Theoretical analysis of Young-type electron interference in $\mathrm{H}{\mathrm{e}}^{2+}+{\mathrm{H}}_2$ collisions using a semiclassical model

A four-body semiclassical model is developed to describe interferences observed in the angular distribution of Auger electrons emitted after double capture in 30-keV $\mathrm{H}{\mathrm{e}}^{2+}+{\mathrm{H}}_2$ collisions. The present model is based on both the corpuscular and wave behaviors of the emitted electron. The corpuscular aspect is used to determine the trajectories of the collision partners, while the wave behavior occurs only in the determination of the phase shift. The results of the calculation are found to reproduce the experiment remarkably well. Series of maxima and minima are found in the angular distribution, with periods that are close to the experimental values. In addition, at a fixed angle, oscillations in the energy distribution are clearly evidenced in both the experiment and calculation.

Figure 1

Oscillation term in the angular distribution of Auger electrons following 30-keV $\mathrm{H}{\mathrm{e}}^{2+}+{\mathrm{H}}_2$ collision. Open circles: experiment. Dashed curve: fit of the experimental oscillations. Full curve: result of the calculation using relation (2).

Figure 2

Fast Fourier transform (FFT) of the angular distributions shown in Fig. 1. Open circles: experiment. Dashed curve: fit of the experiment. Full line: full calculation using relation (2). The latter result is multiplied by 4000 to emphasize the structures. The maxima are depicted by arrows.

Figure 3

Schematic three-dimensional view of initial and final conditions of an emitted electron in the field of two protons $P$₁ and $P$₂. At $t=0$, the angles ${\theta }_e^m$ and ${\phi }_e^m$ characterize the direction of the velocity $\overrightarrow{v_e^m}$ of the electron, which is located at $z_e^m$ from $O$. The angles ${\theta }_{\mathrm{H}}^m$ and ${\phi }_{\mathrm{H}}^m$ characterize the position of the doubly ionized target. At $t=t_{\max }$, the electron is at a distance $r_{\mathrm{e}}$ from $O$ and has a velocity $\overrightarrow{v_e}$. The quantities ${\theta }_e^V$ and ${\theta }_e^r$ are the angles between the incident beam direction and the final electron velocity $\overrightarrow{v_e}$ and position $\overrightarrow{r_e}$, respectively.

Figure 4

Schematic two-dimensional view of two electron trajectories. An electron is detected at $M_{1y}$ on the detector, while the second electron is delayed. The missing distance is ${\delta }_{12}=M_{2y}{M}_y^{\prime }$. The projection of the trajectories on a plan is used to increase the statistics.

Figure 5

Auger electron angular distribution (a) calculated using the 4BSC model (see text), in the range 90°–180°. Series of maxima and minima are clearly observed. (b) presents the FFT of the calculation (full squares). The experimental FFT is also shown for comparison. Three maxima are observed, giving evidence for Young-type interferences in the calculated angular distribution.

Figure 6

Energy distribution of Auger electrons following deexcitation of $2ln{l}^{\prime }$ configurations, calculated using the 4BSC model (a), for a detection angle of 160°. Dashed curve: no interferences between the structures. Full curve: interferences between Auger peaks are taken into account. (b) shows the respective FFT amplitudes. When interferences between Auger peaks are taken into account, a large structure is observed, with an average frequency of the order of $4\mathrm{e}{\mathrm{V}}^{-1}$.

Figure 7

Experimental energy distribution of Auger electrons following deexcitation of $2ln{l}^{\prime }$ configurations [open circles in (a)], at a detection angle of 160°. The full curve fits the experimental structures. (b) shows the respective FFT amplitudes. Full curve: FFT amplitude for the fit. Open circles: FFT amplitude associated with the experiment. The inset in (a) increases the visibility of the oscillating structures. The full line in the inset is used as a guide for the eye.