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 !