Advances in Langmuir probe diagnostics of the plasma potential and electron-energy distribution function in magnetized plasma

Tsv K Popov; M Dimitrova; P Ivanova; J Kovačič; T Gyergyek; R Dejarnac; J Stöckel; M A Pedrosa; D López-Bruna; C Hidalgo

doi:10.1088/0963-0252/25/3/033001

1. Introduction

Among the contact methods of plasma diagnostics, electric probes are the least expensive and still the fastest and most reliable diagnostic tools, allowing one to obtain the local values of the main plasma parameters. The probe technique is seen as a rather simple and attractive scientific instrument, but needs some care and attention in the processing of the measured data if meaningful results are to be obtained. The application of the probe technique has been summarized in numerous reviews and monographs [1–7].

Langmuir demonstrated [8] that, for spherical and cylindrical probes inserted into a weakly ionized low-pressure plasma, the application of a voltage ${{U}_{\text{p}}}$ to the probe and the measurement of the probe current ${{I}_{\text{p}}}$ can provide reliable information about the plasma density $n$ and the electron temperature ${{T}_{\text{e}}}$ . The next important step in the development of probe diagnostics was undertaken by Druyvesteyn [9], who demonstrated that the second derivative of the probe current with respect to the probe potential ${{\text{d}}^{2}}I/\text{d}{{U}^{2}}$ allows the determination of the electron-energy distribution function (EEDF) in the plasma. Following [7], the use of the EEDF, $F\left(t,\vec{r},\varepsilon \right)$ , which gives the number of electrons in a volume element having energies between ε and ε + dε (losing information about angle distribution), has been generally accepted. Then (omitting coordinates and time) the normalization relation to the electron density n in the isotropic plasma is

$\begin{eqnarray}&&{\int}_{0}^{\infty}F(\varepsilon )\text{d}\varepsilon =n.\end{eqnarray} \tag{ 1 }$

On the other hand, if $4\pi {{c}^{2}}{{f}_{0}}(c)\text{d}c$ represents the number of electrons in a unit volume of the plasma having speeds in the range c to c + dc, then

$\begin{eqnarray}&&4\pi {\int}_{0}^{\infty}{{c}^{2}}{{f}_{0}}(c)\text{d}c=n.\end{eqnarray} \tag{ 2 }$

Substituting the velocity c by the energy $\varepsilon =(m/2){{c}^{2}}$ and bearing in mind that ${{c}^{2}}\text{d}c={{(2\varepsilon )}^{1/2}}\,{{m}^{-3/2}}\text{d}\varepsilon$ (m being electron mass), following [7] we get

$\begin{eqnarray}&&{\int}_{0}^{\infty}F(\varepsilon )\text{d}\varepsilon =4\pi {\int}_{0}^{\infty}f(c){{c}^{2}}\text{d}c=\frac{4\pi \sqrt{2}}{{{m}^{3/2}}}{\int}_{0}^{\infty}{{f}_{0}}(\varepsilon )\sqrt{\varepsilon}\text{d}\varepsilon ={\int}_{0}^{\infty}f(\varepsilon )\sqrt{\varepsilon}\text{d}\varepsilon =n.\end{eqnarray} \tag{ 3 }$

The function $f(\varepsilon )\,=4\pi \sqrt{2}{{m}^{-3/2\,}}{{f}_{0}}(\varepsilon )\,=F(\varepsilon )/\sqrt{\varepsilon}$ is frequently referred to as the electron-energy probability function (EEPF)⁷. For isotropic plasma, it has the same information about the electron gas as the EEDF and is frequently used to represent measured probe data. The EEPF, presented on a semi-log scale, allows quick visualization of the departure of the measured EEPF from a Maxwellian distribution, which is a straight line in this representation [7].

The probe technique for evaluation of the EEDF is relatively simple when a number of requirements are satisfied, namely [5]:

(a)
The plasma is isotropic on a scale much larger than the mean free path of the charged particles;
(b)
the mean free paths of the electrons, $\lambda$ , and ions, ${{\lambda}_{\text{i}}}$ , are much larger than the probe radius, R, and the thickness of the probe sheath, d;
(c)
the probe holder does not disturb the plasma in the vicinity of the probe tip;
(d)
there is neither generation nor recombination of charged particles, nor are there chemical reactions in the probe sheath or on the probe surface either;
(e)
the surface area of the reference probe is large enough to sustain all the current collected from the measuring probe without a noticeable potential drop;
(f)
there are no fluctuations in the plasma characteristics;
(g)
the probe surface is free of contamination, such as dielectric films, etc.

Point (b) indicates that electrons originating from the undisturbed plasma and crossing the probe sheath reach the probe surface without collisions. This means that the probe operates at very low gas pressures and in the absence of or in a very weak magnetic field when the electron mean free path $\lambda$ and the Larmor radius ${{R}_{\text{L}}}$ are larger than the probe radius R and the thickness of the sheath d [2–4, 6]:

$\begin{eqnarray}&&\lambda,{{R}_{\text{L}}}\gg R+d.\end{eqnarray} \tag{ 4 }$

Then the electron probe current of the IV characteristic is expressed [1] by

$\begin{eqnarray}&&{{I}_{\text{e}}}(U)=-\frac{2\pi eS}{{{m}^{2}}}{\int}_{eU}^{\infty}(W-eU)\,{{f}_{0}}(W)\text{d}W,\end{eqnarray} \tag{ 5 }$

where e is the electron charge and S is the probe area, $W=(m/2){{c}^{2}}+eU$ is the total electron energy in the probe sheath, and c is the electron velocity at the sheath edge. The probe is negatively biased by a potential ${{U}_{\text{p}}}$ ; U is the probe potential with respect to the plasma potential ${{U}_{\text{pl}}}$ ( $U={{U}_{\text{p}}}-{{U}_{\text{pl}}}$ ).

The EEPF, $f(\varepsilon )$ , can be determined by using the Druyvesteyn formula [9]:

$\begin{eqnarray}&&f(\varepsilon )=\frac{2\sqrt{2m}}{{{e}^{3}}S}\frac{{{\text{d}}^{2}}{{I}_{\text{e}}}(U)}{\text{d}{{U}^{2}}}.\end{eqnarray} \tag{ 6 }$

In many different contemporary technologies, such as plasma chemistry, etching, plasma polymerization, thin dielectric layer deposition, etc, the presence of a magnetic field in the range 0.01–0.1 T is required. It is well known that the presence of a magnetic field affects the measured current–voltage (IV) probe characteristics mostly in their electron parts, while the ion saturation current remains unaffected. In the probe sheath, the electrons are moving along helical trajectories, so that even at low gas pressures the probability of collisions with neutrals increases. As a result, the electron probe current decreases. Figure 1 presents an example of IV results obtained [10] in a low-pressure argon gas discharge (~1 Pa) in the presence of different magnetic fields.

**Figure 1.** IV probe characteristics normalized to the electron densities measured in an Ar gas discharge in the presence of different magnetic fields.
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In these cases, due to the collisions in the probe sheath, the interpretation of the experimental data acquired in order to obtain the correct values of the plasma parameters becomes more complicated. On the other hand, knowledge of the real EEDF is of great importance in understanding the underlying physics of the processes occurring in the magnetized plasma, such as the formation of transport barriers, cross-field diffusion coefficient, plasma–substrate interactions, etc. Moreover, the probe technique is frequently applied to the diagnostics of edge fusion plasmas.

In this paper we review probe measurements in argon and helium DC gas discharges in the presence of a magnetic field in the range 0.01–0.3 T and applications of the probe technique to strongly magnetized (0.5–1 T) turbulent fusion plasmas for the correct evaluation of the plasma potential ${{U}_{\text{pl}}}$ , the EEPF, $f(\varepsilon )$ , and, respectively, the electron temperature and the electron density.

2. Evaluation of the EEDF in the presence of a magnetic field

Swift [11] was the first to take into account the probe size and the effect of collisions in the probe sheath in evaluating the EEDF. He pointed out that the second derivative of the IV probe characteristic is distorted due to the depletion of electrons sinking into the probe surface. Similar results were obtained in [12, 13] on the basis of a non-local approach, when the electrons reach the probe in a diffusion regime. It was shown that the electron probe current is expressed by an extended equation:

$\begin{eqnarray}&&{{I}_{\text{e}}}(U)=-\frac{8\pi eS}{3{{m}^{2}}\gamma}{\int}_{eU}^{\infty}\frac{(W-eU)\,{{f}_{0}}(W)\text{d}W}{1+\frac{(W-eU)}{W}\psi}.\end{eqnarray} \tag{ 7 }$

The value of the geometric factor $\gamma$ varies monotonically from 0.71 to $4/3$ : $\gamma =4/3$ when $\lambda,{{R}_{\text{L}}}\gg R+d$ and $\gamma =0.71$ when $\lambda,{{R}_{\text{L}}}\ll R+d$ [13].

The important parameter in equation (7) is the diffusion parameter $\psi =\psi (B,W)$ . As the number of collisions in the probe sheath increases, so does the value of the diffusion parameter. In the presence of a magnetic field B, $\psi$ depends on the electron free path $\lambda (W)$ and the Larmor radius ${{R}_{\text{L}}}(W,B)$ , as well as on the shape, size and orientation of the probe with respect to the magnetic field.

The problem of evaluating the diffusion parameter in different cases of Langmuir probe application to plasma diagnostics in the presence of a magnetic field needs special attention.

In the absence of a magnetic field, the kinetic equation for an isotropic EEDF is [13]

$\begin{eqnarray}&&{{\nabla}_{r}}D(W){{\nabla}_{r}}f(W,r)=0\end{eqnarray} \tag{ 8 }$

where the diffusion coefficient in the non-local approach in the probe's vicinity is D(W) = cD_r = c²λ /3. The boundary conditions are

$\begin{eqnarray}&&f(r\to \infty,W)={{f}_{\infty}}(W)\end{eqnarray} \tag{ 9 }$

$\begin{eqnarray}&&f\left(R,W>eU\right)=\gamma \,{{f}_{1}}(R,W).\end{eqnarray} \tag{ 10 }$

Here the anisotropic part of the EEDF is f₁(r,W) = $-\lambda {{\nabla}_{r}}f(r,W)=-\lambda \partial f/\partial r.$

To generalize the problem, the probe is considered as an ellipsoid of revolution [14] with dimensions R and $b=L/2$ (L being the probe length). Since the probe surface is an equipotential, the diffusion equation (8) depends on the elliptical coordinate σ determined as

$\begin{eqnarray}&&\frac{{{x}^{2}}+{{y}^{2}}}{{{R}^{2}}\left({{\sigma}^{2}}\pm 1\right)}+\frac{{{z}^{2}}}{{{R}^{2}}{{\sigma}^{2}}}=1.\end{eqnarray} \tag{ 11 }$

At the probe surface $\sigma ={{\sigma}_{0}}=b/\beta$ , $\beta ={{\left(\mid {{b}^{2}}-{{R}^{2}}\mid \right)}^{1/2}}$ and the '+' and '−' signs refer to oblate (b < R) and prolate (b > R) ellipsoids, respectively.

The boundary condition (9) remains the same, but the condition (10) can be written as

$\begin{eqnarray}&&\sigma ={{\sigma}_{0}}\,\,\frac{\lambda}{S}2\pi \beta \left({{\sigma}^{2}}-1\right)\frac{\partial}{\partial \sigma}f=\frac{1}{\gamma}f.\end{eqnarray} \tag{ 12 }$

The solution of equation (8) with boundary conditions (9) and (12) yields the extended equation for electron probe current (7) with diffusion parameter:

$\begin{eqnarray}&&\psi (W)=\frac{S}{4\pi \beta \gamma \lambda (W)}{\int}_{{{\sigma}_{0}}}^{\infty}\frac{D(W)\text{d}\sigma}{\left({{\sigma}^{2}}\mp 1\right)D\left(W-e\phi (\sigma )\right)}.\end{eqnarray} \tag{ 13 }$

Here, D(W) is the diffusion coefficient in the bulk plasma and $D\left(W-e\phi (\sigma )\right)$ is that of the probe sheath ( $\phi (\sigma )$ being the potential variation introduced by the probe). Assuming a constant coefficient of diffusion, for a thin sheath we have

$\begin{eqnarray}&&\psi (W)=\frac{S}{8\pi \beta \lambda (W)\gamma}\ln \left(\frac{{{\sigma}_{0}}+1}{{{\sigma}_{0}}-1}\right)\,\,b~>~R\\ &&\psi (W)=\frac{S}{8\pi \beta \lambda (W)\gamma}\left(\pi -2\arctan {{\sigma}_{0}}\right)\,\,b~ ~R.\end{eqnarray} \tag{ 14 }$

In the presence of a magnetic field, the coefficient of diffusion D(W) in equation (8) becomes a tensor with two components [15], perpendicular ( $\bot$ ) and parallel (||):

$\begin{eqnarray}&&{{D}_{\parallel}}(W)={{c}^{2}}\lambda /3\,\,\text{and}\,\,\,{{D}_{\bot}}(W)={{D}_{\parallel}}(W)/{{\rho}^{2}},\\ &&\,\,\,\text{where}\,\,\rho =\,{{\left[1+{{\left(\frac{\lambda}{{{R}_{\text{L}}}}\right)}^{2}}\right]}^{1/2}}.\end{eqnarray}$

In this case, the kinetic equation for the EEDF has the form of the anisotropic diffusion equation:

$\begin{eqnarray}&&{{D}_{\bot}}(W){{\Delta }_{r\,}}f+{{D}_{\parallel}}(W){{\Delta }_{z}}\,f=0.\end{eqnarray} \tag{ 15 }$

Let us consider a probe placed along the magnetic field. Changing the scale $z\to {{z}^{\prime}}/\rho$ we can reduce the problem to the one solved above so that for the diffusion parameter we have

$\begin{eqnarray}&&{{\psi}_{\parallel}}(B,W)=\frac{S\rho}{8\pi \beta _{\parallel}^{M}\lambda (W)\gamma}\ln \left(\frac{\sigma _{\parallel}^{M}+1}{\sigma _{\parallel}^{M}-1}\right),\,\,{{b}^{\prime}}>\,R\end{eqnarray} \tag{ 16a }$

$\begin{eqnarray}&&{{\psi}_{\parallel}}(B,W)=\frac{S\rho}{8\pi \beta _{\parallel}^{M}\lambda (W)\gamma}\left(\pi -2\arctan \,\sigma _{\parallel}^{M}\right),\,\,{{b}^{\prime}}<\,R\end{eqnarray} \tag{ 16b }$

where

$\begin{eqnarray}&&\beta _{\parallel}^{M}={{\left(\mid {{R}^{2}}-{{b}^{\prime 2}}\mid \right)}^{1/2}},\,\,{{b}^{\prime}}=\frac{b}{\rho},\,\,\sigma _{\parallel}^{M}=\frac{{{b}^{\prime}}}{{{\left(\mid {{R}^{2}}-{{b}^{\prime 2}}\mid \right)}^{1/2}}}=\frac{{{b}^{\prime}}}{\beta _{\parallel}^{M}}.\end{eqnarray}$

For a probe placed across the magnetic field, using this approach with general ellipsoidal coordinates, the diffusion parameter is found to be [13]

$\begin{eqnarray}&&{{\psi}_{\bot}}(B,W)=\frac{S\rho}{4\pi \beta _{\bot}^{M}\lambda (W)\gamma}F\left(\phi /\alpha \right)\end{eqnarray} \tag{ 17 }$

where $F\left(\phi /\alpha \right)$ is an incomplete elliptical integral of the first kind [16] and

$\begin{eqnarray}&&\beta _{\bot}^{M}={{\left(\mid {{R}^{\prime 2}}-{{b}^{2}}\mid \right)}^{1/2}},\,\,{{R}^{\prime}}=\frac{R}{\rho},\,\,\cos (\phi )=\frac{a}{b},\\ &&\qquad \ \,\times \cos (\alpha )=\frac{\left({{R}^{2}}-{{R}^{\prime 2}}\right)}{\beta _{\bot}^{M}}.\end{eqnarray}$

These results were presented at the end of the paper [13] without experimental verification. The equations are important, but for practical use they need to be simplified bearing in mind the actual plasma conditions.

Let us consider a cylindrical probe with radius R = 1 × 10⁻⁴ m and length L = 5 × 10⁻³ m [10]. For a homogeneous argon gas discharge at a gas pressure p ~ 1 Pa and magnetic fields in the range B = 0.01 – 0.1 T,

$\begin{eqnarray}&&\lambda \sim 1\,\text{m,}\,\,{{R}_{\text{L}}}\sim {{10}^{-5}}\,\text{m} ~\text{and} ~ ~\rho ={{\left[1+{{\left(\frac{\lambda}{{{R}_{L}}}\right)}^{2}}\right]}^{1/2}}\approx \frac{\lambda (W)}{{{R}_{\text{L}}}(B,W)}\sim {{10}^{5}}.\end{eqnarray}$

Then for the probe located along the magnetic field $R\gg {{b}^{\prime}}$ ; $\pi \gg 2\arctan \,{{10}^{-3}}$ , $\sigma _{\parallel}^{M}\sim {{10}^{-3}}$ and using equation (16b) we arrive at

$\begin{eqnarray}&&{{\psi}_{\parallel}}(B,W)=\frac{b\rho}{2\lambda (W)\gamma}\pi =\frac{\pi L}{4\gamma {{R}_{\text{L}}}(B,W)}=\frac{\psi _{\,0}^{\,\parallel}}{\sqrt{W}}.\end{eqnarray} \tag{ 18 }$

Here $\psi \,_{0}^{\parallel}$ is constant with respect to the energy part of the diffusion parameter.

When the probe is placed across the magnetic field, using equation (17) we obtain

$\begin{eqnarray}&&{{\psi}_{\bot}}(B,W)=\frac{R}{\gamma {{R}_{\text{L}}}(B,W)}F\left(\phi /\alpha \right).\end{eqnarray} \tag{ 19 }$

Demidov et al [17, 18] used a different approach for the diffusion parameter when the probe is placed across the magnetic field:

$\begin{eqnarray}&&{{\psi}_{\bot}}(B,W)=\frac{R}{\gamma {{R}_{\text{L}}}(B,W)}\ln \left(\frac{\pi L}{4R}\right).\end{eqnarray} \tag{ 20 }$

When $\lambda \gg {{R}_{\text{L}}}$ equation (20) yields a value for the diffusion parameter lower by 6–8% than the one given by equation (19), but this difference practically does not affect the general results for the plasma parameters acquired.

In the case of the probe placed across the magnetic field we can also use the diffusion parameter in the form of

$\begin{eqnarray}&&{{\psi}_{\bot}}(B,W)=\frac{\psi _{0}^{\bot}}{\sqrt{W}}.\end{eqnarray} \tag{ 21 }$

It has to be pointed out that in the equations for the diffusion parameter, the probe radius and length are taken into account. This provides the possibility of using probes of larger size than is usually accepted.

For more complicated inhomogeneous plasmas (flowing, turbulent, chemically active, etc) there are indications [6, 17, 20] that the probe length L in equations (18) and (20) has to be replaced by the characteristic length of the inhomogeneity L'. This concerns the evaluation of the diffusion parameter for probe measurements in strongly turbulent fusion plasmas. Two assumptions were made in the approach presented in [20], as follows:

(i)
For the characteristic length of the inhomogeneity L' we accept the characteristic size of the cross section of the turbulence (blobs), which is typically of the order of 0.01 m.
(ii)
The second important point is the ratio of the coefficients of the global diffusion D(W) to diffusion in the vicinity of the probe D(W – eϕ(r)) in equation (13). In strongly turbulent plasma, the global diffusion is given by the Bohm diffusion and, therefore, we can write [22, 23]
$\begin{eqnarray}&&\frac{D(W)}{D\left(W-e\phi (r)\right)}=\frac{{{D}_{\text{Bohm}}}}{D}=\frac{1}{16}\frac{\lambda (W)}{{{R}_{\text{L}}}(B,W)}\approx \frac{\rho}{16}.\end{eqnarray} \tag{ 22 }$

These assumptions yield different values for the diffusion parameters in the case of fusion plasmas:

$\begin{eqnarray}&&{{\psi}_{\bot}}(B,W)=\frac{R\ln \left(\frac{\pi {{L}^{\prime}}}{4R}\right)}{16\gamma {{R}_{\text{L}}}(B,W)}\,\text{and}\,\,{{\psi}_{\,||}}(B,W)=\frac{\pi {{L}^{\prime}}}{64\gamma {{R}_{\text{L}}}(B,W)}.\end{eqnarray} \tag{ 23 }$

Let us consider now the limiting cases regarding the value of the diffusion parameter:

1.
When $\psi (B,W)\ll 1$ (the absence of or a very weak magnetic field), neglecting the second term $\frac{(W-eU)}{W}\psi$ in the denominator of the integral in equation (7) yields the classical expression (5) for the electron probe current and the EEPF can be determined by using the Druyvesteyn formula (6). Although, as was shown by Swift [11], a drain of electrons to the probe is present at a finite R/λ ratio, the true value of the EEDF at low probe potentials differs by less than 25% from that determined by the classical theory and the EEDF is well characterized by I''(U).
2.
When $\psi (B,W)\sim 1$ (a weak magnetic field) we have to use equation (7). Its second derivative yields the equation
$\begin{eqnarray}&&{{I}^{\prime\prime}}(U)=Cf(eU)-C{\int}_{eU}^{\infty}{{K}^{\prime\prime}}(W,U)\,{{f}_{0}}(W)\text{d}W\end{eqnarray} \tag{ 24 }$
where ${{K}^{\prime\prime}}(W,U)=2\psi {{W}^{2}}/{{\left[W(1+\psi )-\psi eU\right]}^{3}}$ and C = $8\pi {{e}^{3}}S/3{{m}^{2}}\gamma$ .The first term in equation (24) is the well-known Druyvesteyn formula. The second term describes the effect of plasma depletion caused by charged particles sinking on the probe surface.Figure 2 presents model calculations at $\psi \sim 1$ for a Maxwellian EEDF with temperature T = 3 eV. It is seen that the second derivative is not a good representation of the EEPF because ${{K}^{\prime\prime}}(W,U)$ increases indefinitely and I''(U) decreases at small values of U and the electron energy. Only the high-energy part of the EEPF at small R is well characterized by I''(U) because ${{K}^{\prime\prime}}(W,U)$ decreases with an increase in U and in the energy of the electrons. In addition, the plasma potential U_pl does not coincide with the potential U^* at which the second derivative is zero, I''(U^*) = 0, as is usually assumed [5, 19]. Consequently, an additional error will appear if the electron density is obtained by integration over the second derivative according to equation (3).In the case of a Maxwellian EEDF at $\psi (B,W)\sim 1$ , a refined procedure based on obtaining the best fit between model calculations and experimental data was proposed and proved in [19]. The fitting parameters are the electron temperature, T_e, the electron density, n_e, and the plasma potential, U_pl. The temperature is evaluated at high probe potentials, where the second derivative is relatively weakly distorted. The next step is to calculate the second derivative following equation (24) using a Maxwellian EEDF normalized to unity. The results of the model calculations are fitted to the experimental curve using its maximum at low probe potentials: The plasma potential is evaluated by shifting the model curve along the U-axis, while the electron density is estimated by multiplying the model data by a coefficient to achieve the best fit.
3.
When $\psi (B,W)\gg 1$ (high values of the magnetic field) the EEDF is represented by the first derivative instead of the second derivative as was shown in [14, 17, 20]:
$\begin{eqnarray}&&I_{e}^{\prime}(U)=-\frac{8\pi {{e}^{2}}S}{3{{m}^{2}}\gamma}\left(\frac{eU}{\psi}f(eU)\right. \\ &&\left. \qquad \quad \ \;+{\int}_{eU}^{\infty}\frac{W{{f}_{0}}(W)\text{d}W}{(1+\psi )\left[(1+\psi )W-\psi eU\right]}\right).\end{eqnarray} \tag{ 25 }$

**Figure 2.** Second-derivative model curve (dashed line) for a Maxwellian EEDF (T_e = 3 eV) at $\psi \sim 1$ . The solid line presents the second derivative when $\psi \ll 1$ .
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**Figure 2.** Second-derivative model curve (dashed line) for a Maxwellian EEDF (T_e = 3 eV) at $\psi \sim 1$ . The solid line presents the second derivative when $\psi \ll 1$ .
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In the case of very high values of the magnetic field, the contribution of the second term in equation (25) is usually assumed to be small and is therefore neglected. Then, the EEPF is directly represented by the first derivative of the electron probe current:

$\begin{eqnarray}&&f(\varepsilon )=\frac{3\sqrt{2m}\gamma}{2{{e}^{3}}S}\frac{\psi}{U}\frac{\text{d}{{I}_{e}}}{\text{d}U}.\end{eqnarray} \tag{ 26 }$

A comparison of the experimental and model EEPF at electron temperature T_e = 2.5 eV and $\psi (B,W)\gg 1$ is presented in figure 3. The discrepancy in the range from 0 eV to T_e is due to the mathematical approach rather than to physical phenomena [5, 20].

**Figure 3.** Comparison of the experimental (solid line) and model (dashed line) EEPF at electron temperature T_e = 2.5 eV and $\psi \gg 1$ .
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Accurate evaluation of the EEPF requires that the value of the plasma potential U_pl be known. It was shown in [17, 20] that for strong magnetic fields, i.e. for high values of the diffusion parameter $\psi (B,W)\gg 1$ , for a Maxwellian EEDF the minimum of the first derivative of the IV characteristics is shifted to the negative side by 1.1–1.4T^* (the value of T^* being equal to the value of T_e, expressed in volts) with respect to the plasma potential. A more or less pronounced change of the slope of the first derivative is seen at the position of the plasma potential. This is associated with the transition from diffusion of electrons to their drift due to the change in probe potentials from negative to positive with respect to the plasma potential. In [21] we proposed the following procedure for a precise evaluation of the plasma potential. The location of the knee allows us to estimate the position of the plasma potential. Since this position is not very well defined (figure 4(b)), we consider the estimated value as a first approximation for the plasma potential. Then, using equation (26), we evaluate the EEPF and, thus, the electron temperature and density. With these values, a model IV probe characteristic is calculated from equation (7) and compared with the experimental one. If there is a discrepancy, as is shown in figure 4(c), then we make a correction in the value of the plasma potential and calculate the IV probe characteristic repeatedly until the best fit between the calculated and the measured IV is obtained (figure 4(d)).

**Figure 4.** (a) Measured IV probe characteristics in a strong magnetic field. (b) First derivative of the smoothed experimental IV curve (dots) and model curve (solid line) using equation (7). (c) Comparison between the experimental and model curves after the first iteration. (d) Comparison between the experimental and model curves after the final iteration.
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3. Experimental results and discussion

3.1. Probe measurements in the presence of a magnetic field in the range 0.01–0.08 T

We will now present results from the experiments carried out at Jožef Stefan Institute, Ljubljana, Slovenia [10]. The probe measurements were performed in low-pressure (p = 0.8 Pa) argon gas discharges in the presence of a magnetic field in the range 0.010–0.079 T. The plasma was produced in a stainless steel discharge tube with a length of 1.5 m and a diameter of 0.17 m (figure 5) with a hot filament cathode. The discharge tube wall was grounded. A negative potential of −35 V was applied to the cathode, while the gas discharge current was kept constant at 2 A. The axial magnetic field B was created by a solenoid.

**Figure 5.** Schematic representation of the experimental set-up at Jožef Stefan Institute, Ljubljana, Slovenia.
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A cylindrical Langmuir probe with R = 1 × 10⁻⁴ m and length L = 5 × 10⁻³ m was placed axially and radially at the center of the discharge tube. The probe surface was cleaned by ion bombardment by applying a large negative potential before measuring the IV probe characteristics. The measurements were performed with the probe oriented parallel and perpendicular to the magnetic field. The derivatives were calculated numerically. The results of the measurements of the plasma parameters at different values of the magnetic field are presented below.

An example of the results obtained in an argon gas discharge with the probe being oriented perpendicular to a low (0.025 T) magnetic field is presented in figure 6. The constant part of the diffusion parameter (equation (21)) is $\psi _{0}^{\bot}=2$ . Using the extended second-derivative equation (24), the EEDF was found to be Maxwellian with a temperature of 2.9 eV and an electron density of 1.5 × 10¹⁸ m⁻³; the plasma potential was 4 V. The derivatives were calculated numerically. The instrumental function of the differentiation technique is triangular with a half-width equal to the step of the change in probe bias [24]. To take into account the influence of the instrumental function when the experimental and model curves were compared, both of them were smoothed and differentiated in the same way [19]. The accuracy of evaluation of the electron temperature was 10%. Taking into account the uncertainties in all values measured, the uncertainty in the evaluation of electron densities did not exceed 25%. The accuracy of evaluation of the plasma potential was 20%.

**Figure 6.** (a) IV probe characteristic measured in an argon gas discharge in the presence of a magnetic field of 0.025 T. (b) Comparison of the experimental second derivative of the electron probe current (dots) with the model calculations (dashed line) in the case of a Maxwellian EEPF (solid line).
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When the value of the magnetic field was increased, a result was found that was at a first glance unexpected: in measurements with the probe parallel to a magnetic field of 0.035 T and the corresponding $\psi \,_{0}^{\parallel}=30$ , both techniques—the extended second-derivative and the first-derivative one—provided reliable results for the plasma parameters, as shown in figures 7(a) and (b).

**Figure 7.** (a) Comparison of the experimental second derivative of the electron probe current (dots) with the model calculations (dashed line) in the case of a Maxwellian EEPF (solid line). (b) Evaluated EEDF (solid line) in an argon gas discharge in the presence of a magnetic field of 0.035 T and model Maxwellian EEPF (dashed line) at an electron temperature of 2.1 eV.
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The first-derivative probe technique (expression (26)) was used to evaluate the EEPF in a magnetic field B = 0.055 T in an argon gas discharge. Figure 8(a) presents the experimental IV probe characteristic measured with the probe parallel to the magnetic field. Figure 8(b) is a comparison between the experimental first derivative and the model curve with the aim of evaluating the plasma potential. Figure 8(c) shows the evaluated EEPF (solid line). The electron-energy distribution function is generally Maxwellian, although beyond the energy of the first excited state of argon (11.55 eV), the influence of a group of faster electrons can be seen. The effect becomes more pronounced as the value of the applied magnetic field is raised. To clarify this, additional experiments must be performed at different gas pressures and different discharge currents.

Comparing the application of the two techniques, the extended second-derivative and the first-derivative one, we have to point out that the first-derivative probe technique yields the EEPF directly not only in a Maxwellian case, but also when the energy distribution of the electrons deviates from Maxwellian. The application of the extended second-derivative probe technique when the EEDF is non-Maxwellian is more complicated: mathematically speaking, deducing the EEPF from probe measurements under collisional conditions requires that two coupled inverse problems be solved, since the distortion of the second derivative includes an integral over the unknown EEPF and the data are convoluted by the instrumental function [25].

3.2. Probe measurements in the presence of magnetic fields in the range 0.04–0.3 T

Chronologically, Demidov [17, 18] was the first to used the results from the probe theory in a non-local approach for systematic evaluation of the EEDF in the presence of strong magnetic fields. The application of the first-derivative probe technique in the presence of magnetic fields in the range 0.04–0.3 T was studied in detail. Electron-energy distributions in low-pressure (~0.2 Pa) helium plasmas of the Blaamann toroidal device [26] were measured. It was shown that for the conditions investigated, the technique yields the same electron densities and slightly lower temperatures (about 10%) than the kinetic theory for an arbitrary magnetic field strength. Cylindrical probes, which were oriented along and perpendicular to the magnetic field, were used in the measurements. It was shown that these probes give nearly identical results. The cross-field diffusion coefficient of electrons near the probe was estimated and shown to be classical. The results were summarized in the review article [6].

3.3. Probe measurements in high-temperature strongly turbulent fusion plasmas (magnetic fields from 0.45 to 1.3 T)

As was mentioned above, in magnetized plasma the electron part of the IV characteristics above the floating potential, U_fl, is strongly distorted due to the influence of the magnetic field. Therefore, in fusion plasmas, the ion saturation branch of the IV characteristics and the part around the floating potential are usually used when retrieving the plasma parameters, namely, the plasma potential, electron temperature and density [27, 28]. This technique assumes a Maxwellian EEDF, but in fact does not measure the real one. The approximation for the probe current as a function of the probe potential U_p around the floating potential U_fl is given by

$\begin{eqnarray}&&I\left({{U}_{\text{p}}}\right)=I_{\text{s}}^{\text{i}}\left\{1-\exp \left[-\frac{e\left({{U}_{\text{fl}}}-{{U}_{\text{p}}}\right)}{{{T}_{\text{e}}}}\right]\right\}.\end{eqnarray} \tag{ 27 }$

In this expression the ion saturation current is

$\begin{eqnarray}&&I_{\text{s}}^{\text{i}}=0.5e{{n}_{\text{e}}}{{c}_{\text{s}}}\,{{A}_{\text{p}}},\end{eqnarray} \tag{ 28 }$

where A_p is the area of the probe projection in the direction of the magnetic field $\vec{B}$ and ${{c}_{\text{s}}}={{\left[e\left({{T}_{\text{e}}}+{{T}_{\text{i}}}\right)/{{m}_{\text{i}}}\right]}^{1/2}}$ is the ion acoustic velocity. ${{T}_{\text{e}}}$ and ${{T}_{\text{i}}}$ are the electron and ion temperatures in eV.

The difference between the plasma potential and the floating potential ${{U}_{\text{pf}}}$ is given by [29]

$\begin{eqnarray}&&{{U}_{\text{pf}}}=\frac{k{{T}_{\text{e}}}}{2e}\ln \left[2\pi \frac{{{m}_{\text{e}}}}{{{m}_{\text{i}}}}\left(1+\frac{{{T}_{\text{i}}}}{{{T}_{\text{e}}}}\right){{(1-\delta )}^{-2}}\right].\end{eqnarray} \tag{ 29 }$

Here ${{m}_{\text{e}}}$ and ${{m}_{\text{i}}}$ are the electron and ion masses; $\delta$ is the coefficient of secondary emission [30, 31].

Even in the Maxwellian case, to evaluate the plasma potential and electron density one must know the ion temperature ${{T}_{\text{i}}}$ . Additional experiments (usually using a retarding field analyzer, RFA) are required—results obtained in different tokamaks show that in edge fusion plasma the ion temperature can exceed the electron temperature by a factor of 2–4 [32–35].

In fusion plasmas, the assumption of a Maxwellian EEDF is generally valid. However, experimental evidence does exist suggesting non-Maxwellian distributions in tokamak scrape-off layer (SOL) plasmas. In ASDEX, the electron temperatures measured by a Langmuir probe by the technique described above are about twice as high as those determined by Thomson scattering [36, 37]. The discrepancy was explained by the presence of a weakly populated high-temperature electron fraction, together with a predominant electron population with a lower temperature. In this case, the EEDF can be considered as being bi-Maxwellian. As a result of 1D particle-in-cell modeling, Chodura [38], Batishchev [39] and Tskhakaya [40] showed non-Maxwellian distributions in the SOL. It is clear that knowledge of the real EEDF is of importance in understanding the underlying physics of the processes occurring at the SOL, plasma–wall interactions, etc.

3.3.1. Probe measurements in the ISTTOK tokamak.

For illustration, we will first present results obtained by Dimitrova and the ISTTOK tokamak team [41] for a tokamak toroidal magnetic field of 0.45 T. Circular discharges (shots) in hydrogen with a plasma current of 4 kA and a line-averaged electron density of 3.5 × 10¹⁸ m⁻³ were studied. Radial measurements were performed of the current–voltage characteristics with probes with a length of 3 mm and a diameter of 0.75 mm, one oriented parallel and the other perpendicular to the tokamak magnetic field, mounted on a horizontal manipulator. The radial position of the manipulator was changed for different reproducible discharges. The probes were biased with respect to the tokamak chamber wall by a 1 kHz triangular voltage U_p(t) up to ±100 V [42]. The probe bias and probe current versus time were recorded by tokamak DAQ and the IV characteristics were constructed from these data.

An example of the results from the steady-state phase of the discharge at a probe position 80 mm from the tokamak centre is presented in figure 9. The difference in the probe current measured by probes with different orientations is in agreement with equation (28) for the ion saturation current and equation (7) for the electron probe current at different values of the diffusion parameter ( $\psi {{\,}^{||}}(\varepsilon )=65/\sqrt{\varepsilon}$ and ${{\psi}^{\bot}}(\varepsilon )=10/\sqrt{\varepsilon}$ ). Figure 10 presents the ion saturation current densities ${{J}_{\text{sat}}}$ at different radial positions from the center of the tokamak chamber. The positions of the last closed magnetic flux surface (LCFS) at 75 mm and the limiter at 85 mm are indicated by dashed lines.

**Figure 10.** Ion saturation current densities at different radial positions from the tokamak chamber centre.
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The EEPFs were evaluated using the first-derivative probe technique and the measured IV characteristics. The results from the two probes were identical. Examples are presented in figures 11(a) and (b). In the shadow of the limiter (88 mm from the centre), the EEPF was found to be Maxwellian (figure 11(a)) with electron temperature ${{T}_{\text{e}}}=7\pm 0.4$ eV and electron density ${{n}_{\text{e}}}=(4.4\pm 1.1)\times {{10}^{17}}$ m⁻³. In contrast, in the deepest positions, the EEPF deviates from Maxwellian but can be approximated by a bi-Maxwellian one. At 80 mm from the centre, a dominant electron group ( ${{T}_{\text{e}}}=4\pm 0.2$ eV and ${{n}_{\text{e}}}=(7\pm 2)\times {{10}^{17}}$ m⁻³) and a minority higher-energy electron group with temperature ${{T}_{\text{e}}}=12\pm 0.6$ eV and ${{n}_{\text{e}}}=(2.2\pm 0.6)\times {{10}^{17}}$ m⁻³ were registered.

**Figure 11.** (a) Maxwellian EEPF in the limiter shadow at 88 mm and (b) bi-Maxwellian EEPF at a probe position of 80 mm from the centre of the tokamak chamber.
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The radial distribution of the plasma potential is presented in figure 12(a). The results from both probes using the first-derivative probe technique coincide within the experimental error of ±5 V. Figure 12(a) presents a comparison with results from the 'classical' probe technique (CPT) using equation (29). The ion temperatures were measured by the RFA technique [33]. It is clearly seen that the plasma potential increases in the direction from the wall towards the center, reaches a maximum close to the LCFS and then decreases again. This is in agreement with the results reported in [43, 44].

**Figure 12.** (a) Radial distribution of the plasma potential, (b) electron temperatures ${{T}_{\text{e}}}$ and (c) electron densities ${{n}_{\text{e}}}$ obtained by parallel and perpendicular probes in the ISTTOK tokamak.
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Figures 12(b) and (c) present the radial distribution of the electron temperatures T_e and densities n_e retrieved by the first-derivative probe technique. The triangles and squares indicate the low and high temperatures, respectively, of the bi-Maxwellian EEDFs. The dots correspond to the Maxwellian EEDFs. The empty triangles in figure 12(b) represent the electron temperatures obtained by the CPT assuming a Maxwellian distribution (using the approximation of equation (27)). As predicted, a good agreement is observed in ${{T}_{\text{e}}}$ for both techniques when the EEDF is found to be Maxwellian. It should be pointed out that the techniques can be considered as independent: the approaches are different and they use different parts of the IV measured (i.e. ion saturation current and the part around the floating potential for the 'classical' technique, and the electron part of the IV probe characteristics for the first-derivative technique). On the other hand, in the case of a bi-Maxwellian EEDF, the data from the CPT are strongly influenced by the high-energy tail of the electron-energy distribution [20]. It is well known that in this case the part of the IV around the floating potential is determined by the high-energy fraction, although it amounts to a few per cent of the density of the main low-energy electron group [45].

The results presented confirm the consistency of the first-derivative probe technique for precise evaluation of the plasma potential and the real EEDF in tokamak edge plasma.

3.3.2. Probe measurements in the CASTOR tokamak.

One can find a confirmation of this statement in the results obtained [20] in the old Czech tokamak CASTOR, at the Institute of Plasma Physics, Academy of Sciences of the Czech Republic, in Prague. Chronologically, these results were the first evaluation of the real EEDF in fusion plasma using the first-derivative probe technique.

To evaluate the EEPFs, the IV characteristics were measured by using two arrays of cylindrical probe tips (a rake probe) with a length of 2 mm and a radius of 0.35 mm The first rake probe consisted of 16 tips oriented perpendicular to the magnetic field lines and the second of 12 probes oriented parallel to the lines. The rake probes were inserted in the edge plasma from the top of the tokamak. The tips were displaced radially by 2.5 mm from each other and covered a range of about 40 mm in the plasma. All probe tips were biased simultaneously by a triangular voltage U_p(t) with respect to the tokamak chamber wall, which serves as a reference electrode with a temporal resolution of 1 μs. The time necessary to measure a single IV characteristic was typically ~1 ms. The measurements were carried out at the CASTOR tokamak edge hydrogen plasma in two reproducible circular discharges corresponding to probes oriented perpendicular and parallel to the magnetic field. The main plasma parameters of these two discharges were: discharge current 10 kA, line-averaged electron density 8 × 10¹⁸ m⁻³. The magnetic field was 1.3 T. The measurements were performed during the steady-state part of the discharges at a low (2.5 V) loop voltage and with low recycling at the chamber.

The results obtained are presented in figure 13. Hereinafter, the same symbols as in figure 12 are used.

The results obtained in ISTTOK and CASTOR show similar features, which is understandable because the two tokamaks have similar parameters: limiter tokamaks with circular cross section of the chamber; ISTTOK [46] has a major radius of 46 cm, CASTOR 40 cm [47], and they have the same minor radius of 8.5 cm, similar electrical and discharge parameters, etc.

3.3.3. Probe measurements in the COMPASS tokamak.

Research on the CASTOR tokamak was conducted until the end of 2006, when the operation was discontinued due to the installation of the new facility, namely the COMPASS tokamak. COMPASS (COMPact ASSembly) was reinstalled in 2007 at the new tokamak facility in IPP.CR, Prague. The COMPASS tokamak [48] with its size (a major radius of 0.6 m and a minor radius of 0.23 m) ranks among the smaller tokamaks capable of H-mode operation, which represents a reference operation ('standard scenario') for the ITER tokamak. Importantly, due to its size and shape, the COMPASS plasmas correspond to one tenth (in linear scale) of the ITER plasmas. At present, besides COMPASS there are only two operational tokamaks in Europe with ITER-like configuration capable of the regime with the high plasma confinement. They are the German tokamak ASDEX-U (Institut für Plasmaphysik, Garching, Germany) and the Joint European Torus (JET), currently the biggest experimental device of this type in the world.

The COMPASS tokamak is well equipped with diagnostic tools, including a probe set consisting of a vertical reciprocating probe (VRP), a horizontal reciprocating probe (HRP) and 39 divertor probes (DP).

The first measurements of EEPF by the first-derivative probe technique (FDPT) were carried out in reproducible L-mode circular ohmic discharges in hydrogen leaning on the inboard belt limiter with plasma current ${{I}_{\text{pl}}}\sim 120$ kA and a magnetic field B = 1.15 T. A VRP was used to diagnose the radial distribution of the plasma parameters. The semi-cylindrical probe tip, made of graphite, with a diameter of 3 mm and a length of 5 mm, was placed parallel to the magnetic field lines. Figure 14 presents the distribution of the electron temperatures T_e and densities n_e retrieved at a vertical position Z with respect to the LCFS position Z_LCFS for shot #2568, typical of the series of reproducible discharges. It is seen that, in the region 0–20 mm outside the confined plasma, the EEDF is bi-Maxwellian, presenting two populations of electrons with low and high energy/temperature. In the rest of the profile in the far SOL, the EEDF is found to be Maxwellian.

**Figure 14.** Radial distribution of (a) the electron temperatures T_e and (b) electron densities n_e for a COMPASS circular hydrogen discharge. Triangles indicate the temperatures and densities of the low-energy fraction of the EEPF obtained by FDPT, squares the temperatures and densities of the high-energy fraction, and dots the temperatures of the Maxwellian EEPF. The empty triangles represent the electron temperatures obtained by the classical probe technique, assuming a Maxwellian distribution.
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The results for the electron temperatures and densities obtained by the FDPT can be compared with those of the core Thomson scattering (TS) diagnostics [49]. Four ${{n}_{\text{e}}}$ and ${{T}_{\text{e}}}$ TS profiles, taken during the quasi-stationary phase of the discharge, were used for this comparison (figure 15).

**Figure 15.** Comparison of profiles of the TS (asterisks) electron temperatures (a) and densities (b) with the VRP profile (dots, triangles and squares) obtained by the FDPT for a COMPASS circular shot.
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The core TS results presented were obtained from the region of the tokamak chamber mid-plane to 4 mm below the separatrix position at a step of 13 mm. The spatial resolution is about 5 mm. Detailed information about the accuracy of the TS data can be found in [50]. Although the two profiles are complementary, it is seen that the TS temperature measured in the proximity of the LCFS location is close to the temperature of the low-energy electron fraction and differs by about a factor of two from the temperature of the high-energy electron fraction. This is in agreement with the statements in [36, 37], i.e. as the TS experiment assumes a Maxwellian EEDF, the results obtained match well the temperatures and densities of the predominant low-energy electron fraction yielded by the probe measurements.

Next, experiments [21] were performed in D-shape hydrogen L-mode ohmic plasma using the same VRP described above, the HRP and 39 divertor probes. All the probes were made of graphite. The HRP cylindrical pin was oriented perpendicular to the magnetic field lines. It has a diameter of 0.75 mm and length of 1.5 mm. The dome-shaped divertor probes are embedded in the divertor tiles, are oriented parallel to the magnetic field and provide profiles with a spatial resolution down to 5 mm in the poloidal direction.

Figure 16 shows a reconstruction of the magnetic surfaces by the Equilibrium FITing code (EFIT) during the steady-state phase of the discharge #3912. The probes are numbered from #1 at the high-field side (HFS) to #39 at the low-field side (LFS) in the divertor area. The positions of the two strike points are R = 0.417 m and R = 0.503 m, which correspond to the divertors LP #6 and LP #24, respectively. Each probe has similar dimensions to the VRP.

Figure 17 presents the poloidal distribution of the plasma potential in the divertor area. The full dots represent results evaluated by the FDPT, while the data from the CPT calculations are shown by empty dots. To get good agreement between the results from the two approaches, we have to assume that in the private flux region and at the LFS the ion temperature is higher than the electron one by a factor of two. In the vicinity of the HFS strike point, the ion temperature T_i can be assumed to be higher than the temperature of hot electron fraction T_e by a factor of five, i.e. ~60 eV [34, 35].

Figure 18 shows the poloidal distribution of the electron temperatures and electron densities in the divertor area. It is seen that the EEDF is bi-Maxwellian in the inner and outer divertor regions. In the central part of the private flux region, the EEDF is Maxwellian.

Bi-Maxwellian EEDFs in the liquid-lithium divertor area of NSTX were reported also by Jaworski et al [51, 52].

The EEDF results obtained by the VRP and HRP are similar to those obtained in circular discharges: in the vicinity of the LCFS, the EEDF is bi-Maxwellian, while in the far SOL it is Maxwellian. Having acquired results from all probes (VRP, HRP, DP), we can compare the values of the electron temperatures and densities and the plasma potential on the same magnetic surface (figure 16). The open magnetic surface (red line in figure 16) goes through the position of divertor probe #30. Figures 19(a)–(c) present the results of the measurements at three different moments of time during the discharge. As one can see, there is a good agreement for the plasma potential (figure 19(a) between VRP and HRP values obtained. The values for DP#30 are lower because of the drop in sheath potential at the divertor tiles [29].

**Figure 19.** The values of (a) plasma potential, (b) electron temperatures and (c) electron densities obtained by VRP, HRP and DP #30 at magnetic surfaces close to the LCFS at three different moments of time for the discharge as presented in figure 16. Diamonds: plasma potential; triangles: temperature or density of slow electrons; squares: temperature or density of fast electrons. Magenta: VRP; green: HRP; black: DP#30.
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Also, there is a good agreement for the values of the electron temperatures and densities. Similar results were obtained in D-shaped deuterium L-mode ohmic plasmas.

3.3.4. Probe measurements in TJ-II stellarator.

Measurements in the TJ-II stellarator [53] were carried out in deuterium plasmas with magnetic field strength B = 1 T. During the first half of the discharges plasma was heated by an electron cyclotron resonance heating (ECRH, 2 × 250 kW gyrotrons, at 53.2 GHz, second harmonic, X-mode polarization)—the low line-averaged density phase. The second half of the discharge was heated by neutral beam injection heating (NBIH, one beam of 400 kW, port-through (H₀) power at 30 kV)—the high line-averaged density regime. A tungsten Langmuir probe with a length of 2 mm and diameter of 0.75 mm was immersed at different vertical positions with respect to the LCFS (figure 20). To be able to measure at different phases of the discharge (ECR and NBI heating), we chose a different approach than in the COMPASS tokamak: we performed a series of reproducible discharges with continuous gas puffing during the shots (from shot #34509 to shot #34537) in plasma with ι(a)/2π ≈ 1.63. During the separate shots, the probe was immersed in the plasma at different vertical positions Z—from Z = 40 mm with respect to the LCFS in the SOL to Z = −18 mm in the confined plasma (the position of the LCFS being set to zero).

**Figure 20.** Positions of the probe and LCFS indicated by the dotted line in TJ-II.
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The averaged plasma densities for the series of reproducible discharges are presented in figure 21. In the ECRH phase, the densities differ within 18%, while in the NBIH phase, there is a 30% discrepancy. For each probe position, the plasma parameters of the ECRH phase (with a duration of about 60 ms) were evaluated at 1090 ms, and those of the NBIH phase (with a duration of about 80 ms) at 1160 ms, as indicated by dashed lines in figure 8. Again, the probe was biased with respect to the chamber wall by a triangular voltage U_p(t) swept at a frequency of 1 kHz supplied by a KEPCO 100-4M power supply. The probe current signal was filtered by 10 kHz low-pass filter and recorded by the TJ-II DAQ system.

Using the FDPT [54], the radial distributions of the plasma potential for the TJ-II stellarator during the NBIH phase and during the ECRH phase are evaluated and presented in figure 22. In agreement with Langmuir probe measurements and general considerations of neoclassical transport, the NBIH phase is normally characterized by higher densities and all negative radial electric fields, while the ECRH phase, due to the heating system and the typically low densities, has positive electric fields in the core plasma that sometimes reach the edge region [43, 44]. It is seen that during the NBIH phase the plasma potential increases in the SOL, reaching a maximum around the position of the LCFS, and then decreases monotonically to negative values in the confined plasma.

**Figure 22.** Plasma potential profiles in the vertical direction Z with respect to the LCFS position Z_LCFS for the TJ-II stellarator during the high-density NBIH phase (dots) and during the low-density ECRH phase (empty squares).
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During the ECRH phase, the plasma potential increases close to the LCFS position. In the confined plasma it decreases over 5 mm and then starts to increase up to values as high as 80 V. A deviation larger than the experimental error from the smoothed radial distribution of the plasma potential can be explained by the fact that every point is from a different shot. The shots are not exactly identical. The larger difference of 18% during ECRH in the averaged density is for shots where the probe was located around 10 mm above the LCFS. This can explain the large jump in the plasma potential at this position. During NBIH, the larger difference of 30% is for a probe position of 40 mm and the corresponding plasma potential was not taken into account. This is valid also for the other plasma parameters evaluated using data measured in the TJ-II stellarator.

It has to be noted that probe measurements in the TJ-II stellarator during the NBIH phase (high density regime) yield similar results for the electron temperatures T_e and densities n_e as in the COMPASS tokamak—in the far SOL the EEDF is Maxwellian with temperatures below 10 eV. In the vicinity of the LCFS and in the confined plasma, when the electron temperature exceeds 10–15 eV, the profile splits into two branches with a low-temperature electron fraction that has a density higher than that of the high-energy electrons. These results are presented in figure 23.

**Figure 23.** Radial distribution of the electron temperatures T_e (a) and densities n_e (b) for the TJ-II stellarator during the NBI phase (high-density regime).
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In contrast, during the ECRH phase, the EEDF both in SOL and in the confined plasma is bi-Maxwellian (see figure 24)—in the SOL, the high-energy electrons have an almost constant temperature of about 20 eV, which increases up to 35 eV [55] in the confined plasma, while the low-temperature fraction has a temperature of 6–7 eV in the entire region of measurements. The population of the low-temperature fraction is higher by a factor of 2–3 that that of the higher energy one.

3.3.5. Origin of the bi-Maxwellian EEDF.

All results of EEPF measurements in the tokamaks and the TJ-II stellarator presented above show the common feature of the existence of two populations of electrons near the LCFS. We should point out that a more detailed analysis must be performed to clarify the origin of the low-energy fraction in the bi-Maxwellian EEDF. As a result of kinetic 1D particle-in-cell code simulations, Chodura [38], Batishchev [39] and Tskhakaya [40] stated that in the SOL the EEDF can deviate from Maxwellian in high recycling plasmas, while in low recycling plasmas the EEDF is mainly Maxwellian. Ionization of neutrals by thermal electrons penetrating from the bulk plasma into the SOL is considered to be the main cause of the deviation of the EEDF from a Maxwellian in the stationary SOL:

$\begin{eqnarray}&&\text{H}\,+\text{e}\,\to {{\text{H}}^{+}}+2\text{e}\text{.}\end{eqnarray} \tag{ 30 }$

In NSTX, where a bi-Maxwellian EEDF was reported in the liquid-lithium divertor area, a 'heuristic model' accounting for the inelastic collision effects (i.e. excitation and ionization of neutral hydrogen or deuterium) was also proposed to explain this EEDF feature [51].

Indeed, for electrons with temperatures of 15–20 eV, the rate coefficient $\langle \sigma v\rangle$ [29], presented in figure 25, is of the order of 10⁻¹⁴ m³ s⁻¹, and it has a maximum of ~3 × 10⁻¹⁴ m³ s⁻¹ for electron temperatures in the range 120–130 eV. For an electron temperature of 10 eV $\langle \sigma v\rangle$ is ~5 × 10⁻¹⁵ m³ s⁻¹, and at 5 eV it decreases to 9 × 10⁻¹⁶ m³ s⁻¹. Performing a detailed energy balance requires that the ionization through neutral hydrogen excited states must be taken into account as well. On the other hand, the energy of the electrons in the far SOL is below 10 eV and the most probable reactions are dissociation with threshold of 4.5 eV and excitation.

In order to validate the hypothesis that the main reason for the deviation of the EEDF from a Maxwellian in stationary SOL is the ionization of neutrals by thermal electrons penetrating from the bulk plasma into the SOL, a simplified kinetic solver for neutrals included in the ASTRA package [56] was used to calculate the distributions of deuterium atoms and the electron source, S_e, due to ionization of neutral deuterium atoms for the COMPASS diverted configuration. Figure 26 shows a comparison between the radial distribution of the calculated electron source S_e (solid line) due to the ionization of the neutral deuterium atoms and electron densities (symbols) measured for COMPASS diverted deuterium plasma. The radial distribution of the density of deuterium atoms for the same shot is presented in figure 27. A constant density of neutrals (continuous gas puffing during the discharge) in the far SOL (see figure 27) and a 3 cm decay length for the electron density and temperatures are assumed. The source was estimated after fitting the experimental density profile with ad hoc transport coefficients with fixed ${{T}_{\text{e}}}$ and ${{T}_{\text{i}}}$ profiles, and considering a typical particle confinement time τ_p = 30 ms.

**Figure 26.** Comparison between radial distribution of the electron source S_e (solid line) calculated by ASTRA and the electron densities retrieved from the FDPT for COMPASS diverted deuterium plasma.
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**Figure 27.** Radial distribution of the density of deuterium atoms calculated by ASTRA for COMPASS diverted deuterium plasma.
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It is clearly seen that the region in the vicinity of the LCFS where the EEDF is bi-Maxwellian corresponds to the source of electrons. At the same time the radial profile of neutrals corresponds to the profile of the electron source in the confined plasma. These comparisons are consistent with the statement that the main reason for bi-Maxwellian EEDF observed in the vicinity of the LCFS is the ionization of the neutral hydrogen atoms.

In the TJ-II case, the spatial distributions of neutrals and the electron source S_e were calculated using the EIRENE code [57] adapted to the TJ-II geometries [58]. The electron sources and atom density profiles are presented in figures 28 and 29, respectively, for the NBIH and ECRH phases. They were obtained by taking TS profiles [59] in the main plasma complemented with He-beam data near the edge [55], and T_i following [60]. The recycling sources were obtained for a representative TJ-II particle confinement time, τ_p = 8.6 ms.

**Figure 28.** Radial distribution of the electron source S_e calculated by EIRENE for NBIH (dots) and ECRH (dashed line) regimes of TJ-II.
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**Figure 29.** Radial distribution of the atom densities calculated by EIRENE for NBIH (dots) and ECRH (dashed line) regimes of TJ-II.
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Figures 30 and 31 compare the electron source S_e profiles, which account for the radial locations where the ionization processes are important, with the densities of the high- and low-temperature populations obtained by the FDPT. A good agreement is seen in the shape and the position between the calculated and experimentally obtained results for TJ-II.

**Figure 30.** Comparison between the radial distribution of the electron source S_e calculated by EIRENE and experimental electron densities for the NBIH regime of TJ-II.
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**Figure 31.** Comparison between the radial distribution of the electron source S_e calculated by EIRENE and experimental electron densities for the ECRH regime of TJ-II.
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It has to be mentioned that the question of the energetic distribution of the secondary low-temperature electrons remains open. It could be clarified by a comparison with the results of a kinetic 2D massive particle-in-cell modelling of SOL, which is, to our knowledge, in development [40]. Additional confirmation of the interpretation of the bi-Maxwellian distribution as being due to ionization mechanisms can be achieved by comparing the results obtained in hydrogen and deuterium plasmas with the results in helium discharges. Such experiments are foreseen during 2016 experimental campaigns in the COMPASS tokamak and in the TJ-II stellarator.

4. Conclusions

Advanced Langmuir probe techniques for evaluating the plasma potential and electron-energy distribution function in magnetized plasma are reviewed with emphasis on using the extended equation for electron probe current obtained on the basis of a non-local approach when the electrons reach the probe in a diffusion regime. It is shown that:

When the magnetic field applied is very weak and the electrons reach the probe without collisions in the probe sheath, the second-derivative Druyvesteyn formula can be used for EEDF evaluation.
At low values of the magnetic field, an extended second-derivative Druyvesteyn formula yields reliable results.
At high values of the magnetic field, the first-derivative probe technique is applicable for precise evaluation of the plasma potential and the real EEDF. There is an interval of intermediate values of the magnetic field when both techniques—the extended second-derivative and the first-derivative one—can be used.
Experimental results from probe measurements at different ranges of magnetic field are reviewed and discussed: low-pressure argon gas discharges in the presence of a magnetic field in the range 0.01–0.08 T, probe measurements in high-temperature strongly turbulent fusion circular hydrogen plasmas (magnetic fields from 0.45 T to 1.3 T) in the small ISTTOK and CASTOR tokamaks, in D-shape COMPASS tokamak plasmas as well as in the TJ-II stellarator.
In the vicinity of the last closed flux surface in tokamaks and in the TJ-II stellarator, the EEPF obtained is found to be bi-Maxwellian, while close to the tokamak chamber wall it is Maxwellian.
The mechanism for the appearance of a bi-Maxwellian EEDF in the vicinity of the LCFS is discussed. The comparison of the results from probe measurements with those from the ASTRA package and EIRENE code calculations shows that the main reason for the appearance of a bi-Maxwellian EEDF in the vicinity of the LCFS is the ionization of the neutral atoms.
Key works of other authors in the context of the results presented are mentioned.

Acknowledgments

The authors wish to thank the staff of the COMPASS and ISTTOK tokamaks as well as TJ-II teams for technical assistance.

The studies reviewed were partially supported by the bilateral Bulgaria–Slovenia contract BI-BG/11-12-011, European Community under the contract of Association between EURATOM/IPP.CZ and EURATOM/INRNE.BG; by the Bulgarian National Science Fund through the Association EURATOM-INRNE; by the CEEPUS CIII-AT-0063-10-1415 mobility scheme; by the International Atomic Energy Agency (IAEA) Research Contract No 17125/R0, R1 and R2 as a part of the IAEA CRP F13014 on 'Utilisation of a Network of Small Magnetic Confinement Fusion Devices for Mainstream Fusion Research' 5th and 6th IAEA Joint Experiment; by the Joint Research Project between the Institute of Plasma Physics v.v.i., AS CR and the Institute of Electronics BAS BG; by grant project GA CR P205/12/2327; and by MSMT Project # LM2011021 and within the framework of the EUROfusion Consortium, they received funding from the Euratom research and training programme 2014–2018 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.

Advances in Langmuir probe diagnostics of the plasma potential and electron-energy distribution function in magnetized plasma

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Abstract

1. Introduction

2. Evaluation of the EEDF in the presence of a magnetic field

3. Experimental results and discussion

3.1. Probe measurements in the presence of a magnetic field in the range 0.01–0.08 T

3.2. Probe measurements in the presence of magnetic fields in the range 0.04–0.3 T

3.3. Probe measurements in high-temperature strongly turbulent fusion plasmas (magnetic fields from 0.45 to 1.3 T)

3.3.1. Probe measurements in the ISTTOK tokamak.

3.3.2. Probe measurements in the CASTOR tokamak.

3.3.3. Probe measurements in the COMPASS tokamak.

3.3.4. Probe measurements in TJ-II stellarator.

3.3.5. Origin of the bi-Maxwellian EEDF.

4. Conclusions

Acknowledgments

Footnotes

Advances in Langmuir probe diagnostics of the plasma potential and electron-energy distribution function in magnetized plasma

Article metrics

Submit

Permissions

Share this article

Author e-mails

Author affiliations

Dates

Peer review information

Abstract

1. Introduction

2. Evaluation of the EEDF in the presence of a magnetic field

3. Experimental results and discussion

3.1. Probe measurements in the presence of a magnetic field in the range 0.01–0.08 T

3.2. Probe measurements in the presence of magnetic fields in the range 0.04–0.3 T

3.3. Probe measurements in high-temperature strongly turbulent fusion plasmas (magnetic fields from 0.45 to 1.3 T)

3.3.1. Probe measurements in the ISTTOK tokamak.

3.3.2. Probe measurements in the CASTOR tokamak.

3.3.3. Probe measurements in the COMPASS tokamak.

3.3.4. Probe measurements in TJ-II stellarator.

3.3.5. Origin of the bi-Maxwellian EEDF.

4. Conclusions

Acknowledgments

Footnotes