Critical point symmetry for the spherical to triaxially deformed shape phase transition

Article history: Received 18 September 2015 Received in revised form 15 October 2015 Accepted 28 October 2015 Available online 3 November 2015 Editor: J.-P. Blaizot The critical point T(5) symmetry for the spherical to triaxially deformed shape phase transition is introduced from the Bohr Hamiltonian by approximately separating variables at a given γ deformation with 0◦ ≤ γ ≤ 30◦. The resulting spectral and E2 properties have been investigated in detail. The results indicate that the original X(5) and Z(5) critical point symmetries can be naturally realized within the T(5) model in the γ = 0◦ and γ = 30◦ limit, respectively, which thus provides a dynamical connection between the two symmetries. Comparison of the theoretical calculations for 148Ce, 160Yb, 192Pt and 194Pt with the corresponding experimental data is also made, which indicates that, to some extent, possible asymmetric deformation may be involved in these transitional nuclei. © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.


Introduction
Critical point symmetries (CPSs) in nuclear structure have attracted a lot of attention [1][2][3][4][5][6][7][8][9][10], since these models may provide parameter-free (up to an overall scale) predictions about the structural properties of nuclei in the phase transitional region [11,12]. These CPSs include, for example, the critical point of the spherical to γ -unstable shape phase transition E(5) [1], the critical point of the spherical to axially deformed shape phase transition X(5) [2], and the critical point of the prolate to oblate shape phase transition Z(5) [4] (also serving as the CPS for the shape transition from the spherical to the triaxial deformation at γ = 30 • ), etc., which have been widely confirmed in experiment [13][14][15][16][17][18][19][20]. Generally, these CPS models are constructed from the Bohr Hamiltonian by separating the collective β and γ variables, and by making different assumptions about the potentials in β and γ . Specifically, separation of variables in the Bohr Hamiltonian for the X(5) model is achieved by assuming γ = 0 • [2], which represents an axially-symmetric situation, while that in the Z(5) model has been achieved by separating variables at γ = 30 • [4] corresponding to the maximally-triaxial situation. Moreover, in both models the potential in β is assumed to be an infinite square well, while the * Corresponding author.
The purpose of this work is to propose a Bohr Hamiltonian with a potential in γ having minimum at any given γ value with γ ∈ [0 • , 30 • ]. The resulting model, which is called T(5), may provide a connection between the X(5) and Z(5) CPSs [2,4], which thus serves as the CPS for the transition from the spherical to a triaxial shape with 0 • ≤ γ ≤ 30 • since the X(5) and Z(5) symmetries can be used to describe the transitions from the spherical to axially deformed shape at γ = 0 • [2] and the spherical to triaxial-deformed shape at γ = 30 • [4], respectively. It should be mentioned that the spherical to triaxial-deformed shape phase transition can be alternatively analyzed within the interacting boson model [21] by introducing high-order terms in the Hamiltonian [22][23][24][25] since a rigid triaxial structure with a given γ deformation can principally be defined in such cases [24,25].

The model
The original Bohr Hamiltonian [26] is written as where β and γ are the deformation variables, B is the collective mass parameter, and L k (k = 1, 2, 3) are the projections of the angular momentum on the body-fixed k-axis. In the present case, it is assumed [1][2][3][4][5] that (2) in which the potential V (β) is taken to be an infinite square well [2] with whereas the potential V (γ ) is taken to be harmonic around γ = γ e [4] with As the potential has a minimum at γ = γ e , the rotational term of Eq. (1) is approximated as Notably, the approximation used in (5) was also adopted in the X(5) and Z(5) models, but with γ e = 0 • and γ e = 30 • respectively.
By introducing the reduced energy = 2B E/h 2 and reduced po- [2,4], the corresponding Schrödinger equation can be separately written as where r L is the eigenvalue of R L , β 2 is the average of β 2 over η(β) [2], and = β + γ . The total wave function can be constructed as The rotational wave function ϕ L M,s (θ) can be expanded in terms of the Wigner D-functions with where D L M,K (θ i ) is the Wigner D-function, the expanding coefficients C L s,K are determined from the eigenvalue equation (9), and s is used to label the s-th eigenstate for given L and M, which is given as It should be noted that eigenvalue equation (9) can be analytically solved at γ e = 0 • or γ e = 30 • , in which the values of r L are respectively given as [2,4] and with α being the projection of the angular momentum on the body-fixed 1-axis. The resulting models just correspond to the X(5) and Z(5) CPSs, respectively. On the other hand, it should be noted that the last term (13) contributes nothing in the case of K = 0 [2] but an infinite number in the case of K = 0. To avoid such a situation, this term in the original X(5) model was absorbed into Eq. (7) [2]. Therefore, the γ e = 0 situation shown in (13) only applies to the K = 0 case, while K = 0 cases have already been discussed in [2,27]. Generally, the r L values, which are the eigenvalues of a triaxial rotor Hamiltonian, can only be numerically solved from (9) when 0 • < γ e < 30 • . But for a few of the lowest L values, r L s can be solved analytically from (9) as shown in [28,29], which are given as Substituting F (β) = β 3/2 η(β) and z = βk β with k β = √ β [2,4], one can transform Eq. (6) inside the well into the Bessel equation Then the boundary condition η(β W ) = 0 determines the eigenval- and the eigenfunction where x ξ,v is the ξ -th zero of the Bessel function J v (z), and the normalization constants c ξ,v are determined by For the γ -part of the T(5) model, sin(3γ ) in Eq. (7) is approximately replaced by sin(3γ e ) for simplicity since we consider the harmonic oscillator potential having a minimum at γ = γ e [4] and the system behaving a small-amplitude oscillations around the equilibrium point γ e . Then, Eq. (7) can approximately be rewritten with c = 2BC/h 2 . By taking γ = γ − γ e [4], the above equation can be further transformed into the form which is very similar to the corresponding equation in the Z (5) CPS [4] except for that the potential here is taken as u(γ ) ∝ ), nγ = 0, 1, 2, . . . (28) and where the normalization constant is given as One thus obtains the general expression of the total energy E(ξ, s, L, nγ ) where E 0 , A , and B are the corresponding parameters [4]. B(E2) transition rates can be calculated by taking the quadrupole operator T E2 u = tβ[D (2) u,0 (θ i ) cos(γ ) where t is a scale factor. Specifically, B(E2) transition rates are given as In the calculation of reduced matrix elements in (34), the same approximation used for the Hamiltonian, γ ≈ γ e , is also used for the quadrupole operator defined in (33) [2]. Then the integral over γ only contributes δ nγ i nγ f due to the orthonormality of φ nγ (γ ) [4], which indicates that the E2 transitional rate calculated in the present scheme will not be affected by the form of φ(γ ) no matter φ(γ ) is directly solved from (7) or after some approximations solved from (26). The integral over β takes the form while the integral over the Euler angles θ i can be obtained by using the formula involving three Wigner functions [30]. The final result is given as Since the rotational function ϕ L M,s (θ i ) with L = 0, 2, 3, 5 defined in (10) can be analytically solved from (9) [29], one can get some analytical expressions of B(E2) values, such as with [29] e = − . (38) It is clear that the results for the ground band and γ band are given as those with ξ = 1 and nγ = 0, while the results for the β band are those with ξ = 2 and nγ = 0. In the following, we will only consider the low-lying states in the bands with nγ = 0.

Numerical examination
As mentioned previously, the T(5) CPS provides a possible dynamical connection between the X(5) and Z(5) CPSs. To demonstrate the connection, some typical energy ratios and B(E2) ratios in the related models are listed in Table 1. As shown in Table 1, the T(5) results in the γ e = 0 • and γ e = 30 • limits correspond exactly to those of the X(5) and Z(5) CPSs, respectively.
Specifically, the ratios E L 1 /E 2 1 and E L β /E 2 1 in the T(5) model decrease monotonously from the X(5) limit (γ e = 0 • ) to the Z(5) limit (γ e = 30 • ). In addition, it is shown in Table 1 that approximate degeneracy of 6 1 and 0 2 level appears in the T(5) model for all γ e values indicating that the approximate degeneracy may be regarded as a signal of the T(5) CPS. Table 1 Typical energy ratios and B(E2) ratios for the ground band and β band calculated in the T(5) model with different γ e compared with the corresponding quantities in X (5) and Z(5) CPSs [2,4]. On the other hand, the analysis shown in [31,32] indicates that all 0 + bandhead energies, E 0 n , in the CPS models should obey a universal law. However, it can be observed from Table 1 that the ratio E 0 2 /E 2 1 decreases with the increasing of γ e . As a further analysis, the ratios E 0 n /E 2 1 and E 0 n /E 0 2 calculated from the T(5) CPS with several typical γ e values are shown in Fig. 1. As shown in panel (a) of Fig. 1, the bandhead energies E 0 n with n = 2, 3, 4, 5 all monotonically increase with the decreasing of γ e if they are normalized to E 2 1 . However, if these bandhead energies are normalized to E 0 2 as shown in panel (b), they all keep to be a constant independent of γ e , which indeed coincide with the rule [31,32] where A is an overall scale factor independent of γ e . In fact, one can deduce from (15) and (22) Fig. 2, in which the vibrator results are obtained from the U(5) limit (vibrational limit) of the interacting boson model [21], which is generally considered as an algebraic vibrator, while those of the triaxial rotor are obtained from the Davydov-Fillipov rotor model [28], of which the Hamiltonian is proportional to R L defined in (5) [26]. It can be clearly seen from Fig. 2 that the results obtained from the T(5) model all fall in between those from the vibrator and the triaxial rotor with any γ deformation, which indicates that the T(5) model is indeed qualified to be regarded as the CPS of the shape phase transition from the vibrator to the triaxial rotor.
In order to check the sensitiveness of the T(5) model on γ e further, several typical quantities as functions of γ e are calculated, which are shown in Fig. 3. It can be observed in Fig. 3 that the T(5) model generally produces lower energy ratios and larger B(E2) ratios in comparison to the corresponding results of the rigid triaxial rotor model. It is thus confirmed that the T(5) model may behave as a soft triaxial rotor [27]. Particularly, the B(E2) ratio B(E2 : 3 γ → 2 γ )/B(E2; 2 1 → 0 1 ) in the rotor model remains to be a constant independent of γ e , while this quantity in the T(5) model decreases monotonically as a function of γ e . In addition, it is shown that the energy ratio E 2 γ /E 2 1 may be taken as an indicator of γ e value for the T(5) CPS because this quantity is very sensitive to γ e , which is also relatively easily to be measured. Once γ e is fixed by fitting the experimental energy ratio E 2 γ /E 2 1 , the whole spectral structure is determined by the model up to an overall scale factor. It should be emphasized that all the energy ratios and B(E2) ratios in the T(5) model shown in Fig. 3 at γ e = 30 •  coincide with those in the Z(5) CPS [4]. However, the quantities related to the γ band in the X(5) CPS cannot be derived from the present T(5) description in the γ e = 0 • limit. For example, the ratio E 2 γ /E 2 1 tends to infinity in the T(5) model as γ → 0 • , while it is a harmonic strength-dependent quantity in the X(5) model [2,27]. In order to reproduce the X(5)-like γ band in the T(5) model at γ e = 0 • , one has to regroup the quantities in (6) and (7) by absorbing the term K 2 sin 2 (γ e )| γe →0 • into (7) as mentioned previously, which will be further explored in our future work.

Comparison to experiment
As shown in previous section, the T(5) model may provide a dynamical connection between the X(5) CPS [2] and Z(5) CPS [4], which thus offers a more flexible description of nuclei in between the two CPSs, such as either those supposed to be the candidates of the X(5) CPS with the neutron number N = 90 [33], or the candidates of the Z(5) CPS in the Pt isotopic chain [4]. To test the validity of the model, 148