Anisotropy theories

Lipari G. and Szabo A. (1980) Effect of Vibrational Motion on Fluorescence Depolarization and Nuclear Magnetic Resonance Relaxation in Macromolecules and Membranes, Biophys. J. 30, 489—506. Steiner R. F. (1991) Fluorescence Anisotropy Theory and Applications, in Lakowicz J. R. (Ed.), Topics in Fluorescence Spectroscopy, Vol. 2, Principles, Plenum Press, New York, pp. 127-176. 

The application of the optical anisotropy theory of comb-like molecules to graft copolymers permits the quantitative determination of the optical anisotropy and the flexibility of both the main and side chains by using Eq. (77) ... 

Barla, G. (1974). Rock anisotropy Theory and laboratory testing. In Muller, L. (Ed), Rock Mechanics. lAew York Springer Verlag. 

These methods are now strongly suggesting a work-hardening function to account for anisotropy peaks as dislocations lock this feature becomes the dominant one in the most recent developments of hardness anisotropy theory that finally move away from the resolved shear stress models. 

Kawski A (1993) Fluorescence anisotropy—theory and applications of rotational depolarization. CritRev Anal Chem 23(6) 459-529. doi 10.1080/10408349308051654... 

The magnetocrystalline anisotropy is a direct measure of the contributions from individual cations in the lattice. The success of the single ion model of the anisotropy theory (Yoshida and Tachiki, 1957 Wolf, 1957), as applied to the ferrites and garnets, makes the anisotropy constants very useful. 

Another important application of perturbation theory is to molecules with anisotropic interactions. Examples are dipolar hard spheres, in which the anisotropy is due to the polarity of tlie molecule, and liquid crystals in which the anisotropy is due also to the shape of the molecules. The use of an anisotropic reference system is more natural in accounting for molecular shape, but presents difficulties. Hence, we will consider only... 

Berezhkovskii A M and Zitserman V Yu 1991 Comment on diffusion theory of multidimensional activated rate processes the role of anisotropy J. Chem. Phys. 95 1424... 

In certain situations involving coherently interacting pairs of transition dipoles, the initial fluorescence anisotropy value is expected to be larger tlian 0.4. As mdicated by the theory described by Wyime and Hochstrasser [, and by Knox and Gtilen [, ], the initial anisotropy expected for a pair of coupled dipoles oriented 90° apart, as an example. 

One of the major reasons why design should be based on statisties is that material properties vary so widely, and any general theory of reliability must take this into aeeount (Haugen and Wirsehing, 1975). Material properties exhibit variability beeause of anisotropy and inhomogeneity, imperfeetion, impurities and defeets (Bury, 1975). All materials are, of eourse, proeessed in some way so that they are in some useful fabrieation eondition. The level of variability in material properties assoeiated with the level of proeessing ean also be a major eontribution. There are three main kinds of randomness in material properties that are observed (Bolotin, 1994) ... 

Crystallography is a very broad science, stretching from crystal-structure determination to crystal physics (especially the systematic study and mathematical analysis of anisotropy), crystal chemistry and the geometrical study of phase transitions in the solid state, and stretching to the prediction of crystal structures from first principles this last is very active nowadays and is entirely dependent on recent advances in the electron theory of solids. There is also a flourishing field of applied crystallography, encompassing such skills as the determination of preferred orientations, alias textures, in polycrystalline assemblies. It would be fair to say that... 

Gelbart (1974) has reviewed a number of theories of the origins of the depolarized spectrum. One of the simplest models is the isolated binary collision (IBC) model of McTague and Bimbaum (1968). All effects due to the interaction of three or more particles are ignored, and the scattering is due only to diatomic collision processes. In the case that the interacting particles A and B are atoms or highly symmetrical molecules then there are only two unique components of the pair polarizability tensor, and attention focuses on the anisotropy and the incremental mean pair polarizability... 

A problem with studies on inert gas is that the interactions are so weak. Alkali halides are important commercial compounds because of their role in extractive metallurgy. A deal of effort has gone into corresponding calculations on alkali halides such as LiCl, with a view to understanding the structure and properties of ionic melts. Experience suggests that calculations at the Hartree-Fock level of theory are adequate, provided that a reasonable basis set is chosen. Figure 17.7 shows the variation of the anisotropy and incremental mean pair polarizability as a function of distance. 

It is clear that an ah initio calculation of the ground state of AF Cr, based on actual experimental data on the magnetic structure, would be at the moment absolutely unfeasible. That is why most calculations are performed for a vector Q = 2ir/a (1,0,0). In this case Cr has a CsCl unit cell. The local magnetic moments at different atoms are equal in magnitude but opposite in direction. Such an approach is used, in particular, in papers [2, 3, 4], in which the electronic structure of Cr is calculated within the framework of spin density functional theory. Our paper [6] is devoted to the study of the influence of relativistic effects on the electronic structure of chromium. The results of calculations demonstrate that the relativistic effects completely change the structure of the Or electron spectrum, which leads to its anisotropy for the directions being identical in the non-relativistic approach. 

Theories of the oxidation of tantalum in the presence of suboxide have been developed by Stringer. By means of single-crystal studies he has been able to show that a rate anisotropy stems from the orientation of the suboxide which is precipitated in the form of thin plates. Their influence on the oxidation rate is least when they lie parallel to the metal interface, since the stresses set up by their oxidation to the pentoxide are most easily accommodated. By contrast, when the plates are at 45° to the surface, complex stresses are established which create characteristic chevron markings and cracks in the oxide. The cracks in this case follow lines of pores generated by oxidation of the plates. This behaviour is also found with niobium, but surprisingly, these pores are not formed when Ta-Nb alloys are oxidised, and the rate anisotropy disappears. However, the rate remains linear it seems that this is another case in which molecular oxygen travels by sub-microscopic routes. 

This simple relaxation theory becomes invalid, however, if motional anisotropy, or internal motions, or both, are involved. Then, the rotational correlation-time in Eq. 30 is an effective correlation-time, containing contributions from reorientation about the principal axes of the rotational-diffusion tensor. In order to separate these contributions, a physical model to describe the manner by which a molecule tumbles is required. Complete expressions for intramolecular, dipolar relaxation-rates for the three classes of spherical, axially symmetric, and asymmetric top molecules have been evaluated by Werbelow and Grant, in order to incorporate into the relaxation theory the appropriate rotational-diffusion model developed by Woess-ner. Methyl internal motion has been treated in a few instances, by using the equations of Woessner and coworkers to describe internal rotation superimposed on the overall, molecular tumbling. Nevertheless, if motional anisotropy is present, it is wiser not to attempt a quantitative determination of interproton distances from measured, proton relaxation-rates, although semiquantitative conclusions are probably justified by neglecting motional anisotropy, as will be seen in the following Section. 

When other relaxation mechanisms are involved, such as chemical-shift anisotropy or spin-rotation interactions, they cannot be separated by application of the foregoing relaxation theory. Then, the full density-matrix formalism should be employed. 

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