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 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

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

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

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

Symmetric NM: - saturation density ( ~0.16fm -3 ) - energy per nucleon ( MeV) - 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

m*/m W0W0 SLy4 SLy5 SkP SkXc SkM* SIII SkO experiment std. so 90% so SkP SkO SkXc SkM* MSk1 SLy5 SLy4 SkI1 SIII  e(f 7/2 -f 5/2 ) [MeV] W0W0 W0W  e(d 3/2  f 7/2 ) [MeV] SkP SkM* SkXc SLy4 SkI1 SIII SkO MSk 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 *

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

SkP T T 0 =-39(*5);T 1 =-62(*-1.5);SO*0.8 C1C1 J C0C0 J Ca Ni f 7/2 -f 5/2 p 3/2 -p 1/2 f 7/2 -d 3/ 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

OUR VALUES OF COUPLING CONSTANTS: 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, 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, (2006) Brink & Stancu, PRC75, (2007)

M.Zalewski, J.Dobaczewski, WS, T.Werner, PRC77, (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/ 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:

(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, (2008) Otsuka et al., PRL87, (2001); PRL95, (2005)

Z N – 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

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

SkO T’’ : C 0  & 0.99C 1  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 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, (2008)

Ca 48 Ca 56 Ni Ca 48 Ca 56 Ni  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

( ( ) ) 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

Deformation properties in a presence of strong tensor fields

 E [MeV] tensor 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  E tensor [MeV] 22 Rudolph et al. PRL82, 3763 (1999)

SkO SkO TX tensor SkO T’ spin-orbit  E [MeV] 22 80 Zr constrained HFB calculations in spin-saturated 80 Zr

 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, (2005) H.Zduńczuk, W.Satuła, R.Wyss

 E th -  E exp [MeV] 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

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...

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

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