Difference structure factor

As discussed in the following chapter, difference electron density maps, representing Ap = pobs — pcak, are based on the Fourier transform of the complex difference structure factors AF, defined as... 

The difference Ap(r) between the total electron density p(r) and a reference density pref(r), is a measure for the adequacy of the reference density in representing the system. Difference densities Ap(r) are obtained by Fourier summation in which the coefficients AF are equal to the difference between the observed and calculated structure factors. If k is the scale factor, as defined in chapter 4, the difference structure factor AF is given by... 

A better alternative is to use the difference structure factor AF in the summations. The electrostatic properties of the procrystal are rapidly convergent and can therefore be easily evaluated in direct space. Stewart (1991) describes a series of model calculations on the diatomic molecules N2, CO, and SiO, placed in cubic crystal lattices and assigned realistic mean-square amplitudes of vibration. He reports that for an error tolerance level of 1%, (sin 0/2)max = 1-1.1 A-1 is adequate for the deformation electrostatic potential, 1.5 A-1 for the electric field, and 2.0 A 1 for the deformation density and the deformation electric field gradient (which both have Fourier coefficients proportional to H°). 

Using time-resolved crystallographic experiments, molecular structure is eventually linked to kinetics in an elegant fashion. The experiments are of the pump-probe type. Preferentially, the reaction is initiated by an intense laser flash impinging on the crystal and the structure is probed a time delay. At, later by the x-ray pulse. Time-dependent data sets need to be measured at increasing time delays to probe the entire reaction. A time series of structure factor amplitudes, IF, , is obtained, where the measured amplitudes correspond to a vectorial sum of structure factors of all intermediate states, with time-dependent fractional occupancies of these states as coefficients in the summation. Difference electron densities are typically obtained from the time series of structure factor amplitudes using the difference Fourier approximation (Henderson and Moffatt 1971). Difference maps are correct representations of the electron density distribution. The linear relation to concentration of states is restored in these maps. To calculate difference maps, a data set is also collected in the dark as a reference. Structure factor amplitudes from the dark data set, IFqI, are subtracted from those of the time-dependent data sets, IF,I, to get difference structure factor amplitudes, AF,. Using phases from the known, precise reference model (i.e., the structure in the absence of the photoreaction, which may be determined from... 

The second type of stereoisomerism encompasses all other cases in which the three-dimensional structures of two isomers exhibiting the same connectivity among the atoms are not superimposable. Such stereoisomers are referred to as diastereomers. Diastereomers may arise due to different structural factors. One possibility is the presence of more than one chiral moiety. For example, many natural products contain 2 to 10 asymmetric centers per molecule, and molecules of compound classes such as polysaccharides and proteins contain hundreds. Thus, organisms may build large molecules that exhibit highly stereoselective sites, which are important for many biochemical reactions including the transformation of organic pollutants. 

FIGURE 13.3 Difference structure factors AS(Q.) (obtained from the difference between the measured S(Q.) of the samples and S(Q.) of pure D20) of the four isotopically different compositions, H-PEO and H-salt, D-PEO and H-salt, D-PEO and D-salt, and H-PEO and D-salt, with salt concentration c = 0.03 M and volume fraction of PEO v = 0.04. The difference structure factors are shifted vertically and shown on three different scales in parts (a), (b) and (c), for clarity. 

Several different structural factors influence the properties of the mesophase in these polymers, including dipolar effects, the planarity and rigidity of the mesogenic unit, and its length-to-width ratio among others. These factors are difficult to quantify, either absolutely or relatively, but some idea of their influences can be obtained by comparing the properties of polymers with different mesogenic units when combined with the same flexible spacer. This comparison has already been made for the dyad and triad esters in Table 2, and in this section it will be extended to other types of liquid crystalline polymers which contain a common decamethylene spacer. 

These are the usual multiplicity factors. In some crystals, planes having these indices comprise two forms with the same spacing but different structure factor, and the multiplicity factor for each form is half the value given above. In the cubic system, for example, there are some crystals in which permutations of the indices (M/) produce planes which are not structurally equivalent in such crystals (AuBe, discussed in S. 2-7, is an example), the plane (123), for example, belongs to one form and has a certain structure factor, while the plane (321) belongs to another form and has a different structure factor. There are = 24 planes in the first form and 24 planes in the second. This question is discussed more fully by Henry, Lipson, and Wooster [G.8]. 

We have not considered modulations of the displacement (thermal) parameters and occupation parameters. The explicit consideration of these terms, together with 3 + d-dimensional symmetry operators, in the structure factors gives rise to more complex expressions than those written in this section. The reader interested in a deeper knowledge of this subject is invited to check the specialized articles in the existing literature. An excellent review of the different structure factor formalisms is provided by Van Smaalen. ... 

P. T. Beurskens et al.. DIRDIF, a Computer Program System for Crystal Structure Determination by Patterson Methods and Direct Methods applied to Difference Structure Factors. 1992. 

The role of structural factors is determined by the t5 e of the operational loading that provides different structural factors of the required product efficiency level. 

Rim is a common and easily achieved shape for most polymers. Shape memory film or foam has a number of applications in laminated smart fabrics. Various properties of SMP films have been investigated such as thermomechanical properties, different structural factors on the physical and water vapour transport properties, " effects of crystal melting, " molecular weight and influence of different processing temperatures. ... 

Beurkans, P. T. Dirdif Direct Methods for Difference Structure Factors, Crystallography Laboratory Toemooiveld, Nijmegan, Netherlands, 1984. 

With vinyl and related monomers, controlling their polymerizations was more difficult to achieve due to the very high reactivity of propagating carbocations. It was indeed necessary to take into account the effects of different structural factors nature of monomer, nature of solvent, retrogradation of equilibrium between ion pairs and free ions, and nature of counterion. Monomers prone to polymerize under living conditions are also those whose corresponding carbocations are strongly stabilized ... 

If, on the other hand, the domain size is so large that the diffuse component is included in the integration, the modeling of the surface requires an incoherent addition. As an extreme example, consider a surface divided in two halves, each with a different structure factor. The domains are so large that the diffuse component is indistinguishable from the (instrument broadened) Bragg component. The intensity is therefore the incoherent, weighted sum of the separate halves, exactly as expected for two independent parts. 

Therefore, since the is infirmed as the medians of multiple models based on different structural factors give a more significant grouping information this way, consolidating the robustness of the working model of grouping, y, with t=l, r,. .., r=M+ are used for comparing the... 

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