Multiphonon vibrational relaxation

This relative ease of electronic relaxation as compared with multiphonon vibrational relaxation was qualitatively interpreted using phonon Franck-Condon arguments. Change in the vibrational state of the guest molecule requires in general very little change in the equilibrium positions of the lattice atoms, and this results in poor Franck-Condon factors for multiphonon vibrational relaxation. Electronic transitions, on the other hand, are often accompanied by considerable changes in electron density distri-... 

Figure 2 Vibrational energy relaxation (VER) mechanisms in polyatomic molecules, (a) A polyatomic molecule loses energy to the bath (phonons). The bath has a characteristic maximum fundamental frequency D. (b) An excited vibration 2 < D decays by exciting a phonon of frequency ph = 2. (c) An excited vibration >d decays via simultaneous emission of several phonons (multiphonon emission), (d) An excited vibration 2 decays via a ladder process, exciting lower energy vibration a> and a small number of phonons, (e) Intramolecular vibrational relaxation (IVR) where 2 simultaneously excites many lower energy vibrations, (f) A vibrational cascade consisting of many steps down the vibrational ladder. The lowest energy doorway vibration decays directly by exciting phonons. (From Ref. 96.)...

As already discussed in Section 13.1, the multiphonon pathway for vibrational relaxation is a relatively slow relaxation process, and, particularly at low temperatures the system will use other relaxation routes where accessible. Polyatomic molecules take advantage of the existence of relatively small frequency differences, and relax by subsequent medium assisted vibrational energy transfer between molecular modes. Small molecules often find other pathways as demonstrated in Section 13.1 for the relaxation ofthe CN radical. When the concentration of impurity molecules is not too low, intermolecular energy transfer often competes successfully with local multiphonon relaxation. For example, when a population of CO molecules in low temperature rare gas matrices is excited to the v = 1 level, the system takes advantage ofthe molecular anhannomcity by undergoing an intermediate relaxation of the type... 

Picosecond laser pulses in the UV range do not result in better ablation behavior than nanosecond laser pulses. This is different for doped polymers. Experiments with doped PMMA (an IR-absorber, i.e., IR-165 for ablation with near-IR laser and diazomeldrum s acid (DMA) for ablation with UV lasers) with nanosecond and picosecond laser irradiation in the UV (266 nm) and near-IR (1064 nm) range have shown that, in the IR, neat features could be produced with picosecond laser irradiation, while nanosecond irradiation only results in rough surface features [105]. This corresponds well with the different behavior of the two absorbers. With IR-165 the polymer is matrix is heated by a fast vibrational relaxation and multiphonon up-pumping [106]. This leads to a higher temperature jump for the picosecond irradiation, which causes ablation, while for nanosecond pulses only lower temperatures are reached. 

This is clearly the dramatic temperature dependence of the vibrational relaxation rates predicted by theories of multiphonon relaxation. Apparently, in this case one deals with a true multiphonon relaxation process. It is reasonable to expect that also in many other heavier diatomics and in polyatomic molecules with small rotational constants or high barriers to free rotation the multiphonon relaxation mechanism will dominate. 

Matrix-isolated molecules exhibit a surprising facility for interelectronic relaxation processes. Vibrational relaxation in excited electronic states is often dominated by interstate cascades involving other electronic states. The rates of the individual steps of such a cascade are modulated by the intramolecular Franck-Condon factors and exhibit qualitatively an exponential dependence on the size of the energy gap expected by multiphonon relaxation theories. 

The strong dependence of x upon the host lattice and the strong isotopic effect, which are contrary to the predictions of the so-called energy-gap law for multiphonon relaxation processes, indicate a relaxation mechanism with transfer of the vibrational energy into rotational energy of the impurity molecule rather than transfer into phonons of the host lattice. Theoretical models for vibrational relaxation of diatomic molecules in solids, which describe the experimental results for NH and ND, have been presented [4 to 9]. 

This results in the well-known exponential energy gap law that is common to several different types of multiphonon radiationless and vibrational relaxation processes. 

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]. 

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 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. 

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. 

One of the main spectroscopic properties that differentiate fluoride glasses from silica-based glasses is the low multiphonon emission rate. These non-radiative relaxations that may strongly compete with radiative processes in rare-earth ions are nearly three orders of magnitude lower in ZBLAN glass than in silicate, as shown in Fig. 2. This property is directly related to the fundamental vibration modes of the host and, therefore, varies basically in the same manner as the infrared absorption edge. 

Raman spectroscopy or far-IR spectroscopy can determine the fundamental vibration frequencies of the host. However, these methods give information about the whole glass matrix and do not account for the local nature of electron-phonon interactions. So, the fundamental frequencies are preferably determined by recording the phonon-side bands (PSB) of rare-earth transitions or by studying the temperature-dependence of multiphonon relaxations [42,43]. The phonon energies determined by PSB spectroscopy, which is the most direct method, are usually lower (400 cm-1 in ZBLAN) than those measured by other methods ( 500 cm-1) suggesting that weak M—F bonds are coupled to the rare-earth [43]. 

Here, p = AEq o/ eff is the dimensionless energy gap between the upper state and the closest lower-energy state in units of the effective vibrational energy, Veff (cm ). C is the electronic factor, and S is the Huang-Rhys dimensionless excited-state distortion parameter in units of vibrational quanta v ff. As shown in Eq. (2), /c ,p is strongly dependent onp. Additionally, for a given reduced energy gap p, the introduction of even small excited-state distortions, S, can rapidly enhance the radiationless multiphonon relaxation rate such that this dominates the total 0 K relaxation. This model is easily extended to elevated temperatures, where substantial increases in may be observed [7,8]. 

Carrier relaxation due to both optical and nonradiative intraband transitions in silicon quantum dots (QDs) in SiOa matrix is considered. Interaction of confined holes with optical phonons is studied. The Huang-Rhys factor governing intraband multiphonon transitions induced by this interaction is calculated. The new mechanism of nonradiative relaxation based on the interaction with local vibrations in polar glass is studied for electrons confined in Si QDs. 

Fig. 13.2 The relaxation of different vibrational levels of the ground electronic state of 2 in a sohd Ar matrix. Analysis of these results indicates that the relaxation of the v < 9 levels is dominated by radiative decay and possible transfer to impurities. The relaxation ofthe upper levels probably takes place by the multiphonon mechanism discussed here. (From A. Salloum and H. Dubust, Chem. Phys. 189, 179 (1994).)...

The observation that the reaction requires an induction time of tens of picoseconds can be used to differentiate between proposed mechanisms of how shock wave energy localizes to cause chemical reaction. This induction time is expected for mechanisms that involve vibrational energy transfer, such as multiphonon up-pumping [107], where the shock wave excites low frequency phonons that multiply annihilate to excite the higher frequency modes involved in dissociation. It is also consistent with electronic excitation relaxing into highly excited vibrational states before dissociation, and experiments are underway to search for electronic excitations. On the other hand, prompt mechanisms, such as direct high frequency vibrational excitation by the shock wave, or direct electronic excitation and prompt excited state dissociation, should occur on sub-picosecond time scales, in contrast to the data presented here. 

A number of theoretical studies have investigated multiphonon relaxation in solids. Nitzan and Jortner considered a harmonic oscillator coupled to a harmonic lattice the coupling potential was taken to be linear in the vibrational coordinate and of high order in phonon displacements. 

Most of the earlier theoretical studies dealt with the simplest relaxation mechanism where the internal vibrational energy of the guest is dissipated directly into the delocalized and harmonic lattice phonons. The common results of these works " were, as we mentioned above, predictions of a strong temperature dependence for the relaxation and an exponential decrease in the rates with the size of the vibrational frequency. The former result has its origin in stimulated phonon emission the conversion of vibrational energy into lattice phonons is greatly facilitated if some excited phonon states are thermally populated. The energy-gap law is due to the fact that the order of the multiphonon relaxation increases with the size of... 

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