algebraic collective model and its applications gabriela thiamov laboratoire de physique subatomique...

Algebraic collective model and its applications

Gabriela Thiamová

Laboratoire de Physique Subatomique et de CosmologieInstitut National Polytechnique de Grenoble

France

Sofia, October 8-10, 2015

1

1) IBM-BM relationship in various dynamical symmetry limits -in the spherical vibrator U(5) limit -in the O(6) limit -in the rigid-rotor limit

2) The algebraic collective model (ACM)

3) IBM-BM relationship in the triaxial limit

4) ACM applications (single and multi-phonon excitations)

5) Conclusions

Outline of the presentation

2

IBM-BM relationship in various dynamical symmetry limits

In the spherical vibrator U(5) limit the IBM dynamical subgroup chain 

contracts in the N ∞ limit to the BM dynamicalsymmetry chain

D. J. Rowe and G. Thiamova, NPA 760 (2005), 59

Heisenberg-Weyl algebrafor the BM spanned by qm,

m=0, ± 1, ± 2, I

Any developments in the U(5) limit of one modelapply equally to the other

IBM

BM

3

The O(6) basis :

-similarly to the U(5) limit, there is a close correspondence of the physicsof the IBM in its O(6) limit with the BM in its rigid-beta gamma-softWilets-Jean limit.

This correspondence is precise in the limit in which the IBM dynamicalsymmetry group contracts to the chain of the Wilets-Jean model

N ∞ contraction

IBM

BM

4

IBM-BM relationship in various dynamical symmetry limits

5

IBM-BM relationship in various dynamical symmetry limits

The O(6) basis :

-similarly to the U(5) limit, there is a close correspondence of the physicsof the IBM in its O(6) limit with the BM in its rigid-beta gamma-softWilets-Jean limit.

This correspondence is precise in the limit in which the IBM dynamicalsymmetry group contracts to the chain of the Wilets-Jean model

IBM

BM

N ∞ contraction

There is a problem with the BM subgroup chain !

-the delta-function nature of the WJ rigid-beta states-they do not have a convergent expansion in terms of the

U(5) states in the BM.

This problem is circumvented in the ACM model !6

IBM-BM relationship in various dynamical symmetry limits

The O(6) basis :

-similarly to the U(5) limit, there is a close correspondence of the physicsof the IBM in its O(6) limit with the BM in its rigid-beta gamma-softWilets-Jean limit.

This correspondence is precise in the limit in which the IBM dynamicalsymmetry group contracts to the chain of the Wilets-Jean model

IBM

BM

N ∞ contraction

There is a problem with the BM subgroup chain !

-the delta-function nature of the WJ rigid-beta states-they do not have a convergent expansion in terms of the

U(5) states in the BM.

Rigid rotor states

In the BM the beta and gamma rigid subgroup chain

is a submodel of the beta rigid subgroup chain

as much as is a subgroup of

w.f. are delta-func.in β and γ -

a problem !!!

7

IBM-BM relationship in various dynamical symmetry limits

Rigid rotor states

In the BM the beta and gamma rigid subgroup chain

is a submodel of the beta rigid subgroup chain

as much as is a subgroup of

In the IBM

is not a submodel ofbecause SU(3) is not a subgroup of O(6)

However

w.f. are delta-func.in β and γ -

a problem !!!

8

IBM-BM relationship in various dynamical symmetry limits

The rotor-like states of the of the SU(3) limit of the IBM are notrelated to those of its O(6) limit in ways that parallel the relationship

between the rigid-rotor and rigid-beta states in the BM. 9

IBM-BM relationship in various dynamical symmetry limits

Rigid rotor states

In the BM the beta and gamma rigid subgroup chain

is a submodel of the beta rigid subgroup chain

as much as is a subgroup of

In the IBM

is not a submodel ofbecause SU(3) is not a subgroup of O(6)

However

w.f. are delta-func.in β and γ -

a problem !!!

A dynamical subgroup chain is used in the ACM to define a continuous set of basis states for the BM :

dynamical groupfor radial (beta)wave functions

λ= v+5/2 – basis states are those of the harmonic spherical vibrator– convenient for spherical and near spherical nuclei

Deformed nuclei – much more rapid convergence for a suitably chosenmodified SU(1,1) representation – radial w. f. obtained by modifying theSO(5) centrifugal potential.

Davidson basis – λv = 1 + [(v+3/2)2 + β04]1/2

Angular part SO(5) spherical harmonicschar. by seniority v (ang. mom.)

10

Algebraic Collective Model (ACM)

)2()3()3(][)5(][)5()]5([ 55 SOSOSORSORUHW

The rigid rotor states of the BM subgroup chain

are approached with the Hamiltonians

3cosˆˆ H

.)(0

ˆˆˆˆˆ QQQH

in the ACM (beta rigid)

in the IBM

.~~ˆ dssdQ

O(6) quadrupole op.

the SO(5) Casimir inv.

mixes O(5) irreps but preserves O(6) and SO(3) quantum numbers

D. J. Rowe and G. Thiamova, Nucl. Phys. A 760, 59 (2005).

σ ∞ contraction

11

Algebraic Collective Model (ACM)

IBM-BM relationship in triaxial limit

ACM-IBM calculations in the beta-rigid limit

G. Thiamova, D. J. Rowe and M. A. Caprio Nucl. Phys. A 895, 20 (2012). 12

Generic triaxial case

G. Thiamova, D. J. Rowe and M. A. Caprio Nucl. Phys. A 895, 20 (2012). 13

IBM IBM

IBMACM

IBM-BM relationship in triaxial limit

Equlibrium γ = 30° case

A pure potential canonly be reachedin the limit

G. Thiamova, D. J. Rowe and M. A. Caprio Nucl. Phys. A 895, 20 (2012). 14

IBM IBM

IBMACM

IBM-BM relationship in triaxial limit

A more direct measure of thecloseness of the IBM andthe ACM results

The progression ofthe IBM effective γvalues to their ACMcounterparts

γeff non-zero even in theaxially-symmetric case !

G. Thiamova, D. J. Rowe and M. A. Caprio Nucl. Phys. A 895, 20 (2012). 15

IBM-BM relationship in triaxial limit

16

IBM-ACM beta-rigid calculations for N=10, 20 and 40

R1= Eγγ (K=0)/Eγ R2=Eγγ

(K=4)/Eγ

boson number

N 10 20 40 ACM

General triaxial R1 =2.82 R1 =3.40 R1 =3.29 R1 =3.54

R2=2.50 R2=3.09 R2=2.57 R2=2.62

G. Thiamova, D. J. Rowe and M. A. Caprio Nucl. Phys. A 895, 20 (2012).

IBM-BM relationship in triaxial limit

Applications (single and multi-phonon excitations)

17

Various model predictions for double-gamma states are controversial....

18

Applications (single and multi-phonon excitations)

SCCM, MPM, DDM etc. predict K=4 double gamma statesshould be widespread in well-deformed rare-earth region.

Prog. Theor. Phys., 76 (1986), 93 ; 78 (1987) 591NPA 487 (1988) 77

Various model predictions for double-gamma states are controversial....

19

Applications (single and multi-phonon excitations)

SCCM, MPM, DDM etc. predict K=4 double gamma statesshould be widespread in well-deformed rare-earth region.

Prog. Theor. Phys., 76 (1986), 93 ; 78 (1987) 591NPA 487 (1988) 77

QPNM predicts K=4 double gamma statesshould exist only in a few special cases (164Dy, 166Er, 168Er)

PRC 51 (1995) 551

Various model predictions for double-gamma states are controversial....

20

Applications (single and multi-phonon excitations)

Various model predictions for double-gamma states are controversial....

SCCM, MPM, DDM etc. predict K=4 double gamma statesshould be widespread in well-deformed rare-earth region.

Prog. Theor. Phys., 76 (1986), 93 ; 78 (1987) 591NPA 487 (1988) 77

QPNM predicts K=4 double gamma statesshould exist only in a few special cases (164Dy, 166Er, 168Er)

PRC 51 (1995) 551

Pure K=0 double gamma states should not exist.Their position depends on the anharmonicity..

NPA 383 (1982) 205Prog. Theor. Phys., 76 (1986), 93 ; 78 (1987) 591PRC 51 (1995) 551NPA 487 (1988) 77

21

Applications (single and multi-phonon excitations)

22

Anharmonicity of double-gamma vibrations-IBM1 perspective

Applications (single and multi-phonon excitations)

Anharmonicity of double-gamma vibrations-IBM1 perspective « Anharmonicities can only exist for finite boson number

and they are always small if only up to two-body interactions are considered. Thus anharmonicity may be

linked to triaxiality »

J. E. Garcia et al., NPA 637 (1998) 529

23

Applications (single and multi-phonon excitations)

« Anharmonicities can only exist for finite boson number and they are always small if only up to two-body

interactions are considered. Thus anharmonicity may be linked to triaxiality »

J. E. Garcia et al., NPA 637 (1998) 529

J. E. Garcia et al., PRC 61 (2000) 047305

In deformed rare-earth

region χ= -0.4-0.7no substantionalanharmonicity

observed

24

Anharmonicity of double-gamma vibrations-IBM1 perspective

Applications (single and multi-phonon excitations)

Quartic terms needed…J. E. Garcia et al., PRC 62 (2000) 064309

An IBM fit of 166Er

ACM beta-rigid calculations

M. A. Caprio, Phys. Lett. B 672 (2009) 396

25

Anharmonicity of double-gamma vibrations- ACM perspective

Applications (single and multi-phonon excitations)

26

Anharmonicity of double-gamma vibrations- ACM perspective

ACM beta-rigid calculations

M. A. Caprio, Phys. Lett. B 672 (2009) 396

With the onset of triaxiality (increasing ξ)the two-phonon energy anharmonicities evolvefrom slightly negative (Eγγ /Eγ smaller than 2) for

ξ =0 to positive (Eγγ /Eγ larger than 2).

Applications (single and multi-phonon excitations)

With the onset of triaxiality (increasing ξ)the two-phonon energy anharmonicities evolvefrom slightly negative (Eγγ /Eγ smaller than 2) for

ξ =0 to positive (Eγγ /Eγ larger than 2).

The anharmonicity of K=0+ state rises more rapidlythan that for K=4+ state.

27

Anharmonicity of double-gamma vibrations- ACM perspective

ACM beta-rigid calculations

M. A. Caprio, Phys. Lett. B 672 (2009) 396

Applications (single and multi-phonon excitations)

Axially symmetric regime

28

B

Applications (single and multi-phonon excitations)

Negligible anharmonicitiesAppearance of beta vib. state for smaller alpha

Full dynamics (including beta degree of freedom) ACM calculations

B B

D. J. Rowe et al., PRC 79 (2009) 054304

B

Axially symmetric regime Gamma excitation energies increase with increasing α

29SO(5) centrifugal stretching occurs for low-energy beta and gamma bands

B

B

B

Negligible anharmonicitiesAppearance of beta vib. state for smaller alpha

Applications (single and multi-phonon excitations)

D. J. Rowe et al., PRC 79 (2009) 054304

Gamma and beta excitation energies increase with increasing alpha, chi and kappa

(approaching the adiabalic limit of the BM)

B B B

Full dynamics (including beta degree of freedom) ACM calculations

D. J. Rowe et al., PRC 79 (2009) 054304

30

Rigid triaxial description of 106Mo K. Shizuma, Z. Phys. A 311 (1983) 71

XAxially symmetric description of 106Mo A. Guessous, PRL 75 (1995) 2280but soft with respect to triaxial deformation.

Small parameter κ required to describeharmonic double-gamma K=4 state

Gamma band appears higher in energy

106Mo

Applications (single and multi-phonon excitations)

B

K=4 harmonic double-gamma vibration

31

Applications (single and multi-phonon excitations)

106MoX

Small parameter κ required to describeharmonic double-gamma K=4 state

Gamma band appears higher in energy

B

Rigid triaxial description of 106Mo K. Shizuma, Z. Phys. A 311 (1983) 71

Axially symmetric description of 106Mo A. Guessous, PRL 75 (1995) 2280but soft with respect to triaxial deformation.

K=4 harmonic double-gamma vibration

Triaxial regime

32

G. s. centrifugal stretching for low-energygamma bands

Large anharmonicities...

Applications (single and multi-phonon excitations)

Centrifugal stretching effects occurs even in the gamma-stabilized situation

M. A. Caprio., PRC 72 (2005) 054323

B B B

1,0 1,2 1,4 1,6 1,8 2,01,8

2,0

2,2

2,4

2,6

2,8

3,0

3,2

3,4

E(K=0)/E

E(K=4)/E

B=70, =0.8, =3

33

K=4, K=0 anharmonic double-gamma vibrations identified in 166ErP. E. Garrett et al., PRL 78 (1997) 24

A low-lying beta vibration identified in 166ErP. E. Garrett et al., Phys. Lett. B 400 (1997) 250

G. Thiamova, Int. J. of Atomic and Nuclear Physics, to be published…

166Er

Applications (single and multi-phonon excitations)

B

34

166Er

G. Thiamova, Int. J. of Atomic and Nuclear Physics, to be published…

Applications (single and multi-phonon excitations)

B

K=4, K=0 anharmonic double-gamma vibrations identified in 166ErP. E. Garrett et al., PRL 78 (1997) 24

A low-lying beta vibration identified in 166ErP. E. Garrett et al., Phys. Lett. B 400 (1997) 250

35

Applications (single and multi-phonon excitations)164Dy

B

K=4 anharmonic double-gamma vibration identified in 164Dy

B(E2, 2173 2 2γ) uncertain

K=4 is not a pure double-gamma vibration…

F. Corminboeuf et al., PRC 56 (1997) R1201

36

164DyApplications (single and multi-phonon excitations)

B

K=4 anharmonic double-gamma vibration identified in 164Dy

B(E2, 2173 2 2γ) uncertain

K=4 is not a pure double-gamma vibration…

F. Corminboeuf et al., PRC 56 (1997) R1201

-ACM and IBM provide basically identical results for realistic boson numbers.

- ACM is a harmonic model in the axially symmetric regime.

-Large anharmonicities can be accomodated in the ACM in the triaxial regime and similarly in the IBM O(6) beta-rigid model. -A large amount of centrifugal stretching for low-lying gamma/beta bands. As a result, large anharmonicities (as observed in 166Er) are underestimated.

-More ACM calculations for divers Hamiltonians needed. -Details of the description or a fundamental failure of the quadrupole degrees of freedom?

« If the large amount of centrifugal stretching effect were shown to persistfor all reasonable ACM Hamiltonians (giving low-lying beta and gamma bands), itwould call into question the consistency of interpreting low-lying excited bands asbeta and gamma bands, when the corresponding centrifugal stretching effects areobserved to be small. »

Collaborators: D. J. Rowe, M. A. Caprio, J. L. Wood 37

Conclusions