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Chinese Journal of Physics

Transition temperature to the Bose-Einstein condensation of spin polarized tritium, hydrogen, and 4He gases using the second virial coefficient

B.R. Joudeh

Published Dec 12, 2024• 0 citations
Publisher
DOI
10.1016/j.cjph.2024.12.011

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We review the recent activity in the theoretical description of spin-polarized atomic hydrogen and its isotopes at very low temperatures. Spin-polarized hydrogen is the only system in nature that remains stable in the gas phase even in the zero temperature limit due to its small mass and weak interatomic interaction. Hydrogen and its heavier isotope tritium are bosons, the heavier mass of tritium producing a self-bound (liquid) system at zero temperature. The other isotope, deuterium, is a fermion with nuclear spin one making possible the study of three different quantum systems depending on the population of the three degenerate spin states. From the theoretical point of view, spin-polarized hydrogen is specially appealing because its interatomic potential is very accurately known making possible its precise quantum many-body study. The experimental study of atomic hydrogen has been very difficult due to its high recombination rate, but it finally led to its Bose-Einstein condensate state in 1998. Degeneracy has also been observed in thin films of hydrogen adsorbed on the 4He surface allowing for the possibility of observing the Berezinskii–Kosterlitz–Thouless superfluid transition.
Abstract The quantum-mechanical formulations for the second virial, and the acoustic virial, coefficients are derived for spin-polarized atomic hydrogen (H↓). The dependence of these coefficients on the temperature T as well as the nuclear polarization ζ is analyzed. This is done for the first time. The main inputs in these computations are the scattering phase shifts, which are obtained using the Lippmann-Schwinger equation, with the Silvera triplet-state potential. Starting with these phase shifts, comprehensive calculations of the thermophysical properties for this system are performed for T ranging from 1µK to 100 K, and ζ varying from 0 to 1. These properties include, in addition to the quantum second virial coefficient and the acoustic virial coefficient: the pressure-polarization-temperature (P-ζ-T) behavior, the entropy (per atom), the speed of sound, the second virial correction (to the total internal energy per atom and unit density), and the specific heat capacity (per atom and unit density). The Boyle temperature and the Joule inversion temperature are determined. The T-dependence of these thermophysical properties, and related quantities, is explored. As expected, this dependence is most evident in the low-T limit, where quantum effects predominate. The corresponding ζ-dependence becomes noticeable at 80 K and below. It is observed that the virial coefficients tend to decrease with increasing ζ. Comparison of the present results to previous results are included whenever possible. The overall agreement is very good. As T is lowered, the disruptive effects of thermal energy are weakened relative to the attractive interactions between the atoms. Consequently, the H↓ gas makes a transition to a state of higher order and lower entropy – Bose-Einstein condensation (BEC). This is explored carefully.
Properties of spin-polarized hydrogen (${\mathrm{H}}_{\mathrm{\ensuremath{\downarrow}}}$) gas at low densities in two and three dimensions are calculated with the use of a quantum virial expansion. These properties include the second virial coefficients, heat capacities, chemical potentials, and estimates of critical temperatures and densities for Bose-Einstein condensation in three dimensions and for a hypothetical superfluid transition in two dimensions. The scattering phase shifts used are based on a Kolos-Wolniewicz two-body potential. The classical virial expansion and the validity of effective-range theory are discussed.