Symmetries and collective Nuclear excitations PRESENT AND FUTURE EXOTICS IN NUCLEAR PHYSICS In honor...
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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
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Rotation of Molecules
band rotational
oriented
fixed) (nucleiion eigenfunct eapproximat
invariant rotational ),(
nucleielectrons XxH
2
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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
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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
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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
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0
4
8
12
16
20
0.0 0.1 0.2 0.3 0.4 0.5
6
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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
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g-factors
Even for I=2 the angularvelocity is so high thatnucleons respond non-perturbativly.
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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.
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H
HH
H
H
H H
H
CC
NN
FFRotational frequency: 100meV
Chirality of molecules
z
COOH COOH
right left
HH zz 11
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H
HH
H
H
H H
H
CC
NN
FFRotational frequency: 100meV
+-
100meV6000 GHz
Chirality of molecules
z
HH zz 11
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Consequence of static chirality: Two identical rotational bands.
Chi
ral V
ibra
tion
T
unne
ling
2. Dynamic chirality of nuclei
12
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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
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• 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
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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
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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
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135-Nd Transition rates
in-band
cross band
17
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135-Nd Transition rates
in-band
cross band
18
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Small
Dripline
Almehed and Frauendorf
PRC, in review
TAC+RPA inOdd-odd nuclei
19
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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
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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