Microscopic study of evolution of shape change across even-even mass chain of tellurium isotopes using relativistic Hartree-Bogoliubov model

Available online at www.sciencedirect.com

ScienceDirect Nuclear Physics A 988 (2019) 9–23 www.elsevier.com/locate/nuclphysa

Microscopic study of evolution of shape change across even-even mass chain of tellurium isotopes using relativistic Hartree-Bogoliubov model Shivali Sharma a , Rani Devi a,∗ , S.K. Khosa b a Department of Physics, University of Jammu, Jammu-180006, India b Department of Physics and Astronomical Sciences, Central University of Jammu, Jammu-181143, India

Received 2 April 2019; received in revised form 22 April 2019; accepted 9 May 2019 Available online 15 May 2019

Abstract The density dependent point coupling and meson exchange effective interactions are employed in the relativistic Hartree-Bogoliubov (RHB) model to study the development of ground state deformation and occurrence of shape transitions with change of neutron number in 104−144 Te isotopes. Besides this, the systematics of other nuclear structure properties with increasing neutron number are also obtained. The RHB model reproduces well the available experimental data on binding energies, quadrupole deformation parameters, two neutron separation energies and root mean square charge radii. The results of three-dimensional potential energy surfaces predict prolate ground state minima for proton rich and neutron rich tellurium isotopes, oblate minima for 114−124 Te, triaxial minima for 126,128 Te and spherical minima for 130−134 Te. Besides, oblate prolate shape coexistence is predicted in 116−120 Te isotopes. © 2019 Elsevier B.V. All rights reserved. Keywords: Potential energy surfaces; Shape coexistence; Binding energy; Pairing energy; Charge radii

1. Introduction The advent of recent radioactive ion beam facilities and detection technologies, have opened new possibilities of exploring the production of various exotic nuclei and study their ground state * Corresponding author.

E-mail address: [email protected] (R. Devi). https://doi.org/10.1016/j.nuclphysa.2019.05.008 0375-9474/© 2019 Elsevier B.V. All rights reserved.

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properties. Nuclei near Z=50 shell closure are known to exhibit shape coexistence [1,2] and shape transitions with increase in neutron number (N). The authors of ref. [1] have cited 118 Te as one example for coexisting bands. In recent preprints [3,4], it has been suggested that shape coexistence around Z=50 should occur in the region of N=60 to 70. However, in case of Z=52 isotopes, the evidence for shape coexistence in many of these isotopes is not so clear. The low-lying collective excitations of tellurium (Te) isotopes were discussed in terms of quadrupole vibrators [5,6]. However, the available experimental data [7,8] on E2 transition strengths of 114,120−124 Te show a rotational like behavior. The isotopic mass chain of Te isotopes extends from N=54 i.e. proton rich side to the N=92, neutron-rich side. This isotopic mass chain exhibits a variety of shape transitions. The shape transition from prolate to oblate, oblate to triaxial, triaxial to spherical and spherical to prolate deformed. Delaroche et al. [9] have performed a systematic study of even-even nuclei in the framework of Hartree-Fock-Bogoliubov (HFB) theory by using Gogny D1S interaction. They have predicted the values of beta and gamma in the HFB energy minimum for 108−144 Te isotopes. The results on quadrupole deformation parameter (β2 ) and triaxial parameter (γ ) show the prolate to triaxial shape transition as one moves from N=70 to 72. Libert et al. [10] have studied the static and dynamic charge radii of tellurium isotopes in the frame work of a microscopic configuration mixing approach on the basis of Hartree-Fock-Bogoliubov solutions with the D1S Gogny effective interactions. Recently Chong Qi [11] has performed large scale configuration interaction shell-model calculation with a monopole-optimized realistic interaction and obtained the results on spectra and transition probabilities of 104−132 Te. Sabri et al. [12] have investigated the shape coexistence in 118−128 Te isotopes by using a transitional IBM Hamiltonian. It is suggested that gamma-soft rotational feature exist in tellurium isotopes, but with a dominancy of vibrational character. In the present work, a systematic study of ground state properties and occurrence of shape transition in even-even 104−144 Te have been performed. In order to explain the ground state properties of the whole isotopic mass chain of even-even Te isotopes, the three-dimensional relativistic Hartree-Bogoliubov (RHB) model with density dependent meson exchange (DD-ME2) and point coupling (DD-PC1) effective interactions, are employed with separable pairing interaction [13,14]. The systematic constrained triaxial calculations for potential energy surfaces (PES) are performed in the RHB model. Besides, the ground state properties, for example binding energies, two neutron separation energies, pairing energies, root mean square radii and charge radii are also calculated and compared with the available experimental data. The paper is organized as follows: A general overview of the RHB model is presented in Section 2. In Section 3 the results of potential energy surfaces (PES) for the entire isotopic mass chain of Te isotopes are presented and the various ground state properties of Te isotopes are compared with the available experimental data. Section 4 summarizes the results of the present work. 2. Theoretical framework 2.1. DD-ME2 The detailed description of DD-ME2 interaction can be found in Ref. [15]. Here, a brief outline of DD-ME2 interaction is given. The Lagrangian density for DD-ME2 interaction is given by

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1 L = [γ (ι∂ − gω ω − gρ ρ τ − eA) − m − gσ σ ]ψ + (∂σ )2 2 1 2 2 1 1 2 2 1   μν 1 2 2 1 μν − mσ σ − μν + mω ω − Rμν R + mρ ρ − Fμν F μν (1) 2 4 2 4 2 4 where ψ is Dirac spinor and denotes the mass of nucleon ‘m’. mσ , mω and mρ are the masses of σ , ω and ρ mesons, respectively. gσ , gω and gρ are the coupling constants for the σ , ω and ρ mesons to the nucleon. The vector fields of ρ and photon are denoted by μν , R μν and F μν . μν = ∂ μ ων − ∂ ν ωμ R μν = ∂ μ ρ ν − ∂ ν ρ μ F

μν

(2) (3)

=∂ A −∂ A μ

ν

ν

μ

(4)

The coupling of σ and ω mesons to the nucleon field [16–18] in the phenomenological approach is given by gi (ρ) = gi (ρsat )fi (x)

f or

i = σ, ω

(5)

where 1 + bi (x + di )2 (6) 1 + ci (x + di )2 is a function of x = ρ/ρsat , and ρsat denotes the baryon density at saturation in symmetric nuclear matter. The eight real parameters in (6) are not independent. The five constraints-fi (1) = 1, fσ (1) = fω (1), fi (0) = 0, reduce the number of independent parameters to three. Three additional parameters in the isoscalar channel are gσ (ρsat ), gω (ρsat ), and mσ , the mass of the phenomenological σ meson. The calculations of asymmetric nuclear matter [19] in Dirac-Brueckner framework, suggested the functional form of the density dependence of ρ meson couplings as fi (x) = ai

gρ (ρ) = gρ (ρsat )exp[−aρ (x − 1)].

(7)

The isovector channel is parameterized by gρ (ρsat ) and aρ . 2.2. DD-PC1 The effective Lagrangian for the density dependent point coupling interaction [20] is given by 1 L = (ιγ · ∂ − m)ψ − αS (ρ)( ˆ ψψ)( ψψ) 2 1 1 ˆ ψγ μ ψ)( ψγμ ψ) − αT V (ρ)( ˆ ψ τγ μ ψ)( ψ τγμ ψ) − αV (ρ)( 2 2 (1 − τ3 ) 1 ψ (8) − δS (∂ν ψψ)(∂ ν ψψ) − eψγ · A 2 2 Along with the free nucleon Lagrangian and point coupling interaction terms the model also includes the coupling of protons to the electromagnetic field. The derivative terms in (8) elucidate leading effects of finite range interactions that are important for quantitative description of nuclear radii. The functional form of the couplings is given by αi (ρ) = ai + (bi + ci x)e−di x (i ≡ S, V , T V ),

(9)

with x = ρ/ρsat , where ρsat denotes the nucleon density at saturation in symmetric nuclear matter.

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3. Results and discussion 3.1. Position of potential energy minima in 104−144 Te and shape transitions The RHB model is an extension of the conventional Hartree-Bogoliubov framework in which mean field and pairing correlations are treated self consistently. This model gives a unified description of particle hole (ph) and particle-particle (pp) correlations on a mean field level by using the average self consistent mean field potential that encloses the long range ph correlations and a pairing field potential which sums up the pp correlations. The single particle density matrix in the presence of pairing is generalized to two densities, the normal density ρ and pairing tensor κ. ˆ The RHB energy density functional thus depends on both densities ERH B [ρ, ˆ κ] ˆ = ERMF [ρ] ˆ + Epair [κ] ˆ where RMF functional is given by ERMF [ψ, ψ, σ, ωμ , ρ μ , Aμ ] =

(10)

 d 3 r H(r)

The pairing part of the RHB functional is given by  1 ∗  Epair [κ] ˆ = κn1 n n1 n1 | V pp | n2 n2 κn2 n2 1 4  

(11)

(12)

n1 n1 n2 n2

  The n1 n1 | V pp | n2 n2 are the matrix elements of the two body pairing interaction. The pairing interaction is taken of the form: V pp (r 1 , r 2 , r 1 , r 2 ) = −Gδ(R − R  ) P (r)P (r  ),

(13)

where R = √1 (r 1 + r 2 ) and r = √1 (r 1 − r 2 ) denote the center-of-mass and the relative coordi2 2 nates, respectively. P (r) is the Fourier transform of p(κ): P (r) =

1 2 2 e−r /2a 2 3/2 (4πa )

(14)

This pairing force has a finite range and preserves translational invariance because of the factor δ(R − R  ). The RHB equations are solved self consistently. The self consistent solutions of the RHB equations are obtained by expanding the nucleon spinors and the meson fields in the basis of a three-dimensional harmonic oscillator in Cartesian coordinates. The number of major oscillator shells taken in the basis for the present calculation are 12. The RHB calculation has been performed by taking the set of density dependent interactions, DD-ME2 and DD-PC1. The parameters of these interactions are presented in Table 1. In Figs. 1–4, the potential energy surfaces (PES) of twenty-one tellurium isotopes obtained by DD-ME2 and DD-PC1 interactions are displayed. The PES as a function of quadrupole deformation parameter (β2 ) are obtained by imposing constraints on the axial and triaxial quadrupole moments of a given nucleus. The calculations are performed by the method of quadratic constraint along with an unrestricted variation of the function  Hˆ  + C2μ (Qˆ 2μ  − q2μ )2 (15) μ=0,2

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Table 1 The parameters of the effective interactions density dependent meson exchange (DD-ME2) [15] and point coupling (DD-PC1) [20]. Parameter mσ mω mρ gσ (ρsat ) gω (ρsat ) gρ (ρsat ) aσ bσ cσ dσ aω bω cω dω aρ

DD-ME2

Parameter

DD-PC1

550.1238 783.0000 763.0000 10.5396 13.0189 3.6836 1.3881 1.0943 1.0943 0.4421 1.3892 0.9240 1.4620 0.4775 0.5647

aS (fm2 ) bS (fm2 ) cS (fm2 ) dS aV (fm2 ) bV (fm2 ) dV bT V (fm2 ) dT V δS (fm4 )

-10.0462 -9.1504 -6.4273 1.3724 5.9195 8.8637 0.6584 1.8360 0.6403 -0.8149

ˆ 2μ  is the expectation value of the mass quadrupole where Hˆ  represents total energy and Q operators ˆ 22 = x 2 − y 2 Qˆ 20 = 2z2 − x 2 − y 2 and Q

(16)

q2μ is the constrained value of the multipole moment and C2μ , the corresponding stiffness con ˆ 2μ to the system, stant. Moreover, the quadratic constraint adds an extra force term μ=0,2 λμ Q ˆ where λμ = 2C2μ (Q2μ  − q2μ ), for a self-consistent solution. This term is necessary to force the system to a point in deformation space different from a stationary point. The tellurium (Te) isotopic mass chain at present is known to extend from mass number (A) = 104 to 144. Based on the experimental information available near the β-stability line, these isotopes could be grouped into two major categories proton rich and neutron rich isotopes. It would be of immense importance to understand how the deformation systematics develop in this isotopic mass chain with increasing neutron number (N), as we move from proton rich to neutron rich isotopes. Specifically, it would be interesting to find out the shape transitions occurring in this isotopic mass chain as one moves from proton rich to neutron rich members of the mass chain. From the overview of literature, there are some HFB as well as finite range droplet model (FRDM) calculations [21] that have been made by employing D1S Gogny [9] and macroscopic-microscopic interactions, respectively. The values of β2 and γ predicted by D1S interaction and FRDM results are presented in Table 2. Based on D1S Gogny force, it is found that for 108−144 Te, the values of γ are zero except for two sets 112−116 Te and 124−128 Te, the values of γ for the set 112−116 Te are close to zero indicating that this set has negligible triaxiality where as 124−128 Te have value of gamma closer to γ = 60◦ line indicating that they should possess sizeable triaxiality. The β2 is found to increase as one moves from 112 Te to 118 Te, whereafter it continues to decrease till 136 Te. There is another piece of work that is cited in literature [21], according to which 118−128 Te are indicated to have negative β2 indicating that they are oblate in nature whereas the rest of the nuclei are having prolate deformation. The β2 is seen to increase

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Fig. 1. RHB triaxial potential energy surfaces (PES) of the even-even 104−126 Te isotopes in the β − γ plane (0 ≤ γ ≤ 60◦ ) with DD-ME2 interaction. All energies are normalized with respect to the binding energy of the absolute minima. The color bar refers to the energy of each point on the surface relative to the minimum. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)

from 104 Te to 116 Te from a value of 0.0 to 0.216 after which it remains negative till 128 Te. This model, therefore, predicts a sudden prolate-oblate transition as the mass number changes from 116 to 118 and oblate to spherical shape transition at 128 Te which is a slow and gradual shape transition.

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Fig. 2. Same as described in the caption to Fig. 1 but for the isotopes 128−144 Te.

In Table 2, we also present the results of present calculations made for DD-ME2 and DDPC1 interactions under columns 2, 3 and 4, 5, respectively. The corresponding numbers for D1S Gogny force and FRDM results are given in columns 6, 7 and 8 for comparison with numbers presented by us. In Figs. 1–4, the PES have been plotted for 104−144 Te. It is observed from these figures that proton rich 104−112 Te isotopes show weak axial deformation with deformed minima lying in the range β2 ≈ 0.15 − 0.20. For 114−124 Te the figures reveal that these nuclei have oblate minima with β2 ≈ 0.15 − 0.20 and γ =60◦ . The PES for all the Te isotopes have been obtained for two sets of interactions. Both these sets of interactions predict the position of minima in the 104−124 Te and 128−144 Te at the same values of β and γ . However, in the case of 126 Te, the results 2 of two interactions differ in the values of β2 and γ . The DD-ME2 interaction predicts β2 =0.15 and γ = 18◦ whereas DD-PC1 interaction predicts β2 =0.12 and γ =36◦ . In the case of 126,128 Te both the interactions used in the present calculation predict a triaxial ground state. It is revealed by the calculations as one approaches the shell closure N=82, PES tend to approach a spherical

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Fig. 3. RHB triaxial potential energy surfaces (PES) of the even-even 104−126 Te isotopes in the β − γ plane (0 ≤ γ ≤ 60◦ ) with DD-PC1 interaction. All energies are normalized with respect to the binding energy of the absolute minima. The color bar refers to the energy of each point on the surface relative to the minimum.

minimum which is clearly exhibited by PES plots for 130−132 Te. Now, coming to the neutron rich side of Te isotopes for A≥136, the PES show that these neutron rich tellurium isotopes are axially prolate deformed with β2 ≈ 0.15 − 0.20. The PES obtained by using both the DD-ME2 and DD-PC1 interactions show the proton rich tellurium isotopes as axially prolate weakly deformed

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Fig. 4. Same as described in the caption to Fig. 3 but for the isotopes 128−144 Te.

isotopes. The medium mass tellurium isotopes are predicted to be oblate where as 126,128 Te are predicted to have triaxial minima. In addition, the isotopes near the shell closure are predicted to be nearly spherical whereas neutron rich tellurium isotopes are predicted to be axially prolate. From the above discussion, it turns out that four shape transitions can happen in the entire tellurium mass chain. The first shape transition from prolate to oblate shape is predicted to occur as one moves from 112 Te (N=60) to 114 Te (N=62). The second shape transition from oblate to triaxial is predicted to occur as one moves from 124 Te (N=72) to 126 Te (N=74). The third shape transition from triaxial to nearly spherical shape is predicted to occur as one moves from 128 Te (N=76) to 130 Te (N=78). The fourth shape transition from spherical to axially prolate shape is predicted to occur as one moves from 134 Te (N=82) to 136 Te (N=84). From Table 2, it is observed that the present calculations predict 104−112 Te as prolate deformed which is in agreement with the D1S Gogny and FRDM predictions. As one moves from A=112 to 114, the present calculations predict prolate-oblate shape transition whereas D1S Gogny predic-

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Table 2 Calculated values of the β2 and γ deformation parameters for the absolute minima of the potential energy surfaces (PES) of even-even 104−144 Te isotopes. The results of predictions of D1S Gogny force [9] and FRDM [21] are presented in column 6 to 8. The available experimental [22] β2 values are presented in last column. A 104 106 108 110 112 114 116 118 120 122 124 126 128 130 132 134 136 138 140 142 144

DD-ME2

DD-PC1

D1S Gogny

FRDM

Exp.

γ

β2

γ

β2

γ

β2

β2

β2

0.15 0.15 0.20 0.20 0.20 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.10 0 0 0 0.10 0.15 0.15 0.20 0.20

0◦ 0◦ 0◦ 0◦ 0◦

0.15 0.15 0.20 0.20 0.20 0.15 0.15 0.20 0.15 0.15 0.15 0.12 0.10 0 0 0 0.10 0.15 0.15 0.20 0.20

0.156 0.156 0.155 0.286 0.322 0.328 0.203 0.219 0.126 0.116 0.030 0 0 0 0 0.099 0.153 0.155 0.173

0.0 0.119 0.139 0.150 0.183 0.194 0.216 -0.165 -0.176 -0.166 -0.125 -0.105 -0.094 0 0 0 0 0 0.097 0.118 0.161

0◦ 0◦ 0◦ 0◦ 0◦

60◦ 60◦ 60◦ 60◦ 60◦ 60◦ 18◦ 30◦ 0◦ 0◦ 0◦ 0◦ 0◦ 0◦ 0◦ 0◦

60◦ 60◦ 60◦ 60◦ 60◦ 60◦ 36◦ 30◦ 0◦ 0◦ 0◦ 0◦ 0◦ 0◦ 0◦ 0◦

0◦ 0◦ 1◦ 1◦ 4◦ 0◦ 0◦ 0◦ 46◦ 36◦ 54◦ 0◦ 0◦ 0◦ 0◦ 0◦ 0◦ 0◦ 0◦

0.201(21) 0.1847(8) 0.1695(9) 0.1534(16) 0.1363(11) 0.1184(14)

tions show strong prolate to triaxial shape change at A=122. The FRDM predictions show prolate to oblate shape transition at A=116. Thus, the present calculations predict prolate to oblate shape transition between N=60 to 62 and oblate to triaxial transition between N=72 to 74 whereas the Gogny D1S calculations predict direct prolate to triaxial transition at N=72-74, without any oblate region. Further, the FRDM results predict oblate shape for nuclei between N=66 to 76. This result is intermediate between the present calculations and those predicted by Gogny D1S calculations. The calculated values of β2 are in reasonable agreement with the experimental data for such nuclei for which experimental data [22] is available. In addition to our observations made in the preceding paragraphs, we also find that there is a possibility of coexistence of shape in the case of 116−120 Te. The results presented in Figs. 1 and 3 reveal that 116−120 Te show secondary minima. The energies of these minima relative to primary minima for the interaction DD-PC1 are (in MeV) 0.3, 0.5 and 0.5 for 116 Te, 118 Te and 120 Te, respectively whereas the secondary minima are about 1 MeV higher for 118,120 Te when calculations are carried out with DD-ME2 interaction. The secondary minima are seen at (β2, γ ) is equal to (0.32,7◦ ), (0.33,5◦ ) and (0.32,2.5◦ ) for 116−120 Te, respectively. As, the predicted primary minima for 116−120 Te are oblate and the predicted secondary minima are prolate in shape, there is a possibility of oblate prolate shape coexistence in N=64-68 Te isotopes. These results on shape coexistence of N=64-68 Te isotopes are in agreement with the prediction of ref. [1] for 118 Te and recent predictions of preprints [3,4] for N=60 to 70 Te isotopes.

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Fig. 5. Comparison of calculated and experimental average binding energies (B) for the even-even 104−144 Te isotopes obtained with RHB model.

3.2. Discussion of ground state nuclear structure properties After identifying the β2 and γ values for the potential energy minima for each isotope of the mass chain of 104−144 Te isotopes, the results for the various nuclear structure observable quantities obtained from the intrinsic state corresponding to the potential energy minima such as average binding energy (B), two neutron separation energies (S2n ), neutron and total pairing n energies (Epair and Epair ), root mean square charge radii (Rc ), proton and neutron root mean square radii (Rp and Rn ) of Te isotopic mass chain were obtained. 3.2.1. Systematics of average binding energy (B) in tellurium isotopic mass chain In Fig. 5, we present the results of average binding energy calculated by the RHB model by employing DD-ME2 and DD-PC1 interactions. The results on binding energies show good agreement with available experimental data [23]. From the same figure, it can be seen that the B increases as one moves along the isotopic mass chain with neutron number up to N=70. The maximum value of B is found to be 8.7 MeV for N=68, 70 isotopes. From N=70, the B curve shows a decrease in their value up to N=80. Further, as one moves N=80 to N=82, the B shows a sharp decrease with N. Thus, the B curve predicts the medium mass Te isotopes as stable and proton rich and neutron rich isotopes as less bound. 3.2.2. Systematics of two neutron separation energy (S2n ) The S2n are calculated by the formula S2n (Z, N ) = Eb (Z, N ) − Eb (Z, N − 2)

(17)

The calculated S2n obtained by using DD-ME2 and DD-PC1 interactions are compared with experimental data [24] in Fig. 6. From this figure, it is seen that the RHB results reproduce the available experimental data for the entire isotopic mass chain of tellurium isotopes. The S2n values show a decrease with an increase in neutron number and an abrupt decrease is observed around shell closure (N=84) which reproduces the expected shell closure for N=82. For proton rich Te isotopes, the S2n is predicted to vary from 24-21 MeV for 104−110 Te (N=52-58). The S2n values vary from 21-14 MeV for N=60-82 and therefore 112−134 Te isotopes are considered

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Fig. 6. Comparison of calculated and experimental two neutron separation energies (S2n ).

to be reasonably stable. The S2n varies from 5-3 MeV for N=84-92. In these 136−144 Te isotopes neutrons are loosely bound and these are called as neutron rich nuclei. n and Epair ) 3.2.3. Systematics of neutron and total pairing energy (Epair The pairing energy (Epair ) in RHB model is defined as

1 Epair = − T r(κ) 2

(18)

n In Fig. 7, we have presented the neutron and total pairing energies (Epair and Epair ) of even-even n tellurium isotopes obtained from RHB calculations. The calculated results show that the Epair and Epair decrease from the values of zero to nearly -8 MeV as one moves from N=52 to 62 which implies a systematic increase of deformation for 104−114 Te isotopes. Thereafter, the pairing energies increase as one goes from N=62 to 66 and then further decrease as neutron number changes from N=66 to N=76, implying decrease of deformation in 128 Te isotopes and followed n by further increase for N=78. After N=78, the Epair is found to systematically approaching to zero at N=82, thereby giving evidence of shell closure for Te isotopic mass chain at N=82. The Epair also shows a maximum at N=82. Thus, the calculated pairing energies reproduce the experimentally observed shell closure at N=82.

3.2.4. Systematics of root mean square charge (Rc ), neutron (Rn ) and proton (Rp ) radii The root mean square charge radii (Rc ) obtained by the RHB model by taking DD-ME2 and DD-PC1 interactions are compared with the available experimental values [25] in Fig. 8. The two interactions predict almost same value for Rc . The predicted values of Rc are in good accordance with the available experimental data. The systematics of Rp and Rn are presented in Fig. 9. A close study of these results reveals that in the proton rich tellurium isotopes, the values of Rn are less than those of Rp . Beyond N=58, the Rn values show a faster increase as compared to Rp values. The difference between the Rp and Rn at N=92 is 0.338 fm. The larger values for Rn for neutron rich isotopes can be related to the lesser binding energy of neutrons in the neutron rich nuclei. Therefore, the predictions of these results are in agreement with the predictions of the systematics of B and also S2n for these neutron rich nuclei.

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n Fig. 7. The neutron (Epair ) and total (Epair ) pairing energies of even-even 104−144 Te isotopes.

Fig. 8. Comparison of experimental [25] and calculated root mean square charge radii (Rc ) of even-even 104−144 Te isotopes.

4. Summary Summarizing the RHB results for 104−144 Te isotopes, the following broad conclusions can be drawn: 1. The RHB model with DD-ME2 and DD-PC1 effective interactions reproduce well the available experimental data on ground state nuclear structure properties.

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Fig. 9. Neutron (Rn ) and proton (Rp ) radii of even-even 104−144 Te isotopes.

2. The present theoretical results on PES predict four shape transitions in 104−144 Te isotopes. The axial prolate to oblate change of deformation as neutron number changes from N=60 to N=62, oblate to triaxial change of deformation as neutron number changes from N=72 to N=74, triaxial to nearly spherical change of deformation as neutron number changes from N=76 to N=78 and spherical to axially prolate change of deformation as neutron number changes from N=82 to N=84. 3. The oblate-prolate shape coexistence is predicted in N=64 to 68 Te isotopes which is in agreement with the recent preprints [3,4] and ref. [1]. 4. The present theoretical results on nuclear structure properties support the existence of robust shell closure at N=82. 5. On the basis of neutron and proton charge radii calculations, the isotopic mass chain of eveneven Te isotopes can be categorized into neutron rich (136−144 Te), proton-rich (104−112 Te) and stable (114−132 Te) isotopes. Acknowledgements We thank Dr. Sumit Mookerjee of Inter University Accelerator Centre, New Delhi for allocation of computation resources of High performance computing facility. One of the authors, Miss Shivali Sharma acknowledges Council of Scientific and Industrial Research, New Delhi for providing Junior research fellowship vide file no.: 09/100(0227)/2019-EMR-1. References [1] [2] [3] [4] [5] [6] [7] [8]

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