Self Consistently Generated Charge Cylinder in BETA Device

doi:10.4236/jmp.2012.330208

Paper Menu >>

Journal Menu >>

Journal of Modern Physics, 2012, 3, 1697-1702

http://dx.doi.org/10.4236/jmp.2012.330208 Published Online October 2012 (http://www.SciRP.org/journal/jmp)

Self Consistently Generated Charge Cylinder in

BETA Device

Rajwinder Kaur, A. Sarada Sree, Shiban Kishan Mattoo

Institute for Plasma Research, Gandhinagar, India

Email: [email protected]

Received August 22, 2012; revised September 20, 2012; accepted September 27, 2012

ABSTRACT

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 10−6 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 10−4 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-

R. KAUR ET AL.

1698

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 m3. 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

Cylinder

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  10−9 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 m−3 implies that the charge density in the charge

layer is at least four orders of magnitude greater than

R. KAUR ET AL. 1699

Figure 2. Poloidal maps of (a) electron density; (b) floating

potential; (c) plasma potential and (d) electron tempera-

ture.

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

layer.

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

and



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

volume.

Figure 4. Schematic of the toroidal tube of convection cell.

R. KAUR ET AL.

1700

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 m−1 k



 200 m−1. The value of poloidal

wavevector kθ for the dominant mode is ~30 m−1. 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-

ment.

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

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-

ously.





 (1)







 (2)

where



eˆ

eDy























(3)

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

eDis

neV

aB BR



 (4)

where a is the minor radius of plasma torus





(5)

a



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



(6)

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

R. KAUR ET AL.

1702

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.

REFERENCES

[1] S. Roy, B. P. Pandey, J. Poggie and D. V. Gaitonde,

“Modeling Low Pressure Collisional Plasma Sheath with

Space-Charge Effect,” Physics of Plasmas, Vol. 10, No. 6,

2003, pp. 2578-2585. doi:10.1063/1.1572491

[2] R. Iiyoshi, “Numerical Study on Emission Characteristics

of a Point Cathode Electron Gun: Determination of Space

Charge using Random Emitting Conditions,” e-Journal of

Surface Science and Nanotechnology, Vol. 4, 2006, pp.

339-344. doi:10.1380/ejssnt.2006.339

[3] R. Iiyoshi, “Numerical Analysis of Space Charge in a

Point Cathode Thermionic Emission Gun,” e-Journal of

Surface Science and Nanotechnology, Vol. 1, 2003, pp.

147-151. doi:10.1380/ejssnt.2003.147

[4] M. S. Benilov, “Understanding and Modelling Plasma-

Electrode Interaction in High-Pressure Arc Discharges: A

Review,” Journal of Physics D: Applied Physics, Vol. 41,

No. 14, 2008, Article ID: 144001.

doi:10.1088/0022-3727/41/14/144001

[5] M. S. Benilov and A. Marotta, “A Model of the Cathode

Region of Atmospheric Pressure Arcs,” Journal of Phys-

ics D: Applied Physics, Vol. 28, No. 9, 1995, pp. 1869-

1882. doi:10.1088/0022-3727/28/9/015

[6] J. R. Ahn and C. J. Park, “Computer Simulation of Elec-

tron and Ion Trajectories in Electron-Impact Ion Sources

of a Quadrupole Mass Spectrometer,” Nuclear Instru-

ments & Methods in Physics Research Section A: Accel-

erators Spectrometers Detectors and Associated Equip-

ment, Vol. 645, No. 1, 2011, pp. 345-349.

[7] M. R. B. Ghadikolaee, B. Fateh and E. T. Ghadikolaee,

“Design and Study of an Enhanced Filament Ion Source,”

Vacuum, Vol. 86, No. 1, 2011, pp. 44-47.

doi:10.1016/j.vacuum.2011.03.028

[8] H. Somacal, H. Huck, D. E. DiGregario, J. O. F. Niello

and M. Igarzabal, “Simulations of Electron Trajectories

under the Influence of an Array of Permanent Magnets in

a Compact Ion Source,” Nuclear Instruments & Methods

in Physics Research Section A: Accelerators Spectrome-

ters Detectors and Associated Equipment, Vol. 490, No. 1,

2002, pp. 9-15.

[9] R. Kaur and S. K. Mattoo, “Inhomogeneous Electron

Emission from a Hot Filament in a Toroidal Magnetic

Field,” Plasma Sources Science and Technology, Vol. 18,

No. 1, 2009, Article ID: 015993.

[10] R. Kaur, A. S. Sree, A. K. Singh and S. K. Mattoo, “Life

Cycle of Density Structures in a Simple Magnetized To-

rus,” New Journal of Physics, Vol. 12, 2010, pp. 103013-

103018. doi:10.1088/1367-2630/12/10/103013

[11] R. Kaur, A. K. Singh, R. Singh, A. S. Sree and S. K. Mat-

too, “Territorial Characteristics of Low Frequency Elec-

trostatic Fluctuations in a Simple Magnetized Torus,”

Physics of Plasmas, Vol. 18, No. 1, 2011, pp. 12109-

12111. doi:10.1063/1.3551514

[12] R. Singh, S. Mahajan and K. Avinash, “Equilibrium and

Fluctuations in a Plasma Confined in a Pure Toroidal

Field,” Physical Review Letters, Vol. 77, No. 8, 1996, pp.

1504-1507. doi:10.1103/PhysRevLett.77.1504

[13] S. Mahajan, R. Singh and K. Avinash, “Theory of Plasma

Confinement in Devices with Pure Toroidal Field,” Phys-

ics of Plasmas, Vol. 4, No. 7, 1997, pp. 2612-2618.

doi:10.1063/1.872349

[14] K. Rypdal, E. Gronvoll, F. J. Oynes, A. Fredriksen, R. J.

Armstrong, J. Trulsen and H. L. Pecseli, “Confinement

and Turbulent Transport in a Plasma Torus with no Rota-

tional Transform,” Plasma Physics and Controlled Fu-

sion, Vol. 36, No. 7, 1994, pp. 1099-1114.

doi:10.1088/0741-3335/36/7/002

[15] R. Van Nieuwenhove and G. Van Oost, “Charge Neutra-

lization by Poloidal Plasma Rotation,” Plasma Physics

and Controlled Fusion, Vol. 34, No. 5, 1992, pp. 873-880.

doi:10.1088/0741-3335/34/5/015

[16] E. Wagner, et al., “Regime of Improved Confinement and

High Beta in Neutral-Beam-Heated Divertor Discharges

of the ASDEX Tokamak,” Physical Review Letters, Vol.

49, No. 19, 1982, pp. 1408-1412.

doi:10.1103/PhysRevLett.49.1408

[17] E. J. Strait, et al., “Enhanced Confinement and Stability

in DIII-D Discharges with Reversed Magnetic Shear,”

Physical Review Letters, Vol. 75, No. 24, 1995, pp. 4421-

4424. doi:10.1103/PhysRevLett.75.4421

[18] K. K. Jain, “Observation of Improved Behavior by Elec-

trode Biasing of a Toroidal Plasma Having no Poloidal

Magnetic Field,” Physical Review Letters, Vol. 70, No. 6,

1993, pp. 806-809. doi:10.1103/PhysRevLett.70.806

[19] K. K. Jain, “Poloidal Plasma Rotation and Its Effect on

Fluctuations in a Toroidal Plasma,” Nuclear Fusion, Vol.

36, No. 12, 1996, pp. 1661-1669.

doi:10.1088/0029-5515/36/12/I06