X-Git-Url: https://hackdaworld.org/gitweb/?a=blobdiff_plain;f=solid_state_physics%2Ftutorial%2F2_03.tex;h=e268709d862e4d0372e1b6c5c3ac5882c7503659;hb=fec7e52be07515b5dac392facfa0b2022d002c21;hp=e751a496236ebe73862d4727b42067db4b59dc07;hpb=c912342ad8d60890a85fa387abd7a663bed31d32;p=lectures%2Flatex.git

diff --git a/solid_state_physics/tutorial/2_03.tex b/solid_state_physics/tutorial/2_03.tex
index e751a49..e268709 100644
--- a/solid_state_physics/tutorial/2_03.tex
+++ b/solid_state_physics/tutorial/2_03.tex
@@ -46,22 +46,24 @@ 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 internal energy density of the system.
+In the following the contribution to the specific heat due to the
+degrees of freedom of the lattice ions is calculated.
 
 \section{Specific heat in the classical theory of the harmonic crystal -\\
          The law of Dulong and Petit}
 
 In the classical theory of the harmonic crystal equilibrium properties
-can no longer be evaluated by simply assuming that each ion sits quitly at
+can no longer be evaluated by simply assuming that each ion sits quietly at
 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 $d\Gamma=\Pi_{\bf R} d{\bf u}({\bf R})d{\bf P}({\bf R})$
@@ -69,11 +71,11 @@ 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:
        \[
-   u=-\frac{1}{V}\frac{\partial}{\partial \beta} ln \int d\Gamma \exp(-\beta H).
+   w=-\frac{1}{V}\frac{\partial}{\partial \beta} ln \int d\Gamma \exp(-\beta H).
        \]
  \item Show that the potential contribution to the energy
        in the harmonic approximation is given by
@@ -81,7 +83,7 @@ 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}
@@ -91,16 +93,76 @@ $\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 energy contribution) 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}
 
-
 \section{Specific heat in the quantum theory of the harmonic crystal -\\
-         Models of Debye and Einstein}
+         The Debye model}
 
+As found in exercise 1, the specific heat of a classical harmonic crystal
+is not depending on temeprature.
+However, as temperature drops below room temperature
+the specific heat of all solids is decreasing as $T^3$ in insulators
+and $AT+BT^3$ in metals.
+This can be explained in a quantum theory of the specific heat of
+a harmonic crystal, in which the energy density $w$ is given by
+\[
+w=\frac{1}{V}\frac{\sum_i E_i \exp(-\beta E_i)}{\sum_i \exp(-\beta E_i)}.
+\]
 \begin{enumerate}
- \item
- \item
+ \item Show that the energy density can be rewritten to read:
+       \[
+   w=-\frac{1}{V}\frac{\partial}{\partial \beta} ln \sum_i \exp(-\beta E_i).
+       \]
+ \item Evaluate the expression of the energy density.
+       {\bf Hint:}
+       The energy levels of a harmonic crystal of N ions
+       can be regarded as 3N independent oscillators,
+       whose frequencies are those of the 3N classical normal modes.
+       The contribution to the total energy of a particular normal mode
+       with angular frequency $\omega_s({\bf k})$ 
+       ($s$: branch, ${\bf k}$: wave vector) is given by
+       $(n_{{\bf k}s} + \frac{1}{2})\hbar\omega_s({\bf k})$ with the
+       excitation number $n_{{\bf k}s}$ being restricted to integers greater
+       or equal zero.
+       The total energy is given by the sum over the energies of the individual
+       normal modes.
+       Use the totals formula of the geometric series to expcitly calculate
+       the sum of the exponential functions.
+ \item Separate the above result into a term vanishing as $T$ goes to zero and
+       a second term giving the energy of the zero-point vibrations of the
+       normal modes.
+ \item Write down an expression for the specific heat.
+       Consider a large crystal and thus replace the sum over the discrete
+       wave vectors with an integral.
+ \item Debye replaced all branches of the vibrational spectrum with three
+       branches, each of them obeying the dispersion relation
+       $w=ck$.
+       Additionally the integral is cut-off at a radius $k_{\text{D}}$
+       to have a total amount of N allowed wave vectors.
+       Determine $k_{\text{D}}$.
+       Evaluate the simplified integral and introduce the
+       Debye frequency $\omega_{\text{D}}=k_{\text{D}}c$
+       and the Debye temperature $\Theta_{\text{D}}$ which is given by
+       $k_{\text{B}}\Theta_{\text{D}}=\hbar\omega_{\text{D}}$.
+       Write down the resulting expression for the specific heat.
 \end{enumerate}
 
 \end{document}