Some results obtained with Gogny force included in HFB, QRPA as well as in configuration mixing GCM like approach Sophie Péru M. Dupuis, S. Hilaire, F. Lechaftois (CEA, DAM), M. Martini (Ghent University, Belgium), S. Goriely (Université Libre de Bruxelles, Belgium), I. Deloncle (CSNSM, Orsay) DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 Short Reminder Gogny Static mean field (HFB) for Ground State Properties : • Masses • Deformation • (Single particle levels) DAM, DIF, S. Péru Amedee database : http://www-phynu.cea.fr/HFB-Gogny_eng.htm S. Hilaire & M. Girod, EPJ A33 (2007) 237 Beyond static mean field approximation (5DCH or QRPA) for description of Excited State Properties • Low-energy collective levels • Giant Resonances FUSTIPEN, 16th-20th March 2015 Beyond mean field … with 5DCH (2 vibr. + 3 rot.) = 5 Dimension Collective Hamiltonian Potential energy surfaces Wave function of the collective 01+ h2 ˆ H coll = 2 DAM, DIF, S. Péru Triaxial axial 3 ∑ k =1 2 Iˆk h2 − Jk 2 ∑ m, n = 0 and 2 D −1 / 2 ∂ −1 ∂ D 1 / 2 (B mn ) + V (a 0 , a 2 ) − ∆ V (a 0 , a 2 ) ∂a m ∂a n FUSTIPEN, 16th-20th March 2015 Some exploitation of 5DCH with Gogny forces Systematics studies D1S: J. P. Delaroche et al. PRC 81, 014303 (2010) D1M: S. Hilaire et al. PRC 86, 064317 (2012) S. Goriely, S. Hilaire, M. Girod, S. Péru , PRL 102, 242501 (2009) DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 Shell closure studies with 5DCH E (2+1) Theo. = 1.32 MeV E (2+1) Exp. Exp. N=20, 32Mg S. Péru, M. Girod, JF. Berger , Eur. Phys. J. A 9,35-47, (2000) DAM, DIF, S. Péru = 885 keV N=28, 44S E(2+1)Theo. = 1.46 MeV, E(2+1) Exp. = 1.297 (0.018) 0.018)MeV, B(E2,0+gs→2+1) = 420 e2fm4 B(E2,0+gs→2+1) = 314 (88) 88) e2fm4 FUSTIPEN, 16th-20th March 2015 Further analysis : moments of inertia 32Mg DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 Further analysis : moments of inertia 32Mg Sulfur isotopes DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 Shell closure studies with 5DCH N=16 5DCH J. Gibelin et al, PRC 75, 057306 (2007) A. Obertelli, S. Péru, J.-P. Delaroche, A. Gillibert, M. Girod et H. Goutte, Phys. Rev. C 71, 024304 (2005) DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 N=40 Shell closure studies with 5DCH + 5DCH Expt (a) + 5DCH 31 D1S, 5DCH 1.5 + 5DCH 42 4 0.5 2.5x10 3 B(E2) (e 2 fm4) 22 5 E (MeV) E (MeV) 2.5 (b) 1.5 + Expt 22 + Expt 31 + Expt 42 + Expt 5+1 5DCH 51 + 5DCH 62 3 + Expt 62 2 0.5 22 24 26 28 30 32 34 36 38 40 1 42 Z Expt E (MeV) 2 22 24 26 28 30 32 34 36 38 40 42 Z D1S, 5DCH 1.5 1 0.5 First excited 0+ 22 24 26 28 30 32 34 36 38 40 42 Z rotor 0 2.5 2 γ soft -0.5 Qs (eb) E(41+)/E(21+) 3 vibrator Q2+(Expt) 1 -1 Expt 1.5 D1S, 5DCH Q2+(5DCH) 1 Q4+(5DCH) 1 Q6+(5DCH) 1 22 24 26 28 30 32 34 36 38 40 42 22 24 26 28 30 32 34 36 38 40 42 Z DAM, DIF, S. Péru Z L. Gaudefroy et al, Phys.Rev. C 80, 064313, (2009) FUSTIPEN, 16th-20th March 2015 Beyond static mean field … with 5DCH or QRPA 5 Dimension Collective Hamiltonian describes ground state and excited states within configuration mixing : quadrupole vibration and rotational degrees of freedom. HFB+D1S (Q)RPA approaches describe all multipolarities and all parities, collective states and individual ones, low energy and high energy states with the same accuracy. But small amplitude approximation i.e. « harmonic » nuclei E δE/δq=0 SM Exp. δ2E/δq2>0 5DCH S.Péru and M. Martini, EPJA (2014) 50: 88. D. Sohler et al, PRC 66, 054302 (2002) DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 HFB+QRPA versus HFB+5DCH with the same interaction N=16 isotones Sn isotopes (Z=50) 5DCH : A. Obertelli, et al, Phys. Rev. C 71, 024304 (2005) S.Péru and M. Martini, EPJA (2014) 50: 88. DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 HFB+QRPA versus HFB+5DCH with the same interaction Ni isotopes (Z=28) Two shell (N= 28, 50) and one sub-shell (N=40) closures ! For deformed nuclei the first 2+ state is rotational 78Ni is predicted doubly magic S.Péru and M. Martini, EPJA (2014) 50: 88. DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 Beyond mean field… with QRPA RPA approaches describe all multipolarties and all parities, collective states and individual ones, low energy and high energy states with the same accuracy. Within the small amplitude approximation, approximation, i.e. « harmonic » nuclei Spherical RPA with Gogny force E J. Dechargé and L.Sips, Nucl. Phys. A 407,1 (1983) J.P. Blaizot, J.F. Berger, J. Dechargé, M. Girod, Nucl. Phys. A 591, 435 (1995) S. Péru, JF. Berger, PF. Bortignon, Eur. Phys. J. A 26, 25-32, (2005) δE/δq=0 δ2E/δq2>0 Axially symetric deformed QRPA with Gogny force S. Péru, H. Goutte, Phys. Rev. C 77, 044313, (2008) M. Martini, S. Péru and M. Dupuis, Phys. Rev. C 83, 034309 (2011) S. Péru et al, Phys. Rev. C 83, 014314 (2011) RPA approaches are well adapted for describing giant resonances DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 QRPA Formalism 2 p1/2 2 p3/2 p-p RPA Particle-hole excitations + ν ah + a p Qν = ∑ X νph a p+ a h + Y ph ph QRPA 2 quasi-particle excitations + ν + + Qν = ∑ X ij η i η j + Y ijν η jηi ij ηi+ = ∑ uiα aα+ − viα aα α Hartree-Fock Bogoliubov: ε, u, v QRPA: ω, X, Y Ground state properties Excited states properties p-h 1 f7/2 1 d3/2 2 s1/2 Fermi 1 d5/2 1 p1/2 1 p3/2 Neutron’s HF h-h levels 26Ne 1 s1/2 Same interaction (Gogny) in HFB and QRPA DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 RPA in spherical symmetry Giant resonances in exotic nuclei: 100Sn, 132Sn, 78Ni; Monopole S. Péru, J.F. Berger, and P.F. Bortignon, Eur. Phys. Jour. A 26, 26 25-32 (2005) Dipole Quadrupole Such study have shown the role of the consistence between mean field and RPA matrix. matrix. Approach limited to Spherical nuclei with no pairing DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 Axially-symmetric deformed QRPA J Restoration of rotational symmetry for deformed states JM (K ) to calculate: = ~ ˆ 0 Qλµ JM (K ) K 2J +1 J −K J d Ω D ( Ω ) R ( Ω ) θ + ( − ) DMJ − K (Ω ) R (Ω ) θ K MK K ∫ 4π for all QRPA states (K ≤ J) Qˆ λµ ∝ ∑ r λ (Yλµ ) λ 2 r 2Yλµ = ∑ Dµυ r Yλυ υ In intrinsic frame We use rotational approximation and relations for 3j symbols For example: Jπ = 2+ ~ ˆ 0Q JM (K ) = 1 3 3 0 Qˆ 20 θ K δ K , 0 + 0 Qˆ 2−1 θ K δ K , ±1 + 0 Qˆ 22 θ K δ K , ±2 20 5 5 5 Using time reversal symmetry, three independent calculations (Kπ = 0+, 1+, 2+) are needed. Main features: • Possibility to treat axially-symmetric deformed nuclei • Pairing correlations consistently included • Use of an unique nuclear force: force finite range Gogny force - same interaction for all the nuclei - same interaction for ground state and excited states (self-consistency) essential features to treat consistently isotopic chains from drip line to drip line DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 QRPA in axial symmetry IV Dipole dipole Potential Energy Surfaces 0.4 0.4 26 Si 0.4 28 Si 0.3 0.3 0.3 0.2 0.2 0.2 0.1 0.1 0.1 30 Si - K =0 K =1 Kπ=0- K=0 0.4 0.4 22 Mg K=1 J Kπ=10.4 24 Mg 0.4 26 Mg 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 28 Mg K 0 10 20 30 40 MeV 0 10 20 30 40 MeV 0 10 20 30 40 MeV 0 10 20 30 40 MeV S. Péru and H. Goutte, Phys. Rev. C 77, 044313 (2008). DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 Dipole response for Neon isotopes Increasing neutron number •Low energy dipole resonances and shift to low energies •Increasing of fragmentation Ne : B(E1) = 0.49 ± 0.16 e fm 26 18Ne 24Ne 2 2 %STRK = 4.9 ± 1.6 @ 9 MeV J. Gibelin et al, PRL 101, 212503 (2008) 20Ne 22Ne 26Ne 28Ne B(E1)=0.173 e2fm2 @ 10.69 MeV DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 Dipole response for Neon isotopes and N=16 isotones GDR PDR M. Martini, S. Péru and M. Dupuis, Phys. Rev. C 83, 034309 (2011) Increasing |N-Z| number : •Low energy dipole resonances shift to low energies •Increasing of fragmentation and collectivity DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 Multipolar responses for 238U Heavy deformed nucleus 0+ massively parallel computation 1J K 2+ 3- S. Péru et al, PRC 83, 014314 (2011) DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 Beyond the nuclear structure (n, x n) cross section on 238 U Problem of underestimation of n emission cross section at high energy QRPA provides enough collective contribution M. Dupuis, S.Péru, E. Bauge and T. Kawano, 13th International Conference on Nuclear Reaction Mechanisms, Varenna 2012 CERN-Proceedings-2012-002, p 95 DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 Nuclear Masses Comparison with experimental data (2149 nuclei: Audi, Wapstra & Thibault 2003) Gogny D1M r.m.s = 0.798 MeV S. Goriely, S. Hilaire, M. Girod, S. Péru , PRL 102, 242501 (2009) DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 Dipole excitations in QRPA and photoabsorption results D1S versus D1M Spherical nuclei First peak Second peak Deformed nuclei We calculate E1 strength for all the nuclei for which photoabsorption data exist DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 Dipole excitations in QRPA and photoabsorption results Size of the basis Spherical nuclei First peak Second peak Deformed nuclei We calculate E1 strength for all the nuclei for which photoabsorption data exist DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 Dipole excitations in QRPA and photoabsorption results With folding and shift …. 60Ni 28Si 90Zr 94Zr 70Ge 94Mo 74Ge 98Mo 76Ge 116Sn 80Se 120Sn σ [mb] Exp. QRPA 124Te 128Te 138Ba 142Ce 144Nd 148Nd 144Sm 150Sm 158Gd 174Yb 206Pb 232Th DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 Dipole excitations in QRPA and photoabsorption results To take into account complex configurations as well as coupling with phonons, the deformed QRPA strength SE1(w) is folded with a Lorentzian function. The width of the Lorentzian is adjusted on experimental data (RIPL-3 IAEA) while the energy shift is related to the number of 4qp keeping in this way a connection with our microscopic calculation. Calculations performed also for isotopic chains of astrophysical interest (e.g. Sn). S .Goriely, M. Martini, DAM, DIF, S. Péru S. Hilaire, F. Lechaftois and S. Péru FUSTIPEN, 16th-20th March 2015 γ- ray strength functions predictions for exotic nuclei e.g. Sn isotopes DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 Photoneutron cross sections for Mo isotopes DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 Photoabsorption cross sections for Mo isotopes DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 Dipole electric and magnetic excitations for Zr isotopes E1 M1 E1 M1 H. Utsunomiya et al. (2008) H. Utsunomiya et al, PRL 100, 162502 (2008) DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 Nuclear Excitations Photo-absorption Electron scattering e- (N,Z) γ (p,p) or (n,n) p (N,Z) γ e- n or π0 (N,Z) p n p n Charge exchange: β decay e- Neutrino scattering e- ν n (N-1,Z+1) W+ (N,Z) (p,n) or (3He,t) (N-1,Z+1) W+ (N,Z) (N-1,Z+1) π+ (N,Z) p n ν p Spin-isospin nuclear excitations (in particular GT) • Crucial role in nuclear physics, astrophysics and particle physics • Experimentally studied via charge exchange reactions , e.g. (p,n) and β decay • Theoretical models to study the nuclei experimentally inaccessible DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 pnQRPA General expression QRPA p n pnQRPA p DAM, DIF, S. Péru n FUSTIPEN, 16th-20th March 2015 Main charge exchange excitations p Isobaric Analog Resonance (IAR) n isospin flip τ S=0 T=1 Jπ=0+ p n p n Gamow Teller (GT) isospin flip τ spin flip σ DAM, DIF, S. Péru S=1 T=1 Jπ=1+ FUSTIPEN, 16th-20th March 2015 Checks When Coulomb repulsion is out of HFB calculations, the energy of the IAR dives to zero. Sum Rules : Ikeda Fermi DAM, DIF, S. Péru F: 13.9999993456112 114Sn GT/3: 13.9999993456064 F: 32.0000000000001 132Sn GT/3: 32.0000000000001 FUSTIPEN, 16th-20th March 2015 Gogny pnQRPA Strength Distributions M. Martini, S. Péru and S. Goriely, Phys. Rev. C 89, 044306 (2014) Good agreement with experimental data DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 An example of deformed nucleus : 76Ge GT Jπ=1+ distributions obtained by adding twice the Kπ=1+ result to the Kπ=0+ one pnQRPA-D1M HFB D1M 2 β2(min. HFB) = 0.15 β2 (0+1:5DCH) = 0.26 prolate γ(min.HFB) = 0° γ(0+1:5DCH) = 26° Experiment Thies et al., Phys. Rev. C 86, 014304 (2012) • • • The deformation tends to increase the fragmentation Displacements of the peaks Deformation influences the low energy strength hence β decay half-lives are expected to be affected DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 β- decay half-life T1/2 Comparison with experimental data for 145 eveneven-even nuclei • • G.Audi et al. Chinese Phys. C 36, 36 1157 (2012) Deviation with respect to data rarely exceeds one order of magnitude Larger deviations for nuclei close to the valley of β-stability, as found in most models DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 Comparison with other models Our model Other models FRDM: Moller et al., ADNDT, 66,131 (1997) GT2:Tachibana et al. GT2: Prog. Theor. Phys., 84, 641 (1990) DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 β- decay half-lives of deformed isotopic chains DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 β- decay half-lives of the N=82, 126, 184 isotones Relevance for the r-process nucleosynthesis Shell Model : MartinezMartinez-Pinedo et al., PRL 83, 4502 (1999) DAM, DIF, S. Péru DF3+cQRPA : Borzov et al., PRC 62, 035501 (2000) FUSTIPEN, 16th-20th March 2015 Even and odd systems, deformed and spherical nuclei Preliminary results Extension to odd systems in collaboration with Isabelle Deloncle (CSNSM) Orsay 28 Recent experimental results Z.Y. Xu et al, PRL 113, 032505 (2014) β-decay Half lives of 76,77Co, 79,80Ni and 81Cu : Experimental indication of a Doubly Magic 78Ni DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015 To summarize Great successes using the finite range Gogny force: 5DCH : good reproduction of collective low energy spectra and shell effects QRPA : good description of pygmy and giant resonances in spherical or deformed nuclei QRPA and 5DCH complete each other. Some results were shown: * Self-consistent QRPA approach has been applied to the deformed nuclei up to 238U. * IsoScalar-IsoVector mixing for low-lying dipole states in Ne isotopes and N=16 isotones. * QRPA results successfully used in reaction models : (n, x n) , photoabsorption … Extension of QRPA to charge exchange : * For magic spherical nuclei, IAR and GT results in good agreement with data. * The role of the intrinsic deformation has been shown for prolate 76Ge. * Predictions of the β decay half-life time are compatible with experimental data. * Satisfactory agreement with experimental half-lives which justifies the additional study on the exotic neutron-rich N = 82, 126 and 184 isotonic chains (r-process). Promising preliminary results for odd nuclei. DAM, DIF, S. Péru FUSTIPEN, 16th-20th March 2015
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