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Journal of Physics C Solid State Physics

Ion-lattice relaxation processes and spectroscopic linewidths

G E Stedman

Published May 1, 1970• 12 citations
Publisher

Abstract

Raman contributions to the width of a no-phonon line are calculated for a two level system. The existence of the semidiagonal process previously suggested is confirmed, and it connection with the stabilization of the lattice around an electronic excitation is examined. The theory also predicts a new self energy process, in which the Raman transitions eventually return the electronic excitation to its original state. This process demonstrates the inequivalence of the spin-lattice relaxation times as measured by population equilibration and from spectroscopic linewidths. The existence of such linewidth contributions suggests that the correct statement of the well known relation between spectroscopic linewidths and lifetime effects is that the full width at half power of the spectral line equals the sum of the inverse lifetimes of the two states (before and after the photon interaction) with respect to the broadening interaction.

DOI
10.1088/0022-3719/3/5/020

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Dynamic Processes in the Line Shift and Linewidth of CaF₂ and CsCdBr₃ Doped with Er³⁺

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The linewidths and line positions of 4I15/2 to 4I13/2 absorption transitions of trivalent erbium in CsCdBr₃ and CaF₂ were measured as a function of temperature from approximately 10 K to 300 K. By comparing the temperature dependence of these transitions with theoretical models of electron-phonon coupling the primary mechanism involved were determined. For Er3+ doped CaF₂ the Raman scattering processes dominated the line broadening, although the single-phonon direct processes were significant enough to drastically alter the values of the Raman scattering electron-phonon coupling constant. In comparison, the Er³⁺ doped CsCdBr₃ had negligible Raman scattering contributing to the line broadening. This is likely due to the exceedingly low Debye temperature and consequently low number of phonons, reducing the likelihood of two-phonon processes relative to single-phonon processes. The results were then analysed in terms of the bond length with the ligands and compared with other studies showing that as the bond length gets shorter the electron-phonon coupling constant associated with Raman scattering is expected to get smaller. To explain the line shifts in CaF₂:Er³⁺ it is necessary to treat the Debye temperature as a parameter. This indicates that the phonon modes causing the line shifts are different from those causing the line broadening, and in most of the cases are vibronic processes rather than Raman processes. In the case of CsCdBr₃:Er³⁺ there is very little difference between the fits using the fixed Debye temperature and a varied Debye temperature in over half of the cases examined. Given the distribution of phonon modes, it is likely that this small difference arises because combinations of these processes are of similar intensity, meaning that a combination of Raman and optical phonon modes are likely causing the line shift.

•Jan 1, 2012•0 citations
Raman contributions to the width of a no-phonon line are calculated for a two level system. The existence of the semidiagonal process previously suggested is confirmed, and it connection with the stabilization of the lattice around an electronic excitation is examined. The theory also predicts a new self energy process, in which the Raman transitions eventually return the electronic excitation to its original state. This process demonstrates the inequivalence of the spin-lattice relaxation times as measured by population equilibration and from spectroscopic linewidths. The existence of such linewidth contributions suggests that the correct statement of the well known relation between spectroscopic linewidths and lifetime effects is that the full width at half power of the spectral line equals the sum of the inverse lifetimes of the two states (before and after the photon interaction) with respect to the broadening interaction.
By using far-infrared laser spectroscopy we have studied the temperature dependence of the absorption that is due to transitions between the E ((2)E) and 2A ((2)E) levels of excited Cr(3+) ions in ruby. From the temperature dependence of the linewidth we determined the inhomogeneous and the homogeneous contributions to line broadening. From the homogeneous width the lifetime of the 2A level was obtained. We found that, for emperatures up to 70 K, the lifetime is due mainly to one-phonon relaxation, whereas at higher temperatures another relaxation process, probably a two-phonon Raman process, becomes strong.
Phonon-ion interactions have recently been shown to satisfactorily explain the temperature dependence of widths and positions of the RRR lines of some 3d3d3d ions in crystals. These interactions give rise to phonon relaxation processes and are investigated here as the mechanisms responsible for the widths and positions of sharp optical transitions in (4f4f4f) ions, in particular LaF3{\mathrm{F}}_{3}F3​: Pr3+{\mathrm{Pr}}^{3+}Pr3+. The widths of transitions originating or terminating in the metastable 3P0^{3}P_{0}3P0​ state and involving the 3H4^{3}H_{4}3H4​, 3H6^{3}H_{6}3H6​, 3F2^{3}F_{2}3F2​, and 3F4^{3}F_{4}3F4​ states of Pr3+{\mathrm{Pr}}^{3+}Pr3+ in LaF3{\mathrm{F}}_{3}F3​ have been investigated as a function of temperature and are found to be both qualitatively and quantitatively explainable in terms of lifetime broadening of the interacting states via nonradiative processes. Temperature-dependent shifts have also been measured for the above four groups of transitions and are shown to be in qualitative agreement with theoretical predictions. Additional features of the LaF3{\mathrm{F}}_{3}F3​: Pr3+{\mathrm{Pr}}^{3+}Pr3+ spectrum, including several unreported weak transitions and vibrational sidebands, are also reported.
It has been postulated that the state function can be expressed as a linear superposition of eigenstates of Hermitian operators. When a system is subjected to an external impulse in the form of radiation, the probability of finding the system in different eigenstates of the state function changes. The total probability is always unity, which means the change in probabilities for different states occurs at the expense of each other. This change in probability of finding the system in different eigenstates brought about by system-radiation interaction is called the spectroscopic transition. Here, we use a two-level system to develop the details of a general spectroscopic transition, and work further to examine the effects of interruption in the coherent evolution of the state function. The physical meaning of absorption coefficient is discussed in detail. We derive the general forms of relaxation of the excited state, spectral line shape, and line broadening. The origin and the meaning of homogeneous and inhomogeneous line broadening are discussed. We describe the Doppler effect and its use in the interpretation of inhomogeneous line shape.
A comprehensive theory of a new type of three-wave optical mixing (${\ensuremath{\omega}}_{3}=2{\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$) is given for the spectroscopic purpose of determining ultra-short relaxation times (picoseconds or less) associated with excited states of condensed matter. Two incident light frequencies ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$ are chosen so that they are both resonant with an inhomogeneously broadened optical transition and $|{\ensuremath{\omega}}_{2}\ensuremath{-}{\ensuremath{\omega}}_{1}|$ is in the vicinity of inverse relaxation times. The theory is developed for its major application to broad-band electronic transitions. The nonlinear susceptibility ${\ensuremath{\chi}}_{R}^{(3)}$ for this process (named resonant Rayleigh-type mixing) is calculated by a density-matrix formalism with the model of a two-level atomic system incorporating the distribution of resonance frequencies and longitudinal (T1{T}_{1}T1​) and transverse (T2{T}_{2}T2​) relaxation times. The result shows a frequency characteristic depending only on T1{T}_{1}T1​, T2{T}_{2}T2​, and ${\ensuremath{\omega}}_{2}\ensuremath{-}{\ensuremath{\omega}}_{1}$ in a broad-band limit, which serves for the determination of T1{T}_{1}T1​ and T2{T}_{2}T2​ in the frequency domain. The analysis is further extended to include the various effects as follows. The calculation of the saturation effect in the lowest order reveals that it modifies the shape of the frequency characteristic of the nonlinear susceptibility but does not affect seriously the determination of relaxation times. The effect of spectral cross relaxation within the inhomogeneous broadening is incorporated by a generalized density-matrix formalism. The resultant ${\ensuremath{\chi}}_{R}^{(3)}$ consists of two terms. The dominant term is the same as before except that T1{T}_{1}T1​ is replaced by a combined relaxation time ${T}_{1}^{\ensuremath{'}}={({T}_{1}^{\ensuremath{-}1}+{T}_{3}^{\ensuremath{-}1})}^{\ensuremath{-}1}$, where T3{T}_{3}T3​ is the cross-relaxation time. The other term originates from the inverse spectral-diffusion process. The effect of other energy levels located between the two levels under study is analyzed with a simple three-level model. The resultant ${\ensuremath{\chi}}_{R}^{(3)}$ also consists of two terms, the dominant one being the same as that for the two-level model except that ${T}_{1}^{\ensuremath{-}1}$ should be interpreted as total population decay rate of the upper level. For accurate derivation of ${\ensuremath{\chi}}_{R}^{(3)}$ from experiment, we discuss the interference between the resonant and nonresonant susceptibility terms and the light-wave propagation effect. Finally, the close analogy between this type of optical mixing and the photon echo is discussed.
The linewidths and line positions of 4I15/2 to 4I13/2 absorption transitions of trivalent erbium in CsCdBr₃ and CaF₂ were measured as a function of temperature from approximately 10 K to 300 K. By comparing the temperature dependence of these transitions with theoretical models of electron-phonon coupling the primary mechanism involved were determined. For Er3+ doped CaF₂ the Raman scattering processes dominated the line broadening, although the single-phonon direct processes were significant enough to drastically alter the values of the Raman scattering electron-phonon coupling constant. In comparison, the Er³⁺ doped CsCdBr₃ had negligible Raman scattering contributing to the line broadening. This is likely due to the exceedingly low Debye temperature and consequently low number of phonons, reducing the likelihood of two-phonon processes relative to single-phonon processes. The results were then analysed in terms of the bond length with the ligands and compared with other studies showing that as the bond length gets shorter the electron-phonon coupling constant associated with Raman scattering is expected to get smaller. To explain the line shifts in CaF₂:Er³⁺ it is necessary to treat the Debye temperature as a parameter. This indicates that the phonon modes causing the line shifts are different from those causing the line broadening, and in most of the cases are vibronic processes rather than Raman processes. In the case of CsCdBr₃:Er³⁺ there is very little difference between the fits using the fixed Debye temperature and a varied Debye temperature in over half of the cases examined. Given the distribution of phonon modes, it is likely that this small difference arises because combinations of these processes are of similar intensity, meaning that a combination of Raman and optical phonon modes are likely causing the line shift.