Symmetrical features and local phase transitions of ordered solid structures: Tetravalent structures of gas hydrates

Abstract

The interrelation between the topological invariants of the surface of polyhedra and the algebraic invariants has been analyzed. It has been demonstrated that this interrelation allows one to consider the processes of the removal of the geometric degeneracy characterized by the presence of high-symmetry critical points in local regions of the solid system. An analysis has been made of algebraic automorphisms that permit one to replace the stringent crystallographic requirement of the translation invariance by topological constraints which make it possible to determine ordered solid structures and a special type of local phase transitions. It has been established that the eight-dimensional lattice E8, the Galois field GF(11), and the Mathieu group M12 determine the only sequence of constructions of the algebraic geometry that identifies the unique class of 24-vertex tetradecahedral simple polyhedra with tetragonal, pentagonal, and hexagonal faces. The graphs of ten stereohedra of this class, including the Fedorov-Kelvin parallelohedron, have been a priori derived as an example. A model is proposed for local phase transformations in structures of gas hydrates in which the majority of the known tetravalent water frameworks are formed by polyhedra belonging to the class under consideration.