Electron concentration in the non-luminous part of the atmospheric pressure filamentary discharge

Optical emission spectroscopy (OES) is commonly used for non-invasive plasma diagnostics. When good resolution is available, the spectral line broadening can provide important information, e.g. about gas temperature or particle densities. As the contributions of various broadening mechanisms to spectral linewidth are different for each spectral line, some lines are more suitable for this task than others. Electron collision induced Stark broadening of H_α and H_β lines of hydrogen Balmer series is commonly used for electron density measurements in plasmas containing hydrogen [35–37].

Gigosos–Cardeñoso (GC) model [38] evaluates the Stark broadening (FWHM value) of H_β emission line as a function of n_e by a simple relation

$$\begin{equation}{\Delta}{\lambda }_{\mathrm{S}}=4.8\enspace \mathrm{n}\mathrm{m}\cdot {\left(\frac{{n}_{\mathrm{e}}}{1{0}^{23}\enspace {\mathrm{m}}^{-3}}\right)}^{0.68116}.\end{equation} \tag{ 1 }$$

However, other broadening mechanisms, such as instrumental, natural, Doppler and van der Waals broadening, must be taken into account to get correct Stark FWHMs from a measured spectral line profile [36]. During this, the deconvolution of the experimental Voigt profile into its Gaussian and Lorentzian parts is needed. Respective contributions of other mechanism mentioned below were calculated and subtracted from experimental profile in a standard way [41, 42], leaving only the half-width corresponding to Stark broadening which was used for calculation of the electron density.

Random thermal movement of the radiating particle causes a Doppler shift, giving a Gaussian broadening profile with width [35]

$$\begin{equation}{\Delta}{\lambda }_{\mathrm{D}}=7.16{\times}1{0}^{-\mathrm{7}}{\lambda }_{0}\sqrt{\frac{{T}_{\mathrm{g}}}{M}},\end{equation} \tag{ 2 }$$

where λ₀ is the centre of the line, M is the ratio of atomic mass of the emitter and atomic mass unit m_u and T_g is gas temperature.

Impact pressure broadening by neutral particles (van der Waals broadening) has Lorentz profile with simplified width as calculated for Ar–H₂ mixture [35] at atmospheric pressure

$$\begin{equation}{\Delta}{\lambda }_{\mathrm{W}}=5.521{\times}1{0}^{-\mathrm{5}}\enspace \mathrm{n}\mathrm{m}\cdot \frac{p}{{T}_{\mathrm{g}}^{7/10}}\quad \left(\mathrm{P}\mathrm{a};\mathrm{K}\right),\end{equation} \tag{ 3 }$$

where the T_g is gas temperature and p is pressure.

One can estimate T_g to be equal to a rotational temperature T_rot of molecular emission bands. In our case, the C₂ Swan system [40] was used.

Instrumental broadening, which has Gaussian profile, is independent of the plasma properties. It can be estimated by measuring spectral linewidth of the selected line close to H_β under conditions when other broadening mechanisms are negligible [37]. In our case, a mercury calibration lamp is used and the broadened profile is approximated by the Gaussian with a FWHM of 0.024 nm because other broadening mechanisms are below 1 pm.

The natural broadening is commonly [37, 39] neglected due its low value, in order of 10⁻⁵ nm.

The free electron density, n_e, influences the propagation speed of electromagnetic (EM) waves in plasma via the complex relative plasma permittivity ɛ_pl, which depends on probing frequency ω. Usually, the specific formula for plasma induced phase shift in microwave interferometry experiments is derived [20] under certain assumptions, which are normally taken as granted (infinite homogeneous plasma slab, perpendicular incidence transmission experiment, planar travelling wave only, Maxwellian EEDF, constant collision frequency, etc.)

However, real experiments are often rather far from these assumptions. In our case, the first three (geometric) assumptions above are not valid at all. Our plasma has a form of over-dense (over-critical) thin filament of active discharge, surrounded by a lower density (sub-critical) region. The probing wavelength is longer than the plasma filament diameter. The probed volume is in near-field [43] of the waveguides, i.e. the waves are non-planar, even non-TEM. Reflections, evanescent waves and scattering must be included. In effect, a full EM field solution is needed.

Fortunately, the two following assumptions about plasma properties are not so problematic. Ideally, the collision frequency ν_m should be independent of particle velocity, which is also linked to the steady-state Maxwellian EEDF (electron energy distribution function). In real plasmas and high frequency approximation (ω ≫ ν_m) the deviation from the ideal solution is rather negligible, even when the collision frequency depends on velocity and/or EEDF is non Maxwellian [44, 45]. Therefore, one can use the classical formula for the complex relative plasma permittivity

$$\begin{equation}{\varepsilon }_{\text{pl}}=1-\frac{{n}_{\mathrm{e}}{e}^{2}}{{\varepsilon }_{0}{m}_{\mathrm{e}}}\frac{\omega -i{\nu }_{\mathrm{m}}}{\omega \left({\omega }^{2}+{\nu }_{\mathrm{m}}^{2}\right)},\end{equation} \tag{ 4 }$$

where e is the elementary charge, ɛ₀ is the permittivity of vacuum, m_e is the mass of the electron, ω is the angular frequency of probing microwaves and ν_m is the collision frequency for electron-neutral momentum transfer. In collision-less case, the simplified formula ${\varepsilon }_{\text{pl}}=1-\frac{{n}_{\mathrm{e}}{e}^{2}}{{\varepsilon }_{0}{m}_{\mathrm{e}}{\omega }^{2}}=1-\frac{{\omega }_{\text{pl}}^{2}}{{\omega }^{2}}$ with ω_pl being the plasma frequency, directly suggests that for over-dense plasmas, plasma permittivity ɛ_pl would become negative and wave propagation would be impossible.

The more general formula (4) is relevant for both the transparent and over-dense plasma [46], which is crucial in our case. From the wave propagation point of view, one can thus formally describe the plasma as a dielectric (with complex refractive index ${n}_{\text{pl}}=\sqrt{{\varepsilon }_{\mathrm{r},\mathrm{p}\mathrm{l}}}$) even outside the common notion. The plot of such analytic continuation of the refractive index for a typical range of plasma parameters divided by angular frequency of the probing wave is plotted in figure 1.

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Typical [47, 48] magnitude for ν_m for atmospheric pressure discharges is several 10¹⁰ s⁻¹ up to a few 10¹¹ s⁻¹, which means for our frequency 34.5 GHz that ν_m/ω will typically be around 0.1–0.5. While the high frequency assumption ${\omega }^{2}\gg {\nu }_{\mathrm{m}}^{2}$ would be better satisfied for even higher frequencies [49, 50], the squares in the inequality make these values acceptable.

As mentioned above, for dense plasmas, one must consider the wave scattering. With the probing wavelength similar to the scatterer dimensions, the Mie [51] formula exists as an analytic solution for several simple geometries. For our case, the 2D Mie scattering on a cylinder seems to be an obvious choice. Nevertheless, the analytical solution can not fully describe the experiment, as the incident waves are not necessarily planar, the plasma filament does not have exact boundaries, etc. Still, the results of the numerical EM model of scattering on a conductive cylinder in figure 2 are remarkably similar to analytic Mie predictions.

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In the simplest model A (homogeneous dense plasma filament, no plasma surroundings), the only variable parameters are the filament diameter and its plasma density (geometry of the reactor and the interferometer being fixed). A typical pattern of the resulting EM field is shown in figure 7.

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For a rather crude model A, the rough estimation of diameter (2r = 0.7 mm) as experimentally observed FWHM of light emission lateral intensity profile can be considered sufficient. The model then provides figure 8, a dependence of phase shift on filament plasma density. Perhaps unsurprisingly (as the plasma is dense and the interaction region is very small), the graph shows that, across several orders of magnitude of n_e, the predicted phase shifts are below the instrumental error (which is approx. 1°), exceeding this threshold only for very high plasma densities.

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While the absolute value would seem to approach experimentally observed phase shifts at extremely high plasma densities, the sign of these phase shifts is wrong. Moreover, such densities are usually only observed immediately at the plasma nozzle [36], not farther downstream. The results of the model A are therefore dubious at best.

In order to assess them further, the Stark broadening measurement was conducted (as explained above, hydrogen atoms were present due to decomposition of ethanol admixture). At low flowrates of precursor (0–0.5 sccm EtOH) the H_β line was dominant compared to the molecular Swan C₂ band nearby and their overlapping was negligible. Therefore, H_β could be easily used for electron density estimation. Results are in agreement with previous measurements on the same experimental set-up and theoretical model made by [36]. The plasma density was approximately n_e ∼ 10²¹ m⁻³ at the tip of the nozzle. At position 15 mm downstream, where the waveguides are placed, the GC model [38] would yield approximately n_e ∼ 2 × 10¹⁹ m⁻³. While majority of the broadening models were intended for electron densities above 10²⁰ m⁻³ [38, 64, 65] it was shown [66] that the GC model can be used above 4 × 10¹⁹ m⁻³ for H_β . Below this density, the fine structure should be introduced. As it was not included in our measurement and therefore, n_e value is underestimated [37]. However, even this underestimated value is higher than the critical density for 34.5 GHz signal, and so the scattering of the probing microwave beam on the plasma filament is truly important.

The wrong sign of the phase as calculated by the homogeneous model A means that the inhomogeneous model B (which also includes the regions with subcritical plasma) is absolutely necessary.

A determination of the radial profile of plasma density in atmospheric pressure plasma jet can actually turn out quite challenging, both experimentally or theoretically. It is due to large gradients, existence of instabilities and generally insufficient understanding of physics and dynamics of the filamentary plasmas (qualitative or particular so far) [29, 67–69]. Unfortunately, as the probing microwave beam is relatively large, the interferometer itself cannot be used to perform laterally resolved measurements and extract the radial profile (although there are some pioneering attempts to do so [70]). Therefore some other means of profile determination are necessary.

Naively, the easiest option is relating the profile of electron density to that of relative light emission intensity. However, it is clear that the relation between excited species and electron densities is not simple, at least due to the role of electron energy distribution function, which is a priori unknown. The second option is to rely on established methods such as Stark broadening, Thomson scattering, etc. Finally, a fully self-consistent plasma model could be theoretically developed, but the inclusion of all the peculiarities (e.g. column constriction, instabilities, etc) of the atmospheric pressure filamentary discharges is generally still beyond state of the art in plasma modelling.

Stark broadening is an obvious choice for spatial profiling, as it was done and discussed before by many investigators, e.g. [30, 36, 37]. It should be noted that problems mentioned in section 5.1 for averaged plasma density are even worse for its spatial profiling, as the n_e radially decreases and so gets easily out of GC model validity range. In combination with generally lower intensity of H_β emission farther from the axis, the dependable results are limited only to the central region of the filament, not permitting to profile outer region of the filament and especially not the filament surroundings. Assuming that the profile (at least its shape) remains roughly the same along the plasma column, one can carry out the measurement closer to the nozzle, where the electron density is sufficient and the GC model remains appropriate. As an example, a typical lateral electron density profile and light emission profile covering the centre of the filament is shown in figure 9 for axial position 5 mm (i.e. much closer to the nozzle than the position 15 mm of microwave interferometry measurements).

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The light emission profile nicely follows the Gauss curve. For electrons, the interpretation is more questionable, as the shoulders of the spatial profile are experimentally inaccessible. Fortunately, among papers investigating the spatially resolved electron density of atmospheric jet filament, one [31] was measured directly on our experimental device at similar experimental conditions using Thomson scattering. This method is considered superior as it can yield more precise results with decent spatial resolution and lower sensitivity limit. Their published colour maps can be recalculated back to actual profiles, shown in figure 10.

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The bulk of the filament (consider the logarithmic n_e scale) is well described by the Gauss curve, also in agreement with [30]. However, farther from the axis, there is a noticeable deviation from the Gauss curve as the plasma density decrease is slowed down. Also, the data from a self-consistent model, recently developed for free-expanding surfatron discharge simulation [71] suggests that this effect is real and not some unreliability of Thomson scattering for low electron concentrations.

Nevertheless, as the bulk of filament is Gaussian, the model B considers only this analytic shape and ignores any 'slowing down'. The spatial distribution of n_e for varying radius r is therefore parametrised as

$$\begin{equation}{f}_{\text{Gauss}}\left({n}_{\mathrm{e},\mathrm{c}},G\right)={n}_{\mathrm{e},\mathrm{c}}\cdot {e}^{-{\left(r\right)}^{2}/G},\end{equation} \tag{ 6 }$$

where n_e,c (abbrev. for centre) is the central (peak) value of the electron concentration and parameter G controls the width. A series of numerical simulations was run, for several values of G around the value estimated from experimentally observed filament half-widths. The results of these simulations are shown in figure 11.

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Compared to model A, there is some improvement of sensitivity, which has to be attributed to the shoulders of the spatial distribution. Significantly, the interferometric phase shifts according to the model B can be either positive or negative. This depends on which process (the scattering on the over-critical part or the propagation in sub-critical part) dominates. More specifically, during model prototyping and iterating, it became apparent that: (i) the mechanism of over-critical and sub-critical plasma region contributions may be as simple as shown in figure 12 and (ii) the positive contribution of the over-critical central part of the filament is often (smaller radii, lower n_e,c) insignificant. It follows that, in our case, the Gauss function width parameter is not critical and can be safely estimated to be equal to e.g. that from light emission without introducing a significant error.

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Yet again, these numerical results do not reach the experimentally observed phase shifts (<−10°).

Unsatisfactory results of the previous models A and B (as compared to the experimentally observed phase shifts) suggest the existence of extended shoulders of the electron concentration spatial profile. It should be noted that despite much lower electron density than in the central maximum, these shoulders may still play a globally significant role in transport and electron collision processes purely due to geometrical volume factor (scaling with r²).

The Gauss function (while satisfactory for the active filament itself) has shoulders decreasing too fast. However, as it was remarked earlier, during the discussion of figure 10, the reference data indeed hinted on a composed spatial distribution—Gaussian in central bulk and some slower decreasing function in its surroundings. Therefore, the model C includes both these components. Since the results of previous models suggest that interferometric phase shifts are rather insensitive to the innards of the dense plasma filament, many parameters (degrees of freedom) of the model can be fixed, simplifying the modelling and subsequent analysis greatly. More specifically, while the filament presence is critical for the scattering and must be included, the exact width of the filament is not.

The task is then mostly reduced to choosing the appropriate function for the electron concentration profile outside the active plasma region. While the class of possible functions is infinite, we can at least fix its most general properties—it should be continuous, monotonously decreasing and have boundary conditions (it should fall towards zero on the outer boundary, it should have a continuous transition from the Gauss function on the inner radius). Physics, practicality and even certain aesthetics prefer that this function should be 'reasonable'.

The experimental viewpoint is that linearly decreasing shoulders in semilogarithmic plot (figure 10, left) suggest that a simple exponential function (7) could be a suitable candidate. The theoretical viewpoint is that solution of the diffusion equation in cylindrical coordinates is a first order Bessel function (8). We a priori do not know the correct spatial distribution function, but these two candidates are actually quite representative. The exponential is rather fast decreasing, giving prominence to the region in close vicinity of the Gaussian active filament, while the Bessel function is quite flat and so, due to the volumetric r² factor, it effectively accentuates regions farther off-axis.

Among many (again, infinitely many) possibilities of gradual transition near filament edge, we chose to do it by simple stitching the central Gaussian and the outer distribution function together at a certain radius r₀ from the centre. So, for r ⩽ r₀ the spatial distribution is Gaussian (6), while for r > r₀ the profile is exponential or Bessel

$$\begin{equation} {f}_{\mathrm{exp}}\left({n}_{\mathrm{e},\mathrm{s}},F\right)={n}_{\mathrm{e},\mathrm{s}}\cdot {e}^{-\left(r-{r}_{0}\right)/F}\end{equation} \tag{ 7 }$$

$$\begin{equation}{f}_{\text{Bes}}\left({n}_{\mathrm{e},\mathrm{s}},R\right)={n}_{\mathrm{e},\mathrm{s}}\cdot {J}_{0}\left(2.405\left(r-{r}_{0}\right)/R\right).\end{equation} \tag{ 8 }$$

The main parameter of both profiles is the electron concentration n_e,s at radius r = r₀, where they are stitched to the central Gauss profile. The second parameter for Bessel profile (8) is its radius (first zero) R. The second parameter for the exponential profile (7) is its characteristic length F. An initial guess of n_e,s (or equivalently, of r₀) can be made from the limiting condition for surface wave driven discharges [72]. For 2.45 GHz driving power frequency, the n_e,s should be approx. 10¹⁸ m⁻³ (when accounting for collisions). Schematic illustration of both compound profiles is in figure 13.

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It can be argued that also other properties (concave vs convex, (in)equality to zero at a certain distance, etc) of these two functions effectively make them quite distant in the class of permitted profile functions. Therefore, while we are aware of our self-inflicted choice, most conclusions valid for both of these functions should be relevant for other reasonable candidate functions, including the real (albeit unknown) profile.

Finally, the sets of computationally predicted phase shifts for these two compound profiles are plotted in figure 14. There, the half-width of the central filament profile (possible main source of non-linearity) was fixed at 0.35 mm, while different values of n_e,c were tried. For the Bessel profile, the values R = 16 mm and R = 77.5 mm correspond to the distance of waveguides from centre and reactor radius, respectively.

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Following observations can be made: (i) computed phase shifts both qualitatively and quantitatively match the experimentally observed phase shifts, (ii) the central maximum n_e,c influence is noticeable only for the smallest phase shifts (confirming predictions made in section 5.2), (iii) the phase dependence is almost perfectly linear in n_e,s, F and R parameters (giving a solid chance to interpolate and extrapolate these data as needed) and (iv) for quite different shoulder profiles, the results are almost comparable. Therefore, as a more general result, one may conclude that the model C gives good results, the central over-critical filament plays a negligible role despite non-zero skin depth and the actual interferometry takes place in spatial profile shoulders, but the exact choice of shoulder profile is not essential. The insensitivity to shoulder profiles can also be interpreted as that from the microwave interferometry point of view—the true measure of the electron presence being not their exact spatial profile but the integral amount of electrons in the cross-section.

There are two more practical points of this final model C to discuss—a potential azimuthal anisotropy of the spatial profile and a possible influence of experimentally induced geometrical errors.

By default, the computed profiles were considered azimuthally isotropic, despite the presence of the metallic waveguides, where the increased surface recombination could result in anisotropy. Fortunately, in most cases, the electron concentration at the waveguide openings is negligible compared to that in the rest of the modelled domain. The exception being the wider Bessel profile (R = 77.5 mm taken as the full reactor radius). There, the zero of the Bessel profile along the microwave beam should be at the distance of waveguides from centre (16 mm), while in the perpendicular direction, the zero of the profile would occur much farther, at the reactor wall (77.5 mm)—for reference, see figure 3. Modelling has shown that results for such anisotropic profile are almost identical to the isotropic case with zero at 16 mm, and therefore are not listed separately.

Obviously, some mechanical and physical imperfections of the experimental procedure can never be avoided altogether. It is, therefore, important to evaluate the typical cases, e.g. waveguide misalignment due to loose parts of the set-up or plasma filament bending slightly away from the axis, etc. The possibility of such mishaps is the reason why monitoring snapshots of the critical region (plasma and waveguides) were acquired during each measurement run, theoretically permitting us to numerically compensate ex post for such unwanted movements. Fortunately, however, the computations show that these common alignment errors are not critical for the output phase (error under approx. 15%).

The experimental part of this paper has two equally important purposes. Firstly, to determine the electron concentration in plasma torch and its dependence on ethanol abundance during plasma chemical synthesis of graphene. Secondly, to test suitability of numerically enhanced microwave interferometry method for this type of plasma (forming an over-critical filament at atmospheric pressure). In both these tasks, the Stark broadening of H_β was used, too. However, it was found (as discussed above) that Stark broadening is not concurrent but a complementary method. The spectroscopy is more suitable for active dense filament, while the interferometry detects lower electron concentration in the dark surrounding region.

As the main experimental parameters, the microwave power and ethanol admixture (diluted by auxiliary Ar, see section 3) flow rate was selected. In figure 15, the estimation of electron concentration n_e,c from H_β Stark broadening is shown for various ethanol admixtures at constant microwave power. Naturally, the OES gives greater weight to more luminous parts, effectively measuring the active, bright filament only. One can observe that a higher amount of molecular admixture decreases the electron concentration due to heightened energy losses by excitation and dissociation of the precursor molecule and its fragments.

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Related to it is a change of the filament length. The current hypothesis [73] is that such discharges operate in surface wave driven mode, i.e. the excitation microwave power guides itself along the plasma/ambient boundary, gradually transferring its energy into the plasma. The net result is rather long and narrow plasma filament with electron density linearly decreasing with the axial position. The highest electron density is in the wave launching area close to the nozzle tip and the lowest is at the discharge end—the tip of the filament. The surface wave sustaining condition strongly links the plasma length and the microwave excitation power. On a more fundamental level, increased losses due to inelastic electron collisions with molecules (Ar has first excitation level at approx. 11.5 eV) decrease the available power, decreasing the ionisation rate, electron concentration and, by consequence, also the plasma column length.

Actually, the electron density axial profile as calculated from H_β Stark broadening, shown in figure 16, is not linear. It can be explained by the existence of non-linearities and losses by space radiation near the wave launching area (i.e. close to the nozzle). Although this deviation from classical simplified surface wave description is well known, it was recently revisited [73] and is currently under intensive research. Overlapping curves at larger axial distances suggest that amount of ethanol admixture (max 0.1% of total gas flow) does not substantially impact the electron density.

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The results above are values inside the active plasma filament. The microwave interferometry can provide the values of electron density in the dark filament surrounding. The single interferometric measurement consists of manually conducted two-step data acquisition (with and without reference arm), performed within a few seconds by careful adjustment of the attenuator knob. Moreover, the experiment was operated in a switch-on/switch-off regime in order to eliminate the effects of gradual heating of waveguide walls (as they expand, the propagation constant inside the waveguide decreases and phantom phase shifts could occur). Despite being subtle, this effect may add up over a long section of the (thermally conducting) waveguide. It forces us to carry out the interferometry while repeatedly switching the discharge on and off for each experimental setting.

Interestingly, the heating of the waveguides can serve as a validation factor for the measured phase shift sign. During the interferometer set-up and initial adjusting, the correct operation of a variable attenuator is easy to observe on the detector signal. However, with a variable phase shifter, it is not obvious, whether a positive or negative phase shift is being introduced (of course, there is a micrometre dial, but still its operation is error-prone). As the waveguide walls cool down after the plasma measurement, the waveguide walls contract and the phase velocity inside increases (leading to negative phase shift over time). On the contrary, if plasma density is in the expected negative phase shift region (as calculated in figure 14), switching off the plasma must yield a positive phase shift, opposite to that of cooling walls. Indeed, this was observed in the experiment, further confirming the results of the numerical model.

The raw phase shifts from microwave interferometry are shown in figure 17. These values have been compared to precalulated numerical results, giving stitching point electron density n_e,s as shown in figure 18 for different experimental conditions (excitation power, ethanol admixture). The precalculations obviously depend on the choice of a priori unknown spatial profile (stitching point, type of shoulder function, its parameter, etc.)

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Although results of all studied spatial profiles are shown in figure 18, some of them (depicted in red) are somewhat arguable: (i) the reactor wide (R = 77.5 mm) Bessel profile has problematic boundary condition (as already discussed in the paragraph concerning anisotropy in 5.3), as substantial recombination should occur already at the conductive waveguide openings. (ii) The fits of steeply decreasing exponential profile (F = 2 mm) to measured phase shifts yield values of n_e,s often well over 10¹⁸ m⁻³. This is higher than one would expect from knowledge that exciting frequency is 2.45 GHz with corresponding electron density of surface wave excited plasma around 1 × 10¹⁸ m⁻³ (with collisions, ν_m = 5 × 10¹⁰ s⁻¹). Intuitively, the stitching point should coincide with this threshold.

Unfortunately the price paid for indirectness of the method plus having too many free (and pseudofree) parameters is apparent. The uncertainty of the results stem from three main components: (i) experimental error of interferometric phase shift, including short term noise, long term drift and possible imbalance between diodes and their calibrations. We estimate this error to be around ±0.7° for single phase measurement, meaning the error from (two part) phase shift measurement should be approx. ±1°. This fixed value has obviously larger relative effect when the measured phase shift is small. (ii) Geometry errors. Possible lateral shifts, tilting, curving of the filament. Typically, the error should be below ±15%, according to modelling. (iii) Unknown electron density profile. Trying various profiles (shapes, central part electron density, stitching value, radius of main bulk) we arrived at mean error around ±30% of the n_e,s value (as can be roughly seen from the spread of respective black datapoints in figure 18). The error bars shown in graph are our estimation of the combined effect of causes (i)–(iii).

While this error hides possible trends, by comparing with figure 13 one can clearly see, that electron densities as high as n_e > 10¹⁷ m⁻³ extend well into the dark region around the plasma filament.

As discussed before, the total number of electrons seems to be a more dependable parameter than their spatial distribution. Integrating the dark region electron densities from figure 18 over the r, ϕ-plane results in figure 19. Here, the agreement for this total linear electron density (electrons per unit length cylindrical slice) among possible compound profiles is even better.

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As the choice of actual profile plays a lesser role, the error due to cause (iii) gets smaller, around ±15%. Still, the statistical spread is too large to reveal any definite trend in figure 19. It should be noted, that figure 15 depicting Stark broadening data does not provide any such trend, either. Again, the mean values around (6 ± 2) × 10¹³ m⁻¹ seem to be quite reliable.

In the bigger picture, results of all of these tested profiles support two main outcomes: (i) the concentration of free electrons in the outer filament region is in the order of 10¹⁷ m⁻³ for this type of plasma and, at least in the studied range, rather independent of the amount of admixture and discharge excitation power. These values are approximately 2–3 orders of magnitude lower than the electron concentration in active plasma. (ii) The results confirm that extent of these residual electrons is most likely significant (approximately 1 cm away from filament the values in higher order of 10¹⁶ m⁻³ still can be found) suggesting that volume recombination is not as dominant as one might expect outside the active plasma filament.