1. Nuclear structure of lowest 229Th states
and time-dependent fundamental constants
1Elena Litvinova, 2Hans Feldmeier, 3,4Jacek Dobaczewski, 5Victor Flambaum
1National Superconducting Cyclotron Laboratory & MSU, East Lansing, USA
2GSI Helmholtzzenrum für Schwerionenforschung, Darmstadt, Germany
3Institute of Theoretical Physics, University of Warsaw, Poland
4Department of Physics, University of Jyväskylä, Finland
5School of Physics, University of New South Wales, Sydney, Australia
Nuclear isomer clock 2012, EMMI, Sep

2. Outline
General properties and microscopic structure of low-lying states in 229Th.
Low-lying isomeric state in 229Th: a nuclear physics case for testing variations of fundamental constants.
Sensitivity of the 3/2+ -> 5/2+ transition frequency to the potential changes of the fine structure constant α.
Skyrme-Hartree-Fock-Bogoliubov calculations for 229Th lowest states and δα/α amplification factor.
Conclusions & outlook.

3. General properties of 229Th
229Th = 228Th + n
228Th is transitional from octupole to quadrupole deformation:
226Th is octupole deformed, 230Th is vibrational-like,
228Th is „octupole soft“ (strong octupole correlations): Bohr & Mottelson, Leander, Otsuka, Nazarewicz, Jolos, Pashkevich, …
Octupole vibrational 0- state at 338 keV,
β3 ≈ 0 for the ground state (SC mean-field, Macro-Micro models),
β3 ≈ for 7/2[743] and 1/2[770], no deep minima,
E. Ruchowska et al., PRC 73, (2006), Strutinsky method,
β3 ≈ 0.35 at neutron threshold,
Effects of parity nonconservation in scattering of polarized neutrons explained by octupole correlations, V.V. Flambaum, V.G. Zelevinsky, PLB 350 , 8 (1995),
Clear evidence for parity-partner bands, K. Gulda et al., NPA 703, 45 (2002)

4. 229Th structure from QPM calculations
Parity partner bands:
5/2+
5/2-
3/2+
3/2-
K. Gulda et al., Nuclear Physics A 703, 45 (2002)

5. Lowest states in 229Th Idea:
Enchanced sensitivity of 3/2+ -> 5/2+ transition frequency to possible variations of fundamental constants
V.V. Flambaum, PRL 97, Estimation:
Nuclear structure calculations: controversial results for A,
Nilsson model, A.C. Hayes, J.L. Friar, PLB 650, 229 (2007), (no self-consistency
=> no rearrangement): no amplification (A = 0)
Self-consistent spherical mean field (relativistic): X.-T. He, Z.-Z. Ren,
NPA 806, 117 (2008): large amplification A ~ 105.

6. Hellmann-Feynman Theorem
I. Parametrized Hamiltonian
Stationary Schrödinger equation:
II. Energy density functional ε(c,x): J. Goodisman, Phys. Rev. A2, 1 (1970)
Stationary condition:

7. A Amplification factor Defined as: Variation of an Energy
at stationary points: A
Amplification factor m g
Defined as:

8. α – dependence of nucleon masses:
Amplification factor
α – dependence of nucleon masses:
Estimated using: U.-G. Meißner et al., EPJA 31, 357 : Δmnp(EM) = MeV

9. Microscopic calculations with Skyrme density functional: SIII
J. Dobaczewski, J. Dudek, CPC 102, 166 (1997),
J. Dobaczewski, P. Olbratowski, CPC 158, 158 (2004),
J. Dobaczewski, J. Dudek, P. Olbratowski, arXiv:nucl-th/
Deformed mean field + blocking for odd neutron
Rearrangement effects & deformation are taken
into account self-consistently
no pairing
pairing

10. Microscopic calculations with Skyrme density functional: SkM*
J. Dobaczewski, J. Dudek, CPC 102, 166 (1997),
J. Dobaczewski, P. Olbratowski, CPC 158, 158 (2004),
J. Dobaczewski, J. Dudek, P. Olbratowski, arXiv:nucl-th/
Deformed mean field + blocking for odd neutron
Rearrangement effects & deformation are taken
into account self-consistently
no pairing
pairing

11. Microscopic structure: expansion over Nilsson orbits
Overlap integrals: <Φν (Ωπ) | Ωπ [N,Nz,Λ]>
=> Configuration assignment
SIII + HFB: „traditional“ assignment
SkM* + HFB: competing contributions &
changing of nodal structure of the leading components

12. Energies of 229Th structure from Skyrme-HF & HFB calculations
Deformed Skyrme:
SkM* & SIII
Spherical
RMF: NL3
Data
[30] G. Audi et al., NPA 729, 337
[9] B.R. Beck et al., PRL 98,
Larger numbers,
but less realistic
1-2 orders of magnitude effect from
change of nodal structure of the leading
components or/and stronger mixing

13. Conclusions & outlook Magnetic moments-? Data-?
1. 5/2+ and 3/2+ states in 229Th have been studied within self-consistent
Skyrme + HFB model – version HFODD of J. Dobaczewski, J. Dudek et al.
2. Upper limit for the amplification factor of fine structure constant
is A ≈ (6±3)x104 (SkM*), lower limit – zero (SIII).
3. Accuracy is very important because the calculated amplification
is a difference of two big numbers.
4. The source of the amplification is the difference of the nodal structure of
the ground state and isomer wave functions and/or comparable contributions of
several Nilsson orbits to the total wave functions.
5. Similar effect is obtained for differences of the quadrupole moments:
SkM*-HF
SkM*-HFB
SIII-HF
SIII-HFB
Magnetic moments-? Data-?

14. Thank you! Hans Feldmeier Jacek Dobaczewski Victor Flambaum
GSI Helmholtzzenrum für Schwerionenforschung, Darmstadt, Germany
Jacek Dobaczewski
Institute of Theoretical Physics, University of Warsaw, Poland
Department of Physics, University of Jyväskylä, Finland
Victor Flambaum
School of Physics, University of New South Wales, Sydney, Australia
Journal reference: E. Litvinova, H. Feldmeier, J. Dobaczewski, V. Flambaum,
Phys. Rev. C 79, (2009). 14