1. 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 73 Kr – dynamical manifestation of T=0 pn-pairing at high spins? W. Satuła IFT Univ. of Warsaw Probing effective NN interaction at band termination 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... M. Zalewski/saturday H. Zduńczuk/poster

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

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

4. - isovector effective mass (GDR sum-rule enhancement) Fitting the Skyrme force parameters or the nuclear LEDF 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 (???) + Asymmetric infinite nuclear matter and the isovector properties: - symmetry energy ( 30 2MeV) - neutron-matter EOS (Wiringa, Friedmann-Pandharipande) Finite nuclei [masses,radii,sp levels]: - surface properties (semi-infinite nuclear matter) - realistic mean level-density (masses) entire ZOO of parameterizations !!!

5. v(k,k’) = g + g 2 (k-k’) 2 + g 4 (k-k’) 4 +..... (I) Low-momentum transfer expansion in particle-particle channel: TOWARDS EFFECTIVE (local) PAIRING THEORY...in r-space: contact term Gaussian Dirac-delta model leads to Gogny pairing three-point filter 0 0.5 1.0 1.5 2.0 020406080100120140 exp. th. neutron number  (MeV) D1S Gogny offers excellent agreement with data S. Hilaire et al. PLB531 (2002) 61 no cut-off is needed  x  (  c) 2 k F (mc 2 )  >> 1 kFkF 1000MeV 1MeV (200MeVfm) 2 1.4fm -1 interparticle distance However, the so called coherence length i.e. spatial extension of the nucleonic Cooper pair  : 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: Cut-off!!! (otherwise divergent!) dots represent the Gogny gap

7. TOWARDS EFFECTIVE (local) PAIRING THEORY (IIIa) in-medium effects towards the SLDA approach: [Superfluid Local Density Approximation] anomalous density (pairing tensor) ultraviolet divergence Major obstacle in constructing SLDA is: we use cut-off!!! (usual, but not at all satisfactory solution) 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) Formally, gap depends on both the effective (running) coupling constant and on QP cut-off energy In fact, for sufficiently large E c gap is cutoff independent local HFB E c ~p c 2 /2m r ropc~ropc~ &E c ~  2 /mr o 2 ~ 40MeV interaction distance 110 Sn E c =20MeV 30MeV 35MeV 40MeV 45MeV 50MeV

9. E exp -E th [MeV]    (MeV) 22 50 Cr SLy4 0 0.05 0.10 0.15 0.20 0.25 surface mixed volume 1.01.52.0  n [MeV] 0.5 1.0 1.5 2.0 2.5 3.0 3.5 deformed spherical 20406080100120 0 0.2 0.4 0.6 204060 80 14 28 40 50 82 126  (MeV) N,Z nn nn 1.5 2.0 0.5 1.0 Pairing/resolution (1) No selectivity!!! J.Dobaczewski & W.Nazarewicz Prog. of Theor. Phys. Suppl. No. 146 (2002) spherical HFB

10. 22 0.05 0.10 0.15 0.20 EXP 46 Ti SLy4 2.5 3.0 3.5 4.0 4.5 5.0 1.01.52.0  n [MeV] E exp -E th [MeV] surface mixed volume deformed spherical Hilaire et al. PLB531 (2002) 61 M. Kosmulski – licentiate thesis V o (1-  (r)/  i ) Pairing/resolution (2) Beautiful example of selectivity!!!

11. 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  E 8.5MeV const. ~ ~ ~ ~ Hence, the OES: is similar in GS and HSI!!!! no blocking??? We expect: Dracoulis et al. PLB419, 7 (1998) High-spin isomers - new opportunities to study pairing correlations

12. Are Pairing Correlations Reduced in Excited States? (II)  (Z) [MeV] 0 0.5 1.0 1.5 2.0 2.5 3.0 616263646566 Z Z 0 0.4 0.8 61 63  (Z) [MeV] HF-SLy4 TE full GS HF SkO/HSI HF SLy4/HSI GS HSI EXP { { data h 11/2 -2.0 -1.5 -0.5 0.0 0.5 s 1/2 d 3/2 d 5/2 64 SkO SLy4 WS DIPM e sp hole in [402]5/2  -Fermi energy fixed occup. sp contribution to OES time-odd terms (within self-consistent models) Isomerism of the same type! Oblate shapes at HSI (-0.2) Nearly spherical GS Spherical sp spectrum

13.  E HSI [MeV] +10% Strutinsky calculations with pairing: +15% Enhanced pairing is needed (see also Xu et al. PRC60,051301 (1999)) Blocking is too strong!!! GAP/OES

14. Time-odd fields pairing time-odd effects (nuclear magnetism) single-particle effects residual pn interaction (odd-odd) This study reveals that many effects can contribute to OES in particular: Rutz et al. PLB468 (1999) 1 RMF Is blocking under control?...and the question is...

15.  E = f 7/2 n I max E( ) E( ) - d 3/2 f 7/2 n+1 I max 20 d 3/2 f 7/2 p-h energy scale (bulk properties) spin-orbit dominates!!! ~ 0 light ~ ½ heavy nuclei 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 these are ideal for fine tune particle-hole interaction!!!! Consider the energy difference between stretched (terminating) configurations in A~50 mass region

16. SM SkO -0.5 0 0.5 1.0 1.5  E th -  E exp [MeV] 40 Ca 44 Ti 46 V 42 Ca 44 Ca 43 Sc 44 Sc 45 Sc 45 Ti 47 V N=Z 46 Ti 42 Sc 404244 A 1.0 1.4 1.8  E T [MeV] SkO DZ Can be evaluated from mirror-symmetric nuclei e.g.  ph ph T=0 T=1 centroid Isospin symmtery „restoration” in N=Z nuclei:   40 Ca from 40 K and 42 Sc etc.   original HF result for  ph excitation Mean-field versus Shell-Model „isospin symmetry restoration” in N=Z nuclei Shifted by 480keV reduced s-o T=0 pn pairing??? G.Stoitcheva, WS, W.Nazarewicz, D.J.Dean, M.Zalewski, H.Zduńczuk, PRC73, 061304(R) (2006)

17. J. Terasaki, R. Wyss, and P.H. Heenen PLB437, 1 (1998) HFB calculations including T=0 and T=1 pairing 48 Cr  f 7/2  f 7/2 ] 4 4 16 + Collective (prolate) rotation data Non-collective (oblate) rotation isoscalar pairing d 3/2 g 9/2 T=1 collapses Skyrme interaction in p-h DDDI in p-p channel fully self-consistent theory no spherical symmetry no T=0 at low spins (termination)

18. -0.4 -0.2 0 0.2 0.4 44 Ti 46 V 42 Ca 44 Ca 43 Sc 44 Sc 45 Sc 45 Ti 47 V N=Z 46 Ti 42 Sc 40 Ca  E SM -  E EXP [MeV] SM modified SM V phph (JT=0)+3d V phph (JT=1)-d d=175keV Isovector dependence of shell-model matrix elements BFZ mechanism : N = Z (T p =1/2): N = Z (T p =T-1/2):  E=-3b/4  E=b(T-1/2) /2 The SM overestimates b by ~700keV!!! particle contribution T p =T 1/2  E=1/2b[T(T+1)- -T p (T p +1)-T h (T h +1)] single-hole contribution: T h =1/2 + Bansal & French, Phys.Lett. 11, 145 (1964) Zamick, Phys. Lett. 19, 580 (1965) „... 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 v T ~bt 1. t 2 V T=1 -d Single j-shell J-T SM phenomenology: b sym =b sym - 4d V T=0 +3d 2(2j+1) (2j+3)V T=1, =2 -(2j+1)V T=0, =2 -2V T=1,J=0 N. Zeldes, Handbook of Nuclear Properties, Clarendon Press, Oxford, 1996, p.13

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

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

21. Conventional TRS calculations involving only T=1 pairing: -0.5 0.0 0.5 1.0 1.5 2.0 2.5 E  [MeV] (+,+) (,)(,) 0.51.01.5   [MeV] 0.51.01.5 5 10 15 20 25 30 (,)(,) 73 Kr IxIx   [MeV] 0.51.01.5 1qp 5qp 73 Kr positive parity negative parity 3qp 1qp |1qp> = a + (fp) |0> |3qp> = a + g a +  g a +  (fp) |0> ~ 0 (one-body operator) g 40 fp 73 Kr: Kelsall et al., Phys. Rev. C65 044331 (2005) (1) 73 Kr - a fingerprint of T=0 pairing? R.Wyss, P.J. Davis, WS, R. Wadsworth

22. Scattering of a T=0 np pair (fp) g 9/2 (fp) g 9/2 (fp) g 9/2 (fp) g 9/2  (fp)  g 9/2  (fp)  g 9/2  (fp)  g 9/2  (fp)  g 9/2 1qp configuration (fp) (  vacuum g 9/2 (+)  g 9/2  (fp) (  ) 3qp configuration in 73 Kr What makes the 1qp and 3qp configurations alike? 0 5 10 15 20 25 30 0.40.81.21.6 0.20.6 1.0 1.4   [MeV] IxIx 73 Kr theory exp  T=0   0 0.5 1.0  [MeV] TRS involving T=0 and T=1 pairing (2) 73 Kr a fingerprint of T=0 pairing?

23. positive parity negative parity all bands (3) 73 Kr a fingerprint of T=0 pairing? Excellent agreement was obtained in: T z =1 : 74 Kr, 76 Rb, D. Rudolph et al. Phys. Rev. C56, 98 (1997) T z =1/2: 75 Rb, C. Gross et al. Phys. Rev. C56, R591 (1997) T z =1/2: 79 Y, S.D. Paul et al. Phys. Rev. C58, R3037 (1998) Conventional TRS calculations involving only T=1 pairing in neighbouring nuclei: