“WWWWWWWHHIHIWHWIWI HO (JON -'\I I—3 (n [Iii-l E 3 " ‘3 W State Manny This is to certify that the thesis entitled Heavy Ion Dynamics in a TDHF-Based Classical Description presented by Aldo Bonasera has been accepted towards fulfillment of the requirements for M.S. degreein Physics 03w? Major professor\ Date 2/2 0/35~ G. Bertach 0.7639 MS U is an Affirmative Action/Equal Opportunity Institution MSU LIBRARIES M \r RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. HEAVY ION DYNAMICS IN A TDHF-BASED CLASSICAL DESCRIPTION By Aldo Bonasera A THESIS Submitted to Michigan State University in partial fulfillment of the requirements - for the degree of MASTER OF SCIENCE Department of Physics and Astronomy 1985 ABSTRACT HEAVY ION DYNAMICS IN A TDHF-BAS- CLASSICAL DESCRIPTION By Aldo Bonasera we reduce the TDHF equations of two colliding nuclei to a classical form. These equations mimic the actual behavior of finite nucleus TDHF quite well. The calculated fusion cross sections are in good agreement with experimental data for light and heavy nuclei. The fusion threshold for heavy nuclei is well above the interaction barrier, in agreement with the "extra push" systematics. An interesting feature of our ap- proach is the clear distinction of two different states after neck for- mation. The first stage is superfluid, while in the second stage there is a strong damping (superviscosity). The occurrence of superviscosity in the rebounding phase is not sufficient to give cold fusion in heavy systems, but does result in a long interaction time. This could be a signature of fast fission. At higher energies the approaching phase is entirely superfluid and explains the window in the fusion cross section seen in TDHF for light nuclei. TABLE OF CONTENTS List of Figures Chapter I . . . . . . . . . . . . . . . 1.1 Phenomena in heavy ion collisions 1.2 Deep inelastic reactions . . . . 1.3 Fusion cross sections . . . . . Chapter II . . . . . . . . . . . . . . Chapter III . . . . . . . . . . . . . . 3.1 Extension of the model . . . . . 3.2 Fusion cross section . . . . . . 3.3 Comparison with Extra Push Model Chapter IV . . . . . . . . . . . . . 4.1 Deep Inelastic Reactions . . . . 4.2 Comparison with Experimental Data . AP p and 1x 0 O O O O O O O O O O O O O 0 References . . . . . . . . . . . . . . ii . 11 11 17 . 23 43 . 43 . 44 . 51 58 Figure 3.1a,b 3.2 3.3 3.4 3.5a-f 3.6a,b 3.7a-d 3.8a,b 4.1 4.2 4.3 LIST OF FIGURES Experimental isoscalar quadrupole and monopole widths versus mass number compared with Eq. (2), full line. Experimental data are from Ref. 11. . . . Time interval T (see text) versus energy in the C.M. system for the system “°Ca-+“°Ca. E -100 MeV, T is equal to the time delay tD-38 Eg/c . Fusion cross section versus energy for the system l”’Ca-l-“oCa. Data points are from Refs. 16 and 17 . Same as Figure 3.2 for the system 2"Phi-“Ni. The time delay tD is explicitly indicated in the figure. 0 O O O O O O I O O O I O O O O O O O O O O 0 Fusion cross sections versus energy in the C.M. for different systems. Data points are from Ref. 19 I O O O O O O O O O O O O O O O O 0 Same as Figure 3.5. Date are from Ref. 20. . . . Same as Figure 3.5. Experimental data are from Refs. 21 through 25 . . . . . . . . . . . . . . . . . Reaction (full line) and fusion cross sections (dashed line) calculated using the present model, compared to the results of the "extra-push" model (dots and squares, respectively), Refs. 7 and 8 . . . The energy loss spectrum for the system 8"Kr-l- zogBiatELAB-HZMeV............... Interaction time and energy loss versus impact parameter for the system 58Ni-l-197Au. . . . . . . . . . . . . . Same as Figure 4.2 for the system 56Fe-l-lssHo . . . iii Page 13 . 21 . 22 24 . 31 . 34 40 45 48 . 49 CHAPTER I 1.1 Phenomena in heavy ion collision Heavy ion collisions are a powerful tool to study the structure of nuclei. Many phenomena are found in heavy ion collisions, depending on the many variables at the disposal of the experimenter. In the present work we deal with low energy collisions, i.e. Elab< 15 MeV/u. In this energy regime several phenomena occur. We can classify the reaction cross sections into three main types: quasi elastic scattering, deep inelastic and fusion reactions [1-3]. Quasi elastic collisions occurs in reaction in which the surfaces of the two ions have Just been in grazing contact. Thus there is a small energy loss and a few nucleons are transferred from one nucleus to the other. Classically we eXpect that this type of reaction is dominant at large impact parameters. The second major category of reaction, the strongly damped collision, is characterized by a large energy loss, but with the projectile and target still mantaining something of their identity irx the final state. Fusion occurs when the energy of the colliding nuclei is Just above the Coulomb barrier and for small enough impact parameteres (neglect sub barrier fusion). In this work we will present a model that is suitable for describing the last two types of reactions. In the next sections we discuss in a greater detail the experimental signatures of these processes. The formalism is presented in chapter II and results are compared with experimental data in chapters III and IV 1.2 Deep inelastic reactions Extensive experimental results regarding dissipative collisions are presently available. The main features are as follows. First the projectile and the target are mostly heavy nuclei with mass number Duo. The identity of projectile and target is essentially preserved, although a considerable amount of mass ( AA<20 ) can be transferred. This mass transfer occurs during an interaction time, i.e. the time during which the two nuclei are in contact, of the order of 10-21-10-2asec. This time is much smaller than the time needed for a complete rotation of the composite system, Trot”1O-zosec, thus the angular distribution is strongly anisotropic. A large amount of kinetic energy is dissipated, see for example fig. “.1. In the case of 8nKr+20981 at ECM'SOB MeV, the Coulomb energy of two touching spheres is V-300 MeV, using R +R =14.“ fm, where R are 1 2 i the radii of the two nuclei. The energy loss would be 200 MeV, if the two fragments separate with no kinetic energy and only the Coulomb potential energy. Greater energy losses are also observed. This implies that the composite system is greatly elongated before it splits, or that small fragments carry off a great amount of energy. Also a large amount of angular momentum, up to 50h, is transferred from the relative motion into intrinsic excitation. The measured cross-sections are inclusive because not all the reaction products can be observed. Usually only scattering angle, energy and charge of one fragment are detected. Since the excitation energy is large, the channels cannot be resolved and therefore, only averaged quantities are observed. Model calculation and experiments seem to show that deep inelastic collisions cover the range from the fusion to quasi elastic reactions. When the charge of the two nuclei is larger than a critical value, deep inelastic scattering takes also place for zero impact parameter at energies close to the Coulomb barrier. Thus the fusion region is hindered by these processes and possibly for very large charges of both target and projectile, the fusion region completely disappears. The interaction time is a useful concept in discussing the angular distributions, mass asymmetry and energy sharing in the final state of the collisions. For example in the case mentioned above, the interaction time is of the order of the time of rotation of the coalesced system, therefore the angular distribution is almost isotropic and the initial mass asymmetry is equilibrated. This process resembles fission following fusion but with a shorter interaction time, thus the name of fast-fission (FF). The Other extreme case is when the interaction time is very short, say less than lo-Zzsec. Model calculations predict that thermal equilibrium is reached after a time of the order of 10-21 sec. [30]. In this case preequilibrium processes occur, such as prompt emission of light particles: neutrons, protons, alphas or even the breakage of the composite system in three fragments. These processes have recently been observed [4,5]. 1.3 Fusion cross sections The reaction cross section is coincident with the fusion cross section for light nuclei and energies just above the Coulomb barrier. For larger energies and high angular momenta, fusion is replaced by deep inelastic scattering. These features are shown for example in fig. 2.3. For high energies and even zero impact parameter a 'window' in the fusion cross section is predicted by Time Dependent Hartree-Fock theory (TDHF) [6]. But, presently, there is no experimental evidence for such a process. For heavier nuclei, as we saw before, the situation is different. The nuclear attraction increases like R1R2/(R1+Rz), but the Coulomb repulsion increases faster, like 2122. The net result is that for great charges the Coulomb repulsion is very strong and the nuclear attraction is not sufficient to keep the nuclei in a coalesced shape. Thus an extra energy ('extra-push') is needed in order to overcome the Coulomb barrier [7.8]. From the above discussion we expect that a critical value, (Z122) exists below which fusion occurs as soon as the two thr’ nuclei touch. Of course, due to the imprecise knowledge we have of the nucleus-nucleus potential, such threshold value is model dependent. -17-10-19sec. The fused nuclei have a lifetime of the order of 10 During this time a statistical equilibrium is reached, thus concepts like temperature and entropy are very useful. The compound nucleus formed in a heavy-ion collision has an excitation energy 8* dependent on the angular momentum. It cools down mainly by evaporation of neutrons or other light particles until it reaches a certain value of 8* after which photons are emitted in cascade. Since the light particles carry a small fraction of the angular momentum of the compound nucleus, a measurement of the Y- multiplicity gives an estimate of the angular momentum near the yrast line. For high angular momenta where the fission barrier is below 8 MeV, the compound system is more likely to split in two symmetric fragments. For example for the system 208Pb+32 S, it is shown that the entire mass of the projectile is transferred to the composite system which subsequently decays via symmetric fission [9]. CHAPTER II Volume NIB. number l.2 PHYSICS LETTERS 21 June l984 NEWTONIAN DYNAMICS OF TIME-DEPENDENT MEAN FIELD THEORY A. BONASERA, G.F. BERTSCH and EN. ELSAYED cyclotron Laboratory, Michigan State University. East Lanshg. Ml 48824, USA Received 6 February [984 Force equations describing heavy ion collisions are derived from the continuum limit of timedependent mean field theory. The equations mimic the actual behavior of finite nucleus TDHF quite well. and might serve as a starting point for a more complete theory. In this note we investigate a single model for low energy heavy ion collisions [1], based on the dyna- mics of timedependent mean field theory (TDHF). The theory is reduced to the newtonian mechanics of two nuclear centers interacting via classical forces. In view of the many force models that have been put forth already [2-4],it is natural to ask what the point is ofyet another such model. Our objective is to un- derstand the fundamental dynamics of the reactions, and this requires a model that is consistent with our present theoretical framework. At present, the TDHF is the best justified theory at a fundamental level, al- though it is obviously incomplete in several respects and needs to be extended. Because of the computa- tional difficulties of the theory,extensions are un- manageable unless a simple modelling is found for TDl-IF itself. Our efforts are directed to that end. The most important dynamic variable in a simpli- fied description is the separation coordinate between the colliding nuclei, r. We begin with a precise de fini- tion of that quantity. Consider a plane between the two nuclei chosen so that the expectation of nucleon number on each side remains fixed. Calling the two sies A and B, the separation coordinate may be de- fined as the integral over the singleparticle density, r= fp(r')r.'d3r' - fp(r')r'd3r'. (l) A B In a like manner the momentum ofone ofthe nuclei may be defined 0.370-2693/84/5 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) pA = [dwii - i')/2i)p“>(r. HM...- . (2) A . The dynamic equation is found by taking the time de- rivative of eq. (2), using the hamiltonian to evaluate the derivative of the wave function. In TDHF the harniltonian is a single-particle operator and the dyna- mic equation becomes dp .. .. 7f- 51f ; wav - mama, A — (He,-*>(<'é - '6)/2i)¢,1d3r . (3) Here ¢,- are the single-particle wave functions and H is the TDHF hamiltonian. The terms in eq. (3) involv- ing the kinetic energy operator are simplified in the usual way using integration by parts, leaving a surface integral. We assume that the mean field potential de- pends only on the single-particle density, permitting an arbitrary functional dependence on that density. The contribution ofthe short range part of the poten- tial field can then also be expressed as a surface inte- gral on the dividing plane. The resulting equation for the acceleration has the form ch» 7175. {dun-m an: aU/ap — U)! 2 + fd3r fd3r' __pp(r)ppfr')e (r - r') . (4) A B I’ 5' l3 9 Volume l4lB, number 1,2 The first term is a surface integral involving the par- ticle momentum fiux tensor n... - 3"- };mfi — 'i)/2i).((i7' - W20”, . (5) and a potential field contribution, which is expressed in terms of the potential energy density associated with the short-range part of the interaction [5], U. The dividing plane is denoted by S and the normal to this surface by ri. The Coulomb force is given by the second integral in the equation. Eq. (4) is expressed in a form particularly convenient for reduction to classical dynamics. We expect on physical grounds that the force between the two nuclei depends only on the state of the system at the surface of contact, except for the long range Coulomb interaction. Eq.(4) has this form, with the first term in the surface integral arising from the exchange of nucleons between two nuclei. The second term is the force arising from the nuclear potential field. At large separations, the potential dominates because the potential field ex- tends farther than the particle densities. At close con- tact, the two terrns tend to cancel, and the residual force depends sensitively on the momentum distribu- tion of the nucleons and the compressibility modulus associated with the hamiltonian. In principle, the full TDHF wave function is still needed to calculate the right-hand side of eq. (4); our model is a set of simplifying assumptions about how the TDHF behaves. We shall approximate the TDHF density matrix by its value in bulk nuclear matter, together with a surface correction. Parametrizing the contact surface S as a circle of radius r'n , eq. (4) is re- duced to dpA/dr a rrrfifr' [fl+1(p5U/5p - (mm, + 21rornn‘ + (ZlZZe2/r2)ri . (6) The bulk contribution is given by the first term on the right, with the subscript NM denoting a nuclear matter approximation. The surface contribution is proportional to a, which we take to be the empirical surface energy , a = 0.9 MeV/fmz. For the calculations below,we treat the Coulomb interaction in the mono- pole-monopole approximation, which is quite accu- rate for lighter nuclei. In mean field theory, the density matrix for two colliding slabls of nuclear matter is described by two 10 PHYSICS LETTERS 21 June 1984 intersecting Fermi surfaces which may become distort- ed from spherical shape by potential field effects. If the compressibility of nuclear matter is that of a free Fe rrni gas, which seems to be close to the empirical situation, then the surfaces are just the Fermi spheres of the individual slabs. The particle flux tensor was evaluated to lowest order in the relative velocity of the two Fermi spheres, on," by Randrup [6] to give a momentum fiux across the surface iiolrt+1(pw/5p—U)] NM afiqumw-umwnm). (7) Here 01': is the Fermi velocity. Randrup further replaced 0"," by dr/dr, in effect assuming that all parts of the nucleus have the same motion. This reduces the force to a linear friction. However, the Fermi surface in mean field dynamics depends on the motion of the boundary of the nucleus at an earlier time, resulting in a lag [7] between the center of mass motion and the local Fermi surface at S. The delay time is certainly more than the nucleon transit time, and is less than twice the transit time. Within these limits, we shall leave the delay time as a parameter, writing it as (D 3010, (03 ZR/UF , where l < or < 2. The transverse part oft)”, has con- tributions both from the center of mass motion and from the internal angular momentum of the individual nuclei. Our model for on," is then unm(r) = (dr/dr) (r - rD) +‘.i(RAlA/IA - RBIs/13). (8) where R, l, and I are the radii, angular momentum. and moment of inertia of the nuclei, and s is a unit vector in the reaction plane. The angular momentum can be found from the integration of the force equa- tion, so the only need for a finite nucleus TDHF cal- culation is the determination of the evolution of the neck radius. The dynamics of the neck region are quite different when the two nuclei approach each other than when they rebound. The TDHF calculations show that the motion of the nuclei in the approach phase is close to that of rigid spheres [8] , with the overlap region of the spheres defining the neck size. The neck only ex- ceeds the geometric overlap slightly at the closest ap- proach point. The geometric overlap assumption was used in refs. [2] and [3] . We shall improve the para- Volume 1418. number 1.2 meterization by assuming that the nuclear surfaces are sections of spheres with centers separated by the dis- tance r, joined by a cylinder of such a radius to make a constant volume solid. The neck radius is obtained from the equation, ritr - 2r.) - ['ii + W - r0210? ~ '0. (9) where r,‘ I: (R2 — fig)”; We emphasize that this parameterization is strictly for convenience in finding the r“ of TDHF: we do not imply that the system evolves with a constant volume or that it has a particular shape. The neck evolution in the rebound phase is quite different. The neck shrinks rather slowly as the nuclei separate; the TDHF calculation of Dhar and Nilsson [8] obtains a neck shrinkage rate of 1/3 of the separa- tion rate. To establish a function for the neck size in the rebounding phase.we assume a shape of two half spheres connected by two conical surfaces. The radius at the junction of the two cones is determined by fix- ing the volume of the system. The equation for rn in the rebound phase is rn -{[r2R2 - 4r(rR2 - 4R3)] ”2 -— Rr}/2r. Despite the crudeness of these geometric assumptions. the model fits the neck evolution of the TDHF calcu- lation of Dhar and Nilsson quite well, as may be seen from fig. 1. The separation into two nuclei at the end of the rebound phase can take place by two different me- chanisms. If the neck is too narrow in relation to its length, an instability driven by surface tension will cause the neck to pinch off. However, fast hydro- dynamic fiow is possible only when the single-particle fl ‘7 v y r T v 1 v T l- 20an ‘* zoapb 4 7m“ (“1" N u ‘ u 00 C r (fm) Fig. l. Relationship between neck radius rn and separation distance r for m'I’b + m‘i‘b collisions at 800 MeV cm energy and zero impact parameter. The solid line is the present model. and the dashed line is from ref. [8]. PHYSICS LETTERS 21 June 1984 wave functions are nodeless, requiring a small neck radius.We assume scisaion to occur when rn < 1 fm. There is another mechanism for neck breakage when the nuclei rebound at high velocity. An instability develops with respect to density fluctuations when the bulk density falls below a critical value. Accord- ing to ref. [9] this happens in mean field theory when the separation velocity exceeds 0.06c. We shall assume that the neck snaps when the velocity is ex- ceeded. So far we have only considered the forces acting after the nuclei touch. Before they touch, the dyna- mics is well described by potential models, such as the proximity potential [10] or the Bass potential [1 I]. We shall use the Bass potential to describe the interaction before the nuclei touch. Particle exchange will also be included, using the parameterization of ref. [12]. We shall first examine the detailed motion in the collision ‘°Ca + “Ca, to see how well our reduction to force dynamics works. We compare with TDI-IF 0‘s ' _ ‘, “OCo +‘°Co 4 i E/A- LO MN I t '0 o i- I" ‘I “MING d I .| I- s 4 \ '\ apmoacamc -2 n- . wc x i0’2 rlfm) Fig. 2. History of ”Ca + “Ca collision at 80 MeV cm energy and zero impact parameter. The solid line shows the velocity as a function of separation distanc, and the dashed line is a TDHF calculation by Weiss [l3]. The dasheddotted line shows the result of the force model in which the particle mo- mentum flux is treated as a linear friction. ll Volume 1418. number 1.2 calculations by Weiss [13] in which r is evaluated as a function of time. In fig. 2 we plot dr/dr as a function - of r, for a collision under conditions giving fusion. The figure shows the nuclei slowing down as they ap- proach. At r I 10 frn the nuclear attraction is felt and the rate of slowing down diminishes. However, at r - 6.5 fm there is a sudden reversal of velocity, as if the nucleus had hit a hard wall and bounced elastically. The strong repulsive force at that point is due to the particle exchange. Our model reproduces that behavior with a memory time constant a I 1.8 as shown by the dashed line. Under the assumption of extreme dissi- pation, it ~ 0, and the repulsion does not last long enough to keep the nuclei from approaching to essen- tially form a spherical compound nucleus. That is shown as the dasheddotted line in the figure. We next examine the boundaries of the fusion re- gime for light nuclei. The TDHF fusion regime for 28Si + 28Si collisions [14] is shown in fig. 3, compared with our model. The low energy edge is determined by the potential field dynamics. An energy sufficient to surmount the potential barrier will cause the nuclei to touch, and they will remain fused due to the sur- face tension of the neck. At high impact parameter and energy the nuclei may sc'usion after touching. The boundary line is determined essentially by the balance of centrifugai and surface forces, and may be described by a critical angular momentum. Finally, ‘7 2'53 , 2853 N r 9 1 FU$ON REGON ruPacr emanates "mi ‘s‘ $ N T r o :00 em mm Fig. 3. Behavior of ”Si 4' 2‘Si collisions as a function of im- pact parameter and initial cm energy. The outer boundary shows the fusion regionof the force model. The TDHF fusion prediction [14] is shown enclosed in the dashed line. 12 PHYSICS LETTERS 211une1984 I ”00 900 *- 300p IOO' ID 20 so so so ‘enn (nevi Fig. e.‘I'he experimental fudon ere. section for 3“ Ne + 1° Ne collisions compand with the force model. fire decrease in the pndicred eron section at the huh-st energy is due to the opening of the fusion window. there is a fusion window predicted in TDHF, which arises in our model because of the hard bounce beha- vior and the possibility of neck breakage at high re- bound velocities. Other force models do not have this characteristic feature of TDHF. The comparison of TDHF with our model in fig. 3 shows that all the qua- litative features of TDHF can be described with force dynamics. However, actual nuclear collisions probably do nor exhibit fusion windows. In fig. 4 we show the measur- ed fusion cross section [15] for 2(We + 20Ne com- pared with the force model. The experimental cross section is rather constant, even at energies where the fusion window should be evident. Evidently, TDHF is inadequate at the higher enera'es. Nucleon-nucleon collisions will become important and may affect the force dynamics in several ways. For example, the memory time will be shorter if the system is thenna- lized by n-n collisions.This will reduce the magnitude of the bounce, leading to more sticking behavior. We acknowledge the support of the National Science Foundation, and the support of "Fond azione Angelo Della Riccia” for A. Bonasera. 10 Volume 1413, number 1.2 PHYSICS LETTERS 21 June 1984 References [1] G. Bertsch, MSU preprint (1982). [2] D. Gross and H. Kalinowski, Phys. Rep. 45 (1978) 175. [3] F. Beck et al.. Phys. Lett. 76B (1978) 35. [4] D.M. Brink and F1. Stancu, Phys. Rev. C24 (1981) 144. [5] A. Abrikosov and I. Khalatnikov. Rep. Prog. Phys. 22 (1959) 336.eq.(6.3). [6] I. Randrup, Ann. Phys. (NY) 112 (1978) 356. [7] A. Jain and N. Sarma, Phys. Rev. C24 (1981) 1066. [8] A. Dhar and B. Nilsson, Phys. Lett. 77B (1978) 50. [9] G. Bertsch and D. Mundinger. Phys. Rev. C17 (1978) 1646. [10] .I. Blocki et al.. Ann. Phys. (NY) 105 (1977) 427. [11] R. Bass, Phys. Rev. Lett. 39 (1977) 265. [12] C.M. Ko et al.. Phys. Lett. 773(1978) 174. [13] M. Weiss, private communication. [14] P. Bonche et al.. Phys. Rev. C20 (1979) 641. [15] D. Shapiro et al.. Phys. Rev. C28 (1983) 1148. I3 CHAPTER I I I 3.1 Extension of the model Encouraged by the success obtained in fitting T.D.11.F., we will now try to extend the model to asymmetric nuclei and compare predictions with experimental data. The mean field theory has been very useful to suggest a good parametrization for the neck radius and also to give an estimate of the time delay. It is also well known that T.D.H.F. gives a correct qualitative description of nuclear dynamics, but due to computational difficulties and to the imprecise knowledge we have of the mean field potential.there is not always a quantitative agreement with data. For example,using the Skyrme II or Skyrme III potential in the reaction 86Kr+136La, the threshold energy for fusion is shifted by 250 MeV [10]! In the future we will look only for a qualitative explanation of T.D.H.F. and the aim will be to obtain a quantitatively accurate mOdel with few parameters. Let us discuss first the time delay. One body-dissipation relies on the fact that at low excitation energy the mean free path of nucleons is larger than the nuclear diameters. Dissipation arises only when the nucleons hit the surface of the nucleus. We expect the friction is delayed, to allow the nucleons to cross the entire nucleus and hit the surface opposite to the small window formed between the two colliding nuclei. A similar mechanism acts for giant resonances. In that case we 11 12 can imagine a nucleon, excited by the surface oscillatixni. travelling freely in the nuclear medium and interacting with the mean field, after a time t52*R/ (1) where R is the radius of the nucleus and -3/uv is the mean velocity f in a Fermi gas. This is a Characteristic time for the system, and represents the rate of change of the collective energy of the nucleus due to the interaction with different degrees of freedom. According tc> the uncertainty principle the minimum spread of the energy of the resonance, AB, is given by: AEah/td evaluating this formula with standard values for the radius R and the Fermi velocity, we find: /3 MeV (2) AB-17A-1 This functional dependence was suggested in ref.L11] as an experimental fit to the empirical data.The wall formula predicts the same A dependance, but the coefficient is about n times larger than ir1 equaticnl L2) [12]. This simple estimate works quite well for monopole and.quadrupole resonances, as shown in fig.3.1a,b. For higher 13 Figure 3.1a,b. Experimental isoscalar quadrupole and monopole widths versus mass number compared with Eq. (2), full line. Experimental data are from Ref. 11. ma.m amouHm Emmzaz $43 4 14 —-(— —<1-~ —-°"" om. . o.~_. . om . m I _ * o. u. w l I” :_ a}; .. m. w I E L _ + £3 2 8w, . mN( p . .emémég .‘I- :53 muoasmoqsc _ p p r '15 n~.m mmmuHm \I'l .mmmsSz mmSzv < 8. ON. cm 0 4 4 + q 4 \q In _ I II. _ Io. ___- II I. II II. III: 1 I I I .. I . .. -m. v * I h_$ *+ +r+h #__1 II fl 0 n“ w. . _I . c. . 4. «WI I _ lON) m ,mmm Ibo—3 mJOQOZOZ o¢n-vm-3m2 16 multipolarities, in particular the giant octupole resonance, the experimental values are larger than predicted by eq.(2) . However, our main purpose is to show the dissipation via a time delay parameter is reasonable and in agreement with dissipation of very collective motion. In applying the model to asymmetric systems, we treat the nuclei before touching as spheres.The total potential is the same as given before, but we will neglect particle flux through the barrier [31]. The nuclei deform after touching and the simple monopole approximaticni for the Coulomb field is inadequate. A better parametrization of the Coulomb interaction is given in ref. [13] and we will use this in calculations. The evolution of the neck radius requires in general the introduction of the asymmetry degree of freedom. In order to avoid this difficulty and keep the theory at the simplest level possible,we will rely again on TDHF. Looking carefully to figure (2.1) where the neck radius is plotted versus the relative distance for 208Pb+208Pb,we notice that the approaching phase resembles a parabola, while the rebounding phase is very close to a straight line. So we will use the equation: rn-a-sqrt(R1+R2-r) (3) where rn is the neck radius, Ri are the nuclear radii and a-3.1 (fm)1/2 is a parameter fitted to TDHF.Note that the neck radius is zero when the nuclei Just touch.For the rebounding phase we use: 17 rn-rno-B*(r-ro) (u) with rno and r0, respectively, the neck radius and the relative distance at the closest approach.b-.60 is a parameter fitted to the :4 experimental fusion cross section for 6 Ni+208Pb at Ecm-395 MeV.If instead we fit eq (u) to TDHF,we get a larger fusion cross section than experimental data. This is not surprising, in fact, mean field calculations by Stocker et al. show a general overestimate of data for very heavy systems [1a]. The price we have paid to describe nuclear collisions is the introduction of the two parameters in eqs (3) and (14), but still the physical picture beyond the model is simple and clear. 3.2 Fusion cross section The formation of a compound nucleus requires first of all enough initial kinetic energy to overcome the Coulomb barrier. After touching, the surface tension opposes the Coulomb and centrifugal repulsion. Dissipation plays an important role. It is zero in the first stage after neck formation. In the second stage, the nucleon flux produces a strong pressure which suddenly reduces the relative velocity. This 14 ll effect has clearly been showed in the reaction 0Ca+ OCa,(see fig. 2.2). Moroever, this kind of dissipation prevents the two nuclei from reaching a spherical shape. Note that the sudden change between the first stage (superfluidity) and the second stage (superviscidity) can 18 occur in the approaching phase as well as in the rebounding phase. This is very important and explains the occurrence of neck snap and the possibility of "cold fusion"(see below). For light nuclei and low angular momentum the nuclear potential at the touching point is much stronger than the repulsive forces; so the nuclei will fuse once they touch. For high angular momentum, the centrifugal barrier will lead the composite system towards scission. The fusion cross section in this region is then mainly determined by the nuclear potential before the nuclei touch; we find the Bass potential to be in good agreement with data. For higher energies, we observe a rupture of the neck at zero impact parameter as well. This is an effect due to the time delay. At low energy, the superviscidity occurs in the approaching phase:the nuclei move then slowly and are trapped. But at higher energies the approaching phase is entirely superfluid: the relative velocity is very high and the occurrence of friction in the rebounding phase is not enough to avoid the overcoming of the critical velocity for neck snap. This effect is demonstrated in fig.3.2, where the time interval AT-tc-to is plotted versus energy. Here to and to are the time at the turning and at the touching point respectively. The system is uoCa+uOCa at zero impact parameter. We see that the time interval is larger than td up to 100 MeV and in this region we observe fusion. For larger energies the time delay is greater than AT, therefore the approaching phase is entirely superfluid and we get neck snap. For the system 160+16O, TDHF gives a window in the fusion cross section at energies ranging from 50 to 62 Mev in the laboratory,depending on the Skyrme interaction used. Such a prediction 19 ‘ MSU-84-273 AT l T l r I I l («f-cm) ”Co + 4°Co b= Ofm 80 L- .- l 60 .. l . FUSION : I l I 40p 1, NECK SNAP . 20 r '- l l l l L l l 60 80 IOO l20" 140 ISO ISO 200 ECOijeV) Figure 3.2 Time interval AT(see .text) .versus energy in the C.M. system for the system “°Ca+“°Ca. ecu-100 MeV, T is equal to the time delay tD =38 fm/c. ' 20 was tested experimentally at E b'68 MeV but the results are in la disagreement with TDHF [15]. In fig 3.3 we plot the fusion cross section for “0Ca+u00a. Here two different sets of experimental data are plotted [16,17]. Our model is in better agreement with the data by Tomasi et al.and a repetition of the experiment by Barreto et al. confirms this set [18]. Unfortunately there is only one experimental value at high energy and it is underestimated by our model. However if we add the cross section for neck snap to the fusion cross section we get a value of 1057 mb. This estimate is much larger than the experimental value of 720 mb, thus supporting the existence of the low l-window. The conclusion is that there is still a great uncertainty both in theory and in experiments, suggesting the necessity to repeat these experiments at higher energies. For very heavy systems the situation is quite different. The Coulomb repulsion is very strong, and the condition that the nuclei touch is not sufficient to get fusion. In this case the role of 611 friction is changed as is clearly seen in fig.3.u for the system Ni+ 208 Pb. At the energy where the nuclei touch, AT is just equal to the time delay, therefore the approaching phase is entirely superfluid while the rebounding is superviscid. This situation would,in this case, favor fusion; but the repulsion is too strong for a wide neck to be formed and the nuclei reseparate. For this reason, cold fusion as discussed by Swiatecki [7] cannot occur. The only possibility to get it would be for nuclei having a (ZZ/A)eff. close to the threshold value. 21 c_ .mwmm Scum mum mucfioa mama .m~ mam .moosI.muo: amummm mnu u0u xwumam msmum> GOHuomm mmouo cofimam .m.m ouawwm $05.56 O¢ ON. ,4 0 00¢ (aw) ‘9 00m nvnutmuamz 22 .muawwm mnu ma vmumuavafi >Hufioflaaxm ma mu hmamu mafia may .fiz:mI.nmmoN Emumzm mzu pom N.m munmwm mm 08mm .q.m ouswam I>§Iésm . Owc ON? Own O¢m 00m O®N 1 d d d d d I. cc I .. . I I I I NI I 0w 3 m. . . S. I N. I a W. A I I I I I 100 I . I " zoImmIom ._ zoImE . I .. I - I . I I 1 I 1 .205qu 00. E Io H II 55. I .23 + admow aw . I I p p - P4 NNNJvmuamZ 23 In general the fusion observed in this region is due to quantum fluctuations. It is very interesting to notice that in the first region the interaction time is very long, of the order of 700 fm/c. This might be a signature for fast fission. A comparison of this model with experiments performed at GSI using a Pb beam, is shown in fig.3.5a,f [19]. There is an overall good agreement with some discrepancies for the “sea target. For impact parameters where fusion is predicted by experiments, we find an interaction time ranging from 500 to 1000 fm/c and this could explain the difference with experiments. A comparison with experiments performed at MSU is shown in fig 3.6a,b [9,20]. In this case there is a systematic disagreement at higher energies. The interaction time is of the order of 300 fm/c and this suggests that the system could have enough time to relax the initial mass asymmetry. This argument is however not sufficient to affirm that the experimental fusion cross section contains a contribution from fast fission. Finally in fig.3.7a,d a comparison of the model with data for different systems is presented [21-25]. 3.3 Comparison with Extra Push MOdel We showed above how the necessity of an extra energy to get fusion arises quite naturally in our model. This is also a characteristic of Swiatecki's model and it is not surprising since both models are based on one-body dissipation. The main results of the extra push model are confirmed in our picture. A direct comparison between the two models is 24 Figure 3.5a-f. Fusion cross sections versus energy in the C.M. for dif- ferent systems. Data points are from Ref. 19. 25 mm.m meome I>oI2Isam omm . 0mm — _ o ,bl comm (am) 00? mmnuvmuamz 26 nm.m mmawnm I>ozI¢eom 02. co». oom . fi 0 1 oo. o 1 con .. m N. ‘1000 _ £58 + 8% mnmucmnzmi 27 um.m MMDUHK I 2.2 I 5m CON 00¢ 00w onnuvoLIm! {0 (mil) 28 0mm mm.m meuHh 3053.56“. CON 4 3 wow - 11 .._.0n hmnomeawE. CON 00¢ 000 000 MW) ‘0 29 0 .mm museum I>osII .eam .omm _ 0mm 09 I .I . I o l CON 1 00¢. . in . M II 000 m 1 00m 1 000. I I I vmm-¢o-:m: 30 mmqw mmauHm I>o2I.E.om OON 00. o Joov 1.0 loom M In”, loom. I samo~+uzm~ mmnuvwnami Figure 3.6a,b. Same as Figure 3.5. Data are from Ref. 20. 32 mo.m mmauwm I>osII .eam cow .1 oo. 00¢ 000 I I namow + m mm . — .mnm-vo-:m: 33 00M no . m .5505.” £25. 5.0 U 00. I. I _ hum1¢ 0.392 (cm) *0 34 Figure 3.7a-d. Same as Figure 3.5. Experimental data are from Refs. 21 through 25. 35 m~.m mmaon 923 So u OON . 00. I A 0 . 100m {0 (GU-l) 1000. am .14 IN. 0¢.. mNm1¢mnsm2 a: .m mgUHm I>o2I own can A _ I 36 .Ed 00m w amnesia: - (QW‘)-49 37 on.m mmDon, I>osII .Edm 00m 00m 00I - . d q . . o .1000 (0111)}.0 .1000. - - . _ Imm1¢010m2 38 vmtm mmaon I>22Iéem oom .oom 00. I A I I 0. u + loom .m w m. 7 1000. 0400. + I<0¢ . L . _ mmn-¢m-3m2 39 done in fig.3.8a,b for systems near the interesting region of superheavy elements. In the first graph the fusion and total reaction cross section is plotted for the system 2098B“ Cr [7,8J.Fusion for this system has been observed at the energies indicated by the arrows [32]. The Bass and proximity potentials give the same predictions for the total reaction cross section and therefore the slight discrepancies in the fusion region are due to other reasons. The same agreement is found for the reaction 21‘8Cm+“8Ca-->296x. This reaction is very interesting because the compound nucleus has a neutron number close to the magic N-18u and it should be possible to detect experimentally. 40 Figure 3.8a,b. Reaction (full line) and fusion cross sections (dashed line) calculated using the present model, compared to the results of the "extra-push" model (dots and squares, respectively), Refs. 7 and 8. 41 mm.m mmDUHm 9035.0 m oom .1 Irma <5me zoIB§Iéam on» com 08 i 08. Izmaa <5me 6 . Irma <5me It _. 20.03.... -- ZOIh010MeV/u), also 6 particles or even bigger nuclei [26,“). 4.2 Comparison with Experimental Data 20981 . In fig.11.1 we plot the energy spectra for the system “KN The peak at zero energy loss is due to quasielastic scattering. At higher energy there is a broad peak due to deep-inelastic scattering. Our model gives the correct position for the peaks but the magnitude of the cross section is overestimated. This is a limit of any classical model: stochastic processes broaden all sharp structures. At medium energy losses the experimental data are underestimated. In this region of energy losses the nuclei are strongly elongated and there is a large mass transfer. Damping of mass asymmetry becomes important and should explain the discrepancy. Our model is in complete disagreement with experiments for the highest values of energy losses. At angular momenta close to the grazing value, the systems have a short interaction time and the neck snaps. This rapid process might result in more than two particles in the exit channel and this would explain the discrepancy. A clear evidence of direct fragmentation into more than two nuclei 323+58 35 58 has been observed for the reactions Ni and Cl+ Ni at energies 45 26: N: u 93 m an «mmcui.uxso Eoumxm onu you Esuuooam mmoH xwumam one I>02I 000.. >0mwzm oIhuzIx 1.5.0... 00m 00. d It I>mI2 «It Immo~ + Ixem .I.q muswfim n? o w 3 o.) w nH [I m M 0m n¢N1N01xamS 46 above 10 MeV/u. The experimental data suggest that all three fragments are emitted in one step [A]. These features are confirmed from TDHF studies. For the system 136 209 Xe+ Bi at 8.3 MeV/u, a three body break-up was observed [27]. At this relatively low energy the emission of an a-like fragment from the neck at scission occurs only at high angular momentum (i-uxnu. At energies of about 13MeV/u a prompt emission of nucleons is observed for the system 160+93 Nb [28]. A classical model was also proposed to predict the Fermi jet's behavior [26]. The basic idea is that the nucleons in passing through a window from one nucleus to the other, acquire a new velocity whiCh is the sum of the relative velocity plus the Fermi velocity. Thus if the kinetic energy of the nucleon in nucleus B is enough to overcome the nuclear barrier (plus the Coulomb if it is a proton), it will leave the system. This process can be calculated in our moael since it explicitly treats the neck's evolution. But it has the same drawback as in the original model, namely the energy threshold for Fermi jets is half the value predicted in TDHF. This problem needs further investigation. In conclusion let us discuss the results of a recent experiments on 58N1 58 58 +197 the + Ni and Ni Au systems at 15 MeV/u [29]. The purpose of this experiment was to check if thermal equilibrium is attained during the reaction. If this is true, we expect the available excitation energy to be shared in proportion to the masses of the projectile and the target. 58 . 197 . For the system Ni+ Au, the experimental angle-integrated charge distributions show a strong drift of the charge centroids away from symmetry. The light particle evaporation can be calculated under the 47 hypothesis of thermal equilibrium. The values of the predicted charge centroids are larger than those observed. If the excitation energy is assumed to be shared equally between the two fragments, rather than according to the masses, the observed charge distributions are reasonably described. This conclusion is checked for the system 58Ni+58Ni. Since the system is symmetric we expect an evaporation calculation to reproduce the experimental results. This is indeed the case. An explanation of the nonthermal energy sharing is that at high bombarding energy, the system does not have enough time to reach thermal equilibrium. In fig. 11.2 we show the variation of the interaction time and the energy loss with impact parameter. The maximum value for the energy losse agree with the experimental observation. The interaction time is very short,of the order of 10-223ec. This is implies a sudden rupture of the neck and the possibility of a three bOdy process in the exit channel. The same considerations can be repeated for the system 56Fe4~165Ho at 8.5 MeV/u. At this lower energy, the centroids of the charge and neutron distributions are well described by assuming a smooth transition from the limit of equal sharing of the dissipated energy occurring at small energy loss, to the limit of equal temperature at large energy loss. In fig.u.3 we plot the interaction time and energy loss versus impact parameter. For b less than 11.6 fm we get fusion. The fusion cross section is 665 mb which agrees nicely with experimental data. At higher impact parameters, our model gives a very long interaction time, of the order of 10‘21sec. In this case the system has enough time to reach thermal equilibrium. For the highest values of impact parameters, the interaction time is equal 48 >22 03 meow :48. + Izmm 00. 00M AGW 3M1 00¢ .: mmoH mwuoco mam mafia cofiuomuoucH 253 :53 0. . m 0 ¢ N 0. m 0 ¢ N 0v Ihuo¢01002 .N.s eusmIm 49 .ommoII.ommm soumxm ecu you «.8 ouswwm mm mamm .m.¢ muswfim :53 . 2.5.. 0 0 ¢. “w 0 m. ¢ N IJ/ I a a q q I T 0N lazu- 0¢ u 00 noon col mu 000 1 1 m ON. I o! L00b 1.5.0 >22 man. m cm. XII—I... VimUmW—.9Iwnbmme L — LP L P n P P 05N1¢0namz 50 to that for the system 5bNi’r 197Au, resulting in a preequilibrium process. Therefore it is not surprising that the best description of the data is given by a smooth transition as described above. Our model suggests that the time the system takes to reach equilibrium is surely larger than 50 fm/c and smaller than 300 fm/c. Bertsch has determined the local equilibrium time by considering the equilibration of a quadrupole deformation of the Fermi sphere within the Fermi-gas approximation [30]. His numerical result is in good agreement with the above estimate. APPEND IX HEAVY-ION DYNAMICS IN A TURF-BASED CLASSICAL DESCRIPTION Aldo Bonasera Research Institute for Fundamental Physics, Kyoto University, Kyoto 606, Japan and Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA* Abstract: we present a simple classical model based on the mean-field theory. This model is in reasonable agreement with trends in fusion cross-sections for heavy nuclei, including the barrier to fusion at high ZZIA. A critical value of the interaction time, which leads to fast fission, is estimated from the reaction 238U+89Y at Blah-6 MeV/u. Finally, we estimate the equilibration time for energy by comparing the TURF-based classical model with recent-experimental data. I. Introduction In this contribution we investigate lowbenergy heavy-ion collisions using a simple classical model. This model is based on the dynamics of Time-Depend- ent Hartree-Fock (TDHF) theory which at present seems to be the best theory at the microscopic level. Unfortunately, the TDHF theory presents serious compu- tational difficulties and sometimes the results are in disagreement with ex- perimental data.5) Our purpose is to reduce the mean-field theory to classi- cal equations of motion, which can be easily solved. This work is organized in the following way. In section 2 we discuss the equations of motions. In section 3 the results are compared with experimental data on fusion and deep inelastic scattering. Collision times are calculated for a variety of reactions and we estimate a critical value for the interaction 4) time which leads to fast fission and is common to all systems. Also, Compari- 6) suggests that the equilibration son with the results of a recent experiment time for energy lies between 50 fm/c and 250 fm/c. We summarize our main results in section 4. 2. Equations of Motion A convenient way to reduce the TDHF system to a classical form is by tak- ing its Wigner Transform. In the limit.h+0, this gives the Vlasov equation * Present address. 51 52 g? flair) - {h(?,;,t):£(?,3,c)} , (1) where h(?,p,t)- g + w(;) is the Wigner transform of the self-consistent BF hamiltonian and £(¥,;,:) is the phase-space distribution. The curly brackets indicate Poisson brackets. Classically, the most important degrees of freedom are the conjugate va- riables ; and‘; describing the relative motion of the two nuclei. we define these quantities as ; ++ .1: ++ ++ ; ++ ‘+ - I drdp + f(r,p,t) - I drdp + f(r,p,t) , (2) P A P B p where A and B refer to the two colliding nuclei. The Hamilton equations of motion of nucleus A, say, are found by taking the time derivative of eq.(2). Using eq.(l), and after some algebra, we obtain 3 %i‘f m and 9.." . 1rr2 {VG + iffl I 0)] + 21ror n + Coulomb term dt PA N 3p NM N ’ where r is the radius of the necka) formed during the reaction, a is the sur- N 2) face energy , and we assume that U(?)5 d§w(?)£(?,3,t) is a local function of the density. The subscript NM denotes a nuclear-matter approximation. The 7) 3) particle-flux tensor I was evaluated by Randrup. We modify his prescription by‘introducing a time delay in the damping term. Before the nuclei touch the dynamics is well described by potential models and we shall use the Bass potential in calculations.8) Finally, we shall assume that the two nuclei separate into two fragments again in the rebounding phase if r <1 fm or a critical velocity for neck snap N is exceededl). 3. A Comparison with the Expgrimental Data For light nuclei, the reaction cross-section at energies just above the Coulomb barrier is dominated by the fusion cross-section. For very heavy “30°04'33’ 800 55 E b'" 400 40 + 40 C C Ca loo _ O l 1 40 RD 200 300 400 .m.""v’ E s “(New Fig.1. Fusion cross-section versus energy in the C.M. for the systems 40084+ oCa 10)aM208Pb+§8Fe.11) systems, on the other hand, the Coulomb repulsion is very strong, and the condition that the nuclei touch is not sufficient to cause fusion. Thus, an 9) extra injection energy is needed in order to produce fusion. These features 3) are well reproduced by our model. In fig.l we show how our model compares with experimental data on fusion cross-sections for the systems 40Ca+40 Ca 10) ”58 2+208 Pb.ll) sections for several different systems is given in ref.3. A more detailed comparison with data on fusion cross- Recently an interesting new phenomenon, called fast-fission, has been ob- served. Its characteristics are intermediate between fusion and deep inelastic collisions. In this case the binary fragmentation of the intermediate compo- site system is similar to the fission following complete fusion but with a shorter interaction time. We estimate a critical value for the interaction time for fast fissionA) by'comparing our model with the recent experimental data obtained at G.S.I. using a 2‘38U beamlz) incident on several different targets, ranging from 160 to 89Y. For heavy systems the.fusion cross-section is defined as E o + o . (4) 0fusion compound nucleus fast fission 89Y +238 For the system U at energy E1 ab -6 MeV/u, the experimental fusion cross- section is less than 60 mb. Our classical model gives no contribution from compound-nucleus formation. If we assume that interaction times larger than 54 ssg-go-g} . 9‘ \ aY \ 0 \ \ 7.910 ll 121314 ISVUITI Fig.2. Theoretical fusion cross-section (triangles) 2 versus n-(ZT/AT)Ecm/VB (see text). The different targets are explicitly indicated. The experimental data is from ref.12. The dashed lines are drawn in order to guide the eye. 8X10-223ec lead to fast fission, then we obtain a value for the fusion cross- section in agreement with the observed value. This critical value of the interaction time is independent of the system involved, as we show in fig.2 where the fusion cross-section versus the quantity n-(ZglATficm/VB is plotted. ZT and AT refer to the charge and the mass of the target, respectively, and VB is the Coulomb barrier. A comparison of our classical model with other experimental data provides further evidence for the existence of a critical interaction time for fast fission.4) ' The dominant process at higher energies is the deep inelastic scattering. As an example of this type of reactions, we discuss the result of a recent ex- periment on the systems 58Ni+58Ni and 58Ni+197Au, at 15 MeV/u.6) The purpose of this experiment was to see if thermal equilibrium is attained during the reaction. If this is true, then we expect the available excitation energy to be shared between the two nuclei in prOportion to the masses of the projectile and the target. The experimental values of the distributions of charge and mass, however, are in agreement with a calculation performed assuming that the energy is shared equally between the two fragments. One possible explanation of the non-thermal partitioning of excitation energy is that at high bombarding energy, the system has too little time to 55 reach thermal equilibrium.3) In fig.3 we show how the interaction time and the energy loss vary with impact parameter. The maximum value for the energy loss agrees with the experimental observation. The interaction time is very short, of the order of lo-zzs. , The same considerations can be applied to the system 56Pei-165110 at 8.5 MeV/u. At this lower energy, the experimental values on charge and mass dis- tributions agree with calculations performed assuming thermal equilibrium. In fig. 4 we plot the interaction time and energy loss versus impact parameter. For b less than 4.6 fm we see fusion. For higher impact paramr eters, the system has an interaction time of the order of 10-213, and it can ”4'!“ 5%«*”6A “pr ‘°° t...- «hm 70 > 3 “I so i “.1 so 100 40 20'4 s s o 2 4 8 8*76“ bum) bflm) Fig.3. Interaction time and energy loss versus impact parameter for the system 58N1+197Au.6) w - 4- ‘ v T ' ' I r f I T 55Fe0'65He W1 tm- 355 my 700- | . > 500 '- i “ Eeel 300 r- 00 I 40! 100 " 2° " _L l l “I l N:- A 2‘ 4 e Nun) Fig.4. As fig.3 for the system 56Fe+165Ho. 56 attain thermal equilibrium. The above results suggest that the time required by the system to reach thermal equilibrium lies in the range 50 fm/c to 250 fm/c. Bertschla) has determined the time to reach local equilibrium, by con- sidering the equilibration of a quadrupole deformation of the Fermi sphere within the Fermi-gas approximation. Our estimate is in good agreement with his numerical result. 4. Summary In this contribution we have presented a simple classical model based on one-body dissipation. We assumed that during the reaction a neck is formed and we parametrized the radius of the neck by referring to the TDHF calcula- tions.2-3) An interesting feature of this model is the delayed-damping term. During the first stage, after neck formation, this implies that the motion is superfluid, while in the second stage a strong dissipation occurs arising from one-body dissipation (superviscosity). Our model is in good agreement with experimental data on fusion cross- sections and the barrier to fusion at high ZZIA is reproduced as well. Fast fission occurs in our model if a critical value of the interaction time is exceeded. Such a value is common to all systems. Finally, the equilibration time for energy was estimated by referring to an experiment at 15 MeV/u . We plan to extend the model by including fast-particle emissions in the early stage of the reaction and stochastic processes in deep inelastic scattering. Acknowledgements The author wishes to thank the RIFP-Kyoto for the warm hospitality and financial support. 57 References 1) G.F. Bertsch, MSU preprint (1982). 2) A. Bonasera, G.F. Bertsch and E.N. El Sayed, Phys. Lett. 141B (1984) 9. 3) A. Bonasera, submitted to Nucl. Phys. A. 4) A. Bonasera, submitted to Phys. Rev. Lett. 5) H. Stacker, private communication, and H. Stocker et al., 2. Phys. 5396 (1982) 235. 6) T.C. Awes et al., Phys. Rev. Lett. §2§.(1984) 251. 7) J. Randrup, Ann. Phys. (NY) 112 (1978) 356. 8) R. Bass, Phys. Rev. Lett. §2_(1977) 265. 9) W.J. Swiatecki, Nucl. Phys. 5316 (1982) 275. 10) H. Doubre et al., Phys. Lett. 132 (1978) 135, and E. Tomas et al., Nucl. Phys. A313’(1982) 341. 11) R. Bock et al., Nucl. Phys. 5288 (1982) 334. 12) K. Hildebrand, private communication and K. Hildebrand et al., Phys. 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