Shape coexistence in excited $\mathrm{odd}-Z$ proton emitters ${}^{131-136}\mathrm{Eu}$

The excited states of the odd-$Z$ proton rich nuclei ${}^{131-136}\mathrm{Eu}$ are explored theoretically to look for the appearance of shape phase transitions and the phenomena of shape coexistence as a function of temperature and angular momentum. These nuclei remain well deformed with $\beta$ $\approx$ $0.15-0.2$ at the temperatures of the order of $1.5-2.0$ MeV and hence the potential candidates for the experimental probes of nuclear shapes. The phenomenon of shape coexistence is observed with coexisting oblate noncollective and rarely seen prolate noncollective shapes at low temperatures where the effects of rotation are predominant.

Figure 1

Equilibrium deformation $\beta$ vs. angular momentum $M$($\hslash$) at different temperatures for (a) ${}^{131}\mathrm{Eu}$, (b) ${}^{134}\mathrm{Eu}$, (c) ${}^{136}\mathrm{Eu}$ nuclei up to critical temperature. The numbers on the curves denote $T$ values.

Figure 2

$\beta$ vs. $M$($\hslash$) at $T=0.5$ MeV for ${}^{131-136}\mathrm{Eu}$. Shape phase transition from prolate noncollective (n-c) to oblate n-c with angular momentum $M$($\hslash$) is shown.

Figure 3

Free energy minimization curves as a function of deformation parameters ($\beta ,\gamma$) at $T=0.5$ for nearly yrast state of ${}^{131}\mathrm{Eu}$ for (a) $M=20.5\hslash$, (b) $M=42.5\hslash$, (c) $M=43.5\hslash$, (d) $M=50.5\hslash$. Only the curves corresponding to $\gamma$'s close to minima are shown. The rest of the $\gamma$'s at very high energy are avoided to keep the figure from looking too crowded. Shape coexistence is seen with prolate n-c ($\gamma =0^{\circ }$) and oblate n-c ($\gamma =-{180}^{\circ }$) shapes at $M=43.5\hslash$. Shape transition from prolate n-c (a) to coexisting shapes (b),(c) to oblate n-c (d) is shown.

Figure 4

$F$ minima moving from prolate n-c shape at (a) $M=44\hslash$ to coexisting shapes at (b) $48\hslash$ and (c) $50\hslash$ to (d) oblate n-c at $M=54\hslash$. The numbers on the curves denote $\gamma$ values.

Figure 5

Occupation probability $n_i^N$ vs. single particle energies ${\epsilon }_i^N$ for neutrons for the nucleus ${}^{134}\mathrm{Eu}$ at $T=0.5$ MeV and $M=44\hslash$ (a) $48\hslash$ (b), $50\hslash$ (c), and $54\hslash$ (d). The rearrangement of particles near the Fermi level with different shapes is seen. Shapes appearing to coexist in Figs. 4 and 4 with two $F$ minima with a small energy difference have a slightly deeper prolate n-c minima at $48\hslash$ and oblate n-c at $50\hslash$ which is manifested in $n_i$ curves.