Browsing by Author "Gähler, Franz"

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Open Access
Binary tiling quasicrystals and matching rules
(1994) Gähler, Franz; Baake, Michael; Schlottmann, Martin
A general theory on the transfer of perfect matching rules for a quasiperiodic tiling to perfect matching rules for an atomic decoration of the tiling is presented. General conditions on the possibility of such a transfer are discussed, and an upper bound on the range of the matching rules for the atomic structure is derived. This range is identical to the range of interactions needed to stabilize such a quasiperiodic ground state. The main tool in this analysis is the concept of mutual local derivability. The general principles are then applied to two examples of binary tiling quasicrystals. The first one, based on the Tübingen triangle tiling, needs matching rules of a rather long range, whereas the second example, which is a decoration of the Penrose rhombus tiling, has matching rules of reasonable range. Finally, the concepts put forward in this paper are set into a broader context, and we compare them with other theories for the propagation of quasiperiodic order.
Open Access
Classification of space groups for quasicrystals with (2+1)-reducible point group
(1990) Gähler, Franz
Quasicrystalline structures can be viewed as a cut through a periodic structure in a higherdimensional space. In this paper, the crystallography of 5-dimensional periodic structures which can give rise to a quasi crystal in this way is developed. For the five 5d Bravais classes giving rise to quasicrystals periodic in one direction and with symmetry of order 5, 8, 10 or 12 in the plane perpendicular to that direction, a complete classification of point groups and space groups is given. We find a total of 150 distinct space groups, which can be attributed to 44 point groups. For each of these space groups, a set of generators of its point group and the associated non-primitive translations are tabulated, and for the non-symmorphic space groups the indices of the systematic extinctions are given. All the results are expressed in terms of a geometrically natural lattice basis.
Open Access
Comment on "Structure of rapidly quenched Al-Mn"
(1985) Gähler, Franz; Rhyner, Jakob
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Open Access
Comparison of HREM images and contrast simulations for dodecagonal Ni-Cr quasicrystals
(1990) Beeli, Conradin; Gähler, Franz; Nissen, Hans-Ude; Stadelmann, Pierre
ince high-resolution electron micrographs of dodecagonal Ni-Cr quasicrystals are similar in contrast to those of several closely related periodic phases, it has been argued that dodecagonal quasicrystals can be described as a decoration of a dodecagonal quasiperiodic tiling with the same structural units as occur in these periodic phases. In order to corroborate this hypothesis, electron microscopic contrast simulations using such model structures are presented. The models considered are based on different quasiperiodic, twelvefold-symmetric tilings as well as on several periodic tilings leading to the structures of the periodic phases mentioned above. The simulated contrast is in excellent agreement with the experimental electron microscopic images, both for the periodic and for the quasiperiodic structures. It essentially reflects the structure of the underlying tiling, also for the quasiperiodic structures. It is therefore concluded that the usual interpretation of high-resolution structure images is valid not only for periodic but also for the nonperiodic dodecagonal structures, and that therefore the decoration scheme used correctly describes the atomic structure of the dodecagonal quasicrystal.
Open Access
Crystallography of dodecagonal quasicrystals
(1988) Gähler, Franz
A detailed model structure of dodecagonal quasicrystals is proposed which applies to both dodecagonal Ni-Cr and V-Ni or V-Si-Ni. This model structure can be represented as the restriction of a 5d periodic structure to a 3d subspace, which is identified with physical space. The point group and the space group of the 5d periodic structure are determined. The latter is non-symmorphic, containing a set of glide "planes" and a screw axis. These space group elements lead to characteristic extinctions in the Fourier spectrum, which should be experimentally observable. A numeric calculation, which includes multiple scattering effects for electron diffraction, confirms the presence of the extinctions predicted by the space group analysis. The model structure proposed here serves as a very instructive example how crystallographic concepts, such as Bravais type, point group, or space group, can be applied to quasicrystals.
Open Access
The dualisation method revisited : dualisation of product Laguerre complexes as a unifying framework
(1993) Gähler, Franz; Stampfli, Peter
A general framework based on the dualisation of Laguerre cell complexes is presented, which allows to construct and understand a large variety of quasiperiodic tilings, both new ones and well known old ones. The general framework is illustrated with many examples, which are all based on cell complexes which are products of 2-dimensional complexes. The simple structure of these examples makes it particularly easy to understand how the general procedure works. Yet the examples are sufficiently versatile to exploit the power and flexibility of the method.
Open Access
Equivalence of the generalised grid and projection methods for the construction of quasiperiodic tilings
(1986) Gähler, Franz; Rhyner, Jakob
The two main techniques for the generation of quasipcriodic tilings.,de Bruijn's grid method and the projection formalism, are generalised. A very broad class or quasi periodic 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 upplications are discussed.
Open Access
Lattice dynamics of 3-dimensional tilings modelling icosahedral quasicrystals
(1993) Los, Joop; Janssen, Ted; Gähler, Franz
A study of the lattice dynamics of three-dimensional tilings modelling icosahedral quasicrystals is presented. The phonon density of states is calculated, and the character of the eigenstates is determined. Three different types of commensurate approximants are considered, namely symmetrized, perfect and randomized approximants. It appears that the density of states is smoothed by randomization. The participation ratio, which measures the rate of localization of an eigenmode, is given as a function of frequency. Only the states at the very upper end of the frequency spectrum appear to be localized, whereas all other states are extended. The density of states at low frequencies is analyzed in more detail, by applying a Brillouin zone integration over the lowest branches. It is found that these lowest branches scale for successive approximants.
Open Access
Matching rules for quasicrystals : the composition-decomposition method
(1993) Gähler, Franz
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.
Open Access
The phonon spectrum of the octagonal tiling
(1993) Los, Joop; Janssen, Ted; Gähler, Franz
A study of the phonon spectrum of the octagonal tiling is presented, by calculating and analysing the properties of the spectrum of perfect and randomized commensurate approximants with unit cells containing up to 8119 vertices. The total density of states, obtained by numerical integration over the Brillouin zone, exhibits much structure, and in the low frequency range of the spectrum there is deviation from the normal linear behaviour in the form of pseudogaps. For randomized approximants these pseudogaps disappear and the density of states is globally smoothened. It turns out that the widths of the gaps in the dispersion vanish in the low frequency limit. Therefore the scaling behaviour of the lowest branches tends to the behaviour of an absolutely continuous spectrum, which is not the case at higher frequencies. As an application, the vibrational specific heat of the different tiling models is calculated and compared to the specific heat of a square lattice and of a Debye model.
Open Access
Phonons in models for icosahedral quasicrystals : low frequency behaviour and inelastic scattering properties
(1993) Los, Joop; Janssen, Ted; Gähler, Franz
A detailed study of the low frequency behaviour of the phonon spectrum for 3-dimensional tiling models of icosahedral quasicrystals is presented, in commensurate approximations with up to 10 336 atoms per unit cell. The scaling behaviour of the lowest phonon branches shows that the widths of the gaps relative to the bandwidths vanish in the low frequency limit. The density of states at low frequencies is calculated by Brillouin zone integration, using either local linear or local quadratic interpolation of the branch surface. For perfect approximants it appears that there is a deviation from the normal ω2-behaviour already at relatively low frequencies, in the form of pseudogaps. Also randomized approximants are considered, and it turns out that the pseudogaps in the density of states are flattened by randomization. When approaching the quasiperiodic limit, the dispersion of the acoustic branches becomes more and more isotropic, and the two transversal sound velocities tend to the same value. The dynamical structure factor is determined for several approximants, and it is shown that the linearity and the isotropy of the dispersion are extended far beyond the range of the acoustic branches inside the Brillouin zone. A sharply peaked response is observed at low frequencies, and broadening at higher frequencies. To obtain these results, an efficient algorithm based on Lanczos tridiagonalisation is used.
Open Access
Quasicrystal structures from the crystallographic viewpoint
(1988) Gähler, Franz; Fröhlich, J. (Prof. Dr.)
Quasikristalle sind neuartige Phasen, die in schnell abgekühlten Metall-Legierungen vorkommen. Ihre wichtigste Eigenschaft ist, dass Ihre Fourier-Transformierte aus scharfen Bragg-Peaks besteht, deren Positionen und Intensitäten eine Punktsymmetrie haben, die mit einer dreidimensionalen periodischen Struktur nicht verträglich ist. Die Positionen der Bragg-Peaks eines Quasikristalls können jedoch alle als ganzzahlige Linearkombinationen von endlich vielen fundamentalen Wellenvektoren geschrieben werden; dies legt nahe, diese Strukturen als Schnitt durch eine höherdimensionale, periodische Struktur aufzufassen.
Open Access
Quasiperiodic tilings : a generalized grid projection method
(1988) Korepin, Vladimir E.; Gähler, Franz; Rhyner, Jakob
We generalize the grid-projection method for the construction of quasiperiodic tilings. A rather general fundamental domain of the associated higher-dimensional lattice is used for the construction of the acceptance region. The arbitrariness of the fundamental domain allows for a choice which obeys all the symmetries of the lattice, which is important for the construction of tilings with a given non-trivial point-group symmetry in Fourier space. As an illustration, the construction of a two-dimensional quasiperiodic tiling with 12-fold orientational symmetry is described.
Open Access
Scaling properties of vibrational spectra and eigenstates for tilings models of icosahedral quasicrystals
(1993) Los, Joop; Janssen, Ted; Gähler, Franz
A study of the lattice dynamics of 3-dimensional tilings modelling icosahedral quasicrystals is presented, both in commensurate approximations and cluster approximations. In the commensurate approximation this is done for three different types of tilings, namely: perfect, symmetrized and randomized approximants. It turns out that the density of states as a function of frequency is smoothed by randomization. A multifractal analysis of the spectrum shows that mainly at high frequencies the scaling behaviour of the spectrum is different from that for periodic structures. Also the eigenvectors are examined and it appears that only the states at the very upper end of the spectrum have a relatively small participation fraction, i.e. are more localized. The majority of the states scale as normal extended states, as is shown by a multifractal analysis of the eigenvectors for systematic approximants. Also, for most of the states localization is not enhanced by randomization. Throughout the paper the results are compared with those for a 1-dimensional quasicrystal, the Fibonacci chain.
Open Access
Some mathematical problems arising in the study of quasicrystals
(1986) Gähler, Franz
After briefly reviewing the projection method, some properties of quasicrystal structures obtained by projection are discussed. In particular, the connection between the Fourier spectra and the local isomorphism class of a quasicrystal is worked out. It is argued that quasicrystals have no residual entropy at T = 0. Finally, certain self-similarity properties of quasicrystals are discussed.
Open Access
Wave fields in Weyl spaces and conditions for the existence of a preferred pseudo-Riemannian structure
(1984) Audretsch, Jürgen; Gähler, Franz; Straumann, Norbert
Following an axiomatic formulation for general relativity, an equivalent class of pseudo-Riemannian manifolds is derived from the Lorentz conformal structure given by the sets of all possible light rays and all possible free falling objects, the latter set defining the affine geodesics. In particular the set of null geodesics defined by the Lorentz conformal structure characterize a Weyl space-time structure. \par It has been shown by one of the authors that when the classical trajectories of matter fields defined over the Weyl space-time are assumed to coincide with the geodesics of that space, then a set of Lorentzian space-times arise naturally. In the present paper the authors improve this result by use of fiber bundles and the WKB limit to derive the short wave limits to Dirac and Klein-Gordon fields.