Classical chaos in the geometric collective model

Collective vibrations and rotations of atomic nuclei are investigated from the classical viewpoint within the geometric collective model. It is shown that the model, even in its truncated form, exhibits a very sophisticated interplay of regular and chaotic modes of motions. We quantify the proportion of regular and chaotic orbits in the phase space and analyze its sensitive dependence on the model control parameters, energy, and angular momentum. A quasi-regular region is observed at low excitation energies in a bounded domain of the control parameter on the deformed side of the shape phase diagram, and another one at higher energies around the transition between spherical and deformed shapes. We also demonstrate a tendency for overall suppression or enhancement of chaos with angular momentum, depending on the values of control parameters.

Figure 1

Example of the GCM dynamics for Lagrangian parameters $A=-5.05,B=C=K=1$, absolute energy $E=0$, and angular momentum $\mathbf{J}=0$. For $\mathbf{J}=0$, only the coordinates ${\alpha }_0$ and ${\alpha }_2^R$ and the corresponding velocities represent active state variables, while the others can be set to zero; see Sec. 2d. (a) So-called Poincaré section of the four-dimensional phase space at ${\alpha }_2^R=0$. We denote $\eta \equiv {\alpha }_0,v_{\eta }\equiv \stackrel{·}{\eta }$. Each dot corresponds to the passage of one trajectory through the plane (in this figure we show altogether 50,000 such crossings of 52 randomly chosen trajectories). (b)–(e) We exemplify some regular trajectories contributing to the phase portrayal in panel (a), the plane of section being represented by the horizontal line. The evolution time was set to 50 time units (connected with the scaling constant $c$; see Sec. 2b).

Figure 2

GCM phase diagram in the plane of parameters $A$ and $B$ for any $C>0$. Thick curves represent phase separatrices (first order) between spherical (I), prolate (II), and oblate (III) axially symmetric equilibrium shapes; O is the triple point (continuous transition). Thin parabolas (without point O) correspond to dynamically equivalent parameter sets related by scaling transformations (Sec. 2b).

Figure 3

(a) $\xi =0$ Poincaré section (cf. Fig. 1) with 52 randomly chosen trajectories for $A=-2.6$ at $E=24.4$. (b)–(d) Exemples of three regular orbits that evolve for (b), (c) 30 or (d) 75 time units. (e)–(g) The system from (a) is subject to rotations with increasing angular momentum $j=J_z/J_{\max }$ (see Sec. 3b). We see that the patterns (e.g., a pair of “fish”) present in Poincaré section (a) gradually disappear as the $\eta \times \stackrel{·}{\eta }$ plane ceases to uniquely characterize the $\xi =0$ section of the phase space for $j>0$.

Figure 4

Regular fraction (27) as a function of the control parameter $A$ in scaling ($i$) for various values of energy $E$. The shown range of $f_{\mathrm{reg}}$ covers the whole definition interval $[0,1]$.

Figure 5

Regular fraction (27) as a function of energy $E$ for several values of the control parameter $A$ in scaling ($i$). More dependences for $A<0$ were given in Ref. [15]. Minimal energies for $A=0$ and $A\ge \frac{1}{4}$ are $E_{\min }=-0.105$ (symbol a) and $E_{\min }=0$ (symbol b), respectively. The thick solid line represents a logarithmic approximation $f_{\mathrm{reg}}=0.363\log {}_{10}E-0.285$ valid at high excitation energies with $A=0$.

Figure 6

Lower energy bound $E_{\mathrm{ch}}(A)$ for partially chaotic dynamics is compared with the energy $E_{\mathrm{cc}}(A)$ of the convex-concave transition of the border of kinematically accessible area in the plane of deformation parameters. The curve $E_{\mathrm{cc}}^{\text{'}}(A)$ demarcates the change of the concave area back to convex at higher energies.

Figure 7

$\xi =0$ Poincaré sections at $A=-0.84$ for various values of energy displayed in Fig. 8.

Figure 8

Potential (16) for $A=-0.84$ with energy levels corresponding to the Poincaré sections shown in Fig. 7. Level ($i$) corresponds to the intermediate maximum of regularity. The vertical scale is compressed according to $(V+1){}^{1/4}$, which demonstrates the $V\propto {\beta }^4$ high-energy asymptotics.

Figure 9

Comparison of (a) two initially close trajectories in the plane of deformation parameters for zero and nonzero spin and (b), (c) the corresponding plot of the quadrupole tensor eigenvalues. The avoided crossing of the eigenvalues at $t\approx 1$ for $J_z\ne 0$ (b) is in contrast to the real crossing for $\mathbf{J}=0$ (c), which leads to the sudden divergence of both trajectories shown in (a). Calculation done for $A=-1,B=0.55,E=0$ and $J_z=0$ or 5% of the maximal spin $J_{\max }$.

Figure 10

Two sections of the dependence of maximal angular momentum $J_{\max }$ on (a) control parameter $B$ and (b) energy $E$.

Figure 11

Comparison of $f_{\mathrm{reg}}$ (dotted) and $F_{\mathrm{reg}}$ (full line) from Eqs. (27) and (29), respectively, for $\mathbf{J}=0,E=0$, and $A=-1$.

Figure 12

Dependence of the regular fraction $F_{\mathrm{reg}}$ from Eq. (29) on parameter $B$ for $A=-1,E=0$, and various values of the relative angular momentum $j$.