Summary and discussion

The present work reports on the first systematic calculations using the AMP of cranked Hartree-Fock states. The technique used, called the $I_y$ $\rightarrow$$I$ projection scheme, assumes the projection of the angular momentum component $I$ from the one-dimensional (1D) cranked Hartree-Fock solution constrained to $\langle \hat I_y\rangle=I$. The method benefits naturally from such nice physical features of the 1D cranking model as the shape-spin self-consistency or the ability to get a realistic estimate of the nuclear MoI. It is shown that the $I_y$ $\rightarrow$$I$ AMP scheme leads to values of MoI that are much more realistic than those obtained by using the AMP of non-rotating $\langle \hat I_y\rangle = 0$ states, as was the common practice up to now.

In particular, application of the scheme to the rotational band in $^{46}$Ti clearly improves the MoI at the bottom of the band. It also reveals a simple mechanism by which rotational corrections allow for improving upon an incorrect excitation energy of the terminating state, $\Delta E_{I_{\text{max}}}$, obtained in the CHF calculations, which account only for roughly half of the empirical value.

Pairing correlations that are active in the ground state, but do not affect fully aligned terminating state, can heal the situation only partially. Indeed, there is an upper limit for pairing correlation energy in the ground state, which can be sustained by a deformed $A$$\sim$44 system, equal to about 2MeV. Further enhancement of pairing would induce the phase transition from deformed to spherical shape [38], and the rotational band could not have been built, contradicting the experimental data. Together, the rotational and pairing effects can bring $\Delta E_{I_{\text{max}}}$ to about 9MeV, i.e., some $\sim 10$% below the experimental value. Let us mention here that for the $I=14$ state projected from the $\langle\hat{I}_y\rangle=0$ ground state one obtains $\Delta E_{I_{\text{max}}}$$\simeq$19MeV, i.e., the result, which is well above the empirical excitation energy of the terminating state.

For nearly-spherical unfavored-signature $[f_{7/2}^n]_{I_{\text{max}}-1}$ terminating states, our AMP calculations give results in excellent agreement with data, and validate approximate projection methods introduced in Refs. [28,29]. We also show that the onset of collectivity in the $[f_{7/2}^n]_{I_{\text{max}}-2}$ states is quite correctly reproduced by the CHF calculations, on top of which the AMP gives only a small correction going in the right direction in comparison with data. However, details of the isotopic dependence are not reproduced here.

Similar conclusions are obtained for the $[d_{3/2}^{-1} f_{7/2}^{n+1}]$ configurations that involve one-proton ph excitation across the $Z=20$ shell gap. In this case, in both $I_{\text{max}}\!-\!1$ and $I_{\text{max}}\!-\!2$ states near the band termination the collectivity sets in, while the energy differences with respect to terminating $I_{\text{max}}$ states are underestimated in the CHF and AMP calculations.

The AMP of cranked Hartree-Fock states presented in this work was performed by applying the standard projection techniques to the CHF solutions obtained within the EDF method. We have checked that all the results are stable with respect to numerical parameters such as, e.g., numbers of integration points used when integrating kernels over the Euler angles. Nonetheless, one should be aware of potential risks caused by difficult to control, uncompensated poles plaguing projection techniques of states obtained within the EDF method [12,39]. Clearly, the future of projection methods crucially depends on a satisfactory solution of the problem of such singularities.

This work was supported in part by the Polish Ministry of Science and by the Academy of Finland and University of Jyväskylä within the FIDIPRO programme.