Scattering functions density

Even for the resonant transmission through the Sinai billiard, computations show that many eigenfunctions contribute to the scattering wave function as shown in fig. 1. An assumption of a complex RGF for the scattering function (9) means that the joint probability density has the form... 

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... 

The central idea of contrast variation is shown in Fig. 5. The dissolved object is depicted as an assembly of scattering units with different scattering power. The entire object occupies a volume in the system depicted by the shaded area in Fig. 5, left-hand side. The local scattering length density r) is rendered as a product of the shape function T (f) and the local scattering length density inside the object ... 

Hansen, N. K., Study of the Electron Density Distribution in Molecular Crystals by Analysis of X-ray Diffraction Data Using Non-Spherically Symmetric Scattering Functions, Thesis, University of Arhus, Denmark (1978). 

Keywords Local atomic structure Neutron scattering Pair-density function ... 

The most important experiment of this type is thermal neutron scattering A complete determination of the differential scattering cross section for the scattering of neutrons from liquids completely determines the Van Hove scattering function.18 This function is related through a space-time Fourier transform to the autocorrelation function of the number density at two... 

Fig. 2.54 Neutron reflectivity profile for a symmetric PS-dPMMA diblock (Mw 30 kg moP1) as a function of incident wavevector (Russell 1990). The inset shows the scattering length density (b/V, the neutron scattering length per unit volume) profile normal to the film surface that was used to calculate the reflectivity profile shown as the solid line, This is typical of a block copolymer film containing a multilayer stack, with lamellae parallel to the surface.

In calculating the contribution from the density mode given by Eq. (132), the required solvent dynamical variables are F(q, t), the intermediate scattering function, and F0(q, t), the short-time part of the 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... 

Figure 9. Time dependencies of the single-particle and the collective intermediate scattering functions compared for two different solute sizes at a particular wavenumber q = 6.001 at reduced temperature T" = 0.75 and in the normal density regime (p = 0.89). The solid line represents the collective intermediate scattering function. The long-dashed line is the single-particle intermediate scattering function for solute-solvent size ratio 1.0 and the short-dashed line is for solute-solvent size ratio 0.5. The plots show that the decoupling of the solute motion from the solvent dynamics increases as the solute size is decreased. The time is scaled by rsct where TJC = [mo2/kBT] 2.

M.P. Gaigeot et al., Analysis of the structural and vibrational properties of RNA building blocks by means of neutron inelastic scattering and density functional theory calculations. Chem. Phys. 261, 217-237 (2000)... 

The measured reflectivity, R(Q), depends upon the neutron refractive index profile perpendicular to the interface, defined as the z-direction. The neutron refractive index is a function of the scattering length density, Nb, which is the product of the number density N, in units of nuclei per cm3, and the neutron scattering lengths, b, of the nuclei present. Since the neutron scattering length varies from nucleus to nucleus, chances in the nature and composition of the surface result in changes in reflectivity. 

Linder B, Lee KF, Malinowski P, Tanner AC (1980) On the relation between charge-density susceptibility, scattering functions, and Van der Waals forces. Chem Phys 52 353-361... 

Malinowski P, Tanner AC, Lee KF, Linder B (1981) Van der Waals forces, scattering functions and charge density susceptibility. II. Application to the He-He interaction potential. Chem Phys 62 423-438... 

In MCT, the nonergodicity parameter fq(T) is a crucial quantity. It describes the long-time limit of the density correlation function or the so-called intermediate scattering function, in the theory denoted (q, t). In its idealized version, MCT predicts a discontinuous change at a critical temperature Tc. 

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