Order parameter in the quasicrystal-crystal transformation

Some aspects of the relationship between quasicrystal and crystal structures not considered earlier are discussed. It is shown that the atomic positions in a quasicrystal lattice are intermediate between the positions of periodic incommensurate crystal lattice. For the octagonal symmetry, two such generating lattices can be square and equal to one another. Then any quasicrystal position lies exactly between two crystal positions in different lattices. The classical (phason) mechanism of the quasicrystal-crystal phase transition is interpreted as a relative homogeneous deformation of generating crystal lattices leading to the commensurability of these lattices. The order parameter of such a transition is defined.     

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.     

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.     

Al-Cu-Fe quasicrystals: Stability and microstructure

AlCuFe quasicrystal has been considered a novel icosahedral quasicrystal in as much as it can be produced in both thermodynamically stable and metastable state. It is also recognised to be a face centred structure. The system has been studied extensively in the recent past and several contentious results have been obtained. For example, evidences have been presented for both reversible and irreversible transformation of the quasicrystalline phase to the crystalline phases. Similarly, distinct transition metal sites have been proposed based on the Mossbauer studies. A subsequent study of this in fact shows an absence of such distinct sites. It has been shown that in this system, vacancy type of defects of very large concentrations exist. Many of these factors bring into question, the various structural models that have been postulated to explain the icosahedral structure obtained in the system. A critical examination of these various aspects of the AlCuFe quasicrystal is presented.     

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.     

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.     

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.     

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.     

A single-crystal X-ray diffraction study of an Al70.5Pd21Mn8.5 icosahedral quasicrystal

X-ray diffraction data were measured on a single grain of a stable Al_70.5Pd₂₁Mn_8.5 perfect quasicrystal, in order to complement the neutron diffraction data obtained previously. The F-type lattice structure of the six-dimensional periodic arrangement was confirmed along with the very strong chemical order, due basically to an uneven distribution of the Pd atoms among the lattice sites. This distribution might be the origin of the high stability of the icosahedral structure in this system.     

Microstructures of Mg–Zn–Nd alloy including small quasicrystalline grains

Two kinds of Mg-rich and low neodymium Mg–Zn–Nd alloys including icosahedral quasicrystal phase (I-phase) were prepared under conventional casting conditions. The microstructures and phases of Mg–Zn–Nd quasicrystal alloys were investigated by means of X-ray diffraction (XRD), scanning electron microscopy (SEM), energy dispersive spectroscopy (EDS), and transmission electron microscopy (TEM). The results indicate that the I-phase in Mg–Zn–Nd quasicrystal alloy is a simple icosahedral quasicrystal with quasi-lattice of a_R = 0.525 nm. It has been proved that the as-cast Mg_70.8Zn₂₈Nd_1.2 quasicrystal alloy mainly consisted of I-phase and Mg₇Zn₃ matrix phase. While the as-cast Mg_70.5Zn_28.5Nd₁ alloy mainly consisted of I-phase, Mg₇Zn₃ matrix phase, dendrite α-Mg phase and a new rod-like hexagonal phase.     

Interatomic force interaction in an i-AlCuFe quasicrystal

Partial spectra of thermal vibrations of Al, Cu, and Fe atoms in an icosahedral quasicrystal have been obtained by the isotopic-contrast method in inelastic neutron scattering. Joint analysis of these results and the published data on the atomic and electronic structures of the icosahedral i-AlCuFe quasicrystal has been performed. A physical model of the quasicrystal structure is proposed that is in agreement with the existing experimental data and qualitatively describes the peculiarities of interatomic interaction.     

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.     

Plane decagonal quasicrystals with 3-coordinated atoms

A plane decagonal tiling with 3-connected nodes, which is obtained by decorating the Penrose tiling, is proposed as the structure of a two-dimensional covalent quasicrystal. The mean energy of the atoms in the given structure, calculated using the model Tersoff potential for carbon, is found to be close to the energy of the atoms in fullerene. An octagonal tiling containing 3-connected nodes and even cycles only is also proposed.     

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.     

Icosahedral symmetry in clusters

First principles calculations and simulations based on interatomic potentials together with experimental studies of abundance spectrum suggest icosahedral structures to be common for some magic clusters of diverse systems such as rare gases, metals, covalently bonded systems and water. Close packing models obtained from pair potentials are shown to be good representations of the structure of rare gas clusters. However, the electronic structure is found to play the important role in the atomic structure and related properties of other clusters. Results of recent studies of fullerenes, their derivatives as well as some large icosahedral metal clusters containing several thousand atoms are also presented. Further results on doped icosahedral clusters are discussed which hold promise for the development of new materials and for understanding the occurrence of icosahedral order in several aluminum alloys.     

A case for local icosahedral order in undercooled metallic liquids and the influence on the nucleation barrier

An overview of recent experimental and theoretical investigations on undercooled liquids is given, indicating developing short-range order with undercooling. Our in situ synchrotron X-ray diffraction studies of electrostatically levitated droplets of elemental Ti, Zr, and Ni liquids and coordinated undercooling and structural studies of a Ti–Zr–Ni liquid that forms an icosahedral quasicrystal are discussed in the context of these investigations. Icosahedral order is demonstrated to be a dominant component to the structures of all undercooled liquids studied, although it becomes significantly distorted with an increasing importance of the d-band bonding character. The liquid short-range order couples with the nucleation barrier, as predicted by Frank over a half-century ago.     

Chemically driven structural change in quasicrystalline Al---Pd---Mn alloys

Quasicrystals, including face-centered icosahedral (FCI), simple icosahedral (SI) and decagonal (D) phases with periodicities of 1.6 nm and 1.2 nm along periodic axis, are found to form over a wide composition range in the Al---Pd---Mn system. A very strong composition dependence of the chemical order is seen in icosahedral Al_90−xPd_xMn₁₀ alloys alloys. A discontinuous change in the lattice parameter, a_R, with the Pd content of the i-phases suggests that the replacement of Al by Pd is selective. The predominant occupation site for the Pd atoms is icosahedral structure varies between the FCI and SI lattices. It is proposed that the FCI phase can be constructed as a two-cell structure, originating from the Al---Pd and Al---Mn binary systems.     

Search for possible structures of cubic approximants of icosahedral quasicrystals

Admissible structures of the approximants are obtained within the model of the dodecahedral local order (DLO) with due regard for the constraints imposed by the space symmetry. It is shown that the number of cubic approximants of icosahedral quasicrystals having a certain order and dodecahedral local order is finite and that the number of positions that can be occupied by the atoms from the unit cells of an approximant of each order is also finite. The corresponding estimates from above are made. In particular, it is shown that there exists only one structure for the 1/?1 approximant. It is also shown that it is possible to determine all the structures for any approximant of any order. The corresponding algorithm of the exhaustive search for these structures is suggested. The implementation of this algorithm provided the determination of all the structures of the 0/1 approximants and also some possible structures of the 0/1 and 1/1 approximants. The tables of possible approximant structures can be useful in the studies of new phases having approximant structures.     

Modeling of decagonal quasicrystal lattice

The model of the reciprocal lattice of decagonal quasicrystal has been proposed. It is based on the multiplication of the basis sites' groups according to the proposed rules. This model was shown to provide the information not only on the reciprocal sites location, but also on the intensity of the appropriated diffraction reflections. The physical justification of the proposed model correctness has been provided on the base of projection of six-dimensional hyperlattice onto the three-dimensional space. The possibility of three-indices labeling of the diffraction reflections of decagonal quasicrystals has been exhibited.