Structure factor assumption

On the previously made assumption regarding relative reflecting powers, the values for the structure factor of the classes of planes increase in this... 

In order to generate a set of calculated structure factors Fc(Q) from a set of coordinates, it is necessary to introduce a model for the time variation of the electron density. The usual assumptions in macromolecular crystallography include harmonic isotropic motion of the atoms and in addition, the molecular scattering factor is expressed as a superposition of atomic scattering factors. With these assumptions the calculated structure factor (equation III.2) is given by.27... 

There are therefore four adjustable parameters per atom in the refinement (xy, yy, Zj, By). In the computer experiments we have carried out to test the assumptions of the nucleic acid refinement model we have generated sets of observed structure factors F (Q), from the Z-DNA molecular dynamics trajectories. The thermal averaging implicit in Equation III.3 is accomplished by averaging the atomic structure factors obtained from coordinate sets sampled along the molecular dynamics trajectories at each temperature ... 

Under the simplifying assumption that the reflexions are independent of each other, K, can be written as a product over reflexions for which experimental structure factor amplitudes are available. For each of the reflexions, the likelihood gain takes different functional forms, depending on the centric or acentric character, and on the assumptions made for the phase probability distribution used in integrating over the phase circle for a discussion of the crystallographic likelihood functions we refer the reader to the description recently appeared in [51]. 

More general access to the coherent dynamic structure factor was provided by Akcasu and coworkers [93-95], starting from the assumption that the temporal evolution of the densities in Fourier space p(Q,t)... 

At higher Q, however, where the static structure factor reveals the asymptotic power law behavior S (Q, 0) Q 1/v, the assumption of ideal conformation clearly fails. In particular, this is evident for the core (sample 1) and shell contrast conditions (sample 2). 

An alternative way to find out the expression for C is to assume that the same form of the structure factor, Eq. (58), will be valid also locally for the inhomogeneous system, if the average volume fraction <[)0 in Eq. (58) is replaced by its local values < )(r). Guided by this assumption, one can allow the coefficient at the gradient term to be dependent on the local volume fractions and write down C as... 

In protein crystallography we assume that all electron density is real, and does not have an imaginary component. In reciprocal space this observation is known as Friedel s law, which states that a structure factor F(h) and its Friedel mate F(—h) have equal amplitudes, but opposite phases. The correspondence of these two assumptions follows straight from Fourier theory and, in consequence, explicitly constraining all electron density to be real is entirely equivalent to introducing Nadditional equalities of... 

Unlike the treatment described in the previous section, the structure factors are not linear functions of the unknowns Xj, as was the case for the observations described by expression (4.3). The equations of the preceding section can only be retained by the approximation that all the second and higher derivatives are zero. The price paid for this assumption is that the equations are no longer exact, so that a single calculation no longer leads to the minimum of the error function. If the deviations from linearity are not pronounced, the minimum may still be reached, but a number of iterations may be necessary to achieve convergence. 

An X-ray atomic orbital (XAO) [77] method has also been adopted to refine electronic states directly. The method is applicable mainly to analyse the electron-density distribution in ionic solids of transition or rare earth metals, given that it is based on an atomic orbital assumption, neglecting molecular orbitals. The expansion coefficients of each atomic orbital are calculated with a perturbation theory and the coefficients of each orbital are refined to fit the observed structure factors keeping the orthonormal relationships among them. This model is somewhat similar to the valence orbital model (VOM), earlier introduced by Figgis et al. [78] to study transition metal complexes, within the Ligand field theory approach. The VOM could be applied in such complexes, within the assumption that the metal and the... 

From their definitions, one may admit that topological indices may code two structural factors, namely, the size and the shape of the molecule. This assumption is proved by calculating 94) the schemes (B) in Figure 2 for the 36 polyalkylcyclohexanes with 6-10 carbon atoms, and for a subseries of 22 polyalkylcyclohexanes with 10 carbon atoms (resulting in scheme (C)). The schemes (A)-(C) corroborated with the... 

Because Fhkl is a periodic function, it possesses amplitude, frequency, and phase. It is a diffracted X ray, so its frequency is that of the X-ray source. The amplitude of Fhkl is proportional to the square root of the reflection intensity lhkl, so structure amplitudes are directly obtainable from measured reflection intensities. But the phase of Fhkl is not directly obtainable from a single measurement of the reflection intensity. In order to compute p(x,y,z) from the structure factors, we must obtain, in addition to the intensity of each reflection, the phase of each diffracted ray. In Chapter 6,1 will present an expression for p(x,y,z) as a Fourier series in which the phases are explicit, and I will discuss means of obtaining phases. This is one of the most difficult problems in crystallography. For now, on the assumption that the phases can be obtained, and thus that complete structure factors are obtainable, I will consider further the implications of Eqs. (5.15) (structure factors F expressed in terms of atoms), (5.16) [structure factors in terms of p(x,y,z)], and (5.18) [p(x,y,z) in terms of structure factors]. 

Since the structure factor sets provide the greatest contribution to variability, the assumption that all are equally good is worth re-examining. The R-factor was calculated for each refined structure using only the reflections in the set common to all authors. (The refinement was performed on the particular authors full set). This procedure was adopted because one author might include a reflection omitted by the others because of difficulties in measuring its intensity. Had such a reflection been included in the R-factor calculation it would unfairly bias the value. From these calculations, it was clear that the R-factors obtained using the structure factors of Yokouchi et al, were always lower than those for the same combination of factors except that another author s structure factors were used. 

Note that in cases where the large n assumption is not valid, precise form and structure factors should be generated by performing the summations numerically. 

Figure 12. Half-peak time ti,2 of coherent dynamic structure factor for atactic polystyrene plotted vs. Q. Left, CgD solution right, CS2 solution continuous lines, calculated results at T = 30 °C dashed line, at F = 70 °C. Experimental points from ref. 14. [Model assumptions and parameters same as in Figures 10 and 11, tB = 0.012.] (Reprinted with permission from ref. 14, Copyright 1984, American Chemical Society.)...

Another procedure to estimate free enthalpies is based on the assumption that the values for AH and AS can be obtained from additive fragment contributions with some corrections due to two centers contributions or other structural factors [3, 6]. This type of procedure (additivity model) uses for the estimation formulas of the form ... 

In practice, Harker diagrams, like those described in Figure 8.5 are not actually constructed. Instead, the probability of any phase on the circle being correct, given the structure amplitudes of two or more derivatives and the calculated structure factors of the heavy atoms, is computed at increments around the circle, say at every 5 or 10 degrees. Blow and Crick (1958) showed that the probablility formulation, given certain assumptions regarding the distribution of errors, for any reflection is... 

Most of the following facts are only needed in the constructions based on the factoring assumption. There, the basic structure used is the ring of integers modulo n, where n is a chosen number that is hopefully hard to factor. Particular attention is paid to quadratic residues and square roots, because the squaring function plays an important part in the following schemes. 

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