شناسایی ساختار کریستالی سه بعدی با استفاده از شبکه های عصبی فازی 3D Crystal Structure Identification Using Fuzzy Neural Networks

INTRODUCTION Being the fundamental concept of crystallography and having Angstrom-order sizes, Bravais lattices are building blocks for all crystals. Every crystal is constructed of these lattices in various modifications. At the same time, different crystals can have the same lattices. There is a total of 14 such lattices. Depending on special symmetry, all crystals are distributed among seven lattice systems: triclinic, monoclinic, tetragonal, orthorhombic, trigonal, hexagonal, and cubic systems [1]. Figure 1 presents the general arrangements of Bravais lattices (smallest structural cells) for each lattice system. The type of a lattice system is determined by six parameters of a Bravais lattice: the lengths of the three edges and three angles between them (see Table 1). The task of recognizing nano-scale images, which are projections of crystal lattices, can be reduced to the structure identification problem. Among basic lattice structure identification methods are: the comparator of the National Institute of Standards and Technology [2], packing efficiency-based identification [3] and isosurface-based identification [4]. However, these approaches have some drawbacks that restrict their use: the tricky process of crystal preparation (the need for accurate polishing and mounting), low efficiency of comparison of similar lattices, high sensitivity to minor distortions of lattice node coordinates. The major difficulty here is the ambiguity in choosing a two-dimensional basic cell for a particular projection (Fig. 2) [5]. Papers [6, 7] give a careful investigation of this problem representing a Bravais lattice as a unique feature space allowing the resolution of any nano-scale images in elementary structures. One of possible approaches to the determination of crystal lattice type is offered in [8] where previously estimated lattice parameters are compared with predefined reference lattice parameters. The lattice is considered to belong to a particular type if its parameters have the closest match with the parameters of the reference lattice of this type.