Crystallographic calculations

In the following a few selected formulae useful in common crystallographic calculations have been collected. 

The limitations mentioned do not apply if the work can be done on a machine various types of mechanical and electrical analogue machines for Fourier synthesis have been described (see Lipson and Cochran, 1953), culminating in Pepinsky s XRAC (X-ray analogue computer), in which structure amplitudes and phases for a two -dimensional synthesis are put in on an array of dials, and the electron density map appears at once on a cathode ray tube (Pepinsky, 1952). An increasing proportion of Fourier syntheses (and indeed crystallographic calculations of all types) is done on electronic digital computers. 

Crystallographers have been users of computers ever since computers became available for scientific calculations. The nature of crystallographic calculations used in molecular structure determination—large amounts of data to be treated by rather complicated mathematics-makes efficient use of computers essential and led quite early to the development of rather sophisticated techniques for both manual and computer computations. 

The features which make crystallographic calculations somewhat different include 1) the use of symmetry, i.e. space groups,... 

Therefore, symmetrical transformations in the crystal are formalized as algebraic (matrix-vector) operations - an extremely important feature used in all crystallographic calculations in computer software. The partial list of symmetry elements along with the corresponding augmented matrices that are used to represent symmetry operations included in each symmetry element is provided in Table 1.19 and Table 1.20. For a complete list, consult the Intemational Tables for Crystallography, vol. A. 

The location of molybdenum sulfide clusters and M0S2 particles was estimated on the basis of the pore volume of MoSx/zeolite, which was measured by benzene adsorption. Table 3 summarizes the pore volumes of the zeolite before and after the incorporation of molybdenum sulfide clusters. The pore volume of the NaY zeolite is about 0.30 cm g and consistent with the crystallographically calculated volume (0.30 cm g ) of the supercage of NaY zeolite. The pore volumes of USY-Na(3.5) and NaY(630) were slightly lower compared with that of NaY, suggesting a partial destruction of the zeolite framework during dealumination processes. 

Finally, detailed appendices provide additional information (e.g., properties of the pure chemical elements, thermochemical data, crystallographic calculations, radioactivity calculations, prices of metals, industrial minerals and commodities), and an extensive bibliography completes this comprehensive guide. The comprehensive index and handy format of the book enable the reader to locate and extract the relevant information quickly and easily. Charts and tables are all referenced, and tabs are used to denote the different sections of the book. It must be emphasized that the information presented here is taken from several scientific and technical sources and has been meticulously checked and every care has been taken to select the most reliable data. 

Attempts to include heteroatom contribution in the crystallographical calculations, or at least in texture and structure evaluations, began with the pioneering papers by Warren [71,72] and Ergun [12,13] on coals, but they were restricted to hydrogen. 

P, Roberts and G. M. Sheldrick, XANADU Program for Crystallographic Calculations , University of Cambridge, England, 1975. 

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