\sect{Baruch's B12.}
Consider fermions of mass $m$ and spin $\half$ in a gravitational field with constant acceleration $g$ and at uniform temperature $T$.
\begin{itemize} \item[(a)] Assume first that the fermions behave as classical particles and find their density $n(h)$ as function of the height $h\geq 0$ and the density $n(0)$.
\item[(b)] Near the bottom $h=0$ the
fermions are degenerate, i.e. their Fermi energy
$\epsilon_F^0\equiv \epsilon_F(h=0)$ is $\epsilon_F^0\gg k_BT$. Assume here $T=0$ and find the local fermi momentum $p_F(h)$ and the density $n(h)$ in terms of $n(0)$.
\item[(c)] Consider now $T\neq 0$ (but still $\epsilon_F^0\gg k_BT$) and estimate the height $h_c$ where at $h>h_c$ the fermions are non-degenerate. Find n(h) at $h\gg h_c$ in terms of $n(0)$.
\end{itemize}
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