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Journal of Physics B Atomic Molecular and Optical Physics

Hyperfine splitting computation in the 1s1/2(e)2s1/2(mu)state of the exotic (4He2+- mu-e-)0and (3He2+- mu-e-)0atoms

V L Yakhontov, M. Ya. Amusia

Published Aug 28, 1994• 16 citations
Publisher

Abstract

An ab initio calculation of the hyperfine splitting (HFS) intervals in the 2s 1/2 (mu) 1s 1/2 (e) state of the exotic ( 4 He 2+ - mu - e - ) 0 and ( 3 He 2+ - mu - e - ) 0 atoms is carried out. The smallness of beta = m e /m mu approximately 1/200 (the electron and the mu-meson mass ratio) enables us to take into account the non-relativistic interparticle correlational interaction perturbatively. Corresponding corrections, which give the dominant contribution to the HFS, are calculated to logarithmic accuracy analytically and exactly. Together with the leading relativistic and radiative corrections the HFS values obtained turn out to be in accord with the results of other calculations.

DOI
10.1088/0953-4075/27/16/021

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physics.atom-phCross-listedarXiv:2602.18144•Feb 23, 2026
An ab initio calculation of the hyperfine splitting (HFS) intervals in the 2s 1/2 (mu) 1s 1/2 (e) state of the exotic ( 4 He 2+ - mu - e - ) 0 and ( 3 He 2+ - mu - e - ) 0 atoms is carried out. The smallness of beta = m e /m mu approximately 1/200 (the electron and the mu-meson mass ratio) enables us to take into account the non-relativistic interparticle correlational interaction perturbatively. Corresponding corrections, which give the dominant contribution to the HFS, are calculated to logarithmic accuracy analytically and exactly. Together with the leading relativistic and radiative corrections the HFS values obtained turn out to be in accord with the results of other calculations.
The bound state properties of the ground 1 1S(L=0) state and the lowest triplet 2 3S(L=0) state of the 3He, 4He, and infinityHe helium atoms are determined to very high accuracy from the results of direct numerical computations. To compute the bound state properties of these atoms the author applied his exponential variational expansion in relative/perimetric three-body coordinates. For the ground 1 1S(L=0) state and the lowest triplet 2 3S(L=0) state of the 3He, 4He, and infinityHe atoms the author also determined the lowest order QED corrections and the field component of isotopic shift (=field shift). For the 2 3S(L=0) state of the 3He atom the hyperfine structure splitting is evaluated. The considered properties of the ground 1 1S state and the lowest 2 3S state in the 3He and 4He atoms are of great interest in a number of applications.
Hyperfine splittings play an important role in quantum information and spintronics applications. They allow for the readout of the spin qubits, while at the same time providing the dominant mechanism for the detrimental spin decoherence. Their exact knowledge is thus of prior relevance. In this work, we analytically investigate the relativistic effects on the hyperfine splittings of hydrogen-like atoms, including finite-size effects of the nucleis’ structure. We start from exact solutions of Dirac’s equation using different nuclear models, where the nucleus is approximated by (i) a point charge (Coulomb potential), (ii) a homogeneously charged full sphere, and (iii) a homogeneously charged spherical shell. Equivalent modelling has been done for the distribution of the nuclear magnetic moment. For the 1s ground state and 2s excited state of the one-electron systems H1, H2, H3, and He+3, the calculated finite-size related hyperfine shifts are quite similar for the different structure models and in excellent agreement with those estimated by comparing QED and experiment. This holds also in a simplified approach where relativistic wave functions from a Coulomb potential combined with spherical-shell distributed nuclear magnetic moments promises an improved treatment without the need for an explicit solution of Dirac’s equation within the nuclear core. Larger differences between different nuclear structure models are found in the case of the anisotropic 2p3/2 orbitals of hydrogen, rendering these excited states as promising reference systems for exploring the proton structure.
Transitions between hyperfine structure states of antiprotonic 4He^4 \mathrm{He}4He at collisions with medium atoms: interaction \emph{ab initio}
Collisions of metastable antiprotonic helium atoms with atoms of the medium induce, among other processes, transitions between hyperfine structure (HFS) states, as well as shifts and broadening of microwave M1 spectral lines. In order to obtain matrix potential of interaction between (pˉHe+)(\mathrm{\bar{p} He} ^+)(pˉ​He+) and He\mathrm{He}He, we have calculated the potential energy surface (PES) in the framework of unrestricted Hartree-Fock method taking into account electron correlations in the second-order perturbation theory (MP2). With this potential, the system of close-coupling equations for HFS channels is solved numerically. Cross sections and transition rates, shifts and broadening of M1 spectral lines are calculated. They are used to obtain a numerical solution of the master equation that determines the time evolution of the HFS-states density matrix. The results are compared with the experimental data and with the results of model calculation.
Ab initio calculations of the energy level structure of H2+ that include relativistic and radiative corrections to nonrelativistic energies and the diagonal part of the hyperfine interaction have predicted the existence of four bound rovibrational levels [(v = 0, N = 0 − 2) and (v = 1, N = 0)] of the first electronically excited (A+ Σu+2) state of H2+, the (v = 1, N = 0) level having a calculated binding energy of only Eb = 1.082 219 8(4)·10−9 Eh and leading to an extremely large scattering length of 750(5) a0 for the H+ + H collision [J. Carbonell et al., J. Phys. B: At., Mol. Opt. Phys. 37, 2997 (2004)]. We present an investigation of the nonadiabatic coupling between the first two electronic states (X+ Σg+2 and A+ Σu+2) of H2+ induced by the Fermi-contact term of the hyperfine-coupling Hamiltonian. This interaction term, which mixes states of total spin quantum number G = 1/2, is rigorously implemented in a close-coupling approach to solve the spin-rovibronic Schrödinger equation. We show that it mixes states of gerade and ungerade electronic symmetry, that it shifts the positions of all weakly bound rovibrational states of H2+, and that it affects both the positions and widths of its shape resonances. The calculations demonstrate that the G = 1/2 hyperfine component of the A+ (v = 1, N = 0) state does not exist and that, for G = 1/2, the s-wave scattering lengths of the H+ + H(1s) collision are −578(6) a0 and −43(4) a0 for the F = 0 and F = 1 hyperfine components of the H(1s) atom, respectively. The binding energy of the G = 3/2 hyperfine component of the A+ (v = 1, N = 0) state is not significantly affected by the hyperfine interaction and the corresponding scattering length for the H+ + H(1s, F = 1) collision is 757(7) a0.
The study of highly charged electronic and muonic hydrogen-like ions, provides an intriguing way to probe the internal structure of their atomic nuclei. In this work, we use nuclear structure calculations to accurately calculate the hyperfine splitting of electronic and muonic hydrogen-like ions, focusing in particular on the incorporation of finite-volume corrections, such as Bohr-Weisskopf and Breit-Rosenthal, due to the penetration of the electron and muon wavefunction into the nuclear electric charge and magnetic dipole densities. These corrections are essential for refining our understanding of the nuclear magnetic dipole and electric quadrupole moments. Our simulations use a Skyrme-Hartree-Fock-BCS model known for its effectiveness in modeling well-deformed nuclei such as 159Tb64+{}^{159}\mathrm{Tb}^{64+}159Tb64+ and 165Ho66+{}^{165}\mathrm{Ho}^{66+}165Ho66+, with particular emphasis on 161,163Dy65+{}^{161,163}\mathrm{Dy}^{65+}161,163Dy65+ isotopes. It can also be generalised to multi-electron ions by studying the hyperfine anomaly between two isotopes.