Show that, in the case of a degenerate gas of fermions (left(T ll T_{F} ight)), the correlation

Show that, in the case of a degenerate gas of fermions \(\left(T \ll T_{F}\right)\), the correlation function \(g(r)\), for \(r \gg \hbar / p_{F}\), reduces to the expression

\[g(r)-1=-\frac{3(m k T)^{2}}{4 p_{F}^{3} \hbar r^{2}}\left\{\sinh \left(\frac{\pi m k T r}{p_{F} \hbar}\right)\right\}^{-2}\]

Note that, as \(T \rightarrow 0\), this expression tends to the limiting form

\[g(r)-1=-\frac{3 \hbar}{4 \pi^{2} p_{F} r^{4}} \propto \frac{1}{r^{4}}\]