Analytic predictions for nuclear shapes, prolate dominance, and the prolate-oblate shape transition in the proxy-SU(3) model

Using a new approximate analytic parameter-free proxy-SU(3) scheme, we make simple predictions of shape observables for deformed nuclei, namely $\gamma$ and $\beta$ deformation variables, the global feature of prolate dominance, and the locus of the prolate-oblate shape transition. The predictions are compared with empirical results.

Figure 1

Values of the square root of the second order Casimir operator of SU(3), obtained from Eq. (2), vs particle number $M$, for different shells, obtained through proxy-SU(3) or through the particle-hole symmetry assumption. See Sec. 4 for further discussion.

Figure 2

(a) Values of the Elliott quantum number $\lambda$ vs the size of the shell, for different nucleon numbers $M$, as shown in Table 4. (b) Values of the square root of the second order Casimir operator of SU(3), obtained from Eq. (2) and related to $\beta$ through Eq. (3), vs the size of the shell, for different nucleon numbers $M$. See Sec. 5 for further discussion.

Figure 3

Proxy-SU(3) predictions for $\beta$, obtained from Eq. (3), compared with tabulated $\beta$ values [53] and also with predictions from relativistic mean field theory [52]. See Sec. 5 for further discussion.

Figure 4

Proxy-SU(3) predictions for $\beta$. See Sec. 5 for further discussion.

Figure 5

Proxy-SU(3) predictions for $\gamma$, obtained from Eq. (1), compared to empirical values extracted from ratios of the $\gamma$ vibrational bandhead to the first $2^{+}$ state. See Sec. 6 for further discussion.

Figure 6

Proxy-SU(3) predictions for $\gamma$, obtained from Eq. (1), compared with empirical values obtained from Eq. (6) [19, 56], as well as with predictions of Gogny D1S calculations [8]. The small light triangles for the heaviest Pt isotopes at ${30}^{\circ }$ are used because their experimental values for the ratio $R$ of Eq. (5) are a few percent below the limiting value for ${30}^{\circ }$ and hence $\gamma$ cannot be rigorously extracted; however there is abundant evidence that these isotopes have asymmetry values near ${30}^{\circ }$. The corresponding theoretical values are denoted by open circles. See Sec. 6 for further discussion.