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Quantum phase transitions in the shapes of atomic nuclei. J. Jolie, Universität zu Köln

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Quantum phase transitions in the shapes

of atomic nuclei.

J. Jolie, Universität zu Köln

+ Binding energy

3fm

How do complex systems emerge from simple ingredients

Basic ingredients:

two sets of indistinguishable fermions

a complex short range force (Van der Waals typ)

the possibility that one kind of fermions becomes the other kind

The atomic nucleus forms a unique two-component

mesoscopic system, which is hard to manipulate

but genereous in the number of observables it emits.

Collective motion:

nuclear shapes

shell structure:

valence nucleons

Cooper pairing:

N s,d boson system

The interacting boson approximation (IBA)

Once the atomic nucleus is formed effective (in-medium) forces

generate simple collective motions.

Most nuclei are very well described by a very simple

IBA hamiltonian:

generates

deformed

shape

generates

spherical

shape

with

with two structural parameters h and c and a scaling

factor a.

U(5) limit U(6) U(5) O(5) SO(3)

O(6) limit U(6) O(6) O(5) SO(3)

SU(3) limit U(6) SU(3) SO(3)

SU(3) limit U(6) SU(3) SO(3)

SU(3)

O(6)

c = 0

SU(3)

U(5)

The simple hamiltonian has four dynamical symmetries

The rich structure of this simple

hamiltonian are illustrated

by the Casten

triangle

h = 1

h = 0

Nuclear shapes associated with the four dynamical symmetries

The shapes can be studied using the coherent state formalism.

using the intrinsic state (Bohr) variables:

Then the energy functional:

can be evaluated for each value of b and g.

60°

E

U(5)

SU(3)

g

O(6)

0°

b

b

U(5) limit: irrelevant: spherical vibrator

O(6) limit: flat: g-unstable rotor

SU(3) limit: prolate rotor

SU(3) limit: oblate rotor

Experimental example:

196Pt the first O(6) nucleus

110Cd a U(5) nucleus

Discovered in 1978.

Cizewski, Casten, Smith, Stelts, Kane, Börner, Davidson,

Phys. Rev. Lett. 40 (1978) 167

Boerner, Jolie, Robinson, Casten, Cizewski

Phys. Rev. C42 (1990) R2271

M. Bertschy, S. Drissi, P.E. Garrett, J. Jolie, J. Kern,

S.J. Manannal, J.P. Vorlet, N. Warr, J. Suhonen,

Phys. Rev C 51 (1995) 103

Besides the atomic nuclei representing a dynamical symmetry, the

IBA is also able to describe transitional nuclei.

Shape phase transitions in the atomic nucleus.

When studying the changes of the nuclear shape one might observe

shape phase transitions of the groundstate configuration.

They are analogue to phase transitions in crystals

First order phase transition with P = P0= const

F(P0,T,xmin)

xmin

F(P0,T,x)

T>Tc

Tc

T<Tc

Tc

T

Tc

x

T

Second order phase transition

F(P0,T,x)

T>Tc

F(P0,T,xmin)

xmin

Tc

T<Tc

Tc

Tc

x

T

T

Landau theory of continuous phase transitions (1937) describes these

shape phase transitions.

Energy functional:

P

T

L. Landau

J.Jolie, P. Cejnar, R.F. Casten, S. Heinze,

A. Linnemann, V. Werner,

Phys. Rev. Lett. 89 (2002) 182502.

P. Cejnar, S. Heinze, J.Jolie,

Phys. Rev. C 68 (2003) 034326

Thermodynamic potential:

Order

parameter

External

parameters

In the case of our simple hamiltonian Landau theory gives the following

Solution for bmin

c

h

first order transition

second order transition

prolate

deformed

b > 0

oblate

deformed

b < 0

SU(3)

spherical

b = 0

O(6)

c = 0

SU(3)

U(5)

Triple point of

nuclear deformation

The new nuclear shape phase diagram

J. Jolie, R.F. Casten, P. von Brentano, V. Werner, Phys.Rev.Lett.87 (2001)162501

h = 0

h = 1

The shape phase transitions can be seen by the groundstate energies.

U(5)

(N=40)

h

E

O(6)

SU(3)

SU(3)

c

Two-neutron separation energies have been used before to identify

The phase transitions:

S2(N)

MeV

Sm

First order transition

in U(5) to SU(3)

N

The quadrupole moment corresponds to the control parameter b0:

N=10

N=40

A sensitive signature is in particular the B(E2;22+-> 21+)

N=40

N=10

104 106 108 110 112 114 116 118 120 122 124 126

Pb

Hg

Pt

Os

W

Hf

Yb

200Hg

198Hg

194Pt

196Pt

190Os

188Os

192Os

184W

186W

182W

R4/2 B(E2;2+2 ->2+1)[W.u] Q(2+1)[eb]

180Hf

J.Jolie, A. Linnemann

Phys. Rev. C 68 (2003) 031301.

Experimental examples for the prolate-oblate phase transition

X(5)

152Sm

R.F. Casten, V. Zamfir,

Phys. Rev. Lett. 85 (2000)3584

Following a collective model approach F. Iachello introduced new symmetries that

describe certain nuclei at the phase transition: Critical point symmytries, i.e. X(5) and E(5)

F. Iachello, Phys. Rev. Lett.85 (2000) 3580 and 87 (2001) 052502.

Recent plunger lifetime measurements seem to confirm the existence

of X(5) in several nuclei:

N=90

150Nd: R. Krücken et al. Phys. Rev. Lett. 88 (2002) 232501.

154Gd: D. Tonev et al. Phys. Rev. C69 (2004) 034334

156Dy: O. Möller et al. Phys. Rev. C74 (2006) 024313

Where do we expect the shape transitions?

E.A.McCutchan et al., Phys. Rev. C69 (2004) 024308

New examples were indeed found in the Osmium isotopes.

- Dewald et al. AIP Conf. Ser. 831 (2006) 195
- Melon et al. to be publ.

η=0.766

χ=-√7/2

X(5) nuclei and the Interacting Boson Model

IBM 178Os GCM

Gneuss and Greiner

D. Troltenier et al.

Z. Phys. A338,261(1991)

F. Iachello and A. Arima

Cambridge University Press, 1987

H = c[ nd– ( 1- )/N Q·Q] ; Q(χ)

H= T + V(β,γ)

H= T + V(β,γ)

B2=67.47

P3=0.0748

C2=174.9; C3=309.25;C4=3547.4;

C5=0.0; C6=0.0; D6=3712.5

A systematic study allows to place most nuclei in the two parameter

extended Casten triangle.

U(5)

156Dy

176,178Os

154Gd,150Nd

X(5)

X(5)

152Sm

E(5)

O(6)

SU(3)

SU(3)

A. Dewald et al. AIP Conf. Ser. 831 (2006) 195

Level dynamics and phase transitions

Up to now we concentrated only on the lowest states, what happens

with higher excited states and the level density?

U(5)-SU(3) first order shape phase transition

Energies of 0+ states

N=30

SU(3) U(5)

Energy of 0+ states up to 2.5 MeV as a function

of h in the spherical-deformed transition for N = 30.

From P. Cejnar and J. Jolie, Phys. Rev. E (2000) 6237

Can this be experimentally observed?

To excite the 0+ states the ideal and very complete way is using the

(p,t) transfer reaction at the high resolution Q3D spectrometer

(Garching).

Eight nuclei in the rare earth region were systematically studied up to

3 MeV.

(Yale/Köln/Bucarest/ Surrey/LMU-TU Munchen collaboration).

Q3D Spectrometer

Energy resolution: ~4 keV

for 15-20 MeV tritons.

Result:

D.A. Meyer et al, Phys.Lett. B 638(2006) 44

10

184W

180W

# of 0+ states below 2.5 MeV

5

162Dy 168Er 158Gd 176Hf 154Gd 152Gd

0

1.0

0

0.5

h

Level dynamics in the U(5) to O(6) phase transition

v

9

0

6

3

0

6

3

0

3

0

0

S. Heinze, P. Cejnar, J. Jolie, M. Macek, Phys. Rev. C73 (2006) 014306

M. Macek, P. Cejnar, S. Heinze, J. Jolie, Phys. Rev. C73 (2006) 014307

Energy of 0+ states

with

U(6) U(5)

O(5) SO(3)

U(6) O(6) v L

After selection by v for 0+ states (N=80).

Neighbour spacing

Absolute energies

v=0

v=0

v=18

v=18

Overlap with U(5) basis

Here there are very interesting theoretical

issues

0.0 1.0

h

Monodromy (Monodromia)

Energy of 0+ states

P. Cejnar, M. Macek, S. Heinze, J. Jolie and J. Dobes, Journ. of Phys. A 39, L515 (2006).

Conclusions

-) The shapes of atomic nuclei undergo quantum phase transitions which

are smoothed through the finite particle number.

-)The IBM provides a realistic and very rich framework to study shapes

and their relation to quantum phase transitions like Ising models do.

-) New lifetime experiments in ground state bands allow to better identify

X(5) nuclei and to confirm a strong correlation with the P-factor.

-) The finite N and excited states phase transitions form an unknown field.

-) Level dynamics exhibit bunching at E=0 reveals clues to the

fixing of quantum numbers in the particular potential (monodromy).

D.D. Warner,

Nature 420

614 (2002)

- Thanks to:
- Dewald,
- S. Heinze,
- A. Linnemann,
- (V. Werner),
- P. von Brentano, Universität zu Köln;
- R.F. Casten,
- (E. A. McCutchan),
- (V. Zamfir), Yale University;
- P. Cejnar,
- M. Macek Charles University Prague;
- General references:
- P. Cejnar and J. Jolie Prog. Part. Nucl. Phys.62 (2009) 210
- P. Cejnar, J. Jolie, R.F. Casten, to be publ. in Rev. Mod. Phys.
- A. frank, J. Jolie, P. van Isacker, Springer Tracts in Modern Physics Vol 230 (2009)

The analogy with thermodynamics can be further investigated.

Specific heat:

SU(3) -U(5)

O(6) -U(5)

N=

N=80

N=40

N=20

N=10

P. Cejnar, S. Heinze, J. Jolie, Phys. Rev. C68 (2003) 034326.

But also the U(5) wavefunction entropy can be used:

with

SU(3) -U(5)

SU(3) -U(5)

O(6) -U(5)

O(6) -U(5)

with

should be continuous everywhere.

if discontinuous at x0 : first order phase transition.

if discontinuous at x0: second order phase transition.

Solution:

x0

B

-A

First order phase transitions at: Second order at:

or

Energy functional in coherent state formalism

and

So we can absorb it by allowing negative b values !

One obtains then:

when we fix N:

The first order phase transitions should occur when

spherical-deformed

prolate-oblate

The isolated second order transition at:

Landau theory and nuclear shapes.

P

III

I

II

T

: first order transition

: isolated second order transition

Triple point of

nuclear deformation

oblate

deformed

b < 0

prolate

deformed

b > 0

spherical

b = 0

Thermodynamic potential:

Energy functional:

E(N,h,c;b,g)

F(P,T;x)

Order parameter

External parameters

J.Jolie, P. Cejnar, R.F. Casten, S. Heinze, A. Linnemann, V. Werner, Phys. Rev. Lett. 89 (2002)182502

Landau&Lifschitz Statistical Physics §144

Analog system exists for the isotropic-nematic liquid crystal phases

Landau-de Gennes theory for unaxial phases.

when D=E=0

B

Isotropic phase

2

0

Nematic phase

-2

-0.2

0

A