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Physical Review A

Calculation of spin-polarized muonium-muonium and spin-polarized hydrogen-hydrogen scattering

M.-S. Wu, Y. Zhang, Jun-Yi Zhang, Kálmán Varga, Z.-C. Yan

Published Nov 13, 2025• 0 citations
Publisher

Abstract

In this paper, we investigate spin-polarized muonium-muonium and spin-polarized hydrogen-hydrogen scattering for total orbital angular momentum $L=0$ using the non-Born-Oppenheimer confined variational method combined with explicitly correlated Gaussian basis functions. Our approach yielded accurate $S$-wave phase shifts and elastic cross sections at scattering momenta below $0.6{a}_{0}^{\ensuremath{-}1}$, where ${a}_{0}$ is the Bohr radius. By utilizing the effective range formula, we determined the $S$-wave scattering lengths to be $3.70{a}_{0}$ and $1.35{a}_{0}$ for spin-polarized muonium-muonium and spin-polarized hydrogen-hydrogen scattering, respectively.

DOI
10.1103/dn4g-4pp4

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In this paper, we investigate spin-polarized muonium-muonium and spin-polarized hydrogen-hydrogen scattering for total orbital angular momentum L=0L=0L=0 using the non-Born-Oppenheimer confined variational method combined with explicitly correlated Gaussian basis functions. Our approach yielded accurate SSS-wave phase shifts and elastic cross sections at scattering momenta below 0.6a0\ensuremath−10.6{a}_{0}^{\ensuremath{-}1}0.6a0\ensuremath−1​, where a0{a}_{0}a0​ is the Bohr radius. By utilizing the effective range formula, we determined the SSS-wave scattering lengths to be 3.70a03.70{a}_{0}3.70a0​ and 1.35a01.35{a}_{0}1.35a0​ for spin-polarized muonium-muonium and spin-polarized hydrogen-hydrogen scattering, respectively.
The study of muonium-muonium binding and scattering is crucial for precision experiments involving muonium. In this paper, we investigate the bound-state energies and structures of the (muonium)2{(\mathrm{muonium})}_{2}(muonium)2​ molecule for total orbital angular momentum L=0L=0L=0 using the non-Born-Oppenheimer variational method combined with explicitly correlated Gaussian basis functions. Additionally, we employ the confined variational method to explore muonium-muonium scattering at scattering momenta below $0.11{a}_{0}^{\ensuremath{-}1}$, where a0{a}_{0}a0​ is the Bohr radius. Our approach provides accurate SSS-wave phase shifts, as well as the SSS-wave scattering length of 3.71a03.71{a}_{0}3.71a0​ using the effective range formula. In this study, the two positive muons and two electrons are considered in a spin singlet state.
Positronium-hydrogen (Ps-H) scattering is of interest, as it is a fundamental four-body Coulomb problem. We have investigated low-energy Ps-H scattering below the Ps(n=2) excitation threshold using the Kohn variational method and variants of the method with a trial wavefunction that includes highly correlated Hylleraas-type short-range terms. We give an elegant formalism that combines all Kohn-type variational methods into a single form. Along with this, we have also developed a general formalism for Kohn-type matrix elements that allows us to evaluate arbitrary partial waves with a single codebase. Computational strategies we have developed and use in this work will also be discussed.With these methods, we have computed phase shifts for the first six partial waves for both the singlet and triplet states. The 1S and 1P phase shifts are highly accurate results and could potentially be viewed as benchmark results. Resonance positions and widths for the 1S-, 1P-, 1D-, and 1F-waves have been calculated.We present elastic integrated, elastic differential, and momentum transfer cross sections using all six partial waves and note interesting features of each. We use multiple effective range theories, including several that explicitly take into account the long-range van der Waals interaction, to investigate scattering lengths for the 1,3S and 1,3P partial waves and effective ranges for the 1,3S-wave.
Low-energy properties of spin-polarized electron-helium (23S)(2\phantom{\rule{0.16em}{0ex}}^{3}S)(23S) and spin-polarized positronium-helium (23S)(2\phantom{\rule{0.16em}{0ex}}^{3}S)(23S) scattering are calculated using the modified confined variational method. To take the van der Waals interaction into consideration, the van der Waals coefficient between positronium and helium (23S)(2\phantom{\rule{0.16em}{0ex}}^{3}S)(23S) is calculated using the explicitly correlated Gaussian basis. Compared with the ground state noble gases, the larger scattering length and zero pickoff annihilation rate indicates that He(23S)\mathrm{He}(2\phantom{\rule{0.16em}{0ex}}^{3}S)He(23S) may be a more suitable cooling gas. In the energy range we considered, no similarity is found for the SSS-wave cross sections between these two scatterings. A Ramsauer-Townsend minimum is observed in spin-polarized electron-helium(23S)(2\phantom{\rule{0.16em}{0ex}}^{3}S)(23S) scattering due to the large polarization of metastable helium. This may open a new way to find and study the Ramsauer-Townsend effect.
The confined variational method is applied to investigate the low-energy elastic scattering of orthopositronium from H2{\text{H}}_{2}H2​ by first-principles quantum mechanics. Describing the correlations with explicitly correlated Gaussians, we obtain accurate sss-wave phase shifts and pickoff annihilation parameters for different incident momenta. By a least-squares fit of the data to the effective-range theory, we determine the sss-wave scattering length AS=2.02a0{A}_{S}=2.02{a}_{0}AS​=2.02a0​ and the zero-energy value of the pickoff annihilation parameter, 1Zeff=0.1839{}^{1}{Z}_{\text{eff}}=0.18391Zeff​=0.1839. The obtained 1Zeff{}^{1}{Z}_{\text{eff}}1Zeff​ agrees well with the precise experimental value of 0.186(1) [G. L. Wright et al., J. Phys. B 16, 4065 (1983)] and the obtained AS{A}_{S}AS​ agrees well with the value of 2.1(2)a02.1(2){a}_{0}2.1(2)a0​ estimated from the average experimental momentum-transfer cross section for positronium energy below 0.3 eV [F. Saito et al., J. Phys. B 36, 4191 (2003)].
We have continued our investigation into the elastic scattering of positronium atoms by hydrogen atoms, extending the work to include the spin-one (triplet) state of the two electrons. Once again we have extracted the SSS-wave phase shifts from normalizable trial functions by examining the spatial behavior of the center-of-mass wave functions. An effective-range expansion linear in the positronium energy fits the results very well and leads to a scattering length ${a}^{(\ensuremath{-})}\ensuremath{\simeq}2.36{a}_{0}$, compared with the static result ${a}^{(\ensuremath{-})}=2.476{a}_{0}$ of Hara and Fraser. The SSS-wave total and ortho-para conversion cross sections are calculated by making use of both 1S^{1}S1S and 3S^{3}S3S results. In the 3S^{3}S3S case, we see no indication of any resonances below the excitation threshold at 5.1 eV.
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.