Nuclear shape evolution and shape coexistence in Zr and Mo isotopes
Pankaj Kumar, Virender Thakur, Smriti Thakur, Vikesh Kumar, Shashi K. Dhiman
aa r X i v : . [ nu c l - t h ] J u l Nuclear shape evolution and shape coexistence in Zr and Moisotopes
Pankaj Kumar a,1 , Virender Thakur , Smriti Thakur , Vikesh Kumar ,Shashi K. Dhiman Department of Physics, Himachal Pradesh University, Summerhill, Shimla 171005, IndiaReceived: date / Accepted: date
Abstract
The phenomena of shape evolution and shapecoexistence in even-even − Zr and − Mo iso-topes is studied by employing covariant density func-tional theory (CDFT) with density-dependent point cou-pling parameter sets DD-PCX and DD-PC1, and withseparable pairing interaction. The results for rms devia-tion in binding energies, two-neutron separation energy,differential variation of two-neutron separation energy,and rms charge radii, as a function of neutron number,are presented and compared with available experimen-tal data. In addition to the oblate-prolate shape co-existence in − Zr isotopes, the correlation betweenshape transition and discontinuity in the observablesare also examined. A smooth trend of charge radii in Moisotopes is found to be due to the manifestation of triax-iality softness. The observed oblate and prolate minimaare related to the low single-particle energy level densityaround the Fermi level of neutron and proton respec-tively. The present calculations also predict a deformedbubble structure in
Zr isotope.
Keywords
Covariant density functional · Shapecoexistence · Charge radii · Single-particle levels · Bubble nuclei
In nuclear physics, the study of exotic nuclei with largenumbers of protons or neutrons is one of the most ac-tive areas of research, both theoretically as well as ex-perimentally. The radioactive ion beam (RIB) facili-ties and sensitive detection technologies have extendedour knowledge of nuclear physics far away from the β -stability line. The study of nuclear shape evolution in a e-mail: [email protected] atomic nuclei is one of the fundamental quests in nu-clear physics. The polarizing effect of added nucleonsleads to a gradual increase of deformation in the nucleiaway from spherical shell closures. However, an abruptshape transition is seen in neutron-rich nuclei aroundA ≈
100 mass region. In the case of Zr isotopes, theground states from N = 50 up to N = 58 are weaklydeformed and a sudden shape transition is observedas N = 60 is approached. This sudden shape transi-tion in Zr isotopes is evident from the irregularitiesin the two neutron separation energies[1], mean-squarecharge radii[2,3], and excitation energies of 2 +1 statesand B ( E
2) values[4,5]. On the other hand, the shapetransition is rather gradual in Mo isotopes[1,6] showingcharacteristic signatures of triaxiality. This dramaticbehavior makes the mass region A ≈
100 of special in-terest to test various theoretical models.Nuclei at and near closed subshells are known toexhibit shape coexistence[7,8]. Neutron rich zirconium( Z = 40) and molybdenum ( Z = 42) isotopes are goodexamples of shape coexisting bands. The appearance oflow-lying 0 +2 states is a feature of shape coexistence andshape transition in this mass region. Many experimen-tal observations for shape transition and shape coexis-tence have been made in this mass region. The shapecoexistence is seen in closed-subshell Zr nuclei wherethe role of subshells for nuclear collectivity is suggestedto be important[9]. The possibility of shape coexistencehas been suggested in Zr and
Zr nuclei by the anal-ysis of yrast structure from the rotational band andtransition strength through lifetime measurements[8,10,11,12]. The shape coexistence in Mo isotopic chainwas studied experimentally by using the Coulomb exci-tation technique[13,14,15]. Very recently, J. Ha et al. [16]have studied the shape evolution of
Mo,
Mo, and
Mo isotopes through the energies and lifetimes of 2 +1 states. The quadrupole deformation parameter was ob-tained these measurements suggesting the deformationis almost unchanged N = 62 to N = 66 and slightlydecreases at N = 68.Theoretically, various attempts[17,18,19,20,21,22,23,24,25] have been made to study the phenomena ofshape evolution in Zr and Mo isotopes. The structuralevolution in even-even and odd-A neutron-rich Zr andMo isotopes is studied within self-consistent mean-fieldapproximation based on D1S-Gogny interaction[17]. Acorrelation between the isotopic shift of charge radiiand shape transition in these isotopes has been sug-gested. The triple shape coexistence specific for the 0 + states and the evolution of the shape coexistence andmixing in the neutron-rich N = 58 Zr isotopes are stud-ied within the complex excited VAMPIR approach [18].The shape evolution and shape coexistence phenom-ena in neutron-rich nuclei at N ≈
60 is studied usingthe framework the covariant density functional theory(CDFT) with PC-PK1 interaction[19]. They observeda rapid shape transition at N = 60 for Zr nuclei whilea smooth transition in Mo isotopes. Many other the-oretical studies with different formalisms are done tounderstand the evolution of structure in these nuclei,including the relativistic mean-field (RMF) model[20,21,22,23], the interacting boson model (IBM) model[24,25], and the self-consistent mean-field models with theSkyrme and the Gogny force[26,27,28,29].Nuclear density functional theories have been usingto understand nuclear many-body dynamics for appri-ciable description of nuclei near the drip lines[30,31,32,33,34]. Covariant density functional theory (CDFT) isone of the most attractive nuclear density functionaltheories based on energy density functionals. This the-ory has come out to be the most successful theoriesto describe the ground and excited-state properties ofboth spherical and deformed nuclei throughout the nu-clear chart. The CDFT with point-coupling interac-tion has attracted more interest with several param-eter sets, PC-PK1[35], DD-PC1[36] and DD-PCX[37].Among these parameter sets, the DD-PCX was pro-posed very recently. In this work, we have done a sys-tematic study of shape transition and shape coexistencefor − Zr and − Mo isotopes. We have used co-variant density functional theory with a density-dependentpoint coupling effective interaction. The calculationsare performed taking DD-PCX and DD-PC1 parame-ter sets together with the separable pairing force[38,39,40]. Systematic constrained calculations performed forshape evolution and other ground-state properties arecompared with experimental data[41,42,3,6] and withother theoretical models[21]. This paper is structured as follows: In Section. 2, webriefly described the outline of the theoretical frame-work for the relativistic point-coupling model with theseparable pairing interaction. The results of shape evo-lution and other ground-state properties of Zr and Moisotopes are presented in Section 3. Finally, the con-cluding remarks on the present analysis are given insection 4.
Self-consistent mean-field (SCMF) models provide a verysuccessful tool to study and analyze a variety of nu-clear structure phenomena throughout the entire nu-clear chart, from the valley of stability to exotic nu-clei. These models are based on nuclear energy densityfunctionals (EDF) that represents a unified approachto study the static and dynamic properties of finite nu-clei quantitatively. The nucleons are treated as indepen-dent particles moving inside the nucleus under the in-fluence of potentials derived from such functionals[43].The EDF is constructed as a functional of one-bodynucleon density matrices that correspond to a singleproduct state of single-particle states. This approachis analogous to Kohn-Sham density functional theory(DFT), which enables a description of quantum many-body systems in terms of a universal energy densityfunctional.2.1 Lagrangian density for the Point Coupling modelsThe relativistic mean-field representation is formulatedin terms of point-coupling nucleon-nucleon interactions,without the inclusion of meson fields. The medium de-pendence of the interaction can be taken into accountby terms of higher-order in the nucleon fields. The La-grangian for density-dependent point coupling modelsincludes the isoscalar-scalar, isoscalar-vector and isovector-vector four-fermion contact interactions in the isospace-space and can be written as (see Ref.[36,37,44] for moredetails), L = ψ ( iγ.∂ − m ) ψ − α S ( ρ )( ψψ )( ψψ ) − α V ( ρ )( ψγ µ ψ )( ψγ µ ψ ) − α T V ( ρ )( ψ τ γ µ ψ )( ψ τ γ µ ψ ) − δ S ( ∂ ν ψψ )( ∂ ν ψψ ) − eψγ.A − τ ψ, (1)where m is the mass of nucleon, α S , α V and α T V rep-resent the coupling constants for four-fermion contactterms. In addition to free-nucleon terms and point-coupling interaction terms, the Lagangian density in above equa-tion also includes the coupling of the protons to theelectromagnetic field. In Eq. (1), the derivative termsaccounts for the dominating effects of finite-range inter-actions and are necessary for a quantitative descriptionof nuclear properties.The microscopic density-dependent scalar and vectorself-energies are computed by using following functionalform of the couplings. α i ( ρ ) = a i + ( b i + c i x ) e − d i x , ( i = S, V, T V ) (2)where x = ρ/ρ sat denotes the nucleon density in sym-metric nuclear matter at saturation point ρ sat .The point coupling CDFT model involve 10 param-eters, ( a S , b S , c S , d S , a V , b V , d V , b T V , d
T V , δ S ), are pre-sented for DD-PCX and DD-PC1 in Table. 1. The pa-rameters for and PC-PK1 can be found in Ref.[35] Table 1
The parameters of density-dependent point-coupling DD-PCX[37] and DD-PC1[36] interactions in theLagrangian. The value of saturation density is set to 0.152fm − and mass of nucleon m = 939 MeVParameter DD-PCX DD-PC1 Units a S -10.979243836 -10.0432 fm b S -9.038250910 -9.1504 fm c S -5.313008820 -6.4273 fm d S a V fm b V fm d V b T V fm d TV δ S -0.878850922 -0.8149 fm ρ , and pairing densityˆ κ . The relativistic Hartree-Bogoliubov energy densityfunctional can be written as E RHB [ˆ ρ, ˆ κ ] = E RMF [ˆ ρ ] + E pair [ˆ κ ] , (3)where E RMF [ˆ ρ ] is the nuclear energy density functionaland is given by E RMF [ ψ, ψ, σ, ω µ , ρ µ , A µ ] = Z d r H ( r ) . (4)The pairing part of RHB functional is given by E pair [ˆ κ ] = 14 X n n ′ X n n ′ κ ∗ n n ′ (cid:10) n n ′ (cid:12)(cid:12) V P P (cid:12)(cid:12) n n ′ (cid:11) κ n n ′ , (5)where (cid:10) n n ′ | V P P | n n ′ (cid:11) are the matrix elements of thetwo-body pairing interaction. and indices n , n ′ , n , and n ′ denote quantum numbers that specify the Dirac in-dices of the spinor.The pairing force is separable in momentum space andin r-space has the form of V P P ( r , r , r ′ , r ′ ) = − Gδ ( R − R ′ ) P ( r ) P ( r ′ ) , (6)where R = √ ( r + r ) and r = √ ( r − r ) represent thecenter of mass and the relative coordinates, respectivelyand the form factor P ( r ) is of Gussian form written as, P ( r ) = 1(4 πa ) / e − r / a . (7)The pairing force has a finite range and it also conservestranslational invariance due to the presence of the fac-tor δ ( R − R ′ ). Finally, the pairing energy in the nuclearground state is given by E pair = − G X N P ∗ N P N . (8) − Zr and − Mo isotopesShape is one of the most fundamental properties of nu-clei and the quadrupole deformation parameter reflectsthe shape of the nuclei. The classification of deformednuclei depends on the value of the quadrupole deforma-tion parameter ( β ). A positive value of β correspondsto the prolate shape and the negative value to an oblateshape, while β = 0 corresponds to spherical shape nu-clei.Fig. (1) displays the potential energy curves (PECs)of even-even − Zr isotopes as a function of quadrupole deformation parameter β , calculated using CDFT withDD-PCX and DD-PC1 parameters as given in Table.1. The energies in the PECs are normalized to the to-tal energy of global minima. These calculations providethe results for the ground state of the nuclei. The cal-culations are performed systematically by taking con-strained axial symmetry mapped by the quadrupoledeformation parameter β . In both interactions, theisotopes with N = 48 - 52 shows a spherical minimawhich become shallow at N = 54, with -0.2 ≤ β ≤ N = 56. Mean-while, the prolate minima become deeper and the bar-rier height starts increasing on moving from Zr to Zr. In case of DD-PCX interaction, a rapid increasein deformation parameter is observed as one move from Zr ( β = 0 . Zr ( β = 0 . Zr, the ground state is seen to be stabilized withan almost constant prolate deformation, i.e, β ∼ N = 70. A sudden shape transition, from prolateto oblate, is seen at N = 72. On the other hand, the de-formation parameter remains almost same for − Zrisotopes in case of DD-PC1 interaction. A sudden shapetransition is observed on moving from
Zr (prolate) to
Zr (oblate) and the ground state remain oblate from N = 64 to N = 76 with a constant deformation around β ≈ − Zrisotopes. The spherical shape is again restored on ap-proaching neutron shell closure at N = 82. Thus, a dra-matic shape transition is observed in the Zr isotopesbetween two shell closures at N = 50 and N = 82.Fig. (2) presents the PECs of even-even − Moisotopes as a function of β , with axially constrainedCDFT calculations. The shape evolution for Mo iso-topes is similar to that of Zr isotopes. The potentialenergy minima in PECs shows a spherical shape for Moisotopes with N = 48 - 52 and prolate minima around ∼ N = 54 - 56. In the meantime, the oblate min-ima start competing with the prolate minima and be-come deeper, with a small energy difference. In Fig. (2),a clear oblate-prolate shape coexistence is observed for − Mo isotopes for both parameter sets. On movingalong the isotopic chain, there observed a single groundstate oblate minima in − Mo isotopes. The shapeagain becomes spherical on reaching the neutron shellclosure N = 82. However, it has seen in the literaturethat many theoretical models predict a triaxial groundstate between N = 58 and N = 68 in Mo isotopes. Also,the experimental charge radii have reproduced by tak-ing the degree of triaxiality into account. Consideringthis fact, we have performed constrained triaxial calcu-lations, mapped by triaxial parameter, γ , with a fixedvalue of β , in − Mo isotopes.
DD-PCXDD-PC1 - . - . . . - . - . . . - . - . . . - . - . . . - . - . . . E ( M e V ) β Zr Zr Zr Zr Zr Zr Zr Zr Zr Zr Zr Zr Zr Zr Zr Zr Zr Zr Zr Zr Fig. 1 (Color online) The potential energy curves (PEC),calculated using CDFT with DD-PCX and DD-PC1 interac-tions, as function of β for − Zr isotopes. The energiesare normalized with respect to the binding energy of the ab-solute minima.
DD-PCXDD-PC1 - . - . . . - . - . . . - . - . . . - . - . . . - . - . . . E ( M e V ) β Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Mo Fig. 2 (Color online) The potential energy curves (PECs),calculated using CDFT with DD-PCX and DD-PC1 interac-tions, as function of β for − Mo isotopes. The energiesare normalized with respect to the binding energy of the ab-solute minima.
Figs. (3 & 4) show the PECs of − Mo isotopesas a function of triaxial deformation parameter, γ , for afix value of β . The magnitude of β values are taken asthe minima in PECs in Figs. (1 & 2). It can be observedfrom the figures that, the PECs show a triaxial minimaaround γ ≈ o and an oblate minima γ ≈ o for thegiven value of β . In the present study, the triaxial min- β =0.250 β =0.220 β =0.370 β =0.230 β =0.380 β =0.230 β =0.380 β =0.230 γ (in degree)0123456 β =0.380 β =0.230 γ (in degree) β =0.225 E ( M e V ) Mo Mo Mo Mo Mo Mo E ( M e V ) Fig. 3 (Color online) The potential energy curves (PECs) for − Mo isotopes, calculated using CDFT with DD-PCXparameter set, as function of triaxial deformation parameter, γ , for fix value of β . The energies are normalized with respectto the binding energy of the absolute minima. ima in − Mo isotopes are the ground state minimawith the highest total energy among the minimal ob-served in Figs. (1 & 2). The second ground state minimaare oblate with a small energy difference, as presentedin Table. 2 and Table. 3, respectively. Thus, we see thephenomena of shape coexistence in − Mo isotopewith triaxial ground-state that agrees with the predic-tions of various theoretical models[19,22], and the ex-perimental results[6]. It is noted that the triaxial solu-tion Mo isotopes above N = 66 are almost same as theaxial solution for both parameter sets. Hence, we haveused the simple axial deformed calculation for these iso-topes of Mo for the description of ground-state struc-tural observables. Table 2
The location of the ground-state minima ( β, γ ) for − Mo isotopes using DD-PCX parametrization.Nuclei 1 st minima 2 nd minima ∆E tot. (MeV) Mo (0.250,20 o ) (0.220,55 o ) 0.052 Mo (0.370,15 o ) (0.230,60 o ) 0.026 Mo (0.380,20 o ) (0.230,60 o ) 0.233 Mo (0.380,20 o ) (0.230,60 o ) 0.131 Mo (0.380,20 o ) (0.230,60 o ) 0.331 Mo (0.225,60 o ) - - β =0.270 β =0.230 β =0.350 β =0.230 β =0.380 β =0.230 β =0.380 β =0.230 γ (in degree)0123456 β =0.380 β =0.230 γ (in degree) β =0.250 E ( M e V ) Mo Mo Mo Mo Mo Mo E ( M e V ) Fig. 4 (Color online) Same as Fig. 3, but with different aparameter set DD-PC1.
Table 3
The location of the ground-state minima ( β, γ ) for − Mo isotopes using DD-PC1 parametrization.Nuclei 1 st minima 2 nd minima ∆E tot. (MeV) Mo (0.270,20 o ) (0.230,55 o ) 0.067 Mo (0.370,20 o ) (0.230,60 o ) 0.041 Mo (0.380,20 o ) (0.230,60 o ) 0.287 Mo (0.380,20 o ) (0.230,60 o ) 0.224 Mo (0.380,20 o ) (0.230,55 o ) 0.307 Mo (0.250,60 o ) - - σ rms , is calculated us-ing CDFT formalism with DD-PCX and DD-PC1 pa-rameters and compared with NL3 parameter set [21].The rms deviation of binding energies is defined as σ rms = vuut n X i =1 ( E Exp.i − E Cal.i ) n . (9)In Fig (5), the values of rms deviation of binding ener-gies are given for each parameter set. We can see thatthe rms deviation of binding energies with both point-coupling parameter sets (DD-PCX and DD-PC1) arevery close. It indicates that the parameter set DD-PCXis successful as DD-PC1 parameter set in reproducing DD-PCXDD-PC1NL3
48 52 56 60 64 68 72N024 σ ( M e V ) ZrMo σ rms = 2.2987 σ rms = 2.8797 σ rms = 1.9987 σ rms = 2.1397 σ rms = 2.4902 σ rms = 2.1095 Fig. 5 (Color online) The rms deviation of binding energy σ rms in Zr and Mo isotopes with DD-PCX and DD-PC1calculations as a function of the neutron number. Comparisonof calculated rms deviation with other interactions are shown. the ground-state binding energies of the nuclei consid-ered in the present study.Another quantity that is related to the stability of anucleus is two-neutron separation energy, S n , definedas S n ( Z, N ) = E b ( Z, N ) − E b ( Z, N − . (10)This quantity gives information on the stability of anucleus against the emission of two neutrons, and thusdefine the neutron drip lines. Generally, the systematicsof S n are known to decrease with the increasing num-ber of neutrons. This decline in S n behavior is smoothexcept at magic number where a sharp change in theslop of S n is observed due to the presence of neutronshell closures. Fig. (6) presents the trend of S n behav-ior for Zr and Mo isotopic chains as a function of theneutron number. The theoretical results with DD-PCXand DD-PC1 parameter sets are calculated and com-pared with available experimental data[41,42] and withNL3[21] model. An uneven behavior in S n trend at par-ticular neutron number suggests the sudden change inground state shape in Zr isotopic chain. On the otherhand, the Mo isotopic chain shows a smoother behaviordue to the occurrence of triaxiality. An abrupt declineof S n at N = 50 and N = 82 support the existence ofrobust shell closures. In the terminology of energy, wecan say that the energy required to remove two neutronsfrom a magic nucleus is higher than that to remove twoneutrons from the neighboring nucleus ( Z, N magic + 2).
ExpDD-PCXDD-PC1NL3
48 52 56 60 64 68 72 76 80 84 880481216202428 S ( M e V ) N ZrMo
Fig. 6 (Color online) Two-neutron separation energy dS n in Zr and Mo isotopic chains, calculated using CDFT withDD-PCX and DD-PC1 parameter sets, as a function of neu-tron number. Comparison with experimental data[41,42] andtheoretical NL3[21] results are shown. The differential variation of the two-neutron sepa-ration energy ( dS n ( Z, N )) with respect to the neutronnumber ( N ) is defined as dS n ( Z, N ) = S n ( Z, N + 2) − S n ( Z, N )2 . (11)The dS n ( Z, N ) is an important quantity to investi-gate the rate of change of separation energy with re-spect to the neutron number. Here, we have calculatedthe systematics of dS n ( Z, N ) for Zr and Mo isotopicchains with DD-PCX and DD-PC1 parameter sets. Wehave also calculated the dS n ( Z, N ) from the experi-mental two-neutron separation energy. In Fig. (7), wecompared the calculated dS n values for Zr and Mo iso-topes with NL3[21] parameter set and with available ex-perimental data[41,42]. Generally, the large and sharpdeep fall in the dS n in an isotopic chain shows the sig-nature of neutron shell closure. We can say that thedeviation in the general trend may disclose some addi-tional features of nuclear structure, like shape transi-tion. An abnormal behavior in dS n trend at particularneutron number suggests the sudden change in groundstate shape in Zr isotopic chain. On the other hand, theMo isotopic chain shows a smoother behavior due to theoccurrence of triaxiality. A sharp deep fall of dS n at N = 50 and N = 82 support the existence of robust shellclosures. -4-3-2-1012 ExpDD-PCX
48 52 56 60 64 68 72 76 80 84-5-4-3-2-1012
DD-PC1NL3 d S ( M e V ) ZrMo N
Fig. 7 (Color online) Differential variation in two-neutronseparation energy dS n in Zr and Mo isotopic chains, cal-culated using CDFT with DD-PCX and DD-PC1 parametersets, as a function of neutron number. Comparison with ex-perimental data[41,42] and theoretical results[21] are shown. R ch = r R p + h r p i + NZ h r n i , (12)where, R p denotes the rms radii of proton. Here h r p i =0.833(10) fm [45] and h r n i = -0.1161(22) fm are themean-square charge radii of the proton and the neu-tron, respectively. In Figs. (8 & 9), the evolution ofcalculated rms charge radii in even-even − Zr and − Mo isotopes, as a function of neutron number,are plotted corresponding to the spherical, prolate, andoblate local minima in the PECs of these isotopes. Inpanel (a) of Fig. (8), a smooth increase in charge radiiwith almost similar slop is observed for the sphericaland prolate shapes, with an exception from N = 56 to N = 60 in the Zr isotopes. This shows that the defor-mation of prolate minima in Zr isotopic chain is nearlythe same as one move from N = 60 to N = 70. A sud-den increase in charge radii from N = 58 to N = 60 isobserved which is due to the increase in ground-statedeformation from β = 0.290 to β = 0.428 (cf. Fig.(1)). A rapid fall in charge radii is also seen on movingfrom N = 70 to N = 72 which can be related to the sud-den shape transition from prolate to oblate in Zr chain.However, such sharp rise at N = 58 is not seen with DD-PC1 force parameter, as seen in panel (a) of Fig.(9), rather a rapid increase in charge radii is observedon moving from N = 56 to N = 58. This dramatic rise isdue to the shape transition from prolate to oblate as onemove from N = 56 to N = 58. The value of charge radiishows a sharp fall from N = 62 to N = 64 due to theshape transition and then increase smoothly. Chargeradii is again seen to decrease with neutron numberwhich is due to the restoration of spherical shape atneutron shell closure N = 82. The calculated chargeradii for Zr isotopes are well reproducing the experi-mental values[3]. A similar behavior of charge radii isobserved with PC-PK1 point-coupling parameter[19],where a sharp rise in charge radii is seen from N =58 to N = 60 in Zr isotopic chain. On the contrary,the charge radii in and Mo isotopes increase smoothlywith the neutron number, as observed from panel (b) ofFigs. (8 & 9). The experimental charge radii are repro-duced well by taking triaxiality into account[6]. It canbe observed from these figures that the ground-state inMo isotopes become triaxial at N = 58 and an islandof triaxiality is appearing from N = 58 up to N = 66.Similar behavior of charge radii in these isotopic chainswas observed in the many theoretical studies[17,23,19].A prominent characteristics of the isotopic evolution ofcharge radii in this region is probably the appearanceof kink at N = 82 shell closure that was recently re-ported experimentally by laser spectroscopy techniquein Sn isotope[46]. Also, the theoretical DFT calcula-tions produce sharp kink at N = 82 using newly devel-oped Fayans functional[47], which includes a gradientterm in the pairing functional. In the present study, thesharp kink at N = 82 is absent in both Zr and Mo chainsbecause of the non-flexibility of pairing functional in thepresent model, as explained in Ref.[48].3.4 Single-particle energy levelsTo understand the emergence of collectivity and shapecoexistence phenomenon in Zr isotopes around N = 60,we plot the neutron and proton single-particle energy(SPE) levels in Zr, with DD-PCX parameter set, asfunctions of the axial deformation parameter β in Fig.(10). Fermi levels are denoted by thick circles. Gener-ally, the ground-state minima in the potential energycurve are associated with the effect of low-level den-sity around the Fermi energy. According to Jahn-Tellereffect[49], the regions of low-level density favor the on-set of deformation. In the plot of neutron SPEs, a regionof low-level density below the neutron Fermi energylevel is observed around -0.35 ≤ β ≤ -0.15 favoringan oblate shape. The panel (b) of Fig. (10) presentsthe plot of proton single-particle energy levels. A large
48 52 56 60 64 68 72 76 80 844.24.254.34.354.44.454.54.554.64.654.74.75 R c h (f m ) exp.prol.obl.sph.g.s.
48 52 56 60 64 68 72 76 80 84 triaxial
N NZr Mo
Fig. 8 (Color online) Nuclear charge radii in Zr and Moisotopes as a function of neutron number. The calculated val-ues with DD-PCX parameter set corresponding to the spher-ical (squares), prolate (down triangles), oblate (up triangles),and triaxial (diamonds). Experimental data are denoted byfilled circles with error bars[3,6]. Open circles correspond toground-state results.
48 52 56 60 64 68 72 76 80 844.24.254.34.354.44.454.54.554.64.654.74.75 exp.prol.obl.sph.g.s.
48 52 56 60 64 68 72 76 80 84 triaxial R c h (f m ) N Zr MoN
Fig. 9 (Color online) Same as Fig. 8, but with different aparameter set DD-PC1. shell gap is seen around β = 0.45 below proton Fermienergy level which favors the onset of prolate deforma-tion in Zr. This large shell gap is mainly formed bytwo levels split from the degenerate π g / state favor-ing the deformation[49]. A similar energy level diagramcan be obtained using DD-PC1 parameter set. Thus,these ideas propose a simple comprehension of the sev-eral mechanisms leading to deformation in this massregion. -0.4 -0.2 0 0.2 0.4 0.6 β -14-12-10-8-6-4-2 E s . p ( M e V ) -0.4 -0.2 0 0.2 0.4 0.6 β -21-18-15-12-9-6
50 2840
Neutron Proton Zr Fig. 10 (Color online) Neutron and proton single-particleenergy levels in
Zr, calculated using CDFT with DD-PCXinteraction, as a function of axial deformation parameter β .Circles denote the corresponding Fermi energy levels. , , , Zr iso-topes are shown. These densities are plotted in the pos-itive quadrant of the xz plane, with x = y . The nuclearsymmetry axis, z , is along the vertical axis and the co-ordinate x or perpendicular axis is along the horizontalaxis. A clear distinction between the spherical, prolate,and oblate shapes is seen for these isotopes correspond-ing to their ground-state β values in the PECs (cf.Fig. (1 & 2)). In the figures, the red color correspondsto high density ( ∼ − ) and blue color to thelower density or zero density. Similar calculations forneutron and proton densities for , , , Zr isotopesare done and plotted in Ref.[21]. From these figures, weobserve a spherically symmetric neutron density dis-tributions in Zr isotope and a prolate deformationin
Zr indicating a shape transition from sphericalto prolate as one moves away from N = 50. Again, ashape transition from prolate to oblate is observed asone reaches at N = 74 which can be seen from con-tour density plot for Zr. The spherically symmetricdistribution of neutron densities are again restored onapproaching neutron shell closure N = 82, as observedfrom figure. A close inspection of the density profile of Zr in both figures shows that the central part of theneutron density is less dense than the peripheral re-gion, indicating the formation of a bubble. This resultagrees with the study done in a recent paper on bubblestructure nuclei in which a central depletion of neutron Z (f m ) X (fm) Zr Zr Zr Zr b n = 0.0000 b n = 0.0000b n = -0.1750 b n = 0.4270 r n (fm -3 )r n (fm -3 )r n (fm -3 ) r n (fm -3 ) Z (f m ) X (fm) Z (f m ) X (fm) Z (f m ) X (fm)
Fig. 11 (Color online) Contour plots of axially deformedground-state neutron density for , , , Zr isotopesalong symmetry axis, z , and coordinate axis x . The calcu-lations are done with DD-PCX parameter set. density is observed for Zr[50]. The occurrence of thebubble structure can be quantified by defining a termnamed as depletion fraction (DF). The depletion frac-tion for
Zr calculated using CDFT formalism comesout to be 12.65% and 11.73% for DD-PCX and DD-PC1parameter sets, respectively. This central depletion ofdensity could be related to the unoccupency of 3s or-bital. In present work, the occupencies of 3 s / orbitalare 0.0605 and 0.0379 for both parameter sets, respec-tively. However, a detailed study shall be carried out infuture. We have done self-consistent calculations using covari-ant density functional theory with density-dependentpoint-coupling DD-PC1 and recently developed DD-PCX force parameter in even-even − Zr and − Moisotopes. We have studied the systematics of rms devi-ation of binding energy, two-neutron separation energy,differential variation of the two-neutron separation en-ergy, rms charge radii, single-particle energy levels forthese nuclei. The phenomena of structural change andshape coexistence have been observed in these isotopicchains. A sharp rise of charge radii in Zr isotopic chainfrom N = 58 to N = 60 is observed using new param-eter set DD-PCX which can be related to the rapidincrease in deformation parameter as one move from Zr to
Zr. In this chain, a sudden shape transi- Z (f m ) Zr Z (f m ) Z (f m ) Z (f m ) X (fm) Zr Zr Zr X (fm) X (fm)X (fm) r n (fm -3 )r n (fm -3 ) r n (fm -3 )r n (fm -3 )b = 0.000b = -0.177 b = 0.431b = 0.000 Fig. 12
Same as Fig.11, but with DD-PC1 parameter set. tion from prolate to oblate has been seen at N = 70whose signatures can be depicted in the systematics oftwo-neutron separation energies and rms charge radii.On the other hand, a smooth behavior in the evolu-tion of charge radii is found in Mo isotopic chain dueto the occurrence of triaxiality. This smoother increasein charge radii with neutron number is in good agree-ment with the available experimental data. Shape coex-istence and triaxiality softness is seen for − Mo iso-topes. An oblate-prolate shape coexistence with almostdegenerate energies is also seen for − Zr isotopes.The observed oblate and prolate minimum are relatedto the low single-particle energy level density aroundthe Fermi levels of neutron and proton respectively. Wehave also observed the depletion of central density in
Zr isotope indicating a bubble structure. The overallresults of the calculations for the ground-state proper-ties in Zr and Mo isotopic mass chains, with new pa-rameter set, are in good agreement with the availableexperimental data and with results obtained from dif-ferent models.
Acknowledgements
The authors would like to thank Hi-machal Pradesh University for providing computational fa-cilities. One of the authors, Mr. Pankaj Kumar, also thankCouncil of Scientific and Industrial Research (CSIR), NewDelhi for providing financial assistance Senior Research Fel-lowship vide reference no. 09/237(0165)/2018-EMR-I.
References
1. U. Hager et al.,
Phys. Rev. Lett. , 042504 (2006).2. F. Buchinger et al., Phys. Rev. C , 2883 (1990).03. P. Campbell, et al., Phys. Rev. Lett. , 082501 (2002).4. S. Raman, C.W. Nestor Jr. and P. Tikkanen, At. DataNucl. Data Tables , 1-128 (2001).5.
6. F.C. Charlwood, et al.,
Phys. Lett. B
23 (2009).7. J.L. Wood, K. Heyde, W. Nazarewicz, M. Huyse and P.Van Duppen,
Physics Reports , 101 (1992).8. K. Heyde and J.L. Wood,
Rev. Mod. Phys. , 1467(2011).9. A. Chakraborty et al., Phys. Rev. Lett. , 022504(2013).10. P. Singh, et al.,
Phys. Rev. Lett. , 192501 (2018).11. C.Y. Wu, et al.,
Phys. Rev. C , 064312 (2004).12. C.Y. Wu, H. Hua, and D. Cline, Phys. Rev. C , 034322(2003).13. M. Zielinska et al., Acta Phys. Pol. B , 1289 (2005).14. K. Wrzosek-Lipska et al., Int. J. Mod. Phys. E , 443(2011).15. K. Wrzosek et al., Int. J. Mod. Phys. E , 374 (2006).16. J. Ha, et al., Phys. Rev. C , 044311 (2020).17. R. Rodriguez-Guzman, P. Sarriguren, L.M. Robledo, andS. Perez-Martin,
Phys. Lett. B , 202 (2010).18. A. Petrovici,
Phys. Rev. C , 034337 (2012).19. J. Xiang, Z.P. Li, Z.X. Li, J.M. Yao, and J. Meng, Nucl.Phys. A , 1 (2012).20. Bao-Mei Yao and Jian-You Guo,
Mod. Phys. Lett. A ,1177 (2010).21. M. Bhuyan, Phys. Rev. C , 034323 (2015).22. H. Abusara and Shakeb Ahmad, Phys. Rev. C , 064303(2017).23. H. Abusara, Shakeb Ahmad, and S. Othman, Phys. Rev.C , 054302 (2017).24. K. Nomura, R. Rodriguez-Guzman, and L.M. Robledo, Phys. Rev. C , 044314 (2016).25. J.E. Garcia-Ramos and K. Heyde, Phys. Rev. C ,044315 (2019).26. M. Bender, G.F. Bertsch, and P.-H. Heenen,
Phys. Rev.C , 034322 (2006).27. M. Bender, G.F. Bertsch, and P.-H. Heenen, Phys. Rev.C , 054312 (2008).28. Y. El Bassem, and M. Oulne, Nucl. Phys. A , 22(2017).29. H. Mei, J. Xiang, J.M. Yao, Z.P. Li, and J. Meng,
Phys.Rev. C , 034321 (2012).30. J. Meng, S.-G Zhou, and I. Tanihata, Phys. Lett. B ,209-214 (2002).31. J. Meng, H. Toki, J.Y. Zeng, S.Q. Zhang, and S.-G Zhou,
Phys. Rev. C , 041302 (2002).32. Smriti Thakur, and Shashi K. Dhiman, Mod. Phys. Lett.A , , 1950014 (2019).33. Virender Thakur, and Shashi K Dhiman, Nucl. Phys. A , 121623 (2019).34. Virender Thakur, Pankaj Kumar, Suman Thakur, SmritiThakur, Vikesh Kumar, and Shashi K. Dhiman,
Nucl. Phys.A , 121981 (2020).35. P.W. Zhao, Z.P. Li, J.M. Yao, and J. Meng,
Phys. Rev.C , 054319 (2010).36. T. Niksic, D. Vretenar and P. Ring, Phys. Rev. C ,034318 (2008).37. E. Yuksel, T. Marketin and N. Paar, Phys. Rev. C ,034318 (2019).38. Y. Tian, Z.Y. Ma, and P. Ring Phys. Lett. B , 44-50(2009).39. T. Niksic, P. Ring, D. Vretenar, Y. Tian and Z.Y. Ma,
Phys. Rev. C , 054318 (2010).40. T. Niksic, D. Vretenar and P. Ring, Progress in Particleand Nuclear Physics , 519-548 (2011). 41. X.W. Xia, et al., At. Data Nucl. Data Tables , 64(2018).42. M. Wang, G. Audi, F.G. Kondev, W.J. Huang, S. Naimiand X. Xu,
Chin. Phys. C , 030003 (2017).43. M. Bender, P.-H. Heenen, P.-G. Reinhard, Rev. Mod.Phys. , 121 (2003).44. T. Niksic, N. Paar, D. Vretenar and P. Ring, Comput.Phys. Commun. , 1808 (2014).45. N. Bezginov, T. Valdez, M. Horbatsch, A. Marsman, A.C.Vutha, and E.A. Hessels,
Science , 1007-1012 (2019).46. C. Gorges, et al.,
Phys. Rev. Lett. , 192502 (2019.47. P.-G. Reinhard, and W. Nazarewicz, it Phys. Rev. C ,064328 (2017).48. Pankaj Kumar, and Shashi K. Dhiman, Nucl. Phys. A , 121935 (2020).49. P.-G. Reinhard, and E.W. Otten,
Nucl. Phys. A ,173-192 (1984).50. G. Saxena, M. Kumawat, M. Kaushik, S.K. Jain, andMamta Aggarwal,
Phys. Lett. B788