Teoria de grupos aplicada às propriedades de simetria dos nanotubos de Carbono

Abstract

Carbon nanotubes are structures in which carbon form hexagonal chains, which, in turn, are shaped so as to generate a cylindrical surface with a diameter in the order of nanometers and lengths of several micrometers. As these structures have a length much greater than its diameter, they can be treated as 1D systems. Due to its unique structure, carbon nanotubes are the strongest and hardest materials in nature. They have the characteristic of being very resistant to tension and torsion imposed on them. In addition, one of its most interesting features is that many of their mechanical properties are related to their electronic properties, which, in turn, are closely linked to the structural characteristics of the material. For these reasons, these materials have been considered promising for applications as nanoactuators. Because of this relationship between the structure and the properties of the nanotubes, it is interesting to study the possible symmetries of these materials and what they entail. Therefore, it is necessary to use a well-known mathematical theory, the group theory, since it is known that, if a given system has some kind of symmetry, it can be associated with a mathematical entity called a group and their properties related to its symmetry can be analyzed much more easily through this theory. In this work, therefore, the symmetry properties of carbon nanotubes will be studied, that are the properties closely related to the symmetries of these structures. For this, a brief explanation of these materials will be made, focusing on the possible symmetries that they may have. In addition, a detailed theoretical overview on the group theory will be done, focusing on its application in solving physical problems.