Spin squeezed states and wobbling motion in a collective Hamiltonian

A semiclassical approach combined with adiabatic approximation is applied to the particle+triaxial rotor (PTR) model in order to derive the collective potential and mass parameters of a collective Hamiltonian (CH) that depends only on the total angular momentum for the wobbling motion. The CH approximates very well the energies and $E2$ transition probabilities associated with the lowest wobbling states in both even-even and odd-mass systems. The spin squeezed state (SSS) representation with the character of a wave function in orientation angle of the total angular momentum is introduced to illustrate on the structure of the states. The probability distribution for the lowest states of the PTR and the CH agree well. This demonstrates that the method can be used to incorporate collective wobbling into the microscopic tilted axis cranking approach. A classification scheme for the wobbling mode based on the SSS states is discussed.

Figure 1

Schematic illustration of the different states used as basis sets in the $\mathit{j}$ space. Each state is represented by a curve along which the endpoint of angular momentum vector is distributed with equal probability. The $k$ states are circles of constant latitude $k$. The $\varphi$ states are semicircles of constant $\varphi$. The figure displays the pertaining meridians. For the continuous set of SCS only the states $|\theta =0,\varphi =0\rangle$ and $|\theta =\pi /2,\varphi =0\rangle$ are shown. For the continuous set of SSS only the states $|\varphi =0\rangle$ and $|\varphi =-\pi /4\rangle$ are shown.

Figure 2

Upper panel: Classical potential energy $V\left(\varphi \right)$ as a function of $\varphi$ for spin $I=8$. The dots show the quantal energies $E_{\nu }$ calculated by TRM, while the horizontal lines represent the energies obtained by the collective Hamiltonian $H_{\text{CH}}$, which incorporates a $\varphi$-dependent mass parameters $B\left(\varphi \right)$ shown in the lower panel. Note that for the even-$n$ states, they are calculated from $I=8$ states, while for the odd-$n$ states, they are calculated as the $72/90$ of $I=9$ energies.

Figure 3

Probability densities of the SSS for the triaxial rotor states 1–5 of $I=8$ and the state 1 of $I=9$. The smooth solid and dashed lines show the continuous SSS $P{\left(\varphi \right)}_{\nu }$ obtained by the TRM and collective Hamiltonian, respectively. The dots represent the discrete SSS probability $\frac{2\pi }{2I+1}P{\left({\varphi }_n\right)}_{\nu }$.

Figure 4

Collective potential energy $V\left(\varphi \right)$ (upper two panels) and mass parameter $B\left(\varphi \right)$ (lower two panels) as functions of $\varphi$ for the particle-rotor system of ${}^{135}\mathrm{Pr}$. The dots show the PTR energies $E_{\text{PTR}}$, while the horizontal lines show the collective Hamiltonian energies $E_{\text{CH}}$. Their energies are plotted with respect to the minima of the collective potential $E_{\text{ad}}\left(\text{min}\right)$.

Figure 5

Energies of the lowest states $E_{\nu }$ of the PTR Hamiltonian (symbols) and the collective Hamiltonian (full drawn lines), the adiabatic energy (dashed) and the TAC energy (dashed-dotted), for ${}^{135}\mathrm{Pr}$. The energies are shifted by 1.743 MeV, which is the lowest energy from the diagonalization of $h_p\left(\gamma \right)$ in Eq. (25). In the following, the yrast states ($n=0$) are denoted by $11/2_1$, $15/2_1$, $19/2_1,\cdots$, the single wobbling excitations ($n=1$) by $13/2_1$, $17/2_1$, $21/2_1,\cdots$, and the double wobbling excitations ($n=2$) by $15/2_2$, $19/2_2$, $23/2_2,\cdots$.

Figure 6

Upper panel: Calculated in-band $B{\left(E2\right)}_{\text{in}}(I\to I-2)$ of the bands in Fig. 5, where the same line and color conventions are used. Lower panel: Calculated interband $B{\left(E2\right)}_{\text{out}}(I\to I-1)$ transition probabilities between the bands shown in Fig. 5. In order to keep the legend compact, the connecting transitions are labeled by quanta numbers $n$, which are associated with the oscillator states in the TW regime. Beyond, $n$ represents just a counting label for the states.

Figure 7

Full curves show the probability density of the SSS for the $n=0-3$ states calculated by PTR, while the dashed curves display the ones by the CH.

Figure 8

The SSS probability density for the odd particle calculated by PTR for the $n=0-3$ states.

Figure 9

The azimuthal angle $\phi$ of $\mathit{j}$ as a function of azimuthal angle $\varphi$ of $\mathit{J}$ obtained by diagonalization of $H_{\text{ad}}$ for the selected spins $I=13/2$, $21/2$, $29/2$, $37/2$, and $45/2$. The dot on the curve label the potential minimum for each spin.

Figure 10

Same as Fig. 4, but for the case of ${}^{130}\mathrm{Ba}$.

Figure 11

Same as Fig. 5, but for the case of ${}^{130}\mathrm{Ba}$. Note that the energies obtained from the collective Hamiltonian are 0.2 MeV shifted down.

Figure 12

Same as Fig. 6, but for the case of ${}^{130}\mathrm{Ba}$.

Figure 13

Same as Fig. 7, but for the case of ${}^{130}\mathrm{Ba}$.