Tamara Nikšić University of Zagreb

Transcription

1 CONFIGURATION MIXING WITH RELATIVISTIC SCMF MODELS Tamara Nikšić University of Zagreb Supported by the Croatian Foundation for Science Tamara Nikšić (UniZg) Primošten / 3

2 Contents Outline Relativistic nuclear energy density functional Adjusting the model parameters Applications: ground-state properties and giant resonances collective Hamiltonian model based on the SCRMF Applications: 4 Pu isotope Applications: Pt isotopes Applications: Kr isotopes Applications: N = 8 isotones Summary and outlook Tamara Nikšić (UniZg) Primošten / 3

3 Relativistic energy density functional Energy density functional consists of the mean-field and the pairing contribution E = E RMF [j µ, ρ s ] E pp (κ, κ ) Elementary building blocks ( ψoτ Γψ ) O τ {1, τ i } Γ {1, γ µ, γ 5, γ 5 γ µ, σ µν } Isoscalar-scalar density ρ s (r) = occ k ψ k (r)ψ k (r) Tamara Nikšić (UniZg) Primošten / 3

4 Relativistic energy density functional Energy density functional consists of the mean-field and the pairing contribution E = E RMF [j µ, ρ s ] E pp (κ, κ ) Elementary building blocks ( ψoτ Γψ ) O τ {1, τ i } Γ {1, γ µ, γ 5, γ 5 γ µ, σ µν } Isoscalar-vector current j µ (r) = occ k ψ k (r)γ µ ψ k (r) Tamara Nikšić (UniZg) Primošten / 3

5 Relativistic energy density functional Energy density functional consists of the mean-field and the pairing contribution E = E RMF [j µ, ρ s ] E pp (κ, κ ) Elementary building blocks ( ψoτ Γψ ) O τ {1, τ i } Γ {1, γ µ, γ 5, γ 5 γ µ, σ µν } Isovector-scalar density ρ s (r) = occ k ψ k (r) τψ k (r) Tamara Nikšić (UniZg) Primošten / 3

6 Relativistic energy density functional Energy density functional consists of the mean-field and the pairing contribution E = E RMF [j µ, ρ s ] E pp (κ, κ ) Elementary building blocks ( ψoτ Γψ ) O τ {1, τ i } Γ {1, γ µ, γ 5, γ 5 γ µ, σ µν } Isovector-vector current jµ (r) = occ k ψ k (r) τj µ ψ k (r) Tamara Nikšić (UniZg) Primošten / 3

7 Relativistic energy density functional Energy density functional consists of the mean-field and the pairing contribution E = E RMF [j µ, ρ s ] E pp (κ, κ ) Kinetic energy term E kin = i vi ψ i (r) ( γ m) ψ i (r) Tamara Nikšić (UniZg) Primošten / 3

8 Relativistic energy density functional Energy density functional consists of the mean-field and the pairing contribution Second order terms E nd = 1 E = E RMF [j µ, ρ s ] E pp (κ, κ ) [αv (ρ v )ρ v α s (ρ v )ρ s α tv (ρ v )ρ tv] dr Tamara Nikšić (UniZg) Primošten / 3

9 Relativistic energy density functional Energy density functional consists of the mean-field and the pairing contribution E = E RMF [j µ, ρ s ] E pp (κ, κ ) Derivative terms E der = 1 δ s ρ s ρ s dr Tamara Nikšić (UniZg) Primošten / 3

10 Relativistic energy density functional Energy density functional consists of the mean-field and the pairing contribution E = E RMF [j µ, ρ s ] E pp (κ, κ ) Coulomb interaction E coul = e j p µa µ dr Tamara Nikšić (UniZg) Primošten / 3

11 Relativistic energy density functional Energy density functional consists of the mean-field and the pairing contribution E = E RMF [j µ, ρ s ] E pp (κ, κ ) Pairing interaction: finite range separable pairing V (r 1, r, r 1, r ) = Gδ(R R )P(r)P(r ) 1 (1 Pσ ) R = 1 (r 1 r ), r = r 1 r, P(r) = 1 r 4πa e 4a Parameters a and G are adjusted to reproduce the pairing gap in the symmetric nuclear matter calculated using the Gogny force. Tamara Nikšić (UniZg) Primošten / 3

12 Relativistic energy density functional Couplings are density-dependent α i (ρ v ) = a i (b i c i x) e d i x, x = ρ/ρ sat, i s, v, tv Model parameters a s, b s, c s, d s, a v, b v, d v, b tv, d tv, δ s Adjusted to empirical ground-state properties of finite nuclei. Empirical ground-state properties of finite nuclei can only determine a small set of parameters. Tamara Nikšić (UniZg) Primošten / 3

13 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 absorbed in the model parameters heavy deformed systems present best examples of mean-field nuclei high density of states reduces the shell effects Tamara Nikšić (UniZg) Primošten / 3

14 Adjusting the model parameters Empirical mass formula The calculated masses of finite nuclei are primarily sensitive to three leading terms in the empirical mass formula E B = a v A a s A /3 a 4 (N Z ) 4A Fitting strategy generate families of effective interactions that are characterized by different values of a v, a s and the symmetry energy S (.1fm 3 ) determine which parametrization minimizes the deviation from empirical binding energies of a large set of deformed nuclei Tamara Nikšić (UniZg) Primošten / 3

15 Adjusting the model parameters Empirical mass formula The calculated masses of finite nuclei are primarily sensitive to three leading terms in the empirical mass formula E B = a v A a s A /3 a 4 (N Z ) 4A Fitting strategy generate families of effective interactions that are characterized by different values of a v, a s and the symmetry energy S (.1fm 3 ) determine which parametrization minimizes the deviation from empirical binding energies of a large set of deformed nuclei Tamara Nikšić (UniZg) Primošten / 3

16 Adjusting the model parameters!e (MeV)!E (MeV) a v =-16.4 MeV " Sm Gd Dy Er Yb Hf Th U Pu Cm Cf Rare-earth region Sm (Z=6), Gd (Z=64), Dy (Z=66), Er (Z=68), Yb (Z=7), Hf (Z=7) Actinides Th (Z=9), U (Z=9), Pu (Z=94), Cm (Z=96), Cf (Z=98) Total 64 isotopes A Tamara Nikšić (UniZg) Primošten / 3

17 Adjusting the model parameters!e (MeV)!E (MeV) a v =-16.6 MeV " Sm Gd Dy Er Yb Hf Th U Pu Cm Cf Rare-earth region Sm (Z=6), Gd (Z=64), Dy (Z=66), Er (Z=68), Yb (Z=7), Hf (Z=7) Actinides Th (Z=9), U (Z=9), Pu (Z=94), Cm (Z=96), Cf (Z=98) Total 64 isotopes A Tamara Nikšić (UniZg) Primošten / 3

18 Adjusting the model parameters!e (MeV)!E (MeV) a v =-16.8 MeV " Sm Gd Dy Er Yb Hf Th U Pu Cm Cf Rare-earth region Sm (Z=6), Gd (Z=64), Dy (Z=66), Er (Z=68), Yb (Z=7), Hf (Z=7) Actinides Th (Z=9), U (Z=9), Pu (Z=94), Cm (Z=96), Cf (Z=98) Total 64 isotopes A Tamara Nikšić (UniZg) Primošten / 3

19 Adjusting the model parameters!e (MeV)!E (MeV) a v =-16.1 MeV " Sm Gd Dy Er Yb Hf Th U Pu Cm Cf Rare-earth region Sm (Z=6), Gd (Z=64), Dy (Z=66), Er (Z=68), Yb (Z=7), Hf (Z=7) Actinides Th (Z=9), U (Z=9), Pu (Z=94), Cm (Z=96), Cf (Z=98) Total 64 isotopes A Tamara Nikšić (UniZg) Primošten / 3

20 Adjusting the model parameters!e (MeV)!E (MeV) a v =-16.1 MeV " Sm Gd Dy Er Yb Hf Th U Pu Cm Cf Rare-earth region Sm (Z=6), Gd (Z=64), Dy (Z=66), Er (Z=68), Yb (Z=7), Hf (Z=7) Actinides Th (Z=9), U (Z=9), Pu (Z=94), Cm (Z=96), Cf (Z=98) Total 64 isotopes A Tamara Nikšić (UniZg) Primošten / 3

21 Adjusting the model parameters!e (MeV)!E (MeV) a v = MeV " Sm Gd Dy Er Yb Hf Th U Pu Cm Cf Rare-earth region Sm (Z=6), Gd (Z=64), Dy (Z=66), Er (Z=68), Yb (Z=7), Hf (Z=7) Actinides Th (Z=9), U (Z=9), Pu (Z=94), Cm (Z=96), Cf (Z=98) Total 64 isotopes A Tamara Nikšić (UniZg) Primošten / 3

22 Ground-state properties Charge radii Quadrupole deformations r c (fm) Nd Sm! Nd Sm r c (fm) Gd Dy! Gd Dy r c (fm) Er DD-PC1 DD-ME EXP A Yb A! Er A Yb DD-PC1 DD-ME EMP A Tamara Nikšić (UniZg) Primošten / 3

23 Excitation energies of collective modes Pb ISGMR IVGDR 1 ISGMR R (1 3 fm 4 ) 15 1 R (e fm ) Sn 118 Sn 1 Sn 14 Sn E (MeV) E (MeV) R (e fm ) Sn Sn 116 Sn 118 Sn 1 Sn 1 Sn 14 Sn E (MeV) E (MeV) E (MeV) E (MeV) R (1 3 fm 4 ) IVGDR E (MeV) E (MeV) E (MeV) E (MeV) E (MeV) E (MeV) E (MeV) Tamara Nikšić (UniZg) Primošten / 3

24 Implementation of the collective Hamiltonian model based on the SCRMF Collective Hamiltonian Rotational energy H coll = T rot T vib V coll T rot = 1 3 k=1 Ĵ k I k The moments of inertia are calculated by using the Inglis-Belyaev formula. Tamara Nikšić (UniZg) Primošten / 3

25 Implementation of the collective Hamiltonian model based on the SCRMF Collective Hamiltonian Vibrational energy H coll = T rot T vib V coll [ β rw β4 B γγ β β rw β3 B βγ γ ] T vib = β 4 wr [ sin 3γ 1 r wr β γ w sin 3γB βγ β 1 ] r β γ w sin 3γB ββ γ The mass parameters are calculated in the cranking approximation. Tamara Nikšić (UniZg) Primošten / 3

26 Implementation of the collective Hamiltonian model based on the SCRMF Collective Hamiltonian Collective potential H coll = T rot T vib V coll V coll (β, γ) = E tot (β, γ) V vib (β, γ) V rot (β, γ) Corresponds to the mean-field potential energy surface with the zero point energy subtracted. Tamara Nikšić (UniZg) Primošten / 3

27 Applications: 4 Pu isotope Triaxial effects lower the barrier. ND and SD minima are separated by the barrier. Tamara Nikšić (UniZg) Primošten / 3

28 Applications: 4 Pu isotope. Energy (MeV) Theory γ-band g.s. β-band g.s. Experiment β-band γ-band 3 E th /E th = 3.33 E exp /E exp = Pu (6.8) The moments of inertia are renormalized by factor 1.3 to compensate the difference between IB and TV moments of inertia. Tamara Nikšić (UniZg) Primošten / 3

29 Applications: Pt isotopes 19 Pt β B.E. (MeV) γ (deg) B.E. (MeV) Pt β= γ (deg) β Theory Experiment Pt E th /E th =.39 E exp /E exp = Energy (MeV) γ 5 γ 4 γ 3 γ γ (3) Tamara Nikšić (UniZg) Primošten / 3

30 Applications: Pt isotopes 19 Pt β B.E. (MeV) γ (deg) B.E. (MeV) Pt β= γ (deg) β Theory Experiment 19 Pt γ E th /E th =.58 E exp /E exp = Energy (MeV) γ 4 γ 3 γ 17 γ 6 1 7(3) (5) (1) 1 38(9) 19(7) (1).54(4) Tamara Nikšić (UniZg) Primošten / 3

31 Applications: Pt isotopes 194 Pt β B.E. (MeV) γ (deg) B.E. (MeV) Pt β= γ (deg) β Theory Experiment 194 Pt E th /E th =.63 E exp /E exp = Energy (MeV) (9) 6 γ γ 5(14) γ (1) γ γ 85(5) 89(11) (8) (4) Tamara Nikšić (UniZg) Primošten / 3

32 Applications: Pt isotopes 196 Pt β B.E. (MeV) γ (deg) B.E. (MeV) β= Pt γ (deg) β Theory Experiment 196 Pt E th /E th =.66 E exp /E exp = Energy (MeV) γ 5 γ 4 γ 3 γ γ (9) 1 4.6() Tamara Nikšić (UniZg) Primošten / 3

33 Applications: Pt isotopes 198 Pt β B.E. (MeV) γ (deg) B.E. (MeV) Pt β= γ (deg) β Theory 6 γ Experiment 198 Pt γ E th /E th =.49 E exp /E exp = Energy (MeV) γ 3 γ γ (4) 37(7) 1.38(1) 3 () 1 Tamara Nikšić (UniZg) Primošten / 3

34 Applications: Kr isotopes.8 B.E. (MeV).8 B.E. (MeV) 7 Kr Kr (deg) (deg) B.E. (MeV).8 B.E. (MeV) 76 Kr Kr (deg) 3 6 (deg) Tamara Nikšić (UniZg) Primošten / 3

35 74 Kr isotope: level scheme g.s. band predominantely prolate state: γ-vibration, 3 predominantely oblate mixing in the ground state Tamara Nikšić (UniZg) Primošten / 3

36 74 Kr isotope: level scheme Kr 4 4 Collective HamiltonianDDPC1 5 1 DD-PC1 Energy (MeV) Experiment Tamara Nikšić (UniZg) Primošten / 3

37 74 Kr isotope: level scheme Kr 4 4 Collective HamiltonianDDPC1 5 1 DD-PC1 Energy (MeV) Collective HamiltonianGogny D1S Tamara Nikšić (UniZg) Primošten / 3

38 74 Kr isotope: level scheme Kr 4 4 Collective HamiltonianDDPC1 5 1 DD-PC1 Energy (MeV) D AM&PNP Skyrme Tamara Nikšić (UniZg) Primošten / 3

39 74 Kr isotope: collective probability Kr DD-PC1 Energy (MeV) Kr 1 g.s. band (deg) 3 Probability density distribution: ρ Iα (β, γ) = ψαk I (β, γ) β 3 sin 3γ K I Tamara Nikšić (UniZg) Primošten / 3

40 74 Kr isotope: collective probability Kr DD-PC1 Energy (MeV) Kr 1 g.s. band (deg) 3 Probability density distribution: ρ Iα (β, γ) = ψαk I (β, γ) β 3 sin 3γ K I Tamara Nikšić (UniZg) Primošten / 3

41 74 Kr isotope: collective probability Kr DD-PC1 Energy (MeV) Kr 4 1 g.s. band (deg) 3 Probability density distribution: ρ Iα (β, γ) = ψαk I (β, γ) β 3 sin 3γ K I Tamara Nikšić (UniZg) Primošten / 3

42 74 Kr isotope: collective probability Kr DD-PC1 Energy (MeV) oblate deformed band 74 Kr (deg) 3.3 Probability density distribution: ρ Iα (β, γ) = ψαk I (β, γ) β 3 sin 3γ K I Tamara Nikšić (UniZg) Primošten / 3

43 74 Kr isotope: collective probability Kr DD-PC1 Energy (MeV) Probability density distribution: ρ Iα (β, γ) = ψαk I (β, γ) β 3 sin 3γ K I oblate deformed band 74 Kr β 3.3 γ (deg) β Tamara Nikšić (UniZg) Primošten / 3

44 Applications: N = 8 isotones The variation of the mean-field shapes is governed by the evolution of the underlying shell structure of single-nucleon orbitals. Tamara Nikšić (UniZg) Primošten / 3

45 46 Ar isotope: single-particle levels Tamara Nikšić (UniZg) Primošten / 3

46 44 S isotope: single-particle levels Tamara Nikšić (UniZg) Primošten / 3

47 4 Si isotope: single-particle levels Tamara Nikšić (UniZg) Primošten / 3

48 4 Mg isotope: single-particle levels Tamara Nikšić (UniZg) Primošten / 3

49 N 8 observables excitation energies and reduced electric quadrupole transition probabilities full configuration space, no need for effective charges Tamara Nikšić (UniZg) Primošten / 3

50 44 S isotope: level scheme S Energy (MeV) DD-PC1 81 7! (E)*1 3 = B(E): e fm 4 ( ) 4(13) 63(18)! (E)*1 3 = 8.7(7) Exp. Probability density distribution: ρ Iα (β, γ) = ψαk I (β, γ) β 3 K I Tamara Nikšić (UniZg) Primošten / 3

51 44 S isotope: level scheme S Energy (MeV) DD-PC1 81 7! (E)*1 3 = B(E): e fm 4 ( ) 4(13) 63(18)! (E)*1 3 = 8.7(7) Exp. Probability density distribution: ρ Iα (β, γ) = ψαk I (β, γ) β 3 K I Tamara Nikšić (UniZg) Primošten / 3

52 Summary and outlook Summary Unified microscopic description of the structure of stable and nuclei far from stability, and reliable extrapolations toward the drip lines. Summary When extended to take into account collective correlations, it describes deformations and shape-coexistence phenomena associated with shell evolution. Outlook Further improvements of the model and more systematic calculations. Tamara Nikšić (UniZg) Primošten / 3

53 Summary and outlook Summary Unified microscopic description of the structure of stable and nuclei far from stability, and reliable extrapolations toward the drip lines. Summary When extended to take into account collective correlations, it describes deformations and shape-coexistence phenomena associated with shell evolution. Outlook Further improvements of the model and more systematic calculations. Tamara Nikšić (UniZg) Primošten / 3

54 Summary and outlook Summary Unified microscopic description of the structure of stable and nuclei far from stability, and reliable extrapolations toward the drip lines. Summary When extended to take into account collective correlations, it describes deformations and shape-coexistence phenomena associated with shell evolution. Outlook Further improvements of the model and more systematic calculations. Tamara Nikšić (UniZg) Primošten / 3

55 Collaborators Georgios Lalazissis (Aristotle University of Thessaloniki) Zhipan Li (Peking University) Jie Meng (Peking University) Leszek Próchniak (Maria Curie-Sklodowska University, Lublin) Peter Ring (Technical University Munich) Dario Vretenar (University of Zagreb) Tamara Nikšić (UniZg) Primošten / 3