In this section, we give some remarks on our results from mathematical viewpoints. Springer (1959). The projection above “induces” the fibering structure. holds. Let \(\mathcal {S}=\mathcal {S}_{+}\coprod \mathcal {S}_{-}\) be a finite subset of \(\mathbb {E}^{2}\), which is divided into disjoint subsets \(\mathcal {S}_{+}\) and \(\mathcal {S}_{-}\). Algebraically, it is described as additive group (or \(\mathbb {Z}\)-module) by. $$, \(z_{0} \in \mathbb {E}^{2} = \mathbb {C}\), \(\mathbb {E}_{\mathbb {E}^{2}\setminus \{z_{0}\}}\), \(S^{1}_{\mathbb {E}^{2}\setminus \{z_{0}\}}\), \(\sigma _{z_{0}, \gamma } \in \Gamma \left (\mathbb {E}^{2}\setminus \{z_{0}\}, S^{1}_{\mathbb {E}^{2}\setminus \{z_{0}\}}\right)\), $$\sigma_{z_{0}, \gamma}(z) = \left(\gamma\frac{z-z_{0}}{|z-z_{0}|},z\right) \text{ for}~z \in \mathbb{E}^{2} \setminus \{z_{0}\} = \mathbb{C} \setminus \{z_{0}\}. In Section 2, we first introduce certain fibering structures over the plane involving the celebrated exact sequence \(0 \rightarrow \mathbb {Z} \rightarrow \mathbb {R} \rightarrow \mathrm {U}(1) \rightarrow 1\), which will be essential in this article. 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 have another algebraic proof for Lemma 5 as followsFootnote 1. In this subsection, we give five remarks, I–V, on the energy of screw dislocations. Body centered cubic (BCC) lattice. \end{array} $$, $$I\le \frac{4a^{2} y_{0}^{2} } {((n_{1} a -x_{0})^{2} +(n_{2} a -y_{0})^{2})^{2}}, $$, $$I\le \frac{4a^{2} y_{0}^{2} } {((n_{1} a -x_{0})^{2} +(n_{2} a +y_{0})^{2})^{2}}. $$, \(\iota ^{\text {SC}}_{\bar \delta } : {\mathcal {A}}_{p} \hookrightarrow \mathbb {E}^{2}\), \(\iota ^{\text {SC}}_{\bar \delta }({\mathcal {A}}_{p}) \cap \mathcal {S}= \emptyset \), \(\iota ^{\text {SC}}_{\bar \delta } : {\mathcal {A}}_{p} \hookrightarrow \mathbb {E}^{2}\setminus \mathcal {S}\), \(F_{\mathbb {E}^{2}\setminus \mathcal {S}}\), \(\hat \iota ^{\text {SC}}_{\bar \delta }\), \(\gamma =\exp (2\pi \sqrt {-1} \delta _{3}/a) \in S^{1}\), \(\check \sigma _{\gamma } \in \Gamma \left ({\mathcal {A}}_{p}, S^{1}_{{\mathcal {A}}_{p}}\right)\), $$ {\begin{aligned} \hat\iota^{\text{SC}}_{\bar \delta} \left(\widehat\psi^{-1}\left(\check\sigma_{\gamma}({\mathcal{A}}_{p})\right)\right) &=\hat\iota^{\text{SC}}_{\bar \delta}\left(\frac{a}{2\pi\sqrt{-1}} \exp^{-1}\left(\check\sigma_{\gamma}({\mathcal{A}}_{p}) \right)\right) \\ &\subset \mathbb{E}_{\mathbb{E}^{2} \setminus \mathcal{S}} \subset \mathbb{E}^{3} \end{aligned}} $$, \(\iota _{\mathbb {A}_{3},\delta }(\mathbb {A}^{a}_{3})\), \(\delta = (\bar \delta, \delta _{3}) = (\delta _{1}, \delta _{2}, \delta _{3})\), $$\iota_{\mathbb{A}_{3},\delta}(\mathbb{A}^{a}_{3})= \hat\iota^{\text{SC}}_{\bar \delta} \left(\widehat\psi^{-1} \left(\check\sigma_{\gamma}({\mathcal{A}}_{p})\right)\right). -\!x_{0})^{2} \,+\,(n_{2} a\! coincides with \(\iota _{\mathbb {A}_{3},\delta }(\mathbb {A}^{a}_{3})\) as a subset in \(\mathbb {E}^{3}\) for \(\delta = (\bar \delta, \delta _{3}) = (\delta _{1}, \delta _{2}, \delta _{3})\), i.e.. Birkhäuser (1993). Screw dislocation is another type of line defect in which the defect occurs when the planes of atoms in the crystal lattice trace a helical path around the dislocation line. or. We observe the existence of a threshold stress, which results in an abrupt change in the minimum energy path of the kink-nucleation process, and hence, a discontinuity in the activation … is finite even for N→∞ (see Appendix). The forces exerted by a screw dislocation wall on nonparallel edge and screw dislocations are calculated. If we just concentrate ourselves to such results, then the usage of such zeta functions might not be necessary, since divergence and convergence of the relevant power series can be proved rather directly. 4. Hull, D, Bacon, DJ: Introduction to dislocation, 4th ed.Elsevier (2011). Epstein, P: Zur Theorie allgemeiner Zetafunktionen. In other words, the Burgers vector is parallel to the (1,1,1)-direction, or more precisely it coincides with b itself, up to automorphisms of the BCC lattice [21]. which follows directly from the definition of \(\mathbb {Z}_{\mathbb {E}^{2}\setminus \{z_{0}\}, \gamma }\) given in Section 2.2. Hierarchical nanostructures of lead sulfide nanowires resembling pine trees were synthesized by chemical vapor deposition. One of the main objectives of the mathematical modeling of the shear plastic strain is estimation of the shear zone size. Let \(h : \pi _{1}(\mathbb {E}^{2} \setminus \mathcal {S}, x_{0}) \to \mathbb {Z}\) be the homomorphism defined by, Since \(\mathbb {Z}\) is abelian, the kernel H of h contains G′. $$, $$\begin{array}{@{}rcl@{}} \Delta_{n_{1}, n_{2}}^{d(i, \pm)} & = & \pm \frac{1}{\sqrt{2}} \epsilon_{n_{1},n_{2}}^{(i)}+ o\left(\frac{a}{\sqrt{(n_{1} a -x_{0})^{2}+(|n_{2}| a -|y_{0}|)^{2}}}\right), \\ \epsilon_{n_{1},n_{2}}^{(1)}& = & \frac{a}{2\pi} \left(-\frac{ a(n_{2} a -y_{0})} {(n_{1} a -x_{0})^{2} + (n_{2} a - y_{0})^{2}}+\frac{ a(n_{2} a +y_{0})} {(n_{1} a -x_{0})^{2} + (n_{2} a + y_{0})^{2}}\right) \\ & & \quad + o\left(\frac{a}{\sqrt{(n_{1} a -x_{0})^{2}+(|n_{2}| a - |y_{0}|)^{2}}}\right), \\ \epsilon_{n_{1},n_{2}}^{(2)}& = & \frac{a}{2\pi} \left(-\frac{a(n_{1} a -x_{0})} {(n_{1} a -x_{0})^{2} + (n_{2} a - y_{0})^{2}}+\frac{a(n_{1} a -x_{0})} {(n_{1} a -x_{0})^{2} + (n_{2} a + y_{0})^{2}} \right) \\ & & \quad + o\left(\frac{a}{\sqrt{(n_{1} a -x_{0})^{2}+(|n_{2}| a - |y_{0}|)^{2}}}\right). The simple cubic (SC) lattice is usually expressed by the additive free abelian group, which is generated by three elements a1, a2 and a3. \left(\check \sigma_{\mathcal{S}',\gamma}\!\left({\mathcal{A}}_{p}\right)\right)\right) \end{array} $$, \(\mathbb {Z}_{\mathbb {E}^{2}\setminus \mathcal {S},\gamma }\), $${\mathcal{R}_{d}} =\mathbb{C}[\mathbb{A}_{3}]/(\alpha_{1}\alpha_{2}\alpha_{3}-1), $$, \(a_{1}+a_{2}+a_{3} \in \mathbb {A}_{3}^{a} \subset \mathbb {R}^{3}\), $$(a_{1}-a_{3}, a_{1}+a_{2}+a_{3})=0, \quad (a_{2}-a_{3}, a_{1}+a_{2}+a_{3})=0. Then, such an energy is approximately obtained in terms of the truncated Epstein-Hurwitz zeta function (22) in Theorem 1, where the Epstein-Hurwitz zeta function is defined by [9, 10, 24] as. What is Screw Dislocation? Let us now take further quotients. More precisely, in our model, we have the elastic springs on the edges. The results show the core width and Peierls stress in Ta are weakly dependent of the pressure up to 100 GPa when the length and stress are measured separately by the Burgers vector b and shear modulus μ. $$, $${\mathbb{A}_{d}}:=\left\{ \left(\left.\alpha_{1}\alpha_{3}^{-1}\right)^{\ell_{1}} \left(\alpha_{2}\alpha_{3}^{-1}\right)^{\ell_{2}} \, \right| \, \ell_{1}, \ell_{2} \in \mathbb{Z}\right\}, $$, $$\mathbb{C}[{\mathbb{A}_{d}}] = \mathbb{C}\left[\alpha_{1} \alpha_{3}^{-1}, \alpha_{2} \alpha_{3}^{-1}, \alpha_{1}^{-1} \alpha_{3}, \alpha_{2}^{-1} \alpha_{3}\right], $$, $$\mathbb{C}\left[\alpha_{1} \alpha_{2}^{2}, \alpha_{1}^{2} \alpha_{2}, \alpha_{1}^{-1} \alpha_{2}^{-2}, \alpha_{1}^{-2} \alpha_{2}^{-1}\right]. Nabarro, FRN: Theory of crystal dislocations. In fact, the SC lattice is expressed by the graph as depicted in Fig. Furthermore, for later convenience, let us introduce the notation. 10, 3 (2018). Thus, it is expected that \(E_{\rho,N}(\mathcal {S})\) is also finite even for N→∞ due to the reason similar to Proposition 10, although we have not been able to prove this conjecture so far, either. The BCC (body centered cubic) lattice is the lattice in \(\mathbb {R}^{3}\) generated by a1, a2, a3 and (a1+a2+a3)/2. Its projection to the 2-dimensional space corresponds to taking the quotient as. Then, the the principal part of the elastic energy \(E_{\rho,N}^{(\mathrm {p})}(\mathcal {S})\) for the region Bρ,N in the discrete picture converges for N→∞. Then, we will describe screw dislocations in continuum picture, in which certain covering spaces of a punctured complex plane will play important roles. Applying the Schmid law to the actual dislocation trajectory instead of the average {110} glide plane allows the physical interpretation of the phenomenological generalized yield criterion and leads to predictions of T/AT Let us define the section \(\check \sigma _{z_{0}', \gamma } \in \Gamma ({\mathcal {A}}_{p}, S^{1}_{{\mathcal {A}}_{p}})\) by, where \(\gamma =\exp (2\pi \sqrt {-1} \delta _{3}/a) \in S^{1}\). Note that this corresponds exactly to the decomposition mentioned in Remark 7. \end{array}\right. Okada, K, Fujisawa, K, Shirai, T, Wakayama, M, Waki, H, Broadbridge, P, Yamamoto, M: (editors) Study Group Workshop 2015, Abstract, Lecture & Report, MI Lecture Notes, Vol. For each s≥3, there exists a positive constant Ms′ such that. What is Screw Dislocation Gendai-sugakusha(2000). :=\!\{ \ell_{1}(a_{1}\,-\,a_{3}) \,+\,\ell_{2}(a_{2}\,-\,a_{3}) \, | \, \ell_{1}, \ell_{2} \in \mathbb{Z}\},\\ {\mathcal{B}}^{(1)}&\! 1,\!n_{3}\! This is the universal covering space of \(\mathbb {E}^{2} \setminus \mathcal {S}\). Our algebraic approach might enable us to handle such a curve in a lattice locally which could be a part of a curved dislocation line. This means that our spring model, in discrete picture, is consistent with the known dislocation theory in continuum picture, so that our model is plausible in such a sense. In Proposition 4 of Section 3, the discrete picture of such multiple screw dislocations in the SC lattice has been obtained as the pullback of the fiber structure of the multiple screw dislocations in the continuum picture. Physical Review E, 2006. The cube in (a) shows the parallel triangles whose normal direction is (1,1,1) in the simple cubic lattice. We recall that, for \(z_{0} =x_{0} + y_{0} \sqrt {-1}\), \(x_{0}, y_{0} \in \mathbb {R}\), and γ= 1 in Proposition 3. n_{3}\pm 1\! (Reçu le 26 mai 1987, accepté le 14 août 1981) RESUME. for \(z_{k} \in \mathcal {S}\). \right)\!,} \\ \hspace*{6.3cm} (n_{1} a, n_{2} a) \in {\mathcal{A}}_{p}, \end{array}} $$, $$\begin{array}{@{}rcl@{}} & & z_{i}' = z_{i} - \left(\delta_{1} + \delta_{2} \sqrt{-1}\right),\, z_{j}' = z_{j} - \left(\delta_{1} + \delta_{2} \sqrt{-1}\right), \\ & & \mathcal{S}_{+}' = \left\{\left.z_{i}' \,\right|\, z_{i} \in \mathcal{S}_{+}\right\}, \, \mathcal{S}_{-}' = \left\{z_{j}' \,\big|\, z_{j} \in \mathcal{S}_{-}\right\} \end{array} $$, \(\mathcal {S}' = \mathcal {S}_{+}' \coprod \mathcal {S}_{-}'\), \(\mathcal {D}_{\mathcal {S}}^{\text {SC}}\), $$\begin{array}{*{20}l} \mathcal{D}_{\mathcal{S}}^{\text{SC}} &=\hat\iota^{\text{SC}}_{\bar \delta}\left(\widehat{\psi}^{-1}\left(\check \sigma_{\mathcal{S}',\gamma} \! 55, 208–215 (2008). There are two forms of these defects as edge dislocations and screw dislocations. for sufficiently large fixed \(\rho \geq \sqrt {2}\). )]$$, $$ \Delta_{n_{1}, n_{2}}^{d(i, \pm)} =\sqrt{(a \pm \epsilon_{n_{1},n_{2}}^{(i)})^{2} +a^{2}}-\sqrt{2}a, $$, $$[(n_{1}, n_{2},n_{3}),(n_{1}+1, n_{2}\pm1,n_{3})]$$, $$ \Delta_{n_{1}, n_{2}}^{d(\pm)} = \sqrt{2a^{2} + (\epsilon_{n_{1},n_{2}}^{(\pm)})^{2} }-\sqrt{2}a. Vol. There are two types of line defect; they are edge dislocation and screw dislocation, where the atoms of the lattice exist in a misaligned pattern. In the following, we assume that \(\bar \delta \in \mathbb {E}^{2}\) satisfies \(\iota ^{{\text {BCC}},c}_{\bar \delta }\left ({\mathcal {B}}^{(c)}\right) \cap \mathcal {S}= \emptyset \), c=0,1,2. In Section 4, in order to treat the BCC case, we expressed the fiber structure with respect to the (1,1,1)-direction using algebraic methods. For the embedding \(\iota ^{{\text {BCC}},c}_{\bar \delta } : {\mathcal {B}}^{(c)} \hookrightarrow \mathbb {E}^{2}\), c=0,1,2, we have the following: Let \(\mathcal {S}=\mathcal {S}_{+}\coprod \mathcal {S}_{-}\) be a finite subset in \(\mathbb {E}^{2}\) as in Subsection 2.3. Cite this article. In this work, we examine the kink-nucleation process in BCC screw dislocations using atomistic simulation and transition pathway analysis, with a particular focus on the compact core structure. As we have seen in Subsection 5.1, we may concentrate ourselves to \(\Delta _{n_{1}, n_{2}}^{d(i, \pm)}\), which essentially contribute to the energy. \Delta_{n_{1}, n_{2}}^{d(1, -)}\right)^{2} \right. However, in the case of the BCC lattice, the Burgers vector is parallel to the (1,1,1)-direction up to automorphisms of the BCC lattice, and its geometrical structure is a little bit complicated. J. This will be shown by using our algebraic description of a screw dislocation. Therefore, \(\mathbb {Z}_{\mathbb {E}^{2} \setminus \{z_{0}\}, \gamma }\) is path-wise connected. Finally, screw dislocations in the BCC lattice will be described using all these algebraic materials in Propositions 7 and 8. (a) shows the panoramic view of the unit cell of the BCC lattice which contains two triangles whose normal direction is (1,1,1) as in the simple cubic lattice case. The region in which the defect occurs is the dislocation core or area. To visualize a screw dislocation, imagine a block of metal with a shear stress applied across one end so that the metal begins to rip. We have a natural isomorphism as \(\mathbb {C}[{\mathbb {A}_{d}}]\)-modules: First, note that every monomial of \(\mathbb {C}[\mathbb {A}_{3}]\) has its own degree with respect to α1,α2 and α3, each of which has degree 1. J. Diff. The authors would like to express their sincere gratitude 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 [22], and to the participants in the “IMI workshop II: Mathematics of Screw Dislocation”, September 1–2, 2016, and in the “IMI workshop I: Mathematics in Interface, Dislocation and Structure of Crystals”, August 28–30, 2017, both held in the Institute of Mathematics for Industry (IMI), Kyushu University. is sufficiently small, then \(\epsilon ^{(1)}_{n_{1},n_{2}}\), \(\epsilon ^{(2)}_{n_{1},n_{2}}\) and \(\epsilon ^{(\pm)}_{n_{1},n_{2}}\) are approximately given by, respectively, whereas \(\Delta _{n_{1}, n_{2},\! Let us consider the projection which corresponds to the quotient ring. In the following, we often identify \(\mathbb {E}^{2}\) with the complex plane \(\mathbb {C}\). In the following, for γ′∈S1 and \(\tilde \sigma \in \Gamma \left ({\mathcal {B}}, S^{1}_{{\mathcal {B}}}\right)\) expressed as \(\tilde \sigma (x) = (s(x), x)\) for \(x \in {\mathcal {B}}\), we define their multiplication \(\gamma ' \tilde \sigma \in \Gamma \left ({\mathcal {B}}, S^{1}_{{\mathcal {B}}}\right)\) by \((\gamma ' \tilde \sigma)(x) = (\gamma ' s(x), x)\), \(x \in {\mathcal {B}}\), where \(S^{1}_{{\mathcal {B}}}\) is the trivial S1-bundle over \({\mathcal {B}}\). \(0 \rightarrow \mathbb {Z} \rightarrow \mathbb {R} \rightarrow \mathrm {U}(1) \rightarrow 1\), \(\mathcal {F} \rightarrow \mathcal {M}\), \(f:\mathcal {M} \rightarrow \mathcal {F}\), \((\exp 2\pi \sqrt {-1})(x) = \exp (2\pi \sqrt {-1} x)\), $$ {\begin{aligned} &\widetilde\iota_{\delta}:\mathbb{R} \rightarrow \mathbb{E} \text{ defined by } x \mapsto d\cdot x +\delta, \, x \in \mathbb{R}, \text{and}\\ &\iota_{\delta}:\mathrm{U}(1) \to S^{1} \text{ defined by } \exp({\sqrt{-1}\theta}) \mapsto \exp{\sqrt{\!-1}(\theta \,+\,2\pi\delta/d)}, \, \theta \in \mathbb{R}, \end{aligned}} $$, \(\psi (y) = \exp \left (2\pi \sqrt {-1} y/d\right)\), \(\varphi _{\delta } = \widetilde \iota _{\delta } \circ \iota \), $$ \varphi_{\delta}(\mathbb{Z}) = \psi^{-1}(\exp(2\pi\sqrt{-1}\delta/d)), $$, \(\pi _{\mathbb {Z}} : \mathbb {Z}_{\mathbb {E}^{2}} \rightarrow \mathbb {E}^{2}\), \(\pi _{\mathbb {E}} : \mathbb {E}_{\mathbb {E}^{2}} \rightarrow \mathbb {E}^{2}\), \(\pi _{S^{1}} : S^{1}_{\mathbb {E}^{2}} \rightarrow \mathbb {E}^{2}\), \(\mathbb {Z}_{\mathbb {E}^{2}} = \mathbb {Z}\times \mathbb {E}^{2}\), \(\mathbb {E}^{3} = \mathbb {E} \times \mathbb {E}^{2}\), \(\sigma _{\gamma }\in \Gamma (\mathbb {E}^{2}, S^{1}_{\mathbb {E}^{2}})\), $$\sigma_{\gamma}(z) = (\gamma, z) \in S^{1}_{\mathbb{E}^{2}} = S^{1} \times \mathbb{E}^{2} $$, \(\gamma = \exp (2\pi \sqrt {-1}\delta /d)\), $$\mathbb{Z}_{\mathbb{E}^{2}, \gamma}=\widehat\varphi_{\delta}(\mathbb{Z}_{\mathbb{E}^{2}}), $$, $$\mathbb{Z}_{\mathbb{E}^{2}, \gamma}:= {\widehat\psi^{-1}} \left(\sigma_{\gamma}(\mathbb{E}^{2})\right) \subset \mathbb{E}^{3}.

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