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Transcript of Spontaneous symmetry breaking and rotational bands S. Frauendorf Department of Physics University of...
Spontaneous symmetry breaking and rotational bands
S. Frauendorf
Department of Physics
University of Notre Dame
The collective model
yprobabilit moment inertia of
on transitiquadrupolemoment
)(5)2(
)02,2( 212
ebMeV
EBEBQt
420
6
x
Even-even nuclei, low spin
Deformed surface breaks rotational the spherical symmetry band
2
)1()(
IIIE
Collective and single particle degrees of freedom
On each single particle state (configuration) a rotational bandis built (like in molecules).
Single particle and collective degrees of freedom become entangled at high spin and low deformation.
Limitations:scale intrinsic~MeV10~ :scaleenergy rotational 1-
2
Rotationalbands in
Er163
More microscopic approach:
Retains the simple picture of an anisotropic object going round.
Mean field theory + concept of spontaneous symmetry breaking for interpretation.
Rotating mean field (Cranking model):
Start from the Hamiltonian in a rotating frame
zjvtH 12'momentumangular
ninteractiobody - twoeffective
energy kinetic
12
zj
v
t
Mean field approximation:find state |> of (quasi) nucleons moving independently inmean field generated by all nucleons.mfV
(routhian) frame rotating thein nhamiltonia field mean '
},| { :tencyselfconsis , -' , |'|' 12
h
VvJVtheh mfzmf
Selfconsistency : effective interactions, density functionals (Skyrme, Gogny, …), Relativistic mean field, Micro-Macro (Strutinsky method) …….
Reaction of thenucleons to the inertial forces must be taken into account
Low spin: simple droplet.High spin: clockwork of gyroscopes.
Uniform rotation about an axis that is tilted with respect to the principal axes is quite common. New discrete symmetries
Rotational response
Mean field theory:Tilted Axis Cranking TACS. Frauendorf Nuclear Physics A557, 259c (1993)
Quantization of single particlemotion determines relation J().
Spontaneous symmetry breaking
Symmetry operation S and
.|'|'|'
energy same the withsolutions field mean are states All
1||| and ,''
HHE
hh
|SS
|S
|SSS
Full two-body Hamiltonian H’ Mean field approximation
Mean field Hamiltonian h’ and m.f. state h’|>=e’|>.
Symmetry restoration |Siic
'' HH SS
Spontaneous symmetry breaking
Which symmetries can be broken?
Combinations of discrete operations
rotation withreversal time- )(
inversion space-
angleby axis-zabout rotation - )(
y
z
TR
P
R
zJHH ' is invariant under
axis-zabout rotation - )(zRBroken by m.f. rotational
bands
Obeyed by m.f.spinparitysequence
broken by m.f.doublingofstates
zmf jVth '
zJiz e )( axis-z about the Rotation R
peaked.sharply is 1|||
.''but ''
|R
RRRR
z
zzzz hhHH
Rotational degree of freedom and rotational bands.
Deformed charge distribution
nucleons on high-j orbitsspecify orientation
.|2
1I|momentumangular good of State
.energy same thehave )(| nsorientatio All
deiI
z |R
Common bands
by axis-zabout rotation - )(
rotation withreversal time- 1 )(
inversion space - 1
z
y
R
TR
P
Principal Axis CrankingPAC solutions
nI
e iz
2
signature ||)(
R
TAC or planar tilted solutionsMany cases of strongly brokensymmetry, i.e. no signature splitting
Rotationalbands in
Er163
No deformation – no bands?
Dynucleus
medsuperdefor thefrom rays - 152
Pbnucleus
spherical thefrom rays - 199
E2 radiation - electric rotation
M1 radiation - magnetic rotation
I-1/2
19
20
21
22 23 24 2526
27
28
10’
Baldsiefen et al. PLB 275, 252 (1992)
Magnetic rotor composed of two current loops
2 neutron holes
2 pro
ton p
artic
les
22/132/92/13
11619882
)()(
ionconfigurat j-high
Pb
ihi
The nice rotorconsists offour high-jorbitals only!
Why so regular?
repulsive loop-loop interaction
JE
Shears mechanism
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 incoherentlyto staggering.
Staggering in Multiplets!
22/13 )( i )( 2/92/13 hi
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 quadrupolemoments give the calculated small deformation.
||
First clear experimental evidence: Clark et al. PRL 78 , 1868 (1997)
Magnetic rotor Antimagnetic rotor
Anti-FerromagnetFerromagnet
181920
21
22
23
24
18
20
22
24
strongmagneticdipoletransitions
weakelectricquadrupoletransitions
58106
48Cd
A. Simons et al. PRL 91, 162501 (2003) Band termination
J
Degree of orientation (A=180, width of :) |)(|| |R z
Ordinary rotor Magnetic rotor
228 Jo 920 Jo
Many particles 2 particles, 2 holes T
erm
inat
ing
band
s
3.0 1.0Deformation:
ChiralityChiral or aplanar solutions: The rotational axis is out of all principal planes.
rotation withreversal time- 1 )(
by axis-zabout rotation - 1 )(
inversion space - 1
y
z
TR
R
P
20’
Consequence of chirality: Two identical rotational bands.
The prototype of a triaxial chiral rotor
Frauendorf, Meng, Frauendorf, Meng, Nucl. Phys. A617, 131 (1997Nucl. Phys. A617, 131 (1997) )
Composite chiral bands Demonstration of the symmetry concept:It does not matter how the three components of angular momentum are generated.
7513560 Nd 1
2/112
2/11hh 23 0.20 29
6010545 Rh 2
2/1112/9 hg 20 0.22 29
I
Best candidates
S. Zhu et al.Phys. Rev. Lett. 91, 132501 (2003)
Composite chiral band in 7513560 Nd
10 15 20 250.0
0.2
0.4
0.6
0.8
1.0
E
2-E
1
particle rotor model
=30o, jp:j
h=2:1
E2-E
1, [
Me
V]
I
10 15 20 250.0
0.2
0.4
0.6
0.8
1.0
135Nd exp
E2-
E1,
[M
eV
]
I
E
2-E
1
Tunneling between the left- and right-handed configurations causes splitting.
chiral regime
rotEE 3.012
Rotational frequency
Energy difference between chiral sister bands
chiralregime
chiralregime
)|(|2
1| )( lherhI IiChiral sister states:
Transition rates
-+B(-in)B(-out)
Branching B(out)/B(in) sensitive to details.
)|(|2
1| )( lherhI Ii
Robust: B(-in)+B(-out)=B(+in)+B(+out)=B(lh)=B(rh)
Sensitive to details of the system
Rh10560
10545 Rh
22/11
12/9 hg
Chiralregime
J. Timar et al.Phys Lett. B 598178 (2004)
Odd-odd: 1p1h
Even-odd: 2p1h, 1p2h
Even-even: 2p-2h
Best
Chirality
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
Predicted regions of chirality
mass-less particle
p
s
1P
nucleus
New type of chirality 1T
)( zz PR
molecule
Reflection asymmetric shapes
Two mirror planes
Combinations of discrete operations
rotation withreversal time- )(
inversion space-1
by axis-zabout rotation - )(
PTR
P
R
y
z
29’
Good simplex
Several examples in mass 230 region
I
i
z
e
)(parity
simplex |
1)2/(
|S
PRS
Th225
Parity doubling
Only good case.
Tetrahedral shapes
J. Dudek et al. PRL 88 (2002) 25250232a
5.032 a
15.032 a
Which orientation has the rotational axis?
minimum
maximum
Classical no preference
)2/(
zR
P
2/)(parity
12
,2
doublex |
1)2/(
2
signature 1)(
I
i
z
z
e
nI
|D
PRD
R
0
2
4
3
5
7E3 M2 E3 M2
509040 Zr
Prolate ground state
Tetrahedral isomer at 2 MeV
132 MeVp
18 MeVt
Predicted as best case (so far):
Comes down by particle alignment
Summary
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
34’
Thanks to my collaborators! V. Dimitrov, S. Chmel, F. Doenau, N. Schunck, Y. Zhang, S. Zhu
Orientation is generated by the asymmetric distributionquantal orbits near the Fermi surface
Microscopic (“finite system”)
Rotational levels become observable.
eV 10 :scale intrinsiceV10~ :molecules 1-6-2
Spontaneous symmetry breaking=
Appearance of rotational bands.
Energy scale of rotational levels in
momentumangular
]s m kg[eV10~eV10~ levels rotational of distanceenergy
eV10~ levels rotational of scaleenergy
1-215-49-2
49-2
J
JJJ
Tiniest external fields generate a superposition of the |JM>that is oriented in space, which is stable.
Spontaneous symmetry breakingMacroscopic (“infinite”) system
Weinberg’s chair
Hamiltonian rotational invariant
2|),(|),
:ondistributidensity
IM| :momentumangular
good of seigenstate
IMYρ(r,
Why do we see the chair shape?
Symmetry broken state: approximation, superposition of |IM> states: calculate electronic state for given position of nuclei.
3NH
1
2
3
Quadrupole deformation Axial vector deformation
J
Degree of orientation (width of :) |)(|| |R z
o5o15
Orientation is specified by the order parameter
Electric quadrupole moment magnetic dipole moment
Ordinary “electric” rotor Magnetic rotor
Transition rates
-+inout
Branching sensitive to details.
)|(|2
1| )( lherhI Ii
2
2
22
22
||)(||
||)(||
||)(||||)(||
||)(||||)(||
TACTAC
IlhIlh
IIII
IIII
M
M
MM
MM
Robust:
)(yTR
Nuclear chirality