Second-order hyperfine correction to H, D, and $^3$He energy levels

TL;DR

This paper calculates the complete second-order hyperfine corrections for hydrogen, deuterium, and helium-3 energy levels, refining nuclear charge radius measurements and resolving previous discrepancies between muonic and electronic helium determinations.

Contribution

It provides the full second-order hyperfine correction calculations for specific atomic states, improving the accuracy of isotope shift analyses.

Findings

Second-order corrections for $^3$He states are quantified as -2.075 kHz and -0.305 kHz.

Results refine the nuclear charge radius difference from isotope shifts.

The corrections resolve discrepancies between muonic and electronic helium measurements.

Abstract

The complete second-order hyperfine-interaction correction is calculated for centroid energy levels of H, D, and $^3$He atoms. For $^3$He, the corrections of $-2.075$~kHz and $-0.305$~kHz beyond the leading hyperfine-mixing contribution are obtained for the $2^1S$ and $2^3S$ states, respectively. These results shift the nuclear charge radii difference derived from the $^3$He -$^4$He isotope shift and largely resolve the previously reported disagreement between the muonic and electronic helium determinations [van der Werf et al., arXiv:2306.02333 (2023); Schuhmann et al., arXiv:2305.11679 (2023)].