From: hackbard Date: Thu, 6 Dec 2007 11:37:55 +0000 (+0100) Subject: fixes + tutorial 4 X-Git-Url: https://hackdaworld.org/gitweb/?p=lectures%2Flatex.git;a=commitdiff_plain;h=79c30f13ba31a3d5988133db6e6992235b64ca8a fixes + tutorial 4 --- diff --git a/solid_state_physics/tutorial/1_02s.tex b/solid_state_physics/tutorial/1_02s.tex index 2827bb7..6f920e8 100644 --- a/solid_state_physics/tutorial/1_02s.tex +++ b/solid_state_physics/tutorial/1_02s.tex @@ -34,7 +34,7 @@ Prof. B. Stritzker\\ WS 2007/08\\ \vspace{8pt} - {\Large\bf Tutorial 2} + {\Large\bf Tutorial 2 - proposed solutions} \end{center} \section{Phonons 1} @@ -117,11 +117,15 @@ $M_1\ddot{u}_s=C(v_s+v_{s-1}-2u_s)$\\ $M_2\ddot{v}_s=C(u_{s+1}+u_s-2v_s)$ \item Ansatz:\\ - $u_s=u\exp{i(ska-\omega t)}$\\ - $v_s=v\exp{i(ska-\omega t)}$ + $u_s=u\exp(i(ska-\omega t))$\\ + $v_s=v\exp(i(ska-\omega t))$ \item Solution of the equation system:\\ - $-\omega^2M_1u=Cv[1+\exp(-ika)]-2Cu$\\ - $-\omega^2M_2v=Cu[\exp(ika)+1]-2Cv$\\ + $-\omega^2M_1u\exp(i(ska-\omega t))= + C\exp(-i\omega t)[v\exp(iska)+v\exp(i(s-1)ka)-2u\exp(iska)]$\\ + $\Rightarrow -\omega^2M_1u=Cv(1+\exp(-ika))-2Cu$\\ + $-\omega^2M_2v\exp(i(ska-\omega t))= + C\exp(-i\omega t)[u\exp(i(s+1)ka)+u\exp(iska)-2v\exp(iska)]$\\ + $\Rightarrow -\omega^2M_2v=Cu[\exp(ika)+1]-2Cv$\\ Non trivial solution only if determinant of coefficients $u$ and $v$ is zero.\\ $\Rightarrow @@ -131,6 +135,11 @@ -C[1+\exp(ika)] & 2C-M_2\omega^2 \end{array} \right|=0$\\ + $\Rightarrow + 4C^2+M_1M_2\omega^4-2C\omega^2(M_2+M_1)- + \underbrace{C^2(1+\exp(ika))(1+\exp(-ika))}_{ + C^2(\underbrace{1+1+\exp(ika)+\exp(-ika)}_{ + 2+2\cos(ka)=2(1+\cos(ka))})}$\\ $\Rightarrow M_1M_2\omega^4-2C(M_1+M_2)\omega^2+2C^2(1-\cos(ka))=0$ \end{itemize} diff --git a/solid_state_physics/tutorial/1_03s.tex b/solid_state_physics/tutorial/1_03s.tex index 777f575..ac7a51f 100644 --- a/solid_state_physics/tutorial/1_03s.tex +++ b/solid_state_physics/tutorial/1_03s.tex @@ -34,7 +34,7 @@ Prof. B. Stritzker\\ WS 2007/08\\ \vspace{8pt} - {\Large\bf Tutorial 2 - proposed solutions} + {\Large\bf Tutorial 3 - proposed solutions} \end{center} \section{Drude theory of metallic conduction} diff --git a/solid_state_physics/tutorial/1_04.tex b/solid_state_physics/tutorial/1_04.tex new file mode 100644 index 0000000..0fc68a6 --- /dev/null +++ b/solid_state_physics/tutorial/1_04.tex @@ -0,0 +1,99 @@ +\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})} + +\begin{document} + +% header +\begin{center} + {\LARGE {\bf Materials Physics I}\\} + \vspace{8pt} + Prof. B. Stritzker\\ + WS 2007/08\\ + \vspace{8pt} + {\Large\bf Tutorial 4} +\end{center} + +\section{Hall effect and magnetoresistance} +The Hall effect refers to the potential difference (Hall voltage) +on the opposite sides of an electrical conductor +through which an electric current is flowing, +created by a magnetic field applied perpendicular to the current. +Edwin Hall discovered this effect in 1879. + +Consider the following scenario: +An electric field $E_x$ is applied to a wire extending in $x$-direction +and a current density $j_x$ is flowing in that wire. +There is a magnetic field $B$ pointing in the positive $z$-direction. +Electrons are deflected in the negative $y$-direction +due to the Lorentz force $F_L=-evB$ +until they run against the sides of the wire. +An electric field $E_y$ builds up opposing the Lorentz force +and thus preventing further electron accumulation at the sides. +The two quantities of interest are: +\begin{itemize} + \item the magnetoresistance + \[ + \rho(B) = \frac{E_x}{j_x} \textrm{ and} + \] + \item the Hall coefficient + \[ + R_H(B) = \frac{E_y}{j_xB} \textrm{ .} + \] +\end{itemize} +In this tutorial the treatment of the Hall problem is based on a simple +Drude model analysis. +\\\\ +First of all the effect of individual electron collisions can be expressed +by a frictional damping term into the equation of motion for the momentum +per electron. + +\begin{enumerate} + \item Recall the Drude model. + Given the momentum per electron $p(t)$ at time t + calculate the momentum per electron $p(t+dt)$ + an infinitesimal time $dt$ later. + {\bf Hint:} What is the probability of an electron taken at random at + time $t$ to not suffer a collision before time $t+dt$? + If not experiencing a collision it simply evolves under the influence + of the force $f(t)$. + Combine contributions of the order of $(dt)^2$ to the term + $O(dt)^2$. + \item Write down the equation of motion for the momentum per electron + by dividing the above result by $dt$ + and taking the limit $dt\rightarrow 0$. + \item Sketch a schematic view of Hall's experiment. + \item Find an expression for the Hall coefficient. + {\bf Hint:} Insert an appropriate force into the equation of motion + for the momentum per electron. + Consider the steady state and acquire the equations + for the $x$ and $y$ component of the vector equation. + To find an expression for the Hall coefficient use the second equation + and the fact that there must not be transverse current $j_y$ + while determining the Hall field. +\end{enumerate} + +\end{document} diff --git a/solid_state_physics/tutorial/1_04s.tex b/solid_state_physics/tutorial/1_04s.tex new file mode 100644 index 0000000..b5d60e3 --- /dev/null +++ b/solid_state_physics/tutorial/1_04s.tex @@ -0,0 +1,66 @@ +\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})} + +\begin{document} + +% header +\begin{center} + {\LARGE {\bf Materials Physics I}\\} + \vspace{8pt} + Prof. B. Stritzker\\ + WS 2007/08\\ + \vspace{8pt} + {\Large\bf Tutorial 4 - proposed solutions} +\end{center} + +\section{Hall effect and magnetoresistance} +\begin{enumerate} + \item \begin{itemize} + \item probability: $1-\frac{dt}{\tau}$ + \item momentum contribution of non-colliding electrons: + $f(t)dt+O(dt)^2$ + \item momentum per electron at time $t+dt$:\\ + \[ + p(t+dt)=\left(1-\frac{dt}{\tau}\right) + \left[p(t)+f(t)dt+O(dt)^2\right] + =p(t)-\frac{dt}{\tau}p(t)+f(t)+O(dt)^2 + \] + \end{itemize} + \item \[ + p(t+dt)-p(t)=-\frac{dt}{\tau}p(t)+f(t)dt+O(dt)^2 + \] + \[ + \frac{p(t+dt)-p(t)}{dt}=-\frac{p(t)}{\tau}+f(t)+\frac{O(dt)^2}{dt} + \] + \[ + \stackrel{dt\rightarrow 0}{\Rightarrow} \quad + \frac{dp(t)}{dt}=-\frac{p(t)}{\tau}+f(t) + \] + \item +\end{enumerate} + +\end{document}