Error estimates in the determination of rigid body displacements in protein crystals

We report the estimation of random errors in the refinement of the rigid body displacements of the -helices of the enzyme 6-phosphogluconate dehydrogenase. Least-squares refinement of the TLS parameters of the helices has been carried out using X-ray reflection data of 2.1 Å resolution, resulting in anR-factor of 19.5%. Standard deviations were estimated from the normal matrix. The results show that the translational mean-square displacements of nearly all the helices are significant at this resolution. However the libration parameters are only significant when the helices have at least four turns. Screw-rotation tensor values cannot be determined at this resolution.     

Structure of conduction channels in superionic conductors: Crystallochemical modeling

The experimental and theoretical studies of the structure of conduction channels of superionic conductors are analyzed. A crystallochemical method of modeling the conduction channels is proposed in which they are treated as sets of polyhedron Vorono-Dirichlet sections outside of impenetrable spheres of the rigid sublattice. The allowance for both weak flexibility of the spheres and their mean-square displacements relative to the sublattice points yields a channel structure that “wraps” around the Vorono-Dirichlet polyhedron edges. Modeling the interionic potential for α-AgI with the help of equipotential surfaces suggests that the mobile-ion motion slips along the conduction channel walls. For α-AgI, the set of the equipotential surfaces specifies “the minimum energy trajectory” of transition into a superionic state, while the crystallochemical “wrapping” structure corresponds to a saddle point of a multidimensional potential surface. Symmetry selection rules are used for predicting mechanical trajectories as allowed oscillation modes for the tetrahedral and octahedral fragments of α-CuI.     

Theoretical and experimental study of excess quantities in glass-forming systems

The present work deals with the correlation among a macroscopic quantity, i.e. kinematic viscosity, and some microscopic quantities, namely the mean-square displacement and the free volume. In primis, some definitions of ‘fragility’, a quantity which allows to rank different liquids on the basis of the temperature dependence of kinematic and/or thermodynamic quantities, are recalled. Successively, the temperature dependence of viscosity, mean-square displacement and free volume is taken into account for three glass-forming systems and is analyzed in order to highlight the linear relationships between the logarithm of viscosity and the excess mean-square displacement and between the logarithm of viscosity and the excess free volume. Finally, on the basis of the observed correlations, two ‘fragility’ parameters, operatively evaluated by elastic neutron scattering and positron annihilation lifetime spectroscopy, are discussed and compared.     

Study of the microdynamics of liquid lithium and lithium-hydrogen melt by inelastic neutron scattering

The frequency spectra of vibrations of Li atoms at temperatures of 22, 227, 397, and 557°C and the lithium-hydrogen melt (98 at % ⁷Li, 2 at % H) at 557°C have been obtained from the experimental neutron inelastic scattering data. On the basis of the frequency spectra, the temperature dependences of the mean-square displacement of Li atoms, the mean-square amplitudes of atomic vibrations, and the velocity autocorrelation function of atoms have been calculated. The speed of sound in liquid lithium has been estimated within the Debye model. The frequency spectra of lithium-hydrogen melt and solid lithium hydride are compared. A generalized frequency spectrum of vibrations of hydrogen atoms in lithium-hydrogen melt is obtained.     

Theory of fluidity of liquids, glass transition, and melting

This is a presentation of a rigorous theory of fluidity of liquids, glass transition and melting of solids in the frame of an asymmetric double well potential model. Potential wells are doubled time to time by the local density fluctuations caused by the thermal longitudinal waves. The average frequency of doubling of potential wells is equal to the frequency of the most energetic waves which obey a law similar to Wein’s displacement law in black body radiation. Based on the equilibrium thermodynamic theory of fluctuations and the displacement law, a law of linear pre-diffusion mean-square displacement of particles in a solid is derived: the mean-square displacement of molecules within their potential wells increases linearly with temperature. It is shown that when this is broken-down (where the mean-square displacement at a certain temperature rapidly changes its slope as a function of temperature) glass devitrifies and crystal melts, and all possible solid-liquid transitions of a substance occur at the same critical mean-square displacement: any solid (not only crystals) transforms into liquid when the mean-square displacement, as a fraction of the average intermolecular distance, acquires a certain universal critical value - the same for different substances. It is proved that molecules in a liquid perform specific Brownian motion. The average jump distance is a function of temperature and it is much smaller than the nearest intermolecular distances. At a certain temperature, shown to be the Kauzmann temperature, the average jump distance of Brownian motion becomes equal to zero: the super-cooled liquid undergoes glass transition. The transition was proven to be a phase transition of the fourth order: the free energy of the system and its first, second and third derivatives are all continuous functions, but its fourth derivative with respect to temperature is discontinuous. Molecular mobility, diffusion and viscosity are obtained as functions of temperature.     

Magnéli phases in Ti-containing oxides and their solid solutions

The ordered planes of crystallographic shear inherited from titanium dioxide TiO₂ were revealed in complex Ti-containing oxides (Magnéli phases) by powder X-ray diffraction. It is assumed that periodic shear in the structure results in the modulation of titanium and oxygen displacements along the direction normal to the shear planes. A method of calculating the amplitude of the displacements of titanium atoms from the centers of oxygen octahedra is suggested. The method provides good agreement of the calculated and experimental data for single crystals. The degree of A-nonstoichiometry in alkaline earth, cadmium, and lead titanates is evaluated as 2–3%.     

Parameters of anisotropic atomic displacements in chemical compounds as criteria for chemical bond types

Analysis of the parameters of anisotropic atomic displacements in various chemical compounds reveals a feature characteristic of a donor-acceptor bond: the ellipsoids of anisotropic displacements of the atoms involved in this bond are elongated in the direction of the bond. The explanation of the effect observed is proposed.     

Displacements in the cationic motif of nonstoichiometric fluorite phases Ba1 − x R x F2 + x as a result of the formation of {Ba8[R 6F68–69]} clusters: I. The Ba0.78Tm0.22F2.22 crystals

The displacements of Ba²⁺ cations in the cationic motif of Ba_0.78Tm_0.22F_2.22 crystals, which are representatives of nonstoichiometric fluorite phases Ba_{1 ? x} R _x F_{2 + x} , are proved for the first time with the use of precision investigations of the fine atomic structure. It is shown that the cation displacements, like the previously revealed displacements in the anionic motif, reflect the formation of {Ba₈[R ₆F_68–69]} superclusters of structural defects with nanometer linear sizes. The Ba ²⁺ cations are displaced from the fluorite crystallographic positions 4a (space group Fm $ \bar 3 The displacements of Ba²⁺ cations in the cationic motif of Ba_0.78Tm_0.22F_2.22 crystals, which are representatives of nonstoichiometric fluorite phases Ba_{1 − x} R _x F_{2 + x} , are proved for the first time with the use of precision investigations of the fine atomic structure. It is shown that the cation displacements, like the previously revealed displacements in the anionic motif, reflect the formation of {Ba₈[R ₆F_68–69]} superclusters of structural defects with nanometer linear sizes. The Ba ²⁺ cations are displaced from the fluorite crystallographic positions 4a (space group Fm m) to the positions 32f. The static nature of the cation displacements is confirmed by the fact that these displacements are retained at a temperature of 110 K. The correctness of the interpretation of the correlation between the cation displacements and the formation of superclusters of structural defects is supported by the coincidence of the intercationic distances determined in the disordered phase Ba_0.78Tm_0.22F_2.22 with those found in the previously studied ordered phases Ba₄ R ₃F₁₇ (R = Y, Yb). The model with splitting of cationic positions is appropriate for testing in structural investigations of crystals of the fluorite phases M _{1 − x} R _x F_{2 + x} at room temperature. Original Russian Text ? A.M. Golubev, L.P. Otroshchenko, V.N. Molchanov, B.P. Sobolev, 2008, published in Kristallografiya, 2008, Vol. 53, No. 6, pp. 1023–1030.     

Effect of the inhomogeneous structure of freely suspended smectic-A films on the dynamics of fluctuations of smectic-layer displacements

The dynamics of fluctuations of smectic-layer displacements in freely suspended smectic-A films is studied theoretically with due regard for the dependence of the transverse-bending moduli K and extension (or compression) of smectic-B layers on the distance to the free surfaces of the films. The temperature dependences of the dynamic correlation functions are calculated for fluctuations of layer displacements in films of various thicknesses and then used to calculate the correlations between the X-ray intensities scattered by the film at different moments of time. The calculations were performed within a wide temperature range. It is shown that the influence of temperature on the X-ray dynamics of fluctuations of layer displacements in freely suspended smectic-A films can be detected in the experiments on the dynamical scattering by rather thick films (layer number N ≥ 100) at considerably high values of the recoil-momentum component in the film plane (≥10⁶ cm^?1).     

Magnetic state, elastic constants, and longwave static displacements of atoms in γ-FeNi alloys

The experimental results on the concentration and temperature behavior of longwave (both transverse and longitudinal) static displacements of atoms, as well as the data on the elastic constants (derived from neutron and ultrasonic measurements) of γ-FeNi alloys, are reported. The relationship between the alloy magnetic state and the above-mentioned parameters is discussed and these parameters are classified with respect to the premartensitic phenomena and the Invar effect.     

Distortions of the atomic structure of 2/1 approximants of icosahedral quasicrystals

Ideal structures of several 2/1 approximants of icosahedral quasicrystals with the sp. gr. Pm \(\overline 3 \) and Pa \(\overline 3 \) have been constructed. It is shown that most atoms have dodecadhedral local coordination. The structural features of the crystals studied have been analyzed. It is found that displacements of atoms from ideal positions in real crystals tend to increase with an increase in the distance to the center of the approximant projection on the perpendicular space. Large displacements are generally related to the presence of very close neighbors in the ideal structure of the approximant. The importance of the investigation of approximants as a source of information about the structure of polyatomic nanoclusters is indicated.     

Spannungsfelder um Einzelversetzungen in anisotropen Medien

An approximative solution is given for the system of differential-equations of the displacements around a single dislocation in an anisotropic media. The conditions are explained, limiting the accuracy of the approximation.     

Atomic structure and enormous anisotropy of thermal expansion in NiSi single crystals. I. Refinement of structure models

The atomic structure of NiSi single crystals has been studied by the methods of X-ray diffraction analysis at 295 and 418 K. The refinement of the structure models with due regard for anharmonicity of atomic vibrations revealed the pronounced anisotropy of thermal vibrations and the considerable contribution of the fourth-order anharmonicity. The maps of the probability density function of atomic displacements and single-particle potentials of both atoms were constructed.     

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.     

Defect structure of TiS<Subscript>3</Subscript> single crystals of the <Emphasis Type="Italic">A</Emphasis>-ZrSe<Subscript>3</Subscript> type

The defect structure of TiS₃ single crystals of the A-ZrSe₃ type has been determined based on X-ray diffraction data. Shear defects manifest themselves as displacements of ab layers (which can imitate a twin) by ～0.5a. Regular shears facilitate the formation of a superstructure along the c axis. A model of defect in the layer structure is proposed to explain the atomic displacements at an angle to the layer plane.     

Comparison of amorphous and liquid palladium-silicon alloys

The Pd_82,5Si_17,5 alloy has been investigated by X-ray diffraction in an amorphous state obtained by rapid quenching from the liquid state and also in the liquid state at a temperature just above T_f. The differences seen in the interference functions obtained relative to both states are not only due to a temperature effect. From the two atomic radial distribution functions it was possible to evaluate the Pd–Pd distances and the Pd–Si distances as well as the number of nearest neighbours. These distances have the same values, but the displacements of the atoms are more important in the liquid. The contribution of Pd–Pd bonds, for the metal-metal distances equal to 1.65 times the particle diameter, decreases in the liquid.     

Estimation of inter-atomic force constants and phonon dispersion using correlation effects among thermal displacement of atoms in Ge

Neutron diffraction measurements have been performed on powder Ge at 6, 150 and 300 K. Oscillatory diffuse scattering intensity is clearly observed at 150 and 300 K. The diffuse scattering theory including correlation effects among thermal displacements of atoms is applied to background function in the Rietveld analysis. The inter-atomic distance and temperature dependence of the values of correlation effects are discussed. The force constants among the first, second and third nearest neighboring atoms in Ge are obtained from the values of correlation effects and Debye-Waller temperature parameters. A phonon dispersion spectrum in Λ [111] direction for Ge crystal has also been calculated at 80 K using inter-atomic force constants and the crystal structure.     

X-Ray diffraction study of KTiOPO<Subscript>4</Subscript> single crystals doped with hafnium

Single crystals of KTi_{1 − x} Hf _x OPO₄ (x = 0.015(2), 0.035(1), and 0.128(1) are reinvestigated by precision X-ray diffraction at room temperature. It is found that the implantation of hafnium atoms in the crystal structure of KTiOPO₄ does not lead to significant changes in the framework and affects only the positions of the potassium atoms in the channel. Our studies reveal the displacements of the potassium atoms from their main and additional positions in the structure of pure KTP in all three structures studied. The largest displacements from the K1′ and K1″ additional positions are observed in the structure with x = 0.035. At this hafnium concentration, the occupancy of the main positions of potassium atoms decreases and the occupancy of the additional positions increases in relation to those in KTP. This redistribution of potassium atoms enhances the nonuniformity of distribution of the electron density in the vicinity of their positions, which is probably responsible for the increase in the nonlinear susceptibility of KTP crystals that contain 3.5% hafnium in relation to crystals of pure KTP.     

Calculation of the atomic scattering factors of fast electrons at zero angle of incidence of an electron beam and mean inner crystal potentials

The atomic scattering factors of fast electrons at zero angle of incidence of an electron beam, f _el(0), are calculated for neutral atoms with Z = 1?54, using the mean-square radii of electron-density distributions in atoms and ions, derived on the basis of the Hartree-Fock wave functions. The values of f _el(0) have been calculated for the first time for some positive and negative ions. The obtained values of f _el(0) were used to determine the mean inner potentials V₀ of metallic, covalent, and ionic crystals. The calculated values of V₀ are compared with the experimental data in the literature. The values of V₀ calculated for III–V and II–VI compounds are reported.