Chirality of Nuclear Rotation

S. Frauendorf

Department of Physics

University of Notre Dame, USA

IKH, Forschungszentrum Rossendorf

Dresden, Germany

In collaboration withJ. Meng, PKUV. Dimitrov, ISUF. Doenau, FZRU. Garg, NDK. Starosta, MSUS. 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

)( zz PR

R – mintS - caraway

mirror

Chirality of mass-less particles

)( zz PR

z

Triaxial nucleus is achiral.

)(yTR

1PJ

Rotating nucleus

Right-handed Left-handed

)(yTR

New type of chirality

Chirality Changed invariant

MoleculesMassless 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 inversionor 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.

iiiJ )(

on.distributidensity theof axes the withcoincide which

,nsorinertia te theof axes principal for theonly toparallel iJ

The nucleus is not a simple piece of matter,but more like a clockwork of gyroscopes.

Uniform rotation about anaxis that is tilted with respectto the principal axes is quite common.

The prototype of a chiral rotor

Frauendorf, Meng, Frauendorf, Meng, Nucl. Phys. A617, 131 (1997Nucl. Phys. A617, 131 (1997) )

Consequence of chirality: Two identical rotational bands.

band 2 band 1134Pr

h11/2 h11/2

Rotating mean field: Tilted Axis Cranking model

Seek a mean field state |> carrying finite angular momentum,where |> is a Slater determinant (HFB vacuum state)

.0|| zJ

Use the variational principle

with the auxiliary condition

0|| HEi

0||' zJHEi

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)

functions) (wave states particle single

)(routhians frame rotating in energies particle single '

ial)(potentent field mean energy kinetic

(routhian) frame rotating thein nhamiltonia field mean '

|'' -'

i

i

mf

iiizmf

e

Vt

h

ehJVth

tency selfconsis mfi V

Variational principle : Hartree-Fock effective interactionDensity 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

','2

2 '||5

4

basis

potential model shell spherical

kkkk

kkkksph

kkksph

sphsph

cckYrkQcceh

eh

Vth

operator quadrupole ),(5

4

2

202

2

2

YrQ

QQhH sph

Mean field solution

Qq

QqJhheh

QQJhE

zsphiii

zsph

tencyselfconsis

'''

variation

2'

2

2

2

2

Intrinsic frame

Principal axes

2/sincos

00

20

2211

KqKq

qqqq

,ˆ toparallel bemust

tencyselfconsis

cossinsincossin

)('

2200

321

22200332211

JJ

QqQq

QQqQqJJJhh sph

22

222

2220

220

222

|)],,0(),,0([),,0(|4

5

||4

5)2,2(

protonproton

LAB

QDDQD

QIIEB

211

211 |),,0(|

4

3||

4

3)1,1(

v

vLAB DIIMB

Transition probabilities

Spontaneous symmetry breaking

Symmetry operation S

.|'|'|'

energy same the withsolutions field mean are states All

1||| and ,'but ''

HHE

hhHH

|SS

|S

|SSSSS

Symmetries

zJvtH 12'

Broken by m.f. rotationalbands

Principal Axis CrankingPAC solutions

nIe iz 2||)( R

Tilted Axis CrankingTAC or planar tilted solutions

Chiral or aplanar solutionsDoubling of states

Discrete symmetries

Rotationalbands in

Er163

PAC TAC

TAC->PACI=

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 observedand predicts more regions.

First chiral solution for 7513459 Pr

Predictions for different mass regions

Composite chiral bands

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

First chiral solution for 7513459 Pr

Chiral sister bands

Representativenucleus I

observed13 0.21 145910445 Rh 2/11

12/9 hg

13 0.21 4011118877 Ir

2/912/9 gg

447935 Br

12/132/13

ii

13 0.21 14

predicted

predicted

9316269 Tm 1

2/112/13ii predicted45 0.32 26

12/112/11

hh observed13 0.18 267513459 Pr

31/37

5910445 Rh 2/11

12/9 hg

C. Vaman et alPhys. Rev. Lett.92, 032501 (2004)

S. Zhu et al.Phys. Rev. Lett. 91, 132501 (2003)

Composite chiral band in 7513560 Nd

Composite chiral band in 6010545 Rh J. Timar et al.

Phys. Lett. B. subm.

22/11

12/9 hg

0 5 10 15 20 25 30 35 40 45 50 55 60-26-24-22-20-18-16-14-12-10-8-6-4-20

axial

chiral

162

69Tm

93

o

modified oscillator

E[M

eV]

J

0 5 10 15 20 25 30 35 40 45 50 55 60-26-24-22-20-18-16-14-12-10-8-6-4-20

axial

chiral

162

69Tm

93

o

modified oscillator

E[M

eV]

J

0 5 10 15 20 25 30 35 40 45 50 55 60-26-24-22-20-18-16-14-12-10-8-6-4-20

axial

chiral

162

69Tm

93

o

modified oscillator

E[M

eV]

J

band 2 band 1134Pr

h11/2 h11/2

Left-right tunneling

Breaking of chiral symmetry is not very strong.

Particle – Rotor model:

Frauendorf, Meng, Nuclear Physics A617, 131 (1997)Frauendorf, Meng, Nuclear Physics A617, 131 (1997)

Doenau, Frauendorf, Zhang, PRC , in preparation

312 ,

Dynamical (Particle Rotor) calculation

Chiral vibration

Frozen alignment approximation:

They are numbers

One dimensional -very well suited for analysis.

312 44 JJJ

chiralvibration

chiralrotation

jJ crit 3

24

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

Transitionprobabilities

out

in

outout

out

in

in

in

yrast yrare

yrast yrare

outout

inin

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.

ConclusionsChirality 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. 1I

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

I

i

z

parity

e

)(

||

)(

S

PRS

Parity doubling

Th226

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

chirality.chirality.

Tilted rotation

conserved : a.m. of valueabsolute

conserved 2

1

2

1 :energy

)( :inertia of moments

:momentumangular

axes principal 3,2,1 :locityangular ve

22

22

2,3

2,21

i

i

iii

nnnn

iii

i

JJ

JE

xxm

J

i

Triaxial rotor: Classical motion of J

Uniform rotation only aboutthe principal axes!

Small E

Large E

sphere momentumangular 22iJJ

2

1 ellipsoidenergy

2

i

iJE

What is rotating?

HCl

molecules

)( :inertia of moments 2,3

2,21

nnnn xxm

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

134Prband 2 band 1h11/2 h11/2

lsi ,

Theoretical description

Particle – Rotor model:Coupling of the particle and the hole to rotor describedquantum mechanically.

Frauendorf, Meng, Nuclear Physics A617, 131 (1997)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 fieldTilted Axis CrankingFrauendorf, 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.

][ imfV The mean field is a functional of the single particle states determined by an averaging procedure.

The nucleons move independently.

ii

N

c

cc

state in nucleona creates

0|......|tion)(configura statenuclear 1

functions) (wave states particle single

energies particle single

ial)(potentent field mean energy kinetic

i

i

mf

iiimf

e

Vt

ehVth

Total energy is a minimized (stationary) with respect to the single particle states.

with the 12vtH

Calculation of the mean field: Hartree Hartree-Fock density functionals (Skyrme, Gogny, …)

Relativistic mean field Micro-Macro (Strutinsky method) …….

.0|| HEi

.12v

Start from the two-body Hamiltonian

effective interaction

Use the variational principle

nucleus

linear-nonhighly ,,|ˆ|,,),,( 321321321 ii JJ

on.distributidensity theof

axes principal therespect to with tiltedarethat

axesfor possible also toparallel

i

J

molecule inertial ellipsoidiiiJ )(

. axes principal for theonly toparallel iJ

S. Frauendorf Nuclear Physics A557, 259c (1993)

)( Reflection

)( Rotation

inversion Space

zz

yxz

zyx

PR

R

P

Rotating mean field: Cranking model

Seek a mean field solution carrying finite angular momentum.

.0|| zJ

Use the variational principle

with the auxiliary condition

0|| HEi

0||' zJHEi

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

symmetry). rotational (broken 1|||| if ||

zz tJitJi

eet

functions) (wave states particle single

)(routhians frame rotatingin energies particle single '

ial)(potentent fieldmean energy kinetic

(routhian) frame rotating in then hamiltonia fieldmean '

'' -'

i

i

mf

iiizmf

e

Vt

h

ehJVth

tency selfconsis mfi V

0|| HEi

.12vVariational principle : Hartree-Fock effective interaction density functionals (Skyrme, Gogny, …)

Relativistic mean field Micro-Macro (Strutinsky method) …….

Deformed mean field solutions

zJiz e )( axis-z about the Rotation R

.energy same thehave )( nsorientatio All

peaked.sharply is 1|||

.''but ''

|R

|R

RRRR

z

z

zzzz hhHHMeasures orientation.

Rotational degree of freedom: hat) (Mexican

Quantization: band rotational )(2

1|| IIJ z

zmf JVth '

z