Density correlations

As the time separation 11 -, s approaches w tlie second tenn in this correlation vanishes and the remaining tenn is the equilibrium density-density correlation fomuila for an ideal solution. The second possibility is to consider a non-equilibrium initial state, c(r, t) = (r). The averaged solution is [26]... 

Fourier transfonn of the density-density correlation fiinction. Spp(r,t r,t )=(5p(r,t)5p... 

In the next section we discuss linear hydrodynamics and its role in understanding the inelastic light scattering experiments from liquids, by calculating the density-density correlation fiinction,. Spp. 

The tliree equation (A3,3,13). equation (A3,3,14) and equation (A3.3.15) are a usefiil starting point in many hydrodynamic problems. We now apply them to compute the density-density correlation fiinction... 

In order to ensure thermodynamic consistency, in almost all cases these properties are calculated from Tr. and the vapor pressure and liquid density correlation coefficients listed in those tables. This means that there will be slight differences between the values listed here and those in the DIPPR tables. Most of the differences are less than 1%, and almost all the rest are less than the estimated accuracy of the quantity in question. 

Menger P., van der Elsken J. Four time density correlations around a dissolved HC1 molecule in dense argon, krypton and xenon as determined from linewidth data, J. Chem. Phys. 75, 17-21 (1981). 

Wave-like properties cause electrons to be smeared out rather than localized at an exact position. This smeared-out distribution can be described using the notion of electron density Where electrons are most likely to be found, there is high electron density. Low electron density correlates with regions where electrons are least likely to be found. Each electron, rather than being a point charge, is a three-dimensional particle-wave that is distributed over space in... 

Alternatively, NIR spectroscopy has been applied to relate NIR data to mechanical properties [4], A multivariate data analysis was performed on a series of commercial ethene copolymers with 1-butene and 1-octene. For the density correlation, a coefficient of determination better than 99% was obtained, whereas this was 97.7% for the flexural modulus, and only 85% for the tensile strength. 

Several structure sizes caused by microphase separation occurring in the induction period as well as by crystallization were determined as a function of annealing time in order to determine how crystallization proceeds [9,18]. The characteristic wavelength A = 27r/Qm was obtained from the peak positions Qm of SAXS while the average size of the dense domains, probably having a liquid crystalline nematic structure as will be explained later, was estimated as follows. The dense domain size >i for the early stage of SD was calculated from the spatial density correlation function y(r) defined by Debye and Buche[50]... 

Nitrosation, nitration, and halogenation, diffusion control and pre-association in, 16, 1 NMR chemical shift-charge density correlations, 11, 125... 

In the discussion on the dynamics in the bead-spring model, we have observed that the position of the amorphous halo marks the relevant local length scale in the melt structure, and it is also central to the MCT treatment of the dynamics. The structural relaxation time in the super-cooled melt is best defined as the time it takes density correlations of this wave number (i.e., the coherent intermediate scattering function) to decay. In simulations one typically uses the time it takes S(q, t) to decay to a value of 0.3 (or 0.1 for larger (/-values). The temperature dependence of this relaxation time scale, which is shown in Figure 20, provides us with a first assessment of the glass transition... 

This is true only in the low-density limit (9.5.5). At higher solvent densities, correlation between the ligands may occur at a somewhat larger range of distances. 

Fig. 4.36 Scaling representation of NSE data (density correlation function) corresponding to PI at Q=1.92 A [second maximum of S(Q)]. Times have been divided by the KWW time Tpair to obtain a master curve. T=230 (cross), 240 (empty circle), 250 (plus), 264 (empty square), 280 (empty triangle), 320 K (empty diamond). The solid line indicates the fit with the KWW law for 250 K

A more wide-ranging survey of chemical shift-charge density correlations in organic ions is given in the preceding chapter hy D. G. Farnum (Editor). 

Typically, scaling approaches are employed to explain the behavior in the semidilute regime. By examining static correlations near the temperature, Daoud and Jannick( ) have expressed the density-density correlation function in terms of a correlation length that is inversely proportional to concentration. Since the diffusion coefficient is inversely proportional to the correlation length it is directly proportional to the concentration. 

In the experimental geometry used in this study, the distribution of scattered intensity measured by the detector is the two dimensional Fourier transform of the cross section of the electron density correlation function with a plane perpendicular to the extrusion direction (11). 

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