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Scattering of laser light by laboratory plasmas. Chan, Ping Wah 1966

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SCATTERING OF LASER LIGHT BY LABORATORY PLASMAS k  by PING WAH CHAN B.Sc,  U n i v e r s i t y of Hong Kong, 1962  M . S c , U n i v e r s i t y o f B r i t i s h Columbia, 1964  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 required  t h e s i s as conforming  to the  standard  THE UNIVERSITY OF BRITISH COLUMBIA May, 1966  In presenting this thesis  in p a r t i a l  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 ex-  tensive 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 financ i a l gain shall not be allowed without my written permission.  Department The University of B r i t i s h Columbia Vancouver 8, Canada  The .University of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES  PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  of  PING WAH CHAN  B.Sc, The University of Hong Kong, 1962 M.Sc, The University of B r i t i s h Columbia, 1964 IN ROOM 301, HENNINGS BUILDING FRIDAY, JUNE 17, 1966, AT 10:30 A.M. COMMITTEE IN CHARGE Chairman: L. D. Hayward B. Ahlborn A. J . Barnard A. M. Crooker  F. L. Curzon M. M. Z . Kharadly R. A. Nodwell  External Examiner:  S. A. Ramsden  National Research Coundil Ottawa Research Supervisor: R. A. Nodwell  SCATTERING OF LASER LIGHT BY LABORATORY PLASMAS ABSTRACT The  scattering of laser l i g h t by~ laboratory  plasmas has been observed.  When the scattering was  from a plasma formed i n a 6-pinch, with scattering angle of 90° , a nearly Gaussian profile.of. the scattered intensity as a function of wavelengths was observed, corresponding to scattering by non-interacting When the scattering was from a plasma j e t  3  electrons.  with scattering  angle of 45° from the forward direction,, d i s t i n c t s a t e l l i t e peaks were observed on both sides of a narrow central peak at the laser frequency as predicted by theory, J-J  indicating a strong c o l l e c t i v e scattering  effect between the electrons and the ions.  The widths  of the s a t e l l i t e l i n e s were greater than the values predicted by theory.  The discrepancy  i s ascribed to  s p a t i a l variations i n the electron density i n the volume of the observed plasma.  The i n t e n s i t i e s and frequencies  at-which they occur these peaks also vary with the current of the plasma jet i n a manner consistent with theory. The  scattered intensity of the central peak was measured  approximately and i t agrees with t h e o r e t i c a l prediction. Some i n d i c a t i o n of perturbation  of the plasma by the  incident laser l i g h t has also been observed. 1.  E.E; Salpeter, Phys. Rev. 120, 1528(1960).  2.  J.A. Fejer, Can. J . Phys. 38, 1114(1960).  3.  J.P, Dougherty and D.T. Farley, Pro. Roy. Soc. (London) A259, 79(1960).  4.  M.N. Rosenbluth and N. Rostoker, Phys. Fluids 5., 776(1962).  GRADUATE STUDIES  F i e l d of Study:  Physics  Elementary Quantum Mechanics Waves Electromagnetic Theory  W. Opechowski J. Savage G. M„ Volkoff  Plasma Physics  L,  Sobrino  Special R e l a t i v i t y  Ho  Schmidt  Spectroscopy  A. M„ Crooker A o J . Barnard  Advanced Plasma Physics  A o J . Barnard  Plasma Dynamics  F o L o Curzon  PUBLICATION  C o l l e c t i v e Scattering of Laser Light by a PlasmaP. W. Chan & R. A. Nodwell Phys. Rev. Letters 16, 122 (1966).  PING WAH CHAN.  SCATTERING OP LASER LIGHT BY LABORATORY PLASMAS.  S u p e r v i s o r : Dr. R. A. N o d w e l l .  ABSTRACT The s c a t t e r i n g of l a s e r l i g h t by l a b o r a t o r y plasmas has been observed.  When the s c a t t e r i n g was from a plasma formed i n  a 0 - p i n c h , w i t h s c a t t e r i n g angle of 90°, a n e a r l y Gaussian  pro-  f i l e o f the s c a t t e r e d i n t e n s i t y as a f u n c t i o n o f wavelength,  was  observed, c o r r e s p o n d i n g t o s c a t t e r i n g by n o n - i n t e r a c t i n g e l e c t r o n s . When the s c a t t e r i n g was from a plasma j e t , w i t h s c a t t e r i n g  angle  of 45° from the forward d i r e c t i o n , d i s t i n c t s a t e l l i t e peaks were observed on both s i d e s o f a narrow c e n t r a l peak at the l a s e r  fre-  quency as p r e d i c t e d by t h e o r y , ^ ' ^ ' ^ i n d i c a t i n g a s t r o n g c o l l e c t i v e s c a t t e r i n g e f f e c t between the e l e c t r o n s and the i o n s .  The  widths o f the s a t e l l i t e l i n e s were g r e a t e r than the v a l u e s p r e d i c t e d by theory. tions plasma.  The d i s c r e p a n c y i s a s c r i b e d t o s p a t i a l  varia-  i n the e l e c t r o n d e n s i t y i n the volume o f the observed The i n t e n s i t i e s afcthe peaks and f r e q u e n c i e s at which  they occur a l s o vary w i t h the c u r r e n t of the plasma j e t i n a manner c o n s i s t e n t w i t h theory.  The s c a t t e r e d i n t e n s i t y of the  c e n t r a l peak was measured approximately and i t agrees w i t h theoretical prediction.  Some i n d i c a t i o n o f p e r t u r b a t i o n o f the  plasma by the i n c i d e n t l a s e r l i g h t has a l s o been observed.  -iii-  TABLE OF CONTENTS PAGE •i i  Abstract Figure  v  Captions  Acknowledgement  Vii  CHAPTER I  INTRODUCTION  1  CHAPTER II  THEORY  4  (A)  Thomson S c a t t e r i n g from E l e c t r o n s i n a Plasma  4  (B)  C a l c u l a t i o n of  (C)  D i s c u s s i o n of T h e o r e t i c a l R e s u l t s  10  GENERAL EXPERIMENTAL CONSIDERATIONS  18  (A)  S i g n a l of S c a t t e r e d L i g h t  18  (B)  Extraneous S i g n a l s and Noise (i) Stray L i g h t from L a s e r ( i i ) Plasma Luminosity ( i i i ) S t a t i s t i c a l Fluctuations  20  (C)  Signal-to-Nolse 22 (i) Improvement of S c a t t e r e d S i g n a l ( i i ) Reduction of L a s e r Stray L i g h t ( i i i ) Reduction o f Shot No£s© ( i v ) Reduction of Plasma Luminosity (v) Reduction o f E l e c t r i c a l and Magnetic Pick-up ( v i ) Sampling Technique  CHAPTER I I I  CHAPTER IV  <| <n (k,<^.)l*> e  8  SCATTERING FROM 9- PINCH  25  (A)  Apparatus and Procedure (i) The 0-Pinch (ii) The L a s e r ( i i i ) The Monochromator (iv) The P h o t o m u l t i p l i e r (v) The O p t i c s  25  (B)  Experimental Results  35  -ivPAGE CHAPTER V  SCATTERING FROM PLASMA JET  39  (A)  Apparatus and Procedure (i) The Plasma J e t ( i i ) The O p t i c s ( i i i ) Photo-detectors  39  (B)  Experimental R e s u l t s ( i ) CVP 150 P h o t o m u l t i p l i e r Work ( i i ) EMI 9558B P h o t o m u l t i p l i e r Work  45  CONCLUSION  58  CHAPTER VI APPENDIX  60 <|^«Ck,uj>|>  A.  C a l c u l a t i o n of  B.  D i s c u s s i o n o f Case o f P r a c t i c a l I n t e r e s t  65  C.  Spectrum o f S c a t t e r e d R a d i a t i o n f o r °<- ^ I  65  D.  Integrated  67  E.  F o r t r a n Programme f o r c a l c u l a t i o n of S c a t t e r e d Spectrum 68  REFERENCES  t  Intensity Calculations  60  70  - V -  FIGURE CAPTIONS PAGE Figure 1.  Geometry o f S c a t t e r i n g of E l e c t r o m a g n e t i c a Plasma  2.  T h e o r e t i c a l S c a t t e r e d Spectrum f o r °C < | (Hydrogen Plasma at T* = Z5000°<  3.  J  y  2-1  12  14  9 = 90°).  T h e o r e t i c a l S c a t t e r e d Spectrum f o r (Argon plasma  '5a:  9=90" ).  T h e o r e t i c a l S c a t t e r e d Spectrum f o r oc ~ / (Hydrogen Plasma at Te. = Zsooo°K  4.  R a d i a t i o n by  oc > /  16  j T = l35~oo"k; , 9 = 4-5°). e  5.  Schematic Drawing of Theta-Pinch D i s c h a r g e  26  6.  Schematic Drawing o f O p t i c a l Arrangement  27  7.  Block Diagram of T r i g g e r System  29  8.  (a) Current  31  waveform (0-pinch)  (b) Plasma l i g h t 9.  10.  (0-pinch)  31  (a) P h i l i p s CVP 150 P h o t o m u l t i p l i e r C i r c u i t r y -  33  (b) L i n e a r Response of CVP 150 P h o t o m u l t i p l i e r  33  T y p i c a l O s c i l l o s c o p e . Traces (a) L a s e r (b) Plasma (c) L a s e r  ((9-spinch)  36  alone alone and Plasma  11.  Spectrum of S c a t t e r e d L i g h t from 9-pinch at 90°  37-  12.  The Plasma J e t  40  13.  Voltage-Current  14.  Schematic Diagram o f S c a t t e r i n g o f Laser L i g h t from a  C h a r a c t e r i s t i c ; o f Argon Plasma J e t  42 44  Plasma J e t . 15.  (a) EMI 9558B P h o t o m u l t i p l i e r C i r c u i t r y  46  (b) L i n e a r Response o f EMI 9558B P h o t o m u l t i p l i e r  46  - V I -  PAGE 16.  T y p i c a l O s c i l l o s c o p e Traces (Plasma J e t ) taken  48  w i t h CVP 150 P h o t o m u l t i p l i e r . (a) ;V= 7016A  (b) A = 6910A (c) A = 6986A 17.  Averaged O s c i l l o s c o p e Traces (Plasma Jet)  49  18.  Scattered  50  19.  E q u i l i b r i u m Composition o f Argon Plasma at  Spectrum of Laser L i g h t from Plasma J e t  52  Atmospheric P r e s s u r e 20.  (a) I n t e n s i t y of Laser L i g h t s i g n a l at 6943A as a  54  f u n c t i o n o f plasma j e t c u r r e n t (b)  Scattered  light  at 6943A, u s i n g s l i t of 1/2A  54  passband 21.  O s c i l l o s c o p e Traces o f S c a t t e r e d  S i g n a l (Plasma  56  J e t ) taken w i t h EMI 9558B P h o t o m u l t i p l i e r at (a) A=  6900A  (b) ?v= 6910A (c) 7v= 6986A 22.  Scattered  Spectrum o f L a s e r L i g h t by Plasma J e t  57  at 300 amps 23.  Scattered  Spectrum o f L a s e r L i g h t by Plasma J e t  at 250 amps  57 (a)  -vii-  ACKNOWLEDGEMENT I wish t o thank Dr. R.A. Nodwell f o r h i s i n v a l u a b l e superv i s i o n and encouragement d u r i n g the whole course o f t h i s work. Thanks are a l s o due;to Dr. J.H. W i l l i a m s o n f o r h i s s e r v i c e i n programming the t h e o r e t i c a l curves and best f i t s ,  t o Dr. B.  Ahlborn and Mr. R. M o r r i s f o r t h e i r a s s i s t a n c e i n s e t t i n g up the plasma j e t , t o Dr. L. de Sobrino f o r h i s help i n the t h e o r e t i c a l aspects of t h i s work, and t o Dr. J.A. Barnard, Dr. A.M. Crooker, Dr. F.L. Curzpn fipr r e a d i n g and improving t h i s  thesis.  The a s s i s t a n c e of Mr. W. R a t z l a f f and Mr. J . Dooyeweerd i n the work o f e l e c t r o n i c s , and the a s s i s t a n c e o f Mr. J . Lees and Mr. T. Knopp i n the c o n s t r u c t i o n of the apparatus  i s deeply  appreciated. F i n a n c i a l a s s i s t a n c e i n the form o f a NRC Studentship duri n g the course of t h i s work i s a l s o g r a t e f u l l y  acknowledged.  T h i s work i s supported by a grant of the Atomic C o n t r o l Board of Canada.  Energy  -1-  CHAPTER I  INTRODUCTION  In r e c e n t years t h e r e has been a v e r y great i n c r e a s e i n interest  i n the p r o p e r t i e s o f e l e c t r o m a g n e t i c r a d i a t i o n s c a t t e r e d  by a plasma.  T h i s i n t e r e s t was s t i m u l a t e d by an o b s e r v a t i o n made  by Bowles,^ who found that the s p e c t r a l p r o f i l e o f the back s c a t t e r ed r a d i a t i o n from a r a d a r beam i n c i d e n t on the ionosphere has a very narrow width corresponding to D&ppDLer broadening ion v e l o c i t y .  d;ue to the  F o l l o w i n g the p u b l i c a t i o n of these r e s u l t s , the  theory o f s c a t t e r i n g o f e l e c t r o m a g n e t i c waves by a plasma has been developed Rosenbluth  by S a l p e t e r ,  and R o s t o k e r .  4  1  "2 3 F e j e r , Dougherty and F a r l e y , and  T h i s theory p r e d i c t s t h a t , i f the  wavelength of the i n c i d e n t r a d i a t i o n i s l a r g e compared with the Debye l e n g t h of the plasma, the s c a t t e r e d p r o f i l e c o n s i s t s of a c e n t r a l jpeak and two weak s a t e l l i t e l i n e s at approximately the plasma frequency from the c e n t r a l frequency. S a l p e t e r and Yngnessen,® have observed  Recently Perkins,  these s a t e l l i t e  lines  s c a t t e r e d by the ionosphere u s i n g a very powerful radar beam. S t e r n and Tzoar,^ have a l s o observed  the s a t e l l i t e l i n e s by en-  hancing the l o n g i t u d i n a l e l e c t r o s t a t i c o s c i l l a t i o n s i n the plasma w i t h microwaves. Because o f the success of the theory i n e x p l a i n i n g Bowie's r e s u l t s and the p o s s i b i l i t y of d e t e r m i n i n g i n t e r e s t i n g plasma parameters such as e l e c t r o n and i o n temperature; at a s m a l l p o i n t i n space  electron density  and time, i t i s n a t u r a l that plasma  p h y s i c i s t s everywhere have been i n t r i g u e d w i t h t h e p o s s i b i l i t y of u s i n g s c a t t e r i n g of r a d i a t i o n as a d i a g n o s t i c t o o l i n l a b o r a t o r y plasmas.  I t has been found d i f f i c u l t , however, to apply the f  i r \  -2technique to l a b o r a t o r y plasmas mainly because of the extremely small s c a t t e r i n g cross s e c t i o n f o r electrons. the recent  It i s o n l y w i t h  development of l a s e r s which are very powerful monochro-  matic l i g h t sources that experiments of t h i s k i n d i n the l a b o r a t o r y . have become f e a s i b l e . was  reported  The  by F i o c c o and  from an e l e c t r o n beam.  first  successful laboratory  Thompson,® who  scattered laser light  Subsequently s e v e r a l workers have used  t h i s technique f o r d i a g n o s i s of plasma p r o p e r t i e s . Kronast, and  Kunze and  have r e p o r t e d obtained  experiment  K e g e l , ® » ; and Davies and  Funfer, Ramsden,  10  s c a t t e r i n g of l i g h t from a t h e t a p i n c h  and  have  a nearby Gaussian s c a t t e r e d p r o f i l e c o r r e s p o n d i n g to  s c a t t e r i n g by n o n - i n t e r a c t i n g  electrons.  However, u n t i l  r e c e n t l y the numerous attempts, to observe the s a t e l l i t e have not  11  been s u c c e s s f u l .  The  very lines  c h i e f d i f f i c u l t i e s have been the  problem of e l i m i n a t i n g the s t r a y l i g h t from w a l l s , windows and impurities.and  the problem of o b t a i n i n g a uniform and 12  plasma.  S e v e r a l r e p o r t s have been made,  i s an i n d i c a t i o n that a s a t e l l i t e may are ambiguous.  Two  14  ' °'  be present  ' i n which but  there  the r e s u l t s  papers have appeared i n the l i t e r a t u r e  l y i n which the unambiguous o b s e r v a t i o n t o r y experiments are r e p o r t e d . report observing  1 *3  constant  recent-  of s a t e l l i t e s i n l a b o r a -  Ramsden and  Davies,^  the s a t e l l i t e s u s i n g a t h e t a pinch d e v i c e .  The  16  plasma source employed by the author (Chan and Nodwell)  is a  plasma j e t . In t h i s t h e s i s we wish to r e p o r t on the o b s e r v a t i o n s  made  on the p r o f i l e s of the s c a t t e r e d r a d i a t i o n from plasmas formed i n the t h e t a p i n c h and  a l s o i n a high c u r r e n t d.c.  Chapter 2 i s a summary of the theory  plasma j e t .  of the s c a t t e r i n g i n which  -3-  the p h y s i c a l s i g n i f i c a n c e of the s t e p s i n the d e r i v a t i o n s are d i s c u s s e d and a resume of the assumptions  and the r e s u l t s i s g i v e n ,  For those readers i n t e r e s t e d i n a more d e t a i l e d t h e o r e t i c a l c u s s i o n a more complete  d e r i v a t i o n i s given i n the Appendix.  Chapter 3 we d i s c u s s the p r a c t i c a l d i f f i c u l t i e s a s c a t t e r i n g experiment  disIn  i n the d e s i g n of  and g i v e p o s s i b l e experimental techniques  to overcome these d i f f i c u l t i e s .  Chapter 4 i s devoted  to a par-  t i c u l a r d i s c u s s i o n of the s c a t t e r i n g from a t h e t a p i n c h plasma and Chapter 5 d e s c r i b e s the experiment last  on the plasma j e t .  The  chapter (Chapter 6) summarizes the r e s u l t s o b t a i n e d and  d i s c u s s e s the s i g n i f i c a n c e of these r e s u l t s .  The advantages and  disadvantages of the s c a t t e r i n g method as a d i a g n o s t i c t o o l are d i s c u s s e d and s u g g e s t i o n s f o r f u t u r e work g i v e n .  -4-  CHAPTER II  THEORY  The power flux spectrum of electromagnetic  waves scattered  by electrons in a plasma w i l l f i r s t be calculated using the Thomson theory.  in section A  We shall see that this expression  contains a term involving the calculation of the ensemble average of the Fourier transform of the electron density.  This quantity  has been calculated by Salpeter,^ Fejer, Dougherty and Farley,** 2  and Rosenbluth and Rostoker.  4  In calculating this quantity, we  use a method of superposition of dressed test particles suggested by Rosenbluth and Rostoker.^  (A dressed test particle is a charged  particle in which the screening strength of the surrounding plasma is allowed for by postulating an attendant polarization cloud). The detailed mathematical derivations are given in the Appendix, and only the assumptions and principal predictions of this c a l culation^are included here in section B.  A graphical presenta-  tion and a discussion of these results are given in section C. (A) Thomson Scattering from Electrons in a Plasma Fig.  1 shows the geometry of scattering of  wave by a plasma.  electromagnetic  Consider a charge with velocity v under the  influence of a plane electromagnetic  wave of frequencyH„ and  wave vector K|. Assume that the incremental velocity produced on the charge by the f i e l d E is negligible compared with the velocity v and that v/c « 1 where c is the velocity of light, that is we consider a non-relativistic plasma not heated by the incident wave.  Quantum-mechanical effects are also neglected.  Since only a very small part of the incident radiation is scattered, so that we can use the f i r s t Born approximation and  Plane  Incident  F i g . 1:  Radiation  Geometry of S c a t t e r i n g of E l e c t r o m a g n e t i c R a d i a t i o n by a Plasma  -6-  w r i t e the t o t a l f i e l d £  =  The  &  exj> L-p,  at r as the i n c i d e n t 1  field.  k r'J]  t -  r  (1)  a c c e l e r a t i o n of an e l e c t r o n p a r t i c l e at r ' under the  i n f l u e n c e of the plane wave i s  V = Jk Eo * * f > [ - i ( A . t ' - * r l 9 ] where q , m , e  ly.  (2)  are the charge and mass o f the e l e c t r o n  e  ( S c a t t e r i n g due to ions can be n e g l e c t e d  heavy mass)  because o f t h e i r  The r a d i a t i o n f i e l d o f a s i n g l e a c c e l e r a t e d  at an o b s e r v a t i o n  2/ 2  where e /mc f ~  = r +  electron  p o i n t r at time t i s given by  = c l a s s i c a l electron  Q  l £  T i s a unit vector The  respective-  (y  radius  r»r')  i n the d i r e c t i o n of r .  t o t a l f i e l d E,, s c a t t e r e d by the e l e c t r o n  distribution  —s n ( r , t ) i s the v e c t o r sum of the c o n t r i b u t i o n from i n d i v i d u a l e l e c t r o n s and i s given by  B (tj = r ** £* *)jdl±'ri (£:-t')**f>y(n,t'-k r')] c  s  B  0  e  r  where  -»e (!< t) «  Z  £ ( l ~ A <t>)  (5)  where £j i s the p o s i t i o n of the j^ e l e c t r o n 1  At the p o i n t o f o b s e r v a t i o n considered  the s c a t t e r e d wave may be  p l a n e , so that the magnitude of the s c a t t e r e d power  flux i s  Sd)  -  ^ -  E^U)  (6)  To convert t h i s time f u n c t i o n of f l u x t o a frequency f u n c t i o n of f l u x we take the F o u r i e r t r a n s f o r m and apply  )  (4  Parseval's  Theorem  17  S ( n j - ^  (7)  where (8)  —rv  i s the F o u r i e r transform Combining equation  of E ( t ) .  (4) and (8) and e x p r e s s i n g  t i n terms of t * ,  we get  or  EsCtu) - £ e ^  (n,  Vc  n ~n.)  uo>  2  where (ii) (12)  The  power spectrum d e n s i t y i s t h e r e f o r e given by (14) Because the o b s e r v a t i o n  time i s long compared with the  f l u c t u a t i o n time, we are i n t e r e s t e d i n the ensemble average of this quantity, i . e . -  J i l l  ^(Ac'i,  'fli-A)  ^  (15)  F i n a l l y the s c a t t e r e d power spectrum d e n s i t y may be w r i t t e n as  6  = N£  0- " ^ rXI^(^l > s  e  l  z  where N i s the t o t a l number of s c a t t e r i n g e l e c t r o n s , 0 i s the s c a t t e r i n g angle (see F i g . 1)  de)  -8-  ij? the angle between the e l e c t r i c v e c t o r E of t h e i n c i d e n t l i g h t (assumed to be l i n e a r l y p o l a r i z e d ) and the s c a t t e r i n g plane = Ik —a  d e f i n e d by K  and K  -1  *  -  K, -  K  w  =  n, - n  Q  ?  c  and  z  z  <^ f ^ C ^ a j ) ! ^  (B) C a l c u l a t i o n of Consider  2  a plasma w i t h average e l e c t r o n d e n s i t y n , e l e c t r o n Q  temperature T_ and i o n temperature T. and e l e c t r o n Debye l e n g t h  f—J^Js  where e i s the e l e c t r o n charge and K the  )  Boltzmann c o n s t a n t . (i)  The f o l l o w i n g assumptions a r e made:  The e l e c t r o n s and i o n s have Maxwellian v e l o c i t y d i s t r i b u t i o n s ,  but T  e  i s not n e c e s s a r i l y equal to T^.  ( i i ) There i s no e x t e r n a l magnetic ( i i i ) There i s no e l e c t r o n d r i f t (iv)  field.  r e l a t i v e to the i o n s .  We assume that a sphere of r a d i u s equal t o the Debye length c o n t a i n s many e l e c t r o n s , i . e . the q u a n t i t y  A = x m c  The (  3 h  ^  hdSl«  /  (17)  i n q u a l i t y i m p l i e s that the Coulomb i n t e r a c t i o n ) between nearby e l e c t r o n s i s s m a l l compared to the  thermal k i n e t i c energy (v)  »  K. Te •  We f u r t h e r assume that the s c a l i n g l e n g t h k d e f i n e d as ^ = 4K 7C  1  (P^z )  where X i s the i n c i d e n t wavelength and  9 i s the s c a t t e r i n g angle from the forward d i r e c t i o n i s s m a l l compared with  the s m a l l - a n g l e  Coulomb and e l e c t r o n -  n e u t r a l mean f r e e paths so t h a t c o l l i s i o n s may be n e g l e c t e d 18  in the c a l c u l a t i o n .  -9-  The plasma i s c o n s i d e r e d t o c o n s i s t of f u l l y dressed p a r t i c l e s t h a t are uncorrelated,, that i s that the t e s t move w i t h assigned o r b i t s through  test  particles  the plasma w i t h p a r t i c l e  t r i b u t i o n f u n c t i o n d e s c r i b e d by the Vlasov equations.  dis-  We a l s o  assume that the v a r i a t i o n of the p a r t i c l e d i s t r i b u t i o n f u n c t i o n i s n e g l i g i b l e d u r i n g times of the o r d e r of cuy' and .that t h e s p a t i a l inhomogeneity i s n e g l i g i b l e over d i s t a n c e o f t h e o r d e r o f the ttebye l e n g t h ^ ) . Rosenbluth of  More g e n e r a l cases have been c o n s i d e r e d by  and Rostoker*  i n c l u d i n g the c a l c u l a t i o n i n the presence  a constant- magnetic f i e l d and the case when there i s an e l e c t r o n  drift  r e l a t i v e t o the i o n s . The  q u a n t i t y 0'&0>)l \  s  the double F o u r i e r t r a n s f o r m of the  d e n s i t y a u t o c o r r e l a t i o n f u n c t i o n d e f i n e d as (18) as can be  seen,  J>  " < W * ^ > (19)  I t t u r n s out t h a t i t i s e a s i e r t o c a l c u l a t e the q u a n t i t y  CCf,z)  and get the quantity</4tV)j)by  subsequent double  transform.  T h i s i s s u f f i c i e n t f o r a s t a t i o n a r y s p a t i a l l y homogeneous plasma (which we have assumed) because the d e n s i t y a u t o c o r r e l a t i o n funct i o n i s independent of r and t and the s t a t i s t i c a l  avergging can  be interchanged w i t h the r and t i n t e g r a t i o n s . The method of s u p e r p o s i t i o n s of dressed t e s t p a r t i c l e s cons i s t s i n f i n d i n g the e f f e c t on the number d e n s i t y due t o a t e s t p a r t i c l e at a g i v e n p o s i t i o n w i t h a given v e l o c i t y .  By assuming  the t e s t p a r t i c l e s to be u n c o r r e l a t e d , t h e s t a t i s t i c a l  average i s  -10-  calculated  by i n t e g r a t i n g  over the volume of t h e plasma and over  a l l v e l o c i t i e s f o r a l l s p e c i e s of the p a r t i c l e s .  This  calculation  i s c a r r i e d out i n the Appendix and the f i n a l r e s u l t s , given i n S a l p e t e r ' s form, i s  •= . <\4M«t>= where  lf-fr/*^'*fc/ F>;  ff  A  ( 2 0 )  Z = degree of i o n i z a t i o n  Vln  *%•  J t  Q <s») =-</i[i~{k) t  +  ;n*xtop-**-)'],x=-%k, ^ = ( ^ S £ )  -f(x) = 2 x e«f(-x*)j*  i eft 1  where X i s the wavelength of i n c i d e n t (C) D i s c u s s i o n of T h e o r e t i c a l  radiation  Results  We c o n s i d e r the case of p r a c t i c a l i n t e r e s t , that hj^m^ and T ^ T expression  .  I t i s shown i n the Appendix that  the g e n e r a l  (20) can be reduced t o  <lto">r>*«where  i s with  /3* =  7 7c <**  ^fCw^s * ?(-^)>y) ig]  (2i)  -11-  The  e s s e n t i a l f e a t u r e s of the s c a t t e r e d s p e c t r a l d e n s i t y  w i l l be determined by the shape of the f u n c t i o n s /^OOand  T (y)  both being even f u n c t i o n s of x and  shapes  of the f u n c t i o n s v a l u e s of ex.  y respectively.  /^oOhas been given  The  parameter  The  by Salpeter"'' f o r  ~ 4^->^s;»(6A)  *  s  a  ,  fi  various  measure of  the  r a t i o of i n c i d e n t r a d i a t i o n wavelength to the Debye length  and  c o n t r o l s the degree of c o l l e c t i v e s c a t t e r i n g whereas .X - ^(r^Sr) i s a measure of frequency at which s c a t t e r i n g i s observed.  We  s h a l l consider  the shapes of the s c a t t e r e d p r o f i l e f o r  s e v e r a l v a l u e s of ot. «  (i)  I  P h y s i c a l l y i t h i s corresponds roughly to the case when the wavelength of the  i n c i d e n t r a d i a t i o n (A)  Debye l e n g t h  of the plasma.  (X^)  i s smaller  than  the  When \«Xj> , the phases of  the  Wra/K.6rlie;t'Sj s c a t t e r e d  by e l e c t r o n s w i t h i n a Debye sphere i s  uncorrelated,  t h e r e f o r e expect the s c a t t e r e d spectrum to  and  one  be Doppler broadened by the random thermal e l e c t r o n v e l o c i t y . F i g . 2 shows some t h e o r e t i c a l curves of the s c a t t e r e d for T  = 25,000° K f o r s e v e r a l s m a l l v a l u e s o f oi . '  e  zero, fe(x) becomes ~fc<x) = exf>(z-x ) and z  characterised  has  profile  As extends to  a Gaussian shape w i t h a spread  by the e l e c t r o n thermal v e l o c i t y .  The  width  of t h i s spread at h a l f i n t e n s i t y i s by d e f i n i t i o n given X  2  - in 2  by (22)  or  A n = w  2 $s--|(2*& L i f  In wavelength u n i t s  (21)  (23)  becomes (24)  -12-  T h e o r e t i c a l S c a t t e r e d Spectrum f o r di< | c a l c u l a t e d f o r Hydrogen Plasma w i t h Z = 1, T = 25,000°K, 0 = 90° e  -13where the r e l a t i o n A.  (25)  .a  has been used. The t o t a l width at h a l f i n t e n s i t y (twice AA. ) i s t h e r e f o r e (26)  2)*  The c o n t r i b u t i o n from the second term o f (19), i . e . from  2 (j~f^r) Ffit-y)  i  the f i r s t  s  n e g l i g i b l e compared to the c o n t r i b u t i o n from  TZcx) because i t i s a f a c t o r  term  smaller.  ( C e n t r a l peaks i n F i g . 2) (ii)  QC ^ I  Some t y p i c a l s c a t t e r e d p r o f i l e s f o r a T  g  = 25,000°K are  shown i n F i g . 3 where we see that the e l e c t r o n s a t e l l i t e begin t o emerge a s a p p r o a c h e s u n i t y .  The i n t e g r a t e d  peaks  intensities  of the two c o n t r i b u t i o n s from £br,/and ^ ( y ^ c a n be obtained integration  (see Appendix) roe  Therefore  and one o b t a i n s  * 77^  < »>  00 w =  J_</>  2  the t o t a l f . i n t e n s i t y i s p r o p o r t i o n a l t o  For s m a l l values  by d i r e c t  .  2 a  of oc, t h i s i s u n i t y and corresponds to the  c l a s s i c a l Thomson s c a t t e r i n g c r o s s - s e c t i o n . f a c t o r approaches to h a l f a s y m p t o t i c a l l y .  As  increases,  this  The r a t i o o f i n t e g r a t e d  i n t e n s i t y of each s a t e l l i t e t o the c e n t r a l peak i s 1/2 to t J  .  (iii)  . , and approaches 06 »  d  f o r l a r g e ci .  /  In t h i s case the wavelength of the i n c i d e n t r a d i a t i o n i s l a r g e r than the Debye length and the e l e c t r o n s a c t c o l l e c t i v e l y  -14-  20  o  40  20  o  AA(A)  40  T h e o r e t i c a l S c a t t e r e d Spectrum for<*~ I c a l c u l a t e d f o r Hydrogen Plasma with Z = 1, T = 25,000°K, 9 = 90° e  -15and s c a t t e r the i n c i d e n t r a d i a t i o n as a c l o u d . I t i s shown i n the Appendix that  7^c<) has a very sharp  maximum near x=4- x where x i s the s o l u t i o n of — o o Xo*= or  i  6* +3)  u>f +  co,*=  (29)  l  m)  3JL*1L.  m  \Bteich.od?s• • thseliwll >?£jwwa<tt&g{iti£sto*M£t  £3*If6*gitudirial  : h e  e l e c t r o s t a t i c : plasma.coasci?!.!actions. In a d d i t i o n when x i s very near x , I^W o by the L o r e n t z i a n shape, (see Appendix) £(xj  - i o< ex/>C-x^){^y-yj 2i-Ci7!: o/ 4<xf> 2  i  can be  approximated  (32)  C-Xo )] }' 2  2  1  The widths of these maxima are due to the s o - c a l l e d Landau damping of the plasma o s c i l l a t i o n s . each maximum i s given by 7L' 4  j°°  fi(xj  dx  The i n t e g r a t e d i n t e n s i t y of  (see Appendix) =  -L <x~  (32)  2  and decreases r a p i d l y as «- i n c r e a s e s although the h e i g h t i n c r e a s e s very r a p i d l y .  F i g . 4 shows some t h e o r e t i c a l s c a t t e r e d  f o r s e v e r a l v a l u e s of oi>| at a T =13,500°K. e  profiles  I t should be remem-  bered that although the t h e o r e t i c a l widths of these " s a t e l l i t e s " peaks are v e r y narrow, i n p r a c t i c e s m a l l v a r i a t i o n s of the e l e c t r o n d e n s i t y w i l l v a r y the plasma frequency and w i l l broaden spectrum of these ^ s a t e l l i t e s " .  C o l l i s i o n may  a l s o broaden  the these  "satellites." The c o n t r i b u t i o n from the second term i n v o l v i n g important o n l y f o r s m a l l frequency s h i f t s .  ~fe(yj i s  The i n t e g r a t e d i n -  t e n s i t y of t h i s term g i v e s the i n t e n s i t y of the ' c e n t r a l peak* of the s c a t t e r e d spectrum and v a r i e s a s y ^ ^ f ^ i * * ) ( If Z JI03  1,(1=1 as  as w e l l as  oi—oo  s e e  and T7(y)has a f l a t - t o p shape.  , we have / 9 » I and the i o n component  Appendix). I f ?7e>7> J^(yJ  has  -16-  -,6  6,r  oc= 2.0  oc = 1.0  ~4  rO  rO  o  b  I  'o  o  ojo  A X (A)  AXfA) 8  8  <* =  T o  4  8r-  oc = 4.0  3.0  rO  ^  Z i  -ri  o  X  X  421-  2 c.  20  0  A X F i g . 4:  0  40  0  rO  4  i  o  O  X  o  in  0  .8  o  c  cn  2  0  0 0  20  A X  (A)  0  40  ( A )'  T h e o r e t i c a l S c a t t e r e d Spectrum fortf>' c a l c u l a t e d f o r Argon Plasma with Z = 1, T - 13,500°K, 0 = 45° e  a Lorenztian represents  shape l i k e equation (27).  T h i s sharp resonance  the s o - c a l l e d p o s i t i v e - i o n o s c i l l a t i o n s , whose frequency  i s the same as that o f a plasma o s c i l l a t i o n f o r a f i c t i c i o u s p a r t i c l e w i t h i o n charge and mass but w i t h e l e c t r o n The  temperature.  p o s i t i o n s o f these resonances are given by a s i m i l a r d i s p e r -  s i o n r e l a t i o n to (26),  ,2 . where  copi  (Jpi + AUHL  (33)  i s the plasma frequency d e f i n e d  To summarize:  f o r the i o n s .  F o r s m a l l oCcCx^\>), the s c a t t e r e d  shows a Gaussian p r o f i l e whose width i s c h a r a c t e r i z e d e l e c t r o n thermal v e l o c i t y .  spectrum by the  The e f f e c t i v e s c a t t e r i n g c r o s s - s e c t i o n  equals t h e Thomson s c a t t e r i n g c r o s s - s e c t i o n .  F o r l a r g e oi ( x » A £ > )  the s c a t t e r e d spectrum c o n s i s t s o f a c e n t r a l peak whose width i s characterized  by the i o n thermal v e l o c i t y and two " s a t e l l i t e s "  separated from the c e n t r a l peak by approximately uip, the f r e quency o f e l e c t r o s t a t i c o s c i l l a t i o n s i n the plasma. of the " s a t e l l i t e s "  i s c o n t r o l l e d by Landau Damping.  intensity scattered  i s equal t o  r a t i o of the i n t e g r a t e d peak i s  oi' 2.  The width The t o t a l  o f the Thomson v a l u e .  The  i n t e n s i t y o f each s a t e l l i t e t o the c e n t r a l  -18-  CHAPTER I I I  GENERAL EXPERIMENTAL CONSIDERATIONS  A t y p i c a l experimental arrangement fdarxobserving  scattered  l i g h t c o n s i s t s o f a l a s e r as the l i g h t source, a plasma as the s c a t t e r e r , and an a n a l y s e r photomultiplier ful  combination.  t o observe both because o f the s m a l l s c a t t e r -  c r o s s - s e c t i o n and because o f the presence o f l a r g e amount o f  n o i s e due to extraneous l i g h t . desired l i g h t : l i g h t from w a l l s and  D e s p i t e the f a c t that a very power-  l a s e r beam i s used f o r the i n c i d e n t r a d i a t i o n , the s c a t t e r e d  signal i s d i f f i c u l t ing  system such as a monochromator :arid  There are two sources o f t h i s un-  The s t r a y l i g h t due to r e f l e c t i o n o f the i n c i d e n t and windows o f the apparatus and f o r e i g n p a r t i c l e s  the e m i s s i o n from the plasma i t s e l f .  noise  In a d d i t i o n the d e t e c t o r  a l s o s e t s a low l i m i t to the i n t e n s i t y o f l i g h t t h a t can be  measured.  S e c t i o n A i s devoted to e s t i m a t i n g  the s i z e of the  s c a t t e r e d s i g n a l , w h i l e i n S e c t i o n B the main sources of n o i s e are discussed.  In S e c t i o n C p r o p o s a l s f o r o p t i m i z i n g  the s i g n a l to  noise r a t i o ( % ) are discussed. (A) S i g n a l o f S c a t t e r e d  Light  A q u a n t i t a t i v e measure o f s c a t t e r i n g i s c o n v e n i e n t l y  expressed  by the r a t i o o f the power s c a t t e r e d by the plasma i n a given  dir-  e c t i o n to the energy f l u x d e n s i t y o f the i n c i d e n t r a d i a t i o n .  This  r a t i o has the dimension o f area and i s c a l l e d the e f f e c t i v e s c a t t e r ing  c r o s s - s e c t i o n d e f i n e d by dC=E  (1)  where the bar means a time average. dp  i s the energy s c a t t e r e d by the plasma i n t o the s o l i d  dfi  p e r s e c . and I the P o y n t i n g f l u x o f the i n c i d e n t wave.  angle  19-  =  F o r a monochromatic rest,  polarized  beam i n c i d e n t o n a f r e e c h a r g e a t  i  t h e quantitya<r h a s been c a l c u l a t e d  19  t o be  6 f h V da  d<r =  (2)  where / i s t h e a n g l e between t h e s c a t t e r i n g d i r e c t i o n incident If  E  into  T X  = _  If  energy o f the l a s e r p u l s e w i t h a time  an a r e a A,  duration»  then  U  -r— A * —U.  hv where^is  the  field.  U i s the t o t a l  t focused  and  ergs sec  -1-2 cm  photons  sec™^-cm~^  ,o\  w;  At  t h e e n e r g y o f an i n c i d e n t  photon.  t h e s c a t t e r i n g volume ofr=» A . l where 1 i s t h e l e n g t h o f t h e  s c a t t e r i n g p l a s m a volume and n  d.f  G*s»>iV A  »<L dSL  = —LL.  r s,Sy.JL  dJ~L  (4) g i v e s  a n g l e iQ.ftfom a r e moving  i s the e l e c t r o n d e n s i t y ,  = j-j^r h V  Equation  g  one  x  c  photon s e c '  as i n a p l a s m a ,  (4)  1  photons per p u l s e  t h e number o f p h o t o n s s c a t t e r e d  laser pulse.  then  into  the  For a system of e l e c t r o n s  the s c a t t e r e d p r o f i l e w i l l  be  solid  which broadened  d e p e n d i n g on t h e p a r a m e t e r o C a c c o r d i n g t o t h e manner d e s c r i b e d i n Chapter I I . plier will  A detector,  see o n l y  s u c h as a monochromator and  a f r a c t i o n of the photons given  (4) d e p e n d i n g o n t h e b a n d p a s s part of the s c a t t e r e d i s h and of  of the d e t e c t o r  the p h o t o m u l t i p l i e r ,  then the c u r r e n t  i n equation  and t h e  spectrum under o b s e r v a t i o n .  i f q i s t h e quantum e f f i c i e n c y  photomulti-  particular  If this  ( i n electrons per  = -J^— ' A y r  £  2  ?>»^Y>X •  e  photon)  at the photocathode of  the p h o t o m u l t i p l i e r i s  I*  fraction  'A' 9 • MfL . *  amps  (5)  -20Numerical Example Power of L a s e r = 10. MW Area of focused  (1/2  beam = 0.5  j o u l e i n 50 mm  Length of plasma colume = 2 16 E l e c t r o n d e n s i t y = 10 S o l i d angle  =  .01  x 0.5  ns)  mm  mm  "-3 cm  stetadian  Quantum e f f i c i e n c y f o r a S20  ( i . e . f/10  monochromator)  photocathode s u r f a c e  at 7000 A =  3%  (.03  e l e c t r o n per photon) According to (5), the photocathode -7 c u r r e n t i g = hlO amps. Even at the s a t e l l i t e s , f o r an otof 5, (h ^ 1 / 5 0 ) , the photocathode c u r r e n t ip= 2 x 10~® amps. I f the 5 gain of the p h o t o m u l t i p l i e r G = 10 anode i s a few able.  , the output c u r r e n t  tenths of a m i l l i a m p .  The d i f f i c u l t y does not  and  should  at  the  be e a s i l y measur-  l i e so much i n the smallness of  s i g n a l as i n the l a r g e n o i s e background a g a i n s t which one  the  observes  the s i g n a l . (B) Extraneous S i g n a l s and The  Noise  main sources of extraneous s i g n a l s are the s t r a y  from the l a s e r , that i s the l a s e r l i g h t which e n t e r s without being plasma and  s c a t t e r e d by  The  detector  the plasma, the s e l f - r a d i a t i o n of  the n o i s e from the d e t e c t o r  ( i ) Stray L i g h t from  the  light  the  itself.  Laser  source of l i g h t i s a very powerful beam so that  t i o n s from w a l l s and windows and Unless s p e c i a l p r e c a u t i o n s  reflec-  f o r e i g n p a r t i c l e s w i l l be  serious  are taken, extraneous s i g n a l s due  to  the s t r a y l i g h t w i l l mask the s c a t t e r e d s i g n a l which i s t y p i c a l l y a factor 10"  1 2  of the i n c i d e n t  light.  t  -21(ii)  Plasma The  Luminosity  emissions from the plasma i n c l u d e l a r g e l y (a) brems-  t r a h l i i n g r a d i a t i o n s which are t r a n s i t i o n s between f r e e s t a t e of e l e c t r o n s moving i n the Coulomb f i e l d s of the i o n s ( f r e e - f r e e transitions),  (b) recombination r a d i a t i o n s due  t r a n s i t i o n s and In g e n e r a l  to free-bound  (c) l i n e r a d i a t i o n s (bound-bound  these a l l depend i n complicated  ways on the  t u r e , e l e c t r o n d e n s i t y , degree of i o n i z a t i o n and -P  i  4-U  of the plasma. (iii)  transitions). tempera-  the dynamics  20  Statistical  Fluctuations  These are from two  sources:  (a) the shot n o i s e caused  by  the d i s c r e t e number of photons which r e l e a s e e l e c t r o n s from cathode of the p h o t o m u l t i p l i e r s and  the  (b) the shot n o i s e i n the  photocathode c u r r e n t i t s e l f s i n c e the c u r r e n t i s a flow of d i s crete electrons. it  When a p h o t o m u l t i p l i e r i s not exposed to  shows a dark c u r r e n t which i s thermal i n nature,  n o i s e of t h i s c u r r e n t i s given <t >  =  l  3. e  Id  A  by  and  light,  the  shot  21  f  (6)  where e i s the e l e c t r o n charge, *f the bandpass of the a m p l i f i e r and  i ^ the dark c u r r e n t .  In the presence of l i g h t t h i s  expression  becomes  (c )=  ae  7  where i  T  Af  (U +  = ip + i  s  t  r  a  y  (7)  + 1L  i s  t n e  t o t a l l i g h t c u r r e n t at  photocathode, i i and  i  P s  i s the c u r r e n t due t  r  a  y  i s the c u r r e n t due  i s the c u r r e n t due L  to s c a t t e r e d  light,  to s t r a y l i g h t of l a s e r  to the plasma  luminosity.  the  -22= (C) S i g n a l - t o - n o i s e (i)  Improvement of S c a t t e r e d S i g n a l An absolute i n c r e a s e o f s c a t t e r e d s i g n a l n a t u r a l l y  the S/N r a t i o . to  improve  T h i s can be achieved by u s i n g a high power  laser  i n c r e a s e the number of i n c i d e n t ohotons, by a dense plasma to  i n c r e a s e the number of s c a t t e r i n g e l e c t r o n s and a high speed monochromator to i n c r e a s e the s o l i d light of  i s gathered,  angle over which the s c a t t e r e d  (an i n t e r f e r e n c e f i l t e r c o u l d be used i n p l a c e  a monochromator, allows c o l l e c t i o n of more l i g h t ) .  However  s i n c e s c a t t e r i n g i s a f u n c t i o n of angle, the f,-number cannot be increased  indefinitely.  ( i i ) Reduction  of S t r a y L i g h t  S u i t a b l e b a f f l e s and l i g h t reduce the s t r a y l i g h t .  t r a p s should be p r o v i d e d to  I t i s d e s i r a b l e to f i r s t  focus the l a s e r  to  a s m a l l spot and allow the image to pass through  to  e l i m i n a t e extraneous  l a r g e l y an experimental individually. stray l i g h t .  l i g h t from the l a s e r .  a p i n hole  The b a f f l i n g i s  problem and each case has to be t r e a t e d  The eye i s most u s e f u l i n d e t e c t i n g the source of I f windows are present  as i s u s u a l l y the case,  they  should be p l a c e d at the Brewster angle so that the l i n e a r l y p o l a r ised laser light  i s not r e f l e c t e d .  A monochromator with s m a l l  s t r a y r a t i o i s very u s e f u l f o r r e d u c t i o n of s t r a y l i g h t the p a r t o f the spectrum observed l e n g t h o f the l a s e r l i n e .  provided  i s f a r from the c e n t r a l wave-  A double monochromator such  as the  15 one used by Davies  and Ramsden  would i n p r i n c i p l e reduce the  s t r a y l i g h t r a t i o by the square of t h i s r a t i o . red  f i l t e r may be used to prevent  In a d d i t i o n a  second o r d e r s t r a y l i g h t  from  23e n t e r i n g the (iii)  detector.  Reduction of Shot Noise We  observe from ( 7 ) that the r o o t mean square of the  current  shot  at the anode of the p h o t o m u l t i p l i e r i s  (l >*  =  L  S. e Af  Cfl  (C  +< )]  4  T  *  (8)  where G i s the g a i n of the p h o t o m u l t i p l i e r . The  I  t o t a l photon-current at the anode i s  -  T  Q  i  (9)  T  so that the f i g u r e of merit F i s given  p -  />  =  J  by  (10)  „  T h i s r a t i o w i l l improve i f i p i s l a r g e so that a p h o t o m u l t i p l i e r with a h i g h quantum e f f i c i e n c y at the l a s e r wavelength should used.  There i s a l i m i t  i n the r e d u c t i o n of the b a n d w i d t h ^ of  the a m p l i f i e r because the l a s e r p u l s e i s a very f a s t p u l s e . the dark c u r r e n t  i ^ is significant,  dark c u r r e n t should  be  be used, and  a p h o t o m u l t i p l i e r with  the tube cooled with  If low  liquid  n i t r o g e n or dry i c e . (iv)  Reduction of Plasma The  Luminosity  kind of gas used i n the plasma i s q u i t e an  consideration i n designing hydrogen o r helium should r e l a t i v e l y few  a s c a t t e r i n g experiment.  important In  general  be used, f i r s t l y because they have  l i n e s i n the neighbourhood of the l a s e r l i n e .  Secondly, the continuum background which i s mainly bremsstrahlung and  recombination r a d i a t i o n , i n c r e a s e s as the square of the  charge, the h e a v i e r  ionic  elements w i l l produce i n t e n s e continuum at high  temperatures at which m u l t i p l e i o n i z a t i o n o c c u r s .  -24-  C l e a n l i n e s s of the plasma i s a l s o important as any  solid  p a r t i c l e o r dust w i l l have a much l a r g e r s c a t t e r i n g s c r o s s - s e c t i o n than e l e c t r o n s .  These w i l l i n c r e a s e  the s t r a y l i g h t  frequency. S c a t t e r i n g from n e u t r a l atoms and times be s i g n i f i c a n t .  The  at the  molecules may  laser some-  scattering cross-section for neutral 22  atoms and molecules i s given by and  the R a y l e i g h  Scattering  law  i s r e l a t e d to the r e f r a c t i v e index by  where d i s the R a y l e i g h n  scattering cross-section  i s the d e n s i t y of n e u t r a l atoms o r molecules  Q  and  the r e f r a c t i v e index Roughly the c r o s s - s e c t i o n electrons  i s about .003  of that of  and when the i o n i z a t i o n i s not high,  the  this will  cause  s i g n i f i c a n t changes i n the i n t e n s i t y at the l a s e r frequency. (y) Reduction of E l e c t r i c a l and E l e c t r i c a l and  Magnetic Pick-up  magnetic pick-up can  be a s e r i o u s problem  e s p e c i a l l y i n plasma p h y s i c s work where high c u r r e n t  discharges  are common.  properly  shielded and  any  Hence a l l measuring apparatus should be  to prevent e l e c t r i c a l p i c k - u p s from spark gaps, e t c . complete loop  i n e l e c t r i c a l connections should be  avoided  to prevent magnetic pick-up. ( v i ) Sampling Technique I f the s i g n a l - t o - n o i s e r a t i o i s s t i l l nique may  be used.  of r e s u l t s .  The  noise,  tech-  being a random source w i l l tend to average By  sample, the s i g n a l - t o - n o i s e r a t i o may increases  a sampling  E s s e n t i a l l y t h i s technique averages a sample  out whereas the s i g n a l w i l l not.  it  low,  taking a s u f f i c i e n t l y be c o n s i d e r a b l y  as the square r o o t of the number of the  large  improved; samples.  -25-  CHAPTER IV  SCATTERING FROM ©-PINCH  Much work has been done on the ©-pinch  r e c e n t l y because o f  the p o s s i b i l i t i e s o f c r e a t i n g a h i g h temperature plasma o f thermonuclear i n t e r e s t .  & knowledge of the d e n s i t i e s and temperatures  of the e l e c t r o n s and i o n s would  be very u s e f u l toward a b e t t e r  understanding o f the behaviour o f the plasma.  The ©-pinch  i s a l s o very s u i t a b l e f o r demonstrating l i g h t s c a t t e r i n g it  plasma  because  i s r e a s o n a b l y c l e a n and has h i g h e l e c t r o n d e n s i t y and tempera-  ture.  A h i g h e l e c t r o n d e n s i t y w i l l n a t u r a l l y i n c r e a s e the s c a t t e r e d  s i g n a l , whereas a h i g h temperature plasma i s advantageous the s c a t t e r e d p r o f i l e w i l l be Doppler broadened  because  so that observa-  t i o n s can be made at f r e q u e n c i e s where the i n t e n s i t i e s of the s t r a y l i g h t may be expected to be s m a l l . (A) Apparatus and Procedure ( i ) The 9-Pinch A schematic drawing o f the O-pinch d i s c h a r g e i s shown i n F i g . 5.  The condenser bank c o n s i s t s of f i v e 5-uf  20KV NRG low  inductance c a p a c i t o r s normally charged to 12KV through a h i g h v o l t a g e power supply (20KV,50ma).  The v o l t a g e a c r o s s the bank  i s measured to + 250v w i t h a microammeter connected i n s e r i e s w i t h a 500M  resistance.  The d i s c h a r g e v e s s e l  (see F i g . 6) c o n s i s t s o f a 2 i n . diame-  t e r pyrex g l a s s tube two f e e t long.  The ©-pinch c o i l i s made o f  copper 1/32 i n . t h i c k and 5 i n . wide wrapped t i g h t l y round the g l a s s tube.  A s l i t o f 1 cm x 1 mm i s -cut i n the copper c o i l to  a l l o w t r a n s v e r s e o b s e r v a t i o n t o be made.  The v e s s e l i s p r o v i d e d  w i t h two Brewster windows at both ends o f the tube and green  330 K  20 Power  -AAAAAKV Supply 5M^R,  c=  200 K  05uf  25 jjf  -A/W-  i  :500M ^  ,0-Pinch coil  (^j^/hv 50  c s  S  l 2  Fig.  Cond en ser B a n k P*2 Potential Divider Ultra violet I rigger S p a r k 0 P i n c h Bank  Spark  S,  Gap  Gap  Micro a m m e t e r 5:  S c h e m a t i c D r a w i n g o f Q-Pinch D i s c h a r g e  ^  to O I  To h y d r o g e n Gas S u p p l y  e - P i n c h Coil Light  ("To C o n d e n s e r B a n k ) Light ba f f I e s  X -  T _ J Ruby Laser  trap  Focusi n g Lens  Brewste r Win d.ow i  to  I  Mi r ror S y s t e n n Ro t a t i ng SI it 90  for  th roug h Mon o c h r o m a t o r  Degrees Ape r t u re hocusing  F i g . 6:  Len s  Schematic Drawing of S c a t t e r i n g Apparatus  (9-Pinch)  -28glass discs  (with h o l e s of diameter  1/2  i n . i n the c e n t r e )  are  p l a c e d i n s i d e the t u b i n g to reduce the s t r a y l i g h t r e f l e c t e d the g l a s s w a l l .  The v e s s e l i s evacuated  by a fore-pump and  from a  d i f f u s i o n pump so that the system can be pumped down q u i c k l y a f t e r each d i s c h a r g e .  The p r e s s u r e i s measured w i t h a P i r a n i gauge c a l i -  b r a t e d by a Mcloed gauge. F i g . 7 shows a block diagram of the t r i g g e r system. r i s e s y n c h r o n i s a t i o n p u l s e ( +4v,  A ^ast-  10 ps d u r a t i o n ) which o c c u r s at  77 us b e f o r e the l a s e r o p e r a t e s i s f i r s t passed  through  a phase  i n v e r t e r which changes i t i n t o a n e g a t i v e p u l s e f o r t r i g g e r i n g the f o l l o w i n g v a r i a b l e d e l a y u n i t .  The d e l a y time of the d e l a y  u n i t i s v a r i a b l e from 1 ps to about 1 ms. d e l a y u n i t , , which i s a p u l s e of + 40v used to f i r e  a thyratron unit.  The  and  The output  from the  10 ^is d u r a t i o n , i s  t h y r a t r o n t r i g g e r u n i t sends 23  out a 9.5  KV p u l s e i n t o a Theophanis t e r m i n a t i o n  a l a r g e impedance to the p u l s e and doubles w i t h a r i s e time of 40 ns. ultra-violet  (UV)  which p r e s e n t s  i t s amplitude  to 19  T h i s p u l s e i s used to t r i g g e r  t r i g g e r g e n e r a t o r . ( S ^ of F i g . 5). 2 4  KV  an  The  UV  t r i g g e r generator has the advantage that i t i s o l a t e s the Theophanis u n i t from the main condenser bank and pick-up.  thus reduces  electrical  When the UV gap breaks down i t sends out another  p u l s e which breaks down the main spark gap a surge c u r r e n t and  (S  2  of F i g . 5), c a u s i n g  i o n i s i n g the gas i n the pinch-tube.  o v e r a l l t r i g g e r i n g i s a c c u r a t e to few  trigger  tenths of a  Most of the j i t t e r i n g i s from the v a r i a b l e d e l a y  The  microsecond. unit.  The d i s c h a r g e c u r r e n t i s measured w i t h a Rogowski c o i l made from a loop of RG 65 A/U  delay l i n e .  transformer where the secondary  I t i s a form of c u r r e n t  i s the t o r o i d a l l y wound c o i l  Sync Pulse from Laser Head  0-Pinch Coi 1 & Di scharge Vessel  1  Phase 1 n v e r te r  0-Pinch Bank S p a r k Gap  ?2  Fig. 7 :  r  V a r i a bl e TimeDelay  Thyratron  JI  Trigger Unit  Unit  I  Ul t r a Vi o 1 e t  -  hv  r  Theophanis  Trigger Ga p S|  Block Diagram of T r i g g e r  Unit  System  -30-  through which the magnetic f i e l d of the main d i s c h a r g e c u r r e n t (the  primary)  i s threaded.  The maximum value of d l / d t i s  i n times of 10  o r d e r 10  The c u r r e n t waveform of the d i s c h a r g e i s shown  i n F i g , Sa.  us.  The  g i v i n g a peak c u r r e n t of  I t i s o b t a i n e d by i n t e g r a t i n g the s i g n a l from the  Rogowski c o i l  through  an i n t e g r a t o r of RC time constant of  Laser  A TRG-104 g i a n t p u l s e l a s e r w i t h peak power of 10 MW at  69431, and  50ns d u r a t i o n i s used.  The  divergence of 10 m i l l i r a d i a n and i n the v e r t i c a l d i r e c t i o n . 922  f a s t - r i s e photodiode  The  the output l a s e r output  and  centred  g i a n t p u l s e i s achieved  by r o t a t i n g a p r i s m at a speed of 30,000 rpm.  RCA  100'  l i g h t output of the plasma i s a l s o shown i n F i g . Sb.  ( i i ) The  v  1 1  amp/sec. reached amps.  sec,  10  The beam has  i s linearly  a  polarized  i s monitored  by a  the output power i s r e p r o d u c i b l e  to + 10% p r o v i d e d that the l a s e r i s f i r e d at r e g u l a r i n t e r v a l s of about 4 p u l s e s a minute or (iii)  The  less.  Monochromator  A J a r r e l l - A s h Ebert monocbromator (82-010) w i t h a Bausch and Lomb g r a t i n g (35-00-58-38) i s used.  The  g r a t i n g has  a ruled  area of 2 i n x 2 i n and has 30,000 g r o o v e s / i n . , b l a z e d at 7500 A. The  l i n e a r d i s p e r s i o n of the instrument  the speed i s f/10.  corresponds  to 16A/mm and  The wavelength i s c a l i b r a t e d w i t h "a Hg  light  o  source and  i s accurate to h a l f A.  The monochromator, as manu-  f a c t u r e d , has a s t r a y l i g h t background of 1.6%. the i n s i d e w i t h b l a c k f l o c k - p a p e r and to  improve t h i s r a t i o to about .3%.  We  have l i n e d  added a d d i t i o n a l  baffling  This f i g u r e i s obtained  by  measuring the j.strayi l i g h t of, the l a s e r . a t 6943A and at wavelengths o u t s i d e the bandpass of the monochromator s l i t which i s s e t at  8A.  -31-  (b) -* F i g . 8:  time  ( ) Current Waveform Time S c a l e : 5 us/cm Voltage S c a l e : 5 v/cm (b) Plasma L i g h t Output Time S c a l e : 10 us/cm Voltage S c a l e : .1 v/cm a  -32For a 10 u s l i t , .2 A i n the f i r s t (iv)  the instrument  has r e s o l u t i o n at l e a s t equal to  order.  The P h o t o m u l t i p l i e r The p h o t o - d e t e c t o r  i s a P h i l i p s CVP  150 p h o t o m u l t i p l i e r w i t h  an SI photocathode s u r f a c e w i t h a quantum e f f i c i e n c y of about (At  the time that the experiment was  s u r f a c e was  not a v a i l a b l e ) .  shown i n F i g , 9a. under p u l s e d  The  performed a S20  l i n e a r i t y of the tube has been t e s t e d both  (laser) l i g h t  and d.c.  light conditions.  I t i s found  F i g . 0b  that the response  l i n e a r up to 30ma i n the p u l s e d c o n d i t i o n and light.  photocathode  The p h o t o m u l t i p l i e r c i r c u i t r y i s  shows the c a l i b r a t i o n c u r v e s .  d.c.  .4%.  about 2 ma  The p h o t o m u l t i p l i e r i s normally operated  is  with  at 14KV  and  the r.m.s. r i p p l e v o l t a g e of the p h o t o m u l t i p l i e r power supply i s 5mv. (v) The  Optics  With r e f e r e n c e to F i g . 6 again where i s shown the diagram of the o p t i c a l arrangement. through  The  laser light  a l i g h t b a f f l e aperture of diameter  schematic  i s passed  1 cm b e f o r e i t i s  focused by a l e n s of f o c a l l e n g t h 20 i n at the c e n t r e of d i s c h a r g e tube.  (Geometry o f the tube does not permit use of a  s h o r t e r f o c a l l e n g t h lens) about 1 mm. at  one  the  A light  The diameter  of the focused spot i s  t r a p i n the form of a b i g b l a c k box  i s placed  end of the d i s c h a r g e tube o p p o s i t e to the l a s e r to c a t c h  the t r a n s m i t t e d l i g h t  and prevent  i t being r e f l e c t e d i n t o  the  detector. S c a t t e r e d r a d i a t i o n i s p i c k e d up at 90° by a c o l l i m a t i n g l e n s (f = 5.4  in).  The  image of the s l i t  which i s h o r i z o n t a l , i s r o t a t e d through  i n the copper  coil,  90° by a f r o n t - s u r f a c e  •33lo  (a) S2  S3  S4  K Acc SN A  Cathode Accelerating grid Dynode number An ode  S5  Amplifier S I O ,A  S6  47K  1.4 KV  002uf  -L  Percentage Transmissof Neutral Density Filter F i g . 9: (a) P h o t o m u l t i p l i e r ( P h i l i p s CVP 150) Circuitry (b) L i n e a r Response of P h o t o m u l t i p l i e r ;  34-  m i r r o r system to the v e r t i c a l d i r e c t i o n so that i t can be imaged by another  l e n s ( f = 5 . 4 in) on the entrance s l i t  chromator. at  The s l i t  height i s 10 mm and the s l i t width  .5 mm c o r r e s p o n d i n g  useful light  o f the mono-  to 8A passband.  i s set  The maximum amount o f  that can be sent i n t o a monochromator i s l i m i t e d by  the f-number o f the instrument. maximum amount o f u s e f u l l i g h t  F o r a given l i g h t source, the that can be sent i n t o the i n s t r u -  ment, i s o b t a i n e d when the g r a t i n g i s 'flooded* w i t h l i g h t . our present a p p l i c a t i o n , because the l i g h t source ing  (the s c a t t e r -  plasma volume) i s f i n i t e and s m a l l and remote from the en-  trance s l i t to  In  o f the monochromator, a system o f l e n s e s i s r e q u i r e d  image the source onto t h e e s l i t so as to f i l l  g r a t i n g with l i g h t .  the monochromator  The o p t i c a l m a g n i f i c a t i o n i s u n i t y because  t h e r e i s no p o i n t i n o b t a i n i n g g r e a t e r . - i l l u m i n a t i o n at the monochromator s l i t solid  u n l e s s i t i s accompanied by g r e a t e r f l u x per u n i t  angle o r by a g r e a t e r u s e f u l angle.  plasma observed the s l i t  w i l l be determined  The volume of the  by the s i z e o f the image of  t o g e t h e r w i t h the s i z e o f the focused l a s e r spot.  To a l i g n the system, a p i n having approximately  the s i z e  of  the s c a t t e r i n g , p l a s m a volume i s p l a c e d i n s i d e the pinch-tube  to  r e p r e s e n t the plasma which s c a t t e r s the l i g h t .  i s p l a c e d at the e x i t s l i t  A light  source  o f the monochromator and i t s image i s  l i n e d up w i t h the p i n by the method o f no p a r a l l a x .  The l a s e r ,  which can be a d j u s t e d both v e r t i c a l l y and h o r i z o n t a l l y , i s then made t o focus on the p i n . L i g h t b a f f l e s are p r o v i d e d throughout  and i n p a r t i c u l a r an  a p e r t u r e i s p l a c e d i n f r o n t o f the monochromator so that no e x t r a light  i s sent i n t o the instrument,  i n c r e a s i n g the s t r a y l i g h t .  -35-  With these p r e c a u t i o n s , the r a t i o o f the i n c i d e n t l i g h t to the stray l i g h t  i s o f the o r d e r o f 1 0  1 1  .  T h i s r a t i o i s o b t a i n e d by  measuring the i n t e n s i t y of the i n c i d e n t l a s e r l i g h t  (reduced by 12  n e u t r a l d e n s i t y f i l t e r s ) which i s approximately  2 x 10  v and  the i n t e n s i t y o f the s t r a y l i g h t by the same p h o t o m u l t i p l i e r which is  approximately  frequency  2 x lOv.  Even so the s t r a y l i g h t  (20v) i s about 100 times the expected  at the c e n t r e  scattered signal  which i s about , l v , and measurements have to be taken such that the c e n t r a l wavelength o f the l a s e r i s o u t s i d e the bandpass of the monochromator s l i t .  In a d d i t i o n the s t r a y l i g h t  background  p r o f i l e o f the monochromator shows some wavelength dependence probably due to g r a t i n g ghosts.  This stray l i g h t  i s subtracted  from the t o t a l s i g n a l to g i v e the s c a t t e r e d s i g n a l . (B) Experimental R e s u l t s The experimental c o n d i t i o n s are tabulated as f o l l o w s : Gas used  Hydrogen  P r e s s u r e o f gas  150  Stored energy o f condenser bank  1.5 KJ  Time o f o b s e r v a t i o n O b s e r v a t i o n angle  o f Hg  44 us a f t e r i n i t i a t i o n o f d i s c h a r g e , ' i . e . i n the a f t e r - g l o w 90°  S o l i d angle over which l i g h t i s c o l l e c t e d  .01 s t e r a d i a n  Observed plasma volume  lOmmxlmmx  Bandpass o f monochromator s l i t  8A (500 ^u s l i t )  Fig.  o  10 shows some t y p i c a l o s c i l l o s c o p e t r a c e s and i n F i g .  11 the experimental r e s u l t s are p l o t t e d f o r the s c a t t e r e d i n t e n s i t y versus the wavelength s e t t i n g of the monochromator. p o i n t on the graph  Each  ( F i g . 11) i s an average of f o u r experimental  ,5mm  -36-  CC)  Fig. 10:  T y p i c a l O s c i l l o s c o p e Traces (©-pinch) a) Laser alone b) Plasma alone c) L a s e r and Plasma Time S c a l e : 2 us/cm delayed by 70 us Voltage S c a l e : 0.1 v/cm  A = 6930A  Wavelength Fig.  11:  Shift  (A)  Scattered Spectrum of Laser l i g h t by 9-pinch Plasma V e r t i c a l Scale: 4 d i v i s i o n s = .Iv  -38-  determinations deviations  and t h e v e r t i c a l  o f t h e mean.  G a u s s i a n shape w i t h the  n  e  =  (equation  1 6  best  f i t (solid  i s seen that  mental p o i n t s .  there  This  3  Also  of  T  standard  shows a n e a r l y -  II) to the e x p e r i -  = 25,000° K + 6,000° K  shown o n F i g . 11 i s t h e  curve). i s significant  i s thought  s e n s i t i v e toocand  t h e above p a r a m e t e r s c a n n o t  s c a t t e r of the e x p e r i -  t o foe due l a r g e l y t o t h e l o c a l Foro<<l, therefore  be v e r y  arrange experimental c o n d i t i o n s  the s c a t t e r e d the  accurate.  d e t e r m i n e t h i s p a r a m e t e r more a c c u r a t e l y , to  the  A l e a s t squares f i t of  20 o f C h a p t e r  n o n - r e p r o d u c i b i l i t y o f the plasma. trum i s n o t v e r y  profile  top.  .61+. 20,  (1.0+0.3) x 1 0 c i t T .  theoretical It  flat  shows t h a t 0 6 =  mental p o i n t s and  The s c a t t e r e d  a fairly  t h e o r e t i c a l curve  bars represent  spec-  determination In order  to  i t w o u l d be d e s i r a b l e  s u c h t h a t ot> 1 and d e t e r m i n e  them f r o m t h e p o s i t i o n s o f t h e s a t e l l i t e s . A forward  s c a t t e r i n g e x p e r i m e n t h a s been a t t e m p t e d  these s a t e l l i t e the it  signal.  lines,  but the l a r g e  amount o f s t r a y  Although i t i s p o s s i b l e  was d e c i d e d  that  to observe  light  to reduce the s t r a y  masked  light,  a p l a s m a j e t w o u l d be much more s u i t a b l e f o r  the  purpose of seeing  jet  i s described  the s a t e l l i t e s .  i n the subsequent  The work w i t h t h e p l a s m a  chapter.  -39-  CHAPTER V  SCATTERING FROM PLASMA JET  In o r d e r to observe the s a t e l l i t e l i n e s w i t h a O-pinch plasma, one has to make o b s e r v a t i o n of the s c a t t e r e d l i g h t at a very s m a l l angle ( a few degrees ). Because o f the enormous amount of s t r a y l i g h t encounted, i t was  thought that another  plasma might be more s u i t a b l e . A plasma j e t was chosen  because  it  i s f a i r l y simple to o b t a i n a r e p r o d u c i b l e plasma w i t h an 16 17 -3 e l e c t r o n d e n s i t y of 10 to 10 cm at an e l e c t r o n temperature 25  of 1 o r 2 ev i f argon i s used as the working gas  . These  c o n d i t i o n s make i t p o s s i b l e to observe s a t e l l i t e s at a l a r g e s c a t t e r i n g angle. I t i s much e a s i e r to reduce the s t r a y  light  when making o b s e r v a t i o n s at t h i s l a r g e angle as compared with s m a l l angle f o r w a r d - s c a t t e r i n g . In a d d i t i o n , s i n c e the j e t i s operated at atmospheric p r e s s u r e , we need no windows o r w a l l s i n the neighbourhood of the plasma which again reduces s t r a y l i g h t problem. The apparatus can a l s o be arranged to vary the o b s e r v a t i o n angle. (A) Apparatus and Procedure ( i ) The Plasma J e t The b a s i c d e s i g n of the j e t and i t s a s s o c i a t e d power supply i s shown i n F i g . 1 2 . I t c o n s i s t s e s s e n t i a l l y o f a tungsten cathode and a copper anode which i s e l e c t r i c a l l y grounded,  both  c o o l e d by water. The system i s mounted v e r t i c a l l y i n a b r a s s can on an o p t i c a l bench so that i t s p o s i t i o n can be adjusted e a s i l y . A commercial welder u n i t  ( M i l l e r Model SRA-333 ),which  d e l i v e r s a c u r r e n t up to 300 amps, at 40 v o l t s , was as the power s o u r c e . I t was  first  used  found that i t i s u n s a t i s f a c t o r y  because the v o l t a g e r i p p l e i s about 16% of the output v o l t a g e  -40  • The  Jet  Copper Anode  1  Water  Water Brass Can (grounded)  N  Tungsten Cathode  Bakelite  Argon Gas  Battery Supply Rheostat \ Switch  loafer  It R  PLASMA Fig.  12:  JET  The P l a s m a J e t  -41and the c u r r e n t r i p p l e i s about 33% when the j e t i s running at 200  amps. A t o t a l r i p p l e o f 50% i n the l i g h t output was observed.  Such a l a r g e f l u c t u a t i o n would a l t e r the plasma parameters and so a b a t t e r y supply was s u b s t i t u t e d . The b a t t e r y assembly c o n s i s t s of e i g h t 1 2 - v o l t heavy duty l e a d - a c i d b a t t e r i e s each r a t e d at 200 amp-hours w i t h r e s i s t a n c e o f a few m i l l i o h m s . They are arranged  internal  i n a 2x4 array  ( 2 i n p a r a l l e l and 4 i n s e r i e s ) . A high power r h e o s t a t w i t h v a r i a b l e r e s i s t a n c e from 0.1 ohm t o 0.4 ohm i s p l a c e d i n s e r i e s w i t h the b a t t e r i e s . The whole u n i t together i s capable of  de-  l i v e r i n g a c u r r e n t up to 300 amps, at 15 v o l t s . The diameter o f the anode i s 5 mm and the t i p bf the cathode  i s s e t at 7 mm  the top anode s u r f a c e . Argon i s the gas used  i n the j e t and the -1  flow o f the gas i s c o n t r o l l e d by a c a l i b r a t e d  from  at a steady v e l o c i t y of 15 m sec  argon flow-meter.  The v o l t a g e across the j e t was  measured with a v o l t m e t e r and the c u r r e n t by a heavy c u r r e n t shunt of r e s i s t a n c e 0.25 m i l l i o h m i n p a r a l l e l with a v o l t m e t e r . The c u r r e n t - v o l t a g e c h a r a c t e r i s t i c of the j e t i s shown i n Fig.13. The r e a d i n g o f the c u r r e n t was accurate to ± 4 amps. The j e t extends  to a d i s t a n c e o f about 10 cm above the anode s u r f a c e  when the j e t i s running at 280 amps., but the i n t e n s e spot i s about 1 cm above the anode. The l i g h t output  i s a d.c. s i g n a l  w i t h a .'few percent f l u c t u a t i o n . I t should be mentioned that the welder power supply i s used i n running the j e t except when an a c t u a l r e a d i n g i s taken. The welder  i s then switched o f f and the b a t t e r y supply switched on  f o r about 10 seconds to allow the c u r r e n t to reach a steady before a r e a d i n g i s taken. The b a t t e r y supply i s then  state  switched  -42--  100.  200 Current  Fig.  13:  Voltage-Current Jet  300  (Amps)  Characteristic^  o f Argon  Plasma  -43off  and  the welder on u n t i l the next r e a d i n g i s taken,  r e d u c i n g the duty c y c l e of the b a t t e r i e s and keeping in a f a i r l y  steady c o n d i t i o n . The  and r h e o s t a t ) c o s t s about $ 500  the j e t  entire battery unit and has  thus  ( batteries  to be charged o n l y  o c c a s i o n a l l y , making i t both very economic to b u i l d and inconvenient ( i i ) The  to  not  maintain.  Optics  A schematic  diagram of the o p t i c a l arrangement i s shown  i n F i g . 14. Much b e t t e r b a f f l i n g can be obtained  in this  case  because there are no windows o r w a l l s i n the v i c i n i t y of plasma. The  light  from the l a s e r i s f i r s t  focused by a lens of  f o c a l l e n g t h 4 i n . i n t o a p i n - h o l e (diameter then focused by another  the anode s u r f a c e . The box  i n . ) and i s  to.prevent  above the anode s u r f a c e .  the i n c i d e n t beam h i t t i n g  incident light  i n s i d e which i s a f a s t - r i s e RCA  i s trapped  i n a black  922 photodiode which  monitors the l i g h t of the l a s e r output for  1/16  l e n s of f o c a l l e n g t h of 5 i n . at the  c e n t e r of the j e t at a height of 3 mm Diaphragms are provided  the  and p r o v i d e s  triggering  the o s c i l l o s c o p e . With these p r e c a u t i o n s the s t r a y l i g h t  background i s now  about 2 v o l t s at the l a s e r wavelength, -12  r e p r e s e n t i n g a f r a c t i o n of 10  of the i n c i d e n t beam. The  s t r a y l i g h t background of the monochromator has been s t u d i e d c a r e f u l l y to avoid any  instrumental e f f e c t s ,  i n the wavelength range we stray light  are scanning.  i s shown i n a l a t t e r f i g u r e  and  i t i s negligible  The p r o f i l e of ( F i g . 18)  the  together  with  the s c a t t e r e d p r o f i l e . Scattered l i g h t direction. A field and  i s observed  at 45° from the  forward  l e n s of f o c a l l e n g t h 2 i n . c o l l e c t s the  images i t at a b a f f l i n g s l i t . o f  dimensions 2 mm  x 0.5  light mm.  SCATTERING OF LASER LIGHT FROM A PLASMA SCHEMATICS F i g . 14:  OF  JET  APPARATUS  S c a t t e r i n g of Laser L i g h t from a Plasma J e t ( S c h e m a t i c s of Apparatus)  -45-  The l i g h t  i s then passed through a Dove p r i s m which r o t a t e s the  image through 90° and i s focused at the entrance s l i t  of the  monochromator by another l e n s of f o c a l length of 5 i n . The height i s 2 mm to  and the s l i t  slit  width i s s e t at 0.5 mm c o r r e s p o n d i n g  an 8A passband. A g r e a t e r l e n g t h of the plasma i s not s e l e c t e d  because a c c o r d i n g to A h l b o r n ^ , the temperature of the plasma 2  drops r a p i d l y beyond a r a d i a l d i s t a n c e of 1 mm of are  from the c e n t e r  the j e t . A HN 38 p o l a r i o d and a C o r n i n g 52-63 r e d f i l t e r a l s o p l a c e d i n f r o n t of the monochromator.  ( i i i ) Photodetectors Two  p h o t o m u l t i p l i e r s were used, a P h i l i p s CVP  m u l t i p l i e r and an EMI  150 photo-  9558B tube. The l a t t e r has a quantum  e f f i c i e n c y of about 3% at the l a s e r wavelength  as compared  to  0.4% of the former, so that the s i g n a l - t o - n o i s e r a t i o f o r  the  shot n o i s e of the phototube i s expected to improve by a  f a c t o r o f 3. However the CVP before the EMI  tube was  150 was  used i n e a r l i e r  experiments  a v a i l a b l e . The output from the photo-  m u l t i p l i e r i s passed through an e m i t t e r - f o l l o w e r to m a i n t a i n the  time response of the c i r c u i t b e f o r e b e i n g f e d i n t o a d u a l  beam o s c i l l o s c o p e . Fig.15a shows the c i r c u i t diagram of the EMI p h o t o m u l t i p l i e r and F i g . 15b shows i t s response c u r v e . I t is  important to check that the p h o t o m u l t i p l i e r i s not s a t u r a t e d  by the d.c. l i g h t background of the plasma. (B) E x p e r i m e n t a l R e s u l t s (Plasma J e t ) The e x p e r i m e n t a l c o n d i t i o n s are t a b u l a t e d as f o l l o w s :  -46-  G K,iSI  S2  S3  S4  w  \-/  v>  S5  S6  S7  S6  v-'  W  S 9 SIO SM A w  w  DO IOOK  + 22  2 N 1177  AV\r-  to  IK  v  TO  pf  IK|CRO  82K 5 6 K 5 6 K 5 6 K 5 6 K 5 6 K 5 6 K 5 6 K 5 6 K 5 6 K 5 6 K 5 6 K  Mf'-H.T.  K  Cathode  G  Grid  SN A  dynode A n od e  F i g . 15:  lh •002uf  E m i t t e r Follower  1  N  !  (a) P h o t o m u l t i p l i e r (EMI 9558B) c i r c u i t r y (b) L i n e a r Response of P h o t o m u l t i p l i e r  0 10 2 0 3 0 . 4 0 50 6 0 70' 80 9 0 100 P e r c e n t a g e Tran-s-missi on of Nent ral D e n s i t y Fi I ter  -47Gas u s e d  Argon  Rate of Gas Flow  15 m sec -1  Pressure  Atmospheric  Voltage between Anode and Cathode  15 v o l t s  Current  250,280,300 amps.  of J e t  O b s e r v a t i o n angle  45°  Solid  0.01  angle  Bandpass of Monochromator s l i t  steradian  8A (500 u s l i t )  ( i ) CVP 150 P h o t o m u l t i p l i e r Work Fig.  16 shows some t y p i c a l o s c i l l o s c o p e t r a c e s at s e v e r a l  wavelengths. I t i s seen that the s i g n a l - t o - n o i s e r a t i o i s about u n i t y and we found i t necessary to use a sampling technique. At each of the 14 s e l e c t e d monochromator s e t t i n g s , 10 shots were taken and the o s c i l l o s c o p e t r a c e s of the p h o t o m u l t i p l i e r  output  and photodiode were photographed. The wavelength s e t t i n g s f o r successive  shots were s e l e c t e d at random (using random number  t a b l e ) to e l i m i n a t e p o s s i b i l i t y of d r i f t  e f f e c t s i n the plasma  or i n the instruments i n f l u e n c i n g the average r e s u l t at any p a r t i c u l a r wavelength. The s i z e of the- p h o t o m u l t i p l i e r s i g n a l for  each of the shots was measured from the o s c i l l o g r a m s at  seven p o i n t s along the l a s e r p u l s e  the time a x i s near the time occurrence of  and the ten readings  were averaged. The accuracy  of t h i s method i s l i m i t e d by the f i n e n e s s of the o s c i l l o s c o p e t r a c e s and by the number of the t r a c e s Fig. o  taken.  17 shows some averaged traces.There  i s a good s i g n a l  o  6910 A but not one at 6890 A. The r e l a t i v e i n t e n s i t y of the s c a t t e r e d r a d i a t i o n as a f u n c t i o n of wavelength s h i f t i n F i g . 18, where the v e r t i c a l bars represent  i s plotted  the standard  -48-  F i g . 16:  O s c i l l o s c o p e Traces o f S c a t t e r e d S i g n a l (Plasma J e t ) (a) A = 7016A (b) N = 6910A (c) 7v = 6986A Upper t r a c e : P h o t o m u l t i p l i e r s i g n a l (CVP 150) Lower t r a c e : Photo-diode Monitor S i g n a l (long decay time i s due to l o a d i n g by cable) Time s c a l e : ,0.1 jis/cm Voltage s c a l e : 0.1 v/cm (upper trace) 2 v/cm (lower t r a c e ) Plasma Current = 280 amps.  -49  F i g . 17:  Averaged t r a c e s of 10 scope t r a c e s , i n d i c a t i n g a c l e a r s i g n a l at 6910A. The l a s e r p u l s e o c c u r s at 0.26 u s e c .  24-h Light  -Scattered  20  -  (Corrected  for  Stray  Light)  16 -  2  12 8 -  >  Stray  4a>  or  4-  - 7 3 - 6 3 - 5 3 - 4 3 - 3 3 "23 ~I3 Wavelength Fig.  18:  Shift  0  +  13 23 33 4 3 53 6 3 73 (Angstroms)  S c a t t e r e d L i g h t Spectrum from Plasma J e t . A l s o shown i s . the t h e o r e t i c a l curve f o r T. = 13,500 K, an *-= 4.4, an i n s t r u m e n t a l width o f 8 A , and an i n t e n s i t y chosen a r b i t r a r i l y  -51d e v i a t i o n o f the mean f o r the ten experimental p o i n t s . The s a t e l l i t e s occur at ± 38 A from the c e n t r a l l i n e and has a h a l f width o f about  12 A to 14 A. The experimental p o i n t at + 23 A  i s p u r p o s e l y omitted because  the argon at 6965 A s a t u r a t e s  the p h o t o m u l t i p l i e r making i t i m p o s s i b l e to o b t a i n a s i g n a l . 0  An i n t e r e s t i n g e f f e c t i s the n e g a t i v e s i g n a l at .-63 A. I t has been checked that t h i s i s not due to i n s t r u m e n t a l e f f e c t . However we note that t h e r e i s an argon l i n e at about 6880 A and we i n t e r p r e t t h e decrease i n the l i g h t output as due to the dep o p u l a t i o n o f the upper s t a t e of the l i n e . T h i s decrease amounts to about 5% of the d . c . l i g h t  l e v e l . The same e f f e c t  i s observed  i n a l a t e r check at 6965 A . We attempted  a l e a s t square f i t  of the t h e o r e t i c a l curve  to the experimental p o i n t s but were unable to o b t a i n a good f i t because  the observed l i n e w i d t h s are wider than thp.se p r e d i c t e d by  theory. The d i s c r e p a n c y may be accounted f o r by a v a r i a t i o n of the'- e l e c t r o n d e n s i t y i n the plasma volume we observed. A c c o r d i n g 25 n to Ahlborn , the temperature changes from 13,500 K to 11,500°K i n a d i s t a n c e o f 1 mm from the c e n t e r o f the j e t . I f we assume l o c a l thermal e q u i l i b r i u m , then a c c o r d i n g tqcthe curves shown i n F i g . 19  26  , the e l e c t r o n d e n s i t y changes from about  cm" . Assuming  17 10  16 to 5x10  ot i s l a r g e , and u s i n g the r e l a t i o n OJ - cu^t 0  f o r © = 45°, k = 2K s i n 9/2= 0 . 6 7 x l 0  5  cm , -1  f  i t can be c a l c u l a t e d  that the d i f f e r e n c e i n the frequency s h i f t s f o r the two e l e c t r o n —1  1 9  d e n s i t i e s i s Aw = 5.1xlO ^ x  r a d . sec  ° giving a  A A_ of  agreeing w e l l with the width of the observed s a t e l l i t e s . course temporal changes i n n  0  would lead to s i m i l a r  12 A, Of  results  Fig.  19:  Equilibrium  Compositions of A r g o n  Plasma  -53but t h i s experiment  does not d i s t i n g u i s h which v a r i a t i o n i s 27 o c c u r r i n g . R e c e n t l y Nyugen-Quang-Dong suggested that t h i s broadening may a l s o be accounted f o r by c o l l i s i o n . A l s o the i n t e n s i t y o f the l i n e drops by a f a c t o r of h a l f at such s h i f t e d frequency as i t should because o f the drop i n ne. S i n c e we cannot get a good f i t , we cannot determine  n  e  and T . However by t a k i n g Ahlborn'value of 13,500°K f o r the e  temperature  at the cfenter o f the j e t , the t h e o r e t i c a l curve  whose peaks c o i n c i d e with the experimental peaks i s o b t a i n e d with al = 4.4. T h i s v a l u e o f 16 6.5x10  l e a d s to an e l e c t r o n d e n s i t y of  *3  cm"* , a g r e e i n g w e l l with the l o c a l 3  thermal e q u i l i b r i u m  assumption. S c a t t e r i n g from the c e n t r a l peak at the l a s e r frequency i s a l s o observed. T h i s s i g n a l i s l a r g e r than that observed at the s a t e l l i t e s by a f a c t o r of more than 20 as i t should be s i n c e /  '  2  theory r e q u i r e s that i t s i n t e n s i t y i s oL times the i n t e n s i t y of  each s a t e l l i t e . However the s t r a y l i g h t  at the c e n t e r i s  a l s o l a r g e p r i m a r i l y due to the presence o f d u s t - p a r t i c l e s . Fig.  20a shows the l a s e r l i g h t  as a f u n c t i o n o f plasma c u r r e n t .  I f the j e t i s run at a low c u r r e n t o f 50 amps.( which i s the lowest v a l u e c o n v e n i e n t l y reached with the present power s u p p l y ) , the s t r a y l i g h t  i s reduced by a great f a c t o r than when no plasma  i s p r e s e n t . T h i s i s i n t e r p r e t e d to be due to the f a c t  that the  plasma i s c l e a n i n g up the d u s t . p a r t i c l e s . The s c a t t e r e d at  the c e n t r a l frequency i s determined  light  as f o l l o w s . The c u r r e n t  i n the j e t i s s e t at 50 amps, at which the s c a t t e r e d l i g h t the e l e c t r o n s may be n e g l e c t e d , and t h e r e f o r e the l i g h t  from  signal  observed i s equal to s t r a y l i g h t . T h i s v a l u e i s s u b t r a c t e d from  -54-  \ = 6943 A  -I it  I 0  40 80 Pla-s-mia  F i g . 20:  1 2 0 \o0 ZOO' 2 4 0 2 8 0 3 2 0 Jet Current (Amps)  (a) I n t e n s i t y of L a s e r L i g h t as a f u n c t i o n of plasma c u r r e n t  03  ?  !  .03  T3  -02  >  c in  .0  CD  L--4-J  I  Je  6941 6943 694 5 W a v e l e n g t h (A) Fig.  20:  (b) S c a t t e r e d Spectrum of C e n t r a l Peak (x.= 6943A) , u s i n g s l i t of 1/2A passband  -55the s i g n a l o b t a i n e d at 280 amps, to g i v e the s c a t t e r e d at  signal  the c e n t r a l frequency. The width of the c e n t r a l peak i s  found to be narrower  than f A ( F i g . 20b). B e t t e r r e s o l u t i o n  cannot be o b t a i n e d w i t h the present apparatus. (ii)  EMI  9558B P h o t o m u l t i p l i e r Work  Subsequent experiments w i t h the EMI firmed and improved  9558B tube have con-  on the work of the P h i l i p s CVP  150  tube.  A t y p i c a l o s c i l l o s c o p e t r a c e i s shown i n Fig.21 and a s i g n a l t o - n o i s e r a t i o of about 2 o r 3 i s o b t a i n e d at the s a t e l l i t e s so taht we can p i c k out the s i g n a l without the sampling  required  previously. Because of s l i g h t l y d i f f e r e n t e l e c t r o d e c o n d i t i o n s , parameters  have changed a l i t t l e  so that the r e s u l t s may  the not  be compared d i r e c t l y w i t h the r e s u l t s o b t a i n e d w i t h the CVP amps, and  150  t u b e ( i ) . Two  experiments were done, one at 300  the  o t h e r at 250  amps., and the s c a t t e r e d i n t e n s i t i e s are shown i n o  Fig.  22 and i n F i g . 23 r e s p e c t i v e l y . The peaks occur at + 50 A  and at t 42 A c o r r e s p o n d i n g to an n 7.5xl0  1 6  e  of l.lxl0-"-7 m~3 C  and  cm"* r e s p e c t i v e l y i f T =13,500°K. The widths of the 3  e  s a t e l l i t e l i n e s are approximately 12 A at 300 amps, and 15 A at  250 amps. These r e s u l t s cannot be very accurate bec/ause of  the 8 A bandpass of the monochromator s l i t . T h e i n t e n s i t y of the c e n t r a l peak at the l a s e r frequency i s again more than 20  times  g r e a t e r than that of the s a t e l l i t e . The l i n e shape of t h i s cent r a l peak has not been i n v e s t i g a t e d f u r t h e r due to the present l i m i t a t i o n of the o p t i c a l r e s o l u t i o n of the apparatus. We used a s l i t  have  of 25 u wide(§A bandpass) to look at the c e n t r a l  peak, and found that the c e n t r a l l i n e i s at most f A wide, so that we  are probably measuring  the i n s t r u m e n t a l width.  -56-  time F i g . 21:  O s c i l l o s c o p e t r a c e s of S c a t t e r e d  Signal  (Plasma Jet)  (a) A = 6900A (b) > = 6910A (c) X = 6986A Upper Trace: P h o t o m u l t i p l i e r S i g n a l EMI 9558B Lower Trace: Photo-diode Monitor Time S c a l e : 0.1 us/cm Voltage S c a l e : .05 v/cm Plasma Current: 250 amps.  Scattered (Correcter  tO  ra o S3  ct c+ (t)  •i <D a Ul  •o 0  ° < 2 a, B  0  < a>  —  r ^  p an  03 -+ (D -jtr  1  H-  aq  3"  ct-  c «< M  CO  3  S»  c. c+  J» rt W  O  o CO  >o  I  Intensity for  Stray  (Volts) Light^  Scattered ( C o r r e c ted  I n t e n si t y (Volts) for S t ray  Light)  -58-  CHAPTER VI  CONCLUSION  The s c a t t e r i n g method has been s u c c e s s f u l l y a p p l i e d both to the v e r i f i c a t i o n of the theory of s c a t t e r i n g of e l e c t r o magnetic  waves by plasmas and to the d e t e r m i n a t i o n of plasma  parameters.  T h e o r e t i c a l p r e d i c t i o n s have been confirmed by  experimental r e s u l t s . For the case- of s c a t t e r i n g from the 0p i n c h plasma at 90°, where oL i s expected.to be s m a l l ,  the  s c a t t e r e d spectrum shows a n e a r l y Gaussian shape c o r r e s p o n d i n g to n o n - c o l l e c t i v e s c a t t e r i n g from e l e c t r o n s . The temperature  i s determined  to be T = 25,000°K ± 6,000°K and 16  e l e c t r o n d e n s i t y to be n =(  1.0^  e  <*•  = 0.60  ± 0.21.  electron  0.3)  x 10  the  q c i " f o r an  For the case of s c a t t e r i n g from the plasma  j e t at 45° where ot i s expected to be l a r g e , d i s t i n c t  satellite  peaks were observed on both s i d e s of the c e n t r a l frequency, i n d i c a t i n g a s t r o n g c o l l e c t s c a t t e r i n g e f f e c t . By assuming the e l e c t r o n temperature of the j e t to be 13,500°K ( a c c o r d i n g to Ahlborn), an oc of 4.4 g i v i n g an n  e  of 6.5  i s o b t a i n e d f o r the experimental p o i n t s , 16 ^  x 10  cm  . The p o s i t i o n s and  intensities  of the s a t e l l i t e s are found to vary i n an expected manner w i t h the plasma c u r r e n t . The i n t e n s i t y of the c e n t r a l peak i s at l e a s t 20 times the i n t e n s i t y of each s a t e l l i t e ,  again agreeing  w e l l w i t h t h e o r e t i c a l p r e d i c t i o n . Some i n d i c a t i o n s of p e r t u r b a t i o n of the plasma by the l a s e r were observed. The though  s m a l l i n the present case, may  perturbation,  become s i g n i f i c a n t  if a  more powerful l a s e r i s used. A few comments of the s c a t t e r i n g method as a d i a g n o s t i c technique f o r l a b o r a t o r y plasmas are i n o r d e r . The method i s unique amongst o t h e r d i a g n o s t i c methods i n that i t i s able to  - 5 9 -  give both very good time and s p a t i a l r e s o l u t i o n . The method o f f e r s s e v e r a l d i f f i c u l t i e s o f i t s own. F i r s t l y because such a small f r a c t i o n of l i g h t  i s scattered  that even f o r plasmas  of moderate d e n s i t i e s , the s c a t t e r e d r a d i a t i o n i s s m a l l enough to pose d e t e c t i o n problem. Secondly s t r a y l i g h t troublesome and e l a b o r a t e  light  i s very  b a f f l i n g has to be undertaken.  As we have noted, a s m a l l v a r i a t i o n o f e l e c t r o n d e n s i t y broaden the widths o f the s a t e l l i t e peaks, so that  will  observation  should be made over a volume o f plasma i n which the d e n s i t y i s uniform i f p r e c i s e plasma parameters are to be determined. Future work w i l l c e r t a i n l y i n c l u d e a more d e t a i l e d study of the e l e c t r o n s a t e l l i n e peaks as a f u n c t i o n of experimental parameters such as by v a r y i n g T gases, gas flows,  or n  o r by u s i n g d i f f e r e n t  s c a t t e r i n g angles and a d e t a i l e d i n v e s t i g a t i o n  of the centrak peak. The experiment on the c e n t r a l peak w i l l i n v o l v e high o p t i c a l r e s o l u t i o n techniques such as the FabryPerot i n t e r f e r o m e t e r .  Hydrogen o r helium i s p r e f e r a b l e  because  the c e n t r a l peak w i l l be expected to be wider. The s t r a y from the l a s e r has to be f u r t h e r e d building a dust-free scattered  light  light  reduced, f o r example, by  chamber around the j e t so that a b e t t e r  to s t r a y l i g h t  r a t i o may be o b t a i n e d . More  d e t a i l e d s t u d i e s should a l s o be made on the p e r t u r b a t i o n of the l a s e r on the plasma.  effect  -60-  APPENDIX A. C a l c u l a t i o n o f Assume that a p a r t i c l e o f s p e c i e s  i moves i n an  assigned  o r b i t through a plasma c o n s i s t i n g o f e l e c t r o n s aaid ions that i s described  by the Vlasov E q u a t i o n s .  The equations to be s o l v e d are  1  *tt  where  i  ( i  a  i s the unperturbed f i e l d - p a r t i c l e d i s t r i b u t i o n f u n c t i o n to u n i t y ) which i s assumed to be s t a t i o n a r y and s p a t i a l l y eous.  (normalised homogen-  Is the change i n the d i s t r i b u t i n g f u n c t i o n o f the  f i e l d p a r t i c l e s o f species  -j.'produced by the t e s t p a r t i c l e s i at  i s the e f f e c t i v e e l e c t r i c p o t e n t i a l  i>e.0  yi i s the average number d e n s i t y of e l e c t r o n s at thermal 0  equili-  brium qj,mj are the charge and mass of the p a r t i c l e of s p e c i e s j respectively. been s o l v e d 2'g texts.  The time-asymptotic s o l u t i o n ' o f t h i s problem has  by Rostokor Sc Rosenbluth and a l s o given  The method i s to F o u r i e r analyse Equation  i n space and time by the f o l l o w i n g S£/r  v i)*  *)*  <#/U The  ^ Jdk 3  e  -fcJdZ ^  f i n a l r e s u l t s are  ( h  Sij  (i  .  t  .  ( I ) & (2) b o t h  transforms.  V d ^  , r  i n standard  ik.(r-r ) t  *.  (4)  *)  (5)  ^  (6)  where -  j  l< J i  k- (v-v,-t.\)  ,  (7)  -61-  co  The  1  'Fa  electron  particle  ±rnu7g L  a,  /*»» -•  (8)  distribution  i n the neighbourhood  c a n now be c a l c u l a t e d .  electron,  with position  number d e n s i t y  I f the test  and v e l o s i t y  a t ( j , t) when a t e s t  (.!-<•,  particle  W  particle  of a test i s an  , then the e l e c t r o n i s at r  e  and v@ (t=o)  is  w0  and  dk  4  oi  te *" 5 ^ —  3  s i m i l a r l y f o r an i o n t e s t  given  by  p a r t i c l e with position  and v e l o c i t y  ( l i ^ r )  dr  'W  3  To  (9)  simplify  (9) and (10), t h e f o l l o w i n g  - ~3~  substitutions  a r e made  (11)  =Jd±  S(u- v^)  - ^dv dv x  -^dV d  f—^-J]  =  w  (  3/1  )  (V) $[u-  e  ( v , Vy ,  t  -  f^uj  j^d f  Vy f  x  :  y  x  i^J  u)  e  Vy  e  e  (12)  du U  G(kJ= Hence  ^cuj= / -  - tAj  -  t'X  ^r(i3)  f _ _ ^ 2 _ ^  2:  d4)  (9) & (10) become Lf^L  [14 J  A  6b)  J  U- Ue  £&;fl<15) J  -62-  Then  Now  <T  j f^  dt)  Q{f> z)=  G f ' * o- w.j >  <%iCr,t)ruCT*f  y  F«W(16)  >V  J  (i?)  t t)>/^o  ( 1 8 )  r  J  (19)  One gets  Using  I V  dr  (27FJ  e  (20) 2  We get  (21)  Apply F o u r i e r t r a n s f o r m fa;/= One gets  J-  and  27T  -KF (TJ  letting  as before  9  C Cp, ^ , = j §f « ^ 2T». fe;+ -j!rj ^ <  e  (22) -  2 7 r  • i ^ l ' ^ j ^ T ^ ^ ^ I  -T 7  (23)  where  We see t h a t the f i r s t  term i s z e r o except when  T h i s i s a wave w i t h the same frequency  ?~u>=o  ,k=  0  and wave v e c t o r as the  i n c i d e n t wave, t h e r e f o r e i s a t r a n s m i t t e d beam and does not  -63 = contribute  t o the s c a t t e r e d spectrum and s h a l l be omitted from  hereafter. If we put  .:  ?= w  j  */ k  u=  l ^ /  <.\&M>'%{$$\  +  ^ j  ^  T  r  F e ' H  z  (  2  5  )  Since  as given The  by Rosenbluth & R o s t o k e r .  4  s i m p l i c a t i o n of (26) hinges on the c a l c u l a t i o n o f the i n t e g r a l //m.  i^—f'(u')  =f  r  *«'  F'(u') +  (27)  where/denotes the p r i n c i p a l part of the i n t e g r a l . For  an e q u i l i b r i u m plasma,  =ftK~*e  (/3 =""A*T)  ^  2  Making the change U.'-K*C, one can w r i t e f o r an e q u i l i b r i u m plasma  =  Write  /3UBX  .'. I W =  To e v a l u a t e We f i r s t  - 2^  7C-1 J  = oj/co  e  5  - a t  (  u± = [zt'KZ/mJ*)  -2y63X-"|  JC*; s  I  ^ " / c ^  -X  z'^lTdtC^^  dt e  ^L2*t  /Set  (28)  d i f f e r e n t i a t e and o b t a i n  I'oO =  dt c A.  f0D  2  l A  • (29)  -64-  Differentiate  (29)  again (30)  Hence we  have ±  Let  y  '  =  y  z * ( * J  (3D  !'<*)  - ce  X2  where C c a n be o b t a i n e d  .'. 100 =  JB  #  V J ( ^ =  ^  /-  X =» 0  where  '  (32)  C '-o  (  Where 4-to) i s 2/  8  )  ^O  1-***'**j*^*^ ^oJ  2  (33)  <£t~  2  r  -  j W s  c  2X e'* J*  j^^-^  = " =  ^ J  at  / - l7L-*ItrJl  !<*)-> -2 Hence  e  f r o m (29)  -h c/ci  +  2 „ 2. ^ A * 2 v  e  xe~* ] 7  d e f i n e d by e  J  ^  o  (35)  Similarly  Where and  j^cy)  =  2y e ~  y=y^ a -  F i n d we  J  0  £  ^-^  and t h e i o n c h a r g e Z  is  taken  into  account.  have  <|^(W)'>= ? Where  y  Q  »  (37)  ll-<tt|Vc("J+Zfc) M*>> 2  - <rf^,-^W^'^Xtf'^J  ;  *« & B  >  ^K^^f  " ^ J  -65-  2 i n the form given by S a l p e t e r . B. Case of P r a c t i c a l We  Interest  c o n s i d e r the case when  T  & Tz  z  so that  %s^=(™*2*)«l  In t h i s case a good approximation can be g i v e n to the g e n e r a l e x p r e s s i o n given i n ( 3 7 ) . ( i ) C o n s i d e r the f i r s t  term  ' - . — ^  ^  i s of most i n t e r e s t f o r |xf~f .  — ^ - j - -xj>C-y^) and J  the narrow r e g i o n Ix|H>/^ ^ £ we  Disregarding  unity.  e  Where  Ve^J-  term and ^ = JLhJ*—  have|y|»| f o r t h i s  can be n e g l e c t e d compared w i t h £ a n d Hence we  i t involves  have  fck)  =  D+ot - 6LiJ*Q*+  7To< X*«<#  7  }  +  ( i i ) For the second  term where F = ——  if  In the important r e g i o n s we  lyl = |x/ty ' $*  and  -  2  '.  [" I - 4/&) + in*  _  Whe:re T^(y;s  7  X ^C-x^l  (_*^_x  ^ - °<  J*)\ry*){0+f?--piiy)l +  Tol+y  x  z  2  w  e s s e n t i a l f e a t u r e s of the s c a t t e r e d  determined  by  then have |x|<< |  exfC-2y*)\  (39)  17^^  (40)  i s easy  ^ ^ I  as the s c a t t e r e d p r o f i l e w i l l  7*(x) ~ -e.yj>(~x ) and l  g i v e the  profile.  C. Spectrum of S c a t t e r e d R a d i a t i o n f o r ot«l  i s unimportant  2  (40) t o g e t h e r w i t h (38) & (39) w i l l  The case f o r  3 8 )  •«*/> C -  L  <i4cfe^;| >rf -r;cx;4^ + Equation  , and  e  T  <  i s Gaussian.  be  The c o n t r i b u t i o n from  -66-  the second term For of  \,  ^ ( ^ i )p?) w i l l  be n e g l i g i b l e ( o / * s m a l l e r ) .  IZ^) h a s a maximum n e a r  x=x  0  where X i s the s o l u t i o n 0  the d i s p e r s i o n r e l a t o n s h i p . 1(K ) — 0  ti~  | =  (See E q u a t i o n ( 3 8 ) )  2  (41)  V e r y c l o s e t o X„ , f j ( x ) c a n be a p p r o x i m a t e d by t h e shape  CM  because  {  ,  +  4 i x  u x c  _ Now  One  o/  e />  2  x  C-x^J  f o r X » | we  have t h e a s y m p t o t i c e x p a n s i o n  .foo- / *  ( 2 x ^ r ' n + -tr  Using the f i r s t  Lorentzian  +  -J  t e r m as an a p p r o x i m a t i o n  j- (x) - I -  -r -  can w r i t e  F,(x)  1  2X 2  =  .  , ,.  1-—.1, ,  ,—^—rr^r\  Now  is  therefore L o r e n t z i a n near  F r o m (41) and , we  It  the f i r s t  two  x . Q  terms of the a s y m p t o t i c e x p a n s i o n of  have  i s easily varied  that  i.e.  which i s the well-known pla'sma o s c i l l a t i o n . The  The  s h a p e o f Ip(y) for  s a t e l l i t e s occur at d  cu = i 0  the s p e c i a l case of g r e a t e s t i n t e r e s t  o^-» o<5 and t h e r e f o r e yS= / i s a l m o s t f l a t - t o p . shape  (^p.  dispersion longitudinal (electrostatic)  Th©  of ^ M i f f e r s from the Gaussian f o r n o n - i n t e r a c t i n g  ions  - 6 7 -  ions because  e l e c t r o s t a t i c p o t e n t i a l s o f the electrons~fc^are  set up by the requirement t h a t the e l e c t r o n s f o l l o w the charge d e n s i t y of the (slow) i o n s .  %Te>Tx as w e l l as oi>> | , we have^=^p?j>/ and we have the  If  c o r r e s p o n d i n g case of the i o n s a t e l l i t e s whose shape can be s i m i l a r l y shown to be L o r e n t z i a n  and o c c u r r e d s i m i l a r l y at  D. I n t e g r a t e d I n t e n s i t y C a l c u l a t i o n s The i n t e g r a t e d T^Cx)  For  i n t e n s i t i e s of the two c o n t r i b u t i o n s  from  rjyywill now be c a l c u l a t e d .  and oi» /  =  oi~  ( f o r both  satellites)  For g e n e r a l v a l u e s o f oC, the c a l c u l a t i o n i s a l i t t l e more complicated. Now  T  (x) =  Where  -j-Cx) =  2 K -ex^> (-X*)  Let  £ (xJ =  / - 4x>;  e  />;  x-t  Let  y  Now  \,„  .  1  L _  r  -  -J^Jj  ^0  C6  Similarly for  ^ t 2  In  kCR^ ) Z  2  - t zr x «x/> 6-*,>  Re*®  =  I ' JZ ^  *  X  =  ,,  ,  ( f o r both s a t e l l i t e s )  2" = Tx/Te. - /  So that the c o n t r i b u t i o n Ifoby a f a c t o r  «e*j> i  /-fo<  x  •  Tjstyj is  l a r g e r than the c o n t r i b u t i o n  F o r l a r g e o(. t h i s i s approximately <^X/2 and  -68-  hence the r a t i o of the c e n t r a l peak t o each s a t e l l i t e i s E.<_Fortran Program f o r C a l c u l a t i o n o f S c a t t e r e d C C C C C C C  C  C  C  C  cC .  Spectrum.  SUBROUTINE FLUCT(S,WAVE,ATWT,Z,TE,Tl,THETA,ALF,ELDEN,DLAMB) CALCULATION OF THE SPECTRAL DENSITY S/ELDEN UNITS OF WAVE, DLAMB ARE CM UNITS OF ELDEN ARE CM**-3 UNITS OF TE, T l ARE DEG K UNITS OF THETA ARE DEG IF ELDEN IS SPECIFIED,THEN ALPHA IS CALCULATED I F ELDEN IS PUT ZERO, THEN ALPHA IS READ IN REAL K PI = 3.14159265 BT = 3.299E-12 MASS ELECTRON / 2 BOLTZ K= 4.*PI * SIN(THETA*Pl/360.) / WAVE BETAE = SQRT(BT/TE) BETAI - SQRT(BT*1836.*ATWT/TI) DEBL - SQRT(TE/ELDEN) * 6.90 SQRT(BOLTZ / 4 PI)/ELECTRON CHARGE IF (ELDEN.NE.0.0) ALF = l./(DEBL*K) AAE = ALF*ALF AAI = AAE*Z*TE/TI U = DLAMB * 2.*PI*2.998E10 / (WAVE*WAVE*K) U IS OMEGA / K XE = BETAE * U XI = BETAI * U CALL G (FE, PE, QE, XE, AAE) CALL G ( F l , PP, QI, XI, AAI) ZETA => BETAE*FE* ( (1. -PP) * (1. -PP) + QI*QI) + Z*BETAI*FI* 1 (PE*PE+QE*QE) EPS = (l.-PE-PP)*(l.-PE-PP) + (QE+QI)*(QE+QI) S = 3.54490*ZETA/(EPS*K) 2 SQRT(PI) RETURN END  (Continued on next page)  69-  C C C  C 29  28 C 27  26  SUBROUTINE G (F, P, Q, X, AA) P IS THE REAL PART OF G, Q IS THE IMAGINARY PART AA IS ALPHA SQUARED F IS WITHOUT THE FACTOR BETA XX = X*X F = EXP(-XX) Q = 1.77245*X*F*AA SER =1.0 TERM =1.0 AN = 1.0 IF (X.GE.3.0) GO TO 28 ABSOLUTELY CONVERGENT SERIES TERM = - TERM * 2.*XX/AN SER = SER + TERM AN - AN + 2. IF (ABS(TERM),GT.l.E-4) GO TO 29 P = ~SER*AA RETURN SER = 0 . 0 ASYMPTOTIC SERIES TERM = TERM * AN * 0.5 / XX IF (AN*.5.GT.XX) GO TO 26 SER = SER + TERM AN = AN + 2. IF (ABS(TERM).GT.l.E-4) GO TO 27 P = SER*AA RETURN END  -70REFERENCES 1. E . E . S a l p e t e r , Phys. Rev.120, 1528(1960). 2. J . A . F e j e r , Can.J.Phys.  38,1114(1960).  3. J.P.Dougherty and D.T.Farley,Pro.Roy.Soc.(London) A259,79(1960). 4. M.N,Rosenbluth & N.Rostoker,Phys. F l u i d s 5, 776(1962). 5. K.L.Bowles, Phys. R e v . L e t t e r s 1_, 12(1958). 6. F.W.Perkins,E.E.Salpetep,K.O.Yngnessen,Phys.Rev.Letters 14, 15(1965). 7. R.A.Stern & M.Tzoar, P h y s . R e v . L e t t e r s 15, 485(1965). 8. G.Fiocco and E.Thompson,Phys.Rev.Letters 10, 89(1963). 9. E.Funfer,B.Kronast,H-.»J.kunze,Phys.Letters j), 125(1963) . 10. H.J.Kunze,E.Funfer,B.Kronast,W.H.Kegel,Phys.Letters 11, 42(1964). 11. W.E.R.Davies and S.A.Ramsden, P h y s . L e t t e r s 8,179(1964). 12. U . A s c o l i - B a r t o l i , J . K a t z e n s t e i n and L . L o v i s e t t o , Nature 204, 672(1964). 13. A.W.De Silva,D.E.Evans and M.J.Forrest,Nature 203,1321(1964). 14,S.A.Ramsden,W.E.R.Davies,paper presented i n the meeting of the American P h y s i c a l S o c i e t y ( 1 9 6 4 ) . 15.S.A.Ramsden,W.E.R.Davies,Phys.Rev.Letters 16,303(1966). 16. P.W.Chan & R.A.Nodwell,Phys.Rev.Letters 16, 122(1966). 17. see f o r example:J.M.Stone, R a d i a t i o n and O p t i c s (Mcgraw H i l l , 1963) . 18. L . S p i t z e r , P h y s i c s of F u l l y I o n i s e d Gases ( I n t e r s c i e n c e , 1 9 5 6 ) . 19. see f o r example:L.D.Landau 8s E . M . L i f s h i t z , The C l a s s i c a l Theory of F i e l d s (Addison Wesley,1958). 20. see f o r example:H.G.Griem, Plasma Spectroscopy(Mcgraw H i l l , 1964) . 21. A.van der Z i e l , Noise ( P r e n t i c e - H a l l Inc.,1954). 22. Lord R a y l e i g h , P h i l . Mag.  47,375(1899).  23. G.A.Theophanis, R e v . S c i . I n s t . 31^ 427(1960). 24. F.L.Curzon and P.R.Smy,Rev.Sci.Inst.  32,756(1961).  25. B.Ahlborn,Z.Naturforsch. 20,466(1965). 26.IPP R e p o r t , I n s t i t u t e f u r Plasmaphysik,Garching B e i Munchen (1963). 27, Nyugen-Quang-Dong, P h y s . L e t t e r s 21, 159(1966). 28. see f o r example:S.Gartenhaus,Elements of Plasma P h y s i c s , ( H o l t , R i n e h a r t and Winston,1964).  

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