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THE THEORY AND THE DIAGNOSIS OF THE
ELECTRODELESS DISCHARGE
By
DENNIS RALPH KEEFER
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
August, 1967
To my wife
CHARLOTTE
who, having little understanding for the subject,
has shown great understanding for the author.
ACKNOWLEDGEMENTS
The author is deeply indebted to many individuals for
aid and encouragement in the completion of this dissertation.
He wishes to express his gratitude to his fellow students
and co-workers for their aid and suggestions in the course
of the research work.
Special thanks are due to his supervisory committee
chairman, Dr. M. H. Clarkson, who has served with patience
and understanding as teacher, mentor, critic and employer
for these past five years. Without his initial encourage-
ment the project would have never begun. The author wishes
to express his appreciation to the members of his supervisory
committee, Dr. Knox Millsaps, Dr. Orlo E. Myers,
Dr. Thomas L. Bailey and Dr. Earle E. Muschlitz, Jr., for
their guidance and encouragement in the course of his
graduate program.
The author also wishes to thank Mrs. Jacqueline Ward
who graciously and ably prepared this manuscript.
The financial support for this project was provided by
the National Aeronautics and Space Administration under
Grant NsG-542.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS . ...
LIST OF FIGURES . . .
ABSTRACT . . .
CHAPTERS
II.
III.
IV.
V.
VI.
VII.
VIII.
INTRODUCTION .
PROBE MEASUREMENTS . .
TEMPERATURE DISTRIBUTION .
ELECTRON TEMPERATURE AND NUMBER DENSITY
DISTRIBUTION . ..
THE ELECTROMAGNETIC FIELD ..
POWER BALANCE . .
CALCULATIONS . .
CONCLUSIONS . .
REFERENCES . . .
BIOGRAPHICAL SKETCH ... .
iii
V
. vii
. 1
. 4
. 15
S. 29
S. 9. 38
. 49
o 56
S. 68
0 0
o~o
o
LIST OF FIGURES
Figure Page
1. Typical probe characteristic with third probe
trace. 73
2. Electron number density at a pressure of
0.13 Torr. 74
3. Electron number density at a pressure of
0.26 Torr. 75
4. Electron number density at a pressure of
0.40 Torr. 76
5. Comparison of measured number density with
the theory of Eckert. 77
6. Plasma potential. 78
7. Computed average momentum collision frequency. 79
8. Computed average ionization collision frequency. 80
9. Computed average excitation collision frequency. 81
10. Temperature dependence of 2)i/Da. 82
11. Electron temperature as a function of pressure
for a long cylindrical discharge of 2.4 cm
radius. 83
12. Electric and magnetic field for 2)Ca 0 84
13. Electric and magnetic field for )/W = 1 85
14. Electric and magnetic field for /Q = oo 86
15. Average power input at the wall. 87
16. Solution of the power balance equation for a
pressure of 0.5 Torr. 88
Figure
Page
17. Electron number density at the discharge axis
as a function of solenoid current at various
pressures. 89
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
THE THEORY AND THE DIAGNOSIS OF THE
ELECTRODELESS DISCHARGE
By
Dennis Ralph Keefer
August, 1967
Chairman: M. H. Clarkson
Major Department: Aerospace Engineering
An analysis of the asymmetric double probe is described
which permits the measurement of electron temperature, elec-
tron number density and plasma potential in an electrodeless
discharge. Experiments were performed which indicate that
the electron temperature and ionization function are essen-
tially uniform across the discharge radius. These results
are at variance with commonly made assumptions concerning the
discharge. An analysis of the energy transport in the dis-
charge predicts that, due to the thermal conduction of the
electron gas, the temperature is essentially uniform. The
fact that the electron temperature is uniform is used as a
basis for the formulation of a one-dimensional, steady-state
theory for the inductively driven electrodeless discharge.
The results of the theory permit the calculation of electron
temperature, electron number density, the electric and
magnetic fields in the discharge, and the power input to the
discharge as a function of the discharge configuration and
the applied coil current. An important result of the theory
is the prediction that electron temperature is independent
of the input power, and that the electron number density is
a direct function of the solenoid coil current.
vii
CHAPTER I.
INTRODUCTION
An electrodeless discharge is a gaseous electrical dis-
charge which is maintained by high-frequency electric fields.
The fields are applied to either conductive rings or plates
or to helical coils placed outside the discharge container.
These discharges can be maintained in a wide variety of
gases and over a pressure range extending from the region of
.001 Torr to greater than atmospheric.
The discharge was first discovered by Hittorf1 in 1884
and appears to have been widely known and studied in the
early part of the twentieth century. In 1927 and 1928 the
solenoid excited electrodeless discharge received the
attention of Sir James Thomson and J. S. Townsend and their
work resulted in a controversy concerning the nature of the
discharge. The controversy concerned the question of whether
the induced or the electrostatic fields of the solenoid were
responsible for the discharge. Thomson2 gave an analysis of
the induced fields, derived conditions for breakdown, and
reported experiments which indicated that the currents in the
gas were due to the induced fields. Townsend and Donaldson3
reported experiments which showed the breakdown to be caused
by the axial electric field due to the potential drop across
the solenoid. The controversy was resolved in 1929 by
K. A. MacKinnon4 who showed that actually both discharges
were possible, depending on the configuration, and that the
two types of discharge exhibited different visual character-
istics. In 1963, T. B. Reed5 reported two modes of operation
for a low-pressure electrodeless discharge and gave certain
spectrographic data for the two modes of operation which he
designated as low-power and as high-power as an indication of
power absorbed from the RF generator. Clarkson, Field and
Keefer6 showed that these two modes of operation were related
to the same phenomena explained by MacKinnon and gave measure-
ments of electron temperature in the two modes obtained by
use of floating double probes.
Although the electrodeless discharge has been widely
used as a source of laboratory plasma, it has not been widely
studied. A revival of interest occurred in the 1950's as a
result of the need for high-energy gas sources of low con-
tamination for use in wind tunnels designed for testing
re-entering vehicles launched by rockets. The discharge was
not initially successful in these applications and they were
largely discarded in favor of high-energy DC arc jets.
Analytical studies of the electrodeless discharge began
with Thomson who calculated the fields in a cylindrical dis-
charge in a solenoidal field, assuming a uniform conductivity
across the radius. From these calculations, he derived a
breakdown criterion based on the energy gain of an electron
in one mean free path. A more realistic analysis was given
by H. U. Eckert,7 who recognized that the conductivity would
not be uniform across the radius, and that the discharge would
be diffusion controlled. However, Eckert obtained the same
solution for the fields as Thomson since he assumed an
average uniform conductivity in the formulation of Maxwell's
equations. By solving the plasma balance equation, Eckert
was able to give a more realistic breakdown criterion. By
making certain assumptions concerning the spatial variation
of the ionization function and electron temperature, Eckert
was able to calculate the steady-state electric field at the
boundary of the discharge and the spatial variation of the
electron number density.
Development of a highly asymmetric floating double probe
suitable for use in electrodeless discharges by Keefer,
Clarkson and Mathews8 made it possible to measure the spatial
variation of electron number density and temperature within
the electrodeless discharge. The results of these measure-
ments were found to be at variance with the assumptions made
by Eckert. The most striking difference occurred in the
spatial variation of the electron temperature. While Eckert
assumed that the temperature increased exponentially from the
center of the discharge to the wall, the probe measurements
indicate that the electron temperature is essentially uniform.
In addition, the measured spatial variation of the electron
number density implies that the ionization function is nearly
uniform in contrast to the power law variation assumed by
Eckert.
A new theory for the electrodeless discharge has been
formulated as a result of the discrepancy found between the
probe measurements and Eckert's assumptions. The new theory
is in reasonable agreement with the probe measurements, and
provides a description of the plasma properties as a function
of the applied fields. Not all of the predictions of the
theory have been subjected to experimental verification, but
certain qualitative comparisons with observations made in the
course of the experiments indicate that the major physical
processes are described by the theory.
It should be pointed out that, while a relatively small
amount of attention has been given to the electrodeless dis-
charge per se in the past few decades, there has developed a
large body of plasma theory of a general nature which is
applicable to the problem. There has also been a great amount
of work performed on microwave discharges, much of which is
applicable to the electrodeless discharge. Of par-icular
value are the two Handbuch der Physik articles by Allis9 and
Brown.10
CHAPTER II.
PROBE MEASUREMENTS
The electrostatic probe, or Langmuir probe, is one of
the most widely used plasma diagnostic tools. Although its
use dates much earlier, a definitive theory for its behavior
in a plasma was given by I. Langmuir in 1924.11 The tech-
nique is basically simple. A small conductor is placed in
the plasma and a potential applied to it with respect to
some other electrode, usually one of the discharge electrodes.
A curve of current versus voltage is recorded and with the
Langmuir theory the plasma number density, electron tempera-
ture and potential may be determined from this curve. Despite
the fact that the theory is applicable to only a limited
range of plasma conditions, the use of the technique is
nearly universal. The technique, together with its limita-
tions, is adequately described by Loeb.12 It remains virtually
the only way to resolve spatially the properties of a plasma.
The application of this technique to the electrodeless
discharge involves two obvious difficulties. First, there is
no discharge electrode with which the probe may be biased and,
secondly, the plasma potential may vary at the driving
frequency. Johnson and Malter13 describe a system which over-
comes these difficulties. A floating double probe is placed
in the plasma and one probe is biased with respect to the
other. Their experiments utilized probes of equal area, but
it was proposed that probes of unequal area or an asymmetric
probe might be used to increase accuracy. In their analysis,
it is assumed that both probes are placed in a plasma of
uniform potential. As this may not be true in the electrode-
less discharge, it is necessary to extend their analysis to
include this condition.
Whenever a surface is placed in a plasma, a sheath forms
and the surface potential becomes negative with respect to
the local plasma potential. This condition occurs because the
electrons have a higher mobility than the ions and, therefore,
build up more rapidly on the surface. When the surface
potential becomes sufficiently negative, the lower energy
electrons are retarded in their motion toward the surface such
that an equal number of ions and electrons impinge on the
surface and a steady state is established. When a current is
drawn from the surface, the sheath is altered such that the
surface becomes more or less negative with respect to the
local plasma potential. The potential becomes more negative
if a positive current flows from the surface, and less negative
if negative current flows from the surface.
In the use of a floating double probe, two collecting
surfaces are placed in the plasma and a biasing potential
applied between them. The more positive of the probes will
collect a surplus of electrons and the more negative will
collect a surplus of ions such that the net current drawn from
the plasma is zero. As the bias potential is increased, the
current between the probes increases until saturation occurs.
Saturation will occur whenever the maximum possible number of
either the ions or electrons incident upon the sheath boundary
is collected. To insure greater accuracy in the measurement
of plasma properties, it is necessary to operate the probe at
electron saturation. This is the condition where all the
incident electrons are collected. The analysis will show that
for this condition to occur a very large difference in the
surface area of the two probe collectors is required.
The large difference in required probe area will usually
mean that the probes are placed in the plasma at points where
the local plasma potential is not equal. This potential
difference may be established by ambipolar diffusion or by an
external electric field. The quantities which refer to the
two probe surfaces will be designated by superscripts 1 and 2.
Thus, the plasma potential at probes 1 and 2 is V (1) and
(2)
Vp 2, respectively. The potential difference will be assumed
constant and designated Vr where,
(v ) v2
V, vP P 2.1
The flux of ions and electrons impinging on the probe sheath
will be designatedri and P respectively. It is assumed
that these fluxes are a function only of the plasma properties
and independent of the sheath potential. In the electron
retarding region, the current density of electrons reaching a
probe is given by
V.
,e = e 2.2
kTe
where VT = and e is the electron charge, k is Boltzmann's
constant and Te is the electron temperature. Thus the total
current to probe 1 is given by
VP
r)= eA k( l e ) 2.3
and the current to probe 2 is given by
-r V eA(-) ~ V)/V
= eAe- 2.4
where A is the surface area of the probes and V is the poten-
tial of probe 2 with respect to probe 1. The floating probe
system can draw no net current from the plasma and this
requires that
-) = = 2.5
By equating 2.3 to 2.4 and noting that
\ (1) = \(2)
V VVp V, 2.6
one obtains, after some manipulation,
S/0 e? /0 fr
VW AY r1) f ( A (j1) 2.7
This expression relates the local plasma potential at
probe 2 to the applied potential V. If the double probe
system is to behave like the classical Langmuir probe, it is
necessary that the variation of the local plasma potential at
probe 2 be small as the bias potential V is varied. This
requirement yields a criterion for the area ratio of the
probes, namely
A I)e e 1 2.8
A (1) Pe(J
In practice this requirement may be rather severe since the
larger probe (1) would likely be placed near the boundary of
the discharge wheree(l)( re(2). The required area ratio
A(2)/A(1) may be of the order 10-4.
It will be assumed now that the area ratio is sufficiently
small so that the variation of Vp(2) is negligible over the
useful range of bias potential V. The probe current in the
electron retarding region is given by
.) A(2)f~~)(VVp-'/ V)/V'r
z. -- eA ~ ] 2.9
and
r/7 /7 2.10
e/1e)]
[{- ve /
The ratio of ion flux to electron flux is much less than one
and it will be neglected in 2.10. Taking the derivative of
equation 2.10 with respect to V yields the expression
d(isoi) .1 2.11
dV
This is the usual expression for the Langmuir probe from
which the electron temperature may be determined from the
slope of the probe characteristic when it is plotted on semi-
log paper. The electron number density is determined from
the break in the characteristic curve caused by electron
saturation. At electron saturation, Vp(2) = V and 2.9 becomes
eASince = e2 ( 1, then
Since i(2) re(2) 1, then
Se reA 2 2.13
For a Maxwell-Boltzmann distribution of electrons,
e) Te 2.14
where n is the electron number density and m is the electron
mass. Since the electron temperature may be determined from
2.11, the number density may be determined from 2.13 and 2.14.
One additional property of the plasma may be determined
by the probe, namely, the plasma potential. At the break in
the characteristic curve at the onset of electron saturation,
the probe 2 has the same potential as the plasma relative to
probe 1. Thus, if probe 1 is held at a constant potential,
the plasma potential may be determined.
The above analysis shows that a highly asymmetric double
probe behaves like the classical Langmuir probe whenever the
criterion 2.8 is met. An experimental method to determine
whether or not this criterion is being met is described below.
Consider a third probe placed in the plasma at the same
location as probe 2. This probe will be designated probe 3
and the plasma potential and ion and electron flux at its
location will be the same as for probe 2, i.e.,
(3) (2)
VP Vp 2.15
and
(3) ( 2)
e,e ,i 2.16
The current relation for probe 3 is then given by
-v- V] 2.17
Now if no current is allowed to flow from probe 3, it will
assume a floating potential Vf, where
e = 2.18
re
This may be reduced to the expression
vp = YE.+ log 1 2.19
V, V, J
Thus the floating potential differs from the local plasma
potential by a constant amount. By observing the floating
potential of probe 3, it is possible to determine whether
the plasma potential Vp(2) changes as the applied potential V
is varied in accordance with 2.7. If the variation of Vf, and
therefore Vp(2), is negligible then the criteria 2.8 has been
satisfied.
A highly asymmetric double probe was used to measure the
electron temperature and number density in a low-pressure
electrodeless discharge. The discharge was formed in a length
of two-inch diameter Pyrex pipe connected to a vacuum system.
argon was admitted to maintain a prescribed pressure. The
tube was placed inside a structure containing a solenoid coil
connected to the RF power supply and had shielding to prevent
penetration of the axial electric field of the solenoid into
11
the discharge. The RF power supply was a converted television
transmitter operating at approximately 4.5 MHz with a maxi-
mum available power of 5 KW.
The probe system consisted of a large cylindrical elec-
trode placed at the wall inside the discharge tube, and two
small movable probes. The large electrode was constructed of
OFHC copper with a cooling coil soldered to the inside and
the entire assembly silver plated to reduce contamination.
The cylinder was split longitudinally to allow the axial
magnetic field of the solenoid to "penetrate" by acting as a
one-turn secondary. The small electrodes were formed from
0.01 inch diameter tungsten wire with approximately 0.125
inches exposed from a Pyrex sheath. This gave a geometric
area ratio of approximately 7 x 10-5. The probes and Pyrex
sheath were set into a brass tube with an offset to allow
rotation of the probes across the tube radius. Probe voltage
was supplied by a transformer secondary with the primary
connected through a variable autotransformer to the 60 Hz
laboratory power. The current signal was obtained across a
10 ohm shunt and applied to the vertical input of an oscillo-
scope while the probe voltage was connected to the horizontal
input. The resultant current-voltage trace was photographed
from the oscilloscope to provide a permanent trace. When the
third probe was used, it was connected directly to a second
vertical input.
A typical probe characteristic, together with the third
floating probe potential, is shown in Figure 1. The charac-
teristic is typical of a properly functioning Langmuir probe
with a sharp knee and saturation region. The third probe
potential is seen to vary slightly as the bias potential on
the asymmetric probe is varied. The indicated change in
plasma potential is seen to be of the order of 0.4 volt
showing that the change in plasma potential is negligible for
this probe system. Data were taken at pressures of 0.13 Torr,
0.26 Torr and 0.40 Torr in argon at several radial locations.
In all cases the electron temperature was found to be uniform
across the radius of the discharge. At 0.13 Torr the electron
temperature expressed in volts was 2.0 volts, at 0.26 Torr
it was 1.8 volts and at 0.4 Torr it was 1.6 volts. The elec-
tron number density at the three pressures is shown in
Figures 2, 3, and 4. All of these curves have the character-
istic that the number density is greatest at the axis of the
discharge and becomes smaller as the wall is approached. This
is typical of a diffusion controlled discharge.
H. U. Eckert7 has given an approximate analysis for the
electrodeless discharge. In this analysis, the electron pro-
duction term is assumed to obey a power law given by
i = hr 2.20
A?
where i is the average frequency of ionization per electron,
Da is the ambipolar diffusion coefficient, r is the radial
coordinate measured from the discharge axis and h and q are
assumed constant. The solution of the plasma balance equation
for this assumed form is
nD. AF\.4o(^] 70S 2.21
(n D4) 01 [ -I
where n is the electron number density, a is the inner radius
of the discharge tube and Jo is the Bessel function of zero
order. The data from Figures 2, 3, and 4 were normalized and
are shown together with a plot of 2.21 for q = 0, 1, and 2 in
Figure 5. This figure indicates that the ionization function
7)i/Da is approximately uniform (q = 0). In solving the
plasma balance equation to obtain 2,21, it was assumed that
the number density is zero at the boundary. This is not
precisely true. In the actual case, McDaniel14 shows that
the number density has a value at the wall which will extra-
polate to zero at a distance of the order of one mean free
path outside the boundary. This accounts for the fact that
the lower pressure data do not extrapolate to zero in Figure 5.
The plasma potential measured at 0.40 Torr is shown in
Figure 6. The potential gradient is established by the
ambipolar diffusion. The net effect is to establish an
electric field which retards the flow of electrons to the
boundary while accelerating the ions to maintain a quasi-
neutrality in the plasma volume. The plasma potential does
not extrapolate to zero at the boundary even though the wall
electrode is held at zero potential. This is due to the
sheath, mentioned earlier, which covers the electrode. By
an argument similar to that used in deriving 2.19, it can be
shown that the potential drop across the sheath, Vs, is
given by
Vs =-V,3 ) 2.22
By use of Bohm's criteria15 for Pi, one finds that for an
argon discharge
V,= 5.18 V 2.23
The calculated value of V, is plotted in Figure 6 and it
may be seen that the measured potential is readily extra-
polated through this point.
The results of these experiments raise important theo-
retical questions about the operation of the electrodeless
discharge. Due to skin effect at high frequencies, the power
input to the discharge is concentrated in the region near the
wall. It is this fact which led Eckert to assume the form
2.20 for the ionization production term. He also assumed
that the electron temperature increases exponentially with
radius. The probe measurements indicate that these assumptions
are not correct. In the following chapters a theory is given
which describes the operation of the electrodeless discharge
and which is in essential agreement with the experimental
measurements.
CHAPTER III.
TEMPERATURE DISTRIBUTION
The probe measurements described in Chapter II have
shown that the electron temperature is essentially uniform
across the discharge radius. This result is somewhat
surprising since the electric field and power input are
highly dependent on radial position. Elementary theories
and experiment for direct current discharges (see A. von
Engel16) indicate that electron temperature and ionization
rates are directly related to the electric field strength.
These relations are developed on the assumption that the
electric field is uniform. If these relations are applied
to the electrodeless discharge where the electric field
is non-uniform, the electron temperature becomes a strong
function of position. This result is not supported by the
experimental results. In addition to electron temperature,
the relationships would also predict that the ionization
function is a strong function of position. The experimental
results shown in Figure 5 do not support this prediction and
indicate that the ionization frequency is essentially uni-
form. The failure of these relations to predict correctly
the electron temperature and ionization frequency suggests
that they should not be applied to discharges in which the
electric field is not uniform.
The fact that the electron temperature and ionization
frequency are essentially uniform suggests that some mechanism
is operative which quickly transports the energy added by the
non-uniform electric field to the lower field regions of the
plasma. The combination of a uniform temperature and a uni-
form ionization frequency suggests that the ionization may
be thermally produced. That is, the ionization results only
from collisions with the high energy tail of some distribu-
tion of electrons having a characteristic temperature which
is spatially uniform. For this reason the temperature dis-
tribution becomes the most important factor in formulating
a theory of the electrodeless discharge.
It was previously noted that some mechanism must operate
to transport energy from those regions in the discharge where
large amounts of energy are added to those regions where the
energy addition is slight. The majority of the electrical
energy added to a plasma is added through the electron gas
since the ions contribute little to the total plasma conduc-
tivity. Therefore, the transport processes of interest are
those which transport energy from one region to another in
the electron gas. Two processes are considered: diffusion
and conduction. In diffusion an electron transports its
energy to a different spatial location by actually moving
through the plasma to the new location while retaining its
energy. In a slightly ionized plasma this occurs slowly due
to frequent interaction of the electron with the neutral
species in the plasma and with the electrostatic field estab-
lished in ambipolar diffusion. This field is established as
a result of the different mobilities of ions and electrons
and the requirement of a net neutral charge which is
characteristic of a plasma.
In conduction an electron transports its energy by
energy exchange with another particle in an encounter or
collision. In this way energy may be transported from one
region to another without the physical displacement of the
electron to that region. In a slightly ionized gas, en-
counters between electrons and neutral particles are much
more frequent than the electron-electron encounter. Thus,
it might appear that the energy would be transferred from
the electrons to the neutral particles. This process does
occur in fact,but from an elementary study of elastic
collisions it is found that the fractional loss of kinetic
energy per encounter is given by
4 _K (2-aol\ 3.1
where mI and m2 are the masses of the two particles and
Sis the scattering angle in center of mass coordinates.
From 3.2 it is seen that when mi < m2 then
A K, zmP cos 3 .2
and when ml=m2=m then
From 3.2 and 3.3 it is seen that the energy transfer per
encounter is much more effective for particles of equal
mass (electron-electron) than it is for an encounter where
ml1m2 (electron-neutral). Therefore, even though electron-
neutral encounters are more frequent they are less effective
for energy transfer than electron-electron encounters and
the latter must not be neglected.
We shall consider a cylindrical electrodeless dis-
charge to which power is added by a solenoidal field at high
frequency. For such a field, most of the energy is added in
a region near the wall. Thus, if the temperature is to
become uniform, energy must be transported towards the center.
Diffusion, however, will cause a net transport toward the
wall instead of toward the center. This is because the
wall behaves like a sink for charged particles, causing the
number density of charged particles to be a maximum at the
center. This leaves electron conduction as the process
remaining for energy transport toward the center of the dis-
charge and with diffusion as an opposing process.
It is the energy transport through the electron gas
which is of interest and, although the electrons lose some
energy to the neutral gas, in view of the discussion following
3.3 this process will be assumed negligible. Therefore, it
will be assumed that the energy transport can be described by
the equations for a fully ionized gas after taking into
account the smaller diffusion rate associated with ambipolar
diffusion in a slightly ionized gas.
The electron energy transport qe for a fully ionized
plasma with a Maxwell-Boltzmann distribution of electrons is
given by Shkarofsky, Johnston and Bachynski17 (equation 8-107b)
as
( )-> KXTB+n -K T 3.4
n) )e
In 3.4 ,2 and K are tensor coefficients of energy flow due to
electric fields and temperature gradients, respectively. T
is the electron temperature, B is the magnetic induction,
E is the electric field,
k is Boltzmann's constant, je| is the electron charge and
n is the electron number density. For the specific problem
under consideration, it is assumed that the discharge is
cylindrical, infinite in axial extent, and possess axial
symmetry. The discharge is excited by a field produced by
an alternating current flowing in an infinite solenoid placed
outside the discharge. Under the above assumptions the
problem becomes one-dimensional in the radial coordinate. For
the solenoidal field, the magnetic induction B is an alter-
nating vector quantity whose magnitude varies with the radial
coordinate r, and whose direction is axial. For this type of
field, IBI is small and the termv>x B will be neglected
compared to E. When the magnetic induction may be neglected,
the quantities,~and K reduce to the scalar quantities / and
K. Equation 3.4 then becomes
S/ T Vn)
e = ( KVT 35
The first term on the right is the convective energy trans-
port associated with the drift of electrons induced by the
electric field and diffusion. The term KVT represents the
energy transport by conduction through the electron gas due
to temperature gradients.
Consider next an arbitrary volume W bounded by a surface
S within the discharge. An energy flux P from the solenoid
flows into the volume and the electrons transport the energy
qe out of the volume. Conservation of energy requires that
E,dW = (f -1)*ds 3.6
Applying Gauss' theorem and noting that the relation is true
for any arbitrary surface S yields
D = v(f P3.7
ct
For the steady state, the left hand side vanishes and
substituting for e from 3.5 gives
4 '+ i- r) KVr -P 3.8
This constitutes an energy balance equation for the discharge.
The electric field produced by the solenoid is an alternating
field whose magnitude varies with r and whose direction is
azimuthal. This field contributes nothing to the energy flow
in the radial direction so that equation 3.8 may be written
as
+ K -d p 3.9
-n/el dr dr
where Er and Pr are the radial components of electric field
and energy flux respectively.
The radial electric field Er is established as a result
of the ambipolar diffusion in a partially ionized gas. The
electric field which is established retards electron flow to
the boundary and augments the ion flow to the boundary. The
two flows must be the same in a steady state if charge
neutrality is to be preserved. An approximate calculation
of the radial electric field established by ambipolar diffu-
sion can be performed. The flux of electrons and ions is
given by
/7_ = Vn_+ n-,,_ E 3.10a
and
D -D 7rn+ -+ n1+/4 E 3.10b
respectively, where D is the diffusion coefficient and / is
the mobility.
Since the plasma must remain approximately neutral, then
/=, = =n- n 3.11
Equations 3.10 may be solved for the electric field using the
approximations 3.11 to give for the diffusion-induced electric
field
Vn JD+-D- 3.12
Equation 3.12 may be rearranged to give
v n D- 1D+/D- 3.13
3.13
n 1-w- t _
Both the ratios of ion-to-electron mobility and ion-to-
electron diffusion coefficient are small compared to one,
being of the order of the ratio of electron-to-ion mass, and
the term in brackets may be set equal to one. Under the
assumption of a Maxwell-Boltzmann distribution of electrons
the Einstein relation may be used for the ratio of diffusion
coefficient to mobility. The Einstein relation is
D -- AZ- 3.14
Ie e
Therefore, to the order of the approximations used, the ambi-
polar diffusion-induced field becomes
4 7 dn
~ /e/7 d r 3.15
The substitution of 3.15 into 3.9 results in the vanishing
of the convective term leaving only the conductive term
dT
dr Pr 3.16
This result does not mean that no convection occurs in
the plasma. Complete cancellation occurs because of the
approximations used in deriving 3.15. However, the effect
of ambipolar diffusion is to reduce markedly the effective-
ness of the convective process within the discharge. At the
discharge boundary the situation is quite different. Due to
the fact that electrons and ions recombine at the wall the
energy transport from the electrons to the wall is primarily
a convective process. Within the discharge the reduced
effectiveness of the convective process allows the tempera-
ture to approach a uniform value through conduction as will
be shown below.
The radial coordinate and the temperature may be made
non-dimensional by introduction of the variables
r 7-
0 -3.17
where a is the discharge radius and To is the temperature on
the discharge axis. Equation 3.16 may then be written
d =_- a r 3.18
The energy flux Pr may be defined by
Pr = Pw(P)'0 3.19
where P, is the energy flux incident on the discharge at P =1
and 0o gives the variation of energy flux across the radius.
Substitution of the dimensionless variables into 3.16 yields
the differential equation for the temperature
-- -- a P 3.20
de KTo
The electron thermal conductivity K is also a function of the
temperature. For the case of no magnetic field, K is given
in Reference 17 as
K n- A 3.21
The number density n may be made non-dimensional by intro-
duction of the variable ? defined by
)= 3.22
no
where no is the number density on the discharge axis. Equa-
tion 3.21 may be written as
K (=e '17L A K' 3923
where /ZeiS is the average electron-ion collision frequency,
and gK'and g9 are correction factors,depending on the
magnetic field. For a fully ionized gas in a zero magnetic
field their values are given as
= 0.6538 3 = 0.3957 3.24
The average electron-ion collision frequency
a function of temperature which, for a Maxwell-Boltzmann
distribution, is given by
( ___ .
where I is the gamma function, and Yei is defined for a
singly ionized gas in terms of the coulomb logarithm_/A by
Ye = 4-r 7;";', ZOy JA 3.26
where E o is the permittivity of free space. The function./-
is also a function of temperature but log- Ais a slowly
varying function of temperature and its temperature dependence
will be neglected. Introducing the dimensionless variables
into 3.25 and substituting into 3.23 give
-- -F 4 5//I1z/ s/k 7/ZT
Kl w/'R Ye 3.27
The quantity Ce is defined by
K 3.28
Substituting 3.28 into 3.20 yields the differential equation
7 s/ dr a_ 3.29
S7 r 3T2Ce
25
which simplifies to
= -oa() 3.30
Upon integration and application of the boundary condition
q'(o) = 3.31
the temperature distribution becomes
4 ? 7 Ce P d 3.32
The quantity Ce may be written as
=/Ce48 0 kF(S/- .j.L 3.33
7 ^<)^ m e4^ log a
All the terms in Ce are constants except for logA.. There-
fore, the temperature distribution may be written as
^ = 1- 3. 787 x 10s a ?t ( p 3.34
0 7 3.34
where the constants have been evaluated in m.k.s. units.
The function 0( describes the radial dependence of the
input energy flux. It has the value zero at -0 and has a
maximum value of unity at -1. Thus, the maximum value of
the integral must be less than one. The temperature depen-
dence is seen to depend critically on the parameter
7/2
a P, logA /T 7/2 For the discharge for which measurements
are presented in Chapter II,
-2
g-" P.9 )I. / watt/m 335
T f. 1.8 x104 OK
loy /L 10
and
SlgA 9.3 X10 3.36
The equation for the temperature for this case would be
7/Z -.a P50
7 --1- 3.'x Jo ofcdp 3.37
It is clear that the term including the integral may be
neglected compared to one. Thus, for this case, it is seen
that the temperature is essentially uniform which is in
agreement with the experimental data.
The above analysis has been performed for a system in
which the only energy transport mechanism is thermal conduc-
tion as may be seen from equation 3.16. This situation arises
because the diffusive energy transport term is completely
cancelled out when the electric field due to ambipolar
diffusion is substituted into the energy transport equation.
However, since ambipolar diffusion is occurring it is obvious
that some energy transport is involved. The result of nearly
uniform temperature derived above will mean that the tempera-
ture gradient is small, and the propriety of neglecting the
energy transport through diffusion with respect to that due
to conduction is questionable.
An equation similar to 3.9 which includes the energy
transport due to the ambipolar diffusion may be written
Sdn dT P _
Ddr d 3.38
where KD is given by the expression
K = ( kTDa
KD =ZkTD. 3.39
In view of the nearly uniform temperature result which was
obtained above, it is reasonable to assume that KD will vary
only slightly with the radial coordinate.
If KD is assumed to be a constant function of To then
by substitution of the dimensionless quantities 7I and
p equation 3.38 may be written
d (, M) /0a no K d ri
d ) nKD dt 3.40
do T7*Ce 7Ce de
The solution of this equation analogous to 3.34 is
7 = 3.78 7X JO / /L P
3.41
The influence of the diffusive energy transport on the
temperature distribution is given by 3.41. The relative
importance of this term may be determined by considering the
ratio noKD/aPw. The value of no corresponding to the values
given in 3.35 is
no 6. xJ.1 m-3 3.42
and
no KD J0- 3.43
a P,
It is seen that the diffusive energy transport has even less
effect on the temperature distribution than the term involving
the energy input. Thus, the result that the temperature
distribution is essentially uniform is unchanged.
From the previous example it is apparent that for
practical laboratory discharges the electron temperature
should be essentially uniform across the radius, since for
any significant change to occur the parameter aPwlog.A/To 7/2
would have to be increased by an order of magnitude or more.
In the following chapters this important result will be used
as a basis upon which calculations are made for the electro-
magnetic field in the discharge and for the development of a
complete theory of operation of the electrodeless discharge.
The development of the theory will follow the general outline
of theories for the DC glow discharge first given by Schottky.
These theories are discussed in detail by Francis.18
CHAPTER IV.
ELECTRON TEMPERATURE AND NUMBER DENSITY DISTRIBUTION
In the previous chapter, an expression was derived for
the electron temperature distribution. For the discharge
in which the experiments were performed it was found that
the electron temperature was, essentially, constant with
radius. In this chapter, a theory will be given, based on
the solution of the plasma balance equation, which predicts
the discharge temperature. The solution of the plasma
balance equation will also yield an expression for the
electron number density distribution. This expression will
contain another unknown quantity, no, which will be deter-
mined by performing a power balance after the electro-
magnetic field in the discharge is calculated.
Certain assumptions must, again, be made in order to
solve the plasma balance equation. First, as a consequence
of the uniform temperature, it will be assumed that the
diffusion coefficient is constant with radius. Secondly,
it will be assumed that the average ionization collision
frequency is independent of position. This assumption is
based on the experimental results described in Chapter II
and is also a consequence of the uniform temperature.
Essentially, this assumption means that the ionization
process is thermal. The physical picture is that the
energy added to the electron gas in a non-uniform manner by
the electric fields is rapidly transported to the regions
of lower energy addition by thermal conduction. At the same
time the directed velocity of the electrons induced by the
electric field is rapidly randomized by the electron-neutral
collisions. The net result is a distribution of electron
velocities which is random in direction and characterized by
a temperature which is uniform. It is now assumed that the
ionization is due to those electrons in the distribution
having energies in excess of the ionization energy. The
ionization frequency is determined by averaging the ioniza-
tion cross-section over the electron distribution function.
Thus, the ionization function, 1) i/Da, will depend on both
the electron temperature and the form of the distribution
function. Throughout the development of this theory, it
will be assumed that the electron distribution function is
Maxwell-Boltzmann. Justification of this assumption can be
made from a theoretical standpoint (Reference 17) and also
by comparison of the experimental results with calculations
made using other distributions.
The flux of charged particles of species j is given by
/ Dj V + njl 4.1
where Dj is the diffusion coefficient for the jth particle
species and/j is the mobility of the jth particle species.
When the number density of charged particles is small both
ions and electrons will flow independently, each flowing in
accordance with its associated values of Dj andvj. However,
when the number density becomes larger, space charge fields
will be established due to the different flux rates for ions
and electrons. Since the electrons have a higher flux rate
initially, they begin to move out of the discharge, leaving an
excess of ions. The resultant space charge field retards
the electron flux and increases the ion flux. Eventually,
the process attains a steady state where the flux rates of
ions and electrons are the same. This is the condition for
which the diffusion is considered to be ambipolar. Some of
the results of this type of diffusion have been used previ-
ously in Chapter III. The transition from free to ambipolar
diffusion and the criteria for its occurence are studied in
detail by Allis and Rose.19
The flux equations for electrons and ions are
Z=- D_ Vn_ +n-/_, 4.2a
and
= Vn +n + r)4+/z 4.2b
For fully developed ambipolar diffusion, the flux rates must
be equal. Some differences between electron and ion number
densities exist to establish the electric field, but only
small differences in number density may exist in a plasma
and thus,
/ 7 4.3a
and
So n_ 4n 4.3b
Substitution of 4.3 into 4.2 and elimination of E give
r- --- 4.4
The term in parenthesis is called the ambipolar diffusion
coefficient Da. The flux of either ions or electrons is
given by
S-ZDa 7n 4.5
Consider a volume W of surface S where particles are being
produced and from which particles are being lost. Continuity
requires that
n dW = )-S dS = &V-rdW 4.6
dt
Since 4.6 must hold for each volume,
()t ." = DVn 4.7
Equation 4.7 gives the loss rate of ionized particles due to
diffusion. It is assumed that the ionization is produced by
the high energy "tail" of a Maxwell-Boltzmann distribution of
electrons. The ionization rate is given by
dn
dt = ) n 4.8
The average ionization frequency <)i> is determined by
averaging the quantity
z, = QgnS, 4.9
over the distribution function. Qi is the ionization cross-
section, g is the relative velocity between the electron and
a neutral particle and ng is the neutral particle number
density. In a steady state the loss of ionized particles,
represented by 4.7, and the number being created, represented
by 4.8, must balance. Thus, the plasma balance equation is
given by
fn + n = O 4.10
Dq
The solution of this equation, subject to the condition that
the number density vanish at the boundary, yields a number
density distribution dependent on an eigenvalue. The eigen-
value in turn determines the temperature.
To obtain a more general solution to 4.10, the require-
ment of an infinitely long discharge will be relaxed to
allow a finite length L. By separation of variables, the
solution of 4.10 is found to be
S= n cos(z)T r 4.11
where J is the zeroth order Bessel function, z is the axial
coordinate and A i is a separation constant. Applying the
boundary condition at z = L/2 and r = a results in
\A Z(i+i)), L L O, 4,12a
-Al
k = j,, "
4.12b
where the jok are the zeros of Jo. Since the number density
cannot become negative anywhere within the discharge, only
the first zeros have physical significance, and
and
S-
LIZ > J0
4.13a
4.13b
The number density may now be written
n= n, cos( J 3)j(2.4O 4 )0
4.14
where no is the number density at the point r z = 0.
Solving 4.13b,
< -4-=_J8(L
c t A 6-8 7'.
4.15
where AAD is the diffusion length. For a long cylindrical
discharge a/L <1 and
<_> e p.78 4
Da QZ
4.16
The eigenvalue equation 4.16 makes it possible to calcu-
late the discharge temperature. The average ionization
collision frequency is a very strong function of the electron
temperature due to the fact that the cross-section Qi has a
non-zero value only for electron energy greater than the
ionization energy. There is a weak dependence of
the neutral gas temperature through the relative velocity g,
but since the electron velocity is much greater than the
neutral velocity, g is assumed equal to the electron velocity.
With this assumption <(i> is a function only of the electron
temperature for any particular gas.
The ambipolar diffusion coefficient Da is a function of
both the electron and ion temperatures. From equations 4.4
and 4.5 Da is defined as
Dct __ ^D o.
114 4.17
The mobility of ions is generally much smaller than that of
the electrons and
a= + _4 4.18
The Einstein relation
j = k7 4.19
Se
reduces 4.18 to
1Da h T -/ 4.20
DR e
In an active discharge at low pressure the electron tempera-
ture is usually much larger than the ion temperature, and
thus
~e4 4.21
The ambipolar diffusion coefficient is primarily a function
of electron temperature, and the neutral gas pressure through
the ion mobility. The ratiol)i /Da becomes a function of
temperature and pressure and the eigenvalue equation 4.16
determines the electron temperature for a given value of
discharge radius and neutral gas pressure.
The theory given above yields the somewhat surprising
result that the electron temperature is independent of the
power input to the discharge. This result is due to the
assumption that the ionization is not directly related to
the electric field intensity, but is completely accounted
for by averaging the ionization frequency over a Maxwell-
Boltzmann distribution whose characteristic temperature is
uniform. It is this assumption which makes the ionization
function Zi/Da independent of the electromagnetic field and
yields the eigenvalue equation 4.16.
The physical process is one in which power is absorbed
from the electromagnetic field which raises the electron
temperature until a steady state is reached. A further
increase in power causes a transient increase in the ioniza-
tion rate until a new steady state is reached with a higher
level of electron number density.
According to equation 4.14, the radial distribution of
number density will be proportional to Jo. This results from
the fact that, due to the uniform temperature, )i/Da is not
a function of the radial coordinate. The solution due to
37
Eckert, which was discussed in Chapter II, reduces to this
form when q = 0 (equation 2.21). The distribution of number
density becomes very important in the calculation of the
electromagnetic field in the discharge. The electromagnetic
field depends on the plasma conductivity which, in turn,
depends upon the electron number density and temperature.
Since the temperature has been found uniform with radius,
the radial dependence of the conductivity will be determined
by the number density distribution.
CHAPTER V.
THE ELECTROMAGNETIC FIELD
The calculation of the electromagnetic field in the
discharge is a central aspect of any theory for the dis-
charge. The electric field and power input must be known
in order to relate the plasma conditions to the applied
fields. For high-frequency discharges this problem is more
difficult since the plasma conductivity is a function of the
applied frequency. Also, in a high-frequency discharge, the
fields are inherently non-uniform due to the "skin effect."
The term arises from the study of high-frequency fields in
good conductors where the fields are attenuated within a
short distance in the conductor and the current flows in a
thin region near the surface.
An early attempt at an analysis of the electrodeless
discharge by Thomson2 resulted in a one-dimensional calcu-
lation of the field in the discharge. Thomson's model was
that of an infinite cylindrical plasma in an infinite sole-
noid. He assumed the plasma would have a real,uniform
conductivity across the plasma. The conductivity of the
plasma is neither real nor uniform in the discharge, as
recognized by Eckert.7 But Eckert obtained the same solution
as Thomson as a result of his assumption that the reactive
part of the conductivity was small, and by solving the
electromagnetic equations using a uniform conductivity
having the average value of the non-uniform conductivity.
R. J. Sovie20 calculated the fields assuming a uniform
conductivity, but he allowed the conductivity to have a
reactive term so that the conductivity was a complex quantity.
All of the above analyses are characterized by the fact
that the conductivity was assumed to be uniform. In a
diffusion-controlled discharge this will never be true as the
number density must be zero at the discharge boundary. The
analysis of Chapter IV shows that the number density will be
a function of the radius as given by equation 4.14. A solu-
tion for the electromagnetic field may be obtained for a
conductivity which approximates this distribution.
The plasma conductivity 0r is usually defined as
ne2/ 1\
Ors mOV^~Ti) 5.1
where V) is an equivalent collision frequency and WC is the
radian frequency of the applied field. The equivalent
collision frequency is a quantity obtained from certain
averages of the momentum collision frequency )m over the
electron distribution function. The particular function to
be averaged depends upon whether the radian frequency is
large or small compared with the equivalent collision fre-
quency (Reference 17, Section 4-1.2). If the radian fre-
quency is of the same order as the equivalent collision
frequency, certain correction factors are necessary due to
variations of 2) with velocity. These factors modify both
V and CO and have been calculated by Dingle.21 They are
presented in Reference 17, Section 8-1. Including these
correction factors, the conductivity can be written as
e* { +-(I) 5.2
where g and h are the correction factors and (m) is the
equivalent collision frequency averaged for C~>) m)m The
40
conductivity will be assumed to be of the form 5.1 where it
is understood that the correction factors g and h have been
applied. In equation 5.1 the equivalent collision frequency
1) is a function of the electron temperature and the pressure
of the neutral gas. Since the electron temperature is uni-
form ) will be independent of the radial coordinate and,
since (" is directly proportional to n the conductivity
will have the same radial dependence as n From 4.14 it is
seen that the radial dependence will have the form of Jo,
the zeroth order Bessel function. It will again be assumed
that the discharge is infinite in axial extent and therefore
the axial dependence of 0 will vanish. The Jo Bessel
function may be expanded into the series
11/4 2 (l/Z2 (4 -e)..
?7-/)z 4C- ( 4- 5.3
Therefore, a first approximation to n which vanishes at the
boundary is
n(r)= n,[ l- r 5.4
where n0 is the number density on the axis. With this
approximation the conductivity becomes
noe- 5.5
Maxwell's equations for the electromagnetic field can
be reduced to two wave equations in the scalar potential V
and the vector potential A (see Pugh and Pugh22) as
SV P
V'V -Z'E t T 5.6a
and
VtA -- 7 5.6b
where and 6 are the permeability and permittivity of the
medium respectively, t is time, P is the charge density and
J is the current density. Except for the small charge
density which establishes the radial field due to ambipolar
diffusion, the plasma will be assumed to be electrically
neutral, 0 A Faraday cage is installed in the experi-
mental apparatus between the solenoid and plasma to prevent
the axial electric field of the solenoid from being impressed
on the plasma. Therefore, it will be assumed that the
potential within the discharge is everywhere zero, and only
5.6b must be considered. Again, the model assumed is that
of an infinite cylindrical discharge, possessing axial
symmetry and placed within an infinite solenoid. The sole-
noid carries a high-frequency current
I COt ^
Z= 1e 5.7
where Io is the peak value of the solenoid current, C) is
the radian frequency and Qi2 is a unit vector in the azimuthal
direction. Under the above conditions the equation 5.6b
becomes a scalar equation in the azimuthal component of the
vector potential AO namely,
r r ar r2 at + 0
S--'-^A1^+^aJ= 0 5.8
The azimuthal current density JO is given by
J = E= 0 = t 5.9
where the relationship between the electric field and the
vector and scalar potentials,
at- v) 5.10
has been applied. In view of the form assumed for the
applied current, it is reasonable to assume that the vector
potential will have a similar time variation. Assuming that
A=Ae e5.11
equation 5.8 reduces to the ordinary differential equation
A'+ A' A +/ e'dA -t re A = 0 5.12
where primes denote differentiation with respect to r. The
conductivity LT can be written as
r- 60 1- 8I2+co! 5.13
A "skin depth" 6 is defined by
J6 6 no oz j 5.14
The term O' CU becomes
o- ^ (6.)[ I- 5.15
For the plasma we shall assume that the permeability is that
of free space /o and since the plasma is considered to be a
conducting medium, the permittivity is that of free space 6o
(see Holt and Haskel,23 Section 11.2). The term oE46)acan
be written as
Wo6o W2:) (Z7 5.16
where f is the applied frequency, c is the velocity of light
and \ is the free-space wavelength of the applied frequency.
Equation 5.12 becomes
r"A" + rA'
+A (S 2 --~~zj/~ --. 0 5.17
For the frequency and discharge size of interest
and, thus, the first term in brackets will be neglected.
Defining a dimensionless coordinate
P-5
and introducing a new variable
equation 5.17 becomes
ZAl"+2A'+A Z- 7 ,e -) = 0
f (S )e(;z+ U3) JW
5.18
5.19
5.20
5.21
where the primes now denote differentiation with respect to z.
Introducing the complex quantity Keif defined by
5.22
iir a -
Ke = ; f-P +
2 1(_____W_
equation 5.21 reduces to
5.23
A+zA' +A 2 K --- =
From equations 5.20 and 5.22 it is noted that
S= Ke'F 5.24
and the solution of 5.23 will be a function of Kei and .
The parameter K is a real number which gives the ratio of
the discharge radius to the skin depth E*, and f is a param-
eter related to the ratio of collision frequency to applied
radian frequency.
A series solution to equation 5.23 has been given by
B. B. Henriksen.24 The solution which is finite at the origin
is given by
ao e UT 4 (Ker-e 5.25
n (n-i)J
n=!
For convenience, the complicated expression on the right hand
side of equation 5.25 will be represented by
Ro HI 5.26
where a is an arbitrary constant. The electric field Em
and the magnetic induction Bz, are derived from A4 by
E =- 5.27
and
B, (rA ) 5.28
where the quantities Ap E, and Bz are given by
A, Ae tt
E =Ee
BEBeu
5.29a
5.29b
5.29c
It should be noted that E and B are, in general, complex
quantities having both magnitude and phase. The expressions
for E and B are
and
5.30
5.31
E(KeW p) =- icC. H
B(Kee +
At the discharge radius, a, the boundary condition,
5.32
is applied. The quantity Bw is the peak wall value of mag-
netic induction. Since Bw is a real quantity, the phase of
Ba(Ke') = BEw
the fields in the discharge will be relative to the phase of
the magnetic induction at the wall. The quantity Bw can be
related to the current in the solenoid by
Bw = /1o/ N 5.33
where N is the number of turns per unit length, and Io is
the peak value of the solenoid current. The arbitrary
constant ao is found to be
aFdH = Ke 1i 5.34
In Chapter VI,a power balance for the discharge will
be performed to calculate the number density on the discharge
axis. Thus, it will be necessary to calculate the power
input from the electromagnetic field. The instantaneous
energy flux is given by the Poynting vector S where
xS =tF 5,35
This quantity represents the power which goes into changing
the energy of the electromagnetic field plus the energy
dissipated in the plasma. Since the energy in the electro-
magnetic field is alternately absorbed and given up, a time
average of S will give the energy flux which is dissipated
in the plasma. The time average of the Poynting vector
is given by
(S)> FPa,, P Ex E ) 5.36
where H is the complex conjugate of the magnetic intensity
which is related to the magnetic induction by
B = /o H 5.37
Since the electric field is azimuthal and the magnetic
induction is axial the energy flux is a vector in the radial
direction. This energy flux is given by
CPI ?r "r = Ee H, cos o s 5.38
where E is the phase angle between E and Hz and Or is a
unit vector in the radial direction.
CHAPTER VI.
POWER BALANCE
In the preceding chapters expressions have been derived
for the electron temperature distribution, the number density
distribution, and the electromagnetic field. The descrip-
tion of the discharge will be completed upon calculation of
the number density on the discharge axis and this may be
accomplished by consideration of the power balance for the
discharge.
The electrons gain energy from the electromagnetic
field and, subsequently, lose their energy through collision
processes or by diffusion to the discharge wall. Both the
energy gained and the energy lost from the electron gas
depend upon the electron number density. For a given pres-
sure the steady state will be reached when the energy loss
balances the energy gain and this will establish the steady-
state number density.
The power balance will be performed on the electron
gas alone. It will be assumed that all of the electrical
energy is absorbed by the electrons since the ions contrib-
ute little to the conductivity. This absorbed energy is then
lost from the electron gas by diffusion, by ionization, by
excitation of neutral particles with subsequent radiation,
and by heating of the neutral gas. It will be assumed that
the plasma is at low pressure and slightly ionized so that
the energy lost by heating of the ions and by bremsstralung
radiation may be neglected.
The electrical power absorbed by the plasma was given
in Chapter V. The energy flux given by equation 5.38 is a
49
function of Ke and therefore no as well as the radial
coordinate p It will not be necessary to perform a power
balance at each point within the discharge but only for the
discharge as a whole. Thus, it is not necessary to calculate
the average energy flux as a function of the radial coordinate,
but only the total energy flux incident on the discharge at
the discharge boundary. This quantity will be called Pw and
can be calculated from the equations of Chapter V with P = 1.
Since the power input is calculated per unit surface area of
the discharge, the power loss will also be referred to the
unit surface area.
The flux of electrons due to ambipolar diffusion is
given (equation 4.5) as
F -DCVn 6.1
The total loss of electrons from the discharge per unit
surface area is found by evaluating this quantity at the
discharge boundary. Thus the total flux of electrons is
given by
D= D 6.2
The number density is given by equation 4.14 which reduces
for an infinite discharge to
n= onTl (2.4055 r 6.3
The number density gradient is given by
d o.4oS no, (2.40 5 q 6.4
dr Q
where J1 is the first order Bessel function. Evaluation of
this quantity at r = a yields
(dn}l = -_ J 48 no 6.5
\dr/r=a Q
The average energy per electron is given by
'I() -= 3kT= e VT 6.6
where k is Boltzmann's constant, T is the electron temperature,
and VT is the electron temperature expressed in volts as
defined by 6.6. The energy transported to the wall through
diffusion becomes
4 = 1.248 no D eVr 6.7
When an electron suffers an ionizing collision it loses
an amount of energy equal to the ionization energy. This
energy is given by
E e = eVi 6.8
where Vi is the ionization potential of the neutral atom.
The number of ionizing collisions can be found directly by
calculating the average ionization frequency and integrating
over the number density distribution. However, since the
number of ionizing collisions must equal the number of ions
lost by diffusion, the number of ionizing collisions can be
obtained from the diffusion loss. Since the diffusion loss
has been calculated already, the latter method will be used.
From 6.2, 6.5 and 6.8 the total energy loss per unit surface
area due to ionization is given by
S. 28 no D V 6.9
When an electron suffers an inelastic collision in which
a neutral particle is left in an excited state, the neutral
particle radiates this energy. The energy loss per collision
is given by
EX = VX 6.10
where Vx is the excitation potential. The number of these
collisions occurring per unit time per unit volume is given
by
x, n ) 6.11
where (x) is the average excitation collision frequency.
This quantity is obtained by an appropriate average of the
excitation cross-section Qx over the distribution function.
Since it is the total energy loss per unit surface area that
must be obtained, it is necessary to integrate over the
number density and to divide by the surface area. The total
number density in a unit length of the cylinder is given by
G S n (r) r ddr 6.12
0 o
Substituting 6.3 into 6.12 and integrating yields
= 0.2159 no 2Trt 6.13
The surface area per unit length is
S = 2 TTn 6.14
and the total number of electrons per unit surface area is
given by
O. 2159 no 6.15
S
From 6.10, 6.11 and 6.15 it is seen that the energy loss due
to excitation per unit surface area is given by
E3= 0.2159 nea(q eVV 6.16
The loss of energy to heating of the neutrals is
calculated in a similar manner to that for excitation. The
average energy loss of an electron in an elastic collision
with a heavy neutral particle of mass M is given by
UEe rn 6.17
Using equation 6.6, this may be written as
E=-e eVT 6.18
M
By analogy to equation 6.16, the energy loss per unit surface
area due to neutral heating is
: = 0.2159 n a(< ~,> eV 6.19
where is the average momentum collision frequency for
electrons with neutral particles.
The power balance equation is obtained by equating the
power input at the wall per unit surface area to the total
power loss per unit surface area
P'= E6 + E6+ -4_6 6.20
Substituting from 6.7, 6.9, 6.16, and 6.19, the power balance
becomes
PC(n0)=noe [Dlae[V+V1.2
+ O.21599 [ v +<>V, }j 6.21
This equation can be solved for no, thus completing the
theory for the operation of the discharge.
In summary, expressions have been derived for the elec-
tron temperature and number density for a long cylindrical
electrodeless discharge. In addition, the electromagnetic
field in the discharge due to an infinite solenoid has been
derived. All of these quantities will depend on the physical
characteristics of the solenoid and the discharge. These
characteristics are:
1. The gas used in the discharge
2. The discharge pressure
3. The discharge radius
4. The frequency of the applied field
5. The number of turns per unit length of
the solenoid
6. The current carried by the solenoid.
In Chapter VII a sample calculation will be made for the
same discharge in which the experiments of Chapter II were
performed. In addition, the electromagnetic field will be
calculated for a more general range of physical parameters.
CHAPTER VII.
CALCULATIONS
A theory for the electrodeless discharge was developed
in the preceding chapters but no calculations were made
there. The values for the various plasma properties will
depend on the physical characteristics of the particular
discharge under consideration. However, the electromagnetic
field solutions are functions only of Keir and so that
rather complete calculations can be made for the field dis-
tributions which are applicable to a wide range of discharges.
A sample calculation will be made for a discharge having
the physical characteristics of the one in which the experi-
ments of Chapter II were performed. Only certain limited
comparisons of experiment with theory can be made due to the
lack of sufficient experimental data. The discharge for
which the calculations will be performed has the following
characteristics:
1. The gas is argon.
2. The pressure is in the range of 0.1 Torr to 1 Torr.
3. The discharge radius is 2.4 cm.
4. The applied frequency is 4.5 MHz.
5. The number of turns of the solenoid is 139 per meter.
6. The solenoid current is an independent variable.
It has been shown in Chapter III that the temperature
distribution depends upon the parameter Pw logA/T7/2. It
was also shown in that chapter that for the discharge being
considered the variation of the temperature across the dis-
charge radius is negligible and, thus, it is not necessary to
calculate results from equation 3.40.
The determination of the value of the temperature, as
derived in Chapter IV, involves the calculation of the average
ionization collision frequency. Since the determination of
average collision frequencies is required for many of the
calculations, the general procedure will be discussed.
In general, collision frequencies depend upon the elec-
tron energy. To obtain the average collision frequency it
is necessary to average over the electron distribution func-
tion f. The collision frequency for the jth process is
related to the cross-section for that process by
-Qj 9 n 7.1
where Qj is the cross-section for the jth process, g is the
relative velocity between the particle and its target
particle, and ng is the number density of target particles.
The collision frequencies to be calculated are the ionization
collision frequency, the elastic or momentum collision fre-
quency, and the excitation collision frequency. All of these
processes involve the collision of an electron with a neutral
particle. The electron temperature in a low-density plasma
is usually higher than the neutral temperature. For this
reason, and also because the electron mass is much smaller
than the neutral mass, it is assumed that the relative
velocity g is due solely to the electron velocity, i.e.,
g n O 7.2
and Vj becomes
*= Qj l n7
7.3
Two different averages will be considered. The average
momentum frequency0m) is involved in the plasma conductivity.
As pointed out in Chapter V,the type of averaging required
depends upon the applied radian frequency relative to('m).
Following reference 17, it will be assumed that(m) is calcu-
lated for the high-frequency case. Thus, m) is defined by
fd; w3Pdv 7.4
where v is the electron speed, and where it is assumed that
the distribution function f depends only upon v .
In general, the average of a function of electron speed
is given as
< = iT[ ^dV 7.5
Thus, the average collision frequency for the jth process is
given by
(47TS*f y 7.6
From the expression for the collision frequency 7.3, equa-
tions 7.4 and 7.6 become
A)d- 4rr 7.7
and
P n n Qjfy'dy 7.8
The average collision frequencies are seen to be functions
of the appropriate cross-section, the distribution function,
and the neutral number density ng.
The distribution function, as discussed previously, will
be assumed to be Maxwell-Boltzmann. This distribution is
given by
Df= f^/ n "<21Z
f =z Y/.e^ e 7.9
where is the mean square electron speed and is related
to the electron temperature by
Sm (<2 1 kT 7.10
It is convenient for computational purposes to express elec-
tron speeds in terms of volts. The following expressions
serve to define V and VT
m V^= eV 7.11
and
M, V ^VT 7.12
Substitution of 7.9, 7.11 and 7.12 into 7.7 and 7.8 yields
S= (e 3V
gQm Ve 2V~dV 7.13
N0(VT) ~ J
and
3V
Z> = s(3e/rrm)'" n S e 2 dV 7.14
where Qm and Qj are functions of V and the limits of integra-
tion are from 0 to oO as shown.
The calculation of average collision frequency from 7.13
and 7.14 depends on the form of the appropriate cross-section.
The cross-sections which are used are all determined experi-
mentally. The computational scheme used involved finding a
mathematical approximation for the experimental cross-sections
and performing a numerical integration by use of Simpson's
rule on the IBM 7090 or 360/50 computer.
The momentum cross-section Qm for argon was obtained
from Barbiere25 who averaged the angle dependent data of
Ramsauer and Kollath. An analytic approximation to the
Barbiere data was found by use of a curve fitting program
obtained from the University of Florida computing center.
Since the data given by Barbiere extended only to 12.5 volts,
an approximation to data given by Brown26 (who took it from
Brode) was added to the Barbiere approximation to extend the
cross-section to higher values. The final form for the
approximation is
Qn == JO" 606+ .771Va+.244V'- 0.016V) > v
and 7.15
Qe res s f te tt fr >12.7
The results of the computation for(/) are given in Figure 7.
61
The average ionization collision frequency was calculated
from equation 7.14. The cross-section data were obtained from
Kieffer27 who tabulates data from five different investiga-
tions. The data of Smith were chosen as there was little
difference in the data of the various investigators. An
approximation to the Smith data was found by use of the pre-
viously mentioned program. The approximation is given by
Q= O V<15.7
and 7.16
Qi 1o:-0(-3.734 + 0.313V- .(400 XJO'?V
+2e.O2 x1-rV3) V 15s.7
The results of the computation for (6i) are shown in Figure 8.
To calculate a power balance, it is necessary to deter-
mine the losses due to excitation and radiation. It is not
necessary to determine the losses to a particular excited
state since it is the energy loss from the electrons to all
excited states which enters into the power balance. An
experimental determination of the total excitation cross-
section for argon was obtained by Maier-Leibnitz28 and
presented by Brown26 who obtained the data from a paper by
Druyvesteyn and Penning.29 The data cover the range from-
10.8 V. to 19.0 V. and a constant cross-section is assumed
for energies greater than 19.0 V. The approximation for the
excitation cross-section is given by
Q=O V<10.8
Qx = 10 0[.048(V-10.8)1'e 10.8
Q, -10-o V> 19.0
The results of the computation for (7)x) are shown in Figure 9.
In addition to values for the average collision fre-
quencies, it is necessary to determine the ambipolar
diffusion coefficient. The ambipolar diffusion coefficient
was shown in Chapter IV to be related to the ion mobility
and electron and ion temperatures by
e 7.18
In a low-pressure gas discharge the electron temperature is
usually much greater than the ion temperature. In the worst
case the two temperatures would be the same. From 7.18 it
is seen that in the two limiting cases, T+/T_ = 0 and
T+/T_ = 1, the ambipolar diffusion coefficient varies by a
factor of two. Many measurements of ion mobility have been
made and the accepted value for the reduced mobility of
1.5 cm2 V-1 sec-l was obtained from Hasted.30 The reduced
mobility, Ko, is related to the mobility by
.. P 273
Ko 760 7y 7.19
where p is the neutral gas pressure in Torr and Tg is the
neutral gas temperature. Written in terms of the neutral
gas number density ng the expression for the ambipolar
diffusion coefficient for argon becomes
noDa == 2.69X10j Vr(10+ 7) 7.20
It was shown in Chapter IV that the electron temperature
was determined from the eigenvalue equation 4.16 since the
quantity )i)/Da is primarily a function of electron tempera-
ture. The dependence of
the limiting cases T+/T. = 0 and T+/T_ = 1 is shown in
Figure 10. The function
gas number density and, therefore, the neutral gas pressure
and temperature. The neutral gas temperature in a plasma is
difficult to determine accurately. At low pressure only a
small portion of the energy is lost to the neutral gas, and
since the discharge walls are water cooled in the experimental
apparatus it will be assumed that the neutral gas temperature
is 3000K. Thus, the neutral gas number density is given by
n = 3. 219 x P 7.21
where p is the pressure measured in Torr and ng has the units
m-3. Using equation 7.21 for ng, Figure 10 for Ji>/Da,
2.4 cm for the discharge radius and the eigenvalue equation
4.16, it is possible to determine the electron temperature
as a function of the discharge pressure. This curve is
presented in Figure 11. The data points shown in Figure 11
are the electron temperatures as measured by the probe
corresponding to the number density data shown in Figures 2,
3 and 4. It is seen that the electron temperature decreases
as the pressure increases. This is due to the fact that the
diffusion loss becomes less as the pressure is increased, and
a smaller temperature is sufficient to produce the ionization
required for a plasma balance. The measured electron tempera-
tures are consistently lower than the predicted temperature
by approximately 20 per cent with an average discrepancy of
19.6 per cent. This discrepancy is not unreasonable for
probe data and may be due to the lack of purity in the gas
used. The discrepancy could easily be explained by contami-
nants having larger ionization cross-sections or lower
mobility. One such contaminant which might have been present
is mercury, since an untrapped McLeod gauge was used to
measure the discharge pressure.
It was shown in Chapter V that the electromagnetic
field depends only upon Keif and /0. The parameter Kei' is
related to the plasma properties by
Ke S ( P 7.22
where K is a real number given by the ratio of discharge
radius to skin depth. K may be expressed, in terms of the
plasma properties, as
=r w n, e.9
K zJ 7.23
From 7.23 it is seen that K depends on the electron number
density through no and the electron temperature and neutral
pressure through Since the electron temperature-
neutral pressure relationship has been determined for the
discharge, it is possible to calculate K. The parameter
is a function of the ratio of the collision frequency to
the applied radian frequency given by the expression
f=-z+ ra] 7.24
Physically, f is a parameter related to the degree to which
the plasma conductivity is resistive or reactive. It is also
a function of the electron temperature and neutral pressure
through P.
Computer calculations have been performed to obtain the
magnitude of the electric and magnetic fields for three
values of P/w) namely 0, 1 and o which correspond to a
purely reactive, equally reactive and resistive, and a
purely resistive plasma, respectively. These calculations
were performed for integer values of K from 1 through 9. For
convenience, the field quantities have been made non-
dimensional with respect to Bw the value of the magnetic
induction at the wall. The corresponding non-dimensional
quantities are
S and E 7.25
Bw 60 BW
Curves for these quantities are presented in Figures 12, 13,
and 14. In order to perform a power balance it is necessary
to compute the average energy flux into the discharge at the
discharge boundary. This is the quantity Pw of Chapter VI
and it is calculated from the time average of the Poynting
vector given in Chapter V. This quantity is also made non-
dimensional;
P. i^o fPw
P B 7.26
Curves for this quantity as a function of K for various
values of 7/6( are presented in Figure 15. The quantity P*
can be thought of as representing the energy flux per unit
current squared since
7.27
Bw = / N6o
The curves of Figure 15 show that this quantity has a maxi-
mum at some value of K. The maximum occurs because at small
values of K the power input is low due to the small conduc-
tivity, and for values of K greater than that at which the
maximum occurs the fields induced by the solenoid are unable
to penetrate very far into the plasma and the power is
dissipated in a thin region near the surface. Therefore,
the most efficient operating condition for the electrodeless
discharge, in terms of coil current required, occurs where P*
is a maximum.
The electron number density at the center of the dis-
charge will be a function of the power input to the discharge
and,therefore, a function of the coil current. Equation 6.21
can not be solved explicitly for no because of the complex
dependence of Pw upon no. Therefore, a graphical solution
will be obtained. The quantity P*, defined by equation 7.26,
is a function only of Keif. Since for a given discharge the
temperature is a function of pressure, then 7 is determined
for each value of pressure. With 7-determined for a given
pressure P* becomes a function of no. Equation 6.21 can be
made non-dimensional such that
pY Z 7o e f i.248 [,vq7
+ O.159[ P \/Y+ 2)V\j] 7,28
For a given discharge radius, frequency, and number of
solenoid turns per unit length, the right hand side of 7.28
is a function of no and the coil current through the term
2
B It will be convenient to express Bw in terms of the
root mean square value of the coil current such that
B1, = /V; = ',lrms7 7.29
The current Irms will be taken as an independent parameter
and, thus, the right hand side of 7.28 becomes a linear
function of no. For each value of discharge pressure,
equation 7.28 is solved by plotting P* and the right hand
side of 7.28 as a function of n0 for various values of Irms,
A separate graph is required for each value of the pressure.
P* was obtained for the temperature consistent with each
value of pressure by computing the solutions for the elec-
tromagnetic field given in Chapter V. The IBM 360/50 computer
was used to calculate the required series. The series is
complex and the real and imaginary parts oscillate in sign
and acquire large values before they begin to converge.
Therefore, it was necessary to program the computer for
sixteen significant digits to insure sufficient accuracy over
the required range of values for K. A sample of the graphical
solution is shown in Figure 16. This solution is for a dis-
charge pressure of 0.5 Torr.
The solution of the power balance equation as described
above results in a family of curves, one for each discharge
pressure, relating the electron number density at the dis-
charge axis to the applied current in the solenoid. The
family of curves for the sample calculation is shown in
Figure 17.
CHAPTER VIII.
CONCLUSIONS
In the preceding chapters a probe diagnostic technique
was described with which it becomes possible to resolve experi-
mentally the spatial distribution of electron number density
and temperature as well as the plasma potential in an elec-
trodeless discharge. The results of these experiments were
found to disagree with some of the commonly held assumptions
about the electron temperature distribution and the spatial
production of ionization in the discharge. This discrepancy
resulted in the formulation of a new theory for the operation
of the electrodeless discharge which was found to be in
reasonable agreement with the probe measurements. Although
the theory agrees with the measurements, a considerable
number of predictions of discharge behavior have been made
which have not been subject to direct experimental verifica-
tion. However, in the course of the experiments, certain
qualitative observations have been made which can be compared
with the predictions of the theory. These comparisons will
be discussed later.
First, it would seem appropriate to point out some of
the limitations and critical assumptions made in arriving at
the theory for the electrodeless discharge. The model adopted
for the discharge makes the problem one-dimensional. This is
due to the choice of a model which assumed that the discharge
was infinitely long, and possessed axial symmetry.
For an actual laboratory discharge the assumption of
axial symmetry would appear to be quite reasonable. This is
due to the fact that the discharge is usually formed in a
container which is a circular cylinder surrounded by a coil
of circular cross-section. The assumption of a discharge
having infinite length is less reasonable. All laboratory
discharges must be finite in length even though the length
can be made large compared to the radius. This fact imposes
a more serious limitation to the theory due to the axial
variation in the number density which is induced by a dis-
charge of finite length. It is seen from equation 4.14 that
the number density will follow a cosine variation in the
axial direction. For a discharge which is long compared to
its radius,the error in the predicted electron temperature
will be small as can be seen from the eigenvalue equation 4.15.
However, the calculation of power input will be considerably
in error for the discharge as a whole since the value of K,
upon which the power input depends, will vary from a maximum
at the center of the discharge to zero at the ends of the
discharge. The power input as a function of the number den-
sity at the discharge axis should be approximately correct,
however, and a reasonable calculation of total input power
might be made by integration of the power over the length of
the discharge. The input power is a function of the axial
position through K which is, in turn, a function of the number
density on the discharge axis.
A further approximation has been introduced by the
assumption of an infinite solenoid having a purely azimuthal
current flow. In actual practice the coil is a helix of
finite length. For such a coil the electric and magnetic
fields are not purely azimuthal and axial, respectively. The
electric field will have an axial component and the magnetic
field will have an azimuthal component due to the helix
angle and, also, at regions near the end of the coil, large
deviations from the solenoidal field will occur. If the coil
is relatively long with respect to its diameter, the deviation
from a solenoidal field at the center should be small. In
this region the electromagnetic field predicted by the theory
should be a good approximation. The only electromagnetic
quantity involved in the rest of the theory is the power
input and in the region near the center of the coil the result
should be quite accurate.
Due to the limitations imposed on the theory from the
assumptions of infinite length, the prediction of the number
density on the discharge axis as a function of the solenoid
current can only be approximate. However, the general trend
as shown in Figure 17 should be quite representative for a
laboratory discharge.
The assumption of a Maxwell-Boltzmann distribution for
the electrons is, of course, only an approximation. Devia-
tions from this distribution may have a large effect on the
calculated values of the average collision frequencies for
ionization and excitation. This results from the fact that
the cross-sections for these processes are non-zero only
in the high-energy portion of the distribution function. For
the average momentum collision frequency this effect will be
much smaller. This fact is shown by Reference 17 where it
is found that various averages over both a Maxwell-Boltzmann
and Druyvesteyn distribution differ by less than 15 per cent.
In Reference 17 it is shown that the distribution func-
tion may be expanded in spherical harmonics in velocity space.
Due to a spatial effect, integration over the second order
term yields no difference in the calculated average ionization
or excitation collision frequencies. Therefore, only when
higher order approximations are important will the values of
ionization and excitation collision frequencies be affected.
Some calculations of the average ionization collision
frequency were made using the Druyvesteyn distribution and
these led to predicted electron temperatures which were
several electron volts higher than those measured. For this
reason the assumption of a Maxwell-Boltzmann distribution
appears to be justified.
Certain qualitative observations made in the course of
the experimental investigation tend to support the theory.
One striking feature of the theory is that of a constant
electron temperature, for a given discharge pressure, inde-
pendent of the input power. While observing a probe trace
on the oscilloscope it was noticed that a change in input
power did not affect the shape of the probe curve in its
exponential region, which determines electron temperature,
but caused a large change in the current at which saturation
occurred. It is this point which determines the number
density. In fact, a small change in input power could easily
change this point by a factor of ten. This tends to support
the prediction that the temperature is independent of power
input, and that the number density is critically dependent
on input power as shown in Figure 17.
Further evidence that this is true was obtained when an
undergraduate student in Aerospace Engineering, Mr. Don Green,
performed heat transfer measurements at the discharge bound-
ary. The heat transfer to the wall should be proportional to
the number density at the discharge axis. When the heat
transfer rate was plotted as a function of the coil current,
curves were obtained which were in qualitative agreement
with Figure 17.
Another undergraduate student in Aerospace Engineering,
Mr. Paul Bloom, has made measurements of the power input to
the discharge simultaneously with probe measurements of
electron number density and temperature. By using the
measured values of number density and temperature he has
calculated the value of K along the discharge axis, and by
use of the theory integrated the power input over the length
of the discharge to obtain predicted power input. Prelimi-
nary results indicate that good agreement between predicted
and measured total power input is obtained.
The theory for the electrodeless discharge which has
been presented shows reasonable qualitative and quantitative
agreement with the available experiments. Although it is
not an exact description of the discharge it appears to
describe the major physical processes which occur. The
theory should prove useful in determining the general behav-
ior of the discharge and for providing a basis upon which
engineering designs of future discharges could be made.
2.5 Volts/Division
Figure 1. Typical probe characteristic with third probe trace.
74
2.5
2
o
1.5
N'
*r
0
0
P 5
0 .2 .4 .6 .8 1.0
Normalized Radius
0 2. 6. .
+>mlie Rdu
Figure 2. Electron number density at a pressure of 0.13 Torr.
75
3
2.5
'-4
0 2
*4
S1.5
0
0 A
c.
1
0
r'-
.5
0 8 8 I ___
0 .2 .4 .6 .8 1.0
Normalized Radius
Figure 3. Electron number density at a pressure of 0.26 Torr.
2.5A
0
2 .
o 2
4J
s .5
1.5
a)
0
1'
a
-I .5
0 p p a p
0 .2 .4 .6 08 1.0
Normalized Radius
Figure 4. Electron number density at a pressure of 0.40 Torr.
0.13 Torr
0.26 Torr
0.40 Torr
1.0
.2 .4 .6 .8
Normalized Radius
Figure 5. Comparison of measured number density with the
theory of Eckert.
1.0
"A--------A
S Measured
A Calculated
.2 .4 .6 .8 1.0
Normalized Radius
Figure 6. Plasma potential.
15
10
o4
4J
0
04
0
ft
C
10-14
10
1 2 3 4
VT, (volts)
Figure 7. Computed average momentum collision frequency.
80
10-16
10-17
10-18
0)
-19
S10-19
10-20
10-21
10-22
1 2 3 4
VT, (volts)
Figure 8. Computed average ionization collision frequency.
81
10-16 .
10-17
10-18
I -19
M 10
C
i0-20
10-21
10-22
1 2 3 4
VT, (volts)
Figure 9. Computed average excitation collision frequency.
10-38
VT, (volts)
Figure 10. Temperature dependence of z)i> /Da.
-1- (141
lo-
3.4
3.2
3.0
2.8
2.6
2 2.4
> 2.2
2.0 o
1.8 O
1.6 O
1.4
1.2
1.0 -,
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0
Pressure, (Torr)
Figure 11. Electron temperature as a function of pressure for a long cylindrical
discharge of 2.4 cm radius.
1.0 1.0
.8 .8
.7 .7
.6 6
C.54 .5
.4.
.3 .3
.2 .2
.1 .1
.0 0
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1. 0 .1 .2 .3 .4
Figure 12. Electric and magnetic field for i/Yc 0
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.
0 .1, .2 .3 .4 .5 .6 .7 .8 .9 1.
Figure 13. Electric and magnetic field for
Figure 13. Electric and magnetic field for Z/O = 1
On
1.0
.9
.8
.7
.6
.5
.4
.3
.2
.1
0
1.0
.9
.8
.7
.6
00
.5
.4
.3
.2
.1
0
Figure 14. Electric and magnetic field for
Figure 14. Electric and magnetic field for 0)/O ---
I
0.5s
0 1 2 3 4 5 6 7 8
Figure 15. Average power input at the wall.
10-2
10
1018 1019 1020
no, (m-3)
Figure 16. Solution of the power balance equation for a pressure of 0.5 Torr.
OD
aO
89
1020
1019
1018 0
0 1 2 3 4 5 6 7 8 9 10
Irms, (amps)
Figure 17. Electron number density at the discharge axis as
a function of solenoid current at various pressures.
REFERENCES
1. Hittorf, W., "Ueber die Electricitatsleitung der Gase,"
Wiedemann Ann. Phys. Chim. 21, 90 (1884).
2. Thomson, J. J., "The electrodeless discharge through
gases," Phil. Mag. 4, 1128-1160 (1927).
3. Townsend, J. S. and Donaldson, R. H., "Electrodeless
discharges," Phil. Mag. 5, 178-191 (1928).
4. MacKinnon, K. A., "On the origin of the electrodeless
discharge," Phil. Mag. 8, 605-616 (1929).
5. Reed, T. B., "High-power low density induction plasmas,"
J. Appl. Phys. 34, 3146-3147 (1963).
6. Clarkson, M. H., Field, R. E., and Keefer, D. R.,
"Electron temperature in several rf-generated plasmas,"
AIAA Journal 4, 546-547 (1966).
7. Eckert, H. U., "Diffusion theory of the electrodeless
discharge," J. Appl. Phys. 33, 2780-2788 (1962).
8. Keefer, D. R., Clarkson, M. H., and Mathews, B. E.,
"Probe measurements in an electrodeless discharge,"
AIAA Journal 4, 1850-1852 (1966).
9. Allis, W. P., "Motions of ions and electrons," Handbuch
der Physik 21, 383-444, Springer Verlag (1956).
10. Brown, S. C., "Breakdown in gases: alternating and
high-frequency fields," Handbuch der Physik 22, 531-574,
Springer Verlag (1956).
11. Langmuir, I. and Mott-Smith, H., Jr., "Studies of
electrical discharges in gases at low pressures,"
Gen. Elec. Rev. 27, 449-455 (1924).
12. Loeb, L. B., Basic Processes of Gaseous Electronics,
Univ. of Calif. Press (1955).
13. Johnson, E. 0. and Malter, L., "A floating double probe
method for measurements in gas discharges," Phys. Rev.
80, 56-68 (1950).
14. McDaniel, E. W., Collision Phenomena in Ionized Gases,
John Wiley and Sons, Inc., New York (1964).
15. Bohm, D., The Characteristics of Electrical Discharge
in Magnetic Fields, edited by A. Guthrie and R. K.
Wakerling, McGraw-Hill Book Company, Inc., New York
(1949).
16. von Engel, A., "Ionization in gases by electrons in
electric fields," Handbuch der Physik 21, 504-572,
Springer Verlag (1956).
17. Shkarofsky, I. P., Johnston, T. W., and Bachynski, M. P.,
The Particle Kinetics of Plasmas, Addison-Wesley Pub. Co.,
Reading, Mass. (1966).
18. Francis, G., "The glow discharge at low pressure,"
Handbuch der Physik 22, 53-203, Springer Verlag (1956).
19. Allis, W. P. and Rose, D. J., "The transition from free
to ambipolar diffusion," Phys. Rev. 93, 84-93 (1954).
20. Sovie, R. J., Private communication, also presented
APS meeting, Atlantic City, Nov. 1962 and APS meeting,
Boston, Nov. 1966.
21. Dingle, R. B., Appl. Sci. Res. 6B, 144-154, 155-164,
245-2524(1957).
22. Pugh, E. M. and Pugh, E. W., Principles of Electricity
and Magnetism, Addison-Wesley Pub. Co., Inc., Reading,
Mass. (1960).
23. Holt, E. H. and Haskell, R. E., Foundations of Plasma
Dynamics, The Macmillan Co., New York (1965).
24. Henriksen, B. B., Analysis of field distributions in
an electrodeless discharge, Thesis, University of Florida
(1966).
25. Barbiere, D., "Energy distribution, drift velocity, and
temperature of slow electrons in helium and argon,"
Phys. Rev. 84, 653-658 (1951).
26. Brown, S. C., Basic Data of Plasma Physics, Technology
Press of The Massachusetts Institute of Technology and
John Wiley and Sons, Inc., New York (1959).
27. Kieffer, L. J., "A compilation of critically evaluated
electron impact ionization cross section data for atoms
and diatomic molecules," JILA Report No. 30, University
of Colorado (1965).
28. Maier-Leibnitz, H., "Ausbeutemessungen beim Stoss
langsamer Elektronen mit Edelgasatomen," Zeits. Phys.
95, 499-523 (1935).
29. Druyvesteyn, M. J. and Penning, F. M., "The mechanism of
electrical discharges in gases of low pressure," Revs.
Mod. Phys. 12, 87-174 (1940).
30. Hasted, J. B., Physics of Atomic Collisions, Butterworth,
Inc., Washington, D.C. (1964).
BIOGRAPHICAL SKETCH
Dennis Ralph Keefer was born September 22, 1938, at
Winter Haven, Florida. He was graduated from Auburndale
High School in June, 1956. In June, 1962, he received the
degree of Bachelor of Engineering Sciences with High Honors
from the University of Florida. Mr. Keefer enrolled in the
Graduate School of the University of Florida in June, 1962.
He was employed as a research assistant until August, 1963,
when he received the degree of Master of Science in
Engineering. From September, 1963, until the present, he
has pursued his work toward the degree of Doctor of
Philosophy while employed by-the Department of Aerospace
Engineering as a research associate.
Dennis Ralph Keefer is married to the former Charlotte
Ann King and is the father of two children. He is a member
of the American Institute of Aeronautics and Astronautics,
Tau Beta Pi, and Sigma Pi Sigma.
|
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INGEST IEID ELBVJ28I2_SP2BRI INGEST_TIME 2017-07-14T23:11:52Z PACKAGE UF00085807_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
THE THEORY AND THE DIAGNOSIS OF THE
ELECTRODELESS DISCHARGE
By
DENNIS RALPH KEEFER
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
August, 1967
To my wife
CHARLOTTE
who, having little understanding for the subject,
has shown great understanding for the author.
ACKNOWLEDGEMENTS
The author is deeply indebted to many individuals for
aid and encouragement in the completion of this dissertation.
He wishes to express his gratitude to his fellow students
and co-workers for their aid and suggestions in the course
of the research work.
Special thanks are due to his supervisory committee
chairman, Dr. M. H. Clarkson, who has served with patience
and understanding as teacher, mentor, critic and employer
for these past five years. Without his initial encourage
ment the project would have never begun. The author wishes
to express his appreciation to the members of his supervisory
committee, Dr. Knox Millsaps, Dr. Orlo E Myers,
Dr. Thomas L. Bailey and Dr. Earle E. Muschlitz, Jr., for
their guidance and encouragement in the course of his
graduate program.
The author also wishes to thank Mrs. Jacqueline Ward
who graciously and ably prepared this manuscript.
The financial support for this project was provided by
the National Aeronautics and Space Administration under
Grant NsG-542.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS .................... iii
LIST OF FIGURES .................... v
ABSTRACT ............... .... vii
CHAPTERS
I.INTRODUCTION 1
II.PROBE MEASUREMENTS .............. 4
III.TEMPERATURE DISTRIBUTION 15
IV. ELECTRON TEMPERATURE AND NUMBER DENSITY
DISTRIBUTION ................. 29
V. THE ELECTROMAGNETIC FIELD .......... 38
VI. POWER BALANCE ................ 49
VII. CALCULATIONS ................. 56
VIII. CONCLUSIONS ................. 68
REFERENCES ..... .......... 90
BIOGRAPHICAL SKETCH .................. 93
iv
LIST OF FIGURES
Figure Page
1. Typical probe characteristic with third probe
trace. 73
2. Electron number density at a pressure of
0.13 Torr. 74
3. Electron number density at a pressure of
0.26 Torr. 75
4. Electron number density at a pressure of
0.40 Torr. 76
5. Comparison of measured number density with
the theory of Eckert. 77
6. Plasma potential. 78
7. Computed average momentum collision frequency. 79
8. Computed average ionization collision frequency. 80
9. Computed average excitation collision frequency. 81
10. Temperature dependence of <("2^j^/Da. 82
11. Electron temperature as a function of pressure
for a long cylindrical discharge of 2.4 cm
radius. 83
12. Electric and magnetic field for ^/oo~ O 84
13. Electric and magnetic field for tJ/CO J. 85
14. Electric and magnetic field for J^/cO ~ o 86
15. Average power input at the wall. 87
16. Solution of the power balance equation for a
pressure of 0.5 Torr. 88
v
Page
Figure
17.
Electron number density at the discharge axis
as a function of solenoid current at various
pressures.
89
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
THE THEORY AND THE DIAGNOSIS OF THE
ELECTRODELESS DISCHARGE
By
Dennis Ralph Keefer
August, 1967
Chairman: M. H. Clarkson
Major Department: Aerospace Engineering
An analysis of the asymmetric double probe is described
which permits the measurement of electron temperature, elec
tron number density and plasma potential in an electrodeless
discharge. Experiments were performed which indicate that
the electron temperature and ionization function are essen
tially uniform across the discharge radius. These results
are at variance with commonly made assumptions concerning the
discharge. An analysis of the energy transport in the dis
charge predicts that, due to the thermal conduction of the
electron gas, the temperature is essentially uniform. The
fact that the electron temperature is uniform is used as a
basis for the formulation of a one-dimensional, steady-state
theory for the inductively driven electrodeless discharge.
The results of the theory permit the calculation of electron
temperature, electron number density, the electric and
magnetic fields in the discharge, and the power input to the
discharge as a function of the discharge configuration and
the applied coil current. An important result of the theory
is the prediction that electron temperature is independent
of the input power, and that the electron number density is
a direct function of the solenoid coil current.
vii
CHAPTER I,
INTRODUCTION
An electrodeless discharge is a gaseous electrical dis
charge which is maintained by high-frequency electric fields.
The fields are applied to either conductive rings or plates
or to helical coils placed outside the discharge container.
These discharges can be maintained in a wide variety of
gases and over a pressure range extending from the region of
.001 Torr to greater than atmospheric.
The discharge was first discovered by Hittorf^ in 1884
and appears to have been widely known and studied in the
early part of the twentieth century. In 1927 and 1928 the
solenoid excited electrodeless discharge received the
attention of Sir James Thomson and J. S. Townsend and their
work resulted in a controversy concerning the nature of the
discharge. The controversy concerned the question of whether
the induced or the electrostatic fields of the solenoid were
responsible for the discharge, Thomson^ gave an analysis of
the induced fields, derived conditions for breakdown, and
reported experiments which indicated that the currents in the
gas were due to the induced fields. Townsend and Donaldson^
reported experiments which showed the breakdown to be caused
by the axial electric field due to the potential drop across
the solenoid. The controversy was resolved in 1929 by
K. A. MacKinnon^ who showed that actually both discharges
were possible, depending on the configuration, and that the
two types of discharge exhibited different visual character
istics. In 1963, T. B. Reed5 reported two modes of operation
1
2
for a low-pressure electrodeless discharge and gave certain
spectrographic data for the two modes of operation which he
designated as low-power and as high-power as an indication of
power absorbed from the RF generator. Clarkson, Field and
Keefer showed that these two modes of operation were related
to the same phenomena explained by MacKinnon and gave measure
ments of electron temperature in the two modes obtained by
use of floating double probes.
Although the electrodeless discharge has been widely
used as a source of laboratory plasma, it has not been widely
studied. A revival of interest occurred in the 1950s as a
result of the need for high-energy gas sources of low con
tamination for use in wind tunnels designed for testing
re-entering vehicles launched by rockets. The discharge was
not initially successful in these applications and they were
largely discarded in favor of high-energy DC arc jets.
Analytical studies of the electrodeless discharge began
with Thomson who calculated the fields in a cylindrical dis
charge in a solenoidal field, assuming a uniform conductivity
across the radius. From these calculations, he derived a
breakdown criterion based on the energy gain of an electron
in one mean free path. A more realistic analysis was given
by H. U. Eckert,7 who recognized that the conductivity would
not be uniform across the radius, and that the discharge would
be diffusion controlled. However, Eckert obtained the same
solution for the fields as Thomson since he assumed an
average uniform conductivity in the formulation of Maxwell's
equations. By solving the plasma balance equation, Eckert
was able to give a more realistic breakdown criterion. By
making certain assumptions concerning the spatial variation
of the ionization function and electron temperature, Eckert
was able to calculate the steady-state electric field at the
boundary of the discharge and the spatial variation of the
electron number density.
3
Development of a highly asymmetric floating double probe
suitable for use in electrodeless discharges by Keefer,
Q
Clarkson and Mathews made it possible to measure the spatial
variation of electron number density and temperature within
the electrodeless discharge The results of these measure
ments were found to be at. variance with the assumptions made
by Eckert. The most striking difference occurred in the
spatial variation of the electron temperature. While Eckert
assumed that the temperature increased exponentially from the
center of the discharge to the wall, the probe measurements
indicate that the electron temperature is essentially uniform.
In addition, the measured spatial variation of the electron
number density implies that the ionization function is nearly
uniform in contrast to the power law variation assumed by
Eckert.
A new theory for the electrodeless discharge has been
formulated as a result of the discrepancy found between the
probe measurements and Eckerts assumptions. The new theory
is in reasonable agreement with the probe measurements, and
provides a description of the plasma properties as a function
of the applied fields. Not ail of the predictions of the
theory have been subjected to experimental verification, but
certain qualitative comparisons with observations made in the
course of the experiments indicate that the major physical
processes are described by the theory.
It should be pointed out that, while a relatively small
amount of attention has been given to the electrodeless dis
charge per se in the past few decades, there has developed a
large body of plasma theory of a general nature which is
applicable to the problem. There has also been a great amount
of work performed on microwave discharges, much of which is
applicable to the electrodeless discharge. Of particular
value are the two Handbuch der Physik articles by Allis and
Brown.
CHAPTER II.
PROBE MEASUREMENTS
The electrostatic probe, or Langmuir probe, is one of
the most widely used plasma diagnostic tools. Although its
use dates much earlier, a definitive theory for its behavior
1 1
in a plasma was given by I. Langmuir in 1924. The tech
nique is basically simple. A small conductor is placed in
the plasma and a potential applied to it with respect to
some other electrode, usually one of the discharge electrodes.
A curve of current versus voltage is recorded and with the
Langmuir theory the plasma number density, electron tempera
ture and potential may be determined from this curve. Despite
the fact that the theory is applicable to only a limited
range of plasma conditions, the use of the technique is
nearly universal. The technique, together with its limita
tions, is adequately described by Loeb.-1-^ It remains virtually
the only way to resolve spatially the properties of a plasma.
The application of this technique to the electrodeless
discharge involves two obvious difficulties. First, there is
no discharge electrode with which the probe may be biased and,
secondly, the plasma potential may vary at the driving
frequency. Johnson and Malter13 describe a system which over
comes these difficulties. A floating double probe is placed
in the plasma and one probe is biased with respect to the
other. Their experiments utilized probes of equal area, but
it was proposed that probes of unequal area or an asymmetric
probe might be used to increase accuracy. In their analysis,
it is assumed that both probes are placed in a plasma of
4
5
uniform potential. As this may not be true in the electrode
less discharge, it is necessary to extend their analysis to
include this condition.
Whenever a surface is placed in a plasma, a sheath forms
and the surface potential becomes negative with respect to
the local plasma potential. This condition occurs because the
electrons have a higher mobility than the ions and, therefore,
build up more rapidly on the surface. When the surface
potential becomes sufficiently negative, the lower energy
electrons are retarded in their motion toward the surface such
that an equal number of ions and electrons impinge on the
surface and a steady state is established. When a current is
drawn from the surface, the sheath is altered such that the
surface becomes more or less negative with respect to the
local plasma potential. The potential becomes more negative
if a positive current flows from the surface, and less negative
if negative current flows from the surface.
In the use of a floating double probe, two collecting
surfaces are placed in the plasma and a biasing potential
applied between them. The more positive of the probes will
collect a surplus of electrons and the more negative will
collect a surplus of ions such that the net current drawn from
the plasma is zero. As the bias potential is increased, the
current between the probes increases until saturation occurs.
Saturation will occur whenever the maximum possible number of
either the ions or electrons incident upon the sheath boundary
is collected. To insure greater accuracy in the measurement
of plasma properties, it is necessary to operate the probe at
electron saturation. This is the condition where all the
incident electrons are collected. The analysis will show that
for this condition to occur a very large difference in the
surface area of the two probe collectors is required.
6
The large difference in required probe area will usually
mean that the probes are placed in the plasma at points where
the local plasma potential is not equal,, This potential
difference may be established by ambipolar diffusion or by an
external electric field. The quantities which refer to the
two probe surfaces will be designated by superscripts 1 and 2.
Thus, the plasma potential at probes 1 and 2 is V and
(2) ^
VpV respectively. The potential difference will be assumed
constant and designated Vr where,
V, Vp(z) Vp(1> 2.1
The flux of ions and electrons impinging on the probe sheath
will be designated r^ and P e,respectively. It is assumed
that these fluxes are a function only of the plasma properties
and independent of the sheath potential. In the electron
retarding region, the current density of electrons reaching a
probe is given by
2.2
kTe
where V-j. = and e is the electron charge, k is Boltzmann's
constant and Te is the electron temperature. Thus the total
current to probe 1 is given by
2.3
and the current to probe 2 is given by
(V
- n
(2)
2.4
7
where A is the surface area of the probes and V is the poten
tial of probe 2 with respect to probe 1 The floating probe
system can draw no net current from the plasma and this
requires that
> = cz>=
2.5
By equating 2.3 to 2.4 and noting that
vp(1)- vP(z)- vr
2.6
one obtains, after some manipulation.
IL = /oq Jg K- J- ^ '& g
*4 l Vr/3)
Aw rt<
2.7
This expression relates the local plasma potential at
probe 2 to the applied potential V. If the double probe
system is to behave like the classical Langmuir probe, it is
necessary that the variation of the local plasma potential at
probe 2 be small as the bias potential V is varied. This
requirement yields a criterion for the area ratio of the
probes, namely
Aa>r/J)
1
2.8
In practice this requirement may be rather severe since the
larger probe (1) would likely be placed near the boundary of
the discharge wherePe^^ Pe^^. The required area ratio
A^)// may be of the order 10^.
8
It will be assumed now that the area ratio is sufficiently
small so that the variation of Vp^) is negligible over the
useful range of bias potential V. The probe current in the
electron retarding region is given by
- = eA
- W V)/Vr
e ~ri
and
2.9
2.10
The ratio of ion flux to electron flux is much less than one
and it will be neglected in 2.10. Taking the derivative of
equation 2.10 with respect to V yields the expression
d03 0 ^ J_ 2.ii
dV |/r
This is the usual expression for the Langmuir probe from
which the electron temperature may be determined from the
slope of the probe characteristic when it is plotted on semi
log paper. The electron number density is determined from
the break in the characteristic curve caused by electron
saturation. At electron saturation, = V and 2.9 becomes
4 = e Am (rtm-r) 2.12
Since re(2) ^ then
10
The current relation for probe 3 is then given by
-
e
2.17
Now if no current is allowed to flow from probe 3, it will
assume a floating potential Vj, where
-(V-Vf)Ar nw
e = TpQ
This may be reduced to the expression
2.18
Thus the floating potential differs from the local plasma
potential by a constant amount. By observing the floating
potential of probe 3, it is possible to determine whether
the plasma potential Vp(2) changes as the applied potential V
is varied in accordance with 2.7. If the variation of Vf, and
therefore Vp(2)¡, is negligible then the criteria 2.8 has been
satisfied.
A highly asymmetric double probe was used to measure the
electron temperature and number density in a low-pressure
electrodeless discharge. The discharge was formed in a length
of two-inch diameter Pyrex pipe connected to a vacuum system,
argon was admitted to maintain a prescribed pressure. The
tube was placed inside a structure containing a solenoid coil
connected to the RF power supply and had shielding to prevent
penetration of the axial electric field of the solenoid into
9
2.13
For a Maxwell-Boltzmann distribution of electrons
2.14
where n is the electron number density and m is the electron
mass. Since the electron temperature may be determined from
2.11, the number density may be determined from 2.13 and 2.14.
One additional property of the plasma may be determined
by the probe, namely, the plasma potential. At the break in
the characteristic curve at the onset of electron saturation,
the probe 2 has the same potential as the plasma relative to
probe 1. Thus, if probe 1 is held at a constant potential,
the plasma potential may be determined.
The above analysis shows that a highly asymmetric double
probe behaves like the classical Langmuir probe whenever the
criterion 2.8 is met. An experimental method to determine
whether or not this criterion is being met is described below.
Consider a third probe placed in the plasma at the same
location as probe 2. This probe will be designated probe 3
and the plasma potential and ion and electron flux at its
location will be the same as for probe 2, i.e.,
2.15
and
2.16
11
the discharge. The RF power supply was a converted television
transmitter operating at approximately 4.5 MHz with a maxi
mum available power of 5 KW.
The probe system consisted of a large cylindrical elec
trode placed at the wall inside the discharge tube, and two
small movable probes. The large electrode was constructed of
OFHC copper with a cooling coil soldered to the inside and
the entire assembly silver plated to reduce contamination.
The cylinder was split longitudinally to allow the axial
magnetic field of the solenoid to ''penetrate" by acting as a
one-turn secondary. The small electrodes were formed from
0.01 inch diameter tungsten wire with approximately 0.125
inches exposed from a Pyrex sheath. This gave a geometric
area ratio of approximately 7 x 10~5. The probes and Pyrex
sheath were set into a brass tube with an offset to allow
rotation of the probes across the tube radius. Probe voltage
was supplied by a transformer secondary with the primary
connected through a variable autotransformer to the 60 Hz
laboratory power. The current signal was obtained across a
10 ohm shunt and applied to the vertical input of an oscillo
scope while the probe voltage was connected to the horizontal
input. The resultant current-voltage trace was photographed
from the oscilloscope to provide a permanent trace. When the
third probe was used, it was connected directly to a second
vertical input.
A typical probe characteristic, together with the third
floating probe potential, is shown in Figure 1. The charac
teristic is typical of a properly functioning Langmuir probe
with a sharp knee and saturation region. The third probe
potential is seen to vary slightly as the bias potential on
the asymmetric probe is varied. The indicated change in
plasma potential is seen to be of the order of 0.4 volt
showing that the change in plasma potential is negligible for
12
this probe system. Data were taken at pressures of 0.13 Torr,
0.26 Torr and 0.40 Torr in argon at several radial locations.
In all cases the electron temperature was found to be uniform
across the radius of the discharge. At 0.13 Torr the electron
temperature expressed in volts was 2.0 volts, at 0.26 Torr
it was 1.8 volts and at 0.4 Torr it was 1.6 volts. The elec
tron number density at the three pressures is shown in
Figures 2, 3, and 4. All of these curves have the character
istic that the number density is greatest at the axis of the
discharge and becomes smaller as the wall is approached. This
is typical of a diffusion controlled discharge.
H. U. Eckert^ has given an approximate analysis for the
electrodeless discharge. In this analysis, the electron pro
duction term is assumed to obey a power law given by
/, r ^ 2.20
where V^ is the average frequency of ionization per electron,
Da is the ambipolar diffusion coefficient, r is the radial
coordinate measured from the discharge axis and h and q are
assumed constant. The solution of the plasma balance equation
for this assumed form is
sil ,
nho. T \zAOS(-~\ Z 1 2.21
{aJ -I
where n is the electron number density, a is the inner radius
of the discharge tube and J0 is the Bessel function of zero
order. The data from Figures 2, 3, and 4 were normalized and
are shown, together with a plot of 2.21 for q = 0, 1, and 2 in
Figure 5. This figure indicates that the ionization function
13
Vj[/Da is approximately uniform (q 0) In solving the
plasma balance equation to obtain 2.21, it was assumed that
the number density is zero at the boundary. This is not
precisely true. In the actual case, McDaniel^ shows that
the number density has a value at the wall which will extra
polate to zero at a distance of the order of one mean free
path outside the boundary. This accounts for the fact that
the lower pressure data do not extrapolate to zero in Figure 5.
The plasma potential measured at 0.40 Torr is shown in
Figure 6. The potential gradient is established by the
ambipolar diffusion. The net effect is to establish an
electric field which retards the flow of electrons to the
boundary while accelerating the ions to maintain a quasi
neutrality in the plasma volume. The plasma potential does
not extrapolate to zero at the boundary even though the wall
electrode is held at zero potential. This is due to the
sheath, mentioned earlier, which covers the electrode. By
an argument similar to that used in deriving 2.19, it can be
shown that the potential drop across the sheath, Vg, is
given by
By use of Bohmfs criteria^ for ? one finds that for an
argon discharge
Vs = 5.18Vt
The calculated value of V_ is plotted in Figure 6 and it
may be seen that the measured potential is readily extra
polated through this point.
The results of these experiments raise important theo
retical questions about the operation of the electrodeless
14
discharge. Due to skin effect at high frequencies, the power
input to the discharge is concentrated in the region near the
wall. It is this fact which led Eckert to assume the form
2,20 for the ionization production term. He also assumed
that the electron temperature increases exponentially with
radius. The probe measurements indicate that these assumptions
are not correct. In the following chapters a theory is given
which describes the operation of the electrodeless discharge
and which is in essential agreement with the experimental
measurements.
CHAPTER III.
TEMPERATURE DISTRIBUTION
The probe measurements described in Chapter II have
shown that the electron temperature is essentially uniform
across the discharge radius. This result is somewhat
surprising since the electric field and power input are
highly dependent on radial position. Elementary theories
and experiment for direct current discharges (see A. von
Engels-6) indicate that electron temperature and ionization
rates are directly related to the electric field strength.
These relations are developed on the assumption that the
electric field is uniform. If these relations are applied
to the electrodeless discharge where the electric field
is non-uniform, the electron temperature becomes a strong
function of position. This result is not supported by the
experimental results. In addition to electron temperature,
the relationships would also predict that the ionization
function is a strong function of position. The experimental
results shown in Figure 5 do not support this prediction and
indicate that the ionization frequency is essentially uni
form. The failure of these relations to predict correctly
the electron temperature and ionization frequency suggests
that they should not be applied to discharges in which the
electric field is not uniform.
The fact that the electron temperature and ionization
frequency are essentially uniform suggests that some mechanism
is operative which quickly transports the energy added by the
non-uniform electric field to the lower field regions of the
15
16
plasma. The combination of a uniform temperature and a uni
form ionization frequency suggests that the ionization may
be thermally produced. That is, the ionization results only
from collisions with the high energy tail of some distribu
tion of electrons having a characteristic temperature which
is spatially uniform. For this reason the temperature dis
tribution becomes the most important factor in formulating
a theory of the electrodeless discharge.
It was previously noted that some mechanism must operate
to transport energy from those regions in the discharge where
large amounts of energy are added to those regions where the
energy addition is slight. The majority of the electrical
energy added to a plasma is added through the electron gas
since the ions contribute little to the total plasma conduc
tivity. Therefore, the transport processes of interest are
those which transport energy from one region to another in
the electron gas. Two processes are considered: diffusion
and conduction. In diffusion an electron transports its
energy to a different spatial location by actually moving
through the plasma to the new location while retaining its
energy. In a slightly ionized plasma this occurs slowly due
to frequent interaction of the electron with the neutral
species in the plasma and with the electrostatic field estab
lished in ambipolar diffusion. This field is established as
a result of the different mobilities of ions and electrons
and the requirement of a net neutral charge which is
characteristic of a plasma.
In conduction an electron transports its energy by
energy exchange with another particle in an encounter or
collision. In this way energy may be transported from one
region to another without the physical displacement of the
electron to that region. In a slightly ionized gas, en
counters between electrons and neutral particles are much
17
more frequent than the electron-electron encounter Thus,
it might appear that the energy would be transferred from
the electrons to the neutral particles. This process does
occur in fact, but from an elementary study of elastic
collisions it is found that the fractional loss of kinetic
energy per encounter is given by
A A
2 Mi
(l cos %
3,1
where m^ and mg are the masses of the two particles and
')C is the scattering angle in center of mass coordinates
From
3,2 it is seen that when m-^ nig then
A Ki ^ z mi
K, rnz
and when m1=m2=m then
(i -eosX)
3,2
A Ki
Kj
3.3
From 3.2 and 3.3 it is seen that the energy transfer per
encounter is much more effective for particles of equal
mass (electron-electron) than it is for an encounter where
ml<<(m2 (electron-neutral). Therefore, even though electron-
neutral encounters are more frequent they are less effective
for energy transfer than electron-electron encounters and
the latter must not be neglected.
We shall consider a cylindrical electrodeless dis
charge to which power is added by a solenoidal field at high
frequency. For such a field, most of the energy is added in
a region near the wall. Thus, if the temperature is to
become uniform, energy must be transported towards the center
18
Diffusion, however, will cause a net transport toward the
wall instead of toward the center. This is because the
wall behaves like a sink for charged particles, causing the
number density of charged particles to be a maximum at the
center. This leaves electron conduction as the process
remaining for energy transport toward the center of the dis
charge and with diffusion as an opposing process.
It is the energy transport through the electron gas
which is of interest and, although the electrons lose some
energy to the neutral gas, in view of the discussion following
3.3 this process will be assumed negligible. Therefore, it
will be assumed that the energy transport can be described by
the equations for a fully ionized gas after taking into
account the smaller diffusion rate associated with ambipolar
diffusion in a slightly ionized gas.
The electron energy transport for a fully ionized
plasma with a Maxwell-Boltzmann distribution of electrons is
given by Shkarofsky, Johnston and Bachynski1^ (equation 8-107b)
as
% = /*'(£+ X B + j K- VT 3.4
In 3.4 yU and K are tensor coefficients of energy flow due to
electric fields and temperature gradients, respectively, T
is the electron temperature, B is the magnetic induction,
E is the electric field,is the average plasma velocity,
k is Boltzmanns constant, |e| is the electron charge and
n is the electron number density. For the specific problem
under consideration, it is assumed that the discharge is
cylindrical, infinite in axial extent, and posseses axial
symmetry. The discharge is excited by a field produced by
an alternating current flowing in an infinite solenoid placed
outside the discharge. Under the above assumptions the
19
problem becomes one-dimensional in the radial coordinate. For
the solenoidal field, the magnetic induction B is an alter
nating vector quantity whose magnitude varies with the radial
coordinate r, and whose direction is axial. For this type of
field, | BI is small and the termx B will be neglected
compared to E. When the magnetic induction may be neglected,
the quantitiesy~and IT reduce to the scalar quantities yU and
K. Equation 3.4 then becomes
ft)-Kvr 3.5
The first term on the right is the convective energy trans
port associated with the drift of electrons induced by the
electric field and diffusion. The term KVT represents the
energy transport by conduction through the electron gas due
to temperature gradients.
Consider next an arbitrary volume W bounded by a surface
*
S within the discharge. An energy flux P from the solenoid
flows into the volume and the electrons transport the energy
qe out of the volume. Conservation of energy requires that
j~t f EwdW = j(fe-T5>ndS s.e
JW 5
Applying Gauss' theorem and noting that the relation is true
for any arbitrary surface S yields
c>t
-P)
3.7
For the steady state, the left hand side vanishes and
substituting for "q^ from 3.5 gives
20
3.8
This constitutes an energy balance equation for the discharge.
The electric field produced by the solenoid is an alternating
field whose magnitude varies with r and whose direction is
azimuthal. This field contributes nothing to the energy flow
in the radial direction so that equation 3.8 may be written
as
where Er and Pr are the radial components of electric field
and energy flux respectively.
The radial electric field Er is established as a result
of the ambipolar diffusion in a partially ionized gas. The
electric field which is established retards electron flow to
the boundary and augments the ion flow to the boundary. The
two flows must be the same in a steady state if charge
neutrality is to be preserved. An approximate calculation
of the radial electric field established by ambipolar diffu
sion can be performed. The flux of electrons and ions is
given by
/I = £L Vn_+ r\-//_ E
3.10a
and
K = -D+Vn+-+ r\+/A+ E
3.10b
21
respectively, where D is the diffusion coefficient and is
the mobility.
Since the plasma must remain approximately neutral, then
r. n+=n-=n 3.11
Equations 3.10 may be solved for the electric field using the
approximations 3.11 to give for the diffusion-induced electric
field
tr Vn fD+ -D- p 3.12
Equation 3.12 may be rearranged to give
2*= Vn f 3X/D- ~ i ?
n lj
Both the ratios of ion-to~electron mobility and ion-to-
electron diffusion coefficient are small compared to one,
being of the order of the ratio of electron-to-ion mass, and
the term in brackets may be set equal to one. Under the
assumption of a Maxwell-Boltzmann distribution of electrons
the Einstein relation may be used for the ratio of diffusion
coefficient to mobility. The Einstein relation is
E = kT 3.1
m e
Therefore, to the order of the approximations used, the ambi-
polar diffusion-induced field becomes
r da
Lr~~ tein dr
3.15
The substitution of 3.15 into 3.9 results in the vanishing
of the convective term leaving only the conductive term
22
3.16
This result does not mean that no convection occurs in
the plasma. Complete cancellation occurs because of the
approximations used in deriving 3.15. However, the effect
of ambipolar diffusion is to reduce markedly the effective
ness of the convective process within the discharge. At the
discharge boundary the situation is quite different. Due to
the fact that electrons and ions recombine at the wall the
energy transport from the electrons to the wall is primarily
a convective process. Within the discharge the reduced
effectiveness of the convective process allows the tempera
ture to approach a uniform value through conduction as will
be shown below.
The radial coordinate and the temperature may be made
non-dimensional by introduction of the variables
3.17
where a is the discharge radius and TQ is the temperature on
the discharge axis. Equation 3.16 may then be written
d ? _ aPr
dP Kfo
3.18
The energy flux Pr may be defined by
3.19
23
where Pw is the energy flux incident on the discharge at (3 =1
and 0( gives the variation of energy flux across the radius.
Substitution of the dimensionless variables into 3.16 yields
the differential equation for the temperature
dr _ a Pw< 3 o
dp KTo
The electron thermal conductivity K is also a function of the
temperature. For the case of no magnetic field, K is given
in Reference 17 as
Snk*T f l ,jJ
m Vet) l 3k' 3m.
3.21
The number density n may be made non-dimensional by intro
duction of the variable 7£ defined by
3.22
where nQ is the number density on the discharge axis. Equa
tion 3.21 may be written as
i/= T>Z
171
where is averaSe electron-ion collision frequency,
and g^'and g^ are correction factors, depending on the
magnetic field. For a fulJLy ionized gas in a zero magnetic
field their values are given as
3.23
%' = 06 538
3M = -395T
3.24
24
The average electron-ion collision frequency is also
a function of temperature which, for a MaxwelX-Boltzmann
distribution, is given by
3.25
where I is the gamma function, and Yei is defined for a
singly ionized gas in terms of the coulomb logarithmby
Yei = 4-tt
3.26
where £ is the permittivity of free space. The function_/\_
is also a function of temperature but log-A_is a slowly
varying function of temperature and its temperature dependence
will be neglected. Introducing the dimensionless variables
into 3.25 and substituting into 3.23 give
-/mi J/v3-/8
K- Jr^Tz
Yec
3.27
The quantity Ce is defined by
K-ir^r^C,
Substituting 3.28 into 3.20 yields the differential equation
dr= aft*
* de V/zCe
3.29
25
which simplifies to
a Tw
d (rV =
d P Z*C
3.30
Upon integration and application of the boundary condition
r(0) = 1
3,31
the temperature distribution becomes
r7/z 1-fw~Ce JadP
3,32
The quantity Ce may be written as
. 7/ -8
r = 80 rfs/\ //z Tj k
7 1 ( 77
- 3k'
3,33
All the terms in Ce are constants except for log_/\_, There
fore, the temperature distribution may be written as
1-3.767*10* Wdp
3,34
where the constants have been evaluated in mk.s, units.
The function 0( describes the radial dependence of the
input energy flux. It has the value zero at (0 q anij has a
maximum value of unity at ^ -1, Thus, the maximum value of
the integral must be less than one. The temperature depen
dence is seen to depend critically on the parameter
26
7/2
a Pw logV\_ /T For the discharge for which measurements
are presented in Chapter II,
CL /?. 5" m
fZ,~ Z.9 XJO* watt/m* 3 3g
l~1.8 x 10* K
/oj -A ~ 10
and
a R IoqJ\-
T0V*
-J2
9.3 *10
ia/a tt
m k( 7/z
3 o 36
The equation for the temperature for this case would be
r
%=i 3-SxJO
3.37
It is clear that the term including the integral may be
neglected compared to one. Thus, for this case, it is seen
that the temperature is essentially uniform which is in
agreement with the experimental data.
The above analysis has been performed for a system in
which the only energy transport mechanism is thermal conduc
tion as may be seen from equation 3.16. This situation arises
because the diffusive energy transport term is completely
cancelled out when the electric field due to ambipolar
diffusion is substituted into the energy transport equation.
However, since ambipolar diffusion is occurring it is obvious
that some energy transport is involved. The result of nearly
uniform temperature derived above will mean that the tempera
ture gradient is small, and the propriety of neglecting the
27
energy transport through diffusion with respect to that due
to conduction is questionable.
An equation similar to 3.9 which includes the energy
transport due to the ambipolar diffusion may be written
K
dn
D dr
K
dT __
dr
- -ft
where is given by the expression
3.38
3.39
In view of the nearly uniform temperature result which was
obtained above, it is reasonable to assume that will vary
only slightly with the radial coordinate.
If Kp is assumed to be a constant function of T0 then
by substitution of the dimensionless quantities 7" y and
p, equation 3.38 may be written
d('f,/e) _ a & rtc Kd d>2
dP TpCe T07/*Ce dp
The solution of this equation analogous to 3.34 is
3.41
28
The influence of the diffusive energy transport on the
temperature distribution is given by 3.41. The relative
importance of this term may be determined by considering the
ratio n0Kj)/aPw. The value of nQ corresponding to the values
given in 3.35 is
3.42
and
flo Kt>
a Pw
3.43
It is seen that the diffusive energy transport has even less
effect on the temperature distribution than the term involving
the energy input. Thus, the result that the temperature
distribution is essentially uniform is unchanged.
From the previous example it is apparent that for
practical laboratory discharges the electron temperature
should be essentially uniform across the radius, since for
7 / 0
any significant change to occur the parameter aPwlog-A/TQ
would have to be increased by an order of magnitude or more.
In the following chapters this important result will be used
as a basis upon which calculations are made for the electro
magnetic field in the discharge and for the development of a
complete theory of operation of the electrodeless discharge.
The development of the theory will follow the general outline
of theories for the DC glow discharge first given by Schottky.
These theories are discussed in detail by Francis.
CHAPTER IV.
ELECTRON TEMPERATURE AND NUMBER DENSITY DISTRIBUTION
In the previous chapter, an expression was derived for
the electron temperature distribution. For the discharge
in which the experiments were performed it was found that
the electron temperature was, essentially, constant with
radius. In this chapter, a theory will be given, based on
the solution of the plasma balance equation, which predicts
the discharge temperature. The solution of the plasma
balance equation will also yield an expression for the
electron number density distribution. This expression will
contain another unknown quantity, nQ, which will be deter
mined by performing a power balance after the electro
magnetic field in the discharge is calculated.
Certain assumptions must, again, be made in order to
solve the plasma balance equation. First, as a consequence
of the uniform temperature, it will be assumed that the
diffusion coefficient is constant with radius. Secondly,
it will be assumed that the average ionization collision
frequency is independent of position. This assumption is
based on the experimental results described in Chapter II
and is also a consequence of the uniform temperature.
Essentially, this assumption means that the ionization
process is thermal. The physical picture is that the
energy added to the electron gas in a non-uniform manner by
the electric fields is rapidly transported to the regions
of lower energy addition by thermal conduction. At the same
time the directed velocity of the electrons induced by the
29
30
electric field is rapidly randomized by the electron-neutral
collisions. The net result is a distribution of electron
velocities which is random in direction and characterized by
a temperature which is uniform It is now assumed that the
ionization is due to those electrons in the distribution
having energies in excess of the ionization energy The
ionization frequency is determined by averaging the ioniza
tion cross-section over the electron distribution function
Thus, the ionization function, ^/Da, will depend on both
the electron temperature and the form of the distribution
function Throughout the development of this theory, it
will be assumed that the electron distribution function is
Maxwell-Boltzmann Justification of this assumption can be
made from a theoretical standpoint (Reference 17) and also
by comparison of the experimental results with calculations
made using other distributions.
The flux of charged particles of species j is given by
4.1
where Dj is the diffusion coefficient for the particle
species and is the mobility of the particle species.
When the number density of charged particles is small both
ions and electrons will flow independently, each flowing in
when the number density becomes larger, space charge fields
will be established due to the different flux rates for ions
and electrons. Since the electrons have a higher flux rate
initially, they begin to move out of the discharge, leaving an
excess of ions. The resultant space charge field retards
the electron flux and increases the ion flux. Eventually,
the process attains a steady state where the flux rates of
31
ions and electrons are the same. This is the condition for
which the diffusion is considered to be ambipolar. Some of
the results of this type of diffusion have been used previ
ously in Chapter III. The transition from free to ambipolar
diffusion and the criteria for its occurence are studied in
1 Q
detail by Allis and Rose.
The flux equations for electrons and ions are
/! = -£>_ Vn. + n_//_ E
and
/J = -ZWn+ +
For fully developed ambipolar diffusion, the flux rates must
be equal. Some differences between electron and ion number
densities exist to establish the electric field, but only
small differences in number density may exist in a plasma
and thus,
T -T? = 7* 4.
and
H+- =/?_=/?
4.3b
Substitution of 4.3 into 4.2 and elimination of E give
M+- A- J
Vr\
r= -
4.4
32
The term in parenthesis is called the ambipolar diffusion
coefficient Da. The flux of either ions or electrons is
given by
r -Da Vn
Consider a volume W of surface S where particles are being
produced and from which particles are being lost. Continuity
requires that
d_
olt
T-fidS = \ V-fdW
5 vv
4.6
Since 4.6 must hold for each volume,
ff = V-f = 4.7
Equation 4.7 gives the loss rate of ionized particles due to
diffusion. It is assumed that the ionization is produced by
the high energy "tail" of a Maxwell-Boltzmann distribution of
electrons. The ionization rate is given by
dt = <**> n
The average ionization frequency is determined by
averaging the quantity
u =
4.9
33
over the distribution function Qx is the ionization cross-
section, g is the relative velocity between the electron and
a neutral particle and ng is the neutral particle number
density In a steady state the loss of ionized particles,
represented by 4.7, and the number being created, represented
by 4.8, must balance. Thus, the plasma balance equation is
given by
V2n + ^ n = 0 4.i
The solution of this equation, subject to the condition that
the number density vanish at the boundary, yields a number
density distribution dependent on an eigenvalue. The eigen
value in turn determines the temperature.
To obtain a more general solution to 4.10, the require
ment of an infinitely long discharge will be relaxed to
allow a finite length L. By separation of variables, the
solution of 4.10 is found to be
4.11
where JQ is the zeroth order Bessel function, z is the axial
coordinate and ^ is a separation constant. Applying the
boundary condition at z = L/2 and r = a results in
(Zc +l)yr
/= OjljZ
4.12a
34
jo^> A J <2 £= **' 4.12b
where the Jq^ are the zeros of JQ. Since the number density
cannot become negative anywhere within the discharge, only
the first zeros have physical significance, and
A = A 0 4.13a
and
CL
h = *ojr
The number density may now be written
4.13b
n- n0cos(Z- ^J0(.4-os£)
4.14
where n Q is the number density at the point r = z = 0.
Solving 4.13b,
M_ A
D* A*
a*
[*78* H^rfl
4.15
where A d is the diffusion length. For a long cylindrical
discharge a/Ll and
6> £ f.784
4.16
35
The eigenvalue equation 4.16 makes it possible to calcu
late the discharge temperature. The average ionization
collision frequency is a very strong function of the electron
temperature due to the fact that the cross-section has a
non-zero value only for electron energy greater than the
ionization energy. There is a weak dependence of on
the neutral gas temperature through the relative velocity g,
but since the electron velocity is much greater than the
neutral velocity, g is assumed equal to the electron velocity.
With this assumption 0J^ is a function only of the electron
temperature for any particular gas.
The ambipolar diffusion coefficient Da is a function of
both the electron and ion temperatures. From equations 4.4
and 4.5 Da is defined as
A+D- -M-D+
A -yU-
4.17
The mobility of ions is generally much smaller than that of
the electrons and
= T>+( i-
M.n+)
The Einstein relation
4.18
Di 4.19
A] e
reduces 4.18 to
A-hklZ.
e
4.20
36
In an active discharge at low pressure the electron tempera
ture is usually much larger than the ion temperature, and
thus
e
4.21
The ambipolar diffusion coefficient is primarily a function
of electron temperature, and the neutral gas pressure through
the ion mobility. The ratio (j) /Da becomes a function of
temperature and pressure and the eigenvalue equation 4.16
determines the electron temperature for a given value of
discharge radius and neutral gas pressure.
The theory given above yields the somewhat surprising
result that the electron temperature is independent of the
power input to the discharge. This result is due to the
assumption that the ionization is not directly related to
the electric field intensity, but is completely accounted
for by averaging the ionization frequency over a Maxwell-
Boltzmann distribution whose characteristic temperature is
uniform. It is this assumption which makes the ionization
function Z^i/Da indePendent of the electromagnetic field and
yields the eigenvalue equation 4.16.
The physical process is one in which power is absorbed
from the electromagnetic field which raises the electron
temperature until a steady state is reached. A further
increase in power causes a transient increase in the ioniza
tion rate until a new steady state is reached with a higher
level of electron number density.
According to equation 4.14, the radial distribution of
number density will be proportional to J0. This results from
the fact that, due to the uniform temperature, *^^/Da is not
a function of the radial coordinate. The solution due to
37
Eckert, which was discussed in Chapter II, reduces to this
form when q = 0 (equation 2.21). The distribution of number
density becomes very important in the calculation of the
electromagnetic field in the discharge. The electromagnetic
field depends on the plasma conductivity which, in turn,
depends upon the electron number density and temperature.
Since the temperature has been found uniform with radius,
the radial dependence of the conductivity will be determined
by the number density distribution.
CHAPTER V.
THE ELECTROMAGNETIC FIELD
The calculation of the electromagnetic field in the
discharge is a central aspect of any theory for the dis
charge. The electric field and power input must be known
in order to relate the plasma conditions to the applied
fields. For high-frequency discharges this problem is more
difficult since the plasma conductivity is a function of the
applied frequency. Also, in a high-frequency discharge, the
fields are inherently non-uniform due to the "skin effect."
The term arises from the study of high-frequency fields in
good conductors where the fields are attenuated within a
short distance in the conductor and the current flows in a
thin region near the surface.
An early attempt at an analysis of the electrodeless
discharge by Thomson^ resulted in a one-dimensional calcu
lation of the field in the discharge. Thomsons model was
that of an infinite cylindrical plasma in an infinite sole
noid. He assumed the plasma would have a real, uniform
conductivity across the plasma. The conductivity of the
plasma is neither real nor uniform in the discharge, as
recognized by Eckert.^ But Eckert obtained the same solution
as Thomson as a result of his assumption that the reactive
part of the conductivity was small, and by solving the
electromagnetic equations using a uniform conductivity
having the average value of the non-uniform conductivity.
R. J. Sovie^0 calculated the fields assuming a uniform
conductivity, but he allowed the conductivity to have a
reactive term so that the conductivity was a complex quantity.
38
39
All of the above analyses are characterized by the fact
that the conductivity was assumed to be uniform. In a
diffusion-controlled discharge this will never be true as the
number density must be zero at the discharge boundary. The
analysis of Chapter IV shows that the number density will be
a function of the radius as given by equation 4.14. A solu
tion for the electromagnetic field may be obtained for a
conductivity which approximates this distribution.
The plasma conductivity (T is usually defined as
5.1
where 10 is an equivalent collision frequency and u) is the
radian frequency of the applied field. The equivalent
collision frequency is a quantity obtained from certain
averages of the momentum collision frequency over the
electron distribution function. The particular function to
be averaged depends upon whether the radian frequency is
large or small compared with the equivalent collision fre
quency (Reference 17, Section 4-1.2). If the radian fre
quency is of the same order as the equivalent collision
frequency, certain correction factors are necessary due to
variations of pm with velocity. These factors modify both
P and CQ and have been calculated by Dingle.21 They are
presented in Reference 17, Section 8-1. Including these
correction factors, the conductivity can be written as
5.2
where g and h are the correction factors and is the
equivalent collision frequency averaged for CO 'p T!tie
40
conductivity will be assumed to be of the form 51 where it
is understood that the correction factors g and h have been
applied. In equation 5.1 the equivalent collision frequency
^ is a function of the electron temperature and the pressure
of the neutral gas. Since the electron temperature is uni
form will be independent of the radial coordinate and,
since O' is directly proportional to n the conductivity
will have the same radial dependence as n From 4.14 it is
seen that the radial dependence will have the form of JQ,
the zeroth order Bessel function. It will again be assumed
that the discharge is infinite in axial extent and therefore
the axial dependence of (T will vanish. The JQ Bessel
function may be expanded into the series
1/4 2* ,
7W
(Zl)z (3l)2~
5.3
Therefore, a first approximation to n which vanishes at the
boundary is
n(r)= n0
5.4
where nq is the number density on the axis. With this
approximation the conductivity becomes
cr(r) =
r\Qez
m
5.5
Maxwells equations for the electromagnetic field can
be reduced to two wave equations in the scalar potential V
41
and the vector potential A (see Pugh and Pugh^) as
5 6a
and
5.6 b
where and £ are the permeability and permittivity of the
medium respectively, t is time, is the charge density and
J is the current density. Except for the small charge
density which establishes the radial field due to ambipolar
diffusion, the plasma will be assumed to be electrically
neutral, fA = 0 a Faraday cage is installed in the experi
mental apparatus between the solenoid and plasma to prevent
the axial electric field of the solenoid from being impressed
on the plasma. Therefore, it will be assumed that the
potential within the discharge is everywhere zero, and only
5.6b must be considered. Again, the model assumed is that
of an infinite cylindrical discharge, possessing axial
symmetry and placed within an infinite solenoid. The sole
noid carries a high-frequency current
5.7
where IQ is the peak value of the solenoid current, 60 is
the radian frequency and is a unit vector in the azimuthal
direction. Under the above conditions the equation 5.6b
becomes a scalar equation in the azimuthal component of the
vector potential A namely,
42
dzAs> i 1 c>Ai> 1 a £ c>zAa> i j T />
5.8
The azimuthal current density is given by
, t Mi>
0 = cr l < j, ^ dt
5.9
where the relationship between the electric field and the
vector and scalar potentials,
r = _(il +VV)
5.10
has been applied. In view of the form assumed for the
applied current, it is reasonable to assume that the vector
potential will have a similar time variation. Assuming that
A$ = Ae
cot
5.11
equation 5.8 reduces to the ordinary differential equation
A"+j:A'-yz A +/Ueco&A-t
where primes denote differentiation with respect to r. The
conductivity o' can be written as
1-
m v*+coz
5.13
43
A "skin depth" 8 is defined by
8
*-
/ ip p\Vfe i
m {Vz+coz) I
jo co Hoe*- J
5.14
The term
1 1- GO
5 o 15
For the plasma we shall assume that the permeability is that
of free space /o and since the plasma is considered to be a
conducting medium, the permittivity is that of free space £0
(see Holt and Haskell,23 Section 11.2). The term /o;oC0Zcan
be written as
5.16
where f is the applied frequency, c is the velocity of light
and \ is the free-space wavelength of the applied frequency.
Equation 5.12 becomes
rM/y + rA'
44
For the frequency and discharge size of interest
znr
and, thus, the first term in brackets will be neglected
Defining a dimensionless coordinate
1
5 18
and introducing a new variable
Z
-to-IP
(pz+CO^Vz) j
5.19
5.20
equation 5.17 becomes
z*A"+zA' + A
(8*)&\(vz+co&)VzJ
5.21
where the primes now denote differentiation with respect to z.
Introducing the complex quantity Kei? defined by
a_
s*
-A/z
-L-IV 1
equation 5.21 reduces to
5.22
z*A"+A' +A
* 0
5.23
45
From equations 5,20 and 5.22 it is noted that
z = Ke^p
5.24
and the solution of 5.23 will be a function of Ke^ and ^0 .
The parameter K is a real number which gives the ratio of
r*
the discharge radius to the skin depth and J is a param
eter related to the ratio of collision frequency to applied
radian frequency.
A series solution to equation 5.23 has been given by
B. B. Henriksen.^4 solution which is finite at the origin
is given by
A = a,
TO
i)
< He* *)
ni
i
cz
( Kefc)
,Zr\~l
5.25
n= i
For convenience, the complicated expression on the right hand
side of equation 5.25 will be represented by
a0 Hi 5.26
where aQ is an arbitrary constant. The electric field E^>
and the magnetic induction Bz, are derived from A^ by
and
at
5.27
5.28
46
where the quantities A 4, and Bz are given by
5.29a
cot
E e
5.29b
5.29c
It should be noted that E and B are, in general, complex
quantities having both magnitude and phase. The expressions
for E and B are
E(Ke£f,p) =- Hs
5.30
and
KeffM
d z
5.31
At the discharge radius, a, the boundary condition,
Bz(eLU)= Ke 5.32
is applied. The quantity Bw is the peak wall value of mag
netic induction. Since Bw is a real quantity, the phase of
47
the fields in the discharge will be relative to the phase of
the magnetic induction at the wall. The quantity Bw can be
related to the current in the solenoid by
B =/(aNIa
5.33
where N is the number of turns per unit length, and IQ is
the peak value of the solenoid current. The arbitrary
constant aG is found to be
5.34
In Chapter VI,a power balance for the discharge will
be performed to calculate the number density on the discharge
axis. Thus, it will be necessary to calculate the power
input from the electromagnetic field. The instantaneous
energy flux is given by the Poynting vector S where
S = £ H
5.35
This quantity represents the power which goes into changing
the energy of the electromagnetic field plus the energy
dissipated in the plasma. Since the energy in the electro
magnetic field is alternately absorbed and given up, a time
average of S will give the energy flux which is dissipated
in the plasma. The time average of the Poynting vector S
is given by
= ?m
5.36
48
where H* is the complex conjugate of the magnetic intensity
which is related to the magnetic induction by
B* /o H 5.37
Since the electric field is azimuthal and the magnetic
induction is axial the energy flux is a vector in the radial
direction. This energy flux is given by
p£,v = ?r Eq COS 9 5.38
where Q is the phase angle between E ^ and Hz and Clr is a
unit vector in the radial direction.
CHAPTER VI.
POWER BALANCE
In the preceding chapters expressions have been derived
for the electron temperature distribution, the number density
distribution, and the electromagnetic field. The descrip
tion of the discharge will be completed upon calculation of
the number density on the discharge axis and this may be
accomplished by consideration of the power balance for the
discharge.
The electrons gain energy from the electromagnetic
field and, subsequently, lose their energy through collision
processes or by diffusion to the discharge wall. Both the
energy gained and the energy lost from the electron gas
depend upon the electron number density. For a given pres
sure the steady state will be reached when the energy loss
balances the energy gain and this will establish the steady^-
state number density.
The power balance will be performed on the electron
gas alone. It will be assumed that all of the electrical
energy is absorbed by the electrons since the ions contrib
ute little to the conductivity. This absorbed energy is then
lost from the electron gas by diffusion, by ionization, by
excitation of neutral particles with subsequent radiation,
and by heating of the neutral gas. It will be assumed that
the plasma is at low pressure and slightly ionized so that
the energy lost by heating of the ions and by bremsstralung
radiation may be neglected.
The electrical power absorbed by the plasma was given
in Chapter V. The energy flux given by equation 5.38 is a
49
50
function of Kei^
and therefore nQ as well as the radial
coordinate It will not be necessary to perforin a power
balance at each point within the discharge but only for the
discharge as a whole. Thus, it is not necessary to calculate
the average energy flux as a function of the radial coordinate,
but only the total energy flux incident on the discharge at
the discharge boundary. This quantity will be called Pw and
can be calculated from the equations of Chapter V with p = 1.
Since the power input is calculated per unit surface area of
the discharge, the power loss will also be referred to the
unit surface area.
The flux of electrons due to ambipolar diffusion is
given (equation 4.5) as
-DaVn
6.1
The total loss of electrons from the discharge per unit
surface area is found by evaluating this quantity at the
discharge boundary. Thus the total flux of electrons is
given by
6.2
The number density is given by equation 4.14 which reduces
for an infinite discharge to
6.3
The number density gradient is given by
6.4
51
where J-^ is the first order Bessel function. Evaluation of
this quantity at r = a yields
/dn~) =_ 1.24-8 n 6.5
\ d rJr,a a
The average energy per electron is given by
- e\lr 6.6
where k is Boltzmann's constant, T is the electron temperature,
and VT is the electron temperature expressed in volts as
defined by 6.6. The energy transported to the wall through
diffusion becomes
e1 = l^48_n DaeVT 6.7
a 7
When an electron suffers an ionizing collision it loses
an amount of energy equal to the ionization energy. This
energy is given by
E eV[ 6.8
where is the ionization potential of the neutral atom.
The number of ionizing collisions can be found directly by
calculating the average ionization frequency and integrating
over the number density distribution. However, since the
number of ionizing collisions must equal the number of ions
lost by diffusion, the number of ionizing collisions can be
obtained from the diffusion loss. Since the diffusion loss
52
has been calculated already, the latter method will be used.
From 6.2, 6.5 and 6.8 the total energy loss per unit surface
area due to ionization is given by
6.9
When an electron suffers an inelastic collision in which
a neutral particle is left in an excited state, the neutral
particle radiates this energy. The energy loss per collision
is given by
e:* = v
6.10
where Vx is the excitation potential. The number of these
collisions occurring per unit time per unit volume is given
by
6.11
where (Vx/ ^be averaSe excitation collision frequency.
This quantity is obtained by an appropriate average of the
excitation cross-section Qx over the distribution function.
Since it is the total energy loss per unit surface area that
must be obtained, it is necessary to integrate over the
number density and to divide by the surface area. The total
number density in a unit length of the cylinder is given by
2tt
n(r) rd@dr
6.12
0 0
53
Substituting 6.3 into 6.12 and integrating yields
G¡ = 0.2159 n0 Zna;
6.13
The surface area per unit length is
S = Z TTCL
6.14
and the total number of electrons per unit surface area is
given by
-§- = O.Zl59n0a
o
6.15
From 6.10, 6.11 and 6.15 it is seen that the energy loss due
to excitation per unit surface area is given by
65= 0.Z159 n0a(Vx)eVx
6.16
The loss of energy to heating of the neutrals is
calculated in a similar manner to that for excitation. The
average energy loss of an electron in an elastic collision
with a heavy neutral particle of mass M is given by
Ee-i
6.17
Using equation 6.6, this may be written as
F o V
ce M ^ VT
6.18
54
By analogy to equation 6.16, the energy loss per unit surface
area due to neutral heating is
6.19
where is the average momentum collision frequency for
electrons with neutral particles.
The power balance equation is obtained by equating the
power input at the wall per unit surface area to the total
power loss per unit surface area
Rv £ a + + £3 +£4.
6.20
Substituting from 6.7, 6.9, 6.16, and 6.19, the power balance
becomes
6.21
This equation can be solved for n0, thus completing the
theory for the operation of the discharge.
In summary, expressions have been derived for the elec
tron temperature and number density for a long cylindrical
electrodeless discharge. In addition, the electromagnetic
field in the discharge due to an infinite solenoid has been
derived. All of these quantities will depend on the physical
characteristics of the solenoid and the discharge. These
characteristics are:
55
1. The gas used in the discharge
2. The discharge pressure
3. The discharge radius
4. The frequency of the applied field
5. The number of turns per unit length of
the solenoid
6. The current carried by the solenoid.
In Chapter VII a sample calculation will be made for the
same discharge in which the experiments of Chapter II were
performed. In addition, the electromagnetic field will be
calculated for a more general range of physical parameters.
CHAPTER VII.
CALCULATIONS
A theory for the electrodeless discharge was developed
in the preceding chapters but no calculations were made
there. The values for the various plasma properties will
depend on the physical characteristics of the particular
discharge under consideration. However, the electromagnetic
field solutions are functions only of and so that
rather complete calculations can be made for the field dis
tributions which are applicable to a wide range of discharges.
A sample calculation will be made for a discharge having
the physical characteristics of the one in which the experi
ments of Chapter II were performed. Only certain limited
comparisons of experiment with theory can be made due to the
lack of sufficient experimental data. The discharge for
which the calculations will be performed has the following
characteristics:
1. The gas is argon.
2. The pressure is in the range of 0.1 Torr to 1 Torr.
3. The discharge radius is 2.4 cm.
4. The applied frequency is 4.5 MHz.
5. The number of turns of the solenoid is 139 per meter.
6. The solenoid current is an independent variable.
It has been shown in Chapter III that the temperature
distribution depends upon the parameter Pw log-A/T0^^. It
was also shown in that chapter that for the discharge being
considered the variation of the temperature across the dis
charge radius is negligible and, thus, it is not necessary to
calculate results from equation 3.40.
56
57
The determination of the value of the temperature, as
derived in Chapter IV, involves the calculation of the average
ionization collision frequency. Since the determination of
average collision frequencies is required for many of the
calculations, the general procedure will be discussed.
In general, collision frequencies depend upon the elec
tron energy. To obtain the average collision frequency it
is necessary to average over the electron distribution func
tion f. The collision frequency for the process is
related to the cross-section for that process by
Vj = Qj sn9 7.1
where Qj is the cross-section for the process, g is the
relative velocity between the particle and its target
particle, and ng is the number density of target particles.
The collision frequencies to be calculated are the ionization
collision frequency, the elastic or momentum collision fre
quency, and the excitation collision frequency. All of these
processes involve the collision of an electron with a neutral
particle. The electron temperature in a low-density plasma
is usually higher than the neutral temperature. For this
reason, and also because the electron mass is much smaller
than the neutral mass, it is assumed that the relative
velocity g is due solely to the electron velocity, i.e.,
g v-?/3 = v 7.2
and becomes
Tj = QjVHg
7.3
58
Two different averages will be considered. The average
momentum frequency^m^ is involved in the plasma conductivity.
As pointed out in Chapter V, the type of averaging required
depends upon the applied radian frequency relative ,
Following reference 17, it will be assumed that(ym^ is calcu
lated for the high-frequency case. Thus, is defined by
7.4
where v is the electron speed, and where it is assumed that
the distribution function f depends only upon v .
In general, the average of a function of electron speed
is given as
<$) = 7f
7.5
Thus, the average collision frequency for the process is
given by
7.6
From the expression for the collision frequency 7.3, equa
tions 7.4 and 7.6 become
7.7
and
7.8
59
The average collision frequencies are seen to be functions
of the appropriate cross-section, the distribution function,
and the neutral number density rig.
The distribution function, as discussed previously, will
be assumed to be Maxwell-Boltzmann. This distribution is
given by
n /3 fe n Z
T = [ZTTj (y*}3'*. & 7.9
where <(v2^> is the mean square electron speed and is related
to the electron temperature by
-S. m kT
7.10
It is convenient for computational purposes to express elec
tron speeds in terms of volts. The following expressions
serve to define V and V.-p
-L m V* = e V
and
I- m(vz> =eVT
7.12
Substitution of 7.9, 7.11 and 7.12 into 7.7 and 7.8 yields
\V
/V\- 3(3e/nm) r)a
( (Vt)5*
7.13
60
and
(Vj) = s(3e/rrm)Vz
(%)** J.
7.14
where Qm and Qj are functions of V and the limits of integra
tion are from 0 to oO as shown.
The calculation of average collision frequency from 7.13
and 7.14 depends on the form of the appropriate cross-section.
The cross-sections which are used are all determined experi
mentally. The computational scheme used involved finding a
mathematical approximation for the experimental cross-sections
and performing a numerical integration by use of Simpson's
rule on the IBM 7090 or 360/50 computer.
The momentum cross-section Qm for argon was obtained
O C
from Barbiere who averaged the angle dependent data of
Ramsauer and Kollath. An analytic approximation to the
Barbiere data was found by use of a curve fitting program
obtained from the University of Florida computing center.
Since the data given by Barbiere extended only to 12.5 volts,
an approximation to data given by Brown26 (who took it from
Brode) was added to the Barbiere approximation to extend the
cross-section to higher values. The final form for the
approximation is
and
7.15
The results of the computation for^p>m\ are given in Figure 7.
61
The average ionization collision frequency was calculated
from equation 7.14. The cross-section data were obtained from
Kieffer^7 who tabulates data from five different investiga
tions. The data of Smith were chosen as there was little
difference in the data of the various investigators. An
approximation to the Smith data was found by use of the pre
viously mentioned program. The approximation is given by
QL= 0 V < 15.7
and 7.16
Qi= 10 [-3.734 + 0.313V- 4.400 tJO V
+ Z.OZB xlO'^V3) VZIS.7
The results of the computation for are shown in Figure 8.
To calculate a power balance, it is necessary to deter
mine the losses due to excitation and radiation. It is not
necessary to determine the losses to a particular excited
state since it is the energy loss from the electrons to all
excited states which enters into the power balance. An
experimental determination of the total excitation cross-
section for argon was obtained by Maier-Leibnitz^ and
presented by Brown^ who obtained the data from a paper by
Druyvesteyn and Penning. 29 The data cover the range from
10.8 V. to 19.0 V. and a constant cross-section is assumed
for energies greater than 19.0 V. The approximation for the
excitation cross-section is given by
Q*=0 VC10.8
Qx = 10 [o.048(v-10.8) ] JO.8
Q. = t0* V>19.0
The results of the computation for are shown in Figure 9.
62
In addition to values for the average collision fre
quencies, it is necessary to determine the ambipolar
diffusion coefficient. The ambipolar diffusion coefficient
was shown in Chapter IV to be related to the ion mobility
and electron and ion temperatures by
7.18
In a low-pressure gas discharge the electron temperature is
usually much greater than the ion temperature. In the worst
case the two temperatures would be the same. From 7.18 it
is seen that in the two limiting cases, T+/T_ = 0 and
T+/T_ = 1, the ambipolar diffusion coefficient varies by a
factor of two. Many measurements of ion mobility have been
made and the accepted value for the reduced mobility of
O A
1.5 cm2 v-1 sec-1 was obtained from Hasted. u The reduced
mobility, KQ, is related to the mobility by
r, >#+1 273
760 r3
7.19
where p is the neutral gas pressure in Torr and Tg is the
neutral gas temperature. Written in terms of the neutral
gas number density ng the expression for the ambipolar
diffusion coefficient for argon becomes
ndDc? = Z. 69* 10** ZT(l + 7.20
It was shown in Chapter IV that the electron temperature
was determined from the eigenvalue equation 4.16 since the
quantity (y)A/Da is primarily a function of electron tempera
ture. The dependence of/^|\/Da on electron temperature for
63
the limiting cases T+/T_ = 0 and T+/T_ = 1 is shown in
Figure 10. The function <(^i) /Da is a function of the neutral
gas number density and, therefore, the neutral gas pressure
and temperature. The neutral gas temperature in a plasma is
difficult to determine accurately. At low pressure only a
small portion of the energy is lost to the neutral gas, and
since the discharge walls are water cooled in the experimental,
apparatus it will be assumed that the neutral gas temperature
is 300K. Thus, the neutral gas number density is given by
7.21
where p is the pressure measured in Torr and ng has the units
m-3. Using equation 7.21 for ng,
2.4 cm for the discharge radius and the eigenvalue equation
4.16, it is possible to determine the electron temperature
as a function of the discharge pressure. This curve is
presented in Figure 11. The data points shown in Figure 11
are the electron temperatures as measured by the probe
corresponding to the number density data shown in Figures 2,
3 and 4. It is seen that the electron temperature decreases
as the pressure increases. This is due to the fact that the
diffusion loss becomes less as the pressure is increased, and
a smaller temperature is sufficient to produce the ionization
required for a plasma balance. The measured electron tempera
tures are consistently lower than the predicted temperature
by approximately 20 per cent with an average discrepancy of
19.6 per cent. This discrepancy is not unreasonable for
probe data and may be due to the lack of purity in the gas
used. The discrepancy could easily be explained by contami
nants having larger ionization cross-sections or lower
mobility. One such contaminant which might have been present
is mercury, since an untrapped McLeod gauge was used to
measure the discharge pressure.
64
It was shown in Chapter V that the electromagnetic
field depends only upon and p The parameter Ke*-^ is
related to the plasma properties by
~~ C 1 f-nZ ,
r i{VZ-^COZ)yz
7.22
where K is a real number given by the ratio of discharge
radius to skin depth. K may be expressed, in terms of the
plasma properties, as
7.23
From 7.23 it is seen that K depends on the electron number
density through nQ and the electron temperature and neutral
pressure through ^ Since the electron temperature-
neutral pressure relationship has been determined for the
discharge, it is possible to calculate K. The parameter f
is a function of the ratio of the collision frequency to
the applied radian frequency given by the expression
7.24
Physically, is a parameter related to the degree to which
the plasma conductivity is resistive or reactive. It is also
a function of the electron temperature and neutral pressure
through "p .
65
Computer calculations have been performed to obtain the
magnitude of the electric and magnetic fields for three
values of l^/CO namely 0, 1 and which correspond to a
purely reactive, equally reactive and resistive, and a
purely resistive plasma, respectively. These calculations
were performed for integer values of K from 1 through 9. For
convenience, the field quantities have been made non-
dimensional with respect to Bw the value of the magnetic
induction at the wall. The corresponding non-dimensional
quantities are
. B and El 7.25
Bw 0 Byytf
Curves for these quantities are presented in Figures 12, 13,
and 14. In order to perform a power balance it is necessary
to compute the average energy flux into the discharge at the
discharge boundary. This is the quantity Pw of Chapter VI
and it is calculated from the time average of the Poynting
vector given in Chapter V. This quantity is also made non-
dimensional;
r-, # J2 ~Fw
P co B£ a 7-26
Curves for this quantity as a function of K for various
values of 7^/C are presented in Figure 15. The quantity P*
can be thought of as representing the energy flux per unit
current squared since
Bw stio NI0
7.27
66
The curves of Figure 15 show that this quantity has a maxi
mum at some value of K. The maximum occurs because at small
values of K the power input is low due to the small conduc
tivity, and for values of K greater than that at which the
maximum occurs the fields induced by the solenoid are unable
to penetrate very far into the plasma and the power is
dissipated in a thin region near the surface. Therefore,
the most efficient operating condition for the electrodeless
discharge, in terms of coil current required, occurs where P*
is a maximum.
The electron number density at the center of the dis
charge will be a function of the power input to the discharge
and, therefore, a function of the coil current. Equation 6.21
can not be solved explicitly for nQ because of the complex
dependence of Pw upon nQ. Therefore, a graphical solution
will be obtained. The quantity P*, defined by equation 7.26,
is a function only of Ke^f. Since for a given discharge the
temperature is a function of pressure, then is determined
for each value of pressure. With ^determined for a given
pressure P* becomes a function of n Equation 6.21 can be
made non-dimensional such that
P* =
<0 /Q /?Q 6
coB% a
D^[yr+vt]
0. z 15s a VT +
7.28
For a given discharge radius, frequency, and number of
solenoid turns per unit length, the right hand side of 7.28
is a function of nQ and the coil current through the term
2
Bw It will be convenient to express Bw in terms of the
root mean square value of the coil current such that
67
Bw = /So A/I0 /So//1rms 7.29
The current Irms will be taken as an independent parameter
and, thus, the right hand side of 7.28 becomes a linear
function of nQ. For each value of discharge pressure,
equation 7.28 is solved by plotting P* and the right hand
side of 7.28 as a function of nQ for various values of Irms.
A separate graph is required for each value of the pressure.
P* was obtained for the temperature consistent with each
value of pressure by computing the solutions for the elec
tromagnetic field given in Chapter V. The IBM 360/50 computer
was used to calculate the required series. The series is
complex and the real and imaginary parts oscillate in sign
and acquire large values before they begin to converge.
Therefore, it was necessary to program the computer for
sixteen significant digits to insure sufficient accuracy over
the required range of values for K. A sample of the graphical
solution is shown in Figure 16. This solution is for a dis
charge pressure of 0.5 Torr.
The solution of the power balance equation as described
above results in a family of curves, one for each discharge
pressure, relating the electron number density at the dis
charge axis to the applied current in the solenoid. The
family of curves for the sample calculation is shown in
Figure 17.
CHAPTER VIII.
CONCLUSIONS
In the preceding chapters a probe diagnostic technique
was described with which it becomes possible to resolve experi
mentally the spatial distribution of electron number density
and temperature as well as the plasma potential in an elec
trodeless discharge. The results of these experiments were
found to disagree with some of the commonly held assumptions
about the electron temperature distribution and the spatial
production of ionization in the discharge. This discrepancy
resulted in the formulation of a new theory for the operation
of the electrodeless discharge which was found to be in
reasonable agreement with the probe measurements. Although
the theory agrees with the measurements, a considerable
number of predictions of discharge behavior have been made
which have not been subject to direct experimental verifica
tion. However, in the course of the experiments, certain
qualitative observations have been made which can be compared
with the predictions of the theory. These comparisons will
be discussed later.
First, it would seem appropriate to point out some of
the limitations and critical assumptions made in arriving at
the theory for the electrodeless discharge. The model adopted
for the discharge makes the problem one-dimensional. This is
due to the choice of a model which assumed that the discharge
was infinitely long, and possessed axial symmetry.
For an actual laboratory discharge the assumption of
axial symmetry would appear to be quite reasonable. This is
68
69
due to the fact that the discharge is usually formed in a
container which is a circular cylinder surrounded by a coil
of circular cross-section. The assumption of a discharge
having infinite length is less reasonable. All laboratory
discharges must be finite in length even though the length
can be made large compared to the radius. This fact imposes
a more serious limitation to the theory due to the axial
variation in the number density which is induced by a dis
charge of finite length. It is seen from equation 4.14 that
the number density will follow a cosine variation in the
axial direction. For a discharge which is long compared to
its radius, the error in the predicted electron temperature
will be small as can be seen from the eigenvalue equation 4.15.
However, the calculation of power input will be considerably
in error for the discharge as a whole since the value of K,
upon which the power input depends, will vary from a maximum
at the center of the discharge to zero at the ends of the
discharge. The power input as a function of the number den
sity at the discharge axis should be approximately correct,
however, and a reasonable calculation of total input power
might be made by integration of the power over the length of
the discharge. The input power is a function of the axial
position through K which is, in turn, a function of the number
density on the discharge axis.
A further approximation has been introduced by the
assumption of an infinite solenoid having a purely azimuthal
current flow. In actual practice the coil is a helix of
finite length. For such a coil the electric and magnetic
fields are not purely azimuthal and axial, respectively. The
electric field will have an axial component and the magnetic
field will have an azimuthal component due to the helix
angle and, also, at regions near the end of the coil, large
deviations from the solenoidal field will occur. If the coil
70
is relatively long with respect to its diameter, the deviation
from a solenoidal field at the center should be small. In
this region the electromagnetic field predicted by the theory
should be a good approximation. The only electromagnetic
quantity involved in the rest of the theory is the power
input and in the region near the center of the coil the result
should be quite accurate.
Due to the limitations imposed on the theory from the
assumptions of infinite length, the prediction of the number
density on the discharge axis as a function of the solenoid
current can only be approximate. However, the general trend
as shown in Figure 17 should be quite representative for a
laboratory discharge.
The assumption of a Maxwell-Boltzmann distribution for
the electrons is, of course, only an approximation. Devia
tions from this distribution may have a large effect on the
calculated values of the average collision frequencies for
ionization and excitation. This results from the fact that
the cross-sections for these processes are non-zero only
in the high-energy portion of the distribution function. For
the average momentum collision frequency this effect will be
much smaller. This fact is shown by Reference 17 where it
is found that various averages over both a Maxwell-Boltzmann
and Druyvesteyn distribution differ by less than 15 per cent.
In Reference 17 it is shown that the distribution func
tion may be expanded in spherical harmonics in velocity space.
Due to a spatial effect, integration over the second order
term yields no difference in the calculated average ionization
or excitation collision frequencies. Therefore, only when
higher order approximations are important will the values of
ionization and excitation collision frequencies be affected.
Some calculations of the average ionization collision
frequency were made using the Druyvesteyn distribution and
71
these led to predicted electron temperatures which were
several electron volts higher than those measured. For this
reason the assumption of a Maxwell-Boltzmann distribution
appears to be justified.
Certain qualitative observations made in the course of
the experimental investigation tend to support the theory.
One striking feature of the theory is that of a constant
electron temperature, for a given discharge pressure, inde
pendent of the input power. While observing a probe trace
on the oscilloscope it was noticed that a change in input
power did not affect the shape of the probe curve in its
exponential region, which determines electron temperature,
but caused a large change in the current at which saturation
occurred. It is this point which determines the number
density. In fact, a small change in input power could easily
change this point by a factor of ten. This tends to support
the prediction that the temperature is independent of power
input, and that the number density is critically dependent
on input power as shown in Figure 17.
Further evidence that this is true was obtained when an
undergraduate student in Aerospace Engineering, Mr. Don Green,
performed heat transfer measurements at the discharge bound
ary. The heat transfer to the wall should be proportional to
the number density at the discharge axis. When the heat
transfer rate was plotted as a function of the coil current,
curves were obtained which were in qualitative agreement
with Figure 17.
Another undergraduate student in Aerospace Engineering,
Mr. Paul Bloom, has made measurements of the power input to
the discharge simultaneously with probe measurements of
electron number density and temperature. By using the
measured values of number density and temperature he has
calculated the value of K along the discharge axis, and by
72
use of the theory integrated the power input over the length
of the discharge to obtain predicted power input. Prelimi
nary results indicate that good agreement between predicted
and measured total power input is obtained.
The theory for the electrodeless discharge which has
been presented shows reasonable qualitative and quantitative
agreement with the available experiments. Although it is
not an exact description of the discharge it appears to
describe the major physical processes which occur. The
theory should prove useful in determining the general behav
ior of the discharge and for providing a basis upon which
engineering designs of future discharges could be made.
10J Amps/Division
Volts/Division
x N
o
0
2.5 Volts/Division
Figure 1. Typical probe characteristic with third probe trace
co
Electron Number Density, (10^2 cm-^)
74
Figure 2. Electron number density at a pressure of 0.13 Torr.
Electron Number Density, (lO^ cm-3)
75
Figure 3. Electron number density at a pressure of 0.26 Torr.
Electron Number Density, (lO1^ cm
76
co
i
Figure 4. Electron number density at a pressure of 0.40 Torr.
78
Figure 6. Plasma potential.
Normalized Electron Number Density
77
0 0.13 Torr
0.26 Torr
A 0.40 Torr
Figure 5. Comparison of measured number density with the
theory of Eckert.
79
Figure 7. Computed average momentum collision frequency.
- (m3/sec)
80
VT, (volts)
Figure 8. Computed average Ionization collision frequency.
81
Vip, (volts)
Figure 9. Computed average excitation collision frequency.
(volts)
Figure 11.
Electron temperature as a function of pressure for a long cylindrical
discharge of 2.4 cm radius.
oo
u
Figure 12. Electric and magnetic field for
0
p
p
= J
Figure 13. Electric and magnetic field for t^/CO
oo
tn
p
f
Figure 14. Electric and magnetic field for jS/co
oo
05
Z*. Z/co B*
K
00
Figure 15. Average power input at the wall.
Figure 16. Solution of the power balance equation for a pressure of 0.5 Torr.
oo
00
89
Figure 17.
Electron number density at the discharge axis as
a function of solenoid current at various pressures
K ffvN O
REFERENCES
1. Hittorf, Wo, "beber die Electricitatsleitung der Gase,"
Wiedemann Ann Phys. Chim. 21, 90 (1884).
2. Thomson, J. J., "The electrodeless discharge through
gases," Phil. Mag. 4, 1128-1.160 (1927).
3. Townsend, J. S. and Donaldson, R. H., "Electrodeless
discharges," Phil. Mag. 178-191 (1928).
4. MacKinnon, K. A., "On the origin of the electrodeless
discharge," Phil. Mag. £5, 605-616 (1929).
5. Reed, T. B., "High-power low density induction plasmas,"
J. Appl. Phys. 34, 3146-3147 (1963).
6. Clarkson, M. H., Field, R. E., and Keefer, D. R.,
"Electron temperature in several rf-generated plasmas,
AIAA Journal 4, 546-547 (1966).
7. Eckert, H. U., "Diffusion theory of the electrodeless
discharge," J. Appl. Phys. 33, 2780-2788 (1962).
8. Keefer, D. R., Clarkson, M. H., and Mathews, B. E.,
"Probe measurements in an electrodeless discharge,"
AIAA Journal 4, 1850-1852 (1966).
9. Allis, W. P., "Motions of ions and electrons," Handbuch
der Physik 21, 383-444, Springer Verlag (1956).
10.Brown, S. C., "Breakdown in gases: alternating and
high-frequency fields," Handbuch der Physik 22, 531-574,
Springer Verlag (1956).
90
91
11. Langmuir, I. and Mott-Smith, H., Jr., "Studies of
electrical discharges in gases at low pressures,"
Gen. Elec. Rev. 27, 449-455 (1924).
12. Loeb, L. B., Basic Processes of Gaseous Electronics,
Univ. of Calif. Press (1955).
13. Johnson, E. 0. and Malter, L., "A floating double probe
method for measurements in gas discharges," Phys. Rev.
80, 56-68 (1950).
14. McDaniel, E. W., Collision Phenomena in Ionized Gases,
John Wiley and Sons, Inc., New York (1964).
15. Bohm, D., The Characteristics of Electrical Discharge
in Magnetic Fields, edited by A. Guthrie aind R. K.
Wakerling, McGraw-Hill Book Company, Inc., New York
(1949) .
16. von Engel, A., "Ionization in gases by electrons in
electric fields," Handbuch der Physik 21, 504-572,
Springer Verlag (1956).
17. Shkarofsky, I. P., Johnston, T. W., and Bachynski, M. P.,
The Particle Kinetics of Plasmas, Addison-Wesley Pub. Co.,
Reading, Mass. (1966).
18. Francis, G., "The glow discharge at low pressure,"
Handbuch der Physik 22, 53-203, Springer Verlag (1956).
19. Allis, W. P. and Rose, D. J., "The transition from free
to ambipolar diffusion," Phys. Rev. 93, 84-93 (1954).
20. Sovie, R. J., Private communication, also presented
APS meeting, Atlantic City, Nov. 1962 and APS meeting,
Boston, Nov. 1966.
21. Dingle, R. B., Appl. Sci. Res. 6B, 144-154, 155-164,
245-252(1.957) .
92
22. Pugh, E. M. and Pugh, E. W. Principles of Electricity
and Magnetism, Addison-Wesley Pub. Co., Inc., Reading,
Mass. (1960).
23. Holt, E. H. and Haskell, R. E., Foundations of Plasma
Dynamics, The Macmillan Co., New York (1965).
24. Henriksen, B. B., Analysis of field distributions in
an electrodeless discharge, Thesis, University of Florida
(1966).
25. Barbiere, D., "Energy distribution, drift velocity, and
temperature of slow electrons in helium and argon,"
Phys. Rev. 84, 653-658 (1951).
26. Brown, S. C., Basic Data of Plasma Physics, Technology
Press of The Massachusetts Institute of Technology and
John Wiley and Sons, Inc., New York (1959).
27. Kieffer, L. J., "A compilation of critically evaluated
electron impact ionization cross section data for atoms
and diatomic molecules," JILA Report No. 30, University
of Colorado (1965).
28. Maier-Leibnitz, H., "Ausbeutemessungen beim Stoss
langsamer Elektronen mit Edelgasatomen," Zeits. Phys.
95, 499-523 (1935).
29. Druyvesteyn, M. J. and Penning, F. M., "The mechanism of
electrical discharges in gases of low pressure," Revs.
Mod. Phys. 12, 87-174 (1940).
30. Hasted, J. B., Physics of Atomic Collisions, Butterworth,
Inc., Washington, D.C. (1964).
I.
^ ;
BIOGRAPHICAL SKETCH
Dennis Ralph Keefer was born September 22, 1938, at
Winter Haven, Florida. He was graduated from Auburndale
High School in June, 1956. In June, 1962^ he received the
degree of Bachelor of Engineering Sciences with High Honors
from the University of Florida. Mr. Keefer enrolled in the
Graduate School of the University of Florida in June, 1962.
He was employed as a research assistant until August, 1963,
when he received the degree of Master of Science in
Engineering. From September, 1963, until the present, he
has pursued his work toward the degree of Doctor of
Philosophy while employed by the Department of Aerospace
Engineering as a research associate.
Dennis Ralph Keefer is married to the former Charlotte
Ann King and is the father of two children. He is a member
of the American Institute of Aeronautics and Astronautics,
Tau Beta Pi, and Sigma Pi Sigma.
93
This dissertation was prepared under the direction of
the chairman of the candidates supervisory committee and
has been approved by all members of that committee. It was
submitted to the Dean of the College of Engineering and to
the Graduate Council, and was approved as partial fulfill
ment of the requirements for the degree of Doctor of
Philosophy.
August, 1967
Dean, Graduate School
Supervisory Committee:
OrU £
82
Figure 10. Temperature dependence of /Da.
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AUTHOR:
TITLE:
Keefer, Dennis
The theory and the diagnosis of the electrodeless discharge,
(record number: 565597)
PUBLICATION
DATE:
1967
, as copyright holder for the
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Please print, sign and return to:
Cathleen Martyniak
UF Dissertation Project
THE THEORY AND THE DIAGNOSIS OF THE
ELECTRODELESS DISCHARGE
By
DENNIS RALPH KEEFER
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
August, 1967
To my wife
CHARLOTTE
who, having little understanding for the subject,
has shown great understanding for the author.
ACKNOWLEDGEMENTS
The author is deeply indebted to many individuals for
aid and encouragement in the completion of this dissertation.
He wishes to express his gratitude to his fellow students
and co-workers for their aid and suggestions in the course
of the research work.
Special thanks are due to his supervisory committee
chairman, Dr. M. H. Clarkson, who has served with patience
and understanding as teacher, mentor, critic and employer
for these past five years. Without his initial encourage
ment the project would have never begun. The author wishes
to express his appreciation to the members of his supervisory
committee, Dr. Knox Millsaps, Dr. Orlo E Myers,
Dr. Thomas L. Bailey and Dr. Earle E. Muschlitz, Jr., for
their guidance and encouragement in the course of his
graduate program.
The author also wishes to thank Mrs. Jacqueline Ward
who graciously and ably prepared this manuscript.
The financial support for this project was provided by
the National Aeronautics and Space Administration under
Grant NsG-542.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS .................... iii
LIST OF FIGURES .................... v
ABSTRACT ............... .... vii
CHAPTERS
I.INTRODUCTION 1
II.PROBE MEASUREMENTS .............. 4
III.TEMPERATURE DISTRIBUTION 15
IV. ELECTRON TEMPERATURE AND NUMBER DENSITY
DISTRIBUTION ................. 29
V. THE ELECTROMAGNETIC FIELD .......... 38
VI. POWER BALANCE ................ 49
VII. CALCULATIONS ................. 56
VIII. CONCLUSIONS ................. 68
REFERENCES ..... .......... 90
BIOGRAPHICAL SKETCH .................. 93
iv
LIST OF FIGURES
Figure Page
1. Typical probe characteristic with third probe
trace. 73
2. Electron number density at a pressure of
0.13 Torr. 74
3. Electron number density at a pressure of
0.26 Torr. 75
4. Electron number density at a pressure of
0.40 Torr. 76
5. Comparison of measured number density with
the theory of Eckert. 77
6. Plasma potential. 78
7. Computed average momentum collision frequency. 79
8. Computed average ionization collision frequency. 80
9. Computed average excitation collision frequency. 81
10. Temperature dependence of <("2^j^/Da. 82
11. Electron temperature as a function of pressure
for a long cylindrical discharge of 2.4 cm
radius. 83
12. Electric and magnetic field for ^/oo~ O 84
13. Electric and magnetic field for tJ/CO J. 85
14. Electric and magnetic field for J^/cO ~ o 86
15. Average power input at the wall. 87
16. Solution of the power balance equation for a
pressure of 0.5 Torr. 88
v
Page
Figure
17.
Electron number density at the discharge axis
as a function of solenoid current at various
pressures.
89
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
THE THEORY AND THE DIAGNOSIS OF THE
ELECTRODELESS DISCHARGE
By
Dennis Ralph Keefer
August, 1967
Chairman: M. H. Clarkson
Major Department: Aerospace Engineering
An analysis of the asymmetric double probe is described
which permits the measurement of electron temperature, elec
tron number density and plasma potential in an electrodeless
discharge. Experiments were performed which indicate that
the electron temperature and ionization function are essen
tially uniform across the discharge radius. These results
are at variance with commonly made assumptions concerning the
discharge. An analysis of the energy transport in the dis
charge predicts that, due to the thermal conduction of the
electron gas, the temperature is essentially uniform. The
fact that the electron temperature is uniform is used as a
basis for the formulation of a one-dimensional, steady-state
theory for the inductively driven electrodeless discharge.
The results of the theory permit the calculation of electron
temperature, electron number density, the electric and
magnetic fields in the discharge, and the power input to the
discharge as a function of the discharge configuration and
the applied coil current. An important result of the theory
is the prediction that electron temperature is independent
of the input power, and that the electron number density is
a direct function of the solenoid coil current.
vii
CHAPTER I,
INTRODUCTION
An electrodeless discharge is a gaseous electrical dis
charge which is maintained by high-frequency electric fields.
The fields are applied to either conductive rings or plates
or to helical coils placed outside the discharge container.
These discharges can be maintained in a wide variety of
gases and over a pressure range extending from the region of
.001 Torr to greater than atmospheric.
The discharge was first discovered by Hittorf^ in 1884
and appears to have been widely known and studied in the
early part of the twentieth century. In 1927 and 1928 the
solenoid excited electrodeless discharge received the
attention of Sir James Thomson and J. S. Townsend and their
work resulted in a controversy concerning the nature of the
discharge. The controversy concerned the question of whether
the induced or the electrostatic fields of the solenoid were
responsible for the discharge, Thomson^ gave an analysis of
the induced fields, derived conditions for breakdown, and
reported experiments which indicated that the currents in the
gas were due to the induced fields. Townsend and Donaldson^
reported experiments which showed the breakdown to be caused
by the axial electric field due to the potential drop across
the solenoid. The controversy was resolved in 1929 by
K. A. MacKinnon^ who showed that actually both discharges
were possible, depending on the configuration, and that the
two types of discharge exhibited different visual character
istics. In 1963, T. B. Reed5 reported two modes of operation
1
2
for a low-pressure electrodeless discharge and gave certain
spectrographic data for the two modes of operation which he
designated as low-power and as high-power as an indication of
power absorbed from the RF generator. Clarkson, Field and
Keefer showed that these two modes of operation were related
to the same phenomena explained by MacKinnon and gave measure
ments of electron temperature in the two modes obtained by
use of floating double probes.
Although the electrodeless discharge has been widely
used as a source of laboratory plasma, it has not been widely
studied. A revival of interest occurred in the 1950s as a
result of the need for high-energy gas sources of low con
tamination for use in wind tunnels designed for testing
re-entering vehicles launched by rockets. The discharge was
not initially successful in these applications and they were
largely discarded in favor of high-energy DC arc jets.
Analytical studies of the electrodeless discharge began
with Thomson who calculated the fields in a cylindrical dis
charge in a solenoidal field, assuming a uniform conductivity
across the radius. From these calculations, he derived a
breakdown criterion based on the energy gain of an electron
in one mean free path. A more realistic analysis was given
by H. U. Eckert,7 who recognized that the conductivity would
not be uniform across the radius, and that the discharge would
be diffusion controlled. However, Eckert obtained the same
solution for the fields as Thomson since he assumed an
average uniform conductivity in the formulation of Maxwell's
equations. By solving the plasma balance equation, Eckert
was able to give a more realistic breakdown criterion. By
making certain assumptions concerning the spatial variation
of the ionization function and electron temperature, Eckert
was able to calculate the steady-state electric field at the
boundary of the discharge and the spatial variation of the
electron number density.
3
Development of a highly asymmetric floating double probe
suitable for use in electrodeless discharges by Keefer,
Q
Clarkson and Mathews made it possible to measure the spatial
variation of electron number density and temperature within
the electrodeless discharge The results of these measure
ments were found to be at. variance with the assumptions made
by Eckert. The most striking difference occurred in the
spatial variation of the electron temperature. While Eckert
assumed that the temperature increased exponentially from the
center of the discharge to the wall, the probe measurements
indicate that the electron temperature is essentially uniform.
In addition, the measured spatial variation of the electron
number density implies that the ionization function is nearly
uniform in contrast to the power law variation assumed by
Eckert.
A new theory for the electrodeless discharge has been
formulated as a result of the discrepancy found between the
probe measurements and Eckerts assumptions. The new theory
is in reasonable agreement with the probe measurements, and
provides a description of the plasma properties as a function
of the applied fields. Not ail of the predictions of the
theory have been subjected to experimental verification, but
certain qualitative comparisons with observations made in the
course of the experiments indicate that the major physical
processes are described by the theory.
It should be pointed out that, while a relatively small
amount of attention has been given to the electrodeless dis
charge per se in the past few decades, there has developed a
large body of plasma theory of a general nature which is
applicable to the problem. There has also been a great amount
of work performed on microwave discharges, much of which is
applicable to the electrodeless discharge. Of particular
value are the two Handbuch der Physik articles by Allis and
Brown.
CHAPTER II.
PROBE MEASUREMENTS
The electrostatic probe, or Langmuir probe, is one of
the most widely used plasma diagnostic tools. Although its
use dates much earlier, a definitive theory for its behavior
1 1
in a plasma was given by I. Langmuir in 1924. The tech
nique is basically simple. A small conductor is placed in
the plasma and a potential applied to it with respect to
some other electrode, usually one of the discharge electrodes.
A curve of current versus voltage is recorded and with the
Langmuir theory the plasma number density, electron tempera
ture and potential may be determined from this curve. Despite
the fact that the theory is applicable to only a limited
range of plasma conditions, the use of the technique is
nearly universal. The technique, together with its limita
tions, is adequately described by Loeb.-1-^ It remains virtually
the only way to resolve spatially the properties of a plasma.
The application of this technique to the electrodeless
discharge involves two obvious difficulties. First, there is
no discharge electrode with which the probe may be biased and,
secondly, the plasma potential may vary at the driving
frequency. Johnson and Malter13 describe a system which over
comes these difficulties. A floating double probe is placed
in the plasma and one probe is biased with respect to the
other. Their experiments utilized probes of equal area, but
it was proposed that probes of unequal area or an asymmetric
probe might be used to increase accuracy. In their analysis,
it is assumed that both probes are placed in a plasma of
4
5
uniform potential. As this may not be true in the electrode
less discharge, it is necessary to extend their analysis to
include this condition.
Whenever a surface is placed in a plasma, a sheath forms
and the surface potential becomes negative with respect to
the local plasma potential. This condition occurs because the
electrons have a higher mobility than the ions and, therefore,
build up more rapidly on the surface. When the surface
potential becomes sufficiently negative, the lower energy
electrons are retarded in their motion toward the surface such
that an equal number of ions and electrons impinge on the
surface and a steady state is established. When a current is
drawn from the surface, the sheath is altered such that the
surface becomes more or less negative with respect to the
local plasma potential. The potential becomes more negative
if a positive current flows from the surface, and less negative
if negative current flows from the surface.
In the use of a floating double probe, two collecting
surfaces are placed in the plasma and a biasing potential
applied between them. The more positive of the probes will
collect a surplus of electrons and the more negative will
collect a surplus of ions such that the net current drawn from
the plasma is zero. As the bias potential is increased, the
current between the probes increases until saturation occurs.
Saturation will occur whenever the maximum possible number of
either the ions or electrons incident upon the sheath boundary
is collected. To insure greater accuracy in the measurement
of plasma properties, it is necessary to operate the probe at
electron saturation. This is the condition where all the
incident electrons are collected. The analysis will show that
for this condition to occur a very large difference in the
surface area of the two probe collectors is required.
6
The large difference in required probe area will usually
mean that the probes are placed in the plasma at points where
the local plasma potential is not equal,, This potential
difference may be established by ambipolar diffusion or by an
external electric field. The quantities which refer to the
two probe surfaces will be designated by superscripts 1 and 2.
Thus, the plasma potential at probes 1 and 2 is V and
(2) ^
VpV respectively. The potential difference will be assumed
constant and designated Vr where,
V, Vp(z) Vp(1> 2.1
The flux of ions and electrons impinging on the probe sheath
will be designated r^ and P e,respectively. It is assumed
that these fluxes are a function only of the plasma properties
and independent of the sheath potential. In the electron
retarding region, the current density of electrons reaching a
probe is given by
2.2
kTe
where V-j. = and e is the electron charge, k is Boltzmann's
constant and Te is the electron temperature. Thus the total
current to probe 1 is given by
2.3
and the current to probe 2 is given by
(V
- n
(2)
2.4
7
where A is the surface area of the probes and V is the poten
tial of probe 2 with respect to probe 1 The floating probe
system can draw no net current from the plasma and this
requires that
> = cz>=
2.5
By equating 2.3 to 2.4 and noting that
vp(1)- vP(z)- vr
2.6
one obtains, after some manipulation.
IL = /oq Jg K- J- ^ '& g
*4 l Vr/3)
Aw rt<
2.7
This expression relates the local plasma potential at
probe 2 to the applied potential V. If the double probe
system is to behave like the classical Langmuir probe, it is
necessary that the variation of the local plasma potential at
probe 2 be small as the bias potential V is varied. This
requirement yields a criterion for the area ratio of the
probes, namely
Aa>r/J)
1
2.8
In practice this requirement may be rather severe since the
larger probe (1) would likely be placed near the boundary of
the discharge wherePe^^ Pe^^. The required area ratio
A^)// may be of the order 10^.
8
It will be assumed now that the area ratio is sufficiently
small so that the variation of Vp^) is negligible over the
useful range of bias potential V. The probe current in the
electron retarding region is given by
- = eA
- W V)/Vr
e ~ri
and
2.9
2.10
The ratio of ion flux to electron flux is much less than one
and it will be neglected in 2.10. Taking the derivative of
equation 2.10 with respect to V yields the expression
d03 0 ^ J_ 2.ii
dV |/r
This is the usual expression for the Langmuir probe from
which the electron temperature may be determined from the
slope of the probe characteristic when it is plotted on semi
log paper. The electron number density is determined from
the break in the characteristic curve caused by electron
saturation. At electron saturation, = V and 2.9 becomes
4 = e Am (rtm-r) 2.12
Since re(2) ^ then
9
2.13
For a Maxwell-Boltzmann distribution of electrons
2.14
where n is the electron number density and m is the electron
mass. Since the electron temperature may be determined from
2.11, the number density may be determined from 2.13 and 2.14.
One additional property of the plasma may be determined
by the probe, namely, the plasma potential. At the break in
the characteristic curve at the onset of electron saturation,
the probe 2 has the same potential as the plasma relative to
probe 1. Thus, if probe 1 is held at a constant potential,
the plasma potential may be determined.
The above analysis shows that a highly asymmetric double
probe behaves like the classical Langmuir probe whenever the
criterion 2.8 is met. An experimental method to determine
whether or not this criterion is being met is described below.
Consider a third probe placed in the plasma at the same
location as probe 2. This probe will be designated probe 3
and the plasma potential and ion and electron flux at its
location will be the same as for probe 2, i.e.,
2.15
and
2.16
10
The current relation for probe 3 is then given by
-
e
2.17
Now if no current is allowed to flow from probe 3, it will
assume a floating potential Vj, where
-(V-Vf)Ar nw
e = TpQ
This may be reduced to the expression
2.18
Thus the floating potential differs from the local plasma
potential by a constant amount. By observing the floating
potential of probe 3, it is possible to determine whether
the plasma potential Vp(2) changes as the applied potential V
is varied in accordance with 2.7. If the variation of Vf, and
therefore Vp(2)¡, is negligible then the criteria 2.8 has been
satisfied.
A highly asymmetric double probe was used to measure the
electron temperature and number density in a low-pressure
electrodeless discharge. The discharge was formed in a length
of two-inch diameter Pyrex pipe connected to a vacuum system,
argon was admitted to maintain a prescribed pressure. The
tube was placed inside a structure containing a solenoid coil
connected to the RF power supply and had shielding to prevent
penetration of the axial electric field of the solenoid into
11
the discharge. The RF power supply was a converted television
transmitter operating at approximately 4.5 MHz with a maxi
mum available power of 5 KW.
The probe system consisted of a large cylindrical elec
trode placed at the wall inside the discharge tube, and two
small movable probes. The large electrode was constructed of
OFHC copper with a cooling coil soldered to the inside and
the entire assembly silver plated to reduce contamination.
The cylinder was split longitudinally to allow the axial
magnetic field of the solenoid to ''penetrate" by acting as a
one-turn secondary. The small electrodes were formed from
0.01 inch diameter tungsten wire with approximately 0.125
inches exposed from a Pyrex sheath. This gave a geometric
area ratio of approximately 7 x 10~5. The probes and Pyrex
sheath were set into a brass tube with an offset to allow
rotation of the probes across the tube radius. Probe voltage
was supplied by a transformer secondary with the primary
connected through a variable autotransformer to the 60 Hz
laboratory power. The current signal was obtained across a
10 ohm shunt and applied to the vertical input of an oscillo
scope while the probe voltage was connected to the horizontal
input. The resultant current-voltage trace was photographed
from the oscilloscope to provide a permanent trace. When the
third probe was used, it was connected directly to a second
vertical input.
A typical probe characteristic, together with the third
floating probe potential, is shown in Figure 1. The charac
teristic is typical of a properly functioning Langmuir probe
with a sharp knee and saturation region. The third probe
potential is seen to vary slightly as the bias potential on
the asymmetric probe is varied. The indicated change in
plasma potential is seen to be of the order of 0.4 volt
showing that the change in plasma potential is negligible for
12
this probe system. Data were taken at pressures of 0.13 Torr,
0.26 Torr and 0.40 Torr in argon at several radial locations.
In all cases the electron temperature was found to be uniform
across the radius of the discharge. At 0.13 Torr the electron
temperature expressed in volts was 2.0 volts, at 0.26 Torr
it was 1.8 volts and at 0.4 Torr it was 1.6 volts. The elec
tron number density at the three pressures is shown in
Figures 2, 3, and 4. All of these curves have the character
istic that the number density is greatest at the axis of the
discharge and becomes smaller as the wall is approached. This
is typical of a diffusion controlled discharge.
H. U. Eckert^ has given an approximate analysis for the
electrodeless discharge. In this analysis, the electron pro
duction term is assumed to obey a power law given by
/, r ^ 2.20
where V^ is the average frequency of ionization per electron,
Da is the ambipolar diffusion coefficient, r is the radial
coordinate measured from the discharge axis and h and q are
assumed constant. The solution of the plasma balance equation
for this assumed form is
sil ,
nho. T \zAOS(-~\ Z 1 2.21
{aJ -I
where n is the electron number density, a is the inner radius
of the discharge tube and J0 is the Bessel function of zero
order. The data from Figures 2, 3, and 4 were normalized and
are shown, together with a plot of 2.21 for q = 0, 1, and 2 in
Figure 5. This figure indicates that the ionization function
13
Vj[/Da is approximately uniform (q 0) In solving the
plasma balance equation to obtain 2.21, it was assumed that
the number density is zero at the boundary. This is not
precisely true. In the actual case, McDaniel^ shows that
the number density has a value at the wall which will extra
polate to zero at a distance of the order of one mean free
path outside the boundary. This accounts for the fact that
the lower pressure data do not extrapolate to zero in Figure 5.
The plasma potential measured at 0.40 Torr is shown in
Figure 6. The potential gradient is established by the
ambipolar diffusion. The net effect is to establish an
electric field which retards the flow of electrons to the
boundary while accelerating the ions to maintain a quasi
neutrality in the plasma volume. The plasma potential does
not extrapolate to zero at the boundary even though the wall
electrode is held at zero potential. This is due to the
sheath, mentioned earlier, which covers the electrode. By
an argument similar to that used in deriving 2.19, it can be
shown that the potential drop across the sheath, Vg, is
given by
By use of Bohmfs criteria^ for ? one finds that for an
argon discharge
Vs = 5.18Vt
The calculated value of V_ is plotted in Figure 6 and it
may be seen that the measured potential is readily extra
polated through this point.
The results of these experiments raise important theo
retical questions about the operation of the electrodeless
14
discharge. Due to skin effect at high frequencies, the power
input to the discharge is concentrated in the region near the
wall. It is this fact which led Eckert to assume the form
2,20 for the ionization production term. He also assumed
that the electron temperature increases exponentially with
radius. The probe measurements indicate that these assumptions
are not correct. In the following chapters a theory is given
which describes the operation of the electrodeless discharge
and which is in essential agreement with the experimental
measurements.
CHAPTER III.
TEMPERATURE DISTRIBUTION
The probe measurements described in Chapter II have
shown that the electron temperature is essentially uniform
across the discharge radius. This result is somewhat
surprising since the electric field and power input are
highly dependent on radial position. Elementary theories
and experiment for direct current discharges (see A. von
Engels-6) indicate that electron temperature and ionization
rates are directly related to the electric field strength.
These relations are developed on the assumption that the
electric field is uniform. If these relations are applied
to the electrodeless discharge where the electric field
is non-uniform, the electron temperature becomes a strong
function of position. This result is not supported by the
experimental results. In addition to electron temperature,
the relationships would also predict that the ionization
function is a strong function of position. The experimental
results shown in Figure 5 do not support this prediction and
indicate that the ionization frequency is essentially uni
form. The failure of these relations to predict correctly
the electron temperature and ionization frequency suggests
that they should not be applied to discharges in which the
electric field is not uniform.
The fact that the electron temperature and ionization
frequency are essentially uniform suggests that some mechanism
is operative which quickly transports the energy added by the
non-uniform electric field to the lower field regions of the
15
16
plasma. The combination of a uniform temperature and a uni
form ionization frequency suggests that the ionization may
be thermally produced. That is, the ionization results only
from collisions with the high energy tail of some distribu
tion of electrons having a characteristic temperature which
is spatially uniform. For this reason the temperature dis
tribution becomes the most important factor in formulating
a theory of the electrodeless discharge.
It was previously noted that some mechanism must operate
to transport energy from those regions in the discharge where
large amounts of energy are added to those regions where the
energy addition is slight. The majority of the electrical
energy added to a plasma is added through the electron gas
since the ions contribute little to the total plasma conduc
tivity. Therefore, the transport processes of interest are
those which transport energy from one region to another in
the electron gas. Two processes are considered: diffusion
and conduction. In diffusion an electron transports its
energy to a different spatial location by actually moving
through the plasma to the new location while retaining its
energy. In a slightly ionized plasma this occurs slowly due
to frequent interaction of the electron with the neutral
species in the plasma and with the electrostatic field estab
lished in ambipolar diffusion. This field is established as
a result of the different mobilities of ions and electrons
and the requirement of a net neutral charge which is
characteristic of a plasma.
In conduction an electron transports its energy by
energy exchange with another particle in an encounter or
collision. In this way energy may be transported from one
region to another without the physical displacement of the
electron to that region. In a slightly ionized gas, en
counters between electrons and neutral particles are much
17
more frequent than the electron-electron encounter Thus,
it might appear that the energy would be transferred from
the electrons to the neutral particles. This process does
occur in fact, but from an elementary study of elastic
collisions it is found that the fractional loss of kinetic
energy per encounter is given by
A A
2 Mi
(l cos %
3,1
where m^ and mg are the masses of the two particles and
')C is the scattering angle in center of mass coordinates
From
3,2 it is seen that when m-^ nig then
A Ki ^ z mi
K, rnz
and when m1=m2=m then
(i -eosX)
3,2
A Ki
Kj
3.3
From 3.2 and 3.3 it is seen that the energy transfer per
encounter is much more effective for particles of equal
mass (electron-electron) than it is for an encounter where
ml<<(m2 (electron-neutral). Therefore, even though electron-
neutral encounters are more frequent they are less effective
for energy transfer than electron-electron encounters and
the latter must not be neglected.
We shall consider a cylindrical electrodeless dis
charge to which power is added by a solenoidal field at high
frequency. For such a field, most of the energy is added in
a region near the wall. Thus, if the temperature is to
become uniform, energy must be transported towards the center
18
Diffusion, however, will cause a net transport toward the
wall instead of toward the center. This is because the
wall behaves like a sink for charged particles, causing the
number density of charged particles to be a maximum at the
center. This leaves electron conduction as the process
remaining for energy transport toward the center of the dis
charge and with diffusion as an opposing process.
It is the energy transport through the electron gas
which is of interest and, although the electrons lose some
energy to the neutral gas, in view of the discussion following
3.3 this process will be assumed negligible. Therefore, it
will be assumed that the energy transport can be described by
the equations for a fully ionized gas after taking into
account the smaller diffusion rate associated with ambipolar
diffusion in a slightly ionized gas.
The electron energy transport for a fully ionized
plasma with a Maxwell-Boltzmann distribution of electrons is
given by Shkarofsky, Johnston and Bachynski1^ (equation 8-107b)
as
% = /*'(£+ X B + j K- VT 3.4
In 3.4 yU and K are tensor coefficients of energy flow due to
electric fields and temperature gradients, respectively, T
is the electron temperature, B is the magnetic induction,
E is the electric field,is the average plasma velocity,
k is Boltzmanns constant, |e| is the electron charge and
n is the electron number density. For the specific problem
under consideration, it is assumed that the discharge is
cylindrical, infinite in axial extent, and posseses axial
symmetry. The discharge is excited by a field produced by
an alternating current flowing in an infinite solenoid placed
outside the discharge. Under the above assumptions the
19
problem becomes one-dimensional in the radial coordinate. For
the solenoidal field, the magnetic induction B is an alter
nating vector quantity whose magnitude varies with the radial
coordinate r, and whose direction is axial. For this type of
field, | BI is small and the termx B will be neglected
compared to E. When the magnetic induction may be neglected,
the quantitiesy~and IT reduce to the scalar quantities yU and
K. Equation 3.4 then becomes
ft)-Kvr 3.5
The first term on the right is the convective energy trans
port associated with the drift of electrons induced by the
electric field and diffusion. The term KVT represents the
energy transport by conduction through the electron gas due
to temperature gradients.
Consider next an arbitrary volume W bounded by a surface
*
S within the discharge. An energy flux P from the solenoid
flows into the volume and the electrons transport the energy
qe out of the volume. Conservation of energy requires that
j~t f EwdW = j(fe-T5>ndS s.e
JW 5
Applying Gauss' theorem and noting that the relation is true
for any arbitrary surface S yields
c>t
-P)
3.7
For the steady state, the left hand side vanishes and
substituting for "q^ from 3.5 gives
20
3.8
This constitutes an energy balance equation for the discharge.
The electric field produced by the solenoid is an alternating
field whose magnitude varies with r and whose direction is
azimuthal. This field contributes nothing to the energy flow
in the radial direction so that equation 3.8 may be written
as
where Er and Pr are the radial components of electric field
and energy flux respectively.
The radial electric field Er is established as a result
of the ambipolar diffusion in a partially ionized gas. The
electric field which is established retards electron flow to
the boundary and augments the ion flow to the boundary. The
two flows must be the same in a steady state if charge
neutrality is to be preserved. An approximate calculation
of the radial electric field established by ambipolar diffu
sion can be performed. The flux of electrons and ions is
given by
/I = £L Vn_+ r\-//_ E
3.10a
and
K = -D+Vn+-+ r\+/A+ E
3.10b
21
respectively, where D is the diffusion coefficient and is
the mobility.
Since the plasma must remain approximately neutral, then
r. n+=n-=n 3.11
Equations 3.10 may be solved for the electric field using the
approximations 3.11 to give for the diffusion-induced electric
field
tr Vn fD+ -D- p 3.12
Equation 3.12 may be rearranged to give
2*= Vn f 3X/D- ~ i ?
n lj
Both the ratios of ion-to~electron mobility and ion-to-
electron diffusion coefficient are small compared to one,
being of the order of the ratio of electron-to-ion mass, and
the term in brackets may be set equal to one. Under the
assumption of a Maxwell-Boltzmann distribution of electrons
the Einstein relation may be used for the ratio of diffusion
coefficient to mobility. The Einstein relation is
E = kT 3.1
m e
Therefore, to the order of the approximations used, the ambi-
polar diffusion-induced field becomes
r da
Lr~~ tein dr
3.15
The substitution of 3.15 into 3.9 results in the vanishing
of the convective term leaving only the conductive term
22
3.16
This result does not mean that no convection occurs in
the plasma. Complete cancellation occurs because of the
approximations used in deriving 3.15. However, the effect
of ambipolar diffusion is to reduce markedly the effective
ness of the convective process within the discharge. At the
discharge boundary the situation is quite different. Due to
the fact that electrons and ions recombine at the wall the
energy transport from the electrons to the wall is primarily
a convective process. Within the discharge the reduced
effectiveness of the convective process allows the tempera
ture to approach a uniform value through conduction as will
be shown below.
The radial coordinate and the temperature may be made
non-dimensional by introduction of the variables
3.17
where a is the discharge radius and TQ is the temperature on
the discharge axis. Equation 3.16 may then be written
d ? _ aPr
dP Kfo
3.18
The energy flux Pr may be defined by
3.19
23
where Pw is the energy flux incident on the discharge at (3 =1
and 0( gives the variation of energy flux across the radius.
Substitution of the dimensionless variables into 3.16 yields
the differential equation for the temperature
dr _ a Pw< 3 o
dp KTo
The electron thermal conductivity K is also a function of the
temperature. For the case of no magnetic field, K is given
in Reference 17 as
Snk*T f l ,jJ
m Vet) l 3k' 3m.
3.21
The number density n may be made non-dimensional by intro
duction of the variable 7£ defined by
3.22
where nQ is the number density on the discharge axis. Equa
tion 3.21 may be written as
i/= T>Z
171
where is averaSe electron-ion collision frequency,
and g^'and g^ are correction factors, depending on the
magnetic field. For a fulJLy ionized gas in a zero magnetic
field their values are given as
3.23
%' = 06 538
3M = -395T
3.24
24
The average electron-ion collision frequency is also
a function of temperature which, for a MaxwelX-Boltzmann
distribution, is given by
3.25
where I is the gamma function, and Yei is defined for a
singly ionized gas in terms of the coulomb logarithmby
Yei = 4-tt
3.26
where £ is the permittivity of free space. The function_/\_
is also a function of temperature but log-A_is a slowly
varying function of temperature and its temperature dependence
will be neglected. Introducing the dimensionless variables
into 3.25 and substituting into 3.23 give
-/mi J/v3-/8
K- Jr^Tz
Yec
3.27
The quantity Ce is defined by
K-ir^r^C,
Substituting 3.28 into 3.20 yields the differential equation
dr= aft*
* de V/zCe
3.29
25
which simplifies to
a Tw
d (rV =
d P Z*C
3.30
Upon integration and application of the boundary condition
r(0) = 1
3,31
the temperature distribution becomes
r7/z 1-fw~Ce JadP
3,32
The quantity Ce may be written as
. 7/ -8
r = 80 rfs/\ //z Tj k
7 1 ( 77
- 3k'
3,33
All the terms in Ce are constants except for log_/\_, There
fore, the temperature distribution may be written as
1-3.767*10* Wdp
3,34
where the constants have been evaluated in mk.s, units.
The function 0( describes the radial dependence of the
input energy flux. It has the value zero at (0 q anij has a
maximum value of unity at ^ -1, Thus, the maximum value of
the integral must be less than one. The temperature depen
dence is seen to depend critically on the parameter
26
7/2
a Pw logV\_ /T For the discharge for which measurements
are presented in Chapter II,
CL /?. 5" m
fZ,~ Z.9 XJO* watt/m* 3 3g
l~1.8 x 10* K
/oj -A ~ 10
and
a R IoqJ\-
T0V*
-J2
9.3 *10
ia/a tt
m k( 7/z
3 o 36
The equation for the temperature for this case would be
r
%=i 3-SxJO
3.37
It is clear that the term including the integral may be
neglected compared to one. Thus, for this case, it is seen
that the temperature is essentially uniform which is in
agreement with the experimental data.
The above analysis has been performed for a system in
which the only energy transport mechanism is thermal conduc
tion as may be seen from equation 3.16. This situation arises
because the diffusive energy transport term is completely
cancelled out when the electric field due to ambipolar
diffusion is substituted into the energy transport equation.
However, since ambipolar diffusion is occurring it is obvious
that some energy transport is involved. The result of nearly
uniform temperature derived above will mean that the tempera
ture gradient is small, and the propriety of neglecting the
27
energy transport through diffusion with respect to that due
to conduction is questionable.
An equation similar to 3.9 which includes the energy
transport due to the ambipolar diffusion may be written
K
dn
D dr
K
dT __
dr
- -ft
where is given by the expression
3.38
3.39
In view of the nearly uniform temperature result which was
obtained above, it is reasonable to assume that will vary
only slightly with the radial coordinate.
If Kp is assumed to be a constant function of T0 then
by substitution of the dimensionless quantities 7" y and
p, equation 3.38 may be written
d('f,/e) _ a & rtc Kd d>2
dP TpCe T07/*Ce dp
The solution of this equation analogous to 3.34 is
3.41
28
The influence of the diffusive energy transport on the
temperature distribution is given by 3.41. The relative
importance of this term may be determined by considering the
ratio n0Kj)/aPw. The value of nQ corresponding to the values
given in 3.35 is
3.42
and
flo Kt>
a Pw
3.43
It is seen that the diffusive energy transport has even less
effect on the temperature distribution than the term involving
the energy input. Thus, the result that the temperature
distribution is essentially uniform is unchanged.
From the previous example it is apparent that for
practical laboratory discharges the electron temperature
should be essentially uniform across the radius, since for
7 / 0
any significant change to occur the parameter aPwlog-A/TQ
would have to be increased by an order of magnitude or more.
In the following chapters this important result will be used
as a basis upon which calculations are made for the electro
magnetic field in the discharge and for the development of a
complete theory of operation of the electrodeless discharge.
The development of the theory will follow the general outline
of theories for the DC glow discharge first given by Schottky.
These theories are discussed in detail by Francis.
CHAPTER IV.
ELECTRON TEMPERATURE AND NUMBER DENSITY DISTRIBUTION
In the previous chapter, an expression was derived for
the electron temperature distribution. For the discharge
in which the experiments were performed it was found that
the electron temperature was, essentially, constant with
radius. In this chapter, a theory will be given, based on
the solution of the plasma balance equation, which predicts
the discharge temperature. The solution of the plasma
balance equation will also yield an expression for the
electron number density distribution. This expression will
contain another unknown quantity, nQ, which will be deter
mined by performing a power balance after the electro
magnetic field in the discharge is calculated.
Certain assumptions must, again, be made in order to
solve the plasma balance equation. First, as a consequence
of the uniform temperature, it will be assumed that the
diffusion coefficient is constant with radius. Secondly,
it will be assumed that the average ionization collision
frequency is independent of position. This assumption is
based on the experimental results described in Chapter II
and is also a consequence of the uniform temperature.
Essentially, this assumption means that the ionization
process is thermal. The physical picture is that the
energy added to the electron gas in a non-uniform manner by
the electric fields is rapidly transported to the regions
of lower energy addition by thermal conduction. At the same
time the directed velocity of the electrons induced by the
29
30
electric field is rapidly randomized by the electron-neutral
collisions. The net result is a distribution of electron
velocities which is random in direction and characterized by
a temperature which is uniform It is now assumed that the
ionization is due to those electrons in the distribution
having energies in excess of the ionization energy The
ionization frequency is determined by averaging the ioniza
tion cross-section over the electron distribution function
Thus, the ionization function, ^/Da, will depend on both
the electron temperature and the form of the distribution
function Throughout the development of this theory, it
will be assumed that the electron distribution function is
Maxwell-Boltzmann Justification of this assumption can be
made from a theoretical standpoint (Reference 17) and also
by comparison of the experimental results with calculations
made using other distributions.
The flux of charged particles of species j is given by
4.1
where Dj is the diffusion coefficient for the particle
species and is the mobility of the particle species.
When the number density of charged particles is small both
ions and electrons will flow independently, each flowing in
when the number density becomes larger, space charge fields
will be established due to the different flux rates for ions
and electrons. Since the electrons have a higher flux rate
initially, they begin to move out of the discharge, leaving an
excess of ions. The resultant space charge field retards
the electron flux and increases the ion flux. Eventually,
the process attains a steady state where the flux rates of
31
ions and electrons are the same. This is the condition for
which the diffusion is considered to be ambipolar. Some of
the results of this type of diffusion have been used previ
ously in Chapter III. The transition from free to ambipolar
diffusion and the criteria for its occurence are studied in
1 Q
detail by Allis and Rose.
The flux equations for electrons and ions are
/! = -£>_ Vn. + n_//_ E
and
/J = -ZWn+ +
For fully developed ambipolar diffusion, the flux rates must
be equal. Some differences between electron and ion number
densities exist to establish the electric field, but only
small differences in number density may exist in a plasma
and thus,
T -T? = 7* 4.
and
H+- =/?_=/?
4.3b
Substitution of 4.3 into 4.2 and elimination of E give
M+- A- J
Vr\
r= -
4.4
32
The term in parenthesis is called the ambipolar diffusion
coefficient Da. The flux of either ions or electrons is
given by
r -Da Vn
Consider a volume W of surface S where particles are being
produced and from which particles are being lost. Continuity
requires that
d_
olt
T-fidS = \ V-fdW
5 vv
4.6
Since 4.6 must hold for each volume,
ff = V-f = 4.7
Equation 4.7 gives the loss rate of ionized particles due to
diffusion. It is assumed that the ionization is produced by
the high energy "tail" of a Maxwell-Boltzmann distribution of
electrons. The ionization rate is given by
dt = <**> n
The average ionization frequency is determined by
averaging the quantity
u =
4.9
33
over the distribution function Qx is the ionization cross-
section, g is the relative velocity between the electron and
a neutral particle and ng is the neutral particle number
density In a steady state the loss of ionized particles,
represented by 4.7, and the number being created, represented
by 4.8, must balance. Thus, the plasma balance equation is
given by
V2n + ^ n = 0 4.i
The solution of this equation, subject to the condition that
the number density vanish at the boundary, yields a number
density distribution dependent on an eigenvalue. The eigen
value in turn determines the temperature.
To obtain a more general solution to 4.10, the require
ment of an infinitely long discharge will be relaxed to
allow a finite length L. By separation of variables, the
solution of 4.10 is found to be
4.11
where JQ is the zeroth order Bessel function, z is the axial
coordinate and ^ is a separation constant. Applying the
boundary condition at z = L/2 and r = a results in
(Zc +l)yr
/= OjljZ
4.12a
34
jo^> A J <2 £= **' 4.12b
where the Jq^ are the zeros of JQ. Since the number density
cannot become negative anywhere within the discharge, only
the first zeros have physical significance, and
A = A 0 4.13a
and
CL
h = *ojr
The number density may now be written
4.13b
n- n0cos(Z- ^J0(.4-os£)
4.14
where n Q is the number density at the point r = z = 0.
Solving 4.13b,
M_ A
D* A*
a*
[*78* H^rfl
4.15
where A d is the diffusion length. For a long cylindrical
discharge a/Ll and
6> £ f.784
4.16
35
The eigenvalue equation 4.16 makes it possible to calcu
late the discharge temperature. The average ionization
collision frequency is a very strong function of the electron
temperature due to the fact that the cross-section has a
non-zero value only for electron energy greater than the
ionization energy. There is a weak dependence of on
the neutral gas temperature through the relative velocity g,
but since the electron velocity is much greater than the
neutral velocity, g is assumed equal to the electron velocity.
With this assumption 0J^ is a function only of the electron
temperature for any particular gas.
The ambipolar diffusion coefficient Da is a function of
both the electron and ion temperatures. From equations 4.4
and 4.5 Da is defined as
A+D- -M-D+
A -yU-
4.17
The mobility of ions is generally much smaller than that of
the electrons and
= T>+( i-
M.n+)
The Einstein relation
4.18
Di 4.19
A] e
reduces 4.18 to
A-hklZ.
e
4.20
36
In an active discharge at low pressure the electron tempera
ture is usually much larger than the ion temperature, and
thus
e
4.21
The ambipolar diffusion coefficient is primarily a function
of electron temperature, and the neutral gas pressure through
the ion mobility. The ratio (j) /Da becomes a function of
temperature and pressure and the eigenvalue equation 4.16
determines the electron temperature for a given value of
discharge radius and neutral gas pressure.
The theory given above yields the somewhat surprising
result that the electron temperature is independent of the
power input to the discharge. This result is due to the
assumption that the ionization is not directly related to
the electric field intensity, but is completely accounted
for by averaging the ionization frequency over a Maxwell-
Boltzmann distribution whose characteristic temperature is
uniform. It is this assumption which makes the ionization
function Z^i/Da indePendent of the electromagnetic field and
yields the eigenvalue equation 4.16.
The physical process is one in which power is absorbed
from the electromagnetic field which raises the electron
temperature until a steady state is reached. A further
increase in power causes a transient increase in the ioniza
tion rate until a new steady state is reached with a higher
level of electron number density.
According to equation 4.14, the radial distribution of
number density will be proportional to J0. This results from
the fact that, due to the uniform temperature, *^^/Da is not
a function of the radial coordinate. The solution due to
37
Eckert, which was discussed in Chapter II, reduces to this
form when q = 0 (equation 2.21). The distribution of number
density becomes very important in the calculation of the
electromagnetic field in the discharge. The electromagnetic
field depends on the plasma conductivity which, in turn,
depends upon the electron number density and temperature.
Since the temperature has been found uniform with radius,
the radial dependence of the conductivity will be determined
by the number density distribution.
CHAPTER V.
THE ELECTROMAGNETIC FIELD
The calculation of the electromagnetic field in the
discharge is a central aspect of any theory for the dis
charge. The electric field and power input must be known
in order to relate the plasma conditions to the applied
fields. For high-frequency discharges this problem is more
difficult since the plasma conductivity is a function of the
applied frequency. Also, in a high-frequency discharge, the
fields are inherently non-uniform due to the "skin effect."
The term arises from the study of high-frequency fields in
good conductors where the fields are attenuated within a
short distance in the conductor and the current flows in a
thin region near the surface.
An early attempt at an analysis of the electrodeless
discharge by Thomson^ resulted in a one-dimensional calcu
lation of the field in the discharge. Thomsons model was
that of an infinite cylindrical plasma in an infinite sole
noid. He assumed the plasma would have a real, uniform
conductivity across the plasma. The conductivity of the
plasma is neither real nor uniform in the discharge, as
recognized by Eckert.^ But Eckert obtained the same solution
as Thomson as a result of his assumption that the reactive
part of the conductivity was small, and by solving the
electromagnetic equations using a uniform conductivity
having the average value of the non-uniform conductivity.
R. J. Sovie^0 calculated the fields assuming a uniform
conductivity, but he allowed the conductivity to have a
reactive term so that the conductivity was a complex quantity.
38
39
All of the above analyses are characterized by the fact
that the conductivity was assumed to be uniform. In a
diffusion-controlled discharge this will never be true as the
number density must be zero at the discharge boundary. The
analysis of Chapter IV shows that the number density will be
a function of the radius as given by equation 4.14. A solu
tion for the electromagnetic field may be obtained for a
conductivity which approximates this distribution.
The plasma conductivity (T is usually defined as
5.1
where 10 is an equivalent collision frequency and u) is the
radian frequency of the applied field. The equivalent
collision frequency is a quantity obtained from certain
averages of the momentum collision frequency over the
electron distribution function. The particular function to
be averaged depends upon whether the radian frequency is
large or small compared with the equivalent collision fre
quency (Reference 17, Section 4-1.2). If the radian fre
quency is of the same order as the equivalent collision
frequency, certain correction factors are necessary due to
variations of pm with velocity. These factors modify both
P and CQ and have been calculated by Dingle.21 They are
presented in Reference 17, Section 8-1. Including these
correction factors, the conductivity can be written as
5.2
where g and h are the correction factors and is the
equivalent collision frequency averaged for CO 'p T!tie
40
conductivity will be assumed to be of the form 51 where it
is understood that the correction factors g and h have been
applied. In equation 5.1 the equivalent collision frequency
^ is a function of the electron temperature and the pressure
of the neutral gas. Since the electron temperature is uni
form will be independent of the radial coordinate and,
since O' is directly proportional to n the conductivity
will have the same radial dependence as n From 4.14 it is
seen that the radial dependence will have the form of JQ,
the zeroth order Bessel function. It will again be assumed
that the discharge is infinite in axial extent and therefore
the axial dependence of (T will vanish. The JQ Bessel
function may be expanded into the series
1/4 2* ,
7W
(Zl)z (3l)2~
5.3
Therefore, a first approximation to n which vanishes at the
boundary is
n(r)= n0
5.4
where nq is the number density on the axis. With this
approximation the conductivity becomes
cr(r) =
r\Qez
m
5.5
Maxwells equations for the electromagnetic field can
be reduced to two wave equations in the scalar potential V
41
and the vector potential A (see Pugh and Pugh^) as
5 6a
and
5.6 b
where and £ are the permeability and permittivity of the
medium respectively, t is time, is the charge density and
J is the current density. Except for the small charge
density which establishes the radial field due to ambipolar
diffusion, the plasma will be assumed to be electrically
neutral, fA = 0 a Faraday cage is installed in the experi
mental apparatus between the solenoid and plasma to prevent
the axial electric field of the solenoid from being impressed
on the plasma. Therefore, it will be assumed that the
potential within the discharge is everywhere zero, and only
5.6b must be considered. Again, the model assumed is that
of an infinite cylindrical discharge, possessing axial
symmetry and placed within an infinite solenoid. The sole
noid carries a high-frequency current
5.7
where IQ is the peak value of the solenoid current, 60 is
the radian frequency and is a unit vector in the azimuthal
direction. Under the above conditions the equation 5.6b
becomes a scalar equation in the azimuthal component of the
vector potential A namely,
42
dzAs> i 1 c>Ai> 1 a £ c>zAa> i j T />
5.8
The azimuthal current density is given by
, t Mi>
0 = cr l < j, ^ dt
5.9
where the relationship between the electric field and the
vector and scalar potentials,
r = _(il +VV)
5.10
has been applied. In view of the form assumed for the
applied current, it is reasonable to assume that the vector
potential will have a similar time variation. Assuming that
A$ = Ae
cot
5.11
equation 5.8 reduces to the ordinary differential equation
A"+j:A'-yz A +/Ueco&A-t
where primes denote differentiation with respect to r. The
conductivity o' can be written as
1-
m v*+coz
5.13
43
A "skin depth" 8 is defined by
8
*-
/ ip p\Vfe i
m {Vz+coz) I
jo co Hoe*- J
5.14
The term
1 1- GO
5 o 15
For the plasma we shall assume that the permeability is that
of free space /o and since the plasma is considered to be a
conducting medium, the permittivity is that of free space £0
(see Holt and Haskell,23 Section 11.2). The term /o;oC0Zcan
be written as
5.16
where f is the applied frequency, c is the velocity of light
and \ is the free-space wavelength of the applied frequency.
Equation 5.12 becomes
rM/y + rA'
44
For the frequency and discharge size of interest
znr
and, thus, the first term in brackets will be neglected
Defining a dimensionless coordinate
1
5 18
and introducing a new variable
Z
-to-IP
(pz+CO^Vz) j
5.19
5.20
equation 5.17 becomes
z*A"+zA' + A
(8*)&\(vz+co&)VzJ
5.21
where the primes now denote differentiation with respect to z.
Introducing the complex quantity Kei? defined by
a_
s*
-A/z
-L-IV 1
equation 5.21 reduces to
5.22
z*A"+A' +A
* 0
5.23
45
From equations 5,20 and 5.22 it is noted that
z = Ke^p
5.24
and the solution of 5.23 will be a function of Ke^ and ^0 .
The parameter K is a real number which gives the ratio of
r*
the discharge radius to the skin depth and J is a param
eter related to the ratio of collision frequency to applied
radian frequency.
A series solution to equation 5.23 has been given by
B. B. Henriksen.^4 solution which is finite at the origin
is given by
A = a,
TO
i)
< He* *)
ni
i
cz
( Kefc)
,Zr\~l
5.25
n= i
For convenience, the complicated expression on the right hand
side of equation 5.25 will be represented by
a0 Hi 5.26
where aQ is an arbitrary constant. The electric field E^>
and the magnetic induction Bz, are derived from A^ by
and
at
5.27
5.28
46
where the quantities A 4, and Bz are given by
5.29a
cot
E e
5.29b
5.29c
It should be noted that E and B are, in general, complex
quantities having both magnitude and phase. The expressions
for E and B are
E(Ke£f,p) =- Hs
5.30
and
KeffM
d z
5.31
At the discharge radius, a, the boundary condition,
Bz(eLU)= Ke 5.32
is applied. The quantity Bw is the peak wall value of mag
netic induction. Since Bw is a real quantity, the phase of
47
the fields in the discharge will be relative to the phase of
the magnetic induction at the wall. The quantity Bw can be
related to the current in the solenoid by
B =/(aNIa
5.33
where N is the number of turns per unit length, and IQ is
the peak value of the solenoid current. The arbitrary
constant aG is found to be
5.34
In Chapter VI,a power balance for the discharge will
be performed to calculate the number density on the discharge
axis. Thus, it will be necessary to calculate the power
input from the electromagnetic field. The instantaneous
energy flux is given by the Poynting vector S where
S = £ H
5.35
This quantity represents the power which goes into changing
the energy of the electromagnetic field plus the energy
dissipated in the plasma. Since the energy in the electro
magnetic field is alternately absorbed and given up, a time
average of S will give the energy flux which is dissipated
in the plasma. The time average of the Poynting vector S
is given by
= ?m
5.36
48
where H* is the complex conjugate of the magnetic intensity
which is related to the magnetic induction by
B* /o H 5.37
Since the electric field is azimuthal and the magnetic
induction is axial the energy flux is a vector in the radial
direction. This energy flux is given by
p£,v = ?r Eq COS 9 5.38
where Q is the phase angle between E ^ and Hz and Clr is a
unit vector in the radial direction.
CHAPTER VI.
POWER BALANCE
In the preceding chapters expressions have been derived
for the electron temperature distribution, the number density
distribution, and the electromagnetic field. The descrip
tion of the discharge will be completed upon calculation of
the number density on the discharge axis and this may be
accomplished by consideration of the power balance for the
discharge.
The electrons gain energy from the electromagnetic
field and, subsequently, lose their energy through collision
processes or by diffusion to the discharge wall. Both the
energy gained and the energy lost from the electron gas
depend upon the electron number density. For a given pres
sure the steady state will be reached when the energy loss
balances the energy gain and this will establish the steady^-
state number density.
The power balance will be performed on the electron
gas alone. It will be assumed that all of the electrical
energy is absorbed by the electrons since the ions contrib
ute little to the conductivity. This absorbed energy is then
lost from the electron gas by diffusion, by ionization, by
excitation of neutral particles with subsequent radiation,
and by heating of the neutral gas. It will be assumed that
the plasma is at low pressure and slightly ionized so that
the energy lost by heating of the ions and by bremsstralung
radiation may be neglected.
The electrical power absorbed by the plasma was given
in Chapter V. The energy flux given by equation 5.38 is a
49
50
function of Kei^
and therefore nQ as well as the radial
coordinate It will not be necessary to perforin a power
balance at each point within the discharge but only for the
discharge as a whole. Thus, it is not necessary to calculate
the average energy flux as a function of the radial coordinate,
but only the total energy flux incident on the discharge at
the discharge boundary. This quantity will be called Pw and
can be calculated from the equations of Chapter V with p = 1.
Since the power input is calculated per unit surface area of
the discharge, the power loss will also be referred to the
unit surface area.
The flux of electrons due to ambipolar diffusion is
given (equation 4.5) as
-DaVn
6.1
The total loss of electrons from the discharge per unit
surface area is found by evaluating this quantity at the
discharge boundary. Thus the total flux of electrons is
given by
6.2
The number density is given by equation 4.14 which reduces
for an infinite discharge to
6.3
The number density gradient is given by
6.4
51
where J-^ is the first order Bessel function. Evaluation of
this quantity at r = a yields
/dn~) =_ 1.24-8 n 6.5
\ d rJr,a a
The average energy per electron is given by
- e\lr 6.6
where k is Boltzmann's constant, T is the electron temperature,
and VT is the electron temperature expressed in volts as
defined by 6.6. The energy transported to the wall through
diffusion becomes
e1 = l^48_n DaeVT 6.7
a 7
When an electron suffers an ionizing collision it loses
an amount of energy equal to the ionization energy. This
energy is given by
E eV[ 6.8
where is the ionization potential of the neutral atom.
The number of ionizing collisions can be found directly by
calculating the average ionization frequency and integrating
over the number density distribution. However, since the
number of ionizing collisions must equal the number of ions
lost by diffusion, the number of ionizing collisions can be
obtained from the diffusion loss. Since the diffusion loss
52
has been calculated already, the latter method will be used.
From 6.2, 6.5 and 6.8 the total energy loss per unit surface
area due to ionization is given by
6.9
When an electron suffers an inelastic collision in which
a neutral particle is left in an excited state, the neutral
particle radiates this energy. The energy loss per collision
is given by
e:* = v
6.10
where Vx is the excitation potential. The number of these
collisions occurring per unit time per unit volume is given
by
6.11
where (Vx/ ^be averaSe excitation collision frequency.
This quantity is obtained by an appropriate average of the
excitation cross-section Qx over the distribution function.
Since it is the total energy loss per unit surface area that
must be obtained, it is necessary to integrate over the
number density and to divide by the surface area. The total
number density in a unit length of the cylinder is given by
2tt
n(r) rd@dr
6.12
0 0
53
Substituting 6.3 into 6.12 and integrating yields
G¡ = 0.2159 n0 Zna;
6.13
The surface area per unit length is
S = Z TTCL
6.14
and the total number of electrons per unit surface area is
given by
-§- = O.Zl59n0a
o
6.15
From 6.10, 6.11 and 6.15 it is seen that the energy loss due
to excitation per unit surface area is given by
65= 0.Z159 n0a(Vx)eVx
6.16
The loss of energy to heating of the neutrals is
calculated in a similar manner to that for excitation. The
average energy loss of an electron in an elastic collision
with a heavy neutral particle of mass M is given by
Ee-i
6.17
Using equation 6.6, this may be written as
F o V
ce M ^ VT
6.18
54
By analogy to equation 6.16, the energy loss per unit surface
area due to neutral heating is
6.19
where is the average momentum collision frequency for
electrons with neutral particles.
The power balance equation is obtained by equating the
power input at the wall per unit surface area to the total
power loss per unit surface area
Rv £ a + + £3 +£4.
6.20
Substituting from 6.7, 6.9, 6.16, and 6.19, the power balance
becomes
6.21
This equation can be solved for n0, thus completing the
theory for the operation of the discharge.
In summary, expressions have been derived for the elec
tron temperature and number density for a long cylindrical
electrodeless discharge. In addition, the electromagnetic
field in the discharge due to an infinite solenoid has been
derived. All of these quantities will depend on the physical
characteristics of the solenoid and the discharge. These
characteristics are:
55
1. The gas used in the discharge
2. The discharge pressure
3. The discharge radius
4. The frequency of the applied field
5. The number of turns per unit length of
the solenoid
6. The current carried by the solenoid.
In Chapter VII a sample calculation will be made for the
same discharge in which the experiments of Chapter II were
performed. In addition, the electromagnetic field will be
calculated for a more general range of physical parameters.
CHAPTER VII.
CALCULATIONS
A theory for the electrodeless discharge was developed
in the preceding chapters but no calculations were made
there. The values for the various plasma properties will
depend on the physical characteristics of the particular
discharge under consideration. However, the electromagnetic
field solutions are functions only of and so that
rather complete calculations can be made for the field dis
tributions which are applicable to a wide range of discharges.
A sample calculation will be made for a discharge having
the physical characteristics of the one in which the experi
ments of Chapter II were performed. Only certain limited
comparisons of experiment with theory can be made due to the
lack of sufficient experimental data. The discharge for
which the calculations will be performed has the following
characteristics:
1. The gas is argon.
2. The pressure is in the range of 0.1 Torr to 1 Torr.
3. The discharge radius is 2.4 cm.
4. The applied frequency is 4.5 MHz.
5. The number of turns of the solenoid is 139 per meter.
6. The solenoid current is an independent variable.
It has been shown in Chapter III that the temperature
distribution depends upon the parameter Pw log-A/T0^^. It
was also shown in that chapter that for the discharge being
considered the variation of the temperature across the dis
charge radius is negligible and, thus, it is not necessary to
calculate results from equation 3.40.
56
57
The determination of the value of the temperature, as
derived in Chapter IV, involves the calculation of the average
ionization collision frequency. Since the determination of
average collision frequencies is required for many of the
calculations, the general procedure will be discussed.
In general, collision frequencies depend upon the elec
tron energy. To obtain the average collision frequency it
is necessary to average over the electron distribution func
tion f. The collision frequency for the process is
related to the cross-section for that process by
Vj = Qj sn9 7.1
where Qj is the cross-section for the process, g is the
relative velocity between the particle and its target
particle, and ng is the number density of target particles.
The collision frequencies to be calculated are the ionization
collision frequency, the elastic or momentum collision fre
quency, and the excitation collision frequency. All of these
processes involve the collision of an electron with a neutral
particle. The electron temperature in a low-density plasma
is usually higher than the neutral temperature. For this
reason, and also because the electron mass is much smaller
than the neutral mass, it is assumed that the relative
velocity g is due solely to the electron velocity, i.e.,
g v-?/3 = v 7.2
and becomes
Tj = QjVHg
7.3
58
Two different averages will be considered. The average
momentum frequency^m^ is involved in the plasma conductivity.
As pointed out in Chapter V, the type of averaging required
depends upon the applied radian frequency relative ,
Following reference 17, it will be assumed that(ym^ is calcu
lated for the high-frequency case. Thus, is defined by
7.4
where v is the electron speed, and where it is assumed that
the distribution function f depends only upon v .
In general, the average of a function of electron speed
is given as
<$) = 7f
7.5
Thus, the average collision frequency for the process is
given by
7.6
From the expression for the collision frequency 7.3, equa
tions 7.4 and 7.6 become
7.7
and
7.8
59
The average collision frequencies are seen to be functions
of the appropriate cross-section, the distribution function,
and the neutral number density rig.
The distribution function, as discussed previously, will
be assumed to be Maxwell-Boltzmann. This distribution is
given by
n /3 fe n Z
T = [ZTTj (y*}3'*. & 7.9
where <(v2^> is the mean square electron speed and is related
to the electron temperature by
-S. m kT
7.10
It is convenient for computational purposes to express elec
tron speeds in terms of volts. The following expressions
serve to define V and V.-p
-L m V* = e V
and
I- m(vz> =eVT
7.12
Substitution of 7.9, 7.11 and 7.12 into 7.7 and 7.8 yields
\V
/V\- 3(3e/nm) r)a
( (Vt)5*
7.13
60
and
(Vj) = s(3e/rrm)Vz
(%)** J.
7.14
where Qm and Qj are functions of V and the limits of integra
tion are from 0 to oO as shown.
The calculation of average collision frequency from 7.13
and 7.14 depends on the form of the appropriate cross-section.
The cross-sections which are used are all determined experi
mentally. The computational scheme used involved finding a
mathematical approximation for the experimental cross-sections
and performing a numerical integration by use of Simpson's
rule on the IBM 7090 or 360/50 computer.
The momentum cross-section Qm for argon was obtained
O C
from Barbiere who averaged the angle dependent data of
Ramsauer and Kollath. An analytic approximation to the
Barbiere data was found by use of a curve fitting program
obtained from the University of Florida computing center.
Since the data given by Barbiere extended only to 12.5 volts,
an approximation to data given by Brown26 (who took it from
Brode) was added to the Barbiere approximation to extend the
cross-section to higher values. The final form for the
approximation is
and
7.15
The results of the computation for^p>m\ are given in Figure 7.
61
The average ionization collision frequency was calculated
from equation 7.14. The cross-section data were obtained from
Kieffer^7 who tabulates data from five different investiga
tions. The data of Smith were chosen as there was little
difference in the data of the various investigators. An
approximation to the Smith data was found by use of the pre
viously mentioned program. The approximation is given by
QL= 0 V < 15.7
and 7.16
Qi= 10 [-3.734 + 0.313V- 4.400 tJO V
+ Z.OZB xlO'^V3) VZIS.7
The results of the computation for are shown in Figure 8.
To calculate a power balance, it is necessary to deter
mine the losses due to excitation and radiation. It is not
necessary to determine the losses to a particular excited
state since it is the energy loss from the electrons to all
excited states which enters into the power balance. An
experimental determination of the total excitation cross-
section for argon was obtained by Maier-Leibnitz^ and
presented by Brown^ who obtained the data from a paper by
Druyvesteyn and Penning. 29 The data cover the range from
10.8 V. to 19.0 V. and a constant cross-section is assumed
for energies greater than 19.0 V. The approximation for the
excitation cross-section is given by
Q*=0 VC10.8
Qx = 10 [o.048(v-10.8) ] JO.8
Q. = t0* V>19.0
The results of the computation for are shown in Figure 9.
62
In addition to values for the average collision fre
quencies, it is necessary to determine the ambipolar
diffusion coefficient. The ambipolar diffusion coefficient
was shown in Chapter IV to be related to the ion mobility
and electron and ion temperatures by
7.18
In a low-pressure gas discharge the electron temperature is
usually much greater than the ion temperature. In the worst
case the two temperatures would be the same. From 7.18 it
is seen that in the two limiting cases, T+/T_ = 0 and
T+/T_ = 1, the ambipolar diffusion coefficient varies by a
factor of two. Many measurements of ion mobility have been
made and the accepted value for the reduced mobility of
O A
1.5 cm2 v-1 sec-1 was obtained from Hasted. u The reduced
mobility, KQ, is related to the mobility by
r, >#+1 273
760 r3
7.19
where p is the neutral gas pressure in Torr and Tg is the
neutral gas temperature. Written in terms of the neutral
gas number density ng the expression for the ambipolar
diffusion coefficient for argon becomes
ndDc? = Z. 69* 10** ZT(l + 7.20
It was shown in Chapter IV that the electron temperature
was determined from the eigenvalue equation 4.16 since the
quantity (y)A/Da is primarily a function of electron tempera
ture. The dependence of/^|\/Da on electron temperature for
63
the limiting cases T+/T_ = 0 and T+/T_ = 1 is shown in
Figure 10. The function <(^i) /Da is a function of the neutral
gas number density and, therefore, the neutral gas pressure
and temperature. The neutral gas temperature in a plasma is
difficult to determine accurately. At low pressure only a
small portion of the energy is lost to the neutral gas, and
since the discharge walls are water cooled in the experimental,
apparatus it will be assumed that the neutral gas temperature
is 300K. Thus, the neutral gas number density is given by
7.21
where p is the pressure measured in Torr and ng has the units
m-3. Using equation 7.21 for ng,
2.4 cm for the discharge radius and the eigenvalue equation
4.16, it is possible to determine the electron temperature
as a function of the discharge pressure. This curve is
presented in Figure 11. The data points shown in Figure 11
are the electron temperatures as measured by the probe
corresponding to the number density data shown in Figures 2,
3 and 4. It is seen that the electron temperature decreases
as the pressure increases. This is due to the fact that the
diffusion loss becomes less as the pressure is increased, and
a smaller temperature is sufficient to produce the ionization
required for a plasma balance. The measured electron tempera
tures are consistently lower than the predicted temperature
by approximately 20 per cent with an average discrepancy of
19.6 per cent. This discrepancy is not unreasonable for
probe data and may be due to the lack of purity in the gas
used. The discrepancy could easily be explained by contami
nants having larger ionization cross-sections or lower
mobility. One such contaminant which might have been present
is mercury, since an untrapped McLeod gauge was used to
measure the discharge pressure.
64
It was shown in Chapter V that the electromagnetic
field depends only upon and p The parameter Ke*-^ is
related to the plasma properties by
~~ C 1 f-nZ ,
r i{VZ-^COZ)yz
7.22
where K is a real number given by the ratio of discharge
radius to skin depth. K may be expressed, in terms of the
plasma properties, as
7.23
From 7.23 it is seen that K depends on the electron number
density through nQ and the electron temperature and neutral
pressure through ^ Since the electron temperature-
neutral pressure relationship has been determined for the
discharge, it is possible to calculate K. The parameter f
is a function of the ratio of the collision frequency to
the applied radian frequency given by the expression
7.24
Physically, is a parameter related to the degree to which
the plasma conductivity is resistive or reactive. It is also
a function of the electron temperature and neutral pressure
through "p .
65
Computer calculations have been performed to obtain the
magnitude of the electric and magnetic fields for three
values of l^/CO namely 0, 1 and which correspond to a
purely reactive, equally reactive and resistive, and a
purely resistive plasma, respectively. These calculations
were performed for integer values of K from 1 through 9. For
convenience, the field quantities have been made non-
dimensional with respect to Bw the value of the magnetic
induction at the wall. The corresponding non-dimensional
quantities are
. B and El 7.25
Bw 0 Byytf
Curves for these quantities are presented in Figures 12, 13,
and 14. In order to perform a power balance it is necessary
to compute the average energy flux into the discharge at the
discharge boundary. This is the quantity Pw of Chapter VI
and it is calculated from the time average of the Poynting
vector given in Chapter V. This quantity is also made non-
dimensional;
r-, # J2 ~Fw
P co B£ a 7-26
Curves for this quantity as a function of K for various
values of 7^/C are presented in Figure 15. The quantity P*
can be thought of as representing the energy flux per unit
current squared since
Bw stio NI0
7.27
66
The curves of Figure 15 show that this quantity has a maxi
mum at some value of K. The maximum occurs because at small
values of K the power input is low due to the small conduc
tivity, and for values of K greater than that at which the
maximum occurs the fields induced by the solenoid are unable
to penetrate very far into the plasma and the power is
dissipated in a thin region near the surface. Therefore,
the most efficient operating condition for the electrodeless
discharge, in terms of coil current required, occurs where P*
is a maximum.
The electron number density at the center of the dis
charge will be a function of the power input to the discharge
and, therefore, a function of the coil current. Equation 6.21
can not be solved explicitly for nQ because of the complex
dependence of Pw upon nQ. Therefore, a graphical solution
will be obtained. The quantity P*, defined by equation 7.26,
is a function only of Ke^f. Since for a given discharge the
temperature is a function of pressure, then is determined
for each value of pressure. With ^determined for a given
pressure P* becomes a function of n Equation 6.21 can be
made non-dimensional such that
P* =
<0 /Q /?Q 6
coB% a
D^[yr+vt]
0. z 15s a VT +
7.28
For a given discharge radius, frequency, and number of
solenoid turns per unit length, the right hand side of 7.28
is a function of nQ and the coil current through the term
2
Bw It will be convenient to express Bw in terms of the
root mean square value of the coil current such that
67
Bw = /So A/I0 /So//1rms 7.29
The current Irms will be taken as an independent parameter
and, thus, the right hand side of 7.28 becomes a linear
function of nQ. For each value of discharge pressure,
equation 7.28 is solved by plotting P* and the right hand
side of 7.28 as a function of nQ for various values of Irms.
A separate graph is required for each value of the pressure.
P* was obtained for the temperature consistent with each
value of pressure by computing the solutions for the elec
tromagnetic field given in Chapter V. The IBM 360/50 computer
was used to calculate the required series. The series is
complex and the real and imaginary parts oscillate in sign
and acquire large values before they begin to converge.
Therefore, it was necessary to program the computer for
sixteen significant digits to insure sufficient accuracy over
the required range of values for K. A sample of the graphical
solution is shown in Figure 16. This solution is for a dis
charge pressure of 0.5 Torr.
The solution of the power balance equation as described
above results in a family of curves, one for each discharge
pressure, relating the electron number density at the dis
charge axis to the applied current in the solenoid. The
family of curves for the sample calculation is shown in
Figure 17.
CHAPTER VIII.
CONCLUSIONS
In the preceding chapters a probe diagnostic technique
was described with which it becomes possible to resolve experi
mentally the spatial distribution of electron number density
and temperature as well as the plasma potential in an elec
trodeless discharge. The results of these experiments were
found to disagree with some of the commonly held assumptions
about the electron temperature distribution and the spatial
production of ionization in the discharge. This discrepancy
resulted in the formulation of a new theory for the operation
of the electrodeless discharge which was found to be in
reasonable agreement with the probe measurements. Although
the theory agrees with the measurements, a considerable
number of predictions of discharge behavior have been made
which have not been subject to direct experimental verifica
tion. However, in the course of the experiments, certain
qualitative observations have been made which can be compared
with the predictions of the theory. These comparisons will
be discussed later.
First, it would seem appropriate to point out some of
the limitations and critical assumptions made in arriving at
the theory for the electrodeless discharge. The model adopted
for the discharge makes the problem one-dimensional. This is
due to the choice of a model which assumed that the discharge
was infinitely long, and possessed axial symmetry.
For an actual laboratory discharge the assumption of
axial symmetry would appear to be quite reasonable. This is
68
69
due to the fact that the discharge is usually formed in a
container which is a circular cylinder surrounded by a coil
of circular cross-section. The assumption of a discharge
having infinite length is less reasonable. All laboratory
discharges must be finite in length even though the length
can be made large compared to the radius. This fact imposes
a more serious limitation to the theory due to the axial
variation in the number density which is induced by a dis
charge of finite length. It is seen from equation 4.14 that
the number density will follow a cosine variation in the
axial direction. For a discharge which is long compared to
its radius, the error in the predicted electron temperature
will be small as can be seen from the eigenvalue equation 4.15.
However, the calculation of power input will be considerably
in error for the discharge as a whole since the value of K,
upon which the power input depends, will vary from a maximum
at the center of the discharge to zero at the ends of the
discharge. The power input as a function of the number den
sity at the discharge axis should be approximately correct,
however, and a reasonable calculation of total input power
might be made by integration of the power over the length of
the discharge. The input power is a function of the axial
position through K which is, in turn, a function of the number
density on the discharge axis.
A further approximation has been introduced by the
assumption of an infinite solenoid having a purely azimuthal
current flow. In actual practice the coil is a helix of
finite length. For such a coil the electric and magnetic
fields are not purely azimuthal and axial, respectively. The
electric field will have an axial component and the magnetic
field will have an azimuthal component due to the helix
angle and, also, at regions near the end of the coil, large
deviations from the solenoidal field will occur. If the coil
70
is relatively long with respect to its diameter, the deviation
from a solenoidal field at the center should be small. In
this region the electromagnetic field predicted by the theory
should be a good approximation. The only electromagnetic
quantity involved in the rest of the theory is the power
input and in the region near the center of the coil the result
should be quite accurate.
Due to the limitations imposed on the theory from the
assumptions of infinite length, the prediction of the number
density on the discharge axis as a function of the solenoid
current can only be approximate. However, the general trend
as shown in Figure 17 should be quite representative for a
laboratory discharge.
The assumption of a Maxwell-Boltzmann distribution for
the electrons is, of course, only an approximation. Devia
tions from this distribution may have a large effect on the
calculated values of the average collision frequencies for
ionization and excitation. This results from the fact that
the cross-sections for these processes are non-zero only
in the high-energy portion of the distribution function. For
the average momentum collision frequency this effect will be
much smaller. This fact is shown by Reference 17 where it
is found that various averages over both a Maxwell-Boltzmann
and Druyvesteyn distribution differ by less than 15 per cent.
In Reference 17 it is shown that the distribution func
tion may be expanded in spherical harmonics in velocity space.
Due to a spatial effect, integration over the second order
term yields no difference in the calculated average ionization
or excitation collision frequencies. Therefore, only when
higher order approximations are important will the values of
ionization and excitation collision frequencies be affected.
Some calculations of the average ionization collision
frequency were made using the Druyvesteyn distribution and
71
these led to predicted electron temperatures which were
several electron volts higher than those measured. For this
reason the assumption of a Maxwell-Boltzmann distribution
appears to be justified.
Certain qualitative observations made in the course of
the experimental investigation tend to support the theory.
One striking feature of the theory is that of a constant
electron temperature, for a given discharge pressure, inde
pendent of the input power. While observing a probe trace
on the oscilloscope it was noticed that a change in input
power did not affect the shape of the probe curve in its
exponential region, which determines electron temperature,
but caused a large change in the current at which saturation
occurred. It is this point which determines the number
density. In fact, a small change in input power could easily
change this point by a factor of ten. This tends to support
the prediction that the temperature is independent of power
input, and that the number density is critically dependent
on input power as shown in Figure 17.
Further evidence that this is true was obtained when an
undergraduate student in Aerospace Engineering, Mr. Don Green,
performed heat transfer measurements at the discharge bound
ary. The heat transfer to the wall should be proportional to
the number density at the discharge axis. When the heat
transfer rate was plotted as a function of the coil current,
curves were obtained which were in qualitative agreement
with Figure 17.
Another undergraduate student in Aerospace Engineering,
Mr. Paul Bloom, has made measurements of the power input to
the discharge simultaneously with probe measurements of
electron number density and temperature. By using the
measured values of number density and temperature he has
calculated the value of K along the discharge axis, and by
72
use of the theory integrated the power input over the length
of the discharge to obtain predicted power input. Prelimi
nary results indicate that good agreement between predicted
and measured total power input is obtained.
The theory for the electrodeless discharge which has
been presented shows reasonable qualitative and quantitative
agreement with the available experiments. Although it is
not an exact description of the discharge it appears to
describe the major physical processes which occur. The
theory should prove useful in determining the general behav
ior of the discharge and for providing a basis upon which
engineering designs of future discharges could be made.
10J Amps/Division
Volts/Division
x N
o
0
2.5 Volts/Division
Figure 1. Typical probe characteristic with third probe trace
co
Electron Number Density, (10^2 cm-^)
74
Figure 2. Electron number density at a pressure of 0.13 Torr.
Electron Number Density, (lO^ cm-3)
75
Figure 3. Electron number density at a pressure of 0.26 Torr.
Electron Number Density, (lO1^ cm
76
co
i
Figure 4. Electron number density at a pressure of 0.40 Torr.
Normalized Electron Number Density
77
0 0.13 Torr
0.26 Torr
A 0.40 Torr
Figure 5. Comparison of measured number density with the
theory of Eckert.
78
Figure 6. Plasma potential.
79
Figure 7. Computed average momentum collision frequency.
- (m3/sec)
80
VT, (volts)
Figure 8. Computed average Ionization collision frequency.
81
Vip, (volts)
Figure 9. Computed average excitation collision frequency.
(volts)
Figure 11.
Electron temperature as a function of pressure for a long cylindrical
discharge of 2.4 cm radius.
oo
u
Figure 12. Electric and magnetic field for
0
p
p
= J
Figure 13. Electric and magnetic field for t^/CO
oo
tn
p
f
Figure 14. Electric and magnetic field for jS/co
oo
05
Z*. Z/co B*
K
00
Figure 15. Average power input at the wall.
Figure 16. Solution of the power balance equation for a pressure of 0.5 Torr.
oo
00
89
Figure 17.
Electron number density at the discharge axis as
a function of solenoid current at various pressures
K ffvN O
REFERENCES
1. Hittorf, Wo, "beber die Electricitatsleitung der Gase,"
Wiedemann Ann Phys. Chim. 21, 90 (1884).
2. Thomson, J. J., "The electrodeless discharge through
gases," Phil. Mag. 4, 1128-1.160 (1927).
3. Townsend, J. S. and Donaldson, R. H., "Electrodeless
discharges," Phil. Mag. 178-191 (1928).
4. MacKinnon, K. A., "On the origin of the electrodeless
discharge," Phil. Mag. £5, 605-616 (1929).
5. Reed, T. B., "High-power low density induction plasmas,"
J. Appl. Phys. 34, 3146-3147 (1963).
6. Clarkson, M. H., Field, R. E., and Keefer, D. R.,
"Electron temperature in several rf-generated plasmas,
AIAA Journal 4, 546-547 (1966).
7. Eckert, H. U., "Diffusion theory of the electrodeless
discharge," J. Appl. Phys. 33, 2780-2788 (1962).
8. Keefer, D. R., Clarkson, M. H., and Mathews, B. E.,
"Probe measurements in an electrodeless discharge,"
AIAA Journal 4, 1850-1852 (1966).
9. Allis, W. P., "Motions of ions and electrons," Handbuch
der Physik 21, 383-444, Springer Verlag (1956).
10.Brown, S. C., "Breakdown in gases: alternating and
high-frequency fields," Handbuch der Physik 22, 531-574,
Springer Verlag (1956).
90
91
11. Langmuir, I. and Mott-Smith, H., Jr., "Studies of
electrical discharges in gases at low pressures,"
Gen. Elec. Rev. 27, 449-455 (1924).
12. Loeb, L. B., Basic Processes of Gaseous Electronics,
Univ. of Calif. Press (1955).
13. Johnson, E. 0. and Malter, L., "A floating double probe
method for measurements in gas discharges," Phys. Rev.
80, 56-68 (1950).
14. McDaniel, E. W., Collision Phenomena in Ionized Gases,
John Wiley and Sons, Inc., New York (1964).
15. Bohm, D., The Characteristics of Electrical Discharge
in Magnetic Fields, edited by A. Guthrie aind R. K.
Wakerling, McGraw-Hill Book Company, Inc., New York
(1949) .
16. von Engel, A., "Ionization in gases by electrons in
electric fields," Handbuch der Physik 21, 504-572,
Springer Verlag (1956).
17. Shkarofsky, I. P., Johnston, T. W., and Bachynski, M. P.,
The Particle Kinetics of Plasmas, Addison-Wesley Pub. Co.,
Reading, Mass. (1966).
18. Francis, G., "The glow discharge at low pressure,"
Handbuch der Physik 22, 53-203, Springer Verlag (1956).
19. Allis, W. P. and Rose, D. J., "The transition from free
to ambipolar diffusion," Phys. Rev. 93, 84-93 (1954).
20. Sovie, R. J., Private communication, also presented
APS meeting, Atlantic City, Nov. 1962 and APS meeting,
Boston, Nov. 1966.
21. Dingle, R. B., Appl. Sci. Res. 6B, 144-154, 155-164,
245-252(1.957) .
92
22. Pugh, E. M. and Pugh, E. W. Principles of Electricity
and Magnetism, Addison-Wesley Pub. Co., Inc., Reading,
Mass. (1960).
23. Holt, E. H. and Haskell, R. E., Foundations of Plasma
Dynamics, The Macmillan Co., New York (1965).
24. Henriksen, B. B., Analysis of field distributions in
an electrodeless discharge, Thesis, University of Florida
(1966).
25. Barbiere, D., "Energy distribution, drift velocity, and
temperature of slow electrons in helium and argon,"
Phys. Rev. 84, 653-658 (1951).
26. Brown, S. C., Basic Data of Plasma Physics, Technology
Press of The Massachusetts Institute of Technology and
John Wiley and Sons, Inc., New York (1959).
27. Kieffer, L. J., "A compilation of critically evaluated
electron impact ionization cross section data for atoms
and diatomic molecules," JILA Report No. 30, University
of Colorado (1965).
28. Maier-Leibnitz, H., "Ausbeutemessungen beim Stoss
langsamer Elektronen mit Edelgasatomen," Zeits. Phys.
95, 499-523 (1935).
29. Druyvesteyn, M. J. and Penning, F. M., "The mechanism of
electrical discharges in gases of low pressure," Revs.
Mod. Phys. 12, 87-174 (1940).
30. Hasted, J. B., Physics of Atomic Collisions, Butterworth,
Inc., Washington, D.C. (1964).
I.
^ ;
BIOGRAPHICAL SKETCH
Dennis Ralph Keefer was born September 22, 1938, at
Winter Haven, Florida. He was graduated from Auburndale
High School in June, 1956. In June, 1962^ he received the
degree of Bachelor of Engineering Sciences with High Honors
from the University of Florida. Mr. Keefer enrolled in the
Graduate School of the University of Florida in June, 1962.
He was employed as a research assistant until August, 1963,
when he received the degree of Master of Science in
Engineering. From September, 1963, until the present, he
has pursued his work toward the degree of Doctor of
Philosophy while employed by the Department of Aerospace
Engineering as a research associate.
Dennis Ralph Keefer is married to the former Charlotte
Ann King and is the father of two children. He is a member
of the American Institute of Aeronautics and Astronautics,
Tau Beta Pi, and Sigma Pi Sigma.
93
This dissertation was prepared under the direction of
the chairman of the candidates supervisory committee and
has been approved by all members of that committee. It was
submitted to the Dean of the College of Engineering and to
the Graduate Council, and was approved as partial fulfill
ment of the requirements for the degree of Doctor of
Philosophy.
August, 1967
Dean, Graduate School
Supervisory Committee:
OrU £
82
Figure 10. Temperature dependence of /Da.
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The theory and the diagnosis of the electrodeless discharge,
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1967
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