Surface structure of i-Al(68)Pd(23)Mn(9): An analysis based on the T*(2F) tiling decorated by Bergman polytopes

A Fibonacci-like terrace structure along a 5fold axis of i-Al(68)Pd(23)Mn(9) monograins has been observed by T.M. Schaub et al. with scanning tunnelling microscopy (STM). In the planes of the terraces they see patterns of dark pentagonal holes. These holes are well oriented both within and among terraces. In one of 11 planes Schaub et al. obtain the autocorrelation function of the hole pattern. We interpret these experimental findings in terms of the Katz-Gratias-de Boisseu-Elser model. Following the suggestion of Elser that the Bergman clusters are the dominant motive of this model, we decorate the tiling T*(2F) by the Bergman polytopes only. The tiling T*(2F) allows us to use the powerful tools of the projection techniques. The Bergman polytopes can be easily replaced by the Mackay polytopes as the decoration objects. We derive a picture of ``geared'' layers of Bergman polytopes from the projection techniques as well as from a huge patch. Under the assumption that no surface reconstruction takes place, this picture explains the Fibonacci-sequence of the step heights as well as the related structure in the terraces qualitatively and to certain extent even quantitatively. Furthermore, this layer-picture requires that the polytopes are cut in order to allow for the observed step heights. We conclude that Bergman or Mackay clusters have to be considered as geometric building blocks of the i-AlPdMn structure rather than as energetically stable entities

A maximum density rule for surfaces of quasicrystals

A rule due to Bravais of wide validity for crystals is that their surfaces correspond to the densest planes of atoms in the bulk of the material. Comparing a theoretical model of i-AlPdMn with experimental results, we find that this correspondence breaks down and that surfaces parallel to the densest planes in the bulk are not the most stable, i.e. they are not so-called bulk terminations. The correspondence can be restored by recognizing that there is a contribution to the surface not just from one geometrical plane but from a layer of stacked atoms, possibly containing more than one plane. We find that not only does the stability of high-symmetry surfaces match the density of the corresponding layer-like bulk terminations but the exact spacings between surface terraces and their degree of pittedness may be determined by a simple analysis of the density of layers predicted by the bulk geometric model.Comment: 8 pages of ps-file, 3 Figs (jpg

Oxygen adsorption on the Ru (10 bar 1 0) surface: Anomalous coverage dependence

Oxygen adsorption onto Ru (10 bar 1 0) results in the formation of two ordered overlayers, i.e. a c(2 times 4)-2O and a (2 times 1)pg-2O phase, which were analyzed by low-energy electron diffraction (LEED) and density functional theory (DFT) calculation. In addition, the vibrational properties of these overlayers were studied by high-resolution electron loss spectroscopy. In both phases, oxygen occupies the threefold coordinated hcp site along the densely packed rows on an otherwise unreconstructed surface, i.e. the O atoms are attached to two atoms in the first Ru layer Ru(1) and to one Ru atom in the second layer Ru(2), forming zigzag chains along the troughs. While in the low-coverage c(2 times 4)-O phase, the bond lengths of O to Ru(1) and Ru(2) are 2.08 A and 2.03 A, respectively, corresponding bond lengths in the high-coverage (2 times 1)-2O phase are 2.01 A and 2.04 A (LEED). Although the adsorption energy decreases by 220 meV with O coverage (DFT calculations), we observe experimentally a shortening of the Ru(1)-O bond length with O coverage. This effect could not be reconciled with the present DFT-GGA calculations. The nu(Ru-O) stretch mode is found at 67 meV [c(2 times 4)-2O] and 64 meV [(2 times 1)pg-2O].Comment: 10 pages, figures are available as hardcopies on request by mailing [email protected], submitted to Phys. Rev. B (8. Aug. 97), other related publications can be found at http://www.rz-berlin.mpg.de/th/paper.htm

Implicit-explicit timestepping with finite element approximation of reaction-diffusion systems on evolving domains

We present and analyse an implicit-explicit timestepping procedure with finite element spatial approximation for a semilinear reaction-diffusion systems on evolving domains arising from biological models, such as Schnakenberg's (1979). We employ a Lagrangian formulation of the model equations which permits the error analysis for parabolic equations on a fixed domain but introduces technical difficulties, foremost the space-time dependent conductivity and diffusion. We prove optimal-order error estimates in the \Lp{\infty}(0,T;\Lp{2}(\W)) and \Lp{2}(0,T;\Hil{1}(\W)) norms, and a pointwise stability result. We remark that these apply to Eulerian solutions. Details on the implementation of the Lagrangian and the Eulerian scheme are provided. We also report on a numerical experiment for an application to pattern formation on an evolving domain

Robust formation of morphogen gradients

We discuss the formation of graded morphogen profiles in a cell layer by nonlinear transport phenomena, important for patterning developing organisms. We focus on a process termed transcytosis, where morphogen transport results from binding of ligands to receptors on the cell surface, incorporation into the cell and subsequent externalization. Starting from a microscopic model, we derive effective transport equations. We show that, in contrast to morphogen transport by extracellular diffusion, transcytosis leads to robust ligand profiles which are insensitive to the rate of ligand production

Effect of randomness and anisotropy on Turing patterns in reaction-diffusion systems

We study the effect of randomness and anisotropy on Turing patterns in reaction-diffusion systems. For this purpose, the Gierer-Meinhardt model of pattern formation is considered. The cases we study are: (i)randomness in the underlying lattice structure, (ii)the case in which there is a probablity p that at a lattice site both reaction and diffusion occur, otherwise there is only diffusion and lastly, the effect of (iii) anisotropic and (iv) random diffusion coefficients on the formation of Turing patterns. The general conclusion is that the Turing mechanism of pattern formation is fairly robust in the presence of randomness and anisotropy.Comment: 11 pages LaTeX, 14 postscript figures, accepted in Phys. Rev.

Quantitative analysis of cell types during growth and morphogenesis in Hydra

Tissue maceration was used to determine the absolute number and the distribution of cell types in Hydra. It was shown that the total number of cells per animal as well as the distribution of cells vary depending on temperature, feeding conditions, and state of growth. During head and foot regeneration and during budding the first detectable change in the cell distribution is an increase in the number of nerve cells at the site of morphogenesis. These results and the finding that nerve cells are most concentrated in the head region, diminishing in density down the body column, are discussed in relation to tissue polarity

On the Importance of Countergradients for the Development of Retinotopy: Insights from a Generalised Gierer Model

During the development of the topographic map from vertebrate retina to superior colliculus (SC), EphA receptors are expressed in a gradient along the nasotemporal retinal axis. Their ligands, ephrin-As, are expressed in a gradient along the rostrocaudal axis of the SC. Countergradients of ephrin-As in the retina and EphAs in the SC are also expressed. Disruption of any of these gradients leads to mapping errors. Gierer's (1981) model, which uses well-matched pairs of gradients and countergradients to establish the mapping, can account for the formation of wild type maps, but not the double maps found in EphA knock-in experiments. I show that these maps can be explained by models, such as Gierer's (1983), which have gradients and no countergradients, together with a powerful compensatory mechanism that helps to distribute connections evenly over the target region. However, this type of model cannot explain mapping errors found when the countergradients are knocked out partially. I examine the relative importance of countergradients as against compensatory mechanisms by generalising Gierer's (1983) model so that the strength of compensation is adjustable. Either matching gradients and countergradients alone or poorly matching gradients and countergradients together with a strong compensatory mechanism are sufficient to establish an ordered mapping. With a weaker compensatory mechanism, gradients without countergradients lead to a poorer map, but the addition of countergradients improves the mapping. This model produces the double maps in simulated EphA knock-in experiments and a map consistent with the Math5 knock-out phenotype. Simulations of a set of phenotypes from the literature substantiate the finding that countergradients and compensation can be traded off against each other to give similar maps. I conclude that a successful model of retinotopy should contain countergradients and some form of compensation mechanism, but not in the strong form put forward by Gierer

Density functional study of the adsorption of K on the Ag(111) surface

Full-potential gradient corrected density functional calculations of the adsorption of potassium on the Ag(111) surface have been performed. The considered structures are Ag(111) (root 3 x root 3) R30degree-K and Ag(111) (2 x 2)-K. For the lower coverage, fcc, hcp and bridge site; and for the higher coverage all considered sites are practically degenerate. Substrate rumpling is most important for the top adsorption site. The bond length is found to be nearly identical for the two coverages, in agreement with recent experiments. Results from Mulliken populations, bond lengths, core level shifts and work functions consistently indicate a small charge transfer from the potassium atom to the substrate, which is slightly larger for the lower coverage.Comment: to appear in Phys Rev

Stability of cluster solutions in a cooperative consumer chain model

This is the author's accepted manuscript. The final published article is available from the link below. Copyright @ Springer-Verlag Berlin Heidelberg 2012.We study a cooperative consumer chain model which consists of one producer and two consumers. It is an extension of the Schnakenberg model suggested in Gierer and Meinhardt [Kybernetik (Berlin), 12:30-39, 1972] and Schnakenberg (J Theor Biol, 81:389-400, 1979) for which there is only one producer and one consumer. In this consumer chain model there is a middle component which plays a hybrid role: it acts both as consumer and as producer. It is assumed that the producer diffuses much faster than the first consumer and the first consumer much faster than the second consumer. The system also serves as a model for a sequence of irreversible autocatalytic reactions in a container which is in contact with a well-stirred reservoir. In the small diffusion limit we construct cluster solutions in an interval which have the following properties: The spatial profile of the third component is a spike. The profile for the middle component is that of two partial spikes connected by a thin transition layer. The first component in leading order is given by a Green's function. In this profile multiple scales are involved: The spikes for the middle component are on the small scale, the spike for the third on the very small scale, the width of the transition layer for the middle component is between the small and the very small scale. The first component acts on the large scale. To the best of our knowledge, this type of spiky pattern has never before been studied rigorously. It is shown that, if the feedrates are small enough, there exist two such patterns which differ by their amplitudes.We also study the stability properties of these cluster solutions. We use a rigorous analysis to investigate the linearized operator around cluster solutions which is based on nonlocal eigenvalue problems and rigorous asymptotic analysis. The following result is established: If the time-relaxation constants are small enough, one cluster solution is stable and the other one is unstable. The instability arises through large eigenvalues of order O(1). Further, there are small eigenvalues of order o(1) which do not cause any instabilities. Our approach requires some new ideas: (i) The analysis of the large eigenvalues of order O(1) leads to a novel system of nonlocal eigenvalue problems with inhomogeneous Robin boundary conditions whose stability properties have been investigated rigorously. (ii) The analysis of the small eigenvalues of order o(1) needs a careful study of the interaction of two small length scales and is based on a suitable inner/outer expansion with rigorous error analysis. It is found that the order of these small eigenvalues is given by the smallest diffusion constant ε22.RGC of Hong Kon