Recoil corrections with finite nuclear size in hydrogenic systems

TL;DR

This paper derives formulas for nuclear-recoil and finite-nuclear-size effects in hydrogenic systems at high orders without expanding in nuclear charge radius, improving precision in atomic measurements and correcting previous approximations.

Contribution

It provides new formulas valid for both electronic and muonic atoms at orders $(Z\alpha)^5$ and $(Z\alpha)^6$ without nuclear radius expansion, clarifying the nuclear-size contribution in isotope shifts.

Findings

Corrects unphysical linear-in-$r_C$ term in Breit approximation.

Shows full QED treatment yields quadratic $r_C$ dependence.

Highlights importance for high-precision atomic measurements.

Abstract

Formulas for the combined nuclear-recoil and finite-nuclear-size effects of order $(Z\,\alpha)^5$ and $(Z\,\alpha)^6$ are derived without any expansion in the nuclear charge radius $r_C$, making them applicable to both electronic and muonic atoms. The obtained results are particularly relevant for high-precision determinations of root-mean-square charge radii from muonic atom spectroscopy. We demonstrate that calculations of the atomic isotope shift based on the widely used Breit approximation give rise to an unphysical nuclear-size contribution that is linear in the nuclear charge radius $r_C$ at order $(Z\,\alpha)^5$. This spurious term vanishes in a full QED treatment, leaving the correct contribution quadratic in $r_C$. For electronic atoms, this quadratic term is significantly smaller than the spurious linear contribution.