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,
 (1.1) 
The total electric current emitted by the probe is
 (1.2) 
 (1.3) 
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 G_{i}.
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 quasineutral (n_{i} = n_{e}) but the potential perturbation (f) is nonzero. It is, of course, a basic property of plasmas that their selfshielding allows quasineutrality to be violated only over distances of order the Debye length. The sheath, which separates the probe surface from the quasineutral 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.
Continuity,
 (1.4) 
 (1.5) 
If we expand the divergence of the momentum flux, using the continuity equation, and take the vcomponent of the resulting momentum equation, we can quickly obtain:
 (1.6) 
 (1.7) 
We need to use the electron density equation (1.1) to eliminate f. Write it as ef = T_{e} ln(n/n_{¥}) and recall that in the presheath region n_{i} = n_{e} = n. We substitute into the momentum equation and write the velocity in the form of a Mach number M º v/c_{s} where c_{s} º Ö(T_{e}/m_{i}), so that
 (1.8) 
 (1.9) 
The momentum equation can be rearranged in a different way, again using the continuity equation, to get
 (1.10) 
 (1.11) 
Returning then to our solution (Eq. 1.7), which can also be written
 (1.12) 
 (1.13) 
 (1.14) 
It enables us to determine the plasma density, n_{¥}, from the probe characteristic by first getting T_{e} from the slope, as described earlier, then observing the ion saturation current and using
 (1.15) 
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 crossfield diffusion into the presheath flux tube to be sufficient to balance the soundspeed collection flow to the probe. Simple dimensional arguments then give the orderofmagnitude of the presheath length as a^{2} c_{s}/D, where a is the probe radius and D the crossfield diffusion coefficient.

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
Continuity,
 (1.16) 
Parallel Momentum,
 (1.17) 
and Transverse Diffusion,
 (1.18) 
Here we have reintroduced nonzero (but constant) T_{i} (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 S_{m} in the onedimensional parallel fluid equations. To do
this, approximate equations (1.16) to (1.18) by replacing as follows,
 (1.19) 
where ¥ refers to values in the background, and parallel velocity is implied. Then nondimensionalize by the transformations


 (1.20) 
 (1.21) 
A lively debate has taken place over the past few years about what is the `best' value to take for a ( º h/m_{i} n_{¥}D). Stangeby [6] in early treatments of the onedimensional problem took as sources S = const. and S_{m} = m_{i} v_{¥}S, 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
 (1.22) 
 (1.23) 
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 ¹ 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
 (1.24) 
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 fluxratios 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 onedimensional 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 twodimensional fluid models give excellent agreement (within typically 15%) with equivalent onedimensional [8], while full kinetic calculations of onedimensional models, using sources that model crossfield 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:
 (1.25) 
where M_{c} 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 selfconsistent picture was obtained using M_{c} = 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 selfconsistency. 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 M_{c} @ 0.5 to deduce parallel flow from Mach probes.
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 dwelltime 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 photoelectron 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 gyroorbit 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 `Flushmount' probe design embedded in a plasmafacing
tile.
References