Effective ionization rate in nitrogen–oxygen mixtures

Detachment of electrons from negative ions at elevated electric fields mainly occurs in collisions of the ions with neutral molecules. The translational temperature of ions differs from the gas temperature due to the drift of ions in the applied electric field.

The ion drift velocity v_ion can be expressed through the reduced mobility, μ N, and the reduced electric field, E/N, as

$$\begin{equation} v_{\rm ion} = \left(\mu N \right) \left(\frac{E}{N} \right) . \end{equation} \tag{ 12 }$$

The mobility of ions is often assumed to be constant; however, it can significantly vary with electric field. In this work we are interested in the ion behaviour near the breakdown threshold; the values of ion mobility have been taken from [10] at E/N = 100 Td and presented in table 1.

Table 1. Reduced mobility μ N in (m V s)⁻¹ of selected negative ions in pure oxygen and dry air at E/N = 100 Td retrieved from [10].

	O−	${\rm O}_2^-$	${\rm O}_3^-$	${\rm NO}_3^-$
Oxygen	1.2 × 1022	5.0 × 1021	8.1 × 1021	—
Air	1.2 × 1022	7.1 × 1021	7.6 × 1021	6.6 × 1021 (in N2)

An analytical expression for the mean translational energy of ions E_ion has been originally derived by Wannier under the assumption of a constant mean free time between collisions [11, 12]

$$\begin{equation} E_{\rm ion} = \frac{3}{2} T_{\rm gas} + \frac{1}{2} \left(m_{\rm ion} + m_{\rm gas} \right) v_{\rm ion}^2, \end{equation} \tag{ 13 }$$

where m_ion and m_gas are the masses of the ions and gas molecules, v_ion is the ion drift velocity. Here and below, temperatures are considered in energy units, i.e. multiplied by the Boltzmann constant.

Relation (13) has been obtained for elastic collisions without considering charge-exchange collisions between ions and gas. This result is hardly applicable to the drift of ions in parent gas at elevated fields where these types of collisions are known to be dominant.

Fahr and Miller [13, 14] investigated the drift motion of ions when charge-exchange collisions dominate and derived velocity distributions for the extreme cases of very small (q_eEλ ≪ T_gas) and very high reduced electric fields (q_eEλ ≫ T_gas), where λ is the mean free path for charge exchange. At high E/N the ions' thermal motion becomes negligible compared with the drift. In this case, the velocity distribution f(v_z) in field direction z is given by

$$\begin{equation} f(v_z) = \sqrt{\frac{2 m_{\rm ion}}{\pi \theta_z}} \exp \left(- \frac{m_{\rm ion} v_z^2}{2 \theta_z} \right), \end{equation} \tag{ 14 }$$

where θ_z = q_eEλ is the effective ion temperature in field direction; the ion velocity is limited by v_z ⩾ 0. The drift velocity can be deduced from the ion velocity distribution as

$$\begin{equation} v_{\rm ion} = \int_0^\infty v_z f(v_z) \,{\rm d} v_z = \sqrt{\frac{2 \theta_z}{\pi m_{\rm ion}}}. \end{equation} \tag{ 15 }$$

The latter allows one to obtain the ion effective temperature in field direction from the drift velocity

$$\begin{equation} \theta_z = \frac{\pi}{2} m_{\rm ion} v_{\rm ion}^2 + T_{\rm gas}. \end{equation} \tag{ 16 }$$

Note that the last term related to gas temperature T_gas is artificially added to the result in order to get θ_z = T_gas limit at zero field. This allows one to use the relation (16) for the entire range of ion energies with a reasonable accuracy.

Similar to drift velocity, the ion mean translation energy expressed in terms of the drift velocity becomes

$$\begin{eqnarray} &&\fl E_{\rm ion} = \frac{m_{\rm ion}}{2} \int_0^\infty v_z^2 f(v_z) \,{\rm d} v_z + \frac{3}{2} T_{\rm gas} = \frac{\pi}{4} m_{\rm ion} v_{\rm ion}^2 + \frac{3}{2} T_{\rm gas},\nonumber\\ \end{eqnarray} \tag{ 17 }$$

that is approximately 20–25% lower than Wannier's original relation (13).

The calculated translation energies of the most important negative ions in nitrogen–oxygen mixtures (O⁻, ${\rm O}_2^-$, ${\rm O}_3^-$ and ${\rm NO}_3^-)$ are presented in figure 2. The obtained values for ${\rm O}_2^-$ in pure oxygen correspond to those obtained using direct Monte-Carlo modelling of ion transport [15]. Another example of good agreement has been demonstrated in [16] for argon. These results encourage us to extend this approach to other ions and gases.

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It is important to note that the relation θ_z = 2 E_ion between the effective ion temperature (16) and the mean energy (17) remarkably differs from the well-known ideal gas case T_ion = 2/3 E_ion.

Ion velocity distribution (14) allows us to calculate the constant rate k for a process with known cross section σ() as

$$\begin{equation} k = \int_0^\infty \sigma\left(\frac{m_{\rm ion} v_z^2}{2}\right) v_z f(v_z) \,{\rm d} v_z. \end{equation} \tag{ 18 }$$

For an inelastic process with the energy threshold Δ (activation energy), assuming linear variation of the corresponding cross section near the threshold and low gas temperatures (θ_z ≫ T_gas), the following Arrhenius form becomes valid

$$\begin{equation} k = k_0 \exp\left(- \frac{\Delta \epsilon}{\theta_z} \right). \end{equation} \tag{ 19 }$$

Here k₀ is energy-independent part of the reaction constant rate and θ_z is the ion effective temperature in field direction (16).

Strictly speaking, the use of reaction rate (19) is limited by relatively high electric fields when θ_z ≫ T_gas. A commonly used approach (see, for example, [9, 17]) recommends to use instead of θ_z the effective temperature of ion–neutral collisions T_eff defined as

$$\begin{equation} T_{\rm eff} = \frac{m_{\rm gas} T_{\rm ion} + m_{\rm ion} T_{\rm gas}}{m_{\rm gas} + m_{\rm ion}} . \end{equation} \tag{ 20 }$$

This relation, however, assumes isotropic distribution of the velocities of colliding species in space and its use for ion–neutral collisions is questionable. In fact, the motion of ions in parent gas exhibits significant anisotropy (14) with the dominant motion in the field direction. To adapt the above-mentioned relation, one could assume T_ion ≃ 2θ_z, that gives T_eff ≃ θ_z in the case of similar ion and neutral masses. This allows us to use the ion effective temperature in the field direction (16) for ion–neutral reaction rate approximations (19).

The main negative ion created in oxygen at elevated fields (above 20 Td at atmospheric pressure, figure 1) is O⁻. However, it is generally accepted that detachment of electrons in pure oxygen occurs mainly in collisions of ${\rm O}_2^-$ ions with O₂ molecules

$$\begin{equation} {\rm O}_2^- + {\rm O}_2 \Rightarrow \left({\rm O}_2^- \right)^* + {\rm O}_2 \Rightarrow \rme + {\rm O}_2 + {\rm O}_2 \end{equation} \tag{ 21 }$$

and the detachment rate from O⁻ ions in pure oxygen is almost negligible due to the fast reaction of charge transfer to molecules

$$\begin{equation} {\rm O}^- + {\rm O}_2 \Rightarrow \left({\rm O}_3^- \right)^* \Rightarrow {\rm O} + {\rm O}_2^- \end{equation} \tag{ 22 }$$

For the following analysis, we approximate the constant rate of reactions according to (19)

$$\begin{equation} k_{21} = k^0_{21} \exp \left(- \frac{\Delta \epsilon_{21}}{\theta_{{\rm O}_2^-}} \right) \end{equation} \tag{ 23 }$$

for (21), which matches the experiment [23] and the detachment rate calculated on the basis of the statistical approach [24] fairly well with the parameters $k^0_{21}$ and Δ₂₁ presented in table 2.

Table 2. Arrhenius approximation of reaction constant rates k = k⁰exp(−Δ/θ) (21), (22), (27) and k = k⁰ exp(−θ/Δ) (25).

	Proc.	k0	Δ (eV)
${\rm O}_2^- + {M} \Rightarrow {\rm e} + {\rm O}_2 + {M}$	(21)	(1.22 ± 0.07) × 10−11 cm3 s−1	0.78 ± 0.03
${\rm O}^- + {\rm O}_2 \Rightarrow {\rm O} + {\rm O}_2^-$	(22)	(6.9 ± 0.6) × 10−11 cm3 s−1	1.40 ± 0.14
O− + N2 ⇒ N2O + e	(27)	(1.16 ± 0.05) × 10−12 cm3 s−1	0.076 ± 0.005
${\rm O}^- + {\rm O}_2 + {M} \Rightarrow {\rm O}_3^- + {M}$	(25)	(1.3 ± 0.2) × 10−30 cm6 s−1	0.16 ± 0.05

Similarly, fitting the experiments [18–20, 25, 26], the charge transfer reaction rate (22) can be approximated by

$$\begin{equation} k_{22} = k^0_{22} \exp \left(- \frac{\Delta \epsilon_{22}}{\theta_{{\rm O}^-}} \right). \end{equation} \tag{ 24 }$$

Experimentally measured rates of both processes and corresponding Arrhenius rates are presented in figure 4. The fitted parameters are summarized in table 2 and will be used for the determination of detachment rate in air.

Detachment from heavy ions, such as ${\rm O}_3^-$, is hardly observed in oxygen. This ion is easily created at low fields in the following three-body associative reaction

$$\begin{equation} {\rm O}^- + {\rm O}_2 + {M} \Rightarrow \left({\rm O}_3^- \right)^* + {M} \Rightarrow {\rm O}_3^- + {M} \end{equation} \tag{ 25 }$$

when the translation energy of ions is not sufficient for effective collisional detachment (see table 1 and figure 2) and, thus, it can be considered as a stable negative ion. At higher fields, ${\rm O}_3^-$ is not created due to a fast charge transfer reaction similar to (22).

Experimentally measured reaction rate of (25) from [18–22] and its fitted approximation (table 2)

$$\begin{equation} k_{25} = k^0_{25} \exp \left(- \frac{\theta_{{\rm O}^-}}{\Delta \epsilon_{25}} \right) \end{equation} \tag{ 26 }$$

are both presented in figure 3.

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The constant rates k₂₁ (23) and k₂₂ (24) obtained for pure oxygen can be used for description of processes in air using corresponding mobilities for determination of ion temperatures. The detachment reaction rate obtained in this way for air and oxygen is presented in figure 5. As follows from the Figure, the detachment rate in air remains remarkably higher in the entire range of electric fields. It follows from the higher mobility (table 1) and, correspondingly, the higher energy of ${\rm O}_2^-$ ions in air (figure 2).

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An extra process, namely associative detachment, has to be considered in air:

$$\begin{equation} {\rm O}^- + {\rm N}_2 \Rightarrow \left({\rm N}_2{\rm O}^- \right)^* \Rightarrow {\rm N}_2{\rm O} + \rme. \end{equation} \tag{ 27 }$$

It is an exothermic reaction and it contributes to the detachment balance already at low electric fields. This process, however, has a barrier of 0.1–0.2 eV as was pointed out in [27]. In this study, the following approximation was used to fit the measurements [27, 28]

$$\begin{equation} k_{27} = k^0_{27} \exp \left(- \frac{\Delta \epsilon_{27}}{\theta_{{\rm O}^-}} \right). \end{equation} \tag{ 28 }$$

Experimentally measured and approximated constant rates for reactions (21), (22) and (27) are summarized in figure 6. Corresponding constant rates are taken according to (23), (24) and (28) and the corresponding ion temperatures (16) in air.

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The detachment rate from ${\rm O}_2^-$ ions (23) demonstrates fairly good agreement with measurements for E/N = 100–150 Td that corresponds to 1/θ = 2.5–5.0 eV⁻¹ inverse temperature range. As follows from the figure, experimental measurements at higher fields E/N = 150–200 Td (below 2.5 eV⁻¹) are in good agreement with the charge transfer reaction (22) which is the dominant process for O⁻ ions.

The reaction rates for O⁻ ions in air are presented in figure 7 as a function of reduced electric field. Three regions with different detachment mechanisms can be identified using figures 1 and 7. For low electric fields (E/N < 50 Td), the most important process leading to electron detachment is the associative detachment on nitrogen molecules (27), as expected. However, the dominant negative ion at atmospheric pressure at low fields is ${\rm O}_2^-$ which hardly contributes to detachment at these fields.

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For E/N in the range 50–100 Td, the dominant ion is O⁻ and the electron detachment (27) starts contributing to the overall production of electrons (figure 7). This process, however, remains relatively weak at elevated pressures in comparison with the three-body charge transfer to ozone (25).

At higher fields, E/N > 100 Td, the charge transfer (22) with consequent collisional detachment (21) is the dominant process. This can be schematically summarized in the following diagram:

Under conditions of constant electric field, ${\rm O}_3^-$ can be considered as a stable negative ion because its formation occurs only at low electric fields (figure 3). At higher fields (E/N > 70 Td), these ions convert rapidly into O⁻ and contribute to the main chain of electron detachment at high fields (double-line path).

In the relatively simply case of an avalanche development in oxygen, we can use the following reduced set of processes—ionization (5), two-body dissociative attachment (6) and collisional detachment (21) assuming that the ion conversion ${\rm O}^- \Rightarrow \O_2^-$ occurs much faster than the detachment (according to figure 4, the latter is correct for ion temperatures above approximately 0.3 eV). The following partial Townsend coefficients can be used:

$$\begin{eqnarray} &&\fl \alpha = \alpha_{5} \quad \eta = \eta_{6} \quad \,\delta = \,\delta_{21} = k_{21} [{\rm O}_2] / v^{{\rm O}_2^-}_{\rm dr}\nonumber\\ &&\omega = \omega_{25} = k_{25} \left([{\rm O}_2] \right)^2 / v^{{\rm O}^-}_{\rm dr}. \end{eqnarray} \tag{ 35 }$$

The resulting effective ionization coefficient, as well as the ionization and attachment coefficients, in oxygen are presented in figure 8. The most pronounced additional production of elections due to the effect of electron detachment occurs in the range of low electric fields. At low fields, the attachment losses dominate over ionization and the commonly used net ionization coefficient α_eff is negative; that means no avalanche can develop at electric fields below 125 Td. In contrast, $\tilde{\alpha}_{\rm eff}$ in considering electron detachment goes to zero at considerably lower fields, i.e. avalanches can develop in under-critical electric fields where α − η < 0.

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Effective ionization coefficients in different gases, including oxygen, have been measured numerously. For instance, the measured results [32] and the review of six independent publications [2] are presented in the figure 8 for the range 100–190 Td. As follows from comparison, the measured values agree fairly well with the value of net effective ionization coefficient obtained in the present study assuming the gas pressure below 1 bar (that is usually the case for drift-tube experiments [2]). The effect of electron detachment decreases when pressure increases. However, the difference between α_eff and $\tilde{\alpha}_{\rm eff}$ remains remarkable even at pressures above 10 bar (figure 8).

The same approach can be applied for air (more generally, for a nitrogen–oxygen mixture) substituting

$$\begin{eqnarray} &&\fl \alpha = \alpha_{4} + \alpha_{5} \quad \eta = \eta_{6} \tqs \delta = \delta_{21}+\delta_{27} \quad \omega = \omega_{25}. \end{eqnarray} \tag{ 36 }$$

The result obtained in this way and experimentally measured values of the effective ionization rates in air [2, 32, 33] are presented in figure 9. So, the approximation proposed in [33] is an analytic expression matching the experimental values of various authors for dry air and for air under controlled humidity conditions. The results of this approximation are in good agreement with the effective ionization rate α_eff = α − η commonly accepted. At the same time, measurements [2, 32] give much higher values at lower electric fields and agree relatively well with the results of this work.

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In spite of significant scattering of experimental points at low electric fields (below 150 Td) reported by different groups, the general trend remains the same. A systematic measurement of the reduced effective ionization rate in air is reported in [2] for the gas density range 10⁻¹⁹–10⁻²⁰ cm⁻³ and the following fixed values of electric field: 101.5, 103.0 and 104.5 Td. These points and the results of current modelling are presented in figure 10. The maximum discrepancy between the both results reaches approximately two times at low densities and fields but it still remains below the typical uncertainty of experimental measurements at under-critical fields (see figure 9).

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