Rotational states in even atomic nuclei

Z(5): critical point symmetry for the prolate to oblate nuclear shape phase transition

“…When considering the transition from γ = 0 o (prolate) to γ = 60 o (oblate), it is reasonable to expect that the triaxial region (0 o < γ < 60 o ) will be crossed, γ = 30 o lying in its middle. Indeed, there is experimental evidence supporting this assumption [14].2) For γ = 30 o the K quantum number (angular momentum projection on the bodyfixedẑ ′ -axis) is not a good quantum number any more, but α, the angular momentum projection on the body-fixedx ′ -axis is, as found [15] in the study of the triaxial rotator [16,17].3) Assuming an infinite well potential in the β-variable and a harmonic oscillator potential having a minimum at γ = 30 o in the γ-variable, the Z(5) model is obtained.On these choices, the following comments apply: 1) Taking γ = 30 o does not mean that rigid triaxial shapes are prefered. In fact, it has been pointed out [18] that a nucleus in a γ-flat potential [19] (as it should be expected for a prolate to oblate shape phase transition) oscillates uniformly over γ from γ = 0 o to γ = 60 o , having an average value of γ av = 30 o , and, therefore, the triaxial case to which it should be compared is the one with γ = 30 o .…”

“…2) For γ = 30 o the K quantum number (angular momentum projection on the bodyfixedẑ ′ -axis) is not a good quantum number any more, but α, the angular momentum projection on the body-fixedx ′ -axis is, as found [15] in the study of the triaxial rotator [16,17].…”

“…2) Taking an infinite well potential in β (while γ is fixed at 30 o ) corresponds to a transition from a triaxial vibrator to a triaxial rotator [15,16,17], in the same way that an infinite well potential in β in the X(5) model (in which γ = 0 o is assumed [2]) corresponds to a transition from a vibrator [U(5)] to a prolate rotator [SU (3)]. This point will be further discussed in Section 7.…”

“…When moving from SU(3) (prolate) to SU(3) (oblate) on the appropriate side of the extended [10] Casten triangle [20], one then identifies O(6) as the critical point. In the Z(5) model the same pool is crossed in a different way, by fixing γ = 30 o and moving from the triaxial vibrator (close to β = 0) to the triaxial rotator (far from β = 0) [15,16,17].…”

Triaxiality of neutron-rich Ge84,86,88 from low-energy nuclear spectra

“…The γ parameter ranges from 0 • (prolate shape) to 60 • (oblate shape), and maximum triaxiality occurs at 30 • . The rigid triaxial rotor model by Davydov and Filippov [2] considers a well-defined minimum for a certain value of γ in the potential energy surface while the model by Wilets and Jean [3] treats the potential independently of γ , called γ soft. More microscopic models, such as the shell model [4,5], the algebraic interacting boson model (IBM) [6], mean field approaches (e.g., Ref.…”