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 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 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 atoms, which might be experimentally observed.
Using techniques developed by Greenspoon and Pathria, a rigorous, asymptotic analysis of the onset of Bose-Einstein condensation in a finite two-dimensional system at constant pressure is carried out. Some of the results obtained by Imry, Bergman, and Gunther, who first considered this problem, are upheld while others get modified. In particular, a macroscopic occupation of the single-particle ground state at finite temperatures is indeed possible. As the critical region, $T\ensuremath{\simeq}{T}_{c}$, is approached from above, the volume of the system becomes subextensive,
Vc=O(lnNN), which is both necessary and sufficient for the onset of Bose-Einstein condensation in the system. For instance, if the system is now cooled even at constant volume
V(=Vc), the condensate fraction gradually builds up and becomes nonnegligible as
T approaches ${T}_{0}\ensuremath{\simeq}\frac{({T}_{c})}{2}$; below
T0, it grows steadily as $(1\ensuremath{-}\frac{T}{{T}_{0}})$. On the other hand, if we continued to cool the system at constant
P its volume, over a thermodynamically negligible range of temperatures, would reduce to values
O(N21) which, in turn, would be accompanied by an abrupt accumulation of practically all the particles of the system into the single-particle ground state ${\ensuremath{\epsilon}}_{0}$. The specific heat
CP, after having passed through a maximum, would also finally reduce to subextensive values. Phase transitions of this kind may be of relevance to the physical behavior of thin helium films and helium submonolayers.
From a certain temperature on, the molecules “condense” without attractive forces; that is, they accumulate at zero velocity. The theory is pretty, but is there also some truth to it? Albert Einstein, Letter to Ehrenfest (Dec. 1924). Abraham Pais, Subtle Is the Lord: The Science and the Life of Albert Einstein , Oxford University Press, New York (1982) Introduction For over 50 years the low-temperature liquid state of uncharged, spinless He 4 was the only system in which a Bose–Einstein (BE) condensation was considered experimentally realized. In that case, cold (<2.19K) liquid He 4 passes into an extraordinary phase of matter called a superfluid , in which the liquid's viscosity and entropy become zero. With advances in atomic cooling (to ≈ 10 –9 K) the number of Bose systems which demonstrably pass into a condensate has considerably increased. These include several isotopes of alkali gas atoms as well as fermionic atoms that pair into integerspin (boson) composites. Although an ideal Bose gas does exhibit a low-temperature critical instability, the ideal BE gas theory is not, on its own, able to describe the BE condensate wave state. In order for a theory of bosons to account for a condensate wave state, interactions between the bosons must be included. Nevertheless, considerable interesting physics is contained in the ideal Bose gas model.