gogny-hfb nuclear mass model s. goriely (ulb), s. hilaire (cea-dam-dif) et. al. j.-p. ebran...
TRANSCRIPT
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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
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Outline
Gogny-HFB Nuclear Mass Model
I. Energy Density Functional
II. The Gogny Force
III. Results
Relativistic Hartree-Fock-Bogoliubov in Axial Symmetry
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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
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I. Energy Density Functional
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Designed to compute average value of few-body operators
Independent particle picture
I. Energy Density Functional
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Crystal-like G.S.
I. Energy Density Functional
Quantum Liquid-like G.S.
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Particle-Hole and Particle-Particle fields involved in HFB-like equation
I. Energy Density Functional
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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
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Symmetry breaking : take into account additional correlations keeping a single particle picture
I. EDF: Symmetry Breaking
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Restoration of broken symmetries : MR-level
Configuration mixing method : GCM
I. EDF: Symmetry Restoration
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I. EDF: Symmetry Restoration
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Gogny strategy : parametrize both p-h and p-p channels with the same phenomenological finite-range 2-body interaction
II. Gogny Interaction
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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
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Finite range : avoid pathologies “beyond HF” due to unrealistic behavior of 0-range forces at high relative momenta
II. Gogny Interaction
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II. Gogny: Two Fitting Philosophies
14 parameters : (W,B,H,M)1 ; (W,B,H,M)2 ; t3 ; x3 ; a ; WLS ; m1 ; m2
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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
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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
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D1M
Infinite base correction
II. Gogny: Two Fitting Philosophies
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D1M
60Ni
II. Gogny: Two Fitting Philosophies
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D1M
120Sn
II. Gogny: Two Fitting Philosophies
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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
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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
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automatic fit
on masses
D1M
Trial force
New force
Check
properties
Acceptable rms, J, K
II. Gogny: Two Fitting Philosophies
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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III. Results: D1N and the Neutron Matter EOS
F. Chappert, M. Girod & S. Hilaire, Phys. Lett. B668 (2008) 420.
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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
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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
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Results: Masses
Comparison with 2149 Exp. Masses
e = 0.126 MeVr.m.s = 0.798 MeV
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III. Results: Radii
Comparison with 707 Exp. Charge Radii
Rch RHFB2 Rcorr
2
Rcorr Rdyn2 RHFB
2
r.m.s = 0.031 fm
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III. Results: Pairing
Sn
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III. Results: Pairing
Sn
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III. Results: Nuclear Matter
kF=1.346 fm-1 J=28.6 MeV m*/m=0.746 Kinf =225 MeV
Pure Neutron Matter
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III. Results: Nuclear Matter
kF=1.346 fm-1 J=28.6 MeV m*/m=0.746 Kinf =225 MeV
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III. Results: Nuclear Matter
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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
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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
![Page 46: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/46.jpg)
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
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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
![Page 48: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/48.jpg)
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.
![Page 49: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/49.jpg)
Why a Relativstic Approach?
• QCD sum rules Large scalar and time-like self energies with opposite sign
![Page 50: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/50.jpg)
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)
![Page 51: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/51.jpg)
• 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?
![Page 52: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/52.jpg)
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
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Neutron density in the Neon isotopic chain
Results
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Results
N=32 Masses
SLy4 : M.V. Stoitsov et al, Phys. Rev. C68 (2003) 054312
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Results
N=32 static quadrupole deformations
![Page 56: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/56.jpg)
Results
Charge radii
![Page 57: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/57.jpg)
Conclusion & Perspectives
First RHFB model in axial symmetry
Encouraging results but too heavy for triaxial calculations or MR-level
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Thank you
![Page 59: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/59.jpg)
III. Results: Pairing
244Pu
![Page 60: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/60.jpg)
III. Results: Pairing
164Er
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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).
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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)
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III. Results: Nuclear Matter
kF=1.346 fm-1 J=28.6 MeV m*/m=0.746 Kinf =225 MeV
Pure Neutron Matter
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III. Results: Shell Gaps
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III. Results: Shell Gaps
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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)
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D1S Properties
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Results: Masses
Comparison with 2149 Exp. Masses
e = 0.126 MeVr.m.s = 0.798 MeV
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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
![Page 70: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/70.jpg)
• 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?
![Page 71: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/71.jpg)
• 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?
![Page 72: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/72.jpg)
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
![Page 73: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/73.jpg)
Why a Relativstic Approach?
• Pseudo-spin symmetry
2
1,, ljlnr
2
3,2,1 ljlnr
![Page 74: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/74.jpg)
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 )
![Page 75: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/75.jpg)
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
![Page 76: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/76.jpg)
Why a Relativstic Approach?
• pF >> 1 :
Scalar density becomes constant
Vector density diverge
Saturation of nuclear matter
![Page 77: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/77.jpg)
Why a Relativstic Approach?
• First contribution to the expansion:
![Page 78: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/78.jpg)
Why a Relativstic Approach?
Figure from C. Fuchs (LNP 641: 119-146 ,
2004)
![Page 79: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/79.jpg)
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)
![Page 80: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/80.jpg)
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
![Page 81: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/81.jpg)
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)
![Page 82: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/82.jpg)
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
![Page 83: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/83.jpg)
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 :
![Page 84: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/84.jpg)
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
![Page 85: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/85.jpg)
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
![Page 86: 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](https://reader035.vdocuments.us/reader035/viewer/2022062304/56649c775503460f9492c52f/html5/thumbnails/86.jpg)
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
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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
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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
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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
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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
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3) Results A. Ground state observables
Charge radii
DDME2 closer to experimental data Better agreement between PKO2 and DDME2 for
heavier isotopes
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Energy Density Functional