Figure 6.5: An example of theI–V characteristic of a FMP. Original data (light purple line) are fitted by a 3-p. formula (2.34) (pink line) and 4-p. formulas (2.38) (purple full), (2.39), (2.42) and (2.41) (dashed lines, difference with respect to (2.38) is shown, scaled×102). Ion and electron current component shown in light red/blue and electron current fit using (2.32) is shown in blue full line. Red full line shows the ion component from the reference fit (2.38). Input parameters indicated in the title and results of fit of (6.2) are shown in the legend.
6.4 Fit results analysis
The analysis of fit results will be started with a brief review of three-parametric fitting formula results (2.34) presented in figures 6.7 and 6.8. As indicated in fig. 6.5, the accuracy of this fit is quite poor, performing well only for large angles. Only the floating potential is well-identified. Difference of fitted Isat to expected ion saturation current I0 (6.2) surpasses 100 % for αB ≤ 3.5° and the electron temperature is off by 20 % for αB < 30°, meaning that this method is certainly inappropriate for FMP analysis in full bias range.
Four-parametric variants are compared in figs. D.1 to D.8 and these plots share the order of analysed I–V curve expressions. The panel (a) shows results obtained from fitting of the formula (2.38) with full V3/4 term, analogously the panel (b) shows results given by the linearised formula (2.39), panel (c) contains the fitted quantity from fit by Bergmann’s formula (2.41) and finally, the panel (d) represents the Desideri’s function (2.42).
6.4.1 Ion saturation current
If the ion saturation current is to the source of density measurement using either (6.2) or equivalent expression for an actual probe shape, it should be precisely
in the fig. 6.6, where the current drops below the projected current value forV > −3kBTe/e, which was the correspondingVhead. Again, this attenuation depends onξ.
Figure 6.6: Ion component of the ion current for various dpin/rL parameter values from the simulation set B for αB = 2°. Dot-dashed lines show linear fit for Vpin ∈
⟨−10,−8⟩ kBTe/e. Dashed red line is the value of I0 (2.36). Sheath expansion for V < Vhead causes the collection of extra current, while forV > Vhead, less current than I0 is collected. Both parts of the curve are significantly affected bydpin/rL ratio.
known from the fit. However, the fitting methods in the simulation set A shown in fig. D.1 show that the 20 %accuracy is lost for angles below5° even if the sheath expansion is accounted for. This limit is met only when larger magnetic field B > 4 T (fig. D.2) is introduced, suggesting that in high-field machines, FMPs could be operated reasonably well, which is also supported by experimental results [113].
The precision of all fitting methods is similar in the simulated range, with the exception being the Bergmann’s formula (2.41), which is accurate enough down to 3.5°. The overestimation of Isat however indicates that the sheath expansion is more prominent for lower angles than the optical approximation with Child-Langmuir law predict, especially when dpin/rL < 15, since ions with smaller
Figure 6.7: Results obtained via 3-p. fitting in the set A. (a) NormalizationN(Isat, I0), (b) Normalization N(Te, Te, in), (c) SPICE normalization.
6.4. Fit results analysis
Figure 6.8: Results obtained via 3-p. fitting in the set B. (a) NormalizationN(Isat, I0), (b) NormalizationN(Te, Te, in), (c) SPICE normalization.
Larmor radii do not digress from the field line significantly, only upon reaching the sheath edge.
6.4.2 Floating potential
Floating potential results for all four-parametric formulas are shown in fig. D.3 (set A) and fig. D.4 (set B). All presented formulas show matching values of Vfl, even when compared to 3-parametric fitting formula results (figs. 6.7 and 6.8, panel (c)). Bergmann’s formula (2.41) slightly differs for lowest simulated angles and smalldpin/rL due to slightly differentV approximation. A distinct minimum is present for all input combinations, being quite pronounced especially in the set A. This minimum has already been discovered by Chodura [40] (figure 3 in the paper, note the inverted axis) and it was also found in the DITE experiment [114], fig. 6.1. This however contradicts the fluid model, which predicts the value ofVfl as (2.26). Evaluated for the deuterium plasma in this simulation (µ= 3670, τ = 2), the formula gives Vfl = −2.63kBTe/e, which roughly matches the value obtained in simulation with αB = 90°. Both sets actually use µ= 200, however the collected current is scaled to match µ for main ion in the simulation, which is deuterium in this case.
In recent kinetic results [46], the floating potential minimum is not present.
However, in the approximation used in the compared model, the electrons are considered to be subjected to the Boltzmann law. On the contrary, both our and Chodura’s model use full electron orbit. In combination with a slightly different injection schema, this is probably the cause of the difference between our and the kinetic model [46], since this minimum is obtained even if the real µis used (see section 6.5).
6.4.3 Electron temperature
While the floating potential were almost not influenced by the sheath expansion, different result is observed in the case of Te. Results of Te fits are plotted in
figs. D.5 and D.6. In the simulation set A (fig. D.5), the inclusion of V3/4 de-pendency (2.38) proves to describe the I–V characteristic with greater precision, surpassed only by Desideri’s fitting function (2.42). Except several outliers, this formula produces results that are within 20 % accuracy bounds throughout the whole angular scan.
Even better match of the reproducedTeto the input Te, inwas observed in the simulation set B. All of the fitting function eventually fell within the accuracy bounds for some value ofdpin/rL. This again shows that the optical approximation can be a valid tool for plasmas whererLis small enough. The limitrLwas found to be dependent on the approximation ofV dependency and it starts atdpin/rL = 15 for full V3/4 I–V characteristic expression (2.38).
As indicated in the section 6.3, the simulation allows us to analyse the electron current separately using (2.32). Limiting the fit range to ⟨−10, Vfl⟩, we are able to find the value of Te directly from EC2. Resulting angular dependency can be broken into two parts divided by the threshold angle α0. For angles ≤ α0, the overestimation increases with ln sinαB as
Te, fit(αB > α0) = Te, in(1−qlogln sinαB) . (6.3) This function describes upper part of Te overestimation equally well in all simulated angular scans as indicated in fig. 6.10, where the results of the EC fit in set B are added into comparison along with two scans with higher density (n0 = 1×1019m−3). In the analysis of an envelope of maximal values ofTe, fit/Te, in, the value of the slope qlog in (6.3) was found to be qlog = 0.13±0.01.
The threshold angleα0 is given by the maximalTeoverestimation and depends on the magnetization parameter ξ. Averaging of Te results for αB < α0 obtained in the simulation set B has shown the dependency of the maximal overestimation as
Te, fit(αB ≤α0) = Te, in(︂
1 +qξ√︁
ξ)︂
, (6.4)
where qξ = 0.032±0.001. The source data and the fit to (6.4) is shown in the fig. 6.11. Data from the simulation set B are compared to the simulation with higher density with excellent agreement.
Although we are able to separate the electron current in the simulation, we are still not able to recover the input value of the electron temperature completely.
This could be interpreted as a supporting claim for Desideri’s 4-parametric ap-proach [76], although in the derivation of (2.42), the expanding electron-collecting area was ultimately neglected. However, if we treat this value of electron temper-ature given by the fit to the EC as a reference, we are able to compare selected 4-parametric methods. For the comparison to what extent the EC fit Te is recov-ered, we define the performance function as
Pfit = 1 n
n
∑︂
i=1
(︄Te, fiti −Te, ECi Te, in
)︄2
. (6.5)
2We can use this approach for a comparison, however, EC cannot be measured directly by an ordinary LP.
6.4. Fit results analysis
Figure 6.9: Electron temperature given by EC fit. Simulation set A results shown in the color-coded diamonds according to specified input parameters. Red dashed line is given by the overestimation threshold (6.3).
In this definition we iterate through each point in the angular scan looking for a sum of quadratic deviations from the reference temperature, normalized to the inputTe. Dividing thePfit values into three categories for each fitting function as shown in the fig. 6.12, we can see that best option for Te fitting is the Desideri’s function (2.42) with majority of Pfit values being in the first (best) category.
Second closest method is the full V3/4 formula (2.38), implying that the Child-Langmuir approach is rather valid – however, as the comparison between set A and B shows, this is mostly due to correctly fitted results in high-B simulations.
6.4.4 Sheath expansion
Although recovery of other three parameters of theI–V characteristic is straight-forward and limits of operational space can be set, this is not the case of the SE coefficient. Normalized values of appropriate coefficients from selected 4-p. I–V formulas are shown in the fig. D.7 (set A) and fig. D.8 (set B). The normalization was performed according to the table 6.1.
We can see that for small fields in the set A (fig. D.7), there is a region around αB = 5°, where the value of S1 (panel (a)) and a(panel (c)) falls within the20 % margin. Since expected values of the coefficients are rather small for larger αB and thus are easily affected by the noise in the data, the allowed error margin is not kept. It is apparent from the noisy data for αB > 7.8° on specified plots.
This statement also applies to S and a obtained from the set B.
Figure 6.10: Electron temperature given by EC fit for simulations with inputTe= 20 eV.
Sets A (squares) and B (circles) are compared to simulations with n0 = 1×1019m−3 (diamonds). Overestimation is still limited by (6.3) and the maximal magnitude de-pends on ξ (see fig. 6.11.
As for the linear SE coefficient S2 from (2.39), the compliance to selected bounds is found only in the set B. In previous work [72], this disagreement was observed also in the experiment.
Desideri’s coefficientRis however not expressed in absolute value in his work [76], thus that there is no clear reference value for the normalization function (6.1), meaning that the fit reveals only the relative extent of the sheath. Comparing the maximum of R dependency, we can see that other SE coefficients also obey this dependency to certain extent, revealing that probes withdpin/rL ∈ ⟨10,22⟩can be analysed either by full V3/4 formula (2.38) or the Bergmann’s I–V characteristic (2.41).
Bergmann’s treatment of the optical approximation provides two parameters for bidirectional sheath expansion description. Considering the expression for a (2.40), we can analyse the contribution of the perpendicular expansion by fixing c1 to Bergmann’s value of 0.5and performing a fit to c2. In the fig. 6.13, we can see the result of this fit for both sheath boundary shapes (rounded and sharp rectangles, see fig. 2.7b). In the initial article [39], only angles≥5° were studied, thus we have performed the fit in two ranges of αB. First, the full angular scan was used (empty circles in the plot), second the range was limited by αB = 3.5°.
In the limited range, the values match the Bergmann’s constants as shown in the table 2.1. Full fit however shows that this description is not correct for smallest angles. The fig. 6.14 reveals the source of this disproportion – for αB > 3.5°, the values follow the prescribed line for a, while a large difference is observed for αB = 2°. Bergmann’s simulations considered two probe geometries, one having