Some results obtained with Gogny force included in HFB, QRPA as well as in configuration mixing GCM like approach

Transcription

1 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)

2 Short Reminder Gogny Staticmeanfield (HFB) for Ground State Properties : Masses Deformation Amedee database : S. Hilaire & M. Girod, EPJ A33 (2007) 237 (Single particle levels) Beyond static mean field approximation(5dch or QRPA) for description of Excited State Properties Low-energy collective levels Giant Resonances

3 Beyond mean field with 5DCH (2 vibr. + 3 rot.) = 5 Dimension Collective Hamiltonian Wave function of the collective Potential energy surfaces axial Triaxial Hˆ coll h Iˆ k h = 2 J 2 k = 1 k m, n = 0 and 2 D 1/ 2 a m D 1/ 2 a 1 ( B ) + V ( a, a ) V ( a, a ) mn n

4 Some exploitation of 5DCH with Gogny forces Systematics studies D1S: J. P. Delaroche et al. PRC 81, (2010) D1M: S. Hilaire et al. PRC 86, (2012) S. Goriely, S. Hilaire, M. Girod, S. Péru, PRL 102, (2009)

5 Shell closure studies with 5DCH E (2 + 1) Theo. = 1.32 MeV Theo. E (2 + 1) Exp. = 885 kev Exp N=28, 44 S N=20, 32 Mg S. Péru, M. Girod, JF. Berger, Eur. Phys. J. A 9,35-47, (2000) E(2 + 1) Theo. = 1.46 MeV, E(2 + 1) Exp. = ( )MeV, B(E2,0 gs 2 + 1) = 420 e 2 fm 4 B(E2,0 gs 2 + 1) = 314 (88 88) e 2 fm 4,0 + gs,0 + gs

6 Further analysis : moments of inertia 32 Mg

7 Further analysis : moments of inertia 32 Mg Mg Sulfur isotopes

8 Shell closure studies with 5DCH N=16 5DCH J. Gibelin et al, PRC 75, (2007) A. Obertelli, S. Péru, J.-P. Delaroche, A. Gillibert, M. Girod et H. Goutte, Phys. Rev. C 71, (2005)

9 N=40 E (MeV) x10 B(E2) (e 2 fm 4 ) (a) (b) Expt D1S, 5DCH Z Shell closure studies with 5DCH 2 Expt E (MeV) D1S, 5DCH + 5DCH DCH DCH DCH DCH Expt 2 + Expt 1 + Expt 2 + Expt 1 + Expt Z E (MeV) ) + 1 )/E(2 E( rotor γ soft vibrator Z Expt D1S, 5DCH 0.5 First excited Z (eb) Q s Q Q Q Q (Expt) (5DCH) (5DCH) (5DCH) L. Gaudefroy et al, Phys.Rev. C 80, , (2009) Z

10 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)RPAapproaches 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 SM δe/δq=0 δ 2 E/δq 2 >0 S.Péru and M. Martini, EPJA (2014) 50: 88. Exp. 5DCH D. Sohler et al, PRC 66, (2002)

11 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, (2005) S.Péru and M. Martini, EPJA (2014) 50: 88.

12 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 78 Ni is predicted doubly magic S.Péru and M. Martini, EPJA (2014) 50: 88.

13 Beyond mean field with QRPA RPA approaches describe all multipolartiesand and all parities, collectivestates and individualones, low energyand high energystates with the same accuracy. Within the small amplitude approximation, i.e. «harmonic» nuclei E δe/δq=0 δ 2 E/δq 2 >0 Spherical RPA with Gogny force 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) Axially symetric deformed QRPA with Gogny force S. Péru, H. Goutte, Phys. Rev. C 77, , (2008) M. Martini, S. Péru and M. Dupuis, Phys. Rev. C 83, (2011) S. Péru et al, Phys. Rev. C 83, (2011) RPA approaches are well adapted for describing giant resonances

14 QRPA Formalism Q RPA Particle-hole excitations + ν + ν = X ph a p ah + ν Y ph ph a + h a QRPA 2 quasi-particle excitations + ν + + ν Q = X ij ηi η + j Y ij η ν j i ij η + + i = ui αaα vi αaα α Hartree-Fock Bogoliubov: ε, u, v Ground state properties QRPA: ω, X, Y Excited states properties Same interaction (Gogny) in HFB and QRPA p η p-h 2 p1/2 2 p3/2 1 f7/2 p-p 1 d3/2 2 s1/2 Fermi 1 d5/2 1 p1/2 h-h 1 s1/2 1 p3/2 Neutron s HF levels 26 Ne

15 Giant resonances in exotic nuclei: RPA in spherical symmetry 100 Sn, 132 Sn, 78 Ni; S. Péru, J.F. Berger, and P.F. Bortignon, Eur. Phys. Jour. A 26, (2005) Monopole Dipole Quadrupole Such study have shown the role of the consistence between mean field and RPA matrix. Approach limited to Spherical nuclei with no pairing

16 Axially-symmetric deformed QRPA J Restoration of rotational symmetry for deformed states JM 2J + 1 J J K J ( K ) dωd MK ( Ω) R( Ω) θ + ( ) D ( Ω) R( Ω) θ K = K M K 4π K to calculate: 0 Q ˆ λµ JM ( K ) For example: J π = 2 + ~ for all QRPA states (K J) We use rotational approximation and relations for 3j symbols ~ 0 Qˆ 20 JM Qˆ r 1 3 r 2 Y = λ 2 λµ Dµυr Yλυ υ ( K ) = 0 Qˆ 20 θ ˆ ˆ K δ K,0 + 0 Q2 1 θ K δ K, ± Q22 θ K δ K, ± 2 5 Using time reversal symmetry, three independent calculations (K π = 0 +, 1 +, 2 + ) are needed. ( ) λ λµ Yλµ In intrinsic frame Main features: Possibility to treat axially-symmetric deformed nuclei Pairing correlations consistently included Use of an unique nuclear 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

17 QRPA in axial symmetry Potential Energy Surfaces dipole Si IV Dipole Si Si K = 0 - K = K=0 K=1 K π =0 - K π = Mg Mg Mg Mg J K MeV MeV MeV MeV S. Péruand H. Goutte, Phys. Rev. C 77, (2008).

18 Dipole response for Neon isotopes Increasing neutron number Low energy dipole resonances and shift to low energies Increasing of fragmentation 26 Ne : B(E1) = 0.49 ± 0.16 e 2 fm 2 %STRK = 4.9 ± 9 MeV J. Gibelin et al, PRL 101, (2008) 18 Ne 20 Ne 22 Ne 24 Ne 26 Ne 28 Ne B(E1)=0.173 e 2 fm MeV

19 Dipole response for Neon isotopes and N=16 isotones M. Martini, S. Péru and M. Dupuis, Phys. Rev. C 83, (2011) GDR PDR Increasing N-Z number : Low energy dipole resonances shift to low energies Increasing of fragmentation and collectivity

20 Multipolar responses for 238 U Heavy deformed nucleus massively parallel computation J K S. Péru et al, PRC 83, (2011)

21 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 enoughcollective contribution M. Dupuis, S.Péru, E. Bauge and T. Kawano, 13th International Conference on Nuclear Reaction Mechanisms, Varenna 2012 CERN-Proceedings , p 95

22 Nuclear Masses Comparison with experimental data (2149 nuclei: Audi, Wapstra& Thibault 2003) Gogny D1M r.m.s = MeV S. Goriely, S. Hilaire, M. Girod, S. Péru, PRL 102, (2009)

23 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

24 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

25 Dipole excitations in QRPA and photoabsorption results With folding and shift. 28 Si 60 Ni 70 Ge 74 Ge 76 Ge 80 Se 90 Zr 94 Zr 94 Mo 98 Mo 116 Sn 120 Sn Exp. σ [mb] 124 Te 128 Te 138 Ba 142 Ce 144 Nd 148 Nd QRPA 144 Sm 150 Sm 158 Gd 174 Yb 206 Pb 232 Th

26 Dipole excitations in QRPA and photoabsorption results To take into account complex configurations as well as coupling with phonons, the deformed QRPA strength S E1 (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, S. Hilaire, F. Lechaftois and S. Péru

27 γ- ray strength functions predictions for exotic nuclei e.g. Sn isotopes

28 Photoneutron cross sections for Mo isotopes

29 Photoabsorption cross sections for Mo isotopes

30 Dipole electric and magnetic excitations for Zrisotopes E1 M1 E1 M1 H. Utsunomiya et al. (2008) H. Utsunomiya et al, PRL 100, (2008)

31 e - Nuclear Excitations Photo-absorption Electron scattering (p,p) or (n,n) p n γ (N,Z) e - γ (N,Z) p or n π 0 (N,Z) p n e - ν e - Charge exchange: βdecay Neutrino scattering (p,n) or ( 3 He,t) (N-1,Z+1) (N-1,Z+1) (N-1,Z+1) n W + W + π + (N,Z) (N,Z) ν p (N,Z) p Spin-isospinnuclear 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 n

32 pnqrpa General expression QRPA p n pnqrpa p n

33 Main charge exchange excitations p n Isobaric Analog Resonance(IAR) isospin flip τ S=0 T=1 J π =0 + p n Gamow Teller (GT) isospin flip τ spin flip σ S=1 T=1 J π =1 + p n

34 Checks When Coulomb repulsion is out of HFB calculations, the energy of the IAR dives to zero. Sum Rules : Fermi Ikeda 114 Sn 132 Sn F: GT/3: F: GT/3:

35 Gogny pnqrpa Strength Distributions M. Martini, S. Péru and S. Goriely, Phys. Rev. C 89, (2014) Good agreement with experimental data

36 An example of deformed nucleus : 76 Ge GT J π =1 + distributions obtained by adding twice the K π =1 + result to the K π =0 + one pnqrpa-d1m HFB D1M 2 prolate β 2 (min. HFB) = 0.15 γ(min.hfb) = 0 β 2 (0 + 1:5DCH) = 0.26 γ(0 + 1:5DCH) = 26 Experiment Thieset al., Phys. Rev. C 86, (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

37 β - decay half-life T 1/2 Comparison with experimental data for 145 even-even nuclei G.Audi et al. Chinese Phys. C 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

38 Comparison with other models Our model Other models FRDM: Moller et al., ADNDT, 66,131 (1997) GT2:Tachibana et al. Prog. Theor. Phys., 84, 641 (1990)

39 β - decay half-lives of deformed isotopic chains

40 β - decay half-lives of the N=82, 126, 184 isotones Relevance for the r-process nucleosynthesis Shell Model : Martinez-Pinedo Pinedoet et al., PRL 83, 4502 (1999) DF3+cQRPA : Borzovet al., PRC 62, (2000)

41 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, (2014) β-decay Half lives of 76,77 Co, 79,80 Ni and 81 Cu : Experimental indication of a Doubly Magic 78 Ni

42 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 238 U. * 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 76 Ge. * 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.