Problems of dynamical calculation for synthesis of superheavy elements

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Problems of dynamical calculation for synthesis of superheavy elements Yoshihiro Aritomo1,2 , Masahisa Ohta3 , Thomas Materna4 , Francis Hanappe4 and Louise Stuttge5 1 Department

of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo, Japan Laboratory of Nuclear Reactions, JINR, Dubna, Russia 3 Department of Physics, Konan University, 8-9-1 Okamoto, Kobe, Japan 4 Universite Libre de Bruxelles, 1050 Bruxelles, Belgium 5 Institut de Recherches Subatomiques, F-67037 Strasbourg Cedex, France 2 Flerov

We discuss the fusion-fission process in super heavy mass region. In dynamical calculation, we list up the problems and discuss the sensitivity of the parameters. Using the fluctuation-dissipation model we investigate the fusion-fission process and estimate the fusion cross section precisely.

§1.

Introduction

Recently the study on synthesis of superheavy elements is more exciting and heating. Synthesizing of new elements are reported continuously from Dubna, GSI, and Riken.1)–3) Many theoretical studies on the synthesis of superheavy elements have been published and the evaporation residue cross section is estimated. Such calculations show a good agreement with experimental data. However, up to now in the theoretical calculation, many unsolved problems and unknown parameters do exist.4) Here we list up the problem of the calculation, especially dynamical calculation, and discuss on the ambiguity of the model. Our final goal is to estimate the residue cross section, but before do it we need more accurate estimation of fusion probability, which is defined as formation probability of compound nucleus here. Fortunately, we have a lot of relevant experimental data5), 6) which is available in our study. The mass and the total kinetic energy distributions of fission fragments have been extensively measured. The emission of neutrons in coincidence with fission fragments has been studied by DeMon group.6) To estimate the fusion probability and investigate the fusion-fission mechanism, the trajectory calculation are performed under the friction force.7)–9) In the superheavy mass region, the mean trajectory can not overcome the extra potential barrier after the contact only by adding the extra bombarding energy. Therefore, it is necessary to introduce a fluctuation-dissipation dynamics with the Fokker-Plank equation or with the Langevin equation.10)–13)

typeset using P T P TEX.cls hVer.0.89i

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Y. Aritomo, M. Ohta, T. Materna, F. Hanappe and L. Stuttge §2.

Model and problems of calculation

The fusion-fission cross section σER is expressed as; σER =

∞ π~2 X (2l + 1)Tl (Ecm , l)PCN (E ∗ , l)W (E ∗ , l), 2µ0 Ecm

(2.1)

l=0

where µ0 denotes the reduced mass in the entrance channel. Ecm and E ∗ denote the incident energy in the center-of-mass frame and the excitation energy of the composite system, respectively. Tl (Ecm , l) is the capture probability of the lth partial wave. PCN (E ∗ , l) is the probability of forming a compound nucleus in competition with quasi-fission. W (E ∗ , l) denotes the survival probability of compound nuclei during deexitation, which is calculated by statistical model. The value of σER leads to a pico-barn order. Inevitably, an substantial uncertainty becomes to be involved on the each stage. The theoretical framework to estimate them is not confirmed now. Here, we list up the problems to estimate the value of Tl (Ecm , l), PCN (E ∗ , l) and W (E ∗ , l). 2.1. Capture probability Tl (Ecm , l) In order to estimate the capture probability Tl (Ecm , l), mainly two different models are applied, which are Gross-Kalinowski model and the coupled channel model. Gross-Kalinowski model is the classical model,14) so we can not apply to the sub-barrier fusion. When we discuss on the cold fusion reaction, the enhancement of the fusion in the sub-barrier energy region is very important. In this case we should use the coupled channel model. Here, in the cold fusion reaction and the hot fusion reaction, we employ the empirical coupled channel model which is suggested by Zagrebaev.15) 2.2. Formation probability PCN (E ∗ , l) We introduce the fluctuation-dissipation model to calculate the formation probability PCN , and employ the Langevin equation. Which potential energy surface we should use? 1) LDM (liquid drop model), 2) LDM+shell correction energy at T = 0, 3) LDM+shell correction energy at the local temperature Tlocal . In the LDM,14) we can not reproduce the mass distribution of fission fragments at all. When we use the LDM+shell(T =0), the formation probability is smaller than one in the LDM case14) by two order of magnitude, in the reaction 48 Ca+244 Pu at E ∗ =35 MeV. In the calculation, we employ the LDM+shell(T =0). We adopt a three-dimensional nuclear deformation space with two-center parametrization.16), 17) The three collective parameters to be described by the Langevin equation are treated as follows: z0 (distance between two potential centers), δ (deformation of fragments) and α (mass asymmetry of the colliding partner); α = (A1 − A2 )/(A1 + A2 ), where A1 and A2 denote the mass number of target and projectile, respectively. In the two-dimensional calculation (z, α),14) the formation probability is larger than one in the three-dimensional calculation by two-order of

Problems of dynamical calculation for synthesis of SHE

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magnitude. Also two-dimensional calculation can not reproduce the mass distribution of fission fragments at all. Here, a hydrodynamical inertia tensor is adopted with the Werner-Wheeler approximation for the velocity field, and the wall-and-window one-body dissipation is adopted for the dissipation tensor. 2.3. Survival probability W (E ∗ , l) We estimate the survival probability using the statistical code.18), 19) In super heavy mass region, there are many unknown parameters, for example the fission barrier height of the compound nucleus, the level density parameter, the collective enhancement factor, the shell dumping factor and so on. It is very sensitive to estimate the survival probability. When we change the fission barrier height by 1 MeV, the survival probability changes by one-order of magnitude. §3.

Results

3.1. mass distribution of fission fragments and fusion cross section First, we try to reproduce the experimental data of the mass and the total kinetic energy (TKE) distribution of fission fragment.5) By analyzing the mass distribution, we can distinguish between the fusion-fission process (FF) and quasi-fission process (QF). The mechanism of FF and QF are clarified by analyzing the trajectory on the three-dimensional potential energy surface. Figure 1 show the potential energy surface of liquid drop model with shell correction energy in nuclear deformation space for 256 102, which is calculated using the code.20), 21) In order to save a great deal of computation time, we use the scaling and employ the coordinate z, which is described in the reference.22), 23) The position at z = α = 0 corresponds to a spherical compound nucleus. Due to the shell structure of Pb and Sn, we can see the valley on the potential surface. The black arrow denotes the injection point. We can see the FF process is dominant and mass symmetric fission events are detected. The white arrows indicate the fusion-fission paths. However, in the case of 48 Ca+244 Pu the situation changes. The potential energy surface of 292 114 is shown in Fig. 2. In this case, QF process is dominant. Only few probability enters the spherical region. At Dubna the experiments on the fission of superheavy nuclei in the reaction 48 Ca+244 Pu carried out and they present the fusion-fission cross section of compound nuclei which is derived from the mass symmetric fission fragments (A/2 ± 20).5) The subsequent important question is whether all of the mass symmetric fission fragments come from the compound nuclei or not. As the final results of the experiments the mass symmetric fission fragments are detected, but there exists two possibilities where it comes from. One is that the mass symmetric fission fragments come from the compound nuclei and the other is that they come from QF. We try to check them by using three-dimensional trajectory calculation with Langevin equation. In the results, about 90∼99 % of mass symmetric fission frag-

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Y. Aritomo, M. Ohta, T. Materna, F. Hanappe and L. Stuttge

Fig. 1. The potential energy surface of liquid drop model with shell correction energy in nuclear deformation space for 256 102. z and α denote the separation between two potential center and the mass asymmetry, respectively. The white arrows indicate the fusion-fission paths.

Fig. 2. The potential energy surface of 292 114. The QF, DQF and FF are denoted by white arrows.

ments come from QF process, which we call ”deep quasi-fission process (DQF)”. We can see it more precisely in the reference.22), 23) In heavy mass region, due to the large Coulomb repulsion force, potential energy surface has very steep slope in the direction of +δ. Therefore, even if trajectories overcome the fusion barrier, almost all of them flow down to the +δ direction and can not reach the spherical region. Figure. 3 shows the samples of trajectory which

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Fig. 3. Samples of the trajectory which are projected on z − α plane (left) and z − δ plane (right), at E ∗ = 33 MeV in the reaction 48 Ca+244 Pu. The trajectories go to +δ direction, and can not enter the region of spherical nucleus.

are projected on the z − α and z − δ plane, in the reaction 48 Ca+244 Pu at E ∗ = 33 MeV. The cross point (+) denotes the touching point of the system. We can see that trajectories go to +δ direction, and can not enter the region of spherical nucleus. That is reason why the formation of compound nucleus is extremely reduced in heavy mass region. We can say that fusion is hindered by the flow of the trajectory to the δ degree of freedom. Figure 4 (a) shows mass distribution of fission fragments in the reactions 48 Ca+244 Pu at E ∗ = 37 MeV. The mass asymmetric fission events is dominant. The calculation agrees with experimental data. The excitation function of the cross section are shown in Fig. 4 (b). The open and closed diamonds denote the capture cross section σcap and the cross section σA/2±20 which derived by the yield of the mass symmetric fission fragments with A2 ± 20 in the experiments, respectively.5) The calculated σA/2±20 is denoted by the solid line. We can see good agreement with the experimental data and our calculations. The calculated fusion-fission cross section σCN is denoted by the dashed line. This cross section σCN is derived from the trajectory crossing the three-dimensional fusion box. The fusion-fission cross section σCN is one or two order magnitude smaller than the cross section σA/2±20 . We see that the cross section σA/2±20 includes the deep quasi-fission events. Such an information is very important to estimate the evaporation residue cross section. 3.2. pre-scission neutron multiplicity In order to classify the fusion-fission process more precisely, we analyze the prescission neutron multiplicity.5), 6) The neutron multiplicity directly depends on the time scale of trajectory. So using the data, we can investigate the different class of the dynamical process and compare with experimental data directly. We introduce

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Y. Aritomo, M. Ohta, T. Materna, F. Hanappe and L. Stuttge

Fig. 4. (a) Mass distribution of fission fragments in the reaction 48 Ca+244 Pu at E ∗ =37 MeV. The black line and gray line denote the calculation and experiment, respectively. (b) The cross section by the calculation and experiment. Lines and diamonds are given in the text.

the effect of neutron emission in Langevin code.24) Demon group measured the neutron multiplicity in the reaction 58 Ni+208 Pb at ∗ E = 185.9 MeV.6) The neutron multiplicity has two peaks (near 4 and 8 neutron emission), which is shown in Fig. 5 (a). We suppose that the first peak is connected with QF process and second one is connected with FF process. The neutron multiplicity depends on the travelling time of trajectories from the contact point to the scission one. On the FF process, the trajectory is trapped in the pocket around spherical region. It takes a long time to stay in the pocket and it has a large chance to emit neutrons. The calculation is shown in Fig. 5 (b). We can see the two peaks which come from QF process and FF process. Actually the travelling time scale of each process from contact point to scission point are clearly distinguished.25) The time scale of FF process is longer than that of QF process. We investigate the neutron multiplicity in the reactions 48 Ca+208 Pb and 48 Ca+244 Pu.26) The calculation and experimental data are shown in Fig. 6, which excitation energy is about 40 MeV. In the reaction 48 Ca+208 Pb, the neutron multiplicity shows the single peak near 2 neutron emission, which comes from FF process. In the three dimensional Langevin calculation, we can see that the trajectory of FF process is dominant and single peak is reproduced. In the 48 Ca+244 Pu case, in the Langevin calculation, we found mass symmetric fission events come from not only FF process but also DQF process. The experiment

Problems of dynamical calculation for synthesis of SHE

Fig. 5. Pre-scission neutron multiplicity in the reaction experimental data. (b) Calculation.

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Ni+208 Pb at ∗E ∗ =189.5 MeV. (a) The

of neutron multiplicity shows the two peak near 2 and 4 neutron emission. Such two peaks are originated by two different fusion-fission mechanism. It looks that the first peak comes from QF process and the second one comes from mainly DQF or FF. However, the calculation can not reproduce the two peaks. We check the travelling time of both case, and they overlap each other.25) In the calculation, the lifetime of neutron emission is rather short, because the excitation energy is rather low. The time scale of trajectory fluctuation is the same order of the life time of neutron emission. So we can not distinguish QF and DQF process clearly. Also the event of FF process is quit low. We have to investigate the parameters, for example friction tensor, level density parameter and neutron binding energy near di-nucleus configuration etc. We try to use the friction tensor which is derived from liner response theory27) Our analysis is useful for identifying the detail of the fusion-fission process in connection with the neutron multiplicity. The time scale of fusion-fission process depends on the strength of friction tensor. Also the time scale of neutron emission depends on the value of level density parameter and neutron binding energy at each nuclear shapes. The measured neutron multiplicity can be used for the determination of such values. References 1) Yu.Ts. Oganessian et al., Nature 400(1999), 242 ; Yu.Ts. Oganessian et al., Phys. Rev. Lett. 83(1999), 3154 ; Yu.Ts. Oganessian et al., Phys. Rev. C 63(2001), 011301(R). 2) S. Hofmann and G. Munzenberg, Rev. Mod. Phys. 72(2000), 733; S. Hofmann et al., Eur. Phys. J. A 14(2002), 147. 3) K. Morita, Proceedings of the VIII International Conference on Nucleus-Nucleus Collisions (NN2003), Moscow, Russia, 2003, to be published. 4) V. I. Zagrebaev, Y. Aritomo, M. G. Itkis, Yu. Ts. Oganessian, and M. Ohta, Phys. Rev. C65(2002), 014607. 5) M.G. Itkis et al., Proceeding of Fusion Dynamics at the Extremes (World Scientific, Sin-

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Fig. 6. Pre-scission neutron multiplicity in the reaction (a) 48 Ca+208 Pb and (b) 48 Ca+244 Pu at E ∗ =40 MeV. Experiments are carried out by T. Materna. The absolute value of calculations are normalized. gapore, 2001) p93. 6) L. Donadiile et al, Nucl Phys, A 656 (1999), 259. 7) W.J. Swiatecki, Phys. Scripta 24(1981), 113; W.J. Swiatecki, Nucl. Phys. A376(1982), 275. 8) S. Bjørnholm and W. J. Swiatecki, Nucl. Phys. A 391(1982), 471. 9) J.P. Blocki, H. Feldmeier and W.J. Swiatecki, Nucl. Phys.A 459(1986), 145. 10) C.E. Aguiar et al., Nucl. Phys. A 491(1989), 2031; C.E. Aguiar et al. Nucl. Phys. A 514(1990), 205. 11) Y. Aritomo et al, Phys. Rev. C 59 (1999), 796; 12) T. Tokuda, T. Wada and M. Ohta, Prog. Theor. Phys. 101 (1999), 607. 13) C. Schmitt et al, Acta Phy. Polon. B 32(2001), 841 14) C. Shen, G. Kosenko and Y. Abe, Phys. Rev. C66 (2002), 061602. 15) V.I. Zagrebaev, Phys. Rev. C 64(2001) ,034606. 16) J. Maruhn and W. Greiner, Z. Phys. 251(1972), 431. 17) K. Sato, A. Iwamoto, K. Harada, S. Yamaji, and S. Yoshida, Z. Phys. A 288(1978), 383. 18) R. Vandenbosch and J.R. Huizenger, Nuclear Fission (Academic Press, New York, 1973), p233. 19) M. Ohta, Proceeding of Fusion Dynamics at the Extremes, Dubna, May 25 -27, 2000 (World Scientific, Singapore, 2001) p110. 20) S. Suekane, A. Iwamoto, S. Yamaji and K. Harada, JAERI-memo, 5918 (1974). 21) A. Iwamoto, S. Yamaji, S. Suekane and K. Harada, Prog. Theor. Phys. 55(1976), 115. 22) Y. Aritomo, Proceedings of Int. Conf. on Nuclear Physics at Border Lines, Lipari, Italy, 2001 (World Scientific, Singapore, 2002) p38. 23) Y. Aritomo, Proceedings of International Conference on Exotic Nuclei, Baikal, Russia, 2001 (World Scientific, Singapore, 2002) 24) N.D. Mavlitov, P. Frobrich and SI.I. Gonchar, Z. Phys. A 342(1992), 195. 25) Y. Aritomo et. al, Proceedings of Tours Symposium on Nuclear Physics IV, Tours, France, 2003, to be published. 26) T. Materna et. al, Proceedings of the VIII International Conference on Nucleus-Nucleus Collisions (NN2003), Moscow, Russia, 2003, to be published. 27) F.A. Ivanyuk, H. Hofmann, V.V. Pashkevich, S. Yamaji, Phys. Rev. C 55(1997), 1730.