Potential Formation in the Plasma with Two Positive Ion Species

M. Čerček1,2, T. Gyergyek3,1, M. Conţulov4, M. Kršak5
1 “Jožef Stefan” Institute Jamova 39, 1000 Ljubljana, Slovenia
2 University of Maribor, Faculty of Civil Engineering, Smetanova 17, 2000 Maribor, Slovenia
3 University of Ljubljana, Faculty of Electrical Engineering, Tržaška 25, 1000 Ljubljana, Slovenia
4”Ovidius” University, Faculty of Physics, bd. Mamaia 124, 8700 Constanţa, Romania
5”Comenius”University, Faculty of Mathematics, Physics and Informatics, Mlynská dolina, 84248 Bratislava,
Abstract. Potential formation in plasma with two warm positive ion populations has been studied analytically by a onedimensional,
collision-less, fully kinetic model. The results have been verified in a PIC simulation experiment. It has
been found that each ion species enter the sheath region at the floating collector with the same velocity, normalized with
corresponding ion acoustic velocity. For that reason the lighter ions hit the collector with higher velocity as compared to
the case when they are the only ion species in the plasma.
Keywords: plasma, positive ion, presheath, sheath, floating collector
PACS: 52.40.Kh, 52.25.Dg, 52.65 -y
Recently, much attention has been paid to the multiple ion species presheath/sheath problem, which still needs
further investigation [1, 2]. Plasmas with two or more positive ion species are very often found in plasma devices for
technological applications and also in fusion machines. In an investigation [1, and ref. therein] it was found by laserinduced
fluorescence measurements of Ar ion velocity distribution functions within the presheath up to the sheath
edge that the ions reach the sheath edge traveling faster than their individual Bohm speed by more than 75%,
approaching a speed equal to the ion sound speed of the composed (He-Ar) system. The results of an another
investigation of a plasma with two cold ion populations [2] demonstrated that both ion species are closely similar in
terms of obeying their separate “Bohm” criteria and that the spatial distributions of the separate ions species and the
ion fluxes are geometrically similar. There was no evidence of structural separation of plasmas that was found to
exist in electronegative plasmas. In this contribution we present the study on potential formation near a floating
collector in plasma with two warm positive ion populations. In the analytical study we use a collision-less fully
kinetic plasma sheath model, originally developed by Schwager and Birdsall [3] and later extended in order to
include additional particle species like hot electrons [4] and/or negative ions [5]. Further, we complement the results
of the analytical study by a PIC simulation experiment in such bounded plasma system. As shown, good agreement
between both approaches is found and some fundamental findings of [2] were confirmed.
The collision-less one-dimensional presheath/sheath plasma region is bounded by a planar plasma source on one
side and by a floating collector on the other side. From the Maxwellian plasma source two positive ion populations
are injected into the system with an accelerated half-Maxwellian distribution, whereas the electron population is
modelled with a truncated full-Maxwellian distribution. At the floating collector all positive ions are absorbed and
electrons are assumed to be reflected. A small amount of fast electrons are lost at the collector to neutralize the ion
current. Electrons which return to the source are refluxed in the system with their source temperature. By this
process no charge accumulates at the source plane and the electric field at the plasma boundary is zero. Because
equal fluxes of positive and negative particles are injected from the source, the net injected charge density is not zero
and a source sheath forms in order to neutralize the injected plasma. The electric field is therefore constant at the
collector sheath boundary and is chosen to be zero when the source and collector sheaths are many Debye lengths
apart. No additional potential structure is expected to form in the system. The particle densities ni,e and particle
fluxes in the system are obtained by calculating the first and second moment of the corresponding distribution
functions. The collector floating potential and the presheath potential are calculated as functions of various plasma
parameters using three boundary conditions. Setting the net charge at the sheath boundary to zero, we obtain the first
equation which relates the floating potential φC and the presheath (source potential drop) potential φP. The second
equation is obtained from the assumption of zero electric field at the source boundary and at the collector sheath
boundary. Integrating the Poisson equation once within the indicated boundaries gives the desired relation. The third
equation, which enables to express the particle density ratios as a function of the floating potential, is obtained from
the zero net collector current. In Fig. 1 the normalized potentials ΨP = φP/Te and ΨC = φC/Te are shown as functions
of Ar ion density fraction α = ni0Ar/ni0H(He)+ni0Ar, ni0 being the source ion density of the corresponding species. The
ion to electron temperature ratio is chosen to be τH(He) = τAr = τ = Ti/Te = 0.1. On the left plot one can observe that the
presheath potential does not depend on the ion composition of the plasma in accordance with its independence on
ion mass. The change in the potential is less than 1%. On the other hand, the collector floating potential gradually
changes from its value of ΨC = -3.38 in pure hydrogen plasma (-4.08 in He plasma) to ΨC = -5.1 in pure Ar plasma.
From this result one can conclude that in composed collision-less plasma the positive ion population will enter the
sheath region each with its own normalized Bohm velocity, vis/csAr and vis/csH(He) respectively. The lighter ions will
hit the collector surface with higher velocity then in plasma with just light ion population. The velocity of ions
increases for more than 20% in H-Ar plasma and for 10% in He-Ar plasma. In Fig. 2 both potentials are plotted as
functions of relative ion temperature τ for H-Ar and He-Ar plasmas with equal amounts of each positive ion species
(α = 0.5). Both potentials are decreasing functions of τ, whereas on the left plot one can once again notice that the
presheath potential is not sensitive to ion mass. The curves for H-Ar and He-Ar plasma are indistinguishable. In
plasmas with additional hot electron population or in plasmas with negative ions multiple solutions for presheath
potential ΨP were found [4, 5] in certain intervals of additional population density ratio α, which indicated the
existence of one or two potential steps – double layers – in the presheath plasma region. Nevertheless, the collector
potential ΨC remained a continuous single valued function of α. No multiple solutions for ΨP are found in the present
two-positive ion case, indicating that the

In order to complement the theoretical investigation we also studied the plasma system with the XPDP1 particlein-
cell simulation code composed at Berkeley University [3]. The overall set up of the simulation experiment was in
principle the same as in our previous study of collector and source sheaths of finite ion temperature plasma with
additional population of electrons and/or negative ions. In the present case only an additional population of positive
ions was added in the input file. To enable the future experimental comparison, the hydrogen (helium) and argon
ion populations were used. The plasma system was only L = 10 cm long in order to complete the simulations in a
reasonable time span. The chosen system length was typically equivalent to 500 λD. The simulation was run in a
similar run as in our previous studies of simple plasma with one ion component. Initially the simulation region is
empty. Positive ions of both type and electrons are then injected with equal fluxes, j+Ar0+j+H0 = je0. Electrons that
return to the source are refluxed in the system with a velocity characteristic of the source temperature. There is no
charge accumulation at the source, the electric field is zero. When the number of particles in the system becomes
constant, then the simulation run is stopped. The potential, particle density profiles and velocity profiles are
examined and displayed. In Fig. 3 (left) the normalized potential profile Ψ(x/L) is plotted for H – Ar plasma with
argon ion density ratio α = 0.80 and positive ion temperature ratio τ = 0.1. The potential drops from zero value in the
plasma source through the source potential drop to ΨP = -0.85 and remains constant up to the collector sheath edge.
It further decreases steeply through the sheath to the value of ΨC = -4.4, the collector floating potential. No
additional potential structures are observed in the system. Both values of the potentials ΨP and ΨC are in good
accordance with analytically calculated results presented in Fig. 1. In the region where the potential is constant,
Ψ(x/L) = ΨP, the electric field is zero which indicates that the plasma is quasineutral. In the right plot of Fig. 3 the
particle density profiles are shown for this case. One can really observe that the plasma is quasinetral in the central
region and that the sheath is ion reach. From the ion density values read at the source plane (x/L = 0) the ion source
density ratio α is obtained. In Fig. 4 the positive ion velocity profiles are shown. The velocities are normalized with
the corresponding ion acoustic velocity cH(Ar) = (kTe/MH(Ar))1/2. The half-Maxwellian ion populations are first
accelerated in the source sheath potential drop to vx/cH(Ar) = 1.3. With this “Bohm” velocity they enter into the
collector sheath where they are further accelerated to vx/cH(Ar) = 3.0 with which they hit the collector. We can also
notice the apparent “cooling” of ions during the acceleration. Both profiles are identical, which means that each
species of ions satisfy their own “Bohm” criterion at the collector sheath boundary. They enter it with the same
velocity as if they were alone.

We have investigated, theoretically and by PIC simulation, the potential formation in a collision-less plasma with
two warm positive ion species and bounded by a floating collector. The presheath and the floating potential were
calculated as functions of positive ion density ratio and ion temperature. The presheath potential proved to be
independent of the density ratio (ion mass), suggesting that each ion population satisfy its own “Bohm” criterion at
the presheath/sheath boundary. The simulation experiment confirmed this result. No special potential structures were
identified in the system and no additional acceleration of heavier ion species was found. The dependence of floating
potential on ion density ratio could be used for diagnostic purposes.
1. N. Hershkowitz, Phys. Plasmas 12, 055502/1-11 (2005).
2. R. N. Franklin, J. Phys. D: Appl. Phys. 36, 34-38 (2003).
3. L. A. Schwager and C. K. Birdsall, Phys. Fluids B 2, 1057 (1990).
4. M. Čerček, T. Gyergyek and M. Stanojević, Contrib. Plasma Phys. 39, 541 (1999).
5. M. Čerček, T. Gyergyek, “Double layer formation in a negative ion plasma with a bi-Maxwellian electron distribution” in
Europhysics conference abstracts : vol. 29C[S. l.]:, 32nd EPS Plasma Physics Conference, Tarragona, Spain, 27 June-1 July,

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