Multiphonon

In diatomic VER, the frequency Q is often much greater than so VER requires a high-order multiphonon process (see example C3.5.6.1). Because polyatomic molecules have several vibrations ranging from higher to lower frequencies, only lower-order phonon processes are ordinarily needed [34]- The usual practice is to expand the interaction Hamiltonian > in equation (03.5.2) in powers of nonnal coordinates [34, 631,... 

In rare gas crystals [77] and liquids [78], diatomic molecule vibrational and vibronic relaxation have been studied. In crystals, VER occurs by multiphonon emission. Everything else held constant, the VER rate should decrease exponentially with the number of emitted phonons (exponential gap law) [79, 80] The number of emitted phonons scales as, and should be close to, the ratio O/mQ, where is the Debye frequency. A possible complication is the perturbation of the local phonon density of states by the diatomic molecule guest [77]. 

Chen S, Tolbert W A and DIott D D 1994 Direot measurement of ultrafast multiphonon up pumping in high explosives J. Phys. Chem. 98 7759-66... 

Wen X, Tolbert W A and DIott D D 1992 Multiphonon up pumping and moleoular hot spots in superheated polymers studied by ultrafast optioal oalorimetry Chem. Phys. Lett. 192 315-20... 

Thus far we have discussed the direct mechanism of dissipation, when the reaction coordinate is coupled directly to the continuous spectrum of the bath degrees of freedom. For chemical reactions this situation is rather rare, since low-frequency acoustic phonon modes have much larger wavelengths than the size of the reaction complex, and so they cannot cause a considerable relative displacement of the reactants. The direct mechanism may play an essential role in long-distance electron transfer in dielectric media, when the reorganization energy is created by displacement of equilibrium positions of low-frequency polarization phonons. Another cause of friction may be anharmonicity of solids which leads to multiphonon processes. In particular, the Raman processes may provide small energy losses. 

Another conventional simplification is replacing the whole vibration spectrum by a single harmonic vibration with an effective frequency co. In doing so one has to leave the reversibility problem out of consideration. It is again the model of an active oscillator mentioned in section 2.2 and, in fact, it is friction in the active mode that renders the transition irreversible. Such an approach leads to the well known Kubo-Toyozawa problem [Kubo and Toyozava 1955], in which the Franck-Condon factor FC depends on two parameters, the order of multiphonon process N and the coupling parameter S... 

Note that only Er, which is actually the sum of the reorganization energies for all degrees of freedom, enters into the high-temperature rate constant formula (2.62). At low temperature, however, in order to preserve E, one has to fit an additional parameter co, which has no direct physical sense for a real multiphonon problem. 

Description in Terms of a Radiationless Nonadiabatic Multiphonon Process. 93... 

It is thus evident that the experimental results considered in sect. 4 above are fully consistent with the interpretation based on absolute reaction rate theory. Alternatively, consistency is equally well established with the quantum mechanical treatment of Buhks et al. [117] which will be considered in Sect. 6. This treatment considers the spin-state conversion in terms of a radiationless non-adiabatic multiphonon process. Both approaches imply that the predominant geometric changes associated with the spin-state conversion involve a radial compression of the metal-ligand bonds (for the HS -> LS transformation). 

A quantum-mechanical description of spin-state equilibria has been proposed on the basis of a radiationless nonadiabatic multiphonon process [117]. Calculated rate constants of, e.g., k 10 s for iron(II) and iron(III) are in reasonable agreement with the observed values between 10 and 10 s . Here again the quantity of largest influence is the metal-ligand bond length change AR and the consequent variation of stretching vibrations. 

The transition probability for multiphonon, nonadiabatic ET can be formulated in terms of first-order perturbation theory, i.e., by means of the Fermi golden rule, as (2)... 

Figure 5.16 Configurational coordinate diagrams to explain (a) radiative and (b) nonradiative (multiphonon emission) de-excitation process. The sinusoidal arrows indicate the nonradiative pathways.

In Figure 5.16(a), the maximum of the absorption spectrum (at 0 K) corresponds to the line AB, the maximum overlap of the vibrational wavefunctions. This transition terminates in the vibrational level corresponding to point B, which is below the crossover point, X. This proces s is followed by a fast down-relaxation by multiphonon emission to the point C, from which the emission originates. Thus, the emission spectrum has its maximum at an energy corresponding to the line CD. Finally, another multiphonon emission process takes place by down-relaxation from D to the departing point A. 

A very common heating sensing technique used in condensed matter is photoacoustic (PA) spectroscopy, which is based on detection of the acoustic waves that are generated after a pulse of light is absorbed by a luminescent system. These acoustic waves are produced in the whole solid sample and in the coupling medium adjacent to the sample as a result of the heat delivered by multiphonon relaxation processes. 

The nonradiative rate. Am, from a (RE) + ion level is also strongly related to the corresponding energy gap. Systematic studies performed over different (RE) + ions in different host crystals have experimentally shown that the rate of phonon emission, or multiphonon emission rate, from a given energy level decreases exponentially with the corresponding energy gap. This behavior can be expressed as follows ... 

From this expression, we can estimate the multiphonon emission nonradiative rate, from any particular energy level by simply knowing the energy distance to the next lower energy level (the energy gap), AE. 

The exponential decrease in the multiphonon emission rate with an increasing energy gap, given by Equation (6.1), is due to an increase in the number of emitted... 

F ure 6.6 The multiphonon nonradiative rate of (RE) ions as a function of the number of emitted effective phonons for LaCfi (260 cm ), LaEs (350 cm ), and Y2O3 (430-550 cm ). The numbers in brackets indicate the energies of the effective phonons. The shaded area indicates the range of typical radiative rates. 

The very low multiphonon decay rates obtained in Example 6.2 from the Po (Pr +) and p5/2 (Yb +) states are due to the large number of effective phonons that need to be emitted -14 and 38, respectively - and so the high-order perturbation processes. As a consequence, luminescence from these two states is usually observed with a quantum efficiency close to one. On the other hand, from the F3/2 state of Er + ions the energy needed to bridge the short energy gap is almost that corresponding to one effective phonon hence depopulation of this state to the next lower state is fully nonradiative. 

In Table E7.5, the fluorescence lifetimes and quantum efficiencies measured from different excited states of the Pr + ( Po and D2) and Nd + (" Fs ji) ions in a LiNbOs crystal are listed, (a) Determine the multiphonon nonradiative rate from the 19/2 and In/2 states of the Er + ion in LiNbOs. (b) If a fluorescence lifetime of 535 /us is measured from the excited state Fs/2 of the Yb + ion in this crystal, estimate the radiative lifetime from this state. 

In Chapter 5, we discuss in a simple way static (crystalline field) and dynamic (coordinate configuration model) effects on the optically active centers and how they affect their spectra (the peak position, and the shape and intensity of optical bands). We also introduce nonradiative depopulation mechanisms (multiphonon emission and energy transfer) in order to understand the ability of a particular center to emit light in other words, the competition between the mechanisms of radiative de-excitation and nonradiative de-excitation. 

---