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International Journal of Thermophysics

Thermodynamic Properties and Critical Behavior of Spin-Polarized Atomic Hydrogen (H↓) Using the Quantum Second-Virial Coefficient

B.R. Joudeh

Published Jan 24, 2023• 3 citations
Publisher

Abstract

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.

DOI
10.1007/s10765-023-03155-9

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Physical review. B, Condensed matter•Jul 1, 1987•20 citations
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.
The quantum second virial coefficient B[Formula: see text] of 3He[Formula: see text] gas is determined in the temperature range 0.001–4 K from the Beth–Uhlenbeck formula. The corresponding phase shifts are calculated from the Lippmann–Schwinger equation using a highly-accurate matrix-inversion technique. A positive B[Formula: see text] corresponds to an overall repulsive interaction; whereas a negative B[Formula: see text] represents an overall attractive interaction. It is found that in the low-energy limit, B[Formula: see text] tends to increase with increasing spin polarization. The compressibility Z is evaluated as another measure of nonideality of the system. Z becomes most significant at low temperatures and increases with polarization. From the pressure–temperature (P–T) behavior of 3He[Formula: see text] at low T, it is deduced that P decreases with increasing T below 8 mK.
Spin polarized hydrogen and helium are being studied extensively because they are expected to exhibit quantum effects on a macroscopic scale. Here, low density properties of spin polarized hydrogen gas in two and three dimensions are studied using a quantum virial expansion. These properties include heat capacities, chemical potentials, and estimates of critical temperatures and densities for Bose-Einstein condensation in three dimensions and for a possible superfluid transition in two dimensions. The classical virial expansion and the validity of effective-range theory is also discussed. The heat capacity of adsorbed helium films of 3He, 4He and spin polarized 3He is also studied. Virial calculations with a structureless substrate approximation but with substrate mediated corrections to the Lennard-Jones potential are performed as a function of applied magnetic field and magnetization. Effects of substrate potential and geometry on the thermodynamics, low density submonolayer helium films physisorbed on rare gas plated graphite substrates are investigated using a tight-binding formalism and a lattice version of potential scattering theory. A generalization of the formalism to lattices with a basis is made and applied to the honeycomb lattice. The calculations with this theory exhibit systematic trends in the specific heat that are in qualitative agreement with presentlymore » available experimental data.« less
Abstract The thermophysical properties of spin-polarized atomic tritum (T↓) are calculated in the temperature range 0.01 K–10 K using the Galitskii–Migdal–Feynman (GMF) formalism. The calculations are performed using Silvera and Born–Oppenheimer triplet-state potentials. It is concluded that T↓ may form a liquid at very low temperatures. At low T , the effective s-wave scattering length a 0 < 0, which corresponds to a very weak attractive interaction. There is no bound state for the T↓ dimer, but the negative a 0 indicates that it is only slightly short of binding. The transport properties of T↓ increase with increasing temperature. This behavior shows that T↓ remains as a gas in this T -range. The transport properties of 3 He and 4 He are included for comparison purposes. The difference in the transport properties of T↓ and 3 He is due to the fact that T↓ atoms obey Bose statistics while 3 He atoms obey Fermi statistics, whereas the difference with the 4 He (Boson) is due to the mass difference between T↓ and 4 He.
In this paper we use the so-called static fluctuation approximation (SFA) to calculate the thermodynamic properties of spin-polarized atomic hydrogen. This approximation is based on the replacement of the square of the local-field operator with its mean value. A closed set of nonlinear integral equations is derived for neutral many-bosonic systems. This set is solved numerically by an iteration method for two triplet-state potentials: a Morse- and Silvera-type potentials. It is found that the mean internal energy per unit volume, the pressure, the entropy per unit volume, and the specific heat per unit volume increase with temperature and decrease with spin polarization in the low-temperature region (<8 mK ); whereas the condensate fraction increases with an applied magnetic field. It is also found that these quantities are nearly independent of spin polarization for high temperatures (>0.1 K ), and that they are independent of the number density up to 10-3Å-3 in the low-temperature region.
We present a quantum phase-shift calculation of the two-dimensional second virial coefficient B22D{B}_{2}^{2\mathrm{D}}B22D​ and spin-wave characteristic parameters for spin-polarized quantum gases H\ensuremath{\uparrow}, D\ensuremath{\uparrow}, and 3He^{3}\mathrm{He}3He. Emphasis is put on the study of quantum exchange effects which become essential in the 1-K range and below. The dependence of B22D{B}_{2}^{2\mathrm{D}}B22D​ upon the nuclear polarization is analyzed. The calculation of statistics-independent coefficients is extended up to 50 K. The effect of a reduced dimensionality on the second virial and spin-wave coefficients is discussed and demonstrated by a comparison between 2D and 3D results. Two-dimensional bound states of helium dimers are estimated with the best present potential of Aziz et al. to be EB{E}_{B}EB​(3{(}^{3}(3He${\mathrm{\ensuremath{-}}}^{3}$He; 2D)=-0.88\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}5}$ K and EB{E}_{B}EB​(3{(}^{3}(3He${\mathrm{\ensuremath{-}}}^{4}$He; 2D)=-3.38 mK.