Shell structure of potassium isotopes deduced from their magnetic moments

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.

Purpose: Extend our knowledge about the evolution of the $1/2^{+}$ and $3/2^{+}$ states for K isotopes beyond the $N=28$ shell gap.

Method: High-resolution collinear laser spectroscopy on bunched atomic beams.

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 ab initio framework is also presented.

Conclusions: The dominant component of the wave function for the odd-$A$ isotopes up to ${}^{45}\mathrm{K}$ is a $\pi 1d_{3/2}^{-1}$ hole. For ${}^{47,49}\mathrm{K}$, 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 ${}^{51}\mathrm{K}$. 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 ${}^{48}\mathrm{K}$, a significant amount of mixing with $\pi 2s_{1/2}^{-1}\otimes \nu \left(pf\right)$ is observed leading to a $I^{\pi }=1^{-}$ ground state. For ${}^{50}\mathrm{K}$, the ground-state spin-parity is $0^{-}$ with leading configuration $\pi 1d_{3/2}^{-1}\otimes \nu 2p_{3/2}^{-1}$.

Figure 1

Experimental energies for $1/2^{+}$ and $3/2^{+}$ states in odd-$A$ K isotopes. Inversion of the nuclear spin is obtained in ${}^{47,49}\mathrm{K}$ and reinversion back in ${}^{51}\mathrm{K}$. Results are taken from [16, 23, 24, 25]. Ground-state spin for ${}^{49}\mathrm{K}$ and ${}^{51}\mathrm{K}$ were established [22].

Figure 2

Schematic representation of the setup for collinear laser spectroscopy at ISOLDE.

Figure 3

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

Figure 4

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 ${}^{39}\mathrm{K}$ and ${}^{49}\mathrm{K}$.

Figure 5

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 [16, 23, 24, 25] are in good agreement with the shell-model calculations using different effective interactions: SDPF-NR (red dots) and SDPF-U (blue triangles). For ${}^{49}\mathrm{K}$, 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 ${}^{51}\mathrm{K}$.

Figure 6

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

Figure 7

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 ${}^{38\text{–}46}\mathrm{K}$. Only for ${}^{48}\mathrm{K}$ 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.

Figure 8

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 ${}^{40}\mathrm{K}$ and ${}^{42}\mathrm{K}$, which might originate from lack of the excitations across $Z,N=20$, overall reasonable agreement between the experimental and theoretical results is observed.

Figure 9

Experimental energy spectrum of ${}^{48}\mathrm{K}$ adapted from Ref. [15] using the fact that the nuclear spin is firmly established to be $1^{-}$ [26]. Results are compared to the calculated spectra from different effective interactions: SDPF-NR and SDPF-U.

Figure 10

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 $A=38\text{–}46$ and $A=48,50\text{–}51$ 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 ${}^{47\text{–}49}\mathrm{K}$ indicates mixing in the wave function.