Vol. 9, No. 1, 2021

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A novel discrete investigation of screw dislocations in the BCC crystal lattice

Shigeki Matsutani

Vol. 9 (2021), No. 1, 1–32
DOI: 10.2140/memocs.2021.9.1
Abstract

In this paper, we propose a novel method using elementary number theory to investigate the discrete nature of the screw dislocations in crystal lattices, namely the simple cubic (SC) lattice and body-centered cubic (BCC) lattice, by developing the algebraic description of the dislocations in the previous report of Hamada, Matsutani, Nakagawa, Saeki, and Uesaka (Pac. J. Math. Ind. 10 (2018), art. id. 3). Using this method, we showed that the stress energy of the screw dislocations in the BCC lattice and the SC lattice are naturally described; the energy of the BCC lattice was expressed by the truncated Epstein–Hurwitz zeta function of the Eisenstein integers, whereas that of the SC lattice is associated with the truncated Epstein–Hurwitz zeta function of the Gauss integers.

Keywords
crystal lattice, screw dislocation, truncated Epstein–Hurwitz zeta function, Eisenstein integer, Gauss integer, dislocation, algebraic investigation, number theory, BCC lattice
Mathematical Subject Classification
Primary: 08A99, 55R05, 20H15, 11R60, 34M35
Secondary: 82D25, 74E15, 82-10, 74-10
Milestones
Received: 19 June 2020
Revised: 8 October 2020
Accepted: 9 November 2020
Published: 17 March 2021

Communicated by Victor A. Eremeyev
Authors
Shigeki Matsutani
Graduate School of Natural Science and Technology
Kanazawa University, Kakuma
Kanazawa
Japan