\documentclass[11pt,fleqn]{article}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% template that does not use Revtex4
%%% but allows special fonts
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
%%% Please use this template.
%%% Edit it using e.g. Notepad
%%% Ignore the header (do not change it)
%%% Process the file in the Latex site 
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


% Page setup
\topmargin -1.5cm      
\oddsidemargin -0.04cm   
\evensidemargin -0.04cm  
\textwidth 16.59cm
\textheight 24cm 
\setlength{\parindent}{0cm} 
\setlength{\parskip}{0cm} 


% Fonts
\usepackage{latexsym}
\usepackage{amsmath} 
\usepackage{amssymb} 
\usepackage{bm}
\usepackage{graphicx}
\usepackage{float}


% Math symbols I
\newcommand{\sinc}{\mbox{sinc}}
\newcommand{\const}{\mbox{const}}
\newcommand{\trc}{\mbox{trace}}
\newcommand{\intt}{\int\!\!\!\!\int }
\newcommand{\ointt}{\int\!\!\!\!\int\!\!\!\!\!\circ\ }
\newcommand{\ar}{\mathsf r}
\newcommand{\im}{\mbox{Im}}
\newcommand{\re}{\mbox{Re}}

% Math symbols II
\newcommand{\eexp}{\mbox{e}^}
\newcommand{\bra}{\left\langle}
\newcommand{\ket}{\right\rangle}

% Mass symbol
\newcommand{\mass}{\mathsf{m}} 
\newcommand{\Mass}{\mathsf{M}} 

% More math commands
\newcommand{\tbox}[1]{\mbox{\tiny #1}}
\newcommand{\bmsf}[1]{\bm{\mathsf{#1}}} 
\newcommand{\amatrix}[1]{\begin{matrix} #1 \end{matrix}} 
\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}}

% Other commands
\newcommand{\hide}[1]{}
\newcommand{\drawline}{\begin{picture}(500,1)\line(1,0){500}\end{picture}}
\newcommand{\bitem}{$\bullet$ \ \ \ }
\newcommand{\Cn}[1]{\begin{center} #1 \end{center}}
\newcommand{\mpg}[2][1.0\hsize]{\begin{minipage}[b]{#1}{#2}\end{minipage}}
\newcommand{\Dn}{\vspace*{3mm}}

% Figures
\newcommand{\putgraph}[2][0.30\hsize]{\includegraphics[width=#1]{#2}}


% heading
\newcommand{\exnumber}[1]{\newcommand{\exnum}{#1}}
\newcommand{\heading}[1]{\begin{center} {\Large {\bf Ex\exnum:} #1} \end{center}}
\newcommand{\auname}[1]{\begin{center} {\bf Submitted by:} #1 \end{center}}


\begin{document} 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exnumber{3070}
\heading{Magnetization of a ring with electrons} 
\auname{Rami Rahimi and Assaf Voliovich}


{\bf The problem:}

\Dn

A ring with radius $R$ is placed in a uniform magnetic field $B$. In the ring there are 5 electrons with mass $m$, charge $e$ and spin $\dfrac{1}{2}$.\\
The interaction term of the spin with the field is ${g}\dfrac{e}{2m}B{S_Z}$ where ${S_z} = {\pm} \dfrac{1}{2}$.\\
Note that two electrons can\textsc{\char13}t be at the same state.\\
Let $E$ be the ground state energy and $M(B)=-\dfrac{dE}{dB}$ be the magnetization.\\
 $B$, $R$,$m$, $e$, and $g \approx 2.0023$ are given.\\
If the electrons are spin-less, then the magnetization is a periodic function of B with period $B_0$.\\
Later we will assume that $B=l{B_0}+{\bigtriangleup B}$ where $l$ is an integer and $0<{\bigtriangleup B}<<{B_0}$.\\

\Dn

(1) Write an expression for ${B_0}$.\\
(2) Find $M(B)$ for $l=0$ and $\bigtriangleup B \longrightarrow 0$. \\
(3) Write the addition to $M(B)$  for $l=0$ and small ${\bigtriangleup B}$.\\
(4) Find $M(B)$ for large $l$ and $\bigtriangleup B \longrightarrow 0$. \\
(5) Write the addition to $M(B)$  for large $l$ and small ${\bigtriangleup B}$.\\
(6) Define large $l$.\\
\\
Guidance:\\
in sections 2 and 4 it is recommended to draw an occupancy diagram of the energy levels.\\

\Dn\Dn


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf The solution:}

\Dn

(1) 
The Hamiltonian of a spin-less electron in a ring of radius $ R $ inside a homogeneous magnetic field $ B $ is:
\[ H \ = \ \dfrac{1}{2m}{\bigg(}P- \dfrac{e\Phi}{L}{\bigg)}^2 \]
\[ where : \Phi=\pi{R}^{2}B, \qquad L=2\pi{R}\]
The eigen-energies are:
\[ E_{n} \ = \ \dfrac{1}{2m{R^2}}{\bigg(}n- \dfrac{e{R^2}B}{2}{\bigg)}^2 \]
The ground state energy of five spin-less electrons $E$ can be described in the the following graph:\\
\\
\\
\\
\\
\\
\\
\\
\\
\\
\begin{figure}[h!]
\centering
  \includegraphics[width=100mm]{images/ee.jpg}
  \caption{the ground state energy of five spin-less electrons}
  \label{fig:ground energy}
\end{figure}
\\
Each colored route describes the energy of one of the electrons as a function of $B$. For a given $B$ the energy of the ground state is the sum of the energies of the five electrons for that specific $B$.\\
From the graph one can see that the energy of each  electron is periodic and the following equation exists:
\[E_{i}{(n,B)} = E_{i}{(n+1,B+B_{0})} \qquad \Longrightarrow \qquad \dfrac{1}{2m{R^2}}{\bigg(}n- \dfrac{e{R^2}B}{2}{\bigg)}^2 = \dfrac{1}{2m{R^2}}{\bigg(}n+1- \dfrac{e{R^2}}{2}{(B+B_{0})}{\bigg)}^2   \]
Hence:\\
\[B_{0} = \dfrac{2}{e{R^2}}\]\\
Accordingly, $M(B)$ which is received by the taking the derivative of the sum of those energies is also periodic with the same period $B_{0}$.\\

%%%%%%%%%%%%%%%
\Dn
(2)
If the electrons now have a spin, the $Spin-Zeeman$ term is added to the Hamiltonian and now the Hamiltonian is:\\
\[ H \ = \ \dfrac{1}{2m}{\bigg(}P- \dfrac{e\Phi}{L}{\bigg)}^2 +\dfrac{ge}{2m}B{S_Z} \]
Accordingly, the eigen-energies are:\\
\[ E_{n\pm} \ = \ \dfrac{1}{2m{R^2}}{\bigg(}n- \dfrac{e{R^2}B}{2}{\bigg)}^2 \pm \dfrac{ge}{4m}B  \] \\
For small positive $B=\bigtriangleup B$ the electrons will be arranged in the following form:\\
\\
\\
\\
\\
\\
\\
\begin{figure}[h!]
\centering
  \includegraphics[width=100mm]{images/2.jpg}
  \caption{the arrangement in energy states of five electrons for small positive $B=\bigtriangleup B$ }
  \label{fig:five electrons with spin and small B}
\end{figure}
\\

Therefore the ground state energy is:
\[ E=\dfrac{5e}{4m{B_0}}{\bigtriangleup B}^2 - \dfrac{(2+g)e}{4m}{\bigtriangleup B} + \dfrac{3}{2m{R^2}}\] \\
And:
\[ M(B)=-\dfrac{dE}{dB}=-\dfrac{5e}{2m{B_0}}{\bigtriangleup B} + \dfrac{(2+g)e}{4m}\]
For $\bigtriangleup B \longrightarrow 0$: \qquad  $M(B) \longrightarrow \dfrac{(2+g)e}{4m}$

%%%%%%%%%%%%%%
\Dn

(3)
The addition to $M(B)$ for small $\bigtriangleup B$ is:
\[-\dfrac{5e}{2m{B_0}}{\bigtriangleup B}\]

%%%%%%%%%%%%%
\Dn

(4)
For large $B$ (large $l$) the main contribution to the eigen-energies is due to the $Spin-Zeeman$ term. Hence, all the 5 electrons will have spin down $ \left|\downarrow\right\rangle$ and will arrange (for small positive $\bigtriangleup B$)  in the following form:\\
\begin{figure}[H]
\centering
  \includegraphics[width=100mm]{images/4.jpg}
  \caption{the arrangement in energy states of five electrons for large $B$}
  \label{fig:five electrons with spin and large B}
\end{figure}
 $M(B)$ in this case is similar to the case of spin-less electrons with the addition of a constant term. Hence  $M(B)$ is independent of $l$ and W.L.O.G we can choose $l=0$ and then the ground state energy is:
\[ E \ = \ \dfrac{1}{2m{R^2}}{\bigg(}5{{\Big(}\dfrac{\bigtriangleup B}{B_0}{\Big)}^2}+10 {\bigg)} - \dfrac{5g}{2m{R^2}} \dfrac{\bigtriangleup B}{B_0} \]
And:
\[ M(B)=-\dfrac{dE}{dB}=-\dfrac{5 {\bigtriangleup}B}{m{R^2}{{B_0}}^2} + \dfrac{5g}{2m{B_0}{R^2}}\]
So for $\bigtriangleup B \longrightarrow 0$: \qquad  $M(B) \longrightarrow  \dfrac{5g}{2m{B_0}{R^2}}$


%%%%%%%%%%%%%
\Dn

(5)
The addition to $M(B)$ for small $\bigtriangleup B$ is:
\[-\dfrac{5 {\bigtriangleup}B}{m{R^2}{{B_0}}^2}\]



%%%%%%%%%%%%%
\Dn

(6)
Large $l$ is when the electron "prefers" to be in the fifth energy state with spin down $ \left|\downarrow\right\rangle$ than in the lowest energy state with spin up $ \left|\uparrow\right\rangle$.\\ 
For $\bigtriangleup B \longrightarrow 0$, $B\, \widetilde{=}\, l{B_0}$ and $l=n$:
\[  \dfrac{1}{2m{R^2}}{\bigg(}l- \dfrac{l{B_0}}{B_0}{\bigg)}^2 + \dfrac{ge}{4m}l{B_0} > \dfrac{1}{2m{R^2}}{\bigg(}l-2- \dfrac{l{B_0}}{B_0}{\bigg)}^2 - \dfrac{ge}{4m}l{B_0}  \] \\
So:
\[ l > \dfrac{2}{g}\]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document} 
