Research Article | Open Access

# Double Differential Cross-Sections for Electron Impact Ionization of Atoms and Molecules

**Academic Editor:**R. P. S. Chakradhar

#### Abstract

The single ionizing collision between an incident electron and an atom/molecule ends up two kinds of outgoing electrons called scattered and ejected electrons. As features of electron impact ionization, these two types of electrons are indistinguishable. Double differential cross-sections (DDCS) can be obtained by measuring the energy and angular distributions of one of the two outgoing electrons with an electron analyzer. We used He, Ar, H_{2}, and CH_{4} targets in order to understand the ionization mechanisms of atomic and molecular systems. We measured differential cross-sections (DCS) and double differential cross-sections at 250 eV electron impact energy. The elastic DCSs were measured for He, Ar, H_{2}, and CH_{4}, whereas the inelastic DCSs of He were obtained for 2^{1}P excitation level for 200 eV impact electron energy.

#### 1. Introduction

Generally, atomic and molecular physics lead to discoveries about the structure of matter at the atomic or molecular level and explain natural laws. These goals can be achieved with collision methods. Applications of the results from collision physics are of most importance to atmospheric science, laser refinement, and meteorological phenomena. In recent years, an intensive effort of experimental and theoretical work has been devoted to the study of ionization differential cross-sections of atoms and molecules through electron impact. The ionization of rare gas atoms, particularly, the cross-sections obtained with ground state ionization, is considered as benchmark data. Doubly differential cross-sections (DDCS) of ionization, as a function of ejected energy, , and the angle of the ionized electron, , contain valuable information about both the collision dynamics and the internal structure of atomic or molecular systems.

This paper is divided into four main parts. First, the theoretical and experimental studies of elastic DCS and DDCS for Helium, Argon, and Hydrogen molecules and Methane molecules are reviewed. Second, the experimental apparatus and signal processing are described in detail. Then, the results of the elastic DCS and DDCS measurements for He, Ar, H_{2}, and CH_{4} at 200 eV electron impact energy are presented and discussed. Finally, general conclusions are drawn from the experimental results.

#### 2. Review

Absolute elastic differential cross-sections for electron scattering from helium were measured for electron energies from 100 to 200 eV by Kurepa and Vuskovic [1], from threshold to 400 eV by Shyn [2], and from threshold to 200 eV by Trajmar et al. [3] and Fon et al. [4]. Normalized DCS below 100 eV have been measured by several authors [5–10]. Also, DCS calculations have been made using various improved methods [11–22].

The DDCS of helium was calculated at high energies in the framework of eikonal approximation [23, 24] and in the framework of a distorted wave approximation (DWBA) [25, 26]. The literature contains several results from numerous experimental data of DDCS at intermediate electron energies [27–33].

DCS spectra of Ar provide a test for the comparison of experimental and theoretical data, since a critical minimum appears in the spectra, depending both on the incident electron energy and the scattering angle. Understanding the behavior of this critical point has been the interest of studies of DCSs as a function of the scattering angle [34–39]. The literature contains results from several measurements of the DDCS of Ar at intermediate, high [30, 35, 40–42], and low energies [43].

Molecular hydrogen is the most fundamental of all electron-diatomic molecule scattering systems and has been the subject of numerous experimental and theoretical studies of collision physics. For the first time in the literature, the elastic scattering of electrons by H_{2} was measured by van Wingerden et al. [44]. Measurements of elastic scattering cross-sections have been done by several groups [45–51]. The elastic and inelastic DCSs of H_{2} were measured at an electron impact energy of 30 keV [52]. Review articles on electron scattering from H_{2} were presented by Trajmar and McConkey [53] and Morrison et al. [54]. Recently, Anzai et al. have published cross section datasets for electron collisions with H_{2} [55].

In the intermediate electron energy range, a number of studies have been reported on the absolute elastic differential cross-sections for H_{2} [10, 12, 56–63]. Furthermore, the energy- and angle-dependent DCSs of H_{2} by proton impact have been presented in several papers [64, 65].

DDCS of ionized electrons can provide a test of the basic formulation of quantum mechanical theory. The main experimental studies on DDCS of H_{2} by electron impact were reported by several authors [30, 46, 47, 66]. The DDCS of H_{2} was measured at low energies, including with a dissociation channel [67]. Chatterjee et al. presented a dataset for 8 keV electron impact ionization [68]. Their results have been discussed in the frame of Young type interference effects. Recently, Schulz et al. have measured the DDCS and dissociative ionization of H_{2} by 75 keV proton impact using angular-dependent measurements [69].

Mostly in the low incident-energy range, electron collision with methane has been studied both experimentally and theoretically. For instance, experimental data on DCSs for low energy electron impact has been discussed in a number of studies [70–79]. On the theoretical side, the literature on low energy electron-CH_{4} scattering is equally rich. DCSs, momentum transfer cross-sections (MTCSs), and integral cross-sections (ICSs) for elastic electron-methane scattering were calculated at different levels of approximation [80–94].

Comparatively speaking, far fewer studies have been carried out both experimentally and theoretically using intermediate-to-high energies (). The literature for experimental investigations is strongly concentrated on grand total (elastic and inelastic) cross section measurements [95–98]. Some of these works also report the partitioning of the total cross-sections into elastic plus inelastic (ionization and neutral dissociation) cross-sections [96, 98]. Only three sets of absolute measurements of DCSs, ICSs, and MTCSs for elastic electron-CH_{4} scattering were done by Vuskovic and Trajmar [72] at 20, 30, and 200 eV and by Sakae et al. [99] in the 75–750 eV energy range. Theoretically, elastic electron-CH_{4} cross-sections were done by Dhal et al. [100] for incident energies from 205 to 820 eV and by Jain [81] for incident energies up to 500 eV.

The observed cross-sections in electron-methane scattering show a Ramsauer-Townsend minimum around 0.4 eV and a marked increase for higher energies with a maximum at about 8 eV [101–112]. Both of those structures have been well examined by several experiments at different collision energies. Measurements of double differential cross-sections (DDCS) for the ionization of methane molecule are very scarce in the literature [32, 113].

In this paper, we reported double differential cross-sections measured using an apparatus originally developed for coincidence measurements of ejected and scattered electrons (i.e., triple differential cross-sections) [114–118]. The same apparatus can also collect data on the angular distribution of the ejected or scattered electrons, simply disabling the coincident circuit. We considered He, Ar, H_{2}, CH_{4}, as typical atom/molecule couples from which we can understand the ionization process in terms of quantum mechanical description.

#### 3. Experimental Apparatus

In the Electron Collision (e-COL) Laboratory in Turkey, there are three actively working experimental apparatus. A number of modifications to the original spectrometers have been implemented with different projects granted by Afyon Kocatepe University (AKU), the Scientific and Technological Research Council of Turkey (TUBITAK), the State Planning Organization (DPT), and also a donation from Newcastle University by Prof. Albert Crowe. The experiments described here were performed in Afyon with a recently modified version of an electron spectrometer originally developed in Newcastle [119–122].

The electron spectrometer is a crossed-beams type in which the incident electron beam collides orthogonally with the target gas beam. The electron spectrometer was initially designed and constructed to measure angular and energy correlations between outgoing electrons. The apparatus has been used to measure the triple differential cross-sections for the electron-impact double excitation of helium [123, 124], for the electron impact ionization of Argon [115] and H_{2} [114]. And also the apparatus has the capability to measure differential cross-sections of the ionization of the atoms and molecules with electron impact [125].

The electron beam source, electron energy analyzers, and Faraday cup are situated together in a high-vacuum chamber [116]. Figure 1 shows the vacuum chamber which is a nonmagnetic stainless steel cylinder, 72 cm in height, with an internal diameter of 83.5 cm pumped by a 700 Ls^{−1} Pfeiffer turbo molecular pump backed by a 20 m^{3} h^{−1} Pfeiffer rotary pump. All parts of the electron spectrometer are constructed from dural and brass, which are nonmagnetic and easy to process. The vacuum chamber and flanges are sealed with Viton O-rings and a copper ring, respectively. Vacuum pressure is displayed by a Pfeiffer PKR 251 ion gauge connected to a TC 600 controller. The ultimate pressure, which is recorded and displayed on PC, achieved a background residual pressure of ~8 × 10^{−8 }mbar before the target gas beam was let into the vacuum chamber. The target gas is given in vacuum chamber by a gas inlet system consisting of a gas container. The flow of the target gas into interaction region was controlled by a needle valve at entrance to the vacuum chamber and the working pressure was ~6 × l0^{−6} mbar. The target gas beam is shaped through a nozzle, which is also made of brass. The nozzle is 2.5 mm from the interaction region and is a single capillary type with a diameter of 1 mm.

To reduce external magnetic fields, the inside of the vacuum chamber is shielded by 3 mm-thick *μ*-metal and, also, the outside of the vacuum chamber surrounded by three Helmholtz coils sets as it is shown in Figure 1. In the interaction region, the external magnetic fields are reduced to less than 0.5 mG, as measured with a FLUX meter magnetometer.

The DDCS are measured by the electron energy analyzer rotating around the collision centre in a plane. Figure 2 shows the electron spectrometer which consists of an electron gun, two 180° hemispherical electron energy analyzers (both analyzers are used for the electron-electron (e, 2e) coincidence experiments), and a Faraday cup.

The two electron energy analyzers and Faraday cup are all located on concentric tables which can be rotated independently in the horizontal plane from outside the vacuum chamber by manually mechanic feedthroughs. The electron gun is positioned at the same level as the analyzer and faces the interaction region. Faraday cup, the analyzers, and gas nozzle were carefully aligned using a laser beam in the place of the electron gun. We checked whether the electron analyzers rotated correctly in the scattering plane. In principle, each component of electron spectrometer may move through a full 360°, but this is restricted by the presence of the electron gun and Faraday cup. The angular ranges of the analyzers with respect to the incident beam direction were and (we used a small Faraday cup for extending angular range).

Figure 3 is a schematic drawing of the electrostatic lens elements and electric circuits of the gun. The Tungsten Hairpin cathode, which is housed in the first element , is directly heated and gives an energy spread of ~0.5 eV (FWHM).

The beam is accelerated and focused using two groups of cylindrical electrostatic lenses (Figure 3). The beam is collimated by three apertures 0.6 mm (), 0.4 mm (), and 0.6 mm () in diameter [116, 117, 126]. The , which has negative potential, accelerates the energy selected beam of low energy electrons up to the impact energy. The final element is held at ground potential and so the voltage of the element determines the energy of the electron beam. The pairs of and deflector plates are housed in the element , and these deflector plates lead the beam horizontally or vertically to correct misplacement of the filament or for the effect of magnetic fields on the electron beam. The electron energy can be adjusted from 40 to 400 eV, and the electron gun has the capability of focusing to a 1 mm diameter at the interaction region. Figure 4 shows the 3D AutoCAD drawing of the electron gun. The gun and analyzers are shielded by aluminum boxes grounded at the same point with the chamber and the electronic control panels.

The incident beam current is measured by a Faraday cup located in front of the electron gun. Figure 5 shows that the Faraday cup consists of a cylindrical tube with a 5 mm aperture and two separate apertures, a splash plate (2 mm), and a ground aperture (4 mm).

The Faraday cup and Splash plate current are monitored by Keithley picoammeters. The typical electron-beam currents used in these experiments range from 0.3 to 5 * μ*A. The Faraday cup is mounted on a concentric disc and rotated around the interaction region. The Faraday cup can move up and down, out of and into the beam.

The electron analyzers were designed for electron-electron coincidence spectrometers for the ionization of the target atom/molecule with electron impact [116]. These analyzers have been used before in several (e, 2e) experiments by the e-COL group [114, 115, 123]. In this paper, we use one of the analyzers for measuring the cross-sections. Figure 6 shows one of the analyzers with electric and signal circuits. The analyzer consisted of five-element entrance lenses to and a 180° hemispherical deflector. The entrance lens systems focuses on scattered or ejected electrons which are leaving the interaction region to the analyzer, passing energy at the entrance plane. The arrangement of the lens systems made two three-element lenses in a focal mode [118, 127]. The lens systems image the two apertures with a 2 mm diameter at the entrance and exit of the lens system.

The hemispherical energy analyzers consist of two concentric hemispherical surfaces of radii and . A difference of potential, , which is applied to the hemispherical deflector, produces an electrostatic field, so the electrons follow circular orbits. Figure 7 shows that different energy electrons follow different orbits. The low energy electrons pass the inner hemisphere, while the high energy electrons pass the outer hemisphere. The radii of inner and outer hemispheres are mm and mm, and the mean radius is mm. The entrance and exit apertures of the hemisphere are both centered on . Electrons enter the deflector near the centre of the space between the hemispheres and exit after being deflected by 180°. If the electrons with travel in an orbit of radius , the voltages on the inner and outer hemispheres are given by .

Figure 8 shows SIMION simulations of the trajectories of the electrons with different energies. For example, the electrons of 70 eV knock outer sphere, while the electrons of 25, 30, 45 eV knock on the inner sphere, and so the electrons of 50 eV achieve the exit aperture (to detector). Energy resolution of the analyzer depends on the energy selection ability of it. The resolution of the conventional hemispherical analyzer is determined by the size of the analyzer and the fringing fields appeared due to using apertures on the exit plane of the spheres [123, 124, 126].

Following energy selection in the space of the deflector, the electrons are detected by a Photonis-CEM 7018 C WL channel electron multiplier (CEM). The CEMs are located in front of the exit apertures of the deflectors. The CEM has a channel of 2 mm internal diameter and a cone of 5.8 mm diameter, and it is housed in an aluminum grounded box.

A schematic of the electron detection and pulse timing circuits is shown in Figure 9. The signal processing system can measure different kinds of ionization cross-sections of atomic/molecular targets with electron impact. The CEM high voltages are supplied by two ORTEC 5 kV suppliers. The negative pulses from CEM, which has an average amplitude of ~20 mV, are amplified to approximately 200 mV by a Philips Scientific 777 Amplifier. The amplified pulses are fed to a Philips Scientific 705 model discriminator for noise discrimination and 50 ns wide negative rectangular pulses are obtained. The resulting CEM count rates are monitored by an ORTEC 994 dual counter/timer and recorded to a computer using multichannel scalar (MCS) for noncoincidence measurement, such as energy loss spectra, elastic DCS and DDCS. The signal going from the second analyzer is processed in the same way using the same process as Previously mentioned. In coincidence mode, we used an ORTEC 566 time-amplitude converter (TAC) and the ORTEC Maestro PC card. The details of the coincidence technique for electron impact ionization are described in the previous works [114, 115, 120]. High voltage power suppliers, the amplifier, the discriminator, the counter, and the TAC are located in NIM bins and pulses of the electronics are transmitted by RG58A/U coaxial and LEMO cables.

#### 4. Results and Discussion

##### 4.1. Helium

In Figure 10, we presented the measured elastic DCS of Helium at 200 eV in comparison with the previous work of Kurepa and Vuskovic [1], and an excellent agreement with the experimental data was obtained. At small angles scattering dominated and the value of the cross section decreases slowly with increasing scattering angles.

Similar measurements were also done for inelastic DCS at 200 eV to test the reliability of the present results in comparison to the measurements of Fon et al. [4] (Figure 11). The shape of DCSs in both cases had a strong agreement with the present data between 30° and 130°.

In Figure 12, experimental elastic and inelastic DCS of He is shown for 250 eV. On the basis of the shape of DCS, the data of this study are in agreement with the expected results. However, to the best of our knowledge, there is not a comparable data in the literature for the chosen data sets.

**(a)**

**(b)**

Figure 13 shows the DDCS results of He at 250 eV impact electron energy for 20 to 150 eV detection electron energies.

The results showed a smooth systematic variation with energy and had a maximum around 50° at 70 eV. This maximum disappears when the outgoing electron energies increase.

##### 4.2. Argon

We measured cross-sections of elastic scattering from Ar at 250 eV electron impact energy. A critical minimum was observed around 100°, in agreement with the data of Williams and Wills [37] in Figure 14. All noble atoms show parallel structure in the elastic differential cross section at low energies. The DCS of the noble gases show minima at different energies and different angles [127].

Double differential cross-sections (DDCSs) for the single ionization of argon by 250 eV electron impact were measured for the ejection energy range of 10–200 eV (Figure 15).

The high energy ejected electrons are emitted in the forward direction and the lower energy ejected electrons are mostly ejected isotropically in all directions. Ejected electrons with higher energies produce the same structure in the DDCSs due to a binary collision between the incident electron and an electron from the target.

##### 4.3. H_{2}

In this study, we measured the energy and angular distributions of ejected electrons for the ionization of the simplest molecule, H_{2}. Reliable collision cross section data on e-H_{2} are especially needed for the study of planetary atmosphere. The elastic and inelastic DCS and DDCS were measured for the incident energy of 250 eV. In Figure 16, we present the measured elastic DCS of H_{2} at 250 eV. At this energy, we could not find any reliable experimental DCS data to be compared with the findings of our study. The measured DCSs show a maximum at low scattering angles and decrease as the scattering angle increases. However, that increase seen in the He DCS results at high scattering angles was not observed in the H_{2} results.

Figure 17 shows the present measurements of DDCS as a function of the angle of electron detection varying between 40° and 130°, for detected electron energies of 50 and 30 eV, respectively, at 250 eV incident electron energy. Figure 17 also shows the comparison of the data of Shyn and Sharp (1981) for DDCS at 30 and 50 eV electron detection energies. The measurements obtained from our study for these two energies were compared with the data of Shyn and Sharp (1981), and looking at the results of DDCS, there was a broad general agreement. In this case, there was much stronger scattering in the forward direction and a more rapid decrease in the DDCS with increasing angle. A broad maximum around 60° was seen in the results of our study as well as in the data of Shyn and Sharp [46, 47]. The measured values of DDCS were relatively higher for lower detection electron angles and higher detection electron energies since electrons having higher energies were rather scattered in the forward direction.

##### 4.4. CH_{4}

As an experimental confirmation of our results, elastic differential cross section measurements of methane for 200 eV incident electrons were taken and compared with the previous results [72, 90, 99, 100] (Figure 18(a)). The agreement between the present and previous data was good. And also in this study, elastic differential cross section of methane at 250 eV firstly was measured (Figure 18(b)).

**(a)**

**(b)**

The DDCS results for 250 eV incident electrons on a methane molecule are given in Figure 19. The analyzer was adjusted to detect 10–225 eV outgoing electrons after collision. A maximum was observed for 50, 75, and 100 eV detection energies. This is a consequence of the binary character of the collision. Since most of the faster electrons are scattered into the forward direction, the maximum shows that the angle between the scattered and ejected electrons for most of the collision processes is .

#### 5. Conclusion

DDCS experiments give important results about the ionization events in atoms and molecules. DDCS measurements are fundamental studies to which other measurements may be related. As an experimental confirmation of our results, elastic differential cross section measurements of He, Ar, H_{2}, and CH_{4} for 250 eV incident electrons were taken and compared with the previous results. It is expected that these results will further aid our understanding of the ionization mechanisms of small molecules.

#### Acknowledgments

This work was supported by the Scientific and Technological Research Council of Turkey (TUBITAK) through Grants 109T738, 109T722; State Planning Organization (DPT) 2001K120140, and Afyon Kocatepe University Scientific Research Projects Coordination Funds (BAPK).

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