\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 $\lambda_e(T)=(2\pi/\mass_e T)^{1/2}$.