Dynamic structure factor small

For non-interacting, incompressible polymer systems the dynamic structure factors of Eq. (3) may be significantly simplified. The sums, which in Eq. (3) have to be carried out over all atoms or in the small Q limit over all monomers and solvent molecules in the sample, may be restricted to only one average chain yielding so-called form factors. With the exception of semi-dilute solutions in the following, we will always use this restriction. Thus, S(Q, t) and Sinc(Q, t) will be understood as dynamic structure factors of single chains. Under these circumstances the normalized, so-called macroscopic coherent cross section (scattering per unit volume) follows as... 

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. 

Figure 22. Gaussian decay of the dynamic structure factor F(q, t) in the relevant time range. In this figure, the ratio F(q, t)/S(q) has been plotted against the reduced time at a small value of the wavenumber, ql = 0.1206. This figure has been taken from Ref. 186.

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). 

Finally, we comment on the difference between the self part and the full density autocorrelation function. The full density autocorreration function and the dynamical structure factor ire experimentally measured, while in the present MD simulation only the self pairt was studied. However, the difference between both correlation functions (dynamical structure factors) is considered to be rather small except that additional modes associated with sound modes appear in the full density autocorrelation. We have previously computed the full density autocorrelation via MD simulations for the same model as the present one, and found that the general behavior of the a relaxation was little changed. General trends of the relaxation are nearly the same for both full correlation and self part. In addition, from a point of numerical calculations, the self pMt is more easily obtained than the full autocorrelation the statistics of the data obtained from MD simulatons is much higher for the self part than for the full autocorrelation. 

S-wave scattering is the only practical outcome since P-wave final neutron states are not accessible to thermal neutrons, because these wave functions have negligible amplitude at the small radial values that are typical of atomic nuclei. It is convenient to rewrite the equation as a dynamical structure factor (or Scattering Law), which emphasises the dynamics of the sample. 

Q is the scattering vector and t 2 is the half-peak time of the dynamic structure factor, the larger is the chain stiffness, the smaller is the value of which may be as small as 2. 

Inelastic neutron scattering is a technique that has been widely used both in the liquid and in the solid states to measure the stmcture and dynamics at small (that is, molecular) length scales. In an incoherent inelastic neutron-scattering experiment, the measured quantity is the self-dynamic structure factor Ss(Q, (o), which gives information, as in the liquid state, of the self-diSiision coefficient of the water molecules. Ss(Q, (o) is the Fourier transform of the intermediate self-scattering function Fg(Q, t), which is defined by... 

Since rodlike polymers have a large optical anisotropy, they have a significant depolarized light scattering, which is particularly suitable for studying rotational diffusion. In the small-angle regime k L 1, the dynamic structure factor is written as ... 

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],... 

Reptation quantum Monte Carlo (RQMC) [15,16] allows pure sampling to be done directly, albeit in common with DMC, with a bias introduced by the time-step (large, but controllable in DMC e.g. [17]) and the fixed-node approach (small, but not controllable e.g. [18]). Property estimation in this manner is free from population-control bias that plagues calculation of properties in diffusion Monte Carlo (e.g. [19]). Inverse Laplace transforms of the imaginary time correlation functions allow simulation of dynamic structure factors and other properties of physical interest. 

Spontaneous thermal fluctuations of the density, p r,t), the momentum density, g(r,t), and the energy density, e(r,t), are dynamically coupled, and an analysis of their dynamic correlations in the limit of small wave numbers and frequencies can be used to measure a fluid s transport coefficients. In particular, because it is easily measured in dynamic light scattering. X-ray, and neutron scattering experiments, the Fourier transform of the density-density correlation function - the dynamics structure factor - is one of the most widely used vehicles for probing the dynamic and transport properties of liquids [56]. 

The discussion of the large- tail in S(q) in Section 7.4.1.1, which is characteristic for a short-ranged attraction, enables one to formulate a simplified theory of bond formation within MCT with the result that the long-time limit of the dynamic structure factor is controlled by a single interaction parameter, F = fP-(p/b. Bond formation occurs at T, = 3.02... [34]. For small values of F, the dynamic structure factor decays to zero for all wavevectors. Physically, this means that concentration fluctuations decay into equilibrium at long times, just as expected for a colloidal fluid. However, for F > F, the solutions yield a nonzero glass form factor, namely, the system arrests in a metastable state. This simple result requires the approximate expression for S(q) given above and needs to be replaced by a full numerical solution whenever this approximation fails. 

In our example S(Q) (see Eq. 6.21) would be the Leibler structure factor (Eq. 6.12) and Aj(Q) describes the collective dynamics of the diblock copolymer melt. For small momentum transfers Aj(Q) is given by ... 

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