ABSTRACT
The algebraic interacting boson model (IBM) introduced in 1975 by Arima and Iachello has become one of the three standard models of nuclear structure (the other two being the shell model of Mayer and Jensen and the geometric collective models of Bohr and Mottelson). In the first version of IBM, low-lying quadrupole collective states of heavy ( A > 100) even-even nuclei, with valence protons and neutrons occupying different oscillator shells, are described by replacing pairs of valence identical nucleons by ideal bosons and the bosons are assumed to be of two types, the s boson carrying angular momentum ℓπ = 0+
representing pairing and the d boson carrying angular momentum ℓπ = 2+
representing quadrupole deformation. With s and d boson numbers Ns and Nd, the total boson number N = Ns + Nd. The Hamiltonian is assumed to be one plus two-body preserving the boson number N and the total angular momentum carried by the bosons. The SGA for the resulting sdIBM-1 is U(6) and this Lie algebra generates SU(5) , SU(3) and SO(6) subalgebras giving quadrupole vibrational, rotational and γ-unstable spectra respectively [16]. These symmetry limits and the associated quantum numbers are shown in Figure 8.1. A very important development in IBM since 2000 is the recognition that the transitions from one IBM symmetry to other is a quantum phase transitions (QPT) and at the transition points there are critical point symmetries (for example E(5) and X(5) as shown in Figure 8.1) [268, 269]. In addition to SU(3) there is also a SU(3) symmetry generating the same spectrum but oblate shapes. This and also SO(6) play an important role in QPT in nuclei and in chaos studies [268]. See Appendix C for SU(3) and SO(6). Another important development is that 3-body interactions in IBM generate important new structures [270, 271, 272, 273].
