1. 9-10-2008 21:12 (00) Below-barrier paths: multimodal fission & doughnut nuclei A. Staszczak (UMCS, Lublin) FIDIPRO-UNEDF collaboration meeting on nuclear energy-density-functional methods, Jyväskylä, 9-11 Oct. 2008

2. 9-10-2008 21:12 (00) Model The self-consistent HF+BCS equations are solved using the code HFODD v.2.35 that uses the basis expansion method in a 3D Cartesian-deformed HO basis. http://www.fuw.edu.pl/~dobaczew/hfodd/hfodd.html The s.p. basis consists of the lowest 1140/1771 stretched HO states originating from the 31 major oscillator shells. The Skyrme functional SkM* is used in the particle-hole channel and a seniority pairing force is taken in the particle-particle channel.

3. 9-10-2008 21:12 (00) The symmetry reflection planes The self-consistent symmetries:  parity  signature  simplex  time-reversal mass symmetric fission mass asymmetric fission

4. 9-10-2008 21:12 (00) Multimodal fission LORW* ) group * ) LORW group: A. Baran, A. S. (Lublin) W. Nazarewicz (Oak Ridge) J. Dobaczewski (Warszawa )

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8. asymmetric fission (aEF) bimodal fission (sCF & sEF) compact-symmetric fission (sCF)

9. 154 156 158 160 98 100 102 104 106 N Z

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11. Doughnut nuclei C. Y. Wong, A. S.

12. 9-10-2008 21:12 (00) Boltzmann-Nordheim-Vlasov (BNV), Boltzmann-Uehling-Uhlenbeck (BUU) kinetic transport models and Monte Carlo simulations … A. Sochocka, R. Płaneta, N.G. Nicolis, Acta Phys. Pol. B 39, 405 (2008). A. Sochocka et al., Int. J. Mod. Phys. E17,190 (2008). Hartree-Fock-Bogoliubov (HFB) theory with the Gogny D1S force M. Warda, Int. J. Mod. Phys. E16, 452 (2007). Semiclassical extended Thomas-Fermi (ETF) model with the Skyrme SkM* force X. Viñas, M. Centelles, M. Warda, Int. J. Mod. Phys. E17,177 (2008). Liquid drop model (LDM) with Strutinsky shell corrections J.A. Wheeler (unpublished). C.Y. Wong, Phys. Lett. 41B, 446 (1972). C.Y. Wong, Ann. of Phys. (NY) 77, 279 (1973). C.Y. Wong, Proc. Inter. Symp., Lubbock, 1978, (Pergamon Press, 1979), p. 524.

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14. d Potential energy curves for toroidal nuclei

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17. 50 MeV

18. 9-10-2008 21:12 (00) 25 MeV

19. 9-10-2008 21:12 (00) 12 MeV

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24. KONIEC

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45. The constrained HF procedure:  The constraints act as the external fields capable to deform the nucleus in different ways  The collective coordinates can be defined in a natural way by measuring the deformations generated by the various constraints The constrained mean field theory defines the deformed states (BCS- or HFB-type) that solve the variational equation: with the constraint conditions quadratic multipole constraints The multipole constraints prescribe different kinds of deformation characterized by the set of parameters

46. 9-10-2008 21:12 (00) is the many-body nuclear (non-relativistic) Hamiltonian center-of-mass “projection” term (in the VAP technique), to eliminate spurious mode associated with the broken translational symmetry with the broken translational symmetry nuclear effective interaction term (Skyrme, Gogne type forces) To describe the fission process most “important” are the low-multipolarity mass moments, i.e., “nuclear stretching” “reflection-asymmetry” “necking”

47. 9-10-2008 21:12 (00) J. Dudek, et al., J. Phys. G6(1980)447. Seniority pairing: f n = 1.28, f p = 1.11 (for SkM * Skyrme force) In pairing (BCS) window N (or Z) s.p. states are taken, f n/p parameters are chosen to reproduce pairing gaps ∆ n/p for 252 Fm. J. Bartel, et al., Nucl. Phys. A386(1982)79.