Infinitesimal cranking for triaxial angular-momentum-projected configuration-mixing calculations and its application to the $\gamma$ vibrational band

Inclusion of time-odd components into the wave function is important for a reliable description of rotational motion by the angular-momentum-projection method; the cranking procedure with infinitesimal rotational frequency is an efficient way to realize it. In the present work we investigate the effect of this infinitesimal cranking for a triaxially deformed nucleus, where there are three independent cranking axes. It is found that the effects of cranking about three axes on the triaxial energy spectrum are quite different and inclusion of all of them considerably modifies the resultant spectrum from the one obtained without cranking. Employing the Gogny D1S force as an effective interaction, we apply the method to the calculation of the multiple $\gamma$ vibrational bands in ${}^{164}\mathrm{Er}$ as a typical example, where the angular-momentum-projected configuration mixing with respect to the triaxial shape degree of freedom is performed. With this method, both the $K=0$ and the $K=4$ two-phonon $\gamma$ vibrational bands are obtained with considerable anharmonicity. Reasonably good agreement, though not perfect, is obtained for both the spectrum and transition probabilities with rather small average triaxial deformation $\gamma \approx 9^{\circ }$ for the ground-state rotational band. The relation to the wobbling motion at high-spin states is also briefly discussed.

Figure 1

Absolute energy curves as functions of the triaxial deformation parameter $\gamma$ calculated by the angular-momentum projection from the noncranked HFB ground state in ${}^{164}\mathrm{Er}$. Those for the $0^{+},2^{+},4^{+}$, and $6^{+}$ states of the g band and for the $2^{+},3^{+},4^{+},5^{+}$, and $6^{+}$ states of the $\gamma$ band are included.

Figure 2

Excitation energy curves as functions of the triaxial deformation parameter $\gamma$ calculated by the angular-momentum projection from the noncranked HFB ground state in ${}^{164}\mathrm{Er}$. Those for the $2^{+},4^{+}$, and $6^{+}$ states of the g band and for the $2^{+},3^{+},4^{+},5^{+}$, and $6^{+}$ states of the $\gamma$ band are included. Experimental data are also shown by symbols, open circles and squares, at $\gamma =15.4^{\circ }$, where the calculated excitation energy of $2_{\gamma }^{+}$ coincides with the experimental one.

Figure 3

Energy spectrum calculated by the angular-momentum projection from the noncranked HFB state at $\gamma =15.4^{\circ }$ in ${}^{164}\mathrm{Er}$. The experimental data for the g band and the $\gamma$ band are included as open squares.

Figure 4

$E2$ transition probability from the ground state to the second $2^{+}$ state as a function of the triaxial deformation parameter $\gamma$, which is calculated by the angular-momentum projection from the noncranked HFB state in ${}^{164}\mathrm{Er}$ (solid curve). The dotted curve is the prediction of the asymmetric rotor model [16] in Eq. (20) with $B=5.723 \left(e^2{\mathrm{b}}^2\right)$. The experimentally measured values [27], 0.148 and $0.170 \left(e^2{\mathrm{b}}^2\right)$, are shown by two horizontal lines, which are deduced from two different types of reactions.

Figure 5

Energy spectrum calculated by the angular-momentum projection from the infinitesimally cranked HFB state with triaxial deformation, $\gamma ={10}^{\circ }$, for ${}^{164}\mathrm{Er}$. The axis (axes) of cranking is (are) specified in each panel, e.g., “$xyz$ crank” means that the infinitesimal cranking is performed about all the $x,y$, and $z$ axes. The noncranked case is also included as panel (a).

Figure 6

Energy spectrum calculated by the angular-momentum projection from the infinitesimally cranked HFB state with triaxial deformation, $\gamma ={20}^{\circ }$, for ${}^{164}\mathrm{Er}$. The axis (axes) of cranking is (are) specified in each panel, e.g., “$xyz$ crank” means that the infinitesimal cranking is performed about all the $x,y$, and $z$ axes. The noncranked case is also included as panel (a).

Figure 7

Absolute energy curves as functions of the triaxial deformation parameter $\gamma$ calculated by the angular-momentum projection from the $xyz$-cranked HFB ground state in ${}^{164}\mathrm{Er}$. Those for the $0^{+},2^{+},4^{+}$, and $6^{+}$ states of the g band and for the $2^{+},3^{+},4^{+},5^{+}$, and $6^{+}$ states of the $\gamma$ band are included.

Figure 8

Excitation energy curves as functions of the triaxial deformation parameter $\gamma$ calculated by the angular-momentum projection from the $xyz$-cranked HFB ground state in ${}^{164}\mathrm{Er}$. Those for the $2^{+},4^{+}$, and $6^{+}$ states of the g band and for the $2^{+},3^{+},4^{+},5^{+}$, and $6^{+}$ states of the $\gamma$ band are included. Experimental data are also shown by symbols, open circles and squares, at $\gamma =12.8^{\circ }$, where the calculated excitation energy of $2_{\gamma }^{+}$ coincides with the experimental one.

Figure 9

Energy spectrum calculated by the angular-momentum projection from the $xyz$-cranked HFB state at $\gamma =12.8^{\circ }$ in ${}^{164}\mathrm{Er}$. The experimental data for the g band and the $\gamma$ band are included as open squares.

Figure 10

$E2$ transition probability from the ground state to the second $2^{+}$ state as a function of the triaxial deformation parameter $\gamma$, which is calculated by the angular-momentum projection from the infinitesimally cranked HFB state around all the $xyz$ axes in ${}^{164}\mathrm{Er}$ (solid curve). The dotted curve is the prediction of the asymmetric rotor model [16] in Eq. (20), with $B=5.585 \left(e^2{\mathrm{b}}^2\right)$. The experimentally measured values [27], 0.148 and $0.170 \left(e^2{\mathrm{b}}^2\right)$, are shown by two horizontal lines.

Figure 11

Energy spectrum calculated by the angular-momentum-projected configuration-mixing using the five noncranked HFB states with $\gamma =1^{\circ },{10}^{\circ },{20}^{\circ },{30}^{\circ }$, and ${40}^{\circ }$ in ${}^{164}\mathrm{Er}$. The rotational energy $I(I+1)/78$ MeV is subtracted. The experimental data for the g band and the $\gamma$ band are included as open squares.

Figure 12

The probability distributions of $0_1^{+},2_1^{+},4_1^{+}$, and $6_1^{+}$ states in the g band and $2_2^{+},4_2^{+}$, and $6_2^{+}$ states in the $\gamma$ band with respect to the triaxiality parameter $\gamma$ for the configuration-mixing calculation of Fig. 11.

Figure 13

Energy spectrum calculated by the angular-momentum-projected configuration-mixing using the five infinitesimally $xyz$-cranked HFB states with $\gamma =1^{\circ },{10}^{\circ },{20}^{\circ },{30}^{\circ }$, and ${40}^{\circ }$ in ${}^{164}\mathrm{Er}$. The rotational energy $I(I+1)/78$ MeV is subtracted. The experimental data for the g band and the $\gamma$ band are included as open squares.

Figure 14

The probability distributions of the $0_1^{+},2_1^{+},4_1^{+}$, and $6_1^{+}$ states in the g band and the $2_2^{+},4_2^{+}$, and $6_2^{+}$ states in the $\gamma$ band with respect to the triaxiality parameter $\gamma$ for the configuration-mixing calculation of Fig. 13.

Figure 15

Energy spectrum calculated by the angular-momentum projection from one $xyz$-cranked HFB state with $\gamma =11.6^{\circ }$ in ${}^{164}\mathrm{Er}$. The rotational energy $I(I+1)/78$ MeV is subtracted. The experimental data for the g band and the $\gamma$ band are included as open squares.

Figure 16

Energy spectrum calculated by angular-momentum projection from the one $xyz$-cranked HFB state with the triaxial deformation, $\gamma ={20}^{\circ }$, for ${}^{164}\mathrm{Er}$, i.e., the same as Fig. 6 but its higher-spin part is shown. Only the five excited $\left|\mathrm{\Delta }I\right|=2$ bands from the lowest are shown.

Figure 17

The $B\left(E2\right)$ ratio of the out-of-band to the in-band transitions, $B(E2:I\to I\pm 1)/B(E2:I\to I-2)$, for the first and the second excited bands in the wobblinglike structure in Fig. 16. The solid (open) symbols denote the transitions from the first (second) band.

Figure 18

The “expectation values” of the squared $x,y,z$ components of the angular momentum vector in the body-fixed frame, which are defined by Eq. (23), for the yrast band (solid symbols) and for the first excited band (open symbols) in Fig. 16.

Figure 19

Comparison of the two definitions for the “expectation values” of the squared components of the angular momentum vector in the body-fixed frame for the first excited band in Fig. 16, which are calculated by Eq. (23) (solid lines) and by Eq. (A3) (dotted lines). The solid lines are the same as those with the open symbols in Fig. 18.