A. P. Dudka

ASTRA– a program package for accurate structure analysis by the intermeasurement minimization method

ASTRA2.0 is a program package for single-crystal structure research using different experimental conditions and radiation of different types (X-rays, neutrons and electrons). The package includes programs for data collection optimization, data reduction, structural parameter refinement and electron density plotting. The expression that is responsible for minimization of the difference between reduced measurements is included in the goal function of the refinement program. The refinement program is based on an adaptive nonlinear least-squares method. The crystal structure is described by a detailed model for electron density analysis. ASTRA uses a multipole model up to eighth order for the aspherical electron density description, an anharmonic model up to sixth order for the description of atom displacement and a model of mixed atomic positions for the description of atomic replacement. Refinement of the structural parameters using data of precession electron diffraction and the Bloch-wave method to account for multiwave scattering is realized. Tools for refinement automation are included in ASTRA.

New approaches to scaling data measured on a CCD diffractometer

New methods have been developed for scaling the intensities of diffraction reflections measured on a diffractometer with a CCD detector. The algorithm involves cyclic alternation of the refinement of a structure model from experimental data corrected for anisotropic effects with the refinement of anisotropic parameters from initial data for a fixed structure model. The first method consists of refining scale factors for the intensities after they have first been corrected for anisotropic effects of known physical nature (thermal diffuse scattering, absorption, extinction etc.). It has been shown that the majority of the observed biases of intensities are caused by the characteristics of the CCD detector and the technique used for data collection on the diffractometer, rather than by the properties of the sample. The second (alternative) method of fixed scaling has been implemented here. This method does not require refinement of empirical scale factors but uses intensity correction maps obtained by averaging the results derived from several experiments with different crystals. The first map describes the non-uniformity of the response over the surface area of the detector. The second map accounts for distortions dependent on the setting angles of the goniometer. The methods were checked for data measured on an Oxford Diffraction Xcalibur Sapphire 3 CCD diffractometer. The proposed methods significantly improve the results of the refinement and make it possible to obtain a structure model that is reproduced in repeated investigations of samples of the same compound.

Refinement of the λ/2 contribution to CCD detector data

A new method is proposed to improve structure modeling by taking into account the half-wavelength contribution (λ/2) to X-ray data measured on CCD and image-plate diffractometers. Such an effect arises because the reflection intensities contain contributions associated with both the λ and the λ/2 wavelengths. The ratio w of these two contributions is a new parameter refined using the full-matrix least-squares techniques. The new method does not require any additional measurement or information. According to the tests performed, proper account of the λ/2 contribution lowers the wR2(|F|2) factors by a relative percent difference of 4–14% and the atomic displacement parameters by ∼1–4 s.u. Other effects are analyzed that might correlate with the λ/2 contribution, such as absorption, charge-density redistribution, thermal diffuse scattering and extinction. Strong correlation was found between w and the other parameters, but a fixed λ/2 correction reduces this difficulty. The value of w is found to be both characteristic of the diffractometer and a general indicator of the refinement quality (similar to R factors).

Accurate crystal-structure refinement of Ca3Ga2Ge4O14 at 295 and 100 K and analysis of the disorder in the atomic positions

The accurate X-ray diffraction study of a Ca3Ga2Ge4O14 crystal (sp. gr. P321, Z = 1) has been performed using repeated X-ray diffraction data sets collected on a diffractometer equipped with a CCD area detector at 295 and 100 K. The asymmetric disorder in the atomic positions in Ca3Ga2Ge4O14 is described in two alternative ways: with the use of anharmonic atomic displacements (at 295 K R/wR = 0.68/0.60%, 3754 reflections; at 100 K R/wR = 0.90/0.70%, 3632 reflections) and using a split model (SM) (at 295 K R/wR = 0.74/0.67%; at 100 K R/wR = 0.95/0.74%). An analysis of the probability density function that defines the probability of finding an atom at a particular point in space shows that, at 295 K, five of the seven independent atoms in the unit cell are asymmetrically disordered in the vicinity of their sites, whereas only three atoms are disordered at 100 K. At both temperatures the largest disorder is observed at the 3f site on a twofold axis, which is a prerequisite for the formation of helicoidal chains of atoms along the c axis of the crystal and can serve as a structural basis for multiferroic properties of this family of crystals with magnetic ions.

Structure analysis by reduced data. IV. IEM—A new program for refinement of structure models of crystals

A new program for the refinement of models of atomic structures by the X-ray, neutron, and electron diffraction data has been described. The program is based on a special form of the goal function including the term responsible for minimizing the difference between the normalized measurements [the interexperimental minimization (IEM) method] and the adaptive nonlinear algorithm of minimization based on the Lavrent’ev-Levenberg-Marquardt regularization. As a result, it became possible to determine a new solution to the problem different from that obtained by the classical least squares method. To a large extent, the program allows one to overcome the effect of parameter correlations on the procedure of refinement and the results obtained. The test of the program on 17 experimental data sets showed the fast and stable convergence in all the cases. Under the identical initial conditions, the new program provided lower reliability factors for most of the crystals studied.

Refinement of the crystal structures of the La3Ta0.5Ga5.5O14 and La3Nb0.5Ga5.5O14 compounds

This paper reports on the results of accurate X-ray structural investigations of single crystals La3Ta0.5Ga5.5O14 (a = 8.2260(1) Å, c = 5.1207(1) Å, R/Rw = 1.09%/1.10%, 3868 unique reflections) and La3Nb0.5Ga5.5O14 (a = 8.2237(1) Å, c = 5.1247(1) Å, R/Rw = 1.02%/1.03%, 3735 unique reflections) (space group P321, Z = 1, sinϑ/λ|max ≈ 1.34 Å−1 for both compounds). Using sets of data with a large number of high-angle reflections makes it possible to reliably compare the crystal structures by applying statistical tests. The structural differences between the La3Ta0.5Ga5.5O14 and La3Nb0.5Ga5.5O14 crystals are weakly pronounced, which correlates with the closeness of their piezoelectric characteristics.

Structural analysis by reduced data: VI. A new method for refining the parameters of the model describing the absorption of radiation by a single-crystal specimen

An approach to the refinement of crystal structures by reduced experimental data is developed. The anisotropic method of intermeasurement minimization is used to refine the parameters determining the absorption of radiation in the cases when some factors are not known with sufficient accuracy: the specimen sizes, the orientation of ellipsoidal specimens, the degree of perfection of spherical specimens, the elemental composition of the unit cell, etc. Spherical and ellipsoidal crystals and crystals in the form of convex polyhedra are considered.

X-ray diffraction study of a Nd3Ga5SiO14 crystal at 295 and 90 k and the structural basis for chirality

An accurate X-ray diffraction study of a Nd3Ga5SiO14 crystal (sp. gr. P321, Z = 1) has been performed using repeated X-ray diffraction data sets collected on a diffractometer equipped with a CCD area detector at 295 and 90 K. The asymmetry of the disorder in the atomic positions in Nd3Ga5SiO14 is described in two alternative ways: in terms of anharmonic atomic displacements (at 295 K R/wR = 0.80/0.92%; at 90 K R/wR = 0.82/0.76%) and using a split model (at 295 K R/wR = 0.85/0.97%; at 90 K R/wR = 0.80/0.74%). An analysis of the probability density function of finding an atom at a particular point in space shows that most of the atoms in the unit cell are asymmetrically or highly anisotropically disordered in the vicinity of their sites. Electron density regions for cations and anions that form pseudohelices were found. Such regions mimic a screw axis of symmetry and are responsible for the chirality of the structure.

Structural conditionality of the piezoelectric properties of langasite family crystals

The atomic displacements upon isomorphic substitutions in crystals of the langasite family have been analyzed. The thermal parameters are determined and the probability density function of atoms is analyzed. Local potential energy minima are found which can be occupied by atoms under external effects. The contributions of cations in all four independent crystallographic positions and anions in all three such positions to the piezoelectric properties are established. One specific structural feature is the constant (at isomorphic substitutions) or possible (under external effects) but always opposite displacements of two cations along symmetry axis 2. Large cations in eight-vertex polyhedra make the main contribution to the piezoelectric properties. The cations in the tetrahedra on symmetry axis 2 weaken these properties. The cations in the octahedra in the origin of coordinates and in the tetrahedra on symmetry axes 3 only slightly affect the piezoelectricity.

Crystal structure refinement of Sr3TaGa3Si2O14

An accurate X-ray diffraction study of Sr3TaGa3Si2O14 (STGS) crystal (a = 8.3023(10) Å, c = 5.0853(2) Å, sp. gr. P321, Z = 1, R/wR = 0.59/0.52%, 4004 independent reflections) is performed. The use of two independent data sets obtained on diffractometers with point and 2D detectors made it possible to determine the model structure characterized by the best reproducibility of parameters. The ordered distribution of atoms over crystallographic positions and the anharmonic character of displacements of all cations and one oxygen atom are established.