Symmetries of the Cranked Mean Field

S. Frauendorf

Department of Physics

University of Notre Dame

USA

IKH, Forschungszentrum

Rossendorf, Dresden

Germany

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

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).

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)

Spontaneous symmetry breaking

Symmetry operation S

.|'|'|'

energy same the withsolutions field mean are states All

1||| and ,''but ''

HHE

hhHH

|SS

|S

|SSSSS

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

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

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

Consequence of chirality: Two identical rotational bands.

band 2 band 1134Pr

h11/2 h11/2

The prototype of a chiral rotor

Frauendorf, Meng, Frauendorf, Meng, Nucl. Phys. A617, 131 (1997Nucl. Phys. A617, 131 (1997) )

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

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

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?

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

Good simplex

Several examples in mass 230 region

Other regions? Substantial tunneling

I

i

z

e

)(parity

simplex ||

1)(

S

PRS

Th225

Parity doubling

Only good case.Must be better studied!

Substantial tunneling

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

4E3

3

5

7E3

509040 Zr

Prolate ground state

Tetrahedral isomer at 2 MeV

132 MeVp

18 MeVt

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

1t

0t

Isovectorpair fieldbreaks isorotationalinvariance.

Isoscalarpair fieldkeeps isorotationalinvariance.

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

The isovector scenario works well(see poster 161).

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

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?

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

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,

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.

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.

Out of any plane: parity doubling + chiral doubling

,,,10

,,,9

,,,8

Banana shapes

Z=70, N=86,88J. Dudek, priv. comm.

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

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

)(yTR

Rotationalbands in

Er163

1 1’ 2 3 4 7

PAC TAC