Pairing and (9/2)n configuration in nuclei in the 208Pb region

Excited states in low-energy spectra in nuclei near 208 Pb are considered. The pure (j = 9/2) n configuration approximation with delta-force is used for ground state multiplet calculations. The multiplet splitting is determined by the pairing energy, which can be defined from the even-odd straggering of the nuclear masses. For the configurations with more than two valence nucleons, the seniority scheme is used. The results of the calculations agree with the experimental data for both stable and exotic nuclei within 0.06-6.16%. Due to simplicity and absence of the fitted parameters, the model can be easily applied for studies of nature of the excited states in a wide range of nuclei.


Introduction
Pair interaction of identical nucleons significantly effects on the properties of atomic nuclei [1]. A manifestations of the nucleon pairing is the mass surface splitting and the zero angular momentum of the even-even nuclei ground state. The breaking of two valence identical nucleons pair leads to formation of a characteristic set of the excited states with the positive parity and even spin, the socalled ground state multiplet (GSM). The GSM splitting is determined by the pairing energy (or by the doubled even-odd staggering effect): ∆ nn(pp) = 2∆ n(p) [2]. The classical example of GSM is the set of low-lying states in the 210 Pb and 210 Po spectra, which can be considered as the 208 Pb core with the pair of the valence neutrons or protons. Since 208 Pb is the heaviest double magic stable nucleus, this approximation is quite accurate. Therefore, it is interesting to trace the GSM formation in the systematics of isotopes spectra in the (N, Z) = (126, 82) region with sequential filling of the 2g 9/2 subshell with neutrons and the 1h 9/2 subshell with protons. As an example, figure 1 shows the experimental excitation spectra of the even-even isotones N = 128. It can be seen that the neutron pairing energy systematically corresponds to the excited state J P = 8 + , and the proton pairing energy corresponds to the state J P = 10 + . A similar systematic behavior is observed in other chains of isotones and isotopes in the 208 Pb region [6].  [3]. Here and below, the experimental data are from [4,5]. The figure is from [6].
approximation, the energy of the state with a total angular momentum J depends on V 0 and the radial Slater integral F 0 (nl) [7,8]: Since the values of V 0 and F 0 (nl) do not depend on J, the structure of the GSM can be determined from the ratio of the energies of levels with different values of J, to the known experimental value of the excitation energy, for example, E(2 + ) [9] or E(J max ), where J max = 2 j − 1 [10]. However, the GSM structure can be calculated using the ground state energy shift ∆E 0 , defined by the even-odd staggering effect: ∆E 0 = ∆ nn(pp) = 2∆ n(p) [11]. Thus, the energies of all the excited GSM states are determined by the values of the nuclear masses.
The δ-potential approximation allows one to describe GSM in the cases of several pairs or an odd number of valence nucleons using the seniority scheme. The seniority model makes it possible to express multiplets of states in the case of the nucleon number N > 2 on the basis of the known spectrum of states of a nucleon pair through the sets of coefficients of fractional parentage (CFP) ⟨ j N−1 s 1 J 1 ; j|} j N sJ⟩ [12]: where seniority s is the number of unpaired nucleons, s 1 = s − 1, J 1 is the total spin corresponding to s 1 , the sum runs over all values of J 1 . This approach is actively used for the GSM description in the 40 Ca region, because the seniority is a good quantum number for any two-particle interaction for j ≤ 7/2. Moreover this approach can be used for higher values of j for a wide range of two-particle interactions, including the δ-potential [9].  Table 1 shows the possible values of the total angular momentum J for the subshell j = 9/2 for different values of the seniority s. With increasing j, the number of possible states essentially increases. Usually in the experimental spectra the whole set of states for s = 4 is not observed, but the levels with high values of J = 10 and 12 are well identified.

Results
The neutron-rich isotopes of lead are ideal for testing of the properties of nucleon-nucleon interaction and the seniority scheme since they have the closed shell core 208 Pb. Owing to the development of experimental possibilities, it became possible to determine the position of excited states in the neutronrich lead isotopes up to 216 Pb. In figure 2 the results of our calculations for the seniority s = 2 are presented, assuming the filling of the subshell 2g 9/2 by neutrons. Use of ∆ nn as the GSM splitting leads to a good agreement for the state J P = 6 + and allows us to estimate the energy of the state J P = 8 + . The pure-configuration (2g 9/2 ) n approximation with δforce cannot correctly reproduce positions of the J P = 2 + , 4 + levels. The regard for the configuration mixing leads to the substantial improvement of the experimental data description. Figure 2 shows the results of the calculations performed in the shell model (SM) in the (g 9/2 i 11/2 d 3/2 d 5/2 g 7/2 s 1/2 j 15/2 ) n configuration space [13]. The wide configuration space in combination with the Kuo-Herlig potential modified for heavy nuclei [14] makes it possible to describe the experimental spectra with an accuracy of about 100 keV [13].
Isotones with N = 126 represent an excellent example of the applicability of the senority scheme with s > 2 [15]. Calculations in the entire range of proton configurations using the modified Kuo-Herling interaction showed a good agreement with the experimental data and confirmed the seniority conservation for the isotones up to Z = 91. The results were obtained in a wide configuration space, and it would be interesting to verify correlation between the experimental spectra and calculations in the pure (1h 9/2 ) n configuration. In figure 3a the experimental spectrum of the even-odd 211 At isotope are presented. Our calculations in the configuration approximation (1h 9/2 ) 3π show a good agreement with experiment except for the first state (7/2) − . For comparison, in figure 3a the results of the calculations in the shell model using an effective potential based on the Bonn potential A (SM2) [16] are also present. The states shown in the figure were obtained with the configuration (1h 9/2 ) 3π , except for the first state (7/2) − , for which the contribution of the (1h 2 9/2 2 f 7/2 ) π configuration dominates. In figure 3b, the spectra of the even-odd isotopes 213 Fr and 215 Ac calculated assuming the pure (1h 9/2 ) n configuration for seniority s = 1 and 3 are compared with the experiment. The experimental spectra themselves no longer contain all the levels predicted by the seniority scheme. It should be noted that these isotopes belong to the region of neutron-rich short-lived nuclei and the precise determination of their spectra of excited states is a complex experimental task. Low-lying excitations J P = 13/2 − , 17/2 − , 21/2 − are defined with good reliability and interpreted as the (1h 9/2 ) 5π configuration in 213 Fr [17] and the (1h 9/2 ) −3π configuration in 215 Ac [18]. Our calculations show the authenticity of this interpretation. However, the calculated energies of levels with J P = 21/2 − lie below the experimental values. Thus, the deviation of the calculated values from the experimental values is 6%. This value corresponds to the accuracy of our model description of the experimental data considered in this paper.
The results of the calculations for s = 4 are shown in figure 4 for spectra of the 212 Rn and 214 Ra nuclei with 4 protons and 4 proton holes in the 1h 9/2 shell. For comparison, the results of calculations [13,19] and [16] (SM3) are given. One can see that use of pairing energy estimated from the nuclear mass data [3] provide a good description not only for GSM with the senority s = 2, but also for experimental states with J P = 12 + , 10 + with s = 4.

Conclusions
Low-lying excited states of neutron-rich lead isotopes and the isotopes of the N = 126 chain are considered. Basing on the nuclear mass data [3] multiplets for the seniority values s = 2, 3, 4 are calculated in the configuration approximation (2g 9/2 ) n for the valence neutrons and (1h 9/2 ) n for the valence protons. The results are compared with the shell model calculations with the effective potentials and the widest possible set of configurations for valence nucleons.
Since the 208 Pb core is a closed shell double magic nucleus, the approximation of pure configurations for the valence nucleons is appropriate, and the low-lying part of the spectrum obeys the seniority scheme. The approach of δ-potential considered in this paper is a simplified form of residual interaction and can not lead to an accurate description of the spectrum of excited states. Nevertheless, the application of the pairing energy estimated from the nuclear masses as the splitting of GSM allows us to obtain a agreement with accuracy 6% with the experimental data not only for the seniority s = 2, but also for the higher s values without involving fitting parameters.