Partial dynamical symmetry and odd-even staggering in deformed nuclei

Partial dynamical symmetry (PDS) is shown to be relevant for describing the odd-even staggering in the $\gamma$-band of $^{156}$Gd while retaining solvability and good SU(3) symmetry for the ground and $\beta$ bands. Several classes of interacting boson model Hamiltonians with SU(3) PDS are surveyed.

A comparison with the experimental spectrum and E2 rates of 156    γ-band is quite poor. The latter display an odd-even staggering with pronounced deviations from a rigid-rotor pattern, indicative of a triaxial behavior. This effect can be visualized by plotting the quantity [3]  One is thus confronted with the need to select suitable higher-order terms that can break the DS in the γ-band but preserve it in the ground and β bands. These are precisely the defining properties of a partial dynamical symmetry (PDS) [4]. The essential idea is to relax the stringent conditions of complete solvability, so that only part of the eigenspectrum retains all the DS quantum numbers. Various types of bosonic and fermionic PDS are known to be relevant to nuclear spectroscopy [4][5][6][7][8][9][10][11][12][13]. In the present contribution we demonstrate the relevance of PDS to the odd-even staggering in the γ-band of 156 Gd [13].
The method to construct Hamiltonians with PDS is based on an expansion in terms of tensors which annihilate prescribed set of states [12,14] In the present study, the tensors involve n-boson creation and annihilation operatorsB † α ≡B † [n](λ,µ)K; m andB α , with definite character under the SU(3) chain (1), which satisfŷ Interactions as in Eq. (5) can be added to the Hamiltonian (2) without destroying solvability of part of its spectrum. For n = 2, we find two such SU(3) tensorŝ with (λ, µ) = (0, 2) and angular momentum = 0, 2. For n = 3, we find six SU (3) and the fact that W 2m (8b) and W 3m (8d) satisfy Eq. (6) and γ band with higher-lying excited bands. Other approaches advocating the coupling of the γ band to the β band [15] or to the ground band [16] fail to describe the odd-even staggering in 156 Gd. Higher bands exhibit larger SU(3) mixing and their wave functions are spread over many SU(3) irreps, as shown for the K = 0 3 band in Fig. 3. This complex SU(3) decomposition is in marked contrast to the SU(3)purity of the ground (K = 0 1 ) and β (K = 0 2 ) bands.
It should be emphasized that the PDS results for the γ band are obtained without altering the good agreement for the ground and β bands, already achieved with the SU(3) DS calculation. This is further illustrated with the E2 transitions in 156 Gd. The observed B(E2) values between ground, β, and γ bands are shown in Table 1 and compared to the results of the SU(3) DS and PDS calculations. The E2 transitions between ground and β bands can be calculated analytically, and remain valid in SU (3) PDS. Transitions involving γ-band members are different in SU(3) DS and PDS, and are computed numerically for the latter. It is seen from Table 1 that the mixing of the γ band with higher-lying excited bands improves the agreement with the data in most cases.
Another class of Hamiltonians with SU(3) PDS exists which has solvable members of γ k (K = 2k) bands |[N](2N − 4k, 2k)K = 2k, L , with energies E DS (3). This follows from the structure of the relevant Hamiltonian, and the fact that P 0 (7a) and Λ (8e) satisfy Eq. (6) and Two-body Hamiltonians of this class have been shown to play a role in diverse phenomena, including spectroscopy of rare-earth nuclei [5][6][7], quantum phase transitions [17,18] and mixed regular and chaotic dynamics [18,19]. The two classes of SU(3)-PDS Hamiltonians demonstrate the increase in flexibility obtained by generalizing the concept of DS to PDS. In fact, in the IBM more than half of all possible interactions have an SU(3) PDS [13].
In summary, we have presented several classes of IBM Hamiltonians with SU(3) PDS, and obtained an improved description of signature splitting in the γ band of 156 Gd. The analysis serves to highlight the merits gained by using the notion of PDS as a tool for selecting higher-order terms in systems where a prescribed symmetry is not obeyed uniformly. On one hand, the PDS approach allows more flexibility by relaxing the constraints of an exact DS. On the other hand, the PDS picks particular symmetry-breaking terms without destroying results previously obtained with a DS for a segment of the spectrum. The non-solvable states can experience strong symmetry-breaking. These virtues can be exploited in studying the role of higherorder terms in collective Hamiltonians and in attempts to extend beyond-mean-field methods to heavy nuclei.
This work was done in collaboration with J.E. García-Ramos (Huelva) and P. Van Isacker (GANIL) and is supported by the Israel Science Foundation.