21 research outputs found
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 -dimensional nonlinear diffusion equation where , ,
, and are real parameters and is a time-dependent
source. This equation unifies the O'Shaugnessy-Procaccia anomalous diffusion
equation on fractals () and the spherical anomalous diffusion for
porous media (). 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 . 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]
Grating formation in step flow heterogeneous growth and wavelength selection induced by confinement
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