Spontaneous symmetry breaking and rotational bands

S. Frauendorf

Department of Physics

University of Notre Dame

The collective model

yprobabilit moment inertia of

on transitiquadrupolemoment

)(5)2(

)02,2( 212

ebMeV

EBEBQt

420

6

x

Even-even nuclei, low spin

Deformed surface breaks rotational the spherical symmetry band

2

)1()(

IIIE

Collective and single particle degrees of freedom

On each single particle state (configuration) a rotational bandis built (like in molecules).

Single particle and collective degrees of freedom become entangled at high spin and low deformation.

Limitations:scale intrinsic~MeV10~ :scaleenergy rotational 1-

2

Rotationalbands in

Er163

More microscopic approach:

Retains the simple picture of an anisotropic object going round.

Mean field theory + concept of spontaneous symmetry breaking for interpretation.

Rotating mean field (Cranking model):

Start from the Hamiltonian in a rotating frame

zjvtH 12'momentumangular

ninteractiobody - twoeffective

energy kinetic

12

zj

v

t

Mean field approximation:find state |> of (quasi) nucleons moving independently inmean field generated by all nucleons.mfV

(routhian) frame rotating thein nhamiltonia field mean '

},| { :tencyselfconsis , -' , |'|' 12

h

VvJVtheh mfzmf

Selfconsistency : effective interactions, density functionals (Skyrme, Gogny, …), Relativistic mean field, Micro-Macro (Strutinsky method) …….

Reaction of thenucleons to the inertial forces must be taken into account

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

Mean field theory:Tilted Axis Cranking TACS. Frauendorf Nuclear Physics A557, 259c (1993)

Quantization of single particlemotion determines relation J().

Spontaneous symmetry breaking

Symmetry operation S and

.|'|'|'

energy same the withsolutions field mean are states All

1||| and ,''

HHE

hh

|SS

|S

|SSS

Full two-body Hamiltonian H’ Mean field approximation

Mean field Hamiltonian h’ and m.f. state h’|>=e’|>.

Symmetry restoration |Siic

'' HH SS

Spontaneous symmetry breaking

Which symmetries can be broken?

Combinations of discrete operations

rotation withreversal time- )(

inversion space-

angleby axis-zabout rotation - )(

y

z

TR

P

R

zJHH ' is invariant under

axis-zabout rotation - )(zRBroken by m.f. rotational

bands

Obeyed by m.f.spinparitysequence

broken by m.f.doublingofstates

zmf jVth '

zJiz e )( axis-z about the Rotation R

peaked.sharply is 1|||

.''but ''

|R

RRRR

z

zzzz hhHH

Rotational degree of freedom and rotational bands.

Deformed charge distribution

nucleons on high-j orbitsspecify orientation

.|2

1I|momentumangular good of State

.energy same thehave )(| nsorientatio All

deiI

z |R

Common bands

by axis-zabout rotation - )(

rotation withreversal time- 1 )(

inversion space - 1

z

y

R

TR

P

Principal Axis CrankingPAC solutions

nI

e iz

2

signature ||)(

R

TAC or planar tilted solutionsMany cases of strongly brokensymmetry, i.e. no signature splitting

Rotationalbands in

Er163

No deformation – no bands?

Dynucleus

medsuperdefor thefrom rays - 152

Pbnucleus

spherical thefrom rays - 199

E2 radiation - electric rotation

M1 radiation - magnetic rotation

I-1/2

19

20

21

22 23 24 2526

27

28

10’

Baldsiefen et al. PLB 275, 252 (1992)

Magnetic rotor composed of two current loops

2 neutron holes

2 pro

ton p

artic

les

22/132/92/13

11619882

)()(

ionconfigurat j-high

Pb

ihi

The nice rotorconsists offour high-jorbitals only!

Why so regular?

repulsive loop-loop interaction

JE

Shears mechanism

Most of the l-l interaction due to a slight quadrupole polarization of the nucleus.

Keeps two high-j holes/particles in the blades well aligned.

The 4 high-j orbitals contribute incoherentlyto staggering.

Staggering in Multiplets!

22/13 )( i )( 2/92/13 hi

TAC

Long transverse magnetic dipole vectors, strong B(M1)

B(M1) decreases with spin: band termination

Experimental magnetic moment confirms picture.

Experimental B(E2) values and spectroscopic quadrupolemoments give the calculated small deformation.

||

First clear experimental evidence: Clark et al. PRL 78 , 1868 (1997)

Magnetic rotor Antimagnetic rotor

Anti-FerromagnetFerromagnet

181920

21

22

23

24

18

20

22

24

strongmagneticdipoletransitions

weakelectricquadrupoletransitions

58106

48Cd

A. Simons et al. PRL 91, 162501 (2003) Band termination

J

Degree of orientation (A=180, width of :) |)(|| |R z

Ordinary rotor Magnetic rotor

228 Jo 920 Jo

Many particles 2 particles, 2 holes T

erm

inat

ing

band

s

3.0 1.0Deformation:

ChiralityChiral or aplanar solutions: The rotational axis is out of all principal planes.

rotation withreversal time- 1 )(

by axis-zabout rotation - 1 )(

inversion space - 1

y

z

TR

R

P

20’

Consequence of chirality: Two identical rotational bands.

The prototype of a triaxial chiral rotor

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

Composite chiral bands Demonstration of the symmetry concept:It does not matter how the three components of angular momentum are generated.

7513560 Nd 1

2/112

2/11hh 23 0.20 29

6010545 Rh 2

2/1112/9 hg 20 0.22 29

I

Best candidates

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

Composite chiral band in 7513560 Nd

10 15 20 250.0

0.2

0.4

0.6

0.8

1.0

E

2-E

1

particle rotor model

=30o, jp:j

h=2:1

E2-E

1, [

Me

V]

I

10 15 20 250.0

0.2

0.4

0.6

0.8

1.0

135Nd exp

E2-

E1,

[M

eV

]

I

E

2-E

1

Tunneling between the left- and right-handed configurations causes splitting.

chiral regime

rotEE 3.012

Rotational frequency

Energy difference between chiral sister bands

chiralregime

chiralregime

)|(|2

1| )( lherhI IiChiral sister states:

Transition rates

-+B(-in)B(-out)

Branching B(out)/B(in) sensitive to details.

)|(|2

1| )( lherhI Ii

Robust: B(-in)+B(-out)=B(+in)+B(+out)=B(lh)=B(rh)

Sensitive to details of the system

Rh10560

10545 Rh

22/11

12/9 hg

Chiralregime

J. Timar et al.Phys Lett. B 598178 (2004)

Odd-odd: 1p1h

Even-odd: 2p1h, 1p2h

Even-even: 2p-2h

Best

Chirality

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

Predicted regions of chirality

mass-less particle

p

s

1P

nucleus

New type of chirality 1T

)( zz PR

molecule

Reflection asymmetric shapes

Two mirror planes

Combinations of discrete operations

rotation withreversal time- )(

inversion space-1

by axis-zabout rotation - )(

PTR

P

R

y

z

29’

Good simplex

Several examples in mass 230 region

I

i

z

e

)(parity

simplex |

1)2/(

|S

PRS

Th225

Parity doubling

Only good case.

Tetrahedral shapes

J. Dudek et al. PRL 88 (2002) 25250232a

5.032 a

15.032 a

Which orientation has the rotational axis?

minimum

maximum

Classical no preference

)2/(

zR

P

2/)(parity

12

,2

doublex |

1)2/(

2

signature 1)(

I

i

z

z

e

nI

|D

PRD

R

0

2

4

3

5

7E3 M2 E3 M2

509040 Zr

Prolate ground state

Tetrahedral isomer at 2 MeV

132 MeVp

18 MeVt

Predicted as best case (so far):

Comes down by particle alignment

Summary

Orientation does not always mean a deformed charge density:Magnetic rotation – axial vector deformation.

Nuclei can rotate about a tilted axis: New discrete symmetries.

New type of chirality in rotating triaxial nuclei: Time reversal changes left-handed into right handed system.

Bands in nuclei with tetrahedral symmetry predicted

34’

Thanks to my collaborators! V. Dimitrov, S. Chmel, F. Doenau, N. Schunck, Y. Zhang, S. Zhu

Orientation is generated by the asymmetric distributionquantal orbits near the Fermi surface

Microscopic (“finite system”)

Rotational levels become observable.

eV 10 :scale intrinsiceV10~ :molecules 1-6-2

Spontaneous symmetry breaking=

Appearance of rotational bands.

Energy scale of rotational levels in

momentumangular

]s m kg[eV10~eV10~ levels rotational of distanceenergy

eV10~ levels rotational of scaleenergy

1-215-49-2

49-2

J

JJJ

Tiniest external fields generate a superposition of the |JM>that is oriented in space, which is stable.

Spontaneous symmetry breakingMacroscopic (“infinite”) system

Weinberg’s chair

Hamiltonian rotational invariant

2|),(|),

:ondistributidensity

IM| :momentumangular

good of seigenstate

IMYρ(r,

Why do we see the chair shape?

Symmetry broken state: approximation, superposition of |IM> states: calculate electronic state for given position of nuclei.

3NH

1

2

3

Quadrupole deformation Axial vector deformation

J

Degree of orientation (width of :) |)(|| |R z

o5o15

Orientation is specified by the order parameter

Electric quadrupole moment magnetic dipole moment

Ordinary “electric” rotor Magnetic rotor

Transition rates

-+inout

Branching sensitive to details.

)|(|2

1| )( lherhI Ii

2

2

22

22

||)(||

||)(||

||)(||||)(||

||)(||||)(||

TACTAC

IlhIlh

IIII

IIII

M

M

MM

MM

Robust:

)(yTR

Nuclear chirality