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Neutron Drip Line of Z = 9–11 Isotopic Chains* Chinese Physics C Vol. 44, No. 7 (2020) 074101 Neutron drip line of Z = 9–11 isotopic chains* 1 1 1;1) 1,2,3,4;2) Rong An(安荣) Guo-Fang Shen(申国防) Shi-Sheng Zhang(张时声) Li-Sheng Geng(耿立升) 1School of Physics, Beihang University, Beijing 100191, China 2Beijing Key Laboratory of Advanced Nuclear Materials and Physics, Beihang University, Beijing 100191, China 3Beijing Advanced Innovation Center for Big Data-based Precision Medicine, Beihang University, Beijing 100191, China 4School of Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China Abstract: A recent experimental breakthrough identified the last bound neutron-rich nuclei in fluorine and neon iso- topes. Based on this finding, we perform a theoretical study of Z = 9, 10, 11, 12 isotopes in the relativistic mean field (RMF) model. The mean field parameters are assumed from the PK1 parameterization, and the pairing correlation is described by the particle number conservation BCS (FBCS) method recently formulated in the RMF model. We show that the FBCS approach plays an essential role in reproducing experimental results of fluorine and neon isotopes. Fur- thermore, we predict 39Na and 40Mg to be the last bound neutron-rich nuclei in sodium and magnesium isotopes. Keywords: drip line, relativistic mean field theory, particle number conservation DOI: 10.1088/1674-1137/44/7/074101 1 Introduction Numerous theoretical studies on the location of the neutron drip line have been conducted. Table 1 presents a partial list of various theoretical predictions in comparis- Properties of neutron-rich nuclei, in particular, the on with the experimental data. Clearly, not all of the the- location of the neutron drip line, play an important role oretical results agree with those of the experiments and not only in understanding nuclear stability with respect to with each other. This is well understood, as the exact loc- the isospin in hereto unexplored regions of the nuclear ation of the neutron drip line is sensitive to the details of chart [1], but also in numerous related scientific issues of the structure of nuclei, such as the shell evolution, coup- current interest. For instance, the r-process nucleosyn- ling between the continuum and bound states, deforma- thesis of heavy elements in stellar evolution crucially de- tion effect, and three-body force. For example, the au- pends on the values of beta decay rates and neutron cap- thors of Ref. [47] argued that the neutron drip line is re- ture cross sections in neutron-rich nuclei that do not exist lated to closing (sub)shell orbitals, and therefore, the drip in terrestrial conditions [2–4]. The masses of neutron-rich line of fluorine and neon may be due to the closure of the nuclei impose stringent constraints on the equation of 2p3=2 orbital, which results in formation of the drip line state of neutron-rich nuclear matter, which is key to un- nuclei of 31F and 34Ne with N = 24 [14]. In Refs. [48–50], derstanding the properties of neutron stars and super- the extra stability of the neutron-rich isotopes of fluorine novae explosions [5–9]. and neon relative to oxygen isotopes were attributed to New generation facilities, such as the Facility for the emergence of the island of inversion (Z = 10 − 12, Rare Isotope Beams (FRIB) at Michigan State University N = 20 − 22, and their neighbors), where the ground states and the Radioactive Isotope Beam Factory (RIBF) at gain energy by strong deformation. In Ref. [51], the RIKEN, have helped in conducting studies of nuclei up to three-body force was shown to be responsible for mov- extreme isospin symmetry in the last two decades ing the oxygen drip line from 28O to 24O. [10–13]. At the RIBF, the heaviest fluorine and neon iso- At the mean field level, pairing correlations and con- topes were recently determined to be 31F and 34Ne [14], tinuum effects play an important role in describing the extending the neutron drip line from Z = 8 [15–19], de- ground-state properties of drip-line nuclei [23, 52-64]. termined twenty years ago, to Z = 10. Pairing correlations are responsible for scattering nucle- Received 5 February 2020, Published online 26 April 2020 * Partly supported by the National Natural Science Foundation of China (11735003, 11975041, 11775014, 11961141004), the fundamental Research Funds for the Central Universities 1) E-mail: [email protected] 2) E-mail: [email protected] ©2020 Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd 074101-1 Chinese Physics C Vol. 44, No. 7 (2020) 074101 Table 1. Last bound neutron-rich nuclei with Z = 8 − 12 predicted by various theoretical models in comparison with experimental data. theory experiment Z = 8 24O [20, 21], 26O [22], 28O [7, 8, 23-26] 24O [15-18, 27 28] Z = 9 29F [7, 8, 21, 23, 24], 31F [29, 30], 33F [31] 31F [14, 27] Z = 10 30Ne [25], 32Ne [32], 34Ne [7, 8, 20, 21, 24, 29, 31, 33, 34], 38Ne [35], 40Ne [23], 42Ne [26] 34Ne [14] Z = 11 37Na [21, 33, 34], 39Na [25], 45Na [26, 36] 39Na (only one event) [14] Z = 12 40Mg [21, 29, 37-41], 42Mg [25, 42-45], 44Mg [46], 46Mg [26, 35, 45] µ µ on pairs located at the single-particle level below the L = ¯[iγ @µ − M − gσσ − γ (g!!µ + gρ~τ · p~ρµ − τ Fermi surface into the level above with low orbital angu- 1 3 fπ µ −! −! + e Aµ) − γ5γ @µ π · p τ ] lar momentum. Although the Bogoliubov method is argu- 2 mπ ably more appropriate to deal with the pairing correla- 1 µ 1 2 2 1 3 1 4 + @ σ∂µσ − mσσ − g σ − g σ tions in dripline nuclei [57, 65], the BCS method, if 2 2 3 2 4 3 treated properly, can offer a reasonable description 1 µν 1 2 µ 1 µ 2 − Ω Ωµν + m!!µ! + c (! !µ) [66–69]. However, both methods have the drawback that 4 2 4 3 the particle number is not conserved. Around the Fermi 1 ~ ~µν 1 2 µ 1 µ 2 − Rµν · pR + mρ~ρ · p~ρµ + d (~ρ ~ρµ) surface, only few nucleons play an important role in de- 4 2 4 3 termining the properties of neutron-rich nuclei, and the 1 µν − F Fµν; (1) non-conservation of particle number may become partic- 4 ularly relevant. Therefore, the restoration of the particle where M depicts the mass of the nucleon and mσ , m!, and number conservation is preferred, as shown in the present mρ are the masses of the σ, !, and ρ mesons, respectively. 2 study. Here, gσ, g!, gρ, g2, g3, c3, d3, and e =4π are coupling Based on these considerations, in the present work, constants for the σ, !, ρ mesons and photon. is the Dir- we study fluorine, neon, sodium, and magnesium iso- ac spinor for the nucleon. The field tensors for vector topes in the relativistic mean field model and employ the mesons and the photon are defined as Ωµν = @µ!ν − @ν!µ, ~ recently developed particular number conservation BCS Rµν = @µ~ρν − @ν~ρµ − gρ(~ρµ ×~ρν), and Fµν = @µAν − @νAµ. (FBCS) approach to deal with pairing correlations [70]. In the present study, we employ the parameter set We focus on the differences between the results obtained PK1 [80] for the mean-field effective interactions. The by the RMF+BCS and RMF+FBCS approaches to ex- PK1 parameter set is obtained by fitting to the masses of plore the relevance of the latter in the description of neut- a number of spherical nuclei and the compression modu- lus, baryonic density at saturation, and asymmetry en- ron-rich nuclei and in terms of predicting the location of ergy of nuclear matter. It could provide a description of the neutron drip line. both the empirical properties of nuclear matter and the This paper is organized as follows. In Sec. 2, we ground-state properties of finite nuclei better than most briefly describe the relativistic mean field model and the other parameter sets of the same group [80]. Because our FBCS approach. In Sec. 3, we present the results for flu- main interest is to study the impact of the pairing correla- orine, neon, sodium, and magnesium isotopes and dis- tion, particularly the particle number conservation, on the cuss the differences between the BCS and FBCS ap- prediction of the dripline, we should only compare res- proaches. A short summary and outlook are provided in ults obtained with the same mean-field effective force. Sec. 4. Nevertheless, we performed studies using the NL3 para- meter set, as in Ref. [70], and our conclusion remains 2 Theoretical model qualitatively unchanged. In the future, one may evaluate whether the impact of the particle number conservation on drip-line nuclei remains in the point-coupling version The relativistic mean field models or covariant dens- of the RMF model. ity functional theories have made remarkable progress in From the Lagrangian density, by employing the so- describing various nuclear physics phenomena. Refs. called no-sea and mean-field approximations, the Dirac [71–79] provide further detail on this topic. In the present equation can be obtained for the nucleon, and the study, we take the more conventional meson-exchange Klein–Gordon equations can be obtained for the mesons. formulation, starting from the Lagrangian density ex- These equations can be solved self-consistently either in pressed as Eq. (1), which contains the nucleon and the ex- coordinate space or using the basis expansion method.
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