International
Tables for Crystallography Volume A Spacegroup symmetry Edited by M. I. Aroyo © International Union of Crystallography 2016 
International Tables for Crystallography (2016). Vol. A, ch. 1.7, pp. 135136
Section 1.7.2. Relations between Wyckoff positions for group–subgrouprelated space groups
U. Müller^{b}

The crystal structures of two compounds are isotypic if their atoms are arranged in the same way and if they have the same or the enantiomorphic space group. The absolute values of the lattice parameters and interatomic distances may differ and small deviations are permitted for nonfixed coordinates of corresponding atoms. The axial ratios and interaxial angles must be similar. Two structures are homeotypic if the conditions for isotypism are relaxed because (LimadeFaria et al., 1990): (1) their space groups differ, allowing for a group–subgroup relation; (2) the geometric conditions differ (axial ratios, interaxial angles, atomic coordinates); or (3) an atomic position in one structure is occupied in an ordered way by various atomic species in the other structure (substitution derivatives or after a misorder–order phase transition).^{2}
Group–subgroup relations between the space groups of homeotypic crystal structures are particularly suited to disclosing the relationship. A standardized procedure to set forth such relations was developed by Bärnighausen (1980). The concept is to start from a simple, highly symmetrical crystal structure and to derive more complicated structures by distortions and/or substitutions of atoms. A tree of group–subgroup relations between the space groups involved, now called a Bärnighausen tree, serves as the main guideline. The highly symmetrical starting structure is called the aristotype after Megaw (1973) or basic structure after Buerger (1947, 1951) or, in the literature on phase transitions in physics, prototype or parent structure. The derived structures are the hettotypes or derivative structures or, in phasetransition physics, distorted structures or daughter phases. In Megaw's terminology, the structures mentioned in the tree form a family of structures.
Detailed instructions on how to form a Bärnighausen tree, the information that can be drawn from it and some possible pitfalls are given in the second edition of IT A1, Chapter 1.6 and in the book by Müller (2013). In any case, setting up group–subgroup relations requires a thorough monitoring of how the Wyckoff positions develop from a group to a subgroup for every position occupied. The following examples give a concise impression of such relations.
As an example, Fig. 1.7.2.1 shows the simple relation between diamond and zinc blende. This is an example of a substitution derivative. The reduction of the spacegroup symmetry from diamond to zinc blende is depicted by an arrow which points from the highersymmetry space group of diamond to the lowersymmetry space group of zinc blende. The subgroup is translationengleiche of index 2, marked by t2 in the middle of the arrow. Translationengleiche means that the subgroup has the same translational lattice (the same size and dimensions of the primitive unit cell) but its crystal class is of reduced symmetry. The index [i] is the factor by which the total number of symmetry operations has been reduced, i.e. the subgroup has 1/i as many symmetry operations; as mentioned in Section 1.7.1, this is to be understood in the same way as `the number of even numbers is half as many as the number of all integer numbers'.

Group–subgroup relation from the aristotype diamond to its hettotype zinc blende. The numerical values in the boxes are atomic coordinates. 
The consequences of the symmetry reduction on the positions occupied by the atoms are important. As shown in the boxes next to the spacegroup symbols in Fig. 1.7.2.1, the carbon atoms in diamond occupy the Wyckoff position 8a of the space group . Upon transition to zinc blende, this position splits into two independent Wyckoff positions, 4a and 4c, of the subgroup , rendering possible occupation by atoms of the two different species zinc and sulfur. The site symmetry remains unchanged for all atoms.
Further substitutions of atoms require additional symmetry reductions. For example, in chalcopyrite, CuFeS_{2}, the zinc atoms of zinc blende have been substituted by copper and iron atoms. This implies a symmetry reduction from to its subgroup ; this requires one translationengleiche and two steps of klassengleiche group–subgroup relations, including a doubling of the unit cell.
Fig. 1.7.2.2 shows derivatives of the cubic ReO_{3} structure type that result from distortions of this highsymmetry structure. WO_{3} itself does not adopt this structure, only several distorted variants. The first step of symmetry reduction involves a tetragonal distortion of the cubic ReO_{3} structure resulting in the space group P4/mmm; no example with this symmetry is yet known. The second step leads to a klassengleiche subgroup of index 2 (marked k2 in the arrow), resulting in the structure of hightemperature WO_{3}, which is the most symmetrical known modification of WO_{3}. Klassengleiche means that the subgroup belongs to the same crystal class, but it has lost translational symmetry (its primitive unit cell has been enlarged). In this case this is a doubling of the size of the unit cell (a − b, a + b, c) combined with an origin shift of (in the coordinate system of P4/mmm). This cell transformation and origin shift cause a change of the atomic coordinates of the metal atom from 0, 0, 0 to (the decimal value indicates that the coordinate is not fixed by symmetry). Simultaneously, the site symmetry of the metal atom is reduced from 4/mmm to 4mm and the z coordinate becomes independent. In fact, the W atom is shifted from z = 0 to z = 0.066, i.e. it is not situated in the centre of the octahedron of the surrounding O atoms. This shift is the cause of the symmetry reduction. There is no splitting of the Wyckoff positions in this step of symmetry reduction, but a decrease of the site symmetries of all atoms.
When cooled, at 1170 K HTWO_{3} is transformed to αWO_{3}. This involves mutual rotations of the coordination octahedra along c and requires another step of symmetry reduction. Again, the Wyckoff positions do not split in this step of symmetry reduction, but the site symmetries of all atoms are further decreased.
Upon further cooling, WO_{3} undergoes several other phase transitions that involve additional distortions and, in each case, an additional symmetry reduction to another subgroup (not shown in Fig. 1.7.2.2). For more details see Müller (2013), Section 11.6, and references therein.
In the case of phase transitions and of topotactic reactions^{3} that involve a symmetry reduction, the kind of group–subgroup relation determines how many kinds of domains and what domain states can be formed. If the lowersymmetry product results from a translationengleiche group–subgroup relation, twinned crystals are to be expected. A klassengleiche group–subgroup relation will cause antiphase domains. The number of different kinds of twin or antiphase domains corresponds to the index of the symmetry reduction. For example, the phase transition from HTWO_{3} to αWO_{3} involves a klassengleiche group–subgroup relation of index 2 (k2 in Fig. 1.7.2.2); no twins will be formed, but two kinds of antiphase domains can be expected.
1.7.2.5. Presentation of the relations between the Wyckoff positions among group–subgrouprelated space groups
Group–subgroup relations as outlined in the preceding sections can only be correct if all atomic positions of the hettotypes result directly from those of the aristotype.
Every group–subgroup relation between space groups entails specific relations between their Wyckoff positions. If the index of symmetry reduction is 2, a Wyckoff position either splits into two symmetryindependent positions that keep the site symmetry, or there is no splitting and the site symmetry is reduced. If the index is 3 or higher, a Wyckoff position either splits, or its site symmetry is reduced, or both happen. Given the relative settings and origin choices of a space group and its subgroup, there exist unique relations between their Wyckoff positions. Laws governing these relations are considered in Chapter 1.5 of the second edition of IT A1.
Volume A1, Part 3, Relations between the Wyckoff positions, contains tables for all space groups. For every one of them, all maximal subgroups are listed, including the corresponding coordinate transformations. For all Wyckoff positions of a space group the relations to the Wyckoff positions of the subgroups are given. This includes the infinitely many maximal isomorphic subgroups, for which general formulae are given. Isomorphic subgroups are a special kind of klassengleiche subgroup that belong to the same or the enantiomorphic spacegroup type, i.e. group and subgroup have the same or the enantiomorphic spacegroup symbol; the unit cell of the subgroup is increased by some integral factor, which is p, p^{2} or p^{3} (p = prime number) in the case of maximal isomorphic subgroups.
References
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.
Bärnighausen, H. (1980). Group–subgroup relations between space groups: a useful tool in crystal chemistry. MATCH Commun. Math. Chem. 9, 139–175.
LimadeFaria, J., Hellner, E., Liebau, F., Makovicky, E. & Parthé, E. (1990). Nomenclature of inorganic structure types. Report of the International Union of Crystallography Commission on Crystallographic Nomenclature Subcommittee on the Nomenclature of Inorganic Structure Types. Acta Cryst. A46, 1–11.
Megaw, H. D. (1973). Crystal Structures: A Working Approach. Philadelphia: Saunders.
Müller, U. (2013). Symmetry Relationships Between Crystal Structures. Oxford University Press. [German: Symmetriebeziehungen zwischen verwandten Kristallstrukturen; Wiesbaden: Vieweg+Teubner, 2012. Spanish: Relaciones de simetría entre estructuras cristalinas; Madrid: Síntesis, 2013.]