Shelomo I. Ben-Abraham

What is a crystal

In April 1991 the Commission on Aperiodic Crystals [1] adopted the following definition of a crystal:“[In the following] by ‘crystal’ we mean any solid having an essentially discrete diffraction diagram, [and by ‘aperiodic crystal’ we mean any crystal in which three-dimensional lattice periodicity can be considered to be absent.]”This definition has, in my opinion, two flaws. …

Exact solution to the sine-Gordon equation in two space dimensions

Abstract An exact doubly periodic solution to the sine-Gordon equation in two space dimensions is found. The limiting case corresponds to a two-dimensional soliton.

Dodecagonal tilings as maximal cluster coverings

Abstract It is shown that the Socolar tiling, which is quasiperiodic and 12-fold symmetric, can be characterized as the unique tiling which is maximally covered by a suitably pair of clusters. Analogous results can be obtained also for other dodecagonal tilings, among them the shield tiling.

Hybrid quasiperiodic-periodic structures constructed by projection in two stages

A two-stage variant of the cut-and-project method is presented, in which a periodic structure is cut and projected in a high-dimensional space onto three-dimensional physical space so that a second cut and projection onto a plane yields a quasiperiodic structure. The method is applied to the cases of octagonal, dodecagonal and pentagonal/decagonal symmetry. The focus is on the three-dimensional intermediate hybrid structures that are partly quasiperiodic and partly periodic. The method can be generalized to other symmetries as well as to include more intermediate steps.

Generation of Quasiperiodic Order by Maximal Cluster Covering

In quasicrystals, certain structural motifs occur very frequently, and sometimes even cover the entire structure. This property is particularly visible in high-resolution electron micrographs (HREMs) of decagonal quasicrystals. In this chapter, these important structural motifs will be called clusters. On the basis of the observation that at least some quasicrystals can be regarded as being covered by a single kind of cluster, Burkov [1] was one of the first to propose a structure model which was explicitly given as a covering with overlapping copies of a single cluster.

Dodecagonal tilings almost covered by a single cluster

Abstract A single covering cluster cannot produce a rigorously quasiperiodic dodecagonal structure but may almost do so and thus be relevant to real quasicrystals. Such ‘almost covering” patches are shown for the ship, shield, square-triangle and Socolar tiling. The ‘almost covering7rdquo; of the Socolar tiling has a fractal feature.

Curious properties of simple random walks

A simple random walker on the line of integers shows remarkable similarities to relativistic particles.

The 'random' square-triangle tiling : simulation of growth

Abstract Some alloy systems, such as Ni–Cr, V–Ni–Si and Ta–Te, have quasicrystalline phases with 12-fold symmetry. These structures may be described in terms of dodecagonal tilings by equilateral triangles and squares. The formation of quasicrystals still poses a problem, since local information is insufficient for the construction of a perfect quasiperiodic structure. The growth of real quasicrystals may be due to several mechanisms. We have simulated the growth of a quasicrystal from a melt, consisting of squares and equilateral triangles of equal edge length. We are interested in the abundancies of the vertex configurations formed, both regular and defective. Unrestricted random growth tends to result in segregation of triangles from squares. Favoring triangles to attract squares and vice versa brings about nearly perfect patterns with nearly perfect vertex abundancies, as well as realistic defect concentrations. We have also calculated the exact vertex frequencies of the ideal square–triangle tiling by relying on inflation symmetry.

Regular and defective vertex configurations in icosahedral structures

Abstract We have constructed all 5450 combinatorially possible vertex configurations formed by the Ammann rhombohedra. The defectiveness of the vertex configurations is quantified by a penalty functional defined as the minimal sum of squared distances between the dual acceptance domains for the constituent tiles of a given vertex. The vertices are then classified as ideal versus defective, quasiregular versus strictly forbidden, and regular versus singular. The classification is refined by defining rank as the dimension of the dual overlap and degree as the highest dimension of a facet violating the conditions of regularity.

Defective vertices in perfect icosahedral quasicrystals

All combinatorially possible vertex configurations of the two rhombohedral prototiles of the primitive icosahedral tiling can be locally embedded into the perfect tiling. They can be created and relaxed by simpleton flips.