UBC Theses and Dissertations

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UBC Theses and Dissertations

Study of the formative phase of a low pressure, high voltage Z-pinch MacLatchy, Cyrus Shantz 1970

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A STUDY OF THE FORMATIVE PHASE OF A LOW PRESSURE, HIGH VOLTAGE Z-PINCH by CYRUS SHANTZ MacLATCHY B.Sc, Acadia University, 1964 M.Sc, University of B r i t i s h Columbia, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December, 1970 In presenting th i s thes is in p a r t i a l f u l f i lment of the requirements fo r an advanced degree at the Un iver s i t y of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f r e e l y ava i l ab le for reference and study. I fur ther agree that permission for extens ive copying of th i s thes i s fo r scho la r l y purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t i on of th i s thes is fo r f i nanc ia l gain sha l l not be allowed without my wr i t ten permiss ion. Depa rtment The Un ivers i ty o f B r i t i s h Columbia Vancouver 8, Canada Date ABSTRACT The f o r m a t i v e phase o f a 40 kV Z - p i n c h has been i n v e s t i g a t e d i n the p r e s s u r e range f r o m 10 m t o r r to 80 m t o r r i n h y d r o g e n . The e n e r g y s p e c t r u m of the e l e c t r o n s on t h e a x i s of the v e s s e l , the s p a t i a l d i s t r i b u -t i o n o f the c u r r e n t a t the f a c e of t h e anode, the t o t a l d i s c h a r g e c u r r e n t and the v o l t a g e a c r o s s the d i s c h a r g e have a l l been m o n i t o r e d . F o r the p r e s s u r e s examined, the f o r m a t i v e phase l a s t s a few hundred nanoseconds. A t the i n i t i a t i o n o f the d i s c h a r g e , a c u r r e n t o f e l e c t r o n s w i t h e n e r g i e s i n e x c e s s o f 20 keV i s o b s e r v e d on the a x i s . As time p r o g r e s s e s , the a v e r a g e energy of the e l e c t r o n s d e c r e a s e s t o a few keV and the c u r r e n t 2 d e n s i t y i n c r e a s e s t o about 300 amps/cm . A t p r e s s u r e s o f 30 and 50 m t o r r - R ^ j the e l e c t r o n v e l o c i t y d i s t r i b u t i o n a p p e ars t o r e l a x t o a M a x w e l l i a h (T = 2 . 5 keV, n = 3 x 1 0 ^ cm . T h i s r e s u l t i s i n agreement w i t h the e e . t h e o r e t i c a l d e s c r i p t i o n of a w e a k l y i o n i z e d gas i n a s t r o n g e l e c t r i c f i e l d . However, i t s h o u l d be n o t e d t h a t the i n f l u e n c e of p l a s m a - t u r b u l e n c e has been n e g l e c t e d . The c u r r e n t t o the c e n t r a l r e g i o n o f the e l e c t r o d e i n i t i a l l y c a r r i e s more t h a n 507o of the t o t a l c u r r e n t t h r o u g h the d i s c h a r g e . T h i s o b s e r v a -t i o n i n d i c a t e s t h a t i o n i z a t i o n i n i t i a l l y o c c u r s t h r o u g h o u t the e n t i r e d i s c h a r g e v e s s e l . However, s h o r t l y a f t e r the c u r r e n t o f e n e r g e t i c e l e c t r o n s r e a c h e s i t s maximum v a l u e , the c u r r e n t t o the c e n t r a l r e g i o n o f the e l e c t r o d e d i s a p p e a r s . S i m u l t a n e o u s l y , a t r a n s i e n t v o l t a g e o f up t o 20 kV a p p ears a c r o s s the d i s c h a r g e and the r a t e o f i n c r e a s e o f t h e c u r r e n t e x h i b i t s a s h a r p d r o p i n v a l u e . The c o m b i n a t i o n of t h e s e phenomena has been i n t e r p r e t e d as t h e f o r m a t i o n o f the boundary l a y e r . The boundary l a y e r o r c u r r e n t s h e e t i s not completely formed u n t i l the end of the formative phase. Measurements of the pinch time support this conclusion. At low pressures, the time of formation of the boundary layer i s considerably shortened by the enhance-ment of i o n i z a t i o n at the w a l l . This i s caused by the presence of the magnetic f i e l d of the current flowing i n the central region of the v e s s e l . The experiment has led to better comprehension of the mechanism of boundary layer formation i n low pressure Z-pinches. The o v e r a l l under-standing of i o n i z a t i o n i n Z-pinch discharges has been improved and can be extrapolated to higher pressures. In addition, the observations ind i c a t e that the det a i l e d nature of the current sheet collapse can be strongly affected by the formative phase. i i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i v LIST OF TABLES v i i LIST OF ILLUSTRATIONS v i i i ACKNOWLEDGEMENTS x i CHAPTER I INTRODUCTION 1 PART A - THEORY CHAPTER I I CONSIDERATIONS OF IONIZATION I N A Z-PINCH 2:1 I n t r o d u c t i o n 6 2:2 I o n i z a t i o n i n a S i m p l e Spark Gap 7 2:3 The Non-uniform F i e l d o f the Z - p i n c h 9 2:4 Enhancement o f the Boundary L a y e r F o r m a t i o n 13 2:5 Boundary L a y e r F o r m a t i o n and E x t e r n a l l y Measured 14 Para m e t e r s CHAPTER I I I ELECTRON MOTION I N A PARTIALLY IONIZED GAS 3:1 I n t r o d u c t i o n 16 3:2 The Bol t z m a n n E q u a t i o n and the C o l l i s i o n Term 18 3:3 The D i s p l a c e d M a x w e l l i a n 22 3:4 The P e r t u r b e d M a x w e l l i a n 25 3:5 Measurements Made w i t h a R e t a r d i n g G r i d System 28 3:6 The R e t a r d i n g G r i d System and the D i s p l a c e d 29 M a x w e l l i a n 3:7 The R e t a r d i n g G r i d System and the P e r t u r b e d 32 M a x w e l l i a n 3:8 R e l a x a t i o n Time i n the I o n i z e d Gas 35 PART B - EXPERIMENT CHAPTER I V THE EXPERIMENTAL APPARATUS i v Page CHAPTER I V ( c o n t i n u e d ) 4:1 I n t r o d u c t i o n 38 4:2 The Z - P i n c h 39 4:3 The M e a s u r i n g Chamber 46 4:4 The E l e c t r o s t a t i c G r i d s and F a r a d a y Cup 48 Assembly 4:5 R a d i a l Probe Assembly 52 4:6 V o l t a g e and C u r r e n t Measurement 52 4:7 Ph o t o g r a p h s 55 CHAPTER V EXPERIMENTAL OBSERVATIONS AND RESULTS 5:1 I n t r o d u c t i o n 56 G e n e r a l F e a t u r e s of the D i s c h a r g e 57 5:2.1 I n t r o d u c t i o n 58 5:2.2 The Sequence o f Ev e n t s i n the D i s c h a r g e 58 5:2.3 S t a t i s t i c a l and F o r m a t i v e Time Lags 59 5:2.4 C o r r e l a t i o n of V, I and d l / d t 61 5:2.5 The P i n c h Time 67 5:2.6 M a g n e t i c F i e l d I nduced Enhancement o f I o n i z a t i o n 69 The On-- a x i s E l e c t r o n C u r r e n t 73 5:3.1 I n t r o d u c t i o n 73 5:3.2 O s c i l l o g r a p h s o f I 73 5:3.3 Time R e s o l u t i o n and R e p r o d u c i b i l i t y 76 5:3.4 The I Waveforms 77 5:3.5 The Temporal C o r r e l a t i o n o f V, d l / d t and 80 5:3.6 c R e l a x a t i o n o f E l e c t r o n Energy 80 5:3.7 I c ( V g ) f o r 30 and 50 m t o r r - H 2 84 5:3.8 C a l c u l a t i o n o f the E l e c t r o n Temperature 84 5:3.9 C a l c u l a t i o n o f the E l e c t r o n D e n s i t y 87 5:3.10 E v i d e n c e i n Support of the P e r t u r b e d M a x w e l l i a n 89 5:4 The R a d i a l D i s t r i b u t i o n of the C u r r e n t a t the 92 Anode v Page CHAPTER V ( c o n t i n u e d ) 5:4.1 I n t r o d u c t i o n 92 5:4.2 M a g n e t i c F i e l d L i m i t a t i o n s o f t h e 92 O f f - a x i s P r obes 5:4.3 R e p r o d u c i b i l i t y and Time R e s o l u t i o n 94 5:4.4 P r o f i l e s a t the Anode 97 5:4.5 A p p r o x i m a t i o n of I e ( t ) 97 5:4.6 Comparison of I and I 101 CHAPTER VI DISCUSSION AND CONCLUSIONS 6:1 I n t r o d u c t i o n 103 6:2 D e s c r i p t i o n o f the F o r m a t i v e Phase 104 6:3 The P e r t u r b e d M a x w e l l i a n D e s c r i p t i o n o f t h e 106 E l e c t r o n s 6:4 The F a r a d a y Cup as a D i a g n o s t i c T o o l 106 6:5 S u g g e s t i o n s f o r F u r t h e r Work 107 BIBLIOGRAPHY 110 APPENDICES A Problems w i t h E l e c t r o n Energy A n a l y z e r s 112 B P r a c t i c a l Problems w i t h the R i n g E l e c t r o d e 115 System C E l e c t r i c a l C i r c u i t r y 118 D E l e c t r o n Mean F r e e P a t h s 123 E E l e c t r o l y t i c Tank 125 v i L I S T OF TABLES T a b l e Page 3:2-1 C o l l i s i o n F r e q u e n c i e s (T > 200 ev) 21 3:2-2 I n t e r p r e t a t i o n o f p " 23 4:2-1 M a j o r Components and O p e r a t i n g C o n d i t i o n s 40 of the Z - P i n c h 5:1-1 L i s t o f P a r a m e t e r s 57 5:3-1 R e s u l t s o f the F a r a d a y Cup Measurements 90 v i i LIST OF ILLUSTRATIONS F i g u r e Page 2:2-1 F i r s t Townsend C o e f f i c i e n t , <x, i n H2 8 2:3-1 F i e l d D i s t r i b u t i o n i n t h e Z - P i n c h 11 2:3-2 The F i r s t Townsend C o e f f i c i e n t 11 2:3-3 H i g h P r e s s u r e Boundary L a y e r F o r m a t i o n 12 3:3-1 W as a F u n c t i o n o f U 23 3:5-1 F a r a d a y Cup and 29 3:6-1 G e o m e t r i c a l C o l l e c t i o n A n g l e 30 4:2-1 S c h e m a t i c Diagram of the Z - P i n c h 41 4:2-2 E l e c t r i c a l C o n n e c t i o n between the T r a n s m i s s i o n 42 L i n e and the D i s c h a r g e V e s s e l 4:2-3 Spark Gap S w i t c h 44 4:2-4 : Z - P i n c h C i r c u i t 45 4:3-1 Anode H o u s i n g 47 4:4-1 F a r a d a y Cup and R e t a r d i n g G r i d A ssembly 49 4:4-2 R e t a r d i n g G r i d C i r c u i t 50 4:4-3 F a r a d a y Cup C i r c u i t 51 4:5-1 R a d i a l Probe Assembly 53 4:6-1 V o l t a g e Probe Assembly and C i r c u i t 54 5:2-1 V o l t a g e Waveform Showing the S t a t i s t i c a l and 60 F o r m a t i v e Time Lags as w e l l as the C o l l a p s e Phase of the Z - P i n c h 5:2-2(a) S t a t i s t i c a l Time Lag as a F u n c t i o n o f P r e s s u r e 62 5:2-2(b) F o r m a t i v e Time as a F u n c t i o n o f P r e s s u r e 62 5:2-3 V, I and d l / d t a t 80 m t o r r - H 2 63 5:2-4 V, I and d l / d t a t 40 m t o r r - H 2 64 1 5:2-5 V, I and d l / d t a t 20 mtorr-H„ 65 v i i i F i g u r e Page 5:2-6 V, I and d l / d t a t 10 m t o r r - H 2 66 5:2-7 I l l u s t r a t i o n o f the Measurement o f Tp 68 5:2-8 The L o g a r i t h m of T p P l o t t e d as a F u n c t i o n of the 70 P r e s s u r e 5:2-9(a) V a l u e of I a t the I n s t a n t o f V f 71 5:2-9(b) P r e s s u r e Enhancement, p'/p, as a F u n c t i o n of P r e s s u r e 71 5:3-1 I c a t 200 m t o r r - H 2 74 5:3-2 I c a t 10 m t o r r - H 2 75 5:3-3 I n f l u e n c e of the D e l a y L i n e on I c 76 5:3-4 The F a r a d a y Cup C u r r e n t a t 50 m t o r r - H 2 78 5:3-5 The F a r a d a y Cup C u r r e n t a t 30 m t o r r - H 2 78 5:3-6 The F a r a d a y Cup C u r r e n t a t 10 m t o r r - H 2 79 5:3-7 The F a r a d a y Cup C u r r e n t a t 20 m t o r r - H e 79 5:3-8 The C o r r e l a t i o n of V, d l / d t and I 81 5:3-9(a) F a r a d a y Cup C u r r e n t as a F u n c t i o n o f Time 82 (10 m t o r r - H 2 ) 5:3-9(b) F a r a d a y Cup C u r r e n t as a F u n c t i o n o f V 82 5:3-10 N o r m a l i z e d I (V ) f o r t = 0.14 yusec, 0.19 ^ s e c . 83 5 : 3 - l l ( a ) F a r a d a y Cup C u r r e n t as a F u n c t i o n o f Time 85 (50 m t o r r - H 2 ) 5 : 3 - l l ( b ) F a r a d a y Cup C u r r e n t as a F u n c t i o n of the G r i d 85 V o l t a g e (50 m t o r r - H 2 ) 5:3-12(a) F a r a d a y Cup C u r r e n t as a F u n c t i o n of Time 86 (30 m t o r r - H 2 ) 5:3-12(b) F a r a d a y Cup C u r r e n t as a F u n c t i o n o f the G r i d 86 V o l t a g e (30 m t o r r - H 2 ) 5:3-13(a) S e m i - l o g P l o t o f I c ( V g ) a t 50 m t o r r - H 2 88 5:3-13(b) S e m i - l o g P l o t o f I C ( V ) a t 30 m t o r r - H 2 88 5:4-1 E n l a r g e d View of an O f f - a x i s P r o b e 93 i x F i g u r e Page 5:4-2 E l e c t r o n s w i t h C y c l o t r o n R a d i u s G r e a t e r t h a n 1 cm. 95 5:4-3 E l e c t r o n s C o l l e c t e d by the C u r r e n t Probe 95 5:4-4 O s c i l l o g r a m s of I 9 6 5:4-6(a) C u r r e n t Probe S i g n a l s as a F u n c t i o n of Time 98 (40 m t o r r - H 2 ) 5:4-6(b) C u r r e n t P r o f i l e a t t h e Anode (40 m t o r r - H 2 ) 98 5:4-7(a) C u r r e n t Probe S i g n a l s as a F u n c t i o n o f Time 99 (20 m t o r r - H 2 ) 5:4-7(b) C u r r e n t P r o f i l e a t the Anode (20 m t o r r - H 2 ) 99 5:4-8(a) C u r r e n t Probe S i g n a l s as a F u n c t i o n o f Time 100 (10 m t o r r - H 2 ) 5:4-8(b) C u r r e n t P r o f i l e a t the Anode;(10 m t o r r - H 2 ) 100 5:4-9 Comparison of I £ ( t ) and I ( t ) . 102 A - l E l e c t r o n Beam S p r e a d i n g 113 B - l F i e l d D i s t r i b u t i o n i n the H o l e o f the R e t a r d i n g 117 R i n g E l e c t r o d e C - l Probe R e s i s t a n c e as a F u n c t i o n o f the I n p u t 119 V o l t a g e C-2 I n p u t V o l t a g e o f the Probe as a F u n c t i o n o f the 119 Output V o l t a g e C-3 C a l i b r a t i o n Curve Showing the R i n g i n g and Decay of 121 the D i s c h a r g e C u r r e n t C-4 R i s e o f the C u r r e n t to the F i r s t Maximum and 121 d l / d t C-5 A t t e n u a t i o n o f the Dela y L i n e 122 C-6 E f f e c t o f the D e l a y L i n e on a Square Wave 122 D-l Reduced Mean F r e e P a t h i n Hydrogen 124 E - l E l e c t r o l y t i c Tank 126 x ACKNOWLEDGMENTS I am i n d e b t e d to Dr. A. J . B a r n a r d f o r h i s h e l p f u l c r i t i c i s m and p a t i e n t g u i d a n c e d u r i n g t he e x p e r i m e n t a l work and the p r e p a r a t i o n o f t h e t h e s i s . I would a l s o l i k e t o thank D r s . F. L. Curzo n , G. Jones and J . Meyer, f o r t h e i r i n t e r e s t i n the p r e p a r a t i o n of the t h e s i s . The t e c h n i c a l a s s i s t a n c e o f M e s s r s . D. G. S i e b e r g and J . A. Zanganeh of the Plasma P h y s i c s e l e c t r o n i c s shop i s g r a t e f u l l y acknowledged as w e l l as the c o n s i d e r a b l e a s s i s t a n c e r e c e i v e d f r o m a l l o f the members o f . t h e P h y s i c s machine shop s t a f f . As w e l l , I would l i k e t o m e n t i o n Mr. J . Lees and Mr. R. P. H a i n e s . The t h e s i s has been most c a p a b l y typed by Mrs. S h e r r i M a c R a i l d . I n a d d i t i o n , I w i s h to e x p r e s s my a p p r e c i a t i o n f o r the h e l p f u l and s t i m u l a t i n g d i s c u s s i o n s w i t h my f e l l o w g r a d u a t e s t u d e n t s , e s p e c i a l l y t h o s e who have a t t e n d e d t he 'A.J.B. Club' o v e r the p a s t few y e a r s . F i n a l l y , the f i n a n c i a l a s s i s t a n c e o f the N a t i o n a l R e s e a r c h C o u n c i l and the M a c M i l l a n F a m i l y F e l l o w s h i p i s a p p r e c i a t e d . x i CHAPTER I INTRODUCTION A Z - p i n c h i s a h i g h c u r r e n t d i s c h a r g e w h i c h i s c h a r a c t e r i z e d by a c y l i n d r i c a l l y s ymmetric c o l l a p s i n g c u r r e n t sheet'. The d i s c h a r g e o c c u r s between two e l e c t r o d e s w h i c h a r e p l a c e d a t e i t h e r end o f a c y l i n d r i c a l v e s s e l . The c u r r e n t i s r e t u r n e d t o i t s s o u r c e by a c y l i n d r i c a l l y symmetric r e t u r n c o n d u c t o r on the o u t s i d e o f the d i s c h a r g e v e s s e l . The dynamics of the Z - p i n c h i s b e s t d e s c r i b e d i n a system o f c y l i n d r i c a l c o o r d i n a t e s i n w h i c h the Z - a x i s and the a x i s o f the v e s s e l c o i n c i d e . I n the s i m p l e s t model of the d e v i c e , the c u r r e n t f l o w s i n the Z - d i r e c t i o n , i n a t h i n , c y l i n d r i c a l s h e e t . The c u r r e n t s h e e t i n t e r a c t s w i t h i t s own m a g n e t i c f i e l d and c o l l a p s e s r a d i a l l y . The Z - d i r e c t e d c u r r e n t and the p i n c h i n g m o t i o n of the c u r r e n t s h e e t g i v e the d e v i c e i t s name. The s o - c a l l e d p i n c h e f f e c t was f i r s t d i s c u s s e d by Bennet i n 1934. D u r i n g the p a s t few decades, the Z - p i n c h has been a u s e f u l l a b o r a -t o r y plasma s o u r c e and has been examined from many d i f f e r e n t p o i n t s o f view. P a r t i c u l a r i n t e r e s t was a r o u s e d i n the 1950's when the Z - p i n c h was s t u d i e d w i t h t h e r m o n u c l e a r g o a l s i n mind. I t has a l s o been u s e f u l i n the e x a m i n a t i o n of b o t h plasma c o n f i n e m e n t and s t a b i l i t y p r o b l e m s . F o r example, many r e p o r t s d e a l i n g w i t h t h e s e problems i n l a r g e p i n c h d e v i c e s were g i v e n a t the 1958 Geneva C o n f e r e n c e on the P e a c e f u l Uses of A t o m i c Energy. The Z - p i n c h has been an i n t e r e s t i n g case s t u d y i n plasma dynamics from w h i c h the snowplough model ( R o s e n b l u t h , 1954) has e v o l v e d to i t s p r e s e n t form. The snowplough model has been used w i t h good r e s u l t s by b o t h Daughney (1966) and Tarn (1967) . Because o f the r e l a t i v e l y w e l l 1 u n d e r s t o o d b e h a v i o u r o f the Z - p i n c h , i t has been used i n r e c e n t y e a r s as a s o u r c e o f plasma f o r s p e c t r o s c o p i c i n v e s t i g a t i o n s ( R o b e r t s , s u b m i t t e d f o r p u l b i c a t i o n ) and as a t e s t model f o r v a r i o u s d i a g n o s t i c t e c h n i q u e s (Medley, 1970). The p r e s e n t i n v e s t i g a t i o n c o n c e r n s the f o r m a t i v e phase i n a low p r e s s u r e , h i g h v o l t a g e p i n c h - - a phase o f the Z - p i n c h dynamics w h i c h has h i t h e r t o n o t r e c e i v e d a g r e a t d e a l o f a t t e n t i o n . The f o r m a t i v e phase i s the p e r i o d o f time d u r i n g w h i c h the gas b r e a k s down and the boundary l a y e r o r c u r r e n t s h e e t forms. E a r l i e r i n v e s t i g a t i o n s o f the f o r m a t i v e s t a g e may have been hampered by the r a p i d i t y o f the breakdown p r o c e s s and by n o i s e a s s o c i a t e d w i t h the i o n i z a t i o n p r o c e s s and t r i g g e r i n g . I n the e a r l y s t u d i e s o f the dynamics of the p i n c h , i t was n o t n e c e s s a r y t o have p r e c i s e knowledge of the boundary l a y e r f o r m a t i o n . However, r e c e n t measurements of e l e c t r o n d e n s i t y (Medley, 1970) i n d i c a t e t h a t more a c c u r a t e knowledge o f the i n i t i a t i v e s t a g e i s n e c e s s a r y t o improve measurements of l a t e r s t a g e s o f the c o l l a p s e . Some of the n o i s e problems have been e l i m i n a t e d by o p e r a t i n g the d i s c h a r g e near the s t a t i c breakdown p o t e n t i a l . The t e c h n i q u e i n c r e a s e s the t r a n s i e n t v o l t a g e and c u r r e n t f l u c t u a t i o n s d u r i n g the f o r m a t i v e phase. A t the f o r m a t i o n o f the boundary l a y e r , an un e x p e c t e d b u t sub-s t a n t i a l back e.m.f. i s o b s e r v e d and l a r g e f l u c t u a t i o n s appear on the time d e r i v a t i v e o f t h e c u r r e n t . There i s a l s o a s u b s t a n t i a l c u r r e n t o f e n e r g e t i c e l e c t r o n s w h i c h appears a l o n g the a x i s o f the d i s c h a r g e d u r i n g the i n i t i a l s t a g e s o f i o n i z a t i o n . These o b s e r v a t i o n s have l e d to a much b e t t e r u n d e r s t a n d i n g of the growth o f i o n i z a t i o n and boundary l a y e r forma-t i o n i n a Z - p i n c h . 3 I t was a l s o f ound t h a t the a x i a l c u r r e n t c a n n o t be e x p l a i n e d by the w i d e l y a c c e p t e d b e l i e f t h a t i n i t i a l i o n i z a t i o n o c c u r s p r e f e r e n t i a l l y a l o n g the low i n d u c t a n c e p a t h i n the d i s c h a r g e . I n t h i s t h e s i s , i t i s emphasized t h a t the i n i t i a l e l e c t r i c f i e l d d i s t r i b u t i o n s t r o n g l y i n f l u e n c e s the s p a t i a l v a r i a t i o n o f the r a t e o f i o n i z a t i o n . On t h i s b a s i s , b o t h the o c c u r r e n c e and t h e t e m p o r a l b e h a v i o u r o f the e n e r g e t i c e l e c t r o n s can be u n d e r s t o o d . The t r a n s i e n t f l u c t u a t i o n s o f the v o l t a g e and c u r r e n t a r e a l s o e x p l a i n e d . A F a r a d a y cup and r e t a r d i n g g r i d s y s t e m have been used to measure the energy d i s t r i b u t i o n of the e l e c t r o n s . The F a r a d a y cup measurements have been u t i l i z e d i n the c a l c u l a t i o n of the e l e c t r o n d e n s i t y and temp-e r a t u r e . The measurements i n d i c a t e t h a t the v e l o c i t y d i s t r i b u t i o n of the e l e c t r o n s can be r e p r e s e n t e d by a M a x w e l l i a n i n w h i c h the e f f e c t i v e t e m p e r a t u r e i s r e l a t e d to b o t h the s t r e n g t h o f the e l e c t r i c f i e l d and the c o l l i s i o n f r e q u e n c y of the e l e c t r o n s w i t h the n e u t r a l s . T h i s i m p l i e s t h a t the e l e c t r o n v e l o c i t y d i s t r i b u t i o n i s c o n t r o l l e d by the e l e c t r o n -n e u t r a l e l a s t i c c o l l i s i o n s d u r i n g the e a r l y s t a g e of the d i s c h a r g e . T h i s r e s u l t i s i n agreement w i t h the t h e o r e t i c a l d e s c r i p t i o n o f a w e a k l y i o n i z e d gas under the i n f l u e n c e of a s t r o n g e l e c t r i c f i e l d (Tanenbaum, 1967). The p r e s e n t a t i o n o f the t h e s i s i s as f o l l o w s . Chapter I I i s an o u t l i n e o f the b a s i c t h e o r e t i c a l c o n s i d e r a t i o n s n e c e s s a r y to the under-s t a n d i n g of the e v e n t s w h i c h o c c u r d u r i n g the f i r s t few hundred nanoseconds of the d i s c h a r g e . A s i m p l e model based on e l e c t r o n m u l t i p l i c a t i o n i n the gas as i t i s d e s c r i b e d by a s p a t i a l l y inhomogenious Townsend c o e f f i e n t , oi, i s p r o p o s e d . By e x a m i n i n g t h e p r e s s u r e and f i e l d s t r e n g t h dependence of °t, i t i s p o s s i b l e to show t h a t the i n i t i a l i o n i z a t i o n o c c u r s p r e f e r e n -t i a l l y i n the c e n t r a l r e g i o n o f t h e d i s c h a r g e v e s s e l f o r c o n d i t i o n s of low p r e s s u r e and h i g h f i e l d s t r e n g t h . Thus the o c c u r r e n c e o f the e n e r g e t i c e l e c t r o n c u r r e n t i s e x p l a i n e d . The model a l s o l e a d s t o a s l o w e r forma-t i o n of the boundary l a y e r and to m a g n e t i c f i e l d enhancement o f i o n i z a t i o n a t the w a l l of the v e s s e l . The growth o f the boundary l a y e r e x p l a i n s the a b r u p t d i s a p p e a r a n c e of the a x i a l c u r r e n t . Chapter I I I i n v e s t i g a t e s the t h e o r e t i c a l a s p e c t s o f the e l e c t r o n m o t i o n . Two d i s t r i b u t i o n f u n c t i o n s , the d i s p l a c e d M a x w e l l i a n of Dreicer (1959) and the perturbed Maxwellian of Tanenbaum (1967) a r e d i s c u s s e d . The d i s t r i b u t i o n f u n c t i o n s a r e used i n an e x a m i n a t i o n o f the r e l a t i o n s h i p of t h e F a r a d a y cup s i g n a l t o t h e p h y s i c a l p r o p e r t i e s of the i o n i z e d gas. S t a n d a r d methods of c a l c u l a t i n g t h e e l e c t r o n d e n s i t y and t e m p e r a t u r e from r e t a r d i n g g r i d systems have been d i s c u s s e d and c a r e f u l a t t e n t i o n i s g i v e n t o the a p p l i c a t i o n of the t h e o r y t o the d i s c h a r g e o f t h i s e x p e r i m e n t . I n C h a p t e r I V , the e x p e r i m e n t a l a p p a r a t u s i s d i s c u s s e d . Most o f the work has been c a r r i e d o u t w i t h s t a n d a r d t e c h n i q u e s . Where n e c e s s a r y , s p e c i a l a p p l i c a t i o n s a r e d i s c u s s e d i n d e t a i l . C h apter V c o n t a i n s the e x p e r i m e n t a l o b s e r v a t i o n s and r e s u l t s w h i c h a r e p r e s e n t e d w i t h two purposes i n mind. As a means of comprehending the t e m p o r a l b e h a v i o u r o f the c u r r e n t of f a s t e l e c t r o n s , the boundary l a y e r f o r m a t i o n a s p e c t o f the o b s e r v a t i o n s has been s t r e s s e d . I n o r d e r to u n d e r s t a n d the m o t i o n o f the e l e c t r o n s , the F a r a d a y cup measurements and o t h e r c o r r o b o r a t i v e e v i d e n c e have been examined. The c h a p t e r forms a c o n s i s t e n t s e t o f o b s e r v a t i o n s w h i c h agree w i t h the t h o u g h t s p r e s e n t e d i n C h a p t e r s I I and I I I . I n t h e f i n a l c h a p t e r , the r e s u l t s a r e d i s c u s s e d 5 and improvements a r e s u g g e s t e d . A number of A p p e n d i c e s have been i n c l u d e d w h i c h e l u c i d a t e v a r i o u s p o i n t s n o t f u l l y d i s c u s s e d i n the p r e s e n t a t i o n of the t h e s i s . P a r t i c u l a r l y i m p o r t a n t a r e A p p e n d i c e s A and B w h i c h p r e s e n t a d i s c u s s i o n of the energy a n a l y z i n g system. CHAPTER I I CONSIDERATIONS OF IONIZATION I N A Z-PINCH 2:1 I n t r o d u c t i o n I n t h i s c h a p t e r , we s h a l l c o n s i d e r i o n i z a t i o n and boundary l a y e r f o r m a t i o n i n a Z - p i n c h . The p r o c e s s i s c l o s e l y r e l a t e d to i o n i z a t i o n i n a s i m p l e s p a r k gap, e x c e p t t h a t t he c y l i n d r i c a l e l e c t r o d e c o n f i g u r a t i o n causes the c u r r e n t t o fo r m a s h e e t o r boundary l a y e r . Under c o n d i t i o n s o f low p r e s s u r e and h i g h e l e c t r i c f i e l d , the growth o f i o n i z a t i o n by e l e c t r o n m u l t i p l i c a t i o n i n the gas w i l l be g r e a t e r i n t h e c e n t e r o f the d i s c h a r g e v e s s e l t h a n a t the w a l l s . When t h i s o c c u r s , a c u r r e n t o f e n e r g e t i c e l e c t r o n s w i l l appear i n i t i a l l y i n the c e n t r a l r e g i o n o f the v e s s e l . As the con-d u c t i v i t y grows, t h e f i e l d p e n e t r a t i o n d e c r e a s e s and e v e n t u a l l y the r e g i o n of c o n d u c t i o n s h i f t s f rom the c e n t r a l r e g i o n to the w a l l r e g i o n . The f i r s t few s e c t i o n s o f t h i s c h a p t e r w i l l d e s c r i b e t h i s c o u r s e o f e v e n t s . When i o n i z a t i o n o c c u r s i n c r o s s e d e l e c t r i c and m a g n e t i c f i e l d s , c o n s i d e r a b l e enhancement o f the r a t e o f i o n i z a t i o n o v e r the r a t e when no ma g n e t i c f i e l d i s p r e s e n t i s e x p e r i e n c e d a t low p r e s s u r e s . The m a g n e t i c f i e l d of the a x i a l c u r r e n t i s a t r i g h t a n g l e s t o the e l e c t r i c f i e l d i n the d i s c h a r g e v e s s e l . T h e r e f o r e S e c t i o n 2:3 i n v e s t i g a t e s the i n f l u e n c e of t h i s enhancement on the boundary l a y e r f o r m a t i o n . T h i s enhancement i s a low p r e s s u r e phenomenon. The s h i f t i n g o f the c o n d u c t i o n r e g i o n w i l l a f f e c t the c u r r e n t and v o l t a g e o f the d i s c h a r g e . The v a r i a t i o n s o f t h e s e p a r a m e t e r s w i l l a l s o be c o n s i d e r e d . T h i s d e s c r i p t i o n o f the f o r m a t i v e phase i s q u a l i t a -t i v e i n n a t u r e and o f n e c e s s i t y i s based on i d e a l i z a t i o n s w h i c h a r e 6 i n h e r e n t i n the t h e o r y o f gas i o n i z a t i o n . The main c r i t i c i s m o f t h i s p r o p o s a l and i t s a p p l i c a b i l i t y t o the p r e s e n t e x p e r i m e n t i s t h a t t h e t h e o r i s p r o g r e s s i v e l y l e s s r e l i a b l e as the p r e s s u r e i s l o w e r e d . 2:2 I o n i z a t i o n i n a S i m p l e Spark Gap T h i s s e c t i o n w i l l s e r v e t o i n t r o d u c e the b a s i c c o n c e p t s o f i o n i -z a t i o n and breakdown. D u r i n g t h e p r e s e n t d i s c u s s i o n we w i l l assume s i m p l e p l a n a r e l e c t r o d e s w i t h an i n i t i a l l y c o n s t a n t and u n i f o r m e l e c t r i c f i e l d . The growth of i o n i z a t i o n i n the gas i s d e s c r i b e d by the f i r s t Townsend c o e f f i c i e n t , ^, g i v e n by dn <* n (2:2-1) dx e where i s the e l e c t r o n d e n s i t y . The q u a n t i t y , oC, i s the number of i o n p a i r s produced by e l e c t r o n - n e u t r a l c o l l i s i o n s p e r u n i t d i s t a n c e o f d r i f t o f an e l e c t r o n i n the d i r e c t i o n o f the e l e c t r i c f i e l d . The s t e a d y s t a t e r e l a t i o n s h i p between the anode c u r r e n t , I , and the cathode c u r r e n t , I , i s o I exp(ocd) I = r (2:2-2) 1 - [exp(ocd) - lj Here, d r e p r e s e n t s the e l e c t r o d e s e p a r a t i o n and 6j i s the second Townsend c o e f f i c i e n t . = ^ + f + f <2:2-3> I n t h i s e q u a t i o n , X i s the number o f e l e c t r o n s produced a t t h e cathode per i n c i d e n t i o n , 6 i s the number of e l e c t r o n s p r o duced a t t h e cathode per i n c i d e n t photon, and (3 i s the number o f i o n p a i r s p r o duced per u n i t of d i s t a n c e of d r i f t o f an i o n . The f i r s t Townsend c o e f f i c i e n t i s a f u n c t i o n of the k i n d o f gas, the gas p r e s s u r e and the e l e c t r i c f i e l d 8 s t r e n g t h . The second c o e f f i c i e n t depends on t h e t y p e and p r e s s u r e o f the gas, the f i e l d s t r e n g t h , and the geometry and t h e n a t u r e o f the e l e c t r o d e s , Breakdown i s s a i d t o o c c u r f o r an e l e c t r o d e s e p a r a t i o n o f d , f o r w h i c h s 1 - [£->[exp(*d ) - l ] = 0 (2:2-4) The s m a l l e s t v o l t a g e f o r w h i c h t h i s o c c u r s i s the s t a t i c breakdown v o l t a g e , V . The p r i n c i p l e o f s i m i l a r i t y a l l o w s c o m p a r i s o n o f g e o m e t r i -c a l l y s i m i l a r gaps of d i f f e r e n t s i z e and p r e s s u r e i n the same gas. To f a c i l i t a t e such c o m p a r i s o n i t i s c o n v e n i e n t t o p l o t f = fx (E/P) (2:2-5) v s = f 2 ( Pd s) (2:2-6) E/p and p d g a r e r e s p e c t i v e l y c a l l e d the re d u c e d e l e c t r i c f i e l d and the reduced e l e c t r o d e s e p a r a t i o n . E q u a t i o n 2:2-5 has been r e p r e s e n t e d i n . F i g u r e 2:2-1. 10 u u o u I CO 1^ •H nJ Ph c O 10 ex 10 \ \ Von E n g e l (1965) — — — S t u a r t and G e r j u o y (1960) J I 5 2 1000 10,000 100 E/p ( V o l t s / ( c m . t o r r ) ) F i g u r e 2:2-1 F i r s t Townsend C o e f f i c i e n t , c(, i n H„ D a v i e s e t a l . (1965) have examined t h e o r e t i c a l l y the t e m p o r a l growth o f i o n i z a t i o n w h i c h i n v o l v e s t h e s o l u t i o n o f the f o l l o w i n g con-t i n u i t y e q u a t i o n s w i t h the a p p r o p r i a t e boundary c o n d i t i o n s w h i c h i n v o l v e to. a t Here, I , i s the i o n c u r r e n t and I i s the e l e c t r o n c u r r e n t . The q u a n t i t i e s W+ and W a r e the i o n and e l e c t r o n d r i f t v e l o c i t i e s . The con-t i n u i t y e q u a t i o n s h e l p to i l l u s t r a t e the p r o b l e m of a p p l y i n g the s i m p l e t h e o r y of breakdown to normal systems. The b a s i c a s s u m p t i o n g o v e r n i n g the e q u a t i o n s i s t h a t the e l e c t r o n s have a w e l l d e f i n e d d r i f t v e l o c i t y . The s o l u t i o n of e q u a t i o n s 2:2-7 a l s o demands t h a t the i n f l u e n c e of f i e l d d i s t o r t i o n caused by the space charge o f t h e low m o b i l i t y i o n s be t a k e n i n t o a c c o u n t . The i o n i z a t i o n c o e f f i c i e n t must now be r e g a r d e d as a f u n c t i o n of b o t h time and p o s i t i o n i f one i s t o g e t a f i r a 1 s o l u t i o n t o the p r o b l e m of t e m p o r a l growth of i o n i z a t i o n . I n the n e x t s e c t i o n , an a t t e m p t w i l l be made to a p p l y t h e s e b a s i c i d e a s t o a s y s t e m i n w h i c h E/p i s v e r y h i g h , E i s n o n - u n i f o r m and i n w h i c h the concept o f d r i f t v e l o c i t y w i l l n o t always be a p p l i c a b l e . 2:3 The Non-uniform F i e l d of t h e Z - p i n c h The Z - p i n c h d i f f e r s f r o m the s i m p l e model d e s c r i b e d i n S e c t i o n 2:1 . i n one i m p o r t a n t a s p e c t . Though the p r o c e s s of i o n i z a t i o n i s b a s i c a l l y the same, the f i e l d d i s t r i b u t i o n o f the Z - p i n c h i s i n i t i a l l y n o n - u n i f o r m . The e q u i p o t e n t i a l s f o r the f i e l d o f the Z - p i n c h have been drawn i n I + ( x , t ) W + I _ ( x , t ) W = a I ( x , t ) + = a I ( x , t ) -3> I + ( x , t ) ^ I . ( x , t ) 3> x ( 2 : 2 - 7 ( a ) ) ( 2 : 2 - 7 ( b ) ) 10 F i g u r e 2:3-1. The p o t e n t i a l d i s t r i b u t i o n was o b t a i n e d f r o m measurements i n an e l e c t r o l y t i c tank as d e s c r i b e d i n A p p e n d i x E. I t i s p o s s i b l e t o show q u a l i t a t i v e l y t h a t the i o n i z a t i o n p r o c e e d s i n such a way t h a t the e l e c t r o n d e n s i t y i n c r e a s e s most r a p i d l y i n the c e n t r a l r e g i o n of the v e s s e l f o r h i g h v o l t a g e , low p r e s s u r e p i n c h e s and i n the w a l l r e g i o n a t h i g h e r p r e s s u r e s . These arguments a r e made w i t h t h e f o l l o w i n g r e s e r v a t i o n . The p r e s e n c e of space c h a r g e , w h i c h appears as the gas i o n i z e s , has been n e g l e c t e d . Thus, i t i s c o n c e i v a b l e t h a t the f i e l d c o u l d be d i s t o r t e d a f t e r a s h o r t time i n such a way t h a t the p r e s e n t arguments would be i n -v a l i d a t e d . I n o r d e r t h a t i o n i z a t i o n i n the absence of space charge may be d i s c u s s e d , the d a t a of F i g u r e 2:2-1 has been r e p l o t t e d i n F i g u r e 2:3-2 w i t h the p r e s s u r e dependence t r e a t e d as a p a r a m e t e r . I n the low p r e s s u r e range, the r a t e of i o n p a i r p r o d u c t i o n w i l l i n c r e a s e as the f i e l d s t r e n g t h -3 i s l o w e r e d . F o r example, a t 0.01 t o r r i n c r e a s e s f r o m about 2 x 10 _2 i o n p a i r s / c m to 10 i o n p a i r s / c m when the e l e c t r i c f i e l d i s r e d u c e d f r o m 500 V/cm to 100 V/cm. A t h i g h p r e s s u r e s the r e l a t i o n s h i p i s j u s t the o p p o s i t e . V a r i a t i o n s i n p r e s s u r e w i l l change the p a t t e r n o f i o n i z a t i o n i n the p i n c h v e s s e l . A t h i g h p r e s s u r e s where most of the r e g i o n of o p e r a t i o n i s i n the p o s i t i v e s l o p e r e g i o n of the oi c u r v e , one would e x p e c t the i o n i z a t i o n t o p r o c e e d most r a p i d l y i n the h i g h e s t f i e l d r e g i o n . F o r t h e p i n c h v e s s e l i n F i g u r e 2:3-1, t h i s w i l l be a t the edge o f the anode. I o n i z a t i o n w i l l s p r e a d i n a cascade w h i c h w i l l adhere to the w a l l o f the v e s s e l as a plasma s h e e t p r o g r e s s e s a l o n g the i n s i d e o f the g l a s s c o n t a i n e r . The h i g h f i e l d 11 R e t u r n Conductor G l a s s V e s s e l F i g u r e 2:3-1 F i e l d D i s t r i b u t i o n i n t h e Z - p i n c h f o r d.c. C o n d i t i o n s w i t h No Space Charge E(V/cm) F i g u r e 2:3-2 The F i r s t Townsend C o e f f i c i e n t p l o t t e d as a F u n c t i o n o f t h e E l e c t r i c F i e l d S t r e n g t h w i t h t h e P r e s s u r e T r e a t e d as a Parameter 12 r e g i o n p r o g r e s s e s down the tube a t the head o f t h e plasma s h e e t because l i t t l e p o t e n t i a l can be s u p p o r t e d i n s i d e the plasma. F i g u r e 2:3-3 r e p r e s e n t s t h i s type o f breakdown. The boundary l a y e r w i l l be q u i c k l y formed w i t h l i t t l e i o n i z a t i o n i n the i n t e r i o r o f the v e s s e l because o f the s h i e l d i n g e f f e c t o f the c o n d u c t i n g s k i n . However, a t low p r e s s u r e s where o p e r a t i o n i s m a i n l y i n the n e g a t i v e s l o p e r e g i o n of the ct c u r v e , i o n i z a t i o n p r o g r e s s e s most r a p i d l y i n r e g i o n s of low f i e l d s t r e n g t h . From F i g u r e 2:3-1, t h i s would appear to be near the a x i s i n the n e i g h b o u r h o o d of the anode, and i t would appear to c o v e r the f u l l c r o s s s e c t i o n o f the v e s s e l i n the cathode r e g i o n . Thus, i o n i z a t i o n grows more r a p i d l y i n the i n t e r i o r o f the d i s c h a r g e tube t h a n a t the boundary. C u r r e n t d e n s i t y measurements a t the anode d u r i n g the breakdown phase of the p i n c h show a maximum on the a x i s . U n i o n i z e d Gas Anode H i g h E l e c t r i c F i e l d R e t u r n Conductor V e s s e l W a l l F i g u r e 2:3-3 H i g h P r e s s u r e Boundary L a y e r F o r m a t i o n 13 2:4 Enhancement o f the Boundary L a y e r F o r m a t i o n I n the low p r e s s u r e p i n c h , i t has been s u g g e s t e d t h a t the boundary l a y e r f o r m a t i o n i s p r e c e d e d by a s u b s t a n t i a l e l e c t r o n c u r r e n t i n the i n t e r i o r of the v e s s e l . Such a c u r r e n t produces an a z i m u t h a l m a g n e t i c f i e l d normal t o the e l e c t r i c f i e l d l i n e s i n the d i s c h a r g e tube. The e f f e c t o f such f i e l d s i s t o i n c r e a s e t h e v a l u e of the f i r s t Townsend c o e f f i c i e n t , oC. An e x t e n s i v e d i s c u s s i o n o f t h e Townsend c o e f f i c i e n t s i n c r o s s e d e l e c t r i c and m a g n e t i c f i e l d s has been g i v e n by B l e v i n and Haydon (1958). The f i r s t i o n i z a t i o n c o e f f i c i e n t d e s c r i b e s the number o f i o n p a i r s produced per c e n t i m e t e r of d r i f t i n the d i r e c t i o n o f the e l e c t r i c f i e l d . The m a g n e t i c f i e l d , by c a u s i n g the e l e c t r o n s to move i n h e l i c a l p a t h s , i n c r e a s e s the d i s t a n c e t h a t an e l e c t r o n t r a v e l s w h i l e d r i f t i n g one c e n t i -meter i n the d i r e c t i o n o f the e l e c t r i c f i e l d . The chance o f an i o n i z i n g c o l l i s i o n i s t h e r e b y i n c r e a s e d f o r a t r a n s i t between the e l e c t r o d e s . B l e v i n and Haydon have shown t h a t the main e f f e c t of the m a g n e t i c f i e l d on the e l e c t r o n s i s e q u i v a l e n t to a change i n p r e s s u r e f r o m p to p' g i v e n by p' = p 1 + 2 " l v E J -h (2:4-1) where CO, i s the c y c l o t r o n f r e q u e n c y g i v e n by b 00, = — (2:4-2) b m e I n t h i s e q u a t i o n , e, m^, and B a r e r e s p e c t i v e l y the e l e c t r o n c h a r g e , the e l e c t r o n mass, and t h e m a g n e t i c f i e l d s t r e n g t h . The q u a n t i t y , \j , i s E the e l e c t r o n - n e u t r a l e l a s t i c c o l l i s i o n f r e q u e n c y w h i c h , i n the case of R^, i s g i v e n by ( D e l c r o i x , 1964) 14 V E = 6 x 1 0 9 p t o r r " 1 s e c " 1 (2:4-3) The Townsend «. i s n o r m a l l y g i v e n by = A exp (-B f ) (2:4-4) P & where A and B a r e c o n s t a n t s (Von E n g e l , 1965). However, i n the p r e s e n c e of a m a g n e t i c f i e l d normal t o the e l e c t r i c f i e l d , (2:4-5) T h i s i s e q u i v a l e n t t o r e p l a c i n g p by p' as g i v e n i n e q u a t i o n 2:4-1. R e f e r r i n g to F i g u r e 2:3-2, i t can be seen t h a t the p r e s e n c e o f a m a g n e t i c f i e l d a t low p r e s s u r e s i n c a n produce a d r a m a t i c i n c r e a s e i n the e f f e c t i v e v a l u e o f oc. As an example, a f i e l d o f o n l y 20 gauss i s r e q u i r e d to g i v e a p r e s s u r e enhancement o f a f a c t o r o f 10 a t a p r e s s u r e o f 10 m t o r r . The i n c r e a s e i n c<. f o r a f i e l d o f 500 V/cm i s a f a c t o r o f about 150. Thus, a l a r g e a x i a l c u r r e n t w i l l have a d r a m a t i c e f f e c t on the boundary l a y e r f o r m a t i o n t hrough the i n t e r a c t i o n o f the e l e c t r o n s w i t h the m a g n e t i c f i e l d . The r e m a i n i n g Townsend c o e f f i c i e n t s , if, & and fi} are n o t so d r a m a t i c a l l y a f f e c t e d . F o r example, the much lo w e r v e l o c i t y o f the i o n s means t h a t they w i l l n o t e x p e r i e n c e a m a g n e t i c f o r c e t o near the same e x t e n t as the e l e c t r o n s . Thus X and [Z w i l l be e s s e n t i a l l y unchanged. The photon term, 5 , on the o t h e r h a n d , w i l l be a f f e c t e d s i n c e the photon energy r e s u l t s from the e x c i t a t i o n o f t h e gas by the e l e c t r o n s and f r o m B r e m s s t r a h l u n g . 2:5 Boundary L a y e r F o r m a t i o n and E x t e r n a l l y Measured P a r a m e t e r s I t has been p o i n t e d o u t t h a t the low p r e s s u r e p i n c h i s c h a r a c t e r i z e d by the growth o f c u r r e n t i n the c e n t r a l r e g i o n o f the d i s c h a r g e v e s s e l and the e v e n t u a l f o r m a t i o n o f the boundary l a y e r . Such v a r i a t i o n s i n the b e h a v i o u r o f the c u r r e n t c a r r y i n g mechanism s h o u l d be r e f l e c t e d by o t h e r changes i n the d i s c h a r g e . One change w h i c h i s o b s e r v e d e x p e r i m e n t a l l y i s a l a r g e v o l t a g e f l u c t u a t i o n w h i c h o c c u r s when the c u r r e n t o f e n e r g e t i c e l e c t r o n s d i s a p p e a r s . A n o t h e r e f f e c t w h i c h i s o b s e r v e d i s a d r a m a t i c f l u c t u a t i o n i n the time d e r i v a t i v e o f the c u r r e n t . These a r e b o t h c o m p l i c a t e d impedance e f f e c t s w h i c h cannot be h a n d l e d i n a s i m p l e f a s h i o n . The l a r g e v o l t a g e f l u c t u a t i o n w h i c h o c c u r s i s r e l a t e d to the back e.m.f. of the d e c a y i n g m a g n e t i c f i e l d o f the a x i a l c u r r e n t w h i c h i s t r a p p e d i n the i n t e r i o r o f the v e s s e l by the newly formed boundary l a y e r . I t i s n o t the purpose o f t h i s t h e s i s to p r e s e n t a q u a n t i t a t i v e model of low p r e s s u r e breakdown i n a p i n c h . However, two p o i n t s have been made w h i c h a r e n e c e s s a r y to the u n d e r s t a n d i n g o f the o b s e r v a t i o n s of the e x p e r i m e n t . F i r s t l y , the growth o f i o n i z a t i o n depends on the s p a t i a l v a r i a t i o n o f <*. t h r o u g h i t s r e l a t i o n s h i p t o the e l e c t r i c f i e l d v a r i a t i o n . Thus f o r low p r e s s u r e d e v i c e s , i o n i z a t i o n can grow more r a p i d l y i n the c e n t r a l r e g i o n o f the d i s c h a r g e v e s s e l t h a n i t does i n the w a l l r e g i o n where the f i e l d i s i n i t i a l l y h i g h e s t . O b s e r v a t i o n s o f c u r r e n t growth a t the anode show the growth o f c u r r e n t to be i n i t i a l l y a t the c e n t e r o f the e l e c t r o d e . S e c o n d l y , the boundary l a y e r e v e n t u a l l y forms because o f the growth o f c o n d u c t i v i t y and the enhancement o f cC i n the w a l l r e g i o n . When the boundary l a y e r forms and s h i e l d i n g o f the i n t e r i o r o f the v e s s e l i s compl e t e , the c u r r e n t i n the c e n t e r o f the v e s s e l d i s a p p e a r s . E x p e r i m e n t a l o b s e r v a t i o n s s u p p o r t t h i s p o i n t o f v i e w . CHAPTER I I I ELECTRON MOTION I N A PARTIALLY IONIZED GAS 3:1 I n t r o d u c t i o n I n t h i s c h a p t e r , the p r o b l e m o f the e l e c t r o n c u r r e n t i n c i d e n t on the anode of the d i s c h a r g e v e s s e l w i l l be c o n s i d e r e d . The d i s c u s s i o n w i l l d e a l w i t h the e l e c t r o n c u r r e n t d u r i n g the time when the c u r r e n t i n the c e n t r a l r e g i o n of the d i s c h a r g e i s g r o w i n g and w i l l i n c l u d e the p e r i o d of f o r m a t i o n of the boundary l a y e r . C o n c l u s i o n s c o n c e r n i n g the c u r r e n t i n the c e n t r a l r e g i o n of the p i n c h v e s s e l a r e based on the measurement of the e l e c t r o n v e l o c i t y d i s -t r i b u t i o n a t the c e n t e r of the anode of the p i n c h v e s s e l . The i n t e r a c t i o n o f the e l e c t r o n s , b o t h w i t h t h e m s e l v e s and w i t h o t h e r s p e c i e s i n the gas, w i l l be examined. From the measured v e l o c i t y d i s t r i b u t i o n , b o t h t h e e l e c t r o n t e m p e r a t u r e and d e n s i t y can be c a l c u l a t e d . The w o r k i n g gas i n t h i s e x p e r i m e n t has m o s t l y been hydrogen, though some work has been c a r r i e d out w i t h b o t h h e l i u m and a r g o n . Hydrogen has been used most e x t e n s i v e l y because i t produced a more s u b s t a n t i a l c u r r e n t o f f a s t e l e c t r o n s than the o t h e r gases f o r i d e n t i c a l c o n d i t i o n s . The e a r l y s t a t e of i o n i z a t i o n i s i n r a p i d t r a n s i t i o n . A t the b e g i n n i n g o f the d i s c h a r g e sequence, the f i e l d s t r e n g t h i n the gas i s v e r y h i g h . I n i t i a l l y , a v e r y l a r g e amount o f e n e r g y i s i m p a r t e d t o the e l e c t r o n s and c o n s i d e r a b l e a n i s o t r o p y d e v e l o p s i n the v e l o c i t y d i s t r i b u t i o n . One can d e m o n s t r a t e t h i s by e x a m i n i n g the e l e c t r o n mean f r e e p a t h i n the p r e s e n c e of a s t r o n g e l e c t r i c f i e l d , and by n o t i n g t h a t a f a s t e l e c t r o n has an e x c e l l e n t chance of p a s s i n g f r o m one e l e c t r o d e t o the o t h e r w i t h -16 o u t a c o l l i s i o n (see A p p e n d i x D). However, as i o n i z a t i o n p r o c e e d s , the f i e l d w i l l be r e d u c e d because of space charge e f f e c t s , t h e s k i n e f f e c t , and impedance phenomena a s s o c i a t e d w i t h the d i s c h a r g e c i r c u i t . The e l e c t r o n gas f i n d s i t s e l f i n a f i e l d w h i c h i s d e c r e a s i n g a t a r a t e w h i c h i s s l o w compared to the r e l a x a t i o n time o f the e l e c t r o n s . The e l e c t r o n d i s t r i b u t i o n w i l l r e l a x t o a q u a s i - e q u i l i b r i u m s t a t e , the measurement o f w h i c h w i l l a l l o w us t o form some c o n c l u s i o n s c o n c e r n i n g the p r o p e r t i e s o f the i o n i z e d gas. A t the time when the e l e c t r o n v e l o c i t y d i s t r i b u t i o n i s measured, i t i s n e c e s s a r y t o have the e l e c t r i c f i e l d much r e d u c e d f r o m i t s i n i t i a l v a l u e . T h i s c o n d i t i o n must e x i s t so t h a t the e l e c t r o n m o t i o n w i l l be dominated by c o l l i s i o n p r o c e s s e s . I n the e x p e r i m e n t , the e l e c t r o n c u r r e n t t o the anode i s examined. A F a r a d a y cup and r e t a r d i n g g r i d s y s t e m are used t o measure the energy of the e l e c t r o n s and a s i m p l e probe i s used t o measure the c u r r e n t d e n s i t y . From t h e s e measurements, i t i s p o s s i b l e t o c a l c u l a t e b o t h the t e m p e r a t u r e and d e n s i t y of the e l e c t r o n s i f the v e l o c i t y d i s t r i b u t i o n , f ( c ) , i s known. F o r t h i s r e a s o n , the B o l t z m a n n e q u a t i o n and the two d i s t r i b u t i o n s w h i c h most l i k e l y d e s c r i b e the m o t i o n of the e l e c t r o n s i n the d i s c h a r g e a r e c o n s i d e r e d . An i m p o r t a n t a s s u m p t i o n i n the d e r i v a t i o n o f f ( c ) i s based on the concept o f c o l l i s i o n f r e q u e n c y . S i m p l y s t a t e d , the c o l l i s i o n term o f the Boltzmann e q u a t i o n i s dominated by t h a t t y p e o f c o l l i s i o n p r o c e s s w h i c h i s the most f r e q u e n t . T h i s i s a common technique,, By making t h i s a s s u m p t i o n , one g e t s the d i s p l a c e d M a x w e l l i a n i n S e c t i o n 3:3 f o r w h i c h the e l e c t r o n - e l e c t r o n c o l l i s i o n s dominate, and the p e r t u r b e d M a x w e l l i a n i n S e c t i o n 3:4, f o r w h i c h the e l e c t r o n - n e u t r a l c o l l i s i o n s dominate. These d i s t r i b u t i o n s have been p r e v i o u s l y d e r i v e d and i n v e s t i g a t e d by 18 D r e i c e r (1960) and Tanenbaum ( 1 9 6 7 ) . The d i s p l a c e d M a x w e l l i a n d e s c r i b e s an e l e c t r o n f l u i d w h i c h has a s u b s t a n t i a l d r i f t w i t h r e s p e c t t o an i o n - n e u t r a l f l u i d w h i c h i s a t r e s t . The d r i f t a c c o u n t s f o r the c o n d u c t i o n o f current„ The p e r t u r b e d M a x w e l l i a n d e s c r i b e s a s p h e r i c a l l y s y m m e t r i c M a x w e l l i a n d i s t r i b u t i o n w i t h a s m a l l p e r t u r b a t i o n i n the d i r e c t i o n of the e l e c t r i c f i e l d . The p e r -t u r b a t i o n r e p r e s e n t s the c u r r e n t c o n d u c t i o n . By c a l c u l a t i n g the c o l l i s i o n f r e q u e n c i e s f r o m e x p e r i m e n t a l e v i d e n c e , and by e x a m i n i n g t h e f i t o f e x p e r i m e n t a l p o i n t s t o t h e o r e t i c a l d i s t r i b u -t i o n s , the form of f ( c ) can be chosen. I t e v e n t u a t e s t h a t the e l e c t r o n -e l e c t r o n c o l l i s i o n r e l a x a t i o n time i s much too l o n g t o g i v e r i s e t o a d i s p l a c e d M a x w e l l i a n , b u t the e l e c t r o n - n e u t r a l c o l l i s i o n r e l a x a t i o n time i s l e s s than 10 nanoseconds. Thus, i t appears as though f ( c ) has the f o r m of a p e r t u r b e d M a x w e l l i a n . I n S e c t i o n 3:5 the p r o b l e m of d e d u c i n g plasma parameters f r o m the energy s p e c t r u m of the e l e c t r o n s as t h e y pass t h r o u g h a s m a l l h o l e i n the anode i s d i s c u s s e d . From the b a s i c a s s u m p t i o n t h a t f ( c ) i s e i t h e r a d i s p l a c e d o r a p e r t u r b e d M a x w e l l i a n , t h e method of c a l c u l a t i n g the tempera-t u r e and the d e n s i t y of the e l e c t r o n s i s d i s c u s s e d . The f i n a l s e c t i o n examines the r e l a x a t i o n time i n t h e plasma. The comments a r e based on the Krook's r e l a x a t i o n model (see Tanenbaum, 1967). These c o n s i d e r a t i o n s a r e i m p o r t a n t because the c a l c u l a t i o n s a r e based on the a s s u m p t i o n t h a t f ( c ) i s i n the s t e a d y s t a t e . T h i s r e q u i r e s t h a t r e l a x a t i o n times be much l e s s than the time i n w h i c h the e l e c t r i c f i e l d changes. 3:2 The Boltzmann E q u a t i o n and the C o l l i s i o n Term I n t h i s s e c t i o n , we w i l l examine the B o l t z m a n n e q u a t i o n as i t i s 19 to be a p p l i e d to the c o n d i t i o n s o f the p i n c h v e s s e l . A c c o r d i n g t o Chapman and C o w l i n g ( 1 9 6 1 ) , a s e p a r a t e B o l t z m a n n e q u a t i o n can be w r i t t e n f o r each component o f a mu l t i c o m p o n e n t gas m i x t u r e . F o r each component, »t, o f a t h r e e component m i x t u r e o f e l e c t r o n s , i o n s and n e u t r a l s , t he e q u a t i o n i s (3:2-1) I n t h i s e x p r e s s i o n , f r t i s the d i s t r i b u t i o n f u n c t i o n , c i s the p a r t i c l e v e l o c i t y , q<x i s t h e cha r g e , E i s the e l e c t r i c f i e l d and B i s the m a g n e t i c f i e l d . The term on the r i g h t r e p r e s e n t s the c o l l i s i o n p r o c e s s e s and i s to be summed o v e r the t h r e e t y p e s o f p a r t i c l e s assumed to be p r e s e n t i n p a r t i a l l y i o n i z e d hydrogen, namely R^, IT*", and e l e c t r o n s ( n e g l e c t i n g H*, H^, H ) + . I n w r i t i n g e q u a t i o n 3:2-1 we have made the f o l l o w i n g a s s u m p t i o n s . (a) The p a r t i c l e mean f r e e p a t h i s much l e s s t h a n the v e s s e l d i m e n s i o n s . I n the d i s c h a r g e v e s s e l , t he e l e c t r o d e s e p a r a t i o n i s about 5 o r more tim e s the most p r o b a b l e mean f r e e p a t h . (b) The e l e c t r i c f i e l d i s e s s e n t i a l l y c o n s t a n t d u r i n g a mean f r e e c o l l i s i o n t i m e . T h i s s t a t e m e n t i s v a l i d s i n c e the mean f r e e c o l l i s i o n time i s a few nanoseconds compared to f l u c t u a t i o n times o f hundreds o f nanoseconds. We s h a l l make the a d d i t i o n a l a s s u m p t i o n s . (c) The gas m i x t u r e i s homogeneous so t h a t everywhere di^/dr = 0. I n the c e n t r a l r e g i o n o f the v e s s e l t h i s s t a t e m e n t s h o u l d be a p p r o x i m a t e l y t r u e . + H*, H"2 and H a r e n e g l e c t e d because o f t h e i r low d e n s i t y i n t h e gas. T h e i r c r o s s s e c t i o n s a r e l e s s t h a n o r comparable t o t h o s e w h i c h a r e d i s -c ussed i n t h i s s e c t i o n . 20 (d) The e n e r g y t h a t a p a r t i c l e g a i n s between c o l l i s i o n s i s s m a l l compared to the average energy o f the p a r t i c l e s . T h i s s t a t e m e n t i s v a l i d once the e l e c t r i c f i e l d has decayed s u f f i c i e n t l y . (e) The e l e c t r i c f i e l d s t r e n g t h i s i n d e p e n d e n t o f p o s i t i o n . T h i s i s of c o u r s e n o t t r u e . We must assume t h a t the e l e c t r i c f i e l d i s s u f f i c i e n t l y r e l a x e d so t h a t s p a t i a l v a r i a t i o n s a r e n o t i m p o r t a n t t o the average m o t i o n o f the p a r t i c l e s . ( f ) The m a g n e t i c f i e l d can be n e g l e c t e d . I n the c e n t r a l r e g i o n of the d i s c h a r g e t h i s a s s u m p t i o n can be made b u t f o r l a r g e c u r r e n t s t h e a s s u m p t i o n w i l l b r e a k down„ W i t h the use of a s s u m p t i o n s ( c ) a n d ( f ) , e q u a t i o n 3:2-1 becomes S * f> V f St. + q«. E * = /_ — — dt /3 <-> t J (3:2-2) W i t h a s s u m p t i o n (d) c o n c e r n i n g weak f i e l d s , the f o r m of f ^ i s governed by c o l l i s i o n s . A common t e c h n i q u e (see D e l c r o i x , 1962) f o r e v a l u a t i n g the r e l a t i v e i m p o r t a n c e of the c o l l i s i o n terms i s t o examine the f r e -quency of each t y p e of c o l l i s i o n a l i n t e r a c t i o n . I f V r e p r e s e n t s the c o l l i s i o n f r e q u e n c y , the t y p e s o f c o l l i s i o n p r o c e s s w h i c h w i l l be con-s i d e r e d a r e as f o l l o w s , V - e l e c t r o n - e l e c t r o n coulomb s c a t t e r i n g ce y - e l e c t r o n - i o n coulomb s c a t t e r i n g y - e l e c t r o n - n e u t r a l e l a s t i c c o l l i s i o n s E y-j- - e l e c t r o n - n e u t r a l i o n i z i n g c o l l i s i o n s The e x c i t a t i o n c o l l i s i o n f r e q u e n c y i s about the same s i z e as y ( P e r c i v a l , 1966). I n T a b l e 3:2-1, the h i g h energy forms ( e l e c t r o n e nergy > 200 21 e l e c t r o n v o l t s ) o f the c o l l i s i o n f r e q u e n c i e s have been l i s t e d . The s o u r c e of the c o l l i s i o n c r o s s s e c t i o n u sed t o c a l c u l a t e the v a l u e s o f V have a l s o been l i s t e d . TABLE 3:2-1 COLLISION FREQUENCIES (T •> 200 eV) e Type o f C o l l i s i o n V, ce Form n I n A e 0.27 T 3/2 Source S p i t z e r (1962) V i 2 x 1 0 " 7 N. 1.9 x l < f ' -7 logOrrfe) Ni !H 2 D e l c r o i x (1964) P e r c i v a l (1966) 250 x 10 •7 1 3/2 1NH+ D e l c r o i x (1964) I n T a b l e 3:2-1, h g i s the e l e c t r o n d e n s i t y , A i s the r a t i o o f the Debye l e n g t h t o the av e r a g e i m p a c t parameter w h i c h r e s u l t s i n a 90 C d e f l e c t i o n , T g i s the e l e c t r o n t e m p e r a t u r e i n degrees K e l v i n , N^ and N + a r e r e s p e c t i v e l y the n e u t r a l d e n s i t y and t h e i o n d e n s i t y . The H e l e c t r o n - n e u t r a l and t h e e l e c t r o n - i o n c o l l i s i o n f r e q u e n c i e s can be summarized by the f o l l o w i n g e x p r e s s i o n : V E = V T : y = 1 : 0.95 l o B \ l . ^ . x 25 ,3/2 (3:2-3) V V~ where k i s the r a t i o o f the i o n d e n s i t y t o the n e u t r a l d e n s i t y . I n the f o l l o w i n g s e c t i o n s , two type s o f e l e c t r o n d i s t r i b u t i o n w i l l be d i s c u s s e d . T h e i r f o r m w i l l depend on the r e l a t i v e v a l u e s o f y , y , V , and ce E I V • c 22 3:3 The D i s p l a c e d M a x w e l l i a n F o r t h i s model, i t i s assumed t h a t the e l e c t r o n s g a i n energy f r o m the e l e c t r i c f i e l d , r e d i s t r i b u t e t he en e r g y among themselves t h r o u g h e l e c t r o n - e l e c t r o n e n c o u n t e r s , and l o s e a p o r t i o n of t h a t e n e r g y t h r o u g h c o l l i s i o n s w i t h o t h e r s p e c i e s i n the plasma. S y m b o l i c a l l y , t h i s i s e x p r e s s e d by V Q & i>> Vg + y^. W i t h the a s s u m p t i o n t h a t the energy g a i n e d f r o m the e l e c t r i c f i e l d i n one e l e c t r o n - e l e c t r o n mean f r e e p a t h i s much l e s s t h a n the average energy exchange i n an e l e c t r o n - e l e c t r o n c o l l i s i o n , e q u a t i o n 3:2-2 can be w r i t t e n as 6 5 t f & f > e . 5 t , (3:3-1) I n a r e f e r e n c e frame w h i c h moves w i t h the e l e c t r o n f l u i d , the s o l u t i o n o f t h i s e q u a t i o n i s a M a x w e l l i a n . I f the d r i f t v e l o c i t y o f the e l e c t r o n c l o u d i s U ( t ) , t h e n exp f e ( c , t ) 3/2 kT„ (3:3-2) m T h i s e x p r e s s i o n f o r can be fo u n d i n D r e i c e r ( 1 9 5 9 ) . I f f i s p u t i n t o e q u a t i o n 3:2-2, m u l t i p l i e d by m^c and i n t e g r a t e d o v e r v e l o c i t y space, then one has the f i r s t moment e q u a t i o n g i v e n by D r e i c e r . dU m ~7l = e e d t E - I E Y (T ,U) cp 1 p v e' (3:3-3) Some of the p r o p e r t i e s o f t h i s e q u a t i o n have been r e v i e w e d by S m i t h (1964) The term under the summation r e p r e s e n t s the v i s c o u s d r a g e x p e r i e n c e d by the e l e c t r o n c l o u d due to i t s e n c o u n t e r s w i t h h e a v i e r s p e c i e s . E Y cp p i s a f u n c t i o n o f the c o l l i s i o n c r o s s s e c t i o n , 6, w h i c h depends on the 23 r e l a t i v e v e l o c i t y , g, o f the c o l l i d i n g p a i r . S m i th has d i s c u s s e d what happens when the c r o s s s e c t i o n s have the f o l l o w i n g form. 6 * g p (3:3-4) Here, p i s an i n t e g e r . T a b l e 3:3-1 shows the r e l a t i o n s h i p between the v a l u e o f p and the c o l l i s i o n p r o c e s s e s w h i c h have been d i s c u s s e d . TABLE 3:3-1 INTERPRETATION OF p V a l u e of p 0 -1 -2 -4 Type o f I n t e r a c t i o n c o n s t a n t c r o s s s e c t i o n ( h a r d s p h e r e ) M a x w e l l i a n m o l e c u l e ( Vg) I o n i z a t i o n ( h i g h energy, Vj) Coulomb s c a t t e r i n g ( y c ) What Happens? T e r m i n a l d r i f t v e l o c i t y T e r m i n a l d r i f t v e l o c i t y Runaway ( l a r g e E) Runaway ( l a r g e E) F i g u r e 3:3-1 shows the s i z e o f Y as a f u n c t i o n o f U and has been t a k e n f r o m Smith. 24 F o r h a r d s p h e r e s and M a x w e l l i a n m o l e c u l e s , the v i s c o u s d r a g term w i l l i n c r e a s e u n t i l the energy g a i n from the e l e c t r i c f i e l d i s j u s t e q u a l l e d by the energy l o s t t h r o u g h c o l l i s i o n s . However, f o r i o n i z i n g c o l l i s i o n s and coulomb s c a t t e r i n g , the v i s c o u s d r a g term r e a c h e s a maximum v a l u e . I n such c a s e s , a s u f f i c i e n t l y s t r o n g e l e c t r i c f i e l d w i l l i m p a r t more energy to the e l e c t r o n s t h a n they l o s e t h r o u g h c o l l i s i o n s , These e l e c t r o n s w i l l c o n t i n u e t o a c c e l e r a t e and a r e s a i d to r u n away. I f i t i s assumed t h a t t h e p = -1 i n t e r a c t i o n i s dominant among the e l e c t r o n - h e a v y p a r t i c l e i n t e r a c t i o n s , t h e n e q u a t i o n 3:2-6 can be w r i t t e n ft - £ [ E - » ) (3:3-5, I t t u r n s out t h a t K i s the r e c i p r o c a l o f the m o b i l i t y . The s o l u t i o n o f e q u a t i o n 3:3-5 i s U = | [ l - e x p ( - K ^ t ) ] (3:3-6) The concept o f t e r m i n a l d r i f t v e l o c i t y i n a c o n s t a n t e l e c t r i c f i e l d i m p l i e s the d e f i n i t i o n of m o b i l i t y , yu e> as U(« ) = E (3:3-7) As p o i n t e d o u t by D e l c r o i x ( 1 9 6 4 ) , the m o b i l i t y i s r e l a t e d to the c o l l i s i o n f r e q u e n c y by M — 1 6 m e ' V E • < 3 : 3- 8) Thus e q u a t i o n 3:3-6 becomes U ( t ) = y u , E [ l - e x p ( - y g t ) ] (3:3-9) The e l e c t r o n c l o u d w i l l r e a c h a t e r m i n a l d r i f t v e l o c i t y i n a few mean f r e e c o l l i s i o n t i m e s . The v e l o c i t y d i s t r i b u t i o n i s t h e r m a l i z e d by 25 e l e c t r o n - e l e c t r o n e n c o u n t e r s and the d r i f t i s governed by e l e c t r o n -n e u t r a l e n c o u n t e r s . The c u r r e n t d e n s i t y , J , i n such a s i t u a t i o n i s g i v e n by J = n e u e 2 = n 6 .. E (3:3-10) 6 m e E 3:4 The P e r t u r b e d M a x w e l l i a n The p e r t u r b e d M a x w e l l i a n d e s c r i b e s a t y p e o f m o t i o n w h i c h i s c o n s i d e r a b l y d i f f e r e n t from the d r i f t i n g m o t i o n d i s c u s s e d i n the p r e v i o u s s e c t i o n . The p r o p e r t i e s w h i c h dominate the p e r t u r b e d M a x w e l l i a n w i l l be examined as a means of u n d e r s t a n d i n g the t y p e of m o t i o n w h i c h i t d e s c r i b e s , F o r t h i s d i s t r i b u t i o n , i t i s assumed t h a t the c o l l i s i o n s a r e p r e -d o m i n a n t l y o f the e l e c t r o n - n e u t r a l e l a s t i c t y p e . F o r such a s i t u a t i o n ' ^E ^ce' ^i°r V:' D u r i n g the c o l l i s i o n o f a l i g h t e l e c t r o n w i t h a heavy n e u t r a l , the f r a c t i o n a l energy exchange i s v e r y s m a l l b u t the change i n momentum may be v e r y h i g h . I n a d d i t i o n , the average g a i n o f e n e r g y from the e l e c t r i c f i e l d i s assumed t o be j u s t e q u a l t o the ave r a g e e n e r g y l o s t by an e l e c t r o n i n a c o l l i s i o n . That i s , d f e / 3 t i s assumed to be z e r o . These a s s u m p t i o n s a r e v a l i d p r o v i d i n g t h e f i e l d has r e l a x e d t o a s u f f i c i e n t l y low v a l u e . F o l l o w i n g Tanenbaum ( 1 9 6 7 ) , e q u a t i o n 3:2-2 can be w r i t t e n as e E, c» f k e < S f d c k U t } e (3:4-1) The n o t a t i o n has been changed t o conform w i t h the n o t a t i o n used by Tanenbaum. The s u b s c r i p t s , i , j and k, r e f e r t o the t h r e e d i r e c t i o n s i n c o o r d i n a t e space. f can be expanded i n s p h e r i c a l h a r m o n i c s such t h a t 26 c. i c 1 c. c f j c ) = f j c ) - f - ^ f ^ c ) + f ( c ) c 1 J (3:4-2) I n t h i s e q u a t i o n , f Q ( c ) r e p r e s e n t s a s p h e r i c a l l y s ymmetric d i s t r i b u t i o n and the p e r t u r b a t i o n s r e p r e s e n t p r o j e c t i o n s i n the d i r e c t i o n o f the e l e c t r i c f i e l d . The s o l u t i o n of e q u a t i o n 3:4-1 r e q u i r e s t h a t the n e u t r a l s be r e p r e s e n t e d by a M a x w e l l i a n o f t e m p e r a t u r e T^. The c o l l i s i o n term i s t r e a t e d i n d e t a i l by Tanenbaum and the r e s u l t s a r e : e l bf f . = — - 5 E . j m c j a c 9 ^ f Y2 E - r — — 3m c f A 2 d f e o 3 L o M / l g ~ 2 A-g 6C (3:4-3) /) (c) i s the mean f r e e p a t h f o r e l e c t r o n - n e u t r a l c o l l i s i o n s and M i s the E m o l e c u l a r mass. A and Y a r e g i v e n by 2KT r M and E e m„ (3:4-4) The f i n a l form o f f ( c ) i s f = D exp o c'dc' m { ^ — 2 — 3c (3:4-5) F o r e l a s t i c c o l l i s i o n s , X = c/ V N o r m a l i z a t i o n o f f y i e l d s E E o exp m n 3 V 2 E •%m c e KT + H i L n „ s ;2 3 V E' (3:4-6) The d i s t r i b u t i o n i s c l e a r l y M a x w e l l i a n w i t h an e f f e c t i v e t e m p e r a t u r e * 1 M H T = T + ± ^ (3:4-7) n K 3 y 2 The temperature i s a f u n c t i o n o f the e l e c t r i c f i e l d s t r e n g t h t h r o u g h Y. 27 f i s s p h e r i c a l l y s ymmetric b u t i s n o t . From e q u a t i o n 3:4-3, O n m c _£ e e  y „ 3/2 3/2 e exp •%m c e KT* (3:4-8) The c o m b i n a t i o n o f f Q and f i n e q u a t i o n 3:2-2 r e p r e s e n t s an e l e c t r o n gas o f h i g h t e m p e r a t u r e whose v e l o c i t y components a r e about e q u a l l y d i s -t r i b u t e d i n a l l d i r e c t i o n s . T h i s d i s t r i b u t i o n f u n c t i o n w i l l be r e f e r r e d to as the p e r t u r b e d M a x w e l l i a n . I t i s made up of two p a r t s , a s p h e r i c a l l s ymmetric p a r t , f ^ , and a nonsymmetric p a r t , f w h i c h i s o r i e n t e d i n the d i r e c t i o n o f the e l e c t r i c f i e l d . f ' c f e ( c ) m > e_ KT* y n 27TKT* 3 / 2 exp -%m c e KT (3:4-9) m The c u r r e n t d e n s i t y i n the i o n i z e d gas i s r e p r e s e n t e d by the p e r t u r b a t i o n f ^ , t h r o u g h 47Te ( 3, . — 1 C f k d c = e n e n v , e d (3:4-10) The term, Y/V„, r e p r e s e n t s a s m a l l d r i f t , v , i n the d i r e c t i o n o f the E d f i e l d . The r a t i o o f f, t o f s h o u l d be s m a l l . I t i s g i v e n by k o 3m e M c v , (3:4-11) The two d i s t r i b u t i o n f u n c t i o n s w h i c h have been d i s c u s s e d h e r e , and 28 i n the l a s t s e c t i o n , d e s c r i b e q u i t e d i f f e r e n t t y p e s o f m o t i o n . The d i s -p l a c e d M a x w e l l i a n r e p r e s e n t s a p r e d o m i n a n t l y s t r e a m i n g m o t i o n , w h i l e the p e r t u r b e d M a x w e l l i a n r e p r e s e n t s an e l e c t r o n gas w h i c h i s e s s e n t i a l l y a t r e s t w i t h a M a x w e l l i a n d i s t r i b u t i o n , b u t w h i c h has a s m a l l component o f d r i f t . 3:5 Measurements Made w i t h a R e t a r d i n g G r i d System I n t h i s s e c t i o n , assuming t h a t f ( c ) i s known, the method of c a l -c u l a t i n g the e l e c t r o n t e m p e r a t u r e , T e, and the e l e c t r o n d e n s i t y , n^, f r o m F a r a d a y cup measurements w i l l be d e m o n s t r a t e d . The r e t a r d i n g g r i d p r o v i d e s a s i m p l e method f o r o p e r a t i n g on the v e l o c i t y d i s t r i b u t i o n f u n c t i o n w i t h r e s p e c t t o one of i t s v e l o c i t y components. I n the p r e s e n t e x p e r i m e n t , the g r i d s y s t e m i s l o c a t e d b e h i n d the s m a l l h o l e , of a r e a A^, a t the c e n t e r o f the anode o f the Z - p i n c h . A s t r e a m of e l e c t r o n s p a s s e s through the h o l e and i s p a r t i a l l y r e t a r d e d by the g r i d s y s t e m h e l d a t a n e g a t i v e v o l t a g e , V . The k i n e t i c energy o f the e l e c t r o n s can be r e p r e s e n t e d i n e l e c t r o n v o l t s , V', by 2 m c ...e = e V (3:5-1) 2 Only" t h o s e e l e c t r o n s w h i c h have k i n e t i c e n e r g y V' > V w i l l pass t h r o u g h the g r i d to be c o l l e c t e d by the F a r a d a y cup. I n a d d i t i o n , i t i s assumed t h a t the g r i d has a c e r t a i n t r a n s m i s s i o n , Q ( c ) , w h i c h i s l e s s t h a n u n i t y due to s c a t t e r i n g o f e l e c t r o n s o u t of t h e system by c o l l i s i o n s w i t h m o l e c u l e s , g e o m e t r i c a l a p e r t u r e s i z e and e l e c t r o s t a t i c s p r e a d i n g o f the s t r e a m (see A p p e n d i x A ) . The c u r r e n t r e c e i v e d by the cup, I c ( c ) , can-be w r i t t e n as r I c ( c ) = n g e \ Q(c') c ' f ( c , ) d c ' (3:5-2) c 29 The g r i d v o l t a g e i s r e l a t e d to the e l e c t r o n v e l o c i t y by e q u a t i o n 3:5-1. F i g u r e 3:5-1 F a r a d a y Cup and I ( c ) I f the form o f I c(c) i s known, the n i t i s p o s s i b l e to c a l c u l a t e the form of f ( c ) . T h i s can be done by the t e c h n i q u e o f u s i n g the t h e o r e -t i c a l f orm of f ( c ) i n e q u a t i o n 3:5-2 and f i t t i n g the r e s u l t of the i n t e g r a t i o n to the e x p e r i m e n t a l c u r v e I c ( c ) . The d i s t r i b u t i o n can a l s o be f o u n d by d i f f e r e n t i a t i n g the e x p e r i m e n t a l c u r v e 1 ( c ) , and making use of the f o l l o w i n g e x p r e s s i o n . I n S e c t i o n s 3:3 and 3:4, t h e p r o p e r t i e s o f b o t h the d i s p l a c e d and the p e r t u r b e d M a x w e l l i a n were d i s c u s s e d . I n the f o l l o w i n g s e c t i o n s , the f l u x of e l e c t r o n s to the c o l l e c t i n g cup f o r each o f t h e s e d i s t r i b u t i o n s w i l l be c a l c u l a t e d . 3:6 The R e t a r d i n g G r i d System and the D i s p l a c e d M a x w e l l i a n The d i s p l a c e d M a x w e l l i a n can be used i n c o n j u n c t i o n w i t h the 30 Faraday cup signal to calculate T g and n^ i n the ionized gas. The grid system u t i l i z e d has a c o l l e c t i o n angle of 28°, so that a l l of the electrons entering the system do so at an angle of 14° or less to the axis. The exponential term i n equation 3:3-2 can be written as 2 . , „ v2il exp ^ e r ^ . m 2 V Grid Figure 3:6-1 Geometrical C o l l e c t i n g Angle The condition which i s necessary for a p a r t i c l e at a point, p, i n the c o l l e c t i o n cone to pass through the hole i s — = tan -6- , - 14° c ' i s the angle which the v e l o c i t y vector of an electron heading for the hole makes Athe axis of symmetry. Since tan -©• ^ 0.06 for a l l points within the cone, the exponential can be approximated by 2s! exp -^ m kT ( c z - u y Instead of equation 3:3-2 for f e ( c ) , i t i s convenient to use 31 f ( c z ) = £ e x p k T ( c 2 " u ) (3:6-1) m e From e q u a t i o n 3:5-2, the c u r r e n t r e c e i v e d by the c o l l e c t i n g cup i s g i v e n by I c ( c ) = ^ L ^ e ( c z f ( c z ) d c z (3:6-2) -"c where L r e p r e s e n t s a l l t y p e s of l o s s e s b u t i s m a i n l y a g e o m e t r i c a l f a c t o r r e l a t e d t o the a p e r t u r e of the system. F o r the p r e s e n t d i s c u s s i o n , L i s assumed t o be i n d e p e n d e n t o f the e l e c t r o n v e l o c i t y . L i s examined i n more d e t a i l i n A p p e n d i x B. I f the t e m p e r a t u r e of the e l e c t r o n s i s much l e s s t h a n the a v e r a g e k i n e t i c e n e r g y , t h e n I c<°> = % L A h e j c f ( c ) d c z z z n e L e U (3:6-3) Suppose now t h a t the g r i d s ystem i s r e p l a c e d by a cup w h i c h c o l l e c t s a l l of the e l e c t r o n s w h i c h pass t h r o u g h t h e h o l e . The c u r r e n t , I , thus r e c e i v e d w i l l be e q u a l t o the c u r r e n t d e n s i t y i n t h e i o n i z e d gas, From e q u a t i o n 3:3-10, I_o = n g e U (3:6-4) By e q u a t i n g I ( o ) / L t o I , L can be e v a l u a t e d . B o t h U and n can be c o e c a l c u l a t e d f r o m the I (c ) c u r v e . I t i s f i r s t n o t e d t h a t c z oo f ( c ) d c = 1 (3:6-5) z z - CO and U = V c f ( c ) d c (3:6-6) N z z z S e t t i n g Q(c) = L i n e q u a t i o n 3:5-3 y i e l d s the f o l l o w i n g r e s u l t . f ( c ) = ~t — -~ I ( c ) (3:6-7) z L n A, e c dc c z e n z z And from 3:6-5, one has % = TTTI \ f -t^^z^z <3:6-8> n J z z -oo E q u a t i o n 3:6-4 can now be used t o o b t a i n U. An a l t e r n a t i v e method would be to v a r y T and U i n e q u a t i o n 3:6-1 and use the r e s u l t i n g e x p r e s -s i o n f o r f ( c z ) i n e q u a t i o n 3:6-2 t o g e t a f i t t o the e x p e r i m e n t a l p o i n t s . A l o n g w i t h e q u a t i o n 3:3-7, namely U = J^^-y t h e r e i s enough i n f o r m a t i o n to c a l c u l a t e n , U and E. e 3:7 The R e t a r d i n g G r i d System and t h e P e r t u r b e d M a x w e l l i a n I t i s now assumed t h a t the e l e c t r o n v e l o c i t y d i s t r i b u t i o n i n the i o n i z e d gas i s d e s c r i b e d by a p e r t u r b e d M a x w e l l i a n . A method of c a l c u -l a t i n g b o t h T and n i s to be d e m o n s t r a t e d . To do so, the c u r r e n t ° e e d e n s i t y a t the s u r f a c e o f the anode and t h e d i s t r i b u t i o n o f e l e c t r o n e n e r g i e s w i l l be u s e d . S i n c e the c u r r e n t a t t h e anode must e q u a l the c u r r e n t t h r o u g h the c e n t r a l r e g i o n o f the d i s c h a r g e v e s s e l , t h e n J . ^ J (3:7-1) A z v where i s the c u r r e n t d e n s i t y a t the f a c e o f the anode. From e q u a t i o n 3:4-10, J ~ n e v , (3:7-2) A e d An e x p r e s s i o n f o r the e l e c t r o n c u r r e n t r e c e i v e d by the c o l l e c t i n g cup can be w r i t t e n . The v e l o c i t y components o t h e r t h a n c^ a r e f i r s t 33 i n t e g r a t e d o u t . When t h e i n t e g r a t i o n o v e r -G- i s p e r f o r m e d , the c u r r e n t can be r e p r e s e n t e d by -co i I ( c ) = n A , L e \ c f ( c ' ) d c * (3:7-3) c z e h I z z z c z The d i s t r i b u t i o n i s r e p r e s e n t e d by 2 *r x _ J _ , °z , 2 2 M y Z f ( c ) = — — exp (- — r - ) , a = — — — r -rr^a ^ 3 m e y 2 • E As i t t u r n s o u t , the e x p l i c i t f o r m o f L i s n o t r e q u i r e d as l o n g as i t i s in d e p e n d e n t o f v e l o c i t y . L a c c o u n t s f o r the l o s s e s i n the system and the g e o m e t r i c a l i n t e g r a t i o n o v e r F o r s m a l l -O-, i t i s assumed t h a t L i s i n d e p e n d e n t o f c^ as d e s c r i b e d i n A p p e n d i x B. Arguments a r e p r e -s e n t e d t h e r e w h i c h i n d i c a t e t h a t the energy dependence of L f o r e l e c t r o n s of energy l e s s t h a n about one keV i s q u i t e s t r o n g . S i n c e the e l e c t r o n e n e r g i e s measured i n t h i s e x p e r i m e n t a r e g e n e r a l l y l a r g e r t h a n t h i s v a l u e , L i s assumed t o be i n d e p e n d e n t o f c^. I n t e g r a t i o n o f e q u a t i o n 3:7-3 y i e l d s 2 I c ( C z ) = n e L A h e exp (--!-) (3:7-4) c z e h 2 T l f 2 &l I f the v e l o c i t y i s w r i t t e n i n terms o f the ener g y , t h e n t h i s e q u a t i o n becomes T g i s the e l e c t r o n t e m p e r a t u r e e x p r e s s e d i n e l e c t r o n v o l t s . P lasma b o u n d a r i e s a r e c h a r a c t e r i z e d by the f o r m a t i o n o f a s h e a t h over w h i c h a p o t e n t i a l d i f f e r e n c e o f t e n e x i s t s (Langmuir, 1929 and Chen, 1965). I n the r e g i o n o f the anode, c o n s i d e r the f o r m a t i o n o f a n e g a t i v e s h e a t h o f t o t a l p o t e n t i a l drop v g . The p r e s e n c e o f such a s h e a t h i s e q u i v a l e n t t o p l a c i n g an a d d i t i o n a l r e t a r d i n g g r i d i n t h e system. 2 T e V + v I ( V + v ) = n I t - ^ T A / — e x p ( - -) c g s' e Ti \ \l m F V T ' e e (3:7-6) On t a k i n g the n a t u r a l l o g a r i t h m s , one has V + v I n I ( + v ) = S — - — - + I n n L A , c v g s' T e Va + v c a / 2 T e "c g s T g e H i 2^|.% W m e (3:7-7) D i f f e r e n t i a t i o n o f t h i s e q u a t i o n y i e l d s w Cln v v g + vJ - - f <3:7-8> g e Thus the t e m p e r a t u r e of a p e r t u r b e d M a x w e l l i a n i s the r e c i p r o c a l of t h e s l o p e o f the s e m i - l o g p l o t o f I (V ) . Two p o i n t s a r e e v i d e n t . F i r s t l y , c g the c a l c u l a t e d t e m p e r a t u r e i s i n d e p e n d e n t of the s h e a t h p o t e n t i a l and s e c o n d l y , i t i s i n d e p e n d e n t o f L p r o v i d i n g t h e a s s u m p t i o n s r e g a r d i n g L ar e v a l i d . The e f f e c t o f the s h e a t h p o t e n t i a l i s to d e f l e c t the low energy e l e c t r o n s away from the anode r e g i o n . The d r i f t v e l o c i t y , v^, i s r e l a t e d t o t h e e l e c t r o n t e m p e r a t u r e by e q u a t i o n 3:4-7. The e q u a t i o n can be r e a r r a n g e d such t h a t 3 e T v d = ^ j - ^ (3:7-9) I n t h i s e x p r e s s i o n , T has been n e g l e c t e d . T h i s e q u a t i o n can be com-n b i n e d w i t h e q u a t i o n 3:7-2 to y i e l d The c u r r e n t d e n s i t y , J i s measured w i t h a probe o r c o l l e c t i n g cup w h i c h c o l l e c t s a l l of the e l e c t r o n s w h i c h pass t h r o u g h the h o l e . 35 Thus two measurements a r e r e q u i r e d to c a l c u l a t e n . F o r T o n l y the e' e' r e l a t i v e v a l u e s o f I (V ) a r e r e q u i r e d and t h e measurement i s n o t e g a f f e c t e d by t h e p r e s e n c e o f a s h e a t h . 3:8 R e l a x a t i o n Time i n the I o n i z e d Gas I n S e c t i o n s 3:2 to 3:7, the e l e c t r o n v e l o c i t y d i s t r i b u t i o n and i t s r e l a t i o n s h i p t o the f l u x o f e l e c t r o n s t h r o u g h the a n a l y z i n g s y s t e m were d i s c u s s e d . I n t h i s s e c t i o n , the pr o b l e m of t r a n s i e n t c o n d i t i o n s i n the d i s c h a r g e v e s s e l and t h e i r e f f e c t on the e l e c t r o n s w i l l be c o n s i d e r e d . To do t h i s , t he Krook's r e l a x a t i o n model ( s e e Tannenbaum, 1967) w i l l be used. The s i m p l e s t f o r m of t h e c o l l i s i o n term i n the B o l t z m a n n e q u a t i o n i s g i v e n by [ St = V ( f - f Q ) (3:8-1) c I n the absence o f e x t e r n a l f o r c e s , e q u a t i o n 3:2-1 can be w r i t t e n 5 f = \> ( f - f ) (3:8-2) T t The s o l u t i o n o f t h i s e q u a t i o n i s f ( c , t ) - f Q = [ f ( c , o ) - f Q ] e x p ( - V t ) (3:8-3) The d i s t r i b u t i o n f u n c t i o n decays to f w i t h a c h a r a c t e r i s t i c time V 1 o where V i s a c o l l i s i o n f r e q u e n c y . F o r a h o t e l e c t r o n gas i n the background o f n e u t r a l s , t h e r e w i l l be two c h a r a c t e r i s t i c r e l a x a t i o n t i m e s i f the d i s t r i b u t i o n i s r e p r e s e n t e d by a p e r t u r b e d M a x w e l l i a n . I n t h i s c a s e , the c o l l i s i o n s a r e p r e d o m i n a n t l y e l a s t i c e l e c t r o n - n e u t r a l e n c o u n t e r s . 36 R e l a x a t i o n time f o r v e l o c i t y 1/V R e l a x a t i o n time f o r energy The d i f f e r e n c e , as p o i n t e d o u t by Tanenbaum, i s due to t h e e l e c t r o n -n e u t r a l c o l l i s i o n s r e s u l t i n g i n l a r g e d e f l e c t i o n s (r~~~ ^ 1) b u t o n l y A C i n s m a l l changes o f energy ( — — <3< 1 ) . Thus the e l e c t r o n s w i l l e s t a b l i s h an e q u i l i b r i u m v e l o c i t y d i s t r i b u t i o n v e r y q u i c k l y , b u t w i l l r e t a i n t h e i r e nergy f o r a c o n s i d e r a b l y l o n g e r t i m e . S i n c e the tempera-t u r e does n o t r e l a x q u i c k l y by t h i s method, the measurement of T could e n o t r e f l e c t the s t r e n g t h o f the e l e c t r i c f i e l d i f t h i s were the o n l y p r o c e s s w h i c h was i m p o r t a n t . F o r a d i s c h a r g e o f the type c o n s i d e r e d h e r e , i n w h i c h X„ i s comparable t o the v e s s e l d i m e n s i o n s , t h e f a s t e l e c t r o n s can be e x p e c t e d t o d i f f u s e o u t o f the s y s t e m i n a few mean f r e e c o l l i s i o n t i m e s . So, p r o v i d i n g the d i f f u s i o n time i s s m a l l compared to the f l u c t u a t i o n time o f the f i e l d , t he t e m p e r a t u r e o f the e l e c t r o n gas w i l l be i n d i c a t i v e o f the e l e c t r i c f i e l d s t r e n g t h . F o r the d i s p l a c e d M a x w e l l i a n , the c h a r a c t e r i s t i c time f o r b o t h energy and v e l o c i t y r e l a x a t i o n w i l l be governed by the e l e c t r o n - e l e c t r o n c o l l i s i o n f r e q u e n c y . The d r i f t v e l o c i t y w i l l be c o n t r o l l e d by e l e c t r o n -n e u t r a l c o l l i s i o n s . Changes i n the d r i f t energy w i l l be c o n v e r t e d to random t h e r m a l energy o f the e l e c t r o n s . The d r i f t v e l o c i t y w i l l be i n d i c a t i v e o f the e l e c t r i c f i e l d s t r e n g t h , p r o v i d i n g t h e f i e l d v a r i a t i o n i s s l o w compared to 1/V . E T h i s c h a p t e r has i n t r o d u c e d b o t h the d i s p l a c e d M a x w e l l i a n ( e l e c t r o n -e l e c t r o n c o l l i s i o n s dominate) and the p e r t u r b e d M a x w e l l i a n ( e l e c t r o n -n e u t r a l e l a s t i c c o l l i s i o n s d o m i n a t e ) . On t h e b a s i s o f a r e l a x a t i o n time w h i c h i s much l e s s t h a n the time of v a r i a t i o n o f the e l e c t r i c f i e l d , i t 37 has been assumed t h a t the e l e c t r o n v e l o c i t y d i s t r i b u t i o n w i l l r e l a x to a q u a s i - e q u i l i b r i u m s t a t e r e p r e s e n t e d by one o f t h e s e d i s t r i b u t i o n s . Under such c i r c u m s t a n c e s , b o t h and n^ can be c a l c u l a t e d f r o m the F a r a d a y cup and c u r r e n t d e n s i t y measurements. Once I (V ) , n and T have been c a l -J cs g ' e e c u l a t e d , i t w i l l be p o s s i b l e t o examine the c o n s i s t e n c y o f the r e s u l t s w i t h the a s s u m p t i o n t h a t f ( c ) has a p a r t i c u l a r f o r m . From n^ and T , the c o l l i s i o n f r e q u e n c i e s can be c a l c u l a t e d . F o r a p e r t u r b e d M a x w e l l i a n , VU » \J and f o r a d i s p l a c e d M a x w e l l i a n , V w i l l d o minate. B o t h yE vce ce d i s t r i b u t i o n s w i l l y i e l d s t r a i g h t l i n e s on s e m i - l o g p l o t s , though f o r l a r g e d r i f t v e l o c i t i e s , t h e d i s p l a c e d M a x w e l l i a n w i l l e x h i b i t a s h i f t away from the o r i g i n . T h e r e f o r e , the s t r a i g h t l i n e f i t i s n o t enough to d i s t i n g u i s h between the two d i s t r i b u t i o n s . However, because o f the w i d e l y d i f f e r e n t d r i f t v e l o c i t i e s , the e l e c t r o n d e n s i t y c a l c u l a t i o n w i l l r e s u l t i n w i d e l y d i f f e r e n t v a l u e s . CHAPTER I V THE EXPERIMENTAL APPARATUS 4:1 I n t r o d u c t i o n T h i s e x p e r i m e n t has been p e r f o r m e d on a low ener g y , h i g h v o l t a g e Z - p i n c h . The major i n t e r e s t has been i n m e a s u r i n g the c u r r e n t and energy s p e c t r u m o f the e l e c t r o n s as t h e y appear i n the e a r l y s t a g e o f the d i s c h a r g e growth. I n o r d e r t o do t h i s , s m a l l h o l e s w i t h c u r r e n t c o l l e c t o r s were p l a c e d i n the anode w i t h two pu r p o s e s i n mind. F i r s t l y , t he e l e c t r o n energy s p e c t r u m was measured w i t h an e l e c t r o s t a t i c e n ergy a n a l y z e r p o s i t i o n e d a t the c e n t e r o f the anode. S e c o n d l y , the c u r r e n t p r o f i l e was measured w i t h probes l o c a t e d a t d i f f e r e n t r a d i a l p o s i t i o n s . The s e c t i o n of the a p p a r a t u s i n w h i c h the p r o p e r t i e s o f the e l e c t r o n s were measured w i l l be c a l l e d the m e a s u r i n g chamber. D u r i n g t h e o p e r a t i o n o f the p i n c h , the m e a s u r i n g chamber was d i f f e r e n t i a l l y pumped to ensure t h a t the e l e c t r o n s would have a s u b s t a n t i a l mean f r e e p a t h . The v o l t a g e a c r o s s the d i s c h a r g e , t h e c u r r e n t i n the e x t e r n a l c i r c u i t and i t s d e r i v a t i v e have a l l been m o n i t o r e d . The v o l t a g e was measured w i t h a r e s i s t i v e , v o l t a g e d i v i d e r network, and the c u r r e n t was measured w i t h a m a g n e t i c p i c k up c o i l and i n t e g r a t i o n network. These waveforms have been u s e f u l i n d e t e r m i n i n g the dynamic p r o p e r t i e s o f the d i s c h a r g e w h i c h a f f e c t the growth and decay o f the e l e c t r o n c u r r e n t d u r i n g the f o r m a t i v e phase. An image c o n v e r t e r camera was used a l o n g w i t h the c u r r e n t and v o l t a g e waveforms t o examine the p i n c h . The purpose was t o check the u n i f o r m i t y o f the d i s c h a r g e and t o r e l a t e the p i n c h t i m e to the boundary l a y e r f o r m a t i o n . 38 39 4:2 The Z - P i n c h The p i n c h has been d e s i g n e d t o f u n c t i o n a t v o l t a g e s up t o 60 kV but has been o p e r a t e d f o r the most p a r t a t 40 kV. The par a m e t e r s o f i n t e r e s t a r e l i s t e d i n T a b l e 4:2-1 and a s c h e m a t i c o f the p i n c h a p p a r a t u s appears i n F i g u r e 4:2-1. The vacuum v e s s e l i s a p i e c e o f s t a n d a r d 4" I.D. Kimax g l a s s p i p e w i t h 0 - r i n g s e a l s t o b r a s s e l e c t r o d e s a t e i t h e r end. The anode has an extended s e c t i o n w h i c h houses the m e a s u r i n g chamber. Because o f the low base p r e s s u r e used i n t h i s e x p e r i m e n t , i t was con-v e n i e n t to have a c o n t i n u o u s f l o w o f gas t h r o u g h the system. The p r e s s u r e i n the p i n c h v e s s e l was a d j u s t e d by c o n t r o l l i n g t he r a t e o f gas f l o w w i t h a n e e d l e v a l v e . The e f f l u x o f gas was t h r o u g h the s m a l l m o n i t o r i n g h o l e i n the anode and o u t th r o u g h the me a u r i n g chamber vacuum system. A c o n s t a n t p r e s s u r e c o u l d be m a i n t a i n e d f o r h o u r s t h i s way. The Z - p i n c h vacuum sy s t e m f a c i l i t i e s were d e s i g n e d f o r f a s t pump-i n g when such was r e q u i r e d . The sys t e m i s 2" I.D. t h r o u g h o u t , e x c e p t i n g the p i n c h v e s s e l . I n o r d e r to p r e v e n t u n d e s i r a b l e d i s c h a r g e s t o ground through the vacuum system, an e l e c t r i c a l l y i n s u l a t e d , i s o l a t i o n v a l v e was p l a c e d a t the cat h o d e . D u r i n g o p e r a t i o n , the grounded gauges and pumps ar e i n s u l a t e d from the h i g h v o l t a g e cathode by c l o s i n g the i s o l a t i o n v a l v e and pumping on the r e m a i n i n g p a r t o f the system. The p i n c h i s powered by the d i s c h a r g e o f a 2.2 yufd., low i n d u c -tance c a p a c i t o r . I t i s co n n e c t e d t o the p i n c h tube by about one meter of t h r e e c o n d u c t o r , f l a t t r a n s m i s s i o n l i n e . The i n s u l a t i o n between the l e a d s i s 0.020" of m y l a r and t h e i r w i d t h i s 4". The c o n n e c t i o n o f the l e a d s t o the d i s c h a r g e v e s s e l and the t r a n s m i s s i o n l i n e c r o s s s e c t i o n a r e shown i n F i g u r e 4:2-2. L a r g e "C" clamps o f 1" x 1" t u b u l a r s t e e l were TABLE 4:2-1 MAJOR COMPONENTS AND OPERATING CONDITIONS OF THE Z-PINCH C a p a c i t e r ( C o r n e l l - D u b i l i e r , type N.R.G.-363-1A Low I n d u c t a n c e ) C a p a c i t a n c e I n d u c t a n c e V o l t a g e 2.2 ^ f d . 40 nhen. 60 kV Z- P i n c h C i r c u i t Total Inductance 185 nhen. R i n g i n g F r e q u e n c y 250 kHz C h a r g i n g V o l t a g e 40 kV Maximum C u r r e n t 70 kA D i s c h a r g e V e s s e l D i s c h a r g e Tube E l e c t r o d e s E l e c t r o d e S e p a r a t i o n V e s s e l - I.D. V e s s e l - O.D. 4" I.D. Kimax p i p e B r a s s 25 cm 10.2 cm 11.4 cm Z-P i n c h Vacuum System B a l z e r s 2" d i f f u s i o n pump, type D i f f 170 Welch Duo S e a l pump, ty p e 1397 Edwards H i g h Vac. P i r a n i Gauge, type 7-2B Edwards V a c u s t a t Minimum p r e s s u r e <0.1 m t o r r O p e r a t i n g range 5 m t o r r - H ^ < p <C 200 m t o r r - ^ M e a s u r i n g Chamber Vacuum System C o n s o l i d a t e d Vac. L t d . 4" d i f f u s i o n pump, PMC-721 Cenco Hy-Vac 14 C o n s o l i d a t e d E l e c t r o d y n a m i c s P h i l i p s Gauge, PHG-010A - 6 Minimum p r e s s u r e <10 t o r r O p e r a t i n g range < 1 m t o r r Cathode Pumps and Gauges I s o l a t i o n V a l v e S p a r k Gap S w i t c h C o n t i n u o u s F l o w o f Gas Anode F a r a d a y Cup and A n a l y z i n g G r i d 2.2 jxfd. F i g u r e 4:2-1 S c h e m a t i c Diagram o f the Z - P i n c h 4> F i g u r e 4:2-2 E l e c t r i c a l C o n n e c t i o n between t h e T r a n s m i s s i o n L i n e and t h e D i s c h a r g e ^ V e s s e l 43 used to h o l d the l e a d s t o g e t h e r . The t r a c k i n g d i s t a n c e a t 40 kV i s con-s i d e r a b l e . F l a s h o v e r s o f up t o 12" have o c c u r r e d o v e r f l a t s u r f a c e s . T h e r e f o r e , i t has been n e c e s s a r y t o c o v e r open h i g h v o l t a g e p o i n t s w i t h 0.005" of C.I.L. MILROL i n a d d i t i o n t o s e p a r a t i n g them f r o m grounded c o n d u c t o r s . The i n c l u s i o n o f a s h a r p t u r n i n a s u r f a c e w i l l s u b s t a n t i a l l y r e d u c e the t r a c k i n g d i s t a n c e . About 8" s e p a r a t i o n , or even l e s s i n the t r i g g e r can, was found to be adequate under t h e s e c i r c u m s t a n c e s . A d i a g r a m o f t h e s p e c i a l l y d e s i g n e d s p a r k gap w h i c h c o n t r o l s the i n i t i a t i o n o f the d i s c h a r g e i s shown i n F i g u r e 4:2-3. The e l e c t r o d e s and c o n d u c t i n g s e c t i o n s o f the can a r e c o n s t r u c t e d f r o m b r a s s and the i n s u l a t i o n i s epoxy r e s i n . A minimum t h i c k n e s s o f 0.25" o f epoxy and 0.25" deep grooves r u n n i n g normal to the e l e c t r i c f i e l d have p r o v e n e f f e c t i v e i n p r e v e n t i n g breakdown. The d i a m e t e r o f the can i s 3" and the e l e c t r o d e d i a m e t e r i s 2". The e l e c t r o d e s e p a r a t i o n a t 40 kV i s about 2 cm. I n o r d e r t o a c h i e v e good f i r i n g c h a r a c t e r i s t i c s , a f l o w of d r y a i r , a t a p p r o x i m a t e l y one atmosphere, i s m a i n t a i n e d t h r o u g h the s w i t c h . The a i r i s d r i e d by p a s s i n g i t t h r o u g h a D r i e r i t e gas d r y i n g u n i t . The main purpose of the f l o w of a i r i s t o make the s t a t i c breakdown v o l t a g e of the gap i n d e p e n d e n t o f v a r i a t i o n s i n the l a b o r a t o r y a i r and of changes w i t h i n the gap due to the e v a p o r a t i o n o f e l e c t r o d e and i n s u l a t o r m a t e r i a l d u r i n g d i s c h a r g e s . W i t h o u t the f l o w o f a i r , the s p a r k gap had t o be a d j u s t e d on the b a s i s of e v e r y few days. When the a i r f l o w was i n c l u d e d , p e r i o d s o f many weeks passed between r e a d j u s t m e n t s . A d i a g r a m o f the main d i s c h a r g e c i r c u i t i s shown i n F i g u r e 4:2-4. A 1 0 0 0n M o r g a n i t e r e s i s t o r has been p l a c e d i n p a r a l l e l w i t h the p i n c h tube. T h i s r e s i s t o r i s p a r t of the r e s i s t i v e , v o l t a g e d i v i d e r network of the h i g h v o l t a g e p r o b e . I t a l s o l e a k s about 40 amps t o ground t h r o u g h B r a s s Top o f t h e C a p a c i t o r F i g u r e 4:2-3 S p a r k Gap S w i t c h 45 1 k f l Theophanis T e r m i n a t i o n I s o l a t i o n T r a n s f o r m e r 1000 M f L Z - P i n c h Spark Gap 2.2 y u f d . To C.R.O. 1 kn. 1 Dump R e s i s t o r ' S h o r t i n g S w i t c h F i g u r e 4:2-4 Z - P i n c h C i r c u i t 46 the s p a r k gap s w i t c h d u r i n g t h e l o n g s t a t i s t i c a l time l a g o f t h e d i s c h a r g e . D u r i n g t h i s p e r i o d , i t s u s t a i n s a s t r o n g a r c i n t h e gap and t h e r e b y improves the f i r i n g r e l i a b i l i t y o f the s w i t c h . There appears to be l i t t l e d e l a y between the t r i g g e r i n g of the s p a r k gap and the appearance o f the f u l l bank v o l t a g e a c r o s s the d i s c h a r g e v e s s e l . The c h a r a c t e r i s t i c s o f the main d i s c h a r g e c i r c u i t appear i n T a b l e 4:2-1. The t r i g g e r p u l s e i s s u p p l i e d f r o m a t h y r a t r o n (6279 w i t h 8 kV d.c. l e v e l ) u n i t w h i c h can be f i r e d e i t h e r m a n u a l l y o r f r o m an e x t e r n a l p u l s e g e n e r a t o r . A Theophanis (1960) v o l t a g e d o u b l i n g t e r m i n a t o r passes t h e t r i g g e r p u l s e t h r o u g h an i s o l a t i o n t r a n s f o r m e r ( r a t e d a t 120 kV p r o t e c t i o n ) i n t o the s p a r k gap. The p u l s e i s about 16 kV and has a r i s e time o f about 25 nanoseconds a f t e r d o u b l i n g . 4:3 The M e a s u r i n g Chamber The m e a s u r i n g chamber i s l o c a t e d i n s i d e the b r a s s h o u s i n g of t h e anode. A s i m p l i f i e d d i a g r a m of the anode and the e l e c t r o n energy a n a l y z e r assembly i s shown i n F i g u r e 4:3-1. The anode i s c o n s t r u c t e d i n such a way t h a t the complete h o u s i n g shown i n the d i a g r a m can be removed f r o m the p i n c h v e s s e l w i t h o u t d i s t u r b i n g the e l e c t r i c a l c o n n e c t i o n s of the main d i s c h a r g e c i r c u i t . T h i s has been a c o n s i d e r a b l e advantage s i n c e many d e s i g n m o d i f i c a t i o n s of the e l e c t r o n c o l l e c t o r assembly were c a r r i e d o u t over the c o u r s e o f the e x p e r i m e n t . H i g h v o l t a g e and s i g n a l c a b l e f e e d throughs a r e p o s i t i o n e d i n the b a k e l i t e f l a n g e to the r i g h t o f F i g u r e 4:3-1. There a r e f o u r B.N.C. c o n n e c t o r s f o r RG 28/U c o - a x i a l c a b l e and two H.V. f e e d t h r o u g h s . The ground c o n n e c t i o n f o r the main d i s c h a r g e c i r c u i t , and m o n i t o r i n g c i r c u i t s and the r e t a r d i n g g r i d c i r c u i t i s the b r a s s mesh s h i e l d i n g o f the g r i d B r a s s B a k e l i t e (1) B.N.C. E l e c t r i c a l C o n n e c t i o n s f o r S i g n a l C a b l e s (2) H i g h V o l t a g e Feed Through f o r G r i d s B r a s s Mesh S h i e l d i n g ( S y s t e m Ground) I I Vacuum System O-Ring S e a l t o D i s c h a r g e V e s s e l 4" 18" B a k e l i t e F l a n g e 3 (1) (2) 1 S h i e l d i n g , G r i d and C o l l e c t o r A s s e m b l y E l e c t r i c a l C o n t a c t t o M a i n D i s c h a r g e C i r c u i t (See F i g u r e 4:2-2) F i g u r e 4:3-1 Anode H o u s i n g 48 and c o l l e c t o r assembly w h i c h i s f i x e d t o the back o f the anode f a c e p l a t e . The e n t i r e m e a s u r i n g chamber vacuum s y s t e m i s 4" I.D. i n c l u d i n g the c o l d t r a p , v a l v e and d i f f u s i o n pump. P r e s s u r e s were m a i n t a i n e d b e l o w one m t o r r f o r p r e s s u r e s o f 100 m t o r r o r l e s s i n the p i n c h s e c t i o n . The - 6 minimum p r e s s u r e a t t a i n a b l e i n t h i s s e c t i o n was l e s s t h a n 10 t o r r . 4:4 The E l e c t r o s t a t i c G r i d s and F a r a d a y Cup A s s e m b l y The e l e c t r o s t a t i c g r i d s and F a r a d a y cup assembly a r e shown i n F i g u r e 4:4-1. The system i s a d o u b l e , r i n g e l e c t r o d e s t r u c t u r e . I n i t i a l l y a mesh g r i d was used, b u t the f r e q u e n c y o f g r i d to cup d i s -charges was much too h i g h to be a c c e p t a b l e . The r i n g e l e c t r o d e s r e d u c e d t h i s p r o b l e m c o n s i d e r a b l y , though the r e s o l u t i o n of the s y s t e m s u f f e r s (see A p p e n d i x B) . The f u l l a n g l e o f the g e o m e t r i c a l c o l l e c t i o n cone i s 28°„ Because of e l e c t r o s t a t i c f o r c e s i n the e l e c t r o n s t r e a m , t h e e f f e c t i v e a p e r t u r e s i z e i s a f u n c t i o n o f the e l e c t r o n energy. There i s good e v i d e n c e t h a t the system has poor t r a n s m i s s i o n c h a r a c t e r i s t i c s a t low e n e r g i e s . The s m a l l h o l e i n the anode has a d i a m e t e r of 0.080" = 2 mm. The w a l l t h i c k -ness has been r e d u c e d to 0.010" a t the edge o f the h o l e to improve t h e t r a n s m i s s i o n o f e l e c t r o n s . A p p e n d i c e s A and B d i s c u s s the t r a n s m i s s i o n and r e s o l u t i o n o f t h i s system. The r e t a r d i n g v o l t a g e i s a p p l i e d t o the r i n g e l e c t r o d e s by the c i r c u i t shown i n F i g u r e 4:4-2. A s i m i l a r c i r c u i t i s p r o v i d e d f o r each o f the two e l e c t r o d e s . Measurements were made w i t h the v o l t a g e of the e l e c t r o d e n e a r e r to the anode f a c e e q u a l t o o n e - h a l f of t h e v o l t a g e on the second e l e c t r o d e . T h i s arrangement seemed to r e d u c e the number of d i s c h a r g e s to ground from the h i g h v o l t a g e g r i d . The c a p a c i t a n c e i s p r o -Not t o S c a l e F i g u r e 4:4-1 F a r a d a y Cup and R e t a r d i n g G r i d Assembly 50 R i n g E l e c t r o d e f RG - 8/U H.V — Probe 1000 Pf To C.R.O. 200 MfL 10 Mfl 0-20 ZE kV • S h o r t i n g S w i t c h TTftTT F i g u r e 4:4-2 R e t a r d i n g G r i d C i r c u i t v i d e d i n the g r i d c i r c u i t to h o l d the g r i d v o l t a g e r e a s o n a b l y c o n s t a n t d u r i n g the c o l l e c t i o n o f f a s t e l e c t r o n s by the g r i d d u r i n g the f i r i n g o f the p i n c h . A s i m p l e check was r u n on the r e t a r d i n g g r i d v o l t a g e t o see i f the c u r r e n t measured by t h e F a r a d a y cup c o u l d be r e l a t e d t o any v a r i a t i o n i n t he g r i d v o l t a g e . T h i s was done by m o n i t o r i n g the v o l t a g e on the g r i d w i t h a T e k t r o n i x , type P-6015, h i g h v o l t a g e probe. F o r a s u c c e s s f u l s h o t , the g r i d v o l t a g e n o r m a l l y became more n e g a t i v e by about 200-300 v o l t s , i n d i c a t i n g the c o l l e c t i o n o f f a s t e l e c t r o n s by t h e g r i d . I n such i n s t a n c e s the g r i d v o l t a g e was about 5 kV. The d i s c h a r g e o f the g r i d to the cup o r ground was e a s i l y d e t e c t e d by the immediate f a l l o f the g r i d v o l t a g e o r an a b n o r m a l l y h i g h v a l u e o f the cup c u r r e n t . The f r e q u e n c y of such d i s c h a r g e s i n c r e a s e d p r o g r e s s i v e l y as the r e t a r d i n g g r i d v o l t a g e was r a i s e d beyond -15 kV. The F a r a d a y cup c i r c u i t i s shown i n F i g u r e 4:4-3. The ground o f 51 RG-58A/U -3 >• A Fa r a d a y Cup 5on 5 OA 5orL F i g u r e 4:4-3 F a r a d a y Cup C i r c u i t the system i s the b r a s s mesh s h i e l d i n g o f the energy a n a l y z e r . The s i g n a l from the cup i s f e d d i r e c t l y i n t o c o - a x i a l c a b l e , type RG-58 A/U, t e r m i n a t e d a t b o t h t h e i n p u t and o u t p u t by i t s c h a r a c t e r i s t i c impedance ( 5 0 / 1 ) . The c u r r e n t c o l l e c t e d by the cup i s I = 2V c/R, where R i s the t e r m i n a t i n g r e s i s t a n c e o f the c a b l e s ( 5 0 / 1 ) . The f r e q u e n c y r e s p o n s e o f the c i r c u i t i s governed by the RC time o f the cup c a p a c i t a n c e and the impedance of the c i r c u i t ( ^10 n s e c . ) . A s t a n d a r d p a r a l l e l c i r c u i t method i s used t o s u p p r e s s n o i s e p i c k u p . The r e a l s i g n a l i s f e d i n t o c h a n n e l A of a T e k t r o n i x , t y p e G, d i f f e r e n t i a l a m p l i f i e r i n p u t t o the scope. The o u t p u t f r o m an i d e n t i c a l dummy c i r c u i t i s f e d i n t o c h a n n e l B and s u b t r a c t e d f r o m A. I d e n t i c a l n o i s e s i g n a l s w i l l be c a n c e l l e d ( s ee Medley, 1965). F e r r i t e c o r e s have been used t o d e c o u p l e the scope f r o m ground. The c o - a x i a l c a b l e i s s i m p l y wrapped around a t o r o i d a l c o r e ( P h i l i p s -F e r r o x Cube FX 1076 - B8K,4.5" O.D.). N o i s e s i g n a l s on the s h i e l d i n g o f 52 the c a b l e a r e r e d u c e d by t h e l a r g e i n d u c t a n c e of the w i n d i n g . The s i g n a l w h i c h i s t r a n s m i t t e d i n s i d e the c a b l e e x p e r i e n c e s no f l u x l i n k a g e w i t h the f e r r i t e and i s n o t a t t e n u a t e d . The n o i s e r e d u c t i o n i s about a f a c t o r of 3. I n some i n s t a n c e s , AD-YU, typ e 10T5D06, 3 yUsec. d e l a y l i n e s have been used t o d e l a y the r e a l s i g n a l w i t h r e s p e c t t o the n o i s e . The e f f e c t o f the d e l a y l i n e s i s more f u l l y d i s c u s s e d i n A p p e n d i x C. 4:5 R a d i a l Probe Assembly The r a d i a l probes w h i c h were used f o r m o n i t o r i n g the c u r r e n t p r o -f i l e , a r e shown i n F i g u r e 4:5-l„ The f u l l g e o m e t r i c a l c o l l e c t i o n a n g l e i s about 60°. Because o f the s h o r t d i s t a n c e between the probe and the p i n h o l e , the e f f e c t i v e a p e r t u r e s i z e i s n o t a s t r o n g f u n c t i o n o f the e l e c t r o n energy. However, because of the o f f - a x i s p o s i t i o n o f some of the p r o b e s , the e f f i c i e n c y of the probes i s a f f e c t e d by the m a g n e t i c f i e l d o f the d i s c h a r g e c u r r e n t . T h i s problem i s d i s c u s s e d i n d e t a i l a l o n g w i t h the p r e s e n t a t i o n o f the probe r e s u l t s i n S e c t i o n 5:4. The c i r c u i t used i s the same as t h a t used w i t h the F a r a d a y cup shown i n F i g u r e 4:4-3. 4:6 V o l t a g e and C u r r e n t Measurement A s p e c i a l v o l t a g e probe was used to m o n i t o r the Z - p i n c h v o l t a g e . I t c o n s i s t e d o f a p o t e n t i a l d i v i d e r network made w i t h f i v e 200/1 M o r g a n i t e r e s i s t o r s , t y p e 764, and a p a r a l l e l c o m b i n a t i o n o f twenty, 100i~L , 1/2 w a t t c a r b o n r e s i s t o r s . Care was e x e r c i s e d to r e d u c e the i n d u c t a n c e o f the twenty, 100/1 r e s i s t o r s by p u t t i n g them near the w a l l s o f the c y l i n d r i c a l r e t u r n c o n d u c t o r as shown i n F i g u r e 4:6-1. Because of the n o n l i n e a r f e a t u r e o f the M o r g a n i t e r e s i s t o r s , a e s s e l F i g u r e 4:5-1 R a d i a l Probe A s s e m b l y 54 F i g u r e 4:6-1 V o l t a g e Probe Assembly and C i r c u i t c a r e f u l c a l i b r a t i o n p r o c e d u r e had t o be f o l l o w e d . The probe c a l i b r a t i o n i s d i s c u s s e d i n A p p e n d i x C. N o i s e r e d u c t i o n p r o c e d u r e s , s i m i l a r t o t h o s e d i s c u s s e d i n c o n n e c t i o n w i t h the F a r a d a y cup, were used. The c i r c u i t i s shown i n F i g u r e 4:6-1. The c u r r e n t d e r i v a t i v e was m o n i t o r e d w i t h a s i m p l e m a g n e t i c p i c k up c o i l (5 cm. l o n g , 4 mm, d i a m e t e r and 36 t u r n s ) . The c o i l f i t s i n t o a s m a l l c a v i t y between the c e n t e r c o n d u c t o r and the ground of the t r a n s -m i s s i o n l i n e . The c o i l f e e d s d i r e c t l y i n t o RG - 58A/U c o - a x i a l c a b l e t e r m i n a t e d a t the o u t p u t w i t h 50-TL . The probe was c a l i b r a t e d from the c u r r e n t waveform, o b t a i n e d by i n t e g r a t i n g the c o i l o u t p u t w i t h a T e k t r o n i x , type 0, o p e r a t i o n a l a m p l i f i e r p l u g - i n u n i t . The i n t e g r a t i o n time con-s t a n t was 1 w s e c . The d e t a i l s of the c a l i b r a t i o n appear i n A p p e n d i x C. 55 4: 7 P hotographs A T.R.W., Model ID, image c o n v e r t e r camera was used to s t u d y the plasma p h o t o g r a p h i c a l l y . P r i m a r y i n t e r e s t was i n the u n i f o r m i t y o f i o n i z a t i o n , the g e o m e t r i c a l p o s i t i o n o f the p i n c h i n the v e s s e l and the time o f the p i n c h . T i m i n g of t h e s e p hotographs i s s i m p l i f i e d because the camera g i v e s o u t a m o n i t o r i n g p u l s e each time i t makes an e x p o s u r e . The m o n i t o r p u l s e s a r e d i s p l a y e d on a C.R.O. a l o n g w i t h t h e c u r r e n t o r v o l t -age waveform. CHAPTER V EXPERIMENTAL OBSERVATIONS AND RESULTS 5:1 I n t r o d u c t i o n The i n v e s t i g a t i o n o f the e l e c t r o n c u r r e n t w h i c h appears i n the c e n t r a l r e g i o n o f the d i s c h a r g e v e s s e l d u r i n g the f o r m a t i v e phase of the Z- p i n c h i s the main s u b j e c t of t h i s c h a p t e r . The p r i n c i p a l means of i n v e s t i g a t i o n was the F a r a d a y cup and r e t a r d i n g g r i d system w h i c h was used to measure the d i s t r i b u t i o n o f e l e c t r o n e n e r g i e s . I n o r d e r t o u n d e r s t a n d the t e m p o r a l b e h a v i o u r o f the F a r a d a y cup s i g n a l s , i t was n e c e s s a r y to make a d d i t i o n a l measurements. F o r t h i s r e a s o n , the v o l t a g e a c r o s s the d i s c h a r g e , V, the t o t a l c u r r e n t t h r o u g h the d i s -c h arge, I , and i t s time d e r i v a t i v e , d l / d t , were m o n i t o r e d . I n a d d i t i o n , the s p a t i a l d i s t r i b u t i o n o f the c u r r e n t a t the f a c e o f the anode was s t u d i e d w i t h s m a l l probes l o c a t e d a t d i f f e r e n t r a d i a l p o s i t i o n s i n the anode. These o b s e r v a t i o n s form a c o n s i s t e n t p i c t u r e of the e l e c t r o n m o t i o n and the boundary l a y e r f o r m a t i o n i n the low p r e s s u r e , h i g h v o l t a g e p i n c h . I n S e c t i o n 5:2, the V, I and d l / d t waveforms a r e examined. The i n f o r m a t i o n o f t h i s s e c t i o n s e r v e s as a f o u n d a t i o n f o r the d i s c u s s i o n o f o b s e r v a t i o n s and c a l c u l a t i o n s w h i c h f o l l o w i n l a t e r s e c t i o n s o f t h i s c h a p t e r . The waveforms a r e t y p i c a l o f p i n c h d e v i c e s , e x c e p t d u r i n g the f o r m a t i v e phase. T h i s p e r i o d o f the p i n c h i s g i v e n s p e c i a l a t t e n t i o n because of i t s r e l a t i o n s h i p to the e l e c t r o n energy measurements. The F a r a d a y cup measurements a r e p r e s e n t e d i n S e c t i o n 5:3. By measuring the energy d i s t r i b u t i o n o f the e l e c t r o n s , v a l u a b l e i n s i g h t i n t o 56 57 the f o r m a t i o n o f the p i n c h has been g a i n e d . The b e h a v i o u r o f t h e Faraday-cup s i g n a l and the c o r r e l a t i o n o f V, I and d l / d t have l e d t o the model o f boundary l a y e r f o r m a t i o n p r e s e n t e d i n C h a p t e r I I . D u r i n g t h e f o r m a t i v e phase of the d i s c h a r g e , the e x p e r i m e n t a l d i s t r i b u t i o n i s b e s t f i t t e d to a p e r t u r b e d M a x w e l l i a n . Thus the e l e c t r o n m o t i o n i n the d i s c h a r g e i s dominated by e l a s t i c c o l l i s i o n s w i t h the n e u t r a l s . T h i s c o n c l u s i o n i s borne o u t by the c a l c u l a t i o n of the v a r i o u s c o l l i s i o n f r e q u e n c i e s . The t h e o r y o f C h a p t e r I I I appears to be an adequate d e s c r i p t i o n o f the e l e c t r o n m o t i o n . I n S e c t i o n 5:4, the s p a t i a l d i s t r i b u t i o n of the c u r r e n t d u r i n g the f o r m a t i v e phase i s examined. The o b s e r v a t i o n s o f t h i s s e c t i o n add f u r t h e r v e r i f i c a t i o n to the model o f boundary l a y e r f o r m a t i o n . I t has been p o s s i b l e to o b s e r v e the decay o f the c u r r e n t i n the c e n t r a l r e g i o n of the v e s s e l . Throughout t h i s c h a p t e r , a number o f parameters w i l l be used r e p e a t e d l y . Though some o f the p a r a m e t e r s w i l l n o t be s p e c i f i c a l l y i n t r o -duced u n t i l l a t e r , T a b l e 5:1-1 w i l l s e r v e as a r e f e r e n c e . TABLE 5:1-1 L i s t o f P a r a m e t e r s V ( t ) The v o l t a g e between the e l e c t r o d e s o f the d i s c h a r g e . The back e.m.f. p u l s e w h i c h appears a t the f o r m a t i o n of the boundary l a y e r . Vp The v o l t a g e p u l s e w h i c h appears a t the p i n c h . I ( t ) The t o t a l c u r r e n t t h r o u g h the d i s c h a r g e v e s s e l . d l / d t The time d e r i v a t i v e of I ( t ) . Tp The time w h i c h e l a p s e s f r o m the f o r m a t i o n o f the c u r r e n t s h e e t to the c o n s t r i c t i o n o f the plasma column to i t s minimum r a d i u s ; c a l l e d the p i n c h t i m e . I C ( V , t) C u r r e n t t o the F a r a d a y cup as a f u n c t i o n o f the r e t a r d i n g g r i d v o l t a g e , V g , and t i m e , t . I g ( t ) C u r r e n t to the c e n t r a l p o r t i o n o f the anode. 58 5:2 G e n e r a l F e a t u r e s of the D i s c h a r g e 5:2.1 I n t r o d u c t i o n O b s e r v a t i o n s of V, I and d l / d t have been h e l p f u l i n u n d e r s t a n d i n g the growth o f i o n i z a t i o n . The v o l t a g e has been measured w i t h a r e s i s t i v e probe and the c u r r e n t d e r i v a t i v e m o n i t o r e d w i t h a m a g n e t i c p i c k - u p c o i l . B o t h o f t h e s e d e v i c e s have been d e s c r i b e d i n S e c t i o n 4:6 and A p p e n d i x C. The e v i d e n c e p r e s e n t e d i n t h i s s e c t i o n l a r g e l y s u p p o r t s the use of the model of boundary l a y e r f o r m a t i o n s u g g e s t e d i n C h a p t e r I I . There a r e r e l a t i v e l y l o n g s t a t i s t i c a l and f o r m a t i v e time l a g s . These times a r e e a s i l y measured from o b v i o u s changes i n V, I and d l / d t . The f o r m a t i o n o f the boundary l a y e r i s accompanied by a l a r g e back e.m.f., V^, w h i c h i s c o r r e l a t e d w i t h r a p i d f l u c t u a t i o n s i n d l / d t . The phenomenon o c c u r s most s t r o n g l y a t low p r e s s u r e and has a p p a r e n t l y n o t been p r e v i o u s l y o b s e r v e d . The p i n c h t i m e , T , can be e a s i l y measured from the V and d l / d t waveforms a l o n g w i t h image c o n v e r t e r p h o t o g r a p h s . A d i s c r e p a n c y i n the f i t of the e x p e r i m e n t a l v a l u e s o f T^ t o the t h e o r e t i c a l p 2 law put f o r w a r d by Curzon (1963) can be e l i m i n a t e d by assuming t h a t the boundary l a y e r i s n o t f u l l y formed u n t i l the appearance o f V^. The c u r r e n t , I , has been used i n the c a l c u l a t i o n of the enhancement o f i o n i z a t i o n a t the w a l l o f the d i s c h a r g e v e s s e l . A t p r e s s u r e s of l e s s than 50 m t o r r - l ^ , the enhancement i s c o n s i d e r a b l e and must p l a y a s i g n i f i -c a n t r o l e i n the boundary l a y e r f o r m a t i o n . 5:2.2 The Sequence of E v e n t s i n the D i s c h a r g e We s h a l l now d i s c u s s i n a g e n e r a l way, the sequence of e v e n t s w h i c h occurs when the c a p a c i t o r bank i s d i s c h a r g e d t h r o u g h the p i n c h v e s s e l . 59 F i g u r e 5:2-1 shows t y p i c a l waveforms o f V f o r v a r i o u s p r e s s u r e s i n R^. I n o r d e r t o f a c i l i t a t e the d i s c u s s i o n o f the waveforms as t h e y a r e r e l a t e d to the b e h a v i o u r of the d i s c h a r g e , two t i m e s , t and t , have been shown s p on the g r a p h s . As w i l l become c l e a r , t c o r r e s p o n d s a p p r o x i m a t e l y to the breakdown and t t o the i n i t i a t i o n o f the c u r r e n t s h e e t c o l l a p s e . The P v z e r o time has been chosen a t the i n s t a n t when the t h y r a t r o n u n i t i s f i r e d and the t r i g g e r can becomes c o n d u c t i n g . When t h i s happens, the f u l l c a p a c i t o r bank v o l t a g e o f 40 kV appears between the e l e c t r o d e s of the p i n c h v e s s e l . The gas i o n i z e s s l o w l y a t f i r s t . Because of the low p r e s s u r e , i t t a k e s a r e l a t i v e l y l o n g time f o r t h e gas to b r e a k down. D u r i n g t h i s p e r i o d , c a l l e d the s t a t i s t i c a l time l a g , the impedance i s p r e d o m i n a n t l y r e s i s t i v e . A t time t g , the r a t e of growth o f i o n i z a t i o n p a s s e s i n t o an u n s t a b l e domain and the gas b r e a k s down. The p e r i o d between t and t i s u s u a l l y s p J c a l l e d the f o r m a t i v e t i m e . I n t h i s p a r t i c u l a r s i t u a t i o n , i t r e p r e s e n t s the r a p i d growth of i o n i z a t i o n and the f o r m a t i o n of the boundary l a y e r . The v o l t a g e s p i k e , V^, i s r e l a t e d to the boundary l a y e r f o r m a t i o n . , I t i s caused by the decay of the t r a p p e d m a g n e t i c f i e l d when the c u r r e n t s h i f t s from the c e n t r a l r e g i o n to the w a l l . A t time t , the p i n c h i n g c u r r e n t s h e e t i s formed. The dynamics of P the d i s c h a r g e can be d e s c r i b e d by some v a r i a t i o n o f the snowplough model. The v o l t a g e p u l s e , V , i s i n d u c e d by the p i n c h i n g plasma column. The f l u c t u a t i o n s , V , a r e most l i k e l y caused by o s c i l l a t i o n s i n the plasma column. The impedance of the d i s c h a r g e i s m o s t l y i n d u c t i v e , b u t may have a s i g n i f i c a n t r e s i s t i v e p a r t . 5:2.3 S t a t i s t i c a l and F o r m a t i v e Time Lags B o t h the s t a t i s t i c a l time l a g , t 8 , and the f o r m a t i v e t i m e , t p - t s , 60 F i g u r e 5:2-1 V o l t a g e Waveforms f o r V a r i o u s P r e s s u r e s . The p e r i o d c o v e r s the s t a t i s t i c a l and f o r m a t i v e time l a g s and t h e c o l l a p s e phase. The c u r r e n t maximum i s 70 kA. (V, ; I , ) a r e f u n c t i o n s o f p r e s s u r e . T h i s dependence i s shown i n F i g u r e 5:2-2. The shapes of the c u r v e s a r e s i m i l a r t o t h o s e found f o r s i m p l e , p a r a l l e l geo-metry e l e c t r o d e s ( L l e w e l l y n - J o n e s , 1957). The v e r y s t e e p r i s e i n the s t a t i s t i c a l time l a g c u r v e r e p r e s e n t s the approach to the minimum r e d u c e d e l e c t r o d e s e p a r a t i o n a t w h i c h t h e gas w i l l b r eak down f o r 40 kV. The e l e c t r o d e s e p a r a t i o n i s 25 cm. and the minimum p r e s s u r e was n o r m a l l y 10 mtorr-fL^. Thus the minimum r e d u c e d e l e c t r o d e s e p a r a t i o n i s 0.25 t o r r - c m . Under t h e s e c o n d i t i o n s , 40 kV i s a p p r o x i m a t e l y e q u a l to the s t a t i c breakdown p o t e n t i a l . The f o r m a t i v e time i n c r e a s e s as the p r e s s u r e i s r e d u c e d . I t i s d u r i n g the f o r m a t i v e time t h a t the measurements of the e l e c t r o n energy a r e made. Because the growth r a t e i s s m a l l e s t f o r l a r g e f o r m a t i v e t i m e , the b e s t r e s o l v e d o b s e r v a t i o n s c o u l d be made f o r p r e s s u r e s o f 20 m t o r r o r l e s s . 5:2.4 C o r r e l a t i o n of V, I and d l / d t The V, I and d l / d t waveforms, c o v e r i n g t h e f o r m a t i v e phase and the c u r r e n t s h e e t c o l l a p s e phase of the d i s c h a r g e , a r e p r e s e n t e d i n F i g u r e s 5:2-3 to 5:2-6. The c u r v e s r e p r e s e n t t y p i c a l s h o t s a t v a r i o u s p r e s s u r e s i n hydrogen. As p o i n t e d o u t i n S e c t i o n 5:2.2, the p i n c h i s accompanied by a v o l t a g e p u l s e , V . A l s o e v i d e n t i n t h e s e waveforms a r e the t y p i c a l and w e l l known c o r r e s p o n d i n g f l u c t u a t i o n s i n I and d l / d t d u r i n g the p i n c h . These appear beyond the 0.7 ^ u s e c . mark i n the f i g u r e s . The r e l a t i v e l y i n s i g n i f i c a n t s i z e of the f l u c t u a t i o n i n I a t the p i n c h time i s due t o the dominance of the i n d u c t a n c e o f the e x t e r n a l c i r c u i t e l e m e n t s . F o r the purpose of the p r e s e n t t h e s i s , the more i n t e r e s t i n g f e a t u r e s of t h e s e waveforms o c c u r d u r i n g the f o r m a t i v e phase. The f e a t u r e s of V and d l / d t d u r i n g the f o r m a t i v e phase a r e dominated by V f and a l a r g e d i p i n C a p a c i t o r V o l t a g e = 40 kV p ( m t o r r - H 2 ) F i g u r e 5:2-2(a) S t a t i s t i c a l Time Lag as a F u n c t i o n o f P r e s s u r e 0.5 0.4 0.3 0.2 0.1 C a p a c i t o r V o l t a g e = 40 kV 20 40 60 80 p (mtorr-H,,) F i g u r e 5:2-2(b) F o r m a t i v e Time as a F u n c t i o n o f P r e s s u r e F i g u r e 5:2-3 V, I and d l / d t Waveforms a t 80 m t o r r - H and 40 kV on the C a p a c i t o r 64 F i g u r e 5:2-4 V, I and d l / d t Waveforms a t 40 m t o r r - H and 40 kV on the C a p a c i t o r 65 F i g u r e 5:2-5 V, I and d l / d t Waveforms a t 20 m t o r r - H 2 and 40 kV on the C a p a c i t o r 80 . 60 -t ( y u s e c ) F i g u r e 5:2-6 V, I and d l / d t Waveforms a t 10 m t o r r - H 2 and 40 kV on the C a p a c i t o r 67 d l / d t . and the d i p i n d l / d t always o c c u r t o g e t h e r , and g e n e r a l l y s p e a k i n g , the lo w e r the p r e s s u r e the l a r g e r t h e y a r e . Because of the r e l a t i v e l y s l o w r i s e i n I compared t o the f a s t r i s e i n d l / d t , the d l / d t waveform has been used t o c o r r e l a t e e v e n t s i n the d i s -c harge. A l s o , i n the measurement of the e l e c t r o n c u r r e n t p u l s e , the d l / d t waveform has been used as a measure o f the r e p r o d u c i b i l i t y o f the b r e a k -down mechanism because of i t s a p p a r e n t h i g h e r s e n s i t i v i t y to changes i n the boundary l a y e r f o r m a t i o n . F o r example, the breakdown sequences of two s h o t s are c o n s i d e r e d to be the same i f the d i p i n d l / d t o c c u r s a t r o u g h l y the same time and i s a p p r o x i m a t e l y the same s i z e i n the two c a s e s . 5:2.5 The P i n c h Time The p i n c h t i m e , T , as i t i s u n d e r s t o o d h e r e , i s the l e n g t h o f time from the s t a r t o f the c o l l a p s e o f the c u r r e n t s h e e t to the c o m p r e s s i o n of the plasma column t o i t s minimum r a d i u s . The snowplough model can be used t o c a l c u l a t e T . I n t h e s p e c i a l case when I i s a s i n e f u n c t i o n , as P i s a p p r o x i m a t e l y the case i n the p r e s e n t e x p e r i m e n t , k T cxC p P o T h i s r e s u l t i s g i v e n by Curzon ( 1 9 6 3 ) . I n t h i s e q u a t i o n , p^ i s the i n i t i a l p r e s s u r e of the gas i n the p i n c h v e s s e l . The e x p e r i m e n t a l v a l u e of T^ i s e a s i l y measured from image c o n v e r t e r photographs ( S e c t i o n 4:7) and the waveforms o f d l / d t and V. F i g u r e 5:2-7 i l l u s t r a t e s the type o f d a t a used i n the e v a l u a t i o n o f T . The time mark P p u l s e l a b e l e d (b) on the d l / d t time a x i s i s g e n e r a t e d by the image con-v e r t e r camera e l e c t r o n i c s a t the i n s t a n t t h a t frame (b) of the p h o t o g r a p h i c sequence i s t a k e n . Thus, the i n s t a n t o f the p i n c h i s e a s i l y l o c a t e d f r o m F i g u r e 5:2-7 I l l u s t r a t i n g t he Measurement o f the P i n c h Time a-' oo 69 photographs and the c h a r a c t e r i s t i c b e h a v i o u r of V and d l / d t . C h o o s i n g the i n s t a n t a t w h i c h the c u r r e n t s h e e t b e g i n s to c o l l a p s e i s n o t such an easy m a t t e r . L e t us a r b i t r a r i l y choose t as the time of i n i t i a t i o n . T h i s r o u g h l y c o r r e s p o n d s to the i n i t i a l r i s e o f d l / d t . These v a l u e s of T^ appear as c u r v e (a) i n F i g u r e 5:2-8. The p o i n t s above 40 m t o r r - ^ ° n a s t r a : ' - 8 n t l i n e b u t the p o i n t s below 40 m t o r r - H ^ d e v i a t e q u i t e s t r o n g l y from the s t r a i g h t l i n e . I n s t e a d , l e t us measure T f r o m t w h i c h c o r r e s p o n d s t o the f a l l o f P P V . The p o i n t s above 40 m t o r r - ^ a r e n o t g r e a t l y a f f e c t e d by t h i s change because ( t - t )/T 1. b u t p o i n t s f o r v a l u e s o f the p r e s s u r e below P s P 40 m t o r r - ^ a r e s t r o n g l y a f f e c t e d . W i t h i n e x p e r i m e n t a l e r r o r , a l l of the p o i n t s l i e on the same s t r a i g h t l i n e . The s l o p e o f . t h e graph i s 0.43, i n s t e a d o f 0.5 as e x p e c t e d . T h i s d i s c r e p a n c y may be caused by the non-zero c u r r e n t a t the s t a r t of the p i n c h sequence, though the- e x a c t r e a s o n i s n o t known. I t may be t h a t the snow-p l o u g h modeli b r e a k s down f o r the l o n g mean f r e e p a t h s w h i c h a r e p r e s e n t i n t h i s e x p e r i m e n t . However, t h e r e appears to be good r e a s o n t o r e p r e s e n t t as the f o r m a t i o n of the boundary l a y e r . 5:2.6 M a g n e t i c F i e l d Induced Enhancement of I o n i z a t i o n The a z i m u t h a i m a g n e t i c f i e l d o f the e l e c t r o n c u r r e n t f l o w i n g i n the i n t e r i o r o f the d i s c h a r g e v e s s e l enhances the r a t e o f i o n i z a t i o n a t the w a l l of the d i s c h a r g e v e s s e l . The magnitude of the enhancement can be e v a l u a t e d from the t h e o r y o f S e c t i o n 2:4. To f a c i l i t a t e t h i s c a l c u l a t i o n , the measured v a l u e o f the c u r r e n t a t the time o f the boundary l a y e r forma-t i o n has been p l o t t e d as a f u n c t i o n of p r e s s u r e i n F i g u r e 5:2-9 ( a ) . We now assume t h a t the m a g n e t i c f i e l d i s i n d u c e d by t h i s c u r r e n t . The p r e s s u r e 70 F i g u r e 5:2-9(b) P r e s s u r e Enhancement, p'/p, as a F u n c t i o n of P r e s s u r e enhancement, p'/p, i s g i v e n by e q u a t i o n 2:4-1. 21 P "b ll * (2:4-1) where ( ) - J L b m ' 2irr e w I n t h i s e q u a t i o n , the v a l u e o f I has been d i s c u s s e d above and r i s the w w a l l r a d i u s w h i c h i s 5 cm. The v a l u e o f p'/p has been p l o t t e d i n F i g u r e 5:2-9 ( b ) . F o r the l o w e s t p r e s s u r e s o f U.^, the i n f l u e n c e o f the a x i a l c u r r e n t on the r a t e of i o n i z a t i o n i n the boundary l a y e r i s c o n s i d e r a b l e . 5:3 The On-Axis E l e c t r o n C u r r e n t 5:3.1 I n t r o d u c t i o n The purpose of t h i s p a r t of the e x p e r i m e n t i s t o examine the e n e r g y d i s t r i b u t i o n o f the e l e c t r o n s . The measured d i s t r i b u t i o n i s t o be com-pared t o the t h e o r e t i c a l d i s t r i b u t i o n s d i s c u s s e d i n Chapter I I I . The f u n c t i o n w h i c h b e s t f i t s the e x p e r i m e n t a l d a t a i s the p e r t u r b e d M a x w e l l i a n . Once the d i s t r i b u t i o n f u n c t i o n i s known, i t i s a s i m p l e m a t t e r t o c a l c u l a t e b o t h the e l e c t r o n t e m p e r a t u r e and d e n s i t y . The b a s i c d a t a i s the F a r a d a y cup c u r r e n t , I ( t ) . The s u b s c r i p t , C 6 V , i n d i c a t e s t h a t I- ( t ) was c o l l e c t e d on the cup w h i l e the r e t a r d i n g g c> v g g r i d v o l t a g e was s e t a t Vg. The I £ ^ waveforms have been h e l p f u l i n ' g u n d e r s t a n d i n g the i o n i z a t i o n p r o c e s s i n the p i n c h . I n p a r t i c u l a r , the t e m p o r a l b e h a v i o u r o f I c v ( t ) has made i t n e c e s s a r y to re-examine t h e ' g f o r m a t i o n o f the boundary l a y e r i n a Z - p i n c h . The w r i t i n g of C h a p t e r I I , w h i c h d e s c r i b e s the boundary l a y e r f o r m a t i o n , was l a r g e l y a consequence o f t h e s e o b s e r v a t i o n s . I f a p a r t i c u l a r t i m e , t ^ , i s chosen, t h e n the c u r r e n t to the cup can be r e p r e s e n t e d as a f u n c t i o n of the r e t a r d i n g g r i d v o l t a g e , I c , t ^ ( ^ g ) -The e l e c t r o n energy d i s t r i b u t i o n has been measured f r o m I c , t ^ ( V g ) and the e x p e r i m e n t a l o b s e r v a t i o n s a r e b e s t e x p l a i n e d by a p e r t u r b e d M a x w e l l i a n . The t h e o r y o f Chapter I I I has been used i n the c a l c u l a t i o n o f the e l e c t r o n d e n s i t y , t e m p e r a t u r e and d r i f t v e l o c i t y . The c a l c u l a t e d v a l u e s o f the e l e c t r o n c o l l i s i o n f r e q u e n c i e s c o r r o b o r a t e the use of the p e r t u r b e d M a x w e l l i a n . 5:3.2 O s c i l l o g r a p h s o f I £ O s c i l l o g r a p h s o f the F a r a d a y cup c u r r e n t , I , a t p r e s s u r e s up to 74 200 m t o r r - H ^ i n d i c a t e t h a t t h e r e i s on a x i s i o n i z a t i o n w h i c h c o n t r i b u t e s to the b e g i n n i n g o f the c u r r e n t f l o w . I n F i g u r e 5:3-1, o s c i l l o s c o p e t r a c e s of I c f o r 200 m t o r r - H ^ a r e p r e s e n t e d . I n the f i r s t example, the anode h o l e has been c l o s e d by a s m a l l , b r a s s diaphragm. A p a r t from the t r i g g e r n o i s e , t h e r e i s l i t t l e p i c k u p by the cup when t h e r e i s no e l e c t r o n c u r r e n t . I (Anode h o l e c l o s e d ) c I (Maximum = 70 kA) I (Anode h o l e open) r! * i 500 nsec/cm F i g u r e 5:3-1 1^ a t 200 m t o r r - H As the p r e s s u r e i s r e d u c e d , t h e i n i t i a l c u r r e n t c a r r y i n g mechanism appears to be dominated by f a s t e l e c t r o n s i n the c e n t r a l r e g i o n of the d i s c h a r g e v e s s e l . An example of the t y p e of d a t a t a k e n a t 10 m t o r r - F ^ i s shown i n F i g u r e 5:3-2. No d e l a y l i n e has been used f o r t h i s sequence and the t r a c e s were t a k e n s i m u l t a n e o u s l y on two T e k t r o n i x , type 551, dual-beam o s c i l l o -s c o p es. The c u r r e n t d e r i v a t i v e , d l / d t , i s common to b o t h p h o t o g r a p h s f o r the p u r p o s e of c o r r e l a t i n g the waveforms. The r e t a r d i n g g r i d v o l t a g e was h e l d a t -5 k V so t h a t I C j v 3 r e p r e s e n t s a c u r r e n t o f e l e c t r o n s w i t h e n e r g i e s g r e a t e r t h a n 5 keV. 100 nsec/cm (a) (c) (a) (c) 200 nsec/cm (a) dl/dt('-v-19 k A / ^ s e c / c m ) 10 m t o r r - H . (b) I (0.8 amps/cm) (c) V (40 kV maximum, non l i n e a r s c a l e ) g •5 kV V V 12 g g. N Anode F l u x o f E l e c t r o n s < PLASMA E l e c t r i c F i e l d F i g u r e 5:3-2 O s c i l l o g r a p h s o f d l / d t , V and I 76 5:3.3 Time R e s o l u t i o n and R e p r o d u c i b i l i t y A t p r e s s u r e s below 20 m t o r r - H . i t has been p o s s i b l e t o o b s e r v e I 2 c w i t h o u t o b s c u r a t i o n by t r i g g e r n o i s e . Such o b s e r v a t i o n s a r e p o s s i b l e because the s t a t i s t i c a l time l a g f o r t h e s e p r e s s u r e s i s g r e a t e r than the time r e q u i r e d f o r the decay o f the t r i g g e r noise„ S i n c e the n o i s e decays i n about 1 yUsec., i t can be seen f r o m F i g u r e 5:2-2 t h a t p r e s s u r e s below 20 m t o r r - R ^ a r e r e q u i r e d . The time r e s o l u t i o n under t h e s e c i r c u m s t a n c e s i s q u i t e good. I t i s p o s s i b l e t o o b s e r v e the s t r u c t u r e d n a t u r e o f I and to c o r r e l a t e the r i s e and f a l l o f I w i t h f l u c t u a t i o n s on V and d l / d t . c When the p r e s s u r e i s r a i s e d , n o i s e i n t e r f e r e n c e becomes i n t o l e r a b l e so t h a t d e l a y l i n e s have to be i n t r o d u c e d i n t o t he c i r c u i t . The 80 n s e c . r i s e time o f the d e l a y l i n e s r e d u c e s the t e m p o r a l r e s o l u t i o n o f the d e t e c -t i o n system. When the d e l a y l i n e s a r e combined w i t h the f a s t e r growth r a t e of the e l e c t r o n c u r r e n t and the boundary l a y e r a t the h i g h e r p r e s s u r e s , i t becomes p r o g r e s s i v e l y more d i f f i c u l t to o b s e r v e the t e m p o r a l b e h a v i o u r of I c - However, i t i s p o s s i b l e to make a good e s t i m a t e of the maximum v a l u e of 1^. The r e d u c t i o n of t e m p o r a l r e s o l u t i o n due to the p r e s e n c e o f the d e l a y l i n e i s i l l u s t r a t e d i n F i g u r e 5:3-3 and the use o f the d e l a y l i n e i s d i s c u s s e d f u r t h e r i n A p p e n d i x C. 10 m t o r r - H I I c t t D e l a y L i n e No D e l a y L i n e F i g u r e 5:3-3 I n f l u e n c e o f the Dela y L i n e on I The i n i t i a l r a t e o f r i s e o f I has good r e p r o d u c i b i l i t y . The maxi-mum h e i g h t o f the I p u l s e f l u c t u a t e s w i t h i n a range o f about 307o of i t s average v a l u e . About one d i s c h a r g e i n twenty r e s u l t s i n a maximum v a l u e of I wh i c h i s two o r t h r e e times the ave r a g e v a l u e o f the I maximum, c c These l a r g e p u l s e s may be due t o n o n - u n i f o r m i t i e s o f the plasma p a s s i n g i n f r o n t o f the anode h o l e . When the Z - p i n c h v e s s e l i s w e l l o u t g a s s e d , few s h o t s r e s u l t i n a maximum of I much lo w e r t h a n the average. When no d e l a y l i n e i s used, a number of o s c i l l a t i o n s a r e o b s e r v e d a f t e r t he f i r s t maximum o f I , as shown i n F i g u r e 5 : 3 - 3 ( b ) . These f l u c t u a t i o n s u s u a l l y number two o r t h r e e and v a r y i n s i z e . The shape of d l / d t has been used as a measure of the r e p r o d u c i b i l i t y of the plasma f o r m a t i o n . As p o i n t e d o u t i n S e c t i o n 5:2.4, two s h o t s a r e c o n s i d e r e d to be i d e n t i c a l i f t h e i r d l / d t waveforms a r e a p p r o x i m a t e l y the same. The i n i t i a l c o n d i t i o n w h i c h appears t o i n f l u e n c e the r e p r o d u c i b i l i t y most i s t h e o u t g a s s i n g o f the v e s s e l . U s u a l l y a pumping time o f about f o u r o r f i v e h o u r s , combined w i t h a sequence o f about twenty " b u r n i n g o u t " d i s -charges were r e q u i r e d to a c h i e v e r e p r o d u c i b l e waveforms a f t e r the e x p o s u r e of the v e s s e l t o l a b o r a t o r y p o l l u t a n t s . The s e t t i n g o f the i n i t i a l p r e s s u r e and the bank v o l t a g e a f f e c t the shape and d u r a t i o n o f the p u l s e . 5:3.4 The I Waveforms The p u l s e o f h i g h energy e l e c t r o n s has been most e x t e n s i v e l y examined f o r h y drogen. I n F i g u r e s 5:3-4 t o 5:3-6, I c i s r e p r e s e n t e d as a f u n c t i o n of time f o r v a r i o u s p r e s s u r e s i n B.^. The r e t a r d i n g g r i d v o l t a g e i n t h e s e cases i s z e r o . The waveform of d l / d t has a l s o been i n c l u d e d . The e l e c t r o n c u r r e n t i s a l s o p r e s e n t when o t h e r gases a r e used i n the d i s c h a r g e v e s s e l . F i g u r e 5:3-7 shows the r e s u l t when He a t a p r e s s u r e o f 20 m t o r r i s use d . 78 F i g u r e 5:3-5 The F a r a d a y Cup C u r r e n t a t 30 m t o r r - H 79 c d l / d t .2 .4 .6 .8 1.0 t ( j * s e c ) F i g u r e 5:3-7 The F a r a d a y Cup C u r r e n t a t 20 m t o r r - H e 80 5:3.5 The Temporal C o r r e l a t i o n of V, d l / d t and I I n F i g u r e 5:3-8, the t e m p o r a l c o r r e l a t i o n o f V, d l / d t and I i s shown. These waveforms were r e c o r d e d a t 20 m t o r r - R ^ w i t h -3 k V on the r e t a r d i n g g r i d . No d e l a y l i n e has been used i n the m o n i t o r i n g c i r c u i t so t h a t the f i n e s t r u c t u r e o f the waveforms has n o t been o b s c u r e d . The temp-o r a l c o r r e l a t i o n r e p r e s e n t e d h e r e i s o b s e r v e d f o r a l l s h o t s w h i c h were examined. The i n i t i a l f a l l of I c o r r e s p o n d s t o a r i s e i n the v o l t a g e a c r o s s the tube. T h i s v o l t a g e r i s e has been r e f e r r e d t o p r e v i o u s l y as V^. Sub-sequent o s c i l l a t i o n s i n the a x i a l c u r r e n t seem to be r e l a t e d to peaks on Vf . The t e m p o r a l c o r r e l a t i o n between and the d i s a p p e a r a n c e of I s u p p o r t s the v i e w t h a t i s the back e.m.f. caused by the d i s a p p e a r a n c e of the e l e c t r o n c u r r e n t i n the c e n t r a l r e g i o n o f the d i s c h a r g e v e s s e l . These comments are r e l a t e d t o t h e m a t e r i a l of S e c t i o n 2:5. 5:3.6 R e l a x a t i o n of E l e c t r o n Energy The r e l a x a t i o n of the average e l e c t r o n energy as a f u n c t i o n o f time can be o b s e r v e d a t 10 mtorr-R^. The energy d i s t r i b u t i o n i s found f r o m v a l u e s o f I p l o t t e d as a f u n c t i o n of time w i t h the r e t a r d i n g g r i d v o l t a g e t r e a t e d as a parameter. Such c u r v e s have been drawn i n F i g u r e 5 : 3 - 9 ( a ) . I n F i g u r e 5 : 3 - 9 ( b ) , I has been p l o t t e d as a f u n c t i o n of the r e t a r d i n g g r i d v o l t a g e , V , f o r t h r e e times (0.14, 0.19 and 0.24 Jxsec.)„ I n F i g u r e 5:3-10, the c u r v e s have been n o r m a l i z e d to u n i t c u r r e n t f o r V = 0. The s h i f t i n the maximum s l o p e of the c u r v e to l o w e r v o l t a g e s r e p r e s e n t s a drop i n the average energy o f the e l e c t r o n s . T h i s s t a t e m e n t i s based on the r e l a t i o n s h i p o f the energy d i s t r i b u t i o n f u n c t i o n , f ( V ) , to the e l e c t r o n c u r r e n t to the cup, I (V ) , w h i c h can be w r i t t e n as (see 81 82 t ( y u s e c ) F i g u r e 5:3-9(a) F a r a d a y Cup C u r r e n t as a F u n c t i o n o f Time w i t h V as a Parameter F i g u r e 5:3-9(b) F a r a d a y Cup C u r r e n t as a F u n c t i o n o f V w i t h Time as a Parameter g 10 m t o r r - H ^ o 0.14 yusec. + 0.19 yusec. -5 -10 -15 -20 -25 V (kV) g F i g u r e 5:3-10 N o r m a l i z e d I c ( v g ) f o r t = 0.14 y / s e c . , 0.19yusec. and 0.24 yusec. oo 84 e q u a t i o n 3:5-3). f(v ) °^ ITT I (V ) g dV c v g' From F i g u r e 5:3-10, i t i s e v i d e n t t h a t the a v e r a g e e l e c t r o n energy r e l a x e s f r o m a v a l u e of more t h a n 20 keV to about 7 keV d u r i n g the time i n t e r v a l f rom 0.14 jusec. to 0.24 ^usec. 5:3.7 I (V ) f o r 30 and 50 m t o r r - H 0 c v g' 2 D e l a y l i n e s have been used i n the t a k i n g o f d a t a a t b o t h 30 and 50 mtorr-H„. I n F i g u r e s 5 : 3 - l l ( a ) and 5 : 3 - 1 2 ( a ) , I ( t ) has been r e p r e s e n t e d 1 c,V g f o r v a r i o u s v a l u e s of V , The peak v a l u e o f the e l e c t r o n c u r r e n t p u l s e i s chosen and I (V ) i s p l o t t e d i n F i g u r e s 5 : 3 - l l ( b ) and 5:3-12(b). A > t ^ . g smooth cu r v e has been drawn t h r o u g h the p o i n t s t o r e p r e s e n t I (V ) . c > ^ g 5:3.8 C a l c u l a t i o n o f the E l e c t r o n Temperature The e l e c t r o n t e m p e r a t u r e can be c a l c u l a t e d f r o m I (V ). I n o r d e r c 8 t o a c c o m p l i s h t h i s , the I C ( V ) c u r v e s i n F i g u r e s 5 : 3 - l l ( b ) and 5:3-12(b) a r e f i r s t n o r m a l i z e d so t h a t I c ( 0 ) = 1, and t h e n r e p l o t t e d w i t h s e m i - l o g s c a l e s . The e x p e r i m e n t a l p o i n t s appear i n F i g u r e 5:3-13. I f the d i s t r i -b u t i o n of the e l e c t r o n v e l o c i t i e s has r e l a x e d t o a p e r t u r b e d M a x w e l l i a n , t h e n the p o i n t s w i l l f a l l on a s t r a i g h t l i n e . The t e m p e r a t u r e can be c a l c u l a t e d from the s l o p e (see e q u a t i o n 3:7-8). The s t r a i g h t l i n e s w h i c h have been drawn i n F i g u r e 5:3-13 have been f i t -t e d t o the e x p e r i m e n t a l p o i n t s w i t h a s t a n d a r d l e a s t s q u a r e s method. The c e n t r e of g r a v i t y o f the p o i n t s i s g i v e n by i v . V = - — ( 5 : 3 - 1 ) g . n 85 a t ( y U s e c ) F i g u r e 5:3-11(a) F a r a d a y Cup C u r r e n t as a F u n c t i o n of Time V (kV) F i g u r e 5:3-11(b) F a r a d a y Cup C u r r e n t as a F u n c t i o n o f the G r i d V o l t a g e 0 1 h 0.0 -2 kV -4 kV -10 kV d l / d t F i g u r e 5:3-12(a) F a r a d a y Cup C u r r e n t as a F u n c t i o n o f Time 1.0 2 r t = 0.36 jmsec 30 mtorr-H„ to a, to V (kV) g F i g u r e 5:3-12(b) F a r a d a y Cup C u r r e n t as a F u n c t i o n of The G r i d V o l t a g e 87 n 5 1 I n I .(V ) l n I c (y = i n C ' L « (5:3-2) Here, n i s the number of d a t a p o i n t s , I .(V ) . The s l o p e of the b e s t c > 1 g f i t t i n g s t r a i g h t l i n e t h r o u g h (V , l n I (V )) i s g i v e n by g e g n (V . - V ~ ) l n I .(V ) b = i _ _ S ^ 8 ( 5 ; 3 _ 3 ) (v B . - "vJ 2 g j l S The r e c i p r o c a l o f b i s the e l e c t r o n t e m p e r a t u r e , T . Two s l i g h t l y d i f f e r e n t c a l c u l a t i o n s o f T g have been c a r r i e d o u t . I n the f i r s t method, a l l o f the d a t a p o i n t s have been used. The r e s u l t -i n g s t r a i g h t l i n e i s r e p r e s e n t e d by the s o l i d c u r v e i n F i g u r e 5:3-13. The v a l u e s o f T a r e : e 50 mtorr-H„ T = 3.0 keV - 20% 1 e 30 m t o r r - H T = 2.3 keV - 20% 2 e I n t h e second method, we t a k e i n t o c o n s i d e r a t i o n the poor s e n s i t i -v i t y of the c o l l e c t i n g s ystem t o low energy e l e c t r o n s . The low s e n s i t i v i t y i s caused by the d e f l e c t i o n o f low energy e l e c t r o n s o u t of the system by the s e l f - e l e c t r o s t a t i c f o r c e s of the e l e c t r o n s t r e a m as d i s c u s s e d i n A p pendices A and B. There appears t o be a n a t u r a l c u r v e i n the s e m i - l o g graphs of F i g u r e 5:3-13 a t V = 1 kV. T h e r e f o r e , i n t h i s c a l c u l a t i o n o f T g we n e g l e c t the d a t a p o i n t s below 1 kv. The s t r a i g h t l i n e b e s t f i t t i n g t h e s e p o i n t s i s r e p r e s e n t e d by the b r o k e n c u r v e and the t e m p e r a t u r e s a r e : 50 m t o r r - H 0 T = 2.3 keV - 20% 2 e 30 m t o r r - H 0 T = 1 . 8 k e V - 2 0 % 2 e 5:3.9 C a l c u l a t i o n o f the E l e c t r o n D e n s i t y The e l e c t r o n d e n s i t y , n , the e l e c t r o n d r i f t v e l o c i t y , v , and the 88 89 e f f e c t i v e e l e c t r i c f i e l d , E d , can be c a l c u l a t e d from the d i s t r i b u t i o n f u n c t i o n . F o r a p e r t u r b e d M a x w e l l i a n , the most p r o b a b l e v e l o c i t y , v , i s g i v e n by 1 ' 2 •r m v = e T 2 e m e I n the d e f i n i t i o n o f T , i t i s assumed t h a t the t e m p e r a t u r e o f the n e u t r a l s i s n e g l i g i b l e (see e q u a t i o n 3:4-7). The e l e c t r o n d r i f t e n e r g y , V^, can be r e p r e s e n t e d by 3 e m V = - — T d 2 M e w i l l be i n e l e c t r o n v o l t s ( s e e e q u a t i o n 3:7-9). The e l e c t r o n d e n s i t y can be c a l c u l a t e d f r o m the d r i f t v e l o c i t y and the c u r r e n t d e n s i t y . J = n v , e e d The v a l u e o f J i s t a k e n from the c e n t e r probe s i g n a l o f S e c t i o n 5:4. The c u r r e n t d e n s i t y g i v e n t h e r e i s 300 amps/cm^ ^ 307o a t 50 m t o r r - H ^ 2 + and 250 amps/cm - :307o a t 30 m t o r r - I ^ . From the d e f i n i t i o n o f v ^ i t i s p o s s i b l e to c a l c u l a t e an e f f e c t i v e v a l u e o f the e l e c t r i c f i e l d . v _JL_ m _Ze d - v E m ey E The v a l u e s o f T c a l c u l a t e d i n t h e l a s t s e c t i o n and the v a l u e s o f n , e e V,, and E, a r e a l l l i s t e d i n T a b l e 5:3-1. d d 5:3.10 Evidence i n S u p p o r t o f the P e r t u r b e d M a x w e l l i a n The c a l c u l a t i o n o f T , n and E. depend on the v e l o c i t y d i s t r i b u -e' e d r J t i o n b e i n g a p e r t u r b e d M a x w e l l i a n . The v a l u e s o f the c o l l i s i o n f r e -q u e n c i e s V„, YCv V and V have been c a l c u l a t e d to see w h i c h E I x *ce 90 TABLE 5:3-1 RESULTS OF THE FARADAY CUP MEASUREMENTS E l e c t r i c P r e s s u r e Temperature D r i f t D e n s i t y F i e l d ( m t o r r - H 2 ) (ke.V ± 20%) (eV±20%) (cm"3 + 40%) (Volts/cm± 30%) 50 (0 < V<oo) 3.0 1.2 3 x 1 0 1 3 8 (1 < V<co) 2.3 1.0 3.2 x 1 0 1 3 7 30 (0 < V<co) 2.3 1.0 3.4 x 1 0 1 3 4 (1 < v«») 1.8 0.8 3.5 x 1 0 1 3 4 10 (0 < V<co) 4.8 2.0 2.3 x 1 0 1 3 2 (1 < v<co) 4.5 1,9 2.3 x 1 0 1 3 2 91 c o l l i s i o n p r o c e s s i s t h e most i m p o r t a n t . The v a l u e s l i s t e d below a r e c a l c u l a t e d f r o m the f o r m u l a e i n T a b l e 3:2-1 w i t h T = 2.5 keV and n = e e -3 2 x 10 cm . The base p r e s s u r e i s 50 m t o r r - H 9 . E l a s t i c C o l l i s i o n s I o n i z i n g C o l l i s i o n s I o n C o l l i s i o n s E l e c t r o n C o l l i s i o n s The e l a s t i c c o l l i s i o n s a r e f a r more f r e q u e n t t h a n o t h e r c o l l i s i o n s , and the e l e c t r o n - e l e c t r o n e n c o u n t e r s a r e so i n f r e q u e n t t h a t t h e y must be r e g a r d e d as c o m p l e t e l y i n s i g n i f i c a n t . S i n c e the e l e c t r o n - e l e c t r o n c o l l i s i o n s f r e q u e n c y i s so low, the use o f a d i s p l a c e d M a x w e l l i a n f o r t h i s t y p e of d i s c h a r g e would appear to be i n v a l i d . The f i t o f the d a t a p o i n t s by a s t r a i g h t l i n e on the s e m i - l o g g r a p h o f I . ( V ) i s c o n s i s t e n t e g w i t h the p r e s e n c e o f a p e r t u r b e d M a x w e l l i a n d i s t r i b u t i o n o f v e l o c i t i e s . 'The q u a n t i t i e s , 1^, n^, and E, have been c a l c u l a t e d on the assump-t i o n t h a t t h e r e aire no e l e c t r i c f i e l d f l u c t u a t i o n s o r plasma t u r b u l e n c e i n the d i s c h a r g e . There i s some e v i d e n c e t h a t s u c h phenomena may be i m p o r t a n t i n low p r e s s u r e , h i g h f i e l d s t r e n g t h d i s c h a r g e s and t h a t s c a t t e r i n g f r o m such f l u c t u a t i o n s may be more i m p o r t a n t t h a n e l a s t i c s c a t t e r i n g . Anomalous, h i g h r e s i s t i v i t y a t t r i b u t a b l e t o t h i s p r o c e s s has been o b s e r v e d i n Z - p i n c h e s d u r i n g t h e c o l l a p s e phase, and i t i s c u r r e n t l y b e i n g s t u d i e d a t Los Alamos. On t h i s b a s i s , the p e r t u r b e d M a x w e l l i a n model may n o t be a p p l i c a b l e and the c a l c u l a t e d v a l u e s - i n T a b l e 5:3-1 may be of d o u b t f u l v a l u e . The r e l i a b i l i t y of t h e s e measurements can o n l y be e s t a b l i s h e d by an i n d e p e n d e n t measurement o f e i t h e r T o r n . e -e T h i s p a r a g r a p h i s based on remarks by Dr. H. M. S k a r s g a r d ^ f V = 3 x 1 0 8 s e c " 1 h VT = 0.13 x 1 0 8 s e c " 1 3 -1 = 6 x 10 s e c \) = 18 x 1 0 3 s e c " 1 "ce = 1 : 0.04 : 2 x 1 0 - 5 : 6 x 10 92 5:4 The R a d i a l D i s t r i b u t i o n o f the C u r r e n t a t the Anode 5:4.1 I n t r o d u c t i o n We s h a l l now d i s c u s s the r a d i a l d i s t r i b u t i o n of t h e c u r r e n t a t the anode d u r i n g the f o r m a t i v e phase of the d i s c h a r g e . The method i n v o l v e s the use o f e l e c t r o n c o l l e c t o r s p l a c e d b e h i n d s m a l l h o l e s l o c a t e d a t d i f f e r e n t r a d i a l p o s i t i o n s i n the anode. The r a d i a l probes m o n i t o r the t o t a l f l u x o f e l e c t r o n s t h r o u g h the h o l e s . The probe measurements become u n r e l i a b l e f o r r a d i a l p o s i t i o n s g r e a t e r t h a n 2 cm. from the a x i s because of the m a g n e t i c f i e l d o f the d i s c h a r g e c u r r e n t . The c u r r e n t , I ^ C t ) , w h i c h i s d i s c u s s e d i n t h i s s e c t i o n , i s the c u r r e n t of e l e c t r o n s i n c i d e n t on the c e n t r a l r e g i o n s of t h e anode. I t i s d e f i n e d by r I e ( r , t ) = 2?r(J r , J ( r ' , t ) d r ' (5:4-1) w i t h r = r Q ; the r a d i a l p o s i t i o n of the probe f a r t h e s t away from the c e n t e r . C u r r e n t p r o f i l e s , J ( r ) , have been drawn f o r a few s p e c i f i c t i m e s and p r e s s u r e s . An a p p r o x i m a t e form o f I £ ( t ) has been found by assuming J ( r ) t o have a t r i a n g u l a r form. D u r i n g the f o r m a t i v e phase, I g ( t ) con-t r i b u t e s more than h a l f of the t o t a l c u r r e n t , I , a r e s u l t w h i c h i s i n agreement w i t h the model of boundary l a y e r f o r m a t i o n p r e s e n t e d i n C h a p t e r I I . 5:4.2 M a g n e t i c F i e l d L i m i t a t i o n s o f t h e O f f - a x i s P robes To measure I ( t ) , r a d i a l probes a r e p l a c e d 2 mm. b e h i n d the e n t r a n c e h o l e s i n the anode a t r a d i a l p o s i t i o n s of 0, 1, 2 and 3 cm. as d e s c r i b e d i n S e c t i o n 4:5. The e l e c t r o n s p a s s i n g t h r o u g h the h o l e s a t o f f - a x i s p o s i t i o n s a r e d e f l e c t e d by the m a g n e t i c f i e l d o f the d i s c h a r g e c u r r e n t . T h e r e f o r e , the c o l l e c t i o n e f f i c i e n c y o f the probes i s a f f e c t e d by the s t r e n g t h o f the m a g n e t i c f i e l d . We now d i s c u s s t h e i n t e r p r e t a t i o n of the probe c u r r e n t s , I ( t ) . F i g u r e 5:4-1 shows one o f the p r o b e s . F i g u r e 5:4-1 E n l a r g e d View o f an O f f - A x i s Probe F o r s i m p l i c i t y , assume t h a t o n l y t h o s e e l e c t r o n s w i t h c y c l o t r o n r a d i u s g r e a t e r t h a n 1 cm. w i l l r e a c h the probe a f t e r p a s s i n g t h r o u g h the h o l e . L e t the component of v e l o c i t y normal to the m a g n e t i c f i e l d be r e p r e s e n t e d by ( V ) 2 , where V i s the k i n e t i c e n e r g y . I f the probe i s a t a r a d i a l p o s i t i o n r , th e n the e l e c t r o n s must have energy g r e a t e r than V g i v e n by I ( r ) = 16.5 r -JT -^PJL- ( 5 :4-2) c m ( e V ) 2 E q u a t i o n 5:4-1 d e f i n e s I ( r ) . V has been p l o t t e d i n F i g u r e 5:4-2 as a f u n c t i o n o f I ( r ) . e To g e t some i d e a o f the probe l i m i t a t i o n , c o n s i d e r t h a t J ( r ) i s a 94 t r i a n g u l a r d i s t r i b u t i o n g i v e n by J ( r ) = J (1 - — ) (5:4-3) o r o I t t u r n s o u t e x p e r i m e n t a l l y t h a t J ( r ) i s o n l y a s m a l l f r a c t i o n o f J ^ , the c u r r e n t d e n s i t y a t the c e n t e r , f o r r = 3 cm. Thus i t i s assumed t h a t o r = 3 cm. and J ( r ) = 0 i n the c a l c u l a t i o n s , o o The t o t a l c u r r e n t a t r a d i u s , r , i s r I ( r ) = 2 f T j \ r ' ( l - — ) d r ' (5:4-4) e o ) r J o The c u t - o f f energy has been c a l c u l a t e d f o r d i f f e r e n t v a l u e s o f J and f o r o probe p o s i t i o n s o f 1, 2 and 3 cm. The r e s u l t s a r e p l o t t e d i n F i g u r e 5:4-3. The s c a l e on the r i g h t o f the f i g u r e shows the p e r c e n t a g e o f e l e c t r o n s i n the d i s c h a r g e w h i c h have energy g r e a t e r than the energy on the l e f t hand s c a l e . I t has been assumed t h a t the d i s t r i b u t i o n o f the e l e c t r o n v e l o c i t i e s i s d e s c r i b e d by a p e r t u r b e d M a x w e l l i a n w i t h a tempera-t u r e o f 2.5 keV. F i g u r e 5:4-3 can be used to a r r i v e a t an e s t i m a t e o f the p e r c e n t a g e of the c u r r e n t w h i c h a p a r t i c u l a r probe w i l l sample. C o n s i d e r , f o r example, the c u r r e n t probe a t the 2 cm. p o s i t i o n . F o r a c u r r e n t d e n s i t y o f 300 2 amps/cm a t the c e n t e r , l e s s t h a n 30? o of the e l e c t r o n s w i l l be c o l l e c t e d , 2 w h i l e a t 200 amps/cm , about 50?o w i l l be r e c o r d e d . 5:4.3 R e p r o d u c i b i l i t y and Time R e s o l u t i o n S i g n a l s , I ( t ) , from the f o u r probes a r e r e c o r d e d s i m u l t a n e o u s l y P 3 ^  on two T e k t r o n i x 551, dual-beam o s c i l l o s c o p e s . Type G d i f f e r e n t i a l p l u g -i n s and A minus B t e c h n i q u e s as d e s c r i b e d i n S e c t i o n 4:4 a r e used. T i m i n g of the waveforms w i t h r e s p e c t t o o t h e r s i g n a l s has been a c h i e v e d by 95 2 cm. 1 cm. F i g u r e 5:4-2 E l e c t r o n s w i t h C y c l o t r o n R a d i u s G r e a t e r t h a n 1 cm. F i g u r e 5:4-3 E l e c t r o n s C o l l e c t e d by t h e C u r r e n t Probe 96 r e p l a c i n g one o f the probe s i g n a l s w i t h d l / d t . The problems r e g a r d i n g time r e s o l u t i o n and r e p r o d u c i b i l i t y a r e i d e n t i c a l to thos e d i s c u s s e d i n S e c t i o n 5:3.3. I n F i g u r e 5:4-4, examples of I ( t ) f o r a p r e s s u r e o f 10 m t o r r - H - a r e shown. P ™ 1 (a) (b) B CO o 100 nsec/cm N r . . (a) (b) (a) r = 2 cm (b) r = 1 cm F i g u r e 5:4-4 O s c i l l o g r a p h o f I ( t ) P No d e l a y l i n e has been used i n t h e s e p h o t o g r a p h s . The v a r i a t i o n i n t h e s e waveforms i s t y p i c a l o f th e s e measurements. The i n t r o d u c t i o n o f the d e l a y l i n e o b s c u r e s much of the s t r u c t u r e as shown i n F i g u r e 5:4-5. ( c ) (d) / *- / s 100 nsec/cm No D e l a y L i n e 200 nsec/cm D e l a y L i n e (a) r = 2 cm (b) r = 0 (c ) d l / d t (d) r = 0 F i g u r e 5:4-5 E f f e c t o f the D e l a y L i n e on I ( t ) 5:4.4 P r o f i l e s a t the Anode I ( t ) waveforms r e c o r d e d a t 4 0 , 20 and 1 0 mtorr-H_ have been p , r 2 averaged and the r e s u l t a n t c u r v e s have been shown i n F i g u r e s 5 : 4 - 6 ( a ) to 5 : 4 - 8 ( a ) . The probe c u r r e n t , I ( t ) , has been c o n v e r t e d to u n i t s o f p , r c u r r e n t d e n s i t y by u s i n g the e q u a t i o n X n r ( t ) J ( r ) = f — ( 5 : 4 - 5 ) whe r e A^ i s the a r e a of the h o l e . Next, v a l u e s of ( t ) a r e p l o t t e d as a f u n c t i o n of r ^ f o r a f i x e d v a l u e of t . The r e s u l t i s the c u r r e n t d e n s i t y p r o f i l e , J ( r ) , w h i c h i s t o be f o u n d i n F i g u r e s 5 : 4 - 6 ( b ) t o 5 : 4 - 8 ( b ) . The v a l u e o f I ( t ) i s c o n s i d e r e d t o be r e l i a b l e i f 707o of the P e l e c t r o n s , as g i v e n by F i g u r e 5 : 4 - 3 , s t r i k e the probe. I n the J ( r ) p r o -f i l e s , t he s o l i d p a r t s of the c u r v e a r e c o n s i d e r e d t o be r e l i a b l e . From the p r o f i l e s , i t appears t h a t the p u l s e of e l e c t r o n c u r r e n t , I e ( t ) , i s r e s t r i c t e d t o the c e n t r a l p o r t i o n of the d i s c h a r g e v e s s e l i n the n e i g h -bourhood of the anode. F a r t h e r away, t h i s may n o t be t r u e . 5:4.5 A p p r o x i m a t i o n of I e ( t ) S i n c e the e x a c t form of J ( r , t ) i s n o t known, an a p p r o x i m a t e method of c a l c u l a t i n g I e ( t ) w i l l be adopted. The e r r o r i n a d o p t i n g t h i s method i s l e s s t h a n the 407„ e x p e r i m e n t a l e r r o r i n J ( r ) . The c u r r e n t d e n s i t y i s assumed to have a t r i a n g u l a r p r o f i l e as d e s c r i b e d by e q u a t i o n 5:4-3. I f we l e t J q be the c u r r e n t d e n s i t y a t the c e n t e r o f the anode, as g i v e n by the measurements of the p r e v i o u s s e c t i o n , and l e t J ( 3 ) = 0, th e n r 3 I ( t ) = J ( t ) 2ir\ r ( l - ^ ) d r Jo = 9.4 J 0 ( t ) ( 5 : 4 - 6 ) 98 t ( n s e c ) F i g u r e 5:4-6(a) C u r r e n t Probe S i g n a l s as a F u n c t i o n o f Time F i g u r e 5:4-6(b) C u r r e n t P r o f i l e a t the Face of t h e Anode p r i o r t o the F o r m a t i o n of the Boundary L a y e r t ( n s e c ) F i g u r e 5:4-7(a) C u r r e n t Probe S i g n a l s as a F u n c t i o n of Time 3+ ( r) (amps/cm ) 1 .300 20 m t o r r - H ^ / \ -200\ • t / / y • • \ \ \ . 100 \' < \ i i i , i -. u.-i„ 1 ! 1 1 1 5 4 3 2 1 0 1 2 3 4 5 r (cm) F i g u r e 5:4-7(b) C u r r e n t P r o f i l e a t t h e Face o f the Anode p r i o r to the F o r m a t i o n of the Boundary L a y e r 100 300„ 6 o CO 200 U 100 • d l / d t • c e n t e r • 1 cm • 2 cm 3 cm F i g u r e 5:4-8(a) C u r r e n t Probe S i g n a l s as a F u n c t i o n of Time 10 m t o r r - H J ( r ) (amps/cm ) 1 — 4 0 0 • = 320 n s e c + = 200 n s e c 4 5 r (cm) F i g u r e 5:4-8(b) C u r r e n t P r o f i l e a t t h e F a c e o f t h e Anode p r i o r t o the F o r m a t i o n of t h e Boundary L a y e r 101 5:4.6 Comparison of I and I I t i s i n s t r u c t i v e t o compare I ( t ) t o d u r i n g the f o r m a t i v e phase, ^g^*-) i - s c a l c u l a t e d f r o m e q u a t i o n 5:4-6 w i t h v a l u e s of J 0 ( t ) t a k e n from F i g u r e s 5:4-6(b) to 5 : 4 - 8 ( b ) . The c a l c u l a t e d v a l u e s of I ( t ) e a r e r e p r e s e n t e d by the b r o k e n l i n e i n F i g u r e 5:4-9. I ( t ) i s found by a v e r a g i n g I ( t ) waveforms l i k e t h o s e p l o t t e d i n F i g u r e s 5:2-4 to 5:2-6. I t i s e v i d e n t from the f i g u r e s t h a t the major p o r t i o n of the c u r r e n t i s con-d u c t e d t o the c e n t r a l r e g i o n of the. anode up to the time o f the d i s -appearance of I e ( t ) . The d i s a p p e a r a n c e of I e ( t ) has been i n t e r p r e t e d as the s h i f t o f the c u r r e n t t o the w a l l r e g i o n and the f o r m a t i o n o f the boundary l a y e r . The decay time o f 100 n s e c . may be r e l a t e d to the time of f o r m a t i o n of the boundary l a y e r . The b e h a v i o u r of I e ( t ) agrees w i t h the s u g g e s t i o n t h a t marks the f o r m a t i o n of the c u r r e n t boundary l a y e r . The o b s e r v a t i o n of I e ( t ) s u g g e s t s t h a t i o n i z a t i o n i n i t i a l l y o c c u r s t h r o u g h o u t the volume of the v e s s e l . These o b s e r v a t i o n s a r e i n agreement w i t h t h e d i s c u s s i o n i n Chapter I I . 102 F i g u r e 5:4-9 Comparison of I ( t ) and I ( t ) e CHAPTER VI DISCUSSION AND CONCLUSIONS 6:1 I n t r o d u c t i o n The i n v e s t i g a t i o n s r e p o r t e d i n t h i s t h e s i s have l e d to a b e t t e r u n d e r s t a n d i n g o f the f o r m a t i v e phase of a low p r e s s u r e , h i g h v o l t a g e Z - p i n c h . The o b s e r v a t i o n s o f b o t h the s p a t i a l d i s t r i b u t i o n o f the c u r r e n t and the energy d i s t r i b u t i o n o f the e l e c t r o n s a t the f a c e o f the anode, d u r i n g the f o r m a t i v e phase, have n o t been p r e v i o u s l y r e p o r t e d . These o b s e r v a t i o n s i n d i c a t e t h a t i o n i z a t i o n b e g i n s t h r o u g h o u t the volume o f the v e s s e l r a t h e r t h a n a t the w a l l . The r e s u l t can be u n d e r s t o o d on the b a s i s o f a s p a t i a l l y v a r y i n g i o n i z a t i o n c o e f f i c i e n t w h i c h i s dependent on the i n i t i a l f i e l d d i s t r i b u t i o n i n the p i n c h v e s s e l . T h i s e x p e r i m e n t has been performed f o r p r e s s u r e s below 100 m t o r r - H ^ b u t the p r i n c i p l e s can be c a r r i e d to h i g h e r p r e s s u r e s . F o r example, the o b s e r v a t i o n s by Dimoff and Tam (1970) o f r a d i a t i o n from the a x i a l r e g i o n of a Z - p i n c h p r i o r to the measured i n i t i a l r i s e of the c u r r e n t , i s most l i k e l y r e l a t e d to t h i s phenomenon. I n v e s t i g a t o r s who a r e i n t e r e s t e d i n the c o l l a p s e phase o f low p r e s s u r e p i n c h e s s h o u l d t a k e i n t o a c c o u n t the s t r o n g p o s s i b i l i t y o f 1 p r e - i o n i z a t i o n ' o f the ambient gas i n f r o n t o f the c o l l a p s i n g c u r r e n t s h e e t . The F a r a d a y cup measurements o f the e l e c t r o n energy d i s t r i b u t i o n on the a x i s i n d i c a t e t h a t the e l e c t r o n s i n t e r a c t most s t r o n g l y w i t h the n e u t r a l s t h r o u g h e l a s t i c c o l l i s i o n s . T h i s s t a t e m e n t i s based on the con-s i s t e n c y of b o t h t h e f o r m o f I and the c a l c u l a t e d v a l u e s o f n , T and J c e e y w i t h the v a l u e s c a l c u l a t e d u s i n g a p e r t u r b e d M a x w e l l i a n d i s t r i b u t i o n E 103 104 f u n c t i o n . T h i s r e s u l t i s i n agreement w i t h the t h e o r e t i c a l d e s c r i p t i o n of a w e a k l y i o n i z e d gas under the i n f l u e n c e of a s t r o n g e l e c t r i c f i e l d (Tanenbaum, 1967). Thus, the e l e c t r o n s do n o t appear to e i t h e r s t r e a m o r runaway ( d i s p l a c e d M a x w e l l i a n ) b u t s c a t t e r about e q u a l l y i n a l l d i r e c t i o n s due to the e l e c t r o n - n e u t r a l i n t e r a c t i o n s . I t i s c o n c e i v a b l e t h a t b e t t e r t i m e r e s o l u t i o n would show s t r e a m i n g to be i m p o r t a n t p r i o r to the time when the p r e s e n t measurements were made. The boundary l a y e r f o r m a t i o n appears t o be a d r a m a t i c e v e n t . The o b s e r v a t i o n s o f and the f l u c t u a t i o n s i n d l / d t a r e new. The s i z e of t h e s e t r a n s i e n t s i s i n t e n s i f i e d because the p i n c h has been o p e r a t e d near the s t a t i c breakdown p o t e n t i a l . O p e r a t i o n of the d e v i c e i n t h i s r e g i o n , a l t h o u g h i n c r e a s i n g the s i z e of t h e s i g n a l s , does n o t i n v a l i d a t e the g e n e r a l c o n c l u s i o n s w h i c h can be drawn. However, i t s h o u l d be p o i n t e d o u t t h a t enhancement of the boundary l a y e r f o r m a t i o n by the m a g n e t i c f i e l d of the a x i a l c u r r e n t i s a low p r e s s u r e phenomenon. I t cannot be r e g a r d e d as i m p o r t a n t a t p r e s s u r e s much above th o s e i n v e s t i g a t e d i n t h i s e x p e r i m e n t . The e x p e r i e n c e g a i n e d from t h i s e x p e r i m e n t i n d i c a t e s t h a t the s i m p l e F a r a d a y cup and r e t a r d i n g g r i d s y s t e m i s a u s e f u l t o o l i n the s t u d y of the e a r l y s t a g e s of i o n i z a t i o n growth. The p r e s e n t d e s i g n has energy l i m i t a t i o n s w h i c h have been d i s c u s s e d . I f n o i s e s u p p r e s s i o n methods can be improved, t h i s t e c h n i q u e s h o u l d l e a d t o some v e r y i n t e r e s t i n g o b s e r v a -t i o n s o f the t e m p o r a l b e h a v i o u r o f the e l e c t r o n d i s t r i b u t i o n . O b s e r v a t i o n s of the i o n i z a t i o n phase c o u l d l e a d t o f u r t h e r u n d e r s t a n d i n g o f the f u n d a -m e n t a l p r o c e s s e s of i o n i z a t i o n and a b e t t e r d e s c r i p t i o n o f the d i s t r i b u t i o n of p a r t i c l e e n e r g i e s d u r i n g such t r a n s i e n t c o n d i t i o n s . 6:2 D e s c r i p t i o n of the F o r m a t i v e Phase When the v o l t a g e i s i n i t i a l l y a p p l i e d to the e l e c t r o d e s of the d i s -105 charge v e s s e l , i o n i z a t i o n by e l e c t r o n - n e u t r a l c o l l i s i o n s b e g i n s to o c c u r t h r o u g h o u t the v e s s e l . The s p a t i a l v a r i a t i o n o f i o n i z a t i o n depends on the p r e s s u r e and the e l e c t r i c f i e l d d i s t r i b u t i o n i n the gas. F o r a s m a l l , low p r e s s u r e , h i g h v o l t a g e p i n c h , the i o n i z a t i o n p r o g r e s s e s most r a p i d l y i n the i n t e r i o r of the v e s s e l . Thus, a s t r o n g c u r r e n t o f e l e c t r o n s , I , i s i n i t i a l l y o b s e r v e d t o be i n c i d e n t on the c e n t r a l r e g i o n of the anode. I n f a c t , I c o n t r i b u t e s more t h a n 50% of the t o t a l c u r r e n t e th r o u g h the d i s c h a r g e d u r i n g t h i s p e r i o d . The h i g h e l e c t r i c f i e l d i m p a r t s c o n s i d e r a b l e energy t o the e l e c t r o n s a t f i r s t . Because o f the i n c r e a s i n g c o n d u c t i v i t y o f the gas and t h e v o l t a g e d i v i s i o n on the d i s c h a r g e c i r c u i t e l e m e n t s , the e l e c t r i c f i e l d b e g i n s t o d e c r e a s e . The average e l e c t r o n energy i s o b s e r v e d t o d e c r e a s e as w e l l . When the c u r r e n t i s of s u f f i c i e n t s i z e , the m a g n e t i c f i e l d i n the r e g i o n o f the Z - p i n c h v e s s e l w a l l causes enhancement of i o n i z a t i o n , l e a d i n g to the f o r m a t i o n o f a c o n d u c t i n g l a y e r a t the w a l l . C a l c u l a t i o n s i n d i c a t e t h a t such enhancement i s s u b s t a n t i a l f o r p r e s s u r e s below 40 mtorr-R^. The c o n d u c t i v i t y o f the boundary l a y e r a t the w a l l o f the d i s c h a r g e i n c r e a s e and a c u r r e n t s k i n b e g i n s to form. The c u r r e n t I , e i n the c e n t r a l r e g i o n o f the v e s s e l t h e n d i s a p p e a r s w i t h a decay time o f about 100 nanoseconds. Because o f the decay o f the t r a p p e d m a g n e t i c f i e l d due to the a x i a l c u r r e n t , a back e.m.f. i s pro d u c e d . o c c u r s a t the time when I d i s a p p e a r s . The appearance of s u g g e s t s the f o r m a t i o n o f a c u r r e n t s h e e t w h i c h c o l l a p s e s i n the f a m i l i a r snowplough f a s h i o n . The time between the o c c u r r e n c e o f V_ and the f o r m a t i o n o f the p i n c h (T ) f P s c a l e s a p p r o x i m a t e l y as p z - - c o n s i s t e n t w i t h k t h e r e s u l t s o f Curzon ( 1 9 6 3 ) . T h i s r e s u l t s u p p o r t s the a s s u m p t i o n t h a t V o c c u r s when t h e boundary 106 l a y e r forms a t the w a l l of the d i s c h a r g e . 6:3 The P e r t u r b e d M a x w e l l i a n D e s c r i p t i o n o f the E l e c t r o n s The e l e c t r o n e n e r g i e s a r e i n i t i a l l y v e r y h i g h . As the e l e c t r i c f i e l d s t r e n g t h f a l l s , the e l e c t r o n d i s t r i b u t i o n r e l a x e s to a M a x w e l l i a n form. T h i s o b s e r v a t i o n has been made a t gas p r e s s u r e s o f 30 and 50 mtorr-R^. The M a x w e l l i a n d e s c r i b e s t h e d i s t r i b u t i o n o f e l e c t r o n v e l o c i t i e s a t the maximum of I . B e f o r e and a f t e r t h i s s h o r t p e r i o d , the f o r m of c r ' f ( c ) i s n o t known. C a l c u l a t i o n s of c o l l i s i o n f r e q u e n c i e s i n d i c a t e t h a t V i s much E l a r g e r t h a n \ ) ^ , \)^ and Vce' The d i s t r i b u t i o n o f the z-component o f v e l o c i t y has a M a x w e l l i a n form as i n d i c a t e d by the g r a p h of l n 1(V ) , 0 g These o b s e r v a t i o n s a r e c o n s i s t e n t w i t h the v i e w t h a t the e l e c t r o n v e l o c i t y d i s t r i b u t i o n can be r e p r e s e n t e d by a p e r t u r b e d M a x w e l l i a n . On t h i s b a s i s , i t i s c o n c l u d e d t h a t the e l e c t r o n m o t i o n i n s i d e t h e d i s c h a r g e r e l a x e s to an a l m o s t i s o t r o p i c s t a t e . D u r i n g t h i s p e r i o d of the f o r m a t i v e phase & the e l e c t r o n s do n o t r u n away because the e l e c t r o n - n e u t r a l e l a s t i c c o l l i -s i o n s dominate the c o l l i s i o n p r o c e s s e s . These c o n c l u s i o n s a r e based on the t h e o r y of Chapter I I I . 6:4 The F a r a d a y Cup as a D i a g n o s t i c T o o l The F a r a d a y cup has been a u s e f u l d i a g n o s t i c t o o l . I t p r o v i d e s a method of m e a s u r i n g b o t h T^ and n^. Under p r o p e r c i r c u m s t a n c e s , t h e s e measurements can be time r e s o l v e d . The c a l c u l a t i o n of T depends on the r e l a t i v e v a l u e s of I (V ) e c v g' through the s l o p e of the graph o f l n I (V ) as a f u n c t i o n o f V . F o r e g g t h i s r e a s o n , the t e m p e r a t u r e c o u l d be r e l i a b l y c a l c u l a t e d f r o m t h e s l o p e 107 of the h i g h energy p o r t i o n o f the c u r v e . T h i s i s f o r t u n a t e , s i n c e the low energy t r a n s m i s s i o n of the g r i d s i s n o t a s i m p l e g e o m e t r i c a l p r o p e r t y of the system, b u t depends on the e l e c t r o n v e l o c i t y as w e l l . I n the measurement of T , a p o r t i o n o f the d a t a w h i c h i s u n r e l i a b l e can thus be e a v o i d e d . The c a l c u l a t i o n o f n^ depends on t h r e e c o n d i t i o n s . I t i s n e c e s s a r y to know T and the c u r r e n t d e n s i t y a t the f a c e o f the anode. I t i s t h e n e n e c e s s a r y to e s t a b l i s h t h a t the v e l o c i t y d i s t r i b u t i o n i s a p e r t u r b e d M a x w e l l i a n so t h a t n^ can be r e l a t e d to the c u r r e n t d e n s i t y t h r o u g h the e q u a t i o n M J A ne = A/TTT" • T e The e x p e r i m e n t a l e r r o r i n n depends on the measurement of J and t o a 6 A l e s s e r degree on the v a l u e o f T . The compounded e x p e r i m e n t a l e r r o r i s about 407 o. The r e l i a b i l i t y o f the p e r t u r b e d M a x w e l l i a n a s s u m p t i o n i s based on the p a r a m e t e r s mentioned i n the p r e v i o u s s e c t i o n . I n b o t h the measurement o f T g and n^, the g r e a t e s t s o u r c e of e x p e r i m e n t a l e r r o r i s i n the r e p r o d u c i b i l i t y o f the breakdown of the gas. 6:5 S u g g e s t i o n s f o r F u r t h e r Work The r e s u l t s of t h i s e x p e r i m e n t i n d i c a t e t h a t t h e r e i s need f o r a t h e o r e t i c a l i n v e s t i g a t i o n o f t h e breakdown p r o c e s s and boundary l a y e r f o r m a t i o n i n the Z - p i n c h . I t s h o u l d be p o s s i b l e to i n v e s t i g a t e the two extremes of h i g h and low p r e s s u r e breakdown. The o b s e r v a t i o n s of t h i s t h e s i s s h o u l d h e l p t o e s t a b l i s h the p r o c e s s e s w h i c h would be i m p o r t a n t i n such a model. The e x p e r i m e n t s h o u l d be p e rformed a t h i g h e r base p r e s s u r e s and 108 c o n d i t i o n s o f l o w e r e l e c t r i c f i e l d s t r e n g t h i f p o s s i b l e . T h i s would g i v e a much b e t t e r check on the t h e o r y o f b o t h C h a p t e r I I and Cha p t e r I I I . The h i g h e r p r e s s u r e c o n d i t i o n s would e s p e c i a l l y a f f e c t t h e b a s i c assump-t i o n s l e a d i n g t o the use o f b o t h the Boltzmann's e q u a t i o n and the Townsend i o n i z a t i o n c o e f f i c i e n t . However, n o i s e e l i m i n a t i o n problems would have to be s o l v e d . End-on s p e c t r o s c o p i c o b s e r v a t i o n s o f t h e p i n c h d u r i n g t h e f o r m a t i v e phase would perhaps show e v i d e n c e o f the boundary l a y e r f o r m a t i o n . S i d e -on o b s e r v a t i o n s were a t t e m p t e d , b u t though e n c o u r a g i n g , t h e r e s u l t s were n o t c o n c l u s i v e . I t may be w o r t h w h i l e t o map the c u r r e n t d i s t r i b u t i o n t h r o u g h o u t the v e s s e l w i t h m a g n e t i c p r o b e s . Such i n v e s t i g a t i o n s c o u l d l e a d to a t h e o r e t i c a l e x p r e s s i o n f o r V^. M a g n e t i c probe measurements c o u l d a l s o be used to improve b o t h the a c c u r a c y o f , and the c o n c l u s i o n based on I . The probes used i n t h i s e x p e r i m e n t t o measure I were used because e e of e x p e d i e n c y r a t h e r than as an a c c u r a t e method of m o n i t o r i n g 1^. The measurement of I has i n f a c t been v e r y u s e f u l i n the d i s c u s s i o n o f the f o r m a t i v e phase. The e l e c t r o n d i s t r i b u t i o n c o u l d be b e t t e r e v a l u a t e d i f the a n g u l a r dependence of ^ ( V ) c o u l d be measured. T h i s would make i t p o s s i b l e to e v a l u a t e the d i s t r i b u t i o n o f the component o f v e l o c i t y normal to t h e a x i s . These measurements would a l s o y i e l d e v i d e n c e c o n c e r n i n g the runaway o f the e l e c t r o n s . The h i g h f r e q u e n c y o f e l e c t r o n - n e u t r a l i n t e r a c t i o n s would seem to i n d i c a t e a dominance o f i s o t r o p i c s c a t t e r i n g r a t h e r t h a n any s t r e a m i n g m o t i o n w h i c h i s a s s o c i a t e d w i t h runaway. O v e r a l l , i t appears t h a t t h e r e a r e i n t e r e s t i n g problems a s s o c i a t e d w i t h the breakdown i n p i n c h d e v i c e s and t h e r e i s s t i l l a use f o r the b a s i c a l l y s i m p l e r e t a r d i n g g r i d and F a r a d a y cup system. T h i s method of 109 i n v e s t i g a t i o n works w e l l d u r i n g the e a r l y s t a g e o f i o n i z a t i o n growth and may be the e a s i e s t way o f g e t t i n g i n f o r m a t i o n c o n c e r n i n g the f o r m a t i v e phase of a Z - p i n c h . i 110 BIBLIOGRAPHY Bennet, W. (1934, Phys. Rev. 45, 890. B l e v i n , H. A. and Haydon, S. C. ( 1 9 5 8 ) , A u s t . J . Phys. 11, 18. Chapman, S. and C o w l i n g , T. G. ( 1 9 6 1 ) , "The M a t h e m a t i c a l Theory of Non-uniform Gases", Cambridge U n i v . P r e s s . Chen, F. F. ( 1 9 6 5 ) , i n "Plasma D i a g n o s t i c T e c h n i q u e s " (R. H. H u d d l e s t o n e and S. L. 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Rev. 33, 954, . ( 1 9 2 9 ) , Phys. Rev. 34, 876. L l e w e l l y n - J o n e s , F. ( 1 9 5 7 ) , " I o n i z a t i o n and Breakdown i n Gases", Methuen and Co„ Medley, S. S., Curz o n , F. L. and Daughney, C. C. ( 1 9 6 5 ) , Can. J . Phys. 43, 1882. Medley, S. S. ( 1 9 7 0 ) , L a s e r 2^ 33. I l l P e r c i v a l , I . C. ( 1 9 6 6 ) , N u c l . Fus. 6, 182. R o b e r t s , D. E. (To be p u b l i s h e d i n P h y s . F l u i d s ) . R o s e n b l u t h , M. ( 1 9 5 4 ) , R e p o r t LA 1850, Los Alamos S c i e n t i f i c L a b o r a t o r y , Los Alamos, New M e x i c o . S e a r s , F. W. ( 1 9 5 9 ) , "Thermodynamics, the K i n e t i c Theory o f Gases and S t a t i s t i c a l M e c h a n i c s " , A d d i s o n - W e s l y . S m i t h , J . M. ( 1 9 6 5 ) , Phys. F l u i d s 8, 543. S p i t z e r , L. ( 1 9 6 2 ) , " 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 , New Y o r k . Tarn, S. ( 1 9 6 7 ) , Ph.D. T h e s i s , U n i v e r s i t y o f B r i t i s h C o l u mbia. Tanenbaum, B. S. ( 1 9 6 7 ) , "Plasma P h y s i c s " , M c G r a w - H i l l . Theophanis, G. A. ( 1 9 6 0 ) , Rev. S c i . I n s t r . 31, 427. von E n g e l , A. ( 1 9 6 5 ) , " I o n i z e d Gases", O x f o r d U n i v e r s i t y P r e s s . 112 APPENDIX A PROBLEMS WITH ELECTRON ENERGY ANALYZERS There a r e c e r t a i n problems w h i c h a r e common to most e l e c t r o s t a t i c energy a n a l y z i n g systems. A s h o r t d i s c u s s i o n o f some of the problems and how th e y r e l a t e t o t h i s e x p e r i m e n t w i l l now be g i v e n . (1) E l e c t r o n s p a s s i n g t h r o u g h a gas w i l l be s c a t t e r e d by c o l l i -s i o n s w i t h gas m o l e c u l e s o r may even cause breakdown of t h e gas i f t h e r e i s an e l e c t r i c f i e l d . The d e n s i t y r e m a i n i n g i n a s t r e a m o f e l e c t r o n s a f t e r h a v i n g t r a v e l l e d a d i s t a n c e , x, i s g i v e n by the s u r v i v a l e q u a t i o n Here, \ i s the mean f r e e p a t h o f the e l e c t r o n s and n i s the e l e c t r o n o d e n s i t y a t x = 0. V a l u e s f o r X f o r b o t h e l a s t i c and i o n i z i n g c o l l i s i o n s have been p l o t t e d i n F i g u r e D - l . S i n c e A p \ m i n i m i z a t i o n o f p ensures the g r e a t e s t s u r v i v a l o f e l e c t r o n s . I n an e x p e r i m e n t such as t h i s , the p r e s s u r e i s a d j u s t e d so t h a t d<3=:^(p). Here, d i s the d i s t a n c e between the c o l l e c t i n g cup and the h o l e i n the e l e c t r o d e . (2) Because o f the e l e c t r o s t a t i c f o r c e s i n an e l e c t r o n beam, a beam o f h i g h d e n s i t y w i l l s p r e a d as i t t r a v e l s t h r o u g h a f i e l d f r e e s p ace. The r a t e o f i n c r e a s e o f the beam r a d i u s , r ( x ) , as g i v e n by Von En g e l ( 1 9 6 5 ) , i s ^ x (320) ^ < e V ) 3 / t C m " ' r , 3/4 amp^ o k e < ^ > 2 ) '(A-2) o The r a d i u s o f the beam i s i n i t i a l l y r Q and a l l o f the e l e c t r o n s t r a v e l l i n g w i t h c o n s t a n t e nergy k^ a r e assumed t o be moving p a r a l l e l t o the beam a x i s as th e y e n t e r t h e system. The i n i t i a l c u r r e n t d e n s i t y i n t h e beam 113 i s given by j . The rate of spread f o r a beam of t y p i c a l current density i s shown i n Figure A - l . When an accelerating p o t e n t i a l i s included, the analysis i s much more complicated (see Hutter, 1967). Spreading of the beam i s important f o r low energy electrons. A system of grids with a small c o l l e c t i o n angle w i l l have much lower transmission for low energy electrons than f o r high energy ones. j Q = 33 amps/cm r k = 2.5 keV e k = 5 keV e Figure A - l E l e c t r o s t a t i c Beam Spreading (3) The presence of a magnetic f i e l d , normal to the path of the electrons can cause problems by d e f l e c t i n g the electrons out of the measuring system. In this experiment the energy analysis i s not influenced by the magnetic f i e l d because the magnetic f i e l d i s zero on the axis. The e f f e c t of the magnetic f i e l d at o f f - a x i s positions i s discussed i n Section 5:4.2. (4) Fluctuating e l e c t r i c f i e l d s can also cause problems. An e l e c t r i c f i e l d normal to the axis of the c o l l e c t i n g system w i l l d e f l e c t electrons out of the system. Fluctuations of the grid voltage with respect to the source voltage (at the pinhole) w i l l a f f e c t the measured 114 v a l u e s o f t h e energy. Thus, i t i s r e q u i r e d t h a t E • d <sc V (A-3) g where E i s t h e f i e l d imposed f r o m o u t s i d e t h e system and V i s the g r i d s v o l t a g e . The g r i d s y s t e m i s s h i e l d e d t o p r e v e n t such problems and V i s & measured w i t h r e s p e c t to the p i n h o l e p o t e n t i a l . (5) The t r a n s m i s s i o n o f t h e p i n h o l e depends on the h o l e l e n g t h b e i n g much l e s s t h a n the h o l e d i a m e t e r . The h o l e l e n g t h t o d i a m e t e r r a t i o was always l e s s t h a n 0.125. 115 APPENDIX B PRACTICAL PROBLEMS WITH THE RING ELECTRODE SYSTEM (1) The t h e o r y o f S e c t i o n s 3:5, 3:6 and 3:7 has been p r e s e n t e d w i t h the a s s u m p t i o n t h a t L, the t r a n s m i s s i o n o f t h e g r i d system, as i t appears i n the e q u a t i o n 1(c) = n e ^ L ( \ f ( c z ) d c z ( B - l ) i s i n d e p e n d e n t of the e l e c t r o n v e l o c i t y . T h i s s t a t e m e n t i s v a l i d f o r s m a l l v a l u e s of -Q- p r o v i d i n g : (a) the average e l e c t r o n energy i s h i g h i n the r e g i o n of s t u d y (beam s p r e a d i n g and s c a t t e r i n g a r e b o t h s m a l l ) (b) m a g n e t i c f i e l d s a r e a b s e n t and t h e r e a r e no e x t r a n e o u s e l e c t r i c f i e l d s ( c ) the r e t a r d i n g g r i d v o l t a g e i s c o n s t a n t d u r i n g a measurement. I n the measurement o f the e l e c t r o n e n e r g y , we have s i m p l y t a k e n the s l o p e o f the s e m i - l o g a r i t h m i c g r a p h o f the i n t e g r a l of e q u a t i o n B - l . I n so d o i n g , the a s s u m p t i o n s (a) to ( c ) a r e c o n s i d e r e d to be v a l i d . The low energy d a t a p o i n t s a r e e l i m i n a t e d i n t h e c a l c u l a t i o n o f T e because the low energy t r a n s m i s s i o n of the g r i d becomes a s t r o n g f u n c t i o n of the v e l o c i t y . I n e l i m i n a t i n g the low energy data,we a r e making use o f the p r o p e r t y t h a t the t e m p e r a t u r e c a l c u l a t i o n depends o n l y on the r e l a t i v e v a l u e s o f I (V ) . (2) Suppose t h a t t h e r e i s a p r o b l e m i n knowing the t r u e v a l u e of the g r i d v o l t a g e . Assume t h a t V' i s the a c t u a l r e t a r d i n g p o t e n t i a l e x p e r i e n c e d by the e l e c t r o n s and t h a t V i s the g r i d v o l t a g e as i t i s 116 r e a d f r o m the v o l t m e t e r . I f the two p o t e n t i a l s a r e r e l a t e d by V 1 = k V (B-2) g 8 t h e n i t can be e a s i l y d e m o n s t r a t e d t h a t the t r u e t e m p e r a t u r e i s g i v e n by A V T e = k A I n I (V ) = k T e < B " 3 ) c 8 The e f f e c t i s t h a t the graph o f l n I (V ) as a f u n c t i o n of V 1 s u f f e r s a c o n t r a c t i o n and the r e a l t e m p e r a t u r e i s d i f f e r e n t by a f a c t o r o f k f r o m the c a l c u l a t e d t e m p e r a t u r e , T . The p o t e n t i a l d i s t r i b u t i o n i n the h o l e o f the r i n g e l e c t r o d e has a p a r t i c u l a r f o r m w h i c h i s i n f l u e n c e d by the p r e s e n c e of t h e ground p l a n e a t the p i n - h o l e and by the s i z e o f the e l e c t r o d e . An e l e c t r o n p a s s i n g t h r o u g h the c e n t e r of the r i n g r e q u i r e s l e s s k i n e t i c energy t h a n one p a s s i n g near the edge o f t h e e l e c t r o d e . L e t us d e f i n e the f a c t o r k by the f o l l o w i n g e q u a t i o n . 2 TT \ ° V' ( r ) r d r -Jo ^ k = j (B-4) TT r V o g Here, V 1 ( r ) i s the f i e l d d i s t r i b u t i o n i n the p l a n e o f the r i n g w i t h i n the h o l e . Thus, k d e f i n e s the e f f e c t i v e a verage p o t e n t i a l e x p e r i e n c e d by an e l e c t r o n . < V > = k V (B-5) x g ' a r e a _ I f we l e t pV be the v a l u e o f V 1 (0), and r e p r e s e n t V ' ( r ) by a q u a d r a t i c § § § w i t h dV* dr V g = 0 r = 0 = V r = r E o c Then V ' ( r ) = V [ p + (1 - P ) r 2 / r 2 ] 117 (B-6) The a c t u a l f o r m o f V " ( r ) can o n l y be f o u n d by s o l v i n g L a p l a c e ' s e q u a t i o n w i t h mixed boundary c o n d i t i o n s . The a n a l y t i c s o l u t i o n o f t h i s p r o b l e m i s q u i t e d i f f i c u l t . However, i n the p r e s e n t d i s c u s s i o n we need o n l y e s t i m a t e the v a l u e of p. E l e c t r o l y t i c tank measurements i n d i c a t e t h a t p 2? 0.76 (see F i g u r e B - l and A p p e n d i x E ) . When t h i s v a l u e i s s u b s t i t u t e d i n t o e q u a t i o n B-4, k — 0.9. From e q u a t i o n B-3, t h i s means t h e v a l u e o f the te m p e r a t u r e c a l c u l a t e d from the s e m i - l o g g r a p h i s w i t h i n 107o of the t r u e v a l u e . T h i s s i m p l e c a l c u l a t i o n shows t h a t the f i e l d d i s t r i b u t i o n e r r o r i s c o n s i d e r a b l y l e s s t h a n o t h e r e x p e r i m e n t a l e r r o r s w h i c h were e n c o u n t e r e d . r ) 0.9 -0.8 _ 0.7 V g ( r ) f r o m e q u a t i o n B-6 E l e c t r o l y t i c Tank Measurements .6 .8 r / r F i g u r e B - l F i e l d D i s t r i b u t i o n i n the H o l e of the R e t a r d i n g R i n g E l e c t r o d e 118 APPENDIX C ELECTRICAL CIRCUITRY I n t h i s a p p e n d i x , we s h a l l d i s c u s s t h e v o l t a g e p r o b e , the c u r r e n t measurement, the d e l a y l i n e s and t h e f e r r i t e c o r e s . (1) The v o l t a g e probe c i r c u i t i s shown i n F i g u r e 4:6-1. The o u t -put v o l t a g e , V , i n terms of the i n p u t v o l t a g e , V , i s o R(V) V ( L l ) where R ( V ) i s the r e s i s t a n c e o f the M o r g a n i t e r e s i s t o r s . The c a l i b r a t i o n , to f i n d R ( V ) , was a c c o m p l i s h e d by d i s c h a r g i n g t h e main c a p a c i t o r , c harged to known v o l t a g e s , t h r o u g h the pr o b e . E q u a t i o n C - l can be used to c a l -c u l a t e the e f f e c t i v e v a l u e o f R ( V ) f o r known v a l u e s o f V q and V . A p l o t o f R ( V ) appears i n F i g u r e C - l . The e q u a t i o n o f the s t r a i g h t l i n e t h r o u g h the e x p e r i m e n t a l p o i n t s i s R(V) = 1000 (1 - 6.8 x 1 0 " 6 V ) (C-2) And f i n a l l y t he c a l i b r a t i o n e q u a t i o n i s 1000 V V = ~ (C-3) 2.6 + 6.8 x 10 V o E q u a t i o n C-3 has been r e p r e s e n t e d i n F i g u r e C-2. The r e s p o n s e time o f the probe i s c o n s i d e r e d to be g i v e n by the r i s e o f the v o l t a g e waveform when the c a p a c i t o r i s d i s c h a r g e d t h r o u g h the low i n d u c t a n c e Z - p i n c h c i r c u i t . The r i s e time o f such a s i g n a l i s 20 n s e c . (2) The c u r r e n t probe was a s i m p l e m a g n e t i c p i c k - u p c o i l . The o u t p u t o f t h e c o i l was f e d d i r e c t l y i n t o 50JTL , RG 58/U c o a x i a l c a b l e , p r o p e r l y t e r m i n a t e d a t the C.R.O. A T e k t r o n i x , t y p e - 0, o p e r a t i o n a l 100C 500 10 smcBEaJLoen 20 V (kV) 30 40 F i g u r e C - l Probe R e s i s t a n c e as a F u n c t i o n o f the I n p u t V o l t a g e > 150 V ( v o l t s ) o F i g u r e C-2 I n p u t V o l t a g e o f t h e Probe as F u n c t i o n of the Output V o l t a g e 120 ^ a m p l i f i e r , p l u g - i n u n i t was used t o i n t e g r a t e the d l / d t waveform f r o m the c o i l . The time c o n s t a n t was 1 y u s e c . C a l i b r a t i o n was a c h i e v e d by e q u a t i n g the t o t a l a r e a under the i n t e g r a t e d waveform t o the i n i t i a l charge on the c a p a c i t o r . CO I d t = C V (C-4) T h i s method has been t a k e n f r o m Daughney ( 1 9 6 6 ) . The maximum c u r r e n t i n the d i s c h a r g e c i r c u i t i s 70 kA and the r i n g i n g f r e q u e n c y o f t h e c i r c u i t i s 250 kHz. F i g u r e C-3 shows the c a l i b r a t i o n c u r v e and F i g u r e C-4 shows d l / d t and I f o r o n e - q u a r t e r o f a p e r i o d . The i n i t i a l c y c l e i s e l o n g a t e d because of t h e l o n g f o r m a t i v e t i m e . (3) D e l a y l i n e s a r e v e r y e f f e c t i v e i n e l i m i n a t i n g n o i s e p i c k - u p from s i g n a l s , p r o v i d e d t h a t the n o i s e i s c o n d u c t e d on the s h i e l d i n g o f the c a b l e . I n t h i s e x p e r i m e n t , s t a n d a r d Ad-Yu, t y p e 10T5D06, 3 ^ s e c . d e l a y l i n e s were used ( C h a r a c t e r i s t i c Impedance - 50.TL ) . The major p r o b l e m w i t h d e l a y l i n e s i s t h e i r f r e q u e n c y r e s p o n s e . F i g u r e C-5 shows a t y p i c a l r e s p o n s e c u r v e f o r the d e l a y l i n e s used. The m a n u f a c t u r e r ' s s t a t e d r i s e time i s 80 n s e c . F i g u r e C-6 shows the e f f e c t o f the d e l a y l i n e on a square wave o f 5 n s e c . r i s e t ime and v a r y i n g w i d t h s . 10 m t o r r - H 2 C a p a c i t o r V o l t a g e = 40 kV t ( y u s e c ) C a l i b r a t i o n Curve Showing the R i n g i n g and Decay of the D i s c h a r g e C u r r e n t d l / d t I A / v I I . 60 \ \ \ \ J20 \ \ \ 0.4 0.8 1.2 t ( y u s e c ) R i s e of the C u r r e n t t o the F i r s t Maximum and d l / d t 122 F r e q u e n c y (MHz) F i g u r e C-5 A t t e n u a t i o n o f t h e D e l a y L i n e (1) (2) F i g u r e C-6 K f f e c t o f the D e l a y L i n e on a Square Wave APPENDIX D ELECTRON MEAN FREE PATHS I t i s possible to cal c u l a t e the electron mean free paths f o r both e l a s t i c c o l l i s i o n s and i o n i z i n g c o l l i s i o n s in.H^. The mean free path, X . i s given by * = - y — (D-l) If the c o l l i s i o n frequency, y , and the electron v e l o c i t y , <^v ~-^>, are written i n terms of the electron energy, k g, then 3 ~ i ^ / l T ^ c m . (D-2) P o V e ( k V ) i 21 k ~\ _^ e mtorr _ „. ^ 1 - p (In k + 6.7) W C m ' ( D _ 3 ) o e ~X„ and A T are the e l a s t i c c o l l i s i o n mean free path and the i o n i z a t i o n h I mean free path re s p e c t i v e l y . The pressure of the gas i s represented by P q . Expressions f o r the c o l l i s i o n frequencies were derived from cross sections given by Delcroix (1964) i n the case of V„ and by P e r c i v a l (1966) i n the case of V . The value of both / p and } p have been I E o I o plotted i n Figure D-l. In order to get the low energy part of AjP, cross sections were used i n o U I Here, n^ i s the p a r t i c l e density and £" i s the i o n i z a t i o n cross section. 123 124 (a) E l e c t r o n Energy (eV) F i g u r e D-l Reduced Mean F r e e P a t h i n Hydrogen as a F u n c t i o n o f t h e E l e c t r o n Energy APPENDIX E ELECTROLYTIC TANK F i n d i n g the f i e l d d i s t r i b u t i o n i n an e l e c t r o d e gap r e q u i r e s the s o l u t i o n o f Laplace''s e q u a t i o n . V 2 0 = 0 ( E - l ) E q u a t i o n E - l d e s c r i b e s the s i t u a t i o n when t h e r e i s no space c h a r g e . The s o l u t i o n o f L a p l a c e ' s e q u a t i o n i s most commonly a c c o m p l i s h e d by a n a l o g u e methods, the most p o p u l a r of w h i c h i s t h e e l e c t r o l y t i c t a nk. The s o l u t i o n of L a p l a c e ' s e q u a t i o n by a n a l o g u e methods i s d i s c u s s e d i n F r a n c k e n ( 1 9 6 7 ) . I n an e l e c t r o l y t e , the c u r r e n t d e n s i t y j i s g i v e n by ~5 = - 6 V 0 (E-2) where <TJ i s the c o n d u c t i v i t y . I n the s t e a d y s t a t e , charge c o n s e r v a t i o n g i v e s V - T = 0 (E-3) and i f the medium i s homogeneous, e q u a t i o n E - l r e s u l t s f r o m t a k i n g t h e g r a d i e n t o f e q u a t i o n E-2. I n the s p e c i a l case o f an a x i a l l y s y m m e t r i c system, a wedge ta n k i s t he b e s t c h o i c e . The d i e l e c t r i c s u r f a c e s a t the a i r i n t e r f a c e and a t the bottom o f t h e tank a c t as m i r r o r s u r f a c e s . The l i n e o f i n t e r s e c t i o n o f the s u r f a c e s i s the a x i s o f symmetry. I f the wedge a n g l e i s k e p t s m a l l , t h e n a s t r a i g h t w a l l e d tank i s a good a p p r o x i m a t i o n to the c u r v e d s u r f a c e s of t he system. T h i s method i s f a i r l y i n a c c u r a t e near c o n d u c t i n g s u r f a c e s and the a x i s o f symmetry because o f m i n i s c u s e f f e c t s . F i g u r e E - l shows the s t a n d a r d type o f c i r c u i t used t o measure the f i e l d d i s t r i b u t i o n . 125 126 S A P V G S D C r o s s s e c t i o n B through D Figure E - l E l e c t r o l y t i c Tank A and B are e l e c t r o d e s , P i s the probe, V i s a n u l l - r e a d i n g voltmeter, R i s a potentiometer and G i s a s i g n a l generator. The r e s u l t s of t h i s measurement f o r a c o n f i g u r a t i o n s i m i l a r to a Z-pinch are shown i n Figure 2:2-1. I n making the measurement, the i n f l u e n c e of the g l a s s v e s s e l has been neglected. This i s important i n the region of the w a l l o n l y and a f f e c t s the d i s t r i b u t i o n i n the c e n t r a l r e g i o n of the v e s s e l i n the neighbourhood of the electrodes only minimally. The f i e l d d i s t r i b u t i o n f o r the r i n g electrodes of the energy a n a l y s i n g system was a l s o measured. The f i e l d d i s t r i b u t i o n i n the center of the r i n g i s shown i n F i g u r e B - l . 

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