Properties of the $h_{11/2}$ valence nucleons and of the core

We began our study of rotational properties by performing the standard PAC calculations, in which we examined rotations of the found, triaxial HF solutions about their short, medium, and long axes. Let us recall that for the PAC about the principal axis $i=s,m,l$, not only the cranking-frequency vector, $\vec {\omega}$, but all the resulting s.p. and total mean angular-momentum vectors, $\vec {j}$ and $\vec {I}$, have non-zero components, $\omega _i$, $j_i$, $I_i$, only on that axis.

Figure 2: Single-particle Routhians from the HF PAC calculations in $^{132}$La with the SLy4 force and no time-odd fields. Proton (upper half) and neutron (lower half) Routhians for cranking about the short, medium, and long axes are shown. Positive-parity and negative-parity levels are marked with solid and dashed lines, respectively. Negative-parity levels belonging to the $h_{11/2}$ multiplet are marked with crosses. The Routhians occupied by the valence $h_{11/2}$ proton particle and neutron hole are marked with full and open circles, correspondingly.

Figure 2 gives the s.p. Routhians obtained from the PAC calculations in $^{132}$La with the SLy4 force, without the time-odd fields. Routhians for other isotones, forces, and time-odd terms included do not differ in their main features from the shown ones. The upper and lower parts show the proton and neutron levels, respectively, and the left, middle, and right panels correspond to cranking about the short, medium, and long axes. Positive-parity and negative-parity levels are marked with solid and dashed lines. The negative-parity levels belonging to the $h_{11/2}$ multiplet are marked with crosses. The Routhians occupied by the valence $h_{11/2}$ proton particle and neutron hole are marked with full and open circles, respectively.

In both kinds of nucleons, the lowest levels of the $h_{11/2}$ multiplet split strongly for cranking about the short axis, and have almost zero slope for the two remaining axes. The highest $h_{11/2}$ levels behave similarly, but split in the case of the rotation about the long axis. The intermediate levels split for cranking about each axis, but only weakly.

Figure 3: Single-particle angular momentum alignments of the valence $h_{11/2}$ proton particle and neutron hole in $^{132}$La, obtained from the PAC about the short, medium and long axes. The HF results with the SLy4 force are shown for the $G$ variant of calculation defined in Sec. 3.2.

For the PAC about the $i$-th axis, slopes of the s.p. Routhians, $e'$, approximately translate into the s.p. angular-momentum alignments, $j_i$, on that axis according to the well-known formula

$$j_i \approx -\frac{\mathrm{d}e'}{\mathrm{d}\omega_i}~,$$ (5)

which holds almost exactly in the present case, because changes of the mean field with rotational frequency are negligible. Thus, one can infer from Fig. 2 that the valence $h_{11/2}$ proton particle and neutron hole have non-zero alignments, $j^p_i$ and $j^h_i$, only for the PAC about short and long axes, respectively. This can be seen directly in Fig. 3, which shows $j^p_i$ and $j^h_i$ calculated as mean values of the angular-momentum operator for the variant $G$ of the calculations with the SLy4 force. Results for the short, medium, and long axes are plotted with dotted, solid, and dashed lines.

It turns out that further important properties of the valence $h_{11/2}$ particle and hole can be deduced from pure symmetry considerations, which we present in Appendix A. We consider an isolated two-fold degenerate s.p. level in a fixed potential symmetric with respect to the $D_2$ group (rotations by 180$^\circ$ about the three principal axes). The main conclusion relevant for the present case is the following. If in the PAC a s.p. state has a non-zero alignment, $j_i$, only for rotation about one principal axis, then in the TAC its angular-momentum vector, $\vec {j}$, will point along that axis and remain approximately constant regardless of the length and direction of the applied cranking frequency, $\vec {\omega}$. We call it a stiff alignment. Thus, the angular momenta, $\vec {j}^p$ and $\vec {j}^h$, of the valence $h_{11/2}$ proton particle and neutron hole are stiffly aligned along the short and long intrinsic axes, respectively, as expected for the chiral geometry.

In our self-consistent calculations, the s.p. spectrum of the mean-field Hamiltonian, $\hat{h}$ of Eq. (3), exhibits the two-fold Kramers degeneracy only in variant $N$ of the calculations, when no time-odd fields are taken. It seems that the lowest and highest $h_{11/2}$ levels can be indeed treated as isolated: their angular-momentum coupling to other s.p. states is weak, which can be seen from the small curvature of their PAC Routhians in Fig. 2. The assumption about fixed potential is also justified, because in our results the deformation remains nearly constant with rotational frequency.

In the present case, the remnant coupling to other states and changes of the mean field alter the ideal picture in that $\vec {j}^p$ and $\vec {j}^h$ are not strictly constant, but show some remnant dependence on the cranking frequency. As illustrated in Fig. 3, this dependence is to an approximation linear, and the PAC alignments can be written in the form

$$\vec {j}^p\simeq s_s\vec {i}_s+\delta\mathcal{J}^p\vec {\omega}~, \qquad \vec {j}^h\simeq s_l\vec {i}_l+\delta\mathcal{J}^h\vec {\omega}~,$$ (6)

where $\vec {i}_s$ and $\vec {i}_l$ denote the unit vectors along the short and long axes, respectively. The quantities $s_s$ and $s_l$ are initial alignments at vanishing frequency, and $\delta\mathcal{J}^p$ and $\delta\mathcal{J}^h$ are tensor coefficients, representing the s.p. contributions to the total inertia tensor, $\mathcal{J}$, of the nucleus.

The PAC allows to estimate the diagonal components of $\delta\mathcal{J}$, which are slopes of the $j_i(\omega_i)$ curves plotted in Fig. 3. One can see from the Figure that they may attain up to $\sim$4$\hbar^2$/MeV. This is a significant value compared to the total moments of inertia, discussed below and collected in Table 1. For $^{123}$La, the total moments are of the order of $\sim$8-36$\hbar^2$/MeV, depending on the axis.

Figure 4: Total angular momentum alignments obtained from the PAC about the short, medium and long axes in $^{130}$Cs, $^{132}$La, $^{134}$Pr, and $^{136}$Pm. The HF results with the SLy4 and SkM* forces are shown for the $N$, $G$, and $T$ variants of calculation defined in Sec. 3.2.

The same PAC calculations provide the total alignments, $I_s$, $I_m$, $I_l$, on the short, medium, and long axes, respectively. They are plotted in Fig. 4 with dotted, solid, and dashed lines, correspondingly, for all the considered cases. In the nearly oblate minima in $^{134}$Pr and $^{136}$Pm, an attempt to crank around the medium axis leads to such a self-consistent readjustment of the shape that solutions corresponding to the rotation about the long axis are obtained.

The bends in the curves in Fig. 4, like the ones for $I_s$ in $^{130}$Cs, are caused by smooth crossings of the s.p. levels. Otherwise, the $I_i(\omega_i)$ dependence is linear to a good approximation, and therefore the rotation can be called rigid. The corresponding slopes give the collective total moments of inertia, $\mathcal{J}_s$, $\mathcal{J}_m$, and $\mathcal{J}_l$, with respect to the short, medium, and long axes. They contain the valence particle and the valence hole contributions, $\delta\mathcal{J}^p$ and $\delta\mathcal{J}^h$, defined in Eq. (6).

At zero frequency, the cranking around the medium axis gives a vanishing angular momentum, while the cranking around the other two axes give non-zero limiting values, equal to the initial alignments, $s_s$ and $s_l$, of the odd particle and hole. In view of the considerations presented later on in this paper, we have extracted the actual values of the parameters $\mathcal{J}_s$, $\mathcal{J}_m$, $\mathcal{J}_l$, and $s_s$, $s_l$, by fitting straight lines to the calculated alignment vs. frequency curves shown in Fig. 4. Whenever there was a bend in the calculated dependence, the line was fitted in the rotational frequency range below the bend. Since the alignment on the short axis for the oblate minimum in $^{136}$Pm shows a particularly complicated behavior, we have not assigned any value to $\mathcal{J}_s$ in this case. The obtained values of the parameters are listed in Table 1, and discussed in Section 4. These results confirm that the moment of inertia with respect to the medium axis is the largest.

The PAC method allows for estimating the diagonal components, $\mathcal{J}_s$, $\mathcal{J}_m$, and $\mathcal{J}_l$, of the inertia tensor, $\mathcal{J}$. In order to examine, as far as possible, the off-diagonal components we performed a kind of perturbative test within the TAC method. We applied to the non-rotating self-consistent solutions the cranking frequency vector, $\vec {\omega}$, in several directions, performed only one diagonalization of the resulting s.p. Routhian (3), and investigated the response of the mean angular momentum, $\vec {I}$. It turned out that the off-diagonal components of $\mathcal{J}$ are negligibly small in all the considered isotones.

The microscopic results presented so far suggest that the considered system can be modeled by two gyroscopes of spins $s_s$ and $s_l$ rigidly fixed along the short and long axes of a triaxial rigid rotor characterized by the moments of inertia, $\mathcal{J}_s$, $\mathcal{J}_m$, $\mathcal{J}_l$, of which $\mathcal{J}_m$ is the largest. It is instructive to solve the associated problem of motion in the classical framework, which is done in the next Section.