Self-intermediate scattering function

The solute dynamic variables required to calculate the density and the current contribution to the friction are the inertial part of the self-intermediate scattering function, Fs0(q, t), given by... 

The decoupling scenario can be clearly envisaged from. Fig. 9, which shows the time dependence of (a) the self-intermediate scattering function... 

In a recent work, the translational motion of 4- -hexyl-4 -cyanobiphenyl (6CB) was studied in the isotropic phase by atomistic molecular dynamics simulation [134], The mean-square displacement showed evidence of sub-diffusive dynamics, with a plateau that became very apparent at the lowest temperatures. A three-time self-intermediate scattering function revealed that this plateau was connected with a homogeneous dynamics that, at longer times, became heterogeneous and finally exponential. These features, which are shared by, for example, a high-density system of hard spheres, support the universal character of the translational dynamics of liquids in their supercooled regime. 

The quantity Fs(q, t) defined in Eq. (5.4.2) appears so frequently that it has been given the name self-intermediate scattering function. It is related to the probability distribution GS(R, t) for a particle to suffer a displacement R in the time t... 

SuiQ, E) = SiiQ, E) 0 Sr Q, E), that is, a convolution of the translational dynamic structure factor, Si(Q,E), and the rotational one, 5r(<2, ) In addition, for small Q spectra, Q < 1 A the 5r(<2, E) can be made negligibly small, hence 5 h((2, E) Si(Q, E) and its Fourier transform will give the self-intermediate scattering function F Q, t) that have a stretched exponential FniQj) = exp [ - r (g) r] long-time decay. When the T is above the room temperature, P 1. A situation for which the exponential form Eh(Q, t) exp(—r(g)/) can be approximately used, or equivalently, in frequency domain theSnCg, E) of water is approximated as a Lorentzian shape function [67],... 

Figure 7. The self intermediate scattering function for GB particles in the T range of 1050-1150K (defined in the text). The dashed curves are a fit of the stretched exponential relation, Fs q, t) 0 exp[—(t/r) ] to the long-time data, where the short-time decay arises from the inertial atomic dynamics. The inset shows a power fit of r to 7"— 7, where Tq and y are adjustable parameters as in previous measurements and simulations. Figure 7 was originally published in [16], National Academy of Sciences.

Figure 14. The self-intermediate scattering function for interfacial NP dynamics in the Trange of 1300-1375K and for Al= 2899. The dashed curves are a fit using Fs ( , t) a exp [—(t/r) ] and the inset shows a power fit to the /"dependence of the structural relaxation time, r. Figure 14 was originally published in [71], Royal Society of Chemistry.

The latter results neglect any quantum corrections to the self intermediate scattering functions. In the case of the ideal gas, the full quantum-mechanical expression for Ss(Q, o) is obtained from the corresponding classical one simply by multiplying by two exponential factors, i.e. 

This section deals with the fundamental basis of the SCGLE theory. We first describe what is understood here for the GLE and then illustrate its use in the derivation of exact result for the time-dependent friction function A (t), and for the collective and self intermediate scattering functions. In addition, we discuss two additional approximations that convert these exact results into a closed self-consistent system of equations. 

Let us mention that by proceeding in an entirely analogous manner one can also derive a similar result for the self intermediate scattering function F k, t). Such an equation reads... 

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.

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