Quasicrystal Approximant Research Articles

Direct observation of the local atomic structure in two types of Au-Ge-Yb 1/1 approximants

The local atomic structure in two types of Tsai-type 1/1 quasicrystal approximants in the Au-Ge-Yb system were observed using scanning transmission electron microscopy (STEM). The high-angle annular dark-field (HAADF) imaging method in STEM was employed to detect the difference in local atomic structure at the cluster centers for both compounds. Two types of cluster centers were observed in the different compounds: a tetrahedron with various orientations and a single Yb atom. Furthermore, atomic-resolution energy-dispersive X-ray spectroscopy showed direct evidence for the presence of a single Yb atom at the cluster center for the later compound.

Antiferromagnetic order survives in the higher-order quasicrystal approximant

We report the observation of antiferromagnetic (AFM) transitions in 1/1 and 2/1 quasicrystal approximants having slightly different electron-per-atom $(e/a)$ ratios, in the same Au-Ga-Eu system, at N\'eel temperatures of 7.0 and 8.5 K, respectively. This clearly demonstrates that an antiferromagnetic order survives when the degree of approximation to the quasicrystals is raised provided that the $e/a$ ratio is kept nearly the same. Since, unlike the 1/1 approximant, the 2/1 approximant has all the building blocks necessary for the construction of the quasicrystal, this finding can be a significant leap toward the long-sought realization of an AFM quasicrystal.

Whirling spin order in the quasicrystal approximant Au72Al14Tb14

Neutron powder diffraction experiment has been performed on the quasicrystal approximant Au$_{72}$Al$_{14}$Tb$_{14}$, a body-center-cubic crystal of icosahedral spin clusters. The long-range antiferromagnetic order was confirmed at the transition temperature $T_{\rm N} = 10.4$ K. The magnetic structure consists of noncoplanar whirling spins on the icosahedral clusters, arranging antiferroic-manner. A simple icosahedral spin-cluster model with uniaxial anisotropy accounts well the whirling spin order as well as the in-field metamagnetic transition, indicating that the icosahedral symmetry is essential.

Conduction electrons in aperiodic versus periodic structures: An ESR study of quasicrystalline i -Y(Gd)-Cd and its approximant Y(Gd)Cd6

A formal description of collective electronic states in condensed-matter systems lacking long-range periodicity remains a theoretical challenge. To experimentally explore the differences in electronic and magnetic behavior between metallic quasicrystals (QCs) and their conventional crystalline analogs [quasicrystal approximants (QCAs)], we have grown single crystals of ${\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{Gd}}_{x}{\mathrm{Cd}}_{6}$ (QCA) together with their QC counterparts $i\text{\ensuremath{-}}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{Gd}}_{x}\text{\ensuremath{-}}\mathrm{Cd}$ for $x=0.006$, 0.01, 0.1, and 1.00, and we carried out comparative $T$-dependent electron spin resonance (ESR) measurements. On the high Gd concentration side, $x=1.00$, we confirm that ${\mathrm{GdCd}}_{6}$ adopts an antiferromagnetic ground state below ${T}_{N}\ensuremath{\sim}22\phantom{\rule{4pt}{0ex}}\mathrm{K}$, whereas $i$-Gd-Cd presents spin-glass-like behavior showing similar local and dynamical properties from the point of view of ESR. For the diluted samples, our ESR experimental results show similar local conduction electron polarization behavior at the ${\mathrm{Gd}}^{3+}$ site in all QC/QCA pairs investigated, supporting the validity of using QCAs as periodic representations of QCs in terms of short-range electronic interactions. However, there is a measurable difference in the Korringa relaxation rate (spin-flip relaxation process between the localized ${\mathrm{Gd}}^{3+}4f$ electron and the delocalized $s$-type conduction electrons at the Fermi surface) between the QC/QCA pairs probably associated with the lack of periodicity. We expect that our comparative ESR study may provide support and motivation for the development of new theoretical approaches toward a generalized band-structure theory, contemplating condensed-matter systems beyond the scope of traditional periodicity.

Explanation of structural differences and similarities between the AT2Al10 phases (where A=actinide, lanthanide or rare earth element and T=transition metal)

Abstract In the current work we have studied the crystallographic relationship between the AT2Al10 phases (where A = actinide, lanthanide and rare earth element and T = transition metal). It is known that with this stoichiometry two structure types exist: tetragonal CaCr2Al10 and orthorhombic YbFe2Al10. It was found that both CaCr2Al10 and YbFe2Al10 types are structural derivatives of the ThMn12 type structure (which has more general formula of AT x Al12− x , with x > 2). CaCr2Al10 structure has a group-subgroup relationship with the ThMn12 structure, while the relationship of the YbFe2Al10 to the ThMn12 was proved applying the strong reflection approach, suggested initially for approximants of quasi-crystals. Proposed here relationship between the studied structures explains the small difference in total energies, calculated using Density Functional Theory. Understanding the connection between these phases allows regarding AT2Al10 composition as somewhat extension of the AT x Al12− x compositional range. Due to the unique magnetic properties of the AT x Al12− x phases, tunable as a function of crystallographic structure, study of structural stability and crystallographic relationships of related phases are of outmost importance.

Synthesis and Atomic Structure of the Yb-Ga-Au 1/1 Quasicrystal Approximant.

The Yb-Ga-Au 1/1 quasicrystal approximant (AP) composition ranges from Yb14.0Ga20.6Au65.4 to Yb14.8Ga46.3Au38.9, and single crystals of the 1/1 AP having the composition Yb13.8Ga26.1Au60.1 were obtained by the self-flux technique. X-ray structural analysis demonstrated that the atomic structure [space group Im3; a = 14.6889(9) Å] can be described by the body-centered packing of Tsai-type rhombic triacontahedron (RTH) clusters. The positional disorder in these clusters, interpreted as the average of an orientationally disordered tetrahedron and triangle, results in positional disorder in the outer shells. The elemental distributions and positions of mixtures of Au and Ga atoms in the RTH clusters correspond to those in the isostructural Yb15Al36Au49 1/1 AP.

Complex Crystals from Size-Disperse Spheres.

Colloids are rarely perfectly uniform but follow a distribution of sizes, shapes, and charges. This dispersity can be inherent (static) or develop and change over time (dynamic). Despite a long history of research, the conditions under which nonuniform particles crystallize and which crystal forms is still not well understood. Here, we demonstrate that hard spheres with Gaussian radius distribution and dispersity up to 19% always crystallize if compressed slowly enough, and they do so in surprisingly complex ways. This result is obtained by accelerating event-driven simulations with particle swap moves for static dispersity and particle resize moves for dynamic dispersity. Above 6% dispersity, AB_{2} Laves, AB_{13}, and a region of Frank-Kasper phases are found. The Frank-Kasper region includes a quasicrystal approximant with Pearson symbol oS276. Our findings are relevant for ordering phenomena in soft matter and alloys.

A neutron diffraction demonstration of long-range magnetic order in the quasicrystal approximant DyCd6

We have used neutron powder diffraction to demonstrate the existence of long-range antiferromagnetic order of Ising-like Dy moments in the DyCd6 quasicrystal approximant phase. This cubic compound undergoes a slight distortion to a monoclinic cell at low temperatures. The Néel temperature is 18.0(2) K and the magnetic order of the Dy sublattice may be described in the parent cubic Im3¯ structure using a combination of two propagation vectors, k1 = [0 0 0] and k2 = [12 0 12], yielding ‘anti-I’ order. Alternatively, when referred to the monoclinic C2/c cell, the magnetic structure may be described by a single propagation vector: k = [1 0 0].

Energy-dispersive Laue experiments with X-ray tube and PILATUS detector: precise determination of lattice constants

A PILATUS detector in combination with a conventional sealed X-ray tube was used for the development of the energy-dispersive Laue diffraction technique, which can be applied for precise measurements of single-crystal lattice constants in transmission and reflection modes without moving the sample. Exploiting the ability of PILATUS detectors to suppress counting of X-ray photons below a certain energy threshold allows one to recover the wavelength of diffracted Bragg reflections, reconstruct the three-dimensional reciprocal-space pattern, index X-ray diffraction peaks, and find the orientation and lattice parameters of the crystal without any a priori information about the sample. By making some geometrical assumptions and using a set of fast in situ calibration procedures, it is possible to simultaneously refine lattice constants and hardware correction factors, which simplifies the sample preparation and measurement strategies. Several samples [silicon, quartz, fluorite (CaF2), o′-Al13Co4 quasicrystal approximant, Laves (MgZn2) and Bergman (Mg32(Al,Zn)49) phases] were studied with the developed technique, and 0.01 Å and 0.1° precisions were routinely reached for lattice vector lengths and angles, respectively. The use of the developed methods is only limited by the energy resolution of the PILATUS detector, where lattice vectors with >27 Å length cannot be reliably resolved.

Antiferromagnetic order is possible in ternary quasicrystal approximants

Eight years since the discovery of the binary antiferromagnetic (AFM) quasicrystal approximants $\mathrm{C}{\mathrm{d}}_{6}R$ [R. Tamura et al., Phys. Rev. B 82, 220201 (2010)], we report the observation of an AFM transition in quasicrystal approximants Au-Al-R ($R=\mathrm{Gd}$ and Tb), which represent examples of AFM in ternary quasicrystal approximants. Magnetic susceptibility, magnetization, and specific heat coherently show that the Au-Al-R ($R=\mathrm{Gd}$ and Tb) approximants undergo an AFM transition at ${T}_{\mathrm{N}}=9.4$ and 11.8 K, respectively. The AFM transitions are found near the phase boundary of the wide single-phase region where the chemical disorder is substantially low. This finding of ternary AFM approximants has far deeper implications compared with that of the binary AFM approximants: It now indicates the possibility to synthesize new AFM approximants in many other systems by the appropriate replacement of Cd with two other elements, and thus enables exploration of various magnetic orders over AFM spin icosahedra in solids, in particular, their evolution as a function of the electron-per-atom (e/a) ratio. Moreover, the finding provides us a missing route to synthesize an AFM quasicrystal by similar composition tuning.

Previously unknown quasicrystal periodic approximant found in space

We report the discovery of Al34Ni9Fe2, the first natural known periodic crystalline approximant to decagonite (Al71Ni24Fe5), a natural quasicrystal composed of a periodic stack of planes with quasiperiodic atomic order and ten-fold symmetry. The new mineral has been approved by the International Mineralogical Association (IMA 2018-038) and officially named proxidecagonite, which derives from its identity to periodic approximant of decagonite. Both decagonite and proxidecagonite were found in fragments from the Khatyrka meteorite. Proxidecagonite is the first natural quasicrystal approximant to be found in the Al-Ni-Fe system. Within this system, the decagonal quasicrystal phase has been reported to transform at ~940 °C to Al13(Fe,Ni)4, Al3(Fe,Ni)2 and the liquid phase, and between 800 and 850 °C to Al13(Fe,Ni)4, Al3(Fe,Ni) and Al3(Fe,Ni)2. The fact that proxidecagonite has not been observed in the laboratory before and formed in a meteorite exposed to high pressures and temperatures during impact-induced shocks suggests that it might be a thermodynamically stable compound at high pressure. The most prominent structural motifs are pseudo-pentagonal symmetry subunits, such as pentagonal bipyramids, that share edges and corners with trigonal bipyramids and which maximize shortest Ni–Al over Ni–Ni contacts.

Atomic scale analyses of {\bb Z}-module defects in an NiZr alloy.

Some specific structures of intermetallic alloys, like approximants of quasicrystals, have their unit cells and most of their atoms located on a periodic fraction of the nodes of a unique {\bb Z}-module [a set of the irrational projections of the nodes of a (N > 3-dimensional) lattice]. Those hidden internal symmetries generate possible new kinds of defects like coherent twins, translation defects and so-called module dislocations that have already been discussed elsewhere [Quiquandon et al. (2016). Acta Cryst. A72, 55-61; Sirindil et al. (2017). Acta Cryst. A73, 427-437]. Presented here are electron microscopy observations of the orthorhombic phase NiZr - and its low-temperature monoclinic variant - which reveal the existence of such defects based on the underlying {\bb Z}-module generated by the five vertices of the regular pentagon. New high-resolution electron microscopy (HREM) and scanning transmission electron microscopy high-angle annular dark-field (STEM-HAADF) observations demonstrate the agreement between the geometrical description of the structure in five dimensions and the experimental observations of fivefold twins and translation defects.

Stable Frank–Kasper phases of self-assembled, soft matter spheres

Single molecular species can self-assemble into Frank-Kasper (FK) phases, finite approximants of dodecagonal quasicrystals, defying intuitive notions that thermodynamic ground states are maximally symmetric. FK phases are speculated to emerge as the minimal-distortional packings of space-filling spherical domains, but a precise measure of this distortion and how it affects assembly thermodynamics remains ambiguous. We use two complementary approaches to demonstrate that the principles driving FK lattice formation in diblock copolymers emerge directly from the strong-stretching theory of spherical domains, in which a minimal interblock area competes with a minimal stretching of space-filling chains. The relative stability of FK lattices is studied first using a diblock foam model with unconstrained particle volumes and shapes, which correctly predicts not only the equilibrium σ lattice but also the unequal volumes of the equilibrium domains. We then provide a molecular interpretation for these results via self-consistent field theory, illuminating how molecular stiffness increases the sensitivity of the intradomain chain configurations and the asymmetry of local domain packing. These findings shed light on the role of volume exchange on the formation of distinct FK phases in copolymers and suggest a paradigm for formation of FK phases in soft matter systems in which unequal domain volumes are selected by the thermodynamic competition between distinct measures of shape asymmetry.

First-principles study of elastic and slip properties of non-canonical β-based Al–Cu–Fe approximants

ABSTRACTAb initio calculations were carried out to compare the mechanical properties of β-based non-canonical Al–Cu–Fe approximants of quasicrystals with cubic (β), monoclinic (η) and orthorhombic (ξ1, ξ2) structures, which all demonstrate high strengthening. The aim was to elucidate the competitive effects of the η- and ξ-ordering and iron content on deformation behaviour of these phases. We found that the Young’s modulus, polycrystalline shear modulus, mechanical stability and shear elastic modulus G(n,m) for different slip planes decrease for β-Al50Cu1-xFex with lowering iron content, but they grow from β-Al50Cu31.25Fe18.75 to the ordered η-Al50Cu45Fe5, and ξ2-Al45.5Cu50Fe4.5 that indicates a growing resistance to plastic deformation due to ordering and agrees well with our experimental finding. The preferable slip systems were predicted based on the calculated generalised stacking fault (GSF) energies in β-(Cu,Fe)Al and η-(Cu,Fe)Al with similar Fe concentration. The GSF energies confirmed also that the strengthening observed in η-phase is related to ordering rather than the Fe effect in consistence with a stronger covalent bonding in η-phase.

Effects of crystalline electronic field and onsite interorbital interaction in Yb-based quasicrystal and approximant crystal

To get an insight into a new type of quantum critical phenomena recently discovered in the quasicrystal Yb15Al34Au51 and approximant crystal (AC) Yb14Al35Au51 under pressure, we discuss the property of the crystalline electronic field (CEF) at Yb in the AC and show that uneven CEF levels at each Yb site can appear because of the Al/Au mixed sites. Then we construct the minimal model for the electronic state on the AC by introducing the onsite Coulomb repulsion between the 4f and 5d orbitals at Yb. Numerical calculations for the ground state shows that the lattice constant dependence of the Yb valence well explains the recent measurement done by systematic substitution of elements of Al and Au in the quasicrystal and AC, where the quasicrystal Yb15Al34Au51 is just located at the point from where the Yb-valence starts to change drastically. Our calculation convincingly demonstrates that this is indeed the evidence that this material is just located at the quantum critical point of the Yb-valence transition.

Shear localization and size-dependent strength of YCd6 quasicrystal approximant at the micrometer length scale

Mechanical properties of materials are strongly dependent of their atomic arrangement as well as the sample dimension, particularly at the micrometer length scale. In this study, we investigated the small-scale mechanical properties of single-crystalline YCd6, which is a rational approximant of the icosahedral Y-Cd quasicrystal. In situ microcompression tests revealed that shear localization always occurs on {101} planes, but the shear direction is not constrained to any particular crystallographic directions. Furthermore, the yield strengths show the size dependence with a power law exponent of 0.4. Shear localization on {101} planes and size-dependent yield strength are explained in terms of a large interplanar spacing between {101} planes and the energetics of shear localization process, respectively. The mechanical behavior of the icosahedral Y-Cd quasicrystal is also compared to understand the influence of translational symmetry on the shear localization process in both YCd6 and Y-Cd quasicrystal micropillars. The results of this study will provide an important insight in a fundamental understanding of shear localization process in novel complex intermetallic compounds.

Structurally Complex Frank–Kasper Phases and Quasicrystal Approximants: Electronic Origin of Stability

Metal crystals with tetrahedral packing are known as Frank–Kasper phases, with large unit cells with the number of atoms numbering from hundreds to thousands. The main factors of the formation and stability of these phases are the atomic size ratio and the number of valence electrons per atom. The significance of the electronic energy contribution is analyzed within the Fermi sphere–Brillouin zone interaction model for several typical examples: Cu4Cd3, Mg2Al3 with over a thousand atoms per cell, and for icosahedral quasicrystal approximants with 146–168 atoms per cell. Our analysis shows that to minimize the crystal energy, it is important that the Fermi sphere (FS) is in contact with the Brillouin zones that are related to the strong diffraction peaks: the zones either inscribe the FS or are circumscribed by the FS creating contact at edges or vertices.

A complex pseudo-decagonal quasicrystal approximant solved by strong reflections approach

A complex pseudo-decagonal quasicrystal approximant solved by strong reflections approach

Atomic-scale configurations of variant twin boundaries of a pseudo-decagonal quasicrystal approximant in Al60Cr20Fe10Si10 alloy

Twin boundaries (TBs) of decagonal quasicrystal approximants generally exhibit structural characteristics that are peculiar compared with traditional crystals with small lattice parameters. Herein, the TBs of the orthorhombic (3/2, 2/1)2 decagonal approximant in the quaternary Al60Cr20Fe10Si10 alloy were investigated, and four types of (101) TBs were found using atomic-resolution high-angle annular dark-field scanning transmission electron microscopy. The structural features of these TBs at the atomic level were revealed and the corresponding projected structural models along the pseudo-tenfold axis were proposed and discussed. These results demonstrate that the diversifications of the TBs of decagonal approximants are closely related to the rich combinations of substructural blocks, and this may trigger further interest in the planar defects present in other decagonal approximants.

Conformational Asymmetry and Quasicrystal Approximants in Linear Diblock Copolymers.

Small angle x-ray scattering experiments on three model low molar mass diblock copolymer systems containing minority polylactide and majority hydrocarbon blocks demonstrate that conformational asymmetry stabilizes the Frank-Kasper σ phase. Differences in block flexibility compete with space filling at constant density inducing the formation of polyhedral shaped particles that assemble into this low symmetry ordered state with local tetrahedral coordination. These results confirm predictions from self-consistent field theory that establish the origins of symmetry breaking in the ordering of block polymer melts subjected to compositional and conformational asymmetry.