89 lines
3.1 KiB
TeX
Executable File
89 lines
3.1 KiB
TeX
Executable File
\documentclass[a4paper, 11pt]{article}
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\usepackage[utf8]{inputenc}
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\usepackage[
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a4paper,
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headheight = 20pt,
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margin = 1in,
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tmargin = \dimexpr 1in - 10pt \relax
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]{geometry}
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\usepackage{fancyhdr} % for headers and footers
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\usepackage{graphicx} % for including figures
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\usepackage{booktabs} % for professional tables
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\setlength{\headheight}{14pt}
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\fancypagestyle{plain}{
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\fancyhf{}
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\fancyhead[L]{\sffamily Radboud University Nijmegen}
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\fancyhead[R]{\sffamily Superconductivity (NWI-NM117), Q3+Q4}
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\fancyfoot[R]{\sffamily\bfseries\thepage}
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\renewcommand{\headrulewidth}{0.5pt}
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\renewcommand{\footrulewidth}{0.5pt}
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}
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\pagestyle{fancy}
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\usepackage{siunitx}
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\usepackage{hyperref}
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\usepackage{float}
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\usepackage{mathtools}
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\usepackage{amsmath}
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\usepackage{todonotes}
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\setuptodonotes{inline}
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\usepackage{mhchem}
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\newcommand{\pfrac}[2]{\frac{\partial #1}{\partial #2}}
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\title{Superconductivity - Assignment 3}
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\author{
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Kees van Kempen (s4853628)\\
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\texttt{k.vankempen@student.science.ru.nl}
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}
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\AtBeginDocument{\maketitle}
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% Start from 8
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\setcounter{section}{7}
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\begin{document}
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\section{\ce{Nb3Sn} cylinder}
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Consider a cylinder of \ce{Nb3Sb}.
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From lecture 4, we have the following properties for \ce{Nb3Sn}:
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$T_c = \SI{18.2}{\kelvin}$,
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$\xi = \SI{3.6}{\nano\meter}$,
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$\lambda = \SI{124}{\nano\meter}$,
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$\kappa = \frac{\lambda}{\xi} = 34 > \frac{1}{\sqrt{2}}$,
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which means we are indeed dealing with a type-II superconductor.
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As $B_{c1} < B_E < B_{c2}$, the cylinder is in the vortex state.
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From the previous set of assignments, we know what the currents in the cylinder look like.
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From free energy considerations, we have found in lecture 4 that for type-II superconductors, it is favorable to allow flux quanta inside the superconductor in this vortex state.
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In this derivation, the contribution of one flux quantum is considered, but the consideration holds for many vortices, until they start to interact and repel eachother.
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At that point, the vortex-vortex interaction orders the vortices in a lattice.
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When the vortex cores start to overlap, there are no superconducting regions left, thus the material enters the normal conducting state.\footnote{I wanted to paint a complete picture althought it is not needed to answer the question.}
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Minimizing the free energy over the flux shows the energy is lowered for determined thresholds $B_{c1} < B_E < B_{c2}$.
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Let's start with the result from said free energy considerations.
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The average field inside the cylinder is given by the following self-consisting equation as
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\[
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B = B_E - \frac{\phi_0}{8\pi\lambda^2}\ln{\frac{\phi_0}{4\exp{(1)}\xi^2B}}.
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\]
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Plugging in the values for \ce{Nb3Sn}, $B_E = \SI{1}{\tesla}$, and $\phi_0 = \SI{2.0678}{\weber}$, $B$ is found as $B = \SI{.986}{\tesla} \approx B_E$ by intersection.
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%https://www.wolframalpha.com/input?i=B+%3D+1+-+%282.0678*10%5E-15%29%2F%288*pi*%28124*10%5E-9%29%5E2%29+*+ln%282.0678*10%5E-15%2F%284*exp%281%29*%283.65*10%5E-9%29%5E2+*+B%29%29
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\section{Superconducting wire}
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\section{Fine type-II superconducting wire}
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\section{Critical currents}
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\section{A weak junction}
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\bibliographystyle{vancouver}
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\bibliography{references.bib}
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%\appendix
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\end{document}
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