Explicit analysis of chaotic behavior in radial and angular motion in doubly excited helium

Total photoionization cross sections of doubly excited He have been measured up to the single ionization threshold $I_{15}$ of ${\mathrm{He}}^{+}$ and calculated up to $I_{17}$ revealing excellent agreement. The fact that all spectra are dominated by principal Rydberg series is explained by the observation that the quantity $F=N-K$, representing angular motion, is an approximate quantum number for a large fraction of states. While the nearest-neighbor energy spacings between resonances of Rydberg series with small $F$ approach a Wigner distribution, reflecting chaos in radial motion, there is no indication for a transition to full chaos including angular motion.

Figure 1

(Color online) Total photoionization yield of doubly excited He below $I_{15}$ (in units of ${10}^{-3}$ of signal background, s.b.): Experimental data (top, red) and theoretical results (bottom, blue). The vertical bars indicate the positions of the SITs from $I_8$ to $I_{15}$; for details, see text.

Figure 2

(Color online) Calculated $⟨\cos {\theta }_{12}⟩$ values as a function of resonance energy $E$ below $I_{17}$. Each point represents a particular resonance, and the positions of $I_N$ are marked by thick vertical bars in the bottom part. The different colors distinguish resonances below the ionization thresholds $I_N$: Violet $(N=9)$, red (10), cyan (11), green (12), orange (13), blue (14), magenta (15), black (16), and brown (17). The principal Rydberg series $F=N-K=2$ and the series $13,-{12}_n$ are marked by a dashed black line and an encircled dashed ellipse, respectively.

Figure 3

(Color online) Integrated NNS distributions for resonances of those Rydberg series in the region ${\theta }_{12}>120°$ below different $I_N$ that have constant $F$ values within a series. In addition, a Wigner distribution representing a chaotic system (dashed line) and the integrated NNS distribution for the regular Rydberg series $13,-{12}_n$ are shown.