There are two methods to estimate the particle size from X-ray diffraction data: the Debye equation and the Scherrer formula. The main goal of this study is to describe the methodology of particle size estimation on the base of two these methods and to apply it to TiO2 powder to determine the diameters and the mass content of anatase and brookite components. The studied nano-dispersed TiO2 powder was synthesized by the sol-gel method. The proposed method of particle size estimation consists of several steps: 1. Approximation of diffraction peaks by Gaussians and calculation of initial values of particle size with the use of the Scherrer formula; 2. Iterations with the use of the Debye equation to obtain more accurate particle size values; 3. Calculation of the mass content of different components corresponding to the minimum R-factor.

The lower limit of the applicability of the Scherrer formula has been established by calculating the diffraction patterns from model nanoparticles by the Debye formula. Particle size was calculated using the Scherrer formula for different hkl-peaks. The obtained data of particle sizes were compared with "real" sizes of model particles in the same hkl-directions. The form-factor K hkl was analyzed as main correction of Scherrer formula. It was shown that the Scherrer formula error increases nonlinearly at particle sizes less than 4 nm. For any hkl direction, the absolute error of average particle size determination using formula does not exceed 0.3 nm. Analysis shows that average particle size can be determined by Scherrer formula from single diffraction peak of experimental pattern for center-symmetrical particles.

Understanding how topologically protected surface states behave at surfaces and interfaces requires knowledge of the atomic structure. Whether the (111) surfaces of the prototypical topological insulators Bi 2 Se 3 and Bi 2 Te 3 are Bi or chalcogen terminated is the subject of current controversies. We employ photoelectron diffraction and holography, combining the advantages and avoiding the disadvantages of the contesting techniques previously used. We find bulklike chalcogen termination with a very small surface relaxation (<1%) in agreement with density functional theory simulations. We prove the chalcogen termination for cleaved crystals and epitaxial films which shows the robustness of our conclusions.

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