Symmetry of Crystals

A. Pullet, J.-P. Matie Vibration spectra and symmetry of crystals (translation in to Russian) Mir, Moscow, 1974. 

Since the surfaces of crystals have specific symmetries (usually triangular, square, or tetragonal) and indenters have cylindrical, triangular, square, or tetragonal symmetries, the symmetries rarely match, or are rotationally misaligned. Therefore, the indentations are often anisotropic. Also, the surface symmetries of crystals vary with their orientations relative to the crystallographic axes. A result is that crystals cannot be fully characterized by single hardness numbers. 

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. 

The symmetry elements indicated above can be used to describe the external symmetry of crystals. More elements have been described than are actually necessary for the description of all cases. Thus the centre of symmetry is now no longer used as a fundamental element and the inversion axes are used instead. . . d - interplanar distance in A lattice - a network of points used to define the geometry of a crystal... 

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. 

Poinl symiretry of individial molecules to be contrasted with the inuuloiional symmetry of crystals, to be disciisa-d later in this chapter. 

A study of the external symmetry of crystals naturally leads to the idea that a single crystal is a three-dimensional periodic structure i.e., it is built of a basic structural unit that is repeated with regular periodicity in three-dimensional space. Such an infinite periodic structure can be conveniently and completely described in terms of a lattice (or space lattice), which consists of a set of points (mathematical points that are dimensionless) that have identical environments. 

Symmetry of crystal field Symmetry notation Crystal field states Mineral examples... 

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... 

What is less understood is the extent to which the presence of a LP is responsible for orientation effects in crystals and, more generally, for determining the type, compactness, and symmetry of crystal packing. The notion that a stereochemically active LP should have spatial extension implies that significant electron density is associated, locally, with the LP, and that this electron density extends radially and in a particular direction from the parent atom over an appreciable portion of space in the crystal. This effectively excludes other atoms from occupation of this space and creates a void in the crystal structure. 

For a centrosymmetric structure, Uhki can equal only 0 or 180° (i.e. F having signs + or respectively), for a noncentrosymmetric structure it can have any value. Owing to the translational symmetry of crystals, p xyz) can be considered as a periodic function in three-dimensional space and expressed by the Fourier transform of equation (8),... 

Barlow, W. Probable nature of the internal symmetry of crystals. Nature (London) 29, 186-188, 205-207 (1883). 

Friedel, G. Sur les symetries cristallines que pent reveler la diffraction des rayons Rdntgen. [Concerning symmetries of crystals that can be revealed by X-ray diffraction.] Comptes Rendus, Acad. Sci. (Paris) 157, 1533-1536 (1913). 

Friedel, G. Sur les symetries cristallines que peut reveler la diffraction des rayons Rontgen. [Concerning symmetries of crystals that can be revealed by X-ray diffraction.] Comptes Rendus, Acad. Sci. (Pans) 157, 1533-1536 (1913). Bijvoet, J. M., Peerdeman, A. F., and van Bommel, A. J. Determination of the absolute configuration of optically active compounds by means of X rays. Nature London) 168, 271-272 (1951). 

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... 

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