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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 \[
       dF=-pdV-SdT \Rightarrow p=-\left.\frac{\partial}{\partial V}\right|T
       \]
       \[
       dE=TdS-pdV \Rightarrow 
       \]
       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)
       $\frac{1}{B}=-\frac{1}{V}\left.\frac{\partial V}{\partial p}\right|_T$.\\
       $C_p=\left.\frac{\partial E}{\partial T}\right|_p$ 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}

\end{document}
