Multiple floating potentials of an electron emitting electrode immersed in a two-electron temperature plasma with critical electron emission
T. Gyergyek¤ and M. Cˇ ercˇek†
¤University of Ljubljana, Faculty of electrical engineering, Tržaška 25, 1000 Ljubljana, Slovenia,
† Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia,
University of Maribor, Faculty of civil engineering, Smetanova 17, 2000 Maribor, Slovenia
Abstract. A current voltage characteristics of a negatively biased electron emitting electrode immersed in a two-electron
temperature plasma is analyzed by a simple one dimensional fluid model. Based on the assumption that the electron density
in the pre-sheath region obeys the Boltzmann law the Bohm criterion is derived in the form of a transcendental equation for
the Mach number, which can have up to 3 solutions. According to these solutions the ion velocity at the sheath edge can
be determined either by the hot or by the cool electron temperature. When it is determined by the cool electron temperature
and the hot electron temperature is high enough the critical electron emission current from the collector becomes a very
nonmonotonous function of the electrode potential. Because of that the current voltage characteristics of the electrode may
exhibit up to 3 different floating potentials.
Keywords: Plasma, sheath, pre-sheath, Bohm criterion, electron emission, emissive probe, two-electron temperature
PACS: 52.40.Kh, 52.40.Hf, 52.27.Cm
Recently the problem of the space charge limited emission current from a negative electrode has been addressed
Takamura and coworkers [1, 2]. They developed a one-dimensional fluid model, where they treated an infinitely large,
planar, negative electrode (collector) that is immersed in plasma which contains a group of maxwellian electrons
and a group of cold singly charged positive ions. They assumed that the electrode emits monoenergetic electrons
with zero initial velocity and that the flux of emitted electrons is proportional to the flux of incoming electrons. The
proportionality factor is assumed to be the constant emission coefficient.
In a recent paper  we have extended their model by the assumption that the hot electron population is present in
the plasma, but the repulsion of the electrons in the pre-sheath potential drop was not taken into account. The floating
and the current collecting electrode were analyzed. In the next paper  the repulsion of the electrons in the pre-sheath
potential drop was taken into account and the modification of the Bohm criterion in the form of a transcendental
equation for the Mach number was derived. Only the floating electrode was analyzed. In this paper we present some
results for the current collecting electrode with space charge limited electron emission and in the same time taking
into account the repulsion of electrons in the pre-sheath potential drop.
We consider an infinite plane material surface (collector), located at x = 0 in contact with a plasma filling the half-space
x > 0. Far from the collector the plasma is neutral and the plasma potential there is taken as a reference potential, which
is set to zero, F = 0. Also the electric field there is zero. The collector is biased to a certain potential FC, which is
negative with respect to the plasma potential. As one approaches the collector, the plasma potential slowly decreases.
This region of the slow potential drop is called the pre-sheath. There the plasma is assumed to be still neutral, although
a small electric field exists in this region. This electric field accelerates positive ions towards the collector and negative
electrons in the opposite direction. The singly charged positive ions, which are assumed to be cold and at rest at a very
large distance from the collector, are accelerated towards the collector. At a certain distance x = d from the collector,
the plasma potential has a value F0 and there the ions reach the velocity v0 in the direction towards the collector. The
plane at x = d is called the sheath edge. In our model we assume that the sheath edge is a well defined plane parallel
to the electrode. On one side of the sheath edge the plasma is still neutral but immediately beyond this plane (in the
sheath) a strong electric field and an excess of positive space charge exist. In the sheath the potential drops rapidly
from the value F0 at the sheath edge to the value of the collector bias FC.
We assume that there are two electron populations in the plasma, which have two different temperatures, T1 is the
lower temperature and T2 is the higher temperature. We call them the cool and the hot electrons. We also assume that
the collector emits secondary electrons. These electrons are assumed to leave the collector all with the same initial
velocity vC and then they are accelerated by the potential in the sheath away from the collector.
Because the model has already been derived in detail elsewhere [3, 4], we only write down the key equations of the
model. These are the condition of zero electric field at the collector:
to the ion sound velocity
kT1=mi multiplied by the Mach number M, similar is done with the initial velocity of the
emitted electrons vC. The emission coefficient g gives the number of emitted electrons per 1 incident electron and b0
is the ratio between the hot and the cool electron density at a very large distance from the collector.
If the parameters b0, Q, m, g and N are selected, then the current-voltage characteristics of the collector (this means Jt
as a function of YS) can be calculated from equations (2) and (3). The normalized sheath potential drop YS is varied
as an independent variable. For every YS the Mach number M must be calculated from (3) and then Jt is found from
(2). If on the other hand one assumes that the emission of electrons from the collector is critical, i. e. the electric
field at the collector is zero, then g is not selected. Instead g and M must be found simultaneously by solving the
system of equations (1) and (3) for every set of selected parameters b0, Q, m, YC and N. As an example we show a
current-voltage characteristics for the following parameters: m = 1/1836, b0 = 0.15, N = 60 and Q = 50.
FIGURE 2. In the top figures critical emission current Jem to the collector is plotted versus YS for various values of b0, Q and N.
From the system of equations (1) and (3)the solution with M close to 1 is always taken. In the bottom figures solutions of equation
(3) are illustrated.
between of the former two. Jem(YS) has a minimum and maximum, only when the solution for M close to 1 is inserted
into (5) and Q and N have somewhat higher values, roughly above 50
1. M. Y. Ye, T. Shimada, T. Kuwabara, N. Ohno and S. Takamura, Proceedings of the 1998 ICCP&25th EPS Conference on
Controlled Fusion and Plasma Physics, Praha, 29 June – 3 July 1998, ECA vol. 22C, pp. 23, (1998).
2. M. Y. Ye and S. Takamura, Phys. Plasmas, 7, (2000), 3457
3. T. Gyergyek, M. Cˇ ercˇek, Contrib. Plasma Phys., 45, pp. 89–110, (2005).
4. T. Gyergyek, M. Cˇ ercˇek, Contrib. Plasma Phys., 45, pp. 568–581, (2005).
5. Cheoi-Hee Nam, N. Hershkowitz, M. H. Cho, T. Intrator, D. Diebold, J. Appl. Phys., 63, pp. 5674–5677, (1988).
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