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

International Tables for Crystallography (2011). Vol. A1, ch. 1.5, pp. 41-42   | 1 | 2 |

Section 1.5.2. Derivative structures and phase transitions

Ulrich Müllera*

aFachbereich Chemie, Philipps-Universität, D-35032 Marburg, Germany
Correspondence e-mail:

1.5.2. Derivative structures and phase transitions

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In crystal chemistry, structural relations such as the relation diamond–sphalerite are of fundamental interest. Structures that result from a basic structure by the substitution of atoms of one kind for atoms of different elements, the topology being retained, are called derivative structures after Buerger (1947[link], 1951[link]). For the basic structure the term aristotype has also been coined, while its derivative structures are called hettotypes (Megaw, 1973[link]). For more details, see Chapter 1.6[link] . When searching for derivative structures, one must look for space groups that are subgroups of the space group of the aristotype and in which the orbit of the atom(s) to be substituted splits into different orbits.

Similar relations also apply to many phase transitions (cf. Section 1.6.6[link] ). Very often the space group of one of the phases is a subgroup of the space group of the other. For second-order phase transitions this is even mandatory (cf. Section 1.2.7[link] ). The positions of the atoms in one phase are related to those in the other one.


The disorder–order transition of β-brass (CuZn) taking place at 741 K involves a space-group change from the space group [Im\overline{3}m], No. 229, to its subgroup [Pm\overline{3}m], No. 221. In the high-temperature form, Cu and Zn atoms randomly take the orbit of the Wyckoff position [2a] of [Im\overline{3}m]. Upon transition to the ordered form, this position splits into the independent positions [1a] and [1b] of the subgroup [Pm\overline{3}m]. These positions are occupied by the Cu and Zn atoms, respectively. See also Example[link] .

Phase transitions in which a paraelectric crystal becomes ferroelectric occur when atoms that randomly occupy several symmetry-equivalent positions become ordered in a space group with lower symmetry, or when a key atom is displaced to a position with reduced site symmetry, thus allowing a distortion of the structure. In both cases, the space group of the ferroelectric phase is a subgroup of the space group of the paraelectric phase. In the case of ordering, the orbits of the atoms concerned split; in the case of displacement this is not necessary.


In paraelectric NaNO2, space group [Immm], No. 71, Na+ ions randomly occupy two sites close to each other around an inversion centre [(0,\,0,\,{{1}\over{2}})] with half occupation (position [4i] at [0,\,0,\,\pm 0.540)]. The same applies to the nitrite ions, which are disordered in two opposite orientations around the inversion centre at [0,\,0,\,0], with the N atoms at [4i] ([0,\,0,\,\pm 0.072]). At the transition to the ferroelectric phase at 438 K, the space-group symmetry decreases to the subgroup [Imm2], No. 44, and the ions become ordered in one orientation. Each of the [4i] orbits splits into two [2a] orbits, but for every ion only one of the resulting orbits is now fully occupied: Na+ at [2a] [(0,\,0,\,0.540)] and N at [2a] [(0,\,0,\,0.074)].


Paraelectric BaTiO3 crystallizes in the space group [Pm\overline{3}m], No. 221, and the position [1a] of a Ti atom ([0,0,0] with site symmetry [m\overline{3}m]) is in the centre of an octahedron of oxygen atoms. At 393 K, a phase transition to a ferroelectric phase takes place. It has space group [P4mm], No. 99, which is a subgroup of [Pm\overline{3}m]; the Ti atom is now at [0,0,z] ([1a], site symmetry [4mm]) and is displaced from the octahedron centre. The orbit does not split, but the site symmetry is reduced.


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.
Megaw, H. D. (1973). Crystal Structures: A Working Approach. Philadelphia: Saunders.

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