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

Abstract

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 $({E}_{t}={E}_{\mathit{min}})$ for radii near ${R}_{\mathit{min}}=9.95\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$ for ${\mathrm{H}}_{2}$ and $8.48\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$ for $^{4}\mathrm{He}$. 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 $(E\ensuremath{-}{E}_{\mathit{min}})$. 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 ${R}_{\mathit{min}}$.