solid_state_physics/tutorial/1_02s.tex

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  28 \begin{document}
  29
  30 % header
  31 \begin{center}
  32  {\LARGE {\bf Materials Physics I}\\}
  33  \vspace{8pt}
  34  Prof. B. Stritzker\\
  35  WS 2007/08\\
  36  \vspace{8pt}
  37  {\Large\bf Tutorial 2}
  38 \end{center}
  39
  40 \section{Phonons 1}
  41 \begin{enumerate}
  42  \item \begin{itemize}
  43         \item $r_i=r_{i0}+u_i$\\
  44               $\rho=r_2-r_1=r_{20}+u_2-r_{10}-u_1=(r_{20}-r_{10})+(u_2-u_1)
  45                    =\rho_0+\sigma$
  46         \item $\Phi-\Phi_0=\frac{D}{2}(\rho-\rho_0)^2
  47                           =\frac{D}{2}(\rho^2+\rho_0^2-2\rho_0\rho)$\\
  48               $\rho^2=\rho_0^2+\sigma^2+2\rho_0\sigma$
  49               $\Rightarrow$ $\rho=\sqrt{\rho_0^2+\sigma^2+2\rho_0\sigma}$\\
  50               $\Rightarrow$ $\Phi-\Phi_0=\frac{D}{2}
  51                              [2\rho_0^2+\sigma^2+2\rho_0\sigma-
  52                               2\rho_0\sqrt{\rho_0^2+\sigma^2+2\rho_0\sigma}]$
  53        \end{itemize}
  54  \item $\sigma \parallel \rho_0$:
  55        \begin{enumerate}
  56         \item \begin{flushleft}
  57                \includegraphics[height=6cm]{elongation_p01.eps}
  58                \includegraphics[height=6cm]{elongation_p02.eps}
  59                \includegraphics[height=6cm]{elongation_p03.eps}
  60               \end{flushleft}
  61         \item $\sigma = \sigma_{\parallel}$:\\
  62               $\rho_0 \sigma_{\parallel} = |\rho_0| |\sigma_{\parallel}|$\\
  63               $\Phi-\Phi_0=\frac{D}{2}\left(2\rho_0^2+\sigma_{\parallel}^2+
  64                            2\rho_0\sigma_{\parallel}-
  65                            2\rho_0\sqrt{(\rho_0+\sigma_{\parallel})^2}\right)
  66                           =\frac{D}{2}\sigma_{\parallel}^2$
  67        \end{enumerate}
  68  \item $\sigma \perp \sigma_0$:
  69        \begin{enumerate}
  70         \item \begin{flushleft}
  71                \includegraphics[height=5.3cm]{elongation_n01.eps}
  72                \includegraphics[height=5.3cm]{elongation_n02.eps}
  73                \includegraphics[height=5.3cm]{elongation_n03.eps}
  74               \end{flushleft}
  75         \item $\sigma=\sigma_{\perp}$:\\
  76               $\sigma_{\perp} \rho_0 = 0$\\
  77               $\Phi-\Phi_0=\frac{D}{2}\left[2\rho_0^2+\sigma_{\perp}^2-
  78                            2\rho_0\sqrt{\rho_0^2+\sigma_{\perp}^2}\right]$
  79
  80         \item $\sigma_{\perp} = \alpha \rho_0$, $\alpha \ll 1$\\
  81               $\sqrt{\rho_0^2+\sigma_{\perp}^2}=
  82                \sqrt{\rho_0^2+\alpha^2\rho_0^2}=
  83                \rho_0\sqrt{1+\alpha^2}=
  84                \rho_0(1+\frac{\alpha^2}{2}-\frac{\alpha^4}{8}+\ldots)$\\
  85               $\Rightarrow \Phi-\Phi_0=
  86                \frac{D}{2}\left[\rho_0^2\left(2+\alpha^2-
  87                2(1+\frac{\alpha^2}{2}-\frac{\alpha^4}{8}+\ldots)\right)\right]=
  88                \frac{D}{2}\left[\rho_0^2(\frac{\alpha^4}{4}+\ldots)\right]$\\
  89               $\Rightarrow \Phi-\Phi_0\stackrel{\alpha\ll 1}{=}
  90                \frac{D}{2}\rho_0^2\frac{\alpha^4}{4}=
  91                \frac{D}{2}\sigma_{\perp}^2\frac{\alpha^2}{4}$
  92         \item $\sigma_{\parallel}$, $\sigma_{\perp} \ll \rho_0$\\
  93               $\Rightarrow$ potential contribution of $\sigma_{\perp}$
  94               compared to contribution of $\sigma_{\parallel}$
  95               negligible small.
  96        \end{enumerate}
  97  \item \begin{itemize}
  98         \item As long as the displacements and thus the elongation is small
  99               compared to the equilibrium state the change in the potential
 100               due to the perpendicular elongation is negligible small.
 101         \item Regarding a possible existence of perpendicular elongation
 102               the model of the linear chain is unproblematic.
 103         \item In a real crystal couplings in other directions exist.
 104               These can only be neglected if they are small compared to the
 105               coupling of the considered direction.
 106        \end{itemize}
 107 \end{enumerate}
 108
 109 \section{Phonons 2}
 110 \begin{enumerate}
 111 \item Derive the dispersion relation for a linear chain with two different
 112       alternating types of atoms.
 113 \item Discuss the two solutions for $\omega^2$.
 114 \end{enumerate}
 115
 116 \end{document}