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. 4652
Section 1.6.4. The different kinds of symmetry relations among related crystal structures^{a}Fachbereich Chemie, PhilippsUniversität, D35032 Marburg, Germany 
In this section, using a few simple examples, we point out the different kinds of group–subgroup relations that are important among related (homeotypic) crystal structures.
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 dimensions 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: (1) their space groups differ, allowing for a group–subgroup relation; (2) the geometric conditions differ (axial ratios, interaxial angles, atomic coordinates); or (3) corresponding atomic positions are occupied by various atomic species (substitution derivatives). (LimadeFaria et al., 1990.)
The space group Pbca of PdS_{2} is a translationengleiche maximal subgroup of , the space group of pyrite (FeS_{2}; Bärnighausen, 1980). The threefold axes of the cubic space group have been lost, the index of the symmetry reduction is 3. As shown in Fig. 1.6.4.1, the atomic coordinates have not changed much. However, the two structures differ widely, the c axis of PdS_{2} being strongly stretched. This is due to the tendency of bivalent palladium towards squareplanar coordination (electron configuration d^{8}), whereas the iron atoms in pyrite have octahedral coordination.
Strictly speaking, the space groups of FeS_{2} and PdS_{2} are not really translationengleiche because of the different lattice parameters. In the strict sense, however, FeS_{2} at 293.0 and at 293.1 K would not have the same space group either, due to thermal expansion. Such a strict treatment would render it impossible to apply grouptheoretical methods in crystal chemistry and physics. Instead, we use the concept of the parentclamping approximation, i.e. we act as if the translations of the two homeotypic structures were the same (see Section 1.2.7 ). With the parentclamping approximation we also treat isotypic structures with different lattice parameters (like NaCl and MgO) as if they had the same space group with the same translational lattice.
Upon transition from to Pbca none of the occupied Wyckoff positions split, but their site symmetries are reduced. Without the symmetry reduction from to the square coordination of the Pd atoms would not be possible.
If the positions of the sulfur atoms of pyrite and PdS_{2} are substituted by two different kinds of atoms in an ordered 1:1 ratio, this enforces further symmetry reductions. The corresponding subgroups may only be subgroups in which the sulfur positions split into symmetryindependent positions. In the chosen examples NiAsS and PtGeSe the symmetry reductions consist of the loss of the inversion centres of and Pbca (Fig. 1.6.4.1). Coordinate changes are not necessary, but may occur depending on the site symmetries. In our examples there are minor coordinate changes.
To avoid a basis transformation, the nonconventional spacegroup setting has been chosen for PtGeSe; it corresponds to after interchange of the axes a and b. Mind the origin shift from PdS_{2} to PtGeSe; in the conventional description of , and therefore also of , the origin is situated on one of the 2_{1} axes and thus differs from that of Pbca. The origin shift of in the coordinate system of Pbca involves a change of the atomic coordinates by , i.e. with opposite sign.
The substitution derivatives NiAsS and PtGeSe can only be connected by the common supergroup . A direct group–subgroup relation from to does not exist.
Consider two derivatives of the AlB_{2} type as an example of klassengleiche subgroups (Pöttgen & Hoffmann, 2001). AlB_{2} has a simple hexagonal structure in the space group . In the c direction, aluminium atoms and sheets of boron atoms alternate; the boronatom sheets are planar, like in graphite (Fig. 1.6.4.2; Hoffmann & Jäniche, 1935). The ZrBeSi type has a similar structure (Nielsen & Baenziger, 1953, 1954), but the sheets consist of Be and Si atoms. As a consequence, the inversion centres in the middles of the sixmembered rings cannot be retained. This enforces a symmetry reduction to the klassengleiche subgroup with doubled c vector.

The structures of AlB_{2}, ZrBeSi and CaIn_{2}. The mirror planes of perpendicular to c are at and . 
The doubling of c is the essential aspect in the symmetry reduction from the AlB_{2} to the ZrBeSi type. The index is 2: half of all translations are lost, together with half of the inversion centres, half of the symmetry axes perpendicular to c and half of the mirror planes perpendicular to c. The Wyckoff position 2d of the boron atoms of AlB_{2} splits into the two symmetryindependent positions 2c and 2d of the subgroup (Fig. 1.6.4.3, left), rendering possible occupation by atoms of two different species.

Both hettotypes of the AlB_{2} type have the same spacegroup type and a doubled c axis, but the space groups are different due to different origin positions relative to the origin of the aristotype. 
Figs. 1.6.4.2 and 1.6.4.3 show another peculiarity. has two different klassengleiche maximal subgroups of the same type with doubled basis vector c. The second one corresponds to CaIn_{2} (Iandelli, 1964; Wendorff & Roehr, 2005). Here the graphitelike sheets of the AlB_{2} type have become puckered layers of indium atoms; the indium atoms of adjacent layers have shifted parallel to c and have come close to each other in pairs, so that the result is a network as in lonsdaleite (hexagonal diamond). The alternating shift of the atoms no longer permits the existence of mirror planes in the layers; however, neighbouring layers are mutually mirrorsymmetrical. The calcium atoms are on the mirror planes, but no longer on inversion centres. The difference between the two subgroups consists of the selection of the symmetry operations that are being lost with the doubling of c.
The conventional description of the space groups requires an inversion centre to be at the origin of space group . The position of the origin at an Al atom of the AlB_{2} type can be kept when the symmetry is reduced to that of ZrBeSi (i.e. Zr at the origin). The symmetry reduction to CaIn_{2}, however, requires an origin shift to the centre of one of the sixmembered rings. In the coordinate system of the aristotype that is a shift by , as marked in the middle of the group–subgroup arrow in Fig. 1.6.4.3. For the new atomic coordinates (in the coordinate system of the subgroup), the origin shift results in the addition of to the z coordinates; in addition, due to the doubling of c, the z coordinates of the aristotype have to be halved. Therefore, the new z coordinate of the In atom is approximately = . It cannot be exactly this value, because then there would have been no symmetry reduction and the space group would still be . The symmetry reduction requires the atom shift to .
In the relation , the site symmetry of the boron atoms is retained and the Wyckoff position splits. In the relation it is the other way; the position does not split, the atoms remain symmetryequivalent, but their site symmetry is reduced to and the z coordinate becomes variable.
Among klassengleiche subgroups there often exist two and sometimes four or even eight nonconjugate subgroups of the same spacegroup type with different origin positions. It is important to choose the correct one, with the correct origin shift. All of these subgroups are listed in this volume if they are maximal.
Compared to AlB_{2}, ZrBeSi and CaIn_{2} have socalled `superstructures' (they have additional reflections in the Xray diffraction patterns). Whereas the term superstructure gives only a qualitative, informal outline of the facts, the grouptheoretical approach permits a precise treatment.
Isomorphic subgroups comprise a special category of klassengleiche subgroups. Every space group has an infinity of isomorphic maximal subgroups. The index agrees with the factor by which the unit cell has been enlarged. The indices are prime numbers; squares of prime numbers may occur in the case of tetragonal, hexagonal and trigonal space groups, and for cubic space groups only cubes of prime numbers () are possible. For many space groups, not all prime numbers are permitted. The prime number 2 is often excluded, and additional restrictions may apply. In the tables in Parts 2 and 3 of this volume all permitted isomorphic maximal subgroups are listed.
Usually, in accordance with the symmetry principle, only small index values are observed (mostly 2 and 3, sometimes 4, less frequently 5, 7 or 9). However, seemingly curious values like 13, 19, 31 or 37 do occur [for examples with indices of 13 and 37, discovered by Bärnighausen, see Müller (2004)].
A classic example of a relation between isomorphic space groups concerns trirutile (Billiet, 1973). The space group of rutile, , has an isomorphic subgroup of index 3, but none of index 2. By triplication of c it becomes possible to substitute the titaniumatom positions of rutile by two different kinds of atoms in a ratio of 1:2, as for example in ZnSb_{2}O_{6} (Fig. 1.6.4.4; Byström et al., 1941; Ercit et al., 2001). Since the space group has no isomorphic subgroup of index 2, a `dirutile' with this spacegroup type cannot exist.
Note that rutile and trirutile have different space groups of the same spacegroup type.
Two crystal structures can be intimately related even when there is no direct group–subgroup relation between their space groups. Instead, there may exist a common supergroup. The structures of NiAsS and PtGeSe, presented in Section 1.6.4.1, offer an example. In this case, the pyrite type corresponds to the common supergroup. Even if there is no known representative, it can be useful to look for a common supergroup.
βK_{2}CO_{3} and βNa_{2}CO_{3} have similar structures and unit cells (Fig. 1.6.4.5; Jansen & Feldmann, 2000). The planes of the carbonate ions are not aligned exactly perpendicular to c; compared to the perpendicular orientation, in the case of βK_{2}CO_{3}, they are rotated about b by 22.8° and those of βNa_{2}CO_{3} are rotated about a by 27.3°. There is no group–subgroup relation between the space groups and of the two structures (the nonconventional setting has been chosen for βNa_{2}CO_{3} to ensure a correspondence between the cells of both structures). Looking for common minimal supergroups of and one can find two candidates: Cmcm, No. 63, and Cmce, No. 64. Since the atomic coordinates of βK_{2}CO_{3} and βNa_{2}CO_{3} are very similar, any origin shifts in the relations from the common supergroup to as well as must be the same. In the listings of the supergroups the origin shifts are not mentioned either in Volume A or in this volume. Therefore, one has to look up the subgroups of Cmcm and Cmce in this volume and check in which cases the origin shifts coincide. One finds that the relation requires an origin shift of (or ), while all other relations ( , , ) require no origin shifts. As a consequence, only Cmcm and not Cmce can be the common supergroup.

The unit cells of βK_{2}CO_{3} and βNa_{2}CO_{3}. The angles of tilt of the ions are referred relative to a plane perpendicular to c. 
No structure is known that has the space group Cmcm and that can be related to these carbonate structures. Could there be any other structure with even higher symmetry? A supergroup of Cmcm is and, in fact, αK_{2}CO_{3} and αNa_{2}CO_{3} are hightemperature modifications that crystallize in this space group. They have the carbonate groups perpendicular to c. The group–subgroup relations are depicted in Fig. 1.6.4.6. In this case there exists a highersymmetry structure that can be chosen as the common aristotype. In other cases, however, the common supergroup refers to a hypothetical structure; one can speculate why it does not exist or one can try to prepare a compound having this structure.
Among the alkali metal carbonates several other modifications are known which we do not discuss here.
Occasionally a crystal structure shows a pronounced distortion compared to a chosen aristotype and another aristotype can be chosen just as well with a comparable distortion.
When pressure is exerted upon silicon, it first transforms to a modification with the βtin structure (SiII, ). Then it is transformed to siliconXI (McMahon et al., 1994). At even higher pressures it is converted to siliconV forming a simple hexagonal structure (). The space group of SiXI, Imma, is a subgroup of both and , and the structure of SiXI can be related to either SiII or SiV (Fig. 1.6.4.7). Assuming no distortions, the calculated coordinates of a silicon atom of SiXI would be when derived from SiII, and when derived from SiV. The actual coordinates are halfway between. The metric deviations of the lattices are small; taking into account the basis transformations given in Fig. 1.6.4.7, the expected lattice parameters for SiXI, calculated from those of SiV, would be = = 441.5 pm, = = 476.6 pm and = = 254.9 pm.

The structure of the highpressure modification SiXI is related to the structures of both SiII and SiV. 
These phase transitions of silicon involve small atomic displacements and small volume changes. The lattice parameter c of the hexagonal structure is approximately half the value of a of tetragonal SiII. There are two separate experimentally observable phase transitions. In a certain pressure range, the whole crystal actually consists of stable SiXI; it is not just a hypothetical intermediate. Taking all these facts together, a grouptheoretical relation between SiII and SiV exists via the common subgroup of SiXI.
Usually, however, there is no point in relating two structures via a common subgroup. Two space groups always have an infinity of common subgroups, and it may be easy to set up relations via common subgroups in a purely formal manner; this can be quite meaningless unless it is based on well founded physical or chemical evidence. See also the statements on this matter in Section 1.6.4.6, at the end of Section 1.6.6 and in Section 1.6.7.
To comprehend the huge amount of known crystalstructure types, chemists have very successfully developed quite a few concepts. One of them is the widespread description of structures as packings of spheres with occupied interstices. Group–subgroup relations can help to rationalize this. This requires that unoccupied interstices be treated like atoms, i.e. that the occupation of voids is treated like a substitution of `zero atoms' by real atoms.
The crystal structure of FeF_{3} (VF_{3} type) can be derived from the ReO_{3} type by a mutual rotation of the coordination octahedra about the threefold rotation axes parallel to one of the four diagonals of the cubic unit cell of the space group (Fig. 1.6.4.8). This involves a symmetry reduction by two steps to a rhombohedral hettotype with the space group (Fig. 1.6.4.9, left).

The connected coordination octahedra in ReO_{3} and FeF_{3} (VF_{3} type). Light, medium and dark grey refer to octahedron centres at , and , respectively (hexagonal setting). 
FeF_{3} can also be described as a hexagonal closest packing of fluorine atoms in which one third of the octahedral voids have been occupied by Fe atoms (Fig. 1.6.4.9, right). This is a more descriptive and more formal point of view, because the number of atoms and the kind of linkage between them are altered.
The calculated ideal x coordinates of the fluorine atoms of FeF_{3}, assuming no distortions, are x = 0.5 when derived from ReO_{3} and x = 0.333 when derived from the packing of spheres. The actual value at ambient pressure is x = 0.412, i.e. halfway in between. When FeF_{3} is subjected to high pressures, the coordination octahedra experience a mutual rotation which causes the structure to come closer to a hexagonal closest packing of fluorine atoms; at 9 GPa this is a nearly undistorted closest packing with the x parameter and the ratio close to the ideal values of 0.333 and 1.633 (Table 1.6.4.1).

Therefore, at high pressures the hexagonal closest packing of spheres could be regarded as the more appropriate aristotype. This, however, is only true from the descriptive point of view, i.e. if one accepts that the positions of voids can be treated like atoms. If one studies the mutual rotation of the octahedra in FeF_{3} or if phase transitions from the ReO_{3} type to the VF_{3} type are of interest, there is no point in allowing a change of the occupation of the octahedra; in this case only the group–subgroup relations given in the left part of Fig. 1.6.4.9 should be considered.
However, if one wants to derive structures from a hexagonal closest packing of spheres and point out similarities among them, the group–subgroup relations given in the right part of Fig. 1.6.4.9 are useful. For example, the occupation of the octahedral voids in the Wyckoff position 12c instead of 6b of (Fig. 1.6.4.9, lower right) results in the structure of corundum (αAl_{2}O_{3}); this shows that both the VF_{3} type and corundum have the same kind of packing of their anions, even though the linkage of their coordination octahedra is quite different and the number of occupied positions does not coincide.
It should also be kept in mind that the occupation of voids in a packing of atoms can actually be achieved in certain cases. Examples are the intercalation compounds and the large number of metal hydrides MH_{x} that can be prepared by diffusion of hydrogen into the metals. The hydrides often keep the closest packed structures of the metals while the hydrogen atoms occupy the tetrahedral or octahedral voids.
It is not recommended to plot group–subgroup relations in which FeF_{3} is depicted as a common subgroup of both the ReO_{3} type and the hexagonal closest packing of spheres. This would mix up two quite different points of view.
Large trees can be constructed using the modular way to put together Bärnighausen trees, as set forth in Fig. 1.6.3.1 and in the preceding sections. Headed by an aristotype, they show structural relations among many different crystal structures belonging to a family of structures. As an example, Fig. 1.6.4.10 shows structures that can be derived from the ReO_{3} type (Bock & Müller, 2002b). Many other trees of this kind have been set up, for example hettotypes of perovskite (Bärnighausen, 1975, 1980; Bock & Müller, 2002a), rutile (Baur, 1994, 2007; Meyer, 1981), CaF_{2} (Meyer, 1981), hexagonal closest packed structures (Müller, 1998), AlB_{2} (Pöttgen & Hoffmann, 2001), zeolites (Baur & Fischer, 2000, 2002, 2006) and tetraphenylphosphonium salts (Müller, 1980, 2004).
In the left branch of Fig. 1.6.4.10 only one compound is mentioned, WO_{3}. This is an example showing the symmetry relations among different polymorphic forms of a compound. The right branch of the tree shows the relations for substitution derivatives, including distortions caused by the Jahn–Teller effect, hydrogen bonds and different relative sizes of the atoms.
In addition to showing relations between known structure types, one can also find subgroups of an aristotype for which no structures are known. This can be exploited in a systematic manner to search for new structural possibilities, i.e. one can predict crystalstructure types. For this purpose, one starts from an aristotype in conjunction with a structural principle and certain additional restrictions. For example, the aristotype can be a hexagonal closest packing of spheres and the structural principle can be the partial occupation of octahedral voids in this packing. Additional restrictions could be things like the chemical composition, a given molecular configuration or a maximal size of the unit cell. Of course, one can only find such structure types that meet these starting conditions. For every space group appearing in the Bärnighausen tree, one can calculate how many different structure types are possible for a given chemical composition (McLarnan, 1981a,b,c; Müller, 1992).
Examples of studies of structural possibilities include compounds AX_{3}, AX_{6}, A_{a}B_{b}X_{6} (with the X atoms forming the packing of spheres and atoms A and B occupying the octahedral voids; ; Müller, 1998), molecular compounds (MX_{5})_{2} (Müller, 1978), chain structures MX_{4} (Müller, 1981) and MX_{5} (Müller, 1986), wurtzite derivatives (Baur & McLarnan, 1982) and NaCl derivatives with doubled cell (Sens & Müller, 2003).
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