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Outline : towards effective superfluid local density approximation (SLDA)
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  1. Probing effective NN interaction at band termination M. Zalewski/saturday H. Zduńczuk/poster W. Satuła IFT Univ. of Warsaw in collaboration with R.A. Wyss, H. Zduńczuk, M. Zalewski, M. Kosmulski, G. Stoicheva, D. Dean, W. Nazarewicz, H. Sagawa (+exp), A. Bhagwat, J.Meng ... • Outline: • towards effective superfluid local density approximation (SLDA) • - general remarks • pairing: volume-, mixed- or surface-type • - selectivity/resolution of nuclear data • odd-even staggering (OES) in high-spin isomers – new opportunities to • study: • - blocking of pair correlations • - single-particle, time-odd, and residual p-n effects • termination in N~Z and N=Z nuclei in A~50 mass-region • - fine tuning of particle-hole field • - fine tuning of shell-model interaction • 73Kr – dynamical manifestation of T=0 pn-pairing at high spins?

  2. 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

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

  4. Saturation point of symmetric infinite nuclear matter: - saturation density ( ~0.16fm-3) - energy per nucleon (-16 0.2MeV) - incompresibility modulus (210 20MeV) - isoscalar effective mass (???) + + + + Finite nuclei [masses,radii,sp levels]: Asymmetric infinite nuclear matter and the isovector properties: entire ZOO of parameterizations !!! Fitting the Skyrme force parameters or the nuclear LEDF - symmetry energy ( 30 2MeV) • isovector effective mass (GDR sum-rule enhancement) - neutron-matter EOS (Wiringa, Friedmann-Pandharipande) - surface properties (semi-infinite nuclear matter) - realistic mean level-density (masses)

  5. TOWARDS EFFECTIVE (local) PAIRING THEORY (I) Low-momentum transfer expansion in particle-particle channel: v(k,k’) = g + g2(k-k’)2 + g4(k-k’)4+..... 2.0 S. Hilaire et al. PLB531 (2002) 61 ...in r-space: contact term D1S Gogny exp. th. 1.5 Dn (MeV) three-point filter 1.0 Gaussian Dirac-delta model leads to Gogny pairing • offers excellent agreement with data 0.5 (200MeVfm)2 1.4fm-1 • no cut-off is needed 0 20 40 60 80 100 120 140 neutron number (hc)2kF 1 x = dx = >> interparticle distance However, the so called coherence length i.e. spatial extension of the nucleonic Cooper pair x: (mc2)D kF 0 1000MeV 1MeV In this context the use of finite range pairing force can be viewed as rather unnecessary complication.

  6. TOWARDS EFFECTIVE (local) PAIRING THEORY (II) resolution Gogny versus local(DDDI) pp interaction: E. Garrido et al. PRC60, 064312 (1999) PRC63, 037304 (2001) DDDI: dots represent the Gogny gap Cut-off!!! (otherwise divergent!)

  7. TOWARDS EFFECTIVE (local) 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 ;

  8. 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 40MeV 45MeV 110Sn 50MeV 35MeV 30MeV Ec=20MeV 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

  9. Pairing/resolution (1) 0.6 2.0 0.4 0.25 1.5 0.2 0.20 0 0.15 1.0 80 20 40 60 0.10 0.05 0.5 3.5 1.0 1.5 2.0 0 3.0 Dn [MeV] 2.5 2.0 Dn,p(MeV) 1.5 1.0 0.5 spherical HFB b2 surface mixed volume J.Dobaczewski & W.Nazarewicz Prog. of Theor. Phys. Suppl. No. 146 (2002) 50Cr SLy4 No selectivity!!! Dn Dn Eexp-Eth [MeV] dD (MeV) deformed N,Z spherical 40 50 28 14 82 126 N,Z 20 40 60 80 100 120

  10. Pairing/resolution (2) Hilaire et al. PLB531 (2002) 61 5.0 4.5 surface mixed 4.0 volume 3.5 3.0 46Ti 2.5 SLy4 1.0 1.5 2.0 Vo(1-r(r)/ri) 0.20 EXP 0.15 b2 0.10 Beautiful example of selectivity!!! 0.05 deformed Eexp-Eth [MeV] spherical Dn [MeV] M. Kosmulski – licentiate thesis

  11. High-spin isomers - new opportunities to study pairing correlations ~ ~ ~ ~ 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, PRC72, 061303(R), (2005) 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)

  12. 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 fixed occup. 0.5 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 data HF SLy4/HSI HF SkO/HSI { GS EXP HSI • sp contribution to OES • time-odd terms (within • self-consistent models)

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

  14. 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 ...and the question is... Is blocking under control? Time-odd fields Rutz et al. PLB468 (1999) 1

  15. 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!!!!

  16. Mean-field versus Shell-Model Isospin symmtery „restoration” in N=Z nuclei: DZ 1.8 1.5 dET [MeV] SkO 1.4 T=1 1.0 1.0 nph dET DEth-DEexp [MeV] centroid 0.5 pph dET T=0 0 Can be evaluated from mirror-symmetric nuclei e.g. SM -0.5 42Ca 43Sc 45Sc 46Ti 40Ca from 40K and 42Sc etc. A 40 42 44 44Ca 44Sc 45Ti 47V 42Sc 46V 40Ca 44Ti N=Z T=0 pn pairing??? „isospin symmetry restoration” in N=Z nuclei original HF result for pph excitation G.Stoitcheva, WS, W.Nazarewicz, D.J.Dean, M.Zalewski, H.Zduńczuk, PRC73, 061304(R) (2006) SkO Shifted by 480keV reduced s-o

  17. -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

  18. Isovector dependence of shell-model matrix elements BFZ mechanism: „... It is a grave error to assume that the p-h intraction is independent of isotopic spin...” ... isotopic dependence of p-h interaction can be approximated by a monopole poten- tial vT~bt1.t2 Bansal & French, Phys.Lett. 11, 145 (1964) modified SM SM Zamick, Phys. Lett. 19, 580 (1965) DE=1/2b[T(T+1)- -Tp(Tp+1)-Th(Th+1)] 0.4 particle contribution Tp=T 1/2 single-hole contribution: Th=1/2 0.2 DESM-DEEXP [MeV] N = Z (Tp=1/2): + 0 Single j-shell J-T SM phenomenology: DE=-3b/4 VT=0+3d N = Z (Tp=T-1/2): bsym = bsym- 4d -0.2 Vphph(JT=0)+3d Vphph(JT=1)-d DE=b(T-1/2)/2 d=175keV -0.4 The SM overestimates b by ~700keV!!! 42Ca 43Sc 45Sc 46Ti 44Ca 44Sc 45Ti 47V 42Sc 46V 40Ca 44Ti (2j+3)VT=1,n=2-(2j+1)VT=0,n=2-2VT=1,J=0 N=Z 2(2j+1) VT=1-d N. Zeldes, Handbook of Nuclear Properties, Clarendon Press, Oxford, 1996, p.13

  19. modified SM SM 3/2+ 3- 0.8 3/2+ 0.6 3- 3- ESM-EEXP [MeV] 0.4 3- 3- 3- 0.2 3/2+ 3- 3- 3- 3- 3- 3- 3/2+ 3/2+ 2- 3/2+ 0 3/2+ 42Ca 44Sc 44Ti 46Ti 42Sc 47V 40Ca 44Ca 43Sc 45Sc 45Ti 46V Low-spin particle-hole intruder states SM versus modified-SM

  20. OES in high-spin isomeric states: • new opportunities to study blocking, TO terms, • residual-pn, and mean-field (sp-splitting) effects Volume-, mixed- or surface-like local pairing • selective nuclear data exist but must be systematically • identified (and understood) thrughout the nuclear chart Termination in N~Z, A~50 nuclei: 73Kr – possible fingerprint of enhanced T=0 pairing Concluding remarks Consistent superfluid local density approximation is just behind our doors! - excellent laboratory for fine-tuning of ph MF interaction and SM interaction

  21. (1) 73Kr - a fingerprint of T=0 pairing? 3qp 2.5 30 2.0 3qp 1.5 25 1.0 20 0.5 15 0.0 -0.5 10 5 g 40 0.5 0.5 0.5 1.0 1.0 1.0 1.5 1.5 1.5 fp R.Wyss, P.J. Davis, WS, R. Wadsworth Conventional TRS calculations involving only T=1 pairing: negative parity negative parity positive parity Ix (-,-) (+,+) 73Kr 73Kr (-,-) Ew [MeV] 5qp 1qp 1qp 73Kr: Kelsall et al., Phys. Rev. C65 044331 (2005) hw[MeV] hw[MeV] |1qp> = a+n(fp)|0> |3qp> = a+ng a+pg a+p(fp)|0> <1qp|E2|3qp> ~ 0 (one-body operator)

  22. (2) 73Kr a fingerprint of T=0 pairing? TRS involving T=0 and T=1 pairing Dn 1.0 Dp D [MeV] n(fp) p(fp) p(fp) DT=0 0.5 ng9/2 pg9/2 pg9/2 0 p(fp) n(fp) p(fp) 73Kr 30 pg9/2 pg9/2 ng9/2 25 Ix 20 theory 15 exp 10 5 0 hw [MeV] 1.4 0.4 0.8 1.0 1.2 1.6 0.2 0.6 What makes the 1qp and 3qp configurations alike? Scattering of a T=0 np pair in 73Kr n(fp)(-) vacuum 1qp configuration n(fp) ng9/2 n(fp) ng9/2 ng9/2(+) pg9/2 p(fp)(-) 3qp configuration

  23. Ix (-,+) 3qp 75Rb 75Rb (+,+) 2.0 30 1.5 Ew [MeV] 25 1qp 3qp 1.0 20 0.5 1qp 15 0.0 10 hw[MeV] hw[MeV] -0.5 5 0.5 1.0 1.5 0.5 0.5 1.0 1.0 1.5 1.5 (3) 73Kr a fingerprint of T=0 pairing? Conventional TRS calculations involving only T=1 pairing in neighbouring nuclei: negative parity positive parity all bands Excellent agreement was obtained in: Tz=1 : 74Kr,76Rb, D. Rudolph et al. Phys. Rev. C56, 98 (1997) Tz=1/2: 75Rb, C. Gross et al. Phys. Rev. C56, R591 (1997) Tz=1/2: 79Y, S.D. Paul et al. Phys. Rev. C58, R3037 (1998)