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Time-dependent Hartree Fock with full Skyrme Forces in 3 Dimensions

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

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Time-dependent Hartree Fock with full Skyrme Forces in 3 Dimensions

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  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 16O+16O: • 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 12C + 12C 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 16O+16O Reaction in Three Dimensions“, Phys. Rev. Lett. 36, 1166 (1976). • J.A. Maruhn and R.Y. Cusson, „Time-Dependent Hartree-Fock Calculation of 12C + 12C with a Realistic Potential“, Nucl. Phys. A270, 471 (1976). • R.Y. Cusson, J.A. Maruhn, and H.W. Meldner, „Direct Inelastic Scattering of 14N+12C 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 The wave functions have periodic boundary conditions, but for the Coulomb filed interaction with images must be avoided 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) (fictitious) empty space

  6. 16O+48Ca slightly below barrier

  7. 16O+48Ca slightly above barrier

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

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

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

  11. Heavy Systems: 48Ca+208PbImportant 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. 48Ca + 208Pb sligtly above barrier

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

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

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

  16. Deformed partners: 20Ne+20Ne

  17. A curious case: 12C+16O

  18. Energy Loss in 16O+16O • 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 fulfillingleads 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 themean-field Hamiltonian • In the Skyrme energy functional Galilei invaraince requires adding terms likeand similar terms with different isospin dependence. • This leads to contributions in the mean-field Hamiltonian likewithThe 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 alongaxis of largest moment of inertia • If density is low enough, definedividing plane • Determine c. m. distance Rof fragments and ist time derivative • Get relative motion kinetic energy from for central collisions • Point-charge Coulomb energy agrees with full calculation toabout 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-Termsin Central 16O+16O Collisions

  25. The Mechanism

  26. L*S Energy in Central Collision

  27. Impact Parameter Dependence

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

  29. Comparison with previous results

  30. Noncentral Results at Ecm=100 MeV

  31. Late-time behavior shows severe problem!

  32. Use new plot

  33. 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 d-pairing • The solution of the time-dependent Hartree-Fock Bogolyubov problem therefore may have to be attempted

  34. 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.

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