# Langmuir Probe Measurements: The Electron Diffusion Parameter Under Collisional Near-Probe Space Charge Layer Conditions

J L Jauberteau, I Jauberteau

► To cite this version:

J L Jauberteau, I Jauberteau. Langmuir Probe Measurements: The Electron Diffusion Parameter Under Collisional Near-Probe Space Charge Layer Conditions. Contributions to Plasma Physics, 2025, 10.1002/ctpp.70041. hal-05370397

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ORIGINAL ARTICLE

# Langmuir Probe Measurements: The Electron Diffusion Parameter Under Collisional Near-Probe Space Charge Layer Conditions

J. L. Jauberteau | I. Jauberteau

University of Limoges, UMR 7315 CNRS, Limoges, France

Correspondence: J. L. Jauberteau (jean-louis.jauberteau@unilim.fr)

Received: 19 February 2025 | Revised: 16 June 2025 | Accepted: 30 June 2025

**Funding:** The authors received no specific funding for this work.

ABSTRACT

The corrections which are necessary in the determination of the Electron Energy Distribution Function (EEDF) by means of Langmuir probe under near-probe space-charge layer conditions are investigated. Exact calculations of the first and second derivative function of the I(V) probe characteristic are given and approximations are used to determine the electron diffusion parameter (sink parameter) and consequently the EEDF. The results obtained using these different approximations are discussed and compared with those obtained using other methods reported in the literature.

1 | Introduction

The determination of the electron energy distribution function (EEDF) in plasma by means of a cylindrical Langmuir probe using the Druyvesteyn equation [1] is restricted by the requirement of a small probe radius (a) and small sheath length (d) around the cylindrical probe collector compared to the electron and ion mean free path in the sheath ($\lambda_e, \lambda_i$). When the ratio $a/\lambda_e > 1$, the condition of collision near the probe space charge layer is fulfilled, and the electron current collected by the probe under electron retarding potential conditions depends on the electron diffusion through the sheath formed around the probe tip collector and is given by the Swift law. This is the case in high-pressure plasma or in magnetized plasma when the cylindrical probe axis is parallel to the magnetic field. In these conditions, the Langmuir law becomes [2],

$$i_e(V) = g \int_{eV}^{\infty} \frac{\epsilon_e^{-1/2} (\epsilon_e - eV) f_0(\epsilon_e) d\epsilon_e}{\left[1 + \frac{(\epsilon_e - eV)}{\epsilon_e} \Psi(\epsilon_e, V)\right]} \quad (1)$$

Where, $g = \frac{1}{4} \sqrt{\frac{2}{m_e}} eS$, S is the probe area. V is the biased voltage which is the difference between the applied voltage $V_{app}$ and the plasma potential $V_p$ that is, $V = V_{app} - V_p$. $\Psi(\epsilon_e, V)$ is the diffusion parameter (or electron sink parameter) depending on the electron diffusion through the sheath produced around the probe collector. The diffusion parameter changes with the plasma parameters, the probe radius, the shape and the size of the sheath and with the orientation of the probe in the case of the presence of a magnetic field. It depends on collisions of charged particle in the sheath and on the voltage drop profile through the sheath [3–6]. When $\Psi(\epsilon_e, V) < < 1$, in the case of low pressure gas and in the absence of a significant magnetic field, the second term of the denominator in Equation (1) can be neglected and the equation is equivalent to the Langmuir law. In this case, the EEDF can be determined using the Druyvesteyn equation. Under other conditions of $\Psi(\epsilon_e, V)$, the determination of EEDF depends on this parameter.

Regarding the diffusion parameter, different works [3–6] show that it depends on the electron energy and on the drop voltage

<footnote>This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.</footnote>

<footnote>© 2025 The Author(s). Contributions to Plasma Physics published by Wiley-VCH GmbH.</footnote>
through the sheath (potential profile in the sheath around the probe) [5]. However, because of the difficulty in determining the potential profile in the sheath, authors use the approximation $\Psi(\epsilon_e, V) = \Psi(\epsilon_e)$. They assume the diffusion coefficient is constant through the sheath and the sheath profile unchanged, whatever the applied voltage.

In a previous work [7, 8], we have reported a numerical method to determine the diffusion parameter $\Psi(\epsilon_e, V)$ taking into account its dependence on the biased voltage applied to the probe collector. However this method neglect integral terms depending on $\frac{d\Psi(\epsilon_e, V)}{dV}$ in the calculation of the first and second derivative functions of the $I(V)$ curve [3–6].

The purpose of the present work is to calculate the exact values of the first and second derivative functions of Equation (1) and to determine the diffusion parameter $\Psi(\epsilon_e, V)$, even if some approximations are necessary. Results obtained using these approximations are compared and discussed.

## 2 | Theory

Regarding the dependency of the diffusion parameters with the electron energy and the probe biased voltage, the first derivative value of the electron current collected versus $V$ (Equation (1)) is given by,

$$ \frac{dI_e}{dV} = -g \int_{eV_p}^{\infty} \frac{e\epsilon_e^{-1/2} f(\epsilon_e)}{A^2} d(\epsilon_e) - R_1 \quad (2) $$

where $A = \left(1 + \left(\frac{\epsilon_e - eV}{\epsilon_e}\right) \Psi(\epsilon_e, V)\right)$

and $R_1 = g \int_{eV}^{\infty} \frac{\epsilon_e^{-3/2} f(\epsilon_e) (\epsilon_e - eV)^2}{A^2} \frac{d\Psi(\epsilon_e, V)}{dV} d\epsilon_e$

This function is negative whatever the $V$ value.

The second derivative function is given by,

$$ \frac{d^2 I_e}{dV^2} = +g e^2 (eV)^{-1/2} f(eV) - g \int_{eV}^{\infty} \frac{2e^2 \epsilon_e^{-3/2} f(\epsilon_e) \Psi(\epsilon_e, V)}{A^3} d\epsilon_e + R_2 + \frac{d(R_1)}{dV} \quad (3) $$

$$ R_2 = g \int_{eV}^{\infty} \frac{2e\epsilon_e^{-3/2} f(\epsilon_e) (\epsilon_e - eV)}{A^3} \frac{d(\Psi(\epsilon_e, V))}{dV} d\epsilon_e $$

The exact value of the diffusion parameter can be determined by considering these two derivative functions and writing,

$$ \frac{dI_e}{dV} = \frac{dI_e}{d\epsilon_e} \frac{d\epsilon_e}{dV} = g \frac{\epsilon_e^{-1/2} (\epsilon_e - eV) f(\epsilon_e)}{A} \frac{d\epsilon_e}{dV} $$

$$ \frac{d^2 I_e}{dV^2} = \frac{d\left(\frac{dI_e}{dV}\right)}{d\epsilon_e} \frac{d\epsilon_e}{dV} = -g \left[ \frac{e\epsilon_e^{-1/2} f(\epsilon_e)}{A^2} + \frac{\epsilon_e^{-3/2} f(\epsilon_e) (\epsilon_e - eV)^2}{A^2} \frac{d\Psi(\epsilon_e, V)}{dV} \right] \frac{d\epsilon_e}{dV} $$

Writing Rap = $\frac{dI_e}{dV}$, we obtain the first order differential equation,

$$ \Psi(\epsilon_e, V) + \text{Rap} \frac{d\Psi(\epsilon_e, V)}{dV} = - \frac{\epsilon_e}{(\epsilon_e - eV)} \left( \frac{e\text{Rap}}{(\epsilon_e - eV)} + 1 \right) $$

Then using the Lagrange constant variation method [9], the solution of this equation is

$$ \psi(\epsilon_e, V) = C(\epsilon_e, V) \left( \frac{-dI_e}{dV} \right) = - \frac{\epsilon_e}{(\epsilon_e - eV)} + k \left( \frac{-dI_e}{dV} \right) \quad (4) $$

Considering the values $R_1(\epsilon_e, V)$ and $R_2(\epsilon_e, V)$.

$$ \frac{dR_{1,2}(\epsilon_e, V)}{dV} = \frac{dR_{1,2}(\epsilon_e, V)}{d\epsilon_e} \frac{d\epsilon_e}{dV} $$

k is given by,

$$ k = \frac{\frac{dR_1(\epsilon_e, V)}{dV}}{\frac{dR_2(\epsilon_e, V)}{dV}} \frac{2e\epsilon_e}{(\epsilon_e - eV)^2} \left( \frac{dV}{dI_e} \right) $$

and the diffusion parameter is given by,

$$ \psi(\epsilon_e, V) = C(\epsilon_e, V) \left( -\frac{dI_e}{dV} \right) = - \frac{\epsilon_e}{(\epsilon_e - eV)} \left[ 1 + \frac{2e}{(\epsilon_e - eV)} \frac{\frac{dR_1(\epsilon_e, V)}{dV}}{\frac{dR_2(\epsilon_e, V)}{dV}} \right] \quad (5) $$

It is worth noting that a decrease of the collected electron current due to the electron diffusion through the sheath is observed only if $\psi(\epsilon_e, V)$ is positive (see Equation (1)). Consequently, the ratio $\frac{\frac{dR_1(\epsilon_e, V)}{dV}}{\frac{dR_2(\epsilon_e, V)}{dV}}$ must be negative.

For convenience, the three previous Equations (1–3), can be written,

$$ I_e(\epsilon_e, V) = A(\epsilon_e, V) \quad (6) $$

$$ \frac{dI_e(\epsilon_e, V)}{dV} = B(\epsilon_e, V) + R_1(\epsilon_e, V) \quad (7) $$

$$ \frac{d^2 I_e(\epsilon_e, V)}{dV^2} = C(\epsilon_e, V) + R_2(\epsilon_e, V) + \frac{dR_1(\epsilon_e, V)}{dV} \quad (8) $$

Because of the numerical noise in the calculation of $\frac{d\Psi(\epsilon_e, V)}{dV}$, the EEDF is generally obtained by numerical adjustments until $\frac{d^2 I_e}{dV^2 Exp} = C(\epsilon_e, V)$ in Equation (8) [7, 8, 10]. In this case $R_2(\epsilon_e, V) = -\frac{dR_1(\epsilon_e, V)}{dV}$ and the diffusion parameter is,

$$ \psi(\epsilon_e, V) = C(\epsilon_e, V) \left( -\frac{dI_e}{dV} \right) = - \frac{\epsilon_e}{(\epsilon_e - eV)} \left[ 1 - \frac{2e}{(\epsilon_e - eV)} \frac{R_2(\epsilon_e, V)}{\frac{dR_2(\epsilon_e, V)}{dV}} \right] \quad (9) $$
In previous works [7, 8, 10], we had in a first approximation neglected the terms $R_1(\epsilon_e, V)$, $\frac{dR_1(\epsilon_e, V)}{dV}$ and $R_2(\epsilon_e, V)$ in Equations (7) and (8) to calculate the diffusion parameter given by,

$$ \psi(\epsilon_e, V) = -\frac{\epsilon_e}{(\epsilon_e - eV)} \left[ 1 + \frac{e}{(\epsilon_e - eV)} \text{Rap} \right] \quad (10) $$

A great similitude between the three Equations (5, 9, and 10) is observed. They give a dependency of the diffusion parameter with the applied biased voltage and with the electron energy.

The first derivative functions of the three Equations (5, 9, and 10) are given by,

$$ \frac{d(\Psi_1(\epsilon_e, V))}{dV} = - \left[ \frac{e\epsilon_e}{(\epsilon_e - eV)^2} \left( 1 + 2\frac{d(\text{Rap}_1)}{dV} \right) + \frac{4e^2\epsilon_e}{(\epsilon_e - eV)^3} \text{Rap}_1 \right] \quad (11) $$

$$ \frac{d(\Psi_2(\epsilon_e, V))}{dV} = - \left[ \frac{e\epsilon_e}{(\epsilon_e - eV)^2} \left( 1 - 2\frac{d(\text{Rap}_2)}{dV} \right) - \frac{4e^2\epsilon_e}{(\epsilon_e - eV)^3} \text{Rap}_2 \right] \quad (12) $$

$$ \frac{d(\Psi_3(\epsilon_e, V))}{dV} = - \left[ \frac{e\epsilon_e}{(\epsilon_e - eV)^2} \left( 1 + 2\frac{d(\text{Rap}_3)}{dV} \right) + \frac{2e^2\epsilon_e}{(\epsilon_e - eV)^3} \text{Rap}_3 \right] \quad (13) $$

with,

$$ \text{Rap}_1 = \frac{dR_1/dV}{dR_2/dV}, \quad \text{Rap}_2 = \frac{R_2}{dR_2/dV} \quad \text{and} \quad \text{Rap}_3 = \frac{dI}{dV} $$

In these equations the terms $\frac{d(\text{Rap}_1, 3)}{dV}$ depend on the second derivative of $\psi(\epsilon_e, V)$ or on the third derivative of $I_e$, and consequently these terms cannot be correctly determined because of the experimental errors or numerical noise. So, they should be neglected in further calculations using the Equations (6–8), in a first approximation.

In the next part of this article, the different values of $\psi(\epsilon_e, V)$ calculated using the Equations (5, 9, and 10) are tested in the case of theoretical and experimental examples.

### 3 | Results and Discussion

In the case of an arbitrary $I(V)$ probe characteristic and the corresponding first and second derivative functions shown on Figure 1. The EEDF is calculated using the method described in previous works [7, 8, 10]. We first determine the EEDF using Druyvesteyn equation. This is equivalent to neglect the diffusion parameter in

<figure><img src="image_2.png" /><figcaption>FIGURE 1 | The arbitrary $I(V)$ probe characteristic with the corresponding first and second derivative functions.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>FIGURE 2 | The change of diexp, $d^2iexp$, B, C, $R_1$, and $R_2$ versus applied voltage $V_{\text{app}}$; Calculation has been done using Equation (10) in the case of the $I(V)$ probe characteristic shown in Figure 1.</figcaption></figure>

a first time. Then the EEDF is incremented using the difference between the experimental and the calculated value of $\frac{d^2I}{dV^2}$ using Equations (3 or 8). This calculation is done until a good agreement between experimental and calculated values of $\frac{dI}{dV}$ and $\frac{d^2I}{dV^2}$ using Equations (2 and 3) or (7 and 8). For more details see the references cited above [7, 8, 10].

The three Equations (5, 9, and 10) have been tested, and results are compared.

First, the value of the electron diffusion parameter is calculated using Equation (10).

Figure 2 shows the change of terms B, C, $R_1$ and $R_2$ in Equations (7 and 8) versus applied voltage $V_{\text{app}}$ and the value of $B + R_1$ and C are compared to diexp and $d^2iexp$, respectively. Where, diexp = $-di_e/dV$ and $d^2iexp = d^2i_e/dV^2$ are the derivative functions of $I(V)$ shown in Figure 1. A good agreement between diexp and $B + R_1$ and $d^2iexp$ and C is obtained. In spite of the large noise in the value of $\frac{dR_1}{dV}$, the equality $R_2 = -\frac{dR_1}{dV}$ is fulfilled.

Second, calculations have been done using Equation (5) or Equation (9). Similar results are obtained with both Equations (5 and 9). Nevertheless, results are strongly disturbed because of the successive derivations, as shown in Figure 3, when the calculations are performed using Equation (5). In spite of the
<figure><img src="image_4.png" /><figcaption>**FIGURE 3** | Change of diexp, $d^2iexp$, B, C, $R_1$, and $R_2$ versus applied voltage $V_{app}$. Equation (5) has been used in the case of the $I(V)$ probe characteristic shown in Figure 1. The same results are observed using Equation (9).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**FIGURE 4** | Experimental $I(V)$ probe characteristic and the first and second derivative functions versus $V_{app}$. Measurements have been performed in microwave plasma (2.45 GHz) sustained in hydrogen at 0.19 Pa, at 1500 W and confined by means of a magnetic field (97 mT).</figcaption></figure>

strong fluctuations, results are close to those obtained using Equation (10).

Equations (2 and 3) or (7 and 8) have been solved in the case of experiments performed in hydrogen microwave plasma confined by means of a magnetic field. The probe characteristic has been measured by O. D. Cortazar et al. [11] in microwave plasma sustained in hydrogen at 0.19 Pa, working at 2.45 GHz with a power equal to 1500 W. The Langmuir probe is a cylindrical single probe (tungsten wire 6 mm length and 0.5 mm diameter), located in the middle of the reactor along the Z-axis of the cylindrical plasma chamber. It is parallel to the magnetic field $B_z$, which intensity is equal to 97 mT around the probe tip [11]. Figure 4 shows the $I(V)$ probe characteristic and the first and second derivative functions. The potential plasma and the floating potential are equal to 35.47 and 23.5 V, respectively.

The values of B, C, $R_1$ and $R_2$ and $B+R_1$ and $C+R_2+dR_1/dV$ (see Equations 6–8) are calculated considering $\frac{d(R_{ap,2,3})}{dV} = 0$ in the case of the three different values of the diffusion parameter $\psi(\epsilon_e, V)$ calculated using the Equations (5, 9, and 10). They are compared to experimental values of the derivative functions $di \exp = -\frac{di_e}{dV}$ and $d^2i \exp = \frac{d^2i_e}{dV^2}$ of the $I(V)$ curve (see Figure 4). The results are shown in Figures 5–7. As previously, the EEDF is determined using the method detailed in references [7, 8, 10].

<figure><img src="image_6.png" /><figcaption>**FIGURE 5** | $R_1$, $B$, $R_2$, $C$, versus $V_{app}$ by using Equation (5), values are compared with experiments, diexp and $d^2i \exp$.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**FIGURE 6** | $R_1$, $B$, $R_2$, $C$, versus $V_{app}$ by using Equation (9), values are compared with experiments, diexp and $d^2i \exp$.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**FIGURE 7** | $R_1$, $B$, $R_2$, $C$, versus $V_{app}$ by using Equation (10), values are compared with experiments, diexp and $d^2i \exp$.</figcaption></figure>

It can be seen that the results are close to each other. However, the results obtained using Equation (10) are less disturbed than those using the two other Equations (5 and 9). A good agreement is observed between $d^2i \exp$ and $C$ and between diexp and $B+R_1$. Equation (10) is more appropriate to determine the EEDF than Equations (5 and 9).

Values of $\psi(\epsilon_e, V)$ calculated using the three Equations (5, 9, and 10) in the case of the previous experiment have been compared. Results are shown on Figures (8 and 9) versus $E_e = \epsilon_e - eV$, for $V_{app} = 33.1$ and 28.9 V, respectively. These results are compared with the diffusion parameter values calculated using the equation
<figure><img src="image_9.png" /><figcaption>**FIGURE 8** | $\psi(\epsilon_e, V)$ calculated using the three Equations (5, 9, 10, and 14) at $V_{app} = 33.1$ V and the previous experimental $I(V)$ probe characteristic.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**FIGURE 9** | $\psi(\epsilon_e, V)$ calculated using the three Equations (5, 9, 10, and 14) at $V_{app} = 28.9$ V and the previous experimental $I(V)$ probe characteristic.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**FIGURE 10** | Variation of A and n versus $V_{app}$ using the previous experimental $I(V)$ probe characteristic.</figcaption></figure>

reported by Arslanbekov et al. or Popov et al. [6, 12] These authors assume a constant diffusion coefficient for a thin sheath (d), in the case of a cylindrical probe of radius R and length L, localized along the magnetic field, with $R \gg \frac{L}{2} \left(1 + \left(\frac{\lambda}{R_L}\right)^2\right)^{-1/2}$, where $R_L$ and $\lambda$ are the Larmor radius and the electron means free path. They obtain,

$$ \Psi(\epsilon_e) \approx \frac{\pi L}{4\gamma R_L} = \frac{1500B}{\sqrt{\epsilon_e}} \quad (14) $$

The geometric factor is $\gamma = 4/3$ assuming $\lambda, R_L \gg (R + d)$.

A large difference between values proposed by Arslanbekov et al. or Popov et al. [6, 12] Equation (14) and the values calculated using the Equations (5, 9, and 10) is seen. These last ones are close

<figure><img src="image_12.png" /><figcaption>**FIGURE 11** | Results obtained using the method reported in Gorbunov et al. [12], B and C, versus $V_{app}$ in the case of a constant value for $\psi = 5.98 \times 10^{-4}$); Comparison with the experimental values, diexp and $d^2iexp$.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**FIGURE 12** | Results obtained using the method reported in Gorbunov et al. [12], $R_1$, $B$, $R_2$, $C$, and $B + R_1$ versus $V_{app}$, using Equation (10) in Equations (6–8); Comparison with the experimental values, diexp and $d^2iexp$.</figcaption></figure>

to each others. The differences are probably due to numerical errors introduced on the ratio of the differential values. Whatever the applied bias voltage, curves show similar behaviors versus $E_e$ and they are fitted using the exponential law $\psi(\epsilon_e, V) = AE_e^n$ [8]. Figure 10 shows the change of A and n versus $V_{app}$. The results confirm the dependence of the electron diffusion parameter with $\epsilon_e$ and also with $V_{app}$.

According to literature [4, 6, 12], when $\psi(\epsilon_e, V) \ll 1$ the EEDF is proportional to the second derivative $\frac{d^2i_e}{dV^2}$ (Druyvesteyn equation [1]) and when $\psi(\epsilon_e, V) \gg 1$, the EEDF is proportional to the first derivative $\frac{di_e}{dV}$. So, in a previous work, Gorbunov et al. [13], report a simplified solution for the EEDF calculation, using a linear combination of the two derivative functions $\frac{di_e}{dV}$ and $\frac{d^2i_e}{dV^2}$,

$$ f(\epsilon_e) = \frac{3m_e^2\gamma}{8\pi e^3} \left( \frac{d^2i_e}{dV^2} - \frac{\Psi}{V} \frac{di_e}{dV} \right), \quad \text{with} \quad \Psi = \frac{R}{\gamma\lambda} \ln\left(\frac{\pi L}{4R}\right) \quad (15) $$

We have tested this method in the case of the previous experiment (shown in Figure 4). Assuming the electron mean free path $\lambda = 0.4$ m and first with $\psi = \text{constant} = 5.98 \times 10^{-4}$, calculated using Equation (15) then second with $\psi$ calculated using Equation (10). The results obtained using the Equations (7 and 8) are shown in Figures 11 and 12, respectively. The calculations performed using the Equations (5 and 9) are close to those obtained using Equation (10) but they are too noisy and not displayed here.
It can be seen that the method reported by Gorbunov does not work properly in the present case even using Equation (10) in the Equations (7 and 8). It appears that the method detailed in ref. [7–9] gives better results.

## 4 | Conclusion

This work improves the knowledge of the electron diffusion parameter in the sheath formed around a Langmuir probe collector. This parameter has an effect on the electron flux collected by the probe and must be determined to calculate the EEDF in the undisturbed plasma around the probe collector.

The exact equations of the first and second derivative functions of the current-voltage characteristic are given. It is shown that the electron diffusion parameter depends on the second and third derivative functions, which cannot be accurately determined because of experimental or numerical noise. So, approximations must be used. The results obtained using different approximations, which lead to the three equations of the electron diffusion parameter Equation (5, 9, and 10) are compared in the case of theoretical or experimental examples. Also, a comparison with other methods reported in literature [12, 13] is shown. This study confirms that the first method proposed and detailed in previous works [7, 8, 10] using Equation (10) is more appropriate to determine the EEDF, at least in the case of partially ionized plasma. The cases of strongly ionized plasma or with more intense magnetic fields have not been investigated.

However, it is worth noting that the electrostatic probes provide information about the EEDF in unperturbed plasma as long as the disturbed volume by the probe has a characteristic length ($r_d$) much smaller than the electron energy relaxation length $\lambda_e$ which is the characteristic length of the undisturbed EEDF formation. This one depends on the electron collision cross sections with any other species present in the plasma [6, 14, 15]. For larger $r_d$ values, the effect of collision processes within the sheath drastically changes the EEDF. So, these measurements do not reflect the EEDF within the plasma bulk. The probe disturbed length $r_d$ is of the order of the probe sheath radius (probe radius + sheath thickness) only in the case of a collisionless regime in the sheath, and it differs in the collision regime [14, 15].

## Conflicts of Interest

The authors declare no conflicts of interest.

## Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## References

1. M. J. Druyvesteyn, "Der Niedervoltbogen," *Zeitschrift für Physik* 64 (1930): 781.

2. J. D. Swift, "Effects of Finite Probe Size in the Determination of Electron Energy Distribution Functions," *Proceedings of the Physical Society* 79 (1962): 697–701.

3. M. A. Mal'kov "Electron Sink at a Probe in the Conditions of a Collisional Near-Probe Space Charge Layer," *High Temperature* 29 (1991): 614.

4. V. Demidov, R. Armstrong, and A. Friedriksen, "Probe Measurements of Electron Energy Distributions in a Magnetized Plasma of the Toroidal Device," *Contributions to Plasma Physics* 38, no. S1 (1998): 1–6.

5. K. Tsv, M. Popov, P. Dimitrova, et al., "Advances in Langmuir Probe Diagnostics of the Plasma Potential and Electron-Energy Distribution Function in Magnetized Plasma," *Plasma Sources Science and Technology* 25 (2016): 033001.

6. R. R. Arslanbekov, N. A. Khromov, and A. A. Kudryavtsev, "Probe Measurements of Electron Energy Distribution Function at Intermediate and High Pressures and in a Magnetic Field," *Plasma Sources Science and Technology* 3 (1994): 528–538.

7. J. L. Jauberteau, I. Jauberteau, O. D. Cortázar, and A. Megía-Macías, "Langmuir Probe in Magnetized Plasma: Determination of the Electron Diffusion Parameter and of the Electron Energy Distribution Function," *Contributions to Plasma Physics* 60 (2020): e201900067.

8. J. L. Jauberteau, I. Jauberteau, O. D. Cortázar, and A. Megía-Macías, "Langmuir Probe in Magnetized Plasma: Study of the Diffusion Parameter," *Contributions to Plasma Physics* 62 (2022): e202100066.

9. L. Collatz, *Differential Equations an Introduction With Applications* (John Wiley and Sons NY, 1986).

10. J. L. Jauberteau and I. Jauberteau, "Determination of the Electron Energy Distribution Function in Weakly Ionized Plasma by Means of a Langmuir Probe and Numerical Methods," *AIP Advances* 14 (2024): 055326, https://doi.org/10.1063/5.0204161.

11. O. D. Cortázar, J. Komppula, O. Tarvainen, A. Megía-Macías, A. Vizcaíno-de-Julián, and H. Koivisto, "Experimental STUDY of Hydrogen Plasma Breakdown in a 2.45 GHz Microwave Discharge," *Plasma Sources Science & Technology* 22 (2013): 015026.

12. T. K. Popov, P. Ivanova, M. Dimitrova, J. Kovačič, T. Gyergyek, and M. Čerček, "Langmuir Probe Measurements of the Electron Energy Distribution Function in Magnetized Gas Discharge Plasmas," *Plasma Sources Science and Technology* 21 (2012): 025004, https://doi.org/10.1088/0963-0252/21/2/025004.

13. N. A. Gorbunov, A. Grochola, P. Kruk, A. Pietruczuk, and T. Stacewicz, "Studies of Electron Energy Distribution in Plasma Produced by a Resonant Laser Pulse," *Plasma Sources Science and Technology* 11 (2002): 492–497.

14. L. D. Tsendin, "Electron Kinetics in Glows—From Langmuir to the Present," *Plasma Sources Science & Technology* 18 (2009): 014020.

15. U. Kortshagen, "Experimental Evidence on the Nonlocality of the Electron Distribution Function," *Physical Review E* 49, no. 5 (1994): 4369–4380.