Figure 1

Total energy is shown for $I=28$ as a function of ${\Delta }_n$ and ${\lambda }_n$ for the $\gamma \approx -{75}^{°}$ band of ${}^{142}\mathrm{Gd}$ discussed below. The difference between the contour lines is 0.1 MeV.Reuse & Permissions

Figure 2

Illustration of different quantities at $I=28$ for the same configuration as in Fig. 1. In the upper panel, the total energy is shown as a function of the proton (solid line) and neutron (dashed line) Fermi-levels. The BCS vacuum Fermi-level is denoted ${\lambda }_0$. In the middle panel the expectation value of the particle number before projection is shown with the desired particle number subtracted. The lower panel shows the amplitude of the projected states.Reuse & Permissions

Figure 3

Partial level scheme showing the quadrupole bands of ${}^{142}\mathrm{Gd}$ with signature $\alpha =0$. The widths of the arrows represent the relative intensities of the transitions.Reuse & Permissions

Figure 4

Possible configurations for low-energy positive-parity collective bands of ${}^{142}\mathrm{Gd}$ at spin values before excitations across the $N=82$ gap become competitive. The maximum spin values in these configurations are also listed.Reuse & Permissions

Figure 5

The total energy of different configurations of ${}^{142}\mathrm{Gd}$ calculated in the CNS formalism are shown relative to the rotating liquid drop energy, see Ref. [12]. The rotational bands are labeled by the number of $h_{11/2}$ protons and $h_{11/2}$ neutron holes, and in cases with particles excited across the $N=82$ gap, also by the number of $h_{9/2}{f}_{7/2}$ neutrons. Noncollective aligned states are encircled. When a local minimum in the ($\varepsilon ,\gamma$) plane is followed, it is specified by the approximate value of $\gamma$. More realistic calculations including pairing will change the level order somewhat below $I\approx 25$; see Fig. 11.Reuse & Permissions

Figure 6

Alignment for the different groups of orbitals of the (unpaired) [4,2] configuration at a typical deformation (${\varepsilon }_2=0.18,{\varepsilon }_4=0.03$) and for rotation around the longest ($\gamma =-{80}^{°}$), the intermediate ($\gamma =-{40}^{°}$) and the shortest ($\gamma ={40}^{°}$) principal axis. The frequency is expressed in oscillator units and should be multiplied by $~10$ to get $\hslash \omega$ in MeV. The alignment is shown for the four $g_{7/2}{d}_{5/2}$ proton holes and two $d_{3/2}{s}_{1/2}$ neutron holes by a dashed line and then the total alignment of the valence protons and neutrons is shown by a full line. Consequently, the alignment of the 4 $h_{11/2}$ protons and the 2 $h_{11/2}$ neutron holes, respectively, is the difference between the dashed and full lines. To facilitate the comparison between the three cases, the full alignment at $\gamma =-{80}^{°}$ is shown by a dotted line in the $\gamma =-{40}^{°}$ and $\gamma ={40}^{°}$ panels.Reuse & Permissions

Figure 7

Single-neutron routhians at ${\varepsilon }_2=0.18,\gamma =-{80}^{°},{\varepsilon }_4=0.03$, i.e., the deformation that is also used in Fig. 6. Virtual crossings have been removed. The orbitals are labeled by the $N$ shell (or more properly $N_{\mathrm{rot}}$ shell) to which they belong with the ordering within the $N$ shell as a subscript. Lines with dots indicate signature $\alpha =-1/2$ and dashes indicate negative signature, i.e., full lines for ($\pi ,\alpha )=(+,+1/2)$, dotted lines for (+,1/2), dashed lines for (−,+1/2), and dot-dashed lines for ($-,-1/2$). The gap corresponding to the $N=78$ configuration in the $(+,0){}_1$ band of ${}^{142}\mathrm{Gd}$ is indicated and orbitals with holes relative to the closed core $N=82$ configuration are marked by open circles.Reuse & Permissions

Figure 8

The total spin in the [4,2] configuration shown as a function of rotational frequency for the same cases as in Fig. 6, i.e., for rotation around the longer ($\gamma =-{80}^{°}$), the intermediate ($-{40}^{°}$), and the shorter (${40}^{°}$) axis at the deformation ${\varepsilon }_2=0.18,{\varepsilon }_4=0.03$. The different lines are thus simply the sum of the proton and neutron spin contributions in Fig. 6.Reuse & Permissions

Figure 9

Calculated total energy surfaces with pairing included for the $(\pi ,\alpha ){}_p(\pi ,\alpha ){}_n=(+,0){}_p(+,0){}_n$ combination of quantum numbers at $I=10,20$, and $26\hslash$. At each point, the total energy has been minimized with respect to ${\varepsilon }_4,{\lambda }_{p,n}$ and ${\Delta }_{p,n}$ as well as quasiparticle configuration. The contour line separation is 0.25 MeV. The figure is drawn for $\gamma =[-{120}^{°},{60}^{°}]$ with the $\gamma =0^{°}$ and $-{60}^{°}$ axes labeled in the $I=26$ surface.Reuse & Permissions

Figure 10

Experimental energies versus spin of the quadrupole bands of ${}^{142}\mathrm{Gd}$ are shown relative to a rigid rotation reference in the upper panel. The lower panel shows the total energy relative to the same reference for the three minima in Fig. 9. Aligned states with rotation around the symmetry axis are encircled.Reuse & Permissions

Figure 11

Total energies minimized with respect to the pairing parameters (full lines) and in the limit of no pairing (dashed lines), respectively. The energies are obtained with the CNSB formalism and, because of different macroscopic parameters (see Sec. 2) and a different reference, they differ slightly from the CNS energies in Fig. 5.Reuse & Permissions

Figure 12

Observed $B(E2;I\to I-2)$ values for quadrupole bands of ${}^{142}\mathrm{Gd}$ with a signature $\alpha =0$. The calculated values are for the three different minima in Fig. 9.Reuse & Permissions

Figure 13

Different contributions to the pair energy for $Z=64$ at ${\Delta }_p=0.9$ MeV, $\omega =0.5$ MeV, ${\lambda }_p=43.40092$ MeV, ${\varepsilon }_2=0.18,\gamma =4^{°},{\varepsilon }_4=0.026$. The total energy calculated from the $F_1$ term alone is shown with a solid line. The contribution from the $F_2$ term is shown with a dashed line and the sum of both terms is indicated with a dotted line. The inset shows the particle amplitude distribution for two different values of θ. The value at $\Delta =0$ and $\omega =0$, i.e., $-G\frac{N}{2}$, has been subtracted from the $F_2$ term.Reuse & Permissions

Figure 14

The solid line with dots shows the pairing energy $E_2$ from the $F_2$ operator [see Eq. (A2)], with the value at $I=0$ and $\Delta =0$ subtracted. The dashed lines are drawn in steps of $G$. The curve is obtained using cranking around the symmetry axis at an oblate shape (${\varepsilon }_2=0.15,\gamma ={60}^{°}$) and a small value of the pairing gap, $\Delta =0.01$.Reuse & Permissions

Figure 15

Total energies versus spin for deformations in the vicinity of two of the minima in Fig. 9. Solid lines are the result obtained by using the $F_1$ term alone and the dashed lines are the results when adding the $F_2$ term and subtracting $-G\frac{N}{2}$.Reuse & Permissions