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Enhancements observed in the scattered light spectra of a carbon arc plasma Churchland, Mark Trelawny 1972

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ENHANCEMENTS OBSERVED IN THE SCATTERED LIGHT SPECTRA OF A CARBON ARC PLASMA by MARK TRELAWNY CHURCHLAND B.Sc., University of British Columbia. 1967 M.Sc, University of British Columbia. 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH November, 1972. COLUMBIA In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Br i t ish Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Phvsics  The University of Br i t ish Columbia Vancouver 8, Canada Date Dec. 22 . 1972 i i Abstract The spectrum of Ruby laser light scattered from a carbon arc plasma has been studied. The temperature of the plasma i s shown to be much hotter than was previously reported. Enhancements in the spectrum of light scattered were observed. Their frequencies were shown to be a sensitive function of the plasma v/ave scale length K, and the plasma density and temperature. A model i s constructed which produces enhancements in the theoretical scattered spectra at particular frequencies. This model i s f i t t e d to the observed spectra; this f i t t i n g i s discussed. i i i TABLE OF CONTENTS *" Page Abstract i i Table of Contents i i i L i s t of Tables v L i s t of I l l u s t r a t i o n s v i Acknowledgements v i i i CHAPTER I Introduction 1 CHAPTER II Theory 8 A. Summary of Scattering Theory 8 B. C a l c u l a t i o n of Spectral Power Density 11 C. Departures from a Thermal Spectrum 15 D. Speculation on the Source of the Anomalies 17 CHAPTER III The Experiment 2 2 A. The Plasma 22 B. The Power Supply 30 C. The Ruby Laser 32 D. The Light Detection System 33 E. The Experiment 36 F. Recording of Data 39 iv CHAPTER IV Results 45 A. Temperature and Density Profiles 45 B. Symmetry of the Scattered Spectrum 54 C. Enhancement at the Plasma Frequency 58 D. Enhancements at to , . and % O J 59 P P p E. Enhancement as a Function of K 62 F. Anisotropy Check 72 CHAPTER V Conclusions 77 A. Temperature and Density Measurements 77 B. The Enhancements Observed and the Theoretical Model 78 BIBLIOGRAPHY 89 APPENDIX A 92 APPENDIX B 95 APPENDIX C 100 TABLE I Symmetry Check Data TABLE II K Dependence Results v i LIST OF FIGURES Fig . No. T i t l e Page 2A Two Distribution Functions 18 2B Theoretical Profile for f (v,vD) 21 3A Arc Electrode Dimensions 24 3B Arc Apparatus 26 3C Cathode Adjustment Mechanism 28 3D Ballast Resistor Circuit 31 3E Photomultiplier Dynode Circuit 34 3F Photodiode Circuit 35 3G Schematic of Apparatus 40 3H Typical Scope Trace 41 4A Observation Volume for 135° Scattering 47 4B Typical F i t t i n g of Theoretical Curves to the Data 49 4C Electron Temperature vs Axial Position 51 4D Electron Density vs Axial Position 52 4E Electron Temperature and Density vs Radial Position 53 4F Scattered Spectrum Symmetry Check 57 v i i LIST OF FIGURES - (continued) F i g . No, T i t l e Page 4G Enhancement of the Thermal Spectrum as a Function of Wavelength Shift and Electron Density 60 4H Position of the Enhancement vs up 6 2 41 Spectrum Showing Enhancement at ojp, Hap, and ku (for the parameters shown) 64 4J Spectrum Showing Enhancement at OJ , ha , and (for the parameterS shown) p 65 4K 3 Axis Plot of Enhancement vs Wavelength Shift vs K 70 4L Integrated Area of the Enhancements vs K 71 4M Dispersion Plot of the Enhancements 73 4N Scattering Configuration of Anisotropy Check 75 4P Observed Enhancement of Anisotropy Check 76 5A Orientation of Drift Velocities 81 5B Theoretical Spectrum for the two Drift Velocity Distribution Function 83 v i i i ACKNOWLEDGEMENTS I would like to thank Dr. R. A. Nodwell for his constant encouragement and timely suggestions over the course of this work. Thanks also go to many members of the Plasma Physics Group. In particular I would like to thank G. Albach, H, Baldis, R. Morris and J . Meyer for many helpful and interesting discussions during the preparation of this thesis. Thanks go to D. Camm, G. Albach and L. Godfrey for the preparation of the Computer programs used in this work. I wish also to thank Dr. W. B. Thompson for stimulating discussions on the theoretical aspects of this work. Thanks also go to my two typists Mrs. L. M, Churchland and Mrs. J . T. Churchland, 1 INTRODUCTION The scattering of electromagnetic radiation by free electrons in a plasma, Thomson scattering, has been used as a diagnostic tool since 1958. K. L. Bowles, using a 41 MHz pulsed transmitter, observed radiation back scattered from the earth's upper atmosphere. Thomson scattering was f i r s t used as a diagnostic tool on a 2 laboratory plasma in 1963. Fiocco and Thompson scattered the light of a 20 Joule normal mode laser from an arc plasma. The f i r s t well defined spectrum of scattered light showing a wavelength distribution corresponding to the electron spectrum of the plasma was reported by Davies and Ramsden"* (1964) . The development of the high power Q spoiled laser made their work possible. The theory of Thomson scattering in a plasma was developed because the spectra reported by Bowles were not in agreement with the theory used at that time. Such authors 2 as Dougherty and Farley4 (1960), Fejer3 (1960), and Salpeter^(1960) developed the theory of light scattered from an i n f i n i t e homogeneous plasma. The theory was later modified to include the effects of constant magnetic fields7(Salpeter, 1961), gross d r i f t v e l o c i t i e s8 (Rosenbluth and Rostoker, 1962), and other simple departures from isotropic thermal equilibrium^»12 (Perkins and Salpeter, 1965; Kegel, 1970). This theory is in good basic agreement with the observed scattered light p r o f i l e s . However, some of the earliest papers reported deviations from the theory which were not explained. Gerry and Rose1 0( 1966) obtained "spurious peaks" in their scattered spectrum. These peaks were reproducible and significantly above the theoretical curve which best f i t t e d the data. No explanation of these peaks was given. Evans et a l1 1 (1966) reported a scattered light spectrum with two points above the theoretical curve. No comment was made about these points by the authors, but Kegel1 2 (1970) attempted to explain them by postulating that the plasma contained a small fraction of electrons considerably cooler than the bulk of the plasma. He was able to account for one of the anomalous points of the scattered spectrum. No attempt was made 3 to rationalize the existence of the "cool electrons". Data points above the f i t t e d theoretical curve of the scattered light spectrum from a Z pinch plasma have 13 been observed by Kronast . Although small, these deviations are reproducible and under a variety of plasma conditions seem to occur at a shift corresponding to the plasma frequency. No explanation of these phenomena has been proposed. The f i r s t experiment set up with the express purpose of analyzing the anomalies was that of Ringler and Nodwell^"^. Using a D,C«, magnetically stabilized, low pressure hydrogen arc as a plasma, they carefully studied the spectrum of the scattered l i g h t . The results show several deviations from the predictions of the theory for a thermal homogeneous i n f i n i t e plasma. The spectrum of scattered light i s enhanced at wavelength shifts corresponding to integral multiples of the plasma frequency, Wp , The shift corresponding to 1/2 Wp also shows an enhancement. The total scattering cross section integrated over the frequency for a particular scattering vector K does not have the dependence on K that the theory would predict. The scattering cross section i s approximately twice the theoretical cross section at one particular K vector. 4 As K is varied the cross section quickly approaches that predicted by the theory. This departure from theory is of particular importance to those experiments using total intensity (integrated over frequency) to obtain a value of the plasma electron density. The strong K dependence of integrated intensity implies that the anomalies are due to waves in the plasma of a particular K and co Kegel's1 2 theory was developed in an attempt to explain these observations but does not satisfactorily do so. In further experimentation on the same apparatus, Ludwig and Mahn17 found that enhancements occur in the frequency spectrum at integral multiples of l/2„ and that P their occurrence was independent of the K vector orientation. This implies that the anomalies are isotropic and do not depend on the orientation of the magnetic f i e l d or the plasma boundary. Large deviations from the thermal spectrum in a Helium arc are reported by Neufeld4-, An enhancement was found near the central frequency of the spectrum. Possible enhancement at co^ i s also mentioned in this report. Neufeld speculates that the central frequency enhancement might be due to an excess of cold electrons. 5 This i s Kegel's two temperature theory. The origin of such cold electrons i s not discussed. P. K. John et 19 al report the detection of an enhancement of the spectrum of the light scattered from a pulsed plasma. This enhancement occurs at a frequency shift corresponding to the frequency of the electron acoustic wave. In none of the above experiments has a satisfactory explanation of deviations of the spectrum from that predicted by the theory been given. Two authors have tried to explain some of the observed discrepancies, 12 Kegel calculates the effect of having two groups of electrons at different temperatures present in the same plasma. By varying the percentage and temperature of the cold electrons he could obtain a computed profile with appropriate peaks at the central frequency and the plasma frequency. This theory was developed to try to f i t a curve to the data obtained by Ringler and 14 Nodwell , No reason for expecting a two temperature plasma is given. 20 E, Infeld and W, Zakowicz show that i f an undamped osci l l a t i o n at cop i s postulated to exist in the plasma, there w i l l be enhancements at integral multiples of ojp, These occur because the incident laser light i s frequency modulated by the postulated o s c i l l a t i o n at the 6 plasma frequency. The authors do not attempt to explain how such an osci l l a t i o n would come to exist, or why i t would not be damped. Because no satisfactory explanation of the observed anomalies was evident, i t was thought that investigation under different conditions would be useful. The anomalies 11,13,19 had been observed either i n pulsed plasmas or in magnetically stabilized D.C. a r c s ^ ' ^ To eliminate the possible time development effects of the pulsed plasma and magnetic f i e l d effects of the stabilized arc plasmas a non magnetically stabilized D.C. arc was chosen as the plasma to study. The high 21 current carbon arc, similar to one used by Maecker , was chosen because of the wealth of experimental work already done. Chapter II gives a brief summary of the scattering theory for a homogeneous isotropic plasma, A model of the plasma velocity distribution function i s postulated and theoretical scattered light spectra are given. Chapter III gives a description of the experimental apparatus. Chapter IV gives the experimental results, A comparison between the empirical model and the experimental results are made in Chapter V, Appendix A shows that the effect of collisions 7 on the spectrum of light scattered i s negligible. Appendix B contains a brief description of the f i t t i n g of the theoretical spectra to the data. Appendix C gives a brief description of the calculation of S (K, co ) for the postulated distribution function. 8 CHAPTER II THEORY The theory of the scattering of laser light from 4-9 a plasma was developed by several authors . A phenomenological description may be given as follows: An electromagnetic plane wave, in this case ruby laser l i g h t , i s incident on charged particles, the plasma. The E-M wave accelerates each particle and consequently each particle radiates. It can be noted here that the electrons are the only particles which w i l l radiate significantly (because of their low mass and consequent high acceleration). If the electric f i e l d vectors of the E-M waves radiated by the electrons are summed, the power spectrum of the light scattered can be calculated. Thus the scattered profile depends on the position and velocity of the particles in the plasma. The position determines the relative phase between the component electri c f i e l d vectors and the velocity deter-mines the frequency s h i f t . Although the ions do not radiate, their position must be considered because they affect the position and velocity of the electrons. The scattered e l e c t r i c vector (summed over each electron's phase and frequency shift) w i l l add to a non zero value only i f there are variations in the plasma density. Variations may arise from two sources. 9 Microscopic variations occur because of the particle nature of the plasma. These are random and have a magnitude proportional to (N)^where N i s the number of particles in the plasma observed (see for example 22 Bekefi sec. 8.1). Macroscopic fluctuations are caused by the collective excitation of longitudinal plasma waves. If two assumptions are made a simple expression for the power spectrum of the radiation scattered by a plasma can be derived. F i r s t l y , the incident E-M wave and the observed scattered wave are considered to be plane. This i s the Born approximation or Fraunhofer f i e l d condition. Secondly, i t i s assumed that the incident radiation does not accelerate the changes to r e l a t i v i s t i c v e l o c i t i e s . The time averaged scattered power dW per unit solid angle dft, per unit frequency internal dw i s then: dW dt dft dw \ NV I I / = 27 lSi |aT S(-S'w> 10 where: N i s the electron number density V is the observation volume i s the Poynting flux of the incident beam a_ i s the Thomson scattering cross section for an electron and: S(K,w)= lim (2/TVN) | N(K=K„-Ki , w=u) -w • ) I 2 V-»-<» where: K and K. are the scattered and —s — i incident wave vector and M and „ J are the s x scattered and incident frequency and where N (K.co) i s the Fourier transform of the number density N ( r , t ) . Thus, to calculate the scattered power spectrum for a plasma the spectrum of density fluctuations N (K,w) must be calculated. N (K,w) w i l l be derived to show that under certain M M assumptions i t depends only on the velocity distribution function for the plasma. By postulating a particular velocity distribution and calculating S (K,to ) , the theoretical power spectra can be compared to the 11 experimental results. Calculation of £ (K, to ) To calculate N (K,to) we assume the actual distribution dunction fo ( V/£»t) can be described by a function f (v) plus a small correction term f ( V r j ^ t ) fo (Y.»£^) = f (v) + fx (v,r,t) (T) If we assume that the plasma is collisionless and that f i s small compared to f (v) we can use the Boltzraan-Vlasov equation to find f in terms of f (v). The Boltzman-Vlasov equation in linearized form i s : 3f, 3f-, 3f — - + v + (e/m)E« = 0 3t 3r 3v If we write f ^ ( v,r,t) as: Equation (2 ; reduces to: 12 The term v i s introduced here to aid i n evaluating an i n t e g r a l . I t w i l l l a t e r be set to zero. Its physical s i g n i f i c a n c e i s that of a c o l l i s i o n frequency. To complete the c a l c u l a t i o n of f ^  an expression of E i s needed. E i s the e l e c t r i c f i e l d i n the plasma produced by the p a r t i c l e s . I t can be shown ( see, for 22 example Bekefi section 4 . 6 ) that a t e s t charge of charge density: produces an e l e c t r i c f i e l d whose Fourier component i s : P i ( r f t ) = 3*S E(K,to) = |K|£eoKL(K,w) i where: P i ( r , t ) = 2 7 ^ 6 (aj-K-v) e and K L (K,^) i s the lo n g i t u d i n a l d i e l e c t r i c c o e f f i c i e n t . 13 We have completed the construction of f (£»]_>__) using(T\ and ( T ) . The form of f (v) is s t i l l arbitrary. The Fourier spectrum of electron density fluctuations is given by: N(K,w) = 2TT Z 6(co-K-v.) e3-"-1 electrons f 1 (__'_'a>) ^ v The f i r s t term i s the Fourier transform of the charge density p^(r,t) summed over a l l electrons. The second term is the correction for the screened electrons. We have an expression for f (k,v,w) and can now calculate S (K,_)) . 2 Before we form the product |N (K,<_.)| to obtain S (K,u>) we make use of the substitution: G (K, io) co2 K2 K - 3 f ( v ) / 3 v o — — d v w-K • v © and <4 K - 3 f t v ) / 3 v+ G ( K , w ) = — u-K • v d v 14 where the + denotes terms associated with the ions. Remember also the definition of K ( K , c u ) which Li "~ occurs in equation (jf) for E (K,to) K L ( K , C O ) = 1+G+G+ (11 Now substituting (jT) and (V) into (V) and (4^) into (jT) and using C^)t ( i o ) , and ( l l ) we can write (oj as: 1+G+ N(K,w) = 2TT £ 6(w-K«v) e3- £ i 1+G+G+ i (g) G + + 2TT E 6(u-K.v) e3£ *ri 1+G+G i " 2 In this form the product |N (K,<JJ) j i s taken. To do this the s t a t i s t i c a l independence of test particles 22 is invoked (see Bekefi sec. 4.6, for example). The equation: Z x ( V i ) = NV i f(v)X(v)d3v (13) (where X(v) i s any function) i s also used to reduce the product. 15 S(K,W) is then given by; 1+G + 2 (2TT) -1 S(K,w) = f(v) <S (w-K«v)d v 1+G+G' + G 2 + f (v) 6 (co-K. v) d v 1+G+G Given any distribution function f(v) the right hand side of (14) can i n principle be evaluated. Departures from a Thermal Spectrum The anomalies observed in this work are characterized by small enhancements above the theoretical spectra for a Maxwellian plasma. These enhancements have a small frequency band width compared to the thermal spectrum and often occur near the plasma frequency. In this section we w i l l discuss how certain departures from a Maxwellian plasma are treated theoretically. We w i l l then speculate on how the anomalies mentioned above might occur. Departures in the theoretical profiles from those calculated for an isotropic thermal plasma may come from two sources. The acceleration term in the Vlasov equation may include more than the self reaction f i e l d produced by the plasma and the velocity distribution function 16 may not be purely Maxwellian, The effect of including a D.C, magnetic f i e l d in the acceleration term ( A.°_r ) has been calculated 7 ~ 3 V by Salpeter (1961). He shows that the f i e l d produces enhancements at frequency shifts of integral multiples of the cyclotron frequency, co^. These are the Bernstein modes. The effect of the nonlinear coupling of two E-M waves (wave mixing) has been considered by 23 Kroll et al (1964). The accelerating force of the mixed waves i s included in the Vlasov equation and the enhancement 24 of the plasma waves i s calculated. Stansfield (1971) has shown that natural plasma waves can be enhanced using this technique. The effect of the electrons d r i f t i n g with respect to the ions has been calculated by Rosenbluth and g Rostoker (1962). They show that the electron feature is Doppler shifted by the d r i f t velocity and that an asymmetry i s produced in the ion feature. Perkins and 9 Salpeter (196 5) consider the effect of a few superthermal electrons on the scattered spectrum. To the normal Maxwellian distribution a second group of high temperature electrons i s added. The effect of these electrons i s calculated and shown to enhance the electron 12 feature of the scattered spectrum, Kegel (1970) does 17 the above calculation for the more general case where the second group of electrons can have any temperature, hot or cold. His distribution function was the sum of two Maxwellians: _ f(v) = b((m/2TT<T1)?5exp{-mv 2/2KT 1}) + (l-b) ( (m/2TTKT 2) ^exp{-mv 2 /2icT 2 }) The term "b" designates the percentage of electrons that have a temperature T^. When this form of f(v) is used, secondary maxima in the scattered spectrum can be produced. Speculation on the Source of the Anomalies We wish -now .to consider what mechanisms might enhance the scattered spectrum similarly to what has been observed. The addition of a group of high speed electrons in a plasma w i l l enhance certain waves. The waves affected w i l l be those waves whose phase velocity closely matches the velocity of the high speed electrons. A physical description of the effect on the waves of a given frequency,cu, and scale length, K, can be given in terms of Landau damping. Consider that the electron velocity distribution function has been changed by the addition of a group of fast electrons moving in a well defined direction. For example, assume the fast electrons 18 have a Normal distribution function centered about a° given velocity vD: f ^ v ) = (m/2iTKT1J^exp{-m(v-vD) 2/ 2 K T1} where is small compared to the temperature of the main plasma, T. This w i l l change the slope of the total distribution function around the velocity v . -D F i g . 2 A This change in f(v) w i l l change the electron energy distribution around the energy ( _ m ) . The slope of the energy distribution function w i l l be less negative in the region v just less than v^. This means that Landau damping in the region v just less than V p w i l l be less than in the thermal case. Therefore the amplitude of the waves with a phase velocity just less than v^ 19 w i l l be greater than in the thermal case. Conversely, o the slope of the energy distribution function w i l l be more negative in the region v just greater than and the waves of phase velocity w/k just greater than V p w i l l be damped more than in the thermal case. We consider then a distribution function of the type described above: 1 ~ -f(v) = b(m/2TTKT1) 2 exp{ -m(v-y_D) ^ K ^ } 3 9 + (l-b) (m/27ricT2) 2 exp{-mv /2KT2 } If we make "b" small compared to one, the main part of the plasma i s described, by the standard Maxwellian term with a temperature T. If we make small compared to T, we should enhance waves at a phase velocity just less than V p . The choice of a Gaussian distribution for the cold electrons was made because the expression for S C(K,to) has been solved for a Gaussian and the calculation of S (K,to) for the above function, f (v) , can be done using standard integrals. Appendix C gives a brief description of the calculation of S ( K , O J ) for the 20 above distribution function. The actual evaluation of the above i s done on a computer. The function S (K,w) for the above form of f (v) , (l6) , i s calculated for a fixed K and variable w. The value of wis expressed in terms of a wavelength shift AA because this is the form of the experimental data. The values of T, T^, b, Ne, vD x' a n d ® s a r e r e a c* into the computer as parameters. F i g . 2 A shows a result for the parameters stated. This method allows the position of the "bump" to be changed by changing the value of vD^, The height and width of the bump can be changed using various combinations of "b" and T^, The "bump" i s higher and narrower for smaller T^ and i s higher for larger "b". # 8 A A(A) Fig. 2A Theoretical Profile for ftv.v-,) 22 CHAPTER III THE EXPERIMENT The purpose of this chapter i s , f i r s t l y , to describe the components of the experiment and, secondly, to describe the operation of the whole apparatus. The reduction of the data w i l l be discussed at the end of the chapter. The apparatus i s considered i n four sections: the plasma, the plasma power supply, the ruby laser, and the light detection system. The Plasma The plasma producing apparatus constructed for this experiment was a high current carbon arc similar 21 to that used by Maecker (1953), This arc configuration has been well studied by many authors and i s thoroughly documented in the literature. It has been considered a stable plasma in good thermodynamic equilibrium because the results of many different measurements (21,27,28,29) are self consistent. The arc consists of two graphite electrodes set on a vertic a l axis. The lower electrode i s the cathode and i s sharpened to a point. It i s a characteristic of arcs that the arc attachment to the cathode occurs in a small well defined area of high current density. The pointed 23 cathode allows the position of the arc base to be well defined. The upper electrode, the anode, i s a square rod of graphite, five centimeters by five centimeters. The arc attachment to the anode occurs over a large area and i s not well defined. If a large enough area is not provided by the base of the anode the arc w i l l not run in a stable configuration but w i l l attach i t s e l f to the side of the anode. The basic dimensions of the arc are shown in F i g , 3 A, The arc i s operated in open a i r . The s t a b i l i t y of the arc i s maintained by the natural convection of air along the arc column. The convection currents are induced by the heat from the plasma and electrodes. The power dissipated by the arc (60 volts § 400 amps, 24,000 watts) must be carried away by the convected air or lost by radiation as other cooling of the arc apparatus is negligible. In order to help maintain a stable convection flow the arc was enclosed by a chimney. This reduces the effect of cross drafts on the arc. The chimney also protects the surrounding apparatus from the heat radiated by the arc electrodes. Suitable observation ports (6,5 cm in diameter) were cut in the side of the chimney (see F i g , 3B). For each observation port cut in the A Carbon Arc Electrode Dimensions 25 chimney a corresponding port was cut opposite i t so that the observation optics did not look at the inside of the chimney. This kept stray l i g h t to a minimum. Ports were also cut i n the chimney to allow the entrance and e x i t of the la s e r beam. The region around the cathode base was l e f t as open as possible to allow the a i r to form a smooth laminar flow before i t reached the arc column. The supports f o r the anode were made as thi n as possible i n order not to disturb the flow above the anode. Because the 24,000 Watts diss i p a t e d by the arc heated the apparatus to such high temperatures a l l s t r u c t u r a l members of the arc apparatus were made of s t e e l , Metals such as aluminium or copper would melt or anneal i n most regions near the arc. Copper blocks were used as conductors but not as s t r u c t u r a l members. The chimney was made of sheet s t a i n l e s s s t e e l . The rest of the apparatus was made of mild s t e e l . Thick asbestos blocks were used to e l e c t r i c a l l y insulate the anode from the rest of the apparatus. Bakelite or any p l a s t i c could not be used as i n s u l a t i o n because the heat radiated by the arc electrodes would melt or burn them. The asbestos block also served as thermal i n s u l a t i o n f o r the cathode 26 Fig.3 B Carbon Arc Apparatus 27 adjustment machanism which allowed i t to be oiled to ensure smooth operation. During the operation of the arc a considerable amount of carbon i s evaporated into the atmosphere because the graphite electrode surfaces are heated to the boiling point (4827°K), To keep the electrode separation f a i r l y constant, and the observation volume in the arc column at a predetermined position above the cathode, three orthogonal adjustments were built into the cathode mount, A vertical adjustment was bui l t so the cathode could be raised as the carbon evaporated. Because the carbon from the cathode did not always evaporate symmetrically, lateral adjustments were also built into the cathode mount (see Fig, 3 C). This mechanism allowed the position-ing of the cathode while the arc was burning. No adjust-ment was used on the anode. The anode was much larger than the cathode and the rate of erosion was less than that of the cathode. Because of the large area of anode arc attachment, and the inherent i n s t a b i l i t y of the connection flow around the anode, the anode was always set with the same 3° to 5° t i l t from the arc axis. This gave a preferential position of arc attachment to the anode surface and helped the arc to start to run stably. The s t a b i l i t y of the arc was maintained by the depression 28 A D J U S T M E N T Fig. 3 C Cathode Adjustment Mechanism 29 eroded into the anode surface. Although the arc would run stably i n such a depression i t could not be started i n such a depression. This i s probably due to the higher i n i t i a l current density on the cold anode surface. Most observations of the arc were made within one centimeter of the cathode and the plasma could be readi l y positioned using the cathode adjustments. If the arc i s run at a low current the heating and corresponding erosion of the electrodes i s reduced. To minimize the rate of electrode erosion and the corresponding frequency of adjustments of the cathode a current control was used. This allowed the plasma to be useful f o r a longer period of data collection-,- The arc was kept running at a low current of about 200 amps, A large relay then switched i n the ad d i t i o n a l current required (up to 450 amps) during the taking of data (see F i g , 3 D), The 200 amp "stand by" current was the minimum stable operating current for the electrodes used. The time required f o r the current to reach 400 amps from 200 amps was much less than one second. Both the current measured through the arc and the background l i g h t emitted by the arc reached a new steady state i n much less than one second. This meant that the arc only needed to be l e f t on high current long enough to open the scope camera shutter and f i r e the l a s e r . 30 The Power Supply The power to produce the plasma was supplied by a pair of direct coupled motor-generators. The generators were each capable of supplying 300 amps at 150 volts. The two were run in parallel and could therefore supply up to 600 amps. The generators formed a low impedance source. The voltage drop across a load drawing from 0 to 400 amps vaired less than 0,5 volts at 120 volts. The current ripple for 400 amps through a ballast resistor load was 2% at approximately 400 Hz. This ripple produced no measureable fluctuations in the background radiation of the plasma. The resistance of the carbon arc plasma is close to zero. In fact the resistance, dV/dl, can be slightly negative for certain current ranges. If the generators, with their low impedance, were connected directly to the arc the current would be very unstable. In order to prevent t h i s , a positive resistance was put in series with the arc, A series of 2,0 ft , 2000 watt resistors were connected in parallel in such a way that each could be switched in or out of the ci r c u i t (see Fig. 3 D). Because the voltage across the arc was f a i r l y constant over the range of currents used, the inclusion of each 31 1 5 0 V D C From generator RELAY V s-2 ohms 2KW 11 units 2 ohms 2KW 8 units c TO ARC o Fig. 3 D Ballast Resistor Circuit 32 resistor, " i " , gives a current: V ballast i The bank consisted of 20 such resistors, 8 of which could be switched in or out by means of a large relay (see F i g , 3 D), This relay, as mentioned e a r l i e r , made i t possible to run the arc at a low current (^  200 amps) between the data collection times. The major disadvan-tage of this system is the large amount of heat produced in the room by ohmic heating of the ballast resistors. The Ruby Laser The ruby laser used i n this experiment was developed in the plasma physics laboratory and is described in detail in the authors M, Sc. thesis 3 6 (1969), The ruby rod i s 6 inches long by h inch in diameter and has Brewster angle ends. Two linear xenon flashtubes focussed by a double e l l i p t i c a l cavity optically pump the ruby rod, Q spoiling is accomplished with a dye c e l l containing cryptocyanine in methanol. The ruby rod and flashtubes are water cooled. The lasing cavity is formed by a 99.9% r e f l e c t i v i t y dielectric back mirror 33 and a 20% r e f l e c t i v i t y sapphire f l a t front mirror. This configuration r e l i a b l y produces 50 Mw of power with a pumping energy of 4500 Joules. A f a s t charging unit was added to the laser power supply which enabled the la s e r to be discharged every 10 seconds. A reasonable number of data points could then be recorded (about 60 shots of the laser) with each set of arc electrodes. Because the laser capacitor bank could be charged quickly an^automatic shut o f f was i n s t a l l e d i n the charging unit . The voltage of the bank was monitored and at a preset voltage the primary of the charging transformer was disconnected by a relay. Because of the danger of a capacitor breakdown amd explosion due to overcharging, a second voltage monitoring system with a separate relay shut o f f was i n s t a l l e d and preset to the maximum working voltage of the capacitors. The double shut o f f system made the charging both automatic and safe. The Light Detection System To choose a narrow spectral band width and r e j e c t other l i g h t , p a r t i c u l a r l y the stray l a s e r l i g h t , a SPEX 34 monochromator was used. The model used was a 0,75 M grating "monochromator-spectrograph" with an f of 6,5 and a dispersion of 10 ft/mm. The entrance s l i t was used to define the volume of plasma observed; this w i l l be discussed later, A red f i l t e r (Corning No, 29) and a polaroid f i l t e r were used to exclude unwanted light (e.g. second order background l i g h t , and horizontally polarized background l i g h t ) . The light transmitted by the system was detected by an RCA C 31034, Gallium arcinide photocathode, photomultiplier. This tube was chosen for i t s high quantum efficiency (12%) at 6943 8 . The voltage dividing resistor chain is shown in Fi g . 3 E. 50/v C A B L E PHOTO C A T H O D E ANODE -1800V. DY NODES (11 S T A G E S ) 22K 33K 22 K 22K 22K 22 K 22K 22K 22 K 22 K Hf-.05uF .05uF J05UF .1UF . I U F f-HF -4 F i g . 3 E Photomultiplier Dynode Chain 35 The tube was operated at 1800 volts, which produced about 6 M.A. through the resistor chain. Speed up capacitors were used on the last four dynode stages. The power supply for the tube was a Fluke (Model No. 412 B). In order to monitor the laser light output, a Hewlett Packard pin dyode (Lp4203) was employed behind the 99.9% r e f l e c t i v i t y laser back mirror. Neutral density f i l t e r s were used to attenuate the 0.1% of the laser output to a level the diode could respond to linearly. The diode output then gives a relative power output for each shot of the laser. The single sweep action of the oscilloscope also was triggered by the pin diode pulse. F i g . 3 F gives a schematic of the pin diode c i r c u i t . F i g . 3 F Photodiode Circuit 36 The two signals were displayed on a dual trace Textronics 551 oscilloscope. A Polaroid photograph was taken of each pair of single sweep traces, shot by shot, and this data was later reduced. The Experiment When designing an experiment using laser scattering as a diagnostic t o o l , i t i s best to keep the optics as simple as possible. The smaller the number of optical interfaces (lenses, mirrors, windows, dye c e l l s , etc.) the less stray light that can find i t s way through the detection optics. The scattered light measured on any one shot i s the order of 10""^ that of the incident laser beam. Any stray reflection that reaches any part of the detection optics w i l l add a significant level to the real scattered signal. In this experiment the optics were kept very simple. A single lens (an uncoated singlet) of 275 mm focal length was used to focus the laser beam into the plasma. The 2 to 3 milliradian divergence of the laser (due to multimode output) produced a focal spot about 0.5 mm in diameter. After passing through the carbon arc chimney the laser l i g h t i s absorbed by a c e l l (with a 37 Brewster window) containing a saturated copper sulphate solution. Thi3"beam dump" was placed about 50 cm from the plasma. If placed any closer to the plasma the copper sulphate solution would absorb enough energy from the arc to b o i l . A coated lens of 175 mm focal length, f 5.6, was used to focus an image of the entrance s l i t of the monochromator into the plasma. The width and height of the entrance s l i t was varied along with the image and object distance of the lens. This allowed the observation of the desired volume of plasma with the appropriate resolution necessary to construct a scattered spectrum. For example, an entrance s l i t 250 uM by 250 uM focussed with an image to object distance ratio of one would observe a volume of plasma 250 uM by 250 uM by 500 uM. The resolution with 250 uM s l i t s is given from the dispersion of 10 8/mm as 2.5 8 . It should be mentioned here that the solid angle of the light cone entering the monochromator was always kept less than that of the monochromator. In this way the light entering the monochromator was incident only on the mirror surfaces. This helped keep the stray light at a minimum. Ports were cut in both sides of 38 the chimney in line with the observation optics. The wall in line with the observation optics behind the plasma was blackened to reduce stray light problems. The angle between the input laser beam and the observation beam could be varied. The monochromator and observation optics bench were mounted as a single unit. A radius ring was set up so that the unit could be rotated and the scattering angle varied from 105° to 1 5 0 ° . Angles less than 105° could be obtained, but because a was so large the signal scattered at this angle was too small to be of use. A schematic of the apparatus is shown in F i g , 3 G. Another arrange-ment of the laser optics allowed a simple check for large anisotropy of the plasma. The axis of the laser beam was t i l t e d 40° above the normal of the plasma arc column. This produced a component of the K vector along the arc axis proportional to the sine of 20° (see F i g . 3 H). Considerable d i f f i c u l t y was encountered in this configuration while trying to work near the cathode. As the cathode was burned and readjusted the laser would hit the cathode surface. This produced impossibly high stray light levels. It was necessary 39 to take measurements 3.5 mm above the cathode (previous work was done at 2,5 mm) to avoid the above d i f f i c u l t y . Recording of Data The signal proportional to the intensity of the scattered light was recorded i n i t i a l l y on Polaroid f i l m , A scope camera was used to photograph the face of a dual trace Tectronics 551 oscilloscope. One trace corresponded to the output of a photodiode which monitored directly the laser output via the 99,9% re f l e c t i v i t y laser back mirror. The other trace monitored the photomultiplier output. A typical scope trace i s shown in F i g . 3 H, The data was obtained from the photographs by measuring the photomultiplier signal at a position corresponding in time to the maximum of the laser output. The time of the maximum of the laser pulse i s given by the photodiode trace. The diode trace i s used as a time mark because the signal to noise ratio of the photomultiplier output is not always good enough to pick out the scattered signal. There is a time delay between the appearance of the two signals (diode signal and scattered light signal) because of the different cable lengths and the Fig 3 G» Schematic of Apparatus 41 Photomultiplier trace (of scattered signal and plasma noise) Diode trace . (of laser output) 100 n sec/cm Fig . 3 H Typical oscilloscope trace. inherent 60 nsec delay in the photomultiplier dynode chain. The absolute difference in the position of the two pulses can be measured during a stray light check. With no plasma to produce background light the position of the laser signal on the photo-multiplier trace can be seen clearly and the time difference between the appearance of the two signals measured accurately. The photomultiplier signal i s normalized to the laser output as measured by the photodiode signal. This i s necessary because of the large variations, shot to shot, of the laser output power. The laser ! > ! • 1 i i i l 1 1 I • I I I " " M M 1 1 M i i i i M M M M M M - M M M M ( I I I - I I I i 1 1 1 1 1 M i 1 i i I 1 1 1 1 1 f 1 > i l f i t i l l I 1 1 I i i I i M i l 1 M 1 M M M M M M M i l 1 —i—i—i -f- I i 1 1 i i i 1 1 M 1 1 1 1 I I I I I I I I I I I I I 42 power could change 20% from one shot to the next. Over a 50 to 60 shot run i t usually decreased by a factor of two. This was due to the deterioration of the cryptocyanine, methanol solution in the laser Q-switch. Because of the cyclic method of taking data this had no systematic eff e c t . One scattered signal was recorded at each wavelength un t i l the whole spectrum had been covered. This process was repeated n times to obtain n signals at each wavelength. Because the photomultiplier trace contains the background signal of the plasma light an estimate of the average background level must be made for each measurement. The signal to noise ratio varied con-siderably, depending on the band pass of the mono-chromator, the part of the spectrum being analyzed, the scattering angle, and the plasma parameters. For most conditions a signal to noise ratio of about 4 to 1 could be maintained. Such a signal would typically contain 20 photo electrons at the photocathode, This produces a signal, after amplification down the chain, of 0.005 volts into 50 «. Each data point plotted was the average of 3 to 10 such signals normalized to the laser output power. 43 The number of signals used to produce an average signal size was a function of the signal to noise r a t i o . For the case where the S/N ratio was 4 or less, 6 to 10 shots were needed to obtain small error bars. For some experi-mental configurations a S/N of 10 to 15 was realized and only 3 shots were needed to produce small error bars. The error bars shown in graphs of intensity versus wavelength for scattered spectra are in a l l cases the standard deviation of the mean. The reduction of the data, normalization, averaging, and standard deviation calculation was done with the aid of a simple computer program. The results were tabulated and plotted in graph form by the computer output. The data in this form could then be f i t t e d to the 30 theoretical curves. The method due to Kegal (1965) was used to f i t the data to the theory (see Appendix 8)• This method consists of plotting the data as: scattered intensity (normalized to a maximum of unity) y_s log (A X) where AX is the shift of the wavelength of the scattered intensity from the laser wavelength. Kegel provides a set of standardized theoretical curves which we can now f i t to our data. Because of the nature of the theoretical function, S (K,w), the choice of a best f i t curve determines the ratio of N_ to T and the shift 44 along the axis between the experimental plot and the theoretical plot determines the absolute value of N e which allows us to calculate Te. This method i s described in detail in Appendix B . CHAPTER IV RESULTS Temperature and Density Profiles This chapter w i l l f i r s t present results using laser scattering as a diagnostic technique. The temper-ature and density of the arc column, in the region of interest for this work, were mapped using standard scattering techniques. This was useful in later work because a particular temperature and density could be looked at by observing a predetermined region of the arc. In a l l cases the enhancement of the thermal o scattering spectrum at 135 scattering angle i s a very small percentage of the total integrated scattering spectrum. Because of this i t i s possible to f i t the recorded spectrum to theoretical curves and obtain values of electron temperature and density with good accuracy. The temperature and density of the arc column were mapped as a function of position and current. At 200 amps arc current 6 axial positions between 0.1 cm and 1,05 cm above the cathode were observed. At 400 amps arc current 8 axial positions between 0,15 cm and 1,60 cm above the cathode were observed. For the 400 amp arc 6 radial posi-tions from r = 0,0 cm to r = 0,25 cm were observed at 46 the height z = 0.25 cm above the cathode. The volume of plasma observed in the above cases was defined by the width of the focussed laser beam (>,5 mm) and the size of the image of the entrance s l i t . The magnification of the observation optics also affects the observation volume but this was set at unity (image distance equals object distance). The entrance s l i t was set at 0.5 mm wide by 0.2 mm high. This gave an observation volume approximately 0.5 mm on a side and 0.2 mm deep (along the arc axis). This gives excellent axial spatial resolution. This choice of observation volume dimensions i s wider and less deep than i s used in later parts of the work. The diagram (see F i g . 4 A) shows the scattering volume configuration for 135° scattering. The entrance s l i t width of 0.5 mm coupled with an exit s l i t of 0.5 mm gave a pass band of 5 X , This choice of pass band allowed the spectrum to be resolved without the need of deconvolution. The spectral data could be f i t t e d directly to theoretical curves. The 5 8 pass band was wide enough that the anomalous "bumps" in the spectrum were not usually resolved. A typical f i t t i n g of the data to theoretical curves i s shown in F i g . 4 B. Each point and error bar 48 is the average and standard deviation of the mean of three data points. The wavelength i s plotted on a (log^g A X) scale following the f i t t i n g method due to Kegel (see Appendix B ) . The results of similar f i t t i n g s for the positions and currents mentioned earlier are shown in Fig. 4 C, 4 D, and 4 E. The position z = 0 i s the t i p of the cathode and the position r = 0 i s the axis of the arc. The accuracy of the theoretical f i t t i n g to the data should be discussed here. The theoretical curves are characterized by the parameter a . The value of a is proportional to (Ne/Te) 2 where NQ i s the electron density and T is the electron temperature. The theore-e t i c a l curves used were plotted in steps of a of 2,5% in the range of a?s occurring in this experiment. This means the value of N /T i s changed 5% in each step. e e The second parameter used in f i t t i n g the data to the theory is the shift of the wavelength scale of the data with respect to the theoretical wavelength scale. This can be determined with a reproducibility that produces a 5% range in the values of N^. This also produces a 5% range in the value of T obtained. If we e assume these errors are independent the values of both N£ and T£ have a probable error of t 7%, This assumes 1-0 - T H E O R E T I C A L FIT n e = 7:4 x 10 FOR j i c m " 3 / TENSITY o ob - T e = 1-62 x 10 a = 1-85 o E X P T °K J -5 0-6 — T H E O R Y -UJ > < 0 - 4 IJJ tr 0-2 -0 0 i i i i ^ f i 2 0 3 0 4 0 5 0 6 0 8 0 100 AX (A) F i g . 4 B 50 the f i t t i n g is uncertain between only two theoretical curves. This also assumes no systematic errors. The graphs of F i g . 4 C and 4 D show the plasma parameters as a function of position along the arc axis. Because the cathode was constantly being burned and readjusted the observation volume position above the cathode was in doubt in each case. It i s estimated that the position of the observation volume with respect to the cathode t i p could be kept in the range - 0.01 cm. This was accomplished by projecting a magnified image (magnification of 3) of the cathode onto a screen and keeping the position of the image on the screen constant with respect to cross hair l i n e s . Two orthogonal projections were used. The position of the arc could then be kept constant using the three orthogonal adjust-ments mentioned in Chapter I I , The adjustments were made while the arc was burning just prior to each f i r i n g of the laser. The estimated error for each point on the graphs of Figs, 4 C, 4 D, and 4 E i s t 7% on the temperature and density, and t 0,01 cm on the position. It should be noted that the geometry of the arc E L E C T R O N T E M P E R A T U R E T e ( l 0 4 o K ) — ro O J > > <: <: L Fig. 4 D £5 54 was chosen similar to Maecker's because this was a well studied configuration. However, i t is apparent from the results of the scattering technique that the electron temperature of the arc is very much higher than that reported by Maecker. Maecker's results are included in Figs. 4 C and 4 D for comparison. The hot spot at z = 2.5 mm was not resolved by Maecker possibly due to the Abel-unfolding technique used to obtain axial parameters. The electron density is not very different from Maecker's values but shows a peaked high density spot at the same axial position as the temperature p r o f i l e s . The drop in temperature and density near the cathode is quite definite in both the 200 amp and the 400 amp cases. The radial temperature and density profiles show that gradients over the observation volume are small. Symmetry of the Scattered Spectrum 13 15 18 Several authors ' ' ' ' have reported asym-metries in the scattered spectrum between the high frequency side (Blue shifted) and the low frequency side (Red shifted). The theory for an isotropic plasma predicts that the red and blue sides should be mirror images for a fixed value of K. 55 In this section the results of a symmetry check on the profiles for the arc plasma w i l l be presented. The frequency integrated intensity of the scattered spectrum was measured on both sides of the laser frequency. A monochromator entrance s l i t of 100 uM and exit s l i t of 2500 uM were used. This gives an instrument profile of 25 8 that i s essentially square. Intensity measurements were taken at 25 A* intervals on each side starting at the laser wavelength. Excluding the laser wavelength for which no measurements were taken, four intervals of wavelength were measured on each side. Because of the 25 A* steps and the square 25 R pass band, the measurements are totally independent. The total intensity of each side is then obtained by summing the individual measurements. The spectrum scattered from a 200 amp carbon arc 2.5 mm above the cathode was measured using the above configuration. The scattering angle was 1 3 5 ° . Each data point plotted i s the average of 8 shots. The results are plotted in Fig . 4 F, the error bars are standard deviations. The difference in the areas of the two sides i s 22%. The spectral response of the instrument must be calibrated in order to see i f this 22% difference i s a l l or part instrumental. 56 Using the same configuration of optics and s l i t s and photomultiplier voltage as above the light output from a tungston ribbon was measured. The standard temperature light source tungston ribbon lamp was run at 14 amps. This gives a temperature of 1950°K (calibrated by an optical pyrometer), At this temperature the difference in emission between the two major peaks on Graph 4 F (-758 and 75 8) should be about 5%, being brighter on the red side (+75 8); The following table gives the voltage output vs, wavelength for the system using the above light source. Table I Voltage Output vs Wavelength  Wavelength Voltage (IO"1 volts) -100 0,47 i 0.01 - 75 0.46 - 50 0.45 - 25 0.44 0 (6943 8) 0,43 25 0.42 50 0.41 75 0.39 100 0.38 Here the percentage difference between + 75 8 and -75 A C/) 21 •1.0 •I •5-5 -CQ ce < _ ; I/) z LU £ + . 5 •o--100 - 7 5 - 5 0 - 2 5 0 L a s e r A. AA(A) 25 50 75 100 Fig. 4 F Scattered Spectrum Symmetry Check 58 i s 16%. The slope of the emission curve for tungston at 1950°K at this wavelength (A=6943 8) adds another 5% to this which gives a total of 21% variation between the red and the blue side. Within the experimental error this accounts entirely for the observed variation in intensity. We conclude that the scattered spectrum is symmetric within the limits of the detection system. Enhancement at the Plasma Frequency One of the anomalous features which was observed was the enhancement above the thermal spectrum of experimental points at a wavelength shift corresponding to that of the plasma frequency "Wp"• This is perhaps the most commonly observed anomaly. In order to map this feature one scattering angle v/as chosen, 6 = 1 3 5 ° , and the electron density observed was changed from o.93 X 101 7 cm'3 to 1.77 X 1017cm"3 by observing different parts of the arc column at different currents. The observation volume was kept small (200 mm x 250 mm x 500 uM) in order to minimize gradients in Ng and Te« Only a small part of the spectrum was mapped around the wavelength shift corresponding to _)p. This was f i t t e d to the theoretical curve with the aid of the knowledge of the electron density as a function of 59 position obtained from previous measurements. The difference between the thermal theoretical curves, IT (AX), and the observed spectrum, I (AX) was plotted for each wavelength in the spectra. This was done for each electron density measured (see F i g . 4 G). In order to better see the functional relation between the enhancement and the electron density, a plot was made of the wavelength shift of the enhance-ment vs the plasma frequency shift calculated from the value of N obtained previously. The bars on this e graph are the estimated f u l l width at half intensity limits of the enhancement. The straight line i s the theoretical f i t t i n g for the enhancement occurring at io (see Fig, 4 H) . It seems conclusive that for this P particular value of K (for 0 = 135°) an enhancement exists at a shift corresponding closely to the plasma frequency. It is interesting that the enhancement occurs at "wp" for this particular K (0 =135°) over such a range of N^, It w i l l be seen later that the enhancement is a very sensitive function of K, Enhancements at to , -^to^, hap 14 1 It i s noted in the introduction that two authors ' 60 4 G Enhancement of the thermal spectrum as a function of wavelength shift (from 6943 8) and electron density. (The dotted line i s the wavelength shift of. the plasma frequency.) 61 observed enhancement at h w p . Ringler and Nodwell observed h w p "bumps" along with c o p , 2 c o p , and 3 cop bumps. Ludwig and Mahn reported bumps at N/2 cop for n = 1 to 6. Both experiments were done on a magneti-cally stabilized arc. In order to check for the existence of similar enhancements complete spectra of the scattered light were compiled at a scattering angle of 135° with high spectral resolution. In the spectral range u > _p no enhancement could be detected. Because a was approximately 2.0, j cop would be very d i f f i c u l t to see. The position of the -i <_. enhancement would 2 P correspond closely to the already narrow, sharply peaked electron feature. This would make i t almost impossible to resolve any small enhancement close to the peak. The position of a 2 wp enhancement would be beyond the thermal spectrum. In theory for the K observed no waves exist with the frequency 2 cop or near the frequency 2 „p. If waves existed in the region 2 cop they could easily be resolved. No enhance-ment was found at 2 co in this work. P The region co < to was also carefully studied. Fig.4 H Wavelength Shift of Enhancement vs. the Plasma Frequency 63 Not only was an enhancement found at to but also at P h wp and h wp. There was considerable d i f f i c u l t y in resolving the bumps at h t o p and h t o p . The size of the signal was small and the resolution necessary to see the enhancement reduced the signal to noise rat i o . It was also necessary to carefully measure the stray light level and subtract i t from the observed signal. The data required about 10 shots for each point to reduce the error bars to a significant l e v e l . Checks of the spectrum in the region 0 to 10 X were impossible because of a grating ghost at 8 8. Two spectra were recorded at two different densities to check i f the position of the enhancement would move in such a way as to remain at h <op and k wp. The two spectra are shown in F i g . 4 I and 4 J . The line is the theoretical best f i t for the parameters shown. The stray light has been subtracted. Enhancement as a Function of K To this point a l l measurements have been taken at 0 = 1 3 5 ° , This defines a K vector observed in the plasma of 1.67 x 105cm""1, or an observed wavelength of 3760 8. This is determined by the laser wavelength and scattering geometry. To vary the K vector, the INTENSITY (ARBITRARY UNITS) ro O) Co J I ' L V9 8 A R C C U R R E N T = 3 9 0 A M P S ~l 1 i 1 1 — 20 40 a 60 80 100 AA(A) Fig. 4 J 66 scattering angle 0 i s varied. The resulting K is given by: 4 IT Q K = — sin ( \t ) AL where XL i s the laser wavelength. In this experiment the monochromator, photomultiplier and input optics were a l l mounted as a unit and could be moved to any angle from 0 = 140° to 0 = 1 0 0° , This gives a varia-tion in K from 1.47 to 105 cm"1 to 1.70 x 105cm"1. Ten different values of K were chosen and the spectrum around w was carefully studied. The electron density p in the observation volume was kept as constant as possible by carefully controlling the position of the arc. The "hot spot" 2.5 mm above the cathode was chosen as the best place in the plasma to observe because i t was a maximum in temperature and density and the gradients would be at a minimum. This position in the arc column is also easy to keep fixed in space because i t is close to the cathode which is adjustable. The density and temperature at this position had been determined previously by laser light scattering and this information was used in f i t t i n g the data to theory. Because the plasma parameters N and T along 67 with the scattering vector K define a# the spectrum could be f i t t e d to a predetermined a. The f i t t i n g to a predetermined a was necessary because the whole electron feature spectrum could not be obtained easily. Because of the short l i f e of the arc and the high spectral resolution (2 8. band pass) necessary to resolve the anomalies, only about 9 data points could be obtained on each run. These covered only a small section of the spectrum. The part of the spectrum mapped could be f i t t e d to theory well enough to observe slight variations (probably due to slight errors in positioning the observation volume) in plasma parameters. A l i s t i n g of plasma parameters for each scattering angle i s given in Table I I . The difference between the measured spectrum and the best f i t to the theory for each K observed was plotted, A three axis graph i s given containing these results (see Fig, 4 K), Each spectrum plotted (as a function of wavelength shift) i s the difference between the theoretical curve (normalized to unity at the theoretical maximum) and the measured spectrum. The lines joining the points are added to aid the eye in determining which points belong to which spectrum. The third axis is the wavelength of the waves observed 68 in the plasma given by The dotted line is the plasma frequency shift calculated from the electron density. A larger wavelength interval was studied in the region 5 -1 below 1.55 X 10 cm , because bumps began to appear in more places as the wavelength of the plasma waves observed increased (K decreased), The data of F i g . 4 K suggests two other plots. The area under each enhancement can be plotted as a function of K. The units of area used are a function of the maximum of the thermal spectrum f i t t e d to the data. In each case the maxima of the thermal spectrum i s normalized to unity. The difference between the data and the theor-e t i c a l curves is then IE x pU ) - IT n e(A) = where the maxima of the theoretical spectrum (for the electron feature) has been normalized to unity. The width of the enhancement i s in Angstroms. The area measured i s that area under the straight lines joining adjacent points. The area is plotted as a function of K in F i g . 4 L. It would be interesting to continue mapping the size of the enhancement as K is decreased but this would be d i f f i c u l t . The total energy contained in the spectrum 2 of the electron feature is proportional to l /a ; as 69 Table II Plasma Parameters for 135 Scattering 17 3 NgjlO cm" ) _Te(104 °K) 140 2.3 1.57 2.2 135 2.2 1.77 2.74 127 2.3 1.67 2.5 125 2.3 1.64 2.67 122 2.3 1.65 2.65 120 2.3 1.65 2.65 117 2.4 1.57 2.6 113 2.4 1.63 2.6 110 2.4 1.63 2.7 105 2.5 1.63 2.9 © i s decreased a i s increased because K i s decreased; because: a = ( K *_.) * and: K = (4TT/X t) sine/2 70 Fig. 4 K Enhancement of the thermal spectrum as a function of wavelength shift (from (6943 8) and scale length K. (The dotted line is the wavelength shift of the plasma frequency.) X 72 Therefore the total intensity of the electron feature depends on |K|2 . At the same time the width of the electron feature is decreasing. These two factors make i t very d i f f i c u l t to resolve any enhancements in the spectrum. The other interesting plot of the information contained in F i g . 4 K i s the standard to vs K plasma wave dispersion plot. The wavelength s h i f t , AX» i s a direct measure of the frequency of the waves and the scattering angle gives the value of K. The vertical bars in F i g . 4 M are the estimated frequency widths at half intensity of the enhancements. The lines in F i g . 4 M give the position of the enhance-ments produced by the theoretical model described in Chapter V, The significance of this f i t w i l l be discussed in the next chapter. Anisotropy Check A l l measurements done on the arc plasma up to this point have been done with the K vector oriented normal to the axis of the arc. In order to check that the anomalies were not due primarily to very strong axial waves the orientation of the K vector was changed. The laser beam was aimed down into the plasma at an angle o of 50 to the arc axxs. The observation axis was l e f t normal 2.8 2.4-C o 13 -1 10 sec ) 2.0H 1.6 H i i I>"5 model | experiment i • 1.5 1.6 K(105cm ) Fig. 4 M Disperion Plot of the Enhancements — T — 1.7 74 to the arc axis but was positioned so that the o scattering angle was. 135 (see F i g . 4 N) . At this scattering angle previous results show a co enhancement. The component of K along the arc axis with this geometry is 0.34 |K |. Any large difference of amplitude in the anomalous waves in the horizontal and the vertical planes should become apparent in this scattering con-figuration. Because the laser beam was aimed down, measurements could not be taken close to the cathode. The observation volume was moved from 2,5 mm to 3.5 mm above the cathode so that the laser beam would not hit the cathode. This changed the plasma parameters as noted in F i g . 4 P. The data in F i g . 4 P is presented as before. The difference between the measured spectrum and the best f i t to theory i s plotted as a function of wavelength s h i f t . This result i s similar . to the previous results and indicates that the plasma is not strongly anisotropic. 75 Fig. 4 N Scattering Configuration of the Anisotropy Check 76 i £ <—• o O I i 1 X L O II £ £ LO X . o ^. m ° _ CM CO CD 11 CNl QO j j ? CD _3l M L O C D ft. In CV)lI-(V)I 77 CHAPTER V CONCLUSIONS The f i r s t point that should be discussed i s the relative enhancement. The plasma temperature and density were measured in the arc column by f i t t i n g the experimental spectra to theoretical thermal spectra. We can show that the deviations from the thermal spectra are small enough to be of l i t t l e concern in the , general f i t t i n g technique. The area of the enhancements was typically 1% (at 0 = 135°) of the total area of the electron feature. The enhancement exists over a 2 or 3 A" range of the spectrum. If a 5 X pass band in the detection system i s used to measure the shape of the spectrum, the enhancement is distributed over a 5 8 to 10 X range. Over such a large pass band the enhancement increases the signal size by not more than 10% at a particular wavelength. Considering that the error bars on the signals measured were about 10% of the signal size, we would expect the enhancement to have l i t t l e effect on the plasma parameters obtained by the f i t t i n g technique. The data often showed one point about 10% higher than the theoretical curves at a wavelength close to the plasma frequency s h i f t . This did not significantly 78 change the parameters obtained by the f i t t i n g procedure. We also concluded from the results that the scattering spectrum was symmetric about the laser wavelength. From the above two points we conclude that no systematic errors were present and therefore the temperature and density measurements were accurate within the experimental error stated earlier (- 7% on N e and T ) . e The Enhancements Observed and the Theoretical Model In this section we wish to speculate that, using the two distribution function model developed in Chapter I I , theoretical profiles with "anomalous" peaks can be constructed and f i t to the observed spectra. Under certain assumptions a theoretical dispersion curve may be drawn similar to that observed in Chapter IV, The general characteristics of the model w i l l then be discussed. In Appendix C i t i s shown that the spectral power density, S (K,u), can be calculated for a two distribution plasma where one distribution d r i f t s a r b i t r a r i l y with a velocity v^. It is also shown that only the component of vQ along the K vector of the observed plasma wave contributes to S (K,w), A d r i f t velocity (fixed with respect to the arbitrary frame of reference) i s 79 postulated to exist in the plasma. If the scattering angle, 0 , i s varied by moving the observation optics s v/ith respect to the plasma, the direction and magnitude of K w i l l be changed. If the direction of K i s changed, the component of v along K w i l l change. It i s a simple T D "~ matter to calculate the component of v^ along K for each scattering angle, and to construct the scattered p r o f i l e . The frequency shift of the calculated anomalies in the power spectra can then be plotted against K to give a theoretical dispersion curve. The experimental results reported in Chapter iv show a dispersion curve with two secondary branches. In order to obtain a dispersion curve with two branches from this theoretical model i t is necessary to postulate two different d r i f t velocities in two different directions. This requires a distribution function with three components: f (v) = (1-a-b) f (v) • ° — Maxwellian This leads to the construction of the G functions as: G (f , f . , f0) = (1-a-b) G_(f ) + aG (f,) + b G, (f,) o 1 2 O o 1 /. z 8© The r e s t of the c a l c u l a t i o n procedes as b e f o r e . In o r d e r t o o b t a i n a f i t t o the d i s p e r s i o n curve r e p o r t e d i n Chapter IV the d i r e c t i o n and magnitude of the two d r i f t v e l o c i t i e s were chosen as f o l l o w s : | v j = 1.6 X 108 cm/sec vQ^ a t 1 8 0 ° t o (back toward laser,) 8 |v^ = 2,1 X 10 cm/sec v^9 at 1 1 5 ° t o K, The geometry i s shown i n F i g . 5 A; a l l v e c t o r s are i n the same p l a n e . The d o t t e d l i n e s i n F i g . 2 B from v^ and y_2 t o K show the component and alo n g K, The c o o r d i n a t e system i s chosen such t h a t the x a x i s i s p a r a l l e l t o K, U s i n g the above v a l u e s o f v ^ , assuming t h a t the secondary d i s t r i b u t i o n s c o n t a i n .00015 of the e l e c t r o n s (a = b = .00015), and assuming t h a t the secondary,temperature was .005 t h a t of the main plasma. (T-j/T = .005), the f u n c t i o n S(K,w) was c a l c u l a t e d f o r e i g h t v a l u e s o f 0^. The s c a t t e r i n g F i g . 5 A Orientation of Drift Velocities with Respect to the Scattering Geometry 82 angle v/as varied from 105° to 140° in steps of 5 ° . A sample of S (K^co) i s plotted in F i g , 5 B, The dotted line is for a thermal spectrum (a = b = 0) , The position of the secondary peaks was noted in each case and the frequency shift of the peak was plotted vs K (see F i g . 4 M), The frequency plotted here i s to , - to ; the K plotted i s K . - K . This is the i s — — i — s dispersion curve that best f i t s the data presented in Chapter IV. The anomalies studied were found to depend on both the density of the plasma and the scale length ( 2 I T / K ) of the waves in the plasma. We w i l l now look closely at those aspects of the model developed in Chapter II that either agree or disagree with the observations. The model was constructed in such a way that i t would f i t the dispersion curve obtained experimentally. The orientation and magnitude of the two d r i f t velocities were varied u n t i l a good f i t was produced. The percentage of electrons moving with V p and v-^ was chosen along with the temperature T^ and T2 to give a reasonable width and height to the -theoretical peaks. The graph (Fig. 4M) shows that a reasonable f i t t i n g of the model to the results can be A A ( A ) Fig. 5 B Theoretical Profile for two Drift Velocities 84 obtained for two fixed d r i f t velocities v and -DI v . The f i t t i n g is quite sensitive to the choice -D2 of the direction and magnitude of the d r i f t v e l o c i t i e s . Good agreement between the experimental results and the model are obtained for the area of the enhancement as a function of K. This is d i f f i c u l t to measure quantitatively because of the nature of the model. As can be seen in F i g . 2 A there i s no net enhancement of the total spectrum. The model shifts the energy in the waves to a lower frequency, leaving equal regions of enhanced and damped waves. To this point i t has always been considered that the bump has been an excess of waves over and above the normal thermal scattering spectrum. The results have always been f i t t e d to theory on this assumption. If the model we propose i s correct, a small systematic error could occur in the f i t t i n g of the partial spectrum to the normal curves for a thermal plasma. In the region above the secondary peak (Fig. 2 A), the spectrum of our model is slightly below the spectrum of thermal fluctuations. If the experimental spectrum i s shaped like the model but f i t t e d to the thermal theoretical curves, a slightly lower temperature and density would be calculated than really exists. This is due to the error i n estimating the relative frequency 85 s h i f t (see Appendix B). It should be noted that the determination of the temperature and density by f i t t i n g the f u l l spectrum to the curves for the thermal theory would not be susceptible to this systematic error. In this case the main peak of the spectrum can be used to determine the shift relative to the theoretical spectra. The position and height of the main peak is not affected in the model by the addition of the d r i f t . The partial spectra used in determining the K dependence produce a consistently lower temperature and density (see Table II) than is determined from the gross scattering spectra (see F i g , 4 M and 4 N). The difference is quite small ( 5%), but consistent. The possible reasons that the region of damped waves was not noticed as being below the spectrum are: poor resolution due to the large instrument profile (2 8) needed to obtain a signal, and shot to shot plasma variations. These two factors coupled with the high a, ( a = 2.2 - 2 , 4 ) , in the region studied for the K dependence make i t unlikely the damped region would be resolved. However, in the region of lower a ( a= 1,9) the dip is possibly resolved in a few cases. In Fig, 4 G two spectra show a dip on the high frequency side. The 86 The spectrum for N = 1,48 X 10 cm shov/s a distinct dip at 57 8 . The spectrum for Ng = 1.48 X 101 7 cm"3 shows a dip also on the high frequency side. No other spectra on this graph show a significant dip. It i s interesting that there i s no significant dip on the low frequency side of the spectrum for any of the recorded spectra. The model constructed seems to be capable of reproducing the size and position of the anomalies observed in the K dependence spectra. If the model is to predict the size and position of the anomalies in the density dependence spectra, a new factor must be considered. It was shown that for 135° scattering (K=1.6 X 105 cm"1), the position of the bump was a function of the plasma frequency. This means the d r i f t velocity of the secondary electrons v^ must vary linearly with the plasma frequency and therefore must vary as Njs . We are unable to postulate a mechanism that would create electrons with a temperature and d r i f t velocity similar to those used in the model. We are also unable to postulate why the electrons should have a d r i f t velocity dependent on the plasma frequency. It is speculated that the anomalous features are laser induced. This is thought to be the case 87 because of the orientation of the d r i f t v e l o c i t i e s . We can think of no mechanism in the plasma which would produce such d r i f t v e l ocities. Ringler and 15-17 Nodwell concluded that the anomalies observed in their work were not laser induced. This implies that different mechanisms are producing the anomalies in each experiment. It is also speculated that the plasma parameters are important in describing the mechanism that produces the anomalies. The model presented in Chapter II of the thesis is not considered to be the only possible explanation, but does correctly give the functional K dependence of the anomalies. For this model to also include the presence of theh to and %to bumps, hvn and Wp. c p p —D —D velocities would need to be included in the same directions as the f i r s t two. The functional dependence of these could then be checked experimentally. In that this work does not lead conclusively to the origin of the anomalies, a few comments on possible future work w i l l be made. F i r s t l y , the present experiment could be greatly improved. The rate of data acquisition is very low. A multi channel spectral 33 "\& analyzer of the type used by Rohr (1967), Kronasf5*(1971), 88 or Albach (1972) would allow very many more shots to be taken for each wavelength. This would greatly reduce the error bars and allow much more accurate f i t t i n g of the data to theoretical models. The data presented in this work was not complete enough to determine accurately the shape of the anomalies. With the above improvement a greater range of K vectors could be studied. Presently K can be varied only about 15%. A greater range of K would help determine i f the model has the correct angular dependence for the anomalies. Secondly, a great deal more information could be obtained i f a variable frequency Dye laser was msed. If a variable frequency incident light source were used, the dependence of the anomalies on the value w = v /K/ could be checked because the anomaly —D — value of /K/ could be changed. This should change the value of w , , anomaly More information i s needed over as large a range of plasma wave frequencies and K values as i s possible. Without this information i t i s d i f f i c u l t to intelligently postulate mechanisms which could produce anomalous waves in the plasma. 89 Bibliography 1. Bowles, K. L. 1958. Phy. Rev. Lett. 1 (12): 454. 2. Fiocco, G. and Thompson, E. 1963. B u l l . Am. Phys. Soc. 8 (2): 372. 3. Davies, W. E. R. and Ramsden, S. A. 1964. Phys. Lett. 8: 179-180. 4. Dougherty, J . P. and Farley, D. T. 1966. Proc. R. Soc. A 259: 79-99. 5. Fejer, J . A. 1960. Can. J . Phys. 38: 1111-1133. 6. Salpeter, E. E. 1960. Phys. Rev. 120: 1528-1535. 7. Salpeter, E. E. 1961. Phys. Rev. 122: 1663-1674. 8. Rosenbluth, M. N. and Rostoker, N, 1962. Phys. Fluids 5: 776. 9. Perkins, F. and Salpeter, E. E. 1965. Phys. Rev. 139 (IA): A 55-A 62. 10. Gerry, E. T. and Rose, D. J . 1966. J . Appl. Phys. 3_7: 2715-2724. 11. Evans, D. E. et a l . 1966. Nature 211: 23-24. 12. Kegel, W. H. 1970. Plasma Physics 12: 295-304. 13. Kronast, B. 1972. Private communications. 90 14. Ringler, H. and Nodwell, R. A. 1969. Phys. Lett. 29A: 151. 15. Ringler, H. and Nodwell, R. A. 1969. Third Europ. Conf. on Contr, Fusion and Plasma Physics, Utrecht, 16. Ringler, H, and Nodwell, R, A. 1969. Phys. Lett. 30 A: 126. 17. Ludwig, D. and Mann, C. 1971. Phys. Lett. 35A:191. 18. Neufeld, C. R. 1970. Phys. Lett, 31A: 19. 19. John, D. K. et a l . 1971. Phys. Lett. 36 A: 277. 20. Infeld, E. and Zakowicz, W. 1971. Phys. Lett. 32 A: 103. 21. Maecker, H. 1953. Z. Physik 1361 119. 22. Bekefi, G. 1966. Radiation Processes in Plasmas, John Wiley and Sons. 23. K r o l l , N. et a l . 1964. Phys. Rev. Lett. 13: 83. 24. Stansfield, B. L. PhD Thesis (1971). The University of British Columbia. 25. Tanenbaum, B. C. 1967, Plasma Physics, McGraw-Hill, 26. Fried, B, D. and Conte, S, D. 1961. The Plasma  Dispersion Function, Academic Press. 27. Muller, G. et a l . 1962. Z. Physik 16 9: 273. 91 28. Wienecke, R. 1956. Z. Physik 146: 39. 29. Ahlborn, B. and Wienecke, R. 1961. Z. Physik. 16 5: 491. 30. Kegal, W. H. 1965. Internal Report, Institut fur Plasma Physik, IPP 6/34. 31. Grewal, M. S. 1964. Phys. Rev. 134 (lA): 86A? 32. Rose, D. J . and Clark, M, 1961, Plasma and  Controlled Fusion, The M.I.T. Press. 33. Rohr, H. 1967. Institut fur Plasma Physik, IPP 1: 58. 34. Kronast, B, and Pietrzyk, Z. A, 1971. Phys. Rev. Lett. 26 (2): 67-69. 35. Albach, G. G. 1972. M. Sc. Thesis, University of British Columbia, 36. Churchland, M, T. 1969, M. Sc. Thesis, University of British Columbia. 92 APPENDIX A The spectrum of light scattered from a plasma i s calculated in Chapter II under the assumption that the plasma i s collsionless. We must check that this i s a valid assumption for this experiment. The effect of collisions on the spectrum of electron density fluctuation in a plasma has been 31 studied by Grewal (1964). He shows that when an electron in a density wave travels less than ten wavelengths before suffering a c o l l i s i o n , the scattered spectrum i s affected. We must calculate the distance an electron in the density wave travels before suffering a c o l l i s i o n and compare this to the wavelength of the density wave. The most probable c o l l i s i o n in a f u l l y ionized plasma i s an electron-electron interaction due to coulomb forces. A calculation of the relaxation time for multiple coulomb interactions i s given in 32 Rose and Clark. We w i l l use the f i n a l result which they quote. 93 T 9 0 where is the permittivity of free space m i s the mass of an electron K is Boltzman's constant T is electron temperature q is electron charge n i s electron density A i s 9 where i s the number of particles in the Debye sphere 4 o for T = 3 x 10 K 23 -3 and n - 1.7 x 10 M -12 T = 3.9 x 10 sec. 9 o The speed of the electrons that compose the wave is given by the phase velocity, co/K, of the wave. In our case this i s about 2 x 10^ m/sec. Electrons travelling at the above speed w i l l therefore travel 4 o 7.3 x 10 X between c o l l i s i o n s . The wavelength (chosen by the scattering geometry) that we observe is about 4,0 x 103 A*. This means, on the average, 62/TT e2 mJ (<T) 2 q1* n In A 94 that the electrons travel 20 wavelengths between c o l l i s i o n s . According to Grewal the effect of collisions w i l l therefore be negligible. 95 APPENDIX B Fit t i n g Technique Kegel's method of f i t t i n g theoretical curves to the data is used i n this experiment. This method-is published in the internal reports of the Institut Fur Plasma Physik, which are not readily available. There-fore a brief description of the method i s given. The integral in equation ( i o ) (Chapter II) can be 25 put in the form (see for example Tanenbaum , p. 181): K G(K,u) = - ? ~ Kz 1-2C (exp {Z2-C2}dZ + iTrexp{-C2} where: C = a>/K (2<T/m) 2 2 2 and K /K = a D (as defined earlier) The G+ (K,w) term is the same as QA) except for the mass of the ion in C and the replacement of KQ with K^ + The form of S (K,w) i s then given by the sum of terms like fp/j producted with a Gaussian term of the form: (2KT/m)~^exp{-C2} 9 6 If the parameters contained in a are fixed (so that a is fixed) the spectrum S (K,to) can be plotted as a function of C. Each value of a produces a curve S (K,co) which can be plotted on a dimensionless C axis (see for 22 example Bekefi ) . In fact there are two a terms, a and cc+ where ot+ is given by K^+/K, The a+ term affects the part of the spectrum near the central laser frequency and i s related to the ion term in the scattering. We can plot S (K,io) as a function of log^C for a given a. Recall now we can write: to K 2KT m © l o g i o c = I o 9 i o ( u ) ) + I o 9 i o { 2 K T / m ) " " ) S ) ® From this we see that spectra of the same a plotted on a logarithmic scale of frequency w i l l d i f f e r only by a shift of the scale of frequency given by the last term of (jiT) , We can use this to determine the value of ct for an experimentally recorded spectra. The experimental data can be plotted on the same log (co) 10 scale and normalized to the same maxima as a series of theoretical curves covering a range of a. 97 The curve that best f i t s the data, independent of the value of 1°9^Q (u)# gives an experimental value of a. Once this value has been chosen, N and T can e e be calculated. Recall that curves of the same a have a maximum at the same value of the parameter C, We can relate the experimental and theoretical curves by: l o gl 0 f o r a m a x i m u m) =  1O<3IQ *wth* + l o gl 0 2KT, m = l o g1 Q (ue x) + log 10 2KT m 1 K ex We can better relate this last equation to the experiment i f we r e c a l l : © / 2 T r C \ \ X2 j 0 where A X i s the observed wavelength shift and X is the laser wavelength also: K 4 TT 0 X s i n 2 where 0 is the scattering angle. If the theoretical 9d curves are constructed for the same laser wavelength X as used in the experiment then (E) reduces to: ex _ •ch A X ex th [sin2 ( 9e x/ 2 ) ] [sin2 (Gth/2)] ® The value of T is the only unknown factor in H . The axis of the theoretical curves can now be converted to units of A X (from F) which are the experimentally recorded units. We also make use of the definition of a : a = R N. (sin2 1) where R i s a constant of fixed parameters. If we consider that a is by definition the same for both theory and experiment, (tf) and (T) reduce to: N (ex) e Ne(th) A X es A X th J © 99 This gives us experimental values for both NQ and using the known values of 0 and X and the best f i t to theoretical plots on a log^n ^ * scale. The actual evaluation of N and T i s usually e e done using equation (^) and (j^ in the following form (take log^g ° ^ k °t n sides); l o g1 0 Te x - 2 A - l o g 1 0 (sin2 0ex/2) + l o g i o T t h + l o g i o ( s i n 2 0t h/ 2 ) The last two terms can be put into numerical form because both T., and 0,. are fixed. The A i s just th th J the shift on the l°g^Q (A X) scale between the theo-r e t i c a l and experimental spectra. Similarly: l o g10 Nex = l o g Nt h + 2 A Theoretical curves for a in the range 0.1 to 3.0 are given in Kegel's report. 100 APPENDIX C To calculate S (K,w) f o r the d i s t r i b u t i o n given by equation' (16) (Page 19) we use equation (l4) . We assume the ions have a Maxwellian d i s t r i b u t i o n with a temperature equal to the main electron d i s t r i b u t i o n . We chose to calcu l a t e the spectrum f o r the K vectors p a r a l l e l to the x axis (K J| i v , where i i s the unit vector along x) . Recall now equation (9 j for G ( K , t o ) . Because the d i s t r i b u t i o n function f (v) i s written as: G ( K . w ) = — K 2 iop f K.8f ( v)/9v t o - K * v f(v) = (1-b) f Q (v,T) = b f1 ( (v - v D ) , T ) we can write G ( K , t o,f (v)) as: The f i r s t term G ( f Q ) i s given by: 101 G(K,W) = - 2 u z e_ u^K2 a 5 f oo . 2 — 2 \ v' exp(-v£a d3v. v -aC X exp(-v^a~2) dy exp(-v2a""2)dv. where: a = (2<T/m) and: C = (w+iv)/ K a The second term in (IT) can be obtained from (jT) with / N f i of the form of the f i r s t term on the right of (16j : K«3f(v-v )/3v ^ J = D d3y w-K *v We can do this in general i f we recall £||^v x so that K • v = Kv x Now: 8f (v-v^/av = 2(v-vD)a"2f (v-vD) 102 K . 3 f ( V - V Q J / S V This allows us to write in a form similar to (18^  and again the v and v integrals give (a it**) for a nY v« r ; a n <^ G^ becomes: Gl -- 2 w 2 a T r V a3 fTvx"vxD>exP^a"2 <"<vx-vxD)2)} dv vx-ac x where f v and / v have been completed, y z To do this integral we make the substitution V - Vx - vDx dv = dv x - 2 t o 2 f (v)exp{-v2a"2) then: G, = v+(vxD-aC) dv 103 This is the same as (17) with the change of the constant in the denominator. This is a standard integral and 25 can be evaluated (see for example Tanenbaum section 4,5), The integral in manipulated into a form whose value can be closely approximated by a series. The series 2 6 has been summed and tabulated by Fried and Conte (1971) for a range of values of the constants in ( i j ) or (23) , Once values have been obtained for each of i- <y —- ^ — © of S (K,w) can be calculated. the terms G.. in (14) (like (17) and (23) ) the value 


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