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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
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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
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2. TDHFB equation
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Equations of motion of matrices U & V see Ring & Schuck
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Coulomb part is NOT included Gogny-D1S ・basis function:three-dimensional harmonic oscillator wave functions ・space: Gauss part density dependent part L-S part
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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 )
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Energy conservation tdhf
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3. Linear region
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Example: 20 O quadrupole oscillation (small amplitude)
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* 34 – 38 Mg quadrupole (K=0) mode * 18– 22 O quadrupole mode * 44,50,52,54 Ti quadrupole mode
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4-1.Nonlinear region (oscillation type)
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quadrupole oscillation and pairing 52 Ti prolateoblate pairing is zero
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HF “pocket” initial conditions
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52 Ti
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occupation probability in orbital(k) UVUV () k k definition : HFB matrix α : numerical basis label
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initial condition: Q20 = 0 fm^2 (impulse) initial condition: Q20 = 140 fm^2 (constraint)
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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
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( f7/2 members in initial stage) quadrupole moment (fm^2) single-particle energies vs Q 20 0100200 time ( fm ) 44 Ti vibration Fermi energy
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( f7/2 members in initial stage) quadrupole moment (fm^2) single-particle energies vs Q 20 0100200 time ( fm )
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occupation probability p(k) (protons) Time (fm) HFB energies (MeV) HFB eigen energies (MeV)
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4-2. Nonlinear region (relaxation type)
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Q20 (fm ) 2 44 Ti Energy vs Q20 Energy (MeV) 2000 4000 Time (fm) occupation probabilities p(k) ( neutron, minus parity) 0
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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
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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
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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
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