Spectral line shapes in systems undergoing continuous frequency modulation

Abstract

The power spectrum line shapes for oscillators undergoing a continuous modulation of the vibrational frequency are investigated. It is shown that the single, sharp line normally characteristic of such systems broadens and exhibits a wealth of fine structure components. The characteristic fine structure pattern is one of decreasing amplitude and spacing. This continuous frequency modulation (CFM) effect has been examined for a series of model oscillators that includes harmonic systems with linear and exponential variation of the frequency without amplitude damping, a harmonic system with exponential damping of both the resonant frequency and the amplitude, and a Morse oscillator whose kinetic energy is being exponentially damped. An analytic expression for the power spectrum of a harmonic oscillator whose frequency is varying linearly with time is derived. This result demonstrates that the position of the fine structure extrema depends linearly upon the initial oscillator frequency and the square root of the absolute value of the modulation rate. The peak-to-peak spacing is shown to be proportional to the square root of the absolute value of the modulation rate. It is suggested that the CFM effect is the fundamental explanation of many previous empirical observations concerning power spectra. The CFM effect for a harmonic system with an exponentially modulated frequency is very similar to that observed for linear modulation. When amplitude depression is included, there is a significant intensity decrease of many of the spectral lines. Investigation of a Morse oscillator shows that energy transfer in an anharmonic system produces a CFM effect. By assuming that the analytic result for a harmonic oscillator with a linear modulation is transferable to the anharmonic case, an expression is obtained that relates the peak-to-peak fine structure spacing to the Morse potential parameters, the initial oscillator energy and the IVR rate coefficient. An experimental example of a CFM effect is presented by taking an NMR spectrum of H2O and HCCl3 in DCCl3 while the main B0 field is varying with time. The CFM effect is used to extract energy transfer rate coefficients for a diatomic molecule isolated in an argon matrix at 12 K and for total IVR rate coefficients for relaxation of the N=O and O–H local modes in cis-HONO. It is also shown that instantaneous energy transfer rates in small molecules can be determined by using local frequency analysis to compute the temporal variation of the CFM band spacings. It is concluded that line shape analysis can be effectively used as a probe of energy transfer rates.