International
Tables for Crystallography Volume A1 Symmetry relations between space groups Edited by Hans Wondratschek and Ulrich Müller © International Union of Crystallography 2011 
International Tables for Crystallography (2011). Vol. A1, ch. 1.6, pp. 4446
Section 1.6.3. Bärnighausen trees^{a}Fachbereich Chemie, PhilippsUniversität, D35032 Marburg, Germany 
To represent symmetry relations between different crystal structures in a concise manner, we construct a tree of group–subgroup relations in a modular design, beginning with the space group of the aristotype at its top. Each module represents one step of symmetry reduction to a maximal subgroup. Therefore, we have to discuss only one of these modules in detail.
For two structures we want to interrelate, we place their spacegroup symbols one below the other and indicate the direction of the symmetry reduction by an arrow pointing downwards (Fig. 1.6.3.1). Since they are more informative, it is advisable to use only the full Hermann–Mauguin symbols. In the middle of the arrow we insert the kind of maximal subgroup and the index of symmetry reduction, using the abbreviations t for translationengleiche, k for klassengleiche and i for isomorphic. If the unit cell changes, we also insert the new basis vectors expressed as vector sums of the basis vectors of the highersymmetry cell. If there is an origin shift, we enter this as a triplet of numbers which express the coordinates of the new origin referred to the coordinate system of the highersymmetry cell. This is a shorthand notation for the transformation matrices. Any change of the basis vectors and the origin is essential information that should never be omitted.

Scheme of the formulation of the smallest step of symmetry reduction connecting two related crystal structures. 
If the atomic coordinates of two related crystal structures differ because of different settings of their unit cells, the similarities of the structures become less clear and may even be obscured. Therefore, it is recommended to avoid cell transformations whenever possible. If necessary, it is much better to fully exploit the possibilities offered by the Hermann–Mauguin symbolism and to choose nonconventional spacegroup settings [see Chapter 4.3 of International Tables for Crystallography Volume A (2005) and Section 3.1.4 of this volume].
Origin shifts also tend to obscure relations. However, they often cannot be avoided. There is no point in deviating from the standard origin settings, because otherwise much additional information would be required for an unequivocal description. Note: The coordinate triplet specifying the origin shift in the group–subgroup arrow refers to the coordinate system of the highersymmetry space group, whilst the corresponding changes of the atomic coordinates refer to the coordinate system of the subgroup and therefore are different. Details are given in Section 3.1.3 . Also note that in the tables of Parts 2 and 3 of this volume the origin shifts are given in different ways. In Part 2 they refer to the highersymmetry space group. In Part 3 (relations of the Wyckoff positions) they are given only as parts of the coordinate transformations, i.e. in the coordinate systems of the subgroups. As explained in the Appendix , the chosen origin shifts themselves also differ in Parts 2 and 3; an origin transformation taken from Part 3 may be different from the one given in Part 2 for the same group–subgroup relation. If needed, one has to calculate the corresponding values with the formulae given in Section 3.1.3 .
The calculation of coordinate changes due to cell transformations and origin shifts is prone to errors. Some useful hints are given in Section 1.6.5.
For space groups with two possible choices of origin (`origin choice 1' and `origin choice 2'), the choice is specified by a superscript (1) or (2) after the spacegroup symbol, for example . The setting of rhombohedral space groups is specified, if necessary, by superscript (rh) or (hex). Occasionally it may be useful to use a nonconventional rhombohedral `reverse' setting, i.e. with the centring vectors instead of `obverse' with ; this is specified by superscript (rev), for example .
In a Bärnighausen tree containing several group–subgroup relations, it is recommended that the vertical distances between the spacegroup symbols are kept proportional to the logarithms of the corresponding indices. This way all subgroups that are at the same hierarchical distance, i.e. at the same index, from the aristotype appear on the same line.
If several paths can be constructed from one space group to a general subgroup, via different intermediate groups, usually there is no point in depicting all of them. There is no general recipe indicating which of several possible paths should be preferred. However, crystalchemical and physical aspects should be used as a guide. First of all, the chosen intermediate groups should be:
Bärnighausen trees sometimes contain intermediate space groups which, in Howard & Stokes' (2005) opinion, `have no physical significance'. As an example, they cite the phase transition induced by the displacement of the octahedrally coordinated cations in a cubic perovskite along the +z axis. This lowers the symmetry directly from to the nonmaximal subgroup , skipping the intermediate space group . In this particular case, `has no physical significance' in the sense that it cannot actually occur in this kind of phase transition. However, the intermediate space group can occur in other instances (e.g. in order–disorder transitions) and it has been found among several perovskites. In addition, we are not dealing merely with phase transitions. Intermediate space groups do have significance for several reasons: (1) Any step of symmetry reduction reduces the restrictions for every Wyckoff position (either the site symmetry is reduced or the position splits into independent positions, or both happen); every intermediate space group offers new scope for effects with physical significance, even if none have yet been observed. (2) Skipping intermediate space groups in the tree of group–subgroup relations reduces the informative value of the symmetry relations. For example, it is no longer directly evident how many translationengleiche and klassengleiche steps are involved; this is useful to decide how many and what kind of twin domains may appear in a phase transition or topotactic reaction (see Sections 1.2.7 and 1.6.6).
Group–subgroup relations are of little value if the usual crystallographic data are not given for every structure. The mere mention of the space groups is insufficient. The atomic coordinates are of special importance. It is also important to present all structures in such a way that their relations become clearly visible. In particular, all atoms of the asymmetric units should exhibit strict correspondence, so that their positional parameters can immediately be compared.
For all structures, the same coordinate setting and among several symmetryequivalent positions for an atom the same location in the unit cell should be chosen, if possible. For all space groups, except and , one can choose several different equivalent sets of coordinates describing one and the same structure in the same spacegroup setting. It is by no means a simple matter to recognize whether two differently documented structures are alike or not (the literature abounds with examples of `new' structures that really had been well known). One is often forced to transform coordinates from one set to another to attain the necessary correspondence. In Section 15.3 of Volume A (editions 1987–2005) and in a paper by Koch & Fischer (2006) one can find a procedure for and examples of how to interconvert equivalent coordinate sets with the aid of the Euclidean normalizers of the space groups. Note: For enantiomorphic (chiral) space groups like this procedure will yield equivalent sets of coordinates without a change of chirality; for chiral structures in nonenantiomorphic (nonchiral) space groups like the sets of coordinates include the enantiomorphic pairs [for the distinction between chiral and nonchiral space groups see Flack (2003)].
If space permits, it is useful to list the site symmetries and the coordinates of the atoms next to the spacegroup symbols in the Bärnighausen tree, as shown in Fig. 1.6.3.1 and in the following examples. If there is not enough space, this information must be provided in a separate table.
References
Flack, H. D. (2003). Chiral and achiral structures. Helv. Chim. Acta, 86, 905–921.Howard, C. J. & Stokes, H. T. (2005). Structures and phase transitions in perovskites – a grouptheoretical approach. Acta Cryst. A61, 93–111.
International Tables for Crystallography (2005). Vol. A, SpaceGroup Symmetry, edited by Th. Hahn, corrected reprint of 5th ed. Dordrecht: Kluwer Academic Publishers.
Koch, E. & Fischer, W. (2006). Normalizers of space groups: a useful tool in crystal description, comparison and determination. Z. Kristallogr. 221, 1–14.
Stokes, H. T. & Hatch, D. M. (1988). Isotropy Subgroups of the 230 Crystallographic Space Groups. Singapore: World Scientific.