Magnetic moments of short-lived nuclei with part-per-million accuracy: Paving the way for applications of $\beta$-detected NMR in chemistry and biology

We determine the magnetic dipole moment of a short-lived nucleus with part-per-million (ppm) accuracy for the first time. To achieve this two orders of magnitude improvement over previous studies, we implement a number of innovations into our $\beta$-detected Nuclear Magnetic Resonance ($\beta$-NMR) setup at ISOLDE/CERN. Using liquid samples as hosts leads to narrow, sub kHz linewidth, resonances. A simultaneous in-situ stable-isotope NMR measurement allows to calibrate and stabilize the magnetic field to ppm precision. To eliminate a large systematic error, we improve the accuracy of the reference magnetic moment using {\it ab initio} calculations of NMR shielding constants. We demonstrate the potential of this combined approach for the 1.1 s half-life radioactive nucleus $^{26}$Na. Our technique can be extended to other isotopic chains, providing accurate magnetic moments for many short-lived nuclei. This will open the path towards interdisciplinary applications of $\beta$-NMR in biochemistry, where ppm shifts in signals need to be detected.