Chirality of Nuclear Rotation S. Frauendorf Department of Physics University of Notre Dame, USA IKH, Forschungszentrum Rossendorf Dresden, Germany

In collaboration with J. Meng, PKU V. Dimitrov, ISU F. Doenau, FZR U. Garg, ND K. Starosta, MSU S. Zhu, ANL

“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

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

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.

band 2 band Pr  h 11/2 h 11/2

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).

The QQ-model

Mean field solution

Intrinsic frame Principal axes

Transition probabilities

Spontaneous symmetry breaking Symmetry operation S

Symmetries Broken by m.f. rotational bands

Principal Axis Cranking PAC solutions Tilted Axis Cranking TAC or planar tilted solutions Chiral or aplanar solutions Doubling of states Discrete symmetries

Rotational bands in PACTAC TAC->PAC I=I=

The Cranking Model (rotating mean field) provides a reliable description of nuclear rotational bands. It accounts for the discrete symmetries PAC and TAC if the tilt of the rotational axis is taken into account – Tilted Axis Cranking (TAC). TAC gives chiral solutions, where chiral sister bands are observed and predicts more regions. First chiral solution for Predictions for different mass regions Composite chiral bands

V. Dimitrov, S. Frauendorf, F. Doenau, Physical Review Letters 84, 5732 (2000) First chiral solution for

Chiral sister bands Representative nucleus observed predicted observed /37

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

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

J. Timar et al. Phys. Lett. B. subm.

band 2 band Pr  h 11/2 h 11/2

Left-right tunneling Breaking of chiral symmetry is not very strong.

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

Dynamical (Particle Rotor) calculation Chiral vibration

Frozen alignment approximation: They are numbers One dimensional - very well suited for analysis.

chiral vibration chiral rotation

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

Transition probabilities

out in out in yrast  yrare 

yrast  yrare  out in

Tunneling and vibrational motions are manifest in the electromagnetic transitions. Microscopic description of the left-right dynamics needed. The dynamics are being studied in Particle Rotor model.

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, 134. Chirality shows up as a pair of rotational bands. TAC theory accounts for experiment and predicts more cases. Substantial left-right tunneling and chiral vibrations as precursors. Microscopic description of left-right dynamics is needed.

Reflection asymmetric shapes, two reflection planes Simplex quantum number Parity doubling

Thee three components of the angular momentum form two systems of opposite chirality.

Tilted rotation Triaxial rotor: Classical motion of J

Uniform rotation only about the principal axes! Small E Large E

What is rotating? HCl molecules Nuclei: Nucleons are not on fixed positions. More like a liquid, but what kind of?

viscous: “rotational flow ” ideal : “irrotational flow” None is true: complicated flow containing quantal vortices. Microscopic description needed: Rotating mean field

134 Pr band 2 band 1  h 11/2 h 11/2

Theoretical description Particle – Rotor model: Coupling of the particle and the hole to rotor described quantum mechanically. Frauendorf, Meng, Nuclear Physics A617, 131 (1997) Doenau, Frauendorf, Zhang, PRC, in preparation Dynamics of of angular momentum orientation. Chiral vibrations and rotations. Transition probabilities

Are the assumptions about the rotor realized? Is the nucleus triaxial? Is the moment of inertia of the intermediate axis maximal? 20/37 Where can one expect chirality? Microscopic description needed: Rotating mean field Tilted Axis Cranking Frauendorf, Nucl. Phys. A557, 259c (1993) Frauendorf, Rev. Mod. Phys. 73, 462 (2001) Are there more complex chiral configurations?

The mean field concept A nucleon moves in the mean field generated by all nucleons. The mean field is a functional of the single particle states determined by an averaging procedure. The nucleons move independently.

Total energy is a minimized (stationary) with respect to the single particle states. Calculation of the mean field: Hartree Hartree-Fock density functionals (Skyrme, Gogny, …) Relativistic mean field Micro-Macro (Strutinsky method) ……. Start from the two-body Hamiltonian effective interaction Use the variational principle

nucleus molecule inertial ellipsoid S. Frauendorf Nuclear Physics A557, 259c (1993)

Rotating mean field: Cranking model Seek a mean field solution carrying finite angular momentum. 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. In the laboratory frame it corresponds to a uniformly rotating mean field state

Variational principle : Hartree-Fock effective interaction density functionals (Skyrme, Gogny, …) Relativistic mean field Micro-Macro (Strutinsky method) …….

Deformed mean field solutions Measures orientation. Rotational degree of freedom: Quantization: z