Exact and approximate nonlinear least-squares inversion of dielectric relaxation spectra

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Abstract
Three weighted, complex nonlinear least‐squares methods for the deconvolution of dielectric or conducting system frequency‐response data are described and applied to synthetic data and to dielectric data of n‐pentanol alcohol, water, and glycerol. The first method represents a distribution of relaxation times or transition rates by an inherently discrete function. Its inversion accuracy and resolution power are shown to be limited only by the accuracy of the data when the data themselves arise from a discrete distribution involving an arbitrary number of spectral lines. It is shown that those inversion methods employed here which allow the relaxation times to be free variables are much superior to those where these quantities are fixed. Furthermore, free‐τ methods allow unambiguous discrimination between discrete and continuous distributions, even for data with substantial errors. Contrary to previous conclusions, discrete distributions were determined for both n‐pentanol alcohol and water. A complex, continuous distribution estimate was obtained for glycerol. Algorithms for all approaches are incorporated in a readily available computer program. Serious problems with some previous dielectric inversion methods are identified. Finally, several possibilities are mentioned that may allow greater inversion resolution to be obtained for complex nonlinear least‐squares estimation of continuous distributions from noisy data.

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