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.
Highlights
Mathematical descriptions of dislocations in crystal lattices have been studied extensively in the framework of differential geometry or continuum geometry [1, 2, 14, 17, 21]
We describe the continuum geometric property embedded in the euclidean space, whereas the algebraic structures involving group rings enable us to describe the discrete nature of the crystal lattices
We show that the strain energy of a screw dislocation based on the spring model is expressed by the Epstein-Hurwitz zeta function [9, 10] approximately
Summary
Mathematical descriptions of dislocations in crystal lattices have been studied extensively in the framework of differential geometry or continuum geometry [1, 2, 14, 17, 21]. We give algebraic descriptions of screw dislocations in the simple cubic (SC) and the body centered cubic (BCC) crystals in terms of certain “fibrations” involving group rings. Using such fibering structures, we describe the continuum geometric property embedded in the euclidean space, whereas the algebraic structures involving group rings enable us to describe the discrete nature of the crystal lattices. We consider the fibering structure of the SC lattice in terms of the associated group ring and its quotient Using these descriptions, we will describe the screw dislocations in the SC lattice in Propositions 3.
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