Symmetries and collective Nuclear excitations PRESENT AND FUTURE EXOTICS IN NUCLEAR PHYSICS In honor of Geirr Sletten at his 70th birthday

Stefan Frauendorf, Y. Gu, Daniel Almehed

Department of Physics

University of Notre Dame, USA

Institut für Strahlenphysik, Forschungszentrum RossendorfDresden, Germany

Rotation of Molecules

band rotational

oriented

fixed) (nucleiion eigenfunct eapproximat

invariant rotational ),(

nucleielectrons XxH

2

Weak spontaneous symmetry breaking

Hamiltonian has a symmetry approximate eigenstate breaks itcollective mode/doubling

Twofold discrete: Dynamic chirality

Continuous orientation : Condensation of quadrupole phonons -Tidal waves

Combination of the two:Condensation of octupole phonons 3

Yrast line

Mean field:rigid spherical rigid deformedsoft

irregularmulti p-h

regularproportional to I

regularweakly increases with I

-condensation ofquadrupole phonons-very soft rotor

Tidal wave

1.Weakly oriented nuclei – tidal waves

4

E

I

qp. excitations

Tidal wave

s

)]0,()2cos(21[

),,(

2220

YtaR

tR

zLE

2

5

Angular velocity DeformationRotor increases stays constant

Tidal wave

Vibrator stays constant increases

Generation of angular momentum

0

4

8

12

16

20

0.0 0.1 0.2 0.3 0.4 0.5

6

Quadrupole waves: Theoretical method

Cranking model: semiclassical treatment of angular momentumMicro-macro method (Nilsson+fixed pairing).

Find the equilibrium shape for the rotating mean field.

Minimizing at fixed frequency problematic:

).(path on ),,(' constE

Minimizing

well. works),),((with

),),((),),(('),,(

IIJJ

IJIEIE

x

7

S. Frauendorf, Y. Gu, arXiv 0709.0254, PRL, in preparation

g-factors

Even for I=2 the angularvelocity is so high thatnucleons respond non-perturbativly.

Above I=4 collective and single particle motion interwoven

B(E2) more regular than energies.

Z=48, N=60-66: after neutron alignment, smaller deformation -> approach of antimagnetic rotation

Z=46, N=56,60 and Z=44, N=62,64 angular velocitynearly constant during neutron alignment – tidal wave with quasiparticle degrees of freedom

2/11h

2/11h

More B(E2) values to check theory

Strongly anharmonic TW in “vibrational nuclei”

10

Details:

Treating the yrast states of vibrational or transitional nuclei asrunning tidal waves makes microscopic calculations simple.

H

HH

H

H

H H

H

CC

NN

FFRotational frequency: 100meV

Chirality of molecules

z

COOH COOH

right left

HH zz 11

H

HH

H

H

H H

H

CC

NN

FFRotational frequency: 100meV

+-

100meV6000 GHz

Chirality of molecules

z

HH zz 11

Consequence of static chirality: Two identical rotational bands.

Chi

ral V

ibra

tion

T

unne

ling

2. Dynamic chirality of nuclei

12

312 ,

Chiral vibration

Chiral vibration

2D - TAC+RPA

3D - TAC

2D - TAC+RPA

Nuclear chirality - a transient phenomenonLarge amplitude collective motion - tough

Triaxial Rotor+particle+hole

Frauendorf, Frauendorf, Meng, Meng,

NPA 617, NPA 617,

131 (1997131 (1997))

13

• 3D-TAC with spherical Woods-Saxon [Dimitrov et al PRL 84 (2000)]• Modified QQ-force N-dependent in 2 N-shells [Baranger, Kumar NPA 110 (1968)]• Parameters fitted to reproduce Strutinsky results• Pair field adjusted to 80% of odd-even mass difference

H 'hWS (P P)N

2N

QN QN

J

J 1J1 2J2 3J3

1 sin cos 2 sin sin 3 cosMeanfield cos() Q0 sin() Q2 must be parallel to

J

ˆ J ,

RPA - Small amplitude harmonic vibrations around the mean field minimum

H 'RPA ,O ERPA O

Tilted Axis Cranking + RPA

orientationshape

14

Best case of chirality so far: 75

13560 Nd

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

12/11

22/11

hh

15

Chiral vibrations in 135-Nd TAC+RPA calculations

Same inband transition rates - Good agreement with experiment

Mukhopadhyay, Almehed et al.

PRL 99, 172501 (2007)Phonon is mainly orientation fluctuations

16

135-Nd Transition rates

in-band

cross band

17

135-Nd Transition rates

in-band

cross band

18

Small

Dripline

Almehed and Frauendorf

PRC, in review

TAC+RPA inOdd-odd nuclei

19

Orientation amplitudes

Harmonic approximationbut ‘large’ amplitude.

Q1

2Q2

Q 1

Q2 3Q0

Q 2

Q2 3Q0

1||0 mm QQ

Shape amplitudes

0||0

0||0

22

22

0

0

QQQQ

Q

Q

few %

134-Pr=0.4

25keV

20

Rotating triaxial nuclei do become chiralBut chirality is weakly broken.

The observed pairs of bands are manifest of slow motionof angular momentum through the two chiral sectors.

The chiral mode is transitional: strongly anharmonic – strong tunneling

The chiral mode well decouples from the shape modes

I=14I=10 I=12

J1

J2

J3

21