Generalized crystallography

One of the main points in the papers by Mermin and his collaborators [9 - 18] is the insistence on the primacy of reciprocal space. The properties of the Fourier transform of the density rather than the density itself determine those properties which are of importance for "generalized" crystallography. As pointed out by Mermin that point view was stressed in a paper by Bienenstock and Ewald already in 1962 [26]. 

Dermer, W, Schulz, H. and D Amour, H. (1979) Acta Crystallographica. Section A, Crystal Physics, Diffraction, Theoretical and General Crystallography, 35, 360. 

Details about general crystallography of the Laves phases and alternative descriptions and their polytypes and variants are given in 7.4.3.3 whereas some information about their occurrence and thermodynamics is reported in 3.9.3.1.3. 

General crystallography, which includes powder diffraction. Quite sparse in activity but a good place to check for software announcements, jobs on offer and communal problem software... 

The appearance of quasicrystals has caused a minirevolution in crystallography. The lack of periodicity was a major obstacle in applying the traditional terms and approaches to this domain of materials. This was an interesting development also from the point of view of Mackay s suggestions for generalized crystallography (cf. Section 9.7). He truly anticipated the breakdown of the perfect traditional system, which he felt was a little too perfect. 

DE Jong, W.F. (1959) General Crystallography. A Brief Compendium. W.H. Freeman and Co., San Francisco. Donohue, J. (1974) The Structures of the Elements. John Wiley Sons, New York. 

It is beyond the scope of this method to describe the different types of crystalline materials that can be found, or the full description of the classes into which they can fall. However, some general crystallography is provided below to give a foundation to the procedures described. 

Yada, K (1971) Study of microstructure of chrysotile asbestos by high resolution electron microscopy. Acta Crystallographica Section a-Crystal Physics Diffraction Theoretical and General Crystallography, A 2C1,659-. ... 

Billingham, J., Bell, P. S., Lewis, M. H. (1972). Vacancy Short-Range Order in Sub-stoichiometric Transition Metal Carbides and Nitrides withNaCl Structure. I. Electron Diffraction Studies of Short-Range Ordered Corrpounds. Acta Crystallographica. Section A, Crystal Physics, Diffraction, Theoretical and General Crystallography, 28, 602-606. doi 10.1107/S0567739472001524. 

The electron distribution, p(r), has been computed by quantum mechanics for all neutral atoms and many ions and the values off(Q), as well as coefficients for a useful empirical approximation, are tabulated in the International Tables for Crystallography vol C [2]. In general,is a maximum equal to the nuclear charge, Z, lor Q = 0 and decreases monotonically with increasing Q. 

These include rotation axes of orders two, tliree, four and six and mirror planes. They also include screM/ axes, in which a rotation operation is combined witii a translation parallel to the rotation axis in such a way that repeated application becomes a translation of the lattice, and glide planes, where a mirror reflection is combined with a translation parallel to the plane of half of a lattice translation. Each space group has a general position in which the tln-ee position coordinates, x, y and z, are independent, and most also have special positions, in which one or more coordinates are either fixed or constrained to be linear fimctions of other coordinates. The properties of the space groups are tabulated in the International Tables for Crystallography vol A [21]. 

Bahar et al. [46] have used this kind of approach to predict the B-factors of 12 X-ray structures. Elements in the Hessian corresponding to atom pairs separated by a distance of less than 7 A are set to zero, and the remainder have the same value dependent on a single adjustable parameter. Generally B-factor predictions for the a-carbons compare very well with the B-factors measured by X-ray crystallography. Figure 1 shows the result for the subunit A of endodeoxyribonuclease I complexed with actin. 

In general, the advantages of using an automated method may be comparable to those of SA refinement in X-ray crystallography [68], where many of the operations necessary to refine a structure can be done automatically and the remaining manual interventions are easier because the SA refinement usually results in a more easily interpreted electron density map. Automated methods are usually used in combination with manual assignment. However, fully automated assignment of the NOEs is possible (see Eig. 7) [69]. 

Perhaps the last general overview of crystallography in all its many aspects, including crystal chemistry and crystal physics and the history of crystallographic concepts, as well as the basics of crystal structure determination, was a famous book... 

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