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Chirality of Nuclear Rotation
S. Frauendorf
Department of Physics
University of Notre Dame, USA
IKH, Forschungszentrum Rossendorf
Dresden, Germany
In collaboration withJ. Meng, PKUV. Dimitrov, ISUF. Doenau, FZRU. Garg, NDK. Starosta, MSUS. Zhu, ANL
“I call any geometrical figure, or group of points, chiral, and say it has chirality, if its image in a plane mirror, ideally realized, cannot brought to coincide with itself.”Kelvin, 1904, Baltimore lectures on Molecular Dynamics and Wave Theory of Light
Chirality of molecules
mirror
The two enantiomers of 2-iodubutene
)( zz PR
R – mintS - caraway
mirror
Chirality of mass-less particles
)( zz PR
z
Triaxial nucleus is achiral.
)(yTR
1PJ
Rotating nucleus
Right-handed Left-handed
)(yTR
New type of chirality
Chirality Changed invariant
MoleculesMassless particles space inversion time reversal
Nuclei time reversal space inversion
Chirality
“I call a physical object, chiral, and say it has chirality, if its image, generated by space inversionor time reversal, cannot brought to coincide with itself by a rotation.”
11/37
Consequence of chirality: Two identical rotational bands.
Tilted rotation
Classical mechanics: Uniform rotation only about the principal axes.
Condition for uniform rotation: Angular momentum and velocity have the same direction.
iiiJ )(
on.distributidensity theof axes the withcoincide which
,nsorinertia te theof axes principal for theonly toparallel iJ
The nucleus is not a simple piece of matter,but more like a clockwork of gyroscopes.
Uniform rotation about anaxis that is tilted with respectto the principal axes is quite common.
The prototype of a chiral rotor
Frauendorf, Meng, Frauendorf, Meng, Nucl. Phys. A617, 131 (1997Nucl. Phys. A617, 131 (1997) )
Consequence of chirality: Two identical rotational bands.
band 2 band 1134Pr
h11/2 h11/2
Rotating mean field: Tilted Axis Cranking model
Seek a mean field state |> carrying finite angular momentum,where |> is a Slater determinant (HFB vacuum state)
.0|| zJ
Use the variational principle
with the auxiliary condition
0|| HEi
0||' zJHEi
The state |> is the stationary mean field solution in the frame that rotates uniformly with the angular velocity about the z axis.
S. Frauendorf Nuclear Physics A557, 259c (1993)
functions) (wave states particle single
)(routhians frame rotating in energies particle single '
ial)(potentent field mean energy kinetic
(routhian) frame rotating thein nhamiltonia field mean '
|'' -'
i
i
mf
iiizmf
e
Vt
h
ehJVth
tency selfconsis mfi V
Variational principle : Hartree-Fock effective interactionDensity functionals (Skyrme, Gogny, …)Relativistic mean field
Micro-Macro (Strutinsky method) …….
(Pairing+QQ)
X
NEW: The principal axes of the density distribution need not coincide with the rotational axis (z).
The QQ-model
','2
2 '||5
4
basis
potential model shell spherical
kkkk
kkkksph
kkksph
sphsph
cckYrkQcceh
eh
Vth
operator quadrupole ),(5
4
2
202
2
2
YrQ
QQhH sph
Mean field solution
QqJhheh
QQJhE
zsphiii
zsph
tencyselfconsis
'''
variation
2'
2
2
2
2
Intrinsic frame
Principal axes
2/sincos
00
20
2211
KqKq
qqqq
,ˆ toparallel bemust
tencyselfconsis
cossinsincossin
)('
2200
321
22200332211
JJ
QqQq
QQqQqJJJhh sph
22
222
2220
220
222
|)],,0(),,0([),,0(|4
5
||4
5)2,2(
protonproton
LAB
QDDQD
QIIEB
211
211 |),,0(|
4
3||
4
3)1,1(
v
vLAB DIIMB
Transition probabilities
Spontaneous symmetry breaking
Symmetry operation S
.|'|'|'
energy same the withsolutions field mean are states All
1||| and ,'but ''
HHE
hhHH
|SS
|S
|SSSSS
Symmetries
zJvtH 12'
Broken by m.f. rotationalbands
Principal Axis CrankingPAC solutions
nIe iz 2||)( R
Tilted Axis CrankingTAC or planar tilted solutions
Chiral or aplanar solutionsDoubling of states
Discrete symmetries
Rotationalbands in
Er163
PAC TAC
TAC->PACI=
The Cranking Model (rotating mean field) provides a reliable description of nuclear rotational bands.
It accounts for the discrete symmetries PAC and TAC if the tilt of the rotational axis is taken into account –Tilted Axis Cranking (TAC).
TAC gives chiral solutions, where chiral sister bands are observedand predicts more regions.
First chiral solution for 7513459 Pr
Predictions for different mass regions
Composite chiral bands
V. Dimitrov, S. Frauendorf, F. Doenau, Physical Review Letters 84, 5732 (2000)
First chiral solution for 7513459 Pr
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
31/37
5910445 Rh 2/11
12/9 hg
C. Vaman et alPhys. Rev. Lett.92, 032501 (2004)
S. Zhu et al.Phys. Rev. Lett. 91, 132501 (2003)
Composite chiral band in 7513560 Nd
Composite chiral band in 6010545 Rh J. Timar et al.
Phys. Lett. B. subm.
22/11
12/9 hg
0 5 10 15 20 25 30 35 40 45 50 55 60-26-24-22-20-18-16-14-12-10-8-6-4-20
axial
chiral
162
69Tm
93
o
modified oscillator
E[M
eV]
J
0 5 10 15 20 25 30 35 40 45 50 55 60-26-24-22-20-18-16-14-12-10-8-6-4-20
axial
chiral
162
69Tm
93
o
modified oscillator
E[M
eV]
J
0 5 10 15 20 25 30 35 40 45 50 55 60-26-24-22-20-18-16-14-12-10-8-6-4-20
axial
chiral
162
69Tm
93
o
modified oscillator
E[M
eV]
J
band 2 band 1134Pr
h11/2 h11/2
Left-right tunneling
Breaking of chiral symmetry is not very strong.
Particle – Rotor model:
Frauendorf, Meng, Nuclear Physics A617, 131 (1997)Frauendorf, Meng, Nuclear Physics A617, 131 (1997)
Doenau, Frauendorf, Zhang, PRC , in preparation
312 ,
Dynamical (Particle Rotor) calculation
Chiral vibration
Frozen alignment approximation:
They are numbers
One dimensional -very well suited for analysis.
312 44 JJJ
chiralvibration
chiralrotation
jJ crit 3
24
[8] K. Starosta et al., Physical Review Letters 86, 971 (2001)
Transitionprobabilities
out
in
outout
out
in
in
in
yrast yrare
yrast yrare
outout
inin
Tunneling and vibrational motions are manifest in the electromagnetic transitions.
Microscopic description of the left-right dynamics needed.
The dynamics are being studied in Particle Rotor model.
ConclusionsChirality in molecules and massless particles changed by P not by T.
Chirality in rotating nuclei changed by T not by P.
Triaxial nucleus must carry angular momentum along all three axes.
Experimental evidence for chiral sister bands around A=104, 134.
Chirality shows up as a pair of rotational bands. 1I
TAC theory accounts for experiment and predicts more cases.
Substantial left-right tunneling and chiral vibrations as precursors.
Microscopic description of left-right dynamics is needed.
Reflection asymmetric shapes,
two reflection planes
Simplex quantum number
I
i
z
parity
e
)(
||
)(
S
PRS
Parity doubling
Th226
Thee three components of the angular Thee three components of the angular momentum form two systems of opposite momentum form two systems of opposite
chirality.chirality.
Tilted rotation
conserved : a.m. of valueabsolute
conserved 2
1
2
1 :energy
)( :inertia of moments
:momentumangular
axes principal 3,2,1 :locityangular ve
22
22
2,3
2,21
i
i
iii
nnnn
iii
i
JJ
JE
xxm
J
i
Triaxial rotor: Classical motion of J
Uniform rotation only aboutthe principal axes!
Small E
Large E
sphere momentumangular 22iJJ
2
1 ellipsoidenergy
2
i
iJE
What is rotating?
HCl
molecules
)( :inertia of moments 2,3
2,21
nnnn xxm
Nuclei: Nucleons are not on fixed positions.More like a liquid, but what kind of?
viscous: “rotational flow”
ideal : “irrotational flow”
None is true: complicated flow containing quantal vortices.
Microscopic description needed:Rotating mean field
134Prband 2 band 1h11/2 h11/2
lsi ,
Theoretical description
Particle – Rotor model:Coupling of the particle and the hole to rotor describedquantum mechanically.
Frauendorf, Meng, Nuclear Physics A617, 131 (1997)Frauendorf, Meng, Nuclear Physics A617, 131 (1997)
Doenau, Frauendorf, Zhang, PRC , in preparation
Dynamics of of angular momentum orientation.
Chiral vibrations and rotations.
Transition probabilities
Are the assumptions about the rotor realized?Is the nucleus triaxial? Is the moment of inertia of the intermediate axis maximal?
20/37
Where can one expect chirality?
Microscopic description needed:Rotating mean fieldTilted Axis CrankingFrauendorf, Nucl. Phys. A557, 259c (1993)Frauendorf, Rev. Mod. Phys. 73, 462 (2001)
Are there more complex chiral configurations?
The mean field concept
A nucleon moves in the mean field generated by all nucleons.
][ imfV The mean field is a functional of the single particle states determined by an averaging procedure.
The nucleons move independently.
ii
N
c
cc
state in nucleona creates
0|......|tion)(configura statenuclear 1
functions) (wave states particle single
energies particle single
ial)(potentent field mean energy kinetic
i
i
mf
iiimf
e
Vt
ehVth
Total energy is a minimized (stationary) with respect to the single particle states.
with the 12vtH
Calculation of the mean field: Hartree Hartree-Fock density functionals (Skyrme, Gogny, …)
Relativistic mean field Micro-Macro (Strutinsky method) …….
.0|| HEi
.12v
Start from the two-body Hamiltonian
effective interaction
Use the variational principle
nucleus
linear-nonhighly ,,|ˆ|,,),,( 321321321 ii JJ
on.distributidensity theof
axes principal therespect to with tiltedarethat
axesfor possible also toparallel
i
J
molecule inertial ellipsoidiiiJ )(
. axes principal for theonly toparallel iJ
S. Frauendorf Nuclear Physics A557, 259c (1993)
)( Reflection
)( Rotation
inversion Space
zz
yxz
zyx
PR
R
P
Rotating mean field: Cranking model
Seek a mean field solution carrying finite angular momentum.
.0|| zJ
Use the variational principle
with the auxiliary condition
0|| HEi
0||' zJHEi
The state |> is the stationary mean field solution in the frame that rotates uniformly with the angular velocity about the z axis. In the laboratory frame it corresponds to a uniformly rotating mean field state
symmetry). rotational (broken 1|||| if ||
zz tJitJi
eet
functions) (wave states particle single
)(routhians frame rotatingin energies particle single '
ial)(potentent fieldmean energy kinetic
(routhian) frame rotating in then hamiltonia fieldmean '
'' -'
i
i
mf
iiizmf
e
Vt
h
ehJVth
tency selfconsis mfi V
0|| HEi
.12vVariational principle : Hartree-Fock effective interaction density functionals (Skyrme, Gogny, …)
Relativistic mean field Micro-Macro (Strutinsky method) …….
Deformed mean field solutions
zJiz e )( axis-z about the Rotation R
.energy same thehave )( nsorientatio All
peaked.sharply is 1|||
.''but ''
|R
|R
RRRR
z
z
zzzz hhHHMeasures orientation.
Rotational degree of freedom: hat) (Mexican
Quantization: band rotational )(2
1|| IIJ z
zmf JVth '
z