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

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1 Symmetries and collective Nuclear excitations PRESENT AND FUTURE EXOTICS IN NUCLEAR PYSICS In honor of Geirr Sletten at his 70 th birthday Stefan Frauendorf, Y. Gu, Daniel Almehed Department of Physics University of Notre Dame, USA Institut für Strahlenphysik, Forschungszentrum Rossendorf Dresden, Germany

2 Rotation of Molecules r r ( x electrons, X nuclei ) rotational invariant approximate eigenfunction (nuclei fixed) oriented rotational band 2

3 Weak spontaneous symmetry breaking amiltonian has a symmetry approximate eigenstate breaks it collective mode/doubling Continuous orientation : Condensation of quadrupole phonons -Tidal waves Twofold discrete: Dynamic chirality Combination of the two: Condensation of octupole phonons 3

4 1.Weakly oriented nuclei tidal waves Mean field: rigid spherical soft rigid deformed Yrast line irregular multi p-h -condensation of quadrupole phonons -very soft rotor Tidal wave regular w weakly increases with I regular w proportional to I 4

5 E qp. excitations Tidal waves Ω ω = E = ωl z 2 Generation of angular momentum I Angular velocity Deformation Rotor increases stays constant Tidal wave Vibrator stays constant increases R( ϑ, ϕ, t) = R0 [1 + 2a2 cos(2ϕ Ωt) Y22 ( ϑ, ϕ = 0)] 5

6 20 T V R J Nd ω[mev] 6

7 Quadrupole waves: Theoretical method S. Frauendorf, Y. Gu, arxiv , PRL, in preparation Cranking model: semiclassical treatment of angular momentum Micro-macro method (Nilsson+fixed pairing). Find the equilibrium shape for the rotating mean field. Minimizing at fixed frequency problematic: E' ( ω, ε, γ ) const on path γ ( ε ). Minimizing E( I, ε, γ ) = E'( ω( I), ε, γ ) + J ( ω( I), ε, γ ) with < J x >= J ( ω( I), ε, γ ) = I works well. 7

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9 g-factors Even for I=2 the angular velocity is so high that nucleons respond non-perturbativly.

10 Treating the yrast states of vibrational or transitional nuclei as running tidal waves makes microscopic calculations simple. Strongly anharmonic TW in vibrational nuclei Above I=4 collective and single particle motion interwoven B(E2) more regular than energies. Details: h 11/ Z=48, N=60-66: after 2 neutron alignment, smaller deformation -> approach of antimagnetic rotation Z=46, N=56,60 and Z=44, N=62,64 angular velocity nearly constant during h 11/ neutron alignment 2 tidal wave with quasiparticle degrees of freedom More B(E2) values to check theory 10

11 Chirality of molecules σ z C right left C COO N N COO σ = z z F σ Rotational frequency: 100meV 11 F

12 Chirality of molecules σ z C + - C N N σ = z z F 100meV 6000 Gz σ Rotational frequency: 100meV 11 F

13 2. Dynamic chirality of nuclei Chiral Vibration Tunneling 12 Consequence of static chirality: Two identical rotational bands.

14 I 2 > I1,I3 Triaxial Rotor+ particle+hole Frauendorf, Meng, NPA 617, 131 (1997) Chiral vibration 2D - TAC+RPA 3D - TAC Chiral vibration 2D - TAC+RPA Nuclear chirality - a transient phenomenon Large amplitude collective motion - tough 13

15 Tilted Axis Cranking + RPA '= h WS (P + + P) + r ω J r = ω 1 J 1 + ω 2 J 2 + ω 3 J 3 κ N N 2 Q Q r N N ω J r ω 1 = ω sinϑ cosϕ ω 2 = ω sinϑ sinϕ ω 3 = ω cosϑ Meanfield ε cos(γ) = κ < Q 0 > ε sin(γ) = κ < Q 2 > r ω must be parallel to J r =< r ˆ J > ϑ,ϕ 3D-TAC with spherical Woods-Saxon [Dimitrov et al PRL 84 (2000)] Modified QQ-force N-dependent κ 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 shape orientation RPA - Small amplitude harmonic vibrations around the mean field minimum [ + + ' RPA,O λ ]= E RPA O λ 14

16 135 Best case of chirality so far: 60 Nd75 πh νh 2 11/ / 2 S. Zhu et al. Phys. Rev. Lett. 91, (2003) 15

17 Chiral vibrations in 135-Nd TAC+RPA calculations Mukhopadhyay, Almehed et al. Phonon is mainly orientation fluctuations PRL 99, (2007) Same inband transition rates - Good agreement with experiment 16

18 135-Nd Transition rates in-band cross band 17

19 135-Nd Transition rates in-band cross band 18

20 Small ε TAC+RPA in Odd-odd nuclei Almehed and Frauendorf PRC, in review Dripline 19

21 Orientation amplitudes 25 kev 134-Pr ω=0.4 Shape amplitudes Q < 0 Q Q < 0 Q > + Q + Q few % > Q m =< Q 1 > 0 m ϕ = Q 1 2Q 2 θ = armonic approximation but large amplitude. Q 1 Q 2 + 3Q 0 ψ = Q 2 Q 2 3Q 0 20

22 Rotating triaxial nuclei do become chiral But chirality is weakly broken. The observed pairs of bands are manifest of slow motion of angular momentum through the two chiral sectors. The chiral mode well decouples from the shape modes The chiral mode is transitional: strongly anharmonic strong tunneling I=10 I=12 I=14 J 2 J 3 J 1 21