Algorithmic crystal chemistry: A cellular automata approach
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RY OF CRYSTAL STRUCTURES
Algorithmic Crystal Chemistry: A Cellular Automata Approach S. V. Krivovichev St. Petersburg State University, St. Petersburg, 199034 Russia Department for the Study of Micro and Nanoporous Structures, Kola Scientific Center, Russian Academy of Sciences, Apatity, Murmansk oblast, 184209 Russia email: [email protected] Received February 16, 2011
Abstract—Atomic–molecular mechanisms of crystal growth can be modeled based on crystallochemical information using cellular automata (a particular case of finite deterministic automata). In particular, the for mation of heteropolyhedral layered complexes in uranyl selenates can be modeled applying a onedimen sional threecolored cellular automaton. The use of the theory of calculations (in particular, the theory of automata) in crystallography allows one to interpret crystal growth as a computational process (the realiza tion of an algorithm or program with a finite number of steps). DOI: 10.1134/S1063774511060149
INTRODUCTION
the presence of particular fundamental building blocks in its structure, and some other details of this kind are of primary importance This study is devoted to the development of algo rithmic crystal chemistry, in which the construction of a particular crystal structure is considered the result of the realization of a particular algorithm or program. Due to the periodicity of crystal, the algorithm for its construction should be cyclic in essence; i.e., it must imply the implementation of the same program in an almost infinite number of cycles. This program includes a finite number of operations with the attach ment of building particles: atoms, molecules, clusters, and nanoblocks (in the case of nonclassical crystalli zation [12], etc.). Each of these operations can be described in terms of crystal chemistry. Thus, the interpretation of the algorithm for structural growth implies the selection of construction stages in which particular building blocks are attached. Here we develop a method for simulating the growth of crystal structures using cellular automata (CA) as a particular case of finite deterministic automata.
One of the fundamental problems of crystallogra phy is the problem of formation of a threedimen sional ordered periodic structure. N.V. Belov referred to it as the problem of the “501st atom” [1]. In the 1970s–1980s, Belov put this problem before research ers belonging to the mathematical school of B.N. Delone, as a result of which an original theory of the local criterion for regularity of point systems was developed [2–6]. According to this theory, the regular ity of a system of points is set by the equality of the local environment of a point by other points of the sys tem within some region. However, this theory, with all its surprising beauty and mathematical rigor, has not been properly appreciated by “practical” crystal chemists, who dealt with real structures composed of real atoms and molecules rather than mathematical points. A kind of gap arose between the mathematical theory of the f
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