Geometrical structure factor

Fig. 3.1. Left visible pump/X-ray probe scheme for femtosecond TRXRD experiments. Hard X-ray pulses are generated by shining intense femtosecond laser pulses on a metal target (laser plasma X-ray source). Right geometrical structure factor of bismuth as a function of inter-atomic distance for diffraction from (111) and (222) lattice planes. From [1] and [2]...

This is the dispersion of the geometrical structure factor in crystallography. The overall effect of resonant scattering is to cause the breakdown of Friedel s law so that the Bijvoet pairs of reflections S(h) and S(—h) are unequal (see e.g. ). The difference... 

The geometrical factor is the modulus of the normalized geometrical structure factor Sh for the ordered fluorescent-selected atoms ... 

This equation is readily solved for n , and the geometrical structure factor can be factored out ... 

A catalyst is a material that accelerates a reaction rate towards thennodynamic equilibrium conversion without itself being consumed in the reaction. Reactions occur on catalysts at particular sites, called active sites , which may have different electronic and geometric structures than neighbouring sites. Catalytic reactions are at the heart of many chemical industries, and account for a large fraction of worldwide chemical production. Research into fiindamental aspects of catalytic reactions has a strong economic motivating factor a better understanding of the catalytic process... 

Similar to X-Ray and neutron diffraction analysis, electron dilFraction structure analysis consists of such main stages as the obtaining of appropriate diffraction patterns and their geometrical analysis, the precision evaluation of diffraction-reflection intensities, the use of the appropriate formulas for recalculation of the reflection intensities into the structure factors, finally the solution of the phase problem, Fourier-constructions. 

SB. Actually, these tautomeric equilibria are more complex because each of these three forms can exist as two geometric isomers ( and Z for 75A and 75A) or two conformers (rotamers E and Z for 75B). When + H, two diastereomers cis and trans and R ) are possible for the cyclic tautomer 75B. These equilibria have been thoroughly investigated (88TH2) by means of electronic, IR, H-, and C-NMR spectroscopy. The structure of the solid 5-hydroxypyrazoline 74B- (R = Me R = H R = Ph R" = i-Pr) was confirmed by X-ray diffraction (87MI5). An approximate generalization of the influence of the structural factor on these equilibria is shown in Table X. 

In order to illuminate both the phase problem and its solution, I will represent structure factors as vectors on a two-dimensional plane of complex numbers of the form a + ib, where i is the imaginary number (—1)1/2. This allows me to show geometrically how to compute phases. I will begin by introducing complex numbers and their representation as points having coordinates (a,b) on the complex plane. Then I will show how to represent structure factors as vectors on the same plane. Because we will now start thinking of the structure factor as a vector, I will hereafter write it in boldface (FM,Z) instead of the italics used for simple variables and functions. Finally, I will use the vector representation of structure factors to explain a few common methods of obtaining phases. 

If the basis contains atoms of different elements it is useful to separate the phase shifts due to the geometric arrangement from the scattering characteristics of the atoms. If rj is the vector from the origin of the unit cell to the center of atom j (denoting the position of atom j within the unit cell), and nj(r - rj) is the contribution of atom j to the total electron density at r, the structure factor can be written as ... 

The cubic AMO3 perovskite structure consists of an MO3 array of comer-shared MO6/2 octahedra with a large A cation at the body-center position. As is illustrated in tig. 1, this structure allows formation of the Ruddlesden-Popper (1957,1958) rock-salt/perovskite intergrowth structures MO (AMO3 ) . In all these structures, the mismatch between the equilibrium (A-O) and (M-O) bond lengths is given by the deviation from unity of the geometric tolerance factor... 

Magnetic anisotropies xlz (l/3)Tr/ for R = Ce-Yb except Pm, Gd (0.002 < AFj < 0.06, table 9) have been computed with eq. (58) and using five contact contributions Sfj (i = H9, H11-H14) and the geometrical G factors obtained from the crystal structures of (HHH)-[/ Co(L5)3]6+ (R = La, Lu). A qualitative good agreement (AF = 0.23) is obtained between the experimental magnetic anisotropies (scaled to -100 for Dy(III) and corrected for the variation of the crystal-field parameter near the middle of the series (vide supra), table 9) and Bleaney s factors (table 3). Further non-linear least-squares refinements of the molecular... 

First of all the physical properties of optical fibers and some of the factors influencing these properties are briefly discussed. From this it follows that preparation methods for optical fibers have to fulfill extreme requirements with respect to the impurity content of the materials involved as well as the precise optical and geometrical structure of the fiber. 

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