The symmetry of crystals

The crystals of wulfenite appear to be shaped like barrels with square sides crystals of vanadinite have a hexagonal form the crystals of the magnetite specimen are cubic. These differences are a result of differences in the compounds at the atomic level that is, they are produced by the particular arrangements of the atoms. Those arrangements are reproduced as the crystal grows. 

In two dimensions there are five possible units of pattern, unit cells, which can build a repetitive pattern by translation in directions parallel to the edges (Fig. 27.13). The unit cell with the 120° angle is interesting because it permits a threefold or sixfold axis of symmetry at a point in the pattern, which is a permissible type of symmetry in two-dimensional patterns. 

To generate a repetitive pattern in three dimensions, an additional repetition vector out of the plane of the first two must be added. The three vectors define a parallelepiped. Any repetitive pattern in three dimensions has a parallelepiped as a unit cell. There are seven distinct parallelepipeds, those labeled (P) or (R) in Fig. 27.19 (on p. 696), which can generate by translation any repetitive pattern in three dimensions. Crystals are classified 

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It is the arrangement and symmetry of the ensemble of the atomic nuclei in the molecule that is considered to be the geometry and the symmetry of the molecule. The molecules are finite structures with at least one singular point in their symmetry description and, accordingly, point groups are applicable to them. There is no inherent limitation on the available symmetries for molecules whereas severe restrictions apply to the symmetries of crystals, at least in classical crystallography. 

Liquids are difficult to model because, on the one hand, many-body interactions are complicated on the other hand, liquids lack the symmetry of crystals which makes many-body systems tractable [364, 376, 94]. No rigorous solutions currently exist for the many-body problem of the liquid state. Yet the molecular properties of liquids are important for example, most chemistry involves solutions of one kind or another. Significant advances have recently been made through the use of spectroscopy (i.e., infrared, Raman, neutron scattering, nuclear magnetic resonance, dielectric relaxation, etc.) and associated time correlation functions of molecular properties. 

Solids. In general, solids are somewhat more tractable many-body systems than liquids because of the symmetry of crystals. Induced effects are certain to exist because of the density of solids. 

Just as there is a correspondence between the symmetry of crystals and that of their physical properties, there is also a connection between the symmetry exhibited by a crystal at the macroscopic and microscopic length scales, in other words, between the external crystal morphology and tme internal crystal structure. Under favorable circumstances, the point group (but not the space group) to which a crystal belongs can be determined solely by examination of... 

We now come to the second point concerning plane patterns. An isolated object (for example, a polygon) can possess any kind of rotational symmetry but there is an important limitation on the types of rotational symmetry that a plane repeating pattern as a whole may possess. The possession of n-fold rotational symmetry would imply a pattern of -fold rotation axes normal to the plane (or strictly a pattern of -fold rotation points in the plane) since the pattern is a repeating one. In Fig. 2.4 let there be an axis of -fold rotation normal to the plane of the paper at /, and at Q one of the nearest other axes of -fold rotation. The rotation through Ivjn about Q transforms P into F and the same kind of rotation about P transforms Q into Q. It may happen that P and Q coincide, in which case n = 6. n all other cases PQ must be equal to, or an integral multiple of, PQ (since Q was chosen as one of the nearest axes), i.e. 4. The permissible values of n are therefore 1, 2, 3, 4, and 6. Since a 3-dimensional lattice may be regarded as built of plane nets the same restriction on kinds of symmetry applies to the 3-dimensional lattices, and hence to the symmetry of crystals. 

The application of Miller indices allowed crystal faces to be labelled in a consistent fashion. This, together with accurate measurements of the angles between crystal faces, allowed the morphology of crystals to be described in a reproducible way, which, in itself, lead to an appreciation of the symmetry of crystals. Symmetry was broken down into a combination of symmetry elements. These were described as mirror planes, axes of rotation, and so on, that, when taken in combination, accounted for the external shape of the crystal. The crystals of a particular mineral, regardless of its precise morphology, were always found to possess the same symmetry elements. 

At the end of chapter 1, an inherent difficulty became apparent. How is it possible to conveniently specify a crystal structure in which the unit cell may contain hundreds or even thousands of atoms In fact, crystallographers make use of the symmetry of crystals to reduce the list of atom positions to reasonable proportions. However, the application of symmetry to crystals has far more utility than this accountancy task. The purpose of this chapter is to introduce the notions of symmetry, starting with two-dimensional patterns. 

Crystallography31 The symmetry of crystals not only involves the individual point group symmetry of... 

Our book has a simple structure. After the introduction (Chapter 1), the simplest symmetries are presented using chemical and nonchemical examples (Chapter 2). Molecular geometry is then discussed in qualitative terms (Chapter 3). Group-theoretical methods (Chapter 4) are applied in an introductory manner to the symmetries of molecular vibrations (Chapter 5), electronic structure (Chapter 6), and chemical reactions (Chapter 7). These chapters are followed by a descriptive discussion of space-group symmetries (Chapter 8), including the symmetry of crystals (Chapter 9). 

The symmetries of crystals can be divided into six basic types or crystal systems (Table 6) ... 

It is common to use rotation-inversion axes (rather than rotation-reflection axes) to classify the symmetry of crystals. Any S axis is equivalent to a rotation-inversion axis (symbolized by p) whose order p may differ from n. A rotation-inversion operation consists of rotation by 2ir/p radians followed by inversion. Show that... 

Describing the symmetry of crystals is often more complicated than that of solid shapes such as the cube in Figure 5.3. For example, the crystal may have a cubic shape and belong to the cubic crystal system but not have the maximum internal symmetry. 

One could argue that this phase of development in solid state physics was concluded by the publication of Voigt s famous textbook in 1910. The underlying reason is the importance of the symmetry of crystals for all other properties. However, a good imderstanding of the thermodynamics is of utmost importance for the solution of all problems in the physics of crystals. We could say that for a physicist further development of thermodynamics is less important than the development of microscopic models. For all technical appUcations, however, thermodynamics is of special importance. 

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. 

Of course, the main feature of a crystal, and what distinguishes it from other solids, is that it contains order. Conceptually, we describe crystals as being formed from a perfectly ordered array of atoms or molecules. This order has a helpful consequence by describing a small portion of the structure and the symmetry of crystal we can map out the atomic positions of an infinite lattice. The power of this approach is beguiling. It allows us to map all of the atoms in a crystal, which may be metres in... 

Molecular modeling is extensively covered in other articles, and since the interactions between molecules in crystals are the same as in other aggregation states of matter, no really new principles are involved. Therefore, the following discussion is limited to subjects that are direcdy related to the symmetry of crystals. 

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