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

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

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

For spherically symmetric potential Phase shift Potential scattering problem

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

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.

s-wave absorbing potential nuclear potential

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

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)

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

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

E x [ MeV ] E x [ MeV ] C 12 C

E x [ MeV ] O 16 O Prolate

E x [ MeV ] E x [ MeV ] Mg 26 Mg Prolate Triaxial

E x [ MeV ] E x [ MeV ] Si 30 Si Oblate

E x [ MeV ] E x [ MeV ] S 34 S ProlateOblate

E x [ MeV ] Ar Oblate

40 Ca 44 Ca 48 Ca E x [ MeV ] E x [ MeV ] E x [ MeV ] Prolate

Cal. vs. Exp.

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

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

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

Low-energy strength

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

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.