Effects of Nuclear Structure on Quasi-fission
Cédric Simenel, A. Wakhle, Benoît Avez, D. J. Hinde, R. Du Rietz, M. Dasgupta, M. Evers, C.J. Lin, D. H. Luong
aa r X i v : . [ nu c l - t h ] O c t E ff ects of Nuclear Structure on Quasi-fission C. Simenel,
1, 2, ∗ A. Wakhle, B. Avez,
3, 1
D. J. Hinde, R. du Rietz, M. Dasgupta, M. Evers, C. J. Lin, and D. H. Luong CEA, Centre de Saclay, IRFU / Service de Physique Nucl´eaire, F-91191 Gif-sur-Yvette, France Department of Nuclear Physics, Research School of Physics and Engineering,Australian National University, Canberra, ACT 0200, Australia Universit´e Bordeaux 1, CNRS / IN2P3, Centre d’ ´Etudes Nucl´eaires de Bordeaux Gradignan,CENBG, Chemin du Solarium, BP120, 33175 Gradignan, France
The quasi-fission mechanism hinders fusion of heavy systems because of a mass flow between the reactants,leading to a re-separation of more symmetric fragments in the exit channel. A good understanding of the com-petition between fusion and quasi-fission mechanisms is expected to be of great help to optimize the formationand study of heavy and superheavy nuclei. Quantum microscopic models, such as the time-dependent Hartree-Fock approach, allow for a treatment of all degrees of freedom associated to the dynamics of each nucleon. Thisprovides a description of the complex reaction mechanisms, such as quasi-fission, with no parameter adjustedon reaction mechanisms. In particular, the role of the deformation and orientation of a heavy target, as well asthe entrance channel magicity and isospin are investigated with theoretical and experimental approaches.
I. INTRODUCTION
The formation of the heaviest nuclei usually involvesfusion-evaporation reactions [1–4]. The latter are stronglyhindered in the case of heavy ion reactions by two compet-ing mechanisms: ( i ) the quasi-fission (QF) process, and ( ii )the statistical fission of the compound nucleus (CN).Quasi-fission occurs in the early stage of the collision [5–7],when the two reactants form a di-nuclear system, that is, twofragments linked by a neck. An important nucleon transferusually occurs from the heavy fragment toward the light one.The two fragments then re-separate with more mass symme-try than the entrance channel, without forming a compoundnucleus.Typical QF times are shorter than 10 − s [5–8]. These timeshave to be compared with fusion-fission times which can belonger than 10 − s [9]. This shows that the two mechanismsare of very di ff erent nature. In fact, the QF process is a dy-namical mechanism depending on the characteristics of theentrance channel, while CN fission is pure-ly statistical and isdetermined by temperature and angular momentum only.The QF mechanism is responsible for the fusion hindranceobserved in heavy systems [10]. In these reactions, an ad-ditional energy above the Coulomb barrier, sometimes called”extra-push” energy [11], is needed for the system to fuse andform a CN. Note that lighter systems may also exhibit quasi-fission, although with a smaller probability. For instance,quasi-fission has been observed in O, S + U [12–14], andin S + Pb [14].The QF process is known to be a ff ected by nuclear defor-mation and orientation at energies close to the fusion bar-rier [12, 15–20]. The role of isospin has also been investi-gated theoretically [21]. Recently, the influence of entrance-channel magicity and isospin on quasi-fission has been inves-tigated [22]. Magic shells are indeed expected to generate”cold valleys” in the potential energy surface, favouring the ∗ [email protected] formation of a compact CN [23–25]. In addition, magic nu-clei are di ffi cult to excite, reducing energy dissipation, and,then, allowing more compact di-nuclear systems [26, 27].Many experimental data on QF are now available for com-parison with theoretical models in order to test their predictivepower. This is indeed crucial to have reliable theoretical mod-els in order to drive future experiments on heavy elements for-mation. Macroscopic approaches have been thoroughly usedin the past [28, 29]. In addition, the recent increase of compu-tational power allowed micros-copic descriptions of nucleardynamics. For instance, systems as heavy as actinide colli-sions have been studied with microscopic approaches [30–34].In particular, the time-dependent Hartree-Fock (TDHF)theory [35] is particularly well suited at low energies wherea proper treatment of the interplay between reaction mech-anisms and nuclear structure is crucial. Indeed, TDHF cal-culations treat both dynamics and ground-state structure onthe same footing, i.e., with the same energy density func-tional (EDF) as the only phenomenological input. As a result,the TDHF approach has been successful in describing severalreaction mechanisms, such as fusion, nucleon transfer, anddeep-inelastic collisions (see Ref. [36] for a review).We first illustrate the fusion hindrance with TDHF calcula-tions of fusion thresholds in heavy systems. Then we presentan experimental study of the role of magicity and isospin onthe quasi-fission mechanism. Finally, we discuss results of arecent study of quasi-fission with TDHF calculations. II. FUSION HINDRANCE IN HEAVY SYSTEMS
Recent TDHF calculations have been performed to investi-gate collisions of heavy systems leading to quasi-fission andformation of super-heavy compound nuclei [31, 33, 34, 37–40]. In particular, fusion hindrance in heavy systems was pre-dicted [39, 40]. The additional energy needed for fusion tooccur has been found to be of the same order of magnitude asthe extra-push energy [39] determined with the phenomeno-logical approach of Swiatecki [11].Fig. 1 shows the evolution of contact time as a function ofenergy for di ff erent central collisions obtained with the tdhf d E c.m. /B prox. c on t ac t ti m e ( z s ) Ti+ Pb Zn+ Pb Zr+ Sn FIG. 1: Contact time for heavy-ion central collisions as a function ofcenter of mass energy normalised to the proximity barrier [42]. Thearrow indicates a lower limit. code [41]. Here, the contact time is determined arbitrarily asthe time the system spend with a distance between the cen-ters of mass of the fragments smaller than 15 fm. The mostasymmetric reaction, Ti + Pb with Z Z = Zn + Pbreaction with Z Z = ∼ ∼ . B prox . .These calculations do not show any fusion for this system. In-stead, the di-nuclear system always encounters quasi-fission.The Zr + Sn reaction is more symmetric, but with anintermediate charge product Z Z = Zn + Pb case, indicating that quasi-fis-sion occurswithin this energy range. At higher energies, the TDHF cal-culations predict long contact times (greater than 20 zs) whichmay be associated to fusion reactions. As a result, an extra-push E TDHFX ∼
23 MeV is predicted for this system, which isof the same order of magnitude, but slightly larger, than thevalue given by the extra-push model E S w i . X ∼
15 MeV [11].Similar agreements have been obtained for other systems inRef. [39].Detailed investigations on the quasi-fission mechanism aremandatory to understand fusion hindrance in heavy systems.In particular, a good understanding of how these mechanismsare a ff ected by entrance channel properties is of utmost im-portance to optimise the formation of super-heavy elements. III. QUASI-FISSION: INTERPLAY BETWEEN SHELLSAND ISOSPIN
A series of experiments to investigate the quasi-fission pro-cess have been performed recently at the Australian NationalUniversity in Canberra [8, 22, 43–45]. In Ref. [22], we
FIG. 2: Experimental setup for the measurement of MAD. showed that shell e ff ects in the entrance channel may a ff ectthe quasi-fission process. This was done by measuring mass-angle distributions (MAD) of the fragments in several reac-tions at sub-barrier energies. The experimental setup is shownin Fig. 2. The beams were produced by the 14UD electro-static accelerator. Two Multi-Wire Proportional Chambers(MWPC) were used to measure time and positions of both fis-sion fragments in coincidence (see Ref. [22] for more detailson the geometry of the setup). Time of flight (ToF) and po-sitions were converted into fragment masses and angles usingtwo-body kinematics.Fig. 3 shows the resulting MAD (up) and the projectionson the mass-ratio axis (bottom). The mass ratio is defined as M R = m / ( m + m ) where m (resp. m ) is the mass of thefragment in the back (front) detector. Fission and quasi-fissionfragments are located between the bands at extreme M R cor-responding to (quasi-)elastic and deep-inelastic events. Fig. 4sketches the ”trajectory” of the fragments in the MAD. In par-ticular, short scission times induce correlations, i.e. the distri-bution forms a finite angle with the M R = . ff ects in the exitchannel which may enhance the production of magic nucleisuch as Pb and ”delay” the mass drift toward symmetry.In addition, long-time fission of super-heavy systems may oc-cur via asymmetric channels [46, 47], i.e., fusion-fission doesnot necessarily populate the region of the MAD around the M R = . Ca + Pb which has the highest statistics). Suchcorrelations increase the width of the mass ratio distribution(lower panels). We then use the width of the fission frag-ment mass distribution to quantify the amount of ”fast re-separation” which could be associated to quasi-fission events.In particular, the O + U reaction, which is known to ex-hibit only a small amount of quasi-fission [12], gives a widthwhich can be considered as an upper limit for pure fusion-fission events.Fig. 5 shows the width of the fission-like fragment massdistributions as a function of the number of magic numbers inthe entrance channel. We see that, apart for the Ca + Pbcase which is discussed below, there is a clear link between
FIG. 3: (upper panels) Measured MAD. (lower panels) Projected mass ratio spectra. Gaussian fits to the region around M R = σ M R = .
07 (thin red lines) are shown for reference.FIG. 4: Qualitative illustration of the distribution of quasi-fission fragments in the MAD. (a) Non-central collisions induce a rotation of thedi-nuclear system. During the rotation, nucleon transfer occurs toward symmetry. (b) Evolution of the angle as a function of time. The twocolored area corresponds to two di ff erent scission times. (c) Time evolution of the fragment masses toward symmetry. (d) ”trajectory” of thefragments in the MAD. Short scission times (pink area) lead to an angle in the MAD, whereas for longer times (blue area) such correlationsdisappear. Adapted from Ref. [43]. the two quantities, i.e., the more the magicity, the smaller thewidth. This is interpreted as a hindrance of the transfer towardsymmetry process due to shell-e ff ects in the entrance channel.The e ff ect of entrance channel magicity on fusion as beeninvestigated in Ref. [48]. Fig. 6 shows a comparison of fusion-evaporation cross-sections for reactions forming Th ∗ [48].We can see that fusion-evaporation cross-sec-tions decreasewhen the mass asymmetry increases. On the contrary, thesecross-sections increase when shell e ff ects are present in theentrance channel. Indeed, Ca is doubly magic and
Sn hasa magic proton number. Assuming that capture cross-sections, i.e., the sum of QF and fusion cross-sections, are not sensi-tive to shell-e ff ects, we conclude that shell e ff ects in the en-trance channel hinder quasi-fission while it favours fusion ofthe fragments.We see in Fig. 5 that the Ca + Pb does not lie on theglobal trend. In Ref. [22], we interpreted this apparent dis-crepancy as an e ff ect of charge equilibration occurring in theearly stage of the collision. This equilibration process is in-deed very rapid as shown by the TDHF calculations reportedin Fig. 7. The latter are performed at the experimental en-ergy E c . m . = . L to obtain di ff erent FIG. 5: Widths of the mass distributions of the fission-like eventsas a function of the magicity (quantified by the number of magicnumbers N m ) in the entrance channel [22].FIG. 6: Experimental fusion-evaportation cross-sections in reactionsforming Th ∗ . Adapted from [48]. Contact time (zs) ∆ ( N / Z ) f Ca+ Pb Ca+ Pb FIG. 7: TDHF calculations of charge equilibration: the di ff erencebetween final and initial N / Z of the fragments is shown as a functionof the contact time. Adapted from [22]. collision times. ∆ ( N / Z ) f is the di ff erence between the N / Z ratio of the fragments after the collision. The initial value of ∆ ( N / Z ) is 0.54 for Ca + Pb and 0.14 for Ca + Pb. Weobserve a (partial) equilibration (i.e., a reduction of ∆ ( N / Z ) f )within ∼ Ca + Pb system. This time is short ascompared to the quasi-fission time which, according to sim-ple simulations based on the model of Ref. [8], is longer than10 zs. This means that the collision partners change their N and Z at contact and, as far as quasi-fission is concerned, be-have like a non-magical system. This e ff ect is not presentin Ca + Pb which is less N / Z asymmetric and does notencounter a charge equilibration (see the red dashed line inFig. 7). Fusion is then more probable in this reaction whichpreserves its magic nature in the di-nuclear system.We conclude that shell e ff ects in the entrance channel hin-der quasi-fission (and then, favour fusion) only for systemswith small N / Z asymmetry. IV. QUASI-FISSION WITHIN THE TDHF APPROACH
The previous studies showed the importance of the quasi-fission process as a mechanism in competition with fusion,hindering the formation of heavy systems. A quantum andmicroscopic theoretical framework able to describe properlythe quasi-fission properties would be of great importance toget a deep insight into the interplay between structure proper-ties and the QF mechanism.We saw in Section II that quasi-fission was responsible forfusion hindrance in heavy systems. We now use the TDHF ap-proach to investigate the quasi-fission process. Fig. 1 indicatesthat fusion hindrance appears in systems with larger chargeproducts than in the Ti + Pb system (see also Ref. [39]). Inthe following we study the Ca + U reaction ( Z Z = tdhf d code are shown in Fig. 8 for this system at E c . m . =
206 MeV.The left column shows the density for a collision with a tip ofthe
U nucleus at L = ~ , leading to an average exit channel Mn + Fr. The middle and right columns show a collisionwith the side of
U at L = ~ . In this case the exit channelis more symmetric: Rh + Ho in average. These reactionscorrespond to quasi-fission, i.e., an important multi-nucleontransfer from the heavy fragment toward the light one withseveral zs life-time of the di-nuclear system which is typicalfor QF [5–8]. We also see that the mass equilibration (i.e.,the formation of two fragments with symmetric masses) is notcomplete and may depend on the initial conditions.To get a better insight into the mass transfer mechanism,we plot in Fig. 9 the average mass of the light fragment inthe outgoing channel of Ca + U at E c . m . =
206 MeV asa function of the initial angular momentum. Note that themass of the fragments are determined as expectation values ofone-body operators. Distribution of probabilities around thesevalues could be extracted at the TDHF level thanks to particle-number projection techniques [50]. However, the calculationof such distributions is going beyond the scope of this paper.
FIG. 8: Snapshots of the TDHF isodensity at half the saturation den-sity ( ρ / = .
08 fm − ) in the Ca + U system at E c . m . =
206 MeV.(left) Collision with a tip of
U at L = ~ . (middle and right) Col-lision with the side of U at L = ~ . Snapshots are shown every1.5 zs. We see in Fig. 9 that all collisions with the tip lead to quasi-fission with partial mass equilibration. In particular, this ori-entation never leads to fusion, while the other orientation pro-duces long contact time which may lead to fusion at L ≤ ~ .In fact, collisions with the tip favour the production of a heavyfragment in the Pb region up to L = ~ which indicatesthat these QF reactions are strongly a ff ected by shell e ff ects.On the contrary, collisions with the side does not seem to pro-duce an excess of Pb. This may be due to the fact thatmore compact configurations are reached with this orienta-tion, favouring more mass symmetric exit channels for themost central collisions. In addition, QF is observed for col-lisions with the side up to L = ~ only. For larger L , theoverlap between the fragments is too small to allow the forma-tion of a di-nuclear system, leading essentially to quasi-elasticreactions. Orbital Angular momentum M a ss o f t h e li gh t e s t fr a g m e n t FIG. 9: Mass of the light final fragment in Ca + U collisions at E c . m . =
206 MeV as a function of the angular momentum. Collisionswith the tip (red circles, solid line) and with the side (blue squares,dashed line) of
U are considered.
Usual experimental observables to investigate the quasi-fission process include, in addition to the fragment mass dis-tribution, the kinetic energy of the fragments [13] and the scat-tering angle [6, 51]. Fig. 10 shows the evolution of the scatter-ing angle in Ca + U at E c . m . =
206 MeV for collisions withthe tip of
U as a function of angular momentum. For sym-metry reasons, central collisions ( L = ~ ) induce backwardscattering at 180 ◦ , while non-central collisions produce a ro-tation of the di-nuclear system. The higher the angular mo-mentum, the higher the angular velocity. As a result, increas-ing L induces quasi-fission with more forward angle down to ∼ ◦ at L = ~ (see Fig.8 left column for the associateddensity evolution). This indicates almost half a rotation be-fore emission of the fragments at L = ~ . For L ≥ V. CONCLUSIONS AND OUTLOOKS
The quasi-fission process, which is responsible for the fu-sion hindrance in heavy system, strongly depends on the en-trance channel properties. Measured mass angle distribu-tions in several Ca,Ti + Pb,Hg systems show narrower frag-ment mass distributions compatible with fusion when the col-liding partners have strong shell e ff ects and similar N / Z ratio,such as in the Ca + Pb system. Microscopic mean-fieldcalculations with the TDHF approach show that magic shellsalso a ff ect the outgoing channel by favouring the productionof Pb-like fragments in the Ca + U reaction. It is alsoshown that the amount of mass transfer depends strongly onthe orientation of the actinide.TDHF calculations of heavy-ion collisions provide observ-ables, such as the mass of the fragments and the scattering
Orbital Angular momentum S ca tt e r i ng a ng l e ( d e g . ) FIG. 10: Scattering angle as a function of angular momentum incollisions of a Ca with the tip of a
U at E c . m . =
206 MeV. angle, which can be directly compared to experimental data. However, a complete determination of mass-angle distribu-tions requires models which include beyond mean-field fluc-tuations. For instance, extensions of TDHF calculations in-cluding fluctuations at the time-dependent RPA level are nowavailable [52]. Alternatively, a stochastic mean-field approachcould also be considered [53]. These approaches should beapplied to investigate quasi-fission in the near-future.
Acknowledgements
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