Wave vector static structure factor

Figure 19 Static structure factor (minus the trivial self-scattering term) versus wave vector. Circles are X-ray diffraction results for polyethylene at 1 atm and 430 K. Curves are predictions of RISM theory for a RIS model with hard site diameters of 3.70 A (dashed line) and 3.90 A (solid line). The solid square is experimental S(0) - 1. An expanded view of the first peak is shown in the inset. (From Ref. 123.)...

Structures of powdered P-rhombohedral boron and amorphous boron were investigated with pulsed neutron diffraction techniques (Delaplane et al. 1988). To avoid intensive neutron absorption by °B nuclei, samples were "B isotopically enriched up to 97.1% and 99.1%, respectively. Earlier neutron diffraction studies based on nuclear reactor data did not permit the derivation of a meaningful radial distribution of atoms in amorphous material due to limited range of the neutron wave vector (<10.8 A" ). The obtained static structural factor and derived radial distribution function supported a structural model of amorphous boron based on building blocks of B,2 icosahedra resembling those found in p-rhombohedral boron, but with disorder occupying in the linking between ico-sahedral subunits. The intensity data indicated that amorphous samples contained 5% of a mixture of crystalline a- and p-rhombohedral boron. 

Here, the matrices H( ), C q), and W( ) contain the functions h q), c q), and w q) which are the Fourier transformations of the corresponding correlation functions with wave vector q. Having these functions, one can find the partial static structure factors, Saffl), which are the Fourier transformed density-density fluctuation correlation functions and are proportional to the scattering intensities observable in experiments. They are defined as... 

Equation 2.64 illustrates that the static structure factor, and hence the scattering pattern obtained in the light-scattering experiments, is the Fourier transform (see Appendix A2) of the autocorrelation function of the local segment density. 5(k) indicates which wave vector components are present in the correlation function. 

In these experiments the time-averaged scattered intensity /, is measured as a function of the scattering vector q. The net detected intensity can be computed as the superposition of the signals from each scattering center (e.g., each monomer unit). According to the spatial arrangement of the scatterers, the individual scattered waves may interfere constructively or destructively at the detector. Thus, 7f(q) is proportional to the so-called static structure factor 5(q), which sums the waves with different phases from different locations. 5(q), in fact, reflects the spatial Fourier transform of the distribution of scatterers (i.e., the pair correlation function), and... 

Figure 5. Static structure factor of sodium disilicate 5 (9) at room temperature T = 300 K) plotted versus wave vector q. The full curve is the molecular dynamics simulation of Ref. [32], where the experimental neutron scattering lengths for Si, O and Na atoms were used, so there is no adjustable parameter whatsoever. The broken curve represents the corresponding experimental data of Misawa el al. [55]. Fncan Ref. [32]. 

Fig. 6.2 The static structure factor for an undercooled melt in the lattice model. The temperature of the simulation is 570 K the wave vector is given in A. See Section 6.4.

FIGURE 4.2 (a) Wave vector dependence of the static structure factor from left to right < ) = 0.009, 0.016,... 

Fig. 2. Self-intermediate scattering function Fs qo, t) for (a) cations and (b) anions. The wave vector qo is set to 1.24 A in (a) and (b), which corresponds to the position of the first peak in the static structure factor for all ions.

In the last decade however, the dynamics of the exchange and correlation hole became an important topic, in view of the more accurate measurements of the dynamical structure factor at large wave vector, and because sum rules and causality arguments revealed that a static treatment of G[q, w)... 

When the average scattered intensity I q) is measured as a function of the scattering wave vector q, the experiment is usually called static light scattering (SLS). The collected light intensity I q) depends mainly on the particle form factor P(q) and the aggregate structure factor S(q). In these terms, /(g) reads ° as... 

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