1. Time-dependent Hartree Fock with full Skyrme Forces in 3 Dimensions Collaborators: P.-G. Reinhard, U. Erlangen-Nürnberg P. D. Stevenson, U. Surrey, Guildford Topics  The code  Qualitative explorations  Energy loss in 16 O+ 16 O:  Effect of full Skyrme and 3D  The spin excitation mechanism  Accuracy of relative motion energy

2. TDHF in the late `70s  Computer facilities: The 3D code was run on an IBM „supercomputer“ 360/195 with 1MB of memory!  Therefore: no spin, simplified interaction: BKN or g- matrix  Really very few checks of accuracy (!?)  R.Y. Cusson and J.A. Maruhn, „Dynamics of 12 C + 12 C in a Realistic T.D.H.F. Model“, Phys. Lett. 62B, 134 (1976).  R.Y. Cusson, R.K. Smith, and J.A. Maruhn, „Time-dependent Hartree-Fock Calculation of the 16 O+ 16 O Reaction in Three Dimensions“, Phys. Rev. Lett. 36, 1166 (1976).  J.A. Maruhn and R.Y. Cusson, „Time-Dependent Hartree-Fock Calculation of 12 C + 12 C with a Realistic Potential“, Nucl. Phys. A270, 471 (1976).  R.Y. Cusson, J.A. Maruhn, and H.W. Meldner, „Direct Inelastic Scattering of 14 N+ 12 C in a Three- Dimensional Time-Dependent Hartree-Fock Scheme'', Phys. Rev. C18, 2589 (1978).  C.Y. Wong, J.A. Maruhn, and T.A. Welton, „Comparison of Nuclear Hydrodynamics and Time- Dependent-Hartree-Fock Results“. Phys. Lett. 66B, 19 (1977).

3. The New TDHF Code  Three-dimensional Skyrme-force Hartree-Fock, both static and time-dependent  Differencing based on Fast-Fourier-Transform;  Grid spacing typically 1 fm  All variations of modern Skyrme forces can be treated fully  Fourier treatment of Coulomb allows correct solution for isolated charge distribution  Coded fully in Fortran-95  TDHF version can run on message-passing parallel machines

4. The Skyrme Energy Functional

5. Fourier calculation of potential for isolated charge distributions (fictitious) empty space The solution constructed via with two FFToperations in the enlarged region with periodic boundary conditions fulfills the boundary condition for an isolated charge distribution in the physical region J.W. Eastwood and D.R.K. Brownrigg, J. Comp. Phys. 32, 24 (1979) The wave functions have periodic boundary conditions, but for the Coulomb filed interaction with images must be avoided

6. 16 O+ 48 Ca slightly below barrier

7. 16 O+ 48 Ca slightly above barrier

8. 16 O + 48 Ca, Boost 0.3 MeV / nucleon, t=0..450 fm/c

9. 16 O + 48 Ca, Boost 0.3 MeV / nucleon, t=500..950 fm/c

10. Mass Moments in 16 O+ 48 Ca normalized to initial value

11. Heavy Systems: 48 Ca+ 208 Pb Important for Superheavy Element Formation!  Does the interaction dynamics differ dramatically from light system?  12 fm initial distance  4000 time steps of 0.25 fm/c : 1000 fm/c total  Initial boost just sufficient to cause interaction  Needs longer times and systematic variation in boost

12. 48 Ca + 208 Pb sligtly above barrier

13. 48 Ca+ 208 Pb near barrier, t=0..450 fm/c

14. 48 Ca+ 208 Pb near barrier, t=500..950 fm/c

15. Mass Moments in 48 Ca+ 208 Pb normalized to initial value

16. Deformed partners: 20 Ne+ 20 Ne

17. A curious case: 12 C+ 16 O

18. Energy Loss in 16 O+ 16 O  Past experience shows that relaxing symmetries increases the dissipation  Spin orbit coupling is essential for correct shell structure!  Few calculations performed at that time show increased dissipation due to relaxation of symmetries  Now examine energy loss aspects in new directions: Accuracy Effect of 3-D and full modern Skyrme forces Role of time-odd parts in the s.p. Hamiltonian

19. Changes in results  The dissipation is generally increased when symmetries are relaxed and new degrees of freedom enter A.S. Umar, M.R. Strayer, and P.-G. Reinhard, Phys. Rev. Lett. 56, 2793 (1986).

20. Translational Invariance of T.D.H.F.  A ground state nucleus with s.p. wave functions fulfilling leads to a propagating stationary solution with a common phase factor This solves the time-dependent equation (i.e., produces a uniformly translating nucleus), if the s.p. Hamiltonian is Galilei invariant  This is trivial for pure density dependence, but requires adding terms involving currents and spin currents to the density functional (Y. M. Engel, D. M. Brink, K. Goeke, S. J. Krieger, and D. Vautherin, Nucl. Phys. A249, 215 (1975)).

21. The time-odd spin-orbit terms in the mean-field Hamiltonian  In the Skyrme energy functional Galilei invaraince requires adding terms like and similar terms with different isospin dependence.  This leads to contributions in the mean-field Hamiltonian like with The spin-orbit part of these contributions was usually neglected (and is negligible for giant-resonance-type calculations)

22. Determination of relative motion energy  Find minimum of density along axis of largest moment of inertia  If density is low enough, define dividing plane  Determine c. m. distance R of fragments and ist time derivative  Get relative motion kinetic energy from for central collisions  Point-charge Coulomb energy agrees with full calculation to about 0.02 MeV  Accuracy in „trivially“ conserved quantities: total energy 0.1 MeV, particle number 0.01

23. Initial Relative Motion Energy Omission of time-odd l*s terms leads to translational noninvariance of surprisingly strong consequences!

24. Importance of Time-Odd L*S-Terms in Central 16 O+ 16 O Collisions

25. The Mechanism

26. L*S Energy in Central Collision

28. Impact Parameter Dependence

29. Force dependence of reactions: a dynamic test for Skyrme forces J. A. Maruhn, K. T. R. Davies, M. R. Strayer, Phys. Rev. C31 1289 (1985)

30. Comparison with previous results

31. Noncentral Results at E cm =100 MeV

32. Late-time behavior shows severe problem!

33. Use new plot

34. Problem : Pairing  Without pairing, the deformations are still not quantitative and the moments of inertia will be wrong, unless pairing is destroyed rapidly (?)  “Old” calculations did not include pairing, because the BCS formalism with state-independent pairing matrix element produces interaction even between separated fragments  Newer formulations of pairing generate matrix elements from a force such as a  -pairing  The solution of the time-dependent Hartree-Fock Bogolyubov problem therefore may have to be attempted

35. Conclusions  The use of full Skyrme forces brings surprising new effects and problems.  The is a new energy loss mechanism involved with a „spin-twist excitation“  There are problems with a continued loss of relative motion energy for separated fragments, possibly due to cross-boundary interactions. More computational expense may be needed or one has to live with 3 MeV uncertainty.  The energy loss appears to stabilize for several forces  It will be interesting to see how these effects persist in heavier systems.