\pdfoutput=0
\documentclass[a4paper,11pt]{article}
\usepackage[activate]{pdfcprot}
\usepackage{verbatim}
\usepackage{a4}
\usepackage{a4wide}
\usepackage[german]{babel}
\usepackage[latin1]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{amsmath}
\usepackage{ae}
\usepackage{aecompl}
\usepackage[dvips]{graphicx}
\graphicspath{{./img/}}
\usepackage{color}
\usepackage{pstricks}
\usepackage{pst-node}
\usepackage{rotating}

\setlength{\headheight}{0mm} \setlength{\headsep}{0mm}
\setlength{\topskip}{-10mm} \setlength{\textwidth}{17cm}
\setlength{\oddsidemargin}{-10mm}
\setlength{\evensidemargin}{-10mm} \setlength{\topmargin}{-1cm}
\setlength{\textheight}{26cm} \setlength{\headsep}{0cm}

\renewcommand{\labelenumi}{(\alph{enumi})}
\renewcommand{\labelenumii}{\arabic{enumii})}
\renewcommand{\labelenumiii}{\roman{enumiii})}

\begin{document}

% header
\begin{center}
 {\LARGE {\bf Materials Physics II}\\}
 \vspace{8pt}
 Prof. B. Stritzker\\
 SS 2008\\
 \vspace{8pt}
 {\Large\bf Tutorial 1}
\end{center}

\section{Diamagnetism}
There is a linear relationship of the magnetic field ${\bf B}$ and
the magnetization ${\bf M}$ of some material.
The factor of proportionality is called the magnetic suscebtibility $\chi$.
\[
 \chi=\frac{\mu_0 {\bf M}}{{\bf B}}
\]
For negative values of $\chi$ the induced magnetization aligns opposite
to the applied magnetic field.
This behaviour is called diamagnetism.
\\\\
Develop an expression for the diamagnetic contribution to $\chi$ for some
atom or ion.

\begin{enumerate}
 \item {\bf Classical approach:}\\
       Consider the outer electrons of an atom or ion orbiting
       the core with a radius $r$.
       Apply a magnetic field $B$ perpendicular to the orbit plane.
       According to Lenz's law the induced current creates a magnetic
       field that tends to keep the magnetic flux unchanged.
       \begin{enumerate}
        \item Calculate the induced voltage $U$ due to the change in flux.
	      What is the related electric field $E$ along the orbit track?
	      Calculate the corresponding change of the electron velocity
	      due to the change of the magnetic field.
	      What is the resulting angular frequency $\omega_L$
	      (Larmor frequency, named after Joseph Larmor)?
        \item Determine the magnetic momentum $\mu$ caused by the
	      Larmor precession of $Z$ electrons which have a mean square
	      distance $<r^2>$ to the core.
	      {\bf Hint:}
	      The magnetic momentum of a current loop is the product of
	      the current and the area of the loop.
	      The average square of the loop radius $<\rho^2>$ is the average
	      square distance of the electrons perpendicular to the direction
	      of the applied magnetic field ($<\rho^2>=<x^2>+<y^2>$).
	      The average square distance of the electrons to the core is
	      $<r^2>=<x^2>+<y^2>+<z^2>$.
	      Assuming a spherically symmetric charge distribution
	      the equality $<x^2>=<y^2>=<z^2>$ holds.
        \item Write down the magnetic suscebtibility $\chi$.
	      {\bf Hint:} By definition the magnetization is given by $N\mu$,
	      where $N$ is the amount of atoms per unit volume.
       \end{enumerate}
 \item {\bf Quantum mechanical theory:}\\
       In the presence of a magnetic field ${\bf B}=\nabla\times{\bf A}$
       the kinetic part of the Hamiltonian is extended to read
       \[
       H_{kin}=\frac{1}{2m}(-i\hbar\nabla_{r}-e{\bf A})^2
              =H_{kin}^0 + H_{kin}'
       \]
       where ${\bf A}$ is the vector potential and $H_{kin}^0$ is
       the kinetic part of the Hamiltonian without apllied magnetic field.
       \begin{enumerate}
        \item Write down the additional terms $H_{kin}'$ of the kinetic part
	      of the Hamiltonian.
        \item Chose a reasonable vector potential ${\bf A}$ to get a constant
	      magnetic field ${\bf B}$ in $z$-direction.
        \item Rewrite the Hamiltonian 
	      using the definition of the angular momentum operator
	      $L_z=x\frac{\partial}{\partial y}-y\frac{\partial}{\partial x}$.
        \item Calculate the magnetic suscebtibility in a state $\phi$.
	      What term is responsible for the diamagnetic contribution?
	      {\bf Hint:} The magnetic suscebtibility is defined as
	      $\chi=-\frac{1}{V}\frac{\partial^2 E}{\partial B^2}$.
	\item Assuming a spherically symmetric charge distribution the equality
	      $<\phi|x^2|\phi>=<\phi|y^2|\phi>=\frac{1}{3}<\phi|r^2|\phi>$
	      is valid. Rewrite the diamagnetic part of the suscebtibility
	      and compare the result to the one obtained
	      by the classical approach.
       \end{enumerate}
\end{enumerate}

\end{document}
