Mean-Field Dynamics of Nuclear Collisions

Transcription

1 Mean-Field Dynamics of Nuclear Collisions A. Sait Umar Vanderbilt University Nashville, Tennessee, USA Research supported by: US Department of Energy, Division of Nuclear Physics Feza Gürsey Institute, Istanbul, Turkey - 200

2 Overview: General I. Nuclear Many-Body Problem - General introduction - N-N interaction - Effective interactions - Many-body methods - Why/when mean-field is a reliable approximation? - Basic HF-TDHF equations - Numerical Methods II. Time-Dependent Hartree-Fock - Initialization of nuclear collisions - Examples III. Calculation of Potential Barriers and Fusion - Traditional methods - DC-TDHF method - Examples Feza Gürsey Institute, Istanbul, Turkey - 200

3 General introduction I. Nuclear Many-Body Problem Two major challenges A. Interaction - No practical first-principles theory for deriving N-N interaction - Quantum chromodynamics (QCD) Effective field theories - Lattice QCD shows promise - n,p or quarks and gluons - Inside a nucleus medium effects B. Many-body Method - Many-body wavefunction - Many-body equations - Correlations - Center-of-mass motion - Some of these solved for few nucleon systems Feza Gürsey Institute, Istanbul, Turkey - 200

4 N-N Interaction I. Nuclear Many-Body Problem N-N interaction in free space (no medium) Properties deduced from partial-wave analysis of scattering phase shifts and few nucleon systems (e.g. deuteron): - Short range and attractive in the intermediate range - Hard-core (around 0.5fm) (s-wave phase shift) - Spin dependent - Spin-orbit component - Charge independence (almost!) - Tensor component - non-central Argonne V-8: Wiringa, Stoks & Schiavilla, Phys. Rev. C 5, 38 (995) Feza Gürsey Institute, Istanbul, Turkey - 200

5 N-N Interaction I. Nuclear Many-Body Problem Symmetries of N-N interaction Translational invariance Galilean invariance Rotational invariance Invariance under space reflections Time reversal invariance Invariance under particle interchange Isospin invariance (not exact) Okubo and Marshak, Ann. Phys. (NY) 4, 66 (958) { V r = V C V 2 V T } m r 3 3 e S r V L S LS m r m r 2 2 m r m r 2 m r Understanding: Various meson exchange theories (OBE) or chiral perturbation theories provide explanation of this phenomenological N-N interaction π, σ N ρ, ω OBEP 2π + N N πρ N two mesons meson (π) baryon (p / n) baryon (p / n) Feza Gürsey Institute, Istanbul, Turkey - 200

6 Effective Interaction I. Nuclear Many-Body Problem N-N Interaction in Medium (inside nucleus) We usually cannot use the bare N-N interaction inside a nucleus - Hard-core short distance behavior needs renormalization - We only have the on-shell matrix elements of <k V k'> - Presence of other nucleons, NNN interaction, etc. Various theories to achieve this renormalization - G-Matrix - MB perturbation theory - Various truncation methods (model space, momentum) These methods can be utilized in ab initio calculations (MB theories that start from N-N interaction) Feza Gürsey Institute, Istanbul, Turkey - 200

7 Effective Interaction I. Nuclear Many-Body Problem Phenomenological N-N Interaction in Medium Early nuclear structure calculations simply used a delta function v(r)=v0 δ(r-r2) Finite-range implies momentum dependence k v k ' = d r e 3 k k ' r v r Delta-force gives constant. Simplest momentum-dependence which preserves rotational invariance, time-reversal invariance, etc. k v k ' =v 0 v k 2 v k ' 2 v 2 k k ' This corresponds to a v(r) v r =v 0 r v k ' 2 r r k 2 v 2 k ' r k Feza Gürsey Institute, Istanbul, Turkey - 200

8 Effective Interaction I. Nuclear Many-Body Problem Skyrme Interaction v 2=t 0 x 0 P r t x P { r 2 k k r 2 } 2 From above t 2 x 2 P k r 2 k r 2 t 3 x 3 P 6 Phenomenological density dependent term it 4 2 k r 2 k Spin-orbit term Similar form obtained from local density approximation (LDA) t 0,t,t 2,t 3,t 4, x 0, x, x 2, x 3, k= 2 2i (acts to right) Parameters fitted to nuclear properties k = 2 2i (acts to left) 2 P = 2 Feza Gürsey Institute, Istanbul, Turkey - 200

9 I. Nuclear Many-Body Problem Many-Body Methods Many-Body Theories Ab-initio or almost ab-initio methods Perturbative many-body theories Coupled cluster theory Shell-model, no-core shell-model Shell-model Monte-Carlo Density functional theory (ab-initio) Renormalization group Others...(path integrals etc.) Effective methods Non-relativistic mean-field methods (Hartee-Fock, TDHF, ATDHF) Relativistic mean-field (RMF) methods Hartree-Fock Bogoliubov (HFB, TDHFB) Generator coordinate method (configuration mixing) Gaussian overlap approximation (configuration mixing) Energy density functional theory Feza Gürsey Institute, Istanbul, Turkey - 200

10 Many-Body Methods I. Nuclear Many-Body Problem Practical Applications Feza Gürsey Institute, Istanbul, Turkey - 200

11 I. Nuclear Many-Body Problem Many-Body Methods Scope of the problem Solve the many-body Schroedinger equation for A particles x, x2, x3,, x A = E x, x2, x3,, x A H xi= r i, i, i Hamiltonian A A i i j = t v H i ij Differential equation in 3A dimensions! Very difficult to tackle for large A numerically MB wavefunction must satisfy certain conditions (antisymmetry) A We still have the center of mass not fixed ri = 0 i= Feza Gürsey Institute, Istanbul, Turkey - 200

12 I. Nuclear Many-Body Problem Mean-Field (NR) Method General Discussion Zeroth order approximation to nuclear MB problem Assume each nucleon moves in an average effective field generated by all the other nucleons and rarely experience hard collisions First used for describing atoms Well defined center (nucleus) Electrons primarily feel the Coulomb field of the nucleus 2 A A 2 2 p i Z e e = H 2m r 2 i j= r ij i= i= i A Central potential No well defined center! Nucleons seem tightly packed! Does mean-field make sense? 208Pb Feza Gürsey Institute, Istanbul, Turkey - 200

13 Mean-Field (NR) Method I. Nuclear Many-Body Problem Why is mean-field a good approximation? Because of Pauli blocking the only way two nucleons can interact that results in a different final state is if at least one is excited to an unfilled level This violates energy conservation so it can last a limited duration, E t=ℏ The measure is the mean-free path = 0 Which is many times the nuclear size Feza Gürsey Institute, Istanbul, Turkey - 200

14 I. Nuclear Many-Body Problem Basic Hartree-Fock Equations Wavefunction MB wavefunction in the mean-field (independent particle model) is a product x, x2, x 3,, x A = P { x x 2 x 3 x 4 x A } Wavefunction must satisfy certain conditions e.g., x a,, x b, =, x b,, x a,, x a,, x a, =0, x a,, x b, =0 if a= b Slater determinant satisfies these conditions x x2 2 x 2 x2 x, x 2, x 3,, x A = A! A x A x 2 x A 2 x A det x i A! A x A Without loss of generality orthonormalize single-particle states * dx a x b x = ab Feza Gürsey Institute, Istanbul, Turkey - 200

15 I. Nuclear Many-Body Problem Basic Hartree-Fock Equations Time-Dependent Variational Method Use the MB Slater determinant as a trial function for variational method t2 S= dt t H i ℏ t t =0 A A i i j = t v H i ij t Use Skyrme potential which simplifies the exchange term Not too hard since single-particle states are uncorrelated and orthonormal In coordinate space E can be written in terms of Hamiltonian density = d 3 r {H Skyrme,, j, s,t, J ; r H Coulomb p } E = H All defined in terms of single-particle states e.g. A * q r = i r,, q i r,, q i= A Particle density 2 q r = i r,,q Kinetic energy density i= Engel et. al.,nucl. Phys. A249, 25 (975). Chabanat et. al. Nucl. Phys. A635, 23 (998); Nucl. Phys. A643, 44 (998). Feza Gürsey Institute, Istanbul, Turkey - 200

16 I. Nuclear Many-Body Problem Basic Hartree-Fock Equations Skyrme Hamiltonian Density [ ] ] ℏ HS r = t 0 x 0 t 0 x 0 [ p n ] t x t 2 x 2 j 2m [ ] [ ] [ ] t x t 2 x p p n n j 2p j 2n 6 3t x t 2 x [ 2 3t x t 2 x p p n n 2 2 t 3 2 x 3 2p 2n x t 0 x 0 s 2 t 0 s 2n s 2p t 3 x 3 s 2 t 3 s 2n s 2p t 2 3t s q 2 s q t 2 x 2 3 t x s 2 s q Time-odd terms come in pairs! Total is TR invariant + t x t 2 x 2 ( s T J 2 )+ t 2 t ( s q T q J 2q ) 8 8 q t 4 qq ' [ s q j q ' q J ] 2 qq ' (s,j,t) time-odd, vanish for static HF calculations of even-even nuclei non-zero for dynamic calculations, odd mass nuclei, cranking etc. Feza Gürsey Institute, Istanbul, Turkey - 200

17 I. Nuclear Many-Body Problem Numerical Methods Time-Reversal Invariance Without time-reversal invariance Skyrme has many extra terms - They come in pairs/products such that Hamiltonian density is time-even Engel, Brink, Goeke, Krieger, and Vautherin, Nucl. Phys. A249, 25 (975) These terms are zero while fitting the force parameters - To properties of static even-even nuclei Chabanat, Bonche, Haensel, Meyer, and Schaeffer, Nucl. Phys. A635, 23 (998) They are all non-zero for dynamical calculations (also for odd-a) The extra terms are required to satisfy (no new parameters) - Galilean invariance - Local gauge invariance (Dobaczewski and Dudek, Phys. Rev. C52, 827 (995)) In addition, we have a time-even spin-current tensor - Not included in the past due to numerical difficulty J Feza Gürsey Institute, Istanbul, Turkey - 200

18 Basic Hartree-Fock Equations I. Nuclear Many-Body Problem Hartree-Fock Equations Variation with respect to single-particle states gives TDHF equations iℏ =h { } t Static HF equations can be obtained by substituting r,t =e i t /ℏ r A coupled set of differential equations h { } = Single-particle energies Depends on s.p. states! Equations has to be solved self-consistently. Guess a set of orthogonal single-particle states. 2. Compute densities, currents, etc. 3. Compute the Hartree-Fock potential. 4. Solve the Poisson equation for direct Coulomb contribution. 5. Perform an imaginary time step with damping. 6. Do a Gramm-Schmidt orthogonalization of all states. 7. Repeat beginning at step 2 until the convergence criteria are met. Feza Gürsey Institute, Istanbul, Turkey - 200

19 I. Nuclear Many-Body Problem Numerical Methods Time-Evolution and Gradient Damping t =exp[ i h ] t [ Formal solution for a small time-step ] N i h n t t n! n= k k k Numerical approximation k =O { x 0 D [E 0 ] h } [ ][ ][ ] Tx D E 0 = E0 E 0 =20 MeV Ty E0 Tz E0 x 0 =0.05 Damped-Gradient for static solution O=Gramm Schmidt Umar et. al., Comp. Phys. Comm. 63, 79 (99). Umar et, al. Phys. Rev. C 44, 252 (99). Feza Gürsey Institute, Istanbul, Turkey - 200

20 I. Nuclear Many-Body Problem Numerical Methods A New Generation TDHF Code Unrestricted 3-D Cartesian geometry Old version: Umar et al, Phys. Rev. C44, 252 (99) Basis-spline discretization for high accuracy Umar et al, J. Comp. Phys., 93, 426 (99) Coded in Fortran-95 Use of modern Skyrme forces with spin-orbit (SLy4, SkP, etc.) No time-reversal symmetry assumed Thread-safe, runs under OpenMP Umar et. al., Phys. Rev. C 73, (2006). Feza Gürsey Institute, Istanbul, Turkey - 200

21 Numerical Methods I. Nuclear Many-Body Problem Discrete Mathematics Basis-Spline Collocation Method Expand functions in B-splines N M f x = B k x c k k= Lattice-Collocation Method f x f ' [Of x ] O f ' ' b f x dx f a Lattice operators are given by Basis Splines of order M=5 with boundary conditions ' N O [OB k x ] B k ' k = Umar et al, J. Comp. Phys., 93, 426 (99) Feza Gürsey Institute, Istanbul, Turkey - 200

22 I. Nuclear Many-Body Problem Numerical Methods Basis-Spline Collocation Method - Derivation Expand functions in B-splines, discretize on collocation lattice N M k f x = B x c k= N k k f = B M c k k= Solve for expansion coefficients by inverting B k N on lattice c = [B k ] f f x f = Action of an operator on a function N substitute ck M k [Of x ]= [ OB k x ]c N k = Rewrite by defining collocation operator [Of x ] O ' f ' where N [Of x ] = [OB x ] [ B k ' ] f ' k = ' M k N ' O [OB k x ] B '= k ' k = Lattice integration defined in a similar way b f x dx f a with h k c k k Feza Gürsey Institute, Istanbul, Turkey - 200

23 I. Nuclear Many-Body Problem Numerical Methods Discretization of TDHF Equations - Outline Expand single-particle states in B-spline basis ijk x, y, z ; t = Bi x B j y B k z c t ijk Discretize on the collocation lattice before variation S= dt V { [ H i ℏ t ]} After variation local terms are local ' ' ' ' ' ' V = d 3 r 2= 2 Non-local terms look like (matrix-vector multiply) = D ' ' D ' ' j D ' ' k ' ' ' Feza Gürsey Institute, Istanbul, Turkey - 200

24 General Overview Presentation Presentation Overview ) Summary of traditional approaches for ion-ion potentials 2) Implementation of TDHF theory 3) TDHF and density constraint dynamical potentials 4) 3-α states of 8Be 5) Potentials and fusion cross-sections for selected systems 6) Dynamical calculation of excitation energy 7) Application to superheavy formations 8) Conclusions

25 General Overview Ion-Ion Potentials Many-Body Fusion and Fission No ab-initio many-body theory for sub-barrier fusion applicable to heavier nuclei exists R All approaches involve two prongs a) Calculate an ion-ion barrier (usually one-dimensional, V(R)) - Phenomenological (Wood-Saxon, Proximity, Folding, Bass, etc.) using frozen densities. - Microscopic, macroscopic-microscopic methods using collective variables (CHF, ATDHF, empirical methods). b) Employ quantum mechanical tunneling methods for the reduced one-body problem (WKB, IWBC) Incorporate quantum mechanical processes by hand a) Neutron transfer b) Few excitations of the entrance channel nuclei (CC)

26 General Overview Ion-Ion Potentials Traditional Microscopic Methods (CHF-ATDHF) Advantages - Fully microscopic, self-consistent description of nuclear potential energy surface (PES) - Use same microscopic interaction used in ground state. calculation - Gives global information on collective potential (collective subspace) - Include some correlations by restoring broken symmetries Shortcomings - Artificial introduction of constraining operators - Collective motion not necessarily confined in constrained phase space - Static adiabatic approximation - Most energetically favorable state may require sudden rearrangement - No reason why dynamical system should move along the valley of PES - CHF calculations seldom produce the correct saddle-point

27 General Overview Ion-Ion Potentials Phenomenological Fusion Barriers Most information taken from ground state properties of nuclei - These may be correct prior to nuclear overlap but different after Use frozen densities (folding potentials) - Typically Fermi densities fitted to experimental data Phenomenological heavy-ion interaction potentials - Wood-Saxon, Proximity, Bass, double-folding - Several free parameters Use few excited states (2-3) - Coupling potentials for excitations derived in simple models (rigid rotor, harmonic vibrator) - B(EL) values taken from experimental data Rotating frame approximation - To reduce the number of channels Allow for neutron transfer - Based on Q-values

28 General Overview Ion-Ion Potentials Fusion at Low Energies Frozen-density based approximations break down Inner part of the barrier is modified by many effects - Neck formation - Particle transfer - Dynamical rearrangement of the densities Actually different barriers are seen by individual states Similar (but not the same) phenomenon for very heavy systems

29 General Overview Ion-Ion Potentials Desired Improvements Explore collective dynamics in terms of mean-field dynamics - self-organizing system selects its evolutionary path by itself following the microscopic dynamics. Develop dynamical methods for selecting constraining operators - which are not known from the outset nor from the static theory - should it be coordinate or constraint? Go beyond the static adiabatic approximation - Explore nonlinear dynamics between single-particle degrees of freedom and collective motion by going beyond adiabatic approximation Diabatic states - Go beyond single determinant (shape coexistence)

30 Time-Dependent Hartree-Fock Dynamical Approach Initial TDHF Setup Generate HF Slater determinants for each nucleus TDHF equations are translationally invariant Multiply each determinant by a boost, determined from Coulomb trajectory and the asymptotic Ecm, at the initial nuclear separation A for nucleus-j j exp ik j R j and R= r i A j i= j Contains c.m. wavefunction! Combine two determinants into a single one TDHF initial state final state

31 II. Time-Dependent Hartree-Fock Initialization Definition of Final State Validity of mean-field approximation. All or most of Ecm can be transformed to internal excitation! Coulomb trajectory out / 2 = E cm E B Fermi-Gas Fermi-Liquid Coulomb trajectory in TDHF If final stage contains a single fragment FUSION If final stage contains two fragments DEEP INELASTIC SCATTERING Initial approach is determined by Coulomb interaction only

32 II. Time-Dependent Hartree-Fock Examples 6O+28O at Ecm = 43 MeV, SLy5, b = 7.5 fm σfusion = 96 mb

33 II. Time-Dependent Hartree-Fock Examples 6O+28O at Ecm = 43 MeV, SLy5, b = 7.6 fm

34 II. Time-Dependent Hartree-Fock Examples Deformed Nuclei Orientation Dependence Entrance channel Coulomb excitation Diagonal and off-diagonal J f Q2 J i Alignment = Quantum mechanical calculation R Umar, Oberacker, Phys. Rev. C 74, (2006)

35 II. Time-Dependent Hartree-Fock Examples Deformed Nuclei Effect on Fusion Distribution of barrier heights depending on orientation V(R) R Fusion cross-section fus E c.m. = 0 d cos fus E c.m. ; Can lead to enhancement of fusion cross section by orders of magnitude Generalized application in TDHF Umar, Oberacker, Phys. Rev. C 74, (2006)

36 Examples II. Time-Dependent Hartree-Fock 6 8 O + 220Ne (alignment ), Ecm= 95 MeV, b=0 fm

37 Examples 6 8 II. Time-Dependent Hartree-Fock O + 220Ne (alignment 2), Ecm= 95 MeV, b=0 fm

38 Examples 6 8 II. Time-Dependent Hartree-Fock O + 220Ne (alignment ), Ecm= 95 MeV, b=5 fm

39 II. Time-Dependent Hartree-Fock Examples 64Ni+32Sn, Ecm = 76 MeV, b = 4 fm, Deep-Inelastic L initial /ℏ=76.5 L final / ℏ=60.7 E final=42.0 MeV

40 II. Time-Dependent Hartree-Fock Examples 64Ni+32Sn, Ecm = 76 MeV, b = 5 fm, Deep-Inelastic L initial /ℏ=95.6 L final / ℏ=93.0 E final=72.3 MeV

41 Density Constraint and TDHF Dynamical Approach Density-Constraint Project unhindered TDHF evolution onto the dynamical PES - system selects its evolutionary path by itself - constrains all collective degrees of freedom A method to extract internal excitation energy while holding the instantaneous neutron and proton densities constrained. TDHF provides the dynamical densities for calculating V(R) Cusson, Reinhard, Maruhn, Strayer, Greiner, Z. Phys. A320, 475 (985) ρtdhf (r,t) E*(R(t)) Quasi Static Energy Surface

42 Density Constraint and TDHF 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:. Cusson, Reinhard, Maruhn, Strayer, Greiner, Z. Phys. A 320, 475 (985) 2. Umar, Strayer, Cusson, Reinhard, Bromley, Phys. Rev. C 32, 72 (985)

43 Density Constraint and TDHF Dynamical Approach Ion-Ion Potential Total energy in terms of the excitation energy is: E TDHF =T R V E * * V =E TDHF T R E =E DC Conserved quantity EDC contains the binding energies of the two nuclei V R E DC R E A E A Subtract binding energies 2 Asymptotically correct (no normalization needed): E DC R max =E A E A V Coulomb R max 2 V R max =V Coulomb R max Umar, Oberacker, Phys. Rev. C 74, 0260(R) (2006)

44 Density Constraint and TDHF 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 Coulomb tails always accurate to kev

45 Density Constraint and TDHF Dynamical Approach Dynamical Effective Mass 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 2 dr

46 Density Constraint and TDHF Triple Alpha Reaction Formation of 2C In the Universe (also 6O) 8 Be has 0-6s 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 2C (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. 04, (200) See Movie

47 Density Constraint and TDHF Triple Alpha Reaction Dynamics of Transition

48 Density Constraint and TDHF 64Ni Application to Fusion + 32Sn Fusion At low energies fusion x-section orders of magnitude larger than CC Prediction (Liang et al., PRL 9, 5270 (2003)) 3D HF (Sly5) gives oblate deformation for 64Ni: Qzz(n) = b, Qzz(p) = b 64Ni+32Sn, E cm = 76 MeV, SLy5, b = 3 fm

49 Density Constraint and TDHF 64Ni Application to Fusion + 32Sn Limiting Barriers Barrier for β=0o agrees with empirical No parameter/normalization in TDHF VB = 55.8 MeV RB = 2.2 fm Barrier for β=90o lower VB = 50.3 MeV RB = 2.87 fm Umar, Oberacker, Phys. Rev. C 74, 0660(R) (2006) Umar, Oberacker, Phys. Rev. C 76, 0464 (2007)

50 Density Constraint and TDHF 64Ni Application to Fusion + 32Sn Complete Set of Barriers Umar and Oberacker, Phys. Rev. C 76, 0464 (2007)

51 Density Constraint and TDHF 64Ni Application to Fusion + 32Sn Fusion Cross-Section Use IWBC Average over orientations f E c.m. = dcos P E c.m., 0 Umar and Oberacker, Phys. Rev. C 76, 0464 (2007) Exp. Data J.F. Liang et al., PRL 9, 5270 (2003) PRC 75, (2007)

52 Density Constraint and TDHF 64Ni Application to Fusion + 64Ni Fusion (β=90o, β2=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

53 Density Constraint and TDHF 64Ni Application to Fusion + 64Ni Limiting Barriers Variation with Euler angle αi is negligible! Umar, Oberacker, Phys. Rev. C 77, (2008)

54 Density Constraint and TDHF 64Ni Application to Fusion + 64Ni Fusion Cross-Section Problem at low energies (CC) Compression potential Mişicu, Esbensen, PRL 96, 270 (2006) Modify inner turning point Ichikawa, Hagino, Iwamoto, PRC 75, (2007) Exp. Data C.L. Jiang et al., PRL 93, 0270 (2004)

55 Density Constraint and TDHF Application to Fusion Conjecture - Skin versus Core Orientation Neutron rich systems have extended outer skins Orientation of core in 64Ni perpendicular to outer surface Top of the barrier primarily determined by outer surface Lower energies imply larger overlaps for inner turning point Ambiguity in which orientation to choose in angle averaging

56 Density Constraint and TDHF Application to Fusion For Low Energies use Core Orientation Umar, Oberacker, Phys. Rev. C 77, (2008)

57 Density Constraint and TDHF 6O Application to Fusion + 208Pb Fusion Cross-Section Umar, Oberacker, EPJA 39, 243 (2009)

58 Excitation Energy TDHF and Heavy-Ion Dynamics Excitation Energy Excitation energy is an important indicator of reaction dynamics Superheavy formations are very sensitive to excitation energy Indicator of temperature for compound configurations Traditional knowledge based on initial and final reaction products: E *=E c.m. Qgg Time evolution of excitation energy could tell us about the survival of intermediate configurations Density constraint makes this calculation possible via TDHF Umar, Oberacker, Maruhn, Reinhard, Phys.Rev. C 80, 0460(R) (2009)

59 TDHF and Heavy-Ion Dynamics Excitation Energy Excitation Energy via TDHF E * t =E TDHF E coll t, j t Excitation energy E TDHF t = d 3 r H r, t Total TDHF energy (conserved) E coll t =E kin t, j t E DC t Collective energy 2 j t m 3 E kin t, j t = d r 2 t Kinetic energy In terms of ion ion potential: E coll t =E kin t, j t V R t E A E A 2 Umar, Oberacker, Maruhn, Reinhard, Phys.Rev. C 80, 0460(R) (2009)

60 TDHF and Heavy-Ion Dynamics Excitation Energy Example Study for 40Ca+40Ca System Examine E* at the capture point Capture point Collective kinetic energy

61 TDHF and Heavy-Ion Dynamics Excitation Energy Compare with the Traditional Definition

62 TDHF and Heavy-Ion Dynamics Superheavy Systems Dynamics of Heavy Systems

63 Superheavy Systems TDHF and Heavy-Ion Dynamics 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

64 TDHF and Heavy-Ion Dynamics Superheavy Systems Cold and Hot Fusion of Heavy Systems 70 Zn+208Pb 48 Ca+238U (β=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

65 TDHF and Heavy-Ion Dynamics Superheavy Systems Excitation Energies at Capture Point Experimentally cited excitation energy E *exp=e c.m. Q gg We calculate excitation energy as a function of R(t)

66 TDHF and Heavy-Ion Dynamics Superheavy Systems Excitation Energies Should be alignment averaged: E * E c.m. = d sin P E * E c.m., 0

67 TDHF and Heavy-Ion Dynamics Superheavy Systems Potentials - Cores join - Capture - Cores remain distinct - Nucleons exchanged - b>0 deep-inelastic

68 TDHF and Heavy-Ion Dynamics Superheavy Systems Cross-Sections Angle average 238U alignment: - significantly reduces x-section f E c.m. = d sin P E c.m., 0 - x-section falls rapidly for β>0o - sin(β) multiply small angles - P(β) is in the range Experimental data (private communication):. Yu. Ts. Oganessian, Phys. Rev. C 70, (2004) 2. Yuri Oganessian, J. Phys. G: Nucl. Part. Phys. 34, R65 (2007)

69 TDHF and Heavy-Ion Dynamics Superheavy Systems Cross-Sections (tentative) Could not find data for capture x-section Calculated capture x-section (reproducing one σer x-section value.) G. Giardina, S. Hofmann, A.I. Muminov, and A.K. Nasirov, Eur. Phys. J. A 8, 205 (2000) Experiments:. S. Hofmann et al., Rev. Mod. Phys., 72, 733 (2000) 2. S. Hofmann et al., Eur. Phys. J. A 4, 47 (2002)

70 Microscopic Potentials based on TDHF 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.) Although heavy systems pose a greater challenge, such microscopic calculations may provide an insight into these collisions Effort needed to incorporate deformation and scattering information in to the Skyrme parametrization