Orientational parameter, formulas

Depending on the scheme chosen, the birefringence experiments provide [143,144] direct measurements of either Av or (Av)2. To present the theoretical results in a form suitable for comparison with the experimental data, let us consider the orientational oscillations induced in the dipolar suspension by a harmonic held H = Hq cos (at and analyze the frequency dependencies of the spectra of the order parameters (P2) and (Pi)2- As formula (4.371) shows, the latter quantities are directly proportional to Av and (Av)2, respectively. Since the oscillations are steady, let us expand the time-dependent orientational parameters into the Fourier series... 

As a measure of the level of the amorphous orientation, the factor based on sonic measurements, Fas, has been chosen. This value can theoretically vary from 0 for fully unoriented to I for perfectly oriented situations. In practice, this orientation parameter varied from 0.69 to 0.83 in this series of drawn yarns. Experimentally, Fas is determined from the formula... 

In order to simplify the discussion and to keep the derivation of the formulae tractable, the major part of this analysis is limited to a polymer fibre with a single orientation angle 0. This angle is assumed to be a kind of average angle and a characteristic parameter of the orientation distribution of the chain axes. 

For two stacks having the same molecular orientations, the numerical factor 3 (2) in formula (3.3.15) should be substituted by 2 as a result, with

parameters indicated above, the frequency shift yielded in the point-dipole approximation is 2 = - 445 cm"1. On the other hand, the computer simulation of two parallel stacks, each consisting of ten BTCC molecules, gave the frequency... 

The curves 1 in Figs. 4.6a and b show the functions Fr and FA calculated by formulae (4.3.35) and (4.3.38) for the case of normal molecular orientations (e Oz) and plotted versus the argument AQ/( +AQ). The dimensionless argument and functions of this kind normalized with respect to the sum of the resonance and the band widths were introduced so as to depict their behavior in both limiting cases, ACl rj and Af2 77. The deviation of the solid lines from the dotted ones indicates to which degree the one-parameter approximation defined by Eq. (4.3.38) differs from the realistic dispersion law. As seen, this approximation shows excellent adequacy, but for the region AQ r/, where the asymptotic behavior of the approximation (4.3.38) and Eq. (4.3.35) are as follows ... 

The structure of a green rust LDH-type material with the formula [Fe 4Fe 2(OH)i2]S04-ca. 8H2O has been determined by Rietveld analysis [157]. The material exists as a one-layer polytype with the interlayers containing two planes of sulfate and water molecules giving a basal spacing of 1.1011 nm. The sulfate ions are oriented with their C3 axes perpendicular to the layers and alternate anions point up and down (as shown earlier in Fig. 14) and form a superlattice with parameter a = UoV5 = 0.5524 nm (Fig. 39). 

In a quest to increase the efficiency of olefin polymerization catalysts and their selectivity in the orientation of the polymerization, the highly effective Group IV metallocene catalysts, M(Cp)2(L)2, have been studied, since they all display high fluxionality. Following methide abstraction, the metallocene catalysts of general formula M(Cp-derivatives)2(CH3)2 (M= Ti, Zr, Hf), were turned into highly reactive M+-CH3 cationic species. The activation parameters for the methide abstraction, derived from variable temperature NMR experiments, establish a correlation between the enthalpies of methide abstraction, the chemical shift in the resulting cation, and the ethylene polymerization activities [149]. 

The latter is in relation with those proposed by Deeth [30] and Berne [155]. Both involve the d-shell energy as an additional contribution to that of the MM scheme and use the AOM model with interpolated parameters to estimate the latter. In the case of the approach [30] there are two main problems. First is that the AOM parameters involved are assumed to depend only on the separation between the metal and donor atoms. This is obviously an oversimplification since from the formulae Eq. (25) it is clear that the lone pair orientation is of crucial importance. This is taken into account in the EHCF/MM method. Second important flaw is the absence of any correlation in describing the d-shell in the model [30]. This precludes correct description of the switch between different spin states of the open d-shell, although in some situations different spin states can be described uniformly. 

Ellipsometry measures the orientation of polarized light undergoing oblique reflection from a sample surface. Linearly polarized light, when reflected from a surface, will become elliptically polarized, because of presence of the thin layer of the boundary surface between two media. Dependence between optical constants of a layer and parameters of elliptically polarized light can be found on basis of the Fresnel formulas described above. 

The Keesom formula (4.70) is easily derived (Magnasco, 2009a) by taking the Boltzmann average of the dipolar interaction over the angles of relative orientation of the two molecules for small values of the dimensionless parameter ... 

Another most important question in anomalous dielectric relaxation is the physical interpretation of the parameters a and v in the various relaxation formulas and what are the physical conditions that give rise to these parameters. Here we shall give a reasonably convincing derivation of the fractional Smoluckowski equation from the discrete orientation model of dielectric relaxation. In the continuum limit of the orientation sites, such an approach provides a justification for the fractional diffusion equation used in the explanation of the Cole-Cole equation. Moreover, the fundamental solution of that equation for the free rotator will, by appealing to self-similarity, provide some justification for the neglect of spatial derivatives of higher order than the second in the Kramers-Moyal expansion. In order to accomplish this, it is first necessary to explain the concept of the continuous-time random walk (CTRW). 

The parameters PM(r) associated with the scattered radiation may fluctuate in time for instance, as a result of a random rotation motion of the polarizable element. In general, the experimentalist observes the time average of the parameters P. Then, he has to extract from this average, the contribution of the orientation fluctuations, which he considers either as a nuisance or as a studydeserving effect. We shall perform the calculation by using the formula [see (6.1.13) and (6.3.58)]... 

The X-ray powder diffraction study published by Tourne and Tourne is the main reference for the metal-substituted Keggin anionsJ" " These authors studied the K+, Rb+ and NH4+ salts and verified that they crystallized in a small number of structural types, depending on the number of cations in the molecular formula. The water molecules of crystallization also have a small role, as they reinforce the crystal cohesion and influence the orientation of the anions. Loss of water leads to loss of crystallinity and may alter the lattice parameters. 

For molecules, it is also necessary to consider their orientations, which can be monitored using a rotational order parameter. For some systems, such as carbon monoxide or water, complete disorder would be expected in the liquid state at equilibrium. However, if we were simulating a dense fluid of rod-shaped molecules which form a liquid crystalline phase then we might expect that, on average, the molecules would tend to line up in a common direction. The Viellard-Baron rotational order parameter for linear molecules is calculated using the following formula ... 

In orientation to the crystallographic results [152, 168], Kasper et al. (1996) [33, 34] and Seifert et al. [169] used for the model description of the homogeneity range of boron carbide the sublattice description (hi2> BiiC)(CBC, CBB, BVaB). The sublattice model (B)93(B,C)i2 was used to describe the carbon solubility in j8-boron. The Redlich-Kister parameters for the liquid phase and graphite (ss) and general formula descriptions for the solution phases were accepted from [36]. The calculated optimized phase... 

For the following calculation, experimentally determined dielectric functions for silver [30] and for a plasma polymer [31] were taken. The effective dielectric functions e(v) were calculated with the Maxwell Garnett theory for parallel-oriented particles, equation (13). From the effective dielectric function, transmission or extinction spectra can be calculated by using the Fresnel formulas [10] for the optical system air-composite media-quartz substrate. As a further parameter, the thickness of the film with embedded particles and the thickness of other present layers that do not contain metal nanoparticles have to be included. The calculated extinction spectra can be compared with the experimental spectra. 

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