\sect{Cooling by adiabatic pressurization}

%Baruch's  B27

If liquid $^3$He is pressurized adiabatically, it becomes a solid
and the temperature drops. This is a method of cooling by
pressurization. Develop the theory of this process in the
following steps:
\begin{itemize}
\item[(a)] Assume that the liquid state is an ideal Fermi liquid with a
low temperature entropy $S=\half \pi^2Nk_BT/T_F$ where N is the
number of particles and $T_F\approx 5 \,^{\circ}$K is the Fermi
temperature. Find the temperature-pressure relation in an
adiabatic process for $T\ll T_F$.

\item[(b)] At low temperatures the entropy of solid $^3$He comes almost
entirely from the spins while below $10^{-3}\, {^\circ}$K the
spins become antiferromagnetically ordered; assume that at
$T\gtrsim 10^{-3}\, {^\circ}$K the spins are independent. Draw
schematically the entropy of both solid and liquid $^3$He as
function of temperature and draw the adiabatic trajectory for
increasing pressure. Below which temperature $T^*$ must the
initial temperature be for the method to work?

\item[(c)] Of what order is the liquid-solid transition? Evaluate the
jump in the specific heat.

\item[(d)] Use Clapeyron's relation to deduce the shape of the $P(T)$
coexistence solid-liquid curve near $T^*$. Assume that the
difference $\Delta v$ of the specific volumes is temperature
independent and that the solid is denser.

\item[(e)]  Consider an initial pressure that is below the $P(T)$
coexistence line. Draw schematically the adiabatic trajectory on
the P-T plane, using the result (a). What is now the condition on
the initial temperature for the cooling method to work, in terms
of the initial $(P,T)$ and the coexistence line $P(T)$?\\
\end{itemize}

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