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\sect{Fermions in magnetic field - Pauli}

${N}$ electrons are in a box as follows: 
(I) two-dimensional with area ${A}$; 
(II) three  dimensional with volume ${V}$.
The temperature is zero. We create a magnetic field ${B}$. The
electrons behave like an ideal fermi gas. The one particle
hamiltonian is
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\[H_{1}=\frac{\overrightarrow{p}^{2^{}}}{2m}-rB \sigma_{z}\]




\begin {itemize}

\item[(a)]
Show a schematic drawing of the uniparticle states density function.
Distinguish between a spin up conditions and a spin down ones.

\item[(b)]
Determine which of the following graphes describes the magnetization
${M\left(B\right)}$ in each one of the cases (I) (II) and complete
the missing details ${\left(M_{s}=?, B_{c}=?, \chi=?\right)}$.

Use only ${\gamma, m, V, A, N}$.

\end {itemize}



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