Chiral Vibration and Chiral Rotation with a RPA and a RotorHamiltonian Energy splitting and transiti

1
Chiral Vibration and Chiral Rotation with a RPA-
and a Rotor-Hamiltonian Energy splitting and
transition rates in chiral bands
Chiral symmetry

• D. Almehed
• University of Notre Dame

In collaboration with S. Frauendorf, Notre
Dame, FZ Rossendorf
Frauendorf, Meng, Nucl. Phys. A617, 131 (1997)
2
Chirality of molecules
mirror
The two enantiomers of 2-iodubutene
3
Consequence of static chirality Two identical
rotational bands.
4
Triaxial rotor proton particle neutron hole
Chiral vibration
Chiral vibration
5
Chirality?

• What limits the region of chiral vibration
• and chiral rotation?
• Why are there a splitting between the partner
  bands?
• Why do some bands cross and others do not cross?
• What quantum number allows for degenerate
  states
• with same spin and parity?
• Can we understand the splitting?
• Magnitudes? Trends?
• Can we understand the transition rates?

6
Results of experiments
ph11/2 nh11/2
C. Vaman et al. PRL 92,032501 (2004)
K. Starosta et al. Phys. Scr. T125, 18 (2006)
7
Tilted Axis Cranking RPA

• 3D-TAC with spherical Woods-Saxon
• Dimitrov et al PRL 84 (2000)
• Modified QQ-force N-dependent ? in 2 N-shells
• Baranger, Kumar NPA 110 (1968)
• Parameters fitted to reproduce Strutinsky
  results
• Pair field ? adjusted to 80 of odd-even mass
  difference

RPA - Small amplitude harmonic vibrations around
the mean field minimum
8
Chiral vibration
Chiral vibration
Chiral rotation
9
Composite chiral band in
S. Zhu et al. Phys. Rev. Lett. 91, 132501 (2003)
10
Chiral vibrations TACRPA calculations
3D-TAC (WS) Dimitrov et al PRL 84 (2000)
Modified QQ-force Baranger, Kumar NPA 110
(1968) Parameters fitted to reproduce Strutinsky
results
Pairing and QQ interaction can be adjusted to
describe the lower band
Good agreement with transition rates
11
Small ?
Dripline
12
Almost axial
Small ?
13
Orientation amplitudes
Harmonic approximation but large amplitude.
14
Shape amplitudes
15
4th order Rotor Hamiltonian

• Low energy dynamics is dominated orientation
  degrees of freedom
• The adiabatic approximation good when RPA energy
  is small
• The simplest Hamiltonian that can describe our
  tilted system is a forth order rotor Hamiltonian.
  
• where the forth order terms mimic the tilted mean
  field solution.
• This Hamiltonian has the energy surface
• at a constant I which we used to fit the ci
  parameters to the energy surface calculated with
  the full TAC Hamiltonian.

16
134-Pr Energy
17
Rotor wave function 134-Pr
Approximate restoration of K quantum number at
band crossing (I14)! Small (10-40) keV K mixing
matrix elements Soft potential in ? gives small K
mixing matrix elements Restoration of K as a good
quantum number explains the band crossing.
18
Transition Quadrupole moment
change in ? or ? can give a
factor of 2 in B(E2)
19
Transition rates in 134-Pr
Rotor Hamiltonian
Different average K gives different B(E2) in
the two bands
B(E2) inband different
D. Tonev et al. PRL 96, 052501 (2006) EUROBALL
20
Conclusions

• We understand the limits of the chiral regime in
  A135
• (axial) 73ltNlt79 (shell) (shell) 55ltZlt65
  (dripline)
• Evidence for dynamic chirality Chiral vibrations
  
• TAC RPA gives quantitative understanding band
  splitting
• Band cross with little band mixing due to
  restoration of K
• Chiral vibration dominated by large amplitude
  orientation fluctuations
• Harmonic approximation not enough to describe
  detail of data
• Large amplitude motion in orientation degrees of
  freedom can
• explain difference in transition rate between
  partner bands.

21
The prototype of a chiral rotor
Triaxial rotor high-j particle and hole
Left-right tunneling
Frauendorf, Meng, Nucl. Phys. A617, 131 (1997)
Consequence of static chirality Two identical
rotational bands.
22
Transition from chiral vibration to chiral
rotation
Two close bands, same dynamic MoI, 1-2 units
difference in alignment
Cross over of the two bands
Almost no interaction between bands
8 K. Starosta et al., Physical Review Letters
86, 971 (2001)
23
(No Transcript)
24
Potential energy surface 134-Pr
PES is very soft in gamma and phi.
RPA phonon is a mixture of gamma and
orientation vibrations
Non-harmonic terms probably important
Same shape - different orientations
25
(No Transcript)