Experimental and theoretical study of bumped characteristics obtained with cylindrical Langmuir probe in magnetized helium plasma

I(V) characteristics for all inclinations of the probe tip have been plotted for several values of $| | {\bf{B}}| |$ and for a 200 W-RF power input in figure 7. Without magnetic field (figure 7(a)), the 'classical' I(V) is recovered, because the plasma is seen as isotropic by the probe. The slight differences between all six curves come from small variations on the plasma conditions, due to the fact that the change of inclination requested to open the chamber between each measurement (uncertainties within 5% due to the gas pressure gauge, thus the RF coupled power which is sensitive to the pressure may not be exactly the same).

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The shape of the I(V) changes drastically in the presence of a magnetic field as depicted in figures 7(b)–(d). The slope of the exponential part and the electron saturation current one as well as the ratio I_e/I_i are strongly affected by the addition of a magnetic field [11]. Note that the increase of I_e with the magnetic field is due to a better coupling of the RF power and to better confinements. More generally, it can be seen that the overall shape of the characteristics are qualitatively close to the double probe/tanh-shape ones modelled by equation (5). For small angles (ϑ ≤ 12°), the characteristics even display a bump between the exponential and the saturation parts. The bump's overshoot amplitude and the change in the slope between the exponential part and the electron saturation regime is emphasized and steeper with larger $| | {\bf{B}}| |$.

For a probe inclination of 18°, the measured I(V) characteristic looks like the 'tanh-shape' as explained previously. For higher inclination angle, the electron current does not saturate (due to sheath expansion) and for lower inclination angle, there is a bump. The only difference between all these different cases is the width of the flux tube that scales as $\sim {L}_{p}\sin \vartheta$. The probe area facing magnetic field lines (see figure 5) is written as follows:

$$\begin{eqnarray}&&{S}_{\mathrm{face}}=\pi {r}_{p}^{2}\cos \vartheta +\pi {L}_{p}{r}_{p}\sin \vartheta \end{eqnarray} \tag{ 6 }$$

which can be scaled as ${S}_{\mathrm{face}}\sim \sin \vartheta$ because L_p ≫ r_p.

For ϑ = 0° at 100 mT, r_p ≈ $2{\rho }_{{ce}}$, therefore, the probe surface facing the magnetic flux tube is comparable to the 'cyclotron area' (${S}_{{ce}}=\pi {\rho }_{{ce}}^{2}$): in this case of grazing incidence, a bump arises on the measured characteristics. By increasing the angle, the facing surface increases (whereas the cyclotron area remains constant) and the amplitude of the bump decreases, and even disappears for larger angles. One can suggests that the flux tube narrowness comparable to the cyclotron area could explain the bump. However, it remains even if S_face ≫ S_ce (when ϑ > 5°), therefore another mechanism should be invoked in order to explain the presence of the bump.

We performed a series of experiments with a power input in the range 20–200 W in order to quantify the evolution of the characteristics with respect to ϑ. Figure 8 shows the evolution of the current at 70 V, I(V = V_max = 70 V) or the 'end-current', against $\sin \vartheta$. This end value is used, because the plasma potential is actually unknown, so the comparison of the current at plasma potential is not possible for now.

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Without magnetic field (figure 8(a)), the end-current is constant for any inclination as explained previously. Moreover by increasing the RF-power, the overall collected end-current also increases, because the power also increases the plasma density (I ∝ n_e) as expected.

In the presence of a magnetic field of 95 mT (figure 8(b)) two regimes are evidenced: the region where there is a bump ($\vartheta \leqslant 12^\circ \iff \sin \vartheta \leqslant 0.21$) and the region with an asymmetric double probe behaviour (above 12°). In the former region, the end current is proportional to $\sin \vartheta$, as the width of the magnetic flux tube: the sine dependence of the current collection is verified. But in the 'bump region', the collected end-current remains approximately constant with ϑ for any RF-power. Since the collected current is proportional to the product of the density with the collecting surface, n_eS_coll. (assuming $\langle {v}_{e}\rangle \sim \langle {v}_{e,\parallel }\rangle \approx \ \mathrm{CST}$), the increase of the angle also increases S_coll., so to keep constant collected current, the electron density in the flux tube should decrease. This is in a good agreement with figure 9(b): in the presence of a magnetic field, and if there is a bump on the characteristic, the density is lower than in the absence of a bump (going from n_e ≈ 5 × 10¹⁵ m⁻³ with a bump to n_e ≈ 15 × 10¹⁵ m⁻³ without a bump at 200 W RF power). This density difference is enhanced for higher power. However, for lower power the density remains approximately constant at all inclinations.

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However, the evolution of the electron temperature with respect to the inclination angle (figure 10) is impossible to explain straightforwardly. Indeed the electron flow collected by the probe is the combination of two populations: the parallel and the perpendicular to B flow, having each its own temperature (i.e. T_e∥ and T_e⊥ resp.). Our method gives a kind of average of both. Unfortunately, the electron energy distribution function, which could help us to understand the plot, is too noisy to be exploited (even after some filtering such as Stavitzky Golay, or Fourier analysis). The explanation of this plot is still an opened question for further studies.

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As shown in the last subsection, increasing RF-power also increases the overall density. To track the bump evolution with RF-power regardless to the density change, it is convenient to normalize the I(V) to the end-current value I(V)/I(70 V). In figure 11 are depicted the normalized probe characteristics at $| | {\bf{B}}| |$ = 95 mT for all inclinations and for several input RF-power, figures 11(a)–(c).

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Although the end current is proportional to the collecting surface (which is $\propto \sin \vartheta$), the normalization removes this dependence and all angles can be compared. The electron saturation part, directly connected to the sheath extension, is then the same for every angles, as shown in figure 11. In figure 11(a), for 20 W there is no bump at 12°, contrary to figure 11(b) for 80 W. The current at the bump position is also larger than the end-current in figure 11(c). Moreover, the increase of the power increases the amplitude of the bump and its width.

One can suppose the existence of perpendicular (to ${\bf{B}}$) RF currents, pumping the flux tube connected to the probe: this idea is used to derive a fluid model in section 4 to recover the bump analytically. In addition, as depicted in figure 8(b), increasing the power does not increase the end-current in the 'bump region', corroborating the former assumption. These RF currents, when averaged over one RF period, exhibit a net DC perpendicular contribution [31], acting as perpendicular DC currents, which have already been investigated in previous models [22, 23] to explain the depletion and saturation currents of biased flux tubes.

The position of the probe is also an important parameter. It is initially placed relatively far from the antenna to have a homogeneous plasma around the probe. Indeed, near the antenna, the ${\bf{E}}\times {\bf{B}}$ effect is larger and the plasma denser. That is why, there is a thin plasma layer above, and below the antenna (see photograph in figure 12). Moreover, at this RF-pulsation ions do not react to the quick potential change near the antenna, whereas electrons do [32] (ω_pe > ω_RF > ω_pi).

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To make sure that the inclination of the probe scan a homogeneous plasma in a range of $\pm {L}_{p}\sin \vartheta$ around the probing position in the y direction, measurements along the y axis were performed at fixed z = −60 mm position and for ϑ = 0°. Power was fixed at 100 W-RF, $| | {\bf{B}}| |$ = 80 mT in 1.2 Pa He plasma. All characteristics in figure 13 depicted a bump, where the plotted parameters are the floating potential ϕ_fl., the bump potential V_bump and the bump current I_bump. Dote suggested the bump potential to be near the plasma one [15, 19, 20]. According to Dote's assumption and using the combined potential drops in the sheath and the collisionless pre-sheath [1], one can write the potential drop between the plasma and the floating probe potential for cold ions (T_i/T_e → 0) as:

$$\begin{eqnarray}&&{\phi }_{p}-{\phi }_{\mathrm{fl}.}=\displaystyle \frac{{T}_{e}}{2e}\mathrm{ln}\left[\displaystyle \frac{{m}_{i}}{2\pi {m}_{e}}\right]+\displaystyle \frac{{T}_{e}}{2e}=4.03\times {T}_{e},\end{eqnarray} \tag{ 7 }$$

with T_e in eV. For all previous measurements at (z = −60 mm, y = 40 mm), using the approximation ϕ_p ≈ V_bump gives T_e ≈ 1.30 eV (which is a typical value in ALINE magnetized, plasma [26, 27]).

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As seen in figure 13, 1 cm around the y = 40 mm position, the homogeneity hypothesis (almost constant T_e and n_e in the probing area) can be applied in the experimental conditions and the probe inclination scans the same plasma parameters at all angles. Finally, this last figure also highlights the fact that current and temperature (as defined in equation (7)) increases by a factor of ∼7 in the bright regions (see photograph depicted in figure 12), near the antenna.

For a magnetic field of 80 mT, and an input power of 80 W-RF, measurements were performed with a probe parallel to the field line (ϑ = 0°) in a He pressure range from 1.2 to 40 Pa. All characteristics are plotted figure 14.

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When pressure increases, the bump gets narrower and its amplitude diminishes. Above 9.32 Pa, the bumps disappear and the I(V) characteristic turns into an asymmetric double probe one.

That's why one can separate the pressure range in 2 regimes:

The low collisional regime from 1 to 10 Pa. At these pressures the electron–neutral collision frequency ${\nu }_{\mathrm{col}.}^{{eN}}$ is lower than the electron plasma frequency ν_pe, and lower than the electron cyclotron frequency ν_ce (see table 1) and of the same order than the RF frequency. For example at 1 Pa ${\nu }_{\mathrm{col}.}^{{eN}}\approx 17$ MHz [29]. In the same way the ion-neutral collision frequency ${\nu }_{\mathrm{col}.}^{{iN}}$ is much lower than the ion plasma frequency ν_pi, and lower than the ion cyclotron frequency ν_ci up to 4 Pa so that ions are considered as magnetized in the first half of the low collisional pressure range. In this range the classical perpendicular diffusion falls down and perpendicular currents are able to deplete strongly the flux tube while the typical scale length of these current is higher than the radius of the probe, which is the case here because ρ_ci ≫ r_p. In a quiet plasma, as we have in ALINE, such a behaviour can be seen, while in Tokamak edge plasma anomalous transport can still prevent the biased flux tubes to deplete.

In the collisional regime (P > 10 Pa), ${\nu }_{\mathrm{col}.}^{{eN}}$ remains lower than ν_ce and ν_pe, but much higher than the RF frequency. RF electron current are then lowered by collisions. And ions are no more magnetized because ${\nu }_{\mathrm{col}.}^{{iN}}\gt {\nu }_{{ci}}$, which favours their perpendicular diffusion while ion perpendicular current are lowered in the same time, filling the lack of density caused by the probe collection and cancelling the bump on the characteristics. For the highest pressures, the flux tube for ions disappears and the I(V) looks like an unmagnetized one [14].

In the intermediate case of partially magnetized ions, the I(V) looks like a double symmetric probe characteristics. The electron saturation current collected by the probe depends also on the competition between perpendicular DC and RF currents and the cross diffusion of ions due to collisions.

Extracting electron density and temperature from I(V) characteristics is far from simple. But if the measurements are done in the presence of a magnetic field, the exploitation are even more difficult. The challenge lies on the presence of the bump, whose existence, shape, location and amplitude depend on several plasma parameters $(| | {\bf{B}}| | ,\vartheta ,\mathrm{Pwr}.,y,p)$ (see sections 3.1–3.5).

The first problem with bumped characteristics is the uncertainty on the position of the plasma potential. Indeed, the plasma potential is actually the local plasma potential with respect to the perpendicular direction to the magnetic field. It is usually found by assuming that, at the plasma potential V = ϕ_p, ${\rm{d}}I/{\rm{d}}V{| }_{{\phi }_{p}}=\max ({\rm{d}}I/{\rm{d}}V)$, which is equivalent to ${{\rm{d}}}^{2}I/{\rm{d}}{V}^{2}{| }_{{\phi }_{p}}=0$ [2, 3] (this is called the 'classical method' in the following). Another method based on the intersection of the linear fits of the exponential part and the electron saturation one has also been suggested and used in a previous study [17]. It was finally suggested that, in the context of bumped characteristics, the bump potential is at the plasma potential [15, 19]. Thus, three methods are available, in order to determine the plasma potential and we propose to compare them, for different inclinations, in a single 100 W-RF plasma, with $| | {\bf{B}}| |$ = 80 mT and p = 1.2 Pa, whose characteristics are depicted in figure 15(a).

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In the context of the 'intersection method' we linearised the exponential growth as $I(V)\approx {a}_{\exp .}V+{b}_{\exp .}$, and fitted the electron saturation current with the formula:

$$\begin{eqnarray}&&{I}_{e}(V)\approx {a}_{\mathrm{sat}.}V+{b}_{\mathrm{sat}.}+{c}_{\mathrm{sat}.}\sqrt{V}\end{eqnarray} \tag{ 8 }$$

with the $\sqrt{V}$ term similar to one of the OML approach, which gives a relatively good fit with experimental curves. This equation is only able to fit the saturation part, i.e. the end of the I(V)—far from the bump potential range. Only the last 20 volts of each I(V) were used for the fitting, see figure 15(a).

We used an iterative method, in order to determine both n_e and T_e with the plasma potential ϕ_p, the current at plasma potential I_p, floating potential ϕ_fl., magnetic field and probe inclination as input parameters. First, a raw approximation of electron temperature is done, supposing $I\sim {I}_{e}\propto \exp ({eV}/{k}_{{\rm{B}}}{T}_{e})$ for V ≤ ϕ_p in the exponential part. Applying a linear fit on ln I(V) one will find a first value of T_e. From now one starts the iterative loops: this electron temperature value allows a computation of a gross value of n_e since, at plasma potential, ${I}_{p}={{en}}_{e}\langle v\rangle {S}_{e}/4$. The value of S_e is not the probe surface, even at plasma potential (where there is no sheath), because of the cyclotron motion. That is why it is assumed that the collecting surface is the probe surface facing ${\bf{B}}$ plus a layer thick of N_elr.${\rho }_{{ce}}$ (i.e. some Larmor radii—N_elr. being the number of electron Larmor radii connected to the probe):

$$\begin{eqnarray}\begin{array}{rcl}{S}_{e} & = & \pi \cos \vartheta \times {({r}_{p}+{N}_{\mathrm{elr}.}{\rho }_{{ce}})}^{2}\\ & & +\ \pi {L}_{p}\sin \vartheta \times ({r}_{p}+{N}_{\mathrm{elr}.}{\rho }_{{ce}})\end{array}\end{eqnarray} \tag{ 9 }$$

by replacing ${r}_{p}\to {r}_{p}+{N}_{\mathrm{elr}.}{\rho }_{{ce}}$ in equation (6). It is assumed that this equation takes into account the perpendicular motion of electrons along the magnetic field lines connected to the probe.

With T_e and n_e, it is possible to compute the electron Debye length λ_De and the ion sheath thickness using the Child–Langmuir law (since ${\rho }_{{ci}}\gg {r}_{p}\sim {\rho }_{{ce}}$ and that Zhu's corrections [33] for cylindrical geometry only bring minor changes in opposition to its complexity), knowing,

$$\begin{eqnarray}&&{{\ell }}_{\mathrm{CL}}=\displaystyle \frac{\sqrt{2}}{3}{\lambda }_{\mathrm{De}}{\left(2e\displaystyle \frac{| {\phi }_{p}-V| }{{k}_{{\rm{B}}}{T}_{e}}\right)}^{3/4}\end{eqnarray} \tag{ 10 }$$

for V ≤ ϕ_p. Since ions are supposed unmagnetized, the collecting area for ions is

$$\begin{eqnarray}&&{S}_{i}=\pi {({r}_{p}+{{\ell }}_{\mathrm{CL}})}^{2}+2\pi {L}_{p}({r}_{p}+{{\ell }}_{\mathrm{CL}}).\end{eqnarray} \tag{ 11 }$$

It is then possible to compute the ion current for V ≤ ϕ_p, using the Bohm flux formula, I_i = 0.61 × en_ec_sS_i. So, the updated electron current I_e = I(V) − I_i can be calculated. Taking again the log-scale of this new electron current gives a new more accurate value of T_e. The loop starts over again, and ends if temperature values converge (i.e. $| {T}_{e}^{\mathrm{new}}-{T}_{e}^{\mathrm{old}}| \leqslant \varepsilon$, ε being given by the user).

Equations giving S_e and S_i (equations (9) and (11) resp.) take into account the sheath extension for magnetized electrons and unmagnetized ions. To take into account the inclination of the probe, and find reliable plasma parameters, one should also multiply the total current by a geometric factor of $\pi {r}_{p}^{2}/{S}_{\mathrm{face}}$ from equation (6) giving a dimensionless factor of $1/(\cos \vartheta +[{L}_{p}/{r}_{p}]\times \sin \vartheta )$. This allows the recovering of the same amplitude for all bumped I(V). The extracted values of n_e and T_e are plotted in figures 15(b) and (c) using this correction, and plotted in figures 15(d) and (e) without the correction (n_e strongly decreases with the angle). From figures 15(b) and (c) it is clear that the classical ϕ_p-determination method gives the more reliable values of temperature and density (the deviation of T_e values between each inclination is negligible compared to other methods). We have then T_e ≈ 1.2 eV and n_e ≈ 1.3 × 10¹⁶ m⁻³. Since the OML model remains valid in RF-plasmas [34], and that ions are unmagnetized, we extracted ${n}_{i}^{\mathrm{OML}}=1.74\times {10}^{17}$ m⁻³ (which is within the typical errorbar for OML model) and ${T}_{e}^{\mathrm{OML}}=2.69\,\mathrm{eV}$, which are overestimated compared to the previous method.

By comparison, in the absence of magnetic field, the bump method to find the plasma potential makes no sense (since there is no bump) and both classical and intersection methods are alike and give the same value of the plasma potential. Therefore the self-consistent algorithm gives an electron density of the order of 5.32 × 10¹⁵ m⁻³ and an electron temperature of 3.47 eV (for the same discharge parameters as with $| | {\bf{B}}| | \ne 0$).

As suggested by Mihaila and Rozhansky [16, 23], the bump on I(V) characteristics could be induced by density depletion within the flux tube.

The cylindrical flux tube connected from the probe to the reactor's wall is filled by electrons using a single channel, which is the lateral area of the cylinder. Due to magnetic confinement and for grazing incidences, the perpendicular current arises thanks to collisions with neutrals. During a I(V) measurement for V > ϕ_p, one pumps the electrons inside the flux tube, which makes the local electron density decreases. If the pumped electron current is larger than the refill perpendicular one, then the collected current at the probe decreases with phi (the bump origin). But when the probe potential is increased further, the sheath extent around it also increases, which artificially makes the cylindrical flux tube diameter wider. Consequently, when V ≫ ϕ_p, the electron perpendicular current compensate the pumped one and the collected current increases again. Now for larger incidences, the perpendicular current always overcomes the pumped one, which explains the experimentally observed disappearance of the bump for ϑ > 12°.

In the meantime, there is another mechanism involving mainly ions: it is the plasma pumping via perpendicular ion current due to the positive biasing of the flux tube with respect to the surrounding plasma potential. This mechanism has already been invoked to explain the early electron saturation of the I(V) characteristics in the case of planar probe [23] in magnetized plasmas. The typical scale length of these perpendicular ion currents is the ion gyroradius. To explain the bump, this mechanism can be divided in three regimes occurring when the probe potential overcomes the plasma potential:

• 1.  
  When the transverse (perpendicular to ${\bf{B}}$) ion current is lower than the electron saturation current collected by the probe, the space charge of the sheath is electropositive and consequently the flux tube potential 'follows' the probe potential. The density depletion can first appear in that regime.
• 2.  
  When the transverse ion current is exactly equal to the electron saturation current collected by the probe, the sheath between the probe and the flux tube disappears and the collected current can decrease because the flux tube density decreases with the probe potential.
• 3.  
  Finally when the transverse ion current is higher than the electron saturation current collected by the probe, electrons must be accelerated in the sheath to balance the ion current and thus the sheath drop is reversed. The sheath space charge becomes electronegative and the flux tube potential tends to saturate compared to the probe potential. This regime accounts for long and thin flux tube.

Nevertheless, plasma diffusion is more and more efficient as the flux tube is widening. So in the third regime, with the saturation of the flux tube potential, the pumping also saturates and the density depletion can be cancelled resulting in a classical increase of the current in the last part of the I(V) characteristics (beyond the bump).

Finally there is an optimum point for which the pumping is maximum compared to cross diffusion, and this is at this working point the bump appears to be the higher because of the strong negative slope just following the maximum of the bump. Actually, the bump does not mean there is an increase of current compared to an I(V) characteristics with no bump, on the contrary it means a decrease of current.

The complexity of the phenomenon can only be explained by a mass and current conservation taking into account the growing of the flux tube radius with the potential.

The model

The establishment of the potential and density structures in a magnetized plasma is far from simple and out of the scope of this article. To understand the behaviour of a magnetized plasma during a Langmuir probe measurement, we assume a perfectly magnetized plasma for electrons (they do only move along field lines whereas ions can leave the flux tube though transverse conductivity and can join it thanks to transverse diffusion, see figure 16). The flux tube is assumed homogeneous in the magnetic field direction (z-direction) and well delimited in the radial direction from the bulk plasma. Its length is L and delimited by the probe on the one side, and the vessel wall on the other side.

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Rozhansky et al [23] showed that the ion flux tube has a characteristic radius of ${R}_{0}\sim {\rho }_{{ci}}$, and a length L (this ion flux tube connected to the probe also contains the electron flux tube of radius ${\rho }_{{ce}}\ll {\rho }_{{ci}}$). To provide an analytical and comprehensible solution, the case ϑ = 0° is considered, that is, the facing surface of the probe, $\pi {r}_{p}^{2}$ is negligible compared to the ion flux tube section $\pi {R}_{0}^{2}$. For greater inclination angles, the probe radius must be taken into account, making it impossible to provide an analytical resolution.

Due to their cyclotron motion, electrons are trapped in both ion and electron tubes and can only leave them through the ends, producing a parallel net current of ${J}_{e,\mathrm{sat}.}\times \pi {R}_{0}^{2}$. To ensure current and quasi-neutrality conservation in the ion tube, there must be a perpendicular ion flux through the cylindrical surface so that, ${J}_{i,\mathrm{sat}.}\times 2\pi {R}_{0}L\approx {J}_{e,\mathrm{sat}.}\times \pi {R}_{0}^{2}$ (corresponding to Γ_∥e in figure 16). In this regime, where the perpendicular current can be higher than the electron saturation current on the probe, the potential gap can reverse in front of the probe (electronegative sheath) accelerating electrons and repelling ions. Thus, one can neglect the parallel ion flux on the probe side (in the case of an electropositive sheath, the ion current on the probe surface can also be neglected compared to electron current, still assuming that the electron current is close to its saturation value).

In the following we use current continuity for ions in order to obtain a first order ODE that gives the density of the flux tube with respect to the probe potential. Using Laframboise's theory, this tube density (or 'local plasma density') gives the electron fraction that will be collected by the probe regarding its potential V. An analytic expression of the collected current can be then provided.

As shown in the last sections, the pumping is enhanced by perpendicular (to ${\bf{B}}$) RF and DC currents [21, 22]. But periodic RF current can be reduced to an averaged DC over one a period. That is why the model is steady state, and only DC quantities are considered. Finally, to prevent the tube density to drop to zero, we assume the presence of a source term S₀, so that,

$$\begin{eqnarray}&&{\iiint }_{\mathrm{tube}}{S}_{0}\,{{\rm{d}}}^{3}{\bf{r}}=2\times \pi {R}_{0}^{2}\times \displaystyle \frac{1}{2}{n}_{0}\langle {v}_{e}\rangle ,\end{eqnarray} \tag{ 12 }$$

where n₀ is the bulk plasma density (outside the ion flux tube region) and n_t, the ion flux tube density (n₀ ≥ n_t). This term fills the tube at the same rate electrons leave it from both ends (which is an overestimation of the 'real' S₀ source term).

From the stationary ion continuity equation, we have ${\boldsymbol{\nabla }}\cdot {{\boldsymbol{\Gamma }}}_{i}\sim {\boldsymbol{\nabla }}\cdot {{\boldsymbol{\Gamma }}}_{i,\perp }={S}_{0}$. Perpendicular ion flux is separated in two parts (see figure 16): lateral mobility ${{\boldsymbol{\Gamma }}}_{\perp }=-{\mu }_{i}{n}_{t}{\boldsymbol{\nabla }}\phi$ and the diffusion flux ${{\boldsymbol{\Gamma }}}_{\mathrm{diff}.}=-{D}_{\perp }{\boldsymbol{\nabla }}{n}_{t}$. Integration of all ion fluxes through the whole tube using Gauss's law gives:

$$\begin{eqnarray}&&\displaystyle \frac{{n}_{0}\langle {v}_{e}\rangle }{2L}{R}_{0}=-\left({D}_{\perp }{\left.\displaystyle \frac{\partial {n}_{t}}{\partial r}\right|}_{{R}_{0}}+{n}_{t}{\mu }_{i}\times {\left.\displaystyle \frac{\partial \phi }{\partial r}\right|}_{{R}_{0}}\right).\end{eqnarray} \tag{ 13 }$$

In the presence of a strong radial electric field (and especially in a cold plasma), the ion drift velocity is larger than the thermal velocity, thus ${\rho }_{{ci}}={v}_{\perp }/{\omega }_{{ci}}$ = ${({v}_{\mathrm{drift}}^{2}+{v}_{i,\mathrm{Th}.}^{2})}^{1/2}/{\omega }_{{ci}}\,\sim | {v}_{\mathrm{drift}}| /{\omega }_{{ci}}$ = $-{\partial }_{r}\phi /B{\omega }_{{ci}}$ (all at R₀). Recalling that ${R}_{0}\sim {\rho }_{{ci}}$, equation (13) rewrites as,

$$\begin{eqnarray}&&\displaystyle \frac{{n}_{0}\langle {v}_{e}\rangle }{2{LB}{\omega }_{{ci}}}\times {\left.\displaystyle \frac{\partial \phi }{\partial r}\right|}_{{R}_{0}}=\left({D}_{\perp }{\left.\displaystyle \frac{\partial {n}_{t}}{\partial r}\right|}_{{R}_{0}}+{n}_{t}{\mu }_{i}\times {\left.\displaystyle \frac{\partial \phi }{\partial r}\right|}_{{R}_{0}}\right).\end{eqnarray} \tag{ 14 }$$

Now using the chain rule, $\partial {n}_{t}/\partial r{| }_{{R}_{0}}=\partial {n}_{t}/\partial \phi \times \partial \phi /\partial r{| }_{{R}_{0}}$, one will get the following first order ODE at the radius r = R₀:

$$\begin{eqnarray}&&\displaystyle \frac{\partial {n}_{t}}{\partial \phi }=-\displaystyle \frac{{\mu }_{i}}{{D}_{\perp }}{n}_{t}+\displaystyle \frac{{n}_{0}\langle {v}_{e}\rangle }{2{\omega }_{{ci}}{{BLD}}_{\perp }}.\end{eqnarray} \tag{ 15 }$$

The perpendicular mobility can also be written as a conductivity depending on the current nature (collision, inertial, viscosity, anomalous, ...). With the initial condition of n_t(ϕ = ϕ_p) = n_e since there is no sheath nor spatial potential variation at plasma potential, one will get:

$$\begin{eqnarray}&&{n}_{t}(V)={n}_{\infty }+({n}_{0}-{n}_{\infty })\exp \left[\displaystyle \frac{{\mu }_{i}}{{D}_{\perp }}({\phi }_{p}-V)\right],\end{eqnarray} \tag{ 16 }$$

where V is the probe potential and ${n}_{\infty }={n}_{0}\times \langle {v}_{e}\rangle /2{\mu }_{i}B{\omega }_{{ci}}L$. Here we assumed that the flux tube potential equals the probe one. Although generally, ϕ_t = f(V) ≥ V > ϕ_p, with f a cumbersome function [21].

Equation (16) exhibits an exponential decay of the density with V. This strong depletion of the flux tube as soon as the biased potential of the tube is higher than the surrounding plasma potential is needed to see the bump rising. For lower decay (for ex. ∼1/(V − ϕ_p) or $\sim 1/{(V-{\phi }_{p})}^{2}$) the bump does not appear because of the expansion of the sheath which increases the lateral surface of the flux tube and hence the total perpendicular current more rapidly that the density is depleted. This also explains why at higher probe potential value, when the exponential decay saturates, the current rises up again due to sheath expansion. Actually there is a competition between the diffusion D_⊥ across the lateral surface of the tube versus the perpendicular current due to ion mobility μ_i as it can be seen in equation (16).

After that, to fit the sheath expansion above V_p in a magnetic field parallel to the probe, one uses the Laframboise [9] model which showed that the portion of plasma density actually touching a probe and thus collected, n_eff., is given by the relation,

$$\begin{eqnarray}&&{n}_{\mathrm{eff}.}(\xi )=\displaystyle \frac{2{n}_{t}(\xi )}{\sqrt{\pi }}\left[\sqrt{\xi }+\displaystyle \frac{\sqrt{\pi }}{2}{{\rm{e}}}^{\xi }\mathrm{erfc}\sqrt{\xi }\right]\end{eqnarray} \tag{ 17 }$$

for ξ = e(V − ϕ_p)/k_BT_e. Finally, the collected current on the probe is simply given by

$$\begin{eqnarray}&&{I}_{e}(V)=\displaystyle \frac{1}{2}{{en}}_{\mathrm{eff}.}(V)\times \langle {v}_{e}\rangle {S}_{e},\end{eqnarray} \tag{ 18 }$$

where S_e is given by equation (9), and the number of electron Larmor radii (N_elr.) is given as fitting parameter.

This model is compared with the experimental data in figure 17 for a magnetic field of 57, 70 and 100 mT in a 200 W-RF and 1.2 Pa helium plasma (the probe was parallel to ${\bf{B}}$). For V < ϕ_p the exponential ${J}_{e,\mathrm{sat}.}\times \exp (e(V-{\phi }_{p})/{k}_{{\rm{B}}}{T}_{e}){S}_{e}$ part of the electronic current is considered.

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The fitting of bumped characteristics leads to anomalous transport coefficients [2] (see table 2). Indeed, μ_i ≈ 20 m² V⁻¹ s⁻¹ and D_⊥ ≈ 40 m² s⁻¹ are the outputs for the three depicted curves. This highlights the fact that the flux tube evolves in a forced regime: because the parallel current saturates at the electron saturation current, this regime involves anomalous transverse transport coefficients in order to satisfy current conservation in the flux tube.

Table 2.  Outputs values for temperature, density, mobility and diffusion obtained from the fitting of a bumped I(V) (see figure 17), measured with a probe align with magnetic field lines (i.e. ϑ = 0°).

B (mT)	80	70	57
μi (m2 V−1 s−1)	19.5	26	20
D⊥(m2 s−1)	39	52	50
n0 (1015 m−3)	8.0	6.0	3.1
Te (eV)	2.7	2.5	3.5

The number of Larmor radii, N_elr. goes from 0.1 to 5 with increasing $| | {\bf{B}}| |$ in equation (9). The limit density in the flux tube, ${n}_{\infty }$ is close to n₀/10: that means that the measurement heavily depletes the magnetic flux tube. Moreover, the model suggests that the plasma potential is on the top of the bump as proposed by Dote and Mihaila: the pumping mechanism starts when V > ϕ_p according to the theory. Since electrons are way much mobile than ions along B, as soon as the probe potential is above the plasma potential, electrons of the flux tube are flushed towards the probe. Moreover, as pointed out by equation (3), the flux tube itself has its own potential (slightly above the bulk plasma potential) since it is connected to the probe and thus somehow biased.

Finally, according to this theory, the prior parameter is actually the probe surface facing the magnetic field lines (i.e. the width of the magnetic flux tube). Therefore, a bump could appear on a plane probe characteristics or a spherical probe characteristics as well, if the facing surface is small enough corresponding more or less to a disk surface having a radius of the order of ${\rho }_{{ce}}$.