Quadrupole Moments of 29Mg and 33Mg

TL;DR

This paper reports measurements of quadrupole moments for 29Mg and 33Mg using laser spectroscopy, providing data that supports shell-model predictions and enhances understanding of the nuclear structure within the island of inversion.

Contribution

It presents the first precise measurements of quadrupole moments for these isotopes, confirming shell-model predictions and contributing to nuclear structure knowledge.

Findings

Quadrupole moments are consistent with shell-model predictions.

Supports the existence of the island of inversion.

Provides new experimental data for light nuclei.

Abstract

The quadrupole moments of 29Mg and 33Mg have been constrained by collinear laser spectroscopy at CERN-ISOLDE. The values are consistent with shell-model predictions, thus supporting the current understanding of light nuclei associated with the "island of inversion".

Tables

Table 1. Table 1: Hyperfine B 𝐵 B parameters and quadrupole moments from this work, compared with shell-model calculations.

	$I^{\pi }$	$B\left(3p^2{P}_{3/2}\right)$ (MHz)	$Q$ (mb)	$Q_{\text{theory}}$ (mb)
25Mg	$5/2^{+}$	$+24\left(2\right)$	$+199.4\left(20\right)$111Reference quadrupole moment Sundholm and Olsen (1991)	$+217$222Shell-model calculations using the USDB Hamiltonian Brown and Richter (2006)
29Mg	$3/2^{+}$	$-19\left(5\right)$	$-158\left(44\right)$	$-110$222Shell-model calculations using the USDB Hamiltonian Brown and Richter (2006)
33Mg	$3/2^{-}$	$+16\left(11\right)$	$+134\left(92\right)$	$+157$333The two particle-hole value from Ref. Yordanov et al. (2007)

Full text

quadrupole moments, 29Mg, 33Mg, laser spectroscopy, shell model, island of inversion

I Introduction

The deformation in light nuclei in proximity of $N=20$ has been studied extensively since the 1970’s when it was first discovered Thibault et al. (1975); Huber et al. (1978); Détraz et al. (1979). The magnesium isotopes in particular attracted considerable attention, including measurements by collinear laser spectroscopy Neyens et al. (2005); Kowalska et al. (2008); Yordanov et al. (2007, 2012), in an effort to uncover the boundaries of this island of inversion, known as such due to the apparent inversion of states in the relevant shell-model description. As of now, it is well understood that 29Mg Kowalska et al. (2008) follows the expected level ordering in $sd$-shell nuclei, while 33Mg Yordanov et al. (2007) has a ground state determined by particle-hole excitations across the $N=20$ shell gap. There has been much debate Tripathi et al. (2008); Yordanov et al. (2010); Kanungo et al. (2010) as to what is the exact number of particle-hole excitations involved in 33Mg and the associated parity assignment. The experimental evidence supports a $3/2^{-}$ configuration, as outlined in the critical evaluation by Neyens Neyens (2011).

II Experiment

The quadrupole moments presented here are not derived from dedicated measurements as they could be considered a byproduct from the $\beta$-asymmetry detection and NMR work on the two cases Kowalska et al. (2008); Yordanov et al. (2007). Collinear laser spectroscopy was carried out at CERN-ISOLDE with the experimental setup discussed in the original publications. The ions of ${}^{\text{29, 33}}$Mg were excited in the transition $3s\;^{2}S_{1/2}\rightarrow 3p\;^{2}P_{3/2}$ at $280$ nm Kaufman and Martin (1991) which provides quadrupole interaction in the excited state. At the time, the hyperfine structure was utilized primarily as a tool for aiding the NMR measurements by observing the amount of nuclear orientation produced by optical pumping and as a signature of the signs of the electromagnetic moments. Consequently, the quadrupole splitting was not discussed in Refs. Kowalska et al. (2008); Yordanov et al. (2007). This was partly justified by the fact that the hyperfine structure in the $3p\;^{2}P_{3/2}$ state is not resolved, thus suffering a power-dependent cross-pumping effect that needed to be understood. It was later shown Yordanov et al. (2012) that $\beta$-asymmetry measurements of the atomic hyperfine structure could be reproduced quantitatively with the formalism outlined Refs. Keim et al. (2000); Kowalska et al. (2008). We are therefore making at attempt to evaluate the quadrupole moments of ${}^{\text{29, 33}}$Mg from the existing and limited data by using the realistic polarization fit function described in the aforementioned work.

III Results

The hyperfine-structure of 29Mg in Fig. 1 is partly resolved. In the inset (a) the data are fitted with the realistic polarization function discussed in our previous work Yordanov et al. (2007, 2012); Kowalska et al. (2008); Keim et al. (2000). The magnetic hyperfine parameter in the $3p\;^{2}P_{3/2}$ state is substituted with the value reported in Ref. Kowalska et al. (2008), thus the relative position of the resonances is optimized by the variation of the quadrupole hyperfine parameter. In order to obtain a handle on the systematic uncertainty associated with the choice of fit function we have performed in the inset (b) a basic fit using three Lorentzian profiles. The resulting quadrupole hyperfine parameters in the two cases are $-23(2)$ MHz and $-15(2)$ MHz, respectively. As a final result in Tab. 1 we quote the mean value of the two, with half the difference being adopted as the associated systematic uncertainty and being added in quadrature to the statistical ones. The spectrum of 33Mg in Fig. 2, and its fit, have been previously discussed in connection with the corresponding NMR measurement Yordanov et al. (2007), thus fixing the nuclear spin and the sign of the associated magnetic moment. With the magnetic splitting tied to the measured $g$ factor one extracts the quadrupole hyperfine parameter shown in Tab. 1. Both isotopes have been referenced to 25Mg whose hyperfine structure is shown in Fig. 3. The fit therein, using asymmetric Voigt profiles Yordanov et al. (2017), is aided by fixing the magnetic hyperfine parameter of the $3s\;^{2}S_{1/2}$ state to the precise value from Ref. Itano and Wineland (1981). Thus, one determines a ratio of $A(3p\;^{2}P_{3/2})/A(3s\;^{2}S_{1/2})=0.032(1)$ and the $B$ factor presented in the table. The quadrupole moments in Tab. 1 are calculated proportionally to the know value of 25Mg Sundholm and Olsen (1991).

IV Discussion

Shell-model calculations have been carried out as following. For ${}^{\text{25, 29}}$Mg we used the universal $sd$ Hamiltonian USDB Brown and Richter (2006) and the code NuShellX @ MSU Brown and Rae (2014) while the calculated quadrupole moment of 33Mg is adopted from Ref. Yordanov et al. (2007) where the code Oxbash was used Brown et al. (2004) and the $sd\text{-}pf$ interaction from Ref. Nummela et al. (2001). The agreement with experiment for all three cases is generally good. In fairness to the reader there is no specific addition to make to previous discussions, the main reason being not the limited precision, but mostly nature itself. In the case of 33Mg the configurations with one and two particle-hole excitations produce nearly identical quadrupole moments Yordanov et al. (2007). The state with no cross-shell excitations can indeed be excluded on the basis of this work, albeit it is easily done so on the basis of the magnetic moment alone. Both theory and experiment are consistent on the $sd$ cases of ${}^{\text{25, 29}}$Mg, which have been clearly dissociated from the island of inversion by former studies.

V Conclusions

In summary, we have determined the quadrupole moments of 29Mg and 33Mg with an uncertainty of about 30% and 70%, respectively. To our knowledge, these are the first quadrupole moments derived from $\beta$-asymmetry detected hyperfine structure. It should be noted that their accuracy is affected by the unresolved levels of the excited atomic state and by the experimental procedures which were not optimized for higher resolution. Higher precision and accuracy are in principle possible in a dedicated experiment. Comparison with shell-model calculations does not contradict the well-established picture of light nuclei in the region.

Acknowledgements.

This work has been supported by the German Federal Ministry for Education and Research under contract no. 06MZ175 and 06MZ215, the Helmholtz Association (VH-NG-037), the P6-EURONS (RII3-CT-2004-506065), the IUAP project P5/07 of OSCT Belgium, the FWO-Vlaanderen and the Marie Curie IEF program (MEIF-CT-2006-042114). We thank the ISOLDE technical group for their professional assistance.