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Gogny-HFB Nuclear Mass Model

Gogny-HFB Nuclear Mass Model. S. Goriely (ULB), S. Hilaire (CEA-DAM-DIF) et. al. J.-P. Ebran (CEA-DAM-DIF) ECT* 8-12/07/2013 . Outline.  Gogny -HFB Nuclear Mass Model Energy Density Functional The Gogny Force Results.

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Gogny-HFB Nuclear Mass Model

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  1. Gogny-HFB Nuclear Mass Model S. Goriely (ULB), S. Hilaire (CEA-DAM-DIF) et. al. J.-P. Ebran (CEA-DAM-DIF)ECT* 8-12/07/2013

  2. Outline  Gogny-HFB Nuclear Mass Model Energy Density Functional The Gogny Force Results  Relativistic Hartree-Fock-Bogoliubov in Axial Symmetry

  3. Gogny-HFB Mass Model : Motivation  Microscopic Mass Model : as good as possible description of all the properties of all nuclei for both ground and excited states  Feed Reaction model with Structure ingredients  Astrophysical applications : involve nuclei not experimentally accessible  Need for predictive approach

  4. I. Energy Density Functional

  5. I. Energy Density Functional  Designed to compute average value of few-body operators  Independent particle picture   

  6. I. Energy Density Functional Quantum Liquid-like G.S. Crystal-like G.S.

  7. I. Energy Density Functional  Particle-Hole and Particle-Particle fields involved in HFB-like equation

  8. I. EDF: Symmetry Breaking 3+[202] 1+[200] 1+[211] 5+[202] 3+[211] 1+[220] 1d3/2 8 2s1/2 1-[101] 1d5/2 3-[101] 8 8 8 20 20 20 20 20 20 20 20 20 1-[110] 2 1p1/2 1p3/2 1+[000] 2 2 2 2 2 2 2 2 2 1s1/2 8 8 8 8 8 8  Symmetry breaking : take into account additional correlations keeping a single particle picture 1 particule – 1 hole excitations 2 particules – 2 holes excitations 3 particules – 3 holes excitations

  9. I. EDF: Symmetry Breaking  Symmetry breaking : take into account additional correlations keeping a single particle picture

  10. I. EDF: Symmetry Restoration  Restoration of broken symmetries : MR-level  Configuration mixing method : GCM

  11. I. EDF: Symmetry Restoration

  12. II. Gogny Interaction  Gogny strategy : parametrize both p-h and p-p channels with the same phenomenological finite-range 2-body interaction

  13. II. Gogny Interaction  D1 : J. Dechargé & D. Gogny, Phys. Rev. C211568 (1980)  D1S : J.F. Berger, M. Girod& D. Gogny, Comput. Phys. Commun. 63 365 (1991)  D1N : F. Chappert, M. Girod & S. Hilaire, Phys. Lett. B668420 (2008)  D1M : S. Goriely,S. Hilaire, M. Girod& S. Péru, Phys. Rev. Lett. 102 242501 (2009).

  14. II. Gogny Interaction  Finite range : avoid pathologies “beyond HF” due to unrealistic behavior of 0-range forces at high relative momenta

  15. II. Gogny: Two Fitting Philosophies  14 parameters : (W,B,H,M)1 ; (W,B,H,M)2 ; t3 ; x3 ; a ; WLS ; m1 ; m2

  16. II. Gogny: Two Fitting Philosophies Initial Data Inversion B.E., Rc (16O,90Zr)  “Traditional” method involving small set of magic nuclei (!!!) at SR-level 4x4 equations system D1 D1S D1N Pairingconsiderations W1 B1 H1 M1 W2 B2 H2 M2 « Theoretical » data at SR-level 4x4 equations system Symmetryenergy Test in Nuclearmatter: (r, E/A)sat m*/m K t3; x3; a ; WLS ; m1 ; m2 Reject Validation

  17. II. Gogny: Two Fitting Philosophies  Make use of the huge data on masses and incorporate a maximum of physics in the functional  MR-level Parameters kept constant: 4(can be included in the fit) 1=0.7-0.8 ;2=1.2 ; x3=1 ;=1/3 (0.2-0.5 investigated) D1M • Parameters constrained: 3 • J ~ 29 - 32 MeV to reproduce at best neutron matter EoS • K ~ 230 - 240 MeV as expected from exp. breathing mode data • kFkept constant to reproduce charge radii at best (manually adjusted) (av, J, m*, K, kF) (B1, H1, W2, M2, t3) Parameters directly fitted to nuclear masses at MR-level: 7 (av , m*, W1, M1, B2, H2, Wso)

  18. II. Gogny: Two Fitting Philosophies D1M  Infinite base correction

  19. II. Gogny: Two Fitting Philosophies D1M 60Ni

  20. II. Gogny: Two Fitting Philosophies D1M 120Sn

  21. II. Gogny: Two Fitting Philosophies D1M  GCM + GOA  M. Girod and B. Grammaticos, Nucl. Phys. A33040 (1979)  J. Libert, M. Girod and J.-P. Delaroche, Phys. Rev. C60 054301 (1999)

  22. II. Gogny: Two Fitting Philosophies D1M For 1/3 of 2149 exp masses (Audi et al 2003) – N=Z,N=Z±1, N=Z±2 Trial force New force automatic fit on masses

  23. II. Gogny: Two Fitting Philosophies D1M Acceptable rms, J, K Trial force New force automatic fit on masses Check properties

  24. II. Gogny: Two Fitting Philosophies D1M • ~ 200/782 exp. charge radii • with dynamical correction • Play on kF to adjust globally Acceptable rms, J, K Trial force New force automatic fit on masses Check properties

  25. II. Gogny: Two Fitting Philosophies D1M • ~ 200/782 exp. charge radii • with dynamical correction • Play on kF to adjust globally • Nuclear Matter Properties Acceptable rms, J, K • + Landau Parameters (stability, sum rules, G0 ~ 0; G0’~ 0.9-1 (Borzov et al. 1981)) Trial force New force automatic fit on masses Check properties

  26. 244Pu II. Gogny: Two Fitting Philosophies D1M • ~ 200/782 exp. charge radii • with dynamical correction • Play on kF to adjust globally • Nuclear Matter Properties • Energy of 2+ levels Acceptable rms, J, K • Moment of inertia • + Landau Parameters (stability, sum rules, G0 ~ 0; G0’~ 0.9-1 (Borzov et al. 1981)) Trial force New force automatic fit on masses Check properties

  27. II. Gogny: Two Fitting Philosophies D1M Acceptable rms, J, K Acceptable rms, J, K,prop. Trial force New force New Cstr. automatic fit on masses Check properties

  28. II. Gogny: Two Fitting Philosophies D1M Acceptable rms, J, K Acceptable rms, J, K,prop. Trial force New force New Cstr. New D automatic fit on masses Check properties

  29. II. Gogny: Two Fitting Philosophies D1M Acceptable rms, J, K Acceptable rms, J, K,prop. Trial force New force New Cstr. New D New Dquad automatic fit on masses Check properties

  30. II. Gogny: Two Fitting Philosophies D1M Acceptable rms, J, K Acceptable rms, J, K,prop. Trial force New force New Cstr. New D New Dquad automatic fit on masses Check properties

  31. Quadrupole correction to the binding energy

  32. II. Gogny: Two Fitting Philosophies D1M Acceptable rms, J, K Acceptable rms, J, K,prop. Trial force New force New Cstr. New D New Dquad automatic fit on masses Check properties

  33. III. Results: Masses D1S Comparison with 2149 Exp. Masses • Eth = EHFB r.m.s ~ 4.4 MeV • Eth = EHFB - D r.m.s ~ 2.6 MeV • Eth = EHFB - D - Dquad r.m.s ~ 2.9 MeV

  34. III. Results: D1N and the Neutron Matter EOS  F. Chappert, M. Girod &S. Hilaire, Phys. Lett. B668 (2008) 420.

  35. III. Results: Masses D1N Comparison with 2149 Exp. Masses • Eth = EHFB r.m.s ~ 2.5 MeV • Eth = EHFB - D • Eth = EHFB - D - Dquad r.m.s ~ 0.95 MeV

  36. III. Results: Masses Comparison with 2149 Exp. Masses r.m.s ~ 2.5 MeV r.m.s ~ 0.95 MeV e = 0.126 MeV r.m.s = 0.798 MeV

  37. Results: Masses Comparison with 2149 Exp. Masses e = 0.126 MeV r.m.s = 0.798 MeV

  38. III. Results: Radii Comparison with 707 Exp. Charge Radii r.m.s = 0.031 fm

  39. III. Results: Pairing Sn

  40. III. Results: Pairing Sn

  41. III. Results: Nuclear Matter kF=1.346 fm-1J=28.6 MeV m*/m=0.746 Kinf =225 MeV Pure Neutron Matter

  42. III. Results: Nuclear Matter kF=1.346 fm-1J=28.6 MeV m*/m=0.746 Kinf =225 MeV

  43. III. Results: Nuclear Matter

  44. III. Results: Comparison with other Mass Formula D1M – HFB17 D1M – FRDM

  45. Conclusion & Perspectives  First Gogny Mass Model : r.m.s. = 0.798 MeV  With Audi et al 2013, r.m.s.(D1M) better and r.m.s.(D1S) gets worse  Implementation of exact coulomb exchange and (anti-)pairing  Octupole correlations  Development of generalized Gogny interactions (D2, …)

  46. Relativistic Hartree-Fock-Bogoliubov in Axial Symmetry J.-P. Ebran (CEA-DAM-DIF), E. Khan (IPN), D. PeñaArteaga (CEA-DAM-DIF), D. Vretenar (Zagreb University) J.-P. Ebran ECT* 8-12/07/2013

  47. Why a Relativstic Approach? Kinematics • Relevance of covariant approach : not imposed by the need for a relativistic nuclear kinematics, but rather linked to the use of Lorentz symmetry • Microscopic structure model = low-energy effective model of QCD  Many possible formulations but all not as efficient • Relativistic potentials : • S ~ -400 MeV : Scalar attractive potential • V ~ +350 MeV : 4-vector (time-like component) repulsive potential

  48. Why a Relativstic Approach? In medium Chiral Perturbation theory, D. Vretenar et. al. • Modification of the vacuum structure in presence of baryonicmatterat the origin of the S and V self energiesfelt by nucleons

  49. Why a Relativstic Approach? • QCD sumrules Large scalar and time-like self energieswith opposite sign

  50. Why a Relativstic Approach? • Spin-orbitpotentialemergesnaturallywith the empiricalstrenght • Time-oddfields = space-like component of 4-potential • Empiricalpseudospinsymmetry in nuclearspectroscopy • Saturation mechanism of nuclearmatter Figure from C. Fuchs (LNP 641: 119-146 , 2004)

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