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

International Tables for Crystallography (2011). Vol. A1, ch. 1.6, pp. 46-52

Section 1.6.4. The different kinds of symmetry relations among related crystal structures

Ulrich Müllera*

aFachbereich Chemie, Philipps-Universität, D-35032 Marburg, Germany
Correspondence e-mail: mueller@chemie.uni-marburg.de

1.6.4. The different kinds of symmetry relations among related crystal structures

| top | pdf |

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 non-fixed 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). (Lima-de-Faria et al., 1990[link].)

1.6.4.1. Translationengleiche maximal subgroups

| top | pdf |

The space group Pbca of PdS2 is a translationengleiche maximal subgroup of [Pa\overline3], the space group of pyrite (FeS2; Bärnighausen, 1980[link]). 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[link], the atomic coordinates have not changed much. However, the two structures differ widely, the c axis of PdS2 being strongly stretched. This is due to the tendency of bivalent palladium towards square-planar coordination (electron configuration d8), whereas the iron atoms in pyrite have octahedral coordination.

[Figure 1.6.4.1]

Figure 1.6.4.1 | top | pdf |

Bärnighausen tree for the family of structures of pyrite. Coordinates in brackets (not stated normally) refer to symmetry-equivalent positions. Lattice parameters are taken from Ramsdell (1925[link]) and Brostigen & Kjeskhus (1969[link]) for pyrite; from Ramsdell (1925[link]), Steger et al. (1974[link]) and Foecker & Jeitschko (2001[link]) for NiAsS; from Grønvold & Røst (1957[link]) for PdS2; and from Entner & Parthé (1973[link]) for PtGeSe.

Strictly speaking, the space groups of FeS2 and PdS2 are not really translationengleiche because of the different lattice parameters. In the strict sense, however, FeS2 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 group-theoretical methods in crystal chemistry and physics. Instead, we use the concept of the parent-clamping approximation, i.e. we act as if the translations of the two homeotypic structures were the same (see Section 1.2.7[link] ). With the parent-clamping 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 [Pa\overline3] to Pbca none of the occupied Wyckoff positions split, but their site symmetries are reduced. Without the symmetry reduction from [\overline3] to [\overline1] the square coordination of the Pd atoms would not be possible.

If the positions of the sulfur atoms of pyrite and PdS2 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 symmetry-independent positions. In the chosen examples NiAsS and PtGeSe the symmetry reductions consist of the loss of the inversion centres of [Pa\overline3] and Pbca (Fig. 1.6.4.1[link]). 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 space-group setting [Pbc2_1] has been chosen for PtGeSe; it corresponds to [Pca2_1] after interchange of the axes a and b. Mind the origin shift from PdS2 to PtGeSe; in the conventional description of [Pca2_1], and therefore also of [Pbc2_1], the origin is situated on one of the 21 axes and thus differs from that of Pbca. The origin shift of [-\textstyle{1\over 4},\,0,\,0] in the coordinate system of Pbca involves a change of the atomic coordinates by [+\textstyle{1\over 4},\,0,\,0], i.e. with opposite sign.

The substitution derivatives NiAsS and PtGeSe can only be connected by the common supergroup [P2_1/a\overline3]. A direct group–subgroup relation from [P2_13] to [Pbc2_1] does not exist.

1.6.4.2. Klassengleiche maximal subgroups

| top | pdf |

Consider two derivatives of the AlB2 type as an example of klassengleiche subgroups (Pöttgen & Hoffmann, 2001[link]). AlB2 has a simple hexagonal structure in the space group [P6/mmm]. In the c direction, aluminium atoms and sheets of boron atoms alternate; the boron-atom sheets are planar, like in graphite (Fig. 1.6.4.2[link]; Hoffmann & Jäniche, 1935[link]). The ZrBeSi type has a similar structure (Nielsen & Baenziger, 1953[link], 1954[link]), but the sheets consist of Be and Si atoms. As a consequence, the inversion centres in the middles of the six-membered rings cannot be retained. This enforces a symmetry reduction to the klassen­gleiche subgroup [P6_3/mmc] with doubled c vector.

[Figure 1.6.4.2]

Figure 1.6.4.2 | top | pdf |

The structures of AlB2, ZrBeSi and CaIn2. The mirror planes of [P6_3/mmc] perpendicular to c are at [z=\textstyle{1\over 4}] and [z=\textstyle{3\over 4}].

The doubling of c is the essential aspect in the symmetry reduction from the AlB2 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 AlB2 splits into the two symmetry-independent positions 2c and 2d of the subgroup (Fig. 1.6.4.3[link], left), rendering possible occupation by atoms of two different species.

[Figure 1.6.4.3]

Figure 1.6.4.3 | top | pdf |

Both hettotypes of the AlB2 type have the same space-group 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[link] and 1.6.4.3[link] show another peculiarity. [P6/mmm] has two different klassengleiche maximal subgroups of the same type [P6_3/mmc] with doubled basis vector c. The second one corresponds to CaIn2 (Iandelli, 1964[link]; Wendorff & Roehr, 2005[link]). Here the graphite-like sheets of the AlB2 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 mirror-symmetrical. The calcium atoms are on the mirror planes, but no longer on inversion centres. The difference between the two subgroups [P6_3/mmc] 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 [P6_3/mmc]. The position of the origin at an Al atom of the AlB2 type can be kept when the symmetry is reduced to that of ZrBeSi (i.e. Zr at the origin). The symmetry reduction to CaIn2, however, requires an origin shift to the centre of one of the six-membered rings. In the coordinate system of the aristotype that is a shift by [0, 0,-\textstyle{1\over 2}], as marked in the middle of the group–subgroup arrow in Fig. 1.6.4.3[link]. For the new atomic coordinates (in the coordinate system of the subgroup), the origin shift results in the addition of [+\textstyle{1\over 4}] 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 [z'\simeq\textstyle{1\over 2} z+\textstyle{1\over 4}] = [\textstyle{1\over 2}\times\textstyle{1\over 2} +\textstyle{1\over 4}=0.5]. It cannot be exactly this value, because then there would have been no symmetry reduction and the space group would still be [P6/mmm]. The symmetry reduction requires the atom shift to [z'=0.452].

In the relation [{\rm AlB}_2\rightarrow {\rm ZrBeSi}], the site symmetry [\overline6m2] of the boron atoms is retained and the Wyckoff position splits. In the relation [{\rm AlB}_2\rightarrow{\rm CaIn}_2] it is the other way; the position does not split, the atoms remain symmetry-equivalent, but their site symmetry is reduced to [3m1] and the z coordinate becomes variable.

Among klassengleiche subgroups there often exist two and sometimes four or even eight nonconjugate subgroups of the same space-group 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 AlB2, ZrBeSi and CaIn2 have so-called `superstructures' (they have additional reflections in the X-ray dif­fraction patterns). Whereas the term superstructure gives only a qualitative, informal outline of the facts, the group-theoretical approach permits a precise treatment.

1.6.4.3. Isomorphic maximal subgroups

| top | pdf |

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 ([\geq 3^3]) 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[link])].

A classic example of a relation between isomorphic space groups concerns trirutile (Billiet, 1973[link]). The space group of rutile, [P4_2/mnm], has an isomorphic subgroup of index 3, but none of index 2. By triplication of c it becomes possible to substitute the titanium-atom positions of rutile by two different kinds of atoms in a ratio of 1:2, as for example in ZnSb2O6 (Fig. 1.6.4.4[link]; Byström et al., 1941[link]; Ercit et al., 2001[link]). Since the space group [P4_2/mnm] has no isomorphic subgroup of index 2, a `dirutile' with this space-group type cannot exist.

[Figure 1.6.4.4]

Figure 1.6.4.4 | top | pdf |

The group–subgroup relation rutile–trirutile. The twofold rotation axes have been included in the plots of the unit cells, showing that only one third of them are retained upon the symmetry reduction.

Note that rutile and trirutile have different space groups of the same space-group type.

1.6.4.4. The space groups of two structures having a common supergroup

| top | pdf |

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[link], 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.

β-K2CO3 and β-Na2CO3 have similar structures and unit cells (Fig. 1.6.4.5[link]; Jansen & Feldmann, 2000[link]). The planes of the car­b­on­ate ions are not aligned exactly perpendicular to c; compared to the perpendicular orientation, in the case of β-K2CO3, they are rotated about b by 22.8° and those of β-Na2CO3 are rotated about a by 27.3°. There is no group–subgroup relation between the space groups [C12/c1] and [C2/m11] of the two structures (the nonconventional setting [C2/m11] has been chosen for β-Na2CO3 to ensure a correspondence between the cells of both structures). Looking for common minimal supergroups of [C12/c1] and [C2/m11] one can find two candidates: Cmcm, No. 63, and Cmce, No. 64. Since the atomic coordinates of β-K2CO3 and β-Na2CO3 are very similar, any origin shifts in the relations from the common supergroup to [C12/c1] as well as [C2/m11] 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 [Cmce \rightarrow C12/c1] requires an origin shift of [\textstyle{1\over 4},\textstyle{1\over 4},0] (or [-\textstyle{1\over 4},-\textstyle{1\over 4},0]), while all other relations ([Cmcm \rightarrow] [C12/c1], [Cmcm \rightarrow C2/m11], [Cmce \rightarrow C2/m11]) require no origin shifts. As a consequence, only Cmcm and not Cmce can be the common supergroup.

[Figure 1.6.4.5]

Figure 1.6.4.5 | top | pdf |

The unit cells of β-K2CO3 and β-Na2CO3. The angles of tilt of the [{\rm CO}_3^{2-}] 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 [P\,6_3/mmc] and, in fact, α-K2CO3 and α-Na2CO3 are high-temperature 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[link]. In this case there exists a higher-symmetry 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.

[Figure 1.6.4.6]

Figure 1.6.4.6 | top | pdf |

Group–subgroup relations among some modifications of the alkali metal carbonates. Lattice parameters for α- and β-Na2CO3 are taken from Swainson et al. (1995[link]) and those for α- and β-K2CO3 are taken from Becht & Struikmans (1976[link]).

Among the alkali metal carbonates several other modifications are known which we do not discuss here.

1.6.4.5. Can a structure be related to two aristotypes?

| top | pdf |

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 β-tin structure (Si-II, [I4_1/amd]). Then it is transformed to silicon-XI (McMahon et al., 1994[link]). At even higher pressures it is converted to silicon-V forming a simple hexagonal structure ([P6/mmm]). The space group of Si-XI, Imma, is a subgroup of both [I4_1/amd] and [P6/mmm], and the structure of Si-XI can be related to either Si-II or Si-V (Fig. 1.6.4.7[link]). Assuming no distortions, the calculated coordinates of a silicon atom of Si-XI would be [0, {\textstyle{1 \over 4}},-0.125] when derived from Si-II, and [0, {\textstyle{1 \over 4}},0.0] when derived from Si-V. 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[link], the expected lattice parameters for Si-XI, calculated from those of Si-V, would be [a_{\rm XI}] = [a_{\rm V}\sqrt{3}] = 441.5 pm, [b_{\rm XI}] = [2c_{\rm V}] = 476.6 pm and [c_{\rm XI}] = [a_{\rm V}] = 254.9 pm.

[Figure 1.6.4.7]

Figure 1.6.4.7 | top | pdf |

The structure of the high-pressure modification Si-XI is related to the structures of both Si-II and Si-V.

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 Si-II. There are two separate experimentally observable phase transitions. In a certain pressure range, the whole crystal actually consists of stable Si-XI; it is not just a hypothetical intermediate. Taking all these facts together, a group-theoretical relation between Si-II and Si-V exists via the common subgroup of Si-XI.

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[link], at the end of Section 1.6.6[link] and in Section 1.6.7[link].

1.6.4.6. Treating voids like atoms

| top | pdf |

To comprehend the huge amount of known crystal-structure 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 FeF3 (VF3 type) can be derived from the ReO3 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 [Pm\overline3m] (Fig. 1.6.4.8[link]). This involves a symmetry reduction by two steps to a rhombohedral hettotype with the space group [R\overline3c] (Fig. 1.6.4.9[link], left).

[Figure 1.6.4.8]

Figure 1.6.4.8 | top | pdf |

The connected coordination octahedra in ReO3 and FeF3 (VF3 type). Light, medium and dark grey refer to octahedron centres at [z=0], [z=\textstyle{1\over 3}] and [z=\textstyle{2\over 3}], respectively (hexagonal setting).

[Figure 1.6.4.9]

Figure 1.6.4.9 | top | pdf |

Derivation of the FeF3 structure either from the ReO3 type or from the hexagonal closest packing of spheres. The coordinates for FeF3 given in the boxes are ideal values calculated from the aristotypes assuming no distortions. A y coordinate given as x means y = x. The Schottky symbol □ designates an unoccupied octahedral void.

FeF3 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[link], 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 FeF3, assuming no distortions, are x = 0.5 when derived from ReO3 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 FeF3 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 [c/(a\sqrt3)] ratio close to the ideal values of 0.333 and 1.633 (Table 1.6.4.1[link]).

Table 1.6.4.1| top | pdf |
Crystal data for FeF3 at different pressures

The observed lattice parameters a and c, the x coordinates of the F atoms and the angles of rotation of the coordination octahedra (0° = ReO3 type, 30° = hexagonal closest packing) are given (Sowa & Ahsbahs, 1998[link]; Jørgenson & Smith, 2006[link]).

Pressure /GPaa /pmc /pm[c/(a\sqrt3)]xAngle /°
10−4 520.5 1332.1 1.48 0.412 17.0
1.54 503.6 1340.7 1.54 0.385 21.7
4.01 484.7 1348.3 1.61 0.357 26.4
6.42 476 1348 1.64 0.345 28.2
9.0 469.5 1349 1.66 0.335 29.8

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 FeF3 or if phase transitions from the ReO3 type to the VF3 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[link] 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[link] are useful. For example, the occupation of the octahedral voids in the Wyckoff position 12c instead of 6b of [R\overline3c] (Fig. 1.6.4.9[link], lower right) results in the structure of corundum (α-Al2O3); this shows that both the VF3 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 MHx 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 FeF3 is depicted as a common subgroup of both the ReO3 type and the hexagonal closest packing of spheres. This would mix up two quite different points of view.

1.6.4.7. Large families of structures. Prediction of crystal-structure types

| top | pdf |

Large trees can be constructed using the modular way to put together Bärnighausen trees, as set forth in Fig. 1.6.3.1[link] 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[link] shows structures that can be derived from the ReO3 type (Bock & Müller, 2002[link]b). Many other trees of this kind have been set up, for example hettotypes of perovskite (Bärnighausen, 1975[link], 1980[link]; Bock & Müller, 2002[link]a), rutile (Baur, 1994[link], 2007[link]; Meyer, 1981[link]), CaF2 (Meyer, 1981[link]), hexagonal closest packed structures (Müller, 1998[link]), AlB2 (Pöttgen & Hoffmann, 2001[link]), zeolites (Baur & Fischer, 2000, 2002, 2006[link]) and tetraphenylphosphonium salts (Müller, 1980[link], 2004[link]).

[Figure 1.6.4.10]

Figure 1.6.4.10 | top | pdf |

Bärnighausen tree of hettotypes of the ReO3 type. For the atomic parameters and other crystallographic data see Bock & Müller (2002b[link]). [F\overline32/c] and [F\overline3] are nonconventional face-centred settings of [R\overline32/c] and [R\,\overline3], respectively, with rhombohedral-axes setting and nearly cubic metric, [F\overline1] is a nearly cubic setting of [P\overline1]. Every space-group symbol corresponds to one space group (not space-group type) belonging to a specific crystal structure. Note that the vertical distances between space-group symbols are proportional to the logarithms of the indices.

In the left branch of Fig. 1.6.4.10[link] only one compound is mentioned, WO3. 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 crystal-structure 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[link],b[link],c[link]; Müller, 1992[link]).

Examples of studies of structural possibilities include com­pounds AX3, AX6, AaBbX6 (with the X atoms forming the packing of spheres and atoms A and B occupying the octahedral voids; [a+b \,\lt\, 6]; Müller, 1998[link]), molecular compounds (MX5)2 (Müller, 1978[link]), chain structures MX4 (Müller, 1981[link]) and MX5 (Müller, 1986[link]), wurtzite derivatives (Baur & McLarnan, 1982[link]) and NaCl derivatives with doubled cell (Sens & Müller, 2003[link]).

References

Bärnighausen, H. (1975). Group–subgroup relations between space groups as an ordering principle in crystal chemistry: the `family tree' of perovskite-like structures. Acta Cryst. A31, part S3, 01.1–9.
Bärnighausen, H. (1980). Group–subgroup relations between space groups: a useful tool in crystal chemistry. MATCH Commun. Math. Chem. 9, 139–175.
Baur, W. H. (1994). Rutile type derivatives. Z. Kristallogr. 209, 143–150.
Baur, W. H. (2007). The rutile type and its derivatives. Crystallogr. Rev. 13, 65–113.
Baur, W. H. & Fischer, R. X. (2000, 2002, 2006). Landolt–Börnstein, Numerical data and functional relationships in science and technology, New Series. Group IV, Vol. 14, Zeolite type crystal structures and their chemistry. Berlin: Springer.
Baur, W. H. & McLarnan, T. J. (1982). Observed wurtzite derivatives and related tetrahedral structures. J. Solid State Chem. 42, 300–321.
Billiet, Y. (1973). Les sous-groupes isosymboliques des groupes spatiaux. Bull. Soc. Fr. Minéral. Cristallogr. 96, 327–334.
Bock, O. & Müller, U. (2002a). Symmetrieverwandtschaften bei Varianten des Perowskit-Typs. Acta Cryst. B58, 594–606.
Bock, O. & Müller, U. (2002b). Symmetrieverwandtschaften bei Varianten des ReO[_3]-Typs. Z. Anorg. Allg. Chem. 628, 987–992.
Byström, A., Hök, B. & Mason, B. (1941). The crystal structure of zinc metaantimonate and similar compounds. Ark. Kemi Mineral. Geol. 15B, 1–8.
Ercit, T. S., Foord, E. E. & Fitzpatrick, J. J. (2001). Ordoñezite from the Theodoso Soto mine, Mexico: new data and structure refinement. Can. Mineral. 40, 1207–1210.
Hoffmann, W. & Jäniche, W. (1935). Der Strukturtyp von AlB2. Naturwissenschaften, 23, 851.
Iandelli, A. (1964). MX2-Verbindungen der Erdalkali- und seltenen Erdmetalle mit Gallium, Indium und Thallium. Z. Anorg. Allg. Chem. 330, 221–232.
Jansen, M. & Feldmann, C. (2000). Strukturverwandtschaften zwischen cis-Natriumhyponitrit und den Alkalimetallcarbonaten M2CO3 dar­gestellt durch Gruppe-Untergruppe Beziehungen. Z. Kristallogr. 215, 343–345.
Lima-de-Faria, 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.
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.
McMahon, M. I., Nelmes, R. J., Wright, N. G. & Allan, D. R. (1994). Pressure dependence of the Imma phase of silicon. Phys. Rev. B, 50, 739–743.
Meyer, A. (1981). Symmmetriebeziehungen zwischen Kristallstrukturen des Formeltyps AX2, ABX4 und AB2X6 sowie deren Ordnungs- und Leerstellenvarianten. Dissertation, Universität Karlsruhe.
Müller, U. (1978). Strukturmöglichkeiten für Pentahalogenide mit Doppeloktaeder-Molekülen (MX5)2 bei dichtester Packung der Halogen­atome. Acta Cryst. A34, 256–267.
Müller, U. (1980). Strukturverwandtschaften unter den EPh[_4^+]-Salzen. Acta Cryst. B36, 1075–1081.
Müller, U. (1981). MX[_4]-Ketten aus kantenverknüpften Oktaedern: mögliche Kettenkonfigurationen und mögliche Kristallstrukturen. Acta Cryst. B37, 532–545.
Müller, U. (1986). MX[_5]-Ketten aus eckenverknüpften Oktaedern. Mögliche Kettenkonfigurationen und mögliche Kristallstrukturen bei dichtester Packung der X-Atome. Acta Cryst. B42, 557–564.
Müller, U. (1992). Berechnung der Anzahl möglicher Strukturtypen für Verbindungen mit dichtest gepackter Anionenteilstruktur. I. Das Rechenverfahren. Acta Cryst. B48, 172–178.
Müller, U. (1998). Strukturverwandtschaften zwischen trigonalen Verbindungen mit hexagonal-dichtester Anionenteilstruktur und besetzten Oktaederlücken. Z. Anorg. Allg. Chem. 624, 529–532.
Müller, U. (2004). Kristallographische Gruppe-Untergruppe-Beziehungen und ihre Anwendung in der Kristallchemie. Z. Anorg. Allg. Chem. 630, 1519–1537.
Nielsen, J. W. & Baenziger, N. C. (1953). The crystal structures of ZrBeSi and ZrBe2. US Atom. Energy Comm. Rep. pp. 1–5.
Nielsen, J. W. & Baenziger, N. C. (1954). The crystal structures of ZrBeSi and ZrBe2. Acta Cryst. 7, 132–133.
Pöttgen, R. & Hoffmann, R.-D. (2001). AlB[_2]-related intermetallic compounds – a comprehensive view based on a group–subgroup scheme. Z. Kristallogr. 216, 127–145.
Sens, I. & Müller, U. (2003). Die Zahl der Substitutions- und Leerstellenvarianten des NaCl-Typs bei verdoppleter Elementarzelle (a, b, 2c). Z. Anorg. Allg. Chem. 629, 487–492.
Wendorff, M. & Roehr, C. (2005). Reaktionen von Zink- und Cadmiumhalogeniden mit Tris(trimethylsilyl)phosphan und Tris(trimethylsilyl)arsan. Z. Anorg. Allg. Chem. 631, 338–349.








































to end of page
to top of page