Quantum phase transitions in the consistent-QHamiltonian of the interacting boson model
Feng Pan1,3, Tao Wang1,2, Y-S Huo1 and J P Draayer3
1 Department of Physics, Liaoning Normal University, Dalian 116029,People’s Republic of China2 Department of Physics, Tonghua Teachers College, Tonghua, Jilin 134002,People’s Republic of China3 Department of Physics and Astronomy, Louisiana State University, Baton Rouge,LA 70803-4001, USA
Received 15 June 2008Published 3 October 2008Online at stacks.iop.org/JPhysG/35/125105
Abstract
Quantum phase transitional patterns in the whole parameter space of theconsistent-Q Hamiltonian in the interacting boson model are studied basedon an implemented Fortran code for the numerical computation of the matrixelements in the SU(3) Draayer–Akiyama basis. Results with respect to bothground and some excited states of the model Hamiltonian are discussed.Quantum phase transitional behavior in a variety of parameter situationsis shown. It is found that transitional behavior of excited states is morecomplicated. Pt isotopes are taken as examples in illustrating the prolate–oblate shape phase transition.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
Quantum phase transition (QPT) in atomic nuclei is an interesting subject in nuclear structuretheory. Most early theoretical analyses on nuclear shape (phase) transitions in the interactingboson model (IBM) were carried out in [1–3]. It is now widely accepted that the three limitingcases of the IBM correspond to three different geometric shapes of nuclei, referred to asspherical [vibrational, U(5)], axially deformed [SU(3)] and γ -soft [O(6)], respectively, whichis usually described in terms of the Casten triangle [4]. The U(5)–SU(3) transitional descriptionof the rare-earth nuclei reported in [5] included detailed results for most quantities of physicalinterest. Evidence for coexisting phases at low energy in spherically deformed transitionalnuclei was also analyzed by using a U(5)–SU(3) transitional theory, and the results show thatthe two phases coexist in a very narrow range of parameter space around the critical point[6] due to finite N effects [7]. Recently, since the discovery of the X(5) and E(5) symmetryin this region [8–10], the spherical to axially deformed shape transition has attracted further
J. Phys. G: Nucl. Part. Phys. 35 (2008) 125105 F Pan et al
Figure 1. An illustration of the extended Casten triangle.
attention [11–19]. The Q-consistent IBM Hamiltonian approximated by a scalar two-levelinteracting boson description was analyzed in [20]. Furthermore, it was pointed out thatthe oblate shape (phase) of nuclei corresponding to the SU(3) limit situation should also beincluded in the model to elucidate the prolate–γ -soft–oblate shape (phase) transition, whichenlarges the parameter space into the extended Casten triangle [21–23] shown in figure 1.
In this paper, we investigate quantum phase transitional patterns of some quantities in thewhole extended Casten triangle. The results were obtained based on an implemented Fortrancode for numerical diagonalization in the SU(3) Draayer–Akiyama basis [24, 25].
2. The consistent-Q Hamiltonian and its diagonalization
Though there are many different choices of the model Hamiltonian, the simple consistent-Qformalism can be used for characterizing all situations of transitional patterns in the IBM if onlyone- and two-body interactions are taken into consideration. The consistent-Q Hamiltoniancan be written as [26, 27]
H = c
(ηnd +
η − 1
4NQ(ζ ) · Q(ζ )
), (1)
where N is the total number of bosons in a system, Q(ζ ) = s†d + d†s + ζ(d† × d)(2) is thequadrupole operator, c > 0, 0 � η � 1, and −√
7/2 � ζ �√
7/2 are real parameters.Therefore, in figure 1, the U(5) spherical shape (phase) corresponds to the η = 1 case, theO(6)γ -soft shape (phase) corresponds to η = 0 and ζ = 0, the SU(3) prolate shape (phase)corresponds to η = 0 and ζ = −√
7/2, and the SU(3) oblate shape (phase) corresponds toη = 0 and ζ = √
7/2, respectively.Hamiltonian (1) is diagonalized under the U(6) ⊃ SU(3) ⊃ SO(3) basis spanned by
{|N(λμ)χL〉}, where χ is the Draayer–Akiyama quantum number [24, 25] used to resolve thebranching multiplicity occurring in the reduction of SU(3)↓SO(3) of which the basis vectorsare orthonormal, 〈N ′(λ′μ′)χ ′L′|N(λμ)χL〉 = δN ′Nδ(λ′μ′)(λμ)δχ ′χδL′L. Hence, the eigenstatesof (1) can be expressed as
|N,Lξ ; η, ζ 〉 =∑(λμ)χ
CLξ
(λμ)χ (η, ζ )|N(λμ)χL〉, (2)
where ξ is an additional quantum number used to distinguish different eigenstates with thesame angular momentum L and C
Lξ
(λμ)χ (η, ζ ) is the corresponding expansion coefficient.
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Table 1. Amount of CPU time needed to diagonalize Hamiltonian (1) for various N and L cases.
L = 0 L = 2 L = 3 L = 4
N = 10 13 s 1 min 59 s 23 s 3 min 36 sN = 20 1 min 3 s 10 min 21 s 2 min 25 s 24 min 9 sN = 30 2 min 48 s 28 min 26 s 6 min 32 s 1 h 9 min 37 sN = 40 6 min 32 s 59 min 52 s 14 min 12 s 2 h 41 min 57 sN = 50 13 min 36 s 1 h 50 min 27 s 26 min 26 sN = 60 26 min 55 s 46 min 34 sN = 70 49 min 57 s 1 h 17 min 26 s
A Fortran code was implemented for diagonalizing the Hamiltonian (1) with the resultantwavefunction (2) and then for calculating various physical quantities, such as E2 transitionrates and other quantities by using the analytical expressions for U(6) ⊃ SU(3) reducedmatrix elements of the s- and d-boson operators [28] together with Fortran subroutine codesfor evaluating SU(3) ⊃ SO(3) Wigner coefficients in the Draayer–Akiyama basis [24, 25].Since the Fortran code is specially implemented for diagonalizing Hamiltonian (1), one canuse it to study the transitional behavior of the system across the whole of the Casten trianglewithin a reasonable amount of CPU time. Table 1 shows the CPU time used for diagonalizingthe Hamiltonian (1) with arbitrary values for the parameters η and ζ on a Pentium4(R)2.4GHZ/256MB PC for various N and L cases, which suggests that the code, running on amachine of this type, can be used for any case up to N ≈ 100. From these results one can seethat the multiplicity of L in the SU(3) ⊃ SO(3) reductions (typically χmax = 1 for L = 0;χmax = 2 for L = 2; χmax = 4 for L = 4; etc) plays a key role in determining the complexityof the calculation.
3. Quantum phase transitional patterns in the whole extended Casten triangle
In this section, we report some physical quantities varying within the whole extended Castentriangle. Figure 2 shows the 0+
g ground-state energy E0/c as a function of η and ζ for differenttotal numbers of bosons N.
The fractional occupation probability for d-bosons in the ground state [12, 16], knownas the first-order parameter defined by ν1 = ⟨
0+g |nd |0+
g
⟩, as a function of η and ζ is plotted in
figure 3.A common feature of figures 2 and 3 is that the surface is symmetric with ζ reflection:
ζ ←→ −ζ , and is divided into three regions: a spherical (S) region with η > 0.5, a prolate(P) region with η < 0.5 and ζ < 0, and an oblate (O) region with η < 0.5 and ζ > 0. Theintersection or crossing point of the lines that divide these regions is exactly the triple pointcorresponding to the E(5) symmetry of a special Bohr Hamiltonian that describes the S, P andO phases as existing simultaneously [29]. The dividing line between adjacent regions is rathervaguely defined for small values of N, but with increasing N it becomes sharper and sharper.Along these lines there is a rather narrow band where the respective shapes coexist (two-phaseconfigurations) for small values of N. These bands of coexisting shapes grow narrower andnarrower with increasing N. For example, the 0+
g ground-state energy E0/c surface withinthese regions and the respective dividing lines are characterized in figure 4, which agrees withthe fact that the O–P, S–P and S–O are all first-order quantum phase transitions. The factthat there is a band of two-phase configurations along the lines dividing the regions appears
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J. Phys. G: Nucl. Part. Phys. 35 (2008) 125105 F Pan et al
(a)
(c)
(b)
Figure 2. 0+g ground-state energy E0/c as function of η and ζ for different total number of bosons
N, where (a) N = 5, (b) N = 10 and (c) N = 40.
to be a finite N effect; that is, the larger the N the narrower the band and the sharper thetransition.
Besides the quantities for the ground state, more quantities related to some low-lyingexcited states are also useful to reveal the nature of quantum shape (phase) transitions in nuclei.Figure 5 shows excitation energy of the 2+
1 state as a function of ζ and η for N = 5, N = 10and N = 26, respectively.
The isomer shift defined by
δ〈r2〉 = α0(⟨
2+1; η, ζ |nd |2+
1; η, ζ⟩ − ⟨
0+g; η, ζ |nd |0+
g; η, ζ⟩), (3)
where α0 is a constant, is also sensitive to the transitions, which is used to define [6, 16] thesecond-order parameter ν2 = δ〈r2〉/α0N . Figure 6 displays the isomer shift in the extendedCasten triangle.
Similarly, figure 7 shows B(E2; 2+
1 −→ 0+1
)/q2
2 as a function of η and ζ in the extendedCasten triangle, where the E2 operator is taken as simply
T (E2) = q2Q(ζ ) (4)
and q2 is the effective charge.
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J. Phys. G: Nucl. Part. Phys. 35 (2008) 125105 F Pan et al
(a) (b)
(c)
Figure 3. The first-order parameter surface in the extended Casten triangle, where (a) N = 5,(b) N = 10 and (c) N = 40.
Figure 4. 0+g ground-state energy E0/c as function of η and ζ for large N.
The spectroscopic quadrupole moment Q(2+
1
)of the first 2+
1 state can be taken as an orderparameter to measure the prolate–oblate shape (phase) transition. Q
(2+
1
)< 0 in the prolate
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J. Phys. G: Nucl. Part. Phys. 35 (2008) 125105 F Pan et al
(a) (b)
(c)
Figure 5. The excitation energy of 2+1 state in the extended Casten triangle for N = 5 (a), N = 10
(b) and N = 26 (c) , respectively.
(SU(3) limit) case, Q(2+
1
) = 0 in the γ soft (O(6) limit) or spherical (U(5) limit) case, andQ
(2+
1
)> 0 in the oblate (SU(3) limit) case. It can be seen from figure 8 for the N = 5, 10 and
26 case, respectively, that the spectroscopic quadrupole moment Q(2+
1
)of the first 2+
1 changesdrastically from prolate to oblate cases. In addition, the ratio R4/2 of the excitation energies ofthe first 4+
1 and 2+1 states can also be used as an order parameter. In the consistent-Q formalism,
R4/2 = 10/3 for both prolate (SU(3) limit ) and oblate (SU(3) limit) cases, R4/2 = 5/2 for theγ -soft (O(6) limit) case, and R4/2 = 2 for the spherical (U(5) limit) case. Figure 9 shows R4/2
for the N = 5, 10 and 26 cases, respectively. The results shown in figures 7–9 are consistentwith the early observations made in [7, 23].
In addition to the analysis of ground-state properties in the extended Casten triangle, wefound that the quantum phase transition behavior can also be observed in some excited stateswith the same angular momentum when η and ζ assume special values. Here, we only focuson 0+
2 and 0+3 level crossing–repulsion transition occurring near η = 0.3 due to a change in the
ζ parameter. As is clearly shown in figure 10, there is a level crossing point near η = 0.3 whenζ is exactly zero, while it becomes level repulsion when ζ < 0. Figure 10 only shows levelrepulsion curves with ζ = −0.1. Therefore, there is a clear N independent quantum phasetransition when ζ changes from ζ = 0 to ζ < 0. Such a quantum phase transition can also be
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J. Phys. G: Nucl. Part. Phys. 35 (2008) 125105 F Pan et al
(a) (b)
(c)
Figure 6. The isomer shift ν2 = δ〈r2〉/α0N as a function of ζ and η in the extended Castentriangle for N = 5 (a), N = 10 (b) and N = 26 (c), respectively.
observed from the B(E2) values associated with these two levels. Figure 11 shows the ratiosB
(E2; 0+
2 → 2+1
)/B
(E2; 2+
1 → 0+g
)and B
(E2; 0+
3 → 2+1
)/B
(E2; 2+
1 → 0+g
)varying with η
when ζ = 0 and ζ = −0.1. As can be seen, there is a clear crossing point near η = 0.18when ζ = 0, while there is no crossing when ζ = −0.1. Though these ratio changes near thecritical region are rather small, they can be observed for some transitional nuclei.
In order to show relative energy gaps among excited levels, an energy spectrum with oneof the control parameters fixed is plotted, which clearly demonstrates quantum shape (phase)transitional patterns in a two-dimensional plot. Because there are a lot of discussions aboutU(5) ←→ SU(3) transition with ζ = −
√7
2 , and O(6) ←→ U(5) transition with ζ = 0,
figure 12 shows some low-lying levels as functions of η with ζ = −√
74 , where there is a
lowest valley around η ∼ 0.4.Similarly, figure 13 shows the first ten low-lying levels corresponding to
0+g, 0+
2, 0+3, 2+
1, 2+2, 2+
3, 3+1, 4+
1, 4+2, 4+
3 states, as functions of ζ with η = 0.0. These low-lyinglevels are symmetric with ζ ←→ −ζ reflection. The critical point is at ζ = 0 correspondingto the O(6) limit.
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J. Phys. G: Nucl. Part. Phys. 35 (2008) 125105 F Pan et al
(a) (b)
(c)
Figure 7. B(E2; 2+1 −→ 0+
1)/q22 as a function of η and ζ in the extended Casten triangle, where
(a) N = 5, (b) N = 10 and (c) N = 26.
In order to investigate how the level scheme changes in the whole controlparameter space, figure 14 plots some low-lying energy surface cross section at η =0.0, 0.1, 0.2, 0.3, 0.4, 0.45, 0.5, 0.6 for N = 25, which clearly shows the correspondingquantum transitional behavior. These levels are also symmetric with respect to ζ ←→ −ζ
reflection. The critical point is at ζ = 0. With increasing η, sudden change in amplitudeof energy among these levels is gradually diminished. As shown in figure 14 for the case ofN = 25, the SU(3) − O(6) − SU(3) transition disappears when η is near 0.45. It can beinferred that the prolate–oblate shape (phase) transition will totally be diminished when N andη are large enough leading to a vibrational spectrum corresponding to a spherical shape.
Analysis for another quantity β2 related to the quadrupole deformation is also made,which is defined by [30, 31]
β2 = −sign(Q
(2+
1
)) 4π
3ZR20
[B
(E2; 0+
g −→ 2+1
)e2
]1/2
, (5)
where sign(Q
(2+
1
))is the sign of the quadrupole moment of the first 2+ state, Z is the proton
number, R0 is the mean radius of nucleus and e is the charge. In figure 15, the left panel
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J. Phys. G: Nucl. Part. Phys. 35 (2008) 125105 F Pan et al
(a) (b)
(c)
Figure 8. Q(2+1) in the extended Casten triangle for N = 5 (a), N = 10 (b) and N = 26 (c),
respectively.
shows β2 in the extended Casten triangle for N = 25, while the right one provides curves asfunctions of ζ for η = 0.0, 0.1, 0.2, 0.3, 0.35, 0.4, 0.45, 0.5, 0.6, respectively.
4. Application to even–even Pt isotopes
In this section, as an example, we use the consistent-Q Hamiltonian (1) to describe the even–even Pt isotopes. Most of the even–even Pt isotopes were studied using the consistent-QHamiltonian by McCutchan et al [32, 33]. It is well known that quadrupole moments of even–even 174−186Pt are greater than zero, while those of even–even 188−200Pt are less than zero. Thedescription studied in [32–33] was only based on prolate−γ -soft transitions corresponding tothe original Casten triangle. However, it is clear that there is an obvious prolate–γ -soft–oblateshape phase transition occurring around 186−188Pt. Therefore, it is more appropriate to describethese nuclei by using the parameters within the extended Casten triangle. In order to reducethe number of parameters, we first searched for the best set of parameters to fit the energyspectrum of each of these nuclei, and then looked for A-dependent empirical formulae forthese parameters. We found that two sets of mass A dependent results for the three parameters
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J. Phys. G: Nucl. Part. Phys. 35 (2008) 125105 F Pan et al
(a) (b)
(c)
Figure 9. R4/2 in the extended Casten triangle for N = 5 (a), N = 10 (b) and N = 26 (c),respectively.
Figure 10. 0+2 and 0+
3 level crossing–repulsion transition when N = 10.
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J. Phys. G: Nucl. Part. Phys. 35 (2008) 125105 F Pan et al
Figure 11. B(E2; 0+2 → 2+
1)/B(E2; 2+1 → 0+
g) and B(E2; 0+3 → 2+
1)/B(E2; 2+1 → 0+
g) asfunctions of η when ζ = 0, ζ = −0.1 and N = 10.
Figure 12. Some low-lying levels as functions of η with ζ = −√
74 for N = 10 and N = 25.
η, ζ and c yield better fits to the experiment data as long as the prolate–oblate phase transitionaround 186−188Pt is taken into consideration:
η ={−0.012A + 2.638 for A = 174 − 186,
0.0025A2 − 0.94A + 88.41 for A = 188 − 200,
ζ ={
0.05A − 10.0 for A = 174 − 186,
0.06 for A = 188 − 200,(6)
c ={
1.0 for A = 174 − 186,
−0.1A + 21.0 for A = 188 − 200.
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J. Phys. G: Nucl. Part. Phys. 35 (2008) 125105 F Pan et al
Figure 13. The first ten low-lying levels as functions of ζ with η = 0.0 for N = 10, N = 20 andN = 40, respectively.
The discontinuity in these parameters is due mainly to the prolate–oblate phase transitionaround 186−188Pt. A few of the low-lying levels that we fit are shown in figure 16. Thedeviation from the experimental results is reasonable given the fact that a simple three-parameter consistent-Q formula was used. Among these isotopes, levels for A � 194 havebeen well studied [32, 33] by using the same Hamiltonain with ζ < 0. However, in orderto describe the sign change in the quadrupole moments of even–even 174−200Pt, one mustenlarge the parameter space into the extended Casten triangle with −√
7/2 < ζ <√
7/2.Therefore, our fit is different from the situation considered in [32-33], in which only thestandard Casten triangle region with ζ < 0 was considered. Figure 17 shows results ofthe fit to the experiment with the quadrupole deformation parameter β2 given by (5). Theextension of the theory into the whole of the Casten triangle is a key to the quality of thesefits.
Some B(E2) values for 174−200Pt calculated by using the consistent-Q Hamiltonian withparameters given in (6) are compared with the corresponding experimental values in table 2,with the effective charge taken to be q2 = 1.85e. In table 2, the symbol (−) indicatesthat the corresponding experimental value has not been reported up to now. The overallfit seems to be in reasonable agreement to the data. It should be noted that the ratioB
(E2; 0+
2 −→ 2+1
)/B
(E2; 2+
1 −→ 0+g
)is 0.07 for 196Pt, while it is 0.8125 for 198Pt. This
change in the B(E2) ratio is consistent with the prediction shown in figure 11.
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J. Phys. G: Nucl. Part. Phys. 35 (2008) 125105 F Pan et al
Figure 14. Some low-lying levels as functions of ζ for N = 25 and η = 0.0, 0.1, 0.2, 0.3, 0.4,0.45, 0.5, 0.6.
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J. Phys. G: Nucl. Part. Phys. 35 (2008) 125105 F Pan et al
Table 2. Some E2 transition rates (in w.u.) of 174−200Pt, where experimental values are taken from[34–39].
Nucleus Transition Exp. Theory
174Pt 2+1 −→ 0+
g − 53.0176Pt 2+
1 −→ 0+g 87.0 63.6
178Pt 2+1 −→ 0+
g − 77.0180Pt 2+
1 −→ 0+g 153.0 93.5
182Pt 2+1 −→ 0+
g − 112.9184Pt 2+
1 −→ 0+g − 103.9
186Pt 2+1 −→ 0+
g 94.0 92.6188Pt 2+
1 −→ 0+g 82.0 94.7
190Pt 2+1 −→ 0+
g 56.0 78.4192Pt 2+
1 −→ 0+g 57.1 65.7
2+2 −→ 2+
1 109.0 79.32+
2 −→ 0+g 0.54 0.67
4+1 −→ 2+
1 89.0 91.53+
1 −→ 2+1 0.68 1.33
3+1 −→ 2+
2 102.0 63.63+
1 −→ 4+1 38.0 24.9
194Pt 2+1 −→ 0+
g 49.3 51.62+
2 −→ 0+g 0.0158 0.27
0+2 −→ 2+
1 1.34 1.760+
2 −→ 2+2 135.0 104.1
2+2 −→ 2+
1 89.0 64.84+
1 −→ 2+1 85.0 70.7
4+2 −→ 2+
1 0.22 0.00344+
2 −→ 4+1 20.0 33.2
196Pt 2+1 −→ 0+
g 40.57 41.072+
2 −→ 0+g 0.0158 0.029
2+3 −→ 0+
g 0.0025 0.05552+
3 −→ 0+2 5.0 6.03
0+2 −→ 2+
1 2.8 6.10+
3 −→ 2+1 <5.0 3.76
0+2 −→ 2+
2 18.0 98.10+
3 −→ 2+2 <0.41 0.89
4+1 −→ 2+
1 59.9 55.34+
2 −→ 2+1 0.56 0.047
4+2 −→ 2+
2 29.0 32.52+
3 −→ 4+1 0.13 3.0
198Pt 2+1 −→ 0+
g 32.0 28.82+
2 −→ 0+g 0.038 0.00021
2+3 −→ 0+
g 0.05 0.0540+
2 −→ 2+1 26.0 13.2
2+2 −→ 2+
1 37.0 36.82+
3 −→ 2+1 0.6 0.38
2+3 −→ 2+
2 2.2 3.764+
1 −→ 2+1 38.0 38.0
200Pt 2+1 −→ 0+
g − 21.5
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J. Phys. G: Nucl. Part. Phys. 35 (2008) 125105 F Pan et al
Figure 15. β2 in the extended Casten triangle for N = 25.
Figure 16. Some low-lying levels of 174−200Pt fitted by the consistent-Q Hamiltonian, where theexperimental data are taken from [34–39].
Figure 17. The quadrupole deformation parameter β2 of 174−200Pt fitted by the consistent-QHamiltonian with parameters given in (6), where the experimental values of β2 are determined bythe corresponding quadrupole moment Q(2+
1) and B(E2; 0+g −→ 2+
1) taken from [34-39] accordingto (5).
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J. Phys. G: Nucl. Part. Phys. 35 (2008) 125105 F Pan et al
5. Conclusions
In this paper, QPT patterns in the full parameter space of the consistent-Q Hamiltonian inthe IBM are studied based on an implemented Fortran code for numerical computation of thematrix elements in the SU(3) Draayer–Akiyama basis. Results with respect to both groundand some excited states of the model Hamiltonian are considered. Quantum phase transitionalbehavior in a variety of parameter situations is explored. The results suggest that narrowbands of coexisting two-phase regions near the three phase dividing lines for small values ofN may disappear when N becomes sufficiently large, a result that is in agreement with theoblate–prolate, spherical–prolate and spherical–oblate shape (phase) changes being first-ordertransitions. The narrow bands of two-phase coexisting regions seem to be due mainly to thefinite N effect. Also, our results show that the transitional behavior of excited states is morecomplex. As an example, the 0+
2 and 0+3 level crossing−− repulsion transition occurring near
η = 0.3, studied as a function of the ζ parameter, shows changes that may be observed insome transitional nuclei from related B(E2) ratio changes in this region. Finally, the 174−200Ptisotopes illustrate that the prolate–oblate shape (phase) transition is described fairly well bythe consistent-Q Hamiltonian with parameters given in the extended Casten triangle.
Acknowledgments
Support from the U S National Science Foundation (0140 300, 0500 291), the SoutheasternUniversities Research Association, the Natural Science Foundation of China (10 575 047, 10775 064), Liaoning Education Department Research Fund (20 060 464), Jilin EducationDepartment Research Fund (2007–403), and the LSU–LNNU joint research program(C192135) is acknowledged.
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