\sect{Chemical equilibrium A=A+e (Exam 2011)}

$N_0$ atoms of type $A$ are placed in an empty box 
of volume $V$, such that their initial density is $n_0=N_0/V$. 
The ionization energy of the atoms is $\varepsilon_0$.
The box is held in temperature $T$, and eventually 
a chemical equilibrium ${A \rightleftarrows A^{+} + e^{-}}$ 
is reached. The fraction of ionized atoms is ${x=N^{+}/N_0}$.
The masses of the particles are $m_e$ for the electron, 
and $m_{A^+} \approx m_A$ for the atoms and the ions. 
 
\Dn

(1) Define temperature $T_0$ such that ${T \gg T_0}$ is a sufficient 
condition for treating the gas of atoms in the Boltzmann approximation. 

\Dn

(2) Assuming the Boltzmann approximation for both the atoms 
and the electrons, write an equation for $x$. 
Write its {\em approximate} solution assuming ${x\ll1}$. 
Write the condition for the validity of the latter assumption.  

\Dn

(3) Assuming that ${x\ll1}$, write a condition on the density $n_0$, 
that above $T_0$ it was legitimate to treat the electrons 
in the Boltzmann approximation. Note: the condition is a simple 
inequality and should be expressed using $(m_e,m_A,\varepsilon_0)$. 


\Dn

Assume that the condition in (3) breaks down.
It follows that there is a regimes ${T_0 \ll T \ll T_1}$ 
where the atoms can be treated in the Boltzmann approximation, 
while the electrons can be treated as a low temperature quantum gas. 

\Dn

(4) Write an equation for $x$ assuming that the electrons 
can be treated approximately as a zero temperature Fermi gas. 
Exotic functions should not appear.
You are not expected to solve this transcendental equation.

\Dn

(5) What would be the equation for $x$ if the electrons 
were Bosons instead of Fermions.
    
\Dn

{\em Note:} Express all the final answers using $(m_e,m_A,n_0,\varepsilon_0,T)$,  
and {\em elementary} functions. \\ Exotic functions should not appear.
It is allowed to use the notation $\lambda_e(T)=(2\pi/\mass_e T)^{1/2}$.