The half-widths of powder Bragg intensity profiles deduced in reciprocal space. I. Ideal powders

Abstract

New formulas for the full widths at half-maxima (FWHMs) of powder Bragg intensity profiles are deduced in reciprocal space, using the concept for the calculation of the peak width introduced for single-crystal diffractometry by Rossmanith [Acta Cryst. (1992), A48, 596-610]. In paper I, a basic formula for strain-free and prefer.red-orientation-free powders is deduced. Furthermore, it is shown that comparison of experimental widths with theoretical FWHMs calculated with the new expression results in physically significant values for the particle size in the powder. In a forthcoming paper II, the effect of strain on the FWHM will be analysed. IntroductionProfile analysis of powder diffraction diagrams requires knowledge of the height, width and distribution function (i.e. Gauss, Lorentz, pseudo-Voigt etc.) of the intensity profile.© 1994 International Union of Crystallography Printed in Great Britain -all rights reserved Expressions for the calculation of the resolution functions of N-crystal powder spectrometers are given by, for example, Sabine (1987) and Wroblewski (1991). The formulas given by these authors are difficult to handle for two reasons. First, most of the transmission and reflection probability distributions involved in the expressions are not known exactly in routine powder diffraction experiments. Second, evaluation of these expressions requires timeconsuming computations of integrals, even if approximations (for example pseudo-Voigt) for the distribution functions are used.The widths of the Bragg intensity profiles are, therefore, usually calculated using the simplified version of the formula given by Caglioti, Paoletti & Ricci (1958):A202 --U tan 2 0 + V tan 0 + W.(1 a)In Rietveld analysis (Rietveld, 1969), the physically meaningless half-width parameters U, V and W are determined by least-squares fitting of calculated to measured FWHMs.A cta Crystallographica Section A ISSN 0108-7673 ©1994 HALF-WIDTHS OF POWDER BRAGG INTENSITY PROFILES. IRossmanith (1992) introduced a new concept for the calculation of the widths of Bragg intensity profiles for single and multiple diffraction in single-crystal diffractometry. In this paper and in Rossmanith (1993a), it is shown that, with representation of the half-width parameters AA/A (wavelength spread), ~ (divergence), r (mosaic-block radius) and r/ (mosaic spread) as well as the experimental conditions of the diffraction in reciprocal space, the widths of the profiles for single and multiple diffraction can be obtained from purely geometrical considerations. In Rossmanith (1993b), the simple but very effective approximation AOh = +--~ + (AA/A) tan Oh + 2A/(r sin 20h) + ~7 = 6[ --+ 1 + (tan 0h/tan 0M)] + 2A/(r sin 20h) + ~7 (lb) was derived for the width A0 h of the Bragg intensity profile measured with a triple-crystal diffractometer at a synchrotron-radiation source in parallel (-) and antiparallel (+) arrangements of the crystal with respect to the monochromator. Oh and 0M are the kinematical Bragg angles of the sample and monochromator, re...