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Physical Review B

Condensation of helium in interstitial sites of carbon nanotubes bundles

Boris Marcone, Enzo Orlandini, Flavio Toigo, Francesco Ancilotto

Published Aug 21, 2006• 11 citations
Publisher

Abstract

Helium atoms are believed to be strongly bound within the interstitial channels in bundles of carbon nanotubes. In a recent paper [F. Ancilotto et al., Phys. Rev. B 70, 165422 (2004)] inhomogeneity in the size distribution of nanotube radii was shown to make a system of $^{4}\mathrm{He}$ atoms in such an environment effectively a four-dimensional Bose gas, thus permitting a Bose-Einstein condensation (BEC) of the adsorbed atoms into the minimum energy state. This surprising result was obtained for a model of noninteracting atoms in a continuum distribution of (virtually) infinite interstitial channels. Here we investigate how the singular thermal properties of the ideal system and the occurrence of BEC are affected by a more realistic modeling of a bundle of nanotubes where (i) the number of nanotubes is finite and where (ii) $^{4}\mathrm{He}$ atoms adsorbed within the same interstitial channel interact among themselves. Also in this case we observe an anomalous heat capacity close to the ideal condensation temperature, suggesting the persistence of the condensation transition for interacting $^{4}\mathrm{He}$ atoms, which might be experimentally observed.

DOI
10.1103/physrevb.74.085415

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Journal of Physics Conference Series•Feb 1, 2009•2 citations
Helium atoms are believed to be strongly bound within the interstitial channels in bundles of carbon nanotubes. In a recent paper [F. Ancilotto et al., Phys. Rev. B 70, 165422 (2004)] inhomogeneity in the size distribution of nanotube radii was shown to make a system of 4He^{4}\mathrm{He}4He atoms in such an environment effectively a four-dimensional Bose gas, thus permitting a Bose-Einstein condensation (BEC) of the adsorbed atoms into the minimum energy state. This surprising result was obtained for a model of noninteracting atoms in a continuum distribution of (virtually) infinite interstitial channels. Here we investigate how the singular thermal properties of the ideal system and the occurrence of BEC are affected by a more realistic modeling of a bundle of nanotubes where (i) the number of nanotubes is finite and where (ii) 4He^{4}\mathrm{He}4He atoms adsorbed within the same interstitial channel interact among themselves. Also in this case we observe an anomalous heat capacity close to the ideal condensation temperature, suggesting the persistence of the condensation transition for interacting 4He^{4}\mathrm{He}4He atoms, which might be experimentally observed.
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 Et{E}_{t}Et​ (that depends on the radii of neighboring tubes) and a quasicontinuous energy Ez{E}_{z}Ez​ 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 (Et=Emin)({E}_{t}={E}_{\mathit{min}})(Et​=Emin​) for radii near Rmin=9.95A˚{R}_{\mathit{min}}=9.95\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}Rmin​=9.95A˚ for H2{\mathrm{H}}_{2}H2​ and 8.48A˚8.48\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}8.48A˚ for 4He^{4}\mathrm{He}4He. The density of states N(E)N(E)N(E) near the lowest energy is found to vary linearly above this threshold value, i.e., N(E)N(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 Rmin{R}_{\mathit{min}}Rmin​.
We evaluate the effects of heterogeneity on the density of states of H2_22​ molecules inside interstitial channels within bundles of carbon nanotubes. As temperature (T) falls, the density increases within those tubes having the greatest binding energy. At T ~ 10 mK, the molecules undergo Bose-Einstein condensation, exhibiting a singular heat capacity.
Bose-Einstein condensation is a unique phase transition in that it is not driven by interparticle interactions, but can theoretically occur in an ideal gas, purely as a consequence of quantum statistics. This chapter addresses the question, ‘ How is this ideal Bose gas condensation modified in the presence of interactions between the particles?’ This seemingly simple question turns out to be surprisingly difficult to answer. Here we outline the theoretical background to this question and discuss some recent measurements on ultracold atomic Bose gases that have sought to provide some answers. Introduction Unlike the vast majority of phase transitions, Bose-Einstein condensation (BEC) is not driven by interparticle interactions but can theoretically occur in an ideal (noninteracting) gas, purely as a consequence of quantum statistics. However, in reality, interactions are needed for a Bose gas to remain close to thermal equilibrium. It is thus interesting to discuss if something close to ideal gas BEC can be observed in a real system and what happens in the vicinity of the BEC transition in the presence of interparticle interactions. These simple questions have not been easy to answer, either theoretically or experimentally. The theoretical foundations for studying the effect of interactions on Bose condensed systems were laid over half a century ago by Bogoliubov [1], Penrose and Onsager [2], and Belieav [3], among others. These works initially focused on zerotemperature properties and were extended to nonzero temperature in the pioneering papers of Lee, Huang, and Yang [4, 5, 6, 7]. At that time, the main experimental system was liquid He-4 in which the inter-particle interactions are strong, making connections with theory difficult. The realisation in 1995 of BEC in weakly interacting ultracold atomic gases [8, 9] thus opened up the possibility to experimentally revisit some of these long-discussed questions. This was further aided by, among other advances, the use of Feshbach resonances to tune the interaction strength in atomic gases [10, 11]. Thus, in the last twenty years the study of ultracold Bose gases has been very successful (see Chapter 3 for an insightful account of selected topics and surprising developments over this exciting period). However, the fact that, until recently, ultracold atoms were confined using harmonic potentials has hindered the study of the BEC transition itself.
The problem of the transition temperature of liquid helium in narrow channels has been considered by using the ideal Bose-Einstein gas as a model. It has been found that for a channel of square cross section ($D\ifmmode\times\else\texttimes\fi{}D$) and length LLL, the transition temperature Tc{T}_{c}Tc​ is given by the expression $\frac{{T}_{c}}{{T}_{\mathrm{cB}}}=(0.84 {\AA{}}^{\ensuremath{-}1})\frac{{D}^{2}}{L}$ under the conditions that $\frac{{D}^{2}}{L}\ensuremath{\ll}\ensuremath{\lambda}$, where $\ensuremath{\lambda}$ is the thermal de Broglie wavelength, and TcB{T}_{\mathrm{cB}}TcB​ is the transition temperature in the case of an infinite volume.
Motivated by the recent achievement of space-based Bose-Einstein condensates (BEC) with ultracold alkali-metal atoms under microgravity and by the proposal of bubble traps which confine atoms on a thin shell, we investigate the BEC thermodynamics on the surface of a sphere. We determine analytically the critical temperature and the condensate fraction of a noninteracting Bose gas. Then we consider the inclusion of a zero-range interatomic potential, extending the noninteracting results at zero and finite temperature. Both in the noninteracting and interacting cases the crucial role of the finite radius of the sphere is emphasized, showing that in the limit of infinite radius one recovers the familiar two-dimensional results. We also investigate the Berezinski-Kosterlitz-Thouless transition driven by vortical configurations on the surface of the sphere, analyzing the interplay of condensation and superfluidity in this finite-size system.
In the one-dimensional (1D) nanopores of FSM-16 (2.8 nm in diameter) and the 3D pores of HMM-2 (2.7 nm, and 5.5 nm in 3D period), the 4 He films adsorbed on these nanopore walls show the properties of the Bose quantum fluid and superfluidity above 1.4 atomic layers at low temperatures. The boundary of the Bose fluid region was determined from the kink (peak) temperature T C of the heat capacity at each coverage n . The phase diagrams obviously show dependence on the 1D and 3D pore connections, respectively. In the 3D nanopores, the coverage (density) dependence of T C is well reproduced by the Bose-Einstein condensation temperature of the 3D ideal gas which is proportional to ( n - n c ) 2/3 , where n c is the onset coverage. In the case of the 1D pores (channels), T C is proportional to ( n - n c ) at 0.1 < T C < 1K. This T C is likely to correspond to the crossover temperature of the 2D ideal gas heat capacity from the constant heat capacity of the Boltzmann gas to the linear in T , with decreasing temperature. At T C , the thermal de Broglie wavelength of the free particle is still shorter than the pore diameter, while the thermal phonon wavelength is estimated to become longer than the diameter below T C , which indicates the 1D phonon state.