Resonant Electron Capture by Ions into Rydberg States of Atoms

Abstract

Resonant mechanisms of electron–ion recombination accompanied by the formation of Rydberg atoms in a plasma containing atomic and molecular ions are investigated. An analytical approach is developed for the description of three-particle electron capture into a Rydberg state as a result of resonant energy transfer from a free electron to the electronic shell of a quasimolecular ion formed during the collision of an atomic ion with a buffer gas atom. An efficient method is proposed to calculate dissociative recombination rates under thermal excitation of all rotational–vibrational (rovibrational) levels of the molecular ion. The dependence of the cross sections and the rate constants of the processes on the principal quantum number is established, and the relative role of these processes is determined in a wide range of temperatures of the electron, T_e, and gas, T, components of the plasma. Conditions are found under which integral contributions of the continuous spectrum of the molecule and of the whole rovibrational quasicontinuum to the total rate of resonant electron capture are dominant. A specific analysis is carried out by an example of Ne + Xe⁺ + e and Ar + Xe⁺ + e heteronuclear systems with significantly different dissociation energies (D₀ = 33 and 171 meV) of the ground electronic term of the RgXe⁺ (Rg = Ne and Ar) ion. It is shown that the capture rate constants essentially depend on the binding energy |ε_n| of the resulting Xe(n) atom, the temperatures T and T_e, and the relationship between D₀ and the thermal energy k_BT.

1 INTRODUCTION

Recombination of atomic and molecular ions with electrons has been intensively studied for decades [1–5]. This is associated with the role of these processes in gas discharge physics, in active media of plasma lasers, in plasma spectroscopy, and in the physics of stellar and planetary atmospheres. Many recombination processes have been studied in detail. This, in particular, concerns dissociative recombination of electrons with molecular ions [6]. The effectiveness of the “direct mechanism” of this process is due to the fact that it occurs as a result of a nonadiabatic transition between electronic terms of the system and is accompanied by resonant energy exchange with the inner electrons of the BA⁺ + e system.

In this work, we study the resonant mechanisms of electron–ion recombination with the formation of Rydberg atoms in a plasma containing atomic and molecular ions:

$${\text{B}}{{{\text{A}}}^{ + }}(i,{v}J) + e \to {\text{BA}}(f,nl) \to {\text{A}}(nl) + {\text{B}},$$

(1)

$$\begin{gathered} {{{\text{A}}}^{ + }} + e + {\text{B}} \to {\text{B}}{{{\text{A}}}^{ + }}(i) + e \\ \to {\text{BA}}(f,nl) \to {\text{A}}(nl) + {\text{B}}. \\ \end{gathered} $$

(2)

Reaction (1) is a resonant electron capture by a molecular ion into a highly excited atomic level (dissociative recombination). Here i and f denote the initial, U_i(R), and final, U_f(R), electronic terms of the BA⁺ ion; ${v}$ and J are its vibrational and rotational quantum numbers in the initial state i; and n and l are the principal and orbital quantum numbers of the Rydberg atom A(nl). In process (1), the nuclei of the BA⁺ + e system make a bound–free transition. The conditions considered in this work differ significantly from those usually realized in the study of dissociative recombination of strongly bound molecular ions [5, 6], when a small number of rovibrational levels ${v}$J of these ions are populated. We are interested in the conditions under which a very large number of ${v}$J levels contribute to recombination. In the present study, we develop an efficient method for calculating the total contribution of all these levels to the total cross sections and the rate constants of dissociative recombination.

Reaction (2) is an electron capture by an atomic ion into a Rydberg state in a ternary collision with an atom (three-particle recombination). In the traditional collision mechanism [1], which is due to the elastic scattering of a weakly bound electron by an atom B in the field of an ion A⁺, such a process leads to electron capture only into very high levels n. In this paper, we consider a much more efficient resonant mechanism in which a free–bound transition of the outer electron is accompanied by a free–free transition of the nuclei of the BA⁺ + e quasimolecule formed during the collision of the particles B and A⁺. Similar to process (1), the capture mechanism (2) occurs as a result of the intersection of the initial, U_i(R) + ε, and the final, U_f(R) + ε_nl, electronic terms of the BA⁺ + e system, where ε = ${{\hbar }^{2}}$k²/2m_e is the free-electron energy and ε_nl = –Ry/$n_{*}^{2}$ is the energy of the Rydberg electron.

In contrast to dissociative recombination (1), little is known in the literature about channel (2) of three-particle electron capture occurring through the formation of a quasimolecule in the collision of atom B and ion A⁺. In [7], the authors calculated the cross sections and rate constants of the recombination population of Rydberg states H(n) in ternary collisions with helium atoms due to direct transfer of the free-electron energy to the relative motion of the nuclei of a HeH⁺ ion by analogy with the inelastic n → n′ transitions and ionization considered in [8, 9]. In the case of resonant ternary recombination, the rate constants of this process with the formation of H(n) and He(n) atoms were calculated in [10–12] by the semiclassical method for symmetric collisions involving hydrogen and helium under stellar atmospheric conditions (T ~ 5000–10 000 K).

The inverses of reactions (1) and (2) are the reactions of associative and direct ionization of Rydberg atoms A(nl) in collisions with neutral particles B, due to the resonant energy exchange of outer and inner electrons of the BA⁺ + e quasimolecule formed during scattering of the atom B by the atomic core A⁺. Depending on the type of system and the symmetry of electronic terms, ionization can occur as a result of dipole [13] or quadrupole and short-range [14] interaction of the outer electron with BA⁺. Usually, resonant ionization mechanisms are the primary ones for small and intermediate values of n. As the principal quantum number increases, the mechanism of scattering of a quasifree electron by a perturbing atom becomes dominant [15].

Inelastic transitions between Rydberg levels, accompanied by the resonant energy transfer to the outer electron from the electronic subsystem of a homonuclear [16] or a heteronuclear [17] quasimolecular ion, occur similarly. The efficiency of resonant n → n′ transitions turns out to be much higher in a wide range of values of n than that in the case of alternative nonresonant processes of quasielastic l-, nl-, and J-mixing transitions [18–20].

The possibility of a strong increase in the rate of recombination of electrons with atomic ions in ternary collisions with free electrons as a result of resonant n → n′ transitions leading to the de-excitation of the recombining electron by buffer gas atoms was demonstrated in [17]. Estimates made in the recent article [21] indicate the possibility of a sharp increase in the rate of recombination population of relatively low Rydberg levels and an increase in the resulting recombination coefficient due to the contribution of resonant free–bound electron transitions.

The goal of this work is to study process (1) under thermal excitation of a large number of rovibrational levels and to develop a semiquantum theory of resonant ternary recombination (2). This includes the elaboration of a unified approach to the description of these processes and the derivation of formulas for their effective cross sections and rate constants. Specific calculations and the analysis of the efficiencies of reactions (1) and (2) are performed by examples of Rg + Xe⁺ + e (Rg = Ne and Ar) systems for low xenon concentrations, [Xe] ≪ [Rg]. RgXe⁺ heteronuclear ions have relatively low dissociation energies D₀ of the ground term ($D_{0}^{{{\text{HeX}}{{{\text{e}}}^{ + }}}}$ = 13.1 meV, $D_{0}^{{{\text{NeX}}{{{\text{e}}}^{ + }}}}$ = 33 meV, $D_{0}^{{{\text{ArX}}{{{\text{e}}}^{ + }}}}$ = 171 meV, and $D_{0}^{{{\text{KrX}}{{{\text{e}}}^{ + }}}}$ = 400 meV [22, 23]); therefore, the ${v}$J levels of these ions turn out to be strongly excited even at gas temperatures of T ≈ 300–1000 K.

2 INITIAL FORMULAS

The total Hamiltonian of the BA⁺ + e quasimolecule in the center-of-mass system has the form

$$H = {{T}_{{\mathbf{R}}}} + {{H}_{{{\text{B}}{{{\text{A}}}^{ + }}}}} + {{H}_{e}} + {\text{V}},\quad {{H}_{e}} = - \frac{{{{\hbar }^{2}}{{\Delta }_{{\mathbf{r}}}}}}{{2{{m}_{e}}}} - \frac{{{{e}^{2}}}}{r}.$$

(3)

Here T_R = –${{\hbar }^{2}}$Δ_R/2μ is the operator of kinetic energy of the relative motion of nuclei; R = R_A – R_B is the radius vector connecting the nuclei; μ is the reduced mass of the nuclei; H_e is the Hamiltonian of the outer electron; and ${{H}_{{{\text{B}}{{{\text{A}}}^{ + }}}}}$(r_κ, R) is the electron Hamiltonian of the quasimolecular BA⁺ ion in the Born–Oppenheimer approximation. The potential energy

$${\text{V}} = \sum\limits_\kappa ^{} {\frac{{{{e}^{2}}}}{{{\text{|}}{\mathbf{r}} - {{{\mathbf{r}}}_{\kappa }}{\text{|}}}}} - \frac{{{{Z}_{{\text{A}}}}{{e}^{2}}}}{{{\text{|}}{\mathbf{r}} - {{{\mathbf{R}}}_{{\text{A}}}}{\text{|}}}} - \frac{{{{Z}_{{\text{B}}}}{{e}^{2}}}}{{{\text{|}}{\mathbf{r}} - {{{\mathbf{R}}}_{{\text{B}}}}{\text{|}}}} + \frac{{{{e}^{2}}}}{r}$$

(4)

is defined by the Coulomb interaction of the outer electron (r) with all inner electrons (r_κ, κ = 1, … , N) and the nuclei (R_A and R_B are the radius vectors of the nuclei in the center-of-mass system). The last term is included in (4) to compensate for the Coulomb term, which was included in the Hamiltonian H_e.

We will proceed from the general expression for the differential cross section d$\sigma _{{nl \to \varepsilon }}^{{di}}$(E ')/dε [cm² erg^–1] of the resonant direct ionization process accompanied by a nonadiabatic transition between two electronic terms U_i(R) and U_f(R) of the quasimolecular ion BA⁺, which is temporarily formed during the collision of A⁺ and B particles (see formula (11) in [14]), and the detailed balance relation for the effective cross section, $\sigma _{{\varepsilon \to nl}}^{{tr}}$(E) [cm⁴ s], of electron capture into the Rydberg level of the atom A(nl) in the ternary collision (2):

$$\sigma _{{\varepsilon \to nl}}^{{tr}}(E) = \frac{{{{g}_{f}}}}{{{{g}_{i}}}}\frac{{2{{\pi }^{2}}\hbar (2l + 1)}}{{{{k}^{2}}}}\frac{{{{{(q{\kern 1pt} ')}}^{2}}}}{{{{q}^{2}}}}\frac{{d\sigma _{{nl \to \varepsilon }}^{{di}}(E{\kern 1pt} ')}}{{d\varepsilon }},$$

(5)

$$E{\kern 1pt} ' = {{\hbar }^{2}}{{(q{\kern 1pt} ')}^{2}}{\text{/}}2\mu = E + \varepsilon \, + \,{\text{|}}{{\varepsilon }_{{nl}}}{\text{|}}.$$

(6)

Here E ' is the kinetic energy of the colliding particles A⁺ and B in the final channel of reaction (2), E = ${{\hbar }^{2}}$q²/2μ; ε = ${{\hbar }^{2}}$k²/2m_e is the electron energy in the continuous spectrum; and ε_nl = –Ry/$n_{*}^{2}$ is the energy of the Rydberg electron, where $n_{*}^{{}}$ = n – δ_l, and δ_l is the quantum defect of the atomic level nl. The ratio g_f/g_i is expressed as ${{g}_{{{\text{B}}{{{\text{A}}}^{ + }}(f)}}}$/(${{g}_{{{\text{B}}(i)}}}{{g}_{{{{{\text{A}}}^{ + }}(i)}}}$) ≡ g_tr.

Substituting formula (11) from [14] into (5), we obtain an expression for the effective electron capture cross section as a result of the nonadiabatic process (2):

$$\begin{gathered} \sigma _{{\varepsilon \to nl}}^{{{\text{tr}}}}(E) = {{{\tilde {g}}}_{{{\text{tr}}}}}\frac{{8{{\pi }^{2}}\hbar }}{{{{q}^{2}}{{k}^{2}}}}\sum\limits_J^{} {(2J + 1)} \\ \times \sum\limits_{l'm',m}^{} {{\text{|}}\langle \chi _{{EJ}}^{{(i)}}(R){\text{|V}}_{{i,\varepsilon l'm'}}^{{f,nlm}}(R){\text{|}}\chi _{{E'J}}^{{(f)}}(R)\rangle {{{\text{|}}}^{2}}.} \\ \end{gathered} $$

(7)

Here $\chi _{{EJ}}^{{(i)}}$(R) and $\chi _{{E'J}}^{{(f)}}$(R) are the radial wave functions of the relative motion of nuclei in the initial and final states; J is the angular momentum of the BA⁺ ion; the ratio ${{\tilde {g}}_{{{\text{tr}}}}}$ is defined as g_tr/$\mathfrak{s}$, where $\mathfrak{s}$ is equal to 2 for homonuclear systems in the case when U_i or U_f is a Σ‑term, or 1 otherwise; and ${\text{V}}_{{i,\varepsilon l'm'}}^{{f,nlm}}$(R) is the electronic matrix element of the free–bound nonadiabatic transition,

$${\text{V}}_{{i,\varepsilon l'm'}}^{{f,nlm}}({\mathbf{R}}) = \langle {{\psi }_{{\varepsilon l'm'}}}{\text{|}}\langle {{\phi }_{i}}{\text{|V}}({\mathbf{r}},{{{\mathbf{r}}}_{\kappa }},{\mathbf{R}}){\text{|}}{{\phi }_{f}}\rangle {\text{|}}{{\psi }_{{nlm}}}\rangle ,$$

(8)

where ϕ_i(r_κ, R) and ϕ_f(r_κ, R) are the electronic wave functions of the BA⁺ ion that correspond to the terms U_i(R) and U_f (R), respectively; ψ_nlm(r) is the wave function of the Rydberg atom; and ψ_εl 'm'(r) = ${{\mathcal{R}}_{{\varepsilon l'}}}$(r)Y_l 'm'(n_r), where ${{\mathcal{R}}_{{\varepsilon l'}}}$(r) is normalized to δ(ε – ε'). The matrix element (8) is related to the autoionization width of the quasimolecule BA(nl):

$${{\Gamma }_{{nl \to \varepsilon }}}(R) = \frac{{2\pi }}{{2l + 1}}\sum\limits_{m,m'l'}^{} {{\text{|V}}_{{i,\varepsilon l'm'}}^{{f,nlm}}(R){{{\text{|}}}^{2}}.} $$

(9)

From the expression [14] for the partial cross section $\sigma _{{nl \to \varepsilon }}^{{ai}}$(E 'J) of associative ionization of the Rydberg atom A(nl) in a collision with atom B and the detailed balance relation, we obtain a formula for the cross section $\sigma _{{\varepsilon \to nl}}^{{{\text{dr}}}}$(${v}$J) of electron capture into the Rydberg level of the atom A(nl) during dissociative recombination of the BA⁺ ion in the ${v}$J state:

$$\sigma _{{\varepsilon \to nl}}^{{{\text{dr}}}}({v}J) = \frac{{{{g}_{{{\text{B}}{{{\text{A}}}^{ + }}(f)}}}(2l + 1){{q}^{2}}}}{{{{g}_{{{\text{B}}{{{\text{A}}}^{ + }}(i)}}}(2J + 1){{k}^{2}}}}\sigma _{{nl \to \varepsilon }}^{{ai}}(E{\kern 1pt} '{\kern 1pt} J).$$

(10)

This leads to the following result:

$$\begin{gathered} \sigma _{{\varepsilon \to nl}}^{{{\text{dr}}}}({v}J) = {{g}_{{{\text{dr}}}}}\frac{{4{{\pi }^{3}}}}{{{{k}^{2}}}} \\ \, \times S_{{nl}}^{J}\sum\limits_{l'm',m}^{} {{\text{|}}\langle \chi _{{{v}J}}^{{(i)}}(R){\text{|V}}_{{i,\varepsilon l'm'}}^{{f,nlm}}(R){\text{|}}\chi _{{E'J}}^{{(f)}}(R)\rangle {{{\text{|}}}^{2}},} \\ \end{gathered} $$

(11)

where g_dr = ${{g}_{{{\text{B}}{{{\text{A}}}^{ + }}(f)}}}$/${{g}_{{{\text{B}}{{{\text{A}}}^{ + }}(i)}}}$. Note that everywhere from now on we set J′ = J. This is justified because the main contribution to the sums over J in formulas (7) and (23) for the cross sections is made by large values of J, J ≫ 1. We introduced into formula (11) the so-called survival factor $S_{{nl}}^{J}$, which takes into account the effect of the backward decay of the autoionizing Rydberg state of the molecule, BA(f, nl) → BA⁺(i) + e, on the dissociative recombination. It is known that taking into account this process can reduce the resulting cross sections if the value of n is small. In the semiclassical analysis [6, 24], the survival factor $S_{{nl}}^{J}$ is defined by

$$S_{{nl}}^{J} = \exp \left[ { - \int\limits_{{{R}_{\omega }}}^{{{R}_{{{\text{st}}}}}} {\frac{{{{\Gamma }_{{nl \to \varepsilon }}}(R)}}{\hbar }\frac{{dR}}{{V_{{E{\kern 1pt} '{\kern 1pt} J}}^{{(f)}}(R)}}} } \right],$$

(12)

where R_ω is a transition point (see Section 3) and $V_{{E{\kern 1pt} '{\kern 1pt} J}}^{{(f)}}$(R) is the relative velocity of heavy particles in the final reaction channel:

$$V_{{E{\kern 1pt} '{\kern 1pt} J}}^{{(f)}}(R) = \sqrt {\frac{2}{\mu }\left[ {E{\kern 1pt} '\, - \frac{{{{\hbar }^{2}}{{{(J + 1{\kern 1pt} /{\kern 1pt} 2)}}^{2}}}}{{2\mu {{R}^{2}}}}} \right]} .$$

(13)

The quantity R_st ≡ R_nl in (12) is called the stabilization point; outside this point (R > R_st), the autoionization becomes impossible if the exponentially small contribution of the under-barrier region is neglected. This point is determined from the equation

$${{U}_{i}}({{R}_{{{\text{st}}}}}) = {{U}_{f}}({{R}_{{{\text{st}}}}}) - {\text{Ry/}}n_{*}^{2}.$$

(14)

At low electron energies ε → 0, the points R_ω and R_st practically coincide, and the effect of the autoionization decay can be neglected. Calculations show that, for heteronuclear ions of inert gases containing xenon, R_ω ≈ R_st at energies of interest, which allows us to set $S_{{nl}}^{J}$ = 1.

3 CALCULATION OF THE EFFECTIVE CROSS SECTIONS OF RESONANT ELECTRON CAPTURE

For a given transition frequency ω = (ε + |ε_nl|)/$\hbar$, a resonant electron capture event occurs in the vicinity of the crossing point R_ω of the potential curves U_i,ε(R) = U_i(R) + ε and U_f,nl(R) = U_f(R) + ε_nl,

$${{U}_{i}}({{R}_{\omega }}) + \frac{{{{\hbar }^{2}}{{k}^{2}}}}{{2{{m}_{e}}}} = {{U}_{f}}({{R}_{\omega }}) - \frac{{{\text{Ry}}}}{{n_{*}^{2}}}.$$

(15)

At the point R_ω, the energy $\hbar$ω = ε + |ε_nl| transferred by the incident electron upon capture into the level nl becomes equal to the energy splitting ΔU_fi(R_ω) = U_f(R_ω) – U_i(R_ω) of electronic terms. For a self-consistent description of nonadiabatic transitions in classically allowed and forbidden regions of the internuclear distance R, we will use the quantum version of the theory of nonadiabatic transitions for the case of linear crossing of potential energy curves.

3.1 Cross Section of Electron Capture by an Atomic Ion in a Ternary Collision with an Atom

Consider the process of resonant electron capture by atomic ions in the case when the process is accompanied by a free–free transition of the nuclei of the quasimolecule BA⁺ + e. In the approximation of linear crossing of terms, the squared modulus of the transition matrix element with respect to the wave functions of nuclear motion is expressed in terms of the Airy function (see formula (90.22) in [24]),

$$\begin{gathered} \sum\limits_{l'm',m}^{} {{\text{|}}\langle \chi _{{EJ}}^{{(i)}}(R){\text{|V}}_{{i,\varepsilon l'm'}}^{{f,nlm}}(R){\text{|}}\chi _{{E'J}}^{{(f)}}(R)\rangle {{{\text{|}}}^{2}}} \\ = \sqrt {\frac{{2\mu }}{{{{\hbar }^{2}}}}} \frac{{{{\eta }_{{EJ}}}({{R}_{\omega }})}}{{\Delta {{F}_{{fi}}}({{R}_{\omega }})}}\sum\limits_{l'm',m}^{} {{\text{|V}}_{{i,\varepsilon l'm'}}^{{f,nlm}}({{R}_{\omega }}){{{\text{|}}}^{2}},} \\ \end{gathered} $$

(16)

$${{\eta }_{{EJ}}}(R) = \sqrt {{{\xi }_{J}}(R)} \,{\text{A}}{{{\text{i}}}^{2}}[ - {{\xi }_{J}}(R)(E - U_{i}^{J}(R))],$$

(17)

which describes the motion of nuclei in both the over-barrier and the under-barrier regions. Here $U_{i}^{J}$(R) and $U_{f}^{J}$(R) are effective potential energy curves of the BA⁺ ion including the centrifugal energy,

$$\begin{gathered} U_{i}^{J}(R) = {{U}_{i}}(R) + \frac{{{{\hbar }^{2}}{{{(J + 1{\text{/}}2)}}^{2}}}}{{2\mu {{R}^{2}}}}, \\ U_{f}^{J}(R) = {{U}_{f}}(R) + \frac{{{{\hbar }^{2}}{{{(J + 1{\text{/}}2)}}^{2}}}}{{2\mu {{R}^{2}}}}, \\ \end{gathered} $$

(18)

and ΔF_fi(R_ω) is the difference between the slopes of these curves at the crossing point R_ω,

$$\Delta {{F}_{{fi}}}({{R}_{\omega }}) = {{\left| {\frac{{d{{U}_{f}}(R)}}{{dR}} - \frac{{d{{U}_{i}}(R)}}{{dR}}} \right|}_{{R = {{R}_{\omega }}}}}.$$

(19)

According to (16), the nuclear transition matrix element is expressed in terms of the electronic matrix element ${\text{V}}_{{i,\varepsilon l'm'}}^{{f,nlm}}$(R_ω) of interaction at the crossing point R_ω of the potential energy curves, and the Airy function Ai(x). The quantity ξ_J in (17) has the form

$${{\xi }_{J}}(R{{}_{\omega }}) = {{\left( {\frac{{2\mu }}{{{{\hbar }^{2}}}}} \right)}^{{1/3}}}{{\left| {\frac{{\Delta {{F}_{{fi}}}({{R}_{\omega }})}}{{F_{f}^{J}({{R}_{\omega }})F_{i}^{J}({{R}_{\omega }})}}} \right|}^{{2/3}}},$$

(20)

where $F_{i}^{J}$(R_ω) and $F_{f}^{{J'}}$(R_ω) are the slopes of the effective potential energy curves in the initial and final electronic states,

$$\begin{gathered} F_{i}^{J}({{R}_{\omega }}) = - {{\left. {\frac{{dU_{i}^{J}}}{{dR}}} \right|}_{{{{R}_{\omega }}}}}, \\ F_{f}^{J}({{R}_{\omega }}) = - {{\left. {\frac{{dU_{f}^{J}}}{{dR}}} \right|}_{{{{R}_{\omega }}}}}. \\ \end{gathered} $$

(21)

Let us replace the summation over the quantum number J in formula (7) by integration with respect to J. Then, substituting formula (16) into (7), taking into account (9), and introducing the survival factor (12), we obtain an expression for the effective cross section [cm⁴ s] of resonant electron capture into the Rydberg state nl in the ternary collision (2):

$$\begin{gathered} \sigma _{{\varepsilon \to nl}}^{{{\text{tr}}}}(E) = {{{\tilde {g}}}_{{{\text{tr}}}}}(2l + 1)\frac{{4{{\pi }^{4}}{{\Gamma }_{{nl \to \varepsilon }}}({{R}_{\omega }})\sqrt {2\mu } }}{{{{q}^{2}}{{k}^{2}}\Delta {{F}_{{fi}}}({{R}_{\omega }})}} \\ \, \times \int\limits_0^\infty {d({{J}^{2}}){{\eta }_{{EJ}}}({{R}_{\omega }})S_{{nl}}^{J}.} \\ \end{gathered} $$

(22)

3.2 Averaging the Dissociative Recombination Cross Section over the Boltzmann Distribution

For the Boltzmann distribution over the levels ${v}$J, the cross section $\sigma _{{\varepsilon \to nl}}^{{{\text{dr}}}}$(T) [cm²] of dissociative recombination at gas temperature T has the form

$$\begin{gathered} \sigma _{{\varepsilon \to nl}}^{{{\text{dr}}}}(T) = {{(\mathfrak{s}{{Z}_{{{v}r}}})}^{{ - 1}}}\exp \left( { - \frac{{{{D}_{0}}}}{{{{k}_{{\text{B}}}}T}}} \right) \\ \times \sum\limits_{{v}J}^{} {(2J + 1)\sigma _{{\varepsilon \to nl}}^{{{\text{dr}}}}({v}J)\exp \left( { - \frac{{{{E}_{{{v}J}}}}}{{{{k}_{{\text{B}}}}T}}} \right),} \\ \end{gathered} $$

(23)

$${{Z}_{{{v}r}}} = {{\mathfrak{s}}^{{ - 1}}}\sum\limits_{{v}J}^{} {(2J + 1)\exp \left( { - \frac{{{{\mathcal{E}}_{{{v}J}}}}}{{{{k}_{{\text{B}}}}T}}} \right),} $$

(24)

where ${{\mathcal{E}}_{{{v}J}}}$ = ${{E}_{{{v}J}}}$ + D₀. Under the conditions considered, the summation over ${v}$ and J in (23) can be replaced by integration with respect to d${v}$ and dJ. This is justified under the condition of k_BT$\gtrsim$$\hbar$ω_e ($\hbar$ω_e is the lowest vibrational quantum of the BA⁺(i) ion) and corresponds to the quasicontinuum approximation for rovibrational states (${{E}_{{{v}J}}}$ < 0). We use the Bohr–Sommerfeld relation d${v}$ = ${{T}_{{{v}J}}}$d${{E}_{{{v}J}}}$/(2π$\hbar$), according to which integration with respect to ${v}$ can be replaced by integration with respect to energy in the quasicontinuum:

$$\begin{gathered} \sigma _{{\varepsilon \to nl}}^{{{\text{dr}}}}(T) = \frac{{\exp ( - {{D}_{0}}{\text{/}}{{k}_{{\text{B}}}}T)}}{{2\pi \hbar \mathfrak{s}{{Z}_{{{v}r}}}}}\int\limits_{E_{{\min }}^{{{\text{dr}}}}}^0 {d{{E}_{{{v}J}}}\exp \left( { - \frac{{{{E}_{{{v}J}}}}}{{{{k}_{{\text{B}}}}T}}} \right)} \\ \times \int\limits_0^{{{J}_{{\max }}}} {{{T}_{{{v}J}}}\sigma _{{\varepsilon \to nl}}^{{{\text{dr}}}}({v}J)2JdJ.} \\ \end{gathered} $$

(25)

Here $E_{{\min }}^{{{\text{dr}}}}$ = max(–D₀, –ε – Ry/$n_{*}^{2}$), and ${{T}_{{{v}J}}}$ = $\oint {dR}$/${{V}_{{{v}J}}}$(R) is the period of rovibrational motion of nuclei with energy ${{E}_{{{v}J}}}$ < 0 in the effective potential $U_{i}^{J}$(R) = U_i(R) + ${{\hbar }^{2}}$(J + 1/2)²/(2μR²).

Next, let us substitute expressions (9) and (11) into (25) and take into account that the squared modulus of the nuclear matrix element of the bound–free transition differs from expression (16) for the free–free transition only by the normalization constants (${{C}_{{{v}J}}}$ = 2/$\sqrt {{{T}_{{{v}J}}}}$ and C_EJ = $\sqrt {2{\text{/}}\pi \hbar }$) of the wave functions $\chi _{{{v}J}}^{{(i)}}$(R) and $\chi _{{EJ}}^{{(i)}}$(R) and by the replacement of the energy of the relative motion of the nuclei in the continuous spectrum E > 0 by the energy of their rovibrational motion in the discrete spectrum ${{E}_{{{v}J}}}$ < 0. Then we introduce dimensionless variables ν and $\epsilon$ and a dimensionless function Λ_ν(R_ω):

$$\begin{gathered} \nu = \frac{{{{\hbar }^{2}}{{J}^{2}}}}{{2\mu R_{\omega }^{2}{{k}_{{\text{B}}}}T}},\quad \epsilon = \frac{{E - {{U}_{i}}({{R}_{\omega }})}}{{{{k}_{{\text{B}}}}T}}, \\ {{\Lambda }_{\nu }}({{R}_{\omega }}) = {{k}_{{\text{B}}}}T{{\xi }_{{\nu (J)}}}({{R}_{\omega }}) = \frac{{{{k}_{{\text{B}}}}T}}{{{{\hbar }^{{2/3}}}}}{{\left| {\frac{{\Delta {{F}_{{fi}}}({{R}_{\omega }})\sqrt {2\mu } }}{{F_{f}^{{\nu (J)}}({{R}_{\omega }})F_{i}^{{\nu (J)}}({{R}_{\omega }})}}} \right|}^{{2/3}}}. \\ \end{gathered} $$

(26)

Then, after a series of transformations, the final expression, averaged over the Boltzmann distribution, for the electron capture cross section by the molecular ion BA⁺ into a Rydberg level of the atom A(nl) takes the following form:

$$\begin{gathered} \sigma _{{\varepsilon \to nl}}^{{{\text{dr}}}}(T) = \frac{{8{{\pi }^{3}}(2l + 1){{{\tilde {g}}}_{{{\text{dr}}}}}R_{\omega }^{2}}}{{{{k}^{2}}{{Z}_{{{v}r}}}}}\frac{{{{\Gamma }_{{nl \to \varepsilon }}}({{R}_{\omega }})}}{{\Delta {{F}_{{fi}}}({{R}_{\omega }})}} \\ \times {{\left( {\frac{{\mu {{k}_{{\text{B}}}}T}}{{2\pi {{\hbar }^{2}}}}} \right)}^{{3/2}}}\exp \left( { - \frac{{{{D}_{0}} + {{U}_{i}}({{R}_{\omega }})}}{{{{k}_{{\text{B}}}}T}}} \right)\Theta _{T}^{{{\text{dr}}}}({{R}_{\omega }}), \\ \end{gathered} $$

(27)

where ${{\tilde {g}}_{{{\text{dr}}}}}$ = g_dr/$\mathfrak{s}$ and the dimensionless function $\Theta _{T}^{{{\text{dr}}}}$(R_ω) is represented as a double integral:

$$\begin{gathered} \Theta _{T}^{{{\text{dr}}}}({{R}_{\omega }}) = 2\sqrt \pi \int\limits_{\epsilon _{{\min }}^{{{\text{dr}}}}}^{\epsilon _{{\max }}^{{{\text{dr}}}}} {d\epsilon \exp ( - \epsilon )} \\ \times \int\limits_0^{{{\nu }_{{\max }}}} {d\nu \sqrt {{{\Lambda }_{\nu }}({{R}_{\omega }})} } {\text{A}}{{{\text{i}}}^{2}}[ - {{\Lambda }_{\nu }}({{R}_{\omega }})(\epsilon - \nu )]. \\ \end{gathered} $$

(28)

The limits of integration with respect to $d\epsilon$ in (28) are equal to $\epsilon _{{\min }}^{{{\text{dr}}}}$(R_ω) = –U_f(R_ω)/k_BT and $\epsilon _{{\max }}^{{{\text{dr}}}}$(R_ω) = ‒U_i(R_ω)/k_BT, where U_f(R) > U_i(R) (see Fig. 1), and ν_max($\epsilon$) is determined, according to (26), by the maximum possible value of J_max(E) for each fixed value of energy in the quasicontinuum E < 0. The quantity J_max(E) is found from the condition |$U_{i}^{J}$($R_{e}^{J}$)| ≥ |E| ($R_{e}^{J}$ is the equilibrium internuclear distance in the potential well $U_{i}^{J}$, see (18)). This condition implies that the depth of the potential well of the ground electronic term of the molecular ion with regard to the centrifugal term should exceed |E| for 0 ≤ J < J_max.

Fig. 1.

(a) Schematic view of the potential energy curves U_i(R), U_f(R), and U ′(R) of the lower electronic terms X |j_i = 3/2, Ω_i = 1/2〉, A₁ |j_f= 3/2, Ω_f = 3/2〉, and A₂ | j ' = 1/2, Ω' = 1/2〉 of an RgXe⁺ ion (dotted lines) and an RgXe⁺ + e quasimolecule: U_i(R) + ε and U_f(R) – Ry/$n_{*}^{2}$ (solid lines). (b) The structure of Rydberg states nl[K${{]}_{\mathcal{J}}}$ of a Xe atom that converge to the ionization limit 5p⁵(²P_3/2) and are located between two hydrogenlike series n and n – 1.

4 RATE CONSTANTS OF RESONANT ELECTRON CAPTURE

4.1 Rate Constant of Ternary Resonant Recombination

The initial expression for the rate constant $K_{{\varepsilon \to nl}}^{{{\text{tr}}}}$(T) = 〈V_E$\sigma _{{\varepsilon \to nl}}^{{{\text{tr}}}}$(E)〉_T [cm⁵] of ternary recombination (2) for given values of free-electron energy ε and gas temperature T has the form

$$K_{{\varepsilon \to nl}}^{{{\text{tr}}}}(T) = \int\limits_0^\infty {dE{{f}_{T}}(E){{V}_{E}}\sigma _{{\varepsilon \to nl}}^{{{\text{tr}}}}} (E),$$

(29)

where f_T(E) is the distribution function of the kinetic energy E = μV²/2 of the relative motion of particles A⁺ and B. Let us substitute expression (22) into (29) and use relation (17). Just as in Subsection 3.2, we introduce dimensionless variables ν and $\epsilon$ and a function Λ_ν(R_ω). Then, in the case of the Maxwell distribution,

$${{f}_{T}}(E)dE = \frac{2}{{\sqrt \pi }}{{\left( {\frac{E}{{{{k}_{{\text{B}}}}T}}} \right)}^{{1/2}}}\exp \left( { - \frac{E}{{{{k}_{{\text{B}}}}T}}} \right)d\left( {\frac{E}{{{{k}_{{\text{B}}}}T}}} \right)$$

we obtain the following formula for the recombination rate constant (29) for given values of the gas temperature T and electron energy ε:

$$\begin{gathered} K_{{\varepsilon \to nl}}^{{{\text{tr}}}}(T) = (2l + 1){{{\tilde {g}}}_{{{\text{tr}}}}}\frac{{2{{\pi }^{2}}}}{{{{k}^{2}}}}\frac{{4\pi R_{\omega }^{2}{{\Gamma }_{{nl \to \varepsilon }}}({{R}_{\omega }})}}{{\Delta {{F}_{{fi}}}({{R}_{\omega }})}} \\ \times \exp \left( { - \frac{{{{U}_{i}}({{R}_{\omega }})}}{{{{k}_{{\text{B}}}}T}}} \right)\Theta _{T}^{{{\text{tr}}}}({{R}_{\omega }}). \\ \end{gathered} $$

(30)

Here $\Theta _{T}^{{{\text{tr}}}}$(R_ω) is a dimensionless function:

$$\begin{gathered} \Theta _{T}^{{tr}}({{R}_{\omega }}) = 2\sqrt \pi \int\limits_{\epsilon _{{\min }}^{{{\text{tr}}}}}^\infty {d\epsilon \exp ( - \epsilon )} \\ \times \int\limits_0^\infty {d\nu \sqrt {{{\Lambda }_{\nu }}({{R}_{\omega }})} } {\text{A}}{{{\text{i}}}^{2}}[ - {{\Lambda }_{\nu }}({{R}_{\omega }})(\epsilon - \nu )], \\ \end{gathered} $$

(31)

where $\epsilon _{{\min }}^{{{\text{tr}}}}$ = $\epsilon _{{\max }}^{{{\text{dr}}}}$ = –U_i(R_ω)/k_BT.

The rate constant $\beta _{{nl}}^{{{\text{tr}}}}$ = 〈${{{v}}_{\varepsilon }}$〈V_E$\sigma _{{\varepsilon \to nl}}^{{{\text{tr}}}}$(E)〉〉 [cm⁶ s^–1] of resonant three-particle electron capture (2) is calculated by averaging the quantity ${{{v}}_{\varepsilon }}K_{{\varepsilon \to nl}}^{{{\text{tr}}}}$(T) over the free-electron velocity distribution at temperature T_e:

$$\beta _{{nl}}^{{{\text{tr}}}}({{T}_{e}},T) = \int\limits_0^\infty {d\varepsilon {{f}_{{{{T}_{e}}}}}(\varepsilon ){{{v}}_{\varepsilon }}K_{{\varepsilon \to nl}}^{{{\text{tr}}}}(T).} $$

(32)

Let us substitute expression (30) into (32) and replace the integration with respect to the free-electron energy ε by integration with respect to the transition energy, $\hbar$ω, and then, using the relation d($\hbar$ω) = d[ΔU_fi(R_ω)] = ΔF_fi(R_ω)dR_ω, by integration with respect to the internuclear distance dR. As a result, in the case of the Maxwell distribution ${{f}_{{{{T}_{e}}}}}$(ε) of electron energy, we obtain the following expression for the rate constant of resonant three-particle electron capture into a given Rydberg nl level of the atom:

$$\begin{gathered} \beta _{{nl}}^{{{\text{tr}}}}({{T}_{e}},T) = (2l + 1){{{\tilde {g}}}_{{tr}}}{{\left( {\frac{{2\pi {{\hbar }^{2}}}}{{{{m}_{e}}{{k}_{{\text{B}}}}{{T}_{e}}}}} \right)}^{{3/2}}} \\ \times \exp \left( {\frac{{{\text{|}}{{\varepsilon }_{{nl}}}{\text{|}}}}{{{{k}_{{\text{B}}}}{{T}_{e}}}}} \right)\int\limits_0^{{{R}_{{nl}}}} {\frac{{{{\Gamma }_{{nl \to \varepsilon }}}(R)}}{\hbar }\exp \left( { - \frac{{\Delta {{U}_{{fi}}}(R)}}{{{{k}_{{\text{B}}}}{{T}_{e}}}}} \right)} \\ \times \exp \left( { - \frac{{{{U}_{i}}(R)}}{{{{k}_{{\text{B}}}}T}}} \right)\Theta _{T}^{{{\text{tr}}}}(R)4\pi {{R}^{2}}dR, \\ \end{gathered} $$

(33)

where R_nl is given by the condition ΔU_fi(R_nl) = |ε_nl|; i.e., it coincides with the stabilization point (14).

We can obtain simple semiclassical expressions for the rate constants $K_{{\varepsilon \to nl}}^{{{\text{tr}}}}$(T) and $\beta _{{nl}}^{{{\text{tr}}}}$(T_e, T) using the asymptotic expression

$${\text{A}}{{{\text{i}}}^{2}}( - x)\xrightarrow[{x \to + \infty }]{}\frac{1}{{\pi {{x}^{{1/2}}}}}{{\sin }^{2}}\left( {\frac{2}{3}{{x}^{{3/2}}} + \frac{1}{4}\pi } \right)$$

(34)

for the squared Airy function and averaging the result over the oscillation period. In this case, the integration with respect to the squared angular momentum J² of nuclear motion is performed only in the classically allowed region within 0 ≤ J² ≤ $J_{{\max }}^{2}$, where

$${{J}_{{\max }}} = \sqrt {2\mu R_{\omega }^{2}(E - {{U}_{i}}({{R}_{\omega }})){\text{/}}{{\hbar }^{2}}} .$$

(35)

Accordingly, in the semiclassical approximation when the motion of nuclei is taken into account only in the classically allowed region, the upper limit of integration with respect to dν in (31) should be changed from ∞ to $\epsilon$:

$$\begin{gathered} \Theta _{T}^{{{\text{tr}}}}({{R}_{\omega }}) = \frac{1}{{\sqrt \pi }}\int\limits_{\epsilon _{{\min }}^{{{\text{tr}}}}}^\infty {d\epsilon \exp ( - \epsilon )\int\limits_0^\epsilon {\frac{{d\nu }}{{{{{(\epsilon - \nu )}}^{{1/2}}}}}} } \\ = \left\{ \begin{gathered} 1,\quad {{R}_{\omega }} < {{R}_{0}}, \hfill \\ \Gamma (3{\text{/}}2,{\text{|}}{{U}_{i}}({{R}_{\omega }}){\text{|/}}{{k}_{{\text{B}}}}T){\text{/}}\Gamma (3{\text{/}}2),\quad {{R}_{\omega }} \geqslant {{R}_{0}}, \hfill \\ \end{gathered} \right. \\ \end{gathered} $$

(36)

where Γ(3/2, z) = $\int_z^\infty {{{t}^{{1/2}}}}$exp(–t)dt.

4.2 Rate Constant of Dissociative Recombination in a Two-Temperature Plasma

The dissociative recombination rate constant $\alpha _{{nl}}^{{{\text{dr}}}}$ [cm³ s^–1] in a plasma with temperatures T and T_e of its gas and electronic components can be obtained by averaging the quantity ${{{v}}_{\varepsilon }}\sigma _{{\varepsilon \to nl}}^{{{\text{dr}}}}$(T) (see (27)) over the free-electron velocity distribution function ${{f}_{{{{T}_{e}}}}}$(ε):

$$\alpha _{{nl}}^{{{\text{dr}}}}({{T}_{e}},T) = \int\limits_0^\infty {{{{v}}_{\varepsilon }}\sigma _{{\varepsilon \to nl}}^{{{\text{dr}}}}(T){{f}_{{{{T}_{e}}}}}(\varepsilon )d\varepsilon .} $$

(37)

Let us substitute expression (27) into (37) and replace (just as we did when deriving the final formula (33) for the rate constant of resonant three-particle recombination) the integration with respect to the free-electron energy ε by integration with respect to the transition energy $\hbar$ω, and then by integration with respect to the internuclear distance dR. Then, after a series of transformations and transition to dimensionless variables (26), we obtain the following expression for the integral contribution of all rovibrational levels of the molecular ion to the total rate constant of dissociative electron capture into a given Rydberg level of the atom A(nl):

$$\begin{gathered} \alpha _{{nl}}^{{dt}}({{T}_{e}},T) = (2l + 1){{{\tilde {g}}}_{{{\text{dr}}}}}{{\left( {\frac{{2\pi {{\hbar }^{2}}}}{{{{m}_{e}}{{k}_{{\text{B}}}}{{T}_{e}}}}} \right)}^{{3/2}}}\frac{{{{e}^{{ - {{D}_{0}}/{{k}_{{\text{B}}}}T}}}}}{{{{Z}_{{{v}r}}}(T)}} \\ \times {{\left( {\frac{{\mu {{k}_{{\text{B}}}}T}}{{2\pi {{\hbar }^{2}}}}} \right)}^{{3/2}}}\exp \left( {\frac{{{\text{|}}{{\varepsilon }_{{nl}}}{\text{|}}}}{{{{k}_{{\text{B}}}}{{T}_{e}}}}} \right)\int\limits_0^{{{R}_{{nl}}}} {\frac{{{{\Gamma }_{{nl \to \varepsilon }}}(R)}}{\hbar }} \\ \times \exp \left( { - \frac{{\Delta {{U}_{{fi}}}(R)}}{{{{k}_{{\text{B}}}}{{T}_{e}}}}} \right)\exp \left( { - \frac{{{{U}_{i}}(R)}}{{{{k}_{{\text{B}}}}T}}} \right)\Theta _{T}^{{{\text{dr}}}}({{R}_{\omega }})4\pi {{R}^{2}}dR, \\ \end{gathered} $$

(38)

where R_nl is found from the equation ΔU_fi(R_nl) = |ε_nl|. For process (1), the dimensionless coefficient $\Theta _{T}^{{{\text{dr}}}}$ is determined by expression (28).

In the semiclassical approximation, the expression for the coefficient $\Theta _{T}^{{{\text{dr}}}}$(R_ω) and, accordingly, for the rate constant $\alpha _{{nl}}^{{{\text{dr}}}}$(T_e, T) of dissociative electron capture into a given Rydberg level of the atom A(nl), is significantly simplified. According to (26), the lower and upper limits of integration with respect to $\epsilon$ in (28) become equal to $\epsilon _{{\min }}^{{{\text{dr}}}}$ = 0 and $\epsilon _{{\max }}^{{{\text{dr}}}}$ = ‒U_i(R)/k_BT, and the integration with respect to ν should be performed over the range 0 ≤ ν ≤ $\epsilon$, which corresponds to the motion of nuclei only in the classically allowed region. Then, taking into account (34), we have

$$\Theta _{T}^{{{\text{dr}}}}({{R}_{\omega }}) = \left\{ \begin{gathered} 0,\quad {{R}_{\omega }} < {{R}_{0}}, \hfill \\ \frac{{\gamma (3{\text{/}}2,{\text{|}}{{U}_{i}}({{R}_{\omega }}){\text{|/}}{{k}_{{\text{B}}}}T)}}{{\Gamma (3{\text{/}}2)}},\quad {{R}_{\omega }} \geqslant {{R}_{0}}, \hfill \\ \end{gathered} \right.$$

(39)

where γ(3/2, z) = $\int_0^z {{{t}^{{1/2}}}}$exp(–t)dt.

4.3 Total Contribution of the Continuous and Discrete Spectra of a Quasimolecule

In practical calculations, it is often of interest to know the absolute rates of electron capture into Rydberg states, W_nl [s^–1]. These rates naturally depend on the particle concentrations in the initial reaction channels. For processes (1) and (2), the reaction rates W_nl are expressed as

$$\begin{gathered} W_{{nl}}^{{{\text{tr}}}} = \beta _{{nl}}^{{{\text{tr}}}}{{N}_{{\text{B}}}}{{N}_{{{{{\text{A}}}^{ + }}}}},\quad W_{{nl}}^{{{\text{dr}}}} = \alpha _{{nl}}^{{{\text{dr}}}}{{N}_{{{\text{B}}{{{\text{A}}}^{ + }}}}}, \\ {{N}_{{{\text{B}}{{{\text{A}}}^{ + }}}}} = \sum\limits_{{v}J}^{} {N_{{{\text{B}}{{{\text{A}}}^{ + }}}}^{{({v}J)}},} \\ \end{gathered} $$

(40)

and the numbers of the corresponding recombination acts in unit volume per unit time are obtained by multiplying (40) by the concentration of free electrons in the plasma: $\mathfrak{W}_{{nl}}^{{{\text{tr}}}}$ = $W_{{nl}}^{{{\text{tr}}}}$N_e and $\mathfrak{W}_{{nl}}^{{{\text{dr}}}}$ = $W_{{nl}}^{{{\text{dr}}}}$N_e. The total rate of resonant electron capture into an atomic level nl can be determined as

$$W_{{nl}}^{{{\text{res}}}} = \beta _{{nl}}^{{{\text{tr}}}}{{N}_{{\text{B}}}}{{N}_{{{{{\text{A}}}^{ + }}}}} + \alpha _{{nl}}^{{{\text{dr}}}}{{N}_{{{\text{B}}{{{\text{A}}}^{ + }}}}}.$$

(41)

When BA⁺ molecular ions are in equilibrium with the B and A⁺ particles in the continuous spectrum, the law of mass action for the equilibrium concentration ${{N}_{{{\text{B}}{{{\text{A}}}^{ + }}}}}$ of bound ions in the initial state,

$$\frac{{{{N}_{{{\text{B}}{{{\text{A}}}^{ + }}}}}}}{{{{N}_{{\text{B}}}}{{N}_{{{{{\text{A}}}^{ + }}}}}}} = \frac{{{{Z}_{{{\text{vr}}}}}{{g}_{{{\text{B}}{{{\text{A}}}^{ + }}}}}}}{{{{g}_{{\text{B}}}}{{g}_{{{{{\text{A}}}^{ + }}}}}}}{{\left( {\frac{{2\pi {{\hbar }^{2}}}}{{\mu {{k}_{{\text{B}}}}T}}} \right)}^{{3/2}}}\exp \left( {\frac{{{{D}_{0}}}}{{{{k}_{{\text{B}}}}T}}} \right)$$

(42)

allows us to rewrite the expression for $W_{{nl}}^{{{\text{dr}}}}$ as

$$\begin{gathered} W_{{nl}}^{{{\text{dr}}}}({{T}_{e}},T) = \alpha _{{nl}}^{{{\text{dr}}}}({{T}_{e}},T){{N}_{{\text{B}}}}{{N}_{{{{{\text{A}}}^{ + }}}}}{{{\tilde {g}}}_{{tr}}}{{Z}_{{{v}r}}} \\ \times \exp \left( {\frac{{{{D}_{0}}}}{{{{k}_{{\text{B}}}}T}}} \right){{\left( {\frac{{2\pi {{\hbar }^{2}}}}{{\mu {{k}_{{\text{B}}}}T}}} \right)}^{{3/2}}} \equiv \beta _{{nl}}^{{{\text{dr}}}}({{T}_{e}},T){{N}_{{\text{B}}}}{{N}_{{{{{\text{A}}}^{ + }}}}}. \\ \end{gathered} $$

(43)

Here $\beta _{{nl}}^{{{\text{dr}}}}$(T_e, T) [cm⁶ s^–1] is the rate constant of dissociative recombination normalized by the product of the concentrations of atomic ions and neutral atoms. To calculate the partition function (24), we use the following semiclassical formula from [25], which is applicable for k_BT ≳ $\hbar$ω_e:

$$\begin{gathered} {{Z}_{{{v}r}}}(T) = 4\pi {{\mathfrak{s}}^{{ - 1}}}{{\left( {\frac{{\mu {{k}_{{\text{B}}}}T}}{{2\pi {{\hbar }^{2}}}}} \right)}^{{3/2}}}\exp \left( { - \frac{{{{D}_{0}}}}{{{{k}_{{\text{B}}}}T}}} \right) \\ \, \times \int\limits_{{{R}_{0}}}^\infty {\left[ {1 - \frac{2}{{\sqrt \pi }}\Gamma \left( {\frac{3}{2},\frac{{{\text{|}}{{U}_{i}}(R){\text{|}}}}{{{{k}_{{\text{B}}}}T}}} \right)} \right]} \exp \left( { - \frac{{{{U}_{i}}(R)}}{{{{k}_{{\text{B}}}}T}}} \right){{R}^{2}}dR, \\ \end{gathered} $$

(44)

where R₀ is obtained from the equation U_i(R₀) = 0 (see Fig. 1).

Formulas (40) and (43) allow us to introduce the total rate constant $\beta _{{nl}}^{{{\text{res}}}}$ = $W_{{nl}}^{{{\text{res}}}}$/(N_B${{N}_{{{{{\text{A}}}^{ + }}}}}$) of resonant electron capture, which includes the contributions of ternary and dissociative recombination:

$$\beta _{{nl}}^{{{\text{res}}}} = \beta _{{nl}}^{{{\text{tr}}}} + \beta _{{nl}}^{{{\text{dr}}}}.$$

(45)

This expression is especially convenient, in particular, to determine the integral role of resonant processes (1) and (2) as compared to nonresonant ternary recombination processes in collisions with electrons and neutral plasma particles.

In the semiclassical approximation, when nuclear motions are taken into account only in the classically allowed region, we can easily obtain a simplified expression for $\beta _{{nl}}^{{{\text{res}}}}$. Taking into account that $\Theta _{T}^{{{\text{tr}}}}$(R) + $\Theta _{T}^{{{\text{dr}}}}$(R) = 1 in this case and applying formulas (33) and (38), we obtain

$$\begin{gathered} \beta _{{nl}}^{{{\text{res}}}}({{T}_{e}},T) = (2l + 1){{{\tilde {g}}}_{{{\text{tr}}}}}{{\left( {\frac{{2\pi {{\hbar }^{2}}}}{{{{m}_{e}}{{k}_{{\text{B}}}}{{T}_{e}}}}} \right)}^{{3/2}}} \\ \times \exp \left( {\frac{{{\text{|}}{{\varepsilon }_{{nl}}}{\text{|}}}}{{{{k}_{{\text{B}}}}{{T}_{e}}}}} \right)\int\limits_0^{{{R}_{{nl}}}} {\frac{{{{\Gamma }_{{nl \to \varepsilon }}}(R)}}{\hbar }} \\ \times \exp \left( { - \frac{{\Delta {{U}_{{fi}}}(R)}}{{{{k}_{{\text{B}}}}{{T}_{e}}}}} \right)\exp \left( { - \frac{{{{U}_{i}}(R)}}{{{{k}_{{\text{B}}}}T}}} \right)4\pi {{R}^{2}}dR. \\ \end{gathered} $$

(46)

In this paper, we mainly deal with reaction rate constants normalized by the product of the concentrations N_B and ${{N}_{{{{{\text{A}}}^{ + }}}}}$. In thermal equilibrium of heavy particles, relation (42) allows instead the use of the concentrations of molecular ions ${{N}_{{{\text{B}}{{{\text{A}}}^{ + }}}}}$, so that the ratio of the dissociation energy to k_BT is the main factor determining the choice of normalization. In experiments, the distribution functions of molecular ions over rovibrational levels and the distributions of relative particle velocities may significantly differ from equilibrium ones. Therefore, the choice of normalization is often determined by specific physical conditions.

In the case of electron–ion recombination accompanied by the formation of Rydberg atoms, of interest are the total cross sections and rate constants over all values lm of the level n,

$${{\sigma }_{{\varepsilon \to n}}} = \sum\limits_{l = 0}^{n - 1} {{{\sigma }_{{\varepsilon \to nl}}}} ,\quad \beta _{{nl}}^{{{\text{res}}}} = \sum\limits_{l = 0}^{n - 1} {\beta _{{nl}}^{{{\text{res}}}}.} $$

(47)

In formulas (22), (27) and (33), (38), the cross sections $\sigma _{{\varepsilon \to nl}}^{{{\text{res}}}}$ and the rate constants $\beta _{{nl}}^{{{\text{res}}}}$ of resonant electron capture into levels nl are expressed in terms of the autoionization width Γ_{nl→ ε} of Rydberg quasimolecule BA(f, nl). When calculating (47), it is convenient to replace (2l + 1)Γ_{nl→ ε} in expressions (22) and (27) by the effective coupling parameter

$$\begin{gathered} {{{\tilde {\Gamma }}}_{{\varepsilon \to n}}} = \sum\limits_{l = 0}^{n - 1} {(2l + 1){{\Gamma }_{{nl \to \varepsilon }}}({{R}_{\omega }})} \\ = 2\pi \sum\limits_{ml,m'l'}^{} {{\text{|V}}_{{i,\varepsilon l'm'}}^{{f,nlm}}({{R}_{\omega }}){{{\text{|}}}^{2}},} \\ \end{gathered} $$

(48)

which describes the total contribution of individual nl levels to the electron capture into all nlm states with a given n. Note that, for non-hydrogenlike states of atoms with significant quantum defects δ_l, the values of R_ω for each term of the sum over l differ from each other due to the dependence of δ_l on the angular momentum. Therefore, the total electron capture cross sections and the rate constants (47) for a given value of n should, generally speaking, be determined by summing separately calculated contributions from different values of l.

5 AUTOIONIZATION DECAY WIDTHS OF RYDBERG STATES

When calculating the autoionization decay widths Γ_{nl→ ε} and the effective coupling parameter ${{\tilde {\Gamma }}_{{\varepsilon \to n}}}$ of the outer and inner electrons in the quasimolecular system RgXe⁺ + e, one should take into account the complex structure of Rydberg levels of the xenon atom and the specific features of electronic terms of heteronuclear ions of the inert gases RgXe⁺ (Rg = Ne (2s²2p⁶¹S₀), Ar (3s²3p⁶¹S₀)). The electronic configuration 5s²5p⁵ of the Xe⁺ ion causes a strong spin–orbit splitting Δ_3/2,1/2 = 1.3 eV of electronic terms converging to the states ²P_{j = 3/2} and ²P_{j '= 1/2} (see Fig. 1a). In the considered range of electron and gas temperatures, the upper term A₂ |j ' = 1/2, Ω' = 1/2〉, which correlates with the state ²P_{j '= 1/2}, and all higher terms can be ignored. Free–bound transitions of the outer electron of the RgXe⁺ + e system in recombination processes (1) and (2) are thus attributed to the transition X|j_i = 3/2, Ω_i = 1/2〉 → A₁|j_f = 3/2, Ω_f = 3/2〉 between the ground, U_i(R), and the first excited, U_f(R), electronic terms of the RgXe⁺ ion. The Rydberg states Xe[5p⁵(²P_j)nl[K${{]}_{\mathcal{J}}}$] are characterized by the set of quantum numbers n, l, K, and $\mathcal{J}$, where K is the quantum number of the angular momentum K = j + l and $\mathcal{J}$ = K ± 1/2 and j = 3/2, 1/2 are the total momenta of the Rydberg atom and of the atomic core, respectively. Figure 1b shows that the sublevels nl[K${{]}_{\mathcal{J}}}$ with l = 0, 1, 2 have large quantum defects ${{\delta }_{\mathcal{J}}}$, while the sublevels nl[K${{]}_{\mathcal{J}}}$ with l ≥ 3 have an almost hydrogenlike structure [26]. The structure of Rydberg levels of Xe can be approximately taken into account by introducing an effective principal quantum number ${{n}_{*}}$ = n – $\delta _{{nl}}^{{{\text{eff}}}}$. Here $\delta _{{nl}}^{{{\text{eff}}}}$ is the effective quantum defect obtained by averaging over the set of quantum defects ${{\delta }_{{nl[K]}}}_{{_{\mathcal{J}}}}$. This is justified since |$\delta _{{nl}}^{{{\text{eff}}}}$ – ${{\delta }_{{nl[K]}}}_{{_{\mathcal{J}}}}$| ≪ $\delta _{{nl}}^{{{\text{eff}}}}$ for l = 0, 1, 2.

The quantities Γ_{nl→ ε} and ${{\tilde {\Gamma }}_{{\varepsilon \to n}}}$ for the RgXe⁺ + e systems were calculated by the vacancy model [14, 17]. This model reduces the interaction of the outer electron with all inner electrons of a molecular ion to the interaction V = –e²/|r_e – ${{{\mathbf{r}}}_{{v}}}$| of the Rydberg electron (r_e) with a positively charged one-electron vacancy (${{{\mathbf{r}}}_{{v}}}$) in the electronic shell of the Xe⁺ ion (5s²5p⁵²P_3/2) described by the radial wave function ${{\mathcal{R}}_{{5p}}}$(${{r}_{{v}}}$). In the first approximation, the interaction of the Rydberg electron with the electrons of the closed electron shell of the Rg(¹S₀) atom can be neglected. Therefore, the quantities γ_ll' appearing in the expression for the effective coupling parameter,

$${{\tilde {\Gamma }}_{{\varepsilon \to n}}} = \frac{{4\pi }}{{25}}\sum\limits_{ll'}^{} {\frac{{{{\gamma }_{{ll'}}}}}{{n_{*}^{3}}},} $$

(49)

and, accordingly, ${{\tilde {\Gamma }}_{{\varepsilon \to n}}}$, are independent of R. The method for calculating γ_ll ' was described in [14, 17], except that the wave functions of Rydberg states with a significant quantum defect were calculated by the method of [27].

6 RESULTS AND DISCUSSION

6.1 Cross Sections of Resonant Electron Capture

Below we present the results of calculating the cross sections of recombination processes in a plasma of inert gas mixtures Rg/Xe at electron and gas temperatures typical of gas discharge experiments. Here we restrict ourselves to the consideration of resonant processes in heteronuclear systems. It is well known that the plasmas of Rg/Xe mixtures may contain ${\text{Xe}}_{2}^{ + }$ ions whose dissociative recombination is an efficient process. At the same time, a significant number of experimental works have been performed under conditions when the concentration of xenon in a mixture is very small, [Xe] ≪ [Rg]. In this case, the number of ${\text{Xe}}_{2}^{ + }$ ions in the plasma may turn out to be substantially less than the number of RgXe⁺ ions [17, 28, 29].

Figure 2 demonstrates the resonant capture cross sections $\sigma _{{\varepsilon \to n}}^{{{\text{tr}}}}$ as functions of the principal quantum number for the Ar + Xe⁺ + e and Ne + Xe⁺ + e systems. The calculations were performed for the electron and heavy particle energies corresponding to the maximum of the Maxwell distribution (ε = k_BT_e/2 and E = k_BT/2) at an electron temperature of T_e = 2000 K and gas temperatures of T = 300, 600, and 900 K. The cross sections exhibit a pronounced maximum, which defines the range of n predominantly populated by recombination. The maxima are reached at n_max ≈ 10 for Ar + Xe⁺ + e and n_max ≈ 20–35 for Ne + Xe⁺ + e. The values of n_max differ so strongly due to the large difference in the dissociation energies of heteronuclear ions. Figure 2 shows a sharp decrease in the capture cross sections with a decrease in n for n < n_max for the Ne + Xe⁺ + e system, which is attributed to the shift of the resonant transition point R_ω to the classically forbidden region. In the Ar + Xe⁺ + e system, the position of the maximum is almost independent of the collision energy, whereas, in the Ne + Xe⁺ + e system, n_max shifts to smaller n with increasing E = μV²/2. This difference is caused by the fact that, in systems with higher values of D₀, the transition point for small n usually lies near the equilibrium position R_e of the lower term, and transitions become allowed even at low energies E. When the dissociation energy D₀ is low, captures into low n occur at R_ω ≲ R_e in the repulsive part of the term, so that the low energies E are classically unacceptable.

Fig. 2.

(Color online) Cross sections $\sigma _{{\varepsilon \to n}}^{{{\text{tr}}}}$(E) [cm⁴ s] of resonant three-particle electron capture (2) into all lm sublevels of the Rydberg level n (22) for the (a) Ar + Xe⁺ + e and (b) Ne + Xe⁺ + e systems at energies ε = k_BT_e/2 (T_e = 2000 K) and E = k_BT_e/2 (T = 300, 600, and 900 K).

The effective cross sections of dissociative electron capture by ArXe⁺ and NeXe⁺ ions, averaged over the Boltzmann distribution (27) at T = 500 K, are shown in Fig. 3. The electron energy is taken equal to ε = k_BT_e/2 at T_e = 500–1500 K. The cross sections decrease with increasing T_e. Similar to the resonant three-particle capture, the cross sections $\sigma _{{\varepsilon \to n}}^{{{\text{dr}}}}$(T) as a function of n exhibit a maximum that shifts toward lower n with increasing D₀. The position of the maximum is determined by the condition R_ω ≈ R_e. In the case of ArXe⁺, the maximum is located near n_max ≈ 12, and n_max is almost independent of T_e. For NeXe⁺, the maximum lies at n > 20 and quickly shifts to high values of n with increasing T_e.

Fig. 3.

(Color online) Boltzmann distribution (T = 500 K) averaged cross sections $\sigma _{{\varepsilon \to n}}^{{{\text{dr}}}}$(T) [cm²] (27) of dissociative electron capture, into all lm sublevels of the Rydberg level for (a) ArXe⁺ and (b) NeXe⁺ molecular ions. Calculations are performed for the values of electron energy ε = k_BT_e/2 indicated in the figure.

6.2 Rate Constants of Resonant Electron Capture as Functions of n, T, and T_e

Figure 4 shows the total rate constants $\beta _{n}^{{{\text{res}}}}$ = $\beta _{n}^{{{\text{tr}}}}$ + $\beta _{n}^{{{\text{dr}}}}$ of resonant electron capture into Rydberg states as a function of n. The calculations were performed for Ar + Xe⁺ + e and Ne + Xe⁺ + e systems at electron temperatures of T_e = (1–3) × 10³ K and gas temperatures of T = 400 and 800 K. This choice of systems and temperatures allows us to study the behavior of $\beta _{n}^{{{\text{res}}}}$ both in the case of weakly bound ions, D₀ ≲ k_BT (NeXe⁺, D₀ = 33 meV), and in the case of ions with D₀ ≫ k_BT (ArXe⁺, D₀ = 176 meV). The values of the rate constants for these systems are strongly different: the maximum of $\beta _{n}^{{{\text{res}}}}$ for Ar + Xe⁺ + e turns out to be 2 or 3 orders of magnitude higher than that of $\beta _{n}^{{{\text{res}}}}$ for Ne + Xe⁺ + e. The significant difference is due to the fact that, in systems with moderate dissociation energies (such as Ar + Xe⁺), the probability of the formation of bound molecular ions participating in dissociative capture increases. Thus, under quasi-equilibrium conditions, the efficiency of resonant recombination is usually higher in systems with larger D₀.

Fig. 4.

(Color online) The rate constants $\beta _{n}^{{{\text{res}}}}$ = $\beta _{n}^{{{\text{tr}}}}$ + $\beta _{n}^{{{\text{dr}}}}$ of resonant electron capture into all lm sublevels of level n for the (a) Ar + Xe⁺ + e and (b) Ne + Xe⁺ + e systems at gas temperatures T = 400 K (solid curves) and 800 K (dotted curves) and electron temperatures T_e = 1000 (1), 2000 (2), and 3000 K (3).

The capture rate constants $\beta _{n}^{{{\text{res}}}}$(T_e, T) of systems with intermediate and low dissociation energies (Ar + Xe⁺ + e and Ne + Xe⁺ + e) also exhibit qualitatively different behavior. For the Ar + Xe⁺ + e system, a sharp maximum is observed as a function of n at n_max ≈ 11 (Fig. 4). For similar systems with moderate or large values of dissociation energy, the position of n_max is determined from the condition ${{R}_{{{{n}_{{\max }}}}}}$ ≈ R_e and is therefore almost independent of the gas and electron temperatures. This behavior is in accordance with the results shown in Figs. 2a and 3a. In this case, $\beta _{n}^{{{\text{res}}}}$(T_e, T) decreases by an order of magnitude with an increase in T from 400 to 800 K, which is associated with an exponential decrease in the probability of formation of bound molecular ions ArXe⁺, which are required for the realization of the resonant dissociative recombination channel (1).

The rate constants $\beta _{n}^{{{\text{res}}}}$(T_e, T) of systems with low dissociation energies D₀ (HeXe⁺ and NeXe⁺) exhibit a different behavior as a function of n and T. Figure 4b shows that, in this case, the population of lower lying levels with small n is negligible; capture into highly excited levels (n_max ≈ 20 for Ne + Xe⁺ + e) is dominant. Moreover, as T increases, the maximum value of the rate constant changes little, and the effective capture region shifts to smaller values of n (n_max ≈ 15 at T = 800 K). This is in agreement with the results shown in Figs. 2b and 3b and is associated with the fact that, in systems with low dissociation energies, the resonant three-particle capture mechanism is dominant ($\beta _{n}^{{{\text{tr}}}}$ ≫ $\beta _{n}^{{{\text{dr}}}}$), and nonadiabatic transitions occur mainly in the repulsive part of the electronic term (see Fig. 1). An increase in temperature T increases the probability of transitions at small R, which lead to the population of low levels.

The rate constant $\beta _{n}^{{{\text{res}}}}$(T_e, T) in Figs. 4a and 4b demonstrates the same behavior as a function of the electron temperature. In each of the systems considered, the electron capture probability decreases with increasing T_e: a threefold increase in T_e leads to a fourfold decrease in the rate constant $\beta _{n}^{{{\text{res}}}}$(T_e, T). This behavior is determined by an increase in the average energy of free electrons with increasing T_e, which shifts the transition point R_ω to a region with low probability of occurrence of molecular ions.

6.3 Comparative Analysis of the Efficiencies of Resonant Electron Capture Processes

Figure 5 illustrates the comparison of integral contributions of dissociative (1) and three-particle (2) recombination in a two-temperature plasma of an inert gas mixture Rg/Xe ([Xe] ≪ [Rg], Rg = Ar, Ne) to the total rate constant $\beta _{n}^{{{\text{res}}}}$ of electron capture by molecular, RgXe⁺, and atomic, Xe⁺, ions with the formation of Xe(n) atoms. We can see that the relative contribution of three-particle capture increases with decreasing ion dissociation energy due to an increase in the probability that the RgXe⁺(i) system is in the continuous spectrum. For the same reason, the relative efficiency of the three-particle capture mechanism is the higher, the higher the gas temperature T. For the ArXe⁺ + e system, the contribution of mechanism (2) does not exceed 50% even at a sufficiently high temperature of T = 900 K. On the contrary, for the NeXe⁺ + e system with low dissociation energy, the resonant process of three-particle electron capture by an atomic Xe⁺ ion is dominant even at room temperature T = 300 K.

Fig. 5.

(Color online) The relative contribution $\beta _{n}^{{{\text{tr}}}}$/$\beta _{n}^{{{\text{res}}}}$ of three-particle electron capture (2) to the total rate constant $\beta _{n}^{{{\text{res}}}}$ = $\beta _{n}^{{{\text{tr}}}}$ + $\beta _{n}^{{{\text{dr}}}}$ of resonant capture into all sublevels of level n, including the contribution of dissociative capture (1) from all rovibrational levels of the molecular ion. Calculations are performed for (a) Ar + Xe⁺ + e and (b) Ne + Xe⁺ + e at T_e = 2000 K and various gas temperatures T.

Note that, when constructing kinetic models of recombination of inert gas mixtures in plasmas, one usually discusses the channel of dissociative recombination of RgXe⁺ ions, while the alternative channel (2) is assumed to be inefficient (see, for example, [28, 30–32]). As shown above, the contribution of ternary recombination to the total rate constant of resonant capture can be comparable to or even many times exceed the contribution from channel (1). Accordingly, the neglect of process (2) may lead to significant errors in kinetic calculations.

The above results of the comparative analysis of resonant recombination mechanisms were obtained for equilibrium velocity distributions. The deviation from the Maxwellian distribution may significantly change the relationship between the integral contributions of discrete and continuous spectra of a molecular ion to the total recombination rate. For instance, the merged-beam technique [6], in which the detuning energy ε_d between electrons and ions is introduced by the potential difference between the anode and cathode, is an effective way to measure the dissociative recombination coefficient. In such setups, the electron velocity distribution function has the form [6]

$$\begin{gathered} f({{{v}}_{e}},{{{v}}_{d}}) = \frac{{{{m}_{e}}}}{{2\pi {{k}_{{\text{B}}}}{{T}_{ \bot }}}}\exp \left( { - \frac{{{{m}_{e}}{v}_{ \bot }^{2}}}{{2{{k}_{{\text{B}}}}{{T}_{ \bot }}}}} \right)\sqrt {\frac{{{{m}_{e}}}}{{2\pi {{k}_{{\text{B}}}}{{T}_{{||}}}}}} \\ \times \exp \left[ { - \frac{{{{m}_{e}}{{{({{{v}}_{{{\text{||}}}}} - {{{v}}_{d}})}}^{2}}}}{{{{k}_{{\text{B}}}}{{T}_{{||}}}}}} \right]. \\ \end{gathered} $$

(50)

Here, ${{{v}}_{d}}$ is the detuning velocity (ε_d = m_e${v}_{d}^{2}$/2), ${{{v}}_{ \bot }}$ is the velocity of transverse motion, and T_⊥ is the effective temperature of transverse motion, which is usually equal to the electron temperature T_e. The temperature T_|| ≪ T_e describes the motion of an electron along the axis of the setup at velocity ${{{v}}_{{||}}}$ relative to an ion; k_BT_|| ~ 1 meV.

The key effect of detuning on the dynamics of resonant electron capture is associated with the shift of the resonant transition region to smaller R. For large ε_d, transitions mainly occur near the point $R_{n}^{d}$ defined by the condition ΔU_fi($R_{n}^{d}$) = Ry/n² + ε_d. The quantity ε_d usually varies from 0 to 0.3 eV, so that, for ε_d ~ D₀, the characteristic region of transitions that occurred earlier near the equilibrium point R_e may shift significantly toward R ≲ R_e. This leads to an increase in the relative efficiency of the three-particle electron capture mechanism into lower levels n.

This effect is demonstrated in Fig. 6 for the Ar + Xe⁺ + e system at T_e = 2000 K and T = 300 and 600 K. The introduction of the detuning energy ε_d = 0.15 eV increases the relative efficiency of the three-particle process of capture into levels with low n by a factor of up to 2 at T = 600 K and up to 5 at T = 300 K. In this case, the efficiency of three-particle capture into levels with n > 16, conversely, decreases, reaching a plateau, since the transitions that previously occurred near $R_{n}^{d}$ > R_e now occur near the point $R_{n}^{d}$ ≈ R_e, which is defined for large n by the condition ΔU_fi[$R_{n}^{d}$] ≈ ε_d. Note that the effect of the nonequilibrium distribution weakens with increasing T, because the characteristic region of nonadiabatic transitions is “smeared” with respect to R.

Fig. 6.

(Color online) The relative contribution $\beta _{n}^{{{\text{tr}}}}$/$\beta _{n}^{{{\text{res}}}}$ of the three-particle electron capture mechanism (2) to the total rate constant $\beta _{n}^{{{\text{res}}}}$ = $\beta _{n}^{{{\text{tr}}}}$ + $\beta _{n}^{{{\text{dr}}}}$ of resonant capture into all lm sublevels of level n, including the contribution of dissociative recombination (1), $\beta _{n}^{{{\text{dr}}}}$. Calculations are performed for a merged-beam setup with ε_d = 0.15 eV (1) and for the equilibrium case (2) for the Ar + Xe⁺ + e system at T_e = 2000 K and (a) T = 300 K and (b) T = 600 K.

6.4 Comparison of the Results with Semiempirical Models

There are no experimental data on the rate constants of processes (1) and (2) of dissociative and three-particle electron capture into given Rydberg levels of Xe(n) atoms in a plasma of inert gas mixtures. Below we compare the dissociative recombination rate constants, summed over n, that are calculated using the theory developed here with the values of α^dr(T_e, T) obtained by semiempirical formulas proposed in [33, 34] as a result of analysis of experimental data for integral rate constants. Figure 7a demonstrates the temperature dependence of the total recombination rate constants α^dr(T_e, T) in collisions of electrons with NeXe⁺ (T = 300 K) and ArXe⁺ (T = 500 and 900 K) ions. The solid curves show the total rate constant α^dr(T_e, T) = $\sum\nolimits_{nl}^{} {\alpha _{{nl}}^{{{\text{dr}}}}}$(T_e, T) of process (1) for n ≥ 10 and l ≤ n – 1, calculated by formulas (27) and (37) of the present paper for electron capture into individual nl levels. The dotted curve for NeXe⁺ is obtained by the semiempirical formula α^dr(T_e) ~ α₀$T_{e}^{{ - 0.5}}$ [33, 34] and the experimental data of [35, 36]. We can see that the results are reasonably consistent with each other at T_e = 300–2000 K. Note that the dependence $T_{e}^{{ - 0.5}}$ also follows from the results of our work if we limit the transitions to the vicinity of the stabilization point R_ω = R_st ≡ R_n defined by condition (14).

Fig. 7.

(Color online) The rate constants of dissociative recombination of (a) NeXe⁺ and (b, c) ArXe⁺ ions with electrons. Solid curves represent the total rate constant α^dr(T_e, T) = $\sum\nolimits_{nl}^{} {\alpha _{{nl}}^{{{\text{dr}}}}}$(T_e, T), where $\alpha _{{nl}}^{{{\text{dr}}}}$(T_e, T) is obtained by formulas (27) and (37). The dotted curves represent the results calculated by the semiempirical formula α^dr(T, T_e) = α₀$T_{e}^{{ - 0.5}}$[1 – exp(–$\hbar$ω_e/k_BT)] [33, 34], where α₀ is determined from the experimental data of [35, 36]. T = (a) 300, (b) 500, and (c) 900 K.

The dotted curves in Figs. 7b and 7c for ArXe⁺ are defined by the formula of a modified model, α^dr(T_e, T) = α₀$T_{e}^{{ - 0.5}}$[1 – exp(–$\hbar$ω_e/k_BT)], that was proposed in [33, 34] on the basis of the experimental data of [36]. Here, $\hbar$ω_e is the vibrational quantum of the ground term U_i. The dependence on T is obtained by averaging the vibrational levels populations over the Boltzmann distribution under the assumption that the contribution of the ground vibrational state to α^dr(T_e, T) is dominant. Figures 7b and 7c show that such a model provides a reasonable estimate at low values of T; however, like NeXe⁺, it leads to errors for higher values of T_e and T. Note that the deviation of the rate constant α^dr(T_e, T) from $T_{e}^{{ - 0.5 \pm 0.2}}$ to the power-law dependence $T_{e}^{{ - 3/2}}$ at high T_e was also predicted in [37] for the ${\text{Ar}}_{2}^{ + }$ + e system. It should be kept in mind that the estimate of the dissociative recombination coefficient by the direct summation of the rate constants corresponding to all finite values of n is only suitable for a qualitative comparison of the results, which is due to the neglect of the effect of a number of accompanying recombination and relaxation processes.

7 CONCLUSIONS

We have studied complementary processes (1) and (2) of formation of Rydberg atoms in a two-temperature plasma containing atomic and molecular ions. Dissociative recombination (1) has been analyzed under the condition of significant thermal excitation of all rovibrational levels of a molecular ion, which is already the case at temperatures T ≈ 300–1000 K for weakly bound RgXe⁺ ions. We have shown that in this case one should take into account the effect of the process (2) of resonant electron–ion recombination in ternary collisions with buffer gas atoms.

For a unified description of processes (1) and (2), we have developed an original approach that is based on the quantum version of the theory of nonadiabatic transitions and takes into account the form and symmetry of the terms of the BA⁺ ion, the specific features of interaction of outer and inner electrons in the system, and the complex structure of non-hydrogenlike Rydberg states. In calculating the total cross sections and dissociative recombination rate constants, we used the quasicontinuous spectrum approximation for ${v}$J levels of the BA⁺ ion. A similar method of integration with respect to energy E and angular momentum J was used in calculating the reaction rate constants (2). We have obtained for the first time semianalytical expressions for the cross sections and rate constants of processes (1) and (2) applicable in the case of weakly bound quasimolecular systems and correctly described the motion of nuclei in the classically allowed and forbidden regions.

The resulting expressions in Subsection 4.3 describe the integral contributions of the states of discrete, $W_{{nl}}^{{{\text{dr}}}}$, and continuous, $W_{{nl}}^{{{\text{tr}}}}$, spectra of a molecular ion to the total rate $W_{{nl}}^{{{\text{res}}}}$ of resonant electron capture into Rydberg states nl in a plasma with temperatures T_e and T of the electronic and atomic (ionic) components, respectively. The theory can be used for homonuclear and heteronuclear systems, ${\text{A}}_{2}^{ + }$ + e and BA⁺ + e, and, together with expressions for the effective interaction parameter, it allows one to establish the dependence of the rates $W_{{nl}}^{{{\text{dr}}}}$ and $W_{{nl}}^{{{\text{tr}}}}$ on the main parameters of the problem: the temperatures T_e and T; the principal quantum number n; the initial, U_i(R), and final, U_f(R), electronic terms of the BA⁺ ion; and the relationship between the thermal energy k_BT and the dissociation energy D₀ of the BA⁺ ion.

Specific calculations have been carried out for heteronuclear systems Rg + Xe⁺ + e with low (NeXe⁺) and moderate (ArXe⁺) dissociation energies D₀ of the ground term, which are of interest for the kinetics of recombination in a plasma of inert gas mixtures Rg/Xe with small xenon fracture ([Xe] ≪ [Rg]). The analysis of ternary (2) and dissociative (1) recombination has shown for the first time that the fraction of resonant three-particle electron capture by atomic ions in the total rate constant can be close to 100% in the case of weakly bound systems. In this case, channel (2) populates Rydberg states with rather high values of n ≳ 10. The dependence of the corresponding rate constants on n turns out to be sensitive to the gas temperature T and the structure of the terms.

In molecular ions with relatively deep potential wells (D₀ ≫ k_BT), on the contrary, the channel of dissociative recombination involving a large number of rovibrational levels is dominant if the condition k_BT ≫ $\hbar$ω_e is satisfied. This conclusion applies to situations where the Boltzmann distributions over levels ${v}$J and the action mass law (42) for particles in free and bound states are valid. In this study, we have demonstrated that deviations from the Boltzmann and Maxwellian distributions may lead to a different relative role of dissociative and ternary recombination channels and, in particular, may increase the fraction of process (2) in electron capture into Rydberg levels.

The results obtained are of interest for simulating the kinetics of recombination processes in barrier discharges [5], excimer lamps and VUV radiation sources [38, 39], active media of high-power gas lasers [40, 41], and microplasma cells [42].