Simultaneous description of four positive parity bands and four negative parity bands

The extended coherent state model is further extended to describe two dipole bands of different parities. The formalism provides a consistent description of eight rotational bands. A unified description for spherical, transitional, and deformed nuclei is possible. Projecting out the angular momentum and parity from a sole state, the $K^{\pi }=1^{+}$ band acquires a magnetic character, whereas the electric properties prevail for the other band. Signatures for a static octupole deformation in some states of the dipole bands are pointed out. Some properties that distinguish between the dipole band states and states of the same parity but belonging to other bands are mentioned. Interesting features concerning the decay properties of the two bands are found. Numerical applications are made for ${}^{158}\mathrm{Gd}$, ${}^{172}\mathrm{Yb}$, ${}^{228,232}\mathrm{Th}$, ${}^{226}\mathrm{Ra}$, ${}^{238}\mathrm{U}$, and ${}^{238}\mathrm{Pu}$, and the results are compared with the available data.

Figure 1

The structure coefficient ${\mathcal{C}}_1$, determined as explained in the text, is represented as function of $A-0.5*(N-Z)$ (black square). The obtained values are interpolated by a thid-order polynomial (full line curve).

Figure 2

The structure coefficient ${\mathcal{C}}_2$, determined as explained in the text, is represented as function of $A-0.5*(N-Z)$ (black square). The obtained values are interpolated by a third-order polynomial (full line curve).

Figure 3

(Upper left panel) The dynamic moment of inertia is plotted as function of angular frequency. (Upper right panel) The angle between the angular momenta ${\overrightarrow{J}}_2$ and ${\overrightarrow{J}}_3$ is represented as function of angular momentum. (Low-left panel) The first-order energy displacement function is plotted vs. angular momentum. (Low-right panel) The second-order energy displacement is plotted as function of angular momentum. All theoretical results correspond to ${}^{158}\mathrm{Gd}$.

Figure 4

The same as in Fig. 3 but for ${}^{228}\mathrm{Th}$.

Figure 5

The same as in Fig. 3 but for ${}^{232}\mathrm{Th}$.

Figure 6

The same as in Fig. 3 but for ${}^{238}\mathrm{U}$.

Figure 7

The same as in Fig. 3 but for ${}^{238}\mathrm{Pu}$.

Figure 8

The angle between quadrupole and octupole angular momenta in the negative (up triangle) and positive (dagger) ground bands for the nuclei mentioned in the four panels.

Figure 9

The same as described in the legend to Fig. 8 but for ${}^{238}\mathrm{U}$, ${}^{238}\mathrm{Pu}$, and ${}^{226}\mathrm{Ra}$.

Figure 10

The experimental branching ratios for the bands $K^{\pi }=1^{-}$ (black circle) and $K^{\pi }=0^{-}$ (down triangle) in ${}^{172}\mathrm{Yb}$ are given as function of angular momentum. For comparison the calculated branching ratios, represented by stars and up triangles, respectively, are given as function of angular momentum. The transition operator involves the anharmonic term given by Eq. (31).

Figure 11

The same as in Fig. 10 but for ${}^{226}\mathrm{Ra}$. For a better representation the results for the $K^{\pi }=1^{-}$ band are divided by 5. The experimental data for this band are lacking.

Figure 12

The reduced matrix element for the electric dipole transition $J^{-}\to (J-1){}^{+}$ with $J=\mathrm{odd}$ and $J^{+}\to (J-1){}^{-}$ for $J=\mathrm{even}$.

Figure 13

The $B(E2)$ values for the intraband transitions for the $K^{\pi }=1^{+}$ and $K^{\pi }=1^{-}$ divided by the value corresponding to the transition $2_g^{†}\to 0_g^{†}$ are given as function of angular momentum. The ratio is denoted by $R_q$ according to Eq. (35). For comparison, the intraband $B(E2)$ values characterizing the ground band are also given.