aa r X i v : . [ nu c l - t h ] D ec Shape evolution of Zr nuclei and roles of tensor force
S. Miyahara and H. Nakada ∗ Department of Physics,Graduate School of Science, Chiba University,Yayoi-cho 1-33, Inage, Chiba 263-8522, Japan (Dated: December 5, 2018)Shape evolution of Zr nuclei are investigated by the axial Hartree-Fock (HF) calculations usingthe semi-realistic interaction M3Y-P6, with focusing on roles of the tensor force. Deformation at N ≈
40 is reproduced, which has not been easy to describe within the self-consistent mean-fieldcalculations. The spherical shape is obtained in 46 ≤ N ≤
56, and the prolate deformation ispredicted in 58 ≤ N ≤
72, while the shape switches to oblate at N = 74. The sphericity returns at N = 80 and 82. The deformation in 60 < ∼ N < ∼
70 resolves the discrepancy in the previous magic-number prediction based on the spherical mean-field calculations [Prog. Theor. Exp. Phys. ,033D02]. It is found that the deformation at N ≈
40 takes place owing to the tensor force with agood balance. The tensor-force effects significantly depend on the configurations, and are pointedout to be conspicuous when the unique-parity orbit ( e.g. n h / ) is present near the Fermi energy,delaying deformation. These effects are crucial for the magicity at N = 56 and for the predictedshape change at N = 74 and 80. I. INTRODUCTION
Shell structure, which is an obvious quantum effect andis manifested by the magic numbers, is one of the fun-damental concepts in the nuclear structure physics [1].While the spin-orbit ( ℓs ) splitting of the single-particle(s.p.) orbits is significant in medium- to heavy-massnuclei, forming the jj -closed magic numbers ( Z, N =28 , ,
82 and N = 126), it is less important in light nu-clei, where the ℓs -closed magic numbers ( Z, N = 2 , , ℓs -closed magicity is partly maintainedbut not so stiffly, the structure of the Zr ( i.e. Z = 40)isotopes strongly depends on the neutron number N , pro-viding us with a good testing ground of nuclear structuretheories. While the doubly-magic nature of Zr is wellknown, it has been established experimentally that Zrnuclei become deformed both in neutron-deficient ( e.g. Zr) [2] and neutron-rich ( e.g.
Zr) regions [3, 4].This N -dependence is contrasted to the Sn and Pb nu-clei, in which the Z = 50 and 82 magic numbers are rigidin a wide range of N . Moreover, the sudden change ofthe shape from Zr to
Zr was interpreted as a quan-tum phase transition [5]. Shape with the tetrahedralsymmetry was theoretically argued for Zr, Zr and , , Zr [6–8], although no experimental evidence hasbeen reported so far. On the contrary, the measured firstexcitation energy at Zr is significantly higher than insurrounding nuclei [4, 9], implying submagic nature of N = 56 as well as the magicity of Z = 40 [10]. With ex-periments using radioactive beams and progress on ourunderstanding of the nuclear shell structure, it is of in-terest to reinvestigate ground-state (g.s.) properties ofthe Zr nuclei systematically.One of the recent topics in nuclear structure physicsis roles of the tensor force in the Z - or N -dependence ∗ E-mail: [email protected] of the shell structure ( i.e. the shell evolution) [11, 12].Although the tensor force is contained in the nucleonic in-teraction, most of the self-consistent mean-field (MF) cal-culations have been performed without the tensor force.We now face a new problem how the tensor force is in-corporated in the MF framework, and how it alters pic-tures obtained in the conventional calculations. As anattempt to fix this problem, one of the authors (H.N.)developed M3Y-type semi-realistic interactions [13, 14].In Ref. [10], a map of magic numbers was drawn based onthe self-consistent MF calculations assuming the spher-ical symmetry, by adopting the pairing as a measure ofcorrelations. It was shown that the semi-realistic inter-action M3Y-P6 [14] gives a prediction of magic numberscompatible with almost all available data, except N = 20in Mg and Z = 40 in neutron-rich Zr nuclei. The M3Y-P6 interaction contains the realistic tensor force origi-nating from the G -matrix [15], which is profitable in re-producing the shell evolution in some regions [10, 16].By a recent study, it has been suggested that the con-tradiction in Mg mentioned above could be resolvedif the quadrupole deformation is taken into account ex-plicitly [17]. It is noted that the deformation in Mgwas obtained via correlations beyond MF, in the calcu-lations using the Gogny-D1S interaction [18], althoughthe tensor force has certain effects on the shape evolu-tion around N = 20. It is desired to apply deformedMF calculations also to the Zr nuclei, and to reexaminethe Z = 40 magicity with the semi-realistic interactionincluding the realistic tensor force. II. PREDICTION OF DEFORMATION ATGROUND STATE
We have implemented self-consistent HF calculationsassuming the axial symmetry for the even- N Zr iso-topes from Zr to
Zr, and investigate shape evolution -500 0 500 1000 1500 40 45 50 55 60 65 70 75 80 q [f m ] N D1MM3Y-P6
FIG. 1. (Color online) Values of q that give the lowest energyfor the individual nucleus in the axial HF calculations withM3Y-P6 (red line) and D1M (green line). for increasing N . The numerical method is detailed inRef. [19]. Since we apply basis functions having good or-bital angular momentum ℓ , the s.p. space is truncated viathe cut-off value ℓ max . To describe normally-deformeds.p. levels with good precision, ℓ max should be greaterby four than the ℓ value of the corresponding level at thespherical limit [19]. We therefore adopt ℓ max = 9 becauseof the presence of the 0 h / orbit in N ≤
82. The semi-realistic interaction M3Y-P6 [14] is mainly employed. Forcomparison, we have also implemented the axial HF cal-culations using the Gogny-D1M interaction [20], whichis one of the most successful interaction so far but doesnot include tensor force. Whereas the D1M interactionwas developed for calculations in which the pairing andthe additional quadrupole collective degrees of freedom(d.o.f.) are taken into account, it is of interest to com-pare the results within the HF, to examine whether ornot the tensor-force effects can be imitated by the otherchannels at this level.The minimum giving the lowest energy yields a goodcandidate of the g.s. for each nucleus. However, we ig-nore pair correlations, rotational correlations, and tri-axial or odd-parity deformation in the present work. Ifother minima have close energies to the lowest one, cau-tion is needed because the ignored correlations may mixor invert the energies. Keeping this point in mind, weshall look at the predicted g.s. deformation of the Zrisotopes.In Fig. 1 the q values that give the lowest energyat individual N are depicted, which are obtained fromthe axial HF calculations with M3Y-P6 and D1M. Notethat q ≈ corresponds to the deformationparameter β ≈ . A ≈ q =3 A (1 . A / ) β/ √ π .The shape of Zr is indicated to be spherical around N = 50 in the present calculation, as expected. Itis found that the M3Y-P6 interaction predicts prolate shape at N = 40 and 42, which seems consistent with theexperimental data. On the contrary, D1M gives spheri-cal shape at N = 40 at the HF level. At N = 58, theabsolute minimum is prolate with M3Y-P6, while oblatewith D1M. In 60 ≤ N ≤
82, M3Y-P6 and D1M providesimilar deformation except at N = 66. In the M3Y-P6results, well-deformed prolate shape gives the lowest en-ergy for individual nucleus in 58 ≤ N ≤
72. Recall thatmeasured E x (2 +1 )’s are low and close to one another in60 ≤ N ≤
70 [21, 22], suggesting that these nuclei arewell-deformed to a similar degree. The lowest minimumis switched to the oblate side in 74 ≤ N ≤
78, and re-turns to the spherical shape at N = 80. In 64 ≤ N ≤ N = 66 in the D1M result, the energy dif-ference between the prolate and oblate minima is small( ≈ . | q | <
200 fm are identi-fied to have the spherical shape. Though not listed in Ta-ble I to avoid overlaps, Hartree-Fock-Bogolyubov (HFB)results with D1S for − Zr were presented in Ref. [23],where the triaxial deformation was taken into account,and relativistic Hartree-Bogolyubov (RHB) results withDD-PC1 for − Zr in Ref. [8], where octupole andtetrahedral deformations were also considered. The HFBresults with SLy4 were depicted in Ref. [24]. In Ref. [23]prolate shape was predicted in , , Zr. No triaxialground state comes out with D1S, contrary to the RHBresults. All calculations predict deformation in
N > ∼ TABLE I. Brief summary of shapes of Zr nuclei predictedby self-consistent MF calculations; spherical (“sph”), prolate(“pro”), oblate (“obl”) or triaxial (“tri”). For the mean-fieldtype, “(ax.)” indicates that the axial symmetry is assumed.In the row of “EDF”, effective interaction or energy-densityfunctional is specified. For the row of “Ref.”, “PW” standsfor the present work.MF HF (ax.) HF (ax.) HFB HF HFB (ax.) RHB RHBEDF M3Y-P6 D1M D1S SkM ∗ SLy4 NL3 ∗ DD-PC1Ref. PW PW [25] [26] [27] [28] [28] N = 40 pro sph sph pro – – – N = 42 pro sph sph tri – – – N = 44 obl sph sph tri – – – N = 46 sph sph sph pro – – – N = 48 sph sph sph sph – sph sph N = 50 sph sph sph sph – sph sph N = 52 sph sph sph sph – sph sph N = 54 sph sph sph sph – tri tri N = 56 sph sph sph sph – tri tri N = 58 pro obl obl sph – obl pro N = 60 pro pro obl pro – obl pro N = 62 pro pro obl pro pro obl tri N = 64 pro pro obl pro pro obl tri N = 66 pro obl obl pro pro obl obl N = 68 pro pro obl pro pro sph obl N = 70 pro pro obl pro pro sph obl N = 72 pro pro sph pro pro – – N = 74 obl obl sph pro sph – – N = 76 obl obl sph sph sph – – N = 78 obl obl sph sph sph – – N = 80 sph sph sph sph sph – – N = 82 sph sph sph sph sph – – metry restoration [30].In the subsequent section, we shall investigate roles ofthe tensor force in the deformation for selected nuclei,via detailed analyses in terms of the energy curves andthe s.p. levels. III. ENERGY CURVES AND TENSOR-FORCEEFFECTS
In this section, energy curves are depicted for severalnuclei. The HF calculations yield local energy minimadepending on the intrinsic mass quadrupole moment q .The q values at the minima are essentially determined bythe occupied s.p. levels ( i.e. HF configuration), as exem-plified in Ref. [17] and further confirmed in this study. Todraw E ( q ), i.e. energy curve as a function of q , the con-strained HF (CHF) calculations have been carried out.The procedures for the CHF calculations are describedin Ref. [17]. Contribution of the tensor force is evalu- ated by E (TN) ( q ) = h Φ( q ) | ˆ v (TN) | Φ( q ) i , where ˆ v (TN) isthe tensor force and | Φ( q ) i represents the CHF state.We shall compare energies E ( q ), E ( q ) − E (TN) ( q ),both of which are obtained from the M3Y-P6 interac-tion, and E ( q ) obtained from the Gogny-D1M inter-action. It is again emphasized that ˆ v (TN) in M3Y-P6is realistic, which has been derived via the G -matrixwithout adjusting to experimental data, and that thistensor force reproduces variation of relative s.p. ener-gies of p d / and p s / from Ca to Ca remarkablywell [16]. The s.p. energy ε ( ν ) at the minima will beshown as well, where ν denotes the s.p. level in the HF.The s.p. energies are useful for analyzing configurationsthat yield the minima. To visualize contribution of thetensor force to ε ( ν ), ε ( ν ) − ε (TN) ( ν ) is also displayed,where ε (TN) ( ν ) = 2 P ν ′ ( > n ν ′ h νν ′ | v (TN) | νν ′ i with theoccupation probability n ν .We here recall several properties of the tensor force atthe MF level, which have been established in Refs. [11,17, 31].i) The tensor force primarily provides proton-neutroncorrelations.ii) The tensor force acts repulsively.iii) Tensor-force effects are perturbative at the HFlevel, but configuration-dependent. For a fixed con-figuration, E (TN) ( q ) is insensitive to q .iv) The tensor force tends to lower the spherical staterelative to the deformed ones at the ℓs -closed magicnumbers, while the opposite holds at the jj -closedmagic numbers.The point iii) allows us to analyze tensor-force effects interms of the spherical orbits. The tensor force acts repul-sively (attractively) on neutron j = ℓ + 1 / j = ℓ − / j = ℓ + 1 / j = ℓ + 1 / ℓs partner, both for protons and neutrons [17]. The pointiv) takes place because the tensor force works when thespin d.o.f. are active. Its contribution depends on howwell the spin d.o.f. are saturated [17, 31]. As Z = 40forms an ℓs -closed shell at the spherical limit, it is ex-pected that the tensor force tends to favor sphericity inthe Zr nuclei. Owing to the point iii), the HF frameworkis advantageous in investigating tensor-force effects [17],and therefore indispensable in reconstructing the MFscheme including the tensor force. Mixing of the HF con-figurations, which occurs via correlations beyond HF ( e.g. the pairing), leads to obscurity in viewing the tensor-force effects. A. Zr The energy curve E ( q ) for Zr is shown in Fig. 2.While several minima are observed on the prolate and -784-782-780-778-776-774-772-770-768-766-764-762-760-758-756 -1000 -500 0 500 1000 E [ M e V ] q [fm ] Zr D1M M3Y-P6 (w/o v (TN) ) FIG. 2. (Color online) CHF results of E ( q ) (red squares) and E ( q ) − E (TN) ( q ) (blue open diamonds) for Zr, which areobtained with M3Y-P6. E ( q ) with D1M (green circles) arealso plotted. Lines are drawn to guide eyes. For reference, en-ergy obtained by the spherical HFB calculation [10] is shownby the red cross, though invisible here since it is merged withthe HF energy at q = 0. the oblate sides, the HF energy becomes lowest at q = 0.It is confirmed from the spherical HFB energy, which isnot distinguishable from the HF energy at q = 0 inFig. 2, that the pair correlation is so weak at q = 0.This spherical minimum is well developed both in theM3Y-P6 and the D1M results, compared to E − E (TN) .Comparing the full M3Y-P6 energy E with E − E (TN) ,we confirm the points raised above. The tensor force actsrepulsively, and E (TN) does not change much along thecurve connected to each local minimum. Moreover, thetensor force helps the spherical configuration to be thedistinct absolute minimum, as E (TN) comes larger as thedeformation develops. On account of this tensor-force ef-fect, it can be considered that, in the D1M interaction,the tensor-force effects are partly incorporated into theother channels in an effective manner, so that the prop-erty of Zr should be reasonably reproduced.Figure 3 depicts the s.p. levels around the Fermi en-ergy at individual local minima, which are obtained bythe HF calculation with M3Y-P6. The occupied levelsare presented by the filled circles. To examine the tensor-force effects, [ ε ( ν ) − ε (TN) ( ν )]’s are shown by the dashedlines. Since the N = 50 shell gap is large, neutrons arehardly excited at relatively small | q | . What triggers de-formation is excitation of protons from the 1 p / orbitto 0 g / . The tensor force enlarges the shell gap betweenthese orbits, which lowers the energy of the spherical min-imum relative to the deformed ones. In addition, it is seenin Fig. 3 that ε changes more rapidly than ε − ε (TN) , asthe configuration ( i.e. q in the figure) varies. This effectis linked to the degree of the spin saturation. Owing tothe ℓs closure for protons, contribution of the tensor forcecomes small at the spherical limit. As the deformation -12-10-8-6-4-2-1000 -500 0 500 1000 proton p g ε [ M e V ] q [fm ] Zr (1/2) + (3/2) + (5/2) + (7/2) + (9/2) + (1/2) - (3/2) - (5/2) - (9/2) - (11/2) - -14-12-10-8-6-1000 -500 0 500 1000 neutron g d q [fm ] FIG. 3. (Color online) Proton and neutron s.p. energies ε ( ν )in Zr obtained by the axial HF calculations with M3Y-P6,at the minima shown in Fig. 2. Occupied (unoccupied) levelsare represented by the filled (open) circles connected by thesolid lines. Quantum number of each level Ω π is distinguishedby colors, and is indicated in the middle. Dashed lines show ε ( ν ) − ε (TN) ( ν ). Labels for several spherical orbits are givenfor reference. develops, the spin saturation is lost in the proton state,giving rise to the larger E (TN) in Fig. 2. B. Zr We next turn to Zr. See Fig. 4 for the energy curve.Because of Z = N = 40, Zr would be spherical if theshell gap between pf -shell orbits and 0 g / were large. Inpractice, the lowest energy is obtained at q = 0 by theHF calculation with D1M. The same holds in the HFBresult with D1S [25]. In contrast, in the M3Y-P6 resulta prolate and an oblate minima lie lower than the spher-ical minimum. Whereas the pairing lowers the energy ofthe spherical state, the deformed minima are even lowerthan the spherical HFB energy (the red cross in Fig. 4).The lowest energy is given by the prolate minimum with q ≈
800 fm , which corresponds to β ≈ .
5. The dif-ference in the deformation between M3Y-P6 and D1Mis traced back to the N -dependence of the shell gap asdiscussed in Ref. [10]. Even though the tensor-force ef-fects could be partly incorporated in D1M in an effectivemanner, it could be difficult to mimic all aspects of them,particularly in the case that the tensor force directly af-fects the Z - or N -dependence of the shell gap. The D1Minteraction reproduces properties of Zr without the ten-sor force, and this seems to make it difficult to describethe deformation of Zr at the MF level.In E − E (TN) , the prolate minimum has much lowerenergy than the other minima. Because of the ℓs clo-sure both for protons and neutrons, E (TN) is negligiblysmall at the spherical minimum. The repulsive effectof the tensor force is the stronger for the larger deforma-tion, and diminishes energy difference between the spher-ical and the prolate minima. Still, the prolate minimum -670-668-666-664-662-660-658-656-654-652-650-648-646-644-642-640 -1000 -500 0 500 1000 E [ M e V ] q [fm ] Zr D1M M3Y-P6 (w/o v (TN) ) FIG. 4. (Color online) CHF results for Zr. See Fig. 2 forconventions. stays lower than the spherical minimum in the full M3Y-P6 result. Though to less degree, the same mechanismworks for the oblate minimum, also staying lower thanthe spherical minimum. The prolate and the oblate statesare energetically competing in the full M3Y-P6 result.The close energies of the prolate and the oblate statessuggest shape coexistence at low energy.It is mentioned that, as illustrated by the D1M re-sults, the deformation at Zr has not been easy to bereproduced with the interactions that do not contain thetensor force, until beyond-MF effects are taken accountof [32]. Though there exist exceptions ( e.g. the HF resultwith SkM ∗ in Table I), Zr is hardly deformed also withthe Skyrme interactions at the MF level. If we use aninteraction by which the Z = 40 shell gap is kept largeat Zr, this gap becomes even larger at N = 40 if wedo not have the tensor force. The realistic tensor forcereverses this trend, as illustrated in Fig. 9 of Ref. [10].On the other hand, there is another effect of the tensorforce giving the opposite tendency; namely, favoring thespherical shape. The present results imply that the semi-realistic interaction yields the tensor-force effects with anappropriate balance.The s.p. levels are shown in Fig. 5. In this nucleus,deformation is driven by excitation of either protons orneutrons from the pf -shell orbitals to 0 g / . At the low-est minimum with q ≈
800 fm , four protons and fourneutrons are excited to the levels coming down from0 g / , indicating that it is basically an 8 p -8 h state interms of the spherical orbitals. The oblate minimum with q ≈ −
300 fm , which is the second lowest in energy, hasa 4 p -4 h configuration. C. Zr In Ref. [10], N = 56 has been predicted to be sub-magic at Zr, based on the quenched pair correlation -10-8-6-4-2 0-1000 -500 0 500 1000 proton ε [ M e V ] q [fm ] Zr (1/2) + (3/2) + (5/2) + (7/2) + (9/2) + (1/2) - (3/2) - (5/2) - (7/2) - -18-16-14-12-10-8-1000 -500 0 500 1000 neutron q [fm ] FIG. 5. (Color online) Proton and neutron s.p. energies in Zr obtained from the HF calculations with M3Y-P6, at theminima shown in Fig. 4. See Fig. 3 for conventions. -824-822-820-818-816-814-812-810-808-806-804-802-800-798-796 -1000 -500 0 500 1000 E [ M e V ] q [fm ] Zr D1M M3Y-P6 (w/o v (TN) ) FIG. 6. (Color online) CHF results for Zr. See Fig. 2 forconventions. in the spherical HFB result. This submagic nature isin accordance with the high E x (2 +1 ) in measurement [4],and has been accounted for by the enhanced shell gapowing to the tensor force. Figure 6 supports this predic-tion. Even after the deformation is taken into account,M3Y-P6 gives the lowest energy at q = 0, whose energyis distinctly lower than the deformed local minima. Asseen in Fig. 6, the energy difference between HF and HFBat q = 0 is tiny though visible. Although the sphericalconfiguration is the lowest also in the D1M result, energydifference between the spherical and deformed minima issmall and correlations beyond HF could invert them. Inthe HFB result with D1S [25], a shallow minimum around q = 0 was reported.The s.p. levels in Fig. 7 further establish the roleof the tensor force in the magicity of Zr discussed inRef. [10]. For Z = 40, the shell gap between p p / and p g / is enhanced by the tensor force combinedwith occupation of n g / and n d / . For N = 56,the gap between n d / and n g / remains relatively -14-12-10-8-6-4-1000 -500 0 500 1000 proton p g ε [ M e V ] q [fm ] Zr (1/2) + (3/2) + (5/2) + (7/2) + (9/2) + (1/2) - (3/2) - (5/2) - (7/2) - (9/2) - (11/2) - -10-8-6-4-2-1000 -500 0 500 1000 neutron d s q [fm ] FIG. 7. (Color online) Proton and neutron s.p. energies in Zr obtained from the HF calculations with M3Y-P6, at theminima shown in Fig. 6. See Fig. 3 for conventions. large. Thereby n s / becomes the lowest unoccupiedlevel. Similar crossing of the spherical orbitals has beenpointed out for the Ni isotopes [33]. As presented in Figs.6 and 9 of Ref. [10], D1M gives smaller shell gaps thanM3Y-P6 both for protons and neutrons, by which the en-ergies of the deformed minima come close to that of thespherical minimum.We here discuss effects of the tensor force under thepresence of a unique-parity orbit. Because the ℓs part-ner of the unique-parity orbit has much higher energy,the system becomes away from the spin saturation asthe unique-parity orbit is occupied. This makes tensor-force effects stronger, as is observed for the q ≈ minimum in Fig. 6. In the lower part of the majorshell ( e.g. N = 50 − e.g. n h / ) is not occupied at the spherical limit, while be-comes occupied to a greater extent as the deformationgrows. Then the tensor force, which is repulsive, actsmore strongly. Therefore, if the unique-parity orbit is lo-cated above but not distant from the Fermi energy, thetensor force tends to lower energy of the spherical con-figurations relative to those of the deformed ones. As aresult, the q ≈ state lies significantly higherthan the spherical minimum and the doubly-magic na-ture is enhanced in Zr, as recognized by comparing E and E − E (TN) in Fig. 6. In the upper part of the majorshell ( i.e. for larger N ), the unique-parity orbit is alreadyoccupied at the spherical limit to a certain extent. There-fore this mechanism is expected to work weaker than inthe lower part of the major shell. D. Zr Experiments have indicated well-deformed groundstates in N ≥
60. It has been recently argued that thesudden shape change from Zr to
Zr may be inter-preted as a quantum phase transition. It deserves havinga close look at the shape evolution in this particular re- -852-850-848-846-844-842-840-838-836-834-832-830-828-826-824-822 -1000 -500 0 500 1000 E [ M e V ] q [fm ] Zr D1M M3Y-P6 (w/o v (TN) ) FIG. 8. (Color online) CHF results for
Zr. See Fig. 2 forconventions. -16-14-12-10-8-1000 -500 0 500 1000 proton p g ε [ M e V ] q [fm ] Zr (1/2) + (3/2) + (5/2) + (7/2) + (9/2) + (1/2) - (3/2) - (5/2) - (7/2) - (9/2) - (11/2) - -10-8-6-4-2-1000 -500 0 500 1000 neutron d s h q [fm ] FIG. 9. (Color online) Proton and neutron s.p. energies in
Zr obtained from the HF calculations with M3Y-P6, at theminima shown in Fig. 8. The levels at q = 0 are the resultsof the spherical HF calculation. See Fig. 3 for conventions. gion, and we take Zr as an example.Figure 8 shows the energy curve for
Zr. Both in theM3Y-P6 and D1M results, the lowest energy is obtainedby the well-deformed prolate state with q ≈ .Whereas this lowest energy is close to the spherical HFBenergy [10] with the difference less than 0 . q = 0, we observe two other local minima.Several minima are found on the oblate side as well. Fig-ure 9 tells us detailed structure at these minima. Whilewe do not have a minimum at q = 0 in the axial HFcalculations, the s.p. levels obtained from the sphericalHF calculation are presented for reference.We find energy minima at similar q ’s between Zrand
Zr, although their energies are shifted depend-ing on the configurations. It is interesting to comparethe configurations at the minima corresponding to eachother, between these nuclei. At the minima for increas-ing | q | , protons in the pf -shell are excited to 0 g / twoby two, and the minima are characterized by the protonconfigurations in this region. In practice, the first s.p.level dominated by p g / , which has Ω π = (1 / + , isoccupied at the minimum slightly below q = 500 fm ,the second having Ω π = (3 / + above q = 500 fm , andthe third having Ω π = (5 / + at q ≈ for bothnuclei. Analogous variation of proton configurations isseen at the oblate minima. Then neutron configurationsat the minima and their relative energies determine whichminimum provides the lowest energy. Whereas the even-parity levels near the Fermi energy mix one another, theunique-parity orbit n h / stays nearly pure, and occu-pation number on n h / carries important structuralinformation. Owing to the four more neutrons, n h / is closer to the highest occupied level in Zr than in Zr at the spherical limit. This facilitates crossing ofthe s.p. levels one of which is dominated by n h / ,particularly on the prolate side. Two neutrons occupy0 h / at q ≈
800 fm in Zr, while at q ≈ in Zr. At the q ≈ minimum of Zr, fourneutrons occupy the levels corresponding to 0 h / .The repulsive effect of the tensor force is strong at the q ≈ state because of the occupation of n h / ,whereas this state remains to yield the lowest energy,diminishing the energy difference between this and theother minima. This effect becomes more important inthe structure of Zr. Although the lowest minimum hasprolate shape also in Zr in the present HF calculationwith M3Y-P6, it is almost degenerate with an oblate min-imum. The energy difference between the prolate andthe oblate minima is thinner (0 . Zr, which iscompared to 1 . Zr shown in Fig. 8. Becauseof this small difference, these two minima might be in-verted by correlations beyond HF, while such inversionis unlikely to take place in E − E (TN) . We also mentionthat the spherical HFB energy at Zr [10] is lower thanthe lowest axial HF energy by 1 . E. Zr In 60 ≤ N ≤
70, measured E x (2 +1 )’s are steadily low [3,4, 21, 22, 34]. Measured spectroscopic properties are welldescribed by the prolate deformation up to Zr [35]. Ithas been claimed that
Zr has the highest collectivityamong the Zr isotopes, because measured E x (2 +1 ) is thelowest and B ( E
2) is the strongest [36, 37]. We next pickup this nucleus. As seen in Fig. 1, the q value at theg.s. does not change much in 58 ≤ N ≤
72 in the presentcalculations with M3Y-P6. It is observed in Fig. 10 thatthe lowest minimum is well developed in
Zr, having q ≈ , and the second minimum is found at q ≈−
600 fm . It is noticed that these q values are closeto those in Zr shown in Fig. 8. Typified by thesetwo minima, E ( q ) in Zr has a similarity to E ( q ) in Zr, as the q values giving the local minima are not -874-872-870-868-866-864-862-860-858-856-854-852-850-848-846 -1000 -500 0 500 1000 E [ M e V ] q [fm ] Zr D1M M3Y-P6 (w/o v (TN) ) FIG. 10. (Color online) CHF results for
Zr. See Fig. 2 forconventions. -18-16-14-12-10-8-1000 -500 0 500 1000 proton p g ε [ M e V ] q [fm ] Zr (1/2) + (3/2) + (5/2) + (7/2) + (9/2) + (1/2) - (3/2) - (5/2) - (7/2) - (9/2) - (11/2) - -10-8-6-4-2-1000 -500 0 500 1000 neutron d s h q [fm ] FIG. 11. (Color online) Proton and neutron s.p. energies in
Zr obtained from the HF calculations with M3Y-P6, at theminima shown in Fig. 10. See Fig. 3 for conventions. very different.The s.p. levels at the minima of
Zr are depicted inFig. 11. At the absolute minimum with q ≈ ,six neutrons occupy the levels connected to 0 h / . Halfof the levels belonging to n h / (the Ω ≤ / h / . The difference in the occupation on n h / is reflected by E (TN) in Fig. 10. F. Zr In Fig. 1, we have found a sudden shape transitionfrom prolate to oblate at
Zr. The oblate state has q ≈ −
500 fm . The energy curve for Zr is exhibitedin Fig. 12. The tensor force plays a crucial role in theshape change. If there were no tensor force in the effective -912-910-908-906-904-902-900-898-896-894-892-890-888-886-884 -1000 -500 0 500 1000 E [ M e V ] q [fm ] Zr D1M M3Y-P6 (w/o v (TN) ) FIG. 12. (Color online) CHF results for
Zr. See Fig. 2 forconventions. -22-20-18-16-14-1000 -500 0 500 1000 proton ε [ M e V ] q [fm ] Zr (1/2) + (3/2) + (5/2) + (7/2) + (9/2) + (1/2) - (3/2) - (5/2) - (7/2) - (9/2) - (11/2) - -10-8-6-4-2 0-1000 -500 0 500 1000 neutron q [fm ] FIG. 13. (Color online) Proton and neutron s.p. energies in
Zr obtained from the HF calculations with M3Y-P6, at theminima shown in Fig. 12. See Fig. 3 for conventions. interaction, the prolate state with q ≈ shouldbe lower than the oblate minimum as recognized from E − E (TN) . However, the repulsion due to the tensor forceis so strong at the prolate minimum that the energy ofthis state could become higher than the oblate state.The occupied s.p. levels shown in Fig. 13 are useful inanatomizing the shape change in this nucleus. It shouldbe noticed that four neutrons occupy 0 h / at the spher-ical limit, because of N = 74. At the absolute minimumlocated on the oblate side, six neutrons occupy the levelsconnected to 0 h / . The repulsion of the tensor force isstronger than at the spherical configuration, but not sosignificantly. On the contrary, at the prolate minimumwith q ≈ , eight neutrons occupy the levels con-nected to 0 h / . Moreover, two more neutrons occupyan odd-parity level coming down from the n f / orbitin the upper major shell, which further enhance the re-pulsive tensor-force effect.The spherical HFB calculation yields lower energy thanthe deformed HF energies. As presented in Table I, the -928-926-924-922-920-918-916-914-912-910-908-906-904-902-900-898 -1000 -500 0 500 1000 E [ M e V ] q [fm ] Zr D1M M3Y-P6 (w/o v (TN) ) FIG. 14. (Color online) CHF results for
Zr. See Fig. 2 forconventions.
HFB calculations with D1S and SLy4 predict sphericalshape at
Zr. It should be postponed to address fullMF prediction with M3Y-P6 on the shape of
Zr, untilthe deformation and the pairing are simultaneously takeninto account. G. Zr In the present calculation the shape returns to be al-most spherical at
Zr, as shown in Fig. 1. It is foundthat a state in the vicinity of q = 0 is the lowest bothin the M3Y-P6 and the D1M results. However, if E (TN) is subtracted, the oblate state with q ≈ −
500 fm stayslowest. The good doubly magic nature has been pre-dicted for Zr in Ref. [10], which is preserved after thedeformation is taken into account. In E − E (TN) in Zr,the oblate minimum lies relatively close to the sphericalconfiguration, with the difference less than 3 MeV. Thedoubly-magic nature of
Zr is enhanced by the tensorforce to a significant extent.The effect of the tensor force is traced back to the s.p.energies presented in Fig. 15, as typically observed in ε and ε − ε (TN) of the proton Ω π = (9 / + level. Becauseof the energy gain of this level, the oblate configurationbecomes lowest in E − E (TN) . However, the tensor forceraised this level significantly, finally making the sphericalconfiguration the lowest in Fig. 14. Thus, occupation of p g / is primarily responsible for the difference in E (TN) .Role of the n h / orbit is not quite conspicuous for thisnucleus, since this orbit is mostly occupied even at thespherical limit. It is found that excitation across the N = 82 spherical shell gap hardly takes place on theoblate side, while excitation to n f / may occur on theprolate side. -24-22-20-18-16-14-1000 -500 0 500 1000 proton ε [ M e V ] q [fm ] Zr (1/2) + (3/2) + (5/2) + (7/2) + (9/2) + (1/2) - (3/2) - (5/2) - (7/2) - (9/2) - (11/2) - -10-8-6-4-2 0-1000 -500 0 500 1000 neutron q [fm ] FIG. 15. (Color online) Proton and neutron s.p. energies in
Zr obtained from the HF calculations with M3Y-P6, at theminima shown in Fig. 14. See Fig. 3 for conventions.
IV. DISCUSSION AND SUMMARY
In Ref. [10], magic numbers are well predicted by thespherical MF calculations with semi-realistic interactionM3Y-P6. Whereas no deformation d.o.f. were handled,the pairing was used as a representative of correlations.The results were in accordance with the available data formost nuclei all over the nuclear chart. This suggests thatthe description of the nuclear structure may be rathersimplified by the semi-realistic interaction containing therealistic tensor force, as a good first approximation isobtained within the MF framework. However, a dis-crepancy had been found in the Z = 40 magicity in theneutron-rich Zr nuclei. Although it has been known thatthe Zr nuclei are deformed in 60 ≤ N < ∼
70, the spheri-cal HFB calculations provide no signature for the erosionof the Z = 40 magicity. The present axial HF calcula-tions have resolved this problem, showing that Z = 40is not magic in this region because the axially deformedstates lie lower than the spherical state. Although thepair correlation has not been taken into account, the low-est energy of the deformed state is even lower than thespherical HFB energy in most of these nuclei. Thus, to-gether with the result for Mg in Ref. [17], the validity ofM3Y-P6 in predicting magic numbers is reinforced by theaxial HF calculations. It is remarked that deformation at Zr is reproduced at the MF level, which has been diffi-cult in the self-consistent MF calculations so far despite afew exceptions. This deformation can be regarded as anindirect effect of the tensor force, as already discussed.The tensor force acts strongly on the unique-parity or-bit, because of its large ℓ [11]. Moreover, the difference inparity makes the unique-parity orbit almost pure, withlittle admixture of other spherical s.p. orbitals. There-fore the tensor-force effects are more conspicuous on theunique-parity orbit than on others. The repulsive tensorforce effect becomes stronger as the unique-parity orbitis occupied. We have pointed out that, in the lower partof the major shell, the energy of the deformed state goes up by this interplay of the tensor force and the unique-parity orbit, and deformation tends to be delayed as seenin Zr.In the argument of the so-called ‘type-II shell evolu-tion’ [38], it was indicated that deformation could bedriven by the interplay of the unique-parity orbit andthe tensor force, in addition to the central force. An ex-cited state of Ni was raised as an example, in whichprotons are excited from the jj -closed Z = 28 core atthe deformed states. The tensor force makes excitationeasier. In the present case, deformation requires excita-tion of protons from the ℓs -closed Z = 40 core to 0 g / .As in the difference between N = 20 and 28 [17], thetensor-force effects on deformation are opposite betweenthe ℓs -closed nuclei and the jj -closed nuclei. In Ref. [5],the type-II shell evolution was extensively discussed fordeformation around Zr. It could be considered thatthe tensor-force effects involving the unique-parity orbitshown in this article have disclosed a specific effect of thetype-II shell evolution discussed in Ref. [5].As mentioned in Sec. I, the exotic shape with the tetra-hedral symmetry was argued for several Zr nuclei, Zr, Zr and , , Zr [6–8]. Some calculations predictedit take place at the g.s., which have been supported byno experimental data. Although the present calculationsdo not handle the tetrahedral configuration explicitly, weshall give comments on this issue. At Zr, the N = 56magicity as well as the Z = 40 magicity are propped upby the tensor force, which prevent the g.s. deformationincluding the tetrahedral one. Furthermore, the tetra-hedral shape implies admixture of the octupole deforma-tion, which could not be driven without occupation of theunique-parity orbit. The excitation to the unique-parityorbit gives rise to loss of the binding energy as an effectof the tensor force. Therefore, the tensor force will notfavor the tetrahedral shape in the Zr nuclei.In this article we have constrained ourselves to N ≤ N = 82, the n i / orbit may enter, and a biggers.p. space will be desired which includes the ℓ = 10 basis-functions. An interesting subject in N ≥
82 is the gianthalo predicted in Ref. [39]. We leave it for a future study,which should properly take account of pair correlationsand coupling to the continuum [40]. However, it is notedthat the neutron drip line for Zr is located at N = 86,according to the spherical HFB calculation with M3Y-P6 [10]. This implies that, in the prediction with theM3Y-P6 interaction, neutron halos in this region will notbe formed by no more than four neutrons, not being huge.In summary, we have investigated the shape evolu-tion of Zr nuclei and effects of the tensor force on it,by the axial Hartree-Fock calculations with the M3Y-P6semi-realistic interaction. Deformation at N ≈
40 and in60 < ∼ N < ∼
70 is reproduced. The former has not beeneasy for the self-consistent MF calculations so far. Thelatter seems to resolve the discrepancy in the predictionof magic numbers in Ref. [10]. The sudden transitionfrom prolate to oblate is predicted at
Zr, although re-covery of spherical shape via the pairing, shape mixing0or triaxial deformation is not ruled out. The transitionfrom oblate to spherical shape is predicted at
Zr. Forthe tensor-force effects, we have pointed out significantroles of the unique-parity orbit in the shape evolution.These effects could be crucial for the doubly-magic na-ture of Zr and for the predicted shape transitions at
Zr and
Zr.
ACKNOWLEDGMENTS
The authors are grateful to T. Inakura for provid-ing the HF results with SkM ∗ . This work is finan-cially supported in part by JSPS KAKENHI Grant Num-ber 16K05342. Some of the numerical calculations havebeen performed on HITAC SR24000 at Institute of Man-agement and Information Technologies in Chiba Univer-sity. [1] A. Bohr and B.R. Mottelson, Nuclear Structure , vol. 1(Benjamin, New York, 1969).[2] C.J. Lister, et al. , Phys. Rev. Lett. , 1270 (1987).[3] E. Cheifetz, R.C. Jared, S.G. Thompson and J.B. Wil-helmy, Phys. Rev. Lett. , 172502 (2016).[6] S. Tagami, Y.R. Shimizu and J. Dudek, J. Phys. G ,015106 (2015)[7] N. Schunck, J. Dudek, A. G´o´zd´z and P.H.Regan, Phys.Rev. C , 061305(R) (2004); S. Tagami, Y.R. Shimizuand J. Dudek, Phys. Rev. C , 054306 (2013).[8] J. Zhao, B.-N. Lu, E.-G. Zhao and S.-G. Zhou, Phys.Rev. C , 014320 (2017).[9] T.A. Khan, W.D. Lauppe, H.A. Selic, H. Lawin, G.Sadler, M. Shaanan and K. Sistemich, Z. Phys. A ,289 (1975); G. Sadler, et al. , Nucl. Phys. A , 365(1975).[10] H. Nakada and K. Sugiura, Prog. Theor. Exp. Phys. , 033D02.[11] T. Otsuka, T. Suzuki, R. Fujimoto, H. Grawe and Y.Akaishi, Phys. Rev. Lett. , 232502 (2005).[12] H. Sagawa and G. Col`o, Prog. Part. Nucl. Phys. , 76(2014).[13] H. Nakada, Phys. Rev. C , 014316 (2003).[14] H. Nakada, Phys. Rev. C , 014336 (2013).[15] N. Anantaraman, H. Toki and G.F. Bertsch, Nucl. Phys.A , 269 (1983).[16] H. Nakada, K. Sugiura and J. Margueron, Phys. Rev. C , 067305 (2013).[17] Y. Suzuki, H. Nakada and S. Miyahara, Phys. Rev. C ,024343 (2016).[18] R. Rodor´ıguez-Guzm´an, J.L. Egido and L.M. Robledo,Phys. Lett. B , 15 (2000); Phys. Rev. C , 054319(2000).[19] H. Nakada, Nucl. Phys. A , 47 (2008).[20] S. Goriely, S. Hilaire, M. Girod and S. P´eru, Phys. Rev.Lett. , 242501 (2009).[21] T. Sumikama, et al. , Phys. Rev. Lett. , 202501(2011).[22] N. Paul, et al. , Phys. Rev. Lett. , 032501 (2017).[23] R. Rodor´ıguez-Guzm´an, P. Sarriguren, L.M. Robledo andS. Perez-Martin, Phys. Lett. B , 202 (2010). [24] J. Dobaczewski, W. Nazarewicz and M.V. Stoitsov, Eur.Phys. J. A , 21 (2002); M.V. Stoitsov, J. Dobaczewski,W. Nazarewicz, S. Pittel and D.J. Dean, Phys. Rev. C , 054312 (2003).[25] S. Hilaire and M. Girod, Eur. Phys. A , 237(2007); J.-P. Delaroche, M. Girod, J. Libert,H. Goutte, S. Hilaire, S. P´eru, N. Pillet andG.F. Bertsch, Phys. Rev. C (2009) 044301; ibid. (2011) 021302(R); T. Inakura,private communication.[27] A. Blazkiewicz, V.E. Oberacker, A.S. Umar and M.Stoitsov, Phys. Rev. C , 054321 (2005).[28] H. Abusara and S. Ahmad, Phys. Rev. C , 064303(2017).[29] M. Bender, P.-H. Heenen and P.-G. Reinhard, Rev. Mod.Phys. , 121 (2003).[30] J. Dobaczewski, M.V. Stoitsov, W. Nazarewicz and P.-G.Reinhard, Phys. Rev. C
76, 054315 (2007); T. Duguet,M. Bender, K. Bennaceur, D. Lacroix and T. Lesinski,Phys. Rev. C
79, 044320 (2009); L.M. Robledo, J. Phys.G , 064020 (2010).[31] M. Bender, K. Bennaceur, T. Duguet, P.-H. Heenen, T.Lesinski and J. Meyer, Phys. Rev. C , 064302 (2009).[32] T.R. Rodr´ıguez and J.L. Egido, Phys. Lett. B , 255(2011).[33] H. Nakada, Phys. Rev. C , 051302(R) (2010).[34] M.A.C. Hotchkis, et al. , Nucl. Phys. A , 111 (1991).[35] Y.-X. Liu, Y. Sun, X.-H. Zhou, Y.-H. Zhang, S.-Y. Yu,Y.-C. Yang and H. Jin, Nucl. Phys. A , 11 (2011).[36] J.K. Hwang, A.V. Ramayya, J.H. Hamilton, Y.X. Luo,A.V. Daniel, G.M. Ter-Akopian, J.D. Cole and S.J. Zhu,Phys. Rev. C , 044316 (2006).[37] F. Browne, et al. , Phys. Lett. B , 448 (2015).[38] Y. Tsunoda, T. Otsuka, N. Shimizu, M. Honma and Y.Utsuno, Phys. Rev. C , 031301 (2014); T. Otsuka andY. Tsunoda, J. Phys. G , 024009 (2016).[39] J. Meng and P. Ring, Phys. Rev. Lett. , 460 (1998).[40] H. Nakada and K. Takayama, Phys. Rev. C98