algebraic collective model and its applications gabriela thiamov laboratoire de physique subatomique...
DESCRIPTION
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 3TRANSCRIPT
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.)
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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