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

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

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) 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 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

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

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

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

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

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)

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) 429-460. D. Gogny and R. Padjen, Nucl. Phys. A293 (1977) 365-378.

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

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

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

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

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.

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.

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.

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 , 044306 (2005).

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

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.

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

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

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

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.

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)

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

Results without Self-Consistency 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

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

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

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

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

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

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

Variational mpmh configuration mixing applied to pairing… Occupation Probabilities

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

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

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

Matrix elements of Gogny force in sd shell J J

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.

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