Damping constant classical

Finally, we can recover the classical damping constant from Eq. (A3.30) by writing... 

The general relations among the coefficients - and Dy are presented elsewhere [179]. The quantities yj and y2 are the damping constants for the fundamental and second- harmonic modes, respectively. In Eq.(82) we shall restrict ourselves to the case of zero-frequency mismatch between the cavity and the external forces (ff>i — ff> = 0). In this way we exclude the rapidly oscillating terms. Moreover, the time x and the external amplitude have been redefined as follows x = Kf and 8F =. The s ordering in Eq.(80) which is responsible for the operator structure of the Hamiltonian allows us to contrast the classical and quantum dynamics of our system. If the Hamiltonian (77)-(79) is classical (i.e., if it is a c number), then the equation for the probability density has the form of Eq.(80) without the s terms ... 

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... 

The quenching of excited states near metal surfaces has been reviewed extensivelyand only the main results will be summarised here. The classical approach calculates the damping due to near field coupling (for radiation field effects see 38) of an oscillating adsorbate dipole, a distance d above a metal surface, to the metal electrons. The equation for the damping constant is37,39... 

The quantum-mechanical description of the radiation-matter interaction uses the same classical expression (1.46) but assigns new meanings to the quantities involved [41, 44-65]. Thus, o)oy is the frequency of the transition from the ground state 0 to the excited state j separated in energy by hcnoj. The damping constants y are connected with the transition probabilities from the state j to... 

An excited atom can emit its excitation energy as spontaneous radiation (Sect. 2.7). In order to investigate the spectral distribution of this spontaneous emission on a transition Ei Ek, v/t shall describe the excited atomic electron by the classical model of a damped harmonic oscillator with frequency co, mass m, and restoring force constant k. The radiative energy loss results in a damping of the oscillation described by the damping constant y. We shall see, however, that for real atoms the damping is extremely small, which means that y 0). ... 

In Sect. 2.8 we saw that the mean lifetime r, of a molecular level E/, which decays exponentially by spontaneous emission, is related to the Einstein coefficient Ai by Ti = 1/A/. Replacing the classical damping constant y by the spontaneous transition probability A/, we can use the classical formulas (3.9-3.11) as a correct description of the frequency distribution of spontaneous emission and its linewidth. The natural halfwidth of a spectral line spontaneously emitted from the level Ei is, according to (3.11),... 

Fig. 3.5 Plots of the absorption calm, solid curve) and refractive index (n - tig, dashed curve) as functions of angular frequency (m) in the region of an absorption band centered at frequency Q as predicted by the classical theory of dielectric dispersion (Eqs. B3.3.16 and B3.3.17). Frequencies are plotted relative to the damping constant cajm, relative to the factor F = IjcNelfJnigQg, and (n - Ug), relative to the factor F/Ug...

As in classical Brownian motion theoiy, the damping in Eq. (492), characterized by the damping constants Kjk, arises from spontaneous equilibrium fluctuations described by equilibrium time correlation functions. The oscillatory motion of the system is characterized by the frequencies S2y. The influence of microscopic interactions on the overall time evolution of the displacements from equihbrium is buried in the quantities 2y and [Kjk] or equivalently A/fy., A/ y., and Xjk. ... 

In Sect.2.7 we saw that the mean lifetime x. of a molecular level E., which decays exponentially by spontaneous emission, is related to the Einstein coefficient by x. = 1/A. Replacing the classical damping constant... 

The interaction of a light wave and electrons in atoms in a solid was first analysed by H. A. Lorentz using a classical model of a damped harmonic oscillator subject to a force determined by the local electric field in the medium, see Equation (2.28). Since an atom is small compared with the wavelength of the radiation, the electric field can be regarded as constant across the atom, when the equation of motion becomes ... 

F is the bulk collision constant, A is a positive dimensionless factor, Vf is the Fermi velocity and R the particle radius. From a classical point of view, this modification is supported by the fact that, when the radius is smaller than the bulk mean free path of the electrons, there is an additional scattering factor at the particle surface. This phenomenon, known as the mean free path effect, is abundantly discussed in [19]. In a quantum approach, the boundary conditions imposed to the electron wave functions lead to the appearance of individual electron-hole excitations (Fandau damping) [21] resulting in the broadening of the SPR band proportional to the inverse of the particle radius as in Eq. (8) [22]. A chemical interface damping mechanism has also been considered, leading to the l/R dependence of F [23]. 

The observed intensities also depend on the refractive index, vhich in general is frequency dependent [93], This dependence is unkno vn in most cases and has not been considered. We note, however, that for liquid water the refractive index is virtually constant between 300 and 3500 cm i [94]. The dipole autocorrelation function is calculated classically and quantum corrections [95, 96] are introduced through the factor 2/[l+exp(-ko/2 tkBT)]. Eq. (9.16) for the absorption spectrum has previously been applied in calculations of the far- and mid-IR spectra of liquid water [90, 97[ and crystals [85]. The quantum correction damps the intensities of the low frequency motions and more sophisticated schemes [98] may lead to more severe damping of the low frequencies - as found for liquid water [99]. 

The curve of the absolute maximum strain against the frequency of the unloaded MAS (Fig. 6.38) has been calculated and tested taking into account both the field limit and the stress limit at each frequency. It defines a law of current that depends on frequency. This new strain curve is above the classical curve of strain at constant current, based on the maximum current acceptable at resonance. It possesses a large pass band, which might be used in several applications such as active damping, low frequency projectors, etc. 

Brownian dynamics obtain when the molecular system on the energy surface is subject to random impulses, as if from a solvent, and to a frictional damping term. In such a study of butane in water, sufficient transitions of the conformational barrier were obtained during the relaxation of an excess trans population to permit the evaluation of a rate constant. Kramers classical diffusion treatment is in agreement with these results. 

If the thermal equilibration of states 2 and 3 is very slow, the osciUatimis between states 1 and 2 continue indefinitely (Fig. 10.4A). As the time cmistant for conversion of state 2 to state 3 is decreased, the oscillations are damped and state 3 is formed more rapidly (Fig. 10.4B-D). But when Ti becomes much less than hlH2i, the rate of formation of state 3 decreases again (Fig. 10.4E,F) This quantum mechanical effect is completely contrary to what one would expect from a classical kinetic model of a two-step process, where increasing the rate constant for conversion of the intermediate state to the final product can only speed up the overall reaction (Box 10.2). In the stochastic Liouville equation, the slowing of the overall process results from very rapid quenching of the off-diagonal terms of p by the stochastic decay of state 2. This is essentially the same as the slowing of equilibration of two quantum states when is much less than h/Hi2, which we saw in Fig. 10.3. 

FIGURE 1 3D polarization-time-space plot of classical discrete breathers in lithium tantalate system with poling field = 13.9 kv/cm, non-dimensional electrical field = 0.01 and damping = 0.50 and a low level of coupling constant = 0.50. Symmetric breathers are still observed at this damping value that decays further on increasing the damping. 

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