Topological manifolds and vector bundles with applications to crystal physics

Abstract

An adequate description of the phenomena occurring inside a crystal requires an adequate description of the “space” within the crystal characterized by the existence of a regular network of “priveleged” points (the equilibrium positions of the atoms of the crystal). The main purpose of this article is to show that this can be done by using a discrete class of “priveleged” reference systems and notions such as topological manifold, topological group, vector bundle in a way that extends methods well known in the usual formulation of Classical Mechanics or Special Relativity in terms of Differential Geometry. In the case of the crystals having the structure of diamond (the corresponding symmetry group is the crystallographic group O 7 h and it includes semiconductors such as silicon and germanium) each atom has four nearest neighbour atoms and the system of the four corresponding axes is a priviledged reference frame. We indicate some O 7 h-invariant mathematical objects in certain numerical O 7 h-spaces and we prove that they can be associated to the crystal by using the class of privileged reference frames in a way that does not depend on the particular reference frame we use. They have physical meaning; an obvious O 7 h-invariant form and their expressions in the descriptions actually used in Crystal Physics are very intricate and difficult to use. From a mathematical point of view, the present paper contains some interesting examples of topological manifolds, vector bundles and representations of O 7 h.