Quadrupole contribution

The second, and most important, contribution to Raman intensities is the quadrupole contribution, induced by the operator... 

The measured Q.S. values for the surface sites of AU55 give nuclear splittings of 21 mK, 13 mK, and 4 mK for the PPhj coordinated sites, the Cl coordinated sites, arid the bare surface sites respectively. From these values, and the known site occupations, the nuclear quadrupole contribution to the zero field specific heat by these three two-level systems [143] has been calculated directly [144], This value is 5 times as large as that experimentally observed [54]. The maximum value of a linear term in the specific heat of AU55 has been estimate to be no more than one fifth of the bulk value [144]. 

The coordination numbers based on this structure work extremely well for describing the microscopic physical properties of this material, including the Mossbauer I.S.s of the surface sites and of the specific heat of the clusters below about 65 K. No linear electronic term in the specific heat is seen down to 60 mK, due to the still significant T contribution from the center-of-mass motion still present at this temperature. The Schottky tail which develops below 300 mK in magnetic fields above 0.4 T has been quantitatively explained by nuclear quadrupole contributions. 

The above-mentioned nonlinear optical effects can be described by the perturbation of the electromagnetic held intensity under the electric dipole approximation. Actually, this approximation is broken in optical near-helds. Hence, a perturbation effect of multipole such as electric quadrupole or magnetic dipole should also be considered, although such a higher-order effect is normally negligible. Indeed, electric quadrupole contributions can be comparable with electric dipole contributions... 

The operator Zj zj xj + Ea Za zaxa that appears above is the z,x element of the electric quadrupole moment operator Qz x it is for this reason that this particular component is labeled E2 and denoted the electric quadrupole contribution. 

Kligman and Madigosky (j ) extended the work of Chaban to the case of a solid medium. Based on this, it can be shown that when only the lowest order terms in frequency of the monopole and quadrupole contributions are retained (the dipole contribution appears in the effective density), the effective modulus is... 

In the series of / -(+)-1-phenylethylamine, / -(+)-1-phenylethanol and R- +)- -phenylethylthiol Barron et al. (1989) examine the influence of heteroatom Rydberg p-orbitals on the ratio of polarized to depolarized ROA spectra. They explain the intensities of the methyl antisymmetric deformation band as injection of a large electric quadrupole contribution from the Rydberg orbitals, which are, as a crude calculation shows, of sufficient extension. 

The superscript k = 6, 8, and 10 for electric dipole-dipole, dipole-quadrupole, and quadrupole-quadrupole contributions, respectively [20]. Because of the dependence of Wet in Eq. (8) on exp[-2i DA/ o] in the case of exchange, and on in the case of multipole-multipole interactions, ET processes are very concentration dependent. At low dopant concentration levels (<0.1%) in homogeneously-doped crystals, where i i,A is large, Wet will approach zero leaving GSA/ESA as the only viable upconversion mechanism. 

In agreement with the considerations developed above for spherical particles, the first purely local contribution vanishes for perfectly centrosymmetrical particles, and in particular for spheres. In that case, the model developed above coincides with that proposed by J.I. Dadap et al. [39, 40]. In particular, the two non local contributions correspond to the effective dipole p ff and effective quadrupole contributions introduced in that work. The effective dipole arises from retardation effects taken into account at the fundamental frequency and the effective quadrupole to retardation effects at the harmonic frequency only. Contributions with retardation effects at both the fundamental and the harmonic frequencies would be of higher orders of the parameter x. As we shall see later, the different contributions may be distinguished through their angular and polarization patterns. 

The theoretical framework developed above is valid in the electric dipole approximation. In this context, it is assumed that the nonlinear polarization PfL(2 >) is reduced to the electric dipole contribution as given in Eq. (1). This assumption is only valid if the surface susceptibility tensor x (2 > >, a>) is large enough to dwarf the contribution from higher orders of the multipole expansion like the electric quadrupole contribution and is therefore the simplest approximation for the nonlinear polarization. At pure solvent interfaces, this may not be the case, since the nonlinear optical activity of solvent molecules like water, 1,2-dichloroethane (DCE), alcohols, or alkanes is rather low. The magnitude of the molecular hyperpolarizability of water, measured by DC electric field induced second harmonic... 

Question by M. J. Hiza, CEL National Bureau of Standards Were quadrupole contributions considered in the calculation of the second virial coefficient ... 

Answer by Author Analytical calculations were made with and without quadrupole contributions on the second virial coefficients determined from the Lennard-Jones 6-12 and Kihara potentials. However, the correlation between the experimental and analytical results were superior when the quadrupole contributions were neglected, except for the Kihara potential at 190 K. Therefore, the results presented here do not include these contributions. 

The point-charge model can be improved by considering the effects of higher multipole moments in eq. (46). Dipole and quadrupole contributions (terms with k - and 2 in eq. (46)) have been evaluated for PrCL (Hutchings and Ray, 1963). It is possible to include the dipole contributions to the crystal field in a self-consistent manner (Morrison, 1976) in that treatment, point charge and dipole contributions to the electric field at various lattice sites are evaluated, and the dipole moments at these sites... 

The earliest calculations of the dipole and quadrupole contributions to the crystal field used free-ion values for the dipole polarizability, a,-, and for the quadrupole polarizability (Hutchings and Ray, 1963). It is generally recognized today, however, that the polarizability of an ion in a solid is generally much less than the corresponding value for the free ion (Chakrabarti et al., 1976 Bogomolova et al., 1977). How much less, however, is not clear. For this reason, therefore, computations of dipole and quadrupole contributions to the crystal field should be regarded as imprecise, even if they are performed with reduced polarizability values. 

The Ar2 potential curve is shown in Fig. 21. Considering for the present only the attractive state, the Hartree-Fock interaction energy only becomes comparable to that of the polarization potential when the interaction energy is of the order of 0.1 eV. Once again, such considerations restrict the use of the Langevin model to thermal energies. Here, there is no possible ion-quadrupole contribution and in both these cases, dispersion forces are negligible. 

According to Eq. (1.1) the quadrupole contribution to the flexoelec-tric polarization is determined by the gradient of the average quadrupole density, which can be written in the following form using Eq. (1.5) ... 

Thus the quadrupole mechanism yields very simple expressions for the flexo-electric coefficients, which are proportional to the nematic order parameter S in the first approximation (usually the parameter D is much smaller than S). In addition, the difference of the flexoelectric coefficients appears to be equal to zero, i.e. Ae = ei — 03 = 0 if the quadrupole contribution alone is taken into account. 

Let us now discuss the approximate expressions for the flexoelectric coefficients, Eq. (1.31), in more detail. Firstly, note that the expressions for both coefficients and es contain terms proportional to both S and S. It has been assumed in the literature that the dipolar contribution to the flexoelectric coefficients is always proportional to while the quadrupole contribution is proportional to S, and even the method of separation between the dipolar and quadrupolar flexoelectric effect has been proposed based on these preliminary results. The results of the consistent molecular theory presented in this section allow us to conclude that the relation e S for the dipolar contribution is due to the shortcomings of the semi-phenomenological approach. The results of this section also cast some doubt on the quantitative ratio of the dipolar and quadrupole contributions based on a comparison of the two terms in the expression e = eoS + C2S. At the same time, the absence of the linear term in S in the dependence e S) for a number of nematic materials stUl points to the predominant role of the quadrupole flexoeffect for those materials. 

The fiexoelectric polarization is given by the same general Eq. (1.1) where the quadrupole contribution can again be written in the form of Eqs (1.9)-(1.11) with... 

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