Stochastic Theory Of Line Shape Research Articles

Thermally Activated Electron Exchange in Cu12–xFexSb4S13 (x = 1.3, 1.5) Tetrahedrites: A Mössbauer Study

Mixed-valence iron Fe(III/II) tetrahedrites Cu10.7Fe1.3Sb4S13 and Cu10.5Fe1.5Sb4S13 have been prepared, characterized, and studied by Mössbauer spectroscopy. The low-temperature Mössbauer spectra (T ≤ 13 K) of all samples confirm the valence-localized nature of the ground state for the distinct Fe2+ and Fe3+ sites. At elevated temperatures the spectral transformations are characterized by a coexistence of valence-localized states Fe3+/Fe2+ (iron surrounded only by copper ions in the second coordination sphere) and valence-averaged spectral components corresponding to the electron hopping (FeA2+FeB3+) ↔ (FeA3+FeB2+). Our experiments suggest that these tetrahedrites belong to Robin–Day class II mixed-valence systems that display thermally activated charge transfer. Hopping frequencies (Ωhop) have been determined from spectral simulations using the stochastic line shape theory. Analysis of the ⟨Ω⟩(T) temperature dependences reveals the existence of a thermally activated process between (quasi)degenerate iron sites. The activation energies for Cu10.7Fe1.3Sb4S13 (EA ≈ 5.3 meV) and Cu10.5Fe1.5Sb4S13 (EA ≈ 6.6 meV) differ and suggest an important influence of the local crystal surroundings of iron ions on electron hopping. The isomer shift and quadrupole splitting values for the Fe3+ sites reveal a relatively strong and nonmonotonic change with temperature that suggests the occurrence of intrinsic charge delocalization in addition to the intersite electron hopping.

Stochastic Lineshape Theory for Multistate, Non‐Markovian Frequency Fluctuations#

A stochastic lineshape theory is formulated for the non‐Markovian and multistate frequency modulation model. The frequency of the stochastic oscillator undergoes non‐Markovian modulations between multiple states, and the transition processes are described by arbitrary waiting‐time distributions. By incorporating stationarity into the stochastic process in a proper way, we calculate the propagator and the relaxation function. It is shown that a stationary distribution among the oscillator states satisfies a generalized detailed balance condition even in the non‐Markovian case. We find an exact expression for the lineshape function, which contains the Markovian case as a special limit.

Dilution effect on the vibrational frequency and vibrational relaxation of PbCl2–KCl ionic liquids

The Raman spectra of molten xPbCl2–(1−x)KCl mixtures have been measured in the liquid state over a wide composition range from dilute to dense solutions, namely x=0.2, 0.33, 0.53, 0.67, 0.77 and 1mol fraction. We studied the vibrational dephasing and vibrational frequency modulation by calculating time correlation functions of vibrational relaxation by fits in the frequency domain. The fitting method enables us to model the real line profiles intermediate between Lorentzian and Gaussian by a function, which has an analytical counterpart in the time domain. The experimental vibrational data are interpreted in terms of the Kubo model stochastic line-shape theory assuming that the environmental modulation arises from a single relaxation process. Successful application of the formalism proposed by Kubo shows that vibrational dephasing is the dominant relaxation mechanism for the isotropic Raman line shapes studied. The calculated vibrational frequency modulation time and vibrational dephasing correlation time exhibit a continuous variation with opposite trends indicating an interplay between fast (τω<τV) and slow (τω>τV) modulation and is associated to the different coordination polyhedra dominating the structure at low and high PbCl2 contents. The “packing” of the anions around central Pb2+ cation seems to be the key factor for the structure and the dynamics of the melts. The study of picosecond dynamics enables us to discern between instantaneous, short-lived and long-lived species in molten halides. Theoretical models of vibrational second moments and experimental results have been tested and compared by diluting PbCl2 in KCl solvent. The changes in second moments of vibrations suggest the strengthening of attraction between coordination polyhedra in a less crowded environment.

Direct measurements of memory effects in single-molecule kinetics

Statistics and correlations of single-molecule sequences of modulated reactions are explicitly evaluated in the stochastic rate representation. The memory function, introduced through the Gaussian approximation of the stochastic rate expression, characterizes the correlation in single-molecule rate processes in a formalism similar to the stochastic line shape theory. Within this formalism, the on-time correlation is shown to approximate the memory function of the fluctuating rate at discretized effective time separations. A new measurement, the two-event number density, is proposed as a means to map out the memory function over the complete time range. Confirmed by numerical calculations, these relations quantify dynamic disorder caused by conformational fluctuations and hence are useful for analyzing single-molecule kinetics.

Lineshape theory and photon counting statistics for blinking quantum dots: a Lévy walk process

Recent experimental observations have found two different kinds of ``strange kinetic behaviors" in individual semiconductor nanocrystals (or quantum dots). Fluorescence intermittency observed in the quantum dots shows power--law statistics in both {\it on} and {\it off} times. Spectral diffusion of the quantum dots is also described by power--law statistics in the sojourn times. Motivated by these experimental observations we consider two different but related problems: (a) a stochastic lineshape theory for the Kubo-Anderson oscillator whose frequency modulation follows power-law statistics and (b) photon counting statistics of quantum dots whose intensity fluctuation is characterized by power-law kinetics. In the first problem, we derive an analytical expression for the lineshape formula and find rich type of behaviors when compared with the standard theory. For example, new type of resonances and narrowing behavior have been found. We show that the lineshape is extremely sensitive to the way the system is prepared at time $t=0$ and discuss the problem of stationarity. In the second problem, we use semiclassical photon counting statistics to characterize the fluctuation of the photon counts emitted from quantum dots. We show that the photon counting statistics problem can be mapped onto a L\'evy walk process. We find unusually large fluctuations in the photon counts that have not been encountered previously. In particular, we show that $Q$ may increase in time even in the long time limit.

Optical control of molecular dynamics in a liquid

We report the results of a study of the influence of solvent fluctuations on the efficiency of selective population transfer from an initial state to a designated target state of a solute molecule. Our model of the influence of liquid fluctuations on the states of the solute assumes that dephasing is the dominant relaxation process, and utilizes an analog of the Kubo stochastic theory of line shape. The solvent fluctuations are represented as a Gaussian random process that independently modulates each of the energy levels of the solute molecule. For typical liquid densities the maximum amplitude of these fluctuations is taken to be of the order of 150 cm−1, and the correlation time of the fluctuations is taken to be of the order of a few hundred femtoseconds, but we have also explored the effects of varying the fluctuation frequency and correlation time. It is shown that STIRAP (stimulated Raman adiabatic passage) generated population transfer to a designated target state of the solute remains efficient when the frequency of the solvent fluctuations is large or small relative to the inverse of the widths of the pump and Stokes pulses. It is further shown that extended STIRAP generated selective transfer to one of a pair of degenerate states of the solute remains efficient under the same conditions. These results suggest, subject to the accuracy of the representation of the influence of the solvent on the solute, that it should be possible, using coherent superpositions of states generated with picosecond excitation, to control population transfer, hence reactivity, for a class of reactions carried out in the liquid phase.

Influence of extrinsic factors on electron transfer in a mixed-valence Fe(2+)/Fe(3+) complex: experimental results and theoretical considerations.

The crystal structure of the mixed-valence complex (NEt(4))[Fe(2)(salmp)(2)].xMeCN(crystal) (x = 2,3) [1].xMeCN(crystal) was determined at temperatures between 153 and 293 K. The complex shows distinct Fe(2+) and Fe(3+) sites over this temperature interval. Variable temperature Mössbauer spectra confirm the valence-localized character of the complex. In contrast, spectroscopic investigation of powder samples generated from [1].xMeCN(crystal) indicate the presence of a valence-averaged component at temperatures above 150 K. To elucidate this apparent contradiction we have conducted a variable-temperature Mössbauer investigation of different forms of 1, including [1].xMeCN(crystal), [1].2DMF(crystal), [1].yMeCN(powder), and solution samples of 1 in acetonitrile. The low-temperature Mössbauer spectra of all forms are virtually identical and confirm the valence-localized nature of the S = (9)/(2) ground state. The high-temperature spectra reveal a subtle control of electron hopping by the environment of the complexes. Thus, [1].xMeCN(crystal) has valence-localized spectra at all explored temperatures, [1].2DMF(crystal) exhibits a complete collapse into a valence-averaged spectrum over a narrow temperature range, the powder exhibits partial valence averaging over a broad temperature interval, and the solution sample shows at 210 K the presence of a valence-averaged component in a minor proportion. The spectral transformations are characterized by a coexistence of valence-localized and valence-averaged spectral components. This phenomenon cannot be explained by intramolecular electron hopping between the valence-localized states Fe(A)(2+)Fe(B)(3+) and Fe(A)(3+)Fe(B)(2+) in a homogeneous ensemble of complexes, but requires relaxation processes involving at least three distinguishable states of the molecular anion. Hopping rates for [1].2DMF(crystal) and [1].yMeCN(powder) have been determined from spectral simulations, based on stochastic line shape theory. Analysis of the temperature dependences of the transfer rates reveals the existence of thermally activated processes between (quasi) degenerate excited states in both forms. The preexponential factors in the rate law for the hopping processes in the [1].yMeCN(powder) and [1].2DMF(crystal) differ dramatically and suggest an important influence of the asymmetry of the complex environment (crystal) on intramolecular electron hopping. The differences between the spectra for the crystalline sample [1].xMeCN and those for powders generated under vacuum from these crystals indicate that solvate depletion has a profound effect on the dynamic behavior. Finally, two interpretations for the three states involved in the relaxation processes in 1 are given and critically discussed (salmp = bis(salicyledeneamino)-2-methylphenolate(3-)).

Paramagnetic relaxation as seen by nuclear resonant forward scattering of synchrotron radiation

Nuclear forward scattering of synchrotron radiation, which can be regarded as M\"ossbauer spectroscopy in the time domain, is a very promising tool for the investigation of dynamic phenomena. The interpretation of the resulting time-dependent intensity of the scattered radiation can be achieved by incorporating the stochastic theory of line shape into the program package SYNFOS, the theoretical basis of which has been described in an earlier paper [Haas et al., Phys. Rev. B 56, 14 082 (1997)]. This is demonstrated by means of measurements carried out on the paramagnetic iron porphyrin complex $[{\mathrm{F}\mathrm{e}(\mathrm{C}\mathrm{H}}_{3}{\mathrm{C}\mathrm{O}\mathrm{O})(\mathrm{T}\mathrm{P}}_{\mathrm{piv}}\mathrm{P}){]}^{\ensuremath{-}}$ that exhibits dynamic effects due to spin-lattice relaxation at temperatures around 10 K.

A GAUSSIAN, NON-MARKOVIAN APPROACH TO LONG MEMORY WITH APPLICATION TO A STOCHASTIC THEORY OF LINE SHAPE

A new approach to long memory effects is suggested. The main assumption is that the regression of fluctuations occurs via a clustering mechanism. The regression of a scalar variable takes place through a succession of linear decay processes, resulting in a long time tail of the autocorrelation function. The number of decay events is a random variable obeying a self-similar probability distribution whose fractal exponent determines the long time behavior of fluctuation regression. The overall fluctuation dynamics is described by a stationary Gaussian and non-Markovian random process. The theory is applied to the stochastic theory of line shape. The relaxation function ϕ(t) can be exactly evaluated. We show that the memory effects lead to a narrowing of the relaxation function for large time. As the time t tends to infinity, ϕ(t) may be approximated by a ‘contracted’ exponential ϕ(t) ~ exp(-const. t2−H) where 1≥H&gt;0 is the fractal exponent describing the clustering process.

Discrete random walk representations for continuum stochastic dynamics

Complex physical systems are considered, involving both random and deterministic phenomena. We introduce a set of age-dependent transition rates accounting for the random jump processes and a set of deterministic equations describing the evolution of the system between two jumps. The age-state probability density obeys a system of integrodifferential balance equations. Assuming the validity of certain separability conditions, a method is presented for reducing the integration of evolution equations to a discrete random walk problem. For pure jump processes a physical interpretation of the main results is given in terms of the probability density for the waiting time. The approach is applied to heavy ion collisions, to electrodiffusion and to stochastic theory of line shape.

Dynamics of SCN-ions in molten thiocyanates and aqueous solutions by Raman spectroscopy

The Raman spectra of the v 1(CN) stretching mode of SCN- ions have been measured in molten thiocyanates, KSCN and NaSCN, at temperatures 450-600K. The vibrational and the rotational correlation functions are calculated, and the dynamics of SCN- ions in the molten state are compared with those in aqueous solutions of KSCN, NaSCN, and LiSCN at concentrations 1–10 mol dm-3 and at temperatures 303–353 K. The observed vibrational correlation functions are analysed by the stochastic line shape theory of Kubo, in which homogeneous and inhomogeneous broadening are treated simultaneously. Both broadening contributions to the isotropic spectra are extracted. The homogeneous broadening is found to increase with increasing temperature in both melts and aqueous solutions; the inhomogeneous broadening remains constant in molten KSCN while it decreases in aqueous solutions. As the result, the isotropic Raman bandwidth is considered to increase with temperature in the molten state and to decrease in aqueous solutions. Rotational correlation functions of SCN- ions in these molten salts show the behaviour of the short time inertial rotation (t ⪅ 0·15 ps, jump angle 20°), which is a little slower than the free rotation of a single ion. The long time exponential decay of the rotational correlation functions reflects the ultimate diffusional behaviour of the ionic reorientation. The rotational relaxation rate increases with increasing temperature in both melts and aqueous solutions. The vibrational dephasing rate decreases and the rotational relaxation rate increases as the cation size increases in melts. In aqueous solutions, the vibrational dephasing rate follows the same cation dependence as that in melts, while the rotational relaxation rate decreases as the cation size increases. This seems to be a consequence of the specific local structures in aqueous electrolyte solutions.

Stochastic theory of line shape and relaxation.

This paper is a generalization of Kubo's stochastic theory of spectroscopic line shapes which appeared in 1962 under the same title. Using a stochastic, Hermitian Hamiltonian to phenomenologically incorporate absorber-perturber interactions resulting from molecular collisions, we are able to reproduce Kubo's derivation of the simple Lorentzian line shape: I(\ensuremath{\omega})=${\ensuremath{\pi}}^{\mathrm{\ensuremath{-}}1}$\ensuremath{\Gamma}[(\ensuremath{\omega}-${\ensuremath{\omega}}_{0}$${)}^{2}$+\ensuremath{\Gamma} $^{2}$${]}^{\mathrm{\ensuremath{-}}1}$ in which the width \ensuremath{\Gamma} is related to the mean-square strength, ${Q}_{3}$, of a diagonal stochastic Hamiltonian by \ensuremath{\Gamma}=2${Q}_{3}$. Our generalization of this line shape is I(\ensuremath{\omega})=(1/\ensuremath{\pi}){1+(1-\ensuremath{\omega}/\ensuremath{\Omega})[(${\mathrm{\ensuremath{\Gamma}}}_{2}$ ${\mathrm{\ensuremath{-}}\mathit{\ensuremath{\Gamma}}}_{1}$)/2\ensuremath{\Gamma}]}\ensuremath{\Gamma}/[(\ensuremath{\omega}-\ensuremath{\Omega}${)}^{2}$+\ensuremath{\Gamma} $^{2}$${]}^{\mathrm{\ensuremath{-}}1}$, where ${\ensuremath{\Gamma}}_{1}$ and ${\ensuremath{\Gamma}}_{2}$ measure the degree of anisotropy in the off-diagonal part of the stochastic Hamiltonian. With perfect isotropy their contribution vanishes, but we are still left with additional corrections to the simple Lorentzian: \ensuremath{\Gamma}=2Q(1+${\mathrm{\ensuremath{\omega}}}_{0}^{2}$/${\mathrm{\ensuremath{\epsilon}}}^{2}$${)}^{\mathrm{\ensuremath{-}}1}$+ 2${\mathrm{Q}}_{3}$ and \ensuremath{\Omega}=${\ensuremath{\omega}}_{0}$+\ensuremath{\Delta}, where \ensuremath{\Delta}=2Q(${\mathrm{\ensuremath{\omega}}}_{0}$/\ensuremath{\epsilon})(1+${\mathrm{\ensuremath{\omega}}}_{0}^{2}$/\ensuremath{\epsilon} $^{2}$${)}^{\mathrm{\ensuremath{-}}1}$. Our result does reduce to Kubo's in the absence of off-diagonal stochastic terms. The parameter \ensuremath{\epsilon} measures the non-Markovianness of the stochastic Hamiltonian and yields white noise in the limit \ensuremath{\epsilon}\ensuremath{\rightarrow}\ensuremath{\infty}. Our results are valid for this limit regime. The frequency shift \ensuremath{\Delta} is a measure of the non-Markovianness.

Paramagnetic relaxation in high-spin ferrous compounds

Measurements of three high-spin FeIIS4 complexes at various temperatures and applied magnetic fields are reported and the results are compared with those of reduced rubredoxin. Simulations of the measured spectra by means of the fully developed stochastic theory of lineshape prove that intermediate relaxation behavior is present and is determined essentially by the phonon energy distribution and by one single parameter W0, the scaling factor, which contains the strength of spin-lattice coupling.

Raman study of vibrational and rotational dynamics of thiocyanate anion in aqueous solutions

The Raman spectra of the CN stretching mode of linear SCN- ions have been recorded in aqueous LiSCN, NaSCN and KSCN solutions at concentrations 1–10 M and at temperatures 30, 55 and 80°C. The vibrational and rotational correlation functions are calculated. The vibrational width of about 27–50 cm-1 in these thiocyanate solutions is attributed to the vibrational dephasing due to interactions between a SCN- ion and surrounding water molecules and cations. The observed vibrational correlation functions are analysed by the stochastic line shape theory of Kubo, in which homogeneous and inhomogeneous broadening are treated simultaneously, and both contributions to the spectra are extracted. The inhomogeneous contribution is found to increase with decreasing the temperature and in 10 M LiSCN solution at temperatures below 55°C the band is found to be broadened predominantly by inhomogeneous processes. Even in dilute solutions inhomogeneously slow modulation is found to exist besides the fast process which gives t...

Laplace transforms of correlation functions for CARS

Analytic forms and series representations are presented for Laplace transforms of correlation functions derived from Kubo’s stochastic line shape theory. The absolute squares of these expressions describe CARS line shapes; theory real parts describe Raman line shapes.

Raman study of vibrational relaxation of 1-alcynes in solutionv(≡CH) andv(C≡C) modes of 1-hexyne and 1-decyne

The I.R. and Raman bandwidths of the v(≡C-H) and v(C≡C) stretching modes of 1-hexyne and 1-decyne have been measured in dilute solutions in n C7H16 and CCl4 as a function of temperature. The I.R. spectrum is used, by comparison with the Raman spectrum, to verify the generally assumed absence of correlation between the rotational and vibrational relaxations. The important experimental findings is the opposite effect of temperature on the isotropic Raman linewidths of the two modes. The v(≡C-H) mode broadens with decreasing temperature and this is interpreted in a qualitative way by the isolated binary collision model of Fisher and Laubereau and the hydrodynamic model of Oxtoby. On the contrary, these models are unable to interpret the linewidth of the v(C≡C) mode. A possible explanation lies in an intramolecular energy relaxation between Fermi resonance levels. Finally, application of the Kubo stochastic line shape theory to the vibrational correlation function of the v(≡C-H) mode shows that the modulation is fast in n C7H16 specially at high temperature and intermediate between fast and slow limits in CCl4.

EPR linewidth of triplet excitons in molecular crystals. I. Theory and experiment: Pyrene

The EPR linewidths of triplet excitons in single-crystal pyrene at room temperature have been measured in experiments performed at 24 GHz. The data are fitted to a formula first presented by Reineker in a theory based on the Haken-Strobl model of exciton motion but rederived here more directly from Blume's stochastic Liouville formalism. The agreement is excellent provided a small, constant residual width is included. This result implies that the dominant spin-relaxation mechanism in pyrene, as in anthracene and presumably in similar molecular crystals, results from hopping between differently oriented molecules in the unit cell. It is shown that Blume's generalization of the Kubo-Anderson stochastic theory of line shape in its random-phase-approximation specialization is particularly suited as an alternative formulation for the exciton problem, emphasizing the fact that the room-temperature experiment (hopping regime) offers no detailed information on models of exciton transport. The residual width is analyzed in terms of a treatment of orientational disorder that accounted fully for a similar component in a previous analysis of the anthracene lattice. The analysis is not successful in this case, leaving open the possibility of an additional broadening mechanism in pyrene.

Inhomogeneous broadening of vibrational linewidths in polyatomic liquids

The first account of inhomogeneous broadening of vibrational linewidths in non-hydrogen-bonded liquids is presented based on the combined results of isotropic spontaneous Raman studies and picosecond coherent probing experiments. The symmetric CH3-stretching vibrational linewidths studied were found to be inhomogeneously broadened to various degrees. A correlation between the inhomogeneous broadening and the liquid’s local number density distribution width is demonstrated, upon which a theoretical model for inhomogeneous broadening is developed. A stochastic line shape theory is constructed in which homogeneous and inhomogeneous broadening are treated simultaneously in terms of one vibrational correlation function. This treatment unifies the fast and slow modulation approaches to vibrational dephasing and demonstrates how isotropic spontaneous Raman scattering studies and picosecond coherent probing experiments can be used in conjunction to study the inhomogeneous broadening of vibrational linewidths.

Density and temperature effects on vibrational phase relaxation and frequency shifts in liquid isobutylene

The Raman line shapes of the ν4 (A1) symmetric C=CH2 stretching mode at 1657 cm−1 and the ν9 (A1) symmetric C–CH3 stretching mode at 805 cm−1 in isobutylene have been measured as a function of pressure from vapor pressure densities to 0.8 g cm−3 over the temperature range from −25 °C to 75 °C. The frequency shifts of the above isotropic bands have also been determined under the same experimental conditions. The important experimental finding is the opposite effect of density on the isotropic linewidths of the C–CH3 and C=CH2 stretching vibrations; namely, the former band broadens with increasing density whereas the latter band narrows with increasing density under isothermal conditions. Such narrowing is unexpected in view of the theoretical and experimental results on the mechanism of vibrational broadening. A very different behavior is also observed for the frequency shifts of these two stretching modes. There is a large red gas-to-liquid shift for the C=CH2 band while its frequency changes very little with density over the liquid range studied. The opposite is true for the C–CH3 band which has a small gas-to-liquid shift but exhibits a large blue shift with increasing density in the liquid region. A qualitative interpretation of the experimental frequency data is based on consideration of classical vibrational amplitudes of the two vibrations and the infrared activity of the C=CH2 mode. The experimental vibrational widths are discussed in terms of available theoretical models for dephasing and other mechanisms for broadening of isotropic Raman line shapes. It appears that the behavior of the C–CH3 band can be best interpreted considering the isolated binary collision model for dephasing and additional energy relaxation, whereas the unusual density effects on the linewidth of the C=CH2 band are discussed in a more general way using the Kubo stochastic line shape theory.

Theory of Mössbauer effect in intermediate-valence compounds

Using a simple microscopic model, we study the response function corresponding to the M\"ossbauer effect in an intermediate-valence atom. We derive expressions for the spectra which can be used to analyze experimental results at all temperatures, as well as for different hosts in which the intermediate-valence ion may be embedded. For metallic hosts at high temperatures our theory justifies the use of the expressions previously proposed by Bauminger, Nowik, Ofer, Felner, and Mayer, based on Kubo's stochastic theory of line shape. At low temperatures, where the latter cannot be applied due to the non-Markovian character of the fluctuations, our results allow us to interpret experimental data and to obtain from them the parameters characterizing the intermediate-valence condition. For semiconducting hosts, where the phenomenological stochastic theory does not apply either, we can predict spectra which are consistent with experimental results.