Adiabatic damping

Figure 3.7. Schematic showing that associative desorption induced by thermal excitation of the lattice can create e-h pairs (hot electrons) via non-adiabatic damping of nuclear coordinates (described here...

It should be observed that a condition for the GML relations to hold is that the perturbation is time-independent in the SP, apart from the adiabatic damping. With the Fock-space perturbation (27), this condition is fulfilled, but it is NOT true for any time-dependent perturbation, acting in the restricted space. Therefore, in the present formalism, which is based upon the GML theorem, we have to work in the photonic Fock space. 

When operating on unperturbed states with the adiabatic damping, the initial time is to = —oo, which we normally leave out. 

If, instead, the air is damped adiabatically with the wet cloth, so that the state of the air varies, the cloth will settle to a slightly different temperature. Each state of air (0, x) is represented by a certain wet bulb temperature 6, which can be calculated from Eq. (4.116) or its approximation (4.123), when the partial pressures of water vapor are low compared with the total pressure. When the state of air reaches the saturation curve, we have an interesting special case. Now the temperatures of the airflow and the cloth are identical. This equilibrium temperature is called the adiabatic cooling border or the thermodynamic wet bulb temperature (6 ). 

In Fig. 4 we compare the adiabatic (dotted line) and the stabilized standard spectral densities (continuous line) for three values of the anharmonic coupling parameter and for the same damping parameter. Comparison shows that for a0 1, the adiabatic lineshapes are almost the same as those obtained by the exact approach. For aG = 1.5, this lineshape escapes from the exact one. That shows that for ac > 1, the adiabatic corrections becomes sensitive. However, it may be observed by inspection of the bottom spectra of Fig. 4, that if one takes for the adiabatic approach co0o = 165cm 1 and aG = 1.4, the adiabatic lineshape simulates sensitively the standard one obtained with go,, = 150 cm-1 and ac = 1.5. 

In this section we shall give the connections between the nonadiabatic and damped treatments of Fermi resonances [53,73] within the strong anharmonic coupling framework and the former theory of Witkowski and Wojcik [74] which is adiabatic and undamped, involving implicitly the exchange approximation (approximation later defined in Section IV.C). 

It is of importance to note that we shall consider, in the present section, that the fast and bending modes are subject to the same quantitative damping. Indeed, the damping parameter of the fast mode yG and that of the bending mode y will be supposed to be equal, so that we shall use in the following a single parameter, namely y (= yG = y5). This drastic restriction cannot be avoided when going beyond the adiabatic approximation. 

We must stress that the use of a single damping parameter y supposes that the relaxations of the fast and bending modes have the same magnitude. A more general treatment of damping has been proposed [22,23,71,72] however, this treatment (discussed in Section IV.D) requires the use of the adiabatic approximation, so that its application is limited to very weak hydrogen bonds. 

We shall give here a brief summary of our previous work [71,72] that was concerned with the introduction of the relaxation phenomenon within the adiabatic treatment of the Hamiltonian (77), as was done in the undamped case by Witkowski and Wojcik [74]. Following these authors, we applied the adiabatic approximation and then we restricted the representation of the Hamiltonian to the reduced base (89). Within this base, the Hamiltonian that describes a damped H bond involving a Fermi resonance may be split into effective Hamiltonians whose structure is related to the state of the fast and bending modes ... 

Figure 12. Hydrogen bond involving a Fermi resonance damping parameters switching the intensities. The lineshapes were computed within the adiabatic and exchange approximations. Intensities balancing between two sub-bands are observed when modifying the damping parameters (a) with y0 =0.1, and y5 = 0.8 (b) with ya = ys = 0.8 (c) with yB — 0.8 and ys — 0.1. Common parameters oto = 1, A = 150cm, 2g)5 = 2850cm-1, and T = 30 K.

In the full quantum mechanical approach [8], one uses Eq. (22) and considers both the slow and fast mode obeying quantum mechanics. Then, one obtains within the adiabatic approximation the starting equations involving effective Hamiltonians characterizing the slow mode that are at the basis of all principal quantum approaches of the spectral density of weak H bonds [7,24,25,32,33,58, 61,87,91]. It has been shown recently [57] that, for weak H bonds and within direct damping, the theoretical lineshape avoiding the adiabatic approximation, obtained directly from Hamiltonian (22), is the same as that obtained from the RR spectral density (involving adiabatic approximation). 

These conclusions must be considered keeping in mind that the general theoretical spectral density used for the computations, in the absence of the fast mode damping, reduces [8] to the Boulil et al. spectral density and, in the absence of the slow mode damping, reduces to that obtained by Rosch and Ratner one must also rember that these two last spectral densities, in the absence of both dampings [8], reduce to the Franck-Condon progression involving Dirac delta peaks that are the result of the fundamental work of Marechal and Witkowski. Besides, the adiabatic approximation at the basis of the Marechal... 

Attenuation in solids due to viscosity may be treated by a similar analysis. There may well be other damping mechanisms, such as heat conduction (i.e. imperfectly adiabatic conditions) which also gives an f2 law, and other phenomena associated with solid state defects that may have more complicated frequency and temperature dependence. In polycrystalline solids, especially metals and alloys and also ceramics, elastic grain scattering may cause much greater attenuation than any inelastic damping (Papadakis 1968 Stanke and Kino 1984). 

Fortunately, the same limiting conditions that validate the friction approximation can also be used with time-dependent density functional theory to give a theoretical description of rjxx. This expression was originally derived to describe vibrational damping of molecules adsorbed on surfaces [71]. It was later shown to also be applicable to any molecular or external coordinate and at any location on the PES, and thus more generally applicable to non-adiabatic dynamics at surfaces [68,72]. The expression is... 

Both simulations stress that the relaxation rate for the adsorption energy into the lattice is slow, 1—4 ps, and that energy relaxation into e-h pairs, omitted in these molecular dynamics simulations, is likely to be of the same order of magnitude or perhaps even larger. The non-adiabatic relaxation rate is estimated to also be 1 ps from the vibrational damping rate of the parallel mode for H adsorbed on Cu(lll) [150]. The excitation of e-h pairs accompanying H adsorption on Cu has... 

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