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Historical remarks (I)

gap in e-e spectra. Historical remarks (I). this work firmly established that deformed Nilsson plus pairing hamiltonian constitutes. a „minimal” model for nuclear structure. Historical remarks (II). pairing strength. odd-even staggering. moment of inertia. quadrupole moment

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Historical remarks (I)

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  1. gap in e-e spectra • Historical • remarks (I)

  2. this work firmly established that deformed Nilsson plus pairing hamiltonian constitutes a „minimal” model for nuclear structure • Historical remarks (II) pairing strength odd-even staggering moment of inertia quadrupole moment deformation Belyaev: Mat. Fys. Medd. Dan. Vid. Selsk. 31 no. 11 (1959)

  3. Historical remarks/background (III) Pair correlations take place in the vicinity of Fermi energy, in the region: EF-D < p2/2m < EF+D pF2 (pF+dp)2 + vFdp = 2m 2m Uncertainity in momentum space: dp ~ D/vF dx ~ hvF/D (the so called coherence length i.e. spatial extension of the Cooper pair x) (200MeVfm)2 1.4fm-1 • nucleonic Cooper pairs • overlap!!! (hc)2kF 1 x = dx = >> (mc2)D kF • hierarchy!!! 1000MeV 1MeV interparticle distance these considerations pertain to infinite-matter Nuclei are s-wave [predominantly] superconductors in weak-coupling limit!

  4. Pairing in nuclei (selected aspects) W. Satuła IFT Univ. of Warsaw • Outline: • towards effective local pairing theory • - general remarks • odd-even staggering (OES) in high-spin isomers • - blocking of pair correlations • - single-particle, time-odd, and residual p-n effects • termination in N~Z and N=Z nuclei • - fine tuning of particle-hole field • - pn-pairing at high-spins? • from isoscalar pairing to the nuclear symmetry energy (and back!)

  5. Skyrme-force as a particular realization of effective ph interaction Fourier Long-range part of NN interaction (must be treated exactly!!!) local correcting potential infinite number of equivalent effective theories

  6. Gogny: a 0 density-dependent Y | H | Y Slater determinat (number conserving) Skyrme interaction: lim da 10(11) spin-orbit parameters LEDF: 20 parameters

  7. E=-kB ~ 0 2 PHASE-SHIFTS AND 1S0 PAIRING GAP IN IFINITE MATTER Effective range approx: ao=-18.8fm, ro=2.75fm Large negative scattering length, ao=-18.8fm, indicates nearly bound state at zero scattering energy: (pole in NN T-matrix) r0 kF fm-1 1S0 gap in neutron matter from bare NN potentials The inverse scattering problem can be solved for separable potentials Two-particle potential is determined by phase shifts (at all possible energies!!!)

  8. volume contact term effective range corrections....... Can be integrated using DR Gap eq. is divergent! independent on TOWARDS EFFECTIVE PAIRING THEORY (I) pairing in dilute Fermi gases: T.Papenbrock & G.F.Bertsch PRC59, 2052 (1999) v(k,k’) = g + g2(k-k’)2 + g4(k-k’)4+..... Scattering length is also divergent! The divergencies are of the same type and therefore they cancel out!

  9. bare Paris-force ( Gogny) effective G-matrix various corrections must cancel each other!!!! [Let’s leave the proof for purists!!!] with B-HF sp spectra!!! TOWARDS EFFECTIVE PAIRING THEORY (II) in-medium effects towards eff. pp interaction: E. Garrido et al. PRC60, 064312 (1999) PRC63, 037304 (2001) In-medium effects (no consensus is reached so far of how to calculate in medium effects) free spectrum

  10. Dn (MeV) 2.0 exp. th. 1.5 1.0 0.5 0 20 40 60 80 100 120 140 160 0 Neutron number No. of nuclei taken to the average in HFB 10 D1S Gogny 5 three-point filter S. Hilaire et al. PLB531 (2002) 61

  11. DDDI used in Skyrme-HFB calculations by Terasaki et al. NPA621 (1997) 706. Isoscalar pairing Tensor force enhancement Cut-off!!! (otherwise divergent!) ro TOWARDS EFFECTIVE PAIRING THEORY (III) in-medium effects towards local eff. pp interaction: DDDI: E. Garrido et al. PRC60, 064312 (1999) PRC63, 037304 (2001)

  12. TOWARDS EFFECTIVE PAIRING THEORY (IIIa) in-medium effects towards the SLDA approach: [Superfluid Local Density Approximation] Major obstacle in constructing SLDA is: we use cut-off!!! (usual, but not at all satisfactory solution) anomalous density (pairing tensor) ultraviolet divergence In particular, in infinite homogenous system (example): regular „regularize” means in practice simply „remove divergent part” (relate to scattering amplitude; use dimensional regularization; introduce counter-terms [regulators] with explicit cut-off) isolate and regularize divergent term ;

  13. LOCAL EFFECTIVE PAIRING THEORY (IIIb) Bulgac-Yu SLDA approach: A.Bulgac, Y.Yu, PRL88, 042504 (2002) A.Bulgac, PRC65, 051305(R) (2002) In fact, for sufficiently large Ec gap is cutoff independent local HFB 110Sn Formally, gap depends on both the effective (running) coupling constant and on QP cut-off energy interaction distance & ropc~h Ec~h2/mro2 ~ 40MeV Ec~pc2/2mr

  14. ~ ~ ~ ~ We expect: Odd-Even Binding Energy Effect in the High-Spin Isomers: Are Pairing Correlations Reduced in Excited States? A. Odahara, Y. Gono, T. Fukuchi, Y. Wakabayashi, H. Sagawa, WS, W. Nazarewicz Stretched configurations: N=83 DE 8.5MeV const. Hence, the OES: is similar in GS and HSI!!!! no blocking??? Dracoulis et al. PLB419, 7 (1998)

  15. HF-SLy4 GS 0.5 full 3.0 0.8 0.0 D(Z) [MeV] TE p-fermi energy -0.5 2.5 0.4 -1.0 esp D(Z) [MeV] 0 2.0 -1.5 Z d3/2 61 63 -2.0 s1/2 1.5 h11/2 1.0 64 HF SLy4 fixed occup. HF SkO 0.5 { GS data EXP HSI 0 { d5/2 Z SkO WS SLy4 DIPM 61 62 63 64 65 66 hole in [402]5/2 Are Pairing Correlations Reduced in Excited States? (II) Isomerism of the same type! Oblate shapes at HSI (-0.2) Nearly spherical GS Spherical sp spectrum • sp contribution to OES • time-odd terms (within • self-consistent models)

  16. 2.0 OES Excitation energy 1.5 DPES dpn 1.0 0.2 |dpn| [MeV] 0.1 61 62 65 Z 0.25 10 dD[MeV] D(Z) [MeV] 1.5 Z DIPM 63 64 1.0 9 0.20 DPES EXP DPES+dpn DIPM 8 0.15 mean-gap 61 62 63 64 65 66 Z Sm Eu Gd Tb Dy Ho Nd Pm 149 143 145 147 61 63 65 67 144 146 148 150 Blocking is too strong!!! Strutinsky calculations with pairing: DEHSI [MeV] +15% +10% Enhanced pairing is needed (see also Xu et al. PRC60,051301 (1999))

  17. This study reveals that many effects can contribute to OES in particular: pairing single-particle effects time-odd effects (nuclear magnetism) residual pn interaction (odd-odd) RMF Time-odd fields Rutz et al. PLB468 (1999) 1

  18. Consider the energy difference between stretched (terminating) configurations in A=50 mass region • the best examples of almost unperturbed sp motion • uniquely defined (in N=Z) • config. mixing beyond mean-field is expected to be mariginal • (in particular all pairs are broken) • shape-polarization effects included already at the level of the SHF • time-odd mean-fields (badly known) can be tested n f7/2 energy scale (bulk properties) E( ) f7/2 Imax p-h -1 n+1 20 d3/2 f7/2 E( ) - Imax d3/2 spin-orbit dominates!!! ~ 0 light ~ ½ heavy nuclei DE = these are ideal for fine tune particle-hole interaction!!!!

  19. Terminating states in A=50 mass region the results shifted by 475keV includes isospin correction to the pph excitation reduced s-o Isospin symmetry „restoration” in N=Z nuclei: SM(+475keV) 1.5 SkO 1.0 T=1 nph DEexp-DEth [MeV] 0.5 D centroid pph D 0 T=0 -0.5 Can be evaluated from mirror-symmetric nuclei e.g. 42Ca 44Sc 45Ti 47V 44Ti 40Ca from 40K and 42Sc etc. 44Ca 45Sc 46Ti 40Ca 46V original HF result for pph excitation H. Zduńczuk et al. PRC71, 024305 (2005) G.Stoicheva, D.Dean, W.Nazarewicz, WS, M.Zalewski, H.Zduńczuk

  20. -1 d3/2 g9/2 48Cr 4 4 [nf7/2pf7/2] 16+ HFB calculations including T=0 and T=1 pairing J. Terasaki, R. Wyss, and P.H. Heenen PLB437, 1 (1998) • Skyrme interaction in p-h • DDDI in p-p channel • fully self-consistent theory • no spherical symmetry isoscalar pairing no T=0 at low spins Non-collective (oblate) rotation T=1 collapses Collective (prolate) rotation (termination) data

  21. The Wigner effect „mass deffect” within mean-field Experimental data compilation by N. Zeldes, Racah Institute of Physics Hebrew U., Jerusalem (1993)

  22. Energy gain as a functionof T=0/T=1 pairing’s mixing „x” Energy gain: DMass =E(T=0+1)- E(T=1) Thomas-Fermi X=1.1 X=1.2 X=1.3 X=1.4 X= / generalized blocking effect n-excess blocks pn-pairs scattering Wigner term (Myers-Swiatecki) neutrons protons Satuła & Wyss PLB393 (1997) 1

  23. Neutron-proton pairing collectivity (a fit plus three easy steps) (III) ET=1 - ET=0 (even-even) ET=1 - ET=0 (odd-odd) (II) Fit of GT=0/GT=1 • Wigner energy linked to the n-p pairing collectivity • T=2 states in even-even nuclei obtained from isocranking • T=1 states in even-even nuclei obtained as 2qp excitations • T=1 states in odd-odd nuclei obtained from isocranking • T=0 states in odd-odd nuclei obtained bas 2qp excitations (I) W. Satuła & R. Wyss Phys. Rev. Lett., 86, 4488(2001); Phys. Rev. Lett., 87, 052504(2001) ET=2 - ET=0 (even-even)

  24. 3 6 2 1 H=hsp- wT+kTT 2 1 Tx 0 3 DT=0 E=(de+k)T2 2 DT=1 1 1 1 + kT T=2 T=2 T=2 2 2 0 Exc. energy DE 48Cr 0 1 2 3 GS hw [MeV] GS GS TT T=1 + T=0 + Schematic isospin-isospin interaction: see e.g. Bohr & Mottelson „Nuclear Structure” vol. I Neergard PLB572 (2003) 159 mean-field (Hartree) HMF=hsp- (w - k T )T Hartree-Fock D D [MeV]

  25. de*= de (mo/m) * deT2 = DEHF(ro,to,..): 1 2 1.5 1.0 1.5 1.0 2 6 10 14 18 2 6 10 14 18 * isoscalar effective-mass scaled 48 exp SkP SkO 68 SLy4 SkXc SIII MSk3 de [MeV] 88 0.9 0.7 0.5 T T

  26. kT(T+1) 1 1 48 2 2 kT2 1.8 2.4 1.4 2.0 1.6 1.0 68 1.8 1.6 1.6 1.3 1.4 1.4 1.4 1.1 1.2 1.2 1.2 88 0.9 1.0 1.0 0.7 0.8 2 2 6 6 10 10 14 14 18 18 DEHF(full) – DEHF(isoscalar) = k [MeV] SkP SkO SLy4 SkXc SIII MSk3 T T

  27. Satuła & Wyss PLB SLy4 SLy4 e [MeV] 2 118.4 ( 1.45 ) 1- e(eF) = ~ g(eF)-1 ~ 1,5 A A 1/3 * 1 = ~ SM p ... ( 94.5 e 1.56 ) * 0,5 1- ~ (B) ~ 4kF VM A 1/3 A 0 A 0 20 40 60 80 100 120 k , Calculated surface-to-volume ratio for e k 1.56(SLy4); 1.51(SkXc); 1.61(SkP); 1.55(SkO); 1.70(SkM*) Mass-dependence of the nuclear symmetry energy: Theory: S.p. level density for diffuse-wall potential Stocker & Farine, Ann. of Phys. (NY) 159(2):255,1983 1 1 - A 1 1 - A 1.52A-1/3... ro=1.14fm kF=1.36fm-1

  28. Concluding remarks • Local effective pairing theory is knocking to our doors. OES in high-spin isomeric states: (II) - blocking, TO terms, residual-pn, and mean-field effects (IIIa) Termination in N~Z nuclei: - excellent laboratory for fine-tuning of p-h interactions (IIIb) Termination in N=Z nuclei: - possible signal of np T=0 pairing effects at high spins (IV) T=0 pairing Wigner energy symmetry energy

  29. c J.Dobaczewski Mean-field equations

  30. c J.Dobaczewski Complete local energy density Pairing Mean field

  31. c J.Dobaczewski Nuclear densities as composite fields

  32. Band terminating states - an example of 46Ti24 proton neutron 20 (n=6) -1 (n=7) f7/2 d3/2 f7/2 -7/2 -7/2 -5/2 -5/2 -3/2 -3/2 -1/2 -1/2 partially filled f7/2 +1/2 +1/2 +3/2 +3/2 14h p-h +5/2 +5/2 +7/2 across the gap +7/2 +3h -3/2 -3/2 -1/2 -1/2 fully filled d3/2 0h +1/2 +1/2 +3/2 +3/2 cranking

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