New procedure to estimate plasma parameters through the q-Weibull distribution by using a Langmuir probe in a cold plasma

In the present work, the characterization of a nitrogen plasma (N₂) at (2.00 $\pm$ 0.01) Torr (typical value used i.e., in surface treatments) was performed in a stainless steel chamber, powered by a DC potential of (500$\pm$2) V and with a discharge current of (0.455$\,\pm\,$0.005) A.

The voltage-current characteristic curve (I-V curve) was measured with a cylindrical probe constructed with a tungsten wire and a glass support in “V” shape (see Figure 1(a)). The dimensions of the probe are: diameter $d=(30\,\pm\,1)$ $\mu$m and length $l=(6.40\,\pm\,0.02)$ cm. We decided to use this relatively long probe only due to technical reasons, and other probes could also be used. The system under study consists of both a plasma and a probe system. The last one includes a specific probe and the measurement equipment. We have not studied in detail the influence of the probe system in the plasma conditions, and probably with another probe system the obtained curves could have been different. However, this work aims to describe a procedure to use the experimental data from the I-V curve to obtain a precise estimation of the plasma parameters of the whole system, and this procedure is planned to be applied with different probe systems.

The probe was placed at different heights ($h$) starting from the cathode, in order to obtain measurements in the zones of the cathode glow and the cathode dark space. The dimensions of the chamber and the height $h$ are shown in Fig. 2. A total of eight heights were considered at distances of 2.2, 3.4, 4.3, 5.1, 6.2, 7.3, 8.2 and 9.4 mm from the cathode. These positions were set by a stepper motor, which is able to move the probe vertically with enough precision.

A diagram of the probe measurement set-up is shown in Fig. 3. A power source, which is controlled by software, is connected to the probe. This source provides the current and voltage probe as outputs, i.e. the contribution of both electrons and ions. To measure the current, we use a Boxcar Averager SR 250 Gated Integrator, averaging over 200 measurements of the current in a fixed time window starting from 800$\mu s$ after each voltage step. On the other side, the voltage of the probe and the averaged current are measured by a Boxcar Averager SR 245 Computer Interface. Each I-V curve consists of a sweep in voltage of 100 points. Further details about the experimental set-up can be found in Ref. [40], where a similar configuration was used.

For each one of the eight heights considered in this work we carried out three sweeps, and we verified that I-V curves remained constant in time. The measurement results are shown in Fig. 4.

In this last paragraph, we mention the values of some useful parameters to identify the plasma operating regime for the measurements presented. In a neutral gas with $p=2$ Torr and $T=300$ K, we have $N_{g}=6.43\cdot 10^{22}m^{-3}$ molecules per unit volume, and by using the gas kinetic diameter ($d_{g}$), whose value for $N_{2}$ is[41, 42] $d_{g}$ = $3.64\cdot 10^{-10}m$, then the electron mean free path ($\ell_{m}$) can be calculated approximately within the framework of kinetic theory of ionized gases[43, 41, 44, 42] by

$$\ell_{m}=\frac{4}{\pi N_{g}d_{g}^{2}}\approx 150\mu m$$

(1)

The effective dimension of the cylindrical probe, $a$, can be defined by[5]

$$a=\frac{d}{2}\ln{\left(\frac{\pi l}{2d}\right)}\approx 120\,\mu{\rm m}$$

(2)

where $l$ and $d$ are the length and diameter of the probe, it is verified that $\ell_{m}>a$. Therefore, we conclude that the plasma is in the collision-free regime[5, 2]. The Debye length ($\lambda_{D}$) can be calculated by

$$\lambda_{D}=\sqrt{\varepsilon_{0}k_{B}T/e^{2}n}$$

(3)

where $k_{B}$ is the Boltzmann constant and $e$ the electron charge. Using the values of $T$ and $n$ estimated in this work, we obtain $\lambda_{D}$ in the range of $15-45\,\mu{\rm m}$. The plasma sheath thickness, $h_{s}$, is considered[45, 5] from one up to five times greater than the Debye length, finally it is verified that $\ell_{m}>a\gtrapprox h_{s}$ which implies the plasma is in the conventional thin sheath subregime[5].

The second derivative of the curve $d^{2}I/dV^{2}=I^{\prime\prime}$ is usually obtained by the procedure of filtering and numerically differentiating the I-V curve.

There are many possibilities to obtain this second derivative by using different techniques, ranging from analog solutions (e.g. an analog electronic circuit) to digital smoothing (e.g. a digital electronic circuit or a post-processing of the measured data in a software). In the digital smoothing technique, many alternatives are also available, the more commonly used[50] are: Savitzky–Golay filter, the Gaussian filter, polynomial fitting and the Blackman filter. In each of these possibilities, some parameters are needed and may not be obvious the optimal election of them. In this work in particular, we only use the Savitzky-Golay[51, 52] filter (SG), and its parameters are: (M) the degree of the polynomial and (n) the number of points (frame length).

Note that these numerical procedures involve all the measured data, and only after obtaining the second derivative, the restriction $V<V_{p}$ (see Sec. 3) is applied. This is one of the reasons of the effectiveness of the procedure. Figure 5 shows the smoothed $I^{\prime}$ and $I^{\prime\prime}$ curves obtained from experimental data by using SG. It is important to highlight again that there are many possibilities to obtain the second derivative using SG, and these different options lead to different plasma parameters. In the following, we consider:

• (1.) a SG smoothing of the I-V curve and a numerical differentiation to obtain the first derivative, then a new SG smoothing and a numerical differentiation to obtain the second derivative

• (2.) a two-step application of a SG onefold differentiation

• (3.) a one-step application of a SG twofold differentiation

• (4.) a SG smoothing of the I-V curve and an application of (2.) or (3.)

• (5.) a use of (2.) with an intermediate SG smoothing between the steps

We have implemented all these options also using different values to the parameters M and n. To select the optimal values to M and n we have analyzed the final smoothness of the second derivative, also the existence of values $I^{\prime\prime}<0$ which are undesirable (because the EEPF in semilog-scale would have undefined values), and also the plasma parameters $T$ and $n$ obtained from the EEPF. According to our measurements, we have found that options (1.) and (2.) provide similar results of $T$ and $n$ plasma parameters, with the optimal values M=2 and n=19 for the first derivative and M=2 and n=17 for the second derivative.

Figure 6(a) shows the EEPF calculated with Eq. (6) from $I^{\prime\prime}$ to the case (2.). There are some curves that show a “gap” between 1 and 1.5 eV, these are numerical artifacts.

To avoid such “gap” we have tried options (4.) and (5.) but without favourable results. Finally, we implemented option (3.) with M=2 and n=19 and that fixed the “gap” problem. Figure 6(b) shows the EEPF of the last case.

Now that we have solved the “gap” problem, there is another important point to consider: the variations of plasma parameters, principally $T$, with the different options mentioned. We have observed that $T$ is very sensitive to the options considered and even to the specific filter parameters chosen, to be more specific we have found that the different options and filter parameters affect the estimation of the plasma potential $V_{p}$ and this is traduced in a high variation of the T value. For example with option (2.) to the height $h$=2.2 mm, we obtain $V_{p}$=-25.06 V and T=4900 K, and with option (3.) to the same height we obtain $V_{p}$=-25.01 V and T=5750 K. Moreover, if we set $V_{p}$=-25.01 V (obtained from (3.)) and use this value with option (2.) we obtain T=5300 K, so this result shows the importance of the $V_{p}$ value. In the following we consider only option (3.) which does not present the “gap”. We evaluate the plasma parameters using the $V_{p}$ value calculated in two different ways: from the maximum value of $I^{\prime}$, and from $I^{\prime\prime}=0$. The $V_{p}$ value represents the starting potential which sets the 0 $eV$ in the EEPF, and it is also the inferior limit of the integration in Eqs. (7) and (8). When we use the criteria of $V_{p}$ defined as the maximum value of $I^{\prime}$, it is found a more precise estimation of $V_{p}$ because it requires only a unique numerical differentiation. However, the plasma parameters are obtained from the integration of the second derivative $I^{\prime\prime}$, so if we use the $V_{p}$ value obtained from the first derivative to integrate the second derivative we find an inconsistency in the chosen $V_{p}$ potential. Therefore, we consider it is better to use the second criteria, i.e. obtaining $V_{p}$ from $I^{\prime\prime}=0$, because we are being consistent both with the inferior limit of integration chosen and the argument of integration. Finally, the values $V_{p}$, $V_{f}$, $n$, and $T$ (using option (3.)) are summarized in Table 1, for each height considered. The floating potential $V_{f}$ is calculated from $I(V_{f})=0$ by the interpolation of the I-V curves. The uncertainties were obtained taking into account the sensitivity of the results observed using different SG parameters. Specifically, we changed the value of the frame length filter parameter (n), which is n=19 in the optimal case, to n=17 and n=21. Then we evaluate $V_{p}$, $n$ and $T$, and as a result we obtain a bound in the uncertainty of these parameters.

A last comment can be made concerning the depleted shape of the EEPFs in Fig. 6 for energies below 0.5 eV.

Below, we analyze our particular measurements with Nitrogen at a relatively high pressure (2 Torr) and a relatively low discharge current (0.455 A). We define $\Delta$ as the distance between $0$ eV to the maximum of the low energy peak, as it can be seen in Fig. 6 we obtain $\Delta\approx(0.3-0.6)eV$. From practical considerations as it is depicted in Ref. [47] it is useful to compare $\Delta$ with the electron temperature obtained from the I-V curves, and a “good quality experiment” should satisfy $\Delta<eT$. In Table 1, we obtain in the case of $T=5700$ K $\rightarrow\Delta=0.6eV>eT=0.49eV$, and for the lower temperature $T=3700$ K $\rightarrow\Delta=0.3eV<eT=0.32eV$, which implies we are at the limit of “good/not good quality experiment” based on this criterion. This depletion at low electron energies has been attributed in Ref. [47] to probe contamination and low input impedance in the probe system, and plasma density depletion around large probes and high gas pressures. In the case of our measurements, and based on Ref. [53], we consider that probably the predominant effects related to low energy depletion come from the large probe used ($L=6.4$ cm), which can cause an unbalanced electron energy and charge current distribution along the probe, from the high pressure which incentivizes the EEPF to have a more convex shape or to be Druyvesteyn-like, and from the nearness of the probe with the cathode which can favor the low energy electrons to accelerate and also the covering of the probe with the material evaporated by the cathode. Even if we figure out the precise physical mechanism of this depletion, it is known from Ref.[47] that a small “residual” value of $\Delta$ less than $0.5$ eV can not be avoided, probably as a consequence of electron reflection, secondary-electron emission, inhomogeneity of the probe work function along the probe collecting surface and the convolution effect usually accompanying differentiation. For practical considerations, it is common to extrapolate this low electron energy zone to obtain more precise values of the plasma parameters, as it is shown in Refs.  [47] and [54]. However, in this work, in view of the complicated and a priory unknown mechanisms which are involved, we decided to use the experimental I-V curves without modifications. We will obtain plasma parameters which will include the probe and plasma systems as a unique system, with all the previously mentioned effects included in the calculation. It is important to remark that with this procedure we will obtain different plasma parameters when different probe systems are used, and a more precise knowledge of the involved mechanism would allow us, in a future calculation, to correct the plasma parameters in each experimental set-up. This alternative point of view opens the door to studying plasma parameters with homemade probes and avoiding the use of sophisticated probe systems.

Recently, in 2020, Qiu et al. (Eq.(1) in Ref. [39]) obtained an expression for the current from the probe using the theoretical framework of non extensive $q$-statistics. They considered plasma behavior as Maxwellian and replaced the Boltzmann-Gibbs exponential distribution function by the $q$-exponential function; this yields an expression for the current where the $q$ value arises as a new plasma parameter. Based on Ref. [39] we propose to fit the experimental data with the function

$$I_{Q}(V;q^{*},A,B)=A\left(C_{q^{*}}\left(1+(q^{*}-1)B(V-V_{p})\right)^{\frac{1}{q^{*}-1}+\frac{1}{2}}-(1-(q^{*}-1)\frac{1}{2})^{\frac{1}{q^{*}-1}+\frac{1}{2}}\right)\;,$$

(9)

where the fitting parameters are $q^{*}$, $A$ and $B$. The variable $C_{q^{*}}$ depends on $q^{*}$ by

$$C_{q^{*}}=\frac{A_{q^{*}}}{q^{*}}\sqrt{\frac{m_{i}}{2\pi m_{e}}}$$

(10)

where $m_{e}$ and $m_{i}$ are the electron and ion mass, and $A_{q^{*}}$ is a $q^{*}$ dependent normalization parameter whose value is

$$A_{q^{*}}=\begin{cases}\sqrt{1-q^{*}}\frac{\Gamma(\frac{1}{1-q^{*}})}{\Gamma(\frac{1}{1-q^{*}}-\frac{1}{2})}&,\;-1<q^{*}\leq 1\\ \\ \frac{1+q^{*}}{2}\sqrt{q^{*}-1}\frac{\Gamma(\frac{1}{q^{*}-1}+\frac{1}{2})}{\Gamma(\frac{1}{q^{*}-1})}&,\;q\geq 1\end{cases}$$

(11)

The first term in Eq. (9) corresponds to the electron contribution and the second one to the ion contribution. In this equation, the last term is assumed to be a constant in respect to $V$, and it corresponds to the ion saturation current $I_{sat}$. In the notation of Ref. [39] the $A$ and $B$ fitting parameters are given by

$\displaystyle\begin{cases}A=en_{e}A_{p}\sqrt{\frac{k_{B}T_{e}}{m_{i}}}\\ B=\frac{e}{k_{B}T_{e}}\\ \end{cases}$

(12)

where $A_{p}$ is the probe area and $e$ is the electron charge.

Equation (9) is based on the following assumptions (textually cited from Ref. [39])

• •

  No magnetic field;

• •

  The plasma is thin, so the average free path of electrons and ions is much larger than the probe size (collision-free approximation);

• •

  The electrons obey the non-extensive distribution, and the ion temperature is very low ($T_{i}\approx 0$, cold plasma approximation);

• •

  The thickness of the formed sheath is much smaller than the probe size (order of several Debye length $\lambda_{D}$), thus allowing the use of plate approximation rather than relying on the geometry of the probe[55];

• •

  The emission of the secondary electron on the surface of the probe can be ignored, and the charged particles that hit on the probe do not react with the probe, namely the probe is an ideal absorber of charged particles.

Based on the measurements from this work presented in Sec. 2.2, we consider that all the items are satisfied. The fourth item deserves some discussion. As it was analyzed in Sec. 2.2, we have obtained $a\gtrapprox h_{s}$, i.e. the thickness is smaller than the probe size. The fourth item requires $a\gg h_{s}$, therefore, we are aware that the probe geometry can be a subject of future consideration. However, we will consider that Eq. (9) can be applied with our measurements.

By definition, Eq. (9) is valid for $V<V_{p}$ values. In order to find the plasma parameters using this approach, we have to ignore some part of the experimental data from the data set, specifically the $V>V_{p}$ values. To estimate the $V_{p}$ value, we again reside in a numerical procedure such as the one mentioned in Sec. 4.1. This procedure of ignoring some part of the experimental data, as we will see below, conduces to greater errors in the estimated plasma parameters compared to the SG filtering from Sec. 4.1.

To obtain parameters $A$, $B$ and $q^{*}$ from Eq. (9), we do the following procedure: 1. we use the values of $V_{p}$ obtained in Sec. 4.1, then 2. we plot Eq. (9) by evaluating Eqs. (12) with reasonable values on the right side, and compare it with the I-V curve to find the initial values for the fitting parameters for each I-V curve, finally 3. we use the initial parameters to find the convergence of the fit.

The fitted curves using Eq. (9) are shown in Fig. 7 (using only two heights from the cathode). Figure 8 presents the first and second derivatives of Eq. (9) evaluated with the fitted parameters for each height, also the corresponding EEPF in semilog-scale is shown in Fig. 9. The results are summarized in Table 2. The errors from the table were estimated from the uncertainty of the nonlinear fitting and from the uncertainty in the $V_{p}$ value which comes from the SG filtering, both contributing approximately the same value.

To obtain the EEPF, the analytic expression of $I^{\prime\prime}$ is used; this has the advantage, compared to the SG filter, of not having the numerical error given by the differentiation process. The expressions of $I^{\prime\prime}$ and the EEPF (obtained from Eq. (6)) are the following

$$I^{\prime\prime}_{Q}(V)=\frac{AC_{q^{*}}}{4}B^{2}(3-q^{*})(q^{*}+1)(1+B(q^{*}-1)(V-V_{p}))^{\frac{1}{q^{*}-1}-\frac{3}{2}}$$

(13)

$$f_{Q}(V)=\frac{4}{e^{3}S}\sqrt{\frac{m}{2}}I^{\prime\prime}_{Q}(V_{p}-V)\;,\,V>0$$

(14)

In conclusion, the use of Eq. (9) to fit the experimental data has the advantage, compared to the SG filtering procedure, of determining the fitting parameters without ambiguities as a result of the nonlinear fitting, also the analytic expression of $I^{\prime\prime}$ can be used to calculate analytically the EEPF and the other plasma parameters. However, this approach has the disadvantage that it is needed to previously calculate the plasma potential $V_{p}$, and also it dismisses the experimental $V>V_{p}$ values beforehand. The error previously estimated to $V_{p}$ is propagated, and affects greatly the final value of the plasma parameters. The SG filtering method (numerically differentiation), on the other hand, considers all the experimental data but, as we have seen in Sec. 4.1, it can be implemented by different options and this election is not obvious, therefore this method is somehow ambiguous to estimate plasma parameters. In the next section we consider another distribution in order to use the advantages of both methods, i.e. the use of an analytical expression to estimate the plasma parameters without ambiguities, and the use of all the experimental data to obtain smaller errors directly from the fit.

In this section, we consider some distribution functions with enough flexibility to represent the complete I-V curves of this work. We propose a general procedure that can be used in other plasma conditions, and the aim of this approach is to determine the plasma parameters without ambiguities and also to extend the non-extensive $q$-exponential distribution which is not able to fit the complete I-V curve. We first note that the experimental I-V curves in Fig. 4 present a remarkable similarity to a cumulative distribution function (cdf) type. Then, we propose to use a cdf with enough flexibility to fit the I-V curves, the restriction we must include is that the cdf has an inflection point (which corresponds to $V=V_{p}$). The $q$-exponential cdf does not satisfy this criterion, so we propose the $q$-Gaussian and $q$-Weibull distributions, which are natural generalizations of the $q$-exponential distribution[56]. The $q$-Weibull distribution includes an additional $r$ parameter, and in the cases $r=1$ and $r=2$ we recover the $q$-exponential and $q$-Gaussian distributions, respectively. The $q$-Weibull distribution also has been used in a variety of contexts in complex systems[56, 57, 58], and it is closely linked to the Tsallis statistics[59, 60, 61, 62]. We finally propose to fit the I-V curves with the $q$-Weibull cdf, shown in Eq. (15). It is important to note that this cdf is related to the EEPF through a second derivative, this is analyzed in Appendix C.

$$I_{c}(V;V_{p},q,r,A)=A\left(1-\left[1-(q-1)\left(\frac{V}{\lambda}\right)^{r}\right]^{\frac{q-2}{q-1}}\right)$$

(15)

To implement Eq. (15) we propose a slightly modified version which is shown in Eq. (16). To explain these modifications we note first that the $q$-Weibull cdf (Eq. (15)) is a cumulative function, this implies it starts from the origin of coordinates. Therefore, a shift from the origin of coordinates to the initial point of the $I-V$ curve must be made. This shift is shown in Fig. 10. Subplot $(1)$ represents a complete I-V Langmuir curve whose lower voltage is $V^{*}$ and the corresponding current is $I(V^{*})$, it is also shown the theoretical $q$-Weibull cdf as a dashed curve ($I_{c}$). Subplot $(2)$ consists of a shift in the probe current axis of the $I_{c}$ curve, at a value of $I(V^{*})$. This value will be a fitting parameter and will be called $I_{s}$, in the following we will suppose that $I_{s}$ does not depend on $V$, this is a rough approximation that will allow us to neglect this contribution in the derivatives of the function. In the case $V^{*}<<V_{f}$, the $I_{s}$ parameter can represent the ion saturation current. Step $(3)$ consists of a shift in the voltage axis of the $I_{c}$ curve, at a value of $V^{*}$. After step $(3)$, the corresponding curve can represent the complete I-V Langmuir curve. Finally, we replace the $\lambda$ parameter to parameters: $V_{p}$, $q$, and $r$. This is achieved by obtaining the second derivative of the cdf from step $(3)$ and matching to zero, in this value it is verified $V=V_{p}$, then we isolate the variable $\lambda=\lambda(V_{p},q,r)$. Equation (16) summarizes the procedure for the case $r=1$, but the general case including $r=1$ is analyzed in Appendix B.

In summary, Eq. (16) includes $V_{p}$, $q$, $r$, $A$ and $I_{s}$ as fitting parameters. The initial values of these parameters to find the numerical convergence, as we explain in the following, can be easily found.

$$I_{QW}(V;V_{p},q,r,A,I_{s})=A\left(1-\left[1-\frac{(q-1)(r-1)}{(q-1)(r-1)-r}\left(\frac{V-V^{*}}{V_{p}-V^{*}}\right)^{r}\right]^{\frac{q-2}{q-1}}\right)+I_{s}$$

(16)

The initial values of the parameters must be carefully selected to ensure the convergence of the nonlinear fit[63]. The initial values of $V_{p}$ and $A$ can be obtained from the visualization of the I-V curve, the $V_{p}$ value can be estimated from the convexity change, and the $A$ value from the subtraction of the electron saturation current with the ion saturation current, i.e. $A\approx(I(V\gg V_{p})-I(V\ll V_{f}))$. On the other side, the initial values of $q$ and $r$ can be selected in the ranges of $(0.5,1.5)$ and $(1,10)$, respectively. The $I_{s}$ value can be initially chosen as $I(V^{*})$. Taking into account these criteria, we can say that convergence is easily reached. The only error that may appear after convergence is reached consists of the error itself from the fit, because the analytic expression of the second derivative can be used, and the numerical differentiation is not needed. Moreover, an advantage of this procedure is that the $V_{p}$ value can be determined without ambiguities as a result of the fit. The analytic expression of the first and second derivatives, with $r\neq 1$, are shown in Eqs. (17) and (18), respectively.

$$I^{\prime}_{QW}\!(V\!;V_{p},q,r,A)\!=\frac{A(q-2)(r-1)r}{(q-1)(r-1)-r}\frac{(V-V^{*})^{r-1}}{(V_{p}-V^{*})^{r}}\!\left[1\!-\!\frac{(q-1)(r-1)}{(q-1)(r-1)-r}\!\left(\frac{V-V^{*}}{Vp-V^{*}}\right)^{r}\right]^{\frac{1}{1-q}}\;,$$

(17)

	$\displaystyle I^{\prime\prime}_{QW}\!(V\!;V_{p},q,r,A)\!=$	$\displaystyle A\left[1-\frac{(q-1)(r-1)}{(q-1)(r-1)-r}\left(\frac{V-V^{*}}{V_{p}-V^{*}}\right)^{r}\right]^{\frac{q}{1-q}}\cdot$			(18)
		$\displaystyle\frac{r(r-1)^{2}(q-2)}{(q-1)(r-1)-r}\frac{(V-V^{*})^{r-2}}{(V_{p}-V^{*})^{r}}\left(1-\left(\frac{V-V^{*}}{V_{p}-V^{*}}\right)^{r}\right)$

Also, the EEPF can be obtained by the variable change $V$ to $V_{p}-V$, as is required by Druyvesteyn in Eq. (6), obtaining, to $r\neq 1$ and $V>0$, Eq. (19).

	$\displaystyle f_{QW}\!(V\!;V_{p},q,r,A)\!=$	$\displaystyle\frac{4A}{e^{3}S}\sqrt{\frac{m}{2}}\left[1-\frac{(q-1)(r-1)}{(q-1)(r-1)-r}\left(\frac{V_{p}-V^{*}-V}{V_{p}-V^{*}}\right)^{r}\right]^{\frac{q}{1-q}}\cdot$			(19)
		$\displaystyle\frac{r(r-1)^{2}(q-2)}{(q-1)(r-1)-r}\frac{(V_{p}-V^{*}-V)^{r-2}}{(V_{p}-V^{*})^{r}}\left(1-\left(\frac{V_{p}-V^{*}-V}{V_{p}-V^{*}}\right)^{r}\right)\;.$

The fitted curves using Eq. (16) are shown in Fig. 11. Figure 12 presents the first and second derivative of the I-V data evaluated with the fitted parameters for each height, also the corresponding EEPF, obtained from the analytic expression of the second derivative, is shown in Fig. 13. We dedicate a comment for the depleted shape observed at low energies ($\Delta$). With the fit procedure, we have eliminated the problem of the convolution effect, but the effects of electron reflection, secondary-electron emission, inhomogeneity of the probe work function along the probe collecting surface are still present. We observe in Fig. 13 that the values of $\Delta\approx(0.3-0.6)$ eV are similar to those obtained in Sec.4.1, this means that the convolution effect is not responsible for the actual value of $\Delta$ in our measurements.

The results are summarized in Table 3. The plasma parameters $n$ and $T$ are obtained from the analytic expression of the EEPF by using Eqs. (7) and (8). The errors corresponding to $n$ and $T$ were estimated from the uncertainties of the fitting parameters. To be more precise, we have calculated the variations in the values of $n$ and $T$ when the fitting parameters are moved in the range of its uncertainties, then we selected a bound for the uncertainties of $n$ and $T$. In the table, we also show the values of $V_{f}$, which are obtained from Eq. (16) solving for the potential with $I_{QW}=0$.

From Table 3 we observe that $q$-Weibull distribution has a smaller root-mean-square error (RMSE) compared with trimmed data of Sec. 4.2, which implies better results in predicted plasma parameters.

The EEPF can be obtained by the variable change $V$ to $V_{p}-V$, as is required by Druyvesteyn in Eq. (6), obtaining

$$f_{QW}\!(V\!;V_{p},q,r=1,A)\!=\frac{4A(q-2)}{e^{3}S\lambda^{2}}\sqrt{\frac{m}{2}}\left(1-(q-1)\frac{V_{p}-V^{*}-V}{\lambda}\right)^{\frac{q}{1-q}}\;,\;V>0$$

(30)

where $0<V<V_{p}-V^{*}$,

In this case, the EEPF result:

$$f_{QW}\!(V\!;V_{p},q=1,r=1,A)\!=-\frac{4A}{e^{3}S\lambda^{2}}\sqrt{\frac{m}{2}}e^{\frac{V_{p}-V^{*}-V}{\lambda}}\;,\;V>0$$

(33)

where $0<V<V_{p}-V^{*}$.

Defining $\varepsilon:=-e(V_{p}-V^{*}-V)$ , $kT:=e\lambda$ and $B:=-\frac{4A}{eS(kT)^{2}}\sqrt{\frac{m}{2}}$, we obtain

$$f_{BGS}\!(\varepsilon)\!=Be^{-\frac{\varepsilon}{kT}}\;,\;0<\varepsilon<-e(V_{p}-V^{*})$$

(34)

which is the Boltzmann-Gibbs-Shannon EEPF.

The EEPF can be obtained from Eq.(29) by the variable change $V$ to $V_{p}-V$, as is required by Druyvesteyn in Eq. (6), obtaining

$$f_{QW}\!(V\!;V_{p},q=1,r,A)\!=\frac{-Ar(r-1)\left(1-\left(\frac{V_{p}-V^{*}-V}{V_{p}-V^{*}}\right)^{r}\right)}{(V_{p}-V^{*}-V)^{2-r}\lambda^{r}}e^{\left(\frac{V_{p}-V^{*}-V}{\lambda}\right)^{r}}\;,\;V>0$$

(35)

where $0<V<V_{p}-V^{*}$. Defining $\varepsilon:=-e(V_{p}-V^{*}-V)$ and $kT:=(-1)^{\frac{1}{r}}e\lambda$, we obtain

$$f_{W}\!(\varepsilon)\!=\frac{eA_{\varepsilon}}{(kT)^{r}}e^{-\left(\frac{\varepsilon}{kT}\right)^{r}}\;,$$

(36)

which is the Weibull distribution or generalized EEPF. In the case $r=2$, it corresponds to the Druyvesteyn EEPF. For a recent review about these distributions, the reader is referred to [80], and the original works can be found in Refs. [81, 82, 83]