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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 - proposed solutions}
\end{center}

\vspace{4pt}

\section{Legendre transformation and Maxwell relations}

\begin{enumerate}
 \item Legendre transformation:
       \begin{eqnarray}
       dg &=& df - \sum_{i=r+1}^{n} d(u_ix_i)\nonumber\\
          &=& df - \sum_{i=r+1}^{n} (x_idu_i + u_idx_i)\nonumber\\
	  &=& \sum_{i=1}^r u_idx_i - \sum_{i=r+1}^n x_idu_i\nonumber
       \end{eqnarray}
       \[
       \Rightarrow g=g(x_1,\ldots,x_r,u_{r+1},\ldots,u_n)
       \]
 \item Use $T=\left.\frac{\partial E}{\partial S}\right|_V$ and
       $-p=\left.\frac{\partial E}{\partial V}\right|_S$.\\
       Start with internal energy $E=E(S,V)$:
       \[
       \Rightarrow dE=\frac{\partial E}{\partial S}dS +
                      \frac{\partial E}{\partial V}dV =
         	      TdS - pdV
       \]
       Enthalpy $H=E+pV$:
       \[
       \Rightarrow dH=dE+Vdp+pdV=TdS-pdV+Vdp+pdV=TdS+Vdp
       \]
       \[
       \Rightarrow
       \left.\frac{\partial H}{\partial S}\right|_p=T \textrm{ and }
       \left.\frac{\partial H}{\partial p}\right|_S=V
       \]
       Helmholtz free energy $F=E-TS$:
       \[
       \Rightarrow dF=dE-SdT-TdS=TdS-pdV-SdT-TdS=-pdV-SdT
       \]
       \[
       \Rightarrow
       \left.\frac{\partial F}{\partial V}\right|_T=-p \textrm{ and }
       \left.\frac{\partial F}{\partial T}\right|_V=-S
       \]
       Gibbs free energy $G=H-TS=E+pV-TS$:
       \[
       \Rightarrow dG=dH-SdT-TdS=TdS+Vdp-SdT-TdS=Vdp-SdT
       \]
       \[
       \Rightarrow
       \left.\frac{\partial G}{\partial p}\right|_T=V \textrm{ and }
       \left.\frac{\partial G}{\partial T}\right|_p=-S
       \]
 \item Maxwell relations:\\
       Internal energy: $dE=TdS-pdV$
       \[
       \frac{\partial}{\partial S}
       \left(\left.\frac{\partial E}{\partial V}\right|_S\right)_V=
       \frac{\partial}{\partial V}
       \left(\left.\frac{\partial E}{\partial S}\right|_V\right)_S
       \Rightarrow
       \left.-\frac{\partial p}{\partial S}\right|_V=
       \left.\frac{\partial T}{\partial V}\right|_S
       \]
       Enthalpy: $dH=TdS+Vdp$
       \[
       \frac{\partial}{\partial S}
       \left(\left.\frac{\partial H}{\partial p}\right|_S\right)_p=
       \frac{\partial}{\partial p}
       \left(\left.\frac{\partial H}{\partial S}\right|_p\right)_S
       \Rightarrow
       \left.\frac{\partial V}{\partial S}\right|_p=
       \left.\frac{\partial T}{\partial p}\right|_S
       \]
       Helmholtz free energy: $dF=-pdV-SdT$
       \[
       \frac{\partial}{\partial V}
       \left(\left.\frac{\partial F}{\partial T}\right|_V\right)_T=
       \frac{\partial}{\partial T}
       \left(\left.\frac{\partial F}{\partial V}\right|_T\right)_V
       \Rightarrow
       \left.-\frac{\partial S}{\partial V}\right|_T=
       \left.-\frac{\partial p}{\partial T}\right|_V
       \]
       Gibbs free energy: $dG=Vdp-SdT$
       \[
       \frac{\partial}{\partial p}
       \left(\left.\frac{\partial G}{\partial T}\right|_p\right)_T=
       \frac{\partial}{\partial T}
       \left(\left.\frac{\partial G}{\partial p}\right|_T\right)_p
       \Rightarrow
       \left.-\frac{\partial S}{\partial p}\right|_T=
       \left.\frac{\partial V}{\partial T}\right|_p
       \]
\end{enumerate}

\section{Thermal expansion of solids}

\begin{enumerate}
 \item Coefficients of thermal expansion:\\
       Consider a cube with side lengthes $L_1,L_2,L_3$.
       Isotropic material: $\frac{1}{L_1}\frac{\partial L_1}{\partial T}=
                            \frac{1}{L_2}\frac{\partial L_2}{\partial T}=
			    \frac{1}{L_3}\frac{\partial L_3}{\partial T}=
			    \alpha_L$.
       \begin{eqnarray}
       \alpha_V&=&\frac{1}{V}\frac{\partial V}{\partial T}=
       \frac{1}{L_1L_2L_3}\frac{\partial}{\partial T}(L_1L_2L_3)=
       \frac{1}{L_1L_2L_3}\left(L_2L_3\frac{\partial L_1}{\partial T}+
                                L_1L_3\frac{\partial L_2}{\partial T}+
                                L_1L_2\frac{\partial L_3}{\partial T}\right)
				\nonumber\\
       &=&\frac{1}{L_1}\frac{\partial L_1}{\partial T}+
          \frac{1}{L_2}\frac{\partial L_2}{\partial T}+
          \frac{1}{L_3}\frac{\partial L_3}{\partial T}=3\alpha_L\nonumber
       \end{eqnarray}
 \item \[
       dF=-pdV-SdT \Rightarrow p=-\left.\frac{\partial F}{\partial V}\right|T
       \]
       \[
       \left.\frac{\partial E}{\partial T}\right|_V=
       \left.\frac{\partial E}{\partial S}\right|_V
       \left.\frac{\partial S}{\partial T}\right|_V=
       T\left.\frac{\partial S}{\partial T}\right|_V
       \Rightarrow
       \left.\frac{\partial S}{\partial T}\right|_V=
       \frac{1}{T}\left.\frac{\partial E}{\partial T}\right|_V
       \]
       \[
       \textrm{Using } F=E-TS \textrm{ and }
       TS=T\int_0^T\frac{\partial S}{\partial T'}dT'
       \textrm{ (Entropy density vanishes at $T=0$)}
       \]
       \[
       \Rightarrow
       p=-\frac{\partial}{\partial V}\left(
       E-T\int_0^T\frac{dT'}{T'}\frac{\partial E}{\partial T'}
       \right)
       \]
       Harmonic approximation of the internal energy:
       \[
       E=E^{\text{eq}}+\frac{1}{2}\sum_{{\bf k}s}\hbar\omega_s({\bf k})+
       \sum_{{\bf k}s}
       \frac{\hbar\omega_s({\bf k})}{e^{\beta\hbar\omega_s({\bf k})}-1}
       \]
       \[
       \ldots
       \]
       \[
       x=\hbar\omega_s({\bf k})/T'
       \]
       \[
       \ldots
       \]
       \[
       \Rightarrow
       p=-\frac{\partial}{\partial V}\left(
       E^{\text{eq}}+\frac{1}{2}\sum_{{\bf k}s}\hbar\omega_s({\bf k})
       \right)+
       \sum_{{\bf k}s}\left(-\frac{\partial}{\partial V}\hbar\omega_s({\bf k})
       \right)\frac{1}{e^{\beta\hbar\omega_s({\bf k})}-1}
       \]
 \item The pressure depends on temperature
       only if the normal mode frequencies depend on the volume.
       However, the normal mode frequencies of a rigorously harmonic crystal
       are unaffected by a change in volume.\\
       $\Rightarrow$
       The pressure solely depends on the volume.\\
       $\Rightarrow$
       The pressure required to maintain a given volume
       does not vary with temperature.
       \[
       \left.\frac{\partial p}{\partial T}\right|_V=0
       \]
       \[
       \left.\frac{\partial V}{\partial T}\right|_p=
       -\frac{\left.\frac{\partial p}{\partial T}\right|_V}
             {\left.\frac{\partial p}{\partial V}\right|_T}=0
       \]
 \item $\frac{1}{B}=-\frac{1}{V}\left.\frac{\partial V}{\partial p}\right|_T$
       and $\alpha_V=\frac{1}{V}\left.\frac{\partial V}{\partial T}\right|_p$\\
       \[
       C_p=\left.\frac{\partial H}{\partial T}\right|_p=
           \left.\frac{\partial H}{\partial S}\right|_p
           \left.\frac{\partial S}{\partial T}\right|_p=
	   T\left.\frac{\partial S}{\partial T}\right|_p
       \]
       \[
       C_V=\left.\frac{\partial E}{\partial T}\right|_V=
           \left.\frac{\partial E}{\partial S}\right|_V
           \left.\frac{\partial S}{\partial T}\right|_V=
           T\left.\frac{\partial S}{\partial T}\right|_V
       \]
       \[
       \Rightarrow C_p-C_V=
       T\left.\frac{\partial S}{\partial T}\right|_p-
       T\left.\frac{\partial S}{\partial T}\right|_V=
       T\left(
       \left.\frac{\partial S}{\partial T}\right|_p-
       \left.\frac{\partial S}{\partial T}\right|_V
       \right)
       \]
       Using the equality
       \[
       dS=\left.\frac{\partial S}{\partial T}\right|_p dT
       +\left.\frac{\partial S}{\partial p}\right|_T dp
       \Rightarrow
       \left.\frac{\partial S}{\partial T}\right|_V=
       \left.\frac{\partial S}{\partial T}\right|_p+
       \left.\frac{\partial S}{\partial p}\right|_T
       \left.\frac{\partial p}{\partial T}\right|_V
       \]
       and the Maxwell relation
       \[
       \left.\frac{\partial S}{\partial p}\right|_T=
       -\left.\frac{\partial V}{\partial T}\right|_p
       \]
       and the equality
       \[
       dV=\left.\frac{\partial V}{\partial T}\right|_p dT+
       \left.\frac{\partial V}{\partial p}\right|_T dp
       \stackrel{\left.\frac{\partial}{\partial T}\right|_V}{\Rightarrow}
       0=\left.\frac{\partial V}{\partial T}\right|_p+
         \left.\frac{\partial V}{\partial p}\right|_T
	 \left.\frac{\partial p}{\partial T}\right|_V
       \Rightarrow
       \left.\frac{\partial p}{\partial T}\right|_V=
       -\frac{\left.\frac{\partial V}{\partial T}\right|_p}
       {\left.\frac{\partial V}{\partial p}\right|_T}
       \]
       we obtain:
       \[
       C_p-C_V=T\left(
       -\left.\frac{\partial S}{\partial p}\right|_T
       \left.\frac{\partial p}{\partial T}\right|_V
       \right)=T\left(
       \left.\frac{\partial V}{\partial T}\right|_p
       \left.\frac{\partial p}{\partial T}\right|_V
       \right)=T\left(
       \frac{\left.\left.\frac{\partial V}{\partial T}\right|_p\right.^2}
       {-\left.\frac{\partial V}{\partial p}\right|_T}
       \right)=T\left(\frac{V^2\alpha_V^2}{V\frac{1}{B}}\right)=
       TVB\alpha_V^2
       \]
       For a rigorously harmonic potential $C_p=C_V$.
\end{enumerate}

\end{document}
