Symmetry group/subgroup

Since S is a point symmetry group (subgroup of 0(3)), it has the property... 

The most important materials among nonlinear dielectrics are ferroelectrics which can exhibit a spontaneous polarization PI in the absence of an external electric field and which can spHt into spontaneously polarized regions known as domains (5). It is evident that in the ferroelectric the domain states differ in orientation of spontaneous electric polarization, which are in equiUbrium thermodynamically, and that the ferroelectric character is estabUshed when one domain state can be transformed to another by a suitably directed external electric field (6). It is the reorientabiUty of the domain state polarizations that distinguishes ferroelectrics as a subgroup of materials from the 10-polar-point symmetry group of pyroelectric crystals (7—9). 

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. 

Problem 3-3. Find all subgroups of the symmetry group of the water molecule, in two ways (a) For each symmetry element, try to find a specific conformation of a related molecule, or of any geometric object, that has all the symmetry elements except the one considered. 

The symmetry group of that molecule or object is a subgroup of the symmetry group of the water molecule. 

Although the symmetric stretch and the bend have the full symmetry of the molecule, the antisymmetric stretch does not the only covering operations of this mode are E and cr. The symmetry group of each normal mode is either the entire symmetry group of the undistorted molecule, or a subgroup. 

These examples suggest the correct result The possible symmetry types, either for normal modes or electronic wave functions, that are compatible with an overall molecular symmetry, correspond to the full molecular symmetry group or its subgroups. Each normal mode, or electronic state, can be classified... 

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