Symmetries in Crystals

Crystals are characterized by a certain regularity of arrangements called symmetry. 

Center of symmetry—every face has a similar face parallel to it. 

Plane of symmetry—the crystal is divided into two equal portions, and these two portions are mirror images of each other with respect to the plane. 

Axis of symmetry—a line such that after rotation about it through 360°In the crystal assumes a congruent position the value of n determines the degree of the axis. 

A single crystal is a homogeneous solid, which means that all parts within it have identical properties. However, it is not in general isotropic, so that physical properties such as thermal and electrical conductivity, refractive index, and non-linear optical effect generally vary in different directions. 

The symmetry elements and point groups of molecules and ions in the free state have been discussed in Sec. 1.5. For molecules and ions in crystals, however, it is necessary to consider some additional symmetry operations that characterize translational symmetries in the lattice. Addition of these translational operations results in the formation of the space groups that can be used to classify the symmetry of molecules and ions in crystals. 

TABLE 1.17. Seven Crystallographic Systems and 32 Crystallographic Point Groups 

System Axes and Angles 32 Crystallographic Point Groups  

A crystal has a lattice structure that is a repetition of identical units. Bravais has shown that only the 14 different types of lattices shown in Fig. 1.40 are possible. These 

Type of Crystal Structure Number of Lattice Points (LP) 

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Symmetry properties which have so far been successfully treated by the projection operator method, include translational symmetry in crystals, cyclic systems, spin, orbital and total angular momenta, and further applications are in progress. ... 

Based on extensive studies of the symmetry in crystals, it is found that crystals possess one or more of the ten basic symmetry elements (five proper rotation axes 1,2,3, 4,6 and five inversion or improper axes, T = centre of inversion i, 2 = mirror plane m, I, and 5). A set of symmetry elements intersecting at a common point within a crystal is called the point group. The 10 basic symmetry elements along with their 22 possible combinations constitute the 32 crystal classes. There are two additional symmetry... 

It is now time to show how the ideas developed in the previous chapters can be applied to real chemical systems. Apart from a few simple gases, the materials we come across in everyday life are either solids or liquids. A proper understanding of the chemistry of the solid state requires some appreciation of the role of symmetry in crystals and is therefore deferred to Part III. This chapter explores the use of bond valences to understand the simpler chemistry of liquids. Most of this chapter is devoted to the chemistry of aqueous solutions because water is not only the solvent of choice for polar systems but also the most common solvent in our environment. 

It seems that, in its most widely used forms at any rate, the AOM involves such severe approximations and draws on empirical information to such an extent that it cannot be regarded as a proper implementation of quantum mechanics. Nevertheless, as a form of ligand field theory, it possesses distinct advantages and leads to a novel parameterization scheme which promises some degree of transferability of parameters with a metal-ligand bond. This last feature is entirely lacking, at least outside cubic symmetry, in crystal or ordinary ligand field treatments. 

Application of space group symmetry in crystal structure determination... 

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