Novel feature of doubly bubble nuclei in 50 ≤ Z(N) ≤ 82 region along with magicity and weakly bound structure
M. Kumawat, G. Saxena, M. Kaushik, S. K. Jain, J. K. Deegwal, Mamta Aggarwal
aa r X i v : . [ nu c l - t h ] O c t October 20, 2020 1:3 WSPC/INSTRUCTION FILE upper-sd-shell
International Journal of Modern Physics E © World Scientific Publishing Company
Novel feature of doubly bubble nuclei in 50 ≤ Z(N) ≤
82 region alongwith magicity and weakly bound structure
M. Kumawat
Department of Physics, School of Basic Sciences, Manipal University Jaipur, Jaipur-303007,India
G. Saxena
Department of Physics (H & S), Govt. Women Engineering College, Ajmer-305002, [email protected]
M. Kaushik
S. S. Jain Subodh P. G. College, M. C. A. Institute, Rambagh Circle, Jaipur-302004, India
S. K. Jain
Department of Physics, School of Basic Sciences, Manipal University Jaipur, Jaipur-303007,India
J. K. Deegwal
Govt. Women Engineering College, Ajmer-305002, India
Mamta Aggarwal
Department of Physics, University of Mumbai, Kalina Campus, Mumbai-400098, [email protected]
Received Day Month YearRevised Day Month YearIn this work, we identify a unique and novel feature of central density depletion inboth proton and neutron named as doubly bubble nuclei in 50 ≤ Z(N) ≤
82 region. Themajor role of 2d-3s single-particle (s.p.) states in the existence of halo and bubble nucleiis probed. The occupancy in s.p. state 3s / leads to the extended neutron densitydistribution or halo while the unoccupancy results in the central density depletion. Byemploying the Relativistic Mean-Field (RMF) approach along with NL3* parameter,the separation energies, single-particle energies, pairing energies, proton, and neutrondensity profiles along with deformations of even-even nuclei are investigated. Our resultsare in concise with few other theories and available experimental data. Emergence on newshell closure and the magicity of conventional shell closures are explored systematicallyin this yet unknown region. Keywords : Relativistic mean-field plus BCS approach; 50 ≤ Z(N) ≤
82; Shell Closure;Bubble Structure; Weakly bound nuclei. 1 ctober 20, 2020 1:3 WSPC/INSTRUCTION FILE upper-sd-shell M. Kumawat et al.
1. Introduction
Remarkable advancements of various experimental facilities and salient develop-ment in accelerator and detection technology during the last three decades haveprompted the interest to examine various exotic properties viz. new shell closure,weakly bound structure or halo, bubble structure, etc. in the obscure domain ofperiodic chart.
As a result, confirmation of new magicity in sd-shell at neutronnumber N=14,
N=16 and proton number Z=16 as well as in pf-shell atN=32 and N=34 is observed. Interestingly, the break down of conventionalmagicity N=8, 20, 28, etc. is also reported by various experimental investiga-tions. Besides these exotic features, the existence of extended density distributionin C is also observed by recent reaction cross-section measurements. Indeed, ahalo structure is suggested due to the last two neutrons predominantly in the 2s / orbit for C with very weak binding.
20, 21
After recent interaction cross-sectionstudy by Togano et al. and using low-energy limit of neutron-neutron interactionby Suzuki et al. , halo structure in C and O have been confirmed. The roleof 2s / orbit is found crucial as well indicating proton bubble structure in Si inthe sd shell region. Recently, these important features in sd-shell nuclei and pf-shellnuclei are demonstrated using the relativistic mean-field approach.
25, 26
In anticipation with the similar characteristics of 3s / orbit, as mentioned abovefor 2s / orbit, the present theoretical study for nuclei between 50 ≤ Z(N) ≤
82 whichare mainly governed by 2d-3s single-particle states is proposed and aimed. To thebest of our knowledge, this kind of systematic study of these nuclei covering nu-clei between 50 ≤ Z(N) ≤
82 considering exotic phenomenon like new magicity, thedisappearance of conventional magicity, weakly bound structure or halo and withdoubly central depletion as well. However, it is worthy to mention here that protonbubble is indeed visible in
Hg due to substantial role of 3s / orbit as per rela-tivistic mean-field (RMF) approach. It has also been shown that the semibubblestructure in , Hg persists not only in the ground state but also in their ex-cited states. On the other side, the phenomena of giant halo due to filling inof 3s / orbit in neutron-rich even-Ca isotopes is already investigated by relativis-tic continuum Hartree-Bogoliubov (RCHB), non-relativistic Skyrme Hartree-Fock-Bogoliubov (HFB) calculations and also by RMF+BCS approach using TMA andNL-SH parameter. This particular region is also very important for the astrophysical purpose andthere have been few studies which target mainly the astrophysical implication ofthese nuclei. For example, the optical potential was generated folding the nucleardensity with the microscopic nuclear interaction DDM3Y to study low-energy pro-ton reactions for different nuclei in the A ≈ In addition, themass region ranging from A ≈
74 to 196 is explored to calculate astrophysical S-factor for 36 known p-nuclei with (p, γ ) reactions at low energy taking sphericaldensities from RMF calculations. On the other side, a number of studies havebeen completed in the neighborhood of the N=82, A=130 r-process peak. Inctober 20, 2020 1:3 WSPC/INSTRUCTION FILE upper-sd-shell
Novel feature of doubly bubble nuclei in 50 ≤ Z(N) ≤
82 region particular, the first-ever study of the level structure of the waiting-point nucleus Pd and
Pd has been indicated that the shell closure at the neutron numberN=82 is fairly robust.Looking towards the importance of this region between 50 ≤ Z(N) ≤
82 which ex-hibits varieties of exotic features and still scarce with theoretical and systematictreatment, has invoked this present theoretical study. Therefore, in this communi-cation, we describe ground-state properties of even-even nuclei in which neutron orproton lies between Z(N)=50 and 82. To probe magicity, weakly bound structure,and central depletion (bubble structure) we examine various properties viz. bindingenergies, single-particle energies, deformations, separation energies, pairing energiesas well as density distributions, etc. For our calculations, we use Relativistic Mean-Field (RMF) approach with NL3* parameter. We also compare our resultsalong with available experimental data and other theories to testify our outcomes.
2. Theoretical Frameworks
We employ relativistic mean-field approach
25, 26, 39–50, 57 together with a realisticmean-field, which has proved to be very useful and a successful tool as shown inour earlier work .
25, 26, 57
We use the model Lagrangian density with nonlinear termsboth for the σ and ω mesons as described in detail in Refs.
43, 49, 58
For the pairinginteraction, we use a delta force, i.e. V=-V δ ( r ) with the strength V =350 MeVfm same as has been used in Refs.
25, 26, 49, 57 for the description of drip-line nuclei.Based on the single-particle spectrum calculated by the RMF, we perform a statedependent BCS calculations.
59, 60
This approach has proven to be very successfulfor recent extensive study of (i) conventional and new magic nuclei ,
25, 26, 50, 57, 61–64 (ii) describing interdependence of 2p-halo with 2p-radioactivity and (iii) Bubblestructure .
25, 66
Another formalism used for the present study is the triaxially deformed Nil-son Strutinsky (NS) model which treats the delicate interplay of macroscopic bulkproperties of nuclear matter and the microscopic shell effects and has been usedextensively in our earlier works.
67, 68
We evaluate binding energy, separation en-ergy, deformation, and shape by incorporating macroscopic binding energy BE
LDM obtained from the LDM mass formula to the microscopic effects arising dueto nonuniform distribution of nucleons through the Strutinsky’s shell correction δ E shell along with the deformation energy E def obtained from the surface andColumb effects. The detailed description of these theoretical formalisms that have been ade-quately described in our earlier works has not been given in this paper. (Readersmay refer
43, 49, 58, 71 for detailed description of RMF approach and
67, 68 for NSMformalism).ctober 20, 2020 1:3 WSPC/INSTRUCTION FILE upper-sd-shell M. Kumawat et al.
54 56 58 60 62 6481012141618 Z=58(e)
Z N=82
42 48 54 60 66 72-3036912 N=50(d)
N Ni
N=7054 60 66 72 78 84-6-4-20246810 N=70 N=82(a)
N Se
68 72 76 80 84 884681012141618
NL3* TMA RCHB FRDM NSM Expt.
N=82 (c)(b)
N Sn
80 82 84 86 88 90161820222426283054 56 58 60 62 642832364044 S ( M e V ) Z=58(f)
ZN=126
32 36 40 44 48 52-6-4-20246810 Breaking of N=50N=40 S ( M e V ) S ( M e V ) S ( M e V ) S ( M e V ) N Ca S ( M e V ) Figure 1. (Colour online) Two neutron separation energy (S n ) and two proton separation energy(S p ) of various isotopes/isotones.
3. Results and discussions
The 2d-3s shell in the 50 ≤ Z(N) ≤
82 region plays a major role in the formationof halo and bubble like structures and even influences the magicity. Since theseexotic phenomena of halo, bubble and magicity at drip lines have not been muchexplored in this region, we make a systematic study in this work. As per shell modelconvention, 2d-3s shell consists of single-particle states 2d / , 3s / and 2d / sand-wiched between 1g / and 1h / states. This region consisted of two conventionalmagic numbers 50 and 82 whereas two new magic numbers 58 and 70 have beenspeculated to exist. For instance, the measured first excitation energy is foundslightly higher in
Ce than in the surrounding N=82 isotones
Ba and Nd which is one of the indicator of magic character of Z=58, which has been shownin another work to have a sub-magic character at Ce by self-consistent mean-field calculations with the M3Y-P6 interaction. A new magic number has beenreported at N=70 using RMF formalism in the coordinate space in Ni nuclei.To explore the possibility of these new magic numbers, we have plotted twoneutron separation energy (S n ) and two proton separation energy (S p ) of a fewselected isotopes/isotones in Fig. 1. These energies are calculated using the NL3*parameter of the RMF approach and compared with the similar RMF calcula-tions done with TMA parameter. Moreover, we have also calculated S n and S p using the NSM approach
67, 68 which is found in a very good match with our RMFcalculations. We also compare data with that of FRDM, RCHB along with ex-perimental data for the comparison. From Fig. 1(a) and (b) a sharp drop in S n after N= 82 is observed in Se and Sn isotopes which indicates strong magicity ofctober 20, 2020 1:3 WSPC/INSTRUCTION FILE upper-sd-shell Novel feature of doubly bubble nuclei in 50 ≤ Z(N) ≤
82 region N= 82 in accord with the study by Watanabe et al. which stated that the shellclosure at the neutron number N=82 is fairly robust. Fig. 1(c) shows a sharp dropin S n after N=40 in Ca isotopes which indicates a new magic number N=40 ( Ca)at drip line whereas the similar steep decline in S n value is missing in the conven-tional magic number N=50 which is a sign of breaking the conventional magicityof N=50 at the neutron drip-line. Such kind of appearance of new shell closureat N=40 and disappearance of conventional shell closure at N=50 is an importantoutcome for experimental studies such as the recent study done by Tarasov et al. in which discovery of Ca is reported and Ca is anticipated to be a drip-lineof Ca isotopes. However, magicity of N=50 remains strong in Ni (Fig. 1(d)) inaccord with our earlier result. Ni isotopes also exhibit magicity in N=70 with asharp drop in S n after N=70 in agreement with the results of the magicity of N=70using RMF formalism in the coordinate space. Also Se isotopes show significantmagicity at N=70 (Fig. 1(a)). Another new magic number is predicted at Z=58 asseen in the plots of S p vs. Z for N=82, 126 isotones in Fig. 1(e) and (f) inline withRefs. Therefore, Fig. 1 shows the emergence of two new magic numbers Z=58,a new proton magic number and N=70, a new neutron magic number.
32 36 40 44 48 52 56 60 64 68 72 76-16-12-8-40 (c) P a i r i ng E n e r gy ( M e V ) N=82
Proton P. E. Neutron P. E.
28 32 36 40 44 48 52 56 60 64 68-24-16-80 (b)
N=70
Proton P. E. Neutron P. E.
20 24 28 32 36 40 44 48 52-12-8-40 Z (a) Proton P. E. Neutron P. E.
N=50
Figure 2. (Colour online) Neutron and proton pairing energy of isotones with neutron numberN=50, 70 and 82.
It is well-known
49, 62 that the pairing energy vanishes for the magic nuclei andtherefore the analysis of pairing energy offers evidence for the existence of magicityor the shell closures. For this analysis, we show proton and neutron pairing energyin Fig. 2 for the isotonic chain with N=50, 70, and 82. In the case of N=50, themagicity with zero neutron pairing is seen in Fig. 2(a) for mostly isotones except Ca and Ti, in which N=50 does not remain magic and shows the breakdown orctober 20, 2020 1:3 WSPC/INSTRUCTION FILE upper-sd-shell M. Kumawat et al. weakening towards neutron drip-line in accordance with the study done by Yadav et al. Fig. 2(b) shows magic behaviour for N=70 for neutron-rich isotones withZ=28-38 where neutron pairing energy is zero. The neutron pairing energy is foundzero for all the isotones for the case of N=82 as seen in Fig. 2(c) which pointstowards a very strong magicity of N=82 for the full isotonic chain. From protonpairing energy curves in all the panels, one can see that the proton pairing energyfalls in between the two closed shells which indicate non-magic nuclei. Wheneverproton pairing energy is zero along with zero neutron pairing energy, one gets adoubly magic nucleus. The pairing energy increases from zero at shell closure toa maximum value in the mid shell nuclei showing pairing interaction. The protonpairing energy vanishes for Z=20, 28 and 50 in all the plots and we find doublymagic nuclei Ni and
Sn in N=50 isotones, Ni in N=70 isotones, and
Snin N=82 isotones. In addition, the kink due to lower pairing energy comparative totheir neighbourhood isotopes/isotones e.g. at Z=34, Z=40, and Z=58 in Fig. 2(a),(b) and (c), respectively, hint towards the new sub-shell closure in this region.
100 110 120 130 140 150 160 170 180 190-30-20-100 (c)
Z=82
Proton P. E. Neutron P. E. P a i r i ng E n e r gy ( M e V )
60 70 80 90 100 110 120 130-24-16-80 (b)
Proton P. E. Neutron P. E.
Z=58
40 50 60 70 80 90 100 110 120-15-10-50 N (a) Proton P. E. Neutron P. E.
Z=50
Figure 3. (Colour online) Neutron and proton pairing energy of isotopes with proton numberZ=50, 58 and 82.
In Fig. 3, we have shown the pairing energy of the isotopic chains of Z=50,58, and 82. The Z=50 and 82 (Sn and Pb) isotopes are found very strongly magicthroughout the chain as can be seen from Fig. 3(a) and (c). The neutron pairingcontribution of these isotopes shows , Sn and
Pb as the strong doubly magiccandidates with zero neutron and proton pairing energies. We also find new doublymagic nuclei
Sn and
Pb with the emerging new neutron magic number N=112and 184, respectively. As far as the isotopic chain of Z=58 is concerned (shown inFig. 3(b)), it does not show zero pairing energy for most of the isotopes unlikectober 20, 2020 1:3 WSPC/INSTRUCTION FILE upper-sd-shell
Novel feature of doubly bubble nuclei in 50 ≤ Z(N) ≤
82 region Sn and Pb isotopes. However, the proton pairing energy is either close to zero ortowards the lower side for almost throughout the chain than the neutron pairingenergy trend. This behaviour of Z=58 indicates this proton number as a sub-shellclosure and shows a doubly magic character at N=126 in
Ce as is seen in Fig.3(b) where both pairing energy contributions vanish. -20-15-10-50 (b)
RMF(NL3*)
N = 70 Z N e u t r on S . P . E n e r gy ( M e V ) (c) N = 82 N = 50 (a) Figure 4. (Colour online) Neutron single-particle energies of neutron number N=50, 70 and 82isotones.
In order to probe more into magicity, we display neutron single-particle statesof isotones with neutron number N=50, 70, and 82 in Fig. 4(a), (b) and (c), respec-tively. For N=70, we have shown s.p. states of two isotones Ni and
Sn whichhave been found to be spherical in our calculations. In Fig. 4(a), we observe a sig-nificant energy gap between 1g / and 1g / states leading to N=50 shell closurefor proton-rich doubly magic Sn. But this gap decreases from 7 MeV (at
Sn)to 2 MeV (at Ca) as one moves towards proton deficient side. Therefore, theN=50 shell closure weakens for the nuclei with Z ≈
20 ( Ca, Ti, etc.) as is alsoindicated by the pairing energy systematics (Fig. 2(a)). On the other side, in Fig.4(b), the development of a new single-particle gap and hence a new shell closureN=70 is observed while moving towards proton-rich N=70 isotones. This gap arisesbetween the 2d-3s shell and 1h / state which goes to a maximum value of 5.6MeV for Z=28 leading to doubly magic nucleus Ni. For N=82 isotones, from Fig.4(c), it is clear that the gap between 1h / and 1h / remains significantly largearound 4-7 MeV ensuing N=82 with strong magic character.To look for the proton magicity in proton 2d-3s shell nuclei, we show protonsingle-particle states of Z=50, 58, and 82 (Sn, Ce, and Pb) isotopes in Fig. 5(a), (b)and (c), respectively. Fig. 5(a) shows the complete chain of Sn isotopes where thectober 20, 2020 1:3 WSPC/INSTRUCTION FILE upper-sd-shell M. Kumawat et al. (b) P r o t on S . P . E n e r gy ( M e V ) Z = 58 Z=50 (a)
80 90 100 110 120 130 140 150-12-60 (c)
Z = 82 N Figure 5. (Colour online) Proton single-particle energies of proton number Z=50, 58 and 82 iso-topes. gap between proton 1g / and 1g / states persists upto a value of ∼ ≥ / and 2d-3s shell which may give rise to (sub)shell closure at Z=58.Here we have shown two isotopes Ce and
Ce from the Z=58 isotopic chainswhich are found spherical in our calculations in accord with almost zero deformationin them as predicted by our calculations using NSM
67, 68 and data from FRDM, HFB and WS4 available experimental data. These two isotopes show themagic character among Z=58 isotopes in concise with.
72, 73
From Fig. 5(c), thestrong magicity of Pb isotopes (Z=82) is clearly visible and hence we ascertain thestrong magic character of N and Z=82.To get more insight into the weakening of magicity at N=50 and the evolutionof the new magic number at Z=58, we display a density distribution of N=50 andN=126 isotones. One can see from Fig. 6(a) that while moving from the neutrondeficient nuclei
Sn (Z=50) towards neutron richer side Cr (Z=24) and then tovery neutron-rich Ca, neutron density distributions show sharp fall for neutrondeficient
Sn, Ni, up to Cr showing the confinement of the distribution upto smaller distances which indicate the neutron magic character of these isotoneswhich gradually starts extending to longer tails for Ca and Ti. This extendeddistribution at Ca and Ti nuclei prove the disappearance of N=50 shell closureat the neutron drip line side. As mentioned above, this observation of the disap-pearance of shell closure at Ca be very useful and important that can also providean additional impulse for further experimental study after the recent discovery of Ca and interpretation on Ca. From Fig. 6(b), interestingly, the proton den-sity distribution of Z=58 is found to falls off at a smaller distance as comparedctober 20, 2020 1:3 WSPC/INSTRUCTION FILE upper-sd-shell
Novel feature of doubly bubble nuclei in 50 ≤ Z(N) ≤
82 region Cr Ni Sn Ti Ca Z=20 Z=22 Z=24 Z=26 Z=28 Z=50
N = 50
Neutron Density (a) (b)
Z=50 Z=52 Z=54 Z=56 Z=58 D e n s it y ( f m - ) N = 126
RMF(NL3*)
Proton Density
Radius (fm)
Figure 6. (Colour online) Variation of (a) neutron density for N=50 isotones and (b) proton densityfor N=126 isotones with respect to radius. to the other nuclei even with smaller Z. This characteristic brings in the sub-shellcharacter of Z=58 in the nuclei of 2d-3s shell. Sn Gd Radius (fm) N e u t r on D e n s it y ( f m - ) Z=28 Z=50 Z=64
N= 70
RMF(NL3*) Ni Figure 7. (Colour online) Variation of neutron density for N=70 isotones with respect to radius.
To investigate the magicity of N=70 in further details, we show the neutrondensity distribution of isotones of N=70 in Fig. 7. The sharp fall in the neutrondensity of isotones of N=70 shows its magic character. But the density distributionof Ni shows little more spread than other isotones although the energy gap (asseen in Fig. 4) between s.p. states (s-d and h / ) causing the shell closure at N=70is maximum among the other N=70 isotones. To explain this discrepancy, we closelyctober 20, 2020 1:3 WSPC/INSTRUCTION FILE upper-sd-shell M. Kumawat et al. examine the neutron single-particle states along with their occupancy and pairinggap in Fig. 8. As is well known now,
29, 30 that if any low angular momentum stateespecially s-states ( ℓ =0) get occupied by the last filled particles near the Fermilevel, then a large spatial extension of the density distribution is seen because of nocentrifugal barrier of s-state. In case of Ni, the 3s / state get occupied fully byfew of the last neutrons and hence the neutron density displays significant spatialextension due to zero centrifugal barrier. (a) g a p ( M e V ) O cc up a ti on P r ob a b ilit y Occupancy of 3s (b)
Occupancy of 1h
N=70
28 32 36 40 44 48 52 56 60 640.00.81.6
Pairing gap of 1h Z (c) Figure 8. (Colour online) Variation of (a) Occupancy of 3s / , (b) Occupancy of 1h / , and (c)Pairing gap of 1h / with respect to Z are shown. For other isotones of N=70 (
Sn and
Gd) shown in Fig. 7, the occupancyof 3s / state reduces and that of the higher angular momentum state (1h / )increases. The changing occupancies in 3s / and 1h / with increasing Z has beenshown in a very systematic manner in Fig. 8(a) and (b). This figure shows thevariation of occupancy of proton 3s / and 1h / along with the pairing gap of1h / . The occupancy of 3s / is maximum for Ni (Z=28) whereas 1h / doesnot occupy any particle (Fig. 8(a) and (b)). With increasing Z, in particular, afterZ=40, the pairing gap of 1h / (Fig. 8(c)) starts increasing gradually which is asignature of the interaction with lower bound states. Such increased pairing givesrise to higher occupancy in 1h / and the occupancy in 3s / decreases graduallyand becomes a minimum for Gd (Z=64) which is evident in Fig. 8(a). Thisfigure explains the structural aspects of N=70 isotones due to deviation in the s.p.state spectrum. As Z increases, occupancy in 3s / decreases and that in 1h / increases. Occupancy in higher s.p. state results in more of pairing interaction. Thecentrifugal barrier for the last filled valence 1h / state being a higher ℓ (=5) stateis higher which leads to smaller density distribution in (Z=40-64) nuclei. Thereforectober 20, 2020 1:3 WSPC/INSTRUCTION FILE upper-sd-shell Novel feature of doubly bubble nuclei in 50 ≤ Z(N) ≤
82 region the density distribution curve (Fig. 7) is the outcome of the combined effect ofoccupancy in (3s / ,1h / ), pairing energy and the centrifugal barrier.To elaborate more on the difference in the density distribution of Gd and Ni, we have plotted wave-functions of 2d-3s states along with wave-function of1h / in Fig. 9(a) and (b), respectively. The RMF potential energy which is a sumof the scalar and vector potentials is also shown for both the nuclei. A close watchof Fig. 9 gives us a clear picture of 3s / state which is spread over outside thepotential region for Ni causing the poorer overlap with the bound states nearthe Fermi surface and subsequently leading to a larger spatial extension of neutrondensity. On the other hand, for
Gd, wave-function of both 3s / and 1h / statesare confined within the potential range of about 8 fm. Therefore, N=70 stronglypresents its candidature as a new neutron magic number with a unique nucleus Niwhich is a nucleus with doubly magic character (see Fig. 2) and having a weaklybound or halo-like structure (see Fig. 7) at the same time which is reported herefor the first time and is one of the highlights of this work. (b)
Nuclear potential N u c l ea r po t e n ti a l ( M e V ) W a v e f un c ti on Ni Neutron Single Particle states Neutron Single Particle states
Radius (fm) (a) Nuclear potential Gd Figure 9. (Colour online) The RMF potential energy (right scale) and wave-function (left scale)of 2d-3s states and 1h / for the case of Ni and
Gd in N=70 isotones.
On one side, as seen in the case of Ni that the occupancy of s-state (3s / )( ℓ =0) leads to weakly bound or halo-like structure (seen in Fig. 7) whereas on theother side, unoccupancy of s-state may lead to the ”Bubble Structure”. The de-pletion of central density comparative to its value at other radial distance leads tothe phenomena of bubble-like structure. This phenomenon was recently observedin Si and studied by many theoretical works
27, 81–90 along with our recentwork
66, 91, 92 which have established Si as a best candidate showing bubble orcentral depletion in its charge density distribution. Encouraging with these recentctober 20, 2020 1:3 WSPC/INSTRUCTION FILE upper-sd-shell M. Kumawat et al. studies, we have analyzed density distribution at the center for all the isotones ofN=50, 70, 82 and 126 which may be influenced by occupancy in 3s / state.To search for the exotic phenomenon of bubble structure in 2d-3s shell nuclei,we have shown proton and neutron densities of some selected nuclei which are foundwith significant central depletion in Fig. 10. We identify the nuclei Sn,
Gd,
Dy and
Yb as bubble nuclei among the N=50, 70, 82 and 126 isotonic chains.
DF=0.22DF=0.0 n p (d) DF=0.28DF=0.15 pn (c) (b) DF=0.31DF=0.21 pn Gd Yb Dy DF=0.15DF=0.21 p Sn D e n s it y (f m - ) n (a) Radius (fm)
Figure 10. (Colour online) Proton and neutron densities of
Sn,
Gd,
Dy and
Yb fromN=50, 70, 82 and 126 isotonic chain, respectively.
To quantify the depletion of the central density, we have calculated depletionfraction (DF) (( ρ max - ρ c ) /ρ max )
66, 82 which is also mentioned in the Fig. 10. Themost remarkable observation from this figure is that the nuclei
Sn and
Gdare showing the depletion in both the proton and neutron densities which predictthe double bubble structure. These two nuclei along with
Yb also qualify to bethe candidates of doubly bubble nuclei. In
Sn and
Gd nuclei, the s.p. state3s / remains empty for both protons and neutrons whereas for the case of Ybthe proton depletion is due to empty 3s / and neutron depletion is due to empty4s / . In Fig. 10(d), we note that in case of N=82, all the neurons keep 3s / completely filled and therefore no neutron depletion arises in Dy contrary toproton depletion in which Z=66 observes vacant 3s / proton state. The values ofDF mentioned in the figure also explain this feature. From these chains of isotones,it is found that many of the nuclei possess the central depletion in their proton orneutron or both the densities.For a systematic study of both depletion, we have displayed proton and neutronctober 20, 2020 1:3 WSPC/INSTRUCTION FILE upper-sd-shell Novel feature of doubly bubble nuclei in 50 ≤ Z(N) ≤
82 region
30 40 50 60 700.00.10.20.3 (b)
N=70
40 50 60 70 800.00.10.20.3 (c)
Proton DF Neutron DF
N=82
50 60 70 800.00.10.20.3 (d) D e p l e ti on fr ac ti on ( D F ) N=126 Z
20 30 40 500.00.10.20.3
N=50 (a)
Figure 11. (Colour online) Proton and neutron depletion fraction (DF) (( ρ max - ρ c ) /ρ max ) forN=50, 70, 82 and 126 isotonic chain. depletion fraction (DF) for a full isotonic chain of N=50, 70, 82, and 126 in Fig.11. Doubly bubble character is clearly observed for many of the isotones for N=50,70, and 126 in Fig. 11(a), (b) and (d), respectively. In N=82 isotonic chains (Fig.11(c)), zero neutron DF provides evidence of the contribution of 3s / state inbubble formation, which is completely occupied hence the central depletion is zero.One interesting fact which can be seen in Fig. 11(b), (c), and (d) is that DF increasesabruptly (though there is only a hint) at magic number Z=40 and 50 as compared totheir neighbourhood nuclei. In a similar manner, from Fig. 11(a) an abrupt increasein proton DF is found for Z=34 which might be correlated to new proton magicityin Z=34. It is essential to note here that all these hints of magicity on the basis ofDF are very preliminarily and need further investigation and more systematic studywhich is left for our more comprehensive subsequent work on bubble structure. Onthe other side, a fall in DF can also be envisioned from the figure of N=82 and 126while reaching towards Z=82, due to filling of proton 3s / state.To elaborate more about the dependence of proton depletion on the occupancyof proton 3s / state, we have shown a few systematics of N=126 isotones in Fig. 12.Fig. 12(a) shows the variation of proton DF for full chain of N=126 isotones whichis high for Z=50-78 and suddenly drops at Z=82 where the occupancy of 3s / isfull. Fig. 12(b) depicts occupation probability of proton single-particle states 3s / and 1h / as a function of Z. As Z increases the occupancy of 3s / and 1h / states increase gradually. Although, it seems from Fig. 12(b) that the probabilityof occupying 1h / is always lesser than 3s / state but in terms of occupyingctober 20, 2020 1:3 WSPC/INSTRUCTION FILE upper-sd-shell M. Kumawat et al.
50 60 70 800.00.10.20.3 0.0 0.3 0.6 0.90.00.10.20.3 50 60 70 800.00.30.60.9 0.0 0.3 0.6 0.90.00.10.20.3
32 40 48 56 64 72 800.00.20.40.60.81.0 Z D F D F (a) N=126 D F Z Proton deplition fraction (c)
Proton deplition fraction
N=126
Occ. Proability of 3s (b)
N=126 O cc . P r o a b ilit y ( p r o t on ) (d) Proton deplition fraction
N=126
Occ. Proability of 1h
Figure 12. (a) Proton depletion fraction (DF) (( ρ max - ρ c ) /ρ max ), (b) occupation probabilityof 3s / and 1h / states are shown for N=126 isotonic chain. Variation of DF with respect tooccupation probability of (c) 3s / and (d) 1h / states is also shown. the total number of particle the 1h / always has more particle than 3s / forZ >
58. It is worthy to mention here that in addition to the necessary condition ofunoccupied s-state for the bubble formation, the s-state should be surrounded bylarger ℓ states which leads to weaker dynamical correlations consequently enhancesbubble effect. The variation of DF in N=126 isotones indeed fulfills the abovecriteria and therefore appears with maximum proton DF rather than N=50, 70, and82 isotones as can be seen from Fig. 11. The influence of occupancy of both 3s / and 1h / states on proton DF can be easily seen from Fig. 12(c) and (d). As soonas both the states get filled completely for Z=82, the depletion disappears. Hence,both states simultaneously determine central depletion in N=126 isotones. However,after a close watch the influence of 3s / state can be seen more dominating andwider as compared to 1h / state.While discussing the bubble phenomenon in medium or heavy mass region, therole of Coulomb repulsion becomes substantial. The impression of Coulomb repul-sion on bubble formation has been discussed in , Hg. We also emphasized therole of Coulomb forces in superheavy region in our earlier works.
66, 91
To view theeffect of Coulomb repulsion in this region of interest (50 ≤ Z(N) ≤ ≤ Z(N) ≤ Novel feature of doubly bubble nuclei in 50 ≤ Z(N) ≤
82 region
50 60 70 800.000.050.100.150.200.25 A v er ag e D e p l e t i o n F r a c t i o n ( D F ) Particle Number (N or Z)
Proton DF Neutron DF
Figure 13. (Colour online) Average proton and neutron depletion fraction (DF) for the considerednuclei in the range 50 ≤ Z(N) ≤ indicator of effect of Coulomb repulsion. A more detail analysis of effect of Coulombrepulsion on bubble structure would be reported in our subsequent works.
4. Conclusions
Various exotic features are probed in a less-known region which connects two con-ventional and strong magic numbers: Z(N)= 50 and 82. This realm is influenced bythe 2d-3s shell (sometimes referred to as second sd-shell) which leads to new magic-ity, the disappearance of conventional magicity, weakly bound or halo-like structure,and bubble structure. For this investigation, we have used relativistic mean-fieldapproach along with the NL3* parameter to calculate deformation, binding ener-gies, separation energies, pairing energies, single-particle energies as well as densitydistributions, etc. The results are compared with one more calculation using triax-ially deformed Nilson Strutinsky model (NSM) and also FRDM, RCHB along withavailable experimental data and are found in excellent agreement. In this region,N=82 and Z=50, 82 are established as very strong magic nuclei whereas N=50 isfound to disappear towards neutron drip-line for Ca. In addition to this, N=70and Z=58 are found to bear witness to new magicity or sub-shell closure. Amongisotones of N=70, Ni came across with unique behaviour as it is characterizedby its double magicity and weakly bound halo-like structure, simultaneously. Thisweakly bound structure is observed due to occupancy in 3s / state: the state whichalso leads to central depletion in many considered nuclei, if empty. As a very im-portant consequence, many double bubble nuclei are predicted with Sn,
Gdand
Yb as the most effective examples.ctober 20, 2020 1:3 WSPC/INSTRUCTION FILE upper-sd-shell M. Kumawat et al.
5. Acknowledgement
Authors G. Saxena and Mamta Aggarwal acknowledge the support provided bySERB (DST), Govt. of India under CRG/2019/001851 and WOS-A scheme, re-spectively.
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