Interpretation of the high spin states in ${}^{161}\mathrm{Lu}$: A paired and unpaired study

A paired cranked Nilsson-Strutinsky-Bogoliubov (CNSB) model is presented, which employs the same method to calculate the liquid-drop energy and moment of inertia as the unpaired cranked Nilsson-Strutinsky (CNS) model. In the CNSB model, the energy minimization is carried out in the mesh of pairing gaps $\Delta$ and Fermi levels $\lambda$ as well as deformation parameters. The high spin states in ${}^{161}\mathrm{Lu}$ are then investigated with the CNSB and CNS models. The terminating structure shows a striking similarity with these two models. Combining the CNSB and CNS models, a complete understanding of high spin structures, including the normal deformed (ND) and triaxial strongly deformed (TSD) bands and observed side bands in ${}^{161}\mathrm{Lu}$, is achieved. It appears that the only important paired crossings are the first $i_{13/2}$ neutron crossing and the first $h_{11/2}$ proton crossing. For the description of the unpaired high spin crossings, it is important to be able to distinguish between the pseudospin partners in the proton $\mathcal{N}=4$ shell, $(d_{5/2}$,$g_{7/2}$) and $(d_{3/2}$,$s_{1/2}$). The yrast bands are predicted to terminate, which explains the structure of a TSD-like band X2. A band crossing at $I\approx 36.5$ for the TSD band in ${}^{161}\mathrm{Lu}$, unique within the chain of even-$N$ Lu isotopes, is well described by the CNSB model.

Figure 1

Unpaired (a) and paired (b) calculation results and their differences (c) in ${}^{161}\mathrm{Lu}$ for the positive parity configurations. The aligned states at $\gamma ={60}^{\circ }$ are indicated by large open circles.

Figure 2

Same as previous figure but for the negative parity configurations.

Figure 3

Single-particle energies $e_i$ vs aligned spin $m_i$ for protons (a) and neutrons (b) at $({\varepsilon }_2,\gamma )=(0.1,{60}^{\circ })$. Tilted Fermi surfaces for $Z=71$ and 68, and $N=90$ are drawn for the low-energy terminating spin states. For $Z=71$ a proton spin state $I_p=19.5$ with positive or negative parity is formed dependent on if the occupied $m_i=1/2$ orbital belongs to the $h_{11/2}$ or $d_{3/2}$ subshell. It is shown by an arrow how the deexcitation from $m_i=1/2$ to $m_i=-3/2$ in the proton $h_{11/2}$ and $d_{3/2}$ shells leads to proton spin $I_p=17.5^{\pm }$ for $Z=71$ and from $m_i=5/2$ to $m_i=-7/2$ in the neutron $h_{9/2}{f}_{7/2}$ shells leads to neutron spins $I_n={27}^{-}$ and ${24}^{+}$ (starting from the $I_n={33}^{-}$ and ${30}^{+}$ states). Similarly, starting from the proton $19.5^{+}$ state, a deexcitation in the $h_{11/2}$ orbitals, $m_i=1/2\to -11/2$ leads to an $I_p=13.5^{+}$ state corresponding to the tilted Fermi surface drawn by a dashed line.

Figure 4

Calculated PESs for the configuration $(+,-1/2)(-,1)$ with the CNS (top) and CNSB (bottom) models. The global minima are marked by stars. The energy interval of contour lines is 0.2 MeV.

Figure 5

The collective and noncollective states in the CNS (a) and CNSB (b) calculations for the $(+,-1/2)(+,0)$ and $(+,-1/2)(-,1)$ configurations. Aligned states at $\gamma ={60}^{\circ }$ are encircled. The full configuration for the two bands are specified relative to a ${}^{146}\mathrm{Gd}$ core in panel (a). For nonfully aligned particles in the valence space, the spin content is specified by subscript.

Figure 6

The experimental (a) and theoretical (b) energies and their differences (c) for the positive parity bands in ${}^{161}\mathrm{Lu}$. The subscript 2 indicates the theoretical yrare bands. Generally, the same symbols and line types are used in the three panels for the observed bands and the configurations assigned to them. However, the low spin states labeled as [411] 1/2 are compared with the $(+,-1/2)(+,0)$ and $(+,-1/2)(+,0)$ configurations, which are drawn by red squares because the high spin region of the $(+,-1/2)(+,0)$ configuration is assigned to the X2 band. Considering that no linking transition between the TSD1 and ND bands was observed, the energy of the lowest $21/2^{+}$ state of TSD1 was placed at 2.7 MeV so that this band is slightly higher in energy than the ND bands.

Figure 7

Same as previous figure but for the negative parity bands in ${}^{161}\mathrm{Lu}$. The configuration of the experimental yrast bands is assigned as $\pi \left[523\right]7/2$ which is different from the previous assignment $\pi \left[514\right]9/2$ in Ref. [14]; see text for details.

Figure 8

The PESs of the configuration $(-,-1/2)(+,0)$.

Figure 9

The transition quadrupole moments for the ND and TSD band in ${}^{161}\mathrm{Lu}$. The experimental value of the ND bands is taken from Ref. [11] and a 10% error is added.

Figure 10

The single-proton Routhians drawn at a typical deformation of the high spin bands in ${}^{161}\mathrm{Lu}$. At $\omega =0$, the orbitals are labeled according to their main amplitudes, $dg$ $(d_{5/2}$ and $g_{7/2}$), $sd$ $(d_{3/2}$ and $s_{1/2}$), and $h_{11/2}$, respectively, with the ordering within the groups as subscripts. Approximate asymptotic quantum numbers are also specified. The filling of the orbitals in the configurations associated with the X2 and X4 bands is illustrated. Furthermore, the configuration with five $h_{11/2}$ and two $sd$ protons which is predicted to terminate in a low-energy state at $I=49.5^{-}$ is illustrated at $\hslash \omega =0.51$ MeV.

Figure 11

Comparison between the $\left[404\right]7/2$ and X2 bands and the CNS configurations assigned to them. In the middle panel, the filling of the proton orbitals is specified relative to a closed $Z=64$ core, with the notation defined in the Fig. 10 caption. The detailed configurations relative to a ${}^{146}\mathrm{Gd}$${}_{82}$ core, $\pi {\left(dg\right)}^{-p_1}{\left(sd\right)}^{p_2}{\left(h_{11/2}\right)}^{p_3}\nu {\left(f_{7/2}{h}_{9/2}\right)}^{n_1}{\left(i_{13/2}\right)}^{n_2}$, are given in the legend as $[p_1{p}_2{p}_3,n_1{n}_2]$. For an odd number of particles in a group, the subscript + or $-$ defines the signature, $\alpha =1/2$ or $\alpha =-1/2$.

Figure 12

Same as Fig. 11, but for the X4 band and related configurations.

Figure 13

The PES of the $(+,+1/2)(+,0)$ configuration at $I=34.5$ in ${}^{161}\mathrm{Lu}$. The ND, ED, and TSD indicate the normal deformed, enhanced deformed, and triaxial strongly deformed shapes.