Gogny-HFB Nuclear Mass Model

Report
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
I.
Energy Density Functional
II. The Gogny Force
III. Results
 Relativistic Hartree-Fock-Bogoliubov in Axial Symmetry
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
I. Energy Density Functional
I. Energy Density Functional
 Designed to compute average value of few-body operators
 Independent particle picture



I. Energy Density Functional
I. Energy Density Functional
 Particle-Hole and Particle-Particle fields involved in HFB-like equation
I. EDF: Symmetry Breaking
  1  1   2 
2
3 
3
4 
4
5 
5
6 
6
7 
7
8 
8
6 
9
3+[202]
1+[200]
1+[211]
5+[202]
3+[211]
1+[220]
1d3/2
2s1/2
1d5/2
1p1/2
1p3/2
1-[101]
3-[101]
1-[110]
1s1/2
1 particule – 1 hole
excitations
2 particules – 2 holes
excitations
3 particules – 3 holes
excitations
1+[000]
 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 Restoration
 Restoration of broken symmetries : MR-level
 Configuration mixing method : GCM
I. EDF: Symmetry Restoration
II. Gogny Interaction
 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: Two Fitting Philosophies

14 parameters : (W,B,H,M)1 ; (W,B,H,M)2 ; t3 ; x3 ;  ; WLS ; m1 ; m2
II. Gogny: Two Fitting Philosophies
 “Traditional” method involving small set of magic nuclei (!!!) at SR-level
Initial Data
B.E., Rc
(16O, 90Zr)
Pairing
considerations
« Theoretical »
data at SR-level
D1
D1S
D1N
Inversion
4x4 equations
system
W1 B1 H1 M1
W2 B2 H2 M2
4x4 equations
system
Symmetry
energy
Test in Nuclear
matter:
(r, E/A)sat
m*/m K
t3 ; x3 ;  ; WLS ;
m1 ; m2
Reject
Validation
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)
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
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
 GCM + GOA
 M. Girod and B. Grammaticos, Nucl. Phys. A330 40 (1979)
 J. Libert, M. Girod and J.-P. Delaroche, Phys. Rev. C60 054301 (1999)
II. Gogny: Two Fitting Philosophies
For 1/3 of 2149 exp masses (Audi et
al 2003) – N=Z,N=Z±1, N=Z±2
Trial
force
automatic fit
on masses
New
force
D1M
D1M
II. Gogny: Two Fitting Philosophies
Acceptable
rms, J, K
Trial
force
automatic fit
on masses
New
force
Check
properties
D1M
II. Gogny: Two Fitting Philosophies
• ~ 200/782 exp. charge radii
with dynamical correction
Play on kF to adjust globally
Acceptable
rms, J, K
Trial
force
automatic fit
on masses
New
force
Check
properties
D1M
II. Gogny: Two Fitting Philosophies
• Nuclear Matter Properties
• ~ 200/782 exp. charge radii
with dynamical correction
Play on kF to adjust globally
+ Landau Parameters (stability, sum
rules, G0 ~ 0; G0’~ 0.9-1 (Borzov et al.
1981))
Acceptable
rms, J, K
Trial
force
automatic fit
on masses
New
force
Check
properties
D1M
II. Gogny: Two Fitting Philosophies
• Nuclear Matter Properties
• ~ 200/782 exp. charge radii
with dynamical correction
Play on kF to adjust globally
• Energy of 2+ levels
+ Landau Parameters (stability, sum
rules, G0 ~ 0; G0’~ 0.9-1 (Borzov et al.
1981))
Acceptable
rms, J, K
• Moment of inertia
244
Pu
Trial
force
automatic fit
on masses
New
force
Check
properties
II. Gogny: Two Fitting Philosophies
D1M
Acceptable
rms, J, K
Trial
force
automatic fit
on masses
New
force
Acceptable
rms, J,
K,prop.
Check
properties
New
Cstr.
II. Gogny: Two Fitting Philosophies
D1M
Acceptable
rms, J, K
Trial
force
automatic fit
on masses
New
force
Acceptable
rms, J,
K,prop.
Check
properties
New
Cstr.
New D
II. Gogny: Two Fitting Philosophies
D1M
Acceptable
rms, J, K
Trial
force
automatic fit
on masses
New
force
Acceptable
rms, J,
K,prop.
Check
properties
New
Cstr.
New D
New
Dquad
II. Gogny: Two Fitting Philosophies
D1M
Acceptable
rms, J, K
Trial
force
automatic fit
on masses
New
force
Acceptable
rms, J,
K,prop.
Check
properties
New
Cstr.
New D
New
Dquad
Quadrupole correction to the binding energy
6
DE
quad
[M e V ]
5
4
3
2
1
0
0
40
80
120
N
160
200
240
II. Gogny: Two Fitting Philosophies
D1M
Acceptable
rms, J, K
Trial
force
automatic fit
on masses
New
force
Acceptable
rms, J,
K,prop.
Check
properties
New
Cstr.
New D
New
Dquad
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
III. Results: D1N and the Neutron Matter
EOS
 F. Chappert, M. Girod & S. Hilaire, Phys. Lett. B668 (2008) 420.
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
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
Results: Masses
Comparison with 2149 Exp. Masses
e = 0.126 MeV
r.m.s = 0.798 MeV
III. Results: Radii
Comparison with 707 Exp. Charge Radii
r.m.s = 0.031 fm
R ch 
D R corr 
2
2
R HFB  D R corr
2
2
R dyn  R HFB
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
15
D M [M e V ]
10
5
0
-5
-10
D1M – HFB17
-15
0
40
80
D1M – FRDM
120
N
160
200
0
40
80
120
N
160
200
240
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, …)
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
Kinematics
Why a Relativstic Approach?
v( p) 
E ( p )
p
v 
3 pF
4 M eff
1   1 1
v
2
c
2
 2  10 %
•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
Why a Relativstic Approach?
In medium Chiral Perturbation
theory, D. Vretenar et. al.
• Modification of the vacuum structure in presence of baryonic matter at the origin of
the S and V self energies felt by nucleons
Why a Relativstic Approach?
• QCD sum rules  Large scalar and time-like self energies with opposite sign
Why a Relativstic Approach?
 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)
Why Fock Term?
• 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
N
RHB in axial
symmetry
D. Vretenar et al
Phys.Rep. 409:101259,2005
N
N
N
RHFB in spherical
symmetry
W. Long et al
Phys. Rev. C 81,
024308 (2010)
RHFB in axial symmetry
J.-P. Ebran et al
Phys. Rev. C 83, 064323
(2011)
RHFBz Model
• 8 free parameters
Lagrangian
N
Hamiltonian
N
• Mean-field approximation : expectation value in the
HFB ground state
N
EDF
N
N
N
• Minimization
RHFB equations
• Resolution in a deformed harmonic oscillator basis
Observables
Results
Neutron density in the Neon isotopic chain
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
244
Pu
III. Results: Pairing
164
Er
III. Results: Giant Resonances
15.85 MeV
14.25 MeV
GMR
Eexp = 14.17 MeV D. H. Youngblood et al., Phys.
Rev. Lett. 82, 691 (1999).
208Pb
GDR
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
D1S Properties
 Structure properties of ~7000 nuclei + Spectroscopic properties of low energy
collective levels for ~1700 even-even nuclei
 S. Hilaire & M. Girod, Eur. Phys. J A33 237(2007)
D1S Properties
Results: Masses
Comparison with 2149 Exp. Masses
e = 0.126 MeV
r.m.s = 0.798 MeV
Quadrupole correction to the binding energy
6
DE
quad
[M e V ]
5
4
3
2
1
0
0
40
80
120
N
160
200
240
Why a Relativstic Approach?
•Relevance of covariant approach : not imposed by the need of 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
 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 + r) : better description of shell structure

Fully self-consistent beyond mean-field models
N
RHB in axial
symmetry
D. Vretenar et al
Phys.Rep. 409:101259,2005
N
N
N
RHFB in spherical
symmetry
W. Long et al
Phys. Rev. C 81,
024308 (2010)
RHFB in axial symmetry
J.-P. Ebran et al
Phys. Rev. C 83, 064323
(2011)
Why a Relativstic Approach?
Magnetism
Spin-orbit
S and V potentials characterize the essential properties of nuclear systems :
• Central Potential : quasi cancellation of potentials
• Spin-orbit : constructive combination of potentials
• Nuclear systems breaking the time reversal symmetry characterized by currents
which are accounted for through space-like component
of the 4-potentiel :
Why a Relativstic Approach?
• Pseudo-spin symmetry
1

 nr , l, j  l  
2

3

 n r  1, l  2 , j  l  
2

Why a Relativstic Approach?
• Pseudo-spin symmetry
1

 nr , l, j  l  
2

• Relativistic interpretation : comes from
the fact that |V+S|«|S|≈|V|
( J. Ginoccho PR 414(2005) 165-261 )
3

 n r  1, l  2 , j  l  
2

Why a Relativstic Approach?
• Saturation mechanism of nuclear matter
E pot
A
 r
1
0
1
1   g
 s   0     
2
2   m

 r s  g  rb 




 r m  r
0
0 

  

2
2
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
N
N
N
RHB in axial
symmetry
D. Vretenar et al
(Phys.Rep. 409:101259,2005)
RHFB in spherical
symmetry
W. Long et al
(Phys. Rev. C
81:024308, 2010)
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 + 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
2) Approches relativistes
C. Pourquoi une approche relativiste ? Corrections relativistes
Rôle des corrections relativistes dans le mécanisme de saturation
i) Non-relativistic limit :
• Distinction between scalar and vector densities lost :
r s  r   r b r 
E pot

A
1   g
   
2   m


 g   rb
 
 

m  r

    0
2
2
2) Approches relativistes
C. Pourquoi une approche relativiste ? Corrections relativistes
Rôle des corrections relativistes dans le mécanisme de saturation
2
ii) Corrections relativistes cinématiques : Termes d’ordre
M
*
 M
 p 


 M 
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
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
• On développe le spineur sur la base des spineurs de Dirac dans le vide
• Corrections
relativistes
dynamiques
correspondent
à
une
contribution
d’antinucléons.
Petit paramètre (~0.1 dans le modèle de Walecka)
justifiant développement perturbatif
2) Approches relativistes
C. Pourquoi une approche relativiste ? Corrections relativistes
Contenu physique des corrections relativistes dynamiques
• 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
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

similar documents