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

## TL;DR

This paper introduces a new algebraic approach using elementary number theory to analyze the discrete nature of screw dislocations in BCC and SC crystal lattices, linking energy calculations to special zeta functions.

## Contribution

It develops a novel algebraic method to describe dislocation energies in crystal lattices using number theory and special zeta functions, advancing the understanding of lattice defects.

## Key findings

- Dislocation energy in BCC lattice expressed via Epstein-Hurwitz zeta function of Eisenstein integers.
- Dislocation energy in SC lattice linked to Epstein-Hurwitz zeta function of Gauss integers.
- The method provides a natural description of stress energy in discrete crystal lattices.

## Abstract

In this paper, we proposed a novel method using the elementary number theory to investigate the discrete nature of the screw dislocations in crystal lattices, simple cubic (SC) lattice and body centered cubic (BCC) lattice, by developing the algebraic description of the dislocations in the previous report (Hamada, Matsutani, Nakagawa, Saeki, Uesaka, Pacific J. Math.~for Industry {\bf{10}} (2018), 3). Using the 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 SC lattice is associated with the truncated Epstein-Hurwitz zeta function of the Gauss integers.