Vibrationally resolved NO dissociative excitation cross sections by electron impact

A theoretical investigation of the dissociative excitation by electron impact on the NO molecule is presented, aiming to make up for the lack of data for this process in the literature. A full set of vibrationally-resolved cross sections and corresponding rate coefficients are calculated using the Local-Complex-Potential approach and five resonant states of NO^-.

A theoretical investigation of the dissociative excitation by electron impact on the NO molecule is presented, aiming to make up for the lack of data for this process in the literature. A full set of vibrationally-resolved cross sections and corresponding rate coefficients are calculated using the Local-Complex-Potential approach and five resonant states of NO − .

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Nitric oxide (NO) molecule is one of the minor components of terrestrial atmosphere. Generated in atmospheric plasma from chemical reactions of nitrogen with oxygen, NO and its radicals are very important in many industrial technologies [1][2][3][4][5] and play a key role in the combustion of fossil fuels [6,7]. Of the nitrogen oxide compounds, the so-called NOx gasses, the NO molecule has the greater impact on environment and on pollution caused by human activities [8,9].
To fill this gap, we present calculations -based on the formalism used previously [20] -of vibrational state resolved cross sections and the corresponding rate coefficients for dissociative excitation (DE) of NO by electron impact, i.e.: We consider electron collision energies where numerous NO − resonances exist, direct dissociation is negligible and the DE reaction is dominated by resonant processes [28,29]. Our aim is to cover a large range of incident electron energies, so we take into account five resonance states of NO − : the three low-lying states of 3 Σ − , 1 Σ + and 1 ∆ symmetries and two higher ones, with 3 Π and 1 Π symmetry, which lie close to the NO dissociation threshold. In the following, we number these resonances by r = 1, . . . , 5, respectively. The vibrational excited states of which converge on the N( 4 S) + O( 3 P) dissociation limit of Eq. (1), due to their symmetry, have small oscillator strengths and short lifetimes [10,30,31] so their influence on the DE process can be neglected. NO is a stable, open-shell molecule with a 2 Π ground electronic state, and a very accurate theoretical treatment of the electron-NO scattering therefore needs to take into account the spin-dependence of the process. However, the spin-orbit coupling effects are only important at very low energies [23] and, consequently, in the following we will neglect them.
We start by briefly describing the theoretical model used to calculate the cross sections for the process (1), restricting ourselves to the major equations of the Local-Complex-Potential (LCP) model. For a comprehensive treatment of the resonant collisions, we refer to the seminal articles [28,29,32]. The LCP approach was used to calculate the low-energy vibrational excitation of NO [20] and CO [33] by electron impact and the DE of oxygen molecule [34,35], which gave results in good agreement with experiment.
In the LCP model, the DE cross section for an NO molecule initially in vibrational level v by an incident electron of energy is given by [32]: where 2S r + 1 and 2S + 1 are the spin-multiplicities of the resonant anion state and of the neutral target state respectively, g r and g represent the corresponding degeneracy factors, k(k ) are the incoming (   momenta, m is the electron mass, χ c stands for the continuum wave function of NO asymptotically converging to N( 4 S) + O( 3 P), and ξ r v is the resonance nuclear-motion wave function which is obtained for each resonace as the solution of Schrdinger equation: where µ is the NO reduced mass, V − r and Γ r represent the potential energies and the autoionization widths respectively for the five resonant NO − states included in the calculation, χ v is the wave function of the initial vibrational state of NO with energy v , and E = + v is the total energy of the system. Following Ref. [32], the widths Γ r were expressed as: where V 0 is the NO potential energy, H is a Heaviside step-function and l r is the angular momentum of the lowest contributing partial wave associated with the incident electron. Experimental results show that the dominant contribution come from the p-wave [22] so l r was set to one. In order to reproduce the positions and widths of the peaks in the low-energy region of the experimental cross sections, the constants c r in Eq. (4) were introduced as phenomenological external parameters as reported in the paper [20]. Their values are given in Table I. In Eqs. (2) and (3), V r is the discrete-to-continuum coupling given by [21]: where f r is the so-called penetration factor introduced to reproduce the correct dependence of the cross sections on the energy near the excitation threshold, namely, The sum in the cross section formula (2) runs over all the five resonance states of the NO − anion involved in the dissociation process. The integral extends to the continuum part of the NO potential from the dissociation threshold energy th v corresponding to the vibrational level v up to max v = th v + 10 eV. The spin-statistical factors are listed in Table I. In the model shown above, the potentials V 0 (R) and V − r (R) are expressed as a standard Morse function U (R) = D e 1 − e −α(R−Re) 2 +W whose parameters, for the NO molecule and for the three low-lying resonances 3 Σ − , 1 Σ + and 1 ∆, were determined by a fit procedure explained in [20]. In order to take into account the recent results presented in [12], the asymptotes for the singlet states 1 Σ + and 1 ∆ have been shifted to the correct threshold N − ( 3 P) + O( 3 P). We have checked, and the results in [20] are not affected by these changes. Analogously, the 3 Π and 1 Π symmetry parameters were obtained by a fit to the data presented in [11] and the ab-initio R-Matrix results in the Ref. [12]. All Morse parameters are summarized in Table I. The NO ground state potential energy curve, as well as those for the five NO − resonances and their corresponding autoionization widths are reported in Fig. 1. Table II contains the list of the vibrational levels supported by the NO molecule. Figures 2 and 3 summarize the results of the present letter. The cross sections were computed up to 15 eV, at which point they drop off and become negligible, yo cover temperatures up to 50000 K for the reaction rates, relevant for the applications mentioned above.    Figure 2 contains the results of cross sections for the process of dissociation in (1) for three specific vibrational levels of NO molecule. Partial contributions coming from the five resonances as well as the sum are shown. Some features can be noticed: (i) Basically, as expected, for all cases, the major contribution to the total cross section comes from the 3 Π and 1 Π resonances due to its closeness to the dissociation threshold, whereas the 1 Σ + and 1 ∆ resonances, in general, make a minor contribution, in particular for low and middle vibrational levels (v = 0 and v = 10). (ii) As a consequence of the Franck-Condon overlap, the 3 Π and 1 Π contributions to the cross section for v = 0 extends up to 11.5 eV, with a maximum around 9 eV. (iii) Beyond 11.5 eV, the asymptotic behavior is driven by the 3 Σ − resonance. (iv) Sinve high vibrational levels (v = 40) approach the NO dissociation limit, the contributions from 3 Σ − and, in particular, from the 1 Σ + and 1 ∆ states, become comparable to those from the 3 Π and 1 Π resonances at threshold.
Finally, Fig. 3 reports the full set of DE cross section results resolved over the vibrational ladder. By assuming a Maxwellian distribution for the electrons, the corresponding rate coefficient K v is given, as a function of the electron temperature T e , by: where k B is the Maxwell-Boltzmann constant.
Contributions coming from the five resonant states (broken lines, same colors as in Fig. 1) to the total dissociative excitation cross section (solid line) for v = 0 (plot on the left), v = 10 (plot on the middle) and v = 40 (plot on the right). In conclusion, vibrational state-resolved cross sections for dissociation of nitric oxide by electron-impact are computed for the first time using a phenomenological Local-Complex-Potential approach. Among the five resonances we considered in the calculations, the 3 Π symmetry is the one which makes the largest contribution. The full set of data obtained in the present work is available as supplementary material to this letter.