F. C HAPPERT N. P ILLET, M. G IROD AND J.-F. B ERGER CEA, DAM, DIF THE D2 GOGNY INTERACTION F. C HAPPERT ET AL., P HYS. R EV. C 91, (2015)

INTRODUCTION Microscopic description of nuclear structure: - basic constituants: nucleons - mutual interactions Difficulties: - definition of the nuclear interaction - resolution of the quantic N-body problem The method: mean field theory + extensions Widely used for medium weight and heavy nuclei

INTRODUCTION – MEAN FIELD Effective Interaction: - postulated phenomenological form: Skyrme ou Gogny - parameters fitted on nuclear properties This approach has many successes: - it applies to the whole chart of nuclei - it allows the interpretation of many experimental results Beyond mean-field extensions: - independent particules Hartree-Fock, unsufficient - long range correlations HFB, RPA, QRPA, GCM Partial justification : Brueckner-Hartree-Fock theory

INTRODUCTION – OBJECTIVES New study on the Gogny interaction. Two axes: 1.Better description of the isospin degree of freedom with a new set of parameters for the interaction. Mean: try to reproduce the neutrom matter equation of state (Sly4, Lyon research group) 2. give a finite range to the density term of the interaction Goal: to obtain an analytical form suited for beyond mean field approaches.

POTENTIAL ENERGY // TRANSFERRED MOMENTUM k1k1 k2k2 k 1 -q k 2 +q V(q) =

The D2 Gogny interaction: - Analytical form - Fitting procedure The D2 Gogny interaction: - Analytical form - Fitting procedure

ANALYTICAL FORM OF THE GOGNY INTERACTION (D1) 14 parameters: (Wi, Bi, Hi, Mi,  i ) for i=1,2; t 0, x 0,  W ls

A FINITE RANGE DENSITY DEPENDENCE (D2) New analytical form: 17 parameters

17 PARAMETERS: NEW FITTING PROCEDURE (D2) ● Fitting of the new parameters : - global quantities: K , a , m*/m - neutron matter EOS - binding energy and charge radius of 208 Pb - pairing properties, (channel S=0, T=1) - energy in each S,T channel of the nuclear matter ● Great number of solutions for the parameters ● Main difference between the solutions: the contribution of the new density term in the S=0, T=1 channel

Infinite nuclear matter with the D2 Gogny interaction Infinite nuclear matter with the D2 Gogny interaction

SPIN-ISOSPIN S-T CHANNELS BBG: Bethe-Brueckner-Goldstone, Baldo (2005)

NEUTRON AND PROTON EFFECTIVE MASSES ● Asymetry and splitting of the effective masses ● BBG stands for the Bethe-Brueckner-Goldstone method

NEUTRON MATTER ● Equation of state E=f(  /  0 ),  0 : normal density ● FP stands for the Friedman Pandharipande results

PAIRING IN NUCLEAR MATTER ● Amount of density dependence in the pairing channel : - in our fitting procedure, the pairing is adjusted with 1s and 2s matrix elements of the interaction, - a large density dependence (repulsive) in the pairing channel leads to a stronger attraction of the Gaussians (central term) and pairing can appear in magic nuclei ● → The density dependence in the pairing channel has been kept small

PAIRING GAP IN NUCLEAR MATTER ● The Paris force does not give enough pairing when used in nuclei ● With D1S, additional pairing provided in the volume part (k F ~ 1,3 fm -1 ) ● With D2, additional pairing is both in the volume part and the surface part (k F ~ 0,8 fm -1 )

LANDAU PARAMETERS F L ST

A few finite nuclei properties with the D2 Gogny interaction A few finite nuclei properties with the D2 Gogny interaction

CALCULATION TIME WITH A FINITE RANGE DENSITY TERM Calculation of the fields (HFB) with a finite range density term: - numerical integration (density dependence), - exchange component non trivial (finite range).

PAIRING PROPERTIES

● Comparison of the pairing strength of the D1S and D2 interactions ● Neutron pairing energy E app in the even-even Tin isotopes

FISSION BARRIER HEIGHTS FOR 4 ACTINIDES ● HFB calculation under constraint ● Potential energy evolution E HFB according to the axial quadrupole deformation β ● The minima of the potential energy curves have been translated to be the same ● The two wells at β=0,3 and 0,9 correspond to the ground state and the fission isomer ● The height of the first fission barrier is slightly lower with D2

BINDING ENERGIES,  B HFB =B HFB -B EXP ● The drift of binding energies along isotopic chains is corrected with the D1N and D2 interactions

CONCLUSION New analytical form for the density part (finite range), D2 keeps the good isospin properties of D1N, and: a) allows a systematic description beyond mean-field (finite range) b) gives a better ST channel description, especially in S=0, T=0 c) gives the correct splitting of neutron and proton effective masses, d) gives stable nuclear matter at high density (Landau parameters) Points b), c) and d) can be obtained with a zero-range density part (D1M)

PERSPECTIVES Exploration of beyond mean-field properties with D2: RPA, QRPA, 2 nd RPA, MP-MH configuration mixing approach Introduction of a finite range spin-orbit term Introduction of a finite range tensor term With a complete refit of the interaction Link with effective interactions deduced from « Vlow-k » or from chiral effective field theory: more fundamental approaches