117 lines
5.1 KiB
TeX
Executable File
117 lines
5.1 KiB
TeX
Executable File
\documentclass[a4paper, 11pt]{article}
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\usepackage[utf8]{inputenc}
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% Took inspiration from
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% https://www.overleaf.com/latex/templates/tarea-mfm-ii/rkqcbsjyyksm
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% From aga-homework.cls
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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{mathpazo} % use Palation font
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%\usepackage{amsmath} % use AMS math package
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%\usepackage{amssymb} % use AMS symbols
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%\usepackage{amsthm} % for writing proofs
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%\usepackage{array} % for setting up arguments to columns
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\usepackage{booktabs} % for professional tables
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%\usepackage%
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% [tworuled, linesnumbered, noend, noline]%
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% {algorithm2e} % for typesetting pseudo-code
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%\usepackage{xcolor} % for colored text (comments in algorithms)
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%\usepackage{trimspaces, xstring} % for multiple author parsing
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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{href}
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%\usepackage[nottoc,numbib]{tocbibind}
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\usepackage{float}
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\usepackage{mathtools}
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\title{Superconductivity - Assignment 1}
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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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\begin{document}
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\section{Electron-phonon coupling in elements}
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Conventional superconductors (sc) are described by considering Cooper pairs:
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pairs of electrons mediated by electron-phonon coupling.
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This is usually described by BCS theory.
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The hypothesis is that stronger electron-phonon coupling results in enhanced critical temperatures for the superconducting phase transition.
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In order to investigate this, we need a way to determine the electron-phonon coupling strength.
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We will attempt to do this by looking at the room temperature resistivity of superconducting elements.
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For metals, we have the following familiar relation for resistivity $\rho$ over temperature $T$.
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\begin{equation}
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\rho = \underbrace{\rho_0}_\text{{impurities}} + \underbrace{aT^2}_{\text{electron-electron coupling}} + \underbrace{bT^5}_{\text{electron-phonon coupling}}
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\end{equation}
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At $T = 0$, only resistivity due to impurities and lattice defects is left in the material.
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Then, at low temperatures, electron-electron coupling increases resistivity.
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The effect that is the largest at room temperature, is due to electron-phonon interaction, due to the fifth power in temperature.
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The constants $a$ and $b$ differ from material to material.
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If the hypothesis is correct, an increasing trend of critical temperature $T_c$ over room temperature resistivity $\rho_{300K}$ should be observed.
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For a collection of superconducting elements, this relation is plotted in figure \ref{fig:scelements}.
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The data on critical temperatures $T_c$ and (approximately) room temperature resistivity $\rho_{\SI{300}{\kelvin}}$ is from various sources, as can be found in the table in appendix \ref{appendix:scelements}.
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\begin{figure}%[H]
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\label{fig:scelements}
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\includegraphics[width=\textwidth]{sc_elements.pdf}
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\caption{In this plot of the critical temperature $T_c$ versus the room temperature resistivity $\rho_{300K}$ for elemental superconductors, not one clear relation can be distinguished. For most elements, resistivity is taken at room temperature $T = \SI{300}{\kelvin}$. If it was unavailable in consulted references, the value at the temperature closest to \SI{300}{\kelvin} was chosen. See the table in appendix \ref{appendix:scelements} for the raw data including their source. The mess in the left bottom corner was hard to filter out. A log-log plot was attempted and improved separation, but obscured the observed two branches in this linear plot.}
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\end{figure}
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Looking at the plot, there is no obvious positive trend between $T_c$ and $\rho_{300K}$.
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As a way to quantize this (lack of) correlation, we can take a look at the Pearson correlation coefficient: $r = 0.165415$.
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% I used df.corr() to calculate $r$.
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Pearson's $r$ is a measure of linear correlation.
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If $|r| = 1$, there is a perfectly linear relation.
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The lower $|r|$ is, the less correlated the points are.
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The sign of $r$ gives the direction of the trend.
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This slightly positive value found for the superconducting elements suggests a slightly positive but uncertain correlation.
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\section{Exam question electrodynamics in superconductors}
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No idea yet.
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\section{Difference between type-I and type-II superconductors}
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Type-I superconductors are thought to be described by BCS theory.
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They are phonon-mediated.
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They have different phase diagrams from type-II superconductors,
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with two critical fields but one critical temperature.
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There is a superconducting phase and a vortex phase.
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Something about quantized flux by the Meissner-Ochsenfeld effect.
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\bibliographystyle{vancouver}
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\bibliography{references.bib}
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\appendix
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\section{Superconducting elements}
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\label{appendix:scelements}
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\input{sc_elements.tex}
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\end{document}
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