Correlation energy contribution to nuclear masses

1. A study for the influence of collective motion on nuclear ground state energy • To describe many physical observables of many-body systems (e.g. (screened) coulomb interaction in metals, level density at the Fermi energy in atomic nuclei) one is forced to go beyond mean field • Effects due to the coupling of particles and collective vibrations have to be taken into account • The aim of this work is to study the influence zero point fluctuations have on the nuclear masses Correlation energy contribution to nuclear masses Simone Baroni Erice, September 2006 University of Milan, Italy

2. Nuclear masses: the state of the art… The state of the art… rms errors • Liquid drop model 2.970 MeV Weizsacker formula (1935)……………….……….. Finite-range droplet method. (Möller et al.)….….… 0.689 MeV (1654 nuclei fitted) • Microscopic theories (mean field approximation) 0.738 MeV 0.674 MeV HFBCS (Goriely et al.)………………………………… (with Skyrme interaction) HFB (Goriely et al.)…………………………………… (with Skyrme interaction) (1888 nuclei fitted) (2135 nuclei fitted) 0.709 MeV (1719 nuclei fitted) ETFSI (Myers, Swiatecki)…….……………………….. Extended Thomas-Fermi plus Strutinsky integral 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

3. …and the need of a better mass formula …the need of a better formula Sn 7.40 MeV 208Pb S2n 300 keV 11Li An accuracy of about 700 keV is remarkable We need a formula at least a factor of two more accurate than present microscopic ones!! How to improve the prediction?

4. Beyond mean field approximation Beyond MF • vibrations of the surface • change angular momentum • in two (three) units • pairing vibrations • change particle number • in two units Correlation energy associated to zero point fluctuations affects the binding energy of the nuclear system Collective modes of the nucleus:

5. Calculation details Calculation details • HF approximation • box calculations, configuration space • Skyrme interaction MSk7 (rmsd = 0.754 MeV) • Wigner term for  correlations • BCS approximation • Vpair(r,r’)=V0 (r,r’) • energy cutoff at h = 41·A-1/3 MeV Pairing channel: 121 spherical even-even nuclei have been studied Starting point Mean field (MF)

6. Deviation ofcomputed mean field (MF) ground state energies from experimental values for the set of 121 nuclei we studied

7. Calculation details Surface vibrations • QRPA in configuration space • residual interaction derived from MF two-body interaction • 2+ and 3- multipolarities • only most collective low-lying QRPA phonons Pairing vibrations • PV included only in nuclei with N or Z magic number • RPA, based on Woods-Saxon s.p. levels • separable interaction with constant matrix elements  G •  = 0+ (2+) contribution of the lowest (n=1) PV phonon • G calculated in double closed shell nuclei

8. Results Results Correlation energies SV (2+,3-) PV (0+) SV (2+,3-) + PV (0+)

9. Results SV (2+,3-) PV (0+) PV (2+) Correlation energies in Ca and Pb isotopic chains Ca Pb

10. Results rmsd 0.724 MeV 0.667 MeV

11. Summary and conclusions Summary and conclusions • SV: 2+, 3- (QRPA) • PV: 0+ (2+) (RPA) • linear refit HF+BCS MF+SV+PV rmsd = 0.724 MeV rmsd = 0.667 MeV ( - 8 % ) • A systematic analysis of the contribution of medium polarization effects associated with collective surface and pairing vibrations on the binding energies of even-even spherical nuclei. • The contributions of ZPF associated with collective nuclear vibrations (surface and pairing modes) to the nuclear binding energy are important. • Future work: a more systematic refit which is able to make a wider survey • of the parameters space. People working on this project: S.Baroni, F.Barranco, P.F.Bortignon, R.A.Broglia, G.Colò, E.Vigezzi (Baroni et al., J. Phys. G: Nucl. Part. Phys. 30 (2004) 1353) (Baroni et al., Phys. Rev. C 74 (2006) 024305)

16. Other existing work on the subject: (Bender, Bertsch, Heenen PRC 73 (2006) 034322) 605 even-even (spherical and deformed) nuclei have been studied. In the GCM framework, by projection on good angular momentum: • quadrupole Surface Vibration correlations • correlations due to static deformation My work: (Baroni et al., J. Phys. G: Nucl. Part. Phys. 30 (2004) 1353) (Baroni et al., Phys. Rev. C 74 (2006) 024305) 121 spherical even-even nuclei in the QRPA framework: • quadrupole and octupole Surface Vibration correlations • Pairing Vibrations correlations

17. Results Influence of the various contributions full correlations added fluctuations added MF: Mean Field SV: Surface Vibrations PV: Pairing Vibrations • quadrupole PV can give an important contribution. • (however they have to be included only where experimentally observed)

18. Alcuni dati • Variazione % dei coeff. di Skyrme: • MSk7 • MSk7 modified • X0 0,58 0,58 -0,2420% • X1 -0,50 -0,50 0,0000% • X2 -0,50 -0,50 0,0000% • X3 0,79 0,79 -0,0211% • T0 -1828,23 -1828,76 -0,0291% • T1 259,40 255,31 1,5761% • T2 -292,84 -286,71 2,0926% • T3 13421,70 13412,76 0,0666% • W0 118,81 116,14 2,2476% • Alpha 0,33 0,33 0,0000%

20. A brief summary of pairing vibrations PV summary Spatial (quadrupole) deformations Pairing deformations deformation of the surface deformation of the Fermi surface rotational symmetry (3D space) gauge symmetry (gauge space) I N Nilsson wavefunction BCS wavefunction  and  and Euler angles   and gauge angle  Spherical nuclei Normal nuclei Excited by coulomb excitations or inelastic scattering Excited by two-particle transfer reactions surface vibrations pairing vibrations

21. Where corrections… Where are corrections expected to be important? 2+(one phonon state) vibrational spectrum strong B(E2) 0+ (g.s.) 2+ (vibrational) additional rotational structure 6+ 4+ weak B(E2) 2+ 0+ • no pairing distortion • high collectivity of pairing vibrational modes by analogy • permanent pairing distortion (eq  0) • most of collectivity is found in pairing • rotational band In a spherical nucleus In a deformed nucleus In a closed shell nucleus In an open shell nucleus

23. Refit procedure HF+BCS MF binding energies interaction interaction corrected MF b.e. + (Q)RPA SV, PV correlations Refit procedure

24. Results HF+BCS MF binding energies interaction interaction new interaction corrected MF b.e. + (Q)RPA SV, PV correlations REFIT

25. Results HF+BCS MF binding energies interaction interaction new interaction corrected MF b.e. + (Q)RPA SV, PV correlations REFIT corrected & refitted MF b.e. linear

27. Spettro vibraz. Pb208 How was the pairing strength calculated in closed shell nuclei?

28. Spettro vibraz. Pb208 (gs)

29. Spettro vibraz. Pb208 (indep) Spettro vibrazionale di pairing per 208Pb INDEPENDENT PARTICLE MODEL INDEPENDENT PARTICLE MODEL

30. Spettro vibraz. Pb208 (extra energy) INDEPENDENT PARTICLE MODEL INDEPENDENT PARTICLE MODEL

31. Spettro vibraz. Pb208 (due fononi)

32. Curva dispersione Dispersion curve

34. Modello di pairing (0+) Pairing model ( = 0+) { In harmonic approximation: Dispersion relation:

35. Modello di pairing (2+) Pairing model   In harmonic approximation: Dispersion relation: