By Dr. Ganesan Venkataraman, Dr. Debendranath Sahoo, Dr. Venkataraman Balakrishnan (auth.)

ISBN-10: 3642834345

ISBN-13: 9783642834349

ISBN-10: 3642834361

ISBN-13: 9783642834363

Condensed subject indicates a wealthy number of levels. of those, the crystalline nation has, till lately, got so much recognition. this isn't unbelievable, given the geometric regularity of crystals. on the different severe one has amorphous fabrics. In among there are a few of the sorts of liquid crystals, the lately found quasicrystals, and so forth. whereas the absence of the excessive measure of regularity that characterizes the crystalline part is unquestionably an issue, those noncrystalline states have however been receiving a few awareness through the years. in spite of the fact that, it's only over the past few years that anything like a uni fied view of a lot of these stages has began to emerge, via an software of varied subtle thoughts. Geometry and symmetry (and strange realiza tions of the latter) supply a unifying thread during this new and rising perspec tive. This ebook is an try and catch the flavor of a few of those fresh de velopments. The strategy is considerably descriptive, being meant to be available not just to experimental physicists, but additionally to chemists, fabrics scientists, metallurgists and ceramicists, whose paintings borders on physics. the must haves for a learn of this publication are a familiarity with simple solid-state physics and, in locations, the weather of crew thought and statistical mechanics. a number of precise issues are incorporated on the finish to assist those that desire to pur certain extra the subject material taken care of here.

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Our reference states are perfect crystals in various orientations (Fig. 9), and our objective is to describe the spatial pattern of the defective crystal using the reference "templates". Since the reference states are related to each other by rigid rotations, we might as well retain just one reference template and examine what rigid-body operations must be performed on it to bring it into coincidence with the local structure of the defective medium at various space points r. Now, the set of all rigid-body operations of the reference frame forms a group G containing translations as well as rotations.

As in the case ofS 2 , one may regard S3 as being embedded in E 4 , and replace geodesics associated with the tessellations (of S3) by straight chords. One then obtains the 4D version of a regular Euclidean polyhedron. This object is called a regular polytope. A polytope is thus the next in the sequence point, line, polygon, polyhedron, . . 2 Regular Polytopes 51 As in the case ofregular polyhedra, the same SchIatli symbol {p, q, r} is used for both a regular tessellation (ofS 3) and the corresponding regular polytope (in E 4 ).

7 Overview In this chapter we have been concerned mainly with the different ways in which atoms and molecules are arranged in space in various phases of condensed matter. All structures can be characterized in terms of the correlation functions G(r) and Gn(r). Different phases can be distinguished (to some degree) by the extent to which translational and orientational correlations prevail. Some of these distinctions are illustrated in Fig. 12 which compares the radial distribution functions for a 2D hexagonal lattice, a Penrose tiling and a 2D hard disc gas.

### Beyond the Crystalline State: An Emerging Perspective by Dr. Ganesan Venkataraman, Dr. Debendranath Sahoo, Dr. Venkataraman Balakrishnan (auth.)

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