1. a non-adiabatic microscopic description
Tidal Waves –
a non-adiabatic microscopic description
of the yrast states in near-spherical nuclei
Stefan Frauendorf
Yongquin Gu
Jie Sun

2. The soft Quadrupole mode
Deformed nuclei:
Rotors, β, γ-vibrators
Spherical nuclei:
Vibrators
Transitional nuclei
Phenomenological:
Bohr Hamiltonian, Interacting Boson Model
Micoscopic:
Spherical mean field Bohr Hamiltonian deformed rotating mean field
+RPA GCM (Cranking) + RPA
Non-adiabatic Adiabatic Non-adiabatic

3. Only very few vibrational levels are well separated
qp. excitations
n=0
n=1
n=2
Only very few vibrational levels are well separated
from the two quasiparticle excitations.

4. E A second look on quadrupole vibrations I Multiphonon states
qp. excitations
Tidal waves
A second look
on quadrupole
vibrations
Multiphonon states

5. 5

6. Generation of angular momentum
Angular velocity Deformation
Rotor increases stays constant
Tidal wave
Vibrator stays constant increases
(Other degrees
of freedom) 3

7. 6

9. You can describe the Tidal Wave mode
by means of a rotating mean field
The mean field description works for
vibrational, transitional, rotational nuclei

10. Microscopic treatment of the yrast states
Cranking model: Micro-macro method
(Nilsson+ fixed pairing).
Find the equilibrium shape for the rotating mean field. Gives the energy and deformation.
Calculate the E2 transition probability for the deformed charge distribution.
Calculate magnetic moments (g-factors) ….

11. Minimizing at fixed frequency problematic:

12. E
I=8
I=6
I=4
I=2
I=0

14. Diabatic tracing of the configuration

15. A “good” vibrator Strong coupling between qp and quadrupole degrees
F. Corminboeuf et al.
PRC 63,
Strong coupling
between qp and
quadrupole degrees
of freedom.
theory
experiment
“intruder”
arXiv:

16. Remarkable reproduction of data by calculations
Development of tidal waves from vibrational Z=48
toward rotational with decreasing Z at low I.
Energies of “vibrational nuclei” strongly anharmonic,
valence neutrons react non-adiabatically
Z=48, N=60-66: after neutron alignment,
smaller deformation  approach of antimagnetic rotation
Z=46, N=56,60 and Z=44, N=62,64 angular velocity
nearly constant during neutron alignment –
tidal wave with quasiparticle degrees of freedom
More B(E2) values to check theory
Low-lying 0+ (“intruders”) naturally incorporated 9

17. g-factors of the 2+ states
Sensitive to the proton-neutron composition of the state.
Data (new and from literature):
S.K. Chamoli,1 A.E. Stuchbery,1 S. Frauendorf,2 Jie Sun,2 Y. Gu,2 P.T.
Moore,1 A. Wakhle,1 M.C. East,1 T. Kib¶edi,1 A.N. Wilson,1 and Any Others?3
1Department of Nuclear Physics, Research School of Physics and Engineering,
Australian National University, Canberra, ACT 0200, Australia
2Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA
To be published

18. g-factors around A=100
Z/A
theory

19. N-dependence of g-factors
The N-dependence of g-factors not accounted for by the phenomenological collective models.
In the A=100 region it reflects the increase of the neutron fraction.
The increase results from the neutron Fermi level entering the h11/2 shell.
full 92 98 94 96
100
104
102
106
108 19

20. -g is very sensitive to pairing
-for I=6, 8, g-depends
sensitively on the
competing configurations

21. Perspectives Odd mass transitional nuclei
More cases / other regions /predicted the position of 2+, 4+,… far from stability
Better mean field (MM for WS or FY, RMF, S
Problems:
1) missing zero point motion of deformation ) transition operators semiclassical
3) crossings between quasiparticle orbitals
Possible remedy for 1 and 2: generalized density matrix approach,
3 is the toughest
angular momentum projection + diagonalization ?

22. Mass on string/spring
In rotating frame, the spring force balances the centrifugal
force for any l, which thus cannot be found by minimizing
the energy in the rotating frame.

23. A “good” vibrator Strong coupling between qp and quadrupole degrees
F. Corminboeuf et al.
PRC 63,
Strong coupling
between qp and
quadrupole degrees
of freedom.
experiment
theory I 10
0.4
0.2 5
arXiv:

25. g-factors around A=100
Z/A
theory