Accurate Chebyshev Approximations for the Width of the Voigt Profile, Differential Peaks, and Deconvolution of the Lorentzian Width

Abstract

Accurate formulas have been developed using rational Chebyshev approximations for the spectral features/quantities of the Voigt profile and its first two derivatives. The rational polynomial expression for the half width at half maximum (HWHM) of the Voigt profile in terms of the Lorentzian and Gaussian widths is presented in Eq. (7) and provides 12-digit accuracy. The correction expression for the deconvolution of the Gaussian broadening from the Lorentzian broadening in the Voigt profile is presented in Eq. (11) and provides 13-digit accuracy. The expression for the position of the peak in the first derivative of the Voigt profile in terms of the Lorentzian and Gaussian widths is presented in Eq. (15) and provides 14-digit accuracy. The expression for the position of the side peak in the second derivative of the Voigt profile in terms of the Lorentzian and Gaussian widths is presented in Eq. (17) and provides 12-digit accuracy. The expressions for the peak amplitude ratios for the first and second derivatives of the Voigt profile in terms of the Lorentzian and Gaussian widths are presented in Eq. (19) and Eq. (21), respectively, and provide 13-digit accuracy. The reported accuracies are higher than those of earlier work, and the presented expressions provide a reference for the direct calculation of spectral quantities for Voigt profiles. The accuracy of the presented formulas establishes a general foundation for the deconvolution of instrument-based broadening and other broadening mechanisms in the HWHW and the positions of the first- and second-derivative peaks.