From 578f4e5ba230a27e01aca6d6723a7099c18843a3 Mon Sep 17 00:00:00 2001 From: hackbard Date: Mon, 2 Jun 2008 16:11:02 +0200 Subject: [PATCH] finished solutions 2 --- solid_state_physics/tutorial/2_02s.tex | 115 +++++++++++++++ solid_state_physics/tutorial/sc_mfield.gpi | 160 +++++++++++++++++++++ 2 files changed, 275 insertions(+) create mode 100644 solid_state_physics/tutorial/2_02s.tex create mode 100644 solid_state_physics/tutorial/sc_mfield.gpi diff --git a/solid_state_physics/tutorial/2_02s.tex b/solid_state_physics/tutorial/2_02s.tex new file mode 100644 index 0000000..8c12196 --- /dev/null +++ b/solid_state_physics/tutorial/2_02s.tex @@ -0,0 +1,115 @@ +\pdfoutput=0 +\documentclass[a4paper,11pt]{article} +\usepackage[activate]{pdfcprot} +\usepackage{verbatim} +\usepackage{a4} +\usepackage{a4wide} +\usepackage[german]{babel} +\usepackage[latin1]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{amsmath} +\usepackage{ae} +\usepackage{aecompl} +\usepackage[dvips]{graphicx} +\graphicspath{{./img/}} +\usepackage{color} +\usepackage{pstricks} +\usepackage{pst-node} +\usepackage{rotating} + +\setlength{\headheight}{0mm} \setlength{\headsep}{0mm} +\setlength{\topskip}{-10mm} \setlength{\textwidth}{17cm} +\setlength{\oddsidemargin}{-10mm} +\setlength{\evensidemargin}{-10mm} \setlength{\topmargin}{-1cm} +\setlength{\textheight}{26cm} \setlength{\headsep}{0cm} + +\renewcommand{\labelenumi}{(\alph{enumi})} +\renewcommand{\labelenumii}{\arabic{enumii})} +\renewcommand{\labelenumiii}{\roman{enumiii})} + +\begin{document} + +% header +\begin{center} + {\LARGE {\bf Materials Physics II}\\} + \vspace{8pt} + Prof. B. Stritzker\\ + SS 2008\\ + \vspace{8pt} + {\Large\bf Tutorial 2 - proposed solutions} +\end{center} + +\section{Critical current in the surface region of a type 1 superconductor} +\[ + j_s(r)=j_s(R)\exp\left(\frac{-(R-r)}{\lambda}\right) +\] +$R$: radius of the wire, $r$: distance from the cylinder axis, +and $\lambda$: London penetration depth. + +\begin{enumerate} + \item $I_c=\int_0^R dr \int_0^{2\pi} d\phi \, j_c(r) r$\\ + $\Rightarrow I_c=\int_0^R dr \, 2\pi j_c(r) r + = j_c(R)2\pi \int_0^R dr \, r \exp(-(R-r)/\lambda) + = j_c(R)2\pi \exp(-R/\lambda) \int_0^R dr \, r \exp(r/\lambda)$ + $x=\frac{r}{\lambda} \rightarrow \frac{dx}{dr}=\frac{1}{\lambda} + \Rightarrow dr=\lambda dx$, $r=\lambda x$ + $\Rightarrow + I_c=j_c(R)2\pi \lambda^2 \exp(-R/\lambda) \int_0^R d(\frac{r}{\lambda}) + \, \frac{r}{\lambda} \exp(\frac{r}{\lambda})$ + Integration by parts: $\int uv' = uv - \int vu'$\\ + $\int xe^x dx = xe^x-\int e^x dx=xe^x-e^x+c=e^x(x-1)+c$\\ + $\Rightarrow + I_c=j_c(R)2\pi \lambda^2\exp(-R/\lambda) \left[ + \exp(\frac{r}{\lambda})(r\lambda-1) \right]_0^R$\\ + $\left[ \exp(\frac{r}{\lambda})(r\lambda-1) \right]_0^R= + \exp(R/\lambda)(R/\lambda-1)-1(-1)$\\ + $\Rightarrow I_c=j_c(R)2\pi \lambda^2 \left[ + \underbrace{\exp(-R/\lambda)\exp(R/\lambda)}_{=1} + (R/\lambda-1)+ + \underbrace{\exp(-R/\lambda)}_{\approx 0 \text{, since } \lambda\ll R} \right]$\\ + $\Rightarrow + I_c\approx j_c(R)2\pi \lambda^2 + \underbrace{(R/\lambda-1)}_{\approx R/\lambda}\approx + j_c(R)2\pi \lambda R$\\ + $\Rightarrow j_c(R)\approx\frac{I_c}{2\pi R\lambda}$ + \item $j_c(R,T=0)=\frac{I_c(T=0)}{2\pi R\lambda(T=0)} + =7.9\cdot 10^7\frac{A}{cm^2}$ +\end{enumerate} + +\section{Penetration of the magnetic field into a type 1 superconductor} +\[ + {\bf B}_s=\mu_0 \left({\bf H}_a + {\bf M}_s\right) +\] +${\bf H}_a$: strength of the applied magnetic field, +${\bf M}_s$: magnetization of the superconductor. + +\begin{enumerate} + \item $\frac{1}{\mu_0} rot {\bf B}_s = + rot {\bf H}_a + rot {\bf M}_s + \stackrel{rot {\bf H}={\bf j}+\frac{d{\bf D}}{dt}}{=} + \underbrace{{\bf j}_a}_{=0} + +{\bf j}_s$\\ + $\Rightarrow + rot {\bf B}_s=\mu_0 {\bf j}_s \qquad | rot ...$\\ + $\Rightarrow + \underbrace{rot rot {\bf B}_s}_{grad \underbrace{div {\bf B}_s}_{=0} + -\Delta {\bf B}_s} + =\mu_0rot {\bf j}_s\stackrel{\text{London II}}{=} + -\frac{\mu_0}{\Lambda}{\bf B}_s$\\ + $\Rightarrow + \Delta {\bf B}_s=\frac{\mu_0}{\Lambda}{\bf B}_s$ + \item ${\bf B}_a=\mu_0 H_a {\bf e}_z$, ${\bf B}_s=B_{s_z}(x) {\bf e}_x$\\ + Diff. equation: $\frac{d^2}{dx^2}B_{s_z}(x)- + \frac{\mu_0}{\Lambda}B_{s_z}(x)=0$\\ + Solution: $B_{s_z}(x)=B_{s_z}(0)\exp(-\frac{x}{\lambda})$, with + $\lambda=\sqrt{\frac{\Lambda}{\mu_0}}$\\ + \includegraphics[width=16cm]{mfsc.ps} + \item $\mu_0 {\bf j}_s=rot{\bf B}_s= + \frac{-\partial B_{s_z}(x)}{\partial x} {\bf e}_y= + \frac{1}{\lambda} B_a \exp(-\frac{x}{\lambda}){\bf e}_y$\\ + $\Rightarrow$ Direction of screening current is ${\bf e}_y$.\\ + $\Rightarrow$ Exponential decay inside the SC.\\ + Interface: ${\bf j}_s(x=0)=\frac{B_a}{\lambda\mu_0} {\bf e}_y$ +\end{enumerate} + +\end{document} diff --git a/solid_state_physics/tutorial/sc_mfield.gpi b/solid_state_physics/tutorial/sc_mfield.gpi new file mode 100644 index 0000000..072b7da --- /dev/null +++ b/solid_state_physics/tutorial/sc_mfield.gpi @@ -0,0 +1,160 @@ +#!/usr/bin/gnuplot -persist +# +# +# G N U P L O T +# Version 4.2 patchlevel 2 +# last modified 31 Aug 2007 +# System: Linux 2.6.22-2-686 +# +# Copyright (C) 1986 - 1993, 1998, 2004, 2007 +# Thomas Williams, Colin Kelley and many others +# +# Type `help` to access the on-line reference manual. +# The gnuplot FAQ is available from http://www.gnuplot.info/faq/ +# +# Send bug reports and suggestions to +# +# set terminal wxt 0 +# set output +unset clip points +set clip one +unset clip two +set bar 1.000000 +unset border +set xdata +set ydata +set zdata +set x2data +set y2data +set timefmt x "%d/%m/%y,%H:%M" +set timefmt y "%d/%m/%y,%H:%M" +set timefmt z "%d/%m/%y,%H:%M" +set timefmt x2 "%d/%m/%y,%H:%M" +set timefmt y2 "%d/%m/%y,%H:%M" +set timefmt cb "%d/%m/%y,%H:%M" +set boxwidth +set style fill empty border +set style rectangle back fc lt -3 fillstyle solid 1.00 border -1 +set dummy x,y +set format x "% g" +set format y "" +set format x2 "% g" +set format y2 "% g" +set format z "% g" +set format cb "% g" +set angles radians +unset grid +set key title "" +set key inside right top vertical Right noreverse enhanced autotitles nobox +set key noinvert samplen 4 spacing 1 width 0 height 0 +unset label +set label 1 "vacuum" at -0.7, -0.8, 0 left norotate back nopoint offset character 0, 0, 0 +set label 2 "SC" at 0.7, -0.8, 0 left norotate back nopoint offset character 0, 0, 0 +set label 3 "$B_a$" at -0.7, 8, 0 left norotate back nopoint offset character 0, 0, 0 +set label 4 "$B_{s_z}$" at 0.7, 8, 0 left norotate back nopoint offset character 0, 0, 0 +set label 5 "x" at 2.7, -0.8, 0 left norotate back nopoint offset character 0, 0, 0 +set label 6 "z" at -0.1, 11, 0 left norotate back nopoint offset character 0, 0, 0 +set label 7 "$\\lambda$" at 1.8, -0.8, 0 left norotate back nopoint offset character 0, 0, 0 +unset arrow +set arrow 1 from 0, -2, 0 to 0, 12, 0 head back nofilled linetype -1 linewidth 1.000 +set arrow 2 from -1, 0, 0 to 3.1, 0, 0 head back nofilled linetype -1 linewidth 1.000 +set arrow 3 from 0, 1, 0 to 1.8, 1, 0 nohead back nofilled linetype 7 linewidth 1.000 +set arrow 4 from 1.8, 1, 0 to 1.8, 0, 0 nohead back nofilled linetype 7 linewidth 1.000 +set style increment default +unset style line +unset style arrow +set style histogram clustered gap 2 title offset character 0, 0, 0 +unset logscale +set offsets 0, 0, 0, 0 +set pointsize 1 +set encoding default +unset polar +unset parametric +unset decimalsign +set view 60, 30, 1, 1 +set samples 500, 500 +set isosamples 10, 10 +set surface +unset contour +set clabel '%8.3g' +set mapping cartesian +set datafile separator whitespace +unset hidden3d +set cntrparam order 4 +set cntrparam linear +set cntrparam levels auto 5 +set cntrparam points 5 +set size ratio 0 1,1 +set origin 0,0 +set style data points +set style function lines +set xzeroaxis linetype -2 linewidth 1.000 +set yzeroaxis linetype -2 linewidth 1.000 +set zzeroaxis linetype -2 linewidth 1.000 +set x2zeroaxis linetype -2 linewidth 1.000 +set y2zeroaxis linetype -2 linewidth 1.000 +set ticslevel 0.5 +set mxtics default +set mytics default +set mztics default +set mx2tics default +set my2tics default +set mcbtics default +set noxtics +set noytics +set ztics border in scale 1,0.5 nomirror norotate offset character 0, 0, 0 +set ztics autofreq +set nox2tics +set noy2tics +set cbtics border in scale 1,0.5 mirror norotate offset character 0, 0, 0 +set cbtics autofreq +set title "" +set title offset character 0, 0, 0 font "" norotate +set timestamp bottom +set timestamp "" +set timestamp offset character 0, 0, 0 font "" norotate +set rrange [ * : * ] noreverse nowriteback # (currently [0.00000:10.0000] ) +set trange [ * : * ] noreverse nowriteback # (currently [-5.00000:5.00000] ) +set urange [ * : * ] noreverse nowriteback # (currently [-5.00000:5.00000] ) +set vrange [ * : * ] noreverse nowriteback # (currently [-5.00000:5.00000] ) +set xlabel "" +set xlabel offset character 0, 0, 0 font "" textcolor lt -1 norotate +set x2label "" +set x2label offset character 0, 0, 0 font "" textcolor lt -1 norotate +set xrange [ -1.00000 : 3.00000 ] noreverse nowriteback +set x2range [ * : * ] noreverse nowriteback # (currently [-10.0000:10.0000] ) +set ylabel "" +set ylabel offset character 0, 0, 0 font "" textcolor lt -1 rotate by 90 +set y2label "" +set y2label offset character 0, 0, 0 font "" textcolor lt -1 rotate by 90 +set yrange [ -2.00000 : 12.0000 ] noreverse nowriteback +set y2range [ * : * ] noreverse nowriteback # (currently [-10.0000:10.0000] ) +set zlabel "" +set zlabel offset character 0, 0, 0 font "" textcolor lt -1 norotate +set zrange [ * : * ] noreverse nowriteback # (currently [-10.0000:10.0000] ) +set cblabel "" +set cblabel offset character 0, 0, 0 font "" textcolor lt -1 norotate +set cbrange [ * : * ] noreverse nowriteback # (currently [-10.0000:10.0000] ) +set zero 1e-08 +set lmargin -1 +set bmargin -1 +set rmargin -1 +set tmargin -1 +set locale "C" +set pm3d explicit at s +set pm3d scansautomatic +set pm3d interpolate 1,1 flush begin noftriangles nohidden3d corners2color mean +set palette positive nops_allcF maxcolors 0 gamma 1.5 color model RGB +set palette rgbformulae 7, 5, 15 +set colorbox default +set colorbox vertical origin screen 0.9, 0.2, 0 size screen 0.05, 0.6, 0 bdefault +set loadpath +set fontpath +set fit noerrorvariables +f(x)=x>0?B*exp(-x/l):10 +GNUTERM = "wxt" +B = 10 +l = 0.8 +bb = 0 +plot f(x) t "" +# EOF -- 2.39.2