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(color online) Experimental energies for $1/2^{+}$ and $3/2^+$ states in odd-A K isotopes. Inversion of the nuclear spin is obtained in $^{47,49}$K and reinversion back in $^{51}$K. Results are taken from \cite{Measday2006,Huck1980,Weissman2004,Broda2010}. Ground-state spin for $^{49}$K and $^{51}$K were established \cite{Papuga2013}.

(color online) Schematic representation of the setup for collinear laser spectroscopy at ISOLDE.

(color online) The hyperfine spectra of $^{48-51}$K (a-d) obtained by collinear laser spectroscopy. The spectra are shown relative to the centroid of $^{39}$K.

(color online) Experimental magnetic moments (black dots) compared to the shell-model calculation using SDPF-NR (red dashed line) and SDPF-U (blue solid line) interactions and effective $g$ factors (see text for more details). In general a very good agreement between experimental and theoretical results is observed, except for $^{39}$K and $^{49}$K.

(color online) Energy difference between the two lowest states with $I^{\pi}=1/2^{+}$ and $3/2^{+}$ for odd-A K isotopes from $N = 24$ up to $N = 34$. Experimental results (black stars) taken from \cite{Measday2006,Huck1980,Weissman2004,Broda2010} are in good agreement with the shell-model calculations using different effective interactions: SDPF-NR (red dots) and SDPF-U (blue triangles). For $^{49}$K, only the SDPF-NR interaction correctly predicts the spin of the ground state to be $1/2^{+}$. The shaded area represents the expected region based on the measured ground-state spin and the shell-model calculation for the first excited state in $^{51}$K.

(color online) (a) Energy difference between the lowest $1/2^{+}$ and $3/2^{+}$ states obtained in $^{37-53}$K from {\it{ab initio}} Gorkov-Green`s function calculations and experiment. {\it{Ab initio}} results have been shifted by 2.58\,MeV to match the experimental $(1/2^{+}-3/2^{+})$ splitting in $^{47}$K. (b) $\pi d_{3/2}$ and $\pi s_{1/2}$ effective single-particle energies (ESPE) in $^{37-53}$K calculated in Gorkov-Green's functions theory.

(color online) Experimental $g$ factors (black dots) compared to the semi-empirical values (red solid line) calculated from the neighboring isotopes. Based on the good agreement between the experimental and semi-empirical $g$ factors, the dominant component of the wave functions can be easily established for $^{38-46}$K. Only for $^{48}$K a strong mixing between the $\pi 2s_{1/2}^{-1} \otimes \nu 2p_{3/2}$ and the $\pi 1d_{3/2}^{-1} \otimes \nu 2p_{3/2}$ in the wave function is found.

(color online) Measured magnetic moments (black dots) for even-A K isotopes compared to the shell-model calculations using the SDPF-NR (red dashed line) as well as the SDPF-U (blue solid line) effective interaction. Although there is a larger deviation present for $^{40}$K and $^{42}$K, which might originate from lack of the excitations across $Z, N =20$, overall reasonable agreement between the experimental and theoretical results is observed.

Experimental energy spectrum of $^{48}$K adapted from Ref.\,\cite{Krolas2011} using the fact that the nuclear spin is firmly established to be $1^{-}$ \cite{Kreim2014}. Results are compared to the calculated spectra from different effective interactions: SDPF-NR and SDPF-U.

(color online) Proton occupation of the $\pi 2s_{1/2}$ and the $\pi 1d_{3/2}$ orbitals from the shell-model calculations using the SDPF-NR and SDPF-U effective interactions. It is clear that for isotopes from $A = 38 - 46$ and $A =48, 50 - 51$ the dominant component in the configuration is a hole in the $\pi 1d_{3/2}$. In the case of $I^{\pi} = 1/2^{+}$ isotopes, a proton hole is located in the $2s_{1/2}$. Deviation from integer numbers for $^{47-49}$K indicates mixing in the wave function.