Giant resonances in the Skyrme-Hartree-Fock theory P .G. Reinhard - PowerPoint PPT Presentation

Giant resonances in the Skyrme-Hartree-Fock theory P .–G. Reinhard Institut für Theoretische Physik II Universität Erlangen-Nürnberg COMEX05, Krakow 2015 P .–G. Reinhard (Inst.Theor.Physik, Erlangen) Giant resonances in the Skyrme-Hartree-Fock theory COMEX05, Krakow 2015 1 / 36

Acknowledgements Collaborators: W. Nazarewicz, B. Schütrumpf MSU East Lansing J. Erler, P . Klüpfel formerly Univ. Erlangen J. Dobaczewski York (UK), Jyväskyla J. A. Maruhn Frankfurt P . Stevenson Surrey V. Nesterenko, W. Kleinig Dubna J. Speth, S. Krewald Jülich N. Lyutorivich, V. Tselyaev St. Petersburg Support : BMBF contract 05P12RFFTG, DFG Re-322/12-1 P .–G. Reinhard (Inst.Theor.Physik, Erlangen) Giant resonances in the Skyrme-Hartree-Fock theory COMEX05, Krakow 2015 2 / 36

Outline Formal framework 1 The Skyrme energy-density functional Observables Optimization of model parameters by least-squares fits Results 2 RPA: convergence and 2ph effects Giant resonances and nuclear matter parameters (NMP) GDR – trends with mass number Isovector dipole strength at low E Conclusions 3 P .–G. Reinhard (Inst.Theor.Physik, Erlangen) Giant resonances in the Skyrme-Hartree-Fock theory COMEX05, Krakow 2015 3 / 36

Formal framework P .–G. Reinhard (Inst.Theor.Physik, Erlangen) Giant resonances in the Skyrme-Hartree-Fock theory COMEX05, Krakow 2015 4 / 36

The Skyrme energy-density functional (here only time even densities) � d 3 r E Skyrme ( ρ, τ, J ) E tot = ✻ ρ ( r ) = � α v 2 α | ϕ α | 2 density ρ 2 ρ 2 1 2 B ′ 1 2 B 0 + ˜ 0 τ ( r ) = � α v 2 α |∇ ϕ α | 2 kinetic density 2 B 3 ρ 2 + α 1 1 2 B ′ ρ 2 ρ α + + ˜ J ( r ) = − i � α v 2 3 α ϕ † spin-orbit dens. α ∇× σϕ α B ′ + B 1 ρτ + ρ ˜ ˜ τ 1 1 2 B 2 ( ∇ ρ ) 2 1 2 B ′ ρ ) 2 + + 2 ( ∇ ˜ ρ ∇ ˜ 1 1 2 B ′ + 2 B 4 ρ ∇ J + ˜ J total & difference ρ = ρ n + ρ p , ˜ ρ = ρ n − ρ p 4 � �� � � �� � isoscalar isovector isoscalar isovector P .–G. Reinhard (Inst.Theor.Physik, Erlangen) Giant resonances in the Skyrme-Hartree-Fock theory COMEX05, Krakow 2015 5 / 36

The Skyrme energy-density functional (here only time even densities) � � d 3 r E pair ( χ, ρ ) + E Coul − E corr ✛ d 3 r E Skyrme ( ρ, τ, J ) + E tot = E kin + ✻ ✻ ✻ correlations from ✻ low energy modes: p 2 | ϕ α ) ( ϕ α | ˆ � c.m., rotation, vibrat. 2 m N α Coulomb en. (exchange = Slater appr.) kinetic energy � � � � pairing functional ρ V pair χ 2 p + V pair χ 2 1 − only surface effects p n n ρ pair to define open shell nuclei ρ ( r ) = � α v 2 α | ϕ α | 2 density ρ 2 ρ 2 1 2 B ′ 1 2 B 0 + ˜ 0 τ ( r ) = � α v 2 α |∇ ϕ α | 2 kinetic density 1 2 B 3 ρ 2 + α 1 2 B ′ ρ 2 ρ α + + ˜ J ( r ) = − i � α v 2 3 α ϕ † spin-orbit dens. α ∇× σϕ α B ′ + B 1 ρτ + ρ ˜ ˜ τ 1 1 2 B 2 ( ∇ ρ ) 2 2 B ′ 1 ρ ) 2 + + 2 ( ∇ ˜ ρ ∇ ˜ 1 2 B ′ 1 + 2 B 4 ρ ∇ J + ˜ J total & difference ρ = ρ n + ρ p , ˜ ρ = ρ n − ρ p 4 � �� � � �� � isoscalar isovector isoscalar isovector P .–G. Reinhard (Inst.Theor.Physik, Erlangen) Giant resonances in the Skyrme-Hartree-Fock theory COMEX05, Krakow 2015 5 / 36

The Skyrme energy-density functional (here only time even densities) � � d 3 r E pair ( χ, ρ ) + E Coul − E corr ✛ d 3 r E Skyrme ( ρ, τ, J ) + E tot = E kin + ✻ ✻ ✻ correlations from ✻ low energy modes: p 2 | ϕ α ) ( ϕ α | ˆ � c.m., rotation, vibrat. 2 m N α Coulomb en. (exchange = Slater appr.) kinetic energy � � � � pairing functional ρ V pair χ 2 p + V pair χ 2 1 − only surface effects p n n ρ pair to define open shell nuclei ρ ( r ) = � α v 2 α | ϕ α | 2 density ρ 2 ρ 2 1 1 2 B ′ 2 B 0 + ˜ 0 τ ( r ) = � α v 2 α |∇ ϕ α | 2 kinetic density 2 B 3 ρ 2 + α 1 2 B ′ 1 ρ 2 ρ α + + ˜ J ( r ) = − i � α v 2 3 α ϕ † spin-orbit dens. α ∇× σϕ α B ′ + B 1 ρτ + ρ ˜ ˜ τ χ ( r ) = � α u α v α | ϕ α | 2 1 pair density 2 B 2 ( ∇ ρ ) 2 1 1 2 B ′ ρ ) 2 + + 2 ( ∇ ˜ pairing amplit. u α , v α ρ ∇ ˜ 1 2 B ′ 1 + 2 B 4 ρ ∇ J + ˜ J total & difference ρ = ρ n + ρ p , ˜ ρ = ρ n − ρ p 4 � �� � � �� � isoscalar isovector isoscalar isovector B 0 , B ′ 0 , B 1 , B ′ 1 , B 2 , B ′ 2 , B 3 , B ′ 4 , V pair , V pair B 4 , B ′ free parameters: 3 , α, , ρ pair p n � �� � ↔ nuclear matter parameters (NMP) P .–G. Reinhard (Inst.Theor.Physik, Erlangen) Giant resonances in the Skyrme-Hartree-Fock theory COMEX05, Krakow 2015 5 / 36

Observables P .–G. Reinhard (Inst.Theor.Physik, Erlangen) Giant resonances in the Skyrme-Hartree-Fock theory COMEX05, Krakow 2015 6 / 36

The nuclear matter parameters (NMP) given E / A ( ρ ) = energy per particle in symmetric nuclear matter ( ρ = total density) this allows to define basic properties at equilibrium: E / A eq binding energy per particle at equilibrium point ρ eq equilibrium density E K = 9 ρ 2 0 ∂ 2 incompressibility (isoscalar static response) ρ A m ∗ effective mass (isoscalar dynamic response) m J symmetry energy (isovector static response) L = 3 ρ 0 ∂ ρ a sym slope of symmetry energy m ∗ κ TRK TRK sum rule enhancement ↔ isovector m (dynamic response) 1 a surf surface energy a surf , sym surface symmetry energy these 9 NMP are equivalent to parameters of SHF functional (except l*s & pairing) P .–G. Reinhard (Inst.Theor.Physik, Erlangen) Giant resonances in the Skyrme-Hartree-Fock theory COMEX05, Krakow 2015 7 / 36

Ground state properties in the pool of fit data adopted errors Fit Observables: O ∆ O from expected binding energy E B _ + 1MeV correlation effects + _ r.m.s. radius r 0.02 fm large span in mass nmber A sufficient isoscalar information diffraction radius R _ + 0.04 fm surface thickness σ _ + 0.04 fm 82 l*s in doubly magic pairing (even − odd stagg.) "long" chains in proton number Z N − Z direction nonetheless: ↔ only weak ↔ isovector 50 information semi − magic nuclei: spherical 28 small correlations ↔ smallest correlation effects 20 full weight reduced weight 20 28 50 82 126 neutron number N P .–G. Reinhard (Inst.Theor.Physik, Erlangen) Giant resonances in the Skyrme-Hartree-Fock theory COMEX05, Krakow 2015 8 / 36

Further observables Response properties in 208 Pb giant resonances: monopole (GMR), quadrupole (GQR), dipole (GDR) � ∞ d ω S D ( ω ) ω − 1 in 208 Pb dipole polarizability α D = 0 Other: neutron skin r n − r p in 208 Pb neutron “equation of state” (EoS) E / N neut ( ρ ) binding energy E B for exotic nuclei (super-heavy 120/182; neutron rich 148 Sn) P .–G. Reinhard (Inst.Theor.Physik, Erlangen) Giant resonances in the Skyrme-Hartree-Fock theory COMEX05, Krakow 2015 9 / 36

Description of excitation spectra ↔ RPA and beyond RPA: small amplitude limit of time-dependent Hartree-Fock N = � eigenmodes = optimized 1 ph excitation operators ˆ ν b ν ˆ C † A ν � � ˆ , r L + n Y LM , j L ( qr ) Y LM , [ˆ H , r L + n Y LM ] , [ˆ a † ˆ a † ˆ a , ˆ a ˆ A ν ∈ H , j L ( qr ) Y LM ] � �� � � �� � E < 30 MeV global couplings, high E � [ˆ C N , [ˆ H , ˆ C † N ]] � RPA equations from variational formulation: δ b ∗ = 0 � [ˆ C N , ˆ C † ν N ] � numerically handled by commutator algebra on the grid P .–G. Reinhard (Inst.Theor.Physik, Erlangen) Giant resonances in the Skyrme-Hartree-Fock theory COMEX05, Krakow 2015 10 / 36

Description of excitation spectra ↔ RPA and beyond RPA: small amplitude limit of time-dependent Hartree-Fock N = � eigenmodes = optimized 1 ph excitation operators ˆ ν b ν ˆ C † A ν � � ˆ , r L + n Y LM , j L ( qr ) Y LM , [ˆ H , r L + n Y LM ] , [ˆ a † ˆ a † ˆ a , ˆ a ˆ A ν ∈ H , j L ( qr ) Y LM ] � �� � � �� � E < 30 MeV global couplings, high E � [ˆ C N , [ˆ H , ˆ C † N ]] � RPA equations from variational formulation: δ b ∗ = 0 � [ˆ C N , ˆ C † ν N ] � numerically handled by commutator algebra on the grid +phonons: couple basis states ˆ A ν to few low-lying & strong ˆ C † S N = � A ν + � basis of 1 ph & 2 ph operators: ˜ ν b ν ˆ ν, S b ν, S ˆ A ν ˆ C † C † ˜ S approximations: residual interaction ˆ V res from RPA (?: 1 ph -1 ph used for 1 p 1 p -1 ph ) exchange terms in ˆ V res neglected (?: Pauli principle) P .–G. Reinhard (Inst.Theor.Physik, Erlangen) Giant resonances in the Skyrme-Hartree-Fock theory COMEX05, Krakow 2015 10 / 36

Optimization of model parameters P .–G. Reinhard (Inst.Theor.Physik, Erlangen) Giant resonances in the Skyrme-Hartree-Fock theory COMEX05, Krakow 2015 11 / 36

Optimization of model parameters exp. fit data: O exp f least squares error χ 2 ( p ) ✻ ✛ ❄ ✻ feedback fit observables: to minimize f = 1 ... N data labels fit data χ 2 ( p ) → χ 2 ( p 0 ) O f = O f ( p ) ✻ ❄ predicted model: parameters: ✲ ✲ observables: p = ( p 1 ... p F ) SHF A = � ˆ A � = A ( p ) N data � 2 O f ( p ) − O exp � � χ 2 ( p ) = f ∆ O 2 f f = 1 adopted error ∆ O f : χ 2 ( p 0 ) = N data − N params P .–G. Reinhard (Inst.Theor.Physik, Erlangen) Giant resonances in the Skyrme-Hartree-Fock theory COMEX05, Krakow 2015 12 / 36