symmetries of the cranked mean field
DESCRIPTION
Symmetries of the Cranked Mean Field. S. Frauendorf. IKH, Forschungszentrum Rossendorf, Dresden Germany. Department of Physics University of Notre Dame USA. In collaboration with. Afanasjev, UND, USA V. Dimitrov, ISU, USA F. Doenau, FZR, Germany J. Dudek, CRNS, France - PowerPoint PPT PresentationTRANSCRIPT
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Symmetries of the Cranked Mean Field
S. Frauendorf
Department of Physics
University of Notre Dame
USA
IKH, Forschungszentrum
Rossendorf, Dresden
Germany
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In collaboration withA. Afanasjev, UND, USAB. V. Dimitrov, ISU, USAF. Doenau, FZR, GermanyJ. Dudek, CRNS, FranceJ. Meng, PKU, China N. Schunck, US, GB Y.-ye Zhang, UTK, USAS. Zhu, ANL, USA
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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.
TAC: The principal axes of the density distribution need not coincide with the rotational axis (z).
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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
V
t
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
S. Frauendorf Nuclear Physics A557, 259c (1993)
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Spontaneous symmetry breaking
Symmetry operation S
.|'|'|'
energy same the withsolutions field mean are states All
1||| and ,''but ''
HHE
hhHH
|SS
|S
|SSSSS
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Which symmetries
Combinations of discrete operations
rotation withreversal time- )(
inversion space-
2 angleby axis-zabout rotation - )
2(
y
z nn
TR
P
R
leave zJHH ' invariant?
axis-zabout rotation - )(zRBroken by m.f. rotational
bands
Obeyed by m.f.spinparitysequence
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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
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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
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Consequence of chirality: Two identical rotational bands.
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band 2 band 1134Pr
h11/2 h11/2
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The prototype of a chiral rotor
Frauendorf, Meng, Frauendorf, Meng, Nucl. Phys. A617, 131 (1997Nucl. Phys. A617, 131 (1997) )
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10 12 14 16 18 20 220
100
200
300
400
500
600
700
800
900
1000
backbend
134Prexperiment
E2-
E1
I
E2E1 omega1
There is substantial tunneling between the left- and right-handed configurations
chiralregime
Rotational frequency
Energy difference Between the chiral sisters
chiral regime
rotEE 3.012
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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
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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 observed23 0.20 29
6010545 Rh 2
2/1112/9 hg
observed20 0.22 29
I
Is it possible to couple 3 quasiparticles to a chiral configuration?
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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
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Good simplex
Several examples in mass 230 region
Other regions? Substantial tunneling
I
i
z
e
)(parity
simplex ||
1)(
S
PRS
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Th225
Parity doubling
Only good case.Must be better studied!
Substantial tunneling
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Tetrahedral shapes
J. Dudek et al. PRL 88 (2002) 25250232a
5.032 a
15.032 a
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Which orientation has the rotational axis?
minimum
maximum
Classical no preference
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)2/(
zR
P
2/)(parity
12
,2
doublex |
1)2/(
2
signature 1)(
I
i
z
z
e
nI
|D
PRD
R
0
2
4E3
3
5
7E3
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509040 Zr
Prolate ground state
Tetrahedral isomer at 2 MeV
132 MeVp
18 MeVt
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Isospatial analogy
Which symmetries leave
ATHZNHH zpn ' invariant?
axis-zabout nisorotatio - )( zTiz e R
Broken by m.f. isorotationalbands
Proton-neutron pairing: symmetries of the pair-fieldAnalogy between angular momentum J and isospin T
space gauge in rotation- 1D - )( Aig e R
Broken by m.f. Pair-rotationalbands
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1t
0t
Isovectorpair fieldbreaks isorotationalinvariance.
Isoscalarpair fieldkeeps isorotationalinvariance.
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The isovector scenario
02
ˆ
np
ppnn
y
Calculate without np-pair field.
Add isorotational energy.
ionsconfigurat possible restricts for 0
2
)1()0 field, mean( np
ZNT
TTEE
y
iso
preferred axis
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The isovector scenario works well(see poster 161).
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Isorotational energy gives the Wigner term in the binding energies mWigner terenergysymmetry
)(75
2
)1( 2
TTA
MeVTT
iso
Structure of rotational bands in 377437 Rb
nrestrictio ionconfigurat 0yT
reproduced
For the lowest states in odd-odd nuclei with ZN
isoisoTETE
TETE
2/122/1)1(2)0(
)1()0(
No evidence for the presence of an isoscalar pair field
See poster 161
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Isoscalar pairing at high spin?
Isoscalar pairs carry finite angular momentum
iJ z 2
total angular momentum
•A. L. GoodmanPhys. Rev. C 63, 044325 (2001)
Predicted by
Which evidence?
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Adding nn pairs to the condensate does not change the structure.
Pair rotational bands are an evidence for the presence of a pair field.
Ordinary nn pair field
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which symmetries leave ATHH z ' invariant?
1 )( Aig e R Either even or odd A belong to the band.
1 )( Nin e R Even and odd N belong to the band.
1 )( zJiz e R Both signatures belong to the band.
nNI
e ig
nzg
2
gaugeplex ||
1)()(
S
RRS
iJ z 2
total angular momentum
If an isoscalar pair field is present,
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Pair rotational bands for an isoscalar neutron-proton pair field
ZNA 22
2/))()2(( AEAE
Even-even, even I Odd-odd, odd I
Not enough data yet.
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Summary
Symmetries of the mean field are very useful to characterizenuclear rotational bands.
Nuclei can rotate about a tilted axis: New discrete symmetries manifest by the spin and parity sequence in the rotational band:-New type of chirality in nuclei: Time reversal changes left-handed into right handed system.
-Spin-parity sequence for reflection asymmetric (tetrahedral) shapes
The presence of an isovector pair field and isospin conservation explain the binding energies and rotational spectra of N=Z nuclei.
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Out of any plane: parity doubling + chiral doubling
,,,10
,,,9
,,,8
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Banana shapes
Z=70, N=86,88J. Dudek, priv. comm.
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Doublex quantum number
2/)2/(
1
2
)(
)2/(
12
,2
2,2)(
2
||)(||
)2/(
IIi
z
z
iz
i
z
eparity
nI
ee
DRP
RD
RD
PRD
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Restrictions due to the symmetry yT
States with good N, Z –parity are in general no eigenstates of .yT
If they are (T=0) the symmetry restricts the possible configurations, if not (T=1/2) the symmetry does not lead to anything new.
0|:0 yTT
00|)(2
1
00|)(2
1
00|)(2
1
00|)(2
1
00|
00|
jnipjpiny
jnipjpiny
jpipjniny
jpipjniny
inipy
y
T
T
T
T
T
T
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)(yTR
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Rotationalbands in
Er163
1 1’ 2 3 4 7
PAC TAC