Relaxation of a localized vibrational mode

In this paper, the multiphonon relaxation of a local vibrational mode and the non-radiative electronic transitions in molecular systems and in solids are considered using this non-perturbative theory. Results of model calculations are presented. According to the obtained results, the rate of these processes exhibits a critical behavior it sharply increases near specific (critical) value(s) of the interaction. Also the usual increase of the non-radiative transition rate with temperature is reversed at certain value of the non-diagonal interaction and temperature. For a weak interaction, the results coincide with those of the perturbation theory. 

To explain the idea of the method [13,14], let us consider a two-phonon decay of a highly excited local mode caused by the interaction /7irl( = QY.rf V3 A,, where V3 lI/ are the cubic anharmonicity interaction parameters, Q is the coordinate 

The energy of generated phonons is taken from the mode. This allows one to find the rate of the energy loss (relaxation) of the mode as follows  

From the given expression for dE/dt it follows that, in fact, one does not need the full knowledge of the operator transformation, i.e., one does not need to know the complex parameters /r,(f) and, (0- It is sufficient to know I i i(t) 2. To find the latter quantity, one does not necessarily need to diagonalize the Hamiltonian. It is much easier to obtain this parameter from the ocexp(io , t) term of the large time asymptotic of the phonon correlation function Dt(t, r) = (0lx,(t + t)x,(0I0) with t averaged over a vibrational period. Indeed, taking 

For what follows it is essential that equation (3) for the large-time asymptotic of the phonon correlation function holds rather generally, including cases if the interaction Hamiltonian contains more than two phonon operators and the Hamiltonian cannot be diagonalized [13,14]. This allows one to apply the method for the description of multiphononon relaxation processes caused by higher order anharmonicities. 

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A non-perturbative theory of the multiphonon relaxation of a localized vibrational mode, caused by a high-order anharmonic interaction with the nearest atoms of the crystal lattice, is proposed. It relates the rate of the process to the time-dependent non-stationary displacement correlation function of atoms. A non-linear integral equation for this function is derived and solved numerically for 3- and 4-phonon processes. We have found that the rate exhibits a critical behavior it sharply increases near a specific (critical) value(s) of the interaction. 

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