Ground-state rotational band in $^{46}$Ti

As a first example, consider the ground-state rotational band in $^{46}$Ti. According to the CHF model, the shape of this nucleus undergoes a gradual change along the band. Starting from a well elongated ( $\beta_2\sim 0.23$) shape at low spins, the nucleus goes through the alignment processes of the $f_{7/2}$ protons and neutrons, and eventually reaches a nearly spherical ( $\beta_2\sim 0.05$) shape at the terminating spin of $I=14$, as shown in Fig. 1a.

Figure 1: Results of the CHF and AMP calculations for the rotational band in $^{46}$Ti obtained for the SLy4 interaction. The upper panel (a) shows the evolution of quadrupole and hexadecapole shape parameters along the band as calculated using CHF approximation. The middle panel (b) shows the excitation spectra calculated using the CHF approximation (left column) and AMP method with kinematic and dynamic $K$-mixing (middle columns), compared with the empirical data [25] (right column). The lower panel (c) shows deviations $\Delta {E}$ of energy levels with spins $I=2,\ldots ,12$ from the converged values corresponding the plateau condition, as functions of the number $m_{\text{max}}$ of the norm eigenvalues used when solving the HW equation.

Figure: Similar as in Fig. 1b, but shown in the absolute energy scale. Results of the AMP from the $\langle\hat{I}_y\rangle=0$ ground state (first column) are compared to those of the CHF (second column), AMP with kinematic and dynamic $K$-mixing (third and fourth columns), and CHFB calculations (fifth column).

The calculated and experimental $^{46}$Ti rotational bands are shown in Fig. 1b. Results of the CHF calculations (left) are compared to those of AMP (middle) and to experimental data (right). In order to visualize the role of the $K$-mixing, we have depicted results of the AMP calculations separately for the kinematic and dynamic $K$-mixing. For the dynamic $K$-mixing, the AMP excitation energies were calculated from the plateau condition. The stability of results with respect to the number of natural states $m_{\text{max}}$ is shown in Fig. 1c. It can be seen that already at $m_{\text{max}}=2$, a perfect stability is obtained, i.e., here, the difference between the kinematic and dynamic $K$-mixing is related to adding the $m=2$ state to the collective subspace. At higher values of $m_{\text{max}}$, one observes small departures from the converged values, which are due to accidental mixing with spurious solutions (the largest such an effect is seen for $I=4$ at $m_{\text{max}}=5$). Nevertheless, physical converged solutions are clearly seen well beyond the point where the spurious solutions become lower in energy, as is well known for other generator-coordinate-method calculations [26].

Figure 1b clearly shows that within the $I_y$ $\rightarrow$$I$ scheme, the AMP effectively causes a decrease of the mean MoI within the band. This effect is expected to be generic for rotational bands of decreasing collectivity. Indeed, in such cases rotational correction is large at low spins and decreases at higher spins. This is illustrated in Fig. 2, where the calculated bands are shown in the absolute energy scale. In the case of $^{46}$Ti, a net increase in the excitation energy of the terminating state, due to the AMP, amounts to about 2MeV, but it is almost entirely related to the lowering of the $I=0$ state, while the terminating $I=14$ state remains almost unaffected by the AMP.

In the same Figure, we also show for comparison the AMP spectrum obtained by projecting from the $\langle\hat{I}_y\rangle=0$ ground state. In this case, the overall MoI turns out to be much too small and the excitation energy much too high as compared to the CHF or AMP $I_y$ $\rightarrow$$I$ results. This fact once again shows the importance of the structural changes that occur in the system with increasing angular momentum.

Note that the $\langle\hat{I}_y\rangle=0$ state is axial, and thus for all spins contains only the $K=0$ component. Therefore, here, the collective space contains only one state and there is no $K$-mixing at all. Moreover, axial symmetry leads to a tremendous simplification of the AMP method, whereby only one-dimensional integration over the Euler $\beta$ angle is needed. Note also that the CHF states for $\langle\hat{I}_y\rangle\neq0$ are never axial, because the Coriolis coupling always induces some non-zero nonaxiality. Therefore, the full three-dimensional integration over the Euler angles $\alpha\beta\gamma$ is needed in our $I_y$ $\rightarrow$$I$ scheme.

In the right column of Fig. 2, we also show results of the cranked Hartree-Fock-Bogolyubov calculations (CHFB) performed by using a zero-range volume-type (density-independent) interaction in the pairing channel. Its strengths of $V_n=-217.0$ and $V_p=-237.5$MeVfm$^{-3}$ for neutrons and protons, respectively, with the cutoff energy of $\epsilon_{\text{cut}}=50$MeV [27], was adjusted so as to reproduce, on average, the pairing gaps in this region of nuclei. For the $\langle\hat{I}_y\rangle=0$ ground state of $^{46}$Ti, the calculated gaps read $\Delta_n=1.394$ and $\Delta_p=1.632$MeV.

One can see that, in comparison with the CHF results, pairing correlations only affect states at low spins, $I\leq6$, and give about 1MeV of additional binding at $\langle\hat{I}_y\rangle=0$. Pairing correlations vanish in the terminating state, and thus its energy is the same within the CHF and CHFB methods. At present, the code HFODD cannot perform the AMP of paired states, and we are yet unable to evaluate combined effects of pairing and AMP from cranked states.

It is also interesting to observe that, in the case of $^{46}$Ti, the dynamic $K$-mixing is effective essentially only for $I=2$ and 4, see Fig. 1b,c. This result is rather surprising, particularly in view of the fact that the only discontinuity in spatial anisotropy of the CHF solutions can be seen around spins $I\sim 6,8$. This behavior is also qualitatively different from that found in $^{156}$Gd [12], where the dynamic $K$-mixing was effective up to the highest ($I\sim 20$) calculated spins. Apparently, the magnitude of $K$-mixing cannot be inferred solely from the shape but it also depends on individual (alignment) degrees of freedom.

Figure 3: Probabilities $W_I$ (12) and $W_K$ (13) of finding the given $I$ and $K$ components, respectively, in the intrinsic CHF states for different values of the projection on the $y$ axis $\langle \hat I_y\rangle$. Solid lines are drawn only to guide the eye. Note that due to the conserved $y$-signature symmetry, one has $W_I$=0 for odd values of $I$ and $W_{-K}=W_K$.

In Fig. 3, we show probabilities $W_I$ of finding the angular-momentum components in the intrinsic CHF states, i.e.,

$$W_I = \sum_{K=-I}^I \langle \Phi_{I_y} \vert \hat{P}^I_{KK}\vert\Phi_{I_y} \rangle.$$ (12)

The inset also shows probabilities $W_K$ of finding the given $K$ components,

$$W_K = \langle \Phi_{I_y} \vert \hat{P}^I_{KK}\vert\Phi_{I_y} \rangle/W_I,$$ (13)

in one of the intrinsic states, $\langle\hat{I}_y\rangle=6$. At low spins, the $W_I$ probability distributions follow the general pattern of deformed collective states. This pattern suddenly disappears at the termination point of $\langle\hat{I}_y\rangle=14$, where the CHF state becomes totally aligned, and therefore, it contains only the single $I=14$ component. The $W_K$ probabilities do not follow the standard pattern of collective rotation, where they correspond to the Coriolis mixing only, cf. Fig. 2 in Ref. [12]. Here, the aligning state causes the values of $W_K$ to decrease with $I$ and induces the crossing of those corresponding to even and odd $K$ values in function of $I$. Note also, that the angular momentum aligns along the $y$ axis, while the standard projections $K$ are calculated with respect to the $z$ axis. Therefore, we do not expect to see a single $K$ component of the aligned $\langle\hat{I}_y\rangle=14$ state.

Figure: Excitation spectra calculated using the AMP of CHF states constrained to different values of $\langle\hat{I}_y\rangle$, with kinematic $K$-mixing. Solid lines connect states of given projected angular momenta $I$ and are drawn only to guide the eye. Arrows indicate the minimum energies in function of $\langle\hat{I}_y\rangle$. For comparison, the right column shows the CHF spectrum.

We conclude this section by analyzing excitation spectra calculated using the AMP of CHF states constrained to different values of $\langle\hat{I}_y\rangle$, as shown in Fig. 4. It turns out that the AMP energies minimized with respect to $\langle\hat{I}_y\rangle$, or equivalently with respect to the angular frequency $\omega$, significantly differ from those obtained within our $I_y$ $\rightarrow$$I$ scheme. Indeed, although the minimum $I=0$ energy is obtained by projecting the $\langle\hat{I}_y\rangle=0$ CHF state, this is not the case for $I>0$ states. For example, the minimum $I=2$ and 14 energies are obtained by projecting the $\langle\hat{I}_y\rangle=0$ and 10 CHF states, respectively. The resulting excitation spectrum (in Fig. 4 indicated by arrows) is quite irregular and can hardly be associated with the physical result. The reason is the fact that the minimization over $\omega$ does not constitute the full variation-after-projection minimization of energy, which should be performed in the complete parameter space, including, e.g., the deformation parameters. Moreover, the fact that the $I=14$ state may gain energy by using collective correlations on top of the fully aligned (terminating) state points out a possible inadequacy of the EDF method when it is used within the variation-after-projection procedure, and not within the $I_y$ $\rightarrow$$I$ projection-after-variation scheme.