mp2 started, tutorial 1 + modified Makefile

diff --git a/solid_state_physics/tutorial/2_01.tex b/solid_state_physics/tutorial/2_01.tex
new file mode 100644 (file)
index 0000000..2aa6a6d
--- /dev/null
+++ b/solid_state_physics/tutorial/2_01.tex
@@ -0,0 +1,116 @@
+\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}

diff --git a/solid_state_physics/tutorial/Makefile b/solid_state_physics/tutorial/Makefile
index 794214d..8b4e666 100644 (file)
--- a/solid_state_physics/tutorial/Makefile
+++ b/solid_state_physics/tutorial/Makefile
@@ -2,7 +2,7 @@
 LATEX = latex
 DVIPDF = dvipdf
 
-SRC := $(shell ls 1_0*.tex)
+SRC := $(shell ls [12]_0*.tex)
 PDF = $(SRC:%.tex=%.pdf)
 
 all: $(PDF)