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Specific stereochemical features of silicon in oxygen-containing compounds, including hybrid silicates with all oxygen atoms of SiOn groups (n = 4, 5, or 6) entering into the composition of organic anions or molecules, are described by characteristics of Voronoi—Dirichlet polyhedra. It is found that in rutile-like stishovite and post-stishovite phases with the structures similar to those of СаСl2, α-PbO2, or pyrite FeS2, the volume of Voronoi—Dirichlet polyhedra of silicon and oxygen atoms decreases linearly with pressure increasing to 268 GPa. Based on these results, the possibility of formation of new post-stishovite phases is shown, namely, the fluorite-like structure (transition predicted at ~400 GPa) and a body-centered cubic lattice with statistical arrangement of silicon and oxygen atoms (~900 GPa).

Mechanisms of radiation damages in minerals promising for the utilization of highly active radioactive waste (zircon ZrSiO4, monazite LaPO4, orthophosphate YbPO4, lakargiite CaZr0,8Sn0,1Ti0,1O, and Gd2Zr2O7 and Gd2Ti2O7 compounds with the pyrochlore structure) are studied by the computer simulation using the molecular dynamics method. The formation of a damaged region in the structures under study is considered along with the recovery (relaxation) processes in these structures. The number of Frenkel pairs and antistructural defects that form in the structure of these minerals after the pass of the knocked out thorium atom is calculated by the molecular dynamics method. A parameter is proposed that characterizes the mineral tendency to amorphization under the effect of the radiation damage. The obtained results show that one of the main factors determining the radiation stability of minerals is the crystal structure type, compounds with the monazite structure being more radiationally stable than compounds with the zircon structure. It is found that the Gd2Zr2O7 compound and lakargiite have a weaker tendency to amorphization than other minerals such as zircon, monazite, and orthophosphate YbPO4. A high radiation stability of the studied Gd2Zr2O7 compound as well as a solid solution of the composition CaZr0.8Sn0.1Ti0.1O3 makes it possible to propose them as a possible matrix for the utilization of highly active radioactive waste.

The possibilities of studying the local structure in isomorphic systems by methods based on the minimization of the interatomic interaction energy are considered in the work. The original procedure for the theoretical analysis of the local structure, which the authors have been developing for a number of years and have implemented in four computer programs, is discussed in detail. The practical implementation of this procedure is exemplified by the following solid substitution solutions: halite–sylvin (Na,K)Cl, corundum–eskolaite (Al,Cr)2O3, spinel–magnesiochromite Mg(Al,Cr)2O4, chrysoberyl–mariinskite Be(Al,Сr)2O4, grossular–uvarovite Ca3(Al,Cr)2[SiO4]3. Calculations are performed on a SKIF MGU Chebyshev supercomputer for supercells containing several thousands of atoms. The performed analysis of the behavior of changes in the geometric characteristics of CrO6 octahedra in various isomorphic systems makes it possible to reveal the main reasons affecting the alteration of the relaxation parameter. It is found that the key factor is the size of the common structural unit of an isomorphic mixture. Thus, the behavior of the relaxation parameter obeys the isomorphism assistance rule.

Strontianite (SrCO3)-aragonite (CaCO3) solid solutions have been simulated by the interatomic potential method. The composition dependences of the unit cell parameters, the unit cell volume, and bulk modulus have been constructed. It has been shown that the volume of the unit cell and bulk modulus show small negative deviations from additivity. The local structure of solid solutions has been analyzed. It has been established that the enthalpy of mixing is positive and, for the equimolar composition, reaches a maximum of 2.45 kJ/mol. Based on the composition dependences of the Gibbs free energy for the temperature range of 300–650 K, the solvus of the system has been constructed. According to the obtained data, the solubility of aragonite in strontianite under ambient conditions is 5.5 mol %, while that of strontianite in aragonite is 2.8 mol %. The miscibility gap of the system disappears at around 450 K. The calculated results have been compared with the experimental data.

In connection with the 100th anniversary of the discovery of the phenomenon of the diffraction of X-rays by crystals (Max von Laue, 1912) and the basics of X-ray crystallographic analysis (Lawrence Bragg, 1913), the article marks the major milestones in this outstanding event in the history of science. Special emphasis is given to the role of Russian scientists (E. S. Fedorov, G. V. Vulf) in the emergence and early steps in the establishment of this discovery being one of the pillars of the scientific revolution of the first half of the past century.

The coordination number parity law is formulated: the coordination polyhedra (CP) with the odd number of vertices, except for 3 (CP is a triangle), occur far less often than the polyhedra with the even number of vertices. CP with the odd number of vertices, except for the triangle, cannot be regular and have at least two different types of vertices and two sets of interatomic lengths. The main and first c.n. always keeps parity. A CP distortion due to crystal chemical causes tends to be minimum, obeying the pseudo-symmetry laws. A CP distortion, other things being equal, leads to an increase in the average bond length as compared to the regular polyhedron (distortion theorem). The value of the increase in the average interatomic distance depends on the degree of distortion by the linear or (for a very strong distortion) the square dependence. The histograms of the distribution frequencies of the interatomic lengths have a positive deviation from the Gauss normal distribution law. A free molecule or a complex ion having a distortion due to the directed configuration of chemical bonds or to the electronic effects of the Jahn-Teller type for transition metals, maintain it in the crystal structure as well. During the polymerization of radicals there arise specific distortion effects due to a various coordination environment of the bridging and apical ligands. One of the most general rules of the CP distortion is associated with a less symmetric and less homogenous environment of anions in comparison with that of cations and a higher polarizability of anions. An important exception from this rule is the group of compounds with the so called anion-centered tetrahedra in which more symmetrical positions are occupied, as a rule, by anions in the center of the tetrahedra.

The structures of chromium-containing cluster centers in forsterite laser crystals Mg2SiO4 : Cr and Mg2SiO4 : Cr : Li have been simulated using the interatomic potential method. A system of position-dependent parameters of interatomic interaction potentials in forsterite has been developed. In the ionic approximation, this system adequately describes the structure, properties, and defects of the crystal with correct representation of the preferred positional arrangement of chromium ions. The preferred mechanisms of chromium dissolution in forsterite crystals have been evaluated from a comparison of the energies of formation of chromium-containing clusters with different configurations. It has been demonstrated that the results of the simulation of interatomic interaction potentials are consistent with the experimental data obtained from electron paramagnetic resonance and optical spectroscopy studies.

The interatomic potential method in an ionic approximation was used to model the protonation of forsterite crystals. The formation of isolated OH− groups in iron-free and iron-bearing crystals and neutral clusters of protonated cation vacancies was considered. It was shown that the presence of trivalent impurities may significantly facilitate protonation processes owing to their reduction to a divalent state or formation of clusters with cation vacancies. In most cases, charge balancing of hydrogen-bearing defects by magnesium vacancies is energetically favorable over that involving silicon vacancies.

The rule of negative deviation from additivity for bulk modules of solid solutions is considered. The rule is substantiated in terms of the phenomenological theory, which is based on experimental data on the physical properties of crystals. The theory is confirmed by atomistic calculations by the authors for a halite-sylvine solid solution and for various binary oxide, silicate, phosphate, and carbonate systems.

The radiation stability of lakargiite Ca(Zr,Ti,Sn)O3 was investigated by the molecular dynamics method using computer simulation. A nanofragment of the structure of a solid solution of the composition CaZr0.8Sn0.1Ti0.1O3 with sizes 310 × 290 × 315 A, which contains 1.4 million atoms with a maximally disordered distribution of Zr, Ti, and Sn, was used for simulation. It has been established that the susceptibility of lakargiite to amorphization is considerably lower than that of other minerals such as zircon, monazite, and xenotime. The high radiation resistance of the studied solid solution makes it possible to suggest lakargiite as a possible matrix to utilize highly active radioactive wastes.

The radiation resistance of the monazite LaPO4 and the compound YbPO4 (zircon structure type) has been investigated using the computer simulation. The number of Frenkel pairs, which are formed in the structure of these minerals after the passage of a primary knock-on thorium atom with an energy of 30 keV, has been calculated by the molecular dynamics method. The formation of Frenkel pairs and their recombination in the motion of recoil nuclei in the structure of the studied minerals have been discussed. It has been shown that the probability of the “survival” of Frenkel pairs in the LaPO4 monazite is significantly lower than in the YbPO4 compound. The tendency of these minerals toward amorphization under radiation damage has been described numerically. The obtained results have demonstrated that one of the main factors determining the radiation resistance of orthophosphates LnPO4 is the type of crystal structure, and the compounds with the monazite structure are more radiation resistant than the compounds with the zircon structure.

A universal model of the interatomic coupling potentials for structural simulation of borates is developed for correct reproduction and prediction of crystal structures of borates containing BO3 triangles and BO4 tetrahedra. The model is tested on three compounds: CaB2O4, SrB4O7, and CaB6O10. It is concluded that the model can be used in structural simulation and prediction of physical properties of borates containing more complex boron-oxygen radicals.

Structure and thermodynamic properties of zircon, coffinite, and zircon-coffinite solid solutions were subject to semiempirical atomistic simulation on the assumption of compositional disordering of mixed crystals: (Zr1 − x , U x )SiO4, where x = 0.02, 0.05, 0.08, 0.11, 0.14, 0.28, 0.50, 0.72, and 0.86. The solid solutions significantly depart from the Vegard and Retgers laws. The (Zr1 − x , U x )SiO4 mixed crystal structure is characterized by anisotropic expansion (largely in the a and b directions) under growth in x related to anisotropy of structural relaxation degree of the cation-oxygen interatomic spacing in the (Zr, U)O8 polyhedron. Increase in x parameters is accompanied by increase in average size of cation-oxygen polyhedrons and silica-oxygen tetrahedrons and by growth of interatomic spacing dispersion; maximum dispersion values are observed at x = 0.5–0.6. The distortions in local structures of ZrO8 and UO8 dodecahedrons and silica-oxygen tetrahedrons in the solid solution were analyzed on the basis of calculated functions of interatomic distance distribution. The obtained results demonstrate the possibility to assess numerically the structural (geometrical) disordering degree of the compositionally disordered solid solution depending on its composition. The calculated thermodynamic characteristics of solid solutions forecast the following solubility range limits: 2 mol % USiO4 in zircon and 5 mol % ZrSiO4 in coffinite under ∼1750 K.

The radiation resistance of zircon (ZrSiO4) was comparatively tested with computer simulations of four different sets of parameters of interatomic potentials. The energies of Frenkel pairs (FP) for Zr, Si, and O atoms were calculated using the Mott-Littleton method in approximation of isolated defects. The difference between the FP energies calculated for four potentials and the FP energies obtained ab initio is the least in the one of the four calculated potentials that takes into account the ab initio data for Zircon 3. The formation of an atomic displacement cascade (ADC) after the passage of the initially knocked-out Th atom with energy of 20 keV was investigated using the molecular dynamics method. The number of FP in ADC reaches 5300 to 61 900, depending on the chosen potential. The number of FP at the end of the simulation varies from 480 to 4970. The distribution of internodal oxygen atoms in zircon has been studied. It is shown that within a time (t) interval of 0–0.1 ps (ballistic stage), the internodal oxygen atoms knocked-out from the initial equilibrium site are predominant (O1 defects). Thereby the Si-O bond is ruptured. The mean displacement of those defects is 2–3 A. The probability of their survival is 34–73% depending on the chosen potential. After t = 0.1 ps (onset of the thermal stage), many SiO4 tetrahedra in zircon are displaced with formation of a great amount of internodal oxygen atoms, because the Si-O bond is not ruptured. The mean displacement of those O2 defects is 1 A; the probability of their survival is insignificant: 1.5–3.0% depending on the chosen potential. The total amount of internodal oxygen atoms consists of O2 defects (19–25%) and largely of O1 defects. A parameter characterizing that part of the energy of the initially knocked-out atom, which is consumed for FP, has been introduced. The physically acceptable values of this parameter (

The structure of oxygen hole centers in forsterite crystals has been simulated using the interatomic potential method. The energies of isolated oxygen hole centers, as well as the energies of their clusters with intrinsic and extrinsic defects of the crystal, have been estimated for different arrangements of point defects in the structure. It has been shown that the most energetically favorable position for isolated oxygen hole centers is the O3 position, in which the gain in the formation energy is equal to 0.17 eV as compared to the O2 position and 1.66 eV as compared to the O1 position. The maximum energy gain due to the association energy can be achieved when the oxygen hole centers are located at the vertices of the tetrahedron with a silicon vacancy. The presence of chromium in the forsterite crystal can increase the probability of the formation of silicon vacancies. The obtained results have been discussed in terms of the experimental investigations of the color centers generated in the Mg2SiO4 and Mg2SiO4: Cr crystals under ionizing radiation.

The crystal structure of the new mineral magnesioneptunite (K0.8Na0.1−0.1)Na2Li(Ti0.39Mg0.34Fe0.27)2(Ti0.59Mg0.22Fe0.19)2[Si4O11]2(O,OH) from the xenolith of Verkhnechegemskaya caldera (Lakarga Mountain, North Caucasus) has been investigated by X-ray diffraction (XCalibur-S diffractometer, R = 0.0244): a = 16.3271(7) A, b = 12.4788(4) A, c = 9.9666(4) A β = 115.651(5), V = 1830.5(1) A3, sp. gr. C2/c, Z = 4, and ρcalcd = 3.152 g/cm3. The disordered distribution of Ti, Mg, and Fe atoms in the octahedra forming the basis of the cationic framework is established. It is shown that the isomorphic occupation of octahedral positions by cations of three types corresponds to the centrosymmetric crystal structure and is likely caused by the high-temperature crystallization of the mineral.

Chrysoberyl and alexandrite crystals have been grown from solutions in melts based on the Li2CO3-MoO3, Bi2O3-MoO3, PbO-V2O5, Na2B4O7, and K2MoO4-MoO3 systems using natural alexandrite and chrysoberyl debris as the initial BeAl2O4 compound. An analysis of the morphology and homogeneity of the crystals grown has revealed the Bi2O3-MoO3 solvent to be the most appropriate. The optimal color characteristics (“quality” of alexandrite effect) manifest themselves when adding about 5 mol % Cr2O3. The largest crystals (up to 10 mm in size) were obtained from a solution in melt based on PbO-V2O5 at a ratio of the crystal-forming component to the solvent of 9: 91 wt %; These characteristics, along with a relatively low operating temperature (970°C), give grounds to consider this type of solvent promising.

The radiation resistance of the mineral xenotime YPO4 has been investigated using computer simulation methods. The optimal criteria for the appropriate choice of the parameters of the interatomic potentials used for the simulation of α-decay in minerals have been proposed. The formation of a cascade of displaced atoms in the structure of xenotime after the passage of a primarily knocked thorium atom with an energy of 20 keV has been studied using the molecular dynamics method. The specific features of the formation and annealing of individual defects in the structure of xenotime have been considered using the Mott-Littleton method. The dependences of the energy of formation of Frenkel pairs and the probability of their annihilation during the annealing of a cascade of atomic displacements on the distance between the vacancy and the interstitial site have been analyzed within the framework of the supercell approximation.

and M2 of Ci () and Cs (m) symmetry, respectively. The M1 and M2 Mg octahedra are linked in zigzag chains, which alternate with similar chains of M3 and M4 octahedral voids. Silicon occupies 1/8 of the cat� ion tetrahedral sites. Every other chain of filled octa� hedra is linked with S1 Si–O tetrahedra and are sepa� rated by S2 and S3 voids. The computer simulation of the crystal structures, properties, and defects of minerals provides insight into their kinetic properties, including phase transi� tions and diffusion processes, and makes it possible to conduct comparative analysis of the distribution of minor elements between equilibrium phases, which has important implications in the study of differentia� tion processes in the Earth. Finally, this enables one to predict physical and thermodynamic properties of crystals under typical mantle temperatures and pres�

Synthetic analogues of murataite, a very rare mineral—a complex oxide of REE, actinide, Ti, Fe, and other elements,—are of great interest as confinement matrices of radioactive wastes. They are produced by sintering (1200–1300°C) and melting (1500–1600°C) with subsequent crystallization of the melt. Four structural varieties of murataite distinguished by unit-cell parameters have been established by TEM study. All these varieties are derivatives of the fluorite structure designated as murataite 3C, 5C, 7C, and 8C depending on repetition factor of a parameter of the fluorite subcell. The structural features of the synthetic murataite varieties are analyzed in this paper based on data obtained from high-resolution electron microscopy, microdiffraction, and X-ray refinement. The hypothesis of a modular structure of the members of polysomatic pyrochlore-murataite series has been confirmed. At the same time, the structural modules are zero-dimensional rather than two-dimensional as had previously been suggested. The combinations of zero-dimensional modules in 3D space create the entire structural diversity of the polysomatic series.

The thermodynamic and elastic properties of mixing have been calculated and the local structure of Be(Al,Cr,FeIII)2O4 solid solutions has been analyzed using a previously elaborated set of interatomic potentials for atomistic modeling of simple and complex beryllium oxides, which almost exactly reproduced their structural, elastic, and thermodynamic properties. Calculations have been performed for a (4a × 2b × 2c) supercell of the olivine structural type with the relieved nontranslational symmetry (space group P1), which has made it possible to specify the distribution of trivalent atoms over nonequivalent positions and to provide relaxation of the local structure. The mixing properties have been calculated over the entire range of compositions, the changes in the Gibbs free energy with variations in temperature and the regions of stability of the solid solutions have been estimated, and the critical values of the temperature and composition have been determined. The histograms of the distributions of M-M, M-O, and O-O interatomic distances, as well as MO6 octahedron volumes, have been constructed; the compliances of cation positions have been evaluated; and the groups of atoms that are most readily shiftable from their ideal positions have been established. The performed analysis has demonstrated that two different octahedral positions M1 and M2 in the olivine-type structure, namely, chrysoberyl, are extremely sensitive to their own atomic environment. This leads to a preferable incorporation of Al3+ cations into the M1 octahedral position, whereas larger-sized Cr3+ (Fe3+) cations preferentially occupy the M2 octahedral position.

Based on our analysis of factual material (27 various crystals), we have shown the universality of the linear scale composed of Planck’s constant, the density, and Young’s modulus by the dimensional method. This scale is 1 angstrom, i.e., 10−10 m, with a logarithmic accuracy. The quantum effects become significant starting from this scale.

The old issue of the limited number of mineral species is being brought up again and requires explanation. The review of successive calculations of the number of mineral species and their main varieties is given in this paper with allowance for the contemporary view on this problem. The natural selection of mineral species develops under the effect of geochemical differentiation of mineral matter, a multitude of chemical elements and their substitutions, and acid-alkali equilibria. It is noteworthy that water, oxygen, carbon dioxide, and other volatiles make a crucial contribution to the increase in the number of mineral species in the Earth’s crust.

The influence of the oxygen partial pressure P O 2 in the growth atmosphere on the coefficient of chromium distribution between the crystal and the melt of forsterite, the Cr3+ and Cr4+ ion contents in crystals, and the concentration of color centers induced by irradiation has been investigated. It has been established that the crystals grown at low oxygen partial pressures P O 2 (0.01–0.05 kPa) are characterized by low concentrations of Cr4+ ions and color centers. A change in the oxygen partial pressure to P O 2 ∼ 0.85 kPa leads to an increase in the Cr4+ center concentration by a factor of ∼10 and in the color center concentration by a factor of ∼5. A further increase in the oxygen partial pressure to P O 2 to 12 kPa remains the concentration of these centers almost unchanged. A model has been proposed according to which the intrinsic defects formed under conditions of a relative excess of oxygen leads to both the self-oxidation of chromium and the formation of color centers in the forsterite crystals under irradiation.

A set of partially covalent interatomic potentials has been developed with the aim of reproducing the experimentally known crystal structures and predicting the unknown crystal structures and the thermodynamic and elastic properties of bromellite BeO, chrysoberyl BeAl2O4, and its isostructural analogs BeCr2O4 and BeFe2O4. The calculated structural, elastic, and thermodynamic properties of these minerals are in good agreement with the available experimental data. This gives grounds to believe that the prediction of a number of unknown properties of these compounds and the model crystal structure of Fe chrysoberyl are sufficiently reliable. A theoretical analysis of the energy characteristics of the Schottky and Frenkel point defects is carried out for all the substances under investigation.

A detailed historical review is presented and the difficulties arising when defining such a fundamental concept as structural type in crystal chemistry are analyzed. Particular attention is paid to the genetic relationship between different structural types and to the criteria of choosing between them. Many other terms and concepts of modern crystal chemistry defined for completeness.

The dependences of average V-O distances in inorganic compounds of vanadium of different valence on the degree of distortion of coordination polyhedra have been obtained by careful statistical treatment of modern structural data banks. Values of bond lengths in undistorted (regular) polyhedra are recommended. Theoretical analysis of the statistical data made it possible to calculate the most likely values of the parameters of the bond valence model: the interatomic distance for the single (two-electron) bond, corresponding to the single valence, and the bond softness parameter. Calculations of the sums of bond valences for some complicated cases (different coordination numbers, mixed vanadium valence) confirmed reliability of the recommended parameters.

Development of matrices (carriers) for cells with the use of micro- and nanoparticles of various types is one of the key problems in creation of bioartificial organs, because the sizes and structural characteristics of these particles may make them compatible with cultured cellular structures. There are various physicochemical methods of the formation of microheterogeneous mosaic structures imitating the morphological and other characteristics of the surface of biological structures. Therefore, it is important to study the interaction of synthetic and natural (geyserite) opal matrices with cellular systems in order to develop biocompatible materials, including those used in reconstructive plastic surgery. Geyserites (siliceous sinters) are deposits of amorphous silica formed from thermal waters of hot springs, microorganisms being necessarily involved in the process. Specific bacterial communities consisting of both producers (phototrophs and chemotrophs) and reducers occur in places where thermal waters come out of the ground. Bacterial mats varying in the silica content are often formed in these systems; therefore, geyserites are usually classified with biogenic deposits. Several “genetic” types of geyserites are distinguished; they differ from one another in the size and distribution of silica particles and the amount of silicified remnants of microorganisms [1, 2].

A technique for the computer modeling of disordered binary oxide solid solutions MO-M′O in a wide composition range has been developed. The method of atomistic pair potentials was used for 4 × 4 × 4 supercells. The parameters of the potentials are optimized using the structural and elastic properties of pure components MgO, CaO, SrO, and BaO. The temperature dependences of the heat capacity and entropy are calculated for pure components. The excess mixing properties (enthalpy, volume, bulk modulus, vibrational entropy) are found for different compositions of MgxCa(1 − x)O, CaxSr(1 − x)O, and SrxBa(1 − x)O solid solutions. Temperature and composition dependences of the excess Gibbs energy were constructed, which made it possible to approximately estimate the critical decomposition temperatures and limits of component miscibility. Statistical analysis of lattice distortions in the first and second coordination spheres reveals a detailed picture of the solid-solution local structure.

The structural, elastic, and thermodynamic properties of halite NaCl and sylvine KCl and the miscibility properties of the NaCl-KCl solid solution found by computer simulation are in good agreement with the experimental data. Analysis of the relaxation of the solid solution structure suggests that both anion and cation sublattices are distorted; however, the anion sublattice is distorted much more strongly. Calculations of the local bond valence at all types of ions in the solid solution show opposite deviations from the balance at cations, whereas the general balance is retained. The values of the electrostatic potential in the ion positions reflect weakening of bonding in the solid solution with respect to its pure components. In addition, with an increase in the average interatomic distance in the first coordination sphere around cations, the modulus of the electrostatic potential at cations decreases.

An original technique of computer modeling of substitutional solid solutions has been applied to Al2O3-Cr2O3, Al2O3-Fe2O3, and Fe2O3-Cr2O3 binary systems. The parameters of semiempirical interatomic potentials were optimized using the experimentally studied structural, elastic, and thermodynamic properties of pure components. Among point defects, the most energetically favorable ones for all three oxides are Schottky vacancy quintets. To model (Mx1M1 − x2)2O3 solid solutions, 4 × 4 × 1 disordered supercells with M1: M2 cation ratios of 1: 5, 1: 2, 1: 1, 2: 1, and 5: 1 have been constructed in the cation sublattice containing 192 atoms. The mixing enthalpy and volume, interaction parameters, bulk moduli, and vibrational entropy were found by minimizing the interatomic interaction energy in supercells with the symmetry P1. Calculations of the Gibbs energy made it possible to estimate the fields of stability of the Al2O3-Cr2O3 and Al2O3-Fe2O3 solid solutions; these estimates were compared with the experimental data. Histograms of M-M, M-O, and O-O interatomic distances were constructed and the local structure was analyzed for the Al1.0Cr1.0O3, Al1.0Fe1.0O3, and Fe1.0Cr1.0O3 compositions.

Different methods currently employed for choosing and analyzing model configurations of binary systems with isomorphous substitution in the sublattice were considered. A new more efficient algorithm was proposed for determining the most disordered atomic configuration of an arbitrary composition for a cell with any size. The algorithm was implemented in the form of a computer program which makes it possible to approach the most optimum topology of the arrangement of solid solution components over atomic positions. The program was tested using a number of binary systems and can be recommended for ab initio calculations and simulation with semiempirical methods. Moreover, this approach can be used to decompose complex experimental spectra of isomorphous mixtures which are difficult to interpret without recourse to model concepts regarding the local structure of multicomponent systems.

A comprehensive statistical analysis of the symmetry of mineral species leads to a definite conclusion that rare minerals possess lower symmetry than abundant ones, so that the most stable minerals are characterized by higher symmetry. Since all recently discovered new minerals belong to rare and very rare species, their percentage is increasing and the mean symmetry index is decreasing with time. In other words, the average symmetry is gradually decreasing with the growing diversity of mineral species. In general, the irreversible process of rare mineral formation obeys the principle of minimum dissymmetrization. At the same time, the reduced symmetry indices strongly decrease on passing from cosmic materials (meteorites, lunar rocks) to the Earth’s solid substances and from the planetary interior (core, mantle) to the Earth’s crust. This trend of the planet’s evolution is related to the pronounced loss of entropy and increase in ordering of solid substances that compose the lithosphere. This is supplemented by the withdrawal of entropy from the solid to the upper liquid (oceans) and gaseous (atmosphere) shells of the Earth and farther to the surrounding space.

This paper reports on the results of a structural investigation of an important group of synthetic compounds with derivative structures of the mineral murataite, a potential matrix for immobilization of radioactive wastes. A model describing the structure of the entire group of compounds is proposed on the basis of analyzing the results obtained using chemical (electron microprobe) analysis methods, high-resolution transmission electron microscopy (high-resolution electron images and microdiffraction patterns), and X-ray diffractometry of a large array of artificial murataite ceramics. In essence, the proposed structural model is as follows: in the structure, layers of two (pyrochlore and murataite) packing types alternate along the [111] direction of the cubic supercell, so that the resulting structures can undergo substitutional and displacive modulations. Experiments on irradiation of different murataite modifications with heavy ions demonstrate that these modifications have close values of the radiation resistance, which is higher than that of the pyrochlore coexisting with them.

Structure simulation is performed for molybdophates of variable composition A1−x Zr2(PO4)3−x (MoO4)x, where A is Na (0≤x≤0.6), K (0≤x≤0.6), K (0≤x≤0.3), Rb (0≤x≤0.2), or Cs (0≤x≤0.1), using the minimization of the interatomic interaction energy; these molybdophosphates crystallize in the NaZr2(PO4)3 (NZP) structure type. The results of the computer-assisted structure simulation are verified by the synthesis of the molybdophosphates and their characterization by X-ray powder diffraction and IR spectroscopy. The crystallization field of the NZP molybdophosphate shrinks as the alkali cation size increases. The key factors that govern the stability of the NZP structure in alkali zirconium molybdophosphates are determined.

The crystal structure of a synthetic analog of the mineral lipscombite (Fe2.32+Fe4.73+)[PO4]4O2.7(OH)1.3 obtained under hydrothermal conditions in the LiF-Fe2O3-(NH4)2HPO4-H2O system is resolved (R = 0.040) by X-ray diffraction analysis (Bruker Smart diffractometer with a highly sensitive CCD detector, MoKα radiation): a = 14.776(3) A, b = 14.959(3) A, c = 7.394(1) A, β = 119.188(4)°, sp. gr. C2/c, Z = 4, ρexp = 3.8 g/cm3, ρcalcd = 3.9 g/cm3. Fe2+ and Fe3+ cations are statistically distributed in each of four crystallographically independent positions, while occupying the corresponding octahedra with probabilities of 60, 90, 100, and 91%. The ratio Fe2+/Fe3+ in the composition of the crystals was established by Mossbauer spectroscopy. Lipscombite is interpreted as a mineral of variable composition described by the formula (Fex2+Fen−x3+)[PO4]4Oy(OH)4−y. The field of stability is determined as a function of the iron content and the ratio Fe2+/Fe3+. It is shown that at n = 6 iron cations are ordered in octahedra and barbosalite structure is formed. An interpretation of genetically and structurally related members of the lipscombite family within a unified polysomatic series is proposed.