Eigenfrequency model

We now describe how the periodic orbit theory of Section 3.6, which relates the energy levels with the poles of the spectral function g(E), can be extended to two dimensions. For simplicity we shall illustrate this extension by the simplest model, in which the total PES is constructed of two paraboloids that cross at some dividing plane. Each paraboloid is characterized by two eigenfrequencies, dividing plane (symmetric case) or with respect to a point (antisymmetric case). Our discussion draws on Benderskii et al. [1992b]. 

This Bose approximation amounts to adding to each electronic excited state a sequence of equidistant levels, creating a harmonic oscillator that is, it is Thomson s classical model of elastically bound electrons, with this difference that each point dipole presents many eigenfrequencies associated with the transition dipole.23... 

Theoretically 7 was first defined as the variation of the eigenfrequencies of a solid as the volume changed. Equation (6.26) can be derived from this definition, using the model of an elastic continuum. The value of 7 is not predicted, but operationally it can be measured using Equation (6.26). For many solids Cy is not equal to 3k, as in Equation (6.25), because the highest frequencies are not excited. This causes no error, since x is not increased by such frequencies either. 

Further comments should be made before discussing lattice dynamical models. Because the medium in which the phonon waves travel is discrete, there is a minimum wavelength and thus a maximum value of q in any given direction. The planes which bound that region in momentum space define the limits of q and the Brillouin zone. If the eigenfrequencies are summed over the Brillouin zone according to... 

The kinetic ener produced by the absorption of a modulated beam of light in a resonator causes the excitation of acoustic normal modes in the cavity. The rimple theoretical approach considered in this section is that of a lossless gas in a rigid container with perfectly insulating walls. This simple model yields reasonable values for the eigenfrequencies of the resonator. However, for an accurate description of experimentally determined resonance frequencies and the corresponding Q factors, dissipation effects, deviations of the cavity from the ideal geometry, etc. must be taken into account. The advances made in recent years in an understanding of these loss mechanisms will be discussed in the next section. 

Faster computers and development of better numerical algorithms have created the possibility to apply RPA in combination with semiempirical Hamiltonian models to large molecular sterns. Sekino and Bartlett - derived the TDHF expressions for frequency-dependent off-resonant optical polarizabilities using a perturbative expansion of the HF equation (eq 2.8) in powers of external field. This approacii was further applied to conjugated polymer (iialns. The equations of motion for the time-dependent density matrix of a polyenic chain were first derived and solved in refs 149 and 150. The TDHF approach based on the PPP Hamiltonian - was subsequently applied to linear and nonlinear optical response of neutral polyenes (up to 40 repeat units) - and PPV (up to 10 repeat units). " The electronic oscillators (We shall refer to eigenmodes of the linearized TDHF eq with eigenfrequencies Qv as electronic oscillators since they represent collective motions of electrons and holes (see Section II))... 

Table 2 Comparison between SISO models temperature versus eigenfrequency. ...

De Gersem, H., Moens, D., Desmet, W., Vandepitte, D. 2007. Interval and fuzzy eigenfrequency analysis of uncertain substructured FE models. Computer Assisted Mechanics and Engineering Sciences (CAMES), in press. 

Comparison of elgendlsplacements calculated for T0(X) and TA(X) in GaAs with the results of 7 phenomenological models (e.g. "shell model") has shown that their predj.ctlons are extremely scattered, even if the eigenfrequencies m(k) are nearly Identical,... 

In order to describe the dynamic properties of the structure the transfermatrix method is applied. As shown in [5] the advantage of this method over other procedures is the fact that no prior knowlegde about eigenfrequencies or modes is required. Furthermore arbitrary linear damping characteristics can be introduced in the structural model. This is of major importance for the modeling of the soil-structure interaction since an approximation of this mechanism with modal quantities may cauee large errors. In Fig. 2 the equivalent mechanical model for the soil-structure interaction is shown. 

For quantum dots, the frequencies of the surface modes can be calculated for given dielectric constants of the semiconductor and the matrix material. The model assumes a spherical quantum dot with the isotropic dielectric constant e and a hole localized in the center of the dot. With the assumption of a / -wave-vector-independent LO frequency, the SO eigenfrequencies can be calculated as a function of the angular momentum / [155,161,179-182] ... 

Keywords AT-cut quartz crystal Sauerbrey equation Thickness shear mode Oscillation Eigenfrequency Dissipation factor Viscoelasticity Voigt model... 

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