Pegs and Ropes: Geometry at Stonehenge

A recent computer-aided-design investigation of the Neolithic 56 Aubrey Hole circuit at Stonehenge has led to the discovery of an astonishingly simple geometrical construction for drawing an approximately regular 56-sided polygon, feasible with a compass and straightedge. In the present work, we prove analytically that the aforementioned construction yields as a byproduct, an extremely accurate method for approximating a regular heptagon, and we quantify the accuracy that prehistoric surveyors may have ideally attained using simple pegs and ropes. We compare this method with previous approximations, and argue that it is likely to be at the same time the simplest and most accurate. Implications of our findings are discussed

Scaling and Universality in Models of Step Bunching: The "C+-C-" Model

We study further the recently introduced [Ranguelov et al., Comptes Rendus de l'Acad. Bulg. des Sci. 60, 4 (2007) 389] "C+-C-" model of step flow crystal growth over wide range of model parameters. The basic assumption of the model is that the reference ("equilibrium") densities used to compute the supersaturation might be different on either side of a step. We obtain the condition for linear stability of the whole step train in the form CL/CR>1 (L/R stands for left/right in a descending from left to right step train). Further we integrate numerically the equations of step motion to monitor the bunching process in the long times limit. Thus we obtain the exact size- and time- scaling of the step bunches including the numerical prefactors. We show that in a broad range of parameters the morphology is characterized with appearance of the minimal interstep distance in the bunch in the beginning of the bunches (at the trailing edge of the bunch) and may be described by a single universality class, different from those already generated by continuum theories [Krug et al., PRB 71, 045412].Comment: 18 pages, 9 figure

A facet is not an island: step-step interactions and the fluctuations of the boundary of a crystal facet

In a recent paper [Ferrari et al., Phys. Rev. E 69, 035102(R) (2004)], the scaling law of the fluctuations of the step limiting a crystal facet has been computed as a function of the facet size. Ferrari et al. use rigorous, but physically rather obscure, arguments. Approaching the problem from a different perspective, we rederive more transparently the scaling behavior of facet edge fluctuations as a function of time. Such behavior can be scrutinized with STM experiments and with numerical simulations.Comment: 3 page

Capture-zone scaling in island nucleation: phenomenological theory of an example of universal fluctuation behavior

In studies of island nucleation and growth, the distribution of capture zones, essentially proximity cells, can give more insight than island-size distributions. In contrast to the complicated expressions, ad hoc or derived from rate equations, usually used, we find the capture-zone distribution can be described by a simple expression generalizing the Wigner surmise from random matrix theory that accounts for the distribution of spacings in a host of fluctuation phenomena. Furthermore, its single adjustable parameter can be simply related to the critical nucleus of growth models and the substrate dimensionality. We compare with extensive published kinetic Monte Carlo data and limited experimental data. A phenomenological theory sheds light on the result.Comment: 5 pages, 4 figures, originally submitted to Phys. Rev. Lett. on Dec. 15, 2006; revised version v2 tightens and focuses the presentation, emphasizes the importance of universal features of fluctuations, corrects an error for d=1, replaces 2 of the figure

Kinetic step bunching during surface growth

We study the step bunching kinetic instability in a growing crystal surface characterized by anisotropic diffusion. The instability is due to the interplay between the elastic interactions and the alternation of step parameters. This instability is predicted to occur on a vicinal semiconductor surface Si(001) or Ge(001) during epitaxial growth. The maximal growth rate of the step bunching increases like $F^{4}$, where $F$ is the deposition flux. Our results are complemented with numerical simulations which reveals a coarsening behavior on the long time for the nonlinear step dynamics.Comment: 4 pages, 6 figures, submitted to PR

Effect of step stiffness and diffusion anisotropy on the meandering of a growing vicinal surface

We study the step meandering instability on a surface characterized by the alternation of terraces with different properties, as in the case of Si(001). The interplay between diffusion anisotropy and step stiffness induces a finite wavelength instability corresponding to a meandering mode. The instability sets in beyond a threshold value which depends on the relative magnitudes of the destabilizing flux and the stabilizing stiffness difference. The meander dynamics is governed by the conserved Kuramoto-Sivashinsky equation, which display spatiotemporal coarsening.Comment: 4 pages, 3 figures, submitted to Phys. Rev. Lett. (February 2006