FAVORABLE HOT FUSION REACTION FOR SYNTHESIS OF NEW SUPERHEAVY NUCLIDE 272 Ds

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1 9 FAVORABLE HOT FUSION REACTION FOR SYNTHESIS OF NEW SUPERHEAVY NUCLIDE 272 Ds LIU ZU-HUA 1 and BAO JING-DONG 2,3 1 China Institute of Atomic Energy, Beijing , People s Republic of China 2 Department of Physics, Beijing Normal University, Beijing , People s Republic of China 3 Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou , People s Republic of China With the uranium and plutonium isotopes as target materials, we have calculated the formation cross sections of a new superveavy nuclide 272 Ds via the hot fusion reactions 32 S Pu, 34 S Pu, 38 Ar U and 40 Ar U. Among these reactions, 242 Pu( 34 S,4n) 272 Ds is the most favorable one with maximum evaporation residue (ER) cross section of 9 pb. Although mass asymmetry of the projectile-target combination 32 S Pu is somehow larger than that of system 34 S Pu, the maximum ER cross section of the former one is two orders of magnitude smaller than that of the later case. By means of a detail analysis, it is found that the different Q-values of these two reactions bring about this dramatic difference in the formation cross sections. The maximum cross sections of 235 U( 40 Ar,3n) 272 Ds and 238 U( 38 Ar,4n) 272 Ds fusion-evaporation reactions are about 7 and 2 pb respectively. The larger cross section of 235 U( 40 Ar,3n) 272 Ds is mainly due to the larger survival probability of the excited compound nucleus 275 Ds in the 3n evaporation channel. 1. Introduction Since the so-called island of stability was predicted theoretically more than 40 years ago, important progress has been made experimentally in the synthesis of superheavy elements (SHE) by means of cold and hot fusion reactions. In the cold fusion reactions with 208 Pb and 209 Bi targets, the elements with atomic numbers Z = are synthesized. 1,2 Fusion reactions of 48 Ca with actinide targets lead to the formation of compound nuclei with rather higher excitation energy than in cold fusion reactions, hence often refer to those reactions as hot fusion in literature. Elements with Z = are produced in hot fusion reactions of 238 U, 242,244 Pu, 243 Am, 245,248 Cm, 249 Bk, and 249 Cf targets with 48 Ca beams. 3 8 So far,

2 10 these experimental results clearly demonstrate the existence of the island of stability beyond the Z = 82 shell closure. Methods of synthesis SHE with cold and hot fusion reactions have both advantages and disadvantages respectively. In the cold fusion reactions, only one neutron is evaporated in the exit channel, therefore, the survival probability of the compound nucleus is much larger than those of hot fusion reactions. It is well established that the fusion probability (or fusion hindrance factor) is strongly dependent on the mass asymmetry or the product of atomic numbers, Z 1 Z 2 of the reaction partners. The large value of Z 1 Z 2 results in the severe hindrance in the cold fusion reactions, which is the main origin responsible for an exponential decrease in the formation cross section with an increasing atomic number of the nucleus produced in cold fusion reactions (six orders of magnitude from No to element 112). On the contrary, for the hot fusion reactions three or four neutrons are emitted from the hot compound nucleus. The competition of fission with neutron evaporation in each step of decay processes greatly reduces the survival probability of superheavy nuclei. On the other hand, because of the large mass asymmetry of the reaction partners, the fusion probability in the hot fusion reactions is relatively larger as compared with that of the cold fusion reactions. The formation cross section of superheavy nucleus mainly depends on these two factors, i.e., the fusion probability ( hindrance factor) in the entrance channel and the competition of fission with neutron evaporation in the exit channel. It is shown theoretically 9 that for SHE with Z<110, the formation cross section of cold fusion reactions is larger than the one of hot fusion reactions, while for SHE with Z 112, the situation is quite the opposite. Therefore Z=110 or 111 is an ideal nuclear region to investigate the role of these two factors in the process of the SHE synthesis. Superheavy element 110 (Ds) is synthesized by means of the cold fusion reaction 208 Pb( 64 Ni,1n) 271 Ds. 1,10 The corresponding 1n evaporation residue (ER) cross section amounts to 15 pb. Isotopes 273,277,280,281 Ds are observed as daughter nuclei of α decay from the corresponding nuclei of element So far, no nuclide with Z=110 has been synthesized directly via hot fusion reactions. Therefore, it is quite meaningful to synthesize new superheavy nuclide of element 110 with hot fusion reactions. In the following, we propose an experiment for synthesis of new nuclide 272 Ds with the favorable hot fusion reactions. The experiment may be performed using the Heavy Ion Research Facility (HIRFL) at the Institute of Modern Physics (Lanzhou, China) in the near future. We confine our discussion to the reactions with available target materials at the Institute of Modern

3 11 Physics. In this sense, the proposed reaction(s) should be favorable rather than optimal. 2. Model The cross section of a superheavy nucleus produced in a heavy ion fusionevaporation reaction is calculated as follows: σ ER (E) = πλ 2 l=0 (2l + 1)T l (E)P CN (E, l)p xn (E, l). (1) Here T l is the probability that the colliding nuclei penetrate the entrance Coulomb barrier and move up to the contact point. P CN defines the probability that the system will go from the configuration of two nuclei in contact into the configuration of the compound nucleus (CN), and finally, P xn represents the survival probability of the excited compound nucleus after evaporation of x neutrons in the cooling process Capture We calculate T l (E) by means of an approach proposed by Zagrebaev et al. 11,12 In their approach, the coupling between the relative motion of the nuclei and their dynamic deformations is taken into account in terms of a semi-phenomenological barrier distribution function method. The transmission probability is given by T l (E) = f(b){1 + exp ( 2π ω(l) [ ]) B + 2 l(l + 1) 2µRB 2 (l) E } 1 db. (2) Here ω denotes the width of the parabolic barrier, R B the corresponding barrier position. Zagrebaev et al 11,12 assume the barrier distribution to be an asymmetric Gaussian function, ( [ ] ) 2 B Bm f(b) = N exp, (3) where, N is the normalization constant of the distribution, and the average barrier height B m = (B 0 + B s )/2. B 0 is the height of the barrier at zero dynamic deformation, whereas B s is the height of the saddle point, i.e., the lowest barrier due to the dynamic deformations of the colliding nuclei. The widths are taken to be = (B 0 B s )/2 for B B m, and = 1 for B < B m. We set 1 = 2 MeV in all the cases considered below.

4 capt. (mb) S Pu 34 S Pu 32 S Pu 34 S Pu E c.m. (MeV) Fig. 1. The calculated capture cross sections as a function of the center-of-mass energy for the reaction systems 32 S Pu and 34 S Pu. The arrows in the figure illustrate the energy position corresponding to the maximum of ER excitation function. The parameters of the barrier distribution are extracted from the following nucleus-nucleus potential for nuclei with quadrupole deformations V 1,2 (r, β 1, β 2, Θ 1, Θ 2 ) = V C (r, β 1, β 2, Θ 1, Θ 2 ) + V prox (r, β 1, β 2, Θ 1, Θ 2 ) C 1(β 1 β 0 1) C 2(β 2 β 0 2) 2. (4) Here 1 and 2 denote the projectile and the target, β 1,2 are the parameters of the dynamic quadrupole deformation, β1,2 0 are the parameters of the static deformation, Θ 1,2 stand for the orientations of the symmetry axes of the statically deformed nuclei, V prox is the nucleus-nucleus proximity potential, 13 and C 1,2 the stiffness parameters. Figs. 1 and 2 display the capture cross sections as a function of the center-of-mass energy for the reaction systems 32 S Pu, 34 S Pu, 38 Ar U and 40 Ar U. The arrows in the figures illustrate the energy position corresponding to the maximum of ER excitation function. They indicate that the reactions under consideration are near-barrier fusion. For example, the Coulomb barrier heights of the systems 32 S Pu and 34 S Pu are and MeV respectively. The energies of the maximum ER cross section are just at the top of the corresponding barrier Diffusion over the inner barrier For heavy systems, the saddle-point shape shrinks below the length of the contact configuration. After contact a rapid growth of the neck brings the

5 capt. (mb) Ar U 40 Ar U 40 Ar U 38 Ar U E c.m. (MeV) Fig. 2. Same as Fig. 1, but for the 38 Ar U and 40 Ar U reactions system to the injection point in asymmetric fission valley located outside the saddle-point barrier. Hence automatic fusion will no longer take place. As an example, Fig. 3 shows the macroscopic deformation energy along the asymmetric fission valley for the reaction system 32 S Pu as a function of the distance between the surfaces of approaching nuclei, s. We assume that s = 0 is the injection point. The deformation energy as a function of s from the injection point to the saddle-point (the position of the maximum macroscopic deformation energy) defines the conditional saddle-point barrier, or the inner barrier in the fusion process. Starting from the injection point, the system diffuses uphill, and with some probability reaches the compound nucleus configuration due to the thermal shape fluctuations. The equation describing this process is the Smoluchowski partial differential equation which fusion-by-diffusion model is based on. It is assumed in the model that the probability managed to overcome the barrier is given 14,15 by P CN = 1 2 erfc ( B/T ), (5) where B is the height of the inner barrier averaged over the N/Z equilibration distribution, 16,17 and T is an effective temperature and erfc is the error function complement, equal to (1-erf). Figs. 4 and 5 show the calculated fusion probability as a function of the excitation energy of the compound nucleus for the reactions 32 S Pu, 34 S Pu, 38 Ar U and 40 Ar U. Two lines in Fig. 4 are overlapped, which reflects the fact that

6 B sp (MeV) S Pu s (fm) Fig. 3. The macroscopic deformation energy along the asymmetric fission valley for the reaction system 32 S Pu as a function of the distance between the surfaces of approaching nuclei, s S Pu 34 S Pu P CN E 0 * (MeV) Fig. 4. The fusion probability as a function of excitation energy for the reaction systems 32 S Pu and 34 S Pu. the neutron-proton equilibration at contact point of the reaction partners is taken into account in our model Survival from fission competition In the approximation of the Fermi-gas model, the ratio of the partial widths of neutron emission (Γ n ) and fission (Γ f ) for the nucleus after the emission

7 Ar U P CN Ar U E 0 * (MeV) Fig. 5. Same as Fig. 4, but for the 38 Ar U and 40 Ar U reactions. of k neutrons can be expressed as, 20,21 Γ n Γ f (E k, l k ) = 4A 2/3 a f U max n (k) K 0 a n [2 a f U max f (k) 1] [ exp 2 ] a n Un max (k) 2 a f Uf max (k), (6) where A is the mass number of nucleus considered, K 0 = 2 /[2m n r0] 2 9.8MeV with m n and r 0 the neutron mass and nuclear radius parameter, a n and a f are the level density parameters of the daughter nucleus and the fissioning nucleus at the ground and saddle configurations, respectively. The Un max (k) denotes the upper limit of the thermal excitation energy of the (k + 1)th daughter nucleus, Un max (k) = Ek S n (k + 1) p (k + 1) Erot(k gs + 1). (7) Here Ek = E 0 k i=1 (S n(i) + e i ) is the excitation energy of the residual nucleus after the emission of k neutrons, E0 = E cm Q is the excitation energy of compound nucleus, S n (i) and e i are the separation and kinetic energies of the ith evaporated neutron, p (k + 1) and Erot(k gs + 1) are the pairing and the rotational energies of the (k + 1)th daughter nucleus at ground state. Correspondingly, Uf max (k) stands for the upper limit of the thermal excitation energy of the nucleus at the saddle-point after the emission of k neutrons, Uf max (k) = Ek (B LD (k) gs sh (k)) p(k) Erot(k). sd (8)

8 16 Here gs sh, p(k), and Erot(k) sd denote, respectively, microscopic shell correction of the ground state, and the pairing energy as well as the rotational energy at the saddle-point of the kth daughter nucleus. The quantity (B LD (k) gs sh (k)) represents the nuclear mass difference between the saddle point and ground state configurations at zero temperature, and contains the macroscopic liquid drop energy B LD and microscopic shell correction gs sh. We set B LD = 0 for the systems under consideration. The values of gs sh are taken from Ref.22 The shape-dependent level density parameter is given by the following expression: 15 ã = 0.076A A 2/3 F (α) A 1/3 G(α)MeV 1, (9) here the deformation of the nucleus is defined by the parameter α = (R max R)/R, where R max is the semi-major axis of the nucleus with its radius R before deformation. The functions F and G are approximated 15 by, F (α) = 1 + (0.6416α α 2 ) 2, (10) G(α) = 1 + (0.6542α α 2 ) 2. (11) The smooth value of the level density parameter ã is modified due to shell effects according to the formula 21,23 [ ] a = ã 1 + sh Un,f max (1 exp( U max n,f /E D ), (12) where E D is the damping parameter describing the decrease of the influence of the shell effects on the energy level density with increasing excitation energy Un,f max of the nucleus. The survival probability of the excited nucleus is very sensitive to the parameter E D. Unfortunately, the value of E D has not been uniquely determined yet. In this work, E D = 18.5MeV is used. 21,24 Siwek-Wilczyńska and Skwira 21 presented a systematics of the shell corrections at saddle-point sd sh deduced from experimental fission barriers for a wide range of nuclei of 88 Z 100. Their systematics show that the values of sd sh are close to zero for the nuclei with Z 88. Therefore, in the range of superheay nuclei, the level density parameters a f can be safely assumed to be independent on the excitation energy of the nucleus considered. On the other hand, the level density parameter a n increases as the excitation energy increasing due to the damping of the shell correction energy, gs sh of the ground state. Therefore, according to the formalism presented in Refs., 14,15,21 the damping of the shell effects directly influences the decay width of neutron emission, Γ n rather than the fission width Γ f.

9 P xn (E*, =0) n 4n E 0 * (MeV) Fig. 6. Survival probability in 3n and 4n channels of 276 Ds compound nuclei produced in the 32,34 S + 244,242 Pu, 38 Ar U hot fusion reactions with the entrance channel orbital angular momentum l = 0. Zagrebaev et al 11 presented an expression for the formation of a cold residual nucleus after the emission of k neutrons, in which the Maxwell- Boltzmann energy distribution of evaporated neutrons are directly taken into account. We used a similar expression of the survival probability, P kn (E 0, l) = [ ] Γn Γ t (E0, l) E 0 Sn(1) 0 E 1 Sn(2) 0 Γ n Γt (E 1, l 1 )P n (E 0, e 1 )de 1 Γ n Γt (E 2, l 2 )P n (E 1, e 2 )de 2... (13) E k 1 Sn(k) K th P n (E k 1, e k)de k. Here Γ t Γ n + Γ f is the total decay width, in which Γ n and Γ f are calculated with Eq. (6). We have assumed the neutron carries away an angular momentum of 1 in average. P n (E, e) = C e exp[ e/t (E )] is the probability for the evaporated neutron to have energy e, C is the corresponding normalization coefficient. P kn (E 0, l) represents the probability for the compound nucleus to emit just k neutrons rather than fission. After the kth neutron emission, the residual nucleus must neither fission nor emit another neutron. Therefore, the kth neutron must have carried off sufficient energy to bring the system below the thresholds for fission and further neu-

10 Q= -115 MeV 34 S Pu ER (pb) n 32 S Pu 10-3 Q=-106 MeV E 0 * (MeV) Fig. 7. Predicted evaporation residue cross sections for the 4n evaporation channel in the 32 S Pu and 34 S Pu reactions leading to formation of 272 Ds. tron emission. The corresponding lowest kinetic energy of kth neutron is denoted by K th 15 in eq. (13). As an example, Fig. 6 shows the calculated survival probability P xn (E, l = 0) of 276 Ds compound nuclei produced in the 32,34 S + 244,242 Pu, 38 Ar U hot fusion reactions. 3. Results and discussions Displayed in Figs. 7 and 8 are the predictions of evaporation residue (ER) cross sections for the 4n evaporation channel in the 32 S Pu, 34 S Pu and 38 Ar U reactions leading to formation of 272 Ds. Shown in Fig. 9 is the ER cross sections in the 3n evaporation channel of the 235 U( 40 Ar,3n) 272 Ds reaction. Q-values of the relevant reactions are also shown in the figures. It may be seen from Fig. 7 that the maximum ER cross section of the 34 S Pu is two orders of magnitude larger than that of the reaction 32 S Pu. However, as shown in Fig. 1 and Fig. 4 the capture cross sections at the maximum of ER excitation function as well as the fusion probabilities as function of the excitation energy are very similar for these two systems. In addition, the exit channels are the same for both reactions because they all lead to the same nucleus with the 4n evaporation channel. The only difference is the reaction Q-values. They are about 9 MeV in difference. Therefore, the reaction Q-value should be responsible for the large difference in the formation cross sections. As formulated in Eq. (1), the ER cross section is a production of three factors: the transmission coefficient T l, the CN formation probability P CN,

11 U( 38 Ar,4n) 272 Ds ER (pb) Q = -127MeV E 0 * (MeV) Fig. 8. Same as Fig. 8, but for the 38 Ar U reaction U( 40 Ar,3n) 272 Ds ER (pb) Q = -133MeV E 0 * (MeV) Fig. 9. The ER cross sections in the 3n evaporation channel of the 235 U( 40 Ar,3n) 272 Ds reaction. and the survival probability P xn of the compound nucleus. Among these factors, P xn itself contains in principle two constituents, i.e., Γ n /Γ t and G xn, where G xn is the probability of realization of x neutrons evaporation. 20 All three factors, T l, P CN and Γ n /Γ t increase with energy, while G xn decreases exponentially beyond certain threshold energy Eth of which the channel for fission after x neutron emission opens. In order to probe the origin responsible for the dramatic difference in the formation cross sections of the two systems of 32 S Pu and 34 S Pu, we plot the reduced ER cross section σ ER = σ ER /πλ 2, the average values T l P CN Γ n /Γ t and

12 Probability S Pu W surv reduced ER <T L P CN n / t > 32 S Pu E 0 * (MeV) Fig. 10. The reduced ER cross section σ ER, average values T l P CN Γ n /Γ t and survival probability P sur as a function of the CN excitation energy for the reactions 32 S Pu and 34 S Pu. the survival probability P xn as a function of excitation energy, respectively, as solid, dash-dotted and dashed lines in Fig. 10. The maximum of the survival probability as a function of the excitation energy in the 4n channel, P 4n (E) is at E0 = 32.9 MeV. The relevant energies in the center-of mass system are 139 and 148 MeV for the reactions 32 S Pu and 34 S Pu, which are 20 and 10 MeV below the barrier respectively. In the deep barrier energy region, the value of the transmission coefficient T l increases very rapidly, and the decrease of G xn is balanced by this rapid increase so as to push the peak position of the ER cross sections to higher excitation energy. Because of the exponentially decrease of P xn, any slightly increase of excitation energy above threshold energy Eth, i.e., the the energy position of the maximum value of the P xn, will dramatically reduce the ER cross section. Fig. 10 shows that the smaller the absolute Q-value the further the peak position of the ER cross section is pushed away from the threshold energy Eth. This causes the ER cross sections of the reaction 32 S Pu severely reduced. As for the reactions of 235 U( 40 Ar,3n) 272 Ds and 238 U( 38 Ar,4n) 272 Ds, both factors in the entrance and exit channels take important roles in the formation cross sections. In the entrance channel, except the effect of different Q-values, the fusion probability P CN (E) of the reaction 235 U( 40 Ar,3n) 272 Ds is several times smaller than that of 238 U( 38 Ar,4n) 272 Ds. On the other hand, the survival probability P 3n is about one order of magnitude lager than P 4n. These effects bring about the

13 21 ER cross sections of the reaction 235 U( 40 Ar,3n) 272 Ds are almost four times larger than those of 238 U( 38 Ar,4n) 272 Ds. 4. Summary In order to evaluate the possibility for synthesis SHE 110 at the Institute of Modern Physics (Lanzhou, China) in the near future, we have calculated the formation cross section of 272 Ds by means of the fusion-by-diffusion model Based on the available target materials there, the reaction systems 32 S Pu, 34 S Pu, 38 Ar U and 40 Ar U are selected. The results show that the fusion evaporation reaction 242 Pu( 34 S,4n) 272 Ds is the most favorable one among the reactions under consideration. It s maximum ER cross section amounts to 9 pb. However, for the similar reaction 32 S Pu, the maximum ER cross section decreases down to 0.06 pb. To search for the origin responsible for this dramatic decrease, we have made a detail analysis. It is found that the different Q-values bring about this dramatic difference in the formation cross sections between these two systems. The maximum ER cross sections of 235 U( 40 Ar,3n) 272 Ds and 238 U( 38 Ar,4n) 272 Ds are about 7 and 2 pb respectively. The larger maximum ER cross section of 235 U( 40 Ar,3n) 272 Ds is mainly due to the larger survival probability of the excited compound nucleus 275 Ds in the 3n evaporation channel. The SHE 110 has been synthesized by means of the cold fusion reaction 208 Pb( 64 Ni,1n) 271 Ds. 1,10 As discussed in the introduction, it should be quite meaningful if the new nuclide 272 Ds could be synthesized via a hot fusion reaction. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant No , References 1. S. Hofmann, G. Munzenberg, Rev. Mod. Phys (2000) ; S. Hofmann et al., Eur. Phys. J. A 10 5 (2001). 2. K. Morita et al., J. Phys. Soc. Jpn. 76 No (2007); 76 No (2007). 3. Yu. Ts Oganessian et al., Phys. Rev. Lett (1999); Yu. Ts. Oganessian et al., Nature (1999). 4. Yu. Ts. Oganessian et al., Phys. Rev. C 69, (2004). 5. Yu. Ts. Oganessian et al., Phys. Rev. C (2004).

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