Journal of Modern Physics, 2012, 3, 1697-1702 Published Online October 2012 (
Self Consistently Generated Charge Cylinder in
BETA Device
Rajwinder Kaur, A. Sarada Sree, Shiban Kishan Mattoo
Institute for Plasma Research, Gandhinagar, India
Received August 22, 2012; revised September 20, 2012; accepted September 27, 2012
This paper presents a study on near cathode space charge region in BETA (Basic Experiments in Toroidal Assembly), a
toroidal plasma device with purely toroidal magnetic field. A charge cylinder has been found to be embedded in the
plasma center corresponding to the hot filament cathode location in poloidal cross section. This charge cylinder has
been created by the primary electrons emitted from the filament surface, which in turn, leads to the formation of a po-
tential well in the core plasma. We have proposed a model, which shows that a tiny fraction of injected energetic elec-
trons is sufficient to sustain the observed potential well. We have examined the equilibrium of the charge cylinder in
poloidal cross-section and found that it exhibits equilibrium configuration by forming circulation pattern of primary
electrons. The circulation pattern is formed by vertical drift due to toroidal magnetic field and self-consistent poloidal E
B drift. We have concluded that the self-consistency is in adjusting the poloidal drift to the vertical drift of the trapped
primary electrons.
Keywords: Hot Cathode Discharge; Near Cathode Space Charge; Toroidal Magnetic Field
1. Introduction
The near cathode space-charge behavior is of great inter-
est in plasma discharges [1-5]. These studies have at-
tracted lot of interest due to their applications in plasma
surface processing. Analytic and numerical models have
been developed [2-5] to gain the understanding of the
physical processes occurring in the cathode region of dis-
charges. The knowledge of primary electron trajectories
in this region helps in understanding and predicting vari-
ous discharge characteristics. Three dimensional (3D)
computer simulation techniques have been developed for
tracking the primary electrons and applied to numerous
applications [6-9]. The influence of different magnetic
fields on plasma discharge has been studied by tracing
the primary electron trajectories near the filament.
This paper presents an experimental study on near ca-
thode space charge region in a hot filament cathode dis-
charge produced toroidal plasma device BETA. The plas-
ma in BETA device [9-11] is produced by a hot filament
cathode discharge. The hot filament cathode injects ener-
getic primary electrons into the plasma. However, the
observed value of the charge density is much less than
the injected charge density. An anomalous radial cross-
field current transports most of the injected charge [12-
15]. Only a small amount of residual space charge still
exists that is responsible for the creating a charge cylin-
der in plasma center with axis coinciding with the major
axis of the torus. This charge cylinder in turn, sustains
the observed potential well. We have examined the equi-
librium of this charge cylinder and proposed a model to
account for the behavior of trapped primary electrons.
The paper is organized as follows. Section 2 gives an
outline of experimental setup and experimental results
are given in Section 3. Section 4 gives a discussion on
the existence and equilibrium of the charge cylinder. Fi-
nally conclusions drawn from the experimental results
and discussion are given in Section 5.
2. Experimental Setup
The experimental apparatus BETA device (Figure 1)
consists of a toroidal vacuum chamber of major radius 45
cm and minor radius 15 cm pumped down to base pres-
sure of 106 Torr [9-11]. A toroidal magnetic field is pro-
duced by a set of 16 coils and can be varied up to 1 kG.
The plasma is produced by a discharge between a cath-
ode of hot filament and the vacuum vessel at a pressure
of 104 Torr. The working gas is argon. A circular aper-
ture of diameter 18 cm defines the size of the plasma.
The plasma lacks equilibrium in the MHD sense. How-
ever, a steady state plasma can be maintained by the bal-
ance of charge production and their loss. Time averaged
plasma parameters are measured using a radially mov-
opyright © 2012 SciRes. JMP
Figure 1. Schematic of BETA device.
able vertical array of 10 Langmuir probes (l = 5 mm,
1 mm). This array measures the plasma parameters at dif-
ferent vertical locations in one-half (one side of the mid
plane) of the plasma cross section at a fixed radial loca-
tion. The array is, then, rotated for measurements in the
other half of the plasma cross section at the same radial
location. This enables the measurements from Y = 9 to
9 cm at a fixed radial location. The array is moved ra-
dially to create a grid in XY-plane.
3. Experimental Results
The poloidal maps of various plasma parameters meas-
ured in the BETA device are shown in Figure 2. The
typical plasma density is ~1016 m3. The floating potential
maps show the presence of potential well at hot filament
cathode location along the connecting toroidal magnetic
field lines. The equipotential contours are vertically
elongated near the filament. The vertical extent of the
contours (with 1/eth value of the peak) is approximately
of the length of filament i.e., ~14 cm and radial extent ~3
cm. The contours show a net vertically downward shift
~2 cm with respect to the filament. This direction corre-
sponds to the drift direction of primary electrons due to
gradient and curvature of toroidal magnetic field. How-
ever, the contours generally close within the limiter and
do not extend indefinitely in primary electron drift direc-
tion. Beyond the 1/10th value of the peak, the contours
widen in major radius direction and the potential varies
smoothly. The radially inward electric field estimated
from plasma potential contours is ~500 V/m near the fila-
ment. The vertical electric field ~1000 V/m exists only at
the filament ends and is directed towards the filament
ends. As we move away from the potential well, the
shapes of the plasma potential contours change from ver-
tical ellipses to nearly circular ones. The electric field
away from the potential well is directed radially inwards
with magnitude of ~40 V/m. The typical electron tem-
perature in BETA device is ~4 - 10 eV.
4. Discussion
The energetic primary electrons emitted from the hot
filament cathode circulate along toroidal magnetic field
in both the directions, thereby, constitute zero net current.
As mentioned in the preceding section that the floating
potential contours indicates that charge injected by the
hot filament spreads as a 3 cm thick cylindrical charge
layer of 14 cm length, located at the major radius of the
midpoint of the filament, shown schematically in Figure
3. It forms a potential well located at the major radius of
the midpoint of the filament. We call it charge cylinder
and investigate its properties in the succeeding sections.
4.1. Observed Charge Density of Charge
The charge density integrated over the charge layer will
appear as a surface charge density
0 E
. For the
measured electric field value of E ~
500 V/m, the es-
timated surface charge density is
~ 4.43 109 Cou-
lombs/m2. The primary electron density, thus calculated,
varies from ~1.4 1013 electrons/m3 for 2 mm thick (i.e.
filament diameter) charge cylinder to ~1012 electrons/m3
for 3 cm width of the observed nearly rectangular poten-
tial contours. Comparing this value of charge density
with the statistical charge fluctuation in plasma which is
~2 108 m3 implies that the charge density in the charge
layer is at least four orders of magnitude greater than
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R. KAUR ET AL. 1699
Figure 2. Poloidal maps of (a) electron density; (b) floating
potential; (c) plasma potential and (d) electron tempera-
local charge fluctuations. This establishes that charge cy-
linder is significant and should play a crucial role in plas-
ma dynamics of BETA device.
4.2. Injected Charge Density
We have estimated the injected charge by two methods,
viz., by calculating the injected current due to primary
electrons and from the arc current between the filament
cathode and the anode of vessel wall.
4.2.1. Injected Primary Current
The electrons injected by the hot filament cathode form a
shadow-like cylindrical layer along the connecting tor-
oidal magnetic field lines. This cylindrical layer has the
radius of 0.45 m, height of 0.14 m and thickness of 2 mm
corresponding to the location and size of the filament.
The thickness of the cylindrical layer shadow expands to
~3 cm (corresponding to several electron gyroradii) as
shown in Figure 4, thereby increasing the volume of the
The current carried by the primary electrons is Ipe =
npeevA, where npe is the density of primary ionizing elec-
trons, e is the electron charge of 1.6 10–19 C, v = 4.7
106 m/s is the velocity of 120 eV electrons and A is the
area of electron emission. The current from the discharge
power supply is injected into the plasma through emis-
sion of primary electrons by the hot cathode and their
acceleration in the plasma sheath around the filament [9].
Most of the applied discharge voltage drops across the
sheath around the filament. As discussed later, these
primary electrons are not carried all the way from the
source to vessel wall or limiter of BETA, which acts as
the anode. The electrons collected by the anode are the
lower temperature plasma electrons. Nevertheless, the in-
jected current has to have the same magnitude as the an-
ode current. The current due to plasma ions is negligible
[9] and is not taken into account for calculations here.
The discharge volume in BETA device is Voldis ~ 0.2 m3
mfp, the mean free path of ionizing primary electrons
~100 m. For Ipe equal to discharge current of 5A and us-
ing these values for Voldis and
mfp we get the primary
electron density npe ~ 3.3 1015 electrons/m3. This value
is at least two orders of magnitude greater than the elec-
tron density in the charge cylinder (1012 - 1013 electrons/
m3) calculated from the measured electric field E.
Figure 3. Schematic of the charge layer inside the plasma
Figure 4. Schematic of the toroidal tube of convection cell.
Copyright © 2012 SciRes. JMP
4.2.2. Arc Current
One can also estimate the number density of electrons
participating in discharge current by another method. The
primary electrons (injected by hot cathode) of energy
~120 eV are trapped by toroidal magnetic field although
they drift due to curvature and grad B forces. It has been
observed that the primary energetic electrons are not
found beyond 2 cm in radial direction on the inboard and
outboard sides of filament. They are also not found near
the bottom side of the vessel due to downward drift. This
implies that primary electrons do not reach the vessel
wall of BETA. Hence, it is a good assumption to con-
sider that the injected primary electrons are not directly
participating in the discharge current. The functional role
of primary electrons is to create ionization resulting in
the formation of plasma and to excite neutral atoms of Ar.
They spend the major portion of their energy through the
processes of excitation of radiation levels in atoms.
In BETA, it is not possible to compensate the radial
current from charge cylinder to the vessel wall or limiter
due to classical diffusion [12-15]. On the other hand,
electron flow parallel to the toroidal magnetic field form
counter streams due to two sided injection of energetic
electrons on the field lines that intercept filament and
therefore do not contribute to the current. The single par-
ticle drifts (due to gradient and curvature of toroidal
magnetic field and E B drift) are insufficient in trans-
porting the charge from filament region to the vessel wall.
Therefore the cross-field charge transport is carried out
by an anomalous mechanism initiated by the turbulence
in the plasm [12-15].
Once plasma is created, the plasma electrons carry the
discharge current across the toroidal magnetic field, en-
abled by anomalous perpendicular diffusion. The radial
velocity of these electrons vr is ~kθ φ/B, where kθ is the
poloidal wavevector of the dominant plasma fluctuations
mode and φ is amplitude of its electrostatic potential. For
fluctuations in BETA, the range of poloidal wavevectors
[11] is 100 m1 k
200 m1. The value of poloidal
wavevector kθ for the dominant mode is ~30 m1. For the
work reported in this paper, B = 0.02 T and φ ~ 1 V, re-
sulting in radial velocity vr of ~9.4 103 m/s. We note
here that vr is much less that the velocity of the injected
electrons at the plasmasheath boundary around the fila-
For Idis = npeevrAw = 5 A and area of the vessel wall Aw
= 2.7 m2, the calculated value of npe ~ 4.8 1015 elec-
trons/m3. This value of primary electrons density npe is of
the order of value estimated in preceding section by the
method of using the injected current is due to primary
electrons. Further, npe is just about 10% of the peak
plasma density implying that only tail energy distribution
of plasma electrons is sufficient to enable discharge cur-
rent between the hot filament cathode of tungsten and the
anode of vessel wall.
4.3. Resolving Discrepancy between Measured
and Injected Charge Density
The discrepancy between the observed primary electron
density (estimated in Section 4.1) and the calculated in-
jected electron density (estimated in Sections 4.2.1 &
4.2.2) in the charge cylinder implies that only a fraction
of injected energetic electrons is retained as space charge
of the charge cylinder. In fact, electron density in the
charge cylinder is two orders of magnitude smaller than
the density of injected electrons. The argument that most
of the injected electrons, about 99%, are lost due to the
vertical drift is not supported by the observations. It has
been observed experimentally that the negative potential
well has a limited spatial extent in vertical direction and
is receded from the vessel wall. The loss path of primary
electrons to the vessel wall, along the vertically down-
ward direction due to drifts, is not observed in the ex-
periments. This implies that the charge cylinder, made up
of primary electrons, does exist. Therefore, if there is a
massive loss of injected electrons from the charge cylin-
der, it has to be by alternative processes, other than the
drifts introduced by toroidal magnetic field. A reasonable
explanation is that significant charge neutralization takes
place as a consequence of inward ion diffusion towards
the charge cylinder. This leads to significant reduction of
electric field in the charge, i.e., E to 500 V/m as against
the value E ~ 104 V/m for the situation when electric
field is sustained by the injected charge density (esti-
mated in Sections 4.2.1 & 4.2.2).
4.4. Equilibrium of Charge Cylinder
We now consider the dynamics of the observed charge
cylinder that is independent of interaction of primary
electrons with the surrounding plasma and all pervading
neutral atoms. To a large extent, charge neutralization
plays a considerable role in reducing the non-neutral
component of plasma and consequently the magnitude of
electric field inside the charge cylinder. The interaction
with neutrals is through ionization and radiation. How-
ever, these interactions do not explain the existence of
structure, which is isolated from the vessel wall and sus-
pended in the plasma. To achieve isolation from the ves-
sel wall, it is essential that the vertical drift of primary
electrons is either cancelled or is converted into a pol-
oidal drift at the extreme ends of the charge cylinder.
Alternatively, a sink for primary electrons has to be
placed at the bottom end of the cylinder. A sink is con-
ceivable in terms of the filament support holders, which
form a shadow region for the primary electrons. Against
this hypothesis is the bias condition of these holders that
are negatively biased to the same voltage as discharge
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R. KAUR ET AL. 1701
voltage. So there cannot be any direct interception loss
unless a mechanism of energization of primary electrons
to energy greater than discharge voltage is invoked. The
energization of only a minuscule number of primary
electrons is possible by any mechanism. This is because
only a small fraction of primary electrons can gain en-
ergy at the expense of energy residing in the remaining
part of primary electrons through a multiple processes of
exciting electrostatic potential waves in the charge cyl-
inder. So such an explanation would not be suited to the
bulk of primary electrons.
It seems then that explanation has to be in terms of that
there are two drifts in the circulation pattern for primary
ionizing electrons. In the purely toroidal magnetic field,
these electrons exhibit a certain vertical drift due to gra-
dient and curvature of the toroidal magnetic field. This
drift forms the vertical leg of the circulation loop. The
primary electrons also suffer E B-drift along the equi-
potential contours. This E B-drift enables the closing
path of the circulation pattern. The combined effect of
these two drifts is that E B flow of the accumulated
space charge will balance the charge accumulation due to
the vertical drift of primary electrons as shown in Figure
Following the preceding argument, we propose that
there exists a stationary charge density distribution in the
charge cylinder, satisfying the charge continuity equation.
The stationary charge density distribution requires that
the divergence term in continuity equation vanishes ·J
= 0. To obtain a self-consistent solution, the continuity
equation and Poisson’s equation are solved simultane-
 (2)
J is the current density,
is the charge density, npe is the
number density of the primary ionizing electrons in the
observed stationary charge cylinder, vD is the drift of
monoenergetic electrons,
is the space charge potential,
0 is the permittivity of free space and ê
and êy are unit
vectors in toroidal and poloidal directions. First term in
the Equation (3) is the current due to E B drift of the
primary ionizing electrons and second term is their ver-
tical current. The magnitude of the drift for the mono-
energetic electrons leaving the filament is vD = 2Vdis/BR,
where Vdis is the discharge voltage, and R = 0.45 m is the
radius of curvature of toroidal field. In order that ·J = 0
for primary electrons
where a is the minor radius of plasma torus
Combining Equations (4) and (5), we estimate the re-
quired electric field for this stationary charge distribution
constituting the charge cylinder viz.
pe Dis
neV a
For npe ~ 1012 electrons/m3, VDis = 120 V, a = 0.9 m
and R = 0.45 m, the electric field required to set primary
ionizing electrons circulating in the E B direction is
~300 V/m. As described earlier, the electric field esti-
mated from the potential contour maps around the charge
cylinder is ~500 V/m. Thus, these calculations indicate
that the self electric field set up by the charge cylinder
enables a sufficient E B drift for primary electrons to
generate a charge cylinder in equilibrium. For VDis = 120
V, B = 0.02 T and R = 0.45 m, the vertical drift vD ~ 2.67
104 m/s. On the other hand, for E = 500 V/m and B =
0.02 T, the E B drift vE ~ 2.5 104 m/s. This drift is
directed along the equipotential surface. It may be noted
that vD and vE are of the same order and their ratio is in-
dependent of the toroidal magnetic field.
Putting the vertical drift due to toroidal magnetic field
and E B drift of the primary electrons together, a broad
picture that emerges is that the charge cylinder consists
of a toroidal tube of convection cell, as shown in Figure
4, with cylinder as the vertical axis. Since the potential
well depth is well below the discharge voltage (i.e., 120
V), the electrons in the convection cell are those, which
have lost their energy through ionization and radiation
processes by electron-neutral collisions.
5. Summary
The plasma in BETA device is produced by a hot fila-
ment cathode discharge. The hot filament cathode injects
charge into the plasma. Therefore the plasma embeds a
charge cylinder of ~3 cm thickness and ~14 cm height
and located ~2 cm towards corresponding to the filament
location in the poloidal cross-section. The energetic pri-
mary ionizing electrons injected by the hot filament
cathode create this charge cylinder. The comparison of
the injected and observed charge density indicates dis-
crepancy between them. The observed value of charge
density shows that only a small amount of residual space
charge exists that is responsible for the creation of a
charge cylinder. This electric field functions much in the
Copyright © 2012 SciRes. JMP
Copyright © 2012 SciRes. JMP
similar fashion as in biased electrode experiments of to-
kamaks [16,17] and biased ring experiment in BETA
[18,19]. The major difference between the excited elec-
tric field by the filament, aiding in the plasma equilib-
rium in BETA, and enhanced confinement mode experi-
ments on tokamak and BETA is that, in the latter electric
fields are excited from the boundary while in the present
case it is excited from the core of plasma. Examining the
equilibrium configuration of the charge cylinder indi-
cates that it is made up of the self-consistent circulation
pattern formed by the vertically drifting and poloidally
flowing primary electrons. This self-consistency is in ad-
justing the poloidal drift to the vertical toroidal drift of
the trapped primary electrons.
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