Gamma-rigid solution of the Bohr Hamiltonian for gamma = 30 degrees compared to the E(5) critical point symmetry

A gamma-rigid solution of the Bohr Hamiltonian for gamma = 30 degrees is derived, its ground state band being related to the second order Casimir operator of the Euclidean algebra E(4). Parameter-free (up to overall scale factors) predictions for spectra and B(E2) transition rates are in close agreement to the E(5) critical point symmetry, as well as to experimental data in the Xe region around A=130.

In the present work we derive an exact solution of the Bohr Hamiltonian for γ = 30 o , by "freezing" γ (as in Ref. [9]) to this value and taking into account only four degrees of freedom (β and the Euler angles). In accordance to previous terminology, this solution will be called Z (4). It turns out that the Z(4) spectra and B(E2) transition rates are quite similar to the E(5) ones, while in parallel the ground state band of Z(4) is related to the Euclidean algebra E(4), thus offering the first clue of connection between critical point symmetries and Lie algebraic symmetries, in addition to the case of the E(5) model [1]. Experimental examples of Z(4) seem to appear in the Xe region around A = 130.
The Z(4) solution will be introduced in Section 2 and its ground state band will be related to E(4) in Section 3. Numerical results and comparisons to E(5) and experiment will be given in Section 4, while discussion of the present results and plans for further work will appear in Section 5.

The Z(4) model
In the model of Davydov and Chaban [9] it is assumed that the nucleus is rigid with respect to γ-vibrations. Then the Hamiltonian depends on four variables (β, θ i ) and has the form [9] where β and γ are the usual collective coordinates [7], while Q k (k = 1, 2, 3) are the components of angular momentum and B is the mass parameter. In this Hamiltonian γ is treated as a parameter and not as a variable. The kinetic energy term of Eq. (1) is different from the one appearing in the E(5) and X(5) models, because of the different number of degrees of freedom treated in each case (four in the former case, five in the latter).
Introducing [1] reduced energies ǫ = (2B/h 2 )E and reduced potentials u = (2B/h 2 )U, and considering a wave function of the form Ψ(β, θ i ) = φ(β)ψ(θ i ), where θ i ( i = 1, 2, 3) are the Euler angles, separation of variables leads to two equations In the case of γ = π/6, the last equation takes the form This equation has been solved by Meyer-ter-Vehn [10], the eigenfunctions being with where D(θ i ) denote Wigner functions of the Euler angles, L are the eigenvalues of angular momentum, while µ and α are the eigenvalues of the projections of angular momentum on the laboratory fixedẑ-axis and the body-fixedx ′ -axis respectively. α has to be an even integer [10]. Instead of the projection α of the angular momentum on thex ′ -axis, it is customary to introduce the wobbling quantum number [10,11] n w = L − α, which labels a series of bands with L = n w , n w + 2, n w + 4, . . . (with n w > 0) next to the ground state band (with n w = 0) [10]. The "radial" Eq. (2) is exactly soluble in the case of an infinite square well potential

Eq. (2) becomes a Bessel equation
with Then the boundary condition f (β W ) = 0 determines the spectrum, where x s,ν is the sth zero of the Bessel function J ν (z). The eigenfunctions are where the normalization constant c is determined from the condition β W 0 β 3 φ 2 (β)dβ = 1. The notation for the roots has been kept the same as in Ref. [2], while for the energies the notation E s,nw,L will be used. The ground state band corresponds to s = 1, n w = 0. This model will be called the Z(4) model.
The calculation of B(E2)s proceeds as in Ref. [8], the only difference being that the integrals over β have the form since the volume element in the present case corresponds to four dimensions instead of five. A brief discussion of the interrelations among various triaxial models is now in place. In the original triaxial model of Davydov and Filippov [12], the Hamiltonian contains only a rotational term [the second term in Eq. (1)], and is analytically soluble for all values of γ. In contrast, the Hamiltonian of Davydov and Chaban [9] contains both a kinetic energy term [the first term in Eq. (1)] and a rotational term, and is solved numerically. Meyerter-Vehn [10] has shown that triaxial Hamiltonians including both a kinetic energy term and a rotational term are analytically soluble in the special case of γ = 30 o . In the present Z(4) case an analytical solution of the Davydov and Chaban Hamiltonian is obtained for the special case of γ = 30 o , as implied by Meyer-ter-Vehn.

Relation of the ground state band of Z(4) to E(4)
The ground state band of the Z(4) model is related to the second order Casimir operator of E(4), the Euclidean group in four dimensions. In order to see this, one can consider in general the Euclidean algebra in n dimensions, E(n), which is the semidirect sum [13] of the algebra T n of translations in n dimensions, generated by the momenta and the SO(n) algebra of rotations in n dimensions, generated by the angular momenta symbolically written as E(n) = T n ⊕ s SO(n) [14]. The generators of E(n) satisfy the commutation relations From these commutation relations one can see that the square of the total momentum, P 2 , is a second order Casimir operator of the algebra, while the eigenfunctions of this operator satisfy the equation in the left hand side of which the eigenvalues of the Casimir operator of SO(n), ω(ω + n − 2) appear [15]. Putting and Eq. (16) is brought into the form the eigenfunctions of which are the Bessel functions f (r) = J ν (kr) [16]. The similarity between Eqs. (19) and (7) is clear. The ground state band of Z(4) is characterized by n w = 0, which means that α = L. Then Eq. (8) leads to ν = L/2 + 1, while Eq. (18) in the case of E(4) gives ν = ω + 1. Then the two results coincide for L = 2ω, i.e. for even values of L. One can easily see that this coincidence occurs only in four dimensions.

Numerical results and comparisons to E(5) and experiment
The lowest bands of the Z(4) model are given in Table 1. The notation L s,nw is used. All levels are measured from the ground state, 0 1,0 , and are normalized to the first excited state, 2 1,0 . The ground state band is characterized by s = 1, n w = 0, while the even and the odd levels of the γ 1 -band are characterized by s = 1, n w = 2, and s = 1, n w = 1 respectively, and the β 1 -band is characterized by s = 2, n w = 0. These bands are also shown in Fig. 1, labelled by (s, n w ).
The similarity between the spectra and B(E2) values of Z(4) and E(5), for which extensive numerical results can be found in Ref. [17], can be seen in Figs. 2(a) and 2(b), where the spectra of the ground state band and the β 1 band, as well as their intraband B(E2)s are given. One can easily check that the similarity extends to interband transitions between these bands as well, for which the selection rules in the two models are the same.
The main difference between Z(4) and E(5) appears, as expected, in the γ 1 band, the spectrum of which is shown in Fig. 2(c). The predictions of the two models for the odd levels practically coincide, while the predictions for the even levels differ, since in the E(5) model the levels are exactly paired as (3,4), (5,6), (7,8), . . . , as imposed by the underlying SO(5)⊃SO(3) symmetry [1,17], while in the Z(4) model the levels are approximately paired as (4,5), (6,7), (8,9), . . . , which is a hallmark of rigid triaxial models [12]. The latter behaviour is never materialized fully [18], but it is known [19] that γ-unstable models and γ-rigid models yield similar predictions for most observables if γ rms of the former equals γ rigid of the latter, a situation occuring in the Ru-Pd, Xe-Ba (below N = 82), and Os-Pt regions.
Predictions of the Z(4) model are compared to existing experimental data for 128 Xe [20], 130 Xe [21], and 132 Xe [22] in Fig. 3. The reasonable agreement observed is in no contradiction with the characterization of these nuclei as O(6) nuclei [19], since, as mentioned above, the predictions of γ-unstable models (like O(6) [23]) and γ-rigid models (like Z(4)) for most observables are similar if γ rms of the former equals γ rigid of the latter.

Discussion
In the present work an exact solution of the Bohr Hamiltonian with γ "frozen" to 30 o , called Z(4), is obtained. Spectra and B(E2) transition rates of Z(4) resemble these of the critical point symmetry E(5), while the ground state band of Z(4) is related to the Euclidean algebra E(4), thus offering a first clue of connection between critical point symmetries and Lie algebras, in addition to the case of the E(5) model [1]. Empirical evidence for Z(4) in the Xe region around A = 130 has been presented.
It should be emphasized, however, that neither the similarity of spectra and B(E2) values of Z(4) to these of the E(5) model, nor the coincidence of the ground state band of Z(4) to the spectrum of the Casimir operator of the Euclidean algebra E(4) clarify the algebraic structure of the Z(4) model, the symmetry algebra of which has to be constructed explicitly, starting from the fact that γ is fixed to 30 o . The fact that the Bohr Hamiltonian for γ = 30 o possesses "accidentally" a symmetry axis (the body-fixedx ′ -axis) has been early realized [24]. This "accidental" symmetry should also serve as the starting point for clarifying the symmetry underlying other solutions of the Bohr Hamiltonian obtained for γ = 30 o [8,25,26].