line) centres at the local plasma potentialφand the high energy Gaussian function (solid line) at the beam potential φb. Their respective integrals yield the local ion current Iloc
which represents the background ion group, and the nonlocal ion currentInlocrepresenting
the beam ions. The integral of the overall IEDF gives the total ion current Itot which is
equal to the sum of Iloc and Inloc. Additionally, in order to characterize the shape of an
ion beam, the collector current is normally sampled at a chosen high discriminator voltage (about the beam potential) for different spatial positions [86], defined as the delimited current Idel representing the ion group with a mechanical energy higher than the chosen
energy level. When there is no ion beam or the RFEA-orifice orientation is perpendicular to an ion beam, the IEDF presents a single peak at the plasma potential and can be solely fitted with the low energy Gaussian function.
2.4
Chapter Summary
This chapter shows the experimental setup of Chi-Kung reactor. The cylindrical plasma source can be modified into an annular configuration by inserting a glass tube, which allows the positioning of an inner antenna. The diffusion chamber provides multiple interfaces for vacuum-maintenance devices and diagnostic probes. The structure and data interpretation of four electrostatic probes are presented.
Chapter 3
Polytropic Revisit of Nonlocal
Electron Transport
This chapter focuses on the polytropic behavior of electrons in low pressure plasma expan- sion where the electrons are governed by nonlocal electron energy probability functions (EEPFs) [33, 35], represented in the form offpe(εe,r) =fpe[εe−eφ(r)] whereεe,φandr
are the electron kinetic energy, plasma potential and spatial position vector, respectively. It should be noted that the polytrope has also been applied to ions in the solar wind studies [87, 88]; for many laboratory systems, the ions can be approximated as a cold species without thermodynamic behavior [8, 22]. Since the mean velocity of a plasma flow is normally small compared to the electron velocity, the comoving frame attached to the flow can be considered as a stationary frame for electrons on first-order approximation and this setting is used by default unless otherwise specified.
Section 3.1 derives the enthalpy relation for an adiabatic system governed by nonlocal EEPFs. Section 3.2 studies the polytropic relation for electrons using previous EEPF data in a laboratory helicon double layer thruster (HDLT) measured by Takahashi [51], and shows that the use of traditional thermodynamic concepts based on collision-dominated local thermodynamic equilibrium (LTE) can lead to very erroneous conclusions regarding the thermal conductivity for non-LTE plasmas governed by nonlocal particle dynamics. Section 3.3 focuses on a new theoretical perspective of how nonlocal EEPFs determine the polytropic index of electrons through three different bi-Maxwellian distributions, and hypothesizes a new scenario of electron transport in the solar wind by considering the interrelation between the solar wind and laboratory plasmas. Additionally, the energy conversion mechanism behind ion acceleration is briefly discussed in sections 3.2 and 3.3.
3.1
Enthalpy Relation for Nonlocal Electrons
When electrons move nonlocally along decreasing potentials, the total mechanical energy of electrons is conserved with the electrons bound back and forth within the potential structure [8], hence their transport is a self-consistent adiabatic process. Electron enthalpy he is defined using formula:
dhe= dqe+ dpe ne = dpe ne (3.1) 21
z [ cm ] -30 -20 -10 0 10 20 30 r [c m ] -20 -10 0 10 20 z [ cm ] -30 -20 -10 0 10 20 30 B [G ] 0 40 80 120 160 ( a ) ( b ) CP -28.6 -9 top pump gas exit RF solenoid solenoid
Figure 3.1: (a) Helicon double layer thruster experiment, showing major components, the RF compensated Langmuir probe (CP) and magnetic field lines. (b) Magnetic flux density B on the central axis.
where the heat term dqe is omitted due to adiabaticity. The electron pressurepe and elec-
tron densityneare obtained from the EEPFs usingpe= 2/3·
R∞ 0 ε 3/2 e fpe(εe−eφ) dεeand ne = R∞ 0 ε 1/2
e fpe(εe−eφ) dεe, respectively. Integrating formula (3.1) along the potential
path yields: he= Z −eφ −eφ0 2 3 R∞ 0 ε 3 2 e f 0 pe(εe−eφx) dεe R∞ 0 ε 1 2 e fpe(εe−eφx) dεe
d (−eφx) +he0=eφ−eφ0+he0 (3.2)
for which limεe→∞ ε3e/2fpe(εe−eφx) = 0 has been used when applying integration by
parts to the numerator. φ0 and he0 are the plasma potential and electron enthalpy at a
reference position, respectively. The mathematical deduction of equation (3.2) is detailed in Appendix A.
Rearranging equation (3.2) yields a conservation relation:
∆he+ ∆ (−eφ) = 0 (3.3)
which shows that the electrons transfer their enthalpy into the potential energy in an adiabatic process. It should be noted that this conservation relation is a typical form of the Bernoulli integral [49, 87] where the macro (convective) kinetic energy of the plasma flow is omitted due to the approximation of a stationary comoving frame as stated earlier. Equation (3.3) is a generalized result independent of the specific form of nonlocal EEPFs, and its differentiation with respect to the plasma transport path z yields:
dpe
dz +neEz = 0 (3.4)
where Ez is the electric field along z direction. Equation (3.4) has a consistent form
of momentum balance with a LTE system, but the pressure term here is an effective parameter determined by nonlocal motion of electrons rather than local collisions. It should be noted that, for astrophysical electrons covering large distances, the electron enthalpy will be partially consumed to overcome the gravitational barrier.
3.2 POLYTROPIC RELATION IN HELICON DOUBLE LAYER THRUSTER 23