fermions cj

N bosons

A nucleons

valence nucleons

N nucleon pairs

L = 0 and 2 pairss,d

even-even nuclei

2.2 The Interacting Boson Approximation

A. Arima, F. Iachello, T. Otsuka, I Talmi

Schrödinger equation in second quantisation

N s,d boson system

with

N=cte

The hamiltonian is written in terms of the 36 generators of U(6):

])~

(,)~

[( '''"

†"'

† kll

kll bbbb

(2k 1)(2k' 1)( 1)k k' (kk '' k"")k ", "

[( 1)k k ' k "k k' k"

l"' l l'

l' l"(bl

† ˜ b l"' ) "k "

k k' k"

l" l' l

ll"' (bl "

† ˜ b l' ) "k " ]

.

kll bb )

~( '

†

Construction of a dynamical symmetry as an example U(5) limit:

[(d† ˜ d )k ,(d† ˜ d ) '

k ' ]

kk ddbb )~

()~

( †2

†2

(2k 1)(2k' 1)( 1)k k' (kk '' k"")k ", "

k k' k"

2 2 2

(d† ˜ d )"k" [( 1)k k ' k " 1]

Now, consider the 10 odd k generators only and then the angular momentum ones.

LnN

SUU

d

)3(0)5(0)5()6(

U(5)

SU(3)

SO(6)196Pt

156Gd

110Cd

U(5): Vibrational nuclei

SO(6): -unstable nuclei

SU(3): Rotational nuclei (prolate)

Dynamical symmetries of a N s,d boson system

U(5) SO(5) SO(3) SO(2) {nd} () L M

U(6) SO(6) SO(5) SO(3) SO(2)[N] <> () L M SU(3) SO(3) SO(2)

() L M

Analytic solutions are associated with each dynamical symmetry:

U(5): H = 1C1[U(5)] + ’1C2[U(5)] + 4C2[O(5) ] +5C2[ SO(3)] E(nd,L) = 1nd+ ’1nd( nd+ 4) + 4(+3) +5L(L+1)

H = 1C1[U(5)] + ’1C2[U(5)] + 2C2[SU(3) ] +3C2[SO(6) ] +4C2[SO(5) ] + 5C2[ SO(3)]

Again using first and second order Casimir operators a six parameter hamiltonian results::

which needs to be solved numerically.

SU(3): H = 2C2[SU(3)] +5C2[ SO(3)] E(,L) = 2+5L(L+1)

SO(6): H = 3C2[SO(6)] + 4C2[O(5) ] +5C2[ SO(3)] E(,L) = 3(+4) + 4(+3) +5L(L+1)

A dynamical symmetry leads to very strict selection rules that can be used to test it.

If an operator is a generator of a subalgebra G then due to the property:

[Gi,Gj]=kcijkGk.and

E = a f() =>Ek = a f() k withk {Gk}

it cannot connect states having a different quantum number with respect to G.

Example: 196Pt and its E2 properties.

)2(2 )~

(ˆ sddsT E is an SO(6) generator

E2 transitions between different SO(6) representations are forbidden.

Also quadrupole moments are equal to zero because of SO(5) (seniority) and the d-boson number changing E2 operator ||=1

Experimentally: Q(2+1) = +0.66(12) eb

B(E2)= 0?

H.G. Borner, J. Jolie, S.J. Robinson, R.F. Casten, J.A. Cizewski, Phys. Rev. C42(1990) R2271

195Pt(n,)196Pt + GRID method

Nuclear shapes associated with the four dynamical symmetries

The shapes can be studied using the coherent state formalism.

01

;,,

N

dsN

N

using the intrinsic state (Bohr) variables:

0)(2

sincos

1,;,, 220

N

dddsN

N

Then the energy functional:

,;,,,;,,

,;,,ˆ,;,,),;,,(

NN

NHNNE

can be evaluated for each value of and

U(5) limit: irrelevant: spherical vibrator

SO(6) limit: flat: -unstable rotor

SU(3) limit: prolate rotor

SU(3) limit: oblate rotor

SU(3)

E U(5)

SO(6)

00 ;0 00 ;0

0;0 00 60;0 00

0°

60°

Shape phases and critical point solutions.

QQN

naH d ˆ.ˆ)1(ˆˆ

Most nuclei are very well described by a very simple IBA hamiltonianof Ising form:

with two structural parameters and and a scaling factor a

)2()2( )~

()~

(ˆ ddsddsQ with

generatessphericalshape

generatesdeformedshape

U(5)

The rich structure of this simplehamiltonian are illustrated by the extended Castentriangle

SU(3)

2/7

1 U(5) limit U(6) U(5) SO(5) SO(3) 0,0 SO(6) limit U(6) SO(6) SO(5) SO(3)

27,0 SU(3) limit U(6) SU(3) SO(3)

The simple hamiltonian has four dynamical symmetries

QQN

naH d ˆ.ˆ)1(ˆˆ

SO(6)

2/7SU(3)

27,0 SU(3) limit U(6) SU(3) SO(3)

60000:)3cos( 003

Energy functional in coherent state formalism

Shape phase transitions in the atomic nucleus.

When studying the changes of the nuclear shape one might observeshape phase transitions of the groundstate configuration.

They are analogue to phase transitions in crystals

Thermodynamic potential: );,( TP

Externalparameters

Orderparameter

Energy functional: ),;,,( NE

L. Landau

Landau theory of continuous phase transitions (1937) describes theseshape phase transitions.

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

PT

),( 11min TP ),( 21min TP

T

c

c

c

min

Tc

P0,T,min

Tc

P0,T,

First order phase transition with P = P0= const

c

c

c

min

c

P0,T,min

Tc

P0,T,Second order phase transition

...),(),(),();,( 4320 TPCTPBTPATP

);,( 0TP

);,( TP

should be continuous everywhere.

if discontinuous at 0 : first order phase transition.

2

2 );,(

TP if discontinuous at 0: second order phase transition.

with TPTPB ,;0),(

Extremum are at:

0]),(4),(3),(2[ 2 TPCTPBTPA0);,(

TP

C

ACBB

8

3293

0

2

Our case: TPTPCTPB ,0),(,0),(

0);,(

2

2

TP

0),(12),(6),(2 2 TPCTPBTPA

0),(0 TPAalways:

cATPATPB ),(,0),(0

cATPATPB ),(,0),(0

0),( TPC

and

),(4

),( 2

TPC

TPBAc

...),(),(),();,( 4320 TPCTPBTPATP

0)),(),(),(( 22 TPCTPBTPA

Both minima become degenerated at:

),(),(4),( 2 TPCTPATPB

or at

),(2

),(

TPC

TPBc

c 0),( TPA

),(4),(

),(2

TPCTPB

TPA

To fullfill this equation and the one for 0 ),(32

),(9),(

2

TPC

TPBTPA

00

0),( TPB 0),( TPB

B

-A

0

C

ACBB

8

3293 2

0

C

ACBB

8

3293 2

0

Solution:

First order phase transitions at: Second order at:

),(4

),(),(

2

TPC

TPBTPA

),(4),(

),(2

TPCTPB

TPA

0),(),( TPBTPA

0

),( TPA

...4321)1(

1 64222

So we can absorb it by allowing negative values !

60000:)3cos( 003

and

06000 00 ßß

Energy functional in coherent state formalism

One obtains then:

...),,(),,(

),,(),(),;,,(

43

20

NCNB

NANENE

when we fix N: ...),(),(),();,( 4320 TPCTPBTPATP

72

)1)(1(4),,( NNB

)}84)(1({),,( 2 NNNA

54

85

840),,(

2

2

0

N

NNA

prolate-oblate00),,( 0 NB

N

The first order phase transitions should occur when

spherical-deformed

The isolated second order transition at:

8584

0,0),,(),,( 00

NN

andNBNA

spherical = 0

prolatedeformed > 0

oblatedeformed < 0

Triple point ofnuclear deformation

: first order transition

: isolated second order transition

00

00

00 I

III

II

P

T

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

Landau theory and nuclear shapes.

Thermodynamic potential:

External parameters

Order parameter

Energy functional:

(P,T;) E(N,)

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

E

SU(3)

O(6)

SU(3)

U(5) (N=40)

The quadrupole moment corresponds to the control parameter 0:

N=10 N=40

N=10 N=40

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

+)

J. Jolie, P. Cejnar, R.F. Casten, S. Heinze, A. Linnemann, V. Werner, Phys.Rev.Lett. 89(2002)182502P. Cejnar and J. Jolie, Rep. on Progress in Part. and Nucl. Phys. 62 (2009) 210.

dynamical symmetry

First order phasetransition

Second order phasetransition (isolated)

sphericalprolatedeformed

oblatedeformed

Conclusion: the following shapes phase transitions are obtained:

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

Examples for the prolate-oblate and sherical-prolate phase transition

PbHgPtOsWHfYb

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

200Hg198Hg

196Pt194Pt

192Os190Os188Os

186W184W182W

180Hf

Samarium isotopes

N=10N=20

50

82

Normal states Intruder states

110Cd

2p-2hpair + Q+ ...

K. Heyde, et al. Nucl. Phys. A466 (1987) 189.

inmix

mixn

HH

HHˆˆ

ˆˆThis can be described in the IBM by a N (normal) plus N+2 (intruder) systemwhich might mix. :

with

and)0(||)0(|| )

~~()( ddddssssHmix

2.2.4 Core excitations

Iz = + 1/2

Iz = - 1/2

Up(6)

Uh(6)

U(6)

Also new kinds of symmetries are possible: Intruder or I-spin

always fulfilled[H,Iz] =0

[H,I2] = 0

[H,I+] = [H,I-] = 0 intruder-analog state

good intruder Spin

K. Heyde, C. De Coster, J. Jolie, J.L. Wood, Phys. Rev. C 46 (1992), 541

I=0

I=1/2

I=1

I=2

I=3/2

+1/2 +3/2-1/2-3/2 +1 +2-1-2 0

Iz

H. Lehmann, J. Jolie, C. De Coster, B. Decroix, K. Heyde, J.L. Wood, Nucl. Phys. A 621 (1997) 767

Intruder analog states

The cadmium isotopes are unique in several respects.

-) protons nearly fill the Z=50 (Sn) shell;

-) neutrons are near mid-shell (N=66)

-) there are 8 stable isotopes of Cd.

This is clearly the mass region where we can learn about nuclear structure.

2.3 A case study: 112Cd

M. Délèze, S. Drissi, J. Jolie, J. Kern, J.P. Vorlet, Nucl. Phys, A554 (1993) 1

Pd (,2n) Cd reaction allowedthe establishment of multiphononstates up to nd=6.

5.39ps

0.68ps0.73ps

But, above 1.2 MeV additional states exist and build asecond collective structure.

< 2.1ps

< 2.8ps

0.7ps

and

The additional states are intruder states presenting 2 particle- 2 hole excitations across Z=50. They lead to shape coexistence and can be described using the SO(6) limit.

E[N,nd,,L] = nd + (+3) +L(L+1)

intrL] = (+3) + (+3) +L(L+1)

U(6)

U

U(5) O(6)

U

U

O(5)

O(3)

J. Jolie and H. Lehmann, Phys. Lett B342 (1995)

normal states

112Cd

Intruder states

)0(||)0(|| )~~

()( ddddssssHmix is a O(5) scalar.

Symmetries can play a dominant role in shape coexistence.

Wavefunctions with O(5) symmetry have fixed seniority of d-bosons.

)0(

)0(

5

1

5

1

6

555

6

1

ddssddss

)0(||)0(|| )~~

()( ddddssssHmix

)0)(0(0]2[ T

0)0(2]2[0)0(0]2[

5

1

6

55

6

1

TT

Cannot connect intruder withnormal states

=N+2

max=N0)0(2]2[

5

1

6

5

TH effmix

Moreover one can rewrite:

)0(||)0(|| )~~

()( ddddssssHmix

Inelastic Neutron Scattering (INS) experiment at the Van de Graaff Accelerator of the University of Kentucky (Prof S.W. Yates, Lexington USA).

(n,n’ ) Elevel J and placements of E from excitation function varying En

from angular distributions

n = 1.25ps1.200.42

= 1.16ps0.490.27

= 0.67ps0.210.13

= 1.20ps0.830.35

= 0.42ps0.100.07

= 0.51ps0.170.10

(n,n´) with 3.4 and 4 MeV neutrons for lifetimes and coincidences allowed theextension to higher low-spin states (P.E. Garrett et al. Phys. Rev. C75 (2007)054310

There it becomes difficult to describe the details.

Harmonic vibrator (collective model)

+ finite N effects (IBM in the U(5)-limit)

+ intruder states within a U(5)-O(6) model

+ neutron-proton degree of freedom and symmetry breaking

Absolute B(E2) values for the decay of three phonon states in 110Cd

Confirmation of the U(5)-O(6) picture.

F. Corminboeuf, T.B. Brown, L. Genilloud, C.D. Hannant, J. Jolie, J. Kern, N. Warr and S.W. Yates, Phys. Rev. C 63 (2001) 014305.

Fribourg-Kentucky-Köln Data B(E2) Values in Six Valence Proton Configurations

Neutron Cd Intruder Ru Ba

Number B(E2;230A) B(E2;2101) B(E2;2101)

27182362

558175664

1415457086166

61167748668 2430

169870

It works well only in a given shell.

M. Kadi, N. Warr, P.E. Garrett, J. Jolie, S.W. Yates, Phys. Rev. C68 (2003) 031306(R).