Photoionization of air species as impurities in atmospheric pressure helium plasma

We revisit the problem of photoionization of small admixtures of nitrogen and oxygen molecules in atmospheric pressure helium plasma originally formulated in the pioneering work of Naidis (2010 J. Phys. D: Appl. Phys. 43 402001). The radiation trapping of resonance emission lines in atomic helium is quantified, and it is demonstrated that photoionization occurs due to radiative decay of the electronic A state of helium molecules. The collisions and atomic precursors that populate the excited A state of the helium molecule are clearly identified. The Einstein probabilities for the transition from bound and quasi-bound rovibrational levels of the A state to the continuum of the ground X state are provided. A kinetic scheme for the production of the fast component of ultraviolet emissions in atmospheric pressure helium plasma is proposed. The photoionization of molecular oxygen and molecular nitrogen as impurities in 99.9% and 99.99% purity helium is studied.


Introduction
Helium plasma jets injected into atmospheric pressure air represent an effective means of production of chemically active non-thermal plasmas that are of significant practical interest (Babaeva and Naidis 2021, Babaeva et al 2023a, and extensive list of references therein).The photoionization of small admixtures of argon and molecules of nitrogen and oxygen due * Author to whom any correspondence should be addressed.
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to vacuum ultraviolet (VUV) emissions from excited helium molecules (He 2 ) in the context of helium plasma jets was originally studied by Naidis (2010).These VUV emissions are due to the dipole-allowed transition between the A 1 Σ + u excited state and the X 1 Σ + g repulsive ground state of the helium molecule, and have been frequently observed (e.g.Hill 1989, and references therein) and extensively studied (e.g.Tanaka et al 1958, Tanaka and Yoshino 1963, Sando 1971, 1972, Sando and Dalgarno 1971).
He 2 (A) emissions suggest that two mechanisms with distinct temporal signatures populate the rovibrational levels (v ′ , J ′ ) of He 2 (A) (Carman et al 2016): (i) radiative cascade from He * 2 (n = 3) excited molecular states leads to the fast component of emission from relatively low vibrational levels of the A state; (ii) slow decay of He(2 1 S) (relative to the fast component) via two-and three-body collisions with groundstate He(1 1 S) atoms populates the higher vibrational levels of He 2 (A), whereas several alternative mechanisms involving a precursor other than He(2 1 S) have been proposed for pumping the low vibrational levels.We note that rate constants for two-and three-body collisions of He(2 1 S) with He(1 1 S) were measured in e.g.(Payne et al 1975a(Payne et al , 1975b) ) and (Nayfeh et al 1976), and the first measurement of the rate constant for the two-body reaction by Phelps (1955) motivated further studies of the collision-induced radiative decay of He(2 1 S) through interaction with He(1 1 S) (Burhop and Marriott 1956, Allison et al 1966, Sando 1972, Zygelman and Dalgarno 1988, Dehnbostel et al 1990, Zygelman 1991).
In part of the literature on high-pressure helium plasma there is a discrepancy in the identification of the species and processes through which the excimer is produced.For instance, Golubovskii et al (2002) identified He(2 3 P) as the atomic species responsible for the production of He 2 (A) through three-body reactions with ground-state He.Naidis (2010) also refers to (Golubovskii et al 2002) and states (without mention of spin multiplicity) that both the S and P states of atomic helium contribute to He 2 (A) production.Fierro et al (2018) identified He(2 3 P) as the atomic precursor to He 2 (A) while Fierro et al (2021) identified both He(2 3 P) and He(2 3 S).In the limit of large internuclear distance r, i.e. the dissociation limit, He 2 (A) dissociates into He(1 1 S)+He(2 1 S) atoms (e.g.Ginter and Battino 1970).From the previous paragraph, one infers that He(2 1 S) is the atomic precursor to He 2 (A).Collisions of He(2 1 P), He(2 3 S), and He(2 3 P) with He(1 1 S) may create the B 1 Π g , a 3 Σ + u , and b 3 Π g states of He 2 , respectively (Ginter and Battino 1970, Yarkony 1989, Carbone et al 2016).
Previous attempts to quantify the density of He 2 (A) assume that an arbitrary fraction 0.1 ⩽ ζ ⩽ 1 of the atomic precursor to He 2 (A) (which here we identify as He(2 1 S)) converts to the excimer (e.g.Naidis 2010, Bourdon et al 2016).Specifically, Naidis (2010) used a relatively slow timescale for the production of He 2 (A) through three-body collisions (µs timescale at atmospheric pressure (e.g.Nayfeh et al 1976)), which corresponds to the slow VUV component, to argue that during streamer propagation (on the tens/hundreds of ns timescale) only a fraction of atomic metastable states participate in this process to convert to He 2 (A).In the present work, we reiterate that during the formation/ignition phase of helium plasma, radiative cascade from higher-energy molecular levels contributes significantly to the production of He 2 (A) (e.g.Villarejo et al 1966, Gand et al 1979).In particular, we infer from (Carbone et al 2016, Carman et al 2016, Schregel et al 2016, and references therein) a consistent kinetic picture of fast He 2 (A) production.
In the remainder of the paper and in agreement with previous work (e.g.Mies and Smith 1966, Michels et al 1987, Carman et al 2016) we demonstrate that emissions from He 2 (A) span the ∼60 − 120 nm interval and therefore, a fraction of photons emitted from He 2 (A) is capable of ionizing impurities such as molecular oxygen O 2 (with an ionization threshold of ∼102.72 nm) and molecular nitrogen N 2 (with an ionization threshold of ∼79.57nm) that are present in the high-purity (i.e.99.9% and 99.99%) helium gas considered in this study.Although the formulation of the photoionization problem presented in the next section closely follows previous work (e.g.Janalizadeh and Pasko 2019), peculiarities of the current problem, such as the radiation trapping of resonance emission lines of atomic helium, and necessary quantities for the modeling of photoionization, are introduced and thoroughly discussed in the paper and accompanying supplementary file.In particular, section 2 accommodates readers who are mainly interested in the implementation aspect of the model and provides all the ingredients for that purpose.The effect of radiation trapping on the resonance emission lines of atomic helium is quantified in section 3. Section 4 presents details of the derivations resulting in the photoionization model outlined in section 2. All calculations are performed for the neutral density n corresponding to atmospheric pressure p = 760 Torr.Section 5 includes a discussion of fast He 2 (A) production and proposes a related kinetic scheme.The concluding remarks are found in section 6.

Model outline and implementation
This section focuses exclusively on the implementation of the photoionization model, whereas sections 3 and 4 present the details of arguments that result in the sequential construction of this model.As derived in section 4, the rate of photoionization (i.e.number of photoionization events per unit volume per unit time) of impurity k with density n im k at position ⃗ r due to I(⃗ r ′ ) photons per unit volume per unit time emitted from point ⃗ r ′ of a source comprising volumetric elements dV ′ may be expressed via the integral representation (e.g.Janalizadeh and Pasko 2019) as where ⃗ R =⃗ r −⃗ r ′ and R = | ⃗ R|.In section 3 we demonstrate that emissions from He 2 (A) (as opposed to atomic helium resonance emission lines) are responsible for the photoionization of impurities considered here, exclusively.As such, I = A avg A n A , where A avg A = 0.4 × 10 9 s −1 as calculated in section 4 and n A ≃ 6.6 × 10 12 cm −3 (Gand et al 1979, equation ( 3)) (also see section 5).Bourdon et al (2007) fit g(pR)/(pR), where g(pR) is the photon propagator (see figure 3) to exponential functions in order to convert the integral representation above to a set of differential equations (e.g.Janalizadeh and Pasko 2020).Tables 1 and 2 respectively provide the fit parameters for propagators of N 2 and O 2 photoionization in 99.9% pure helium mixed with air, where the fractional composition of N 2 and O 2 is respectively α N2 = 0.8 and α O2 = 0.2 of the 0.1% impurity.Similarly, tables 7-12 in section 4 summarize the fit parameters for O 2 and N 2 photon propagators in 99.9% pure helium mixed with pure O 2 and pure N 2 , and for 99.99% pure helium mixed with air, pure O 2 , and pure N 2 .At the end of the current section we list the procedure for quantifying the rate of photoionization given in equation (1): 1.The density of impurity k (i.e.either O 2 or N 2 ) may be calculated as n im k = α im k n where n denotes the total density and is proportional to the atmospheric pressure p = 760 Torr.In air, we have α O2 = 0.2α im and α N2 = 0.8α im for a given total impurity fraction α im = 1 − α He , where α He = 0.999 for the case chosen for illustration in this section.2. The number of He 2 (A) photons emitted per unit time per unit volume of the source is I = A avg A n A = 2.64 × 10 21 cm −3 s −1 .3. For a given distance R, the quantity pR defines the photon propagator g(pR), where the interested reader may use the fit parameters from tables 1, 2, and 7-12 to calculate g(pR) efficiently.Also, following Bourdon et al (2007) and Janalizadeh and Pasko (2019), one can convert the integral representation in equation (1) to a set of differential equations corresponding to a set of tabulated fit parameters to obtain the photoionization rate S im k .4. The calculated photoionization rate S im k may then be used as a source term in the drift-diffusion-reaction equation governing the density of electrons and positive ions (e.g.Naidis 2010, equation ( 2)).

Radiation trapping of atomic helium emission lines
Both resonance emission lines of atomic helium corresponding to j 1 P → 1 1 S transitions for j > 1 (e.g.Wiese and Fuhr 2009, table 14) and photons emitted by He 2 (A) may ionize air species.However, the relative population of these two sources of photons has not yet been accurately quantified.Specifically, the theory of radiation trapping (Holstein 1947, 1951, Walsh 1959) already applied to resonance emission lines of atomic helium (Drawin and Emard 1973) has not been used in previous studies of photoionization in helium plasma at atmospheric pressure (e.g.Fierro et al 2021, Babaeva et al 2023b).
Here, we follow Srivastava and Ghosh (1981), and calculate (radiation) escape factors Λ 1j = A app 1j /A 1j , in which A app 1j is the 'apparent' Einstein transition probability A 1j for the j 1 P → 1 1 S transition in the presence of radiation trapping, assuming the source of photons is a sphere with radius r jet ∈ [0.1, 0.5, 1] cm and electron density n e ∈ [10 12 , 10 15 ] cm −3 corresponding to the plasma gun discharge/streamer head properties in e.g.(Naidis 2010, Bourdon et al 2016, Babaeva et al 2023b) (see supplement for details).Tables 3 and 4 present the Λ 1j and A app 1j values for atmospheric pressure helium gas at room temperature, and n e = 10 12 and 10 15 cm −3 , respectively.The numerical values for A 1j , where j = 2 − 10, are taken from (Wiese and Fuhr 2009, table 14).
The relative photon emission rate from atomic and molecular helium may be expressed via where A avg A ≃ 0.4 × 10 9 s −1 is the Einstein transition probability for photons emitted from He 2 (A) calculated in section 4, n A ≃ 6.6 × 10 12 cm −3 is the density of He 2 (A) (defined in section 2), and n j 1 P is the density of atomic helium state j 1 P. We note that accurate quantification of n A time dynamics requires a clear kinetic picture of fast He 2 (A) production, which is absent from the current literature (see section 5).Also, to the best knowledge of the authors, the densities n j 1 P have not been calculated/measured for all j considered here.Nevertheless, for the range of pressures in (Emmert et al 1988), including 1 bar, the He(2 1 P) densities are n 2 1 P ≃ 10 13 − 10 14 cm −3 in the early stages of discharge (and independent of pressure).Consequently, for the lowest values of j, where Λ 1j ≪ 1, the ratio I j 1 P /I A ≪ 1.
At high electron density and large j we have Λ 1j → 1 (see table 4).However, in these cases n j 1 P ≪ n 2 1 P due to the higher energy thresholds and smaller cross-sections for electron impact excitation (e.g.Ralchenko et al 2008).Thus, one may expect lower densities for j > 2 during the plasma breakdown and discharge phase, where atomic helium states are produced by electron collisions (as opposed to recombination processes) (e.g.Carbone et al 2016).Thus, I j 1 P /I A remains negligible and the only source of photoionizing radiation is He 2 (A).

Model formulation
The rate of photoionization (i.e. the number of photoionization events per unit time per unit volume) of impurity k with density n im k at position ⃗ r due to He 2 (A, v ′ , J ′ ) photons emitted from a source comprising volumetric elements dV ′ located at ⃗ r ′ is given by  where n A v ′ J ′ is the density of the He 2 (A, v ′ , J ′ ) rovibrational level.Considering the discharge volume of related studies (e.g.Naidis 2010, Bourdon et al 2016, Fierro et al 2018, Babaeva et al 2023b), we ignore the photon time of flight R/c in this work, such that photoionization occurs instantly after a photon is produced.In addition, photoabsorption occurs only due to impurities enumerated with index k in the equation above.Photoabsorption due to the abundant atomic helium occurs at wavelengths λ ⩽ 58.4 nm (corresponding to the 1 1 S ↔ 2 1 P transition (e.g.Wiese and Fuhr 2009)) (e.g. Lee and Weissler 1955), and, as will be demonstrated below, it is irrelevant to the current study since the onset of He 2 VUV emissions is at ∼60 nm.Equation (2) follows from (Riley and Alford 1995) and is derived in the supplementary file.Here, we assume that the neutral density (i.e.pressure) is constant, n im k = α im k n ≪ n ≃ n He , and that only n A v ′ J ′ , which emit photoionizing radiation, may vary as a function of time.In equation ( 2), σ pa (λ) and σ pi (λ) denote photoabsorption and photoionization cross-sections, respectively.Einstein's probability of transition from He 2 (A, v ′ , J ′ ) to the continuum of He 2 (X) per unit wavelength A v ′ J ′ (λ) is derived in the supplement and is given by where ϵ 0 and h are respectively the permittivity of free space and Planck's constant, and ε denotes energy, where for a photon ε = hc/λ.In equation ( 3), M e is the electric dipole moment for the A ↔ X transition as a function of the ).Bottom inset depicts the small minimum of V X (≃ 1 meV) at re ≃ 5.6a 0 , where a 0 denotes the Bohr radius.Top inset shows V A in addition to probability of radial functions , 10, 17] and J ′ = 0. De ≃ 2.47 eV is the dissociation energy of the A 1 Σ + u state and Te denotes the energy difference between the minimum of the two potential curves.In the limit of r → ∞ both molecules approach the separated atom limit where the difference in energy between the curves is equivalent to the excitation threshold of He(2 1 S) = 20.6157eV (e.g.Kramida et al 2022).internuclear distance r, ψ v ′ J ′ satisfies the radial Schrödinger equation corresponding to the interatomic potential of He 2 (A) (Komasa 2006) and is obtained using the LEVEL16 program (Le Roy 2017).The turning point r t of the He 2 (X) interatomic potential V X is defined through ε = V X (r t ).
In the separated atom limit (where r → ∞), He 2 (X) → He(1s 2 1 S) + He(1s 2 1 S) and He 2 (A) → He(2s 1 S) + He(1s 2 1 S) (e.g.Ginter and Battino 1970).Thus, in this limit, the energy difference between the two potentials is equivalent to the excitation threshold of He(2s 1 S), i.e. 20.6157 eV (e.g.Kramida et al 2022).The dissociation energy for He 2 (A) is D e ≃ 2.47 eV (e.g.Komasa 2006) while V X has a minimum of ∼ − 0.9476 meV (e.g.Przybytek et al 2010) at r e = 0.296 83 nm (e.g.Aziz et al 1995) with respect to 0 eV at r → ∞ .Thus, the energy difference between the minima of the two potential curves is T e = 18.1432 eV (see figure 1).
The top inset in figure 1 depicts a zoomed-in version of the potential for the He 2 (A) electronic state provided in (Komasa 2006).We have imported this potential in the LEVEL16 program to obtain ψ v ′ J ′ and ε v ′ J ′ for various pairs of (v ′ , J ′ ).The latter is depicted for J ′ = 0 with dashed lines in the potential well.Details of this calculation are provided in the supplementary file and ψ v ′ J ′ radial functions calculated for v ′ = 0 − 17 and J ′ = 0 − 30 (excluding dissociated rovibrational levels) may be found in the online repository linked in the data availability statement.Here, we show |ψ v ′ J ′ | 2 for v ′ = [0, 5, 10, 17] and J ′ = 0 for illustration purposes.Note that the unit of the vertical axis in the top inset does not correspond to the unit of |ψ v ′ J ′ | 2 but only to the potential curve of the A 1 Σ + u state V A .The |ψ v ′ J ′ | 2 values have been plotted at the energy corresponding to level (v ′ , J ′ ), and have been arbitrarily scaled such that the structure of the function is clearly seen without overlap of depicted Using ψ v ′ J ′ outputted from LEVEL16 we calculate the Einstein transition probabilities per unit wavelength (in nm) for all (v ′ , J ′ ) levels considered via equation (3).The transition dipole moment in (Komasa 2006, table 5) is used for this calculation.For illustration purposes we show A v ′ J ′ (λ) in figure 2 for J ′ = 0 and v ′ = 0 − 14 in increments of ∆v ′ = 2.One can clearly see the shift in peak emission toward 60 nm in addition to the band-like structure of the spectrum with the increase of v ′ (e.g.Mies and Smith 1966).Note that the ∼60 nm (∼20.66403 eV) onset of VUV emissions corresponds to the transition of He 2 (A) from the quasi-bound rotational levels of v ′ = 17 to the continuum of X 1 Σ + g .
Once A v ′ J ′ (λ) is quantified one may calculate A v ′ J ′ = ´λ A v ′ J ′ (λ)dλ to obtain the total Einstein transition probability (per unit time and independent of wavelength) for each rovibrational level.Table 5 presents A v ′ J ′ (in units of 10 9 s −1 ) for rotational levels J ′ = 0 − 30 of v ′ = 0 − 17.The obtained values are in satisfactory agreement with (Yarkony 1989, table 8) (see table 6) where rotation was neglected (i.e.J ′ = 0) and v ′ = 0 − 10.We note that the minor between results may be attributed to the fact that following Schneider and Cohen (1974), and as opposed to e.g.van Dishoeck and Dalgarno (1983), Le Roy (1989), and Yarkony (1989), we do not solve the radial Schrödinger equation for the continuum function ψ ε ′ ′ J ′ ′ (see supplement).Instead, under the approximation that the major contribution to the integral over the internuclear coordinate comes near the classical turning point of the lower potential curve, we assume ψ ε ′ ′ J ′ ′ is proportional to the Dirac delta function (e.g.Carman et al 2016, and references therein).Although the A v ′ J ′ values presented here may have been calculated before (e.g.Michels et al 1987, Carman et al 2016), the values in table 5 provide the most comprehensive set of Einstein transition probabilities available in the literature.
Defining s v ′ J ′ (λ) ≡ A v ′ J ′ (λ)/A v ′ J ′ as the shape function for emission from He 2 (A, v ′ , J ′ ) such that ´λ s v ′ J ′ (λ)dλ = 1, equation (2) may be written as where we have now removed the explicit dependence of quantities on time, and is the photon propagator that describes the propagation of photons emitted by He 2 (A, v ′ , J ′ ) in a given medium in addition to their ionization of impurity k.The rate of photon emission A v ′ J ′ n A v ′ J ′ only includes information about the production of the photons as opposed to photon interaction with plasma.Note that, as demonstrated in (Janalizadeh and Pasko 2019), one can infer the pR dependence of the propagator through n im k = nα im k , where n ∝ p.The calculation of photon propagators similar to g im k v ′ J ′ (pR) has been carried out in previous work (Janalizadeh and Pasko 2019, 2020).As such, for the sake of brevity, we include plots of g im k v ′ J ′ (pR) for a limited number of (v ′ , J ′ ) in the supplement.Note that in the case of negligible photoabsorption (e.g. at small distances from the source of emission) one obtains Janalizadeh and Pasko (2020) demonstrate how one can take advantage of this result in efficient calculation of photoionization rates in the case of negligible photoabsorption.
Once n A v ′ J ′ in equation ( 4) is quantified, one can proceed to calculate the rate of photoionization due to each   He is S N = 0, such that the antisymmetric (a) levels are not populated, i.e. d(S N ) = 0 for a levels whereas d(S N ) = 1 for symmetric (s) levels (Tatum 1967).As such, n A v ′ J ′ = 0 for J ′ = 0, 2, 4, . . .(e.g.Bernath 2015, pp 267-70, 374-5), and the relative population of all rovibrational levels may be quantified with respect to v ′ = 0, J ′ = 1.In other words,

Einstein transition probabilities
for rotational levels with odd J. Consequently, for a given He 2 (A) total density where

the rate of photon emission in which
396 × 10 9 s −1 is the average Einstein transition probability calculated from table 5 for all N v ′ J = 497 rovibrational levels (excluding dissociated levels), and where We note that, technically, the summation should be carried out over all non-dissociated rovibrational levels for which A v ′ J ′ ̸ = 0. From the slow variation of A v ′ J ′ with J ′ (see table 5), one infers that for lower vibrational levels this calculation may require A v ′ J ′ for very high J ′ not presented in table 5.
Nevertheless, the robustness of the results with respect to the number of included rovibrational levels was verified, and it was observed that both (v ′ = 17, J ′ = 0) and (v ′ = 0, J ′ = 30) levels already had a negligible population (i.e.χ v ′ J ′ ≪ 1) as a result of T vib and T rot .
Figures 3(a) and (b) depict the calculated photon propagators of O 2 and N 2 in 99.9% and 99.99% pure helium, respectively.The absorbing medium has been varied among pure N 2 , pure O 2 , and conventional air.In the case of air, O 2 and N 2 constitute 0.2 and 0.8 of the impurity fraction, respectively.The photoabsorption and photoionization cross-sections of O 2 and N 2 have been compiled from (Fennelly and Torr 1992).Tables 1, 2, and 7-12 provide parameters of analytical fits to g(pR)/(pR) depicted in figure 3 for conversion of the integral solution of the photoionization rate to a differential form (e.g.Bourdon et al 2007).The process of fit development is detailed in (Janalizadeh and Pasko 2020).For the sake of brevity, plots comparing the accuracy of the fits with the calculated propagators are presented in the supple mentary file.
For 99.99% pure helium at atmospheric pressure and short distances from the streamer head in e.g.(Naidis 2010 and Babaeva et al 2023b) (i.e.pR ≃ 10 3 Torr cm), one infers that photoabsorption is practically negligible (see figure 3(b)).In that case, , where one can implement the differential representation of the problem using the fit for (pR) −1 provided in (Janalizadeh and Pasko 2020, section 3.1).Note that in this case information about the composition of the propagation medium is absent from the propagator.On the other hand, from figure 3(a) one may infer that in 99.9% pure helium and on a scale of a few cm, the photoabsorption is considerable.Thus, for pR ≃ 10 4 Torr cm, where the photoabsorption becomes significant, changing the absorbing medium varies the propagator of a given impurity.

Discussion
We acknowledge that in addition to He 2 (A), the He 2 (D 1 Σ + u ) state also contributes to the VUV signature of the helium molecule (Michels et al 1987, Carman et al 2016).However, in this work, we consider VUV emissions solely due to the A → X transition since, as demonstrated in (Carman et al 2016), the However, panels mainly depict range including significant photoabsorption with an onset of pR ≃ 10 2 − 10 3 Torr cm.In the case of air, α O2 and α N2 respectively constitute 0.2 and 0.8 of the impurity percentage, such that α O2 + α N2 = 10 −3 and 10 −4 in panels (a) and (b), respectively.fast component data do not match the calculated spectral profile of the D 1 Σ + u → X 1 Σ + g transition exclusively.Carman et al (2016) achieved a close match between the calculated spectra and experimental data by treating the fast component as a combination of emission mainly from the A 1 Σ + u state with a minor contribution (∼6%) from the D 1 Σ + u state.The authors circumvented the unavailability of the electric dipole moment for the D ↔ X transition by assuming a transition dipole moment with a radial dependence identical to the A ↔ X transition dipole moment.Similarly, Michels et al (1987) assumed a constant electronic transition moment.Thus, conclusions drawn  by Carman et al (2016) and Michels et al (1987) are predicated upon their approach to overcome the lack of transition dipole moment(s) necessary for modeling He 2 VUV emission spectra.
As mentioned above, the dynamics of the density of He 2 (A) molecules are the single remaining unknown for the accurate quantification of the photoionization rate.Here, we used (Gand et al 1979, equation (3)) to calculate n A ≃ 6.6 × 10 12 cm −3 .The kinetics of the slow component were described in section 1, and it is seen from the measurements of Carman et al (2016) that during the discharge period t ⪅ 0.5 µs, the  contribution of slow and fast components to the VUV intensity, proportional to n A , is comparable.Thus, as an estimate of n A produced via the fast component, we calculate the population of He 2 (A) (in the absence of a fast component) considering electron impact excitation of He(2 1 S) (at electron temperature T e ≃ 3.5 eV (e.g.Schregel et al 2016)) with a rate constant of 5 × 10 −12 cm 3 s −1 (e.g.Kato and Nakazaki 1989) and the conversion of He(2 1 S) to He 2 (A) with a frequency ∼2 × 10 6 s −1 corresponding to the slow component (Carman et al 2016).For n e (t) similar to that in (Carbone et al 2016, Schregel et al 2016), with a peak value of ∼10 14 cm −3 , one obtains n A ≃ 10 12 − 10 13 cm −3 in agreement with (Gand et al 1979).Note that in this estimation we ignored the conversion of 2 1 P to 2 1 S and the radiative decay of imprisoned He(j 1 P) following Carbone et al (2016).Below, we propose a kinetic scheme for the fast production of He 2 (A) (i.e.quantification of n A ) that merges the observations of Carman et al (2016) and Carbone et al (2016).One may quantify n A by adopting this kinetic scheme after experimental verification.Carman et al (2016) suggested that rapid radiative decay of He * 2 (n = 3) populates He 2 (A) and leads to the fast component of VUV emissions because the temporal evolution of He * 2 (n = 3) emissions, similar to the fast component of He 2 (A) VUV, closely follows the power deposition in the plasma excitation phase, where the electron temperature k B T e ≃ 3 − 4 eV (see e.g.Schregel et al 2016, figure 8) is high.However, since Carman et al (2016) assume that recombination should proceed only at low electron temperatures, they do not attribute the production of He * 2 (n = 3) to electrons recombining with He + , He + 2 , He + 3 .Meanwhile, Carbone et al (2016) point out that according to the experimental data, the main loss channels of He 2 + (Carbone et al 2016, reactions (10) 8).The sudden rise of these optical emissions suggests that the process responsible for their production depends strongly on the electron temperature.This is indeed the case for the recombination of He 3 + with electrons (Carbone et al 2016, reaction (12)) such that the authors label it as the main source of the visible He 2 molecular bands.In summary, the recombination of He 2 + produces molecules that should radiatively decay to He 2 (A) to produce the fast VUV emissions.He 2 (E 1 Π g ) is the main upper state candidate (Villarejo et al 1966, Carman et al 2016).On the other hand, the recombination of He 3 + produces He 2 (e 3 Π g ) and He 2 (d 3 Σ + u ), which subsequently produce emissions in the visible range.Table 13 summarizes the recombination, collisional, and radiative processes resulting in the visible and slow and fast components of VUV emissions discussed above.Readers are referred to (Carbone et al 2016, and references therein), for the detailed kinetics of atmospheric pressure helium plasma, and to e.g.(Carman et al 2016, figure 3) for the potential energy curves of A and E states of He 2 as part of a comprehensive compilation of potential energy curves of He 2 electronic states.
Finally, we note that the results of the present work are predicated on the consistent vibrational and rotational temperatures assumed in (Emmert et al 1988) and (Carman et al 2016) for the best reproduction of experimental results in helium plasma at atmospheric pressure.We share the necessary tools and data for expanding the presented work to a custom problem in the data repository linked to this work (Janalizadeh and Pasko 2023).(Rosen 1970, p 194) a The E 1 Πg state.
b The e 3 Πg and d 3 Σ + u states.

Conclusions
The photoionization of air species as impurities in helium plasma has been revisited.The spectroscopic properties/constants of the A 1 Σ + u state of the helium molecule, such as rovibrational energy levels and Einstein probabilities corresponding to the A ↔ X transition, have been presented.The effect of radiation trapping on the resonance emission lines of atomic helium is quantified, and a new recipe for the calculation of the density of He 2 (A 1 Σ + u ) molecules in accordance with recent kinetic and experimental studies of high-pressure helium plasma is introduced.Following previous work on the modeling of photoionization, numerical fits to photon propagators for the efficient calculation of photoionization rates are provided.It may be inferred from the literature that the kinetics of the fast component of the helium excimer VUV emissions are not yet understood.A complete understanding of the fast VUV production will enhance the modeling of the time dynamics of impurity photoionization.

Figure 1 .
Figure 1.Potential energy curves for He 2 (A)(Komasa 2006)  and He 2 (X) (solid lines)(Przybytek et al 2010).Note that the energy scale for the A state is on the right vertical axis.Dashed lines in the potential well of He 2 (A) show the energies of the 18 rotationless vibrational levels (e.g.Sando 1972).Bottom inset depicts the small minimum of V X (≃ 1 meV) at re ≃ 5.6a 0 , where a 0 denotes the Bohr radius.Top inset shows V A in addition to probability of radial functions|ψ v ′ J ′ (r)| 2 ( ´r |ψ v ′ J ′ (r)| 2 dr = 1, [r] = Å) for v ′ = [0,5, 10, 17]  and J ′ = 0. De ≃ 2.47 eV is the dissociation energy of the A 1 Σ + u state and Te denotes the energy difference between the minimum of the two potential curves.In the limit of r → ∞ both molecules approach the separated atom limit where the difference in energy between the curves is equivalent to the excitation threshold of He(2 1 S) = 20.6157eV (e.g.Kramida et al 2022).

Table 1 .
Parameters for the fit

Table 2 .
Parameters for the fit

Table 6 .
Comparison of Einstein transition probabilities A v ′ J ′ (10 9 s −1 ) for v ′ = 0 − 10, J ′ = 0 with (Yarkony 1989, table8).′ = 0 − 17 and J ′ = 0 − 30, but more importantly, to the best of the authors' knowledge and based on the current literature, the development of an elaborate kinetic scheme that quantifies the relative population of He 2 (A) rovibrational levels vs time is not feasible.In what follows, we focus on the fast component of emissions since it captures the setting of related work (e.g.Naidis 2010) as it coexists with the source of excitation during the ignition of the plasma and precedes the early afterglow period during which the input power is terminated.Previous attempts to explain the observed emission from atmospheric pressure helium discharge have resulted in estimates of vibrational temperatures T (Carman et al 2016) et al 2016)and k B T vib = 180 meV (2088 K) (Emmert et al 1988) with a minor effect of the rotational temperature T rot on the emission spectra.As such, here we also choose T vib = 1500 K and T rot = 300 K in a Boltzmann distribution that includes the effect of nuclear spin S N degeneracy d(S N ) (e.g.Tatum 1967) in the statistics of each rovibrational level.The nuclear spin of 4

Table 7 .
Parameters for the fit

Table 8 .
Parameters for the fit

Table 9 .
Parameters for the fit

Table 10 .
Parameters for the fit

Table 11 .
Parameters for the fit

Table 12 .
Parameters for the fit (Carman et al 2016)) andSchregel et al (2016)electron temperature.Specifically,He + 2 + He + e → He Ry 2 + He(Carbone et al 2016, reaction  (11)) produces Rydberg molecules He Ry 2 that, as seen in identical results shared byCarbone et al (2016) andSchregel et al (2016), have a temporal rise in density similar to electrons and He 2 + ions, that promptly imitate the input power.This addresses the concern ofCarman et al (2016), i.e. expecting recombination to occur only at low electron temperatures since the decay of He 2 + into Rydberg molecules through recombination is not significantly sensitive to electron temperature.In addition, the time dynamics of Rydberg molecules in(Carbone et al 2016)and (Schregel et al 2016) are in agreement with those of (Gand et al 1979) and(Carman et al 2016)in that higher-energy (Rydberg) molecules populate He 2 (A).We also note that a radiative channel to He 2 (A) is absent from the kinetic scheme in (Emmert et al 1988), which initiates with the fast conversion of He + ions produced during ignition to He + 2 (e.g.Carbone et al 2016, equation (8)).In(Emmert  et al 1988), the subsequent dynamics of the plasma follow the recombination of He + 2 .Carbone et al (2016) also reported the observation of radiative helium molecules He 2 (e 3 Π g ) and He 2 (d 3 Σ + u ) participating in the e 3 Π g → a 3 Σ + u and d 3 Σ + u → b 3 Π g transitions at 465 and 640 nm, respectively (e.g.Rosen 1970, p 194).Specifically, they observed a fast rise in these emissions only after the input power is shut down such that the electron temperature becomes equal to room temperature (i.e.T e is low) (seeCarbone et al 2016, figure 9 and Schregel et al 2016,  figure