Left-handed Nuclei S. Frauendorf Department of Physics University of Notre Dame, USA IKH, Forschungszentrum Rossendorf Dresden, Germany

“I call any geometrical figure, or group of points, chiral, and say it has chirality, if its image in a plane mirror, ideally realized, cannot brought to coincide with itself.” Kelvin, 1904, Baltimore lectures on Molecular Dynamics and Wave Theory of Light

Chirality of molecules mirror The two enantiomers of 2-iodubutene

R – mint S - caraway Thalidomide

mirror Chirality of mass-less particles z

Triaxial nucleus is achiral.

Rotating nucleus Right-handed Left-handed

New type of chirality Chirality Changed invariant Molecules Massless particles space inversion time reversal Nuclei time reversal space inversion

Generalized definition of Chirality “I call a physical object, chiral, and say it has chirality, if its image, generated by space inversion or time reversal, cannot brought to coincide with itself by a rotation.” 11/37

Consequence of chirality: Two identical rotational bands.

Tilted rotation Classical mechanics: Uniform rotation only about the principal axes. Condition for uniform rotation: Angular momentum and velocity have the same direction.

The nucleus is not a simple piece of matter, but more like a clockwork of gyroscopes. Uniform rotation about an axis that is tilted with respect to the principal axes is quite common.

The prototype of a chiral rotor Frauendorf, Meng, Nucl. Phys. A617, 131 (1997 )

Consequence of chirality: Two identical rotational bands (Energies and EM transition rates).

Rotating mean field: Tilted Axis Cranking model Seek a mean field state |> carrying finite angular momentum, where |> is a Slater determinant (HFB vacuum state) Use the variational principle with the auxiliary condition The state |> is the stationary mean field solution in the frame that rotates uniformly with the angular velocity  about the z axis. S. Frauendorf Nuclear Physics A557, 259c (1993)

Variational principle : Hartree-Fock effective interaction Density functionals (Skyrme, Gogny, …) Relativistic mean field Micro-Macro (Strutinsky method) ……. (Pairing+QQ) X NEW: The principal axes of the density distribution need not coincide with the rotational axis (z).

sister bands Representative nucleus observed predicted observed observed

band 2 band Pr  h 11/2 h 11/2

C. Vaman et al Phys. Rev. Lett. 92, (2004)

PLB, submitted

Composite chiral bands Demonstration of the symmetry concept: It does not matter how the three components of angular momentum are generated. observed observed Is it possible to couple 3 quasiparticles to a chiral configuration?

S. Zhu et al. Phys. Rev. Lett. 91, (2003) Composite chiral band in

The dynamical nature of nuclear chirality Molecules stay in their left- or right handed configuration for ever. Easy to detect chirality Nuclei rapidly oscillate between the left- and right-handed configurations. For the cases we have encountered so far: Problem to detect chirality.

Left-right tunneling Observed bands are super positions of the left-handed and right-handed configurations.

Signatures for chirality: Two very similar D I =1 bands of the same parity. Small energy separation. Very similar electromagnetic transition rates in and between the sister bands. Prototype General: Angular momentum/rotational frequency=constant. Staggering of the M1-transition strength

Particle – Rotor model: Frauendorf, Meng, Nuclear Physics A617, 131 (1997) Doenau, Frauendorf, Zhang, PRC, in preparation

Nuclear chirality - a transient phenomenon Chiral vibration w I

chiral regime

There is substantial tunneling between the left- and right-handed configurations chiral regime Rotational frequency Energy difference Between the chiral sisters chiral regime

[8] K. Starosta et al., Physical Review Letters 86, 971 (2001)

Composite chiral band in J. Timar et al. Phys. Lett. B. 598, 178 (2004)

Chiral regime Angular momentum/rotational frequency =constant. experiment frequency

C. Vaman et al Phys. Rev. Lett. 92, (2004)

S. Zhou et al. PRL 91, (2003)

Transition probabilities

Triaxial Rotor with microscopic moments of inertia Rigid shape IBFFM Soft shape PRL, submitted

Axial Shape Additional left-right coupling

Conclusions Chirality in molecules and massless particles changed by P not by T. Chirality in rotating nuclei changed by T not by P. Triaxial nucleus must carry angular momentum along all three axes. Experimental evidence for chiral sister bands around A=104, 110, 134 from energies. Chirality shows up as a pair of rotational bands of same parity. Substantial left-right tunneling and chiral vibrations as precursors. EM transition rates differ between sisters – coupling to shape vibrations ?