Electromagnetic properties of $^{21}$O for benchmarking nuclear Hamiltonians

The structure of exotic nuclei provides valuable tests for state-of-the-art nuclear theory. In particular electromagnetic transition rates are more sensitive to aspects of nuclear forces and many-body physics than excitation energies alone. We report the first lifetime measurement of excited states in $^{21}$O, finding $\tau_{1/2^+}=420^{+35}_{-32}\text{(stat)}^{+34}_{-12}\text{(sys)}$\,ps. This result together with the deduced level scheme and branching ratio of several $\gamma$-ray decays are compared to both phenomenological shell-model and ab initio calculations based on two- and three-nucleon forces derived from chiral effective field theory. We find that the electric quadrupole reduced transition probability of $\rm B(E2;1/2^+ \rightarrow 5/2^+_{g.s.}) = 0.71^{+0.07\ +0.02}_{-0.06\ -0.06}$~e$^2$fm$^4$, derived from the lifetime of the $1/2^+$ state, is smaller than the phenomenological result where standard effective charges are employed, suggesting the need for modifications of the latter in neutron-rich oxygen isotopes. We compare this result to both large-space and valence-space ab initio calculations, and by using multiple input interactions we explore the sensitivity of this observable to underlying details of nuclear forces.

derived from the lifetime of the 1/2 + state, is smaller than the phenomenological result where standard effective charges are employed, suggesting the need for modifications of the latter in neutron-rich oxygen isotopes. We compare this result to both large-space and valence-space ab initio calculations, and by using multiple input interactions we explore the sensitivity of this observable to underlying details of nuclear forces.
Keywords: lifetime measurement, exotic nuclei, ab initio calculations, effective charges Understanding nuclear structure and dynamics in terms of the fundamental interactions between protons and neutrons is one of the overarching goals of nuclear science. To this end, nuclear theory is developing chiral effective field theory (EFT) [1,2], a unified approach to nuclear forces, where two-nucleon (NN), three-nucleon (3N) and higher-body forces are derived within a consistent, systematically improvable framework. This approach coupled with parallel advances in ab initio many-body theory [3,4,5,6,7] provides the possibility to link the structure of nuclei to the underlying symmetries of quantum chromodynamics.
Neutron-rich oxygen isotopes are particularly fruitful candidates to test ab initio theory at the interface of the light-and medium-mass regions. Due to their semi-magic nature, most oxygen isotopes are accessible to many-body approaches amenable to heavier systems, while still being light enough to be treated in quasi-exact methods, such as extensions of the no-core shell model (NCSM). First valence-space calculations with NN+3N forces were able to explain, for the first time, the location of the oxygen dripline at 24 O [8]. More recently, large-space ab initio calculations, where all nucleons are treated as explicit degrees of freedom, have confirmed those early results [9,10,5] and new calculations have even extended dripline predictions to the entire region [11]. Furthermore excitation spectra in oxygen have also been obtained with NN+3N forces, generally yielding agreement with experiment approaching that of state-of-the-art phenomenology [12,13,14,15]. An important next step is to benchmark ab initio theory against other observables which are sensitive to physics beyond what is relevant for excitation energies alone. For instance the long-standing problem of quenching of beta decays across the nuclear chart has recently been explained [16], but electromagnetic properties have only been intermittently studied [17,18,19,20]. In particular, limited data exists for transition rates in the neutron-rich oxygen isotopes. In 21 O no experimental information is available on transition strengths, while γ decays from bound excited states beyond the first have been reported in [21] with limited statistics.
In this Letter, we report first electromagnetic transition rates from low-lying excited states of 21 O. We compare our results to predictions from phenomenological shell model and two ab initio many-body methods, the in-medium (IM-) NCSM and the valence-space in-medium similarity renormalization group (VS-IMSRG). Using a number of chiral EFT NN+3N forces, we study the sensitivity of electromagnetic transitions to details of nuclear interactions. It should be noted that electromagnetic two-body currents, currently under development, are not included in the ab initio calculations.
The experiment was performed at the National Superconducting Cyclotron Laboratory (NSCL) at Michigan State University. A 24 F secondary beam was produced by fragmenting a 140 MeV/nucleon 48 Ca beam on a 893 mg/cm 2 9 Be production target. The A1900 separator [22,23] was used to select and transport the 24 F ions (with an energy of 95 MeV/nucleon, a 2.5% momentum dispersion and 95% purity) to the experimental vault where they underwent reactions on a secondary 2 mm 9 Be target, located at the target position of the S800 spectrograph [24]. 21 O was produced via the 9 Be( 24 F, 21 O+γ)X multi-nucleon removal reaction and identified on an event-by-event basis via energy-loss and time-of-flight measurements. Emitted γ rays were detected with the Gamma-Ray Energy Tracking In-beam Nuclear Array (GRETINA) [25,26]. To determine the lifetime of the 21 O 1/2 + state, its γ-ray decay to the 5/2 + ground state has been analysed using data from settings I and II. In setting I, the lifetime is inferred from the low-energy tail that is generated in the γ-ray spectrum, see Fig. 1(a). This tail results from nuclear levels decaying farther away from the target, but being Doppler corrected as if they decay promptly at the target position, see, e.g., [28,29,30,31]. In setting II, the Recoil Distance Method (RDM) was employed, where γ rays emitted before or after the degrader experience different Doppler shifts leading to two laboratory energies, typically called the fast and slow component of the peak in the γ-ray spectrum, as can be seen in the insets of Fig. 1(b) and (c). The ratio of the number of γ rays in the fast and slow peak infers the lifetime of the state. The RDM for fast beams and its implementation at the NSCL is described in [31,32,33,34,35,36].
Since reactions populating the state of interest occur also in the degrader, the degrader-reaction ratio (DRR) has to be taken into account when evaluating the fast and slow components in an RDM measurement. The DRR has been determined by evaluating γ-ray transitions with much shorter lifetimes. Since the production mechanism is a multi-nucleon removal reaction, the DRR is assumed to be similar for all transitions.
To determine the lifetime, branching ratios and DRR, the measurements are compared to simulations obtained with Geant4 [37], simulating all relevant properties, i.e., detector geometry and response, γ-ray cascades, lifetimes, beam profile [32]. The energy of the 1/2 + state is taken as 1221.5 ± 2.2 keV, the average from two β-decay experiments [38,39]. The energies of the remain- −12 (sys) ps in an equivalent minimization process as described for the DRR, resulting in the χ 2 distribution shown in Fig. 1(d). The χ 2 is calculated between 1050 keV and 1260 keV, covering both the full peak and tail for setting I, as well as both fast and slow peaks for settings II. The statistical uncertainty is given by the un-normalized χ 2 min + 1 range, while the systematic uncertainty is dominated by the uncertainty of the DRR and the energy of the first excited state. To construct a consistent level scheme from the measured γ-ray energies, the lifetime of the 3/2 + state is determined via the centroid-shift method, see, e.g., [30,40], to τ 3/2 + = 8 +21 −8 ps. We compare the experimental results to phenomenological shell model, and two ab initio methods using chiral EFT NN+3N interactions, the VS-IMSRG and the IM-NCSM, see Figs. 2, 3 and Table 1. 2s 1/2 , and 1d 3/2 single-particle orbitals [41]. The E2 matrix elements are calculated with harmonic-oscillator radial wavefunctions, and with an effective neutron charge e n = 0.45, obtained from a fit to experimental B(E2) values of sd-shell nuclei (A = 17 − 38) [42]. In addition, the results for A = 17 − 22 are compared to experiment in Fig. 3. For nuclei close to 16 O, and in particular for 18 O, the experimental B(E2) are larger than those calculated, due to the mixing with low-lying states coming from the excitation of protons from the p shell to the sd shell [43]. These core-excited states move to a higher excitation energy for heavier nuclei, as their mixing with the sd-shell states becomes smaller. Thus 21 O, free from such state mixing, is ideal to examine the shell-model effective charge. The overestimated B(E2) in 21 O compared to experiment (see Table 1) shows that the neutron effective charge is smaller than the average value for the sd shell (e n = 0.45). The reason for this was discussed in [43] in terms of microscopic theories for the effective charge, the latter being orbital and mass dependent. The average for the effective charges for 1d − 1d and 1d − 2s for A = 17 (Z = 8) and A = 40 (Z = 20) is e n = 0.50 [43], which is close to the average empirical value of e n = 0.45 from [42] used for the calculations. For 21 O the 1/2 + to 5/2 + transition is dominated by the 1d − 2s one-body transition density. The effective charges for Z = 8 are e n = 0.374 for 1d − 1d and e n = 0.248 for 1d − 2s. The reason for the smaller 1d − 2s effective charge is that in this case the valence transition density has a node near the maximum value of the core-polarization density k(r) in Eq. (8) of [43]. This results in some cancellation in the integral whose integrand is a product of these two densities. If these values are used for calculations, the transition strength for 21 O is reduced to B(E2) = 0.82 e 2 fm 4 and agrees with experiment (see Fig. 3). It is interesting to confirm whether these reduced effective charges can reproduce the B(E2) for 22 O. The latter has been measured in [44] with large uncertainties.
New experiments aiming at constraining this value, e.g., [45], will thus add to our understanding of the oxygen isotopes.
The VS-IMSRG [6,7,48,49,50,51] provides a framework to produce ab initio valence-space Hamiltonians, based on NN+3N forces derived from chiral EFT. Working in a Hartree-Fock basis, we use the Magnus formulation of the IMSRG [6,7,52], to first decouple the 16 O core energy. Then we decouple an sdshell Hamiltonian, using the ensemble normal ordering procedure described in [50], to include effects of 3N forces between valence nucleons, specifically the five valence neutrons for the 21 O energies and transition rates. Finally, we use the approximate unitary transformation from the Magnus framework to additionally decouple an M1 or E2 valence-space operator consistent with the valence-space Hamiltonian [17]. In this framework, effective charges are thus not needed, but the effective operator is calculated consistently. Unless otherwise specified, all other technical details are the same as in [17,50]. The particular input NN+3N interaction used here, EM 1.8/2.0 developed in [53,54], begins from a chiral NN interaction at next-to-next-to-next-to-leading order (N 3 LO) [55] and 3N forces at N 2 LO. This Hamiltonian, fit to few-body data, has been shown to reproduce ground-and excited-state energies across the nuclear chart from the p shell to the tin region [5,11,54,56,57]. Indeed, very good agreement between the experimental and VS-IMSRG excited-state energies is observed in Fig. 2.
While E2 transition rates in the sd-shell are generally systematically below experiment, owing to the difficulty in capturing the highly collective physics of this transition, the trends typically agree well with experiment [19]. This can also be seen qualitatively in Fig. 3, where the staggering of E2 strength resembles experiment. For the odd mass cases, in particular 21 O, the agreement with experiment is rather good, while for the more collective transitions in evenmass isotopes the VS-IMSRG largely underestimates experiment.
We perform ab initio IM-NCSM calculations in the framework introduced in [58]. This novel method is a combination of the NCSM with a multireference IMSRG evolution of the many-body Hamiltonian that decouples a multi-determinantal reference state, typically an NCSM eigenstate from a small  [58] were necessary: the consistent multi-reference in-medium evolution of the electromagnetic operators, as well as an extension to odd particle numbers via a particle-attachment or particleremoval scheme. The details of these extensions are presented in [59]. We ). We use the particle-removed calculation at the largest available N max = 6 as nominal result and the difference to the particle-attached calculations and the residual N max -dependence to quantify the uncertainty of the many-body calculation. Fig. 2 shows that the IM-NCSM calculations mostly provide a consistent description of the low-lying spectrum in very good agreement with experiment, except for the N 2 LO SAT interaction. While the latter includes information beyond the few-body sector, particularly oxygen energies and radii, into the fit, it nonetheless produces a 1/2 + state over 1 MeV higher than experiment. The B(E2) transition strength from the first excited 1/2 + to the ground state, shown in Table 1, indicate interesting differences, even among the interactions that provide a consistent excitation spectrum. The This shows that the E2 observables measured here provide a good test for chiral interactions that goes beyond the aspects probed by the excitation energies alone. A systematic study of the oxygen isotopes in the IM-NCSM is under way [63].