75 lines
2.2 KiB
TeX
Executable File
75 lines
2.2 KiB
TeX
Executable File
\documentclass[a4paper, 11pt]{article}
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\usepackage[utf8]{inputenc}
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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{booktabs} % for professional tables
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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{float}
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\usepackage{mathtools}
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\usepackage{amsmath}
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\usepackage{todonotes}
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\setuptodonotes{inline}
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\usepackage{mhchem}
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\usepackage{listings}
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\newcommand{\pfrac}[2]{\frac{\partial #1}{\partial #2}}
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\title{Superconductivity - Assignment 5}
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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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% Start from 8
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\setcounter{section}{16}
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\begin{document}
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\section{$T_c$ upper limit in BCS}
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In BCS theory, the formation of Cooper pairs is mediated by phonons.
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There is a phonon-electron interaction quantified by the dimensionless quantity
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\[
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\lambda := Vg(\epsilon_F)
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\]
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with $V$ Cooper's approximate potential and $g(\epsilon_F)$ the density of states near the Fermi surface for the electrons.
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A thorough discussion can be found in Annett's book \cite[chapter 6]{annett} and in the slides of week 6 of this course.
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The binding energy of the Cooper pairs (i.e. the energy gain of forming these pairs) is
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\[
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-E = 2\hbar\omega_De^{-1/\lambda}.
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\]
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In the weak coupling limit of the BCS theory, the case we have considered so far, it is assumed that $\lambda << 1$.
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It is when this assumption breaks down, BCS does not work and we find an upper limit to the critical temperature $T_c$.
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We will look at a way to express the critical temperature in terms we can derive, and than look at the values that maximize this critical temperature whilst still following BCS theory.
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\section{Energy gap $\Delta$ et al.}
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\bibliographystyle{vancouver}
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\bibliography{references.bib}
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\end{document}
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