International Tables for Crystallography (2013). Vol. D. ch. 3.3, pp. 413-487
https://doi.org/10.1107/97809553602060000917

Chapter 3.3. Twinning of crystals

Contents

3.3. Twinning of crystals (pp. 413-487) | html | pdf | chapter contents |
Th. Hahn and H. Klapper
- 3.3.1. Crystal aggregates and intergrowths (pp. 413-414) | html | pdf |
- 3.3.2. Basic concepts and definitions of twinning (pp. 414-418) | html | pdf |
  - 3.3.2.1. Definition of a twin (p. 415) | html | pdf |
  - 3.3.2.2. Essential addenda to the definition (p. 415) | html | pdf |
  - 3.3.2.3. Specifications and extensions of the orientation relations (p. 417) | html | pdf |
    - 3.3.2.3.1. Binary twin operations (twin elements) (pp. 415-417) | html | pdf |
    - 3.3.2.3.2. Pseudo n-fold twin rotations (twin axes) with $[n\ge 3]$ (pp. 416-417) | html | pdf |
  - 3.3.2.4. Notes on the definition of twinning (pp. 417-418) | html | pdf |
- 3.3.3. Morphological classification, simple and multiple twinning (pp. 418-419) | html | pdf |
  - 3.3.3.1. Morphological classification (p. 419) | html | pdf |
- 3.3.4. Composite symmetry and the twin law (pp. 419-423) | html | pdf |
  - 3.3.4.1. Composite symmetry (pp. 419-420) | html | pdf |
  - 3.3.4.2. Equivalent twin laws (p. 421) | html | pdf |
  - 3.3.4.3. Classification of composite symmetries (pp. 421-422) | html | pdf |
  - 3.3.4.4. Categories of composite symmetries (pp. 422-423) | html | pdf |
- 3.3.5. Description of the twin law by black–white symmetry (p. 423) | html | pdf |
- 3.3.6. Examples of twinned crystals (pp. 423-436) | html | pdf |
  - 3.3.6.1. Macroscopic identification of twins and of twin laws (p. 423) | html | pdf |
  - 3.3.6.2. Inversion twins in orthorhombic crystals (pp. 425-426) | html | pdf |
  - 3.3.6.3. Twinning of gypsum, CaSO₄·2H₂O (pp. 426-427) | html | pdf |
  - 3.3.6.4. Twinning of low-temperature quartz (α-quartz, SiO₂) (pp. 427-428) | html | pdf |
    - 3.3.6.4.1. Dauphiné twins (p. 427) | html | pdf |
    - 3.3.6.4.2. Brazil twins (pp. 427-428) | html | pdf |
    - 3.3.6.4.3. Combined Dauphiné–Brazil (Leydolt, Liebisch) twins (p. 428) | html | pdf |
    - 3.3.6.4.4. Japanese twins (or La Gardette twins) (p. 428) | html | pdf |
  - 3.3.6.5. Twinning of high-temperature quartz (β-quartz) (pp. 428-429) | html | pdf |
  - 3.3.6.6. Twinning of rhombohedral crystals (p. 429) | html | pdf |
  - 3.3.6.7. Spinel twins (pp. 429-431) | html | pdf |
  - 3.3.6.8. Growth and transformation twins of K₂SO₄ (p. 431) | html | pdf |
  - 3.3.6.9. Pentagonal–decagonal twins (pp. 431-432) | html | pdf |
  - 3.3.6.10. Multiple twins of rutile, TiO₂ (pp. 432-433) | html | pdf |
  - 3.3.6.11. Variety of twinning in gibbsite, Al(OH)₃ (p. 433) | html | pdf |
  - 3.3.6.12. Plagioclase twins (pp. 433-434) | html | pdf |
  - 3.3.6.13. Staurolite (pp. 434-435) | html | pdf |
  - 3.3.6.14. BaTiO₃ transformation twins (p. 435) | html | pdf |
  - 3.3.6.15. Merohedral growth twinning of pentaerythrite (p. 435) | html | pdf |
  - 3.3.6.16. Twins of twins (pp. 435-436) | html | pdf |
- 3.3.7. Genetic classification of twins (pp. 436-440) | html | pdf |
  - 3.3.7.1. Growth twinning (pp. 436-438) | html | pdf |
    - 3.3.7.1.1. Twinning by nucleation (pp. 436-437) | html | pdf |
    - 3.3.7.1.2. Twinning during crystal growth (pp. 437-438) | html | pdf |
  - 3.3.7.2. Transformation twinning (pp. 438-439) | html | pdf |
  - 3.3.7.3. Mechanical twinning (pp. 439-440) | html | pdf |
- 3.3.8. Lattice aspects of twinning (pp. 440-446) | html | pdf |
  - 3.3.8.1. Basic concepts of Friedel's lattice theory (p. 441) | html | pdf |
  - 3.3.8.2. Lattice coincidences, twin lattice, twin lattice index (pp. 441-442) | html | pdf |
  - 3.3.8.3. Twins with three-dimensional twin lattices (`triperiodic' twins) (pp. 442-443) | html | pdf |
  - 3.3.8.4. Approximate (pseudo-)coincidences of two or more lattices (p. 444) | html | pdf |
  - 3.3.8.5. Twin obliquity and lattice pseudosymmetry (pp. 444-446) | html | pdf |
  - 3.3.8.6. Twinning of isostructural crystals (p. 446) | html | pdf |
  - 3.3.8.7. Conclusions (p. 446) | html | pdf |
- 3.3.9. Twinning by merohedry and pseudo-merohedry (pp. 446-450) | html | pdf |
  - 3.3.9.1. Definitions of merohedry (p. 447) | html | pdf |
  - 3.3.9.2. Types of twins by merohedry and pseudo-merohedry (pp. 447-450) | html | pdf |
    - 3.3.9.2.1. Merohedral twins of lattice index (p. 447) | html | pdf |
    - 3.3.9.2.2. Pseudo-merohedral twins of lattice index (pp. 447-448) | html | pdf |
    - 3.3.9.2.3. Twinning with partial lattice coincidence (lattice index ) (p. 448) | html | pdf |
    - 3.3.9.2.4. Twinning with partial lattice pseudo-coincidence (lattice index ) (pp. 448-450) | html | pdf |
  - 3.3.9.3. Pseudo-merohedry and ferroelasticity (p. 450) | html | pdf |
- 3.3.10. Twin boundaries (pp. 450-469) | html | pdf |
  - 3.3.10.1. Contact relations in twinning (pp. 450-451) | html | pdf |
  - 3.3.10.2. Strain compatibility of interfaces (pp. 451-455) | html | pdf |
    - 3.3.10.2.1. Sapriel approach to permissible (compatible) boundaries in ferroelastic (non-merohedral) transformation twins (p. 452) | html | pdf |
    - 3.3.10.2.2. Extension to non-merohedral growth and mechanical twins (pp. 452-454) | html | pdf |
    - 3.3.10.2.3. Permissible boundaries in merohedral twins (lattice index [j] = 1) (p. 454) | html | pdf |
    - 3.3.10.2.4. Permissible twin boundaries in twins with lattice index $[[j]\, \gt\, 1]$ (pp. 454-455) | html | pdf |
  - 3.3.10.3. Electrical constraints of twin interfaces (pp. 455-456) | html | pdf |
    - 3.3.10.3.1. Merohedral twins (pp. 455-456) | html | pdf |
    - 3.3.10.3.2. Non-merohedral twins (p. 456) | html | pdf |
    - 3.3.10.3.3. Non-pyroelectric noncentrosymmetric crystals (p. 456) | html | pdf |
  - 3.3.10.4. Displacement and fault vectors of twin boundaries (pp. 456-459) | html | pdf |
    - 3.3.10.4.1. Twin displacement vector $[{\bf t}]$ (pp. 457-458) | html | pdf |
    - 3.3.10.4.2. Fault vectors of twin boundaries in merohedral twins (p. 458) | html | pdf |
    - 3.3.10.4.3. Examples of fault-vector determinations (pp. 458-459) | html | pdf |
  - 3.3.10.5. Examples of structural models of twin boundaries (pp. 459-461) | html | pdf |
    - 3.3.10.5.1. Aragonite, CaCO₃ (p. 459) | html | pdf |
    - 3.3.10.5.2. Dauphiné twins of $[\alpha]$ -quartz (pp. 459-460) | html | pdf |
    - 3.3.10.5.3. Potassium lithium sulfate, KLiSO₄ (pp. 460-461) | html | pdf |
    - 3.3.10.5.4. Twin models of molecular crystals (p. 461) | html | pdf |
  - 3.3.10.6. Observations of twin boundaries by transmission electron microscopy (pp. 461-464) | html | pdf |
    - 3.3.10.6.1. Anatase, TiO₂ (Penn & Banfield, 1998, 1999) (pp. 461-462) | html | pdf |
    - 3.3.10.6.2. Rutile, TiO₂ (p. 462) | html | pdf |
    - 3.3.10.6.3. Cassiterite, SnO₂ (rutile structure) (p. 462) | html | pdf |
    - 3.3.10.6.4. Σ3 (111) twin interface in BaTiO₃ [cf. Section 3.3.8.3(iii)] (pp. 462-463) | html | pdf |
    - 3.3.10.6.5. Σ = 3 bicrystal boundaries in Cu and Ag (p. 463) | html | pdf |
    - 3.3.10.6.6. Fivefold cyclic twins in nanocrystalline materials (pp. 463-464) | html | pdf |
  - 3.3.10.7. Twin textures (pp. 464-467) | html | pdf |
    - 3.3.10.7.1. Merohedral (non-ferroelastic) twins (see Sections 3.3.9 and 3.3.10.2.3) (p. 464) | html | pdf |
    - 3.3.10.7.2. Non-merohedral (ferroelastic) twins (p. 464) | html | pdf |
    - 3.3.10.7.3. Fitting problems of ferroelastic twins (pp. 464-466) | html | pdf |
    - 3.3.10.7.4. Tweed microstructures (pp. 466-467) | html | pdf |
    - 3.3.10.7.5. Twin textures in polycrystalline aggregates (p. 467) | html | pdf |
  - 3.3.10.8. Twinning dislocations (pp. 467-468) | html | pdf |
  - 3.3.10.9. Coherent and incoherent twin interfaces (pp. 468-469) | html | pdf |
- 3.3.11. Effect of twinning in reciprocal space (pp. 469-477) | html | pdf |
  - 3.3.11.1. Basic features of twin diffraction records (pp. 469-470) | html | pdf |
  - 3.3.11.2. General (non-merohedral, inclined-lattice) twins (p. 470) | html | pdf |
  - 3.3.11.3. Twinning by (strict) merohedry (parallel-lattice twins, Σ1 merohedral twins) (pp. 470-471) | html | pdf |
  - 3.3.11.4. Twinning by reticular merohedry (partially-parallel-lattice twins, Σ > 1 merohedral twins) (pp. 471-474) | html | pdf |
    - 3.3.11.4.1. General survey (pp. 471-472) | html | pdf |
    - 3.3.11.4.2. The four Σm merohedral twin families (pp. 472-474) | html | pdf |
  - 3.3.11.5. Pseudo-merohedral twins (pp. 474-477) | html | pdf |
    - 3.3.11.5.1. General remarks (pp. 474-475) | html | pdf |
    - 3.3.11.5.2. Example: pseudohexagonal (cyclic) twins of orthorhombic crystals (pseudo-coincident Σ3 twins) (pp. 475-477) | html | pdf |
  - 3.3.11.6. Programs for structure determinations with twinned crystals (p. 477) | html | pdf |
- 3.3.12. Domain structures (by V. Janovec) (p. 477) | html | pdf |
- 3.3.13. Glossary (pp. 477-478) | html | pdf |
- References | html | pdf |
- Figures
  - Fig. 3.3.1.1. Parallel intergrowth (a) of spinel octahedra and (b) of hexagonal quartz prisms (p. 413) | html | pdf |
  - Fig. 3.3.1.2. (a) Optical anomaly of a cubic mixed (K,NH₄)-alum crystal grown from aqueous solution, as revealed by polarized light between crossed polarizers (p. 413) | html | pdf |
  - Fig. 3.3.2.1. Schematic illustration of the orientation relations of noncentrosymmetric triclinic twin partners, see Section 3.3.2.3.1 (p. 416) | html | pdf |
  - Fig. 3.3.2.2. As Fig. 3.3.2.1, (a) for twin element (iii) `rational twofold twin axis' and (b) for twin element (iv) `irrational twin mirror plane' (see text); common lattice row [uvw] for both cases (p. 416) | html | pdf |
  - Fig. 3.3.2.3. Illustration of the Kantennormalengesetz (complex twin) for twin elements (v) and (vi) (see text); common lattice row [uvw] for both cases (p. 416) | html | pdf |
  - Fig. 3.3.2.4. (a) Triple growth twin of orthorhombic aragonite, CaCO₃, with pseudo-threefold twin axis (p. 417) | html | pdf |
  - Fig. 3.3.3.1. Schematic illustration of simple (polysynthetic) and multiple (cyclic) twins (p. 418) | html | pdf |
  - Fig. 3.3.4.1. Gypsum dovetail twin: schematic illustration of the coset of alternative twin operations (p. 420) | html | pdf |
  - Fig. 3.3.4.2. Twinning of an orthorhombic crystal (eigensymmetry 2/m 2/m 2/m) with equivalent twin mirror planes (110) and $[({\bar 1}10)]$ (p. 421) | html | pdf |
  - Fig. 3.3.6.1. X-ray topographs (Mo Kα radiation) of a $[(11 \bar 2 0)]$ plate (about 12 × 12 mm, about 1 mm thick) cut from a hexagonal KLiSO₄ (phase III) crystal (point group 6; cf (p. 426) | html | pdf |
  - Fig. 3.3.6.2. X-ray transmission topograph of a (110) crystal plate (2.4 mm thick, width about 15 mm) cut from a very perfect crystal of polar lithium formate monohydrate, HCOOLi·H₂O, point group mm2, grown from aqueous solution (reflection 111, Mo Kα radiation) (p. 426) | html | pdf |
  - Fig. 3.3.6.3. Dovetail twin (a) and Montmartre twin (b) of gypsum (p. 427) | html | pdf |
  - Fig. 3.3.6.4. Distinction of the four different domain states generated by the three merohedral twin laws of low-quartz and of quartz homeotypes (p. 428) | html | pdf |
  - Fig. 3.3.6.5. The four variants of Japanese twins of quartz (p. 429) | html | pdf |
  - Fig. 3.3.6.6. Twin intergrowth of `obverse' and `reverse' rhombohedra of rhombohedral FeBO₃ (p. 430) | html | pdf |
  - Fig. 3.3.6.7. Contact growth twin of calcite with the same twin law as FeBO₃ in Fig. 3.3.6.6 (p. 430) | html | pdf |
  - Fig. 3.3.6.8. Spinel (111) twins of cubic crystals (two orientation states) (p. 430) | html | pdf |
  - Fig. 3.3.6.9. Pseudo-hexagonal growth twin of K₂SO₄ showing six sector domains in three orientation states (p. 431) | html | pdf |
  - Fig. 3.3.6.10. Pentagonal–decagonal twins (p. 431) | html | pdf |
  - Fig. 3.3.6.11. Various forms of rutile (TiO₂) twins with one or several equivalent twin reflection planes {011} (p. 432) | html | pdf |
  - Fig. 3.3.6.12. Sixfold reflection twin of gibbsite, Al(OH)₃, with equivalent (110) and $[({\bar 1}10]$ ), both as twin mirror and composition planes (p. 433) | html | pdf |
  - Fig. 3.3.6.13. Polysynthetic albite twin aggregate of triclinic feldspar, twin reflection and composition plane (010) (p. 434) | html | pdf |
  - Fig. 3.3.6.14. Pericline twin of triclinic feldspar (p. 434) | html | pdf |
  - Fig. 3.3.6.15. Twinning of staurolite (p. 435) | html | pdf |
  - Fig. 3.3.6.16. The various steps (idealized) of phillipsite growth twins: (a) monoclinic untwinned, (b) orthorhombic, (c) tetragonal, (d) cubic, and (e) `filled-in' pseudo rhomb-dodecahedron (after Ramdohr & Strunz, 1967, p (p. 436) | html | pdf |
  - Fig. 3.3.7.1. Orthoclase (monoclinic K-feldspar) (p. 437) | html | pdf |
  - Fig. 3.3.7.2. Photographs of (001) plates ( $[\approx]$ 20 mm diameter, $[\approx]$ 1 mm thick) of NH₄LiSO₄ between crossed polarizers (p. 437) | html | pdf |
  - Fig. 3.3.7.3. Mechanical twins of calcite, CaCO₃ (p. 439) | html | pdf |
  - Fig. 3.3.8.1. Lattice relations of $[\Sigma 5]$ twins of tetragonal crystals with primitive lattice (p. 443) | html | pdf |
  - Fig. 3.3.8.2. (a) A (110) silicon slice (10 cm diameter, 0.3 mm thick), cut from a Czochralski-grown tricrystal for solar-cell applications (p. 443) | html | pdf |
  - Fig. 3.3.10.1. (a) Classical description of mechanical twinning by homogeneous shear deformation (p. 451) | html | pdf |
  - Fig. 3.3.10.2. Boundaries B–B between 180° domains (merohedral twins) of pyroelectric crystals (p. 456) | html | pdf |
  - Fig. 3.3.10.3. Charged and uncharged boundaries B–B of non-merohedral twins of pseudo-hexagonal NH₄LiSO₄ (p. 456) | html | pdf |
  - Fig. 3.3.10.4. Lattice representation of twin displacement vectors (p. 457) | html | pdf |
  - Fig. 3.3.10.5. HRTEM micrograph of anatase, TiO₂, with a (112) reflection twin boundary (arrows), viewed edge-on along $[[1{\bar 3}1]]$ (p. 458) | html | pdf |
  - Fig. 3.3.10.6. Structural model of the (110) twin boundary of aragonite (after Bragg, 1924), projected along the pseudo-hexagonal c axis (p. 460) | html | pdf |
  - Fig. 3.3.10.7. Simplified structural model of a $[\{10{\bar 1}0\}]$ Dauphiné twin boundary in quartz (p. 460) | html | pdf |
  - Fig. 3.3.10.8. KLiSO₄: Bulk tetrahedral framework structures and models of (0001) twin boundary structures of phases III and IV (p. 461) | html | pdf |
  - Fig. 3.3.10.9. (a) HRTEM micrograph of a coherent (111) twin boundary in BaTiO₃, projected along $[[1{\bar 1}0]]$ (p. 462) | html | pdf |
  - Fig. 3.3.10.10. (a) Schematic block diagram of a $[\Sigma = 3]$ bicrystal (spinel twin) for $[\varphi[{\bar 1}10] = 0^\circ]$ (p. 463) | html | pdf |
  - Fig. 3.3.10.11. HRTEM micrograph of a fivefold-twinned Ge nanocrystal (right) in an amorphous Ge film formed by vapour deposition on an NaCl cleavage plane (p. 464) | html | pdf |
  - Fig. 3.3.10.12. Illustration of space-filling problems of domains for a (ferroelastic) orthorhombic $[\longrightarrow]$ monoclinic phase transition with an angle $[\varepsilon]$ (exaggerated) of spontaneous shear (p. 465) | html | pdf |
  - Fig. 3.3.10.13. Thin section of tetragonal leucite, K(AlSi₂O₆), between crossed polarizers (p. 465) | html | pdf |
  - Fig. 3.3.10.14. Twin textures generated by the two different hexagonal-to-orthorhombic phase transitions of KLiSO₄ (p. 466) | html | pdf |
  - Fig. 3.3.10.15. Definition of the Burgers vector b of a twinning dislocation TD (p. 467) | html | pdf |
  - Fig. 3.3.11.1. Rhombohedral Σ3 (reverse/obverse) twins in direct space, described with hexagonal axes, viewed down the common c axis (p. 472) | html | pdf |
  - Fig. 3.3.11.2. Reciprocal lattice of the rhombohedral Σ3 twins for the layers l = 3n (a), l = 3n + 1 (b), l = 3n + 2 (c), described with hexagonal axes (p. 473) | html | pdf |
  - Fig. 3.3.11.3. Tetragonal lattices (a–b planes, common c axis pointing upwards) of twin domain I (start domain, lattice points small circles, right-handed green unit cell a₁, b₁, c₁), of the Σ5 twin-related domain II (small crosses, left-handed blue unit cell a₂, b₂, c₂) and the Σ5 coincidence lattice (large black points, right-handed red unit cell a_T, b_T, c_T) (p. 473) | html | pdf |
  - Fig. 3.3.11.4. Reciprocal tetragonal lattices (hk0 lattice planes) of twin domain I (start domain, lattice points small circles) and of the Σ5 twin-related domain II (small crosses) (p. 473) | html | pdf |
  - Fig. 3.3.11.5. Hexagonal lattices (a–b planes, common c axis pointing upwards) of twin domain I (start domain, lattice points small circles, right-handed green unit cell a₁, b₁, c₁), of the Σ7 twin-related domain II (small crosses, left-handed blue unit cell a₂, b₂, c₂) and of the Σ7 coincidence lattice (large black points, right-handed red unit cell a_T, b_T, c_T) (p. 474) | html | pdf |
  - Fig. 3.3.11.6. Reciprocal hexagonal lattices (hk0 lattice planes) of twin domain I (start domain, lattice points small circles) and of the Σ7 twin-related domain II (small crosses) (p. 474) | html | pdf |
  - Fig. 3.3.11.7. (a) Morphologically idealized pseudo-hexagonal threefold sector twin of an orthorhombic crystal with b/a = tan 58.5°, generated by the two symmetry-equivalent twin mirror planes (110) and $[(1 \bar 1 0)]$ (p. 475) | html | pdf |
  - Fig. 3.3.11.8. (a) (001) plate of ammonium lithium sulfate, NH₄LiSO₄, (about 11 mm diameter, 0.8 mm thick) between crossed polarizers, exhibiting sectorial pseudo-hexagonal growth twinning (p. 476) | html | pdf |
- Tables
  - Table 3.3.4.1. Gypsum, dovetail twins: coset of alternative twin operations (twin law), given in orthorhombic axes of the composite symmetry $[{\cal K}_D]$ (p. 420) | html | pdf |
  - Table 3.3.4.2. Reduced composite symmetries $[{\cal K}^*(1,2)={\cal H}^*\cup k_2\times{\cal H}^*]$ and $[{\cal K}^*(1,3)={\cal H}^*\cup k_3 \times {\cal H}^*]$ for the orthorhombic example in Fig. 3.3.4.2 (p. 422) | html | pdf |
  - Table 3.3.6.1. Types of X-ray reflections generating (`yes') or not generating (`no') X-ray topographic domain contrast (yes/no) for the $[6\rightarrow 6mm]$ growth twins of KLiSO₄ (p. 425) | html | pdf |
  - Table 3.3.6.2. Orthorhombic inversion twins: coset of alternative twin operations (twin law) (p. 426) | html | pdf |
  - Table 3.3.6.3. Gypsum: cosets of alternative twin operations of the dovetail and the Montmartre twins, referred to their specific orthorhombic axes (subscripts D and M) (p. 427) | html | pdf |
  - Table 3.3.6.4. Dauphiné twins of α-quartz: coset of alternative twin operations (twin law) (p. 428) | html | pdf |
  - Table 3.3.6.5. The four different variants of Japanese twins according to Frondel (1962) (p. 429) | html | pdf |
  - Table 3.3.6.6. Plagioclase: albite and pericline twins (p. 434) | html | pdf |
  - Table 3.3.8.1. Lattice planes (hkl) and lattice rows [uvw] that are mutually perpendicular (after Koch, 2004) (p. 441) | html | pdf |
  - Table 3.3.8.2. Examples of calculated obliquities $[\omega]$ and lattice indices [j] for selected (hkl)[uvw] combinations and their relation to twinning (p. 445) | html | pdf |
  - Table 3.3.9.1. Staurolite, 60° and 90° twins (p. 449) | html | pdf |
  - Table 3.3.10.1. Examples of permissible twin boundaries for higher-order merohedral twins ([j]> 1) (p. 455) | html | pdf |
  - Table 3.3.11.1. Relative frequencies of the four coincidence cases (i)–(iv) for the general Σm twins and the specific twins Σ3, Σ5 and Σ7 treated in this chapter. (p. 472) | html | pdf |