24E11
Energy distribution of charged particles in a magnetically expanding plasma
and their application to electric propulsion
発散磁場配位下プラズマ中の粒子エネルギー分布と電気推進への応用
Kazunori Takahashi1,2,3, Christine Charles1, Rod Boswell1 and Tamiya Fujiwara2,3
高橋和貴1,2,3, Christine Charles1，Rod Boswell1，藤原民也2,31Space Plasma, Power and Propulsion Group, Research School of Physics and Engineering,
The Australian National University
Canberra, ACT 0200, Australia
2

Department of Electrical Engineering and Computer Science, Iwate University,
4-3-5 Ueda, Morioka, Iwate 020-8551, Japan
岩手大学工学部 電気電子・情報システム工学科 〒020-8551 岩手県盛岡市上田4-3-53Soft-Path Engineering Research Center (SPERC), Faculty of Engineering, Iwate University,
4-3-5 Ueda, Morioka, Iwate 020-8551, Japan
岩手大学ソフトパス工学総合研究センター 〒020-8551 岩手県盛岡市上田4-3-5

Experimental and theoretical investigations of electron energy distribution functions (EEDF) in amagnetically expanding plasma system are reported. The experimental EEDF has a depleted tail above theenergy corresponding to the double layer (DL) potential drop, instead of the electron beam or hot electron
tail required for DL formation in the previous theoretical models. Here, an analytical particle-balance
model using the Druyvesteyn distribution is successfully compared to the experimental results. Both theupstream particle balance and the DL formation are sustained by a Druyvesteyn-like EEDF, and thedepleted tail of electrons overcome the DL and neutralize the ion beam. Direct measurements of the thrustfrom the permanent magnets helicon double layer plasma is also presented.

1. Introduction
It is well known that electric fields of electric
double layers (DLs) spontaneously formed inplasmas accelerates or trap charged particles.
Recently, the investigations of current-free doublelayers (CFDLs) in magnetically expanding rf
helicon-wave and/or inductive-mode plasmas areprogressed in a number of laboratory experiments,
via analytical models, and computer simulations [1].
The electrical and optical diagnoses of the ion
energy distributions in these experiments haveshown that the ions produced inside the source tube
are accelerated by the CFDLs and the supersonicion beams are generated in the downstream
diffusion chamber. Some theories of the CFDLs
include the electron beam accelerated by theCFDLs from the downstream to upstream plasmas,
or take into account of a bi-Maxwellian electronenergy distribution with a hot tail for finding CFDL
solutions [2,3]. However, only a few experiments
have shown the experimental electron energy
distributions, and its feature is the depleted tail atthe break energy corresponding to the potential
drop of the DL [4]. Hence, it is important to model
the electron energy distribution of the CFDL

plasmas.
Here the electron energy distribution upstream of
the helicon CFDL is analytically modeled: theEEPFs with the depleted tail are approximately
fitted by a Druyvesteyn distribution and theeffective electron temperatures are analyticallyobtained. The theoretical results are successfullycompared with the experimental results obtained fortwo system geometries and two operating gases.
Furthermore, the direct measurement of the thrustfrom the permanent magnets helicon CFDL plasma
is also reported.
2. Particle balance for Druyvesteyn distribution
The balance equation between volume ionizationand surface loss of the charged particles is a simple
and powerful method to determine the electrontemperature. It is written as()( )where Kiz, ng, R, l, uB are the ionization rate
constant, the gas density, the plasma radius, theplasma length, and the bohm velocity, respectively.
hR and hl are the axial and radial center-to-edge
density ratios and described in Ref.[5]. The

generalized electron energy probability function(EEPF) gp can be expressed as( )

[() ]

( )

then the electron energy distribution function(EEDF) ge can be given by ge = 1/2gp, and Teff is theeffective electron temperature in units of volts. x is
the parameter giving the shape of the EEPF/EEDF,
and x = 1 and x = 2 correspond to the Maxwellian
and Druyvesteyn distribution functions. The Bohm
velocity and the ionization rate constant for thegeneralized EEPF can be obtained as reported inRef. [6]. Equation (1) and the generalized Bohm
velocity and ionization rate constant can give theeffective electron temperature.
The experimental EEPFs observed in two
different machines (Chi-Kung at the Australian
National University and PMPI at Iwate University)
have shown similar distributions with the depletedtails. The shapes of the EEPFs are very close to thecalculated EEPF for x = 2, i.e., the Druyvesteyn
distribution function. Figure 1 shows theexperimental and theoretical effective electron
temperatures in two types of gases and in twodifferent machines. It is found that the theoreticalresults are successfully compared with theexperimental ones. Although the previous modelsof the CFDLs have assumed the electron beam
accelerated by the DL or the bi-Maxwellian
EEPF/EEDF with the hot tail, the present results doshow that the particle balance in the source tube canbe sustained by the Druyvesteyn-like EEPF with thedepleted tail. This CFDL solution for theDruyvesteyn-approximated distribution is not inaccordance with the result of the previous models.
Hence, this seems to indicate that the CFDL in themagnetically expanding plasmas is a new class ofCFDLs.
3. Thrust measurement
The results of the EEPFs have shown that thesupersonic ion beams accelerated by the CFDLs areneutralized by the electrons overcoming the CFDLs.
Hence, this type of plasma system can be useful forthe development of electrodeless and neutralizerfree electric propulsion systems, because equal
fluxes of ions and electrons are emitted by thesystem. Since the physical mechanisms of thrustgeneration by electrodeless plasma thrusters are notclarified and verified experimentally yet. Here we
report the direct measurement of the thrust from thepermanent magnets helicon double layer thrust(PM-HDLT). The pendulum thrust balance is used
to measure the thrust and the detailed plasma
parameters are measured by a Langmuir probe anda retarding field energy analyzer. It is found that thethrust from the PM-HDLT is about 3 mN for 700 W
effective rf power. This value agrees well with thecross-sectional electron pressure estimated from theupstream probe diagnosis. Moreover, the supersonicion beam is detected in the diffusion chamber. In
this sense, the plasma momentum is conserved andthe momentum is transferred from the upstream
axial electron pressure into the downstream axial
ion momentum through the CFDL [7].
References

Fig.1. Theoretical and experimental effective electron
temperature Teff in argon in PMPI (crosses with solidline and filled squares), in argon in Chi-Kung (opentriangles with solid line and open squares), and in xenon
in Chi-Kung (open diamonds with slid line and opencircles) as a function of the operating gas pressure. The
dashed line is the analytical results for Maxwellian
distribution in argon in Chi-Kung for comparison.

[1] C. Charles: Plasma Sources Sci. Technol. 16 (2007)
R1, and references therein.
[2] M. A. Lieberman and C. Charles: Phys. Rev. Lett. 97(2006) 045003.
[3] E. Ahedo and M. M. Sanchez: Phys. Rev. Lett. 103(2009) 135002.
[4] K.Takahashi et al.: Phys. Plasmas 14 (2007) 114503.
[5] M. A. Lieberman and A. J. Lichtenberg: Principlesof Plasma Discharges and Materials Processing(Wiley, Hoboken, NJ, 2005) 2nd ed.
[6] K. Takahashi et al.: Phys. Rev. Lett. 107 (2011)
035002, and references therein.
[7] K. Takahashi et al.: Appl. Phys. Lett. 98 (2011)
141503.