Fundamental parameters line profile fitting in laboratory diffractometers

Abstract

The fundamental parameters approach to line profile fitting uses physically based models to generate the line profile shapes. The instrument profile shape K(2θ) is first synthesised by convoluting together the geometrical instrument function J(2θ) with the wavelength profile W(2θ) at the Bragg angle 2θ B of the peak, K(2θ) = ∫W(2θ -2ϕ)J(2ϕ)d2ϕ = W(2θ) ⊗ J(2θ) (1) where the function J(2θ) itself is a convolution of the various instrument aberration functions associated with the diffractometer, ie., J(2θ) = J 1 (2θ) ⊗ J 2 (2θ) ⊗…⊗ J i (2θ)…..⊗ J N (2θ).Diffraction broadening is incorporated into the profile function I(2θ) by convoluting the broadening function B(2θ) into the instrument profile function as shown I(2θ) = K(2θ) ⊗ B(2θ).This technique of profile synthesis was first introduced 50 years ago by Alexander [1], but has only been implemented as a standard fitting procedure during the last ten years [2,3,4]. More recently, freeware and commer- FPPF has been used to synthesise and fit data from both parallel beam and divergent beam diffractometers. The refined parameters are determined by the diffractometer configuration. In a divergent beam diffractometer these include the angular aperture of the divergence slit, the width and axial length of the receiving slit, the angular apertures of the axial Soller slits, the length and projected width of the x-ray source, the absorption coefficient and axial length of the sample. In a parallel beam system the principal parameters are the angular aperture of the equatorial analyser/Soller slits and the angular apertures of the axial Soller slits. The presence of a monochromator in the beam path is normally accommodated by modifying the wavelength spectrum and/or by changing one or more of the axial divergence parameters. Flat analyser crystals have been incorporated into FPPF as a Lorentzian shaped angular acceptance function.One of the intrinsic benefits of the fundamental parameters approach is its adaptability to any laboratory diffractometer. Good fits can normally be obtained over the whole 2θ range without refinement using the known properties of the diffractometer, such as the slit sizes and diffractometer radius, and the emission profile. Fine tuning is sometimes necessary to accommodate a monochromator or to compensate for the fact that certain aberrations are not completely independent [8]. Under these conditions some of the instrument parameters need to be refined, but the refined values normally are within ±10 % of the actual values. Correlation between refined instrument parameters can occur when fitting to data over a restricted 2θ range. Such correlation occurs between the axial divergence parameters and absorption as both of these aberrations can produce similar forms of asymmetric profiles between 2θ = 50° and 100° in diverging beam diffractometers. Correlation is minimised by using data with a large 2θ range so that the unique angular dependence of individual aberrations becomes evident. When a set of instrument profiles cannot be fitted by FPPF, this...

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