1. E1 strength distribution in even-even nuclei studied with the time-dependent density functional calculations Takashi NAKATSUKASA Theoretical Nuclear Physics Laboratory RIKEN Nishina Center 2008.9.25-26 Workshop “New Era of Nuclear Physics in the Cosmos” Mass, Size, Shapes → DFT (Hohenberg-Kohn) Dynamics, response → TDDFT (Runge-Gross)

2. Basic equations Time-dep. Schroedinger eq. Time-dep. Kohn-Sham eq. dx/dt = Ax Energy resolution ΔE 〜 ћ/T All energies Boundary Condition Approximate boundary condition Easy for complex systems Basic equations Time-indep. Schroedinger eq. Static Kohn-Sham eq. Ax=ax (Eigenvalue problem) Ax=b (Linear equation) Energy resolution ΔE 〜 0 A single energy point Boundary condition Exact scattering boundary condition is possible Difficult for complex systems Time Domain Energy Domain

3. How to incorporate scattering boundary conditions ? It is automatic in real time ! Absorbing boundary condition γ n A neutron in the continuum

4. For spherically symmetric potential Phase shift Potential scattering problem

5. Scattering wave Time-dependent picture (Initial wave packet) (Propagation) Time-dependent scattering wave Projection on E :

6. Boundary Condition Finite time period up to T Absorbing boundary condition (ABC) Absorb all outgoing waves outside the interacting region How is this justified? Time evolution can stop when all the outgoing waves are absorbed.

7. s-wave absorbing potential nuclear potential

8. 3D lattice space calculation Skyrme-TDDFT Mostly the functional is local in density →Appropriate for coordinate-space representation Kinetic energy, current densities, etc. are estimated with the finite difference method

9. Skyrme TDDFT in real space X [ fm ] y [ fm ] 3D space is discretized in lattice Single-particle orbital: N: Number of particles Mr: Number of mesh points Mt: Number of time slices Time-dependent Kohn-Sham equation Spatial mesh size is about 1 fm. Time step is about 0.2 fm/c Nakatsukasa, Yabana, Phys. Rev. C71 (2005) 024301

10. Real-time calculation of response functions 1.Weak instantaneous external perturbation 2.Calculate time evolution of 3.Fourier transform to energy domain ω [ MeV ]

11. Nuclear photo- absorption cross section (IV-GDR) 4 He E x [ MeV ] 0 50 100 Skyrme functional with the SGII parameter set Γ=1 MeV Cross section [ Mb ]

12. E x [ MeV ] 10402030 E x [ MeV ] 10402030 14 C 12 C

13. E x [ MeV ] 10 4020 30 18 O 16 O Prolate

14. E x [ MeV ] 10402030 E x [ MeV ] 10402030 24 Mg 26 Mg Prolate Triaxial

15. E x [ MeV ] 10402030 E x [ MeV ] 10402030 28 Si 30 Si Oblate

16. E x [ MeV ] 10402030 E x [ MeV ] 10402030 32 S 34 S ProlateOblate

17. E x [ MeV ] 10 40 2030 40 Ar Oblate

18. 40 Ca 44 Ca 48 Ca E x [ MeV ] 10402030 E x [ MeV ] 10402030 E x [ MeV ] 102030 Prolate

19. Cal. vs. Exp.

20. Electric dipole strengths SkM* R box = 15 fm  = 1 MeV Numerical calculations by T.Inakura (Univ. of Tsukuba) Z N

21. Peak splitting by deformation 3D H.O. model Bohr-Mottelson, text book.

22. He O Si Be Ne Si C Mg Centroid energy of IVGDR Ar Fe Ti Cr Ca

23. Low-energy strength

24. Low-lying strengths Be C He O Ne MgSi S Ar Ca Ti Cr Fe Low-energy strengths quickly rise up beyond N=14, 28

25. Summary Small-amplitude TDDFT with the continuum Fully self-consistent Skyrme continuum RPA for deformed nuclei Theoretical Nuclear Data Tables including nuclei far away from the stability line → Nuclear structure information, and a variety of applications; astrophysics, nuclear power, etc. Photoabsorption cross section for light nuclei Qualitatively OK, but peak is slightly low, high energy tail is too low For heavy nuclei, the agreement is better.