Tetrahedral structures with icosahedral order and their relation to quasicrystals

The possible existence of quasicrystals in tetrahedral phases is considered. It is shown that one of the well-known crystalline silicon phases (the BC8 phase or silicon III) is characterized by the icosahedral local order with three-quarters of the interatomic bonds being directed along the fivefold axes of an icosahedron. This crystal is considered as an approximant of an icosahedral quasicrystal. Higher order approximants and other tetrahedral structures related to quasicrystals are also constructed. It is shown that in these structures, the formation of the intrinsic phason disorder with the preservation of the energetically favorable coordination number four is possible. The ab initio quantum-mechanical calculations for carbon and silicon show that, although all the considered phases are metastable, their energies only slightly differ from the energies of the corresponding stable phases.     

Approximants of icosahedral quasicrystals: Atomic structure, inherent defects, and superstructural ordering

The structural features of approximants of icosahedral and decagonal quasicrystals and new unusual approximants (rhombohedral AlPd and cubic Al₆₈Pd₂₀Ru₁₂) are considered. It is shown that most approximants can be described in terms of universal local ordering of atoms, in which the nearest neighbors of each atom occupy the vertices of an almost ideal dodecahedron: the so-called dodecahedral local ordering. A set of general atomic motifs in the approximants of different orders is found for quasicrystals of different types. It is shown that the dodecahedral local ordering can be easily described by the project method, in which the basis vectors directed along icosahedral threefold axes are used. Different types of defects inherent in dodecahedral local ordering are analyzed.     

Gorsky-Bragg-Williams theory of phase transitions in the approximants of icosahedral quasicrystals

The Gorsky-Bragg-Williams (GBW) theory has been modified to describe phase transitions caused by the ordering of two kinds of atoms in the even and the odd sublattices of the structures of icosahedral quasicrystals with the dodecahedral local order (DLO) and their approximants. The modified method is exemplified by numerical calculations of such phase transitions in the structures of several simplest approximants.     

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.     

Experimental investigation of the atomic dynamics of amorphous and icosahedral alloys

Most recent results from the experimental investigation of the single particle dynamics and the collective atomic motions in amorphous metals and in icosahedral alloys using neutron inelastic scattering techniques are briefly presented. The extension of the dispersion curve of short wavelength collective motions into the first pseudo Brillouin zone by neutron Brillouin scattering and the investigation of crystal field excitations in an amorphous metal and its dependence on structural relaxation are discussed. Comparative studies of the atomic dynamics of polydomain icosahedral alloys, approximant polycrystals, and polycrystals with similar chemical composition but different structure are discussed. It is shown that the generalized vibrational density of states and the total dynamic structure factor of the icosahedral alloys investigated so far are rather different from any crystalline sample except for the case of higher order approximants with very large units cells.     

Generation of the structures of molecular crystals with two molecules related by the center of inversion in a primitive unit cell

An algorithm for the generation of possible crystal structures with two inversion-related molecules of known shape in a primitive unit cell is proposed within the method of discrete packing modeling in molecular crystals. The algorithm is based on the replacement of molecules by polycubes (geometric figures composed of identical cubes) and looking through a finite number of all the possible periodic packings of these polycubes with a given coefficient of packing. A program package for personal computers is developed on the basis of the proposed algorithm and is approved by the example of several crystal structures that were determined earlier by X-ray diffraction.     

Synthesis and characterization of single-crystalline PrCO3OH dodecahedral microrods and its thermal conversion to Pr6O11

Single-crystalline PrCO₃OH dodecahedral microrods with an orthorhombic structure have been successfully synthesized by the hydrothermal method used urea as the precipitator. Pr₆O₁₁ dodecahedral microrods have been obtained by thermal conversion of PrCO₃OH dodecahedral microrods at 600 °C in air for 6 h. The as-synthesized products were characterized by X-ray powder diffraction, field-emission scanning electron microscope, transmission electron microscopy, high-resolution transmission electron microscopy, selected-area electron diffraction, X-ray photoelectron spectra, fourier transform infrared spectroscopy and thermogravimetry–differential thermal analysis. The effect of the reaction parameters on the morphology of the product has been investigated. The dodecahedral microrods with larger size and better crystallinity can be obtained under the higher reaction temperature. The possible formation mechanism of PrCO₃OH microrods was discussed.     

Calculation of vibrational spectra of an icosahedral quasicrystal AlCuFe

Vibrational spectra of an icosahedral quasicrystal AlCuFe have been calculated on the basis of a crystalline 1/1 approximant by the recurrence method. To describe the interaction of atoms in a quasicrystal, the semiempirical EAM model was used. It is shown that the calculated spectra are in satisfactory agreement with the experimental neutron inelastic scattering data.     

On the structure of amorphous germanium and silicon

The simple microcrystallite model is used to calculate the diffraction and radial distribution functions for a variety of tetrahedrally coordinated crystal structures: diamond, wurtzite, Ge III and Si III — two high-pressure polytypes of Ge and Si — and two clathrate structures based on pentagonal dodecahedral units. Comparison with data for sputtered amorphous Ge suggests that the simple microcrystallite model is inadequate to fit diffraction data. A statistical or combination microcrystallite model may be more promising. Recent electron microscopic investigations of Rudee and Howie are also discussed.     

Sol–gel simulation—II: Mechanical response

A novel computational procedure is proposed to predict the outstanding mechanical properties of sol–gel structures. An aggregation algorithm incorporating Brownian motion and chemical reactions is used to recreate the sol–gel structures at molecular scale. Just like in the physical colloidal aggregation process, the computational aggregation process produces structures with fractal features. Such fractal character leads to a recursion algorithm for calculating mechanical properties at any scale using a recursive multiscale approach. The mechanical properties are then predicted at each scale by calculating the effective properties using the Finite Element Method. It is shown that Young's modulus naturally follows a power law relationship with density, whereas Poisson's ratio displays more complicated behavior. Also, it is shown that Young's modulus and Poisson's ratio depend on a) the mass distribution of the structure, which is influenced by the Brownian motion and chemical reactivity during the aggregation process, and b) the connectivity, which is also influenced by additional processes as sintering and/or aging. Finally, it is shown that the Young's modulus and Poisson's ratio can be correlated to scattering intensity of sintered and/or aged structures.     

Continuous periodic to quasiperiodic structural transformations in crystalline and superlattice structures

Attention is focussed on the fact that the β—Mn, the σ—phase and the Al₅Ti₃ and the Al₁₁Ti₇ superlattice derivatives of the δ—AlTi phase all belong to a category of structures which comprises an identical periodic arrangement of square and rhombic tiles — the shapes of the rhombic tiles being different in each of these cases. While the first and the second structures constitute periodic approximants of the octagonal and the dodecagonal quasicrystals respectively, the last two (superlattice) structures are the approximants of a hypothetical quasiperiodic superlattice structure. The strip projection method has been used here to model the intermediate structures which arise during the continuous periodic to quasiperiodic transitions between the relevant structures and was found to be quite successful in explaining many of the related experimental findings.     

Formation of structure in small lead clusters under thermal effect

The thermal effect on lead clusters with radii up to 5.5 nm has been investigated by the molecular dynamics method using a modified tight-binding potential TB-SMA. The melting of Pb nanoparticles of these sizes is strictly homogeneous, without the formation of a surface liquidlike layer. The primary fcc phase in the particles is retained upon heating in the overwhelming majority of model experiments. An analysis of the structure formation during crystallization has shown that structures with pentagonal symmetry are preferred for lead clusters in this case. It is noted that an increase in the nanoparticle size leads to the dominance of the dodecahedral structure over the icosahedral one.     

In situ diffraction studies of the phase selection in undercooled Ti–Fe–Si–O melts

A time-resolved in situ determination of the phase formation sequence in undercooled Ti–Fe–Si–O melts was performed using the combination of electromagnetic levitation and energy-dispersive X-ray diffraction of synchrotron radiation. Ti–Fe–Si–O alloys are known to form a large variety of stable and metastable phases, including quasicrystals and approximant phases. Different solidification pathways are found as a function of undercooling, with the primary nucleation of the 1/1 crystal approximant phase from the liquid occurring at the largest undercoolings. The results on phase selection are discussed with respect to the influence of the short-range order in the undercooled liquid on the nucleation behavior of solid phases.     

Symmetry old and new: Some recent examples of the role of symmetry in X-ray analysis

Two recent examples in X-ray structure analysis at opposite ends of the molecular weight range, in which very careful consideration of space group and diffraction symmetry was necessary to resolve all the associated problems, are described. Case 1 involves the structures of (1) transdiBromo (1,4,8,11-tetraazacyclotetradecane) chromium (III) bromide and (2) the corresponding trans-bromo-chloro compound. Both crystallise in space groupP4₂/m with unit cell dimensions sufficiently close to suggest that the structures are isomorphous. It is shown however that the structures are not superimposable or even related by simple rotation or inversion, but that it is necessary to apply a hypothetical glide operation in order to bring the structures into coincidence. The organic moieties are thus structural enantiomorphs in spite of existing in a centrosymmetric space group. Case 2 involves studies on a co-crystallised derivative of the enzyme RNase with deoxycytidylyl-3–5-deoxyguanosine (dCpdG). Preliminary X-ray precession photographs, from low quality crystals, indicated an orthogonal C-centred unit cell, but were unable to define the true diffraction symmetry, which wasnot orthorhombic. The true symmetry was discovered only after solving the structure in a second (erroneous) cell and careful inspection of the lattice and diffraction symmetry from the measured intensity values.     

The short-range order in ternary ZrF4---BaF2---ErF3 glasses

The short-range order structures in ternary ZrF₄---BaF₂---ErF₃ glasses have been studied by X-ray absorption fine structure (XAFS). The near-neighbor structures around Zr, Ba and Er have been determined and compared with that in binary ZrF₄---BaF₂ glasses. The near-neighbor structures around Zr and Ba are scarcely affected by doping with ErF₃. The structural parameters of near neighbors around Er in the glasses are near to that in ErF₃, but the corresponding distance and coordination number of near neighbors in the glasses decrease slightly. It is also observed that the intensities of the white lines at the Er L and L absorption edges in these glasses are different from that in c-ErF₃. Meanwhile, compared with that of the corresponding fluoride, the changes of the ‘white line’ at the Er L_,-edges in the glasses differ from the case of Nd. The causes leading to these differences are discussed.     

New approach to simulation of radiative-conductive heat transfer in semi-transparent crystals being pulled from a melt

A method is suggested for the solution of multi-dimensional radiative heat transfer problems arising in growing crystals. The basic idea of the approach lies in the construction of a special division of the unit sphere into a set of solid angles (cells) and in the approximation of the radiation intensity in each solid angle by the P₁ approximation. The radiant transport equation is satisfied in the mean over each elementary cell and the system of partial differential equations of the second order relative to the local zeroth-moments of radiation intensity are obtained. It is shown that the solid angle subdivision can be carried out in different ways with respect to specific features of the heat transfer problem under consideration. As a result even a very rough partition permits satisfactory accuracy of the numerical solutions. One of the main advantages of the method consists in using solid angle subdivisions which can be varied from point to point of the spatial domain. The latter gave possibility to simulate the radiative heat transfer in a circular cylinder of finite length with specular side surface. On this basis the calculation of the temperature field in a cylindrical sapphire crystal being pulled from the melt has been carried out without any restriction on the size of the crystal     

Block Lattices with Asymmetric Interaction Forces

A series of 2D block lattices constructed by identical building elements and having asymmetric interaction forces is proposed as a generalisation of the well‐known Kossel crystal. Different polymorph structures are created depending on the mutual orientation of the neighbour elements. The lattice energy for some of them is calculated at low temperatures using the repeatable step of blocks (monomers, dimers, quadrumers). A variety of “surface” structures (including alternating “surfaces”) is established. The corresponding coexisting equilibrium forms are determined following the method of Stranski and Kaischew. The thermodynamic stability of these polymorph structures is discussed. An orientation order/disorder surface transition is expected to appear as a novel phenomenon at high temperatures.     

Stable Al-Cu-Co decagonal quasicrystals

Stable Al₆₅Cu₂₀Co₁₅ and Al₆₅Cu₁₅Co₂₀ decagonal quasicrystals, sometimes using a few percent of Si to replace Al, can be grown directly from the melt as single grain decaprisms of up to 1 cm in length. Some salient features of their solidification processes, growth morphologies, surface structures, high resolution electron microscopic images, two dimensional quasicrystalline structure, relationship to one-dimensional quasicrystals, crystalline approximants, and dislocations are presented.     

Structural modelling of icosahedral and simple cubic approximant phase in Al-Mn(Fe)-Si system

In spite of various models being available for the icosahedral quasicrystal structure in Al-Mn(Fe)-Si system, a consistent model for describing the simple cubic approximant and icosahahedral structure in terms of rational and irrational sections of the same six dimensional cubic crystal seems to be lacking. Such a model is presented and the subtle difference between the present model and the previous ones is discussed. The necessity of such considerations is spelt out while accomplishing the partitioning of the tricontahedral motif (with extension in three dimensional pseudo direct space) based on the chemistry of the alloy. An algorithm for generating the required polyhedra for partitioning is also presented.     

Bifurcation analysis of the thermocapillary convection in cylindrical liquid bridges

We consider the instability of the steady, axisymmetric thermocapillary convection in cylindrical liquid bridges. Finite-difference method is applied to compute the steady axisymmetric basic solutions, and to examine their linear instability to three-dimensional modal perturbations. The numerical results show that for liquid bridges of O(1) aspect ratio Γ (= length/radius) the first instability of the basic state is through either a regular bifurcation (stationary) or Hopf bifurcation (oscillatory), depending on the Prandtl number of the liquid. The bifurcation points and the corresponding eigenfunctions are predicted precisely by solving appropriate extended systems of equations. For very small Prandtl numbers, i.e. Pr < 0.06, the instability is of hydrodynamical origin that breaks the azimuthal symmetry of the basic state. The critical Reynolds number, for unit aspect ratio and insulated free surface, tends to be constant, Re_c → 1784, as Pr → 0, the most dangerous mode being m = 2. While for Pr 0.1, the instability takes the form of a pair of hydrothermal waves traveling azimuthally. The most dangerous mode is m = 3 for 0.1 Pr 0.8 and m = 2 for Pr 0.9. Dependence of the critical Reynolds number on other parameters is also presented. Our results confirm in large part the recent linear-theory results of Wanschura et al. [7] and provide a more complete stability diagram for the finite half-zone with a nondeformable free surface.