Spectroscopy of ^{33}Mg with knockout reactions
D. Bazin, N. Aoi, H. Baba, J. Chen, H. Crawford, P. Doornenbal, P. Fallon, K. Li, J. Lee, M. Matsushita, T. Motobayashi, H. Sakurai, H. Scheit, D. Steppenbeck, R. Stroberg, S. Takeuchi, H. Wang, K. Yoneda, C. X. Yuan
SSpectroscopy of Mg with knockout reactions
D. Bazin,
1, 2, ∗ N. Aoi,
3, 4
H. Baba, J. Chen, H. Crawford, P. Doornenbal, P. Fallon, K. Li, J. Lee,
3, 6
M. Matsushita, T. Motobayashi, H. Sakurai,
3, 7
H. Scheit,
3, 8
D.Steppenbeck, R. Stroberg, S. Takeuchi, H. Wang, K. Yoneda, and C.X. Yuan National Superconducting Cyclotron Laboratory, 640 S. Shaw Lane, East Lansing, MI 48824-1321 Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824-1321, United States RIKEN Nishina Center, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan Research Center for Nuclear Physics, 10-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA Department of Physics, University of Hong Kong, Pokfulam Road, Hong Kong, China Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan Technische Universit¨at Darmstadt, 64289 Darmstadt, Germany Department of Physics, University of Washington, Seattle, WA 98195-1560, USA Sino-French Institute of Nuclear Engineering and Technology, Sun Yat-Sen University, Zhuhai 519082, China (Dated: January 29, 2021)The structure of Mg was investigated by means of two knockout reactions, one-neutron removalfrom Mg and one-proton removal from Al. Using comparative analysis of the population ofobserved excited states in the residual Mg, the nature of these states can be deciphered. Inaddition, the long-standing controversy about the parity of the Mg ground state is resolved usingmomentum distribution analysis, showing a clear signature for negative parity. Partial cross sectionmeasurements are compared with the results of eikonal reaction theory combined with large-scaleshell model calculations of this complex nucleus located in the island of inversion, where configurationmixing plays a major role.
PACS numbers:I. INTRODUCTION
The structure of the nucleus Mg has been the subjectof much debate over the past few years. It is located inthe so-called island of inversion [1] where intruder con-figurations arise due to the quenching of the N=20 shellgap. Numerous experimental evidence point to this re-gion as strongly deformed, from the lowering of the first2 + excited states in Mg [2–6] and more recently Ne [7, 8] as well as large B(E2) transition strengths in Na [9] and Ne [6, 10].The origin of the deformation in this region is now wellestablished and revealed from the evolution of the effec-tive single-particle energies leading to the disappearanceof the N=20 shell gap in neutron-rich isotopes, and theresulting enhancement of multi-particle multi-hole exci-tations across the narrowed gap. Shell model calculationscan now reproduce the narrowing of the gap between the1 d / and (1 f / , 2 p / ) orbitals, and the appearance ofintruder states [11]. In a mean-field picture [12], this de-formation can be understood in terms of a degeneracy ofthe 1 f / and 2 p / orbits.The case of Mg however remains less clear. β -decaymeasurements from Na seemed to indicate a spin-parityof 3 / + based on log(ft) values [13], that was interpretedas a 1p-1h excitation across an inversion of the 1 d / and 1 f / orbitals. This interpretation was further re-inforced by a Coulomb excitation experiment where the ∗ [email protected] observed state at 485 keV was assigned a spin-parity of5 / + [14]. On the other hand, the measured negativemagnetic moment of Mg is in direct contradiction withthis interpretation, suggesting a spin-parity of 3 / − anda 2p-2h excitation [15]. A year after the magnetic mo-ment measurement, results on the β -decay from Mgheated the debate again by reporting a large branchingratio to the 5 / + ground state of Al, and tried to rec-oncile the overall picture by proposing a mixing of 1p-1hand 3p-3h configurations [16–18]. A subsequent mea-surement of the inclusive momentum distribution in theone-neutron knockout from Mg indicated a large occu-pation of the 2 p / orbital, an indication of its lowering[19]. Although a fit of spectroscopic factors assuming a3 / − ground state seemed to be closer to Monte-CarloShell Model predictions, no conclusion on the spin-paritywas reached in that paper. A Coulomb dissociation ex-periment [20] unveiled some evidence of multiparticle-hole ground state configuration involving the 1 s / and1 p / orbitals. Finally, excited states of Mg were popu-lated in the fragmentation reaction of a Ar radioactivebeam, and detected using the GRETINA γ -ray trackingarray [21]. The energies of the populated states weremeasured, and from the energy differences the presenceof a rotational band based on a 3 / − ground state wasinferred [22].In this work, the ground and excited states configura-tions of Mg are investigated by means of one-neutronand one-proton knockout reactions from Mg and Al,respectively. The selectivity of these two reactions playsan important role in the identification of the final statespopulated in Mg. From the 0 + ground state of Mg, a r X i v : . [ nu c l - e x ] J a n only one partial wave can contribute to the direct feed-ing a given excited state in Mg, therefore an analysisof the momentum distribution shape reveals the orbitalangular momentum, and by deduction, the parity of thatstate. In contrast, the removal of a proton from Al canproceed via a number of partial waves, depending on thespin-parity of its ground state, but as this valence protonmost probably occupies the 1 d / orbital, the populationof final states in Mg is expected to be more selectiveand favor states of the same parity as the ground stateof Al for a (cid:96) =2 proton removal.
II. EXPERIMENT
The experiment was conducted at the Radioactive IonBeam Factory (RIBF) located on the RIKEN campus ofTokyo, Japan, and operated jointly by RIKEN and theCenter of Nuclear Study from the University of Tokyo.The Mg and Al radioactive beams were producedfrom the fragmentation of a 345 MeV/u Ca primarybeam on a 15 mm thick Be target. The radioactivebeams were selected and filtered by the BigRIPS frag-ment separator [23] up to the F8 focal plane, where the1032 mg/cm
Be reaction target was placed. Surround-ing this target was the NaI(Tl) γ -ray array DALI2 [24],which recorded the Doppler-shifted γ -rays emitted by thereaction residues. The Mg residues produced in theknockout reactions were then selected and collected bythe ZeroDegree Spectrometer (ZDS) operated in disper-sive mode in order to measure their momentum. The in-coming energies of the Mg and Al radioactive beamswere 242.5 MeV/u and 229.6 MeV/u, and their averageintensities 4.5 × pps and 6.5 × pps respectively.One of the main goals of this experiment was to mea-sure partial and inclusive cross sections populating thevarious states of the Mg residue, therefore careful cal-ibrations of the incoming fluxes of the Mg and Alprojectiles were performed by setting the ZDS on themagnetic rigidity of these projectiles after slowing downin the target, and measuring the ratios of Mg and Aldetected at the focal plane of the ZDS relative to thetotal number of particles counted by the plastic scintilla-tor located upstream of the target at the F7 focal planeof BigRIPS. The purities deduced from these measure-ments are 69% and 60% for Mg and Al, respectively.In addition, the acceptance and momentum dispersion ofthe ZDS were measured by scanning the unreacted Mgbeam across the focal plane of the ZDS by varying itsmagnetic rigidity. The momentum dispersion was deter-mined to be 4.5 cm/%, close to the expected value fromoptics calculations.The particle identification prior to the reaction targetwas performed using the detectors located at F3 and F7of the BigRIPS fragment separator. The time-of-flightmeasured between the plastic scintillators at F3 and F7was used in combination with the energy loss signal fromthe F7 ion chamber. However, because the contami- F E n e r g y l o ss ( a r b i t r a r y un i t ) Mg incoming gate605040302010 F E n e r g y l o ss ( a r b i t r a r y un i t ) Mg outgoing gate
FIG. 1. Particle identification spectra using time-of-flight ver-sus energy loss before (a) and after (b) the reaction target.The plots are shown for the case of the Mg radioactive beam.Plot (b) is gated on the Mg incoming gate shown in plot (a).The double structure observed at the Mg location corre-sponds to inelastic scattering of incoming Mg beam eventsthrough the Be target. Due to the dispersive mode of theZDS, these particles are transmitted on the high side of themomentum acceptance, and hit the F11 scintillator close toits edge near where the light guide is located and producedifferent energy losses. nants were clearly separated in time-of-flight from the Mg/ Al, and there was significant pile-up in the ionchamber at higher rate, the incoming Mg gate was lim-ited to the time-of-flight only. The momentum width ofthe incoming radioactive beam was set to 0 . Mg beam.The parallel momentum of the residues was deducedfrom the dispersive position measured at the F11 focalplane and the measured dispersion of 4 . / %. Thebroadening effect due to the momentum width of the in-coming beam, was canceled using the dispersive positionmeasurement at the F5 focal plane. A resulting momen-tum resolution of 0 . Mg residues, the 4% momentum acceptance of the ZDSwas large enough to collect all residues with negligiblelosses.
III. RESULTS
The quantities measured during this experiment werethe parallel momentum distribution of the Mg residues,the energy of the γ -rays emitted following nucleon re-moval, and the cross sections of the reactions. Due tothe large background coming from the breakup of the Bereaction target and the Bremsstrahlung radiation, the γ -ray multiplicity M recorded by the DALI2 array wasvery large, and some of the γ -ray transitions could onlybe observed in the M=1 spectra. For this reason, theM=1 and M=2 spectra were used to identify the tran-sitions and possible γ - γ coincidences. However, becausethe Geant4 simulations are not able to simulate the back-ground, the intensities of the γ -ray transitions were fittedusing the M ≤ A. One-neutron knockout from Mg The inclusive cross section of this reaction was mea-sured using the incoming beam normalization methoddescribed in the previous section. The cross section ob-tained is 93(2) mb, where the error bar is dominated bythe systematic errors associated with the determinationof the number of incoming Mg particles.The Doppler-corrected γ -ray spectrum measured incoincidence with the one-neutron knockout of Mg isshown in Fig. 2. This spectrum was generated usingan add-back procedure to reduce the peak-to-backgroundratio, and gated on multiplicity one (M=1) to further en-hance the identification of peaks.Eight transitions were observed and their energies fit-ted using simulated line-shapes from a Geant4 simulationof the array. The inset of Fig. 2 shows the γ spectrumin coincidence with the 484(6) keV transition. A veryclear coincidence is observed between the 484(6) keVand 295(7) keV transitions, in agreement with the re-ported observations in [13] and [22]. The weaker 219(8)keV transition corresponding to the 221 keV transitionobserved in [13] and [22] could only be resolved in thecoincidence spectrum due to the large Bremsstrahlungbackground present at this beam energy. Although its C o un t s / k e V In coincidence with 484 keV Mg - 1n
295 484 549 703 779 1258 1850
FIG. 2. M=1 γ -ray spectrum observed in coincidence with Mg residues produced from one-neutron knockout off Mgprojectiles. Seven γ -ray transitions are identified, and theirenergies determined from fitting the spectrum with calculatedline-shapes from Geant4 simulations and a double-exponentialbackground (all shown in dashed lines). The inset shows the γ - γ coincidence spectrum gating on the 484 keV transitionwhere a large coincidence is observed with the 295 keV γ -rayline, as well as a weaker transition at 219 keV. presence cannot be ruled out because of the limited en-ergy resolution, the 759 keV transition reported in [13]in coincidence with the 484 keV transition is only hintedin this neutron knockout data. A transition at 1857 keVwas reported in [13] but not assigned to any of the β -decay daughters of Na. The energy however is veryclose to the 1850(40) keV transition observed in our data,so it likely corresponds to the same transition (the one-neutron separation energy of Mg is 2.07 MeV).The M=1 γ -ray spectrum was used to extract the mo-mentum distributions corresponding to the populated ex-cited final states in Mg. Due to the strong overlap be-tween some of the peaks, the shapes of the momentumdistributions may be cross-contaminated. Also, the back-ground subtraction could only be performed using thedata around 1 MeV and above, since at lower energiesthe spectrum is dominated by the numerous transitionsand their Compton contributions. In order the minimizethe cross-contamination where peaks overlap, the gatesused to extract the momentum distributions were shiftedfrom the centroid of the peak away from the contami-nating peak. The resulting momentum distributions aredisplayed in Fig. 3, each labeled with the transition en-ergy.The top left panel of the figure shows the inclusive mo-mentum distribution, while the top middle panel showsthe momentum distribution extracted for the direct feed-ing of the ground state. This distribution is calculatedby subtracting the contributions from the excited feederstates weighted from the branchings deduced from the γ -ray spectrum without multiplicity cut. This procedureis necessary because of the inability to accurately simu-late the γ background in the Geant4 simulations. The x -400 -200 0 200 400p lab (MeV/c) inclusive Mg 6050403020100 x -400 -200 0 200 400p lab (MeV/c) gs * data 2s lab (MeV/c) 295 keV-400 -200 0 200 400p lab (MeV/c)140120100806040200 484 keV * lab (MeV/c) 549 keV 200150100500 -400 -200 0 200 400p lab (MeV/c) 703 keV100806040200 -400 -200 0 200 400p lab (MeV/c) 779 keV 100806040200 -400 -200 0 200 400p lab (MeV/c) 1258 keV 706050403020100 -400 -200 0 200 400p lab (MeV/c) 1850 keV FIG. 3. Mg residues momentum distributions observed incoincidence with γ -ray transitions, each labeled with the cor-responding energy. The top left panel show the inclusive mo-mentum distribution as well as the unreacted Mg distri-bution obtained with the ZDS spectrometer centered on itsenergy after the reaction target. The latter illustrates the mo-mentum resolution obtained in this experiment with the ZDSspectrometer set in dispersive mode. Unlike the momentumdistributions gated on the γ -ray peaks from the M=1 spec-trum, the ground state and 484 keV distributions (indicatedby an asterisk) are obtained by subtracting the contributionsfrom their feeding states, using the γ -ray spectrum for all mul-tiplicities and the branchings deduced for each of the feederstates. The distributions are compared to calculations fromthe eikonal model (see text). observed cascades between excited states are also takeninto account in the subtraction. The same procedure isused to deduce the momentum distribution correspond-ing to the direct feeding of the 484 keV state, althoughthe error bars are large because of the strong feeding fromthe 779 keV state (see section IV A below).Each experimental momentum distribution is com-pared to calculated distributions using the eikonal model[25], assuming the removed neutron belongs to the or-bitals 2 s / , 2 p / , 1 d / , or 1 f / . To convert theminto the laboratory reference frame, these distributionsare scaled by the relativistic factor γ =1.243 correspond-ing to the velocity of the Mg projectiles at mid-target.For most distributions, the angular momentum of theremoved neutron is clearly identified by comparing thecalculated shapes to the experimental data. The groundstate momentum distribution, although displaying theusual low momentum tail usually observed in knockoutreactions, is best reproduced by the 2 p / distribution.This establishes without ambiguity the negative parityof the ground state of Mg, in agreement with the mag-netic moment measurement [15] and conclusions from theinclusive measurement of neutron knockout from Mg [19].The shapes of the momentum distributions associatedwith other transitions are identified, with the excep-tion of the one associated with the 484 keV state forwhich the error bars are large due to the feeding sub-traction, and the one associated with the 703 keV forwhich both p-wave and d-wave components seem to bepresent. This latter observation is most likely due tocross-contamination from the 779 keV d-wave componentdue to the proximity of the γ -ray transitions, and indi-cates that the momentum distribution associated withthe 703 keV transition is most likely a p-wave. Dueto the limited statistics, the shapes associated with thehigher energy transitions at 1258 keV and 1850 keV arealso ambiguous between a s-wave and p-wave assignment,especially since the difference between the two theoreti-cal distributions is relatively small and their widths notmuch larger than the resolution. Noteworthy is the non-observation of any transition associated with an (cid:96) =3 mo-mentum distribution. This is investigated in the nextsection. B. Search for the (cid:96) =3 strength
The parallel momentum distributions shown in fig. 3clearly exclude the observation of any (cid:96) =3 strength asso-ciated with any of the prompt γ -rays detected in coinci-dence with the Mg residues, whereas the lowest 7/2 − state corresponding to the 0p0h configuration is expectedwithin all shell model calculations to be populated witha significant strength. This surprising result could beexplained if this state is long-lived and therefore decaysafter the residues have left the sensitive area of the γ -rayarray.In order to test this hypothesis and evaluate thebranching ratio of this missing strength, two fits of the in-clusive parallel momentum distribution were performed,using calculated momentum distributions for (cid:96) =1,2 and (cid:96) =0,1,2,3, respectively. The inclusive momentum distri-bution has a much larger statistics because it is not sub-ject to the γ -ray array detection efficiency. The results ofthe fits are shown in fig. 4. All momentum distributionsare centered on the mean momentum in the laboratoryframe. The only fitting parameters are the strengths ofeach component, and only the high momentum portionof the data is used. This restriction is necessary becausethe calculated momentum distributions from the eikonalmodel are symmetrical, whereas the measured ones showan asymmetry due to the low momentum tail. This ef-fect is well known and originates from the lack of energyconservation in the eikonal model [25], as well as dissipa-tive contributions from inelastic excitations of the coreresidue [26]. The reduced χ values (displayed on thefigure) indicate a much better agreement with the exper-iment in the second fit (b). The residuals obtained inthe first fit (a) indeed clearly indicate that a third com-ponent with a shape similar to the (cid:96) =3 distribution is C o un t s x -400 -200 0 200 400Momentum (MeV/c)-400400 Data Fit Residual p (44%) d (56%) χ = 4.08 (a) C o un t s x -400 -200 0 200 400Momentum (MeV/c)2000-200 Data Fit Residual s (6.9%) p (53%) d (20%) f (20%) χ = 1.29 (b) R e l a t i v e s t r e n g t h ( % ) -200 -150 -100 -50 0 50 100Low momentum cutoff (MeV/c) s p d f (c) FIG. 4. Fits to the inclusive momentum distribution of one-neutron knockout from Mg. The top panel (a) shows theresults of a fit using only (cid:96) =1 and (cid:96) =2 calculated momentumdistributions, whereas the middle panel (b) shows the samefit adding (cid:96) =3 and (cid:96) =0 components. Panel (c) shows theevolution of the various (cid:96) components with respect to the lowmomentum cutoff in the fit. See text for details. needed. The (cid:96) =0 component is added for completeness,as it should be present as well in the inclusive data, al-though it is not distinguishable from the (cid:96) =1 componentdue to the resolution of the parallel momentum measure-ment. In addition, the percentages that represent therelative strengths of each component show that the firstfit (a) gives almost equal strength for the (cid:96) =1 and (cid:96) =2components, which is in contradiction with the resultsobtained from the γ -ray branching ratios in the previoussection, where the (cid:96) =1 component clearly dominates.The relative strength of the f / component is 20 ± − state makes it impossible to subtract its com-ponent from the inclusive distribution in order to extractthe ground state momentum distribution, as is done infig. 3, therefore it is composed of both the ground andisomer states components. Indeed, a fit to this groundstate momentum distribution using both p-wave and f-wave components also gives a much better match to the data, although its accuracy is limited by the relativelylarge error bars. Based on the 49 ±
1% branching ratiofor the ground and isomer states deduced from the ob-served γ -ray intensities, this fit gives a relative strengthof 15 ±
5% for the f-wave component, which is compatiblewith the 20 ±
2% determination based on the inclusive dis-tribution fit. However, it should be noted that this 49%is likely an upper limit because some of the weaker γ -raytransitions are probably missed due to the limited resolu-tion. A reduction of the deduced intensity to the groundand isomer states would reconcile better the two f-waverelative strength determinations outlined above. Becauseof its much smaller error bars, the determination basedon the inclusive momentum distribution is kept as thefinal result.To further test the validity of this extraction of the (cid:96) =3 strength, the evolution of the various (cid:96) componentsreturned by the fit shown in (b) is plotted as a function ofthe low momentum cutoff (c). A clear plateau is observedbetween cutoff values of -60 MeV/c and 20 MeV/c, indi-cating a robust determination of the various components.The shaded area around the curves shows the evolutionof the errors on the relative strength determination. C. One-proton knockout from Al The γ spectrum observed in coincidence with Mgresidues produced from the one-proton knockout of Alare shown in Fig. 5, superimposed on the spectrum ob-served from the one-neutron knockout of Mg. Althoughthe limited energy resolution of the DALI2 array doesn’tallow to resolve all the transitions, the comparison be-tween the two spectra clearly shows the disappearance ofthe 549 keV and 703 keV transitions in the one-protonknockout data. Unlike in the one-neutron knockout reac-tion, the population of final states in Mg from this re-action involves several partial waves because the groundstate of the odd-odd Al is not a 0 + , and is confirmedto be a 4 − with a close-by 1 + isomeric state at 47 keVwith a half-life of 21.6(15) ms [27, 28]. It likely thatboth the normal 4 − and intruder 1 + long-lived states arepopulated in the fragmentation reaction used to produce Al from Ca, although it is not known in what propor-tions. For these reasons, the momentum distributions ofthe Mg residues from this reaction are not as valuableas in the neutron knockout case and were not analyzed.The proton knockout reaction will preferably populateproton-hole states in Mg, which are likely to correspondto the removal of a proton from the 1 d / orbital. How-ever, because both fundamental and isomeric states of Al are likely present in the incoming beam, it is notpossible to use the selectivity of this reaction to pinpointthe parity of the states populated.Nevertheless, there are clear differences between thetwo observed γ -ray spectra. The two transitions that areclearly suppressed in the proton knockout data comparedto the neutron knockout are the 549 keV and the 703 keV, C o un t s / k e V in coincidencewith 484 keV Al - 1p Mg - 1n (÷10)549 keV703 keV
FIG. 5. M=1 γ -ray spectra from one-proton knockout off Al (solid red) and one-neutron knockout off Mg (dashedgreen) reduced by a factor 10. The difference in populationof transitions between the two reactions is clearly illustrated.The inset shows the γ - γ coincidence spectrum gated on the484 keV transition. In addition to the already observed coinci-dence with the 219 keV and 295 keV transitions, an additionalweaker coincidence is observed at around 780 keV, althoughit is not visible in the coincidence spectrum from the neutronknockout data. that both correspond to the feeding of negative paritystates, as identified from the momentum distributions ofthe Mg-1n reaction. This could indicate, as suggestedby [13], that the 549 keV transition originates from thestate at 703 keV and feeds the unobserved 7/2 − isomericstate, which would then be placed at 154 keV. This hy-pothesis is however at odds with the non-observation ofthe 549 keV transition in [22] whereas the 703 keV isclearly populated by that reaction.The coincidence spectrum shown in the inset of fig.5 reveals the same cascades going through the 484 keVstate as in the neutron knockout data, with the additionof a weak coincidence peak at around 780 keV. The en-ergy sum of this third cascade is 1264 keV, very closeto the 1258 keV transition observed in both neutron andproton knockout data. The number of counts in the re-gion of the 1258 keV transition is much larger in the pro-ton knockout data than in the neutron knockout data,which could explain why this cascade is not visible inthe latter coincidence spectrum. It is however surprisingthat the negative parity state observed at 1258 keV inthe neutron knockout data would be more strongly pop-ulated in the proton knockout reaction. Unfortunatelythe limited resolution of the γ -ray array does not allowto determine whether another close-by state is present inthis region. For this reason, the possible cascade transi-tion at 780 keV is ignored in the level scheme presentedbelow.The inclusive cross section measured for this reactionis 3.1(2) mb. IV. DISCUSSIONA. Spin assignments and branching ratios
Based on the results and analysis presented above, alevel scheme corresponding to the best hypothesis ex-tracted from this data is presented in fig. 6. The transi-tions observed in this work are indicated by the arrows.The placement of the 7/2 − state at 154 keV is basedon indirect evidence only, with the assumption that thisstate has a long lifetime that prevented its decay frombeing detected as a prompt γ -ray. E x c it a ti on e n e r gy ( k e V ) EXP IMSRG SDPF-MU SDPF-U-SI + - - - - + - + - - + - - - - - + + - - + - - + - - - + + + + - - (3/2,5/2) + - (3/2) - (3/2,5/2) + (1/2,3/2) - (1/2,3/2) -
484 219 295703 779 12581850549 (1/2) + (1/2) + <>root:Mg33level:jpiioi[0]}<>root:Mg33level:jpiioi[1]}<>root:Mg33level:jpiioi[2]}<>root:Mg33level:jpiioi[3]}<>root:Mg33level:jpiioi[4]}<>root:Mg33level:jpiioi[5]}<>root:Mg33level:jpiioi[6]} FIG. 6. Experimental and calculated level schemes for Mg.The experimental parity assignments are based on the shapesof the momentum distributions observed in coincidence withthe transitions. The tentative spin assignments are chosenfrom comparisons with earlier works as well as shell modelcalculations. The calculated level schemes use the variousshell model interactions described in the text.
The data clearly indicates that the state at 779 keV haspositive parity, and likely correspond to either the 3 / + or 5 / + state calculated from shell model. This resultis in contradiction with the hypothesis that this state ispart of a rotational band built on the 3 / − ground state,as reported in [22]. The strong feeding to the 484 keVstate, also observed in [22], indicates that the 484 keV haslikely also a positive parity, as also suggested by shellmodel calculations (see section IV B below). However,the present data does not allow a firm assignment due tothe large error bars and fluctuations in the momentumdistribution obtained after subtracting the feeding.The three states at 703 keV, 1258 keV and 1850 keVthat have negative parity from their associated momen-tum distributions, could correspond to the negative par-ity states calculated from the shell model, although it E level (keV) E γ (keV) I γ (%) b γ (%) b (%) σ (mb) (cid:96) < < < > > > γ -ray energies, intensities, deduced branch-ing ratios and partial cross sections. The b γ branching ratiosresult from the γ -ray intensity analysis, whereas the b branch-ing ratios take into account the presence of the isomeric levelat 154 keV, which is speculative and not directly observed.The last column indicates the (cid:96) -value observed from the gatedmomentum distributions, as well as the fit to the inclusivemomentum distribution. is not possible from the low level of statistics to firmlyassign an angular momentum of either (cid:96) =0 or 1 fromthe momentum distributions in coincidence with the 1258keV and 1850 keV transitions.The results showing intensities, branching ratios andpartial cross sections are summarized in Tab. I. The ta-ble is ordered by increasing level energy, with feedingtransitions from each level listed immediately below theground state transition. One complication arises fromthe non-observation of the isomeric state in the γ -raybranching ratio analysis, for which the branching ratiois included in the ground state. For this reason the col-umn labeled b γ differs from the final branching ratio b fortransitions involving the isomeric and ground states.Under the assumption that the first 7/2 − state ismostly populated and isomeric, the relative strength de-duced in sec. III B corresponds to unobserved strengthin the branching ratios deduced from the γ -ray analysispresented in Tab. I, where it is assigned to the groundstate. Based on that assumption, the branching ratiosfor the ground state and the 7/2 − state are revised to29 ±
3% and 20 ± γ -ray intensities, therefore its intensity isunknown. Since this transition originates from the 703keV level, the resulting branching ratios and partial crosssections for the 484 keV and 703 keV levels are taken aslower and upper limits, respectively. Therefore, only thelargest feeding contribution from the 295 keV transitionhas been subtracted to obtain the momentum distribu-tion corresponding to the 484 keV level displayed in fig.3. B. Comparison to shell model
The partial cross sections are compared to calcu-lated ones using spectroscopic factors from different shellmodel interactions. The single-particle cross sections arecalculated using the eikonal model [25] relevant at theseenergies. The usual prescriptions for the sizes of the bod-ies used in these calculations are followed: the Be targetnucleus is modeled with a Gaussian density distributionof width 2.36 fm, while the projectile and residue nucleidensities are calculated from Hartree-Fock calculationsusing the SkX force.The shell model interactions used to calculate the spec-troscopic factors are a IMSRG-derived interaction withan O core, and 2 interactions using an O core with0-5 (cid:126) ω excitations from sd to pf shell for valence neutrons.Valence protons are constrained in the sd shell only, be-cause their cross-shell excitations are less important inthe neutron-rich Mg isotopes. Also, the inclusion of suchexcitations demands much larger calculation resource.The IMSRG interaction was generated by the VS-IMSRG method with ensemble normal ordering, as de-scribed in [29], using the Magnus formulation [30]. Theinput Hamiltonian is the 1.8/2.0(EM) interaction de-scribed in [31], evaluated in an oscillator basis frequencyhw=16 MeV, with truncations e ≡ n + (cid:96) ≤ e max = 12and e + e + e ≤ E max = 14. A Gloeckner-Lawson [32]center-of-mass term β cm ( H cm − / (cid:126) ω ) is added to pushspurious states out of the spectrum. For more details onthe treatment of the center of mass in this context, see[33]. The resulting valence space interactions were diag-onalized using the code NuShellX [34]. To compute thespectroscopic factors, we do not consistently-evolve the a † operator, (work on implementing this is in progress).Since the spectroscopic factors are only used for a qual-itative comparison, we expect this to be sufficient in thepresent context. When computing the spectroscopic fac-tor, we perform all calculations using the interaction de-rived with the Mg reference.The SDPF-MU [35] and SDPF-U-SI [36] interac-tions are widely used Hamiltonians for the sdpf region.They are both constructed to describe the properties ofneutron-rich Si isotopes. It is therefore reasonable to usethem to study Mg and nearby nuclei.A spin-parity assignment of 3 / + of the 484 keV statewould imply that the Coulex experiment [14] performedon Mg measured an E1 transition. In this paper how-ever, the assumptions were that the J π of the gs and 484keV state are reversed. The B(E1) they deduced fromthe Coulex cross section is 0.035(10) e fm using thatassumption. Even though the parity assignment usedin [14] is incorrect, the deduced B(E1) is unchanged byswapping the parities of the two states, because the spinsof both inital and final states are the same, and the cor-rection factor for a transition going in the opposite direc-tion is (2J+1)/(2J’+1). This large value of the B(E1) issimilar to the one observed in Ne [37], and could indi-cate similar effects of core excitation and deformation as C r o ss s ec ti on ( m b ) C r o ss s ec ti on ( m b ) C r o ss s ec ti on ( m b ) C r o ss s ec ti on ( m b ) / - ( / , / ) + / - ( / ) - ( / , / ) + ( / , / ) - ( / , / ) - - - + + - - + - - + - - - + + + + - + - - - - + - + - - + - - - ( / ) + ( / ) + EXP IMSRG SDPF-MU SDPF-U-SI
FIG. 7. Comparison between the measured and deduced par-tial cross sections and calculations, for the one-neutron knock-out reaction from Mg. The calculated cross sections arededuced from the shell model spectroscopic factors using var-ious interactions, and the eikonal model single-particle crosssections. the source of this large dipole strength. The shell modelcalculation based on the SDPF-MU interaction gives amuch smaller value of 5.06e-6 e fm , but it is normallydifficult for shell model to reproduce B(E1) values in alimited model space.The SDPF-MU and IMSRG calculations seem to bestmatch the data, with the correct level ordering and qual-itative reproduction of partial cross sections. It is clearhowever that the low-lying levels of Mg and nearby nu-clei in the island of inversion are very sensitive to thelocal single particle energies, namely, the effective singleparticle energies (ESPE). These were recently explored innearby isotopes using a newly developed interaction [38],that showed the extent of ph mixing in the configurationof Mg low-lying states, and the role that three-nucleonand tensor forces play in the evolution of the ESPE. Itwould be interesting to compare calculations using thisnew interaction to the results presented here.The non-observation of a 154 keV γ -ray in this experi-ment corresponding to the decay from the inferred 7/2 − state to the ground state is likely due to two factors: thehigh background at low energy due to Bremsstrahlungradiation, and the possible isomeric nature of this state. Some estimates of the lifetime from shell model calcula-tions follow. The 7/2 − is not a very pure 0h0p state, es-pecially in SDPF-MU. In SDPF-U-SI, the probability ofpure π (d5/2) ν (f7/2) configuration is 32.44%, while inSDPF-MU, it is just 12.82%, where several 2p2h ν (f7/2) configurations contribute. The B(E2) values are 88.4 and74.5 e fm (effective charges e p =1.5, e n =0.5) for SDPF-MU and SDPF-U-SI, respectively, and the correspondinghalf-lives are 100 and 119 ns. With the IMSRG interac-tion, the prediction gives a lifetime of 180 ns assumingan energy of 150 keV and B(E2) of 10 Weisskopf unit,which is close to the value obtained for Mg [39]. Theseestimates indicate that it would be impossible to detectthis transition as a prompt γ -ray, and challenging as adelayed one because the lifetime is similar to the time-of-flight from the reaction target to a possible decay stationat the end of the ZDS.Finally the inclusive cross sections measured for bothone-neutron and one-proton knockout reactions allow todetermine the so-called reduction factor [40] that relatethem to theoretical cross sections calculated from shellmodel spectroscopic factors and single-particle cross sec-tions from the eikonal model. The interaction used inthe shell model calculations is SDPF-MU, which seemsto match the present experimental data the best. The Mg to Mg asymmetry energy is ∆S = S n -S p = -18.19MeV. A theoretical cross section of 135.6 mb is obtainedby summing all partial cross sections to individual statescalculated in the shell model. With the inclusive crosssection of 93(2) mb measured in this experiment, the re-duction factor obtained is 0.69(2). For the one-protonknockout the situation is complicated by the fact thatthe relative population of the 1 + isomer in the Al beamis unknown. Nevertheless, the theoretical cross sectionscalculated using either the 4 − or 1 + are very close, 11.61mb and 12.54 mb, respectively. The Al to Mg asym-metry energy is ∆S = S p -S n = 12.65 MeV. From themeasured inclusive cross section of 3.1(2) mb, a reduc-tion factor of 0.26(2) in obtained where the average valueof the theoretical cross section is used, and the error bartakes into account the experimental error and differencebetween the theoretical values. These two reduction fac-tor determination fall within the systematics presented in[40], although somewhat on the low side of the observedband. They show once again a strong reduction of thecross section when a deeply bound nucleon is removed,compared to a calculation using an independent-particlemodel. V. CONCLUSION
In this work we have studied the structure of Mg bymeans of one-neutron and one-proton removal reactionsfrom radioactive beams of Mg and Al, respectively.Momentum distributions in coincidence with prompt γ -rays recorded at the reaction target were analyzed and re-vealed the parity assignment of the populated final statesin Mg. The ground state momentum distribution ob-tained by subtraction from the inclusive momentum dis-tribution is compatible with a p-wave shape, thereby con-firming the 3/2 − spin assignment of the Mg groundstate without ambiguity.The state at 779 keV previously assigned as part of arotational band built on top of the 3/2 − ground state[22] has a positive parity, an observation in contradic-tion with this hypothesis. The strong feeding to the 484keV state points to a similar parity, and comparison withshell model calculations give tentative spin assignmentsof 3/2 + and 5/2 + . The states observed at 703 keV, 1258keV and 1850 keV on the other hand have negative parity.The spin assignments for those states becomes more diffi-cult in part due to the limited resolution of the Doppler-reconstructed γ -ray measurements, and the inability todistinguish between (cid:96) =0 and (cid:96) =1 shapes due to the finiteresolution of the momentum reconstruction.With a spin-parity assignment of the 484 keV stateto 3/2 + , the strength measured in the Coulomb excita-tion experiment [9] would correspond to an E1 transition,with a B(E1) of 0.035(10) e f m . This large electricdipole strength is similar to the one observed in Ne[37], although the weak binding and low angular momen-tum conditions are not as extreme as in the case of Mg.This may indicate that strong deformation and core ex-citations play an important role in this nucleus.The absence of observation of any (cid:96) =3 momentum dis-tribution, which would reveal the feeding of the nor-mal 0p0h configuration 7/2 − state, is surprising. Thislikely indicates that this state has an isomeric nature,and therefore decayed too far from the γ -ray detectionarray. This assumption is further reinforced by the fit-ting of the inclusive momentum distribution which indi-cate that an (cid:96) =3 component is needed. A fit includingthe (cid:96) =3 component, combined with the deduced groundstate branching, allows to indirectly determine the par-tial cross section for this elusive 7/2 − state.A possible indication of the location of the 7/2 − stateis provided by the comparison of the populated statesin Mg from the one-neutron and one-proton removalreactions. The differences observed in the one-proton re-moval reaction can reveal the composition of the statesnot populated in that reaction. The most important dif-ference is the disappearance of both 549 keV and 703 keVtransitions. Since the removal of a d / proton from Alis expected to populate mainly positive parity states in Al, the disappearance of both transitions could indicatethat the 549 keV transitions corresponds to the popula-tion of the unobserved 7/2 − isomeric state from the 703keV state, which would place this state at around 154 keV, reaching a similar conclusion to the work in [13]where the presence of this isomer was also inferred.Partial cross sections are compared to theoretical onesbased on the eikonal reaction model and shell model cal-culations using 3 modern interactions (IMSRG, SDPF-MU and SDPF-U-SI). They reveal the complicated na-ture of the states in Mg where large configuration mix-ing with p-h excitations play a very important role. Boththe IMSRG and SDPF-MU reproduce the observed crosssections rather well, especially when taking into accountthe strength feeding the indirectly observed 7/2 − stateat 154 keV.Clearly additional data is needed on the spectroscopyof this nucleus. The one-neutron removal reaction from Mg would be advantageously performed using a high-resolution γ -ray tracking array, in order to better resolvethe numerous transitions and γ - γ coincidences. How-ever, the non-observation of the prompt decay from theinferred isomeric 7/2 − state would remain in such animproved experiment. From the estimates of shell modelcalculations, the delayed 154 keV γ -ray from the decay ofthe isomer could be detected in a setup with γ -ray detec-tion around the implantation site of the Mg residues. Adirect search for this isomer could also be performed froma decay experiment using a beam of Mg produced viaprojectile fragmentation. Another possibly more promis-ing venue would be using a (d,p) transfer reaction on a Mg beam in inverse kinematics, that would be able tomeasure and identify the (cid:96) =3 strength directly, as wellas to other populated states. The parity of the 484 keVstate in particular, could not be determined in this ex-periment, but would be easily identified in a transfer re-action study. Such a Mg beam will be available in theearly days of FRIB operations, produced via projectilefragmentation and re-accelerated by the ReA6 linac toenergies close to 10 MeV/u.
VI. ACKNOWLEDGEMENTS
The authors would like to thank the RIKEN NishinaRIBF accelerator staff and the BigRIPS team for pro-viding the high intensity Ca primary beam and theresulting high quality Mg and Al radioactive beams.DB acknowledges fruitful discussions with A. Gade. Thiswork is supported in part by the U.S. National ScienceFoundation under grant PHY-0606007 and the U.S. De-partment of Energy, Office of Science, Office of NuclearPhysics under grant DE-SC0020451. SRS is supportedby the U. S. Department of Energy under Contract DE-FG02-97ER41014. [1] E. K. Warburton, J. A. Becker, and B. A. Brown, Phys.Rev. C , 1147 (1990). [2] C. D´etraz, D. Guillemaud, G. Huber, R. Klapisch,M. Langevin, F. Naulin, C. Thibault, L. C. Carraz, andF. Touchard, Phys. Rev. C , 164 (1979). [3] K. Yoneda, H. Sakurai, T. Gomi, T. Motobayashi,N. Aoi, N. Fukuda, U. Futakami, Z. Gacsi, Y. Higurashi,N. Imai, N. Iwasa, H. Iwasaki, T. Kubo, M. Kunibu,M. Kurokawa, Z. Liu, T. Minemura, A. Saito, M. Ser-ata, S. Shimoura, S. Takeuchi, Y. Watanabe, K. Yamada,Y. Yanagisawa, K. Yogo, A. Yoshida, and M. Ishihara,Physics Letters B , 233 (2001).[4] A. Gade, P. Adrich, D. Bazin, M. D. Bowen, B. A. Brown,C. M. Campbell, J. M. Cook, S. Ettenauer, T. Glas-macher, K. W. Kemper, S. McDaniel, A. Obertelli,T. Otsuka, A. Ratkiewicz, K. Siwek, J. R. Terry, J. A.Tostevin, Y. Utsuno, and D. Weisshaar, Phys. Rev. Lett. , 072502 (2007).[5] P. Doornenbal, H. Scheit, S. Takeuchi, N. Aoi, K. Li,M. Matsushita, D. Steppenbeck, H. Wang, H. Baba,H. Crawford, C. R. Hoffman, R. Hughes, E. Ideguchi,N. Kobayashi, Y. Kondo, J. Lee, S. Michimasa, T. Moto-bayashi, H. Sakurai, M. Takechi, Y. Togano, R. Winkler,and K. Yoneda, Phys. Rev. Lett. , 212502 (2013).[6] S. Michimasa, Y. Yanagisawa, K. Inafuku, N. Aoi,Z. Elekes, Z. F¨ul¨op, Y. Ichikawa, N. Iwasa, K. Kurita,M. Kurokawa, T. Machida, T. Motobayashi, T. Naka-mura, T. Nakabayashi, M. Notani, H. J. Ong, T. K. On-ishi, H. Otsu, H. Sakurai, M. Shinohara, T. Sumikama,S. Takeuchi, K. Tanaka, Y. Togano, K. Yamada, M. Yam-aguchi, and K. Yoneda, Phys. Rev. C , 054307 (2014).[7] P. Doornenbal, H. Scheit, N. Aoi, S. Takeuchi, K. Li,E. Takeshita, H. Wang, H. Baba, S. Deguchi, N. Fukuda,H. Geissel, R. Gernh¨auser, J. Gibelin, I. Hachiuma,Y. Hara, C. Hinke, N. Inabe, K. Itahashi, S. Itoh,D. Kameda, S. Kanno, Y. Kawada, N. Kobayashi,Y. Kondo, R. Kr¨ucken, T. Kubo, T. Kuboki, K. Kusaka,M. Lantz, S. Michimasa, T. Motobayashi, T. Naka-mura, T. Nakao, K. Namihira, S. Nishimura, T. Ohnishi,M. Ohtake, N. A. Orr, H. Otsu, K. Ozeki, Y. Satou,S. Shimoura, T. Sumikama, M. Takechi, H. Takeda, K. N.Tanaka, K. Tanaka, Y. Togano, M. Winkler, Y. Yanagi-sawa, K. Yoneda, A. Yoshida, K. Yoshida, and H. Saku-rai, Phys. Rev. Lett. , 032501 (2009).[8] I. Murray, M. MacCormick, D. Bazin, P. Doornenbal,N. Aoi, H. Baba, H. Crawford, P. Fallon, K. Li, J. Lee,M. Matsushita, T. Motobayashi, T. Otsuka, H. Sakurai,H. Scheit, D. Steppenbeck, S. Takeuchi, J. A. Tostevin,N. Tsunoda, Y. Utsuno, H. Wang, and K. Yoneda, Phys.Rev. C , 011302 (2019).[9] B. V. Pritychenko, T. Glasmacher, B. A. Brown, P. D.Cottle, R. W. Ibbotson, K. W. Kemper, L. A. Riley, andH. Scheit, Phys. Rev. C , 011305 (2000).[10] P. Doornenbal, H. Scheit, S. Takeuchi, N. Aoi, K. Li,M. Matsushita, D. Steppenbeck, H. Wang, H. Baba,E. Ideguchi, N. Kobayashi, Y. Kondo, J. Lee, S. Michi-masa, T. Motobayashi, A. Poves, H. Sakurai, M. Takechi,Y. Togano, and K. Yoneda, Phys. Rev. C , 044306(2016).[11] T. Otsuka, A. Gade, O. Sorlin, T. Suzuki, and Y. Ut-suno, Rev. Mod. Phys. , 015002 (2020).[12] I. Hamamoto, Phys. Rev. C , 064329 (2012).[13] S. Nummela, F. Nowacki, P. Baumann, E. Caurier,J. Cederk¨all, S. Courtin, P. Dessagne, A. Jokinen,A. Knipper, G. Le Scornet, L. G. Lyapin, C. Mieh´e,M. Oinonen, E. Poirier, Z. Radivojevic, M. Ramdhane,W. H. Trzaska, G. Walter, J. ¨Ayst¨o, and I. Collabora-tion, Phys. Rev. C , 054313 (2001). [14] B. V. Pritychenko, T. Glasmacher, P. D. Cottle, R. W.Ibbotson, K. W. Kemper, L. A. Riley, A. Sakharuk,H. Scheit, M. Steiner, and V. Zelevinsky, Phys. Rev.C , 061304 (2002).[15] D. T. Yordanov, M. Kowalska, K. Blaum, M. De Rydt,K. T. Flanagan, P. Lievens, R. Neugart, G. Neyens, andH. H. Stroke, Phys. Rev. Lett. , 212501 (2007).[16] V. Tripathi, S. L. Tabor, P. F. Mantica, Y. Utsuno,P. Bender, J. Cook, C. R. Hoffman, S. Lee, T. Otsuka,J. Pereira, M. Perry, K. Pepper, J. S. Pinter, J. Stoker,A. Volya, and D. Weisshaar, Phys. Rev. Lett. ,142504 (2008).[17] D. T. Yordanov, K. Blaum, M. De Rydt, M. Kowalska,R. Neugart, G. Neyens, and I. Hamamoto, Phys. Rev.Lett. , 129201 (2010).[18] V. Tripathi, S. L. Tabor, P. F. Mantica, Y. Utsuno,P. Bender, J. Cook, C. R. Hoffman, S. Lee, T. Otsuka,J. Pereira, M. Perry, K. Pepper, J. S. Pinter, J. Stoker,A. Volya, and D. Weisshaar, Phys. Rev. Lett. ,129202 (2010).[19] R. Kanungo, C. Nociforo, A. Prochazka, Y. Utsuno,T. Aumann, D. Boutin, D. Cortina-Gil, B. Davids, M. Di-akaki, F. Farinon, H. Geissel, R. Gernh¨auser, J. Gerl,R. Janik, B. Jonson, B. Kindler, R. Kn¨obel, R. Kr¨ucken,M. Lantz, H. Lenske, Y. Litvinov, K. Mahata, P. Maier-beck, A. Musumarra, T. Nilsson, T. Otsuka, C. Perro,C. Scheidenberger, B. Sitar, P. Strmen, B. Sun, I. Szarka,I. Tanihata, H. Weick, and M. Winkler, Physics LettersB , 253 (2010).[20] U. Datta, A. Rahaman, T. Aumann, S. Beceiro-Novo,K. Boretzky, C. Caesar, B. V. Carlson, W. N. Catford,S. Chakraborty, M. Chartier, D. Cortina-Gil, G. de An-gelis, P. Diaz Fernandez, H. Emling, O. Ershova, L. M.Fraile, H. Geissel, D. Gonzalez-Diaz, B. Jonson, H. Jo-hansson, N. Kalantar-Nayestanaki, T. Kr¨oll, R. Kr¨ucken,J. Kurcewicz, C. Langer, T. Le Bleis, Y. Leifels, J. Mar-ganiec, G. M¨unzenberg, M. A. Najafi, T. Nilsson, C. No-ciforo, V. Panin, S. Paschalis, R. Plag, R. Reifarth,V. Ricciardi, D. Rossi, H. Scheit, C. Scheidenberger,H. Simon, J. T. Taylor, Y. Togano, S. Typel, V. Volkov,A. Wagner, F. Wamers, H. Weick, M. Weigand, J. S.Winfield, D. Yakorev, and M. Zoric, Phys. Rev. C ,034304 (2016).[21] S. Paschalis, I. Lee, A. Macchiavelli, C. Campbell,M. Cromaz, S. Gros, J. Pavan, J. Qian, R. Clark,H. Crawford, D. Doering, P. Fallon, C. Lionberger,T. Loew, M. Petri, T. Stezelberger, S. Zimmermann,D. Radford, K. Lagergren, D. Weisshaar, R. Winkler,T. Glasmacher, J. Anderson, and C. Beausang, NuclearInstruments and Methods in Physics Research Section A:Accelerators, Spectrometers, Detectors and AssociatedEquipment , 44 (2013).[22] A. L. Richard, H. L. Crawford, P. Fallon, A. O. Macchi-avelli, V. M. Bader, D. Bazin, M. Bowry, C. M. Camp-bell, M. P. Carpenter, R. M. Clark, M. Cromaz, A. Gade,E. Ideguchi, H. Iwasaki, M. D. Jones, C. Langer, I. Y.Lee, C. Loelius, E. Lunderberg, C. Morse, J. Rissanen,M. Salathe, D. Smalley, S. R. Stroberg, D. Weisshaar,K. Whitmore, A. Wiens, S. J. Williams, K. Wimmer,and T. Yamamato, Phys. Rev. C , 011303 (2017).[23] T. Kubo, D. Kameda, H. Suzuki, N. Fukuda, H. Takeda,Y. Yanagisawa, M. Ohtake, K. Kusaka, K. Yoshida,N. Inabe, T. Ohnishi, A. Yoshida, K. Tanaka, andY. Mizoi, Progress of Theoretical and Experimen- tal Physics (2012), 10.1093/ptep/pts064,03C003, https://academic.oup.com/ptep/article-pdf/2012/1/03C003/11595011/pts064.pdf.[24] S. Takeuchi, T. Motobayashi, Y. Togano, M. Matsushita,N. Aoi, K. Demichi, H. Hasegawa, and H. Murakami,Nuclear Instruments and Methods in Physics ResearchSection A: Accelerators, Spectrometers, Detectors andAssociated Equipment , 596 (2014).[25] P. Hansen and J. Tostevin, Annual Review ofNuclear and Particle Science , 219 (2003),https://doi.org/10.1146/annurev.nucl.53.041002.110406.[26] S. R. Stroberg, A. Gade, J. A. Tostevin, V. M. Bader,T. Baugher, D. Bazin, J. S. Berryman, B. A. Brown,C. M. Campbell, K. W. Kemper, C. Langer, E. Lun-derberg, A. Lemasson, S. Noji, F. Recchia, C. Walz,D. Weisshaar, and S. J. Williams, Phys. Rev. C ,034301 (2014).[27] R. Lic˘a, F. Rotaru, M. J. G. Borge, S. Gr´evy, F. Negoit¸˘a,A. Poves, O. Sorlin, A. N. Andreyev, R. Borcea,C. Costache, H. De Witte, L. M. Fraile, P. T. Greenlees,M. Huyse, A. Ionescu, S. Kisyov, J. Konki, I. Lazarus,M. Madurga, N. M˘arginean, R. M˘arginean, C. Mi-hai, R. E. Mihai, A. Negret, R. D. Page, J. Pakari-nen, S. Pascu, V. Pucknell, P. Rahkila, E. Rapisarda,A. S¸erban, C. O. Sotty, L. Stan, M. St˘anoiu, O. Teng-blad, A. Turturic˘a, P. Van Duppen, R. Wadsworth, andN. Warr (IDS Collaboration), Phys. Rev. C , 021301(2017).[28] R. Han, X. Li, W. Jiang, Z. Li, H. Hua, S. Zhang,C. Yuan, D. Jiang, Y. Ye, J. Li, Z. Li, F. Xu, Q. Chen,J. Meng, J. Wang, C. Xu, Y. Sun, C. Wang, H. Wu,C. Niu, C. Li, C. He, W. Jiang, P. Li, H. Zang, J. Feng,S. Chen, Q. Liu, X. Chen, H. Xu, Z. Hu, Y. Yang, P. Ma,J. Ma, S. Jin, Z. Bai, M. Huang, Y. Zhou, W. Ma, Y. Li,X. Zhou, Y. Zhang, G. Xiao, and W. Zhan, Physics Let-ters B , 529 (2017). [29] S. R. Stroberg, A. Calci, H. Hergert, J. D. Holt, S. K.Bogner, R. Roth, and A. Schwenk, Phys. Rev. Lett. ,032502 (2017).[30] T. D. Morris, N. M. Parzuchowski, and S. K. Bogner,Phys. Rev. C , 034331 (2015).[31] K. Hebeler, S. K. Bogner, R. J. Furnstahl, A. Nogga, andA. Schwenk, Phys. Rev. C , 031301 (2011).[32] D. Gloeckner and R. Lawson, Physics Letters B , 313(1974).[33] T. Miyagi, S. R. Stroberg, J. D. Holt, and N. Shimizu,“Ab initio multi-shell valence-space hamiltonians and theisland of inversion,” (2020), arXiv:2004.12969 [nucl-th].[34] B. Brown and W. Rae, Nuclear Data Sheets , 115(2014).[35] Y. Utsuno, T. Otsuka, B. A. Brown, M. Honma,T. Mizusaki, and N. Shimizu, Phys. Rev. C , 051301(2012).[36] F. Nowacki and A. Poves, Phys. Rev. C , 014310(2009).[37] C. Loelius, N. Kobayashi, H. Iwasaki, D. Bazin, J. Be-large, P. C. Bender, B. A. Brown, R. Elder, B. Elman,A. Gade, M. Grinder, S. Heil, A. Hufnagel, B. Longfellow,E. Lunderberg, M. Mathy, T. Otsuka, M. Petri, I. Syn-dikus, N. Tsunoda, D. Weisshaar, and K. Whitmore,Phys. Rev. Lett. , 262501 (2018).[38] N. Tsunoda, T. Otsuka, N. Shimizu, M. Hjorth-Jensen,K. Takayanagi, and T. Suzuki, Phys. Rev. C , 021304(2017).[39] M. Seidlitz, D. M¨ucher, P. Reiter, V. Bildstein,A. Blazhev, N. Bree, B. Bruyneel, J. Cederk¨all,E. Clement, T. Davinson, P. Van Duppen, A. Ekstr¨om,F. Finke, L. Fraile, K. Geibel, R. Gernh¨auser, H. Hess,A. Holler, M. Huyse, O. Ivanov, J. Jolie, M. Kalk¨uhler,T. Kotthaus, R. Kr¨ucken, R. Lutter, E. Piselli, H. Scheit,I. Stefanescu, J. Van de Walle, D. Voulot, N. Warr,F. Wenander, and A. Wiens, Physics Letters B ,181 (2011).[40] J. A. Tostevin and A. Gade, Phys. Rev. C90