1. Gogny-TDHFB calculation of nonlinear vibrations in 44,52 Ti Yukio Hashimoto Graduate school of pure and applied sciences, University of Tsukuba 1.Introduction 2.TDHFB equation 3.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. Coulomb part is NOT included Gogny-D1S ・basis function：three-dimensional harmonic oscillator wave functions ・space： Gauss part density dependent part L-S part

6. Q 0 ： matrix representation of multipole operator initial U & V HFB ground state U, V initial conditions: ・ Q 20 type impulse on ground state （ impulse type ） ・ constrained state with quadrupole operator （ constraint type ）

7. Energy conservation tdhf

8. 3. Linear region

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

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

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

12. quadrupole oscillation and pairing 52 Ti prolateoblate pairing is zero

13. HF “pocket” initial conditions

14. 52 Ti

15. occupation probability in orbital(k) UVUV () k k definition ： 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 = 140 fm^2 initial condition: Q20 = 0 fm^2 (impulse) initial condition: Q20 = - 165 fm^2

18. ( f7/2 members in initial stage) quadrupole moment (fm^2) single-particle energies vs Q 20 0100200 time （ fm ） 44 Ti vibration Fermi energy

19. ( f7/2 members in initial stage) quadrupole moment (fm^2) single-particle energies vs Q 20 0100200 time （ fm ）

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

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

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

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

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

25. summary 1.(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