m. rafalski, w. satuła, j. dobaczewski
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
M. Rafalski, W. Satuła, J. DobaczewskiInstitute of Theoretical Physics, University of Warsaw, Poland
Isospin mixing of isospin projected Slater determinants: formalism and preliminary applications
XV Nuclear Physics Workshop, Kazimierz 24-28.09.2008
Outline
1. Isospin symmetry breaking2. Procedure of isospin projection3. Preliminary results4. Summary
Isospin projection procedure presented here, has been implemented into HFODD code. All presented results are obtained using this code.
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)
Isospin symmetry breaking
Cognition of the isospin symmetry breaking mechanism is cruitial e. g. for understanding of super-allowed β decay.
Broken symmetry Hartree-Fock state:
Procedure of the isospin projection
Energy of the projected state:
Projection operator in the spectral representation:
Projection operator defined by isospin rotation operator:
Procedure of the isospin projection
Two components of the Hamiltonian:
Isospin invariant Isospin breaking
Skyrme energy of the projected state:
HF state rotated in the isospace:
Procedure of the isospin projection
Isospin invariant Isospin breaking
Rediagonalization !!!
iso-scalar iso-vector iso-tensor
Coulomb interaction consists of three components:
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
Isospin mixing along the N=Z line
Isospin mixing rises with A,from ~0% for light nuclei up to ~5% for A=100.
A
1
2
3
4
5
30 50 70 90
SIII 12 shellsN=Z nuclei
afterrediagonalization
beforerediagonalizationIs
osp
in m
ixin
g [%
]
0
0.4
0.8
1.2
Isos
pin
mix
ing
[%]
30 50 70 90A
SLy4 relative to SIII
SkP relative to SIII afterrediagonalization
beforerediagonalization
Isospin mixing along the N=Z line
Results strongly depend on the Skyrme force parametrization.
before rediagonalization, Z=const.
Isos
pin
mix
ing
[%]
-2 0 2 4 6 8 Tz
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)
Isospin mixing as a function of Tz
Unclear situation, rediagonalization is necessary,
Before rediagonalization for light nuclei isospin mixing is lower for Tz=0 than for neighbor nuclei (Tz≠0). For heavy nuclei situation is opposite.
-2 0 2 4 6 8
Tz
Isos
pin
mix
ing
[%]
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)
Isos
pin
mix
ing
[%]
6
5
4
3
2
1
0 -2 0 2 4 6 8 Tz
after rediagonalization, Z=const.
Isospin mixing as a function of Tz
Isospin mixing is lower in light nuclei than in heavy ones.
We observe quenching of the isospin mixing when |Tz| increases,
A=104A= 88A= 72A= 56A= 32
after rediagonalization, A=constIs
ospi
n m
ixin
g [%
]
6
5
4
3
2
1
0 -2 0 2 4 6 8 Tz
Isospin mixing as a function of Tz
We can see the same dependance, as for Z=const.
A=104A= 88A= 72A= 56A= 32
-2 0 2 4 6 8 Tz
Isos
pin
mix
ing
/ Iso
spin
mix
ing(
Tz =
0) 1.0
0.8
0.6
0.4
0.2
0
normalized isospin mixing, A=const
Isospin mixing as a function of Tz
Quenching of the isospin mixing as a function of Tz is similar for different mass numbers.
A
0
0.5
1.0
1.5
30 50 70 90
E-E
HF [
MeV
]
SIII 12 shellsN=Z nuclei
beforerediagonalization
afterrediagonalization
Impact of the isospin projection on the energy
HF energy is almost good:it is only ~30 keV above energy after rediagonalization.
Results as a function of schells numberResults as a function of schells number
Competition between accuracy and saving CPU time:N0=12 seems to be a good choice.
For N0 >12 changes in energy are relatively small.
Results as a function of schells number
Results as a function of schells number
Appropriate number of schells: N0= 9.
SII Skyrme force
Results as a function of schells number
Appropriate number of schells: N0= 10.
SII Skyrme force
Results as a function of schells number
Appropriate number of schells: N0= 11.
SII Skyrme force
Results as a function of schells number
SII Skyrme force
Appropriate number of schells: N0= 12.
• 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 to 100Sn, number of shells N0= 12 is sufficient.
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 to 100Sn, number of shells N0= 12 is sufficient.
Summary
Thanks for your attention !