Geometric attractor of an electron beam passing through a crystal

Abstract

The Lemmlein algorithm assigns a cyclic interaction of a mathematical point with other (n+1) points of the n - dimensional Euclidean space. In this paper generalization A of the Lemmlein algorithm for an arbitrary number of points m situated in the n - dimensional Riemann space is proposed. Algorithm A generates a Markovian chain consisting of the finite number of combinatorially different strongly convergent attractors. Algorithm A is generalized to describe the interaction of mass points, e.g., the motion of electrons in the real crystal medium. The strong convergence of the attractors provides stability of the electron trajectories in the vicinity of an attractor in the unit cell of the crystal after electron scattering at the crystal defects.