Iluzija nagiba mjeseca

The moon tilt illusion is the startling discrepancy between the direction of the light beam illuminating the moon and the direction of the sun. The illusion arises because the observer erroneously expects a light ray between sun and moon to appear as a line of constant slope according to the positions of the sun and the moon in the sky. This expectation does not correspond to the reality that observation by direct vision or a camera is according to perspective projection, for which the observed slope of a straight line in three-dimensional space changes according to the direction of observation. Comparing the observed and expected directions of incoming light at the moon, we derive a quantitative expression for the magnitude of the moon tilt illusion that can be applied to all configurations of sun and moon in the sky.Iluzija nagiba mjeseca zapanjujući je raskorak između svjetlosne zrake koja osvjetljava mjesec i smjera sunca. Ona se povećava, jer promatrač pogrešno očekuje da zraka svjetla između sunca i mjeseca bude pravac konstantnog koeficijenta smjera s obzirom na položaj sunca i mjeseca na nebu. Ovakvo očekivanje ne odgovara stvarnosti kod koje je promatranje s direktnom osi pogleda, ili s kamerom, u skladu s perspektivom (centralnim projiciranjem) za koju se promatrani koeficijent smjera pravca u trodimenzionalnom prostoru mijenja s obzirom na os pogleda. Uspoređujući promatrane i očekivane smjerove zrake usmjerene na mjesec, izvodimo kvantitativan izraz za veličinu iluzije nagiba mjeseca koji se može primijeniti na sve položaje sunca i mjeseca na nebu

Surface Kinetics and Generation of Different Terms in a Conservative Growth Equation

A method based on the kinetics of adatoms on a growing surface under epitaxial growth at low temperature in (1+1) dimensions is proposed to obtain a closed form of local growth equation. It can be generalized to any growth problem as long as diffusion of adatoms govern the surface morphology. The method can be easily extended to higher dimensions. The kinetic processes contributing to various terms in the growth equation (GE) are identified from the analysis of in-plane and downward hops. In particular, processes corresponding to the (h -> -h) symmetry breaking term and curvature dependent term are discussed. Consequence of these terms on the stable and unstable transition in (1+1) dimensions is analyzed. In (2+1) dimensions it is shown that an additional (h -> -h) symmetry breaking term is generated due to the in-plane curvature associated with the mound like structures. This term is independent of any diffusion barrier differences between in-plane and out of-plane migration. It is argued that terms generated in the presence of downward hops are the relevant terms in a GE. Growth equation in the closed form is obtained for various growth models introduced to capture most of the processes in experimental Molecular Beam Epitaxial growth. Effect of dissociation is also considered and is seen to have stabilizing effect on the growth. It is shown that for uphill current the GE approach fails to describe the growth since a given GE is not valid over the entire substrate.Comment: 14 pages, 7 figure

Analytical solution of generalized Burton--Cabrera--Frank equations for growth and post--growth equilibration on vicinal surfaces

We investigate growth on vicinal surfaces by molecular beam epitaxy making use of a generalized Burton--Cabrera--Frank model. Our primary aim is to propose and implement a novel analytical program based on a perturbative solution of the non--linear equations describing the coupled adatom and dimer kinetics. These equations are considered as originating from a fully microscopic description that allows the step boundary conditions to be directly formulated in terms of the sticking coefficients at each step. As an example, we study the importance of diffusion barriers for adatoms hopping down descending steps (Schwoebel effect) during growth and post-growth equilibration of the surface.Comment: 16 pages, REVTeX 3.0, IC-DDV-94-00

Nonlinear anomalous diffusion equation and fractal dimension: Exact generalized gaussian solution

In this work we incorporate, in a unified way, two anomalous behaviors, the power law and stretched exponential ones, by considering the radial dependence of the $N$-dimensional nonlinear diffusion equation $\partial\rho /\partial{t}={\bf \nabla} \cdot (K{\bf \nabla} \rho^{\nu})-{\bf \nabla}\cdot(\mu{\bf F} \rho)-\alpha \rho ,$ where $K=D r^{-\theta}$, $\nu$, $\theta$, $\mu$ and $D$ are real parameters and $\alpha$ is a time-dependent source. This equation unifies the O'Shaugnessy-Procaccia anomalous diffusion equation on fractals ($\nu =1$) and the spherical anomalous diffusion for porous media ($\theta=0$). An exact spherical symmetric solution of this nonlinear Fokker-Planck equation is obtained, leading to a large class of anomalous behaviors. Stationary solutions for this Fokker-Planck-like equation are also discussed by introducing an effective potential.Comment: Latex, 6 pages. To appear in Phys. Rev.

Superpositions of Probability Distributions

Probability distributions which can be obtained from superpositions of Gaussian distributions of different variances v = \sigma ^2 play a favored role in quantum theory and financial markets. Such superpositions need not necessarily obey the Chapman-Kolmogorov semigroup relation for Markovian processes because they may introduce memory effects. We derive the general form of the smearing distributions in v which do not destroy the semigroup property. The smearing technique has two immediate applications. It permits simplifying the system of Kramers-Moyal equations for smeared and unsmeared conditional probabilities, and can be conveniently implemented in the path integral calculus. In many cases, the superposition of path integrals can be evaluated much easier than the initial path integral. Three simple examples are presented, and it is shown how the technique is extended to quantum mechanics.Comment: 23 pages, RevTeX, minor changes, accepted to Phys. Rev.

A Simple Model for Anisotropic Step Growth

We consider a simple model for the growth of isolated steps on a vicinal crystal surface. It incorporates diffusion and drift of adatoms on the terrace, and strong step and kink edge barriers. Using a combination of analytic methods and Monte Carlo simulations, we study the morphology of growing steps in detail. In particular, under typical Molecular Beam Epitaxy conditions the step morphology is linearly unstable in the model and develops fingers separated by deep cracks. The vertical roughness of the step grows linearly in time, while horizontally the fingers coarsen proportional to $t^{0.33}$. We develop scaling arguments to study the saturation of the ledge morphology for a finite width and length of the terrace.Comment: 20 pages, 12 figures; [email protected]

Instabilities in crystal growth by atomic or molecular beams

The planar front of a growing a crystal is often destroyed by instabilities. In the case of growth from a condensed phase, the most frequent ones are diffusion instabilities, which will be but briefly discussed in simple terms in chapter II. The present review is mainly devoted to instabilities which arise in ballistic growth, especially Molecular Beam Epitaxy (MBE). The reasons of the instabilities can be geometric (shadowing effect), but they are mostly kinetic or thermodynamic. The kinetic instabilities which will be studied in detail in chapters IV and V result from the fact that adatoms diffusing on a surface do not easily cross steps (Ehrlich-Schwoebel or ES effect). When the growth front is a high symmetry surface, the ES effect produces mounds which often coarsen in time according to power laws. When the growth front is a stepped surface, the ES effect initially produces a meandering of the steps, which eventually may also give rise to mounds. Kinetic instabilities can usually be avoided by raising the temperature, but this favours thermodynamic instabilities. Concerning these ones, the attention will be focussed on the instabilities resulting from slightly different lattice constants of the substrate and the adsorbate. They can take the following forms. i) Formation of misfit dislocations (chapter VIII). ii) Formation of isolated epitaxial clusters which, at least in their earliest form, are `coherent' with the substrate, i.e. dislocation-free (chapter X). iii) Wavy deformation of the surface, which is presumably the incipient stage of (ii) (chapter IX). The theories and the experiments are critically reviewed and their comparison is qualitatively satisfactory although some important questions have not yet received a complete answer.Comment: 90 pages in revtex, 45 figures mainly in gif format. Review paper to be published in Physics Reports. Postscript versions for all the figures can be found at http://www.theo-phys.uni-essen.de/tp/u/politi