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[link], p. 212); Chalmers (1959[link], p. 125); Van Bueren (1961[link], pp. 251 and 450), Friedel (1964[link], p. 173); Klassen-Neklyudova (1964[link], p. 156); Kelly & Groves (1970[link], p. 308).

  • (2) Christian (1965[link], 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[link], p. 493) and Sutton & Balluffi (1995[link], 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[link], Chapter 3.4).

  • (3) Putnis (1992[link], 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[link], 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[link], 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.[link]

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[link] and 3.3.10.3[link]) and on the notion of twin displacement vector (Section 3.3.10.4[link]).

  • (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[link]), 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[link], 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 [[j] = 1]) 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[link]). The same holds for merohedral twins of lattice index [[j]> 1] (Section 3.3.10.2.4[link]). 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 [3m] 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)[link] 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[link]). Several powerful computer programs for the solution of these problems exist. For a case study, see Herbst-Irmer & Sheldrick (1998[link]).

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