Algebraic collective model and its applications Gabriela Thiamová Laboratoire de Physique Subatomique et de Cosmologie Institut National Polytechnique de Grenoble France Sofia, October 8-10,

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 ∞ l imit to the BM dynamical symmetry chain D. J. Rowe and G. Thiamova, NPA 760 (2005), 59 Heisenberg-Weyl algebra for the BM spanned by q m, m=0, ±1, ±2, I Any developments in the U(5) limit of one model apply equally to the other IBM BM 3

The O(6) basis : -similarly to the U(5) limit, there is a close correspondence of the physics of the IBM in its O(6) limit with the BM in its rigid-beta gamma-soft Wilets-Jean limit. This correspondence is precise in the limit in which the IBM dynamical symmetry group contracts to the chain of the Wilets-Jean model N∞ contraction IBM BM 4 IBM-BM relationship in various dynamical symmetry limits

5 The O(6) basis : -similarly to the U(5) limit, there is a close correspondence of the physics of the IBM in its O(6) limit with the BM in its rigid-beta gamma-soft Wilets-Jean limit. This correspondence is precise in the limit in which the IBM dynamical symmetry 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 physics of the IBM in its O(6) limit with the BM in its rigid-beta gamma-soft Wilets-Jean limit. This correspondence is precise in the limit in which the IBM dynamical symmetry 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 of because 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 not related 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 of because 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 group for 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 chosen modified SU(1,1) representation – radial w. f. obtained by modifying the SO(5) centrifugal potential. Davidson basis – λ v = 1 + [(v+3/2) 2 + β 0 4 ] 1/2 Angular part SO(5) spherical harmonics char. by seniority v (ang. mom.) 10 Algebraic Collective Model (ACM)

The rigid rotor states of the BM subgroup chain are approached with the Hamiltonians in the ACM (beta rigid) in the IBM 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 ACM IBM-BM relationship in triaxial limit

Equlibrium γ = 30° case A pure potential can only be reached in the limit G. Thiamova, D. J. Rowe and M. A. Caprio Nucl. Phys. A 895, 20 (2012). 14 IBM ACM IBM-BM relationship in triaxial limit

A more direct measure of the closeness of the IBM and the ACM results The progression of the IBM effective γ values to their ACM counterparts γ eff non-zero even in the axially-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 R 1 = E γγ (K=0)/ E γ R 2 = E γγ (K=4)/ E γ boson number N ACM General triaxial R 1 =2.82 R 1 =3.40 R 1 =3.29 R 1 =3.54 R 2 =2.50 R 2 =3.09 R 2 =2.57 R 2 =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 Applications (single and multi-phonon excitations)

SCCM, MPM, DDM etc. predict K=4 double gamma states should be widespread in well-deformed rare-earth region. Prog. Theor. Phys., 76 (1986), 93 ; 78 (1987) 591 NPA 487 (1988) 77 Various model predictions for double-gamma states are controversial Applications (single and multi-phonon excitations)

SCCM, MPM, DDM etc. predict K=4 double gamma states should be widespread in well-deformed rare-earth region. Prog. Theor. Phys., 76 (1986), 93 ; 78 (1987) 591 NPA 487 (1988) 77 QPNM predicts K=4 double gamma states should exist only in a few special cases ( 164 Dy, 166 Er, 168 Er) PRC 51 (1995) 551 Various model predictions for double-gamma states are controversial Applications (single and multi-phonon excitations)

Various model predictions for double-gamma states are controversial.... SCCM, MPM, DDM etc. predict K=4 double gamma states should be widespread in well-deformed rare-earth region. Prog. Theor. Phys., 76 (1986), 93 ; 78 (1987) 591 NPA 487 (1988) 77 QPNM predicts K=4 double gamma states should exist only in a few special cases ( 164 Dy, 166 Er, 168 Er) PRC 51 (1995) 551 Pure K=0 double gamma states should not exist. Their position depends on the anharmonicity.. NPA 383 (1982) 205 Prog. Theor. Phys., 76 (1986), 93 ; 78 (1987) 591 PRC 51 (1995) 551 NPA 487 (1988) 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) 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) In deformed rare-earth region χ= no substantional anharmonicity 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) An IBM fit of 166 Er

ACM beta-rigid calculations M. A. Caprio, Phys. Lett. B 672 (2009) 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 evolve from slightly negative ( E γγ /E γ smaller than 2) for ξ =0 to positive ( E γγ /E γ larger than 2). Applications (single and multi-phonon excitations)

With the onset of triaxiality (increasing ξ) the two-phonon energy anharmonicities evolve from slightly negative ( E γγ /E γ smaller than 2) for ξ =0 to positive ( E γγ /E γ larger than 2). The anharmonicity of K=0 + state rises more rapidly than 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 anharmonicities Appearance 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) B

Axially symmetric regime Gamma excitation energies increase with increasing α 29 SO(5) centrifugal stretching occurs for low-energy beta and gamma bands B B B Negligible anharmonicities Appearance of beta vib. state for smaller alpha Applications (single and multi-phonon excitations) D. J. Rowe et al., PRC 79 (2009) 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)

30 Rigid triaxial description of 106Mo K. Shizuma, Z. Phys. A 311 (1983) 71 X Axially symmetric description of 106Mo A. Guessous, PRL 75 (1995) 2280 but soft with respect to triaxial deformation. Small parameter κ required to describe harmonic double-gamma K=4 state Gamma band appears higher in energy 106 Mo Applications (single and multi-phonon excitations) B K=4 harmonic double-gamma vibration

31 Applications (single and multi-phonon excitations) 106 Mo X Small parameter κ required to describe harmonic 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) 2280 but soft with respect to triaxial deformation. K=4 harmonic double-gamma vibration

Triaxial regime 32 G. s. centrifugal stretching for low-energy gamma 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) B B B

33 K=4, K=0 anharmonic double-gamma vibrations identified in 166Er P. E. Garrett et al., PRL 78 (1997) 24 A low-lying beta vibration identified in 166Er P. E. Garrett et al., Phys. Lett. B 400 (1997) 250 G.Thiamova, Int. J. of Atomic and Nuclear Physics, to be published… 166 Er Applications (single and multi-phonon excitations) B

Er 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 166Er P. E. Garrett et al., PRL 78 (1997) 24 A low-lying beta vibration identified in 166Er P. E. Garrett et al., Phys. Lett. B 400 (1997) 250

35 Applications (single and multi-phonon excitations) 164 Dy B K=4 anharmonic double-gamma vibration identified in 164Dy B(E2, γ ) uncertain K=4 is not a pure double-gamma vibration… F. Corminboeuf et al., PRC 56 (1997) R1201

Dy Applications (single and multi-phonon excitations) B K=4 anharmonic double-gamma vibration identified in 164Dy B(E2, γ ) 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 persist for all reasonable ACM Hamiltonians (giving low-lying beta and gamma bands), it would call into question the consistency of interpreting low-lying excited bands as beta and gamma bands, when the corresponding centrifugal stretching effects are observed to be small. » Collaborators: D. J. Rowe, M. A. Caprio, J. L. Wood 37 Conclusions