Role of tensor terms of the Skyrme energy-density functional on neutron deformed magic numbers in the rare-earth region
RRole of tensor terms of the Skyrme energy-density functional on neutron deformedmagic numbers in the rare-earth region
Kai-Wen Kelvin-Lee ( 李 凯 文 ), Meng-Hock Koh ( 辜 明 福 ),
1, 2, ∗ and L. Bonneau † Department of Physics, Faculty of Science, Universiti Teknologi Malaysia, 81310 Johor Bahru, Johor, Malaysia UTM Centre for Industrial and Applied Mathematics, 81310 Johor Bahru, Johor, Malaysia CENBG, UMR 5797, Universit´e de Bordeaux, CNRS, F-33170, Gradignan, France (Dated: February 26, 2021)The role of the tensor part of the nuclear interaction is actively investigated in recent years dueto experimental advancement yielding new data in nuclei far from the β -stability line. In this articlewe study the effect of this part of the nuclear interaction on deformed neutron magic numbers in therare-earth region within the Skyrme energy-density functional for various TIJ [1] parametrizations.Two quantities signaling magic numbers are considered: two-neutron separation energies and single-particle energies. They are calculated in isotopic series involving well-deformed rare-earth nucleiranging from Z = 64 to Z = 72 in the N = 100 region. Obtained results show that, whereas theneutron-proton tensor contribution to binding energies is important to reproduce neutron sub-shellclosure at N = 104 in heavier rare earths Yb ( Z = 70) and Hf ( Z = 72) isotopes, like-particle tensoralso plays a role in the single-particle spectrum around Fermi level and is even favored in lighter Gd( Z = 64) and Dy ( Z = 66) rare-earth isotopes. I. INTRODUCTION
The discovery of gravitational-wave signal GW170817coming from binary neutron stars (BNS) merger showsthat BNS merger is more likely the candidate site forrapid neutron capture process (known as the r pro-cess) [2]. The nucleosynthesis through r process occursthrough rapid capture of free neutrons forming neutron-rich elements away from the beta-stability line. In thesolar abundances, the r process is responsible for thesecond and third peaks around A = 130 ∼
138 and A = 195 ∼ A = 165 in the rare-earth region. This peak was ex-plained in terms of increasing nuclear deformation whichstabilizes the nucleus similarly to the role of neutronclosed shell [4]. Substantial effort has been made by theexperimental nuclear physics community to uncover thisso-called deformed magic numbers. However, as we willsee shortly, these magic numbers remain somewhat elu-sive.On the experimental side, considerable effort has beenmade resulting in many proposals for new deformedmagic numbers. In 1999, Asai and collaborators showedtwo minimum in the first 2 + energies in Dy isotopes [5].They proposed that the second minima at N = 104 co-incides with the location of maximum deformation. Thefirst minimum at N = 98 was however dismissed as aris-ing due to some local effect which enhanced the defor-mation around this isotope. This conclusion on N = 98was also agreed upon in the work of [6] about a decadelater based on systematic studies on yrast levels of Dy ∗ [email protected] † [email protected] isotopes and the 4 + → + transition in Dy. Subse-quent work by Patel et al. [7] in 2014 showed that adeformed magic number exists for neutrons at N = 100in elements with proton number Z ≤
66, namely in Nd( Z = 60), Sm ( Z = 62), Gd ( Z = 64) and Dy ( Z = 66)isotopes. This magic-number character of N = 100 inthis region was, however, challenged three years later byWu and collaborators [8]. They reported to find no ev-idence of deformed subshell gap at N = 100 from theanalyses of β - decay half-lives of Pm ( Z = 61) isotopes.Instead, they proposed two different magic numbers at N = 96 for Z = 58 to Z = 62 and N = 104 for Z = 63 to Z = 66. The more recent works of Hartley and collabora-tors [9, 10] on the other hand showed that N = 98 couldinstead be a candidate neutron subshell closure around Z = 64, contradicting the findings of Wu et al. [8] that N = 104 should be the deformed magic number in thisisotope. Going to heavier rare-earth nuclei, [11] reportedemergence of new sub-shell closure at N = 108 in Hf( Z = 72), W ( Z = 74) and Os ( Z = 76) isotopic series.On the theoretical side, there are rather limited studiesto uncover the possible deformed magic numbers in therare-earth region. To the best of our knowledge, all of thestudies supported the magicity of N = 100 in the lightrare-earth elements Sm and/or Dy [12–14]. The possi-bility of different subshell closures in heavier rare-earthisotopes ( Z >
66) as indicated by some experimentaldata was not explored. There were however calculationson K isomers for example the 6 + in Dy [15, 16] andyrast levels in Dy isotopes [17]. Interestingly, Yadav etal. [17] reported that N = 102 is more likely the magicnumber in Dy isotopes instead of N = 104 based on en-ergies of the ground-state and first 2 + states obtainedwithin cranked Hartree-Fock-Bogoliubov calculations.One of the current major theme in nuclear theory isrelated to the impact of tensor two-nucleon interaction.While pioneering work on tensor effective potential wasperformed in 1977 by Stancu et al. [18], there was not a r X i v : . [ nu c l - t h ] F e b much follow up of this work until mid 2000s when accessto exotic nuclei was made possible through technologicaland experimental breakthroughs. Within the mean-fieldapproach based on Skyrme energy-density functional, ef-forts have been made to design new parametrizationsthrough either a perturbative or a full fitting procedure.In the perturbative approach, only the two zero-rangetensor terms are adjusted while all other Skyrme pa-rameters are kept constant. This is the case for theSIII+tensor (SIII+T) parameter sets of Refs. [18, 19] andthe SLy5+tensor (SLy5+T) parametrization of Ref. [20].On the other hand, a fit of all parameters has beenperformed by Lesinski et al. [1] yielding a set of TIJparametrizations which were applied to the investiga-tions of spherical nuclei [1], nuclear deformation [21] andtime-odd systems [22]. Investigations on fit protocols oftensor effective potential components was also studiedby Zalewski et al. [23] who proposed that the single-particle levels should be considered instead of the usualbulk properties like the binding energies.Within the Gogny mean-field approach, similar efforthas been made by Anguiano et al. [24] highlighting theneed for inclusion of tensor effective potential. In thiswork a density-independant, finite-range tensor interac-tion term is added to the D1S parametrization, yield-ing the parametrization called D1ST2a. In this pertur-bative approach, independant like-nucleon and neutron-proton contributions are present in the effective two-nucleon potential, as in the zero-range Skyrme tensor po-tential, and the fitting protocol–involving neutron single-particle energies of 1f / and 1f / –yielded a strength of −
20 MeV for the like-nucleon term and a much largerstrength of 115 MeV for the neutron-proton term. Sub-sequently Grasso and Anguiano [25] studied the appro-priate range for the strength of the tensor terms withinthe Skyrme and Gogny energy density functionals (EDF)while Ref. [26] showed that tensor effective potential isimportant to explain magicity at N = 32 and N = 34in the Ca and Ca nuclei. More recently, Bernardand collaborators [27] investigated the role in fission ofthe tensor terms of the Gogny EDF through a thoroughcomparison of several fission-related quantities–rangingfrom fission-barrier heights and paths to fission-fragmentneutron emission–obtained with the D1S and D1ST2aparametrizations. One of the most important conclusionis that the added tensor terms are able to account for thenew compact-symmetric fission configuration experimen-tally observed during the 2012 SOFIA campaign at GSIDarmstadt [28].From the rich literature showing that a tensor effectivepotential affects the single-particle levels ordering, we areinterested to investigate if inclusion of such a potentialwithin our Skyrme EDF would allow us to explain de-formed magic numbers suggested by experiment in therare-earth region. We shift our attention to the heavierrare-earth nuclei which have not gained much attentionfrom theorists as compared to their lighter counterpartswith particular interest in the N = 104 sub-shell closures. After a brief presentation of the relevant theoretical in-gredients, we address successively in sections III to V theeffect of the Skyrme tensor effective potential on chargequadrupole moments, two-neutron separation energies,and single-particle spectra. We give concluding remarksin section VI. II. THEORETICAL APPROACH
We considered several Skyrme fully refitted TIJparametrizations namely • T22, T24 and T26 (pure like-particle coupling) • T22, T42 and T62 (pure neutron-proton coupling) • T41 and T44 (mixed coupling).The TIJ forces are labelled in such a way that I and J values are related to the proton-neutron β , and like-particle α tensor coupling, respectively [1] α =60( J −
2) MeV fm β =60( I −
2) MeV fm with α = α C + α T and β = β C + β T . The subscriptC and T refer to the central and tensor contributions,respectively.We have also included the original SIII [29] and SLy5[30] parametrizations and their counterparts in whichtensor effective potential components are added pertur-batively for comparison. The seniority force is used toapproximate the residual pairing interaction whereby theneutron and proton pairing strengths were adjusted suchthat the BCS pairing gap yields the empirical Jensen for-mula [31]. The single-particle wave function is expandedon a deformed harmonic oscillator basis with a basis sizeof 16. The oscillator parameters b and q have been op-timized to yield the lowest ground-state energy for eachnucleus [32]. We have limited ourselves to axial and par-ity symmetric nuclear shapes.Within the Skyrme EDF, in addition to the strengthparameters of the central and spin parts of the effectivepotential, the tensor parameters t e and t o enter the totalbinding energy in the B , B , B and B couplingconstants given by B = − t x + t x t e + t o ) B = t − t −
14 ( t e − t o ) B = −
38 ( t e + t o ) B = 38 ( t e − t o ) . In order to isolate the contribution from the tensorparameters to the binding energy, we separate the con-tributions coming from B and B into two parts, suchthat: E CB = − (cid:16) t x + t x (cid:17) J µν J µν E TB = 14 ( t e + t o ) z (cid:88) µ,ν = x J µν J µν E CB = (cid:16) t − t (cid:17) z (cid:88) µ,ν = x J q,µν J q,µν E TB = −
14 ( t e − t o ) z (cid:88) µ,ν = x J q,µν J q,µν where J µν is the spin-current density with µ, ν = { r, z, φ } and J q,µν is the spin-current density for the charge state q (see Ref. [33] for their definition). The contributionsfrom the B and B terms to the binding energy are E B = B z (cid:88) µ = x (cid:16) J µµ (cid:17) E B = B z (cid:88) µ = x (cid:16) J q,µµ (cid:17) . Separating these terms in such a way allows us to drawout the contribution of the tensor part alone from allother non-tensor related terms to the binding energy.This means that the binding energy can be partitionedinto E = E C + E T where E C = E kin + E Coul + E pair + E B x + E CB + E CB with the contribution to the E B x term comes from allthe Skyrme coupling constants except for B , B , B and B . The contribution to the E T comes from termsrelated to the tensor effective potential parameters t e and t o such that E T = E TB + E TB + E CB + E CB . III. CHARGE INTRINSIC QUADRUPOLEMOMENT
We first present our ground-state intrinsic chargequadrupole moment for isotopic series of Gd, Dy, Er, Yb and Hf and compared to experiment [34]in Figure 1. Calculations with the various TIJ forcesgive good agreement with available experimental data.More importantly, we find a peak around N ∼ N ∼ IV. TWO-NEUTRON SEPARATION ENERGIES
We compute the two-neutron separation energy S n and two-neutron separation energy differential ∆ S n us-ing the expression S n = E ( N − , Z ) − E ( N, Z )∆ S n = S n ( N, Z ) − S n ( N + 2 , Z ) . The calculated ∆ S n are plotted in Figure 2 togetherwith experimental data taken from AME2016 [35].Let us first discuss the results for the three heavier el-ements considered in our study namely Er, Yb and Hf. The experimental data show a peak at N = 104[35] in these elements. To compare the theoretical resultswith data, we take the T22 as the reference parametriza-tion because it is such that α = β = 0, although theSkyrme parameters t e , t o are not zero. The T22 pa-rameter set manages to produce a pronounced peak at N = 104 especially in Er and Yb. The peak at thisneutron number is even more enhanced when increas-ing β by considering the T42 and T62 parametrizations.This shows that neutron-proton tensor coupling constant β is essential to reproduce the neutron N = 104 sub-shellclosure in these rare-earth nuclei. This behavior of the N = 104 peak with β is even more marked in Hf iso-topes.On the contrary, increasing like-particle tensor cou-pling constant α with a vanishing β contribution, i.e.in the sequence T22 → T24 → T26, results in largerdips, instead of peaks, at N = 104. With non-vanishing β and still increasing α in the sequence T41 → T42 → T44 parametrizations, we see that the pronounced peakat N = 104 obtained with T41 decreases when usingthe T42 parametrization, and then vanished totally withT44. This clearly shows that like-particle tensor couplingtends to remove the N = 104 peak in heavy rare-earthnuclei. Therefore the reproduction of this peak requiressmall α values and positive, sizeable β values.Concerning the two parametrizations obtained fromperturbative fits of the tensor effective potential, we findthat the SLy5+T improves the results as compared tothe original SLy5 parametrization. Indeed a significantpeak is found with SLy5+T in the ∆ S n plot at N = 104for Yb isotopes instead of a minute peak at N = 102with SLy5. However, neither SLy5 nor SLy5+T are ableto reproduce the magicity of N = 104 in Hf isotopes.Before moving on to lighter elements of the rare-earthregion, we make a remark on the ∆ S n at N = 108 in Hf isotopes. In this element, the experimental point at N = 108 is higher than the one at N = 104. We did notmanage to reproduce this pattern in our calculations. Inspite of this, we see that the ∆ S n between N = 106and N = 108 exhibits a positive slope when using T42and T62 forces, while all other TIJ forces give a negativeslope. This reinforces the conjecture that neutron-protontensor coupling is more favored in heavy rare-earth nucleiand can, at the very least, reproduce the experimentaltrend qualitatively.In Dy isotopes, peaks are seen at N = 98 ,
102 andpossibly N = 106 in experimental ∆ S n [35]. Calcu-lations with TIJ parametrizations are not able to re-produce this experimental trend. Instead, the SIII andSIII+T parametrizations performed better there. TheSIII parametrization generates peaks at N = 98 and 102while the SIII+T calculations yield a peak at N = 102and follow the experimental trend at N = 106. Resultswith the TIJ parameter sets, however, yield two peaks at N = 100 and N = 104. The former peak is enhancedwhen increasing the like-particle coupling constant α ,while the latter is more pronounced when the neutron-proton coupling constant β is larger. Indeed, T22 cal-culations serving as a reference, we see that the peak at N = 100 is more pronounced when going to T24 and T26forces. Conversely, the peak at the same neutron numberis reduced when going from T22 to T42 and then to T62forces. The reverse is seen at N = 104 where going fromT22 to T24 induced a sharp drop in the ∆ S n . Com-paring the results obtained with T41, T42 and T44 alsoindicates that strong like-particle tensor coupling is un-desirable to produce a peak at N = 104, similar to whatis found in heavier rare-earth elements. Before closingthis discussion of Dy results, we would like to draw thereader’s attention to the fact that while our TIJ calcu-lations do not reproduce experimental data of Wang etal. [35], the TI2 results are however in agreement withWu et al. [8] who showed that N = 104 forms a sub-shell closure. Clearly more experimental data in lightrare-earth nuclei are needed to resolve this discrepancy.Finally, we comment on the results for the lightest rare-earth element considered in our study. In the Gd iso-topes, the experimental ∆ S n is almost constant overthe range 94–98 of N . It makes a dip at N = 100 beforeforming a peak at N = 102. Our TIJ calculations fails toreproduce this trend, and the same pattern seen in thevariation of the above N = 100 and N = 104 peaks with α and β is obtained here.To conclude this section, we can say that the TIJparametrizations produce two persistent peaks at N =100 and N = 104. The peaks can be obtained in particu-lar with the T22 parameter set for which the central andtensor contributions cancel, yielding α = β = 0. Whenswitching to TIJ forces with α >
0, we find an enhance-ment for the N = 100 peak while TIJ forces with β > N = 104. In order to understand the role of the tensor effectivepotential, we plot in Figure 3 the contributions of E C and E T terms of the Skyrme energy-density to ∆ S n asa function of N for T26, T22 and T62 parametrizations.In all considered nuclei, the tensor contribution is smallas compared to the sum of all other terms, but it playsan important role in shaping the fine structure of thepatterns seen in Figure 2.In Hf, the tensor contribution is particularly crucialwhen using T22 and T62 parameter sets. Indeed thecontribution from all other terms to ∆ S n does not yielda peak at N = 104, which can only be obtained thanksto the tensor component.In Yb, non-tensor terms alone do produce two peaksat N = 100 and N = 104 with the T22 and T26 forces.When including tensor, the calculated ∆ S n at N = 100is decreased while the point at N = 104 is push upwardsyielding only one peak at N = 104. A similar effect isobserved in in Gd and Dy elements.
V. NEUTRON SINGLE-PARTICLE ENERGYSPECTRA
We now turn our attention to the neutron single-particle levels for some nuclei in Figure 4. The vari-ation in the calculated ∆ S n with different Skyrmeparametrizations in Figure 2 coincide with the variationin the single-particle energy gap.In Dy isotopes, two pieces of information can belearnt. On the one hand pure like-particle ( α ) tensorcoupling (as in T24 and T26) favors sub-shell closure at N = 100. In fact, a substantial single-particle energy gapappears at N = 96 only with T26 force. Pure neutron-proton ( β ) tensor coupling, on the other hand, favorssub-shell closure at N = 104. The N = 98 sub-shell clo-sure, while not reproduced by any TIJ forces, seems tobe accounted for by a strong neutron-proton rather thanlike-particle tensor component. This is correlated withthe decreasing trend in the single-particle energy gap at N = 98 when going from T24 to T26. However, accord-ing to Hartley et al in Ref. [9], “1 / / Eu”. This suggests to explore refinements to existingparametrizations.Let us move to the single-particle states of Yb and Hf isotopes. The peak in ∆ S n at N = 104 is relatedto the widening of the single-particle energy gap betweenthe 7 / − and 7 / + states seen in Figure 4. The energygap between the two states increases with α (along thesequence of calculations T22 → T42 → T62), while itdecreasing from T24 to T26, that is to say when α = 0and β increases. With T22, we see that the 7 / + stateis located below a 5 / − state and remains almost at thesame energy when switching on neutron-proton tensorcoupling and keeping α = 0. In contrast the 7 / − statekeeps beeing shifted higher in energy, giving rise to a verylarge energy gap when increasing neutron-proton tensorstrength.Moreover an important observation is made regardingthe like-particle tensor coupling by comparing the resultsobtained with T42 and T44 forces. A slight increase oflike-particle tensor coupling in T44 causes tremendouslowering of the 7 / − state, while elevating the 7 / + stateabove 5 / − . Consequently, the N = 104 is not a sub-shell closure for T44 while it is so for T42. This sug-gests that strong neutron-proton coupling is importantto reproduce this deformed magic number in the heavyrare-earth region. However, the increasing energy gap at N = 106 sub-shell in Yb and Hf isotopes with T24and T26 could indicate the importance of like-particletensor coupling.
VI. CONCLUSION
In conclusion, we have performed Skyrme Hartree–Fock-BCS calculations for even-even rare-earth nucleiwith Z = 64 up to Z = 72. We have found a maxi-mum deformation around N ∼
100 which confirms thatthe neutron deformed magic numbers could be found inthis neutron-number region.Then we have calculated two-neutron separation ener-gies and studied their difference ∆ S n between two con-secutive even- N values. with several TIJ parametriza-tions of the Skyrme energy-density functional. Two per-sistent peaks have been found at N = 100 and N = 104.These peaks have been obtained in all considered nucleiwith the T22 parameter set (for which α = β = 0) ex-cept for Hf. While the N = 100 peak is even morepronounced when switching on like-particle tensor terms(driven by the α coupling constant) the N = 104 is en-hanced by neutron-proton tensor terms (driven by the β coupling constant). Comparison with experimental data of Ref. [35] suggests that neutron-proton tensor termsare favored in heavy rare-earth nuclei to reproduce the N = 104 peak. In contrast, increasing like-particle ten-sor strength with a fixed, positive β coupling constant(in T41, T42, T44 parametrizations) has the detrimentaleffect of decreasing the N = 104 peak. In the lighter rare-earth elements Gd and Dy however, the situation isnot so clear. In these nuclei the like-particle tensor termscan produce a peak in ∆ S n at some neutron numbersdepending on the parametrization.To better understand the role of the tensor terms onthis observable, we have studied the contribution to ∆ S n arising solely from the t e and t o parameters of the tensoreffective potential and shown that, while being small, thiscontribution is important to produce the ∆ S n peaks.We have also studied the neutron single-particle spectrafor various parametrizations of the Skyrme EDF. A neatcorrelation between the peak structure of ∆ S n and largesingle-particle energy gaps around Femi level could thusbe evidenced.Overall the present work indicates that N = 104 canbe considered as a “deformed” magic neutron numberthanks to neutron-proton tensor coupling in heavy rare-earth elements and that like-particle tensor coupling isnot desirable in this region. However, in lighter rare-earth elements the situation is less clear and further workis called for to better understand the intricate role oftensor terms of the effective nucleon-nucleon potential. ACKNOWLEDGMENTS
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