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The CDFT+PRM provides a microscopic and quantal approach to study the wobbling motion. In the following, a classical view on the wobbling motion is provided by investigating the time evolution of wobbling motion. To study the time evolution of wobbling motion, we start from the triaxial rotor Hamiltonian, which is written as[1]
$$ \begin{aligned} H_{ \rm{TR}} = \sum_{i = 1,2,3}\frac{\hat{J}_i^2}{2\mathcal{J}_i}, \end{aligned} $$ (1) where $ \hat{J}_i $ are the components of the collective angular momentum with respect to the principal axes of the rotor and $ \mathcal{J}_i $ the corresponding moments of inertia. Classically, the angular momentum components are expressed, in the spherical coordinate frame for a given spin value I, by the polar angle $ \theta $ and azimuthal angle $ \phi $ as
$$ \begin{aligned} & J_1 = {\sqrt{I(I+1)}}\sin \theta\cos \phi,\\ & J_2 = {\sqrt{I(I+1)}}\sin \theta\sin \phi,\\ & J_3 = {\sqrt{I(I+1)}}\cos\theta. \end{aligned} $$ (2) Here, $ \theta $ is defined as the angle between J and the long($ l $) axis, and $ \phi $ is the angle between the projection of J onto the short-intermediate ($ sm $) plane and the s-axis.
The orbits of the angular momentum on the unit angular momentum sphere are determined by the implicit equation[3]
$$ \begin{aligned} E& = \frac{J_3^2}{2B(\phi)}+V(\phi),\qquad\quad\; \end{aligned} $$ (3) $$ \begin{aligned} V(\phi)& = I(I+1)\left(\frac{\sin^2\phi}{2\mathcal{J}_2} +\frac{\cos^2\phi}{2\mathcal{J}_1}\right), \end{aligned} $$ (4) $$ \begin{aligned} \frac{1}{2B(\phi)}& = \left(\frac{1}{2\mathcal{J}_3}- \frac{\sin^2\phi}{2\mathcal{J}_2} -\frac{\cos^2\phi}{2\mathcal{J}_1}\right) ,\;\;\;\; \end{aligned} $$ (5) which are obtained by the intersection lines between the sphere of constant angular momentum ${\boldsymbol{J}}^2 = J_1^2+ J_2^2+J_3^2 = I(I+1)$ and the ellipsoid of constant energy given in Eq. (1).
According to the Euler equation for a classical rotor, the angular velocity tangential to the orbit is[3]
$$ \begin{aligned} \omega_\parallel(\phi) = \frac{{\rm{d}}l_\parallel}{{\rm{d}}t} = \sqrt{\left(\frac{{\rm{d}}\phi}{{\rm{d}}t}\right)^2 +\left(\frac{{\rm{d}}\theta}{{\rm{d}}t}\right)^2},\qquad\qquad\qquad\quad\; \end{aligned} $$ (6) $$ \begin{aligned} \frac{{\rm{d}}\phi}{{\rm{d}}t} & = \sqrt{I(I+1)}\cos\theta\left(\frac{\cos^2\phi}{\mathcal{J}_1} +\frac{\sin^2\phi}{\mathcal{J}_2}-\frac{1}{\mathcal{J}_3}\right),\, \end{aligned} $$ (7) $$ \begin{aligned} \frac{{\rm{d}}\theta}{{\rm{d}}t} & = \sqrt{I(I+1)}\sin\theta\left(\frac{1}{\mathcal{J}_1} -\frac{1}{\mathcal{J}_2}\right)\sin\phi\cos\phi, \;\;\; \end{aligned} $$ (8) where $ \theta(\phi) $ can be derived from the Eq. (3) for a given energy value $ E_\nu $, i.e.,
$$ \begin{aligned} \cos^2\theta(\phi) = \frac{2B(\phi)}{I(I+1)}[E_\nu-V(\phi)]. \end{aligned} $$ (9) Therefore, once the moment of inertia and the initial condition of the triaxial rotor are known, the time evolution equation Eqs. (7)~(8) can be solved.
In Fig. 4, an example of the orientation angles $ \theta $ and $ \phi $ as well as the corresponding angular momentum components along the intermediate($ J_m $), short($ J_s $), and long($ J_l $) principal axis of the triaxial rotor are shown as functions of $ t $ for given values of $I = 8\,\hbar$ and $\mathcal{J}_m = 30\, \hbar^2/ \rm{MeV}$, $\mathcal{J}_s = 10\, \hbar^2/ \rm{MeV}$, and $\mathcal{J}_l = 5\, \hbar^2/ \rm{MeV}$. This system has been studied in the framework of triaxial rotor model[3] and has been revealed as a good example to show the wobbling picture. The initial condition is chosen as $(\theta = \pi/2, \; \phi = 2)$, which corresponding to the lowest energy state orbit calculated by Eq. (9) with $ E_1 = 1.616\; \rm{MeV} $. The $ \theta $ and $ \phi $ oscillate with respect to $ \theta = \pi/2 $ and $ \phi = \pi/2 $, respectively. Correspondingly, the angular momentum aligns mainly along the m-axis with $J_m \approx 8\,\hbar$. The $ J_s $ and $ J_l $ are small but not negligible. They drive the rotational axis precess and wobble around the axis with the largest moment of inertia. All of these characteristic present the wobbling motion with respect to the m-axis.
Figure 4. (a) The orientation angles $ \phi $ and $ \theta $ as a function of time t at $I = 8\,\hbar$ obtained by solving the Euler equation Eq. (6). (b) The corresponding angular momentum components along the intermediate($ J_m $), short($ J_s $), and long($ J_l $) principal axis of the triaxial rotor as a function of t . (color online)
Both $ \theta $ and $ \phi $ develop periodically with time under the same period of $T = 7.5\, \hbar/ \rm{MeV}$. The periodic characters are also seen for the angular momentum components. The period $ T $ will vary as the initial condition of $(\theta,\; \phi)$ or say the excitation energy $ E $. In Fig. 5, we delineate the variation of the wobbling motion periods $ T $ as a function of excitation energy, with respect to both the m-axis (the eigen-states of TRM with $n = 0\sim 4$) and the l-axis (the eigen-states of TRM with $n = 5\sim 8$) for $I = 8\,\hbar$. Our analysis reveals a distinctive behavior in these two regimes. Specifically, within the domain corresponding to m-axis wobbling, there is a noticeable increment in the period of wobbling motion as the excitation energy increases. This trend suggests that the dynamical stability associated with rotational motion around the m-axis diminishes progressively with higher excitation energies. Conversely, in the context of l-axis wobbling, an inverse relationship is observed: the period of wobbling motion exhibits a decrement with an increase in excitation energy. Such a trend indicates an enhanced dynamical favorability for rotational motion around the l-axis at higher excitation energies. Moreover, T will become infinite when the initial condition is chosen as $(\theta = \pi/2,\; \phi = 0)$, i.e., starting from the s-axis. Namely, the wobbling motion with respect to the s-axis (intermediate moment of inertia axis) does not exist. The behavior of the period can be also seen for the other spins. The period decreases with spin, indicating the faster rotation. More detailed results will be published elsewhere[68].
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摘要: 本工作简要介绍了摇摆运动的近期研究进展。目前在质量数为100, 130, 160, 190核区的奇A核与偶偶核中已报导了17个摇摆候选核。以130Ba中的两准粒子组态摇摆及三轴转子的摇摆运动为例说明偶偶核中的摇摆运动。对于130Ba,采用协变密度泛函理论(CDFT)和粒子转子模型(PRM)来进行研究。CDFT为观察到的能带的组态和变形参数提供了关键信息,并为PRM计算提供了输入量。在130Ba中,理论计算再现了实验能谱和电磁跃迁概率。角动量几何的分析揭示了两准粒子组态的横向摇摆相比单准粒子组态的横向摇摆更加稳定。对于三轴转子,利用欧拉方程来研究时间演化问题,给出了方位角($\phi $和$\theta $)以及角动量分量随时间的演化图像。研究表明,三轴转子的低能激发态主要展示围绕中间轴的摇摆运动。随着激发能量的增加,中间轴摇摆运动的周期延长。相反,在长轴摇摆的情况下呈现出一种截然不同的趋势,即激发能量的增加导致摇摆周期减少。Abstract: The recent progresses on the wobbling motion are briefly introduced. So far 17 wobbling candidates have been reported in odd-A and even-even nuclei that spread over A≈100, 130, 160 and 190 mass regions. The two-quasiparticle configuration wobbling in 130Ba and the wobbling motion in a triaxial rotor are taken as examples in this paper to show the wobbling motion in even-even nuclei. For the 130Ba, the wobbling are investigated based on the combination of the covariant density functional theory (CDFT) and the particle rotor model (PRM). The CDFT provides crucial information on the configuration and deformation parameters of observed bands, serving as input for PRM calculations. The corresponding experimental energy spectra and electromagnetic transition probabilities are reproduced. An analysis of the angular momentum geometry reveals the enhanced stability of transverse wobbling of a two-quasiparticle configuration compared to a single-quasiparticle one. For the triaxial rotor, the time evolution of wobbling motion is explored through the solution of Euler equations. This investigation yields valuable insights into the evolution of orientation angles ($\phi $ and $\theta $) and angular momentum components. Notably, the study reveals that low-energy states of a triaxial rotor predominantly exhibit wobbling motion around the intermediate axis. Moreover, an increase in excitation energy corresponds to a prolonged period of intermediate axis wobbling motion. Conversely, a contrasting trend is observed in the case of long axis wobbling, where an increase in excitation energy leads to a decrease in the wobbling period.
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Key words:
- wobbling motion /
- even-even nucleus /
- two-quasiparticle /
- time evolution
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Figure 2. The potential energy as a function of deformation $ \beta $ in adiabatic (open circles) and configuration-fixed(lines) constrained triaxial CDFT calculations with the PC-PK1 effective interaction for 130Ba. The local minima in the energy surfaces for fixed configuration are represented as stars and labeled, respectively, as A-D and a. (color online)
Figure 3. Distributions of the probability $ P(\theta, \phi) $ for the orientation of the angular momentum $ {\boldsymbol{J}} $ with respect to the body-fixed frame at $ I = 14 $ ($ n = 0 $ and $ 2 $) and 15 ($ n = 1 $ and 3). Brown indicates maximal and blue minimal probability. Here, $ \theta $ is the angle between $ {\boldsymbol{J}} $ and the $ l $-axis, and $ \phi $ is the angle between the projection of $ {\boldsymbol{J}} $ onto the $ sm $-plane and the s-axis. The panels are labeled by the wobbling phonon $ n $. (color online)
Figure 4. (a) The orientation angles $ \phi $ and $ \theta $ as a function of time t at $I = 8\,\hbar$ obtained by solving the Euler equation Eq. (6). (b) The corresponding angular momentum components along the intermediate($ J_m $), short($ J_s $), and long($ J_l $) principal axis of the triaxial rotor as a function of t . (color online)
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