
\sect{Chemical equilibrium for misc reactions}

%Baruch's  B21

\begin{itemize}
\item [(a)]
The following reaction occurs inside a star
\[\ \gamma+ \gamma\leftrightarrow     e^{+}+e^{-}\]
where ${\gamma}$  is a photon and ${e^{\pm}}$ are the positron and
electron, respectively. Assume overall charge neutrality and that
the system is in equilibrium at temperature ${T}$. Find an
expression for the densities of ${e^{\pm}}$. (In general
${e^{\pm}}$ with mass $m$ are relativistic). Find an explicit
result for these densities in the limit ${k_{B}T<<mc^{2}}$. (Hint:
no conservation law for photons).
\item [(b)]
Repeat (a) for the reaction
\[\ \gamma + \gamma\leftrightarrow \pi^{+}+\pi ^{-}\]
where ${\pi^{\pm}}$ are bosons with mass $m_{\pi}$.
Can these bosons become Bose-condensed if the temperature is
sufficiently lowered? What are the densities at $T=0$?
\item[(c)] The photons in (b) are now eliminated from the system, but the
density of the bosons is maintained. If these bosons are cooled,
is there a temperature at which Bose condensation is possible?
[Assume that the initial temperature satisfies $k_BT<<
m_{\pi}c^2$]. What are the densities at $T=0$?\\
\item[(d)] Consider initial equal densities ${\bar n}$ for $e^+$ and for $e^-$. The
$e^{\pm}$  are allowed to equilibrate with $\pi^{\pm}$ via the reaction
\[ e^{+}+e^{-} \leftrightarrow \pi^{+}+\pi^{-}\]
Write equations that determine the chemical potentials at $T\neq 0$.
Consider now $T=0$ and determine if
Bose Einstein condensation is possible and if so what is the condition for that. Consider both $m_e>m_{\pi}$ and $m_e<m_{\pi}$ situations.
\end{itemize}

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