Angular momentum projection of cranked Hartree-Fock states: Application to terminating bands in $A~44$ nuclei

We present the first systematic calculations based on the angular momentum projection of cranked Slater determinants. We propose the $I_y\to I$ scheme, by which one projects the angular momentum $I$ from the one-dimensional cranked state constrained to the average spin projection of $\langle {Î}_y\rangle =I$. Calculations performed for the rotational band in ${}^{46}\mathrm{Ti}$ show that the AMP $I_y\to I$ scheme offers a natural mechanism for correcting the cranking moment of inertia at low spins and shifting the terminating state up by $~2$ MeV, in accordance with data. We also apply this scheme to high-spin states near the band termination in $A~44$ nuclei and compare results thereof with experimental data, shell-model calculations, and results of the approximate analytical symmetry-restoration method proposed previously.

Figure 1

Results of the CHF and AMP calculations for the rotational band in ${}^{46}\mathrm{Ti}$ obtained for the SLy4 interaction. (a) The evolution of quadrupole and hexadecapole shape parameters along the band as calculated using the CHF approximation. (b) 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 [27] (right column). (c) Deviations $\Delta E$ of energy levels with spins $I=2,\dots ,12$ from the converged values corresponding to the plateau condition, as functions of the number $m_{\max }$ of the norm eigenvalues used when solving the HW equation.

Figure 2

Similar to 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).

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 because of the conserved $y$-signature symmetry, one has $W_I=0$ for odd values of $I$ and $W_{-K}=W_K$.

Figure 4

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 as a function of $\langle {\hat{I}}_y\rangle$. For comparison, the right column shows the CHF spectrum.

Figure 5

(a) Probabilities of finding the $I_{\max }-1$ spin components in the $|{\Phi }_{I_{\max }-1}^{(\pi )}\rangle$ CHF solutions. Squares and dots show results calculated using the exact AMP and approximate method of Ref. [30], respectively. (b) Energy differences between the favored- and unfavored-signature terminating states, $[f_{7/2}^n]{}_{I_{\max }}$ and $[f_{7/2}^n]{}_{I_{\max }-1}$, respectively. Values calculated using the AMP method (diamonds) are compared to those obtained within the shell model [30] (circles) and empirical data [33, 34, 35, 36, 37, 38] (dots).

Figure 6

Energy differences between the $[f_{7/2}^n]{}_{I_{\max }}$ terminating states and the lowest $[f_{7/2}^n]{}_{I_{\max }-2}$ states. Filled and unfilled circles represent the empirical data [33, 34, 35, 36, 37, 38, 39] and SM results of Ref. [30], respectively. Unfilled diamonds label the lowest CHF solutions, which are collective except for the cases of ${}^{42}\mathrm{Ca}$ and ${}^{42,43}\mathrm{Sc}$. Filled diamonds represent the AMP results. Relative energies among the $I_{\max }$ reference state, noncollective ph $I_{\max }-1$ and $I_{\max }-2$ excitations (thin lines), SSB effect in the $I_{\max }-2$ state (thick dashed line), and final AMP configuration mixing (solid lines) are schematically illustrated in the inset.

Figure 7

Energy differences between the $[d_{3/2}^{-1}{f}_{7/2}^{n+1}]{}_{I_{\max }}$ terminating states and the lowest $[d_{3/2}^{-1}{f}_{7/2}^{n+1}]{}_{I_{\max }-2}$ states. Filled and unfilled circles represent empirical data [33, 34, 35, 36, 37, 38, 39] and SM results of Ref. [30], respectively. Filled and unfilled diamonds label the AMP results and the collective CHF solutions, respectively.

Figure 8

Energy differences between the $[d_{3/2}^{-1}{f}_{7/2}^{n+1}]{}_{I_{\max }}$ terminating states and the lowest unfavored-signature terminating $d_{3/2}^{-1}{f}_{7/2}^{n+1}]{}_{I_{\max }-1}$ states. Filled circles represent empirical data [34, 35, 36, 37, 38, 39] and unfilled and filled diamonds represent the CHF and AMP results, respectively. The inset shows the probabilities of finding the $I_{\max }-1$ components within the lowest CHF solution $|{\Phi }_{I_{\max }-1}\rangle$.