
\sect{Beta decay n==p+e+nu}

%Baruch's  B28

The neutrino is a massless fermion with spin $\half$.
\begin{itemize}
\item[(a)] Assuming neutrinos in thermal equilibrium, write the density in
terms of the fugacity $\zeta$ and the temperature $T$ (take in
account the quantum nature of the neutrinos). Evaluate the
integral in the limit of high $T$ and at $T=0$. What is the
condition for high $T$?

\item[(b)]
b. Given an initial density $n_0$ of neutrons. Some of these
neutrons $\beta$ decay to protons, electrons and antineutrinos via
the reaction
\[n\rightarrow p+e^{-}+{\bar \nu}.\]
All the particles are fermions with spin $\half$. The masses
$m_{n},\, m_{p},\,m_{e}$ are given and the antineutrino is
massless. Assuming quantum ideal gases with $p,n,e^-$
nonrelativistic, write the equilibrium equations for the proton
density ${n_p}$. Consider only the following two cases (i) a high
temperature limit, and (ii) $T=0$. In both cases solve explicitly
in the limit $n_p\ll n_0$.\\
\end{itemize}

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