Translationengleiche subgroup

Translationengleiche subgroups have an unaltered translation lattice, i.e. the translation vectors and therefore the size of the primitive unit cells of group and subgroup coincide. The symmetry reduction in this case is accomplished by the loss of other symmetry operations, for example by the reduction of the multiplicity of symmetry axes. This implies a transition to a different crystal class. The example on the right in Fig. 18.1 shows how a fourfold rotation axis is converted to a twofold rotation axis when four symmetry-equivalent atoms are replaced by two pairs of different atoms the translation vectors are not affected. 

WC is the technically most important transition metal carbide. It adopts a very simple structure (Fig. 11, space group P6m2) [44,45], which may be described as a defect-AlB2 structure in which every other boron position is unoccupied. This results in a symmetry reduction. In space group P6m2 (a translationengleiche subgroup of index 2 (t2) of P6/mmm) the two-fold 2d position of P6/mmm is reduced to two one-fold positions. This allows the introduction of ordered defects. The... 

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... 

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. 

The data are a small selection from Volume A of the "International Tables for Crystallography". Type I represents translationengleiche or t subgroups, type II klassengleiche or /c subgroups. 

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