1. Variational multiparticle-multihole configuration mixing approach
using Gogny force
Nathalie Pillet
CEA Bruyères-le-Châtel, France
Collaborators: JF. Berger(1) , E. Caurier(2) , D. Gogny(3), H. Goutte(1)
(1) CEA, Bruyères-le- Châtel (2) IPHC, Strasbourg (3) LLNL, Livermore (4) IPN, Orsay
“Convergence of Particle-Hole expansions for the description of nuclear correlations”
Collaboration with N. Sandulescu(5), N. Van Giai(6) and JF Berger
(5) NIPNE, Bucharest (6) IPN, Orsay

2. “Advances in Nuclear Physics”, vol.9, 1977
Introduction…
“Advances in Nuclear Physics”, vol.9, 1977
Computational methods for shell-model calculations

3. Brief history of … HTDA approach
Introduction…
Brief history of …
HTDA approach
N. Pillet, P. Quentin and J. Libert, Nucl. Phys. A697 (2002) 141.
High-K isomers in 178Hf
P. Quentin, H. Laftchiev, D. Samsoen et al., Nucl. Phys. A734 (2004) 477.
Rotating nuclei (kinetic and dynamical moments of inertia in 192Hg and 194Pb)
L. Bonneau, P. Quentin and K. Sieja, Phys. Rev. C76 (2007)
Ground state properties of even-even N=Z nuclei for [56Ni;100Sn]
L. Bonneau, J. Bartel and P. Quentin, arXiv: v1 [nucl-th].
Isospin mixing
L. Bonneau Talk
Variational mpmh configuration mixing approach
N. Pillet, JF Berger and E. Caurier, to be submitted to PRC.
Pairing correlations in Sn isotopes with D1S Gogny force

4. Plan Formalism of the Variational mpmh Configuration Mixing
Introduction…
Plan
Formalism of the Variational mpmh Configuration Mixing
Motivations
Wave functions and symmetries
Variational principle- Derivation of equations
Applications to Pairing-type correlations
Exactly solvable model- test of truncations
Description of ground states of even-even Sn isotopes with variational mpmh configuration mixing using D1S Gogny force
Summary and Outlook

5. beyond mean-field methods
Motivations
Towards a unified description of long range correlations in the context of
beyond mean-field methods
“Take advantage of both Mean-field and Shell-Model approaches”
Description of all correlations (Pairing, RPA, particle vibration coupling)
All nucleons are considered for the description of states
Conservation of particle numbers + Pauli principle
Description on the same footing of even-even, odd and odd-odd nuclei
Description of both ground states and excited states
Symmetries
Axial symmetry:
Eigen-solutions are specified by K quantum number
(K projection of total angular momentum J)
Parity

6. Formalism Trial wave function + + … Slater determinant vacuum 0p0h
(a priori for ground and yrast states) +
+ …
0p0h
1p1h
2p2h
mpmh
Slater determinant
and
vacuum
Variational parameters:
- Mixing coefficients Aαπαν
- Orbitals a+i

7. Variational Principle applied to…
Functional:
Present Prescription for one-body density:
Determination of Mixing Coefficients
Determination of Optimized orbitals

8. Variational Principle applied to… Determination of Mixing Coefficients
“Secular equation” equivalent to the diagonalization of H(ρ)+δH(ρ) in the multiconfiguration space
Highly non-linear equation because of δH(ρ)

9. Variational Principle applied to… Determination of Mixing Coefficients
Two-body correlation function
with
Residual interaction: two-body matrix elements + rearrangement terms
Example: for α≠α’
Importance of consistency
JP. Blaizot and D. Gogny, Nucl. Phys. A284 (1977)
D. Gogny and R. Padjen, Nucl. Phys. A293 (1977)

10. Variational Principle applied to…
Pairing, RPA
Particle-vibration
RPA
Pairing

11. Variational Principle applied to… Determination of Optimized orbitals
Variation of the functional
with
Definition of projectors associated with the multiconfiguration space I
Inside I
Outside I

12. Variational Principle applied to… Variational Principle applied to…
Thouless’ theorem
with
At first order:
with
G(σ) antisymmetric

13. Variational Principle applied to… Solution of the mpmh approach
Solution of both equations
Aαπαν
orbitals
In present application: neglect of σ

14. Test of Truncations in the mpmh wave function…
Richardson exact solution of Pairing Hamiltonian(*)
Test of Truncations in the mpmh wave function…
Pairing Hamiltonian
(*) R.W. Richardson and N. Sherman, Nucl. Phys. 52 (1964) 221.

15. Richardson exact solution… Exact solution of Pairing Hamiltonian
Similarity between the many-fermion-pair system with pairing forces and the many-boson system with one-body forces
Exact wave function : mpmh wave function including all the configurations built as pair excitations
Exact solution obtained from a coupled system of algebraic equations deduced from variational principle
Test of the importance of the different terms in the mpmh wave function expansion (1 pair, 2pairs…)
(*) R.W. Richardson, Phys.Rev. 141 (1966) 949.

16. Richardson exact solution…
Picket fence model (*) g εi
εi+1 d
System of 2N particles in 2N equispaced and doubly-degenerated levels
System of identical fermions
Constant pairing interaction strength
Prototype of axially deformed nuclei
(*) R.W. Richardson, Phys.Rev. 141 (1966) 949.

17. Ground state Correlation energy
Richardson exact solution…
Ground state Correlation energy
Ecorr=E(g≠0)-E(g=0)
ΔEcorr = Ecorr (exact) – Ecorr (mpmh)
Truncation in mpmh order of excitation
g (Pairing interaction strength)
Truncation in excitation energy
N. Pillet, N. Sandulescu, Nguyen Van Giai and JF. Berger , Phys.Rev. C71 , (2005).

18. Richardson exact solution… Ground state occupation probabilities
N. Pillet, N. Sandulescu, Nguyen Van Giai and JF. Berger , Phys.Rev. C71 , (2005).

19. Variational mpmh configuration mixing applied to Pairing-type correlations in Sn isotopes using D1S Gogny force (*)
mpmh wave function
Usual pairing-type correlations (pp and nn)
No residual proton-neutron interaction
Ground states of even-even spherical nuclei
Kp=Jp=0+
Configurations with:
Kp=0+
One excited pair of nucleons . …
Several excited Pairs of nucleons
Kp=0+
(*) N. Pillet, JF Berger and E. Caurier, to be submitted to PRC.

20. Link between mpmh and PBCS wave functions
Variational mpmh configuration mixing applied to pairing…
Link between mpmh and PBCS wave functions
BCS wave function
Projected BCS (PBCS) wave function
Component of |BCS> with 2N nucleons
HF-type reference state

21. Link between mpmh and PBCS wave functions
Variational mpmh configuration mixing applied to pairing…
Link between mpmh and PBCS wave functions
PBCS wave function
with
mpmh wave function:
mpmh wave function similar to PBCS one with more general mixing coefficients

22. Variational mpmh configuration mixing applied to pairing…
D1S Gogny force
Parameterization
central
Spin-orbite
Contributions in Spin-Isospin ST channels for D1S
Residual interaction

23. Variational mpmh configuration mixing applied to pairing…
Three pairing regimes
Weak pairing
100Sn
Medium pairing
106Sn
Strong pairing
116Sn
S. Hilaire and M. Girod, EPJ A33 (2007) 237.

24. Results without Self-Consistency
Variational mpmh configuration mixing applied to pairing…
Results without Self-Consistency
Convergence properties
Some Dimensions
11 shell harmonic oscillator basis (286 neutron +286 proton states)
Number of configurations
Shell-model “Standard” dimensions
(E. Caurier)

25. Results without Self-Consistency
Variational mpmh configuration mixing applied to pairing…
Results without Self-Consistency
Correlation energy (MeV)
Configurations with 1 and 2 excited pairs are required
Configurations with 3 excited pairs are negligible

26. Results without Self-Consistency
Variational mpmh configuration mixing applied to pairing…
Results without Self-Consistency
Correlation energy (MeV)
1 pair : MeV
2 pairs : MeV
100Sn
1 pair : MeV
2 pairs : MeV
116Sn
Contributions associated with:
Protons ~ 1.7 MeV
Coulomb ~ 700 keV
S=0 T=1 (Central+ s.o.) ~ 99% of Ecorr without Coulomb

27. Results without Self-Consistency
Variational mpmh configuration mixing applied to pairing…
Results without Self-Consistency
Structure of correlated wave functions
3s1/2→ 1d3/2
3s1/2→ 1h11/2
116Sn
106Sn
2d5/2→ 1g7/2
100Sn
No specific configurations
65% ~ (92%)π x (71%)ν

28. Results without Self-Consistency
Variational mpmh configuration mixing applied to pairing…
Results without Self-Consistency
~141 neutron states
~ 98 proton states
Effect of a truncated space
Total
Truncated
Nucleus
|Ecorr|
116Sn
5.44
3.45
106Sn
4.62
3.54
100Sn
3.67
2.79
Total
Truncated

29. Effect of Approximate Self-Consistency
Variational mpmh configuration mixing applied to pairing…
Effect of Approximate Self-Consistency
First step: neglecting of the two-body correlation matrix σ
Use of the truncated space for: the number of valence orbitals
the order of excitation
Correlation energy

30. Effect of Approximate Self-Consistency
Variational mpmh configuration mixing applied to pairing…
Effect of Approximate Self-Consistency
Correlated wave function
PBCS after variation

31. Polarization of single particle states
Variational mpmh configuration mixing applied to pairing…
Polarization of single particle states
7/2
7/2

32. Variational mpmh configuration mixing applied to pairing…
Effects of Pairing correlations on proton and neutron single particle spectrum

33. Variational mpmh configuration mixing applied to pairing…
Occupation
Probabilities

34. Variational mpmh configuration mixing applied to pairing…
Neutron Skin
Charge Radii
Exp.

35. mpmh configuration mixing: Summary and outlook
Formalism for the description of ground states and yrast states
Still a lot of work to do !
First applications to nuclear superfluidity quite encouraging
Specify the fundamental nature of correlations induced in our study
Study of the effect of pn pairing-type correlations on Sn ground states
Study of different prescriptions for ρ
Study the effect of the two body correlation function σ

36. More general correlations…
Challenge for Gogny density-dependent interaction
Unique interaction for both mean-field and residual part
pairing, RPA and particle-vibration correlations
Interaction with good properties in T=1 and T=0 residual channels
Shell-Model interactions: a guide for effective interactions?
Different valence spaces
Different truncations in excitation order of the wave function?
Shell-model matrix elements: fitted to reproduced excited states

37. Matrix elements of Gogny force in sd shellJ J

38. Matrix elements of Gogny force in sd shellJ J J J
D2: Gogny force with a finite range density-dependent term, PhD thesis of F. Chappert.

39. Matrix elements of Gogny force in sd shellJ J J J J J