Constant-energy ellipsoid

A non-variational method has also been used by [25] to determine the donor energy levels in uniaxial crystals, with an application to 4 //-SiC. It considers first a constant-energy ellipsoid with three different electron effective masses mi, my and m-z along three mutually orthogonal axes, which... 

Fig. 8.1. Multi-valley structure of the CB minimum of silicon showing the constant-energy ellipsoids along <100>. For convenience, the direction of an applied compressive force F is defined with respect to the orthogonal axes chosen along the <100> direction (after [140]). Copyright 1972 by the American Physical Society...

The conduction band is characterized by six equivalent minima along the [100] axes of the Brillouin zone located at about ko = 0.85(2jr/a) (symmetry Ai). The surfaces of constant energy are ellipsoids of revolution with their major axes along [100]. Higher minima are located at F and along the [111] axes about 1 eV above the [100] minima. 

Various components of the interactions are calculated using different formalisms. In fact, the shape and size of the cavity are defined differently in various versions of the continuum models. It is generally accepted that the cavity shape should reproduce that of the molecule. The simplest cavity is spherical or ellipsoidal. Computations are simpler and faster when simple molecular shapes are used. In Bom model, with simplest spherical reaction held, the free energy of difference between vacuum and a medium with a dielectric constant is given as [16]... 

By means of this combination of the cross section for an ellipsoid with the Drude dielectric function we arrive at resonance absorption where there is no comparable structure in the bulk metal absorption. The absorption cross section is a maximum at co = ojs and falls to approximately one-half its maximum value at the frequencies = us y/2 (provided that v2 ). That is, the surface mode frequency is us or, in quantum-mechanical language, the surface plasmon energy is hcos. We have assumed that the dielectric function of the surrounding medium is constant or weakly dependent on frequency. 

The solvent effects are often described within a semiempirical selfconsistent reaction field theory (SCRF)248. In this theory the free energy of solvation is obtained from a set of selfconsistent equations describing the interaction of the solute (denoted by S) with the solvent modeled by a polarizable continuum characterized by a dielectric constant e. In the SCRF formalism, as developed by Rivail and collaborators249- 250 the solute-solvent system is modeled by a polarizable continuum (characterized by a dielectric constant e) in which the solvent molecule is immersed within an ellipsoidal cavity251,252. The Hamiltonian describing the solute in the cavity is given by,... 

It is well known that in isotropic and purdy dielectric systems (i.e. in the absence of condudivity and didectric loss) ellipsoidal particles always tend to be aligned with their longest axis parallel to the applied field. This is in prindple not true, however, if the conductivity of the material involved must also be taken into account Nevertheless, it can be shown that still one of the prindpal axes of the ellipsoid is oriented parallel to the field in the state of minimum energy (setting aside very abnormal cases where the complex dielectric constant of the particle is described by a tensor whose principal axes do not coindde with those of the ellipsoid). 

There are claims that also aluminum produces enhancements. Liao and Stern " saw Raman scattering of p-nitrobenzoic acid adsorbed on oxide-covered ( 3-nm thickness) aluminum ellipsoids, which was 500-fold weaker than exhibited on silver. The intensities on aluminum increased by a factor of 3 changing the exciting energy from 2.2 to 2.7 eV, though the optical parameters of aluminum are rather constant in this region. Lopez-Rios et also reported surface Raman spectra from (perhaps) molecular oxygen and carbon oxides adsorbed on aluminum in UHV conditions. No SERS of pyridine was seen. 

Therefore, the best estimate of a solute cavity size can be found by minimizing the difference between the calculated and experimental solvation free energies at different cavity sizes. However, a common scheme for the calculation of the cavity size for different compounds is necessary in order to minimize the number of empirical parameters involved in the theory. For convex cavity shapes (sphere and ellipsoid), it is feasible to assume that the solute cavity is formed by adding the contributions from both the solute and solvent molecules. Such scheme allows to account for the dielectric saturation at the vicinity of the solute molecule and for the change of the cavity size in the solvents with molecules of different size. Thus, in the present study the radius of a spherical cavity was calculated as the sum of the intrinsic radius of the solute molecule, aos, and a constant additive term by solvent (water), aw- In the case of the ellipsoidal cavity, the additive solvent term was added to each semi-axis of the intrinsic ellipsoid of the solute molecule. 

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