From 6755c668d425e5d27c0c4b8f504d1eac24d0c9ba Mon Sep 17 00:00:00 2001
From: hackbard <hackbard@sage.physik.uni-augsburg.de>
Date: Wed, 21 May 2008 00:23:36 +0200
Subject: [PATCH] finished excercise 1

---
 solid_state_physics/tutorial/2_03.tex | 31 +++++++++++++++++++++------
 1 file changed, 24 insertions(+), 7 deletions(-)

diff --git a/solid_state_physics/tutorial/2_03.tex b/solid_state_physics/tutorial/2_03.tex
index e751a49..bddd192 100644
--- a/solid_state_physics/tutorial/2_03.tex
+++ b/solid_state_physics/tutorial/2_03.tex
@@ -46,9 +46,9 @@ required to increase the temperature of a unit quantity of a substance
 by a certain temperature interval.
 Thus, the specific heat at constant volume $V$ is given by
 \[
-c_V = \frac{\partial u}{\partial T}
+c_V = \frac{\partial w}{\partial T}
 \]
-in which $u$ is the energy density of the system.
+in which $w$ is the energy density of the system.
 
 \section{Specific heat in the classical theory of the harmonic crystal -\\
          The law of Dulong and Petit}
@@ -59,9 +59,9 @@ its Bravais lattice site {\bf R}.
 From now on expectation values have to be claculated by
 integrating over all possible ionic configurations weighted by
 $\exp(-E/k_{\text{B}}T)$, where $E$ is the energy of the configuration.
-Thus, the energy density $u$ is given by
+Thus, the energy density $w$ is given by
 \[
-u=\frac{1}{V} \frac{\int d\Gamma\exp(-\beta H)H}{\int d\Gamma\exp(-\beta H)},
+w=\frac{1}{V} \frac{\int d\Gamma\exp(-\beta H)H}{\int d\Gamma\exp(-\beta H)},
 \qquad \beta=\frac{1}{k_{\text{B}}T}
 \]
 in which $d\Gamma=\Pi_{\bf R} d{\bf u}({\bf R})d{\bf P}({\bf R})$
@@ -69,7 +69,7 @@ is the volume elemnt in crystal phase space.
 ${\bf u}({\bf R})$ and ${\bf P}({\bf R})$  are the 3N canonical coordinates
 (here: deviations from equlibrium sites)
 and 3N canonical momenta
-of the ion whose equlibrium site is ${\bf R}$.
+of the ions whose equlibrium sites are ${\bf R}$.
 \begin{enumerate}
  \item Show that the energy density can be rewritten to read:
        \[
@@ -81,7 +81,7 @@ of the ion whose equlibrium site is ${\bf R}$.
        U&=&U_{\text{eq}}+U_{\text{harm}} \nonumber \\
        U_{\text{eq}}&=&\frac{1}{2}\sum_{{\bf R R'}} \Phi({\bf R}-{\bf R'})
        \nonumber \\
-       U_{\text{harm}}&=&\frac{1}{2}\sum_{\stackrel{{\bf R R'}}{\mu,v=x,y,z}}
+       U_{\text{harm}}&=&\frac{1}{4}\sum_{\stackrel{{\bf R R'}}{\mu,v=x,y,z}}
        [u_{\mu}({\bf R})-u_{\mu}({\bf R'})]\Phi_{\mu v}({\bf R}-{\bf R'})
        [u_v({\bf R})-u_v({\bf R'})] \nonumber
        \end{eqnarray}
@@ -91,7 +91,24 @@ $\Phi_{\mu v}({\bf r})=
        and $\Phi({\bf r})$ is the potential contribution of two atoms
        separated by ${\bf r}$.
        {\bf Hint:}
-
+       Write down the potential energy for the instantaneous positions
+       ${\bf r}({\bf R})$, with ${\bf u}({\bf R})={\bf r}({\bf R})-{\bf R}$.
+       Apply Taylor approximation to $\Phi({\bf r}+{\bf a})$ with
+       ${\bf r}={\bf R}-{\bf R'}$ and
+       ${\bf a}={\bf u}({\bf R})-{\bf u}({\bf R'})$
+       and only retain terms quadratic in $u$.
+ \item Use the evaluated potential to calculate the energy density
+       (do not forget the kinetic contribution to energy) and
+       the specific heat $c_{\text{V}}$.
+       {\bf Hint:}
+       Use the following change of variables
+       \[
+       {\bf u}({\bf R})=\beta^{-1/2}\bar{{\bf u}}({\bf R}), \qquad
+       {\bf P}({\bf R})=\beta^{-1/2}\bar{{\bf P}}({\bf R})
+       \]
+       to extract the temperature dependence of the integral.
+       Does this also work for anharmonic terms?
+       Which parts of the integral do not contribute to $w$ and why?
 \end{enumerate}
 
 
-- 
2.20.1