The Voigt Profile as a Sum of a Gaussian and a Lorentzian Functions, when the Weight Coefficient Depends on the Widths Ratio and the Independent Variable

Abstract

Assuming that V (x) ≈ (1 − µ) G₁(x) + µL₁(x) is a very good approximation of the Voigt function, in this work we analytically find µ from mathematical properties of V (x). G₁(x) and L₁(x) represent a Gaussian and a Lorentzian function, respectively, with the same height and HWHM as V (x), the Voigt function, x being the distance from the function center. In this paper we extend the analysis that we have done in a previous paper, where µ is only a function of a; a being the ratio of the Lorentz width to the Gaussian width. Using one of the differential equation that V (x) satisfies, in the present paper we obtain µ as a function, not only of a, but also of x. Kielkopf first proposed µ(a, x) based on numerical arguments. We find that the Voigt function calculated with the expression µ(a, x) we have obtained in this paper, deviates from the exact value less than µ(a) does, specially for high |x| values.