Generalized Partial Dynamical Symmetries in Nuclear Spectroscopy

Explicit forms of IBM Hamiltonians with a generalized partial dynamical O(6) symmetry are presented and compared with empirical data in 162Dy. A dynamical symmetry corresponds to a situation in which the Hamiltonian is written in terms of the Casimir operators of a chain of nested algebras G1 ⊃ G2 ⊃ . . . ⊃ Gn , (1) and has the following properties. (i) Solvability. (ii) Quantum numbers related to irreducible representations (irreps) of the algebras in the chain. (iii) Symmetry-dictated structure of wave functions independent of the Hamiltonian’s parameters. The merits of a dynamical symmetry are self-evident, however, in most applications to realistic systems, one is compelled to break it. Partial dynamical symmetry (PDS) corresponds to a particular symmetry breaking for which some (but not all) of the above virtues of a dynamical symmetry are retained. Two types of partial symmetries were encountered so far. The first type correspond to a situation for which part of the states preserve all the dynamical symmetry. This is the case for the SU(3) PDS found in the IBM-1 [1, 2] and the Symplectic Shell Model [3, 4], and for the F-spin PDS in the IBM-2 [5]. The corresponding PDS Hamiltonians have a subset of solvable states with good symmetry while other eigenstates are mixed. A second type of partial symmetries correspond to a situation for which all the states preserve part of the dynamical symmetry. This occurs, for example, when the Hamiltonian preserves only some of the symmetries Gi in the chain (1) and only their irreps are unmixed [6, 7]. In this case there are no analytic solutions, yet selected quantum numbers (of the conserved symmetries) are retained. In the present contribution we show that it is possible to combine both types of partial symmetries, namely, to construct a Hamiltonian for which part of the states have part of the dynamical symmetry. We refer to such a structure as a generalized partial dynamical symmetry [8]. Partial symmetry of the second kind was recently considered in [7] in relation to the chain U(6)⊃ O(6) ⊃ O(5) ⊃ O(3) . (2) The Hamiltonian employed has twoand three-body interactions of the form H1 = κ0P † 0 P0 +κ2 ( Π(2)×Π(2) (2) ·Π(2) . (3) The κ0 term is the O(6) pairing term defined in terms of monopole (s) and quadrupole (d) bosons, P† 0 = d † ·d†−(s†)2. It is diagonal in the dynamical symmetry basis |[N],σ ,τ,L〉 of Eq. (2) with eigenvalues κ0(N−σ)(N +σ +4). The κ2 term is composed only of the O(6) generator: Π(2) = d†s + s†d̃, which is not a generator of O(5). Consequently, H1 cannot connect different O(6) irreps but can induce O(5) mixing. The eigenstates have good σ but not good τ quantum numbers. To consider a generalized O(6) PDS, we introduce the following IBM-1 Hamiltonian, H2 = h0P † 0 P0 +h2P † 2 · P̃2 . (4) The h0 term is identical to the κ0 term of Eq. (3), and the h2 term is defined in terms of the boson pair P† 2,μ = √ 2s†d† μ + √ 7(d†d†)(2) μ with P̃2,μ = (−)P2,−μ . The latter term can induce both O(6) and O(5) mixing. Although H2 is not an O(6) scalar, it has an exactly solvable ground band with good O(6) symmetry. This arises from the fact that the O(6) intrinsic state for the ground band |c; N〉 = (N!)(bc)|0〉 , bc = (d† 0 + s †)/ √ 2 , (5) has σ = N and is an exact zero energy eigenstate of H2. Since H2 is rotational invariant, states of good angular momentum L projected from |c; N〉 are also zero-energy eigenstates of H2 with good O(6) symmetry, and form the ground band of H2. It follows that H2 has a subset of solvable states with good O(6) symmetry (σ = N), which is not preserved by other states. All eigenstates of H2 break the O(5) symmetry but preserve the O(3) symmetry. These are precisely the required features of a generalized partial dynamical symmetry as defined above for the chain of Eq. (2). In Fig. 1 we show the experimental spectrum of 162Dy and compare with the calculated spectra of H1 and H2. The spectra display rotational bands of an axially-deformed nucleus, in particular, a ground band (K = 01) and excited K = 21 and K = 02 bands. An L ·L term was added to both Hamiltonians, which contributes to the rotational splitting but has no effect on wave functions. The parameters were chosen to reproduce the excitation energies of the 2+K=01 , 2+K=21 and 0+K=02 levels. The O(6) decomposition of selected bands is shown in Fig. 2. For H2, the solvable K = 01 ground band has σ = N and exhibits an exact L(L + 1) splitting. The K = 21 band is almost pure with only 0.15% admixture of σ = N − 2 into the dominant σ = N component. The K = 02 band has components with σ = N (85.50%), σ = N−2(14.45%), and σ = N−4(0.05%). Higher bands exhibit stronger mixing, e.g., the K = 23 band shown in Fig. 2, has components with σ = N (50.36%), σ = N−2(49.25%), σ = N−4(0.38%), and σ = N−6(0.01%). The O(6) mixing in excited bands of H2 depends critically on the ratio h2/h0 in Eq. (4) or equivalently on the ratio of the K = 02 and K = 21 bandhead energies. In contrast, all bands of H1 are pure with respect to O(6). Specifically, the K = 01,21,23 bands shown in Fig. 2 have σ = N and the K = 02 band has σ = N −2. In this case the diagonal κ0 term in Eq. (3) simply shifts each band as a whole in accord with its σ assignment. All eigenstates of both H1 and H2 are mixed with respect to O(5). To gain more insight into the underlying band structure of H2 we perform a bandmixing calculation by taking its matrix elements between large-N intrinsic states. The