Signature of Ericson fluctuations in helium inelastic scattering cross sections near the double ionization threshold

We calculated the inelastic electron impact excitation cross sections of ${\text{He}}^{+}$ by electrons for a model helium atom to examine the onset of the signature of quantum chaotic scattering in this simple system. We find Ericson fluctuations (EF) in the calculated inelastic scattering cross sections only when the impact energies lie within about 0.21 eV below the double ionization threshold. We also discuss the stringent requirements and the proper methods for analyzing the inelastic scattering cross sections in order to observe EF experimentally.

Figure 1

(Color online) (a) Adiabatic hyperspherical potential curves for $s^2$-model helium for states that are associated with the ${\text{He}}^{+}\left(N=6\right)$ to ${\text{He}}^{+}\left(N=9\right)$ thresholds. (b) Energies of the bound states associated with each potential curve calculated in the single-channel approximation. (c) Partial $1s\to 5s$ inelastic scattering cross sections vs the total energy of the two electrons. The energy region covered is between the $N=6$ and $N=7$ thresholds of ${\text{He}}^{+}$.

Figure 2

(Color online) Same as Fig. 1 but for states associated with $I_{21}$ to $I_{27}$ thresholds of ${\text{He}}^{+}$. In (c), inelastic scattering cross sections from 18$\mathit{s}$ to 19$\mathit{s}$ are calculated for the energy region between the $N=21$ and $N=22$ thresholds, vs the total energy of the two electrons.

Figure 3

Variation of inelastic scattering cross sections with the number of “intruder” channels. The cross sections shown are for the $10s\to 15s$ transitions in the energy region of $I_{16}$ and $I_{17}$ thresholds. (a) The uppermost channel included is $N=17$. The spectra show regular autoionizing Rydberg series. By adding the $N=18$ channel in the calculations, the intruder channel modifies the spectra locally, but also shifts the spectra (b). By adding one more channel $\left(N=19\right)$ (c) and two more channels $\left(N=19,20\right)$, the calculated spectra change significantly and the resulting spectra appear rather erratic, or chaotic. Note the regular sharp features as in (a) and (b) change into complicated “real” spectra as shown in (d), as the number of intruder channels increases.

Figure 4

Autocorrelation function extracted from cross sections on the energy scale for (a) $10s\to 15s$ excitation in the energy region between $I_{16}$ and $I_{17}$. (b) Same for the $1s\to 5s$ excitation in the $I_6\text{−}{I}_7$ region. Note that the autocorrelation functions in the two energy regimes do not differ much, even though the spectra for the former is chaotic while for the latter it is quite regular.

Figure 5

Inelastic scattering cross sections expressed on the $n$-scale (left column) and the corresponding autocorrelation functions (right column). (a) $18s\to 19s$ excitation between $I_{21}$ and $I_{22}$. (b) $10s\to 15s$ excitation between $I_{16}$ and $I_{17}$. (c) $3s\to 5s$ excitation between $I_9$ and $I_{10}$. (d) $1s\to 5s$ excitation between $I_6$ and $I_7$. (e) $1s\to 2s$ excitation between $I_3$ and $I_4$.

Figure 6

(Color online) Illustration of how the “chaos” in the electron spectra between $I_{16}$ and $I_{17}$ and the corresponding autocorrelation functions are washed out by the energy resolution of the electron spectrometer. Gaussian functions with FWHMs of 0.3, 0.7, 1.4 meV are used to convolute the spectra, respectively.