Fusion Barrier of Super-heavy Elements in a Generalized Liquid Drop Model

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1 Commun. Theor. Phys. (Beijing, China) 42 (2004) pp c International Academic Publishers Vol. 42, No. 4, October 15, 2004 Fusion Barrier of Super-heavy Elements in a Generalized Liquid Drop Model CHEN Bao-Qiu 1,2 and MA Zhong-Yu 1,2,3 1 China Institute of Atomic Energy, Beijing , China 2 Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator of Lanzhou, Lanzhou , China 3 Institute of Theoretical Physics, the Chinese Academy of Sciences, Beijing , China (Received March 3, 2004) Abstract The macroscopic deformed potential energies for super-heavy elements Z = 110, 112, 114, 116, 118 are determined within a generalized liquid drop model (GLDM). A quasi-molecular mechanism is introduced to describe the deformation of a nucleus in the GLDM and the shell model simultaneously. The macroscopic energy of a twocenter nuclear system in the GLDM includes the volume-, surface-, and Coulomb-energies, the proximity effect at each mass asymmetry, and accurate nuclear radius. The shell correction is calculated by the Strutinsky method and the microscopic single particle energies are derived from a shell model in an axially deformed Woods Saxon potential with the quasi-molecular shape. The total potential energy of a nucleus can be calculated by the macro-microscopic method as the summation of the liquid-drop energy and the Strutinsky shell correction. The theory is applied to predict the fusion barriers of the cold reactions 64 Ni Pb , 70 Zn Pb , 76 Ge Pb , 82 Se Pb , 86 Kr Pb It is found that the neck in the quasi-molecular shape is responsible for the deep valley of the fusion barrier. In the cold fusion path, double-hump fusion barriers could be predicted by the shell corrections and complete fusion events may occur. PACS numbers: Pj, Jj, b Key words: generalized liquid drop model, super-heavy elements, fusion barrier 1 Introduction More than 30 years ago, theoretical study has predicted that there is the possible existence of a stable island of super-heavy elements (SHE) in the periodic table. [1] Since then, the search for super-heavy elements becomes one of the most interesting subjects in nuclear physics. The synthesis of super-heavy elements has apparently been strongly advanced in recent years using both cold (Zn on Pb) [2] and warm (Ca on U, Pu and Cm) [3] fusion reactions at GSI, Dubna, and Berkeley. The experimental data analysis was also discussed. [4] In the cold fusion, the synthesis of the SHE is produced by the reaction type of X +(Pb, Bi) SHE+1n at subbarrier energies. In order to calculate the formation cross section of deformed SHE in the cold fusion, several models have been proposed. Adamian et al. [5] assumed that after the full dissipation of the collision kinetic energy, a dinuclear system is formed. After that such a system evolves to the compound nucleus by the nucleon transfer from one nucleus to another. Denisov et al. [6] developed a dinuclear system model in which low energy surface vibrations and a transfer of few nucleons are taken into account. The model is able to describe experimental data for some reactions such as 58 Fe + 207,208,210 Pb 264,265,267 Hs + 1n. They found that the contribution of the surface vibrations to the fusion cross section is larger than that of the nucleon transfer. The best fits are obtained by considering both transfer and vibrations simultaneously. Smolanczuk [7] proposed a simple model to describe the formation of super-heavy elements in the cold fusion reaction in which the compound nucleus is formed by quantum tunnelling through pure Coulomb barrier or the phenomenological fusion barrier. It is worth while noting that although Smolanczuk s model can describe the process of the fusion reaction, the fusion barrier used in that model is either the pure Coulomb barrier or the phenomenological one. The recent investigation pointed out that the height, position, and width of the potential barrier are the main ingredients in fission and fusion reactions. [8] Very recently, Sahu et al. [9] constructed a fusion barrier that is an analytically solvable, smooth, short-ranged, realistic and composite barrier potential with parameters controlling the flatness at the top, the range and the asymmetry of the barrier. They found that the asymmetry of the barrier provides a correct explanation of the sub-barrier enhancement of experimental data in the fusion cross section and the oscillatory structure in the barrier distribution function can be described by the flatness of the barrier near the top. Therefore, in order to provide reasonable predictions, it is important to derive the potential barrier by theoretical models instead of an input or adjustable parameters. The generalized liquid drop model (GLDM) [8] is one of the most successful macroscopic models. The model is able The project supported by National Natural Science Foundation of China under Grant Nos , , , and and the State Key Basic Research Program of China under Grant No. G

2 No. 4 Fusion Barrier of Super-heavy Elements in a Generalized Liquid Drop Model 595 to describe processes of both fusion and fission reactions through quasi-molecular deformation mechanism. The aim of this work is to study the potential barriers for several cold fusion reactions in the GLDM. The total fusion potential energy of a nucleus can be calculated in the macro-microscopic method as the summation of the macro energy and the Strutinsky shell correction energy. The macro energy is calculated in the GLDM and the shell correction energy is calculated by the Strutinsky method, while the microscopic single particle energies are derived from a shell model in an axially deformed Woods Saxon potential with the quasi-molecular shape. The paper is arranged as follows. The generalized liquid drop model is briefly introduced in Sec. 2. The quasimolecular mechanism and shell correction method are elucidated in Secs. 3 and 4, respectively. The prediction of fusion potential barriers in several cold fusion reactions is presented and discussed in Sec. 5. At the end we give a short summary. 2 Generalized Liquid Drop Model The main formalism of the GDLM can be described as follows. For a deformed nucleus, the macroscopic total energy is defined as [8] E GLDM = E LDM + E N, (1) where E LDM and E N are the liquid drop model energy and the nuclear proximity energy, respectively. The energy of the liquid drop model includes the volume, surface, and Coulomb energies, E LDM = E v + E s + E c. (2) For the one-body shape, the volume energy E v, surface energy E s, and Coulomb energy E c read E v = a v (1 k v I 2 )A, (3) E s = a s (1 k s I 2 )A 2/3 S 4πR0 2, (4) E c = 3 5 e2( Z 2 ) 1 V (θ) ( R(θ) ) 3 sin θd θ, (5) R 0 2 R(θ) where A, Z, and I = (N Z)/A are mass, charge, and relative neutron excess of the compound nucleus, respectively. V (θ) is the electrostatic potential at the surface of the shape and V 0 is the surface potential of the sphere. After the separation, the volume, surface, and Coulombenergies in the GLDM read V 0 E v = a v [(1 1.8I 2 1 A 1 + A 2 ], (6) E s = a s [(1 2.6I 2 1 )A 2/3 1 + A 2/3 2 ], (7) E c = 3 5 e2 Z R 1 5 e2 Z2 2 + Z 1Z 2. (8) R 2 r The volume- and surface-coefficients have been chosen to be a v = and a s = , respectively. In order to reproduce the increase of the ratio r 0 = R 0 /A 1/3, the effective shape radius of the radioactive emitter is defined by R 0 = 1.28A 1/ A 1/3. To ensure volume conservation, the radii R 1 and R 2 of the daughter and alpha nuclei are given by R 1 = R 0 (1 + β 3 ) 1/3, R 2 = R 0 (1 + β 3 ) 1/3, where β = (1.28A 1/ A 1/3 2 )/(1.28A 1/ A 1/3 1 ). The nuclear proximity energy can be written as hmax D ) E n = 2γ Φ( 2πhd h, (9) h min b where h is the ring radius in the plane perpendicular to the longitudinal deformed axis and D is the distance between the opposite infinitesimal surface, [8] b is the surface width, φ is called the Feldmeier function. [10] The surface parameter γ = (1 2.6I 2 1 )(1 2.6I2 2 ) MeV fm 2. 3 Quasi-molecular Mechanism It is well known that in the early fission studies, researchers assumed that the repulsive Coulomb force and attractive surface tension force control the evolution of nuclear shapes. In order to describe the evolution process of nuclear shapes, the radius of nuclear shape expanded by Legendre polynomials led to elongated one-body configuration, which was able to explain the bulk properties of the nuclear fission. However, this description of a deformed radius fails to reproduce the deep and narrow necks of nuclear deformed shape in the fusion reaction. It has been found that [8] the quasi-molecular mechanism can provide deep and narrow necks of the nuclear deformed shape in the fusion path. A two-parameter shape sequence has been defined [8] to describe the continuous transition from one initial spherical nucleus to two tangent spherical fragments, { a 2 sin 2 θ + c 2 1 cos 2 θ, 0 θ π 2 R(θ) = a 2 sin 2 θ + c 2 2 cos 2 π, (10) θ, 2 θ π where c 1 and c 2 are the elongations of two interacting nuclei and a is the neck radius. Assuming the volume conservation, we can completely define the shape by the two parameters s 1 = a/c 1 and s 2 = a/c 2. For a given decay channel, the ratio η = R 2 /R 1 between radii of the future fragments allows to connect s 1 and s 2, s 2 2 = s 2 1 s (1 s2 1 )η2 (0 s 1, s 2, η 1), (11) where s 1 decreases from 1 to 0, the shape evolves continuously from one sphere to two touching spheres with the formation of a neck while keeping spherical ends. This behavior is illustrated in Fig. 1, where a typical evolution of shape is displayed for a very asymmetric system.

3 596 CHEN Bao-Qiu and MA Zhong-Yu Vol. 42 Fig. 1 sphere). Shape evolutions in the fusion reaction 64 Ni Pb with s 1 from 0.0 (contact point) to 1.0 (one 4 Shell Correction Method The macro-microscopic model can be used to describe potential energy of a nucleus with arbitrary shape. In the macro-microscopic model the total nuclear potential energy E def, as a function of the proton number Z, neutron number N and the shape of a nucleus, is given by E def = E macro + E micro, (12) where E macro is macroscopic energy given by Eq. (1), and the microscopic energy E micro is a shell correction energy E sc. The shell correction arises from fluctuations in the distribution of single particle levels with shell structure relative to uniform distribution of levels in the liquid drop model. This correction is calculated for a given nucleus with a given deformation by standard Strutinsky method, [11] E sc = N part ɛ n Npart ɛ n, (13) where ɛ n are the single particle energies. The first term in Eq. (13) is a staircase function of the particle number while the second one is a smooth term. The smooth term is calculated as Npart λ ɛ n = ɛ g(ɛ)dɛ, (14) where g(ɛ) is the smoothly varying part of the exact level density g(ɛ) and λ is the Fermi energy of the smooth distribution of levels. The exact level density g(ɛ) is obtained by expanding the δ function in a series of Hermite polynomials and the separating the terms into a smoothly varying part and a fluctuating part. The expressions can be found in Ref. [11] in more details. It is well known that the shell model with an axially deformed Woods Saxon potential [12] has been very successful in many early calculations for an explanation of not only the ground state properties but also excited state properties of nuclei. It turns out that the shell model with an axially deformed Woods Saxon potential is a useful tool in predicting and explaining the structure of high spin particle-hole excitations, the fission barriers and a number of the single particle effects for strongly deformed nuclei. We calculate the single particle energies and wave functions in the shell model with an axially deformed Woods Saxon potential while the nuclear shapes are described by the quasi-molecular mechanism. The deformed Woods Saxon potential is defined in the following way V 0 V (r, s 1 ) = 1 + exp[dist Σ (r, s 1 )/a], (15) where dist Σ (r, s 1 ) is the distance between the point r and the nuclear surface represented by Eq. (11), where the minus sign is taken inside the surface. The diffuseness parameter is denoted by a. The depth of the central potential is parametrized as an isospin-dependent quantity, V 0 = v[1 ± κ(n Z)/(N + Z)] for proton and neutron, respectively. The spin-orbit potential is assumed to be in the form of ( h ) V so = λ V (r, s 1 ) (σ p), (16) 2MC where λ denotes the strength of the spin-orbit potential. M is the nucleonic mass; σ and p are the Pauli matrix and the linear momentum operator, respectively. The Coulomb potential for proton, V c, is calculated with the assumption of a uniformed charge distribution inside the surface Σ. The universal parameter set of the Woos Saxon potential [12] is adopted in our calculations. The complete set of 9 parameters are: r 0 (n) = fm, r 0 (p) = fm, V 0 = 49.6 MeV, κ = 0.86, a = 0.70 fm, λ(n) = 35, λ(p) = 36, r0 so (n) = fm, r0 so (p) = fm. It has been tested that for nuclei from light to heavy and throughout the periodic table, a very good agreement to the experimental data for both ground-state and high-spin states could be obtained. It is known that the nuclear shape expanded by Legendre polynomials failed to reproduce the deep and narrow necks of a nuclear deformed shape in fusion reactions. On

4 No. 4 Fusion Barrier of Super-heavy Elements in a Generalized Liquid Drop Model 597 the other hand, it has been shown that the quasi molecular mechanism in the GLDM gives a reasonable description of the formation of the deep and narrow neck in the fusion reaction which may play an important role in the formation of super-heavy elements. In order to describe nuclear shapes with the deep and narrow neck in fusion path, therefore, in our calculations the quasi-molecular shape is not only introduced in the macroscopic model but also in microscopic one. For the shell correction, we develop the deformed shell model in a proper deformation with the same quasi-molecular shape used in the macroscopic model. 5 Results and Discussions The deformed potential barriers of cold fusion reactions: 64 Ni+ 208 Pb , 70 Zn+ 208 Pb , 76 Ge Pb , 82 Se Pb , 86 Kr Pb are studied in the macroscopicmicroscopic model as discussed above. The macroscopic deformed potential energy is obtained in the GLDM. The shell correction energy is calculated in a microscopic shell model. A same quasi-molecular shape is used in both macroscopic- and microscopic-model calculations. In the shell model an axially deformed Woods Saxon potential is assumed. Actually those reactions were recently analyzed by Royer et al. [13] in an inconsistent way, where they used different deformed nuclear shapes in macroscopic and microscopic calculations. The fusion barriers versus the distances R between two mass-centers in the cold fusion reactions mentioned above are displayed in Fig. 2. The dashed curves denote pure macroscopic potential energies given in the GLDM. In ordinary fusion studies, it is often only those barriers that are taken into account. The dashed curve shows a wide macroscopic potential pocket due to a proximity energy at large deformations and the energy is almost constant till the spherical compound nucleus. In a generalized liquid drop model, the surface energy E s takes into account only the effect of the surface tension force and does not include the contribution of the attractive nuclear forces between surfaces in regard to the neck or the gap between the nascent fragments. The nuclear proximity energy is adopted to take into account these additional surface effects in the deformation path. At the contact point, the proximity energy reaches maximum while it decreases rapidly on both sides till to zero. The proximity energy reduce the barrier height by several MeV and moves the peak position of the barrier outside, which corresponds to two separated fragments in unstable equilibrium by the balance between the attractive nuclear proximity force and the repulsive Coulomb force in the GDLM. The solid curve is the summation of the GLDM macroscopic potential energy and the shell correction energy. From Fig. 2, one can see that went over the contact point the barrier drops down rapidly and reaches a deformation with a deep and narrow neck, which corresponds to s 1 values from 0 to 0.1. This deep and narrow neck is formed in the fusion process, which can be described by the quasi-molecular shape. Very recently, Sahu et al. [9] constructed a phenomenological fusion barrier, which is an analytically solvable, smooth, short-ranged one. A set of parameters in the phenomenological fusion barrier controls the flatness at the top, the range and asymmetry of the barrier. They found that the asymmetry of the barrier could correctively explain the sub-barrier enhancement of the fusion cross section in the experimental data. The potential barrier obtained in the GLDM shows a similarity with the phenomenological one. Fig. 2 Fusion potential energies for the reactions to form the supper-heavy elements of Z = 110, 112, 114, 116, and 118 as functions of the distances between two centers. The dashed and solid curves correspond to the macroscopic and total potential energies, respectively. It is observed in Fig. 2 that double-hump potentials appear in the total fusion barriers, which are produced by the shell correction. Full microscopic corrections create a plateau. The hight of the inner barrier becomes higher and the plateau becomes wider from Z = 110 to Z = 118. This behavior implies that the probability of the synthesis of super-heavy elements decreases rapidly when the charge number Z increases. It is consistent with the experimental observation. The incomplete fusion and quasi fission might appear and exchanges of nucleons from one nucleus to another one occur at the minimum between two maximum barriers, where a deep neck between two nuclei is

5 598 CHEN Bao-Qiu and MA Zhong-Yu Vol. 42 formed. The compound nucleus is formed by a quantum tunnelling through the external barrier and a spherical nucleus is located at the minimal valley. The second minimal valley corresponds to a large deformed shape. In summary, the generalized liquid drop model with a quasi-molecular shape is able to predict the potential energy of fusion reactions, which takes account of an accurate radius, a proximity and the mass asymmetry. The quasi-molecular shape can describe nuclear deformations with deep and narrow necks. It plays an important role in the formation of super-heavy elements. The single particle energies and the wave functions have been derived from the shell model with an axially deformed Woods Saxon potential and the quasi-molecular shape. The shell corrections are calculated in the Strutinsky method at each shape. It is found that the neck in the quasi-molecular shape is responsible for the deep valley of the fusion barrier. An double-hump fusion barrier in the cold fusion path is produced by the shell correction. References [1] F.A. Gareev, B.N. Kalinkin, and A. Sobiczewski, Phys. Lett. 22 (1966) 500. [2] Y.T. Oganessian, et al., Phys. Rev. Lett. 83 (1999) [3] Y.T. Oganessian, et al., Phys. Rev. C62 (2000) (R). [4] S. Hofmann and G. Muenzenberg, Rev. Mod. Phys. 72 (2000) 733; P. Armbruster, Eur. Phys. J. A7 (2000) 7. [5] G.G. Adamian, et al., Nucl. Phys. A633 (1998) 409; G.G. Adamian, et al., Nucl. Phys. A678 (2000) 24. [6] V.Yu Denisov and S. Hofmann, Phys. Rev. C61 (2000) [7] R. Smolanczuk, Phys. Rev. C59 (1999) 2634; Phys. Rev. C63 (2001) [8] G. Royer and B. Remaud, Nucl. Phys. A444 (1985) 477; G. Royer, J. Phys. G26 (2000) [9] S. Sahu, S.K. Agarwalla, and C.S. Shastry, Nucl. Phys. A713 (2003) 45. [10] H. Feldmeier, 12th Summer School on Nuclear Physics, Mikolajki, Poland (1979). [11] V.M. Strutinsky, Nucl. Phys. A95 (1967) 420; R.A. Herghescu, D.N. Poenaru, and W. Greiner, J. Phys.: Nucl. Part. Phys. G23 (1997) [12] J. Dudek, et al., Phys. Rev. C26 (1982) [13] G. Royer and R.A. Gherghesu, Nucl. Phys. A699 (2002) 479.