Small structure factor function

There are two approaches to map crystal charge density from the measured structure factors by inverse Fourier transform or by the multipole method [32]. Direct Fourier transform of experimental structure factors was not useful due to the missing reflections in the collected data set, so a multipole refinement is a better approach to map charge density from the measured structure factors. In the multipole method, the crystal charge density is expanded as a sum of non-spherical pseudo-atomic densities. These consist of a spherical-atom (or ion) charge density obtained from multi-configuration Dirac-Fock (MCDF) calculations [33] with variable orbital occupation factors to allow for charge transfer, and a small non-spherical part in which local symmetry-adapted spherical harmonic functions were used. 

In Fig. 3.16 dynamic structure factor data from a A =36 kg/mol PE melt are displayed as a function of the Rouse variable VWt (Eq. 3.25) [4]. In Fig. 3.6 the scaled data followed a common master curve but here they spht into different branches which come close together only at small values of the scahng variable. This splitting is a consequence of the existing dynamic length scale, which invalidates the Rouse scaling properties. We note that this length is of purely dynamic character and cannot be observed in static equilibrium experiments. 

There is an important case which is intermediate between small bounded systems and macroscopic fully extended systems, namely the description of the surface region of a macroscopic metal. The correlation functions which describe density fluctuations in the surface region are extremely anisotropic and of long range, very unlike their counterparts in the bulk, and the thermodynamic limit must be taken with sufficient care. Consider the static structure factor for a large system of N particles contained within a volume Q,... 

It was Ziman [77] who has noted that there is little hope, at least at present, to develop an experimental technique permitting the direct measurement of these correlation functions. The only exception are the joint densities x / (r> ) information about which could be learned from the diffraction structural factors of inhomogeneous systems. On the other hand, optical spectroscopy allows estimation of concentrations of such aggregate defects in alkali halide crystals as Fn (n = 1,2,3,4) centres, i.e., n nearest anion vacancies trapped n electrons [80]. That is, we can find x mK m = 1 to 4, but at small r only. Along with the difficulties known in interpretating structure factors of binary equilibrium systems (gases or liquids), obvious specific complications arise for a system of recombining particles in condensed media which, in its turn, are characterized by their own structure factors. 

In Fig. 22, the normalized dynamic structure factor F(q, t) /S(q) is plotted at a small wavevector value, ql = 0.1206, in the time domain where the VACF shows pronounced t 3 decay. Circles show the simulated values, and the full line is the Gaussian fit. As seen from the figure, F(q, t) is a Gaussian function of time in the q — 0 limit. This is an important observation because it provides the key to the physical origin for the slow decay of Cv t). 

Because the electron density we seek is a complicated periodic function, it can be described as a Fourier series. Do the many structure-factor equations, each a sum of wave equations describing one reflection in the diffraction pattern, have any connection with the Fourier series that describes the electron density As mentioned earlier, each structure-factor equation can be written as a sum in which each term describes diffraction from one atom in the unit cell. But this is only one of many ways to write a structure-factor equation. Another way is to imagine dividing the electron density in the unit cell into many small volume elements by inserting planes parallel to the cell edges (Fig. 2.16). 

These volume elements can be as small and numerous as desired. Now because the true diffractors are the clouds of electrons, each structure-factor equation can be written as a sum in which each term describes diffraction by the electrons in one volume element. In this sum, each term contains the average numerical value of the desired electron density function p x,y,z) within... 

Since experiments for Kr have been performed at small angle neutron scattering for some low density states, we present the results of the Fourier transform of the direct correlation function, c(q) = (S(q) — 1 )/p.S (7/), rather than those of the structure factor S(q). Figure 20 shows the curves of c q). As it can be seen, the theoretical results, obtained by HMSA+WCA and MD with the AS plus AT potentials, are in excellent agreement with the experimental data [12]. While the AT contribution is included by means of an effective pair potential in the SCIET, it is used under its original form owing to Eqs.(l 19) and... 

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