Gogny-HFB Nuclear Mass Model

Gogny-HFB Nuclear Mass ModelS. 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 ModelI. Energy Density FunctionalII. The Gogny ForceIII. Results

Relativistic Hartree-Fock-Bogoliubov in Axial Symmetry

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 Need for predictive approach

I. Energy Density Functional

Designed to compute average value of few-body operators Independent particle picture


Crystal-like G.S.


Quantum Liquid-like G.S.

Particle-Hole and Particle-Particle fields involved in HFB-like equation


1 particule – 1 holeexcitations

2 particules – 2 holesexcitations

3 particules – 3 holesexcitations

1d5/2

2s1/21d3/2

8

20

8

20

8

20

8

20

8

20

8

20

8

20

8

20

8

20

968877665544332211

1s1/2

1p3/21p1/2

2 2 2 2 2 2 2 2 2

1+[000]

3-[101]1-[101]

1+[220]

1+[211]1+[200]

2

8

1-[110]

3+[211]5+[202]

3+[202]

Symmetry breaking : take into account additional correlations keeping a single particle picture

I. EDF: Symmetry Breaking

Symmetry breaking : take into account additional correlations keeping a single particle picture

I. EDF: Symmetry Breaking

Restoration of broken symmetries : MR-level

Configuration mixing method : GCM

I. EDF: Symmetry Restoration

I. EDF: Symmetry Restoration

Gogny strategy : parametrize both p-h and p-p channels with the same phenomenological finite-range 2-body interaction

II. Gogny Interaction

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


Finite range : avoid pathologies “beyond HF” due to unrealistic behavior of 0-range forces at high relative momenta


II. Gogny: Two Fitting Philosophies

14 parameters : (W,B,H,M)1 ; (W,B,H,M)2 ; t3 ; x3 ; a ; WLS ; m1 ; m2

Inversion

4x4 equations system

4x4 equations system

W1 B1 H1 M1W2 B2 H2 M2

Test in Nuclear matter:

(r, E/A)sat m*/m K

B.E., Rc

(16O, 90Zr)

Pairing consideration

s

Symmetry energy

Initial Data

t3 ; x3 ; ; WLS ; m1 ; m2

Reject Validation

« Theoretical » data at SR-

level

D1 D1S D1N

“Traditional” method involving small set of magic nuclei (!!!) at SR-level


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)m1=0.7-0.8 ; m2=1.2 ; x3=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• kF kept 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)


D1M

Infinite base correction


D1M

60Ni


D1M

120Sn


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


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


automatic fit

on masses

D1M

Trial force

New force

Check

properties

Acceptable rms, J, K


automatic fit

on masses

D1M

Trial force

New force

Check

properties


• ~ 200/782 exp. charge radii with dynamical correction Play on kF to adjust globally


automatic fit

on masses

D1M

Trial force

New force

Check

properties



• Nuclear Matter Properties

+ Landau Parameters (stability, sum rules, G0 ~ 0; G0’~ 0.9-1 (Borzov et al. 1981))


244Puautomatic fit

on masses

D1M

Trial force

New force

Check

properties



• Energy of 2+ levels

• Nuclear Matter Properties

+ Landau Parameters (stability, sum rules, G0 ~ 0; G0’~ 0.9-1 (Borzov et al. 1981))

• Moment of inertia


automatic fit

on masses

Trial force

New force

Check

properties


New Cstr.

Acceptable rms, J, K,prop.

D1M


automatic fit

on masses

D1M

Trial force

New force

Check

properties


New Cstr.


New D


automatic fit

on masses

D1M

Trial force

New force

Check

properties


New Cstr.


New DNew Dquad


automatic fit

on masses

D1M

Trial force

New force

Check

properties


New Cstr.


New DNew Dquad


Quadrupole correction to the binding energy

0

1

2

3

4

5

6

0 40 80 120 160 200 240

DE qu

ad [M

eV]

N

automatic fit

on masses

D1M

Trial force

New force

Check

properties


New Cstr.


New DNew Dquad


III. Results: MassesComparison with 2149 Exp. Masses

D1S

r.m.s ~ 4.4 MeV

• Eth = EHFB

r.m.s ~ 2.6 MeV

• Eth = EHFB - D

r.m.s ~ 2.9 MeV

• Eth = EHFB - D - Dquad

III. Results: D1N and the Neutron Matter EOS

F. Chappert, M. Girod & S. Hilaire, Phys. Lett. B668 (2008) 420.


D1N

r.m.s ~ 2.5 MeV

r.m.s ~ 0.95 MeV

• Eth = EHFB

• Eth = EHFB - D

• Eth = EHFB - D - Dquad


r.m.s ~ 2.5 MeV

e = 0.126 MeVr.m.s = 0.798 MeV

r.m.s ~ 0.95 MeV

Results: MassesComparison with 2149 Exp. Masses

e = 0.126 MeVr.m.s = 0.798 MeV

III. Results: RadiiComparison with 707 Exp. Charge Radii

Rch RHFB2 DRcorr

2

DRcorr Rdyn2 RHFB

2

r.m.s = 0.031 fm

III. Results: Pairing

Sn


Sn

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

Pure Neutron Matter


III. Results: Nuclear Matter

III. Results: Comparison with other Mass Formula

0 40 80 120 160 200 240N

-15

-10

-5

0

5

10

15

0 40 80 120 160 200

DM

[MeV

]

N

D1M – HFB17 D1M – FRDM

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

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

Why a Relativstic Approach? p

pEpv

)()(eff

F

Mpv

43

Kine

mati

cs

%102111 2

2

cv

•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 potentialV ~ +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

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.


• QCD sum rules Large scalar and time-like self energies with opposite sign

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


Figure from C. Fuchs (LNP 641: 119-146 ,

2004)

• 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 + r) : 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?

Hamiltonian

Observables

• Resolution in a deformed harmonic oscillator basis

EDF

• Mean-field approximation : expectation value in the HFB ground state

N NN

N

RHFB equations

• Minimization

N N

Lagrangian • 8 free parameters

RHFBz Model

Neutron density in the Neon isotopic chain

Results

ResultsN=32 Masses

SLy4 : M.V. Stoitsov et al, Phys. Rev. C68 (2003) 054312

Results

N=32 static quadrupole deformations

ResultsCharge radii

Conclusion & Perspectives

First RHFB model in axial symmetry

Encouraging results but too heavy for triaxial calculations or MR-level

Thank you


244Pu


164Er

III. Results: Giant Resonances

14.25 MeV

GMR GDR

208Pb15.85 MeV

Eexp = 14.17 MeV D. H. Youngblood et al., Phys. Rev. Lett. 82, 691 (1999).

Eexp = 13.43 MeV B. L. Berman and S. C. Fultz, Rev. Mod. Phys. 47, 713 (1975).

III. Results: SpectroscopyExcitation 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)


Pure Neutron Matter

III. Results: Shell Gaps

III. Results: Shell Gaps

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)

D1S Properties

Results: MassesComparison with 2149 Exp. Masses

e = 0.126 MeVr.m.s = 0.798 MeV

Quadrupole correction to the binding energy

0

1

2

3

4

5

6

0 40 80 120 160 200 240

DE qu

ad [M

eV]

N

• Relativistic potentials :S ~ -400 MeV : Scalar attractive potentialV ~ +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


• 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 + r) : better description of shell

structure Fully self-consistent beyond mean-field models


D. Vretenar et al Phys.Rep. 409:101-

259,2005


W. Long et al Phys. Rev. C 81, 024308 (2010)

N

N

N

N





S and V potentials characterize the essential properties of nuclear systems :

• Central Potential : quasi cancellation of potentials

• Spin-orbit : constructive combination of potentialsSpin

-orb

it

• Nuclear systems breaking the time reversal symmetry characterized by currents

which are accounted for through space-like component of the 4-potentiel :

Mag

netis

m


• Pseudo-spin symmetry

21,, ljlnr

23,2,1 ljlnr


• Pseudo-spin symmetry

21,, ljlnr

23,2,1 ljlnr

• Relativistic interpretation : comes from the fact that |V+S|«|S|≈|V| ( J. Ginoccho PR 414(2005) 165-261 )


• Saturation mechanism of nuclear matter

0

2

0

2

01

0 21

21

rr

rrr

bss

pot

mg

mg

AE


• pF >> 1 : Scalar density becomes constant Vector density diverge Saturation of nuclear matter


• First contribution to the expansion:


Figure from C. Fuchs (LNP 641: 119-146 ,

2004)

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


D. Vretenar et al (Phys.Rep. 409:101-

259,2005)


W. Long et al (Phys. Rev. C

81:024308, 2010)

N

N

N

N



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

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 + rN tensor coupling• rN 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)

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

Rôle des corrections relativistes dans le mécanisme de saturation

• Distinction between scalar and vector densities lost :

rs r rb r

0

22

21

rr

bpot

mg

mg

AE

2) Approches relativistes C. Pourquoi une approche relativiste ? Corrections relativistes

i) Non-relativistic limit :


ii) Corrections relativistes cinématiques : Termes d’ordre dans lesquels

pM

2

M* M

• 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



• Saturation de la matière nucléaire retrouvée à l’échelle du champ moyen!!• Mais à une énergie et à un moment de fermi irréalistes



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


Contenu physique des corrections relativistes dynamiques


• 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


• 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

3) Results A. Ground state observables

Two-neutron drip-line

• Two-neutron separation energy E : S2n = Etot(Z,N) – Etot(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)

• S2n < 0 (Z,N) Nucleus can spontaneously and simultaneously emit two neutrons it is beyond the two neutrons drip-line


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


Charge radii

DDME2 closer to experimental data Better agreement between PKO2 and DDME2 for

heavier isotopes

Energy Density Functional

Documents

Gogny-HFB Nuclear Mass Model