2.3 Electric probes
2.3.2 Flush-mounted probe
Consider a collecting probe that is actually only a segment of the wall in the direct contact with the plasma, as can be seen in the fig. 2.7. Then, if the probe operates with bias voltages V in ion saturation part of the I–V curve, there is a notable change of sheath size ds, since it depends on V (2.23). Ions that reach the space influenced by the probing voltage are collected from further distances than in the case of the floating surface, while others are scraped by the sheath off the surrounding tiles.
In the optical approximation shown in the fig. 2.7, the collecting area of the probe is divided into two parts. Ions coming through the projected area A⊘
make up the most of the collected current when αB is perpendicular, while ions
collected through Aexp contribute the most for small αB. Expression of these areas differ not only by the probing electrode shape (and type), but also on the model of ds(αB, V). In general, the extra current termISE is introduced into the I–V characteristic expression (2.34) and it is given by the ion saturation current density as
ISE(αB, V) =JsatAexp(αB, V) .
In the magnetic field with inclination angle αB, the optical approximation in two dimensions gives the extra area as
Aexp = (ds(V)−ds(Vfl)) cosαB,
provided that all ions reaching the sheath boundary are collected. The term in brackets is the difference of sheath sizes in front of the probing element and in front of the surrounding tile, since all ions crossing ds(Vfl) are thought to be collected by the surrounding PFCs.
Several approaches were taken to express the value ofAexp and all of them are eventually based on the Child-Langmuir sheath size formula (2.23). For example, let’s consider the ds to be dependent only on the probe potential in the simple rectangular sheath approximation (fig. 2.7a). It gives the extra area as
Aexp =lx2√4 2 3
(︄⃓
⃓
⃓
⃓ eV kBTe
⃓
⃓
⃓
⃓
3 4
−
⃓
⃓
⃓
⃓ eVfl kBTe
⃓
⃓
⃓
⃓
3 4
)︄
λDcosαB and subsequent expression for I–V characteristic is
I(V) = JsatAsinαB
⏞ ⏟⏟ ⏞
Isat
× [︄
1−exp
(︃e(V −Vfl) kBTe
)︃
+lx A
2√4 2
3 λDcotαB
⏞ ⏟⏟ ⏞
qSE
(︄⃓
⃓
⃓
⃓ eV kBTe
⃓
⃓
⃓
⃓
3 4
−
⃓
⃓
⃓
⃓ eVfl kBTe
⃓
⃓
⃓
⃓
3 4
)︄
⏞ ⏟⏟ ⏞
Φ(V, Vfl)
]︄
, (2.36)
where qSE is the sheath expansion coefficient and Φ(V, Vfl) substitutes for V de-pendency. This is one of possible expressions of the four-parametric I–V curve description. Other forms can be obtained using different approximations of sheath shape and V dependency.
Sheath expansion term
To describe the sheath expansion term, several approaches were taken in the past.
All were eventually based on the sheath expansion description using an optically approximated sheath. We will provide a short overview of the prominent ones based on the 2D model of a flush-mounted probe (FMP). ForI–V description, we will consider the probe oriented as shown in the fig. 2.8, using only they–z plane.
Variants of I–V curve descriptions will be used for further analysis of simulated I–V characteristics in Chapter 6.
2.3. Electric probes
Figure 2.8: Definition of axes for FMP sheath expansion coefficient derivation. Magnetic field vectorB lies iny–zplane.
Already presented I–V curve description in (2.36) did not consider the local change of Debye length in the oblique sheath. Updating the sheath expansion coefficient expression using (2.25), we define S1 as
qSE= cosαB
2√ sinαB
λD
dpin ≡S1 (2.37)
to be used in the full V3/4 form ofI–V characteristic description I(V) =Isat
[︄
1−exp
(︃e(V −Vfl) kBTe
)︃
+S1
(︂
|V|34 − |Vfl|34)︂(︃
e kBTe
)︃34]︄
. (2.38) It is useful to approximate the V3/4 term. Most common approximation is the linearisation of the term, which reads V3/4 ∼ −V /3. Other approach is to treat the full term as V3/4−Vfl3/4 ∼ (V −Vfl)3/4. In the potential range usually used in the experiment, both ways perform equally well if the qualitative trend is considered. Comparison of these approximations is shown in the fig. 2.9. However, in the absolute value, difference up to 20 % is observed.
Linearising the V term in (2.38), we get I(V) = Isat
[︃
1−exp
(︃e(V −Vfl) kBTe
)︃
−S2e(V −Vfl) kBTe
]︃
, (2.39)
which is the description of I–V characteristic commonly used for experimental data fitting fitting [72, 74]. The sheath expansion coefficient S2 = S1/3, since the linearisation gives additional factor of 1/3. However, usually no assumption is made aboutS2, since it has been found that the rectangular approximation of the sheath is not precise enough to describe the change ofS2 with magnetization parameter ξ [42, 72].
Bergmann extended the sheath expansion approximation with addition of the lateral direction as shown in the fig. 2.7b. Together with ds affected by the local Debye length change, the description ofqSE can be written in general as
a= c1+c2g(αB)
√sinαB
λD
dpin, (2.40)
Figure 2.9: A comparison of possible approximations of |eV /kBTe|3/4− |eVfl/kBTe|3/4. Plot normalized by electron temperature.
using lateral (c1, in y direction) and perpendicular (c2, in z direction) sheath expansion coefficients and a geometric factor g(αB), which depends on the shape of the probe. The sheath in the shape of a round-cornered rectangle was used in [39] to describe qSE for probes with small ratio ofdpin/rL≈4. In later work [75], behaviour of probes probes with dpin/rL > 10 was found to be described better by a sharp-cornered rectangular sheath. A summary of possible parameters of a can be found in table 2.1. Together with ϕ(V, Vfl) approximation, modified I–V curve expression is
I(V) =Isat [︄
1−exp
(︃e(V −Vfl) kBTe
)︃
+a
(︃e(V −Vfl) kBTe
)︃34]︄
. (2.41) A slightly different approach was taken by Desideri and Serianni [76]. In their derivation, they have treated the expanded current to be composed from both ions and electrons. Instead of the additive term, this approach leads to a multiplicative one:
I(V) = Isat(1 +R(V −Vfl)) [︃
1−exp
(︃e(V −Vfl) kBTe
)︃]︃
, (2.42)
whereR describes the sheath expansion. To relate it with the sheath size, the co-efficient αD is defined in the expression (2.43), which relates the bias-dependent
Corners c1 c2 g(αB) Rounded (a1) 0.5 0.4 (sinαB)−1−1
Sharp (a2) 0.5 0.6 cotαB
Table 2.1: Bergmann’s sheath expansion description.
2.3. Electric probes collecting area of the probe Acoll(V)to the collecting area of the probe biased to a floating potential Afl.
Acoll(V) =Afl [︃
1 +αDds(V)−ds(Vfl) λD
]︃
.
Value of αD can be calculated from the value of R given by the fit of (2.42) to the measured data as
αD = 1.1λDR√︁
Jsat, (2.43)
where the ion saturation current density is given by (2.31).
Sheath expansion in 3D
The sheath expansion is a three-dimensional phenomenon. An attempt to de-scribe it by a simple V3/4 Child-Langmuir formula (2.22) was made by Daube [77] based on fluid model. For a rectangular probe with dimensionsly and lx, the total ion current can be written using the sheath size difference∆ =ds(V)−ds(Vfl) as
I+=JsatsinαB[lxlyg1(αB) +lx∆g2(αB) +ly∆g3(αB)] , (2.44) where gi are geometric factors calculated from ion speed change in the sheath:
g1(αB) = 1
g2(αB) = 2(1−sinαB) cosαB g3(αB) =
√︄
−2 ln sinαB−4
(︃1−sinαB cosαB
)︃2
, (2.45)
which is similar to Bergmann’s a expression, however it has yielded different results for a FMP in [39] (in 2D,g3 = 0) and for a rectangular probe we get the sheath expansion coefficient as
qSE ≡qD = ∆
(︃g2(αB)
ly + g3(αB) lx
)︃
= 0.5λD
√sinαB
(︃g2(αB)
ly +g3(αB) lx
)︃
(2.46) If the 3D probe with circular cross section is treated in the optical approxi-mation similar to the one shown in the fig. 2.7b, a sheath of a cylindrical shape is formed. If the expansion length in x and y is identical (δx = δy = δ), the Bergmann’s coefficient a can be written as
a3 = π
(︂dpin
2 +δ )︂2
+ ∆(dpin+ 2δ) cosαB− πd
2 pin
4 sinαB πd2pin
4 sinαB
= (2.47)
= 4
√sinαB (︃λD
dpin )︃2[︃
c1 (︃λD
dpin + c1
√sinαB )︃
+1 π
(︃λD
dpin + 2c1
√sinαB )︃
c2cotαB ]︃
, where c1 and c2 have the same interpretation as in (2.40). In the Chapter 6, we will compare both 2D and 3D simulations to either (2.40) or (2.47).
3. Thesis motivation
In order to create a viable, energy producing fusion reactor, the control of the burning plasma has to be mastered. Since the processes taking place in the edge region as well as its properties determine the behaviour of the core plasma, it is necessary to improve its understanding and develop techniques for control of edge plasma parameters. This cannot be done without proper understanding and interpretation of the applied diagnostic methods and tools.
Among them, the Langmuir probe plays an important role as it represents one of the most versatile and used plasma diagnostic method. Many of plasma parameters, such as electron temperature Te, floating potential Vfl, the plasma density n [65] as well as the electron energy distribution function (EEDF) [78]
can be derived from current–voltage characteristic of the probe. Magnetic field, however, affects the measurement in various ways and to correctly interpret the experimental results, this influence should be understood. Despite of a large effort devoted over last decades to understanding of the physics related to the interpretation of the measure data, some uncertainties still persist and they are addressed within this thesis.
First, the probe in so called flush-mounted arrangement will be studied. The emphasis will be given on determination of the operational range and on assess-ment of accuracy of probe measureassess-ment in conditions present in a real device. In such environment, the probe is subjected to significant sheath expansion, which causes the measured parameters to be over- or underestimated.
Additionally, the ion collection by the standard Langmuir probe in a form of a protruding pin into magnetized plasma will be investigated with respect to the probe actual (effective) collecting area. The knowledge of the area size is especially important for calculation of the electron density and usually, it is estimated only by the comparison with experimental data. We will extend this approach by the description of additional effects which influence the effective collecting area.
Both these topics can be effectively studied by the means of computer mod-elling as they are closely linked together. The main tool used in the thesis is the PIC plasma simulation code family SPICE (sheathparticle-in-cell). It is able to precisely simulate small selected regions of the SOL, which is the region where the probe measurements are performed. In addition, it is equipped with various diagnostics allowing to study the current collection by biased electrodes immersed in the plasma.
As shown before [50, 52], SPICE codes also provide valuable input for inves-tigation of power and particle deposition on PFCs. Applying these data on real cases for the ITER tokamak showed interesting consequences even with a sim-ple heat conduction model extension. While this extension has been developed within this thesis, only results of probe simulations will be presented here. Re-sults obtained by the heat conduction model were however an important part of publication [53].
Moreover, a significant development of the SPICE2 code has been performed within work on the thesis in order to enable larger scale simulations. This has been achieved by implementation of the parallel solver of the Poisson equation, which is the core concept behind PIC technique.
4. Computer modeling of plasmas
Computer models and numeric methods are nowadays the prime tool for theoret-ical calculation in various research branches and plasma physics employs various techniques to tackle issues that can be found in all its topics. Having introduced basic plasma characterizations in the Chapter 2, we can now elaborate on the simulations with respect to the quantities and qualities they are aiming to assess.
The brief classification of plasma models and commentary with respect to sheath modeling is to be given and more attention will be paid to one specific technique, the particle-in-cell model.
4.1 Classification of models
To classify the models used in the plasma physics research, multiple approaches can be utilized. In this section, we will make the division by the basic laws behind each model type.