Crystalline Approximants Research Articles

Pseudo-Mackay clusters in the Al-Pd-Co System

A great number of complicated intermetallic compounds were reported in binary and ternary alloy systems of Al with transition metals. Al–Co–Pd is one of the most interesting alloys, since a variety of crystalline phases associated with quasicrystals has been reported [1]. Among these crystalline phases, the structures of ε-phases are closely associated with Al3Pd, which is an important crystalline approximant for the decagonal phase with a period of 1.6 nm. W-phase is another approximant for the decagonal phase and its structural information is useful for understanding the columnar unit in the decagonal phase with a periodicity of 0.8 nm [2]. On the other hand, some crystalline phases associated with the icosahedral phase were also found in this Al-Pd-Co system. C2–phase, R-phase (R¯3: a = 2.91 nm, c = 1.32 nm) and F-phase (Pa3: a = 2.44 nm) are classified into this category. In particular, the structures of R-phase and F-phase consist of a variety of pseudo-Mackay clusters similar to those found in 1/1-AlCuRu and the trigonal χ-AlPdRe. As an example, the structure of R-phase shows two types of pMCs. These pMCs can be ranked by their atomic arrangements of the first shells, nevertheless every outer shell is a harmony of an Al-icosidodecahedron and a Co/Pd-icosahedron. These pMCs interpenetrate each other by sharing edges of Co/Pd-icosahedra and the interstitial space is subsequently filled by the smaller Al-icosahedra around Pd/Al sites [3]. The characteristic structural motifs for R-phase and F-phase readily suggest the importance of pMC as a fundamental structural unit for icosahedral quasicrystals.

Magnetic quasicrystals: Current Status and Open Questions

The magnetic properties of quasicrystals have been the subject of intense theoretical and experimental interest since Dan Shechtman's discovery of icosahedral quasicrystals in 1982 [1]. The first generation of Al-TM (TM = 3d transition metal) quasicrystals with moment bearing elements, such as Al-Mn and Al-Pd-Mn, provided some insight into the fate of magnetism in quasiperiodic systems: only a small fraction of the Mn ions carry a moment and these moments appear to be distributed randomly on the quasilattice. The discovery of the Zn-Mg-R and Cd-Mg-R (R = rare earth) quasicrystals with sizeable local moments that interact via indirect exchange (i.e. the RKKY interaction) reinvigorated the search for long-range magnetic order in the icosahedral phase. However, to date, all of the known quasicrystals with moment bearing elements exhibit frustration and spin-glass-like behavior at low temperature. The recent discovery of binary magnetic quasicrystals in the R-Cd system [2] is particularly exciting because their closely related crystalline approximants, RCd6, exhibit long-range antiferromagnetic order at low temperature [3] permitting a detailed investigation of the effects of aperiodicity on magnetic interactions in this system. Here, I will describe the status of investigations of the R-Cd quasicrystals and RCd6 approximants with regard to their structural and magnetic properties. This research was supported by the Office of the Basic Energy Sciences, Materials Sciences Division, US Department of Energy (DOE).

Compositions of Al-Based Quasicrystals Interpreted by Cluster Formulae

It is known previously that bulk metallic glass compositions satisfy cluster formulae [cluster](glue atom)1,3 of 24 valence electrons as deduced from our cluster-resonance model. In the present work, it is further shown that compositions of Al-based binary and ternary quasicrystals are also explained by 24-electron cluster formulae of the types [icosahedron](glue atom)0,1, where the icosahedral cluster is identi ed from a corresponding crystalline approximant according to dense atomic packing and cluster isolation criteria, and the glue atom site is either vacant for an icosahedral quasicrystal or equal to one for a decagonal quasicrystal. Ternary quasicrystals are formulated with the same formulae as their basic binary ones but the icosahedron shell sites are substituted by third elements. The 24-electron cluster formulae are then the chemical and electronic structural units of quasicrystals, mimicking the molecular formulae of chemical substances.

Transport Properties of the Au-Al-Yb Quasicrystal and Approximant under Hydrostatic Pressure

The mixed-valence Au Al Yb quasicrystal exhibits unusual quantum criticality. Its crystalline approximant behaves like a conventional heavy fermion metal. Due to these novel features, they have attracted much attention recently. To examine the sample dependence of this new type of materials, we have prepared several samples and measured the electrical resistivity both at ambient and high pressures as well as the magnetic susceptibility at ambient pressure. The residual resistivity ratio of the quasicrystals is almost sample independent while the power law exponent n of the electrical resistivity varies from sample to sample. The e ective magnetic moment peff of the quasicrystals is also sample dependent, ranging from about 3.4 to 3.9 μB/Yb, and these values are all smaller than the free Yb ion value, con rming the mixed-valence nature. Although the magnitude of n and peff is sample-dependent, the principal feature of the mixed-valence is con rmed in all the samples studied thus far. A combination of these results indicates correlation between n and peff , suggesting that the 4f electrons may contribute to a scattering mechanism of the conduction electrons. External pressure increases the magnitude of the electrical resistivity and decreases the index n of both the quasicrystal and the approximant. Up to the highest pressure measured in the present study no experimental evidence for a phase transition was found.

Numerical methods for quasicrystals

Quasicrystals are one kind of space-filling structures. The traditional crystalline approximant method utilizes periodic structures to approximate quasicrystals. The errors of this approach come fr...

Ferromagnetism and re-entrant spin-glass transition in quasicrystal approximants Au–SM–Gd (SM = Si, Ge)

Magnetic susceptibility and specific heat measurements on quasicrystalline approximants Au–Si–Gd and Au–Ge–Gd reveal that a ferromagnetic (FM) transition occurs at Tc = 22.5(5) K for Au–Si–Gd and at Tc = 13(1) K for Au–Ge–Gd, which are the first examples of ferromagnetism in crystalline approximants. In addition, a re-entrant spin-glass (RSG) transition is observed at TRSG = 3.3 K for Au–Ge–Gd in contrast to Au–Si–Gd. The different behaviors are understood based on the recent structural models reported by Gebresenbut et al (2013 J. Phys.: Condens. Matter 25 135402). The RSG transition in Au–Ge–Gd is attributed to a random occupation of the center of the Gd12 icosahedron by Gd atoms; a central Gd spin hinders the long-range FM order.

Crystal–quasicrystal transition depending on cooling rates in directionally solidified Al–3Mn–7Be (at.%) alloy

The transition of primary phases from the crystalline approximant phase to icosahedral quasicrystal phase (I-phase) is observed in the directional solidified Al–3Mn–7Be (at.%) alloy. The structures of I-phase and approximant phases are characterized by SEM, TEM and HRTEM. The addition of Be significantly enhances the formation ability of I-phase, combining with the directional solidification method where the growth rates and temperature gradients can be separately controlled, giving rise to a promising way to prepare quasicrystal-reinforced Al-matrix composite.

Numerical methods for quasicrystals

Quasicrystals are one kind of space-filling structures. The traditional crystalline approximant method utilizes periodic structures to approximate quasicrystals. The errors of this approach come from two parts: the numerical discretization, and the approximate error of Simultaneous Diophantine Approximation which also determines the size of the domain necessary for accurate solution. As the approximate error decreases, the computational complexity grows rapidly, and moreover, the approximate error always exits unless the computational region is the full space. In this work we focus on the development of numerical method to compute quasicrystals with high accuracy. With the help of higher-dimensional reciprocal space, a new projection method is developed to compute quasicrystals. The approach enables us to calculate quasicrystals rather than crystalline approximants. Compared with the crystalline approximant method, the projection method overcomes the restrictions of the Simultaneous Diophantine Approximation, and can also use periodic boundary conditions conveniently. Meanwhile, the proposed method efficiently reduces the computational complexity through implementing in a unit cell and using pseudospectral method. For illustrative purpose we work with the Lifshitz–Petrich model, though our present algorithm will apply to more general systems including quasicrystals. We find that the projection method can maintain the rotational symmetry accurately. More significantly, the algorithm can calculate the free energy density to high precision.

Computing Phase Diagrams for a Quasicrystal-Forming Patchy-Particle System

We introduce an approach to computing the free energy of quasicrystals, which we use to calculate phase diagrams for systems of two-dimensional patchy particles with five regularly arranged patches that have previously been shown to form dodecagonal quasicrystals. We find that the quasicrystal is a thermodynamically stable phase for a wide range of conditions and remains a robust feature of the system as the potential's parameters are varied. We also demonstrate that the quasicrystal is entropically stabilized over its crystalline approximants.

A family of binary magnetic icosahedral quasicrystals based on rare earths and cadmium

Examples of stable binary icosahedral quasicrystals are relatively rare, and at present there are no known examples featuring localized magnetic moments. These would represent an ideal model system for attaining a deeper understanding of the nature of magnetic interactions in aperiodic lattices. Here we report the discovery of a family of at least seven rare earth icosahedral binary quasicrystals, i-R-Cd (R = GdtoTm, Y), six of which bear localized magnetic moments. Our work highlights the importance of carefully motivated searches through phase space and supports the proposal that, like icosahedral Sc12Zn88 (ref.), binary quasicrystalline phases may well exist nearby known crystalline approximants, perhaps as peritectically forming compounds with very limited liquidus surfaces, offering very limited ranges of composition/temperature for primary solidification.

Low-temperature superstructures of a series of Cd6M (M = Ca, Y, Sr, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm, Yb and Lu) crystalline approximants

The low-temperature (LT) superstructure and the phase transition temperature have been investigated for a series of Cd6M crystalline approximants by transmission electron microscopy as well as electrical resistivity measurements. Except for M = Lu, Cd6M is found to undergo a phase transition to a monoclinic phase at a low temperature and the transition temperature (Tc) scales well with the size of the M atom. For M = Ca, Y, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er and Tm the LT superstructure is explained by a lattice with the space group C2/c, and for M = Sr and Yb a monoclinic lattice with P2/m. On the other hand, no phase transition is observed for M = Lu, indicating that a Cd4 tetrahedron at the cluster center remains disordered down to the lowest temperature, i.e. 16 K. It is shown that the volume inside the Cd20 dodecahedron plays a crucial role in the occurrence of the phase transition, and long-term aging in particular promotes the phase transition for late rare-earth elements such as Ho, Er and Tm, suggesting that the transition is sensitive to and is even hindered by disorder such as atomic vacancies. The absence of the transition for M = Lu is attributed to the highest activation energy for the transition due to the smallest volume inside the Cd20 dodecahedron.

Quantum critical state in a magnetic quasicrystal

Quasicrystals are metallic alloys that possess long-range, aperiodic structures with diffraction symmetries forbidden to conventional crystals. Since the discovery of quasicrystals by Schechtman et al. in 1984, there has been considerable progress in resolving their geometric structure. For example, it is well known that the golden ratio of mathematics and art occurs over and over again in their crystal structure. However, the characteristic properties of the electronic states--whether they are extended as in periodic crystals or localized as in amorphous materials--are still unresolved. Here we report the first observation of quantum (T = 0) critical phenomena of the Au-Al-Yb quasicrystal--the magnetic susceptibility and the electronic specific heat coefficient arising from strongly correlated 4f electrons of the Yb atoms diverge as T→0. Furthermore, we observe that this quantum critical phenomenon is robust against hydrostatic pressure. By contrast, there is no such divergence in a crystalline approximant, a phase whose composition is close to that of the quasicrystal and whose unit cell has atomic decorations (that is, icosahedral clusters of atoms) that look like the quasicrystal. These results clearly indicate that the quantum criticality is associated with the unique electronic state of the quasicrystal, that is, a spatially confined critical state. Finally we discuss the possibility that there is a general law underlying the conventional crystals and the quasicrystals.

Intermediate-valence icosahedral Au-Al-Yb quasicrystal

A quasiperiodic intermediate-valence (IV) system is realized in an icosahedral Au-Al-Yb quasicrystal. X-ray absorption spectroscopy near the Yb ${L}_{3}$ edge indicates that quasiperiodically arranged Yb ions assume a mean valence of 2.61, between a divalent state ($4{f}^{14}$, $J=0$) and a trivalent one ($4{f}^{13}$, $J=7/2$). Magnetization measurements demonstrate that the $4f$ holes in this quasicrystal have a localized character. The magnetic susceptibility shows a Curie-Weiss behavior above $\ensuremath{\sim}100$ K with an effective magnetic moment of 3.81${\ensuremath{\mu}}_{\mathrm{B}}$ per Yb. Moreover, a crystalline approximant to this quasicrystal is an IV compound. We propose a heterogeneous IV model for the quasicrystal, whereas the crystalline approximant is most likely a homogeneous IV system. At temperatures below $\ensuremath{\sim}10$ K, specific heat and magnetization measurements reveal non-Fermi-liquid behavior in both the quasicrystal and its crystalline approximant without either doping, pressure, or field tuning.

Structural and magnetic ordering in Cd-based crystalline approximants

Structural and magnetic ordering in Cd-based crystalline approximants

Structural and magnetic transitions in the crystalline approximant Cd6Sm

Electrical resistivity, specific heat, magnetic susceptibility, and magnetization of the crystalline approximant Cd${}_{6}$Sm are investigated in a temperature range 1.8\ensuremath{-}300 K. Both structural and magnetic transitions are clearly evidenced. A sharp structural phase transition is observed at ${T}_{\mathrm{c}}$ $=$ 179 K in the electrical resistivity as well as in the specific heat, which is attributed to orientational ordering of the Cd${}_{4}$ tetrahedron located at the center of the icosahedral cluster. In addition, three successive magnetic transitions are found at ${T}_{\mathrm{M}1}$ $=$ 12.5, ${T}_{\mathrm{M}2}$ $=$ 10.2, and ${T}_{\mathrm{M}3}$ $=$ 6.5 K. The anomaly at ${T}_{\mathrm{M}1}$ is attributed to an antiferromagnetic transition, whereas those at ${T}_{\mathrm{M}2}$ and ${T}_{\mathrm{M}3}$ are to ferrimagnetic transitions. The observation of the ferrimagnetic orders indicates that (pseudo) icosahedra made of localized Sm spins have a nonzero net magnetic moment, and the moments of the magnetic clusters form a long-range magnetic order.

Structural characterization of highly defective Al-Pd-Fe crystalline approximant

Structural characterization of highly defective Al-Pd-Fe crystalline approximant

Mechanical properties of Al–Cu–Fe alloys sintered at high pressure

The structure and mechanical behavior of bulk Al–Cu–Fe materials sintered at high hydrostatic pressure are studied at room temperature. Quasicrystalline Al63Cu25Fe12 and Al62.7Cu25Fe12Sc0.3 powders and Al66Cu18Fe8Cr8, powder, which is the quasicrystal approximant, are sintered at high pressure. The powders are obtained by high-pressure water atomization. At an optimal hot compaction pressure of 4.5 GPa, the porosity of the compact materials is lower than 2%. Sintering is conducted at 700°C. The stress-strain curves of quasicrystals and their approximants that are brittle in standard test conditions are obtained using indentation method. It is shown that the crystalline approximants of quasicrystals are much closer in their mechanical behavior to quasicrystals than to crystalline solids.

Diffuse scattering in the Zn6Sc 1/1 cubic approximant

The low-temperature phase transition of the Zn6Sc cubic 1/1 approximant [1] has been investigated by X-ray diffraction. It is a crystalline approximant to a recently discovered quasicrystal Zn88Sc1 ...

Composition rule for Al–transition metal binary quasicrystals

Compositions of Al–transition metal (TM) binary quasicrystals are revisited using the cluster-plus-glue-atom model. These compositions all conform to a unified cluster formula [icosahedron](glue)1, where the icosahedron is taken from a crystalline approximant. In a given binary system, icosahedral and decagonal quasicrystals are expressed by the same icosahedron but glued, respectively, by one Al and one TM, and their e/a ratios fall close to 1.86 and 1.71–1.79, respectively. Their relative stabilities are discussed by referring to compositions satisfying the fully filled TM-3d states according to Häussler's resonance model.

Intermediate-valence quasicrystal of a Cd-Yb alloy under pressure

A quasiperiodic intermediate-valence system was prepared by applying pressure to an icosahedral Cd-Yb quasicrystal. In this system, each quasiperiodic lattice point has a charge degree of freedom. $\text{Yb}\text{ }{L}_{\text{III}}$ x-ray absorption spectroscopy experiments demonstrate that the Yb valence increases continuously upon compression from a divalent state at ambient pressure and reaches 2.33 at 31.7 GPa. A comparison study to the crystalline approximant suggests that the quasiperiodically arranged Yb ions have a homogeneous intermediate valence that takes a valence fluctuation state under pressure.