\sect{Baruch's B25.}
Given ${N}$ fermions (with spin $\half$) in a volume $V$ of type
${F}$ which can decay into a
boson ${B}$ (of spin ${0}$) and a fermion of type ${A}$ in the
reaction ${F\rightarrow A+B}$. The reaction has an energy gain of
${\epsilon_0}$ (i.e. ${A+B}$ have lower energy than ${F}$ ) and
the masses are $m_{F},m_{A},m_{B}$ respectively.
\begin{itemize}
\item [(a)]
Assuming ideal gases at temperature ${T}$, write the equations
which determine the densities ${n_{F}, n_{A}, n_{B}}$ in
equilibrium.
\item [(b)]
Write the equations of (a) at ${T=0}$ and plot (qualitatively) the
densities as functions of ${\epsilon_{ 0}}$. Find $\epsilon_c$
such that for $\epsilon_0>\epsilon_c$ the number of F fermions
vanishes.
\item[(c)] Assume that the condensation of bosons B occurs at
$T_c$ such that $k_BT_c\ll \frac{p_F^2}{2m_A}$
where $p_F$ is the Fermi momentum of fermions A. Evaluate $T_c$
and rewrite the condition $k_BT_c\ll \frac{p_F^2}{2m_A}$ in terms
of the given parameters.
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%