Group-subgroup relations

Examples for translationengleiche group-subgroup relations left, loss of reflection planes right, reduction of the multiplicity of a rotation axis from 4 to 2. The circles of the same type, O and , designate symmetry-equivalent positions... 

A suitable way to represent group-subgroup relations is by means of family trees which show the relations from space groups to their maximal subgroups by arrows pointing downwards. In the middle of each arrow the kind of the relation and the index of the symmetry reduction are labeled, for example ... 

In all cases we start from a simple structure which has high symmetry. Every arrow (= -) in the preceding examples marks a reduction of symmetry, i.e. a group-subgroup relation. Since these are well-defined mathematically, they are an ideal tool for revealing structural relationships in a systematic way. Changes that may be the reason for symmetry reductions include ... 

The group-subgroup relation of the symmetry reduction from diamond to zinc blende is shown in Fig. 18.3. Some comments concerning the terminology have been included. In both structures the atoms have identical coordinates and site symmetries. The unit cell of diamond contains eight C atoms in symmetry-equivalent positions (Wyckoff position 8a). With the symmetry reduction the atomic positions split to two independent positions (4a and 4c) which are occupied in zinc blende by zinc and sulfur atoms. The space groups are translationengleiche the dimensions of the unit cells correspond to each other. The index of the symmetry reduction is 2 exactly half of all symmetry operations is lost. This includes the inversion centers which in diamond are present in the centers of the C-C bonds. 

Group-subgroup relation diamond-zinc blende... 

Group-subgroup relations from hexagonal closest-packing of spheres to some MX3 and M2X3 structures. The boxes represent octahedral voids, with the coordinates as given at the top left. The positions of the octahedron centers are labeled by their Wyckoff letters. Gray boxes refer to occupied voids. The dots indicate how the atoms Ru, P and N are shifted from the octahedron centers parallel to c... 

Group-subgroup relation between the modifications of calcium chloride (cf Fig. 4.1, p. 33)... 

Group-subgroup relations and the emergence of twins in the phase transition /3-quartz — a-quartz (Dauphine twins)... 

H. Barnighausen, Group-subgroup relations between space groups a useful tool in crystal chemistry. MATCH, Commun. Math. Chem. 9 (1980) 139. 

GROUP-SUBGROUP RELATIONS BETWEEN SPACE GROUPS FOR THE REPRESENTATION OF CRYSTAL-CHEMICAL RELATIONSHIPS... 

Phase transitions. Examples BaTiO (> 120°C, cubic perovskite type) -y BaTiOj (< 120°C, tetragonal), cf. Fig. 19.5, p. 230 CaCl2 (> 217°C, rutile type) CaCl2 (< 217°C), cf. Fig. 4.1, p. 33. For second-order phase transitions it is mandatory that there is a group-subgroup relation between the involved space groups (Section 18.4). 

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