Anisotropy statistics

Definition and Uses of Standards. In the context of this paper, the term "standard" denotes a well-characterized material for which a physical parameter or concentration of chemical constituent has been determined with a known precision and accuracy. These standards can be used to check or determine (a) instrumental parameters such as wavelength accuracy, detection-system spectral responsivity, and stability (b) the instrument response to specific fluorescent species and (c) the accuracy of measurements made by specific Instruments or measurement procedures (assess whether the analytical measurement process is in statistical control and whether it exhibits bias). Once the luminescence instrumentation has been calibrated, it can be used to measure the luminescence characteristics of chemical systems, including corrected excitation and emission spectra, quantum yields, decay times, emission anisotropies, energy transfer, and, with appropriate standards, the concentrations of chemical constituents in complex S2unples. 

Hashimoto T., Shibayama M., Kawai H., and Meier D. J. Confined chain statistics of block polymers and estimation of optical anisotropy and domain size. Macromolecules, 18, 1855, 1985. 

Our finding that linewidth anisotropy in biomolecular EPR spectra can be described by a statistical theory in which the random variables that cause the broadening are fully correlated, does not only make analysis by simulation practical it also holds a message on the nature of the ultimate source of the broadening if the three principal elements of the p-tensor are fully correlated, then they should find their cause in a single, scalar quantity. 

In order to characterize quantitatively the polydisperse morphology, the shape and the size distribution functions are constructed. The size distribution function gives the probability to find a droplet of a given area (or volume), while the shape distribution function specified the probability to find a droplet of given compactness. The separation of the disconnected objects has to be performed in order to collect the data for such statistics. It is sometimes convenient to use the quantity v1/3 = [Kiropiet/ ]1 3 as a dimensionless measure of the droplet size. Each droplet itself can be further analyzed by calculating the mass center and principal inertia momenta from the scalar field distribution inside the droplet [110]. These data describe the droplet anisotropy. 

The apparently statistical distribution observed in the calculations for low J arose from several kinds of trajectories that combined to produce the observed distribution. The existence, position and magnitude of the rotational rainbow is sensitive to the orientational anisotropy of the attractive interaction potential (washed out by averaging), and does not result from anisotropies in the repulsive interaction region. 

For most astronomers, the solution to these cosmological problems resides in a combination of various methods. The luminosity-redshift test must be combined with independent techniques, such as anisotropies in the cosmic background radiation and statistical study of gravitational lenses. 

Additionally, since the acceptor is excited as a result of FRET, those acceptors that are fluorescent will emit photons (proportional to their quantum efficiency) also when FRET occurs. This is called sensitized emission and can also be a good measure of FRET (see Fig. 1). To quantitate FRET efficiency in practice, several approaches have been evolved so far. In flow cytometric FRET (7), we can obtain cell-averaged statistics for large cell populations, while the subcellular details can be investigated with various microscopic approaches. Jares-Erijman and Jovin have classified 22 different approaches that can be used to quantify energy transfer (8). Most of them are based on donor quenching and/or acceptor sensitization, and a few on measuring emission anisotropy of either the donor or the acceptor. Some of these methods can be combined to extend the information content of the measurement, for example two-sided FRET (9) involves both acceptor depletion (10) and... 

In many luminescence centers the intensity is a function of a specific orientation in relation to the crystallographic directions in the mineral. Even if a center consists of one atom or ion, such luminescence anisotropy may be produced by a compensating impurity or an intrinsic defect. In the case of cubic crystals this fact does not disrupt optical isotropy since anisotropic centers are oriented statistically uniformly over different crystallographic directions. However, in excitation of luminescence by polarized fight the hidden anisotropy may be revealed and the orientation of centers can be determined. 

Evidence of only a low barrier to inversion in Si—O—Si sequences could be important with regard to the interpretation of the statistical properties of silicone polymers. The effects are estimated for the temperature coefficients of the unperturbed dimensions, dipole moments, and the optical anisotropy for PDMS. 

The influence of the chain expansion produced by excluded volume on the mean-square optical anisotropy is studied in six types of polymers (PE, PVC, PVB, PS, polylp-chlorostyrene), polylp-bromostyrenel. RIS models are used for the configuration statistics of the unperturbed chains. The mean-square optical anisotropy of PE is found to be insensitive to excluded volume. The mean-square optical anisotropy of the five other polymers, on the other hand, is sensitive to the imposition of the excluded volume if the stereochemical composition is exclusively racemic. Much smaller effects are seen in meso chains and in chains with Bernoullian statistics and an equal probability for meso and racemic diads. 

In addition, the optical anisotropy of the statistical segment Aa calculated [50, 51] from the stress-optical coefficient C,... 

Unlike Xi, which in principle cannot be evaluated analytically at arbitrary a [90] for Xjnt an exact solution is possible for arbitrary values of the anisotropy parameter. Two ways were proposed to obtain quadrature formulas for Tjnt. One method [91] implies a direct integration of the Fokker-Planck equation. Another method [58] involves solving three-term recurrence relations for the statistical moments of W. The emerging solution for Tjnt can be expressed in a finite form in terms of hypergeometric (Kummer s) functions. Equivalence of both approaches was proved in Ref. 92. 

From Eq. (4.324), taking into account that for a solid random system the equilibrium statistical moments of the particle anisotropy axes are... 

In the absence of external fields the suspension under consideration is macroscopically isotropic (W = const). The applied field h (we denote it in the same way as above but imply the electric field and dipoles as well as the magnetic ones), orienting, statically or dynamically, the particles, thus induces a uniaxial anisotropy, which is conventionally characterized by the orientational order parameter tensor (Piin h)) defined by Eq. (4.358). (We remind the reader that for rigid dipolar particles there is no difference between the unit vectors e and .) As in the case of the internal order parameter S2, [see Eq. (4.81)], one may define the set of quantities (Pi(n h)) for an arbitrary l. Of those, the first statistical moment (Pi) is proportional to the polarization (magnetization) of the medium, and the moments with / > 2, although not having meanings of directly observable quantities, determine those via the chain-linked set [see Eq. (4.369)]. 

Atomic processes are very fast, so that intrinsic properties obey equilibrium statistics. An intermediate regime is characterized by typical magnetostatic and anisotropy energies per atom, about 0.1 meV, which correspond to times of order t0 0.1 ns. Examples are ferromagnetic resonance and related precession and damping phenomena. When energy barriers are involved, thermal excitations lead to a relatively slow relaxation governed by the Boltzmann-Arrhenius law [99, 133-137]... 

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