Evolution of the N=20 and 28 Shell Gaps and 2-particle-2-hole states in the FSU Interaction
R. S. Lubna, K. Kravvaris, S. L. Tabor. Vandana Tripathi, E. Rubino, A. Volya
aa r X i v : . [ nu c l - e x ] N ov Evolution of the N=20 and 28 Shell Gaps and 2-particle-2-hole states in the FSUInteraction.
R. S. Lubna, ∗ K. Kravvaris,
S. L. Tabor, Vandana Tripathi, E. Rubino, and A. Volya Department of Physics, Florida State University, Tallahassee, FL 32306, USA Lawrence Livermore National Laboratory, Livermore, CA 94550, USA (Dated: November 17, 2020)The connection between fundamental nucleon-nucleon forces and the observed many-body struc-ture of nuclei is a main question of modern nuclear physics. Evolution of the mean field, inversionof traditional shell structures and structure of high spin states in nuclei with extreme proton toneutron ratios are at the center of numerous recent experimental investigations targeting the matrixelements of the effective nuclear Hamiltonian that is responsible for these phenomena. The FSU spsdfp cross-shell interaction for the shell model was successfully fitted to a wide range of mostlyintruder negative parity states of the sd shell nuclei. In this paper we explore the evolution ofnuclear structure in and around the “Island of Inversion” (IoI) where low-lying states involve cross-shell particle-hole excitations. We apply the FSU interaction to systematically trace out the relativepositions of the effective single-particle energies (ESPE) of the 0 f / and 1 p / orbitals forming the N = 20 and 28 shell gaps. We find that above a proton number of about 13 the 0 f / neutron orbitallies below that of 1 p / , which is considered normal ordering, but systematically, for more exoticnuclei with lower Z = 12 and 10 the order of orbitals reversed. The crossing of the neutron orbitalshappens right near the neutron separation threshold. Our Hamiltonian reproduces remarkably wellthe absolute binding energies for a broad range of nuclei, and the inversion in the configurations ofnuclei inside the IoI. The new effective interaction accounts well for the energies and variations withmass number A of aligned high-spin states that involve nucleon pairs prompted across the shell gap.This work puts forward an empirically determined effective Hamiltonian where data from manyrecent experiments allowed us to significantly improve our knowledge about cross shell nuclearinteraction matrix elements. The quality, with which this Hamiltonian describes the two-particletwo-hole (2p2h) cross shell excitations, binding energies, and the physics of aligned states that werenot a part of the fit, is remarkable, making the FSU interaction an important tool for the futureexploration of exotic nuclei. I. INTRODUCTION
Recent experimental works in the 1 s d shell withlarge γ detector arrays and heavy-ion fusion reactionshave substantially extended the knowledge of relativelyhigh spin states. However, these do not form well-behaved rotational bands amenable to study by collec-tive models because rotational energies are comparableto single-particle energies. On the other hand micro-scopic configuration-interaction model calculations arefeasible in these lighter nuclei. The USD family of ef-fective interactions [1, 2] have been very successful indescribing most lower-lying positive-parity states of nu-clei with 8 ≤ ( N, Z ) ≤
20. However, higher spin statesinvolve excitations into the f p shell where orbitals con-tributing larger values of angular momentum are occu-pied, which is beyond the scope of the USD interaction.Also, neutron-rich isotopes quickly move beyond the sd shell boundaries [3–8].Over the years, several configuration interaction mod-els have made significant contribution towards explainingcross-shell excitations [9–13]. A case in point is the “Is-land of Inversion” (IoI). Perhaps in an inverse way thefirst contribution came from the failure of the otherwise ∗ Present address: TRIUMF, Vancouver, BC V6T 2A3, Canada. very successful pure sd interactions [1, 2] to reproduce thestronger binding energy measured for Na [14], pointingto the importance of effects outside the sd shell. TheWBMB [15] interaction, which was designed for the nu-clei near Ca was successful in reproducing the inversionof some nuclei within the IoI. More recent shell modelcalculations using interactions like SDPF-M [12], SDPF-U-MIX [13] have shown that the IoI phenomenon can beaccounted for by a reduction of the N = 20 shell gap.Recently, a significant theoretical result was reported,see Ref. [16], showing the emergence of IoI effect fromnucleon-nucleon forces stemming from the fundamentalprinciples of QCD. This highlights the importance of cer-tain cross sd - f p interaction terms that we assess in thiswork using experimental systematics.In search for a single cross-shell interaction whichworks well over a wide range of nuclei, we have developeda new interaction [17] with parallel treatment of protonsand neutrons by fitting the energies of 270 states in nu-clei from C through Ti and V originated from theWBP interaction [18] using well-established techniques.The present report is organized as follows: First we willdiscuss the development of the new FSU shell model in-teraction. The trend of the effective single particle en-ergies (ESPEs) of the 0 f / and 1 p / orbitals for the sd -shell nuclei will be examined along with the compari-son to the experimental data. Then we will move to theIoI region and test some predictions of the FSU interac-tion in this region. Finally, the experimentally observedfully aligned states with the f / configuration will beinterpreted with the new shell model interaction. II. DEVELOPMENT OF THE FSU EMPIRICALSHELL MODEL INTERACTION
A modified version of the WBP [18] interaction hasbeen used as the starting point of the data fitting pro-cedure. The WBP interaction was developed in order toaddress the cross-shell structure around A = 20. Whilethe sd - f p cross-shell matrix elements of the WBP weretaken from the WBMB [15] interaction which was devel-oped for the nuclei around Ca, the different single par-ticle energies and different implementation make WBPnot good for the upper sd -shell nuclei. Yet, the WBP isa perfect starting point for a more modern, much broaderassessment of the nuclear matrix elements. Our data setincluded nuclei from the upper mass region of the p , thefull sd , and lower mass region of the f p shells; wherewe systematically looked at states that involve a particlepromotion across the harmonic oscillator shell, referredto as 1 particle- 1 hole excited states (1p1h). In the sd -shell region of most interest and most data, the combined0p0h and 1p1h space considered in the fit is equivalent to0 ~ ω and 1 ~ ω often referred to as the N max = 1 harmonicoscillator basis truncation. The resulting fit seamlesslyspans from the A = 20 region, where the low-lying in-truder states are predominantly those with holes in thelower p -shell, to the island of inversion around A = 40where particles are promoted to f p shell. The ability toseparate the center of mass exactly within the N max = 1harmonic oscillator basis truncation is an additional ben-efit of this strategy.Before the current effort of developing the FSU inter-action, a number of attempts have been made to mod-ify the WBP interaction, mainly by changing the singleparticle energies (SPEs) of the f p -shell orbitals for a par-ticular sd -shell nucleus and applying that for the nearbyisotopes. For example, in the WBP-A [19] version, theSPEs of the f / and p / orbitals were lowered in orderto better explain the negative parity intruder states of P. This adjustment was quite successful in explainingthe energy levels of P and P, however, WBP-A failedto predict the intruder states of Si. Hence, another ver-sion of the WBP, called the WBP-B was introduced [20]by changing the SPEs of the f / , p / and p / orbitals.In a different version, named WBP-M [21], all the SPEsof the f p shell orbitals were changed in order to repro-duce the energies and the ordering of the 3 / − and 7 / − states of Ne which eventually fixed the ordering of thesame levels in Ne and Mg. However, none of thesemodified versions was able to reproduce the experimen-tal data for a large range of the nearby nuclei, and hencewe have taken a step forward towards building a moregeneral effective shell model Hamiltonian.The model space for the WBP interaction and for the newly developed one consists of four major oscilla-tor shells; 0 s , 0 p , 1 s d , and 0 f p . The following stepsbriefly describe the development of the FSU interaction. • The newly developed interaction starts from theWBP framework, the model consists of four majoroscillator shells: 0 s , 0 p , 1 s d , and 0 f p . Isospin in-variance is assumed and Coulomb corrections to thebinding energies are implemented using the stan-dard procedures as discussed in Refs. [1, 2, 15]. • The single particle energies (SPE) and the twobody matrix elements (TBME) of the 0 s and 0 p shells and across 0 s - 0 p are same as those of theoriginal WBP interaction and are not a part of thefit. • The TBMEs within the sd shell are taken from theUSDB [2] interaction and also are not a part of thefit. • The 6 monopoles between the orbitals of the 0 p shell and sd shell are modified simultaneously withthe sd shell single particle energies, thus changingthe shell gap but ensuring that excitation energiesof all 0 ~ ω states in the sd shell are identical to thosefrom the USDB calculations. • sd - f p cross-shell matrix elements:1. 1 p / orbital in the f p shell is relatively highand not very sensitive to our data set. Wethus fitted only one monopole term betweenthe 1 p / and the sd orbitals. This amountsto two fit parameters because we have alloweddifferent strengths for isospin T=0 and T=1.2. Only the monopole terms between 0 f / -0 d / and 1 p / - 0 d / were considered since d / is deeply bound for sd - f p cross-shell nu-clei. A total of 4 parameters were varied forT=0 and T=1.3. For the remaining 0 f / - 1 s / , 0 f / - 0 d / ,1 p / - 0 s / , and 1 p / - 0 d / all multipole-multipole density terms were fitted. A totalof 24 parameters were varied. • For the f p shell, GXPF1A [22] was used as a start-ing Hamiltonian and all the TBMEs associatedwith only 0 f / and 1 p / were fitted; a total 30TBMEs and hence 30 parameters were adjustedwithin the f p shell. • All the matrix elements within the sd and f p shellsas well as the sd - f p cross shell were scaled with A − . . However, no scaling was adopted for thecross shell interactions between the lower p and the sd shells. • A total of 70 parameters were fitted using 270 ex-perimentally observed states compiled in Ref. [23]and [24]. The experimental data was compiled fromfour groups1. Intruder states sensitive to p - sd shell gap.This group consists of pure p shell C and N iso-topes and nuclei between O to Si with statesthat have strong spectroscopic factors if pop-ulated via ( p, d ) reactions.2. Negative parity states in sd -shell populatedvia ( d, p ) reactions which are sensitive to par-ticle promotion from sd to f p . High spinstates, that gain spin from the promotion of aparticle to 0 f / are of particular importance.3. Neutron rich cross shell nuclei with Z <
N >
20 where both 0 ~ ω and 1 ~ ω types ofstates were included in the fit.4. Nuclei in f p shell with Z ≥
20 and N ≥ ~ ω states in these nuclei are critical fortuning the 0 f / - 1 p / gap. • The fitting procedure followed the method de-scribed in [2], with 40 linear combinations of pa-rameters being selected at each iteration. Wereached the convergence after 6 iterations with anoverall rms deviation from experiment of 190 keV. • All calculations were carried out within N max = 1truncation thus including 0 ~ ω and 1 ~ ω types of ex-citations that due to different parities do not mix.This truncation allows for exact identification andseparation of the spurious center-of-mass excita-tions. • Tables of the matrix elements can be found in theThesis publication of Lubna, Rebeka Sultana [23].Users are encouraged to contact the authors forhelp with the calculations, further details and up-dates.All the shell model calculations were performed withthe shell model code CoSMo [25]. A histogram of thedifferences between the experimental states included inthe fit and those predicted with the FSU interaction isshown in Figure 1.
III. EFFECTIVE SINGLE PARTICLE ENERGY(ESPE)
The evolution of the mean field, which is describedby the position of the single particle levels and howthey change with number of protons and neutrons, isa particularly interesting and non-trivial question in thestrongly-interacting two-component many-body systemsof atomic nuclei. In most nuclei the single-particlestrength is distributed over many states. Systematicstudies have been performed before with other shellmodel interactions [10, 12, 26] to understand the evo-lution of the ESPE. An experimental approach of deter-mining the ESPEs has been to measure and sum up theenergies of appropriate states weighted by the reaction -0.4 -0.2 0 0.2 0.4
Difference (MeV) N u m b e r FIG. 1. Histogram of the differences in excitation energybetween experiment and the FSU interaction fit. The root-mean-square deviation is 190 keV. spectroscopic factors. This process is limited by decreas-ing cross sections for higher lying states and by difficul-ties in making spin assignments and in determining whatfraction of the cross sections come from direct reactioncomponents.Theoretical approaches do not suffer from most of theseexperimental limitations, but have their own uncertain-ties. Perhaps chief among them being the uncertaintyin the interaction Hamiltonian. The bare single-particleenergies tell only a part of the story of the effective shellpositions. The TBME have a major influence on thepositions of the orbitals. In fact, the TBMEs shift theorbitals based on the number of particles in shells, andare the major reason that one interaction could fit sucha wide range of nuclei.How the newly developed FSU interaction describesthe shell evolution is among the most interesting imme-diate questions that can be addressed. While the FSUinteraction was fitted to the negative-party states in sd nuclei, the study of the ESPE extrapolates to a muchbroad spectrum of configurations not limited by thoseexperimentally reachable with single-nucleon transfer re-actions.The evaluation of the ESPE relies on 0 ~ ω and 1 ~ ω cal-culations. In order to determine the ESPE of the 0 f / and 1 p / orbitals, we have followed a procedure similarto the experimental approach, but using the theoreti-cally computed energies and spectroscopic factors in thefollowing formulaESPE = P i=1 SF i × E ∗ i P i=1 SF i (1)In the above formula, SF i is the spectroscopic factor for A → A +1 where a particle is placed onto a single-particleorbit of interest above an even-even A core. The E ∗ i is theexcitation energy of the i-th state in A +1 with the match-ing quantum numbers measured relative to the groundstate energy of the core A. It has been observed from thecalculations that it is enough to consider 30 lowest statesin the sum (1), by then the SF reach to a saturationand the ESPE converges. From the formal theoreticalperspective, Eq. (1) represents single particle energies ofthe mean field arising from the shell model Hamiltonian.The ESPEs obtained from the above formula acrossthe sd shell are plotted in Figure 2 as a function of pro-ton number Z . The points represent the ESPEs of the0 f / and 1 p / . The systematic crossing of the ESPEsof the 0 f / and 1 p / orbitals with increasing neutronnumber is evident in the figure. The crossing occursbetween Z = 10 and 12, suggesting that the N = 28shell gap shifts to N = 24 with lower Z , which points tothe inversion of 0 f / and 1 p / neutron orbitals. Theground state of Ne is tentatively assigned 3 / − as isthe first excited state in Ne [24]. In Mg the lowest3 / − and 7 / − states are essentially degenerate [24]. -8-6-4-202 -6-4-2024-8-6-4-202 -6-4-20240f Z Z E S P E ( M e V ) E S P E ( M e V ) N = 20N = 18N = 16 N = 14N = 12N = 10
FIG. 2. Neutron Effective Single Particle Energies (ESPEs) of0 f / and 1 p / orbitals calculated with the FSU interaction.They represent the theoretical centroids of the energies of the0 f / and 1 p / orbitals. In the “normal” ordering the reddiamonds (1 p / ) lie above the black circles (0 f / ). This inversion of the 1 p / and 0 f / ESPE is relatedto the 2-body interactions between nucleons in the sd and f p shells; the effect of this interaction is density depen-dent and varies as a function of the shell fillings. In theFSU interaction these TBME emerge as a consequenceof fitting the energies of the states in a wide range ofnuclei. Over half a century ago Talmi and Unna [27]attributed the inversion of the 1 s / and 0 p / orbitalsto the same principle. Alternate explanations, especiallyfor the 1 s / and 0 p / case, have been given in termsof the effects of weak binding on the mean field of low ℓ orbitals. Hoffman et al . [28] have explored the weakbinding effect for pure single-particle shells in a Woods-Saxon potential and have shown that it is large near thethreshold for neutron s states. While much smaller for p states, there is still a crossing between the 0 p / and 0 d / orbitals at the threshold. A similar effect for the1 p / and 0 f / appears to be a contributing factor to theinversion shown in Figure 2. Nearly all crossings occuraround ESPE= 0 indicating that the levels become un-bound. Indeed, the centrifugal barrier for ℓ = 3 f orbitalis high which would make a transition into the continuumsmooth, while for the ℓ = 1 p -wave the interaction withthe continuum is strong and is pushing the level downas discussed in Ref. [29]. It appears that the continuumeffect is incorporated through the fitting of the effectiveinteraction, but this can be a challenge for theoreticalmethods that do not take continuum of reaction statesinto account. This inversion of the 1 p / and 0 f / ESPEat high neutron excess also has implications for the IoIphenomenon discussed in the next section.Another way of examining the systematics of shell evo-lution, which is closer to experiment, is from the posi-tions of the states carrying the largest part of the single-particle strength. Such a comparison is shown in TableI which lists the experimental and theoretical excitationenergies of the lowest 3 / + , 7 / − , and 3 / − states, ofthe even Z odd mass nuclei, along with the predictedand measured ( d, p ) reaction spectroscopic factors (SF).As mentioned before, there is more uncertainty in mea-suring the values of SF than excitation energies and insome cases the SF cannot (lack for appropriate targets)or have not been measured. With this in mind, the agree-ment between experiment and predictions using the FSUinteraction for both excitation energies and SF is gener-ally good. Also, the relatively large values of the SF showthat these states represent the dominant single-particlestates.Figure 3(a) provides a pictorial summary of the relativepositions between the 7 / − and 3 / − states as a functionof the proton number Z . The black circles and red linesshow the average values from Table I for experiment andtheory, respectively, while the black error bars representthe variation of the experimental differences. The ob-served trends are reproduced by theory, see Figure 3(a).This graph agrees qualitatively with those in Figure 2.It demonstrates that the evolution of the separation be-tween the 7 / − and 3 / − states is largely a function ofthe proton number Z and that the 3 / − energies dropbelow the 7 / − ones between Z = 14 and 12. In contrastto the ESPEs which approximate the positions of the0 f / and 1 p / orbitals the crossing between 0 f / and1 p / happens between Z = 10 and 12. Together theseresults show that the trend is robust, but the question ofthe relative position of the orbitals is more complex andnuanced than was expected earlier. IV. EVOLUTION OF THE N=20 SHELL GAPAND THE ISLAND OF INVERSION (IOI)
One of the first indications that the pure sd shell modelcould not represent low-lying states in all sd nuclei camefrom the experimentally measured mass of Na [14]. The
TABLE I. Comparison of the experimentally observed 7 / − ,3 / − and 3 / + states of even Z odd mass sd -shell nuclei tothe predictions by the FSU interaction. The measured spec-troscopic factors were taken from the NNDC [24]. All theexperimental spectroscopic factors were compiled from the( d, p ) reactions.Nucleus J π Energy (2J+1)SFEXP Th EXP Th Ne 7 / − / − / + Ne 7 / − / −
765 858 2.6 2.43 / + Mg 7 / − / − /
2+ 974 1098 0.8 0.9 Mg 7 / − / − /
2+ 984 994 2.4 1.56 Mg 7 / − / − / + Si 7 / − / − / + Si 7 / − / − / + Si 7 / − / − / + Si 7 / − / −
910 909 2.8 3.73 / +
974 936 S 7 / − / − / + S 7 / − / − / + S 7 / − / −
646 573 1.8 3.53 / + Ar 7 / − / − / + Ar 7 / − / − / +
10 12 14 16 18
Proton Number Z N e u t r o n O cc up a n c y f G.S.p
G.S.f
Excp
Exc -1-0.500.511.5 E ( / - ) - E ( / - ) ( M e V ) Exp.FSU (a)(b)
FIG. 3. (a) Average energy differences between the lowest7 / − and 3 / − experimental levels in Table I. The error barsgive an indication of the range of values for different neutronnumbers. Positive (negative) values of the ordinate corre-spond to the 3 / − state above (below) the 7 / − one. (b)Occupancies of the neutron 0 f / and 1 p / orbitals in neu-tron number N = 20 nuclei as a function of proton number Z for the lowest 2p2h states. The values are shown as filledcircles for the cases where the lowest 2p2h state is the groundstate (IoI) and as open circles where the lowest 2p2h state isexcited above the ground state. experimental mass was about 1.6 MeV lower than thatpredicted from the USD interaction [1]. This was furtherclarified by the USDA, USDB showing that states in thehighest N - Z nuclei can not be fitted. A consistent over-prediction of 1 to 2 MeV of the ground state energies ofthese nuclei can be seen in Figure 9 of Ref. [2]. This re-gion of nuclei is now known as the “Island of Inversion”(IoI) and its origin has been discussed a lot. Most expla-nations center around the filled or almost filled neutron sd shell and f p intruder configurations leading, counter-intuitively, to lowering the energy of the 2 particle- 2 hole(2p2h) state, with two nucleons being promoted from sd to f p shell, below that of the “normal” 0p0h. Such low-ering is associated with increased correlation energy orhigher deformation, lowering Nilsson orbitals. Howeverthe effect fades away with filling of the proton sd shell.While a number of shell model calculations in the pasthave reproduced many aspects of the IoI, as discussedin the Introduction, here we study what the FSU inter-action predicts for the Iol nuclei. Concentrating on theIoI region, we consider the states where two nucleons arepromoted from sd to f p referring to them as 2p2h states.These states were not a part of the fit and for this ex-trapolation to be meaningful the additional 2p2h statescannot be allowed to directly mix and renormalize thepreviously fitted 0p0h states. Due to the valence spacelimitation the full 2 ~ ω excitations from the sd space can-not be considered. Moreover, our tests have shown thatexcitations from the 0 s and 0 p are nearly irrelevant forthe validity of this discussion, thus we did not includethose states into our definition of 2p2h excitations. Italso has been verified that the inevitable center-of-masscontamination in this truncation scheme is very low. Weestimate that the errors from truncation and center-of-mass contamination amount to less then 200 keV uncer-tainty in the energies, which is of the same order as therms deviation in the fit.
0, 3/2 (+) + )1162, (7/2 + ) 0, 3/2 + + + EXPFSU, 0p0h FSU, 2p2h + + + Na + + + - FSU, 1p1h - FIG. 4. The experimentally known levels of Na compared tothe lowest ones predicted using the FSU interaction for 0p0h,1p1h, and 2p2h configurations. The experimental levels agreewell with the 2p2h results while the 0p0h states start almost2.5 MeV higher in excitation energy. Only the two lowestcalculated 1p1h states are labeled because of the high leveldensity above this.
We first discuss the case of Na ( N = 20) [14]. Asshown in Figure 4, the total binding energies for the firstfour 2p2h states were found to be below that of the lowest0p0h state. The first three 2p2h states agree well withwhat is so far known experimentally, whereas the lowest0p0h state (5 / + ) appears much higher in energy and hasa different spin from the experimentally observed groundstate of Na.While the experimental information is limited, it isclear that the FSU interaction has depicted the correctpicture of Na as one with the inverted configuration.As mentioned above, only the low Z and N ≈
20 nu-clei exhibit the IoI or inverted 2p2h - 0p0h behavior.To explore the transition from IoI to “normal” behav-ior, Figure 5 compares experimentally measured energieswith our calculations for the lowest levels in a sequenceof N = 20 even- A sd nuclei. For Z = 10 and 12, not only do the lowest states have 2p2h character, but thewhole 0 + , + , + Z = 14 Si, where the0p0h 0 + state is substantially lower than the 2p2h one.The experimental second 0 + and first 2 + states are muchcloser to the 2p2h ones, while the second experimental 2 + level corresponds well with the 0p0h one. This shows theshape coexistence, also discussed in Ref. [30]. For Z = 16and 18 both the first experimental 0 + and 2 + states cor-respond with the 0p0h calculations. The second 0 + statesin both the nuclei were discussed to have 2p2h dominantconfigurations [31–33] and are in very good agreementwith the FSU predictions. The 4 + states of S and Arlie much closer to the calculated 2p2h ones. Note, thatthe FSU cross-shell interaction describes the transitionfrom inverted 0p0h-2p2h order to normal as a functionof Z despite not having been fitted to any of these states.This emergence of the IoI does not involve any f p or-bitals dropping below the sd shell, at least not for spher-ical shape. The lowering in energy of the 2p2h configu-rations does not extend so much to 1p1h ones, as shownfor Na in Figure 4. The lowest 1p1h state (3579 keV,3 / − ) lies over an MeV above the lowest 0p0h state. Soit is the promotion of a neutron pair to the f p shell whichfavors the 2p2h configuration so much. The promotionof a neutron pair to the f p orbital appears to lower itsenergy because of correlation energy in the shell model.Clearly, collective behaviors such as pairing and deforma-tion and intricate interplay between them are central forthe IoI phenomenon. Representing a mesoscopic phasetransition, the picture is highly sensitive to the matrixelements of the effective Hamiltonian and in particularto the components describing short and long range limitsof nucleon-nucleon in-medium interaction.In a geometrical picture IoI can be associated with in-creased prolate deformation due to the promotion of apair into a down-sloping Nilsson orbital whose excita-tion energy decreases rapidly with increasing deforma-tion. An indication of this difference in deformation isshown in the lower panel of Figure 6. For Ne and Mgthe calculated B(E2) transition strengths from the lowest2 + to ground states (both of which have 2p2h configu-rations) are relatively large at over 400 e fm , consistentwith relatively high deformation, and agree well with ex-periment. In contrast the B(E2) strengths for S and Ar are rather low, consistent with near spherical shape.Figure 7 portrays the differences between experimentand theory of the binding energies around the IoI whichare sensitive to pairing correlations. The calculated totalbinding energies are compared with the measured groundstate masses from the 2016 mass evaluation [34]. TheCoulomb corrections to the total binding energies areincluded following procedures in Refs. [1, 2, 15]. The N = 21 0p0h (2p2h) configurations have 1(3) nucleonsin f p , and, N = 22 2p2h actually have 4 f p nucleons sothe f p matrix elements are tested along with the cross- +
0, 0 +
0, 0 + + + + + + + + + +
0, 0 + + + +
0, 0 + + +
0, 0 + + +
0, 0 + + + + + + +
0, 0 + + + + +
0, 0 + + + +
0, 0 +
0, 0 + + + + + + + + Ne Mg Si S Ar + + FIG. 5. The lowest experimental energy levels of N = 20 sd -shell nuclei compared to those calculated using the FSU shellmodel interaction for 0p0h and 2p2h configurations. The levels of the known IoI nuclei Ne and Mg agree well with the 2p2hresults while the lowest states in the higher Z nuclei agree much better with the 0p0h results. shell ones. Looking at the N = 20 isotonic chain, theagreement is quite good with an RMS deviation of 276keV comparing the experimental binding energies withthe 2p2h results below Z = 13 and with 0p0h for higher Z . For 10 ≤ Z ≤
12 and 19 ≤ N ≤
21 the 2p2h invertedconfiguration is lower in energy and agrees better withexperiment. Outside this range the 0p0h configuration islower, which again agrees with experiment. For N = 22it appears that promoting a second neutron pair to f p isnot energetically favorable.A similar approach of calculating the 2p2h states wastaken in Ref. [15] using the WBMB interaction. Asmentioned earlier, the WBMB interaction was developedfor the mass region near Ca by fitting the 1p1h stateswithin the sdf p model space. We have compared the dif-ferences in the 0p0h and 2p2h ground states calculated byusing the WBMB and the FSU interactions for N = 20isotones in Table II. The predictions with the WBMBinteraction were taken from Ref. [15]. From Table II,we see that both the interactions predict Ne, Na and Mg having their 2p2h ground state more tightly boundthan that calculated for the 0p0h configurations, mean- ing that these nuclei are the members of the IoI. TheFSU interaction predicts F also as an IoI nucleus, whichwas suggested recently by the Ref. [35]. The differencebetween the first two 0 + states in Si is known experi-mentally as 2719 keV [30]. The FSU interaction predictsit better as 2432 keV. The experimentally observed 0 +2 states in S and Ar are at 3346 and 3376 keV respec-tively, which are presumably 2p2h in nature. The FSUinteraction predicts them at 3373 and 3140 keV respec-tively. In Cl the first 2p2h state was identified at 3708keV energy [36], whereas the FSU prediction is 3538 keV.The better predictability of the FSU interaction comesfrom a more extensive fit for a wide range of cross shelldata as well as the use of a better Hamiltonian for the f p shell, we believe.Since the IoI involves excitations into the f p shell, thequestion arises how the inversion of the 0 f / and 1 p / single particle energies at low Z , discussed above, affectsour understanding of the IoI. The answer, within the con-text of the FSU interaction is shown in Figure 8. Thisfigure shows some of the f p shell occupancies calculatedfor the lowest 2p2h states in Figure 5. Occupancy here E ( + ) k e V EXPFSU Interaction30 32 34 36 38 A B ( E : + → + ) e f m EXPFSU Interaction
FIG. 6. Experimental E(2 +1 ) and B(E2: 0 +1 → +1 ) valuesfor the N = 20 isotones are compared to those calculated byusing the FSU interaction. The B(E2: 0 +1 → +1 ) value of Sihas not been calculated because of the different configurationsassociated with the 0 +1 and 2 +1 states.
17 18 19 20 21 22 23N9101112131415 Z
131 1451 2662 3112 1648-121 798 2384 3319 1851-12 2938 2232 28143396-181 87 97 71220 75 -149 -56 17 18 19 20 21 22 23N 9101112131415 Z
850 208 -161 955 2165121 -63 1030965 23051976 677 566 1718 31483656 2896 1019 15633560 2283 24350 - 750 keV (Over-bound)0 - 750 keV (Under-bound)750 - 1250 keV (Under-bound)> 1250 keV (Under-bound) 5153
FIG. 7. The number displayed inside a box correspondingto an isotope is the difference in binding energy between ex-periment and shell model predictions using the FSU inter-action with 0 or 2 particle-hole configurations. We call thestates over-bound where the calculated states are more tightlybound than that of the experimental ones and under-boundwhen it is otherwise. is defined as the average number of nucleons in a givenorbital. There is almost no proton f p occupancy calcu-lated for these nuclei and there is a relatively constant ν p / occupancy of about 0.1 neutron. For Z = 10 Ne, which is the most strongly inverted, the ν p / oc-cupancy is about twice that of ν f / . With increasing Z , the ratio of ν p / to ν f / decreases steadily fromabout 2 to about 0.2 across this region. Of course, theenergies of the 2p2h configurations rise above that of the0p0h ones around Z = 14.We note that considering that the degeneracy of the f / is twice that of p / , at the level crossing or in the TABLE II. The ground state energies with 2p2h configura-tions are calculated with respect to those with the 0p0h con-figurations using the WBMB [15] and the FSU interactions.The symbols W, F and T in the WBMB calculations standfor weak coupling, full WBMB space and the truncated spacerespectively. Nucleus WBMB a FSU O 3038: W -7552956: F F 1286: W -22011338: F Ne -698: W -2823-788: F Na -502: W -2452-764: T Mg -926: W -1666-966: T Al 854: W 922 Si 1816: W 24321554: T P 2698: W 3264 S 3146: W 33733009: T Cl 3195: W 35383091: T Ar 2701: F 3140 a Ref. [15] limit of strong pairing the ratio of occupancies of ν p / to ν f / should be about 0.5. This indeed happens ataround Z = 14; however significant deviation from 0.5suggests that pairing, or at least pair transfer between f / and p / is weak. Pair transfer and pair vibration,collective pairing condensation, interplay of paring anddeformation in the IoI region, as well as the connectionof these collective effects with the underlying matrix el-ements of the FSU Hamiltonian, all require more studyand remain a challenge for the future. The occupancytrend is perhaps illustrated more clearly in Figure 3(b)which shows the ν p / and ν f / occupancies of thelowest 2p2h states in the N = 20 nuclei as a function ofproton number Z . Note, that for Si, the 2p2h 0 + statelies 2432 keV above the 0p0h ground state but the 2p2h2 + level lies close in energy with the lowest experimen-tal 2 + state. Together these calculations imply that the ν p / orbital plays a larger role in the IoI phenomenonthan does the ν f / one. + +
10 12 14 16 18 Z + ν f Occupancy ν p Ocuupancy
FIG. 8. 2p2h occupancies of the ν f / and ν p / orbitalsfor the first 0 + , 2 + and 4 + calculated states using the FSUinteraction for nuclei with N = 20 and Z between 10 and 18. V. FULLY ALIGNED STATES
In describing the states used in the fit of the FSU in-teraction, we included only 0p0h(1p1h) configurations fornatural(unnatural) parity sectors. In particular, no 2p2hconfigurations were used to adjust the interaction param-eters. After the fitting, two early tests were performedto explore the predictive properties of the FSU interac-tion for 2p2h configurations. One was the calculation ofthe lowest 2p2h 7 + states in Cl and Cl [17]. Theseagreed within 200 keV with the experimental states. Theother test was performed on Ar [37], since experimentalstates up to 8 + and (10 + ) are known. Calculations usingthe USD family of interactions agree within 200 keV withthe excitation energy of the lowest 2 + state of Ar, butover-predict the lowest experimental 4 + level by over 3MeV. With only two holes in the sd shell, the maximumspin from coupling two 0 d / protons is 2 ~ . The veryhigh 4 + energy represents the cost of promoting a 0 d / proton to 0 d / , but nature finds another less energeticway of achieving 4 + . This must be by promoting an sd nucleon pair to the f p shell. A 2p2h calculation withthe FSU interaction predicts the lowest 4 + level only 300keV above the experimental one, and it predicts the 6+state 200 keV below experiment, while the predicted 8 + state is 100 keV above experiment as shown in Ref. [37].With this success we have searched for other stateswith confirmed 2p2h structure to compare with theory.One such group of excited states across the sd shell areoften called the “fully aligned” states. One subgroup offully-aligned states is the lowest J π = 7 + states. Thesestates have been suggested to have both odd nucleons in the highest spin orbital around - 0 f / - and withtheir spins fully aligned, which, from the Pauli princi-ple, is only possible for non-identical nucleons. For thesecalculations it is critical that the FSU interaction treatsprotons and neutrons on an equivalent basis. These fully-aligned πf / ⊗ νf / are yrast and strongly populated inhigh-spin γ -decay sequences. Stronger evidence of theirunique nature comes from ( α, d ) reactions [38–44] wherethey are the most strongly populated states with an or-bital angular momentum transfer of ℓ = 6. In mostcases such states involve two nucleons beyond those inthe dominant ground state configuration outside the sd shell. The energies of these 7 + states (including those in Cl and Cl mentioned above) are graphed in Figure9 along with calculated results using the FSU interac-tion. The agreement is excellent both in value and inthe trend which extends from 10 MeV for the lightestnuclei down to 2 MeV for the heaviest and from 2p2h to1p1h excitations relative to the ground state. The cal-culations also indirectly confirm the spin alignment withapproximately equal proton and neutron occupancies inthe 0 f / orbitals, even though most 2p2h states in theseneutron-rich nuclei as discussed in the IoI section involvepredominantly two neutron configurations.Fully aligned states are also known for some odd- A nuclei where an sd nucleon is also aligned in spin withthe aligned 0 f / nucleons. Five such cases in Figure 9are known experimentally as the strongest states popu-lated in ( α, d ) reactions. They have an unpaired nucleonin the 0 d / orbital which contributes an extra spin of3 / ~ . Again the 2p2h and 1p1h calculations with theFSU interaction agree well. In lighter odd-A nuclei thealigned sd nucleon could be in the 1 s / or 0 d / orbitals,leading to total spins of 15/2 or 19/2 and higher excita-tion energies. Their calculated energies are also shownin Figure 9, but none have been seen in ( α, d ) reactions.A (11 / + , / + ) state which decays only to the low-est 13 / + state and is very likely the 15 / + fully alignedstate has been reported [24] in P, as shown in the figurewould agree well with the predictions.The last category of aligned states in the sd shell con-sists those in even-even nuclei. Their excitations involvethe breaking of a proton and a neutron pair and pro-motion of one of each nucleon to the 0 f / orbital. Forexample, all 4 unpaired nucleons coupled to maximumspin of 10 + if both unpaired sd nucleons are in the 0 d / orbital. No ( α, d ) reactions to the fully aligned state ineven-even nuclei are known because of the absence of sta-ble odd- Z odd- N targets in the sd shell. However, thelowest experimentally known 10 + state in Ar observedby other reactions does compare well with a 2p2h calcula-tion using the FSU interaction, as shown in Figure 9. Inthe case of Ca the analogous state would involve break-ing a πd / pair, promoting one proton to 0 f / , breakingthe νf / pair and coupling them to maximum spin fora total of 11 − . This state has been seen in γ decay fol-lowing fusion-evaporation and its energy agrees well withthe FSU calculation. We hope that future experiments0
24 26 28 30 32 34 36 38 40 42 44 A E n er gy ( M e V ) seen in ( α , d) reaction and γ decayfrom 2p-2h FSU Shell Modelseen in γ decay onlyfrom 1p-1h FSU Shell Model + + + + 42 K K K Cl Cl Cl P P P Al Al Ca Ca Ar Ar Cl S P Al Si Mg Ar10 +31 Si Ca11 - FIG. 9. Comparisons of the energies of fully aligned states in sd -shell nuclei with those predicted employing the FSUinteraction. Many of the experimental points are confirmed by both selective population in ( α, d ) reactions and in high-spin γ decay sequences and are displayed with solid black circles, while dotted black circles are used to represent states observed byonly one of the two signatures. The structure of many of these aligned states involve the promotion of two (extra) nucleons tothe 0 f / orbital and are shown with solid red diamonds. Those with at least one nucleon in the 0 f / orbital may require onlyone more promotion (1p1h excitation) and are shown with dotted red diamond symbols. in the FRIB age will be able to test these predictions.This study of aligned states targets cross shell matrix el-ements of high angular momentum channels that describelong-range effective in-medium nucleon nucleon interac-tions and play key role in determining nuclear shape anddeformation. VI. SUMMARY
In this work we present an effective nuclear interactionHamiltonian for shell model calculations, named FSU in-teraction. The interaction targets a broad range of nucleifrom p to f p shells with a particular emphasis on exoticnuclei with extreme proton to neutron ratio and on statesthat involve cross shell excitations. The interaction wasfitted using binding energies and 1 ~ ω states that probecross-shell matrix elements in nuclei from C through Ti and V. Additional details of the fit can be foundin Refs. [17, 23]. This report provides a comprehensivestudy of nuclei in the region of the Island of Inversion,namely those nuclei between sd and f p shells whose low-lying structure is dominated by cross shell excitations. We use the newly obtained FSU interaction to inferinformation about the mean field and evolution of theeffective single particle energies (ESPE). The ESPEs ofthe 0 f / and 1 p / show the expected normal ordering,where 0 f / is below 1 p / for Z >
12 and a consistenttrend of a decreasing separation with decreasing Z un-til the energy order reverses around Z = 10 to 12. Itis remarkable that the inversion happens near zero en-ergy associated with the decay threshold. The interac-tion with the continuum is not explicitly included butmaybe captured as a part of the fit. While there havebeen many indications of inverted shell ordering in thepast, these results present a more systematic picture froma model very firmly rooted in data. Perhaps somewhatsurprisingly, over the range explored here, the inversionappears to depend more on the proton number than onthe neutron excess. The lowest 3 / + , 7 / − , and 3 / − ex-perimental states are surveyed for a complementary viewof shell evolution. These energies are compared with pre-dictions of the FSU interaction in excitation energies andspectroscopic factors. They present a similar picture ofthe 0 f / - 1 p / shell evolution as a function of protonnumber.1The success of the FSU interaction in reproducing thenegative parity states of the sd -shell nuclei with the 1 ~ ω configuration suggests that improved, over those in Ref.[45], calculations of the rp process rates can be performedin the future.In this report, the FSU interaction was taken a stepforward and applied to configurations involving promo-tion of two nucleons from sd to f p (2p2h) in the regionof IoI. In this region the nuclei are more tightly boundthan predicted within the pure sd model space (0p0h).The 2p2h configurations have lower binding energies andagree well with the measured ground state masses in therange 10 ≤ Z ≤
12 and 19 ≤ N ≤
21, while the 0p0h con-figurations are lower in energy and agree better with themeasured masses elsewhere. The lowest 2 + states agreewell with the 2p2h calculations in the region Z = 14 andwith 0p0h for Z = 16 −
18. The results of the FSU inter-action which was not fitted to these states reproduce wellboth the IoI and the transition to normal behavior. Siwith Z = 14 emerges as transitional with a 0p0h groundstate and a 2p2h lowest 2 + state. It would be interestingto locate the experimentally 4 +1 state which is predictedas 2p2h at 5523 keV. Another implication of the FSUshell model calculations is that ν p / pairs dominateover ν f / ones in the IoI, but ν f / pairs dominatethe lowest 2p2h states beyond the IoI. This is an indi-cation of a relative weakness of pairing that would actto equilibrate occupancy. Interestingly the IoI coincidesrelatively well with the region where the ν p / orbitalfalls below the ν f / one.Another success of the FSU interaction has been thecalculation of the energies and occupancies of the fullyaligned states, first identified in the early 1960’s in ( α, d )reactions and frequently observed in high-spin γ -decay cascades (most involve 2p2h excitations relative to theground state). Their energies are reproduced very wellacross the mass range, and their occupancies prove theexcitation of both protons and neutrons, even thoughpure neutron excitations are more common in otherstates. This is an important result that establishes val-ues for the specific cross shell high angular momentummatrix elements that are responsible for long range ef-fective nucleon-nucleon interaction and are particularlychallenging to obtain from fundamental principles.This work brings forward an interesting comparisonbetween traditional shell model interactions with thosearising from first principles methods. While the formerare obtained from simply fitting SPEs and TBMEs toexperimental data, the latter require renormalizations,many-body forces and explicit inclusion of the reactioncontinuum to achieve agreement with experiment. Thisdichotomy, presents a modern challenge to nuclear theoryand deserves a full investigation.The capability of the FSU interaction to explain theexotic phenomena of the nuclei carries the prospect thatthe interaction will be successful for more exotic nucleior states. It is hoped that the interaction will provevaluable in the coming FRIB age. ACKNOWLEDGMENTS
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