THE UNIVERSIT Y OF MICHIGAN COLLEGE OF ENGINEERING Department of Nuclear Engineering Technical Report CONTINUOUS CROSS-MODULATION OF MICROWAVES IN A HELIUM PLASMA Kenneth D. Ware ORA Project 07599 sponsored by: Advanced Research Projects Agency Project DEFENDER ARPA Order No. 675 under contract with: U.S. ARMY RESEARCH OFFICE-DURHAM CONTRACT NO. DA-31-124-ARO(D)-403 DURHAM, NORTH CAROLINA administered through: OFFICE OF RESEARCH ADMINISTRATION ANN ARBOR February 1968

This report was also a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 1968.

ACKNOWLEDGMENTS I am grateful for the willing support and direct assistance of several people. To Professor David R. Bach, project director, I am deeply indebted for his patience and guidance *throughout the extent of this investigation. To Professors Ziya Akcasu (who suggested this study) and Richard K. Osborn, I wish to express sincere gratitude for directing my theoretical analysis. To Professor William N. Lawrence I am very thankful for many late hour sessions of technical assistance and personal counsel. I also acknowledge Professors John S, King and Andrejs Olte for their helpful suggestions during the early stages of the experiment, in particular, for their understanding of other research in this field, and for their critical evaluation of the results. I wish to thank Professor Edward A. Martin for his guidance in the area of gaseous discharges and Mr. Ronald R. Rickwald for uncovering some of the properties of the glow discharge used in this investigation. Experimental assistance from students Mr. Douglas Kreifels and Mr. Thomas Leonard is gratefully acknowledged. Several discussions with Dr. Edmund K. Miller proved valuable to the project, and for his time and suggestions I am grateful. In addition to the friendships of the persons above, my thanks go to Drs. Ralph R. Rudder, Leonard H. Wald, Carl M. Penney, and James F. Lafferty, and Mr. Erol Oktay. Discussion of problems with these friends was very helpful to my progress. I would especially like to thank Professor William Kerr, Chairman of the Nuclear Engineering Department, for his personal encouragement and the availability of departmental facilities and services. His continued interest in keeping the paths to thesis progress clear of unscientific obstacles is gratefully acknowledged. The love and understanding of my wife, Judy, have been invaluable through. out this study. For her encouragement, as well as our parents', I ill always be thankful. In preparation of the manuscript my thanks go to Miss Julie Calver and to the typing, technical illustration, photography, and reproduction staffs of the Office of Research Administration. I appreciate the Ford Motor Company Technical Computer Center for giving me valuable time with their GE-2355 computer during the course of this project. The major portion of this work was supported by the Advanced Research Projects Agency (Project DEFENDER) and monitored by the U. S. Army Research OfficeDurham under Contract No. DA-31-124-ARO(D)-403. ii

TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS v ABSTRACT vii Io INTRODUCTION 1 IIo THEORY OF CONTINUOUS CROSS-MODULATION 5 A. Sensing Microwave Response to Time Varying Conductivity 7 B. Formulation of Electron Density and Temperature Equations 16 C, Variations in the Electron Temperature 29 D. Variations in the Electron Density 36 E, Relative Importance of Temperature and Density Variations 40 III. EXPERIMENTAL APPARATUS 45 A. Vacuum System 45 B. Discharge System 47 C. Langmuir Probe Circuit 48 D. Microwave Circuitry 53 E, Electronic Noise Suppression 57 IVo EXPERIMENTAL TECHNIQUES AND DATA ANALYSIS 59 A, Helium Glow Discharge 59 B. Cross-Modulation Response 72 V. RESULTS AND DISCUSSION 80 APPENDIX A. THE HIGH FREQUENCY AND DC PLASMA CONDUCTIVITIES 90 1. The High Frequency Conductivity 91 2. The DC Conductivity 92 APPENDIX B. NORMALIZED VARIATION OF THE DC CONDUCTIVITY WITH THE ELECTRON TEMPERATURE 95 APPENDIX Co PRINCIPLES OF REFLECTED PLANE WAVE INTERACTION WITH SEMI-INFINITE TIME VARYING PLASMA 100 APPENDIX D, COMPUTER PROGRAM 104 ii

TABLE OF CONTENTS (Concluded) Page APPENDIX E. TABULATION OF THE DATA 112 REFERENCES 116 iv

LIST OF ILLUSTRATIONS TABLE Page 5.1. Evaluation of the Probability of Collision for Momentum Transfer, Pm(cm-l Torr-1 at O~C), and the Electron-Ion Recombination Coefficient, ao(cm3 sec-1) 87 5.2. Helium Collision Probability 88 5o53 Helium Electron-Ion Recombination Coefficient 89 El. Data 115 FIGURE 35oo Photographic view of the apparatus showing: the high voltage power supply; the movable disturbing microwave circuit; the sensing microwave circuit; the bell jar discharge with some of the microwave absorber removed; the pumping station; and the receiving and recording instrumentation on the right. 46 3.2. High voltage discharge circuit. 47 3.35 Detail of the H.V. cathode supporting post and its vacuum seal construction, 48 354. Photographic view of electrode structure with bell jar removedo As shown, the entire discharge was enclosed within a microwave absorbing "box", 49 355. Construction of the Langmuir hairpin probe and the movable probe vacuum sealo 51 3.6. Langmuir probe circuit. 52 3.7. Schematic of the microwave circuit. 54 358. Calibration curve or frequency response of the receiving instrumentation, 56 3.9. Low pass filter for 110 V, 60 Hz power lines. 58 4,1o Photographic view of the helium glow discharge. 61 v

LIST OF ILLUSTRATIONS (Concluded) FIGURE Page 4.2. Current-voltage-pressure dependence of the helium glow discharge with an aluminum cathode. 63 4,35 Typical Langmuir probe trace. 64 4.4. Electron density variation with discharge current and helium gas pressure, on axis 9-1/2 in, above the anode. 66 4.5o Electron temperature variation with the discharge current and helium gas pressure, on axis 9-1/2 in. above the anode. 67 4.6. Electron density and temperature radial variations, 9-1/2 in. above the anode. 69 4.7. Electron density and temperature axial variations, 70 4.8. Axial variation of the plasma potential. 71 4.9. Sample trace of the cross-modulation depth on the sensing microwave as recorded by the X-Y recordero 74 4.10. Four sets of normalized cross-modulation data, with computer solution. 76 4.11. "Frequency distribution" plots of the measured break frequencieso 79 Bl1. Variation of p0(y). 97 B.2. Variation of [l-y2g(y)] with y. 99 vi

ABSTRACT The purpose of this study was to investigate continuous cross-modulation of microwaves in a, steady state plasma. Earlier explorations of the technique in the ionosphere demonstrated its usefulness in plasma diagnostics, however, the measurements indicated that the phenomenon was not completely understood. The theoretical analysis is extended in this study to include effects arising from the dc conductivity, electron-ion collisions, and electron density variations, as well as those due to electron temperature variations which were considered in the previous work. Due to the heating effect of a modulated disturbing microwave, all of these plasma properties are disturbed and can cause variations in a second sensing microwave. The analysis describes the relative importance of the properties in continuous cross-modulation. One of the most favorable features of cross-modulation is that only a relative measurement of the transferred modulation depth as a function of modulation frequency is required. This suggests that the technique might be useful for geometrically unwieldy plasmas since the analysis of standard microwave absorption experiments become very difficult due to the complex scattering pattern resulting from plasma inhomogeneity. The microwave scattering patterns are most complicated when the free space wavelength is comparable to the plasma dimensions, but they only contribute to the normalization of the cross-modulation coefficient in a cross-modulation experiment. Measurements were made in a large volume (4 cu ft) helium glow discharge V.;,. 10 cm microwaves. In this case, cross-modulation depended primarily on electron density variations. Using independent measurements of electron temperature and density, the transferred modulation depth led to a value for the probability of collision for electron momentum transfer, Pm(18.7 ~ 2.1 cm-1 Torr-l at 0~C), and to a value for the electron-ion recombination coefficient, ao(356 + 2.2 x 10-8 cm3 sec-'). These quantities are in good agreement with other published values. vii

I. INTRODUCTION This experimental study was designed to measure the continuous crossmodulation between two microwaves interacting in a steady state helium plasma. The transfer of modulation is possible because of the nonlinear character of the interaction of microwaves with a plasma. A variable amplitude "disturbing" microwave can transfer energy to the plasma and cause time variations in the macroscopic properties of the plasma, In particular, the disturbing microwave can produce slow variations in the plasma conductivity, which will cause variations in the amplitude of a second "sensing" microwave interacting with the plasma. The first report of cross-modulation was presented by Tellegen in 1933 for radio waves in the ionosphere. He reported that Luxembourg radio programs, originally transmitted from Luxembourg on a carrier wave of 250 K Hz, were detected on a 650 K Hz carrier wave. This latter signal was transmitted from Beromunster, Germany to Einhoven, Holland via ionospheric reflection. Luxembourg was located nearly midway between the other two sites. After a year of speculation on the source of the interaction, Bailey and Martyn presented a theory based on the nonlinear absorption of the two carrier waves in the ionosphere to explain the phenomenon. Their analysis demonstrated the relationship between the amplitude of the transferred modulation and the electron temperature relaxation time constants Huxley and Ratcliffe, in a survey article on cross-modulation, reported measurements made with two radio waves in the ionosphere which determined the electron-neutral collision frequency 1

2 using the relative transferred modulation dependence on frequency. They noticed that the transferred modulation depth consistently decreased faster with modulation frequency than the theory predicted. The first extensive application of cross-modulation to laboratory studies 4 of plasmas was reported by Goldstein et al,, beginning in 1953. They measured the transmission of a sensing microwave passed through a plasma disturbed by a pulsed microwave signal~ In these experiments, the afterglow plasmas were enclosed within the waveguide structureo They reported measurements of the electron-neutral and electron-ion collision cross-sections and the electron thermal conductivity. For the studies cited above, the authors considered only the effects resulting from changes in the electron temperature, due to the continuously modulated or pulsed disturbing microwave. However, for some experimental conditions, such as in a dc discharge, the disturbing electric field can cause changes in the electron density through changes in the electron temperature, The magnitude of the electron density fluctuation depends on the electron density loss rate, as well as the electron temperature relaxation rate, compared to the modulation frequency. Therefore, the electron density variations exhibit a, double frequency correlation with the modulation and the electron temperature variations exhibit primarily a single frequency correlation. The relative importance of these variations in cross-modulation will also depend on the plasma conductivity for the sensing microwave. If the collision frequency for electrons is much less than the carrier microwave frequency, the imaginary part of the conductivity will be proportional to the electron density

5 and nearly independent of the electron temperatureo However, for this condition, the real part of the conductivity depends strongly on both the electron density and the electron collision frequency which is a function of the electron temperature. When the sensing microwave interacts with the plasma primarily through the imaginary part of the conductivity, cross-modulation will depend strongly on the electron density variation, Furthermore, there are experimental conditions in which the change of th(e real part of the conductivity with temperature is zero, and then the complex conductivity is sensitive only to the variation in the electron densityc Our experiment was dominated by electron density variations. We measured the continuous cross-modulation coefficient between two microwaves interacting with a steady state large volume (4 cu ft) helium glow discharge as a function of the modulation frequency The measured frequency dependence of the relative cross-modulation depth was fitted numerically to the theoretical transfer function. This yielded an averaged electron-ion recombination rate and electron temperature relaxation ratec These measurements, and Langmuir probe measurements of the electron temperature and density, led to values for the electronion recombination coefficient and the electron-neutral cross-section. To the best of our knowledge, the electron recombination rate had not been observed before in cross-modulation measurements. In Chapter II a thery for t he continuous cross-modulation of microwaves is presentedc Electron temperature and density balance equations obtained from the Boltzmann equation were taken as a, suitable description of the plasma. These equations related the variations of the macroscopic properties of the

plasma to the effects of the disturbing high frequency electric field. Our analysis was complicated by the need to consider gradients in the density, the effect of the dc electric field, and the electron-ion collision rate. The balance equations, along with Maxwell's wave equation, were solved to determine the sensing microwave signal seen by an antenna located outside the plasma. In Chapter III the experimental apparatus is described. Chapter IV contains a descript.cn of the experimental techniques and resulting data with cross-modulation and probe measurements in the large volume helium glow discharge. The results are presented and discussed in Chapter V.

II. THEORY OF CONTINUOUS CROSS-MODULATION The task in this chapter is to describe analytically the continuous crossmodulation between two microwave signals interacting with a steady state helium glow discharge. In particular, we solve for the time varying output voltage of an antenna receiving the "sensing" microwave. The amplitude modulation of the sensing microwave results from its interaction with a plasma having slowly varying macroscopic properties. The variations in the plasma are due to nonlinear heating by an amplitude modulated "disturbing" microwave. The problem is nonlinear in space and time, and the general solution is not easily found even for the simplest of real experimental conditions. We begin the search for a useful solution, which can be approximated experimentally, by considering small signal interactions. This approximation serves several simplifying purposes. First, it allows us to separate the analysis into two meaningful problems: the effect of an amplitude modulated disturbing microwave on the macroscopic plasma properties and the effect of the disturbed plasma on a small sensing microwave signal. That is, we can neglect selfmodulation of the microwaves and effects due to the coupling between the high frequency components of the two microwave signals. Also, this small signal assumption allows us to treat the leading high frequency components of the electric field in the plasma as those corresponding to the applied frequencies. However, we do not assume that the amplitude of the microwave electric field vectors are constant over the plasma volume. Furthermore, we can assume that 5

6 the variations in the plasma properties due to the disturbing microwave are small compared to the steady state values In Section A we solve for the small, time varying detector voltage output of an antenna receiving the sensing microwave signal. Using Maxwell's wave equation, we relate this response to the variations in the electron temperature and density through the complex conductivity of the plasma. Guided by the results of Sections B, C, and D, we consider linear balance equations relating the small variations in the electron temperature and density to the heating effect of the amplitude modulated disturbing microwaveo We solve these coupled equations to obtain the functional dependence of the time varying detector voltage on the modulation frequency. In Section B, we derive balance equations for the time rate of change of the electron temperature and density due to the presence of the disturbing microwave electric field and a dc electric field. For this analysis we use the conventional approach of solving the Boltzmann equation for the plasma state by expanding the electron distribution function f (x,v,t) using spherical harmonics. We use the P -approximation by considering only the first two terms of the expansion. We solve for the slow variations in the plasma properties by averaging over the period of the high frequency disturbing electric field, In Sections C and D, we consider reduced forms for the electron temperature and density balance equations, respectively, pertaining to our experimental conditionsO The nonlinear equations are solved by expanding the averaged variables about their steady state values for the first order linear

7 variations in the plasma properties. Careful consideration is given to describing the restoring rates of the variations in the electron temperature and density, as well as the coupling rates between the properties. As will be shown, it is these rates compared to the rate of energy input, the modulation frequency, which determine the relative magnitu.de of the cross-modulation coefficient of the sensing microwave detector voltage output. In Section E, we discuss qualitatively a facet of our experimental result, that i.s, the dominance of the electron density variations. We present possible explanations for this observed response. However, these arguments are not necessary for quantitative interpretation of our results, but do hopefully shed light on understanding the response. Our analysis contains all the response to continuous cross-modulation considered by previous authors with the additional effects due to dc conductivity, electron-ion collisions, and variations in the electron density. The response due to these latter effects is quite important to the interpretation of our experiment. Ao SENSING.MICROWAVE RESPONSE TO TIME VARYING CONDUCTIVITY We consider here the sensing microwave, given by E(x )e as a fixed -t input signal at the transmitting antenna located at, interacting with a plasma which can be characterized by the complex conductivity a. In order to calculate the output field evaluated at the receiving antenna, we need to solve Maxwells wave equation which can be written, in the absence of space charge effects, as

8 (v2 t'_E - (201) where C2,. ) (2.2) We want to consider a plasma with properties which are varying slowly about some steady state value. Describing these variations by 5k (x,t) about the value k (x), we let 5E(x,t) be the desired change in the field about the value E (x). Then assuming that E (x) is a, known solution of Eq. (2.1) with 2 k (x)^ we can find 6E(x,t) by solving Solving ( )h(2.5) Solving Eq, (2o3) and evaluating the field at the receiving antenna located at x, we obtain (2.4) where x1 corresponds to points in the plasma and the Green's function is 6 defined by

9 ( 7 ) i = 6 - ) (2.5) where the right hand side is a, Dirac delta function. We now consider the measurable quantity. Since there is no mixing of the polarizations, the vector signs are of no significance and may be dropped. When the complex electric field [E (x ) + bE(x,t)]e, is detected and measured with standard microwave equipment (e.g., a diode and integrating circuit with ) << 1/RC <<, where X is the rate of variations of 5k ) the rem s m sulting voltage is of the form V(c) -,1 + ~ V = I E ( 1'(2.6) where c is the "power law" of the detecting circuit. The vertical bars represent the absolute value of the complex quantity. V is proportional to IE (x )| Since the second term of Eq. (2.6) is small, we expand V(t) as oV(+)V( 4 (2.7) n terms of the normalized perturbation, be = E/ only the real part is reIn terms of the normalized perturbation, 5e = 6E/E, only the real part is required for the first order variation in the detector voltage. We write the normalized variation in the detector voltage, 6v(t) = bV(t)/V, in terms of the variation in the conductivity, using Eq. (2.2), as

10 6 ( ~-/: j& - - /' ^crj,' LI(t' O'/ - - (2.8) The high frequency complex conductivity of the plasma for the sensing microwave is given by (see Appendix A),n Y /,:. d"..,,. 1': -'';,- (,4, (2.9) 2 2 where we have assumed that 2 >> v. The effective collision frequency for s electron-neutral atom collisions is':::1~L,' r- 4 / a 18' I e, (2.10) where q is the total cross-section and for electron-ion collisions is en'/i......72 —,,i 2 e) 2 i/.7.,I 2e'b Ir. 11 (2.11) where N and n are the density of scattering centers and the remaining variables are properties of the electrons. (We write the temperature as e = kT throughout this report.) The complex conductivity is essentially a function of only

11 the electron temperature and density. We Taylor expand a(x,t) about its steady state value. Keeping only the first order terms, we take ~ TC'x -6) -i.- i eL ID C,-,40 5 - isi,)N b )de:' (2.12) in terms of the variations in the electron temperature and density. We have in mind letting 0 (xt) = ~e(x) + be(xt) and n (x,t) = n (x) + bn (x,t). O - -- O - -- Further, treating the normalized variations in the electron temperature and density as slowly varying functions of space, we expand them about the point x and neglect the gradients of 3 e/e and bne/n. We shall discuss some of "-p / o o the implications of this operation and the identification of x in Chapter V. We can now write Eq. (2.8) as-p We can now write Eq. (2.8) as 8tOff) CLt C r -.5 / /,'-~ /11!- I,' (2.13) where and (".1 2I' 2 2 a - - c) F (A,<,, (:<a A(I,; e. A - - f /',. $17~~~~~~~~~~~~~~~~~~~~~~ (2.14) (2.15)

12 To continue, we need to know the time variations of the electron temperature and density about their steady state values. We consider the effect of an amplitude modulated electric field, with a leading term in the plasma, given by (x _,;.._.. -') (i?'. + v* C c r > / _ "^ (2.16) on the electron temperature and density of an independently sustained plasma. The modulation frequency W is much less than the carrier frequency DW and m D the modulation depth 5 is less than unity. This electric field produces slow 0 variations in the plasma, properties of electron temperature and density. As we shall show in the following sections, they are, to first order, solutions of the linear equations,,~IL'c%' v e....... Liz r,.r._..'': i 11_ o I'l I ~a,L)C.u, i~'" -7 I / -- i (2.17) and I I - I t a I -3 - 1 ( f. - t... f Il,, ). o:j - C e (2.18) where

13./. a1- ).Ct (2.19) The frequencies a b' X and d in Eqs. (2.17) and (2.18) will be discussed in detail in Sections C and D of this chapter. As was discussed before Eq. (2.13), these frequencies, as well as Pi, are to be evaluated at the particular point in the plasma, denoted by x. We now solve the coupled set of equations, Eqs. (2.13), (2.17), and (2.18), for the time variation in the detector output voltage. We will neglect the harmonic term in the source P1, since it varies like 6 /4 compared to the first term. We obtain K i,i"'' ) (2.20) where,,'. 2'e.^,, | "";"'. +_"/! |,' ). —-,- ^ l,) ) -'.* —-.,.... and 0 is the phase corresponding to the complex function in the last factor. In our experiment, we measured the cross-modulation coefficient 6(wt ) as a m function of the modulation frequency w o To evaluate Eq. (2.21) we define m

14.12 2 -- (2.22) Waa) = ~ 2 a.- OW)2 _ _cI Wd 2W, ^2y 2 (2 23) and CJ - Cob -+ b od (2.24) Using the relations discussed in the following sections for the four frequencies (subscripts a, b, c and d), we found that 1 and 2 are real for our experi1 2 mental conditionso Hence, we consider only the case in which the normalized cross-modulation coefficient is of the form c3 ( ) /(/ (m/I 2)( (o), / ) (2.25) We wish to point out an important feature pertaining to the real variables (a) and (b), defined in Eqs. (2o14) and (2o15)o From Eq. (2.13), the relative values of (a) and (b) determine the importance of the variations in the electron temperature and density, respectively, to the cross-modulation coefficiento

15 Some authors have neglected the effect due to the density variation as a primary response. From Eq. (2.18), this is equivalent to taking ad as zero, d since we 6e is the source term for the density variation. For this case o( ), d m from Eq. (2.21), reduces to S (^) /-,,/, S(0) 7 / o /4)2 (2.26) This is the form considered by Huxley and Ratcliffe for continuous crossmodulation of radio waves in the ionosphere and equivalent to the form con4 sidered by Goldstein et alo, for the pulsed cross-modulation of microwaves in 8 an afterglow plasma, (Langberg and Siegel allowed electron density variation due to a, disturbing microwave, however their report deals primarily with radiation emission.) In the two cases cited, neglecting the electron density variation appears justified, However, there are experimental conditions, such as ours as we shall discuss in Section E of this chapter and in Chapter V, in which (wdb/a) dominates w3. Therefore, we use Eq. (2.25) rather than the simplified Eqo (2.26) to interpret our experimental results, to allow for density variation arising from temperature variationo We now have the form of the continuous cross-modulation coefficient with which we analyze our measurements, The remaining theoretical task is to derive the linear electron temperature and density balance equations, Eqs. (2.17) and (2.18), in order to describe the functions c., and o This probal b' c d lem is considered in the next three sections of this chapter.

16 B. FORMULATION OF ELECTRON DENSITY AND TEMPERATURE EQUATIONS In this section, we are interested in developing suitable equations for describing the slow variations in the electron density and temperature due to the presence of an amplitude modulated high frequency electric field. The slightly ionized plasma is sustained by a dc electric field. Let the singlet electron velocity distribution function be defined by fe(x,v,t), such that f d3xd3v is the number of electrons in the six dimensional element of volume d xd v about x and v at time t. It follows that the electron density is given by eX J = i -^Ir,) (2.27) and the electron temperature is defined by I7 e( ) j ( 7i: / V 2/ ) - -' e.? _1' - - - /l 6r (2.28) We take as a, description of the variations of the singlet electron distribution function in the presence of an electric field the Boltzmann equation9 a7. L - (2.29) where V and V are gradient operators in velocity space and configuration - -x space, respectively. The magnetic field effects are explicitly neglected.

17 The term on the right hand side represents the net gains per unit time in the distribution function due to collisions. That is, C L I S / A/S (2.30) where the summation is for collisions with particles of all kind B. We begin by expanding f using spherical harmonicso Keeping only the first two terms of the expansion (Pl-approximation), we have - 7=vQ' #) b T(J ~ y 77 ( 2.31) Substituting Eq. (2.31) into Eqo (2,29) and integrating over the solid angle dO, where 0 = /v, we obtain -t~ v~ g l - e a (ra I) =a J7Q (2.32) Multiplying Eqo (2o29) by Q and then integrating we obtain _t g _ err a i =(2.33)

18 The collision integrals on the right hand sides of Eqs. (2.32) and (2.33) have been discussed in great detail for this expansion by several authors. 10 11 Some of the earlier work was done by Lorentz, Morse et al., Chapman and 12 15 14 15 Cowling, and, more recently by, Dreicer,3 Desloge, and Bowe. Although the functional dependences of these terms are quite important to our analytical solution, we shall make no effort to improve on the previous authors' results. Under the approximation of m << M the collision integrals can be shown to be dn2' = S) + / ~( l i (~ i _e a 9r3) V (2.34) and Itjdn 3 =-/9 If ) S.^J (2.35) where 0 is the temperature, (kT ), of the neutral atoms and ions. The collision rate for momentum transfer g is defined in terms of the differential cross sections for electron-neutral atom and electron-ion elastic scattering collisions, dq and dq., respectively, by1 en el -G = Fu/ i I-Cos A (I/c i (Pi (.) (2.36)

19 i where N and n are the density of scattering centers, and 0 is the angle cm through which the electron is scattered in the CMCS. Electron-electron collisions are explicitly neglected, however they are implicitly considered to be responsible for keeping the electrons in equilibrium with themselves. In Eq. (2.35), gl is the total collision rate for inelastic collisions suffered by the electronso Note that g = g + gl is in general a function of x and t, as well as v, due to its dependence on n (x,t) ~ n (x,t). The term in Eq. (2.34) identified as S(f ) represents collectively the dependence on inelastic collisions, including ionization and recombination collisions, where it is sufficient at this point to note that it is simply a linear functional of f, after making the assumption that the inelastic collisions are isotropic in velocity space. We now substitute Eq. (2.35) into Eq. (2.33) in order to solve for J(x,v,t) in terms of the distribution function f (x,v,t). We obtain O Je X ttJ U= Z_ tr. ~ >-a Yu., -ix 5+) = Jam L ( xt-M) Ji- | (J C V-') (2.37) where -3X14- ir 3 -l (2.38) Substituting this result into Eq. (2.32) yields

20 /9-7- 1 CF ( -)).L/- P(+-'F2.59) Recall that we are interested mainly in the electron density and temperature as they are defined in Eqs. (2.27) and (b2a28). In the P -approximation used here they become oO f 2 (2.40) and terxi 6^}x) = ri 3 ^ (W}t ) ~o (2.41) Therefore, to obtain an electron density balance equation from Eq. (2.39), we multiply through by v and integrate over the electron speed. The result of this operation is 9 JoO -le _ V ic r zL S(') = - x da I~o s2du L' - At-2 (-) y[ (2.42)

21 The terms from Eq. (23.9) due to elastic scattering integrate to zero, which is to be expected from consideration of the conservation of particles. The last term in Eq. (2.39) also becomes a perfect differential, and therefore is identically zero for the functions considered here. To obtain an electron temperature balance equation we multiply Eq. (23.9) by mv /3 and integrate over the speed v. We obtain t d(2) -3M ~2( t- +6 - 00 -I 2 3 d u. L / (|-# ( _ -) f' J 2 2~ t> u 0 0~0 o oO 20oy 2 E L_) e - -'J (2.43) where we have substituted the term 5ane/at from Eq. (2.42). We now take the leading components of the electric field in the plasma as ~( X ) =EDi(/ &0 Cos w -)Cosc D (X Ej) (2.44) which represents a modulated signal with a modulation frequency ( and depth m, plus a dc electric field. Note that there are two distinct time scales, one associated with variations on the order of wc) and one associated with the D

22 much slower variations on the order of X. As mentioned earlier, we are interm ested explicitly in the slow variations of the electron density and temperae e ture. Therefore, consider the slow variations in n and 0 by averaging Eqs. (2.42) and (2.43) with respect to t over the period of the high frequency electric field. During the extent of this time integration, (t - t/% ) to (t + r/w ), we neglect the changes in slowly varying functions of cos c t and D m f (t). With this reasoning we find that the only surviving contribution of the high frequency electric field is the first term of 2 Dc (-.2 45) where J) = ( /- ~) -2co(Co5 cJ + ~ Cos 2 ) (2.46) (2.46) in the last term of Eq. (2.43). We now perform the indicated integration over the time variable u, after noting that f (t - u) and g(t - u') can be approximated by f (t) and g(t). This approximation is justified by the fact that the function exp(-gu) will make the entire integrand very small for u greater than the collision period l/g, which is much smaller than the period of variations of interest in f or any macroscopic average property such as n (t). o The balance equation for slow variations in the electron density is

23 __U. e- f d~~a S(V' = and in the electron temperature is (2.47) 2e 2m 3/1 0c 0 l<>(o +3 ( j-) =:ay^ - h lltr + 9 U -?e t~ 2Y W ^4r V ^ __ /2 L roO 9- {^. 26 e.v% L dc. d [J'( (r3e g C 3r i ~~~ 3M J |d 3 Cf } L 92 _ z 2 9 o~~o (2.48) We now make an approximation by assuming that the several averages, defined in the two balance equations above, over the electron speed are adequately described by taking f (v) as the Maxwellian distribution function v(l i6i% = 47T 4 fiy+ J (ze/)f 2 (e t/j 3 (2.49)

24 The assumption of Eq. (2.49) has been discussed by Cahn 7 and Margenau8 to be a good approximation, even in the presence of electric fields, when taking averages over the electron speed. In order to evaluate the collision rate g (v), we take as the differential cross-sections for the electron-neutral atom and ion collisions, for a helium gas, respectively (2.50) and /eL. 4_/n.2 4V c/-^fcAl 2 nY0,, v~ ( < 6~ (2.51) The cut-off angle o is taken as that scattering angle screened Coulomb potential of separation Xd where corresponding to a - 2~r2 e e (&)/ 2 e3 e r n e, (2.52) Relating d to the impact parameter of a two-body Coulomb collision we find, since 0 is small,9,k7'LI /e T )'I ^ ~~ ) L^f " m v1 \ QH- J (2.53)

25 Substituting into Eq. (2.36) we obtain for the collision rate tr- L I Ii, &r~ 3 (2.54) where Se f- sL) ( ) e 3 = ^{-^1 ^(jT^/^s /~~~~~~~~~~~~~~~ (2.55) We evaluate Fei at the average energy by letting v = (3 e/m) 1/2 and will treat both qe and i as constants throughout the remainder of this analysis for a helium plasma. Also, we assume that g can be replaced by simply g, since gl is expected to be relatively small. Further, we neglect V 0e and will discuss -x.. this approximation in Chapter IV when we consider the data in Fig. 4.7. We can now write the electron density balance equation as a9 aL fdlS( = go. V p ne + e edJ > L -,K e (25 0 (2.56) where - 32 0e, oe) s 91-ir m *[,. (2.57) The effective collision frequencies v and v are defined as en ei

26 - = 3 7 /s,7 (2.58) and d, "nn e177 I? e / 7 (2.59) for singly ionized ions. The term g(y) is a, tabulated auxiliary exponential integral function defined as (see Appendix B) _ yX Psy7 d X e - I0 XZ If / ) ( ^> O ) (2.60) for the case considered here, y is a real variable given by 2 I3i 2 IC~ A~ (2.61) When evaluating the high frequency conductivity term in Eq. (2.48), first term on the right hand side, we take the carrier frequency, (D, to be much 2 2 greater than the collision rate, g, and therefore neglect g compared to wD With this additional assumption the electron temperature balance equation, Eq. (2.48), becomes

00 e /M -; C ~~ 2 (i ee 3L 2 ) Sc) J.I U, - V e 3,, e 2 M') 2t -t — i~ i0~. cc 3 2 (fcl, 3 —c 3 n2e )e 4 ne. -f l - -e ee Vnee s (2.62) conductivity, normalized e by n, is where the real part of the high frequency $ ((^c) e2 M CJD ( l'>i^i) X;; i )(2.63) e, (see Appendix A) normalized by the dc conductivity is (Jd —'c' A 32 e2 q 77-n -~ I - Y2y) (2.64) and -- y''y.dc (2.65) Inns o this chapter, we e xpand In the fO11 -We ec s(t) and e o about their steady state values as e +e) n and ne(t) By neglecting

28 products of the small variables, we obtain linear equations for the electron temperature and density time variations and nonlinear equations for the steady e e state values. We take 0 and n to be known solutions of the steady state o o e e equations and solve for the variations b0 (t) and 6n (t) in terms of them. For the electron temperature balance equation, treated in the next section, we reduce Eq. (2.62) in order to obtain a useful solution. We noted that the measured steady state electron temperature and dc electric field (using Langmuir probes as discussed in Chapter IV) correlate very well by equat'ng the volume loss rate, third term on the left hand side of Eq. (2.62) e and the dc conductivity term. Hence, we neglect the variations in 0 due to particle flow rates by dropping the last two terms of Eq. (2.62). Furthermore, we neglect the variance-like term resulting from the inelastic collisions, the second term on the left hand side of Eq. (2.62). For the electron density balance equation, treated in Section D, we consider the effects due to electron-ion recombination and electron-neutral atom ionization collisions by taking oo i/v 2 n - (n) /0 (2.66) where 5 is the ionization rate and a is the recombination coefficient. We assume that the coefficients are not explicit functions of n, although they e may depend on 0 o The entire right hand side of Eq. (2.56) is retained since for the steady state discharge used in this experiment the dominant contributions

29 e to n, both in magnitude and spatial variation, come from the effects of diffusion and mobility. Note that the dc electric field considered here is the actual field in the plasma, which is to be measured if needed. C. VARIATIONS IN THE ELECTRON TEMPERATURE In the previous section we derived a fairly general electron temperature balance equation, A reduced form of the equation, which neglects all gradients in configuration space but retains the space dependence of all the variables, can be written as )30 _() = G( ) -- P(x ) x I ) D~ (2.67) where L = 2rnef'- ee (2.68) ~ - 6teE C' - y2oslyj7 (2.69) P e- 3; (e' (2.70) and

30 / j g7 e)S(Co a t ) yCo 2 ) (2.71) The terms in Eq. (2.67) are given the following physical interpretations: L is the electron temperature loss rate due to collisions with the neutral atoms and ions; G is the electron temperature gain rate due to the dc electric field; and (P + P1 ) is the gain rate to the electron temperature due to the amplitude modulated high frequency electric field. We now linearize Eq. (2.67) for small variations in the electron temperature and density due to the slowly varying source term P1. That is, we expand e e e e e e e 0 and n about their steady state value as 0 = + 6 and n = n + 6n, o o e e and neglect all products in the small quantities of Pi, 60e and 6n. The e equation for the time variations of 6e is / D'e s0ene 9 e a An 8S 2 Lc ne 3i (C C (2.72) where = F S9L _ - i~- = 9 aO 9e ( (2.73)

and tC6 = / - _ P andoe an' a;,qe (2.74) The coupling term in Eq. (2.72) proportional to w results from the electronion collision frequency through its dependence on the ion density. We are assuming that the density of scattering centers for the electron-ion collision e i frequency can be taken as n instead of n at each point in the plasma. Furthermore, for the slow variations considered in this analysis, we have replaced n /n by n e/n in Eq. (2.72). The steady state electron temperature is o o taken as the solution of the nonlinear equation /(en t=- Pj (;( e P& m /}(64 134) (2.75) or more explicitly _~ = e2L62 / ({ i ) + c 2 N2 /-2_y uo ~e2 C ( 2 3-W, (JL2 2f wi (2.76) e 2 Note that the last term is a nonlinear function of e through y and v o en We will digress slightly to consider the plasma conditions discussed by 3 4 previous authors on cross-modulation in the ionosphere and afterglow plasmas. In these examples, the dc heating9 ioe,, G, was not present or was taken to be independent of the electron temperature. Also, for these experiments,

52 e the variations in n were neglected entirely. Under these conditions our electron balance Eq. (2.72) reduces to _6 Vite A f-ife = C6 4('. L I6) co,, C os j (2.77) and Eqs. (2.76) and (2.73) reduce to Ge -a * _ ( 2 s =6 o-e 2 (-U 2)(2.78) (2.78) and F D, —) /IC ol") p,::') /;)I 1 - Ge(2.79) " 1 *(2.79) or simply (2.80) This expression for c agrees with the decay rate for pulsed cross-modulation a 4 20 derived by Dougal and Goldstein and by Bloch. The latter author's results were for a, general velocity dependence for the cross-section for electron mo2 2 mentumn transfer, when 2 >> v total. D

33 We can readily solve Eq. (2.77) for the time dependent variation in the electron temperature. We obtain, after allowing the transients to decay, 6e e=Ei ___)o_ _~__ _ - _-_LL CJ^ -C~ CO t oese Co (2 C)2-,7;,v;' 4 (2.81) This result for the electron temperature will be identical to that of Huxley and Ratcliffe, if we further neglect the effects due to the electron-ion collisions. This approximation appears reasonable in the ionosphere, as was considered by Huxley and Ratcliffe, However, in our plasma the effects of the electron-ion collisions, de electric field, and the variations in the electron density are found to be important to interpreting the experimental resultsO Therefore, we will treat the entire effect on the electron temperature as given by Eq. (2-72), along with Eqs, (2073) and (2774). The electron temperature of the steady state plasma considered in this experiment is larger than the neutral atom and ion temperature due to the dc electric field, For this case we can neglect the small effect due to the high frequency heating, P. We take 9 as the solution of 0 j(e,*e) / (2.82)

34 and C as a C _. L e / -e -e =(2.8) (2.85) where in writing Eq. (2.83) we have used Eq. (2.82) and defined the factor; -G \a G_ I - - - 1 - L 2 ~~~~~~~~, - ~ 4 d Y S -~ 3 j), ) f L L. iz''f, 2 i, 6 j C ( Y() (2.84) The values to +1.5 as derivative for p~ are plotted in Appendix B, Fig. Be.1 and vary from -0.5 2 y = 2v ei/v varies from 0 to o. After performing the indicated en in Eq. (2.83) we can express the break frequency as, = 2 f - (2 f- [,- -^/^ /'A(_ ^2 ^<9^': 2oe) 0a - k+ ) - (3 + *> ) J (2.85) A detailed study of the break frequency in Eq. (2.85) shows that the dependence of the dc conductivity on the electron-ion collision frequency has an e n important effect on the value of = At the equilibrium value of 0 = a o o O Eq. (2.85) reduces to Eq. (2.80) and la increases with the electron density through the electron-ion collision frequency. However, at electron temperatures greater than 3 /2 the effect due to the electron-ion collision frequency 0

355 is eventually to decrease C as the electron density increases. In fact, at a quite modest electron densities the break frequency will pass through zero and take on negative values.* For example, at the electron temperature of 0.06 eV, in a helium plasma at a, gas pressure of 0.378 Torr and n = 0.025 eV, we find 0 that X is zero at an electron density of about 6.2 x 101 elec/cc. This a e -5 condition corresponds to a fractional ionization, n /N, of only 4.5 x 10 Another interesting extreme condition for w 9 in contrast to the thermoa dynamic equilibrium condition, is for high electron temperatures. When 0 >~ =, Eq. (2085) becomes o o o 2L' L(2 + A 4 )uea- Hi(2.86) We have written p~ as (-0.5 + 6p ) since at these higher electron temperatures and modest electron densities, 6p~ is very small. Neglecting the electronion contribution, the break frequency ) is about twice as large in Eq. (2.86) a as for the equilibrium condition in Eq. (2.80). *The condition of negative values for wa, corresponds to the electron temperature runaway phenomenon in the presence of neutral atoms. This phenomenon results from energy being absorbed by the electron gas from the dc electric field at a rate exceeding the temperature loss rate. Since the electron-ion collision frequency varies inversely with the electron temperature this condition can be obtained when the loss rate is dominated by the electron-ion collisions. The phenomenon in a, fully ionized gas is discussed by Dreicer21 or for more references see Delcro.ix22

36 To evaluate the coupling rate ), we can also neglect the small effect due to the high frequency heating, P. We can then write it as TVM - Z8_ )- (2)er t if?) 4 (2.87) The pe is the same as given in Eq. (2.84), and recall that it varies from -0,5 to +1.5 as the ratio of the electron-ion to electron-neutral atom collision frequencies increases from zero. We see that D is always a positive quantity and varies approximately like (2m/M) v i for high electron temperatures and/or low electron densities. In the range of electron temperatures and densities of interest in this experiment (see Fig. 4.6), is found to be c of the same order as the frequency CD a, D. VARIATIONS IN THE ELECTRON DENSITY In Section B we derived an electron density balance equation, neglecting the gradients of the electron temperature, which can be written = ne e - (ne)Z + S (2.88) e where a is the volume recombination coefficient and pn is the volume source rate for electrons due to ionizations. The "leakage" term is given by S = DV ne +D- e neo- e <^ ee -/C (2.89)

37 where 32 e - 2 ~) (2.90) e D is the electron diffusion coefficient and eD/8e is the electron mobility. Note that E is the actual steady state electric field in the plasma, at any -dc point due to the applied electrode voltage and space charge effects. Often, Poisson's equation is used to eliminate the radial space charge electric field and then the diffusion coefficient becomes the ambipolar diffusion coefficient.2 This will not be necessary for our analysiso e We take the steady state electron density, n, to be the solution of (jef = ) e t S2 (2.91) e e e e We then make the small variable expansion of n and 0, about n and e, to o o obtain an equation for the slow variations in the electron density. We neglect the gradients of 6n /n o as well as 68 /0, and the products in the small e e variations bn and 5e to obtain / 0S,~e t, 6) = gd e (2.92) where

38 - 2 ocor e -— ~ 09 a- D,, PS n~ (2.93) and ee U~~i =~6O 7ne - (/ne)2 g 3e^ -+ &S 1 DSg pe D3ee (2.94) We rewrite the break frequency (b by introducing the factor b | e a S 2rq()- Y cii'S at LO= J1 I y - 2?Y) (2.95) which in terms of the previously discussed factor p0 is simply p== - j 3 Y (2.96) Subtracting p S from Eq. (2o93), using Eq. (2o91), we obtain n o Wb o= OC ne(- e-) /- e) (2.97)

39 2 As the ratio y = 2v e/v increases from 0, the factor p varies from 1 to 0. ei en n' e Therefore, the break frequency o ranges from a n,when vi is small compared b o o0 ei e to v, to (2a n - a ) when v e is the dominant collision frequency. The en o o o ei term -P is generally quite small and can be neglected compared to a n o o o We can rewrite wd by subtracting p S and neglecting the gradient of d~ o P., as OCDG = oC. n: (- 1 C3 [ - 9 )[ 1 G e (2.98) If we let a have an electron temperature dependence of (e ) 3/2 then the coefficient of the recombination rate in Eqo (2.98) varies from 1 to 3, as 2v./v ei en 23 increases from O. The diffusion term is negative in most of the regions of our plasma, and, due to the large diameter of the plasma, is on the order of a n or less. The last term in Eqo (2.98) is dominated by the derivative term o o and is a positive contribution to wdo However, since P is quite small the entire term can be neglected, except when the other two terms nearly cancel each other. Clearly this brief discussion is not conclusive, but is presented e a,s an indication that cd can be a small positive number, about a n. It will be calculated later in Chapter V when we discuss the evaluation of the experimental responseo However, the assumption that the frequencies e1 and w2 are real numbers, Eqs. (2o22) and (2o23), is supported by the above estimate and is found later to be consistent with the experimental results.

40 E. RELATIVE IMPORTANCE OF TEMPERATURE AND DENSITY VARIATIONS The relative contributions to the detector output voltage of the electron temperature and density variations is predominantly determined by the values of (a) and (b) in Eqo (2.13)o The results of this experiment indicate that the ratio (a/b) was less than 1/20, and hence that the detector response was primarily due to the variations in the electron density. Since this result may be surprising in contrast to the results and assumptions of other experiments with cross-modulation, we present a qualitative discussion of the parameters (a) and (b). From Eqs. (2o14) and (2.15) a- oj^^ ^ ^^ ^ j s C2 E (g\) cI ~ - 6L(~'y~~) Q~(6j) L~J ~ (2.99) and _ d(2.100) The ratio is written as fa f,1 \rC'G OvO:el i] ~I b -l ( J'JG Mo ~ C X - } E^e~a aO *jk (2.101)

41 The complex conductivity in these equations is given explicitly by, Eq. (2.9), Eq. (A.9), or 2 (2.102) where v = v + v. The imaginary part of o = R - ic is independent of e en ei R I 2 2 as a result of the approximation of v << For the purpose of this discussion, we assume the logarithmic derivatives, such as (2.103) to be slowly varying functions of position as compared to the remaining factors in the integrandso This approximation enables us to treat quantities like Eq. (2.103) as constants and take them outside the integrals. To be more specific, we evaluate these slowly varying quantities at a certain point x in -P the plasma. Clearly, this approximation is equivalent to assuming that the sensing microwave interacts locally with the plasma at the point x. Other -p authors have demonstrated that when the collision frequency is much less than the carrier frequency, the microwave begins to interact with the plasma, quite strongly when 2 /c2 is near unity.4 The plasma frequency is defined by P s

42 2 _ee COp -= 4/77 n (2.104) and we use this to help define the point x. This condition for the interaction is particularly applicable for the reflected signal, which appeared to be the major contribution to cross-modulation in our experiment. With this reasoning, we define 1< -_ (xt) 3 " 4 -, E- ( 6:, (x. (,) a ~ y' 0 J (2.105) and iL ~(),I / - (X.-)Ix ( O )C-,( j CT (2.106) Equation (2.101) is now written as ge ra, 0 pla ean / + &(2)f (9-cY i/97n 7 - ic^ I) 7 a 1/;\l e/) (2.107) Evaluating the partial derivatives using Eqo (2o102), we obtain

43 Pr -_/ b (ien Lj / (Xi/f ) i/ ) (2o108) Although the first factor in Eqo (2o108) vanishes when v e= 3i (which is en ei satisfied in a, cylindrical region at abouts half of the plasma radius under the conditions of Figo 4.6), the main reason for (a,/b) to be small is perhaps due to the large denominator, second factor in the equation. It is difficult to evaluate the terms Re(K1) and Re(K ) appearing in Eq. (2.108) quantitatively to demonstrate this point because they involve the Green's function for the steady state sensing microwave. However, we can still gain some insight to the magnitude of the second factor by considering a plane wave reflected from a, semi-infinite plasma. For this case, we find that the denominator (see Appendix C for some of the details of this calculation) can be written as + _(1 -0_ t29/7)j/- /7) i 3 127jj/ (2.109) 22 2 22 where T = (C / 2 ) < 1 and d (v /2) << 1. It can be shown that for low P s S electron densities, i.e., << 1i Eqo (2,109) reduces to approximately 3 (2110) J (2.110) I

44 For higher electron densities, such that (1 - ) < 1, Eq. (2.109) reduces to approximately 2 C(-C)d2 (2o111) Both of the limits in Eqso (2.101) and (2.111) make the ratio (a/b) very small. From the preceding discussion, we conclude that there are experimental conditions in which the variations in the electron density dominate the crossmodulation measurement. This has been found experimentally to be the case for our measurements.

III. EXPERIMENTAL APPARATUS The experimental apparatus consisted of a, vacuum station and discharge, microwave, and Langmuir probe circuits. A photographic view of the entire system is shown in Fig. 3.1o Ao VACUUM SYSTEM The discharge volume was a bell jar 18 ino in diam and 30 ino in height, supported by a 20 in, stainless steel base plate. A Kinney PV-400 vacuum station proved satisfactory for this experiment. To obtain a stable vacuumdischarge system it was necessary to cycle the discharge and pumping periods. This technique served to outgas the bell jar and "age" the cathode surface. After several weeks of this conditioning procedure, the vacuum was stable for discharge times over 12 hr with no apparent outgassingo The pressure rise rate after the outgassing procedure was less than 105 Torr/hr or an equivalent leak rate of 3 x 107 Torr-liter/sec. At an operating pressure of.5 Torr and for a discharge time of 12 hr this was less than a.03% change in the pressure. Helium (10 parts/million impurity) was used throughout this study as the discharge gas. The gas was admitted to the bell jar through an inlet port passing through the base plate (anode). The system pressure was recorded by a thermocouple gauge, which was also mounted on the base plate. The helium pressure response of the thermocouple gauge was calibrated using a, Cenco McCleod gaugeo The standard deviation for reproducing a given pressure was less than 45

I:"Iv/ ii; I Fig. 3.1. Photographic view of the apparatus showing: the high voltage power supply; the movable disturbing microwave circuit; the sensing microwave circuit; the bell jar discharge with some of the microwave absorber removed; the pumping station; and the receiving and recording instrumentation on the right.

47 B. DISCHARGE SYSTEM The high voltage discharge circuit is illustrated in Fig. 3.2. The power supply was a Universal Voltronic BAL 6-300-M, with maximum output power of 1200 w. This power supply had a rated 2% rms ripple which resulted in a comparable oscillation in the discharge and an additional 30 Hz low pass filter was necessary. This resulted in less than.02% rms ripple for a 20 kohm loading. r - - @7.5 hy 1,0 0002-1 i —— IDc I I H.V. I POWER I @ I V SUPPLY 1 7.5 fd I _.+ IL _ LOW PASS FILTER _ _ Fig. 3.2. High voltage discharge circuit. The stainless steel base plate provided the grounded anode for the glow discharge. The high voltage cathode was made of a 12 in. aluminum disc placed 24 in. above the anode. The cathode was supported by a 1/2 in. copper rod which passed through the base plate at a radial point of 7 in. and was connected to the top of the cathode. The edge and top of the cathode disc, as well as the supporting rod, were insulated by teflon. The feed through supporting post was made of low density polyethylene, which insulated and sealed the electrode to the base plate (see Figs. 3.3 and 3.4).

48 I" S.S. Base Plate __ M L-_- _r__~~~~~ Gx-^AL. Sleeve I r Coppicr y | _ _ _ Thi fContrut < ^ X _! PLNuts (1/8") Copper Ro\ TRing Seal J This Construction 1/2Dia Insulating Prevented Elongation Torr - Seal of the Threaded Polyethylene Section Fig. 3.3. Detail of the H.V. cathode supporting post and its vacuum seal construction. Aluminum was selected as the cathode material since it reportedly had a relatively low sputtering rate. Nevertheless, aluminum was evolved from the cathode and depositions of the powder were noted on the bell jar walls and supporting structure near the cathode. C. LANGMUIR PROBE CIRCUIT A very important diagnostic tool used in this experiment was the single element Langmuir probe. Properly used, this probe can yield quantitative information on the free electron density and temperature with reasonable spatial resolution. Two different probes were used. The first was a 1 cm length of 10 mil tungsten wire and the second was a hairpin loop of 10 mil tungsten wire which had a, total exposed length of 1.8 cm. Both probes were mounted on a, 4 ft

49 Fig. 3.4. Photographic view of electrode structure with bell jar removed. As shown, the entire discharge was enclosed within a microwave absorbing "box. I

50 length of 3/8 in. diam pyrex tubing. This support was sealed to the base plate at a radius of 4-3/4 in. by a dual sealing rubber stoppero This seal allowed the probe to be placed at any vertical position and at representative radial positions. The probe leads were vacuum sealed by a Torr-Seal plug moulded around the l.eads at the bottom of the glass tubing (see Fig. 355), The complete probe circuit is shown in Fig. 3.6 and consisted of four individual circuits. Circuit A provided a dc current of about 3 amperes to heat the hairpin probe to a dull redO This feature of the loop probe yielded a more stable probe surface condition. The single element probe used earlier showed a continuous change in its response (on a time scale of seconds) after heating by electron bombardment. This heating was provided by circuit B. When the continuously heated hairpin probe was used9 only long time changes over several minutes were noticed in the probe responses and these could be "rezeroed" by flash heating the probe by electron bombardment. The primary probe circuit C provided a, time varying voltage t.o the probe through a decaying RC circuito The resulting probe voltage was recorded directly on the Xaxis of a Mosely 2-D X-Y recorder. The probe current passed through a 1% precision 200 ohm resistoro This latter voltage was amplified with a gain from 2 to 10 and then the log of this voltage was recorded on the Y-axis of the recorder. This resulted in a semi-log trace of the current-voltage response of the probe in the plasma, Because of the very low electron temperature, the circuit D was used to obtain a more sensitive measurement of the slope of the semi-log probe response. The circuit consisted of a, transistor controlled square-wave current sourceo

51. 4-3/ I Torr- Seal with Venting Port 8mm. (10 Mils) Teflon Probe Coated Leads 1.8mm. Top View of Probe Element Pyrex Tubing(3/8"dio.) Base Plate Gasket Vacuum Seal Probe Leads Probe Leads Sealing Gasket Torr-Seal Pluga -~ \ Fig. 355. Construction of the Langmuir hairpin probe and the movable probe vacuum seal.

+ 470 QI N2726 200 v. Probe, 2750 v. 75t 0/ D,8 -- 12 v. 2N2338 Amp. 25 K r 0.6 SI 2N2338 3Amps. T-250v 7r Vp - 4U v. -20v. t200Q 1% \J1 Po Anode "ty" Scope Ver. Defl. (AV) -I"mx Fig. 5.6. Langmuir probe circuit.

The resistors R1 and R2 were chosen such that (R1 + R2)/R1 = 2P72. Since the voltage change in the electron transition region was exponential (see Eq. (4o2)), this choice of current ratio led to a direct calibration of resulting change in probe voltage AV s in terms of the electron temperature in electron volts. The oscillating voltage was recorded directly on an oscilloscopeo Do MICROWAVE CIRCUITRY The microwave equipment and related instrumentation were composed of two independent systemso The first system generated and transmitted the disturbing microwave signal to the plasma, This signal was produced by a 2K41 Sperry klystron powered by a FXR Z-815B power supply with a supplementary power supply to provide the grid currento The operating conditions were: beam voltage 1250 V, reflector voltage from 440 to 600 V, beam current 50 mA and grid current 15 mA. The signal frequency was 2.7 G Hz and klystron output power was approximately 0~6 Wo The sinusoidal. modulation was provided by a HewlettPackard signal generator model 200CDo The disturbing signal was transmitted to the plasma by S-band rectangular wave guides and horn, and partially collimated by Bo F. Goodrich microwave absorbero The sensing microwave was generated by a, 2K42 Sperry klystron powered by a FXR Z-815B power supplyo The operating conditions were: beam voltage 100 V, reflector voltage 550 V9 beam current 40 mA, and grid current 0 mAo The signal frequency was 350 G Hz with klystron output power of Ool w. The transmitted and received sensing microwave signals were guided by H-band rectangular wave guides and hornso A schematic drawing of the microwave circuit is shown in Figo 3507

ISOLATOR E-TUNER ISOLATOR SCOPE DISPLAY FARIABLE SLOTTED ATTEN. SECTION rI PS. IF7 SLOTTED SECTION E-TUNER VARIABLE ATTEN. ISOLATOR Y" (AC) \1 "x" (AC) I Fig. 3.7. Schematic of the microwave circuit.

55 The purpose of the receiving circuit, in the upper right of Fig. 357, was to detect and record the modulation depth of the audio modulation transferred to the sensing wave from the disturbing wave via the plasma. It was important not to detect any of the disturbing wave directlyo The H-band receiving horn and wave guides damped the 2.7 G Hz signal relative to the 350 G Hz signal. Then the received signals were separated completely by the 20 M Hz band width cavity tuned to 350 G Hzo This signal was rectified by a 1N23B diode and integrated, so that 1/RC was much greater than the modulation frequency, Wc m and much less than the carrier frequency9 t o The resulting sigII S nal was amplified and displayed on a Type 545A Tektronix oscilloscope with a 1A7 plug-in unit using a 1 K Hz to 500 K Hz band pass filtero The filter was necessary to help eliminate modulation of the sensing wave due to plasma disturbances which included 60 Hz variations from the cathode voltage source. The modulation of the received sensing microwave was a maximum of about 0.ol and the entire signa was 0 db less thn te riginal transmitted signal, Much of the signal loss was due to plasma reflection. The oscilloscope signal a,c output was amplified by a factor greater than 20 and fed into the a,c (full wave rectifier) vertical i.:nput of the X~-Y recorder amplifier. The recording was completed by sweeping the horizontal response with a, voltage related to the modulation frequency. The calibration curve in Fig. 358 was the response, from the circuit point marked "Cal" in Fig. 357'to the recorder, of the sinusoidal signal from the oscillator.

z 0 IZ,>.7 / uJ 5. 9 Z.4 - M uJ CIL -J w.i or D I fm. MODULATION FREQUENCY (K Hz) 0. (D Fig. 3.8. Calibration curve or frequency response of the receiving instrumentation.

57 E ELECTRONIC NOISE SUPPRESSION Even though the discharge was basically a dc power system, very small 60 Hz ripples and other oscillations on the power supply voltage affected the measurements. Also, weak electric fields, such as local radio station signals, were troublesome. In this experiment the disturbing microwave was an unwanted local electric field since any direct pickup in the receiving instrumentation of this field could affect the measurement, As can be seen in Fig. 3.1^ tbhe entire system was enclosed in screen cages. There were actually six compartments constructed of slotted-sections of angle iron with complete screen coveringo All the joints were soldered including the screen to angle iron contactso The floor of each compartment was a copper sheet with a, heavy copper braided wire leading to the building ground. The six compartments were: the power supply filter9 movable disturbing microwave system, sensing microwave system, discharge region, and two receiving systems. The 110 V, 60 Hz power leads entering each compartment passed through a power line filter, as shown in Figo 35.9 These 10 ampere filters worked quite well for blocking the local radio station signals and high frequency transients due to other research projects within the building. Coaxial cable was used for most leads and many had an additional braided copper shield. All instruments were grounded by copper straps to the common grcund planeo Extra care was also taken with the microwave circuits. The klystrons were mounted in heavy Narda tube mountso The mounts were enclosed in a copper

58. fiJ_"" __JL__ 7.5/hy 7.5/Lhy _..lfd 0.5/fd 0. IJIfd Fig. 3.9. Low pass filter for 110 V, 60 Hz power lines. box and the power leads were all coaxial cables. In both microwave circuits 20 decibel isolators were used to prevent nonlinear mixing of the signals within the klystrons. Mechanical vibrations were a, noise source since any motion of the suspended cathode caused small variations in the plasma. Therefore, mechanical pumps had to be turned off during the measurements.

IV. EXPERIMENTAL TECHNIQUES AND DATA ANALYSIS This chapter has two parts discussing the experimental measurements and analysis of the data with the helium discharge system. Section A concerns basic current-voltage-pressure and Langmuir probe measurements and Section B concerns the cross-modulation measurements. A, HELIUM GLOW DISCHARGE The glow discharge used as the plasma source in this experiment was unique due to the large volume and the length to diameter ratio of only 2:1. Glow discharges are generally classified as "normal" or "abnormal," depending on whether or not the entire cathode is used for electron emission. The discharge becomes abnormal when the ion bombardment covers the entire cathode. With geometries having a larger length to diameter ratio,, the abnormal discharge consists of well defined transition layers near the cathode and a low field, nearly constant density9 positive column to the anode.25 The glow discharge used in this experiment was abnormal as defined above. In fact, the normal discharge did not appear to be stable at any current for gas pressures less than 1 Torro Because of the large diameter, diffusion losses were low and the discharge did not appear to have a, positive column except at very low currents. The main voslume of plasma was partially controlled by recombination. losses and had some advantages over the positive column. It did not generate the n.-ijse which i-s found wit;h moving striations in. a typical positive columnr The electron density was somewhat higher, over 1011 electrons/ cu cm, and the electron temperature was lower, about.04 to.10 eV, than 59

60 usually obtained with abnormal glow discharge positive columns. A disadvantage was that the plasma properties were more sensitive to the cathode surface condition than were the positive column plasmas Figure 41l is a, photograph of the glow discharge. The cathode was at the top of the discharge. The discharge was usually a fairly uniform bluewhite and somewhat brighter in the inner half of the volume. Near the cathode was a, narrow region of emerald green with a dark transition region to the cathode. It was this latter region, less than. in, thick, which supported the main voltage drop of the dischargee (This voltage drop accelerates the ions for bombarding the cathode and liberation of the electronso) At lower discharge currents the lower region was dark and was supposedly the Faraday 25 dark space. As mentioned earlier a newly evacuated bell jar and a, clean cathode produced a discharge with unstable properties. In particular, the currentvoltage-pressure relationship continued to change during the early stages of the discharge. It was found after several weeks of cycling the discharge and evacuating periods that the prcperties stabilizedo The experimental approach to stability was accomplished by producing a, plasma with the helium gas pressure and discharge current held consta:nt and recording the discharge voltage. At first the voltage changed as much as a factor of two during a, three hour period. After a, few weeks the voltage would change only during the first hour and asymptotically approach a constant -value. Eventually this voltage was stable within ~5% for discharge'times exceeding 12 hr. The asymptotic voltage was also reproducible for different discharge runs to within ~10%.

::::j: ~-:a::i:-::::: i..-:::..:.:::::::-:::::::::::_ j::::-::.::i: i i Fig. 4.1. Photographic view of the helium glow discharge.

62 Figure 4.2 is a, plot of the asymptotic voltage drop across the helium glow discharge versus the discharge current over the pressure squared, (I/P ). The solid curve matching the data is from the expression 4(voLrs) -= 170 -+63 0 7Am - -R- (4.1) During the conditioning period, Langmuir probe traces were recorded. A typical trace with the hairpin probe is shown in Fig. 4.03 The theory for the current-voltage properties of a, conducting element in an ionized gas was first developed by Langmuir and Mott-Smith in 1924 and has been discussed by 26 several authors. The relationship needed in this experiment was for the e probe current, I, due to electron flow to the probe against a retarding pr field. It is given by f e r-Ie- _L __ ( _9 (4.2) where V is the potential of the probe and V is the plasma potential surpr p1 e rounding the probe, both with respect to the anode. I is the saturation sa electron current given by sI e St4 - PTj72 (4.3)

2000 1000 o 1/2 500 VDC= 170+630P 2 ID1C o50 v DC\''ji"^ 0.630 Torr X.378 0.325 A.260 o.190 200 0.11 0 ioo —---------------- I I I I I i -.01.02.05.1.2.5 I 2 5 10 IDc(AMPERES) p2(TORR2) Fig. 4.2. Current-voltage-pressure dependence of the helium glow discharge with an aluminum cathode. 3-1\

ee =.06 E ne= 2.1 x 64 - 3 SLOPE e/&e - " ^- - lsa ~a0 lectron Volts -6 H. 10oElectron/cc 4 o d I Z2 Ij: — 04 ^ o-.02 ~o oA.06 -40 11-35 -30 > ^ - 20 02 0 Vf P I -.01 -40 1-35 -30 -20 -01 Vpr. PROBE POTENTIAL RELATIVE TO ANODE (VOLTS) Fig. 4.3. Typical Langmuir probe trace.

65 e e where n is the local electron density, e is the local electron temperature (KT ), and m is the mass of an electron. A semi-log plot of I vs. V has pr pr a slope of e/ e, which yields the electron temperature. A measurement of I s a, yields the electron density. Another parameter of interest was the probe floating potential, Vfo This open circuit voltage is maintained by the probe when the electron and ion flow to the probe are equal. Vf differs from Vpl due to the difference in the random speeds of the equally dense electrons and ions. Measurements of Vf, corrected for the electron temperature, vSo probe position can yield information on the dc electric field in the plasma. Electron temperature measurements were also made using the oscillating square-wave current source discussed in IV-C. From Eqo (4.2) X ThAzV - by,2 -l = e(4.4) so that in the electron transition region the resul.ting oscillating probe voltage will be proportional to the electron temperatureo With the choice of current ratio of 2.72, the voltage change is directly calibrated in terms of e 27, in electron voltso Figures 4.4 and 4.5 are plots of the electroon densities and temperatures measured by the Langmuir probe at the asymptotic current-voltage-pressure conditions. All these measurements were taken at the microwave cross-over point, i.e., 9-1/2 in. above the anode and on the center line of the discharge.

~appoue aq. aAoqe -UT. /T-6 sTx- uo'aanssaid s-e umTrIa; pu'e quLaJ:nJ aa'eLqosTp q;TX uoT;TwIeJ'A qX;Fsusup uoja;c oT'T1* iT (S383dldV1I91IW) IN38d nD 30dVHOSIQa'0I 00~ OOZ 001 0 z01 \4/$/ / 7/ / / Fi JQo 029' Ji0106 I.0000 -A / p / / / T rI / / / I. / I I I I,2 3 ID m r z m01 m C) 0 C m z -I 601 m m 3101 -- m 1:01 / / I L 2 - %o\ ~g-E'- ZX _'! "01 8ZL~'_ I / 99

67 5 2lh C) I0 z 0 ILJ I aLLJ bJ 0 nQLUi...j LJ.5.630 Torr — 7T-.190 Torr.2.1.378 Torr 0e Recorded During Cross- Modulotion Meosurements.02.010 0 100 200 IDCDISCHARGE CURRENT (MILLIAMPERES) 300 Fig. 4.5. and helium Electron temperature variation with the discharge current gas pressure, on axis 9-1/2 in. above the anode.

68 Note from Fig. 4.4 that a near maximum electron density is obtained here for a pressure of about.378 Torr and for currents between 150 to 250 mA. Nearly all cross-modulation measurements were made at these conditions. The electron density decreases for higher or lower discharge currents or pressures. Figure 4.5 shows that the high electron density region corresponds to the lowest electron temperature region. The uncertainties indicated on the figures include the reproducibility of the measurements as well as an indication of their absoe lute valueo This latter uncertainty is based on locating the point I and sa e measuring the slope e/9 The radial distribution of the electron density and temperature are shown in Fig. 4.6. It can be seen that the density variation is close to a zero-order Bessel function. The rise in the electron temperature toward the bell jar wall is due in part to the relatively negative potential of the wall which repels the lower energy electronso The electric field measured along the radial path was o0038 V/in. directed outwardlys In Fig. 4.7 the axial variation of the electron density and temperature are plotted. The axial plasma potential is shown in Figo 4.8. Note that the minimum electron temperature, for the conditions indicated, occurred near the region of zero axial field. The electric field which heated the electrons in the lower region and contributed to the electron flow to the anode was 01356 V/in. and directed toward the cathode. Because of the slow variation of 0 e with position, we have neglected the gradient of e in the balance equations, Eqs (2o056) and (2062)o Eqs. (2.56) and (2.62).

1.0 u 0 LU'.8ig 0 L z L0. LJ L.2 Ur 00 LLJ LJ o CL lLJ az 0 FCU 2 3 4 5 6 7 8 9 r, RADIUS (INCHES) Fig. 4.6. Electron density and temperature radial variation, 9-1/2 in. above the anode.

1.4 K LJ -J LU >o I LLJ 0 z 0 cr n-J LU C 1.2 H 1.0 -.8K LU LJ LJ.068 z 0 04: *06 LJ I rJ LLJ a) Xb -- 0.6.4 -I IDC= 150 Milliamperes.2 VDC=1055 Volts Po =.378 Torr 1.02 0 ANODE 2 4 6 8 10 12 14 16 18 20 22 24 CATHODE Z,DISTANCE ABOVE ANODE (INCHES) Fig. 4.7. Electron density and temperature axial variations.

71 -1055 Volts -40 -35L / I 0 Volts 4 / / _o_ o...I I I I c0 I Fr LJ 0 (I) -i 0. x >~ 30H -25 / I ~ X Xx I / lOx Exponsion x I X of Voltage Scale x Slope -0.136 Volts/Inch I I I I I -20H 15 IDC= 150 Milliamperes VDC= 1055 Volts PO =.378 Torr -IOH - 5 Radial Slope at Z=9.5 Inches -0.038 Volts/Inch I I I 11 NO ANODE I I I I I I I 2 4 6 8 10 12 14 16 18 20 22 24 Z,DISTANCE ABOVE ANODE (INCHES) CATHODE Fig. 4.8. Axial variation of the plasma potential.

72 Bo CROSS-MODULATION RESPONSE 24 An analysis of the microwa;ve absorption theory indicated that the discharge conditions of o37& Torr and 1.50: 50 mA would probably yield a measurable cross-modul>.a, =ti- n be=t.ween t he carrier wave frequencies eof 217 and 350 G Hzo A calculation -of CD +The electron tiemperaature restoring rate, indicated that fl = Wl/2n would ie between 5 and 12 K Hz for these discharge parameters. Theref:re9 The helium gas pressure was held at o378 Torr and the modulation frequency set at- 2 K Hz during the crcoss -modul.a,tiol-,n tuning procedureo At these conditio.':ns and with -the disturb:lFng and sensing microwaves turned on, the discharge curren+t was varied over t'he:ndicated rangeo The modulation on the sensing microwave was displayed on. an oscilloscopeo To facilitate the observat-ion of'he cr;ss-modulation', the oscilloscope was phase locked with the modula+tion signal from th- oscil..ato,.ro When cross-modulation was observed o:n the oscilloscope9 the task was to stab.il.ize t.,he discharge a,t, the current which yiel.ded h.e maximum crossmodulati:,-n signLa,l. Whenever the current was increased or decreased from a stable conditiorn it.' required about.- 30 min t';, restabilize. This discharge current had t1 be co,,n inuousl,y a,djusted for the maximum cross-modulation signal as the stabil.zed condtioin was approached asymptoticallyo Also during thie power a,djust'mentj peri od9 +the microwa-ve wave guide equipment was tuned in the presence'of Pta-h plasma to yi.eld thee maximum cross-modulation signal to noise ratio. This noise was caused by macroscopic oscillations in the plasma propert:ies, mecharical v..bratiojns in tkhe microwave receiving instrumenta;tion, and direct modola, on;.n t he sensing wave such a,s that due to the cooling fan

vibrations in the klystron. During the final stage of this tuning period, the X-Y recorder response to the ac modulation on the sensing microwave was used to obtain the maximum signal to noise ratio. Once the ent-ire system was tuned for maximum cross-modulation response, the modulation frequency was varied from O05 to 50 K Hz and the ac modulation was recorded over this frequency rangeo An example of this recording is shown in Figo 4o9~ Each hI-ri.zn.ta t:ravxerse of the recorder required about 2 to 3 sec because.f the.low frequencies in the responseo One or more traces were ma,de over the two frequency ranges in order to average out the fluctuations in the traces, as well as t make sure that there were no changes in the response occurring during the single tracingo Often, the recording did not retrace and the data were disregardedo The discontinuity occurring between 2 and 2~5 K Hz in this example was calused by a, short duration change in the discharge which apparently restabilizedo These were accompanied by very tiny momentary arc spots on the cathode, The response from Y = 0 to the Cross-Modulation Zero Line represents the various sources o:f modulation o;n tthe sensing microwave which were independent of the modulation frequency, f o At one time this level was several factors m larger than shown in Figo 4o9 due to the 60 Hz ripple on the dc power supply and other stray noise sourceso The filter shown in Figo 352 and the shielding explained in III-E part.ially corrected for the noise levelo The C-M Zero Line represents the asympt;ot ic cross-modulatiron high frequency response, as well as the level of the respor.nse wi.th the disturbing microwave modulation turned off,

+2% E %' C0 x +38% z +3% i Cross-Modulalion Zero Line F-I U'J Z CO 0 5 6 7 8 10 12 14 17 20 25 30 Jr I8%.5.88 1.2 14 1.7 2 2.5 4 5 6 ~132%.5.8 11.2 1.4 1.7 2 2.5 3 4 5 6 fm, MODULATION FREQUENCY (K Hz) Fig. 4.9. Sample trace of the cross-modulation depth on the sensing microwave as recorded by the X-Y recorder.

75 The desired response was taken as the vertical displacement between the C-M Zero Line and an average smooth curve indicated by the broken line in Fig. 4.9. In this example the individual uncertainties of each data point range from:t 2% to I 32%o These data had to be corrected for the frequency response of the instrumentat:icon C(f )o This correction function is shown in m Fig. 3508 Data were used for all the discrete frequencies indicated in Fig. 4o9 up to 17 K Hzo Each set of corrected data was fit, using the methods of least-squares (see Appendix D), by a computer to the relation (4~5) This technique yielded the normalization 6(o) and the break frequencies fl, f2, and f3 In all cases analyzed f3 was much greater than fl and f. In Figo 4o10 the solid curve is the computer solution to 6(f )/S(o) for the average of the four individual sets of normalized data shown. The four sets were taken at a constant discharge condition over about a 1/2 hr period. The data with the error bars were from the recording in Fig. 4.9. For illustration9 the two functions (4.6) and

1.0 T LI 1 t z.9 I hif1____(i (f) 2) qE^o Z 8(0) fm\2\ fs ^.6.2.512 5 10 20 50, MODULATION FREQUENCY (KHz) <::21 Fig. 4.10. Four sets of normalized cross-modulation data with computer solution.

77 f ^// #r ( 1/f ~) ~(4.7) have been plotted in Figo 4o10. The data points surrounding the upper curve were simply the original data divided by the corresponding values of Eqo (4o7) and therefore they represent the "information" in the original data which determines the break frequency f1o Similar reasoning applies to the other curve, although the da;ta were not plotted in this caseo A few of the individual error bars have been indicated on the upper curve The individual values for fl determined by the four sets of data ranged from 5.8 to 7.2 K Hz. This indicates the reproducibility of the measuring and analyzing technique. The fractional standard deviation of the mean value between the data and the matching transfer function2 Eq. (4o5), for this example was about ~ 2% (standard deviation for one more data point was about ~ 10%). Over a, six week period3 107 sets of data were taken and analyzed in the preceding manner. These data are tabulated in Appendix E along with other information on each measurement. Al,.though the data appear to vary over a broad range of values for the break frequencies no correlation was found during the extent of the experiment between the variatizons and the average plasma properties or microwave circuitry alignment. Therefore, the sets of data were treated collectively by finding their average and root-mean-square deviation, As an illustration of this interpretation the number of break frequency measure

78 ments for f and f occurring in a fixed frequency interval vs. the frequencies 1 2 are plotted in Fig. 4.11. The distribution plots are compared with the normaldistribution function based on the averages and root-mean-square deviations indicated abveo The area under the normal-distribution curve and the bar graphs are equal for each of the two plots. The results of this analysis were fl = 6034 ~ 53K Hz (4.8) f = 1.65 53% K Hz (4.9) 2 with the average plasma conditions of I = 170 ~ 8% mA (4.10) DC VD = 965 ~ 8% V (411) P = ~ 378 ~ 3% Torr (4o12) 0 The indicated uncertainties are the standard deviations of the mean values for fl and f2, the spread in the values of IDC and VDC and the reproducibility of P o 0

< f = — 1.65 KHz cc S.D.= 0.46KHz S.D. 2.01 KHz z Q-4 i. oI La_ LU LLI 0 0 2 4 6 8 10 12 14 16 f2 OR fBREAK FREQUENCY (K Hz) Fig - 4. 11. "Frequency distribution" plots of the measured break frequencies.

V. RESULTS AND DISCUSSION The results of the theory for cross-modulation from Chapter II showed that the normalized variation in the detector output voltage, in response to the received sensing microwave signal, is given by* Eq. (2.21), Su-C 67wf) Cos (zT7fmr - -) (5.1) In this experiment only the normalized cross-modulation depth, 6(f )/6(O), was measured. From Eq. (2.25) it is given by I-(O) __ / / 1 (f/r;(-/ 4 (7 /W ))(/ + R( /)) (5.2) where f and f are 1 2 J2 JI I - l 2'c L (5.3) and *In this chapter all frequencies are in Hertz, cps, instead of rad/sec, i.e., let f = o/2io 80

81 af (5.4) The restoring rate for the electron temperature variations, f from Eq. (2.84), ~~~~~~~is~~a is pa 2 rT \/- / 2a 7Tr M (2 -e) 1 2 ee - - t.{ + ee - 3 - 6) veo ii (5.5) The restoring rate for the electron density variations, fb from Eq. (2.96), is X f= ~ h b 2 7T qS5 (5.6) The coupling frequencies between the electron temperature and density variations, Eqs. (2.87) and (2.98), are 2 7T 2 M 2,)e ti e LI) -/ [ 2 -(I -( / 1 (5.7) and

82 - oc n fe 0, 1 - _ o7 ZeW W 2 7-F 2 2T7 -i1e (5.8) neglecting terms in the ionization coefficient e and the gradient of the diffusion coefficient D. The following functions of the electron temperature, e e, electron density, n, and the cross-section for electron-neutral atom collio o sions, q, were defined. The effective electron-neutral atom collision frequency is ^/7 ^ 3 / IT M^ i (5.9) The effective electron-ion collision frequency is -/ n e em ve 3 7(y0e)32 ( rz e 3 - / (5.10) The electron diffusion coefficient is My) ^ 3^ e Lr -Y 9 ) 32Q 1) -7)e (5.11)

83 where the bracketed term is plotted in Appendix B vs. the variable y = (2v /v ) /2 The factor p$ is the logarithmic derivative of the dc conei en e ductivity with respect to the electron temperature, and has been evaluated and plotted in Appendix B as a function of yo a is the electron-ion recombination o coefficient The method of leas't-squares, as discussed in Chapter IV, Section B, and Appendix D, was used to fit the function in Eq. (5.2) to the measured transferred modulation depth (arbitrarily normalized). For all the data analyzed the term (1 + f /f ) / was found to be close to unity. That is, the experim 3 ment showed that the cross-modulation depth varied as 1/f for the largest f /m m Using calculations for fb and fd, we find that this response requires that the ratio (b/a) in Eqo (5.4) be greater than 20. Furthermore, from Eq. (2.13) we can conclude that the cross-modulation measurement is primarily due to the electron density variations. Since this result may be surprising in contrast to the results and assumptions of other cross-modulation experiments, we have presented in Chapter II, Section E, a qualitative discussion of the parameters (a,) and (b)o A large value of (b/a) appears reasonable for a sensing micro29 wave signal which is primarily reflected by the plasma and this was apparently the case in this experimento As an experimental result, therefore, the crossmodulation depth is given by S(o) = _ P s(# (m1/12);)(h 0 ( /)7' ) (5.12)

In Chapter IV, we discussed the techniques used to measure s(f )/6(0) and to obtain the experimental values for the two break frequencies, f and f 107 sets of data were measured and analyzed for the pair of break frequencies with a fairly constant helium plasma (ID = 170 mA with an 8% spread and P = o378 Torr with a, 3% uncertainty). The average values are o f~ = 6.34 ~ o19 K Hz (5.15) and f = o165 + o04 K Hz (5.14) 2 where the uncertainties are the standard deviations of the mean values. We now relate the measured values of the break frequencies to the functions defined in Eq. (5.3) as f, = f ( 0 e ) (5.15) and *The standard error is defined for a normal distribution by S. E. h -) /

85 fee ~e Qrn c?e 2 T (c/c^ ~) get J1 he (5.16) In order to solve Eqs. (5.15) and (5.16) for the electron-neutral atom collision cross-section and the electron-ion recombination coefficient, the electron temperature, the electron density, and the plasma "buckling" V2n /n were required. In particular, we needed these variables at the point (x ) of -p cross-modulationo As discussed in Chapter II, Section E, cross-modulation was expected to occur for p /(2) close to 1, where the plasma frequency is defined P s by _2 _ /ee2 (5.17) and c is the sensing microwave carrier frequency. For the conditions in our s plasma this ratio was approximately one at the center of the discharge (9-1/2 in. above the anode) and decreased along the radius. Therefore, the interaction was greatest in the inner region of the discharge. We consider points in a cylindrical volume 4 in. in height and 6 in.o in diam, centered about the axial point 9-1/2 ino above the anodeo The sensing microwave free-space wavelength is about 4 ino From Fig. 4.6, we ha;ve

86 V _ - =. (0062 ~ 11//%) -2 Mfe l(5.18) The set of equations [Eqs. (5.5), (56), (5.7), (5.8), (5.15), and (5.16)] can now be solved using iteration techniques for the values of q and a deteren o mined by the measurements of f (x )f f2(x ), 0(x ), and n (x ). This has been done for the four locations indicated in Table 5.1. The electron temperature and density at x1 corresponds to the average discharge current, IDC using Figs. 4o4 and 4.5. The other temperatures and densities have been scaled to the values of xl using the variations in Figs. e e 4.6 and 4.7. The uncertainties in e and n are a measure of the absolute o value and reproducibility of the probe measurements. The uncertainties for e e P and a are due to those in e and n, assuming that these variables are inm o o o dependent. The total spread of P in this volume, about the central value, is only ~6% compared to the uncertainty of:~9% at each point. We consider the point x1 to represent the most likely interacting region, however we combine the uncertainties to obtain as our experimental results g3? = (l8.-7 TO R 0 0 C (5.19) and o=(3.6 - 6- ) / 3 - (5.20)

87 TABLE 5 1 EVALUATION OF THE PROBABILITY OF COLLISION FOR MOMENTUM TRANSFER, Pm(cm-1 Torr-1 AT O~C), AND THE ELECTRON-ION RECOMBINATION COEFFICIENT, 0o(cm3 sec-1) 1, 1 1 x_ P _ 1(_ 0) X2(1i20) (923) 2X4(7o0) 0e(eV) ~ 15%.052.048.055.057 O 0_. __ __ _ _......-.... n (elec/cc)+22% 11x 0 l 1.27x101 o91x1l01.87x1011 O 8 8 8 8 v (sec ) 1.28x10 o 28xl0. ol9xlO 1.18x1O en_ v,(sec ) 1.71x1 8 2253xl08 1 32xl8 1.5x108 ei______________ f (Hz) ~ 6% 7140 7400 7000 6950 a, fb (Hz) + 49% 860 750 1000 1040 f (Hz) 6270 8100 4260 4100 C fd(Hz) 690 640 790 800 d...... _ _.____._.. P (cm-lTorr-) +-9% 18.7 19.4 17.5 17.3 m -(cmsec 468 -8 -8. -8 a (cm sec )~54% 5.6x10o 2.6x10" 5.1x1O 5.5x10 0_ _ _ _ _ _ _ _ 2 / f-e ii. I 00,~~~~r v

88 for = (.(052 -- is ) eV (5. 21) The published values ef P 9 the probability of collision for momentum transfer, for helium are given in Table 5.2 together with the value obtained -1 -1 in this experiment. A simple average of the values listed is 19.8 cm Torr TABLE 5.2 HELIUM COLLISION PROBABILITY -l -1 P (cm Torr ) 19 18.3 ~ 2% 24 21 18.7 20 ~ 5% 18.5 ~ 3% 20 -: 11% 18-.7 ~ 11% 3 ee (eV) 2 o 0539 0 to. 75.039 < 2 0 to.1.039 ~.l.058 o078 o o78 Reference 30 —Phelps (1951) 351-Gould (1954) 32 —Anderson (1956) 33 —Phelps (1960) 34 —Pack (1961) 35 —Chen (1961) 36 —Golden (1965)* 37 —Wald (1966) This experiment The value for the electron-ion recombination is not as well known as P e m The experimental values for this and other experiments are shown in Table 5.3. *This is actually Pc the total probability of collision, measured by the Ramsa;uer techniqueO We have estimated the value at O.l eV, which is probably accurate within the indicated erroro

89 TABLE 5o3 HELIUM ELECTRON-ION RECOMBINATION COEFFICIENT 3 -1l aC(cm sec Reference 1.7 x 10 358 —Biondi (1949).o0 x 10 39 —Johnson (1950) 8.9 x 109 33 —Chen (1961) (356 6 61%) x 108 This experiment In conclusion, the experiment has shown that continuous cross-modulation of microwaves can be measured in a large steady state plasma. For a glow discharge with an electron collision frequency much less than the carrier wave frequency, the interaction is observable primarily for a reflected sensing microwave signal due to electron density variationso For this case, the crossmodulation depth depends on both the restoring rates for electron temperature and density variationso These rates or break frequencies were measured for a helium glow discharge and led to values for the probability of collision for electron momentum transfer and the electron-ion recombination coefficient. These quantities agree favorably with other published values.

APPENDIX A THE HIGH FREQUENCY AND DC PLASMA CONDUCTIVITIES We present the usual derivation for the complex high frequency conductivity including the effects due'to both electron-neutral atom and electronion collisions, where the carrier frequency is much larger than the total collision frequency8 Then we present a less common solution for the dc conductivity including both collision processes. The electron current- density is defined in terms of the electron singlet distribution function as (3 ee J ^ - =-e _rfCx r^) (A.1) For the P1-approximation, Eq. (2o31), this becomes 00 I(k X ) = - e -, (_x ) (A.2) in terms of the function J(x9v3+t) which is related to the electric field by Eq. (2.37) and (2.38)~ Substituting J(x,v,t) from Eq. (2.37) into (A.2) and considering the contribution of electric field only, we get 0C$ e2 d 39~ ctt (( (A.3) 90

91 As before, we are treating f and g as slowly varying functions of time and will not be concerned with this feature here. 1, THE HIGH FREQUENCY CONDUCTIVITY We consider an electric field given by E(x)e. For this case the electron current density becomes (x = - e -() e -I^f 4" (% - s ) DL 0 C + 2 9C L (A.4) From Eq. (Ao4) the complex conductivity is defined as C2 Cdv' IWs-L f c = 3n Mats gr The collision rate with neutral atoms and ions is given by m = "/Vy r + /-t m I^eA tr- 4 Ile su' (A.5) (A.6) where -7 - nze i {2e / (A.7)

92 i 19 and N and n are the scattering centers.9 We treat both q and F.i as conFen ei stants for a helium plasma. We assume that f is the Maxwellian distribution function;70 (, My f ) = 4,7T te (i 7T 23 G (A.8) We substitute Eqs. (A.6) and (A,8) into Eq. (A,5), which can be readily solved 2 2 for the case when c >> g considered here and obtain s gm = e 2 (,e.2)e4 ) - 2ne (/9C 23 w37 r w5 (A.9) where ~"^ - Y V' r Y (A.10) and 2) (A.11) 2. THE DC CONDUCTIVITY For a constant electric field the conductivity is defined from Eq. (A.3) a,s

93 3 -mL2 3m7 00 /-3 r^ "g (A.12) Substituting Eqs. (Ao6) and (A 8) for g and f, we want to solve O m o dc DC -177 e2/' -y 3- -/ ~0n e)J, mv 2 7 nr3 a-3 (A.13) Changing the variable of integration using X2 _ /7 I =_ SS CEY (A.14) we have vDC 32 e 7y4n z r2d 12 X -e x2 4 / 7 (A.15) where 2 _ 23), ^ ^~~~~~e (A.16)

94 3 2 using the definitions in Eqs. (A.10) and (A.11). Letting x = x(x + 1) - x, we express the bracketed term as / f2 ('1)7 (A.17) where g(y) is a tabulated auxiliary exponential integral function.4

APPENDIX B NORMALIZED VARIATION OF THE DC CONDUCTIVITY WITH THE ELECTRON TEMPERATURE During the analysis presented in Chapter II, it was found convenient to define the factor p Eqo (2083) which is 9 PC C, I ] = oe (-_ - y2(I- 29 zl)+) t y [ - 2 cy) (B.1) where 00 -y x / e o 2 t. / dX y>o) (B.2),.i.r ~~ _^ / + e / 0 I Ryo (B.3) and y z = 2, t' C 1Srt T 9nXe CoST^A/T x/ (B.4) 95

96 Since we did not find tabulations for the functions g(y) and f(y) directly, we 40 defined them in terms of the available sine and cosine integrals as4 gc( yi = -G C( y) 0C5 )- (X($) 2 ) Ash by) (B.5) and 62p = OG (a) W y) -Xy)- 2 ) Ch(B.y) (6) where -j x K By~~~~~(B.7) and G(y) =X 7 j+44 i' a-1 x (B.8) The factor y is Euler's constant and equals 0.57721 56649. * We have numerically evaluated p9 over the range of y equal 0.1 to 10 and have plotted the results in Fig. Bol. For the electron temperature of.05 eV -16 2 (q = 5.66 x 10 cm and P =.378 Torr), this range corresponds to the denen o 8 12 sity range from about 5 x 10 to 10 elec/cc. We have labeled the value corresponding to e 11 responding to 0 o.052 eV and n 1.1 x 10 elec/cc which was used for the o o majority of our analysis in Chapter V,

97 y y 8 7 6 5 4 3 2 1.5 16 2'qen = 5.66 x 10-6 cm lec./cc Po=.378 TORR.1.2.3 1.1 9 I..'rr ~ ~ ~ ~ ~ ~ * I-f w f * 2 yJ' Z J'I'-''T C + y )3f I - I - gCy) J 9(Y) = - j. () C S y) - (.4.(y) _7r)S Any) = Ci(y) e' (yJ - (L cy) - ) Cos CY) 2 2I~ n 3 = 7 A/. ^ 3 j/r^y-i;). - it/ih't e f/2 3,' 3e -(/"~. ^ F- 1.12 e.9 F T p-/ I L2~i'lrrti J Fig. B.1. Variation of p (y).

98 We have also plotted, in Fig. B.2, the variation of (1 - y g(y)) as a function of y. From Chapter II Eq. (2.69) or Appendix A Eq. (A.17), this is the factor which determines the importance of the electron-ion collision frequency in dc conductivity. For large y it varies like (6/y ) or (5v /v ). en ei

99 y 3 1.5.9 -.8 -.7 -.6 -.5 -.4.2.3.4.5 [I-y2g (y)] —.15 Fig. B.2. Variation of [l-y2g(y)] with y.

APPENDIX C PRINCIPLES OF REFLECTED PLANE WAVE INTERACTION WITH SEMI-INFINITE TIME VARYING PLASMA Consider a plane electromagnetic wave, E te (s, propagating in the x direction in vacuum with the angular frequency W and propagation constant k, reflected at the plane boundary of a semi-infinite plasma at the point x. The amplitude of the reflected signal seen by an antenna located at x, in terms r of the reflection coefficient, R, is given by E(rx,) = Rcx), (Xp) (C.1) If the plasma properties are slowly changing, then the change in the received signal is given by SE CX^ =6 Rxp) ~ up) (C.2) Therefore, the normalized detector voltage output is expressed by 6r+ = 6 ~ Rm ) (c.3) where c is the power law of the detecting circuit. In terms of the normalized variations in the electron temperature and density we obtain 100

101 S(t-) a ee 60o 4, b he I (c.4) where a [6e 7a a =R Ce 0 L-~~~~ 0 (c.5) and The reflection coefficient for The reflection coefficient for Fe% aRP this case is given by this case is given by4 (C.6) g6 R R.~r R o(-L~~ (c.7) where (c.8) is given in Eq. (2.2) in terms of the complex conductivity. We can express the phase and attenuation constants, a and P, in terms of the real and imaginary parts of the conductivity, a = aR i'Ia as

102 0 /7(1 L- 7 ) 2 C4 J 2 )Y2 i~77' + / (c.9) and? = 0 pL Ag7F ( W2j LI T 2 ~4 )2 I's (C.10) where k = wa /c. We approximate c and 3 for the condition when aI >> aR. 0o s corresponds to the approximation in Appendix A of v << c which led to s This ca e ne PI) W; ( 4e/ I 4J -' ne e6 /e _(' 7 ------ (C.11) We obtain 0 ~77 - (C.12) and k 77 / W 26/' -/7r0. j-I /_ (a 1 (c.13)

103 Notice that for this approximation the phase constant a is independent of the electron temperature. We are now in a position to evaluate the ratio (a/b). We want a b ee (I/fo\e) rt (l\l 2/ a tq% (C.14) Using Eq. (CG7) and neglecting the partial derivative of a with 0 we obtain a,eRia _) f b n);'/' [ (k- 4'/z)0 (I~oc/arle) 7 Oe (-;/I /la/e I (C.15) 2 We define normalized variables as T = (U /nD ) and d = v/. Now (a/b) is written in terms of and d, using Eqs (C12) and (C13 as written in terms of T and d, using Eqs. (C.12) and (C.15), as a ge (I Ra9e)e h 0(9O/9nr fL- (/- qL - [L I d' /(2-?)(-O -n d/? 4 4 2(c.16 (a.16) We now have the form which can be readily compared with the general case for (a/b) given in Eq. (5o13). Further discussion of Eq. (C.16) can be found in Chapter II, Section E.

APPENDIX D COMPUTER PROGRAM The computer program was designed to find the "best fit" y(x,A,B) to the measured values of yi (x,) which had a first order fit of y(x,A,B ) The obs principle of Least-Squares indicates that if the deviations d = y - y. are i 1 1 statistical in nature then y. is the best fit when 1 E d minimum (D.1) i Therefore, we want to find a and b, of A = A + a and B = B + b, such that o o0 2 Z d is a minimum, and then i xi (- R B) = jos _ t. g, ef, A, B ) (D.2) Treating a and b as small corrections to A and B, we attempt to find a o o solution to Eq. (D.1) by Taylor expanding y(x,A,B) about y (x,Ao,B ) and keeping only the first correction terms. That is, take y,. (~.,,8) s~ y. (, o) a/,. y cx,~. n, R) rL yI tt. SPqB.iou 7" "ia+ a(D.3) Then d. in (Do2) becomes 1 i -- (D 64 (JO9 (D/4) 104

105 We now perform the summation of a 9/t 2 Z d. and minimize the sum by setting d = 0 (D.5) and 9D 98 Z o = 0 ~ (D.6) This yields two equations for a and b which can be written as a ___l_ a 0 I bY7 (2^^i )7y (>^bs 0 )( ay(;.7) a3 a aA? i (Dc7) and 6LW( ^9 ( Y ) Y bI ( ~) L2 X(bs~ y 9)( (D.8) Solving these equations simultaneously for a and b gives the desired results, Usually the resulting A and B will not satisfy the condition of Eq. (D.1) exactly, unless Eq. (Do3) is an exact expansion, and one must iterate the solution by replacing A by A + a and B by B + b. If the iterations converge, one can obtain the values for A and B to within the accuracy justified by the obs measured values of yo and x. 1 i The method can readilay be extended for n number of independent parameters. This would result in n equations like Eq. (Do7) and Eq. (D.8) with n variables.

106 With the program described below we were interested in fitting the function 7-T /#,L x-y xc) = Ai,, (D.9) to the measurements of y, and xi(= fm ) For this experiment A = f, -2 -2 B = f, C = f. An earlier used program, with A = 0, was based on a three 1' 2 parameter (T, B, and C) fit, but would not converge for the cases in which B and C were nearly equal. This appeared to be due to the finite accuracy of the computer such that the coefficients in the "normal equations" (qs. (D.7) and Do8)) became nearly identical and the solution indeterminate. That is, once the values of B and C became nearly equal the computation was approximately a two parameter solution with three equations. If the denominator was treated 1/2 as (1 + Bx) instead of ((l+Bx)(l+Cx)) / the program converged. For this reason, the program was set up as only a two parameter fit during any given iteration cycleo During the first iteration the solution was for T T + t 1 o and B = B + b with C, and then during the next iteration T = T + t and 1 B o o 2 1 C = C + c with B This approach converged for all cases tested, and was 2 Co 1 actually fastest for the cases when B was approximately equal to C. When A was allowed to be non-zeros it was iterated with B (or C) with T held constant. The program below was written in the BASIC computer language and was used with an on-line teletype contact with a, GE-2355 computer.

107 100 DIM F(25), T(25), X(25), G(25), V(25) 110 READ D,M,N,E,T,W1,W2 120 FOR R = 1 TO N 130 READ F(R), T(R) 140 NEXT R 150 LET B = 1/Wlt2 160 LET C = 1/W2t2 170 PRINT "DATA FILE NOO "D;",DO YOU WANT IT?"; 180 INPUT El. 190 IF El = 0 THEN 100 200 LET V4 = O 210 LET A5 = 10 220 LET A6 = 10 230 LET U = 0 240 IF El = 2 THEN 890 250 PRINT "A ="; 260 INPUT A 270 PRINT "INPUT VALUES OF T="T;"B="B; "C="C 280 LET V5 = O 290 FOR J = 1 TO M 300 FOR K = 1 TO 10 310 LET S(K) = 0 320 NEXT K 330 FOR I = 1 TO N 340 LET X(I) = F(I)t2 350 LET Z5 = 1 + X(I)*B 360 LET Z6 = 1 + X(I)*C 370 LET Z7 = 1 + X(I)*A 380 LET Z1 = SQR(Z7)/SQR(Z5*Z6) 390 LET Z2 = -T*Z1*X(I)/(2*Z5) 400 LET Z4 = T*ZI - T(I) 410 LET S(9) = S(9) + Z4*Z4/((T*Z1)*(T*Z1)) 420 LET V6 = V4 + V5 430 IF V6 < > 444 THEN 450 440 LET Z1 = T*Z1*X(I)/(2*Z7) 450 LET S(1) = S(1) + Z1l2 460 LET S(2) = S(2) + Z1*Z4 470 LET S(3) = S(3) + Z1*Z2 480 LET S(4) = S(4) + Z2*Z2 490 LET S(5) = S(5) + Z2*Z4 500 NEXT I 510 LET P4 = S(3)*S(3) - S(4)*S(1) 520 IF P4 < > 0 THEN 550 5530 PRINT "P4 IS ZERO" 540 GO TO 860 550 LET D2 = -(S(2)*S(3) - S())*S( 1))/P4 560 LET D1 = -(D2*S(3) + S(2))/S(1)

108 570 LET D5 = S(8)/SQR(N*(N-1)) 580 LET D6 = SQR(S(9)/(N*(N-1))) 590 LET U = U + 1 600 IF V6 < > 444 THEN 660 610 LET A = A + D1*1.2 620 LET A5 = D1/A 630 IF A >= 0 THEN 680 640 LET A =.0001 650 GO TO 680 660 LET T = T + Dl1*1.2 670 LET A5 = D1/T 680 LET G = 1.5 690 LET B = B + D2*G 700 IF B >= 0 THEN 720 710 LET B =.0001 720 LET A6 = D2/B 730 LET C1 = B 740 LET B = C 750 LET C = C1 760 IF J < 4 THEN 820 770 IF ABS(A6) + ABS(A5) > E/100 THEN 810 780 PRINT ABS(A6) + ABS(A5) 790 IF J > = M-2 THEN 820 800 LET J = M-2 810 IF J < M-2 THEN 830 820 PRINT U; T; A; B; C; D6*100; J 830 NEXT J 840 PRINT 850 PRINT "W3 =" 1/SQR(A + 1E-6) 860 PRINT "AFTER J ="U;"T ="T;"W1 ="SQR(1/B);';W2 ="SQR(1/C) 870 PRINT 880 PRINT 890 INPUT V 900 IF V < > 303 THEN 940 910 PRINT "FOR T(CAL) LET C ="; 920 INPUT C1 930 GO TO 1080 940 IF V < > 0 THEN 990 950 PRINT "M,E" 960 INPUT ME 970 IF V > 400 THEN 250 980 GO TO 290 990 IF V < > 444 THEN 1020 1000 LET V4 = V 1010 GO TO 950 1020 IF V < > 441 THEN 1050 1030 LET V4 = 0

109 1040 GO TO 950 1050 IF V = 333 THEN 1080 1060 PRINT "NEXT SET OF DATA - PLEASE." 1070 GO TO 100 1080 PRINT "DATA FILE NOo"D;"NORMALIZED BY "; 1090 INPUT V1 1100 PRINT 1110 PRINT "F(KHZ) T(DATA) T(CAL) RATIO" 1120 PRINT 1130 FOR K = 1 TO N 1140 LET G(K) = T*SQR(l+A*X(K))/SQR((l+B*X(K))*(l+Cl*X(K))) 1150 PRINT F(K),T(K)/V1,G(K)/V1,T(K)*T/(G(K)*Vl) 1160 NEXT K 1170 PRINT 1180 GO TO 890 A feature of the on-line teletypewriter computer application is that it allows the operator to observe the progress of the program and make decisions at any pre-determined point. The program above makes use of this important feature. After giving the program a set of data and permission to proceed, the operator must give a value for A, usually letting A = 0 initially. The program then makes from six to M iterations with T and alternately B and C. The statements 760 to 820 control the printing. Only six sets of results are printed, the first and last three. If the error condition in 770 is not satisfied then only two more iterations will be madeo Using statement 890 one can either: (1) continue with a new M and E; (2) continue with T held constant and allowing A to vary; (3) read in the next set of data; (4) print input data, Ti and f, along with the calculated Ti(f,T,AB,C) and the ratio of i Mi the T~ to T; or (5) by using statement 910 and 920 with Cl = 0 the Ti be1 1 1 obs comes a calculation based only on T (f,T,A,B) and the ratio of Ti to this i m, i 1 T, is effectively the fraction of the data which was used to predict C (see Fig. 4.10), of course Cl can be either B or C.

110 Below is the data taken from the sample trace in Fig. 4.9 as it is supplied to the program. Then the computer solution is presented along with tabulated data and calculation of the function T(f,T,A,B,C) and the ratio T ob/T. This result is plotted in Fig. 4.10; in the upper curve it is represented by the symbol "o". 1200 DATA 414.406,20,20,.01,2.3,2,7 1201 DATA.5,2.31,.6,2.28,.7,2.21,.8,2.14,1,1.96,1.2,1.77 1202 DATA 1.4,1.58,1.71.39,2,1.22,2.5,1.01,35.85354,.65 1203 DATA 5,.51,6,.39,7, 31,8,.239,10,,.145,12,.104.14,. 083,17,.074 2000 END RUN DATA FILE No. 414.406, DO YOU WANT IT? 1 A =? 0 INPUT VALUES 1 2.46902 2 2.50822 3 2.54496 18 2.57422 5.28146 E-5 19 2.574355 20 2.57424 OF T 0 0 0 0 = 2.3 2. 04082.622071 3.12037.748957 B =.25 C = 2 E-2.622071 3.12037 E-2 E-2.682793 1.89057 E-2.04082 E-2 5.98698 1.78691 1.75484 1.64799 1 2 3 18 19 20 0 1.89057 E-2 0.748969.748969 1.64017 1.88545 E-2 1.64032 W3 = 1000 AFTER J = 20 T = 2.5r? 333 DATA FILE NO. 414.406 F(KHZ).5.6.7.8 1 1.2 1.4 1.7 2 2.5 4 4 7424 W1 = 1.1555 W2 = 7.2827 NORMALIZED BY? 2.57424 T(DATA).897352.885698.858506.831313.76139.687582.613773.539965.473926.392349.33136.252502 T(CAL).915608.88449.851373.793173.749124.684396.625101.547427.482399.396823.332335.243251 RATIO.980062 1.00137 1.00838 1.04808 1.01637 1.00465.981878.986369.982436.988725.997068 1.03803

ll F(K Z) T(DATA) 5 6 7 8 10 12 14 17.198117.151501.120424 9.28429 5.63273 4.04003 3.22425 2.87463 T(CAL).185627.145954.117421 9.62334 6.75745 4.97282 3.79601.026704 RATIO 1. 06728 1.038 1.02557.964768.833558.812423.849379 1.07648 E-2 E-2 E-2 E-2 E-2 E-2 E-2 E-2 E-2

APPENDIX E TABULATION OF THE DATA Below is a, complete table of the data and plasma conditions which were used in the distribution plot of Fig. 4.11. They were taken over a four month period (January 23, to April 17, 1967). The 107 sets of data represent 93% of the data analyzed during this period and about 25% of the traces (see Fig. 4.9) recorded. In the table there is some special notation. The data designation, D in the program of Appendix D, is best described by an example: 223.135 means that the data was taken on the 23rd day of February with a, helium glow discharge which had been on (P =.378 Torr, V = 888 V, and I = 182 mA) for 1 hr and 0o DC DC 35 min after the evacuated condition. All the data below was taken at a helium pressure (room temperature) of.378 Torr. The other plasma conditions of voltage and current are in the table. The rest of the columns are: the discharge total run time (hr); the signal to noise ratio; and from the computer N obs 2/ 1/2)28 analysis: the "standard error" (100[ Z ((y - yN(N-1) i=l and the two break frequencies f and fl (K Hz). 2 1 112

113 TABLE E.1 DATA D VDC IDC Time SNR SE f2 DC DC ~~~~~2 f 223 o135 223.154 223 440 223.453 223.545 223.713 227.131 227 137 227.037 228.515 300.311 301.111 301.117 301o243 301.306 302.057 302.101 302.105 302.109 305.111 305.115 305.123 305.215 305.222 305.415 305.424 305.431 305.451 305o501 305 505 306.321 306.336 306.421 307.704 309.331 309.335 309.345 309.358 309.359 309.409 888 935 902 888 950 936 1030 990 1010 960 968 972 965 962 1020 978 978 968 976 935 928 928 928 928 980 978 970 968 981 968 925 928 925 940 918 917 917 932 932 935 182 200 183 192 210 204 175 170 170 170 182 168 170 168 180 172 172 172 174 155 152 152 152 152 170 172 170 172 174 172 155 157 160 160 177 178 178 188 188 185 1.6 1.9 4.7 4.9 5.8 7.2 1.5 1.6 0.5 5.2 3.2 1.2 1.3 2.7 3.1 0.9 1.0 1.1 1.2 1.2 103 1.5 2.3 2.4 4.3 4.4 4.5 4.9 5.0 5.1 3.4 3.6 4.4 7.1 3.5 3.6 3.7 3.9 4.0 4.1 2.4 3.2 2.0 2.2 2.4 0.9 1.0 0.8 1.7 2.4 2.1 3.6 3 3 1.8 1.6 1.7 1.7 2.0 2.0 1.2 1.3 1.0 1.2 0.9 1.4 1.1 1.1 0.8 0.9 0.9 0.8 1.0 0.8 0.9 2.5 1.9 1.6 1.7 1.7 2.1 3.1 3.0 2.5 2.8 2.4 1.5 2.8 4.9 3.4 3 9 3 3 4.5 4.1 4.9 4.3 4.2 3.4 3.7 3.1 2.0 1.9 2.9 3.0 2.6 2.7 2.5 1.6 3.9 2.3 2.6 2.6 3.2 4.0 4.1 3.8 3.8 3.6 2.9 2.3 3.1 1.15 1.32 1.09 1.04 1.60 2.71 2,68 1.90 2.18 2.02 1.70 1.83 1,94 1.60 2.01 2.41 2.07 1.98 1.87 1.49 1.40 1.44 1.50 1.49 2.02 1.87 2.16 2.09 2.07 1.82 1.19 1.30 1.18 1.24 0.99 1.04 0.89 1.00 1.12 1.06 9.24 9.74 8.08 8.08 2.80 2.97 2.97 6.91 7.05 7.19 11.00 10.09 7.07 6 69 5.81 7.81 8.25 7.84 6 69 7.12 5.41 5.43 5.17 5.95 6.34 5.27 5.24 6.69 6.39 6.03 6.48 6.90 6.17 9.47 8.46 6.96 8.38 7.08 8.40

TABLE E.1 (Continued) D VDC IDC Time SNR SE f2 1 DC DC 2 309.411 935 185 4.2 2.1 3.1 1.o6 8.40 309.435 930 190 3.7 1.9 2.3 1.10 8.15 309.437 930 190 3.7 1.0 1.5 1.24 5.84 320.230 1160 240 2.5 1.4 2.9 3.54 3.55 320.241 1194 246 2.7 1.7 2.8 3.00 4.37 320.243 1194 246 2.7 3.6 2.8 2.22 9.12 320.245 1194 246 2.8 1.3 2.7 2.39 5.02 320.249 1194 246 2.8 3.8 3.7 2.14 8.76 322.124 1072 183 1.4 1.0 1.7 2.06 4.72 322.126 1072 183 1.5 1.6 2.4 1.54 6.63 322.128 1072 183 1.5 1.0 2.2 2.05 4.46 322.131 1072 183 1.5 1.0 6.6 1.14 13.12 322.133 1072 183 1.5 1.2 3.1 1.26 6.96 411.306 923 172 3.0 2.0 3.2 1.22 10.08 412.132 928 172 1.5 1.6 2.5 1.27 7.96 412.137 928 172 1.5 1.9 3.1 1.04 7.88 412.156 915 170 1.9 2.2 3.1 1.39 8o01 412.158 915 170 2.0 2.3 4.5 1.49 7.36 412.211 915 170 2.1 2.3 3.3 1.11 7.78 412.214 915 170 2.2 2.5 3.9 1.08 7.98 412.238 920 170 2.6 2.0 2.3 1.44 7.90 412.242 920 170 2.8 2.7 2.2 1.12 8.88 414.337 940 170 3.6 0.7 4.7 1.09 8.47 414.347 937 168 3.8 1.6 2.2 1.20 6.76 Used in 414.406 937 168 4.1 1.6 1.6 1.16 7 28 Fig. 414.432 965 170 4.5 1,0 1.8 1.18 5.91 4.10 414 437 965 170 4.6 1.0 2.4 1.22 5. 93 415.242 900 150 2.7 1.1 5.0 1.00 6.32 415253 93 4 166 2.9 2.3 3.4 1.13 8.97 415.408 945 170 4.1 2.1 2.3 1.23 7.99 415.1025 980 170 10.4 0.9 1.7 1.27 6.94 415.1028 980 170 10.6 0.9 0.8 1.39 5.80 415.1031 980 170 10.6 0.8 3.4 1.52 4.06 415.1039 979 171 10.7 4.0 2.0 1.34 5.93 415.1052 860 125 10.9 1.1 4.7 1.58 5.77 415.1111 1031 189 11.2 1.1 2.7 1.62 6.06 415.1113 1031 189 11.2 0.8 0.9 1.52 7.33 415.1122 1026 179 11,4 2.2 3.7 2.02 3.64 415,1128 1025 175 11 5 0.5 1.7 1.43 4.50 415.1144 1020 175 11.7 0.6 3.8 1.61 3.11 416.1918 903 142 19.3 0.5 6.0 2.51 2.51 416.1923 905 140 19.4 1.7 4.5 1.64 6.50

115 TABLE Eol (Concluded) D VDC IDC Time SNR SE f2 1 416.1925 416.1928 416.2031 416.2036 416.2048 416.20553 416,2055 416.2100 416o2106 416 2142 416o2154 416.2200 416.2207 416.2225 416 2241 416.2311 416 2325 416.2330 41.6.2335 416.2355 417.3402 417.3425 417.3426 417o3428 417.3430 905 905 913 913 905 905 915 914 924 1029 1029 1034 1035 1052 1076 960 965 963 963 999 970 993 1000 997 997 140 140 142 142 140 140 146 146 146 161 161 161 162 168 174 150 145 147 147 151 120 157 160 153 153 19.4 19.5 20.5 20.6 20.8 20.9 20.9 21.0 21.1 21,7 21.9 22.0 22.1 22.4 22.7 23.2 23.4 23.5 23.6 23.9 3400 54.4 34.4 34.5 34.5 1.1 1.4 1.1 0.9 0.8 0.9 1.1 1.4 1.5 1.0 1.0 0.8 0.8 0.9 1.0 1.0 0.9 1.4 1.5 0.7 1.3 1.3 1.2 1.6 1.5 2.8 5.3 2.6 2.7 1.7 4.1 1.2 0.8 2.3 3.5 4.5 1.9 1.6 2.2 2.6 3.2 2.4 2.4 2.6 1.2 2.9 1.9 1.0 1,1 1.9 1.59 1.56 1.58 1.45 2.15 2.93 2.09 1.80 2.355 1.71 1.96 2.07 1.75 2.65 2.44 1.50 1.34 1.51 1.27 1.60 2.54 1.61 1.54 1.90 1.65 5.88 5.79 5.19 5.13 3.55 2.97 5.48 6.86 4.83 4.49 3.6o 2.85 4.28 3.24 35.86 4.61 5.38 4.47 7.31 3.93 2.56 4.76 5.47 5.55 7.08

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