Lowest excited 2 + and 3 − states in even-A N = 82 and N = 126 isotones from relativistic quasiparticle random-phase approximation

Excitation energy, reduced transition rates, and g factors of the lowest 2+ and 3− states of even-even N = 82 and N = 126 isotones are studied in relativistic QRPA formalism. These values are compared to available experimental data and some of the recent theoretical results. In the following the obtained results are presented first for N = 82 isotones with Z = 36–72, that is very neutron-rich to neutron-deficient nuclei with N/Z varying from 2.28 to 1.14 and then for N = 126 isotones with Z = 76–92. These calculations are performed employing NL3, as well as its recently revised version NL3∗ set of the RMF Lagrangian parameters. It is found that both the interaction sets produce almost similar results. The agreement with available data is rather satisfactory in view of such a wide range of nuclei considered here, though there are serious disagreements like in the case of Ce and properties of 3− states of N = 126 isotones.


I. INTRODUCTION
Since the early 1980s there have been attempts to study the structure of N = 82 isotones (for example, Refs.[1,2]).As early as 1971, Maier et al. [3] studied the excited levels of the N = 126 isotones with Z = 85-88 (Z > 82) by in-beam γ -ray spectroscopy with heavy ion reactions.However, for Z < 82 isotones the experimental information is even now very scarce.Some spectroscopic information for 206 Hg could be obtained only in 2001 [4] and for 204 Pt only in 2008 [5] and, as far I am aware, there are no data available for 202 Os.By now it is well known from light nuclei that for very neutron-rich nuclei nearing the neutron-drip line new magic numbers may appear or usual closed-shell energy gaps may get reduced (quenched).For the type of isotones considered here some recent studies [6][7][8] put forward arguments that for Z < 50 shell quenching should occur at N = 82.In a self-consistent mean-field calculation Dobaczewski et al. [6] demonstrated that for an isobar of A = 120 (Z = 38) the N = 82 shell gap decreases dramatically.Based on high Q β value (8.34 MeV) for 130 Cd Dillmann et al. [7] believed that it is a direct signature of N = 82 shell quenching.Taprogge et al. [8] in their study of 2p 3/2 proton-hole state in 132 Sn conclude that their study provided a robust evidence for the disappearance of the Z = 38 and 40 proton subshell closure at N = 82.As a consequence, a significant reduction of the N = 82 gap in the region of the r-process path for nucleosynthesis is expected.In fact, based on this observation the present study has been extended down to Z = 36.
However, Jungclaus et al. concluded in Ref. [9] that their observation of isomeric decays in 130 Cd and their interpretation in terms of shell-model calculations show no evidence of shell Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
quenching at N = 82.That is, there is need to go down to even lighter N = 82 nuclei for their mass measurements and spectroscopic investigations.Similar conclusions were drawn by Watanabe et al. [10] in their study of isomers in 128 Pd (Z = 46).
Regarding shell quenching for Z < 82 in N = 126 nuclei, again a similar conclusion was drawn by Steer et al. [5] studying the isomeric decays in 204 Pt emphasizing the need for investigations towards Z 72.
In view of these, and to further extend our earlier studies, I have undertaken to study the structure of 2 + and 3 − excited states of these isotones in the relativistic quasiparticle random phase approximation (RQRPA) which has been quite successful for the description of tin and lead isotopic chains [11,12].Here it may be useful to discuss briefly as to how successful has been the RQRPA predictions regarding the experimental data, say, on B(E2; 0 + → 2 + ) [in short B(E2) ↑] of the long chain of Sn isotopes before proceeding further to present and discuss the results on N = 82 and N = 126 isotones.In Fig. 1 for A = 100-134 tin isotopes the curve labeled "Expt-a" represents the experimental data [13] as reported a decade ago in Ref. [11] and that labeled "NL3" indicates the RQRPA predictions for the whole range of N = 50-84 isotopes with a shallow minimum at N = 66 (the midshell nucleus).For the stable isotopes A = 112-122 the agreement with experimental data (Expt-a) is quite poor, though for A > 124 it is very satisfactory.The shell-model results [14] are in good agreement for A = 116-130, predicting a symmetric distribution with a maximum at the midshell (N = 66).On the other hand, RQRPA predicted a very asymmetric distribution with a maximum at A = 106, a very neutrondeficient isotope.Recently, Doornenball et al. [15] emphasized that our predicted low values for stable isotopes triggered re-measurements of these isotopes, including extension to a lower mass region.As can be seen, the latest remeasured values for A = 112-124 (Expt-b), all are lowered relative to the earlier values.Thus, the latest values [15][16][17][18][19] are now in good agreement with the predictions done a decade ago, including the minimum at N = 66.In view of the above indicated 1. Variation of B(E2) ↑ of tin isotopes with mass number A. Experimental data "Expt-a" represent the adopted values taken from [13] for A = 112-124 and from Ref. [16] for A = 126-134 (as reported in Ref. [11]).Expt-b indicates the latest values taken from Refs.[15][16][17][18][19].The label NL3 indicates the RQRPA results [11] using the corresponding force parameters listed in Table 1.The shell model (SM) numbers are taken from Ref. [14].success, the same scheme of calculation is employed here to study the structure of the N = 82 and N = 126 isotones.

II. CALCULATIONAL DETAILS
As far the calculational details, as discussed in Refs.[11,12], the full RQRPA formulation is presented in Ref. [20].For the effective interaction the NL3 [21] and NL3 * [22] parameter sets are used for the relativistic mean field (RMF) Lagrangian.The pairing part of the Gogny finite-range interaction D1S [23] is used as a phenomenological pairing interaction.For the basis space 20 or 16 harmonic oscillator shells are considered depending on the mass number along with hole-particle (or two quasiparticles) energy cutoff of 150 MeV, and that of hole-antiparticle energy cutoff of 1800 MeV (for details see Ref. [12]).The NL3 and NL3 * parameter sets for the RMF Lagrangian are presented in Table I.The table also shows how well the nuclear matter properties are reproduced using these parameters, the main difference being seen on the values of the incompressibility parameter K and the symmetry energy J .

A. N = 82 isotones
It may be useful to present the RMF [relativistic Hartree-Bogoliubov(RHB)] results for all the isotones considered here to see how well the ground-state binding energies are reproduced.Out of 19 nuclei considered here only 132 Sn is included in the list of the spherical nuclei that are used for fitting these force parameters.The binding energy per particle (B/A), taken as positive, is listed for NL3 as well as NL3 * along with the available experimental data [24] in Table II.Also shown are the pairing energy (in MeV) due to protons corresponding to the NL3 force.This last would be qualitatively indicative of the density of states near the Fermi surface, which have implications, at least, on the low-lying excitation properties.As can be seen, the B/A values are slightly higher for NL3 compared to that for NL3 * till A = 142 and slightly lower for heavier ones.For most of the nuclei the calculated values show slight overbinding.To be specific it may be mentioned that for 132 Sn itself there is overbinding by about 1.0 MeV and that for 154 Hf it is about 3.7 MeV (≈2.4%).Another noticeable thing is the disappearance of the pairing energy for 140 Ce, which does not seem to be realistic as it is not known to be a doubly closed shell nucleus (g 7/2 subshell closure with energy gap = 3.664 MeV).To highlight these differences in B/A values in a more visual manner these are displayed as a B/A versus A plot in Fig. 2.
In passing, it may be mentioned that the excitation energies are usually not so sensitive to the absolute value of the groundstate binding energy of a nucleus as compared to the singleparticle levels in the vicinity the Fermi surface.Now in Fig. 3 are presented the main results for the J = 2 + excited states of 118 Kr to 154 Hf nuclei.The calculated numbers for E 2 , B(E2) ↑ and g factor g 2 using NL3 as well as NL3 * sets of the force parameters are compared to the available experimental data, and some of the other theoretical model results.An obvious result to be noticed is that the NL3 and NL3* sets produce almost similar results for all the three quantities shown here, except that for A 126 the g 2 values obtained with NL3* are somewhat smaller than those with the NL3 set.
In Fig. 3(a) excitation energies E 2 are compared to the experimental data [25].For A 132 the agreement is satisfactory except for 140 Ce and 146 Gd.In fact, for Z = 66-72 the ageement is quite good.For Z < 50 the calculated numbers are higher with a small bump at Z = 40, as should be expected.
Other theoretical results are not displayed, as most of these calculations adjust the interaction parameters to reproduce these, at least for some of the nuclei in the considered mass range.However, it may still be useful to discuss briefly the findings of two recent theoretical calculations.
(i) Recently Terasaki, Engel, and Bertsch [26] performed QRPA calculations employing the Skyrme interaction SkM*.They made a very elaborate systematic study of the 2 + excitation energy in 178 spherical nuclei.In the context of the present results for N = 82 isotones, even the excitation energies, E 2 are not as well reproduced as here in the RQRPA.Surprisingly, even they find at Z = 58 a shell closure effect with sudden increase in the E 2 value and for Z > 58 they obtain higher values compared to the experimental ones.Also 3. Variation of E 2 , B(E2; 0 → 2) and g factor g 2 as a function of mass number for N = 82 isotones.The labels Holt [2] and SMc [29] for B(E2) and SMa [28], SMb [31] for g 2 represent shell model values and "expt" is the experimental data from Refs.[25,30].
at Z = 64 they do not obtain a rise as seen in the experimental data.
(ii) A generator coordinate method (GCM) calculation with particle number and angular momentum projected Hartree-Fock-Bogoliubov wave function using quadrupole deformation parameter as the generator coordinate was performed recently by Rodriguez et al. [27].They studied the evolution of 2 + excitation energies in a long chain of Cd isotopes for N = 50 to 82 employing the finite-range density-dependent Gogny interaction [23].The trend of the variation of E 2 with N comes out quite well, but the calculated energies are higher by a couple of hundred keV throughout, so much so that at N = 50 and N = 82, respectively, the calculated values are about 1.0 and 2.0 MeV higher.In the present RQRPA calculation it comes to be higher by about 800 keV for 130 Cd.
The most important quantities sensitive to the singleparticle structure near the Fermi level are the B(E2) transition rates and the g factors.In Fig. 3(b) the B(E2) ↑ rates are displayed with experimental data available only for Z = 50-62 [25].The RQRPA results are in quite close proximity to the experimental data except for 140 Ce and 144 Sm.However, a sharp drop of the values at Z = 60 and 62 relative to the value at Z = 58 is very significant, which none of the models including RQRPA is able to reproduce.For B(E2) rates three shell-model curves are displayed indicated by SMa [28], SMc [29], and Holt [2].In SMa the authors considered a so-called nucleon-pair approximation (NPA) of active protons in the-major shell, which includes the 1g 7/2 ,2d 5/2 ,2d 3/2 ,3s 1/2 ,1h 11/2 orbitals for even-A Z = 52-60 nuclei.If all the possible pairs are considered, the NPA space is equivalent to the full shell-model space; if only a few important pairs are considered, it provides a truncated model space.This is the main theme of this scheme of calculations, the considered nuclei being 134 Te to 142 Nd.The employed twobody interaction is a model monopole and quadrupole pairing + quadrupole-quadrupole and octupole-octupole interaction between the valence protons.The effective charge for protons is 1.7 e.For such a limited number of nuclei the agreement with data are rather good, including a small drop for 142 Nd.
In SMc 144 Sm is also included within the same singleparticle shell-model space as in SMa, but using a realistic effective Hamiltonian.The single-particle (sp) energies as well as the two-body effective interaction are derived from the high precision CD-Bonn potential with a cutoff momentum = 2.6 fm −1 .Though the sp energies are not taken from experiments, these are computed adjusting this to be close to the experimental values.Without discussing further on the model, it can be seen from the plot that the B(E2) rates are only rising without any decrease after A = 140.
The third plot by Holt et al. [2] also shows only an increasing trend for the considered nuclei with Z = 52-62.Here also the spherical sp model space is the same 3s,2d,1g 7/2 ,1h 11/2 proton orbitals with energies taken from experimental spectrum of 133 Sb and for 2s 1/2 from Sagawa et al. [1].The effective two-body interaction is the G matrix derived from meson-exchange potential models.The proton effective charge is taken to be 1.4 e.
We may add that Sagawa et al. [1] also computed B(E2) rates for even Z = 52-68 nuclei following shell model + core polarization and a generalized seniority scheme including core polarization.The effective Hamiltonian was determined by a least squares fit to the well-determined combination of twobody matrix elements to experimental energy-level data in the A = 133-148 region with the initial values calculated from a surface-delta parametrization.The shell-model results show a kink at 144 Sm similar to the experimental value, while the results of the generalized seniority scheme show a nearly linear increase.
Coming back to RQRPA, the curves (for NL3 and NL3* both) depicting a rapid increase of the B(E2) values (for A > 142) appear rather surprising, which can be settled only when some experimental data are available for this mass region (very neutron-deficient).However, one point to be noted is that E 2 values for this region are small and gradually decreasing with the increase of Z, which is quite well reproduced by the RQRPA calculation.The predictions here for the lower mass region (A < 132) may also turn out to be important in view of expected quenching of the N = 82 shell gap with the decrease of Z values.
Before moving to the discussion on g factors it may be useful to make some comments on the results for 140 Ce.Since in the present calculation the proton pairing energy is coming out to be zero (see Table II), to see the effect of pairing, the calculation has been repeated increasing the pairing interaction strengh by 10% by multiplying all the pairing channel matrix elements of the Gogny-D1S interaction by a constant factor, V fac = 1.10 corresponding to the NL3 parameters.This leads to some improvement in the right direction with E P pair = 6.595MeV and B/A = 8.412 MeV.The values of E 2 and B(E2) presented in Fig. 3 change to E 2 = 2.910 MeV (about 100.0 keV lower), and B(E2) ↑ = 0.275 e 2 b 2 (about three-fold higher).
However, here would not like to indulge much in such type of playing with parameters.Moreover, even if one finds some other force parameter set which is, say, good for 140 Ce and 146 Gd, there may not be any guarantee that it would be good for all other nuclei too.In such a scenario one option, though not quite desirable, may be to have more than one version of the NL3 (or some other) set, like several versions of Skyrme interactions in the nonrelativistic approach, by inclusion of the relevant spherical nuclei in a given mass region while fitting the force parameters.Now coming to g factor values of these nuclei, Fig. 3(c) exhibits the RQRPA results along with experimental data [30] and some model calculations SMa [28] and SMb [31].RQRPA results using NL3 set of parameter for Z = 52-70 have already appeared in Ref. [32] and compared to the available data for Z = 54-62.Now NL3* is also used for the calculation and the number of nuclei is extended further, particularly to lower Z values.The RQRPA results are in good agreement with the data for four nuclei, but are significantly higher for Z = 60 and 62, predicting almost a constant value for higher masses.After using V fac = 1.10 for 140 Ce the g 2 value has slightly decreased to 0.933 (compared to 0.991).The shell-model results show only an increasing trend besides the agreement at 134 Te.It is important to notice that the curve depicting Z/A values, a signature of collectivity, is far below the experimental as well as the theoretical points, except for 132 Sn which is due to a large negative contribution from neutrons.
To have some understanding of the contributions of the neutrons and protons to the physical quantities computed, in Fig. 4 is shown the contribution of neutrons I n to the total QRPA wave-function normalization I = 1 for J = 2 + as well as J = 3 − for both the parameter sets.As can be seen the variation of I n with A is almost similar for NL3 (upper panel) as well as NL3 * (lower panel).Concentrating on the variation of I n with A for J = 2, it may be noticed that at Z = 50 ( 132 Sn) the neutron contribution is almost 80%, the remaining 20% coming from protons.In the plot each point shows the summed up contributions from pairs of two quasiparticles or hole particles.As an illustration we list here the contribution to I n from one hole-particle (or two quasiparticle) pair of (1h 11/2 ,2f 7/2 ), nearest to the Fermi level, from Z = 48 down to 36: 0.038, 0.072, 0.108, 0.177, 0.297, 0.291, and 0.352.An increase in the value of I n shows an increasing contribution of neutrons to the 2 + excitation energy, B(E2) and g 2 which is clearly reflected in their values in the previous figure.It may be a sign of quenching of the N = 82 shell gap as the proton number decreases.To look for quenching of the shell gap at N = 82 an RHB calculation was performed with a monopole pairing adjusting the interaction strength to reproduce almost the same pairing energy as obtained by the use of the Gogny interaction.The energy gap at N = 82 in 132 Sn is 6.314 MeV, i.e., between the sp states 2f 7/2 and 1h 11/2 .This gap goes decreasing with the decrease of Z.For example, for 130 Cd, 128 Pd, 122 Zr, and 120 Sr the energy gaps (in MeV) at N = 82 are 5.817, 5.320, 3.910, and 3.665, respectively.Also it is found that there is no proton subshell closure at Z = 38,40 as discussed by the authors of Ref. [8].In 120 Sr the energy gap between 2p 3/2 and 2p 1/2 levels is 1.224 MeV, whereas at Z = 50 the shell gap is 5.657 MeV.
As Fig. 4 shows, the contribution of neutrons to I n is very small for Z > 50, and so the protons contribute more than 80% leading to a fast increase in the B(E2) rates.The rise for g 2 is not so steep as it depends sensitively on the sp orbitals around the Fermi level.The variation of I n for J = 3 − will be discussed after presentation of the corresponding energy, transition rates and g factors in the next paragraph.The results for the J = 3 − states of the N = 82 isotones are presented in Fig. 5. Figure 5(a) shows the variation of E 3 − versus A. Qualitatively, the shape of the experimental curve [33] is reproduced with NL3 as well as NL3* parameters including a dip around 146 Gd and 148 Dy (proton midshell nucleus) contrary to the fact that for J = 2 + the small hump at 146 Gd was not reproduced.However, quantitatively for most of the nuclei the computed numbers are quite off.The RQRPA calculation is predicting a similar shape and curvature in the lower mass region with a dip at Z = 40.The other theoretical curve presented is by Holt et al. [2] for Z = 52-62 in the QRPA approach, and for the limited number of nuclei considered by them the agreement with the experiment is really good.
Figure 5(b) displays the B(E3) ↑ results where the experimental data are known only for Z = 54-64.Except for 140 Ce and 146 Gd, the RQRPA numbers are close to the experimental points for the other four nuclei.For the lower part, Z 50, our numbers are 0.2 e 2 b 3 which rise sharply for Z > 60 with a peak at Z = 66 (midshell number), where the NL3 and NL3* numbers are somewhat different from each other.It may be noticed that corresponding to dips in the E 3 curve there are peaks in the B(E3) curve.The QRPA results of Holt et al. [2] in this case are not as close to experimental points as for E 3 − , but are still reasonably close.
In Fig. 5(c) g 3 values are displayed( with V fac = 1.0).Unfortunately, there are no experimental or even theoretical results available to compare.Except for 140 Ce the numbers for other nuclei should be reliable, at least qualitatively.At Z = 66 here too NL3 and NL3* values of g 3 are not very close to each other.At Z = 50 and 52 the value of g 3 is the lowest as the neutrons contribute about 60% to the total QRPA wave function normalization (see Fig. 4).

B. N = 126 isotones
Here the number of nuclei considered is rather small in view of quite scarce experimental information.As already mentioned, the considered nuclei are even-A with Z = 76-92; three nuclei below Z = 82 and five with Z > 82.Excitation energy for the lowest 2 + state is known only for Z = 78-90, and B(E2) transition rate only for 208 Pb and 210 Po.Information on 3 − excited states is even more scarce.
Like in Table II the binding energy per particle and the pairing energy (due to protons) are presented in Table III.As for N = 82 isotones, here too NL3 shows a slight overbinding compared to NL3* values, and both give a little overbinding compared to the experimental value.Regarding the pairing energy, it vanishes for Z = 92 due to a shell gap above 1h 9/2 sp state.For a visual benefit these are also displayed in a B/A versus A plot in Fig. 6.The difference in theoretical and experimental values is clearly visible.Specifically for 208 Pb which is used in the fitting of the force parameters the binding energy reproduction is within about 1% and that for 216 Th it is within about 4%. the Fig. 7 are displayed the results for E 2 , B(E2) ↑ and g 2 of the N = 126 isotones.In Fig. 7(a) the RQRPA calculated energies are in reasonably good agreement with the data for A = 208-216.For 218 U the computed value is very high, reflecting the subshell closure at Z = 92 which may not be realistic.However, with V fac = 1.10 it becomes 2.977 MeV, lowered by about 500 keV.The curve for A = 202-206 is flat at ≈2.0 MeV, about an MeV higher than the experimental value for 204 Pt.
In Fig. 7(b) the B(E2) rates are displayed as a function of A which vary in an oscillatory manner.As reported awhile ago [12], there is good agreement with data for 208 Pb, but the steep drop at Z = 84 in the experimental value seems a puzzle.Considering 208 Pb as an inert core there are two protons in 1h 9/2 orbitals, which should lead to a reasonably large transition quadrupole moment.The calculated number shows slight rise for 210 Po.When the E 2 significantly drops relative to 208 Pb, the drop in B(E2) as well seems very unusual.Another point to be noted is that E 2 is changing very linearly from Z = 84-90, whereas the calculated B(E2) ↑ values show an inverted parabola type variation.However, if V fac = 1.10The variation of g 2 with A is shown in Fig. 7(c) with no experimental or theoretical data available with which to compare.The Z/A curve is far below these RQRPA values.The sudden drop of g 2 from A = 206 to 210 indicates a sudden change in the single-particle structure around the Fermi level.For 218 U here again the value of g 2 changes in the presence of pairing to 0.880 with an increase of about 20%.It seems difficult to understand the behavior of B(E2) and g 2 in terms of the variation of I n with A (see Fig. 8).Except for 208 Pb the contribution of neutrons for J = 2 is negligible.For Z < 80 there is only a slight tendency of increase with the decrease of Z. So, first of all there seems no shell quenching at N = 126 (see the discussions in Ref. [5]).The value of B(E2) is increasing in going down from 206 Hg to 202 Os, whereas the contribution of protons to the total QRPA wave function normalization (I = 1), I p shows conversely a slight decreasing trend with values as 0.983, 0.959, and 0.937, respectively.Looking at the contribution to I p of some of the two quasiparticle pairs, like (1h 11/2 ,1h 11/2 ) and (2d 5/2 ,2d 3/2 ) it is found that these go on increasing from Hg to Os.The former pair gives 0.012, 0.096, 0.258 and the latter gives 0.006, 0.032, 0.057, respectively.The quadrupole matrix elements of high-l pairs should be larger than those of the low-l pairs and this may explain the rise of B(E2).A more detailed analysis is difficult in such a large basis of states and complex involved calculations in a transformed canonical basis.
Regarding some understanding of the dependence of g 2 on A, one may look at some two quasiparticle states near the Fermi level (smallest quasiparticle energy).For Po to Th when I p is about 98% the value of I p from a single quasiparticle pair (1h 11/2 ,1h 9/2 ) is about 95%, thereby explaining the almost same value of g 2 for all of these nuclei.For the lighter mass nuclei three to four pairs contribute with some varying degrees: (2d 3/2 ,3s 1/2 ), (2d 5/2 ,3s 1/2 ), (1h 11/2 ,1h 11/2 ), and so on.
Finally, in Fig. 9 is displayed the variation of E 3 , B(E3) and g 3 for J = 3 − states of these nuclei on which experimental data available are only a few.Also, for J = 3 − no other recent theoretical result seems to be available.As the figure shows, in Fig. 9(a) is displayed the variation of E 3 − with A. Experimental information is available only for Pb, Po, and Th.While the agreement with Pb is good, it looks totally off for the other two nuclei.Corresponding to NL3 and NL3* forces the numbers show some differences beyond 206 Hg.
In Fig. 9(b) the variation of B(E3) transition rates versus A is shown which looks like an inverted parabola from A = 204 to 218, with a maximum at around 212 Rn (orbital 1h 9/2 half filled).The values for 202 Os and 204 Pt are almost the same.The agreement with only two experimental points for 208 Pb and 210 Po is not so bad (higher by about 0.1 unit of e 2 b 3 ) keeping in mind that there is no freedom to play with any parameter.Another way to look at is that it is almost a constant varying between 0.75 to 0.80 units for A = 208-216 with a sudden drop at A = 218.The I p decreases almost linearly from 0.77 at A = 202 to 0.42 at A = 208, but B(E3) shows jump.So, as Z increases and the proton Fermi level goes up the contribution from high-l two quasiparticles increases for I p like: (2d 3/2 ,h 9/2 ), (1h 11/ ,1i 13/2 ), (1h 9/2 ,1i 13/2 ), and so on.This should be the reason for the increase of these B(E3) rates.
The curve showing the variation of g 3 in Fig. 9(c) looks almost like a mirror image of the B(E3) plot above.Again as Fig. 8 shows, due to the increasing role of the neutrons with the increase of Z the g factor shows a decrease except at the two ends with the lowest and highest Z values.Certainly it cannot just be a smooth function of I n or I p , it has to depend on the sp levels near the Fermi surface including that of neutrons.
Again in the presence of nonzero pairing with V fac = 1.10 the values for the 3 − state of 218 U get changed to the following numbers: E 3 = 2.437 MeV (about 500 keV decrease), B(E3) = 0.8368 e 2 b 3 (about 70% increase), and g 3 = 0.803 (a decrease by about 20%).

IV. CONCLUSION
Excitation energies, reduced transition rates, and g factors of lowest 2 + and 3 − states of several even-even N = 82 and N = 126 isotones were studied following RQRPA formalism for spherical nuclei.These calculations are performed employing the well-known NL3, as well as its recently revised version NL3 * set of the RMF Lagrangian parameters.In all, 28 nuclei for two angular momentum cases were considered and it turns out that the use of NL3 * shows hardly any significant difference on the values that were obtained with the use of the original version.
As far as N = 82 isotones are concerned, the experimental data on B(E2), B(E3), g 2 , and g 3 are limited only to some nuclei in the Z = 50-64 region.
In view of there being no free adjustable parameter, the RQRPA numbers are over all in reasonable agreement with these data, with exceptions at Z = 58, 62 for B(E2), Z = 60, 62 for g 2 and Z = 58, 64 for B(E3).In the case of 140 Ce a good improvement is obtained if the pairing interaction strength is increased by about 10%.Furthermore, it may be noted that there is no other theoretical model calculation that reproduces these properties in a better agreement than what is obtained here.
The experimental data for N = 126 isotones are even more scarce.Out of the nine nuclei considered here, E 2 excitation energies are known for seven nuclei, and B(E2) transition rates are available only for two, 208 Pb and 210 Po and none for g factors.The excitation energies for Z 82 are well reproduced in RQRPA, though for 204 Pt and 206 Hg the agreement is not as good.The B(E2) value for 208 Pb is in good agreement, but the case of 210 Po seems quite puzzling.The situation with E 3 − and B(E3) is still worse as far as the experimental information is concerned.
A simultaneous reproduction of spectroscopic properties of 2 + as well as 3 − states of these nuclei seems to be a very challenging task.Thus, there is a need for more efforts on experimental measurements as well as theoretical calculations.
Another aspect discussed above has been the quenching of N = 82 and N = 126 shell gaps for very neutron-rich nuclei.The present calculation seems to support the idea of shell quenching at N = 82 for Z < 44.On the other hand, this does not seem to be the case at N = 126.

3 FIG. 4 .
FIG. 4. Variation of the neutron contribution I n to the total overlap I normalized to unity as a function of the mass number for N = 82 isotones.The upper panel (a) corresponds to the NL3 set of the parameters and the lower one to NL3 * .

TABLE I .
[22]meter sets NL3[21]and NL3*[22]of the effective forces used in the present calculation including the nucleon mass m.

TABLE II
[24] 2. Variation of binding energy per nucleon (B/A) with massnumber A for N = 82 isotones.Experimental data are taken from Ref.[24].

TABLE III .
Same as in TableII, but for N = 126 isotones.Like for 140 Ce, if V fac = 1.10 is used for 218 U then the proton pairing energy becomes 9.343 MeV and B/A = 7.691 MeV.