Rational functions as profile models in powder diffraction

Rational functions, the ratio of two polynomials, are shown to be good approximations to powder diffraction profiles. These functions are generalizations of the Lorentzian, the modified Lorentzian, and the profile model of Parrish [Parrish, Huang & Ayers (1976). Trans. Am. Crystallogr. Assoc. 12, 55–73]. The simplest of these functions is of the form f(x) = 1/(1 + A₁x² + A₂x⁴) with constants A₁ and A₂ that describe the shape of the profile, x = 2θ − 2θ₀, and 2θ₀ the position of the peak maximum. This function approximates very well Pearson VII distributions with exponents between 1 and 3. An asymmetric profile model with different A₁, A₂ parameters for the two halves of the peaks was fitted to silicon X-ray powder diffraction profiles and gave unweighted agreement factors from R₂ = 0.02 to 0.04 for peaks varying from 28 to 137° 2θ.