Eigenfrequency equation

Eigenfrequency Equation and Elementary Excitations of the Matter-Radiation System... 

The eigenfrequency spectrum of the surface modes of a hollow sphere with gas inside is well known (e.g., see Ref. [109] as well as our Appendix A). If we pretend for a moment that the surface tension coefficient a is curvature independent, the possible values of the eigenfrequency oo are found by solving the following equation ... 

The exact approach to the problem of dynamic (linear) stability is based on the solution of the equations for small perturbations, and finding eigenvalues and eigenfunctions of these equations. In a conservative system a variational principle may be derived, which determines the exact value of eigenfrequency... 

The corresponding vibrational eigenmodes and eigenfrequencies at a given point q on the constraint surface are given by the solutions of the eigenvalue equation... 

Balian, R. and Bloch, C. (1972). Distribution of eigenfrequencies for the wave equation in a finite domain III. Eigenfrequency density oscillations, Annals of Physics 69, 76-160. 

The quantity eoo represents the optical permittivity, which is determined by the electronic polarizability. The second term represents the ionic crystal lattice as a sum of N classical harmonic oscillators with eigenfrequencies ujj, damping constants Tj and oscillators strengths Sj. In order to fit Equation (5.7) to the observed far infrared spectra, these parameters are used... 

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. 

To find the coefficients zfjf and, one has to solve an infinite set of coupled equations (18) (k = 1,2,...) with time-dependent coefficients moreover, each equation also contains an infinite number of terms. However, the problem can be essentially simplified, if the walls perform small oscillations at the frequency [Pg.323]

This problem can be solved by means of a mathematical procedure applied to the damped free oscillation data. This procedure is called a Fourier transform and can easily be performed with an electronic computer. For the mechanical example with on eigenfrequency, the basic equations of the Fourier transform read... 

The index n is used to label the successive roots of this equation. The values of the first roots a are listed in Table 2. The eigenfrequencies of the radial modes of a spherical resoottra cn be cafculated with these a valtms using the equation ... 

The equation of motion for the molecular oscillator with oscillation amplitude q, mass m, and vibrational eigenfrequency [Pg.163]

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))... 

Figure 2.25a illustrates this transition probability as a function of the detuning Aco = co—coba. Equation (2.99) shows that b(t) depends on the absolute value of the detuning Aco = co—coba of the field frequency co from the eigenfrequency cota When tuning the frequency co into resonance with the atomic system (co coba), the second factor in (2.99) approaches the value t because lim c o[(sin xt)/x ] = t. The transition probability at resonance,... 

The inhomogeneous equation (2.109a) shows that the induced dipole moment of the atom interacting with a monochromatic radiation field behaves like a driven damped harmonic oscillator with coha = (Et - Ea)/ h for the eigenfrequency and Yab = Ya + Yb) for the damping constant oscillating at the driving field frequency a>. 

The solution of this equation is a vibration with eigenfrequency... 

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