Spontaneous symmetry breaking and rotational bands
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Spontaneous symmetry breaking and rotational bands. S. Frauendorf. Department of Physics University of Notre Dame. x. The collective model. Even-even nuclei, low spin. Deformed surface breaks rotational

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Spontaneous symmetry breaking and rotational bands

Spontaneous symmetry breaking and rotational bands

S. Frauendorf

Department of Physics

University of Notre Dame


The collective model

x

The collective model

Even-even nuclei, low spin

Deformed surface breaks rotational

the spherical symmetry band


Spontaneous symmetry breaking and rotational bands

Collective and single particle degrees of freedom

On each single particle state

(configuration) a rotational band

is built (like in molecules).


Spontaneous symmetry breaking and rotational bands

Limitations:

Single particle and collective degrees of freedom

become entangled at high spin and low deformation.

Rotational

bands in


Spontaneous symmetry breaking and rotational bands

More microscopic approach:

Mean field theory

+

concept of spontaneous symmetry

breaking for interpretation.

Retains the simple picture of an anisotropic object going round.


Rotating mean field cranking model

Reaction of the

nucleons to

the inertial

forces must be

taken into account

Rotating mean field (Cranking model):

Start from the Hamiltonian in a rotating frame

Mean field approximation:

find state |> of (quasi) nucleons moving independently in

mean field generated by all nucleons.

Selfconsistency : effective interactions, density functionals (Skyrme, Gogny, …),

Relativistic mean field, Micro-Macro (Strutinsky method)

…….


Rotational response

Rotational response

Low spin: simple droplet.

High spin: clockwork of gyroscopes.

Quantization of single particle

motion determines relation J(w).

Uniform rotation about an axis that

is tilted with respect to the principal

axes is quite common.

New discrete symmetries

Mean field theory:

Tilted Axis Cranking TAC

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


Spontaneous symmetry breaking

Spontaneous symmetry breaking

Full two-body Hamiltonian H’

Mean field approximation

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

Symmetry operation S and

Spontaneous symmetry breaking

Symmetry restoration


Which symmetries can be broken

Broken by m.f.

rotational

bands

Combinations of discrete operations

spin

parity

sequence

Obeyed by m.f.

doubling

of

states

broken by m.f.

Which symmetries can be broken?

is invariant under


Spontaneous symmetry breaking and rotational bands

nucleons on

high-j orbits

specify orientation

Deformed charge distribution

Rotational degree of freedom and rotational bands.


Common bands

Common bands

Principal Axis Cranking

PAC solutions

TAC or planar tilted solutions

Many cases of strongly broken

symmetry, i.e.

no signature splitting


Spontaneous symmetry breaking and rotational bands

Rotational

bands in


Spontaneous symmetry breaking and rotational bands

E2 radiation - electric rotation

I-1/2

23

24

25

22

26

27

21

M1 radiation - magnetic rotation

28

20

19

No deformation – no bands?

10’

Baldsiefen et al. PLB 275, 252 (1992)


Spontaneous symmetry breaking and rotational bands

2 proton particles

2 neutron holes

Magnetic rotor composed of two current loops

The nice rotor

consists of

four high-j

orbitals only!


Spontaneous symmetry breaking and rotational bands

repulsive loop-loop

interaction

E

J

Shears mechanism

Staggering in Multiplets!

Why so regular?

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 incoherently

to staggering.


Spontaneous symmetry breaking and rotational bands

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

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 quadrupole

moments give the calculated small deformation.


Spontaneous symmetry breaking and rotational bands

Anti-Ferromagnet

Ferromagnet

Magnetic rotor

Antimagnetic rotor

24

24

23

22

22

21

20

20

19

18

18

weak

electric

quadrupole

transitions

strong

magnetic

dipole

transitions


Spontaneous symmetry breaking and rotational bands

Band termination

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


Spontaneous symmetry breaking and rotational bands

Ordinary rotor

Magnetic rotor

J

Terminating bands

Degree of orientation (A=180, width of

Many particles

2 particles, 2 holes

Deformation:


Chirality

20’

Chirality

Chiral or aplanar solutions:

The rotational axis is out of all principal planes.


Spontaneous symmetry breaking and rotational bands

Consequence of chirality: Two identical rotational bands.


The prototype of a triaxial chiral rotor

The prototype of a triaxial chiral rotor

Frauendorf, Meng,

Nucl. Phys. A617, 131 (1997)


Spontaneous symmetry breaking and rotational bands

20 0.22 29

23 0.20 29

Composite chiral bands

Demonstration of the symmetry concept:

It does not matter how the three components

of angular momentum are generated.

Best candidates


Spontaneous symmetry breaking and rotational bands

Composite chiral band in

S. Zhu et al.

Phys. Rev. Lett.

91, 132501 (2003)


Spontaneous symmetry breaking and rotational bands

chiral regime

chiral

regime

chiral

regime

Chiral sister states:

Tunneling between the left- and

right-handed configurations

causes splitting.

Rotationalfrequency

Energy difference between chiral sister bands


Transition rates

Transition rates

-

+

B(-out)

B(-in)

Sensitive to details of the system

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

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


Rh105

Rh105

Chiral

regime

J. Timar et al.

Phys Lett. B 598

178 (2004)


Spontaneous symmetry breaking and rotational bands

Chirality

Odd-odd: 1p1h

Even-odd: 2p1h, 1p2h

Even-even: 2p-2h

Best


Predicted regions of chirality

13 0.18 26

observed

13 0.21 14

observed

predicted

13 0.21 40

13 0.21 14

predicted

predicted

45 0.32 26

Predicted regions of chirality

Chiral

sister bands

Representative

nucleus


Spontaneous symmetry breaking and rotational bands

nucleus

mass-less particle

molecule

New type of

chirality


Reflection asymmetric shapes

29’

Reflection asymmetric shapes

Two mirror planes

Combinations of discrete operations


Spontaneous symmetry breaking and rotational bands

Good simplex

Several examples in mass 230 region


Spontaneous symmetry breaking and rotational bands

Parity doubling

Only good case.


Tetrahedral shapes

Tetrahedral shapes

J. Dudek et al. PRL 88 (2002) 252502


Spontaneous symmetry breaking and rotational bands

minimum

maximum

Which orientation has the rotational axis?

Classical no preference


Spontaneous symmetry breaking and rotational bands

E3 M2

E3 M2


Spontaneous symmetry breaking and rotational bands

Predicted as best case (so far):

Prolate ground state

Tetrahedral isomer at 2 MeV

Comes down by particle alignment


Summary

Summary

34’

Orientation is generated by the asymmetric distribution

quantal orbits near the Fermi surface

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

Thanks to my collaborators!

V. Dimitrov, S. Chmel, F. Doenau, N. Schunck,

Y. Zhang, S. Zhu


Spontaneous symmetry breaking and rotational bands

Microscopic (“finite system”)

Rotational levels become observable.

Spontaneous symmetry breaking

=

Appearance of rotational bands.

Energy scale of rotational levels in


Spontaneous symmetry breaking and rotational bands

Tiniest external fields generate a superposition of the |JM>

that is oriented in space, which is stable.

Spontaneous symmetry breaking

Macroscopic (“infinite”) system


Weinberg s chair

Weinberg’s chair

Hamiltonian rotational invariant

Why do we see the chair shape?


Spontaneous symmetry breaking and rotational bands

3

2

1

Symmetry broken state:

approximation, superposition of |IM> states:

calculate electronic state for given position of nuclei.


Spontaneous symmetry breaking and rotational bands

Quadrupole deformation

Axial vector deformation

J

Degree of orientation (width of

Orientation is specified by the order parameter

Electric quadrupole moment magnetic dipole moment

Ordinary “electric” rotor

Magnetic rotor


Transition rates1

Robust:

Transition rates

-

+

out

in

Branching sensitive to details.


Nuclear chirality

Nuclear chirality


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