Effective grain potential in a plasma with external sources of ionization
A new approach is proposed for analytical description of effective grain potentials in dusty plasmas. The basic idea is to describe absorption of electrons and ions by grain in terms of effective point sinks introduced into the equations of plasma dynamics. The proposed approach makes it possible...
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2006
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| Cite this: | Effective grain potential in a plasma with external sources of ionization / A.G. Zagorodny, A.V. Filippov, A.F. Pal’, A.N. Starostin, A.I. Momot // Вопросы атомной науки и техники. — 2006. — № 6. — С. 99-103. — Бібліогр.: 12 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-817922025-02-09T13:10:21Z Effective grain potential in a plasma with external sources of ionization Zagorodny, A.G. Filippov, A.V. Pal’, A.F. Starostin, A.N. Momot, A.I. Basic plasma physics A new approach is proposed for analytical description of effective grain potentials in dusty plasmas. The basic idea is to describe absorption of electrons and ions by grain in terms of effective point sinks introduced into the equations of plasma dynamics. The proposed approach makes it possible to find explicit relations for the potential and particle densities distributions. The example of grain screening is considered for the case of weakly ionized plasma in the presence of the external sources of plasma ionization. The present work was partially supported by the Program of Fundamental Research of the Department of Physics and Astronomy of the National Academy of Sciences of Ukraine and the Russian Fund for Fundamental Research (grant 05-02-11258-a and 05-02- 08158 ofi-a). 2006 Article Effective grain potential in a plasma with external sources of ionization / A.G. Zagorodny, A.V. Filippov, A.F. Pal’, A.N. Starostin, A.I. Momot // Вопросы атомной науки и техники. — 2006. — № 6. — С. 99-103. — Бібліогр.: 12 назв. — англ. 1562-6016 PACS: 52.27 Lw https://nasplib.isofts.kiev.ua/handle/123456789/81792 en Вопросы атомной науки и техники application/pdf Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Basic plasma physics Basic plasma physics Zagorodny, A.G. Filippov, A.V. Pal’, A.F. Starostin, A.N. Momot, A.I. Effective grain potential in a plasma with external sources of ionization Вопросы атомной науки и техники |
| description |
A new approach is proposed for analytical description of effective grain potentials in dusty plasmas. The basic idea
is to describe absorption of electrons and ions by grain in terms of effective point sinks introduced into the equations of
plasma dynamics. The proposed approach makes it possible to find explicit relations for the potential and particle
densities distributions. The example of grain screening is considered for the case of weakly ionized plasma in the
presence of the external sources of plasma ionization. |
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Article |
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Zagorodny, A.G. Filippov, A.V. Pal’, A.F. Starostin, A.N. Momot, A.I. |
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Zagorodny, A.G. Filippov, A.V. Pal’, A.F. Starostin, A.N. Momot, A.I. |
| author_sort |
Zagorodny, A.G. |
| title |
Effective grain potential in a plasma with external sources of ionization |
| title_short |
Effective grain potential in a plasma with external sources of ionization |
| title_full |
Effective grain potential in a plasma with external sources of ionization |
| title_fullStr |
Effective grain potential in a plasma with external sources of ionization |
| title_full_unstemmed |
Effective grain potential in a plasma with external sources of ionization |
| title_sort |
effective grain potential in a plasma with external sources of ionization |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2006 |
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Basic plasma physics |
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https://nasplib.isofts.kiev.ua/handle/123456789/81792 |
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Effective grain potential in a plasma with external sources of ionization / A.G. Zagorodny, A.V. Filippov, A.F. Pal’, A.N. Starostin, A.I. Momot // Вопросы атомной науки и техники. — 2006. — № 6. — С. 99-103. — Бібліогр.: 12 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT zagorodnyag effectivegrainpotentialinaplasmawithexternalsourcesofionization AT filippovav effectivegrainpotentialinaplasmawithexternalsourcesofionization AT palaf effectivegrainpotentialinaplasmawithexternalsourcesofionization AT starostinan effectivegrainpotentialinaplasmawithexternalsourcesofionization AT momotai effectivegrainpotentialinaplasmawithexternalsourcesofionization |
| first_indexed |
2025-11-26T02:01:52Z |
| last_indexed |
2025-11-26T02:01:52Z |
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1849816539390279680 |
| fulltext |
Problems of Atomic Science and Technology. 2006, 6. Series: Plasma Physics (12), p. 99-103 99
EFFECTIVE GRAIN POTENTIAL IN A PLASMA WITH EXTERNAL
SOURCES OF IONIZATION
A.G. Zagorodny 1, A.V. Filippov2, A.F. Pal’ 2, A.N. Starostin 2, A.I. Momot 3
1Bogolyubov Institute for Theoretical Physics, Kyiv, Ukraine;
2Troitsk Institute for Innovation and Fusion Research, Troitsk, Russia;
3Kyiv National Taras Shevchenko University, Kyiv, Ukraine
A new approach is proposed for analytical description of effective grain potentials in dusty plasmas. The basic idea
is to describe absorption of electrons and ions by grain in terms of effective point sinks introduced into the equations of
plasma dynamics. The proposed approach makes it possible to find explicit relations for the potential and particle
densities distributions. The example of grain screening is considered for the case of weakly ionized plasma in the
presence of the external sources of plasma ionization.
PACS: 52.27 Lw
1. INTRODUCTION
The problem of nonlinear grain screening still remains
one of the most important issue of dusty plasma theory. It
is of grate importance for description of such interesting
phenomena as dusty crystal formation, excitation of dust
acoustic waves, evolution of dust structures etc. In spite
of the fact that the screened potentials have been studied
during many years the problem of macroparticle
screening still requires the further studies. First of all this
concerns the grain screening in low-temperature plasma
for which even the screening length is not determined in
the general case [1]. The problem is that in the case of
grain charging due the absorption of electrons and ions
from plasma the effective potentials are crucially
dependent on the processes in plasma background, in
particular, on the details of plasma regeneration. In view
of the importance of nonlinear effects and self-consistent
description of charging processes the problem of grain
(probe) screening usually is solving numerically [2-9].
Giving many details of the screening numerical results,
however, can not be used for obtaining approximate
analytical relations. At the same time it is very important
to have such analytical approximations for the solution of
the problems mentioned above. The purpose of the
present paper is to work out analytical description of the
screened potentials of grains charged by plasma current in
low-temperature plasma.
We start from the formulation of the model (point-sink
model) which provides the analytical solution of the
problem, if the grain charge and the intensities of plasma
particle fluxes are known (Section 2). Then we perform
necessary analysis of the obtained relations and compare
the results with the numerical solutions (Section 3). In
Section 4 we apply the proposed model for treatment of
screening in the collisionless Vlasov plasma.
2. POINT-SINK-MODEL. BASIC SET OF
EQUATIONS FOR WEAKLY-IONIZED
PLASMA
Let us consider single grain embedded into infinite
weakly-ionized plasma. We assume that the grain absorbs
all encountered electrons and ions. In the stationary case
the grain charge is maintained by electron and ion fluxes
which are equal to each other. In the case under
consideration plasma dynamics can be described by the
continuity equations which have the following form:
0div ( ) ( ) ( )e ir I n r n rσ βΓ = −
ur
, (1)
where
( ) ( ) ( ) ( )r e n r r D n rσ σ σ σ σ σµΓ = − ∇Φ − ∇
ur
, (2)
σµ is the plasma particle mobility, Dσ is the diffusion
coefficient, subscript σ (σ=e, i) labels plasma particle
species, 0I is the intensity of ionization sources, if
present, β is the coefficient of the electron-ion
recombination, the rest of notation is traditional.
The electric potential ( )rΦ
r
satisfies the Poisson
equation
( ) 4 ( )r e n rσ σ
σ
π∆Φ = − ∑ . (3)
Eqs. (1) – (3) should be supplemented by the following
boundary conditions
( ) | 0,r an rσ = = 2
( ) | ,r a
r q
r a=
∂Φ
= −
∂
(4)
( ) |rn r nσ σ→∞ = , ( ) | 0rr →∞Φ = ,
where q is the stationary grain charge which is determined
by the equation
| 0i i e e r ae e =Γ + Γ = , (5)
a is the grain radius , nσ is the unperturbed density of
particles of σ species. The later quantities are determined
by the conditions of plasma regeneration. They are given
by the relation
0 0/e in n I nβ= = ≡ , (6)
if external sources of ionization are present, or they are
assumed to be given, if the sources of plasma regeneration
are located at large distance from the grain.
100
(12)
(13)
In view of the nonlinearity of the basic set of
equations it is usually solved numerically (see, for
example, Refs. [2, 3, 5-9]). However, it is quite difficult
to obtain analytical relations for the screened potentials
and plasma particle distribution on the basis of the
numerical solutions. Therefore, it would be highly
desirable to have approximate analytical expressions for
such quantities, at least for their asymptotic behavior.
This problem can be solved taking into account that in
many cases the nonlinearity produces considerable effects
at small distances from the grain surface, only. Such
conclusion follows from the analysis of the nonlinear
numerical solutions for equilibrium screening (no particle
fluxes through the grain surface [10]) and for the
screening of grains charged by plasma currents in the
collisionless plasmas [4]. Moreover, in the case of grains
of small sizes the nonlinearity does not lead to
considerable deviation from the linear solution even in the
vicinity of the grain, but on the other hand, the plasma
particle fluxes toward the grain contributes to the changes
of the potential asymptotics [4]. Probably, it could be
explained by the fact that in the case of grain absorbing
plasma particles the electron and ion densities near the
grain are small and thus the influence of nonlinearity is
not well pronounced. The possibility to use linearized
equation for the analytic description of the asymptotic
behavior of the effective grain potentials in weakly-
ionized plasma was also confirmed by the estimates
presented in Ref. [11].
With regard to these arguments we propose to
describe screened potentials on the basis of the linearized
version of the point-sink model. In terms of such model it
is assumed that the effects associated with plasma particle
absorption by the finite-size grain can be satisfactory
approximated by the effective point sinks, i.e. instead of
Eq. (1) and the boundary conditions (4), (5) we propose to
use the equation:
0div ( ) ( ) ( ) ( )e ir I n r n r S rσ σβ δΓ = − −
ur r
, (7)
where Sσ is the intensity of the point sink
S ds σσ = − Γ∫
ur
Ñ . (8)
The linearized version of Eqs. (2) and (7) is the
following:
0 0 ( )
( )
e
e i
e
e n nn n n
T D
S r
D
σ
σ
σ
σ
σ
β
δ δ δ
δ
− ∆Φ − ∆ = − +
−
r
(9)
4 ( ) 4 ( )e e i ie n e n q rπ δ δ π δ∆Φ = − + −
r
, (10)
where ( )n rσδ is the density perturbation. We also take
into account that in the absence of the grain the plasma
density is given by Eq. (6) and the sink intensities for
electrons and ions are equal to each other, i.e.
e iS S S= = . (11)
3. SCREENED POTENTIAL AND CHARGE
DENSITY DISTRIBUTION
The solution of Eqs. (9), (10) can be easily obtained in
the k
r
-representation
2 2 4 2 2
0
1 ( )i Di i De e Diekn k S k q qk S k S k
e
δ = − + − − − ∆
r % % % ,
2 2 4 2 2
0
1 [ ( ) ]e De i De e Diikn k S k q qk S k S k
e
δ = − + + − −
∆
r % % % ,
where
4 2 2 2 4
0
4 2 2 2 2 2 2 2
0
| | ,
( ) 2 ;
; ;
e i
D S
Se Di Si Di S Si Se
e e e
k k k k k
k k k k k k k k
= =
∆ = + + +
= + = +
(14)
2 0 ,S
nk
Dσ
σ
β
=
eSS
Dσ
σ
=% .
The potential kΦ r is given by
2 2 2
4 2 2 2 4
0
( )4
( ) 2
S D
k
D S
q k k k S
k k k k k
π
+ −
Φ =
+ + +
r
%
, (15)
where
2
1 1
D i e
eSS
k D D
= −
% . (16)
In its turn Eq. (15) can be rewritten as
1 2
2 2 2 2
1 2
4 4
k
Q Q
k k k k
π π
Φ = +
+ +
r , (17)
where
( )
2 2 2
1,2
1,2 2 2
1 2
2 2 2 2 2 2 2
1,2 0
( )
1 ( ) 8
2
S D
D S D S
q k k k S
Q
k k
k k k k k k
− +
=
−
= + ± + −
%
. (18)
In the coordinate representation
1 2
1 2( )
k r k re er Q Q
r r
− −
Φ = + , (19)
i.e. the effective potential is given by the superposition of
two screened potentials with different screening lengths.
Notice, that these screening lengths were obtained for the
first time in Ref. [11]. At large distances the potential is
determined by the term with the larger screening length. This
result is in agreement with the numerical simulation [5-8].
The induced charge density is
1 2
1 2( )
k r k re er Q Q
r r
ρ
− −
= +% % . (20)
101
Here
4 2 2
0 1,2
1,2 2 2
1 2
2 ( )Dqk k k q S
Q
k k
− +
=
−
%
% . (21)
If the plasma sources are located at large distance
from the grain and recombination processes could be
neglected ( 0β = )
( ) ( )
Dk re Sr q S
r r
−
Φ = + −
%
% (22)
and
2( ) ( )
4
Dk r
D
er k q S
r
ρ
π
−
= − + % . (23)
As is seen, in the case under consideration the
effective potential has the Coulomb-like asymptotics, but
the induced charge is determined by the screened part.
The quantity S% can be treated as an effective charge
which generates the Coulomb part of the potential. It can
not be found within the present theoretical treatment.
Therefore, in what follows we take this quantity from the
numerical solution (see Fig. 1), which is taken from the
numerical data obtained in Ref. [6], assuming that S% is
proportional to real grain charge, i.e. S qα= −% .
0 1 2 3 4 5 6
0,00
0,05
0,10
0,15
0,20
0,25
0,30
0,35
0,40
0,45
0,50
R
el
at
iv
e
un
sc
re
en
ed
c
ha
rg
e
A
Fig.1. Dependence of dimensionless effective
unscreened charge S q− % on dimensionless grain radius
DakA =
To compare the obtained results with the numerical
simulation [6] it is convenient to use the following
quantity
0
( ) ' ( ')
r
Q r q d r rρ= + ∫
r
, (24)
which is the total charge distributed in the sphere of the
radius r. Substitution of Eq. (23) into Eq. (24) gives
0( ) (1 ) (1 )k r
DQ r q q e k rα α −= + − + . (25)
The dependences of the quantity Q(r) on the
dimensionless distances Ra=r/a for different grain sizes
are presented in Fig. 2. They are in a good agreement with
those obtained in Ref. [6]. The accuracy of the analytical
estimates is seen from Fig. 3 in which the charge
distributions Q(r) calculated analytically (thin curve) and
numerically (bold curve) are presented. As is seen, in the
case under consideration (β=0, I=0) at large distances
Q(r) approaches some constant values which are nothing
but the dimensionless effective charge α.
0 50 100 150 200
0,0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1,0
1,1
6.25
2
0.5
0.1
0.05Q
(r
)/
r
R
a
Fig.2 Relative charge distribution corresponding to
Eq. (25), at various values of A (0.05, 0.1, 0.5, 2, 6.25)
0 50 100 150 200
0,4
0,5
0,6
0,7
0,8
0,9
1,0
Q
(r
)/
q
R
a
Fig.3. Relative charge distribution at A=0.05, bold line
presents the data from [6], thin line corresponds to
Eq. (25)
The situation is considerably changed, if the ionization
sources are present. In this case
1
2
1
1
22
2
( ) 1 (1 )
1 (1 )
i
D
k r
i
k r
QQ r q e k r
k
Q e k r
k
−
−
= + − + −
− − +
%
%
. (26)
102
Dependencies of this quantity on the dimensionless
distance for the different values of ionization intensity are
presented in Fig. 4. In this case the analytical estimates
also are in a good agreement with the exact solution of the
nonlinear equations (see, Fig. 5).
0 10 20 30 40 50
0,01
0,1
1
4
3
21
Q
(r
)/
q
R
Fig.4. Relative charge distribution at A=0.158 and
various values of iDIai 5
0 = (1) 1.15·10-2, (2) 2.5·10-3,
(3) 5·10-4, (4) 10-4, dashed line corresponds to the case of
absence of bulk ionization and recombination
0 10 20 30 40 50
0,01
0,1
1
Q
(r
)/
q
R
Fig.5. Relative charge distribution at A=0.158 and
i0=5⋅10-4, bold line presents the data from [6],
thin line corresponds to Eq. (26)
4. POINT-SINK-MODEL FOR
COLLISIONLESS PLASMA
In the case of collisionless plasma its dynamics is
described by the Vlasov equation. With regard to the
presence of the point-sink the stationary version of this
equation has the form
( ) ( , )
( ) ( ) ( , )
ev r f r v
mr v
r v vf r v
σ
σ
σ
σ σ
φ
δ σ
∂ ∂
− ∇ = ∂ ∂
= −
r r r r
r r
r r r
, (27)
where ( )vσσ is the charging cross-section
2
2 2
2 2( ) 1 1qe qev a
mv a mv a
σ σ
σσ π θ = − −
. (28)
Assuming that the sink produces small effect the equation
for perturbation reduces to
0
0
( , ) ( ) ( )
( ) ( ) ( )
ev f r v r f v
mr v
r v vf v
σ
σ σ
σ
σ σ
δ φ
δ σ
∂ ∂
− ∇ =
∂ ∂
= −
r r r r
r r
r
(28’)
The potential Φ(r) satisfies the Poisson equation (10) with
( )n n d v f vσ σ σδ δ= ∫
r r
. Using k
r
-representation it is
easy to show that
0
0
( ) ( )( ) ( )
0k k
e i v vf vf v f v
T kv i
σ σ σ
σσ
σ
σ
δ φ= − +
−
r r
r
rr , (29)
2
2
0
2
( ) ( )
k k
e nn n
T k
v v f v dv
σ σ
σσ
σ
σ σ
π
δ φ
σ
= − − ×
×∫
r r
(30)
and thus,
2 ( )( ) ( )Dk r i e
D
D
q C Cr e g k r
r rk
π
φ − +
= − , (31)
where
2
0
( ) ( ) ( ),
( ) ( ) .
x xg x e Ei x e Ei x
C e n v v f v dvσ σ σ σ σσ
−= − −
= ∫
(32)
It follows from Eq. (31) that at kDr >> 1
2
4 ( )( )
( )
i e
D
C Cr
rk
π
φ
+
≈ −
in agreement with the result obtained by other methods [12].
CONCLUSIONS
The model of point-like sink is proposed to describe
electric grain potential with regard to plasma particle
absorption by grain. This makes possible to obtain
analytical solution of the problem of grain screening and
to find explicitly asymptotic behavior of the potential.
With the appropriate choice of the parameters of the
model (grain charge q and sink intensity S) the proposed
approximation recovers numerical solutions of the
consistent nonlinear problems.
ACKNOWLEDGEMENTS
The present work was partially supported by the
Program of Fundamental Research of the Department of
Physics and Astronomy of the National Academy of
103
Sciences of Ukraine and the Russian Fund for
Fundamental Research (grant 05-02-11258-a and 05-02-
08158 ofi-a).
REFERENCES
1. Yu.P. Raizer. Physics of Gas Discharge: Moscow:
”Nauka”, 1987 (in Russian).
2. I.I. Ivanov, A.F. Pal’, A.N. Starostin// Zh. Eksp. Teor. Fiz.
2004, v. 126, p. 75; A.F. Pal’, A.N. Starostin, A.V. Filippov//
Fizika Plazmy. 2001, v. 27, p. 155 (in Russian).
3. A.F. Pal’, D.V. Sivokhin, A.N. Starostin, A.V. Filippov,
V.E. Fortov// Fizika Plazmy. 2002, v. 28, p. 32 (in Russian).
4. T. Bystrenko, A. Zagorodny// Phys. Lett. A. 2002,
v. 299, p. 383.
5. A.V. Filippov, N.A. Dyatko, A.F. Pal’,
A.N. Starostin// Fizika Plazmy. 2003, v. 29, p. 214 (in
Russian).
6. T. Bystrenko, A. Zagorodny// Phys. Rev. E. 2003,
v. 67, p. 066403.
7. A.G. Leonov, A.F. Pal’, A.N. Starostin,
A.F. Filippov// Pis’ma ZhETP. 2003, v. 77, p. 577 (in
Russian).
8. A.G. Leonov, A.F. Pal’, A.N. Starostin,
A.F. Filippov// ZhETP. 2004, v. 126, p. 75 (in Russian).
9. A. Zagorodny, V. Mal’nev, S. Rumyantsev// Ukr. J.
Phys. 2005, v. 50, p. 448.
10. T. Bystrenko, A. Zagorodny// Phys. Lett. A. 1999,
v. 255, p. 325.
11. A.V. Filippov, A.G. Zagorodny, A.F. Pal’,
A.N. Starostin// JETP Lett. 2005, v. 81, p. 146.
12. V.N. Tsytovich, Yu.K. Khodataev. R. Bingham//
Comments Plasma Phys. Controlled Fusion. 1996, v. 17,
p. 249.
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