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% header
\begin{center}
 {\LARGE {\bf Materials Physics II}\\}
 \vspace{8pt}
 Prof. B. Stritzker\\
 SS 2008\\
 \vspace{8pt}
 {\Large\bf Tutorial 4}
\end{center}

\vspace{4pt}

\section{Legendre transformation and Maxwell relations}

\begin{enumerate}
 \item Consider the total differential
       \[
       df= \sum_{i=1}^{n} u_i dx_i
       \]
       with the state function $f=f(x_1,\ldots,x_n)$ and its partial derivatives
       $u_i=\frac{\partial f}{\partial x_i}$.
       Rewrite the total differential of the function $g$ defined as
       \[
       g=f-\sum_{i=r+1}^{n} u_i x_i
       \]
       in such a way that $g$ is immediately identified to be a function of
       the variables $x_1,\ldots,x_r$ and $u_{r+1},\ldots,u_n$,
       where $u_i$ is called the conjugate variable of $x_i$.
       This transformation is called Legendre transformation.
 \item By taking the derivatives of transformed thermodynamic potentials
       with respect to the variables they depend on,
       relations between intensive and extensive variables can be gained.

       Start with the internal energy $E=E(S,V)$.
       Write down the total differential using the equalities
       $T=\left.\frac{\partial E}{\partial S}\right|_V$ and
       $-p=\left.\frac{\partial E}{\partial V}\right|_S$.
       Apply Legendre transformation to the following potentials
       \begin{itemize}
        \item $H=E+pV$ (Enthalpy)
        \item $F=E-TS$ (Helmholtz free energy)
        \item $G=H-TS=E+pV-TS$ (Gibbs free energy)
       \end{itemize}
       and find more relations by taking the appropriate derivatives.
 \item For a thermodynamic potential $\Phi(X,Y)$ the following identity
       expressing the permutability of derivatives holds:
       \[
       \frac{\partial^2 \Phi}{\partial X \partial Y} =
       \frac{\partial^2 \Phi}{\partial Y \partial X}
       \]
       Derive the Maxwell relations by taking the mixed derivatives of the
       potentials in (b) with respect to the variables they depend on.
       Exchange the sequence of derivation and use the identities gained in (b).
\end{enumerate}

\section{Thermal expansion of solids}

It is well known that solids change their length $L$ and volume $V$ respectively
if there is a change in temperature $T$ or in pressure $p$ of the system.
The following exercise shows that
thermal expansion cannot be described by rigorously harmonic crystals.

\begin{enumerate}
 \item The coefficient of thermal expansion of a solid is given by
       $\alpha_L=\frac{1}{L}\left.\frac{\partial L}{\partial T}\right|_p$.
       Show that the coefficient of thermal expansion of the volume
       $\alpha_V=\frac{1}{V}\left.\frac{\partial V}{\partial T}\right|_p$
       equals $3\alpha_L$ for isotropic materials.
 \item Find an expression for the pressure as a function of the free energy
       $F=E-TS$.
       Rewrite this equation to express the pressure entirely in terms of
       the internal energy $E$.
       Evaluate the pressure by using the harmonic form of the internal energy.
       {\bf Hint:}
       Step 2 introduced an integral over the temperature $T'$.
       Change the integration variable $T'$ to $x=\hbar\omega_s({\bf k})/T'$.
       Use integration by parts with respect to $x$.
 \item The normal mode frequencies of a rigorously harmonic crystal
       are unaffected by a change in volume.
       What does this imply for the pressure
       (Which variables does the pressure depend on)?
       Draw conclusions for the coefficient of thermal expansion.
 \item Find an expression for $C_p-C_V$ in terms of temperature $T$,
       volume $V$, the coefficient of thermal expansion $\alpha_V$ and
       the inverse bulk modulus (isothermal compressibility)
       $\frac{1}{B}=-\frac{1}{V}\left.\frac{\partial V}{\partial p}\right|_T$.\\
       $C_p=\left.\frac{\partial H}{\partial T}\right|_p$ is the heat capacity
       for constant pressure and
       $C_V=\left.\frac{\partial E}{\partial T}\right|_V$ is the heat capacity
       for constant volume.
\end{enumerate}

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