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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, 034312 (2015)
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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
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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
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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.
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POTENTIAL ENERGY // TRANSFERRED MOMENTUM k1k1 k2k2 k 1 -q k 2 +q V(q) =
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The D2 Gogny interaction: - Analytical form - Fitting procedure The D2 Gogny interaction: - Analytical form - Fitting procedure
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ANALYTICAL FORM OF THE GOGNY INTERACTION (D1) 14 parameters: (Wi, Bi, Hi, Mi, i ) for i=1,2; t 0, x 0, W ls
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A FINITE RANGE DENSITY DEPENDENCE (D2) New analytical form: 17 parameters
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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
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Infinite nuclear matter with the D2 Gogny interaction Infinite nuclear matter with the D2 Gogny interaction
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SPIN-ISOSPIN S-T CHANNELS BBG: Bethe-Brueckner-Goldstone, Baldo (2005)
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NEUTRON AND PROTON EFFECTIVE MASSES ● Asymetry and splitting of the effective masses ● BBG stands for the Bethe-Brueckner-Goldstone method
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NEUTRON MATTER ● Equation of state E=f( / 0 ), 0 : normal density ● FP stands for the Friedman Pandharipande results
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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
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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 )
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LANDAU PARAMETERS F L ST
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A few finite nuclei properties with the D2 Gogny interaction A few finite nuclei properties with the D2 Gogny interaction
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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).
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PAIRING PROPERTIES
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● Comparison of the pairing strength of the D1S and D2 interactions ● Neutron pairing energy E app in the even-even Tin isotopes
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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
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BINDING ENERGIES, B HFB =B HFB -B EXP ● The drift of binding energies along isotopic chains is corrected with the D1N and D2 interactions
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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)
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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
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