Subgroup of space group

SELECTION OF MAXIMAL SUBGROUPS OF SPACE GROUPS RELEVANT FOR THE EXAMPLES PRESENTED IN THE TEXT. 

As an example we consider the case of the naphthalene crystal, G ft being its space group. The point group C-2h (see Table 2.1) has only one-dimensional representations. Since any subgroup of the group C h can also have only onedimensional representations, it is clear that in crystals of naphthalene type the compulsory degeneracy for excitonic states inside the first Brillouin zone is not possible. 

In this and the next subsections we consider induced reps of space groups generated by the irreps (3 of the site-symmetry subgroup Gq of point q in the direct lattice space (/ 1 G). This type of induced rep of space groups was considered in [13]. In [39] a more general concept of band reps was introduced. All elements of the space group G can be written in the form VB+a , where Vr is a fractional lattice translation associated with the F point-group element R and a is a direct lattice translation vector. 

A further complication is that an icosahedron sharing four faces loses its symmetry operations of the second sort and becomes chiral, D or L. The eclipsed configuration described above requires that D and l alternate. Hence, in a ring of five icosahedra one DD or LL bond must occur, with a strain in the ring of 40°, which is a rotation of 4 per shared face in each icosahedron. The space-group symmetry is thus changed from that of diamond to a subgroup. 

With the chemical structure of PbTX-1 finally known and coordinates for the molecule available from the dimethyl acetal structure, we wanted to return to the natural product crystal structure. From the similarities in unit cells, we assumed that the structures were nearly isomorphous. Structures that are isomorphous are crystallographically similar in all respects, except where they differ chemically. The difference between the derivative structure in space group C2 and the natural product structure in P2. (a subgroup of C2) was that the C-centering translational symmetry was obeyed by most, but not all atoms in the natural product crystal. We proceeded from the beginning with direct methods, using the known orientation of the PbTX-1 dimethyl acetal skeleton (assuming isomorphism) to estimate phase... 

The similarity of the structures of rutile, CaCl2 and marcasite also comes to light by comparison of their crystal structure data (Table 17.2). The space groups of CaCl2 and marcasite (both Pnnm) are subgroups of the space group of rutile. The tetragonal sym-... 

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 ... 

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