Multiphonon order

Having looked for the smallest tItq has to be increased as nearly as possible to unity. From eq. (129) and a given AE, ho) has to be minimized for two reasons (i) in order to increase the multiphonon order N=AEIhcOm), and (ii) because cte w 82 s is the highest for materials with smallest hco (Auzel 1978). 

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,... 

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. 

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)... 

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 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. 

When the frequency of a laser falls fully into an absorption band, multiple phonon processes start to appear. Leite et al 2° ) observed /7 h order ( = 1, 2. 9) Raman scattering in CdS under conditions of resonance between the laser frequency and the band gap or the associated exciton states. The scattered light spectrum shows a mixture of fluorescent emission and Raman scattering. Klein and Porto 207) associated the multiphonon resonance Raman effect with the fluorescent emission spectrum, and suggested a possible theoretical approach to this effect. 

The multiphonon problem involves a complex many-body (both many-ion and many-electron) Hamiltonian and, in order to induce transitions, some perturbation. Essentially, headway on this problem has been possible only by (1) approximating the unperturbed Hamiltonian, (2) assuming some sensible wave functions for this unperturbed Hamiltonian, and (3) assuming some perturbation. 

The effect of temperature on the photoinduced electron transfer from [Ru(bpy)3]2+ to methyl viologen solubilized in cellophane has been investigated 98 K The first-order rate constant which depends exponentially on the distance between the reactants shows a non-Arrhenius type of behavior in the temperature interval from 77 to 294 K. This phenomenon, previously found to be of great importance in biological systems, is quantitatively interpreted in terms of a nonadiabatic multiphonon non-radiative process. 

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. 

We stress that only the rates satisfying the condition yk kr are consistent with the assumptions of the theory. In the case of a high-order multiphonon relaxation (k 1) for large wk (larger than wk cY) the rate very slowly decreases... 

To sum up, we have developed a general non-perturbative method that allows one to calculate the rate of relaxation processes in conditions when perturbation theory is not applicable. Theories describing non-radiative electronic transitions and multiphonon relaxation of a local mode, caused by a high-order anharmonic interaction have been developed on the basis of this method. In the weak coupling limit the obtained results agree with the predictions of the standard perturbation theory. 

Diatomic molecules are the simplest condensed phase VER systems, for example, a dilute solution of a diatomic such as I2 or XeF in an atomic (e.g., Ar or Xe) liquid or crystal. Other simple systems include neat diatomic liquids or crystals, or a diatomic molecule bound to a surface. VER of a diatomic molecule can occur only by energy transfer to the collective vibrations of the bath, i.e., the phonons. Ordinarily VER is a high-order multiphonon process. Consequently there is an enormous variability in VER lifetimes, which may range from 56 s [liquid N2 (20)] to 1 ps [e.g., XeF in Ar (21)], and a high level of sensitivity to environment. Diatomic molecules have simple structures but complex VER mechanisms. 

An alternative approach widely used in polyatomic molecule studies is based on the Golden Rule and a perturbative treatment of the anharmonic coupling (57,62). This approach is not much used for diatomic molecules. In the liquid O2 example cited above, the Hamiltonian must be expanded to 30th order or so to calculate the multiphonon emission rate. But for vibrations of polyatomic molecules, which can always find relatively low-order VER pathways for each VER step, perturbation theory is very useful. In the perturbation approach, the molecule s entire ladder of vibrational excitations is the system and the phonons are the bath. Only lower-order processes are ordinarily needed (57) because polyatomic molecules have many vibrations ranging from higher to lower frequencies and only a small number of phonons, usually one or two, are excited in each VER step. The usual practice is to expand the interaction Hamiltonian (qn, Q) in Equation (2) in powers of normal coordinates (57,62) ... 

Equation (5), VER involves a higher-order anharmonic coupling matrix element, which gives rise to decay via simultaneous emission of several phonons nftjph (multiphonon emission). In the ACN case, three phonons must be emitted simultaneously via quartic anharmonic coupling (or four phonons via fifth-order coupling, etc.). 

Doorway vibration decay is particularly interesting because it is the one situation where polyatomic molecule VER looks just like diatomic molecule VER. The doorway vibrations of polyatomic molecules decay by exactly the same multiphonon mechanism as the VER of a diatomic molecule. Diatomic molecules have been extensively studied (7). One prediction for diatomic molecules is an exponential energy-gap law (2). As the vibrational frequency is increased, with everything else held constant, the number of emitted phonons increases (the order of the multiphonon process increases) and the VER rate should decrease exponentially with increasing vibrational frequency. 

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. 

The minimum prerequisite for generation of upconversion luminescence by any material is the presence of at least two metastable excited states. In order for upconversion to be efficient, these states must have lifetimes sufficiently long for ions to participate in either luminescence or other photophysical processes with reasonably high probabilities, as opposed to relaxing through nonradiative multiphonon pathways. The observed decay of an excited state in the simplest case scenario, as probed for example by monitoring its luminescence intensity I, behaves as an exponential ... 

We have shown, in later sections, how precise INS measurements of the DOS provide the most stringent means of testing the model potential functions that lie at the heart of any LD or MD simulation. In the last a few years, we have systematically studied the vibrational dynamics of a large verity of phases of ice using above instruments at ISIS. These spectra were obtained at very low temperatures (< 15 K) on the recoverable high-pressure phases of ice and a few forms of amorphous forms of ice, in order to reduce the Debye-Waller factor and avoid multiphonon excitations. Hence the one-phonon spectra, g(co), can be extracted from the experimental data for the theoretical simulations. 

The Time Scales and Mechanism of Quasi-coherent Excitation Hopping Within B850/B875 Rings. This appears to be an area where simple theory cannot apply. It will be a challenge for experimentalists and theorists to address this issue collaboratively. For example, it is not clear whether linear coupling to a harmonic bath is adequate to describe such systems. For example, it may be necessary to include multiphonon and Duschinsky effects on the dynamics in order to describe the influence of temperature on such systems. 

Single-ion nonradiative decay for Ln3+ diluted into transparent host elpa-solite crystals, where the energy gap is greater than the Debye cutoff, is primarily due to multiphonon relaxation (with rate kmp). In some cases, first order selection rules restrict phonon relaxation between states, such as between Tig and T4g, or between T2g and T5g, CF states for MX63- systems. The dependence of the multiphonon relaxation rate, kmp, upon the energy gap to the next-lowest state (AE) has been investigated for other systems and is given by a relation such as [353, 354]... 

---