From fff954979dc1aa60eec2d373637dda7a1e64db70 Mon Sep 17 00:00:00 2001 From: hackbard Date: Mon, 21 Apr 2008 18:04:34 +0200 Subject: [PATCH] mp2 started, tutorial 1 + modified Makefile --- solid_state_physics/tutorial/2_01.tex | 116 ++++++++++++++++++++++++++ solid_state_physics/tutorial/Makefile | 2 +- 2 files changed, 117 insertions(+), 1 deletion(-) create mode 100644 solid_state_physics/tutorial/2_01.tex diff --git a/solid_state_physics/tutorial/2_01.tex b/solid_state_physics/tutorial/2_01.tex new file mode 100644 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 $$ 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 ($<\rho^2>=+$). + The average square distance of the electrons to the core is + $=++$. + Assuming a spherically symmetric charge distribution + the equality $==$ 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 --- 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) -- 2.20.1