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\sect{Fermions in magnetic field - Pauli}

$N$ 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.
Consider the following cases:

\begin {itemize}
\item[(a)] one-dimensional box with length $L$. \\
\item[(b)] two-dimensional box with area $A$.\\
\item[(c)] three dimensional box with volume $V$.\\
\begin {itemize}

Answer the following questions.
Express your results using $\gamma$, $m$, $N$, $L$, $A$, $V$.

\begin {itemize}
\item[(1)]
What is the single particle density of states. 
Distinguish between a spin up and spin down particles.

\item[(2)]
Which is the graph that describes the magnetization $M(B)$ 
of each case $(a)$,$(b)$,$(c)$. Complete the missing 
details: what are $M_s$, $B_c$ ,$\chi$.

\end {itemize}

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