Exponential Energy Gap Law Research Articles

Anomalous nonradiative decay in Dy-doped glasses and crystals.

Nonradiative relaxation in Dy-doped fluoroindate glass is measured and characterized in terms of the well-known exponential energy gap law. It is found that although three low-lying Dy transitions fit well with an exponential dependence on an energy gap, the overall magnitude of the Dy decay rate is an order of magnitude higher than that of other rare earth ions. It is also found that the nonradiative decay rates in InF3 glass are comparable to those in ZrF4 glass, contrary to expectations based on maximum phonon energy. These unexpected results have important implications for a potential 4 μm Dy:InF3 fiber laser.

Large cross section for super energy transfer from hyperthermal atoms to ambient molecules

The experimentally measured cross section for super energy transfer collisions between a hyperthermal H atom and an ambient molecule is presented here. This measurement substantiates an emerging energy transfer mechanism with significant cross section, whereby a major fraction of atomic translational energy is converted into molecular vibrational energy through a transient collision-induced reactive complex. Specifically, using nanosecond time-resolved infrared emission spectroscopy, it is revealed that collisions between hyperthermal hydrogen atoms (with 59 kcal/mol of kinetic energy) and ambient ${\mathrm{SO}}_{2}$ result in the production of vibrationally highly excited ${\mathrm{SO}}_{2}$ with $g14\phantom{\rule{0.16em}{0ex}}000\phantom{\rule{0.28em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$ of internal energy. The lower limit of the cross section for this super energy transfer process is determined to be $0.53\ifmmode\pm\else\textpm\fi{}0.05$ \AA{}${}^{2}$, i.e., 2% of all hard-sphere collisions. This cross section is orders of magnitude greater than that predicted by the exponential energy gap law, which is commonly used for describing collisional energy transfer through repulsive interactions.

Study on rotational energy transfer in diatomic molecules in the separable kernel approximation: An analytical approach

In this article, we present a new technique for determining the transition probabilities per collision using a set of transition rates per unit time. The problem is solved by introducing a normalization function, which is readily found from a recursion equation for the case of separable kernels. A classical approach to rotation of molecules allows the line-broadening coefficients, the moments of energy transfer, and the mean number of collisions to be calculated analytically in terms of the exponential energy-gap law.

Solidlike coherent vibronic dynamics in a room temperature liquid: Resonant Raman and absorption spectroscopy of liquid bromine

UV-visible absorption and resonance Raman (RR) spectra of liquid bromine are presented and rigorously interpreted. The RR spectra, which show an anharmonic vibrational progression of up to 30 overtones, define the ground state potential in the range 2.05 A<r<3.06 A. The attractive branch of the X-state potential is softened and apparent dissociation limit of the molecule dramatically reduced by approximately 30% in the liquid phase, indicating an attractive cage-molecule interaction. The excited state potentials (A('), B, and C) are extracted from the absorption spectrum. The spectrum is first inverted under assumption of the classical reflection approximation, then corrected by forward simulations through quantum time correlations. The extrapolated B and C potentials are used to simulate RR spectra. Their validity is cross-checked by the interference pattern of the polarized spectra due to two-channel RR scattering. The discrepancy between calculated and observed intensities can be entirely assigned to vibrational dephasing, which is observed to follow the exponential energy gap law-dephasing rates perfectly trace the Birge-Sponer plot of the vibrational progression-suggesting that vibrational dissipation controls the decay of coherence. Despite strong intermolecular electronic interactions and vibrational energy gaps of approximately kT, vibrational coherences are long lived: Coherence times range from >or=25 to >or=2.4 ps between v=1 and v=25. Remarkably, the RR line shapes are skewed toward the red, indicating upchirp in frequencies that develop over a period of 400 fs. Evidently, the molecular vibrations adiabatically follow the solvent cage, which is impulsively driven into expansion during the approximately 20 fs evolution on the electronically excited state. Liquid bromine retains coherence in ordered sluggish local cages with quadrupolar interactions-dynamics akin to molecules isolated in structured cryogenic rare gas solids.

Time-dependent photoionization of azulene: Competition between ionization and relaxation in highly excited states

Pump-probe photoionization has been used to map the relaxation processes taking place from highly vibrationally excited levels of the S(2) state of azulene, populated directly or via internal conversion from the S(4) state. Photoelectron spectra obtained by 1+2(') two-color time-resolved photoelectron imaging are invariant (apart from in intensity) to the pump-probe time delay and to the pump wavelength. This reveals a photoionization process which is driven by an unstable electronic state (e.g., doubly excited state) lying below the ionization potential. This state is postulated to be populated by a probe transition from S(2) and to rapidly relax via an Auger-like process onto highly vibrationally excited Rydberg states. This accounts for the time invariance of the photoelectron spectrum. The intensity of the photoelectron spectrum is proportional to the population in S(2). An exponential energy gap law is used to describe the internal conversion rate from S(2) to S(0). The vibronic coupling strength is found to be larger than 60+/-5 microeV.

Hopping and Jumping between Potential Energy Surfaces

The crossing of potential energy surfaces (and the flow of nuclear wave function amplitudes between them) becomes important in many different contexts in chemical physics. Photochemical processes often involve dynamics on surfaces that cross (or can be made to cross in an appropriate diabatic representation). Photophysical processes involving radiationless transitions between different Born-Oppenheimer potential energy surfaces are extremely common. Finally, once we “dress” an initial potential energy surface with the energy of a photon, radiative processes involving absorption or emission of a photon are also seen to involve transitions between crossing or closelying potential energy surfaces. Some years ago, Tully and Preston 1 introduced an approach for approximate treatment of dynamics at regions of close-lying potential energy surfaces. In many subsequent trials and refinements, it has proved a worthy computational tool, simple to implement and also very intuitive. The “surface hopping” method has its roots in the Landau-Zener theory 2 but goes beyond it in dealing with multiple crossings and with many degrees of freedom. Related theories include the exponential energy gap law of radiationless transitions 3 and the exponential momentum gap law of Ewing 4 for vibrational predissociation (in which a highfrequency vibrational mode plays the role of the electronic state, relative to a low-frequency van der Waals mode). In these studies, considerable effort has been devoted to developing new intuition for nonclassical Franck-Condon factors. The emphasis has been on the mode competition problem and the overall dependence of rates on the energy or quantum number gaps. Our emphasis is instead directly on the features of the potential energy surfaces, which control the Franck-Condon factors, which in turn control the rates. The fundamental idea that permits a classical treatment of surface hopping is contained in the Landau-Zener-Stuckelberg model, 2 which shows that the hops from one surface to another are localized to the region where the surfaces cross or almost cross. For weak coupling, the probability for hopping is given in terms of the overall coupling, W, between the surfaces, the difference in slopes, j¢F┴j, normal to the intersection of the two potentials at the crossing, and the speed, p┴/m, with which the trajectory goes through the crossing region:

Excitation energy transfer between J-aggregates in layer-by-layer alternate assemblies

The layer-by-layer alternate assemblies incorporating two kinds of cyanine dyes have been fabricated by the alternate adsorption technique. In order to examine excitation energy transfer between J-aggregates, a thiacyanine dye has been employed as the donor and thiacarbocyanine dyes as the acceptor. It is confirmed that these dye combinations form the mixed J-aggregate in the alternate assemblies. From steady-state fluorescence spectra of the molecular assemblies, the occurrence of efficient energy transfer from the donor J-aggregate to the acceptor J-aggregate has been demonstrated. The experimentally obtained rate constant of energy transfer kET in the Stern–Volmer kinetics of the order of 1013nm2molecules−1s−1 is interpreted in terms of exciton migration and trapping in the mixed J-aggregate. The difference in kET between various combinations of cyanine dyes is not inconsistent with an exponential energy gap law.

Rotational state-to-state energy transfer of NH2(Ã 2A1) in beam-gas condition

The detailed study on the rotational state-to-state energy transfer (RSET) of NH2(Ã 2A1) at a relative translational energy of about 360 cm−1 has been reported herein, by combining self-breakdown pulsed dc discharge and a photon counting technique. The transferred populations from the parent levels to the daughter levels have been obtained through the spectral simulations. It is clear that the rotational state distributions depend not only on the rotational quantum numbers of the initial and collisionally populated states, but also on the quantum number Ka. It is interesting to find that the probability of RSET behaves differently depending on the rotational quantum numbers of the parent levels, i.e., the exponential angular momentum transfer law (AMT) is preferred with the relatively low rotational levels, while the exponential energy gap law is preferred with the relatively high levels. According to the sudden approximation and the theory of Osborne et al., the experimental results are explained qualitatively.

Excitation effects in non-radiative multiphonon decays of rare earth doped laser materials

The non-radiative decay process of rare earth ions in solids, which are known to constitute a whole class of laser materials, has been considered up to now to be independent of excitation intensity. We shall show here that the multiphonon non-radiative decay probabilities of rare earth ions in glasses are reduced at high excitation state densities for the larger energy gaps. The classical exponential energy gap law is shown to ‘rotate’ at higher excitation around the 3.2-phonon point. The observed effect is described in term of a spatial saturation of the accepting mode term with effective diffusion lengths ranging between 54 and 20 Å for excited state density from 10 17 to 10 19 cm −3. This effect which is predicted to be general for multiphonon processes at large gives hints for the obtention of laser effects from otherwise non-radiative transitions provided the excitation density is sufficient to allow the coupling between ions through a common phonon diffusion volume.

Bottleneck in multiphonon nonradiative transitions

The nonradiative decay process of rare-earth ions in solids has been considered up to now to be independent of excitation intensity. In this paper we show that the multiphonon nonradiative decay probabilities of rare-earth ions in a germanate glass are reduced at high excitation state densities for the larger energy gaps. The classical exponential energy gap law is shown to 'rotate' at higher excitation around the 3.2-phonon point. The observed effect is described in terms of a spatial saturation of the accepting mode term with an effective diffusion length ranging between 45 and 20 \AA{} for excited state density from 2\ifmmode\times\else\texttimes\fi{}${10}^{17}$ to 8\ifmmode\times\else\texttimes\fi{}${10}^{18}$ ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}3}$ at an active ion concentration of 2.5\ifmmode\times\else\texttimes\fi{}${10}^{19}$ ${\mathrm{cm}}^{\mathrm{\ensuremath{-}}3}$.

Collisional quenching of OH (A2Σ+, ′=0) by N2, O2 and CO2 between 204 and 294 K. Implications for atmospheric measurements of OH by laser-induced fluorescence

A. E. Bailey, D. E. Heard, P. H. Paul and M. J. Pilling, J. Chem. Soc., Faraday Trans., 1997, 93, 2915 DOI: 10.1039/A701582H

Effects of solvent induced modulation of energy gaps on electronic relaxation of excited hydrogen bonded complexes of some aromatic carbonyl compounds

Abstract A detailed study of the influence of solvent polarity and temperature dependence (T ≤ 300 K) on the radiationless transitions of hydrogen bonded complexes of 2-naphthaldehyde (1), 2-acetonaphthone (2), methyl 2-naphthoate (3) and 1,2-dihydro-3H-benz[e]inden-3-one (4) is presented. The hydrogen bonded complexes are strongly fluorescent. The energy gaps between S1 and S0 and between S1 and T1 could be varied by using various 1,4-dioxane/water mixtures as the solvent. In the case of the complex of 1 intersystem crossing and internal conversion from S1 have both been found to proceed through a direct process as well as by way of a proces involving thermal excitation to S2. The conversion of S1 to T1 proceeds only through thermal excitation to S2 in the case of 2 and 4, whereas in the case of 3 a contribution from a thermally activated process could not be detected. An inverse exponential energy gap law has been found for the temperature independent intersystem crossing from S1 in the case of the complexes of 1 and 3. This is shown theoretically to be in accordance with a nuclear tunneling process. The tunneling appears to proceed along the C-O stretching mode. The internal conversion from the state S1 of the complex of 3 satisfies the regular exponential energy gap law.

Counterpropagating pulsed molecular beam scattering of NH3–Ar. I. State resolved integral cross sections

A new approach to molecular beam scattering is described. The method uses counterpropagating molecular beam pulses to define a scattering geometry of cylindrical symmetry while resonance enhanced multiphoton ionization is applied for the state specific product detection. The simple correlation of laboratory and center-of-mass quantities allows a straightforward determination of differential cross sections from measured ion time-of-flight distributions. In addition, the short duration of the pulses causes a delay dependence of the scattering signal which is used as an additional control parameter to define the size of the scattering volume. The method is applied to the rotational excitation of NH3 in collisions with Ar at a collision energy of 158 meV. Delay and depletion studies yield an effective mean free path of 60 cm, confirming single collision condition. While parity averaged integral cross sections are determined for the para modification of NH3, fully state resolved integral cross sections are determined for o-NH3. The general behavior of the integral cross sections for both modifications is well described by an exponential energy gap law. Deviations of individual cross sections from the scaling law confirm the propensity for inelastic collisions with Δk=3. Transitions to parity levels, which are forbidden in the centrifugal sudden approximation, show significantly less intensity.

Electron transfer and intramolecular radiationless transitions

We outline the conceptual framework for the description of long-range photoinduced electron transfer in isolated solvent-free supermolecules and explore the dependence of the intramolecular electron transfer rates from photoselected states on the molecular parameters, i.e. energy gaps and mode-specific intramolecular reorganization energies. An energy gap law for the microscopic electron transfer rate was derived which, for relaxation from the electronic origins, exhibits a poissonian energy gap dependence, with an exponential energy gap law for large gaps. The exponential energy gap law manifests the dominant role of nuclear tunnelling in the weak electron—nuclear coupling limit, being quantitatively distinct from the thermal gaussian energy gap dependence.

Coherent anti-Stokes Raman spectroscopy study of collisional broadening in the O2–H2O Q branch

The fundamental isotropic Raman Q branch of oxygen perturbed by collisions with water vapor has been studied at pressures up to 1.5 atm and for temperatures between 446 and 990 K. The spectra have been recorded by using coherent anti-Stokes Raman spectroscopy (CARS) which has been preferred to stimulated Raman spectroscopy (SRS) in order to obtain more signal and higher sensitivity as the mixture has a small percentage of oxygen. The high resolution CARS spectrometer uses a seeded Nd:YAG laser actively stabilized on an external Fabry–Perot interferometer to prevent any frequency drift during the course of the experiment. The line broadening coefficients have been determined for several rotational quantum numbers (up to N=31 at 990 K). The effect of the splitting into triplets at lower pressure and the effect of interferences between neighboring lines at higher pressure have been taken into account. The influence of Dicke narrowing has also been considered and special care has been taken to avoid Stark broadening. The line broadening coefficients have been calculated according to a semiclassical model. The rotational quantum number and temperature dependencies of the experimental line broadening coefficients have also been studied with another approach based on fitting and scaling laws. Among several laws, the modified exponential energy gap law (MEG), the statistical power-exponential gap law (SPEG), and the energy corrected sudden law with basis rate constants taken as a hybrid exponential-power law (ECS-EP) have given good results. We have used the fitting and scaling laws to extrapolate in temperature the linewidths at 2000 K.

Rotationally and vibrationally inelastic scattering of 4 1 D 2CO

Results of quantum calculations on rotationally and vibrationally inelastic scattering of S 1 ( 1Ã) D 2CO by He and Ar are presented. The azimuthal and vibrationally close-coupled, infinite-order sudden method is used. The extremely anharmonic ν 4 bending mode of S 1 D 2CO is modelled using a double minimum potential. Large cross sections are predicted for collision-induced relaxation of the ν 4 mode. Also, a large steric effect is predicted for energy transfer to this mode: collisions with the D 2-end of the molecule are far more effective in promoting vibrational energy transfer than collisions with the O-end. It should be possible to measure this steric effect in an experiment which employs a beam of oriented D 2CO molecules. The cross sections for pure rotational excitation are fitted well to an exponential energy gap law for both collision partners. In contrast, the rotationally resolved cross sections for vibrational deactivation show considerable structure. The results presented here can serve as predictions for experiments that are currently being carried out.

On the exponential energy gap law in He–I2 vibrational relaxation

A comparison between coupled states, infinite order sudden, and classical path calculations is used to elucidate the origin of an exponential energy gap law recently observed for vibrational relaxation from highly excited states in the B 0+u state of I2 due to collisions with He. All three methods provide relaxation cross sections in good agreement with experiment. Anharmonic effects play an important role, with accurate results obtained with a Morse, but not harmonic, oscillator description of the I*2 molecule. The nearly exact agreement between rotationally summed coupled states cross sections and the IOSA is consistent with the view that the I*2 molecule does not rotate significantly during a collision. A closed form solution of the forced harmonic oscillator, valid for highly excited states, predicts a J2‖Δv‖ distribution of vibrationally relaxed states at a given collision angle and impact parameter. The vibrationally close coupled-infinite order sudden (VCC-IOSA) results bear this out and show that the observed exponential scaling law arises from a superposition of such distributions over θ and b.

Vibrationally (and rotationally) inelastic scattering characteristics for the He+I*2 system

An analysis is provided for the state-resolved vibrationally inelastic scattering cross sections σ(Δυ) for He interacting with I2 B0+u molecules in either υ′=15, 25, or 35. The collision energy for these crossed molecular beam data is 720 cm−1 (89 meV), whereas the local I*2 vibrational quantum size varies from about 100 to 60 cm−1. The σ(Δυ) encompass scattering events with Δυ ranging to ±3 for υ′=15 and to ±7 for υ′=35. The sets of σ(Δυ) for each initial υ′ scale with an exponential energy gap law, and the scaling is identical for all initial υ′ levels. Additionally, σ(Δυ) values for conjugate T→V and V→T transitions (i.e., pairs of Δυ=±n for UP vs DOWN transitions) are nearly equal so that the single scaling law σ(Δυ)∝exp(−‖ΔEvib ‖/110 cm−1) describes the entire set of data. The scaling for the He target beam is identical to that for D2 but different from H2 indicating that the pattern of vibrational energy flow is determined mainly by the mass of the target gas and collision energy as opposed to subtle details of the interaction potential. 1D and 3D classical trajectory calculations replicate the principal characteristics of the scattering, particularly the common exponential scaling and UP–DOWN symmetry of conjugate σ(Δυ), but fail to account quantitatively for processes with large Δυ. The vibrational flow pattern is not markedly influenced by big variations in the rotational energy content of the initial υ′ level. The competition between rotationally and vibrationally inelastic scattering is about the same for each initial υ′. The rotational cross section is only about 2.5× larger than σ(Δυ=−1), the largest vibrational cross section. The total vibrational cross section, however, actually equals or exceeds that for pure rotationally inelastic scattering for all initial υ′ levels. Comparisons are made with the vibrational and rotational energy transfer characteristics observed in 300 K bulb experiments.

Four-wave mixing spectroscopy of Cr-doped laser crystals: Emerald, alexandrite, garnets and ruby

Four-wave mixing techniques were used to establish and probe population gratings of Cr 3+ ions in different types of laser host crystals between 6 and 300 K. The variation of the signal decay rate with grating spacing was used to elucidate the properties of long-range spatial energy migration in these samples. These properties were found to be quite different for the various samples investigated, especially with respect to the role played by phonon processes. The variation of the signal intensity with write beam crossing angle was used to determine the pump band to metastable state radiationless decay rate. These were found to be in the ps time range and to approximately follow an exponential energy gap law with energy difference between the zero-phonon lines of the 4T 2 and 2E levels.

IOS and ECS line coupling calculation for the CO–He system: Influence on the vibration–rotation band shapes

Line coupling coefficients resulting from rotational excitation of CO perturbed by He are computed within the infinite order sudden approximation (IOSA) and within the energy corrected sudden approximation (ECSA). The influence of this line coupling on the 1–0 CO–He vibration–rotation band shape is then computed for the case of weakly overlapping lines in the 292–78 K temperature range. The IOS and ECS results differ only at 78 K by a weak amount at high frequencies. Comparison with an additive superposition of lorentzian lines shows strong modifications in the troughs between the lines. These calculated modifications are in excellent quantitative agreement with recent experimental data for all the temperatures considered. The applicability of previous approaches to CO–He system, based on either the strong collision model or exponential energy gap law, is also discussed.