solid_state_physics/tutorial/2_01s.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 1 - proposed solutions}
  40 \end{center}
  41
  42 \section{Diamagnetism}
  43 \begin{itemize}
  44  \item Magnetic field ${\bf B}$
  45  \item Magnetization ${\bf M}$
  46  \item Suscebtibility $\chi=\frac{\mu_0 {\bf M}}{{\bf B}}$
  47 \end{itemize}
  48
  49 \begin{enumerate}
  50  \item {\bf Classical approach:}
  51        \begin{enumerate}
  52         \item Maxwell: $\oint_{\partial A} E \, ds
  53                         = -\frac{d}{dt}(\int_A B \, dA)
  54                         \stackrel{B(r)=B}{=}-\frac{d}{dt}(BA)$\\
  55               $-\frac{d(BA)}{dt}=-\pi r^2 \dot{B}=U_{ind}$\\
  56               $U_{ind}=\oint_{\partial A} E \, ds
  57                \stackrel{E(s)=E}{\Rightarrow}
  58                E=-\frac{\pi r^2}{2\pi r}\dot{B}$\\
  59               $\dot{v}=a=\frac{e}{m}E=-\frac{e}{2m}r\dot{B}
  60                \Rightarrow v=-\frac{e}{2m}rB$\\
  61               $\omega_L=\frac{v}{r}=-\frac{e}{2m}B$
  62         \item $I = (\textrm{charge}) \cdot (\textrm{loops per time})
  63                \stackrel{1/T=\omega_L/2\pi}{=}
  64                (Ze)(\frac{1}{2\pi}\frac{-e}{2m}B)$\\
  65               $\mu=IA=I2\pi<\rho^2>=-\frac{Ze^2B}{4m}<\rho^2>$\\
  66               $<x^2>=<y^2>=<z^2> \Rightarrow <r^2>=3<x^2>=3<y^2>$\\
  67               $<\rho^2>=<x^2>+<y^2>=\frac{2}{3}<r^2>$\\
  68               $\mu=-\frac{Ze^2B}{6m}$
  69         \item $\chi=\frac{\mu_0N\mu}{B}=-\frac{\mu_0NZe^2}{6m}<r^2>$
  70        \end{enumerate}
  71  \item {\bf Quantum mechanical theory:}
  72        \begin{itemize}
  73         \item vector potential ${\bf A}$
  74         \item ${\bf B}=\nabla\times{\bf A}$
  75         \item $
  76               H_{kin}=\frac{1}{2m}(-i\hbar\nabla_{r}-e{\bf A})^2
  77               =H_{kin}^0 + H_{kin}'
  78               $
  79        \end{itemize}
  80        \begin{enumerate}
  81         \item \begin{eqnarray}
  82               H_{kin}&=&\frac{1}{2m}(-\hbar^2\nabla_{r}^2+e^2{\bf A}^2
  83                                      +i\hbar \nabla_{r}e{\bf A}
  84                                      +e{\bf A}i\hbar \nabla_{r})\nonumber\\
  85               H_{kin}^0&=&\frac{-\hbar^2}{2m}\nabla_r^2\nonumber\\
  86               H_{kin}'&=&\frac{i\hbar e}{2m}(\nabla_r{\bf A}+{\bf A}\nabla_r)+
  87                          \frac{e^2{\bf A}^2}{2m}\nonumber
  88               \end{eqnarray}
  89               Terms in $H_{kin}'$ can be treated as small perturbation.
  90         \item ${\bf A}=\left(-\frac{1}{2}yB,\frac{1}{2}xB,0\right)$, since:
  91               $\nabla_r\times{\bf A}=\left(0,0,\frac{1}{2}B+\frac{1}{2}B\right)=
  92                \left(0,0,B\right)$\\
  93               Note: $\nabla_r{\bf A}=0$
  94         \item $
  95               H_{kin}'=\frac{i\hbar e}{2m}\frac{B}{2}\left(
  96                x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}
  97                                           \right)+\frac{e^2B^2}{8m}(x^2+y^2)
  98               $\\
  99               $
 100                L_z=x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}
 101                \Rightarrow  H_{kin}'=\frac{i\hbar e}{2m}\frac{B}{2}L_z+
 102                \frac{e^2B^2}{8m}(x^2+y^2)
 103               $
 104         \item $\chi=-\frac{1}{V}\mu_0\frac{\partial^2 E}{\partial B^2}
 105                \Rightarrow$
 106               only second term contributes to $\chi$!\\
 107               $\chi=-\frac{1}{V}\mu_0\frac{e^2}{4m}<\phi|(x^2+y^2)|\phi>$
 108         \item $<\phi|x^2|\phi>=<\phi|y^2|\phi>=\frac{1}{3}<\phi|r^2|\phi>$\\
 109               $\Rightarrow \chi=-\frac{1}{V}\mu_0\frac{e^2}{6m}
 110                <\phi|r^2|\phi>$\\
 111               Consider all $Z$ electrons and all atoms per volume:\\
 112               $\chi=-\frac{\mu_0NZe^2}{6m}<\phi|r^2|\phi>$
 113        \end{enumerate}
 114 \end{enumerate}
 115
 116 \end{document}