Wobbling motion in ${}^{135}\mathrm{Pr}$ within a collective Hamiltonian

The recently reported wobbling bands in ${}^{135}\mathrm{Pr}$ are investigated by the collective Hamiltonian, in which the collective parameters, including the collective potential and the mass parameter, are respectively determined from the tilted axis cranking (TAC) model and the harmonic frozen alignment (HFA) formula. It is shown that the experimental energy spectra of both yrast and wobbling bands are well reproduced by the collective Hamiltonian. It is confirmed that the wobbling mode in ${}^{135}\mathrm{Pr}$ changes from transverse to longitudinal with the rotational frequency. The mechanism of this transition is revealed by analyzing the effective moments of inertia of the three principal axes, and the corresponding variation trend of the wobbling frequency is determined by the softness and shapes of the collective potential.

Figure 1

Sketch of the angular momentum vector of the proton particle with respect to the principal axis frame.

Figure 2

Contour plots of the total Routhian surface calculation $E^{\prime }(\theta ,\phi )$ for ${}^{135}\mathrm{Pr}$ at the frequencies $\hslash \omega =0.10$, 0.30, 0.50, and $0.70\mathrm{MeV}$. All energies at each rotational frequency are normalized with respect to the absolute minimum.

Figure 3

${\phi }_{min}$, i.e., the $\phi$ which minimizes the total Routhian surface, as a function of rotational frequency and the extracted collective potential $V\left(\phi \right)$ at $\hslash \omega =0.30$, 0.50, and $0.70\mathrm{MeV}$.

Figure 4

Rotational frequency (upper panel) and rotational energy spectra (lower panel) of the yrast band in ${}^{135}\mathrm{Pr}$ as functions of the angular momentum calculated by TAC (open squares) in comparison with the data (solid dots) of Ref. [11]. In the TAC calculations, the quantal correction $1/2$ has been extracted for the angular momenta [35].

Figure 5

The calculated mass parameter (upper panel) as well as the effective MOIs of the three principal axes ${\mathcal{J}}_1^{*}, {\mathcal{J}}_2^{*}$, and ${\mathcal{J}}_3$ (lower panel) as a function of rotational frequency $\hslash \omega$. A diagrammatic sketch of the angular momentum vector of the proton particle with respect to the principal axis frame is also shown.

Figure 6

Energy spectra of the yrast and wobbling bands (a) and the corresponding wobbling frequency (b) in ${}^{135}\mathrm{Pr}$ as functions of the angular momentum calculated by the collective Hamiltonian in comparison with the data of Ref. [11]. In the collective Hamiltonian results, the angular momenta are calculated from the TAC model. Similar comparisons with PRM are shown in (c) and (d).

Figure 7

Collective potential calculated by TAC model and two lowest collective energy levels obtained from the collective Hamiltonian at rotational frequencies $\hslash \omega =0.20-0.70\mathrm{MeV}$. The wobbling frequency $E_{\mathrm{wob}}$ for each rotational frequency is also shown.

Figure 8

Collective wave functions obtained from the collective Hamiltonian at rotational frequency $\hslash \omega =0.20-0.70\mathrm{MeV}$.