The diffraction methods commonly used to determine the crystalline structure have specific features with respect to definite materials. High dispersion and weak scattering ability of the objects having coherent scattering domains (CSD) of several nanometers produce a strong broadening of the diffraction peaks (up to their disappearing into the background). The contribution of diffuse scattering associated with a surface structure or defects is also significant for nanocrystalline materials. Therefore, it is reasonable to use methods that can consider information over the whole diffraction pattern. For nanocrystalline powders with well-defined diffraction maxima, standard approaches based on the Rietveld method can be applied. When it is necessary to take into account diffuse scattering or an anisotropic broadening of the peaks caused by the shape of the particles such approaches are usually not applicable. To solve these tasks more general algorithms can be used. Among such methods are the Debye function analysis (DFA)  for determining the long-range order of atoms, and the radial distribution function (RDF) for analyzing the local environment of atoms. The DFA method makes it possible to calculate diffraction patterns using the Debye scattering equation , , . The RDF method is based on the Fourier transform of diffraction pattern. Naturally, a combination of DFA and RDF methods creates additional advantages in determining the phase composition of nanomaterials and revealing their structural features.
The main task in the application of DFA and RDF methods is to construct the atomic model of an object of nanometer sizes. DFA and RDF can be employed to verify and optimize the proposed structural models of ultradisperse systems, which take into account the structure and shape of the object.
At present, several software suites implementing the DFA method , ,  are available. Our work aims to create the software product with a friendly graphical interface for operating systems of the Windows family, which should be universal, suitable for a wide range of conventional PC users, provide intelligible interactive approaches to the development of atomic models of nanoparticles, and ensure treatment of the acquired data. The paper describes the main currently available features of the software and gives examples of its application for solving some problems.
Calculation of the diffraction profile with the use of the Debye scattering equation
The whole profile calculations can be performed by the Debye scattering equation (DSE) through the enumeration of all the interatomic distances rij in the structure (which is not necessarily periodic). The formula can be written as follows:
where s=4π·sin(θ)/λ is the current reciprocal space coordinate; P(s) is the instrumental correction; and fi(s), fj(s) are the scattering factors of atoms i and j with regard to thermal vibrations, respectively.
The DFA method is more general in comparison with the Rietveld-like methods . This considers each atom in the entire object and allows analyzing the diffraction pattern not only at the Bragg angles. It takes into account diffuse scattering, which is a source of additional structural information on the presence of defects in the crystalline structure or specific nanostructure.
It should be noted that, since formula (1) is applicable to any set of atoms, this approach makes it possible to calculate the diffraction also from non-periodic structures.
Calculation of simulated RDF curves
It is necessary to indicate that the DFA method implies calculation of all interatomic distances in a crystal from structural data. Interatomic distances can be used also to calculate the RDF function ,  with regard to the specified size and geometrical shape of crystallites. This method is employed for investigation of the local structure (the nearest coordination spheres of atoms in the range of 1–2 nm).
The RDF method gives a 1D description of the order of atoms as the ρ(r) function. Positions of peaks on the RDF curve directly correspond to interatomic distances, and areas are related to coordination numbers (the number of atoms at a given distance).
The theoretical curve is a set of peaks where each peak corresponds to the Gaussian function:
where ri is the position, Ai is the amplitude, and bi is the root-mean-square deviation of the i-th maximum. Experimental RDF curves are analyzed by comparison with the simulated curves.
Since even the highly disperse and structurally disordered phases are characterized by the presence of the short-range order of atoms, the method is efficiently employed to detect phases with the crystallite sizes of several nm, which are difficult to identify by conventional X-ray diffraction analysis.
The DIANNA software was developed using the universal DFA and RDF methods . DIANNA is intended to determine features of the atomic structure and parameters of the nanostructure (the shape and sizes of crystallites) by calculating and comparing powder X-ray diffraction patterns with experimental data.
For each calculation, the structure of the object model should be specified. The model is determined by the full array of atomic coordinates, which can be loaded from a separately prepared XYZ file or generated by the program itself. In the latter case, it is necessary to introduce into the program the unit cell parameters and coordinates of atoms, space group (standard structural files in the .CIF and .CEL formats can be used as input), as well as shape and sizes of crystallites. The program allows a modelling of nanoparticles of various shapes (sphere, ellipse, parallelepiped or polyhedron) with sizes starting from the size of one unit cell (Figure 1a).
Furthermore, the program is based on the algorithms that implement different approaches to describe the surface of particle; among them is the option, which allows the construction of a model where only a certain type of atoms constitute the outer shell. For example, for oxides the model can be specified in such a way that only the oxygen atoms will reside on the surface (Figure 1c). To this end, metal atoms can be marked in the program belonging to the framework, and then the oxygen atoms will be constructed as their environment.
The structural data are used to calculate all the interatomic distances in objects. Distribution of interatomic distances (which can be represented as a histogram of interatomic distances) can be used for diffraction pattern calculations according to the Debye equation (1) and RDF by equation (2), taking into account the size and geometric shapes of crystallites. Since a particle size distribution always takes place, the program allows specifying the analytical distribution function with variable parameters (or a histogram).
To compare the calculated diffraction pattern with an experimental data, the background can be modelled by a polynomial, and the normalization coefficient can be specified. The program refines (find a local minimum) the normalization coefficient of the plot and polynomial coefficients using χ-square (the least square method) or the Rp factor (the least modulus method) optimization. The target function minimum is found by the Levenberg-Marquardt algorithm , , which combines the Newton and gradient descent methods.
The program system uses an easy-to-use graphical user interface (GUI) which controls the calculations on a single standalone PC, and can be run efficiently on a single computer having more than one CPU or CPU with multiple cores.
Main features of the software:
Calculations for the particles of different shape: sphere, cylinder, oblique (in the general case) parallelepiped or polyhedron with arbitrary cutting.
Development of a model for crystal particles of any syngony.
Data on the atomic structure of nanoparticles are entered directly via a structural interface of the program system and from files, particularly those formed by the ICSD crystal structure database; displacement parameters of atoms and possible isomorphic substitutions of crystallographic positions are taken into account.
Application of 3D graphics to visualize and export the model images.
Direct (ab initio) calculation of the X-ray scattering intensity at each point of the powder diffraction pattern using the model that takes into account the atomic structure of nanoparticles, their size and shape, without additional hypotheses on the form of diffraction peak profiles.
Calculation of numerical criteria for fitting the diffraction patterns (different R-factors, χ-criterion).
Automatic approximation of the background as a polynomial and normalization of calculations (in particular by areas) to experimental data.
Showing the particle size distribution (as a histogram of weight coefficients, Gaussian function or lognormal distribution function).
Choosing the instrumental characteristics (monochromator, radiation wavelength) for correct estimation of the effect exerted by the form factor and radiation polarization on the intensity of X-ray scattering.
Graphical visualization of the results obtained and comparison of the calculated diffraction patterns with experimental data and other models.
The results of calculation (theoretical diffraction patterns) are obtained in formats that allow them to be used in other graphical editors and programs for visualization and treatment of experimental data.
The instrumental broadening is taken into account by convolution of calculated curve with an instrumental function which can be determined using reference samples.
Computation time depends mostly on the number of atoms; such computations were made on PC AMD Athlon X4 740 quad core processor 3.2 Ghz.
DIANNA was written in C++ and GUI in C# that requires Microsoft.NET Framework 4.0 (or higher) to be installed. The program itself runs under MS Windows. Distribution of the new version of DIANNA software is available from www.sourceforge.net/projects/dianna.
Optimization of calculations by the Debye equation. Calculation of interatomic distances
Since the number of interatomic distances is proportional to the third degree of the number of atoms, calculation of interatomic distances is the most time-consuming procedure. The problem of decreasing the calculation time was always topical for the DSE. So, of interest are some algorithms that can reduce the computation time. Sorting of interatomic distances gives the interatomic distance distribution [the multiplicity function of distances М(rij)], which makes it possible to decrease the number of terms in the DSE , , , , . This function describes all interatomic distances in crystallites and calculates the number of pairs for each distance. Using such an approach, one can rewrite the Debye equation as
The goal of this algorithm is to calculate all interatomic distances in a crystal and to find the repetitive ones by sorting, thus allowing the distribution of interatomic distances to be constructed as a pair of numbers ri↔ni, where ri is the interatomic distance value (nm), and ni is the number of such distances (items). Such operation makes it possible to reduce the number of terms in the Debye equation, which strongly decreases the computational cost per each point of the 2θ diffraction pattern. The form of such a distribution can be represented by a table or histogram. Plotting of a histogram is related to additional difficulties caused by rounding-off and comparing the computer data upon sorting. The algorithm used in the program is described in the Supplementary materials section.
Calculation of sine in formula (1) also substantially extends the computation time. Therefore, to speed up the operation, the values of sine are taken from a prepared table.
Examples of DIANNA software application
Some examples of DIANNA software application for revealing the structure of nanocrystalline samples of aluminum hydroxide (pseudoboehmite) and gallium oxide can be found in Refs. , , respectively.
A combination of DFA and RDF methods was used to identify phases in ultradisperse samples of iron oxide . For a sample with the CSD size of 1–2 nm and a specific surface area of 350 m2/g, the best model, satisfactorily describing both the diffraction pattern and the RDF curve, corresponds to the structure of ferrihydrite (Figure 2). The insert on Figure 2 shows the RDF curves that were calculated in the approximation of an infinite crystal size and taking into account the particle size of 2 nm. The particle size was found by modeling the profile by the DFA method.
In order to demonstrate the capability to simulate particles of strongly anisotropic shape, Figure 3 shows experimental and theoretical diffraction patterns of nanosized magnesium hydroxide powder Mg(OH)2. The structure was specified using the ICSD card No. 95475: the hexagonal space group parameters a=0.3144 and c=0.4752 nm.
Some peaks (100 and 110) are much narrower than others. The size along directions  and  can be estimated from the 100 peak using the Scherrer formula , which gives ca. 20 nm. The size along direction  estimated form the first 001 peak is ca. 5 nm.
The model represented by spherical particles 15 nm in diameter is displayed on Figure 3a. The best model refined by DFA is represented by the plate-shaped particles with the dimensions 23×23×5.5 nm (Figure 3b). The diffraction was calculated using the DIANNA program. Satellite peaks appeared on the model curve because size distribution was not taken into account in this example.
The paper describes the DIANNA software that allows one to develop atomic models of nanocrystals with different shape and size and to calculate interatomic distances, RDF curves and diffraction patterns.
It was demonstrated that the DFA method can be used to validate and optimize the proposed structural models of ultradisperse systems with due regard not only for the structure but also for the shape and size of nanoparticles.
The program is focused on a wide range of users with an ordinary PC and takes user-friendly interactive approaches both for creating atomic models of nanoparticles and for treating with the obtained data.
The software has been developed within the basic budgetary funding (project 0303-2016-002).
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The online version of this article (DOI: https://doi.org/10.1515/zkri-2017-2056) offers supplementary material, available to authorized users.
About the article
Published Online: 2017-05-20
Published in Print: 2018-01-26