n-p pairing in N=Z nuclei reality or fiction? W. Satuła University of Warsaw Motivation & fingerprints (basic concepts): symmetric nuclear matter calculations binding energies - mean-field crisis around N~Z line the Wigner energy and the generalized blocking phenomenon elementary isobaric exciatations in N~Z nuclei – a need for isosopin symmetry restoration cranking in isospace - response of t=0 pairing against rotations in isospace high-spin signatures of pn-pairing

Structure of nucleonic pairs N=Z  nucleons start to occupy „identical” spatial orbitals Nuclear interaction favoures L=0 coupling Pair-structure is governed by the Pauli principle: Isovector (or 1S0) pairs T=1, S=0 - Isoscalar (deuteron-like or 3S1) pairs T=0, S=1 Neutron Proton + Tz +1 0 -1 - Sz +1 0 -1

3S1-3D1 (coupled) pairing gap in symmetric Nuclear Matter from Paris VNN free-space sp spectrum BHF sp spectrum (in-medium corrections) tensor-force enhancement From M. Baldo et al. Phys. Rev. C52, 975 (1995)

Gaps from local effective pairing interaction DDDI: E. Garrido et al. PRC60, 064312 (1999) PRC63, 037304 (2001) DDDI used in Skyrme-HFB calculations by Terasaki et al. NPA621 (1997) 706. Isoscalar pairing Tensor force enhancement ro Cut-off!!! (otherwise divergent!)

3S1-3D1 (coupled) pairing gap in symmetric Nuclear Matter including relativistic corrections Saturation density O. Elgaroy, L.Engvik, M.Hjorth-Jensen, E.Osnes, Phys.Rev.C. 57 (1998) R1069 Includes relativistic in-medium corrections (sp levels from Dirac-Brueckner-HF)

Empirical NN interaction in N~Z T=1 channel: J=0 coupling dominates T=0 channel: J=1 and J=2j are similar T=0 is, on the average, stronger than T=1 by a factor of ~1.3 N.Anantaraman & J.P. Schiffer PL37B (1971) 229 Dufour & Zucker, Phys. Rev. C54, 1641 (1996)

The model: deformed mean-field plus pairing: Pairs p-n and p-ñ Pairs: ~ Pairs ñ-n and p-p « usual » ; T=1 ~ Pairs: p-ñ + n-p; T=1 0 0 ~ Pairs: p-ñ – n-p ; T=0 Hamiltonian BCS: N.Anantaraman and J.P. Schiffer PL37B (1971) 229

Comparison with delta-force towards a local theory M.Moinester, J.P. Schiffer, W.P. Alford, PR179 (1969) 984

BCS transformation BCS transformation takes the following form : real A.L.Goodman Nucl. Phys. A186 (1972) 475 BCS transformation takes the following form : real complex where the variational parameters are: i 2 Density matrix (occupation) and the pairing tensor Generalization: BCSHFB UiU & ViV matrices of dimension 4N

BCS Solution Energy (Routhian) Variational equation in N=Z system (without Coulomb) Occupation probabilities ; quasiparticle energies: Pair gaps:  n-ñ, p-p T=1 n-p + p-ñ ~ Gap T=0 aã Gap T=0 aa ~

T=0/T=1 (no)mixing 48Ca X= / Incomplete mixing? T=1, Tz=+/-1 andTz=0 W.S. & R.Wyss PLB 393 (1997) 1

Energy gain as a function of 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 from Myers & Swiatecki protons neutrons Satuła & Wyss PLB393 (1997) 1

Wigner effect from self-consistent Skyrme-HF N=Z Exp. HFBCS T=1 Sph. HFBCS T=1 Def. (SIII) Defficiency of conventional self-consistent models: HF or HFB including standard T=1, |Tz|=1 ~ p-p & n-n pairs: (N-Z)2 ~ T2 term is OK! no (or very weak) |N-Z| ~ term |N-Z|=2,4 (black) A.S. Jensen, P.G.Hansen, B.Jonson, Nucl.Phys. A431(1984) 393 o-o e-e

DE= asymT(T+x) The Wigner effect 1 2 w / w X= Jmax 0? 1?? 25 A=48 20 B (MeV) 15 48Cr 10 1.0 5 DE= asymT(T+x) 2 1 w w / w total 0.8 -4 4 N-Z 0.6 0.4 0? 1?? 1.25??? exp. in N~Z 4 ???? Wigner SU(4) 0.2 24Mg X= 0.0 1 2 3 4 5 6 7 Jmax

Isobaric excitations in N~Z nuclei 0.5 1.0 1.5 2.0 47/A [MeV] W(A) [MeV] 30 40 0.6 1.4 GT=0 GT=1 A The lowest: T=0, T=1 & T=2 in e-e nuclei T=0 & T=1 states in o-o nuclei The model needs to be extended to include isospin projection  isospin cranking strong T=0 pairing limit! J.Janecke, Nucl. Phys. A73 (1965) 73 A. Macchiavelli et al. Phys. Rev. C61 (2000) 041303(R) P.Vogel, Nucl. Phys. A662 (2000) 148

„inertia” defined through The extreme s.p.model: 4-fold degenerated equidistant s.p. spectrum Eigen-states (routhians) are 2-fold (Kramers) degene- rated „stright lines”: Crossings form simple arithmetic serie:   Energy: „inertia” defined through mean level spacing !!!

5 10 15 20 DET=2 [MeV] T=2 iso-cranking 20 30 40 50 A DE= deT2 1 2 14 T=2 states in e-e nuclei 5 10 15 20 20 28 DET=2 [MeV] hWS+HT=1 +HT=0-wtx T=2 hWS+HT=1 -wtx iso-cranking 20 30 40 50 A Iso-cranking gives excitation energy which goes like: DE= deT2 1 2 + Epair vacuum mean level spaceing at the Fermi energy

(iso)Coriolis antipairing effect iso-MoI D [MeV] hw [MeV] 48Cr 6 Tx 3 0.5 1.0 1.5 iso-moment of inertia D/e = 0.001 D [MeV] 1 2 3 hw [MeV] DT=0 DT=1 48Cr 6 Tx e=1 D/e = 0.5;1.0;1.5 0.7 0.6 iso-moment of inertia Tz 1 2 3 4 0.5 0.4 0.3 1 2 3 hw

T=1 states in e-e N=Z nuclei T=1 states: 2qp + isocranking

Isocranking N=Z odd-odd nuclei odd-T sequence T de 5 de de 2de 4de 6de 4 hw even-T sequence 3 de 2 de de 1 hw de 3de 5de iso-signature selection rule Eeven-T = 1/2deTx2 Eodd-T = 1/2deTx2 - 1/2de

T=0 vs T=1 states in o-o N=Z nuclei 1.0 2qp cranking vacuum 0.5 DET=1 - DET=0 [MeV] 0.0 -0.5 exp th 20 30 40 50 60 70 A

Neutron-proton pairing collectivity (a fit plus three easy steps) (III) ET=1 - ET=0 (even-even) ET=1 - ET=0 (odd-odd) (II) 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 as 2qp excitations Fit of GT=0 /GT=1 ET=2 - ET=0 (even-even) (I) W. Satuła & R. Wyss Phys. Rev. Lett., 86, 4488 (2001); Phys. Rev. Lett., 87, 052504 (2001)

see e.g. Bohr & Mottelson „Nuclear Structure” vol. I Schematic isospin-isospin interaction: extreme sp model de 3de hw de+k 3(de+k) l even-even vacuum see e.g. Bohr & Mottelson „Nuclear Structure” vol. I Neergard PLB572 (2003) 159 1 H=hsp- wT+ kTT 2 mean - field (Hartree) HMF =hsp- (w - k T )T iso-cranking with isospin-dependent frequency!!! E= (de+k)T2 1 2 Hartree E= (de+k)T2 + kT 1 2 1 2 Hartree- -Fock

Pairing in fast rotating nuclei Muller et al., Nucl. Phys. A383 (1982) 233 Resistance of nucleonic paires against fast rotation:

48Cr ; HFB calculations including T=0 & 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 two-classes of solutions: d3/2 g9/2 -1 - T=0 dominated at I=0 - T=1 dominated at I=0 isoscalar pairing Non-collective (oblate) rotation no T=0 at low spins Collective (prolate) rotation T=1 collapses [nf7/2 pf7/2] 4 16+ (termination) exp

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

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

(3) 73Kr – manifestation of (dynamical) T=0 pairing? Conventional TRS calculations involving only T=1 pairing in neighbouring nuclei: all bands positive parity negative parity Ix (-,+) 75Rb 75Rb 3qp 5 10 15 20 25 30 -0.5 0.0 0.5 1.0 1.5 2.0 (+,+) Ew [MeV] 1qp 3qp 1qp 0.5 1.0 1.5 0.5 1.0 1.5 0.5 1.0 1.5 hw [MeV] hw [MeV] 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)

SUMMARY Part of T=0 correlations in N~Z nuclei is definitely beyond standard formulation of mean-field (Wigner energy) Adding T=0 pairing helps but cannot solve the problem of the Wigner energy (symmetry energy) in N~Z nuclei which seems to be beyond mean-field There is no convincing arguments for coherency of the T=0 phase Theoretical treatment of T=1 states in e-e nuclei and T=0 states o-o nuclei requires angular momentum and isospin projections

Independent least-square fits of: the Wigner energy strength: aw|N-Z|/Aa the symmetry energy strength: as(N-Z)2/Aa 4asT(T+x); x=aw/2as a aw 2as sn-1 x (*) (**) 0.95 39 0.196 31 0.106 1.26 1/2 8 0.239 6 0.153 1.33 2/3 14 0.213 11 0.125 1.27 1 47 0.196 38 0.107 1.24 Głowacz, Satuła, Wyss, J. Phys. A19, 33 (2004) very consistent with: Janecke, Nucl. Phys. (1965) 97 Fit includes N~Z nuclei with: Z>10; 1<Tz<3 excluding odd-odd Tz=1 nuclei - - - (*) See: Satuła et al. Phys. Lett. B407 (1997) 103 (**) Based on double-difference formula: J.-Y Zhang et al. Phys. Lett. B227 (1989) 1