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

%Based on Baruch's  B28

In this problem we regard (anti)neutrino as massless spin 1/2 fermions.
If we place neutrons in some confined region, some of them $\beta$-decay 
into protons, electrons and antineutrinos via the reaction
${n \Leftrightarrow  p + e^{-} + \bar{\nu}}$. 
The masses are ${\mass_{n}, \mass_{p}, \mass_{e} }$. 
The objective is to estimate the final equilibrium density of~$n$ 
of the neutrons, given that that initially density was~$n_0$.   

\begin{itemize}

\item[(a)]
Find the equations of states for gas of massless spin 1/2 fermions.
Namely, express the density of particles and the density of energy as a function
of the chemical potential $\mu$, the temperature $T$, and the speed of light $c$.
Evaluate the integrals for $T=0$ and for large $T$. Define what does it mean large $T$.


\item[(b)]
Write equations for the neutron density $n$, 
given that initially there were only neutrons with density $n_0$, 
and the temperature is $T$. 
Consider appriximation under the following assumptions:

\item[(b1)]
The particles are non-relativistic (except the nutrino)

\item[(b2)]
The particles are hyper relativistic (negligible mass).

\item[(b3)]
The temperature is zero.

\item[(b4)]
The temperature is high (Boltzmann approximation).


\end{itemize}


