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I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA

Calculation of the Wigner Term in the Binding Energies by Diagonalization of the Isovector Pairing Hamiltonian. I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA. Important for p/rp process near the N=Z line. Quantify X. Subtract the Coulomb energy. A=44.

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I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA

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  1. Calculation of the Wigner Term in the Binding Energies by Diagonalization of the Isovector Pairing Hamiltonian I. Bentley and S. Frauendorf Department of Physics University of Notre Dame, USA

  2. Important for p/rp process near the N=Z line

  3. Quantify X Subtract the Coulomb energy

  4. A=44 A=68 A=56

  5. “Experimental” Wigner X Substantial scatter caused by shell effects Mean value ~1 for A<70 Mean value ~4 for 80<A<90 Contains shell effects! Separation is problematic.

  6. Phenomenological treatment: Micro-Macro

  7. Micro-Macro with Nilsson potential

  8. Density functionals Skyrme–Hartree–Fock–Bogoliubov mass formula by N. Chamel, S. Goriely, J.M. Pearson, Nuclear Physics A 812 (2008) 72–98: Skyrme HFB give parameter dependent values of X, substantially smaller than 1, sensitive to effective mass (Satula, Wyss, Rep. Prog. Phys. 68, 131 (05) Unsatisfactory! Relativistic Mean Field gives X approximately 1 (Ban et al., Phys. Lett. B 633, 231 (06)

  9. What is the origin of X? There is a well founded mechanism, which has to be there: Isovector Proton-Neutron Pairing. Strength is fixed by isospin invariance of strong interaction. It gives X approximately 1 by symmetry. (Frauendorf, Sheikh, Nucl. Phys. A 645, 509, (99) 1) Fixing the isovector pairing strength to the standard value for pp, nn pairing, obtained from even-odd mass differences, we quantitatively reproduce the experimental X. 2) Possibilities for implementation into density functional approaches (ongoing)

  10. Isovector Pairing Hamiltonian Solve the pairing problem by diagonalization: -Isospin is good -No problems with instabilities of the pair field Generate all configurations by lifting pp, nn, pn pairs and diagonalize. 6 or 7 levels around the Fermi level -> dimension ~ 10000 Few cases with 8 levels -> no significant change if G is scaled.

  11. Why X=1? Strong pairing limit Spontaneous breaking of isorotational symmetry Frauendorf SG, Sheikh JACranked shell model and isospin symmetry near N=ZNUCLEAR PHYSICS A 645, 509 (1999)

  12. Isorotations (strong symmetry breaking) Bayman, Bes, Broglia, PRL 23 (1969) 1299 ( 2 particle transfer) Frauendorf, Sheikh, NPA 645, 509 (1999) Frauendorf, Sheikh, Physica Scripta T88, 162 (2000) Afanasjev AV, Frauendorf S, PRC 71, 064318   (2005) Afanasjev AV, Frauendorf S, NPA 746, 575C (2004 ) Kelsall NS, Svensson CE, Fischer S, et al. EURO. PHYS. J. A 20, 131 (2004) ….

  13. level spacing dominates pair field dominates T 1 3

  14. spherical deformed spherical

  15. Wigner X with AutoTAC Deformations • Not perfect, but promising. • Two problems : 44≤A≤58 too strong scatter 74≤A≤88 Xc~1 Xe~4 • Why? • Calculated deformations • not good enough

  16. Optimize the deformation • Nilsson calculated • Woods Saxon calculated • Folded Yukawa calculated • Experimental (BE2(2->0) • Experimental yrast energies “adopted deformations” Rotational response small medium large

  17. Adjusted deformations

  18. Isovector proton neutron pairing with the strength fixed by isospin conservation gives the correct X • Mean field treatment (HFB) is insufficient – violates isospin conservation • In devising approximations beyond mean field it is decisive to incorporate restoration of isospin

  19. Isovector and isoscalar pairing

  20. Indication for weak isoscalar pairing correlations?

  21. Isoscalar pairing attenuates the staggering between the even-even and odd-odd N=Z nuclei: some indication from experiment • Small isoscalar pair correlation would only slightly increase the X values: within the tolerance range of the isovector scenario • What is GS/GV ?

  22. Implementation into mean field approaches • 8 levels around the Fermi level is not enough-> dimensions explode->approximations. • Iso-cranking approximation • HFB + RPA • HFB + SCRPA • T-,N-,Z- projected HFB • BCS-truncation

  23. Iso-cranking Frauendorf, Sheikh, NPA 645, 509 (1999) For spatial rotations of well deformed nuclei do HFB with: In analogy do HBF with:

  24. Problem: It works only for a sufficiently strong pair field. HFB+Lipkin-Nogami may mend the problem.

  25. HFB+QRPA K. Neergard PLB 537, 287 (2002); PLB 572, 159 (2003); PRC 80, 044313 (2009)

  26. Equidistant levels Iso-cranking

  27. HFB+QRPA unreliable near the critical G. We are close by.

  28. nn pairing SCQRPA is not worked out for full isovector pairing. Hung & Dang RIKEN working on it.

  29. T-, N-, Z-, projected HFB

  30. BCS Only nn pairing BCS

  31. Generalization to full isovector pairing OK • Not implemented yet • ?

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