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## Material Information- Title:
- The theory and the diagnosis of the electrodeless discharge
- Added title page title:
- Electrodeless discharge
- Creator:
- Keefer, Dennis Ralph, 1938-
- Publication Date:
- 1967
- Language:
- English
- Physical Description:
- vii, 93 leaves : illus. ; 28 cm.
## Subjects- Subjects / Keywords:
- Electric fields ( jstor )
Electrodeless discharges ( jstor ) Electron energy ( jstor ) Electrons ( jstor ) Energy ( jstor ) Ionization ( jstor ) Ions ( jstor ) Magnetic fields ( jstor ) Plasmas ( jstor ) Solenoids ( jstor ) Aerospace Engineering thesis Ph. D Dissertations, Academic -- Aerospace Engineering -- UF Electric discharges through gases ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Thesis:
- Thesis - University of Florida.
- Bibliography:
- Bibliography: leaves 90-92.
- General Note:
- Manuscript copy.
- General Note:
- Vita.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 022267181 ( ALEPH )
13556760 ( OCLC ) ACZ2015 ( NOTIS )
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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, 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 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 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 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 pressureand 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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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(KeÂ£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 Electron Number Density, (lO1^ cm 76 co i Figure 4. Electron number density at a pressure of 0.40 Torr. 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 % = /*'(Â£+ 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 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. 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. 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 Â£ a + + Â£3 +Â£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: 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 Â£ 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 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 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 82 Figure 10. Temperature dependence of /Da. 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 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 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 The results of the computation for are shown in Figure 9. 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 53 Substituting 6.3 into 6.12 and integrating yields GÂ¡ = 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 -Â§- = 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 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 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 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 term 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 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. 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. 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^. 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 pÂ£,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. 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 p f Figure 14. Electric and magnetic field for jS/co oo 05 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. 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 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). 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 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. 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 Â£ 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. 78 Figure 6. Plasma potential. Electron Number Density, (lO^ cm-3) 75 Figure 3. Electron number density at a pressure of 0.26 Torr. 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 = Â£L Vn_+ r\-//_ E 3.10a and K = -D+Vn+-+ r\+/A+ E 3.10b 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 10J Amps/Division Volts/Division x N o 0 2.5 Volts/Division Figure 1. Typical probe characteristic with third probe trace co To my wife CHARLOTTE who, having little understanding for the subject, has shown great understanding for the author. 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 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 (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 - 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)Â¡, 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 % = /*'(Â£+ 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 term 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 = Â£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 7Â£ 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 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 Â£ 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 /! = -Â£>_ 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 Â£= **' 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-osÂ£) 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> Â£ 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 Â£ 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 42 dzAs> i 1 c>Ai> 1 a Â£ 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 conductivity o' can be written as 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 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 Â£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(KeÂ£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 = Â£ 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 pÂ£,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 GÂ¡ = 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 -Â§- = 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 Â£ a + + Â£3 +Â£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 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 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 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 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 BÂ£ 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. Â£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 Â£ 82 Figure 10. Temperature dependence of /Da. Internet Distribution Consent Agreement In reference to the following dissertation: AUTHOR: TITLE: Keefer, Dennis The theory and the diagnosis of the electrodeless discharge, (record number: 565597) PUBLICATION DATE: 1967 , as copyright holder for the aforementioned dissertation, hereby grant specific and limited archive and distribution rights to the Board of Trustees of the University of Florida and its agents. I authorize the University of Florida to digitize and distribute the dissertation described above for nonprofit, educational purposes via the Internet or successive technologies. This is a non-exclusive grant of permissions for specific off-line and on-line uses for an indefinite term. Off-line uses shall be limited to those specifically allowed by "Fair Use" as prescribed by the terms of United States copyright legislation (cf, Title 17, U.S. Code) as well as to the maintenance and preservation of a digital archive copy. Digitization allows the University of Florida to generate image- and text-based versions as appropriate and to provide and enhance access using search software. This grant of permissions prohibits use of the digitized versions for commercial use or profit. Printed or Typed Name of Copyright Holder/Licensee Personal information blurred Date of Signature S/t-S/d' Please print, sign and return to: Cathleen Martyniak UF Dissertation Project Electron Number Density, (10^2 cm-^) 74 Figure 2. Electron number density at a pressure of 0.13 Torr. 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 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 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 41 and the vector potential A (see Pugh and Pugh^) as 5 6a and 5.6 b where and Â£ 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 10 The current relation for probe 3 is then given by - 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)Â¡, 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 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 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 p p = J Figure 13. Electric and magnetic field for t^/CO oo tn 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 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 Â£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' 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 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 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 (volts) Figure 11. Electron temperature as a function of pressure for a long cylindrical discharge of 2.4 cm radius. oo u 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 7Â£ 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 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 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 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 42 dzAs> i 1 c>Ai> 1 a Â£ 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 conductivity o' can be written as m v*+coz 5.13 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 /! = -Â£>_ 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 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 xml version 1.0 encoding UTF-8 REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID ELBVJ28I2_SP2BRI INGEST_TIME 2017-07-14T23:11:52Z PACKAGE UF00085807_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES 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. 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 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 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 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 60 and (Vj) = s(3e/rrm)Vz 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. 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 Internet Distribution Consent Agreement In reference to the following dissertation: AUTHOR: TITLE: Keefer, Dennis The theory and the diagnosis of the electrodeless discharge, (record number: 565597) PUBLICATION DATE: 1967 , as copyright holder for the aforementioned dissertation, hereby grant specific and limited archive and distribution rights to the Board of Trustees of the University of Florida and its agents. I authorize the University of Florida to digitize and distribute the dissertation described above for nonprofit, educational purposes via the Internet or successive technologies. This is a non-exclusive grant of permissions for specific off-line and on-line uses for an indefinite term. Off-line uses shall be limited to those specifically allowed by "Fair Use" as prescribed by the terms of United States copyright legislation (cf, Title 17, U.S. Code) as well as to the maintenance and preservation of a digital archive copy. Digitization allows the University of Florida to generate image- and text-based versions as appropriate and to provide and enhance access using search software. This grant of permissions prohibits use of the digitized versions for commercial use or profit. Printed or Typed Name of Copyright Holder/Licensee Personal information blurred Date of Signature S/t-S/d' Please print, sign and return to: Cathleen Martyniak UF Dissertation Project 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 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 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 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. Â£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 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 = Â£ 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 Z*. Z/co B* K 00 Figure 15. Average power input at the wall. Figure 12. Electric and magnetic field for 0 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 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 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 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 . - (m3/sec) 80 VT, (volts) Figure 8. Computed average Ionization collision frequency. 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 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 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 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 BÂ£ 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 34 jo^> A J <2 Â£= **' 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-osÂ£) 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> Â£ f.784 4.16 Figure 16. Solution of the power balance equation for a pressure of 0.5 Torr. oo 00 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 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 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 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 Page Figure 17. Electron number density at the discharge axis as a function of solenoid current at various pressures. 89 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 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. 79 Figure 7. Computed average momentum collision frequency. 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 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 (2) 2.4 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. 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) . 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 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 89 Figure 17. Electron number density at the discharge axis as a function of solenoid current at various pressures K ffvN O 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 81 Vip, (volts) Figure 9. Computed average excitation collision frequency. 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 |