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Rational functions as profile models in powder diffraction
Author(s) -
Pyrros N. P.,
Hubbard C. R.
Publication year - 1983
Publication title -
journal of applied crystallography
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.429
H-Index - 162
ISSN - 1600-5767
DOI - 10.1107/s0021889883010468
Subject(s) - diffraction , position (finance) , rational function , function (biology) , powder diffraction , silicon , mathematics , materials science , physics , mathematical analysis , computational physics , crystallography , optics , chemistry , finance , evolutionary biology , metallurgy , economics , biology
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 + A 1 x 2 + A 2 x 4 ) with constants A 1 and A 2 that describe the shape of the profile, x = 2 θ − 2 θ 0 , and 2 θ 0 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 A 1 , A 2 parameters for the two halves of the peaks was fitted to silicon X‐ray powder diffraction profiles and gave unweighted agreement factors from R 2 = 0.02 to 0.04 for peaks varying from 28 to 137° 2 θ .