\documentclass[a4paper, 11pt]{article}
\usepackage[utf8]{inputenc}

% Took inspiration from 
% https://www.overleaf.com/latex/templates/tarea-mfm-ii/rkqcbsjyyksm

% From aga-homework.cls
\usepackage[
	a4paper,
	headheight = 20pt,
	margin = 1in,
	tmargin = \dimexpr 1in - 10pt \relax
]{geometry}

\usepackage{fancyhdr} % for headers and footers
\usepackage{graphicx} % for including figures
%\usepackage{mathpazo} % use Palation font
%\usepackage{amsmath} % use AMS math package
%\usepackage{amssymb} % use AMS symbols
%\usepackage{amsthm} % for writing proofs
%\usepackage{array} % for setting up arguments to columns
\usepackage{booktabs} % for professional tables
%\usepackage%
%  [tworuled, linesnumbered, noend, noline]%
%  {algorithm2e} % for typesetting pseudo-code
%\usepackage{xcolor} % for colored text (comments in algorithms)
%\usepackage{trimspaces, xstring} % for multiple author parsing
\setlength{\headheight}{14pt}


\fancypagestyle{plain}{
	\fancyhf{}
	\fancyhead[L]{\sffamily Radboud University Nijmegen}
	\fancyhead[R]{\sffamily Superconductivity (NWI-NM117), Q3+Q4}
	\fancyfoot[R]{\sffamily\bfseries\thepage}
	\renewcommand{\headrulewidth}{0.5pt}
	\renewcommand{\footrulewidth}{0.5pt}
}
\pagestyle{fancy}

\usepackage{siunitx}
\usepackage{hyperref}
%\usepackage{href}
%\usepackage[nottoc,numbib]{tocbibind}
\usepackage{float}
\usepackage{mathtools}

\title{Superconductivity - Assignment 1}
\author{
	Kees van Kempen (s4853628)\\
	\texttt{k.vankempen@student.science.ru.nl}
}
\AtBeginDocument{\maketitle}


\begin{document}

\section{Electron-phonon coupling in elements}
Conventional superconductors (sc) are described by considering Cooper pairs:
pairs of electrons mediated by electron-phonon coupling.
This is usually described by BCS theory.
The hypothesis is that stronger electron-phonon coupling results in enhanced critical temperatures for the superconducting phase transition.
In order to investigate this, we need a way to determine the electron-phonon coupling strength.
We will attempt to do this by looking at the room temperature resistivity of superconducting elements.

For metals, we have the following familiar relation for resistivity $\rho$ over temperature $T$.

\begin{equation}
	\rho = \underbrace{\rho_0}_\text{{impurities}} + \underbrace{aT^2}_{\text{electron-electron coupling}} + \underbrace{bT^5}_{\text{electron-phonon coupling}}
\end{equation}

At $T = 0$, only resistivity due to impurities and lattice defects is left in the material.
Then, at low temperatures, electron-electron coupling increases resistivity.
The effect that is the largest at room temperature, is due to electron-phonon interaction, due to the fifth power in temperature.
The constants $a$ and $b$ differ from material to material.
If the hypothesis is correct, an increasing trend of critical temperature $T_c$ over room temperature resistivity $\rho_{300K}$ should be observed.

For a collection of superconducting elements, this relation is plotted in figure \ref{fig:scelements}.
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}.

\begin{figure}%[H]
	\label{fig:scelements}
	\includegraphics[width=\textwidth]{sc_elements.pdf}
	\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.}
\end{figure}

Looking at the plot, there is no obvious positive trend between $T_c$ and $\rho_{300K}$.
As a way to quantize this (lack of) correlation, we can take a look at the Pearson correlation coefficient: $r = 0.165415$.
% I used df.corr() to calculate $r$.
Pearson's $r$ is a measure of linear correlation.
If $|r| = 1$, there is a perfectly linear relation.
The lower $|r|$ is, the less correlated the points are.
The sign of $r$ gives the direction of the trend.
This slightly positive value found for the superconducting elements suggests a slightly positive but uncertain correlation.

\section{Exam question electrodynamics in superconductors}
No idea yet.

\section{Difference between type-I and type-II superconductors}
Type-I superconductors are thought to be described by BCS theory.
They are phonon-mediated.
They have different phase diagrams from type-II superconductors,
with two critical fields but one critical temperature.
There is a superconducting phase and a vortex phase.

Something about quantized flux by the Meissner-Ochsenfeld effect.

\bibliographystyle{vancouver}
\bibliography{references.bib}

\appendix
\section{Superconducting elements}
\label{appendix:scelements}
\input{sc_elements.tex}

\end{document}