Variance and distribution of residuals

One of the main outcomes of this study is the set of effective s.p. moments $q_{20,\alpha}^{\mbox{\rm\scriptsize {eff}}}$, $q_{22,\alpha}^{\mbox{\rm\scriptsize {eff}}}$, $q_{t,\alpha}^{\mbox{\rm\scriptsize {eff}}}$, and alignments $j_\alpha^{\mbox{\rm\scriptsize {eff}}}$. The quality of the additivity principle can be assessed by studying the distribution of first moments of residuals (25), i.e., differences between the self-consistently calculated values of physical observables and those obtained from the additivity principle. For instance, for the quadrupole moment $Q_{20}$, the quantity of interest is

$$\Delta Q_{20}= \sum_{\alpha}^{} { c_{\alpha}(k) q_{20,\alpha}^{\mbox{\rm\scriptsize {eff}}} } - \delta Q_{20} (k).$$ (34)

Deviations $\Delta Q_{22}$, $\Delta Q_t$, and $\Delta J$ are given by similar expressions. Figures 5 and 6 show distributions of these deviations. The quality of the additivity principle for $Q_{20}$ is shown in the top two panels of Fig. 5. In the CHF model, the majority of $\Delta Q_{20}$ values (more than 97.8% of the total number) fall comfortably within the interval of $\pm$0.1eb. This corresponds to a relative distribution width of about 1.3%. In CRMF, the distribution is even narrower, with more than 90% of $\Delta Q_{20}$ values falling within the $\pm$0.05eb interval, or less than 0.7% of the total value.

Figure 5: Histogram of differences between self-consistent values obtained in CHF and CRMF and those given by the additivity formula [see e.g. Eq. (34)]. The results for $Q_{20}$ are shown in the two upper panels and those for the total angular momentum are displayed in the two lower panels.

Figure 6: Similar to Fig. 5 except for $Q_{22}$.

The results for the total angular momentum are shown in the bottom panels of Fig. 5. In CRMF, the distribution of deviations is very narrow, with only 10% of the cases differing by more than $\pm\hbar/2$. The CHF histogram is somewhat wider, but more than 90% of deviations fall within the $\pm\hbar/2$ interval. Taking into consideration that the experimental spins of highly deformed and SD bands are often assigned with uncertainties that are multiples of $\hbar$, our results give considerable encouragement for theoretical interpretations based on the method of relative (effective) alignments [13,47,8].

In CHF and CRMF, the distributions of deviations of charge quadrupole moments $Q_{22}$ (Fig. 6) are relatively narrow. Again, for CRMF, nearly 95% of deviations fall within $\pm$0.025eb, and 98% fall within $\pm$0.1eb. For CHF, the distribution of deviations is somewhat wider, with more than 90% of deviations falling within the $\pm$0.2eb interval.

We interpret these results as a strong indication that the additivity principle works fairly well in self-consistent cranked theories. While distributions of deviations in $Q_{20}$ and $Q_{22}$ are rather similar in the CHF+SLy4 and CRMF+NL1 models (see top of Fig. 5 and Fig. 6), deviations in angular momentum differ between these two approaches. Considering that (i) the uncertainties in $j_{\alpha}^{\mbox{\rm\scriptsize {eff}}}$ are similar in both methods (Sec. 3.3), and (ii) shape polarization effects are not that different (Sec. 3.2), one can conclude that the observed difference is due to the polarization of time-odd mean fields. However, the detailed investigation of this effect is beyond the scope of this study.