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

International Tables for Crystallography (2006). Vol. D, ch. 3.3, pp. 444-445

Section 3.3.11. Glossary

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:

3.3.11. Glossary

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(hkl) crystal face, lattice plane, net plane (Miller indices)
{hkl} crystal form, set of symmetrically equivalent lattice (net) planes
[uvw] zone axis, crystal edge, lattice direction, lattice row (direction indices)
[\langle uvw\rangle] set of symmetrically equivalent lattice directions (rows)
[{\cal G}] symmetry group of the (real or hypothetical) `parent structure' or high-symmetry modification or `prototype phase' of a crystal; group in general
[{\cal H}] eigensymmetry group of an (untwinned) crystal; symmetry group of the deformed (`daughter') phase of a crystal; subgroup
[{\cal H}_1], [{\cal H}_2], [\ldots], [{\cal H}_j] oriented eigensymmetries of domain states 1, 2, [\ldots], j
[{\cal H}_{1,2}^*], [{\cal H}^*] intersection symmetry group of the pair of oriented eigensymmetries [{\cal H}_1] and [{\cal H}_2], reduced eigensymmetry of a domain
[{\cal K}] composite symmetry group of a twinned crystal (domain pair); twin symmetry
[{\cal K}_{1,2}^*], [ {\cal K}^*] reduced composite symmetry of the domain pair (1, 2)
[{\cal K}(n)] extended composite symmetry of a twinned crystal with a pseudo n-fold twin axis
k, [k_1], [k_2], [\ldots], [k_i] twin operations ([k_1] = identity)
[2^{\prime}], [m^{\prime}], [{\bar 1}{^{\prime}}], [4'(2)], [6'(3)], [{\bar 3}{^\prime}(3)], [{\bar 6}{^\prime}(3)] twin operations of order two in colour-changing (black–white) symmetry notation
[\vert {\cal G}\vert], [\vert {\cal H}\vert], [\vert {\cal K}\vert] order of group [{\cal G}], [{\cal H}], [{\cal K}]
[i] index of [{\cal H}] in [{\cal G}], or of [{\cal H}] in [{\cal K}]
[j], [\Sigma] index of coincidence-site lattice (twin lattice, sublattice) with respect to crystal lattice
[\omega] twin obliquity
[{\bf b}_t] Burgers vector of twinning dislocations
f fault vector of a merohedral twin boundary
t twin displacement vector
[{\cal G}F{\cal H}] Aizu (1970a[link]) symbol of a ferroic phase transition (ferroic species); F = ferroic
W, [W^{\prime}] designation of non-merohedral ferroelastic twin boundaries (according to Sapriel, 1975[link])
[F_{hkl}] structure factor of reflection hkl
[{\bf g}_{hkl}] diffraction vector (reciprocal-lattice vector) of reflection hkl
[\varphi_{hkl}] phase angle of structure factor [F_{hkl}]
[\Psi_{hkl}], [\Phi_{hkl}] difference of phase angles (`phase jump') across twin boundary
[\rho] charge density of a ferroelectric twin boundary
P spontaneous polarization


Aizu, K. (1970a). Possible species of ferromagnetic, ferroelectric and ferroelastic crystals. Phys. Rev. B, 2, 754–772.
Barrett, C. S. & Massalski, T. B. (1966). Structure of metals, 3rd edition, especially pp. 406–414. New York: McGraw-Hill.
Baumhauer, H. (1879). Über künstliche Kalkspath-Zwillinge nach -1/2R. Z. Kristallogr. 3, 588–591.
Bragg, W. L. (1924). The structure of aragonite. Proc. R. Soc. London Ser. A, 105, 16–39.
Ellner, M. (1995). Polymorphic phase transformation of Fe4Al13 causing multiple twinning with decagonal pseudo-symmetry. Acta Cryst. B51, 31–36.
Ellner, M. & Burkhardt, U. (1993). Zur Bildung von Drehmehrlingen mit pentagonaler Pseudosymmetrie beim Erstarrungsvorgang des Fe4Al13. J. Alloy. Compd. 198, 91–100.
Engel, G., Klapper, H., Krempl, P. & Mang, H. (1989). Growth twinning in quartz-homeotypic gallium orthophosphate crystals. J. Cryst. Growth, 94, 597–606.
Ernst, F., Finnis, M. W., Koch, A., Schmidt, C., Straumal, B. & Gust, W. (1996). Structure and energy of twin boundaries in copper. Z. Metallkd. 87, 911–922.
Friedel, J. (1964). Dislocations, especially ch. 6. Oxford: Pergamon.
Frondel, C. (1962). The system of mineralogy, 7th edition, Vol. III. Silica minerals, especially pp. 75–99. New York: Wiley.
Heide, F. (1928). Die Japaner-Zwillinge des Quarzes und ihr Auftreten im Quarzporphyr von Saubach i. V. Z. Kristallogr. 66, 239–281.
Hofmeister, H. (1998). Forty years study of fivefold twinned structures in small particles and thin films. Cryst. Res. Technol. 33, 3–25, especially Section 4.
Hofmeister, H. & Junghans, T. (1993). Multiple twinning in the solid phase. Crystallisation of amorphous germanium. Mater. Sci. Forum, 113–115, 631–636.
Hornstra, J. (1959). Models of grain boundaries in the diamond lattice I. Physica, 25, 409–422.
Hornstra, J. (1960). Models of grain boundaries in the diamond lattice II. Physica, 26, 198–208.
Klassen-Neklyudova, M. V. (1964). Mechanical twinning of crystals. New York: Consultants Bureau.
Koch, E. (2004). Twinning. In International tables for crystallography, Vol. C. Mathematical, physical and chemical tables, edited by E. Prince, 3rd ed., ch. 1.3. Dordrecht: Kluwer Academic Publishers.
Kohn, J. A. (1956). Twinning in diamond-type structures: high-order twinning in silicon. Am. Mineral. 41, 778–784.
Kohn, J. A. (1958). Twinning in diamond-type structures: a proposed boundary-structure model. Am. Mineral. 43, 263–284.
Le Page, Y. (1999). Low obliquity in pseudo-symmetry of lattices and structures, and in twinning by pseudo-merohedry. Acta Cryst. A55, Supplement. Abstract M12.CC001.
Le Page, Y. (2002). Mallard's law recast as a Diophantine system: fast and complete enumeration of possible twin laws by [reticular] [pseudo] merohedry. J. Appl. Cryst. 35, 175–181.
Liebisch, Th. (1891). Physikalische Kristallographie. Leipzig: Veit & Comp.
Mügge, O. (1883). Beiträge zur Kenntnis der Structurflächen des Kalkspathes. Neues Jahrb. Mineral. 81, 32–54.
Niggli, P. (1920/1924/1941). Lehrbuch der Mineralogie und Kristallchemie, 1st edition 1920, 2nd edition 1924, 3rd edition, Part I, 1941, especially pp. 136–153, 401–414. Berlin: Gebrüder Borntraeger.
Penn, R. L. & Banfield, J. F. (1998). Oriented attachment and growth, twinning, polytypism, and formation of metastable phases: insights from nano-crystalline TiO2. Am. Mineral. 83, 1077–1082.
Phillips, F. C. (1971). An introduction to crystallography, 4th ed. London: Longman.
Queisser, H. J. (1963). Properties of twin boundaries in silicon. J. Electrochem. Soc. 110, 52–56.
Ramdohr, P. & Strunz, H. (1967). Klockmann's Lehrbuch der Mineralogie, 15th edition, especially p. 512. Stuttgart: Enke.
Rečnik, A., Brulay, J., Mader, W., Kolar, D. & Rühle, M. (1994). Structural and spectroscopic investigation of the (111) twins in barium titanite. Philos. Mag. B, 70, 1021–1034.
Sapriel, J. (1975). Domain-wall orientations in ferroelastics. Phys. Rev. B, 12, 5128–5140.
Scherf, Ch., Hahn, Th., Heger, G., Becker, R. A., Wunderlich, W. & Klapper, H. (1997). Optical and synchrotron radiation white-beam topographic investigation during the high-temperature phase transition of KLiSO4. Ferroelectrics, 191, 171–177.
Seifert, H. (1928). Über Schiebungen am Bleiglanz. Neues Jahrb. Mineral. Geol. Palaeontol. 57, Beilage-Band, Abteilung A, Mineralogie und Petrographie, pp. 665–742.
Shtukenberg, A. G., Punin, Yu. O., Haegele, E. & Klapper, H. (2001). On the origin of inhomogeneity of anomalous birefringence in mixed crystals: an example of alums. Phys. Chem. Miner. 28, 665–674.
Tschermak, G. & Becke, F. (1915). Lehrbuch der Mineralogie, 7th edition, pp. 93–114. Wien: Alfred Hölder.

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