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

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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

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

Quantify X Subtract the Coulomb energy

A=44 A=56 A=68

“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.

Phenomenological treatment: Micro-Macro

Micro-Macro with Nilsson potential

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)

What is the origin of X? 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) There is a well founded mechanism, which has to be there:

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

Why X=1? Strong pairing limit Spontaneous breaking of isorotational symmetry Frauendorf SG, Sheikh JA Cranked shell model and isospin symmetry near N=Z NUCLEAR PHYSICS A 645, 509 (1999) Cranked shell model and isospin symmetry near N=Z

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, (2005) Afanasjev AV, Frauendorf S, NPA 746, 575C (2004 ) Kelsall NS, Svensson CE, Fischer S, et al. EURO. PHYS. J. A 20, 131 (2004) ….

level spacing dominates pair field dominates 13 T

spherical deformed

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

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

Adjusted deformations

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

Isovector and isoscalar pairing

Indication for weak isoscalar pairing correlations?

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 G S /G V ?

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

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

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

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

Equidistant levels Iso-cranking

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

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

T-, N-, Z-, projected HFB

Only nn pairing BCS

Generalization to full isovector pairing OK Not implemented yet ?