Moments of inertia

The moments of inertia may constitute another independent indicator of the shell structure. This is true under the assumption that their values, similarly as for the odd-even staggering, are dictated by varying pairing correlations. Then, larger shell gaps would induce weaker pairing and thus larger moments of inertia.

The kinematical moments of inertia were calculated as

$$J^{(1)} \equiv \frac{\langle \hat{J}_\perp \rangle} {\omega_\perp} ,$$ (10)

where $\omega_\perp$ is the value of the constrained rotational frequency and $\langle \hat{J}_\perp \rangle$ is the expectation value of the component of angular momentum, both in the directions perpendicular to the axial-symmetry axis. The dynamic moment of inertia is not considered due to larger experimental uncertainties consequence of higher order derivatives in its definition. Moreover, contrary to the kinematic moment of inertia the dynamic one does not depend on spin. Thus, the kinematic moment of inertia provides stricter constrain of model predictions.

All calculations of moments of inertia were performed at $\hbar\omega_\perp=20$keV. In Figs. 13 and 14, they are compared to experimental values determined as

$$J^{(1)} \equiv \frac{3\hbar^2}{E_{2^+}} .$$ (11)

In nuclei where at the bottom of rotational bands the experimental $2^+$ levels have not been seen, we used values extrapolated [78] by the Harris formula. In this region of nuclei, the low-spin moments of inertia turn out to be very weakly dependent on the angular frequency, and, therefore, a specific method of extracting them from experiment is not essential.

Figure 13: Kinematical moments of inertia $\mathcal{J}^{(1)}$ of yrast rotational bands of even-even nuclei in the nobelium region plotted in the isotonic chains as functions of $Z$. Theoretical values, calculated at the angular frequency of $\hbar\omega_\perp=20$keV, Eq. (10), are compared to experimental values extracted from energies of the 2$^+$ states, Eq. (11). Experimental energies are taken from Ref. [67] except for $^{252}$Fm, which is taken from Ref. [79].

FigSHE13.eps

Figure 14: Same as in Fig. 13, but plotted along the isotopic chains as functions of $N$.

FigSHE14.eps

The experimental values shown in panels (e) of Figs. 13 and 14 show clear maxima of $J^{(1)}$ in function of $Z$ at $Z=100$ in the $N=152$ isotonic chain, and at $Z=98$ in the $N=150$ isotonic chain, as well as function of $N$ at $N=152$ in the $Z=100$ isotopic chain. These maxima only partly appear in those nuclei that show maxima of the two-particle staggering indicators, discussed in Sec. 3.4.

In our theoretical calculations, weak maxima of $J^{(1)}$ are obtained at $Z=98$ for $N=146$-150 isotonic chains (D1M¹, UNEDF2, and SLy4 EDFs) and a stronger maxima at $Z=96$ for $N=146$-152 isotonic chains (NL1 and NL3* EDFs), whereas in heavier isotonic chains we see only a gradual increase, without indications of increased shell gaps. Similarly, in the isotopic chains maxima appear at $N=148$ (NL1 EDF) and merely kinks appear at $N=150$ (D1M, UNEDF2, and SLy4 EDFs)

Comparing moments of inertia, Figs. 13 and 14, with the Nilsson diagrams, Figs. 1-3, we see that our calculations with different models and forces seem to exhibit rather nice correspondence between the respective proton (neutron) single-particle shell gaps and peaks/kinks in the moments of inertia along the isotonic (isotopic) chains.

In Fig. 13, for the covariant EDFs NL1 (a) and NL3* (d), the peaks at $Z=96$ obtained for the $N=144$-152 isotonic chains can be associated with the shell gap that in Fig. 3 opens up at $Z=96$. The peak moves to $Z=104$ for $N=154$ and 156 chains, see Fig. 3. For the Skyrme EDFs SLy4 (c) and UNEDF2 (f), the peak/kink at $Z=98$ may be associated with a shell gap at $Z=98$ visible in the Nilsson diagram of Fig. 1. For the Gogny EDF D1M, the correspondence is not as clear as that visible in other cases. It is to be noted that all our calculations predict that for most of the isotonic chains, values of $\mathcal{J}^{(1)}$ peak at $Z=104$. This corresponds to the proton gap at $Z=104$ that is clearly visible in Figs. 1-3.

In Fig. 14, for the covariant EDF NL3* (d), values of $\mathcal{J}^{(1)}$ show pronounced maxima at $N=148$ for isotopic chains of $Z=96$-104. For NL1 (a), the maximum becomes a kink occurring at $N=150$. Neutron numbers of $N=148$ and 150 correspond nicely to the shell gaps shown in Fig. 3. Non-relativistic functionals predict either a peak at $N=150$ or a plateau for $N=150$ and 152, particularly for the $Z=100$-104 isotopic chains, shown in Fig. 14.

It is necessary to recognize that for a proper reproduction of experimental moments of inertia, the inclusion of the LN method into the calculations appears to be more important in covariant [80] than in non-relativistic EDFs. The LN method renders the values of calculated moments of inertia much closer to the data [20,81] and at the same time much less sensitive to the underlying shell structure, at variance with the data, cf. the discussion in Ref. [21].