Dynamical Approach Microscopic Fusion Dynamics Based on TDHF FISSION FUSION Calculate PES as a function of nuclear shape Microscopic HF, HFB, RMF + constraints e.g. Q20, Q30, Q40 as H + lql0 Macroscopic-Microscopic models parametrize the nuclear shape Dynamics of Fission open problem Densities along fusion path given by TDHF Find static energy along fusion path Constraint on density H + lnrn + lprp Extracts E* Provides E1+2(R) along fusion path V(R) = E1+2(R) E1 - E2 ρtdhf (r,t) E*(R(t)) V(R) H. Goutte, et al. Nucl. Phys. A734 (2004) 217 Moller, Sierk, and Iwamoto, Nucl. Phys. A734 (2004) 217
Dynamical Approach Density-Constraint Static collective state corresponding to TDHF state Φ(t)>: (t)> Solution of Hartree-Fock equations TDHF Subject to constraints EDC contains binding energies V ( R(t ))= E DC (ρ(r, t )) E A E A 1 V coll E DC 2 Subtract binding energies Label it with ion ion separation at time t Excitation Energy Approximate expression for collective kinetic energy
Dynamical Approach Implementation Generalize the ordinary method of constraints - for a single constraint H Q H - for a set of constraints H Q H i i i - for density constraint H d 3 r r r H q q Works as accurately as a single constraint - numerical method for steering the solution to TDHF density is given in: 1. Cusson, Reinhard, Maruhn, Strayer, Greiner, Z. Phys. A 320, 475 (1985) 2. Umar, Strayer, Cusson, Reinhard, Bromley, Phys. Rev. C 32, 172 (1985)
Dynamical Approach DC-TDHF Method for Ion-Ion Potentials Completely microscopic description of ion-ion potential barriers No adjustable parameters, no normalization Includes all dynamical entrance channel effect through density (neck formation, particle transfer, surface vibrations, giant resonances) Obtain fusion cross-sections via IWBC Average over all orientations for deformed nuclei V ( R(t ))= E DC (ρ(r, t )) E A E A 1 DC-TDHF E* - PRC 74, - PRC 80, 64Ni + 132Sn - PRC 74, PRC 76, 64Ni + 64Ni - PRC 77, 16O + 208Pb - EPJA 39, 4He + 8Be - PRL 104, 132Sn + 96Zr - PRC 82, 124Sn + 96Zr 48Ca + 238U - PRC 81, 70Zn + 208Pb 132Sn + 48Ca - PRC 85, 124Sn + 40Ca Ca+Ca - PRC 85, O+O, C+O - PRC 85, 2 021601(R) (2006) 041601(R) (2009) 06160(R) (2006) 014614 (2007) 064605 (2008) 243 (2009) 212503 (2010) 034603 (2010) 064607 (2010) 034609 (2012) 044606 (2012) 055801 (2012)
Dynamical Approach Comparison to Empirical Fusion Potentials DC-TDHF potential contains no parameters and normalization Double folding: M3Y effective NN interaction densities from electron scattering Energy dependence: For light systems energy dependence is small Umar, Oberacker, Phys. Rev. C 74, 021601(R) (2006) Coulomb tails always accurate to 50-150 kev
Dynamical Approach Dynamical Effective Mass 1 2 E c.m.= M R R V R 2 TDHF DC TDHF Typical CHF type peak Because we are over the barrier! M R = 2 E c.m. V R R 2 Transform effect to V(R) = dr M R 1 2 dr
64Ni + 132Sn Fusion At low energies fusion x-section orders of magnitude larger than CC Prediction (Liang et al., PRL 91, 152701 (2003)) 3D HF (Sly5) gives oblate deformation for 64Ni: Qzz(n) = -0.85 b, Qzz(p) = -0.59 b 64Ni+132Sn, E cm = 176 MeV, SLy5, b = 3 fm
64Ni + 132Sn Limiting Barriers Barrier for b=0o agrees with empirical No parameter/normalization in TDHF VB = 155.81 MeV RB = 12.12 fm Barrier for b=90o lower VB = 150.13 MeV RB = 12.87 fm Umar, Oberacker, Phys. Rev. C 74, 061601(R) (2006) Umar, Oberacker, Phys. Rev. C 76, 014614 (2007)
64Ni + 132Sn Complete Set of Barriers Umar and Oberacker, Phys. Rev. C 76, 014614 (2007)
64Ni + 132Sn Fusion Cross-Section Use IWBC Average over orientations 1 f E c.m. = dcos P E c.m., 0 Umar and Oberacker, Phys. Rev. C 76, 014614 (2007) Exp. Data J.F. Liang et al., PRL 91, 152701 (2003) PRC 75, 054607 (2007)
64Ni + 64Ni Fusion (b1=90o, b2=90o) Interesting neutron rich identical system (a) outer turning point (b) inner turning point (c) reorientation of the core (d) Density at the minimum Turning points for Ecm = 86 MeV
64Ni + 64Ni Limiting Barriers Variation with Euler angle ai is negligible! Umar, Oberacker, Phys. Rev. C 77, 064605 (2008)
64Ni + 64Ni Fusion Cross-Section Problem at low energies (CC) Compression potential Mişicu, Esbensen, PRL 96, 112701 (2006) Modify inner turning point Ichikawa, Hagino, Iwamoto, PRC 75, 064612 (2007) Exp. Data C.L. Jiang et al., PRL 93, 012701 (2004)
16O + 208Pb Fusion Cross-Section Umar, Oberacker, EPJA 39, 243 (2009)
C+O, O+O Sub-barrier fusion calculations for the neutron star crust: 12 16,24 C + 16,24O and O + 16,24,28O A. S. Umar, V. E. Oberacker, and C. J. Horowitz, Phys. Rev. C 85, 055801 (2012)
Potentials for Oxygen Isotopes barrier heights(mev) 10.00 9.24 8.54 7.95
Density contours: 16O + 24O peak of potential barrier (VB = 8.54 MeV) inner turning point of potential (Ecm = 6.0 MeV) 5 5 0 0 5 R=9.7 fm 5 R=8.5 fm inner turning point of potential (Ecm = 4.0 MeV) inner turning point of potential (Ecm = 3.0 MeV) 5 5 0 0 5 R=8.2 fm R=8.1 fm 5 10 5 0 5 10
Fusion Cross-Sections for O + O Exp.: J. Thomas et al., PRC 33, 1679 (1986)
Fusion Cross-Sections for C + O barrier heights (MeV) 7.77 6.64 P. R. Christensen et al., Nucl. Phys. A280, 189 (1977)
Ca + Ca Sub-barrier fusion calculations for calcium isotopes 40 Ca + 40Ca, 48Ca + 48Ca, 40Ca + 48Ca R. Keser, A. S. Umar, and V. E. Oberacker, Phys. Rev. C 85, 044606 (2012)
Ca + Ca Fusion 40Ca and 48Ca usually in Skyrme fits (here SLy4) but shell structure not that good Small deviations due to small c.m. energy dependence of V(R) High E part of 40Ca+48Ca and low E part of 48Ca+48Ca show larger deviations exp. data: G. Montagnoli et al., PRC 85, 024607 (2012) A. M. Stefanini et al., Phys. Lett. B 679, 95 (2009) C. L. Jiang et al., PRC 82, 041601(R) (2010)
Ca + Ca Fusion UNEDF1 improves the low energy part of 48Ca+48Ca Transfer and pre-equilibrium GDR excitation included in 40Ca+48Ca High E part of 40Ca+48Ca may be due to other inelastic/transfer processes
Superheavy Systems Dynamics of Heavy Systems
Superheavy Systems Factors Influencing Superheavy Formations Excitation energy - high excitation at the capture configuration quasi-fission - high excitation of compound nucleus - fusion-fission Nuclear deformation and alignment Shell effects Mass asymmetry in the entrance channel Impact parameter dependence... ER= capture P CN P survival Form compound system Survive FF process Capture in ion ion potential pocket
Superheavy Systems Cold and Hot Fusion of Heavy Systems 70 Zn+208Pb 48 Ca+238U (b=45o) Heavy nuclei exhibit a very different behavior in forming a composite system Light-Medium Mass Systems capture ER fusion - Fission and quasi-fission negligible - Simple V(R) for composite system Heavy Systems capture = QF FF ER - Quasi-fission dominant - Di-nuclear composites common - A multi-stage V(R) - QF may masquerade as DI
Superheavy Systems Potentials - Cores join - Capture - Cores remain distinct - Nucleons exchanged - b>0 deep-inelastic
Superheavy Systems Cross-Sections Angle average 238U alignment: - significantly reduces x-section 1 f E c.m. = d sin P E c.m., 0 - x-section falls rapidly for β>10o - sin(β) multiply small angles - P(β) is in the range 0.4-0.6 Experimental data (private communication): 1. Yu. Ts. Oganessian, Phys. Rev. C 70, 064609 (2004) 2. Yuri Oganessian, J. Phys. G: Nucl. Part. Phys. 34, R165 (2007)
Triple Alpha Reaction Formation of 12C In the Universe (also 16O) 8 Be has 10-16s lifetime and not found in nature In stars due to 4He abundance small amount of 8Be always present 4 He+8Be combine to form resonant state of 12C (Hoyle state) Excited state decays to ground state via an intermediate state Use TDHF to study the dynamics of this process Umar, Maruhn, Itagaki, Oberaker Phys. Rev. Lett. 104, 212503 (2010) See Movie
Triple Alpha Reaction Dynamics of Transition
Summary Conclusions There is mounting evidence that TDHF dynamics give a good description of the early-stages of low-energy HI collisions We have developed powerful methods for extracting more information from the TDHF dynamical evolution (V(R), M(R), E*, etc.) Effort needed to incorporate deformation and scattering information in to the Skyrme parametrization to test the limits of mean-field dynamics