Takashi NAKATSUKASA Theoretical Nuclear Physics Laboratory RIKEN Nishina Center Walter Kohns biography Bohr-Mottelson Kohn-Sham Walter Kohn Nobel Prize in Chemistry 1998 Harvard (Van Vleck, Schwinger) While I did not yet know in what subfield of physics I wanted to do my thesis, I was sure it would not be in solid state physics. Green's function variational method for low-energy neutron-deuteron scattering ( Failed?) QED, Field theory of nucleons and mesons ( Feel completely useless) Polaroid Laboratory charged particles falling on a photographic plate lead to a photographic image ( Solid-state physics, Van Vleck) Since you are familiar with solid state physics,..." Copenhagen No one, including Niels Bohr, had even heard the expression "Solid State Physics. Very exciting work was going on in Copenhagen, which eventually led to the great "Collective Model of the Nucleus" of A. Bohr and B. Mottelson, both of whom had become close friends. cole Normale Suprieure (Nozires) I knew that there was a 1-to-1 correspondence between a weak perturbing potential v(r) and the corresponding small change n(r) of the density distribution. It seemed such a remarkable result that I did not trust myself. Nucleonic single-particle motion in nucleus Neutron # N Neutron Separation energy Bohr & Mottelson, Nuclear Structure Vol.1 Mayer-Jensens Shell Model Harmonic oscillator potential + spin-orbit force Correct magic numbers. Nucleons as free gas in the potential. Nucleus is gas? Liquid-drop Model Bethe-Weizscker mass formula Binding energy B/A 8 MeV Density 0.14 fm -3 d 2 fm Nuclear fission Bohr & Mottelson Nuclear Structure Vol.1 Nucleus as a quantum liquid Classical vs Quantum Strength of interaction vs Zero-point kinetic energy Nuclear force vs molecular force Bohr, Mottelson, Nucl. Str. Vol.1 Crystallized at low temperature Classical MD Liquid at low temperature Quantum Collective (Unified) Model Nucleons are independently moving in a potential that slowly changes. Collective motion oscillation/rotation of the potential. The fluctuation of the potential changes the nucleonic single-particle motion. In modern approaches, microscopic construction by Bohr & Mottelson Time-dependent density functional theory Nucleons are independently moving in a potential that depends on nucleon densities Collective motion Density oscillation/rotation The density oscillation induces oscillation of the potential that affects the nucleonic motion. ( ) Identical to the time-dependent mean-field theory Kohn-Sham Hohenberg-Kohn, Kohn-Sham, Runge-Gross exact GeV ~ keV Existence of one-to-one mapping Hohenberg-Kohn Theorem A system with a one-body potential The theorem tells us that Density (r) determines v(r), Strictly speaking, one-to-one or one-to-none v-representative except for arbitrary choice of zero point. Hohenberg & Kohn (1964) Kohn-Sham Scheme: Ground state TD state TD density TD state TD density Real interacting system Virtual non-interacting system Kohn-Sham theory Assuming non-interacting v-representability Since we know the ground state of the non-interacting system (Slater det.), we obtain the exact density as solving the Kohn-Sham (KS) equation Ab-inito Energy functional 500 keV Odd-even mass difference (Pairing channel functional ) Filling approx. Jaguar Cray XT4 at ORNL No. 2 on Top500 11,706 processor nodes Each compute/service node contains 2.6 GHz dual-core AMD Opteron processor and 4 GB/8 GB of memory Peak performance of over 119 Teraflops 250 Teraflops after Dec.'07 upgrade 600 TB of scratch disk space Collective Hamiltonian in 5-dim. quadrupole collective coordinates constructed by the constrained HFB calculation J k (a 0, a 2 ): moment of inertia Thouless-Valatin ZPE pot neglected B mn (a 0, a 2 ): collective mass (vibration) Cranking D(a 0, a 2 ): metric Delaroche et al, 2009 Pack Forest Meeting Gogny-HFB Time-dependent density functional theory (3D lattice simulation for Skyrme functionals) Mostly the functional is local in density Appropriate for coordinate-space representation Kinetic energy, current densities, etc. are estimated with the finite difference method Time-dependen Kohn-Sham Scheme TD state TD density TD state TD density Real interacting system Virtual non-interacting system Time-dependent Kohn-Sham theory Assuming non-interacting v-representability Solving the TDKS equation, in principle, we can obtain the exact time evolution of many-body systems. The functional depends on (r,t and the initial state 0. Time-dependent Kohn-Sham (TDKS) equation 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 [ MeV ] Neutrons Protons > 0 < 0 16 O Time-dep. transition density E x [ MeV ] O 16 O Prolate Finite Amplitude Method A method to avoid the explicit calculation of the residual fields (interactions) Residual fields can be estimated by the finite difference method: (1) Programming of the RPA code becomes very much trivial, because we only need calculation of the single-particle potential, with different bras and kets. T.N., Inakura, Yabana, PRC76 (2007) Starting from initial amplitudes X (0) and Y (0), one can use an iterative method to solve eq. (1). Skyrme FAM in 3D real space x z 3D space is discretized in lattice F & B amplitudes: N: Number of particles Mr: Number of mesh points ME: Number of energy points Linear response equations Spatial mesh size is about 0.8 fm. Energy mesh size is about 0.3 MeV Inakura, Nakatsukasa, Yabana, arXive: y Electric dipole responses Finite amplitude method to Skyrme-HF+RPA SkM* interaction 3D mesh R box = 15 fm Inakura, Nakatsukasa, Yabana, arXive: 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 PDR: impact on the r-process S. Goriely, Phys. Lett. B436, 10. E x [ MeV ] E x [ MeV ] Mg 26 Mg Prolate Triaxial E x [ MeV ] E x [ MeV ] Si 30 Si Oblate E x [ MeV ] Ar Oblate 40 Ca 44 Ca 48 Ca E x [ MeV ] E x [ MeV ] E x [ MeV ] Prolate Cal. vs. Exp. Summary Kohn-Sham