5. Exotic modes of nuclear rotation

Tilted Axis Cranking -TAC

Cranking Model

Seek a mean field solution carrying finite angular momentum.

.0|| zJ

Use the variational principle

with the auxillary 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

tency selfconsis mfi V

functions) (wave states particle single

routhians) p. (s. 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

Low spin: simple droplet.High spin: clockwork of gyroscopes.

Uniform rotation about an axis that is tilted with respect to the principal axes is quite common. New discrete symmetries

Rotational response

Quantization of single particlemotion determines relation J().

)ˆ(xR

)/))ˆ(exp((1)(

)ˆ(1)ˆ(

0

2

2 20

axRr

VxV

xYRxR

mf

Quadrupole deformation

0)( VxV

0)( xVa

Intrinsic frame

Principal axes

2/sincos

00

20

2211

q

,ˆ toparallel bemust

,

:tencyselfconsis

cossinsincossin

)('

20

321

332211

JJ

JJJxVth mf

Symmetries

zJvtH 12'

Broken by m.f. rotationalbands

Discrete symmetries

Combinations of discrete operations

rotation withreversal time- )(

inversion space-

angleby axis-zabout rotation - )(

y

z

TR

P

R

Common bands

by axis-zabout rotation - )(

rotation withreversal time- 1 )(

inversion space - 1

z

y

R

TR

P

PAC solutions(Principal Axis Cranking)

nI

e iz

2

signature ||)(

R

TAC solutions (planar)(Tilted Axis Cranking) Many cases of strongly brokensymmetry, i.e. no signature splitting

Rotationalbands in

Er163

Chiral bands

Chirality of molecules

mirror

It is impossible to transform one configurationinto the other by rotation.

mirror

Chirality of mass-less particles

Only left-handed neutrinos:Parity violation in weak interaction

Consequence of static chirality: Two identical rotational bands with the same parity.

Best example of chirality so far 7513560 Nd

Chi

ral V

ibra

tion

T

unne

ling

Weak

symmetr

y breaking

Reflection asymmetric shapes,

two reflection planes

Simplex quantum number

I

i

z

parity

e

)(

||

)(

S

PRS

Parity doubling

20/23

Th223

Weak

symmetr

y breaking

Summary

• The different discrete symmetries of the m.f. are manifest by different level sequences in the rotational bands.

• For reflection symmetric shapes, a band has fixed parity and one has:

• Rotation about a principal axis (signature selects every second I)

• Rotation about an axis in a principal plane (all I)

• Rotation about an axis not in a principal plane (all I, for each I a pair of states – chiral doubling)

• For reflection asymmetric shapes, a band contains both parities.

• If the rotational axis is normal to one of two reflection planes the bands contain all I and the levels have alternating parity.

• For reflection asymmetric shapes exists 16 different symmetry types.

5. Emergence of bands

Orienteded mean field solutions

zJiz e )( axis-z about the Rotation R

This is clearly the case for a well deformed nucleus.Deformed nuclei show regular rotational bands.Spherical nuclei have irregular spectra.

n.orientatiodifferent

of states theseall of ionsuperposita is state rotational The

.energy same thehave )( | states field mean All

peaked.sharply is 1|||

.but

|R

|R

RRRR

z

z

zzzz hhHH

deformed

Er163

Isotropybroken

spherical

Pb200

Isotropyconserved

The rotating nucleus: A Spinning clockwork of gyroscopes

Nucleonicorbitals –Highly tropicgyroscopes

Orbitals with high nodal structureat the Fermi surface generate orientation

How does orientation come about?

Orientation of the gyroscopes

Deformed density / potential

Deformed potential aligns thepartially filled orbitals

Partially filled orbitals are highly tropic

Nucleus is oriented – rotational band

Well deformed Hf174 -90 0 90 180 2700.0

0.2

0.4

0.6

0.8

1.0

over

lap

5

Angular momentum is generated by alignment of the spin of the orbitals with the rotational axisGradual – rotational bandAbrupt – band crossing, no bands

7

M1 band in the sphericalNucleus Magnetic rotation –orientation specified byfew orbitals

SD

Pb199

Magnetic Rotation

-90 0 90 180 2700.0

0.2

0.4

0.6

0.8

1.0

over

lap

Weakly deformed Pb199

8

TAC

Measurements confirmed the length of the parallelcomponent of the magnetic moment.

Soft deformation:Terminating bands

A. Afanasjev et al. Phys. Rep. 322, 1 (99)

Orientation of the gyroscopes

Deformed density / potential

],[],[ 2/112/112/9lki hhglik

termination

5810951 Sb

The nature of nuclear rotational bands

The experimentalist’s definition of rotational bands:

Requirements for the mean field:

norientatio peaked.sharply is 1||| |Rz

smoothness 1. toclose is 1|)1(|)(| DII

deformation super normal weak

axes ratio (

1:2 (0.6) 1:1.5 (0.3) 1:1.1 (0.1)

mass 150 180 200

1 1/2 1/7

2 4 20

4 8 20

60 30 8

D 0.005 0.03 0.05

rig /)2(

irrot /)2(

][o

][J

4,, 2 J

R

Rrigirrot

Transition to the classical limit

Classical periodic orbits in a deformed potential

Summary

• Breaking of rotational symmetry does not always mean substantial deviation of the density distribution from sphericity.

• Magnetic rotors have a non-spherical arrangement of current loops. They represent the quantized rotation of a magnetic dipole.

• The angular momentum is generated by the shears mechanism.

• Antimagnetic rotors are like magnetic ones, without a net magnetic moment and signature symmetry.

• Bands terminate when all angular momentum of the valence nucleons is aligned.

• The current loops of the valence orbits determine the current pattern and the moment of inertia.

deformed

Er163

spherical

Pb200

Isotropybroken

Isotropyconserved

Summary

• The mean field may spontaneously break symmetries. • The non-spherical mean field defines orientation and the rotational

degrees of freedom.• The rotating mean field (cranking model) describes the response of the

nucleonic motion to rotation.• The inertial forces align the angular momentum of the orbits with the

rotational axis. • The bands are classified as single particle configurations in the

rotating mean field. The cranked shell model (fixed shape) is a very handy tool.

• At moderate spin one must take into account pair correlations. The bands are classified as quasiparticle configurations.

• Band crossings (backbends) are well accounted for. • Nuclei may rotate about a tilted axis• New types of discrete symmetries of the mean field.