Fusion–Fission Dynamics and Perspectives of Future Experiments*

Home , Atomic nucleus, Fission barrier

Physics of Atomic Nuclei, Vol. 66, No. 6, 2003, pp. 1033–1041. From Yadernaya Fizika, Vol. 66, No. 6, 2003, pp. 1069–1077. Original English Text Copyright c 2003 by Zagrebaev, Itkis, Oganessian.

V. I. Zagrebaev **, M.G.Itkis,andYu.Ts.Oganessian Flerov Laboratory of Nuclear Reactions, Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia Received September 2, 2002

Abstract—The paper is focused on reaction dynamics of superheavy-nucleus formation and decay at beam energies near the Coulomb barrier. The aim is to review the things we have learned from recent experiments on fusion–fission reactions leading to the formation of compound nuclei with Z ≥ 102 and from their extensive theoretical analysis. Major attention is paid to the dynamics of formation of very heavy compound nuclei taking place in strong competition with the process of fast fission (quasifission). The choice of collective degrees of freedom playing a fundamental role and finding the multidimensional driving potential and the corresponding dynamic equation regulating the whole process are discussed. A possibility of deriving the fission barriers of superheavy nuclei directly from performed experiments is of particular interest here. In conclusion, the results of a detailed theoretical analysis of available experimental data on the “cold” and “hot” fusion–fission reactions are presented. Perspectives of future experiments are discussed along with additional theoretical studies in this field needed for deeper understanding of the fusion–fission processes of very heavy nuclear systems. c 2003 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION mononucleus. For light or very asymmetric nuclear systems, this evolution occurs with a probability close The interest in the synthesis of superheavy nuclei to unity. Two touching heavy nuclei after dynamic has grown lately due to new experimental results [1– deformation and exchange by several nucleons may 3] demonstrating a real possibility of producing and reseparate into PLF and TLF or may go directly to investigating the nuclei in the region of the so-called fission channels without formation of a compound “island of stability.” The new reality demands more nucleus. The later process is usually called quasifis- substantial theoretical support of these expensive ex- sion. Denote a probability for two touching nuclei to periments, which will allow a more reasonable choice form the compound nucleus (CN) as PCN(l, E).At of fusing nuclei and collision energies as well as a bet- the third reaction stage, the CN emits neutrons and γ ter estimation of the cross sections and unambiguous rays, lowering its excitation energy and finally forming identification of evaporation residues (ERs). the residual nucleus in its ground state. This process takes place in strong competition with fission (normal A whole process of superheavy-nucleus formation fission), and the corresponding survival probability can be divided into three reaction stages. At the first ∗ P (l, E ) is usually much less than unity even for a stage, colliding nuclei overcome the Coulomb bar- xn weakly excited superheavy nucleus. rier and approach the point of contact R = R + cont 1 Thus, the production cross section of a cold resid- R2. Quasielastic and deep-inelastic reaction channels dominate at this stage, leading to formation of ual nucleus B,whichistheproductofneutronevapo- projectile-like and target-like fragments (PLF and ration and γ emission from an excited compound nu- TLF) in the exit channel. At subbarier energies, only cleus C, formed in the fusion process of two heavy nu- A + A → C → B + xn + Nγ E a small part of incoming flux with low partial waves clei 1 2 at c.m. energy reaches the point of contact. Denote the correspond- close to the Coulomb barrier in the entrance channel, can be decomposed over partial waves and written as P (l, E) ing probability as cont . Experiments on deep- ∞ inelastic collisions and our knowledge about nuclear π2 σxn (E) ≈ (2l +1)P (E,l) (1.1) friction forces allow us to conclude that, at the contact ER 2µE cont point, nuclei have almost zero kinetic energy. At the l=0 ∗ second reaction stage, touching nuclei evolve into × PCN(A1 + A2 → C; E,l)Pxn(C → B; E ,l). the configuration of an almost spherical compound Different theoretical approaches are used for an- ∗This article was submitted by the authors in English. alyzing all three reaction stages. However, the dy- **e-mail: [email protected] namics of the intermediate stage of the CN forma-

1063-7788/03/6606-1033$24.00 c 2003 MAIK “Nauka/Interperiodica” 1034 ZAGREBAEV et al.

(a) Capture cross section, mb Oblate Prolate 16O + 208Pb 48Ca + 208Pb 48Ca + 238U 3 Spherical 10 B1 B2 Saddle 180 B1 B2 B1 B2 140 101 100 1 β 10 –1 15 20 25 0 10 r, fm (b) Deformation

10–3 ~ ~ ~ ~ 200 65 70 165 170 185 190 195 200 Potential energy, MeV Ec.m, MeV 180 16 208 160 Fig. 2. Capture cross sections in the O+ Pb [9], 48Ca + 208Pb [10], and 48Ca + 238U[3]fusionreac- 140 tions. Dashed curves represent one-dimensional barrier penetration calculations with the Bass barriers. Solid 10 90 14 18 –90 0 curves show the eﬀect of dynamic deformation of nuclear surfaces (ﬁrst two reactions) and orientation of statically r, fm Orientation, deg deformed nuclei (48Ca + 238U case). The arrows marked

48 208 by B1 and B2 show the positions of the corresponding Fig. 1. (a) Potential energy of Ca + Pb depending Coulomb barriers (see the text). on distance and quadrupole dynamic deformations of both nuclei. (b) Potential energy of 48Ca + 238Udepending 238 g.s on orientation of statically deformed U nucleus (β2 = 0.215). the height of the potential barrier. Coupling with the excitation of nuclear collective states (surface vibrations and/or rotation of deformed nuclei) and tion is the most vague. It is due to the fact that, in the fusion of light and medium nuclei, in which with nucleon transfer channels significantly influ- the fissility of the CN is not very high, the colliding ences the capture cross section at near-barrier en- nuclei having overcome the Coulomb barrier form ergies. In Fig. 1, the potential energy is shown depending on dynamic deformation of spherical nuclei a CN with a probability PCN ≈ 1. Thus, this reaction stage does not influence the yield of ER at all. 48Ca + 208Pb and on mutual orientation of deformed 48 238 g.s However, in the fusion of heavy nuclei, it is the fission nuclei Ca + U(β2 =0.215). The incoming channels (normal and quasifission) that substantially flux has to overcome, in fact, a multidimensional determine the dynamics of the whole process; the ridge with its height depending on orientation and/or PCN value can be much smaller than unity, while its dynamic deformation. This means that we have to talk accurate calculation is very difficult. Setting Pxn =1 not about one barrier B but rather about a “barrier in (1.1), we get the cross section of CN formation distribution.” σCN, which can be measured by detection of ERs and fission fragments forming in normal fission (if In [5, 6], a semiempirical approach was proposed they are distinguished from quasifission fragments for calculating the penetration probability of such and from products of deep-inelastic collision). Setting multidimensional potential barriers. Calculating the P =1 in addition CN in (1.1), we get the capture cross barriers B1 and B2 for two limit configurations [in the section σcap, which can be measured by detection of case of statically deformed nuclei, they correspond to all fission fragments (if they are distinguished from the tip and side orientations, otherwise they corre- products of deep inelastic collision). It is clear that, spond to the so-called saddle dynamic deformation for symmetric fusion reactions, σCN and σcap cannot (see Fig. 1a)andsphericalconfiguration], we may be measured experimentally. approximate the barrier distribution function [7] by an asymmetric Gaussian centered at B0 =(B1 + B2)/2. 2. CAPTURE CROSS SECTION Approximating the radial dependence of the barrier The Bass approximation of the potential energy by a parabola and using the Hill–Wheeler formula [8] of the interaction between two heavy spherical nu- for the penetration probability of the one-dimensional clei [4] is widely used and reproduces rather well potential barrier, we may estimate the quantum pen-

PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 6 2003 FUSION–FISSION DYNAMICS 1035 etrability of the multidimensional barrier as follows: V (r = R ), MeV fus–fis cont 2π 210 (a) P (E,l)= f(B) 1+exp (2.1) Spherical cont ω(l) 200 190 −1 2 fus × − 180 Qgg B + 2 l(l +1) E dB. Deformed 2µRB(l, B) 170

Here, ωB is defined by the width of the parabolic 160 R barrier, B defines the position of the barrier, and the Z (b) 1 barrier distribution function satisfies the normaliza- Sn Ge Sn Sn Sn Dy tion condition f(B)dB =1. 100 Xe Kr Xe Xe Xe Sm The capture cross sections calculated within this approach are shown in Fig. 2 for the three reac- 80 tions (solid curves). They are compared with the- Quasi-fission oretical calculations made within a model of one- CN formation dimensional barrier penetrability for spherical nuclei 60 Regular fission (dashed curves). In all three cases, a substantial increase in the barrier penetrability is observed in the 208Pb + 88Se 40 subbarrier energy region. However, the character of Compound nucleus area this increase significantly changes: the shift of the 248Cm + 48Ca barrier and the distribution width, in particular, grow 20 with the increase in the masses of fusing nuclei. An 296116 additional decrease in the 48Ca + 238Ucapturecross section at above-barrier energies as compared with 20 40 60 80 100 Z2 its geometrical limit is explained by a much shallower potential pocket and, thus, by a much smaller value of Fig. 3. Driving potential Vfus–fis(Z1, Z2) of the nuclear the critical angular momentum. system consisting of 116 protons and 180 neutrons. (a) Good agreement between the calculated and ex- Potential energy of two touching nuclei at A1 + A2 = ACN, ∆A =0, i.e., along the diagonal of the lower figure. perimental capture cross sections allows us to be- The thick line corresponds to the case of spherical nuclei, lieve that we may get a rather reliable estimation whereas the thin line corresponds to δ1 + δ2 =0.3.(b) of the capture cross section for a given projectile– Topographical landscape of the driving potential on the target combination if there are no experimental data plane (Z1, Z2) (zero deformations). The dark regions or if these data cannot be obtained at all (symmet- correspond to the lower potential energies (more compact ric combinations). However, we should realize that configurations). some uncertainty nevertheless remains in choosing the parameters defining the multidimensional potential barrier and the capture cross section [6]. The role consensus for the answers and for the mechanism of the neutron exchange is also not clear yet. Thus, in of the compound nucleus formation itself, and quite the cases of fusion of very heavy nuclei and especially different, sometimes opposite in their physics sense, for symmetric fusion reactions, the accuracy of our models are used for its description. current predictions of the capture cross sections in In [5, 11], a new approach was proposed for de- the subbarrier energy region is about one order of magnitude. At above-barrier energies, this accuracy scription of fusion–fission dynamics based on a sim- is much better. plified semiempirical version of the two-center shell model idea [12]. It is assumed that, on a path from the initial configuration of two touching nuclei to 3. FUSION–FISSION DYNAMICS the CN configuration and on a reverse path to the The processes of the CN formation and quasifission channels, the nuclear system consists of two fission are the least studied stages of the heavy-ion cores (Z1,N1)and(Z2,N2) surrounded by a cer- fusion reaction. To solve this problem, we have to tain number of common (shared) nucleons, ∆A = − − answer very fundamental questions. What are the ACN A1 A2, moving in the whole volume oc- main degrees of freedom playing most important role cupied by the two cores. The processes of CN for- at this reaction stage? What is the corresponding mation, fission, and quasifission take place in the driving potential and what is an appropriate equa- space (Z1,N1,δ1; Z2,N2,δ2), where δ1 and δ2 are the tion of motion for description of time evolution of dynamic deformations of the cores. The compound the nuclear system at this stage? Today, there is no nucleus is finally formed when two fragments A1 and

PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 6 2003 1036 ZAGREBAEV et al.

(a) (a) 300 ) 2 0.8 Cm + Ca A 116 296 + 1 250 A Pb + Se

)/( 0.4 QF1 2 1 1 A Dy + Sn 200 – CN 48 248 1 QF2 Ca Cm A 0 Fission 150 Compound nucleus area

–0.4 Total kinetic energy, MeV 100 50 100 150 200 250 Fragment mass number –0.8 Mass asymmetry ( 5 (b) 8 10 12 14 238U 296116 R12, fm 4 (b) Coulomb barrier Contact point 3 Saddle point

CN Yield, % 2

200 1 180 0 160 0.8 80 100 120 140 160 180

Potential energy, MeV Fragment mass number 8 0.4 Elongation, fm 10 0 Fig. 5. (a)Two-dimensionalTKE–mass plot. The asymmetric quasifission process (QF1 path in Fig. 4) con- 12 –0.4 tributes mainly to the regions marked by 1.Normalfission 14 Mass asymmetry –0.8 and near-symmetric quasifission (QF2 path in Fig. 4) contribute to the region marked by dashed quadrangle. (b) Mass distribution of near-symmetric fission frag- Fig. 4. Driving potential Vfus–fis as a function of mass ments [dashed quadrangle on panel (a)] detected in the asymmetry and distance between centers of two nu- 48Ca + 248Cm reaction at excitation energy of E∗ = clei with the deformations δ + δ =0.3, topographical 1 2 33 MeV compared with the fission of 238Umeasuredat landscape (a) and three-dimensional plot (b). The black approximately the same excitation energy [13]. solid curve in (a) shows the contact configurations. The paths QF1 and QF2 lead to the asymmetric and near- symmetric quasifission channels, the dashed curve shows the most probable way to formation of the compound itisacontinuousfunctionatr = Rcont,anditgives nucleus, and the dotted curve corresponds to normal a realistic Coulomb barrier at r = RB >Rcont.At (regular) fission. See the conformity with Fig. 3. last, instead of using the variables (Z1, N1; Z2, N2), we may easily recalculate the driving potential as a function of mass asymmetry (A1 − A2)/(A1 + A2) A2 go into its volume, i.e., at R(A1)+R(A2)=RCN 1/3 1/3 and elongation R12 = r0(A + A ) (at r ≥ Rcont, or at A1/3 + A1/3 = A1/3. 1 2 1 2 CN R12 = r = s + R1 + R2,wheres is the distance be- The corresponding driving potential Vfus–fis(r, Z1, tween nuclear surfaces). These variables along with δ + δ N1, δ1; Z2, N2, δ2) was derived in [5] and is shown in deformation 1 2 are commonly used for description of the fission process. The corresponding driving Fig. 3 as a function of Z1, Z2 (minimized over N1, N2 potential is shown in Fig. 4. and at fixed values of δ1 + δ2). It was found that the microscopic two-center shell model calculations give As can be seen from Figs. 3 and 4, the shell very close values of potential energy, though slightly structure, clearly revealing itself in the contact of less structural. There are several advantages of the two nuclei (Fig. 3a), is also retained at ∆A =0 proposed approach. The driving potential is derived (R12

PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 6 2003 FUSION–FISSION DYNAMICS 1037

48 246 Probability of CN formation PCN Ca + Cf 0 (a) 102 10 48Ca + 208Pb Saddle point Z = 118

110 112 CN 10–2 266 114 58Fe + 208Pb → 108 116 118 48Ca + 232Th 64Ni + 208Pb → 272110 300 10–4 280 DIP 0.8 48Ca + 238U 260 48 244 Potential Ca + Pu energy, MeV 0.4 48 248 QF Ca + Cm 8 86Kr + 208Pb 48Ca + 249Cf Elongation, fm 10–6 10 0 0203010 E*, MeV Yield of fragments, rel. units 12 –0.4 100 (b) Mass asymmetry 48 248 → 14 Ca + Cm f1 + f2 –0.8

Fig. 7. Fusion–fission driving potential for the nucleus 294118 formed in the 86Kr + 208Pb and 48Ca + 246Cf fusion reactions. 50 understandable qualitatively in terms of multidimensional potential energy surface shown in Figs. 3 and 4. Using the driving potential Vfus–fis(Z1, N1, Z2, N2, δ1, δ2), we may determine the probability of CN 30 50 70 90 Z f formation PCN(A1 + A2 → C), being part of expres- sion (1.1) for the cross section of the synthesis of Fig. 6. (a) Probability of compound-nucleus formation superheavy nuclei. It can be done, for example, by for the “hot” (solid curves) and “cold” (dashed curves) fusion reactions. (b) Charge distribution of quasifission solving the master equation [14] for the distribution fragments in the 48Ca + 248Cm fusion reaction at E∗ = function F (Z1, N1, Z2, N2, δ1, δ2; t). The probability 40 MeV (linear scale, relative units). The main peaks cor- of CN formation is determined as an integral of the ≤ respond to the path QF1 in Fig. 4 (see region 1 in Fig. 5a), distribution function over the region R1 + R2 RCN. whereas the small near-symmetric peaks correspond to Similarly, one can define the probabilities of finding the path QF2. the system in different channels of quasifission, i.e., the charge and mass distribution of fission fragments measured experimentally. intermediate minima correspond to the shape isomer states. From our analysis, we may definitely conclude Results of such calculations performed with a re- stricted number of variables are shown in Fig. 6. For that these isomeric states are nothing else but two- 48 cluster configurations with magic or semimagic cores the “hot” fusion reactions, based on using Ca as a (see Fig. 3b). projectile, the probability of CN formation at first falls very sharply with increasing ZCN, but then it remains As regards the superheavy compound nucleus for- −3 at the level of 10 for ZCN = 114–118 at excitation 48 248 ∗ mation in the fusion reaction Ca + Cm, one can energies E ≥ 30 MeV. Such behavior of PCN re- see that, after the contact, the nuclear system may flects the fact of insignificant changes of Vfus–fis for all easily decay into the quasifission channels (mainly these reactions. In contrast with that, for the “cold” asymmetric: Se + Pb, Kr + Hg; also near-symmetric: fusion reactions, based on using 208Pb as a target, Sn + Dy, Te + Gd)—solid arrow lines in Figs. 3b and the probability of CN formation decreases very fast 4. Only a small part of the incoming flux reaches a CN with increasing ZCN (see dashed curves in Fig. 6a). configuration (dashed-arrow line). An experimental A qualitative explanation of that can be made again two-dimensional total kinetic energy (TKE) mass in terms of potential energy surface. In Fig. 7, the plot for the 48Ca + 248Cm fusion–fission reaction [3] driving potential is shown for the synthesis of nucleus is shown in Fig. 5. The experimental data are quite 294118 in the “cold” fusion reaction 86Kr + 208Pb. Due

PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 6 2003 1038 ZAGREBAEV et al.

Cross section, mb to reach the CN configuration. It is possible only at –3 (a) sufficiently high excitation energy. Disappearance of 10 2n 48 206 → 254 Ca + Pb 102No the locked Coulomb barrier makes the system unsta- –4 ble against reseparation into the deep-inelastic and 10 quasielastic channels, which have to dominate in this 1n 3n reaction at low energies. In contrast with that, in more –5 10 asymmetric case of 48Ca + 246Cf fusion reaction leading to the same CN, the potential energy at the 10–6 point of contact is above the ground state of the CN, 4n 1n and the nuclear system evolves down along the po- 2n –7 γ –1 tential energy surface. Of course, the main flux goes 10 3n D = 0.05 MeV 0.06 4n to the quasifission valley (Kr + Pb), but nevertheless a 0.08 small part of it reaches the CN configuration (dashed 10–8 10 15 20 25 30 35 40 45 line in Fig. 7). (b) Exploration of the multidimensional fusion–fis- 3 10 B1 B2 sion driving potential itself and of the corresponding evolution of a heavy-nuclear system along its surface is a very promising and fruitful experimental problem. For that purpose, one may perform, for exam- σ σ 100 capt fus ple, fusion–fission and ER measurements on formation of the same easily fissile 224U nucleus in differ- 20 204 48 238 → 286 ent projectile–target combinations: Ne + Pb Ca + U 112 (very asymmetric, behind the Businaro–Gallone bar- 10–3 rier), 64Ni + 160Gd (less asymmetric, in front of the Businaro–Gallone barrier), 88Sr + 136Xe and 100Mo + 124Sn (symmetric, inverse to fission pro- 86 138 –6 cess), Kr + Ba (closed shell nuclei, suppressed 10 76 148 Bf = 6.5 MeV deep-inelastic channels), and Ge + Nd (de- 5.5 4.5 formed, dependence on orientation). σER –9 3n 10 4. FISSION BARRIERS OF SUPERHEAVY NUCLEI 5 15253545 * ∗ E , MeV The survival probability Pxn(l, E ) of a cooling excited compound nucleus can be calculated within Fig. 8. (a) Cross sections of the evaporation residue 48 206 a statistical model [6, 15]. The most uncertain pa- production for different xn channels in the Ca+ Pb rameter here is the fission barrier. For nuclei with fusion reaction. The experimental data are from [17]. Z>100 The solid curves correspond to the calculations with the , which cannot be used as a target material, −1 experimental measurement of the fission barriers is damping factor γD =0.06 MeV , whereas the dashed and dotted curves for the 4n channel are calculated with not possible. Calculating the fission barrier for the −1 γD =0.05 and 0.08 MeV , respectively.(b)Thecapture atomic nucleus (mainly its microscopic component) cross section (all fission fragments, open circles), the is also a very complicated puzzle faced with the ne- total yield of near-symmetric fission fragments with A = cessity of solving a many-body quantum problem. A /2 ± 20 CN (solid circles), and the evaporation residue The exact solution to that problem is currently un- production cross section in the 3n channel of the 48Ca + 238U reaction. The arrows show the Coulomb barriers for obtainable, and the accuracy of the approximations two ultimate orientations of the deformed target nucleus. inuseisratherdifficult to estimate. As a result, the The cross section of evaporation residue formation was fission barriers for superheavy nuclei calculated with- calculated with a fission barrier of 4.5 (dotted curve), 5.5 in the different approaches differ greatly (by several (solid curve), and 6.5 MeV (dashed curve). megaelectronvolts). Any experimental information on the fission barriers of those nuclei seems to be highly valuable. to dynamic deformation of both nuclei, the potential An important property of the fission barrier is that energy at the point of contact is even lower than the it has a pronounced effect on the survival probabil- energy of the CN ground state. The nuclear system ity of an excited nucleus in its cooling by emitting has to evolve upward on the potential energy surface neutrons and γ rays in competition with fission. It

PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 6 2003 FUSION–FISSION DYNAMICS 1039 is this property that may be taken advantage of to The lower limits of the heights of fission barriers make an estimate of the fission barrier of a super- ∗ heavy nucleus if it is impossible to measure the fission σ , σ , σER, Nucleus E , capt fus Bf , barrier directly. Higher sensitivity may be obtained if MeV mb mb pb MeV such competition is tested several times during the 286 ≤ +6.3 ≥ evaporation cascade (“hot” fusion reactions). In this 112 31.5 40 5 5.0−3.2 (3n) 5.5 case, the cross section σxn (E), which is proportional ER 292 +0.8 x 114 36.5 30 ≤ 4 0.5− (4n) ≥ 6.7 roughly to (Γn/Γf ) , happens to be more sensitive 0.3 to the value of the fission barrier since it increases +0.8 296116 34.8 30 ≤ 2 0.5 (4n) ≥ 6.4 in importance by a factor of x. For the experimental −0.3 value of the survival probability of the superheavy nucleus to be deduced, it is necessary to measure the simpler, the same value B was used for these nuclei. cross section of weakly excited CN production in the f The typical sensitivity of the calculated production near-barrier fusion of heavy ions as well as the cross cross section for the ER to a change in the value of section for the yield of a heavy evaporation residue. It was experiments of this kind that were carried out the fission barrier is shown in Fig. 8. It is the fact at FLNR (JINR, Dubna) recently [1–3] as part of that this sensitivity is high which allows one to expect a series of experiments on the production of nuclei the value of the fission barrier to be deduced to an accuracy of the order of ±0.5 MeV with allowance with Z = 112, 114,and116 formed in the 3n and 4n evaporation channels. made for the experimental error in measuring this cross section and the uncertainty of some parame- The fission barriers are usually calculated accord- − ∗ ters used in the calculations [6]. Since, as discussed − γDE ing to the formula Bf (J =0)=BLD δWe , above in Section 3, the production probability for a where BLD is the liquid-drop fission barrier [16], true compound nucleus may really be less than the which is negligibly small for nuclei with Z>112; value of σexp/σexp , then comparing the measured and δW is the shell correction for the ground-state CN capt calculated cross sections for the evaporation residues energy; and γD is the damping parameter, which accounts for the fact that shell effects fall off as the allows one to deduce in fact the lower limits for the fis- excitation energy of the CN increases. The value of sion barriers of the corresponding nuclei. Final results this parameter is especially important in the case of are presented in the table. superheavy nuclei, whose fission barriers are mainly The analysis of the available experimental data on determined just by the shell corrections for their the fusion and fission of the nuclei of 286112, 292114, ground states. In the literature, one can find close but and 296116 produced in the reactions 48Ca + 238U, slightly different values for the damping parameter, 48Ca + 244Pu, and 48Ca + 248Cm [3], as well as and we paid special attention to the sensitivity of the experimental data on the survival probability of those calculated cross sections to this parameter. Figure 8a nuclei in evaporation channels of three- and four- shows how much the cross section for the 4n channel neutron emission [1, 2], enables the quite reliable is sensitive to a change in the damping parameter. conclusion that the fission barriers of those nuclei are Simultaneous analysis of a great number of “hot” really quite high, which results in their relatively high fusion reactions used for producing heavy elements stability. The lower limits that we have obtained for allows the conclusion that the value of this parameter 283–286 288–292 − the fission barriers of nuclei of 112, 114, 1 292–296 lies in the range γD =14–18 MeV. and 116 are 5.5, 6.7, and 6.4 MeV, respec- After calculating the value of Pcont(E,l) in such tively [18]. a way as for the measured capture cross section to be reproduced and parametrizing the CN production probability P in such a way as for σexp to 5. CROSS SECTIONS CN CN OF SUPERHEAVY-ELEMENT PRODUCTION be reproduced, fission barriers for the nuclei of the evaporation cascade can be chosen in such a way Calculating the capture cross sections and the as for the corresponding measured cross section of probability for CN formation as described above and the yield of a heavy evaporation residue nucleus to using the fission barriers based on the ground-state be reproduced with the help of (1.1). The calcu- shell corrections of Moller¨ et al. [19], we estimated lated results are shown in Fig. 8 for the case of the the cross sections of superheavy element formation in 48Ca + 238U → 286112 fusion reaction. Taking ac- the “hot” and “cold” fusion reactions leading to heavy count of the fact that fission barriers vary not so much nuclei with ZCN ≥ 102 (Fig. 9). The cross sections from nucleus to nucleus in an evaporation cascade, for formation of superheavy nuclei with Z = 114–118 as well as making the procedure for assessing them in the 3n and 4n evaporation channels of the “hot”

PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 6 2003 1040 ZAGREBAEV et al.

Cross section, pb its way from the ground state to the scission point 1012 passes through the optimal configurations with min- σ * capt(E ~ 35 MeV) imal potential energies (shape isomer states), which are nothing else but the two-cluster configurations σ * σ * capt(E = 15 MeV) CN(E ~ 35 MeV) with magic (closed shell) cores. Analysis of the ex- 109 perimental data on the fusion and fission of the nuclei of 286112, 292114,and296116, as well as experimen- σ * tal data on the survival probability of those nuclei 6 CN(E = 15 MeV) 10 48Ca + 208Pb in evaporation channels of three- and four-neutron emission, enables us to make the quite reliable conclusion that the fission barriers of those nuclei are 103 really quite high, which results in their relatively high stability. The lower limits that we have obtained for the fission barrier heights of 283–286112, 288–292114, and 292–296116 nuclei are 5.5, 6.7, and 6.4 MeV, re- 100 σ3n ER spectively. This makes the synthesis of superheavy nuclei with 112 ≤ Z ≤ 120 in asymmetric fusion re- σ1n σ4n ER ER actions experimentally attainable (see Fig. 9). The 10–3 choice of appropriate projectile–target combination is very important here. For example, using the fis- 102 106 110 114 118 sion barriers predicted in [19], we found that the ER Z CN cross section for production of element 116 in the 48 247 Fig. 9. Capture, fusion, and evaporation residue forma- 3n evaporation channel of the Ca + Cm fusion tion cross sections. For “hot” fusion reactions, the trian- reaction should be about 1.5 pb at 35 MeV of initial gles show maximal values of σER in the 3n evaporation excitation energy of the CN. That is due to more channel, and the diamonds correspond to the 4n evap- favorable evaporation of two odd neutrons with lower oration channel. Evaporation residue cross sections for separation energies compared with a synthesis of the the “cold” fusion reactions (1n evaporation channel) are 48 248 shown by squares. The closed symbols correspond to the same element in the Ca + Cm reaction. experimental values, whereas the open ones correspond to the calculated cross sections. For the “hot” fusion reactions the following projectile–target combinations ACKNOWLEDGMENTS are used: 48Ca + 208Pb, 12C + 249Cf, 18O + 249Cf, 26 248 48 The work was supported by INTAS, grant no. 00- Mg + Cm; for 110 ≤ ZCN ≤ 118, Ca is used as a projectile and 232Th, 231Pa, 238U, 237Np, 244Pu, 243Am, 655. 248Cm, 247Bk, and 249Cf are the targets. The last com- 58 244 bination leading to ZCN = 120 is Fe + Pu. For the “cold” fusion reactions, 208Pb is used as a target and the REFERENCES projectiles are the heaviest isotopes of the corresponding 1.Yu.Ts.Oganessian,A.V.Yeremin,A.G.Popeko, stable nuclei (from 48Ca to 86Kr). et al.,Nature400, 242 (1999). 2.Yu.Ts.Oganessian,V.K.Utyonkov,Yu.V.Lobanov, et al.,Yad.Fiz.63, 1769 (2000) [Phys. At. Nucl. 63, fusion reactions were found to be at the level of 0.1– 1679 (2000)]. 1.0 pb. For the available experimentally “cold” fusion 3.M.G.Itkis,Yu.Ts.Oganessian,A.A.Bogatchev, reactions, the cross sections for formation of the same et al.,inProceedings of the International Work- elements in the 1n evaporation channel are much shop on Fusion Dynamics at the Extremes, Dubna, lower. A gain of about three orders of magnitude in the 2000, Ed. by Yu. Ts. Oganessian and V. I. Zagrebaev ∗ ∗ survival probability, P1n(E ≈ 15 MeV)/P3n(E ≈ (World Sci., Singapore, 2001), p. 93. 35 MeV) ≈ 103,iscompensatedherebyalossoftwo 4. R. Bass, Nuclear Reactions with Heavy Ions (Springer-Verlag, Berlin, 1980), p. 326. orders of magnitude in the capture cross sections and 5. V.I. Zagrebaev, Phys.Rev. C 64, 034606 (2001). more than two orders of magnitude in the probability 6. V. I. Zagrebaev, Y. Aritomo, M. G. Itkis, et al.,Phys. of CN formation. Rev. C 65, 014607 (2002). 7. N. Rowley, G. R. Satchler, and P. H. Stelson, Phys. 6. CONCLUSION Lett. B 254, 25 (1991). 8. D. L. Hill and J. A. Wheeler, Phys. Rev. 89, 1102 From the analysis of the multidimensional fusion– (1953). fission driving potential, we may conclude that, in the 9. C. R. Morton, D. J. Hinde, J. R. Leigh, et al.,Phys. fission process, a weakly excited heavy nucleus on Rev. C 52, 243 (1995).

PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 6 2003 FUSION–FISSION DYNAMICS 1041

10. M.G.Itkis,Yu.Ts.Oganessian,E.M.Kozulin,et al., 15. A. V. Ignatyuk, Statistical Properties of Excited Nuovo Cimento A 111, 783 (1998). Atomic Nuclei (Energoatomizdat, Moscow, 1983). 11. V. I. Zagrebaev, J. Nucl. Radiochem. Sci. 3,13 16. W. D. Myers and W. J. Swiatecki, Ark. Fys. 36, 343 (2002). (1967). 12. U. Mosel, J. Maruhn, and W. Greiner, Phys. Lett. B 17. Yu.Ts. Oganessian, V. K. Utyonkov, Yu. V. Lobanov, 34B, 587 (1971); J. Maruhn and W. Greiner, Z. Phys. et al., Phys. Rev. C 64, 054606 (2001). 251, 431 (1972). 18. M.G.Itkis,Yu.Ts.Oganessian,andV.I.Zagrebaev, 13. A. Goverdovski, private communication. Phys. Rev. C 65, 044602 (2002). 14. L. G. Moretto and J. S. Sventek, Phys. Lett. B 58B, 19. P.Moller,¨ J. R. Nix, W.D. Myers, and W.J. Swiatecki, 26 (1975). At. Data Nucl. Data Tables 59, 185 (1995).

PHYSICS OF ATOMIC NUCLEI Vol. 66 No. 6 2003