Congresso del Dipartimento di Fisica Highlights in Physics 2005

1. 208Pb • stable nucleus, lying along the stability valley • one-neutron separation energy = Sn 7.40 MeV 11Li • halo nucleus, lying near or at the n-drip line • two-neutron separation energy = S2n 300 keV Our mean field calculation • HF-BCS approximation • Skyrme-type interaction MSk73 • particle-particle channel: • Wigner term • Finite proton correction (rms = 0.754 when fitted to 1768 nuclei) • -pairing force • pp and nn channel • state dependent matrix elements • energy cutoff at 1 h=41A-1/3 • different pairing strength for  and  (correcting the absence of T=0 np pairing in the model) The largest deviations from experiment are associated to closed shell nuclei For a better prediction one has to go beyond static mean field approximation. 3 developed by Goriely et al. • vibrations of the nuclear surface • pairing vibrations One has to consider collective degrees of freedom like: Dynamic vibrations of the surface Oscillations in the shape of the nucleus a change in the binding field of each particle (i.e. with a field which conserves the number of particles and arising from ph residual interaction)… • Microscopic description, Random Phase Approximation (RPA) • Vibrations: coherent particle-hole excitations are associated with The correlation energy associated to zero point fluctuations has the expression: where Yki() are the backwards-going amplitudes of the RPA wavefunctions  Analogy between Experimental observation spatial (quadrupole) deformations and pairing deformations Some details of our calculation: (t,p) and (p,t) reactions are excellent tools for probing pairing correlations Deformation of the surface of the nucleus. Distortion of the Fermi surface (superfluid state). • Skyrme-type interaction MSk7 with a  pairing force • 2+ and 3- multipolarities are taken into account • states with h < 10 MeV and with B(E)  2% (neutron) pairing vibrations in even Ca nuclei The associated average field is not invariant under rotations in three dimensions, gauge transformations, Where are correlation energies expected to be important? whose generator is the (nr, na) are pair removal and pair addition quanta particle number operator N. total angular momentum operator I. In a spherical nucleus One can parametrize the deformation of the potential in terms of vibrational spectrum (e.g. of quadrupole type)  and  and of the Euler angles  the BCS gap parameter  and the gauge angle  that defines an orientation of the intrinsic frame of reference in ordinary 3D space. in gauge space. In a deformed nucleus Going from a physical state with an additional rotational structure is displayed 2+ (one phonon state) total angular momentum I1 particle number N1 (neutron) pairing rotations in even Sn nuclei to another physical state with strong B(E2) due to high collectivity total angular momentum I2 , particle number N2 , exp. values 0+ (g.s.) there is a change in the energy along the a permanent (shape) deformation makes the system more rigid to oscillations surface vibrations are more important in spherical nuclei 2+ (vibrational) relative cross sections display a linear dependence on the number of pairs added/removed from N=28 shell exp. values In short: rotational band. pairing rotational band. neutron closed shell nucleus harmonic model For small values of the interaction parameter, the system has rotational band: it “absorbs” most of collectivity } g.s.  g.s. cross sections are much larger than g.s.  p.v. cross sections 6+ 4+ Q0=0 (spherical nucleus) =0 (normal nucleus) by analogy weak B(E2) 2+ 0+ and displays a typical phonon spectrum • no stable pairing distortion • high collectivity of pairing vibrational modes (surface vibrations). (pairing vibrations). In a closed shell nucleus It corresponds to oscillations of the energy gap around eq = 0, of the surface around spherical shape, In an open shell nucleus • permanent pairing deformation (eq  0) • most of the pairing collectivity is found in • pairing rotational bands the excited states being states with different angular momentum. particle number. pairing vibrations are more important in closed shell nuclei In short: doubly closed shell nuclei neutron closed shell nuclei Pairing vibration calculations details • calculations carried out in the RPA • separable pairing interaction with constant matrix elements • L = 0+, 2+ multipolarities taken into account (only L = 0+ for lightest nuclei) • pairing interaction parameter calculated in double closed shell nuclei, • solving a dispersion relation and reproducing the experimental extra binding • energies observed in X02 systems, X0 being the magic neutron (N0) or proton • (Z0) number associated with the closed shell system Oxygen Calcium Lead (magic) Z = 8 (magic) Z = 20 (magic) Z = 82 Tin Argon Titanium (magic) Z = 50 Z = 18 Z = 22 Congresso del Dipartimento di Fisica Highlights in Physics 2005 11–14 October 2005, Dipartimento di Fisica, Università di Milano Contribution to nuclear binding energies arising from surface and pairing vibrations S.Baroni*†, F.Barranco#, P.F.Bortignon*†, R.A.Broglia*†x, G.Colò*†, E.Vigezzi† * Dipartimento di Fisica, Università di Milano † INFN – Sezione di Milano # Escuela de Ingenieros, Sevilla, Spain xThe Niels Bohr Institute, Copenhagen, Denmark (S. Baroni et al., J. Phys. G: Nucl. Part. Phys. 30 (2004) 1353) Nuclear masses: the state of the art… In the table of nuclei one can encounter very different systems: Describing the nucleus like a liquid drop Weizsacker formula (1935)…………………………………. 2.970 MeV Finite-range droplet method1………………………………. 0.689 MeV 1654 nuclei fitted Using microscopically grounded methods (mean field approximation) The r-processes nucleosynthesis path evolves along the neutron drip line region rms error  HF-BCS calculation with Skyrme interaction2…….. 0.674 MeV The need of a mass formula able to predict nuclear masses with an accuracy of the order of magnitude of S2n  300 keV seems quite natural 2135 nuclei fitted Hartree-Fock Bardeen-Cooper-Schrieffer ETFSI….…………..……………………………..………………….. 0.709 MeV Extended Thomas-Fermi plus Strutinsky integral  1719 nuclei fitted We need a formula at least a factor of two more accurate than present microscopic ones!! 1 P.Möller et al., At. Data Nucl. Data Tables 59 (1995) 185 2 S.Goriely et al., Phys. Rev. C 66 (2002) 024326-1 A remarkable accuracy, but one is still not satisfied!! What are pairing vibrations? …there exist vibrational modes based on fields which create or annihilate pairs of particles the corresponding collective mode is called pairing vibration Results Calculations have been carried out for 52 spherical nuclei in different regions of the mass table • clear reduction of rms errors in closed shell nuclei (all data in MeV) a factor of nearly 5 better!! • extension to open shell nuclei: (all data in MeV)