updated tutorial 6 + added gnu plot save file

[lectures/latex.git] / solid_state_physics / tutorial / 1_06.tex

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}