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 \[
 155        dF=-pdV-SdT \Rightarrow p=-\left.\frac{\partial}{\partial V}\right|T
 156        \]
 157        \[
 158        dE=TdS-pdV \Rightarrow
 159        \]
 160        Find an expression for the pressure as a function of the free energy
 161        $F=E-TS$.
 162        Rewrite this equation to express the pressure entirely in terms of
 163        the internal energy $E$.
 164        Evaluate the pressure by using the harmonic form of the internal energy.
 165        {\bf Hint:}
 166        Step 2 introduced an integral over the temperature $T'$.
 167        Change the integration variable $T'$ to $x=\hbar\omega_s({\bf k})/T'$.
 168        Use integration by parts with respect to $x$.
 169  \item The normal mode frequencies of a rigorously harmonic crystal
 170        are unaffected by a change in volume.
 171        What does this imply for the pressure
 172        (Which variables does the pressure depend on)?
 173        Draw conclusions for the coefficient of thermal expansion.
 174  \item Find an expression for $C_p-C_V$ in terms of temperature $T$,
 175        volume $V$, the coefficient of thermal expansion $\alpha_V$ and
 176        the inverse bulk modulus (isothermal compressibility)
 177        $\frac{1}{B}=-\frac{1}{V}\left.\frac{\partial V}{\partial p}\right|_T$.\\
 178        $C_p=\left.\frac{\partial E}{\partial T}\right|_p$ is the heat capacity
 179        for constant pressure and
 180        $C_V=\left.\frac{\partial E}{\partial T}\right|_V$ is the heat capacity
 181        for constant volume.
 182 \end{enumerate}
 183
 184 \end{document}