Interference effects for intermediate energy electron-impact ionization of H2 and N2 molecules

We have studied electron impact ionization of H2 and N2 molecules at intermediate energies to look for possible two center interference effects experimentally and theoretically. Here we report a study of the interference factor I for 250 eV electron-impact ionization. The experimental measurements are performed using a crossed-beam-type electron-electron coincidence spectrometer and theoretical calculations are obtained using the Molecular Three Body Distorted Wave Approximation (M3DW). We found that the I-factor demonstrated strong evidence for two-center interference effects for both H2 and N2. We also found that the I-factor is more sensitive to projectile angular scans than to ejected electron energy scans which indicate that for the present set of kinematics the diffraction of the projectile from two scattering centers is more important than interference between electron waves emitted from two different centers.


Introduction
Electron impact ionization cross sections have been measured and calculated since the early days of collision physics due to the wealth of information that can be obtained about the collision dynamics and also due to their relevance in many application areas. Because of the importance of the cross sections in biological applications, much emphasis has been devoted recently to the experimental studies of electron-impact ionization cross sections of molecules and radicals. The study of the energy and angular distributions of electrons ejected by electron impact is a sensitive means of testing the theory of collision processes.
The possibility of a diatomic molecule acting as a microscopic double slit for photon impact was suggested by Cohen and Fano several decades ago [1]. Since quantum particles are also waves, the next natural question was whether Young's type interference effects could also be observable in triple differential cross section (TDCS) spectra for electron-impact ionization of diatomic molecules. Young type interference effects resulting from the coherent superposition of the scattered waves from two atomic centers were predicted by Stia et al. [2][3] and by Gao et al. [4] for electron impact ionization of molecular H2 and molecular N2, respectively. However, due to the small cross sections, the experimental investigations of the ionization of small molecules by this technique are limited. There have been two experimental studies, in coplanar asymmetric geometry, presented for electron impact ionization of H2 (Milne-Brownlie et al. in 2006 [5] and Casagrande et al. in 2008 [6]). These studies found evidence for interference effects by comparing the relative sizes of the binary and recoil peaks.
The theoretical basis for these studies lies in the interference factor which is defined to be the ratio of the molecular cross section divided by the cross section for two atoms. The idea is that dividing by the atomic cross section should isolate the molecular two-center effects. Cohen and Fano [1] showed that, for photon impact on H2, this factor could be approximated as where ρ0 denotes the equilibrium internuclear vector of the molecular target and χ is the momentum transfer. Stia et al. [2][3] showed that the same approximation could be used for electron impact ionization of H2. This factor predicts that the recoil peaks for atomic H should be either larger or smaller than molecular H2 depending on the kinematics. Due to the difficulty in measuring atomic H cross sections, the experimentalists substituted He for two atomic H and found the predicted enhancement/suppression in the recoil peak.
Gao et al. [4] predicted similar interference effects for ionization of the 3σg state of N2. Murray et al. [7] and Hargreaves et al. [8] performed experiments for ionization of N2 for both asymmetric and symmetric geometries. However, their results did not provide strong evidence for these interference effects.
There are three different types of two center interference effects for electron-impact scattering: (1) Incident electron being diffracted by two scattering centers; (2) scattered electron wave being emitted from two centers; and (3) ejected electron wave being emitted from two centers. We performed calculations for the three different types of possible interference effects for ionization of H2 and we found that the most important contribution comes from the incident projectile diffracting from two scattering centers [9][10]. As an extension of this study, we have now examined electron impact ionization of N2 molecules at intermediate energies to look for interference effects both experimentally and theoretically. The experimental measurements are performed using a crossed-beam-type electronelectron coincidence spectrometer and theoretical calculations are obtained using the Molecular Three Body Distorted Wave Approximation (M3DW) [11].

Experimental Apparatus
The measurements were performed at e-COL laboratory (Afyon, Turkey) using an electron spectrometer that is designed for electron-electron coincidence (e,2e) experiments. The details of the electron spectrometer are described in detail in previous papers [9,[12][13][14]. Briefly, the electron spectrometer consists of a electron gun producing a beam of electrons which passes through the gas target perpendicularly, two hemispherical electrostatic energy analyzers, a Faraday cup and a data acquisition system (see in figure 1a). The spectrometer is contained in a cylindrical stainless steel vacuum chamber and the pressure in the chamber is around ~2x10 -6 mbar while the experiment is running. The spectrometer operated at an electron current of ~2 µA with a resolution of ~0.7 eV. The (e,2e) technique is used to detect two outgoing electrons in coincidence after the ionization of the target. Two electrons of the desired energies are detected and amplified using Channel Electron Multipliers (CEM). This technique has an advantage of obtaining single ionization events meaning the outgoing electrons have originated from the same ionization event. To do this, time correlation between the detected electrons are taken into consideration and time delay between the electrons is converted to a signal that is measured by computer, and a narrow coincidence peak in the timing spectrum is observed. Coincidence electronics are shown in figure 1b.
The overall energy resolution of the coincidence system was limited by both the thermal spread of the electrons emitted from the tungsten hairpin cathode and the analyzer system. By measuring the full width at half-maximum (FWHM) of the binding energy spectra, the coincidence energy resolution was found to be ≈1.5 eV. This resolution is good enough to separate the 3σg orbital of the N2 molecule.

Theory
Al-Hagan et al. [11] showed that the molecular three-body distorted wave approximation (M3DW) coupled with an orientation-averaged molecular orbital approximation [15] yielded a good agreement with experimental TDCS data for H2. The molecular 3-body distorted wave (M3DW) approximation has been presented elsewhere [16][17] so only a brief description of the theory will be presented. The triple differential cross section (TDCS) for the M3DW is given by: where i k r , a k r , and b k r are the wave vectors for the initial, scattered and ejected electrons, and dir T is the direct scattering amplitude given by: Coulomb interaction between the two final state electrons. We have found that using the exact interaction can over-estimate the strength of the interaction for atoms so we have used the Ward-Macek approximation [18] for atoms and the exact interaction for molecules. The initial state interaction potential between the incident electron and the neutral molecule is V, and i U is a spherically symmetric approximation for V. This potential is called the distorting potential and it is used to calculate the initial-state distorted wave for the incident electron the charge density is summed over the seven occupied orbitals. The nuclear contribution to s U is the interaction between the projectile electron and the 2 nuclei averaged over all orientations which means that the net charge of the two nuclei is placed on a thin shell whose radius is the distance of the nuclei from the center-of-mass.
The potential E U is the exchange potential of Furness-McCarthy [19] which approximates the effect of the continuum electron exchanging with the passive bound electrons in the molecule, and CP U is the correlation-polarization potential of Perdew and Zunger [20] (see also Padial and Norcross [21]). The final state distorted waves are obtained the same as the initial state except that the final state charge density for the ion is used to calculate s U . The final state charge density is obtained the same as the initial state except that the occupancy number for the active electron is unity.

Results and Discussion
In a previous paper, we examined the TDCS for electron-impact ionization of the H2 molecule in comparison to atomic He. We found that there was an overall good agreement between experiment and theory [9]. Both experiment and theory predict a much more complicated interference pattern, particularly in the binary peak region, than is given by the elementary Cohen-Fano interference factor, I CF . We also found that the I-factor is more sensitive to projectile angular scans than to ejectedelectron energy scans which indicate that for the present set of kinematics the diffraction of the projectile from two scattering centers is more important than the interference between electron waves emitted from two different centers.
In this work, we have extended our TDCS measurements to N2 molecules for the same kinematics. for atomic N. All the TDCS have been normalized to unity at the peak. The lower right panel contains the corresponding I-factors. Since there were no experimental TDCS available for atomic N, the experimental I-factors for N2 represent the experimental N2 TDCS divided by the theoretical atomic N TDCS. Again it is seen that the experimental and M3DW I-factors have a more complicated structure than I CF . While the I-factors for H2 and N2 are similar, the I-factor for H2 exhibits a double peak structure within the angular range of binary peak which is supported by the experimental data while N2 has only a single peak in this angular range with a much smaller peak outside the angular range of the experiment. The M3DW I-factor for N2 is in very good agreement with experiment for all the measured points. The logic behind the I-factor is that dividing the molecular TDCS by the atomic TDCS removes all atomic effects in the TDCS and leaves two-center (molecular) effects. The Cohen-Fano I CF factor shows what this interference is predicted to look like for photon impact where the interference would result from the target electron being ejected from two centers. The fact that the experiment and theory qualitatively have the shape of the Cohen-Fano factor indicates that this type of interference is present. The fact that the actual structure is much more complicated indicates that the other two possible modes of interference (i.e. incident electron being diffracted from two scattering centers and the scattered electron wave being emitted from two centers) is more important. In summary, we see that there is significant interference at the quantum level that it is not amenable to a simple classical interpretation for lower energy incident electrons. These results demonstrate for the first time that Young's type double slit interference effects are present for ionization of N2 as well as H2. These are preliminary results. We are in the process of measuring more TDCS for N2 at other projectile scattering angles and ejected electron energies and will present a more complete set of results in the near future.