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 I. Energy Density Functional II. The Gogny Force III. Results Fock  Relativistic Hartree-Fock-Bogoliubov in Axial Symmetry

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

4. I. Energy Density Functional

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

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

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

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

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

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

13. D1  D1 : J. Dechargé & D. Gogny, Phys. Rev. C21 1568 (1980) D1S  D1S : J.F. Berger, M. Girod & D. Gogny, Comput. Phys. Commun. 63 365 (1991) D1N  D1N : F. Chappert, M. Girod & S. Hilaire, Phys. Lett. B668 420 (2008) D1M  D1M : S. Goriely, S. Hilaire, M. Girod & S. Péru, Phys. Rev. Lett. 102 242501 (2009). II. Gogny Interaction

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

15. II. Gogny: Two Fitting Philosophies  14 parameters : (W,B,H,M) 1 ; (W,B,H,M) 2 ; t 3 ; x 3 ;  ; W LS ;  1 ;  2

16. Inversion 4x4 equations system W 1 B 1 H 1 M 1 W 2 B 2 H 2 M 2 Test in Nuclear matter: ( , E/A) sat m*/m K  B.E., R c ( 16 O, 90 Zr) Pairing considerations Symmetry energy Initial Data t 3 ; x 3 ;  ; W LS ;  1 ;  2 Reject Validation « Theoretical » data at SR-level D1 D1S D1N  “Traditional” method involving small set of magic nuclei (!!!) at SR-level II. Gogny: Two Fitting Philosophies

17. D1M  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 ; x 3 =1 ;  =1/3 (0.2-0.5 investigated) 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 k F kept constant to reproduce charge radii at best (manually adjusted) (a v, J, m *, K, k F )(B 1, H 1, W 2, M 2, t 3 ) Parameters directly fitted to nuclear masses at MR-level: 7 (a v, m *, W 1, M 1, B 2, H 2, W so ) II. Gogny: Two Fitting Philosophies

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

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

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

21. D1M  M. Girod and B. Grammaticos, Nucl. Phys. A330 40 (1979)  J. Libert, M. Girod and J.-P. Delaroche, Phys. Rev. C60 054301 (1999)  GCM + GOA II. Gogny: Two Fitting Philosophies

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

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

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

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

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

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

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

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

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

31. Quadrupole correction to the binding energy

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

33. III. Results: Masses Comparison with 2149 Exp. Masses D1S r.m.s ~ 4.4 MeV E th = E HFB E th = E HFB r.m.s ~ 2.6 MeV E th = E HFB -   E th = E HFB -   r.m.s ~ 2.9 MeV E th = E HFB -   -  quad E th = E HFB -   -  quad

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

35. III. Results: Masses Comparison with 2149 Exp. Masses D1N r.m.s ~ 2.5 MeV r.m.s ~ 0.95 MeV E th = E HFB E th = E HFB E th = E HFB -   E th = E HFB -   E th = E HFB -   -  quad E th = E HFB -   -  quad

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

37. Results: Masses Comparison with 2149 Exp. Masses  = 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 k F =1.346 fm -1 J=28.6 MeV m*/m=0.746 K inf =225 MeV Pure Neutron Matter

42. III. Results: Nuclear Matter k F =1.346 fm -1 J=28.6 MeV m*/m=0.746 K inf =225 MeV

43. III. Results: Nuclear Matter

44. III. Results: Comparison with other Mass Formula D1M – HFB17D1M – 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  Development of generalized Gogny interactions (D2, …)  Octupole correlations

46. Relativistic Hartree-Fock-Bogoliubov in Axial Symmetry J.-P. Ebran (CEA-DAM-DIF), E. Khan (IPN), D. Peña Arteaga (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 Relativistic potentials : S ~ -400 MeV : Scalar attractive potential V ~ +350 MeV : 4-vector (time-like component) repulsive potential Microscopic structure model = low-energy effective model of QCD  Many possible formulations but all not as efficient

48. Why a Relativstic Approach? Modification of the vacuum structure in presence of baryonic matter at the origin of the S and V self energies felt by nucleons In medium Chiral Perturbation theory, D. Vretenar et. al.

49. Why a Relativstic Approach? QCD sum rules  Large scalar and time-like self energies with opposite sign

50.  Spin-orbit potential emerges naturally with the empirical strenght  Time-odd fields = space-like component of 4-potential  Empirical pseudospin symmetry in nuclear spectroscopy  Saturation mechanism of nuclear matter Why a Relativstic Approach? Figure from C. Fuchs (LNP 641: 119-146, 2004)

51. Relativistic mean field models (RMF) treat implicitly Fock terms through fit of model parameters to data Relativistic Hartree-Fock models (RHF): more involved approaches which take explicitly into account the Fock contributions  Description of nuclear matter in better agreement with DBHF calculations  Tensor contribution to the NN force (pion +  ) : better description of shell structure  Fully self-consistent beyond mean-field models RHB in axial symmetry D. Vretenar et al Phys.Rep. 409:101- 259,2005 RHFB in spherical symmetry W. Long et al Phys. Rev. C 81, 024308 (2010) N N N N RHFB in axial symmetry J.-P. Ebran et al Phys. Rev. C 83, 064323 (2011) Why Fock Term?

52. Hamiltonian Observables Resolution in a deformed harmonic oscillator basis EDF Mean-field approximation : expectation value in the HFB ground state N N N N RHFB equations Minimization N N Lagrangian 8 free parameters RHFBz Model

53. Neutron density in the Neon isotopic chain Results

54. N=32 Masses SLy4 : M.V. Stoitsov et al, Phys. Rev. C68 (2003) 054312

55. Results N=32 static quadrupole deformations

56. Results Charge radii

57. Conclusion & Perspectives  First RHFB model in axial symmetry  Encouraging results but too heavy for triaxial calculations or MR-level

58. Thank you

59. III. Results: Pairing 244 Pu

60. III. Results: Pairing 164 Er

61. III. Results: Giant Resonances 14.25 MeV GMRGDR 208 Pb 15.85 MeV E exp = 14.17 MeV D. H. Youngblood et al., Phys. Rev. Lett. 82, 691 (1999). E exp = 13.43 MeV B. L. Berman and S. C. Fultz, Rev. Mod. Phys. 47, 713 (1975).

62. III. Results: Spectroscopy Excitation energies of the first 2 + for 519 e-e nuclei  J.P. Delaroche et al., Phys. Rev. C81 (2010) 014303.  S. Hilaire & M. Girod, Eur. Phys. J A33 237(2007)

63. III. Results: Nuclear Matter k F =1.346 fm -1 J=28.6 MeV m*/m=0.746 K inf =225 MeV Pure Neutron Matter

64. III. Results: Shell Gaps

66.  Structure properties of ~7000 nuclei + Spectroscopic properties of low energy collective levels for ~1700 even-even nuclei D1S Properties  S. Hilaire & M. Girod, Eur. Phys. J A33 237(2007)

67. D1S Properties

68. Results: Masses Comparison with 2149 Exp. Masses  = 0.126 MeV r.m.s = 0.798 MeV

69. Quadrupole correction to the binding energy

70. Relativistic potentials : S ~ -400 MeV : Scalar attractive potential V ~ +350 MeV : 4-vector (time-like component) repulsive potential Relevance of covariant approach : not imposed by the need of a relativistic nuclear kinematics, but rather linked to the use of Lorentz symmetry  Spin-orbit potential emerges naturally with the empirical strenght  Time-odd fields = space-like component of 4-potential  Empirical pseudospin symmetry in nuclear spectroscopy  Saturation mechanism of nuclear matter Why a Relativstic Approach?

71. Relativistic mean field models (RMF) treat implicitly Fock terms through fit of model parameters to data Relativistic Hartree-Fock models (RHF): more involved approaches which take explicitly into account the Fock contributions  Description of nuclear matter in better agreement with DBHF calculations  Tensor contribution to the NN force (pion +  ) : better description of shell structure  Fully self-consistent beyond mean-field models RHB in axial symmetry D. Vretenar et al Phys.Rep. 409:101- 259,2005 RHFB in spherical symmetry W. Long et al Phys. Rev. C 81, 024308 (2010) N N N N RHFB in axial symmetry J.-P. Ebran et al Phys. Rev. C 83, 064323 (2011) Why a Relativstic Approach?

72. S and V potentials characterize the essential properties of nuclear systems : Central Potential : quasi cancellation of potentials Spin-orbit : constructive combination of potentials Spin-orbit Nuclear systems breaking the time reversal symmetry characterized by currents which are accounted for through space-like component of the 4-potentiel : Magnetism

73. Why a Relativstic Approach? Pseudo-spin symmetry

74. Why a Relativstic Approach? Pseudo-spin symmetry Relativistic interpretation : comes from the fact that |V+S|«|S|≈|V| ( J. Ginoccho PR 414(2005) 165-261 )

75. Why a Relativstic Approach? Saturation mechanism of nuclear matter

76. Why a Relativstic Approach? p F >> 1 :  Scalar density becomes constant  Vector density diverge  Saturation of nuclear matter

77. Why a Relativstic Approach? First contribution to the expansion:

78. Why a Relativstic Approach? Figure from C. Fuchs (LNP 641: 119-146, 2004)

79. Why Fock terms? Relativistic mean field models (RMF) treat implicitly Fock terms through fit of model parameters to data Relativistic Hartree-Fock models (RHF): more involved approaches which take explicitly into account the Fock contributions RHB in axial symmetry D. Vretenar et al (Phys.Rep. 409:101- 259,2005) RHFB in spherical symmetry W. Long et al (Phys. Rev. C 81:024308, 2010) N N N N RHFB in axial symmetry J.-P. Ebran et al Phys. Rev. C 83, 064323 (2011)

80. Why Fock terms? Effective Mass Figure from W. Long et al (Phys.Lett.B 640:150, 2006) Effective mass in symmetric nuclear matter obtained with the PKO1 interaction

81. Why Fock terms? Shell Structure Figure from N. van Giai (International Conference Nuclear Structure and Related Topics, Dubna, 2009) Explicit treatment of the Fock term  introduction of pion +  N tensor coupling  N tensor coupling (accounted for in PKA1 interaction) leads to a better description of the shell structure of nuclei: artificial shell closure are cured (N,Z=92 for example)

82. Why Fock terms? RPA : Charge exchange excitation Figure from H. Liang et al. (Phys.Rev.Lett. 101:122502, 2008) RHF+RPA model fully self-consistent contrary to RH+RPA model

83. Rôle des corrections relativistes dans le mécanisme de saturation Distinction between scalar and vector densities lost : 2) Approches relativistes C. Pourquoi une approche relativiste ? Corrections relativistes i) Non-relativistic limit :

84. Rôle des corrections relativistes dans le mécanisme de saturation ii) Corrections relativistes cinématiques : Termes d’ordre dans lesquels Corrections cinématiques peuvent être rajoutées dans n’importe quel potentiel NN non-relativiste Distinction entre densité scalaire et densité vecteur retrouvée, mais brisure de l’auto-cohérence caractérisant l’évaluation de la densité scalaire 2) Approches relativistes C. Pourquoi une approche relativiste ? Corrections relativistes

85. Rôle des corrections relativistes dans le mécanisme de saturation Saturation de la matière nucléaire retrouvée à l’échelle du champ moyen!! Mais à une énergie et à un moment de fermi irréalistes 2) Approches relativistes C. Pourquoi une approche relativiste ? Corrections relativistes

86. Rôle des corrections relativistes dans le mécanisme de saturation iii) Corrections relativistes dynamiques : corrections générées par le spineur habillé par rapport au spineur libre  Saturation de la matière nucléaire plus proche du point empirique 2) Approches relativistes C. Pourquoi une approche relativiste ? Corrections relativistes

87. Contenu physique des corrections relativistes dynamiques 2) Approches relativistes C. Pourquoi une approche relativiste ? Corrections relativistes Corrections relativistes dynamiques correspondent à une contribution d’antinucléons. Petit paramètre (~0.1 dans le modèle de Walecka) justifiant développement perturbatif On développe le spineur sur la base des spineurs de Dirac dans le vide

88. 2) Approches relativistes C. Pourquoi une approche relativiste ? Corrections relativistes Première contribution non-nulle du développement : Contribution interprétée comme une contribution à 3 corps, ne pouvant pas être ajoutée comme correction dans un potentiel NN non-relativiste Contenu physique des corrections relativistes dynamiques

89. 3) Results A. Ground state observables Two-neutron drip-line Two-neutron separation energy E : S 2n = E tot (Z,N) – E tot (Z,N-2). Gives global information on the Q-value of an hypothetical simultaneous transfer of 2 neutrons in the ground state of (Z,N-2) S 2n < 0  (Z,N) Nucleus can spontaneously and simultaneously emit two neutrons  it is beyond the two neutrons drip-line

90. 3) Results A. Ground state observables Axial deformation  For Ne et Mg, PKO2 deformation’s behaviour qualitatively the same than the other interactions PKO2 β systematically weaker than DDME2 and Gogny D1S one

91. 3) Results A. Ground state observables Charge radii  DDME2 closer to experimental data Better agreement between PKO2 and DDME2 for heavier isotopes

92. Energy Density Functional