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M. Rafalski, W. Satuła, J. Dobaczewski

Isospin mixing of isospin projected Slater determinants: formalism and preliminary applications. M. Rafalski, W. Satuła, J. Dobaczewski Institute of Theoretical Physics, University of Warsaw , Poland. XV Nuclear Physics Workshop, Kazimierz 24-28.09.2008. Outline. Isospin symmetry breaking

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M. Rafalski, W. Satuła, J. Dobaczewski

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  1. Isospin mixing of isospin projected Slater determinants: formalism and preliminary applications M. Rafalski, W. Satuła, J. Dobaczewski Institute of Theoretical Physics, University of Warsaw, Poland XV Nuclear Physics Workshop, Kazimierz 24-28.09.2008

  2. Outline Isospin symmetry breaking Procedure of isospin projection Preliminary results Summary Isospin projection procedure presented here, has been implemented into HFODD code. All presented results are obtained using this code.

  3. Isospinsymmetrybreaking Cognition of the isospin symmetry breaking mechanism is cruitial e. g. for understanding of super-allowed β decay. • There are two sources of the isospin symmetry breaking: • unphysical, related to the HF procedure • physical, caused mostly by Coulomb interaction • (also by the strong force isospin non-invariance) Broken symmetry Hartree-Fock state:

  4. Procedure of theisospinprojection Projection operator in the spectral representation: Projection operator defined by isospin rotation operator: Energy of the projected state:

  5. Procedure of theisospinprojection Two components of the Hamiltonian: Isospin invariant Isospin breaking HF state rotated in the isospace: Skyrme energy of the projected state:

  6. Procedure of theisospinprojection Isospin invariant Isospin breaking Coulomb interaction consists of three components: iso-tensor iso-scalar iso-vector Two last terms mix states with different isospin – produce nondiagonal elements of the Hamiltonian. Because of this, to obtain proper eigenstates, we have to perform Rediagonalization!!!

  7. 5 4 3 2 1 30 50 70 90 IsospinmixingalongtheN=Zline SIII 12 shells N=Z nuclei after rediagonalization Isospin mixing [%] before rediagonalization A Isospin mixing rises with A, from ~0% for light nuclei up to ~5% for A=100.

  8. SLy4 relative to SIII 1.2 SkP relative to SIII 0.8 0.4 0 30 50 70 90 IsospinmixingalongtheN=Zline after rediagonalization Isospin mixing [%] before rediagonalization A Results strongly depend on the Skyrme force parametrization.

  9. Isospinmixing as a function of Tz before rediagonalization, Z=const. Te (Z=52) Ru (Z=44) Kr (Z=36) Ni (Z=28) Ca (Z=20) O (Z= 8) 6 5 4 3 2 1 0 Isospin mixing [%] -2 0 2 4 6 8 Tz Before rediagonalization for light nuclei isospin mixing is lower for Tz=0 than for neighbor nuclei (Tz≠0). For heavy nuclei situation is opposite. Unclear situation, rediagonalization is necessary,

  10. Isospinmixing as a function of Tz after rediagonalization, Z=const. 6 5 4 3 2 1 0 Te (Z=52) Ru (Z=44) Kr (Z=36) Ni (Z=28) Ca (Z=20) O (Z= 8) 6 5 4 3 2 1 0 Isospin mixing [%] Isospin mixing [%] -2 0 2 4 6 8 Tz -2 0 2 4 6 8 Tz Isospin mixing is lower in light nuclei than in heavy ones. We observe quenching of the isospin mixing when |Tz| increases,

  11. Isospinmixing as a function of Tz after rediagonalization, A=const A=104 A= 88 A= 72 A= 56 A= 32 6 5 4 3 2 1 0 Isospin mixing [%] -20 24 68 Tz We can see the same dependance, as for Z=const.

  12. Isospinmixing as a function of Tz normalized isospin mixing, A=const A=104 A= 88 A= 72 A= 56 A= 32 1.0 0.8 0.6 0.4 0.2 0 Isospin mixing / Isospin mixing(Tz =0) -2 0 2 4 6 8 Tz Quenching of the isospin mixing as a function of Tz is similar for different mass numbers.

  13. 1.5 1.0 0.5 0 30 50 70 90 Impact of theisospinprojection on the energy SIII 12 shells N=Z nuclei before rediagonalization E-EHF [MeV] after rediagonalization A HF energy is almost good: it is only ~30 keV above energy after rediagonalization.

  14. Results as a function of schellsnumber Results as a function of schellsnumber Competition between accuracy and saving CPU time: N0=12 seems to be a good choice.

  15. Results as a function of schellsnumber For N0>12 changes in energy are relatively small.

  16. Results as a function of schellsnumber SII Skyrme force Appropriate number of schells: N0= 9.

  17. Results as a function of schellsnumber SII Skyrme force Appropriate number of schells: N0= 10.

  18. Results as a function of schellsnumber SII Skyrme force Appropriate number of schells: N0= 11.

  19. Results as a function of schellsnumber SII Skyrme force Appropriate number of schells: N0= 12.

  20. Summary • Theoretical tool to performing isospin projection has been developed, • Isospin mixing is lower in light nuclei than in heavy ones, • We observe quenching of the isospin mixing when |Tz| increases, • HF energy (before projection) is almost good: it is only ~30 keV above energy after rediagonalization, • Results strongly depend on the Skyrme force parametrization, • Up to100Sn, number of shells N0= 12 is sufficient.

  21. Summary • Theoretical tool to performing isospin projection has been developed, • Isospin mixing is lower in light nuclei than in heavy ones, • We observe quenching of the isospin mixing when |Tz| increases, • HF energy (before projection) is almost good: it is only ~30 keV above energy after rediagonalization, • Results strongly depend on the Skyrme force parametrization, • Up to100Sn, number of shells N0=12 is sufficient. Thanks for your attention !

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