Analysing stacking disorder with a genetic algorithm

Abstract

A disordered stacking of layers is described in terms of probabilities governing the addition of new layers to a stack. The diffuse scattering exhibited by such stacks takes the form of diffuse lines perpendicular to the layers and can be represented as a function of the stacking probabilities in closed form [1]. The corresponding expression becomes increasingly unwieldy, as the number of layers affecting the probabilities increases. In this work we have determined the probabilities with a numerical procedure based on a genetic algorithm. The crystal structure of tris(bicyclo[2.1.1ηexeno)-benzene, 1, shows layers of coplanar molecules with layer symmetry p6bar2m. Consecutive layers may assume one of three positions with probabilities depending on the position of the three preceding layers. Initially the probabilities have been estimated by visually comparing measured diffuse intensities with those calculated from a complicated analytical function of the probabilities [2]. We have redetermined these probabilities using an adaptation of a genetic algorithm described earlier [3]. The mean values and their standard deviations from 20 sets of three probabilities obtained after 113 generations of optimization are 0.08(1), 0.60(4) and 0.46(4) in good agreement with the values derived from the analytical approach, 0.08(2), 0.56(6) and 0.46(10). Although the general structure of maxima andminima is reproducedwell (Fig.1), their relative heights indicate room for further improvement of the structuralmodel, e.g. by tilting themolecules out of the plane and adjusting the displacement parameters. These quantities are presently optimized with the genetic algorithm.