arXiv:cond-mat/0310067 · Physical Review B

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

Francesco Ancilotto, M. Mercedes Calbi, Silvina M. Gatica, Milton W. Cole

Published Oct 29, 2004• 18 citations

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 ${R}_{\mathit{min}}$.

DOI: 10.1103/physrevb.70.165422

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arXiv:cond-mat/0310067 · Physical Review B

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

Francesco Ancilotto, M. Mercedes Calbi, Silvina M. Gatica, Milton W. Cole

Published Oct 29, 2004• 18 citations

arXiv HTML PDF

Abstract

{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}}

DOI: 10.1103/physrevb.70.165422

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Oldenburg, Jakob Felix

Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum eﬀects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. 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More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for suﬃciently low densities as predicted in the physics literature first by Wu and subsequently others. $\textit{Full-text embargoed until: 2026-01-01}$ Quantum systems consisting of many particles are both interesting from the physics and the mathematics perspective. One aspect that makes them fascinating from the physics point of view is that they can exhibit quantum eﬀects on macroscopic scales such as Bose-Einstein condensation. This phenomenon, first predicted 100 years ago by Bose [19] and Einstein [44], occurs when a dilute gas of bosons is cooled down to very low temperatures. Below a certain critical temperature a condensate forms, meaning that a macroscopic fraction of the particles is in the same one-particle state, and thus, quantum phenomena become relevant on length scales much larger than the individual atom. The occurrence of Bose-Einstein condensation is also related to superfluidity as suggested in [88]. Bose-Einstein condensation has been achieved experimentally independently by the groups of Cornell and Wieman [3] and Ketterle [40] earning them a Nobel prize in 2001. While Bose and Einstein analyzed theoretically the non-interacting gas, an eﬀective theory describing weakly interacting Bose gases was proposed by Bogoliubov in 1947 with the goal of explaining the occurrence of superfluidity in liquid helium [18]. Using assumptions such as that condensation indeed occurs, his theory yields predictions for the low-energy states of the system. Seen from the mathematics perspective, describing the Bose gas is an interesting challenge due to the strong correlations between particles introduced by the interaction making this case much harder to study than the non-interacting one. In particular, a proof that Bose-Einstein condensation occurs in the thermodynamic limit for continuous systems is still out of reach. However, there has been a lot of work regarding the rigorous analysis of bosonic systems in various relevant regimes. This thesis contains three new contributions establishing rigorous results on bosonic systems in diﬀerent, physically relevant settings. It is based on the articles [37, 27, 31]. The first result concerns the time-evolution of initially trapped Bose gases in the ultra-dilute Gross-Pitaevskii regime. Under some assumptions on the initial data satisfied in particular by a class of states that are expected to approximate the ground state in a trap, we derive a quasi-free approximation that is close to the true solution in $L^2$-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for suﬃciently low densities as predicted in the physics literature first by Wu and subsequently others. $\textit{Full-text embargoed until: 2026-01-01}$

Dec 9, 20250 citations

{E}_{t}

(that depends on the radii of neighboring tubes) and a quasicontinuous energy

{E}_{z}

({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}})

{R}_{\mathit{min}}

^{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.

We evaluate the effects of heterogeneity on the density of states of H

_2

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.

L^2

-norm. In particular, we show that this approximate wave function is described in terms of a Bose-Einstein condensate in a one-particle state following the time-dependent Gross-Pitaevskii equation together with pair excitations with respect to this condensate. As a corollary, we establish a central limit theorem for bounded one-particle observables. The second result is a simple proof of Bose-Einstein condensation for interacting, ultra-dilute gases. More specifically, we consider scalings interpolating between the Gross-Pitaevskii regime and the thermodynamic limit. While condensation had been shown to hold in these regimes before, the advantage of the method used here is its simplicity. Lastly, the third result establishes an upper bound on the ground state energy of a dilute Bose gas in the thermodynamic limit. It captures rigorously the third order correction to the energy per unit volume for suﬃciently low densities as predicted in the physics literature first by Wu and subsequently others.

\textit{Full-text embargoed until: 2026-01-01}