Consider an ideal Bose gas in a uniform gravitational field of acceleration \(g\). Show that the phenomenon of Bose-Einstein condensation in this gas sets in at a temperature \(T_{c}\) given by

\[
T_{c} \simeq T_{c}^{0}\left[1+\frac{8}{9} \frac{1}{\zeta\left(\frac{3}{2}ight)}\left(\frac{\pi m g L}{k T_{c}^{0}}ight)^{1 / 2}ight]
\]

where \(L\) is the height of the container and \(m g L \ll k T_{c}^{0}\). Also show that the condensation here is accompanied by a discontinuity in the specific heat \(C_{V}\) of the gas:

\[
\left(\Delta C_{V}ight)_{T=T_{c}} \simeq-\frac{9}{8 \pi} \zeta\left(\frac{3}{2}ight) N k\left(\frac{\pi m g L}{k T_{c}^{0}}ight)^{1 / 2} ;
\]

see Eisenschitz (1958).