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Improved isospin dependent quantum molecular dynamics model and its application to fusion reactions near Coulomb barrier
Nuclear Physics A 750 (2005) 232–244
Improved isospin dependent quantum molecular dynamics model and its application to fusion reactions near Coulomb barrier Zhao-Qing Feng a,b,e,∗ , Feng-Shou Zhang a,c,d , Gen-Ming Jin a,b , Xi Huang a,b,e a Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China b Institute of Modern Physics, Chinese Academy of Sciences, PO Box 31, Lanzhou 730000, China c Institute of Low Energy Nuclear Physics, Beijing Normal University, Beijing 100875, China d CCAST (Word Laboratory), PO Box 8730, Beijing 100080, China e Graduate School, Chinese Academy of Sciences, Beijing 100039, China
Received 15 November 2004; received in revised form 14 December 2004; accepted 3 January 2005 Available online 7 January 2005
Abstract The isospin dependent quantum molecular dynamics model is developed by introducing switch function method which deals with correctly the surface interaction and shell effect in the process of projectile and target approaching. The fusion excitation functions for 40 Ca + 40 Ca, 40 Ca + 48 Ca and 48 Ca + 48 Ca at energies in the vicinity of the Coulomb barrier are studied. The experimental data of the fusion cross sections for these three systems can be regenerated very well. It is found that the fusion cross sections for neutron-rich system increase obviously. The static and dynamical Coulomb barriers are studied in order to clarify the phenomena. The neutron to proton ratio (N/Z ratio) at neck region is also studied, which apparently presents isospin effects of projectile-target combinations. 2005 Elsevier B.V. All rights reserved. PACS: 25.60.Pj; 25.70.Jj; 24.10.-i
* Corresponding author.
E-mail address: [email protected] (Z.-Q. Feng). 0375-9474/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2005.01.001
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1. Introduction In recent years, with the establishment of secondary beam facilities at many laboratories around the world, radioactive beams of nuclei with large neutron or proton excess can be produced [1]. Consequently, one can investigate the properties of nuclei far from the β stability and the nuclear reactions induced by radioactive nuclear beam, especially for the synthesis of superheavy elements (SHEs). The shell model predicts that the next doubly magic nucleus in the sequence will contain either 114, 120, 124 or 126 protons and 184 neutrons [2–4]. Other studies predict a whole superheavy “island of stability” around these proton and neutron numbers. It needs very neutron-rich projectile-target combinations to reach the island, such as by using radioactive nuclear beams. Theoretically, the mere existence of the heaviest elements with Z > 102 is due to quantal shell effects [5]. The properties of SHEs were studied theoretically using Strutinsky method [6,7], where the ground-state deformation, the fission barrier, binding energy and competition between various possible decay modes were also investigated. However, the dynamical process of the formation of SHEs is not understood well enough [8]. There are mainly two methods for the synthesis of SHEs experimentally, cold fusion reaction and hot fusion reaction [8,9]. Recently, elements 115 and 113 were discovered in Dubna by bombarding the americium-243 target with a beam of energetic calcium-48 (evaporation 3 or 4 neutrons) [10]. Recently, a few attempts were undertaken to develop models for describing the fusion process and for reproducing the measured cross section data based on dinuclear system [9,11]. Microscopic transport theory such as isospin dependent quantum molecular dynamics (IQMD) model or isospin dependent Boltzmann–Uehling–Uhlenbeck (IBUU) model is a suitable model for describing the dynamical process of the SHE formation, which has been used successfully for studying isospin effects of heavy ion collisions at intermediate energies [12,13]. But these models have difficulty to study the fusion reactions near the Coulomb barrier, since some unphysical nucleon emissions in the process of projectile and target approaching. Therefore, it is necessary to develop these models for investigating the fusion reactions near Coulomb barrier, especially for the synthesis of SHEs. In this paper, IQMD model based on the improved QMD model [14] is further developed by introducing shell effect and switch function method. The shell correction energy of the system in the process of projectile and target approaching is calculated by using deformed two center shell model (DTCSM) [15,16]. As a test for our improved model, the fusion cross sections of 40 Ca + 40 Ca, 40 Ca + 48 Ca and 48 Ca + 48 Ca systems near the Coulomb barrier are calculated. Our calculated results are consistent with the experimental data quantitatively. The dynamical and static Coulomb barriers are also studied to clarify the subbarrier fusion reactions. We find that the dynamical Coulomb barrier is obviously lower than the static case. The time evolution of the N/Z ratio at the neck region is presented systematically, a large enhancement of the N/Z ratio for neutron-rich system is found. 2. Model description The same as the conventional IQMD model [17], in the developed IQMD model, each nucleon is represented by a Gaussian wave packet 1 2 e−[r−ri (t)] /4L eipi (t)·r/h¯ , (1) Ψi (r, t) = (2πL)3/4
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where ri (t), pi (t) are the centers of ith wave packet in the coordinate and momentum space, respectively, which satisfy the canonical equation of motion. L is square of the Gaussian wave packet width (here L = 1.82 fm2 ),which is size dependent [18]. The total N -body wave function is assumed to be the direct product of these coherent states. After Wigner transforming for Eq. (1), we get the nucleon’s Wigner density distribution in the N -body phase space and the one-body distribution function f (r, p, t) can be expressed as fi (r, p, t), (2) f (r, p, t) = i
1 [r − ri (t)]2 [p − pi (t)]2 · 2L . − exp − fi (r, p, t) = 2L (π h) h¯ 2 ¯ 3
(3)
Here, we have considered the minimum uncertainty relation. The density distribution in coordinate and momentum space are given by ρ(r, t) = f (r, p, t) d 3 p = ρi (r, t), (4) i
g(p, t) =
f (r, p, t) d 3 r =
gi (p, t),
(5)
i
respectively, where the sum runs over all particles in the system, ρi (r, t) and gi (r, t) are the ith density and momentum distribution 1 (r − ri (t))2 , (6) ρi (r, t) = exp − 2L (2πL)3/2 1 (p − pi (t))2 · 2L gi (p, t) = . (7) exp − (π h¯ 2 /2L)3/2 2h¯ 2 Considering nucleon’s fermionic nature, as an approximative treatment of antisymmetrization, phase space constraint method [19] is introduced in our model. It is requested by the constraint that the occupation number in a volume h3 of the one-body phase space around the point of (ri (t), pi (t)) which is the centroid of ith wave packet should always be not larger than 1 according to the Pauli principle. The one-body occupation number is given by δτi τj δsi sj fj (r, p, t) d 3 r d 3 p, (8) f¯i = j
h3
where si and i represents the projection quantum number of spin and isospin, the integration is performed in the one-body phase space volume h3 . The method can efficiently prevent the phase space distribution from evolving to be a classical distribution from the initial ground state distribution. The time evolutions of the nucleons in the system under the self-consistently generated mean-field is governed by Hamiltonian equations of motion, which are derived from the time dependent variational principle and is expressed as ∂H ∂pi =− , ∂t ∂ri
∂H ∂ri = . ∂t ∂pi
(9)
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The Hamiltonian H consists of the kinetic energy and the effective interaction potential energy: H = T + U, p2 i . T= 2m
(10) (11)
i
The effective interaction potential energy is composed of the Coulomb interaction potential energy and the local interaction potential energy: U = Ucoul + Uloc .
(12)
The Coulomb interaction potential energy is expressed as: Ucoul =
√ e2 1 (1 + tiz )(1 + tj z ) erf(rij / 4L ), 4 rij
(13)
j =i
i
where the tiz is the zth component of the isospin degree of freedom for the ith nucleon, which is equal to 1 and −1 for proton and neutron, respectively. rij = |ri −rj | is the relative distance. The local interaction potential energy includes the volume term, symmetry term and surface term: Uloc = Uvol + Usym + Usurf . The above terms can be written as α ρij β ρij γ + , Uvol = 2 ρ 1+γ ρ i j =i 0 i j =i 0 Csym ρij ri − rj 2 3 1 − ksym , − tiz tj z Usym = 2 ρ0 2L 2L i j =i ri − rj 2 ρij gsurf 3 Usurf = − , 2 2L 2L ρ0 i
(14)
(15)
(16)
(17)
j =i
where ρij is given by ρij =
(ri − rj )2 1 . exp − 4L (4πL)3/2
(18)
The symmetry energy term has included the surface-symmetry energy term which is proposed by Wang et al. [20] in order to correctly describe the neck dynamics for neutronrich combinations, it reads as 2 Csym ksym (19) ∇ρ(r) δ(r) dr. Usurf-sym = − 2ρ0 The parameters of Eqs. (15)–(17) are showed in Table 1, which are corresponding to the soft equation of state which gives the modulus of the incompressibility of 200 MeV for a saturation density 0.165 fm−3 .
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Table 1 The parameter sets of the model α [MeV]
β [MeV]
γ
Csym [MeV]
ksym [fm2 ]
gsurf [MeV fm2 ]
ρ0 [fm−3 ]
−356.0
303.0
7/6
32.0
0.08
18.0
0.165
The surface term of the system is improved by switch function method which connects the surface energies of projectile and target with the one of the compound nucleus. The switch function is expressed as R − Rlow 2 R − Rlow 3 R − Rlow S = C 0 + C1 + C2 + C3 Rup − Rlow Rup − Rlow Rup − Rlow 4 5 R − Rlow R − Rlow + C4 + C5 , (20) Rup − Rlow Rup − Rlow where R is the distance of the centers between projectile and target, Rup and Rlow are the distance between centers at initial time and final time at which the compound nucleus is formed, respectively. Here, the two parameters are set to be 20 and 0 fm, respectively, if R is larger than Rup , then we set R equal to Rup . The parameters C0 , C1 , C2 , C3 , C4 and C5 are taken to be 0, 0, 0, 10, −15 and 6, respectively, which assure the continuity of the surface energy and its first derivative. Thus, the surface energy of the system is written as
surf surf surf Usys = Uproj + Utsurf (21) arg S + Ucomp (1 − S), where Uisurf denotes the surface energy term of the ith system. Various experiments have shown the importance of static deformations and of complex surface vibrations [21–23]. However, the influence on sub-barrier fusion processes such as transfer and breakup reaction is not yet clear [24,25]. Moreover, the effect of structures such as halos and skins is currently being studied [26,27], although so far just restricted to light nuclei. Therefore, it is necessary to consider shell effect in the dynamical process to correctly describe the fusion process near the Coulomb barrier, especially for studying the synthesis of superheavy elements [28]. The levels in the process of projectile and target approaching are calculated by the deformed two center shell model (DTCSM) [15], then using Strutinsky method [29], the shell correction energy of the system is calculated. As we know that shell effect is the diversity of shell model (shell structure) and macroscopic model (bulk property). Thus, the shell correction energy can be obtained from the variance of shell levels and uniformed levels, which is written as following, Eshell = E − E¯ = 2
[N/2] i=1
λ¯ ei − 2
eg(e) ¯ de,
(22)
−∞
where the factor 2 is the degeneracy of the deformed levels (Nilsson levels), the smooth level density is given by 1 e − ei (e − ei )2 g(e) ¯ = √ f . (23) exp − γ γ2 γ π i
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Fig. 1. Proton levels calculated by DTCSM and Fermi surface (left panel) versus the smoothed levels calculated by using Strutinsky method and the smoothed Fermi surface (right panel).
Here γ is Gaussian distribution width, which is usually taken to be 1.2h¯ ω, hω ¯ is equal i to 41A−1/3 . The term of f ( e−e γ ) is the correction function which keeps the long-range variation over energies much larger than hω ¯ unchanged, in the calculation, we select the 3rd-order Laguerre polynomial. ei is the ith level which is calculated by DTCSM. ¯ which can be expressed From the nucleon number N , we can obtain the Fermi energy λ, as following λ¯ N =2
g(e) ¯ de.
(24)
−∞
As a typical example, we have calculated the proton and the smoothed levels of doubly magic nucleus 48 Ca using the above method, which are showed in Fig. 1. The left part is the levels calculated by DTCSM, the right part corresponds to the smoothed levels calculated by using Strutinsky method. The Fermi surfaces shown in the figure for the both kinds of levels are 42.54 and 41.77 MeV, respectively. Using Eqs. (22)–(24), we can obtain the proton shell correction energy −3.74 MeV for the doubly magic nucleus. With the smooth level density increasing (towards right), we can see that the smoothed levels become more and more densely. From Eq. (9), we can see that, it is paying more attention to the effective force imposing on each nucleon derived from the self-consistently generated mean field in the model. However, it is difficult to directly impose the force on each nucleon derived from the shell
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correction energy Eshell . As an approximative treatment, the shell correction energy in the model is written as IQMD Eshell =
Eshell . 1 + exp[(r − R)/a]
(25)
Taking derivative on the equation, we can easily obtain the force derived from the shell correction energy as IQMD Fshell =
exp[(r − R)/a] Eshell . a (1 + exp[(r − R)/a])2
(26)
Inserting expression (22), one obtains the force of each nucleon derived from the shell correction energy: i Fshell =
exp[(ri − R)/a] ei − e¯i . a (1 + exp[(ri − R)/a])2
(27)
Here the ei and the e¯i are the shell level and the smoothed level, respectively. The ordering of filling in the levels is taken according to angular momentum value as well as single nucleon energy (for the same angular momentum). The ri , R and are the coordinate of the center of mass for ith nucleon, the nucleus radius and the dispersion width, respectively. For the R and the a, we take 1.2A−1/3 fm and 0.55 fm respectively, where A is the total nucleon number.
3. Results and discussions In this section we examine the dependability of the improvements of the model. The fusion reactions around Coulomb barrier of the doubly magic nuclei 40 Ca + 40 Ca, 40 Ca + 48 Ca and 48 Ca + 48 Ca which have strong shell effects are studied systematically by using the improved IQMD model. The static and dynamical barriers are studied to clarify the sub-barrier fusion reactions. The N/Z ratio is also investigated at the neck region below and over the Coulomb barrier, which may explain the reason for dynamical lowering of the barrier. 3.1. The Coulomb barrier The interaction potential V (R) is defined as usual [14], V (R) = Ept (R) − Ep − Et .
(28)
Here R is the distance between centers of projectile and target, Ept (R) is the total energy of whole system, Ep and Et are the energies of projectile and target, respectively. The Thomas–Fermi approximation [30] is adopted for calculating the total kinetic energies as 3 h¯ 2 3π 2 ρi 2/3 , (29) E¯ kin = 5 2m 2 i
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Fig. 2. The static Coulomb barriers of 40 Ca + 40 Ca, 40 Ca + 48 Ca and 48 Ca + 48 Ca.
where ρi is the interaction density, which is given by (ri − rj )2 1 . exp − ρi = 4L (4πL)3/2
(30)
j
By Eq. (28), both the static and dynamical Coulomb barrier can be calculated. For the static Coulomb barrier, the density distribution is the same as the initial density distribution. The static Coulomb barrier is shown in Fig. 2. One can see that the barriers are 56.28, 54.42 and 50.61 MeV for the systems 40 Ca + 40 Ca, 40 Ca + 48 Ca and 48 Ca + 48 Ca, respectively. The values are slightly different from the ones used in Ref. [31] which are 54.9, 53.2 and 51.7 MeV, respectively. The lines with solid circles, open circles and solid squares denote the interaction potentials of the combinations 40 Ca + 40 Ca, 40 Ca + 48 Ca and 48 Ca + 48 Ca, respectively. Also included in Fig. 2 is the interaction shell correction energies which are defined as pt
p
t . Vshell = Eshell − Eshell − Eshell pt
p
(31)
t are the projectile-target combinations, projectile and target Here Eshell , Eshell and Eshell shell correction energies which are calculated by DTCSM, respectively. If only considering shell effect, it is obviously not propitious to form the compound nucleus because the projectiles and targets of the three systems are doubly magic nuclei. However, for the dynamical Coulomb barrier, the density distribution changes dynamically from time to time and their shapes get deformed due to the interaction between the reaction partners, which not only depends on the incident energy, but also depends on the initial sampling as well as the impact parameter. As a typical fusion event, the dynamical barriers of the three systems can be seen in Fig. 3 for head on collisions at incident energies 50 MeV in the centers of mass below the static barriers. We note that the dynamical barriers which are 48.70, 47.23 and 45.68 MeV for the systems 40 Ca + 40 Ca, 40 Ca + 48 Ca and
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Fig. 3. The dynamical Coulomb barriers for head on collisions at incident energies 50 MeV in the center of mass system below the static Coulomb barriers.
+ 48 Ca, respectively, are obviously lower than the static barriers. These phenomena are perhaps resulted from the Coulomb repulsion at the neck region leading to the N/Z ratio increasing which is more obviously for neutron-rich nuclear fusion reactions. The sign denotations are the same as the Fig. 2. In order to systematically explaining the dynamical lowering of the barrier, in Fig. 4 we show the time evolution of the N/Z ratio at the neck region for head on collisions of the three systems at the incident energies of the center of mass 50 MeV (below the static barrier) and 60 MeV (above the static barrier), in which the time is started from beginning of the neck formation when the density at the touching point reaches 0.02ρ0 . The neck region is defined as a sphere with radius 2 fm for these systems in the paper. The solid, dashed and dotted lines represent the N/Z ratios of the combinations 40 Ca + 40 Ca, 40 Ca + 48 Ca and 48 Ca + 48 Ca, respectively. From this figure, we can see that the N/Z ratio at the neck region increases with the time increasing at first, then reaches a maximum value and whereafter decreases, and finally it approaches the average N/Z ratio value of the combinations. Simultaneously, the time evolution of the N/Z ratio strongly depends on the isospin of the initial system, especially at the maximum value, which is also studied in detail in Ref. [20]. The reasons for the fluctuation may be understood as following. At the beginning time when the neck is formed, the neutrons move to the neck region drived by the symmetry potential which makes nuclear system towards more symmetrical combinations of proton and neutron, with the neck growing, the Coulomb repulsion plays a important role since the enhancement of the proton number. The interplay of the Coulomb repulsion force and the symmetry potential results in the fluctuation in the time evolution of the N/Z ratio at the neck region. At the same time, nuclear potential may play an adjustable role until the compound nucleus is formed finally. The enhancement of the N/Z ratio at the neck region may lead to the decrease of the dynamical barrier. 48 Ca
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Fig. 4. The N/Z ratio at the neck region at the incident energies in the center of mass system below (left panel) and above (right panel) the barriers.
3.2. Fusion cross sections In the model, it is indispensable to construct the stable nucleus, which assure that there is no nucleon emission before projectile and target touching each other. We elaborately select the stable projectile and target nuclei from several thousands of sampling nuclei randomly, which can retain the stability over 1000 fm/c. That is to say, there is no nucleon emission in the time step if we observe the time evolution of one nucleus. After making the preparation of initial nuclei, the fusion reaction can be calculated. The simulation events are set to be 200 for each incident energy E and impact parameter b, for sub-barrier reactions, they are set to be 400 for enhancing statistics. The fusion cross section is calculated by bmax
σfus (E) = 2π
bpfus (E, b) db = 2π
bpfus (E, b) b,
(32)
0
where Pfus (E, b) stands for the fusion probability for the incident energy E and the impact parameter b. As a test for the model, we systematically study the fusion reaction of doubly magic nuclei near the Coulomb barrier. In Fig. 5, we show the fusion excitation functions of 40 Ca + 40 Ca (a), 40 Ca + 48 Ca (b) and 48 Ca + 48 Ca (c) at energies near the Coulomb barriers. The experimental data are presented for comparison, and contrast of the three systems (d) is also shown. In the figure, the solid circles and solid squares denote the experimental data measured by Trotta et al. [31] and Aljuwair et al. [32], respectively, the lines with open circles and crosses represent the results calculated by Wang et al. in which included events with emitting several nucleons prior to the formation of the compound nucleus in the calculation of fusion events [14,18], and the improved model,
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Fig. 5. The fusion excitation functions of 40 Ca + 40 Ca (a), 40 Ca + 48 Ca (b), 48 Ca + 48 Ca (c) and comparison of the three systems (d) at energies near Coulomb barriers. The solid circles and squares denote the experimental data in Ref. [31] and in Ref. [32], respectively, the lines with open circles and crosses denote the results calculated by Wang et al. [14] and the improved model, respectively, and the line with open squares represents the results calculated by the model in which the shell effect and switch function method are not included.
respectively, and the line with open squares represents the results calculated by the model in which the shell effect and switch function method are not included. In the calculation of the fusion cross section with and without including the shell effect and switch function method, we only consider complete fusion events. Quantitatively, our calculated results are consistent with the experimental data within the error range, especially interesting for the sub-barrier fusion reactions. The sub-barrier fusion reaction process can be understood as following: at the early stage, the enhancement of the N/Z ratio at the neck region results in the dynamical lowering of the Coulomb barrier, and contrasting to the static barrier, that is what we called sub-barrier fusion reactions. Simultaneously, in the figure (d) we note that the calculated fusion cross sections become more larger for the neutron-rich combinations, which are perhaps caused by the more rapid enhancement of the N/Z ratio at the neck region for the neutron-rich system. However, at incident energy below 51 MeV, the phenomena are not obvious because the statistical fusion events in the calculation of fusion probability are very small. We can understand microscopically the extra-push energy in the fusion reactions which is more obviously for heavy systems [33].
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4. Summary In summary, the IQMD model has been developed, in which the switch function method and the shell effect are introduced for studying the fusion reactions near Coulomb barrier. The fusion reactions of doubly magic nuclei 40 Ca + 40 Ca, 40 Ca + 48 Ca and 48 Ca + 48 Ca are studied systematically using the improved IQMD model. The experimental data can be reproduced quantitatively. Our calculated results show a slight enhancement of the fusion cross sections for the neutron-rich combinations. The reason is that the dynamical and static Coulomb barriers for neutron-rich system are lower than the ones for non-neutronrich case. The time evolution of the N/Z ratio at the neck region strongly depends on the isospins of projectile-target combinations at the incident energies below or above the Coulomb barriers. For neutron-rich system, the maximum value of the N/Z ratio at the neck region is larger than that for non-neutron-rich system. The N/Z ratio at the neck region strongly affects the dynamical barriers, consequently, affects the fusion cross sections.
Acknowledgements We are very grateful to Dr. R.A. Gherghescu for providing us the levels of the systems 40 Ca + 40 Ca, 40 Ca + 48 Ca and 48 Ca + 48 Ca calculated by DTCSM and for fruitful discussions.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
I. Tanihata, Prog. Part. Nucl. Phys. 35 (1995) 505. K. Rutz, M. Bender, T. Bürvenich, et al., Phys. Rev. C 56 (1997) 238. A.T. Kruppa, M. Bender, W. Nazarewicz, et al., Phys. Rev. C 61 (2000) 034313. S.K. Patra, C. L Wu, C.R. Praharaj, et al., Nucl. Phys. A 651 (1999) 117. W. Nazarewicz, M. Bender, S. Cwiok, et al., Nucl. Phys. A 710 (2002) 165. V.M. Strutinsky, Nucl. Phys. A 122 (1968) 1. R. Smolanczuk, Phys. Rev. C 59 (1999) 2634. V.Yu. Denisov, S. Hofmann, Phys. Rev. C 61 (2000) 034606. R. Smolanczuk, Phys. Rev. C 63 (2001) 044607. Yu.Ts. Oganessian, V.K. Utyonkoy, Yu.V. Lobanov, et al., Phys. Rev. C 69 (2004) R021601. Y. Abe, Y. Aritomo, T. Wada, M. Ohta, J. Phys. G 23 (1997) 1275. F.S. Zhang, L.W. Chen, Z.Y. Ming, Z.Y. Zhu, Phys. Rev. C 60 (1999) 064604. R. Pak, B.A. Li, W. Benenson, et al., Phys. Rev. Lett. 78 (1997) 1026. N. Wang, Z.X. Li, X.Z. Wu, et al., Phys. Rev. C 69 (2004) 034608. R.A. Gherghescu, Phys. Rev. C 67 (2003) 014309. R.A. Gherghescu, W. Greiner, G. Müzenberg, Phys. Rev. C 68 (2003) 054314. L.W. Chen, F.S. Zhang, G.M. Jin, Phys. Rev. C 58 (1998) 2283. N. Wang, Z.X. Li, X.Z. Wu, Phys. Rev. C 65 (2002) 064608. M. Papa, T. Maruyama, A. Bonasera, Phys. Rev. C 64 (2001) 024612. N. Wang, X.Z. Wu, Z.X. Li, Phys. Rev. C 67 (2003) 024604. C.R. Morton, A.C. Berriman, R.D. Butt, et al., Phys. Rev. C 64 (2001) 034604. A.M. Stefanini, D. Ackermann, L. Corradi, et al., Phys. Rev. Lett. 74 (1995) 864. A.M. Stefanini, L. Corradi, A.M. Vinodkumar, et al., Phys. Rev. C 62 (2000) 014601. M.S. Hussein, M.P. Pato, A.F.R. de Toledo Piza, Phys. Rev. C 51 (1995) 846.
244
[25] [26] [27] [28] [29] [30] [31] [32] [33]
Z.-Q. Feng et al. / Nuclear Physics A 750 (2005) 232–244
C.H. Dosso, A. Vitturi, Phys. Rev. C 50 (1994) R12. J.J. Kolata, et al., Phys. Rev. Lett. 81 (1998) 4580. M. Trotta, et al., Phys. Rev. Lett. 84 (2000) 2342. R.A. Gherghescu, J. Skalski, Z. Patyk, A. Sobiczewski, Nucl. Phys. A 651 (1999) 237. V.M. Strutinsky, Nucl. Phys. A 95 (1967) 420. M. Brack, C. Guet, H.B. Hakansson, Phys. Rep. 123 (1985) 276. M. Trotta, A.M. Stefanini, L. Corradi, et al., Phys. Rev. C 65 (2001) R011601. H.A. Aljuwair, R.J. Ledoux, M. Beckerman, et al., Phys. Rev. C 30 (1984) 1223. K. Siwek-Wilczynska, J. Wilczynski, Phys. Rev. C 64 (2001) 024611.
Improved isospin dependent quantum molecular dynamics model and its application to fusion reactions near Coulomb barrier Zhao-Qing Feng a,b,e,∗ , Feng-Shou Zhang a,c,d , Gen-Ming Jin a,b , Xi Huang a,b,e a Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China b Institute of Modern Physics, Chinese Academy of Sciences, PO Box 31, Lanzhou 730000, China c Institute of Low Energy Nuclear Physics, Beijing Normal University, Beijing 100875, China d CCAST (Word Laboratory), PO Box 8730, Beijing 100080, China e Graduate School, Chinese Academy of Sciences, Beijing 100039, China
Received 15 November 2004; received in revised form 14 December 2004; accepted 3 January 2005 Available online 7 January 2005
Abstract The isospin dependent quantum molecular dynamics model is developed by introducing switch function method which deals with correctly the surface interaction and shell effect in the process of projectile and target approaching. The fusion excitation functions for 40 Ca + 40 Ca, 40 Ca + 48 Ca and 48 Ca + 48 Ca at energies in the vicinity of the Coulomb barrier are studied. The experimental data of the fusion cross sections for these three systems can be regenerated very well. It is found that the fusion cross sections for neutron-rich system increase obviously. The static and dynamical Coulomb barriers are studied in order to clarify the phenomena. The neutron to proton ratio (N/Z ratio) at neck region is also studied, which apparently presents isospin effects of projectile-target combinations. 2005 Elsevier B.V. All rights reserved. PACS: 25.60.Pj; 25.70.Jj; 24.10.-i
* Corresponding author.
E-mail address: [email protected] (Z.-Q. Feng). 0375-9474/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2005.01.001
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1. Introduction In recent years, with the establishment of secondary beam facilities at many laboratories around the world, radioactive beams of nuclei with large neutron or proton excess can be produced [1]. Consequently, one can investigate the properties of nuclei far from the β stability and the nuclear reactions induced by radioactive nuclear beam, especially for the synthesis of superheavy elements (SHEs). The shell model predicts that the next doubly magic nucleus in the sequence will contain either 114, 120, 124 or 126 protons and 184 neutrons [2–4]. Other studies predict a whole superheavy “island of stability” around these proton and neutron numbers. It needs very neutron-rich projectile-target combinations to reach the island, such as by using radioactive nuclear beams. Theoretically, the mere existence of the heaviest elements with Z > 102 is due to quantal shell effects [5]. The properties of SHEs were studied theoretically using Strutinsky method [6,7], where the ground-state deformation, the fission barrier, binding energy and competition between various possible decay modes were also investigated. However, the dynamical process of the formation of SHEs is not understood well enough [8]. There are mainly two methods for the synthesis of SHEs experimentally, cold fusion reaction and hot fusion reaction [8,9]. Recently, elements 115 and 113 were discovered in Dubna by bombarding the americium-243 target with a beam of energetic calcium-48 (evaporation 3 or 4 neutrons) [10]. Recently, a few attempts were undertaken to develop models for describing the fusion process and for reproducing the measured cross section data based on dinuclear system [9,11]. Microscopic transport theory such as isospin dependent quantum molecular dynamics (IQMD) model or isospin dependent Boltzmann–Uehling–Uhlenbeck (IBUU) model is a suitable model for describing the dynamical process of the SHE formation, which has been used successfully for studying isospin effects of heavy ion collisions at intermediate energies [12,13]. But these models have difficulty to study the fusion reactions near the Coulomb barrier, since some unphysical nucleon emissions in the process of projectile and target approaching. Therefore, it is necessary to develop these models for investigating the fusion reactions near Coulomb barrier, especially for the synthesis of SHEs. In this paper, IQMD model based on the improved QMD model [14] is further developed by introducing shell effect and switch function method. The shell correction energy of the system in the process of projectile and target approaching is calculated by using deformed two center shell model (DTCSM) [15,16]. As a test for our improved model, the fusion cross sections of 40 Ca + 40 Ca, 40 Ca + 48 Ca and 48 Ca + 48 Ca systems near the Coulomb barrier are calculated. Our calculated results are consistent with the experimental data quantitatively. The dynamical and static Coulomb barriers are also studied to clarify the subbarrier fusion reactions. We find that the dynamical Coulomb barrier is obviously lower than the static case. The time evolution of the N/Z ratio at the neck region is presented systematically, a large enhancement of the N/Z ratio for neutron-rich system is found. 2. Model description The same as the conventional IQMD model [17], in the developed IQMD model, each nucleon is represented by a Gaussian wave packet 1 2 e−[r−ri (t)] /4L eipi (t)·r/h¯ , (1) Ψi (r, t) = (2πL)3/4
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where ri (t), pi (t) are the centers of ith wave packet in the coordinate and momentum space, respectively, which satisfy the canonical equation of motion. L is square of the Gaussian wave packet width (here L = 1.82 fm2 ),which is size dependent [18]. The total N -body wave function is assumed to be the direct product of these coherent states. After Wigner transforming for Eq. (1), we get the nucleon’s Wigner density distribution in the N -body phase space and the one-body distribution function f (r, p, t) can be expressed as fi (r, p, t), (2) f (r, p, t) = i
1 [r − ri (t)]2 [p − pi (t)]2 · 2L . − exp − fi (r, p, t) = 2L (π h) h¯ 2 ¯ 3
(3)
Here, we have considered the minimum uncertainty relation. The density distribution in coordinate and momentum space are given by ρ(r, t) = f (r, p, t) d 3 p = ρi (r, t), (4) i
g(p, t) =
f (r, p, t) d 3 r =
gi (p, t),
(5)
i
respectively, where the sum runs over all particles in the system, ρi (r, t) and gi (r, t) are the ith density and momentum distribution 1 (r − ri (t))2 , (6) ρi (r, t) = exp − 2L (2πL)3/2 1 (p − pi (t))2 · 2L gi (p, t) = . (7) exp − (π h¯ 2 /2L)3/2 2h¯ 2 Considering nucleon’s fermionic nature, as an approximative treatment of antisymmetrization, phase space constraint method [19] is introduced in our model. It is requested by the constraint that the occupation number in a volume h3 of the one-body phase space around the point of (ri (t), pi (t)) which is the centroid of ith wave packet should always be not larger than 1 according to the Pauli principle. The one-body occupation number is given by δτi τj δsi sj fj (r, p, t) d 3 r d 3 p, (8) f¯i = j
h3
where si and i represents the projection quantum number of spin and isospin, the integration is performed in the one-body phase space volume h3 . The method can efficiently prevent the phase space distribution from evolving to be a classical distribution from the initial ground state distribution. The time evolutions of the nucleons in the system under the self-consistently generated mean-field is governed by Hamiltonian equations of motion, which are derived from the time dependent variational principle and is expressed as ∂H ∂pi =− , ∂t ∂ri
∂H ∂ri = . ∂t ∂pi
(9)
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The Hamiltonian H consists of the kinetic energy and the effective interaction potential energy: H = T + U, p2 i . T= 2m
(10) (11)
i
The effective interaction potential energy is composed of the Coulomb interaction potential energy and the local interaction potential energy: U = Ucoul + Uloc .
(12)
The Coulomb interaction potential energy is expressed as: Ucoul =
√ e2 1 (1 + tiz )(1 + tj z ) erf(rij / 4L ), 4 rij
(13)
j =i
i
where the tiz is the zth component of the isospin degree of freedom for the ith nucleon, which is equal to 1 and −1 for proton and neutron, respectively. rij = |ri −rj | is the relative distance. The local interaction potential energy includes the volume term, symmetry term and surface term: Uloc = Uvol + Usym + Usurf . The above terms can be written as α ρij β ρij γ + , Uvol = 2 ρ 1+γ ρ i j =i 0 i j =i 0 Csym ρij ri − rj 2 3 1 − ksym , − tiz tj z Usym = 2 ρ0 2L 2L i j =i ri − rj 2 ρij gsurf 3 Usurf = − , 2 2L 2L ρ0 i
(14)
(15)
(16)
(17)
j =i
where ρij is given by ρij =
(ri − rj )2 1 . exp − 4L (4πL)3/2
(18)
The symmetry energy term has included the surface-symmetry energy term which is proposed by Wang et al. [20] in order to correctly describe the neck dynamics for neutronrich combinations, it reads as 2 Csym ksym (19) ∇ρ(r) δ(r) dr. Usurf-sym = − 2ρ0 The parameters of Eqs. (15)–(17) are showed in Table 1, which are corresponding to the soft equation of state which gives the modulus of the incompressibility of 200 MeV for a saturation density 0.165 fm−3 .
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Table 1 The parameter sets of the model α [MeV]
β [MeV]
γ
Csym [MeV]
ksym [fm2 ]
gsurf [MeV fm2 ]
ρ0 [fm−3 ]
−356.0
303.0
7/6
32.0
0.08
18.0
0.165
The surface term of the system is improved by switch function method which connects the surface energies of projectile and target with the one of the compound nucleus. The switch function is expressed as R − Rlow 2 R − Rlow 3 R − Rlow S = C 0 + C1 + C2 + C3 Rup − Rlow Rup − Rlow Rup − Rlow 4 5 R − Rlow R − Rlow + C4 + C5 , (20) Rup − Rlow Rup − Rlow where R is the distance of the centers between projectile and target, Rup and Rlow are the distance between centers at initial time and final time at which the compound nucleus is formed, respectively. Here, the two parameters are set to be 20 and 0 fm, respectively, if R is larger than Rup , then we set R equal to Rup . The parameters C0 , C1 , C2 , C3 , C4 and C5 are taken to be 0, 0, 0, 10, −15 and 6, respectively, which assure the continuity of the surface energy and its first derivative. Thus, the surface energy of the system is written as
surf surf surf Usys = Uproj + Utsurf (21) arg S + Ucomp (1 − S), where Uisurf denotes the surface energy term of the ith system. Various experiments have shown the importance of static deformations and of complex surface vibrations [21–23]. However, the influence on sub-barrier fusion processes such as transfer and breakup reaction is not yet clear [24,25]. Moreover, the effect of structures such as halos and skins is currently being studied [26,27], although so far just restricted to light nuclei. Therefore, it is necessary to consider shell effect in the dynamical process to correctly describe the fusion process near the Coulomb barrier, especially for studying the synthesis of superheavy elements [28]. The levels in the process of projectile and target approaching are calculated by the deformed two center shell model (DTCSM) [15], then using Strutinsky method [29], the shell correction energy of the system is calculated. As we know that shell effect is the diversity of shell model (shell structure) and macroscopic model (bulk property). Thus, the shell correction energy can be obtained from the variance of shell levels and uniformed levels, which is written as following, Eshell = E − E¯ = 2
[N/2] i=1
λ¯ ei − 2
eg(e) ¯ de,
(22)
−∞
where the factor 2 is the degeneracy of the deformed levels (Nilsson levels), the smooth level density is given by 1 e − ei (e − ei )2 g(e) ¯ = √ f . (23) exp − γ γ2 γ π i
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Fig. 1. Proton levels calculated by DTCSM and Fermi surface (left panel) versus the smoothed levels calculated by using Strutinsky method and the smoothed Fermi surface (right panel).
Here γ is Gaussian distribution width, which is usually taken to be 1.2h¯ ω, hω ¯ is equal i to 41A−1/3 . The term of f ( e−e γ ) is the correction function which keeps the long-range variation over energies much larger than hω ¯ unchanged, in the calculation, we select the 3rd-order Laguerre polynomial. ei is the ith level which is calculated by DTCSM. ¯ which can be expressed From the nucleon number N , we can obtain the Fermi energy λ, as following λ¯ N =2
g(e) ¯ de.
(24)
−∞
As a typical example, we have calculated the proton and the smoothed levels of doubly magic nucleus 48 Ca using the above method, which are showed in Fig. 1. The left part is the levels calculated by DTCSM, the right part corresponds to the smoothed levels calculated by using Strutinsky method. The Fermi surfaces shown in the figure for the both kinds of levels are 42.54 and 41.77 MeV, respectively. Using Eqs. (22)–(24), we can obtain the proton shell correction energy −3.74 MeV for the doubly magic nucleus. With the smooth level density increasing (towards right), we can see that the smoothed levels become more and more densely. From Eq. (9), we can see that, it is paying more attention to the effective force imposing on each nucleon derived from the self-consistently generated mean field in the model. However, it is difficult to directly impose the force on each nucleon derived from the shell
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correction energy Eshell . As an approximative treatment, the shell correction energy in the model is written as IQMD Eshell =
Eshell . 1 + exp[(r − R)/a]
(25)
Taking derivative on the equation, we can easily obtain the force derived from the shell correction energy as IQMD Fshell =
exp[(r − R)/a] Eshell . a (1 + exp[(r − R)/a])2
(26)
Inserting expression (22), one obtains the force of each nucleon derived from the shell correction energy: i Fshell =
exp[(ri − R)/a] ei − e¯i . a (1 + exp[(ri − R)/a])2
(27)
Here the ei and the e¯i are the shell level and the smoothed level, respectively. The ordering of filling in the levels is taken according to angular momentum value as well as single nucleon energy (for the same angular momentum). The ri , R and are the coordinate of the center of mass for ith nucleon, the nucleus radius and the dispersion width, respectively. For the R and the a, we take 1.2A−1/3 fm and 0.55 fm respectively, where A is the total nucleon number.
3. Results and discussions In this section we examine the dependability of the improvements of the model. The fusion reactions around Coulomb barrier of the doubly magic nuclei 40 Ca + 40 Ca, 40 Ca + 48 Ca and 48 Ca + 48 Ca which have strong shell effects are studied systematically by using the improved IQMD model. The static and dynamical barriers are studied to clarify the sub-barrier fusion reactions. The N/Z ratio is also investigated at the neck region below and over the Coulomb barrier, which may explain the reason for dynamical lowering of the barrier. 3.1. The Coulomb barrier The interaction potential V (R) is defined as usual [14], V (R) = Ept (R) − Ep − Et .
(28)
Here R is the distance between centers of projectile and target, Ept (R) is the total energy of whole system, Ep and Et are the energies of projectile and target, respectively. The Thomas–Fermi approximation [30] is adopted for calculating the total kinetic energies as 3 h¯ 2 3π 2 ρi 2/3 , (29) E¯ kin = 5 2m 2 i
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Fig. 2. The static Coulomb barriers of 40 Ca + 40 Ca, 40 Ca + 48 Ca and 48 Ca + 48 Ca.
where ρi is the interaction density, which is given by (ri − rj )2 1 . exp − ρi = 4L (4πL)3/2
(30)
j
By Eq. (28), both the static and dynamical Coulomb barrier can be calculated. For the static Coulomb barrier, the density distribution is the same as the initial density distribution. The static Coulomb barrier is shown in Fig. 2. One can see that the barriers are 56.28, 54.42 and 50.61 MeV for the systems 40 Ca + 40 Ca, 40 Ca + 48 Ca and 48 Ca + 48 Ca, respectively. The values are slightly different from the ones used in Ref. [31] which are 54.9, 53.2 and 51.7 MeV, respectively. The lines with solid circles, open circles and solid squares denote the interaction potentials of the combinations 40 Ca + 40 Ca, 40 Ca + 48 Ca and 48 Ca + 48 Ca, respectively. Also included in Fig. 2 is the interaction shell correction energies which are defined as pt
p
t . Vshell = Eshell − Eshell − Eshell pt
p
(31)
t are the projectile-target combinations, projectile and target Here Eshell , Eshell and Eshell shell correction energies which are calculated by DTCSM, respectively. If only considering shell effect, it is obviously not propitious to form the compound nucleus because the projectiles and targets of the three systems are doubly magic nuclei. However, for the dynamical Coulomb barrier, the density distribution changes dynamically from time to time and their shapes get deformed due to the interaction between the reaction partners, which not only depends on the incident energy, but also depends on the initial sampling as well as the impact parameter. As a typical fusion event, the dynamical barriers of the three systems can be seen in Fig. 3 for head on collisions at incident energies 50 MeV in the centers of mass below the static barriers. We note that the dynamical barriers which are 48.70, 47.23 and 45.68 MeV for the systems 40 Ca + 40 Ca, 40 Ca + 48 Ca and
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Fig. 3. The dynamical Coulomb barriers for head on collisions at incident energies 50 MeV in the center of mass system below the static Coulomb barriers.
+ 48 Ca, respectively, are obviously lower than the static barriers. These phenomena are perhaps resulted from the Coulomb repulsion at the neck region leading to the N/Z ratio increasing which is more obviously for neutron-rich nuclear fusion reactions. The sign denotations are the same as the Fig. 2. In order to systematically explaining the dynamical lowering of the barrier, in Fig. 4 we show the time evolution of the N/Z ratio at the neck region for head on collisions of the three systems at the incident energies of the center of mass 50 MeV (below the static barrier) and 60 MeV (above the static barrier), in which the time is started from beginning of the neck formation when the density at the touching point reaches 0.02ρ0 . The neck region is defined as a sphere with radius 2 fm for these systems in the paper. The solid, dashed and dotted lines represent the N/Z ratios of the combinations 40 Ca + 40 Ca, 40 Ca + 48 Ca and 48 Ca + 48 Ca, respectively. From this figure, we can see that the N/Z ratio at the neck region increases with the time increasing at first, then reaches a maximum value and whereafter decreases, and finally it approaches the average N/Z ratio value of the combinations. Simultaneously, the time evolution of the N/Z ratio strongly depends on the isospin of the initial system, especially at the maximum value, which is also studied in detail in Ref. [20]. The reasons for the fluctuation may be understood as following. At the beginning time when the neck is formed, the neutrons move to the neck region drived by the symmetry potential which makes nuclear system towards more symmetrical combinations of proton and neutron, with the neck growing, the Coulomb repulsion plays a important role since the enhancement of the proton number. The interplay of the Coulomb repulsion force and the symmetry potential results in the fluctuation in the time evolution of the N/Z ratio at the neck region. At the same time, nuclear potential may play an adjustable role until the compound nucleus is formed finally. The enhancement of the N/Z ratio at the neck region may lead to the decrease of the dynamical barrier. 48 Ca
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Fig. 4. The N/Z ratio at the neck region at the incident energies in the center of mass system below (left panel) and above (right panel) the barriers.
3.2. Fusion cross sections In the model, it is indispensable to construct the stable nucleus, which assure that there is no nucleon emission before projectile and target touching each other. We elaborately select the stable projectile and target nuclei from several thousands of sampling nuclei randomly, which can retain the stability over 1000 fm/c. That is to say, there is no nucleon emission in the time step if we observe the time evolution of one nucleus. After making the preparation of initial nuclei, the fusion reaction can be calculated. The simulation events are set to be 200 for each incident energy E and impact parameter b, for sub-barrier reactions, they are set to be 400 for enhancing statistics. The fusion cross section is calculated by bmax
σfus (E) = 2π
bpfus (E, b) db = 2π
bpfus (E, b) b,
(32)
0
where Pfus (E, b) stands for the fusion probability for the incident energy E and the impact parameter b. As a test for the model, we systematically study the fusion reaction of doubly magic nuclei near the Coulomb barrier. In Fig. 5, we show the fusion excitation functions of 40 Ca + 40 Ca (a), 40 Ca + 48 Ca (b) and 48 Ca + 48 Ca (c) at energies near the Coulomb barriers. The experimental data are presented for comparison, and contrast of the three systems (d) is also shown. In the figure, the solid circles and solid squares denote the experimental data measured by Trotta et al. [31] and Aljuwair et al. [32], respectively, the lines with open circles and crosses represent the results calculated by Wang et al. in which included events with emitting several nucleons prior to the formation of the compound nucleus in the calculation of fusion events [14,18], and the improved model,
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Fig. 5. The fusion excitation functions of 40 Ca + 40 Ca (a), 40 Ca + 48 Ca (b), 48 Ca + 48 Ca (c) and comparison of the three systems (d) at energies near Coulomb barriers. The solid circles and squares denote the experimental data in Ref. [31] and in Ref. [32], respectively, the lines with open circles and crosses denote the results calculated by Wang et al. [14] and the improved model, respectively, and the line with open squares represents the results calculated by the model in which the shell effect and switch function method are not included.
respectively, and the line with open squares represents the results calculated by the model in which the shell effect and switch function method are not included. In the calculation of the fusion cross section with and without including the shell effect and switch function method, we only consider complete fusion events. Quantitatively, our calculated results are consistent with the experimental data within the error range, especially interesting for the sub-barrier fusion reactions. The sub-barrier fusion reaction process can be understood as following: at the early stage, the enhancement of the N/Z ratio at the neck region results in the dynamical lowering of the Coulomb barrier, and contrasting to the static barrier, that is what we called sub-barrier fusion reactions. Simultaneously, in the figure (d) we note that the calculated fusion cross sections become more larger for the neutron-rich combinations, which are perhaps caused by the more rapid enhancement of the N/Z ratio at the neck region for the neutron-rich system. However, at incident energy below 51 MeV, the phenomena are not obvious because the statistical fusion events in the calculation of fusion probability are very small. We can understand microscopically the extra-push energy in the fusion reactions which is more obviously for heavy systems [33].
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4. Summary In summary, the IQMD model has been developed, in which the switch function method and the shell effect are introduced for studying the fusion reactions near Coulomb barrier. The fusion reactions of doubly magic nuclei 40 Ca + 40 Ca, 40 Ca + 48 Ca and 48 Ca + 48 Ca are studied systematically using the improved IQMD model. The experimental data can be reproduced quantitatively. Our calculated results show a slight enhancement of the fusion cross sections for the neutron-rich combinations. The reason is that the dynamical and static Coulomb barriers for neutron-rich system are lower than the ones for non-neutronrich case. The time evolution of the N/Z ratio at the neck region strongly depends on the isospins of projectile-target combinations at the incident energies below or above the Coulomb barriers. For neutron-rich system, the maximum value of the N/Z ratio at the neck region is larger than that for non-neutron-rich system. The N/Z ratio at the neck region strongly affects the dynamical barriers, consequently, affects the fusion cross sections.
Acknowledgements We are very grateful to Dr. R.A. Gherghescu for providing us the levels of the systems 40 Ca + 40 Ca, 40 Ca + 48 Ca and 48 Ca + 48 Ca calculated by DTCSM and for fruitful discussions.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
I. Tanihata, Prog. Part. Nucl. Phys. 35 (1995) 505. K. Rutz, M. Bender, T. Bürvenich, et al., Phys. Rev. C 56 (1997) 238. A.T. Kruppa, M. Bender, W. Nazarewicz, et al., Phys. Rev. C 61 (2000) 034313. S.K. Patra, C. L Wu, C.R. Praharaj, et al., Nucl. Phys. A 651 (1999) 117. W. Nazarewicz, M. Bender, S. Cwiok, et al., Nucl. Phys. A 710 (2002) 165. V.M. Strutinsky, Nucl. Phys. A 122 (1968) 1. R. Smolanczuk, Phys. Rev. C 59 (1999) 2634. V.Yu. Denisov, S. Hofmann, Phys. Rev. C 61 (2000) 034606. R. Smolanczuk, Phys. Rev. C 63 (2001) 044607. Yu.Ts. Oganessian, V.K. Utyonkoy, Yu.V. Lobanov, et al., Phys. Rev. C 69 (2004) R021601. Y. Abe, Y. Aritomo, T. Wada, M. Ohta, J. Phys. G 23 (1997) 1275. F.S. Zhang, L.W. Chen, Z.Y. Ming, Z.Y. Zhu, Phys. Rev. C 60 (1999) 064604. R. Pak, B.A. Li, W. Benenson, et al., Phys. Rev. Lett. 78 (1997) 1026. N. Wang, Z.X. Li, X.Z. Wu, et al., Phys. Rev. C 69 (2004) 034608. R.A. Gherghescu, Phys. Rev. C 67 (2003) 014309. R.A. Gherghescu, W. Greiner, G. Müzenberg, Phys. Rev. C 68 (2003) 054314. L.W. Chen, F.S. Zhang, G.M. Jin, Phys. Rev. C 58 (1998) 2283. N. Wang, Z.X. Li, X.Z. Wu, Phys. Rev. C 65 (2002) 064608. M. Papa, T. Maruyama, A. Bonasera, Phys. Rev. C 64 (2001) 024612. N. Wang, X.Z. Wu, Z.X. Li, Phys. Rev. C 67 (2003) 024604. C.R. Morton, A.C. Berriman, R.D. Butt, et al., Phys. Rev. C 64 (2001) 034604. A.M. Stefanini, D. Ackermann, L. Corradi, et al., Phys. Rev. Lett. 74 (1995) 864. A.M. Stefanini, L. Corradi, A.M. Vinodkumar, et al., Phys. Rev. C 62 (2000) 014601. M.S. Hussein, M.P. Pato, A.F.R. de Toledo Piza, Phys. Rev. C 51 (1995) 846.
244
[25] [26] [27] [28] [29] [30] [31] [32] [33]
Z.-Q. Feng et al. / Nuclear Physics A 750 (2005) 232–244
C.H. Dosso, A. Vitturi, Phys. Rev. C 50 (1994) R12. J.J. Kolata, et al., Phys. Rev. Lett. 81 (1998) 4580. M. Trotta, et al., Phys. Rev. Lett. 84 (2000) 2342. R.A. Gherghescu, J. Skalski, Z. Patyk, A. Sobiczewski, Nucl. Phys. A 651 (1999) 237. V.M. Strutinsky, Nucl. Phys. A 95 (1967) 420. M. Brack, C. Guet, H.B. Hakansson, Phys. Rep. 123 (1985) 276. M. Trotta, A.M. Stefanini, L. Corradi, et al., Phys. Rev. C 65 (2001) R011601. H.A. Aljuwair, R.J. Ledoux, M. Beckerman, et al., Phys. Rev. C 30 (1984) 1223. K. Siwek-Wilczynska, J. Wilczynski, Phys. Rev. C 64 (2001) 024611.