\sect{Adiabatic cooling}
Consider an ideal gas whose $N$ atoms have mass $m$, spin $1/2$ and a magnetic moment $\gamma$.
The kinetic energy of a particle is $p^2/(2m)$ and the interaction with the
magnetic field $B$ is $\pm \gamma B$ for up/down spins.
\begin{itemize}
\item[(a)] Calculate the entropy as $S(T,B)=S_{kinetic}+S_{spin}$.
\item[(b)] Consider an adiabatic process in which the magnetic field is varied
from $B$ to zero. Show that the initial and final temperatures $T_i$ and $T_f$
are related by the equation:
\[ \ln \frac{T_f}{T_i} = \frac{2}{3N}[S_{spin}(T_i, B)-S_{spin}(T_f, 0)]\]
\item[(c)] Find the solution for $\frac{T_f}{T_i}$ in the large $B$ limit.
\item[(d)] Extend (c) to the case of space dimensionality $d$ and general spin $S$.
\end{itemize}
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