Faces, crystallographic notation

As a complement to this section on crystallography, some remarks on the old notations are given. In the nineteenth-century treatises, the crystallographic notations are varied and difficult to understand. However C. Friedel (1893) recorded all notations used from Haiy onward. For all the simple forms, he compared five different modes (Levy, Weiss, Naumann, Dana, Miller) which could be reduced substantially to two different schemes. Some crystallographers assigned a symbol to a simple form (e.g., p is cube in Levy, O is octahedron in Naumann), whereas others referred to the intercepts of the faces on axes and assigned some numbers to each face in comparison with a reference face, with direct (Weiss) or inverse (Miller) proportionality. The following table, which reports the various symbols for some simple forms of the cubic system, clarifies these differences. 

Obviously, much of the development of crystallography predates the discovery of diffraction of X-rays by crystals. Early studies of crystal structures were concerned with external features of crystals and the angles between faces. Descriptions and notations used were based on these external features of crystals. Crystallographers using X-ray diffraction are concerned with the unit cells and use the notation based on the symmetry of the 230 space groups established earlier. 

Tetrakis(hydroxymethyl)methane, [C(CH20H)4, also known as pen-taerythritol], is one of the simpler substitution products of methane for which structures have been determined. The tetragonal cell contains two molecules, S4,14, a0 6.083, c0 = 6.726 A. The structure shown in Figure 11.10 is body centered, 3 2PTOT(t). At higher temperatures, between 179.5 and 260.5°C, this body-centered structure is transformed to a face-centered cubic structure. The cubic cell (a0 = 8.963 A at 230°C) contains four molecules. It has been proposed that at the higher temperature the molecules achieve a statistical orientation that makes them effectively spherical crystallographically. The notation is 3P. 

The generally accepted notation for Miller indices is that Qikl) represents a crystal face or lattice plane, while hkl represents a crystallographic form comprising all faces that can be derived from hkl by symmetry operations of the crystal. 

Miller indices The notations usually used for naming crystal faces they have the form hkl. These notations are based on the assignment of crystallographic axes and on an expression of the intercepts of the face on the three axes (hexagonal has four.)... 

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