The use of direct convolution products in profile and pattern fitting algorithms. I. Development of algorithms

Convolution products Were obtained by folding a specimen-related function into another function representing the intrinsic profile of the diffractometer used in this study. The instrumental contributions were modeled with three split-Pearson VII functions: one for each of the α₁, α₂ and α₃ components in the Cu Kα spectral distribution. The positions and intensities of the α₂ and α₃ lines were based on those of the α₁ line while their shapes were constrained to follow that of the α₁. Values of the variable parameters of these functions, obtained from a `defect-free' specimen, were fit with polynomials to establish four discrete curves from which the instrument profiles could be synthesized at any diffraction angle. Both a normalized Lorentzian and a Gaussian function were evaluated for use in representing the specimen contributions. The integral breadth (β) of the specimen function was adjusted until the instrument-specimen convolution product best matched the observed profile. In specimens with small crystallite size, the angular dependence of β for the specimen profile followed the Scherrer relation while, in a strained specimen, the angular dependence followed the simple 4[epsilon] tan θ relation. In both cases, the specimen contributions were best modeled by a Lorentzian-type function.