Charge Transfer Excitation of NeAr⁺ Ions in Collisions with Electrons

Abstract

We study the resonant processes of the excitation of weakly bound ${\mathrm{NeAr}}^{+}$ ions in collisions with the free electrons of Ne/Ar mixture plasma under conditions typical of the active media of gas lasers and plasma-based UV radiation sources. The transitions leading to the population of charge transfer electronic terms are considered. Using an original theoretical approach developed recently, we study the dependences of the cross-sections of several competing resonant processes on the incident electron energy and the gas temperature of the plasma. The role of the continuous states of internuclear motion is discussed. We highlight the specific features of the processes considered that stem from the low binding energy of ${\mathrm{NeAr}}^{+}$ ions and demonstrate that, in the weakly bound systems, the efficiencies of different charge transfer excitation channels differ greatly from those obtained for ions with moderate dissociation energies.

1. Introduction

The binary and ternary mixtures of noble gases are regular constituents of the active media of high-power lasers [1,2,3] and various plasma-based UV and IR range radiation sources [3,4,5,6,7]. In addition to atoms and atomic ions, the low-temperature plasma of such mixtures contains homonuclear and heteronuclear molecular cations. The latter play an important role in the kinetics of the upper state population of the working optical transitions of the aforementioned light sources [3,8]. The heteronuclear rare gas cations are characterized by small to moderate dissociation energies, $D_0$, of the ground electronic state (from 13.1 meV for ${\mathrm{HeXe}}^{+}$ to 647 meV for ${\mathrm{HeNe}}^{+}$ [9,10,11,12]), so that even at room gas temperatures of the plasma, the radiative and collisional processes involving these ions occur with the participation of the highly excited bound rovibrational states, as well as of the states in the continuum.

Distinct feature of the heteronuclear ${\mathrm{BA}}^{+}$ ions is the presence of the excited electronic terms of charge transfer (CT) character. Unlike the ground term and the regular excited states which dissociate to B + ${\mathrm{A}}^{+}$, the CT terms dissociate to the combination of ${\mathrm{B}}^{+}$ + A. The electronic transitions originating from the regular states and leading to the population of the CT states are often called charge transfer excitations. The electronic and spectroscopic properties of the CT states and radiative transitions between them and the lowermost electronic terms of the heteronuclear rare gas cations have been studied experimentally [13,14,15,16,17,18,19,20,21] and theoretically [9,10,11,17,18,19,22]. The energies $\hslash \omega$ of the transitions between the CT and the ground states are primarily determined by the difference of ionization energies of A and B atoms and the magnitudes of the spin–orbit splitting in ${\mathrm{A}}^{+}$ and ${\mathrm{B}}^{+}$ ions, and they range between 2.5 eV and 12 eV [13,22] for different noble gas A and B atoms.

In the experiments [3,23] and in the kinetic models of the operation of plasma-based light sources [2,8,24], the charge transfer electronic terms of heteronuclear rare gas ions are assumed to be populated via three-body conversion reactions. However, CT terms can also be populated via the electron–impact excitation of the ${\mathrm{BA}}^{+}$ ion in the ground state. Reactions like

$${\mathrm{BA}}^{+}\left(i,E_i\right)+e\left(\epsilon \right)\to {\mathrm{AB}}^{+}\left(f,E_f\right)+e\left({\epsilon }^{\prime }\right),$$

(1)

where i and f denote the initial and the final states of the molecular ion, $\epsilon$ and ${\epsilon }^{\prime }$ are the electron energies before and after the collision, while $E_i$ and $E_f$ are the energies of the states of internuclear motion in the initial and final channels of the reaction ($E_{i,f}<0$ and $E_{i,f}>0$ correspond to bound and free states, correspondingly), have been studied for ions of astrophysical importance (${\mathrm{HeH}}^{+}$, ${\mathrm{LiH}}^{+}$, etc.) [25,26,27,28,29]. On the contrary, for low-temperature plasmas of rare gas mixtures, the role of such processes, to the best of our knowledge, has only been considered in our recent work [30] for ${\mathrm{ArXe}}^{+}$ and ${\mathrm{KrXe}}^{+}$ ions and was briefly discussed in [8], where an estimate for a rate constant of the bound–bound process inverse to Reaction (1) was given. Here, we note that the the scale of electron energy threshold of Reaction (1), primarily determined by the difference of the ionization energies of the noble gas atoms forming the heteronuclear ion, falls into the range of the energies of the electrons in the typical setups of the lasers pumped by the electric discharges and the electron beam ionization. Thus, some channels of Reaction (1) may play an important role in the kinetics of the energy relaxation in the active media of various plasma-based sources of UV and IR range radiation.

In our recent work [30], we discussed the efficiency of three different channels of Reaction (1) in Ar + ${\mathrm{Xe}}^{+}+e$ and Kr + ${\mathrm{Xe}}^{+}+e$ systems, namely

$${\mathrm{BA}}^{+}\left(i,vJ\right)+e\left(\epsilon \right)\to {\mathrm{AB}}^{+}\left(f,E^{\prime }\right)+e\left({\epsilon }^{\prime }\right)\to \mathrm{A}+{\mathrm{B}}^{+}+e\left({\epsilon }^{\prime }\right),$$

(2)

$${\mathrm{BA}}^{+}\left(i,vJ\right)+e\left(\epsilon \right)\to {\mathrm{AB}}^{+}\left(f,v^{\prime }{J}^{\prime }\right)+e\left({\epsilon }^{\prime }\right),$$

(3)

$${\mathrm{BA}}^{+}\left(i,vJ\right)+e\left(\epsilon \right)\to \mathrm{AB}\left(f,E^{\prime },nl\right)\to \mathrm{B}\left(nl\right)+\mathrm{A},$$

(4)

where $vJ$ and $v^{\prime }{J}^{\prime }$ denote the initial and final bound rovibrational states of the ${\mathrm{BA}}^{+}$ ion, $E^{\prime }$ is the energy of the internuclear motion in the continuum in the final channels of Reactions (2) and (4), and $nl$ denotes the Rydberg state of B atom populated in the Reaction (4). Reactions (2)–(4) represent the resonant dissociative excitation of the ${\mathrm{BA}}^{+}$ ion, electron impact bound–bound excitation and dissociative recombination populating atomic Rydberg states, respectively. Unlike the commonly studied variants of these processes, Reactions (2)–(4) are accompanied by the transitions between the ground and the CT electronic terms. The efficiency of such transitions have strong dependence on the specifics of the potential energy curves of the initial and final states, and decline exponentially with the increase in the internuclear distance R.

In the initial channels of Reactions (2) and (3), the molecular ${\mathrm{BA}}^{+}$ ions occupy a bound state of nuclear motion. However, the charge transfer excitations may also occur during the three-body collisions of an atom, an ion and an electron, $\mathrm{B}+{\mathrm{A}}^{+}+e$. The non-adiabatic transitions that happen in the course of such ternary collisions give rise to free–free

$$\mathrm{B}+{\mathrm{A}}^{+}+e\left(\epsilon \right)\to {\mathrm{BA}}^{+}\left(i,E\right)+e\left(\epsilon \right)\to {\mathrm{AB}}^{+}\left(f,E^{\prime }\right)+e\left({\epsilon }^{\prime }\right)\to \mathrm{A}+{\mathrm{B}}^{+}+e\left({\epsilon }^{\prime }\right),$$

(5)

and free–bound charge transfer excitations

$$\mathrm{B}+{\mathrm{A}}^{+}+e\left(\epsilon \right)\to {\mathrm{BA}}^{+}\left(i,E\right)+e\left(\epsilon \right)\to {\mathrm{AB}}^{+}\left(f,v^{\prime }{J}^{\prime }\right)+e\left({\epsilon }^{\prime }\right),$$

(6)

where $E>0$ is the kinetic energy of the internuclear motion in the initial channel of the reactions. Under the quasi-equilibrium plasma conditions, when the electrons and the heavy particles are thermalized to different temperatures, $T_e$ and T, the ratio of the concentrations of the molecular ${\mathrm{BA}}^{+}$ ions in a bound state, $N_{{\mathrm{BA}}^{+}}$, and the product of the concentrations of the neutral atoms $\mathrm{B}$, $N_{\mathrm{B}}$, and the atomic ions ${\mathrm{A}}^{+}$, $N_{{\mathrm{A}}^{+}}$, obeys the mass action law

$${\displaystyle \frac{\left[N_{{\mathrm{BA}}^{+}}\right]}{\left[N_{\mathrm{B}}\right]\left[N_{{\mathrm{A}}^{+}}\right]}}={\displaystyle \frac{g_{{\mathrm{BA}}^{+}\left(i\right)}}{g_{\mathrm{B}}{g}_{{\mathrm{A}}^{+}}}}{Z}_{\mathrm{vr}}{\left({\displaystyle \frac{2\pi {\hslash }^2}{\mu k_{\mathrm{B}}T}}\right)}^{3/2}{e}^{D_0/k_{\mathrm{B}}T},$$

(7)

where $Z_{\mathrm{vr}}$ is the rovibrational partition function, $\mu$ is the reduced mass of ${\mathrm{BA}}^{+}$ ion, $k_{\mathrm{B}}$ is the Boltzmann constant, $g_{{\mathrm{BA}}^{+}\left(i\right)}$ is the electronic statistical weight of the molecular ion in the initial electronic state, while $g_{\mathrm{B}}$ and $g_{{\mathrm{A}}^{+}}$ are the statistical weights of the atom B and ion ${\mathrm{A}}^{+}$, respectively. Therefore, for moderately bound heteronuclear molecular ions, like ${\mathrm{ArXe}}^{+}$ and ${\mathrm{KrXe}}^{+}$ considered previously [30], the roles of Processes (5) and (6) are exponentially small as $D_0\gg k_{\mathrm{B}}T$. The situation may, however, be qualitatively different in systems with small binding energies $D_0\lesssim 100$ meV, where the impact of the exponential factor in (7) is not so severe.

The ground states of ${\mathrm{ArXe}}^{+}$ and ${\mathrm{KrXe}}^{+}$ systems discussed in [30] have potential energy wells, $D_e$, of 184 and 396 meV, respectively, while their CT excited terms are weakly bound with $D_e$ in the range of 80–150 meV depending on the specific state. In the present work, we consider the efficiency of different channels of Reaction (1) in collisions involving the ${\mathrm{NeAr}}^{+}$ molecular ion in Ne/Ar mixture plasma. The mixture is used in various gas lasers and plasma applications [2,8,31], including the active media of ArF excimer laser [2,32]. The ${\mathrm{NeAr}}^{+}$ ion is characterized by the inverse situation: its ground state is weakly bound with $D_e\approx$ 90 meV, and the binding energy of the CT states is predicted [22] to exceed 150 meV. Therefore, one can expect quite different roles of the channels of Reaction (1) in the kinetics of the radiation and collisional energy relaxation of Ne/Ar mixture as compared to Ar/Xe and Kr/Xe plasmas.

The main goal of this work is to study the dynamics of the resonant charge transfer excitation processes in the collisions of ${\mathrm{NeAr}}^{+}$ ions with the free electrons of the plasma. Using the recently developed theoretical approach [11,33], we carry out the calculations of the cross-sections of different channels of Reaction (1) realized via bound–bound, bound–free, free–free and free–bound non-adiabatic transitions. The calculations are performed for physical conditions typical of the active media of gas lasers, excimer lamps and microplasma cells. We demonstrate the efficiency of the processes studied for the system under consideration, highlight the differences in the dynamics of Reaction (1) stemming from the small binding energy of the heteronuclear ion, and specifically discuss the role of the continuous states of the internuclear motion in the initial and final channels of the reaction. It is important to note that in the present work, only the processes of electron–impact excitation are considered, so Reaction (4) is outside the scope of this study.

This paper is organized as follows. In Section 2, we give a brief summary of the theoretical method and main formulas used. The behavior of the electronic terms of the ground and several excited electronic terms of the ${\mathrm{NeAr}}^{+}$ ion and the oscillator strengths of the dipole transitions between these states are discussed in Section 3. The results of the calculations of the cross-sections of different channels of Process (1) are presented in Section 4. The main results of this paper are summarized in Section 5.

2. Theoretical Approach

Here, we only briefly outline the theoretical approach used. For more details, the reader is referred to our recent works [11,30,33,34,35].

All channels of Reaction (1) are described as non-adiabatic transitions in $\mathrm{Ne}+{\mathrm{Ar}}^{+}+e$ system. The transitions occur in the vicinity of the crossing point of the effective potential energy curves of the system given by the sum of the energy of the ground, $U_i\left(R\right)$, or the excited charge transfer, $U_f\left(R\right)$, electronic terms of ${\mathrm{NeAr}}^{+}$ and the energy of the incident electron before, $\epsilon$, or after, ${\epsilon }^{\prime }$, the collision. For processes of dissociative excitation (2), bound–bound electron impact excitation (3), as well as for the processes of the electron–impact induced charge transfer (5) and free–bound electron impact excitation (6), the magnitude of ${\epsilon }^{\prime }$ is positive.

For a given $\epsilon$ and ${\epsilon }^{\prime }$, the position of the potential energy curve crossing, $R_{\omega }$, is given by (see Figure 1)

$$U_f\left(R_{\omega }\right)-U_i\left(R_{\omega }\right)\equiv \Delta U_{fi}\left(R_{\omega }\right)=\epsilon -{\epsilon }^{\prime }\equiv \hslash \omega .$$

(8)

Here, we note that for transitions to CT states the magnitudes of the electronic terms $U_i\left(R\right)$ and $U_f\left(R\right)$ in the dissociation limit (i.e., at $R\to \infty$) differ substantially (by ${\Delta }_{\infty }\approx$ 5.7 eV for ${\mathrm{NeAr}}^{+}$), so that all channels of Reaction (1) have thresholds for the incident electron energy. Depending on the energy of the internuclear motion in the initial channel, the non-adiabatic transition will lead to one of the processes, (2), (3), (5), or (6), as depicted by the colored filled areas in Figure 1. The energy of internuclear motion in the final channel is, of course, determined by the energy conservation law.

Equations (2)–(6) describe processes which occur at given initial and final states of internuclear motion (either bound or free). When one is interested in the efficiency of a process in plasma, it is preferable to obtain the cross-sections of the process averaged over the distribution of the particles in the initial channel. For the weakly bound heteronuclear ions considered in the present work the entire manifold of the rovibrational states turns out to be populated even at room temperatures, so that the calculations of the individual contributions from all possible initial states become impractical. To address this problem, we made use of the quasi-continuum approximation for rovibrational states [36,37] and obtained [30,33,35] semiclassical expressions for the cross-sections of Processes (2)–(3), which describe the integral contributions from the entire spectrum of $vJ$ levels.

Assuming a Boltzmann distribution over the bound states of internuclear motion at given gas temperature T, the integral cross-sections of Processes (2) and (3) are given by

$${\sigma }_T^{\left(\mathrm{ch}\right)}\left(\epsilon \right)={\displaystyle \frac{g_{{\mathrm{BA}}^{+}\left(f\right)}}{g_{{\mathrm{BA}}^{+}\left(i\right)}}}{\displaystyle \frac{8{\pi }^3}{k^2{Z}_{\mathrm{vr}}\left(T\right)}}{\left({\displaystyle \frac{\mu k_{\mathrm{B}}T}{2\pi {\hslash }^2}}\right)}^{3/2}exp\left(-{\displaystyle \frac{D_0^{\left(i\right)}}{k_{\mathrm{B}}T}}\right)\underset{R_{min}\left(\epsilon \right)}{\overset{R_{max}\left(\epsilon \right)}{\int }}{\Gamma }_{\epsilon \to {\epsilon }^{\prime }}\left(R_{\omega }\right)exp\left(-{\displaystyle \frac{U_i\left(R_{\omega }\right)}{k_{\mathrm{B}}T}}\right){\Theta }_T^{\left(\mathrm{ch}\right)}\left(R_{\omega }\right)R_{\omega }^{2}d{R}_{\omega },$$

(9)

where index (ch) equal to either ’de’ or ’bbe’ denotes the following reaction type: dissociative excitation (2) or bound–bound excitation (3). In (9), the terms of $g_{{\mathrm{BA}}^{+}\left(i\right)}$ and $g_{{\mathrm{BA}}^{+}\left(f\right)}$ are the statistical weights of the initial and final electronic terms, k is the wavenumber of the incident electron, $k=\sqrt{2m_e\epsilon /{\hslash }^2}$, $m_e$ is the electron mass, and $D_0^{\left(i\right)}$ is the dissociation energy of the initial term. The quantity of ${\Gamma }_{\epsilon \to {\epsilon }^{\prime }}\left(R_{\omega }\right)$ is the effective coupling parameter, which describes the interaction of the incident electron with the electronic shell of the molecular ion. In a general case, the evaluation of the coupling parameter is a rather complex problem, which can be solved on the basis of ab initio calculations and multi-channel quantum defect theory [38,39]. However, when one considers the initial and final electronic terms for which transitions are dipole-allowed, ${\Gamma }_{\epsilon \to {\epsilon }^{\prime }}\left(R_{\omega }\right)$ can be obtained using a simple expression, as follows:

$${\Gamma }_{\epsilon \to {\epsilon }^{\prime }}\left(R_{\omega }\right)={\displaystyle \frac{2{\hslash }^2{G}_{\epsilon \to {\epsilon }^{\prime }}}{m_e{a}_0^2\left(\epsilon -{\epsilon }^{\prime }\right)\sqrt{3}}}{\mathsf{f}}_{fi}\left(R_{\omega }\right),$$

(10)

where $G_{\epsilon \to {\epsilon }^{\prime }}$ is the Gaunt factor [40], ${\mathsf{f}}_{fi}\left(R_{\omega }\right)$ is the electronic oscillator strength of the dipole transition between the electronic terms of the cation, and $a_0$ is the Bohr radius. Note that since ${\mathsf{f}}_{fi}\left(R_{\omega }\right)\sim \epsilon -{\epsilon }^{\prime }$, the total value of the effective coupling parameter ${\Gamma }_{\epsilon \to {\epsilon }^{\prime }}\left(R_{\omega }\right)$ depends only on resonant internuclear distance $R_{\omega }$ and not on values of energies $\epsilon$ and ${\epsilon }^{\prime }$.

The lower integration limit $R_{min}\left(\epsilon \right)$ in (9) is the solution of equation $\Delta U_{fi}\left(R_{min}\left(\epsilon \right)\right)=\epsilon$, while the upper limit $R_{max}\left(\epsilon \right)$ is the largest solution of the same equation $\Delta U_{fi}\left(R_{max}\left(\epsilon \right)\right)=\epsilon$ if it has two solutions (if $\epsilon <\Delta U_{fi}(R\to \infty )$, see Figure 1b) and is equal to the infinity if that equation has only one solution. If the energy of the incident electron equals the minimal energy possible for resonant non-adiabatic transitions, $\epsilon =min\left[\Delta U_{fi}\left(R_{\omega }\right)\right]$, then $R_{min}\left(\epsilon \right)=R_{max}\left(\epsilon \right)$, such that ${\sigma }_T^{\left(\mathrm{ch}\right)}\left(\epsilon \right)=0$.

The dimensionless factor ${\Theta }_T^{\left(\mathrm{ch}\right)}\left(R_{\omega }\right)$ in (9) describes the relative roles of Processes (2) and (3). In the semiclassical approximation, it is given by the expressions obtained in [33] for dipole-allowed photoabsorption transitions:

$${\Theta }_T^{\mathrm{de}}\left(R_{\omega }\right)=\left\{\begin{array}{cc}0,\hfill & \hfill R_{\omega }<R_0^{\left(i\right)},\\ {\displaystyle \frac{\gamma \left(3/2,\left|U_i\left(R_{\omega }\right)\right|/k_{\mathrm{B}}T\right)}{\Gamma (3/2)}},\hfill & \hfill R_0^{\left(i\right)}\le R_{\omega }\le R_0^{\left(f\right)},\\ {\displaystyle \frac{\gamma \left(3/2,\left|U_i\left(R_{\omega }\right)\right|/k_{\mathrm{B}}T\right)}{\Gamma (3/2)}}-{\displaystyle \frac{\gamma (3/2,|{\tilde{U}}_f\left(R_{\omega }\right)|/k_{\mathrm{B}}T)}{\Gamma (3/2)}},\hfill & \hfill R_x^{\left(f\right)}>R_{\omega }\ge R_0^{\left(f\right)},\\ 0,\hfill & \hfill R_{\omega }\ge R_x^{\left(f\right)}.\end{array}\right.$$

(11)

$${\Theta }_T^{\mathrm{bbe}}\left(R_{\omega }\right)=\left\{\begin{array}{cc}0,\hfill & \hfill R_{\omega }<R_0^{\left(f\right)},\\ {\displaystyle \frac{\gamma (3/2,|{\tilde{U}}_f\left(R_{\omega }\right)|/k_{\mathrm{B}}T)}{\Gamma \left(3/2\right)}},\hfill & \hfill R_x^{\left(f\right)}\ge R_{\omega }\ge R_0^{\left(f\right)},\\ {\displaystyle \frac{\gamma (3/2,|U_i\left(R_{\omega }\right)|/k_{\mathrm{B}}T)}{\Gamma \left(3/2\right)}},\hfill & \hfill R_{\omega }\ge R_x^{\left(f\right)}.\end{array}\right.$$

(12)

Here, ${\tilde{U}}_f\left(R_{\omega }\right)=U_f\left(R_{\omega }\right)-\Delta U_{fi}(R\to \infty )$, $\Gamma \left(x\right)\equiv {\int }_0^{\infty }{t}^{x-1}exp(-t)dt$ is the Gamma function, and $\gamma (x,y)\equiv {\int }_0^y{t}^{x-1}exp(-t)dt$ is the lower incomplete gamma function. The quantities of $R_0^{\left(i\right)}$, $R_0^{\left(f\right)}$ and $R_x^{\left(f\right)}$ are determined by the following equations (see also Figure 1):

$$U_i(R_0^{\left(i\right)})=0,{\tilde{U}}_f(R_0^{\left(f\right)})=0,U_i(R_x^{\left(f\right)})={\tilde{U}}_f(R_x^{\left(f\right)}).$$

(13)

The effectiveness of three-body processes (5) and (6) is described by rate coefficients $K_T^{\left(\mathrm{ch}\right)}\left(\epsilon \right)\left[{\mathrm{cm}}^2\right]$ defined as

$$K_T^{\left(\mathrm{ch}\right)}\left(\epsilon \right)={〈V_E{\sigma }^{\left(\mathrm{ch}\right)}(E,\epsilon )〉}_T,\left(\mathrm{ch}\right)=\{\mathrm{ff},\mathrm{fb}\}$$

(14)

where index (ch) equal to either ’ff’ or ’fb’ denotes the following reaction type: free–free (5) or free–bound (6) electron impact excitation. Here, $V_E=\sqrt{2E/\mu }$ is the velocity of internuclear motion, ${\sigma }^{\left(\mathrm{ch}\right)}(E,\epsilon )\left[{\mathrm{cm}}^4·\mathrm{s}\right]$ is the effective cross-section of Processes (5) and (6), and angular brackets ${\langle ...\rangle }_T$ represent the averaging operation over the internuclear motion corresponding to gas temperature T. Assuming quasi-classical approximation for non-adiabatic transitions and Maxwellian distribution over E at given gas temperature T, the rate constants of Processes (5) and (6) can be written as

$$K_T^{\left(\mathrm{ch}\right)}\left(\epsilon \right)={\displaystyle \frac{g_{{\mathrm{BA}}^{+}\left(f\right)}}{g_{\mathrm{B}\left(i\right)}{g}_{{\mathrm{A}}^{+}\left(i\right)}}}{\displaystyle \frac{8{\pi }^3}{k^2{Z}_{\mathrm{vr}}\left(T\right)}}\underset{R_{min}\left(\epsilon \right)}{\overset{\infty }{\int }}{\Gamma }_{\epsilon \to {\epsilon }^{\prime }}\left(R_{\omega }\right)exp\left(-{\displaystyle \frac{U_i\left(R_{\omega }\right)}{k_{\mathrm{B}}T}}\right){\Theta }_T^{\left(\mathrm{ch}\right)}\left(R_{\omega }\right)R_{\omega }^{2}d{R}_{\omega }.$$

(15)

The dimensionless factor ${\Theta }_T^{\left(\mathrm{ch}\right)}\left(R_{\omega }\right)$, (ch) = {ff, fb} in (15), describing the relative efficiency of the Processes (5) and (6), is given by [33]

$${\Theta }_T^{\mathrm{ff}}\left(R_{\omega }\right)=\left\{\begin{array}{cc}1,\hfill & \hfill R_{\omega }<R_0^{\left(i\right)},\\ {\displaystyle \frac{\Gamma (3/2,|U_i\left(R_{\omega }\right)|/k_{\mathrm{B}}T)}{\Gamma \left(3/2\right)}},\hfill & \hfill R_0^{\left(i\right)}\le R_{\omega }<R_x^{\left(f\right)},\\ {\displaystyle \frac{\Gamma (3/2,|{\tilde{U}}_f\left(R_{\omega }\right)|/k_{\mathrm{B}}T)}{\Gamma \left(3/2\right)}},\hfill & \hfill R_{\omega }\ge R_x^{\left(f\right)}.\end{array}\right.$$

(16)

$${\Theta }_T^{\mathrm{fb}}\left(R_{\omega }\right)=\left\{\begin{array}{cc}0,\hfill & \hfill R_{\omega }<R_x^{\left(f\right)},\\ {\displaystyle \frac{\gamma (3/2,|{\tilde{U}}_f\left(R_{\omega }\right)|/k_{\mathrm{B}}T)}{\Gamma \left(3/2\right)}}-{\displaystyle \frac{\gamma \left(3/2,\left|U_i\left(R_{\omega }\right)\right|/k_{\mathrm{B}}T\right)}{\Gamma \left(3/2\right)}},\hfill & \hfill R_{\omega }\ge R_x^{\left(f\right)},\end{array}\right.$$

(17)

where $\Gamma (x,y)\equiv {\int }_y^{\infty }{t}^{x-1}exp(-t)dt$ is the upper incomplete Euler gamma function.

It is important to notice that collision rate coefficients $K_T^{\left(\mathrm{ch}\right)}\left(\epsilon \right)$ of free–free (5) and free–bound transitions (6) given by (15) are normalized to the product $N_{\mathrm{B}}{N}_{{\mathrm{A}}^{+}}$ of the concentrations of atoms and atomic ions in the case of equilibrium over internuclear motion. On the other hand, the cross-sections (9) of dissociative (2) and bound–bound excitation (3) are normalized to the total concentration $N_{{\mathrm{BA}}^{+}}^{\left(i\right)}$ of bound molecular ions in the ground electronic state $\left|i\right.〉$ averaged over the Boltzmann distribution. Under LTE conditions, the efficiency of free–free (5) and free–bound (6) processes can be expressed through cross-section ${\sigma }_T^{\left(\mathrm{ch}\right)}\left(\epsilon \right)$, normalized to the concentration $N_{{\mathrm{BA}}^{+}}^{\left(i\right)}$ of bound molecular ions:

$${\sigma }_T^{\left(\mathrm{ch}\right)}\left(\epsilon \right)=K_T^{\left(\mathrm{ch}\right)}\left[N_{\mathrm{B}}\right]\left[N_{{\mathrm{A}}^{+}}\right]/\left[N_{{\mathrm{BA}}^{+}}\right],\left(\mathrm{ch}\right)=\{\mathrm{ff},\mathrm{fb}\}.$$

(18)

Using Formulas (15) and (7), it is seen that cross-sections ${\sigma }_T^{\left(\mathrm{ch}\right)}\left(\epsilon \right)$ for all studied processes (2)–(3), (5)–(6) are given by the same Equation (9), with the only difference in the dimensionless factors ${\Theta }_T^{\left(\mathrm{ch}\right)}\left(R_{\omega }\right)$ given by Equations (11), (12), (16) and (17), correspondingly. Therefore, for the sake of simplicity of the comparison, the calculations’ results will be expressed in terms of cross-sections ${\sigma }_T^{\left(\mathrm{ch}\right)}\left(\epsilon \right)$ [${\mathrm{cm}}^2$], (ch) = {de, bbe, ff, fb}.

3. Electronic Terms and Oscillator Strengths of Dipole Transitions

Most heteronuclear rare gas cations are characterized by relatively small dissociation energies of the ground electronic term, and fairly large spin–orbit interaction energies, which often exceed the energy splitting between neighboring non-relativistic terms. As such, they belong to the “c” type of Hund’s angular momentum coupling scheme. This is also true for the ${\mathrm{NeAr}}^{+}$ considered herein. In the “c” type of coupling, the electronic terms have only one good quantum number, $\Omega$, which is the projection of the total angular momentum of the electrons onto the internuclear axis.

Both a small dissociation energy and large spin–orbit coupling make the calculation of the electronic terms quite challenging. In a recent work [41], the potential energy curves of 36 lowermost electronic states of ${\mathrm{NeAr}}^{+}$ and the oscillator strengths of the dipole electronic transitions originating from three lowest terms, $X1/2$, $A_{1}3/2$ and $A_{2}1/2$, have been obtained using a hybrid approach based on the coupled cluster method with iterative single and double and non-iterative triple excitations (CCSD(T)) and the complete active space self-consistent field method (CASSCF) with the active space of 13 electrons in 12 orbitals, followed by n-electron valence perturbation theory method (NEVPT2) [42,43] and quasi-degenerate perturbation theory corrections (QDNEVPT2) [44]. The results of the calculations led to certain improvements in the previous ab initio data on the three lowest electronic states [10] and to the self-consistent description of the charge transfer terms, which were only treated using a model approach [22]. Most notably, the oscillator strengths of the electronic transitions were reported for the first time. The ab initio calculations presented in [41] were carried out using Orca computer program suite [45] version 6.0.

The results of electronic terms’ calculation of ${\mathrm{NeAr}}^{+}$ relevant to the present study are plotted in Figure 2. In addition to the three lowest terms, $X1/2$, $A_{1}3/2$ and $A_{2}1/2$, mentioned above, the figure features the potential energy curves of three terms with the charge transfer character, $B1/2$, $C_{1}3/2$ and $C_{2}1/2$. For all six states, numbers of 1/2 or 3/2 are the magnitudes of $\Omega$. Two lowest terms, $X1/2$ and $A_{1}3/2$, converge to $\mathrm{Ne}(2p^6 {}^{1}S)+{\mathrm{Ar}}^{+}(3p^5 {}^{2}P_{3/2})$ configuration in the dissociation limit, while $A_{2}1/2$ correlates with $\mathrm{Ne}(2p^6 {}^{1}S)+{\mathrm{Ar}}^{+}(3p^5 {}^{2}P_{1/2})$ configuration separated by the fine-structure splitting of ${\mathrm{Ar}}^{+}$ ion. Similarly, $B1/2$ and $C_{1}3/2$ CT terms dissociate to ${\mathrm{Ne}}^{+}(2p^5 {}^{2}P_{3/2})+\mathrm{Ar}\left(3p^6{}^{1}S\right)$ system, while $C_{2}1/2$ correlates with the ${\mathrm{Ne}}^{+}(2p^5 {}^{2}P_{1/2})+\mathrm{Ar}(3p^6 {}^{1}S)$ configuration, such that, at $R\to \infty$, they are separated by the fine-structure splitting of the ${\mathrm{Ne}}^{+}$ ion.

In the present work, we consider reactions originating from the ground electronic state of ${\mathrm{NeAr}}^{+}$ (although the initial states of the internuclear motion can be either bound or free) and resulting in the population of a CT term. The oscillator strengths of the dipole electronic transitions for such reactions are presented in Figure 3. It is seen from the figure that the magnitudes of ${\mathsf{f}}_{fi}\left(R\right)$ decline rapidly with the increase in R after passing the equilibrium distances of the CT terms. The decrease in ${\mathsf{f}}_{fi}\left(R\right)$ has clear exponential behavior until $R\approx$ 5 Å. The deviation from the exponential trend stems from the numerical errors caused by the near-degeneracies of two pairs of electronic states at a large R. One can see that transitions from $X1/2$ to $B1/2$ and $C_{2}1/2$ are dominant, while the transition from $X1/2$ to $C_{1}3/2$ becomes important only at very small or very large internuclear distances. The comparison with Figure 2 indicates that the respective small-R region corresponds to the strongly repulsive part of CT states, so the transitions which occur at such R bring negligible contributions to the integral cross-sections of the processes considered. In a large-R regime, when the $X1/2\to C_{1}3/2$ transition becomes comparable to $X1/2\to B1/2$ and $X1/2\to C_{2}1/2$, the typical values of ${\mathsf{f}}_{fi}\left(R\right)$ are more than two orders of magnitude smaller than their maximal values. Therefore, it is reasonable to neglect the role of the $X1/2\to C_{1}3/2$ transition. Below, we only consider the Processes (2)–(3) and (5)–(6), which occur via excitations from the ground state to $C_{2}1/2$ and $B1/2$ terms.

4. Results and Discussion

4.1. Dissociative Excitation and Bound–Bound Charge Transfer Excitation

The calculated cross-sections of the charge transfer excitation processes in collisions of electrons with ${\mathrm{NeAr}}^{+}$ ions in bound X 1/2 state are plotted in Figure 4. The cross-sections of dissociative excitation ${\sigma }_T^{\mathrm{de}}\left(\epsilon \right)$ and bound–bound electron impact charge transfer excitation ${\sigma }_T^{\mathrm{bbe}}\left(\epsilon \right)$ that occur due to the $X1/2\to B1/2$ transitions, resulting in the formation of the reaction product of Ar + ${\mathrm{Ne}}^{+}{(}^2{P}_{3/2})$, are presented in Figure 4a,b, correspondingly. Similarly, the cross-sections of Processes (2) and (3) that occur due to the $X1/2\to C1/2$ transitions are plotted in Figure 4c,d. The minimal electron energies at which the corresponding non-adiabatic transitions can occur (or, in other words, the curves $U_i\left(R\right)+\epsilon$ and $U_f\left(R\right)$ have intersections) are denoted by ${\Delta }_{min}$ and are shown by thin solid vertical lines.

As it is seen from Figure 4a,c, cross-sections ${\sigma }_T^{\mathrm{de}}\left(\epsilon \right)$ have zero values in the energy range from ${\Delta }_{min}$ to ${\Delta }_{\infty }$ (${\Delta }_{\infty }$ being the limit $\Delta U_{fi}(R\to \infty )$), which has the width of 0.07–0.08 eV. This is due to the fact that processes of dissociative excitation are energetically forbidden at energy deposits of $\hslash \omega =\epsilon -{\epsilon }^{\prime }\le {\Delta }_{\infty }$[11,30,33]. In a region of higher incident electron energies $\epsilon >{\Delta }_{\infty }$, the exponential rise of the cross-sections ${\sigma }_T^{\mathrm{de}}\left(\epsilon \right)$ begins. The cross-sections increase rapidly in the range of energies, for which the non-adiabatic transitions mainly occur at lower internuclear distances $R\approx R_e^{\left(i\right)}$, where the transition dipole moments reach higher values and, most importantly, molecular ions ${\mathrm{NeAr}}^{+}$ have higher equilibrium integral probability densities of the rovibrational states. The positions of the maxima of ${\sigma }_T^{\mathrm{de}}\left(\epsilon \right)$ correspond to the transitions that occur mostly in the vicinity of the ${\mathrm{NeAr}}^{+}$ equilibrium distance $R_{\omega }\lesssim R_e^{\left(i\right)}$ and can be roughly estimated as $\approx \Delta U_{fi}(R_e^{\left(i\right)})+0.5$ eV. With the further increase in $\epsilon$, the cross-sections start to slowly decline since the corresponding non-adiabatic transitions shift to the domain of even smaller internuclear distances and thus make minor contributions to the total value of the integral in (9). At energies $\epsilon -{\epsilon }^{\prime }>\Delta U_{fi}(R_0^{\left(i\right)})$ (or, in other words, at distances $R<R_0^{\left(i\right)}$), Process (2) is classically forbidden so the corresponding transitions make zero contribution to integral in (9), and at high $\epsilon$, the values of ${\sigma }_T^{\mathrm{de}}\left(\epsilon \right)$ decrease strictly as $1/\epsilon$.

According to Figure 4a,c, the maximal value $max\left[{\sigma }_T^{\mathrm{de}}\left(\epsilon \right)\right]\approx 7·{10}^{-18}{\mathrm{cm}}^2$ for transitions $X1/2\to B1/2$ at $T=300$ K, while for transitions $X1/2\to C1/2$, the cross-sections reach $max\left[{\sigma }_T^{\mathrm{de}}\left(\epsilon \right)\right]\approx 1.2·{10}^{-16}{\mathrm{cm}}^2$ at the same gas temperature. With a further increase in T, maximal and asymptotic values of ${\sigma }_T^{\mathrm{de}}\left(\epsilon \right)$ for both types of transitions slowly decrease. The main reason of the significant difference in maximal values is that the transitions responsible for Process (2) are the most efficient in the area of small internuclear distances $R\lesssim R_e^{\left(i\right)}$ ($R_e^{\left(i\right)}=4.63$ a.u. ≡ 2.45 Å), and the oscillator strengths for $X1/2\to C1/2$ transitions are far bigger in this area than for $X1/2\to B1/2$ (see Figure 3). The appreciable magnitudes of the cross-sections indicate the importance of the process of the dissociative excitation accompanied by the transitions to CT terms.

The results for cross-sections of bound–bound excitation ${\sigma }_T^{\mathrm{bbe}}\left(\epsilon \right)$ are shown by solid lines in Figure 4b,d. It is seen that the values of ${\sigma }_T^{\mathrm{bbe}}\left(\epsilon \right)$ demonstrate a sharp increase in the limited electron energy interval $({\Delta }_{min},\Delta U_{fi}(R_0^{\left(f\right)}))$ of typical width $\approx 0.2$ eV. This is the classically allowed range for the incident electron energy deposit in the bound–bound excitation process (3) [11,30,33], and it corresponds to the area of internuclear distances $R_{\omega }>R_0^{\left(f\right)}$ in which the dimensionless factor ${\Theta }_T^{\left(\mathrm{bbe}\right)}\left(R_{\omega }\right)$ is greater than zero. At higher energies $\epsilon$, the cross-sections ${\sigma }_T^{\mathrm{bbe}}\left(\epsilon \right)$ behave as $\sim 1/\epsilon$, as the contributions of transitions with high energy deposits to the total integral value in (9) are equal to zero in accordance with (12). Since the curves for ${\sigma }_T^{\mathrm{de}}\left(\epsilon \right)$ and ${\sigma }_T^{\mathrm{bbe}}\left(\epsilon \right)$ show a rapid increase in the different energy intervals, there is a range of incident electron energies which can be used to selectively activate the channel of bound–bound excitations leading to the formation of ${\mathrm{ArNe}}^{+}$ molecular ions.

The maximal values of ${\sigma }_T^{\mathrm{bbe}}\left(\epsilon \right)$ at $T=300$ K are approximately $5·{10}^{-17}{\mathrm{cm}}^2$ for $X1/2\to B1/2$ transitions and $1.5·{10}^{-16}{\mathrm{cm}}^2$ for $X1/2\to C1/2$ transitions. This difference is mostly explained by the higher values of transition dipole moments for the $X1/2\to C1/2$ channel in the range between $R_x^{\left(f\right)}$ and $R_e^{\left(f\right)}$ (equilibrium distance of an upper term, see Figure 1), where the bound–bound excitation processes turn out to be the most efficient (see Figure 3, $R_x^{\left(f\right)}\approx 2.50$ Å and $R_e^{\left(f\right)}\approx 2.73$ Å for both $B1/2$ and $C1/2$ states). The comparison of maximal values of ${\sigma }_T^{\mathrm{bbe}}\left(\epsilon \right)$ with ${\sigma }_T^{\mathrm{de}}\left(\epsilon \right)$ (dashed lines in panels b and d) shows that bound–bound excitation Processes (3) are dominant for $X1/2\to B1/2$ and have comparable efficiency with dissociative excitation Processes (2) for $X1/2\to C1/2$ transitions. This fact indicates that bound–bound processes have a significant contribution to the dynamics of collisional non-adiabatic processes involving ${\mathrm{NeAr}}^{+}$ molecular ions, which is quite different from the case of moderately bound heteronuclear ions (${\mathrm{ArXe}}^{+}$, ${\mathrm{KrXe}}^{+}$) [30,35]. The main reason for a high relative efficiency of a bound–bound channel (3) is the presence of the deeper potential wells in the upper terms ($B1/2$ and $C1/2$) of ${\mathrm{NeAr}}^{+}$ that have higher $D_e$ than lower term $X1/2$ does. This also explains why the relative efficiency of Processes (3) decreases with an increase in the gas temperature T: a higher T increases the probability of populating the free internuclear motion states of the upper terms.

Overall, we note that despite the small dissociative energy of the ground term, the charge transfer excitation processes originating from a bound state of ${\mathrm{NeAr}}^{+}$ are not negligible. To an extent, this stems from the fact that the respective transitions primarily occur at relatively small internuclear distances, where the oscillator strengths of the dipole transitions are quite high (see Figure 3).

4.2. Free–Free and Free–Bound Charge Transfer Excitation

Figure 5 shows the results of calculations of the cross-sections of free–free ${\sigma }_T^{\mathrm{ff}}\left(\epsilon \right)$ [${\mathrm{cm}}^2$] and free–bound ${\sigma }_T^{\mathrm{fb}}\left(\epsilon \right)$ [${\mathrm{cm}}^2$] charge transfer excitation processes. Although Processes (5) and (6) are three-body collisions and thus should have a dimension of [${\mathrm{cm}}^5$] for their rate constants $K_T^{\left(\mathrm{ch}\right)}\left(\epsilon \right)$, the cross-sections ${\sigma }_T^{\left(\mathrm{ch}\right)}\left(\epsilon \right)$, (ch) = {ff, fb}, have a dimension of [${\mathrm{cm}}^2$] since they are normalized to concentrations of molecular ${\mathrm{NeAr}}^{+}$ ions in the case of quasi-equilibrium over nuclear motion:

$$K_T^{\left(\mathrm{ch}\right)}\left(\epsilon \right)v_e\left[N_{\mathrm{Ne}}\right]\left[N_{{\mathrm{Ar}}^{+}}\right]N_e={\sigma }_T^{\left(\mathrm{ch}\right)}\left(\epsilon \right)v_e\left[N_{{\mathrm{NeAr}}^{+}}\right]N_e,v_e=\sqrt{2\epsilon /m_e}.$$

(19)

The results for the Processes (5)–(6) involving $X1/2\to B1/2$ transitions are presented in Figure 5a,c, whereas the cross-sections corresponding to $X1/2\to C1/2$ transitions are plotted in Figure 5b,d.

As is seen from Figure 5, solid curves for cross-sections ${\sigma }_T^{\mathrm{ff}}\left(\epsilon \right)$ [${\mathrm{cm}}^2$] of the free–free excitation (5) have the widest profile among all four processes studied since this is the only process that is energetically allowed in the entire region of $\epsilon \ge {\Delta }_{min}$ beyond the threshold. The maximal value of ${\sigma }_T^{\mathrm{ff}}\left(\epsilon \right)$ at $T=300$ K equals $\sim 2.1·{10}^{-18}{\mathrm{cm}}^2$ for $X1/2\to B1/2$ and $\sim 1.8·{10}^{-17}{\mathrm{cm}}^2$ for $X1/2\to C1/2$ transitions. A comparison with the results from Figure 4 shows that the efficiency of free–free channel (5) is several times lower than that of dissociative excitation at low gas temperatures. This means that Process (5) makes a relatively low contribution into the total dynamics of the charge transfer excitation. However, its efficiency increases exponentially with an increase in the gas temperature so that, at $T=600$ K, the contribution of (5) becomes higher than that of the dissociative excitation and comparable with the bound–bound excitation. A further increase in T to 900 K makes the free–free excitation the strongest channel of the electron impact charge transfer excitation processes in ${\mathrm{NeAr}}^{+}$, reaching a maximal value of $\sim 4.5·{10}^{-17}{\mathrm{cm}}^2$ for $X1/2\to B1/2$ and $\sim 2.25·{10}^{-16}{\mathrm{cm}}^2$ for $X1/2\to C1/2$ transitions. Thus, Reaction (5) may play a significant role in the kinetics of the energy relaxation in the plasmas discussed and may lead to a decrease in the mean electron energy.

It should be noted that the free–free charge transfer excitation process considered here as a molecular process can also be viewed as a charge transfer from Ne to ${\mathrm{Ar}}^{+}$ induced by the collisions with electrons. Such a process is analogous to the photoinduced charge transfer reactions which are actively studied in the fields of solar energy harvesting and ultracold collisions (see, for example, [46,47,48] and references therein). However, unlike radiative charge exchange, Reaction (5) received little attention in the literature, and, to our knowledge, was not studied systematically.

The results for cross-sections ${\sigma }_T^{\mathrm{fb}}\left(\epsilon \right)$ [${\mathrm{cm}}^2$] of free–bound processes (6) are indicated in Figure 5c,d by dashed curves. Like the cross-sections of bound–bound excitation (3), ${\sigma }_T^{\mathrm{fb}}\left(\epsilon \right)$ rapidly increase in a narrow energetically allowed interval only and after they decrease following $1/\epsilon$ dependence. The energetically allowed interval for free–bound excitation is (${\Delta }_{min},{\Delta }_{\infty }$) and has an approximate width of 0.07–0.08 eV. As one can see from the comparison of solid and dashed curves in Figure 5, the relative efficiency of free–bound processes (5) is comparable to that of free–free excitation (5) at $T=300$ K; however, it decreases with a increase in T. This decline is explained by an increase in the probability of populating the free internuclear motion states of the upper terms. Still, the channel of free–bound excitation (5) for $X1/2\to C1/2$ transitions remains significant and notable even at $T=900$ K (Figure 5c,d).

To sum up, the results presented in this subsection demonstrate that in the case of Ne/Ar mixture plasmas, owing to the relatively low dissociation energy of ${\mathrm{NeAr}}^{+}$, the process of free–free excitation (5) makes a crucial or even dominant contribution into collisional dynamics at elevated gas temperatures. This distinguishes the dynamics of the charge transfer excitation processes in ${\mathrm{NeAr}}^{+}$ ions from the previously studied moderately bound inert gas molecular ions. Moreover, we demonstrated that the process of the electron–impact association of the heteronuclear rare gas cations (6) also has a relatively high efficiency in Ne/Ar gas mixtures, due to the high dissociation energy of the “charge-transferred” ${\mathrm{ArNe}}^{+}$ ion.

4.3. Total Cross-Section of All Charge Transfer Excitation Channels

The results of total cross-sections’ calculations, including all four possible channels of the charge transfer excitation induced by the electron impact, taking into account both types of electronic transitions $X1/2\to B1/2$ and $X1/2\to C1/2$ in ${\mathrm{NeAr}}^{+}$, are presented in Figure 6a,b. Figure 6a features a smaller scale of the incident electron energy, which allows one to compare the roles of different Channels (2)–(3) and (5)–(6) in the transitions of type B. It is seen that the profile of the total cross-section of Reaction (1) in the $\mathrm{Ne}+{\mathrm{Ar}}^{+}+e$ system essentially consists of two parts: the small low-energy one, determined by processes accompanied by $X1/2\to B1/2$ transitions, and the high-energy one, mostly dominated by the processes which occur via $X1/2\to C1/2$ excitation. At $T=$ 300 K, the magnitudes of the total cross-sections are very close to the sums of ${\sigma }_T^{\mathrm{de}}\left(\epsilon \right)$ and ${\sigma }_T^{\mathrm{bbe}}\left(\epsilon \right)$, so charge transfer excitations primarily originate from the bound rovibrational states of the ${\mathrm{NeAr}}^{+}$ ion. The increase in the gas temperature to 900 K leads to reduced values of the cross-sections of (2), (3) and (6) channels. This reduction is, however, fully suppressed by the rise in the efficiency of free–free transitions (5). As a result, the total cross-section increases by a factor of almost 2. Therefore, the role of charge transfer excitation processes in collisions of ${\mathrm{NeAr}}^{+}$ with electrons becomes even more important at higher gas temperatures.

Such a behavior strongly contrasts with the case of moderately bound ${\mathrm{ArXe}}^{+}$ and ${\mathrm{KrXe}}^{+}$ ions previously studied in [30], where the total cross-sections of the electron–impact charge transfer decreased with the increase in T. Plotted in Figure 6c,d are the cross-sections of Processes (2) and (3) for ${\mathrm{ArXe}}^{+}$ at a different T. These two channels provide dominant contributions to Reaction (1) in the moderately bound heteronuclear rare gas cations [30]. One can see that the cross-sections of both channels decrease with the increase in T, which is similar to the results presented in Figure 4, although the decrease in ${\sigma }_T^{\mathrm{de}}\left(\epsilon \right)$ and ${\sigma }_T^{\mathrm{bbe}}\left(\epsilon \right)$ is somewhat stronger for ${\mathrm{ArXe}}^{+}$, which has a larger dissociation energy of the ground state. The difference in the behavior of ${\sigma }_T^{\mathrm{tot}}={\sigma }_T^{\mathrm{de}}+{\sigma }_T^{\mathrm{bbe}}+{\sigma }_T^{\mathrm{ff}}+{\sigma }_T^{\mathrm{fb}}$ in ${\mathrm{NeAr}}^{+}$ and ${\mathrm{ArXe}}^{+}$ is a direct result of the smaller binding energy of the system considered in the present work. A lower $D_0^{\left(i\right)}$ reduces the impact of the exponential term in (7), allowing the processes originating from the free states of the internuclear motion to make a significant contribution to the total cross-section of Reaction (1). Thus, for the proper description of the charge transfer excitation processes in the weakly bound systems, one should account for a wider range of reaction channels as compared to systems with higher dissociation energies.

Here, it is important to stress that we consider the case of the equilibrium conditions for the nuclear motion. However, depending on the specific experimental setup, the plasma of the rare gas mixture may exhibit some deviations from the equilibrium. In such a situation, the relative roles of the channels of Reaction (1) may, of course, be different. The theoretical approach used in the present work can be directly generalized to the cases of different distributions of the heavy particles and the electrons over the energy states, which was demonstrated in [34,49].

4.4. Rate Constant of Bound–Bound Quenching

In this section, we present the results of the calculations of rate constant $K_T^{\mathrm{bbq}}\left(T_e\right)$$\left[{\mathrm{cm}}^3·{\mathrm{s}}^{-1}\right]$ of bound–bound quenching (bbq) of ${\mathrm{ArNe}}^{+}$ ions in collisions with electrons and compare them to the estimations made in [8]. Bound–bound quenching (or de-excitation) is a process inverse to the bound–bound excitation Reaction (3), and its rate constant is defined as

$$K_T^{\mathrm{bbq}}\left(T_e\right)={〈v_{{\epsilon }^{\prime }}{\sigma }_T^{\mathrm{bbq}}\left({\epsilon }^{\prime }\right)〉}_{T_e},$$

(20)

where ${\sigma }_T^{\mathrm{bbq}}\left({\epsilon }^{\prime }\right)\left[{\mathrm{cm}}^2\right]$ is a Boltzmann-averaged cross-section of bound–bound quenching induced by collisions with free electrons with energies ${\epsilon }^{\prime }\equiv m_e{v}_{{\epsilon }^{\prime }}^2/2$, and brackets ${\langle ...\rangle }_{T_e}$ denote averaging over the Maxwellian distribution of the electron energies corresponding to electron temperature $T_e$. It should be stressed that charge transfer de-excitation processes have some specifics as compared to Reactions (2)–(3) and (5)–(6) (for example, they do not have a threshold on the incident electron energy). Here, we only briefly discuss bound–bound quenching in the context of the comparison of our results with the existing data. A systematic study of the electron–impact de-excitation reactions accompanied by the transitions originating from the charge transfer electronic terms is outside the scope of the present work and will be carried out in one of our upcoming publications.

Following the same theoretical approach as was used for description of Processes (2)–(3) and (5)–(6) in Section 2, we obtain the expression for cross-section ${\sigma }_T^{\mathrm{bbq}}\left({\epsilon }^{\prime }\right)$:

$${\sigma }_T^{\mathrm{bbq}}\left({\epsilon }^{\prime }\right)={\displaystyle \frac{g_{{\mathrm{BA}}^{+}\left(i\right)}}{g_{{\mathrm{BA}}^{+}\left(f\right)}}}{\displaystyle \frac{4{\pi }^3{\hslash }^2}{m_e{\epsilon }^{\prime }{Z}_{\mathrm{vr}}^{\left(f\right)}\left(T\right)}}{\left({\displaystyle \frac{\mu k_{\mathrm{B}}T}{2\pi {\hslash }^2}}\right)}^{3/2}exp\left(-{\displaystyle \frac{D_0^{\left(f\right)}}{k_{\mathrm{B}}T}}\right)\underset{R_0^{\left(f\right)}}{\overset{\infty }{\int }}{\Gamma }_{\epsilon \to {\epsilon }^{\prime }}\left(R_{\omega }\right)exp\left(-{\displaystyle \frac{{\tilde{U}}_f\left(R_{\omega }\right)}{k_{\mathrm{B}}T}}\right){\Theta }_T^{\mathrm{bbe}}\left(R_{\omega }\right)R_{\omega }^{2}d{R}_{\omega }.$$

(21)

Here, like in previous formulas, indices $\left(i\right)$ and $\left(f\right)$ correspond to the lower and higher electronic terms of ${\mathrm{NeAr}}^{+}$ ion, respectively; $Z_{\mathrm{vr}}^{\left(f\right)}\left(T\right)$ and $D_0^{\left(f\right)}$ represent a rovibrational partition function and a dissociation energy of an upper term ($C1/2$ or $B1/2$), respectively; the value of $R_0^{\left(f\right)}$ is defined by Equation (13), and the dimensionless function ${\Theta }_T^{\mathrm{bbe}}\left(R_{\omega }\right)$ is given by (12). Note that, unlike the case of the excitation cross-sections (9), the integration limits in the expression for the cross-section ${\sigma }_T^{\mathrm{bbq}}\left({\epsilon }^{\prime }\right)$ of bound–bound quenching do not depend on incident electron energy. This means that the dependence of ${\sigma }_T^{\mathrm{bbq}}\left({\epsilon }^{\prime }\right)$ on ${\epsilon }^{\prime }$ is given by the factor of $1/{\epsilon }^{\prime }$ only. The averaging over the Maxwellian distribution (20) results in a common dependence on electron temperature, $T_e^{-1/2}$:

$$K_T^{\mathrm{bbq}}\left(T_e\right)={\displaystyle \frac{g_{{\mathrm{BA}}^{+}\left(i\right)}}{g_{{\mathrm{BA}}^{+}\left(f\right)}}}{\left({\displaystyle \frac{\mu T}{m_e{T}_e}}\right)}^{3/2}{\displaystyle \frac{k_{\mathrm{B}}{T}_e}{\hslash }}{\displaystyle \frac{4\pi }{Z_{\mathrm{vr}}^{\left(f\right)}\left(T\right)}}exp\left(-{\displaystyle \frac{D_0^{\left(f\right)}}{k_{\mathrm{B}}T}}\right){\int }_{R_0^{\left(f\right)}}^{\infty }{\Gamma }_{\epsilon \to {\epsilon }^{\prime }}\left(R_{\omega }\right)exp\left(-{\displaystyle \frac{{\tilde{U}}_f\left(R_{\omega }\right)}{k_{\mathrm{B}}T}}\right){\Theta }_T^{\mathrm{bbe}}\left(R_{\omega }\right)R_{\omega }^{2}d{R}_{\omega }.$$

(22)

The calculated rate constants $K_T^{\mathrm{bbq}}\left(T_e\right)$ of bound–bound quenching of ${\mathrm{ArNe}}^{+}$ realized via electronic transitions $C1/2\to X1/2$ and $B1/2\to X1/2$ at $T=300$, 600 and 900 K are presented in Figure 7a and Figure 7b, respectively, and the estimation from [8] is shown by a dashed line in both panels. It is seen from a comparison of Figure 7a,b that, at $T=300$ K, the efficiency of bound–bound quenching accompanied by transitions $C1/2\to X1/2$ is $\approx 2$ times higher than that of $B1/2\to X1/2$ transitions. This is primarily determined by the difference of the corresponding oscillator strengths ${\mathsf{f}}_{fi}\left(R_{\omega }\right)$ in the vicinity of equilibrium points $R_e^{\left(f\right)}$ of upper terms ($R_e^C\approx R_e^B\approx 5.15$ Bohr radii). The estimations made in [8] turn out to be $\approx 1.5$ times higher than the rate constant, $K_T^{\mathrm{bbq}}\left(T_e\right)$, for $B1/2\to X1/2$ transitions and $\approx 1.5$ times lower than that for $C1/2\to X1/2$ under standard conditions, $T=T_e=300$ K.

The authors of [8] clearly state that the value of $K_T^{\mathrm{bbq}}\left(T_e\right)$, which they used ($2·{10}^{-7}{\mathrm{cm}}^3{\mathrm{s}}^{-1}$), is merely an estimate. Our calculations indicate that it has the correct order of magnitude but may lead to significant errors depending on the gas and electron temperature, and on the specific electronic transition considered.

5. Summary and Conclusions

We have carried out a theoretical study of of the electron–impact excitation of the weakly bound ${\mathrm{NeAr}}^{+}$ ions in the ground state accompanied by the transitions to the charge transfer terms. Four channels of the general reaction (1) were considered: dissociative (2), bound–bound (3), free–free (5) and free–bound (6) excitations. We demonstrated that as compared to the previously studied moderately bound ${\mathrm{ArXe}}^{+}$ and ${\mathrm{KrXe}}^{+}$ ions, where only two channels, (2) and (3), were important, in ${\mathrm{NeAr}}^{+}$, which has much lower dissociation energy $D_0^i$ of the ground state and much deeper potential wells in the CT terms, all four processes considered play significant roles. The calculations were performed on the basis of the semianalytic approach [11,34,50] that allows one to give a unified self-consistent description of the contributions from the entire quasi-continuous spectra of rovibrational levels and the continuum of the free internuclear motion to the cross-sections and rate constants of all channels of the Reaction (1).

The cross-sections of the processes studied, (2)–(3) and (5)–(6), were presented for ranges of the incident electron energies $\epsilon =$ 5.5–8 eV and gas temperatures $T=$ 300–900 K for two types of possible electronic transitions, $X1/2\to C1/2$ (type A) and $X1/2\to B1/2$ (type B). The total cross-sections taking into account both types of transitions have a clearly distinguishable overlap, but the efficiency and overall contribution of A-type transitions is several times higher compared to B-type transitions. This results from the significant difference in the oscillator strengths of the electronic dipole transitions. The comparison of dissociative excitation (2) and bound–bound excitation (3) shows that maximal efficiency of the latter process is higher at $T=300$ K, so that in contrast to the cases of ${\mathrm{ArXe}}^{+}$ and ${\mathrm{KrXe}}^{+}$, where dissociative excitation was dominant, collisions of ${\mathrm{NeAr}}^{+}$ ions favor the production of ${\mathrm{ArNe}}^{+}$ ions in the bound CT states. The high efficiency of the bound–bound channel is due to high dissociation energies of upper CT electronic states of ${\mathrm{NeAr}}^{+}$ ions.

We demonstrated that the cross-sections of free–free (5) and free–bound (6) processes, which were not previously discussed in the context of the rare gas mixtures, increase rapidly with the increase in T, so that these reactions become predominant at elevated temperatures $T=900$ K. This is owing to the relatively low dissociation energy of the ground state of the ${\mathrm{NeAr}}^{+}$ ion. Overall, a comparative analysis of four studied channels (2)–(3) and (5)–(6) suggests that all of them are presented equally in collisional dynamics in the considered range of plasma parameters, which clearly distinguishes the weakly bound heteronuclear noble gas cations from their moderately bound counterparts. The calculation of the total cross-section of electron–impact excitation to the charge transfer states (1) exhibited the rise with the increase in the gas temperatures. Such a behavior is somewhat unusual and results from the concurrent reduction in the efficiency of the channels originating from a bound state, (2)–(3), and a sharp increase in the efficiency of free–free excitation (5).

A formula was obtained, and the calculations were carried out for the rate constant of the process of bound–bound quenching of ${\mathrm{ArNe}}^{+}$ ions in collisions with free electrons of the plasma. This process is an inverse to (3) and was previously discussed in [8]. We have shown that the estimation of the rate constant given in [8] provides a correct order of magnitude for the rate constant but may overestimate or underestimate it by up to a factor of eight, depending on plasma conditions.

The results of the present study demonstrate the important role of the charge transfer excitation processes in collisions with electrons in the dynamics of rare gas mixture plasmas. They also highlight the specifics of the processes considered in the collisions involving weakly bound molecular cations. Finally, the reactions discussed are additional paths of the energy relaxation which are not yet included in most of the kinetic models of the plasma-based UV and IR radiation sources, so the results of this work may be used in the further optimization of such devices.