\sect{Chemical potential, reaction in ideal quantum gases}

${N}$ fermions of type ${F}$ that have spin $1/2$ are placed in a box of volume ${V}$. 
Each fermion can decay into a boson of type ${B}$ that have spin ${0}$, 
and a fermion of type ${A}$ that has spin $1/2$. 
The reaction is ${F\rightarrow A+B}$, and it has an energy gain ${\epsilon_0}$. 
This means that ${A+B}$ has a lower binding energy than ${F}$. 
The masses of the particles are $m_{F},m_{A},m_{B}$ respectively.

\Dn

(1) Assuming ideal gases at temperature ${T}$, write the 
chemical equilibrium condition 
that determine the densities ${n_{F}, n_{A}, n_{B}}$ at equilibrium.

\Dn

(2) Write the chemical equilibrium condition at ${T=0}$. 
Describe the dependece of the densities on ${\epsilon_{0}}$. 
Find $\epsilon_c$ such that for $\epsilon_0>\epsilon_c$ 
the number of F fermions vanishes.

\Dn

(3) Assume that the condensation of bosons B occurs at~$T_c$ 
such that ${T_c \ll p_F^2/(2m_A) < \epsilon_0}$, 
where $p_F$ is the Fermi momentum of fermions A. 
Evaluate $T_c$ and rewrite the condition on $T_c$ in terms of the given parameters.


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