Franz Gähler

Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings

The two main techniques for the generation of quasiperiodic tilings, de Bruijns grid method (1981), and the projection formalism, are generalised. A very broad class of quasiperiodic tilings is obtained in this way. The two generalised methods are shown to be equivalent. The standard calculation of Fourier spectra is extended to the whole general class of tilings. Various applications are discussed.

Potfit: effective potentials from ab initio data

We present a program called potfit which generates an effective atomic interaction potential by matching it to a set of reference data computed in first-principles calculations. It thus allows to perform large-scale atomistic simulations of materials with physically justified potentials. We describe the fundamental principles behind the program, emphasizing its flexibility in adapting to different systems and potential models, while also discussing its limitations. The program has been used successfully in creating effective potentials for a number of complex intermetallic alloys, notably quasicrystals.

Effective potentials for quasicrystals from ab-initio data

Classical effective potentials are indispensable for any large-scale atomistic simulations, and the relevance of simulation results crucially depends on the quality of the potentials used. For complex alloys such as quasicrystals, however, realistic effective potentials are almost non-existent. We report here our efforts to develop effective potentials especially for quasicrystalline alloy systems. We use the so-called force-matching method, in which the potential parameters are adapted so as to reproduce the forces and energies optimally in a set of suitably chosen reference configurations. These reference data are calculated with ab-initio methods. As a first application, embedded-atom method potentials for decagonal Al–Ni–Co, icosahedral Ca–Cd, and both icosahedral and decagonal Mg–Zn quasicrystals have been constructed. The influence of the potential range and degree of specialization on the accuracy and other properties is discussed and compared.

A MOLECULAR DYNAMICS RUN WITH 5 180 116 000 PARTICLES

We report on the development of IMD, a scalable program for classical molecular dynamics simulations on massively parallel supercomputers. New features like online-visualization and metacomputing are described.

Matching rules for quasicrystals: the composition-decomposition method

A general method is presented which proves that an appropriately chosen set of matching rules for a quasiperiodic tiling enforces quasiperiodicity. This method, which is based on self-similarity, is formulated in general terms to make it applicable to many different situations. The method is then illustrated with two examples, one of which is a new set of matching rules for a dodecagonal tiling.

Phason elastic constants of a binary tiling quasicrystal

For a two-dimensional binary tiling model quasicrystal, the full set of (zero temperature) elastic constants is determined. It is found that the elastic energy is a perfect quadratic form in the phonon and phason strains. One of the phason elastic constants turns out to be negative, implying that the quasicrystal is only metastable at zero temperature.

Spectral and topological properties of a family of generalised Thue-Morse sequences

The classic middle-thirds Cantor set leads to a singular continuous measure via a distribution function that is known as the Devils staircase. The support of the Cantor measure is a set of zero Lebesgue measure. Here, we discuss a class of singular continuous measures that emerge in mathematical diffraction theory and lead to somewhat similar distribution functions, yet with significant differences. Various properties of these measures are derived. In particular, these measures have supports of full Lebesgue measure and possess strictly increasing distribution functions. In this sense, they mark the opposite end of what is possible for singular continuous measures. For each member of the family, the underlying dynamical system possesses a topological factor with maximal pure point spectrum, and a close relation to a solenoid, which is the Kronecker factor of the system. The inflation action on the continuous hull is sufficiently explicit to permit the calculation of the corresponding dynamical zeta funct...

Direct Wolf summation of a polarizable force field for silica

We extend the Wolf direct, pairwise r(-1) summation method with spherical truncation to dipolar interactions in silica. The Tangney-Scandolo interatomic force field for silica takes regard of polarizable oxygen atoms whose dipole moments are determined by iteration to a self-consistent solution. With Wolf summation, the computational effort scales linearly in the system size and can easily be distributed among many processors, thus making large-scale simulations of dipoles possible. The details of the implementation are explained. The approach is validated by estimations of the error term and simulations of microstructural and thermodynamic properties of silica.

Wave fields in Weyl spaces and conditions for the existence of a preferred pseudo-Riemannian structure

Previous axiomatic approaches to general relativity which led to a Weylian structure of space-time are supplemented by a physical condition which implies the existence of a preferred pseudo-Riemannian structure. It is stipulated that the trajectories of the short wave limit of classical massive fields agree with the geodesics of the Weyl connection and it is shown that this is equivalent to the vanishing of the covariant derivative of a “mass function” of nontrivial Weyl type. This in turn is proven to be equivalent to the existence of a preferred metric of the conformal structure such that the Weyl connection is reducible to a connection of the bundle of orthonormal frames belonging to this distinguished metric.

Cluster model of decagonal tilings

A relaxed version of Gummelts 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, such as those produced by Gummelts perfect covering rules.