Non-thermal plasma instabilities induced by deformation of the electron energy distribution function

The NDC was observed for a number of molecules: CH₄ [29–31]; C₂H₄ [32]; CF₄ [33–34]. The perfluoroalkanes C₂F₆ and C₃F₈ exhibit the pronounced NDC similar to but smaller in magnitude than that in CH₄ [34]. The electron drift velocities in BF₃ and SiF₄ demonstrate the NDC in some ranges of the electric field strength similar to but smaller in magnitude than those in CH₄ and CF₄ [35]. The momentum-transfer cross sections for selected atoms and molecules possessing the R–T minimum are shown in figure 2. Figure 3 shows the experimentally measured W(E/N) in pure molecular gases (CF₄, CH₄, C₂H₄) in comparison with the results of numerical solutions of the BE at room gas temperature.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The authors of [41–43] have analyzed what special features in the electron collision cross sections are responsible for the NDC appearance. The NDC can occur when an increase in the E/N parameter leads to an abnormally large increase in v_m value. Then, the enhanced randomization of directions of the velocity vectors can decrease the drift velocity even though the mean electron energy increases. The combination of the rapidly increasing elastic cross section or Ar, Kr and Xe above the R–T minimum and the vibrational excitation cross section for some molecules is known to produce the NDC in molecular–rare-gas mixtures at a small fraction of the molecular gas [44–45, 19, 14, 46]. The R–T minimum is most deep for the transport cross section of Ar (see figure 2). Therefore, the appearance of NDC in mixtures of molecular gases with Ar is more probable.

Figure 4(a) shows W(E/N) for mixtures CO–Ar of variable composition calculated in this work, including limits of pure CO and Ar gases. It is seen that the NDC occurs within certain ranges of CO concentrations and of E/N values. Theoretical curves for CO fractions 1%, 2%, and 5% are compared with the swarm data [19] for these mixtures. Good agreement between calculated and measured data proves that the self-consistent set of cross sections for CO is rather accurate. The regions, where the NDC effect is predicted for Y–Ar mixtures (Y denotes CO, N₂ or O₂), are shown in figure 4(b) along with available experimental data. Results presented in figure 4(b) are calculated by taking into account loss and gain of the electron energy in the elastic scattering from atoms and molecules, excitation and de-excitation of rotational states, excitation of vibrational and electronic states and ionization by electron impact. Electron scattering cross section sets were taken from [49] for N₂, CO and from [50] for O₂. Vibrational excitation of molecules was assumed to be low, and electron collisions with vibrational-excited molecules were neglected. The concentrations of molecular additives, below which the NDC disappeared, and corresponding E/N values (figure 4) are as follows: 66 ppm N₂ at E/N = 1.1 Td; 66 ppm CO at E/N = 0.65 Td; 400 ppm O₂ at E/N = 0.054 Td.

Zoom In Zoom Out Reset image size

In fact, electrons in non-thermal plasma effectively excite vibrational and electronic states of molecules. The stepwise and second-kind collisions can remarkably change the EEDF shape [11]. In [51] it was shown numerically that growth of vibrational temperature of N₂ molecules leads to the disappearance of the NDC effect in the mixture of N₂ : Ar = 1 : 99. A similar effect takes place in gas mixtures CO : Ar. A specific feature of molecular vibration kinetics for N₂ and CO is the strong difference in vibrational relaxation and vibration–vibration exchange rates. The latter is much faster. As a result, at a moderate vibration excitation rate and low gas temperature vibrational distribution function (VDF) in N₂ and CO strongly deviates from the Boltzmann distribution. The VDF under these conditions has a long plateau at intermediate vibrational levels. The analytical theory [52] for this case provides an approximate expression for the VDF. In the theory, a parameter T_V defined by formula n(v = 1)/n(v = 0) = exp(−E₁/kT_V) serves to characterize the strongly non-equilibrium VDF (E₁ is the vibration quantum energy, v is the number of vibration level). Within the frame of this theory, we have made numerical calculations by taking into account electron energy exchange with vibrational excited molecules. Figure 5(a) shows how the W(E/N) curve for CO : Ar = 2 : 98 mixture transforms with the growth of an effective vibrational temperature, T_V, at the fixed translational gas temperature equal to 300 K.

Zoom In Zoom Out Reset image size

Figure 5(b) shows the regions on the (E/N–T_V) plane for mixture CO : Ar = 2 : 98 where the NDC is predicted. It is instructive to compare the results found for non-equilibrium (solid line) and equilibrium (dashed line) VDFs. It is seen that the vibrational non-equilibrium causes the region to diminish with the NDC.

With an increase in the ionization degree, α = n_e/N, Coulomb collisions can influence the EEDF shape and, accordingly, the NDC appearance. Figure 6 illustrates the influence of e–e and e–i collisions on the NDC region on the (E/N–α) plane (solid-line boundary) for a mixture N₂ : Ar = 2 : 98 at T_v = T = 300 K. It is seen that at α ⩾ 3 × 10⁻⁴ the NDC disappeared. To distinguish between effects produced by e–e and e–i collisions, calculations were performed neglecting e–i collisions (dashed line). In this case, the NDC is predicted to exist even at α > 0.1. This indicates the importance of the e–i collision input into q_m amplitude [12].

Zoom In Zoom Out Reset image size

It was revealed in swarm experiments [22] that in the CO₂–Ar mixtures, at certain concentrations of CO₂, two separate NDC regions may appear. Figure 7 (curve 1) shows two regions, in which measured W(E/N) decreases as a function of E/N in Ar with 0.203% of CO₂ admixture. Results of these measurements along with swarm data for pure CO₂ were used in [22] to correct both the vibrational excitation and transport cross sections in CO₂, as well as the cross sections for excitation of CO₂ electronic states. The set of cross sections proposed in [22] is, obviously, the most reliable at the present time.

Zoom In Zoom Out Reset image size

Our BE calculations of swarm data with variation of shapes of vibrational excitation cross sections allow us to associate the appearance of two humps in the W(E/N) with a relatively slow decrease in the excitation cross section for the CO₂ (0 0 1) mode at the high-energy side [53]. This finding is confirmed by the fact that in H₂O–Ar mixtures the double-humped shape of the W(E/N) disappeared. The faster decrease in the vibrational excitation cross sections at higher electron energies is a cause of its disappearance [53].

It was shown in [53] that the increase in CO₂ concentration from 0.203% to 10% results in noticeable growth of the electron drift velocity at E/N > 0.1 Td (compare curves 1 and 2 in figure 7). Note that the double-humped shape of the W(E/N) curve remains but the amplitude of wavy structure diminishes. Numerical calculations [53], by taking into account the second-kind electron collisions with vibrationally excited CO₂ molecules, have shown that in the CO₂–Ar mixtures the degree of vibrational excitation negligibly affects the double-humped shape of the drift velocity (figure 7, lines 2 and 3). In the calculations, the temperatures of all vibrational modes of the CO₂ molecule were assumed to be equal to the gas temperature, T_v = T.

For Kr and Xe, the R–T minimum is located at higher energies (figure 2) in comparison with Ar. This feature explains why the predicted double-humped shape of the W(E/N) is less pronounced in CO₂–Kr mixtures and disappears in the CO₂–Xe mixtures [53]. It was shown theoretically that the double-humped shape of W is most impressive in 10% CO₂–Ar and 15% CO₂–Kr mixtures. Thus, among binary mixtures of rare–molecular gases the double-humped curve of W(E/N) was found only for the CO₂–Ar (Kr) mixtures.

A three-component mixture of two molecular gases and a heavy inert gas provides more variety in behavior of energy loss rate as a function of electron energy and better chances to realize double-humped behavior of W(E/N). As an example, the double-humped shape of the W(E/N) curve predicted theoretically [53] for the CH₄–H₂–Ar mixture is shown in (figure 8). In these mixtures, the drift velocity changes not so much in a wide range of E/N from 0.2 to 10 Td.

Zoom In Zoom Out Reset image size

The difference in W(E/N) values for the two NDC regions calculated [53] was found to be quite large in the H₂O–N₂–Ar and SF₆–N₂–Ar mixtures. The drift velocities differ by a factor of 2–3. Since the high-field layers propagate with the velocity nearly equal to the electron drift velocity, the appearance of oscillations in two NDC regions can be easily detected experimentally due to a notable difference in oscillation frequencies. Verification of these theoretical predictions can be easily carried out by respective experiments with an EBSD in gas mixtures predicted.

The double-humped shape of the W(E/N) curve is a rather fine effect, which requires specific shapes of both the transport cross section for electron scattering from inert gas atoms and cross sections for inelastic scattering from molecules. For binary mixtures, this effect is found only in the CO₂–Ar (Kr) mixtures.

For ternary mixtures, the required energy dependence of inelastic processes rates can be realized by changing the concentrations of the molecular components.

The NDC can take place also in mixtures of Xe with He [54–55]. The elastic collisions with He atoms are accompanied by energy exchange, which plays a role similar to inelastic collisions in molecular gas–rare gas mixtures. Results of measured swarm data on W(E/N) for a variety of He concentrations in the mixtures He–Xe [56] are in qualitative agreement with that predicted by [54].

The use of external ionization, such as a fast electron beam or an x-ray source, allows one to control the ionization degree and E/N, independently. If an EBSD is implemented for pumping gas lasers or in plasma-chemical reactors, the excitation degree is not negligible and can exert an influence on electron transport in rare gas mixtures. It means that the values of ionization degree (α) and of electronic excitation degree δ = N^*/N(N^* is the number density of excited atoms) are correlated and both have an influence on the EEDF shape and electron transport coefficients. To get a deeper insight into the mechanisms of NDC formation, it is convenient to analyze roles of these parameters as independent ones. The authors of [57] have carried out detailed numerical studies on the NDC in He–Xe mixtures. It should be emphasized that the set of electron scattering cross sections used in [57] enable an accurate reproduction of swarm data [56]. The influence of He percentage, α and δ on the NDC was analyzed. Figure 9 demonstrates that at low He percentage Coulomb collisions extend the E/N range with the NDC at δ = 0. At [He] > 58% the NDC does not exist. Figure 10 shows the region of the NDC existence on the plane E/N − α (at δ = 0). Starting from α = 2 × 10⁻⁹ the E/N range with the NDC appears and grows at [He]⩽10% including the limit of pure Xe. It was revealed that the disappearance of the NDC at high ionization degree is associated with a change in the EEDF shape at energies in the range of the R–T minimum. It was found that e–i collisions do not exert an influence on the NDC range.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Electron collisions with excited atoms at a high δ change the shape of the EEDF. To take into account this effect the drift velocity of electrons is calculated as a function of the excitation degree δ = N^*/N, which is defined as a fraction of Xe atoms in the lowest metastable level. In the absence of relevant experimental data for a rare-gas-mixture discharge, such an approach gives an idea about the opportunity to find conditions for NDC realization.

The authors of [58] have revealed that the Coulomb collisions can induce the NDC in pure rare gases (Ar, Kr, Xe) at a sufficiently high ionization degree α. These results are surprising since it was previously believed that the NDC occurs only in mixtures of rare gases possessing R–T minimum in q_m with gases introducing electron energy losses [14, 41–42, 54–55]. Figure 11 shows W(E/N) dependences calculated for pure Xe for a few values of α and δ. The processes taken into account were the following: loss and gain of the electron energy in the elastic scattering from atoms, excitation of electronic states and ionization by electron impact from the ground state, stepwise excitation, second kind collisions with excited atoms, e–e and e–i collisions. It should be stressed that in pure Xe the NDC effect appears even at δ = 0 (curve 2 in figure 11). It can be explained by an influence of Coulomb collisions on the EEDF shape.

Zoom In Zoom Out Reset image size

At low energies, collisional energy relaxation at energy up to the R–T minimum in the momentum-transfer cross section is very slow. Therefore, at lower energies, the EEDF is almost constant. Due to the u⁻² dependence of the Coulomb collision cross sections, e–e collisions are most effective at low electron energies. Hence, e–e collisions drive the EEDF toward the Maxwellian in the low-energy region. Then, this region with the quasi-Maxwellian EEDF extends to higher energy with α growing. This change in the EEDF shape is accompanied by diminishing mean energy, and the growth of W may be explained in two ways: as a result of an increase in the integrand (−(∂f₀/∂u)) in formula (7) for W or due to decrease in q_m at the mean energy (see figure 2). The NDC takes place, if the increment in W is greater at the lower electric field. Numerical calculations demonstrate that this is the case. According to our calculations, for a high degree of ionization, α = 10⁻⁵, the EEDF is close to the Maxwellian in the energy range of a few eV. This is also the range that contributes the most to the integral in (7). In this limit, W is a monotonically growing function of E/N.

Figure 11 demonstrates that in the presence of excited Xe atoms the NDC is more pronounced. Their role, because of the relatively low threshold for the stepwise excitation process Xe^* + e → Xe^** + e (1.16 eV), is similar to the role of molecular additives. The influence of second-kind collisions with excited atoms on the NDC was found to be insignificant [58].

By numerical calculations, the regions in the α–E/N parameter space were identified where the NDC exists. The calculations were made for Xe (figure 12), Kr and Ar (figure 13) [58].

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In figures 12 and 13 results for δ = 0, δ = α and δ = 10 α(δ is the fraction of atoms on lowest metastable state) are demonstrated. The populations of higher electronic states were neglected.

The role of electron–ion (e–i) collisions was also studied. It should be noted that in contrast to e–e collisions, where electron momentum is conserved, the e–i collisions cause a notable change in the electron momentum, while their impact on the electron energy distribution is much less. We therefore neglect electron energy change in the e–i collisions. The momentum-transfer cross section for the e–i collisions in Ar is about 10⁻¹¹ cm² at u = 0.25 eV resulting in the electron momentum relaxation rate of same order as at the electron–atom collisions at α = 10⁻⁷. Calculations show that for Xe and Kr, the role of e–i collisions can be neglected, while for Ar these collisions reduce the regions for NDC existence. To illustrate this effect, curve 1 in figure 13 relates to the case when e–i collisions were neglected. All other calculations were made including the effect of e–i collisions.

Let us discuss the data presented in figures 12 and 13 in more detail. We have found that NDC behavior at variation of α and δ is similar for Xe and Kr. Therefore, we focus our attention on the comparison between the NDC effect in Xe and Ar. The greater the electric field strength, the greater the characteristic electron energy and higher ionization degree is necessary for the e–e collisions to compete with the electron–atom collisions. The high sensitivity of the NDC effect to molecular additives makes it difficult to identify the NDC effect in pure Ar, Kr and Xe associated with the e–e collisions. The difference in the ranges of E/N where NDC can be observed (compare figures 12 and 13) is explained by the difference in positions of the minimum of the momentum-transfer cross section for Ar and Xe (see figure 2). The smaller size of the NDC region for Ar in comparison with Xe is due to more gradual growth of the momentum-transfer cross section with energy.

Comparison of figures 12 and 13 reveals a new unexpected feature: deformation of the NDC region with the increase in metastable state populations for Xe and for Ar proceeds in different ways. The broadening of the NDC region with N* growth for Xe can be easily explained by electron energy loss in stepwise excitation of atoms, which is similar to inelastic collisions with molecular species. To understand why the same collisions in Ar give opposite results, it is necessary to look at the processes in more depth. The R–T minimum for Ar is much deeper and located at a lower energy than for Xe. As a result, the e–i collisions come into play at much lower α, and effectively modify the momentum-transfer cross section near the minimum.

Let us discuss whether it is possible to experimentally observe the predicted NDC in pure rare gases. The EBSD allows one to change gas pressure, E/N and electric current density independently. The first observation of discharge current oscillation in the EBSD in 'pure' Ar [24] was explained by the authors as a manifestation of the Gunn effect. Our calculations for conditions of this experiment predict that the NDC exists at E/N < 0.6 Td. Estimations of E/N value from the measured discharge voltage and discharge spacing gave the E/N ≈ 5–6 Td. Such a discrepancy between predicted and estimated values of E/N might be explained by the presence of impurities in Ar.

It is well known [59] that in the glow discharge there exist regions with rather low electric field. These regions are the negative glow and the Faraday dark space. The electron kinetics in these regions is quite similar to electron kinetics in EBSDs. Ranges of plasma parameters where one may expect appearance of the NDC in pure Ar and Xe are shown in figures 10–13. Current oscillations associated with the NDC reflect formation of high-field layers in plasma, which propagate from the cathode layer to the anode. Conditions for their formation depend, among others, on the length of the plasma layer with the NDC. It means that the local existence of NDC is a necessary but not sufficient condition for the appearance of electric current oscillations.

The possibility of observing NDC phenomena in pure Xe, as induced by Coulomb collisions, is searched in detail by self-consistent numerical modeling [60]. The self-consistency of the model gives credence to the predictions. It was shown that the NDC can be observed experimentally in the EBSD in pure Xe provided the admixture of N₂ concentration is lower than 10 ppm.

This terminology corresponds to the situation when the variation of the EEDF shape is much faster than the variation of the electron number density. To the best of our knowledge, the transient negative electron mobility was introduced for the first time in [65, 66]. In these papers the time evolution of the EEDF in an electric field in Ar, Kr and Xe was studied theoretically for the limiting case when the initial EEDF was approximated by the delta function located at a given energy u₀. It was shown that at the appropriate choice of u₀ the negative electron mobility in xenon appears within a short time interval. Later [67], a similar effect was predicted in calculations of the EEDF relaxation in Ar plasma after switching off the external electric field. In contrast to papers [65, 66], the EEDF at the switching off moment was taken in [67] equal to the steady-state EEDF in the initial external electric field. In this case, the initial f₀(u) is not the inverse one, and the inverse EEDF is formed in the course of the relaxation process. It was demonstrated that for the electric field E/N ⩾ 0.15 Td electron mobility in argon is negative in a time interval during EEDF relaxation.

The author of [67] performed calculations for Ar and indicated that the transient negative electron mobility should take place in Kr and Xe, as well. Calculations for Xe were performed in [68, 69], and it was shown that after the electric field stepwise switching from relatively high (5 Td, 1 Td) to a low (0.01 Td) E/N value there exists a time interval where the electron drift velocity is negative. Transient negative electron mobility in Xe was also predicted in [70], where initial EEDF was assumed to be a narrow Gaussian distribution with a center at u = 15 eV.

To illustrate the situation described above we have calculated the time variation of the EEDF in homogeneous low-ionized Ar plasma under conditions when the electric field is switched by a single step from E/N = 1 Td to E/N = 0.01 Td. Calculations were performed by numerically solving time-dependent BE. The following processes were taken into account: loss and gain of the electron energy in the elastic scattering from atoms and excitation of electronic states. The momentum-transfer cross section was taken from [36] and cross sections for the excitation of electronic states were taken from [71]. Electron concentration was assumed to be constant during EEDF relaxation and quite low, so the Coulomb collisions were not taken into consideration. The calculated EEDF is shown in figure 16 for various time moments. In the course of evolution, the shape of the EEDF changes significantly. There exists a large time interval where f₀(u) is the inverse one. To give an explanation for this effect let us recall that the q_m for Ar atoms increases in magnitude in the energy interval 0.25–10 eV (figure 2). At the initial moment, many electrons have relatively high energies (>1 eV). These electrons gradually lose their energy in elastic collisions, and, as a result, the probability of the elastic scattering falls down, which means a decrease in the rate of energy losses. For this reason, the flux of electrons along energy axis from high to low energies results in the formation of the inverse EEDF. Figure 17 shows the calculated electron drift velocity as a function of time. One can see that the drift velocity is a non-monotonic function of time and becomes negative within a time interval (0.06–0.35 ms).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The influence of e–e collisions on the EEDF shape and the value of transient electron mobility was studied in [68] for the case of Xe. It was shown that the transient negative mobility could be realized at the ionization degree of plasma α ⩽ 10⁻⁸.

In experiments [61], the ANC was detected on a nanosecond time scale in relaxing Xe plasma (gas pressure P = 20 atm) produced by a short (10 ns) x-ray pulse (figure 18). Electric field applied across the plasma was as low as 1.16 × 10⁻³ Td. The observed effect was explained in terms of transient negative electron mobility [72]. It was shown that the behavior of the measured plasma conductivity and calculated μ_e(t) as functions of time are rather similar in a case when the initial f₀(u) in calculations is taken as a delta function at u₀ = 0.864 eV.

Zoom In Zoom Out Reset image size

In paper [73] transient ANC was theoretically predicted for Xe : Cs mixtures. The authors suggested ionizing Cs atoms by a laser pulse with an appropriate wavelength, which is simpler than an x-ray pulse.

Figure 15 demonstrates that in some electronegative gases the electron attachment rate grows at lower electron energy. Such behavior is beneficial for the formation of the inverse EEDF and to the ANC. Experimental conditions for the appearance of the ANC are expected to be realized in the EBSD in rare gases with admixtures of electronegative gases such as those shown in figure 15. A number of papers are devoted to searching conditions for realization of the ANC in the EBSD (Ar : CCl₄ [74, 75], Ar : F₂ [76–78] and Xe : F₂ [79–80]).

Electron energy spectrum in a gas irradiated by a beam of fast electrons stretches from zero to the energy of fast electrons. The low-energy range of this spectrum, i.e. the range of energies from 0 to ionization potential of atoms or molecules (I), is of a particular interest to finding conditions for the ANC. Respectively, in this subsection the term EEDF will be used precisely for this range of the electron energy spectrum. For uniform and steady-state plasma BE for the EEDF can be written as

$$\begin{eqnarray} &&\fl qS(u)+\sqrt {\frac{2}{m}} \frac{Nn_{{\rm e}}}{3}\left( {\frac{eE}{N}} \right)^{2}\frac{{\rm d}}{{\rm d}u}\left( {\frac{u}{q_{{\rm m}} (u)}\frac{{\rm d}f_{0}}{{\rm d}u}} \right) \nonumber\\ &&+\sqrt u St(n_{{\rm e}} f_{0} )=0. \end{eqnarray} \tag{ 10 }$$

The term qS(u) is the source of electrons in the energy range 0 ⩽ u ⩽ I with a spectrum $S(u) (\int_0^\infty {S(u)} \,{\rm d}u=1)$ and total production rate q. This term accounts for ionization of atoms and molecules by fast electrons.

For the first time, the inverse EEDF and the negative plasma conductivity in the EBSD were theoretically predicted in [74–75] for the case of the Ar : CCl₄ mixture. The EEDF and plasma parameters were calculated by solving equation (10) by taking into account loss and gain of the electron energy in the elastic scattering from atoms and molecules, inelastic processes (excitation of electronic states of Ar atoms and CCL₄ molecules, electron attachment to CCL₄ molecules) and electron–electron collisions. Spectrum S(u) was calculated using a fast electron degradation spectrum in argon [81]. The main channel in plasma electron losses is electron attachment to CCl₄ molecules. The EEDF and electron concentration were calculated in a self-consistent manner.

It was shown in [74] that, under certain conditions, the EEDF in Ar + 0.1% CCl₄ plasma is inverted, but the possibility of the existence of negative plasma conductivity was not discussed. The ANC in the EBSD was then studied in [75] for the same gas mixture. Figure 19 shows the EEDFs calculated in [75] for various q values at E/N = 0. One can see that EEDFs have the pronounced maximum at energies ∼1.5 eV. A decrease in f₀(u) in the energy range u < 1.5 eV is explained by the electron attachment to CCl₄ molecules. Calculated electron drift velocity and electron current density as functions of E/N value are shown in figures 20(a) and (b). One can see that electron drift velocity and current density are negative in the range of E/N from 0 to a certain value, which depends on the partial concentration of CCl₄ molecules and on the production rate of electrons. As the electric field increased, the negative electron mobility vanishes. This effect can be explained by a flattening of the EEDF with electric field growth.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In the Ar : CCl₄ discharge, negative ion number density can essentially exceed electron number density. Plasma conductivity is negative, if the negative electron current exceeds (by magnitude) positive and negative ion currents. According to our estimations for the conditions of curves 2 and 3 in figure 20(b) at E/N < 0.4 Td conductivity of plasma is negative. This prediction is not yet verified experimentally.

In conclusion, so far there was no experimental observation of negative electron conductivity in the EBSD. In [82] the electron number density was measured by laser interferometry at wavelength 9.6 µm in Ar : F₂ electron beam-sustained discharge in a wide range of parameters (percentage of F₂ admixture, electron beam current density and applied electric field). The electron drift velocity values were calculated as a ratio of the discharge current to electron number density and the elementary charge. These measurements showed that electron drift velocity is positive in the whole range of parameters under study. In [77] the range of parameters for the existence of ANC in Ar : F₂ EBSD was theoretically determined and compared with that considered in [82]. It was found that the predicted region of the ANC appearance on a plane (n_e, %F₂) touched closely the region with experimental data but had no common part. The authors of [78] revealed that the appearance of the ANC for Ar : F₂ is very sensitive to the shape of the Ar transport cross section. Using two different versions of Ar transport cross sections ([36] and [83]), in the first case, the ANC existence was predicted, while in the second case the ANC was absent at the same EBSD parameters. Present analysis with Ar transport cross sections from [36] allows us to conclude that in a whole range of n_e, %F₂ parameters explored experimentally [82] showed the ANC does not exist.

In [84] the negative electron mobility was predicted for pure Xe EBSD. The authors claimed that the steady-state inverse EEDF is formed due to the dissociative recombination of low-energy electrons with ${\rm Xe}_{2}^{+}$ ions. In the following paper [75] it was shown that taking into account e–e collisions results in the disappearance of the ANC under conditions studied in [84]. It is noteworthy that the omission of e–e collisions in the model of [75] leads to the appearance of the ANC under conditions analyzed in [84].

This subsection is devoted to studies of decay of homogeneous plasma of heavy rare gas with electronegative admixture in the presence of external electric field. The initial electron and positive ion densities are taken equal to each other. Plasma with such initial conditions can be created by a short ionizing pulse. If the applied electric field on the post discharge phase is sufficiently low, the rate of electron attachment to electronegative molecules is higher than the ionization rate, so plasma starts to decay. In this case, electron density is decreasing and, simultaneously, the EEDF shape deviates from the initial one.

In this case, it is convenient to analyze the evolution of function F(u, t) = n_e(t)f₀(u, t) with the initial condition F(u, 0) = n_e(0)f₀(u, 0). In a plasma decaying in a steady-state electric field, after a certain time interval, the EEDF shape is established, and electron number density exponentially diminishes with time. As a result, the function F(u, t) is factorized as F(u, t) = n_e(t) × f₀(u), where n_e(t) ∼ exp(−ν_att), and ν_at corresponds to the attachment frequency calculated from the established f₀(u). Such a scenario for F(u, t) evolution was confirmed in [78, 85–88]. For a given E/N, f₀(u) and ν_at do not depend on the initial conditions. It was shown that under certain conditions established f₀(u) has the inverse form, and electron mobility is negative (Ar : NF₃ [85, 86], Ar : F₂ [78, 87, 88]). Various mathematical techniques were implemented in these studies: solving the BE in two-term approximation [78, 85–87]; Monte Carlo simulations [87] and solving the BE in the eight-term approximation for the distribution function [88].

It is worth mentioning [89, 90], where the role of negative electron mobility under RF electric field conditions was studied by the Monte Carlo method in decaying Ar : F₂ plasma.

For illustration purposes, figures 21 and 22 show the established f₀(u) and drift velocity in decaying Ar : F₂ = 1 : 0.005 plasma calculated in [87] by solving the BE. In the computations, the following processes were taken into consideration: loss and gain of the electron energy in the elastic scattering from atoms and molecules, the excitation of the vibrational levels and electronic states of F₂ molecules; and the electron attachment to F₂ molecules. Figure 22 demonstrates quite unexpected behavior W as a function of E/N. An increase in drift velocity in the range E/N from 0.01 to 0.053 Td is followed by an abrupt transition from a positive value at 0.053 Td to a negative one at 0.055 Td. In the range from 0.055 to 0.55 Td the drift velocity is negative, and its minimum value is about −5 × 10⁴ cms⁻¹. For E/N > 0.55 Td the drift velocity becomes positive again and increases with E/N.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Such an unusual dependence of electron drift velocity on the electric field was explained in [86, 87] in the following way. The plasma electrons can be divided into two groups: electrons with energies u < u_m, where u_m = 0.08 eV is the energy at which the electron attachment cross section is maximum (see figure 15) and electrons with energies u > u_m. Particle exchange between these groups proceeds through a number of collisions resulting in high probability for the electron to be attached. As the plasma decays, the electron density in every group decreases almost independently, and the shape of the established distribution function f₀(u) is governed by the ratio between the decay rates of these electron populations. If the electric field is weak, the rate of the electron loss in the first group is low since the attachment cross section is small in the low-energy range. Owing to elastic and inelastic collisions the energy of electrons from the second group becomes lower than 0.4 eV. Since the attachment cross section is large in the energy range u_m < u < 0.4 eV the electron loss rate is very high. As a result, the established f₀(u) is governed by electrons from the first group. In this case f₀(u) is a monotonically decreasing function (see curve 1 in figure 21). The corresponding value of the electron drift velocity is positive.

As the electric field increases, the mean electron energy grows and the attachment for the electrons from the first group rises, since the attachment cross section increases with energy in the range u < u_m. For E/N > 0.55 Td the rate of electron loss for the first group is higher than that for the second group. In this case, the established f₀(u) is mainly governed by the electrons from the second group and has an inverse form (see curves 2–3 in figure 21). As a result, drift velocity becomes negative. With further electric field increase the electrons from the first group acquire sufficient energy (during their lifetime) to overcome the attachment barrier in the energy space. In this case, a weakly inverse or monotonic function f₀(u) is formed (see curves 4–5 in figure 21), and the drift velocity becomes positive.

In decaying plasmas, the electron number density falls down because of the electron attachment to molecules. Hence, the total negative plasma conductivity can be observed provided the electron number density is sufficient for dominance of electron current component. Moreover, it was shown in [78] (for Ar : F₂) and [86] (for Ar : NF₃) that evolution of the EEDF shape is rather sensitive to the detailed shape of the initial EEDF, f(u, 0). It was concluded that the best way to observe the negative plasma conductivity in experiments is to use a high-voltage short pulse in order to form the initial EEDF with a high temperature.

In a SST experiment a steady flux of electrons is emitted from the cathode and electrons move in the uniform electric field between electrodes. The electron concentration and the shape of the EEDF change with distance from the cathode. In the case of electronegative gas and sufficiently low electric field strength electron concentration decreases with distance due to attachment of electrons to electronegative molecules. The space variation of function F(u, x) = n_e(x)f₀(u, x) can be found by solving the steady-state BE with the boundary condition F(u, 0) = n_e (0)f₀(u, 0). In this case, the EEDF shape is established at some distance from the cathode, the function f(u, 0) is factorized, and the electron number density falls down exponentially with coordinate x:

$$\begin{equation} F\left( {u, x} \right)=n_{{\rm e}} (x)\times f_{0} (u),\;\;\;n_{{\rm e}} (x)\sim {\rm exp}(-\gamma_{{\rm at}} x). \end{equation} \tag{ 11 }$$

Here γ_at is the attachment coefficient at the established EEDF shape. Such a scenario for F(u, x) spatial evolution was predicted in [91] by Monte Carlo calculations.

In this limit, the energy dependent factor, f₀(u), satisfies the steady-state BE for the function [92]:

$$\begin{eqnarray} &&\fl \sqrt {\frac{2}{m}} \frac{uf_{0}}{3Nq_{{\rm m}}}\gamma_{{\rm at}}^{2} +\sqrt {\frac{2}{m}} \frac{N}{3}\left( {\frac{eE}{N}} \right)^{2}\frac{\partial}{\partial u}\left( {\frac{u}{q_{{\rm m}}}\frac{\partial f_{0}}{\partial u}} \right) \nonumber\\ &&+\sqrt {\frac{2}{m}} \frac{eE}{3N}\gamma_{{\rm at}} \left( {\frac{u}{q_{{\rm m}}}\frac{\partial f_{0}}{\partial u}+\frac{\partial}{\partial u}\left( {\frac{uf_{0}}{q_{{\rm m}}}} \right)} \right)+\sqrt u St(f_{0} )=0.\nonumber\\ \end{eqnarray} \tag{ 12 }$$

Integrating (12), the following equation for the attachment coefficient reads

$$\begin{equation} \gamma_{{\rm at}}^{2} D+\gamma_{{\rm at}} W-N_{{\rm at}} k_{{\rm at}} =0. \end{equation} \tag{ 13 }$$

Here N_at is the number density of an electronegative additive, k_at is the attachment rate coefficient.

By solving equations (12) and (13) it was shown [91] that in an Ar : NF₃ mixture under SST conditions the inverse EEDF can be formed and electron mobility can be negative (see figures 23, 24). The following processes were taken into account in calculations: loss and gain of the electron energy in the elastic scattering from Ar atoms and NF₃ molecules, the excitation of the electronic states of Ar atoms, the excitation of the vibrational levels of NF₃ and the attachment of electrons to NF₃ molecules.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The dip in the EEDF at low-energy (see figure 23) was explained in [91] as follows. For pure Ar the EEDF at low E/N is established as a result of exact compensation of forward and the backward flows of electrons along energy axis. The forward flow is due to energy gain in the electric field, and the backward flow is due to its loss in elastic collisions with atoms. The result is the monotonic EEDF (see dashed curve in figure 23). The NF₃ admixture introduces the attachment process, the cross section of which has a maximum at an energy value of about 1.5 eV (see figure 15). It is instructive to consider the EEDF formation in vicinity of zero energy. Evidently, the heating of electrons by the electric field results in electrons moving these to a higher energy range. Both, electron heating by the electric field and elastic collisions are conservative processes with respect to electron number density. However, when electrons approach an energy value, where the attachment is probable, they disappear due to the attachment processes. This disturbs the balance of forward–backward electron flows along energy axis. Namely, the backward flow diminishes. Due to the conservative nature of the energy gain-loss processes, this leads to electron number density diminishing the with zero energy.

If the electron energy is relatively high, the probability of electron attachment is comparable, or higher, than the probability for removal of electrons from the energy interval by virtue of elastic collisions. As a result, the EEDF with a lack of low-energy electrons is formed. The location of the EEDF's maximum on the energy axis depends on both the value of the reduced electric field and the NF₃ concentration.

It was also demonstrated that at all NF₃ concentrations under analysis W is a non-monotonic function of E/N. Moreover, in all of the cases, a range of E/N parameter values exists where the drift velocity is negative (see figure 24). Under SST conditions the flux of electrons from the cathode to the anode is formed by two processes: drift and diffusion. The total electron flux is controlled by the so-called diffusion-modified drift velocity V = W + γ_atD (see [92–94]). The authors of [91] proved that V ⩾ 0 at all reasonable values of plasma parameters. In other words, the negative drift velocity W is compensated by the electron diffusion caused by descending spatial profile of electron number density. As a result, the plasma conductivity is positive.

In experimental work [95] the Q-factor of a microwave resonator filled with a Xe : Hg : Cs : CO gas mixture was measured. The Q-factor was found to increase when the gas mixture was ionized by UV radiation. In [95], the increase in a Q-factor was attributed to the negative conductivity of the plasma created in the resonator. However, in [96] this statement was disproved.

In later papers [97–102] the possibility of negative electron mobility in optically excited plasma was analyzed theoretically by solving the BE. The following gas mixtures were considered: Na : Ar (Xe, Kr) [97, 98], Na : N₂ : Ar (Xe, Kr) [98, 99], Li : N₂ : Ar (Xe, Kr) [100–101]. In these papers, the EEDF and related plasma parameters were calculated for a high concentration of excited Na* (or Li*) atoms in plasma. The atom excitation degree, δ = [Na*]/[Na] (or [Li*]/[Li]), and ionization degree of plasma, α, were considered as independent parameters (except for paper [102]). The specific feature of the electron energy balance at low E/N values in these plasmas is that electrons gain the energy mostly in the second-kind collisions with excited atoms:

$$\begin{equation} {\rm Na}({\rm 3p}^{{\rm 2}}{\rm P}^{{\rm o}}) +{\rm e}\to {\rm Na}({\rm 3s}^{{\rm 2}}{\rm S})+{\rm e}(+{\rm 2}.{\rm 1}\,{\rm eV}), \end{equation} \tag{ 14 }$$

$$\begin{equation} {\rm Li}({\rm 2p}^{{\rm 2}}{\rm P}^{{\rm o}})+{\rm e}\to {\rm Na}({\rm 2s}^{{\rm 2}}{\rm S})+{\rm e}(+{\rm 1}.{\rm 8}\,{\rm eV}). \end{equation} \tag{ 15 }$$

For a fixed excitation degree, processes (14) and (15) are steady sources of electrons with energies u > 2.1 eV and u > 1.8 eV, respectively. It was shown in [97–102] that in certain ranges of parameters (E/N, δ, α and the partial concentration of Na or Li atoms), calculated EEDFs are inverse ones, and electron mobility is negative.

The predicted effect exists at very low ionization degrees (a < 10⁻¹⁰–10⁻⁹) and this is in contradiction to the required high excitation degree of plasma (δ ⩾ 0.1). The possibility of the creation of steady-state plasma with such characteristics is rather slim. In [102], time evolution of electron concentration and conductivity in optically excited Li : N₂ : Xe plasma was calculated starting from the exposure to radiation at zero electron concentration. It was shown that conductivity is negative at the earlier phase of the electron multiplication and becomes positive as soon as n_e exceeds a certain value.

For a limit of the Maxwellian EEDF, the bi-stability was predicted in [103] (see figure 26). An object of study in [103] was plasma of rare gases possessing R–T minimum in momentum-transfer cross sections. It was found that, in a certain range of E/N values, the right-hand side of equation (16) has three roots. Taking into account the striking similarity between the electron temperature behavior for Ar, Kr and Xe, further discussion concerns Xe gas. The respective H(T_e), L(T_e) and Φ(T_e) functions calculated for pure Xe at E/N = 0.035 Td and gas temperature T = 300 K are shown in figure 27. The Φ(T_e) curve is just of the type shown in figure 25. The N-type curve for H(T_e) in figure 27 is a result of crossing two wavy curves for L(T_e) and Φ(T_e) functions. Wavy behavior in the opposite direction of L(T_e) and Φ(T_e) curves is explained by variation of q_m around R–T minimum. The point is that L(T_e) is proportional to q_m(u), while H(T_e) varies proportionally to [q_m(u)]⁻¹. The competition between these two terms derives the bi-stability phenomenon.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In [104] the bi-stability effect was predicted for positron swarms in He. The positron energy distribution function was approximated by the Maxwellian one.

In fact, approximation of the realistic EEDF by the Maxwellian can be approved under rather specific conditions. Evidently, just e–e collisions drive the EEDF to the Maxwellian function. Moreover, e–e collisions introduce non-linear terms into the BE [2]. It seems probable that this non-linearity may result in the appearance of the bi-stability effect. A possible way to find numerically different stable solutions to the BE at fixed plasma parameters is to solve time-dependent BE (relaxation method) with different initial f₀(u). If the iteration method is used to solve steady-state BE, then different initial f₀(u) should also be exploited. It is convenient to take as an initial f₀(u) the Maxwellian with a given electron temperature T_e0, with the following variation of its value in a series of calculations.

The bi-stability effect in plasma of heavy rare gases was studied in [105] by solving the BE by taking into account loss and gain of the electron energy in the elastic scattering from atoms, excitation of electronic states by electron impact and e–e collisions. This effect was revisited by numerically solving the BE [106, 107] with the use of updated data on momentum-transfer cross sections [36]. It follows from [105, 106] that, in a certain range of parameters (namely, the reduced electric field, E/N, and the degree of ionization, α), the BE has two stable solutions. It means that in such a plasma two different steady states with different plasma parameters can be realized.

The EEDFs in Xe plasma calculated at E/N = 0.035 Td and different α values [107] are shown in figure 28. According to calculations, for 10⁻⁷ > α > 10⁻¹⁰ there are two stable solutions to the BE. Let us note that for the ionization degree α = 10⁻⁹ two calculated f₀(u) functions (curves 3 and 3' in figure 28) differ markedly from the Maxwellian, while for α = 10⁻⁷ EEDFs are close to the Maxwellian.

Zoom In Zoom Out Reset image size

Electron drift velocity in pure Kr and Xe calculated at low electric fields are shown in figures 29(a) and (b), respectively. As follows from these figures, bi-stability effect takes place for a > 10⁻¹⁰ in the E/N intervals E/N ≈ 0.027–0.043 Td (Xe) and E/N ≈ 0.01–0.015 Td (Kr).

Zoom In Zoom Out Reset image size

Electric field strengths where the bi-stability effect has been found are rather low, so that the rate of electron-impact ionization is negligible under bi-stability conditions. Experimentally, such conditions can be realized, for example, in decaying plasma in the low electric field. In [103] the time variation of the current in xenon afterglow plasma with applied probing electric field was measured. It was observed that at some point there is a jump-like decrease in the current value (t = 0.6 ms, see figure 30). The authors of [103] consider this abrupt decrease as the manifestation of the bi-stability effect.

Zoom In Zoom Out Reset image size

In the alternating electric field the bi-stability leads to hysteresis in cyclic time variation of electron drift velocity (and, respectively, current). In [106] the electron drift velocity in Xe under the influence of the alternating electric field was studied by numerically solving time-dependent BE. The amplitude of the electric field (E₀ = 0.06 Td at P = 1 Torr and T = 300 K) was chosen in order to cover the E/N interval in which bi-stability takes place (see figure 29(b)). Under these conditions n_e varies slowly in comparison with the electric field variation, and the ionization degree was taken as a time-independent parameter.

Figure 31 shows the established time variation of the electron drift velocity within half a cycle, calculated for P = 1 Torr, α = 10⁻⁸ and different electric field circular frequencies, ω. It is seen that for ω ⩽ 10³ rad/s an abrupt increase in drift velocity took place at the cycle phase ω t ≈ 0.35π at growing electric field. This abrupt increase corresponds to the transition from the lower to upper branch of W_e(E/N) dependence, when E/N reaches a value about 0.042 Td (see curve 4 in figure 29(b)). The transition from the upper to lower branch of W(E/N) dependence, which occurs when decreasing electric field reaches a value of about 0.025 Td, is remarkably smooth and does not have any peculiarities. It was pointed out in [106], that the difference between lower-to-upper and upper-to-lower transitions is explained by the specific shape of the momentum-transfer cross section of Xe.

Zoom In Zoom Out Reset image size

According to our estimations, under considered conditions effective frequency of the EEDF relaxation, ν_u, is about 2 × 10³ s⁻¹. If the electric field frequency ω≫ν_u (and ω ≪ ν_m), then the EEDF and electron transport coefficients vary slightly within the period, and their values averaged over the period are close to those in the steady the electric field with $\tilde{{E}}/N=\left( {E_{0} /\sqrt 2} \right)/N$ [108]. If the effective $\tilde{{E}}/N$ value is that at which bi-stability takes place, then there could be two different solutions to time-dependent BE at ω ≫ ν_u. Calculations performed for Xe (P = 1 Torr, T = 300 K, α = 10⁻⁸) under alternating electric field conditions E(t)/N [Td] = 0.05^*sin(ωt) show that at ω < 2 × 10³ rad/s there is a unique established solution to time-dependent BE. At ω > 2 × 10³ rad/s two different established dynamic solutions are observed. To illustrate this fact, figure 32 shows the calculated mean electron energy (averaged over a period) as a function of the electric field frequency. It is noteworthy that under alternating electric field bi-stability appears just at the electric field frequency about the frequency of the EEDF relaxation, ω ∼ ν_u.

Zoom In Zoom Out Reset image size

In the studies described above the electron number density was considered as an independent parameter. Since the bi-stability effect has been found at rather low electric fields, the situation considered can be realized in the decaying plasma or in non-self-sustained discharge. This opportunity was examined in [109, 110], in which the EEDFs in plasma of steady-state EBSD in heavy rare gases (Xe [109] and Ar, Kr [110]) were calculated.

The BE for the EEDF in plasma of steady-state EBSD was discussed in section 4.2. Under conditions considered in [109, 110] the secondary electrons lose their energy in elastic and inelastic collisions and disappear when recombining with positive molecular ions ( ${\rm Xe}_{2}^{+}$, ${\rm Ar}_{2}^{+}$, ${\rm Kr}_{2}^{+}$), which are the dominant ions in plasma. In [109, 110] recombination processes were taken into account in BE, and the EEDF and the electron concentration were calculated in a self-consistent manner.

It was found for Xe [109] and Kr [110] that in a certain range of q (production rate of electrons in the energy range 0 ⩽ u ⩽ I) and E/N values two different stable plasma states exist, the effect is more pronounced in the case of Xe. For an Ar plasma [110], the BE has a unique solution over the entire parameter range under study.

Figure 33 demonstrates calculated mean electron energy and plasma electron current density for non-self-sustained discharge in Xe at atmospheric pressure as functions of E/N [109]. Within the interval of q values 10¹⁴ cm⁻³ s⁻¹ < q < 10¹⁷ cm⁻³ s⁻¹ the bi-stability is clearly seen. The difference in discharge currents in the bi-stability range is almost ten-fold, and can be easily measured by common diagnostics. Production rate q = 10¹⁶ cm⁻³ s⁻¹ corresponds to the electron beam current density ≈2µAcm⁻² with the energy of fast electrons ∼300 keV [109]. The electron beam with such parameters can be easily realized.

Zoom In Zoom Out Reset image size

A possible experimental study is to measure the ampere–volt characteristics of the EBSD in Xe under described conditions. The required magnitudes of the voltage applied across the discharge gap should provide the E/N values in the range 0.01–0.06Td. When the bi-stability exists, the hysteresis effect should be observed manifesting itself as stepwise changes of the discharge current at certain values of the discharge voltage.

In [111] an unusual time variation of the electron temperature in Ar+1% N₂ post discharge plasma was observed (see curves 2 and 3 in figure 34) and an explanation was given in terms of the EEDF bi-stability effect.

Zoom In Zoom Out Reset image size

In experiments [111] the discharge ran in a cylindrical tube of 3 cm diameter in a pulse repetition mode with pulse current I = 200–700 mA. Pulse duration was 40µs, and the pulse repetition frequency was 1 kHz. The EEDF and the vibrational temperature, T_v, of N₂ molecules were measured at various delay times after the discharge pulse. The electron temperature was estimated from measured EEDF. Moreover, the time dependence of the electron concentration was estimated. Results of measurements performed at P = 0.5 Torr are shown in figure 34. It was found that measured T_v value is almost independent of pulse current and no noticeable decrease in T_v occurs during the post discharge period. In contrast, the electron temperature evolution strongly depends on the value of pulse current. For the pulse current 700mA T_e gradually decreases with time (see curve 1 in figure 34(a)). At lower pulse currents a rather sharp bend appears in the T_e(t) curves (curves 2 and 3 in figure 34(a)).

A specific feature of the EEDF formation in nitrogen (and nitrogen containing gas mixtures) afterglow is that heating of electrons is due to the super-elastic collisions of electrons with vibrational-excited molecules [11, 112–118]. Then it is the strong coupling between the vibrational excitation degree of the nitrogen molecules and the EEDF shape. For this reason, at first thought the jump-like T_e(t) variation is in contradiction with the constant T_v(t) value (see figure 34(a)). The conditions under study in [111] are quasi-stationary for the formation of the EEDF. Therefore, the origin of T_e(t) changing is the variation of the plasma parameters influencing it. Of these parameters only electron number density shows noticeable time dependence in the afterglow (see figure 34(b)). The n_e value determines the frequency of e–e collisions, and these processes influence the EEDF formation in the afterglow plasma (see, for example, [116]).

A theoretical study [111] was made by numerically solving the steady-state BE for the EEDF. The following processes were taken into account: loss and gain of the electron energy in the elastic scattering from atoms and molecules, excitation and de-excitation of rotational, vibrational and electronic states of nitrogen molecules by electron impact, and e–e collisions. The distribution of nitrogen molecules over vibrational levels has been supposed to be a Boltzmann one with a given temperature T_v. The electron concentration was taken as an independent parameter. The population of all electronically excited states except $A^{\,3}\Sigma_{{\rm u}}^{+}$ was neglected. Some results of calculations are shown in figures 35(a) and (b).

Zoom In Zoom Out Reset image size

For a given T_v the calculated T_e strongly depends on the electron concentration (see figure 35(a)). In the case of high electron concentrations T_e is close to T_v and slightly decreases as n_e falls down. Then the sharp transition from high to low T_e takes place at some critical n_e value. The lower the vibrational temperature the higher the critical n_e value and the more abruptly the transition occurs. Moreover, for lower T_v (for example, T_v = 3000 K, see figure 35(a)) the range of n_e exists, in which there are two different stable solutions to the BE (see figure 35(b)). The possibility of bi-stability appearance is connected with comparatively narrow maximum in the cross sections for energy exchange between electrons and molecular vibrations.

Electron density decays in the afterglow. Then the jump-like T_e(t) variation observed in experiments can be explained as a transition from high to low T_e values after the electron density falls down to some critical value. The experimental results described can be considered as implicit evidence of the possibility of the bi-stability effect in Ar : N₂ afterglow plasma.

The possibility to the observe the bi-stability effect in pure nitrogen afterglow plasma was discussed in [117–119].