Finding the best-fit background function for whole-powder-pattern fitting using LASSO combined with tree search

A new linear function for modelling the background in whole-powder-pattern fitting has been derived by applying LASSO (least absolute shrinkage and selection operator) and the technique of tree search. The background function (BGF) consists of terms b n L(2θ/180)−n/2 and b n H(1 − 2θ/180)−n/2 for the low- and high-angle sides, respectively. Some variable parameters of the BGF should be fixed at zero while others should be varied in order to find the best fit for a given data set without inducing overfitting. The LASSO algorithm can automatically select the variables in linear regression analysis. However, it finds the best-fit BGF with a set of adjustable parameters for a given data set while it derives a different set of parameters for a different data set. Thus, LASSO derives multiple solutions depending on the data set used. By regarding the individual solutions from LASSO as nodes of trees, tree structures were constructed from these solutions. The root node has the maximum number of adjustable parameters, P. P decreases with descending levels of the tree one by one, and leaf nodes have just one parameter. By evaluating individual solutions (nodes) by their χ2 index, the best-fit single path from a root node to a leaf node was found. The present BGF can be used simply by varying P in the range 1–10. The BGF thus derived as a final single solution was incorporated into computer programs for Pawley-based whole-powder-pattern decomposition and Rietveld refinement, and the performance of the BGF was tested in comparison with the polynomials currently widely used as the BGF. The present BGF has been demonstrated to be stable and to give an excellent fit, comparable to polynomials but with a smaller number of adjustable parameters and without introducing undulation into the calculated background curve. Basic algorithms used in statistics and machine learning have been demonstrated to be useful in developing an analytical model in X-ray crystallography.

Powder X-ray diffraction of azelastine hydrochloride, C22H25ClN3O·Cl

Commercial azelastine hydrochloride crystallizes in the monoclinic space group P21/n (#14) with a = 13.7844(5), b = 16.39920(14), c = 9.41231(22) Å, β = 97.5340(20)°, V = 2109.32(4) Å3, and Z = 4. The lattice parameters differ by −0.02, +0.04, and +0.04% from those in the previous determination (reflecting differences in the temperature and the sample source), and are more precise, from the use of synchrotron radiation. The experimental powder pattern is included in the Powder Diffraction File™ (PDF®) as entry 00-070-1219.

A practical approach to the direct-derivation method for QPA: use of observed powder patterns of individual components without background subtraction in whole-powder-pattern fitting

The direct-derivation (DD) method for quantitative phase analysis (QPA) can be used to derive weight fractions of individual phases in a mixture from the sums of observed intensities along with the chemical composition data [Toraya (2016). J. Appl. Cryst. 49, 1508–1516]. The whole-powder-pattern fitting (WPPF) technique can be used as one of the tools for deriving the observed intensities of individual phases. In WPPF, the observed powder pattern of a single-phase sample after background (BG) subtraction can be used as the fitting function in combination with the fitting functions widely used in Pawley and Rietveld refinements. The direct fitting of the observed pattern is a very useful technique when the target component is a low-crystallinity or amorphous material [Toraya (2018). J. Appl. Cryst. 51, 446–455]. Technical problems in utilizing the BG-subtracted pattern are the uncertainty associated with the determination of BG height and the parameter interaction between the BG function (BGF) and the BG-subtracted pattern in the least-squares fit. In this study, a practical approach in which single-phase observed patterns are used for the direct fitting without subtracting their BG intensities is proposed. In QPA, the contribution of BG intensities can be neutralized by converting the sum of BG-included intensities into the sum of BG-subtracted intensities by multiplying by a conversion factor. When the magnitudes of the conversion factors are almost identical for all components, they can be canceled out under the normalization condition in deriving weight fractions, and they are not required in QPA. The magnitude of the conversion factor for each component can be determined by one of two experimental techniques: using a single-phase powder of the target component or a mixture containing the target component in a known weight ratio. The theoretical basis of the present procedure is given, and the procedure is experimentally verified. In this procedure, the interaction between the BGF and the BG-included observed pattern is negligibly small. Least-squares fitting with a few adjustable parameters is very fast and stable. Accurate QPA could be conducted, as indicated by the average deviation of 0.05% from weighed values in QPA of α-Al2O3 + γ-Al2O3 mixtures with five different weight ratios and 0.4% in QPA of an α-SiO2 + SiO2 glass mixture

X-RAY POWDER DIFFRACTION DATA FOR A NEW PIRAZOLINE COMPOUND

Pyrazolines are important agents in medicinal chemistry as a promising scaffold for structural modification and drug development studies due to their wide range of biological activities such as anticancer, antifungal, antibacterial, antidepressant, anticonvulsant, antitubercular, antioxidant, antileishmanial and antiinflammatory activity. These heterocyclic compounds can be prepared by refluxing chalcone with hydrazine hydrate and anhydrous sodium acetate in the presence of glacial acetic acid. The structural characterization, molecular and crystalline structure, of these organic compounds, allows studying their biological properties to know their potential applications. Hence the use of XRPD is very important because it allows obtaining a record to be used as a method of identification. The aim of this investigation was to obtain and reported good quality Xray powder diffraction data the pyrazoline compound 1-(3-(4-iodophenyl)-5-(3-methyl thiophen-2-yl)-4,5-dihydro-1H-pyrazol-1-yl)ethan-1-one, which could be used as potential anti-microbial and anti-cancer agent. The powder pattern was indexed in the monoclinic space group I2/a with unit cell parameters a = 25.440(5) Å, b = 5.124(2) Å, c = 26.261(6) Å, b = 105.75(2)° and figures of merit M20= 38.2 and F20= 66.6 (0.00573, 53). All measured lines were indexed and are consistent with the monoclinic space group. The powder pattern will be included in the Powder Diffraction File database to be used as a reference.

Whole powder pattern modelling macros for TOPAS

Macros implementing the main concepts of the whole powder pattern modelling approach have been written for TOPAS. Size and strain broadening components of the diffraction line profiles can be convolved with the instrumental profile already available among the standard commands of TOPAS. Specific macros are presented with examples of applications including plastically deformed powders and atomistic simulations. A macro is presented for the modelling of surface relaxation effects in spherical nanocrystals.