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. Fusion–Fission Dynamics and Perspectives of Future Experiments* 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 chan- nels 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 reac- tion 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 in- crease 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.