1. Global nuclear structure aspects of tensor interaction Wojciech Satuła in collaboration with J.Dobaczewski, P. Olbratowski, M.Rafalski, T.R. Werner, R.A. Wyss, M.Zalewski Kazimierz Dolny 2008 + NNN +.... tens of MeV ab initio Principles of low-energy nuclear physics  effective theories Coupling constants & fitting strategies Single-particle fingerprints of tensor interaction - SO splittings & magic gaps Influence of tensor fields on: - nuclear deformability - the total binding energy & S 2n - high-spin (terminating) states Summary

2. Modern Mean-Field Theory  Energy Density Functional  j, , , J,  T,  s,  F, Hohenberg-Kohn-Sham Effective theories for low-energy nuclear physics:

3. Fourier local correcting potential hierarchy of scales: 2r o A 1/3 roro ~ 2A 1/3 is based on a simple and very intuitive assumption that low-energy nuclear theory is independent on high-energy dynamics ~ 10 The nuclear effective theory Long-range part of the NN interaction (must be treated exactly!!!) where regularization Coulomb ultraviolet cut-off denotes an arbitrary Dirac-delta model Gogny interaction przykład There exist an „infinite” number of equivalent realizations of effective theories

4. lim  a a 0 Skyrme interaction - specific (local) realization of the nuclear effective interaction: spin-orbit density dependence 10(11) parameters  | v(1,2) |  Slater determinant (s.p. HF states are equivalent to the Kohn-Sham states) Spin-force inspired local energy density functional local energy density functional relative momenta spin exchange

5. Symmetric NM: - saturation density ( ~0.16fm -3 ) - energy per nucleon (-16 0.2MeV) - incompresibility modulus (210 20MeV) + - isoscalar effective mass (0.8) + Asymmetric NM: - isovector effective mass (GDR sum-rule enhancement) - symmetry energy ( 30 2MeV) + - neutron-matter EOS (Wiringa, Friedmann-Pandharipande) Finite, double-magic nuclei [masses,radii, rarely sp levels]: -surface properties -ZOO– 20 parameters are fitted to: density   dependent CC Skyrme-inspired functional is a second order expansion in densities and currents: tensor spin-orbit

6. 150 160 170 180 190 0.70.80.9 1 m*/m W0W0 SLy4 SLy5 SkP SkXc SkM* SIII SkO 5.5 6.5 7.5 8.5 140150160170180190 experiment std. so 90% so SkP SkO SkXc SkM* MSk1 SLy5 SLy4 SkI1 SIII  e(f 7/2 -f 5/2 ) [MeV] W0W0 W0W0 120130140150160170  e(d 3/2  f 7/2 ) [MeV] SkP SkM* SkXc SLy4 SkI1 SIII SkO MSk1 5 6 7 experiment * std. so 90% so scales with W o (two-body SO interaction) Binding energy-dictated fit: superficial m* dependence in the spin-orbit strength: and contradicting scalings in the single-particle splittings scales with W o * (W o * = W o ) m momo *

7. Fitting strategies of the tensorial coupling constants (I)  e(f 5/2 -f 7/2 ) [MeV] 5 6 7 8 5 6 7 8 40 Ca 48 Ca 56 Ni a) b) neutrons protons bare SkO spectra

8. SkP T T 0 =-39(*5);T 1 =-62(*-1.5);SO*0.8 C1C1 J C0C0 J 1 3 5 7 0.70.80.91 40 Ca 1 3 5 7 -40-30-20-100 56 Ni f 7/2 -f 5/2 p 3/2 -p 1/2 f 7/2 -d 3/2 2 4 6 8 -80-60-40-200 f 7/2 -f 5/2 f 7/2 -d 3/2 from binding energies 48 Ca f 7/2 -f 5/2 f 7/2 -d 3/2 f 7/2 -p 3/2 p 3/2 -p 1/2 Single-particle energies [MeV] Fitting strategies of the tensorial coupling constants (II) 1) Fit of the isoscalar SO strength 48 Ca 56 Ni 40 Ca 2) Fit of the isoscalar tensor strength: 3) Fit of the isovector tensor strength or, more precisely, C 1 J /C 1 j<j< j>j> FF j>j> FF j<j< - the details -  J 48 Ni or 78 Ni are needed in order to fix SO-tensor sector f 7/2 f 5/2 splittings around

9. OUR VALUES OF COUPLING CONSTANTS: -100 -50 0 50 -4004080 Colo BSF triangle C 1 [MeV fm 5 ] J Brink & C 0 [MeV fm 5 ] J SLy4 SkP SLy5 Skxc SkO’ MSk1 SkO T SLy4 T SkP T Stancu Skxta Skxtb et al. C0∇JC0∇J C0JC0J C1JC1J m* SLy4 SKO SKP SIII SkM* 0,69 0,90 1,00 0,76 0,67 -60 -45 -60 -62 -33 -92 -60 -38 -61 -58-51-65 -56 -42 -68 all CC are in [MeV fm 5 ] „World” CC overview - strategy dependence - Colo et al. PLB646, 227 (2007) C0∇JC0∇J C1∇JC1∇J = 3 Standard: SkO: = -0,78 Brown et al. PRC74, 061303 (2006) Brink & Stancu, PRC75, 064311 (2007)

10. M.Zalewski, J.Dobaczewski, WS, T.Werner, PRC77, 024316 (2008) Spin-orbit splittings [MeV] SLy4 T T 0 =-45;T 1 =-60; SO*0.65 n 1h1h 1i1i f 7/2 -f 5/2 g 9/2 -g 7/2 1 3 5 7 1 3 5 7 16 O 40 Ca 48 Ca 56 Ni 90 Zr 132 Sn 208 Pb p 1h1h f 7/2 -f 5/2 g 9/2 -g 7/2 16 O 40 Ca 48 Ca 56 Ni 90 Zr 132 Sn 208 Pb SLy4 T (I) spin-orbit splittings Selected single-particle fingerprints of tensor interaction:

11. (II) magic-gap energies Selected single-particle fingerprints of tensor interaction: (III) „Otsuka mechanism”: Neutrons filling j > ’ subshell influence proton s.p. energies: M.Zalewski et al., PRC77, 024316 (2008) Otsuka et al., PRL87, 082502 (2001); PRL95, 232502 (2005)

12. Z N 14 32 56 90 14 – d 5/2 32 – f 7/2 p 3/2 56 – g 9/2 d 5/2 90 – h 11/2 f 7/2 total isoscalar Z N isoscvector Z N The tensorial „magic structure” N=Z

13. Z~14, N~32 Baumann et al. Nature Vol 449, 1022 (2007) 40 Mg, 42 Al Z~32 N~56 Z~56 N~90 known nuclei Tensor forces in neutron rich nuclei

14. SkO T’’ : 1.00015C 0  & 0.99C 1  -2 0 1 2 3 16 O 40 Ca 48 Ca 56 Ni 80 Zr 90 Zr 100 Sn 132 Sn 208 Pb E TH – E EXP [MeV] E>0 SLy4 SkO T’ SkO T’’ 20 shells SkO T’ : SO reduced by 15% C 0 J =-44.1MeVfm 5 C 1 J =-91.6MeVfm 5 -20 -15 -10 -5 0 5 40 Ca 48 Ca 56 Ni 90 Zr 132 Sn 208 Pb SLy4 SLy4 T SLy4 Tmin E TH – E EXP [MeV] M.Zalewski et al., PRC77, 024316 (2008)

15. 5 6 7 8 5 6 7 8 40 Ca 48 Ca 56 Ni 5 6 7 8 40 Ca 48 Ca 56 Ni 5 6 7 8 5 6 7 8 5 6 7 8  e( f 5/2 - f 7/2 ) [MeV]  e(  f 5/2 -  f 7/2 ) [MeV] bare Polarisation effects in a presence of strong tensor fields SkO versus SkO T’ time-even TE&TO

16. 0 5 10 18202224262830 ( ( ) ) -2 0 1 18202224262830 A S 2n [MeV]  S 2n [MeV] oxygen SkO SkO T’ AME03 d 5/2 d 3/2 s 1/2 Influence of tensor on two-neutron separation energy in oxygen isotopes

17. Deformation properties in a presence of strong tensor fields

18. 0 2 4 6 8 10 12 00.10.20.30.4 00.10.20.30.4  E [MeV] tensor 00.10.20.30.4 spin-orbit deformacja  2 SkO SkO TX SkO T’ f 7/2 f 5/2 p 3/2 neutrons protons 4p-4h [303]7/2 [321]1/2 Nilsson -6 -5 -4 -3  E tensor [MeV] 00.10.20.30.4 22 Rudolph et al. PRL82, 3763 (1999)

19. 0 1 2 3 4 5 6 0.10.20.30.40.5 SkO SkO TX tensor SkO T’ spin-orbit  E [MeV] 22 80 Zr constrained HFB calculations in spin-saturated 80 Zr

20.  E = f 7/2 n I max E( ) E( ) - d 3/2 f 7/2 n+1 I max Further tests in simple-situations: terminating states around A~50: across the gap 46 Ti 24 protons neutrons +3/2 +1/2 -1/2 -3/2 +7/2 +5/2 +3/2 +1/2 -1/2 -3/2 -5/2 -7/2 p-h +3  (n=7) f 7/2 d 3/2 00 14  f 7/2 +3/2 +1/2 -1/2 -3/2 d 3/2 +7/2 +5/2 +3/2 +1/2 -1/2 -3/2 -5/2 -7/2 partially f 7/2 (n=6) 20 filled fully 28 cranking: -  j z PRC71, 024305 (2005) H.Zduńczuk, W.Satuła, R.Wyss

21.  E th -  E exp [MeV] -2.0 -1.5 -0.5 42 Ca 44 Ca 44 Sc 45 Sc 45 Ti 46 Ti 47 V  E th -  E exp [MeV] SIII SkM* SkP SkO SLy4 SLy5 SkXc Spin-orbit and tensor modified parameterizations Standard parameterizations: „spectroscopic-quality” functionals must have large (>0.9) effective mass!!! ~ 20 d 3/2 f 7/2 p-h ~5MeV

22. SUMMARY & OUTLOOK Simple three-step procedure is proposed in order to fit the SO & tensor CC The method leads to strong attractive tensor fields and week SO potentials:  improvement of the s.p. properties The tensor interaction influences:  binding energies („magic structure”)  S 2n energies  nuclear deformability (novel mechanisms)  high-spin properties...... in an extremely neat and robust manner... Amenable to further generalizations...

23. mean-field averaging From two-body, zero-range tensor interaction towards the EDF:

24. Local Density Functional Theory for Superfluid Fermionic Systems: The Unitary Gas Aurel Bulgac, Phys. Rev. A 76, 040502 (2007) ab initio calculations by: Chang & Bertsch Phys. Rev. A76, 021603 von Stecher, Greene & Blume, E-print:0705.0671v1 running coupling constant in order to renormalize.... ultraviolet divergence in pairing tensor