Electric Probes in Magnetized and Flowing Plasmas

We simplify our treatment of the electrons by making the very standard assumption that the electron density is everywhere given by a Boltzmann factor for thermodynamic equilibrium,

n(r) = n exp


where Te is the electron temperature and f is the electric potential in the plasma, measured relative to its value at `infinity' (i.e., far from the perturbing effects of the probe), where the density is n. This equation will be an excellent approximation for most cases because: (1) electrons move much faster than ions and so they readily reach thermal equilibrium; and (2) the case usually of most interest is where the electrons are being repelled, i.e., the probe potential, fp, is negative. (Discussion of this assumption is given, e.g., in [1].)

The total electric current emitted by the probe is

I = eA ( Ge - Gi )   ,
where A is the collection area, the ions are assumed singly charged, and Ge, Gi are the electron and ion flux densities to the probe. Generally, the ion flux variation with probe potential is minor provided the probe is much bigger than the Debye length. Therefore, writing Isi = -eAGi as the ion saturation current, the probe current characteristics is

(I - Isi)        Ge         exp


as long as Eq. (1.1) is valid. This will be the case provided we are away from the `electron saturation' region of the characteristic, i.e., provided the probe potential is substantially negative. Thus, the shape of the characteristic gives directly the electron temperature either by fitting the slope of ln(I-Isi) or directly graphically as Fig. 1.2 indicates.

Figure 1.2: Typical probe characteristic: I vs V. The electron temperature can be obtained from the logarithmic slope which may be obtained as indicated.

It is easy to see physically that the magnitude of the probe current will be proportional to the plasma density. The hard part of probe theory is to determine the exact constant of proportionality. Most often, the ion saturation current is the quantity that we can measure unambiguously in an experiment. Therefore, our theoretical attention focuses on calculating Gi.

A final preliminary remark before we begin this calculation is that one must recognize that the perturbed plasma round the probe splits up into two regions: (1) the Sheath, in which charge neutrality is violated and potential variations are very rapid; and (2) the `Presheath', in which the plasma is quasi-neutral (ni = ne) but the potential perturbation (f) is non-zero. It is, of course, a basic property of plasmas that their self-shielding allows quasi-neutrality to be violated only over distances of order the Debye length. The sheath, which separates the probe surface from the quasi-neutral plasma region, is therefore only a few (typically ~ 4) Debye lengths thick. We shall assume this sheath thickness is much smaller than the probe size. The presheath, however, has dimensions of order a few probe radii in unmagnetized plasmas and often very much larger in magnetized plasmas. It is the analysis of the presheath that dominates the problem.

1.2  Unmagnetized Plasma Presheath

We analyze the plasma ion flow to the probe using the (two-fluid) equations for the ion fluid. In their simplest form they are:


. (n v) = 0
and Momentum,

. (nmivv) + en f = 0    .
In using such simplified equations we are ignoring ion viscosity and pressure. This can be justified either by discussion of the particle kinetics or by simply assuming that Ti << Te.

If we expand the divergence of the momentum flux, using the continuity equation, and take the v-component of the resulting momentum equation, we can quickly obtain:

n v .

mv2 + e f

= 0   ,
whose solution is

m v2 + e f = const.
This, of course, is just the energy conservation equation for ion flow along streamlines. If the velocity is zero at , where f = 0, then the constant is zero.

We need to use the electron density equation (1.1) to eliminate f. Write it as ef = Te ln(n/n) and recall that in the presheath region ni = ne = n. We substitute into the momentum equation and write the velocity in the form of a Mach number M v/cs where cs (Te/mi), so that

.( n M ) = 0

.(n M M + n 1 ) = 0   .

The momentum equation can be rearranged in a different way, again using the continuity equation, to get

(nM)(nM). 1
+ n + (M. )(nM) = 0  .
So if we take the M-component of this equation and note that (1/n) = -(1/n2)n we get:

[-M2 + 1]M. n = -M[(M.)(nM)]   .
This equation shows that there is a singularity in the solution (i.e., |n| ) at any point where M2 = 1, provided that the right hand side is not zero. (The R.H.S. is not zero if v has divergence, e.g., spherically or cylindrically symmetric flow.) Thus, extremely general analysis shows that the plasma presheath equations break down where the ion flow speed reaches the sound speed. This represents the formation of a shock; or, in the probe context, it says that quasineutrality breaks down, i.e., that we enter the sheath at |M| = 1. In fluid terminology the sheath is a shock.

Returning then to our solution (Eq. 1.7), which can also be written

M2 = - ln|n/n|   ,
on the basis of the discussion of Eq(1.1), we now have the boundary condition Ms = 1 to apply at the sheath edge, giving immediately

ns = n exp

- 1

The ion flux into the sheath (and hence probe if the sheath is thin) is therefore

Gi = ns Ms cs = n exp

- 1

( Te/mi )   .
This is the standard result, more usually obtained by particle kinetic analysis.

It enables us to determine the plasma density, n, from the probe characteristic by first getting Te from the slope, as described earlier, then observing the ion saturation current and using

n = Isi/{e A exp

- 1

(Te/mi )}
to deduce the density. Notice, though, that ion temperature does not substantially affect the observed current (at least when Ti << Te). So Ti cannot be obtained from Langmuir probe measurements.

1.3  Magnetized and Flowing Plasmas

When the ion Larmor radius is smaller than the probe size, the ions are no longer free to be drawn to the probe directly along the density (and hence potential) gradient. Instead they can move freely only parallel to the magnetic field. Ion flow perpendicular to the field is governed by diffusion, generally anomalous diffusion, and not by the simple momentum equation (1.5). Analysis of the parallel momentum equation shows, in essentially the same way as for the unmagnetized plasma, that the ions enter the sheath at Mach 1; although now their velocity is parallel to the field so the collection area, A, is just the projection of the probe surface in the parallel direction.

The density at the sheath edge, which then determines the ion flux, is not as given by the previous analysis (although the quantitative difference is relatively small). The situation is illustrated in Fig. 1.3 (see ref.[1] for a detailed discussion). The presheath becomes highly elongated along the magnetic field. Its length is determined by requiring the inward cross-field diffusion into the presheath flux tube to be sufficient to balance the sound-speed collection flow to the probe. Simple dimensional arguments then give the order-of-magnitude of the presheath length as a2 cs/D, where a is the probe radius and D the cross-field diffusion coefficient.

Figure 1.3:  In a strong magnetic field, the     Figure 1.4:  A Mach probe measures
presheath becomes elongated and parallel flow     the ion flux with separate upstream
to the probe is supported by cross-field     and downstream collectors

Before we go into the mathematics of how to analyze this situation, it is extremely valuable to introduce a further topic, namely parallel plasma flow. Notice that the ion collection is into two different presheaths extending in either direction along the field. The plasma ions are accelerated in each presheath from whatever velocity they had when they entered it up to the sound speed at the probe. If there is a net ion parallel flow velocity outside the presheath, in the background plasma, one can see that there will tend to be more ion collection on the upstream side and less on the downstream side. This suggests the idea of what has come to be called a `Mach probe'. As illustrated in Fig. 1.4, the Mach probe is designed to collect current separately to either side. When biassed to draw ion saturation current, an imbalance in the currents to the two sides is interpreted as an indication of parallel flow velocity in the background plasma. Specifically, the ratio, R, of upstream to downstream ion current should tell us the Mach number of the flow.

Clearly the analysis of this problem is inherently (at least) two dimensional. We write down the ion fluid equations, separated into parallel and perpendicular parts, for


|| ( n v|| ) = ^ . ( n v^ ) S     ,

Parallel Momentum,

|| ( n mi v|| v|| ) + ( Ti + Te ) || n = -^ . ( n mi v^ v|| ) + ^ . ( h^ v|| ) Sm    ,

and Transverse Diffusion,

n v^ = - D ^ n   .

Here we have reintroduced nonzero (but constant) Ti (which could also have been done in the previous section) and nonzero shear viscosity, h (which would not have changed the unmagnetized calculation in the usual case where there is no velocity shear). The approach now is to regard the transverse divergences as sources S and Sm in the one-dimensional parallel fluid equations. To do this, approximate equations (1.16) to (1.18) by replacing as follows,




  ;  ^2




where refers to values in the background, and parallel velocity is implied. Then nondimensionalize by the transformations

x/a x   ,   y/a y   ,   
dz z   ,

n/n n   ,   v||/cs M   ,
where z is the parallel coordinate and now cs = [(Te + Ti)/mi]. The equations that result, once v^ is eliminated are:

M dn
+ n dM
= 1 - n

+ nM dM
= ( M - M ) {1 - n + h
mi n D
}    .
For a given flow (M), the only undetermined parameter here is (a h/(mi nD), which is the ratio of the perpendicular diffusivities of parallel momentum and of particles. The absolute value of D determines only the parallel extend of the presheath, not the density and velocities that determine the ion flux.

A lively debate has taken place over the past few years about what is the `best' value to take for a  ( h/mi nD). Stangeby [6] in early treatments of the one-dimensional problem took as sources S = const. and Sm = mi vS, leading to equations that are essentially equivalent to adopting a = 0. The convenience of this choice is that the equations can then be solved analytically, giving

n = n
1 - M M + M2
and consequently ion flux density into the sheath (where M = 1) on the upstream and downstream sides is

G updown = n cs
2 M

On the other hand, I have argued that a @ 1 is a more plausible value [7], and have shown by a series of calculations at different a, that the a = 0 solution is actually a singular case [8]. Solutions of Eqs(1.20) and (1.21) for n as a function of M are illustrated in Fig. 1.5.

Figure 1.5: Example of the solution of the presheath equations for n as a function of M. An n = 1, M = M: the external flow speed. M increases to 1 at the sheath edge. The density there then gives G. The a = 0 case (a) is qualitatively different from a 0, e.g., a = 1 (b).

This question has great practical importance. For density measurements, the difference lies in the coefficient, f, in the expression for the ion flux density to the probe

G = f . n cs   .
In a stationary (M = 0) plasma, one finds f = 0.5 for a = 0 and f = 0.35 for a = 1. (Compared with f = exp(-1/2) = 0.61 for the unmagnetized calculation.) This 30% difference (with a) in the density one would deduce from the ion saturation current, is substantial but not overwhelming, since electric probe measurements tend to be subject to uncertainties that are anyway in the 10-20% range.

For velocity measurements using Mach probes, however, the discrepancies are extremely large. Fig. 1.6 shows the sheath edge density (and hence G) versus flow Mach number (M) for various values of a. Negative M means flow away from the probe (i.e., the downstream side) while positive means towards it (upstream). The discrepancy in predicted G for a = 0 versus a = 1 on the downstream side can be a factor of 3. The corresponding ratio of ion collection fluxes, R, is shown in Fig. 1.7. The way such curves would be used is, knowing or choosing the `best' value of a, observe the ratio R using a Mach probe; then read off the value of M to which it corresponds. The value deduced using a = 0 is higher by a factor of 2 or more than that using a = 1. So we must somehow decide roughly what a is appropriate, or else we can have no confidence in the interpretation of Mach probe measurements.

Figure 1.6: The normalized density at the sheath edge (giving G) plotted as a function of Flow Mach number, M.

Figure 1.7: The flux-ratios as a function of M for two fluid calculations (Hutchinson and Stangeby) compared with kinetic calculations [11].

One might also be concerned about the other approximations and simplifications of this model, such as the one-dimensional approximation or the validity of a fluid treatment. It turns out that these effects are much weaker than the viscosity (a) problem. Calculations of fully two-dimensional fluid models give excellent agreement (within typically 15%) with equivalent one-dimensional [8], while full kinetic calculations of one-dimensional models, using sources that model cross-field transport, also agree well with corresponding fluid calculations [11]. (This demonstrates, for example, that parallel viscosity is not very important in determining G.) It has been found from these numerical studies that the ratio of upstream to downstream ion collection flux can be well fitted by the functional form:

R = exp(M/Mc)

where Mc is a calibration factor that varies from ~ 0.45 for a = 1 to ~ 1.0 for the a = 0 fluid model.

To resolve the question of what value to take for a, experiments are needed. Harbour, Proudfoot and others [9,10] had been making Mach probe measurements in the edges of tokamaks for some years before the theory described here was developed. They found that the a = 0 model of Stangeby gave unreasonably high Mach numbers when used to interpret their data. Independent flow velocity measurements were not available but a reasonably self-consistent picture was obtained using Mc = 0.6 in Eq. (1.25).

In the interests of deciding a, Chung, et al. [12] have done a series of experiments on the PISCES linear plasma. Again truly independent velocity measurements were lacking; however a value of a @ 0.5 gave the best self-consistency. Recently, other experiments with laser induced fluorescence measurements of velocity have also shown a ~ 1 to give reasonable interpretation [13]. Finally, it should be remarked that the interior of tokamaks generally show momentum diffusivity comparable to heat and particle diffusivity, again suggesting a ~ 1.

There is therefore mounting evidence that the viscous (a ~ 1) theory is basically correct and that Eq. (1.25) can be used with Mc @ 0.5 to deduce parallel flow from Mach probes.

1.4  Practical Considerations and Remaining Problems

Although electric probes are relatively `low tech' diagnostics to implement, there are many practical problems. Valuable reviews of these practical considerations have been given by Manos and McCracken [14] for fusion research applications and by Hershkowitz [15] in the context of plasmas for materials processing. We shall just touch on a few topics.

Perhaps the most obvious problem is that in hot, dense plasmas the probe must survive the heat load. This places a limitation on the plasmas that can be studied. However there are also steps that can be taken to help. One is to build big probes. Probes in large tokamaks are often very bulky by comparison with the fine, light probes of the past, and are made from graphite and other heat resistant materials. A second step is to avoid drawing large electron currents, which can cause much greater heating than ion currents. This is done by keeping the probe biassed to roughly floating potential or below. This causes no great loss of information because in magnetized plasmas the electron collection region is unreliable for diagnosis because of perturbations to the ideal Boltzmann law (1.1). One should therefore avoid fitting to the characteristic much above the floating potential anyway. A third step is to limit the time duration of the probe's exposure. This can be done by moving it rapidly in and out of the plasma. Pneumatic drives can reduce the dwell-time to only ten milliseconds or so, thus reducing the energy deposited. Finally, probes are often built directly into limiter and divertor plates to benefit from the `solidarity' of the plate, which again minimizes the heat load.

Numerous complicating factors may have to be considered. These include: determining the effective collection area, which may be changed by erosion or sputtered films; secondary- and photo-electron emission from the probe; effects of nearby atomic processes such as ionization and charge exchange. In addition, many more elaborate types of probe exist, which are used to try to obtain additional information (see e.g., [14,15]).

Two specific problems of current interest concern situations frequently encountered. The first is when the length of the presheath is greater than the distance along the field to the nearest solid surface. The presheath is then said to be `connected' and the theory of the previous section for a `free' (i.e., unconnected) presheath probably requires modification. The second is that when probes are mounted in divertor or limiter surfaces, as illustrated in Fig. 1.8, which are almost tangential to the field, complicated gyro-orbit effects occur that change the effective collision area. Both of these are active theoretical and experimental research areas.

Thus, electric probe diagnostics are a research area where substantial development is still taking place. The area requires close collaboration between theoretical analysis and experiment and offers excellent opportunities for important contributions to be made by relatively small facilities.

Figure 1.8: Example of a `Flush-mount' probe design embedded in a plasma-facing tile.


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