Fractal structure of the equilibrium crystal shape

Abstract

The equilibrium crystal shape of the classical crystal at zero temperature is found in the framework of the model suppressing the atomic distorsions. In this model the interaction between two atoms of a pair is supposed to decrease at large distances more rapidly than r −4 . The crystal surface is shown to have a fractal structure: it consists of an infinite number of facets and of solitary edges and corners, which form the Cantor set. The crystal is completely faceted (i.e. there is no rounded area), but there is no sharp edge (i.e. the slope is continuous). The boundary of a facet is shown to have a fractal structure too. It consists of an infinite number of «smooth» edges and of the Cantor set. The microscopic configuration of steps and kinks on every facet is also found Obtention de la forme d'equilibre d'un cristal classique a temperature nulle dans le cadre d'un modele negligeant les distorsions atomiques. Dans ce modele, l'interaction entre deux atomes d'une meme paire decroit avec la distance plus rapidement que r −4 . Il est montre que la surface du cristal a une structure fractale qui consiste en un nombre infini de facettes, d'aretes et de sommets qui forme un ensemble de Cantor. La surface consiste uniquement en facettes (il n'y a pas de zone arrondie) mais la pente reste continue aux aretes. Le bord d'une face a egalement une structure fractale. Obtention egalement de la surface microscopique de chaque facette