A Simple Numerical Method for Calculating Gaussian NMR Spectral Line Shapes Partially Narrowed Due to a Motion with the Exponential Spectral Autocorrelation Function

A simple numerical method for calculating NMR spectral line shapes resulting from a Gaussian line by a partial narrowing due to a motion with the exponential spectral autocorrelation function of the form exp (-|τ|/τc) was developed. It was found that the partially narrowed line is narrower not only than the parent Gaussian line with the second moment of ωp2 but also than the Lorentzian line with the half-width of 2ωp2τc obtained from the extreme narrowing approximation. The central intensity increase compared with the closer of these two lines is less than 50.2%. Asymptotic developments for large values of ω - ω0 and for large values of ωpτc were derived. Two-term approximation applied to the extreme narrowing case led to a very simple modification of the Lorentzian line having the correct second moment ωp2. Analysis of this modified Lorentzian line showed that attempts to estimate ωp2 from truncated second moments of Lorentzian lines without knowledge of τc are hopeless. The case of the polyexponential spectral autocorrelation function with all but one correlation times fast enough to allow for the extreme narrowing, modelling the case of an anisotropic motion, is also considered.