000 (2018) 1–18
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Time-dependent HF approach to SHE dynamics
A.S. Umar a , V.E. Oberacker a a Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235, USA
Abstract
We employ the time-dependent Hartree-Fock (TDHF) method to study various aspects of the reactions uti-lized in searches for superheavy elements. These include capture cross-sections, quasifission, prediction of P CN , and other interesting dynamical quantities. We show that the microscopic TDHF approach provides animportant tool to shed some light on the nuclear dynamics leading to the formation of superheavy elements. Keywords:
TDHF; TDDFT; SHE; Superheavy nuclei, Quasifission
1. Introduction
The search for new elements is one of the most novel and challenging research areas of nu-clear physics [1–4]. The discovery of a region of the nuclear chart that can sustain the so called superheavy elements (SHE) has lead to intense experimental activity resulting in the discoveryof elements with atomic numbers as large as Z =
117 [5–7]. The theoretically predicted is-land of stability is the result of new proton and neutron shell-closures, whose location is notprecisely known [8–13]. The experiments to discover these new elements are notoriously di ffi -cult, with production cross-sections in pico-barns. Of primary importance for the experimentalinvestigations appear to be the choice of target-projectile combinations that have the highestprobability for forming a compound nucleus that results in the production of the desired ele-ment. Experimentally, two approaches have been used for the synthesis of these elements, oneutilizing doubly-magic Pb targets or
Bi (cold-fusion) [3, 14, 15], the other utilizing de-formed actinide targets with neutron-rich projectiles (hot-fusion), such as Ca [4, 16, 17]. Whileboth methods have been successful in synthesizing new elements the evaporation residue cross-sections for hot-fusion were found to be several orders of magnitude larger than those for coldfusion. To pinpoint the root of this di ff erence it is important to understand the details of the re-action dynamics of these systems. For light and medium mass systems the capture cross-sectionmay be considered to be the same as that for complete fusion, whereas for heavy systems leading Email address: [email protected] (A.S. Umar) a r X i v : . [ nu c l - t h ] D ec uthor /
00 (2018) 1–18 to superheavy formations the evaporation residue cross-section is dramatically reduced due tothe quasifission (QF) and fusion-fission processes [18–24] thus making the capture cross-sectionto be essentially the sum of these two cross-sections, with QF occurring at a much shorter time-scale.. Consequently, quasifission is the primary reaction mechanism that limits the formationof superheavy nuclei. Various theoretical models have been developed to study the quasifissionprocess [25–31].In many branches of science, highly complex many-body systems are often described inmacroscopic terms, which is particularly true in the case of non-relativistic heavy-ion collisions.For example, the time evolution of the collective nuclear surface variables α (cid:96) m ( t ) and the cor-responding geometrical nuclear shape R ( θ, φ, t ) provides a very useful set of parameters to helporganize experimental data. Using this approach numerous evolutionary models have been de-veloped to explain particular aspects of experimental data. These methods provide a useful andproductive means for quantifying multitudinous reaction data. In practice, they require a quan-titative understanding of the data as well as a clear physical picture of the important aspectsof the reaction dynamics. The depiction of the collision must be given at the onset, includingthe choice of coordinates which govern the evolution of the reaction. Guessing the correct de-grees of freedom is extremely hard, without a full understanding of the dynamics, and can easilylead to misbegotten results. More importantly, it is most often not possible to connect thesemacroscopic classical parameters, describing nuclear matter under extreme excitation and rear-rangement, with the more fundamental properties of the nuclear force. Such di ffi culties can onlybe overcome with a fully microscopic theory of the collision dynamics.The theoretical formalism for the microscopic description of complex many-body quantumsystems and the understanding of the nuclear interactions that result in self-bound, compositenuclei possessing the observed properties are the underlying challenges for studying low energynuclear physics. The Hartree-Fock approximation and its time-dependent generalization, thetime-dependent Hartree-Fock theory, has provided a possible means to study the diverse phe-nomena observed in low energy nuclear physics [32, 33]. In general TDHF theory provides auseful foundation for a fully microscopic many-body theory of large amplitude collective mo-tion including collective surface vibrations and giant resonances [34–42], nuclear reactions in thevicinity of the Coulomb barrier, such as fusion [43–47], deep-inelastic reactions and transfer [48–53], and dynamics of fission fragments [54]. As a result of theoretical approximations (singleSlater determinant), TDHF does not describe individual reaction channels; rather, it describes thetime-evolution of the dominant reaction channel. In other words TDHF is a deterministic theory.To obtain multiple reaction channels or widths of observables one must go beyond TDHF [55–57]. In connection with superheavy element formation, the theory predicts best the cross-sectionfor a particular process which dominates the reaction mechanism. This is certainly the case forstudying capture cross-sections and quasifission.In recent years has it become numerically feasible to perform TDHF calculations on a 3DCartesian grid without any symmetry restrictions and with much more accurate numerical meth-ods [33, 58–60]. In addition, the quality of e ff ective interactions has been substantially im-proved [61–63]. While ordinary TDHF calculations can be used for fusion or capture at energiesabove the barrier they cannot be used directly at sub-barrier energies. During the past severalyears, a novel approach based on TDHF called the density constrained time-dependent Hartree-Fock (DC-TDHF) method was developed to compute heavy-ion potentials [64] and excitationenergies [65] directly from TDHF time-evolution. This method was applied to calculate fusionand capture cross sections above and below the barrier, ranging from light and medium mass sys-tems [45, 66–70] to hot and cold fusion reactions leading to superheavy element Z =
112 [43].2 uthor /
00 (2018) 1–18 In all cases a good agreement between the measured fusion cross sections and the DC-TDHFresults was found. This is rather remarkable given the fact that the only input in DC-TDHFis the Skyrme e ff ective N-N interaction, whose parameters are determined from static structureinformation and there are no adjustable parameters.Within the last few years the TDHF approach has been utilized for studying the dynamics ofquasifission [71, 72] and scission [54]. Particularly, the study of quasifission is showing a greatpromise to provide insight based on very favorable comparisons with experimental data. In thisarticle we will focus on the TDHF studies of capture cross-sections, quasifission observables,and related quantities.
2. Theory
We now give a brief outline of the TDHF method and some of the recent extensions usedin the calculations presented [32, 33]. Given a many-body Hamiltonian H containing two andthree-body interactions the time-dependent action S can be constructed as S = (cid:90) t t dt < Φ ( t ) | H − i (cid:126) ∂ t | Φ ( t ) > . (1)Here, Φ denotes the time-dependent many-body wavefunction Φ ( r , r , . . . , r A ; t ). General varia-tion of S recovers the time-dependent Schr¨odinger equation. In TDHF approximation the many-body wavefunction is replaced by a single Slater determinant and this form is preserved at alltimes. The determinental form guarantees the antisymmetry required by the Pauli principle fora system of fermions. In this limit, the variation of the action yields the most probable time-dependent path between points t and t in the multi-dimensional space-time phase space δ S = → Φ ( t ) = Φ ( t ) . (2)In practice Φ ( t ) is chosen to be a Slater determinant comprised of single-particle states φ λ ( r , t )with quantum numbers λ . If the variation in Eq.(2) is performed with respect to the single-particle states φ ∗ λ we obtain a set of coupled, nonlinear, self-consistent initial value equations forthe single-particle states h (cid:16)(cid:110) φ µ (cid:111)(cid:17) φ λ = i (cid:126) ˙ φ λ λ = , ..., N . (3)These are the fully microscopic time-dependent Hartree-Fock equations which preserve the ma-jor conservation laws such as the particle number, total energy, total angular momentum, etc.As we see from Eq.(3), each single-particle state evolves in the mean-field h generated by theconcerted action of all the other single-particle states.In TDHF, the initial nuclei are calculated by solving the static Hartree-Fock (HF) equationsusing the damped-relaxation method [73, 74]. The resulting Slater determinants for each nu-cleus comprise the larger Slater determinant describing the colliding system during the TDHFevolution. Nuclei are assumed to move on a pure Coulomb trajectory until the initial separationbetween the nuclear centers used in TDHF evolution. Using the Coulomb trajectory we computethe relative kinetic energy at this separation and the associated translational momenta for eachnucleus. The nuclei are than boosted by multiplying the HF states with a phase factor Φ j → exp( ı k j · R ) Φ j , (4)3 uthor /
00 (2018) 1–18 where Φ j is the HF state for nucleus j and R is the corresponding center of mass coordinate R = A j A j (cid:88) i = r i . (5)The Galilean invariance of the TDHF equations assures the evolution of the system withoutspreading and the conservation of the total energy for the system. In TDHF, the many-body stateremains a Slater determinant at all times. The concept of using density as a constraint for calculating collective states from TDHFtime-evolution was first introduced in [75], and used in calculating collective energy surfaces inconnection with nuclear molecular resonances in [76].In this approach we assume that a collective state is characterized only by density ρ andcurrent j . This state can be constructed by solving the static Hartree-Fock equations < Φ ρ, j | a † h a p ˆ H | Φ ρ, j > = , (6)subject to constraints on density and current < Φ ρ, j | ˆ ρ ( r ) | Φ ρ, j > = ρ ( r , t ) < Φ ρ, j | ˆ ( r ) | Φ ρ, j > = j ( r , t ) . Choosing ρ ( r , t ) and j ( r , t ) to be the instantaneous TDHF density and current results in the lowestenergy collective state corresponding to the instantaneous TDHF state | Φ ( t ) > , with the corre-sponding energy E coll ( ρ ( t ) , j ( t )) = < Φ ρ, j | ˆ H | Φ ρ, j > . (7)This collective energy di ff ers from the conserved TDHF energy only by the amount of internalexcitation present in the TDHF state, namely E ∗ ( t ) = E TDHF − E coll ( t ) . (8)However, in practical calculations the constraint on the current is di ffi cult to implement but wecan define instead a static adiabatic collective state | Φ ρ > subject to the constraints < Φ ρ | ˆ ρ ( r ) | Φ ρ > = ρ ( r , t ) < Φ ρ | ˆ ( r ) | Φ ρ > = . In terms of this state one can write the collective energy as E coll = E kin ( ρ ( t ) , j ( t )) + E DC ( ρ ( r , t )) , (9)where the density-constrained energy E DC , and the collective kinetic energy E kin are defined as E DC = < Φ ρ | ˆ H | Φ ρ > E kin ≈ m (cid:88) q (cid:90) d r j q ( t ) /ρ q ( t ) , uthor /
00 (2018) 1–18 where the index q is the isospin index for neutrons and protons ( q = n , p ). From Eq. 9 is isclear that the density-constrained energy E DC plays the role of a collective potential. In fact thisis exactly the case except for the fact that it contains the binding energies of the two collidingnuclei. One can thus define the ion-ion potential as [64] V = E DC ( ρ ( r , t )) − E A − E A , (10)where E A and E A are the binding energies of two nuclei obtained from a static Hartree-Fockcalculation with the same e ff ective interaction. For describing a collision of two nuclei one canlabel the above potential with ion-ion separation distance R ( t ) obtained during the TDHF time-evolution. This ion-ion potential V ( R ) is asymptotically correct since at large initial separations itexactly reproduces V Coulomb ( R max ). In addition to the ion-ion potential it is also possible to obtaincoordinate dependent mass parameters. One can compute the “e ff ective mass” M ( R ) using theconservation of energy in a central collision M ( R ) = E c . m . − V ( R )]˙ R , (11)where the collective velocity ˙ R is directly obtained from the TDHF time evolution and the poten-tial V ( R ) from the density constraint calculations.In practice, the potential barrier penetrabilities T L at E c . m . energies below and above the bar-rier are obtained by numerical integration of the Schr¨odinger equation for the relative coordinate R using the well-established Incoming Wave Boundary Condition (IWBC) method [77].
Almost all TDHF calculations have been done using the Skyrme energy density functional.The Skyrme energy density functional contains terms which depend on the nuclear density, ρ ,kinetic-energy density, τ , spin density, s , spin kinetic energy density, T , and the full spin-currentpseudotensor, J , as E = (cid:90) d r H ( ρ, τ, j , s , T , J ; r ) . (12)The time-odd terms ( j , s , T ) vanish for static calculations of even-even nuclei, while they arepresent for odd mass nuclei, in cranking calculations, as well as in TDHF. The spin-currentpseudotensor, J , is time-even and does not vanish for static calculations of even-even nuclei. Ithas been shown [40, 59, 78–81] that the presence of these extra terms are necessary for preservingthe Galilean invariance and make an appreciable contribution to the dissipative properties of thecollision. Our TDHF program includes all of the appropriate combinations of time-odd terms inthe Skyrme interaction. In addition, commonly a pairing force is added to incorporate pairinginteractions for nuclei. The implementation of pairing for time-dependent collisions is currentlyan unresolved problem although small amplitude implementations exist [82, 83]. However, forreactions with relatively high excitation this is not expected to be a problem.
3. Capture cross-sections
For the reactions of heavy systems the process of overcoming the interaction barrier is com-monly referred to as capture . After capture a number of possibilities exist [84] σ capture = σ fusion + σ quasifission + σ fastfission σ fusion = σ evaporation residue + σ fusion − fission uthor /
00 (2018) 1–18 Understanding each of these cross-sections is vital for selecting reaction partners that have thehighest probability for producing a superheavy element. As we have discussed above one ofthe applications of the DC-TDHF method is the calculation of microscopic potential barriersfor reactions leading to superheavy formations. This allows the calculation of capture-crosssections and the excitation energy of the system at the capture point. In this section we brieflydescribe capture cross-section calculations and in the following sections we will show results forquasifission and the possibility of calculating some of the ingredients for the calculation of P CN .In connection with superheavy element production, we have studied the hot fusion reaction Ca + U and the cold fusion reaction Zn + Pb [43]. Considering hot fusion, Ca is aspherical nucleus whereas
U has a large axial deformation. The deformation of
U stronglyinfluences the interaction barrier for this system. This is shown in the left panel of Fig. 1, whichshows the interaction barriers, V ( R ), calculated using the DC-TDHF method as a function ofc.m. energy and for three di ff erent orientations of the U nucleus. The alignment angle β isthe angle between the symmetry axis of the U nucleus and the collision axis. Also, shown inthe left panel of Fig. 1 are the experimental energies [4, 17] for this reaction. We observe that allof the experimental energies are above the barriers obtained for the polar alignment of the
Unucleus. V ( R ) ( M e V ) } Ca + U E c.m. =250 MeVE c.m. =220 MeVE c.m. =200 MeVE c.m. =185 MeV Exp. Energies } DC-TDHF
Point Coulomb β = 45 o
180 185 190 195 200 205E c.m. (MeV)10 -2 -1 σ ca p t u r e ( m b ) Ca + U Figure 1. Left: Potential barriers, V ( R ), obtained from DC-TDHF calculations [43] as a function of E c . m . energy andorientation angle β of the U nucleus. Also shown are the experimental c.m. energies. Right: Capture cross-sectionsas a function of E c . m . energy (black circles). Also shown are the experimental cross-sections [4, 85, 86] (red squares). The barriers for the polar orientation ( β = o ) of the U nucleus are much lower and peak atlarger ion-ion separation distance R . On the other hand, the barriers for the equatorial orientation( β = o ) are higher and peak at smaller R values. We observe that at lower energies the polarorientation results in sticking of the two nuclei, while the equatorial orientation results in a deep-inelastic collision. We have also calculated the excitation energy E ∗ ( R ) as a function of c.m.energy and orientation angle β of the U nucleus. The system is excited much earlier duringthe collision process for the polar orientation and has a higher excitation than the correspondingcollision for the equatorial orientation.To obtain the capture cross-section, we calculate potential barriers V ( R , β ) for a set of initialorientations β of the U nucleus. Then we determine partial cross sections σ ( β ) and perform anangle-average. However, as a result of long-range Coulomb excitation, not all initial orientation6 uthor /
00 (2018) 1–18 angles occur with the same probability. Rather, the dominant excitation of the ground staterotational band in deformed nuclei leads to a preferential alignment which is calculated in aseparate semiclassical Coulomb excitation code [66]. This code is only used to determine thealignment probability d P / (d β sin β ) in Eq. 13 for the angle averaging of the cross-section σ capture ( E c . m . ) = (cid:90) π d β sin β d P d β sin β σ ( E c . m . , β ) , (13)and σ ( E c . m . , β ) is the capture cross-section associated with a particular alignment. In the rightpanel of Fig. 1 we show our results for the capture cross-sections which are in remarkably goodagreement with experimental data.
4. Quasifission
The feasibility of using TDHF for quasifission has only been recognized recently [33, 71, 72].By virtue of long contact-times for quasifission and the energy and impact parameter dependencethese calculations require extremely long CPU times and numerical accuracy [58–60, 73, 79, 80].In the present TDHF calculations we use the Skyrme SLy4d interaction [87] including all ofthe relevant time-odd terms in the mean-field Hamiltonian. First we generate very accurate staticHF wave functions for the two nuclei on the 3D grid. The initial separation of the two nuclei
Figure 2. Quasifission in the reaction Ca + U at E c . m . =
209 MeV with impact parameter b = .
103 fm ( L = is 30 fm. In the second step, we apply a boost operator to the single-particle wave functions.The time-propagation is carried out using a Taylor series expansion (up to orders 10 −
12) of the7 uthor /
00 (2018) 1–18 unitary mean-field propagator, with a time step ∆ t = . / c. In Fig. 2 we show contour plots ofthe mass density for the Ca + U reaction at E c . m . =
209 MeV as a function of time. The impactparameter b = .
103 fm corresponds to an orbital angular momentum quantum number L = × ×
30) fm. As the nuclei approach each other, a neckforms between the two fragments which grows in size as the system begins to rotate. Due to thecentrifugal forces the dinuclear system elongates and forms a very long neck which eventuallyruptures leading to two separated fragments. The
U nucleus exhibits both quadrupole andhexadecupole deformation; in the present study, its symmetry axis was oriented initially at 90 ◦ to the internuclear axis. This orientation leads to the largest “contact time” [33] which is definedas the time interval between the time t when the two nuclear surfaces first touch and the time t when the dinuclear system splits up again. In this case, we find a contact time ∆ t = .
35 zs (1 zs = − s) and substantial mass transfer (66 nucleons to the light fragment). The event has all thecharacteristics of quasifission. The orientation of the U symmetry axis at 0 ◦ to the internuclearaxis also result in QF but with much shorter contact-times and consequently with smaller masstransfer. These contribute more to the large asymmetric part of the mass distribution [71] andwill not be considered for the purposes of this study. t (zs) b sidetip Figure 3. TDHF results showing the time-dependence of the deformation β for a central collision of Ca + U at E c . m . =
203 MeV. The two curves show the two orientations of the
U nucleus with respect to the collision axis.
Another interesting observable is the time-evolution of the quasifissioning system. In orderto have the correct quadrupole moment for a changing nuclear density one has to diagonalize thequadrupole tensor matrix Q i j = (cid:90) d r ρ TDHF ( r , t )(3 x i x j − r δ i j ) . (14)The largest eigenvalue gives the quadrupole moment calculated along the principal axis for thenucleus (after multiplying with √ / π ). The other two eigenvalues allow the calculation of Q uthor /
00 (2018) 1–18 as well. From these one can construct the deformation parameter β β = π Q AR , (15)where R = . A / . The changing quadrupole moment during the collision, particularly towardsthe last stages of the quasifission process shows not only the elongation of the nucleus but the rateof change also shows the velocity by which the quasifission event is taking place. In Fig. 3 weplot the TDHF results showing the time-dependence of the deformation β for a central collisionof Ca + U at E c . m . =
203 MeV. The two curves show the two orientations of the
Unucleus with respect to the collision axis. One clearly observes that the neutron-rich systemstays at a compact shape much longer than the neutron-poor system and the actual quasifissionevent happens relatively quickly. c on t a c t t i m e ( zs ) A L , Z L
200 205 210 215 220 E c.m. (MeV) E * ( M e V ) Ca +
U, b = 0 fm Z L A L E* L E* H (a)(b)(c) Figure 4. Several observables as a function of center-of-mass energy for a central collision of Ca + U. (a) contacttime, (b) mass and charge of the light fragment, and (c) excitation energy of the light and heavy fragments. Quasifissiondominates in the energy region E c . m . = −
220 MeV.
Next we consider central collisions of Ca + U. In the energy interval E c . m . = −
220 MeV we always observe two fragments in the exit channel, i.e. these events are either a9 uthor /
00 (2018) 1–18 deep-inelastic reaction or quasifission. In Fig. 4a we display the contact time as a function ofthe center-of-mass energy. We observe that in the energy interval E c . m . = −
205 MeV thecontact time increases slowly with increasing energy. Between E c . m . = −
208 MeV there isa steep increase in the contact time. In the energy range E c . m . = −
220 MeV the contact timevaries between (21.7 − −
24) times larger than the contact time observed at E c . m . =
200 MeV. The contact times observed in our TDHF calculations are of similar magnitudeas those obtained by Simenel (see Fig. 38 of Ref. [33] and Fig. 8 of Ref. [88]). c on t a c t t i m e ( zs ) b (fm) A L , Z L Ca +
U, E c.m. = 209 MeVZ L A L (a)(b) Figure 5. (a) contact time and (b) mass and charge of the light fragment as a function of impact parameter.
In Fig. 4b we show the corresponding masses A L and charges Z L of the light fragment. Acomparison with Fig. 4a reveals that mass and charge transfer are proportional to the contacttime. For example, at E c . m . =
200 MeV there is very little mass transfer ( A L = .
4) and somecharge pickup ( Z L = . E c . m . =
208 MeV, the dramatic increase in contact time results ina large amount of both mass and charge transfer, A L = . Z L = .
5. At E c . m . =
220 MeVwe find a light fragment mass A L = . Z L = .
4. Based on these results, weconclude that the energy region E c . m . = −
207 MeV is likely dominated by deep-inelasticreactions while the energy region E c . m . = −
220 MeV is dominated by quasifission. TDHFcalculations carried out at higher energy, E c . m . = −
225 MeV show one fragment in the10 uthor /
00 (2018) 1–18 exit channel; this is the fusion region. Naturally, quasifission is still possible at higher energies,but only for a certain range of non-zero impact parameters. Recently, we have developed an c on t a c t t i m e ( zs ) A L , Z L
198 200 202 204 E c.m. (MeV) E * ( M e V ) Ca +
U, b = 0 fm Z L A L E* H E* L (a)(b)(c) Figure 6. Several observables as a function of center-of-mass energy for a central collision of Ca + U. (a) contacttime, (b) mass and charge of the light fragment, and (c) excitation energy of the light and heavy fragments. Quasifissiondominates in a very narrow energy window, E c . m . = − . extension to TDHF theory via the use of a density constraint to calculate fragment excitationenergies directly from the TDHF time evolution [65]. In Fig. 4c we show the excitation energiesof the light and heavy fragments as a function of the center-of-mass energy. For Ca + U, wefind excitation energies in the QF region to be as high as 60 MeV for the heavy fragment and40 MeV for the light fragment.Figure 5a shows the impact parameter dependence of the contact time, and Fig. 5b exhibitsthe corresponding masses A L and charges Z L of the light fragment for Ca + U at a fixedcenter-of-mass energy E c . m . =
209 MeV. We observe that the contact time decreases from itsmaximum around 21 zs for the central collision to about 5 zs for the impact parameter of 2.2 fm,where the last QF events are observed. The light fragment mass and charge stay flat around A L = . − . Z L = . − . − uthor /
00 (2018) 1–18 are strongly bound with a large prolate deformation around β = . ff ects, these configurations may be energetically favorable during the QF dynamics. However,since TDHF theory does not include fluctuations experimentally a distribution of masses will beobserved.We have repeated the same set of TDHF calculations for the neutron-rich system Ca + U,with the purpose of comparing the two systems. In Figures 6a and 6b we display the contacttime and the light fragment mass / charge as a function of the center-of-mass energy for centralcollisions. These results are dramatically di ff erent as compared to the Ca + U system: thequasifission region, as evidenced by long contact time and large mass transfer, is confined to avery narrow center-of-mass energy window, E c . m . = − . − A L = . − . Z L = . − .
8. Using our microscopic approach wehave also calculated the excitation energy of the emerging fragments for the neutron-rich system Ca + U as shown in Fig. 6c. We find excitation energies in the QF region up to 40 MeV forthe heavy fragment as compared to 60 MeV in the Ca + U reaction.
50 100 150 200 250 A T K E ( M e V ) Ca + U Figure 7. TKE of both the light and heavy fragments formed in Ca + U central collisions. The filled circles representresults from TDHF calculations, and the solid line represents TKE values based on the Viola formula [93].
The contact times discussed above are long enough to enable the conversion of the initialrelative kinetic energy into internal excitations. The total kinetic energy (TKE) distribution ofthe reaction products is one of the indicators of the source of the observed fragments. For quasi-fission, the TKE distribution is expected to be described by the Viola systematics [23, 94]. Thisindicates the TKE’s of final fragments are primarily due to their Coulomb repulsion and do notcarry a substantial portion of the initial relative kinetic energy. Experimentally, the measuredtotal kinetic energy of the quasifission fragments in , Ca + U reactions is in relatively goodagreement with the Viola systematics. The TDHF approach contains one-body dissipation mech-anisms which are dominant at near-barrier energies and can be used to predict the final TKE ofthe fragments. The TKE of the fragments formed in Ca + U have been computed for a range12 uthor /
00 (2018) 1–18 of central collisions up to 10% above the barrier. Figure 7 shows that the TDHF predictions ofTKE are in excellent agreement with the Viola systematics. This is a further confirmation thatthe TDHF dynamics is providing a plausible description of the quasifission process.An important observation of the above results is that the neutron-rich Ca + U systemshows considerably less QF in comparison to the stable Ca + U system. Similarly, the exci-tation energies of the emerging QF fragments have considerably less intrinsic excitation. Theseresults point to the conclusion that the neutron-rich system has a higher probability for leadingto the formation of a superheavy element, as it was discovered experimentally.Another observable that can be studied using TDHF is the mass-angle distribution (MAD) fora quasifission reaction. Experimental MAD’s show the yield of mass-ratio, M R = m / ( m + m ),as a function of the c.m. angle of the quasifission products with masses m and m . In the leftplot of Fig. 8 we show the TDHF calculations of quasifission MADs for the reaction Cr + Wat E c . m . = . W nucleus. In the plotshown in the right side of Fig. 8 corresponding experimental MADs are shown [95]. The regionsof MAD’s near M R = . M R = . Ca + U system in Ref. [71]. M R = m /(m +m ) θ c . m . ( deg ) Cr + WE c.m. =218.6 MeV R M0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 [ deg ] θ Cr54W186_e282_Run30
Figure 8. Left: TDHF calculation of MAD for the quasifission products of the reaction Cr + W at E c . m . = .
5. Ingredients for evaluating P CN In this section we will discuss the possibility of calculating some of the ingredients that gointo the computation of P CN which is the probability that the system evolves into a fused systemrather than quasifission. The two main references used in the discussion of P CN for quasifissionare [84, 96]. 13 uthor /
00 (2018) 1–18 During the collision process the nuclear densities, as described by TDHF time-evolution, un-dergo complicated shape changes, rotations, etc. finally leading to two separated final fragmentsidentified as quasifission due to the long contact-time for the reaction as well as the mass / chargeof the fragments.We have realized that the proper way to calculate the moment-of-inertia for such time-dependent densities is to directly diagonalize the moment-of-inertia tensor (cid:61) i j / m = (cid:90) d r ρ TDHF ( r , t )( r δ i j − x i x j ) , (16)where ρ TDHF is the number-density in units of ( N / f m ), m is the nucleon mass, and x i de-note the Cartesian coordinates. The TDHF calculations are done in three-dimensional Cartesiangeometry [59]. Numerical diagonalization of this 3 × (cid:61) (cid:107) and (cid:61) ⊥ . Naturally, for triaxial density distributions the two perpendicular components arenot exactly equal but for practical calculations they are close enough and always larger than theparallel component.Using the time-dependent moment-of-inertial obtained from the TDHF collision one cancalculate the so-called e ff ective moment-of-inertia1 (cid:61) e f f = (cid:61) (cid:107) − (cid:61) ⊥ . (17)In literature [84, 96] what is usually given is the ratio (cid:61) / (cid:61) e f f at the saddle point of the fissionbarrier, where (cid:61) is the moment-of-inertia of spherical nucleus with the same mass. This ratio isto be constant for impact parameters leading to quasifission ( J > J CN ), where J CN is the largest J value resulting in compound nucleus formation. The expression for the moment-of-inertia fora rigid sphere is given by (cid:61) / m = / AR , which in units of (cid:126) MeV − can also be written as (cid:61) = (cid:126) (2 / AR ) / ( (cid:126) / m ) (18)In Ref. [96] the R was chosen to be R = . A / . With the choice of (cid:126) / m = .
471 MeV · fm , corresponding to the value used in the Skyrme SLy4d interaction, for A = (cid:61) = . (cid:126) · MeV − In Fig. 9 we show the time-evolution of the ratio (cid:61) / (cid:61) e f f for the TDHF collision of the Ca + U system for central collision at an energy E c . m . =
203 MeV. We see that during thetimes that possibly correspond to the vicinity of the saddle point the ratio appears to be smallerthan the traditionally used value of 1 .
5. We will come back to the discussion of finding the saddlepoint later in this document. Before we end this section we should also point out that we havingthe numerical values for (cid:61) ⊥ also allows the computation of the rotational energy E rot = (cid:96) ( (cid:96) + (cid:126) (cid:61) ⊥ . (19)14 uthor /
00 (2018) 1–18 t (zs) ` / ` e ff ( T DH F ) sidetip ` = 179.693 h _ MeV -1 ` = h _ (2/5AR )/(h _ /m)R = 1.225 A Ca + U Figure 9. TDHF results showing the time-dependence of the ratio (cid:61) / (cid:61) ef f for the Ca + U system at energy E c . m . =
203 MeV and zero impact parameter.
Recently, the newly developed density-constrained TDHF method has proven to be a power-ful method of obtaining fusion barriers microscopically from TDHF time-evolution of the nucleardensities. This is a parameter free way of obtaining these barriers.In principle, the same approach can be used to compute the underlying barrier during thequasifission dynamics. From this we may be able to get most of the other ingredients of comput-ing K ,(cid:96) = T (cid:61) e f f / (cid:126) , (20)where T is the nuclear temperature involving various quantities such as E ∗ , barrier height, andothers. In practice the (cid:96) dependence of this expression is ignored, which may be a reasonableapproximation. The computation of the barrier will be very time-consuming but it may give us abetter understanding of the quasifission process.
6. Conclusions
Recent TDHF calculations of phenomena related to SHE searches show that TDHF can bea valuable tool for elucidating some of the underlying physics problems encountered. As amicroscopic theory with no free parameters, where the e ff ective nucleon-nucleon interaction isonly fitted to the static properties of a few nuclei, these results are very promising. Acknowledgments
We gratefully acknowledge discussions with C. Simenel, Z. Kohley, and W. Loveland. Thiswork has been supported by the U.S. Department of Energy under Grant No. DE-FG02-96ER40975 with Vanderbilt University. 15 uthor /
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