Reduced structural factor

With increasing time, the bell-shaped curve for the reduced structure factor S k,i) is markedly sharpened and its peak shifts to lower k. The latter behavior is one of the non-linear effects incorporated in eq 2.16. The overall features of... 

Fig. 10-7. Reduced structure factor S(k,i) for a critical spinodal decomposition at various reduced times i, where k is a reduced wavenumber. The insert shows pi at two values of Hi5, 10]. 

In addition to the orientational information gained from the analysis of observed dichroic ratios, Glenz and Peterlin s results have also yielded valuable information concerning changes in the degree of crystallinity and chain conformation upon drawing. The latter information comes from the evaluation of the so-called reduced structural factor (A ) given by... 

Several structural factors have been considered as possible causes of the anomeric effect. In localized valence bond terminology, it can be recognized that there will be a dipole-dipole repulsion between the polar bonds at the anomeric carbon in the equatorial conformation. This dipole-dipole interaction is reduced in the axial conformation, and this factor probably contributes to the solvent dependence of the anomeric effect. 

The reflecting powers of Mn and Fe are nearly the same, and may be taken equal without serious error. This reduces the number of distinct structures to three namely, 1 ab, %abc, and 3, of which 1 ab depends on two parameters and the others on one. It is possible to decide among them in the following way. Let us assume that the contribution of oxygen atoms to the intensity of reflection in various orders from (100) is small compared with the maximum possible contribution of the metal atoms that is, with 32M. The metal atom structure factor for structure 1 for (/a 00) is... 

These results thus show that whereas the flashpoint was only moderately influenced by the compound structure (their chemical functionality but especially their atomic composition and vapour), autoignition temperatures seem to be closely linked to the structural factors that affect the chain. So additivity rules for estimation of AIT should be sought. Every time a chemical or physical property is highly influenced by the structure, chemists tried to establish rules that enable one to reduce a molecule to characteristic groups for which the contribution to the value of this property is known. This was done for instance by Kinney for boiling points and Benson2 for thermochemical properties. 

Cluster analysis (which is covered extensively in Chapter 30) can be performed on the factor scores of a data table using a reduced number of factors (Section 31.1.4) rather than on the data table itself. This way, one can apply cluster analysis on the structural information only, while disregarding the noise or artefacts in the data. The number of structural factors may be determined by means of internal... 

The terms involving the subscript j represents the contribution of atom j to the computed structure factor, where nj is the occupancy, fj is the atomic scattering factor, and Ris the coordinate of atom i. In Eq. (13-4) the thermal effects are treated as anisotropic harmonic vibrational motion and U =< U U. > is the mean-square atomic displacement tensor when the thermal motion is treated as isotropic, Eq. (13-4) reduces to ... 

Under these conditions, Eq. (32) indicates the maximum extent to which a particular mode p can reduce S(Q,t) as a function of the momentum transfer Q. Figure 10 presents the Q-dependence of the mode contributions for PE of molecular weights Mw = 2000 and Mw = 4800 used in the experiments to be described later. Vertical lines mark the experimentally examined momentum transfers. Let us begin with the short chain. For the smaller Q the internal modes do not influence the dynamic structure factor. There, only the translational diffusion is observed. With increasing Q, the first mode begins to play a role. If Q is further increased, higher relaxation modes also begin to influence the... 

Obviously, in the case of PS these discrepancies are more and more reduced if the probed dimensions, characterized by 2ti/Q, are enlarged from microscopic to macroscopic scales. Using extremely high molecular masses the internal modes can also be studied by photon correlation spectroscopy [111,112], Corresponding measurements show that - at two orders of magnitude smaller Q-values than those tested with NSE - the line shape of the spectra is also well described by the dynamic structure factor of the Zimm model (see Table 1). The characteristic frequencies QZ(Q) also vary with Q3. Flowever, their absolute values are only 10-15% below the prediction. 

The salient feature of the experimental results is the observation of a pronounced minimum in the Q(Q)/Q3 vs. z plot. It occurs at the same position, where the static structure factor in its Kratky representation exhibits its maximum. Furthermore, the reduced line width scales with the scaling variable z in the same way that the static structure factor does. Thus, the occurrence of the minimum is directly related to peculiarities of the star architecture. 

Fig. 56a-c. Reduced relaxation rates Q (Q)/Q3 for diblock PIP stars under various conditions and theoretical predictions using the partial structure factors of Gaussian stars (dashed lines) or the experimental determined ones (solid line), a core contrast, b shell contrast, c average contrast. (Reprinted with permission from [154]. Copyright 1990 American Chemical Society, Washington)... 

CO hydrogenation activity was found to depend on both the amount of available surface cobalt metal and the environment of the reduced cobalt. Factors influencing these cobalt properties included the type of support, the type of cobalt species present on a given support, and the presence of additives. Due to the interrelationship of these properties, a detailed examination of the activity and structure of each catalyst is needed in order to determine the importance of each factor for CO hydrogenation activity. 

The crossover between the Kirkwood-Riseman-Zimm behavior and the Rouse behavior requires a better understanding, in terms of the contributing factors for the occurrence of a maximum in the plot of reduced viscosity against polyelectrolyte concentration at low salt concentrations. A firm understanding of the structure factor of polyelectrolyte solutions at concentrations comparable to the overlap concentration is necessary. 

The multipole model reduces the crystal electron density to a number of parameters, which can be fitted to experimental structure factors. For CU2O, structure factors for the (531) and higher-order reflections out to (14,4,2) were taken from X-ray measurements. Weak (ooe) (with o for odd and e for even) and very weak (eeo) reflections were also taken from X-ray work. Fig. 6 shows a three-dimensional plot of the difference between the static crystal charge density obtained from the multipole fitting to... 

According to Ref. [11], the dynamical scattering effect was reduced by forcing the integral amplitudes of reflections to be equal to the corresponding structure factor amplitudes of the corresponding perfect... 

Thus, each measured intensity (Ihkl) can be reduced to structure factor amplitude (Fhkl) with unknown phases, where is proportional to the square root of fhkl- Each structure factor amplitude (Fhkl) its... 

Data processing occurs in two stages. Initially, diffraction images are reduced to a tabulation of reflection indices and intensities or, after truncation, structure factors. The second stage involves conversion of the observed structure factors into an experimental electron density map. The choice of how to execute the latter step depends on the method used to determine the phase for each reflection. The software packages enumerated above generally focus on exploitation of anomalous signals to overcome the phase problem for a protein of unknown structure. 

As anticipated, the multipolar model is not the only technique available to refine electron density from a set of measured X-ray diffracted intensities. Alternative methods are possible, for example the direct refinement of reduced density matrix elements [73, 74] or even a wave function constrained to X-ray structure factor (XRCW) [75, 76]. Of course, in all these models an increasing amount of physical information is used from theoretical chemistry methods and of course one should carefully consider how experimental is the information obtained. 

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