A Solvable Model for Nuclear Shape Phase Transitions

There has been considerable interest recently in phase transitions that occur between some well-defined nuclear shapes, e.g. the spherical vibrator, the axially deformed rotor and the γ-unstable rotor, which are assigned to the U(5), SU(3) and 0(6) symmetries. These shape phase transitions occur through critical points of the IBM phase diagram and correspond to rapid structural changes. The first transition of this type describes transition form the spherical to the γ-unstable phase and has been associated with an E(5) symmetry. Later further critical point symmetries e.g. X(5) and Y(5) have also been proposed for transitions between other nuclear shape phases. In another application the chain of even Ru isotopes was considered from A 98 to 112 [2]. The parameters were extracted from a fit to the low-lying energy spectrum of each nucleus and were used to plot the corresponding potential. It was found that up to A =102 the potential is essentially an harmonic oscillator, while at A =104 a rather flat potential was seen, in accordance with the expected phase transition and E(5) symmetry there. With increasing A then the minimum got increasingly deeper and moved away from β = 0. We discuss the possibility of generalizing the formalism in two ways: first by including dependence on the 7 variable allowing for the approximate description of nuclei close to the X(5) symmetry, and second, including higher-lying energy levels in the quasi-exactly solvable formalism