Effect of dust particles on electron energydistribution in glow and afterglow plasmas
Analytical expressions describing electron energy probability functions (EEPFs) in glow and afterglow dusty plasmas are obtained from the homogeneous Boltzmann equation for electrons. At large energies in a glow dusty plasma, the quasiclassical approach for calculation of the EEPF is applied. Cons...
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nasplib_isofts_kiev_ua-123456789-1154452025-02-09T17:01:25Z Effect of dust particles on electron energydistribution in glow and afterglow plasmas Влияние пылевых частиц на распределение электронов по энергии в плазме в режимах свечения и послесвечения Вплив пилових частинок на розподіл електронів за енергією в плазмі в режимах світіння та післясвітіння Denysenko, I.B. Azarenkov, N.A. Ivko, S. Burmaka, G. Glazkov, A. Iter and fusion reactor aspects Analytical expressions describing electron energy probability functions (EEPFs) in glow and afterglow dusty plasmas are obtained from the homogeneous Boltzmann equation for electrons. At large energies in a glow dusty plasma, the quasiclassical approach for calculation of the EEPF is applied. Considering the afterglow case, the analytical expressions are obtained assuming that the electron energy loss is mainly due to momentum-transfer electronneutral collisions and due to deposition of electrons on dust particles. Effect of dust particles on the EEPF is analyzed. Из однородного уравнения Больцмана получены аналитические выражения для функции распределения электронов по энергии (ФРЭЭ) в пылевой плазме в режимах свечения и послесвечения. Для расчёта ФРЭЭ в режиме свечения при больших электронных энергиях применён квазиклассический подход. Для плазмы в режиме послесвечения аналитические выражения получены в предположении, что потери энергии электронов происходят в основном за счёт упругих электрон-нейтральных столкновений и благодаря осаждению электронов на пылевые частицы. Проанализировано влияние пылевых частиц на ФРЭЭ. З однорідного рівняння Больцмана отримано аналітичні вирази для функції розподілу електронів за енергією (ФРЕЕ) в запорошеної плазмi в режимах світіння та післясвітіння. Для розрахунку ФРЕЕ в режимі світіння при великих електронних енергіях застосовано квазікласичний підхід. Для плазми в режимі післясвітіння аналітичні вирази отримані в припущенні, що втрати енергії електронів відбуваються в основному за рахунок пружних електрон-нейтральних зіткнень та завдяки осадженню електронів на пилові частинки. Проаналізовано вплив пилових частинок на ФРЕЕ. This work was supported by the State Fund for Fundamental Research of Ukraine. 2016 Article Effect of dust particles on electron energydistribution in glow and afterglow plasmas / I.B. Denysenko, N.A. Azarenkov, S. Ivko, G. Burmaka, A. Glazkov // Вопросы атомной науки и техники. — 2016. — № 6. — С. 179-182. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 52.25.Vy, 52.27.Lw, 51.50.+v, 52.80.Pi https://nasplib.isofts.kiev.ua/handle/123456789/115445 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Iter and fusion reactor aspects Iter and fusion reactor aspects |
| spellingShingle |
Iter and fusion reactor aspects Iter and fusion reactor aspects Denysenko, I.B. Azarenkov, N.A. Ivko, S. Burmaka, G. Glazkov, A. Effect of dust particles on electron energydistribution in glow and afterglow plasmas Вопросы атомной науки и техники |
| description |
Analytical expressions describing electron energy probability functions (EEPFs) in glow and afterglow dusty
plasmas are obtained from the homogeneous Boltzmann equation for electrons. At large energies in a glow dusty
plasma, the quasiclassical approach for calculation of the EEPF is applied. Considering the afterglow case, the analytical
expressions are obtained assuming that the electron energy loss is mainly due to momentum-transfer electronneutral
collisions and due to deposition of electrons on dust particles. Effect of dust particles on the EEPF is analyzed. |
| format |
Article |
| author |
Denysenko, I.B. Azarenkov, N.A. Ivko, S. Burmaka, G. Glazkov, A. |
| author_facet |
Denysenko, I.B. Azarenkov, N.A. Ivko, S. Burmaka, G. Glazkov, A. |
| author_sort |
Denysenko, I.B. |
| title |
Effect of dust particles on electron energydistribution in glow and afterglow plasmas |
| title_short |
Effect of dust particles on electron energydistribution in glow and afterglow plasmas |
| title_full |
Effect of dust particles on electron energydistribution in glow and afterglow plasmas |
| title_fullStr |
Effect of dust particles on electron energydistribution in glow and afterglow plasmas |
| title_full_unstemmed |
Effect of dust particles on electron energydistribution in glow and afterglow plasmas |
| title_sort |
effect of dust particles on electron energydistribution in glow and afterglow plasmas |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2016 |
| topic_facet |
Iter and fusion reactor aspects |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/115445 |
| citation_txt |
Effect of dust particles on electron energydistribution in glow and afterglow plasmas / I.B. Denysenko, N.A. Azarenkov, S. Ivko, G. Burmaka, A. Glazkov // Вопросы атомной науки и техники. — 2016. — № 6. — С. 179-182. — Бібліогр.: 9 назв. — англ. |
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Вопросы атомной науки и техники |
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ISSN 1562-6016. ВАНТ. 2016. №6(106)
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2016, № 6. Series: Plasma Physics (22), p. 179-182. 179
EFFECT OF DUST PARTICLES ON ELECTRON
ENERGYDISTRIBUTION IN GLOW AND AFTERGLOW PLASMAS
I.B. Denysenko, N.A. Azarenkov,
S. Ivko, G. Burmaka, A. Glazkov
School of Physics and Technology, V.N. Karazin Kharkiv National University,
Kharkov, Ukraine
E-mail: idenysenko@yahoo.com
Analytical expressions describing electron energy probability functions (EEPFs) in glow and afterglow dusty
plasmas are obtained from the homogeneous Boltzmann equation for electrons. At large energies in a glow dusty
plasma, the quasiclassical approach for calculation of the EEPF is applied. Considering the afterglow case, the ana-
lytical expressions are obtained assuming that the electron energy loss is mainly due to momentum-transfer electron-
neutral collisions and due to deposition of electrons on dust particles. Effect of dust particles on the EEPF is ana-
lyzed.
PACS: 52.25.Vy, 52.27.Lw, 51.50.+v, 52.80.Pi
INTRODUCTION
Dusty plasmas have been extensively studied in the
last three decades because these complex ionized gas
systems are of great interest in different fields [1-5]. At
theoretical description of dusty plasmas, one usually
assumes that electrons are in Maxwellian equilibrium
[1, 2]. However, for most of industrial and laboratory
plasmas, the electron energy probability function
(EEPF) often deviates from Maxwellian because of the
many different electron collision processes [3]. The
profile of electron energy probability function affects
different plasma parameters. Therefore, determination
of the EEPF profile is very important in studying of
different plasmas. For calculation of the EEPF in dusty
plasmas, usually different numerical approaches are
used. Here, we present analytical expressions describing
the EEPF in glow and afterglow dusty plasmas.
1. MAIN EQUATIONS AND ASSUMPTIONS
We consider an argon dusty plasma maintained by
an electric field )(tE . Plasma consists of electrons, ions
with number densities ne and ni, respectively, and of
dust particles of submicron size with density dn and
radius da . The plasma is assumed to be quasineutral,
i.e., idde nZnn , where dZ is the dust charge (in
units of electron charge e). It is also supposed that the
ions have Maxwellian distribution with temperature iT
(= 0.026eV), but the electron energy probability func-
tion F in general is not Maxwellian and satisfies to the
homogeneous Boltzmann equation [6]:
FS
u
tuF
E
v
u
uum
e
t
tuF
me
),(
3
2),( 2
2/3
, (1)
where u is the electron energy (in eV), t is the time, me
is the electron mass,
22
22
p2
)(
)(
2
uv
uvE
E
m
m for the RF
case, and pEE is the external electric field in the DC
case. Here, pE is the RF field amplitude, Ef 2 ,
and Ef is the RF frequency. em
e
edm , where
e
ed and em are the frequencies for momentum-transfer
electron-dust and electron-atom collisions. FS =Sea(F)
+Sed(F), where Sea(F) and Sed(F) are the terms describ-
ing electron-atom and electron-dust collisions, respec-
tively [3, 4]. We consider low-ionized dusty plasma,
therefore, electron-electron collisions are not accounted
for in the model. The homogeneous Boltzmann equation
(1) may be used if the energy relaxation length is small
compared to the spatial inhomogeneity scale of the dis-
charge. For the case of a glow dusty plasma and for the
afterglow with low dust density, the EEPF can be pre-
sented in the form F0 = nef0(u). The function f0 is nor-
malized by 1)(
0
0
duuuf .
It is assumed that the electron (Ie ) and ion ( Ii ) cur-
rents to a floating dust particle are in balance, or Ie +
Ii=0. The electron and ion currents to a dust particle are
calculated using the orbit-motion-limited (OML) theory,
taking into account the ion-neutral collisions in the
sheath around a dust particle [1, 2, 4]. In general, the
EEPF from Eq. (1) may be found only numerically.
However, as it will be shown below, approximate ana-
lytical solutions of Eq. (1) also exist.
2. THE APPROXIMATE ANALYTICAL SO-
LUTIONS FOR THE EEPF IN GLOW DUSTY
PLASMA
Assuming that the number of electrons with energy
larger than the first excitation energy threshold
( *u 5.11 eV for Ar) is small, one can neglect by trans-
formation of electrons with large energy into low-
energetic electrons, and the term describing inelastic
electron-atom collisions can be presented in the follow-
ing form [6]
k
k
ea
exc
ea uuFuvFS ,2/1 where v
k
ea is
the collision frequency of the k-th inelastic process with
a threshold energy Vk..
As a result, the homogeneous Boltzmann equation
(1) can be written as
3/2 0
0
1 0
( )
( ( ( )
( ))
1
) ( ))
( ( ) ( ) ,
e
D ed em
c
ed
f u
u f u
u
u f
u u
u
u u u
(2)
180 ISSN 1562-6016. ВАНТ. 2016. №6(106)
where =2me/ma, D =2me/md, am and dm are the
masses of neutral gas atoms and dust particles, respec-
tively
)(3
2
2
1
em
e
edDme
g
E
m
e
T
,
k
k
ea uvu)(1 is the total frequency for inelastic
electron-atom collisions including processes of excita-
tion and ionization, c
ed is the frequency describing dep-
osition of electrons on dust particles [4].
At large electron energies ( 5.11**
1 uuu eV, and
by taking *
1u = 20 eV), the electron energy probability
function decreases rapidly with an increase of u. There-
fore, the quasiclassical approach can be applied for cal-
culation of the EEPF at large electron energies, and [6]:
))(exp()( 20 uSCuf , (3)
where 2C is a constant, and //
1
*
1
)()( duuuS
u
u
with
)()(
)]()([1
)( 1
1
uu
uu
u
u
em
e
edD
c
ed
.
At large electron energies the EEPF in a dusty plasma
decreases faster than in a dust-free plasma because the
ratio of 1 in a dusty plasma to that in a dust-free plas-
ma for the same electric field sustaining the plasmas is
approximately
)(
)()(
1
1
u
uuc
ed
. The decrease increas-
es with an increase of nd or ad.
To calculate the EEPF at moderate and low energies
)( *
1uu , we move from u to the new variable
uuy *
1 . In this case, Eq. (3) can be presented in the
following form
)4()()]()([
)(1
)()(
*
101
0
0
yuyfyy
y
yf
yfy
y
c
ed
where ))()(()( 2/3 uuuy em
e
edD
with
yuu *
1 . It follows from Eq. (4) that
)5(,)(exp
)(exp)()()(
0
//
/
0 0
//////
20
/
y
y y
dyy
dydyyyyCyf
where
)(
)()]()([
)(
/
0
/*
1
/
0
/
1
/
y
dyyuyfyyA
y
y
c
ed
.
To obtain Eq. (5), we assumed that the EEPF and
it’s derivative on energy are continious at *
1uu . The
constant C2 in Eqs. (3) and (5) can be found from the
normalization condition
1)(
0
0 duuuf . Dusty
plasma parameters (the EEPF, effective electron tem-
perature and dust charge) obtained using Eqs. (3) and
(5) were compared with those calculated numerically by
a finite-difference method with accounting for electron-
electron collisions and transformation of high-energy
electrons into low-energetic electrons at inelastic elec-
tron-neutral collisions. It was found that the analytical
expressions (3) and (5) can be used for calculation of
the EEPF and dusty plasma parameters at typical exper-
imental conditions [7], in particular, in the positive col-
umn of a direct-current glow discharge and in the case
of an RF plasma maintained by an electric field with
frequency f=13.56 MHz. Moreover, in a 13.56 MHz
plasma, the EEPF may become close to the Maxwellian
distribution at an increase of dust density. That is in a
qualitative agreement with numerical and experimental
results of previous authors [4, 7].
3. THE APPROXIMATE ANALYTICAL
SOLUTION FOR THE EEPF IN DUSTY
PLASMA AFTERGLOW
Now consider a dusty plasma afterglow. The after-
glow has the two characteristic times [6]:
1
1 )( um and 1
12 )()(
uu c
ed , where
... denotes the averaging on time. The time 2 de-
termines the energy relaxation of electrons with large
energy ( *uu ). 1 is the time for energy relaxation of
electrons in the EEPF core (with *uu ). For *uu ,
)()(1 uu m , and, therefore, 12 [6]. In the
afterglow, the number of electrons decreases first in the
tail of the EEPF, and essential changes in the EEPF core
take place at larger afterglow times ( 2t ).
Here, we will consider the afterglow times larger
than 2 . For these times, the effect of inelastic electron-
atom collisions on the EEPF is small, and
u
F
TFuvu
m
m
uu
FS gm
a
e
ea
2/321
, (6)
where Tg is the neutral gas temperature, which is as-
sumed to be equal to 300 K (0.026 eV).
It is supposed that the average electron energy is
larger than the neutral gas temperature. We also assume
that for the most of electron energies considered here
the electron-atom elastic collisions dominate over the
electron-dust momentum-transfer collisions. Therefore,
the term in Eq. (1) describing electron-dust collisions
simplifies to FtuvFS c
eded ),( , where
e
c
edd
c
ed meutuntu /2),(),( is the frequency describing
collection of electrons by dust particles with the cross-
section σ
c
ed( u ) = πa
2
d(1– φs(t) / u ) for u ≥
φs(t) and 0 for u < φs(t). Here, φs(t) is the absolute val-
ue of dust surface potential. Taking into account these
assumptions, Eq. (1) simplifies to the following equa-
tion:
ISSN 1562-6016. ВАНТ. 2016. №6(106) 181
.0),(),(
),()(
1),( 2/3
tutuF
tuFuu
uut
tuF
c
ed
m
(7)
Here, we assume that the time-dependencies for dust
charge tZd and dust surface potential ts are expo-
nential: /exp0 tZtZ dd , /exp0 tt ss ,
where 00 tss , 00 tZZ dd , and is the
time characterizing the dust charge decrease in after-
glow. In this case, uuttu s
c
ed )//exp1(),( 0
for tu s , where edd man /22 . In this study, it
is also assumed that the frequency for electron-atom
collisions does not depend on electron energy
( constm ) and the EEPF for t = 0 can be pre-
sented in the following form [8]
)exp()0,( 21
xuAAuF , (8)
where x is a number. For Maxwellian and Druyvesteyn
EEPFs, x = 1 and x = 2, correspondingly.
x
xu
A
)(
)(1
1
2
2
,
where )2/(31 x , )2/(52 x , dttt )exp()(
0
1
with >0 is the gamma function, and u is the mean
energy of electrons, which is connected with the effec-
tive electron temperature by expression effTu
2
3
. If
u << *u , then
2/5
1
2/3
2
2/31
)]([
)]([
u
x
A .
Taking into account these assumptions and using the
method of characteristics [9], one gets from Eq. (7) for
tu s the following expression for the EEPF:
)2/3exp(),( 21
txxeuAtAtuF . (9)
If the initial EEPF is Maxwellian, the previous ex-
pression coincides with that presented in [6]:
)2/3exp()exp(exp),( tt
T
u
CtuF
eff
. (10)
To find the EEPF at tu s , we also apply the
method of characteristics and get from Eq. (7):
)11(,)1(2
)1(2
1
0
1
0
1/01
21
)2(
)1(4
2
2
2
2
3
exp),(
t
s
s
ts
txx
e
u
u
e
u
uA
t
AtuF e
where , and esdd man /2 0
2
1 .
If the EEPF at tu s is Maxwellian, it follows
from Eq. (11) the following expression for the EEPF at
tu s :
Normalized EEPF for t = 0 (a) and )/(1.0 t (b);
ad=50 nm, ne=10
9
сm
-3
and different dust densities: nd
=5.0 10
7
сm
-3
(solid curve), 3.0 10
7
сm
-3
(dashed
curve ), 10
7
сm
-3
(dotted curve). For ts , the
EEPF is Maxwellian with Теff =2 еВ for t=0. Here,
)/(1.0 . The same for nd =0 (c) and different af-
terglow times: t = 0 (dash-dotted curve);
0.1 )/( (dotted curve); 0.3 )/( (dashed curve);
0.5 )/( (solid curve)
a
b
c
182 ISSN 1562-6016. ВАНТ. 2016. №6(106)
.
)2(
)1(4
2
2
2
2
3
exp),(
)1(2
)1(2
1
0
1
0
1/01
1
t
s
s
ts
eff
t
e
u
u
e
u
T
uet
AtuF
(12)
Using Eqs. (10) and (12), the EEPF was calculated
for different dust densities (see Figs.a,b). The case nd =0
was also considered (Figure c).
One can see in Figure that the EEPFs for ts
in a dusty plasma differ essentially from those in a dust-
free plasma. With an increase of dust density, the differ-
ence on Maxwellian distribution in the region increases
which is accompanied by a decrease of a number of
electrons at large energies ( ts ) and their increase
at small energies ( ts ).
CONCLUSIONS
Thus, we have shown that the EEPF in dusty glow
and afterglow plasmas can be described analytically.
The results of analytical studies presented here are in a
good agreement with numerical and experimental re-
sults on dusty plasma of previous authors. It has been
shown that dust particles affect essentially the EEPF in
glow and afterglow plasmas, decreasing the number of
electrons at energies larger than the dust surface poten-
tial, while increasing their number at energies smaller
than the dust surface potential.
ACKNOWLEDGEMENTS
This work was supported by the State Fund for Fun-
damental Research of Ukraine.
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Article received 19.10.2016
ВЛИЯНИЕ ПЫЛЕВЫХ ЧАСТИЦ НА РАСПРЕДЕЛЕНИЕ ЭЛЕКТРОНОВ ПО ЭНЕРГИИ
В ПЛАЗМЕ В РЕЖИМАХ СВЕЧЕНИЯ И ПОСЛЕСВЕЧЕНИЯ
И.Б. Денисенко, Н.А. Азаренков, С. Ивко, Г. Бурмака, А. Глазков
Из однородного уравнения Больцмана получены аналитические выражения для функции распределения
электронов по энергии (ФРЭЭ) в пылевой плазме в режимах свечения и послесвечения. Для расчёта ФРЭЭ в
режиме свечения при больших электронных энергиях применён квазиклассический подход. Для плазмы в
режиме послесвечения аналитические выражения получены в предположении, что потери энергии электро-
нов происходят в основном за счёт упругих электрон-нейтральных столкновений и благодаря осаждению
электронов на пылевые частицы. Проанализировано влияние пылевых частиц на ФРЭЭ.
ВПЛИВ ПИЛОВИХ ЧАСТИНОК НА РОЗПОДІЛ ЕЛЕКТРОНІВ ЗА ЕНЕРГІЄЮ В ПЛАЗМІ
В РЕЖИМАХ СВІТІННЯ ТА ПІСЛЯСВІТІННЯ
І.Б. Денисенко, М.О. Азарєнков, С. Івко, Г. Бурмака, А. Глазков
З однорідного рівняння Больцмана отримано аналітичні вирази для функції розподілу електронів за енер-
гією (ФРЕЕ) в запорошеної плазмi в режимах світіння та післясвітіння. Для розрахунку ФРЕЕ в режимі сві-
тіння при великих електронних енергіях застосовано квазікласичний підхід. Для плазми в режимі післясві-
тіння аналітичні вирази отримані в припущенні, що втрати енергії електронів відбуваються в основному за
рахунок пружних електрон-нейтральних зіткнень та завдяки осадженню електронів на пилові частинки.
Проаналізовано вплив пилових частинок на ФРЕЕ.
http://scitation.aip.org/search?value1=I.+B.+Denysenko&option1=author&option912=resultCategory&value912=ResearchPublicationContent
http://scitation.aip.org/search?value1=H.+Kersten&option1=author&option912=resultCategory&value912=ResearchPublicationContent
http://scitation.aip.org/search?value1=N.+A.+Azarenkov&option1=author&option912=resultCategory&value912=ResearchPublicationContent
https://archive.org/search.php?query=creator%3A%22L.+Elsgolts%22
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