Shell structure of potassium isotopes deduced from their magnetic moments

$\textbf{Background:}$ Ground-state spins and magnetic moments are sensitive to the nuclear wave function, thus they are powerful probes to study the nuclear structure of isotopes far from stability. \\ \\ $\textbf{Purpose:}$ Extend our knowledge about the evolution of the $1/2^+$ and $3/2^+$ states for K isotopes beyond the $N = 28$ shell gap. \\ \\ $\textbf{Method:}$ High-resolution collinear laser spectroscopy on bunched atomic beams. \\ \\ $\textbf{Results:}$ From measured hyperfine structure spectra of K isotopes, nuclear spins and magnetic moments of the ground states were obtained for isotopes from $N = 19$ up to $N = 32$. In order to draw conclusions about the composition of the wave functions and the occupation of the levels, the experimental data were compared to shell-model calculations using SDPF-NR and SDPF-U effective interactions. In addition, a detailed discussion about the evolution of the gap between proton $1d_{3/2}$ and $2s_{1/2}$ in the shell model and $\textit{ab initio}$ framework is also presented. \\ \\ $\textbf{Conclusions:}$ The dominant component of the wave function for the odd-$A$ isotopes up to $^{45}$K is a $\pi 1d_{3/2}^{-1}$ hole. For $^{47,49}$K, the main component originates from a $\pi 2s_{1/2}^{-1}$ hole configuration and it inverts back to the $\pi 1d_{3/2}^{-1}$ in $^{51}$K. For all even-$A$ isotopes, the dominant configuration arises from a $\pi 1d_{3/2}^{-1}$ hole coupled to a neutron in the $\nu 1f_{7/2}$ or $\nu 2p_{3/2}$ orbitals. Only for $^{48}$K, a significant amount of mixing with $\pi 2s_{1/2}^{-1} \otimes \nu (pf)$ is observed leading to a $I^{\pi}=1^{-}$ ground state. For $^{50}$K, the ground-state spin-parity is $0^-$ with leading configuration $\pi 1d_{3/2}^{-1} \otimes \nu 2p_{3/2}^{-1}$.