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

sc_elements.py

@@ -54,8 +54,8 @@ from matplotlib import pyplot as plt
 fig, ax = plt.subplots(figsize=(10, 5))
 #df.plot('rho_300', 'T_c', kind='scatter', ax=ax, loglog=True)
 df.plot('rho_300', 'T_c', kind='scatter', ax=ax)
-ax.set_xlabel('$\\rho_{300K}$')
-ax.set_ylabel('$T_c$')
+ax.set_xlabel('$\\rho_{300K} (n \Omega m)$')
+ax.set_ylabel('$T_c (K)$')
 
 for k, v in df.iterrows():
     #ax.annotate(v['element'], (v['rho_300']*.95, v['T_c']*1.05))

superconductivity_assignment1_kvkempen.tex

@@ -43,6 +43,7 @@
 %\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.