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ass1: Describe T_c vs \rho_{300K} and Pearson's r

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%\usepackage{href}
%\usepackage[nottoc,numbib]{tocbibind}
\usepackage{float}
\usepackage{mathtools}
\title{Superconductivity - Assignment 1}
\author{
@ -55,15 +56,42 @@
\begin{document}
\section{Electron-phonon coupling in elements}
The data on critical temperatures $T_c$ and (approximately) room temperature resistivity $\rho_{\SI{300}{\kelvin}}$ is from varies sources, as can be found in the table in appendix \ref{appendix:scelements}.
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.
\begin{figure}[H]
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 form.}
\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}
As a way to quantize the (lack of) linear correlation, the calculated Pearson correlation coefficient $r = 0.165415$, suggesting a slightly positive but uncertain correlation.
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.