J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)429–432
000–000

Reduction of Sheath Potential and Dust Ion-Acoustic Waveby Negatively Charged Dust Particles
TOMITA Yukihiro, SMIRNOV Roman1, CHUTOV Yuriy2,
TAKAYAMA Arimichi and TAKIZUKA Tomonori3
National Institute for Fusion Science, Toki, 509-5292, JapanThe Graduate University for Advanced Studies, Toki, 509-5292, Japan
2
Taras Shevchenko Kiev University, Volodymyrska 64, Kiev, 01033 Ukraine
3Naka Fusion Research Establishment, Japan Atomic Energy Research Institute, Ibaraki, 311-0193, Japan1(Received: 9 December 2003 / Accepted: 3 April 2004)

Abstract
Formation of an electrostatic potential due to negatively charged dust particles near a target plate was studied by
one-dimensional kinetic analysis. Immovable dust particles are distributed uniformly inside a system. A thresholddust density makes the electric field at the target plate zero. In case of dust density below the threshold a chargeneutral point separates into two between positions of a plasma injection and a wall. An electron sheath near theinjection point changes to an ion sheath in front of the target plate through the charge-neutral point. It is clarified thatthe negatively charged dust particles reduce the ion-sheath potential drop from the neutral point. The stationary
oscillation appears in the electrostatic potential for the dust density above the threshold. We confirmed that thisoscillation is the Dust Ion-Acoustic Wave (DIAW) with a wavelength of the order of Debye length, where dust charges
are effective to determine the behavior. The effective dust density to produce the oscillation is as high as 1013-14 m–3
for plasma density ~ 1018 m–3 with plasma temperature ~ 10 eV.
Keywords:
dust particle, electrostatic potential, target plate, dust charging, sheath, dust ion-acoustic wave

1. Introduction

2. ModelIn recent years there has been much interest in charged
dust particles in fusion plasmas as well as the space andlaboratory plasmas. Dust particles were observed after adisruption of plasma current in a tokamak device [1]. In
several fusion devices (TEXTOR-94, ASDEX-U, LHD), thedust particles were collected and analyzed their characteristics[2,3], where radii of dust particles widely range from 10 nm
to 100 µm. These dust particles were composed mainly ofcarbon and constituents of stainless steel, which materials areused for most plasma-facing surfaces like the divertor plates
and the vacuum vessels. With prolonged discharges in fusionplasma experiments, the effect of dust particles on thepotential formation near target plates and the impurity for abulk plasma might be important. Despite the small amount of
dust particles, the charge of a dust particle is large enough tostrongly affect the potential formation and a radiation loss
from the bulk plasma. Understandings of the basic phenomena
between dust and plasma particles are quite important. In thisstudy we investigate the formation of an electrostatic potential
with dust particles near the target plate by using onedimensional kinetic analysis.

In the model for one-dimensional analysis, plasma
particles with half-Maxwellian distribution functions are
injected from an injection position (x = 0) toward a perfectly
absorbing target plate (x = xw) with a floating potential, whereimmovable dust particles are distributed uniformly. Theabsorption of plasma particles by dust particles and Coulombinteractions between particles are neglected because the meanfree-paths are much longer than the system length, which isassumed to be of the order of the Debye length. The
negatively charged dust particles affect the potential formation
through Poisson equation. In a system without particle sources
and sinks, the local energy distribution function of the plasma
particles fj (x, εjx ) [ j = i, e], is the same as at the injection
point fj (x = 0, εjx ). Here εjx is the total energy of a particle(εjx = mj v2 + qj φ (x), mj and qj are the mass and electric chargeof j-th particles.) in the local electrostatic potential φ (x). The
local macroscopic quantities, such as plasma densities andflow velocities, are obtained as a function of a potential byusing the local energy distribution function [4]. The potential
at the floating wall φ w is determined by the equality of particle©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchCorresponding author’s e-mail: tomita@nifs.ac.jp1429

Tomita Y. et al., Reduction of Sheath Potential and Dust Ion-Acoustic Wave by Negatively Charged Dust Particles

fluxes of ions and electrons at the wall:

The spatial potential distribution is calculated fromPoisson equation by using the plasma charge densities andthe dust charge density according to the OML theory [5]. The
spatial potential profiles for several values of the normalized
dust density n*d R*d (= nd Rd λ2Des) are shown in Fig. 1, where thefloating target is located at the position xw = 50 λDes , thetemperature ratio at the injection point τs is 10, the density
ratio δs is set as 0.501, and a hydrogen plasma is considered.
Near the injection point an electron sheath is formed and themiddle region of this system a wide quasi-neutral region isformed. In front of the floating target plate an ion-sheath isformed. Note that the wall potential is not disturbed by theimmovable dust particles. As the dust density increases, thepotential difference between the wall and the position of
charge neutrality decreases. As the ion flow velocity Vi at thecharge-neutral point exceeds the sound speed cs (Vi /cs ~ 2.0)
corresponding the Bohm criterion, the potential between thecharge-neutral point and the plate monotonically decreases.
This potential difference corresponds to the potential drop of
the Debye-sheath. As the dust density increases further, theelectric field at the wall approaches zero. At the thresholddust density the electric field at the wall becomes zero,
whereas the wall potential φ w still remains the same value. In
the case mi /me = 1836.1, δs = 0.501, and τs = 10, the thresholdof the normalized dust density is as high as 0.022, whichcorresponds to the dust density as high as 1013-14 m–3 forplasma density and temperature of 1018 m–3 and 10 eV,respectively.
At dust densities above the threshold, a stationary
oscillation appears in the potential (Fig. 2), where the plasma
parameters are the same as in Fig. 1. The higher dust density,
the more the minimal potential, which is the same as the wall
potential φ w , is close to the injection point and the deeper
potential well becomes. At the further point from the potential
minimum φ w , both of the particle fluxes of ions and electrons
are the same, which means the floating plate can be locatedanywhere far from the minimal potential. The wavelength of
this steady oscillation is presented as a function of the dust
density by the solid line in Fig. 3, where the wavelength isthe order of Debye length. At the higher dust density than thethreshold, ions are accelerated enough to be a monoenergetic
distribution, where the ion flow velocity is determined fromthe energy conservation in the electrostatic potential φ. As
electrons are almost repelled by the highly negative dustcharge, their contribution to the potential formation isneglected. The currents of plasma particles to dust particles

Fig. 1Fig. 2

φw = −⎛ 2 m ⎞Tesln ⎜⎜δ sτ s m i ⎟⎟ .
⎝e⎠
2e(1)

Here e is the unsigned charge on an electron, δs and τs denotethe density ratio and temperature ratio at the injection point:(+) (+)δs ≡ nes/nis and τs ≡ Tes /Tis respectively, where njs(+) and Tjs isthe density of the particles with positive velocities and plasma
temperature at the injection point. Note that a dust particle
does not contribute to the wall potential because they areimmovable.
In this analysis the density ratio δs is determined so as
to satisfy the charge-neutrality condition at the injection point(x = 0) without the dust particles. The steady state dust charge
Zd is determined from the condition of zero net currents ofplasma particles to dust particles. These currents are obtained
from the local distribution function of the plasma particles
and the absorption cross-section, which is adopted from theOML (Orbital Motion Limited) theory [5]. The dust chargeZd (> 0) is normalized as βd ≡ Zd e2 /ε0 Rd Tes , where ε0 is thevacuum permittivity and Rd denotes the dust radius. This
normalization gives the normalized charge density of dust(+)
= n*d R*d βd , where nd is the dust density,
particles as Zd e nd /e nes
3n*d ≡ nd λDes and R*d ≡ Rd /λDes . Here λDes ≡ ε 0 Tes / e 2 n (+)
ises
the electron Debye length associated with the positive-velocity
electron density at the injection point.

3. Results and discussions

Spatial distributions of the normalized electrostatic
potential for several values of dust density below thethreshold, where the temperature, the density ratio at
the injection point, and the mass ratio are τs = 10, δs =0.501, and mi /me = 1836.1. The normalized thresholddust density is 0.022 in these plasma parameters.

2
430

Spatial distributions of the normalized electrostatic
potential for the dust density above the threshold,which is 0.022 in the plasma parameters, which are thesame as in Fig. 1. The minimal potential is equal to thewall potential φw in case of dust density below thethreshold.

Tomita Y. et al., Reduction of Sheath Potential and Dust Ion-Acoustic Wave by Negatively Charged Dust Particles()

c lw = 3 2π βdw / α lw n∗d R∗d th − n ew ,α lw ≡(5)

2π 3/ 2 ( Z i eφw / Tes − 1/ 2 τ s − β dw / 4π )1 − Z i eφw / Tes + 1 / 2τ s + β dw / 4π

[ β /4π + π(1/2 −β /4π)exp(β /4π)erfc( β /4π)], (6)dw

Fig. 3dwdw

dw

where erfc is the complementary error function. Thewavelength near threshold is presented by the dotted line inFig. 3.
In this analysis the absorption of plasma particles by adust particle is not taken into account. In case of the larger(+)= 1018 m–3, thedust radius than 1 µm in Tes = 10 eV and nes
absorption length to the wavelength is not ignorable where
the dumping of the oscillation may be appreciable.

The wavelength as a function of normalized dust
density: solid line. The approximated wavelengths, eqs.
(3) and (4), are presented by the dashed and the dottedlines, respectively. The parameters of the plasma are
the same as in Fig. 1.

4. Summary
are almost the same in the highly negative potential. This
result gives us the dust charge is nearly proportional to thesquare of the ion flow velocity according to the charging
cross-section. From these conditions, the wave number k isobtained by Poisson equation with respect to fluctuations of~potential φ :⎞⎛ Z i n iw
ε 0 k2φ ≅+ Z dw n d ⎟⎟ φ ,Z i n i – Zd nd = ⎜⎜e−φ⎝ − 2 φww ⎠The effects of dust particles have been investigated on
the potential formation in the near-target region. The local
charge of the dust particles is calculated according to theabsorption cross-section from OML theory [5]. We found thatthe uniformly distributed immovable dust particles in front of
the target plate reduce the potential drop in the sheatheffectively, so that the bombardment of the plate by ions withhigh energies is avoided. Above the threshold dust density,where the electric field at the plate becomes zero, thestationary oscillation in a potential distribution appears. Thisoscillation is confirmed to be a dust ion-acoustic wave, where
the dust charge determines the wavelength and electrons do
not contribute to the oscillation except for the vicinity of thethreshold. This comes from that electrons are repelled by thehighly negative dust charge. The effective dust density(~ 10 13-14 m–3) is so high compared to the observed andexpected dust density in fusion devices, which means thecollective effect of dust particles in fusion plasma may beneglected. In this simple analytical study there are several
issues left for the understanding the dust-particle effect in arealistic divertor plasma. The presheath such as the geometric,
collisional, ionizing, or magnetic presheath is important todetermine the spatial distribution of dust particles. The
magnetic field obliquely penetrating the divertor plate affects
the potential formation in the existence of dust particles. The
Coulomb coupling might be important to investigate dynamics
of dust particles in this high dust density. From the results
the absorption effect of plasma particles by the dust particles
with large radii (> 1 µm) may not be ignored. For a dust
particle with a radius smaller (< 100 nm) the OML theory is
not applicable, because of a short Debye shielding length of
a dust particle. These effects are needed to incorporate to themodel.

(2)

where the ion atomic number is denoted by Z i. The
corresponding wavelength is expressed by the normalized
quantities at the wall/minimal potential φ w:

λ wave / λ Des =2 π − eφ w / Tes,
Z i n iw / 2 + β dw n∗d R∗d(3)(+)
where the particle density is normalized by nes, which is theelectron density with positive velocities at the injection point.
The dashed line in Fig. 3 shows the obtained wavelength,
where the difference from the real value comes from theassumption of the proportionality of Z d to V i2 . The
conventional DIAW is assumed the fluctuation of the charging
effect of dust particles is neglected and the fluctuation ofelectrons determined the dispersion relation [6]. On the other
hand in our case the fluctuation of the electron density isneglected and the effect of the dust charging is dominant todetermine the wavelength.
In the vicinity of the threshold density the contributionof electrons to a wavy potential cannot be neglected. After
troublesome algebra one can obtain the wavelength withrespect to the lowest order of the difference of the dust densityfrom the threshold, n*d R*d – n*d R*d⏐th ,:λ wave / λ Des ≅ c lw n∗d R∗d − n∗d R∗d th ,(4)
Referenceswhere the coefficient c lw is expressed by the physicalquantities at the wall/minimal potential:[1] K. Narihara et al., Nucl. Fusion 37, 1177 (1997).
[2] J. Winter, Plasma Phys. Control. Fusion 40, 1201 (1998).
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Tomita Y. et al., Reduction of Sheath Potential and Dust Ion-Acoustic Wave by Negatively Charged Dust Particles[6] P.K. Shukla and V.P. Silin, Physica Scripta 45, 508(1992).

[3] J. Sharpe et al., J. Nucl. Mater. 313-316, 455 (2003).
[4] Y. Tomita et al., Contr. Plasma Phys. 44, 138 (2004).
[5] J.E. Allen, Physica Scripta 45, 497 (1992).

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