176 research outputs found
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 comments on the inverse problem of pure point diffraction
In a recent paper, Lenz and Moody (arXiv:1111.3617) presented a method for
constructing families of real solutions to the inverse problem for a given pure
point diffraction measure. Applying their technique and discussing some
possible extensions, we present, in a non-technical manner, some examples of
homometric structures.Comment: 6 pages, contribution to Aperiodic 201
What is a crystal?
Almost 25 years have passed since Shechtman discovered quasicrystals, and 15
years since the Commission on Aperiodic Crystals of the International Union of
Crystallography put forth a provisional definition of the term crystal to mean
``any solid having an essentially discrete diffraction diagram.'' Have we
learned enough about crystallinity in the last 25 years, or do we need more
time to explore additional physical systems? There is much confusion and
contradiction in the literature in using the term crystal. Are we ready now to
propose a permanent definition for crystal to be used by all? I argue that time
has come to put a sense of order in all the confusion.Comment: Submitted to Zeitschrift fuer Kristallographi
An aperiodic hexagonal tile
We show that a single prototile can fill space uniformly but not admit a
periodic tiling. A two-dimensional, hexagonal prototile with markings that
enforce local matching rules is proven to be aperiodic by two independent
methods. The space--filling tiling that can be built from copies of the
prototile has the structure of a union of honeycombs with lattice constants of
, where sets the scale of the most dense lattice and takes all
positive integer values. There are two local isomorphism classes consistent
with the matching rules and there is a nontrivial relation between these
tilings and a previous construction by Penrose. Alternative forms of the
prototile enforce the local matching rules by shape alone, one using a
prototile that is not a connected region and the other using a
three--dimensional prototile.Comment: 32 pages, 24 figures; submitted to Journal of Combinatorial Theory
Series A. Version 2 is a major revision. Parts of Version 1 have been
expanded and parts have been moved to a separate article (arXiv:1003.4279
Space-Time Approach to Scattering from Many Body Systems
We present scattering from many body systems in a new light. In place of the
usual van Hove treatment, (applicable to a wide range of scattering processes
using both photons and massive particles) based on plane waves, we calculate
the scattering amplitude as a space-time integral over the scattering sample
for an incident wave characterized by its correlation function which results
from the shaping of the wave field by the apparatus. Instrument resolution
effects - seen as due to the loss of correlation caused by the path differences
in the different arms of the instrument are automatically included and analytic
forms of the resolution function for different instruments are obtained. The
intersection of the moving correlation volumes (those regions where the
correlation functions are significant) associated with the different elements
of the apparatus determines the maximum correlation lengths (times) that can be
observed in a sample, and hence, the momentum (energy) resolution of the
measurement. This geometrical picture of moving correlation volumes derived by
our technique shows how the interaction of the scatterer with the wave field
shaped by the apparatus proceeds in space and time. Matching of the correlation
volumes so as to maximize the intersection region yields a transparent,
graphical method of instrument design. PACS: 03.65.Nk, 3.80 +r, 03.75, 61.12.BComment: Latex document with 6 fig
A time lens for high resolution neutron time of flight spectrometers
We examine in analytic and numeric ways the imaging effects of temporal
neutron lenses created by traveling magnetic fields. For fields of parabolic
shape we derive the imaging equations, investigate the time-magnification, the
evolution of the phase space element, the gain factor and the effect of finite
beam size. The main aberration effects are calculated numerically. The system
is technologically feasible and should convert neutron time of flight
instruments from pinhole- to imaging configuration in time, thus enhancing
intensity and/or time resolution. New fields of application for high resolution
spectrometry may be opened.Comment: 8 pages, 11 figure
Overlapping Unit Cells in 3d Quasicrystal Structure
A 3-dimensional quasiperiodic lattice, with overlapping unit cells and
periodic in one direction, is constructed using grid and projection methods
pioneered by de Bruijn. Each unit cell consists of 26 points, of which 22 are
the vertices of a convex polytope P, and 4 are interior points also shared with
other neighboring unit cells. Using Kronecker's theorem the frequencies of all
possible types of overlapping are found.Comment: LaTeX2e, 11 pages, 5 figures (8 eps files), uses iopart.class. Final
versio
Pattern equivariant functions and cohomology
The cohomology of a tiling or a point pattern has originally been defined via
the construction of the hull or the groupoid associated with the tiling or the
pattern. Here we present a construction which is more direct and therefore
easier accessible. It is based on generalizing the notion of equivariance from
lattices to point patterns of finite local complexity.Comment: 8 pages including 2 figure
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