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Instabilities of a Z-pinch discharge Hodgson, Rodney Trevor 1964

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INSTABILITIES OF A Z-PTNCH DISCHARGE  by RODNEY TREVOR HODGSON B.Sc,  The U n i v e r s i t y o.f B r i t i s h Columbia, I960  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS  We accept t h i s thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COUJMBIA A p r i l , 196U  In the  r e q u i r e m e n t s f o r an  British  mission  for reference  for extensive  p u r p o s e s may  be  without  advanced  of my  written  and  by  It  this thesis  that  study.  the  in partial  degree at  the  copying of  granted  representativesc,  cation  this thesis  Columbia, I agree  available  his  presenting  Library  this thesis  Head o f my  i s understood  permission.  Department Columbia,  of  University  of  s h a l l make i t f r e e l y  I further  agree for  that  or  c o p y i n g or  shall  per-  scholarly  Department  that  for f i n a n c i a l gain  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada.  the  fulfilment  not  be  by publi-  allowed  The U n i v e r s i t y  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  RODNEY TREVOR HODGSON  B.'Sc, The U n i v e r s i t y  MONDAY  s  of British  Columbia  FEBRUARY 3, 1964, AT 3:30 P.M.  IN ROOM 303, HENNINGS BUILDING (PHYSICS)  COMMITTEE IN CHARGE Chairman: F.H. Soward F.L. Curzon A.J. Barnard F.W. Dalby External  . C.P.S. T a y l o r W.A.G. Voss T. Watanabe  Examiner: A. F o l k i e r s k i Imperial  College  U n i v e r s i t y o f London  INSTABILITIES OF A Z-PINCH DISCHARGE ABSTRACT  The first  c y l i n d r i c a l column o f plasma produced  i n the  stage o f a z - p i n c h d i s c h a r g e i s t h e o r e t i c a l l y  unstable.  F o r one p a r t i c u l a r  the amplitude  type o f i n s t a b i l i t y ,  of a surface p e r t u r b a t i o n increases at  a r a t e dependent on the a c c e l e r a t i o n o f the s u r f a c e (Rayleigh-Taylor  instabilities).  An e x p e r i m e n t a l study o f these i n s t a b i l i t i e s has been c a r r i e d out by photographing column w i t h a high-speed Simple  the d i s c h a r g e  framing-camera.  r o t a t i o n a l l y symmetric i n s t a b i l i t i e s  been e x c i t e d i n the n o r m a l l y s t a b l e i n i t i a l  have  stage o f  an argon z - p i n c h d i s c h a r g e by means o f a s e t o f e q u a l l y spaced g l a s s r i n g s . photographs  show that the i n s t a b i l i t i e s  p r o x i m a t e l y i n accordance theory.  The framing camera develope ap-  w i t h the R a y l e i g h - T a y l o r  No a x i a l d r i f t o f the i n s t a b i l i t i e s  i s ob-  served, but the new technique o f s t u d y i n g i n s t a b i l i t i e s r e v e a l s that the a c c e l e r a t i o n o f the d i s c h a r g e boundary changes a p p r e c i a b l y t h r e e o r f o u r times d u r i n g the i n i t i a l  stage o f the d i s c h a r g e .  GRADUATE STUDIES  F i e l d o f Study:  Physics  Electromagnetic Waves  Theory  1...  G.M. V o l k o f f R,W.  Quantum Theory o f S o l i d s Plasma P h y s i c s Advanced Plasma  Stewart  R, B a r r i e L.G. de S o b r i n o  Physics  Magneto-hydrodynamics  F.L. Curzon F.  . C.urzon P.R. Smy  CA.  Swans on  Related Studies: D i f f e r e n t i a l Equations  PUBLICATIONS  - Delayed Implosion of the Z-Pinch i n Nitrogen= F.L, Curzon, R.T. Hodgson and R.J. C h u r c h i l l ; Can.Jour.Phys. 41, 1547, (1963). - E x c i t a t i o n o f m=0 I n s t a b i l i t i e s i n a Z-Pinch DischargeF.L. Curzon, R.T, Hodgson and R.J. C h u r c h i l l ; Accepted f o r p u b l i c a t i o n i n J o u r . N u c l . E n e r g y (Part C ) .  ABSTRACT  Simple r o t a t i o n a l l y symmetric i n s t a b i l i t y modes have been excited i n the normally stable i n i t i a l stage of an argon Z-pinch discharge by means of a set of equally spaced glass rings.  High speed framing camera photographs  show that the  i n s t a b i l i t i e s develop approximately i n accordance with the Rayleigh-Taylor theory.  No a x i a l d r i f t of the i n s t a b i l i t i e s  i s observed but the new technique of studying i n s t a b i l i t i e s reveals that the acceleration of the discharge boundary changes appreciably three or four times during the course o f the f i r s t stage of the discharge.  F. L. Curzon  -viii-  ACKNOWLEDGEMENTS  I would l i k e t o t h a n k D r . F. L. C u r z o n f o r h i s e x c e l l e n t d i r e c t i o n and a d v i c e t h r o u g h o u t t h e c o u r s e o f t h e e x p e r i m e n t s reported here. I would a l s o l i k e t o thank t h e o t h e r members o f t h e plasma p h y s i c s group and D r . F. W. D a l b y f o r many s t i m u l a t i n g discussions. The i n v a l u a b l e a s s i s t a n c e o f M r . J . H. T u r n e r , Mr. J o h n L e e s , and t h e d e p a r t m e n t a l work shop s t a f f i n t h e c o n s t r u c t i o n of apparatus i s gr&tdTully  acknowledged.  -iii-  TABLE OF CONTENTS  Abstract  i i  L i s t of I l l u s t r a t i o n s  v  L i s t of Tables  vi  L i s t o f Plates  vii  Acknowledgements CHAPTER CHAPTER  viii  I - Introduction I I - Theoretical Background Section A - Unperturbed Discharge Section B - Perturbed Discharge  CHAPTER I I I - Experimental Apparatus and Data Measurement Section A - Apparatus  5 3> 9  19 19  1.  Discharge C i r c u i t  22  2.  Triggering Operation  25  3.  Discharge Perturbation  29  Section B - Measurement Devices and Data Measurement  CHAPTER  1  30  1.  Current Measurement  30  2.  High Speed Framing Camera  3k  IV - Observations and Results .  37  Section A - Unperturbed Discharge  37  Section B - Perturbed Discharge  li2  1.  Observations  li2  2. Treatment of Data 3o  Error Analysis  U»  Results  5.  Discussion of Results  6. Suggestions for Further Work CHAPTER  V - Conclusions  Appendix A Appendix B ... Bibliography  LIST OF ILLUSTRATIONS  FIG.  FIG.  FIG.  I . 1 - Gas and E l e c t r o d e s o f Z - P i n c h  I I . 1 = Diagram o f F i r s t S t a g e o f a Z - P i n c h D i s c h a r g e  2  6  2 - Diagram o f P e r t u r b e d Plasma S u r f a c e  12  3 - "Long" Wavelength I n s t a b i l i t y  17  k - " S h o r t " Wavelength I n s t a b i l i t y  17  I I I . l = B l o c k Diagram o f A p p a r a t u s  20  2 - S c h e m a t i c Diagram o f D i s c h a r g e C i r c u i t  22  3 - Condenser Bank  23  k «=• M a i n Spark Gap S w i t c h  2^  $ - Z - P i n c h D i s c h a r g e Tube  26  6 - C i r c u i t Diagram o f I s o l a t e d D i s c h a r g e C i r c u i t and T r i g g e r G e n e r a t o r  28  7 ~ Diagram o f G l a s s R i n g s U s e d t o P e r t u r b D i s c h a r g e  30  8 - Cut-away D r a w i n g o f Rogowski C o i l  31  9 - C i r c u i t Diagram o f Rogowski C o i l and I n t e g r a t o r  31  10,» C u r r e n t T r a c e s f r o m Rogowski C o i l  33  11 •= Framing C a t ^ r a and C o n t r o l Equipment  3h  12 - Diagram o f One Frame o f a P e r t u r b e d D i s c h a r g e Photograph  FIG.  36  I V . 1 - R a d i u s V e r s u s Time and C u r r e n t V e r s u s Time i n A and N 2 D i s c h a r g e s Under Comparable D e n s i t y C o n d i t i o n s  ho  _vi~  FIG.  IV.2 - Pinch Times f o r Various Discharges Versus '  o  3 - Plots of I n A versus t , and "r" versus t I4. ~ LP Versus / a f o r a)  ^ —  2 cm  b)  ^ ~-  3 cm  c)  ^  ~= k cm  LIST OF TABLES  TABLE I I I . l » E s s e n t i a l Characteristics of Apparatus TABLE  IV.1 - Dependence o f Measuring Errors i n a and uO Number of Frames (n) 2 - Dependence of cO on i/a i n Argon  for Instabilities  -vii-  LIST OF PLATES  PLATE I V .  I  S i n g l e Frames o f Unperturbed  Discharge i n  SOOp/Kg A r g o n II  38  Framing Camera Photograph o f 500^vHg Perturbed Argon Discharge  III  hi  S i n g l e Frames o f P e r t u r b e d D i s c h a r g e i n  1*8  500^Hg Argon IV  S i n g l e Frame o f P e r t u r b e d D i s c h a r g e Rings  PLATE  A.  I II  with  9 cm A p a r t  53  F i r s t Stage o f U n p e r t u r b e d  N i t r o g e n Discharge  63  S i n g l e Frames o f N i t r o g e n Showings a) U n p e r t u r b e d  F i r s t Stage  b.) P o s t - p i n c h I n s t a b i l i t i e s  i n Unperturbed  D i s charge c) P e r t u r b e d P r e - P i n c h D i s c h a r g e d) P e r t u r b e d P o s t - P i n c h D i s c h a r g e  65  I I I - a and b - P e r t u r b e d P o s t - P i n c h I n s t a b i l i t i e s i n Argon;  X i 2 era and 3 cm  c and d - P e r t u r b e d Hydrogen D i s c h a r g e ; and P o s t - P i n c h  Pre-Pinch 66  CHAPTER I  INTRODUCTION  I n r e c e n t y e a r s , t h e r e has been a surge o f i n t e r e s t i n plasma p h y s i c s , s i n c e s t u d i e s o f t h e g r o s s p r o p e r t i e s o f plasmas a r e v i t a l l y necessary t o research i n the f i e l d s o f c o n t r o l l e d nuclear f u s i o n ( B i s h o p , 1958), a t m o s p h e r i c p h y s i c s ( E l l i o t , 1962), and space r e s e a r c h  ( E n g e l , 1959). S t e l l a r and i n t e r s t e l l a r plasmas a r e c o n t r o l l e d i n e q u i l i b r i u m b y n a t u r a l l y o c c u r r i n g g r a v i t a t i o n a l and magnetic f i e l d s .  However,  l a b o r a t o r y plasmas c a n e x i s t o n l y a s h o r t t i m e , s i n c e c o n t a c t w i t h any " c o l d " substance r a p i d l y c o o l s and d e - i o n i z e s them.  For this  reason,  magnetic f i e l d s a r e u s e d t o " c o n t a i n " many l a b o r a t o r y plasmas and keep them away f r o m s o l i d o b j e c t s w h i c h serve as h e a t s i n k s . F o r such l a b o r a t o r y p l a s m a s , r a d i a t i o n i s t h e most i m p o r t a n t l o s s mechanism ( S p i t z e r ,  heat  1956). The r a d i a t i o n r a t e i n c r e a s e s r a p i d l y  w i t h i n c r e a s i n g plasma t e m p e r a t u r e so t h a t h i g h e r t e m p e r a t u r e s c a n be achieved  o n l y i f t h e power i n p u t r a t e exceeds t h e power l o s s r a t e . One  way o f a c h i e v i n g a h i g h power i n p u t r a t e i s t o d i s c h a r g e t h e s t o r e d e l e c t r i c a l e n e r g y o f a condenser i n t o t h e plasma.  One o f t h e d e v i c e s  u s i n g t h i s method i s t h e so c a l l e d Z - p i n c h . A Z - p i n c h d e v i c e r e l e a s e s t h e e n e r g y o f t h e condenser d i r e c t l y i n t o a gas column b y d i s c h a r g i n g a h i g h c u r r e n t t h r o u g h i t  (see F i g . l ) .  _2-  The  ad-vantage o f t h i s d e v i c e i s t h a t t h e magnetic f i e l d generated b y  t h e h i g h c u r r e n t s i m u l t a n e o u s l y h e a t s t h e plasma,and keeps i t away from h e a t s i n k s .  F i g . 1.1.  Gas and E l e c t r o d e s o f Z - p i n c h .  F i g u r e 1 shows t h e c y l i n d r i c a l l y symmetric system o f gas and e l e c t r o d e s t h a t forms t h e e s s e n t i a l p a r t o f a Z - p i n c h d e v i c e . a x i a l c u r r e n t ( I ) produces an a z i m u t h a l magnetic f i e l d  The  (B* ) . The e  c u r r e n t c a r r y i n g e l e c t r o n s have an a x i a l d r i f t v e l o c i t y v and experience  a Lorentz force  e ( v x B) a c t i n g r a d i a l l y i n w a r d .  The  e l e c t r o n s t r e a m t h e r e f o r e c o n t r a c t s r a d i a l l y and draws t h e i o n s i n w i t h i t b y means o f space charge f i e l d s .  T h i s p r o c e s s i s known as  the " P i n c h E f f e c t " , and, when t h e plasma column reaches i t s minimum diameter,  i t i s s a i d t o be  "pinched".  A p i n c h e d plasma column may be m a i n t a i n e d  i n equilibrium i f  t h e magnetic and k i n e t i c p r e s s u r e s a r e e q u a l .  However, i t c a n be shown  t h a t t h i s e q u i l i b r i u m i s u n s t a b l e , s i n c e any s m a l l p e r t u r b a t i o n o f t h e equilibrium c o n f i g u r a t i o n w i l l increase with time.  T h e o r e t i c a l con-  s i d e r a t i o n s show t h a t t h e plasma s u r f a c e i s a l s o u n s t a b l e even d u r i n g t h e i n i t i a l s t a g e when t h e d i s c h a r g e column i s c o n t r a c t i n g r a d i a l l y . The t i m e dependence  o f the i n s t a b i l i t i e s  i s determined by the p h y s i c a l  p r o p e r t i e s o f t h e d i s c h a r g e and o f t h e plasma.  For t h i s reason, t h e  s t u d y o f plasma i n s t a b i l i t i e s c a n p r o v i d e i m p o r t a n t i n f o r m a t i o n about t h e plasma i t s e l f . Plasma s u r f a c e i n s t a b i l i t i e s have been e x t e n s i v e l y t r e a t e d t h e o r e t i c a l l y (Summer S c h o o l o f Plasma P h y s i c s , R i s o , I960), b u t because o f e x p e r i m e n t a l d i f f i c u l t i e s v e r y l i t t l e e x p e r i m e n t a l work has been r e p o r t e d (Curzon e t . a l , 1960| G r e e n and N i b l e t t , I960; C u r z o n and C h u r c h i l l , 1962; H e r t z , 1962). C o m p a r i s i o n o f t h e e x p e r i m e n t a l work w i t h t h e a v a i l a b l e t h e o r i e s i s a l s o extremely d i f f i c u l t , since t h e o r i e s postulate s i n u s o i d a l ( s i n g l e mode) s u r f a c e p e r t u r b a t i o n s , whereas i n t y p i c a l c o n d i t i o n s t h e plasma s u r f a c e i s h i g h l y i r r e g u l a r .  experimental  A F o u r i e r decompo-  s i t i o n o f t h e s u r f a c e would i n p r i n c i p l e be p o s s i b l e , e n a b l i n g a c o m p a r i s o n o f t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s .  However, w i t h  p r e s e n t e x p e r i m e n t a l t e c h n i q u e s F o u r i e r a n a l y s i s o f t h e plasma s u r f a c e is  impractical. T h i s t h e s i s p r e s e n t s f o r t h e f i r s t t i m e a n a l t e r n a t i v e method o f  s t u d y i n g plasma s u r f a c e i n s t a b i l i t i e s i n t h a t t h e t y p e s o f i n s t a b i l i t i e s a r e c o n t r o l l e d b y t h e e x p e r i m e n t a l d e s i g n i n s t e a d o f d e v e l o p i n g randomly. T h i s r e p r e s e n t s a s i g n i f i c a n t advance over e a r l i e r e x p e r i m e n t a l  techniques  s i n c e measurements c a n be made more a c c u r a t e l y and s i n c e t h e experimental  and t h e o r e t i c a l r e s u l t s c a n b e compared without r e s o r t i n g t o  F o u r i e r a n a l y s i s o f t h e plasma s u r f a c e . The o b s e r v a t i o n s on plasma s u r f a c e i n s t a b i l i t i e s d e s c r i b e d i n t h i s t h e s i s were made on t h e i n i t i a l discharge.  stage o f a 5>00p Hg argon  Z-pinch  Under t h e s e c i r c u m s t a n c e s , t h e e x p e r i m e n t a l c o n d i t i o n s  conform most c l o s e l y t o t h o s e assumed i n s i m p l e t h e o r e t i c a l models. The t h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s ( f o r a s t a b l e d i s c h a r g e ) needed t o d i s c u s s t h e s e i n s t a b i l i t i e s C h a p t e r s I I and I V .  The apparatus  are given i n t h e f i r s t  parts of  and measurement methods a r e out-  l i n e d i n Chapter I I I , and t h e main r e s u l t s o f t h e o r y and experiment on plasma s u r f a c e i n s t a b i l i t i e s Chapters I I and I V .  are reported i n the l a t t e r parts o f  CHAPTER I I  THEORETICAL BACKGROUND  This chapter i s d i v i d e d i n t o  two s e c t i o n s c o r r e s p o n d i n g t o t h e  d i v i s i o n o f t h e e x p e r i m e n t a l r e s u l t s p r e s e n t e d i n Chapter I V . The m a t e r i a l i n S e c t i o n A c o v e r s t h e f i r s t  stage o f t h e Z - p i n c h  d i s c h a r g e when t h e plasma s u r f a c e o f t h e d i s c h a r g e column i s unperturbed.  A simple model ( o f t h e d i s c h a r g e ) which g i v e s r e s u l t s  agreeing with experimental r e s u l t s i s t r e a t e d . In  S e c t i o n B, t h e development o f r o t a t i o n a l l y symmetric p e r -  t u r b a t i o n s o f t h e plasma s u r f a c e i s c a l c u l a t e d .  S e c t i o n A - Unperturbed The is  Discharge  r a d i u s o f the Z - p i n c h d i s c h a r g e column as a f u n c t i o n o f time  one o f t h e e a s i e s t o f i t s p h y s i c a l parameters  tally.  S e v e r a l authors  t o measure experimen-  (Roseribluth, 195>1|J Kuwabara, 1963)  have  f o r m u l a t e d models o f t h e i n i t i a l stage o f t h e Z-pinch i n which t h e p h y s i c a l conditions o f the discharge are s p e c i f i e d .  From these models,  the r a d i u s o f t h e d i s c h a r g e column may be c a l c u l a t e d and compared t o the e x p e r i m e n t a l l y measured r a d i u s . The r e s u l t s p r o v i d e d b y t h e v a r i o u s t h e o r i e s do not d i f f e r appreciably.  F o r t h i s reason, o n l y t h e s i m p l e s t model, t h e snow-plow  model i n t r o d u c e d b y Rosenbluth  (195k),  w i l l be c o n s i d e r e d i n d e t a i l i n  -6.  this thesis.  The shock wave model t r e a t e d b y Kuwabara ( 1 9 6 3 ) i s more  c o m p l i c a t e d , w h i l e t h e f r e e p a r t i c l e model i n t r o d u c e d a l s o b y Rosenbluth  (l9<h)  }  i s n o t a p p l i c a b l e t o h i g h d e n s i t y plasmas such as  those t r e a t e d i n t h i s t h e s i s . The snow-plow model i s a p p l i c a b l e i f t h e d i s c h a r g e c u r r e n t f l o w s i n a t h i n c y l i n d r i c a l s h e l l a t t h e boundary o f t h e plasma column.  The  model assumes t h a t t h i s s h e l l sweeps up a l l t h e gas i n t o i t s e l f as i t moves i n w a r d .  I t i s a l s o assumed t h a t t h e k i n e t i c gas p r e s s u r e i s  s m a l l compared w i t h t h e magnetic p r e s s u r e a r i s i n g from L o r e n t z f o r c e s . Suppose t h a t t h e gas column has r a d i u s R  0  and R ( t ) (meters) a f t e r a t i m e t ( s e e F i g . 1 ) .  (meters) a t t i m e t - 0 The mass o f gas swept  up i n t o t h e i n w a r d moving s h e l l i s j u s t t h e mass o f t h e gas which i n i t i a l l y o c c u p i e d t h e c y l i n d r i c a l volume between r a d i i R  Q  and R ( t ) .  The mass p e r u n i t l e n g t h (M) o f t h e s h e l l i s g i v e n b y e q u a t i o n  (1).  Return current conductor G l a s s tube  Current carrying mass s h e l l  U n d i s t u r b e d gas  Vacuum  Fig. I I . 1 .  Diagram o f F i r s t Stage o f a Z - P i n c h D i s c h a r g e .  -7-  M  (1)  =  7T p  [R  „ R (t)3  2  2  O  M has u n i t s o f k i l o g r a m s p e r u n i t l e n g t h o f t h e d i s c h a r g e column, and p  i s t h e i n i t i a l gas d e n s i t y i n KG/rrrl  The momentum (P) a s s o c i a t e d  w i t h t h i s mass o f gas i s g i v e n b y  P = 7* P  (2)  R  L  - R (t)j  2  d R(t)/dt  2  n  0  The r a t e o f change o f t h i s momentum must be e q u a l t o t h e t o t a l L o r e n t z F o r c e ( F ) a c t i n g o n t h e s u r f a c e o f t h e plasma.  F  (3)  =  - _2KR(t)] p  F i s given by  I ( t ) / 2 \2T\H{t)\ 2  n  where l ( t ) i s t h e t o t a l d i s c h a r g e c u r r e n t ( i n amperes) as a f u n c t i o n of time. U s u a l l y , i t i s c o n v e n i e n t t o work i n terms o f t h e d i m e n s i o n l e s s r a d i u s v a r i a b l e (y) d e f i n e d b y  y(t)  (h)  =  R(t)/R  c  B y e q u a t i n g t h e t i m e d e r i v a t i v e o f t h e momentum ( e q u a t i o n 2) w i t h t h e f o r c e ( e q u a t i o n 3) and u s i n g e q u a t i o n (I4), we may d e r i v e t h e following equation.  (5)  (1 - y ) 2  dt  § dt  -8-  E q u a t i o n {$) i s r e f e r r e d t o as R o s e n b l u t h ' s  snow-plow e q u a t i o n . (5)  I f t h e t o t a l c u r r e n t l ( t ) i s known as a f u n c t i o n o f t i m e , e q u a t i o n c a n be s o l v e d . A p a r t i c u l a r l y simple s o l u t i o n occurs i f  (6)  I  where I  0  =  IQ  and t p a r e c o n s t a n t s .  i n t o e q u a t i o n (5)  s i n ( TT t / t p )  By d i r e c t l y s u b s t i t u t i n g equation  (6)  and i n t e g r a t i n g , t h e f o l l o w i n g s o l u t i o n f o r y i s  obtained:  (7)  y  = . cos ( TT  t/2t ) p  where  Now, when t  —  tp, y(t) =  0 , and t p r e p r e s e n t s t h e " p i n c h t i m e "  f o r t h e snow-plow model w i t h a s i n u s o i d a l l y t i m e dependent c u r r e n t . The p i n c h t i m e i s t h e t i m e w h i c h e l a p s e s between t h e s t a r t o f t h e d i s c h a r g e and t h e " p i n c h " ( i . e . minimum d i s c h a r g e r a d i u s ) . I n p r a c t i c e , t h e d i s c h a r g e c u r r e n t does n o t have t h e t i m e , dependence p r e s c r i b e d b y e q u a t i o n ( 6 ) .  However, when t h e t i m e depend-  ence i s a p p r o x i m a t e l y s i n u s o i d a l , one would e x p e c t t h e o b s e r v e d p i n c h t i m e s t p t o depend on t h e p a r a m e t e r s R , /° , and I 0  given by equation  (8).  0  i n t h e manner  „9~  The  s o l u t i o n f o r y g i v e n b y e q u a t i o n (7)  i s not a p p l i c a b l e t o  h i g h energy f a s t discharges s i n c e the current v a r i e s l i n e a r l y w i t h time r a t h e r than s i n u s o i d a l l y . equation  (5)  However,, i f I i s p r o p o r t i o n a l to t ,  has t h e power s e r i e s s o l u t i o n *  (9) 1  where t p , i n t h i s c a s e , i s p r o p o r t i o n a l t o  R ^ . 4  Q  These v a r i o u s r e s u l t s d e r i v e d above may  be checked e x p e r i m e n t a l l y  w i t h o u t d i s t u r b i n g t h e d i s c h a r g e , s i n c e the d i s c h a r g e r a d i u s R ( t ) be measured p h o t o g r a p h i c a l l y and the d i s c h a r g e c u r r e n t l ( t )  can  can  be  measured w i t h a Rogowski c o i l . I n t h i s s e c t i o n , complete a x i a l and r o t a t i o n a l symmetry o f t h e d i s c h a r g e has been i m p l i e d . where t h e d i s c h a r g e  I n s e c t i o n B , we s h a l l c o n s i d e r the case  r a d i u s depends a l s o on t h e a x i a l v a r i a b l e Z.  Section B - Perturbed  Discharge  I n s t a b i l i t i e s , o r d i s t u r b a n c e s which i n c r e a s e w i t h t i m e , are  not  g e n e r a l l y o b s e r v e d on t h e plasma s u r f a c e i n the i n i t i a l stage o f t h e Z-pinch.  Y e t , a l l o f t h e proposed models a r e t h e o r e t i c a l l y u n s t a b l e  to a r b i t r a r y p e r t u r b a t i o n s o f t h e plasma s u r f a c e . As an a n a l y t i c a l l y s i m p l e example, c o n s i d e r the snow-plow model o f t h e p i n c h i n w h i c h t h e d i s c h a r g e c u r r e n t ( I ) i s p r o p o r t i o n a l t o the t i m e ( t ) from t h e s t a r t o f the d i s c h a r g e .  Each p a r t o f the plasma  s u r f a c e obeys e q u a t i o n (£) and moves, i n d e p e n d e n t l y o f any o t h e r p a r t , to  the a x i s .  I f we a p p l y a s m a l l i n i t i a l s i n u s o i d a l p e r t u r b a t i o n t o  -10-  t h e plasma s u r f a c e so t h a t t h e d i s c h a r g e r a d i u s a t time t — 0 (R ) becomes a f u n c t i o n o f Z g i v e n b y e q u a t i o n ( 1 0 ) , t h e n t h e r a d i i Q  o f two d i f f e r e n t p a r t s o f t h e plasma s u r f a c e ( a t Z]_ and Z^ say) a r e given b y equations  (11) and ( 1 2 ) .  (10)  R (Z)=  ~% - k  (11)  R(Z ,t)  = R (Z_) j  (12)  a(z ,t)  = R (z (  x  where  D  =  mean i n i t i a l r a d i u s  A  =  constant  *p(Zl)  °<  M  Z  [ 1 -  2  R o Q  y  2  )  R  I f we choose Z i  [t/t (z )] ,| 2  p  2  t  l ) >  )(see z  l - [t/tp(Z_)]?...}  0  2  s i n kZ  0  0  o( 2)  e q u a t i o n (9) e t seq.)  )  z  H/2k and Z  2  —  ~ 7 y 2 k , t h e maximum d i f f e r e n c e  o f t h e r a d i i ( o r t h e " a m p l i t u d e " o f t h e d i s t u r b a n c e ) has t h e time dependence shown b y e q u a t i o n (13)•  (13)  R(Z_ t ) - R(Z ,t) - 2 A 2  Q  [ l +K(t/t ) p  where K  ^  1  and  ) tp  -  t (z ) p  x  ^  t (z ) p  2  2 >  ]  -11-  On t h e b a s i s o f t h i s s i m p l e model, t h e n , t h e p a r t i c u l a r d i s t u r b a n c e c o n s i d e r e d above w i l l i n c r e a s e q u a d r a t i c a l l y w i t h t i m e . I n t h e more g e n e r a l c a s e , t h e r a d i a l p o s i t i o n o f t h e plasma s u r f a c e o f a Z - p i n c h d i s c h a r g e may be w r i t t e n a s a F o u r i e r s u p e r p o s i t i o n o f terms l i k e  (lk)  A ( k , m , to) exp i ( k Z +  m.9) e x p e c t  I f m "= 0, t h e d i s c h a r g e i s r o t a t i o n a l l y symmetric and t h e space v a r i a t i o n o f t h e plasma s u r f a c e depends o n l y on t h e a x i a l v a r i a b l e Z. S u r f a c e s o f t h i s t y p e (m = 0) a r e t h e e a s i e s t t o observe i n t h e Z - p i n c h d i s c h a r g e , and f o r t h i s r e a s o n a r e c o n s i d e r e d i n more d e t a i l below. The  time dependence o f t h e plasma s u r f a c e c o o r d i n a t e s may be  c a l c u l a t e d b y s o l v i n g t h e r e l e v a n t e q u a t i o n s o f m o t i o n and c o n s e r v a t i o n . I f an e q u i l i b r i u m s o l u t i o n t o t h e s e e q u a t i o n s i s found, t h e s t a b i l i t y o f t h e s o l u t i o n may be checked b y p o s t u l a t i n g an a r b i t r a r y , s m a l l disturbance o f the e q u i l i b r i u m s i t u a t i o n .  I f the disturbance increases  with time, the e q u i l i b r i u m s o l u t i o n i s unstable. The a r b i t r a r y , s m a l l p e r t u r b a t i o n o f t h e plasma s u r f a c e i s u s u a l l y p o s t u l a t e d i n t h e f o r m o f t h e e x p r e s s i o n (lU),  s i n c e any p e r t u r b a t i o n  o f t h e s u r f a c e may be made up b y combining e x p r e s s i o n s o f t h i s t y p e . When t h e e q u a t i o n s o f m o t i o n a r e s o l v e d , c e r t a i n r e l a t i o n s a r e f o u n d between m, k, and oo. If  These r e l a t i o n s a r e c a l l e d d i s p e r s i o n r e l a t i o n s .  CU has a r e a l , p o s i t i v e p a r t , t h e d i s t u r b a n c e w i l l grow  e x p o n e n t i a l l y w i t h t i m e , and t h e e q u i l i b r i u m s o l u t i o n i s u n s t a b l e .  -12-  ¥e s h a l l now c a l c u l a t e t h e d i s p e r s i o n r e l a t i o n between k and U-> f o r r o t a t i o n a l l y symmetric (m -= 0) plasma s u r f a c e s i n a Z - p i n c h discharge.  A p h y s i c a l l y r e a l i s t i c model o f t h e d i s c h a r g e w i l l be u s e d ,  where t h e assumptions a r e : 1) The c u r r e n t f l o w s i n a t h i n l a y e r on t h e s u r f a c e o f t h e plasma. 2 ) The plasma mass s h e l l i s o f f i n i t e t h i c k n e s s . 3) The plasma s h e l l h a s a c o n s t a n t a x i a l l y d i r e c t e d r a d i a l acceleration. For a n a l y t i c a l  s i m p l i c i t y , we may l e t t h e r a d i u s o f t h e d i s c h a r g e  (R) t e n d t o i n f i n i t y so t h a t t h e c y l i n d r i c a l s u r f a c e becomes pllanar (see F i g .  2).  4  Z  -»  f>  a  Fig. I I . 2 .  Diagram o f P e r t u r b e d Plasma S u r f a c e .  -13-  T h i s problem has been t r e a t e d b y K r u s k a l and S c h w a r z s c h i l d  (I95h)•  A s i m p l i f i e d form o f t h e i r t r e a t m e n t f o l l o w s . C o n s i d e r an i n f i n i t e l y c o n d u c t i n g non-viscous' plasma supported i n e q u i l i b r i u m a g a i n s t g r a v i t y o r a n a c c e l e r a t i o n " a " b y a magnetic f i e l d B . The i n t e r i o r o f t h e plasma i s d e s c r i b e d b y t h e e q u a t i o n o f m o t i o n ( 1 5 ) , the e q u a t i o n f o r c o n s e r v a t i o n o f mass ( 1 6 ) and t h e c o n s e r v a t i o n o f entropy equation (17). by Maxwell's equations  The vacuum e x t e r i o r t o t h e plasma i s d e s c r i b e d (18) and ( 1 9 ) .  The s u p p o r t i n g magnetic  pressure  i s p r o v i d e d b y a d i s c o n t i n u i t y i n t h e magnetic f i e l d w h i c h a r i s e s f r o m a sheet c u r r e n t on t h e plasma s u r f a c e . Let  "v* = plasma v e l o c i t y i n m/sec p  = plasma d e n s i t y i n KG/m^  P  — plasma p r e s s u r e i n newtons/m  2  —» a  2 =. d i r e c t e d a c c e l e r a t i o n i n m/sec  Then  (15)  (16)  (17)  P  ~  =  " §  A  D  P  div(pv)  1 dP p ^ -  div B  (19)  curllT  ~ P  a  ^  at  p  (18)  The  R  a t  =0  =  e q u i l i b r i u m s o l u t i o n o f equations  0  ( 1 5 ) - (19) i s g i v e n b y  I v  (20)  =  Q  0  =hy  (21)  (22)  P  Q  (23)  (  2  =  P  B , B  W  a  (25)  P r - o  P  where h i s an " a t m o s p h e r i c "  T  '-  y  =  -  =  0  e-*r  y x 0  T  Q  h  B  X  y - o  P  0  2  A  constant,  -w a p p l y a s m a l l s i n g l e mode p e r t u r b a t i o n t o t h e plasma  r  s u r f a c e so t h a t i t s y c o o r d i n a t e ( s a y ^ )  (26)  ^ ( t - 0) -  where A  / 2  A  i s a c o n s t a n t and t h e w a v e l e n g t h  o  Q  i s given by equation  e  (26)  i k z  ()\) of the perturbation i s  given by  (27)  ^  Suppose t h a t ^ Let  ^  =  2 A/k  s t a n d s f o r any o f t h e t i m e dependent v a r i a b l e s .  be t h e e q u i l i b r i u m v a l u e o f  ^ , and  0  -f i  be t h e p e r t u r b e d  value.  (28)  I f we assume t h a t  \ ^  s  and t h e n s u b s t i t u t e  Y  i  t ) =  ^  |  ^  e" ^ e k  -j- ^-j.  Q  e  i k z  w  t  e q u a t i o n s (l£) - (19), we  i n t o  o b t a i n , on n e g l e c t i n g second o r d e r t e r m s , t h e f o l l o w i n g s e t o f e q u a t i o n s  (29)  /?  (30)  - k p  ?  *7 l  1 -  0  P  v  k  " ®a  + p  0  i  k  P  l "  f l  * \  (-h/> > =  a  e  -  0  - U J ^  o E q u a t i o n s (29), (30) and (31) r e p r e s e n t a s e t o f l i n e a r homogeneous equations f o r the "amplitude f a c t o r s "  % \»  A non-trivial solution of  t h i s s e t o f e q u a t i o n s e x i s t s i f and o n l y i f t h e d e t e r m i n a n t o f t h e coefficients vanishes.  If h  <<  k ( i . e . t h e p r e s s u r e and d e n s i t y  e q u i l i b r i u m v a l u e s change l i t t l e over a d i s t a n c e "X ) , t h e n t h e d e t e r m i n a n t v a n i s h e s i f and o n l y i f  LJO  (32)  2  -  ak  T h i s r e l a t i o n between t h e e x p o n e n t i a l growth r a t e  <-0 and t h e wave  number k ( t h e d i s p e r s i o n r e l a t i o n ) i s s i m i l a r t o t h e d i s p e r s i o n r e l a t i o n o f t h e w e l l known i n s t a b i l i t y o f a l i g h t f l u i d s u p p o r t i n g a h e a v i e r one against g r a v i t y (Taylor  5  1950).  I n s t a b i l i t i e s o f t h i s kind are generally  -16-  called Rayleigh-Taylor  instabilities.  The p o s i t i o n o f t h e plasma s u r f a c e i s g i v e n by t h e  (33)  ?  =  A  to t +  (  exp  0  equation  ikz)  T h i s means t h a t the plasma s u r f a c e w i l l r e t a i n i t s s i n u s o i d a l shape, but the a m p l i t u d e  (A)  o f t h e disturbances, g i v e n i n i t i a l l y b y A  equation ( 2 6 ) , w i l l i n c r e a s e e x p o n e n t i a l l y i n time  (3U)  A(t)  =  A  D  exp(  0  of  since  wt)  where  (35)  tjl  =/  ak  S t r i c t l y s p e a k i n g , t h e t h e o r i e s g i v e n above can o n l y be a p p l i e d t o the Z - p i n c h d i s c h a r g e when c e r t a i n l i m i t i n g  c o n d i t i o n s are f u l f i l l e d .  These c o n d i t i o n s are d i s c u s s e d below. I n t h e f i r s t case o f t h e snow-plow i n s t a b i l i t y , we assume i m p l i c i t l y t h a t t h e wavelength o f t h e i n s t a b i l i t y i s much l o n g e r t h e plasma l a y e r t h i c k n e s s (J) as drawn i n F i g . 3 . j u s t completed a l s o h o l d s f o r a away i n t o the plasma o n l y as  than  S i n c e the a n a l y s i s  t h i n mass s h e l l , the d i s t u r b a n c e d i e s  e x p ( - k y ) ( e q u a t i o n ( 2 8 ) ) , so t h a t t h e  i n s i d e s u r f a c e o f the plasma l a y e r w i l l be d i s t o r t e d as w e l l as t h e outside surface.  The  s h e l l moving i n sweeps up gas i n t o i t s e l f and  the  mass i n f l u x t e n d s t o damp the growth o f t h e i n s t a b i l i t y from e x p o n e n t i a l  -17-  F i g . II.3.  "Long" Wavelength I n s t a b i l i t y .  Fig. II.U.  " S h o r t " Wavelength I n s t a b i l i t y .  •18-  ( R a y l e i g h - T a y l o r ) time dependence t o a q u a d r a t i c t i m e dependence. I n t h e second case d i e s away r a p i d l y , and t h e o u t s i d e s u r f a c e has v i r t u a l l y no knowledge o f t h e mass i n f l u x  occurring at the i n n e r surface o f the s h e l l .  On t h e b a s i s o f t h e s e arguments^ we would expect an e x p o n e n t i a l t i m e dependence f o r s h o r t wavelength i n s t a b i l i t i e s , and a q u a d r a t i c t i m e dependence f o r l o n g wavelength ones. s h e l l i s the c r i t i c a l  dimension.  The t h i c k n e s s o f t h e plasma  -19-  CHAPTER I I I  EXPERIMENTAL APPARATUS AND DATA MEASUREMENT  T h i s c h a p t e r i s d i v i d e d i n t o two s e c t i o n s . The m a t e r i a l i n S e c t i o n A d e s c r i b e s t h e apparatus u s e d t o produce a Z-pinch  discharge.  S e c t i o n B c o n t a i n s a d e s c r i p t i o n o f t h e measurement d e v i c e s u s e d t o study i t ,  and an o u t l i n e o f t h e methods u s e d i n d a t a measurement.  S e c t i o n A - Apparatus Of t h e two p r i n c i p a l d i a g n o s t i c t e c h n i q u e s u s e d t o s t u d y d i s c h a r g e plasmas, t h e magnetic probes employed b y B u r k h a r d t and Lovberg d i s t u r b t h e d i s c h a r g e w h i l e t h e h i g h speed p h o t o g r a p h i c employed b y C u r z o n et, a l , ( i 9 6 0 ) do n o t .  (1958)  methods  The l a t t e r t e c h n i q u e i s t h e  more s u i t a b l e f o r t h e xrork d e s c r i b e d i n t h i s t h e s i s , so t h e equipment was d e s i g n e d f o r p h o t o g r a p h i c a n a l y s i s o f t h e Z - p i n c h . Such a d e s i g n s h o u l d have t h e f o l l o w i n g f e a t u r e s : a) The d i s c h a r g e i s v i s i b l e . b) E v a p o r a t i o n o f m a t e r i a l from t h e w a l l s o f t h e d i s c h a r g e v e s s e l i s m i n i m i z e d t o reduce c o n t a m i n a t i o n o f t h e p l a s m a . c) The temperature  i s h i g h enough t o enable comparison o f  experimental r e s u l t s with the t h e o r e t i c a l r e s u l t s derived from h i g h c o n d u c t i v i t y plasma models. I n o r d e r t o s a t i s f y t h e s e s p e c i f i c a t i o n s , t h e e n e r g y put i n t o t h e d i s c h a r g e must be kept low, w h i l e t h e r a t e o f c u r r e n t r i s e s (dl/dt)  (and temperature r i s e dT/dt) i n t h e gas must be as h i g h as p o s s i b l e . T h e r e f o r e the i n d u c t a n c e o f the condenser bank, l e a d s and s w i t c h e s must be as low as p o s s i b l e . to  The  equipment must a l s o be s u f f i c i e n t l y rugged  w i t h s t a n d h i g h v o l t a g e s (20,000 v o l t s ) ' a n d h i g h c u r r e n t s (200,000  amperes). The b l o c k d i a g r a m b e l o w shows t h e r e l a t i o n o f t h e p r i n c i p a l p i e c e s o f equipment needed t o produce Z=pinch d i s c h a r g e a t a p r e c i s e l y known time. The  condenser bank ( (B) o f F i g . l ) d i s c h a r g e s i t s energy i n t o t h e  gas i n t h e d i s c h a r g e t u b e ^ C ) when the main spark gap s w i t c h (A) c l o s e s . The t r i g g e r i n g o p e r a t i o n ( o r sequence o f o p e r a t i o n s t o c l o s e the  spark  gap switch)' s t a r t s w i t h a p u l s e f r o m t h e h i g h speed f r a m i n g camera ( D ) . Each major p i e c e o f equipment i s d e s c r i b e d below and a resume'' o f t h e e s s e n t i a l c h a r a c t e r i s t i c s o f t h e apparatus  I Framing Camera D  Pulse Generator  Trigger Generator  E  F  Discharge Tube  i s given i n Table  III.l.  1  \f Main Spark Gap  IK  Diagnostic Equipment  Charging Unit  Condenser Bank B  Fig.  III.l.  B l o c k Diagram o f  Apparatus.  TABLE I I I . A . l E s s e n t i a l C h a r a c t e r i s t i c s o f Apparatus  C a p a c i t o o ? Bank p l u s Leads  50 /Jf 10 KV 0.12 pE 130 KA  Capacity C Working v o l t a g e Inductance Maximum d i s c h a r g e c u r r e n t ( I ) Maximum d l / d t Ringing frequency  2 x 10 A/sec 180 Kc/sec 1 0  D i s c h a r g e Tube Inter-electrode distance I n s i d e diameter Outside diameter Return conductor Vacuum S y s t e m  62 cm 15' cm 16.2 cm B r a s s mesh  U o i l d i f f u s i o n pump w i t h K i n n e y KS-13 b a c k i n g pump M  -'vl^Hg -/vjlj^Hg/hour  Base p r e s s u r e Leak r a t e Framing Camera " 60 frames Average exposure t i m e A v e r a g e t i m e between exposures  -  0.25/'sec  -  0.25/Jsec  Gases and P r e s s u r e s U s e d Argon CuHg)  50 100 250 500  Nitrogen (/JHg)  37.5 75 15 o 375 750  Hydrogen (mmHg)  "675  1.0  2 5 10  M a j o r p a r t o f work i s i n 500 jj Hg argon  22  Discharge The  Circuit  d i s c h a r g e c i r c u i t c o n s i s t s o f t h e condenser bank, main spark  gap s w i t c h , h i g h c u r r e n t l e a d s and d i s c h a r g e t u b e .  They a r e d e p i c t e d  s c h e m a t i c a l l y by F i g . 2.  Electrodes  Condenser Bank  Spark Gap Switch A Fig. III.2.  .Discharge Tube C  [  .Triggering Apparatus  Schematic Diagram o f D i s c h a r g e Circuit„  a) Condenser bank and l e a d s The  condenser bank  (B) o f Figs„ 1 and 2  i n d u c t a n c e N.R.G. t y p e 201 condensers connected  consists of t e n low  i n p a r a l l e l and  a r r a n g e d i n two p i l e s on a s t o u t D e x i o n t r o l l e y ( F i g . 3)» capacitor i s rated at  Each  SJO f a r a d c a p a c i t y and i s o p e r a t e d a t a c h a r g i n g  v o l t a g e o f 10 kV, so t h a t t h e t o t a l s t o r e d e n e r g y o f t h i s bank (^ CV^) i s 2500 J o u l e s . The h i g h c u r r e n t l e a d s c o n s i s t o f p a r a l l e l p l a t e s o f copper, k i n c h e s , w i d e , l / l 6 o f an i n c h t h i c k and k f e e t l o n g . closely  They a r e clamped  t o g e t h e r t o m i n i m i z e i n d u c t a n c e , and have 16 s h e e t s o f 5 m i l  p o l y t h e n e between them t o p r o v i d e adequate  insulation.  -23-  Fig. III.3.  Condenser  Bank  -22*.  k ) M a i n Spark Gap The d e s i g n shown i n F i g . k was chosen f o r t h e main spark  gap  s w i t c h s i n c e t h i s d e s i g n has been p r o v e n v e r s a t i l e , , r e l i a b l e and demountable.  The i n d u c t a n c e i s n o t as l o w as some o t h e r t y p e s o f  s w i t c h e s (Cormack and B a r n a r d , 1962) b u t i t i s s t i l l s m a l l compared w i t h t h e mean i n d u c t a n c e o f t h e Z=pinch d i s c h a r g e .  To "ground" t e r m i n a l o f condenser bank  F i g . III.k>  From t o p e l e c t r o d e o f discharge tube  M a i n S p a r k Gap S w i t c h , (Scale 1:1)  c) D i s c h a r g e Tube A s e c t i o n t h r o u g h the r o t a t i o n a l l y symmetric, v e r t i c a l l y mounted d i s c h a r g e tube i s shown by t h e diagram o f F i g . 5>. o f a p y r e x g l a s s c y l i n d e r 75 cm l o n g , 15 a w a l l t h i c k n e s s o f 0.6  I t consists  cm i n t e r n a l d i a m e t e r , w i t h  cm.  P l a n e b r a s s e l e c t r o d e s a r e l o c a t e d a t e i t h e r end w i t h p r o v i s i o n f o r e v a c u a t i n g t h e tube and i n t r o d u c i n g t h e t e s t gas.  The atmospheric  p r e s s u r e a c t i n g on t h e e l e c t r o d e s compresses t h e O - r i n g s a g a i n s t t h e tube w i t h s u f f i c i e n t f o r c e t o ensure a good vacuum s e a l  ( ^10~  3  mm  Hg).  The b r a s s gauze w h i c h r e t u r n s t h e c u r r e n t f r o m t h e t o p e l e c t r o d e t o t h e ground o f t h e condenser bank e n a b l e s t h e d i s c h a r g e t o be viewed e a s i l y and m i n i m i z e s i t s i n d u c t a n c e .  2.  Triggering Operation The d i s c h a r g e t u b e must be f i r e d i n s y n c h r o n i s m w i t h t h e f r a m i n g  camera i n much t h e same way as a f l a s h b u l b must be f i r e d when an o r d i n a r y camera s h u t t e r opens.  The f r a m i n g camera p u t s out a p u l s e  when i t i s r e a d y t o r e c o r d ( i . e . " s h u t t e r open") and t h e discharge,must be f i r e d from t h i s p u l s e a t a p r e c i s e l y known t i m e ( - 0.1  JU s e c ) .  The main s p a r k gap s w i t c h has been p r o v e n t o b r e a k down . 3 y t / s e e t .01 yjsec  a f t e r a f a s t t r i g g e r i n g p u l s e o f 10 kV i s a p p l i e d  t o t h e t r i g g e r p i n . T h e r e f o r e , t h e t r i g g e r i n g system must p r o v i d e a h i g h v o l t a g e p u l s e a t a known t i m e a f t e r r e c e i v i n g t h e l o w v o l t a g e p u l s e from t h e f r a m i n g camera. The sequence o f o p e r a t i o n s t o do t h i s i s shown b y t h e b l o c k diagram  -26^ Exhaust and gas i n l e t  Pressure  contact  Upper e l e c t r o d e  Gas column  G l a s s tube B r a s s gauze Discharge current I Returning  current  To condenser  bank ground From s p a r k gap s w i t c h  Fig. Ill.5.  Z - p i n c h D i s c h a r g e Tube.  ( F i g . 1,  page 20).  A low v o l t a g e p u l s e from t h e f r a m i n g camera (D) i s  put i n t o a s t a n d a r d t h y r a t r o n (l;C33>) p u l s e g e n e r a t o r ( E ) .  The  single  o u t p u t p u l s e o f 9 kV has i t s a m p l i t u d e doubled by r e f l e c t i o n f r o m t h e open end o f a charged c o a x i a l c a b l e (Theophanis,  i960).  T h i s h i g h v o l t a g e p u l s e c o u l d be used t o t r i g g e r t h e main spark gap s w i t c h .  However, another s t a g e i n t h e t r i g g e r i n g o p e r a t i o n ( t h e  t r i g g e r g e n e r a t o r F o f F i g . l ) has been i n t r o d u c e d i n o r d e r t o : a) i n c r e a s e t h e e n e r g y o f t h e t r i g g e r spark f r o m 0.1  to 3  joules,  b) i s o l a t e t h e h i g h e n e r g y d i s c h a r g e c i r c u i t - f r o m t h e t r i g g e r i n g system and measuring i n s t r u m e n t s . The f i r s t f e a t u r e improves t h e r e l i a b i l i t y o f t h e main spark gap° The  second f e a t u r e makes t h e d i s c h a r g e equipment much s a f e r and  reduces  n o i s e s i g n a l s i n t h e measuring d e v i c e s b y e l i m i n a t i n g s p u r i o u s e l e c t r i c a l c o u p l i n g between t h e h i g h e n e r g y d i s c h a r g e c i r c u i t and t h e measuring devices. F i g u r e 6 shows a c i r c u i t diagram o f t h e i s o l a t e d d i s c h a r g e and t r i g g e r g e n e r a t o r .  circuit  The d e s i g n o f t h e t r i g g e r g e n e r a t o r i s adopted  f r o m one d e s c r i b e d by C u r z o n and Smy  (I96l)  where the h i g h v o l t a g e  p u l s e f r o m t h e t h y r a t r o n p u l s e g e n e r a t o r i s u s e d t o o p e r a t e an u l t r a v i o l e t l i g h t source.  Photons from t h i s source pass through a q u a r t z  b u l b and produce s u f f i c i e n t e l e c t r o n s i n t h e spark gap o f t h e t r i g g e r g e n e r a t o r (T-j o f F i g . 6) t o cause i t t o b r e a k down.  The  resultant pulse  a c r o s s R^ causes the main spark (T-^) t o b r e a k down. C u r z o n and Smy  (I96l) have shown t h a t t h e u l t r a v i o l e t  triggering  p r o c e s s works w e l l p r o v i d e d t h e v o l t a g e a c r o s s t h e a u x i l i a r y spark To i s w i t h i n h% o f t h e o v e r v o l t i n g p o t e n t i a l .  gap  I n t h e d e s i g n shown i n  28-  >  >  > \  R/  > «  Fig. I l l . 6 .  R  1  R£  = =  100 KJl;  a  9-  C i r c u i t Diagram o f I s o l a t e d D i s c h a r g e C i r c u i t and T r i g g e r G e n e r a t o r .  R  2  UO M J l j  R  6  100 KJ2; =  2lj0 M SL  C-j_ : C a p a c i t e r bank, SO /J f c a p a c i t e r .06 /J f ; T  2  : Discharge tube;  R  ,10 kV;  £0 J L  |  R  C  2  =  7  :  ; 100  K JL.  Trigger  T]_ s main spark gap; T^ : U.V. t r i g g e r e d s p a r k gap  ;  F i g . 6 the main condenser bank (C]_) i s used to stabilize the potential across the auxiliary spark gap (T^).  The potential difference remains  within the k% tolerance for a reasonable time after high voltage power supply leads are removed ( A j 1 min.).  3.  Discharge Perturbation The equipment described i n the previous parts of this section i s  necessary to produce a simple Z-pinch discharge.  For the results  presented i n Chapter IV, Section B, however, the gas i n the discharge tube must be perturbed so that the radius of the plasma surface varies sinusoidally^with wavelength A along the axis. The discharge i s given an i n i t i a l perturbation by a series of glass rings supported at regular intervals inside the discharge tube by four 1/2 inch wide glass tapes.  Each ring i s made by bending a 3/l6 inch  diameter glass rod around a cylindrical form, and fits snugly against the inner wall of the discharge tube with i t s plane perpendicular to the axis of the tube.  When the rings are i n position inside the  discharge tube, there i s nearly complete cylindrical symmetry about the discharge tube axis, except for the four glass tapes and a small gap (  ^  l / U inch) in each ring (see F i g . 7).  The rings are threaded through the glass tapes at the proper intervals and introduced into the discharge tube by means of a wooden jig.  After the rings are inserted into the tube and the end rings are  firmly wedged, this j i g can be collapsed and removed.  -30-  Fig. III.7.  Diagram of Glass Rings Used to Perturb Discharge.  Section B - Measurement Devices and Data Measurement 1.  Current Measurements - Rogowski Coil The total current (I) through the discharge is measured by a  Rogowski c o i l "current transformer".  A three inch length of RG 65 A/U  delay line with i t s outer ground shield removed is placed between the parallel plate high current leads (Fig. 8).  v  I f the total current (I) i s uniformly distributed over the h inch wide leads, the magnetic flux (B) through the c o i l L i i s proportional to (I). The voltage (V) induced i n the c o i l (Li) i s proportional to dB/dt and therefore to dl/dt. The output signal of the c o i l is integrated by the passive integration.circuit (Fig. 9) so as to produce an output signal ( V proportional to I .  This signal i s displayed on a Tektronix 5U5 A  Qut  )  -31-  Fig. III.9.  C i r c u i t Diagram o f E o g p w s k i C o i l and I n t e g r a t o r .  L i ; Rogowski c o i l , 11 R  2  -  100  Kj  R3  —  h; R^  —  R  i~  180  50-0- 1  C-L  j -  .005  JOf.  •32-  o s c i l l o s c o p e s c r e e n and photographed w i t h a P o l a r o i d camera. The  c o i l i s p r o t e c t e d against e l e c t r o s t a t i c pickup by a  grounded s p l i t r i n g s h i e l d .  (1963)  S e v e r a l techniques  carefully  suggested b y Daughney  are i n c o r p o r a t e d i n t o t h e c u r r e n t measuring d e v i c e i n o r d e r t o  i n c r e a s e the s i g n a l t o n o i s e r a t i o ( i . e . grounding and s h i e l d i n g procedures).  The  f r e q u e n c y r e s p o n s e o f the c o i l and i n t e g r a t o r network  are o p i t i m i z e d b y f o l l o w i n g the p r o c e d u r e d e s c r i b e d b y S e g r e and A l l e n  (i960).  Figure  10.a i s a t y p i c a l example o f a c u r r e n t t r a c e o b t a i n e d  w i t h t h i s system. F i g u r e 10.b T h i s t r a c e was scope.  shows a n o t h e r c u r r e n t t r a c e on a l o n g e r t i m e s c a l e . measured a c c u r a t e l y u s i n g a Z e i s s X - Y t r a c k i n g m i c r o -  measurements^ t a k e n e v e r y l / 3 y U s e c , were used t o i n t e g r a t e  The  the t r a c e and c a l i b r a t e the c o i l and i n t e g r a t o r c i r c u i t .  The  calibration  was  done b y c a l c u l a t i n g t h e t o t a l charge o f the condenser bank (Q)  the  relation  (1)  Q =  and e q u a t i n g  CV  i t t o the i n t e g r a l o f the discharge c u r r e n t , i . e .  (2)  Q '0  o where  C  =  c a p a c i t y o f condenser bank  V  -  charging  voltage  S i n c e t h e o u t p u t v o l t a g e o f t h e i n t e g r a t o r network o f F i g . (V ^.) i s p r o p o r t i o n a l t o the t o t a l c u r r e n t ( I ) ou  10  from  -33-  Flg. III.10.  Current  Traces  a.  1 Aj sec/cm h o r i z o n t a l 5 volts/cm v e r t i c a l  b.  10jjsec/cm  horizontal  5> v o l t s / c m v e r t i c a l  (3)  V t  =  *  cx  1  —- c o n s t a n t  then  (h)  V  The  c a l i b r a t i o n constant  o<  =  out  d t  =  W  c< o b t a i n e d i n t h i s way  1.07  x 10™^  volt/amp  was  Three independent c a l i b r a t i o n s o f . t h e c o i l - i n t e g r a t o r network agreed to w i t h i n 1$.  2  *  H i g h Speed Framing Camera The B a r r and S t r o u d M o d e l CP 5 h i g h speed r o t a t i n g m i r r o r  camera c a n p h o t o g r a p h 60 frames a t a maximum f r a m i n g  framing  r a t e o f 8 x 10^  frames/sec. I t c o n s i s t s e s s e n t i a l l y o f an o b j e c t i v e l e n s L-^ (see F i g . 11} and a m a g n i f i e r , l ^ , which f o c u s e s mirror.  t h e image on t h e s u r f a c e o f t h e r o t a t i n g  A l e n s q u a d r a n t , L3, f o c u s e s t h e m i r r o r s u r f a c e on t h e curved  film track. The  p o l i s h e d s t a i n l e s s s t e e l m i r r o r runs i n f l e x i b l y mounted w h i t e  metal bearings.  I t i s d r i v e n a t a maximum r a t e o f  330,000  r.p.m. b y an  a i r t u r b i n e o p e r a t e d a t 180 P„S.I„G.  PM2  Fig. Ill.11.  £  Framing Camera and C o n t r o l Equipment.  The  event t o be photographed must be s y n c h r o n i z e d w i t h t h e p o s i t i o n '  o f the r o t a t i n g m i r r o r , s i n c e r e c o r d i n g can o n l y t a k e p l a c e during o f each h a l f r e v o l u t i o n o f t h e m i r r o r .  15>°  A pulse t o i n i t i a t e the high  c u r r e n t d i s c h a r g e i s p r o v i d e d b y one o f t h e two p h o t o m u l t i p l i e r s , PM o r PM 2.  1  L i g h t f r o m a 6 v o l t lamp i s r e f l e c t e d from a p a r t o f the m i r r o r  s u r f a c e and d i r e c t e d b y a p r i s m t o t h e two p h o t o m u l t i p l i e r s , so t h e y g i v e p u l s e s s y n c h r o n i z e d w i t h the m i r r o r p o s i t i o n . The m i r r o r speed i s m o n i t o r e d d i r e c t l y b y a f r e q u e n c y meter which r e c e i v e s a s i g n a l from a s e a r c h c o i l a d j a c e n t t o a m a g n e t i z e d c o l l a r  on  the m i r r o r s h a f t . I n o r d e r t o determine  t h e e x a c t frame exposure t i m e , t h e p u l s e s  from one p h o t o m u l t i p l i e r are counted is  f o r l / l O s e c . j u s t when t h e  discharge  fired. The d i s c h a r g e was photographed f r o m t h e s i d e .  U s u a l l y , about t e n  d i s c h a r g e s were photographed and t h e n the f i l m ( I l f o r d HP 3) was (10 m i n u t e s i n I l f o r d d e v e l o p e r I.D.  11).  camera f i l m o f each e x p e r i m e n t a l r u n was  A f t e r development, t h e f r a m i n g numbered and f i l e d .  The images o f t h e d i s c h a r g e on t h e developed 8 mm.  developed  f i l m a r e about £ mm  by  They were m a g n i f i e d 10 t i m e s b y t h e o p t i c a l system o f a J a r r e l l -  Ash m i c r o d e n s i t o m e t e r , which p r o j e c t e d each image onto a ground g l a s s screen.  The  d i s c h a r g e d i a m e t e r was t h e n measured from t h e s e  enlarged  images w i t h an o r d i n a r y r u l e . The photographs o f t h e p e r t u r b e d d i s c h a r g e appear as d e p i c t e d i n Fig. D  12.  The maximum and minimum diameter o f t h e d i s c h a r g e ( D  .„ o f F i g . 12) min °  were measured f o r each frame.  i n a random o r d e r t o p r e v e n t s y s t e m a t i c e r r o r s .  m a x  and  The frames were measured  -36-  Diagram of one frame of perturbed discharge photograph.  F i g . TII.13.  The mean discharge radius r and the amplitude of the instability A were calculated from the relations  r A  =  ( max  =  ( max " min)A  D  D  +  D  min)A  D  The errors i n r and A estimated from the standard deviation of a number of independent measurements of the same film are  1.5  (?)  A  —  ™ l  0.8 mm  The results obtained i n this way are treated in detail i n the next chapter.  -37-  CHAPTER I V  OBSERVATIONS AND RESULTS  T h i s c h a p t e r i s d i v i d e d i n t o two s e c t i o n s . r e s u l t s o f o b s e r v a t i o n s made on t h e u n p e r t u r b e d a r g o n and n i t r o g e n a r e p r e s e n t e d .  I n s e c t i o n A some Z-pinch d i s c h a r g e i n  The purpose o f t h i s s e c t i o n i s t o  e s t a b l i s h those p r o p e r t i e s of t h e unperturbed  discharge which are  required i n t h e d i s c u s s i o n o f the c h a r a c t e r i s t i c s o f the perturbed discharge. S e c t i o n B c o n t a i n s a f u l l d i s c u s s i o n o f t h e r e s u l t s o b t a i n e d byperturbing the discharge.  S e c t i o n A.  Unperturbed  T h i s work forms t h e main body o f t h e t h e s i s .  Discharge  A d e t a i l e d s t u d y o f t h e Z - p i n c h i n a r g o n and n i t r o g e n was c a r r i e d out i n o r d e r t o determine t h e b e s t c o n d i t i o n s f o r t h e experiments cribed i nSection B o f this  des-  chapter.  Photographs o f t h e n i t r o g e n d i s c h a r g e column (see A p p e n d i x , P l a t e A.I, page 64 ) r e v e a l many p e c u l i a r f e a t u r e s i n t h e e a r l y s t a g e s o f t h e d i s c h a r g e (see a l s o C u r z o n and C h u r c h i l l , 1962, Hodgson and C h u r c h i l l , 1963)• A p p e n d i x ( P l a t e A.IKx/page65)  The a x i a l streamers  and C u r z o n ,  shown i n t h e  demonstrate t h a t t h e d i s c h a r g e column i s  b y no means r o t a t i o n a l l y s y m m e t r i c , and t h e r e i s t h e r e f o r e l i t t l e p o i n t i n u s i n g such d i s c h a r g e s t o i n v e s t i g a t e r o t a t i o n a l l y symmetric d i s c h a r g e s , instabilities.  -38-  On t h e o t h e r hand, t h e d i s c h a r g e i n a r g o n appears q u i t e r e g u l a r (see P l a t e I . a ) .  There i s no marked i r r e g u l a r i t y i n t h e c u r v e o f  d i s c h a r g e r a d i u s as a f u n c t i o n o f t i m e , and t h e a x i a l streamers do n o t appear.  However, end on photographs r e v e a l t h a t t h e d i s c h a r g e gas ( i n  b o t h a r g o n and n i t r o g e n d i s c h a r g e s ) forms a t h i c k luminous s h e l l ( s e e Plate I,b). -  . .  • - —  - —*— . - -  - j Plate IV.1.  S i n g l e frames o f u n p e r t u r b e d d i s c h a r g e i n 500/jHg a r g o n . Time: 3 psec a f t e r s t a r t o f d i s c h a r g e j Exposure t i m e : 0.25 p s e c . a) s i d e - o n b) end-on ( t h r o u g h c a t h o d e ) .  S i n c e t h e snow-plow model i s p a r t i a l l y based o n t h e a s s u m p t i o n t h a t t h e c u r r e n t and mass s h e l l i s i n f i n i t e s i m a l l y t h i c k , i t i s n o t s t r i c t l y applicable.  However, t h e t h e o r y ( C h a p t e r I I , S e c t i o n A,  pages 6-9) was t e s t e d b y measuring t h e d i m e n s i o n l e s s r a d i u s y =  R(t)/R  Q  -39-  photographically for discharges i n the entire range of pressures i n nitrogen and argon.  The discharge current I(t) was measured as a  function of time, and the measured values bf I were used i n integrating the Rosenbluth snow-plow equation (equation ( *> )> page 7 ) numerically with an I.B.M. 1 6 2 0 computor. 'The (smoothed) measured radius, the c a l culated radius,, and the current were plotted on the same graphs as functions of time (see F i g . l ) .  In order to check the scaling laws for  a Z-pinch with a sinusoidally time dependent current (Chapter I I , equation ( 8 ) , page ( 8 ) ) the pinch times were taken from these graphs and plotted versus R  0  p /l z  0  (see F i g . 2 ) .  In view of the differences between the theoretical model and the experimental system,, the agreement between the theoretical and experimental y versus t curves i s satisfactory, while the agreement of the measured pinch times with the theory i s surprisingly good. The observations of the unperturbed discharge column i n argon therefore show that i t i s rotationally symmetric and that i t s dynamic characteristics agree well with the predictions of the snow-plow theory. Because of these features, argon was chosen as the discharge gas for the experiments on perturbed plasma surfaces described i n the next section.  F i g . I V . 1.  Radius v e r s u s t i m e a n d C u r r e n t v e r s u s time i n A and N d i s c h a r g e s under comparable d e n s i t y c o n d i t i o n s . P l o t s o f d i m e n s i o n l e s s r a d i u s y a g a i n s t t i m e ; b r o k e n curves c a l c u l a t e d from "snow-plow" t h e o r y ; f u l l c u r v e s measured from f r a m i n g camera photographs; d i s c h a r g e c u r r e n t waveform p l o t t e d as measured b y a Rogowski c o i l , (a) 25>Hg, (b) 50-//Hg, ( c ) 100//Hg, (d) 2$Cy/Hg, .(e) 500/;Hg, ( f ) 37.5/VHg, (g).75^Hg, ( h ) l ^ H g , 2  ( i ) 375/jHg, (j) 750yUHg.  .1  -U2-  Section B. 1.  Perturbed Discharge  Observations Side-on photographs of the discharge i n argon (SOOjjKg  pressure)  with the perturbing rings i n place show that each ring makes a "dent" i n the plasma surface.  The disturbance on each side of the "dent"  spreads out i n the axial direction u n t i l i t meets the disturbance from the next ring.  Then the surface appears sinusoidal.  Plate II (page Ifi) shows a typical sequence of frames of a Z-pinch discharge (taken side-on) with rings 2 cm apart.  For ring  spacings of 2 , 3 and h cm the perturbed surface of the plasma appears sinusoidal with wavelength  "X equal to the ring spacing (see Plate  I l i a and I l l b , page I i 8 ) . For larger ring spacing (6 and 9 cm) the plasma surface does not become sinusoidal (see Plate IV, page 53 ) since the disturbances from each ring do not spread rapidly enough to meet before the f i r s t pinch.  2»  Treatment of Data Attempts to relate  the  measured  values  of  the discharge  radius (r~) and instability amplitude (A(t)) to each other were made on the  basis  of  the following considerations.  In section A of  this chapter, i t was shown that the snow-plow theory is a satisfactory model for the dynamic properties of the Z-pinch i n argon.  This theory  predicts that the acceleration "a" of the discharge boundary should be directed radially inward, and should be approximately constant (equation (9) , page 9 )•  Therefore the conditions favour growth of  Rayleigh-Taylor instabilities i n the f i r s t stage of an argon Z-pinch  •U3-  discharge.  From Chapter H , Section B, then, a radial perturbation of  the discharge surface of the form A(t) sin 2J\Z/ X is expected to be unstable, and the amplitude A(t) should increase according to equation (3U) derived on page 1 6 . For convenience, the equation is repeated here  A(t) - A exp(cot)  (1)  0  where  ud  (2)  X  - Jl-Ka/  Hence, i f "a" i s constant, then plots of l n A(t) versus t should be straight lines of slope  , where the "growth rate"  oJ i s given  by equation ( 2 ) . Logarithmic plots of A(t) versus t were therefore made, and i n fact consist of several linear segments of different slopes, page 5 0 ).  (see F i g . 3&>  The time intervals over which co remains constant,  say, vary i n length from about 1 to 2 /jsec. frames (n) for which  ^ T( ui)  Hence, the number of  remains constant, or the number of points  (t, l n A) which define the line segment, varies from n = 3 to n •= 1 0 . In order to verify equation (2) the acceleration "a" was assumed to be constant during the time intervals A T ( i O ) .  This assumption is  qualitatively justified by the appearance of the ~r versus t curves (see, for example, F i g . 3 b , page £o). The acceleration for each time range  L\ T( cO) was calculated by  -kh'  fitting parabolic arcs to the-r versus t curves i n each time range. The least squares method used i s described i n detail i n Worthing and Geffner (19U3). •In order to check for the linear correlation of OJ with v T predicted by equation (2),  uOwas plotted versus  /"a for ring spacings  /\ , of 2, 3, and k cm. However, before discussing the qualitative and quantitative results the errors involved i n calculations of "a" and  must be mentioned,  since a l l "least squares" f i t s use the points weighted according to thei accuracy.  3 o Error Analysis The errors i n >/a  and uJ may be calculated from the measured errors  i n A(t) and r(t) by using the standard formula for propagation of errors According to this formula, the expected error ( <£ f) i n an arbitrary function f of M variables (XJ_, say) is given by the expression  (3)  (  where  <5 x^ i s the expected error i n the variable Xj_.  The growth rate (  ) depends on the n values of In A i n the range  A T( cJ ), and i s given by equation (k) 2 A j 2 A J In A j ( j . A t ) - 1 A j ( i A t ) (  }  ^  2 A  ±  where i A t i s the l^  2 Ai(i 1  At)  2  -  [1  Ai(i * t ) J  point i n the range  2  A j In A j  se7^~  AT( uj ) on the In A versus •  curve.  A t i s the time interval between points on this curve and  the interval between frames ( ^0.2$^f The estimated error i n uJ ( i . e .  sec). & u) )  c a  n be derived by applying  the error propagation formula (equation ( 3 ) ) to the expression for (equation (U)).  (?)  ^  The following approximate result is obtained,,  Su3 _  ^  T  /  I  -sec-  2  1  A ( A t ) 7 (n-l)(n)(n l) where o A i s the estimated error i n A and A i s the mean value of A over the range  A T ( oJ ). as a function of n i s proportional to <^ (n) of Table I ,  and i s shown as the error flags of F i g . )j (page E>1). The acceleration (a) of the discharge-boundary may be calculated from the points (i  &t  5  r±) i n a time range  CiT( <^) by using the f o l -  lowing expression  '  (  " 7 7 ^  („-2)(n-:i)(n)(„ l)<„*2)  ^  f  The estimated error  c^a derived from equations (3) and (6) i s  given by equation (7)  £  m U  a  = _AjL  /  —  2  where  Vsec"  2  +  ^ r  is the estimated error i n ~r  ^ a  is proportional to  t  The corresponding error i n Fig. k'  _  (n-2)(n-l)(n)(iHl)(n 2)  ( L\t) V  ;  360  P(n) of Table 1.  >v  Va"is given by the error flags of  Ja of  -U6-  TABLE IV.1  Dependence o f measuring e r r o r s i n a LO  2  3  U  <£(n)  U0  20  13  9  7  ^(n)  «°  80  33  18  11  n  00 a  on number o f frames  v  5  6  and  (n)  7  8  9  5.5  1+.5  3-7  7  5  li  I t i s e v i d e n t from T a b l e 1 t h a t v a l u e s o f " a " f o r s m a l l v a l u e s o f n are not v e r y r e l i a b l e .  F o r t h i s r e a s o n , " a " was  f o r n l e s s t h a n 6 ( i . e . f o r AT(  not  calculated  ) l e s s than 1.5 y f s e c ) .  T h i s means  i n e f f e c t , t h a t t h e a c c e l e r a t i o n o f t h e d i s c h a r g e boundary cannot be measured a c c u r a t e l y u n l e s s i t i s c o n s t a n t f o r a t l e a s t 1 . 5 /) sec (6  frames).  U.  Results S i n g l e frames ( P l a t e s I I I . a and I l l . b ) t a k e n from a sequence o f  f r a m i n g camera photographs (such as P l a t e II)  show t h a t the p e r t u r b e d  s u r f a c e o f t h e plasma i s o f t h e form A ( t ) s i n ( 2 / T Z / ^ . ) , where e q u a l t o the r i n g s p a c i n g .  A  S i n g l e mode m — 0 plasma s u r f a c e  i n s t a b i l i t i e s are t h u s a v a i l a b l e f o r s t u d y .  Next page;  Plate IV.II. Framing Camera P h o t o g r a p h o f a r g o n d i s c h a r g e ( 5 0 0 / J Hg p r e s s u r e ) w i t h p e r t u r b i n g r i n g s i n p l a c e . Time i n c r e a s e s l e f t t o r i g h t ; frame p o s i t i o n s a r e s t a g g e r e d ; exposure time  is  Plate IV.III.  S i n g l e frames o f p e r t u r b e d d i s c h a r g e i n a r g o n (exposure time 0.25/Osec) a) R i n g s p a c i n g 2 cm, l + J s e c a f t e r s t a r t of discharge b) R i n g s p a c i n g U cm, £^sec a f t e r s t a r t of discharge.  -U9-  Graphs of the logarithmic amplitude of the instability (In A) as a function of time show clearly the exponential time dependence of the amplitude A (see F i g . 3a)• The linear correlation between uJ and ~Ja i s well demonstrated by F i g . Ua for  A - 2 cm. The "least squares" straight line fitted to  these points gives the relation  (8)  <jJ  where  </& ~  10  — to I O  (5.0 t l ) 5  /a  +  om' s e c s  (O.l; t 1) x IO sec" 5  (see  _J  page  1  51).  The Rayleigh-Taylor relations ^equations (l) and (2;) predict that uJ  versus Ja  should be a straight line through the origin.  Since  the relation (8) agrees with the theory within experimental error, i t i s concluded that the observed instabilities for "X - 2 cm are RayleighTaylor i n s t a b i l i t i e s . The evidence i s not conclusive that the 3 and k cm wavelength instabilities are Rayleigh-Taylor i n s t a b i l i t i e s .  However, on the basis  of the theory and the results for X - 2 cm, i t is reasonable to assume that the ")\ = 3 and "X = k cm perturbations are also Rayleigh-Taylor instabilities.  Therefore, the straight lines on F i g . iia, Ub and kc were  fitted using "least squares" procedures i n which the lines are constrained to pass through the origin of the by the theory. i n Table u.2.  The n u m e r i c a l values of  ^  and  cO / / a  /a" - axes,as required  so obtained are given  -50-  1  °  TIME  I  I  I  I  FROM START OF DISCHARGE - MICROSEC.  ( a )  500 p Hg., ARGON RING SPACING 3 Cffl.  J 0  1  TIME  F i g . 17.3.  2  I  I  I  I  3  4  5  6  FROM START OF DISCHARGE - MICROSEC. ( b) Plots of:  a) In A versus t b.) T versus t.  -51-  Fig. IV.a.  °^ versus /IT for: a) "X - 2 cm b) >> - 3 cm c ) > - h cm  .CI-  TABLE I V . 2 Dependence o f UJ o n f o r i n s t a b i l i t i e s i n argon ( i n i t i a l p r e s s u r e 500/fHg)  >\(cm)  bJ/fa  (cm"2)  l i J / </ 2 Tfa/>  2  5-7 t 1  0.32 t 0 . 0 6  3  5.1  t1  0 . 3 5 t 0.07  ii.3 - 0 . 5  0 . 3 k t 0.0k  The e x p e r i m e n t a l e r r o r s a r e t o o g r e a t t o make d e f i n i t e c o n c l u s i o n s about t h e w a v e l e n g t h dependence o f t h e g r o w t h , r a t e (equation (2)) p r e d i c t s t h a t  UJ/ J2 IT a/A  Uj .  The t h e o r y  s h o u l d be a c o n s t a n t and  T a b l e 2 v e r i f i e s t h i s r e l a t i o n , w i t h i n e x p e r i m e n t a l e r r o r , f o r )\ - 2 , 3 and k cm. One i n t e r e s t i n g f e a t u r e o f t h e r e s u l t s i s t h a t t h e a c c e l e r a t i o n (a) o f t h e plasma boundary i s n o t c o n s t a n t as p r e d i c t e d b y t h e s i m p l e t h e o r y o f Rosenbluth  (195U) ( i . e . e q u a t i o n ( 9 ) , page 9)»  The a c c e l e r a t i o n ,  i n f e r r e d from t h e p l o t s o f I n A v e r s u s t , appears t o change t h r e e o r four times during the discharge. F i n a l l y , i t s h o u l d be n o t e d t h a t t h e p e r t u r b a t i o n s do n o t d r i f t a l o n g t h e d i s c h a r g e tube a x i s t o any measureable e x t e n t .  -53-  P l a t e IV.IV.  5o  S i n g l e frame o f p e r t u r b e d d i s c h a r g e i n 500 Hg a r g o n k/J sec a f t e r s t a r t o f d i s c h a r g e . R i n g s p a c i n g : 9 cm, exposure t i m e : 0.25 J) s e c .  D i s c u s s i o n of Results  The  experimental conditions described e a r l i e r i n t h i s t h e s i s  d i f f e r c o n s i d e r a b l y f r o m those s p e c i f i e d by the' s i m p l e R a y l e i g h - T a y l o r i n s t a b i l i t y theory.  However, a c o m p a r i s o n o f the measured r e s u l t s w i t h  t h e o r y can be p a r t i a l l y j u s t i f i e d by t h e f o l l o w i n g d i s c u s s i o n . F o r t h e s i m p l e R a y l e i g h - T a y l o r t h e o r y o f plasma i n s t a b i l i t i e s , K r u s k a l and S c h w a r z s c h i l d (1951;) assume t h a t a s e m i - i n f i n i t e , p e r f e c t l y c o n d u c t i n g , non-viscous  plasma i s s u p p o r t e d b y a magnetic f i e l d a g a i n s t  a -gravitational f i e l d or acceleration ( a ) .  -5U-  The plasiria s u r f a c e i n a Z - p i n c h i s c y l i n d r i c a l , . n o t p l a n e as assumed b y K r u s k a l and SchwarzschLld.  However, H a r r i s (1962) p o i n t s o u t t h a t t h e  c y l i n d r i c a l case approximates t h e p l a n e case i f 2 7TR/ X  >>  1* where  R i s t h e r a d i u s o f c u r v a t u r e o f t h e c y l i n d r i c a l plasma s u r f a c e . c o n d i t i o n i s f u l f i l l e d i n a l l experiments reported i n t h i s  1  This  thesis.  The f i n i t e t h i c k n e s s o f t h e plasma l a y e r i n t h e Z - p i n c h does n o t • p r o h i b i t comparison o f t h e experimental w i t h t h e t h e o r e t i c a l r e s u l t s , s i n c e T a y l o r (195>0) has shown t h a t t h e s e m i - i n f i n i t e f l u i d t h e o r y i s v a l i d p r o v i d e d t h e t h i c k n e s s o f t h e plasma l a y e r i s g r e a t e r t h a n  A/3  S  where  N i s t h e wavelength o f t h e p e r t u r b a t i o n a p p l i e d t o t h e s u r f a c e . P l a t e l b (page 38) shows t h a t * f o r t h e t i m e s o f i n t e r e s t i n o u r e x p e r i ment, t h i s c o n d i t i o n h o l d s . A n o t h e r r e s t r i c t i o n w h i c h t h e e x p e r i m e n t a l c o n d i t i o n s must s a t i s f y i n o r d e r t o conform t o t h e f i r s t o r d e r K r u s k a l - S c h w a r z s c h i l d t h e o r y has been e s t a b l i s h e d b y L e w i s (195>0) f o r a s i m i l a r p r o b l e m i n hydrodynamics. He demonstrates t h a t t h e f i r s t o r d e r i n s t a b i l i t y t h e o r y i s v a l i d A  ~  until  /N/2. T h i s r e s t r i c t i o n i s s a t i s f i e d i n a l l experiments r e p o r t e d  here. However, t h e r e m a i n i n g  assumptions o f i n f i n i t e e l e c t r i c a l conduct-  i v i t y and zero v i s c o s i t y made i n t h e K r u s k a l - S c h w a r z s c h i l d t h e o r y a r e not v a l i d , and may a f f e c t t h e c o m p a r i s o n between e x p e r i m e n t a l and theoretical  results.  I n a d d i t i o n t o t h o s e assumptions made i n t h e K r u s k a l - S c h w a r z s c h i l d t h e o r y , one o t h e r assumption has a l s o been made i n e v a l u a t i n g t h e d a t a . I n a p p l y i n g " l e a s t squares" curve f i t t i n g t e c h n i q u e s , one i m p l i e s t h a t e r r o r s i n t h e measurements o f F, A, uJ , and  are normally  distributed.  S i n c e t h e number o f measurements i s i n s u f f i c i e n t  v e r i f y t h i s assumption,  to  the use o f " l e a s t s q u a r e s " procedures must be  r e g a r d e d as a c o n v e n i e n t n o n - s u b j e c t i v e method o f t r e a t i n g the d a t a . The  e x p e r i m e n t a l r e s u l t s d e r i v e d i n t h i s way  may,  however, be  compared w i t h t h o s e r e p o r t e d b y o t h e r o b s e r v e r s . The (1960)5  e x p e r i m e n t a l work o f Curzon and C u r z o n  and C h u r c h i l l  i n s t a b i l i t y modes, observed  e t a l ( l ° 6 o ) ; Green and  (1962)  Niblett  shows t h a t t h e f a s t e s t  growing  a f t e r the p i n c h , have growth r a t e s r o u g h l y  one h a l f the v a l u e p r e d i c t e d b y simple t h e o r y .  The  results  presented  here are a l s o o f t h i s o r d e r o f magnitude. However, the e x p e r i m e n t a l t e c h n i q u e s adopted  f o r the work  d e s c r i b e d i n t h i s t h e s i s r e p r e s e n t a c o n s i d e r a b l e improvement on used b y e a r l i e r workers. UJ  ,  "X , and a  F o r example, Curzon e t a l ( I 9 6 0 )  b y t a k i n g s t a t i s t i c a l averages  ^  evaluated  o f measurements made  on a l a r g e number o f s i n g l e K e r r c e l l photographs. was  those  O n l y one  photograph  o b t a i n e d from each d i s c h a r g e , so t h a t t h e s t a t i s t i c a l averages 3  "X and  /a  can not be  r e a d i l y r e l a t e d t o each o t h e r .  I n a s i m i l a r s t u d y w i t h a f r a m i n g camera G r e e n and N i b l e t t have e s t i m a t e d the growth r a t e o f the i n s t a b i l i t i e s o f a 9-pinch. order" of unity.  From t h i s  m  ~  They f i n d t h a t e s t i m a t e , and  (i960)  ^5 cJ//2T1a/>s  i s " o f the  from t h e s i m i l a r i t y between  photographs o f the l a t e r s t a g e s o f t h e i r i n s t a b i l i t i e s graphed  of  b y Lewis (195>0), t h e y have c o n c l u d e d t h a t these  and those  photo-  9-pinch  i n s t a b i l i t i e s are o f the R a y l e i g h - T a y l o r t y p e . F u r t h e r comparison  o f t h e i r work w i t h the r e s u l t s p r e s e n t e d  i s not p o s s i b l e , however,.because o f t h e l a c k o f d e t a i l i n t h e i r  here paper.  -56.  They do not state, f o r example, whether they established that the i n s t a b i l i t i e s grow exponentially with time or not.  I n addition, i t i s  d i f f i c u l t to compare t h e i r measured r e s u l t s with the Kruskal Sehwarzsc h i l d theory because of the considerable differences between the plasma geometry assumed i n the theory and that observed i n p r a c t i c e . Curzon and C h u r c h i l l (1962) have also used a framing camera to study post-pinch i n s t a b i l i t i e s (possibly Rayleigh-Taylor i n s t a b i l i t i e s ) i n argon and nitrogen Z-pinches. single t r a i n o f m =  I n an argon discharge they observed a  0 sinusoidal i n s t a b i l i t i e s , three wavelengths long.  However, the acceleration of the discharge boundary was calculated from only f i v e frames so that i t may be very inaccurate.  Furthermore, since  the occurrence of such t r a i n s i s fortuitous ( i . e . one discharge i n f i f t y ) , the measuring accuracy can not be greatly  improved.  The only report on i n s t a b i l i t i e s i n the f i r s t stage of the Z-pinch has been given by. Hertz ( 1 9 6 3 ) .  He uses the n a t u r a l l y occurring m - 0  i n s t a b i l i t i e s found i n a discharge i n hydrogen, and photographs them with an image converter. Hertz reports l i n e a r l y time dependent amplitudes and exponentially increasing wavelengths f o r these i n s t a b i l i t i e s .  He  then compares h i s data with some theories of i n s t a b i l i t i e s i n the f i r s t stage of the Z-pinch put forward by Wyld ( 1 9 5 8 ) . Hertz' published photographs  show that the discharge surface i s  not at a l l sinusoidal and i t i s d i f f i c u l t to see how he can (with h i s equipment) measure the i n s t a b i l i t i e s to w i t h i n the accuracy he claims. Although the observations referred to above demonstrate a c o r r e l a t i o n between the growth rate  oo and the acceleration, a, none  of the observers has made s u f f i c i e n t l y accurate measurements to comment  •57-  on the wavelength dependence of (AJ , of the wavelength  X  The growth rate, as a function  has been calculated f o r f l u i d s of f i n i t e  v i s c o s i t y (Chandresekhar,  1955)  K i l l e e n , and Roseribluth, 1963)•  and f i n i t e conductivity (Furth, S u f f i c i e n t l y accurate  ^  versus  A  curves could be compared to the results of these theories i n order to estimate the plasma conductivity and v i s c o s i t y . Whereas the i n s t a b i l i t i e s on the discharge column have been investigated by other workers, no one has reported fluctuations i n the acceleration of the discharge boundary. i s caused by magneto-acoustic  Perhaps t h i s phenomenon  waves propagating i n the layer of shock  heated gas trapped between the shock front and the d r i v i n g magnetic field.  Further discussion of t h i s phenomenon must be deferred, how-  ever, u n t i l further experimental data i s a v a i l a b l e .  6.  Suggestions f o r Further Work The work described i n t h i s thesis can be extended i n two ways.  F i r s t l y , the techniques of exciting and studying i n s t a b i l i t i e s during the i n i t i a l stage of the Z-pinch can be improved and, secondly, i t i s possible to use the techniques developed to study i n s t a b i l i t i e s produced during the l a t e r stages of the discharge (e.g. post-pinch). The possible developments along these l i n e s are considered below. I n i t i a l Stage of the Discharge The three main defects of the work described i n t h i s thesis are: a) The technique employed f a i l s to excite longer wavelength (  X  >  k cm)  instabilities,  b) L i t t l e i s known at present about the physical properties of  '  the discharge  -58-  plasma.  c) The measuring a c c u r a c y i s low. The w a v e l e n g t h l i m i t a t i o n s t u d i e d b y the t e c h n i q u e s  on the i n s t a b i l i t i e s which can  be  d e s c r i b e d above i s s e r i o u s because i n s t a b i -  l i t i e s cannot be produced i n c o n d i t i o n s where t h e snow-plow model i s applicable.  S i n c e no e x p e r i m e n t a l v e r i f i c a t i o n o f the snow-plow t h e o r y  o f i n s t a b i l i t i e s i s a t p r e s e n t a v a i l a b l e , the p r o b l e m o f e x t e n d i n g  the  range o f i n s t a b i l i t y - w a v e l e n g t h s i s o b v i o u s l y o f c o n s i d e r a b l e i m p o r t a n c e . One  way o f i m p r o v i n g t h e p r e s e n t  s i t u a t i o n i s t o use b e t t e r methods  o f c o r r u g a t i n g t h e i n i t i a l s u r f a c e o f the d i s c h a r g e column. u s i n g g l a s s r i n g s one  Instead  of  c o u l d , f o r example, use a d i s c h a r g e v e s s e l i n  w h i c h the r a d i u s o f i t s c i r c u l a r c r o s s - s e c t i o n v a r i e s s i n u s o i d a l l y as a f u n c t i o n of the a x i a l p o s i t i o n .  Unfortunately, f o r l a r g e r tubes,  t h e c o s t o f p r o d u c t i o n would be too h i g h .  However, an experiment u s i n g  a s m a l l e r system i s now b e i n g c o m p l e t e d i n t h e Plasma P h y s i c s L a b o r a t o r i e s at t h i s u n i v e r s i t y . More i n f o r m a t i o n about t h e p h y s i c a l p r o p e r t i e s o f t h e  discharge  plasma i t s e l f may  be o b t a i n e d b y measuring t h e c u r r e n t d i s t r i b u t i o n  magnetic p r o b e s .  The  e l e c t r o n and i o n d e n s i t i e s and t e m p e r a t u r e s can  measured u s i n g Langmuir probes and t i m e r e s o l v e d The  with be  spectroscopy.  a c c u r a c y o f the measurements, as shown by T a b l e I , page k6 ,  depends c r i t i c a l l y on the number o f f r a m i n g camera photographs (n) w h i c h can be o b t a i n e d w h i l e t h e a c c e l e r a t i o n o f the d i s c h a r g e boundary i s constanto I f t h e d i s c h a r g e c o n d i t i o n s a r e not changed, n can be o n l y b y d e c r e a s i n g t h e frame exposure t i m e .  Unfortunately,  increased this  -5>  procedure would considerably reduce the l i f e of the bearings i n the framing camera turbine, and therefore a compromise between maximum accuracy and maximum turbine l i f e must be sought.  However, n can be  r e a d i l y increased by 2%% with a corresponding reduction i n errors i n and CO . For values of /a" calculated from n - 6  n = 8  instead of  points, f o r example, the estimated absolute error i n / a  drop from t 5 x 10 ^  sec"  1  to t 3,5 x 10^  afi s e c "  1  would  . Similarly  the absolute error i n ( J J would be changed from i " 1.3 x 10^ s e c " to 1  11x10  ^ sec" . 1  The s t a t i s t i c a l errors could be reduced by taking more photographs. Here again, the f i n i t e l i f e o f the camera turbine must be considered. A l t e r n a t i v e l y , n may be increased by changing the discharge conditions and maintaining the frame exposure time constant.  The collapse  of the discharge column could be slowed down either by increasing the i n i t i a l argon gas density or reducing the peak discharge current.  Both  methods would slow down the dynamic c h a r a c t e r i s t i c s and hence increase n and the measuring accuracy.  Unfortunately, such procedures result i n  an undesirable reduction i n the plasma  temperature.  To slow down the collapse stage of the discharge column and maintain high temperatures, could be used.  gases of high atomic number such as cesium o r xenon  These two gases seem to o f f e r the best prospects of  improving the experiments described i n t h i s t h e s i s .  I t must o f course  f i r s t be established that the i n i t i a l stages of the Z-pinches i n these gases do not possess the undesirable c h a r a c t e r i s t i c s observed i n the Z-pinch i n nitrogen (see Plate A . l ) .  60-  Post-Pinch I n s t a b i l i t i e s The e x i s t i n g photographs o f t h e d i s c h a r g e s i n a r g o n f o r 2, 3,  h  and 6 cm wavelengths show r e a s o n a b l y s i n u s o i d a l p o s t - p i n c h i n s t a b i l i t i e s (see P l a t e A . I l i b , p a g e 66).  These i n s t a b i l i t i e s have not been e v a l u a t e d  e x t e n s i v e l y , s i n c e t h e y do not conform t o t h e t h e o r e t i c a l models. example, t h e r a t i o o f w a v e l e n g t h t o d i s c h a r g e r a d i u s ( X /R) o r d e r o f u n i t y and p l a n e geometry c a n not be assumed.  For  i s of the  Furthermore,  p i n c h i n s t a b i l i t i e s are l i k e l y t o be c o m p l i c a t e d by unknown f a c t o r s ( s u c h as plasma c o n t a m i n a t i o n b y w a l l m a t e r i a l s ) .  However, f o r a  c o n t i n u i n g c o n s i s t e n t r e s e a r c h program t h e s e i n s t a b i l i t i e s s h o u l d be s t u d i e d i n more d e t a i l .  post-  -61-  CHAPTER V  CONCLUSIONS  The original results presented i n this thesis ares 1. Single mode m = 0 instabilities can be produced on the normallystable plasma surface of an argon Z-pinch.  The experimental conditions  are such that i t i s possible to compare the development of the instabil i t i e s with the predictions of the linearized magneto hydrodynamic theory. 2.  Within experimental error the amplitude of these instabilities  grows exponentially with time. 3.  For the wavelength  X -  2 cm, the growth rate ( ^ ) of these  instabilities i s correlated with the acceleration of the plasma boundary i n accordance with the simple Rayleigh-Taylor theory, i . e .  ^ k»  =  (5.7 t 1)  sTa" sec"  1  The acceleration of the plasma boundary does not appear to be  constant during the f i r s t stage of the Z-pinch i n argon.  -62  APPENDIX  A  T h i s s e c t i o n p r e s e n t s i n t e r e s t i n g r e s u l t s which a r e not to t h e main body o f t h e t h e s i s .  necessary  These i n c l u d e photographs o f " p o s t -  p i n c h " i n s t a b i l i t i e s i n argon d i s c h a r g e s , as w e l l as photographs o f n i t r o g e n and hydrogen d i s c h a r g e s . The The  s i n g l e frames shown have an average exposure t i m e o f 0 . 2 5 / J s e c *  time (T) a t w h i c h t h e p h o t o g r a p h i s t a k e n , t h e gas, t h e  initial  p r e s s u r e , and t h e r i n g s p a c i n g a r e g i v e n i n the p l a t e c a p t i o n s .  The  t i m e T i s measured f r o m t h e s t a r t o f t h e d i s c h a r g e . P l a t e I shows a sequence o f frames o f an u n p e r t u r b e d nitrogen.  The  discharge i n  e n l a r g e d frame ( P l a t e I I . a ) t a k e n f r o m t h i s sequence  shows t h e s t r e a m e r s  and l a c k o f u n i f o r m i t y q u i t e c l e a r l y (compare w i t h  a r g o n d i s c h a r g e P l a t e I V . I . a , page  38).  A n o t h e r s i n g l e frame t a k e n from t h i s same s e t o f photographs shows a t y p i c a l example o f p o s t - p i n c h i n s t a b i l i t i e s i n d i s c h a r g e s o f most gases (see P l a t e I I . b , T  =  11  sec).  The b e s t example o f a p e r t u r b e d p r e - p i n c h i n s t a b i l i t y i n n i t r o g e n i s g i v e n i n P l a t e I I . c . I n no case are t h e s e i n s t a b i l i t i e s as u n i f o r m as t h o s e i n t h e a r g o n d i s c h a r g e s . P l a t e I I . d may be compared w i t h P l a t e I I , b .  The  e f f e c t of the  r i n g s on t h e p o s t - p i n c h i n s t a b i l i t i e s i s q u i t e c l e a r . The  s i n g l e frames o f t h e p e r t u r b e d SOpEg  a r g o n d i s c h a r g e shown i n  P l a t e I I I . a and I l l . b i n d i c a t e t h a t t h e t e c h n i q u e o f s t u d y i n g  •63-  instabilities  can be e a s i l y extended t o  the r e g i o n a f t e r t h e p i n c h ,  F o r completeness, examples o f a p e r t u r b e d d i s c h a r g e i n hydrogen are i n c l u d e d (see P l a t e I I I . c and  Next Pages  PLATE A.I.  First  d)  stage o f a n i t r o g e n d i s c h a r g e  ( i n i t i a l p r e s s u r e 375//  Hg).  -6U-  c PIATE A . I I .  d N i t r o g e n d i s c h a r g e ( i n i t i a l p r e s s u r e 375  Hg)  a) U n p e r t u r b e d pre-pinch s t r e a m e r s (T - U/Jsec) b) P o s t - p i n c h n a t u r a l l y o c c u r r i n g i n s t a b i l i t i e s (T r 10yusec ) c) P e r t u r b e d p r e - p i n c h i n s t a b i l i t i e s (T - U/Jsec, r i n g s p a c i n g u cm) d) P e r t u r b e d p o s t - p i n c h i n s t a b i l i t i e s (T lOpsec, r i n g s p a c i n g k cm)  d  c Plate A . I l l < > a) P e r t u r b e d p o s t - p i n c h (T - 6/^ s e c , 50^Hg, b,) P e r t u r b e d p o s t - p i n c h (T "^y^sec, £ 0 / j H g , c) P e r t u r b e d p o s t - p i n c h  instabilities r i n g spacing instabilities ring spacing instabilities  i n argon 3 cn) i n argon 2 cm) in  hydrogen (T - hy s e c , 0.5" mm Hg, r i n g s p a c i n g 2 d) P e r t u r b e d p r e - p i n c h i n s t a b i l i t i e s i n hydrogen (T - 5 ^ s e c 2 mm Hg, r i n g s p a c i n g 2 cm). 3  cm)  APPENDIX B  The v a l u e s o f mean r a d i u s (F) and i n s t a b i l i t y a m p l i t u d e (A.) ( f o r each frame) a r e t a b u l a t e d below f o r w a v e l e n g t h s ( ^ ) o f .2, 3 and k cm. r". and A  are i n m i l l i m e t e r s , while t h e time i n t e r v a l  g i v e n b e l o w each column i n m i c r o s e c o n d s .  a)  >  -  2 cm A  66.6 63.0 6U.3 63.0  61.9  59.5 57.2 56.0 56.6  51.8 U7.6  1.68 1.68 2.52 2.52 2.52 2.52 3.36 3.36 2.9k 2.9k  2.52 2.52 kh.5 3.78 1*2.1 3.78 38.5 2.9U 36.7 3.36 U6.ii  A t —0.25  1.68 1.68 2.52 2.52 2.52 3.36 U.20 3.36 3.36 3.78 -3.36 k2.l 3 -78 1,2.1 36.7 3 . 3 6 3.78 35.1 32.7 3 . 7 8 30.3 3 . 7 8  61.9 62.5 62.5 60.7 57.2 5U.8 52.1, 51.2 50.0 U5.7  0  4t -  0.25  6U.3 1.68  6U.3 1.68 6U.3 2.52 63.1 2.52 61.9 2.52 58.9  2.9U  56.0 3.36 5U.2 2.9k 52. k 2.52 1+8.8 3.36 U7.0 2.9U U3.9 3.36 kl.k 2 . 9 U 38.5 U.62 36.7 U.20  At  -0.25  65.5 65.5 6U.3  59.6  57.8 57.2 5U.2 51.2 U7.6 U6.U U2.1 38.5  At  1.68 1.68 1.68 2.52 2.10 2.52 2.9k  3.36 3.36 3.78 3.78 3.78  = 0.30  6 t is  b)  72.6 72.6 71.5 71.5 69.3 66.0, 6L*.9 61.6 59.U 56.1 55.0 50.6 1*8.2 U5.1 1*2.9 39.6 35.2  2.2 2.2 3.3 3.3 h.h 5.5 h •5 5 5 .6 6 7 6 7 8.8 ll.o  At-0.33  )\  72.6 72.6 71.5 70.1* 69.3 67.1 63.8 62.7 59.1* 58.3 55.0 50.6 1*8.1* 1*7.3 1*5.1 1*2.9 39.6 A  t  3.3 h.h h.h h.h .h 3..3 h..1* h..1* 5..5 5.5 6 6. 66 6.6 7.7  =0.31  3 cm  72.6 72.6 72.6 70.1* 70.1* 68.2 61*. 9 61.6 59.1* 57.2 53.9 52.8 1*7.3 1*5.1 U5.1  2.2 2.2 3.3 3.3 3.8 h.h 6.0 3.8 3.3 h.h h.h 5.5 6.0 5.5 6.0  70.1* 68.2 62.7 6U.9 62.7 58.3 57.2 52.8 50.6 1*8.1* U6.2 1*2.9 1*1.8  At -  0.33  At  - 0.31  69.  X=  c)  U cm  A  70.3 70.3 69.1 69.1  66.7  66.7 6U.3 63.1 58.3  1.68 1.6.8 2.52 2.52 2.52 2.52 2.52 3.36 3.36 2.52 3.36  57.2 53.6 52.ii U.20 U8.8 3.36 U8.8 3.36 U3.9 5.0U Uo.3 5.88 39.1 5.0U 35.7 5.88 37.9 7.56  71.5  70.3 69.1 69ol 69.1 67.9  65.5 65.5 63.1 58.3 56.0 53.6 51.2 U8.8 U8.8  Ul.U Ul.U  Uo.3 39.1 35.7  3U.5  At  = o.2U5  At  0.8U 1.68 2.52 2.52 2.52 3.36 3.36 3.36 5.0U 3.36 3.36 3.36 3.36 3.36 3.36 5.0U 5.0U U.20 3.78 U.20 3.36  ^0.2U8  73.9 1.26 69.1 :2.52 69.1 2.52 70.3 , 1.68  70.3 67.9  67.9  6U.3 6U.3 59.6 57.2 5U.8 53.6 51.2 U5.1 U2.7  1.68 1.68 3.36  U.20 U.20 2.52 2.52 2.52 3.36 3.36  5.88 5.88  Ul.U 5.0U  At  =o.2U5  71.5 1.68 72.7 2.52  71.5 2.52 70.3 3.36 67.9 3.36 66.1 3.78 63.7 U.62 61.9 U.20 58.9 5.U6 57.8 5.U6 56.0 5.0U 53.0 U.62 51.8 6.30 U8.2 5.U6 U7.6 5.88 U5.7 7.1U U3.9 7.56 U2.1 7.1U 39.1 7.56 35.1 7.1U 31.5 7. Hi 30.9 7.56 At  = o.2U5  70.9 2.9U 70.9 2.9U 69.1 3.36 66.1 U.62 6U.9 U.62 63.1 U.20 61.9 U.20 59.6 U.20 56.6 •U.62 5U.2 U.62 53.0 5.U6 50.0 5.88 U6.U 5.88 1*2.1 7.56 39.7 6.30 37.3 6.30 33.9 7.1U 32.7 7.1U 32.1 7.98  6t  1.68 69.1 3.36 2.9U 68.5 3.36 67.9 2.9U 67.3 3.36 6U.3 62.5 3.78 U.20 59.6 U.62 57.8 3.78 56.6 U.20 56.0 3.78 5U.2 50.6 U.62 U9.U U.62 71.5  At = 0o262  = 0.218  BIBLIOGRAPHY  B i s h o p , A., ,(195"8) P r o j e c t Sherwood, A d d i s o n Wesleyo B u r k h a r d t , L . C. and L o v b e r g , R. H.  So (1955)  Chandresekhar,  (1958) P r o c . Geneva Co rife re nc  Proc. Camb. P h i l . Soc. 5 l 5 162.  Cormaek, G„ D . and B a r n a r d , A.  J. (1962)  C u r z o n , F. L . and C h u r c h i l l , R. J .  Rev. S c i . I n s t . _33 6 o 6 . 5  (1962) Can. J . Phys. iuO, 1191.  C u r z o n , F. L„, F c l k i e r s k i , A., Latham, R. and N a t i o n , J.A. P r o c , Roy, S o c . A 257,  (i960)  386.  C u r z o n , F. L . , Hodgson, R. T. and C h u r c h i l l , R. J .  (1963) Can. J .  P h y s . iO., I 5 u 7 . Daughney, C. C . E l l i o t , H.  (1963) M a s t e r ' s T h e s i s , U.B.C.  (1962) Rep. P r o g . P h y s .  E n g e i , A, v o n  (1959) N a t u r e  26,  11*5.  573.  183,  F u r t h , H. P., K i l l e e n , J . and R o s e n b l u t h , M. N.  (1963) P h y s . F l u .  6, Ii59. Green, T. S, and N i b L e t t , G. B.  (i960) Nuc. F u s i o n 1, u2.  H a r r i s , E , G» (1962) P h y s . F l u , 5  5  1057.  H e r t z , W. (1962) Z. N a t u r f . I T a ^ 6 8 1 . K r u s k a l , M, and S e h w a r z s c h i l d , M. A 2 3 3 , 3)48,  J.  (195U) P r o c . Roy. S o c . Lond,  Kuwabara, S,  (1963)  L e w i s , D. J .  (1950) P r o c . Roy, S o c . L o n d .  Phys. S o c . J a p . 18, 713« A202,  81.  -71-  Rosenblufch, M. N . G a r w i n , R... and R o s e r i b l u t h , A. (195U) Los Alamos #  Report 18?0. S e g r e , 3. E, and A l l e n , J , E. ( i 9 6 0 ) J , S c i . I n s t . 3 7 , 3 6 9 . S p i t z e r , L . (1956) P h y s i c s o f F u l l y I o n i z e d Gases, I n t e r s c i e n c e Summer S c h o o l o f Plasma P h y s i c s  Ris8  (i960).  T a y l o r , G. (1950) P r o c , Roy. S o c Lond. 201_A, 193. Theophanis,  C  A. ( i 9 6 0 ) Rev, S c i * I n s t  c  3 1 , U27.  W o r t h i n g , A. G. and G e f f n e r , J . (19U3) Treatment o f E x p e r i m e n t a l D a t a , J o h n W i l e y and S o n s  0  

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