Variational multiparticle-multihole configuration mixing approach using Gogny force

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

3. Introduction… L. Bonneau Talk 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) 014304. Ground state properties of even-even N=Z nuclei for [56Ni;100Sn] • L. Bonneau, J. Bartel and P. Quentin, arXiv: 0705.2587v1 [nucl-th]. Isospin mixing • 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. 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. Symmetries • Axial symmetry: Eigen-solutions are specified by K quantum number (K projection of total angular momentum J) • Parity Motivations Towards a unified description of long rangecorrelations 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

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

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

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… • Residual interaction:two-body matrix elements + rearrangement terms Example: for α≠α’ • Importance of consistency JP. Blaizot and D. Gogny, Nucl. Phys. A284 (1977) 429-460. D. Gogny and R. Padjen, Nucl. Phys. A293 (1977) 365-378. Determination of Mixing Coefficients Two-body correlation function with

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

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

14. 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… Test of the importance of the different terms in the mpmh wave function expansion (1 pair, 2pairs…) 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 (*) R.W. Richardson, Phys.Rev. 141 (1966) 949.

16. Richardson exact solution… εi+1 g εi d Picket fence model (*) • 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. Richardson exact solution… g (Pairing interaction strength) Truncation in excitation energy Truncation in mpmh order of excitation Ground state Correlation energy Ecorr=E(g≠0)-E(g=0) ΔEcorr= Ecorr (exact) – Ecorr(mpmh) N. Pillet, N. Sandulescu, Nguyen Van Giai and JF. Berger , Phys.Rev. C71 , 044306 (2005).

18. Richardson exact solution… Ground state occupation probabilities N. Pillet, N. Sandulescu, Nguyen Van Giai and JF. Berger , Phys.Rev. C71 , 044306 (2005).

19. No residual proton-neutron interaction Kp=Jp=0+ • Ground states of even-even spherical nuclei Kp=0+ . … Several excited Pairs of nucleons Kp=0+ Kp=0+ Kp=0+ 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) • Configurations with: One excited pair of nucleons (*) N. Pillet, JF Berger and E. Caurier, to be submitted to PRC.

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

21. 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 http://www-phynu.cea.fr/HFB-Gogny.htm S. Hilaire and M. Girod, EPJ A33 (2007) 237.

24. 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. 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. Variational mpmh configuration mixing applied to pairing… Results without Self-Consistency • Correlation energy (MeV) 1 pair : 3.397 MeV 2 pairs : 0.275 MeV 100Sn 1 pair : 4.474 MeV 2 pairs : 0.967 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. 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. Variational mpmh configuration mixing applied to pairing… Results without Self-Consistency ~141 neutron states ~ 98 proton states • Effect of a truncated space Total Truncated Total Truncated

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

31. 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… Exp. Neutron Skin Charge Radii

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 shell J J

38. Matrix elements of Gogny force in sd shell J 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 shell J J J J J J