solid_state_physics/tutorial/2_04s.tex

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  30 \begin{document}
  31
  32 % header
  33 \begin{center}
  34  {\LARGE {\bf Materials Physics II}\\}
  35  \vspace{8pt}
  36  Prof. B. Stritzker\\
  37  SS 2008\\
  38  \vspace{8pt}
  39  {\Large\bf Tutorial 4 - proposed solutions}
  40 \end{center}
  41
  42 \vspace{4pt}
  43
  44 \section{Legendre transformation and Maxwell relations}
  45
  46 \begin{enumerate}
  47  \item Legendre transformation:
  48        \begin{eqnarray}
  49        dg &=& df - \sum_{i=r+1}^{n} d(u_ix_i)\nonumber\\
  50           &=& df - \sum_{i=r+1}^{n} (x_idu_i + u_idx_i)\nonumber\\
  51           &=& \sum_{i=1}^r u_idx_i - \sum_{i=r+1}^n x_idu_i\nonumber
  52        \end{eqnarray}
  53        \[
  54        \Rightarrow g=g(x_1,\ldots,x_r,u_{r+1},\ldots,u_n)
  55        \]
  56  \item Use $T=\left.\frac{\partial E}{\partial S}\right|_V$ and
  57        $-p=\left.\frac{\partial E}{\partial V}\right|_S$.\\
  58        Start with internal energy $E=E(S,V)$:
  59        \[
  60        \Rightarrow dE=\frac{\partial E}{\partial S}dS +
  61                       \frac{\partial E}{\partial V}dV =
  62                       TdS - pdV
  63        \]
  64        Enthalpy $H=E+pV$:
  65        \[
  66        \Rightarrow dH=dE+Vdp+pdV=TdS-pdV+Vdp+pdV=TdS+Vdp
  67        \]
  68        \[
  69        \Rightarrow
  70        \left.\frac{\partial H}{\partial S}\right|_p=T \textrm{ and }
  71        \left.\frac{\partial H}{\partial p}\right|_S=V
  72        \]
  73        Helmholtz free energy $F=E-TS$:
  74        \[
  75        \Rightarrow dF=dE-SdT-TdS=TdS-pdV-SdT-TdS=-pdV-SdT
  76        \]
  77        \[
  78        \Rightarrow
  79        \left.\frac{\partial F}{\partial V}\right|_T=-p \textrm{ and }
  80        \left.\frac{\partial F}{\partial T}\right|_V=-S
  81        \]
  82        Gibbs free energy $G=H-TS=E+pV-TS$:
  83        \[
  84        \Rightarrow dG=dH-SdT-TdS=TdS+Vdp-SdT-TdS=Vdp-SdT
  85        \]
  86        \[
  87        \Rightarrow
  88        \left.\frac{\partial G}{\partial p}\right|_T=V \textrm{ and }
  89        \left.\frac{\partial G}{\partial T}\right|_p=-S
  90        \]
  91  \item Maxwell relations:\\
  92        Internal energy: $dE=TdS-pdV$
  93        \[
  94        \frac{\partial}{\partial S}
  95        \left(\left.\frac{\partial E}{\partial V}\right|_S\right)_V=
  96        \frac{\partial}{\partial V}
  97        \left(\left.\frac{\partial E}{\partial S}\right|_V\right)_S
  98        \Rightarrow
  99        \left.-\frac{\partial p}{\partial S}\right|_V=
 100        \left.\frac{\partial T}{\partial V}\right|_S
 101        \]
 102        Enthalpy: $dH=TdS+Vdp$
 103        \[
 104        \frac{\partial}{\partial S}
 105        \left(\left.\frac{\partial H}{\partial p}\right|_S\right)_p=
 106        \frac{\partial}{\partial p}
 107        \left(\left.\frac{\partial H}{\partial S}\right|_p\right)_S
 108        \Rightarrow
 109        \left.\frac{\partial V}{\partial S}\right|_p=
 110        \left.\frac{\partial T}{\partial p}\right|_S
 111        \]
 112        Helmholtz free energy: $dF=-pdV-SdT$
 113        \[
 114        \frac{\partial}{\partial V}
 115        \left(\left.\frac{\partial F}{\partial T}\right|_V\right)_T=
 116        \frac{\partial}{\partial T}
 117        \left(\left.\frac{\partial F}{\partial V}\right|_T\right)_V
 118        \Rightarrow
 119        \left.-\frac{\partial S}{\partial V}\right|_T=
 120        \left.-\frac{\partial p}{\partial T}\right|_V
 121        \]
 122        Gibbs free energy: $dG=Vdp-SdT$
 123        \[
 124        \frac{\partial}{\partial p}
 125        \left(\left.\frac{\partial G}{\partial T}\right|_p\right)_T=
 126        \frac{\partial}{\partial T}
 127        \left(\left.\frac{\partial G}{\partial p}\right|_T\right)_p
 128        \Rightarrow
 129        \left.-\frac{\partial S}{\partial p}\right|_T=
 130        \left.\frac{\partial V}{\partial T}\right|_p
 131        \]
 132 \end{enumerate}
 133
 134 \section{Thermal expansion of solids}
 135
 136 \begin{enumerate}
 137  \item Coefficients of thermal expansion:\\
 138        Consider a cube with side lengthes $L_1,L_2,L_3$.
 139        Isotropic material: $\frac{1}{L_1}\frac{\partial L_1}{\partial T}=
 140                             \frac{1}{L_2}\frac{\partial L_2}{\partial T}=
 141                             \frac{1}{L_3}\frac{\partial L_3}{\partial T}=
 142                             \alpha_L$.
 143        \begin{eqnarray}
 144        \alpha_V&=&\frac{1}{V}\frac{\partial V}{\partial T}=
 145        \frac{1}{L_1L_2L_3}\frac{\partial}{\partial T}(L_1L_2L_3)=
 146        \frac{1}{L_1L_2L_3}\left(L_2L_3\frac{\partial L_1}{\partial T}+
 147                                 L_1L_3\frac{\partial L_2}{\partial T}+
 148                                 L_1L_2\frac{\partial L_3}{\partial T}\right)
 149                                 \nonumber\\
 150        &=&\frac{1}{L_1}\frac{\partial L_1}{\partial T}+
 151           \frac{1}{L_2}\frac{\partial L_2}{\partial T}+
 152           \frac{1}{L_3}\frac{\partial L_3}{\partial T}=3\alpha_L\nonumber
 153        \end{eqnarray}
 154  \item
 155        Find an expression for the pressure as a function of the free energy
 156        $F=E-TS$.
 157        Rewrite this equation to express the pressure entirely in terms of
 158        the internal energy $E$.
 159        Evaluate the pressure by using the harmonic form of the internal energy.
 160        {\bf Hint:}
 161        Step 2 introduced an integral over the temperature $T'$.
 162        Change the integration variable $T'$ to $x=\hbar\omega_s({\bf k})/T'$.
 163        Use integration by parts with respect to $x$.
 164  \item The normal mode frequencies of a rigorously harmonic crystal
 165        are unaffected by a change in volume.
 166        What does this imply for the pressure
 167        (Which variables does the pressure depend on)?
 168        Draw conclusions for the coefficient of thermal expansion.
 169  \item Find an expression for $C_p-C_V$ in terms of temperature $T$,
 170        volume $V$, the coefficient of thermal expansion $\alpha_V$ and
 171        the inverse bulk modulus (isothermal compressibility)
 172        $\frac{1}{B}=-\frac{1}{V}\left.\frac{\partial V}{\partial p}\right|_T$.\\
 173        $C_p=\left.\frac{\partial E}{\partial T}\right|_p$ is the heat capacity
 174        for constant pressure and
 175        $C_V=\left.\frac{\partial E}{\partial T}\right|_V$ is the heat capacity
 176        for constant volume.
 177 \end{enumerate}
 178
 179 \end{document}