Selection of independent-particle configurations

In both CHF and CRMF calculations, the set of independent-particle configurations in nuclei around $^{131}$Ce was considered. The final sets used in additivity analysis consisted of 183 and 105 configurations in the CHF and CRMF variants, respectively. All ambiguous cases, due to crossings, convergence difficulties, etc., were removed from those sets. Since the CRMF calculations are more time-consuming than the CHF ones, the CRMF set is smaller. Nonetheless, the adopted CRMF set is sufficiently large to provide reliable results. To put things in perspective, in Ref. [5], where the CHF analysis of additivity principle in the SD bands of the $A\sim 150$ mass region was carried out, 74 calculated SD configurations were considered.

Every calculated product-state configuration was labeled using the standard notation in terms of parity-signature blocks $\left[ N_{+,+i}, N_{+,-i}, N_{-,+i}, N_{-,-i} \right]$, where $N_{\pi,r}$ are the numbers of occupied s.p. orbitals having parity $\pi$ and signature $r$. In addition, the s.p. states were labeled by the Nilsson quantum numbers and signature $[{\cal N}n_z\Lambda]\Omega^{r}$ of the active orbitals at zero frequency. The orbital identification is relatively straightforward when the s.p. levels do not cross (or cross with a small interaction matrix element), but it can become ambiguous when the crossings with strong mixing occur. In some cases, it was necessary to construct diabatic routhians by removing weak interaction at crossing points. Even with these precautions, a reliable configuration assignment was not always possible; the exceptional cases were excluded from the additivity analysis. Clearly, the likelihood of the occurrence of level crossings is reduced when the s.p. level density is small, e.g., in the vicinity of large shell gaps.

Large deformed energy gaps develop at high rotational velocity for $Z$=58 and $N$=73 (see Figs. 2 and 3). Therefore, the lowest SD band ($\nu i_{13/2}$ band) in $^{131}$Ce is a natural choice for the highly deformed core configuration in the $A$$\sim$130 mass region. The additivity analysis was performed at a large rotational frequency of $\hbar\omega$=0.65MeV. This choice was dictated by the fact that (i) at this frequency the pairing is already considerably quenched, and (ii) no level crossings appear in the core configuration around this frequency (cf. Figs. 2 and 3). Moreover, at this frequency, the lowest neutron $i_{13/2}$ intruder orbital already appears below the $N$=73 neutron gap (see Fig. 2). The choice of an odd-even core, strongly motivated by its doubly closed character at large deformations/spins, does not impact the additivity scheme, which is insensitive to the selection of the reference system.

The highly deformed core configuration in $^{131}$Ce ([18,19,18,18]$_n\otimes$ [14,14,15,15]$_p$) has the following orbital structure:

$$\left\vert {\mbox{\bf core}} \right> = \left\vert {\mbox{\bf core}} \right>_\nu \otimes \left\vert {\mbox{\bf core}} \right>_\pi \equiv$$

$$\begin{array}{r} (\nu (1i_{13/2}) {\mbox{\bf 6}}_1^{-i} ) \\ [-1... ...\pm i})^2 \\ [1pt] (~\cdots~) \left\vert{\mbox{\bf0}}\right>_{\nu} \end{array} \otimes \begin{array}{r} (\pi (1g_{9/2}) [404]9/2^{\pm i})^2 \\ %[1pt] ... ...pm i})^2 \\ [1pt] (~\cdots~) \left\vert{\mbox{\bf0}}\right>_{\pi}, \end{array}$$

where dots denote the deeply bound states and $\left\vert{\mbox{\bf0}}\right>_{\nu}$ and $\left\vert{\mbox{\bf0}}\right>_{\pi}$ are the neutron and proton vacua, respectively. The spherical subshells from which the deformed s.p. orbitals emerge (cf. Fig. 1) are indicated in the front of the Nilsson labels.

The Nilsson orbital content of an excited configuration is given in terms of particle and hole excitations with respect to the core configuration through the action of particle/hole operators with quantum labels corresponding to the occupied or emptied Nilsson orbitals. The character of the orbital (particle or hole) is defined by the position of the orbital with respect to the Fermi level of the core configuration. It is clear from Fig. 2 that the neutron states $\nu [523]7/2^{\pm i}$, $\nu [411]1/2^{\pm i}$, $\nu [413]5/2^{\pm i}$, $\nu [541]1/2^{\pm i}$, $\nu [532]5/2^{\pm i}$ and $\nu 6^{-i}_1$ have hole character, while $\nu 6^{+i}_2$, $\nu 6^{-i}_3$, $\nu [530]1/2^{\pm i}$, $\nu [402]5/2^{\pm i}$, $\nu [532]3/2^{\pm i}$, and $\nu [514]9/2^{\pm i}$ have particle character. In a similar way, the proton orbitals $\pi [541]3/2^{\pm i}$, $\pi [422]3/2^{\pm i}$, $\pi [301]1/2^{\pm i}$, and $\pi [420]1/2^{\pm i}$ and $\pi [404]9/2^{\pm i}$ can be viewed as holes, while $\pi [532]5/2^{\pm i}$, $\pi [411]3/2^{\pm i}$, $\pi [541]1/2^{\pm i}$, and $\pi [413]5/2^{\pm}$ have particle character (see Fig. 3).