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Physical review. B, Condensed matter

Virial calculations for H↓ in two and three dimensions

E. V. L. de Mello, John J. Rehr, O. E. Vilches

Published Oct 1, 1983• 19 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.28.3759

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Physical review. B, Condensed matter•Jul 1, 1987•20 citations
Properties of spin-polarized hydrogen (H\ensuremath↓{\mathrm{H}}_{\mathrm{\ensuremath{\downarrow}}}H\ensuremath↓​) 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.
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 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.
The quantum theory of the third virial coefficient CCC is discussed. Three types of intermolecular pair forces must be distinguished. (1) No bound or low-lying two- and/or three-body states exist. The first four terms of the low-temperature expansion of CBE{C}_{\mathrm{BE}}CBE​ are obtained. They depend on the scattering length, the effective range, and a third length which cannot be inferred from scattering data. The limitations of the applicability of such expansions are discussed, both for He4{\mathrm{He}}^{4}He4 and He3{\mathrm{He}}^{3}He3, by means of a comparison of the corresponding expansion for the second virial coefficient BBB with detailed numerical results known for specific potentials. (2) Existence of a near zero energy level both for the two- and the three-body system. It is shown how in this case the actual potentials may be replaced by suitably matched boundary conditions on the two- and the three-body wave functions near the respective coordinate origins. It is first explained how the method applies to BBB. Then the leading term of CCC is explicitly determined. (3) Existence of strongly bound two- and three-body states. An approximate expression for CCC is given by treating the single atoms and the binary and ternary compounds as a system of three ideal gases in chemical equilibrium.
We study the virial expansion for three-dimensional Bose and Fermi gases at finite temperature using an approximation that only considers two-body processes and is valid for high temperatures and low densities. The first virial coefficients are computed and the second is exact. The results are obtained for the full range of values of the scattering length, and the unitary limit is recovered as a particular case. A weak coupling expansion is performed and the free case is also obtained as a proper limit. The influence of an anisotropic harmonic trap is considered using the local density approximation (LDA), analytical results are obtained, and the special case of the isotropic trap is discussed in detail.
Accurate scattering lengths and effective ranges for hydrogen atomic collisions are determined from effective range theory. The precise ab initio b3Σ+u potential energy curve values of H2 calculated by Kolos and Wolniewicz and C6, C8, C10 long-range expansion coefficients are used in obtaining accurate representation of the two-body interaction potential. The s-wave scattering length obtained for this potential (A =1/33455 ao) is used to calculate the zero temperature energy as a function of density of electron spin-polarized ("spin-aligned") atomic hydrogen (H+) using previous many-body perturbation theory results for bosons. Neglecting small potential shape dependent terms, one obtains E/N (energy per particle ) as a function of density, which may be compared with similar results obtained from Monte-Carlo or variational calculations. The scattering lengths and effective ranges for the other hydrogen isotopic systems are also presented. It is shown that the first bound state for the b3Σ+u potential (within the Born-Oppenheimer approximation) only exists for a hypothetical isotopic species with mass larger than a value m = 3.258176 g/mole (where the scattering length becomes infinite). In terms of the quantum parameter, η ≡ [MATH]2/ (2µεσ2), two-body binding occurs in the b3Σ+u potential only for η < 0.17094. The second order liquid-to-gas transition (corresponding to the onset of infinite-body binding) occurs at the accurately determined critical value of the quantum parameter ηCB = 0.43891. This corresponds to zero scattering length.
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.