Effective angular momenta $j^{\mbox{\rm\scriptsize {eff}}}_{\alpha}$ (s.p. alignments)

In this section, we evaluate and interpret the effective s.p. contributions to the total angular momentum. Table 3 displays effective s.p. angular momenta $j^{\mbox{\rm\scriptsize {eff}}}_{\alpha}$ for the s.p. orbitals of interest. The relative uncertainties in calculated $j^{\mbox{\rm\scriptsize {eff}}}_{\alpha}$ values are on average larger than those for effective s.p. quadrupole moments. This is due to the fact that, on the mean field level, polarization effects pertaining to the angular momentum are more complex than those for quadrupole moments: they involve not only shape changes but also the variations of time-odd mean fields [41,42,43]. For eight proton states calculated in both approaches, the mean uncertainties are $0.19\hbar$ and $0.18\hbar$ in CHF+SLy4 and CRMF+NL1, respectively. The same holds also for the set of 18 neutron states, where the average uncertainties are $0.25\hbar$ and $0.20\hbar$ in CHF+SLy4 and CRMF+NL1, respectively.

Table 3 also compares CHF+SLy4 expectation values of the s.p. angular momentum $j^{\mbox{\rm\scriptsize {bare}}}_{\alpha}=\left<\hat{j}\right>_{\alpha}$ with their effective counterparts $j^{\mbox{\rm\scriptsize {eff}}}_{\alpha}$. It is seen that these two quantities differ considerably. As discussed in Ref. [42], this is due to both shape polarization and time-odd mean-field effects. It is also important to remember that, unlike the cranked Nilsson scheme, in self-consistent models the expectation value of the projection of the s.p. angular momentum on the rotation axis $j^{\mbox{\rm\scriptsize {bare}}}_{\alpha}$ cannot be extracted from the slope of its s.p. routhian versus rotation frequency [44].

Our results indicate that the additivity principle for angular momentum alignment does not work as precisely as it does for quadrupole moments. This conclusion is in line with a similar analysis in the $A\sim 60$ region of superdeformation [45,46]. A configuration assignment based on relative alignments depends on how accurately these alignments can be predicted. For example, the application of effective (relative) alignment method in the $A\sim 140-150$ region of superdeformation requires an accuracy in the prediction of relative angular momenta on the level of $\sim 0.3\hbar$ and $\sim 0.5\hbar$ for non-intruder and intruder orbitals, respectively [13,47,8]. In the highly deformed and SD nuclei from the $A\sim 60-80$ mass region, these requirements for accuracy are somewhat relaxed (see Refs. [48,49]). We expect that in the $A\sim130$ region, the relative alignments should be predicted with a precision similar to that in the $A\sim 140-150$ region. However, for a number of orbitals (for example, $\nu [411]3/2^{\pm i}$, $\nu [532]5/2^{\pm i}$, $\nu 6_3^{-i}$, $\pi [301]1/2^{+i}$, $\pi [413]5/2^{-i}$, $\pi [550]1/2^{-i}$), the calculated uncertainties in $j^{\mbox{\rm\scriptsize {eff}}}_{\alpha}$ are close to $0.5\hbar$, and this probably prevents reliable assignments based on the additivity principle for the configurations involving these orbitals. The situation becomes even more uncertain if several orbitals with high uncertainties in $j^{\mbox{\rm\scriptsize {eff}}}_{\alpha}$ are occupied.

Let us also remark that while theory provides effective alignments at a fixed rotational frequency, relative alignments extracted from experimental data may show appreciable frequency dependence (see for instance Ref. [45]). Therefore, for reliable configuration assignments, measured relative alignments should be compared with calculated ones over a wide frequency range.