From 1cb8623412b971196e43b13d85f14f71f8f217b0 Mon Sep 17 00:00:00 2001 From: Kees van Kempen Date: Thu, 17 Feb 2022 18:23:05 +0100 Subject: [PATCH] ass1: Describe T_c vs \rho_{300K} and Pearson's r --- sc_elements.py | 4 +-- superconductivity_assignment1_kvkempen.tex | 36 +++++++++++++++++++--- 2 files changed, 34 insertions(+), 6 deletions(-) mode change 100644 => 100755 superconductivity_assignment1_kvkempen.tex diff --git a/sc_elements.py b/sc_elements.py index 71fd576..b99d613 100755 --- a/sc_elements.py +++ b/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)) diff --git a/superconductivity_assignment1_kvkempen.tex b/superconductivity_assignment1_kvkempen.tex old mode 100644 new mode 100755 index 691b7af..449e39d --- a/superconductivity_assignment1_kvkempen.tex +++ b/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.