Point group of symmetry

A. Cayley was also the creator of the matrix theory he made an essential contribution to the development of the group theory, i.e. those branches of mathematics which were later extensively used in physics and mathematics. Moreover, Cayley was the first to indicate the relationship between the point groups of symmetry and the permutation groups (see Chapter 6). 

Each molecule (or conformation) belongs to a definite point group of symmetry and each point group of symmetry includes a set of symmetry operations which are transformations leaving the whole system in a position equivalent to the initial one identity, rotation, mirror reflection, inversion, mirror rotation. The various groups of symmetry are ... 

Molecular symmetry originates in the fact that there exist symmetry operations (transformations of the nuclear coordinates) which transform the molecule into a nuclear configuration identical with an initial one. The symmetry elements (axis, plane, inversion centre) remain unchanged. Molecules belong to the point groups of symmetry as all the symmetry operations have at least one point in common (this point is not necessarily identified with any atom of the molecule) [5, 12, 19-38]. 

Hoi = number of orientation domains go = order of the point group of symmetry gi Si = structures indicated by subscript... 

The fluorite structure and the C type have a common cubic lattice which is a superlattice of the fluorite structure, and which is the lattice for the C structure, so that the symmetry elements of C are also symmetry elements of the fluorite structure. This is the situation of the well-known order-disorder transformation with a common lattice. In this case the number of orientation domains is fioi = order of go/order of g, = 48/24 = 2, a relationship which can be deduced from the more general one noi = order of go/order of ga fl gi because in this case goH gi = gi, where go and gi are the point groups of symmetry, respectively, of... 

The maximum numbers of hgand-field (or Stark) sublevels depend on the point group of symmetry they are given in Table 3 versus values of J. This can be exploited for the determination of the symmetry point group from f-f absorption or emission spectra, at least when J is integer. 

The symmetry of crystals is completely described by the set of symmetry elements m, 1,2,3,4,6,1,3,4,6. They can be combined into 32 classes of point groups of symmetry, see Table 2.1. 

Table 2.1 Symbols and names of 32 point groups of symmetry...

The set of operations of symmetry of a crystal forms its group of symmetry G called a space group of symmetry. Group G includes both translations, operations from point groups of symmetry, and also the combined operations. The structure of space groups of symmetry of crystals and their irreducible representations is much... 

The transition vector cannot belong to a degenerate representation of the point group of symmetry of the transition state. 

The collection of symmetry elements at any point of the lattice is termed as the point symmetry or the point group of symmetry. 

Conclusion The point group of symmetry of a crystal is that collection of macroscopic symmetry elements which occurs at every lattice point of the space lattice of the crystal taking into consideration that point group of a lattice may be different from the point symmetry of the actual crystal itself as a consequence of the shape of the motif (atoms or molecules). 

A detailed analysis of the compatibility of the ten sjmimetry elements in all possible combinations is beyond the scope of this book. It can be analyzed that out of a large number of possible groupings of these ten synunetry elements, only 32 are compatible combinations of one or more of them and thus there exist only 32 point groups of symmetry (Fig. 4.1). 

Thirty-Two Point Groups of Symmetries in Hermann—Mauguin Notations... 

Note The group of SI. No. 32 is the most highly symmetrical of all the point group of symmetry. 

A particular crystal system has some definite number of point groups and for this monoclinic system it has symmetry operations like 2, m, and 2/m, that is, twofold rotation, a mirror plane, and twofold with mirror plane of symmetries. Now, for three-dimensional crystal the possible symmetry elements will include also screw axes and glide planes, and when screw axes and glide planes are added to the point group of symmetries for this system, we can say that different possibilities that may exist are 2, 2i, m, c, 2/m, 2i/m, 2/c, and 2j/c. Now each of these symmetry groups are repeated by lattice translation of the Bravais lattices of that system. As monoclinic system has only primitive P and C, all the symmetry possibilities may be associated with both P and C. Therefore, if they are worked out, they come out to be 13 in number and they are Pm, Pc, Cm, Cc, P2, P2i, C2, P2/m, P2i/m, C2/m, P2/c, P2i/c, C2/c, etc. 

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