
\sect{Baruch's  B05.}

Consider an ideal Bose gas of particles with mass $m$
in a uniform gravitational field of accelaration ${g}$.
\begin{itemize}
\item [(a)]
Show that
the critical temperature for the Bose-Einstein condensation is
\[\ T_{c} = T_{c}^{0}\left[1+\frac{8}{9}\frac{1}{\zeta  \left(3/2\right) }  \left(  \frac{\pi mgL}{k_{B}T_{c}^{0}} \right)^{1/2}
\right]\]
where L is the height of the container, $mgL\ll k_BT_c^0$ and $T_c^0 =T_c(g=0).$

[Hint: $g_{3/2}(\zeta )=g_{3/2}(1)- 2\sqrt{-\pi\ln\zeta}+
O(\ln\zeta ).$]
\item [(b)]
Show that the condensation is accompanied by a discontinuity in
the specific heat at ${T_{c}}$,
\[\ \Delta C_{V} = -\frac{9}{16\pi} \zeta(3/2)Nk_{B}
\left(\frac{\pi mgL}{k_{B}T_{c}^{0}}  \right)^{1/2}.\]
[Hint:  $\Delta C_V$ is due to discontinuity in
${(\partial\zeta /\partial T)_{N,V}}$].\\
\end{itemize}

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