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.     

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.     

Hydrodynamics of layered and cubic quasicrystals

The dissipative processes associated with sound absorption in layered and cubic quasicrystals have been studied within the framework of the hydrodynamic approach. The decagonal, octagonal, and dodecagonal quasicrystals are considered as layered quasicrystals. Proceeding from the general theory, the complex law of dispersion is calculated for various directions of the propagation and polarization of sonic waves in layered and cubic quasicrystals. The effect of the phason modes on the process of sound absorption and the role of symmetry constraints are analyzed. The validity of the macroscopic consideration is discussed in terms of the theory of the generalized color symmetry.     

Stability, atomic and electronic structure of icosahedral quasicrystals

Ab-initio investigations of the structure, stability, and of the electronic spectrum of isosahedral quasicrystals are presented.     

Partial spectra of atomic thermal vibrations in decagonal and icosahedral quasicrystals

The atomic dynamics of an Al-Ni-Fe decagonal quasicrystal and an Al-Cu-Fe icosahedral quasicrystal are investigated experimentally using the isotopic contrast method in inelastic neutron scattering. The partial spectra of thermal vibrations of copper, nickel, iron, and aluminum atoms in the decagonal and icosahedral quasicrystals are reconstructed directly from the experimental data without recourse to model concepts. The limiting energies and positions of the main features in the partial spectra of atomic thermal vibrations in decagonal and icosahedral quasicrystals are determined. It is established that, in the quasicrystals under investigation, the copper and nickel atoms are bound more weakly than the iron atoms and that the partial vibrational spectrum of aluminum atoms in the quasicrystals is considerably harder than the spectrum of pure metallic aluminum.     

Formation of quasicrystals and metallic glasses in relation to icosahedral clusters

It has been widely accepted that quasicrystals and at least some metallic glasses are built up with icosahedral clusters. Information about the cluster structures can be obtained from crystalline counterparts. In this paper, we will describe the formation rules of bulk metallic glasses, originally developed for quasicrystals, by combining cluster structures with phase diagram features. We will introduce the e/a-constant and e/a-variant criteria for ternary systems, and e/a-constant and atomic size constant criteria for quaternary systems. We will show how the glass forming composition optimization is realized by applying these rules in the Zr–Al–Ni and Zr–Al–Ni–Cu systems. In both systems the optimized glass-forming composition is related to a common binary icosahedral cluster Zr₉Ni₄ derived from the fcc Zr₂Ni phase. A novel route to reach amorphous forming composition is also attempted by mixing Al-based quasicrystal-forming compositions with Zr.     

Formation of icosahedral quasicrystals in (Ag,Au)-based ternary systems

We investigated the phase constitution of rapidly solidified (Ag,Au)-based ternary alloys at various compositions of (M,X)_85.7RE_14.3 (M = Ag, Au, X = Al,Ga,In, RE = Er,Lu). It is found that an icosahedral phase is formed at an average electrons per atom ratio close to 2.0 as well as at the composition ratio of (M,X)₆RE, suggesting that these factors have crucial roles in the stability of the icosahedral phases. It is also found that the trend of the icosahedral phase formation in the Au-based alloys is quite different from that in the Ag-based alloys.     

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.     

Intensity and width of bragg reflections from imperfect icosahedral quasicrystals obtained by simulating atomic growth

The Bragg reflections from icosahedral quasicrystals obtained in the course of a computer experiment on the simulation of crystal growth have been studied. The computer experiments based on the theory developed earlier allow one to “grow” in a computer the imperfect quasicrystals of nanometer dimensions. It is shown that the absolute value of the structure factor can be close to the maximum possible one for crystals, i.e., to the structure factor in the case where all the atoms scatter in phase. The spectral width of Bragg reflections is studied, and it is shown that the reflection width depends not only on the quasicrystal dimensions in the physical space but, also, on the perpendicular component of the reciprocal-lattice vectors. The data obtained are compared with the known experimental data.     

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.     

Calculations of the percolation thresholds of a three-dimensional (icosahedral) Penrose tiling by the cubic approximant method

The percolation thresholds of a three-dimensional Penrose tiling with icosahedral symmetry are determined using the cubic approximant method. The percolation thresholds of the three-dimensional Penrose tiling for the site problem and the bond problem are calculated with an accuracy of 0.001.     

Shapes of quasicrystals

In the present article, we review the present status of our understanding of the shapes of quasicrystals. Although theoretical efforts exist to determine the equilibrium shape of quasicrystals, very little experimental information exists. We report some preliminary results of our work in this direction. On the other hand, beautiful growth shapes of quasicrystals are reported in the literature. However, only recently theoretical attention is paid to the problem of atomistic mechanism of growth. In spite of limited studies, comparison of theory and experiments reveal a difference in growth behaviour between crystal and quasicrystal.     

EXAFS spectroscopy of quasicrystals

The results of the investigation of the features of the local structure of quasicrystalline materials by extended X-ray absorption fine structure (EXAFS) spectroscopy with the use of synchrotron radiation are analyzed. The advantages of this method from the point of view of deriving information about the local shifts of the atoms forming an icosahedral structure are demonstrated. The rearrangement of the local environment of copper and iron in Al-Fe-Cu ternary alloys at a transition from the crystalline to the quasicrystalline phase has been investigated. It is established that the nearest copper coordination retains the symmetry characteristic of the crystal; however, rotation and small displacements of copper matrix atoms lead to significant rearrangement of aluminum atoms around iron atoms. As a result, icosahedral clusters with pentagonal symmetry are formed around iron atoms and violation of the translational symmetry is accompanied by the transition of Al-Fe-Cu to the quasicrystalline state.     

Twinning of quasicrystals and related crystals

Electron microscopy and electron diffraction have been employed to bring out the intimate relationship between icosahedral quasicrystal and various types of crystalline phases with emphasis on aluminium transition metal alloys. The relationship between quasicrystal and twinned rational approximant structures will be treated in the context of irrational twinning. The twinned aggregates show an icosahedral like symmetry in electron diffraction patterns. The crystallography of irrational twins has been discussed in terms of various ways of representing a twin relationship in a cubic lattice.     

Magnetic properties of Al-based icosahedral alloys

One of the fundamental questions which has not been answered sufficiently well till today is whether a nonclassical symmetry such as an icosahedral one would induce any special and distinctive features in the physical properties of the systems. Most theoretical investigations indicate that there are unique features in the electronic structure on account of icosahedral symmetry which could for example, influence the magnetic properties significantly. Experimental studies, particularly, on Al-based icosahedral alloys, have shown a large variation in magnetic properties ranging from diamagnetic to Curie like and Pauli paramagnetism on one hand and ferromagnetic and spin glass like behaviour on the other. In the present article, we will concentrate on some aspects of moment formation and variation in magnetic properties in Al-Mn based icosahedral systems and their dependence on the method of preparation which presumably controls and introduces the variable disorder. We will also try to answer some pertinent questions related to magnetism in icosahedral systems such as extent and origin of moment formation existence of magnetic and nonmagnetic Mn sites and distribution of interatomic distances reponsible for variation in magnetic properties.     

Raman study of lattice dynamics in quasicrystals

We present the first Raman investigation of icosahedral quasicrystals. Broad structured bands in the energy range up to ~500 cm^− 1 have been observed in a series of AlCuFe, AlPdMn and AlPdRe systems. Original information on the vibrational density of states g(ω) was obtained for AlPdRe; for AlCuFe and AlPdMn estimated g(ω) shows a good agreement with the previous neutron results, but demonstrates finer structure. Strong increase in the parameter of electron-vibrational coupling for the low-energy vibrations and its correlation with changes in electronic conductivity has been observed in the series from AlCuFe to AlPdRe. This suggests the increase of the degree of localization for these vibrational excitations and involved electronic states.     

Computer-aided evaluation of kossel patterns obtained from quasicrystals

For the evaluation of Kossel experiments on quasicrystals such as the decagonal phase of Al Co Ni or Al Co Cu Si a computer program called KOQUA has been written, which allows to simulate Kossel patterns of two- and three-dimensional quasicrystals as well as of every conventional crystal system. In some examples the application of the program is demonstrated.     

An interpretation of three intersections of cubic and rhombohedral structures based on the conservative nature of force skeletons

A crystallographic analysis of the structures of inorganic compounds has shown the uniformity of ??force skeletons??: pseudotranslational sublattices of atoms determining the structure of different classes of materials. At three specific c _hex/a _hex ratios, structures with a rhombohedral cell can be considered in the pseudocubic representation with P, I, and F cells (c _hex/a _hex is, respectively, 1.225, 0.612, and 2.45). The large fraction of structures with c _hex/a _hex ratios close to these values indicates the stability of these atomic constructions. Force skeletons with a typical hexagonal subcell (different for cations and anions), which reflects the relationship of these structures with the high-symmetry structural types of sphalerite (ZnS) and galena (PbS), have been established for the representative group of sulfides structures with a cell parameter ??4 ? in the direction perpendicular to the mirror plane.     

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.