1. 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 x The Niels Bohr Institute, Copenhagen, Denmark In the table of nuclei one can encounter very different systems: stable nucleus, lying along the stability valley one-neutron separation energy = S n  7.40 MeV 11 Li 208 Pb halo nucleus, lying near or at the n-drip line two-neutron separation energy = S 2n  300 keV The r-processes nucleosynthesis path evolves along the neutron drip line region  The need of a mass formula able to predict nuclear masses with an accuracy of the order of magnitude of S 2n  300 keV seems quite natural  We need a formula at least a factor of two more accurate than present microscopic ones!! For a better prediction one has to go beyond static mean field approximation. One has to consider collective degrees of freedom like: vibrations of the nuclear surface vibrations of the nuclear surface pairing vibrations pairing vibrations 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 X 0  2 systems, X 0 being the magic neutron (N 0 ) or proton (Z 0 ) number associated with the closed shell system Calculations have been carried out for 52 spherical nuclei in different regions of the mass table Results extension to open shell nuclei: extension to open shell nuclei: 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 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)… are associated with Oxygen (magic) Z = 8 Calcium (magic) Z = 20 Lead (magic) Z = 82 Tin (magic) Z = 50 Argon Z = 18 Titanium Z = 22 clear reduction of rms errors in closed shell nuclei doubly closed shell nuclei neutron closed shell nuclei a factor of nearly 5 better!! (all data in MeV) Our mean field calculation HF-BCS approximation Skyrme-type interaction MSk7 3 particle-particle channel: Wigner term Finite proton correction 3 developed by Goriely et al. (correcting the absence of T=0 np pairing in the model) -  -pairing force - pp and nn channel - state dependent matrix elements - energy cutoff at 1 h  =41A -1/3 - different pairing strength for and  (rms = 0.754 when fitted to 1768 nuclei) The largest deviations from experiment are associated to closed shell nuclei Where are correlation energies expected to be important? In a spherical nucleus vibrational spectrum (e.g. of quadrupole type) In a deformed nucleus an additional rotational structure is displayed 0 + (g.s.) 2 + (one phonon state) strong B(E2) due to high collectivity a permanent (shape) deformation makes the system more rigid to oscillations surface vibrations are more important in spherical nuclei In short: In a closed shell nucleus by analogy no stable pairing distortion high collectivity of pairing vibrational modes In an open shell nucleus permanent pairing deformation (  eq  0) most of the pairing collectivity is found in pairing rotational bands In short: pairing vibrations are more important in closed shell nuclei 0+0+ 2 + (vibrational) } rotational band: it “absorbs” most of collectivity 2+2+ 4+4+ 6+6+ weak B(E2) Q 0 =0 (spherical nucleus) (surface vibrations). The associated average field is not invariant under whose generator is the One can parametrize the deformation of the potential in terms of that defines an orientation of the intrinsic frame of reference there is a change in the energy along the For small values of the interaction parameter, the system has to another physical state with and displays a typical phonon spectrum It corresponds to oscillations Going from a physical state with Analogy between Deformation of the surface of the nucleus. Distortion of the Fermi surface (superfluid state). rotations in three dimensions,gauge transformations,  and  and of the Euler angles  the BCS gap parameter  and the gauge angle  total angular momentum I 1 particle number N 1 total angular momentum I 2,particle number N 2, rotational band.pairing rotational band.  =0 (normal nucleus) (pairing vibrations). of the surface around spherical shape, of the energy gap around  eq = 0, in ordinary 3D space.in gauge space. the excited states being states with different angular momentum.particle number. total angular momentum operator I. particle number operator N. spatial (quadrupole) deformations and pairing deformations Nuclear masses: the state of the art… A remarkable accuracy, but one is still not satisfied!! rms error Weizsacker formula (1935) …………………………………. Describing the nucleus like a liquid drop Finite-range droplet method 1 ………………………………. 0.689 MeV 1654 nuclei fitted 2.970 MeV 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 Using microscopically grounded methods mean field approximation (mean field approximation) HF-BCS calculation with Skyrme interaction 2 …….. ETFSI ….…………..……………………………..………………….. 0.674 MeV 2135 nuclei fitted Extended Thomas-Fermi plus Strutinsky integral Hartree-Fock Bardeen-Cooper-Schrieffer 0.709 MeV 1719 nuclei fitted Experimental observation (t,p) and (p,t) reactions are excellent tools for probing pairing correlations (neutron) pairing vibrations in even Ca nuclei (neutron) pairing rotations in even Sn nuclei exp. values harmonic model relative cross sections display a linear dependence on the number of pairs added/removed from N=28 shell neutron closed shell nucleus (n r, n a ) are pair removal and pair addition quanta exp. values g.s.  g.s. cross sections are much larger than g.s.  p.v. cross sections (S. Baroni et al., J. Phys. G: Nucl. Part. Phys. 30 (2004) 1353) Dynamic vibrations of the surface The correlation energy associated to zero point fluctuations has the expression:  Some details of our calculation: where Y ki ( ) are the backwards-going amplitudes of the RPA wavefunctions Microscopic description, Random Phase Approximation (RPA) Vibrations: coherent particle-hole excitations 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%