Double differential distributions of e-emission in ionization of N₂ by 3, 4 and 5 keV electron impact

The present ionization cross sections are calculated within the 1st Born approximation framework by using the partial-wave expansion formalism recently employed for describing the electron-induced ionization of isolated biomolecules [36–39]. In this approach, the incident/scattered electron is described by a plane wave whereas the ejected electron is modelled by a Coulomb wave associated to an effective target charge ${Z}_{\mathrm{T}}^{{\ast}}=\sqrt{-2{n}^{2}{\epsilon}}$ where n refers to the principal quantum number of each atomic orbital component used in the molecular target description (see hereafter) and the active electron orbital energy is related to the ionization energies B of each occupied molecular orbital by = −B. Besides, it will be assumed that the passive (not ionized) electrons remain as frozen in their molecular orbitals during the collision, which permits us to reduce the electron target interaction potential to a one-active electron potential (see hereafter).

Under these conditions, the triply differential cross sections—hereafter denoted σ⁽³⁾(Ω_S, Ω_e, E_e)—differential in the direction of the scattered electron Ω_S, differential in the direction of the ejected electron Ω_e and differential in the ejected energy E_e may be written as

$$\begin{equation}{\sigma }^{\left(3\right)}\left({{\Omega}}_{\mathrm{S}},{{\Omega}}_{\mathrm{e}},{E}_{\mathrm{e}}\right)=\sum _{j=1}^{N}{\sigma }_{j}^{\left(3\right)}\left({{\Omega}}_{\mathrm{S}},{{\Omega}}_{\mathrm{e}},{E}_{\mathrm{e}}\right)\end{equation} \tag{ 1 }$$

where N is the number of molecular orbitals used in the description of the target and with ${\sigma }_{j}^{\left(3\right)}$ expressed as a weighted sum of the atomic triply differential cross sections ${\sigma }_{\mathrm{a}\mathrm{t},i}^{\left(3\right)}\left({{\Omega}}_{\mathrm{S}},{{\Omega}}_{\mathrm{e}},{E}_{\mathrm{e}}\right)$ corresponding to the different components involved in the description of the N₂ target (N_1s , N_2s , and N_2p orbitals), namely,

$$\begin{equation}{\sigma }_{j}^{\left(3\right)}\left({{\Omega}}_{\mathrm{S}},{{\Omega}}_{\mathrm{e}},{E}_{\mathrm{e}}\right)=\sum _{i}{\xi }_{j,i}.{\sigma }_{\mathrm{a}\mathrm{t},i}^{\left(3\right)}\left({{\Omega}}_{\mathrm{S}},{{\Omega}}_{\mathrm{e}},{E}_{\mathrm{e}}\right)\end{equation} \tag{ 2 }$$

where the effective number of electrons ξ_j,i as well as the corresponding binding energy are calculated in the gas phase with the Gaussian 09 software (see Frisch et al [46]). In order to assess the possible impact of different descriptions of N₂ molecular wave functions on the DDCS, ab initio calculations were carried out at both the RHF/6-311G and CCSD/cc-pVTZ levels of theory. The RHF/6-311G is a rather simple restricted Hartree–Fock description of the target with the medium accuracy Pople basis set 6-311G, while the CCSD/cc-pVTZ description is a more accurate coupled cluster calculation using both single and double substitutions from the Hartree–Fock determinant along with a much larger Dunning's correlation-consistent polarized basis set with triple-zeta. The first ionization energy corrected for zero-point vibrational energy (ZPE) was 16.900 eV with RHF/6-311G and 16.527 eV with CCSD/cc-pVTZ using Koopman's theorem. The ionization energy was further constrained to match the experimental value of 15.581 ± 0.008 for N₂ in the gas phase [47].

Thus, in the laboratory framework, the atomic triply differential cross sections ${\sigma }_{\mathrm{a}\mathrm{t},\mathrm{i}}^{\left(3\right)}$ were calculated from the atomic transition matrix element between the ground state to the 1st ionized level of the target. Then, by using the well-known frozen-core approximation which reduces the present multi-electron problem to a one active electron problem and considering the well-known partial-wave expansion of the plane wave as well as that of the Coulomb wave, the DDCS could be analytically expressed for each molecular orbital, the target ionization cross sections being simply obtained by summing up all the subshell contributions. Finally, singly differential and total cross sections were obtained after the numerical integrations over the scattering direction and the ejected energy spectrum, respectively.

The CTMC method is a non-perturbative method, where classical equations of motions are solved numerically [48–51]. In the present work the CTMC simulations were made in the three-body approximation, i.e. the many-electron target atom was replaced by a one-electron atom and the projectile ion was taken into account as one particle [52, 53]. For the target atom a central model potential has been used which is based on the Hartree–Fock method as developed by Green [54]. The potential can be written as:

$$\begin{equation}V\left(r\right)=q\frac{Z-\left(N-1\right)\left(1-{{\Omega}}^{-1}\left(r\right)\right)}{r}=q\frac{Z\left(r\right)}{r},\end{equation} \tag{ 3 }$$

where Z is the nuclear charge, N is the total number of electrons in the atom or ion, r is the distance between the nucleus and the test charge q, and

$$\begin{equation}{\Omega}\left(r\right)=\frac{\eta }{\xi }\left({\mathrm{e}}^{r\xi }-1\right)+1.\end{equation} \tag{ 4 }$$

The potential parameters ξ and η can be obtained in such a way that they minimize the energy for a given atom or ion [55]. We treat the N₂ molecule as two N atoms in our simulation and accordingly we use ξ = 1.179 a.u. and η = 2.27 a.u for the N-atom. Further, this type of potential has certain advantages, because it has a correct asymptotic form for both the small and large values of r.

In the present CTMC approach, Newton's classical non-relativistic equations of motions for a three-body system are solved numerically for a statistically large number of trajectories for given initial conditions. We have used an ensemble of 5 × 10⁷ trajectories. The equations of motion were solved using a standard Runge–Kutta method. A three-body, three-dimensional CTMC calculation is performed as described by Tőkési and Kövér [53]. The initial conditions of the individual collisions are chosen at sufficiently large internuclear separations from the collision center, where the interactions among the particles are negligible. These are selected in a similar fashion as described by Reinhold and Falcon [56] for non-Coulombic systems. A microcanonical ensemble characterizes the initial state of the target. The initial conditions were taken from this ensemble in such a way that initial binding energies of the N(2p) level (E_b = −0.5343 a.u.) and N(2s) level (E_b = −1.371 a.u.) were constrained. For ionization channel the energy and the scattering angles of the particles were recorded. These parameters were calculated at large separation of the projectile and the target nucleus.

The total and double differential cross-sections were computed using the following formulas:

$$\begin{equation}\sigma =\frac{2\pi {b}_{\mathrm{max}}}{{T}_{\mathrm{N}}}\sum _{j}{b}_{j}^{\left(i\right)},\end{equation} \tag{ 5 }$$

$$\begin{equation}\frac{{\mathrm{d}}^{2}\sigma }{\mathrm{d}E\mathrm{d}{\Omega}}=\frac{2\pi {b}_{\mathrm{max}}}{{T}_{\mathrm{N}}{\Delta}E{\Delta}{\Omega}}\sum _{j}{b}_{j}^{\left(i\right)}.\end{equation} \tag{ 6 }$$

In equations (5)–(6) T_N is the total number of trajectories calculated for impact parameters less than b_max, and ${b}_{j}^{\left(i\right)}$ is the actual impact parameter for the trajectory corresponding to the ionization process under consideration in the energy interval ΔE and the emission angle interval ΔΩ of the electron.

In addition to comparing the data with the CB1 and CTMC models, the total ionization cross section (TCS) data have also been compared with a semi-empirical model, namely the CSP-ic model [22, 57] which is used for calculating the ionization and excitation of varieties of molecules under electron impact. Since it has already been discussed in earlier papers we are providing a brief outline. Initially the total inelastic cross sections are calculated based on a complex scattering potential V_CSC, constructed using the target molecular charge density which is obtained by a linear combination of the atomic charge densities. This potential can be expressed as:

$$\begin{align}\hfill {V}_{\text{CSC}}\left({E}_{\mathrm{i}},r\right)& ={V}_{\mathrm{S}\mathrm{T}}\left(r\right)+{V}_{\mathrm{E}\mathrm{X}}\left({E}_{\mathrm{i}},r\right)+{V}_{\text{POL}}\left({E}_{\mathrm{i}},r\right)\hfill \\ \hfill & +\enspace \mathrm{i}{V}_{\text{ABS}}\left({E}_{\mathrm{i}},r\right)\hfill \end{align} \tag{ 7 }$$

While the real part of this interaction potential takes into account the static effect (V_ST), the exchange (V_EX) between the projectile and a target electron and the polarization (V_POL) of the charge density cloud, the imaginary part is an absorption term (V_ABS) [59], responsible for the loss of scattered flux into the allowed channels of electronic excitation and ionization [58]. The partial wave approach under the spherical approximation is used to calculate the complex phase shifts δ_l (k) which carry the signature of the interaction between the incident electrons and the molecule [60]. To compute the TCS we define a ratio between total ionization cross section and total inelastic cross sections, R(E_i) [57], such that,

$$\begin{equation}R\left({E}_{\mathrm{i}}\right)=1-{c}_{1}\left(\frac{{c}_{2}}{U+a}+\frac{\mathrm{ln}\enspace U}{U}\right)\end{equation} \tag{ 8 }$$

where $U=\frac{{E}_{\mathrm{i}}}{I}$, E_i is impact energy and I is the ionization potential of the target. We evaluate the three constants c₁, c₂ and a to obtain R(E_i) and hence the total ionization cross sections.

Figure 1 displays the absolute DDCS of the secondary electrons emitted from N₂ in collisions with 3 keV electrons. The DDCS spectra are shown for six different emission angles. The spectra fall by about four orders of magnitude in the measured emission energy range between 1 and 500 eV. The cross section is maximum in the lowest energy range corresponding to the soft collision mechanism and then falls rapidly with the increase in the emission energies. The soft collision mechanism involves very little momentum transfer from the projectile to the bound electrons of the target and hence these electrons are emitted for large impact parameter collisions. The intermediate part of the spectrum is normally dominated by the two-centre effect where the ejected electron is under the influence of both the projectile and the positively charged recoil target-ion. However, in case of electrons as projectiles, two centre effect does not play a major role unlike the case for a typical ion-molecule collision. The sharp peak observed at ∼350 eV corresponds to the K-LL Auger electron emission. The measured DDCS have been compared with the CB1 model calculations. The calculations have been performed using two different descriptions of the target wave functions at the RHF/6-311G and CCSD/cc-pVTZ levels of the theory. Both the calculations have been shown in figure 1 (solid and dash-dot-dot lines) and it is seen that in the log–log scale, the two models almost merge with each other. This suggests that the description of the target wave functions has very limited impact on the DCCS in the energy range considered here. In case of the forward angles, the CB1 model calculations underestimate the data upto about 60 eV, beyond which it shows overall good agreement with the data. For the intermediate angles, around 90°, the theory matches qualitatively and quantitatively with the measured DDCS above 50 eV. However, in case of backward angles, the model underestimates the data over the entire spectra, with maximum discrepancy occurring in the low emission energies. The ratio between the CB1 predictions using two different wave functions is found to vary very little i.e. from 1.01 in the forward angles to 1.04 for the higher backward angles (see insets in figure 1). The experimental data have also been compared with CTMC model calculations for twice the atomic nitrogen or 2N (red dashed line). Overall an excellent agreement is observed with this model over the entire energy regime. In figures 2 and 3, we have shown the energy dependence of the e-DDCS for the projectile energies 4 and 5 keV, respectively. In both the cases the CTMC model is seen again to match well with the data points except for the lowest energy electrons, where it predicts slightly higher cross sections. On the other hand, the CB1 model show a good qualitative agreement reproducing the shape of the energy distribution accurately. However, this model quantitatively underestimates the data below 70 eV for all the angles. For higher beam energy i.e. at 5 keV (figure 3) the CB1 model although predicts somewhat different cross sections from the experiment, but the difference is quite less compared to that for the 3 keV and 4 keV electron beam. The insets in each panel in figures 2 and 3 show the magnified view of the K-LL Auger peak.

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Figure 4 displays the theoretical (CB1) DDCS values as a function of the ejected electron energy for various emission angles. The calculated values are shown for incident electron energy of 3 keV. It may be noticed that in the low energy region, all the curves corresponding to the different emission angles bunch together. This region is dominated by the soft collision mechanism for which the DDCS remains almost independent of the emission angles. With the increase in the emission energies, the spectra corresponding to the intermediate angles, (i.e., 75°, 80° and 90°) start going up whereas the spectra for the forward and backward angles show a steady fall. The separation among different lines represent the angular distributions.

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To have a better understanding of the features seen in figure 4, the absolute DDCS of the ejected electrons as a function of different emission angles have been displayed in figure 5 corresponding to the projectile beam energy of 3 keV. The six plots shown in figure 5 expand over the entire emission energy range revealing the signature of different features at different parts of the spectrum. In figure 5(a) an almost flat distribution is observed corresponding to the soft collision mechanism which is dominated by large impact parameter events. For higher electron emission energies, a peak like structure starts appearing around 80° which sharpens further with the increase in the emission energies. This peak is due to the binary nature of collision i.e. the direct two-body free-electron scattering between the incident electron and the target electron while the recoil-ion remains passive. The CB1 model (black solid and magenta dash-dot-dot lines) show a qualitative agreement only but quantitatively underestimates the data, except in the peak region, where it matches well with the measured quantities. With the increase in the emission energies, it is seen that the DDCS values for the forward angles are slightly higher compared to those for the backward angles. For ejected electron energy 160 eV (figure 5(d)), the measured DDCS for forward angle is 1.6 times higher compared to the backward angle, whereas for 260 eV, the difference goes up to 2.6 times (figure 5(f)). These numbers, i.e. forward–backward angular asymmetry parameters, are close to that predicted by the CB1 model.

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It is seen that the CTMC model show very good agreement with the data at 7 eV (figure 5(a)). However, with increase in emission energies, it is observed that the CTMC model predicts a higher cross section for the backward angles compared to that for the forward angles. Thereby the forward-backward angular asymmetry is not reproduced properly by the CTMC model unlike the cases for the experimental measurements and the CB1 model. Similar features are also observed for the impact energies of 4 keV (figure 6) and 5 keV (figure 7). In order to understand and correct this behavior we made some initial tests on the strength of the projectile electron and target electron interactions as modelled in the CTMC approach. As a result of the standard calculations (shown in the figures), the interaction between the two electrons is kept 'ON' during the entire motion of the particles till the asymptotic limit. However, as a initial test we also performed the simulations by switching off the e–e interaction in the exit channel. This indicated certain improvement in the distribution. However, further systematic calculations are required to be performed to improve the angular distribution. We conclude that the present CTMC model overestimate the strength of the electron–electron interactions particularly in the exit channel. The projectile electron sweep out the ejected target electron from the forward angles to the backward ones. The detailed analysis of this effect is in progress and will be published elsewhere.

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Integrating the e-DDCS spectrum over the emission energy or emission angle gives us the SDCS. Integrating over the emission energies, we obtain the SDCS i.e. dσ/dΩ_e, as a function of angles which is given by:

$$\begin{equation}\frac{\mathrm{d}\sigma }{\mathrm{d}{{\Omega}}_{\mathrm{e}}}=\int \frac{{\mathrm{d}}^{2}\sigma }{\mathrm{d}{{\Omega}}_{\mathrm{e}}\mathrm{d}{{\epsilon}}_{\mathrm{e}}}\mathrm{d}{{\epsilon}}_{\mathrm{e}}.\end{equation} \tag{ 9 }$$

Similarly, integrating over the emission angles, we get the SDCS as a function of the emission energy:

$$\begin{equation}\frac{\mathrm{d}\sigma }{\mathrm{d}{{\epsilon}}_{\mathrm{e}}}=\int \frac{{\mathrm{d}}^{2}\sigma }{\mathrm{d}{{\epsilon}}_{\mathrm{e}}\mathrm{d}{{\Omega}}_{\mathrm{e}}}\mathrm{d}{{\Omega}}_{\mathrm{e}}.\end{equation} \tag{ 10 }$$

Figure 8 displays the SDCS i.e., dσ/d_e as a function of the emission energies corresponding to 3, 4 and 5 keV incident energies. For all the three beam energies, the CB1 model predicts lower cross sections compared to the data upto ∼50–60 eV, beyond which one can observe a very good agreement. The discrepancy is largest for incident energy 3 keV and least for the 5 keV electrons. The measured DDCS have also been compared with the CTMC model (red dashed line). Overall an excellent agreement is observed with the CTMC model over the entire energy regime for all the three beam energies under investigation. However, in case of 4 and 5 keV (see figures 8(b) and (c)), the model overestimates the data for the lowest energy electrons only by a little amount.

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Integrating the SDCS over the emission energies or emission angles gives the total ionization cross section. The TCS were obtained by integrating over the electron energies between 1 and 500 eV and over the emission angles from θ = 0° to θ = 180°. The data points below 30° and above 145° were estimated by extrapolation to obtain the total cross section and the difference was found to be about 11%–13%. It was observed that the TCS values derived by integrating the SDCS over the emission angles and energies varied very little i.e. only by ∼0.3%–0.4%. In figure 9, the experimental and theoretical TCS values have been displayed which includes the data obtained for the incident energies of 7 keV [61] as well as for the 6 and 8 keV [62]. The total contribution of the K-shell ionization (σ_K−LL) for 3, 4 and 5 keV beam energies are 0.23 Mb, 0.21 Mb and 0.19 Mb, respectively, whereas, the TCS at these three energies were found to be 26.3 Mb, 16.8 Mb and 14.2 Mb.

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From figure 9 it is seen that the CB1 model falls well below the present experimental data but provides an excellent qualitative agreement with the observed energy-dependence. The CTMC model, on the other hand, falls a bit higher compared to most of the data points but mostly within the experimental uncertainties which are about 22%–27%. The present TCS values are found to be lower than the existing data [7] by about 20% to 40% (not shown). The CTMC calculations also fall below these existing data and the difference increases for higher energy. The CB1 model is closer to the present measurements and has a large deviation from the existing data. The TCS values predicted by the CSP-ic model overestimates the measured data for all the energies, but provides a good qualitative behavior regarding the energy dependence. This discrepancy could be due to the consideration of the spherical charge density of the N₂ molecule and other approximations [63] used in the semi-empirical model. It is to be noted that the TCS calculations obtained using the two ab initio models, (CB1 and CTMC), lie below and just above the experimental values, respectively. The CTMC model provides closest agreement to the present data.