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Physical Review A

Scattering lengths and effective ranges for He-He and spin-polarized H-H and D-D scattering

Michael J. Jamieson, A. Dalgarno, M. Kimura

Published Mar 1, 1995• 59 citations
Publisher

Abstract

The modifications of effective-range theory, necessitated by the long-range nature of interatomic dispersion and relativistic retardation forces, are discussed. A numerical comparison is given of parameters describing $^{3}\mathrm{\ensuremath{-}}^{3}$He, $^{3}\mathrm{\ensuremath{-}}^{4}$He, and $^{4}\mathrm{\ensuremath{-}}^{4}$He scattering at very low energies calculated with several recently published interaction potentials. The effect of relativistic retardation on these parameters and on the calculated binding energies of the $^{4}\mathrm{\ensuremath{-}}^{4}$He dimer is shown. It is demonstrated that all of these quantities are extremely sensitive to the potential. Scattering lengths and effective ranges for the lowest triplet interactions of H-H and D-D are also reported.

DOI
10.1103/physreva.51.2626

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•May 15, 1982•0 citations
The modifications of effective-range theory, necessitated by the long-range nature of interatomic dispersion and relativistic retardation forces, are discussed. A numerical comparison is given of parameters describing 3\ensuremath−3^{3}\mathrm{\ensuremath{-}}^{3}3\ensuremath−3He, 3\ensuremath−4^{3}\mathrm{\ensuremath{-}}^{4}3\ensuremath−4He, and 4\ensuremath−4^{4}\mathrm{\ensuremath{-}}^{4}4\ensuremath−4He scattering at very low energies calculated with several recently published interaction potentials. The effect of relativistic retardation on these parameters and on the calculated binding energies of the 4\ensuremath−4^{4}\mathrm{\ensuremath{-}}^{4}4\ensuremath−4He dimer is shown. It is demonstrated that all of these quantities are extremely sensitive to the potential. Scattering lengths and effective ranges for the lowest triplet interactions of H-H and D-D are also reported.
We compare a number of helium–helium potentials with respect to their predictions of dimer binding energy, scattering length, effective range and Efimov states. We also study the effect of retardation on the ‘‘best’’ potential. All realistic potentials support a weakly bound dimer, while none supports an Efimov state. We agree with other authors that retardation decreases the binding energy by about 10%. Finally, we investigated the effect on the binding energy from the application of retardation over different ranges of separation. The precise effects of retardation at short range in realistic potentials require further study.
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.
The best available intermolecular potential for helium by Aziz, Nain, Carley, Taylor, and McConville is shown here for the first time to have a 4He2 bound state. Two numerical analyses, namely, eigenvalue solution and effective range theory, are used to support this conclusion. Unlike usual chemically bound species, the binding energy of this very weakly bound level is found to be only 8.3×10−4 K, which is four orders of magnitude smaller than the potential well depth ε = 10.8 K. The scattering length for He+He collisions, determined from effective range theory, is used to calculate the elastic cross section in the very low energy limit. The results (1.878×105 Å2 for 4He+4He and 6.035×102 Å for 3He+3He) are consistent with measurements at the lowest velocities yet attained. In terms of the estimated uncertainties of the parameters of the potential of Aziz and co-workers, it is shown that it is very likely that a bound state of the 4He2 molecule does in fact exist.
Instead of calculating phase shifts at small scattering energies for a given two-body central potential and from these deduce by extrapolation to zero energy the associated scattering length and effective range, we write those parameters as integrals which are evaluated numerically to high accuracy for the Lennard-Jones and Aziz et al. interatomic potentials for both 3He^{3}\mathrm{He}3He and 4He^{4}\mathrm{He}4He, as well as for the Kolos-Wolniewicz potential between hydrogen isotopes.
The two-body scattering length a for 4He{}^{4}\mathrm{He}4He atoms is much larger than their effective range rs.{r}_{s}.rs​. As a consequence, low-energy few-body observables have universal characteristics that are independent of the interaction potential. Universality implies that, up to corrections suppressed by rs/a,{r}_{s}/a,rs​/a, all low-energy three-body observables are determined by a and a three-body parameter ${\ensuremath{\Lambda}}_{*}.$ We give simple expressions in terms of a and ${\ensuremath{\Lambda}}_{*}$ for the trimer binding-energy equation, the atom-dimer scattering phase shifts, and the rate for three-body recombination at threshold. We determine ${\ensuremath{\Lambda}}_{*}$ for several 4He{}^{4}\mathrm{He}4He potentials from the calculated binding energy of the excited state of the trimer and use it to obtain the universality predictions for the other low-energy observables. We also use the calculated values for one potential to estimate the effective range corrections for the other potentials.
The best available intermolecular potential for helium by Aziz, Nain, Carley, Taylor, and McConville is shown here for the first time to have a /sup 4/He/sub 2/ bound state. Two numerical analyses, namely, eigenvalue solution and effective range theory, are used to support this conclusion. Unlike usual chemically bound species, the binding energy of this very weakly bound level is found to be only 8.3 x 10/sup -4/ K, which is four orders of magnitude smaller than the potential well depth epsilon = 10.8 K. The scattering length for He+He collisions, determined from effective range theory, is used to calculate the elastic cross section in the very low energy limit. The results (1.878 x 10/sup 5/ A/sup 2/ for /sup 4/He+/sup 4/He and 6.035 x 10/sup 2/ A for /sup 3/He+/sup 3/He) are consistent with measurements at the lowest velocities yet attained. In terms of the estimated uncertainties of the parameters of the potential of Aziz and co-workers, it is shown that it is very likely that a bound state of the /sup 4/He/sub 2/ molecule does in fact exist.