\sect{Liquid He3, cooling by adiabatic pressurization}

%modified versions of Baruch's B27

The phase-diagram of $^3$He is displayed in the figure below. 
Assume that the liquid phase can be described as a low temperature  
ideal Fermi gas with ${T_F \approx 5 \,^{\circ}}$K, 
while the solid phase is described as a set of spins.  
Assume that $T>0.01\, {^\circ}$K, such that antiferromagnetic 
order is diminished and the spins can be treated as non-interacting. 
In this region the $P(T)$ coexistence solid-liquid curve
exhibits a minimum at temperature~$T^*$.  
This implies an anomaly:
as the temperature is raised, keeping the pressure constant, 
the liquid is ``frozen" into solid, and only later 
melts back into a liquid. 
This is the regime of interest for this question.
 
\includegraphics[width=7cm]{PhasesHe3}

In particular we are interested in method of cooling by pressurization. 
Consider $N$ Helium particles in the the liquid phase with temperature lower than~$T^*$.
The system is pressurized adiabatically. 
Once the liquid becomes a solid, the temperature drops down to a lower value.

\begin{itemize}

\item[(a)]
Express the entropy $S_{\ell}(T,P)$ in the liquid phase 
and the entropy $S_{s}(T)$ in the solid phase.   
Determine the temperature~$T^*$ 
for which both phases have the same entropy. 

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

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

\item[(d)] 
Find the trajectory $T(P)$ for an
adiabatic process in the liquid phase region,
starting at some initial state $(T_0,P_0)$. 
Assuming that $(T^*,P^*)$ are known,  
determine the condition that this process 
leads to cooling via solidification. 

\item[(e)] 
If the condition in (d) is satisfied,
what will happen to the $T(P)$ trajectory 
after it hits the $P(T)$ coexistence curve? 

\end{itemize}




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