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September 25, 2009
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March 31, 2008
Abstract
In the present study, an icosahedral single phase bulk quasicrystalline material based on Al 62 Cu 25.5 Fe 12.5 has been synthesised by a spray forming route. Microstructural characterization showed an average grain size of 10 μm. The oversprayed fine powder showed the presence of β - and λ -phases, whereas, the deposit consited of the fully single phase bulk quasicrystalline material with compositional homogeneity. The hardness and fracture toughness measurements were carried out at different indentation loads of 50–500 g. The hardness values varied in the range 10.4–8.1 GPa and fracture toughness was seen to decrease with increasing load. The varionan of hardness with load, which is known as indentation size effect (ISE), has been established clearly. Fracture toughness value was constant in the load range from 200 to 500 g at 1.2 MPa m 1/2 . The cracking pattern after indentation at higher load has been observed to be intergranular as well as transgranular. The evolution of the single phase bulk quasicrystalline material has been discussed in light of the unique combination of atomization and deposition process elements in spray forming technique.
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September 25, 2009
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In the present study, a low-temperature synthesis of (Cu,Fe)Al 2 O 4 and (Cu,Cr,Fe)Al 2 O 4 spinels from the quasicrystalline phases was investigated with the variation of process parameters during milling and annealing. The milling of the quasicrystalline materials was carried out in an attritor mill at 400 rpm for 40 hours with ball to powder ratio of 40 : 1 in hexane medium. Subsequently, annealing was performed in an air ambience for 10, 20, and 40 h at 400, 500, 600 and 700 °C inside a furnace in order to oxidize the mechanically milled quasicrystalline phase for the possible formation of the spinel phase. It was found that after annealing at higher temperatures (about 500 °C), mechanically milled quasicrystalline alloy transformed to a spinel phase whereas annealing at lower temperatures (<500 °C) led to the formation of B2 phase as a major one along withminor amount of oxide phase. The X-ray diffraction and transmission electron microscopy of the annealed samples confirmed the formation of the spinel phase with an average grain size of about 20–40 nm. It is interesting to note that the nanospinel phases showed the different colors during various annealing time and temperature. The optical properties of nanospinel materials, investigated employing UV visible spectrometer exhibited absorption characteristics.
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September 25, 2009
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It has recently been shown that identical, isotropic particles can form complex crystals and quasicrystals. In order to understand the relation between the particle interaction and the structure, which it stabilizes, the phase behavior of a class of two-scale potentials is studied. In two dimensions, the phase diagram features many phases previously observed in experiment and simulation. The three-dimensional system includes the sigma phase with 30 particles per unit cell, not grown in simulations before, and an amorphous state, which we found impossible to crystallize in molecular dynamics. We suggest that the appearance of structural complexity in monodisperse systems is related to competing nearest neighbor distances and discuss implications of our result for the self-assembly of macromolecules.
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September 25, 2009
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Formation of the nanoscale icosahedral quasicrystalline phase (I-phase) in the melt-spun Zr 70 Pd 30 and Zr 80 Pt 20 binary metallic glasses were reported. Local atomic structure in the glassy and quasicrystal (QC)-formed states were also analyzed by XRD and EXAFS measurements in order to investigate the formation mechanism of QC phase. The distorted icosahedral-like local structure can be identified around Zr atom in the Zr 70 Pd 30 metallic glass. In the QC formation process, a change of local environment around Zr is detected, in which the approximately one Zr atom substitutes for one Pd atom. In contrast, since the local environment around Pt atom is remaining during the QC precipitation, it is suggested that the stable icosahedral local structure is mainly formed around a center Pt atom in the glassy state in Zr 80 Pt 20 . We also found that the local environment around Zr atom significantly changes during the quasicrystallization in the alloy. These results differ from those in the Zr 70 Pd 30 metallic glass.
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September 25, 2009
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The present research aimed to characterize the microstructure of icosahedral quasicrystalline phase ( i -phase) in different chemical compositions of Al–Cu–Fe alloys. The ternary alloys with the nominal compositions of Al 70 Cu 20 Fe 10 , Al 65 Cu 20 Fe 15 and Al 63 Cu 25 Fe 12 (at%) were rapidly solidified by melt-spinning technique. The melt-spun ribbons were characterized by X-ray diffraction (XRD), scanning electron microscopy (SEM) and differential thermal analysis (DTA). According to the XRD results, diffraction patterns of Al 70 Cu 20 Fe 10 alloy showed the presence of the i -phase together with tetragonal θ -Al 2 Cu phase. However, XRD pattern of the melt-spun Al 63 Cu 25 Fe 12 alloy showed the formation of i -phase together with a small amount of β -phase. The single quasicrystalline phase was observed only in the rapidly solidified Al 65 Cu 20 Fe 15 alloy. The SEM examinations showed that the melt-spun Al 70 Cu 20 Fe 10 and Al 63 Cu 25 Fe 12 alloys composed of i -phase with an average grain size of 1–5 μm. However, for the Al 65 Cu 20 Fe 15 alloy, the quasicrystalline particles in the star-shape dendritic formations were observed. The melting point of i -phase was determined as 890 °C by DTA examinations.
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September 25, 2009
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An Al 94 Mn 2 Be 2 Cu 2 cast alloy was developed displaying increased quasicrystalline formation ability at moderate cooling rates. The as-cast microstructure consisted of a mainly icosahedral phase in the Al-matrix. The microstructure remained stable during uniform heating to 580 °C and isothermal annealing at 400 °C. Most of the icosahedral phase was preserved even after 24 h annealing at 500 °C. For that reason, this alloy presents a promising basis for further development of cast Al-alloys containing quasicrystals.
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September 25, 2009
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Electronic structures of Al-TM (transition metal) compounds with rather simple B2, L1 2 and A15 structures are studied. Focusing on the energy bands, which are not hybridized strongly with the TM-d bands, we estimate the number of electrons occupying the states below the gap induced by periodic potentials and hence obtain the average number of valence electrons per atom e / a . Although the values of e / a obtained for the different structures are scattered in spite of combination of the same chemical species, these values are obtained if one assumes similar negative values for effective valence of TM. Roles of the interference effect opening the gap in the nearly-free-electron bands are discussed.
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September 25, 2009
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The origin and stability of quasicrystals are still heavily debated issues. One may ask the same questions about aperiodic crystals in general. For some other classes, there are answers to these questions, and it is useful to compare the situations. A short discussion is given of what is known for the various classes of aperiodic crystals.
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March 31, 2008
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Several Al-alloys strengthened by quasicrystalline phases have been developed over the last few years showing the considerable potential for practical application. Therefore there is a strong need for developing new metallographic methods or adapting the traditional ones in order to identify and characterize quasicrystalline phases in a reliable, quick and economical way. This paper describes different techniques: the classical metallographic method, deep etching, particle extraction technique and cross-sectioning using focused ion beam (FIB), and discusses their advantages and disadvantages when identifying quasicrystalline particles. It was discovered that particle extraction techniques are very powerful methods for the identification of phases according to their morphology, and preparation of quality samples for X-ray diffraction (XRD). Transmission electron microscopy (TEM) analyses are also possible provided the extracted particles are thin enough.
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September 25, 2009
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Mechanical alloying was applied to form a decagonal phase in the Al–Cu–Cr system. Elemental powders of Al 67 Cu 20 Cr 13 , Al 75 Cu 10 Cr 15 , Al 65 Cu 24 Cr 11 , Al 69 Cu 21 Cr 10 and Al 78 Cu 10 Cr 12 compositions (in at%) were mechanically alloyed in a planetary ball mill. Annealing in the temperature range of 500 to 550 °C results in the formation of binary and ternary compounds including the decagonal quasicrystalline phase that was found to be stable at least up to 800 °C and was present in various amounts in all investigated alloys. No icosahedral quasicrystalline phase was found in the samples. The maximum content (95 vol%;) of decagonal quasicrystalline phase was observed for Al 69 Cu 21 Cr 10 compound annealed at 635 °C. Further increase in the annealing temperature results in the quasicrystalline phase transformation into ψ -Al 65 Cu 25 Cr 10 phase, which is an approximant phase to icosahedral quasicrystal.
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September 25, 2009
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We address the problem of uniquely reconstructing F-type icosahedral quasicrystals from few images produced by quantitative high resolution transmission electron microscopy and explain recent results in the discrete tomography of these sets.
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September 25, 2009
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Restricted square-triangle tilings are tilings of regular squares and triangles where the number of vertex types has been limited to a subset of all possible vertex types. Such structures have recently been observed in several places: for example as strip phases in colloidal patterns on laser fields with pentagonal symmetry, and as square-pair-free tilings in patterns formed by dipolar magnetic binary colloidal mixtures. In general there are four vertex configurations: A (4 squares, 4 4 ), Z (6 triangles, 3 6 ), H (4 2 3 3 ), and S (43433). Strip phases belong to type AHZ, square-pair-free phases to SZ, standard inflation generated quasicrystals to HSZ, and unrestricted dodecagonal random tilings to AHSZ. We study several vertex type combinations and describe the properties of the patterns generated. AHZ tilings can be mapped to binary necklaces and classified completely. A similar mapping is possible for many HS tilings. Defect-free SZ tilings can be transformed into a triangle-square-hexagon (TSH) supertiling with matching rules. Monte-Carlo (MC) simulations are used to examine the properties of several classes of restricted tilings.
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September 25, 2009
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There is some confusion in the literature what “modulated quasicrystals” are: Some people treat “modulated quasicrystals” and “deformed model sets” as exchangeable termini (compare [6, 9, 5]), others claim that “[…] the projection method becomes powerless against incommensurate modulated structures” ( e.g. , [12 p. 148]). We use a mathematical approach and propose the following classification: While deformed model sets are characterised by a deformation that does not change the location of the Bragg peaks, a modulated phase yields “satellites” in the diffraction pattern and is achieved “by enlarging the internal space” with a torus.
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March 31, 2008
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The groups of similarity and coincidence rotations of an arbitrary lattice Γ in d -dimensional Euclidean space are considered. It is shown that the group of similarity rotations contains the coincidence rotations as a normal subgroup. Furthermore, the structure of the corresponding factor group is examined. If the dimension d is a prime number, this factor group is an elementary Abelian d -group. Moreover, if Γ is a rational lattice, the factor group is trivial ( d odd) or an elementary Abelian 2-group ( d even).
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September 25, 2009
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A first step in investigating colour symmetries of periodic and nonperiodic patterns is determining the number of colours which allow perfect colourings of the pattern under consideration. A perfect colouring is one where each symmetry of the uncoloured pattern induces a global permutation of the colours. Two cases are distinguished: Either perfect colourings with respect to all symmetries, or with respect to orientation preserving symmetries only (no reflections). For the important class of colourings of regular tilings (and some Laves tilings) of the Euclidean or hyperbolic plane, this mainly combinatorial question is addressed here using group theoretical methods.
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September 25, 2009
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The inverse problem of diffraction theory in essence amounts to the reconstruction of the atomic positions of a solid from its diffraction image. From a mathematical perspective, this is a notoriously difficult problem, even in the idealised situation of perfect diffraction from an infinite structure. Here, the problem is analysed via the autocorrelation measure of the underlying point set, where two point sets are called homometric when they share the same autocorrelation. For the class of mathematical quasicrystals within a given cut and project scheme, the homometry problem becomes equivalent to Matheron´s covariogram problem, in the sense of determining the window from its covariogram. Although certain uniqueness results are known for convex windows, interesting examples of distinct homometric model sets already emerge in the plane. The uncertainty level increases in the presence of diffuse scattering. Already in one dimension, a mixed spectrum can be compatible with structures of different entropy. We expand on this example by constructing a family of mixed systems with fixed diffraction image but varying entropy. We also outline how this generalises to higher dimension.
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September 25, 2009
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Colour symmetries with ten colours are presented for different tilings. In many cases, the existence of these colourings were predicted by group theoretical methods. Only in a few cases explicit constructions were known, sometimes using combination of two-colour and five-colour symmetries. Here we present explicit constructions of several of the predicted colourings for the first time, and discuss them in contrast to already known colourings with ten colours.
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September 25, 2009
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In this work we study the color symmetries pertaining to colorings of M n = Z [ ξ ], where ξ = exp (2π i / n ) for n ∈ {5,8,12} which yield standard symmetries of quasicrystals. The first part of the paper treats M n as a four dimensional lattice Λ with symmetry group G and a result is provided on sublattices of Λ which are invariant under the point group of G . The second part of the paper characterizes the color symmetry groups and color fixing groups corresponding to Bravais colorings of M n using an approach involving ideals.
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September 25, 2009
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The generation of random tilings with arbitrarily high rotational symmetry is discussed from the point of view of substitutions. The methods are based on the analysis of geometric constructions with different allowed orientations increasing in this way the possible types of inflation rules for which the entropy can be computed in an exact way.
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September 25, 2009
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This is a survey of some recent results in the mathematics of diffraction, concentrating on the use of dynamical systems and stochastic point processes.
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September 25, 2009
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The integer Cech cohomology of canonical projection tilings of dimension three and codimension three is derived. These formulae are then evaluated for several icosahedral tilings known from the literature. Rather surprisingly, the cohomologies of all these tilings turn out to have torsion. This is the case even for the Danzer tiling, which is, in some sense, the simplest of all icosahedral tilings. This result is in contrast to the case of two-dimensional canonical projection tilings, where many examples without torsion are known.
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September 25, 2009
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Based on the standard inflation-deflation method to generate substitution tilings, we discuss two types of generalizations in the plane. Firstly, we study substitution rules which do not preserve volume and/or shape of the building blocks. Secondly, we consider prototiles with two or more different possibilities for replacement which can be chosen randomly according to certain probabilities. For illustration, we give various examples of generalized tile substitutions derived from Penrose tiles.
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September 25, 2009
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A material is spherically uniform when its generalized Hooke´s law at each material point referred to a spherical coordinate system is the same everywhere. We study a sphere of spherically uniform linear anisotropic one-dimensional quasicrystalline material subjected to radial phonon and phason strains on its surface. For a certain range of material parameters the phonon as well as the phason stress diverges at the center. The applied phason strain does not affect the phonon field but the phason field is disturbed by the phonon strain. “White hole” at the center may occur when the sphere is subjected to uniform tension while uniform pressure may cause a “black hole”. This behavior depends only on a dimensionless material parameter and the kind of applied strain but not on its magnitude.
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September 25, 2009
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We have analyzed the electrical conductivities of Al–Cu–Fe and Al–Cu–Ru quasicrystals in a low temperature region, taking into account the conducting mechanism of the quasicrystal-like system. The present analysis suggests strongly that not only the weak localization and the electron-electron interaction contributions but also the conducting effect of the quasicrystal-like system should be considered.
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September 25, 2009
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A natural way to describe the Penrose tiling employs the projection method on the basis of the root lattice A 4 or its dual. Properties of these lattices are thus related to properties of the Penrose tiling. Moreover, the root lattice A 4 appears in various other contexts such as sphere packings, efficient coding schemes and lattice quantizers. Here, the lattice A 4 is considered within the icosian ring, whose rich arithmetic structure leads to parametrisations of the similar sublattices and the coincidence rotations of A 4 and its dual lattice. These parametrisations, both in terms of a single icosian, imply an index formula for the corresponding sublattices. The results are encapsulated in Dirichlet series generating functions. For every index, they provide the number of distinct similar sublattices as well as the number of coincidence rotations of A 4 and its dual.
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September 25, 2009
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A recent structural model [1] for the F -type icosahedral phases i -AlPdMn and i -AlCuFe based on the KG cell decomposition [2] of the Mackay clusters has shown that they both enter in an unique model of general composition Al 61.8 (Pd Ι Cu) 21.35 (Mn Ι Fe) 8.29 (Al Ι Fe Ι Mn) 4.28 (Al Ι Cu) 4.28 where ( X Ι Y ) means X or Y species. Three new alloys satisfying this composition rule, two ternaries Al 66.08 Cu 21.35 Mn 12.57 and Al 70.36 Pd 21.35 Fe 8.29 and one quaternary Al 66.08 Cu 21.35 Mn 8.29 Fe 4.28 have been prepared by rapid quenching and annealing and studied by X-ray diffraction and electron microscopy. All three phases are quasicrystalline after quench: Al 66.08 Cu 21.35 Mn 12.57 is decagonal, and the two others are F-icosahedral phases. After annealing, the quaternary alloy transforms into a decagonal phase. The present results are compared to those of previous studies in the litterature of samples with close compositions.
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March 31, 2008
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Complex metallic alloys frequently exhibit atomic clusters as structural units. In quasicrystals, they are densely arranged retaining an overall long-range quasiperiodic translational order. Thus, clusters are geometrical entities. The often posed question is, whether these are also responsible for physical properties. Fracture experiments and simulations strongly point towards a positive answer. However, a recent analysis of experimental fracture surfaces questions a signature of the clusters. From the self-affine behavior of a model system it is shown that the findings cannot vitiate the role of the clusters in the fracture process.
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March 31, 2008
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Complex cubic compounds Zn 6 Sc and Cd 6 M (M = Yb, Ca, Y, rare earth) are approximant crystals of Zn- and Cd-based quasicrystals. A low-temperature structural transition has been observed for these compounds universally and is considered to be due to orientational ordering of an atomic shell in the icosahedral cluster. To get microscopic information on structure and orientational ordering of the atomic shell in the icosahedral cluster, the first-principles structural relaxation for Zn 6 Sc is carried out. We obtain variety of favorable structures for different lattice constants. The results are compared with those for isostructural Cd 6 Ca obtained in the previous work. We stress differences between Zn 6 Sc and Cd 6 Ca and discuss their possible consequences on phase diagrams.
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September 25, 2009
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The influence of limited experimental reciprocal space resolution on the appearance of diffraction patterns from quasicrystals is discussed. Using simulated data from a perfectly ordered one-dimensional Fibonacci chain it is shown that realistic experimental resolution effects may lead to artificial ‘diffuse scattering’ of significant intensity. Two different diffuse features are identified in simulations of macroscopically sized crystals. A broad and almost smooth diffuse background is found, which strongly inceases with decreasing reciprocal space resolution. In addition, a set of of narrow diffuse peaks having profiles similar to thermal and phasonic diffuse scattering is present, which is only seen in high resolution patterns. The former diffuse scattering represents unresolved Bragg reflections, while the latter is due to truncation effects coming from the finite crystal size. The relevance of the findings for the understanding of real experimental diffraction patterns is discussed on the examples of diffuse scattering from d -Al 70 Co 12 Ni 18 and i -Al 64 Cu 23 Fe 13 . Finally, some criteria for distinguishing disorder diffuse scattering from artificial diffuse scattering are given.
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September 25, 2009
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The iron dynamics in the icosahedral quasicrystal i -AlCuFe and two cubic 1/1 approximants as well as the non-approximant cubic B2 phase has been studied by Mössbauer spectroscopy. In the i -phase the iron Lamb-Mössbauer recoilless fraction decreases below that predicted for lattice vibrations alone above about 550 K. This decrease is correlated with the onset of a quasielastic signal at high Doppler velocities, demonstrating local iron jump processes at temperatures above this point. In the two cubic approximants, a loss in the recoilless fraction also occurs above 550 K, revealing the same kind of Fe dynamics as in the i -phase but the effect is smaller. On the other hand, no anomalous Fe dynamics (other than lattice vibrations) is detected in the B2-AlCuFe phase. A comparison is made to literature Mössbauer and neutron scattering results in i -AlCuFe.
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September 25, 2009
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We consider a model decagonal quasicrystal of composition Al 80.1 Co 19.9 – closely related to actual structures, and using realistic pair potentials – on a quasilattice of candidate sites. Its ground state, according to simulations, is a Hexagon-Boat-Star tiling satisfying Penrose´s matching rules. In this note, we rationalize these results in terms of the potentials; the Al–Co second-neighbor potential well is crucial.
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September 25, 2009
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For the first time, the refinement of a decagonal Al 72 Ni 20 Co 8 quasicrystal was conducted in physical space. We used a structure factor obtained in statistical approach. As a reference lattice we took a rhombic Penrose tiling to which we applied overlapping rules of clusters. The resulting structure gives R-factor = 8.0% and R w - factor = 6.1%. The density of the refined structure is very close to the experimental one.
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September 25, 2009
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The electronic structure and atomic arrangement of a decagonal Al–Co–Ni quasicrystal have been investigated in terms of a model unit cluster by using a discrete variational X α (DV- Xα ) cluster calculation. The partial electronic density of states (DOS) and overlap population analysis for local clusters around the constituent transition-metals (TM´s) show the energy reduction of both the Co and Ni 3 d states but the increase in the anti-bonding character for their TM-Al bonds at the second inner ring TM site of the model cluster in comparison to the most inner ring TM site. Comparing the calculated DOS with the observed valence-band photoelectron and TM Lα X-ray emission spectra, we discuss the occupation at the transition metal sites by Co and Ni.
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September 25, 2009
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An analytical expression of the diffuse scattering intensity due to short-range order in quasicrystals is derived. The intensity is calculated on the basis of higher-dimensional cluster models and correlation functions between statistically occupied sites. This expression is applied to the phason flip disordering in the Penrose pattern. The analytical and numerical calculations show quite similar diffuse scattering intensity distribution, suggesting the validity of the derived formula.
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September 25, 2009
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Structure refinement of a water quenched alloy of Al 72 Ni 20 Co 8 , which is known as a highly ordered decagonal quasicrystal, has been carried out by quantitative comparison of experimental and simulated intensities of convergent-beam electron diffraction (CBED) patterns. In the present study, higher-order Laue zone reflections, whose intensities are senstively changed with atom positions, have been used for the refinement of atom position parameters in the quasiperiodic directions. The refined parameters have been compared with the results obtained by X-ray diffraction.
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September 25, 2009
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The Co—Ni distribution in d -Al 69.7(4) Co 10.0(4) Ni 20.3(4) was investigated based on X-ray and neutron diffraction data. The structure was modelled in higher dimensional space using the ‘charge-flipping’ and ‘low-density elimination’ methods and it was quantitatively refined in three-dimensional space employing a pseudo-approximant approach. In higher-dimensional description, the Co atoms are found at the centre of one of the two symmetry independent occupation domains, enclosed by regions mainly occupied by Ni. The other occupation domain is mostly occupied by Al. In physical space Co atoms are located in the centres of small Al pentagons and form pentagonal units, which are arranged in decagonal rings. On these sites Co is partly substituted by Ni, while all other transition metal sites are occupied by Ni and to a minor degree by Al. The fraction of Co found on transition metal sites decreases with decreasing Co–Co distances, whereby Co is replaced by Ni.