Various crystallite size estimation methods were used to analyze X-ray diffractograms of spherical cerium dioxide and titanium dioxide anatase nanoparticles aiming to evaluate their reliability and limitations. The microstructural parameters were estimated from several integral breadth methods such as Scherrer, Monshi, Williamson–Hall, and their variants: (i) uniform deformation model, (ii) uniform strain deformation model, and (iii) uniform deformation energy density model. We also employed the size–strain plot and Halder–Wagner method. For this purpose, an instrumental resolution function of an Al2O3 standard was used to subtract the instrumental broadening to estimate the crystallite sizes and strain, and the linear regression analysis was used to compare all the models based on the coefficient of determination. The Rietveld whole powder pattern decomposition method was introduced for comparison purposes, being the best candidate to fit the X-ray diffraction data of metal-oxide nanoparticles. Refined microstructural parameters were obtained using the anisotropic spherical harmonic size approach and correlated with the above estimation methods and transmission electron microscopy images. In addition, μ-Raman spectra were recorded for each material, estimating the mean crystallite size for comparison by means of a phonon confinement model.

Zeolites 5A were obtained by ion exchange of a zeolite 4A, previously synthesized by the hydrothermal method from precursor kaolin, with the aim of removal As(V) from drinking water and Pb(II) from irrigation surface water. Zeolite 5A was characterized before and after adsorption by X-ray diffraction, scanning electron microscopy, energy dispersive spectroscopy, and X-ray photoelectron spectroscopy. To find the adsorption mechanisms of both As and Pb in real waters, experiments on adsorption kinetics, optimum pH, adsorbent dose, and adsorption isotherms were developed. Adsorption kinetics and adsorption isotherm models were evaluated, and the selection criterion for the appropriate model was made using the residual sum of squares (RSS) and Bayesian information criterion (BIC). As a result, removal adsorption of As(V) and Pb(II) was higher than 95% in 9 and 12 h, respectively. The estimated maximum adsorption capacities for As and Pb were 36.35 mg g−1 and 46.67 mg g−1, respectively. Zeolite 5A is a low-cost adsorbent, through which a concentration of less than 0.01 mg L−1 of As and Pb was obtained in drinking and irrigation water, which is below the permissible limit established by the World Health Organization (WHO).

Various crystallite size estimation methods were used to analyze X-ray diffractograms of spherical cerium dioxide and donut-like titanium dioxide anatase nanoparticles aiming to evaluate their reliability and limitations. The microstructural parameters were estimated from Scherrer, Monshi, Williamson-Hall, and their variants: i) uniform deformation model, ii) uniform strain deformation model, and iii) uniform deformation energy density model, and also size-strain plot, and Halder-Wagner method. For that, and improved systematic Matlab code was developed to estimate the crystallite sizes and strain, and the linear regression analysis was used to compare all the models based on the coefficient of determination, where the Halder Wagner method gave the highest value (close to 1). Therefore, being the best candidate to fit the X-ray Diffraction data of metal-oxide nanoparticles. Advanced Rietveld was introduced for comparison purposes. Refined microstructural parameters were obtained from a nanostructured 40.5 nm Lanthanum hexaboride nanoparticles and correlated with the above estimation methods and transmission electron microscopy images. In addition, electron density modelling was also studied for final refined nanostructures, and μ-Raman spectra were recorded for each material estimating the mean crystallite size and comparing by means of a phonon confinement model.