92 lines
2.6 KiB
TeX
Executable File
92 lines
2.6 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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\newcommand{\pfrac}[2]{\frac{\partial #1}{\partial #2}}
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\title{Superconductivity - Assignment 2}
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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 4
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\setcounter{section}{3}
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\begin{document}
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\section{Temperature dependence in Landau model}
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In the Landau model, free energy is given as function of order parameter $\psi$ and temperature $T$ as
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\[
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\mathcal{F} = a(T - T_c) \psi^2 + \frac{\beta}{2}\psi^4.
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\]
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The equilibrium state as function of temperature $T$ is the state of minimal free energy with respect to the order parameter $\psi(T)$.
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This point we call $F_0(T)$ with order parameter $\psi_0(T)$.
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For this, we will take the derivative of $F$ with respect to $\psi$ and equate it to zero.
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\[
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0 = \pfrac{\mathcal{F}}{\psi} = \pfrac{}{\psi} \left[ a(T-T_c)\psi^2 \right] = 2a(T-T_c)\psi + 2\beta\psi^3
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\]
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Extreme points are found at $\psi = 0$ and $\psi = \pm\sqrt{\frac{-a}{\beta}(T-T_c)}$.
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For $T \geq T_c$, $\psi_0(T \geq T_c) = 0$ gives the minimum, i.e. $\mathcal{F}_0(T \geq T_c) = 0$.
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For $T \leq T_c$, $\psi_0(T \leq T_c) = \sqrt{\frac{-a}{\beta}(T-T_c)}$ is the minimum,
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giving free energy
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\[
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\mathcal{F}_0(T \leq T_c) = \frac{-a^2}{\beta}(T-T_c)^2 + \frac{a^2}{2\beta}(T-T_c)^2 = \frac{-a^2}{2\beta}(T-T_c)^2 \leq \mathcal{F}_0(T \geq T_c)
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\]
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where we chose the positive of the $\pm$ as the order parameter is understood to increase from finite at the phase transition.
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For the specific heat, we find
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\[
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C(T) = -T\pfrac{^2\mathcal{F}}{T^2} =
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\begin{cases}
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0 & T > T_c \\
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\frac{a^2}{\beta}T & T < T_c
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\end{cases}.
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\]
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There is thus a discontinuity in $C(T)$ at $T = T_c$ with size $\Delta C(T) = \frac{a^2}{\beta}T_c$.
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\section{Type-I superconducting foil}
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\section{Type II superconductors and the vortex lattice}
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\section{Currents inside type-II superconducting cylinder}
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\bibliographystyle{vancouver}
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\bibliography{references.bib}
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%\appendix
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\end{document}
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