Bose-Einstein condensation of helium and hydrogen inside bundles of carbon nanotubes [95]

Helium atoms or hydrogen molecules are believed to be strongly bound within the interstitial channels (between three carbon nanotubes) within a bundle of many nanotubes. The effects on adsorption of a nonuniform distribution of tubes are evaluated. The energy of a single-particle state is the sum of a discrete transverse energy E_t (that depends on the radii of neighboring tubes) and a quasicontinuous energy E_z of relatively free motion parallel to the axis of the tubes. At low temperature, the particles occupy the lowest-energy states, the focus of this study. The transverse energy attains a global minimum value (E_t=E_min) for radii near R _min =9.95 Å for H₂ and 8.48 Å for ⁴He. The density of states N(E) near the lowest energy is found to vary linearly above this threshold value, i.e., N(E) is proportional to (E-E_min). As a result, there occurs a Bose-Einstein condensation of the molecules into the channel with the lowest transverse energy. The transition is characterized approximately as that of a four-dimensional gas, neglecting the interactions between the adsorbed particles. The phenomenon is observable, in principle, from a singular heat capacity. The existence of this transition depends on the sample having a relatively broad distribution of radii values that include some near R_min.