0^{+} states in the large boson number limit of the Interacting Boson Approximation model

states in the large boson number limit of the Interacting Boson Approximation model

Dennis Bonatsos    E. A. McCutchan    R. F. Casten
Abstract

Studies of the Interacting Boson Approximation (IBA) model for large boson numbers have been triggered by the discovery of shape/phase transitions between different limiting symmetries of the model. These transitions become sharper in the large boson number limit, revealing previously unnoticed regularities, which also survive to a large extent for finite boson numbers, corresponding to valence nucleon pairs in collective nuclei. It is shown that energies of states grow linearly with their ordinal number in all three limiting symmetries of IBA [U(5), SU(3), and O(6)]. Furthermore, it is proved that the narrow transition region separating the symmetry triangle of the IBA into a spherical and a deformed region is described quite well by the degeneracies , , , while the energy ratio turns out to be a simple, empirical, easy-to-measure effective order parameter, distinguishing between first- and second-order transitions. The energies of states near the point of the first order shape/phase transition between U(5) and SU(3) are shown to grow as n(n+3), in agreement with the rule dictated by the relevant critical point symmetries resulting in the framework of special solutions of the Bohr Hamiltonian. The underlying partial dynamical symmetries and quasi-dynamical symmetries are also discussed.

Interacting Boson Approximation model, geometric collective model, shape/phase transitions, order parameter
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address=Institute of Nuclear Physics, National Centre for Scientific Research “Demokritos”, GR-15310 Aghia Paraskevi, Attiki, Greece

address=Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA

address=Wright Nuclear Structure Laboratory, Yale University, New Haven, Connecticut 06520-8124, USA

The Interacting Boson Approximation (IBA) model [1], describing collective phenomena in atomic nuclei in terms of and bosons (of angular momentum 0 and 2, respectively), is known to possess an overall U(6) symmetry, containing three different dynamical symmetries, U(5), SU(3), and O(6), corresponding to near-spherical (vibrational), axially symmetric prolate deformed (rotational), and soft with respect to axial asymmetry (-unstable) nuclei respectively. These limiting symmetries are shown at the vertices of the symmetry triangle [2] of the model, shown in Fig. 1.

Figure 1: IBA symmetry triangle with the three dynamical symmetries. The critical point models E(5) and X(5) are placed close to the phase transition region (slanted lines). The solid curve indicates the Alhassid-Whelan arc of regularity. Adopted from Ref. [3].

An energy functional can be obtained [4] in the classical limit of the model, through the use of the coherent state formalism [5, 6]. Studying this energy functional in the framework of catastrophe theory one can see [7] that a first order phase transition (in the Ehrenfest classification) is predicted to occur between the limiting symmetries U(5) and SU(3), while a second order phase transition is expected between U(5) and O(6). We refer to these transitions as shape/phase transitions. A narrow shape coexistence region is then predicted [4] in the symmetry triangle of the IBA, separating the spherical and deformed phases. The shape coexistence region shrinks into the point of second order phase transition as the U(5)-O(6) line is approached, as shown in Fig. 1.

Shape/phase transitions have been considered recently in the framework of the geometric collective model [8] as well, in which the collective variables and are used. The second order transition between U(5) and O(6) has been described by the E(5) critical point symmetry [9], using in the Bohr Hamiltonian [10] a potential independent of and having the shape of an infinite square well potential in , while the first order transition between U(5) and SU(3) has been described by the X(5) critical point symmetry [11], using in the Bohr Hamiltonian a potential of the form , with having again the shape of an infinite square well potential in , and being a steep harmonic oscillator centered around . E(5) and X(5) are shown in Fig. 1, close to the points of the second and first order phase transitions of the IBA, respectively.

The IBA model predictions have also been studied using different measures of chaotic behavior. It has been found that chaotic behavior prevails over most of the symmetry triangle, with the noticeable exception of two regions, in which highly regular behavior is observed [12]. One of them is located along the U(5)-O(6) line, and its existence is due to the O(5) symmetry known to survive along the whole line [13]. The other regular region is connecting U(5) and SU(3) through a narrow path within the triangle, called the Alhassid–Whelan arc of regularity [12] (also shown in Fig. 1), the symmetry implying its existence being yet unknown.

SU(3) O(6)
Irrep (,) () Irrep (,) () Irrep () ()
(2,0) 0 () 0
(2-4,2) 1 (-2) 1
(2-8,4) (4-6)/(2-1) (2-6,0) (4-3)/(2-1) (-4) 2
(2-12,6) (6-15)/(2-1) (2-10,2) (6-10)/(2-1) (-6) 3-(3/)
(2-16,8) (8-28)/(2-1) (2-14,4) (8-21)/(2-1) (-8) 4-(8/)
Table 1: Irreducible representations (irreps) of SU(3) and O(6) and the corresponding energy of the excited bandheads. stands for the boson number, . Adopted from Ref. [3].

States with zero angular momentum are of particular interest, since centrifugal effects are absent, facilitating the detection of underlying symmetries. In the U(5) limit of the IBA, the energies of states increase linearly with the number of bosons, corresponding to their phonon number. In the SU(3) limit, the energies of bandheads are determined by the eigenvalues of the second order Casimir invariant of SU(3), the results shown in Table 1. In the O(6) limit, they are determined by the second order Casimir invariant of O(6), the results also shown in Table 1. From Table 1 it is clear that in the limit of large boson numbers , the energies of bandheads in all three dynamical symmetries of the IBA grow linearly, .

Figure 2: (Left) Line of degeneracy between the and levels for = 10, 40, 100, and 250 in the IBA triangle. (Right) Line of degeneracy between the and levels for = 250 (top) and between the and levels for = 250 (bottom) in the IBA triangle. The dashed lines denote the critical region in the IBA obtained in the large limit from the intrinsic state formalism. Adopted from Ref. [17].

IBA numerical calculations have been performed using the recently developed code IBAR [14, 15], which can handle large boson numbers. The standard two-parameter IBA Hamiltonian [16] has been used, depending on the parameters and . Examining the degeneracy , a hallmark of the X(5) critical point symmetry, we find [17] that its locus in the IBA triangle is a straight line, falling withing the coexistence region for large boson numbers, as seen in Fig. 2. Similar results are obtained for the degeneracies and , also shown in Fig. 2.

Figure 3: The ratio ()/() as a function of for three values of for (a) = 15 and (b) = 100. The inset to (a) shows the corresponding behavior for the order parameter  [18]. Adopted from Ref. [17].
Figure 4: (a) Experimental ()/() ratio as a function of neutron number for the Nd, Sm, Gd, and Dy isotopes. (b) Same for the Xe and Ba isotopes. For smaller neutron numbers, the state was taken in the ratio if its (2) decay was consistent with the = state. This corresponds to = 74 in Xe and = 76,78 in Ba. Valence (hole) neutron number increases to the left. Adopted from Ref. [17].

Plotting the ratio [17], related to the first of the degeneracies mentioned above, we see in Fig. 3 that it exhibits the behavior expected [18] for an order parameter of a first (second) order phase transition for (). It is remarkable that experimental data around the isotones, which are known to be the best empirical examples of X(5) [19, 20, 21, 22, 23], shown in Fig. 4, exhibits a behavior very similar to the one expected for a first order phase transition, while data around Ba, the best example of E(5) [24, 23], shows the behavior expected for a second order transition.

E(5) Z(5) X(5) Norm IBA-Norm Z(4) Norm X(3) Norm
0 0 0 0 0 0 0 0 0
3.03 3.91 5.65 1.0 1.0 2.95 1.0 2.87 1.0
7.58 9.78 14.12 2.50 2.48 7.60 2.57 7.65 2.67
13.64 17.61 25.41 4.50 4.62 13.93 4.71 14.34 5.00
21.22 27.39 39.53 7.00 7.13 21.95 7.43 22.95 8.00
30.31 39.12 56.47 10.00 9.85 31.65 10.72 33.47 11.67
Table 2: (Left) Energies of states in the E(5), Z(5), and X(5) models. Energies on the left are in units of () = 1.0, while in the column Norm, in units () = 1.0. The normalized results are identical for each of the models. The column IBA-Norm gives the normalized energies for a large IBA calculation near the critical point. (Middle) Same for the Z(4) model. (Right) Same for the X(3) model. Adopted from Ref. [3].
Model D Model D
E(5) + 5
X(5) 5 Z(5) 5
X(3) 3 Z(4) 4 1
Table 3: Order , dimension, , of the model space and for states in the geometrical models E(5), X(5), Z(5), Z(4), and X(3). is the spin of the level, , and is the wobbling quantum number [8] which is zero for states.

Now we turn attention to energies of states within the critical point symmetries E(5) and X(5), mentioned above, as well as within Z(5) [25], a similar solution of the Bohr Hamiltonian, also using an infinite square well potential in , with . As seen in Table 2, although the relevant energies look different in each model when normalized to the energy of the state, they become identical if normalized to the energy of [3]. Moreover, they increase as . Similarly, energies of states within Z(4) [26] [similar to Z(5), but with fixed to ] increase as . Within X(3) [27] [similar to X(5), but with fixed to 0], they increase as , as also shown in Table 2. These results can be easily interpreted [3] by taking into account the order of the Bessel functions appearing as eigenfunctions in these models, given in Table 3, as well as the fact that the spectra of the Bessel functions are found to increase approximately as , this result being exact only for [3]. Consideration of the second order Casimir operator of E() [28, 26], the Euclidean algebra in dimensions, shows [3] that the present situation is a partial dynamical symmetry [29] of Type I [30], in which some of the states (the states in the present case) preserve all the relevant symmetry. The recent conjecture [31] of a partial SU(3) dynamical symmetry underlying the Alhassid–Whelan arc of regularity is also receiving attention.

It is a nontrivial result [3] that the IBA near the critical point of the first order transition also yields energies of states increasing as , i.e. in the same way as critical point symmetries based on infinite square well potentials in 5 dimensions (degrees of freedom) predict, as shown in Table 2.

In conclusion, states in the large boson number limit of the IBA exhibit many interesting properties, as well as degeneracies to non-zero angular momentum states, which invite further investigations into determining of the symmetries underlying these regularities.

Acknowledgements.
Work supported in part U.S. DOE Grant No. DE-FG02-91ER-40609 and under Contract DE-AC02-06CH11357.

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