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34 {\LARGE {\bf Materials Physics II}\\}
39 {\Large\bf Tutorial 1}
42 \section{Diamagnetism}
43 There is a linear relationship of the magnetic field ${\bf B}$ and
44 the magnetization ${\bf M}$ of some material.
45 The factor of proportionality is called the magnetic suscebtibility $\chi$.
47 \chi=\frac{\mu_0 {\bf M}}{{\bf B}}
49 For negative values of $\chi$ the induced magnetization aligns opposite
50 to the applied magnetic field.
51 This behaviour is called diamagnetism.
53 Develop an expression for the diamagnetic contribution to $\chi$ for some
57 \item {\bf Classical approach:}\\
58 Consider the outer electrons of an atom or ion orbiting
59 the core with a radius $r$.
60 Apply a magnetic field $B$ perpendicular to the orbit plane.
61 According to Lenz's law the induced current creates a magnetic
62 field that tends to keep the magnetic flux unchanged.
64 \item Calculate the induced voltage $U$ due to the change in flux.
65 What is the related electric field $E$ along the orbit track?
66 Calculate the corresponding change of the electron velocity
67 due to the change of the magnetic field.
68 What is the resulting angular frequency $\omega_L$
69 (Larmor frequency, named after Joseph Larmor)?
70 \item Determine the magnetic momentum $\mu$ caused by the
71 Larmor precession of $Z$ electrons which have a mean square
72 distance $<r^2>$ to the core.
74 The magnetic momentum of a current loop is the product of
75 the current and the area of the loop.
76 The average square of the loop radius $<\rho^2>$ is the average
77 square distance of the electrons perpendicular to the direction
78 of the applied magnetic field ($<\rho^2>=<x^2>+<y^2>$).
79 The average square distance of the electrons to the core is
80 $<r^2>=<x^2>+<y^2>+<z^2>$.
81 Assuming a spherically symmetric charge distribution
82 the equality $<x^2>=<y^2>=<z^2>$ holds.
83 \item Write down the magnetic suscebtibility $\chi$.
84 {\bf Hint:} By definition the magnetization is given by $N\mu$,
85 where $N$ is the amount of atoms per unit volume.
87 \item {\bf Quantum mechanical theory:}\\
88 In the presence of a magnetic field ${\bf B}=\nabla\times{\bf A}$
89 the kinetic part of the Hamiltonian is extended to read
91 H_{kin}=\frac{1}{2m}(-i\hbar\nabla_{r}-e{\bf A})^2
94 where ${\bf A}$ is the vector potential and $H_{kin}^0$ is
95 the kinetic part of the Hamiltonian without apllied magnetic field.
97 \item Write down the additional terms $H_{kin}'$ of the kinetic part
99 \item Choose a reasonable vector potential ${\bf A}$ to get a constant
100 magnetic field ${\bf B}$ in $z$-direction.
101 \item Rewrite the Hamiltonian
102 using the definition of the angular momentum operator
103 $L_z=x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}$.
104 \item Calculate the magnetic suscebtibility in a state $\phi$.
105 What term is responsible for the diamagnetic contribution?
106 {\bf Hint:} The magnetic suscebtibility is defined as
107 $\chi=-\frac{1}{V}\mu_0\frac{\partial^2 E}{\partial B^2}$.
108 \item Assuming a spherically symmetric charge distribution the equality
109 $<\phi|x^2|\phi>=<\phi|y^2|\phi>=\frac{1}{3}<\phi|r^2|\phi>$
110 is valid. Rewrite the diamagnetic part of the suscebtibility
111 and compare the result to the one obtained
112 by the classical approach.