The atomic structure of quasicrystals1 — solids with long-range order, but non-periodic atomic lattice structure — is often described as the three-dimensional generalization of the planar two-tile Penrose pattern2. Recently, an alternative model has been proposed3,4,5 that describes such structures in terms of a single repeating unit3,4,5 — the three-dimensional generalization of a pattern composed of identical decagons. This model is similar in concept to the unit-cell description of periodic crystals, with the decagon playing the role of a ‘quasi-unit cell’. But, unlike the unit cells in periodic crystals, these quasi-unit cells overlap their neighbours, in the sense that they share atoms. Nevertheless, the basic concept of unit cells in both periodic crystals and quasicrystals is essentially the same: solving the entire atomic structure of the solid reduces to determining the distribution of atoms in the unit cell. Here we report experimental evidence for the quasi-unit-cell model by solving the structure of the decagonal quasicrystal Al72Ni20Co8. The resulting structure is consistent with images obtained by electron and X-ray diffraction, and agrees with the measured stoichiometry, density and symmetry of the compound. The quasi-unit-cell model provides a significantly better fit to these results than all previous alternative models, including Penrose tiling.

Experimental verification of the quasi-unit-cell model of quasicrystal structure

The structure of a decagonal Al_{72}Ni_{20}Co_{8} quasicrystal with space group P10_{5}/mmc has been determined on the basis of a single-crystal X-ray data set using the five-dimensional description. The best-fit model structure based on a cluster model having lower symmetry than the decagonal symmetry with 103 parameters gives wR = 0.045 and R = 0.063 for 449 reflections. The structure is well described by the hexagon, boat and star tiling with an edge length of 6.36 A and is very consistent with recent high-resolution electron-microscopy images. The refined structure is compared with previously discussed model structures including cluster-based models having 20 A tenfold symmetric clusters.

The structure of a decagonal Al72Ni20Co8 quasicrystal

The atomic surfaces modelling technique has been used to solve the structure of the basic Ni-rich Al–Co–Ni decagonal phase. Formula Al70.6Co6.7Ni22.7, space group P\overline{10}, five-dimensional unit-cell parameters: d1 = d4 = 4.752 (3) A, d2 = d3 = 3.360 (2) A, d5 = 8.1710 (2) A; α12 = α34 = 69.295°, α13 = α24 = 45°, α14 = 41.410°, α23 = αi5 = 90° (i = 1–4), V = 291.2 (7) A5; Dx = 3.887 Mg m−3. Refinement based on |F|; 2767 unique reflections (|F| > 0), 749 parameters, R = 0.17, wR = 0.06. Describing the structure of quasicrystals embedded in n-dimensional superspace in principle takes advantage of n-dimensional periodicity to select the minimal set of degrees of freedom for the structure. The method of modelling of the atomic surfaces yielded the first fully detailed structure solution of this phase. Comparison with numerous former, less accurate models confirms several features already derived, but adds a new essential insight of the structure and its complexity. The atoms fill the space forming recurrent structure motifs, which we will (generically) refer to as clusters. However, no unique cluster exists, although differences are small. Each cluster shows a high degree of structural disorder. This gives rise to a large configurational entropy, as much as expected in a phase which is stable at high temperature. On the other side, the cluster spatial arrangement is perfectly quasiperiodic. These considerations, corroborated by analysis of the structural relationship with neighbouring periodic phases, strongly suggest the existence of a non-local, long-range interaction term in the total energy which may be essential to the stability.

Structure solution of the basic decagonal Al-Co-Ni phase by the atomic surfaces modelling method.

Detailed investigations of the atomic and electronic structures of decagonal AlNiCo alloys have been performed. Several different models for the decagonal structure have been investigated: A model based on a rhombic-hexagon tiling proposed by Henley and models based on a cluster decoration of the Penrose tiling with large rhombus edge. The topology of the structural models has been refined on the basis of the existing x-ray-diffraction data which, however, do not allow us to specify the chemical decoration uniquely. The chemical order on the decagonal lattice has been optimized via the comparison of the calculated electronic spectra with photoemission and soft-x-ray data and using total-energy calculations. The electronic structure calculations for large periodic approximants with up to 1276 atoms/cell have been performed self-consistently using a real-space tight-binding linear muffin-tin orbital technique. The best agreement with the experimental spectra is achieved for a large-rhombus-tiling model with the innermost ring of the pentagonal columnar clusters occupied by Ni atoms only. This configuration also has the lowest total energy. As in decagonal AlCuCo we find a high density of states at the Fermi level, but the chemical ordering is very different: whereas in d-AlCuCo direct Cu-Cu neighbors are suppressed and there is a slight preference for Co-Co homocoordination, in $d\ensuremath{-}\mathrm{AlNiCo}$ a strong Ni-Ni interaction stabilizes the innermost Ni ring, direct Co-Co neighbors are suppressed and there is a strong Co-Al interaction.

Atomic and electronic structure of decagonal Al-Ni-Co alloys and approximant phases

After a brief historical survey, this paper introduces the notion of entropic model sets (cut and project sets), and, more generally, the notion of diffractive point sets with entropy. Such sets may be thought of as generalizations of lattice gases. We show that taking the site occupation of a model set stochastically results, with probabilistic certainty, in well-defined diffractive properties augmented by a constant diffuse background. We discuss both the case of independent, but identically distributed (i.i.d.) random variables and that of independent, but different (i.e. site dependent) random variables. Several examples are shown.

Diffractive point sets with entropy

In a decagonal quasicrystal with composition AI70Co15Ni15 two different types of superstructure have been found by electron microscopy techniques. One of these superstructures has previously been observed. The other newly characterized superstructure causes a ring-like zone of low diffraction intensities and, additionally, reflections at positions having half-integer indices. Both types of superstructure have a doubled period of 8·2 A along the unique tenfold axis, which is suggested here to be due to Al/TM (TM= Co, Ni) ordering on different layers. The main structural elements are columns with a diameter of approximately 20 A. During irradiation with a 300 keV electron beam both superstructures are transformed to a ‘basic’ decagonal structure with a 4·1 A period, in which the wheel-like contrast features with 20 A diameter are absent. For the more disordered basic decagonal structure no conspicuous tiling can be found which can be superimposed onto the electron micrographs, while the new superst...

Two different superstructures of the decagonal Al-Co-Ni quasicrystal

Reference CIME-ARTICLE-1996-012doi:10.1103/PhysRevLett.76.2507View record in Web of Science Record created on 2007-02-15, modified on 2017-05-12

Phason related stacking disorder in decagonal Al-Co-Ni.

A stable one-dimensional quasicrystal has been found for Al-Co-Ni which is related closely to the decagonal phase in this system. It exists as a high-temperature state above 900 C in a range of compositions between approximately 71 and 71.5at.% Al between 18at.% and 22at.% Co. The modulation length along the periodic direction in its pseudo tenfold diffraction patterns is approximately 61A. We show that the 20A edge length defective rhomb tiling superimposed on to a corresponding high-resolution image, defined by centres of atomic clusters, can be considered as a subset of a phason-strained, 2.5A edge length Penrose tiling that has the observed period. An ideal 20A edge length tiling based on special configurations of this Penrose tiling can be chosen as a prototype for the observed experimental tiling.

A stable one-dimensional quasicrystal related to decagonal Al-Co-Ni

A one-dimensionally periodic pentagonal quasicrystal with a superstructure has been found in the Al-Co-Ni system in a sample quenched from 1160oC at the composition Al71.5Co25.5Ni3. Its electron diffraction patterns are characterized by a fivefold rotation axis as a unique symmetry element as well as superstructure reflections similar to those of a related decagonal phase. The presence of this symmetry in electron diffraction patterns originates from strong dynamical scattering effects of electrons in relatively thick regions of the sample, in combination with the unique parallel arrangement of fivefold symmetric atom clusters. Differently, the superstructure reflections are caused by the long-range order of this new state which can be described by a random variant of a Penrose rhomb tiling.

Pentagonal Al-Co-Ni quasicrystal with a superstructure

A subdivision of the stability region of decagonal Al-Co-Ni quasicrystals into eight different structural modifications is presented for the temperature against composition section along the maximum extension of the decagonal phase. The only methods identifying this subdivision are transmission electron microscopy techniques. Besides two decagonal superstructures (one of which also exists as a high-temperature modification), two variants of the basic decagonal state were observed. Also, a one-dimensionally periodic fivefold quasicrystal (together with its high-temperature superstructure modification) and a one-dimensional quasicrystal are parts of the quasicrystalline stability region. Both have a close structural relationship to the decagonal states. All samples investigated were obtained by quenching from the corresponding temperatures. The eight modifications can be differentiated by a set of attributes, especially by their characteristic diffraction patterns taken with the electron beam parallel to the unique periodic axis.

The existence regions of structural modifications in decagonal Al-Co-Ni

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