Three-dimensional Penrose Lattice Research Articles

Electronic spectrum of the three-dimensional Penrose lattice

The electronic spectrum of the three-dimensional Penrose lattice with “central” decoration by atoms is investigated using the tight binding model with nearest-neighbor interaction. Inverse participation ratios, higher moments of density probabilities, and fractal dimensions of the system are determined. The WFs are critical (they have a power-law dependence on the distance) at all energies in the band and are multifractal measures leading to the entire spectrum of the exponents. The results show that the system is in the critical state of the metal-insulator transition. On critical WFs, the cubic root temperature dependence of the conductivity is obtained.

Arrangement of atomic clusters in a 2/1 cubic approximant in the Al-Zn-Mg alloy system

An arrangement of the so-called Bergman clusters of atoms in a 2/1 cubic crystalline approximant phase, which is closely related to Frank-Kasper-type icosahedral quasicrystals, in an Al-Zn-Mg alloy system is discussed in detail. The 2/1 cubic structure has eight Bergman clusters in its unit cell and the positions of the clusters can be understood as twelvefold sites in the 2/1 cubic approximant of the three-dimensional Penrose lattice. The atomic clusters are located at vertices of two types of rhombohedron, which are formed by six rhombic planes with an edge length of 3 1/2 a 2 /( 2 + 1) 1/2 and a shorter diagonal of 2 a 2 /( 2 + 1) 1/2 ( is the golden ratio and a is a quasilattice constant), which correspond to half a body diagonal and an edge length respectively of the 1/1 cubic approximant unit cell. The present results provide an important key for understanding the arrangements of atomic clusters in icosahedral quasicrystals.

Transport properties of the three-dimensional Penrose lattice

Transport properties of the three-dimensional (3-D) quasiperiodic systems have been studied for the 3-D Penrose lattice at zero temperature. The linear scaling with respect to the sample width (cross-section) is observed. With increasing sample width, the conductance fluctuation as a function of the Fermi energy is suppressed and a smooth energy dependence is obtained for larger samples. This suggests that the transport properties of the 3-D quasiperiodic systems are less anomalous than those of the low-dimensional quasiperiodic systems. The sample-length dependence of the conductance is, on the other hand, not that expected for extended Bloch waves and the conductance decreases with increasing sample length.

A Large Dodecahedral Cluster Containing about 480 Atoms in a 2/1 Cubic Crystalline Approximant

A dodecahedral atom cluster, which is the largest atom cluster with icosahedral symmetry found in crystalline alloy phases, was observed in the cubic crystalline phase of β-(AlPdMnSi), referred to as a 2/1 approximant of an icosahedral quasicrystal, with a lattice constant of 2.0211 nm. The large dodecahedral cluster has a diameter of about 2.6 nm and contains about 480 atoms. The atomic arrangement of the dodecahedral cluster is divided into atomic shells with icosahedral symmetry, and most of the atoms in the shells are located at special positions associated with the golden mean in the three-dimensional Penrose lattice. A small icosahedral atom cluster, formed with twelve Al atoms at the vertices and a Pd atom at the center, is located at each vertex of a dodecahedral shell with a diameter of about 2 nm. The large dodecahedral atom cluster is considered to be an important cluster for understanding the structure of an Al–Pd–Mn icosahedral quasicrystal.

An Atom Cluster Model ofF-Type Icosahedral Quasicrystals, Derived from the Structure of 1/1 Cubic Approximant Phase (α-(AlPdMnSi))

We propose an atom cluster model of F -type icosahedral quasicrystals based on the structure of an α-(AlPdMnSi) phase, which is a p / q =1/1 cubic approximant of icosahedral quasicrystals. In the model, large dodecahedral and small icosahedral atom clusters, which are located at the (1/2, 1/2, 1/2) and (0, 0, 0) positions in the α-(AlPdMnSi) structure, respectively, are placed at twelvefold sites classified into two groups in the three-dimensional Penrose lattice. A diffraction pattern of this model reproduces well the characteristic features of an electron diffraction pattern of the F -type Al–Pd–Mn icosahedral quasicrystal.

Localized modes and topological frustration in rational approximants to quasicrystals.

The existence of localized vibrational states is demonstrated for rational approximants of icosahedral Al-Zn-Mg quasicrystals. It is shown that the origin of localized modes is in a local topological frustration, i.e., in a local deviation from ideal icosahedral packing. As the localized mode involves only atoms in a single tile of the three-dimensional Penrose lattice, it can be expected to exist in the infinite quasicrystal.

Quasi-crystal structure: cold water on the Penrose tiling scheme

Quasi-crystalline structures are related to periodic distributions of A3perp volumes in a higher-dimensional space. If these A3perp volumes are actually confined inside the complementary space, three-dimensional Penrose lattice can be generated by the six-dimensional cut method. Diffraction data show that the actual A3perp must have components in the physical space, which make the three-dimensional structure description in terms of Penrose tiling not really reliable. Such 'parallel' components have been experimentally observed for the first time.

Electronic structure in three-dimensional quasicrystals

Electronic structure in three-dimensional quasicrystals is discussed both for simple tight-binding models of Penrose lattice and rationally approximated AlMn alloys. In three-dimensional Penrose lattice, there are several infinitelly degenerated states though they are not generic in quasi-periodicity. The electronic structure in the latter system is calculated by the LMTO-ASA method with the local density functional theory. The density of states in AlMn alloys consists of a set of spiky peaks, and the stability and the role of the vacant center of the Mackay icosahedron are discussed.

Strictly localized eigenstates on a three-dimensional Penrose lattice.

Strictly localized eigenstates on a three-dimensional Penrose lattice.

Structure of an icosahedral Al-Mn quasicrystal.

A structure model of an icosahedral Al-Mn quasicrystal is proposed based on the structure-factor calculation and the x-ray and neutron powder-diffraction patterns of rapidly solidified Al-Mn alloys. The model has a local atomic arrangement similar to that of the cubic \ensuremath{\alpha}--Al-Mn-Si and has icosahedral point symmetry. To derive the model six-dimensional description of the three-dimensional Penrose lattice is extended to more general quasicrystals. The model explains the x-ray and neutron diffraction intensity, chemical composition, and density of the icosahedral quasicrystal.