Anharmonic double - γ vibrations in nuclei and their description in the interacting boson model

Double-γ vibrations in deformed nuclei are studied in the context of the interacting boson model with special reference to their anharmonic character. It is shown that large anharmonicities can be obtained with interactions that are (at least) of three-body nature between the bosons. As an example the γ vibrations of the nucleus 166 68Er98 are studied in detail. Typeset using REVTEX 1 Nuclear quadrupole shape oscillations can be of two types: β or γ vibrations [1]. The β vibration preserves axial symmetry and a one-quantum excitation gives rise to a K = 0 band where K is the projection of the angular momentum on the axis of symmetry of the nucleus. A γ vibration breaks axial symmetry and leads to a K = 2 band. Although their existence has been conjectured a long time ago [2], the observation and interpretation of β-vibrational K = 0 bands is still fraught with questions and difficulties. In contrast, γ-vibrational K = 2 bands are systematically observed in deformed nuclei and their properties are correspondingly better understood. Since single-γ vibrations are so well established, it is natural to search for double-γ vibrations and to examine their harmonic nature (i.e., whether they occur at twice the energy of the single vibration.) Two intrinsic K = 2 quanta can be combined parallel or antiparallel and hence lead to two bands: one with K = 0 and another with K = 4. The experimental identification of double-γ vibrations in deformed nuclei is difficult since they are expected to lie above the pairing gap and to mix with two-quasiparticle excitations, resulting in fragmentation and a corresponding reduction in the collectivity of the states. During the last few years, however, a steady improvement of experimental techniques has allowed the measurement of low-spin states in the energy region of interest [3–5]. This possibility has reopened the old debate on the existence of two-phonon (β or γ) vibrational states and their properties. Experiments have been reported recently pointing out the existence of double-γ vibrations in several deformed nuclei with a wide range of anharmonicities [6–11]. In particular, in Refs. [7,8] the first observation of the K = 0 and K = 4 double-γ states in one nucleus, Er, is reported. They are observed at 1.949 MeV and 2.029 MeV, respectively. This information is of great interest since it provides a stringent test of nuclear models; for instance, the Quasiphonon Nuclear Model (QPNM) predicts no K = 0 twophonon state below 2.5 MeV in Er [12]. Several calculations of two-phonon states, using either phenomenological or microscopic models, are available, particularly for Er and Er [12–17]. One of the models employed is the Interacting Boson Model (IBM) [18]. In the simplest version of this model, referred to as IBM-1, an even-even nucleus with n valence 2 nucleons is treated as a system of N = n 2 bosons with l = 0 (s bosons) or l = 2 (d bosons). In the usual formulation of the model only up to two-body interactions between the bosons are taken. What are the predictions of IBM with regard to two-phonon states in deformed nuclei and their (an)harmonic nature? It was pointed out some time ago by Bohr and Mottelson [13] that the IBM-1 is unable to accommodate large anharmonicities, as observed for instance in Er. Subsequently, it was shown that these can be described but require g bosons with l = 4 in addition to the s and d bosons (sdg-IBM) [16]. More recently, we reported a study of two-phonon states in IBM-1 treated in the intrinsic frame [19] and showed that the IBM-1 is a harmonic model in the limit of large boson number. Anharmonicities can only exist for finite boson number and they are always small if only up to two-body interactions are considered. It was also suggested that anharmonicity in the model is linked to triaxiality. Since it is known that IBM-1 with only up to two-body interactions cannot give rise to a stable triaxial minimum, the model’s capability for describing anharmonicities depends on the inclusion in the Hamiltonian of higher-order interactions, some of which are known to induce triaxial shapes [20,21]. In this article the relation between three-body interactions in IBM-1 and the anharmonicity of γ vibrations in deformed nuclei is investigated. Although the analysis presented is not exhaustive, it is shown that anharmonic behavior can be obtained with reasonable three-body interactions. As an example, the energy and E2 transition properties of the γ vibrations of the nucleus Er are studied in detail. In addition, the nature of the 02 state in the same nucleus, which has been the subject of an intense debate in the last few years [22–25], is briefly discussed. The Hamiltonian adopted in the following includes a quadrupole-quadrupole term, a rotational L̂ term, and three-body interactions between the d bosons, Ĥ = −κQ̂ · Q̂+ κL̂ · L̂+ ∑