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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 B r i t i s h Columbia. 1967  M.Sc,  University of B r i t i s h Columbia. 1969  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS We accept t h i s thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA November, 1972.  In presenting this thesis in partial  fulfilment  of the requirements for  an advanced degree at the University of B r i t i s h 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 B r i t i s h Columbia Vancouver 8, Canada  Date  Dec.  22  .  1972  ii  Abstract The spectrum of Ruby l a s e r l i g h t 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 i n the spectrum of l i g h t scattered were observed.  Their frequencies were shown  to be a s e n s i t i v e function of the plasma v/ave scale length K, and the plasma density and temperature.  A model i s constructed which  produces enhancements i n the t h e o r e t i c a l scattered spectra at p a r t i c u l a r frequencies. This model i s f i t t e d to the observed spectra; t h i s f i t t i n g i s discussed.  iii  TABLE OF CONTENTS *"  Page  Abstract  i i  Table o f Contents L i s t of Tables L i s t of I l l u s t r a t i o n s Acknowledgements CHAPTER I  Introduction  CHAPTER I I  Theory  i i i v vi viii 1 8 8  A.  Summary o f S c a t t e r i n g Theory  B.  C a l c u l a t i o n of Spectral Power D e n s i t y  11  Departures from a Thermal Spectrum  15  S p e c u l a t i o n on t h e Source o f the Anomalies  17  C. D.  CHAPTER I I I The Experiment  22  A.  The Plasma  22  B.  The Power Supply  30  C.  The Ruby L a s e r  32  D.  The L i g h t D e t e c t i o n System  33  E.  The Experiment  36  F.  Recording o f Data  39  iv  CHAPTER IV  Results  45  A.  Temperature and Density P r o f i l e s  45  B.  Symmetry of the Scattered Spectrum  54  C.  Enhancement at the Plasma Frequency  58  D.  Enhancements at to ,  59  E.  Enhancement as a Function of K  62  F.  Anisotropy Check  72  P  CHAPTER V A. B.  . and %OJ P  Conclusions  p  77  Temperature and Density Measurements  77  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  vi  LIST OF FIGURES F i g . No.  Title  Page  2A  Two D i s t r i b u t i o n Functions  18  2B  Theoretical P r o f i l e f o r f (v,v D )  21  3A  Arc Electrode Dimensions  24  3B  Arc Apparatus  26  3C  Cathode Adjustment Mechanism  28  3D  B a l l a s t Resistor C i r c u i t  31  3E  Photomultiplier Dynode C i r c u i t  34  3F  Photodiode C i r c u i t  35  3G  Schematic of Apparatus  40  3H  T y p i c a l Scope Trace  41  4A  Observation Volume f o r 1 3 5 ° Scattering  47  4B  T y p i c a l F i t t i n g of Theoretical Curves to the Data  49  4C  Electron Temperature vs A x i a l Position  51  4D  Electron Density vs A x i a l Position  52  4E  Electron Temperature and Density vs Radial P o s i t i o n  53  Scattered Spectrum Symmetry Check  57  4F  vii  LIST OF FIGURES - (continued) F i g . No,  Title  Page  Enhancement of the Thermal Spectrum as a Function of Wavelength S h i f t and Electron Density  60  4H  Position of the Enhancement vs u p  62  41  Spectrum Showing Enhancement at ojp, Hap, and ku (for the parameters shown)  64  Spectrum Showing Enhancement at OJ , ha , and (for the p parameterS shown)  65  3 Axis Plot of Enhancement vs Wavelength S h i f t 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 D r i f t V e l o c i t i e s  81  5B  Theoretical Spectrum f o r the two D r i f t V e l o c i t y D i s t r i b u t i o n Function  83  4G  4J  4K  viii  ACKNOWLEDGEMENTS  I would l i k e to thank Dr. R. A. Nodwell for h i s constant encouragement and timely suggestions over the course of t h i s work.  Thanks also go to  many members of the Plasma Physics Group. In p a r t i c u l a r I would l i k e to thank G. Albach, H, B a l d i s , R. Morris and J . Meyer f o r many h e l p f u l and i n t e r e s t i n g discussions during the preparation of t h i s t h e s i s . Thanks go to D. Camm, G. Albach and L. Godfrey f o r the  preparation of the Computer programs used i n t h i s  work.  I wish also to thank Dr. W. B. Thompson f o r  stimulating discussions on the t h e o r e t i c a l aspects of t h i s work.  Thanks also go to my two t y p i s t s  Mrs. L . M, Churchland and Mrs. J . T. Churchland,  1  INTRODUCTION The s c a t t e r i n g of electromagnetic  radiation  by free electrons i n a plasma, Thomson s c a t t e r i n g , has been used as a diagnostic t o o l since 1958. K. L. Bowles, using a 41 MHz pulsed transmitter, observed r a d i a t i o n back scattered from the earth's upper atmosphere.  Thomson s c a t t e r i n g  was f i r s t used as a diagnostic t o o l on a  2 laboratory plasma i n 1963. Fiocco and Thompson scattered the l i g h t of a 20 Joule normal mode l a s e r from an arc plasma.  The f i r s t  well defined spectrum of scattered l i g h t showing a wavelength d i s t r i b u t i o n 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 t h e i r work p o s s i b l e . The theory of Thomson s c a t t e r i n g i n a plasma was developed because the spectra reported by Bowles were not i n agreement with the theory used at that time.  Such authors  2  as Dougherty and F a r l e y  4  3  (1960), F e j e r (1960), and  Salpeter^(1960) developed the theory of l i g h t scattered from an i n f i n i t e homogeneous plasma.  The theory was  l a t e r modified to include the e f f e c t s of constant magnetic 7  f i e l d s ( S a l p e t e r , 1961), gross d r i f t v e l o c i t i e s  8  (Rosenbluth and Rostoker, 1962), and other simple departures from i s o t r o p i c thermal e q u i l i b r i u m ^ » 1 2 (Perkins and Salpeter, 1965; Kegel, 1970).  This theory  i s i n good basic agreement with the observed scattered l i g h t p r o f i l e s .  However, some of the  e a r l i e s t papers reported deviations from the theory which were not explained. 10  Gerry and Rose ( 1966) obtained "spurious peaks" i n t h e i r scattered spectrum.  These peaks were reproducible  and s i g n i f i c a n t l y above the t h e o r e t i c a l curve which best f i t t e d the data. was  given.  No explanation of these peaks  Evans et a l  1 1  (1966) reported a scattered  l i g h t spectrum with two points above the t h e o r e t i c a l curve.  No comment was made about these points by the  authors, but K e g e l  12  (1970) attempted to explain them  by postulating that the plasma contained a small f r a c t i o n of electrons considerably cooler than the bulk of the plasma.  He was able to account f o r one of the anomalous  points of the scattered spectrum.  No attempt was made  3  to r a t i o n a l i z e the existence of the "cool e l e c t r o n s " . Data points above the f i t t e d t h e o r e t i c a l curve of the scattered l i g h t spectrum from a Z pinch plasma have 13 been observed by Kronast  .  Although small, these  deviations are reproducible and under a v a r i e t y of plasma conditions seem to occur at a s h i f t 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 N o d w e l l ^ " ^ .  Using a D , C « , magnetically  stabilized,  low pressure hydrogen arc as a plasma, they c a r e f u l l y studied the spectrum of the scattered l i g h t .  The  r e s u l t s show several deviations from the predictions of the theory f o r a thermal homogeneous i n f i n i t e plasma. The spectrum of scattered l i g h t i s enhanced at wavelength s h i f t s corresponding to i n t e g r a l multiples of the plasma frequency, W p  ,  The s h i f t corresponding t o  1/2 Wp also shows an enhancement.  The t o t a l s c a t t e r i n g  cross section integrated over the frequency f o r a p a r t i c u l a r scattering vector K does not have the dependence on K that the theory would p r e d i c t . The scattering cross section i s approximately twice the t h e o r e t i c a l cross section at one p a r t i c u l a r K vector.  4  As K i s varied the cross section quickly approaches that predicted by the theory.  This departure from theory  i s of p a r t i c u l a r importance to those experiments using t o t a l i n t e n s i t y (integrated over frequency) to obtain a value of the plasma electron density.  The strong K  dependence of integrated i n t e n s i t y implies that the anomalies are due to waves i n the plasma of a p a r t i c u l a r K and co  Kegel's  12  theory was  developed  i n an attempt to explain these observations not s a t i s f a c t o r i l y do  but does  so.  In further experimentation on the same apparatus, Ludwig and Mahn  17  found that enhancements occur i n the  frequency spectrum at i n t e g r a l multiples of l/2„ and  that  P t h e i r occurrence was orientation.  independent of the K vector  This implies that the anomalies are i s o t r o p i c  and do not depend on the o r i e n t a t i o n of the magnetic f i e l d or the plasma boundary. Large deviations from the thermal spectrum i n 4  a Helium arc are reported by Neufeld -, was  An enhancement  found near the c e n t r a l frequency of the spectrum.  Possible enhancement at report.  co^ i s also mentioned i n t h i s  Neufeld speculates that the c e n t r a l frequency  enhancement might be due to an excess of cold e l e c t r o n s .  5  This i s Kegel's  two temperature theory.  of such cold electrons i s not discussed. 19 al  The o r i g i n P. K. John et  report the detection of an enhancement of the spectrum  of the l i g h t scattered from a pulsed plasma. enhancement occurs at a frequency s h i f t  This  corresponding  to the frequency of the electron acoustic wave. In none of the above experiments has a s a t i s f a c t o r y explanation of deviations of the spectrum from that predicted by the theory been given. Two authors have t r i e d to explain some of the observed discrepancies, 12 Kegel c a l c u l a t e s the e f f e c t of having two groups of electrons at d i f f e r e n t temperatures present i n the same plasma.  By varying the percentage and temperature  of the cold electrons he could obtain a computed p r o f i l e with appropriate peaks at the c e n t r a l frequency and the plasma frequency. This theory was developed to t r y to f i t a curve to the data obtained by Ringler and 14 Nodwell  , No reason f o r expecting a two temperature  plasma i s given. 20 E, I n f e l d and W, Zakowicz  show that i f an undamped  o s c i l l a t i o n at cop i s postulated to e x i s t i n the plasma, there w i l l be enhancements at i n t e g r a l multiples of ojp,  These occur because the incident laser l i g h t 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 o s c i l l a t i o n would come to e x i s t , or why i t would not be damped. Because no s a t i s f a c t o r y explanation of the observed anomalies was evident, i t was thought that i n v e s t i g a t i o n under d i f f e r e n t conditions would be u s e f u l . The anomalies 11,13,19 had been observed e i t h e r i n pulsed plasmas or i n magnetically s t a b i l i z e d D.C. a r c s ^ ' ^ To eliminate the possible time development e f f e c t s of the pulsed plasma and magnetic f i e l d e f f e c t s of the s t a b i l i z e d arc plasmas a non magnetically s t a b i l i z e d D.C. arc was chosen as the plasma to study.  The high 21  current carbon a r c , s i m i l a r to one used by Maecker  ,  was chosen because of the wealth of experimental work already done. Chapter II gives a b r i e f summary of the scattering theory f o r a homogeneous i s o t r o p i c plasma,  A model  of the plasma v e l o c i t y d i s t r i b u t i o n function i s postulated and t h e o r e t i c a l scattered l i g h t spectra are given. Chapter I I I gives a description of the experimental apparatus.  Chapter IV gives the experimental r e s u l t s ,  A comparison between the empirical model and the experimental r e s u l t s are made i n Chapter V, Appendix A shows that the e f f e c t of c o l l i s i o n s  7  on the spectrum of l i g h t scattered i s n e g l i g i b l e . Appendix B contains a b r i e f description of the f i t t i n g of the t h e o r e t i c a l spectra to the data.  Appendix C  gives a b r i e f description of the c a l c u l a t i o n of S (K, co ) f o r the postulated d i s t r i b u t i o n  function.  8  CHAPTER II  THEORY  The theory of the scattering of laser l i g h t from 4-9 a plasma was  developed by several authors  phenomenological description may  .  A  be given as follows:  An electromagnetic plane wave, i n t h i s case ruby l a s e r l i g h t , i s incident on charged p a r t i c l e s , the plasma. The E-M  wave accelerates each p a r t i c l e and consequently  each p a r t i c l e r a d i a t e s .  I t can be noted here that the  electrons are the only p a r t i c l e s which w i l l radiate s i g n i f i c a n t l y (because of t h e i r low mass and consequent high a c c e l e r a t i o n ) . the E-M  I f the e l e c t r i c f i e l d vectors of  waves radiated by the electrons are summed,  the power spectrum of the l i g h t scattered can calculated.  be  Thus the scattered p r o f i l e depends on  the  p o s i t i o n and v e l o c i t y of the p a r t i c l e s i n the plasma. The p o s i t i o n determines the r e l a t i v e phase between the component e l e c t r i c f i e l d vectors and the v e l o c i t y determines the frequency s h i f t .  Although the ions do  not  r a d i a t e , t h e i r p o s i t i o n must be considered because they a f f e c t the p o s i t i o n and v e l o c i t y of the e l e c t r o n s . The scattered e l e c t r i c vector  (summed over each  electron's phase and frequency s h i f t ) w i l l add to a non zero value only i f there are v a r i a t i o n s i n the plasma density.  Variations may  arise from two  sources.  9  Microscopic v a r i a t i o n s occur because of the p a r t i c l e nature of the plasma.  These are random and have a  magnitude proportional to (N)^where N i s the number of p a r t i c l e s i n the plasma observed (see f o r example  22 Bekefi  s e c . 8.1).  Macroscopic f l u c t u a t i o n s are  caused by the c o l l e c t i v e e x c i t a t i o n of l o n g i t u d i n a l plasma waves. If two assumptions are made a simple expression for the power spectrum of the r a d i a t i o n 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 r a d i a t i o n does not accelerate the changes to relativistic velocities.  The time averaged scattered  power dW per unit s o l i d angle dft, per unit frequency i n t e r n a l dw  i s then:  dW dt dft dw  \ /  =  NV IS I a 27 l i | T  S(  w  -S' >  10  where:  N  i s the electron number density  V  i s the observation volume i s the Poynting f l u x of the incident beam  a_ i s the Thomson s c a t t e r i n g cross section f o r an electron and: S(K,w)= l i m (2/TVN) | N ( K = K „ - K i , 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 f o r a plasma the spectrum of density fluctuations N (K,w) must be c a l c u l a t e d . N (K,w) w i l l be derived to show that under c e r t a i n MM  assumptions i t depends only on the v e l o c i t y d i s t r i b u t i o n function f o r the plasma.  By postulating a p a r t i c u l a r  v e l o c i t y d i s t r i b u t i o n and c a l c u l a t i n g S (K,to ) , the t h e o r e t i c a l power spectra can be compared to the  11  experimental r e s u l t s . C a l c u l a t i o n of £ (K, to ) To calculate N (K,to) we assume the actual d i s t r i b u t i o n dunction fo ( V / £ » t ) can be described by a function f (v) plus a small correction term f  f  o  (Y.»£^) =  f (v) +  (Vrj^t)  f x (v,r,t)  If we assume that the plasma i s c o l l i s i o n l e s s and that f  i s small compared to f (v) we can use the  Boltzraan-Vlasov equation to f i n d f f  i n terms of  ( v ) . The Boltzman-Vlasov equation i n l i n e a r i z e d  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 t o :  (T)  12  The term v i s i n t r o d u c e d here t o a i d i n e v a l u a t i n g an integral.  I t w i l l l a t e r be s e t t o z e r o .  Its  p h y s i c a l 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 e x p r e s s i o n o f E i s needed. E i s the e l e c t r i c f i e l d i n t h e plasma produced by the p a r t i c l e s .  I t can be shown ( see, f o r  22 s e c t i o n 4 . 6 ) t h a t a t e s t charge o f  example B e k e f i charge  density:  P (r t) = i  f  produces an e l e c t r i c f i e l d whose F o u r i e r component i s :  E(K,to) =  3*S |K| e K (K,w) i £  o  L  where: P ( r , t ) = 2 7 ^ 6 (aj-K-v) e i  and K  L  (K,^) i s the l o n g i t u d i n a l d i e l e c t r i c  coefficient.  13  We have completed the construction  using(T\  and ( T ) .  of f  (£»]_>__)  The form of f (v)  is s t i l l arbitrary. The Fourier spectrum of electron  density  fluctuations i s given by: 3  N(K,w) = 2TT Z 6(co-K-v.) e -"-  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 e l e c t r o n s .  The second term  i s the correction f o r the screened e l e c t r o n s . an expression f o r f  We have  (k,v,w) and can now calculate S (K,_)) . 2  Before we form the product |N (K,<_.)| t o obtain S (K,u>) we make use of the s u b s t i t u t i o n :  co 2 G(K,io)  K  2  K-3f(v)/3v  —  o  —  d v  w-K • v  and G  (K,w) =  <4  K-3ftv)/3v  —  u-K • v  +  d v  ©  14  where the + denotes terms associated with the ions. Remember also the d e f i n i t i o n of K  ( K , c u ) which  Li  "~  occurs i n equation (jf) for E (K,to) +  K (K,CO)  =  L  1+G+G  (11  Now s u b s t i t u t i n g (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  +  1+G+G  3  £ 6(w-K«v) i  e -£i (g)  G  3  + 2TT 1+G+G  r+  E 6(u-K.v) e £ * i i "  In t h i s form the product |N  (K,<JJ)  j  2 i s taken.  To do t h i s the s t a t i s t i c a l independence of t e s t p a r t i c l e s 22 i s invoked (see Bekefi  sec. 4.6, f o r example).  The equation:  Z i  x(Vi)  (where X(v)  = NV  i s any function)  the product.  3  f(v)X(v)d v  (13)  i s also used to reduce  15  S(K, W ) i s then given by; (2TT)  -1 S(K,w) =  1+G+  f(v) < S (w-K«v)d v  + 1+G+G'  G +  2  1+G+G  2  f (v) 6 (co-K. v) d v  Given any d i s t r i b u t i o n function f(v) the right hand side of (14) can i n p r i n c i p l e be evaluated. Departures from a Thermal Spectrum The anomalies observed i n t h i s work are characterized by small enhancements above the t h e o r e t i c a l spectra f o r a Maxwellian plasma.  These enhancements  have a small frequency band width compared to the thermal spectrum and often occur near the plasma frequency. In t h i s section we w i l l discuss how c e r t a i n departures from a Maxwellian plasma are treated t h e o r e t i c a l l y . We w i l l then speculate on how the anomalies mentioned above might occur. Departures i n the t h e o r e t i c a l p r o f i l e s from those calculated f o r an i s o t r o p i c thermal plasma may come from two sources. The acceleration term i n the Vlasov equation may include more than the s e l f reaction f i e l d produced by the plasma and the v e l o c i t y d i s t r i b u t i o n function  16  may not be purely Maxwellian, The e f f e c t of including a D.C, magnetic f i e l d i n the acceleration term ( A.°_r ) has been calculated 3 V 7 ~ by Salpeter (1961). He shows that the f i e l d produces enhancements at frequency s h i f t s of i n t e g r a l multiples of the cyclotron frequency, co^. Bernstein modes.  These are the  The e f f e c t of the nonlinear  coupling  of two E-M waves (wave mixing) has been considered by 23 K r o l l e t a l (1964). The accelerating force of the mixed waves i s included i n the Vlasov equation and the enhancement 24 of the plasma waves i s c a l c u l a t e d .  Stansfield  (1971)  has shown that natural plasma waves can be enhanced using t h i s technique. The e f f e c t 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 i s Doppler s h i f t e d by the d r i f t v e l o c i t y and that an asymmetry i s produced i n the i o n feature. Perkins and 9 Salpeter (196 5) consider the e f f e c t of a few superthermal electrons on the scattered spectrum.  To the normal  Maxwellian d i s t r i b u t i o n a second group of high temperature electrons i s added.  The e f f e c t of these electrons  i s calculated and shown to enhance the electron 12 feature of the scattered spectrum, Kegel (1970) does  17  the above c a l c u l a t i o n f o r the more general case where the second group of electrons can have any temperature, hot or c o l d .  His d i s t r i b u t i o n function  was the sum of two Maxwellians:  f(v)  ?5  _  = b((m/2TT<T 1 ) exp{-mv /2KT }) 2  1  + (l-b) ( (m/2TTKT ) ^ e x p { - m v / 2 i c T } ) 2  2  2  The term "b" designates the percentage of electrons that have a temperature T^.  When t h i s form of f(v) i s used,  secondary maxima i n 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 s i m i l a r l y to what has been observed.  The addition of a group of high speed  electrons i n a plasma w i l l enhance c e r t a i n waves.  The  waves affected w i l l be those waves whose phase v e l o c i t y c l o s e l y matches the v e l o c i t y of the high speed e l e c t r o n s . A physical d e s c r i p t i o n of the e f f e c t on the waves of a given frequency,cu, and scale length, K, can be given i n terms of Landau damping.  Consider that the electron  v e l o c i t y d i s t r i b u t i o n function has been changed by the addition of a group of f a s t electrons moving i n a well defined d i r e c t i o n . For example, assume the f a s t electrons  18  have a Normal d i s t r i b u t i o n function centered about a° given v e l o c i t y  f^v)  where  =  vD:  2  (m/2iTKT1J^exp{-m(v-vD) / 2 K T 1 }  i s small compared to the temperature of  main plasma, T.  This w i l l change the slope of  the  the  t o t a l d i s t r i b u t i o n function around the v e l o c i t y v . -D  Fig. 2 A  This change i n f(v) w i l l change the electron energy d i s t r i b u t i o n around the energy ( _ m  ).  The  slope of  the energy d i s t r i b u t i o n function w i l l be less negative i n the region v just less than v^.  This means that  Landau damping i n the region v j u s t less than V p w i l l be less than i n the thermal case. Therefore the amplitude of the waves with a phase v e l o c i t y just less than v^  19  w i l l be greater than i n the thermal case.  Conversely, o  the slope of the energy d i s t r i b u t i o n function w i l l be more negative i n the region v j u s t greater than and the waves of phase v e l o c i t y w/k just greater than V p w i l l be damped more than i n the thermal case. We consider then a d i s t r i b u t i o n function of the type described above: 1  ~-  f(v) = b(m/2TTKT ) 2 exp{ -m(v-y_D) ^ K ^ } 1  3  9  + (l-b) (m/27ricT2) 2 exp{-mv / 2 K T 2  }  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.  I f we make  small  compared to T, we should enhance waves at a phase v e l o c i t y just less than V p . The choice of a Gaussian d i s t r i b u t i o n f o r the cold electrons was made because the expression f o r S  C  (K,to)  has been solved f o r a Gaussian and the c a l c u l a t i o n of S  (K,to)  f o r the above f u n c t i o n , f (v) , can be done  using standard i n t e g r a l s .  Appendix C gives a b r i e f  d e s c r i p t i o n of the c a l c u l a t i o n of  S  (K,OJ)  f o r the  20  above d i s t r i b u t i o n f u n c t i o n . The actual evaluation of the above i s done on a computer.  The function S (K,w) f o r the above  form of f (v) , (l6) , i s calculated for a fixed K and variable w.  The value of wis expressed i n  terms of a wavelength s h i f t AA because t h i s i s the form of the experimental data. b, Ne,  v D x  '  a n d  parameters. stated.  ®s  a r e reac  The values of T, T^,  * i n t o the computer as  F i g . 2 A shows a r e s u l t f o r the parameters  This method allows the p o s i t i o n of the  "bump" t o 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 f o r smaller T^ and i s higher f o r larger "b".  #  8  A A(A) F i g . 2A  Theoretical P r o f i l e f o r ftv.v-,)  22  CHAPTER III  THE EXPERIMENT  The purpose of t h i s 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 l a s e r , and the l i g h t detection system. The Plasma The plasma producing apparatus constructed f o r t h i s experiment was a high current carbon arc s i m i l a r 21 to that used by Maecker  (1953),  This arc configuration  has been w e l l studied by many authors and i s thoroughly documented i n the l i t e r a t u r e .  I t has been considered  a stable plasma i n good thermodynamic equilibrium because the r e s u l t s of many d i f f e r e n t measurements (21,27,28,29) are s e l f consistent. The arc consists of two graphite electrodes set on a v e r t i c a l a x i s .  The lower electrode i s the cathode and  i s sharpened to a p o i n t .  I t i s a c h a r a c t e r i s t i c of arcs  that the arc attachment to the cathode occurs i n a small well defined area of high current density.  The pointed  23  cathode allows the p o s i t i o n of the arc base to be well defined.  The upper electrode, the anode, i s a square  rod of graphite, f i v e centimeters by f i v e  centimeters.  The arc attachment to the anode occurs over a large area and i s not well defined.  If a large enough area i s not  provided by the base of the anode the arc w i l l  not  run i n 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 i n F i g , 3 A,  The arc i s operated i n 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 a i r along the arc column.  The  convection  currents are induced by the heat from the plasma and electrodes.  The power dissipated by the arc  (60  v o l t s § 400 amps, 24,000 watts) must be c a r r i e d away by the convected a i r or l o s t by r a d i a t i o n as other cooling of the arc apparatus i s n e g l i g i b l e . In order to help maintain a stable convection enclosed by a chimney.  flow the arc  was  This reduces the e f f e c t of  cross drafts on the a r c .  The chimney also protects  the surrounding apparatus from the heat radiated by arc electrodes.  Suitable observation ports  (6,5  cm  i n diameter) were cut i n the side of the chimney (see F i g , 3B).  For each observation port cut i n the  the  A  Carbon Arc Electrode Dimensions  25  chimney a c o r r e s p o n d i n g p o r t was c u t o p p o s i t e i t so t h a t the o b s e r v a t i o n o p t i c s d i d not look a t t h e i n s i d e of the chimney.  T h i s kept s t r a y l i g h t t o a minimum.  P o r t s were a l s o c u t i n the chimney t o a l l o w the e n t r a n c e and e x i t o f the l a s e r beam.  The r e g i o n around t h e  cathode base was l e f t as open as p o s s i b l e t o a l l o w t h e a i r t o form a smooth l a m i n a r f l o w b e f o r e i t reached t h e a r c column.  The supports f o r t h e anode were made as  t h i n as p o s s i b l e i n o r d e r n o t t o d i s t u r b t h e flow above the anode. Because  the 24,000 Watts d i s s i p a t e d by the a r c  heated the apparatus t o such h i g h temperatures a l l s t r u c t u r a l members o f t h e a r c apparatus were made o f steel,  Metals such as aluminium o r copper would melt o r  anneal i n most r e g i o n s near the a r c .  Copper b l o c k s were  used as conductors but n o t 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 apparatus was made o f m i l d s t e e l . b l o c k s were used t o e l e c t r i c a l l y the r e s t o f t h e apparatus.  The r e s t o f  T h i c k asbestos  i n s u l a t e the anode from  B a k e l i t e o r any p l a s t i c  could  not be used as i n s u l a t i o n because t h e heat r a d i a t e d by the a r c e l e c t r o d e s would melt o r burn them.  The asbestos  b l o c k a l s o s e r v e d as thermal i n s u l a t i o n f o r t h e cathode  26  Fig.3 B  Carbon Arc Apparatus  27  adjustment machanism which allowed i t to be o i l e d 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 b o i l i n g point (4827°K),  To keep the electrode separation f a i r l y  constant, and the observation volume i n the arc column at a predetermined p o s i t i o n above the cathode, three orthogonal adjustments were b u i l t into the cathode mount, A v e r t i c a l adjustment was b u i l t so the cathode could be raised as the carbon evaporated.  Because the carbon from  the cathode did not always evaporate symmetrically, l a t e r a l adjustments were also b u i l t into the cathode mount (see F i g , 3 C ) .  This mechanism allowed the p o s i t i o n -  ing of the cathode while the arc was burning. No adjustment 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 a x i s .  This  gave a p r e f e r e n t i a l p o s i t i o n of arc attachment to the anode surface and helped the arc to s t a r t to run s t a b l y . The s t a b i l i t y of the arc was maintained by the depression  28  ADJUSTMENT  Fig. 3 C  Cathode Adjustment Mechanism  29  eroded i n t o the anode s u r f a c e .  Although the a r c would  run s t a b l y i n such a d e p r e s s i o n  i t c o u l d n o t be s t a r t e d  in  such a d e p r e s s i o n .  higher i n i t i a l  T h i s i s probably  c u r r e n t d e n s i t y on the c o l d anode s u r f a c e .  Most o b s e r v a t i o n s centimeter  due t o the  of the a r c were made w i t h i n one  o f the cathode and t h e plasma c o u l d be r e a d i l y  p o s i t i o n e d u s i n g t h e cathode adjustments. i s run a t a low c u r r e n t the h e a t i n g and e r o s i o n o f the e l e c t r o d e s i s reduced.  I f the a r c corresponding  To minimize the  r a t e o f e l e c t r o d e e r o s i o n and the c o r r e s p o n d i n g  frequency  of adjustments o f the cathode a c u r r e n t c o n t r o l was used. T h i s allowed  t h e plasma t o be u s e f u l f o r a l o n g e r  of data collection-,-  The a r c was kept running  c u r r e n t o f about 200 amps,  d u r i n g the t a k i n g o f data  a t a low  A l a r g e r e l a y then  i n the a d d i t i o n a l c u r r e n t r e q u i r e d  period  switched  (up t o 450 amps)  (see F i g , 3 D),  "stand by" c u r r e n t was the minimum s t a b l e  The 200 amp operating  c u r r e n t f o r t h e e l e c t r o d e s used. The  time r e q u i r e d f o r the c u r r e n t t o reach  from 200 amps was much l e s s than one second.  400 amps  Both the  c u r r e n t measured through t h e a r c and t h e background l i g h t emitted  by the a r c reached a new steady s t a t e i n  much l e s s than one second.  T h i s meant t h a t the a r c o n l y  needed t o be l e f t on h i g h c u r r e n t  long enough t o open  the scope camera s h u t t e r 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 p a i r of d i r e c t coupled motor-generators.  The generators  were each capable of supplying 300 amps at 150 v o l t s . The two were run i n p a r a l l e l 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 v o l t s at 120 v o l t s . The current r i p p l e f o r 400 amps through a b a l l a s t r e s i s t o r load was 2% at approximately  400 Hz. This  r i p p l e produced no measureable fluctuations i n the background r a d i a t i o n of the plasma. The resistance of the carbon arc plasma i s close to zero.  In fact the r e s i s t a n c e , dV/dl, can be s l i g h t l y  negative f o r c e r t a i n current ranges.  I f the generators,  with t h e i r low impedance, were connected d i r e c t l y to the arc the current would be very unstable.  In order  to prevent t h i s , a p o s i t i v e resistance was put i n series with the a r c ,  A series of 2,0 ft , 2000 watt r e s i s t o r s  were connected i n p a r a l l e l i n such a way that each could be switched i n or out of the c i r c u i t  (see F i g . 3 D).  Because the voltage across the arc was f a i r l y  constant  over the range of currents used, the i n c l u s i o n of each  31  150VDC F r o m generator 2 ohms 11 units  2KW  2 ohms 8 units  2KW  RELAY  Vs  c o Fig. 3 D  B a l l a s t Resistor C i r c u i t  TO A R C  32  resistor, " i " ,  gives a current: V ballast  i The bank consisted of 20 such r e s i s t o r s , 8 of which could be switched i n or out by means of a large relay (see F i g , 3 D),  This r e l a y , 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 c o l l e c t i o n times.  The major disadvan-  tage of t h i s system i s the large amount of heat produced i n the room by ohmic heating of the b a l l a s t r e s i s t o r s . The Ruby Laser The ruby laser used i n t h i s experiment was  developed  i n the plasma physics laboratory and i s described i n detail  i n the authors M, Sc. thesis  3 6  (1969),  The  ruby rod i s 6 inches long by h inch i n diameter and Brewster angle ends.  Two  has  l i n e a r xenon flashtubes  focussed by a double e l l i p t i c a l cavity o p t i c a l l y pump the ruby rod,  Q s p o i l i n g i s accomplished with a dye  c e l l containing cryptocyanine i n methanol. and flashtubes are water cooled. formed by a 99.9%  The ruby rod  The l a s i n g cavity i s  r e f l e c t i v i t y d i e l e c t r i c 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.  T h i s c o n f i g u r a t i o n r e l i a b l y produces 50 Mw o f power w i t h a pumping energy  o f 4500 J o u l e s .  A f a s t c h a r g i n g u n i t was added t o the l a s e r power supply which enabled the l a s e r t o be d i s c h a r g e d every 10 seconds.  A r e a s o n a b l e number o f d a t a p o i n t s c o u l d  then be r e c o r d e d  (about 60 shots o f t h e l a s e r ) w i t h  each s e t o f a r c e l e c t r o d e s . Because the l a s e r c a p a c i t o r bank c o u l d be charged q u i c k l y an^automatic charging u n i t .  shut o f f was i n s t a l l e d i n the  The v o l t a g e o f the bank was monitored  and a t a p r e s e t v o l t a g e t h e primary o f t h e c h a r g i n g t r a n s f o r m e r was d i s c o n n e c t e d by a r e l a y .  Because o f  the danger o f a c a p a c i t o r breakdown amd e x p l o s i o n due t o o v e r c h a r g i n g , a second v o l t a g e m o n i t o r i n g system w i t h a separate r e l a y shut o f f was i n s t a l l e d and p r e s e t t o t h e maximum working v o l t a g e o f t h e c a p a c i t o r s . The double  shut o f f system made the c h a r g i n g both  automatic  and s a f e .  The  L i g h t D e t e c t i o n System To choose a narrow s p e c t r a l band width  other l i g h t , p a r t i c u l a r l y  and r e j e c t  the s t r a y 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; t h i s w i l l be discussed l a t e r ,  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 l i g h t (e.g.  second order background l i g h t , and h o r i z o n t a l l y  polarized background  light).  The l i g h t transmitted by the system was detected by an RCA C 31034, Gallium arcinide photocathode, photomultiplier.  This tube was chosen f o r i t s high  quantum e f f i c i e n c y (12%) at 6943 8 .  The voltage  d i v i d i n g r e s i s t o r chain i s shown i n F i g .  3 E.  50/v CABLE PHOTO CATHODE  ANODE DY NODES (11 S T A G E S )  -1800V.  22K  33K 22 K 22K  22K  22 K 22 K 22K 22K  Hf-  22 K  f-HF  .05uF .05uF J05UF .1UF . I U F  Fig.  3 E  -4  Photomultiplier Dynode Chain  35  The tube was operated at 1800 v o l t s , which produced about 6 M.A.  through the r e s i s t o r chain.  Speed up  capacitors were used on the l a s t four dynode stages. The power supply f o r the tube was a Fluke (Model No. 412 B ) . In order to monitor the laser l i g h t 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 m i r r o r .  Neutral  density f i l t e r s were used to attenuate the 0.1% of the laser output to a l e v e l the diode could respond to linearly.  The diode output then gives a r e l a t i v e power  output f o r each shot of the l a s e r .  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 .  Fig. 3 F  Photodiode C i r c u i t  36  The  two signals were displayed on a dual trace  Textronics 551 o s c i l l o s c o p e .  A Polaroid photograph  was taken of each p a i r of single sweep t r a c e s , shot by shot, and t h i s data was l a t e r reduced.  The  Experiment When designing an experiment using l a s e r scattering  as a diagnostic  t o o l , i t i s best to keep the optics as  simple as p o s s i b l e . interfaces  The smaller the number of o p t i c a l  (lenses, m i r r o r s , windows, dye c e l l s , etc.)  the less stray l i g h t that can f i n d i t s way through the detection one  optics.  The scattered  l i g h t measured on any  shot i s the order of 10""^ that of the incident  l a s e r beam. Any stray r e f l e c t i o n that reaches any part of the detection optics w i l l add a s i g n i f i c a n t l e v e l to the r e a l scattered  signal.  In t h i s experiment the optics were kept very simple. A single lens (an uncoated singlet) of 275 mm f o c a l length was used to focus the laser beam into the plasma. The  2 to 3 m i l l i r a d i a n divergence of the laser (due to  multimode output) produced a f o c a l spot about 0.5 mm i n 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.  I f placed any c l o s e r 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 f o c a l length, f 5.6, was used to focus an image of the entrance s l i t of the monochromator into the plasma. of  The width and height  the entrance s l i t was varied along with the image  and object distance of the l e n s .  This allowed the  observation of the desired volume of plasma with the appropriate r e s o l u t i o n 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 r a t i o of one would observe a volume of plasma 250 uM by 250 uM by 500 uM.  The r e s o l u t i o n with 250 uM  s l i t s i s given from the dispersion of 10 8/mm as 2.5 8 . It should be mentioned here that the s o l i d angle of the l i g h t cone entering the monochromator was always kept less than that of the monochromator.  In t h i s way  the l i g h t entering the monochromator was incident only on the mirror surfaces. l i g h t at a minimum.  This helped keep the stray  Ports were cut i n both sides of  38  the chimney i n l i n e with the observation o p t i c s . The wall i n l i n e with the observation optics behind the plasma was blackened to reduce stray l i g h t problems. The angle between the input laser beam and the observation beam could be v a r i e d .  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 1 0 5 ° to 1 5 0 ° .  Angles less than 1 0 5 ° could be  but because a was t h i s angle was  obtained,  so large the signal scattered at  too small to be of use.  the apparatus i s shown i n F i g , 3 G.  A schematic of  Another arrange-  ment of the laser optics allowed a simple check f o r large anisotropy of the plasma. beam was  The axis of the laser  t i l t e d 4 0 ° 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 2 0 ° (see F i g . 3 H).  Considerable d i f f i c u l t y was  encountered  i n t h i s configuration while t r y i n g to work near the cathode.  As the cathode was burned and readjusted the  laser would h i t the cathode surface. impossibly high stray l i g h t l e v e l s .  This produced 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 i n t e n s i t y of the scattered l i g h t was recorded i n i t i a l l y on Polaroid film,  A scope camera was used to photograph the face  of a dual trace Tectronics 551 o s c i l l o s c o p e .  One trace  corresponded to the output of a photodiode which monitored d i r e c t l y the l a s e r output v i a the 99,9% r e f l e c t i v i t y l a s e r back mirror.  The other trace  monitored the photomultiplier output. trace i s shown i n F i g . 3 H,  A t y p i c a l scope  The data was obtained  from the photographs by measuring the photomultiplier signal at a p o s i t i o n corresponding i n time to the maximum of the l a s e r output.  The time of the maximum  of the l a s e r pulse i s given by the photodiode t r a c e . The diode trace i s used as a time mark because the signal to noise r a t i o of the photomultiplier output i s not always good enough to pick out the scattered signal.  There i s a time delay between the appearance  of the two signals (diode s i g n a l and scattered l i g h t signal) because of the d i f f e r e n t cable lengths and the  F i g 3 G» Schematic of Apparatus  41  ! > ! •  M  M  f 1 M i l 1  1 i i i M M  - M M  1  > i l  f  i  t i l l —i—i—i -f-  l 1 1 I (III-  • I I  I 1 1 I I i 1  i i I i i i i 1  1  III" " M M i 1 1 1 1  I  M i l 1 M 1  1 1  M  1 M 1 1 1 1 I  i i i i 1 M i  M M 1 i i I  M M 1 1 1 1  M I I M I I  M I I M I I  M I I M I I  Photomultiplier trace (of scattered signal and plasma noise)  Diode trace . (of laser output)  100 n sec/cm Fig.  3 H  Typical oscilloscope t r a c e .  inherent 60 nsec delay i n the photomultiplier dynode chain.  The  absolute difference i n the p o s i t i o n of  the two pulses can be measured during a stray l i g h t check.  With no plasma to produce background l i g h t  the p o s i t i o n of the laser s i g n a l on the photom u l t i p l i e r trace can be seen c l e a r l y and the time difference between the appearance of the two  signals  measured accurately. The photomultiplier  s i g n a l i s normalized to  the laser output as measured by the photodiode s i g n a l . This i s necessary because of the large v a r i a t i o n s , shot to shot, of the l a s e r output power. The  laser  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 i n the laser Qswitch.  Because of the c y c l i c method of taking data  t h i s had no systematic e f f e c t .  One scattered s i g n a l  was recorded at each wavelength u n 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 l i g h t an estimate of the average background l e v e l must be made f o r each measurement. The signal to noise r a t i o varied cons i d e r a b l y , depending on the band pass of the monochromator, the part of the spectrum being analyzed, the scattering angle, and the plasma parameters. For most conditions a signal to noise r a t i o of about 4 to 1 could be maintained.  Such a signal would t y p i c a l l y  contain 20 photo electrons at the photocathode,  This  produces a s i g n a l , a f t e r amplification down the chain, of 0.005 v o l t s into 50 « . Each data point plotted was the average of 3 t o 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 . the case where the S/N r a t i o was  4 or l e s s , 6 to 10 shots  were needed to obtain small error bars. mental configurations  For  For some experi-  a S/N of 10 to 15 was  realized  and only 3 shots were needed to produce small error bars. The error bars shown i n graphs of i n t e n s i t y versus wavelength f o r scattered spectra are i n a l l cases the standard deviation of the mean. The reduction of the data, normalization, averaging, and standard deviation c a l c u l a t i o n was done with the a i d of a simple computer program.  The r e s u l t s were tabulated and plotted i n graph  form by the computer output. The data i n t h i s form could then be f i t t e d to the 30 t h e o r e t i c a l curves.  The method due to Kegal  (1965) was  used to f i t the data to the theory (see Appendix This method consists of p l o t t i n g the data as:  8)•  scattered  i n t e n s i t y (normalized to a maximum of unity) y_s log (A X) where  AX  i s the s h i f t of the wavelength of the  scattered i n t e n s i t y from the laser wavelength.  Kegel  provides a set of standardized t h e o r e t i c a l curves which we can now  f i t to our data.  Because of the nature of  the t h e o r e t i c a l f u n c t i o n , S (K,w), the choice of a best f i t curve determines the r a t i o of N_ to T  and the s h i f t  44  along the axis between the experimental p l o t and the t h e o r e t i c a l plot determines the absolute value of N e which allows us to calculate T e .  This method i s  described i n d e t a i l i n Appendix B .  CHAPTER IV  RESULTS  Temperature and Density P r o f i l e s This chapter w i l l f i r s t present r e s u l t s using laser scattering as a diagnostic technique.  The temper-  ature and density of the arc column, i n the region of i n t e r e s t for t h i s work, were mapped using scattering techniques.  This was  standard  useful i n l a t e r work  because a p a r t i c u l a r temperature and density could be looked at by observing a predetermined region of the a r c . 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 t o t a l integrated scattering spectrum.  Because of t h i s i t i s possible to f i t the  recorded spectrum to t h e o r e t i c a l curves and obtain values of e l e c t r o n temperature and density with good accuracy. The temperature and density of the arc column were mapped as a function of p o s i t i o n and current.  At 200 amps  arc current 6 a x i a l positions between 0.1 cm and 1,05 above the cathode were observed. 8 a x i a l positions between 0,15 cathode were observed.  At 400 amps  cm and 1,60  For the 400 amp  tions from r = 0,0 cm to r = 0,25  cm  arc current  cm above the  arc 6 r a d i a l posi-  cm were observed at  46  the height z = 0.25 cm above the cathode. The volume of plasma observed i n 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 a f f e c t s the observation volume but t h i s was set at unity (image distance equals object d i s t a n c e ) . 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 a x i s ) . axial spatial resolution.  This gives excellent  This choice of observation  volume dimensions i s wider and less deep than i s used i n l a t e r parts of the work. The diagram (see F i g . 4 A) shows the scattering volume configuration f o r 1 3 5 ° scattering.  The entrance s l i t width of 0.5 mm  coupled  with an e x i t 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 s p e c t r a l  data could be f i t t e d d i r e c t l y to t h e o r e t i c a l curves. The 5 8 pass band was wide enough that the anomalous "bumps" i n the spectrum were not usually resolved. A t y p i c a l f i t t i n g of the data to t h e o r e t i c a l curves i s shown i n F i g . 4 B.  Each point and error bar  48  i s the average and standard deviation of the mean of three data p o i n t s .  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 r e s u l t s of s i m i l a r f i t t i n g s f o r the positions and currents mentioned e a r l i e r are shown i n F i g . 4 C, 4 D, and 4 E.  The p o s i t i o n z = 0 i s the t i p of the  cathode and the p o s i t i o n r = 0 i s the axis of the a r c . The accuracy of the t h e o r e t i c a l f i t t i n g to the data should be discussed here.  The t h e o r e t i c a l curves  are characterized by the parameter a . i s proportional to ( density and T  e  N e  /  T e  )  2  The value of a  where NQ i s the electron  i s the electron temperature.  The theore-  t i c a l curves used were plotted i n steps of a of 2,5% i n the range of a?s occurring i n t h i s experiment. This means the value of N /T i s changed 5% i n each step. e e The second parameter used i n f i t t i n g the data to the theory i s the s h i f t of the wavelength scale of the data with respect to the t h e o r e t i c a l wavelength s c a l e . This can be determined with a r e p r o d u c i b i l i t y that produces a 5% range i n the values of N^. produces a 5% range i n the value of T e  This also  obtained.  I f we  assume these errors are independent the values of both N £ and T £ have a probable error of t 7%,  This assumes  TENSITY o ob  1-0 -  5  0-6  -  THEORETICAL  FIT FOR  n  e  = 7:4  T  e  = 1-62 x 10  a  =  o  EXPT  —  x 10  cm"  ji 3  J  °K  /  -  1-85  -  THEORY  UJ  > <0-4 IJJ  tr  -  0-2  00  i  20  i  30  AX  40 (A)  Fig. 4 B  50  i  60  i ^  f  80  i  100  50  the f i t t i n g i s uncertain between only two curves.  theoretical  This also assumes no systematic e r r o r s .  The graphs of F i g . 4 C and 4 D show the plasma parameters as a function of p o s i t i o n along the arc axis.  Because the cathode was  constantly being burned  and readjusted the observation volume p o s i t i o n above the cathode was  i n doubt i n each case.  I t i s estimated  that the p o s i t i o n of the observation volume with respect to the cathode t i p could be kept i n 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 p o s i t i o n of the image on the screen constant with respect to cross h a i r l i n e s . projections were used.  Two  orthogonal  The p o s i t i o n of the arc could  then be kept constant using the three orthogonal adjustments mentioned i n Chapter I I , The adjustments were made while the arc was burning just p r i o r to each f i r i n g of the l a s e r .  The estimated error f o r each  point on the graphs of F i g s , 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  ELECTRON —  TEMPERATURE ro  T  e  (l0  4 o  K) OJ  > >  <: <:  L  Fig. 4 D  £5  54  was  chosen s i m i l a r to Maecker's  because t h i s was  a  well studied configuration. However, i t i s apparent from the r e s u l t s of the scattering technique that the electron temperature of the arc i s very much higher than that reported by Maecker.  Maecker's r e s u l t s are  included i n F i g s . 4 C and 4 D f o r comparison.  The  hot  spot at z = 2.5 mm was not resolved by Maecker possibly due to the Abel-unfolding technique used to obtain a x i a l parameters.  The electron density i s not very  d i f f e r e n t from Maecker's values but shows a peaked high density spot at the same a x i a l p o s i t i o n as the temperature p r o f i l e s .  The drop i n temperature and  density near the cathode i s quite d e f i n i t e i n both the 200 amp  and the 400 amp  cases.  The r a d i a l temperature and density p r o f i l e s show that gradients over the observation volume are small. Symmetry of the Scattered Spectrum 13 15 18 Several authors  '  ' ' ' have reported asym-  metries i n the scattered spectrum between the high frequency side (Blue shifted) and the low frequency side (Red s h i f t e d ) .  The theory for an i s o t r o p i c plasma  predicts that the red and blue sides should be mirror images for a fixed value of K.  55  In t h i s section the r e s u l t s of a symmetry check on the p r o f i l e s f o r the arc plasma w i l l be presented. The frequency integrated i n t e n s i t y of the scattered spectrum was measured on both sides of the laser frequency.  A monochromator entrance s l i t of 100 uM  and e x i t s l i t of 2500 uM were used.  This gives an  instrument p r o f i l e of 25 8 that i s e s s e n t i a l l y square. Intensity measurements were taken at 25 A* i n t e r v a l s on each side s t a r t i n g at the laser wavelength. the  Excluding  laser wavelength f o r which no measurements were  taken, four i n t e r v a l s of wavelength were measured on each s i d e .  Because of the 25 A* steps and the square  25 R pass band, the measurements are t o t a l l y independent. The t o t a l i n t e n s i t y of each side i s then obtained by summing the i n d i v i d u a l measurements.  The spectrum  scattered from a 200 amp carbon arc 2.5 mm above the cathode was measured using the above c o n f i g u r a t i o n . The scattering angle was 1 3 5 ° . i s the average Fig.  of 8 shots.  Each data point p l o t t e d  The r e s u l t s are plotted i n  4 F , the error bars are standard deviations.  difference i n the areas of the two sides i s 22%.  The The  spectral response of the instrument must be c a l i b r a t e d in order to see i f t h i s 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 l i g h t output from a tungston ribbon was measured.  The standard  temperature l i g h t source tungston ribbon lamp was run at 14 amps. This gives a temperature of 1950°K (calibrated by an o p t i c a l pyrometer),  At t h i s  temperature the difference i n 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 v s , wavelength for the system using the above l i g h t source. Table I  Voltage Output vs Wavelength  Wavelength  Voltage (IO"  -100  0,47  - 75  0.46  - 50  0.45  - 25  0.44  0 (6943 8 )  1  volts)  i 0.01  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  CQ  •I  ce < _;  I/)  •5-  z LU  £+.5  5-100  •o-75  -50  -25  0 L a s e r A.  25  50  75  AA(A) Fig. 4 F  Scattered Spectrum Symmetry Check  100  58  i s 16%.  The slope of the emission curve f o r tungston  at 1950°K at t h i s wavelength  (A=6943 8 ) adds another  5% to t h i s which gives a t o t a l of 21% v a r i a t i o n between the red and the blue s i d e .  Within the experimental  error t h i s accounts e n t i r e l y f o r the observed v a r i a t i o n in intensity.  We conclude that the scattered spectrum  i s symmetric within the l i m i t s 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 s h i f t corresponding to that of the plasma frequency "Wp"• the most commonly observed anomaly.  This i s perhaps In order to map  t h i s feature one scattering angle v/as chosen, 6 = 1 3 5 ° , and the electron density observed was changed from o.93 X 10  17  3  cm'  to 1.77  17  X 10 cm"  3  by observing  d i f f e r e n t parts of the arc column at d i f f e r e n t currents. The observation volume was kept small (200 mm x 250 mm x 500 uM) i n order to minimize gradients i n N g and T e «  Only a small part of the spectrum was mapped  around the wavelength s h i f t corresponding to _ ) p .  This  was f i t t e d to the t h e o r e t i c a l curve with the a i d of the knowledge of the electron density as a function of  59  p o s i t i o n obtained from previous measurements.  The  difference between the thermal t h e o r e t i c a l curves, I T (AX), and the observed spectrum, I (AX) was plotted f o r each wavelength i n the spectra. This was done f o r each electron density measured (see F i g . 4 G). In order to better see the functional r e l a t i o n between the enhancement and the electron d e n s i t y , a plot was made of the wavelength s h i f t of the enhancement vs the plasma frequency s h i f t calculated from the value of N  e  obtained previously.  The bars on t h i s  graph are the estimated f u l l width at h a l f i n t e n s i t y l i m i t s of the enhancement.  The straight l i n e i s the  t h e o r e t i c a l f i t t i n g f o r the enhancement occurring at io (see F i g , 4 H) . P  I t seems conclusive that f o r t h i s  p a r t i c u l a r value of K (for 0 = 135°) an enhancement e x i s t s at a s h i f t corresponding c l o s e l y to the plasma frequency.  I t i s i n t e r e s t i n g that the enhancement  occurs at "wp"  f o r t h i s p a r t i c u l a r K (0 =135°) over  such a range of N^,  I t w i l l be seen l a t e r that the  enhancement i s a very s e n s i t i v e function of K, Enhancements at to , -^to^, hap It i s noted i n the introduction that two authors  14 '  1  60  4 G  Enhancement of the thermal spectrum as a function of wavelength s h i f t (from 6943 8) and electron density. (The dotted l i n e i s the wavelength of. the plasma frequency.)  shift  61  observed enhancement at h w . p  observed h w  p  Ringler and Nodwell  "bumps" along with c o , 2 c o , and 3 co p  p  bumps. Ludwig and Mahn reported bumps at N/2 n = 1 to 6.  p  co f o r p  Both experiments were done on a magneti-  cally stabilized arc. In order to check for the existence of s i m i l a r enhancements complete spectra of the scattered l i g h t were compiled at a scattering angle of 1 3 5 ° with high spectral r e s o l u t i o n .  In the spectral range u > _ p no  enhancement could be detected. approximately see.  2.0, j co  Because a was  would be very d i f f i c u l t to  p  The p o s i t i o n of the -i <_. enhancement would 2  P  correspond closely to the already narrow, sharply peaked e l e c t r o n f e a t u r e .  This would make i t almost  impossible to resolve any small enhancement close to the peak.  The p o s i t i o n of a 2 wp  be beyond the thermal spectrum.  enhancement would In theory for the K  observed no waves e x i s t with the frequency 2 co or p  near the frequency 2 „  p  . If waves existed i n the  region 2 co they could e a s i l y be resolved. p  ment was  No enhance-  found at 2 co i n t h i s work. P  The region co < to was  also c a r e f u l l y studied.  Fig.4 H  Wavelength S h i f t of Enhancement v s . 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 i n  resolving the bumps at h t o p and h t o p . The s i z e of the s i g n a l was small and the r e s o l u t i o n necessary to see the enhancement reduced the s i g n a l to noise r a t i o . I t was also necessary to c a r e f u l l y measure the stray l i g h t l e v e l and subtract i t from the observed s i g n a l . The data required about 10 shots f o r each point to reduce the error bars to a s i g n i f i c a n t l e v e l .  Checks  of the spectrum i n the region 0 to 10 X were impossible because of a grating ghost at 8 8. Two spectra were recorded at two d i f f e r e n t densities to check i f the p o s i t i o n of the enhancement would move i n such a way as to remain at h <op and k wp. The two spectra are shown i n F i g . 4 I and 4 J . The l i n e i s the t h e o r e t i c a l best f i t f o r the parameters shown. The stray l i g h t has been subtracted. Enhancement as a Function of K To t h i s point a l l measurements have been taken at 0 = 135°,  This defines a K vector observed i n the 5  1  plasma of 1.67 x 10 cm"" , or an observed wavelength of 3760 8.  This i s determined by the laser wavelength and  scattering geometry.  To vary the K vector, the  INTENSITY ( A R B I T R A R Y ro  J  I  V9  UNITS) O)  '  Co  L  8  ARC CURRENT = 390 AMPS  ~l  20  i  1  40  a  AA(A)  Fig. 4 J  60  1  80  1 —  100  66  scattering angle 0 i s v a r i e d .  The r e s u l t i n g K i s given  by: K =  4 IT —A L  Q  s i n ( \t )  where XL i s the laser wavelength.  In t h i s experiment  the monochromator, photomultiplier and input optics were a l l mounted as a unit and could be moved to any angle from 0 = 1 4 0 ° to 0 = 1 0 0 ° , t i o n i n K from 1.47 to 10  5  1  cm"  This gives a v a r i a 5  1  to 1.70 x 10 cm" .  Ten d i f f e r e n t values of K were chosen and the spectrum around wp was c a r e f u l l y studied.  The electron density  i n the observation volume was kept as constant as possible by c a r e f u l l y c o n t r o l l i n g the p o s i t i o n of the arc.  The "hot spot" 2.5 mm above the cathode was  chosen as the best place i n the plasma to observe because i t was a maximum i n temperature and density and the gradients would be at a minimum.  This p o s i t i o n  i n the arc column i s also easy to keep fixed i n space because i t i s close to the cathode which i s adjustable. The density and temperature at t h i s p o s i t i o n had been determined previously by laser l i g h t scattering and t h i s information was used i n 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 e a s i l y . 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. the  spectrum.  These covered only a small section of The part of the spectrum mapped  could be f i t t e d to theory w e l l enough to observe s l i g h t variations (probably due to s l i g h t errors i n p o s i t i o n i n g the  observation volume) i n plasma parameters.  A listing  of plasma parameters f o r each s c a t t e r i n g angle i s given i n Table I I . The difference between the measured spectrum and the best f i t to the theory f o r each K observed was p l o t t e d ,  A three axis graph i s given containing  these r e s u l t s (see F i g , 4 K ) ,  Each spectrum plotted  (as a function of wavelength s h i f t ) i s the difference between the t h e o r e t i c a l curve (normalized to unity at the t h e o r e t i c a l maximum) and the measured spectrum. The l i n e s j o i n i n g the points are added to a i d the eye i n determining which points belong to which spectrum. The t h i r d axis i s the wavelength of the waves observed  68  i n the plasma given by  The dotted l i n e i s the  plasma frequency s h i f t calculated from the electron d e n s i t y . A larger wavelength i n t e r v a l was studied i n the region 5 -1 below 1.55 X 10 cm , because bumps began to appear i n more places as the wavelength of the plasma waves observed increased (K decreased), The data of F i g . 4 K suggests two other p l o t s . 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 u n i t y .  The difference between the data and the theor-  e t i c a l curves i s then I E x p U ) - I T n e ( A )  =  where  the maxima of the t h e o r e t i c a l spectrum (for the electron feature) has been normalized to u n i t y . enhancement i s i n Angstroms.  The width of the  The area measured i s that  area under the straight l i n e s j o i n i n g adjacent p o i n t s . The area i s plotted as a function of K i n F i g . 4 L. I t would be i n t e r e s t i n g to continue mapping the size of the enhancement as K i s decreased but t h i s would be d i f f i c u l t .  The t o t a l energy contained i n the spectrum 2 of the electron feature i s proportional t o l / a ; as  69  Table II  Plasma Parameters f o r 135  NgjlO  17 cm"  3  Scattering  ) _T e (10  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:  and:  a = ( K *_.) *  K  =  (4TT/X t ) sine/2  4  °K)  70  Fig. 4 K  Enhancement of the thermal spectrum as a function of wavelength s h i f t (from (6943 8) and scale length K. (The dotted l i n e i s the wavelength of the plasma frequency.)  shift  X  72  Therefore the t o t a l i n t e n s i t y of the electron feature 2  depends on |K| . At the same time the width of the electron feature i s decreasing.  These two factors make i t very  d i f f i c u l t to resolve any enhancements i n the spectrum. The other i n t e r e s t i n g plot of the information contained i n F i g . 4 K i s the standard to vs K plasma wave dispersion p l o t .  The wavelength s h i f t , AX» i s  a d i r e c t measure of the frequency of the waves and the scattering angle gives the value of K. The v e r t i c a l bars i n F i g . 4 M are the estimated frequency widths at half i n t e n s i t y of the enhancements. The l i n e s i n F i g . 4 M give the p o s i t i o n of the enhancements produced by the t h e o r e t i c a l model described i n Chapter V,  The s i g n i f i c a n c e of t h i s f i t w i l l be  discussed i n the next chapter. Anisotropy Check A l l measurements done on the arc plasma up to t h i s point have been done with the K vector oriented normal to the axis of the a r c .  In order to check that the  anomalies were not due primarily to very strong a x i a l waves the orientation of the K vector was changed. The laser beam was aimed down i n t o 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-  i  Co  13 -1 10 sec )  2.0H  I>"  5  model  i  | experiment  1.6 H  i  •  1.5  1.6 K(10 cm ) 5  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) .  scattering angle previous r e s u l t s show a co  At t h i s enhancement.  The component of K along the arc axis with t h i s geometry i s 0.34 |K |.  Any large difference of amplitude i n the  anomalous waves i n the horizontal and the v e r t i c a l planes should become apparent i n t h i s scattering configuration.  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 h i t the cathode.  This changed the plasma parameters  as noted i n F i g . 4 P. presented as before.  The data i n F i g . 4 P i s The difference between the measured  spectrum and the best f i t to theory i s p l o t t e d as a function of wavelength s h i f t .  This r e s u l t i s s i m i l a r .  to the previous r e s u l t s and indicates that the plasma i s not strongly a n i s o t r o p i c .  75  Fig. 4 N  Scattering Configuration of the Anisotropy Check  76  i  £  I  i  1  <—• o O X LO II  X  . o  ^.  m°_  £ £  CO CD L O 11 CNl QO j j ? CD _3l M  CM  LO CD  ft.  In  CV)lI-(V)I  77  CHAPTER V  CONCLUSIONS  The f i r s t point that should be discussed i s the r e l a t i v e enhancement.  The plasma temperature and density  were measured i n the arc column by f i t t i n g the experimental spectra to t h e o r e t i c a l 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 i n the ,  general f i t t i n g technique. The area of the enhancements was t y p i c a l l y 1% (at 0 = 135°) electron feature.  of the t o t a l area of the  The enhancement e x i s t s over a 2 or  3 A" range of the spectrum.  If a 5 X pass band i n the  detection system i s used to measure the shape of the spectrum, the enhancement i s d i s t r i b u t e d 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 p a r t i c u l a r wavelength.  Considering that the  error bars on the signals measured were about 10% of the  signal s i z e , we would expect the enhancement to  have l i t t l e e f f e c t 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 t h e o r e t i c a l curves at a wavelength close to the  plasma frequency s h i f t .  This d i d not s i g n i f i c a n t l y  78  change the parameters obtained by the f i t t i n g procedure. We also concluded from the r e s u l t s that the scattering spectrum was wavelength.  symmetric about the laser  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 e a r l i e r  (- 7% on N e  and T ) . e The Enhancements Observed and the Theoretical Model In t h i s section we wish to speculate t h a t , using the two d i s t r i b u t i o n function model developed i n Chapter I I , t h e o r e t i c a l p r o f i l e s with "anomalous" peaks can be constructed and f i t to the observed spectra. Under certain assumptions a t h e o r e t i c a l dispersion curve may  be drawn s i m i l a r to that observed i n  Chapter IV,  The general c h a r a c t e r i s t i c s 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 c a l c u l a t e d f o r a two  distribution  plasma where one d i s t r i b u t i o n d r i f t s a r b i t r a r i l y with a v e l o c i t y v^.  It i s also shown that only the component  of v Q 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 a r b i t r a r y frame of reference) i s  79  postulated to e x i s t i n the plasma.  If the scattering  angle, 0 , i s varied by moving the observation optics s v/ith respect to the plasma, the d i r e c t i o n and magnitude of K w i l l be changed. the component of v TD  If the d i r e c t i o n of K i s changed,  along K w i l l change.  It i s a simple  "~  matter to calculate the component of v^ along K f o r each s c a t t e r i n g angle, and to construct the scattered p r o f i l e .  The frequency s h i f t of the calculated  anomalies i n the power spectra can then be plotted against K to give a t h e o r e t i c a l dispersion curve. The experimental r e s u l t s reported i n Chapter iv show a dispersion curve with two secondary  branches.  In order to obtain a dispersion curve with two branches from t h i s t h e o r e t i c a l model i t i s necessary to postulate two d i f f e r e n t d r i f t v e l o c i t i e s i n two d i f f e r e n t d i r e c t i o n s . This requires a d i s t r i b u t i o n 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 . , f 0 ) = (1-a-b) G_(f ) + aG o 1 2 O o  (f,) + b G, (f,) 1 /. z  8©  The  r e s t of the c a l c u l a t i o n In o r d e r  to obtain  curve reported of  t h e two  p r o c e d e s as  before.  a f i t t o the d i s p e r s i o n  i n Chapter IV the d i r e c t i o n  drift  |vj  v e l o c i t i e s were c h o s e n  = 1.6  vQ^at  X 10  8  1 8 0 °to  and  as  magnitude  follows:  cm/sec  (back t o w a r d  laser,)  8 |v^  = 2,1  v^9  X 10  at 1 1 5 ° to  cm/sec  K,  The  g e o m e t r y i s shown i n F i g . 5 A;  the  same p l a n e .  The  dotted  lines  a l lvectors in Fig. 2 B  v ^ and y_2 t o K show t h e component along  K,  The  coordinate  x axis i s p a r a l l e l v^,  assuming  contain assuming of was  t o K,  system Using  that the secondary  .00015 o f t h e e l e c t r o n s that  (T-j/T =  from  and  i s chosen  such t h a t  t h e above v a l u e s  the  of  distributions (a = b =  .00015),  t h e s e c o n d a r y , t e m p e r a t u r e was  t h e main p l a s m a .  are i n  .005  and that  . 0 0 5 ) , t h e f u n c t i o n S(K,w)  c a l c u l a t e d f o r eight values  o f 0^.  The s c a t t e r i n g  Fig. 5 A  Orientation of D r i f t V e l o c i t i e s with Respect to the Scattering Geometry  82  angle v/as varied from 1 0 5 ° to 1 4 0 ° i n steps of 5 ° . A sample of S (K^co) i s plotted i n F i g , 5 B, dotted l i n e i s f o r a thermal spectrum  The  (a = b = 0) ,  The p o s i t i o n of the secondary peaks was noted i n each case and the frequency s h i f t of the peak was vs K (see F i g . 4 M),  plotted  The frequency p l o t t e d here  i s to, - to ; the K plotted i s K . - K . i s — — i —s  This i s the  dispersion curve that best f i t s the data presented i n Chapter IV.  The anomalies studied were found to depend  on both the density of the plasma and the scale length (2IT/K)  of the waves i n the plasma.  We w i l l now look  c l o s e l y at those aspects of the model developed i n Chapter II that either agree or disagree with the observations. The model was constructed i n such a way that i t would f i t the dispersion curve obtained experimentally. The o r i e n t a 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 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 t h e o r e t i c a l peaks.  The graph  ( F i g . 4M) shows that a  reasonable f i t t i n g of the model to the r e s u l t s can be  AA(A)  Fig. 5 B  Theoretical P r o f i l e f o r two D r i f t  Velocities  84  obtained f o r two f i x e d d r i f t v e l o c i t i e s v and -DI v . The f i t t i n g i s quite s e n s i t i v e to the choice -D2  of the d i r e c t i o n and magnitude of the d r i f t v e l o c i t i e s . Good agreement between the experimental r e s u l t s and the model are obtained f o r the area of the enhancement as a function of K.  This i s d i f f i c u l t to measure  q u a n t i t a t i v e l y because of the nature of the model. As can be seen i n F i g . 2 A there i s no net enhancement of the t o t a l spectrum.  The model s h i f t s the energy i n  the waves to a lower frequency, leaving equal regions of enhanced and damped waves.  To t h i s 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 r e s u l t s have always been f i t t e d to theory  on t h i s assumption.  I f the model we propose i s c o r r e c t ,  a small systematic error could occur i n the f i t t i n g of the p a r t i a l spectrum to the normal curves f o r a thermal plasma.  In the region above the secondary peak ( F i g . 2 A ) ,  the spectrum of our model i s s l i g h t l y below the spectrum of thermal f l u c t u a t i o n s .  If the experimental spectrum  i s shaped l i k e the model but f i t t e d to the thermal t h e o r e t i c a l curves, a s l i g h t l y lower temperature and density would be calculated than r e a l l y e x i s t s .  This  i s due to the error i n estimating the r e l a t i v e frequency  85  shift  (see Appendix B ) .  determination  I t should be noted that the  of the temperature and density by  fitting  the f u l l spectrum to the curves for the thermal theory would not be susceptible to t h i s systematic e r r o r . In t h i s case the main peak of the spectrum can be used to determine the s h i f t r e l a t i v e to the t h e o r e t i c a l spectra.  The p o s i t i o n and height of the main peak i s  not affected i n the model by the addition of the  drift.  The p a r t i a l spectra used i n determining the K dependence produce a consistently lower temperature and density (see Table II) than i s determined from the gross s c a t t e r i n g spectra (see F i g , 4 M and 4 N).  The d i f f e r e n c e i s quite  small ( 5 % ) , but consistent. The possible reasons that the region of damped waves was not noticed as being below the spectrum are: poor r e s o l u t i o n due to the large instrument p r o f i l e (2 8 ) needed to obtain a s i g n a l , and shot to shot plasma v a r i a t i o n s .  These two  factors coupled with the  high a , ( a = 2.2 - 2 , 4 ) , i n the region studied for the K dependence make i t u n l i k e l y the damped region would be resolved.  However, i n the region of lower a  the dip i s possibly resolved i n a few cases.  ( a=  1,9)  In F i g , 4 G  two spectra show a dip on the high frequency s i d e .  The  86  The spectrum for N dip at 57 8 .  = 1,48  X 10  The spectrum for Ng  cm  shov/s a d i s t i n c t  = 1.48  X 10  17  shows a dip also on the high frequency s i d e .  3  cm"  No  spectra on t h i s graph show a s i g n i f i c a n t d i p .  other  It i s  i n t e r e s t i n g that there i s no s i g n i f i c a n t dip on the low frequency side of the spectrum f o r any of the recorded  spectra.  The model constructed seems to be capable of  reproducing  the size and p o s i t i o n of the anomalies observed i n the K dependence spectra.  I f the model i s to predict the  size and p o s i t i o n of the anomalies i n the density dependence spectra, a new  f a c t o r must be considered.  that for 1 3 5 ° s c a t t e r i n g (K=1.6 X 10 p o s i t i o n of the bump was  5  I t was  shown  1  cm" ), the  a function of the plasma frequency.  This means the d r i f t v e l o c i t y of the secondary electrons v^ must vary l i n e a r l y 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 v e l o c i t y s i m i l a r to those used in the model.  We are also unable to postulate why  the  electrons should have a d r i f t v e l o c i t y dependent on the plasma frequency. It i s speculated that the anomalous features are laser induced.  This i s thought to be the case  87  because of the o r i e n t a t i o n of the d r i f t v e l o c i t i e s . We can think of no mechanism i n the plasma which would produce such d r i f t v e l o c i t i e s .  Ringler and  15-17 Nodwell  concluded that the anomalies observed  i n t h e i r work were not laser induced.  This implies that  d i f f e r e n t mechanisms are producing the anomalies i n each experiment. It i s also speculated that the plasma parameters are important i n describing the mechanism that produces the anomalies. The model presented i n Chapter II of the thesis i s not considered to be the only possible explanation, but does c o r r e c t l y give the functional K dependence of the anomalies. For t h i s model to also include the c presence of theh to and %to bumps, hvn and Wp. p p —D —D v e l o c i t i e s would need to be included i n the same directions as the f i r s t two.  The functional dependence  of these could then be checked experimentally. In that t h i s work does not lead conclusively to the o r i g i n 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. acquisition i s very low.  The rate of data  A multi channel spectral 33 "5\& analyzer of the type used by Rohr (1967), Kronasf *(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 t h e o r e t i c a l models. presented i n t h i s work was  The  data  not complete enough to  determine accurately the shape of the anomalies.  With  the above improvement a greater range of K vectors could be studied. 15%.  Presently K can be varied only about  A greater range of K would help determine i f the  model has the correct angular dependence f o r the anomalies. Secondly, a great deal more information could be obtained i f a v a r i a b l e frequency Dye laser was msed. If a v a r i a b l e frequency incident l i g h t 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 p o s s i b l e . Without t h i s information i t i s d i f f i c u l t to i n t e l l i g e n t l y postulate mechanisms which could produce anomalous waves i n the plasma.  89  Bibliography  1.  Bowles, K. L.  1958.  Phy. Rev. L e t t . 1 (12): 454.  2.  F i o c c o , 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.  L e t t . 8: 179-180. 4.  Dougherty, J . P. and F a r l e y , D. T.  1966.  Proc. R.  Soc. A 259: 79-99. 5.  F e j e r , 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.  F l u i d s 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.  12.  Kegel, W. H.  13.  Kronast, B.  1970. 1972.  Nature 211: 23-24.  Plasma Physics 12: 295-304. Private communications.  90  14.  Ringler, H. and Nodwell, R. A.  1969.  Phys.  1969.  Third  L e t t . 29A: 151. 15.  Ringler, H. and Nodwell, R. A.  Europ. Conf. on Contr, Fusion and Plasma Physics, Utrecht, 16.  Ringler, H, and Nodwell, R, A.  1969.  Phys. L e t t .  30 A: 126. 17.  Ludwig, D. and Mann, C.  18.  Neufeld, C. R.  19.  John, D. K. et a l .  20.  I n f e l d , E. and Zakowicz, W.  1970.  1971.  Phys. L e t t . 35A:191.  Phys. L e t t , 31A: 19.  1971.  Phys. L e t t . 36 A: 277. 1971.  Phys. L e t t .  32 A: 103. 21.  Maecker, H.  22.  Bekefi, G.  1953. 1966.  Z. Physik 1361 119. Radiation Processes i n Plasmas,  John Wiley and Sons. 23.  K r o l l , N. et a l .  24.  S t a n s f i e l d , B. L.  1964.  Phys. Rev. L e t t . 13: 83.  PhD Thesis (1971). The University  of B r i t i s h Columbia. 25.  Tanenbaum, B. C.  1967,  Plasma Physics, McGraw-Hill,  26.  F r i e d , 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, I n s t i t u t  fur Plasma Physik, IPP 6/34. 31.  Grewal, M. S. 1964.  Phys. Rev. 134 ( l A ) : 86A?  32.  Rose, D. J . and C l a r k , M,  1961,  Plasma and  Controlled Fusion, The M.I.T. Press. 33.  Rohr, H.  1967.  I n s t i t u t fur Plasma Physik, IPP  1: 58. 34.  Kronast, B, and Pietrzyk, Z. A,  1971.  Phys. Rev.  L e t t . 26 (2): 67-69. 35.  Albach, G. G.  1972.  M. Sc. Thesis, University  of B r i t i s h Columbia, 36.  Churchland, M, T.  1969,  of B r i t i s h Columbia.  M. Sc. Thesis, University  92  APPENDIX A The spectrum of l i g h t scattered from a plasma i s calculated i n Chapter II under the assumption that the plasma i s c o l l s i o n l e s s .  We must check that t h i s  i s a v a l i d assumption f o r t h i s experiment. The e f f e c t  of c o l l i s i o n s on the spectrum of  electron density f l u c t u a t i o n i n a plasma has been 31 studied by Grewal (1964). He shows that when an electron i n a density wave t r a v e l s l e s s than ten wavelengths before s u f f e r i n g a c o l l i s i o n , the scattered spectrum i s a f f e c t e d .  We must c a l c u l a t e  the distance an electron i n the density wave t r a v e l s before s u f f e r i n g a c o l l i s i o n and compare t h i s to the wavelength of the density wave. The most probable c o l l i s i o n i n a f u l l y ionized plasma i s an electron-electron i n t e r a c t i o n due to coulomb f o r c e s .  A c a l c u l a t i o n of the relaxation  time f o r multiple coulomb interactions i s given i n 32 Rose and C l a r k . they quote.  We w i l l use the f i n a l r e s u l t which  93  62/TT e  T  2  J  m  (<T)  2  9 0 1  q* n  where  In A  i s the p e r m i t t i v i t y of free space m i s the mass of an electron K i s Boltzman's constant T i s electron temperature q i s electron charge n i s electron density A is 9  where  i s the number of p a r t i c l e s  i n the Debye sphere 4 o K 23 -3 and n - 1.7 x 10 M -12 T = 3.9 x 10 sec. for T = 3 x 10  9 o  The speed of the electrons that compose the wave i s given by the phase v e l o c i t y , co/K, of the wave. In our case t h i s i s about 2 x 10^ m/sec.  Electrons  t r a v e l l i n g at the above speed w i l l therefore t r a v e l 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 i s about 4,0 x 10  3  A*. This means, on the average,  94  that the electrons t r a v e l 20 wavelengths between collisions.  According to Grewal the e f f e c t of  c o l l i s i o n s w i l l therefore be n e g l i g i b l e .  95  APPENDIX B F i t t i n g Technique Kegel's method of f i t t i n g t h e o r e t i c a l curves to the data i s used i n t h i s experiment.  This method-is  published i n the i n t e r n a l reports of the I n s t i t u t Fur Plasma Physik, which are not readily a v a i l a b l e .  There-  fore a b r i e f description of the method i s given. The i n t e g r a l i n equation ( i o ) (Chapter II) can be 25 put i n the form (see f o r example Tanenbaum , p. 181):  K G(K,u) = - ?z ~ K where:  and  2  2  2  1-2C (exp {Z -C }dZ + iTrexp{-C }  C = a>/K (2<T/m) 2  K /K D  2  =a  2  (as defined e a r l i e r )  The G+ (K,w) term i s the same as QA) except f o r the mass of the i o n i n C and the replacement of K Q with K^+ The form of S (K,w) i s then given by the sum of terms l i k e fp/j producted with a Gaussian term of the form:  2  (2KT/m)~^exp{-C }  96  If the parameters contained i n a are fixed (so that a i s 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 22 example Bekefi  (see f o r  ) . In fact there are two a terms, a and +  cc+ where ot+ i s given by K^ /K,  The a+ term a f f e c t s the  part of the spectrum near the central l a s e r frequency and i s related to the i o n term i n the s c a t t e r i n g .  We  can plot S (K,io) as a function of l o g ^ C f o r a given a . Recall now we can write:  l o g  io  c  =  I o 9  to  2KT  K  m  io  ( u ) )  +  © I o 9  io  { 2 K T  /m)"")S)  ®  From t h i s 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 s h i f t of the scale of frequency given by the l a s t term of (jiT) , We can use t h i s to determine the value of ct f o r an experimentally recorded spectra.  The  experimental data can be plotted on the same l o g (co) 10 scale and normalized to the same maxima as a series of t h e o r e t i c a l curves covering a range of a .  97  The curve that best f i t s the data, independent of u  the value of 1 ° 9 ^ Q ( )# gives an experimental value of a . Once t h i s value has been chosen, N and T can e e be c a l c u l a t e d .  Recall that curves of the same a  have a maximum at the same value of the parameter C, We can r e l a t e the experimental and t h e o r e t i c a l curves by: 2KT, l o g  l0  f o ra  m a x i m u m  =  )  1O<3  IQ  w  * th*  2KT  = log1Q  (uex) + log 10  m  +  l o g  l0  m  1  K  ©  ex  We can better r e l a t e t h i s l a s t equation to the experiment i f we r e c a l l : / \  2T r C 2  0  \  X j  where A X i s the observed wavelength s h i f t and X i s the laser wavelength also:  K  4 TT s X  i n  0 2  where 0 i s the scattering angle.  I f the t h e o r e t i c a l  9d  curves are constructed f o r the same laser wavelength X as used i n the experiment then (E) reduces t o :  ex _  AX  ex th  •ch  The value of T  [sin [sin  2  2  (9ex/2)]  ®  (G th /2)]  i s the only unknown factor i n H  .  The axis of the t h e o r e t i c a l curves can now be converted to units of A X (from F) which are the experimentally recorded u n i t s .  We also make use of the d e f i n i t i o n  of a : N. a = R  (sin  2  1)  where R i s a constant of f i x e d parameters.  I f we  consider that a i s by d e f i n i t i o n the same f o r both theory and experiment, (tf) and (T) reduce t o :  N (ex) e  AX  N e (th)  A X  es th J  ©  99  This gives us experimental values f o r both N Q and using the known values of 0 and X and the best f i t to t h e o r e t i c a l plots on a log^n ^ * s c a l e . The actual evaluation of N and T i s usually e e done using equation (^) and (j^ i n the following (take log^g ° ^ k °  t n  sides);  log10 Tex - 2 A - log10  +  l o g  form  io  T  t  h  +  (sin  l o g  io  2  0 e x /2)  ( s i n 2  0  th  / 2 )  The l a s t two terms can be put into numerical form because both T., and 0,. are f i x e d . th th  The A i sJ just  the s h i f t on the l°g^Q (A X) scale between the theor e t i c a l and experimental spectra. log  Theoretical  N  10 ex  =  l o gN  th  +  2  Similarly:  A  curves f o r a i n the range 0.1 to 3.0 are  given i n Kegel's report.  100  APPENDIX C To c a l c u l a t e S (K,w) f o r the d i s t r i b u t i o n by e q u a t i o n ' (16)  (Page 19) we use e q u a t i o n  assume the i o n s have a Maxwellian  given  (l4) .  We  d i s t r i b u t i o n with  a temperature equal t o the main e l e c t r o n d i s t r i b u t i o n . We chose t o c a l c u l a t e t h e spectrum f o r t h e K v e c t o r s p a r a l l e l t o the x a x i s ( K v e c t o r a l o n g x) .  =  , where i i s the u n i t  R e c a l l now e q u a t i o n (9 j f o r  iop G(K.w)  J| i v  G  (K,to).  f K.8f(v)/9v  — K2  to-K* v  Because the d i s t r i b u t i o n f u n c t i o n f (v) i s w r i t t e n a s : f ( v ) = (1-b) f  we can w r i t e  The  first  G  (K,to,f  term G  Q  (v,T) = b f  (v)) a s :  ( f ) i s g i v e n by: Q  1  ( (v - v ) , T ) D  101  G(K,W) =  -2u  f oo  z  2e_  u^K a  .  2  — 2 \  v' e x p ( - v £ a 5  2  2  3  2  d v. exp(-v^a~ ) dy exp(-v a"" )dv.  X  v -aC  where:  a = (2<T/m)  and:  C = (w+iv)/ K a  The second term i n (IT) can be obtained from (jT) with /  fi  N  of the form of the f i r s t term on the r i g h t of (16j :  K « 3 f ( v - v )/3v ^J=D  d  3  y  w-K *v  We can do t h i s i n general i f we r e c a l l so that K • v = Kv  x  Now: 2  8f (v-v^/av = 2 ( v - v D ) a " f (v-v D )  £||^  v  x  102  K.3f  (V-VQJ/SV  This allows us to write and again the v an  Y  v  «r;  a n <  l -  where  i n t e g r a l s give (a it**) f o r  G  ^ ^ becomes:  -2w G  and v  i n a form s i m i l a r to  Tv  2  a  TrVa  v  ex  a  2  v  -v  f x" xD> P^ " <"< x xD  )2)}  3  fv y  v x -ac  and / v have been completed, z  To do t h i s i n t e g r a l we make the s u b s t i t u t i o n V  V  v  - x - Dx  dv = dv  x 2  -2to  then:  G, =  2  2  f (v)exp{-v a" ) v+(v xD -aC)  dv  dv  x  (18^  103  This i s the same as  (17) with the change of the constant  i n the denominator.  This i s a standard i n t e g r a l and 25  can be evaluated (see f o r example Tanenbaum The i n t e g r a l i n manipulated  section  into a form whose value  can be c l o s e l y approximated by a s e r i e s .  The  series 26  has been summed and tabulated by F r i e d and Conte (1971) f o r a range of values of the constants i n ( i j ) or Once have (17) been and obtained of the (23) terms, G.. i n values (14) (like (23) )for the each value  iof S (K,w)  <y —-  can be  ^  calculated.  4,5),  —  ©  

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