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 Model
I. Energy Density Functional
II. The Gogny Force
III. 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
I. Energy Density Functional
Crystal-like G.S.
I. Energy Density Functional
Quantum Liquid-like G.S.
Particle-Hole and Particle-Particle fields involved in HFB-like equation
I. Energy Density Functional
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).
II. Gogny Interaction
Finite range : avoid pathologies “beyond HF” due to unrealistic behavior of 0-range forces at high relative momenta
II. Gogny Interaction
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 M1
W2 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 ; a ; WLS ; m1 ; m2
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
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 ; 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)
II. Gogny: Two Fitting Philosophies
D1M
Infinite base correction
II. Gogny: Two Fitting Philosophies
D1M
60Ni
II. Gogny: Two Fitting Philosophies
D1M
120Sn
II. Gogny: Two Fitting Philosophies
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
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
automatic fit
on masses
D1M
Trial force
New force
Check
properties
Acceptable rms, J, K
II. Gogny: Two Fitting Philosophies
automatic fit
on masses
D1M
Trial force
New force
Check
properties
Acceptable rms, J, K
• ~ 200/782 exp. charge radii with dynamical correction Play on kF to adjust globally
II. Gogny: Two Fitting Philosophies
automatic fit
on masses
D1M
Trial force
New force
Check
properties
Acceptable rms, J, K
• ~ 200/782 exp. charge radii with dynamical correction Play on kF to adjust globally
• Nuclear Matter Properties
+ Landau Parameters (stability, sum rules, G0 ~ 0; G0’~ 0.9-1 (Borzov et al. 1981))
II. Gogny: Two Fitting Philosophies
244Puautomatic fit
on masses
D1M
Trial force
New force
Check
properties
Acceptable rms, J, K
• ~ 200/782 exp. charge radii with dynamical correction Play on kF to adjust globally
• 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
II. Gogny: Two Fitting Philosophies
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
automatic fit
on masses
D1M
Trial force
New force
Check
properties
Acceptable rms, J, K
New Cstr.
Acceptable rms, J, K,prop.
New D
II. Gogny: Two Fitting Philosophies
automatic fit
on masses
D1M
Trial force
New force
Check
properties
Acceptable rms, J, K
New Cstr.
Acceptable rms, J, K,prop.
New DNew Dquad
II. Gogny: Two Fitting Philosophies
automatic fit
on masses
D1M
Trial force
New force
Check
properties
Acceptable rms, J, K
New Cstr.
Acceptable rms, J, K,prop.
New DNew Dquad
II. Gogny: Two Fitting Philosophies
Quadrupole correction to the binding energy
0
1
2
3
4
5
6
0 40 80 120 160 200 240
E
quad
[M
eV]
N
automatic fit
on masses
D1M
Trial force
New force
Check
properties
Acceptable rms, J, K
New Cstr.
Acceptable rms, J, K,prop.
New DNew Dquad
II. Gogny: Two Fitting Philosophies
III. Results: Masses
Comparison 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.
III. Results: Masses
Comparison with 2149 Exp. Masses D1N
r.m.s ~ 2.5 MeV
r.m.s ~ 0.95 MeV
• Eth = EHFB
• Eth = EHFB - D
• Eth = EHFB - D - Dquad
III. Results: Masses
Comparison with 2149 Exp. Masses
r.m.s ~ 2.5 MeV
e = 0.126 MeVr.m.s = 0.798 MeV
r.m.s ~ 0.95 MeV
Results: Masses
Comparison with 2149 Exp. Masses
e = 0.126 MeVr.m.s = 0.798 MeV
III. Results: Radii
Comparison with 707 Exp. Charge Radii
Rch RHFB2 Rcorr
2
Rcorr Rdyn2 RHFB
2
r.m.s = 0.031 fm
III. Results: Pairing
Sn
III. Results: Pairing
Sn
III. Results: Nuclear Matter
kF=1.346 fm-1 J=28.6 MeV m*/m=0.746 Kinf =225 MeV
Pure Neutron Matter
III. Results: Nuclear Matter
kF=1.346 fm-1 J=28.6 MeV m*/m=0.746 Kinf =225 MeV
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
M
[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
M
pv
4
3
Kine
mati
cs
%1021112
2
c
v
•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
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.
Why a Relativstic Approach?
• 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
Why a Relativstic Approach?
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 + ) : 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
Results
N=32 Masses
SLy4 : M.V. Stoitsov et al, Phys. Rev. C68 (2003) 054312
Results
N=32 static quadrupole deformations
Results
Charge radii
Conclusion & Perspectives
First RHFB model in axial symmetry
Encouraging results but too heavy for triaxial calculations or MR-level
Thank you
III. Results: Pairing
244Pu
III. Results: Pairing
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: 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)
III. Results: Nuclear Matter
kF=1.346 fm-1 J=28.6 MeV m*/m=0.746 Kinf =225 MeV
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: Masses
Comparison 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
E
quad
[M
eV]
N
• 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?
• 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?
Why a Relativstic Approach?
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
Why a Relativstic Approach?
• Pseudo-spin symmetry
2
1,, ljlnr
2
3,2,1 ljlnr
Why a Relativstic Approach?
• Pseudo-spin symmetry
2
1,, ljlnr
2
3,2,1 ljlnr
• Relativistic interpretation : comes from
the fact that |V+S|«|S|≈|V|
( J. Ginoccho PR 414(2005) 165-261 )
Why a Relativstic Approach?
• Saturation mechanism of nuclear matter
0
2
0
2
01
0 2
1
2
1
bss
pot
m
g
m
g
A
E
Why a Relativstic Approach?
• pF >> 1 :
Scalar density becomes constant
Vector density diverge
Saturation of nuclear matter
Why a Relativstic Approach?
• First contribution to the expansion:
Why a Relativstic Approach?
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
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 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 + 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)
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 :
s r b r
0
22
2
1
bpot
m
g
m
g
A
E
2) Approches relativistes C. Pourquoi une approche relativiste ? Corrections relativistes
i) Non-relativistic limit :
Rôle des corrections relativistes dans le mécanisme de saturation
ii) Corrections relativistes cinématiques : Termes d’ordre dans lesquels
p
M
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
2) Approches relativistes C. Pourquoi une approche relativiste ? Corrections relativistes
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
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
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
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
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
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
3) Results A. Ground state observables
Charge radii
DDME2 closer to experimental data Better agreement between PKO2 and DDME2 for
heavier isotopes
Energy Density Functional