Structure evolution and phase transition in odd-mass nuclei

The evolution of level structures due to the unique parity orbitals $g_{9/2}$, $h_{11/2}$, and $i_{13/2}$ in odd-mass nuclei from Zn to Am is studied within a unified framework, by correlations between ratios of excitation energies in both odd-mass nuclei and their even-even core nuclei. These plots reveal regularities that can be understood in terms of the particle-plus-rotor model, as evolutions along its three limiting coupling schemes: weak coupling, decoupling, and strong coupling, and transitions between them. Peculiar transitions between the decoupling and strong coupling schemes are found in both $i_{13/2}$ structures of neutron-odd nuclei and $h_{11/2}$ structures of proton-odd nuclei, at neutron numbers around 90 and 70, respectively. These are correlated with the critical shape phase transitions from vibrator to rotor from the even-even nuclei in the same regions and are characterized as critical phase transitions too. This behavior is corroborated with a nonmonotonic behavior of the differential variation of the two-neutron separation energies in the same nuclear regions.

Figure 1

Colored contours approximately showing the regions where the three coupling schemes of the PRM are expected to occur in the $R_{j+4/j+2}$ vs $R_{4/2}$ representation. Equality of the two quantities is indicated by the dashed line. The three crosses indicate the theoretical limits of the strong coupling for the three UPOs (see discussion in Sec. 2).

Figure 2

Correlation between the energy ratios $R_{j+4/j+2}$ of UPO favored sequences and $R_{4/2}$ of the even-even cores. The dashed line indicates equality of the two quantities. The symbols disclose the UPO and the type (proton or neutron, particle or hole type) of the odd particle. See text for the meaning of the numbered contours.

Figure 3

Panels (a)–(c) are the same as Fig. 1, but separately for each of the three UPOs. Panels (d)–(f) present evolution trajectories of some selected isotopic chains from the respective left-side panels (a)–(c).

Figure 4

Similar to Fig. 1, but for the signature splitting index $R_j^s$ (see text). Symbol meanings are as in Fig. 1. The three numbered contours correspond to those from Fig. 1.

Figure 5

Detail of Fig. 1 highlighting the nuclei with unpaired hole-type particles which realize a decoupling scheme. For each isotope chain the first and last points are marked with the mass number.

Figure 6

Different correlations illustrating the critical shape phase transition between the decoupling and strong coupling schemes of $\nu i_{13/2}$ structures (contours 1 and 3; see Figs. 2 and 3). (a) Rapid change of $E(j+2)$ evolution around the value $E_c\left(2_1^{+}\right)\approx 140$ keV corresponding to the X(5) critical point of the core nuclei. (b) Turning point behavior in the $E(j+4)$ vs $E(j+2)$ graph. (c) Evolution of the signature splitting index $R_j^s$. The dashed lines in (a)–(c) are just drawn through the data points. Filled symbols mark the nuclei closest to the critical point (see text). (d) Derivative of the trajectory from (b) (see text for details).

Figure 7

Evolution of the derivative $d{S}_{2n}$ of the two-neutron separation energy $S_{2n}$ for (a) even-even nuclei, (b) neutron-odd nuclei, and (c) proton-odd nuclei, respectively, from the $Z=54–74$ region. The dashed circles indicate the region of shape phase transition at $N\approx 90$ (see text).

Figure 8

Similar to Fig. 6, but for the transition region from decoupling to strong coupling [contours 1 and 3 in Fig. 2; see also plots (b) and (e) in Fig. 3] of $\pi h_{11/2}$ structures (nuclei from La to Eu). The analog of Fig. 6 is missing here because data on the signature splitting index are too scarce. The nuclei for which $\pi h_{11/2}$ structure data are available (Cs to Tb isotopes) have neutron numbers below 82, the collectivity increasing with decreasing $N$.

Figure 9

Similar to Fig. 6, but for $h_{11/2}$ structures of the proton-hole odd-mass nuclei evolving on trajectories within contour 2 in Fig. 2; see also Figs. 3 and 3. The dashed line in (a) denotes $E(j+2)=E\left(2^{+}\right)$, the dotted line in (b) is for the strong coupling limit $E(j+4)=2.286E(j+2)$, and the dashed line in (c) is also for the strong coupling limit, $R_s^j=0.54$. The insets in (b) and (c) show details of the evolution pattern around $N=98$ of Lu isotopes with $N$ from 88 to 106, which are similar to those of Tm, Ta, and Re isotopes.

Figure 10

Same as Fig. 9, but for $h_{11/2}$ structures of the neutron-hole odd-mass nuclei evolving on trajectories within contour 2 in Fig. 2; see also Figs. 3 and 3.