\sect{Adiabatic cooling of spins}

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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