26 research outputs found
Rules for Computing Symmetry, Density and Stoichiometry in a Quasi-Unit-Cell Model of Quasicrystals
The quasi-unit cell picture describes the atomic structure of quasicrystals
in terms of a single, repeating cluster which overlaps neighbors according to
specific overlap rules. In this paper, we discuss the precise relationship
between a general atomic decoration in the quasi-unit cell picture atomic
decorations in the Penrose tiling and in related tiling pictures. Using these
relations, we obtain a simple, practical method for determining the density,
stoichiometry and symmetry of a quasicrystal based on the atomic decoration of
the quasi-unit cell taking proper account of the sharing of atoms between
clusters.Comment: 14 pages, 8 figure
Cluster Model of Decagonal Tilings
A relaxed version of Gummelt's covering rules for the aperiodic decagon is
considered, which produces certain random-tiling-type structures. These
structures are precisely characterized, along with their relationships to
various other random tiling ensembles. The relaxed covering rule has a natural
realization in terms of a vertex cluster in the Penrose pentagon tiling. Using
Monte Carlo simulations, it is shown that the structures obtained by maximizing
the density of this cluster are the same as those produced by the corresponding
covering rules. The entropy density of the covering ensemble is determined
using the entropic sampling algorithm. If the model is extended by an
additional coupling between neighboring clusters, perfectly ordered structures
are obtained, like those produced by Gummelt's perfect covering rules.Comment: 10 pages, 20 figures, RevTeX; minor changes; to be published in Phys.
Rev.
Some Recent Results on Pair Correlation Functions and Susceptibilities in Exactly Solvable Models
Using detailed exact results on pair-correlation functions of Z-invariant
Ising models, we can write and run algorithms of polynomial complexity to
obtain wavevector-dependent susceptibilities for a variety of Ising systems.
Reviewing recent work we compare various periodic and quasiperiodic models,
where the couplings and/or the lattice may be aperiodic, and where the Ising
couplings may be either ferromagnetic, or antiferromagnetic, or of mixed sign.
We present some of our results on the square-lattice fully-frustrated Ising
model. Finally, we make a few remarks on our recent works on the pentagrid
Ising model and on overlapping unit cells in three dimensions and how these
works can be utilized once more detailed results for pair correlations in,
e.g., the eight-vertex model or the chiral Potts model or even
three-dimensional Yang-Baxter integrable models become available.Comment: LaTeX2e using iopart.cls, 10 pages, 5 figures (5 eps files), Dunk
Island conference in honor of 60th birthday of A.J. Guttman
Quasicrystals: Atomic coverings and windows are dual projects
In the window approach to quasicrystals, the atomic position space E_parallel
is embedded into a space E^n = E_parallel + E_perp. Windows are attached to
points of a lattice Lambda \in E^n. For standard 5fold and icosahedral tiling
models, the windows are perpendicular projections of dual Voronoi and Delone
cells from Lambda. Their cuts by the position space E_parallel mark tiles and
atomic positions. In the alternative covering approach, the position space is
covered by overlapping copies of a quasi-unit cell which carries a fixed atomic
configuration. The covering and window approach to quasicrystals are shown to
be dual projects: D- and V- clusters are defined as projections to position
space E_parallel of Delone or Voronoi cells. Decagonal V-clusters in the
Penrose tiling, related to the decagon covering, and two types of pentagonal
D-clusters in the triangle tiling of 5fold point symmetry with their windows
are analyzed. They are linked, cover position space and have definite windows.
For functions compatible with the tilings they form domains of definition. For
icosahedral tilings the V-clusters are Kepler triacontahedra, the D-clusters
are two icosahedra and one dodecahedron.Comment: 15 pages, 7 figures, see also
http://homepages.uni-tuebingen.de/peter.kramer/ corrections, appendix A,B ne
Icosahedral multi-component model sets
A quasiperiodic packing Q of interpenetrating copies of C, most of them only
partially occupied, can be defined in terms of the strip projection method for
any icosahedral cluster C. We show that in the case when the coordinates of the
vectors of C belong to the quadratic field Q[\sqrt{5}] the dimension of the
superspace can be reduced, namely, Q can be re-defined as a multi-component
model set by using a 6-dimensional superspace.Comment: 7 pages, LaTeX2e in IOP styl
A quasi-unit cell model for Al-Ni-Co Ideal Quasicrystal based on clusters with broken 10-fold symmetry
We present new evidence supporting the quasi-unit cell description of the
decagonal quasicrystal which shows that the solid is
composed of repeating, overlapping decagonal cluster columns with broken
10-fold symmetry. We propose an atomic model which gives a significantly
improved fit to electron microscopy experiments compared to a previous proposal
by us and to alternative proposals with 10-fold symmetric clusters.Comment: 4 pages, 4 eps figures, use epsfig.sty and revtex revised text and
figure