Tables for
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

International Tables for Crystallography (2006). Vol. A1, ch. 1.1, pp. 2-5   | 1 | 2 |
doi: 10.1107/97809553602060000537

Chapter 1.1. Historical introduction

Mois I. Aroyo,a* Ulrich Müllerb and Hans Wondratschekc

aDepartamento de Física de la Materia Condensada, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain,bFachbereich Chemie, Philipps-Universität, D-35032 Marburg, Germany, and cInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail:


Aroyo, M. I. & Perez-Mato, J. M. (1998). Symmetry-mode analysis of displacive phase transitions using International Tables for Crystallography. Acta Cryst. A54, 19–30.
Ascher, E. (1966). Role of particular maximal subgroups in continuous phase transitions. Phys. Lett. 20, 352–354.
Ascher, E. (1967). Symmetry changes in continuous transitions. A simplified theory applied to V3Si. Chem. Phys. Lett. 1, 69–72.
Ascher, E. (1968). Lattices of equi-translation subgroups of the space groups. Geneva: Battelle Institute.
Ascher, E., Gramlich, V. & Wondratschek, H. (1969). Corrections to the sections `Untergruppen' of the space groups in Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935) Vol. I. Acta Cryst. B25, 2154–2156.
Astbury, W. T. & Yardley, K. (1924). Tabulated data for the examination of the 230 space-groups by homogeneous X-rays. Philos. Trans. R. Soc. London, 224, 221–257.
Barlow, W. (1894). Über die geometrischen Eigenschaften homogener starrer Strukturen. Z. Kristallogr. Mineral. 23, 1–63.
Bärnighausen, H. (1980). Group–subgroup relations between space groups: a useful tool in crystal chemistry. MATCH Comm. Math. Chem. 9, 139–175.
Baur, W. H. & Kassner, D. (1992). The perils of Cc: comparing the frequencies of falsely assigned space groups with their general population. Acta Cryst. B48, 356–369.
Bertaut, E. F. & Billiet, Y. (1979). On equivalent subgroups and supergroups of the space groups. Acta Cryst. A35, 733–745.
Billiet, Y. (1973). Les sous-groupes isosymboliques des groupes spatiaux. Bull. Soc. Fr. Minéral. Cristallogr. 96, 327–334.
Billiet, Y. (1980). The subgroups of finite index of the space groups: determination via conventional coordinate systems. MATCH Comm. Math. Chem. 9, 127–190.
Billiet, Y. & Bertaut, E. F. (2005). Isomorphic subgroups of space groups. International tables for crystallography, Vol. A, Space-group symmetry, edited by Th. Hahn, Part 13. Dordrecht: Kluwer Academic Publishers.
Billiet, Y. & Sayari, A. (1984). Les sous-groupes isomorphes d'un group d'espace de type p4. I. Détermination univoque. Acta Cryst. A40, 624–631.
Biltz, W. (1934). Raumchemie der festen Stoffe. Leipzig: L. Voss.
Birman, J. L. (1966a). Full group and subgroup methods in crystal physics. Phys. Rev. 150, 771–782.
Birman, J. L. (1966b). Simplified theory of symmetry change in second–order phase transitions: application to V3Si. Phys. Rev. Lett. 17, 1216–1219.
Bravais, A. (1850). Mémoire sur les systèmes formés par les points distribués régulièrement sur un plan ou dans l'espace. J. Ecole Polytech. 19, 1–128. (English: Memoir 1, Crystallographic Society of America, 1949.)
Buerger, M. J. (1947). Derivative crystal structures. J. Chem. Phys. 15, 1–16.
Buerger, M. J. (1951). Phase transformations in solids, ch. 6. New York: Wiley.
Burckhardt, J. J. (1988). Die Symmetrie der Kristalle. Basel: Birkhäuser.
Byström, A., Hök, B. & Mason, B. (1941). The crystal structure of zinc metaantimonate and similar compounds. Ark. Kemi Mineral. Geol. 15B, 1–8.
Chapuis, G. C. (1992). Symmetry relationships between crystal structures and their practical applications. Modern perspectives in inorganic chemistry, edited by E. Parthé, pp. 1–16. Dordrecht: Kluwer Academic Publishers.
Cracknell, A. P. (1975). Group theory in solid state physics. New York: Taylor and Francis Ltd. and Pergamon.
Deonarine, S. & Birman, J. L. (1983). Group–subgroup phase transitions, Hermann's space group decomposition theorem, and chain subduction criterion in crystals. Phys. Rev. B, 27, 4261–4273.
Fedorov, E. (1891). Symmetry of regular systems and figures. Zap. Mineral. Obshch. (2), 28, 1–46. (In Russian.) (English: Symmetry of crystals, American Crystallographic Association, 1971.)
Fischer, W. & Koch, E. (1983). On the equivalence of point configurations due to Euclidean normalizers (Cheshire groups) of space groups. Acta Cryst. A39, 907–915.
Goldschmidt, V. M. (1926). Untersuchungen über den Bau und Eigenschaften von Krystallen. Skr. Nor. Vidensk. Akad. Oslo Mat.-Nat. Kl. 1926 No. 2 and 1927 No. 8.
Haag, F. (1887). Die regulären Krystallkörper. Programm des Königlichen Gymnasiums in Rottweil zum Schlusse des Schul­jahres 1886–1887.
Heesch, H. (1930). Zur systematischen Strukturtheorie. II. Z. Kristallogr. 72, 177–201.
Hermann, C. (1929). Zur systematischen Strukturtheorie. IV. Untergruppen. Z. Kristallogr. 69, 533–555.
International Tables for Crystallography (1983). Vol. A, Space-group symmetry, edited by Th. Hahn, 1st ed. Dordrecht: Kluwer Academic Publishers. (Abbreviated IT A.)
International Tables for Crystallography (2002). Vol. A, Space-group symmetry, edited by Th. Hahn, 5th ed. Dordrecht: Kluwer Academic Publishers. (Abbreviated IT A.)
International Tables for X-ray Crystallography (1952, 1965, 1969). Vol. I, Symmetry groups, edited by N. F. M. Henry & K. Lonsdale. Birmingham: Kynoch Press.
Internationale Tabellen zur Bestimmung von Kristallstrukturen (1935). 1. Bd. Edited by C. Hermann. Berlin: Borntraeger. (In German, English and French.) (Abbreviated IT 35.)
Janovec, V. & Přívratská, J. (2003). Domain structures. International tables for crystallography, Vol. D, edited by A. Authier, Physical properties of crystals, ch. 3.4. Dordrecht: Kluwer Academic Publishers.
Koch, E. (1984). The implications of normalizers on group–subgroup relations between space groups. Acta Cryst. A40, 593–600.
Kroumova, E., Aroyo, M. I. & Pérez-Mato, J. M. (2002). Prediction of new displacive ferroelectrics through systematic pseudosymmetry search. Results for materials with Pba2 and [Pmc2_1] symmetry. Acta Cryst. B58, 921–933.
Landau, L. D. (1937). Theory of phase transitions. Zh. Eksp. Teoret. Fiz. 7, pp. 19–32, 627–636. (In Russian.) (German: Phys. Z. Sowjetunion, 11, pp. 20–47, 545–555.)
Landau, L. D. & Lifshitz, E. M. (1980). Statistical physics, Part 1, 3rd ed., pp. 459–471. London: Pergamon. (Russian: Stati­sticheskaya Fizika, chast 1. Moskva: Nauka, 1976; German: Lehrbuch der theoretischen Physik, 6. Aufl., Bd. 5, Teil 1, S. 436–447. Berlin: Akademie-Verlag, 1984).
Levanyuk, A. P. & Sannikov, D. G. (1971). Phenomenological theory of dielectric anomalies in ferroelectric materials with several phase transitions at temperatures close together. Sov. Phys. JETP, 11, 600–604.
McLarnan, T. J. (1981a). Mathematical tools for counting polytypes. Z. Kristallogr. 155, 227–245.
McLarnan, T. J. (1981b). The numbers of polytypes in close packings and related structures. Z. Kristallogr. 155, 269–291.
McLarnan, T. J. (1981c). The combinatorics of cation-deficient close-packed structures. J. Solid State Chem. 26, 235–244.
Marsh, R. E., Kapon, M., Hu, S. & Herbstein, F. H. (2002). Some 60 new space-group corrections. Acta Cryst. B58, 62–77.
Megaw, H. D. (1973). Crystal structures: A working approach. Philadelphia: Saunders.
Müller, U. (1992). Berechnung der Anzahl möglicher Strukturtypen für Verbindungen mit dichtest gepackter Anionenteilstruktur. I. Das Rechenverfahren. Acta Cryst. B48, 172–178.
Müller, U. (1993). Inorganic structural chemistry, pp. 233–246. Chichester, New York: Wiley. (German: Anorganische Strukturchemie, 3. Aufl., 1996, S. 296–309. Stuttgart: Teubner.)
Müller, U. (1998). Strukturverwandtschaften zwischen trigonalen Verbindungen mit hexagonal-dichtester Anionenteilstruktur und besetzten Oktaederlücken. Z. Anorg. Allg. Chem. 624, 529–532.
Müller, U. (2003). Die Zahl der Substitutions- und Leerstellenvarianten des NaCl-Typs bei verdoppelter Elementarzelle (a, b, 2c). Z. Anorg. Allg. Chem. 629, 487–492.
Müller, U. & Brelle, A. (1995). Über isomorphe Untergruppen von Raumgruppen der Kristallklassen [4], [\bar{4}], [4/m], [3], [\bar{3}], [6], [\bar{6}] und [6/m]. Acta Cryst. A51, 300–304.
Nespolo, M. & Ferraris, G. (2004). Applied geminography – symmetry analysis of twinned crystals and definition of twinning by reticular polyholohedry. Acta Cryst. A60, 89–95.
Neubüser, J. & Wondratschek, H. (1966). Untergruppen der Raumgruppen. Krist. Tech. 1, 529–543.
Niggli, P. (1919). Geometrische Kristallographie des Diskontinuums. Leipzig: Gebrüder Borntraeger. Reprint (1973) Wiesbaden: Dr M. Saendig.
Pauling, L. (1928). The coordination theory of the structure of ionic compounds. Probleme der modernen Physik (Sommerfeld-Festschrift), p. 11. Leipzig: S. Hirzel.
Pauling, L. (1929). The principles determining the structures of complex ionic crystals. J. Am. Chem. Soc. 51, 1010–1026.
Sayari, A. (1976). Contribution à l'étude des groupes d'espace bidimensionnels, dérivation des sous-groupes et des surstructures. Thesis, University of Tunis, Tunisia.
Schoenflies, A. M. (1891). Krystallsysteme und Krystallstruktur. Leipzig: Teubner. Reprint 1984: Springer.
Sohncke, L. (1879). Entwickelung einer Theorie der Krystallstruktur. Leipzig: Teubner.
Sowa, H. (2001). On the transition from the wurtzite to the NaCl type. Acta Cryst. A57, 176–182.
Stokes, H. T. & Hatch, D. M. (1988). Isotropy subgroups of the 230 crystallographic space groups. Singapore: World Scientific.
Stokes, H. T. & Hatch, D. M. (2002). Procedure for obtaining microscopic mechanisms of reconstructive phase transitions in crystalline solids. Phys. Rev. B, 65, 144114–144123.
Strukturbericht 1913–1928 (1931). Edited by P. P. Ewald & C. Hermann. Leipzig: Akademische Verlagsgesellschaft. Continued by Structure Reports.
Tolédano, J.-C. & Tolédano, P. (1987). The Landau theory of phase transitions. Singapore: World Scientific.
Wiener, C. (1863). Grundzüge der Weltordnung. I. Atomlehre, p. 82. Leipzig: Heidelberg.
Wondratschek, H. & Aroyo, M. I. (2001). The application of Hermann's group [\cal{M}] in group–subgroup relations between space groups. Acta Cryst. A57, 311–320.
Wyckoff, R. W. G. (1922). The analytical expression of the results of the theory of space groups. Washington: Carnegie Institution.
Wyckoff, R. W. G. (1965). Crystal structures, 2nd ed., Vol. 3, pp. 361–362. New York: Interscience.