Relaxation multiphonon

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. 

Emission of d-f type in the divalent lanthanides is usually quenched by multiphonon relaxation from the 5d level to levels of the 4/ configuration. There are a few exceptions. When the red shift and Stokes shift place the 5d level of Sm " near to or below the Dq state (Fig. 5.3) - df emission may occur. The same applies for Eu when the level is shifted to near to or below 7/2. When for Tm the 5d level is not shifted too much towards the F j2 level, df emis-... 

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. 

As an example, we consider the multiphonon relaxation of a local mode caused by an anharmonic interaction with a narrow phonon band. We suppose that the mode is localized on an atom and take into account two diagonal elements of the Green function which stand for the contribution of two nearest atoms of the lattice to the interaction the non-diagonal elements are usually much smaller [16] and approximate the density of states of the phonon band by the parabolic distribution... 

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. 

With the assumption that the phonons involved are of equal energy, a commonly used expression for the temperature-dependent multiphonon relaxation rate is (Riseberg and Moos, 1968)... 

The longer lifetime (1.84 ms) of 5Do in Eu Cb/ZnO is most possibly due to the non-solid medium surrounding the nanoparticles that changes the effective index of refraction. The filling factor is estimated to be approximately 58%, smaller than that of Eu Y203/Al2C>3 (72%). The much shorter 5Di lifetime (27 ps at RT) compared to the bulk counterpart (90-120 ps) is most possibly related to the enhanced nonradiative multiphonon relaxation induced by surface effects of nanocrystals. 

Reisfeld, R. Multiphonon Relaxation in Glasses in Radiationless Processes (B. DiBartolo, V. Goldberg, eds.) p. 489. New York, Plenum Press 1980... 

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]. 

Up-converted UV and visible emissions occur also from lower-lying levels such as 2Pi/2, (2D3/2 2G9/2 4G11/2 2K15/2), 4G7/2 and 4Gs/2 with excitation at 532 nm or 801 nm [75,76], Excitation of 4D3/2 requires a 3-step process with 801 nm pumping. Due to the predominance of multiphonon relaxations from these levels, experimental lifetimes are very short, in the tens of nanosecond range. 

Three-fold up-conversion of 637 nm light results in excitation of the 2P3/2 level via (4Iis/2 - 4F9/2) + (4113/2 -> 4F5/2) + (4S3/2 -> 4[Pg.254]

The (3H4 - 3F4 3H6 - 3F4) cross-relaxation is an efficient energy transfer mechanism for Tm3+ concentration greater than 0.1 mol% in indium-based fluoride glasses [124], Therefore, the intermediate 3F4 level can be populated not only by successive multiphonon relaxations from 3F3, but also through this cross-relaxation process [125],... 

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]. 

Such differences in multiphonon relaxation rates are believed to be largely responsible for the strongly enhanced visible luminescence observed in low-... 

Table 3. Energetic parameters of oxide and halide lattices relating to upconversion involving the I9/2 state of Er +, compiled from various literature sources. A (cm 0 is the average Er + " 19/2 energy gap, (cm ) is the highest-energy lattice phonon energy,p is the reduced energy gap, /c p is the estimated multiphonon-relaxation rate constant, is the estimated range of radiative rate constants, and /Ctot = Kad + /c p. Adapted from [26]...

Carrier interaction with optical phonons in non-polar Si takes place only for holes via deformation potential. Since most of the gaps between the energy levels in QD are larger than a hundred of meV we consider multiphonon relaxation of holes. These processes are controlled by Huang-Rhys factor which has been calculated as a function of QD diameter in the range of 2-4 nm for various transitions and changes from 0.5 to 0.01, respectively. The rate of multiphonon relaxation promoted by a single acoustic phonon emission can be... 

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