BEYOND THE RELATIVISTIC MEAN-FIELD
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1 BEYOND THE RELATIVISTIC MEAN-FIELD APPROXIMATION: CONFIGURATION MIXING CALCULATIONS Tamara Nikšić University of Zagreb Arctic FIDIPRO-EFES Workshop Saariselkä, Finland April 20-24, 2009 Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
2 Outline Introduction Theoretical framework relativistic energy density functional (point-coupling implementation) symmetry restoration and generator coordinate method collective Hamiltonian model Applications Summary and outlook Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
3 Outline Introduction Theoretical framework relativistic energy density functional (point-coupling implementation) symmetry restoration and generator coordinate method collective Hamiltonian model Applications Summary and outlook Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
4 Outline Introduction Theoretical framework relativistic energy density functional (point-coupling implementation) symmetry restoration and generator coordinate method collective Hamiltonian model Applications Summary and outlook Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
5 Outline Introduction Theoretical framework relativistic energy density functional (point-coupling implementation) symmetry restoration and generator coordinate method collective Hamiltonian model Applications Summary and outlook Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
6 Introduction Successfull applications of the RMFSC models ground states of both stable as well as the exotic nuclei giant multipole resonances low-energy multipole response in weakly-bound nuclei description of rotational bands particle-vibration coupling modeling of β-decay rates of neutron-rich nuclei Next step: spectroscopy of stable and exotic nuclei large amount of the experimental data constraints on the energy density functional Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
7 Introduction Successfull applications of the RMFSC models ground states of both stable as well as the exotic nuclei giant multipole resonances low-energy multipole response in weakly-bound nuclei description of rotational bands particle-vibration coupling modeling of β-decay rates of neutron-rich nuclei Next step: spectroscopy of stable and exotic nuclei large amount of the experimental data constraints on the energy density functional Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
8 Introduction Successfull applications of the RMFSC models ground states of both stable as well as the exotic nuclei giant multipole resonances low-energy multipole response in weakly-bound nuclei description of rotational bands particle-vibration coupling modeling of β-decay rates of neutron-rich nuclei Next step: spectroscopy of stable and exotic nuclei large amount of the experimental data constraints on the energy density functional Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
9 Introduction Successfull applications of the RMFSC models ground states of both stable as well as the exotic nuclei giant multipole resonances low-energy multipole response in weakly-bound nuclei description of rotational bands particle-vibration coupling modeling of β-decay rates of neutron-rich nuclei Next step: spectroscopy of stable and exotic nuclei large amount of the experimental data constraints on the energy density functional Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
10 Introduction Successfull applications of the RMFSC models ground states of both stable as well as the exotic nuclei giant multipole resonances low-energy multipole response in weakly-bound nuclei description of rotational bands particle-vibration coupling modeling of β-decay rates of neutron-rich nuclei Next step: spectroscopy of stable and exotic nuclei large amount of the experimental data constraints on the energy density functional Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
11 Introduction Successfull applications of the RMFSC models ground states of both stable as well as the exotic nuclei giant multipole resonances low-energy multipole response in weakly-bound nuclei description of rotational bands particle-vibration coupling modeling of β-decay rates of neutron-rich nuclei Next step: spectroscopy of stable and exotic nuclei large amount of the experimental data constraints on the energy density functional Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
12 Introduction Successfull applications of the RMFSC models ground states of both stable as well as the exotic nuclei giant multipole resonances low-energy multipole response in weakly-bound nuclei description of rotational bands particle-vibration coupling modeling of β-decay rates of neutron-rich nuclei Next step: spectroscopy of stable and exotic nuclei large amount of the experimental data constraints on the energy density functional Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
13 Introduction Successfull applications of the RMFSC models ground states of both stable as well as the exotic nuclei giant multipole resonances low-energy multipole response in weakly-bound nuclei description of rotational bands particle-vibration coupling modeling of β-decay rates of neutron-rich nuclei Next step: spectroscopy of stable and exotic nuclei large amount of the experimental data constraints on the energy density functional Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
14 Introduction Successfull applications of the RMFSC models ground states of both stable as well as the exotic nuclei giant multipole resonances low-energy multipole response in weakly-bound nuclei description of rotational bands particle-vibration coupling modeling of β-decay rates of neutron-rich nuclei Next step: spectroscopy of stable and exotic nuclei large amount of the experimental data constraints on the energy density functional Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
15 Introduction Successfull applications of the RMFSC models ground states of both stable as well as the exotic nuclei giant multipole resonances low-energy multipole response in weakly-bound nuclei description of rotational bands particle-vibration coupling modeling of β-decay rates of neutron-rich nuclei Next step: spectroscopy of stable and exotic nuclei large amount of the experimental data constraints on the energy density functional Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
16 Relativistic energy density functional Relativistic EDF: point-coupling implementation EDF consists of the mean-field and the pairing contribution E = E RMF [j µ, ρ s ] E pp (κ, κ, ρ v ) Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
17 Relativistic energy density functional Relativistic EDF: point-coupling implementation EDF consists of the mean-field and the pairing contribution E = E RMF [j µ, ρ s ] E pp (κ, κ, ρ v ) Mean field contribution: kinetic energy E kin = i v 2 i ψ i (r) ( γ m)ψ i (r) Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
18 Relativistic energy density functional Relativistic EDF: point-coupling implementation EDF consists of the mean-field and the pairing contribution E = E RMF [j µ, ρ s ] E pp (κ, κ, ρ v ) Mean field contribution: second order terms E 2nd = 1 2 (αs ρ 2 s α v j µ j µ α tv (j tv ) µ j µ ) tv Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
19 Relativistic energy density functional Relativistic EDF: point-coupling implementation EDF consists of the mean-field and the pairing contribution Mean field contribution: E = E RMF [j µ, ρ s ] E pp (κ, κ, ρ v ) E hot = higher order terms ( βs 3 ρ3 s γ s 4 ρ4 s γ ) v 4 (j µj µ ) 2 Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
20 Relativistic energy density functional Relativistic EDF: point-coupling implementation EDF consists of the mean-field and the pairing contribution E = E RMF [j µ, ρ s ] E pp (κ, κ, ρ v ) Mean field contribution: derivative terms E der = 1 (δ s ρ s ρ s δ v j µ j µ δ tv (j tv ) µ j µ tv 2 ) Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
21 Relativistic energy density functional Relativistic EDF: point-coupling implementation EDF consists of the mean-field and the pairing contribution E = E RMF [j µ, ρ s ] E pp (κ, κ, ρ v ) Mean field contribution: E coul = e j p 2 µa µ Coulomb interaction Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
22 Relativistic energy density functional Relativistic EDF: point-coupling implementation EDF consists of the mean-field and the pairing contribution E = E RMF [j µ, ρ s ] E pp (κ, κ, ρ v ) Pairing contribution: density-independent δ-interaction BCS approximation V(r 1, r 2 ) = V p,n δ(r 1 r 2 ) Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
23 Relativistic energy density functional Relativistic EDF: point-coupling implementation EDF consists of the mean-field and the pairing contribution E = E RMF [j µ, ρ s ] E pp (κ, κ, ρ v ) Model parameters: particle-hole channel 9 parameters: α s, β s, γ s, δ s, α v, γ v, δ v, α tv, δ tv Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
24 Relativistic energy density functional Relativistic EDF: point-coupling implementation EDF consists of the mean-field and the pairing contribution E = E RMF [j µ, ρ s ] E pp (κ, κ, ρ v ) Model parameters: particle-particle channel 2 parameters: V p, V n Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
25 Relativistic energy density functional Relativistic EDF: point-coupling implementation EDF consists of the mean-field and the pairing contribution Model parameters: E = E RMF [j µ, ρ s ] E pp (κ, κ, ρ v ) the parameters have been adjusted to ground state properties of finite nuclei PC-F1 effective interaction, T. Bürvenich et al., PRC 65 (2002) Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
26 Nuclear many-body correlations Implicitly included in the EDF short-range hard repulsive core of the NN-interaction long-range mediated by nuclear resonance modes (giant resonances) the corresponding corrections vary smoothly with the number of nucleons Correlations treated explicitly restoration of broken symmetries (particle number, rotational symmetry) fluctuation of the quadrupole moment the corresponding corrections are sensitive to shell-effects and vary rapidly with the number of nucleons Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
27 Nuclear many-body correlations Implicitly included in the EDF short-range hard repulsive core of the NN-interaction long-range mediated by nuclear resonance modes (giant resonances) the corresponding corrections vary smoothly with the number of nucleons Correlations treated explicitly restoration of broken symmetries (particle number, rotational symmetry) fluctuation of the quadrupole moment the corresponding corrections are sensitive to shell-effects and vary rapidly with the number of nucleons Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
28 (PN)AMP GCM model Calculation procedure mean-field Lipkin-Nogami BCS equations, with a constraint on the quadrupole moment angular-momentum and particle-number projection generator coordinate 0 method -2 configuration mixing q (b) E (MeV) S MFLNBCS Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
29 (PN)AMP GCM model Calculation procedure mean-field Lipkin-Nogami BCS equations, with a constraint on the quadrupole moment angular-momentum and particle-number projection generator coordinate method configuration mixing q (b) E (MeV) S J π =0 J π =2 J π =4 J π =6 MFLNBCS Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
30 (PN)AMP GCM model Calculation procedure mean-field Lipkin-Nogami BCS equations, with a constraint on the quadrupole moment angular-momentum and particle-number projection generator coordinate method configuration mixing q (b) E (MeV) S J π =0 J π =2 J π =4 J π =6 J π =8 MFLNBCS Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
31 Implementation of the collective Hamiltonian model based on the SCRMF Collective Hamiltonian Rotational energy H coll = T rot T vib V coll the moments of inertia are calculated by using the Inglis-Belyaev formula the mass parameters are calculated in the cranking approximation corresponds to the mean-field potential energy surface with the zero point energy subtracted Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
32 Implementation of the collective Hamiltonian model based on the SCRMF Collective Hamiltonian Vibrational energy H coll = T rot T vib V coll the moments of inertia are calculated by using the Inglis-Belyaev formula the mass parameters are calculated in the cranking approximation corresponds to the mean-field potential energy surface with the zero point energy subtracted Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
33 Implementation of the collective Hamiltonian model based on the SCRMF Collective Hamiltonian Collective potential H coll = T rot T vib V coll the moments of inertia are calculated by using the Inglis-Belyaev formula the mass parameters are calculated in the cranking approximation corresponds to the mean-field potential energy surface with the zero point energy subtracted Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
34 Applications: 32 Mg (AMP GCM) The ratio between the neutron spherical gap and the gap at the deformation q = 1.5 b is large s. p. energy (MeV) neutrons f 7/2 d 3/2 s 1/2 d 5/2 7.2 MeV 2.9 MeV protons f 7/2 d 3/2 s 1/2 d 5/2-230 N sh = Mg Energy (MeV) MeV Mg q (b) q (b) q (b) There is no deformed minimum, but only a shoulder located 3.5 MeV above the ground state. Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
35 Applications: 32 Mg (AMP GCM) Angular momentum projection does not provide enough correlation energy to bring the prolate shoulder to the ground state. The effective interaction has to be improved. Energy (MeV) J π =0 J π =2 J π =4 J π =6 MF The average quadrupole moment of the GCM state is defined as q J k = X j g J k (q j) 2 q j N sh = q (b) 2 1 no gain in energy at q=0 32 Mg Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
36 Applications: Nd isotopes 16 Mean-field potential energy curves for the chain of the Nd isotopes Single-particle energy (MeV) i 13/2 3p 1/2 3p3/2 2f5/2 2f 7/2 1h 9/2 1h 11/2 3s 1/2 2d 3/2 2d 5/2 1g 7/2 82 Neutron 0.2 β 3s 1/2 1h11/2 2d 3/2 2d 5/2 1g 7/2 1g 9/2 50 2p 1/2 2p 3/2 Proton Energy (MeV) β Nd 144 Nd 146 Nd 148 Nd 150 Nd 152 Nd q (b) Neutron and proton single-particle levels in 150 Nd as functions of the axial deformation parameter β Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
37 Applications: 150 Nd (PNAMP GCM) The particle-number projected GCM spectrum of 150 Nd, compared with the data and the X5-model predictions for the excitation energies, intraband and interband B(E2) values in Weisskopf units. Configuration mixing is performed only on the prolate side. Energy (MeV) GCM EXP X(5) Nd (12) (13) 2 170(51) 2 278(25) (23) (2) 17(3) (2) (2) (2) s=1 s=2 Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
38 Applications: 150 Nd (triaxial calculation) Nd β β γ E (MeV) β= γ 150 Nd γ dependence of the potential energy surface at the deformation β = 0.2 Potential energy surface for 150 Nd. Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
39 Applications: 150 Nd (collective model) The 150 Nd spectrum calculated by using the collective Hamiltonian model. Energy (MeV) Collective model EXP X(5) (12) (13) (51) 278(25) (23) Nd 210(2) 17(3) (2) (2) (2) s=1 s=2 Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
40 Applications: Nd isotopes (collective model) Evolution of the characteristic observables R 4/2 and B(E2; ) with neutron number in Nd isotopes. E(4 1 )/ E(2 1 ) (a) Exp. PC- F1 (b) B(E2; ) (W.u.) N N 0 Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
41 Applications: Nd isotopes (collective model) Evolution of the first excited 0 state and the ratio E(6 1 )/E(0 2 ) with neutron number in Nd isotopes Exp. PC- F E(0 2 ) (MeV) E(6 1 )/ E(0 2 ) (a) N (b) N Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
42 Applications: Gd isotopes (collective model) E(J)/E(2 1 ) E(J)/E(2 1 ) E(J)/E(2 1 ) E(J)/E(2 1 ) E(J)/E(2 1 ) Gd 158 Gd 156 Gd 154 Gd 152 Gd g.s. band g.s. band J PC-F1 EXP E(J)/E(2 1 ) E(J)/E(2 1 ) E(J)/E(2 1 ) E(J)/E(2 1 ) E(J)/E(2 1 ) Gd 158 Gd 156 Gd 154 Gd 152 Gd β band β band J PC-F1 EXP E(J)/E(2 1 ) E(J)/E(2 1 ) E(J)/E(2 1 ) E(J)/E(2 1 ) E(J)/E(2 1 ) Gd 158 Gd 156 Gd 154 Gd 152 Gd γ band γ band Excitation energies for the ground state, β and γ bands in comparison with the empirical data. J PC-F1 EXP Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
43 Applications: Gd isotopes (collective model) S(J) S(J) S(J) S(J) S(J) Gd 158 Gd 156 Gd 154 Gd PC-F1 exp 152 Gd S(J) = J N K N K N K N K N K K Gd Gd Gd Gd Gd E(J) E(J 1) E(J 1) E(J 2) E(2 1 ) E(2 1 ) Distribution of the K components in the collective wave functions. K N K N K N K N K N K K Gd Gd Gd Gd Gd K Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
44 Summary and outlook RMFSC models can be used to study spectroscopy of atomic nuclei the model should be developed further improved treatment of the paring correlations: RHB instead of the BCS improved moments of inertia in the collective Hamiltonian model particle number projection (variation after projection instead of the projected Lipkin-Nogami) gamma degree of freedom and crancked wave functions in the GCM calculations better effective interactions are needed some problems are common to standard SCRMF and Skyrme interactions (e.g. Z 82 shell) Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
45 Summary and outlook RMFSC models can be used to study spectroscopy of atomic nuclei the model should be developed further improved treatment of the paring correlations: RHB instead of the BCS improved moments of inertia in the collective Hamiltonian model particle number projection (variation after projection instead of the projected Lipkin-Nogami) gamma degree of freedom and crancked wave functions in the GCM calculations better effective interactions are needed some problems are common to standard SCRMF and Skyrme interactions (e.g. Z 82 shell) Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
46 Summary and outlook RMFSC models can be used to study spectroscopy of atomic nuclei the model should be developed further improved treatment of the paring correlations: RHB instead of the BCS improved moments of inertia in the collective Hamiltonian model particle number projection (variation after projection instead of the projected Lipkin-Nogami) gamma degree of freedom and crancked wave functions in the GCM calculations better effective interactions are needed some problems are common to standard SCRMF and Skyrme interactions (e.g. Z 82 shell) Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
47 List of collaborators People involved... Peter Ring Dario Vretenar Georgios Lalazissis Zhipan Li Leszek Prochniak (Bohr Hamiltonian code) Tamara Nikšić (University of Zagreb) FIDIPRO-EFES Workshop 23 April / 19
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