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\sect{Bose gas in a uniform gravitational field}

Consider an ideal Bose gas of particles with mass ${\mathsf{m}}$ in
a uniform gravitational field of acceleration ${g}$.  Show that the
critical temperature for the Bose-Einstein condensation is \, ${
T_{c} =
T_{c0}\left[1+\left(\frac{8}{9}\right)\left(1/\zeta\left(\frac{3}{2}\right)\right)\left(\frac{\pi
mgL}{T_{c0}}\right)1/2\right]}$,
where ${L}$ is the height of the tank, \,${ mgL<<kBT_{c0}}$   and \,
${T_{c0} =T_{c0}\left(g=0\right)}$.

Hint:
${g_{\frac{3}{2}}\left(z\right)=g_{\frac{3}{2}}\left(1\right)-2\left(-\pi
\ln\left(z\right)\right)^{\frac{1}{2}} +
O\left(\ln\left(z\right)\right)}$.

Show that the condensation is
accompanied by a discontinuity in the specific heat at ${T_{c}}$
with the result  ${\triangle CV = -\left(\frac{9}{16\pi}
\right)\zeta\left(\frac{3}{2}\right) N\left(\frac{\pi
mgL}{Tc0}\right)_{\frac{1}{2}}}$.

Hint: ${\triangle CV}$ is due to
discontinuity in ${\left(\frac{\partial z}{\partial
T}\right)_{N,V}}$



