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\begin{document}
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\heading{E3550: Magnetic properties of T=0 electrons (Pauli)}
\auname{Gitit Feingold}
{\bf The problem:}\\
\Dn
$N$ electrons with mass $m$ and spin $\frac{1}{2}$ are placed in a box at zero temperature. A magnetic field B is applied, such that the interaction is $-\gamma B \sigma_{z}$ where $\gamma$ is the gyromagnetic ratio.
Solve the question for:
(I) one-dimensional box with length L. \\
(II) two-dimensional box with area A.\\
(III) three dimensional box with volume V.\\
\begin {itemize}
\item[(a)]
Describe the single particle density of states function, differ spins up from spins down.
\item[(b)]
Determine which graph describes the magnetization $M(B)$ of each case $(I)$,$(II)$,$(III)$ and complete the missing details ($M_s=$?,$B_c=$?,$\chi=$?)
\end {itemize}
Use only $\gamma$,m,N,L,A,V.
\begin{figure}[h]
\begin{center}
\includegraphics[width=4in]{Ex3550}
\end{center}
\end{figure}
\Dn\Dn
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{\bf The solution:}
\begin {itemize}
\item[(a)] One can easily see that the system can occupy more spin up states.
\end {itemize}
%%%%%%%% 1D %%%%%%%%
\textbf{1D:}\\
For $B0
\end{equation}
The critical magnetic field has the same condition as in the 2D case:
\begin{equation}
\frac{\gamma B}{2}=E_f \Longrightarrow B_c=\frac{1}{m\gamma}(\frac{3\pi ^2 N(0)}{V})^{2/3}
\end{equation}
\begin {itemize}
\item[(b)]
\end {itemize}
Using the above calculations we are able to match the right graph to the magnetization functions first graph from the left represent the $1D$ case, the second one refers to the $3D$ case and the third refers to the $2D$ case.
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\end{document}