updated tutorial 6 + added gnu plot save file

diff --git a/solid_state_physics/tutorial/1_06.tex b/solid_state_physics/tutorial/1_06.tex
index a8b8987..3452519 100644 (file)
--- 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 (file)
--- 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 (file)
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  <gnuplot-info@lists.sourceforge.net>
+#      Send bug reports and suggestions to <gnuplot-bugs@lists.sourceforge.net>
+#    
+# 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