TOPAS and its academic variant TOPAS‐Academic are nonlinear least‐squares optimization programs written in the C++ programming language. This paper describes their functionality and architecture. The latter is of benefit to developers seeking to reduce development time. TOPAS allows linear and nonlinear constraints through the use of computer algebra, with parameter dependencies, required for parameter derivatives, automatically determined. In addition, the objective function can include restraints and penalties, which again are defined using computer algebra. Of importance is a conjugate gradient solution routine with bounding constraints which guide refinements to convergence. Much of the functionality of TOPAS is achieved through the use of generic functionality; for example, flexible peak‐shape generation allows neutron time‐of‐flight (TOF) peak shapes to be described using generic functions. The kernel of TOPAS can be run from the command line for batch mode operation or from a closely integrated graphical user interface. The functionality of TOPAS includes peak fitting, Pawley and Le Bail refinement, Rietveld refinement, single‐crystal refinement, pair distribution function refinement, magnetic structures, constant wavelength neutron refinement, TOF refinement, stacking‐fault analysis, Laue refinement, indexing, charge flipping, and structure solution through simulated annealing.

A convolution approach to X‐ray powder line‐profile fitting is developed in which the line shape is synthesized from the Cu Kα emission profile, the dimensions of the diffractometer and the physical variables of the specimen. In addition to the integrated intensities and 2θ positions of the line profiles, the parameters that may be fitted include the receiving‐slit width, the receiving‐slit length, the X‐ray‐source size, the angle of divergence of the incident beam, the X‐ray attenuation coefficient of the specimen and the crystallite size. This is a self‐consistent approach to fitting as the instrumental parameters are usually known by direct measurement. To minimize correlation between refined instrumental parameters, profiles at high and low 2θ values should be fitted simultaneously. The Cu Kα emission profile used in this work is based on recent monolithic double‐crystal spectrometer measurements that have identified a doublet structure in both the Kα1 and Kα2 components. Fast and accurate convolution procedures have been developed and a mixture of multilinear regression and Gauss–Newton non‐linear least squares with numerical differentials is used for fitting the profiles. The method is evaluated by fitting powder diffraction data from well crystallized specimens of MgO and Y3Al5O12 (YAG). Testing has also been carried out by examining the changes in the fitted values after altering various instrumental parameters (e.g. receiving‐slit width, detector defocus, receiving‐slit length and inclusion of a monochromator).

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...

A fast method for indexing powder diffraction patterns has been developed for large and small lattices of all symmetries. The method is relatively insensitive to impurity peaks and missing high d‐spacings: on simulated data, little effect in terms of successful indexing has been observed when one in three d‐spacings are randomly removed. Comparison with three of the most popular indexing programs, namely ITO, DICVOL91 and TREOR90, has shown that the present method as implemented in the program TOPAS is more successful at indexing simulated data. Also significant is that the present method performs well on typically noisy data with large diffractometer zero errors. Critical to its success, the present method uses singular value decomposition in an iterative manner for solving linear equations relating hkl values to d‐spacings.

Techniques and methods to facilitate the solution of structures by simulated annealing have been developed from the starting point of a space group and lattice parameters. The simulated-annealing control parameters have been systematically investigated and optimum values characterized and determined. Most signi®cant is the inclusion of electrostatic-potential penalty functions in a non-linear least-squares Rietveld re®nement procedure. The long-range electrostatic potentials are calculated using a general real-space summation which can be used for all space groups. In addition, a general weighting scheme for penalty functions negates the need to determine weighting schemes experimentally. Also investigated and improved is the non-linear least-squares minimization procedure used in the re®nement of structural parameters. The behaviour and success of the techniques have been tested on X-ray diffraction powder data against the known structures of AlVO 4 in P1 with 18 atoms in the asymmetric unit, K 2 HCr 2 AsO 10 in P3 1 with 15 atoms in the asymmetric unit excluding hydrogen, and [Co(NH 3 ) 5 CO 3 ]NO 3 .H 2 O in P12 1 with 15 atoms in the asymmetric unit excluding hydrogen.