Investigations on Nuclei near Z = 82 in Relativistic Mean Field Theory with FSUGold

Plasma Science and Technology, Vol.14, No.6, Jun. 2012 Investigations on Nuclei near Z = 82 in Relativistic Mean Field Theory with FSUGold SHENG Zongqiang ( ) 1,2, REN Zhongzhou ( ) 1,2,3 1 Department of Physics, Nanjing University, Nanjing 210093, China 2 Kavli Institute for Theoretical Physics China, Beijing 100190, China 3 Center of Theoretical Nuclear Physics, National Laboratory of Heavy-Ion Accelerator, Lanzhou 730000, China Abstract In this work, the ground-state properties of Pt, Hg, Pb, and Po isotopes have been systematically investigated in the deformed relativistic mean field (RMF) theory with the new parameter set FSUGold. The calculated results show that FSUGold is as successful as NL3 in reproducing the ground-state binding energies of the nuclei in this region. The calculated twoneutron separation energies, quadrupole deformations, and root-mean-square charge radii are in agreement with the experimental data. The parameter set FSUGold can successfully describe the shell effect of the neutron magic number N = 126 and give smaller neutron skin thicknesses than NL3 for all the nuclei considered. Keywords: relativistic mean field theory, parameter set FSUGold, binding energy, deformation PACS: 21.10.-k, 21.10.Dr, 21.60.-n DOI: 10.1088/1009-0630/14/6/23 1 Introduction In recent decades, the relativistic mean field (RMF) model has witnessed great success in describing many nuclear phenomena [1 8]. There are some successful RMF parameter sets, such as NL3 [9,10], NL-SH [11], TMA [12,13] etc. The parameter set NL3 was presented by LALAZISSIS et al. in 1997 [9]. It has been widely used to describe the properties of nuclear matter and finite nuclei and got remarkable success. With NL3, the calculated neutron skin thickness in 208 Pb is 0.28 fm [10,14]. However, various models give very different predictions for the neutron skin thickness in 208 Pb, ranging from 0.1 fm to 0.28 fm [15 19]. This uncertainty may be attributed to the poor knowledge of the density dependence of the symmetry energy. In the RMF model, the isoscalar-isovector was introduced by HOROWITZ et al. in 2001 to simulate the various density dependences of the symmetry energy [14]. The role of the isoscalar-isovector coupling has been extensively explored in finite nuclei and neutron star matter [19 23]. In 2005, TODD-RUTEL and PIEKAREWICZ proposed a new parameter set FSUGold [24] which includes an isoscalar-isovector coupling term to simulate appropriately the density dependence of the symmetry energy. With this parameter set, the neutron skin thickness in 208 Pb is reduced to 0.21 fm, which is in agreement with the one extracted from the isospin diffusion data from intermediate-energy heavy-ion collisions [15]. The parameter set FSUGold has been successfully used to investigate nuclear matter and some spherical nuclei [25 27]. It is of significance to use it to calculate the properties of some deformed nuclei. In our previous work, we have systematically investigated the ground-state properties of rare-earth even-even nuclei and nuclei near Z = 50 in the deformed RMF model with FSUGold, and its validity and reliability have been tested for spherical and deformed nuclei [28,29]. In this work, we will extend our investigation to another region Z = 82. It is a very important region for the purpose of investigating nuclear structure. It is well known that Z = 82 is a typical proton magic number. There are plenty of experimental data available and many theoretical models have been employed to investigate the nuclei in this region. We intend to validate the deformed RMF calculations with FSUGold on the structure and properties of both the spherical and deformed nuclei in this region. It is interesting to examine the effect of the isoscalar-isovector coupling on the structural properties of nuclei. The paper is organized as follows. In section 2, a brief review of the RMF theory is provided. Numerical results and discussions are presented in section 3, and summary is given in the last section. 2 Theoretical framework In the RMF model, the nuclear interaction is described via the exchange of three mesons: the isoscalar meson φ, which provides the medium-range attraction between the nucleons, the isoscalar-vector meson V µ, which offers the short-range repulsion, and the isovector-vector meson b µ, which gives the isospin de- supported by National Natural Science Foundation of China (Nos. 10535010, 10675090, 10775068, 10735010, 10975072, 11035001); 973 National Major State Basic Research and Development of China (Nos. 2007CB815004, 2010CB327803); CAS Knowledge Innovation Project (No. KJCX2-SW-N02); Research Fund of Doctoral Point (RFDP) (No. 20070284016); Science Foundation of Educational Committee of Anhui Province(No. KJ2012A083)

SHENG Zongqiang et al.: Investigations on Nuclei near Z = 82 in RMF Theory with FSUGold pendence of the nuclear force. A µ is the photon field, which is responsible for the electromagnetic interaction. The effective Lagrangian density is given as follows [20,24] L = ψ[γ µ (i µ g v V µ g ρ 2 τ b µ e 2 (1 + τ 3)A µ ) (M g s φ)]ψ + 1 2 µ φ µ φ 1 2 m2 sφ 2 1 4 V µν V µν + 1 2 m2 ωv µ V µ 1 4 bµν b µν + 1 2 m2 ρ b µ b µ 1 4 Fµν F µν U eff (φ, V µ, b µ ). (1) The various field tensors are given as V µ ν = µ V ν ν V µ, b µ ν = µ b ν ν b µ, F µ ν = µ A ν ν A µ. (2) The self-interacting terms of φ, V µ mesons and the isoscalar-isovector coupling one are expressed in the following form U eff (φ, V µ, b µ ) = κ 3! (g s φ) 3 + λ 4! (g s φ) 4 ς 4! (g2 v V µ V µ ) 4 Λ v (g 2 ρ b µ b µ ) (g 2 v V µ V µ ). (3) Here, the new additional isoscalar-isovector coupling term (Λ v ) is used to modify the density dependence of the symmetry energy and the neutron skin thicknesses of heavy nuclei [24]. Because of the time reversal symmetry in even-even nuclei, the spatial components of the vector meson fields can be disregarded. In the RMF approximation only the 0th-component of the ρ meson is retained according to the charge conservation. By using the classical variation principle, the Dirac equation for nucleon spinors reads { iα + β [M S(r)] + U(r)} ψ i (r) = ε i ψ i (r), (4) where the scalar and vector potentials are respectively taken as { S(r) = gs φ(r), U(r) = g v V 0 (r) + g ρ 2 τ 3 b 0 (r) + e 2 (1 + τ 3)A 0 (r). (5) The Klein-Gordon equations for the mesons and photon are ( + m 2 s) φ(r) = g s ρ s (r) κ 2 g3 s φ 2 (r) λ 6 g4 s φ 3 (r), ( + m 2 ω) V 0 (r) = g v ρ v (r) ς 6 g4 v V0 3 (r) 2Λ v gρ 2 gv 2 b 2 0(r) V 0 (r), ( + m 2 ρ) b 0 (r) = g ρ 2 ρ 3(r) 2Λ v gv 2 gρ 2 V0 2 (r) b 0 (r), A 0 (r) = e ρ p (r). (6) Where ρ s, ρ v, ρ p are respectively the densities of scalar, baryon and proton, and ρ 3 is the difference between the neutron and proton densities. These source densities are written as ρ s = A ψ i (r) ψ i (r), ρ v = A ψ + i (r) ψ i(r), ρ 3 = Z ψ p + (r) ψ p (r) N ψ n + (r) ψ n (r), p=1 ρ p = Z ψ p + (r) ψ p (r). p=1 n=1 (7) Eqs. (4) and (6) can be self-consistently solved by iteration. We list the parameters of the parameter set FSUGold [24] in Table 1, and the parameter set NL3 is also shown there for comparison. 3 Numerical results and discussions In this work, we apply the deformed RMF model with FSUGold to study the nuclei near Z = 82. The calculations are focused on even-even isotopes of Pt (Z = 78), Hg (Z = 80), Pb (Z = 82), and Po (Z = 84) with known experimental binding energies in the nuclear mass table [30]. The method of oscillator bases expansions is applied to solve the coupled RMF equations. The numbers of bases are chosen as N f = N b = 14. The pairing correlations are simply treated by BCS approximation. The inputs of pairing gaps are n = p = 11.2 A MeV. (8) The cut-off energy for the pairing window is set as ε i λ 2 (41A 1/3 ) MeV [31]. An axial deformation is initially assumed in calculations. The whole calculations are performed with the inputs of initial guess deformations β 0 = 0.2, 0.1, 0.0, 0.1, 0.2, 0.3. We choose the biggest calculated binding energy as the groundstate binding energy for a nucleus. The charge radii are taken as R c = Rp 2 + 0.862 2 0.336 2 (N/Z) fm [32]. The center-of-mass correction to binding energy is given as E c.m. = 0.75 41 A 1/3 MeV. For the details of calculations, please refer to Refs. [1,2,31]. Table 1. Parameters of the parameter sets FSUGold and NL3. The nucleon, omega, and rho masses are kept fixed at M = 939 MeV, m ω = 782.5 MeV, and m ρ = 763 MeV, respectively Force m s (MeV) gs 2 gv 2 gρ 2 κ (MeV) λ ζ Λ v FSUGold 491.5000 112.1996 204.5469 138.4701 1.4203 +0.0238 0.0600 0.0300 NL3 508.1940 104.3871 165.5854 79.6000 3.8599 0.0159 0.0000 0.0000 535

Plasma Science and Technology, Vol.14, No.6, Jun. 2012 The number of the nuclei under investigation is 70 in this region. Since there are many calculated data, only the results of Hg isotopes are given as an example. The RMF results on binding energies, quadrupole deformations, and charge radii for Hg isotopes are listed in Table 2. The available experimental data are also given for comparison [30,33,34]. Table 2. The ground-state properties of Hg isotopes used in the RMF model with FSUGold. The available experimental data are also listed for comparison [30,33,34] Nuclei FSUGold Expt. B (MeV) β 2 R c (fm) B (MeV) β 2 R c (fm) 172 Hg 1334.300 0.000 5.315 1326.766 - - 174 Hg 1354.668 0.000 5.327 1348.469 - - 176 Hg 1374.020-0.057 5.339 1369.743 - - 178 Hg 1393.621-0.114 5.360 1390.424 - - 180 Hg 1413.983 0.289 5.433 1410.494 - - 182 Hg 1434.282 0.305 5.456 1429.968-5.384 184 Hg 1453.778 0.314 5.476 1448.884 0.160 5.395 186 Hg 1471.507 0.293 5.477 1467.217 0.132 5.401 188 Hg 1488.734 0.307 5.505 1485.022-5.408 190 Hg 1504.033 0.281 5.504 1502.333-5.416 192 Hg 1518.954-0.151 5.447 1519.116-5.423 194 Hg 1535.109-0.142 5.455 1535.441-5.431 196 Hg 1550.686-0.133 5.463 1551.217-0.116 5.439 198 Hg 1565.811-0.119 5.470 1566.487-0.107 5.447 200 Hg 1580.509-0.096 5.476 1581.180-0.098 5.455 202 Hg 1594.755-0.067 5.483 1595.164-0.083 5.463 204 Hg 1608.667-0.001 5.489 1608.651-0.069 5.474 206 Hg 1622.065 0.001 5.498 1621.049-5.485 In Table 2, the symbols B, β 2, and R c denote binding energies, quadrupole deformations, and charge radii, respectively. The first column marks the isotopes. Columns 2 4 correspond to the calculated values for the nuclei tabulated in the first column. The last three columns are respectively the experimental data for the binding energies, quadrupole deformations and charge radii which are taken from the nuclear mass table [30] and Refs. [33,34]. From Table 2, it can be seen that the calculated quantities are in good agreement with the experimental data. For better comparing the theoretical results and experimental ones, we will make some detailed discussions on the calculated results with the help of some figures. Binding energy is a very important quantity for nuclear physics. Whether a theoretical model can quantitatively reproduce the experimental binding energy is a crucial criterion to judge its validity. Seventy even-even nuclei are calculated ranging from Z = 78 to Z = 84. The deviation between the calculated binding energy and the experimental one is defined as B = B Cal B Expt. The deviations for Pt, Hg, Pb, and Po isotopic chains are plotted in Fig. 1 as a function of the neutron number N. In Ref. [10], the nuclei considered in this work are calculated with NL3. Their results are also given for comparison. In Fig. 1, the filled and open circles denote the calculated results with FSUGold and NL3, respectively. Fig.1 The deviations between the calculated binding energies and the experimental ones for Pt, Hg, Pb, and Po isotopic chains. Here, the deviation is defined as B = B Cal B Expt. The filled and open circles denote the calculated results with FSUGold and NL3, respectively It is seen from Fig. 1 that most calculated binding energies with FSUGold are slightly larger than the experimental values in neutron-deficient region. For Pt, Hg and Po isotopic chains, the results with FSUGold are very similar to the ones with NL3. For Pb isotopic chain, the results with FSUGold are different from the ones with NL3 in neutron-deficient region. The deviations obtained with FSUGold are generally less than 5 MeV. The largest deviation is 7.543 MeV in 172 Hg, while the relative deviation is less than 0.6%. The average deviation and the root-mean-square (rms) deviation of the binding energies with FSUGold are defined as σ = 70 B Cal i BExpt i /70 = 2.932 MeV, (9) [ 70 1/2 σ2 = (BCal i BExpt) /70] i 2 = 3.402 MeV. (10) We can see that both the σ and σ 2 are small. The σ and σ 2 obtained with NL3 are 2.577 MeV and 3.059 MeV, respectively, which are very close to our results with FSUGold. It indicates that the parameter set FSUGold is as successful as NL3 in reproducing the ground-state binding energies of the nuclei considered here. Two-neutron separation energy S 2n is also a very important quantity for testing the stability of a model. It is defined as S 2n (Z, N) = B(Z, N) B(Z, N 2). (11) The experimental two-neutron separation energies [30] and the calculated results with FSUGold are shown in Fig. 2. For an isotopic chain, the two-neutron separation energies should be getting smaller and smaller as the neutron number N increases. Except for several nuclei, one can see this phenomenon for all the four isotopic chains in Fig. 2. For Pt and Hg isotopic chains, the experimental curve is nearly a line, but there are some 536

SHENG Zongqiang et al.: Investigations on Nuclei near Z = 82 in RMF Theory with FSUGold Fig.2 The two-neutron separation energies S2n for Pt, Hg, Pb, and Po isotopic chains. The filled and open circles denote the experimental values and the calculated results with FSUGold, respectively fluctuations in the theoretical curve in neutron-deficient region. For Pb and Po isotopic chains, the calculated two-neutron separation energies are in good agreement with the experimental values. It can be clearly seen that there is sudden decline at N = 126 both in the experimental and theoretical curves for Pb and Po isotopic chains, which indicates that N = 126 is a shell closure. It is well known that N = 126 is a typical neutron magic number. The above result shows that the parameter set FSUGold can successfully reflect the shell effect of the neutron magic number N = 126. On the whole, the RMF theory with FSUGold can indeed be used to calculate the two-neutron separation energies of the nuclei here. Quadrupole deformation β2 is also very important for describing structure and shape. The calculated and experimental [33] β2 values are shown in Fig. 3, where the calculated β2 obtained with FSUGold and the experimental values are plotted as a function of the neutron number N. Except for a few Pt and Hg isotopes, the calculated values with FSUGold are in good agreement with the experimental data. The proton number of Pb isotopes is Z = 82, which is a typical proton magic number. The RMF theory predicts that all the isotopes from N = 122 to N = 132 are spherical, while the experimental β2 values show that the Pb isotopes from N = 122 to N = 128 exhibit weak deformation. It must be pointed out that the experimental β2 values are usually extracted from experimental B(E2) values by using the Bohr model, which is valid only for well-deformed nuclei. For this reason, even the doubly closed shell nuclei and obviously spherical nuclei, such as 16 O or 208 Pb, would exhibit small deformations. It is also seen from Fig. 3 that the shapes and changing trends of the theoretical curves are very similar to those of experimental results. In general, the parameter set FSUGold is really capable of describing the quadrupole deformations in this region. Next, we discuss rms charge radii. There are 43 experimental data in this region [34]. The deviation between the calculated and experimental charge radii is defined as Rc = RcCal RcExpt. The deviations are shown in Fig. 4. It is seen from Fig. 4 that the calcu- Fig.3 The calculated quadrupole deformations β2 for Pt, Hg, Pb, and Po isotopic chains obtained with FSUGold. The experimental β2 are also included for comparison Fig.4 The deviations between the calculated and experimental charge radii for the nuclei studied in this work. Here, the deviation is defined as Rc = RcCal RcExpt lated charge radii are in agreement with the experimental data for most nuclei. Most Rc are less than 0.06 fm. On the whole, the agreement between the calculated results and the experimental data is satisfactory. As the uncertainty of the neutron skins ( R = Rn Rp ) for heavy nuclei is unsettled [14 16], for instance, for 208 Pb, it is significant to investigate the neutron skin thickness with FSUGold that differs from NL3 for the density dependence of the symmetry energy. The neutron skin thicknesses for Pt, Hg, Pb, and Po isotopic chains with FSUGold and NL3 are plotted in Fig. 5. It is seen from Fig. 5 that the neutron skins increase monotonously with increasing neutron number for all the isotopic chains with both FSUGold and NL3. The neutron skin thicknesses obtained with NL3 are obviously larger than the ones with FSUGold for all nuclei. With increasing neutron number, the difference between them is getting larger and larger. A typical feature of the parameter set FSUGold is that it can soften the symmetry energy at high densities and offer a smaller neutron skin through adding an additional isoscalar-isovector coupling term. Our results of neutron skin thicknesses are in accordance with the expectations of the parameter set FSUGold. It must be stressed that the neutron skin thicknesses in 208 Pb obtained with FSUGold is 0.209 fm, and is thus closer 537

Plasma Science and Technology, Vol.14, No.6, Jun. 2012 to the experimental value than the one obtained with NL3, which is 0.277 fm [10]. 4 5 6 7 8 9 10 11 Fig.5 The neutron skin thicknesses ( R = Rn Rp ) of Pt, Hg, Pb, and Po isotopic chains in the RMF theory with FSUGold and NL3, respectively 4 Conclusions 12 13 14 15 In this work, the ground-state properties of Pt, Hg, Pb, and Po isotopic chains have been systematically investigated in the deformed RMF theory with FSUGold. The deviations between the calculated binding energies and the experimental ones are less than 5 MeV, and the average deviation and rms deviation of the binding energies with FSUGold are respectively 2.932 MeV and 3.402 MeV, which are very close to the results with NL3. It can thus be concluded that the parameter set FSUGold is as successful as NL3 in reproducing the ground-state binding energies of the nuclei. The calculated two-neutron separation energies are in good agreement with the experimental data. The parameter set FSUGold can successfully reflect the shell effect of the neutron magic number N = 126. Except for a few Pt and Hg isotopes, the calculated quadrupole deformations with FSUGold are in good agreement with the experimental data. The calculated charge radii are also consistent with the experimental data for most nuclei. The neutron skin thicknesses obtained with NL3 are obviously larger than the ones with FSUGold for all nuclei. The neutron skin thicknesses in 208 Pb obtained with FSUGold is 0.209 fm, and is thus closer to the experimental value than the one obtained with NL3, which is 0.277 fm. On the whole, the parameter set FSUGold can be used to describe the nuclei near Z = 82. 16 References 34 1 2 3 538 Serot B D, Walecka J D. 1986, Adv. Nucl. Phys., 16: 1 Ring P. 1996, Prog. Part. Nucl. Phys., 37: 193 Lalazissis G A, Ring P. 1998, Phys. Lett. B., 427: 225 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 Lalazissis G A, Niks ic T, Vretenar D, et al. 2005, Phys. Rev. C, 71: 024312 Ren Z Z, Faessler A, Bobyk A. 1998, Phys. Rev. C, 57: 2752 Ren Z Z, Tai F, Chen D H. 2002, Phys. Rev. C, 66: 064306 Ren Z Z, Zhu Z Y, Cai Y H, et al. 1996, Phys. Lett. B, 380: 241 Jiang W Z, Li B A, Chen L W. 2007, Phys. Rev. C, 76: 054314 Lalazissis G A, Ko nig J, Ring P. 1997, Phys. Rev. C, 55: 540 Lalazissis G A, Raman S, Ring P. 1999, At. Data Nucl. Data Tables, 71: 1 Sharma M M, Nagarajan M A, Ring P. 1993, Phys. Lett. B, 312: 377 Toki H, Hirata D, Sugahara Y, et al. 1995, Nucl. Phys. A, 588: c357 Ren Z Z, Toki H. 2001, Nucl. Phys. A, 689: 691 Horowitz C J, Piekarewicz J. 2001, Phys. Rev. Lett., 86: 5647 Chen L W, Ko C M, Li B A. 2005, Phys. Rev. C, 72: 064309 Piekarewicz J, Weppner S P. 2006, Nucl. Phys. A, 778: 10 Agrawal B K, Shlomo S, Kim Au V. 2003, Phys. Rev. C, 68: 031304(R) Furnstahl R J. 2002, Nucl. Phys. A, 706: 85 Steiner A W, Prakash M, Lattimer J M, et al. 2005, Phys. Rep., 411: 325 Todd B G, Piekarewicz J. 2003, Phys. Rev. C, 67: 044317 Jiang W Z, Zhao Y L, Zhu Z Y, et al. 2005, Phys. Rev. C, 72: 024313 Jiang W Z. 2006, Phys. Lett. B, 642: 28 Jiang W Z, Ren Z Z, Sheng Z Q, et al. 2010, Eur. Phys. J. A, 44: 465 Todd-Rutel B G, Piekarewicz J. 2005, Phys. Rev. Lett., 95: 122501 Piekarewicz J. 2006, Phys. Rev. C, 73: 044325 Piekarewicz J. 2007, Phys. Rev. C, 76: 031301(R) Piekarewicz J. 2007, Phys. Rev. C, 76: 064310 Sheng Z Q, Ren Z Z, Jiang W Z. 2010, Nucl. Phys. A, 832: 049 Sheng Z Q, Ren Z Z. 2010, Eur. Phys. J. A, 46: 241 Audi G, Wapstra A H, Thibault C. 2003, Nucl. Phys. A, 729: 337 Gambhir Y K, Ring P, Thimet A. 1990, Ann. Phys. (N.Y.), 198: 132 Sugahara Y, Toki H. 1994, Nucl. Phys. A, 579: 557 Raman S, Nestor jr C W, Tikkanen P. 2001, At. Data Nucl. Data Tables, 78: 1 Angeli I. 2004, At. Data Nucl. Data Tables, 87: 185 (Manuscript received 17 May 2011) (Manuscript accepted 5 July 2011) E-mail address of SHENG Zongqiang: shengzongq@gmail.com