Pitfalls in the interpretations of Langmuir probe currents

Langmuir probes^1–7 are versatile diagnostic tools for studies of low⁸ and medium temperature plasmas.⁹ Several numerical simulations have been carried out in order to understand the functioning of such probes.^10–15 A probe characteristic is generally obtained by counting the number of charged particles that cross the probe surface per unit time. Although this procedure gives the correct result after averaging, it is argued in the present study that this method can lead to incorrect conclusions when results are compared to the physical performance of these probes, the probe noise in particular. In this simple model, a charge e moving with velocity vector $w$ directed at an angle $θ$ with respect to the normal to a small area element $dA$ contributes with $e w cos θ δ(r−wt)$⁠, see Fig. 1. For this velocity component, the average charge per unit time arriving at $dA$ is $⟨I⟩=e|w|n cos θ dA$ with n being the density of these particles. For particles with an isotropic velocity distribution $f(w)$⁠, the net result becomes $⟨I⟩=e dA∫0π/2∫0∞|w|f(w) cos θ dw dσ/4π$⁠, where $dσ/4π$ is the fraction of particles that are directed toward $dA$⁠, crossing a narrow belt around the normal vector¹⁶ (see Fig. 1), where $dσ=2πdθ sin θ$⁠. With a Maxwellian velocity distribution for charged particles with mass m and temperature T, the result is $⟨I⟩=e n dAT/2πm$⁠. Plasma perturbations such as ion acoustic waves are manifested by time variations of the density, $n=n(t)$⁠.

When individual particle current contributions are modeled by $δ$-pulses, the associated temporal correlation function will also become a $δ$-function. The associated noise becomes a standard “shot noise” with a flat or “white” power spectrum,¹⁷ irrespective of the angle of impact. Only charges arriving at the probe surface will contribute, while passing charges are irrelevant in this standard model. These comments apply solely to the probe–plasma interaction. The observed noise characteristic can be strongly modified by the detecting circuit.⁴ A characteristic frequency $Ω$ related to an inflow of independent charged particles with mass m taken from a distribution with density n can be constructed by a typical velocity $T/m$ and a length scale given by the average particles separation $∼1/n1/3$⁠, giving $Ω≈Np1/3ωp$ in terms of a plasma frequency $ωp$ and the plasma parameter $Np≡λD3n$⁠, where $λD$ is the Debye length. No meaningful signal in the probe current can be detected at frequencies larger than $Ω$⁠. Note that $Np∼103$ is usually considered large, but then $Np1/3∼10$ gives a moderate $Ω$-value.

The present study illustrates the problem by a simple, yet physically realistic and relevant model considering the induced charges in an ideally conducting sphere. Infinitely long cylindrical probes can also be analyzed, but these results will add little to the discussion. The noise contributions from the charged particles differ from the usual “shot noise,” and results are found for the correlation functions and power spectra. It is demonstrated that both collected and passing particles contribute to this noise level, which sets the lower limit to signals that can be identified by such probes.

The model outlined before fails by ignoring the induced charge caused by an approaching particle before it reaches the conducting object. In fact, it does not even need to reach the surface. The following analysis illustrates this process using the method of images. This is a tool explicitly assuming electrostatic conditions by relying on a uniqueness theorem for static electric fields with given boundary conditions.^18,19 Strictly speaking, it fails if charges are in motion, since then the electric field need no longer be determined by the boundary conditions alone. For small charge velocities and charge accelerations, where electromagnetic radiation can be ignored, the method is applicable as an approximation similar to Kirchhoff's laws in electronics.²⁰ These latter laws can be understood as corollaries of Maxwell's equations in the low-frequency limit. They are accurate for direct current (DC) circuits, and for alternating current (AC) electronics at frequencies where the wavelengths of electromagnetic radiation are very large compared to the circuits. This limit is also assumed in the following. The arguments are made more detailed in the  Appendix.

Consider an ideally conducting spherical surface as illustrated in Fig. 2. Two point particles, a) and b), with charge e moving with velocity w are shown as well. Particle b) represents one crossing $dσ$ in the direction of $dA$ in Fig. 1. The sphere can be electrically floating or be biased through a thin conducting wire. The first part of the following discussion assumes the latter case, where the reference potential, or “ground,” is taken to be the plasma potential when many charges constituting a plasma surrounding the probe. For this case, the sphere will not disturb the orbits of approaching charges and their velocity vectors will be assumed to be constant. The reference particle charge can have either sign. For an illustration, it is an advantage to use a vertical “cut” in the figure as given in Fig. 3. The moving reference charged particle induces a time varying charge distribution on the surface of the conducting sphere.^18,19,21 This charge is mediated by a current $i(t)$ through a connecting wire also shown in Fig. 3. The solution of the problem in terms of image charges is found in many textbooks on electrodynamics,^18,19 and only the relevant results need to be reproduced here. A summary figure is included in Fig. 4 with explanations in the caption. Results for many non-interacting charges can be found by superposition. The total net charge (unevenly) distributed over the conducting sphere is $q=−eR/r=−eR/p2+x2$ with symbols explained in Fig. 3, see also Fig. 4.

The time origin is here chosen so that the charge e is at the smallest distance to the origin of the sphere at $t=0$⁠. The discussion assumed that the change in the charging of the conducting sphere is mediated by currents in a connecting wire illustrated in Fig. 3. With the present current definition and sign convention, we find $i(t)>0$ for an approaching positive particle, $e>0$⁠, where $t<0$ and $i(t)<0$ for a receding one when $t>0$⁠.

In the case of a floating sphere, the net charge on the sphere is constant while its potential will be changing as a charged particle approaches. Then, the current $i(t)$ will be compensated by the $ε0 ∂E/∂t$ term (Maxwell's displacement current) in Maxwell's equations. The two problems, floating and grounded/biased spheres, can be treated the same way as far as single-particle motions are concerned. The difference will be manifested when many charges are included: for a floating conductor, the already absorbed charges will affect the motion of those approaching at a later time, while the potential remains fixed for the grounded case.

The current $i(t)$ is illustrated in Fig. 5 for a range of p-values. When the charged reference particle reaches the conducting surface, it neutralizes the image charge (or rather the induced surface charge), but it does not give any particular contribution to the current $i(t)$ at that time. The time variation of the current pulse depends on the normal and tangential velocity components of the local surface element.

A complete solution to the problem requires the standard averaging of the distribution of particle velocities and directions for all species as well as all impact parameters p. For relevant conditions, the light species (electrons) will have an RMS velocity exceeding that of the heavy ones and the probe will carry a net current.⁶ Additional corrections will be induced by inhomogeneities and bulk relative particle motions.

If the record for $I(t)$ is used for estimating the mean square fluctuations of the current, the result will be different from Eq. (10) because of the missing contribution from Eq. (6). The interpretation of the statistical average of the steady state, or DC, current is correct, even though the interpretation of the individual charge contributions is more complicated than commonly assumed.

For $p<R$⁠, we have the angle $θ=arcsin(p/R)$ between the sphere's radius vector and the velocity vector of the particle at the time of impact, see also Fig. 1. The variation of $i(t)$ shown in Fig. 5 for varying p can also be seen as representing the change in current for varying $θ$⁠, i.e., varying direction of the velocity vector at a fixed point of impact on the sphere. When $⟨I⟩$ in Eq. (9) is independent of p and R, it is also independent of $θ$⁠. The variable $γ$ in $F(γ)$⁠, see Fig. 6, is directly related to $θ$⁠.

Using the extensions of the Campbell theorem,^24,27,28 we can obtain the auto-correlation function for the current $⟨I(τ)I(τ+t)⟩$ for the model (4). Again, the two limits $p≤R$ and $p>R$ can be identified.

Results for the correlations $⟨I(τ)I(τ+t)⟩$ are shown in Figs. 7 and 8 for $p>R$ and $p<R$⁠, respectively, omitting the $(∫−∞∞i(τ)dτ)2$ constant part for the case $p≤R$⁠.

The power spectra are obtained by Fourier transforming the auto-correlation functions, with an example shown for $p>R$ in Fig. 9. The spectrum has a broad peak at $ω∼2πw/p$ for this case. This result is consistent with a characteristic interaction time²² of $p/w$⁠. Nearby particles (small p) contribute with large frequencies to the net power spectrum and have the largest influence due to the $p−3$ factor. The number of incoming particles for increasing p increases as $2πp dp$⁠, implying that the net noise contribution from distant particles reduces as $p−2$⁠. A dominant frequency will be $∼2π⟨w2⟩1/2/R$⁠.

The system considered here, taken as a whole, is not in thermal equilibrium, so the fluctuation–dissipation theorem^24,29,30 does not apply. For the present conditions, the noise induced by image charges will have two distinct contributions from $p>R$ and from $p≤R$⁠.

The particle's contribution to the power spectrum of the noise when $p<R$ can be found similarly by Fourier transform of the auto-correlation function in Fig. 8, see Fig. 10 for the result. The shape of the power spectrum depends on the angle of impact with respect to the sphere's local radius vector at the time the particle reaches the surface. The spectra differ significantly from the “white noise” resulting from the classical shot noise¹⁷ in both cases $p<R$ and $p>R$⁠. There appears to be no analytical expression for the power spectra in Fig. 10. An averaging over the distribution of particle velocities and directions for each species as well as all impact parameters has to be performed numerically. The noise spectra for $p<R$ shown in Fig. 10 do not have local maxima.

The analysis assumes that the image charges respond instantaneously to changes in the position of the moving reference charge, i.e., assuming perfectly conducting surfaces. This assumption is expected to work best for slowly moving charges. Some basic features of finite conductivity can be modeled by inserting an impedance in the connecting wire, e.g., a self-inductance for the finite inertia of the conducting electrons in the metal, and a resistor to model the internal collisional friction. The electrical network connected to the probe is essential for its overall performance,⁴ but this aspect is not addressed here.

A bias on the sphere at a potential $Φ$ different from the reference ground can be modeled by introducing a charge $Q=4πε0R Φ$ at the center. This charge will create an electric field that perturbs the orbits of an approaching charge. At large distances $r≫R$⁠, the resulting force will in general dominate the contribution from the image charge. The straight-line orbits in Figs. 2 and 3 will then be changed³¹ into parabolic or hyperbolic orbits in Eq. (14). For simple symmetries and geometries, also this more general problem may be solvable, but the general case requires a numerical analysis.

Sections II and III addressed spherical surfaces at a fixed potential with a fluctuating current. An alternative, and also relevant problem, deals with a floating conductor where the potential fluctuates and the net average current is vanishing, $⟨I(t)⟩=0$⁠. For reasons outlined in the following, this is a more complicated problem. The fluctuating potential gives here rise to a fluctuating electric field, which gives $ε0∂E/∂t≠0$ in Ampère's law to compensate the current to the surface caused by the randomly arriving charged particles.

The problem of a single particle arriving at a floating charge neutral sphere can be analyzed along the same lines as for the grounded or biased spherical conductor. For a grounded probe, the image charge will make the surface an equipotential. For the corresponding floating problem, an additional charge $eR/r$ similar in magnitude but with opposite sign has to be introduced at the center to make the sphere charge neutral. This central charge taken alone will make the spherical surface equipotential. The surface potential will vary with time in case the reference particle is moving. An illustrative solution of the problem is given in Fig. 11. The two image charges inside the spherical surface form a dipole displaced from the center: a somewhat counter-intuitive result. A passing particle with charge e, velocity w and impact parameter p gives rise to a time varying potential $ϕ(t)=e/(4πε0p2+w2t2 )$ at the surface of the sphere, independent its radius. The magnitude of the dipole moment $Di$ is the product of the image charge and the separation distance between the inner charges in Fig. 11, i.e., $Di(t)=|e|R3/(p2+w2t2)$⁠. The direction of the dipole moment vector varies from $−π$ to $π$ during the passage of the charged particle.

When more particles are introduced, it will be found that the arrival of the next one will be influenced by the potential induced by the previous ones, etc., as discussed before. Campbell's theorem is no longer directly applicable for the present problem since the particle arrivals can no longer be considered independent. The process is no longer time stationary. Asymptotically, when the sphere is saturated at the floating potential, it will again be possible to assume stationary processes. The analysis of this many particle problem is likely to be carried out best by a combination of analytical and numerical methods.

The presence of dipolar charge distributions has been discussed and illustrated by numerical simulations also in relation to conducting dust particles moving at high speeds relative to a fully ionized plasma.³⁵ In such cases, a permanent dipole will be due to an asymmetrical distribution of the charges absorbed on the surface of the conducting dust grain or a similar finite size object.

The general problem with many collisions is studied best by numerical simulations, with results depending critically on the collisional model applied. The average currents remain the same also in collisional plasmas, but the power spectrum of the noise can be significantly different from the collisionless case.

So far, the analysis did not impose any restrictions on the impact parameter p. For relevant plasma conditions, it will be argued that charges with $p>λD$ will be shielded and not generate any image charges in the conducting surface. Considering one spherical probe with radius R as a reference case two limits can be distinguished, sheath limitation where the sheath is narrow compared to R and orbit limitation³⁶ where the sheath is wider, having in mind⁶ that the sheath can be somewhat wider than $λD$⁠. A commonly accepted sheath-width is $Dsw∼15λD$⁠, depending weakly on applied voltages.^6,34 In the former case with R much larger than the sheath-width, almost all incoming charges will reach the conducting surface and the contribution (9) is dominant. For the latter orbit limited case,³¹ the fluctuating currents Eq. (2) induced by passing particles can be noticeable. Using a simple, and often applied model, we can assume that charges more than a sheath-width from the conducting surface are completely shielded and has no effect the probe. Charges that are closer will have an effect almost as in vacuum. The interaction between the charge and its image charge is restricted to the time spent inside the sheath. The effective time of interaction is estimated by $Δt≈Dsw/w$⁠. For large probes, the current signal will only be a fraction of the time variation shown in Fig. 5.

The effects of the sheath can be modeled by formally introducing a charge moving with $w>0$ at the sheath edge at time $T≡−(1/w)(R+Dsw)2−p2 <0$ with $0<p<R+Dsw$⁠, so that $q(t)=−e(R+Dsw) S(t−T)/p2+w2t2$⁠. The remaining part of the discussion becomes similar to the collisional problem, and again, we find the result $⟨I(t)⟩=μe$⁠, while the correlation function (and thereby the power spectrum of the noise) becomes complicated, amenable only to a numerical solution.³⁷ 

A biased sphere can be modeled as discussed in Sec. III. A probe with a swept bias can be analyzed in the same way as one at a fixed potential provided the sweep is slow when measured on a timescale of $Dsw/w$⁠. Then, the sheath changes only little during a particle transit, and the problem can be analyzed as a series of quasi-stationary conditions. A fast sweep of the bias is likely to require a numerical approach.

It is an advantage to use normalized variables and parameters as $p/R$⁠, $ζ/R$⁠, $ωceR/w$⁠, and $wt/R$⁠. The expression (19) reproduces Eq. (2) for $ζ=0$⁠, where the gyrocenter coincides with the particle position and the motion is along a straight line given by $B$⁠. The special case for $p=0$ and $ζ>R$ is independent of $ωc$⁠.

An illustrative example for $i(t)$ given by Eq. (19) is shown in Fig. 13. It can be shown that $⟨I(t)⟩=0$ also for this case, while $⟨I2(t)⟩$ does not have a simple analytical expression. The signal contains the cyclotron frequency, but it is rich also in its harmonics. The figure assumes, as mentioned, the initial gyro-phase to vanish. This phase is however important for large velocities w when $R+ζ>p$ since in this case, some particles can avoid absorption, see Fig. 14. Here, the contribution to the net current will be vanishing, but the contribution to the RMS-fluctuation level remains. For small w, all particles will be absorbed in this case, but the position on the surface where this happens will depend on the initial phase. Even if the moving charge misses the sphere, it can give a significant contribution to the noise by coming close to its surface.

As for the unmagnetized case, we can also here construct the correlation functions of signals like Eq. (19) to the fluctuation spectra detected by the biased spherical probe. This problem has many parameters so an illustration is given only for a selected case where $p/R=1.75$⁠, $ζ/R=0.5$⁠, and $ωcR/w=5.0$⁠, see Fig. 15. The conspicuous feature is the cyclotron harmonics in the power spectrum. Harmonic generation is often associated with physically nonlinear effects, but in the present case, it is a consequence of simple geometrical effects.

The rare event where a charged particle is trapped in a circular orbit around a biased spherical probe will not give rise to an induced current, while an elliptical orbit will.

The present analysis gives details of the current contributions of charges arriving at the surface of spherical conductors. When compared to simple models where arriving charges are assumed to contribute by $δ$-pulses, it is found here that the signals become more complicated, depending on the angle of incidence. Particular attention is given to the noise associated with the discrete particle arrivals and it is demonstrated that also passing particles contribute here. For the simple model summarized in Sec. I, they would not contribute at all.

For scales comparable to or larger than the Debye length $λD$⁠, we have $R dA≥λD3$ in Eq. (20). In that case, $⟨I2(t)⟩/⟨I(t)⟩2≲1/Np≪1$ in terms of the plasma parameter $Np$ introduced before. An RMS value for the relative fluctuations scales as $Np−1/2$⁠. For a typical Q-machine plasma⁸ with $Te=2.200$ K and $n=1015$ m⁻³, we have $Np≈300$ for comparison. Due to the density dependence of $λD$⁠, the plasma parameter is large for hot and dilute plasmas. The arguments outlined here can be repeated using Eq. (10) instead of Eq. (6) giving the same conclusions. In the “Vlasov limit” of large plasma parameters, we can expect contributions Eqs. (6) and (10) to be small but important by setting a minimum noise level for the measurements. For conditions in space plasmas as well as in related numerical simulations with metallic dust grains, it can be relevant to include conducting surfaces on scales smaller than the Debye length, and in that case, the ratio $⟨I2(t)⟩/⟨I(t)⟩2$ need not be small. All charges contribute to the noise. When a Langmuir probe is used for identifying a subset of particles, such as a small density plasma beam component, it can happen that the signal is masked by the noise caused by all the other particles.

We estimate the parameters for conditions where passing charges give dominant contributions compared to those absorbed. The sheath width will be taken to be simply $λD$⁠, ignoring numerical coefficients. For a grounded metallic object, the integrated noise contribution from the two regions $p<R$ and $R+λD>p>R$ can be compared by use of Eqs. (6), (10), and (11) giving a mean square noise ratio $(1+R/λD)$ ignoring some numerical factors and taking $F(γ)≈1/4$⁠, see Fig. 6. The noise induced by particles reaching the surface will dominate when $R≳λD$⁠, while the passing particle noise will become comparable when $R≲λD$⁠. With the possible exception of some space applications, plasma probes will usually have $R>λD$⁠.

The discussions in the preceding sections apply to one conducting surface. Taking as an example a double probe or a combination of two or more surfaces close together,^15,38 it will be found that a moving charge contributes to the signal in several of them, although it is ultimately absorbed by only one. A Langmuir probe with a guard-ring^5,39 is another example. Two nearby probes can be used to estimate an electric field component.^9,15,40,41 Prior to absorption on one surface, a moving charged particle will induce currents also in the other probes with an intensity depending on the relevant impact parameter and probe separations. There seems to be no analytical solution to this formulation of the problem, in particular also because image charges induced by a moving charge in one surface induce image charges in the other surfaces, etc.

The foregoing discussion can be extended to other probe forms. In some cases, the analysis can be simplified by use of conformal mappings. The case of a large plane probe needs, however, special attention.⁴² 

Addition of a nearby insulating surface is in principle straightforward. In this case, it is physically realistic that charges are accumulating there and these will induce increasing image charges on the conductor. A steady increase in the accumulated charge $∑jqj=Q(t)∼t$ will give rise to a current $dQ(t)/dt∼$ const. to the conducting sphere, although no charges actually reach the metallic surface.

Contamination of a probe surface⁴³ need not give problems for the analysis as long as it contributes with a uniform conducting layer. The only change will be in the work function at the surface. A layer with varying thickness and composition over the surface is problematic. An imposed bias means that the Fermi level of the conductor is constant and a varying composition implies that the work function is then varying over the surface, so it is no longer equipotential. The present results will have to be modified, but the basic ideas of currents induced by image charges remain applicable. The distinction between absorbed and passing charge contributions can also be maintained. Coatings by insulating materials would have no effect for the first charged particle arriving. It will bind an image charge that gives rise to a corresponding current through the connecting wire in Figs. 2 and 3. Charges arriving later will pile up until an equilibrium is reached, making the surface potential different from the one imposed by the external connection. The analysis will be reminiscent of the one for a floating conducting surface.

The effects of passing particles can be tested experimentally by placing the spherical probe in a glass tube that cancels the direct current. Results from different tube diameters can be compared to test the effects of varying impact parameters. For instance, a density modulated narrow collimated beam of charged particles passing by the probe will induce a time varying current in the wire connected to the probe.⁴² The signal will be indistinguishable for a similar case in which the particles reach the probe surface. The difficulty in the experiment comes from the time-varying accumulation of charges on the outer glass surface. An externally imposed magnetic field can reduce the charging and allow also a qualitative comparison with Fig. 15, keeping in mind³⁰ that harmonic generation can also be due to plasma gradients $⊥B0$ that give perturbations of the cyclotron orbits.

Parts of Figs. 4 and 11 are based on a program made public by Masatsugu Sei Suzuki, Department of Physics, SUNY at Binghamton.

The author has no conflicts to disclose.

Hans L. Pécseli: Conceptualization (equal); Formal analysis (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

The method of images rests on a uniqueness theorem valid for electrostatics. In principle, it breaks down when the charges are in motion. It is argued in the following that the method can be applied as a good approximation for realistic plasma conditions.

Numerical solutions of Eq. (A2) are shown in Fig. 16 for selected parameters K. For small K, the particle can be attracted by its image charge to reach the conducting surface. However, for the given impact parameter and reference velocity, even moderate parameters as $K>15$ give only small deflections of the particle. For $K≥100$⁠, the accelerations are negligible and the assumption of a constant velocity w used in the main analysis is justified. When the charged particle moves without acceleration, it will not give rise to any electromagnetic radiation. When $K≫1$⁠, the use of the method of images will be justified also for the present dynamic problem.

A floating conductor that has reached a saturated floating potential given by Eq. (15) can be modeled by introducing a charge $Qf=4πε0RΦf$ at the center of the sphere. When the external moving charge comes very close to the conducting surface, the electric forces on it will be dominated by the image charge. At larger distances, however, the force from the dipole in Fig. 11 decreases as $r−3$ compared to the $r−2$ force from $Qf$⁠. In this case, the image charge force can be ignored under all circumstances, retaining only the force from $Qf$⁠.

In the past, it has been speculated that charged particles could be attracted by metallic dust particles in space and by their acceleration contribute to low energy electromagnetic background radiation. The idea may be worth pursuing, but it needs an approach extending the one summarized here. For instance, the radiation losses⁴⁴ of the accelerated charges will need attention. Phenomenological models for this can be found in the literature.^24,44