Averaged structure factor

The inverse of the typical domain size is proportional to the location fcm(x) of the peak of the spherically averaged structure factor. 

Fig. 19 Spherically averaged structure factor of annealed charged chains (N=1000) (/)=0.040, Ad=16b (circles) (/)=0.083, XB 64b (squares) (/)= 0.125, XD=256b (triangles), additionally the result of an uncharged chain (diamonds). Thin lines indicate asymptotic scaling laws...

Therefore, the three significant parameters for the evaluation of the structure factors of glasses are a) porosity, b) polydispersity, and c) the size of solid entities. Fig. 3a illustrates the calculated averaged structure factors. Note that as the size of the hard spheres is increased the maxima are shifted to the lower Q region corresponding to the increasing distance between pore centroids. Based on the setting parameters for the structure factors, the form factor is calculated for a polydisperse pore size distribution ftmction, D (Rp), that states the number of pores defined by the radius parameter Rp (Fig. 3b). 

For typical measure of domain size, one usually considers either the location qmax(t) of the peak of the spherically averaged structure factor or some moment of S(q,t). Here only the first moment ql will be considered as a measure of average domain size, since qmax cannot be precisely determined due to the discretization of the scattering vector q in a finite lattice. Five independent runs are performed for each case, and all the results are reported by averaging the data from five independent runs. 

Finally, we note that the (angle-averaged) structure factor of the adsorbed fluid S(k), which is accessible in scattering experiments, contains both a blocked and a connected part. To see this we start from the expression... 

Figure 7. A vector diagram illustrating the effects of heavy atom anomalous scattering on the reflections hkl (denoted +) and filci (denoted —). Fpn is the average structure factor amplitude for the heavy atom derivative of the protein. FJ Ih imaginary part of the heavy atom structure factor amplitude which arises from anomalous scattering. Because FJl always advances the phase by fI/2, FpH (+) and Fpn (—) are no longer equal. The measured difference between these amplitudes can be used fbr phase determination.

In the calculation, a model of the averaged structure factor for a hard-sphere (HS) interaction potential, S(g) is used [47, 48], which considers the Gaussian distribution of the interaction radius cr for individual monodisperse systems for polydispersity m, and a Percus-Yevick (PY) closure relation to solve Omstein-Zernike (OZ) equation. The detailed theoretical description on the method has been reported elsewhere [49-51]. 

Inverse Fourier transformation of the corrected, averaged structure factors yields a map which is a projection of the structure onto a plane perpendicular to the incident electron beam, at a resolution determined by the diffraction order of the Fourier terms contributing to it. 

For N> 10, the parameter Rg c can be replaced by Rg without significant effect. The corresponding curves are compared with the averaged structure factor of DLCA... 

FIGURE 6.11 SAXS results for 5% Brij 35/ethanol/water ternary systems (black) with the result of the reference binary 5% Brij 35/water binary system (gray) at 25°C. (a) Experimental SAXS spectra on an absolute scale, (b) the GIFT results of some curves from Figure 6.11a p r) functions and (c) averaged structure factor Inset Parameters of... 

The second equality in Eq. (3.7) follows from Eq. (2.2). The summations in Eq. (3.6) run over the -independent sites making up the monomer. Assuming for the moment that the scattering cross sections of each site are equal, then the scattering intensity of a three-site vinyl polymer melt is proportional to the average structure factor defined according to... 

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