Variational approach to isospin symmetry breaking in medium mass nuclei

A. PETROVICI

Institute for Physics and Nuclear Engineering, Bucharest, Romania Institut für Theoretische Physik, Universität Tübingen, Germany

Characteristics of N~Z nuclei in the region :

• shape coexistence and mixing• competition between T=0 and T=1 pairing correlations, isospin mixing• rapid structural changes with particle number, angular momentum and excitation energy

Isospin symmetry breaking effects in the A~70 mass region on

- Superallowed Fermi beta decay

- Mirror Energy Differences

Self-consistent description of proton-rich A~70 nuclei based on:

- beyond mean field variational VAMPIR approaches - realistic effective interactions in large model spaces

Complex EXCITED VAMPIR approach

• the the model spacemodel space is defined by a finite dimensional set of is defined by a finite dimensional set of spherical single particlespherical single particle states states

•• the effective many-bodythe effective many-body Hamiltonian Hamiltonian is represented as a sum of is represented as a sum of one- and two-bodyone- and two-body

termsterms•• the basic the basic building blocksbuilding blocks are Hartree-Fock-Bogoliubov ( are Hartree-Fock-Bogoliubov (HFBHFB) vacua) vacua

•• the HFB transformations are the HFB transformations are essentially essentially complexcomplex and allow for and allow for proton-neutron, parity proton-neutron, parity

and angular momentum mixingand angular momentum mixing being restricted by time-reversal and axial symmetry being restricted by time-reversal and axial symmetry

• the broken symmetries (the broken symmetries (s=N, Z, I, ps=N, Z, I, p) are restored by ) are restored by projection before variationprojection before variation

Beyond mean field variational procedure

complex VAMPIR

complex EXCITED VAMPIR

A ~ 70 mass region

<1p 1d5/2; T=0|G| 1p 1d5/2;T=0>

renormalized G-matrix (OBEP, Bonn A/Bonn CD)

• short range Gaussians in pp, np, nn channels

• monopole shifts:

<0g9/2 0f ; T=0|G| 0g9/2 0f ;T=0>

model space for both protons and neutrons:

1p1/2 1p3/2 0f5/2 0f7/2 1d5/2 0g9/2

40Ca - core

(charge-symmetric basis + Coulomb contributions to the π-spe from the core)

Superallowed Fermi β decay between 0+ T=1 analog states

δc – isospin-symmetry-breaking correction

test of the CVC hypothesis

test of the unitarity of CKM matrix

Isospin-symmetry-breaking contributions:

electromagnetic interaction • Coulomb contribution to the single particle energies resulting from the Ca core • Coulomb two-body matrix elements charge-dependent strong interaction • Bonn CD potential

Charge-symmetric effective Hamiltonian: • same single particle energies for and •• Bonn A potential

Isospin-symmetry-breaking effective Hamiltonians: Bonn A + Coulomb

Bonn CD + Coulomb

Isospin operator

absingle particle state of the model space )

MF = <f |+ |i>

Strategy:

Petrovici et al. Nucl. Phys. A747 (2005) 44

br ms

GANIL, J. Garces Narro et al, PRC63 (2001) 044307

A. Petrovici et al, Phys. Rev. C78 (2008) 064311

A. M. Hurst et al, Phys. Rev. Lett.98 (2007) 072501 (70Se: No evidence for oblate shapes)

J. Ljungvall et al, Phys. Rev. Lett. 100 (2008) 102502 (70Se: Evidence for oblate shapes)

G. de Angelis et al, Eur. Phys. J. A12 (2001) 51 (70Br)

Complex Excited Vampir predictions: oblate-prolate mixing specific for each nucleus (varying with increasing spin).

Mirror Energy Differences A= 70, 82, 86

Strong oblate-prolate mixing up to spin 6+ : oblate components dominate the yrast states of 70Se, but the yrare states of 70Br

Precise quadrupole moments for low spin states could clarify the open problem.

High spin states obtained within the Excited Vampir approach (Bonn A + Coulomb)

Analog states obtained within the complex Excited Vampir approach

A. Petrovici et al, Phys. Rev. C78 (2008) 064311

A. Petrovici, J. Phys.G: Nucl. Part. Phys 37 (2010) 064036

Summary and outlook

• strategy for calculating the isospin-symmetry-breaking effects on the superallowed Fermi β decay based on a self-consistent description of the analog and non-analog branches (experimental branching ratios are needed)

• in progress: systematic derivation of improved effective interaction in larger model spaces

• trends in mirror energy differences correlated with shape mixing in the structure of the analog states ( precise quadrupole moment measurements are required)

complex Excited Vampir variational approach to coexistence phenomena

In collaboration with:

K. W. Schmid, Amand Faessler

Tuebingen University, Germany

O. Andrei

National Institute for Physics and Nuclear Engineering, Bucharest, Romania