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

Chapter 3.3. Twinning of crystals

Contents

  • 3.3. Twinning of crystals  (pp. 413-487) | html | pdf | chapter contents |
    • 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, CaSO 4 ·2H 2 O  (pp. 426-427) | html | pdf |
      • 3.3.6.4. Twinning of low-temperature quartz (α-quartz, SiO 2 )  (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 K 2 SO 4   (p. 431) | html | pdf |
      • 3.3.6.9. Pentagonal–decagonal twins  (pp. 431-432) | html | pdf |
      • 3.3.6.10. Multiple twins of rutile, TiO 2   (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. BaTiO 3 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 [[j] = 1]  (p. 447) | html | pdf |
        • 3.3.9.2.2. Pseudo-merohedral twins of lattice index [[j] = 1]  (pp. 447-448) | html | pdf |
        • 3.3.9.2.3. Twinning with partial lattice coincidence (lattice index [[j]> 1])  (p. 448) | html | pdf |
        • 3.3.9.2.4. Twinning with partial lattice pseudo-coincidence (lattice index [[j]>1])  (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, CaCO 3   (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, KLiSO 4   (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, TiO 2 (Penn & Banfield, 1998, 1999)  (pp. 461-462) | html | pdf |
        • 3.3.10.6.2. Rutile, TiO 2   (p. 462) | html | pdf |
        • 3.3.10.6.3. Cassiterite, SnO 2 (rutile structure)  (p. 462) | html | pdf |
        • 3.3.10.6.4. Σ3 (111) twin interface in BaTiO 3 [ 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 Σ m 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,NH 4 )-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, CaCO 3 , 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 K α radiation) of a [(11 \bar 2 0)] plate (about 12 × 12 mm, about 1 mm thick) cut from a hexagonal KLiSO 4 (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·H 2 O, point group mm 2, grown from aqueous solution (reflection 111, Mo K α 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 FeBO 3   (p. 430) | html | pdf |
      • Fig. 3.3.6.7. Contact growth twin of calcite with the same twin law as FeBO 3 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 K 2 SO 4 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 (TiO 2 ) 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 NH 4 LiSO 4 between crossed polarizers  (p. 437) | html | pdf |
      • Fig. 3.3.7.3. Mechanical twins of calcite, CaCO 3   (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 NH 4 LiSO 4   (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, TiO 2 , 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. KLiSO 4 : 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 BaTiO 3 , 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(AlSi 2 O 6 ), 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 KLiSO 4   (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 = 3 n ( a ), l = 3 n + 1 ( b ), l = 3 n + 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 a 1 , b 1 , c 1 ), of the Σ5 twin-related domain II (small crosses, left-handed blue unit cell a 2 , b 2 , c 2 ) 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 ( hk 0 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 a 1 , b 1 , c 1 ), of the Σ7 twin-related domain II (small crosses, left-handed blue unit cell a 2 , b 2 , c 2 ) 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 ( hk 0 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, NH 4 LiSO 4 , (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 KLiSO 4   (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 Σ m twins and the specific twins Σ3, Σ5 and Σ7 treated in this chapter.   (p. 472) | html | pdf |