An algebraic description of screw dislocations in SC and BCC crystal lattices

Abstract

We give an algebraic description of screw dislocations in a crystal, especially simple cubic (SC) and body centered cubic (BCC) crystals, using free abelian groups and fibering structures. We also show that the strain energy of a screw dislocation based on the spring model is expressed by the Epstein-Hurwitz zeta function approximately. Crystal lattice and screw dislocation and topological defect and monodromy and group ring of abelian group and dislocation energy and Epstein-Hurwitz zeta function 1.1. Notations and Conventions. Throughout the article, we distinguish the euclidean space E from the real vector space R: in particular, R is endowed with an algebraic structure, while E is not. We often identify the 2-dimensional euclidean space E 2 with the complex plane C. The group U(1) acts on the set S 1 simply transitively. Given a fiber bundle F → M over a base space M, we denote the set of all continuous sections f : M → F by Γ(M, F ). Screw Dislocations in Continuum PictureLet us consider the exact sequence of groups (see [B]) (2.1) 0

A Novel Discrete Theory of a Screw Dislocation in the BCC Crystal Lattice

“…This paper is organized as follows. Section 2 and 3 review the previous report [9]. In Section 2, we show the screw dislocation in the continuum picture.…”

“…It means that the lattice with dislocations is not stable for the crystal group in general but is stable for its subgroup, at least, approximately. In the previous report [9] by Hamada, Matsutani, Nakagawa, Saeki and Uesaka, we focused on the fiber structure of the dislocations as an essential of dislocations. The algebraic approach describes well the fiber structure of the screw dislocation of the SC (simple cubic) and BCC lattices because these approaches are natural tools of the algebraic geometry even for both the continuum picture (characteristic 0) and the discrete picture (non-vanishing characteristic).…”

“…Further as the crystal group represents the perfect crystals, by considering the fiber structure, we describe an ideal property of the discrete nature of the dislocations from a viewpoint of (classical) statistical mechanics. Under a certain assumption, the stress energy of the dislocation in the SC lattice was also obtained in [9].…”

“…In this paper, we propose a novel method to investigate the discrete nature of the screw dislocation in crystal lattices, the SC and the BCC lattices, using the elementary number theory since the crystal lattice is closely related to the number theory. This method is based on the algebraic description in the previous report [9]. Though the crystallography has been for the perfect crystals, the crystal lattice is a mathematical object and is surely connected with the profound mathematics [8,26].…”

“…Following the previous approaches in [9]. we go on to focus on the fiber structure; it is one of the essentials of the dislocations, at least, as the first step of an approximation of the discrete nature of the dislocation.…”