Shape phase mixing in critical point nuclei

Spectral properties of nuclei near the critical point of the quantum phase transition between spherical and axially symmetric shapes are studied in a hybrid collective model which combines the $\gamma$-stable and $\gamma$-rigid collective conditions through a rigidity parameter. The model in the lower and upper limits of the rigidity parameter recovers the $\text{X}\left(5\right)$ and $\text{X}\left(3\right)$ solutions respectively, while in the equally mixed case it corresponds to the $\text{X}\left(4\right)$ critical point symmetry. Numerical applications of the model on nuclei from regions known for critical behavior reveal a sizable shape phase mixing and its evolution with neutron or proton numbers. The model also enables a better description of energy spectra and electromagnetic transitions for these nuclei.

Figure 1

The low-lying energy spectrum of ground and first two $\beta$ excited bands (a) and few $\mathrm{\Delta }K=0 B\left(E2\right)$ transition probabilities [(b) and (c)] are given as functions of the rigidity parameter $\chi$.

Figure 2

Theoretical ground state to ground state $E2$ transition probabilities normalized to the $2_g^{+}\to 0_g^{+}$ transition are compared with the available experimental data corresponding to all considered nuclei and with the $\text{X}\left(3\right)$ [25], $\text{X}\left(4\right)$ [24], and $\text{X}\left(5\right)$ [21] predictions. All the data are gathered from Nuclear Data Sheets [58, 59, 60, 61, 62, 63, 65, 66, 68, 69], with the exception of ${}^{176,178}\mathrm{Os}$ and ${}^{180}\mathrm{Pt}$ nuclei, whose experimental values are extracted from Refs. [38, 70] and [71] respectively. Also the $E2$ rate for the $2_g^{+}\to 0_g^{+}$ transition of ${}^{178}\mathrm{Pt}$ was taken from Ref. [72].

Figure 3

The fitted values of $\chi$ from Table 1 for the $N=90$ nuclei as well as the two $\chi =0$ values obtained for two other $N=90 \text{X}\left(5\right)$ nuclei are plotted as function of atomic number $Z$.

Figure 4

The fitted values of $\chi$ from Tables 2 and 3 for the Os and Pt isotopes as well as the two $\chi =1$ values corresponding to the $\text{X}\left(3\right)$ nuclei ${}^{172}\mathrm{Os}$ and ${}^{186}\mathrm{Pt}$ are plotted as functions of neutron number $N$.

Figure 5

The quadrupole deformation ${\beta }_2$, calculated with RMF and the ground state average of $\beta$ (3.19) scaled to reproduce the relativistic mean-field value for the nucleus with the best rms value, is given as a function of $Z$ for $N=90$ nuclei (a) and as functions of $N$ for Os (b) and Pt (c) isotopes.