Anisotropy fluctuations

The existence of molecules often creates permanent intramolecular optical anisotropy. The optical anisotropy of the liquid is then due to fluctuations in the orientations of the molecules or molecular subunits. If we assign a symmetric traceless anisotropy tensor a to each molecule or molecular subunit in the scattering volume, then the relaxation function for collective optical anisotropy fluctuations can be expressed as... 

As long as the concentration of the small molecule is low (<5%), the scattered intensity due to concentration fluctuations will be negligible relative to the density or anisotropy fluctuations. In polystyrene, the HV spectrum will not have any contribution due to concentration fluctuations, but in principle there could be a contribution due to the diluent anisotropy. The average relaxation time will be determined by the longest time processes and thus should reflect only the polymer fluctuations. The data were collected near the end of the thermal polymerization of styrene. Average relaxation times were determined as a function of elapsed time during the final stages of the reaction... 

The anisotropy fluctuations (AK) discussed above typically result from fluctuating orientations of the magnetic easy axis which varies from one grain to another in conventional polycrystalline microstructures, though Eq. (1) is not limited to this mechanism of anisotropy fluctuations. Hence, for conventional polycrystalline materials where 4 > Lcx, the parameter AK introduced into Eq. (1) can be well approximated by the magnetocrystalline anisotropy constant Kh However, as we discuss in the subsequent section, the approximation of AK by K is no longer applicable for small structural correlation lengths. 

The X-ray scattering is only sensitive to density flnctnations whereas seatler-ing of polarized light (so-called // -component) reacts to anisotropy fluctuations as well. It has been observed that the // -component does not show aity angular dependence for amorphons polymers in eqitihbrirrm. It demonstrates that there are no anisotropic domains resulting from alignment or order of ehains. 

The above presentation illustrates the similar bases of the electro-optical methods yl]. The components of the scattering matrix can differ in their sensitivity to the various sources of particle anisotropy, but they yield similar information on the induced anisotropy fluctuations, particularly on their relaxation. This scheme is true in so far as noninteracting particles are concerned. Though the correlations in particle orientation [42] have been accounted for in recent electro-optical theories, the influence of the applied field on the space distribution of the scattering elements is generally neglected. 

In the general case, light scattering in matter is caused not only by density fluctuations but also by anisotropy fluctuations and fluctuations of the optical axis orientation of anisotropic areas. Such fluctuations arise as a result of mutual orientation of anisotropic molecules, or of their aggregates, or owing to the internal stresses in solid matter. In this case, the polarizability of scattering elements is represented by a polarizability tensor and two correlation functions (density correlation auid orientation correlation) are introduced (Goldstein and Michalik, 1955 van Aartsen, 1972). 

Besides the triplet, a wider spectrum (due to the anisotropy fluctuations) is observed in liquids (Pike, 1974 Berne and Pecora, 1976 Vuks, 1977). Its halfwidth gives the anisotropy relaxation time Tr which is of 10 s order of magnitude. This time is sensitive to molecular associations, eg. to H-bond formation (Vuks, 1977). 

The mechanism for coercivity in the Cr—Co—Fe alloys appears to be pinning of domain walls. The magnetic domains extend through particles of both phases. The evidence from transmission electron microscopy studies and measurement of JT, and anisotropy vs T is that the walls are trapped locally by fluctuations in saturation magnetization. 

Here, ojr is the rate of spinner rotation. I is the proton spin number, 8 is the chemical shift anisotropy (CSA) and q is the asymmetric parameter of the CSA tensor. Thus, the line broadening occurs when an incoherent fluctuation frequency is very close to the coherent amplitude of proton decoupling monotonously decreased values without such interference in Figure 1. 

The main goal of the Planck instrument is to improve the accuracy of the measurement of the cosmic microwave background (CMB), in order to extract cosmological parameters that remain poorly constrained after the results of WMAP (Wilkinson microwave anisotropy probe) and of the best ground-based experiments. The basic idea of HFI-Planck is to use all the information contained in the CMB radiation, i.e. to perform a radiometric measurement limited by the quantum fluctuations of the CMB radiation itself. In these conditions, the accuracy is only limited by the number of detectors and by the duration of the observation. 

NMR spin relaxation is not a spontaneous process, it requires stimulation by a suitable fluctuating field to induce an appropriate spin transition to reestablish equilibrium magnetization. There are four main mechanisms for obtaining relaxation dipole-dipole (most significant relaxation mechanism for I = 1/2 nuclei), chemical shift anisotropy, spin rotation, and quadrupolar (most significant relaxation mechanism for I > 1/2 nuclei) (Claridge, 1999). 

In general, fluctuations in any electron Hamiltonian terms, due to Brownian motions, can induce relaxation. Fluctuations of anisotropic g, ZFS, or anisotropic A tensors may provide relaxation mechanisms. The g tensor is in fact introduced to describe the interaction energy between the magnetic field and the electron spin, in the presence of spin orbit coupling, which also causes static ZFS in S > 1/2 systems. The A tensor describes the hyperfine coupling of the unpaired electron(s) with the metal nuclear-spin. Stochastic fluctuations can arise from molecular reorientation (with correlation time Tji) and/or from molecular distortions, e.g., due to collisions (with correlation time t ) (18), the latter mechanism being usually dominant. The electron relaxation time is obtained (15) as a function of the squared anisotropies of the tensors and of the correlation time, with a field dependence due to the term x /(l + x ). 

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