1
Dipartimento di sica
Corso di laurea magistrale in sica della materia Indirizzo sica del plasma
Plasma characterization in Hall thruster by triple Langmuir probe
Candidato Anna Giacobbe
Relatori
Dott. A. Macchi Prof. T. Andreussi
Contents
0.1 Introduction . . . 5
1 Electric propulsion and Hall thrusters 7 1.1 Thrusters principles . . . 7
1.1.1 The rocket equation . . . 7
1.2 Thrust and thruster e ciency . . . 9
1.2.1 Thrust . . . 9
1.2.2 Thruster e ciency . . . 11
2 Hall thrusters 13 2.1 Hall Thrusters . . . 13
2.1.1 E cross B devices . . . 15
2.1.2 Force transfer in Hall thrusters . . . 15
2.1.3 Thruster limits and improvement of operation: the magnetic shielding . . 16
3 Bayesian data analysis 21 3.1 Bayes rule . . . 21
3.2 Bayesian analysis . . . 22
3.3 Bayesian computation . . . 23
3.4 Nested sampling . . . 24
3.4.1 Generating new object by random sampling . . . 26
3.4.2 Posterior distribution . . . 26
3.4.3 Terminating the iterations . . . 27
4 Langmuir probe diagnostics 29 4.1 Langmuir probes . . . 30
4.1.1 Theory of the planar sheaths . . . 30
4.1.2 The I − V curve . . . 32
4.2 Triple Langmuir probes . . . 38
4.3 Laframboise theory for cylindrical Langmuir probe analysis . . . 40
4.3.1 Laframboise theory . . . 40
4.4 Mausbach parametrization of Laframboise results . . . 42
4.5 Utilization of Mausbach parametrization for probe analysis . . . 44
4.5.1 Set of equations . . . 44
4.6 Bohm theory . . . 46 3
4 CONTENTS
5 Technical characteristics of applied devices 47
5.1 Experimental campaign and devices . . . 47
5.2 The thruster . . . 48
5.3 The cathode . . . 48
5.4 The probes . . . 49
5.5 Studied magnetic con gurations . . . 52
5.6 Vacuum facilities . . . 55
5.6.1 IV 4 vacuum facility . . . 55
5.6.2 IV 10 vacuum facility . . . 56
6 Plasma investigation and obtained results 59 6.1 Magnetic con gurations and probe switching . . . 59
6.1.1 Magnetic con gurations . . . 59
6.1.2 Probe switching . . . 60
6.2 Deviations from homogeneity . . . 60
6.2.1 Case δn . . . 61
6.2.2 Case δT . . . 62
6.2.3 Case δV . . . 63
6.3 Data processing . . . 64
6.3.1 Introduction to the processing procedure . . . 64
6.4 Results relative to the δn case . . . 66
6.5 Results relative to the δT case . . . 69
6.6 Results relative to the δV case . . . 72
6.7 Comments and considerations . . . 75
6.8 Drifting electrons collection from a simple cylindrical Langmuir probe . . . 77
6.8.1 Extension to the cylindrical probe case . . . 78
0.1. INTRODUCTION 5
0.1 Introduction
This thesis is in the eld of plasma electric propulsion for space applications. Plasma propulsion systems work by electrically expelling an ionized propellant at high velocity.
Amongst the several dierent electric propulsion systems, we focus on the Hall thruster (HT). The one considered in the present work is a high power HT designed for operating in the 2.5 − 7.5 kW power range, and the exhaust velocity reached by the ions is of the order of 20 km/s. In order to optimise the thruster operations, it is important to characterize the plasma generated in them. In the present case, we use triple Langmuir probes.
Given the harsh plasma environment present inside the thruster, the triple probe was inserted for a short time (a few ms), to avoid any damage to the probe. Several measurements were made for the same thruster operative conditions using dierent electrode con gurations and bias voltages. In order to calculate the plasma properties from the measured current and potential data, it is necessary to consider how the probe interacts with the plasma, so a model of the plasma sheath around the electrodes is needed. We use an empirical parametrization of numerical simulations based on a steady state Vlasov-Poisson model.
Initial measurements showed an unexpected dependence of the results on the electrodes con- guration, as if the plasma parameters were varying in space on a scale comparable to the distance between the electrodes, contrary to assumption of a locally homogeneous plasmas. The need to analyse data coming from dierent electrode con gurations, as well as the addition of parameters to model inhomogeneities in the plasma, make it di cult to reliably calculate the plasma properties. For these reasons, we use a Bayesian probabilistic approach to combine data from dierent con gurations and to obtain a "most likely" plasma state, while keeping track of the uncertainty in the estimations of the plasma properties.
We explain the detected plasma inhomogeneities as an eect of the presence of drifting elec-trons inside the thruster in the azimuthal direction. In fact, electron drift causes a dierence in the potential measured by the electrodes due to the "screening" of the ow by the upstream electrode with respect to the downstream ones. This eect can be modelled by extending the plasma probe theory to a drifting electron distribution.
Chapter 1
Electric propulsion and Hall
thrusters
In general, electric propulsion (EP ) encompasses any propulsion technology in which electricity is used to increase the propellant exhaust velocity. It is a technology aimed at achieving thrust with high exhaust velocities, which results in a reduction in the amount of required propellant. This is very important for space missions, in fact reduced propellant mass can signi cantly decrease the launch mass of a spacecraft or satellite. It leads to lower costs from the use of smaller launch vehicles to deliver a desired mass into a given orbit or to a deep-space target. This technology was known and investigated starting from 19500s.
1.1 Thrusters principles
Electric thrusters propel the spacecraft using the same basic principle as chemical rockets accel-erating mass and ejecting it from the vehicle. The ejected mass from electric thrusters, however, is primarily in the form of energetic charged particles. This changes the performance of the propulsion system compared to other types of thrusters and modi es the conventional way of calculating some of the thruster parameters, such as speci c impulse and e ciency. Chemical rockets generally have exhaust velocities of 3 to 4 km/s, while the exhaust velocity of electric thrusters can approach 100 km/s for heavy propellant such as xenon atoms, and 1000 km/s for light propellants such as helium.
1.1.1 The rocket equation
The mass ejected to provide thrust to the spacecraft is the propellant, which is carried on board the vehicle and expended during thrusting. From conservation of momentum, the ejected propellant mass times its velocity is equal to the spacecraft mass times its change in velocity. The rocket equations describes the relationship between the spacecraft velocity and the mass of the system. Considering a one-dimensional problem, the rocket equation of motion in a gravitational eld can be written as:
M ˙v = − ˙mvex+ Fg (1.1)
where M is the instantaneous mass of the vehicle, ˙v is the rocket acceleration, ˙m is the mass ow rate, vex is the exhaust velocity (assumed to be constant and spatially uniform) i.e. the velocity
at which the propellant leaves the the rocket with respect to it and nally Fg is the gravitational
8 CHAPTER 1. ELECTRIC PROPULSION AND HALL THRUSTERS force. Moreover the rst term on the right hand side of the relation (1.1) represents the thrust acting on the vehicle
T = − ˙mvex . (1.2)
Considering Fgto be negligible respect to the given thrust and to have a constant vex parallel to
rocket initial velocity, the equation of motion takes a simpli ed form that can be easily integrated that results in
∆v = vexln(
M0
Mf
) (1.3)
where M0 and Mf are initial and nal vehicle mass respectively and ∆v is the magnitude of
velocity increment due to the ejection of a certain mass ∆M = M0− Mf of propellant.
Another important parameter traditionally used to characterize thruster performance is the speci c impulse
Isp=
vex
g , (1.4)
in which g is the gravitational acceleration. In terms of this new parameter the relation above can be written as
∆v = Ispg ln(
M0
Mf
) , (1.5)
which in turn can be rearranged in Mf
M0
= exp(− ∆v Isp g
) . (1.6)
This last formulation highlights the rule of the speci c impulse in order to minimize the needed propellant mass to accomplish a mission.
From this simple analysis it is clear that the thrust should be achieved via a high vex rather
than a large ejection of mass in order to reduce the amount of needed propellant. The exhaust velocity of chemical rockets is limited by the energy contained in the chemical bonds of the propellant used; typical values are up to 4 km/s. Electric thrusters, however, separate the propellant from the energy source (which is now a power supply) and thus are not subject to the same limitations. Modern Hall thrusters operating on xenon propellant have exhaust velocities in the range of 10 − 20 km/s.
1.2. THRUST AND THRUSTER EFFICIENCY 9
1.2 Thrust and thruster e ciency
1.2.1 Thrust
Thrust is the force supplied by the engine to the spacecraft. Since the spacecraft mass changes with time due to the propellant consumption, the thrust is given by the time rate of change of the momentum, so, as said before, it can be written as
T = d
dt(mvex) = dm
dt vex= ˙mvex , (1.7)
where ˙m is the propellant mass ow rate in kg/s and that can also be de ned as ˙
m = Q m (1.8)
where Q is the propellant particle ow rate (in particles/s) and m is the particle mass. The kinetic thrust power of the ejected beam, called the beam power, is de ned as
Pbeam= 1 2mv˙ 2 ex = T2 2 ˙m . (1.9)
This expression shows that techniques that increase the thrust without increasing the propellant ow rate will result in an increase in the beam power. Since we are dealing with a thruster in which the thrust is realized by an electric eld that accelerates ions, from the energy conservation we can write the ion exhaust velocity
vi,ex=
r 2qV
m , (1.10)
where V is the net voltage through which the ion was accelerated, q is the charge, and m is the particle mass. The mass ow rate of ions is related to the ion beam current, Ibeam , by
˙ mi=
mIbeam
q , (1.11)
so we can write for thrust
T = r 2m e Ibeam √ V [N ] . (1.12)
If the propellant is xenon,p2m/e = 1.65 · 10−3, the thrust is given by
T = 1.65 Ibeam
√
V [mN ] , (1.13)
where Ibeam is in Amperes and V is in Volts. This equation is the basic thrust equation that
applies for a unidirectional, singly ionized, mono energetic beam of ions. The equation must be modi ed to account for the divergence of the ion beam and the presence of multiply charged ions commonly observed in electric thrusters. The correction to the thrust equation for beam divergence is straightforward for a beam that diverges uniformly upon exiting from the thruster. For a thruster with a constant ion current density pro le accelerated by uniform electric elds, the correction to the force due to the eective thrust-vector angle is simply
Ft= cos θ , (1.14)
where θ is the average half-angle divergence of the beam. If the thrust half angle is 10 deg, then cos θ = 0.985, which represents a 1.5% loss in thrust. If the plasma source is not uniform
10 CHAPTER 1. ELECTRIC PROPULSION AND HALL THRUSTERS the thrust correction must be integrated over the beam pro les. For cylindrical thrusters, the correction factor is then
Ft=
Rr
0 2πrJ (r) cos θdr
Ibeam
, (1.15)
where J(r) is the ion current density which is a function of the radius. The ion current density is usually determined from direct measurement of the current distribution in the plume by plasma probes. For a constant value of J(r) the two relations become the same one.
1.2. THRUST AND THRUSTER EFFICIENCY 11
1.2.2 Thruster e ciency
The thruster mass utilization e ciency, which accounts for the ionized versus not ionized pro-pellant, is de ned for singly charged ions as
ηm= ˙ mi ˙ m = Ibeam e m ˙ m . (1.16)
The mass utilization e ciency describes the fraction of the input propellant mass that is con-verted into ions and accelerated in the electric thruster.
The electrical e ciency of the thruster is de ned as the beam power out of the thruster, Pbeam, divided by the total input power, PT :
ηe= Pbeam PT = IbeamV IbeamV + P0 , (1.17)
where P0 represents the other power input to the thruster required to create the thrust beam.
The cost of producing ions is described by an ion production e ciency term, sometimes called the discharge loss:
ηd=
Pd
Ibeam
, (1.18)
where ηd has units of Watts per Ampere (W/A), Pd is the power to produce ions and Ibeam is
the current of the ions produced. Contrary to most e ciency terms, it is desirable to have ηd as
small as possible since this represents a power loss.
The total e ciency of an electrically powered thruster is de ned as the beam power divided by the total electrical power into the thruster:
ηT =
Pbeam
PT
(1.19) so, the e ciency of any electric propulsion thruster is
ηT =
T2
2 ˙mPT
. (1.20)
From this relation we can say that the electric power needed to produce thrust is
P = T Isp g 2ηT
. (1.21)
Given a reasonable speci c impulse for an electric device of about 2000 s and a thrust e ciency of about 50% (a slightly optimistic value), the actual power levels available in space do not allow to obtain high levels of thrust. For this reason the usage of electric propulsion is con ned to low thrust applications.
The power into a thruster that does not result in thrust must be dissipated primarily by radiating the unused power into space. If the thruster electrical e ciency is accurately known, the dissipated power is
Chapter 2
Hall thrusters
There are many types of electric thrusters and they are usually described in terms of the accel-eration method used to produce the thrust. The one on which we will focus is the Hall thruster.
2.1 Hall Thrusters
The Hall thruster (HT ) is an electric thruster that uses both electric and magnetic elds to generate plasma inside the thruster channel and to extract from it ions to provide the thrust. Its typical geometry is given by two concentric dielectric cylinders. The plasma is generated in a channel between these cylinders. The dielectric material must have a low ion sputtering yield and good properties with respect to secondary electron emission due to electron impact to avoid as much as possible channel erosion. Ceramic materials such as BN (boron nitride) or BN-SiO2
are often used for the channel walls. At the beginning of the channel there is the anode from where neutral gas atoms exit while the cathode is outside the thruster channel. Between anode and cathode an electrostatic eld E is set. The channel is surrounded by a magnetic circuit that generates a quasi radial magnetic eld B into the channel. Electrons, owing out from the cathode into the channel, attempting to reach the anode, encounter a transverse radial magnetic eld, which reduces their mobility in the axial direction and inhibits their ow to the anode. The electrons tend to spiral around the thruster axis in the E cross B direction forming the Hall current from which the device derives its name. Electron con nement is very important because thanks to their presence inside the channel is formed a ionization zone where by collision with electrons neutral atom are ionized; as a result, the potential drop at the end of the channel forces the ions, which are not magnetized and are therefore free to move along the channel, to accelerate down the potential gradient to high exhaust velocities (namely in the order of 20 km/s, much higher than in chemical propulsion devices). This process is triggered by a small fraction of the electron ow leaving the cathode, but most of the electron current is sustained by the electrons produced by the ionization of the propellant gas. An equivalent amount of electrons represents the rest of the cathode ow which neutralizes the ion beam outside the channel. The positioning of the cathode breaks the axisymmetric con guration of the thruster; however, because electrons are emitted in a region of low electric potential (essentially a zero electric eld region), their injection point is nearly indierent for performance evaluation purposes. These thrusters can be powered with lots of dierent propellants but the most used is xenon thanks to its high molecular weight and its low ionization energy.
14 CHAPTER 2. HALL THRUSTERS
Figure 2.1: Schematic section of the Hall thruster [22]
Figure 2.2: Schematic external view of the
2.1. HALL THRUSTERS 15
2.1.1 E cross B devices
Hall thruster are examples of cross eld devices; in these devices an external magnetic eld B is placed perpendicularly to an applied electric eld E and an electron drift is generated in the E cross B direction. The E × B drift velocity is of the order of E/B if the electron collision frequency ν is small respect to the electron angular cyclotron frequency Ωce = qB/m. It is
essential that the cross B electron current associated with the electron ow in this direction, also called Hall current, does not ow to a wall. This would induce charge polarization leading to the generation of another electric eld in the E × B direction (Hall eect), the Hall electric eld EH, that would oppose the current. This Hall electric eld would in turn generate an EH cross
B ux antiparallel to the applied electric eld, therefore destroy the electron con nement by the magnetic eld and enhance the cross eld transport. There should not be any Hall eect in a Hall thruster to ensure a good con nement of electrons and a lowering of the electron transport in the axial direction. The best way to ensure that Hall current could not reach thruster walls is to use a cylindrical geometry for the thruster, with E cross B in the azimuthal direction.
Figure 2.4: E cross B mechanism [22]
2.1.2 Force transfer in Hall thrusters
In Hall thrusters, ions are generated in a plasma volume and accelerated by an electric eld in the plasma. However, the presence of the transverse magnetic eld responsible for the rotational Hall current modi es the force transfer mechanism. Assume for argument that the Hall thruster plasma is locally quasi neutral (qni ∼= qne) in the acceleration region, where ne and ni are
respectively the electron and the ion plasma density, and that in the acceleration zone the electric and magnetic elds are uniform. The ions are essentially unmagnetized and feel the force of the local electric eld, so the force on the ions is
~ Fi= 2π
Z Z
qniErdrdz .~ (2.1)
The electrons in the plasma feel an E cross B force and circulate in the system transverse to the electric and magnetic elds with the velocity
~ ve=
~ E × ~B
16 CHAPTER 2. HALL THRUSTERS The electrostatic force on the ions is the opposite of the electrostatic force on the electrons due to their sign dierences. The electrons are constrained not to move axially by the transverse magnetic eld, so the force per unit area on the electrons (to the left) is balanced by the Lorentz force: ~ Fe= −2π Z Z qneErdrdz − 2π~ Z Z ene~ve× ~Brdrdz = 0 . (2.3)
Using quasi-neutrality and the de nition of the Hall current density, JH = −eneve, the force on
the ions is shown to be equal to the ~v × ~B part of the Lorentz force on the electrons: ~ Fe= −2π Z Z qniErdrdz + 2π~ Z Z ~ JH× ~Brdrdz = − ~Fi+ 2π Z Z ~ JH× ~Brdrdz = 0 (2.4)
so, the force on the ions is then ~ Fi= 2π
Z Z ~
JH× ~Brdrdz . (2.5)
By Newton second law, the Hall current force on the magnets is equal and opposite to the Hall current force on the electrons and, therefore, is also equal and opposite to the force on the ions
~ T = −2π
Z Z ~
JH× ~Brdrdz = − ~Fi . (2.6)
In Hall thrusters the thrust is transferred from the ions to the thruster body through the electro-magnetic Lorentz force. These thrusters are sometimes called electroelectro-magnetic thrusters because the force is transferred through the magnetic eld. However, since the ion acceleration mechanism is by the electrostatic eld, we choose to call them electrostatic thrusters.
2.1.3 Thruster limits and improvement of operation: the magnetic
shielding
The main limiting factor for good operation and thruster life is the erosion due to ion bombard-ment on channel walls covering the magnetic circuit. The best way to limit this phenomenon is to work with a so called magnetic shielded con guration in which magnetic lines will remain as parallel as possible to channel walls.
Magnetic shielding is a new concept in Hall thruster technology that is being currently inves-tigated for the design of long life, low erosion, HTs.
Isothermal magnetic eld lines and thermalized potential
In order to be able to understand qualitatively the erosion saturation and the magnetically shielded con gurations, it is rst needed to have a general brief discussion about electrons be-haviour along magnetic eld lines.
Magnetic eld lines are isothermal, in fact electrons rapidly reach isothermal conditions along them because of low collisionality of the plasma, making the electrons free to ow along the magnetic eld lines. It is therefore possible to assume Te= Te0along a magnetic eld line, where
Te0 is the value of the electron temperature at a reference position on the magnetic line. The
second important property we have to take into account is the thermalized potential. First of all we have to consider the second order moment of the Boltzmann equation for the electrons
mene[
∂ ~ue
2.1. HALL THRUSTERS 17 where me is the electron mass, ne is the electron density, ue is the electron velocity, e is the
electron charge, Pe is the electron pressure and C is the collisional exchange of momentum. We
introduce now some simplifying hypothesis, considering typical HT operational parameters and neglecting those terms not relevant for an intuitive understanding of the concept:
• negligible electron inertia;
• negligible collisional momentum exchange for electrons;
• stationary conditions, therefore ~E = − 5 φ, with φ electric potential.
We then project the vectorial equation along a magnetic eld line, over which a curvilinear abscissa s is de ned, obtaining
0 = ene
∂φ ∂s −
∂Pe
∂s . (2.8)
If we nally assume that
• electrons are in thermal equilibrium (Maxwellian distribution function), so Pe = kneTe,
with k Boltzmann constant;
• the electron temperature is constant along the magnetic eld line, i.e. Te = TeM where
TeM is the value of the electron temperature at the intersection of the magnetic line with
the channel centerline.
Then, the previous equation can be written as
ene
∂φ
∂s = kTeM ∂ne
∂s , (2.9)
that can be simply integrated from the centerline to the general abscissa s to obtain
φ = φM + kTeM e ln ( ne neM ) , (2.10)
in which φM and neM are the values of the electric potential and electron density at the
inter-section of the considered magnetic line with the channel centerline respectively.
This formula states that the plasma potential along a magnetic eld line is close to constant as long as the electron temperature and the density gradients along that particular line are small. This implies that the electric eld is close to be orthogonal to the magnetic eld and that the deviations from this condition can be reduced decreasing the local electron temperature. This property, already recognized by Morozov [29], is called thermalized potential. It is extremely relevant because if the temperature is kept su ciently low, one can simply design the electric eld in the channel by properly designing the magnetic eld that is externally imposed.
Channel erosion
In a standard magnetic con guration the magnetic eld is almost radial and magnetic eld lines orthogonally intercept the channel walls; the peak of Br on the channel centerline is close to
the channel exit. Following the magnetic eld topology, the electrons can reach the channel walls moving along the magnetic lines. Here a plasma sheath is formed and, in order to balance the ow of charged particles, electron ow is reduced by the sheath voltage barrier, while ions are accelerated toward the walls. The ion bombardment of the walls produces sputtering of the walls, generating erosion. As emerged from the analysis of [1] and [30], the energy of the incoming
18 CHAPTER 2. HALL THRUSTERS
Figure 2.5: HT channel: a) standard con guration at the beginning of operative life; b) standard con guration at the end of operative life; c) shielded con guration to avoid erosion [9]
ions has a direct dependence on the local electron temperature, since the region of maximum temperature is near the exit of the channel, this is where the erosion is greater. As said before, erosion of the channel walls is one of the main issues for thruster operative life limitation. This phenomenon prompted the analysis of a non-standard magnetic con guration for increasing the thruster lifetime, in fact recent mission analyses have shown the need for longer thruster lifetimes for both Earth orbit raising and deep space applications.
Magnetic shielded con guration
To avoid these problems the magnetic shielded con guration is set. Considering a thruster with a chamfered channel ceramics, the magnetic eld is shaped to be almost tangent to the walls. The magnetic peak is moved outside of the channel while the grazing line (the magnetic line that touches the chamfer edge) extended deeply into the channel. In this con guration with the magnetic eld lines that do not cross the channel walls directly but that are made tangent to it, the erosion should be drastically reduced. In fact, in this con guration, the electron temperature near the walls is close to the temperature of the grazing line, that is signi cantly lower than the peak temperature. Electrons following this line are considerably cooler than before and so the sheath potential drop decreases and the ions acquire less kinetic energy before striking the wall. An experimental assessment of the magnetically shielded thruster [34] showed that the erosion can be reduced by more than two orders of magnitude, whereas performances remain close to those of the standard con guration; so these analyses demonstrate that the shielded con gurations manage to reduce the interaction between plasma and channel walls.
Figure 2.6: on top: unshielded and shielded magnetic con guration. On bottom: axial pro le of the electric potential and electron temperature close to the wall [22]
2.1. HALL THRUSTERS 19
Chapter 3
Bayesian data analysis
In the present work, we use a Bayesian probabilistic approach to combine data from dierent diagnostics con gurations (we will explain this in details in the next chapter) and to obtain a "most likely" plasma state, while keeping track of the uncertainty in the estimations of the plasma properties. In addition, this approach allows to deal with underdetermined systems as ours will reveal to be. Therefore, before entering into the details of this work, it is advisable to dwell brie y on the characteristics of Bayesian inference and its analysis methods.
3.1 Bayes rule
Ideally, the diagnostic model would provide the exact function D = f(x), which gives the vector of data that would be observed D if the state of the entire system was exactly the one described by the parameters vector x. This is never practically possible, since there are always some details of the physical state too subtle or impossible to assign parameters to, as the noise created by the thermal uctuations. For that reason, instead of a f(x) the diagnostic model produces a probability density function (P DF ) of the data that would be observed P (D|x), known as the likelihood distribution. It gives the uncertainty in D for an exact x. For a real experiment the observed data D are known exactly and the parameters vector x is uncertain. The P DF in this case is P (x|D) and it is known as the posterior P DF . The posterior describes the probability that the system was in a state x, given that the data D was observed, and it can be obtained from the Bayes rule:
p(x|D, I) =p(D|x, I)p(x|I)
p(D|I) . (3.1)
Since D is known exactly, P (D|I) is constant and it is called the evidence of D, where I are the informations on which our model is based on, while P (x|I) is known as the prior probability. The prior encodes any knowledge, or lack of knowledge, of the plasma state before the measurement took place. So, nally, the posterior represents everything that is known about the plasma state from the diagnostic observation and the prior assumptions, including all uncertainties.
22 CHAPTER 3. BAYESIAN DATA ANALYSIS
3.2 Bayesian analysis
As said before, Bayesian inference is the process of tting a probability model to a set of data and summarizing the result by a probability distribution of the parameters and on unobserved quantities. The essential characteristic of Bayesian methods is their explicit use of probability for quantifying uncertainty in inferences based on statistical data analysis.
The process of Bayesian data analysis can be idealized by dividing it into three steps: • Setting up a full probability model, a joint probability distribution for all observable and
unobservable quantities in a problem. The model should be consistent with knowledge about the problem we are dealing with and the data collection process;
• Conditioning on observed data: calculating and interpreting the appropriate posterior dis-tribution, the conditional probability distribution of the unobserved quantities of ultimate interest, given the observed data;
• Evaluating the t of the model and the implications of the resulting posterior distribution: how well does the model t the data, are the conclusions reasonable, and how sensitive are the results to the modelling assumptions in step 1? In response, one can alter or expand the model and repeat the three steps.
Checking the model is crucial to statistical analysis. Bayesian prior-to-posterior inferences assume the whole structure of a probability model and can yield misleading inferences when the model is poor. A good Bayesian analysis, therefore, should include at least some check of the adequacy of the t of the model to the data and the plausibility of the model for the purposes for which the model will be used. This is sometimes discussed as a problem of sensitivity to the prior distribution. The basic question of a sensitivity analysis is: how much do posterior inferences change when other reasonable probability models are used in place of the present model? Other reasonable models may dier substantially from the present model in the prior speci cation, the sampling distribution, or in what information is included. It is possible that the present model provides an adequate agreement to the data, but that posterior inferences dier under plausible alternative models.
External validation
More formally, we can check a model by external validation using the model to make predictions about future data, and then collecting those data and comparing to their predictions. If the model ts, then replicated data generated under the model should look similar to observed data. Any systematic dierences between the simulations and the data indicate potential failings of the model. We measure the discrepancy between model and data by de ning test quantities, the aspects of the data we wish to check.
3.3. BAYESIAN COMPUTATION 23
3.3 Bayesian computation
There are a lot of modern numerical techniques that are useful for doing Bayesian calculations when analytical approximations are inadequate; in particular, we will focus on the one named nested sampling. The inputs to our computation are the prior π(x) and the likelihood L(x), while the desired outputs are the evidence Z and the posterior P (x). The normalization required for probability means that the total masses (accumulated amounts of probability) of both prior and posterior are unity
Z Z ... Z π(x)dx = 1 and Z Z ... Z P (x)dx = 1. (3.2)
From the second relation of the (3.2) we can separate Z from P (x), obtaining Z =
Z Z ...
Z
L(x)π(x)dx . (3.3)
Our aim is to evaluate these outputs for problems too large for brute-force enumeration of all x or for approximation by any algebraic form. Even if we can not do it in practice, we can imagine to evaluate Z from (3.3) by the direct summation of small elements, and moreover we can imagine sorting these elements into decreasing order of likelihood value.
Figure 3.1: Sorting of the elements into decreasing order of likelihood on a logarithmic scale [21] So let us de ne this new quantity ξ(λ) that is equal to the proportion of prior with likelihood grater than λ. ξ(λ) = Z Z L(x)>λ ... Z π(x)dx , (3.4)
in which the element of prior mass is dξ = π(x)dx. Because the restriction on likelihood becomes tighter as λ increases, ξ is a decreasing function of the likelihood limit λ. The extreme values are λ ≥ 0 at ξ = 1, because likelihood values can not be negative, and λ=Lmax(if the maximum
exists) at ξ = 0. It is more convenient to use these quantities linked into the inverse form
L(ξ(λ)) = λ , (3.5)
but we have to be careful to confuse not L(ξ) with the likelihood L(x), because the rst has a scalar argument ξ, while the second has the vector argument x. The new likelihood function L(ξ) underlines both the evidence Z and the posterior π(x). In fact, each element dξ comes from a source volume of that same prior mass π(x)dx. Hence the evidence, being a sum over these elements, is simply the enclosed area shown in gure (3.2 a),
Z = Z 1
0
24 CHAPTER 3. BAYESIAN DATA ANALYSIS
Figure 3.2: (a) Likelihood function with area Z. (b) Posterior samples are randomly scattered over the area Z [21]
Any point taken randomly from this area, also shown in gure, yields a random sample from the posterior distribution
P (ξ) = L(ξ)
Z , (3.7)
which by the same argument gives equivalently a random sample from the posterior distribution P (x). Thus the sorted likelihood L(ξ) is the key to obtain both the evidence Z and the posterior P (x)as a set of random samples.
3.4 Nested sampling
The technique of nested sampling uses a collection of n objects, randomly sampled with respect to the prior π(x), but also subject to an evolving constraint L(x) > L∗, preventing the likelihood
to exceed the current limiting value L∗. In terms of ξ the objects are uniformly sampled subject
to the constraint ξ < ξ∗, where ξ∗corresponds to L∗. Then the iteration continues inward in ξ ,
so upward in L, in order to locate and quantify the tiny region of high likelihood where most of the joint distribution is to be found. The evidence of (3.3) is evaluated by associating with each object in the sequence a width h = ∆ξ and hence a vertical strip of area A=hL; we can thus write
Z ≈ ΣkAk, where Ak=hkLk (3.8)
and the simplest de nition for the width is hk = ξk−1− ξk. The usual behaviour of the areas A
is that they start by rising with the likelihood L increasing faster than the widths h decrease. At some point, L attens o su ciently that decreasing width dominates increasing likelihood, so that the areas pass across a maximum and start to fall away.
log Z ≈ log(ΣkAk) . (3.9)
The summation of (3.9) is performed in this way: at the beginning we set an iteration counter k = 0and N active (or live) points drawn from the full prior π(x), so the initial prior volume is ξ0= 1. The samples are then sorted in order of their likelihood and the smallest (likelihood L0)
3.4. NESTED SAMPLING 25
Figure 3.3: Objects sampled uniformly in ξ < ξ∗ [21]
Figure 3.4: An iteration replaces the worst ob-ject with a new one inside the shrunken do-main [21]
is removed from the active set, becoming inactive, and replaced with a new point drawn from the prior now subject to the constraint that the point has a likelihood L > L0. Now the prior
volume contained within the corresponding iso-likelihood contour will be ξ1= t1ξ0, with t1 that
follows the distribution P (t) = NtN −1, with mean and standard deviation log t = (−1 ± 1)/N.
This iteration is repeated until the entire prior volume has been explored, or more commonly on determining the evidence Z to some speci ed precision previously decided. Thus the algorithm goes through nested levels of likelihood as the prior volume is reduced. Successive iterations generate a sequence of discarded objects on the edges of progressively smaller nested domains. At iterate k we have Lk = L∗ and ξk= ξ∗= k Y j=1 tj , (3.10)
in which each shrinkage ratio tk is independently distributed with its probability distribution
function and statistics, so it follows that
log ξk= (−k ±
√
k)/N (3.11)
that, if we ignore the standard deviation on t and consider log t = −1/N, becomes ξk =
exp −k/N. We can also de ne the negative entropy as the (positive) amount of information in the posterior H = Z P (ξ) log[ P (ξ) ] dξ ≈X k Ak Z log[ Lk Z ] , (3.12)
26 CHAPTER 3. BAYESIAN DATA ANALYSIS
Figure 3.5: Discrete sequence of the sorted likelihood function L(ξ) [21]
3.4.1 Generating new object by random sampling
Continuing a nested sampling calculation involves generating a new object from the prior, subject to the likelihood constraint L>L*. The prior domain obeying this constraint shrinks geomet-rically as the calculation proceeds and we do not expect to be able to nd an object in this tiny domain ab initio. Instead we need to learn from experience and use guidance from previous iterates. A sequence of iterates in which each use the previous ones is called Markov Chain and the method that do this in the context of random sampling is called Markov Chain Monte Carlo, MCMC.
3.4.2 Posterior distribution
All the information from a Bayesian analysis of a given set of experimental data is contained in the posterior distribution of the parameters. The expectation value of the posterior quantity of interest can be written in the following form
hQ(x)i = 1 Z
Z
Q(x)P (x|I)P (D|x, I)dx , (3.13)
where Z = R P (x|I)P (D, |x, I)dx. The expectation value of Q(x) is therefore the average value of Q(x) over the normalized posterior distribution of the parameters x.
Marginalization of the posterior distribution
The posterior distribution will be a multivariate distribution in accordance to the number of parameters associated to it. In order to arrive at the probability distribution function of any quantity x of the model, marginalization of the multidimensional posterior distribution function can be regard as a projection of the complete probability distribution function onto that quantity. Marginalization is performed by integration over the quantities one want to get rid of. After obtaining by marginalization the probability distribution function of a single parameter x we can associate to it its mean value and standard deviation. Obviously the generation of marginal densities and the calculation of moments from the posterior distribution require to carry out
3.4. NESTED SAMPLING 27
Figure 3.6: The two-dimensional joint probability P (D, θ|I) together with the marginal densities [10]
integral of high dimensionality, so numerical methods are needed and consequently considerable computational power is required.
Summarizing, a formal recipe of the Bayesian approach consists of formulation of the model with the inclusion of prior knowledge, quanti cation of the resulting likelihood, determination of the posterior and nally marginalization of the quantity of interest in the posterior distribution. The marginal distribution P (xk|D, I) is then the complete result of the Bayesian inference on
the parameters xk.
3.4.3 Terminating the iterations
To decide which one would be the last step of a Bayesian investigation has always been an issue. The usual behaviour of the areas A is that they start by rising with the likelihood L increasing faster than the widths h decrease. At some point L attens o su ciently that decreasing width dominates increasing likelihood, so the areas pass across a maximum and start to fall away. Most of the total area is found in the region of this maximum, which occurs in the region of ξ ≈ exp−H.
Remembering that ξk ≈ exp−k/n suggests a termination condition like 'continue iterating until
the count k signi cantly exceeds nH'. Unfortunately there is no rigorous criterion to ensure the validity of any such termination condition for the iteration process; sometimes termination is crudely imposed by stopping after su cient number of iterates.
Chapter 4
Langmuir probe diagnostics
Electrostatic plasma probes were invented by Irving Langmuir in 1924. Of all the ways to measure a plasma, the Langmuir probe is probably the simplest, since it consists of sticking a wire into the plasma and measuring the current to it at various applied voltages.
In this way it is possible to measure electron and ion density, electron temperature and plasma local potential. Obviously, their accuracy in measuring these plasma properties depends on the particular situation in which they are used. However, Langmuir probing is an intrusive technique and the probe must be carefully designed so as not to interfere with the plasma nor be destroyed by it. Moreover, despite their simple construction and operation, the interpretation of their data can be quite complex due to several deviations from ideal conditions.
One of the most complex issue to take into account is that electrodes walls form bounding surfaces for the plasma and thus cause the formation of a sheath in which particles motion is governed by dierent equations with respect to the rest of the plasma. Moreover, the perturbation that a probe can generate inside the plasma, and thus the possibility that its presence could signi cantly alter measurements, is linked to the conditions of the surrounding plasma.
We de ne low pressure plasma the plasma regime in which the mean free paths of ions and electron are bigger respect to the probe size and in which they feel the in uence of the electric eld generated by the probe at a distance lower than a mean free path from the probe walls. Low pressure plasma is the one considered originally by Langmuir and the one we will consider in this work. In addition, this kind of plasma is typically characterized by Te Ti. If, instead,
we are dealing with high pressure plasma, we have that the mean free path of charged particles is comparable or less than the probe size and that the electric eld generated by the probe penetrates into the plasma for a distance much greater than a mean free path. Typically, in high pressure plasma regime Ti ≈ Te. Knowing in which plasma condition we are is important to
interface the plasma with a probe of the appropriate size.
30 CHAPTER 4. LANGMUIR PROBE DIAGNOSTICS
4.1 Langmuir probes
Langmuir probes generally consist in one or more conducting electrodes inserted into a plasma. The simplest Langmuir probe is made by a single electrode. The probe consists in a conducting wire typically made out of high temperature material, such tungsten or nickel (0.1 − 1 mm in diameter), surrounded by an insulating sleeve, usually of a ceramic such as alumina. These materials can be exposed to low-temperature laboratory plasmas without melting or excessive sputtering. To avoid disturbing the plasma, the ceramic tube should be as thin as possible, preferably < 1 mm in diameter but usually several times this value. The conductor wire extends out of the insulator for a short distance, forming the probe collecting area, without touching the ceramic tube, so that it would not be in electrical contact with any conducting coating that may deposit onto the insulator. The assembly is encased in a vacuum jacket, which could be a stainless steel or glass tube. The thickness of the probe represents a very important characteristic because it can determine probe operation. In fact, if the probe is too thin and we have to work with a dense plasma, it will not resist to high temperature and, draining ion current, it can be eroded by sputtering, changing the collecting area. In fact, even if we deal with low pressure plasma, we must keep in mind that probe electrodes have to face with a temperature of about tens of eV (1 eV ≈ 11000 K) that still represents a fairly harsh environment for the devices. On the other hand, as said before, if the probe is too thick, it can perturb the measurements. A
Figure 4.1: Schematic section of a Langmuir probe [17]
grounded power supply is used to bias the probe to potentials both above and below the plasma potential, while the current collected by the probe is measured. In this way, a characteristic curve with some distinct regions is extracted and each of these regions allows one or more of the plasma parameters to be determined. This is the I − V probe characteristic.
4.1.1 Theory of the planar sheaths
The non neutral region between the plasma and the probe surface is called sheath. Its simplest modelling is the planar collisionless sheath model.
Assumptions for the model
• Boltzmann equilibrium at Tefor electrons;
4.1. LANGMUIR PROBES 31
• quasi-neutrality (ne=ni) at the plasma-sheath edge.
We set the plasma-sheath edge at x = 0. Moreover, we de ne the zero of the potential V (0) = 0 and take the ions to have a velocity vsthere.
Let us consider now ion energy conservation and continuity of ion ux respectively: 1 2miv 2(x) = 1 2miv 2 s− eV (x) , (4.1) ni(x)v(x) = nisvs . (4.2)
Solving the rst equation for v(x) and substituting into (4.2) we obtain
ni(x) = nis(1 −
2eV (x) mivs2
)−12 . (4.3)
Instead, the electron density neis given by the Boltzmann relation
ne(x) = nese
V (x)
Te , (4.4)
with nes= nis= nsdensity at the sheath edge, because there the quasi-neutrality of the plasma
is still valid. So, the Poisson equation is d2V dx2 = e 0 (ne− ni) = ens 0 [eTeV − (1 − 2eV mivs2 )−12] . (4.5)
This is the basic nonlinear equation governing the sheath potential and both ion and electron densities. However it has solutions only for su ciently large vs, created in an essentially neutral
pre-sheath region that we will introduce later.
32 CHAPTER 4. LANGMUIR PROBE DIAGNOSTICS Bohm sheath criterion
As mentioned before, to have a sheath solution of the Poisson equation we need a condition on vs. In order to nd the condition on the edge velocity vs, we rst multiply both sides
of the equation (4.5) by dV/dx and after we integrate over x respect to V . This leads to
1 2( dV dx) 2 = ens 0 [Tee V /Te − T e+ mivs2 e (1 − V (miv2s/e)) 1/2− mivs2
e ]. Since the left hand part of the
relation is a squared term, it is always positive and so, for the right hand term, we must have:
[TeeV /Te− Te+ mivs2 e (1 − V (miv2s/e) )1/2−miv 2 s e ] ≥ 0 ; (4.6)
for V → 0 we expand it to the second order of Taylor series: 1 2 V2 Te −1 2 V2 (mivs2/e) ≥ 0 . (4.7)
The potential terms disappear and the existence condition becomes
v2s≥ eTe/mi⇒ vs≥ (eTe/mi)1/2≡ vB . (4.8)
So we nd that the existence condition for Poisson equation solution is vs ≥ vB. vB is called
Bohm velocity, while this existence condition is known as the Bohm sheath criterion. As marginal condition for the sheath we assume v(0) = vB.
The pre-sheath region
To give the ions the velocity vB there must be a nite electric eld in the plasma over some
region, typically much wider than the sheath, called pre-sheath.
Hence the pre-sheath region is not strictly eld free, even if E is very small there. At the sheath/pre-sheath interface there is a transition from subsonic (vi ≤ vB) to supersonic
(vi ≥ vB) ion ow, where the condition of charge quasi-neutrality must break down. The
potential drop across a collisionless pre-sheath, which accelerates ions to Bohm velocity, is given by 1
2miv 2
B = eV where V is the plasma potential respect to the potential at the
sheath/pre-sheath edge. Substituting the value of vB, it becomes V = Te/2. Because uid equations present
a singularity for vi → vB, no continuous connection solution between sheath and pre-sheath
regions can be found. In this model we x vi = vB as matching condition at the edge between
the quasi-neutral zone and the sheath [4].
4.1.2 The I − V curve
A Langmuir I − V curve shows the trend of the current collected by the probe with respect to an applied voltage V . The curve is shown in gure (4.3). The point at which the curve crosses the VB axis (applied voltage) is called the oating potential Vf. For a negative biased
probe (to the left of the Vf point), the probe drains ion current and the curve soon attens
out to a more or less constant value called ion saturation current Ii,sat. Instead, for a positive
biased probe (to the right of the Vf point), electron current is drained and the curve goes into
an exponential part, also called the transition region. The transition region ends when the space potential Vs is reached. Here the curve takes a sharp turn, known as the knee, and after the
collected current saturates at the electron saturation current Ie,sat. The curve drawn in gure
4.1. LANGMUIR PROBES 33
Figure 4.3: Typical I − V characteristic for a Langmuir probe [17]
Electron and ion saturation current
If the electron population is Maxwellian, the electron current to the probe is expected to vary exponentially with the probe voltage until it reaches its saturation regime:
Ie(Vp) = ( Ie,sate Vp Te f or Vp< 0 , Ie,sat f or Vp≥ 0 . (4.9) where Ie,sat= neevtheAp (4.10)
in which vthe is the electron thermal velocity and it is equal to vthe =pkTe/2πme; Ap is the
probe collection area and Vp is the probe voltage respect to the space potential Vs. The space
potential is reached when the electron current to the probe saturates.
Substituting all the terms into the relation for Ie, the electron current can be written as
Ie(Vp) = eneAp q kTe 2πmee Vp Te f or Vp< 0 , eneAp q kTe 2πme f or Vp≥ 0 . (4.11)
The equation for the electron current shows that the slope of the (ln I)−V curve is exactly 1/Te
and it is a good de nition for the electron temperature. In fact ln( Ie Ie,sat ) = Vp Te , (4.12)
from which we see that the inverse slope of the logarithmic electron probe current with respect to Vpgives directly Te. However, the above simple interpretation is limited by the dynamic range
34 CHAPTER 4. LANGMUIR PROBE DIAGNOSTICS on which Ie= I − Ii can be measured with su cient accuracy. In fact, for Ietoo small, adding
to Ii the measured I can introduce errors in the determination of Ie. On the other hand, for Vp
too large, the Boltzmann exponential is no longer accurate as electron saturation approaches. The nominal useful range of voltages over which the slope can be measured is then [3]
|∆Vp| Te ≈ ln(Ie,sat Ii,sat ) = ln( mi 2πme )1/2 . (4.13)
Moreover, before Ie can be obtained from I one has to subtract the ion current Ii. This
can be done approximately by drawing a straight line through Ii,sat and extrapolating it to the
electron region. In fact for what concerns ions we can neglect the kinetic energy (Ti Te) and
so the velocity they reach at the sheath edge is just the Bohm velocity vB, due to the potential
drop of the pre-sheath equal to −Te
2. So, considering ns and substituting into it the potential
drop we have Ii,sat= IB = 0, 61eniAp r kTe mi , (4.14) where 0, 61 is e−1 2.
So, under the thin sheath conditions the ion saturation current does not depend on probe potential respect to plasma.
Floating potential
The oating potential Vf is reached at the point in which the curve crosses the VB axis, so it is
de ned by Ii= IB= Ie. Equating the relations given for the currents we have nally
Vf = Vs+ Teln(0, 61
r 2πme
mi ) . (4.15)
Eects of a magnetic eld
Until now all the analyses have assumed that no magnetic led was present and so that the particle dynamics were determined only by the electric eld. What we want to consider now is how the presence of magnetic eld aect the results. The main eect of the B eld is to cause the electrons and ions to move no longer in straight lines but to orbit around the magnetic lines with Larmor radius ρ = mv/eB. The particle motion across the magnetic eld is thus greatly restricted. The importance of the magnetic eld eects is obviously determined by the ratio of ρ to the radius r of the probe. Clearly, the electron Larmor radius is smaller than the ion one by the factorp(me/mi). As a result, the electrons are more strongly aected than the
ions; the rst thing that happens to the probe characteristic when a magnetic eld is present is that the electron saturation current is decreased, since the electron ow is impeded. This will be most immediately evident as a reduction in the ratio of electron to ion saturation currents. Instead, the ion dynamic, for ρi r, is relatively unaected by the magnetic eld and thus
the ions satisfy the same equation (4.14) for the current. The ion current, in fact, depends only on n and ion dynamics, so it is just the same. However, the electron current will maintain its exponential dependence on Vp, so that analysis of the current slope will again provide Te. In
summary, if ρi r and most of electrons are repelled, the previously discussed interpretation
of the probe characteristic should provide accurate results. Instead, when the magnetic eld is su ciently strong that the ion Larmor radius is smaller than the probe size ρi< r, considerable
modi cations to the ion collection occur and it is no longer possible to formulate a completely collisionless theory. In the thruster con gurations we considered, the magnetic peak was always low enough to avoid this situation.
4.1. LANGMUIR PROBES 35 From an ideal to a real I − V characteristic
Between the ideal curve and the one extrapolated from data there are a lot of dierences. First of all, the sharp knee of the ideal curve at Vs is smoothed and rounded and that creates some
di culties in the evaluation of the plasma potential. In fact, now, we have not a precise point for the knee and it is necessary to nd it with some approximation methods as the linear in-terception. This is the standard method of determining plasma potential from probe data: the electron current is rst calculated from the probe current by subtracting the ion current, typi-cally assumed to be constant, from the data. The natural logarithm of the electron current is then plotted as a function of the bias voltage. So, in the transition region, below the plasma potential, a line is tted to the electron current. A line is also tted to the electron saturation region above the plasma potential. The probe voltage at which the two lines intersect is taken as the local plasma potential.
Figure 4.4: Linear intersection method for determining the local plasma potential [19]
Moreover, for what concerns the saturation currents, in a real I − V curve they are not constant but they slowly increase changing the potential; respectively, Ie,satgrows with increasing
potential while Ii,sat grows with decreasing potential. It is caused by the sheath expansion, that
changes the probe collection area Ap. We will treat this phenomenon in details in the next
section.
Sheath expansion
Until now we have suppose to deal with an ideal planar sheath, considering it as a good approx-imation to the sheath formed around our cylindrical probe. In our assumption we consider the sheath thickness to be small compared to the probe radius, so, as a rst approximation, we take the sheath surface AS to be equal to the probe surface area Ap. In the thin sheath model the ion
saturation current is theoretically independent to the probe potential. However,in practice we nd a quite dierent situation. In fact, when ion saturation current is drained, it generally shows a slow increase in magnitude with increasing negative potential (the same happens for Ie,satthat
slowly increases with the increasing positive potential). The reason of this behaviour is that the sheath thickness increases as the probe potential is made more negative (or more positive for
36 CHAPTER 4. LANGMUIR PROBE DIAGNOSTICS
Figure 4.5: Schematic representation of the sheath with repelled electrons
electrons), resulting in an increase of AS and consequently in an increase of the probe collecting
area Ap. We can obtain an approximation to the sheath thickness by analysing it under the
as-sumption that electron density is negligible, i.e. for very negative probes. In fact, if the potential drop across the sheath is su ciently large, electrons are repelled over the majority of the sheath thickness (eVs Te). This means that the electron density goes essentially to zero relatively
close to the sheath edge and the electron space charge does not signi cantly aect the sheath thickness.
There are two dierent kinds of approximation for evaluating the change in sheath thickness: the Matrix sheath and the Child law sheath and we will now explain both [3].
Matrix sheath
The matrix sheath is the simplest sheath model and it considers a uniform ion density. So, letting ni = ns constant within the sheath thickness s and considering x = 0 at the sheath edge the
Gauss law gives
dE dx =
ens
0 (4.16)
which yields a linear variation of E with x:
E(x) = ens 0
x , (4.17)
integrating dV/dx = −E(x) to have the relation in terms of the potential we obtain a parabolic pro le
V (x) = −ens 0
x2
2 . (4.18)
Setting V = V0 at x = s we have the sheath thickness
s = (20 V0 ens
4.1. LANGMUIR PROBES 37 In terms of Debye length at the sheath edge λDs= (0Te/ens)1/2 we can see that
s = λDs(
2V0
Te
)1/2 . (4.20)
Hence the sheath thickness for the matrix sheath approximation can be tens of Debye length.
Child law sheath
Dierently from the matrix sheath model, it takes into account the decrease in ion density as the ions accelerate across the sheath. In the limit that the initial ion energy is small compared to the potential, the ion energy ux and ion current density conservation equations reduce to
1 2miv
2(x) = −eV , (4.21)
en(x)v(x) = J0 , (4.22)
where J0 is constant and so can be given explicitly as ensvB. Solving the two equations above
for n(x) we have n(x) = J0 e (− 2eV mi )−1/2 (4.23)
and using this in Poisson equation we obtain d2V dx2 = − J0 0 (−2eV mi )−1/2 . (4.24)
Multiplying this by dV/dx and integrating from 0 to x the relation, it becomes 1 2( dV dx) 2= 2J0 0 (2e mi )−1/2 (−V )1/2 (4.25)
where we have chosen dV/dx = 0 at x = 0. Integrating again in the same domain we nally have −V3/4=3 2( J0 0 )1/2(2e mi )−1/4 x . (4.26)
Letting V = V0at x = s and substituting the Bohm current density we obtain a sheath thickness
s = √ 2 3 λDs( 2V0 Te )3/4. (4.27)
Comparing this to the matrix sheath width, we see that Child law sheath is larger by a factor of the order (V0/Te)1/4and still proportional to the Debye length.
The dependence of ion saturation current on voltage is determined by the expansion of the sheath size.
An appropriate approximation for a spherical probe is
AS≈ Ap(1 + s/r)2 , (4.28)
while for a cylindrical probe it is
AS ≈ Ap(1 + s/r) , (4.29)
where s is given by the previous equation and r is the probe radius. Allen, Boyd and Reynolds (1957) proceeded to solve the Poisson equation (4.5) numerically to obtain the probe character-istics for a spherical probe with cold ions. Their results con rm that the sheath approximations are reasonably accurate [35].
38 CHAPTER 4. LANGMUIR PROBE DIAGNOSTICS
4.2 Triple Langmuir probes
A triple Langmuir probe provides an advantage over single and double probes, allowing simul-taneous measurement of plasma parameters without the need of many interpolation of data, increasing, respect to other methods, the experimental results accuracy. These characteristics make the triple probe useful for plasma measurement where the probe can not be maintained within the plasma for long time. The triple probe consists into three electrodes con gured to
Figure 4.6: Schematic representation of electrodes potential
measure electron temperature, plasma density and plasma local potential by the value of the collected current I and the potential V measured on electrodes surfaces. If all the electrodes have the same geometry and are exposed to plasma with the same collecting area then the probe can be considered as symmetric. In accordance with the gure (4.6) the electrode 1 is called Common. It is connected to ground through a voltmeter, so it measures VGC, the potential drop
between the ground reference and the common potential. The electrode number 3 is biased with a xed voltage VBC with respect to the common electrode. It is thus obviously called Bias. It
measures VGB = VGC− VBC. Having a voltage bias, a net current ows through these two
elec-trodes, which is measured with an ampere-meter. Finally the electrode number 2, being isolated from the rest of the circuit, drains no net current and its potential is free to oat, so it is called
4.2. TRIPLE LANGMUIR PROBES 39
Figure 4.7: Triple Langmuir probe
Float and its potential is VF. A voltmeter is set between oat and common. The entire circuit
is globally oating, so no net current is transferred between the probe and the plasma, i.e. the sum of the currents that ow in each electrodes is equal to zero IC+ IF + IB = 0. Because
IF = 0, we have that IC = −IB and so we can call them I and −I respectively. Moreover all
the electrodes are at a negative potential respect to plasma. Set of equations for the triple probe
For each electrode the current that ows is given by the sum of the electron and ion current
In= An[Je,se
(Vn−Vplasma)
Te − Ji,s] , (4.30)
where Je,s is the electron current density due to the thermal stirring out of the sheath, that
is equal to Je,s = en
q
kTe
2πme; instead, Ji,s is the ion current density that, in the thin sheath
approximation, is independent to plasma potential, so it is the same for each electrodes and it is in turn equal to Ji,s = en
q
kTe
mie
−1
2. After these considerations the set of equation for the triple
probe is IC= I = ACne r kTe mi [ r m i 2πme e(VGC−VGP)/Te− e−12], (4.31) IB = −I = ABne r kTe mi [ r m i 2πme e(VGC−VBC−VGP)/Te− e−12], (4.32) IF = 0 = r m i 2πme e(VGC−VF C−VGP)/Te− e−12. (4.33)
This set of equation is based on the planar sheath model without taking into account the sheath expansion. It is thus related to the Bohm theory.
Electron temperature
Consider now for simplicity that all the electrodes have the same collecting area A. Let us examine the ratio between the dierence of the rst and the third equation of the set, IC− IF,
and the dierence of the rst and the second equation, IC− IB:
IC− IF IC− IB = q m i 2πmee (VGC−VGP)/Te− e−12 −q mi 2πmee (VGC−VF C−VGP)/Te+ e−12 q m i 2πmee (VGC−VGP)/Te− e−12 − q m i 2πmee (VGC−VBC−VGP)/Te+ e−12 , (4.34)
40 CHAPTER 4. LANGMUIR PROBE DIAGNOSTICS that becomes 1 2 = 1 − eVCF/Te 1 − eVCB/Te . (4.35)
This gives an implicit form of the relation for the electron temperature; in fact, knowing the potential drop at the electrodes we can have a numeric resolution for Te.
Electron density
Knowing Tefrom the relation (4.35), we can use it to nd the electron density ne. From IF in
(4.33) we know that JiF = Je,se(VGC−VF C−VGP)/Te and, substituting it into the equation (4.31)
for IC, we have Je,s= I/A e(VGC−VGP)/Te− e(VGC−VF C−VGP)/Te = I/A e(VGC−VGP)/Te(1 − e−VF C/Te) . (4.36)
Finally, substituting Je,s, we can write the electron density as
ne= I/A q kTe mie (VGC−VGP)/Te(1 − e−VF C/Te) . (4.37)
4.3 Laframboise theory for cylindrical
Langmuir probe analysis
However, the simple Bohm theory model, based on the planar sheath assumption, does not provide a good description for the experimental data, in other words it is not accurate enough to represent them. So, in order to take into account the evolution of the sheath thickness with respect to the potential, we use a parametrization [15] of the results of the numerical simulations of Laframboise [18], which were based on a two species, steady state Vlasov-Poisson model.
4.3.1 Laframboise theory
Laframboise model
The equations that describe the model that Laframboise used in his work are Dfi Dt = ∂fi ∂~r · ~v + ∂fi ∂~p · ~Fi= 0 , (4.38) Dfe Dt = ∂fe ∂~r · ~v + ∂fe ∂~p · ~Fe= 0 , (4.39) ~ Fi = −Zie ∂Vs ∂~r , (4.40) ~ Fe= −Zee ∂Vs ∂~r , (4.41) 52V s= −ρ/ , (4.42) ρ = e(Zini+ Zene) , (4.43)
4.3. LAFRAMBOISE THEORY FOR CYLINDRICAL LANGMUIR PROBE ANALYSIS 41 ne(~r) = Z fe(~r, ~v)d3v , (4.44) ni(~r) = Z fi(~r, ~v)d3v , (4.45)
where ~r is the position vector in physical space and ~p is the momentum vector. fiand feare the
distribution functions in position-momentum space for ions and electrons respectively; ~v is the velocity vector and t is the time. ~Fi and ~Fe are the forces exerted by the electric eld on ions
and electrons. Vs is the plasma potential; ρ is the net density of electric charge and ne and ni
are the number densities for electrons and ions. Numerical resolution
In his studies on Langmuir probes [18], Laframboise formulated a numerical method for com-puting the functional relationship between the probe potential and the current drained by the probe (both normalized) in ion and electron saturation regions for a Maxwellian plasma. The normalized potential is de ned as
χ = (Vp− Vs) Te
, (4.46)
where Vp and Vs are the probe and the plasma potential respectively . The normalized probe
current is thus described as a function f (χ). Considering a Maxwellian plasma, Laframboise computed f (χ) for an electrostatic cylindrical probe using the following assumptions:
• The radius of the probe is much smaller than the probe length;
• The plasma consists of electrons and single ionized ions and both species, far from the probe, should have a Maxwellian velocity distribution;
• Far from the probe the net charge density vanishes;
• Within the sheath we can neglect collisions in comparison to collective phenomena; • Finite collection of both species is allowed to occur to obtain the entire probe characteristic; • A steady state exists.
Moreover he did not consider the presence of any magnetic eld.
For each species f (χ) was calculated for −25 < χ < 25, with f (χ) normalized to the saturation current at χ = 0. Additionally, he distinguished the cases Ti/Te = 1 and Ti/Te = 0for r/λD
varying from 0 to 100, with r probe radius.
Because of the negative values of χ in the ion saturation region, the corresponding function will be fi(−χ) for ions, while for electrons it will be simply fe(χ), because the electron saturation
zone has positive values of χ.
Therefore, the probe characteristic for a Maxwellian plasma can be described as
I = Ii0fi(−χ) + Ie0eχ f or χ < 0 , (4.47) I = Ie0fe(χ) f or χ ≥ 0 , (4.48) where Ie0= eneA( kTe 2πme )1/2 (4.49)
42 CHAPTER 4. LANGMUIR PROBE DIAGNOSTICS and Ii0= eniAvi= ( eniA(2πmkTei)1/2 f or cold ions, Ti/Te= 0 , eniA(2πmkTi i) 1/2 f or hot ions, T i/Te> 0 , (4.50) in which A is the probe area and vi is the ion velocity, that can be substituted by the thermal
ion velocity for hot ions, while it is proportional to the Bohm velocity for cold ones (∝ pTe/mi).
4.4 Mausbach parametrization of Laframboise results
Figure 4.8: Fitted parameters for electrons and ions as function of r/λDand Ti/Te [15]
To obtain plasma parameters from numerical results Mausbach proposed a generalized root function of the form
f (χ) = a(b + χ)c . (4.51)
ais a scaling factor, b is the zero point and c is the power of the root function.
Figure (4.8) gives the tted values of a, b and c as function of r/λD and Ti/Te as well as the
relative errors.
In gure (4.10) are displayed the dierent t functions with the corresponding numerical results of Laframboise. As can be easily seen, the t is very good, with dierences usually < 3%. Fit parameters a, b and c are also shown in gure (4.9) over r/λDand for the two dierent values
4.4. MAUSBACH PARAMETRIZATION OF LAFRAMBOISE RESULTS 43
Figure 4.9: a, b and c as function of r/λD[15]
Figure 4.10: Comparison of the numerical results of Laframboise with the tted function f(χ) [15]
