From 80df9b92f696a4b82e0cf46054c14efdfd8c1bce Mon Sep 17 00:00:00 2001 From: hackbard Date: Tue, 22 Jan 2008 11:41:53 +0100 Subject: [PATCH] updated tutorial 6 + added gnu plot save file --- solid_state_physics/tutorial/1_06.tex | 28 +++- solid_state_physics/tutorial/1_06s.tex | 33 ++++- solid_state_physics/tutorial/dielectric.gpi | 150 ++++++++++++++++++++ 3 files changed, 201 insertions(+), 10 deletions(-) create mode 100644 solid_state_physics/tutorial/dielectric.gpi diff --git a/solid_state_physics/tutorial/1_06.tex b/solid_state_physics/tutorial/1_06.tex index a8b8987..3452519 100644 --- a/solid_state_physics/tutorial/1_06.tex +++ b/solid_state_physics/tutorial/1_06.tex @@ -59,13 +59,31 @@ Some facts about silicon: \item Draw conclusions concerning optical applications. \end{enumerate} -\section{\ldots} - -\ldots +\section{Dielectric function of the free electron gas} \begin{enumerate} - \item \ldots - \item \ldots + \item Derive an expression for the dieletric function $\epsilon(\omega)$ + of the free electron gas. + {\bf Hint:} The equation of motion for a free electron + (position vector $x$) in an electric field $E$ is given by + $m\frac{d^2x}{dt^2}=-eE$. + For an electric field which has a + $e^{-i\omega t}$ dependance on time + the ansatz $x=x_0 e^{-i\omega t}$ is suitable + to solve the equation of motion. + What is the dipole moment of that electron? + Now write down the polarization $P$ which is defined as + the dipole moment of all electrons per volume. + As known from electro statics the polarization is connected + to the dielectric constant by + $\epsilon\epsilon_0E=\epsilon_0E+P$. + \item Rewrite $\epsilon(\omega)$ using the plasma frequency $\omega_p$ + defined as $\omega_p^2=\frac{ne^2}{\epsilon_0m}$ + ($n$: electron density, $e$: electron charge, + $\epsilon_0$: vacuum premitivity, $m$: electron mass). + Sketch $\epsilon(\omega)$ against $\frac{\omega}{\omega_p}$. + Explain what is happening to electromagnetic waves in the regions + $\frac{\omega}{\omega_p}<1$ and $\frac{\omega}{\omega_p}>1$. \end{enumerate} \end{document} diff --git a/solid_state_physics/tutorial/1_06s.tex b/solid_state_physics/tutorial/1_06s.tex index 4e965e3..7d55113 100644 --- a/solid_state_physics/tutorial/1_06s.tex +++ b/solid_state_physics/tutorial/1_06s.tex @@ -16,6 +16,8 @@ \usepackage{pstricks} \usepackage{pst-node} \usepackage{rotating} +\usepackage{epic} +\usepackage{eepic} \setlength{\headheight}{0mm} \setlength{\headsep}{0mm} \setlength{\topskip}{-10mm} \setlength{\textwidth}{17cm} @@ -66,13 +68,34 @@ \end{itemize} \end{enumerate} -\section{\ldots} - -\ldots +\section{Dielectric function of the free electron gas} \begin{enumerate} - \item \ldots - \item \ldots + \item \begin{itemize} + \item Equation of motion: $m\frac{d^2x}{dt^2}=-eE$ + \item Ansatz: $x(t)=x_0 e^{-i\omega t}$ + \item Solution of the equation of motion: $-m\omega^2x=-eE$ + \item Dipole moment: $p=-ex=\frac{-e^2E}{m\omega^2}$ + \item Polarization: $P=np=\frac{-ne^2E}{m\omega^2}$ + \item Dielectric function: + $\epsilon(\omega)=1+\frac{P}{\epsilon_0E} + =1+\frac{-ne^2E}{m\omega^2\epsilon_0E} + =1-\frac{ne^2}{\epsilon_0m\omega^2}$ + \end{itemize} + \item \begin{itemize} + \item Using $\omega_p^2=\frac{ne^2}{\epsilon_0m}$\\ + $\Rightarrow \epsilon(\omega)=1-\frac{\omega_p^2}{\omega^2}$ + \item Sketch of dielectric function:\\ + (page 2) + \item Influence on electromagnetic waves:\\ + $\frac{\omega}{\omega_p}>1\Leftrightarrow \omega>\omega_p$: + $\Rightarrow \epsilon=n^2>0$ + $\Rightarrow$ transparent region\\ + $\frac{\omega}{\omega_p}<1\Leftrightarrow \omega<\omega_p$: + $\Rightarrow \epsilon=n^2<0$ + $\Rightarrow$ reflective region\\ + \end{itemize} +\input{dielectric_pslatex} \end{enumerate} \end{document} diff --git a/solid_state_physics/tutorial/dielectric.gpi b/solid_state_physics/tutorial/dielectric.gpi new file mode 100644 index 0000000..55c1ae6 --- /dev/null +++ b/solid_state_physics/tutorial/dielectric.gpi @@ -0,0 +1,150 @@ +#!/usr/bin/gnuplot -persist +# +# +# G N U P L O T +# Version 4.2 patchlevel 0 +# last modified March 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 comments and help requests to +# Send bug reports and suggestions to +# +# set terminal pslatex rotate leveldefault color colortext \ + dashed dashlength 1.0 linewidth 1.0 butt \ + palfuncparam 2000,0.003 \ + rotate noauxfile +# set output 'dielectric_pslatex.tex' +unset clip points +set clip one +unset clip two +set bar 1.000000 +set border 31 front linetype -1 linewidth 1.000 +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 "% g" +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 +unset arrow +set style increment default +unset style line +unset style arrow +set style histogram clustered gap 2 title offset character 0, 0, 0 +set object 1 rect from 0, -2, 0 to 1, 1, 0 behind lw 1.0 fc rgb "gray" fillstyle solid 1.00 border -1 +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 100, 100 +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 0 linewidth 1.000 +set yzeroaxis linetype 0 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 xtics border in scale 1,0.5 mirror norotate offset character 0, 0, 0 autofreq +set ytics border in scale 1,0.5 mirror norotate offset character 0, 0, 0 autofreq +set ztics border in scale 1,0.5 nomirror norotate offset character 0, 0, 0 autofreq +set nox2tics +set noy2tics +set cbtics border in scale 1,0.5 mirror norotate offset character 0, 0, 0 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 "$\\frac{\\omega}{\\omega_p}$" +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 [ 0.00000 : 2.00000 ] noreverse nowriteback +set x2range [ * : * ] noreverse nowriteback # (currently [-10.0000:10.0000] ) +set ylabel "$\\epsilon(\\omega)$" +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 : 1.00000 ] 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 +e(x)=1-1/x**2 +GNUTERM = "wxt" +plot e(x) t "" +# EOF -- 2.20.1