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29
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               $\mu=IA=I\pi<\rho^2>=-\frac{Ze^2B}{4m}<\rho^2>$\\
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}