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

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), “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

2. TDHFB equation

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

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

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 ）

Energy conservation tdhf

3. Linear region

Example: 20 O quadrupole oscillation (small amplitude)

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

4-1.Nonlinear region (oscillation type)

quadrupole oscillation and pairing 52 Ti prolateoblate pairing is zero

HF “pocket” initial conditions

52 Ti

occupation probability in orbital(k) UVUV () k k definition ： HFB matrix α ： numerical basis label

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

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

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

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

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

4-2. Nonlinear region (relaxation type)

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

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

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

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