# SYSTEMATICS OF COINCIDENCE SITE LATTICES OF CRYSTALS

### Abstract

The object of this study is polycrystalline materials containing intercrystalline boundaries as one of the accompanying components of the defective structure. A wide variety of geometric characteristics that determine the atomic structure and physical properties of intergranular boundaries, gives rise to the expediency of systematization of the entire set of possible intergranular boundaries. The intercrystalline boundaries obtained on the basis of the coincidence of nodes are an important family of defects in polycrystals, so these defects are given a lot of research, but currently there is no complete systematization. The construction of such systematization was the purpose of this paper.

The basis of classification is the coordination polyhedron of the simple cubic lattice. Seven different polytopes with different topological characteristics are given. As parameters defining the geometry of the lattice matching of the nodes of the selected pair of vectors the vertices of these polyhedra and the lattice vector that defines the axis of rotation was used. Using algebraic methods used in studies of solid rotation, formulas are obtained for the crystallographic plane in which all possible axes of turns and vectors of the lattice cell of matching nodes should lie. As a result, it was shown that the coincidence lattices of General form belong to monoclinic crystal structure. In particular cases they may belong to orthorhombic and tetragonal crystal structure. It is shown that such lattices of coinciding nodes are possible, which are obtained at different parameters of rotation, so polycrystals on the basis of the common lattice of coincidences are possible. A table is constructed in which the data are systematized for the vectors of the cell of the lattice of coincidences, rotation angles, the number of atoms in the cell at different initial characteristics of rotation.

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**ACKNOWLEDGMENTS**

*The work was carried out within the framework of the state task of the Ministry of Education and Science of Russia to higher educational institutions and scientific organizations in the field of scientific activity (project No. 4.7972.2017 / 8.9).*

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