\sect{Cooling by demagnetization: heat flow between two subsystems}

%Baruch's  A15

Consider a solid with ${N}$ non-magnetic atoms and ${N_{i}}$
non-interacting magnetic impurities with spin ${s}$. There is a
weak spin-phonon interaction which allows energy transfer between
the impurities and the non-magnetic atoms.

\begin{itemize}
\item [(a)]
A magnetic field is applied to the system at a constant
temperature ${T}$. The field is strong enough to line up the spins
completely. What is the change in entropy of the system due to the
applied field? (neglect here the spin-phonon interaction).
\item [(b)]
Now the magnetic field is reduced to zero adiabatically. What is
the qualitative effect on the temperature of the solid? Why is the
spin-phonon interaction relevant?
\item [(c)]
Assume that the heat capacity
of the solid is ${C_{V}=3Nk_{B}}$ in the relevant temperature
range. What is the temperature change produced by the process (b)?
(assume the process is at constant volume).\\
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%