Anomalously Large Deformation in Some Medium and Heavy Nuclei

Commun. Theor. Phys. (Beijing, China) 44 (2005) pp. 687 696 c International Academic Publishers Vol. 44, No. 4, October 15, 2005 Anomalously Large Deformation in Some Medium and Heavy Nuclei CHEN Ding-Han, 1 REN Zhong-Zhou, 1,2, and TAI Fei 1 1 Department of Physics, Nanjing University, Nanjing 210008, China 2 Center of Theoretical Nuclear Physics, National Laboratory of Heavy-Ion Accelerator at Lanzhou, Lanzhou 730000, China (Received March 4, 2005) Abstract We investigate the properties of the Ce isotopes with neutron number N = 60 90 and the properties of the heavy nuclei near 242 Am within the framework of deformed relativistic mean-field (RMF) theory. A systematic comparison between theoretical results and experimental data is made. The calculated binding energies, two-neutron separation energies, and two-proton separation energies are in good agreement with experimental ones. The variation trend of experimental quadrupole deformation parameters on the Ce isotopes can be approximately reproduced by the RMF model. It is found that there exists an abnormally large deformation in the ground state of proton-rich Ce isotopes. This phenomenon can be the general behavior of proton-rich nuclei on the neighboring isotopic chains such as Nd and Sm isotopes. For the heavy nuclei near 242 Am the properties of the ground state and superdeformed isomeric state can be approximately reproduced by the RMF model. The mechanism of the appearance of anomalously large deformation or superdeformation is analyzed and its influence on nuclear properties is discussed. Further experiments to study the anomalously large deformation in some proton-rich nuclei are suggested. PACS numbers: 21.60.Jz, 27.90.+b, 21.10.Dr, 21.10.Tg Key words: superdeformation, proton-rich Ce nuclei, isomeric state of 242 Am, constraint relativistic meanfield calculationn 1 Introduction The existence of deformation in nuclei is a well-known fact in nuclear physics. The related phenomenon on nuclear deformation is discussed in the famous textbook, Nuclear Structure by Bohr and Mottelson. [1] Actually deformation is a common phenomenon for many nuclei and spherical shape is just a special case that occurs for some nuclei. Systematic measurements on electric quadrupole moments and quadrupole deformation parameters clearly show that many nuclei have deformation in their ground state or in their excited states. [2,3] Usually nuclei have moderate deformation in their ground states, where the absolute value of quadrupole deformation parameter lies in the range of β 2 = 0.1 0.3 for many nuclei. [2,3] For the excited states of nuclei the deformation parameter can be as large as β 2 = 0.4 1.0, where the terminology such as superdeformation or hyperdeformation is used. These are well studied in high-spin physics and they form an important branch in nuclear physics. Besides the known superdeformation and hyperdeformation that occur in the high-spin rotational band of nuclei, it is interesting to investigate whether there exists anomalously large deformation or superdeformation in the ground state of nuclei and in the low excited state of nuclei from both experimental and theoretical sides. In light nuclei there are experimental evidences on anomalously large deformation or superdeformation in the ground state and in the low excited states [1 3] because various exotic shapes are easily observed for light nuclei. Theoretical calculations on the deformation of light nuclei agree reasonably with experimental values. [4 8] On medium nuclei there are a few indications on the existence of abnormally large deformation in the ground state of some medium nuclei. [2,3] For example the experimental quadrupole deformation parameters of the nuclei near 122 Ce are extraordinarily large: β 2 0.354 for 122 Ba, β 2 0.385 for 124 Ce, β 2 0.37 for 130 Nd, and β 2 0.366 for 134 Sm. [2,3] However a detailed theoretical study on the abnormally large deformation in this mass range is rare in publications. For heavy nuclei near 242 Am it has been known for a long period that their isomeric states are superdeformed. [9 13] A few nonrelativistic calculations were carried out to explain the superdeformation in the isomeric state of 242 Am. [1,12 14] In view of the experimental indications that there are anomalously large deformation in the ground state of some medium nuclei and superdeformation in the isomeric state of the heavy nuclei near 242 Am, it is interesting to test the predicting ability of the relativistic mean-field (RMF) model on an extremely large deformation in the ground states of some medium nuclei or in the isomeric state of the heavy nuclei. This is very important for the self-consistent RMF model because the effective force parameters in the The project supported by National Natural Science Foundation of China under Grant No. 10125521, the 973 National Major State Basic Research and Development of China under Grant No. G2000077400, the CAS Knowledge Innovation Project No. KJCX2-SW-N02, and the Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20010284036 E-mail: zren@nju.edu.cn

688 CHEN Ding-Han, REN Zhong-Zhou, and TAI Fei Vol. 44 model are usually obtained by fitting the ground state properties of some spherical nuclei and some quantities of infinite nuclear matter. Although the deformation parameters of nuclei have not been included in fitting the effective forces, the deformation parameters can be given automatically in the mean-field model by solving the coupled mean-field equations in axially deformed bases. It is also important to explore whether the above large deformation is a systematic behavior in the mass range. In this paper we investigate theoretically if there is anomalously large deformation in the ground state of medium nuclei within the framework of the RMF model. We also investigate the superdeformed isomeric state of the heavy nuclei near 242 Am by the model. We will show the energy surface of some medium and heavy nuclei by the constraint relativistic mean-field (CRMF) calculations in order to obtain the theoretical results reliably. For medium-mass nuclei we choose the Ce isotopes and its neighboring nuclei as the candidate of our calculations. This is because there are rare publications on RMF results of the Ce isotopes although there are the RMF calculations on the Nd, Sm, Gd, Dy, Er, and Yb isotopes with N = 70 130. [5,15] Another reason is that the properties of some Ce isotopes near the proton drip line are studied very recently by experimental physicists. [16 19] A few proton-rich Ce isotopes will be the good candidates for future experiments. This paper is organized in the following way. Section 2 is the formalism of the RMF model. The numerical results and discussions of the Ce isotopes and its neighboring nuclei are given in Sec. 3. The results of the heavy nuclei near 242 Am are presented in Sec. 4. Section 5 is a summary. 2 Formalism of Relativistic Mean-Field Theory The relativistic mean-field model is from the quantum hadron dynamics (QHD) by Walecka and Serot. [20] There are four kinds of mesons in QHD. [20] The most important meson in QHD is π meson but the mean-field of π meson breaks parity on the Hartree level. [5,8,20] It is assumed that parity conserves for even-even nuclei and therefore π meson does not enter the equations in the RMF model. [5,8,20] In the RMF approach, we start from the local Lagrangian density for interacting nucleons, σ, ω, and ρ mesons, and photons, [4 8,15,20 24] L = Ψ(iγ µ µ M)Ψ g σ ΨσΨ gω Ψγ µ ω µ Ψ g ρ Ψγ µ ρ a µτ a Ψ + 1 2 µ σ µ σ 1 2 m2 σσ 2 1 3 g 2σ 3 1 4 g 3σ 4 + 1 4 c 3(ω µ ω µ ) 2 1 4 Ωµν Ω µν + 1 2 m2 ωω µ ω µ 1 4 Raµν R a µν + 1 2 m2 ρρ aµ ρ a µ with 1 4 F µν F µν e Ψγ µ A µ 1 2 (1 τ 3 )Ψ (1) Ω µν = µ ω ν ν ω µ, (2) R aµν = µ ρ aν ν ρ aµ, (3) F µν = µ A ν ν A µ, (4) where the meson fields are denoted by σ, ω µ, and ρ a µ and their masses are denoted by m σ, m ω, and m ρ, respectively. The nucleon field and rest mass are denoted by Ψ and M. A µ is the photon field, which is responsible for the electromagnetic interaction, e 2 /4π = 1/137. The effective strengths of the coupling between the mesons and nucleons are, respectively, g σ, g ω, and g ρ. g 2 and g 3 are the nonlinear coupling strengths of the σ meson. c 3 is the self-coupling term of the ω field. The isospin Pauli matrices are written as τ a with τ 3 being the third component of τ a. The equations of motion for the fields are easily obtained from the variational principle. [5,6,8,20,22,25] In order to describe the static properties of nuclei we need static solution of the above Lagrangian. For this case the meson field and photon fields are assumed to be classical fields and they are time-independent (c-numbers). The nucleons move in classical fields as independent particles (mean-field approximations). The Dirac field operator can be expanded in terms of single particle wave functions Ψ = i φ ia i, where a i is a particle creation operator [5,8,20] and φ i is the single particle wave function. For actual calculations, we omit the contribution of the Fermi-sea under no-sea approximations. The sum on single particles states runs on physical bound states, i.e. the occupied shell model states. Symmetries will simplify the calculations considerably. Time reversal symmetry is used and therefore the spacial vector components for ω µ, ρ a µ, and A µ are zero. This is a good approximation for the ground state properties of nuclei. Charge conservation guarantees that only the third-component of the isovector field (ρ 0 0) survives. [5,8,20,22] We denote simply ρ 0 0 as ρ 0. Finally we have the following Dirac equations for nucleons and the Klein Gordon equations for meson fields (for details: see Refs. [5], [8], [20], and [22]): [ iα + βm (r) + V (r)]φ i (r) = ɛ i φ i (r), (5) where the effective mass M (r) = M + g σ σ(r). The potential V (r) is a timelike component of a Lorentz vector, V (r) = g ω ω 0 (r) + g ρ τ a ρ a 0(r) + e 1 τ 3 A 0 (r), (6) 2 ( + m 2 σ)σ(r) = g σ ρ s (r) g 2 σ 2 (r) g 3 σ 3 (r), (7) ( + m 2 ω)ω 0 (r) = g ω ρ v (r) c 3 ω 3 0(r), (8) ( + m 2 ρ)ρ 0 (r) = g ρ ρ 3 (r), (9) A 0 (r) = eρ p (r), (10)

No. 4 Anomalously Large Deformation in Some Medium and Heavy Nuclei 689 where ρ s, ρ v, and ρ p are, respectively, the densities of scalar, baryon, and proton. ρ 3 is the difference between the neutron and proton densities. They will be c-numbers by taking expectation values. Their expressions are as follows: A ρ s (r) = φ i (r)φ i (r), (11) ρ v (r) = ρ 3 (r) = ρ p (r) = i=1 A φ i (r)φ i(r), (12) i=1 A φ i (r)τ 3 φ i (r), (13) i=1 A i=1 φ i (r)1 τ 3 φ i (r). (14) 2 Now we have a set of coupled equations for mesons and nucleons and they will be solved consistently by iterations. After a final solution is obtained, we can calculate the binding energies, root-mean-square radii of proton and neutron density distributions, single particle levels, and quadrupole deformation. An axial deformation is assumed in our calculations for medium and heavy nuclei. The details of numerical calculations are described in Refs. [5], [8], [20], [22], and [25]. The main presentation of this paper is based on deformed RMF calculation. Some constraint RMF results (CRMF) are given as an explanation to the deformed RMF results. In the RMF model the mean-field has even parity after the π meson is dropped off based on parity conservation of even-even nuclei. As the parity conservation and invariance of time reversal are assumed in the RMF model, there are only evenmultipolarity deformations such as quadrupole and hexadecupole deformations in the RMF model. It is impossible to obtain the odd-multipolarity deformations such as octupole deformation in the standard RMF model with σ, ω, and ρ mesons. This is because that odd-multipolarity deformation is zero when there is parity conservation or there is time reversal invariance. [26] 3 Numerical Results and Discussions of Medium Nuclei We carry out RMF calculations with many sets of force parameters, TMA, NL-SH, NL1, NL3, NLZ2. [5,15,25] They are typical forces in the RMF model. The method of harmonic basis expansions [5 7,24,25] is used in solving the coupled RMF equations. The number of bases is chosen as N f = N b = 18 for medium nuclei and N f = N b = 20 for heavy nuclei. This space is enough for the convergence of the binding energy and deformation of our calculations. Some constraint RMF calculations are carried out with N f = N b = 18 in order to save computational time. The inputs of pairing gaps are n = p = 11.2/ A MeV and this is a standard input in nuclear structure calculations. We do not make any adjustments on the current force parameters or on the pairing gaps. An axial deformation is assumed in calculations. For the details of calculations please see relevant publications. [5 7,25] 3.1 Binding Energy, Two-neutron Separation Energies, and Two-proton Separation Energies of Medium Nuclei At first let us focus on the global behavior of the RMF model. We calculate the total binding energy and wave function of even-even nuclei on the Ce isotopic chain and its neighboring nuclei. The theoretical binding energies, quadrupole deformation parameters of protons and neutrons, two-neutron separation energies, two-proton separation energies, and root-mean-square radii are listed in Tables 1 3, together with the available data of the binding energies, of two-nucleon separation energies, and of root-mean-square radii. The RMF results of the Ce isotopes with TMA are listed in Table 1. In Table 1, the first column is the isotopes. B is the theoretical binding energy. The symbols β n and β p in Table 1 denote the quadrupole deformations of neutrons and protons, respectively. Further, the symbols S 2n and S 2p are the calculated two-neutron separation energy and two-proton separation energy. The experimental binding energy, quadrupole deformation parameter, two-neutron separation energy, and two-proton separation energy are listed in columns 2 5 of the table. Experimental binding energies and separation energies are obtained from the nuclear mass table, [27,28] where the values with the symbol # are the estimated values by Audi et al. [27,28] Experimental quadrupole deformation parameters are taken from Raman et al. [2,3] At first let us see the general agreement between the theoretical results and experimental data. It is seen from Table 1 that the theoretical binding energies are very close to the experimental data. The average difference between the theoretical binding energy and experimental one is approximately 2 MeV. This shows that the RMF model can give a reliable binding energy for the nuclei here. The relative difference between the theoretical binding energy and the data is approximately 0.2%. This difference is very small. Therefore we can say that the experimental binding energy can be reasonably reproduced by the RMF model. Although the deviation of the energies is slightly large around neutron number N = 82, the deviation decreases much with the decrease of neutron number. For the proton-rich nuclei with N = 60 76 ( 118 134 Ce) the deviation on the total binding energy is less than 1 MeV. Especially we do not introduce any adjustments on the pairing strength and on the effective force parameters when we carry out calculations. So we can conclude

690 CHEN Ding-Han, REN Zhong-Zhou, and TAI Fei Vol. 44 that the above agreement on the total binding energies is satisfying. It is seen from columns 3 and 4 and columns 7 and 8 of Table 1 that the theoretical two-neutron separation energies and two-proton separation ones are also close to the experimental data. The good agreement between the theoretical results and the data is obtained. Although a slightly large deviation exists around N = 82, the deviation is very small for the interesting range with N = 60 76. This shows clearly that the RMF model has a good predicting precision for the binding energies and separation energies of proton-rich nuclei. The slight deviation of the binding energies and separation energies occurs around N = 82 and the cause of the deviation is due to that the spherical shell effect of N = 82 in the RMF model is stronger than the experimental one. Therefore the RMF model overestimates the binding energies and separation energies near N = 82. When it is away from neutron number N = 82, this overestimate on the binding energies or separation energies disappears. We will discuss this later by analyzing the variation of quadrupole deformation. Table 1 The binding energies, two-neutron separation energies, two-proton separation energies, and deformations of even-even Ce isotopes with N = 60 100. The columns 2 5 are experimental data. The columns 6 10 are theoretical results. The units of the separation energies are MeV. Nuclei B (expt.) (MeV) S 2n (expt.) S 2p (expt.) β 2 (expt.) B (theor.) (MeV) S 2n S 2p β n β p 118 Ce 949.4 28.9 1.7 0.37 0.38 120 Ce 974.8 25.4 2.9 0.37 0.37 122 Ce 997.1# 3.4# 998.8 24.0 4.5 0.37 0.37 124 Ce 1020.2# 23.1# 5.0# 0.39 1021.0 22.2 5.6 0.35 0.35 126 Ce 1042.3# 22.1# 6.2# 0.33 1042.0 21.0 6.2 0.31 0.31 128 Ce 1063.3# 21.0# 7.5# 0.30 1063.3 21.3 7.3 0.26 0.25 130 Ce 1083.4# 20.0# 8.6# 0.26 1084.1 20.8 8.6 0.23 0.23 132 Ce 1102.5# 19.1# 9.8# 0.26 1103.4 19.3 9.4 0.20 0.20 134 Ce 1120.9 18.4# 10.9# 0.20 1121.9 18.5 10.0 0.15 0.15 136 Ce 1138.8 17.9 12.1 0.17 1141.1 19.2 12.1 0.00 0.00 138 Ce 1156.0 17.2 13.3 0.13 1161.2 20.1 15.0 0.00 0.00 140 Ce 1172.7 16.7 14.4 0.10 1179.5 18.3 17.1 0.00 0.00 142 Ce 1185.3 12.6 15.8 0.13 1190.6 11.1 17.8 0.00 0.00 144 Ce 1197.3 12.0 17.2 0.17 1200.3 9.7 18.3 0.01 0.01 146 Ce 1208.7 11.4 18.5 0.19 1210.2 9.9 19.0 0.06 0.05 148 Ce 1219.6 10.9 20.0 0.25 1219.7 9.5 20.0 0.05 0.04 150 Ce 1230.3 10.7 21.5 0.32 1228.3 8.6 20.7 0.21 0.19 152 Ce 1240.5 10.2# 22.9# 1238.9 10.6 24.0 0.31 0.31 154 Ce 1248.0 9.1 24.9 0.33 0.31 156 Ce 1256.5 8.5 26.0 0.34 0.32 158 Ce 1264.4 7.9 27.1 0.35 0.33 3.2 Quadrupole Deformation in Ground State of Medium Nuclei The quadrupole deformation parameters of the Ce isotopes are listed in columns 5, 9, and 10 of Table 1. Column 5 is the experimental data. Columns 9 and 10 are theoretical quadrupole deformation parameters of neutrons and protons, respectively. The experimental data of the quadrupole deformation parameter in column 5 can be compared with those in column 10. The experimental trend of the variation of quadrupole deformation parameters with neutron number agrees with the theoretical one. The quadrupole deformation is the smallest around neutron number N = 82 and it increases when it is away from the magic number N = 82. For a few nuclei near N = 82 the experimental deformation is approximately β 2 = 0.15 but the theoretical deformation is approximately zero. Therefore the spherical shell effect around N = 82 in the RMF model is stronger than the experimental one. This leads to the theoretical overestimate of the binding energies and separation energies near N = 82. Now let us discuss the variation of quadrupole deformation of proton-rich nuclei with neutron number

No. 4 Anomalously Large Deformation in Some Medium and Heavy Nuclei 691 N = 60 76 ( 118 134 Ce). With the decrease of neutron number the quadrupole deformation increases for the Ce isotopes. This is seen clearly from columns 5 and 10 in Table 1. For 124 Ce the experimental deformation parameter is as large as β 2 = 0.39. This extremely large deformation is close to the superdeformation in nuclei. In general the variation of the quadrupole deformation parameter with neutron number in the RMF model can follow the experimental trend qualitatively but the RMF model underestimates the experimental value. In view of the qualitative agreement of the trend between theory and experiment and the theoretical underestimate on experimental data in quantity, it is reasonably expected that the deformation of proton-rich nuclei with N = 60 64 ( 118 122 Ce) is also extremely large because their theoretical deformation parameters are as large as β 2 = 0.35 0.37. This can suggest that some nuclei in this range can form a superdeformed area on the chart of nuclides. In order to confirm whether this is possible we also carry out further RMF calculations with other forces or for neighboring nuclei. β 2 = 0.37 0.44 for different sets of force parameters. This shows clearly that the appearance of abnormally large deformation or superdeformation in the ground state of the proton-rich nucleus is common for different effective forces in the RMF model. Fig. 2 The variation of the quadrupole deformation parameter of the proton-rich Ce and Nd isotopes with nucleon number. The square points are experimental data and they are connected by a dotted curve. The circle points are theoretical results and they are connected by a solid curve. Fig. 1 The variation of the energy of 122 Ce with quadrupole deformation parameter in the RMF model, where three sets of force parameters are used. The points are numerical results and they are connected by a curve for each set of force parameters. A minimum appears around the deformation parameter β 2 = 0.37 0.44 and it corresponds to the extremely large deformation of the ground state. In Fig. 1 we draw the curve of the ground state energy with the quadrupole deformation parameter for the nucleus 122 Ce. This is the constraint RMF (CRMF) calculation with three sets of force parameters NL3, NLZ2, and TMA. The points are calculated values and we connect them by a curve. The variation of the energy with quadrupole deformation parameter is very similar for three sets of forces. The minimum of the energy corresponds to an extremely large deformation around In Table 2 we list the RMF results with TMA for proton-rich Nd and Sm isotopes. The quantities in Table 2 have the similar meaning to those in Table 1. It is seen from Table 2 that the theoretical binding energies of Nd and Sm isotopes are very close to experimental data. The deviation between model and data is less than 1 MeV for many nuclei considered here. This corresponds to a relative deviation of 0.1% because the total binding energy is as high as 1000 MeV. For the two-neutron separation energies and two-proton separation energies, the difference between model and data is also less than 1 MeV for many nuclei. So the RMF model is very reliable for the binding energies and separation energies of the proton-rich Nd and Sm isotopes. On the quadrupole deformation of Nd isotopes it is concluded from the data of Table 2 that the quadrupole

692 CHEN Ding-Han, REN Zhong-Zhou, and TAI Fei Vol. 44 deformation parameter increases with the decrease of neutron number. For 130 Nd the experimental quadrupole deformation is as large as β 2 = 0.37. In general the theoretical deformation parameter can follow the experimental trend of deformation parameter qualitatively but it still underestimates the data in quantity. The experimental data of quadrupole deformation of 128 120 Nd are unknown and the theoretical values of β p lie between 0.36 and 0.41. It is expected that the experimental deformation of 128 120 Nd will be extremely large or it is near superdeformation. Table 2 The binding energies, two-neutron separation energies, two-proton separation energies, and deformations of even-even Ce and Sm isotopes with N = 60 80. The columns 2 5 are experimental data. The columns 6 10 are theoretical results. The units of the separation energies are MeV. Nuclei B (expt.)(mev) S 2n (expt.) S 2p (expt.) β 2 (expt.) B (theor.)(mev) S 2n S 2p β n β p 120 Nd 948.9 0.39 0.41 122 Nd 975.8 26.9 1.0 0.39 0.40 124 Nd 1001.5 25.7 2.7 0.38 0.40 126 Nd 1224.9 23.4 3.9 0.38 0.39 128 Nd 1046.4# 4.1# 1047.0 22.1 5.0 0.35 0.36 130 Nd 1068.7# 22.3# 5.4# 0.37 1068.9 21.9 5.6 0.29 0.30 132 Nd 1090.1# 21.4# 6.7# 0.35 1090.7 21.8 6.6 0.26 0.27 134 Nd 1110.4# 20.3# 7.9# 0.25 1110.6 19.9 7.2 0.22 0.24 136 Nd 1129.9 19.5# 9.0# 1129.7 19.1 7.8 0.18 0.19 138 Nd 1148.9# 19.0# 10.1# 1149.4 19.7 8.3 0.05 0.05 128 Sm 1026.2 1.3 0.37 0.39 130 Sm 1050.1 23.9 3.1 0.36 0.37 132 Sm 1073.3 23.2 4.4 0.33 0.34 134 Sm 1094.5# 4.4# 0.37 1095.9 22.6 5.2 0.29 0.31 136 Sm 1116.0# 21.5# 5.6# 0.29 1116.9 21.0 6.3 0.32 0.34 138 Sm 1136.6# 20.6# 6.6# 0.21 1136.6 19.7 6.9 0.31 0.32 140 Sm 1156.5# 19.9# 7.5# 1155.5 18.9 6.1 0.27 0.30 The variation of the quadrupole deformation parameters of the Ce and Nd isotopes with nucleon number (A) is plotted in Fig. 2. The upper part of Fig. 2 is for Ce isotopes and the lower one is for Nd isotopes. In Fig. 2 the solid curve is the theoretical deformation and the dotted curve is the experimental one. The trend of variation of quadrupole deformation parameters with nucleon number is similar for two isotopic chains. The quadrupole deformation parameter increases to a large value near the proton drip line. The theoretical curve is slightly lower than the experimental one. This suggests that the experimental deformation of the proton-rich nuclei is larger than the calculated value in the RMF model. We expect that the ground state of some proton-rich Ce and Nd isotopes can be superdeformed. In the lower part of Table 2 we list the ground state properties of the proton-rich Sm isotopes. The experimental binding energies and separation energies are approximately reproduced by the model. The experimental deformation parameter of 134 Sm is abnormally large β 2 = 0.37. The theoretical deformation parameter of 134 Sm is slightly smaller than this value. The theoretical deformation parameters of 128 132 Sm are abnormally large in the RMF model. If we combine the theoretical deformation parameters and available data of Ce, Nd, and Sm isotopes together, we conclude that some proton-rich nuclei on these isotopic chains can form a superdeformed range in the chart of the nuclides. It is strongly suggested that more experiments are carried out to confirm the theoretical prediction on the existence of the superdeformation in this mass range. 3.3 Root-Mean-Square Radii of Ce Isotopes As experimental study on the root-mean-square (RMS) radii of density distributions of nuclear matter is an important topics in nuclear physics, we list in Table 3 the RMS radii of neutron and proton density distributions in Ce isotopes. The experimental RMS radii of charge distributions of Ce isotopes are unknown up to now. For 140,142 Ce they are stable nuclei and their charge radii can be measured by electron-scattering experiments in the future. For the radii of neutron and proton den-

No. 4 Anomalously Large Deformation in Some Medium and Heavy Nuclei 693 sity distributions physicists can extract the radii by the measurement of reaction cross sections of nucleus-nucleus collisions. Therefore with the development of radioactive beams the measurement on these radii will be available. So one can compare the theoretical radii of Ce isotopes with future experimental results. Table 3 The root-mean-square radii of even-even Ce isotopes with N = 60 100. Nuclei R n (fm) R p (fm) Nuclei R n (fm) R p (fm) 118 Ce 4.74 4.80 140 Ce 5.02 4.83 120 Ce 4.77 4.80 142 Ce 5.08 4.85 122 Ce 4.80 4.81 144 Ce 5.12 4.88 124 Ce 4.84 4.81 146 Ce 5.16 4.90 126 Ce 4.86 4.81 148 Ce 5.20 4.92 128 Ce 4.87 4.81 150 Ce 5.26 4.95 130 Ce 4.90 4.81 152 Ce 5.33 5.01 132 Ce 4.93 4.81 154 Ce 5.37 5.03 134 Ce 4.95 4.82 156 Ce 5.41 5.05 136 Ce 4.96 4.82 158 Ce 5.45 5.07 138 Ce 5.00 4.82 Because the RMF model can approximately reproduce the experimental data of binding energies, two-nucleon separation energies, and deformation parameters of the Ce isotopes and its neighboring isotopes, it is concluded that the RMF model is reliable for the ground state properties of these proton-rich nuclei. The RMF predictions on the existence of superdeformation in the proton-rich nuclei of this range are valuable for future experiments. 4 Numerical Results and Discussions of Some Heavy Nuclei Near 242 Am After we discuss the ground state properties of eveneven Ce, Nd, and Sm nuclei, we present in this section some numerical results for nuclei near 242 Am. The existence of the superdeformed isomeric state in 242 Am was reported for a long time [9] but there is no RMF study on this. The RMF calculations on the ground state properties of the Am isotopes are also very rare in publications. Here we report the first RMF result on the isomeric state of 242 Am and on the ground states of the Am isotopes. The RMF results on the ground state and the isomeric state of 242 Am and 242 Pu are listed in Table 4, where three sets of the RMF forces (TMA, NL3, NLSH) are used. In Table 4, the first column denotes the nucleus. The second column and the third one are the experimental energy and quadrupole deformation parameter, respectively. The experimental data are taken from the nuclear data table and other publications. [1 3,13,27 29] Columns 4 9 are the theoretical results and the input force parameters in the RMF model. Table 4 The energies, quadrupole deformation parameters, and radii of density distribution of protons and neutrons for the ground state (g.s.) and the isomeric state (i.s.) of nuclei 242 Am and 242 Pu. The columns 2 and 3 are experimental data. The columns 4 10 are theoretical results. The units of the separation energies are MeV. In numerical calculations we used the several sets of RMF forces such as TMA, NLSH, and NL3. Nuclei E(exp.) (MeV) β 2 (exp.) β n (theor.) β p (theor.) E (theor.) (MeV) R n (fm) R p (fm) Force 242 Am (g.s.) 1823.47 0.29 0.28 0.28 1825.41 6.11 5.88 TMA 242 Am (i.s.) 1820.57 0.6 0.82 0.84 1822.24 6.54 6.36 TMA 242 Am (g.s.) 1823.47 0.29 0.29 0.30 1826.02 6.14 5.88 NL3 242 Am (i.s.) 1820.57 0.6 0.86 0.88 1823.89 6.60 6.40 NL3 242 Am (g.s.) 1823.47 0.29 0.28 0.29 1831.41 6.09 5.86 NLSH 242 Am (i.s.) 1820.57 0.6 0.83 0.85 1829.71 6.53 6.34 NLSH 242 Pu (g.s.) 1825.01 0.29 0.28 0.29 1825.51 6.12 5.88 TMA 242 Pu (i.s.) 1822.81 0.6 0.81 0.82 1822.30 6.54 6.34 TMA 242 P u (g.s.) 1825.01 0.29 0.29 0.30 1825.51 6.15 5.88 NL3 242 Pu (i.s.) 1822.81 0.6 0.85 0.86 1823.38 6.60 6.38 NL3 242 Pu (g.s.) 1825.01 0.29 0.29 0.29 1830.84 6.11 5.86 NLSH 242 Pu (i.s.) 1822.81 0.6 0.81 0.83 1829.49 6.53 6.33 NLSH It is seen from the first two rows of Table 4 that the theoretical energy of 242 Am with TMA is very close to the experimental one. The experimental ground state energy of 242 Am is E(exp.) = 1823.47 MeV (the ground state energy is the minus of the binding energy). The theoretical one is E(theor.) = 1825.41 MeV (see row 2 in Table 4). The deviation between theory and data is approximately 0.1%. Good agreement of the ground state energy between theory and data is obtained. The theoretical energy for the isomeric state (E(theor.) = 1822.24 MeV) also

694 CHEN Ding-Han, REN Zhong-Zhou, and TAI Fei Vol. 44 agrees with the data (E(exp.) = 1820.57 MeV). The experimental excited energy of the isomeric state of 242 Am is E (exp.) = 1823.47 1820.57 = 2.9 MeV. The theoretical one is E (exp.) = 1825.41 1822.24 = 3.17 MeV (see rows 2 and 3 in Table 4). It is close to the experimental one. Here it should be stressed again that no adjustment on force parameters or on pairing gaps is introduced in numerical calculations. Therefore we can say the agreement of the energy between the model and the data is good. On the quadrupole deformation parameter of 242 Am the theoretical value agrees with the data for the ground state. [29] For the isomeric state of 242 Am the experimental deformation parameter is β 2 0.6 and the theoretical one with TMA is approximately β 2 = 0.83. The theoretical value is slightly larger than the experimental one. The model can qualitatively reproduce the superdeformation in the isomeric state although there is a small discrepancy of the quadrupole deformation in quantity. Because the current force parameters in the RMF model are obtained by fitting the properties of several spherical nuclei, no information on nuclear deformation is used during the process of fitting effective force parameters in the self-consistent RMF model. Based on this we consider that the agreement of the deformation between the model and the data is accepted. Of course better agreement on deformation can be achieved if some adjustments are introduced in the model. But we do not think that perfect agreement between model and data is important in physics. The root-mean-square radii of neutron-density distribution and proton-density distribution are also listed in the table. It is seen that the nucleus in the isomeric state has larger radii than that in the ground state. This large radius can be related to the superdeformation in the isomeric state. It is seen from rows 3 and 4 of Table 4 that the RMF results with another set of forces NL3 also agree with the available data of 242 Am. The theoretical energies are very close to the experimental ones. The theoretical deformation for the ground state is close to the experimental one but the theoretical deformation for the isomeric state is slightly larger than the experimental value. This is very similar to that in the RMF model with TMA force. Rows 5 and 6 of Table 4 are the RMF results with force NLSH. The theoretical energy with NLSH is slightly lower than the experimental one but it is still accepted. The theoretical deformation with NLSH is close to that with TMA and with NL3. Here we do not repeat the discussion. When we compare the RMF results with three sets of forces, we find that the RMF results are very stable for the properties of the ground state and of the isomeric state of 242 Am. This shows that the RMF model is reliable for 242 Am. In the lower part of Table 4 we list the numerical results of 242 Pu, where three sets of force parameters are used in the RMF model. Reasonable agreement of the energy and quadrupole deformation between the model and the data is seen again for the nucleus 242 Pu. Previous conclusions on 242 Am still hold true for 242 Pu and here we do not repeat them. Fig. 3 The variation of the energy of 242 Am with quadrupole deformation parameter in the RMF model, where three sets of force parameters are used. The points are numerical results and they are connected by a curve for each set of force parameters. There are some minimums in the curve. The lowest minimum appears around the deformation parameter β 2 = 0.28 0.30 and it corresponds to the moderate deformation of the ground state. Another minimum occurs at β 2 = 0.82 0.88 and it corresponds to the superdeformation of the isomeric state. Fig. 4 The variation of the energy of 242 Pu with quadrupole deformation parameter in the RMF model where three sets of force parameters are used. The points are numerical results and they are connected by a curve for each set of force parameters. There are some minimums in the curve. The lowest minimum appears around the deformation parameter β 2 = 0.28 0.30 and it corresponds to the moderate deformation of the ground state. Another minimum occurs at β 2 = 0.82 0.88 and it corresponds to the superdeformation of the isomeric state.

No. 4 Anomalously Large Deformation in Some Medium and Heavy Nuclei 695 In order to show the reliability of the RMF model further we carry out a constraint RMF calculation for 242 Am and its neighboring nucleus 242 Pu. The results of the constraint RMF calculation is drawn in Figs. 3 and 4, where three sets of forces are used. In Fig. 3 the X-axis is the quadrupole deformation parameter and the Y -axis the energy of the nucleus. It is seen clearly from the energy curve of Figs. 3 and 4 that the ground state of 242 Am and 242 Pu corresponds to a moderate prolate deformation with β 2 0.28 0.30. Its isomeric state corresponds to a superdeformation. This agrees well with the data in Table 4. In view of the fact that the RMF calculations on Am isotopes are very rare in publications, we give in Table 5 the RMF results on the ground state properties of Am isotopes. We hope to show that the agreement between the model and the data is a general behavior in this mass range (i.e. it is not an accident agreement for special nuclei 242 Am and 242 Pu or for special forces in the model). In Table 5 we list the RMF results with TMA and NLZ2 forces. It is seen from Table 5 that the theoretical binding energies agree well with the data, where the data with symbol # are the estimated values by Audi et al. [27,28] The theoretical two-neutron separation energy also agrees with the data well. Importantly the theoretical results with two sets of forces are close and this shows again the stability of the model in this mass range. Table 5 The binding energies, two-neutron separation energies, and deformations of odd-even Am isotopes in RMF model with two sets of force parameters TMA and NL-Z2. Nuclei B (MeV) S 2n (MeV) β n β p B (MeV) S 2n (MeV) β n β p B (expt.) S 2n (expt.) 237 Am 1793.11 12.74 0.25 0.26 1791.60 13.96 0.29 0.30 1791.76# 14.06# 239 Am 1806.46 13.35 0.26 0.27 1804.70 13.10 0.30 0.31 1805.34 13.58# 241 Am 1819.26 12.80 0.27 0.28 1817.06 12.36 0.30 0.31 1817.94 12.60 243 Am 1831.52 12.26 0.28 0.29 1828.93 11.87 0.31 0.31 1829.84 11.90 245 Am 1842.57 11.05 0.28 0.28 1840.04 11.11 0.31 0.31 1841.26 11.38 247 Am 1853.11 10.54 0.27 0.27 1850.18 10.14 0.31 0.31 1852.07# 10.81# 249 Am 1863.41 10.30 0.26 0.26 1859.79 9.61 0.31 0.31 251 Am 1872.97 9.56 0.24 0.23 1869.36 9.57 0.30 0.30 253 Am 1882.57 9.60 0.21 0.20 1878.82 9.46 0.29 0.29 255 Am 1891.87 9.30 0.20 0.19 1887.84 9.02 0.28 0.28 257 Am 1901.05 9.18 0.19 0.18 1896.22 8.38 0.26 0.25 259 Am 1909.84 8.79 0.17 0.15 1904.54 8.32 0.23 0.22 261 Am 1919.24 9.40 0.10 0.09 1912.93 8.39 0.20 0.20 263 Am 1928.47 9.23 0.11 0.09 1921.39 8.46 0.16 0.15 265 Am 1937.41 8.94 0.11 0.09 1930.48 9.09 0.15 0.13 267 Am 1945.85 8.44 0.11 0.09 1939.25 8.77 0.13 0.11 269 Am 1953.58 7.73 0.10 0.08 1947.77 8.52 0.11 0.10 271 Am 1961.09 7.51 0.08 0.07 1956.19 8.42 0.09 0.08 273 Am 1968.25 7.16 0.07 0.06 1964.19 8.00 0.07 0.06 275 Am 1974.90 6.65 0.03 0.03 1971.76 7.57 0.02 0.02 277 Am 1981.76 6.86 0.00 0.00 1979.84 8.08 0.00 0.00 279 Am 1988.02 6.26 0.00 0.00 1987.12 7.28 0.00 0.00 281 Am 1992.04 4.02 0.00 0.00 1990.96 3.84 0.00 0.00 5 Summary We have investigated the properties of the Ce nuclei and the nuclei near 242 Am in the RMF model. The theoretical binding energies, separation energies, and deformation parameters agree reasonably with the experimental data of Ce isotopes. The energy curve of 122 Ce is obtained by a constraint RMF (CRMF) calculation. It is found that the proton-rich Ce nuclei have an abnormally large deformation in their ground states. Its neighboring isotopes such as Nd and Sm also exhibit similar behavior of deformation. So the proton-rich nuclei in this mass range can form a largely deformed range or a superdeformed range. For nuclei 242 Am and 242 Pu both the ground state properties and the isomeric state properties

696 CHEN Ding-Han, REN Zhong-Zhou, and TAI Fei Vol. 44 can reasonably reproduced by the model. The experimental superdeformation of the isomeric state of 242 Am and 242 Pu is qualitatively reproduced by the model although a small discrepancy exists in quantity. The constraint RMF calculations clearly show the existence of this isomeric state where no adjustment on forces is introduced in the model. This is the first RMF calculation on the superdeformation of the isomeric state of 242 Am. The RMF model can well reproduce the binding energies and separation energies of Am isotopes. In numerical calculations, even-multipolarity deformations are taken into account in the standard RMF model. The mean-field of the standard RMF model is from the contributions of σ, ω, and ρ mesons when the π meson is dropped off. The mean-field of the standard RMF model has even-parity. As the parity conservation and invariance of time reversal are assumed in the RMF model, there are only even-multipolarity deformations such as quadrupole and hexadecupole deformations in the RMF model. It is impossible to obtain the odd-multipolarity deformations such as octupole deformation in the standard RMF model with σ, ω, and ρ mesons. This is because that odd-multipolarity deformation is zero when there is parity conservation or there is time reversal invariance. [26] By comparing the theoretical results with the available data we test the reliability of the RMF model for the abnormally large deformation in the ground state or superdeformation in the isomeric state in medium and heavy nuclei. We expect that the anomalously large deformation or superdeformation can exist in some nuclei of other mass range. The model can be applied to investigate the extremely large deformation of other mass range. [30 32] Acknowledgments Z. Ren thanks Profs. T. Otsuka, H. Toki, H.Q. Zhang, Z.Y. Zhang, Z.Y. Ma, and B.Q. Chen for discussions on superdeformation in nuclei. References [1] A. Bohr and B.R. Mottelson, Nuclear Structure, Vol. II, Benjamin, Reading, Massachusetts (1975). [2] S. Raman, C.H. Malarkey, W.T. Milner, C.W. Nestor, Jr., and P.H. Stelson, Atomic Data and Nuclar Data Tables 36 (1987) 1. [3] S. Raman, C.W. Nestor, Jr., and P. Tikkanen, Atomic Data and Nuclar Data Tables 78 (2001) 1. [4] Zhong-Zhou Ren, Z.Y. Zhu, Y.H. Cai, and Gong-Ou Xu, Phys. Lett. B 380 (1996) 241. [5] P. Ring, Prog. Part. Nucl. Phys. 37 (1996) 193. See p. 198 and p. 203. [6] D. Hirata, H. Toki, I. Tanihata, and P. Ring, Phys. Lett. B 314 (1993) 168. [7] T.R. Werner, J.A. Sheikh, W. Nazarewicz, M.R. Strayer, A.S. Umar, and M. Misu, Phys. Lett. B 333 (1994) 303. [8] P.G. Reinhard, Rep. Prog. Phys. 52 (1989) 439. [9] S.M. Polikanov, et al., Soviet Phys. JETP 15 (1962) 1016. [10] V.M. Strutinsky, Nucl. Phys. A 95 (1967) 420. [11] D. Habs, V. Metag, H.J. Spetht, and G. Ulfert, Phys. Rev. Lett. 38 (1977) 387. [12] H.J. Specht, Rev. Mod. Phys. 46 (1974) 773. [13] R. Vandenbosch, Ann. Rev. Nucl. Sci. 27 (1977) 1. [14] S.A.E. Johansson, Nucl. Phys. A 12 (1959) 449. [15] G.A. Lalazissis, M.M. Sharma, and P. Ring, Nucl. Phys. A 597 (1996) 35. [16] G.M. Petrache, et al., Eur. Phys. J. A 16 (2002) 439. [17] E.S. Paul, et al., Phys. Rev. C 58 (1998) 801. [18] K.L. Ying, et al., J. Phys. G 12 (1996) L211. [19] Zhan-Gui Li, Shu-Wei Xu, et al., Phys. Rev. C 56 (1997) 1157. [20] B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. 16 (1986) 1. [21] Zhong-Zhou Ren, et al., Commun. Theor. Phys. (Beijing, China) 42 (2004) 851. [22] S. Marcos, N. Van Giai, and L.N. Savushkin, Nucl. Phys. A 549 (1992) 143. [23] Zhong-Zhou Ren, Amand Faessler, and A. Bobyk, Phys. Rev. C 57 (1998) 2752. [24] B.Q. Chen, Z.Y. Ma, F. Gruemmer, and S. Krewald, Phys. Lett. B 455 (1999) 13. [25] Zhong-Zhou Ren and H. Toki, Nucl. Phys. A 689 (2001) 691. [26] M.A. Preston and R.K. Bhaduri, Structure of the Nucleus, Addison-Wesley, Reading, MA (1975) p. 21 and p. 69. [27] G. Audi and A.H. Wapstra, Nucl. Phys. A 565 (1993) 1. [28] G. Audi, O. Bersilion, J. Blachot, and A.H. Wapstra, Nucl. Phys. A 624 (1997) 1. [29] The experimental ground deformation of 242 Am is taken from the value of the neighboring even-even nuclei β 2 = 0.29. It is generally accepted that the superdeformation on the isomeric state is approximately β 2 = 0.6. [30] Zhong-Zhou Ren, Phys. Rev. C 65 (2002) 051304(R). [31] P. Möller, J.R. Nix, and K.L. Kratz, Atomic Data and Nuclear Data Table 66 (1997) 131. [32] W.D. Myers and W.J. Swiatecki, Phys. Rev. C 58 (1998) 3668.