Structure factors 264 INDEX

In the same paper (Yamamoto 1996) an authoritative description is given of several interrelated topics such as super-space group determination, structure determination, indexing of diffraction patterns of quasicrystals, polygonal tiling, icosahedral tiling, structure factor calculation, description of quasicrystal structures, cluster models of quasicrystals. 

Extracted magnitudes of structure factors Fhki (or Fhkif) for indexed... 

The direction and the periodicity of each cosine wave are given by its index u = hkl), the amplitude of the cosine wave is 2 F(u), proportional to the structure factor amplitude F(u). More importantly, the positions of the maxima and minima of the cosine wave (in relation to the unit cell origin) are determined by the structure factor phase ( )(u). If both the amplitudes F(u) and the phases ( )(u) of the structure factors for all reflections u are known, the potential cp(r) can be obtained by adding a series of such cosine waves. 

This review is intended to cover most of the aspects of NMR in phosphorus chemistry, especially stressing those aspects directly related to electronic and structural factors. Some subjects will only be briefly reviewed, e.g. resonances in solid compounds, in metal complexes [this subject has been treated in a previous article in this seriesa969,1)] and biological applications. All the compounds reviewed are tabulated in either Formula Index I (inorganics) or Formula Index II (organics), the whole comprising of more than four thousand five hundred entries. These indicesf list all available chemical shifts for phosphorus and fluorine those for other nuclei (except protons) are reported in the text of this review. 

It follows from a comparison of Fig. 5(b) and (c) with Fig. 6(b) and (c) that the electrocatalytic reduction of NOs" ions depends strongly on structural factors, reflected by the voltammetric behavior in pure FI2SO4 solution. This observation is in accordance with earlier findings [173, 233, 234] that (100) crystal faces have an excellent catalytic activity toward the reduction of N03 ions compared with the other low index planes. 

The residual index, or R factor, gives a measure of the difference between the observed and calculated structure factors and therefore of how well the structure has refined. It is defined as... 

Structure refinements of the AH 200 and AH 300 samples were conducted in the same way. Unit cell constants, final atomic parameters, and R indexes are given in Table I. (Observed and calculated structure factors are available from the authors.) Interatomic distances and angles are given in Table II. Estimated errors on the population and position of the cations may in some cases be greatly underestimated especially for atoms with low occupancy factors. 

Next, we examine the term i2. In a gas-like single segment approximation, this term can be replaced by 1212. The molecular conformation statistics are independent of each other. This might be due to the fact that in the absence of a three-dimensional lattice-potential, nematic shifts of neighboring segments are very likely to occur. In this approximation the configuration does not depend on which individual pair of molecules k, 1 is picked out The molecular structure factor is independent of the indexes k and L Hence 1 inter, d can be written as... 

The progress of iterative real- and reciprocal-space refinement is monitored by comparing the measured structure-factor amplitudes IFobsl (which are proportional to (/obs ) /2) with amplitudes IFca(c I from the current model. In calculating the new phases at each stage, we learn what intensities our current model, if correct, would yield. As we converge to the correct structure, the measured Fs and the calculated Fs should also converge. The most widely used measure of convergence is the residual index, or R-factor (Chapter 6, Section V.E). 

The agreement index from the observed single crystal electron diffraction structure factors (28 separate reflections) compared to the calculated structure factors for the Yokouchi at. model was 0.33. This is far from the values of 0.24 and 0.135 reported for the two published x-ray fiber analyses (14,15). ... 

Toraya s WPPD approach is quite similar to the Rietveld method it requires knowledge of the chemical composition of the individual phases (mass absorption coefficients of phases of the sample), and their unit cell parameters from indexing. The benefit of this method is that it does not require the structural model required by the Rietveld method. Furthermore, if the quality of the crystallographic structure is poor and contains disordered pharmaceutical or poorly refined solvent molecules, quantification by the WPPD approach will be unbiased by an inadequate structural model, in contrast to the Rietveld method. If an appropriate internal standard of known quantity is introduced to the sample, the method can be applied to determine the amorphous phase composition as well as the crystalline components.9 The Rietveld method uses structural-based parameters such as atomic coordinates and atomic site occupancies are required for the calculation of the structure factor, in addition to the parameters refined by the WPPD method of Toraya. The additional complexity of the Rietveld method affords a greater amount of information to be extracted from the data set, due to the increased number of refinable parameters. Furthermore, the method is commonly referred to as a standardless method, since the structural model serves the role of a standard crystalline phase. It is generally best to minimize the effect of preferred orientation through sample preparation. In certain instances models of its influence on the powder pattern can be used to improve the refinement.12... 

The structure of the y phase is characterized by diffraction lines which are indexed by mixed Miller indices, such as (110), (112), (310), (114), etc. The expression for the structure factor is ... 

T o demonstrate more clearly the effect of structural factors on polymer flammability, we considered the relationship between the limiting oxygen index and the specific... 

IdealMicroscope. A Macintosh program helps index diffraction patterns, find excitation errors and structure factors, draw crystal structures, K-hnes, stereograms, CBED geometry, etc. Contact sharon emlahsoftware.com. 

There is a variety of software that can be used to process pseudo-single crystal experimental diffraction data represented in a form of individual structure factors or their squares (consult http //www.iucr.org and/or http //www.ccpl4.ac.uk). The most commonly used software products are SHELXS-90 and SHELXL-97 (author G.M. Sheldrick), which are distributed free for academic use (consult SHELX home page at http //shelx.uni-ac.gwdg.de/SHELX/index.html for details). Unless noted otherwise, processing of the individual structure factor data, direct phase determination, Patterson-, Fourier- and E-map calculations shown in this chapter were performed using WinCSD software, which is available from STOE Cie, Gmbh (http //www.stoe.com/products/index.htm). 

To save paper, we are here compressing notation to the vector forms for position x = x, y, z and indexes h = h,k t, and using Euler s form of the complex numbers. We will expand them back to their full forms later. Again, it is unnecessary to actually multiply by NP, which we do not even know, since it is a constant and doesn t change the relative intensities or phases of the structure factors. 

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