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.

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.

Equations

$${\iota }_{\delta }:{\mathbb{R}}^3\hookrightarrow {\mathbb{E}}^3,(x\mapsto x+\delta ),\text{mbox}or$$

$${\iota }_{\delta }:\mathbb{C}\times \mathbb{R}\hookrightarrow {\mathbb{E}}_{\mathbb{C}}\times \mathbb{E},((x_1+\sqrt{-1}{x}_2,x_3)\mapsto (x_1+\sqrt{-1}{x}_2+{\delta }_{\mathbb{C}},x_3+{\delta }_3),$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\tilde{i}}_{d,\delta }:\mathbb{R}\to \mathbb{E},$$

$$\mathbb{E}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathbb{Z}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\phi }_{\delta }(\mathbb{Z})={\psi }_d^{-1}(\exp (2\pi \sqrt{-1}{\delta }_3/d)).$$

$$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 9.11667pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\crcr}}}\ignorespaces{\hbox{\kern-9.11667pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\mathbb{Z}}_{{\mathbb{E}}_{\mathbb{C}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 14.45pt\raise 7.30556pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.91667pt\hbox{$\scriptstyle{\widehat{\varphi}_{\delta}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 33.11667pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 33.11667pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{{\mathbb{E}}_{{\mathbb{E}}_{\mathbb{C}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 57.22571pt\raise 7.30556pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.91667pt\hbox{$\scriptstyle{\widehat{\psi}_{d}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 76.04446pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 76.04446pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{S^{1}_{{\mathbb{E}}_{\mathbb{C}}}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$$

$${\iota }_{\delta }:\mathbb{C}\hookrightarrow {\mathbb{E}}_{\mathbb{C}},(x+\sqrt{-1}y\mapsto x+\sqrt{-1}y+{\delta }_{\mathbb{C}}).$$

$${\mathfrak{u}}_{\delta }(z)={\gamma }_{\delta }\in S_{{\mathbb{E}}_{\mathbb{C}}}^1{|}_z=S^1\times {\mathbb{E}}_{\mathbb{C}}{|}_z,$$

$${\mathbb{Z}}_{{\mathbb{E}}_{\mathbb{C}},\delta }={\hat{\phi }}_{\delta }({\mathbb{Z}}_{{\mathbb{E}}_{\mathbb{C}}}),$$

$${\mathbb{Z}}_{{\mathbb{E}}_{\mathbb{C}},\delta }:={\hat{\psi }}_d^{-1}\left({\mathfrak{u}}_{\delta }({\mathbb{E}}_{\mathbb{C}})\right)\subset {\mathbb{E}}^3={\mathbb{E}}_{{\mathbb{E}}_{\mathbb{C}}}.$$

$${\mathfrak{u}}_{z_0,\delta }(z)={\gamma }_{\delta }\frac{z-z_0}{|z-z_0|}\text{mbox}forz\in {\mathbb{E}}_{\mathbb{C}}\setminus \{z_0\},$$

$${\mathbb{Z}}_{{\mathbb{E}}_{\mathbb{C}}\setminus \{z_0\},\delta }:={\hat{\psi }}_d^{-1}\left({\mathfrak{u}}_{z_0,\delta }({\mathbb{E}}_{\mathbb{C}}\setminus \{z_0\})\right)\subset {\mathbb{E}}_{{\mathbb{E}}_{\mathbb{C}}\setminus \{z_0\}}\subset {\mathbb{E}}^3$$

$${\pi }_{\mathbb{E}}^{-1}{U}_p\cong \mathbb{Z}\times U_p,$$

$${\mathbb{Z}}_{\mathrm{SC},\delta }:=\{({\ell }_{1}a,{\ell }_{2}a,{\ell }_{3}a)+\delta |{\ell }_1,{\ell }_2,{\ell }_3\in \mathbb{Z}\},$$

$${\mathcal{Z}}_{\mathrm{SC}}=\mathbb{Z}[\sqrt{-1}]a\subset \mathbb{C}.$$

$${\iota }_{\delta }^{\mathrm{SC}}:{\mathcal{Z}}_{\mathrm{SC}}\to {\mathcal{Z}}_{\mathrm{SC}}+{\delta }_{\mathbb{C}}\subset {\mathbb{E}}_{\mathbb{C}},$$

$${\mathbb{Z}}_{{\mathcal{Z}}_{\mathrm{SC}},\delta }={\hat{\psi }}_a^{-1}\left({\mathfrak{u}}_{\delta }({\iota }_{\delta }^{\mathrm{SC}}({\mathcal{Z}}_{\mathrm{SC}}))\right),$$

$$\sum _{\ell =0}^3(\sqrt{-1}{)}^{\ell }=0,$$

$$\begin{array}{rl}& [(n_1,n_2,n_3),(n_1\pm 1,n_2,n_3)],[(n_1,n_2,n_3),(n_1,n_2\pm 1,n_3)],[(n_1,n_2,n_3),(n_1,n_2,n_3\pm 1)],\\ & [(n_1,n_2,n_3),(n_1,n_2\pm 1,n_3\pm 1)],[(n_1,n_2,n_3),(n_1\pm 1,n_2,n_3\pm 1)],\\ & [(n_1,n_2,n_3),(n_1\pm 1,n_2\pm 1,n_3)]\end{array}$$

$${\mathfrak{u}}_{z_0,\delta }^{\mathrm{SC}}:={\iota }_{{\delta }_{\mathbb{C}}}^{\mathrm{SC}\ast }{\mathfrak{u}}_{z_0,\delta }={\mathfrak{u}}_{z_0,\delta }\circ {\iota }_{{\delta }_{\mathbb{C}}}^{\mathrm{SC}},$$

$${\mathfrak{u}}_{z_0,\delta }^{\mathrm{SC}}(na)=\left({\gamma }_{\delta }\frac{na+{\delta }_{\mathbb{C}}-z_0}{|na+{\delta }_{\mathbb{C}}-z_0|}\right),na\in {\mathcal{Z}}_{\mathrm{SC}}=\mathbb{Z}[\sqrt{-1}]a.$$

$${\mathbb{Z}}_{{\mathcal{Z}}_{\mathrm{SC}},z_0,\delta }^{\mathrm{SC}}:=\left({\hat{\psi }}_a^{-1}({\mathfrak{u}}_{z_0,\delta }^{\mathrm{SC}}({\mathcal{Z}}_{\mathrm{SC}}))\right)=\left(\frac{a}{2\pi \sqrt{-1}}{\exp }^{-1}\left({\mathfrak{u}}_{z_0,\delta }^{\mathrm{SC}}({\mathcal{Z}}_{\mathrm{SC}})\right)\right),$$

$${\pi }_{{\mathcal{Z}}_{\mathrm{SC}}}:{\mathbb{Z}}_{{\mathcal{Z}}_{\mathrm{SC}},z_0,\delta }^{\mathrm{SC}}\to {\mathcal{Z}}_{\mathrm{SC}}.$$

$$\begin{array}[]{rl}\displaystyle{\varepsilon_{\ell}^{(1)}}&\displaystyle{=\frac{a}{2\pi{\sqrt{-1}}}\left(\log(\mathfrak{u}^{\mathrm{SC}}_{z_{0}}((\ell+1)a)-\log(\mathfrak{u}^{\mathrm{SC}}_{z_{0}}(\ell a))\right),}\\ \displaystyle{\varepsilon_{\ell}^{(2)}}&\displaystyle{=\frac{a}{2\pi{\sqrt{-1}}}\left(\log(\mathfrak{u}^{\mathrm{SC}}_{z_{0}}(\ell a+{\sqrt{-1}}a))-\log(\mathfrak{u}^{\mathrm{SC}}_{z_{0}}(\ell a))\right),}\\ \displaystyle{\varepsilon_{\ell}^{(\pm)}}&\displaystyle{=\frac{a}{2\pi{\sqrt{-1}}}\left(\log(\mathfrak{u}^{\mathrm{SC}}_{z_{0}}((\ell+1)a\pm{\sqrt{-1}}a))-\log(\mathfrak{u}^{\mathrm{SC}}_{z_{0}}(\ell a)\right),}\\ \end{array}$$

$$\begin{array}[]{rl}\displaystyle{\varepsilon_{\ell}^{(1)}}&\displaystyle{=\frac{a}{4\pi{\sqrt{-1}}}\left(\log(1+a/(\ell a-z_{0}))-\log(1+\overline{a/(\ell a-z_{0})})\right)},\\ \displaystyle{\varepsilon_{\ell}^{(2)}}&\displaystyle{=\frac{a}{4\pi{\sqrt{-1}}}\left(\log(1+a{\sqrt{-1}}/(\ell a-z_{0}))-\log(1+\overline{a{\sqrt{-1}}/(\ell a-z_{0})})\right)},\\ \displaystyle{\varepsilon_{\ell}^{(\pm)}}&\displaystyle{=\frac{a}{4\pi{\sqrt{-1}}}\left(\log(1+a(1\pm{\sqrt{-1}})/(\ell a-z_{0}))-\log(1+\overline{a(1\pm{\sqrt{-1}})/(\ell a-z_{0})})\right)}.\end{array}$$

$${\Delta }_{\ell }^{(i)}=\sqrt{a^2+({\epsilon }_{\ell }^{(i)}{)}^2}-a,(i=1,2),$$

$${\Delta }_{\ell }^{d(i,\pm )}=\sqrt{(a\pm {\epsilon }_{\ell }^{(i)}{)}^2+a^2}-\sqrt{2}a,(i=1,2),$$

$${\Delta }_{\ell }^{d(\pm )}=\sqrt{2a^2+({\epsilon }_{\ell }^{(\pm )}{)}^2}-\sqrt{2}a\text{mbox}and$$

Full text

MSC2020: 08A99; 55R05; 20H15; 11R60: 34M35; 82D25; 74E15; 82-10; 74-10;

1. Introduction

Since the dislocations in crystal lattices have effects on the properties of the materials, i.e., elasticity, plasticity and fracture, the screw dislocations have been studied from several viewpoints [AHL, HB, Na]. Recently the progress of technology in material industry, especially steel industry, requires much higher spec of the properties of material than those decades before and requires to control the production processes of materials more highly from various viewpoints. It implies that in a next couple of decades, it will be necessary to control the dislocations much more precisely than current quality. The rapid development of technology also influences the experimental equipments and thus recently we can directly observe micro-scopic and meso-scopic features of materials even in crystal scale [ISCKI, TH1, TH2]; the observation scheme could meet such expectations. However there do not exist proper tools to represent such phenomena in discrete nature in complex system; the tools must be mathematical tools, which might be quite different from the current approaches.

In order to prepare for the drastic change in material science from the viewpoint of mathematical science, we have had serial conferences for these five years in which mathematicians and material scientists including researchers in steel industry have discussed to provide the novel mathematical tools for next material science (see Acknowledgments); we provided a novel tool to describe the discrete nature in dislocation in terms of algebraic language in the previous paper [HMNSU]. In this paper, we develop the previous result to describe the symmetry in the discrete nature of the screw dislocations well in terms of elementary number theory. Using the elementary number theory, we focus on the expression of the difference between the screw dislocations in the simple cubic (SC) lattices and the body centered cubic (BCC) lattices. Though some of them are represented by other methods, the number theoretic approach turns out to be a good and natural tool for the description of the discrete systems, which will be the basic tool to investigate much more complex systems.

Though the origin of the screw dislocations is a discrete nature of crystals, the dislocations have been studied in the continuum picture because 1) there was no proper method to describe their discrete structure and 2) the continuum picture is appropriate for the behavior of macro-scale of the dislocations. In the geometrical description of the dislocations as a continuum picture [Me, KE, KF, LA], which Kondo and Amari started to investigate [Ko, A1, A2], the global behavior of the dislocations is expressed well. Even in continuum picture of the dislocations including phenomenological models, there are so many crucial mathematical problems which are effective for the material science, e.g, [CS, ENA, NN, SG].

However as mentioned above, we cannot avoid to understand micro- and meso-scopic feature of dislocations and in order to understand them, mathematics also plays important roles. Since the positions of atoms in the crystal in the micro-scopic scale are fluctuated, the micro-scopic properties of the dislocation have been investigated by means of the molecular dynamics or molecular mechanics in classical level and in the level of the first principle, e.g, [Cl, GD, IKY]. It is a crucial problem, in mathematics, how we introduce links to consider the topological properties for given position of atoms in our euclidean space.

We are concerned with the properties in the meso-scopic scale, which cannot be represented by the continuum picture neither by the molecular mechanics nor the first principle approaches. One of our purposes in this paper is to investigate the dependence of the dislocations on the type of crystals mathematically. Recently Ponsiglione [P] and Alicandro, Cicalese and Ponsiglione [ACP] investigated the behavior of dislocations in the meso-scopic scale in the framework of $\Gamma$-convergence. Hudson and Ortner [HO] and Braun, Buze, and Ortner [BBO] considered the discrete picture of dislocations. Ariza and Ortiz [AO], Ramasubramaniam, Ariza and Ortiz [ARO], and Ariza, Tellechea, Menguiano and Ortiz [AATMO] studied the discrete nature of the dislocations in terms of modern mathematics, i.e., homology theory, graph theory, group theory and so on. Especially Ariza and Ortiz [AO] and Hudson and Ortner [HO] provided geometrical methods to reveal the discrete nature of dislocations and studied the core energy of the dislocations of the BCC lattice.

We recall that the crystal lattices have high symmetries such as translational and rotational symmetries governed by the crystal groups, which are studied in the framework of crystallography. These symmetries are described well in terms of algebraic language and algebraic tools in wider meaning [CS, S]. Representation of finite groups is representation of their group rings and modules in the module theory. The lattice ${\mathbb{Z}}^{n}$ in Euclidean space ${\mathbb{E}}^{n}$ has been studied in the number theory, which is known as Minkowski arithmetic geometry and related to the quadratic fields and the harmonic analysis such as the Epstein zeta function [Te]. The two dimensional lattice, ${\mathbb{Z}}+{\mathbb{Z}}\tau(\subset{\mathbb{C}})$, ($\tau\in{\mathbb{H}}:=\{x+y{\sqrt{-1}}\in{\mathbb{C}}\ |\ y>0\}$), associated with the elliptic curves has been studied well in the study of modular forms [Kn, IR]. The action of $\mathrm{SL}(2,{\mathbb{Z}})$ on the lattice and its subgroup show the symmetry of the lattice ${\mathbb{Z}}+{\mathbb{Z}}\tau$. When $\tau={\sqrt{-1}}$ and $\tau=\omega_{6}$ (or $\omega_{3}$) for $\omega_{p}=\mathrm{e}^{2\pi{\sqrt{-1}}/p}$, they are known as the Gauss integers and the Eisenstein integers respectively [IR, Tr]. They have been studied well in the framework of the algebra and the algebraic number theory.

It is emphasized that the crystal lattices even with defects and their interfaces still have higher symmetries. They should be regarded as a kind of symmetry breaking of the group [W]. It means that they are not stable for the crystal group in general but are stable for its subgroup, at least, approximately, and should be described by algebraic theory cooperated with analytic and geometric theories.

The interfaces of two crystal lattices are described well by the quadratic fields in the elementary number theory [ISCKI, KAKT]. Thus even for the dislocations, we should express their symmetry properly. In the previous report [HMNSU] with Hamada, Nakagawa, Saeki and Uesaka, we focused on the fiber structure of the screw dislocations as an essential of the screw dislocations. The bundle map in the Cartesian square realizes the screw dislocations in the SC and the BCC lattices induced from the continuum picture. The fiber structure shows the translational symmetry of the fiber direction, which is the survived symmetry in the these crystal lattices even if the screw dislocation exists. On the other hand, the vertical to the fiber direction (the direction of Burgers vector) there are other symmetries which are induced from the crystal group for the perfect crystals, i.e., the two-dimensional crystal lattices. Though we did not argue the analytic properties in [HMNSU], when we consider the minimal point of the configuration of the atoms, their initial configuration should be indexed by natural indices reflecting the symmetry of the dislocation.

In this paper, we extend the method in previous report to express the difference between the screw dislocations in the SC and the BCC lattices algebraically. We propose a novel method to investigate the algebraic nature of the screw dislocation in crystal lattices, the SC and the BCC lattices, using the elementary number theory; the Gauss integers ${\mathbb{Z}}[{\sqrt{-1}}]$ and the Eisenstein integers ${\mathbb{Z}}[\omega_{3}]={\mathbb{Z}}[\omega_{6}]$ correspond to the vertical two-dimensional lattices for the screw dislocations of the SC and the BCC lattices. Our method shows the natural indices of configurations of atoms, which must be useful even when we consider their analytic properties. For examples, as in Remarks 3.1 and 4.6, and Lemmas 4.1 and 4.2, the ring of integers ${\mathbb{Z}}[\tau]$ of the cyclotomic field ${\mathbb{Q}}[\tau]$ show the algebraic properties in these lattices. Especially, we investigate the symmetry of the two-dimensional crystal lattice in terms of ${\mathbb{Z}}[\tau]$ to show the critical relations between the energy of the dislocations and the Epstein-Hurwitz zeta functions, in the SC and the BCC lattices as we show in Theorems 3.8 and 4.14. The number theoretic approach shows the symmetry of these systems well.

This paper is organized as follows. Section 2 and 3 review the previous report [HMNSU]. In Section 2, we show the screw dislocation in the continuum picture. Section 3 reviews the results of the SC lattice case in [HMNSU] in terms of Gauss integers ${\mathbb{Z}}[{\sqrt{-1}}]$. In Section 4, after we also show the configuration of the screw dislocation in the BCC lattice in terms of the Eisenstein integers ${\mathbb{Z}}[\omega_{6}]$ following [HMNSU], we provide the algebraic expression of the stress energy of the screw dislocation in the BCC lattice, which is our main result in this paper. In Section 5, we discuss these results and in Section 6, we summarize our results.

2. Screw Dislocations in Continuum Picture

In this section, we review the previous report [HMNSU] and show the algebraic expression of the screw dislocations in continuum picture.

2.1. Notations and Conventions

Since the translational symmetry is crucial in physics [W], in this paper, we distinguish the euclidean space ${\mathbb{E}}$ from the real vector space ${\mathbb{R}}$: we regard that ${\mathbb{R}}$ is a vector space, whereas ${\mathbb{E}}$ is the space consisting of the position vectors with translational symmetry, though both ${\mathbb{E}}^{n}$ and ${\mathbb{R}}^{n}$ are topological spaces with the ordinary euclidean topology. Similarly we distinguish the set of the complex position vector, the affine space ${\mathbb{E}}_{\mathbb{C}}$, from the complex vector space ${\mathbb{C}}$. We basically identify the $2$-dimensional euclidean space ${\mathbb{E}}^{2}$ with ${\mathbb{E}}_{\mathbb{C}}$, and ${\mathbb{R}}^{2}$ with ${\mathbb{C}}$. The group $U(1)$ naturally acts on the circle $S^{1}$. ${\mathbb{Z}}$ and ${\mathbb{Q}}$ are the sets of the rational integers and the rational numbers respectively. For a fiber bundle ${\mathcal{F}}\rightarrow{\mathcal{M}}$ over a base space ${\mathcal{M}}$, the set of continuous sections $f:{\mathcal{M}}\to{\mathcal{F}}$ is denoted by $\Gamma({\mathcal{M}},{\mathcal{F}})$.

In this paper, for $\delta=(\delta_{1},\delta_{2},\delta_{3})\in{\mathbb{E}}^{3}$, we consider an embedding $\iota_{\delta}$ of the vector space ${\mathbb{R}}^{3}={\mathbb{C}}\times{\mathbb{R}}$ into ${\mathbb{E}}^{3}={\mathbb{E}}_{\mathbb{C}}\times{\mathbb{E}}$ by

[TABLE]

[TABLE]

where $\delta_{\mathbb{C}}:=\delta_{1}+{\sqrt{-1}}\delta_{2}$.

Further we employ some conventions listed up in Appendix.

2.2. Exact Sequence and Sequence of Maps

We consider the exact sequence of groups (see [Br]),

[TABLE]

which is essential in this paper. ${\mathbb{Z}}$ and ${\mathbb{R}}$ are additive groups, ${\mathrm{U}}(1)$ is a multiplicative group, $i(n)=n\in{\mathbb{R}}$ for $n\in{\mathbb{Z}}$, and $(\exp 2\pi{\sqrt{-1}})(x)=\exp(2\pi{\sqrt{-1}}x)$ for $x\in{\mathbb{R}}$.

In our description of the screw dislocations, we fix the third axis as the direction of the Burgers vector. For $\delta_{3}$ in $\delta=(\delta_{1},\delta_{2},\delta_{3})\in{\mathbb{E}}^{3}$ and a certain positive number $d>0$ which is given as $d=a$ in Section 3 and $d=\sqrt{3}a/2$ in Section 4, we define the shifted maps,

[TABLE]

satisfying the commutative diagram,

[TABLE]

where $\psi_{d}(y)=\exp(2\pi{\sqrt{-1}}y/d)$, $y\in{\mathbb{E}}$, and $\varphi_{\delta}=\widetilde{i}_{d,\delta}\circ i$. It means that we have the sequence of maps

[TABLE]

where

[TABLE]

2.3. Fiber Structures of Crystals in Continuum Picture

Let us consider some trivial bundles over ${\mathbb{E}}_{\mathbb{C}}$; ${\mathbb{Z}}$-bundle $\pi_{{\mathbb{Z}}}:{\mathbb{Z}}_{{\mathbb{E}}_{\mathbb{C}}}\to{\mathbb{E}}_{\mathbb{C}}$, ${\mathbb{E}}$-bundle $\pi_{{\mathbb{E}}}:{\mathbb{E}}_{{\mathbb{E}}_{\mathbb{C}}}\to{\mathbb{E}}_{\mathbb{C}}$ and $S^{1}$-bundle $\pi_{S^{1}}:S^{1}_{{\mathbb{E}}_{\mathbb{C}}}\to{\mathbb{E}}_{\mathbb{C}}$. The sequence of maps (2.2) induces the sequence of bundle maps $\widehat{\varphi}_{\delta}$ and $\widehat{\psi}_{d}$,

[TABLE]

It is obvious that ${\mathbb{E}}_{{\mathbb{E}}_{\mathbb{C}}}$ is identified with our three-dimensional euclidean space ${\mathbb{E}}^{3}={\mathbb{E}}\times{\mathbb{E}}_{\mathbb{C}}$ whereas ${\mathbb{Z}}_{{\mathbb{E}}_{\mathbb{C}}}={\mathbb{Z}}\times{\mathbb{E}}_{\mathbb{C}}$ is a covering space of ${\mathbb{E}}_{\mathbb{C}}$. ${\mathbb{Z}}_{{\mathbb{E}}_{\mathbb{C}}}$ expresses the geometrical objects which consist of parallel sheets over ${\mathbb{E}}_{\mathbb{C}}$.

Let $\delta=(\delta_{\mathbb{C}}=\delta_{1}+{\sqrt{-1}}\delta_{2},\delta_{3})\in{\mathbb{E}}^{3}$. We consider the embedding

[TABLE]

This $\iota_{\delta}$ plays important roles in the following sections by restricting the domain ${\mathbb{C}}$ into its discrete sets ${\mathbb{Z}}[{\sqrt{-1}}]$ and ${\mathbb{Z}}[\omega_{6}]$, and thus we set $z=x+{\sqrt{-1}}y+\delta_{\mathbb{C}}\in{\mathbb{E}}_{\mathbb{C}}$.

Let us consider the image of the bundle map $\widehat{\varphi}_{\delta}$. For $\gamma_{\delta}=\exp(2\pi{\sqrt{-1}}\delta_{3}/d)\in S^{1}$, we define the global constant section $\mathfrak{u}_{\delta}\in\Gamma({\mathbb{E}}_{\mathbb{C}},S^{1}_{{\mathbb{E}}_{\mathbb{C}}})$ of $S^{1}_{{\mathbb{E}}_{\mathbb{C}}}$ by

[TABLE]

for $z\in{\mathbb{E}}_{\mathbb{C}}$. The following lemma is naturally obtained:

Lemma 2.1**.**

For $\gamma_{\delta}=\exp(2\pi{\sqrt{-1}}\delta_{3}/d)$, we have

[TABLE]

where

[TABLE]

Here we note that $\widehat{\varphi}_{\delta}({\mathbb{Z}}_{{\mathbb{E}}_{\mathbb{C}}})$ is the system consisting of parallel equi-interval sheets realized in the three-euclidean space ${\mathbb{E}}^{3}={\mathbb{E}}_{{\mathbb{E}}_{\mathbb{C}}}$.

2.4. Single Screw Dislocation in Continuum Picture

For $z_{0}\in{\mathbb{E}}_{\mathbb{C}}$, let us consider the non-trivial bundles ${\mathbb{E}}_{{\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\}}$ and $S^{1}_{{\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\}}$ over ${\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\}$. In other words, we consider the section $\mathfrak{u}_{z_{0},\delta}\in\Gamma({\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\},S^{1}_{{\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\}})$ defined by

[TABLE]

and a natural universal covering of ${\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\}$,

[TABLE]

by letting the restriction $\pi_{z_{0},\delta}=\pi_{{\mathbb{E}}}|_{{\mathbb{Z}}_{{\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\},\delta}}$, i.e., $\pi_{z_{0},\delta}:{\mathbb{Z}}_{{\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\},\delta}\to{\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\}$. In this paper, we call this covering a screw dislocation in a continuum picture which is realized as a subset of ${\mathbb{E}}^{3}$ following [AHL, HB, Na]; In these textbooks [AHL, HB, Na], ${\mathbb{Z}}_{{\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\},\delta}$ is given by geometrical consideration as a screw dislocation, which is mentioned in Remark 2.3, whereas it should be noted that our construction of ${\mathbb{Z}}_{{\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\},\delta}$ is purely algebraic.

As in [HMNSU], it is not difficult to extend this expression of the single screw dislocation to one of multi-screw dislocations.

Remark 2.2**.**

For the simply connected neighborhood $U_{p}\subset{\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\}$ of a point $p$ of ${\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\}$,

[TABLE]

as a covering space of $U_{p}$.

Remark 2.3**.**

${\mathbb{Z}}_{{\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\},\delta}$ can be obtained by the following the operation on the trivial covering ${\mathbb{Z}}\times({\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\})$ with the embedding $\iota_{{\mathbb{E}}}:{\mathbb{Z}}\times({\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\})\hookrightarrow{\mathbb{E}}^{3}$, such that $\pi_{{\mathbb{E}}}:\iota_{{\mathbb{E}}}({\mathbb{Z}}\times({\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\}))\to{\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\}$. We regard it as the set of sheets indexed by the integers $n$. The third position component of the $n$-th sheet is given by $nd+\delta_{3}$. Let us consider a half line $L:=\{x+{\sqrt{-1}}y_{0}\ |\ x\geq x_{0}\}$ for $z_{0}=x_{0}+{\sqrt{-1}}y_{0}$ and ${\mathbb{E}}_{\mathbb{C}}\setminus L$ as a simply connected open set of ${\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\}$. First we cut $\iota_{{\mathbb{E}}}({\mathbb{Z}}\times({\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\}))$ at the inverse $\pi_{{\mathbb{E}}}^{-1}(L)\subset{\mathbb{E}}^{3}$. In other words, we consider $\pi_{{\mathbb{E}}}^{-1}({\mathbb{E}}_{\mathbb{C}}\setminus L)$ noting Remark 2.2. We deform the $n$-th sheet in ${\mathbb{E}}^{3}$ such that the third component is given by $nd+\delta_{3}+\displaystyle{\frac{d}{2\pi}\mathrm{arg}\frac{z-z_{0}}{|z-z_{0}|}}$. After then, we connect the $n$-th sheet to the $(n+1)$-th sheet at the place $\pi_{{\mathbb{E}}}^{-1}(L)$. Then we obtain ${\mathbb{Z}}_{{\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\},\delta}$ in (2.6). It means that this is a construction of ${\mathbb{Z}}_{{\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\},\delta}$ as a discontinuous deformation of ${\mathbb{Z}}_{{\mathbb{E}}_{\mathbb{C}},\delta}$, which is the standard geometrical description of the dislocation [AHL, HB, Na]. **

3. Screw Dislocation in Simple Cubic Lattice

In this section, we show the algebraic description of the screw dislocation in the SC lattice and its stress energy in terms of the Gauss integers ${\mathbb{Z}}[{\sqrt{-1}}]\subset{\mathbb{C}}$ (see Appendix).

3.1. SC Lattice as Covering Space of ${\mathbb{Z}}[{\sqrt{-1}}]$

For the SC lattice in the three euclidean space ${\mathbb{E}}^{3}$,

[TABLE]

where $\delta=(\delta_{1},\delta_{2},\delta_{3})\in{\mathbb{E}}^{3}$, and $a$ is the lattice length $(a>0)$, we find its fiber structure as in the previous section. Let ${\mathcal{Z}}_{\mathrm{SC}}:=\{n_{1}a+n_{2}a{\sqrt{-1}}\ |\ n_{1},n_{2}\in{\mathbb{Z}}\}\subset{\mathbb{C}},$ which can be expressed by the Gauss integers ${\mathbb{Z}}[{\sqrt{-1}}]={\mathbb{Z}}+{\mathbb{Z}}{\sqrt{-1}}$,

[TABLE]

For $\delta=(\delta_{1},\delta_{2},\delta_{3})\in{\mathbb{E}}^{3}$, we define the embedding,

[TABLE]

where, $\delta_{\mathbb{C}}=(\delta_{1}+\delta_{2}{\sqrt{-1}})\in{\mathbb{E}}_{\mathbb{C}}$. The embedding $\iota^{\mathrm{SC}}_{\delta}$ induces the bundle map $\widehat{\iota}^{\mathrm{SC}}_{\delta}$.

Using $\mathfrak{u}_{\delta}(z)$ in Lemma 2.1 of $\gamma_{\delta}:=\exp(2\pi{\sqrt{-1}}\delta_{3}/a)$ for the position $\delta\in{\mathbb{E}}^{3}$, we reconstruct the SC lattice ${\mathbb{Z}}_{{\mathrm{SC}},\delta}$ by

[TABLE]

which is realized in ${\mathbb{E}}^{3}$, ${\mathbb{Z}}_{{\mathrm{SC}},\delta}={\mathbb{Z}}_{{\mathcal{Z}}_{\mathrm{SC}},\delta}\subset{\mathbb{E}}^{3}$. Here we set $d=a$ in $\psi_{d}$ in the previous section.

Remark 3.1**.**

Corresponding to Lemmas 4.1 and 4.2, and Remark 4.6 for the BCC lattice case, we have the formula in ${\mathbb{Z}}[{\sqrt{-1}}]$,

[TABLE]

which is known as the cyclotomic symmetry of ${\mathbb{Z}}[{\sqrt{-1}}]$ or the cyclic group ${\mathfrak{C}}_{4}$ action of the order 4 on ${\mathbb{Z}}[{\sqrt{-1}}]$. This relation makes the formula (3.13) simply described and connected with the Epstein-Hurwitz zeta function as in Theorem 3.8. **

3.2. Graph related to ${\mathbb{Z}}_{{\mathcal{Z}}_{\mathrm{SC}},\delta}$

We introduce the infinite graph $G^{\mathrm{SC}}_{\delta}$ whose nodes are given by ${\mathbb{Z}}_{{\mathcal{Z}}_{\mathrm{SC}},\delta}\cong{\mathbb{Z}}^{3}$.

We consider the edges among the nodes in $G^{\mathrm{SC}}_{\delta}$. As $G^{\mathrm{SC}}_{\delta}$ is parameterized by ${\mathbb{Z}}^{3}$, we consider the edges

[TABLE]

for every point $(n_{1},n_{2},n_{3})\in{\mathbb{Z}}^{3}\cong{\mathbb{Z}}_{{\mathcal{Z}}_{\mathrm{SC}},\delta}$. The first and the second components correspond to the horizontal directions whereas the third one does to the vertical direction.

3.3. Dislocation in SC Lattice as Covering Space of ${\mathbb{Z}}[{\sqrt{-1}}]$

A screw dislocation in the simple cubic lattice appears along the $(0,0,1)$-direction [Na] up to automorphisms of the SC lattice. The Burgers vector is parallel to the $(0,0,1)$-direction.

Using the fibering structure of ${\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\}$, we can describe a single screw dislocation in the SC lattice as in [HMNSU].

For $\delta=(\delta_{1},\delta_{2},\delta_{3})\in{\mathbb{E}}^{3}$, we also let $\gamma_{\delta}=\exp(2\pi{\sqrt{-1}}\delta_{3}/a)\in S^{1}$ and $\delta_{\mathbb{C}}=(\delta_{1}+\delta_{2}{\sqrt{-1}})$. Using (3.1), let us define the section $\mathfrak{u}_{z_{0},\delta}^{\mathrm{SC}}\in\Gamma({\mathcal{Z}}_{\mathrm{SC}},S^{1}_{{\mathcal{Z}}_{\mathrm{SC}}})$ by

[TABLE]

[TABLE]

Using this $\mathfrak{u}_{z_{0},\delta}^{\mathrm{SC}}$, we define its screw dislocation in the SC lattice, which is realized in ${\mathbb{E}}^{3}$:

Proposition 3.2**.**

For a point $z_{0}\in{\mathbb{E}}_{\mathbb{C}}$ and $\delta=(\delta_{1},\delta_{2},\delta_{3})\in{\mathbb{E}}^{3}$ such that the image of the embedding $\iota^{\mathrm{SC}}_{\delta}:{\mathcal{Z}}_{\mathrm{SC}}\to{\mathcal{Z}}_{\mathrm{SC}}+\delta_{\mathbb{C}}$ is a subset of ${\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\}$, $\iota^{\mathrm{SC}}_{\delta}({\mathcal{Z}}_{\mathrm{SC}})\subset{\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\}$, the screw dislocation around $z_{0}$ given by,

[TABLE]

is realized in ${\mathbb{E}}^{3}$, where $\gamma_{\delta}=\exp(2\pi{\sqrt{-1}}\delta_{3}/a)$ and $\delta_{\mathbb{C}}=(\delta_{1}+\delta_{2}{\sqrt{-1}})$.

It is worth while noting that ${\mathbb{Z}}_{{\mathcal{Z}}_{\mathrm{SC}},z_{0},\delta}^{\mathrm{SC}}$ can be regarded as a ‘covering space’ of the lattice ${\mathcal{Z}}_{\mathrm{SC}}$ and thus there is a natural projection,

[TABLE]

Here each fiber is ${\mathbb{Z}}=\pi_{{\mathcal{Z}}_{\mathrm{SC}}}^{-1}(\ell)$ for every $\ell\in{\mathcal{Z}}_{\mathrm{SC}}$.

3.4. Graph of Screw Dislocation in SC Lattice

We basically consider the local structure of ${\mathbb{Z}}_{{\mathcal{Z}}_{\mathrm{SC}},z0,\delta}^{\mathrm{SC}}$, i.e., ${\mathbb{Z}}_{{\mathcal{Z}}_{\mathrm{SC}},z0,\delta}^{\mathrm{SC}}\bigcap\pi_{{\mathcal{Z}}_{\mathrm{SC}}}^{-1}U_{\ell a}$ for a simply connected neighborhood $U_{\iota_{\delta_{\mathbb{C}}}(\ell a)}$ of $\iota_{\delta_{\mathbb{C}}}(\ell a)\in{\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\}$ and $\ell\in{\mathbb{Z}}[{\sqrt{-1}}]$. The $\pi_{{\mathcal{Z}}_{\mathrm{SC}}}^{-1}U_{\iota_{\delta_{\mathbb{C}}}(\ell a)}$ can be regarded as a “trivial covering” as in the sense of Remark 2.2. We can continue to consider the edges as in Subsection 3.2. The horizontal edges in (3.3) can be determined as a set on the same sheet as in Remark 2.2. Thus we can consider the graph $G^{\mathrm{SC}}_{z0,\delta}$ for ${\mathbb{Z}}_{{\mathcal{Z}}_{\mathrm{SC}},z0,\delta}^{\mathrm{SC}}$ as a natural extension of $G^{\mathrm{SC}}_{\delta}$.

3.5. Energy of Screw Dislocation in SC Lattice

Let us consider the graphs $G^{\mathrm{SC}}_{z0,\delta}$ and $G^{\mathrm{SC}}_{\delta}$ as the subsets of ${\mathbb{E}}^{3}$. Due to the dislocation, the length of each edge in $G^{\mathrm{SC}}_{z0,\delta}$ is different from that in $G^{\mathrm{SC}}_{\delta}$. Since $G^{\mathrm{SC}}_{\delta}$ is stable mechanically, the energy of $G^{\mathrm{SC}}_{z0,\delta}$ is higher than that of $G^{\mathrm{SC}}$. We compute the energy difference following [HMNSU], which is called the stress energy of screw dislocation or the stress energy simply. Further we basically consider the local structure of ${\mathbb{Z}}_{{\mathcal{Z}}_{\mathrm{SC}},z0,\delta}^{\mathrm{SC}}$ in this section.

In the following, we also assume that $\delta=(0,0,0)$ and $\gamma_{\delta}=1$, and identify ${\mathcal{Z}}_{\mathrm{SC}}$ and its image of $\widehat{\iota}^{\mathrm{SC}}_{\delta}$ for simplicity. Further we denote $\mathfrak{u}^{\mathrm{SC}}_{z_{0},\delta}$ etc.  by $\mathfrak{u}^{\mathrm{SC}}_{z_{0}}$ etc. by suppressing $\delta$.

For $\ell\in{\mathbb{Z}}[{\sqrt{-1}}]$, we define the relative height differences $\varepsilon_{\ell}^{(1)}$, $\varepsilon_{\ell}^{(2)}$ and $\varepsilon_{\ell}^{(\pm)}$ by

[TABLE]

respectively. It is obvious that for this dislocation of the simple cubic lattice, $-a/2<\varepsilon_{\ell}^{(i)}<a/2$ for $i=1,2$ and $\pm$. It is easy to obtain

[TABLE]

Here $\overline{z}$ is the complex conjugate of $z$. The difference of length $\Delta$ in each segment from the natural length of $G^{\mathrm{SC}}_{\delta}$ is obtained by,

1. (1)

for $[(\ell\pm 1,\ell_{3})a,(\ell,\ell_{3})a]$ and $[(\ell,\ell_{3})a,(\ell+{\sqrt{-1}},\ell_{3})a]$,

[TABLE] 2. (2)

for $[(\ell,\ell_{3}),(\ell+1,\ell_{3}\pm 1)]$ or $[(\ell,\ell_{3}),(\ell\pm{\sqrt{-1}},\ell_{3}\pm 1)]$

[TABLE] 3. (3)

for $[(\ell,\ell_{3}),(\ell+1\pm{\sqrt{-1}},\ell_{3})]$,

[TABLE] 4. (4)

for $[(\ell,\ell_{3}),(\ell,\ell_{3}+1)]$, $\Delta_{\ell}^{(3)}=0$.

Remark 3.3**.**

By letting $\displaystyle{w:=\frac{a}{\ell a-z_{0}}}$ for $\ell\in{\mathbb{Z}}[{\sqrt{-1}}]$, these $\varepsilon$’s are real valued functions of $w$ and $\overline{w}$, i.e,

[TABLE]

As these expressions looks simple, the origin of the simplicity is the description based on the elementary number theory which we employ.

Since the square root function $\sqrt{1+x}$ at $x=0$ is also a real analytic function of $x$, these properties are succeeded to these $\Delta$’s. **

Lemma 3.4**.**

For $\ell a\in{\mathcal{Z}}_{\mathrm{SC}}={\mathbb{Z}}[{\sqrt{-1}}]a$ satisfying that $\displaystyle{\frac{a}{\sqrt{|\ell a-z_{0}|^{2}}}}\ll 1$, $\varepsilon^{(1)}_{\ell}$, $\varepsilon^{(2)}_{\ell}$ and $\varepsilon^{(\pm)}_{\ell}$ are approximated by

[TABLE]

respectively, whereas $\Delta_{\ell}^{(i)}$ $\Delta_{\ell}^{d(i,\pm)}$ and $\Delta_{\ell}^{d(\pm)}$ are approximated by

[TABLE]

respectively, $i=1,2$.

Proof.

Using $\log(1+z)=z+o(z^{2})$, we have the leading terms in (3.4). Remark 3.3 shows the estimation in (3.10). However we can also estimate them directly; for example, $(\varepsilon_{\ell}^{(i)})^{2}$ is estimated by

[TABLE]

since $|\ell a-z_{0}|^{2}=(\ell_{1}a-x_{0})^{2}+(\ell_{2}a-y_{0})^{2}$. Further the relation $\displaystyle{\sqrt{1+z}-1=\frac{1}{2}z+o(z^{2})}$ shows (3.10). ∎

Lemma 3.5**.**

We let $\varepsilon_{\ell}^{(c)}:=\varepsilon_{\ell}^{(2)}+{\sqrt{-1}}\varepsilon_{\ell}^{(1)}$ and we have the following:

[TABLE]

Remark 3.6**.**

As the term in Lemma 3.5 is the leading term in the stress energy in Theorem 3.8, it represents the symmetry of the dislocation in the SC-lattice as follows.

1. (1)

Since the lattice points are expressed in terms of the Gauss integers ${\mathbb{Z}}[{\sqrt{-1}}]$, the edges in the graph $G^{\mathrm{SC}}_{z_{0},\delta}$ are described by $(\ell,\ell+d)$ for $d=\{\pm 1,\pm{\sqrt{-1}}\}$, and 2. (2)

the real valued analytic function in $f(w,\overline{w})$ in the complex structure in plane $w\in{\mathbb{C}}$ has the property, $f(w,\overline{w})=\overline{f(w,\overline{w})}$.

Our expression in terms of ${\mathbb{Z}}[{\sqrt{-1}}]$ shows the property manifestly.

Further in the computations, we use the relation $1+{\sqrt{-1}}^{2}=0$, which comes from the cyclotomic symmetry as mentioned in Remark 3.1. **

Following [HMNSU], let us introduce the subsets of ${\mathbb{Z}}[{\sqrt{-1}}]$,

[TABLE]

for $N>\rho$, which is bounded and is a finite set, and the core region $C^{\sqrt{-1}}_{\rho}$,

[TABLE]

Let $A_{\rho}^{{\sqrt{-1}}}:=\displaystyle{\lim_{N\to\infty}A_{\rho,N}^{{\sqrt{-1}}}}$.

Let us evaluate the stress energy, the elastic energy caused by the screw dislocation in the meso-scopic scale. Since the screw dislocation is invariant under the translation from $\ell_{3}$ to $\ell_{3}+1$, we compute the energy density for unit length in the $(0,0,1)$-direction using Remark 2.3 and call it simply the stress energy of dislocation again.

Let $k_{p}$ and $k_{d}$ be the spring constants of the horizontal springs and the diagonal springs respectively. Then, the stress energy of dislocation in the annulus region $A_{\rho,N}^{\sqrt{-1}}$ is given by

[TABLE]

where ${\mathcal{E}}^{\mathrm{SC}}_{\ell}$ is the energy density defined by

[TABLE]

We recall Proposition 9 in [HMNSU] in terms of our convention, which are also directly obtained via Lemmas 3.4 and 3.5, and Remarks 3.3 and 3.6:

Proposition 3.7**.**

• $(1)$

For $\ell\in A_{\rho}^{\sqrt{-1}}$, the energy density ${\mathcal{E}}^{\mathrm{SC}}_{\ell}$ is expressed by a real analytic function ${\mathcal{E}}^{\mathrm{SC}}(w,\overline{w})$ of $w$ and $\bar{w}\in{\mathbb{C}}$ with $|w|<1/\sqrt{2}$ in such a way that

[TABLE]

• $(2)$

For the power series expansion

[TABLE]

with $C_{i,j}\in{\mathbb{C}}$, the following holds

• (a)

${\mathcal{E}}_{\mathrm{SC}}^{(0)}(w,\overline{w})={\mathcal{E}}_{\mathrm{SC}}^{(1)}(w,\overline{w})=0$,

• (b)

the leading term is given by

[TABLE]

• (c)

$C_{i,j}=\overline{C_{j,i}}$, and

• (d)

for every $s\geq 2$, there is a constant $M_{s}>0$ such that

[TABLE]

As the summation in (3.12) is finite, we have

[TABLE]

As mentioned in Lemmas 3.4, following [HMNSU], the “principal part” of the stress energy of the screw dislocation in the SC lattice is given by the following theorem:

Theorem 3.8**.**

The principal part of the stress energy $E_{\rho,N}(z_{0})$, defined by

[TABLE]

is given by the truncated Epstein-Hurwitz zeta function (see Appendix),

[TABLE]

As mentioned in Remark 3.1, it is noted that we obtain (3.13) and this theorem due to the cyclotomic symmetry (3.2).

By Proposition 3.7 (2) (d), we can estimate each of the other terms appearing in the power series expansion (3.14) by the truncated Epstein-Hurwitz zeta function as follows.

Proposition 3.9**.**

For each $s\geq 3$, there exists a positive constant $M_{s}^{\prime}$ such that

[TABLE]

4. Screw Dislocation in BCC Lattice and its energy

In this section, we consider the screw dislocation in the BCC lattice. The studies of the screw dislocations in the BCC crystal lattices have long history, e.g., [KV, Ta], and still attract attentions; some of the studies are based on the first principle approaches, e.g. [Cl, GD, IKY], others are via the continuum approaches, e.g, [Me, KE, KF, LA], and geometrical approaches [P, AO, AATMO, HO].

However in this paper, we concentrate ourselves on the number theoretic approaches based on the previous report [HMNSU]. We summarize the algebraic descriptions of the BCC lattice and its screw dislocation in [HMNSU]. In this paper, we employ the novel description of the screw dislocation in terms of the elementary number theory. We show that the screw dislocation of the BCC lattice is expressed well in terms of the Eisenstein integers. Using this description, we compute its stress or the stress energy like the case of the SC lattice.

4.1. Preliminary: Eisenstein Integers

We show the basic properties of the Eisenstein integers ${\mathbb{Z}}[\omega_{6}]={\mathbb{Z}}[\omega_{3}]$ [Tr] (see Appendix). For the primitive sixth root of unit, $\omega_{6}$, we have the following relations:

Lemma 4.1**.**

[TABLE]

We introduce $\nu_{i}$ and $\mu_{i}$ by

[TABLE]

It is noted that they belong to ${\displaystyle\frac{1}{3}}{\mathbb{Z}}[\omega_{6}]:=\{\ell_{1}+\ell_{2}\omega_{6}\ |\ 3\ell_{a}\in{\mathbb{Z}}\}$ and have the properties in the following lemma:

Lemma 4.2**.**

1. (1)

${\mathbb{Z}}[\omega_{6}]={\mathbb{Z}}\oplus{\mathbb{Z}}\omega_{6}={\mathbb{Z}}[\omega_{3}]$, 2. (2)

${\mathbb{Z}}[\omega_{6}]+\nu_{0}\ni\nu_{2},\nu_{4}$, 3. (3)

${\mathbb{Z}}[\omega_{6}]+\nu_{1}\ni\nu_{3},\nu_{5}$, and 4. (4)

for $z\in{\mathbb{E}}_{\mathbb{C}}$,

[TABLE]

[TABLE]

Proof.

The relations (1)-(3) are geometrically obvious but it can, also, be proved by the cyclotomic properties in Lemma 4.1. In (4), the left hand side is equal to

[TABLE]

From Lemma 4.1, we have

[TABLE]

and thus the left hand side gives the right hand side. ∎

Remark 4.3**.**

As mentioned in Remark 3.1, ${\mathbb{Z}}[{\sqrt{-1}}]$ has the cyclic group ${\mathfrak{C}}_{4}$ action of the order 4 as the cyclotomic symmetry of ${\mathbb{Z}}[{\sqrt{-1}}]$. The Eisenstein integers ${\mathbb{Z}}[\omega_{6}]$ has the cyclic group ${\mathfrak{C}}_{6}$ action of the order 6 as the cyclotomic symmetry of ${\mathbb{Z}}[\omega_{6}]$. Since the cyclic group ${\mathfrak{C}}_{3}$ of the order 3 is a subgroup of ${\mathfrak{C}}_{6}$, there is the ${\mathfrak{C}}_{3}$ action on ${\mathbb{Z}}[\omega_{6}]$. Lemma 4.2 (4) is based on the symmetry.

As the cyclotomic symmetry of ${\mathbb{Z}}[{\sqrt{-1}}]$ plays the important role in Lemma 3.5 and Remark 3.6, and thus in Proposition 3.7 and Theorem 3.8, the cyclotomic symmetry in ${\mathbb{Z}}[\omega_{6}]$ also plays an important role in the evaluation of the stress energy of the dislocation in BCC lattice as in Lemma 4.11, Remark 4.12, Proposition 4.13, and Theorem 4.14. **

4.2. Algebraic Structure of BCC Lattice

Though there are several algebraic descriptions of the BCC lattice (see [CS, p. 116], for example), we recall the algebraic descriptions of the BCC lattice [HMNSU]. We assume that $a_{1}=(a,0,0)$, $a_{2}=(0,a,0)$, $a_{3}=(0,0,a)$ in ${\mathbb{R}}^{3}$ for a positive real number $a$ as shown in Figure 1. The generator $b$ corresponds to the center point of the cube generated by $a_{1}$, $a_{2}$ and $a_{3}$. The BCC lattice is the lattice in ${\mathbb{R}}^{3}$ generated by $a_{1}$, $a_{2}$, $a_{3}$ and $b=(a_{1}+a_{2}+a_{3})/2$. Algebraically, it is described as an additive group (or a ${\mathbb{Z}}$-module) by

[TABLE]

where $\langle 2b-a_{1}-a_{2}-a_{3}\rangle_{\mathbb{Z}}$ is the subgroup generated by $2b-a_{1}-a_{2}-a_{3}$. The lattice point in ${\mathbb{B}}^{a}$ is given by $\ell_{1}a_{1}+\ell_{2}a_{2}+\ell_{3}a_{3}+\ell_{b}b$ for a certain $\ell_{i}\in{\mathbb{Z}}$ ($i=1,2,3$) and $\ell_{b}\in\{0,1\}$.

The lattice ${\mathbb{B}}^{a}$ is group-isomorphic to the multiplicative group

[TABLE]

Let us denote by ${\mathbb{A}}_{4}$ the multiplicative free abelian group of rank $4$ generated by $\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$ and $\beta$, i.e.,

[TABLE]

Then, ${\mathbb{B}}$ is also described as the quotient group

[TABLE]

where $\langle\beta^{2}\alpha_{1}^{-1}\alpha_{2}^{-1}\alpha_{3}^{-1}\rangle$ is the (normal) subgroup generated by $\beta^{2}\alpha_{1}^{-1}\alpha_{2}^{-1}\alpha_{3}^{-1}$. We shall consider the group ring ${\mathbb{C}}[{\mathbb{B}}]$ of ${\mathbb{B}}$,

[TABLE]

4.3. Algebraic Structure of BCC Lattice for $(1,1,1)$-Direction

It is known that a screw dislocation in the BCC lattice is basically given by the $(1,1,1)$-direction since the Burgers vector is parallel to the $(1,1,1)$-direction [Na].

In this subsection, we consider the algebraic structure of the BCC lattice of (111)-direction to describe its fibering structure by noting Figure 2 (a) and (b). Let us consider the subgroup of ${\mathbb{B}}$, which corresponds to the translation in the plane vertical to $(1,1,1)$-direction,

[TABLE]

and ${\mathbb{C}}[{\mathbb{B}}_{H}]$-modules.

Lemma 4.4**.**

There are isomorphisms as ${\mathbb{C}}[{\mathbb{B}}_{H}]$-modules:

[TABLE]

[TABLE]

These decompositions mean that the BCC lattice has the triple different fiber structures of three sheets. We should note that ${\mathcal{R}}_{6}$ can be regarded as a double covering of ${\mathcal{R}}_{3}$. The interval between the sheets is now given by $\sqrt{3}a/6$, and let us denote ${\mathcal{R}}_{3}$ as a set, the image of the forgetful functor to the category of set, by ${\mathbb{B}}^{a}$ as a subset of the vector space ${\mathbb{R}}^{3}$ corresponding to the three sheets:

Lemma 4.5**.**

As a set, ${\mathbb{B}}^{a}$ is also decomposed as

[TABLE]

where

[TABLE]

4.4. Fiber Structure of BCC Lattice

and Eisenstein Integers

We can regard ${\mathbb{B}}^{(a)}$ as trivial covering space of $\ell_{3}$-direction. On the other hand, the additive group of ${\mathbb{B}}_{H}$,

[TABLE]

can be expressed by the the Eisenstein integers.

We define

[TABLE]

and ${\mathcal{Z}_{\mathrm{BCC}}^{(a)}}:=({\mathbb{Z}}[\omega_{6}]+\mu_{c})d_{1}$ $(a=0,1,2)$ using (4.1), i.e.,

[TABLE]

which correspond to ${\mathbb{B}}^{(0)}$, ${\mathbb{B}}^{(1)}$ and ${\mathbb{B}}^{(2)}$ respectively as in Figure 3, i.e., there are natural projections for $b$-direction, $\pi_{{\mathrm{BCC}}}^{(a)}:{\mathbb{B}}^{(a)}\to{\mathcal{Z}_{\mathrm{BCC}}^{(a)}}$.

Remark 4.6**.**

The projection $\pi_{{\mathrm{BCC}}}^{(a)}$ of these ${\mathbb{B}}^{(0)}$, ${\mathbb{B}}^{(1)}$ and ${\mathbb{B}}^{(2)}$ are essentially equal to ${\mathbb{Z}}[\omega_{6}]$ up to translation and dilatation $d_{1}$. Further $\mu_{c}$ and $\nu_{i}$ are third points in the lattice $\nu_{i}\in\frac{1}{3}{\mathbb{Z}}[\omega_{6}]$. They have the algebraic properties of Lemmas 4.1 and 4.2 and Remark 4.3, whose origin in the ring of integers of the cyclotomic field ${\mathbb{Q}}[\omega_{6}]$. They have studied in the number theory and algebraic geometry [FKMPA, Appendix], and their application to physics [Ma].

As shown in the following, the $z_{3}$ position of each sheet in the screw dislocations along the $(1,1,1)$-direction and the local energy due to the dislocation can be regarded as functions on ${\mathbb{Z}}[\omega_{6}]$, more precisely on ${\mathcal{Z}_{\mathrm{BCC}}^{(c)}}:=({\mathbb{Z}}[\omega_{6}]+\mu_{c})d_{1}$ $(c=0,1,2)$. Thus Lemmas 4.1 and 4.2 govern the computations of the functions in Lemma 4.11 and make them very simple and connected with the Epstein-Hurwitz zeta function as in Theorem 4.14.

Though in the previous works including [HMNSU], such properties had not been mentioned, these properties show algebraic nature of the BCC lattice and the screw dislocation in the BCC lattice. **

For a point $\delta=(\delta_{1},\delta_{2},\delta_{3})\in{\mathbb{E}}^{3}$, we consider the embedding $\iota_{\delta}:{\mathbb{B}}^{a}\to{\mathbb{E}}^{3}$ and its image $\iota_{\delta}({\mathbb{B}}^{a})$. Corresponding to $\iota_{\delta}$, for the point $\delta_{\mathbb{C}}=\delta_{1}+{\sqrt{-1}}\delta_{2}\in{\mathbb{E}}_{\mathbb{C}}$, let us also consider the embedding $\iota_{\delta_{\mathbb{C}}}:{\mathcal{Z}_{\mathrm{BCC}}^{(c)}}\to{\mathcal{Z}_{\mathrm{BCC}}^{(c)}}+\delta_{\mathbb{C}}\in{\mathbb{E}}_{\mathbb{C}}$ and the bundle maps $\widehat{\iota}_{\delta_{\mathbb{C}}}$. Further for $\gamma_{\delta}=\mathrm{e}^{{\sqrt{-1}}\delta_{3}/d_{0}}\in S^{1}$ and a constant section $\mathfrak{u}_{\delta}\in\Gamma({\mathbb{E}}_{\mathbb{C}},S^{1}_{{\mathbb{E}}_{\mathbb{C}}})$ $(\mathfrak{u}_{\delta}(z)=\gamma_{\delta})$, we consider $\mathfrak{u}^{\mathrm{BCC}}_{\delta}\in\Gamma({\mathcal{Z}_{\mathrm{BCC}}},S^{1}_{{\mathcal{Z}_{\mathrm{BCC}}}})$ as

[TABLE]

where $S^{1}_{{\mathcal{Z}_{\mathrm{BCC}}}}$ is the trivial $S^{1}$ bundle over ${\mathcal{Z}_{\mathrm{BCC}}}$.

Proposition 4.7**.**

The BCC lattice $\iota_{\delta}({\mathbb{B}}^{a})$ is expressed by

[TABLE]

Here we set $d=d_{0}=|b|$ of $\psi_{d}$ in Section 2.

4.5. Spiral Structure in Graph of BCC Lattice

In the BCC lattice $\iota_{\delta}({\mathbb{B}}^{a})$, let us consider the graph $G^{\mathrm{BCC}}_{\delta}$ whose nodes are given as the lattice points of the BCC lattice and edges are given as the shortest connections of the nodes as shown in Figure 4 (a). We regard $G^{\mathrm{BCC}}_{\delta}$ as a subset of ${\mathbb{E}}^{3}$. The 0-th sheet $\iota_{\delta}({\mathbb{B}}^{(0)})$ ($\iota_{\delta_{\mathbb{C}}}({\mathcal{Z}_{\mathrm{BCC}}^{(0)}})$) whose nodes are denoted by the black dots is connected with the first sheet $\iota_{\delta}({\mathbb{B}}^{(1)})$ ($\iota_{\delta_{\mathbb{C}}}{\mathcal{Z}_{\mathrm{BCC}}^{(1)}})$) which corresponds to the gray dots, via the long dots lines in Figure 4 (a). The short dot line connects $\iota_{\delta}({\mathbb{B}}^{(2)})$ ($\iota_{\delta_{\mathbb{C}}}{\mathcal{Z}_{\mathrm{BCC}}^{(2)}})$) and $\iota_{\delta}({\mathbb{B}}^{(0)})$ ($\iota_{\delta_{\mathbb{C}}}{\mathcal{Z}_{\mathrm{BCC}}^{(0)}})$) whereas the black lines connect $\iota_{\delta}({\mathbb{B}}^{(1)})$ ($\iota_{\delta_{\mathbb{C}}}{\mathcal{Z}_{\mathrm{BCC}}^{(1)}})$) and $\iota_{\delta}({\mathbb{B}}^{(2)})$ ($\iota_{\delta_{\mathbb{C}}}{\mathcal{Z}_{\mathrm{BCC}}^{(2)}})$).

The graph $G^{\mathrm{BCC}}_{\delta}$ has a projection to a plane ${\mathbb{E}}_{\mathbb{C}}$ as in Figure 4 (b): $\pi_{G}:G^{\mathrm{BCC}}_{\delta}\to{\mathbb{E}}_{\mathbb{C}}$, These edges give the paths which connect these covering sheets. As in Figure 4 (b), let us consider the path whose projection is a cycle $\pi_{G}(G^{\mathrm{BCC}}_{\delta})$ consisting of tree edges, which is called ”spiral path” because the end point $p$ and the start point $q$ exist on the different covering sheets but $q\in\pi_{G}^{-1}(\pi_{G}(p))$; if the start point is $(\ell_{1},\ell_{2},\ell_{3})$ in $\iota_{\delta}({\mathbb{B}}^{(a)})$, the end point is given as $(\ell_{1},\ell_{2},\ell_{3}^{\prime})$ in $\iota_{\delta}({\mathbb{B}}^{(a)})$ for $|\ell_{3}^{\prime}-\ell_{3}|=d_{0}=|b|$. Thus the path shows the spiral curve in ${\mathbb{E}}^{3}$.

The set of the spiral paths is classified by two types. We assign an orientation on ${\mathbb{E}}_{\mathbb{C}}$ and the orientation of the arrowed graph $[\pi_{G}(G^{\mathrm{BCC}}_{\delta})]$ is naturally induced from it. For the oriental cycle in $\pi_{G}(G^{\mathrm{BCC}}_{\delta})$, the spiral path is ascendant or decedent with respect to $\ell_{3}$. We call these triangle cells ascendant cell and descendant cell respectively. They are illustrated in Figure 5 (a) and (b) respectively.

For a center point $z_{c}$ of a ascendant triangle cell of $\pi_{G}(G^{\mathrm{BCC}}_{\delta})$, the nodes in $G^{\mathrm{BCC}}_{\delta}$ are given by

[TABLE]

whereas for a center point $z_{c}$ of a descendant triangle cell of $\pi_{G}(G^{\mathrm{BCC}}_{\delta})$, the nodes are expressed by

[TABLE]

These pictures are well-described in the works of Ramasubramaniam, Ariza and Ortiz [ARO] and [AO] using the homological investigations more precisely.

4.6. Algebraic Description of Screw Dislocations in BCC Lattice

As defined in Subsection 4.5, for $\delta\in{\mathbb{E}}^{3}$ and $z_{0}\in{\mathbb{E}}_{\mathbb{C}}$, we use the embedding $\iota_{\delta_{\mathbb{C}}}:{\mathcal{Z}_{\mathrm{BCC}}^{(i)}}\to{\mathcal{Z}_{\mathrm{BCC}}^{(i)}}+\delta_{\mathbb{C}}\in{\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\}$, $\delta_{\mathbb{C}}=\delta_{1}+{\sqrt{-1}}\delta_{2}\in{\mathbb{E}}_{\mathbb{C}}$, $\gamma_{\delta}=\mathrm{e}^{{\sqrt{-1}}\delta_{3}/d_{0}}\in S^{1}$ and the bundle map $\widehat{\psi}_{d_{0}}$ so that the description of the screw dislocation is obtained as follows. Let us consider the non-trivial $S^{1}$-bundle over ${\mathcal{Z}_{\mathrm{BCC}}}$ induced from the embedding $\iota_{\delta_{\mathbb{C}}}$. Using the section of $\mathfrak{u}_{z_{0},\delta}\in\Gamma({\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\},S^{1}_{{\mathbb{E}}_{\mathbb{C}}\setminus\{z_{0}\}})$ in (2.5), we define the section $\widehat{\mathfrak{u}}_{z_{0},\delta}$ in $\Gamma({\mathcal{Z}_{\mathrm{BCC}}},S^{1}_{{\mathcal{Z}_{\mathrm{BCC}}}})$,

[TABLE]

It implies that

[TABLE]

Proposition 4.8**.**

The single screw dislocation around $\pi_{{\mathbb{E}}}^{-1}(z_{0})\subset{\mathbb{E}}^{3}={\mathbb{E}}_{{\mathbb{E}}_{\mathbb{C}}}$ expressed by

[TABLE]

is a subset of ${\mathbb{E}}^{3}$.

Proof.

${\mathbb{E}}_{{\mathbb{E}}\setminus\{z_{0}\}}$ is obviously a subset of ${\mathbb{E}}^{3}={\mathbb{E}}_{{\mathbb{E}}_{\mathbb{C}}}$. ∎

Remark 4.9**.**

Though it is obvious that the screw dislocation exists in ${\mathbb{E}}^{3}$ in physics, it is not obvious that a geometrical object constructed in algebraic topology is realized in ${\mathbb{E}}^{3}$, e.g, the Klein bottle. Proposition 4.8 is crucial in the description of the physical object in terms of algebraic language. **

As in the SC lattice, we also consider the graph $G_{z_{0},\delta}^{\mathrm{BCC}}$. In the following, we also assume that $\gamma=1$ and $\delta_{\mathbb{C}}=0$ for simplicity.

4.7. Note on the Core Region of Screw Dislocations in BCC Lattice

Though the core structure in the screw dislocations in the BCC lattices has been studied well, e.g., in [Ta, ARO, AATMO] and using the first principle approach [Cl, GD, IKY], we show the description of the core region in terms of our framework.

Noting the equations (4.2) and (4.3), we consider the core region of the screw dislocation. The core region is the cells neighborhood of $z_{0}$.

If $z_{0}$ is the center of the ascendant triangle, for a vertex $z$ of the triangle, the set of the fiber direction is

[TABLE]

On the other hand, if $z_{0}$ is the center of the decedent triangle, the set of the fiber direction is

[TABLE]

for each vertex $z$ of the triangle. They are illustrated in Figure 5 (c) and (d) respectively. In the former case, there might exist different connections illustrated by the dotted lines. Thus the screw dislocation in the BCC lattice shows the quite different aspect from the case of the SC lattice.

The operation in Remark 2.3 can be applied to this system so that we have Figure 5 (c) and (d). By the operation, the connected spiral paths are deformed the disjointed paths. The disjoint subgraphs characterize the direction of the screw dislocations.

4.8. Energy of Screw Dislocation in BCC Lattice

In this section, we estimate the stress energy of the screw dislocation in the BCC lattice in the meso-scopic scale. We basically investigate the energy in parallel with the computations in the SC lattice.

For simply convention, we denote $\mathfrak{u}^{\mathrm{BCC}}_{z_{0},\delta}$ etc. simply by $\mathfrak{u}^{\mathrm{BCC}}_{z_{0}}$ etc. by suppressing $\delta$.

For $\ell\in{\mathbb{Z}}[\omega_{6}]+\mu_{c}$ $(c=0,1,2)$, we define the relative height differences $\varepsilon_{\ell}^{(c,j)}$ $(j=0,1,\cdots,5)$,

[TABLE]

Here we require that $-d_{3}/2<\varepsilon_{\ell}^{(c,j)}<d_{3}/2$. Let us introduce a parameter $\varepsilon>0$ and using it, we define the core region $C^{{\mathrm{BCC}}(j)}_{\varepsilon,I}$ of type I,

[TABLE]

Assume that $\varepsilon<d_{3}/2$. The difference of length in each segment between $\ell\in{\mathcal{Z}_{\mathrm{BCC}}^{(c)}}\setminus d_{1}C^{{\mathrm{BCC}}(c)}_{\varepsilon,\mathrm{I}}$ and its nearest neighbor lattice points is given by

[TABLE]

for $j=0,1,\cdots,5$. Here we note that $\sqrt{d_{3}^{2}+d_{2}^{2}}=\sqrt{3}a/2=d_{0}$.

We have the following.

Lemma 4.10**.**

If $\displaystyle{\frac{d_{1}}{\sqrt{|\ell d_{1}-z_{0}|^{2}}}}$ for an $\ell\in{\mathbb{Z}}[\omega_{6}]+\mu_{c}$ $(c=0,1,2)$ is sufficiently small, $\varepsilon^{(c,j)}_{\ell}$’s are approximated by

[TABLE]

respectively, whereas $\Delta_{\ell}^{(c,j)}$ are approximated by

[TABLE]

Proof.

Noting $\log(1+z)=z+o(z^{2})$, $\displaystyle{\sqrt{1+z}-1=\frac{1}{2}z+o(z^{2})}$, $\sqrt{d_{3}^{2}+d_{2}^{3}}=\sqrt{3}a/2=d_{0}$, the direct computations show them. ∎

As mentioned in Remark 4.6, due to the properties of the Eisenstein integers ${\mathbb{Z}}[\omega_{g}]$ in Lemmas 4.1 and 4.2, we have the simple expression:

Lemma 4.11**.**

If $\displaystyle{\frac{d_{1}}{\sqrt{|\ell d_{1}-z_{0}|^{2}}}}$ for an $\ell\in{\mathbb{Z}}[\omega_{6}]+\mu_{c}$ $(c=0,1,2)$ is sufficiently small,

[TABLE]

Proof.

The left hand side is equal to

[TABLE]

and thus using Lemmas 4.1 and 4.2, it becomes

[TABLE]

Here the extra terms are canceled due to the properties of $\frac{1}{3}{\mathbb{Z}}[\omega_{6}]$. ∎

Remark 4.12**.**

As we show in Remark 3.3, $\varepsilon_{\ell}^{(c,j)}$ is a real analytic function of $\displaystyle{w_{j}:=\frac{d_{1}\nu_{j}}{\ell d_{1}-z_{0}}}$ and $\overline{w}_{j}$ for $|w_{j}|\ll 1$, i.e.,

[TABLE]

and thus $\Delta_{\ell}^{(c,j)}$ is also a real analytic function of $w$ and $\overline{w}$.

As we mentioned in Remark 4.3, we have the cyclotomic symmetry in ${\mathbb{Z}}[\omega_{6}]$, or ${\mathfrak{C}}_{6}$ action on ${\mathbb{Z}}[\omega_{6}]$; there is the element $g\in{\mathfrak{C}}_{6}$ such that $g\nu_{j}=\nu_{j+1}$ whose index is given modulo 6. The action induces the action on the function $f$ over ${\mathbb{Z}}[\omega_{6}]$, i.e., $g^{*}f(x)=f(gx)$. By fixing $\ell d_{1}-z_{0}$, the action is given as

[TABLE]

and $\displaystyle{\sum_{j=0}^{5}(\Delta_{\ell}^{(c,j)})^{2}}$ in Lemma 4.11 is invariant for the action $g\in{\mathfrak{C}}_{6}$. **

For a positive number $\rho$, let us define another core region $C^{{\mathrm{BCC}}(c)}_{\rho,\mathrm{II}}$ of type II,

[TABLE]

In order to avoid to count doubly, we should concentrate one of ${\mathcal{Z}_{\mathrm{BCC}}^{(c)}}$’s and choose ${\mathcal{Z}_{\mathrm{BCC}}^{(0)}}$ in this paper. Let the core region of type III and its compliments be

[TABLE]

and

[TABLE]

where for a node $v$ in the plane graph $\pi_{G}(G_{z_{0}}^{\mathrm{BCC}})$, we denote the set of the adjacent nodes of $v$ by $\mathrm{Ad}(v)$ and we define

[TABLE]

For the case that $\rho$ is sufficiently large for given $\varepsilon\displaystyle{\left(<\frac{d_{3}}{2}\right)}$ so that $C^{{\mathrm{BCC}}(0)}_{\varepsilon,\rho,\mathrm{III}}\subset C^{{\mathrm{BCC}}(0)}_{\rho,\mathrm{II}}$, we also define

[TABLE]

We compute the stress energy caused by the screw dislocation in the BCC lattice as in the SC lattice case. We compute the energy density for unit length in the $(1,1,1)$-direction, and call it simply the stress energy of dislocation again.

Let $k_{d}$ be the spring constant of the edges. The stress energy of dislocation in the annulus region $A^{\omega_{6}}_{\varepsilon,\rho,N}$ is given by

[TABLE]

where ${\mathcal{E}}^{\mathrm{BCC}}_{\ell}$ for every $\ell\in{\mathbb{Z}}[\omega_{6}]$ is the energy density defined by

[TABLE]

As in Proposition 3.7 in the SC lattice, we summarize the above results as the stress energy of the dislocation of the BCC lattice case:

Proposition 4.13**.**

• $(1)$

For $\ell\in A^{\omega_{6}}_{\varepsilon,\rho,N}$, the energy density ${\mathcal{E}}_{\ell}$ is expressed by a real analytic function ${\mathcal{E}}^{\mathrm{BCC}}(w,\overline{w})$ of $w$ and $\bar{w}\in{\mathbb{C}}$ with $|w|<1/\sqrt{2}$ in such a way that

[TABLE]

• $(2)$

Let us consider the power series expansion

[TABLE]

for some $C_{i,j}\in{\mathbb{C}}$. Then, we have the following:

• (a)

${\mathcal{E}}_{\mathrm{BCC}}^{(0)}(w,\overline{w})={\mathcal{E}}_{\mathrm{BCC}}^{(1)}(w,\overline{w})=0$,

• (b)

the leading term is given by

[TABLE]

• (c)

$C_{i,j}=\overline{C_{j,i}}$, and

• (d)

for every $s\geq 2$, there is a constant $M_{s}>0$ such that

[TABLE]

Proof.

Remark 4.12 Lemmas 4.10 and 4.11 show (1) and (2) (a), (b). Since the energy density is a real number, we obtain the relation in item (c). The analyticity in item (1) implies (d). This completes the proof. ∎

As the summation in (4.8) is finite, we have

[TABLE]

In particular, we have the following theorem for the “principal part” of the stress energy.

Theorem 4.14**.**

For the case that $\rho$ is sufficiently large for given $\varepsilon\displaystyle{\left(<\frac{d_{3}}{2}\right)}$ so that $C^{{\mathrm{BCC}}(0)}_{\varepsilon,\rho,\mathrm{III}}\subset C^{{\mathrm{BCC}}(0)}_{\rho,\mathrm{II}}$. Let $A^{\omega_{6}}_{\rho,N}:=A^{\omega_{6}}_{\varepsilon,\rho,N}$. The principal part of the stress energy $E_{\rho,N}(z_{0})$, defined by

[TABLE]

is given by the truncated Epstein-Hurwitz zeta function (see Appendix),

[TABLE]

It is noted that this theorem is obtained due to the properties of $\displaystyle{\frac{1}{3}{\mathbb{Z}}[\omega_{6}]}$ cf. Remark 4.6.

By Proposition 4.13 (2) (d), we can estimate each of the other terms appearing in the power series expansion (4.11) by the truncated Epstein-Hurwitz zeta function for Eisenstein integers as follows.

Proposition 4.15**.**

For each $s\geq 3$, there exists a positive constant $M_{s}^{\prime}$ such that

[TABLE]

5. Discussion

In this paper, we investigated the screw dislocations of the SC lattice and the BCC lattice using the number theoretic descriptions in terms of the Gauss integers ${\mathbb{Z}}[{\sqrt{-1}}]$ and the Eisenstein integers ${\mathbb{Z}}[\omega_{6}]$.

As mentioned in Remark 4.6, using the properties of the Eisenstein integers ${\mathbb{Z}}[\omega_{6}]$, e.g., Lemmas 4.1 and 4.2, we obtain the simple description of the stress energy for the screw dislocation for the finite region except the core region. It reflects the symmetry of the screw dislocations. It is quite natural to investigate the symmetry of a mathematical object using algebraic language. Without the representation of the dislocation in terms of the Eisenstein integers ${\mathbb{Z}}[\omega_{6}]$, it is very difficult to obtain the result because the dislocation in the BCC lattice is very complicate. Even for the core region, we can investigate it as in Subsection 4.7 using the properties of ${\mathbb{Z}}[\omega_{6}]$. Our description is natural even when we consider the analytic property like the energy minimum point of the screw dislocations because the symmetry is built in the descriptions.

We could estimate the dislocation of meso-scopic scale because the stress energy $E_{\mathrm{total}}$ is given by

[TABLE]

The effect in the core region should be investigated by the first principle computations but the meso-scopic energy could not be obtained. Even though we need more precise investigation for the estimation because there are some parameters, we have the formula to evaluate the meso-scopic energy. It is noted that the core energy is determined by the local data whereas the meso-scopic energy is determined by the meso-scopic data.

The energy of the meso-scale essentially diverges and thus it is important to determine the cut-off parameter $N$. Let $\zeta_{\rho,N}^{\tau}(s,z_{0}):=\zeta_{A_{\rho,N}^{\tau}}^{\tau}(s,z_{0})$. As we show the behaviors of the $\zeta_{\rho,N}^{\tau}(2,0)$ for $\tau={\sqrt{-1}}$ and $\omega_{6}$ in Figure 6, they are approximated well by the logarithmic function. It is natural since the continuum theory, in which the dislocation energy $E_{\mathrm{total}}(R)$ in the inner region $\{z\in{\mathbb{C}}\ ||z-z_{0}|<R\}$ is written by the logarithmic function with respect to the radius from the dislocation line; $E_{\mathrm{total}}(R)\propto\log R$.

Further we show the density of the $\zeta_{\rho,N}^{\tau}(2,x+y\tau^{\prime})$ as in Figure 7 by numerical computations; the region of $z_{0}/d$ is divided by $20\times 20$ blocks. These aspects in the regions are different though the differences are not large due to the divergent properties like the logarithmic function.

As we computed the double dislocations case in the SC lattice in the previous work [HMNSU, Appendix], they are described well by the Green function in the statistical field theory like vortexes as in [AO, ID]. In the computations of the Green function, there appear the quadratic form $k\ell\in{\mathbb{C}}$ modulo $2\pi{\mathbb{Z}}$, where $k\in{\mathbb{Q}}(\tau)\pi/d$ and $\ell\in{\mathbb{Z}}(\tau)d$ for $d=a$ or $d=d_{1}$. These computations are very crucial in the quadratic number theory [Tr]. If the distance between the dislocation is larger enough, the behavior of the dislocations are determined by the continuum theory. However otherwise, it implies that the prime numbers in the Gauss integer or the Eisenstein integer (Gauss primes or Eisenstein primes) might have effects on the configurations of the dislocations if the meso-scopic energy plays crucial in the total energy.

6. Conclusion

In this paper, we show a discrete investigation on the screw dislocations of the SC lattices and the BCC lattices in terms of the elementary number theory. It is well-known that the two-dimensional lattices in the SC lattice perpendicular to $(0,0,1)$-direction, and in the BCC lattices perpendicular to $(1,1,1)$-direction are described in terms of the Gauss integers ${\mathbb{Z}}[{\sqrt{-1}}]$ and the Eisenstein integers ${\mathbb{Z}}[\omega_{6}]$ respectively. Since the Burger vectors in the screw-dislocation are the $(0,0,1)$ and the $(1,1,1)$ directions for the SC and the BCC lattices respectively, we use the facts and show the following:

1. (1)

The displacement caused by the screw dislocations are expressed by the functions of the Gauss integers ${\mathbb{Z}}[{\sqrt{-1}}]$ in the SC lattice case, and of the Eisenstein integers ${\mathbb{Z}}[\omega_{6}]$ in the BCC lattice case respectively, as in Propositions 3.2 and 4.8. 2. (2)

As mentioned in Remarks 3.1 and 4.3, the cyclic groups ${\mathfrak{C}}_{4}$ and ${\mathfrak{C}}_{6}$ act on the dislocations via ${\mathbb{Z}}[{\sqrt{-1}}]$ and ${\mathbb{Z}}[\omega_{6}]$ respectively, and the action plays important roles in the evaluation of the stress energy of the dislocations in these lattices as in Remarks 3.3 and 4.12. 3. (3)

Due to the symmetry, we explicitly evaluate the stress energy density of the screw dislocations in the meso-scopic scale in terms of the truncated Epstein-Hurwitz zeta functions as in Propositions 3.7 and 4.13. We remark that without number theoretic descriptions, it is difficult to obtain these expressions of the energy in terms of the truncated Epstein-Hurwitz zeta functions. 4. (4)

With knowledge in the elementary number theory, we can explicitly evaluate the leading term of the stress energy density in the meso-scopic scale using the truncated Epstein-Hurwitz zeta functions as in Theorems 3.8 and 4.14, and Figures 6 and 7 in Discussion. They show the meso-scopic contributions of the stress density in the screw dislocations.

As mentioned in Introduction, since the crystal lattices with dislocations even have high symmetries, we should investigate the dislocations by considering the symmetries. The number theoretic approach is a practical tool to describe their symmetries, translations and rotations. We demonstrated that the number theoretic approach reveals the properties of the dislocations and recovers the stress energy in the continuum picture using the Epstein-Hurwitz zeta functions. Further since the Gauss integers, the Eisenstein integers and the Epstein-Hurwitz zeta functions have interesting properties, we might find more crucial phenomenon by cooperating with analytic considerations in the future. Thus we expect that our method shed light on novel investigations on dislocations.

Acknowledgments

The author thanks to all those who participated in the problem session “Mathematical description of disordered structures in crystal” in the Study Group Workshop 2015 held in Kyushu University and in the University of Tokyo during July 29–August 4, 2015, and to the participants in the “IMI workshop II: Mathematics of Screw Dislocation”, September 1–2, 2016, in the “IMI workshop I: Mathematics in Interface, Dislocation and Structure of Crystals”, August 28–30, 2017, “IMI workshop I: Advanced Mathematical Investigation of Screw Dislocation”, September 10–11, 2018” held in Institute of Mathematics for Industry (IMI), in Kyushu University, especially Shun-ichi Amari, Toshikazu Sunada, Tetsuji Tokihiro, Kenji Higashida, Hiroyuki Ochiai, and Kazutoshi Inoue for variable discussions and comments. He is also grateful to the authors in [HMNSU], Hiroyasu Hamada, Junichi Nakagawa, Osamu Saeki and Masaaki Uesaka for helpful discussions and comments. The author has been supported by JSPS KAKENHI Grant Number 15K13438 and by Takahashi Industrial and Economic Research Foundation 2018-2019, 08-003-181. He also thanks to anonymous referees for critical and helpful comments.

Appendix A Two-dimensional lattices and the Epstein-Hurwitz zeta function

When we regard two dimensional lattice $L_{(a_{1},a_{2})}$ as the free ${\mathbb{Z}}$-modules, $L_{(a_{1},a_{2})}={\mathbb{Z}}a_{1}+{\mathbb{Z}}a_{2}(\subset{\mathbb{R}}^{2}),$ for unit vectors $a_{1},a_{2}\in{\mathbb{C}}$, where $a_{1}$ and $a_{2}$ are linear independent. It is obvious that the lattice has the unit cell. When we consider the classification of $L_{(a_{1},a_{2})}$, or its moduli space (its parameter space), it is natural to introduce the normalized lattice $L_{\tau}={\mathbb{Z}}+{\mathbb{Z}}\tau,$ for $(1,\tau:=a_{2}/a_{1})$. We assume $\tau\in{\mathbb{H}}:=\{x+{\sqrt{-1}}y\in{\mathbb{C}}\ |\ y>0\}$ without loss of generality. However there are ambiguities which ones are regarded as the unit vectors. There is an action of $\mathrm{SL}(2,{\mathbb{Z}})$ as an automorphism on $L_{\tau}\times L_{\tau}$; for $(\ell_{1},\ell_{2})$ and ${g:=\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\mathrm{SL}(2,{\mathbb{Z}})}$ $\displaystyle{g(\ell_{1},\ell_{2})={}^{t}(g\ {}^{t}(\ell_{1},\ell_{2}))=(a\ell_{1}+b\ell_{2}\tau,a\ell_{1}+d\ell_{2}\tau)}$ so that the area of the parallelogram generated by $\ell_{1}$ and $\ell_{2}$ preserves. Here for the parallelogram generated by $z_{1}=x_{1}+y_{1}{\sqrt{-1}}$ and $z_{2}=x_{2}+y_{2}{\sqrt{-1}}$, its area is equal to $x_{1}y_{2}-x_{2}y_{1}$. Thus we regard every element $g(1,\tau)$ in $\mathrm{SL}(2,{\mathbb{Z}})(1,\tau)=\{g(1,\tau)\ |\ g\in\mathrm{SL}(2,{\mathbb{Z}})\}$ as the unit vector in $L_{\tau}$.

Therefore the Möbius transformation (for $g\in\mathrm{SL}(2,{\mathbb{Z}})$, $g(z_{1}:z_{2}):=(az_{1}+bz_{2}:cz_{1}+dz_{2})$) is also introduced, which is denoted by $\mathrm{PSL}(2,{\mathbb{Z}})$. By regarding $g\tau:=g(1:\tau)$, it induces a natural group action of $\mathrm{PSL}(2,{\mathbb{Z}})$ on ${\mathbb{H}}$. The fundamental domain as the moduli of $L_{\tau}$ turns out to be ${\mathbb{H}}/\mathrm{PSL}(2,{\mathbb{Z}})$.

The following are well-known facts: e.g.,[Kn, Theorem 8.5]

Lemma A.1**.**

For a point $\tau\in{\mathbb{H}}/\mathrm{PSL}(2,{\mathbb{Z}})$, the stabilizer subgroup $G_{\tau}$ of $\mathrm{SL}(2,{\mathbb{Z}})$, $G_{\tau}:=\{g\in\mathrm{SL}(2,{\mathbb{Z}})|g(1,\tau)=(1,\tau)\}$, becomes a cyclic group ${\mathfrak{C}}_{n}:=\{t^{\ell}\ |\ \ell=0,1,\ldots,n-1,\ t^{n}=t^{0}\}$ of the order $n$, i.e.,

1. (1)

$\tau={\sqrt{-1}}=\omega_{4}$, $G_{\tau}={\mathfrak{C}}_{4}$ 2. (2)

$\tau=\omega_{6}$, $G_{\tau}={\mathfrak{C}}_{6}$, and 3. (3)

otherwise, $G_{\tau}={\mathfrak{C}}_{2}$,

where $\omega_{p}:=\mathrm{e}^{2\pi{\sqrt{-1}}/p}$.

In this paper, the both $\omega_{4}$ and $\omega_{6}$ play the crucial role. Let ${\mathbb{Z}}[\tau]:=\{\ell_{1}+\ell_{2}\tau\ |\ \ell_{1},\ell_{2}\in{\mathbb{Z}}\}$ as a discrete subset of ${\mathbb{R}}^{2}$ and ${\mathbb{C}}$. The set of the Gauss integers is denoted by ${\mathbb{Z}}[{\sqrt{-1}}]$ and the set of the Eisenstein integers is by ${\mathbb{Z}}[\omega_{6}]={\mathbb{Z}}[\omega_{3}]$ for $\omega_{3}=\omega_{6}^{2}$ noting $\omega_{3}+1=\omega_{6}$.

The truncated Epstein-Hurwitz zeta function of $\tau\in{\mathbb{H}}$, $\zeta_{A}^{\tau}(s,z_{0})$ is defined by [Te]

[TABLE]

where $z_{0}:=x_{0}+y_{0}{\sqrt{-1}}\in{\mathbb{C}}$ and $A$ is a subset of ${\mathbb{Z}}[\tau]$.