Phonon broadening theory

According to the quantum transition state theory [108], and ignoring damping, at a temperature T h(S) /Inks — a/ i )To/2n, the wall motion will typically be classically activated. This temperature lies within the plateau in thermal conductivity [19]. This estimate will be lowered if damping, which becomes considerable also at these temperatures, is included in the treatment. Indeed, as shown later in this section, interaction with phonons results in the usual phenomena of frequency shift and level broadening in an internal resonance. Also, activated motion necessarily implies that the system is multilevel. While a complete characterization of all the states does not seem realistic at present, we can extract at least the spectrum of their important subset, namely, those that correspond to the vibrational excitations of the mosaic, whose spectraFspatial density will turn out to be sufficiently high to account for the existence of the boson peak. 

We found a correct account of the broadening magnitude and increase with temperature, out of reach of the renormalized perturbation theory. Exact numerical calculations by Schreiber and Toyozawa53 agree with our data and confirm our values of the exciton-phonon coupling strengths. 

The theories of the Urbach edge are based on the idea that a sharp absorption edge is broadened by some mechanism. In ionic crystals there is little doubt that optical phonons are responsible for the Urbach edges. If their frequency is then by a general argument given below... 

Any temperature dependence of the CEF halfwidth can be attributed to either s-f interaction or the interaction of f states with phonons, s-f line broadening for simple metals has been attributed to the effects of the carrier spin dynamics on the CEF-state lifetimes and has been described by the Fermi liquid theory, introduced by Becker et al. (1977). The s-f interaction Hamiltonian describing the coupling between the 4f electrons and the conduction electrons can be written as... 

We present a derivation of the broadening due to the solvent according to a system/ bath quantum approach, originally worked out in the field of solid-state physics to treat the effect of electron/phonon couplings in the electronic transitions of electron traps in crystals [67, 68]. This approach has the advantage to treat all the nuclear degrees of freedom of the system solute/medium on the same foot, namely as coupled oscillators. The same type of approach has been adopted by Jortner and co-workers [69] to derive a quantum theory of thermal electron transfer in polar solvents. In that case, the solvent outside the first solvation shell was treated as a dielectric continuum and, in the frame of the polaron theory, the vibrational modes of the outer medium, that is, the polar modes, play the same role as the lattice optical modes of the crystal investigated elsewhere [67,68]. The total Hamiltonian of the solute (5) and the medium (m) can be formally written as... 

Here, o is the resonant electronic frequency when the interaction with phonons is absent. The cumulant functions q> (t) describe the influence of the electron-phonon interaction on the resonant frequency. In contrast to the dassical Eq. (2), the quantum mechanical dynamical theory yidds different expressions for the absorption and the emission band. A value of the resonant frequency o is determined solely by electrostatic interaction between the chromophore and its neighbours. AH the dynamical interactions only influence the cumulant functions q> (t) which cause a homogeneous broadening. 

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