X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=blobdiff_plain;f=solid_state_physics%2Ftutorial%2F1_06.tex;h=3452519cb94c1e9bcd528bd55fe66fa60717c725;hp=a8b8987c4a24f675d6ae9b83beaa8f37e09950ae;hb=80df9b92f696a4b82e0cf46054c14efdfd8c1bce;hpb=6c4e0f74452033badf606ba05abc9c77b2331566

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}