Testing microscopically derived descriptions of nuclear collectivity: Coulomb excitation of 22Mg

Many-body nuclear theory utilizing microscopic or chiral potentials has developed to the point that collectivity might be dealt with in an {\it ab initio} framework without the use of effective charges; for example with the proper evolution of operators, or alternatively, through the use of an appropriate and manageable subset of particle-hole excitations. We present a precise determination of $E2$ strength in $^{22}$Mg and its mirror $^{22}$Ne by Coulomb excitation, allowing for rigorous comparisons with theory. No-core symplectic shell-model calculations were performed and agree with the new $B(E2)$ values while in-medium similarity-renormalization-group calculations consistently underpredict the absolute strength, with the missing strength found to have both isoscalar and isovector components.


Introduction
Recent developments in many-body nuclear theory have seen a great advance in the number of nuclei accessible to microscopically derived theoretical modelsincluding those constructed in an ab initio framework [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. As these models increasingly reach regions of the nuclear landscape inaccessible to experiment, it is essential that their performance is scrutinized in detail using less-exotic systems where high-precision experimental data are available. The sd-shell lies between the traditional shell-model proton and neutron magic numbers of 8 and 20 and is an ideal laboratory for testing new models. The region contains examples of many phenomena found across the nuclear landscape, ranging from α-clustering [16] and Borromean-nuclei [17], to shell evolution [18] and high degrees of collective deformation [19]. In particular, the sd-shell provides an excellent opportunity for investigations of collectivity through the probing of first-excited 2 + states in midshell even-even nuclei, which are typically dominated by collective degrees of freedom. By probing transitions to such states in mirror nuclei, one is additionally sensitive to charge-dependent effects in the interaction.
Historically, the phenomenological shell model has proved a successful tool in the modeling of this mass region, with empirically fit interactions typically wellreproducing experimental data [20]. A particular limitation in the model, however, lies in the reproduction of nuclear collectivity -the bulk motion of many nucleons -and especially the electric-quadrupole (E2) strength commonly associated with it. As the shell model be-gins with an assumption of sphericity, collective E2 strength is generated through a coherent sum of many small-amplitude multi-particle multi-hole (mp-mh) excitations. A model space and interaction that achieve good reproduction of level energies does not necessarily reproduce transition strength. This strength is often underpredicted as the inclusion of a sufficiently large number of mp-mh excitations is in practice unfeasible. The typical approach is to explicitly compensate for this missing physics through an artificial inflation of the nucleon charges with phenomenological effective charges. It is therefore of considerable interest to determine whether modern microscopically derived nuclear theories are able to reproduce the experimentally observed collectivity in this region without the need for the phenomenologically derived corrections required in the shell model.
Accurate calculation of collective E2 strengths without the use of effective charges is currently being pursued within several theoretical frameworks. For example, the no-core symplectic shell model (NCSpM) has in recent years determined B(E2) values of nuclei within the sd shell, without resorting to such phenomenological corrections [21]. This model provides the capability to reach large shell-model spaces using a microscopic interaction, while being in agreement with ab initio symmetry-adapted no-core shellmodel [13] (SA-NCSM) calculations in smaller, more feasible model spaces that use the N2LO opt chiral potential [22]. A suite of ab initio many-body techniques are also able to perform calculations in the sd-shell with, for example, coupled-cluster (CC) [23], no-core shell model (NCSM) [24] and in-medium similarityrenormalization-group (IM-SRG) [25,15] methodologies demonstrating promising results in terms of levelenergy calculations. CC techniques reproduced transition strengths in self-conjugate 20 Ne and 24 Mg with precision comparable to the available experimental data [23]; however, this required the use of effective charges.
Two previous measurements of the 2 + 1 state lifetime in 22 Mg have been reported resulting in an evaluated B(E2; 2 + 1 → 0 + 1 ) of 95 ± 40 e 2 fm 4 [26][27][28]. The stable nuclide 22 Ne has been well measured, with a precisely known lifetime yielding a B(E2; 2 + 1 → 0 + 1 ) value of 46.72 ± 0.66 e 2 fm 4 [28]. Furthermore, the diagonal matrix element, 2 + 1 | E2 |2 + 1 , and thus the spectroscopic quadrupole moment of the 2 + 1 state, Q s (2 + 1 ) has also been measured in 22 Ne, yielding an evaluated value of Q s (2 + 1 ) = −0.19±0.04 b [29]. In this Letter we present a Coulomb-excitation measurement of the A = 22 mirror pair, 22 Mg- 22 Ne, through which we have significantly improved the precision of the 22 Mg B(E2) and Q s (2 + ) values. This represents the first measurement of Q s (2 + ) in an even-even T z = 1 2 (N − Z) = −1 nuclide, where Z (N) is the number of protons (neutrons). The new data are now of sufficient quality to test state-of-the-art microscopically derived theoretical calculations. It is found that NCSpM predictions for this A = 22 mirror pair are in excellent agreement with experimental data.

Experimental details
The first-excited 2 + states in 22 Mg and its stable mirror 22 Ne were populated through Coulomb excitation in normal kinematics at the TRIUMF-ISAC-II facility. 22 Mg was produced using a 50 µA, 480-MeV proton beam impinged on a SiC target coupled to an ion guide laser ion source (IG-LIS) [30,31]. With laser resonance ionization and suppression of isobaric contamination from surface ionization a 22 Na suppression in excess of 10 6 compared to the conventional hot cavity-laser ion source was achieved [32]. It was therefore possible to accelerate a clean beam of 22 Mg ions through the ISAC accelerator chain to the TIGRESS facility [33]. Two 22 Mg beam energies were used for the present measurement: 92.4 MeV and 83.4 MeV. Beam intensities at TI-GRESS were maintained at approximately 1 · 10 4 pps throughout the experiment. The 22 Ne beam was provided by the offline ion-source (OLIS) and accelerated by the ISAC and ISAC-II accelerators to a final energy of 54.8 MeV with a mean intensity of approximately 5 ppA.
The 22 Mg ( 22 Ne) beam was impinged onto a 97.6-% enriched, 2.6-mg/cm 2 (1.6-mg/cm 2 ) thick 110 Pd target within the BAMBINO setup at the center of the TIGRESS array. For the present measurements BAM-BINO consisted of a pair of Micron S3-type silicon detectors [34] covering angles of 20 • to 49.4 • and 131.6 • to 160 • in the laboratory frame. Scattered beam-like particles were detected in the BAMBINO S3 detectors and γ-rays de-exciting states populated in the beamand target-like nuclei were detected with TIGRESS. TIGRESS was operated in its high-efficiency configuration [35], with fourteen HPGe clover detectors at a target-to-detector distance of 11 cm. Data were acquired through the TIGRESS digital data acquisition system [36] using a single hit in one of the silicon detectors as the experimental trigger for the 22 Mg portion of the experiment, and with a particle-γ trigger for the higher-rate 22 Ne beam. A timing signal from the laser ion source was acquired with the experimental data and made it possible to distinguish prompt laserionized 22 Mg from time-random surface-ionized 22 Na events. This method of continuously monitoring surface ionized contamination was verified by periodically redirecting the beam into a Bragg detector [37] and yielded a 22 Na: 22 Mg ratio over the course of the experiment of approximately 2%.

Analysis
Data were sorted using the in-house GRSISort [38] software package, built on the ROOT [39] data analysis framework. Particle-gated γ-ray spectra were Doppler corrected for beam-like and target-like scattering kinematics on an event-by-event basis, determined by the trajectory of the detected particle in the S3 detectors. Gamma-ray spectra, Doppler corrected for 22 Mg, 22 Ne and 110 Pd are shown in Fig. 1. Due to the higher beam energies used for the 22 Mg beams, the upstream S3 detector was excluded from the analysis as a result of lying in an "unsafe" Coulomb excitation regime, i.e. the distance of closest approach was less than 5 fm [40]. In the 22 Mg analysis the data were split into six angular bins, while the 22 Ne data were analyzed on a ring-byring basis to maximize sensitivity. The data were corrected for offsets in the x-and y-directions relative to 22 Mg 22 Ne the beam axis on the basis of asymmetries in the particle distributions on the S3 detectors. Addback was applied to the TIGRESS γ-ray spectra on the basis of the subcrystal segmentation within the HPGe clover detectors. Gamma-ray detection efficiencies in TIGRESS were determined using 152 Eu, 133 Ba and 60 Co sources. Efficiency-corrected 22 Mg, 22 Ne and 110 Pd Coulomb excitation yields were then evaluated using the GOSIA and GOSIA2 software packages [41], allowing for simultaneous analysis of both beam-like and target-like excitation. As described in Ref. [42], χ 2 surface distributions could thus be created for the 0 + | E2 |2 + and 2 + | E2 |2 + matrix elements in both 22 Ne and 22 Mg, based on excitation relative to the well-known low-lying matrix elements in 110 Pd which were included in the GOSIA analysis, with yields corrected to account for the degree of enrichment of the target and the contamination in the beam. Literature 0 + 1 | E2 |2 + 1 and 2 + 1 | E2 |2 + 1 matrix elements for 22 Ne and 22 Mg were not included as experimental inputs in the analysis. The levels and transitions included in the analysis for 22 Ne and 22 Mg are shown in Fig. 2. Figures 3 and 4 show the total and 1σ χ 2 surface distributions plotted for 22 Mg, and the 1σ χ 2 surface for 22 Ne, respectively. Based on these analyses, values for the matrix elements were extracted and are summarized in Table 1 Figure 3: χ 2 surfaces in 22 Mg determined through a comparison of calculated Coulomb-excitation yields and experimental yields using GOSIA2 [41]. (a) Total χ 2 surface for the 0 + 1 | E2 |2 + 1 and 2 + 1 | E2 |2 + 1 matrix elements. (b) As (a) but within the χ 2 min + 1 (1σ) limit, demonstrating the preference for a negative 2 + 1 | E2 |2 + 1 matrix element.  Figure 4: χ 2 surface at the χ 2 min + 1 (1σ) limit for the 0 + 1 | E2 |2 + 1 and 2 + 1 | E2 |2 + 1 matrix elements in 22 Ne. where available.

Discussion
The determined B(E2; 2 + 1 → 0 + 1 ) value in 22 Mg is approximately 20% lower than the evaluated value reported in the literature [28]. The present value lies within the 1σ uncertainties of the literature value but is considerably more precise. Taking a weighted average of the 22 Mg literature values [26,27] and present values yields B(E2; 2 + 1 → 0 + 1 ) = 76.5± 9.9 7.4 e 2 fm 4 . Asymmetric uncertainties were combined using the method outlined in Ref. [44]. The extracted 2 + 1 | E2 |2 + 1 matrix element is negative, indicating a preference for prolate deformation. The 22 Mg B(E2; 2 + 1 → 0 + 1 ) value now has uncertainties comparable to the other T z = −1 nuclei, as shown in Fig. 5 in which the updated data are plotted with theory.
For 22 Ne good agreement is obtained with the wellknown literature transition matrix elements, confirming the validity of the analysis. While agreeing at approximately the 2σ limit with the evaluated 2 + 1 | E2 |2 + 1 value, the present result is in best agreement with the values obtained in Ref. [43]. The present 2 + 1 | E2 |2 + 1 matrix element is more than a factor of two more precise than the evaluated values (see Tab. 1). Incorporat-  Fig. 5, the NCSpM reproduces the A = 22 data well. For comparison phenomenological shellmodel calculations were performed using the USDB interaction using NuShellX [45] with some of the common combinations of effective charge [20,45,46]. NC-SpM calculations are performed with a harmonic oscillator frequency, ω = 15 MeV in a model space of 15 major shells. NCSpM calculations agree with the corresponding ab initio SA-NCSM results using the N2LO opt in smaller model spaces where ab initio calculations are feasible [47] (e.g., for 22 Mg in 9 shells, B(E2) strengths differ by 0.4%). We note that to achieve the converged B(E2) values shown in Fig. 5, it is important to include mp-mh excitations to very high shells, as achieved in the NCSpM. An underprediction is found in the B(E2) value in the A = 30 case where a smaller model-space selection had to be made, with an improved calculation of these heavier nuclei being under way.
Also shown in Fig. 5 are ab initio calculations performed using the valence-space IM-SRG formalism [48,49,25,15] using a consistently evolved E2 operator (see Ref. [50] for details of the operator evolution) without incorporating effective charges. These calculations were performed using the SRG-renormalized [51] 1.8/2.0 chiral interaction [52][53][54] with a harmonic oscillator basis of ω = 20 MeV. Clearly, these values significantly underpredict the B(E2; 2 + 1 → 0 + 1 ) strength. It should be noted, however, that the IM-SRG calculations do provide a good qualitative description of the evolution of E2 strength. Furthermore, the new data indicate that, while the phenomenological shell-model is able to reproduce the A = 22 case with a given choice of effective charge, no single combination of effective charges is able to reproduce the entire sd-shell, with notable deviations at T z = −1, A = 26 and T z = +1, A = 34.
In order to assess the nature of the missing strength in the IM-SRG calculations, the B(E2) data were normalized according to the ratio of theoretical and experimental values of their mirror partner. For example, a B(E2) strength for the proton-rich mirror was projected as: This analysis was performed for both IM-SRG and shell-model calculations and the projected B(E2) values were compared with experiment. It is found that, with the exception of mirror-pairs containing a magic number, the IM-SRG results are highly consistent, overprojecting the proton-rich strength by a factor of approximately 15%. If the missing strength were purely isoscalar, a common scaling between theory and experiment would be expected for the T z = +1 and T z = −1 members of the mirror pair. The common 15% discrepancy therefore indicates that the missing strength is not purely isoscalar, and that a non-negligible isovector component must also be incorporated. Shell-model calculations -both with and without effective chargeson the other hand, exhibit no such consistent behavior in this analysis.

Conclusions
In conclusion, we present an improved measurement of the low-lying E2 strength in the |T z | = 1, A = 22 mirror pair. A first Coulomb-excitation measurement of 22 Mg has been performed, indicating its prolate deformation at the first-excited J π = 2 + state and significantly improving the uncertainty of the B(E2; 2 + 1 → 0 + 1 ) value. This represents the first spectroscopic quadrupole moment measurement for an even-even N < Z nuclide. Comparison with the state-of-the-art no-core symplectic shell model calculations, validated in smaller model spaces by the ab initio SA-NCSM, show excellent agreement in the A = 22 and A = 26 cases without a reliance on effective charges. On the other hand, the ab initio valence-space IM-SRG, provides good qualitative agreement of the evolution of E2 strength, but underpredicts the absolute values. These agreements provide some promise for reaching descriptions of enhanced collectivity in sd-shell nuclei in the framework of the ab initio theory starting with chiral potentials.

Acknowledgements
The authors would like to thank the TRIUMF beam delivery group for their efforts in providing high-quality stable and radioactive beams. This work has been supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), The Canada Foundation for Innovation and the British Columbia Knowledge Development Fund. TRIUMF receives federal funding via a contribution agreement through the National Research Council of Canada. This work has been partly supported by the U.S. NSF ACI -1516338 and also benefited from computing resources provided by Blue Waters and LSU; K.D.L. acknowledges useful discussions with J. P. Draayer. The work at LLNL is under contract DE-AC52-07NA27344. This work was supported in part by the UK STFC under grant number ST/L005735/1. Computations were performed with an allocation of computing resources at the Jülich Supercomputing Center (JURECA).