Effect Of Negative-Ion Flux On Ion Distribution Around A Spherical Probe In Electronegative Plasmas
N. A. Azarenkov1, I. Denysenko1, M. Y. Yu2
1School of Physics and Technology, V. N. Karazin Kharkiv National University, Svobody sq.4, 61077 Kharkiv,
Ukraine; 2Theoretical Physics I, Ruhr University, D-44780 Bochum, Germany
Abstract. The electric potential and the electron- and ion-density profiles near a spherical probe in electronegative
plasmas containing negative as well as positive ions are studied. It is shown that for relatively large probe radius and
neutral-gas pressure, and/or small electron-to-ion temperature ratio, the density profile of the negative ions can differ
from that of Boltzmann if their flux near the probe is outwards. The existence of stationary states is considered and the
corresponding maximum negative-ion flux is obtained.
Keywords: gas discharge, plasma, probes.
Consider a collisional electronegative plasma consisting of electrons, positive and negative ions, as well as
neutral particles. A spherical probe is immersed in the plasma. The distributions of the positive and negative ions
around the probe are described by the following equations [1,2]:
( 2 ) 0
r −2d n r = r i nυi n , (1)
miniυidrυi = eniE −ν iaminiυi −Tidrni , (2)
mnnnυndrυn = −ennE −ν namnnnυn −Tndrnn , (3)
where r is the distance from the probe center. The subscripts e, i, and n denote electrons, positive and negative
ions, respectively, nα ,υα ,ναa ,mα and Tα (α = e, i, n) are the corresponding densities, velocities, ion-neutral
collision frequencies, masses and temperatures, respectively. E = −drV is the electric field and V the potential.
The electrons are in local thermodynamic equilibrium and their density is thus Boltzmann distributed. For simplicity
we shall also assume that Ti = Tn = Tg, where Tg is the temperature of the neutral gas.
The electric field is determined by the Poisson’s equation
2 ( 2 ) 4 ( )
r dr r E = e ni − ne − nn − π , (4)
except in the presheath or bulk-plasma, where the plasma can be assumed to be uniform. In these regions the
ni = ne + nn (5)
It is convenient to introduce the normalizations nα = nα / ne0 , uα =υα /υB , ξ = r /λDe ,
δαa =ναaλDe /υ B , η = −eV /Te , γ = Te /Tn and α = nn0 / ne0 . Here ne0 and n0 n are the electron and
negative ion densities at r →∞, υB is the Bohm velocity, and λDe is the electron Debye length at r →∞. The
normalized ion flux through a spherical surface of radius r is Jα = nα uαξ 2 . We note that ui ≤ 0 and ≥ 0 n u .
From Eqs. (1) – (4) we obtain
niui = Ji ξ 2 ,
nnun = Jn ξ 2 ,
niuidξ ui =εni −δ ianiui , (6)
dξε = ni − nn − exp(−η ) − 2ε /ξ ,
kmnnundξ un = −εnn − kmδ nannun −γ −1dξ nn , (7)
where ε = dη / dξ and km = mn / mi .
At the probe surface ξ =ξ p (= rp /λDe ) the ion fluxes balance the electron flux, or
− Ji (ξ p ) − Jn (ξ p ) + Je (ξ p ) = 0 , (8)
where ( ) ( / 2 ) 2 exp( )
Je ξ p = − mi πme ξ p −η p is the normalized electron flux and η p is the probe potential.
The system (6) and (7) of ordinary differential equations is solved numerically by a 4th-order Adams-Bashforth-
The Negative-Ion Flux
The density of the negative ions can differ from Boltzmann if there is a negative-ion flux outwards from the
probe . We now analyze how the flux Jn affects the negative-ion density profile. Because of the negative-ion flux,
the terms m n n n k n u d u ξ and kmδ nannun in Eq. (7) can become important. Far away from the probe the negativeion
velocity is small or zero and the convection (former) term is negligible. In the presence of collisions, far from the
probe we have kmδ nannun >>εnn , so that the friction force on the negative ions by the neutrals is balanced by
the pressure force. The solution of Eq.(7) is then
nn =γδ nakmJn /ξ +α , (9)
which shows that the negative-ion density increases towards the probe.
For nearly collisionless ( kmδ nannun <<εnn ) plasmas, the density of the negative ions is given by
nn =α exp[−γ (kmun2 / 2 +η )] , (10)
which can be obtained from Eq. (7). It follows from (10) that the negative-ion density is decreased because of the
negative-ion flux. This decrease can be significant when the negative-ion velocity un is large. Note that un can be
large only near the probe, where nn is small. For the near-collisionless case, it also follows from Eq. (7) that
z k J z m n exp(2 ) 1 ln
= , (11)
where z = (α / nn )exp(−γη) . Equation (11) can be presented in the form
A z2 ln z
2 = , (12)
where 2 exp(2 ) /(2 2 4 )
A2 =γkmJn γη α ξ .
Since the function z2 grows faster than ln z , we have from (12) that A2 < 1, or
n nb k
2ξ 2α exp(−γη )
< = , (13)
from which one sees that the negative-ion flux around a spherical probe in a near-collisionless electronegative
plasma cannot exceed Jnb. The latter depends on the ratio γ of the electron to negative-ion temperature as well as
the spatial coordinate ξ . For constant Jn, the minimum of ξ is clearly ξ p = rp /λDe . This means that the flux Jn
can not exceed Jnb (ξ =ξ p ) . From the criterion (13), it follows that the flux Jn can be rather large for small γ
and large ξ p . In most low-density low-temperature laboratory experiments one has γ >>1.
In next section we shall solve the equations (6)- (8) numerically for different external parameters. We shall find
the maximum Jn that the time-independent equations (6)- (8) have a solution for.
First we consider the effect of neutral gas pressure on the potential as well as electron and ion density
distributions around the spherical probe. In Fig.1 the dependencies of the normalized electron and positive- and
negative-ion densities on x = 1/ξ are shown for Jn =0 and Jn ≠ 0 . In the study for Jn ≠ 0 the negative ion
flux Jn is about the maximal magnitude when the system (6)-(8) has a solution. The solid curves in Fig.1 are
calculated for Jn = 0 and Ji = -27.4, -21.5, and -15.1 for the neutral gas pressures p0 = 0.75, 1, and 1.5 Torr,
respectively. The dashed curves are for Jn = 0.01, 4.026, and 16.0 and Ji = – 27.4,-22.85, and -21.05 for p0 =
0.75, 1, and 1.5 Torr, respectively. The magnitudes of Jn are much larger than ( ) nb p J ξ . For the parameters
considered ( ) nb p J ξ is very small ( < 10-48 ). One can see from Fig. 1 (b) that the profiles of e n and i n for
Boltzmann negative ions (when Jn=0) at p0 =1 Torr are close to that calculated for Jn ≠ 0 . The differences in the
profiles of e n and i n for Jn=0 and Jn ≠ 0 are small because of small negative-ion flux compared with that of the
positive ions ( |Ji/Jn|~ 5.7 ). However, even for small Jn the profile of nn differs noticeably from that for Jn = 0.
The negative-ion density at x ≥ 0.015 for Jn ≠ 0 is larger than that for Jn = 0. The increase of n n can be
attributed to ion-neutral collisions [see Eq. (9)]. At p0 = 0.75 Torr the profiles of e n and i n obtained for
Boltzmann negative ions coincide with that for Jn ≠ 0 because the ratio |Ji/Jn| is very large. At relatively high
pressures the maximum Jn can be much larger than ( ) nb p J ξ , and the density profiles obtained for Jn ≠ 0 differ
significantly from that calculated for Boltzmann negative ions [see Fig. 1(c)]. As a result, the difference in the
potential profiles for Jn=0 and Jn ≠ 0 increases with p0 , and for large (~ 1.5 Torr) gas pressures the difference
may be near 10% (see Fig. 2). At smaller pressures the negative ion flux does not affect the potential profile around
the probe because |Ji/Jn|<<1. We also note that (i) the magnitude of the positive-ion flux at Jn = 0 decreases and
the probe potential increases with p0, in agreement with Bryant ; (ii) the probe potential decreases with Jn , which
can be attributed to the removal of negative charge from the probe by the negative-ion flux.
The effect of the negative ion flux increases with an increase of the probe radius (see Fig.3) and with a
decrease of the electron temperature with respect to that of ions .
Thus, it is found that the density profile of the negative ions near a spherical probe can differ from that of
Boltzmann if there is a negative-ion flux from the probe. Furthermore, in the stationary state, the negative-ion flux in
a collisionless plasma sheath cannot be larger than the value given by (13). As a result, the negative-ion density
profile differs from Boltzmann only for relatively large probe radius and ion-to-electron temperature ratio.
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