Gogny-TDHFB calculation of nonlinear vibrations in 44,52 Ti

1. Gogny-TDHFB calculation of nonlinear vibrations in 44,52 Ti Yukio Hashimoto Graduate school of pure and applied sciences, University of Tsukuba • Introduction • TDHFB equation • Linear region • 4-1. Nonlinear region (vibration type) • 4-2. Nonlinear region (relaxation type) • 5. summary

2. 1. Introduction ☆ random phase approximation (RPA) on a large scale T. Inakura, from “Report of KEK Ohgata Simulation Program (2010)” ☆ S. Ebata et al., Phys. Rev. C 82 (2010), 034306. “canonical-basis TDHFB” with Skyrme force ☆ in this talk, Gogny force is used in TDHFB calculations Gogny force: ph channel  pp channel role of pairing correlation in vibration / relaxation

3. 2. TDHFB equation

4. Equations of motion of matrices U & V see Ring & Schuck

5. Gogny-D1S Gauss part density dependent part L-S part Coulomb part is NOT included ・basis function：three-dimensional harmonic oscillator wave functions ・space：

6. initial conditions: 　・Q20 type impulse on ground state（impulse type） 　・constrained state with quadrupole operator（constraint type） initial U & V HFB ground state U, V Q0：matrix representation of multipole operator

7. Energy conservation tdhf

8. 3. Linear region

9. Example:20O quadrupole oscillation (small amplitude)

10. ＊ 18– 22O quadrupole mode ＊ 34 – 38Mg quadrupole (K=0) mode ＊ 44,50,52,54Ti quadrupole mode

11. 4-1.Nonlinear region(oscillation type)

12. quadrupole oscillation and pairing52Ti pairing is zero oblate prolate

13. initial conditions HF “pocket”

14. 52Ti

15. U V ( ) k k definition occupation probability in orbital(k) ：HFB matrix α：numerical basis label

16. initial condition: Q20 = 0 fm^2 (impulse) initial condition: Q20 = 140 fm^2 (constraint)

17. initial condition: Q20 = 140 fm^2 initial condition: Q20 = - 165 fm^2 initial condition: Q20 = 0 fm^2 (impulse) initial condition: Q20 = 140 fm^2

18. 44Ti vibration ( f7/2 members in initial stage) single-particle energies vs Q20 time （fm） Fermi energy 0 100 200 quadrupole moment (fm^2)

19. ( f7/2 members in initial stage) single-particle energies vs Q20 time （fm） 0 100 200 quadrupole moment (fm^2)

20. occupation probability p(k) (protons) HFB eigen energies (MeV) HFB energies (MeV) Time (fm)

21. 4-2. Nonlinear region(relaxation type)

22. occupation probabilities p(k) ( neutron, minus parity) 44Ti Energy vs Q20 Energy (MeV) 2 Q20 (fm ) 4000 0 2000 Time (fm)

23. relaxation of quadrupole oscillation（ 44Ti ） 2 fm occupation probability p(k) (protons) time （fm） Fermi energy occupation probability p(k) single particle energy （MeV） time（fm） quadrupole moment

24. relaxation of quadrupole oscillation（ 44Ti ） occupation probability p(k) (protons) time （fm） occupation probability p(k) single particle energy （MeV） time（fm） quadrupole moment 2 fm

25. summary (small amplitude case) RPA linear response  strength functions 2. (nonlinear case) i) long period oscillation accompanied with “adiabatic” configuration around single-particle level crossing region ii) relaxation together with adiabatic configuration across single-particle level crossing