UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

The emission characteristics of a Z-pinch plasma in a vacuum spark discharge Fong, Kenneth Sau-Kin 1982

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1982_A1 F66.pdf [ 8.2MB ]
Metadata
JSON: 831-1.0085775.json
JSON-LD: 831-1.0085775-ld.json
RDF/XML (Pretty): 831-1.0085775-rdf.xml
RDF/JSON: 831-1.0085775-rdf.json
Turtle: 831-1.0085775-turtle.txt
N-Triples: 831-1.0085775-rdf-ntriples.txt
Original Record: 831-1.0085775-source.json
Full Text
831-1.0085775-fulltext.txt
Citation
831-1.0085775.ris

Full Text

c./ THE EMISSION CHARACTERISTICS O f A Z-PINCH PLASMA IN A VACUUM SPARK DISCHARGE by KENNETH SAU-KIN FONG B . S c , The U n i v e r s i t y of B r i t i s h C olumbia, 1975 M . S c , The U n i v e r s i t y of B r i t i s h C o lumbia, 1978 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n • THE FACULTY OF GRADUATE STUDIES (Department of P h y s i c s ) We a c c e p t t h i s t h e s i s as c o n f i r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA June 1982 © Kenneth Sau-Kin Fong, 1982 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The University of B r i t i s h Columbia 1956 Main M a l l Vancouver, Canada V6T 1Y3 DE-6 (3/81) i i ABSTRACT The Z-pinches i n a vacuum spark can be c l a s s i f i e d i n t o s low, f a s t and s u p e r f a s t a c c o r d i n g t o t h e i r p i n c h d u r a t i o n s . T h e i r e m i s s i o n c h a r a c t e r i s t i c s a r e i n v e s t i g a t e d i n the v i s i b l e , u l t r a v i o l e t and the X-ray w a v e l e n g t h s . The plasma d u r i n g a f a s t p i n c h was found t o have . an e l e c t r o n t e mperature between 100 and 600eV. The s u p e r f a s t p i n c h r e s u l t e d i n a minute c y l i n d r i c a l plasma a p p r o x i m a t e l y 40um i n d i a m e t e r , w i t h an e l e c t r o n temperature of 1 t o 4keV and a l i f e t i m e of l e s s than 4ns. The slow and the f a s t p i n c h were found t o be i n agreement w i t h the t h e o r e t i c a l r e s u l t s p r e d i c t e d by a shock wave model. The f o r m a t i o n of the s u p e r f a s t p i n c h and i t s a s s o c i a t e d h i g h d e n s i t y and t e mperature were e x p l a i n e d as the r e s u l t s of magnetohydrodynamic i n s t a b i l i t y . i i i TABLE OF CONTENTS Chapter 1 I n t r o d u c t i o n 1 1 . 1 H i s t o r y 3 1.2 M o t i v a t i o n and Scope of Study 4 Chapter 2 Theory of Z-pinch 6 2.1 Breakdown Phase 7 2.2 R e a c t i v e Phase and the Dynamics of an Im p l o d i n g Z-pinch 9 2.2.1 Free P a r t i c l e Model 15 2.2.2 Shock Wave Model 19 2.3 D i s c u s s i o n of Assumptions i n the Models 26 Chapter 3 D e s c r i p t i o n of App a r a t u s 34 3.1 D i s c h a r g e V e s s e l and Power Supply 35 3.2 T r i g g e r i n g A p p a r a t u s 38 3.2.1 E l e c t r i c a l D i s c h a r g e T r i g g e r i n g 40 3.2.2 L a s e r T r i g g e r i n g 43 3.3 Measurement of E l e c t r i c a l Parameters 46 3.4 F a s t Photography 48 3.5 V i s i b l e and U l t r a v i o l e t S p e c t r o s c o p y 48 3.6 X-ray Measurements 51 3.6.1 T o t a l F l u x 51 3.6.2 S p e c t r a l Measurement u s i n g F o i l A b s o r b e r s 52 3.6.3 P i n h o l e Photography 54 i v C hapter 4 D e s c r i p t i o n and D i s c u s s i o n of E x p e r i m e n t s 57 4.1 C o m p a r i s i o n of T r i g g e r i n g Methods 57 4.2 G e n e r a l O b s e r v a t i o n s of Vacuum Spark 62 4.3 R e p r o d u c i b i l i t y 69 4.4 E m i s s i o n s i n the V i s i b l e and U l t r a v i o l e t Region ....71 4.4.1 Framing Camera P i c t u r e s 71 4.4.2 V i s i b l e and UV S p e c t r o s c o p y 73 4.5 E m i s s i o n i n the X-ray Region 86 4.5.1 Time R e s o l v e d Absorber F o i l Measurements 86 4.5.2 X-ray P i n h o l e Photographs 95 4.6 Comparison of E x p e r i m e n t a l R e s u l t s and Shock Wave Model i 96 Chapter 5 I n s t a b i l i t y of Z-pinch 103 5.1 S t a b i l i t y of F r e e P a r t i c l e Model 108 5.2 I n s t a b i l i t y of Snowplow P i n c h 110 5.3 S t a b i l i t y of C y l i n d r i c a l C o n v e r g i n g Shock F r o n t ....112 5.4 R a y l e i g h - T a y l o r I n s t a b i l i t y of a P i n c h Column 115 5.5 N o n l i n e a r E v o l u t i o n of m=0 R a y l e i g h - T a y l o r I n s t a b i l i t y 118 5.6 The G e n e r a t i o n of a S u p e r f a s t P i n c h .119 Chapter 6 C o n c l u s i o n and S u g g e s t i o n s 123 B i b l i o g r a p h y 128 Appendix A D e t e r m i n a t i o n of E l e c t r o n Temperature 131 Appendix B Shock Dynamics 143 V LIST OF FIGURES 1- 1 Vacuum Spark 2 2- 1 Space-time Diagram of Z-pinch 11 2-2 I d e a l i z e d P i n c h Parameters 14 2-3(a) N u m e r i c a l R e s u l t s - D i f f e r e n t E n t h a l p y C o e f f i c i e n t s .26 2-3(b) N u m e r c i a l R e s u l t s - D i f f e r e n t E f f e c t i v e R a d i i 27 2-3(c) N u m e r i c a l R e s u l t s - D i f f e r e n t I n i t i a l P r e s s u r e s 28 2- 4 Dependence of F i n a l P i n c h e d R a d i u s 29 3- 1 The D i s c h a r g e V e s s e l 36 3-2 Transformed D i s c h a r g e T r i g e r r i n g C i r c u i t 41 3-3 D i r e c t D i s c h a r g e T r i g g e r i n g C i r c u i t 43 3-4 L a s e r T r i g g e r i n g Schematic Diagram 44 3-5 S p e c t r o s c o p i c A pparatus Arrangement 49 3-6 X-ray D e t e c t o r 52 3- 7 X-ray P i n h o l e Camera 54 4- 1 J i t t e r i n Transformed D i s c h a r g e T r i g g e r i n g 59 4-2 D i r e c t D i s c h a r g e T r i g g e r Spark C u r r e n t Trace 61 4-3(a) d l / d t of Vacuum Spark 63 4-3(b) C u r r e n t of Vacuum Spark 63 4-3(c) V o l t a g e of Vacuum Spark 63 4-4(a) d l / d t and X-ray T r a c e of a Spark 65 4-4(b) d l / d t and X-ray Trace of a S u p e r f a s t P i n c h 66 4-5 Framing Camera Photographs of a Vacuum Spark 72 4-6(a) T i m e - i n t e g r a t e d Cu-Cu l i n e P r o f i l e ... .' 77 v i 4-6(b) T i m e - i n t e g r a t e d Cu-Cu l i n e P r o f i l e ( F i r s t 1us) ....78 4-7 R a t i o of L i n e t o Continuum i n Cu-Cu D i s c h a r g e 79 4-8(a) T i m e - i n t e g r a t e d C-Cu L i n e P r o f i l e 80 4-8(b) d l / d t and C IV L i n e E m i s s i o n P u l s e Shape 80 4-9 D e n s i t y D i s t r i b u t i o n a l o n g I n t e r - E l e c t r o d e Gap ....81 4-10 e - f o l d i n g Times f o r Copper Plasma 84 4-11(a) I o n i c D e n s i t i e s f o r Copper Plasma 85 4-11(b) I o n i c D e n s i t i e s f o r Carbon Plasma 85 4-12 d l / d t and X-ray Trace of Cu-Cu D i s c h a r g e 88 4-13(a) S u p e r f a s t P i n c h and X-ray E m i s s i o n 89 4-13(b) X-ray E m i s s i o n through d i f f e r e n t F o i l s ( C u Plasma) .89 4-14 X-ray E m i s s i o n t h r o u g h d i f f e r e n t F o i l s ( C u Plasma) .90 4-15 X-ray E m i s s i o n t h r o u g h d i f f e r e n t F o i l s ( C Plasma) ..93 4-16 X-ray E m i s s i o n t h r o u g h d i f f e r e n t F o i l s ( C Plasma) ..94 4-17 X-ray P i n h o l e Photographs 96 4- 18 Dependence of P i n c h D u r a t i o n 102 5- 1 C o o r d i n a t e Systems i n S t a b i l i t y A n a l y s i s 106 5-2 Plasma Column of d i f f e r e n t m modes 107 5-3(a) Shock S t a b i l i t y ( S h o r t Wavelength) 114 5-3(b) Shock I n s t a b i l i t y (Long Wavelength) 114 5-4 Dimensions of S u p e r f a s t P i n c h e d Plasma 121 A-1 T r a n s m i s s i o n of Copper Plasma 135 A-2 T r a n s m i s s i o n of Carbon Plasma .136 A-3 R e l a t i v e E m i s s i v i t y of Copper and Carbon Plasmas ..137 B-1 V a r i a t i o n of E n t h a l p y C o e f f i c i e n t s 148 v i i LIST OF TABLES 3.1 Vacuum Spark Parameters 38 3.2 T r i g g e r i n g A p paratus Parameters 46 4.1 C l a s s i f i c a t i o n of P i n c h e s 68 4.2 E m i s s i o n Timings of Cu and C L i n e s 75 v i i i LIST OF SYMBOLS a a c o u s t i c speed A Angstrom A r a d i u s of plasma column B magnetic i n d u c t i o n c speed of l i g h t e e l e c t r o n i c charge E e l e c t r i c f i e l d i n t e n s i t y eV e l e c t r o n v o l t g e n t h a l p y c o e f f i c i e n t g g r a v i t a t i o n a l a c c e l e r a t i o n h e n t h a l p y i c u r r e n t j c u r r e n t d e n s i t y k Boltzmann's c o n s t a n t k wave number I l e n g t h of d i s c h a r g e gap L i n d u c t a n c e In A Coulomb l o g a r i t h m m a z i m u t h a l number M Mach number of shock f r o n t N number d e n s i t y of i o n n e number d e n s i t y of e l e c t r o n magnetic p r e s s u r e Rc plasma k i n e t i c p r e s s u r e R e f f e c t i v e r a d i u s of r e t u r n c o n d u c t o r '*0 r a d i u s of r e t u r n c o n d u c t o r s r a d i u s of shock f r o n t t time T temp e r a t u r e u i n t e r n a l energy V v o l t a g e v<z A l f v e n wave speed V f l o w v e l o c i t y v a plasma column i m p l o s i o n v e l o c i t y v 4 shock f r o n t v e l o c i t y a d i m e n s i o n l e s s r a d i u s of plasma column $ p e r t u r b a t i o n i n r a d i u s n =£nR/A 0 Nordheim f u n c t i o n K. magnetodynamic wave number X wavelength V growth r a t e 5 p e r t u r b a t i o n d i s p l a c e m e n t P d e n s i t y a c o n d u c t i v i t y a d i m e n s i o n l e s s r a d i u s of shock f r o n t T d i m e n s i o n l e s s time T .' d i m e n s i o n l e s s i m p l o s i o n time rmp T d u r d i m e n s i o n l e s s p i n c h d u r a t i o n a z i m u t h a l a n g l e * work f u n c t i o n * c u r r e n t f r e e magnetic p o t e n t i a l 01 complex f r e q u e n c y X ACKNOWLEDGEMENT I w i s h t o s i n c e r e l y thank my s u p e r v i s o r , Dr. A . J . B a r n a r d , f o r s u g g e s t i n g t h i s work and f o r h i s c o n s t a n t g u i d a n c e . I a l s o l i k e t o thank Dr. B. A h l b o r n f o r b e i n g my s u p e r v i s o r d u r i n g Dr. B a r n a r d ' s s a b b a t i c a l l e a v e . H i s e n t h u s i a s m and s u g g e s t i o n s a r e most a p p r e c i a t e d . I l i k e t o e x p r e s s my g r a d i t u d e t o Dr. A. Ng, f o r h i s v a l u a b l e s u g g e s t i o n s d u r i n g the c o u r s e of t h i s work, and t o Dr. F. Curzon, f o r h i s many comments on t h i s t h e s i s . I l i k e t o e x p r e s s my a p p r e c i a t i o n t o a l l members of the UBC Plasma P h y s i c s Group f o r t h e i r s t i m u l a t i n g d i s c u s s i o n s and comradeship. I have had many v a l u a b l e d i s c u s s i o n s w i t h Dr. W. L i e s e , Dr. J . Kwan, Mr. R. P o p i l and Mr. D. P a s i n i . I am i n d e b t e d t o Mr. H. Houtman f o r h i s a s s i s t a n c e w i t h the equipments and t o Mr. A. Cheuck f o r h i s a s s i s t a n c e w i t h the e l e c t r o n i c s . I e s p e c i a l l y l i k e t o thank my f i a n c e e , M i s s R. Tseng. She has not o n l y a s s i s t e d i n t a k i n g d a t a , t y p i n g , p r o o f r e a d i n g i n the p r e p a r a t i o n of t h i s t h e s i s , but above a l l , she has g i v e n me moral support and put up w i t h a l l the d e s e r t e d weekends d u r i n g the y e a r s . T h i s t h e s i s i s d e d i c a t e d t o my p a r e n t s . 1 CHAPTER 1  INTRODUCTION Recent advances i n s o l a r p h y s i c s , plasma p h y s i c s and the quest f o r c o n t r o l l e d t h e r m o n u c l e a r f u s i o n have caused c o n s i d e r a b l e i n t e r e s t i n the s p e c t r o s c o p y i n the vacuum u l t r a v i o l e t and the X-ray r e g i o n . T h i s has l e d t o the s e a r c h f o r s p e c t r o s c o p i c l i g h t s o u r c e s i n t h i s r e g i o n of the e l e c t r o m a g n e t i c spectrum. A vacuum spark i s c a p a b l e of p r o d u c i n g minute r e g i o n s of hot plasma w i t h t e m p e r a t u r e s of the o r d e r of keV, d e n s i t i e s of the o r d e r of l 0 2 O c m " 3 . T h i s hot plasma e m i t s i n t e n s e r a d i a t i o n s i n the u l t r a v i o l e t and the X-ray w a v e l e n g t h s . The b a s i c elements of a vacuum spark a p p a r a t u s a r e shown i n F i g . 1 - 1 . A h i g h v o l t a g e i s a p p l i e d a c r o s s the e l e c t r o d e s and the d i s c h a r g e i s i n i t i a t e d by the i n j e c t i o n of charge c a r r i e r s ( e l e c t r o n and/or i o n s ) i n t o the i n t e r - e l e c t r o d e gap. A t r i g g e r e l e c t r o d e i s shown i n the f i g u r e f o r t h i s purpose. The t r i g g e r e d vacuum spark d i s c h a r g e o c c u r s a l o n g a spark channnel i n which the c u r r e n t i s c a r r i e d by a plasma produced from the e l e c t r o d e m a t e r i a l . Because of i t s s i m p l i c i t y and ease of o p e r a t i o n , the vacuum spark i s an a t t r a c t i v e t o o l t o study and t o d e v e l o p d i a g n o s t i c t e c h n i q u e s f o r plasma of such h i g h temperature and d e n s i t y regime f o r o b v i o u s a p p l i c a t i o n s t o the r m o n u c l e a r f u s i o n r e s e a r c h . F u r t h e r m o r e , s i n c e the o p e r a t i o n of the spark i s i n vacuum, d i f f i c u l t i e s u s u a l l y e n c o u n t e r e d i n the 2 •-trigger pin * —vacuum * V ; * 5 i s >) = * \* 1 s ^ ::::x::::::; X ^-insulator + F i g . 1-1 B a s i c elements of a vacuum spark a p p a r a t u s vacuum u l t r a v i o l e t and s o f t X-ray s p e c t r o s c o p y such as background gas and window m a t e r i a l s a r e a v o i d e d . 3 1.1 HISTORY The vacuum spark was f i r s t produced by Wood[5l] i n 1897, who noted t h a t the d e v i c e was a new way f o r p r o d u c i n g X - r a y s . In 1918, M i l l i k a n et §_1[31] used the vacuum spark as a s p e c t r o s c o p i c source i n the vacuum u l t r a v i o l e t r e g i o n . They were a b l e t o observe s p e c t r a w i t h r i c h s t r u c t u r e s down to t h e " r e f l e c t i v i t y l i m i t of t h e i r i n s t r u m e n t , a t about 150A. These s t r u c t u r e s were i n t e r p r e t a t e d as e m i s s i o n s from f o u r or f i v e t i m e s i o n i z e d e l e c t r o d e m a t e r i a l s . Then t h e r e was a f l u r r y of a c t i v i t y u s i n g the vacuum spark as a s p e c t r o s c o p i c source i n the vacuum u l t r a v i o l e t r e g i o n . H i g h e r i o n i z a t i o n s t a g e s were r e p o r t e d by A l f v e n and Sanner[4] u s i n g a low i n d u c t a n c e d i s c h a r g e c i r c u i t r y . F u r t h e r s p e c t r o s c o p i c s t u d i e s were done by Flemberg i n the X-ray r e g i o n . H e l i u m - l i k e i o n s of f l u o r i n e , magnesium, and aluminum were o b s e r v e d . Cohen et a _ l [ l 3 ] i n 1968 r e p o r t e d the o b s e r v a t i o n of h e l i u m - l i k e i o n s f o r h e a v i e r i o n s such as i r o n , t i t a n i u m and copper. H y d r o g e n - l i k e s p e c t r a of i r o n and t i t a n i u m were o b s e r v e d by L i e and E l t o n [ 2 8 ] . They a l s o noted the s i m i l a r i t y between the vacuum spark and s o l a r f l a r e s . S i m u l t a n e o u s work done by Turechek[47] c o n f i r m e d t h a t the p r o d u c t i o n of such h i g h t emperature X-ray b u r s t was time c o i n c i d e n t w i t h the o c c u r r e n c e of a Z-pinch i n the plasma column. T h i s Z-pinch i s a r a d i a l c ompression of the plasma column due t o the i n t e r a c t i o n of the c u r r e n t i n the d i s c h a r g e w i t h the a s s o c i a t e d magnetic f i e l d . P r e s e n t l y , t r i g g e r e d vacuum s p a r k s a re used as s p e c t r o s c o p i c s o u r c e s t o 4 stud y the s p e c t r a of h i g h l y s t r i p p e d i o n s . 1.2 MOTIVATION AND SCOPE OF STUDY A l t h o u g h many workers[27,28,40,47] have r e p o r t e d t h a t h i g h e s t t emperature and d e n s i t y a r e a t t a i n e d d u r i n g a Z- p i n c h , the d e t a i l s of the h e a t i n g and the compression mechanisms are not w e l l u n d e r s t o o d due t o the s h o r t l i f e - t i m e of the hot plasma. F u r t h e r m o r e , most of the s p e c t r o s c o p i c measurements of the vacuum spark a re t i m e - i n t e g r a t e d over many d i s c h a r g e s . These r e s u l t s do not y i e l d any t i m e - r e s o l v e d i n f o r m a t i o n r e g a r d i n g the plasma c o n d i t i o n s d u r i n g the Z-pinches which produce the h i g h l y s t r i p p e d i o n s t h a t a r e ob s e r v e d i n the s p e c t r a . The m o t i v a t i o n of t h i s study i s t o p r o v i d e some i n f o r m a t i o n on the p r o d u c t i o n of the h i g h temperature and d e n s i t y by a Z- p i n c h , and t o the e m i s s i o n c h a r a c t e r i s t i c s of the spark by making t i m e - r e s o l v e d measurements. Another m o t i v a t i o n i s t o make the vacuum spark more r e p r o d u c i b l e by improvements on the t r i g g e r i n g methods. A t r i g g e r e d vacuum spark i s used i n our i n v e s t i g a t i o n and measurements a r e made i n the f i r s t q u a r t e r c y c l e of the spark d i s c h a r g e , d u r i n g which the plasma e x p e r i e n c e s m u l t i p l e Z-pinches. S e v e r a l t r i g g e r i n g methods a r e proposed and t h e i r r e s u l t s a r e r e p o r t e d . These methods i n c l u d e e l e c t r i c a l d i s c h a r g e t r i g g e r i n g and l a s e r 5 t r i g g e r i n g . In the vacuum spark experiment i t s e l f , copper c a t h o d e , copper and carbon anodes are used i n the i n v e s t i g a t i o n . T h e i r s u i t a b i l i t y t o s p e c t r o s c o p y i s a l s o n o t e d . The t i m e - r e s o l v e d e m i s s i o n s of the vacuum spark a r e measured i n the v i s i b l e , u l t r a v i o l e t and the X-ray w a v e l e n g t h s . In p a r t i c u l a r , the X-ray e m i s s i o n c h a r a c t e r i s t i c s a r e s t u d i e d i n d e t a i l , as t h i s g i v e s an i n s i g h t i n t o the h e a t i n g mechanism and a l s o the i n s t a b i l i t i e s of the plasma. Chapter 2 d e s c r i b e s the b a s i c t h e o r i e s of a Z-pinch i n a vacuum s p a r k . The e x p e r i m e n t a l s e t u p i s d e s c r i b e d i n Chapter 3 . The d i a g n o s t i c equipment, which i n c l u d e s c u r r e n t and v o l t a g e probes, f a s t f r a m i n g camera, v i s i b l e - U V s p e c t r o m e t e r and X-ray d e t e c t o r s a r e a l s o d e s c r i b e d i n the same c h a p t e r . The r e s u l t s and t h e i r i n t e r p r e t a t i o n s a r e p r e s e n t e d i n Chapter 4 . Chapter 5 i n v e s t i g a t e s the i n s t a b i l i t i e s and t h e i r growth r a t e s i n a Z-pinched plasma. I t a l s o g i v e s a p l a u s i b l e e x p l a n a t i o n t o the mechanism f o r the p r o d u c t i o n of h i g h temperature and d e n s i t y o b s e r v e d . Chapter 6 g i v e s a c o n c l u s i o n and s u g g e s t i o n s f o r f u r t h e r work. Appendix A g i v e s a d e s c r i p t i o n of the measurement of temperature by f o i l a b s o r p t i o n t e c h n i q u e , and Appendix B v d e r i v e s the shock dynamics e q u a t i o n s t h a t a r e used i n Chapter 2 . 6 CHAPTER 2  THEORY OF Z-PINCH In any Z-pinched d i s c h a r g e , i n c l u d i n g the vacuum s p a r k , the d i s c h a r g e can be r o u g h l y d i v i d e d i n t o two phases. The f i r s t phase, namely the breakdown phase, i n v o l v e s the f o r m a t i o n of a plasma d i s c h a r g e c h a n n e l i n which the c u r r e n t i s d e t e r m i n e d by the v a r i a t i o n of r e s i s t i v i t y of the plasma i n the i n t e r - e l e c t r o d e gap. T h i s phase i s of i n t e r e s t t o vacuum i n s u l a t i o n , and a s h o r t r e v i e w i s g i v e n i n S e c t i o n 2.1. When the plasma column i s w e l l e s t a b l i s h e d and i t s r e s i s t i v i t y becomes n e g l i g i b l e , the c u r r e n t i s governed by the e x t e r n a l c a p a c i t a n c e , e x t e r n a l i n d u c t a n c e and gap i n d u c t a n c e . The d i s c h a r g e i s s a i d t o be i n the r e a c t i v e phase. D u r i n g t h i s phase the d i s c h a r g e c u r r e n t i n c r e a s e s r a p i d l y and i t s accompanying magnetic f i e l d i n t e r a c t s w i t h the c u r r e n t c a r r y i n g plasma i n such a way t h a t the plasma i s compressed r a d i a l l y , c a u s i n g the column t o p i n c h . S e c t i o n 2.2 d e s c r i b e s the dynamics of a p i n c h i n g plasma column. A review of the f r e e - p a r t i c l e model i s g i v e n i n S u b - S e c t i o n 2.2.1. We have d e v e l o p e d a c y l i n d r i c a l shock wave model i n S u b - S e c t i o n 2.2.2. Our model t a k e s i n t o c o n s i d e r a t i o n the e f f e c t s of a shock wave t h a t i s g e n e r a t e d by the time v a r y i n g magnetic f o r c e . The v a l i d i t y of the assumptions of 7 the s e models i s examined i n S e c t i o n 2.3. 2.1 BREAKDOWN PHASE The breakdown mechanisms have been i n v e s t i g a t e d by a number of workers. However, because of the s h o r t time s c a l e and the number of p r o c e s s e s i n v o l v e d , u n d e r s t a n d i n g of the breakdown p r o c e s s e s i s s t i l l not c o m p l e t e . For an e l e c t r i c a l l y t r i g g e r e d s p a r k , the f i e l d e m i s s i o n c u r r e n t d e n s i t y j produced by the e l e c t r i c f i e l d of the h i g h v o l t a g e t r i g g e r spark i s g i v e n by the Fowler-Nordheim e q u a t i o n f 42] J • C | 2 - P ( ^ ) - (2.1» w i t h C=1.54x10" 1statamp, N=-6.38x10 7cm" 1. $ i s the work f u n c t i o n of the e l e c t r o d e m a t e r i a l , Q(y) w i t h y = 2 . 1 9x1 0" 5 E 1 / 2 / ( J > ; ( s t a t v o l f 1 1 2cm) i s the Nordheim f u n c t i o n . T h i s f u n c t i o n i s t a b u l a t e d i n S l i v k o v [ 4 2 ] . The t r i g g e r spark produces a dense plasma composed of e l e c t r o d e and i n s u l a t i o n m a t e r i a l s . These propagate i n t o the i n t e r - e l e c t r o d e gap w i t h a v e l o c i t y of the o r d e r of I0 6cm/s. A f t e r a time i n t e r v a l c o r r e s p o n d i n g t o the e l e c t r o n t r a n s i t time of the i n t e r - e l e c t r o d e gap, a s i m i l a r c l o u d appears near the anode and s t a r t s t o move outward. These two a d v a n c i n g c l o u d s were obse r v e d i n a number of t i m e - r e s o l v e d p h o t o g r a p h i c s t u d i e s [ 1 1 , 2 8 , 4 2 ] . The e m i s s i v i t y p r o p e r t i e s 8 of t h e s e c l o u d s a r e the s u b j e c t of study by Suprunenko[46] et a l . These c l o u d s have f i n i t e c o n d u c t i v i t y and e l e c t r i c f i e l d s a r e a b l e t o p e n e t r a t e i n t o them. I f the e l e c t r i c f i e l d exceeds the runaway t h r e s h o l d *e 3n /kT, runaway e l e c t r o n s a r e produced. X-ray e m i s s i o n s o b s e r v e d d u r i n g t h i s i n t e r v a l a r e b e l i e v e d t o be produced by the s e runaway e l e c t r o n beams c o l l i d i n g w i t h the e l e c t r o d e s . The c u r r e n t of the runaway e l e c t r o n s i s , however, l i m i t e d by beam-plasma i n s t a b i l i t y . The c r i t i c a l v a l u e of D c r i t i s g i v e n by R o s t o k e r [ 3 6 ] J „ i t • " • ¥ b f e 1 ) " p f o ) • ( 2 - 2 ) where n i s the e l e c t r o n d e n s i t y , v^ i s the beam v e l o c i t y , e b J v i s the beam t h e r m a l v e l o c i t y , i s the t h e r m a l v e l o c i t y of the background e l e c t r o n s . E x c e s s energy above t h i s c r i t i c a l v a l u e i s t r a n s f e r r e d t o the background e l e c t r o n s t h r ough e l e c t r o n - e l e c t r o n i n t e r a c t i o n , w h i l e p r e s e r v i n g the average e l e c t r o n d r i f t v e l o c i t y . Buneman[lO] showed t h a t the energy of the e l e c t r o n d r i f t motion i s t r a n s f e r r e d i n t o l o n g i t u d i n a l plasma o s c i l l a t i o n t h r ough e l e c t r o n - i o n i n t e r a c t i o n i n a p p r o x i m a t e l y 100 plasma o s c i l l a t i o n s . The i o n s a r e heated by the o s c i l l a t i o n s , which f u r t h e r reduce the c r i t i c a l f i e l d v a l u e and cause more energy t r a n s f e r and h e a t i n g . As a r e s u l t , the e l e c t r o n beam c u r r e n t drops and the temperature of the i o n s r i s e s r a p i d l y , w i t h a c o r r e s p o n d i n g v a r i a t i o n i n 9 the time d e r i v a t i v e of the c u r r e n t . At the end of the breakdown p r o c e s s , the d e n s i t y and the t emperature of the plasma i n c r e a s e t o such a p o i n t t h a t the r e s i s t i v i t y of the plasma becomes n e g l i g i b l e when compared w i t h the impedance of the r e s t of the c i r c u i t r y . The d i s c h a r g e c u r r e n t i s now d e t e r m i n e d by the e x t e r n a l c a p a c i t a n c e , the i n d u c t a n c e of the e x t e r n a l c i r c u i t and the gap. We c a l l t h i s the r e a c t i v e phase. The c u r r e n t i n c r e a s e s r a p i d l y . I t s r a t e i s l i m i t e d o n l y by the i n d u c t a n c e . In the o r d e r of t e n s of nanoseconds ( a p p r o x i m a t e l y 14ns i n our c a s e ) , the magnetic p r e s s u r e produced by the c u r r e n t w i l l exceed the plasma k i n e t i c p r e s s u r e . The magnetic f i e l d t h u s w i l l t e n d t o compress the plasma and cause the column t o p i n c h . 2.2 REACTIVE PHASE AND THE DYNAMICS OF AN IMPLODING Z-PINCH T h i s phase i s c h a r a c t e r i z e d by the low r e s i s t i v i t y of the plasma. The dynamics of the plasma column a r e i n f l u e n c e d by the v a r y i n g i n d u c t a n c e of the d i s c h a r g e . D u r i n g t h i s phase, the r a t e of i n c r e a s e of the c u r r e n t i s c o n t r o l l e d by the e x t e r n a l i n d u c t a n c e . T h i s d e t e r m i n e s the r i s e i n magnetic p r e s s u r e and hence the o v e r a l l i m p l o s i o n t i m e . To c o n s i d e r the hydrodynamics of the i m p l o s i o n , we s h a l l assume the plasma t o be i n f i n i t e l y c o n d u c t i v e . C o n s e q u e n t l y Ohmic h e a t i n g i s n e g l e c t e d and the c u r r e n t can 10 be c o n s i d e r e d t o f l o w o n l y on a t h i n s heath on the o u t e r boundary of the plasma column. F u r t h e r m o r e , we s h a l l n e g l e c t the power l o s s due t o r a d i a t i o n , and hence the o n l y form of energy c o u p l i n g between the plasma and the e x t e r n a l enviroment i s t h rough e l e c t r o m a g n e t i c f o r c e s . C o n s i d e r a plasma column of r a d i u s A c a r r y i n g an a x i a l c u r r e n t I i n the Z - d i r e c t i o n . T h i s c u r r e n t g e n e r a t e s an a z i m u t h a l magnetic f i e l d B, which i n t e r a c t s w i t h the c u r r e n t c a r r y i n g plasma i n such a way t h a t i t tends t o compress the plasma r a d i a l l y . T h i s c o m p r e s s i o n a l f o r c e can be e x p r e s s e d as the magnetic p r e s s u r e R2 p = i - (2.3) ^mag 8ir ,2 O 2 A 2 2irc A where the p a r t i c l e c o l l i s i o n s a r e s u f f i c i e n t l y f r e q u e n t t h a t the p r e s s u r e can be e x p r e s s e d as a s c a l a r q u a n t i t y . In t h i s d e s c r i p t i o n , the magnetic p r e s s u r e due t o the t h i n c u r r e n t sheath can be c o n s i d e r e d as a s o l i d "magnetic p i s t o n " d r i v i n g i n t o the plasma. T h i s i m p l o s i o n scheme i s i l l u s t r a t e d i n F i g . 2 - 1 . I n i t i a l l y , the magnetic p i s t o n s l o w l y compresses the o u t e r boundary of the plasma column. The r i s e i n p r e s s u r e , d e n s i t y and temperature i n the b e g i n n i n g , due t o the c o m p r e s s i o n , a r e n e g l i g i b l e . The p i s t o n a c c e l e r a t e s due t o the d e c r e a s i n g r a d i u s and the i n c r e a s i n g c u r r e n t . When the p i s t o n reaches s u p e r s o n i c radius —> F i g . 2-1 Space-time diagram of Z-pinch 1 2 speed, a shock wave i s d e v e l o p e d i n f r o n t of the p i s t o n . P h y s i c a l l y , the shock wave p r o v i d e s a mechanism which r e d i s t r i b u t e s p a r t of the m e c h a n i c a l energy i m p a r t e d by the magnetic p i s t o n i n t o t h e r m a l energy. The plasma i s swept up by the shock f r o n t as i t advances towards the c e n t e r . The p r e s s u r e of the shocked plasma a c q u i r e s a magnitude comparable t o t h a t of the d r i v i n g magnetic p i s t o n , which meanwhile i n c r e a s e s i n s t r e n g t h as the r a d i u s of the plasma column c o l l a p s e s . T h i s r e s u l t s i n a l a r g e d i s c o n t i n u i t y i n the d e n s i t y and temperature of the plasma a c r o s s the shock f r o n t . D u r i n g the f i n a l s t age of the i m p l o s i o n as the shock f r o n t converges towards the c e n t e r , the c o l l a p s i n g shock f r o n t i s s t r e n g t h e n e d by the volume convergence of the shocked plasma. The i n c r e a s e s i n p r e s s u r e , d e n s i t y and temperature of the shocked plasma, due t o convergence, p r o p a g a t e towards the shock f r o n t v i a the C+ c h a r a c t e r i s t i c s 1 . T h i s causes the s t r e n g t h of the shock t o 1 The c h a r a c t e r i s t i c s a r e c u r v e s i n a space-time diagram, a l o n g which the s m a l l d i s t u r b a n c e s p r o p a g a t e . The c h a r a c t e r i s t i c which i s i n the same d i r e c t i o n as the f l o w v e l o c i t y of the medium i s c a l l e d the C+ c h a r a c t e r i s t i c . The c h a r a c t e r i s t i c which i s i n the o p p o s i t e d i r e c t i o n from the f l o w v e l o c i t y i s c a l l e d the C- c h a r a c t e r i s t i c . The s l o p e of a c h a r a c t e r i s t i c i n the space-time diagram i s n u m e r i c a l l y e q u a l t o the r e c i p r o c a l of the a l g e b r i c sum of the f l o w v e l o c i t y and the a c o u s t i c v e l o c i t y of the medium. 13 i n c r e a s e . T h i s i n f o r m a t i o n a l s o t r a v e l s back t o the magnetic p i s t o n v i a the C- c h a r a c t e r i s t i c s and causes the speed of the p i s t o n t o v a r y , i n a d d i t i o n t o the p i s t o n ' s own v a r i a t i o n due t o c h a n g i n g plasma r a d i u s and c u r r e n t . The change i n the speed of the magnetic p i s t o n causes c o r r e s p o n d i n g changes i n p r e s s u r e , d e n s i t y and t e m p e r a t u r e , which propagate t o the e n t i r e plasma r e g i o n and the shock f r o n t v i a the C+ c h a r a c t e r i s i t i c s . U s u a l l y the d i s c h a r g e c u r r e n t i s s u f f i c i e n t l y s t r o n g such t h a t the d e c e l e r a t i o n of the magnetic p i s t o n does not cause the shock wave t o decay b e f o r e i t c o l l a p s e s a t the c e n t e r . As the c y l i n d r i c a l i m p l o d i n g shock f r o n t c o l l i d e s a t the c e n t e r of convergence, a r e f l e c t i o n shock i s g e n e r a t e d . T h i s p r o p a g a t e s back i n t o the shocked plasma, s t o p p i n g the convergence f l o w and c o n v e r t i n g the m e c h a n i c a l energy of the i n f l o w i n g plasma i n t o t h e r m a l energy. F i n a l l y , as the r e f l e c t e d shock passes t h r o u g h the c u r r e n t l a y e r , i t d i s s i p a t e s i n the vacuum. The e n t i r e plasma column now has e x p e r i e n c e d two shocks and has a k i n e t i c p r e s s u r e h i g h e r than the magnetic p r e s s u r e t h a t i s g e n e r a t e d by the o u t e r c u r r e n t l a y e r . As a r e s u l t , the hot plasma w i l l expand i n t o the s u r r o u n d i n g vacuum, p u s h i n g the c u r r e n t c a r r y i n g o u t e r l a y e r w i t h i t . A r a r e f a c t i o n fan i s g e n e r a t e d t h r o u g h t h i s e x p a n s i o n and p r o p a g a t e s back i n t o the plasma. However, t o o b t a i n an a n a l y t i c a l s o l u t i o n t o t h i s d e s c r i p t i o n i s seemingly i m p o s s i b l e . To o b t a i n a n u m e r i c a l s o l u t i o n w i l l r e q u i r e the use of hydrodynamic and p a r t i c l e codes and l o n g c o m p u t a t i o n t i m e . We f i n d t h a t by u s i n g v a r i o u s 14 a p p r o x i m a t i o n s , the problem can be reduced t o a system of c o u p l e d o r d i n a r y d i f f e r e n t i a l e q u a t i o n which can then be s o l v e d n u m e r i c a l l y . However, we s h a l l f i r s t r e v i ew the p i n c h dynamics by the f r e e p a r t i c l e m o d e l [ 3 5 ] , which i s a n a l y t i c a l l y easy and more a p p l i c a b l e i n the e a r l y s t a g e s of the p i n c h . //A/*>////srzWn Cylinder t FL A, plasma column F i g . 2-2 I d e a l i z e d p i n c h parameters 15 2.2.1 FREE PARTI CLE MODEL C o n s i d e r the i d e a l i z e d s i t u a t i o n d e p i c t e d i n F i g . 2 - 2 . We assume a c o m p l e t e l y i o n i z e d plasma column of l e n g t h I and r a d i u s A w i t h i n f i n i t e c o n d u c t i v i t y i n s i d e a c y l i n d e r of r a d i u s R Q, the w a l l s of which a l s o has i n f i n i t e c o n d u c t i v i t y . L e t the i n i t i a l r a d i u s of the plasma column be A Q and the i n i t i a l d e n s i t y be P Q. I f the v o l t a g e a c r o s s the plasma column L i s V, the c i r c u i t e q u a t i o n i s V - i - L I , (2.4) d t i n which the i n d u c t a n c e L i s a time v a r y i n g q u a n t i t y g i v e n by k - ~j £ n | , (2.5) c where R i s the " e f f e c t i v e r a d i u s " of the r e t u r n c u r r e n t . Any e x t e r n a l i n d u c t a n c e i n the c i r c u i t can be taken i n t o c o n s i d e r a t i o n by a d j u s t i n g the v a l u e of R. For a vacuum spark p i n c h , the e x t e r n a l i n d u c t a n c e i s u s u a l l y the d o m i n a t i n g f a c t o r i n the t o t a l i n d u c t a n c e because of the s m a l l s e p a r a t i o n between the e l e c t r o d e s . 16 Assuming t h a t the c a p a c i t a n c e i s v e r y l a r g e so t h a t the a p p l i e d v o l t a g e V i s a p p r o x i m a t e l y c o n s t a n t , we can e x p r e s s the c u r r e n t i n terms of the v o l t a g e as T V f c 2 , I = -5- • (2.6) 2£.£n£ A In the f r e e p a r t i c l e model by R o s e n b l u t h [ 3 5 ] , the plasma p a r t i c l e s a r e assumed t o be r e f l e c t e d by the c o l l a p s i n g magnetic p i s t o n and hence the r a t e of change of momentum a t the plasma s u r f a c e i s P = 2 p o \ d T / ' (2.7) where t o t a l sweep-up of the p a r t i c l e s i s assumed. S u b s t i t u t e Eq.2.3 and 2.6 i n t o Eq.2.7 and i n t r o d u c e the d i m e n s i o n l e s s v a r i a b l e s « = , (2.8) Ao and T = (2.9) 17 One o b t a i n s the d i m e n s i o n l e s s e q u a t i o n g o v e r n i n g the r a d i u s of the plasma column da - T dT * 2a(n-lrui) * (2.10) R where n = tn — o For a vacuum s p a r k , the i n t e r n a l i n d u c t a n c e can be: n e g l e c t e d . Eq.2.10 can then be i n t e g r a t e d t o g i v e the d i m e n s i o n l e s s r a d i u s as a f u n c t i o n of the d i m e n s i o n l e s s time 11) D e f i n e the d i m e n s i o n l e s s i m p l o s i o n time x- as the d i m e m s i o n l e s s time between a=1 and a=0. I t i s g i v e n by (2.12) The d i m e n s i o n l e s s i m p l o s i o n time i n c r e a s e s w i t h i n c r e a s i n g e x t e r n a l i n d u c t a n c e s i n c e the r a t e of c u r r e n t v a r i a t i o n i s l i m i t e d by the e x t e r n a l i n d u c t a n c e . 18 One can a l s o d e f i n e the d i m e n s i o n l e s s p i n c h d u r a t i o n as the d i m e n s i o n l e s s time bewteen (1/a)(da/dx)=-1.0 and a=0. For n>>1, i t can be shown from Eq.2.11 t h a t . x du.fi = 0.5 . (2.13) The p i n c h d u r a t i o n i s r e l a t e d d i r e c t l y t o d l / d t and can be measured e x p e r i m e n t a l l y . I t i s a more u s e f u l q u a n t i t y than the i m p l o s i o n t i m e , s i n c e i n a m u l t i p l e p i n c h the l a t t e r q u a n t i t y cannot be d e t e r m i n e d i n the subsequent p i n c h e s . I t s h o u l d be noted t h a t depends on the i n i t i a l d e n s i t y p Q , the a p p l i e d v o l t a g e V, and t h a t i t i s i n v e r s e l y p r o p o r t i o n a l t o A q , the i n i t i a l r a d i u s . I t can be shown t h a t Eq.2.13 h o l d s even under a t r a n s l a t i o n i n t i m e . T h i s means t h a t the p i n c h d u r a t i o n s c a l e s as x , r e g a r d l e s s of when the p i n c h o c c u r s . From t h i s r e l a t i o n s h i p the i n i t i a l r a d i u s A can be d e t e r m i n e d even i f the v a l u e of p i s not o o a c c u r a t e l y known as i n the c a s e s of m u l t i p l e p i n c h e s . T h i s model p r e d i c t s t h a t the r a d i u s of the p i n c h e d column c o l l a p s e s t o z e r o , which i s of c o u r s e i m p o s s i b l e . Of c o u r s e the model has not t a k e n i n t o c o n s i d e r a t i o n the subsequent r i s e i n p r e s s u r e and temperature of the p i n c h e d plasma. To a l l o w f o r t h e s e e f f e c t s , one must c o n s i d e r the shock wave t h a t i s g e n e r a t e d by the magnetic p i s t o n a t a l a t e r t i m e . 19 2.2.2 SHOCK WAVE MODEL As mentioned b e f o r e , the e x a c t d e s c r i p t i o n of t h i s model i s a n a l y t i c a l l y e x t r e m e l y d i f f i c u l t . Moreover, i t i s not a p p l i c a b l e i n the e a r l y s t a g e s of a p i n c h , when the magnetic p i s t o n speed i s slowe r than the a c o u s t i c speed of the plasma under c o n s i d e r a t i o n . However, i t i s a p p l i c a b l e a t l a t e r s t a g e s of the p i n c h , when the magnetic p i s t o n has a c q u i r e d a s u p e r s o n i c speed. We w i l l l o o k a t the dynamics of a p l a n e shock and then adapt them t o a c y l i n d r i c a l c o n v e r g i n g geometry. 20 C o n s i d e r a p l a n e shock p r o p a g a t i n g w i t h v e l o c i t y v^ i n t o a s t a t i o n a r y gas of p r e s s u r e p Q and d e n s i t y p Q . The c o r r e s p o n d i n g p r e s s u r e , d e n s i t y P j and p a r t i c l e v e l o c i t y of the gas beh i n d the shock f r o n t a r e g i v e n by (see Appendix B) p l ~ p o = p o v a V 4 ' (2.14a) P 8 i 6 o p g M 2 * 2 £ (2.14b) 2 £ a o and v _ = a. u 2 a M 6 o (2.14c) where g - g m 2 g • ! V 4 M i s the Mach number, M - —— a o and a Q i s the a c o u s t i c speed of the unshocked gas, a * . bl° In the above e q u a t i o n s , g Q , g 1 a r e the e n t h a l p y c o e f f i c i e n t s ( r a t i o s of e n t h a l p y t o i n t e r n a l energy) of the gases i n f r o n t of and b e h i n d the shock. These c o e f f i c i e n t s , i n t r o d u c e d by A h l b o r n [ l ] , t a k e c a r e of the 21 thermodynamics t h a t o c c u r d u r i n g the shock t r a n s i t i o n and the d e s c r i p t i o n i s not r e s t r i c t e d t o f u l l y i o n i z e d plasma. For an i d e a l gas the e n t h a l p y c o e f f i c i e n t g i s e q u a l t o the r a t i o of the s p e c i f i c h e a t s . For a r e a l gas g ranges between the upper l i m i t of 5/3 a t v e r y h i g h t e m p e r a t u r e s ( of the o r d e r of 1MeV) and the lower l i m i t of 1. In p r i n c i p l e g can be c a l c u l a t e d f o r plasmas i n LTE or steady s t a t e c o r o n a l e q u i l i b r i u m and t h i s i s done i n Appendix B. But i t i s a time v a r y i n g q u a n t i t y f o r the t r a n s i e n t , m u l t i p l y i o n i z e d plasma t h a t e x i s t s i n a vacuum sp a r k . However, i n a l l the c a l c u l a t i o n s t h a t f o l l o w s , the g's are assumed t o be known pa r a m e t e r s . One n otes t h a t Eq.2.14a can be w r i t t e n as (2.15) which i s Newton's second law. The f i r s t f a c t o r i n s i d e the square b r a c k e t on the r i g h t i s the mass of m a t t e r t h a t i s swept up by the f r o n t . The second f a c t o r i s t h e i r change i n v e l o c i t y . I f one assumes t h a t a u n i f o r m p a r t i c l e v e l o c i t y e x i s t s b e h i n d the shock f r o n t , then f o r a c y l i n d r i c a l shock wave the p r e s s u r e e q u a t i o n i s 2 i r ( A P i - S p o ) » d_ dt 0 d t . (2.16) 22 where S i s the d i s t a n c e of the shock f r o n t f r o m the c e n t e r of i m p l o s i o n . The e f f e c t s of volume convergence on the fl o w b e h i n d the shock i s n e g l e c t e d and the co m p r e s s i o n shock t h i c k n e s s i s assumed t o be s m a l l compared t o t h e r a d i u s of the shock f r o n t . The r a d i u s of the magnetic p i s t o n i s g i v e n as a f u n c t i o n of the r a d i u s of the shock f r o n t a c c o r d i n g t o Eq.2.14c l £ o f l dx 4 **J*2L\\ 1 6 g i a 2 . d x \ W g 0 ( g x * D 2 0 (2.17) where a i s the magnetic p i s t o n r a d i u s , a i s the shock f r o n t r a d i u s and a Q i s the a c o u s t i c speed of the i n i t i a l plasma i n the d i m e n s i o n l e s s c o o r d i n a t e s . Under our e x p e r i m e n t a l c o n d i t i o n s , an a Q=l w i l l c o r r e s p o n d s a p p r o x i m a t e l y t o an a c o u s t i c speed of 1.25x10 7cm/s. S u b s t i t u t i n g Eq.2.3, 2.6, and 2.8 i n t o 2.16, one g e t s the d i m e n s i o n l e s s e q u a t i o n f o r an i m p l o d i n g shock i _ [ (1 . 0 2 )10 . 1 . _ _ i i + 2 t f V _ ^ ( 2 > 1 8 ) dx L d-rj 2n 2 g o v a l i d when T > / — a n. V t . ' • The c o n d i t i o n f o r the v a l i d i t y of Eq.2.18 a r i s e s from the requirement t h a t the magnetic p r e s s u r e s h o u l d exceed the plasma k i n e t i c p r e s s u r e . The r e s u l t thus o b t a i n e d from s o l v i n g t h e s e two c o u p l e d o r d i n a r y d i f f e r e n t i a l e q u a t i o n s w i l l g i v e :a lower bound of the mininum r a d i u s a c h i e v e d by 23 the i m p l o s i o n For the l i m i t i n g case of g^=1(infinitely c o m p r e s s i b l e ) and a o = 0 ( s t r o n g s h o c k ) , the shock f r o n t v e l o c i t y i s the same as the p a r t i c l e v e l o c i t y , and the shock wave model reduces t o the snowplow model of R o s e n b l u t h [ 3 5 ] . Furthermore the v a r y i n g i n d u c t a n c e term can be n e g l e c t e d f o r a vacuum spark due t o the h i g h e x t e r n a l i n d u c t a n c e and one o b t a i n s the power s e r i e s s o l u t i o n . . 1 - - i T 2 . ILL-Lll'lV + ... (2.19) 6n 36 n 2 \ 11 n / A f t e r the i m p l o d i n g shock has c o l l a p s e d a t the c e n t e r , a r e f l e c t i o n shock w i l l be g e n e r a t e d t o s t o p the c o n v e r g i n g f l o w of plasma and c o n v e r t t h e i r m e c h a n i c a l energy i n t o t h e r m a l energy. We now proceed t o e v a l u a t e the v e l o c i t y of t h i s r e f l e c t i o n shock. For the sake of s i m p l i c i t y we w i l l n e g l e c t the e f f e c t s of volume convergence on the i m p l o d i n g shock and of volume d i v e r g e n c e on the r e f l e c t e d shock, s i n c e t h e s e two e f f e c t s w i l l be p a r t i a l l y s e l f - c a n c e l l e d . The boundary c o n d i t i o n r e q u i r e s the p a r t i c l e v e l o c i t y v ^ of the plasma be h i n d the r e f l e c t e d shock be z e r o . 24 Whitham[49] has shown t h a t , f o r i d e a l gas, the speed of the r e f l e c t i o n shock wave v^, the p r e s s u r e p 2 and the d e n s i t y p 2 of the gas b e h i n d the r e f l e c t e d shock g e n e r a t e d by the i n c i d e n t s t r o n g shock a r e E 2 - l v» = 2 * P, " —^-TPI • (2.20) 2 f ! 2 « > i 1 P S - — — p 2 where v , p , p a r e the f i n a l i m p l o d i n g shock speed, 4 1 1 p r e s s u r e and d e n s i t y r e s p e c t i v e l y a c h i e v e d by the gas beh i n d the i m p l o d i n g shock wave. The n u m e r i c a l r e s u l t s of the i m p l o s i o n s and the r e f l e c t i o n s a re shown i n F i g . 2 - 3 ( a ) f o r i d e a l gas of d i f f e r e n t g's. The r e s u l t s f o r d i f f e r e n t R and a 0 a r e shown i n F i g . 2 - 3 ( b ) and (c) r e s p e c t i v e l y . They show t h a t the i m p l o s i o n time i s m a i n l y a f u n c t i o n of the e x t e r n a l i n d u c t a n c e , and o n l y v a r y s l i g h t l y over the e n t i r e a l l o w a b l e range of g. The f i n a l p i n c h e d r a d i i a r e d e t e r m i n e d by t h e i r f i n a l g v a l u e s . A lower v a l u e of g r e s u l t s i n a s m a l l e r p i n c h e d r a d i u s , i n a d d i t i o n t o a s l i g h t l y s h o r t e r i m p l o s i o n t i m e . 25 The e q u i v a l e n t forms of Eq.2.12 an 2.13 a r e now T A.mp (2.21 ) « 0 . 7 (2.22) The i n i t i a l a c o u s t i c speed a 0 , which i s a measure of the i n i t i a l p r e s s u r e , a f f e c t s o n l y s l i g h t l y the i m p l o s i o n time and the f i n a l r a d i u s , w i t h a h i g h e r a f 0 g i v i n g a s l i g h t l y l o n g e r i m p l o s i o n time and a l a r g e r r a d i u s . A graph of t h i s r e l a t i o n s h i p i s p l o t t e d i n F i g . 2 - 4 . The c a l c u l a t i o n w i l l g i v e an e r r o n e o u s l y h i g h f i n a l r a d i u s f o r h i g h i n i t i a l a c o u s t i c speed a f 0 . T h i s i s because the p r e s s u r e wave a t the b e g i n n i n g of the shock f r o n t w i l l o n l y be an a c o u s t i c p r e s s u r e wave. C o n s e q u e n t l y , the a c t u a l r e f l e c t i o n shock w i l l be g e n e r a t e d l a t e r than the time i n d i c a t e d i n the c a l c u l a t i o n , and t h i s w i l l g i v e the compressed s h e l l more time t o implode. A l t h o u g h the shock wave model i s s t r i c t l y s p e a k i n g not a p p l i c a b l e d u r i n g the e a r l y s t a g e of a p i n c h due t o the absence of a w e l l formed shock, y e t comparison between the f r e e p a r t i c l e model and the shock wave model i n d i c a t e s t h a t one can s a f e l y use the shock wave model even i n the e a r l y s t a g e , b e a r i n g i n mind t h a t the dynamics a t t h i s s tage a r e q u i t e i n s e n s i t i v e t o the models b e i n g used. 26 0 3 6 9 12 15 dimensionless time f F i g . 2-3(a) N u m e r i c a l s o l u t i o n s of shock model f o r d i f f e r e n t e n t h a l p y c o e f f i c i e n t g. 2.3 DISCUSSION OF ASSUMPTIONS IN THE MODELS We have made s e v e r a l i m p o r t a n t assumptions i n the attempt t o s o l v e the h y d r o d y n a m i c a l e q u a t i o n s . At the r i s k of b e i n g r e p e t i t i v e , t h e s e assumptions a r e o u t l i n e d here a g a i n and t h e i r consequences examined more c l o s e l y . Eq.2.15 i s v a l i d o u t s i d e the shock t r a n s i t i o n r e g i o n , which c o n s i s t s of the compression shock and the r e l a x a t i o n l a y e r . The compression shock i s the r e g i o n i n which the plasma p a r t i c l e s a r e r a p i d l y compressed t o a h i g h e r d e n s i t y , 27 F i g . 2-3(b) N u m e r i c a l s o l u t i o n s of shock model f o r d i f f e r e n t e f f e c t i v e r a d i u s R of r e t u r n c y l i n d e r . t r a n s f o r m i n g t h e i r m e c h a n i c a l k i n e t i c energy i n t o random t h e r m a l k i n e t i c energy. As the compression shock t h i c k n e s s i s u s u a l l y of the o r d e r of a few mean f r e e p a t h s [ 5 3 ] , i t p r e s e n t s no p a r t i c u l a r problem i n e x p e r i m e n t s of our d i m e n s i o n s and d e n s i t i e s . The r e l a x a t i o n l a y e r i s the r e g i o n i n which the shock-compressed plasma r e o r g a n i z e s i t s energy by e s t a b l i s h i n g an e q u i l i b r i u m i n v a r i o u s , e x c i t a t i o n l e v e l and i o n i z a t i o n s t a g e s . These a r e u s u a l l y slow p r o c e s s e s and t h e i r d i m e n s i o n s exceed the d i m e n s i o n of our a p p a r a t u s . When such i s the c a s e , then i m m e d i a t e l y b e h i n d 28 F i g . 2-3(c) N u m e r i c a l s o l u t i o n s of shock model f o r d i f f e r e n t i n i t i a l a c o u s t i c speed a0. the c o m p r e s s i o n shock the plasma can be t r e a t e d as a monatomic i d e a l gas w i t h g=5/3, when n e i t h e r e x c i t a t i o n nor i o n i z a t i o n have time t o p r o c e e d . The v a l u e of g v a r i e s downstream a l o n g the shocked plasma, where r e o r g a n i z a t i o n of e n e r g i e s t a k e s p l a c e . T h i s means t h a t i n the case where the t h i c k n e s s of shock-compressed r e g i o n i s s m a l l e r than the r e q u i r e d shock t h i c k n e s s , one can s t i l l use Eq.2.18 a l b e i t w i t h a h i g h e r and v a r y i n g g v a l u e . 29 0.1 10 3 T J O l _ c o g 0.01| E T J O C 0.001 c 5-003 \ 0 D 2 ^ O D K Q 0 0 5 -0.0 oy lit /# I 1 1D 1.1 12 13 g — 1.4 15 1.6 F i g . 2-4 Dependence of f i n a l p i n c h e d r a d i u s on e n t h a l p y c o e f f i c i e n t g and i n i t i a l a c o u s t i c speed a",, . In the model, we have a l s o n e g l e c t e d the s e l f - s t r e n g t h e n i n g of the c o n v e r g i n g shock wave. To see when t h i s e f f e c t becomes i m p o r t a n t , one l o o k s a t the shock s t r e n g t h e n i n g due t o a c y l i n d r i c a l i m p l o d i n g s t r o n g shock g i v e n by Whitham[49] n M A , (2.23) where n 1 + «i V«i-= 0 as g 1 — 1 = 0.23 as g..=5/3 30 Thus t h i s e f f e c t i s o n l y i m p o r t a n t a t the f i n a l s t age of the i m p l o s i o n , when g i s l a r g e r than 1. S i n c e g i s a l s o r e l a t e d t o the c o m p r e s s i b i l i t y of the plasma, t h e r e f o r e i n i g n o r i n g the s t r e n g t h e n i n g due t o convergence, we have assumed t h a t the plasma i s i n f i n i t e l y c o m p r e s s i b l e i n the a z i m u t h a l d i r e c t i o n . T h i s r e s u l t s i n a s m a l l e r f i n a l p i n c h e d r a d i u s . The assumption t h a t a u n i f o r m f l o w v e l o c i t y e x i s t s b e h i n d the shock f r o n t i n Eq.2.16 i s e q u i v a l e n t t o the a s s e r t i o n t h a t the shock-compressed plasma becomes i n c o m p r e s s i b l e i n the r a d i a l d i r e c t i o n . T h i s r e s u l t s i n a l a r g e r f i n a l p i n c h e d r a d i u s . I t i s d i f f i c u l t t o determine which of the above two assumptions has a d o m i n a t i n g e f f e c t . However, as a g e n e r a l g u i d e l i n e one can use a o=0.05 as the v a l u e when the two e f f e c t s a r e b a l a n c e d . For a<><0.05 the c a l c u l a t i o n then g i v e s a lower bound f o r the f i n a l p i n c h e d r a d i u s and v i c e v e r s a . S i n c e i n our experiment a'o < <0.05 the r e s u l t s of the c a l c u l a t i o n s h o u l d be t r e a t e d as the lower bound f o r the r a d i u s of convergence of the p i n c h e d plasma. We have assumed the i m p l o d i n g column t o be c y l i n d r i c a l l y symmetric and u n i f o r m a l o n g the z - a x i s . We s h a l l see i n Chapter 5 t h a t the plasma column i s u n s t a b l e t o p e r t u r b a t i o n s . However, the growth r a t e s of these p e r t u r b a t i o n s a re s m a l l d u r i n g the i n i t i a l and the i n t e r m e d i a t e s t a g e s of the i m p l o s i o n . However, these growth r a t e s a re l a r g e d u r i n g the f i n a l moments of the 31 i m p l o s i o n and any n o n - u n i f o r m i t y a l o n g the z - a x i s w i l l enhance the growth of t h e s e p e r t u r b a t i o n s . The developments of t h e s e p e r t u r b a t i o n can cause the n u m e r i c a l s o l u t i o n s t o break down at the f i n a l moments of the i m p l o s i o n . The assumption t h a t the magnetic f i e l d i n s i d e the plasma i s z e r o i s q u e s t i o n a b l e . The magnetic f i e l d from the r e s i s t i v e phase of the d i s c h a r g e may be t r a p p e d i n s i d e the plasma and the magnetic f i e l d from the vacuum can d i f f u s e i n , s i n c e the plasma does not have i n f i n i t e c o n d u c t i v i t y . T h i s changes the p i c t u r e of a ' s o l i d ' p i s t o n t o a 'porous' one. P r e v i o u s e x p e r i m e n t s on g a s - f i l l e d Z-pinches had shown t h a t the magnetic f i e l d i s non-zero i n s i d e the plasma, a l t h o u g h i t has a maximum v a l u e a t the o u t e r plasma s u r f a c e . The t r a p p e d magnetic f i e l d p r e s e n t s no p a r t i c u l a r problems s i n c e t h i s w i l l be v e r y s m a l l due t o the s m a l l amount of c u r r e n t t h a t f l o w s t h r o u g h the plasma d u r i n g the r e s i s t i v e phase. The d i f f u s i o n of the magnetic f i e l d does not change the o v e r a l l p r e s s u r e a c t i n g on the column, but i t now e n a b l e s e l e c t r i c a l energy t o be c o u p l e d i n t o the plasma by Ohmic h e a t i n g i n s t e a d of t h r o u g h magnetic comp r e s s i o n a l o n e . The c o n d u c t i v i t y of a plasma i s g i v e n by S p i t z e r [ 3 9 ] 2(2kT ) 3 / 2 « = 3 / 2 / ^ * 2 , . fW , (2.24) where f(Z)=0.58 f o r Z=1 and->l/Z f o r l a r g e Z, where Z i s the 32 average charge s t a t e of the i o n s i n the plasma under c o n s i d e r a t i o n . In o r d e r f o r our models t o be v a l i d , the energy i n p u t t h r o u g h magnetic c o m p r e s s i o n s h o u l d be much h i g h e r than t h a t t h r o u g h Ohmic h e a t i n g . The magnetic c o m p r e s s i o n power per u n i t l e n g t h i s i i i ^ (2.25) 4 d t ' w h i l e the Ohmic h e a t i n g per u n i t l e n g t h i s 1 2 (2.26) i r A 2 a Thus the above c r i t e r i o n can be w r i t t e n as A i i » si d t -aa (2.27) » 4 D H c 2 where = 4TTO i s the magnetic f i e l d d i f f u s i o n c o e f f i c i e n t [ 2 0 ] . T h i s means the r a t e of magnetic f i e l d d i f f u s i o n s h o u l d be s m a l l compared t o the r a t e of i m p l o s i o n . T h i s c o n d i t i o n i s o b v i o u s l y not s a t i s f i e d d u r i n g the e a r l y s t a g e s of the i m p l o s i o n , when the magnetic p i s t o n i s moving v e r y s l o w l y . D u r i n g t h i s time most of the e l e c t r i c a l energy i s d e l i v e r e d 33 t o the plasma as Ohmic h e a t i n g . As the t emperature of the plasma i n c r e a s e s i t s c o n d u c t i v i t y i n c r e a s e s . Meanwhile the speed of i m p l o s i o n i n c r e a s e s and Eq.2.27 i s q u i c k l y s a t i s f i e d . The i n t e g r a l form of Eq.2.27 i s iroA 2 — ~ > > 1> (2.28) 2 c 2 t , dur which s h o u l d be s a t i s f i e d by the models i n o r d e r f o r t h e i r o v e r a l l d e s c r i p t i o n s t o be v a l i d . Energy l o s s e s t h r o u g h r a d i a t i o n i s a n other i m p o r t a n t p r o c e s s which has not been c o n s i d e r e d i n the models. T h i s i s s i g n i f i c a n t i n a vacuum spark because of the h i g h Z of the plasma and the abundance of i n c o m p l e t e l y s t r i p p e d i o n s g i v i n g r i s e t o s t r o n g l i n e r a d i a t i o n . T h i s w i l l lower the temperature of the plasma as w e l l as p r e h e a t i n g the uncompressed plasma. We s h a l l examine these energy terms i n Chapter 4 u s i n g some of the e x p e r i m e n t a l r e s u l t s o b t a i n e d . 34 CHAPTER 3  DESCRIPTION OF APPARATUS The p r i n c i p a l elements of a vacuum spark a r e two e l e c t r o d e s i n s i d e a vacuum chamber w i t h a h i g h v o l t a g e a p p l i e d a c r o s s them. Breakdown i s t r i g g e r e d e i t h e r e l e c t r i c a l l y (by a h i g h v o l t a g e p u l s e on a t h i r d e l e c t r o d e ) or o p t i c a l l y (by a h i g h power p u l s e d ruby l a s e r ) . The r e m a i n i n g a p p a r a t u s c o n s i s t s of the d i a g n o s t i c equipment. A v o l t a g e probe i s used t o mo n i t o r the d i s c h a r g e v o l t a g e and a Rogowski c o i l i s used t o measure the d i s c h a r g e c u r r e n t d e r i v a t i v e . The v i s i b l e and u l t r a v i o l e t p a r t of the spark e m i s s i o n a r e a n a l y z e d w i t h a f a s t f r a m i n g camera and a monochromator w i t h e i t h e r an o p t i c a l m u l t i - c h a n n e l a n a l y z e r f o r s p e c t r a l r e s o l u t i o n or a p h o t o m u l t i p l i e r f o r tempor a l r e s o l u t i o n . T i m e - r e s o l v e d X-ray e m i s s i o n i s measured by a PIN X-ray d i o d e and a 4-channel p l a s t i c s c i n t i l l a t o r - p h o t o m u l t i p l i e r assembly. The s p a t i a l l y r e s o l v e d measurements i n the X-ray r e g i o n a r e done u s i n g a p i n h o l e camera. A more d e t a i l e d d e s c r i p t i o n of the system f o l l o w s . 35 3.1 DISCHARGE VESSEL AND POWER SUPPLY The vacuum spark chamber i s shown i n F i g . 3-1. I t has a t r i a x i a l s t r u c t u r e , w i t h the outermost . c y l i n d e r as the vacuum chamber. The second c y l i n d e r w i t h a d i a m e t e r of 3.175cm, p r o v i d e s a low-impedance c u r r e n t r e t u r n p a t h . The c e n t r a l c o n d u c t o r has two p h y s i c a l l y s e p a r a t e d p a r t s . The lower p a r t has a d i a m e t e r of 1.05cm, which makes up the main body of the anode c a r r i e r . I t i s mounted d i r e c t l y onto the p o s i t i v e t e r m i n a l of a c a p a c i t o r , and e l e c t r i c a l l y i n s u l a t e d from the r e s t of the chamber by D e l r i n . A Rogowski c o i l , w hich w i l l be d e s c r i b e d l a t e r , s l i p s over t h i s c o n d u c t o r and measures the d i s c h a r g e c u r r e n t d e r i v a t i v e d l / d t . V a r i o u s e l e c t r o d e t i p s , t y p i c a l l y w i t h d i a m e t e r s of 0.635cm, can be f i t t e d on t o p of t h i s c o n d u c t o r . The upper p a r t of the c e n t r a l c o n d u c t o r c o n s t i t u t e s the c a t h o d e , which i s i n d i r e c t e l e c t r i c a l c o n t a c t w i t h the second c y l i n d e r . The cathode has a d iameter of 1.27cm. I t i s d r i l l e d out i n the c e n t e r f o r the placement of v a r i o u s t r i g g e r e l e c t r o d e s f o r e l e c t r i c a l t r i g g e r i n g o p e r a t i o n s , or f o r the passage of l a s e r l i g h t f o r l a s e r t r i g g e r i n g o p e r a t i o n . The two e l e c t r o d e s a r e n o m i n a l l y s e p a r a t e d by 5mm, a c r o s s which the d i s c h a r g e o c c u r s . By r a i s i n g or l o w e r i n g the e n t i r e chamber, and hence the c a t h o d e , the s e p a r a t i o n can be e a s i l y a d j u s t e d , w h i l e the anode remains f i x e d by the c a p a c i t o r bank. V a r i o u s e l e c t r o d e g e o m e t r i e s were t r i e d , but the f i n a l d e s i g n had round e l e c t r o d e t i p s on both the anode and the cathode f o r l o n g e v i t y and a c c e p t a b l e r e p r o d u c i b i l i t y . to vacuura-pump copper central conductor glass insulating jacket brass outer conductor O-ring cathode trigger electrode anode scale Ddrin to capacitor -ve terminal Fcbgowski con to capacitor +ve terminal F i g . 3-1 The d i s c h a r g e v e s s e l 37 The t o t a l l e n g t h of the i n n e r c o a x i a l tube i n c l u d i n g the d i s c h a r g e gap i s 14cm, g i v i n g an i n d u c t a n c e of 38nH. However, the gap i n d u c t a n c e c o n t r i b u t e s l e s s than 2nH s i n c e the d i s c h a r g e gap i s o n l y 5mm l o n g . The o u t e r and the m i d d l e c y l i n d e r each has f o u r p o r t s , 90 degrees a p a r t , f o r d i a g n o s t i c a c c e s s t o the plasma r e g i o n . The whole system i s e v a c u a t e d by an o i l d i f f u s i o n pump t o a r e s i d u a l p r e s s u r e of 5x10" 6 t o r r as measured by an i o n i z a t i o n gauge. The d i s c h a r g e i s powered by a 1OuF c a p a c i t o r , w i t h a maximum c h a r g i n g v o l t a g e of 25kV. The i n d u c t a n c e of the c a p a c i t o r i s 40nH, w h i l e the i n d u c t a n c e of the r e t u r n p a t h from the c o a x i a l chamber t o the n e g a t i v e t e r m i n a l of the c a p a c i t o r i s 72nH. Together w i t h the i n d u c t a n c e of the c o a x i a l l i n e , t h i s g i v e s a t o t a l e x t e r n a l i n d u c t a n c e of l50nH. D u r i n g the breakdown phase and the r e a c t i v e phase, the d i s c h a r g e e m i t s a broad spectrum of r a d i o n o i s e which can cause e l e c t r i c a l i n t e r f e r e n c e t o nearby measuring i n s t r u m e n t s . For t h i s reason the e n t i r e chamber and the c a p a c i t o r bank a r e e n c l o s e d i n s i d e a copper s h i e l d . The h i g h v o l t a g e c h a r g i n g c a b l e s coming out of the s h i e l d are f i l t e r e d by p i - n e t w o r k s of c o a x i a l c a p a c i t o r and f e r r i t e - w o u n d i n d u c t o r . T a b l e 3.1 g i v e s a summary of the i m p o r t a n t parameters of the vacuum s p a r k . 38 Vacuum spark parameters: r e t u r n c y l i n d e r c e n t r a l c o n d u c t o r anode cathode i n t e r e l e c t r o d e s e p a r a t i o n background p r e s s u r e c a p a c i t o r c h a r g i n g v o l t a g e s t o r e d energy c i r c u i t i n d u c t a n c e r i n g i n g f requency q u a r t e r c y c l e time copper 3 . 175cm I.D. 14.0cm l o n g c o p p e r , c a r b o n 0.635cm O.D. copper 1.275cm O.D. 0. 5cm < 5 x 1 0 " 6 t o r r 10.47UF 1 OkV 523.5J 1 50nH 127KHZ 1 .96JJS T a b l e 3.1 Vacuum spark parameters 3.2 TRIGGERING APPARATUS S i n c e the s e l f - b r e a k d o w n v o l t a g e f o r vacuum i s much h i g h e r than the a p p l i e d v o l t a g e a c r o s s the e l e c t r o d e s , the d i s c h a r g e w i l l not be s e l f - i n i t i a t e d . The breakdown i s a r t i f i c i a l l y i nduced by the i n j e c t i o n of charge c a r r i e r s ( e l e c t r o n s and i o n s ) i n t o the i n t e r - e l e c t r o d e gap. To a c h i e v e t h i s , one can use a t h i r d e l e c t r o d e . A s m a l l t r i g g e r e l e c t r o d e i s i n s e r t e d i n t o the c e n t e r of the cat h o d e , s e p a r a t e d by an i n s u l a t i n g r i n g . When a h i g h 39 v o l t a g e i s p u l s e d onto the t r i g g e r e l e c t r o d e , breakdown o c c u r s a t the e l e c t r o d e and a l o n g the i n s u l a t o r , by f i e l d e m i s s i o n . I t produces a plasma c o n s i s t i n g of the e l e c t r o d e m a t e r i a l s and the i n s u l a t o r . T h i s plasma then e n t e r s i n t o the i n t e r - e l e c t r o d e gap, p r o v i d i n g e l e c t r o n s f o r the main breakdown. Another way t o produce e l e c t r o n s i n the i n t e r - e l e c t r o d e gap i s by f o c u s s i n g an i n t e n s e l a s e r beam onto the e l e c t r o d e s u r f a c e s . The i n t e n s e l a s e r p u l s e produces a h i g h l y i o n i z e d plasma c o n s i s t i n g of the e v a p o r a t e d e l e c t r o d e m a t e r i a l s which then f i l l s up the i n t e r - e l e c t r o d e gap and s u p p l i e s the charge c a r r i e r r e q u i r e d f o r the breakdown of the main gap. At the o u t s e t of t h i s study i t was not c l e a r which of the t r i g g e r i n g methods would g i v e more r e p r o d u c i b l e r e s u l t s . As t h i s had t o be d e t e r m i n e d e x p e r i m e n t a l l y , these d i f f e r e n t t r i g g e r i n g a p p a r a t u s were c o n s t r u c t e d . They a r e d e s c r i b e d below and t h e i r performances a r e e v a l u a t e d i n Chapter 4. We found t h a t the d i r e c t d i s c h a r g e t r i g g e r i n g d e s c r i b e d i n the f o l l o w i n g S u b - S e c t i o n g i v e s the be s t performance. 40 3.2.1 ELECTRICAL DISCHARGE TRIGGERING The r e p r o d u c i b i l i t y of the main d i s c h a r g e depends on the r e l i a b l e f o r m a t i o n of a spark c h a n n e l d u r i n g the breakdown phase. The i n j e c t i o n of a l a r g e amount of plasma i n t o the i n t e r - e l e c t r o d e gap by the t r i g g e r spark w i l l improve the r e p r o d u c i b i l i t y of the main d i s c h a r g e . To a c h i e v e t h i s , the t r i g g e r gap i s r e c e s s e d 0.5cm i n s i d e the c a t h o d e , but connected t o the i n t e r - e l e c t r o d e gap by a 0.15cm dia m e t e r h o l e . A w e l l - d e f i n e d plasma j e t i s formed when the expanding plasma from the t r i g g e r spark escapes th r o u g h t h i s h o l e i n t o the i n t e r - e l e c t r o d e gap. Two methods were used t o produce the t r i g g e r s p a r k . In the f i r s t method, the t r a n s f o r m e d d i s c h a r g e t r i g g e r i n g , a h i g h v o l t a g e p u l s e i s g e n e r a t e d i n the secondary c o i l of a s t e p - u p t r a n s f o r m e r by d i s c h a r g i n g a c a p a c i t o r i n t o the p r i m a r y c o i l of the t r a n s f o r m e r . T h i s h i g h v o l t a g e causes the t r i g g e r gap t o break down, p r o d u c i n g a t r i g g e r s p a r k . In the second method, the d i r e c t d i s c h a r g e t r i g g e r i n g , a t r i g g e r spark i s g e n e r a t e d d i r e c t l y from a h i g h v o l t a g e c a p a c i t o r d i s c h a r g e t h r o u g h the t r i g g e r gap. The c i r c u i t of the t r a n s f o r m e d d i s c h a r g e t r i g g e r i n g system i s shown i n F i g . 3 - 2 . I t c o n s i s t s of a 1uF c a p a c i t o r , a SCR s w i t c h and a step-up (1:50) t r a n s f o r m e r . The c a p a c i t o r i s c h a r g e d t o 500V- DC, s t o r i n g 125mJ of energy. A low v o l t a g e p u l s e a p p l i e d t o the gate of the SCR causes i t t o conduct, thus s h o r t i n g the c a p a c i t o r a c r o s s the 41 r*' 5 0 0 V supply 1juf A A / V i n J ^ - A A A Jl?pF I spark shield _ J F i g . 3-2 C i r c u i t diagram f o r t r a n s f o r m e d d i s c h a r g e t r i g g e r i n g p r i m a r y c i r c u i t of the t r a n s f o r m e r . The h i g h v o l t a g e • a c r o s s the secondary c i r c u i t causes the breakdown of the t r i g g e r gap. The e n t i r e c i r c u i t i s c o m p l e t e l y s h i e l d e d t o p r e v e n t i t from c a u s i n g e l e c t r i c a l i n t e r f e r e n c e t o nearby measuring equipment. T h i s s i m p l e c i r c u i t , however, has one major d i s a d v a n t a g e . When breakdown o c c u r s , the impedance of the t r i g g e r gap f a l l s s h a r p l y . The v o l t a g e a c r o s s the gap a l s o f a l l s s h a r p l y due t o the h i g h impedance of the secondary c o i l of the t r a n s f o r m e r . The t r i g g e r spark i s quenched when the v o l t a g e f a l l s below the t h r e s h o l d f o r s e l f - s u s t a i n e d d i s c h a r g e and the vacuum i n s u l a t i o n s t r e n g t h i s r a p i d l y r e s t o r e d . Only a s m a l l f r a c t i o n of the s t o r e d 42 energy i s r e l e a s e d t o the plasma, and u s u a l l y the amount of plasma r e l e a s e d d u r i n g t h i s s h o r t time may not be s u f f i c i e n t t o i n i t i a t e the breakdown of the main gap. Another breakdown of the t r i g g e r gap w i l l o c c u r a t the next v o l t a g e maximum, h a l f a c y c l e l a t e r . S e v e r a l of t h e s e d i s c h a r g e s may be n e c e s s a r y t o produce a s u f f i c i e n t number of e l e c t r o n s t o cause the main gap t o breakdown. T h i s causes u n a c c e p t a b l e j i t t e r and i r r e p r o d u c i b i l i t y i n the main d i s c h a r g e . Attempts t o i n c r e a s e the power of the t r i g g e r spark and d e c r e a s e the r i n g i n g time have not been s u c c e s s f u l because of the i n t r i n s i c l i m i t a t i o n s of the t r a n s f o r m e r . We t h e r e f o r e r e s o r t e d t o d i r e c t d i s c h a r g e t r i g g e r i n g . In t h i s method a low i n d u c t a n c e c a p a c i t o r bank i s charged t o a h i g h v o l t a g e , which i s p u l s e d d i r e c t l y t o the t r i g g e r i n g p i n . The step-up t r a n s f o r m e r w i t h i t s a s s o c i a t e d problems i s thus e l i m i n a t e d . The c i r c u i t i s shown i n F i g 3-3. The c a p a c i t o r bank has a t o t a l c a p a c i t a n c e of l 6 . 2 n F . I t i s charged t o 9kV g i v i n g a bank energy of 656mJ, and i s o l a t e d from the t r i g g e r p i n by a c o a x i a l p r e s s u r i z e d spark gap s w i t c h . The s w i t c h i s c l o s e d by f i r i n g an a u x i l i a r l y d i s c h a r g e , which i s powered by a k r y t r o n - c a p a c i t o r u n i t . A h i g h v o l t a g e p u l s e then p r o p a g a t e s down the c o a x i a l l i n e , w i t h the t r i g g e r e l e c t r o d e as the c e n t r a l c o n d u c t o r and the cathode as the o u t e r c o n d u c t o r . Breakdown by f i e l d e m i s s i o n then o c c u r s a t the t r i g g e r p i n and the i n s u l a t o r . The d i s c h a r g e e j e c t s a l a r g e q u a n t i t y of plasma i n t o the i n t e r - e l e c t r o d e gap t o 43 6x 2.72nF MM H H 100Mf Rogowski coil 9KV power supply pressurized spark gap - switch 1 4 : 1 spark _gap F i g . 3-3 C i r c u i t diagram f o r d i r e c t d i s c h a r g e t r i g g e r i n g i n i t i a t e the main d i s c h a r g e . 3.2.2 LASER TRIGGERING L a s e r t r i g g e r i n g has been demonstrated as a v e r y r e l i a b l e method of t r i g g e r i n g i n gas d i s c h a r g e s . The l a s e r produces a l o n g spark c h a n n e l a l o n g i t s beam p a t h and t h i s method i s noted f o r i t s s h o r t f o r m a t i v e time l a g ' and the r e p r o d u c i b i l i t y of t h e d i s c h a r g e . However, t h e s i t u a t i o n i s d i f f e r e n t i n a vacuum s p a r k , where the l a s e r produced plasma i s formed a t the s u r f a c e s of the e l e c t r o d e s and the spark c h a n n e l i s then formed from t h e e x p a n s i o n of t h e s e plasmas i n t o the i n t e r e l e c t r o d e gap. 44 A schematic diagram of the l a s e r system i s shown i n F i g . 3 - 4 . A two-stage, Q-switched ruby l a s e r i s used. The 2 E 8 §j £ 2. oscillator - s -pockels eel r^KVUH (x2) o m flash lamps 6 0 u H 2 4 0 u F amplifier ( x 4 ) I2KV1 | 3 6 ^ H 580UF F i g . 3-4 Schematic diagram of the ruby l a s e r t r i g g e r o s c i l l a t o r and the a m p l i f i e r s t a g e both c o n s i s t s of Bre w s t e r cu t ruby r o d s , 15.24cm i n l e n g t h and 1.27cm i n d i a m e t e r . They a r e each pumped by f o u r l i n e a r f l a s h lamp i n double e l l i p t i c a l c a v i t i e s . The f l a s h lamp energy i s 8kJ f o r the o s c i l l a t o r and 9.2kJ f o r the a m p l i f i e r . The ruby rods a r e water c o o l e d and the f l a s h lamps a r e a i r c o o l e d . Q - s w i t c h i n g i s done by a P o c k e l s c e l l and a c r y s t a l p r i s m p o l a r i z e r p l a c e d between the ruby rod and the r e a r m i r r o r i n 45 the o s c i l l a t o r c a v i t y . The f l a s h l a m p c u r r e n t s a r e a p p r o x i m a t e l y r e c t a n g u l a r p u l s e s of 1 ms d u r a t i o n f o r the o s c i l l a t o r and 0.6ms f o r the a m p l i f i e r . At the end of the m i l l i s e c o n d the P o c k e l s c e l l b i a s i n g h i g h v o l t a g e i s cro w b a r r e d t o ground, t h e r e b y c o m p l e t i n g the o p t i c a l c a v i t y . A t h r e e - c h a n n e l d e l a y g e n e r a t o r i s used t o c o o r d i n a t e the t i m i n g of thes e e v e n t s . The Q-switched l a s e r l i g h t appears a p p r o x i m a t e l y 300ns a f t e r the c r o w b a r r i n g of the P o c k e l s c e l l b i a s v o l t a g e . The o s c i l l a t o r p u l s e , w i t h FWHM of 80ns, i s a m p l i f i e d t o produce a 50ns FWHM, 15 j o u l e s output p u l s e . The l a s e r l i g h t e n t e r s the d i s c h a r g e v e s s e l a t the t o p of the vacuum chamber t h r o u g h a 15cm f o c a l l e n g t h plano-convex l e n s , s i t u a t e d 15cm from the t i p of the anode. The cathode has a c e n t r a l l y t a p e r e d h o l e , w i t h a diameter of 0.32cm on the e l e c t r o d e s i d e and 1.27cm on the t o p s i d e , f o r the passage of the l a s e r l i g h t . The beam i s f o c u s s e d down t o a spot w i t h a t h e o r e t i c a l spot s i z e of 150pm a t the t i p of the anode. The a c t u a l spot s i z e i s l a r g e r because the d i s t a n c e between the anode and the l e n s i s v a r i e d t o account f o r t he pr o p e r s e p a r a t i o n between the e l e c t r o d e s . When p r o p e r l y f o c u s e d , the power f l u x a t the e l e c t r o d e s u r f a c e i s 4X!0 1 1W/cm 2. 46 t r a n s f o r m e d d i scharge HV d i r e c t d i scharge l a s e r s t o r e d energy 1 25mJ 656mJ 17.2kJ i n c i d e n t power f l u x 4x10 1 1W/cm 2 breakdown mechani sm f i e l d e m i s s i o n f i e l d emi s s i o n mult i p h o t o n i o n i z a t i o n main d i s c h a r g e ' d e l a y 500ns 200ns 300ns j i t t e r 500ns 50ns 1 00ns reproduc i b i l i t y poor f a i r poor T a b l e 3.2 Parameters of t r i g g e r i n g a p p a r a t u s 3.3 MEASUREMENT OF ELECTRICAL PARAMETERS A f a s t response Rogowski c o i l i s used t o monitor d l / d t i n the plasma. I t i s wound i n the u s u a l manner but o n l y w i t h 4 t u r n s and a damping r e s i s t o r i n s i d e an e l e c t r o s t a t i c s h i e l d . D e t a i l s of the c o n s t r u c t i o n of the e l e c t r o s t a t i c s h i e l d i s g i v e n by P e l l i n e n f 3 2 ] . The v a l u e of the r e s i s t o r i s e m p i r i c a l l y chosen such t h a t the c o i l i s c r i t i c a l l y 47 damped when t e r m i n a t e d w i t h 50 ohms. T h i s e n s u r e s t h a t the c o i l has both f a s t r i s e time and f a l l t i m e . The r i s e time of the c o i l i s 1ns w i t h a c o i l c o n s t a n t of 1.25Vus/kA. The c o i l i s p l a c e d around t h e anode j u s t u nderneath t h e vacuum chamber t o measure the time d e r i v a t i v e of the d i s c h a r g e c u r r e n t . An example of the measured d l / d t i s shown i n F i g . 4 - 3 ( a ) . The d i s c h a r g e v o l t a g e i s measured by a x 2 . 5 x l 0 " 7 d i f f e r e n t i a l a t t e n u a t o r c a p a c i t a t i v e l y c o u p l e d t o the p o s i t i v e and n e g a t i v e t e r m i n a l s of the c a p a c i t o r . The l e a d s from the t e r m i n a l s t o the a t t e n u a t o r a r e h i g h v o l t a g e r e s i s t o r c a b l e s a r r a n g e d as a t w i s t e d p a i r t o reduce n o i s e p i c k u p and r e f l e c t i o n s of the v o l t a g e s i g n a l due t o impedance mismatches a t the t e r m i n a l s and a t the a t t e n u a t o r . The output of the d i f f e r e n t i a l a t t e n u a t o r i s f e d t o a T e k t r o n i x 7A24 d i f f e r e n t i a l v e r t i c a l a m p l i f i e r p l u g - i n by a t w i s t e d p a i r of c o a x i a l c a b l e s . I t s h o u l d be noted t h a t t h i s v o l t a g e i s d i f f e r e n t from the v o l t a g e a c r o s s the plasma due t o i n d u c t i v e e f f e c t . 48 3.4 FAST PHOTOGRAPHY A f a s t f r a m i n g camera (TRW model 1-D) i s used t o study the shape and l o c a t i o n of the plasma a t d i f f e r e n t t i m e s and to look f o r e v i d e n c e of hydrodynamic i n s t a b i l i t i e s . I t r e c o r d s the v i s i b l e l i g h t e m i s s i o n from the plasma d u r i n g the r e a c t i v e phase of the d i s c h a r g e i n which p i n c h e s o c c u r . The camera w i t h the t h r e e - f r a m e p l u g - i n (TRW model 6D) has an a p e r t u r e time of 5ns and a minimum i n t e r f r a m e time of 100ns. However, due t o j i t t e r and the s h o r t d u r a t i o n s of the p i n c h e s i t i s not p o s s i b l e t o observe the p i n c h i n g sequence d u r i n g one s i n g l e d i s c h a r g e . P o l a r o i d type 667 f i l m w i t h an ASA of 3000 i s used f o r q u i c k o b s e r v a t i o n of the r e s u l t s and Kodak R o y a l Pan RX-120 f i l m w i t h an ASA of 2450 i s used f o r e n l a r g e m e n t s . 3.5 VISIBLE AND ULTRAVIOLET SPECTROSCOPY F i g . 3 - 5 shows the arrangement f o r s p e c t r o s c o p i c measurements i n the v i s i b l e and the u l t r a v i o l e t r e g i o n . The l i g h t e m i t t e d from t h e spark i n the i n t e r - e l e c t r o d e gap i s c o l l e c t e d by a f / 8 , 20cm f o c a l l e n g t h q u a r t z ~ a q u a achromat l e n s s i t u a t e d 54cm away from the sp a r k . A h o r i z o n t a l s l i t i s p l a c e d a t the image p l a n e of the l e n s . The h e i g h t of the s l i t i s a d j u s t a b l e t o s e l e c t a r e g i o n of the i n t e r e l e c t r o d e gap f o r o b s e r v a t i o n . The l i g h t emerging 49 pin hoi<2 camera PIN .diode? quartz -aqua lens LJ vacuum spark plastic scintillator cluster <Z2> •horizontal slit OMA d .monochrome tor pm tube F i g . 3-5 E x p e r i m e n t a l arrangement f o r s p e c t r o s c o p i c measurement i n the v i s i b l e and the u l t r a v i o l e t r e g i o n . 50 from the s l i t i s f u r t h e r c o l l e c t e d by a f / 1 . 5 , 6cm f o c a l l e n g t h q u a r t z l e n s and imaged onto the e n t r a n c e s l i t of a Spex 1702 3/4 meter monochromator. I t has f/6.8 o p t i c s and a 1200 lines/mm g r a t i n g b l a z e d a t 5000A t o g i v e a p l a t e f a c t o r of 10A/mm. E i t h e r a p h o t o m u l t i p l i e r tube(RCA C-31034) or an o p t i c a l m u l t i - c h a n n e l a n a l y z e r ( P r i n c e t o n A p p l i e d Research S e r i e s 1200, r e f e r r e d t o as OMA from now on) can be mounted on the e x i t p o r t of the monochromator. The p h o t o m u l t i p l i e r i s used t o o b t a i n e d t i m e - r e s o l v e d i n t e n s i t y measurement of a p a r t i c u l a r w a v elength w h i l e the OMA t o g e t h e r w i t h the h o r i z o n t a l s l i t a r e used t o o b t a i n s p a t i a l l y r e s o l v e d s p e c t r a . The RCA C-31034 PM tube has a Ga-As photocathode w i t h type 128 response, which has a u s e f u l s p e c t r a l range from 2000A t o 9300A. I t i s b i a s e d a t 1.5kV w i t h a dynode c h a i n c i r c u i t as recommended by RCA and g i v e s a r i s e time of 2.5ns. The P r i n c e t o n A p p l i e d R esearch model 1205A OMA c o n s o l e w i t h the 1205D d e t e c t o r head i s a 500 c h a n n e l v i d i c o n d e v i c e w i t h a f i r s t s t age g a t e a b l e image i n t e n s i f i e r . We have c o a t e d the d e t e c t o r w i t h a t h i n l a y e r of sodium s a l i c y l a t e [ 3 7 ] t o e x t e n d the s p e c t r a l response from v i s i b l e t o u l t r a v i o l e t , w i t h the s h o r t wavelength response a t a p p r o x i m a t e l y 2100A l i m i t e d by the q u a r t z o p t i c s . The d e t e c t o r i s mounted at the e x i t p o r t of a monochromator i n such a way t h a t the image p l a n e of the image i n t e n s i f i e r i s c o i n c i d e n t w i t h the e x i t f o c a l p l a n e of the monochromator. The r e s o l u t i o n of the d e t e c t o r i s 25pm 51 per c h a n n e l . T h e r e f o r e when c o u p l e d t o the monochromator the t h e o r e t i c a l i n s t r u m e n t a l w i d t h i s 0.25A. However, due t o c r o s s t a l k between c h a n n e l s the i n s t r u m e n t a l w i d t h i s i n c r e a s e d t o 0.4A. The c o a t i n g of the s c i n t i l l a t o r m a t e r i a l f u r t h e r i n c r e a s e s the i n s t r u m e n t a l w i d t h t o 0.8A, as measured w i t h a He-Ne l a s e r and a low p r e s s u r e mercury vapour lamp. The f i r s t s t age of the OMA d e t e c t o r i s ga t e d by a 1.2kV p u l s e g e n e r a t e d by a K r y t r o n p u l s e r . The 1us p u l s e i s a p p l i e d 200ns a f t e r the t r i g g e r spark t o e n a b l e the g a t i n g c i r c u i t r y of the OMA. T h i s e l i m i n a t e s the e m i s s i o n due t o the i n i t i a l breakdown phase and the l a t e r a f t e r g l o w of the plasma. 3.6 X-RAY MEASUREMENTS 3.6.1 TOTAL FLUX The e m i s s i o n of s o f t X -rays i s m o n i t o r e d u s i n g a s o l i d - s t a t e PIN d i o d e ( Q u a n t r a d 100-PIN-250N) s i t u a t e d i n s i d e the vacuum chamber, 20cm from the s o u r c e . S i n c e the d i o d e i s a l s o s e n s i t i v e t o v i s i b l e and vacuum u l t r a v i o l e t l i g h t , t h e s e a r e f i l t e r e d by a 25um A l f i l t e r l o c a t e d i n f r o n t of the d e t e c t o r . The s e n s i t i v e a p e r t u r e of the di o d e i s 52 100mm2 and the r i s e time i s 3ns when b i a s e d w i t h 200V. 3.6.2 SPECTRAL MEASUREMENT USING FOIL ABSORBERS Time- and s p e c t r a l l y - r e s o l v e d X-ray e m i s s i o n s were measured by a p l a s t i c s c i n t i l l a t o r ( N E 1 0 2 ) - p h o t o m u l t i p l i e r assembly as i l l u s t r a t e d i n F i g . 3 - 6 . The assembly houses 0= ^-fibre optic 1 2 i i scale 3 cm ill! aluminum foil NE102 scintillator O-ring F i g . 3-6 Four c h a n n e l X-ray d e t e c t o r assembly f o u r independent c h a n n e l s . Each c h a n n e l c o n s i s t s of a 0.80cm d i a m e t e r , 0.635mm t h i c k NE102 p l a s t i c s c i n t i l l a t o r d i s c , a 0.635cm d i a m e t e r , 1 meter l o n g f i b r e o p t i c c a b l e , and a RCA-931A p h o t o m u l t i p l i e r t u b e . C o r n i n g S i l i c o n 200 53 o p t i c a l f l u i d i s smeared between each i n t e r f a c e t o improve o p t i c a l c o u p l i n g . ' The s c i n t i l l a t o r c l u s t e r i s l o c a t e d 15cm from the plasma, and the a b s o r b e r f o i l s of aluminum w i t h t h i c k n e s s e s r a n g i n g from 6 t o 400um a r e p l a c e d i n f r o n t of the s c i n t i l l a t o r d i s c . The d e t e c t o r assembly i s s i t u a t e d 17cm from the s p a r k . T h i s d i s t a n c e i s n e c e s s a r y t o p r e v e n t the plasma b l o w - o f f from c o a t i n g the aluminum f o i l s and t o a c h i e v e more e q u i v a l e n t g e o m e t r i e s f o r the f o u r d e t e c t o r s . Permanent magnets were p l a c e d i n f r o n t of the s c i n t i l l a t o r s t o p r e v e n t any f a s t e l e c t r o n s from h i t t i n g the d e t e c t o r and c a u s i n g e r r o n e o u s r e s u l t s . However, these magnets a r e found t o be unnecessary due t o the h i g h magnetic f i e l d t h a t e x i s t s d u r i n g a p i n c h . The p h o t o m u l t i p l i e r tubes a r e l o c a t e d 1 meter away from the spark chamber and a r e c o m p l e t e l y s h i e l d e d i n a copper h o u s i n g t o m i n i m i z e n o i s e p i c k u p . N e u t r a l d e n s i t y f i l t e r s a r e i n s e r t e d between the PM tubes and the f i b r e o p t i c s t o p r e v e n t the tubes from s a t u r a t i n g 2 . The f o u r PM tubes a r e b i a s e d by 2.5mA dynode c h a i n a t 900V powered from a s i n g l e power s u p p l y w i t h d e c o u p l i n g r e s i s t o r s and c a p a c i t o r s between i n d i v i d u a l PM t u b e s . The PM tube has a s i g n a l r i s e time of 2ns and an e l e c t r o n t r a n s i t time of 20ns, w h i l e the s c i n t i l l a t o r has a 2 RCA quoted t h a t the space-charge s a t u r a t i o n c u r r e n t w i l l cause the 931-A PM tube t o have an a p p r o x i m a t e l y 5% d e v i a t i o n from l i n e a r i t y a t 25mA. T h i s means the maximum s i g n a l s h o u l d be l e s s than 2V a t 50 ohm l o a d . 54 decay time of 2.2ns. As the response of the NE102 s c i n t i l l a t o r i s l i n e a r t o X-ray e n e r g i e s between 1.5 and 8.1KeV[30], the PM tube output v o l t a g e i s d i r e c t l y p r o p o r t i o n a l t o i n t e n s i t y of the°transmitted X - r a y s . 3,6.3 PINHOLE PHOTOGRAPHY A c r o s s - s e c t i o n of the X-ray p i n h o l e camera i s shown i n F i g . 3 - 7 . The s t a i n l e s s s t e e l p i n h o l e has an a p e r t u r e of F i g . 3-7 X-ray p i n h o l e camera 25pm, and aluminum f o i l s of v a r i o u s t h i c k n e s s e s a r e p l a c e d i n f r o n t of the p i n h o l e t o g i v e d i f f e r e n t . X - r a y energy 55 c u t o f f s . The d i s t a n c e of the f i l m h o l d e r t o the p i n - h o l e i s 15cm. The p i n - h o l e i s p l a c e d 15cm away from the s p a r k , r e s u l t i n g i n a 1X m a g n i f i c a t i o n . Kodak SC-5 f i l m i s used because of i t s s e n s i t i v i t y and f i n e g r a i n s i z e . The exposed f i l m i s d e v e l o p e d i n Kodak X-ray d e v e l o p e r f o r 7 minutes and then f i x e d i n Kodak X-ray f i x e r f o r 2 m i n u t e s . The X-ray photograph thus o b t a i n e d g i v e s a t i m e - i n t e g r a t e d p i c t u r e of the X-ray e m i t t i n g r e g i o n s i n the plasma. However, due t o the l a r g e f number ( f / 6 0 0 0 ) , o n l y the v e r y i n t e n s e e m i s s i o n from a d e n s e l y p i n c h e d plasma has s u f f i c i e n t l y h i g h i n t e n s i t y t o s e n s i t i z e the f i l m . The s i g n a l s from the p h o t o m u l t i p l i e r s , the X-ray d i o d e s as w e l l as from the Rogowski c o i l and the v o l t a g e probe are r o u t e d by 50 ohm c o a x i a l c a b l e s (RG58-U) t o e i t h e r a T e k t r o n i x 7804 or a 7704 o s c i l l o s c o p e . The c a b l e s a l l have e x t r a b r a i d e d s h i e l d s t o reduce n o i s e p i c k u p and a r e t e r m i n a t e d w i t h 50ohms a t the o s c i l l o s c o p e s . Up t o t h r e e s i g n a l s can be m o n i t o r e d s i m u l t a n e o u s l y w i t h the two o s c i l l o s c o p e s . S i n c e the PM tubes have l o n g e l e c t r o n t r a n s i t t i m e s compared w i t h the d u r a t i o n of the e v e n t s under o b s e r v a t i o n , a l o n g c o a x i a l c a b l e a c t i n g as a 40ns d e l a y l i n e can be hooked up t o any o t h e r s i g n a l c a b l e s t o o f f s e t the t i m i n g . T h e r e f o r e i n the o s c i l l o g r a m s the PM tube s i g n a l w i l l have a l e a d time of 20ns over the o t h e r s i g n a l s . W i t h the p r e c a u t i o n of s h i e l d i n g the s p a r k , the measuring equipment, the c a b l e s , and e x t r a a t t e n t i o n t o the e l i m i n a t i o n of any ground l o o p i n the whole system, the 56 s i g n a l - t o - n o i s e r a t i o i s reduced t o an a c c e p t a b l e l e v e l f o r s i g n a l measurements. The l o w e s t l e v e l s i g n a l i s the PM tube o u t p u t , which i s the o r d e r of 1V. I t a l s o has a t y p i c a l d i s c h a r g e n o i s e of 50mV rms, g i v i n g a worst case s i g n a l - t o - n o i s e r a t i o of 15:1. T h i s i s measured u s i n g two methods, by p l a c i n g a 50mm copper b l o c k i n f r o n t of the s c i n t i l l a t o r , t h e r e b y b l o c k i n g o f f the X - r a y s , and by removing the f i b r e o p t i c c a b l e from the PM tube assembly. 57 CHAPTER 4 DESCRIPTION AND DISCUSSION OF EXPERIMENTS In t h i s c h a p t e r the e x p e r i m e n t s a r e d e s c r i b e d and the r e s u l t s d i s c u s s e d . The performances of the t h r e e t r i g g e r i n g methods a r e r e p o r t e d i n S e c t i o n 4.1 and the g e n e r a l o b s e r v a t i o n s of the vacuum spark are then d e s c r i b e d i n S e c t i o n 4.2. The r e p r o d u c i b i l i t y of the spark i s d i s c u s s e d i n S e c t i o n 4.3. S e c t i o n 4.4 r e p o r t s the r e s u l t of the v i s i b l e and the UV e m i s s i o n measurements. The r e s u l t s on X-ray e m i s s i o n s i s then d i s c u s s e d i n S e c t i o n 4.5 and f i n a l l y t hese r e s u l t s a r e compared w i t h t h e o r y i n S e c t i o n 4.6. 4.1 COMPARISION OF TRIGGERING METHODS In an e f f o r t t o a c h i e v e a more r e p r o d u c i b l e vacuum sp a r k , the t h r e e t r i g g e r i n g methods were t r i e d and t h e i r p erformances e v a l u a t e d . In the t r a n s f o r m e d d i s c h a r g e t r i g g e r i n g method, the s w i t c h i n g on of the SCR causes a 25kV p u l s e t o appear on the secondary c i r c u i t of the t r a n s f o r m e r w i t h a r i n g i n g time of 5.5us. I t r i s e s above the breakdown t h r e s h o l d 1us l a t e r , p r o d u c i n g a s m a l l spark a t the t r i g g e r p i n and a l o n g the i n s u l a t o r . T h i s spark l a s t s f o r 50ns, a f t e r which i t i s s e l f - e x t i n g u i s h e d . The e m i t t e d plasma 58 causes the main gap t o break down 500ns a f t e r the t r i g g e r s p a r k . A f t e r a s u c c e s s f u l breakdown, the r a t e of i n c r e a s e of the main gap c u r r e n t reaches a maximum i n 50 t o 100ns, w h i l e the c u r r e n t reaches i t s maximum i n 2.2us. A major o p e r a t i o n a l problem w i t h t h i s t r i g g e r i n g method i s t h a t the spark i s s e l f - q u e n c h i n g , r e s u l t i n g i n o n l y a s m a l l amount of plasma b e i n g e m i t t e d . Sometimes t h i s i s not s u f f i c i e n t t o cause the breakdown of the main gap, and even i f breakdown does o c c u r on the f i r s t t r i g g e r s p a r k , the j i t t e r i s about 500ns. F i g . 4 - 1 i l l u s t r a t e s the j i t t e r i n 10 main d i s c h a r g e d l / d t t r a c e s . Another u n d e s i r a b l e f e a t u r e i s t h a t the t r i g g e r spark r e - i g n i t e s a t i t s next v o l t a g e maximum. In d o i n g so i t g e n e r a t e s a l a r g e amount of RF n o i s e and i t a l s o i n t e r f e r e s w i t h the o p e r a t i o n of the main d i s c h a r g e . The second method t r i e d i s l a s e r t r i g g e r i n g which has been used s u c e s s f u l l y i n gas d i s c h a r g e s by p r e v i o u s w o r k e r s . The two stage ruby l a s e r produces a 50ns FWHM l i g h t p u l s e a t i t s a m p l i f i e r o u t p u t . The subsequent breakdown of the main gap depends c r i t i c a l l y on the s p o t s i z e and hence on the al i g n m e n t of the beam. Under optimum c o n d i t i o n the breakdown o c c u r s 300ns a f t e r the l a s e r p u l s e , w i t h a j i t t e r of 100ns. However, s l i g h t m i s a l i g n m e n t of the l a s e r w i l l cause the j i t t e r t o i n c r e a s e t o 1us. The main d i s c h a r g e has a maximum r a t e of i n c r e a s e of c u r r e n t 100ns a f t e r the breakdown, and a t t a i n s maximum c u r r e n t i n 2.2us. 59 trigger spark spread in L_timing J 500ns 10 shots superposition F i g . 4-1 Trace of vacuum spark t r i g g e r e d by t r a n s f o r m e d d i s c h a r g e showing j i t t e r i n breakdown t i m i n g The advantages of l a s e r t r i g g e r i n g a re as f o l l o w s : 1) . I t does not cause e l e c t r i c a l i n t e r f e r e n c e d u r i n g the o b s e r v a t i o n t i m e . 2) . I t produces a h i g h l y i o n i z e d plasma. 3) . No i m p u r i t i e s a r e i n t r o d u c e d i n t o the spark gap. I t s d i s a d v a n t a g e s a r e : 1). The amount of power d e l i v e r e d i n t o the spark gap depends c r i t i c a l l y on the a l i g n m e n t of the beam and on the c l e a n l i n e s s of the e n t r a n c e window. S i n c e the window i s always c o a t e d a f t e r each d i s c h a r g e w i t h a t h i n l a y e r of e l e c t r o d e m a t e r i a l , the power of l a s e r energy d e l i v e r e d t o the gap d e c r e a s e s g r a d u a l l y . 60 2) . A l t h o u g h the l a s e r produces plasma w i t h h i g h l y s t r i p p e d i o n s , t h e s e have ample time t o recombine when the l a s e r i s e x t i n g u i s h e d and the main d i s c h a r g e f i r e s some 300ns l a t e r . In view of t h i s , the temperature of the t r i g g e r plasma i s not i m p o r t a n t t o the r e l i a b i l i t y of t h e breakdown w h i l e the amount of t r i g g e r plasma i s of p r i m a r y importance. 3) . S i n c e the l a s e r plasma i t s e l f i s i r r e p r o d u c i b l e , the l a s e r t r i g g e r method does not r e s u l t i n r e p r o d u c i b l e p i n c h i n g . The most r e l i a b l e of the t h r e e methods i s the HV d i r e c t d i s c h a r g e t r i g g e r . The 9kV p u l s e of the t r i g g e r i n g c a p a c i t o r produces an e l e c t r i c f i e l d of 6xl0 6V/m a t the t r i g g e r p i n . Breakdown o c c u r s 60ns l a t e r a t the e l e c t r o d e and the i n s u l a t o r s u r f a c e s . S i n c e the l i n e impedance i s d i f f e r e n t from both the impedance a t the c a p a c i t o r bank and at the t r i g g e r gap, m u l t i p l e r e f l e c t i o n s o c c u r a t these p o i n t s . The d l / d t t r a c e of the t r i g g e r s p a r k ( F i g . 4 - 2 ) has the c h a r a c t e r i s t i c s of two damped o s c i l l a t i o n s . The LC r i n g time of 300ns c o n t r i b u t e s t o the lower frequency o s c i l l a t i o n , w h i l e the c a b l e round t r i p time of 3ns c o n t r i b u t e s t o the h i g h e r f r e q u e n c y o s c i l l a t i o n . The main d i s c h a r g e s t a r t s 200ns a f t e r the t r i g g e r s p a r k , a f t e r about 1/3 of the s t o r e d energy i n t h e t r i g g e r c a p a c i t o r bank has d i s s i p a t e d i n t o the plasma. J i t t e r time i s i n the r e g i o n of 50ns, and maximum d l / d t i s reached 100ns l a t e r . The r e d u c t i o n i n j i t t e r time r e p r e s e n t s a 10 ti m e s improvement over the t r a n s f o r m e r t r i g g e r i n g and a 2 times improvement 61 : t » I '! ! T~ tr i |cje Tggp DTcaRs. down F i g . 4-2 C u r r e n t d e r i v a t i v e of the t r i g g e r spark i n the d i r e c t d i s c h a r g e t r i g g e r i n g method. over the l a s e r t r i g g e r i n g . The time l a g between the t r i g g e r spark i n i t i a t i o n f o r the t h r e e methods and the breakdown of the main gap a r e not too d i s s i m i l a r . T h i s time l a g i s c o n s i s t e n t w i t h the s e p a r a t i o n between the e l e c t r o d e s and the v e l o c i t i e s of t h e e l e c t r o d e c l o u d [ l l ] . There i s no o b s e r v a b l e d i f f e r e n c e i n the main d i s c h a r g e s i n the t h r e e t y p e s of t r i g g e r . S i n c e t h e r e i s a mininum of 200ns d e l a y between t h e t r i g g e r s p a r k and t h e main d i s c h a r g e , most of the hot components of t h e t r i g g e r spark plasma have time t o c o o l o f f when the main gap b r e a k s 62 down. F u r t h e r m o r e , the energy d e l i v e r e d by the main spark i s many o r d e r s of magnitude l a r g e r than t h a t of the t r i g g e r s p a r k . T h e r e f o r e a t a l a t e r s t a g e of the main d i s c h a r g e , the plasma i s c h a r a c t e r i z e d by the parameters p e r t i n e n t t o the main gap o n l y . 4.2 GENERAL OBSERVATIONS OF VACUUM SPARK F i g . 4 - 3 ( a ) shows the d l / d t t r a c e of a copper-copper e l e c t r o d e d i s c h a r g e . The c u r r e n t i s shown i n F i g . 4 - 3 ( b ) and the v o l t a g e a c r o s s the c a p a c i t o r and a c r o s s the e l e c t r o d e gap i n F i g . 4 ~ 3 ( c ) . D u r a t i o n of the breakdown phase depends upon the i n t e r e l e c t r o d e s e p a r a t i o n and the type of t r i g g e r i n g . T y p i c a l l y , t h i s d u r a t i o n i s 500ns f o r t r a n s f o r m e d d i s c h a r g e t r i g g e r i n g , and i s 200ns f o r d i r e c t d i s c h a r g e t r i g g e r i n g w i t h a gap of 5mm. At t h i s i n t e r v a l , the v o l t a g e a c r o s s the gap f a l l s r a p i d l y , from the i n i t i a l c h a r g i n g v o l t a g e t o a minimum of a p p r o x i m a t e l y 2.8kV. X-ray e m i s s i o n i s a l s o o b s e r v e d d u r i n g t h i s phase. The i n i t i a l breakdown phase i s f o l l o w e d by the r e a c t i v e phase. The f i r s t q u a r t e r c y c l e of the r e a c t i v e phase l a s t s f o r 2us. I t i s d u r i n g t h i s i n t e r v a l t h a t Z-pinches of the plasma column o c c u r . The d l / d t t r a c e i n t h i s phase i s s i n u s o i d a l i n b e h a v i o r , but p u n c t u a t e d by v a r i o u s d i p s caused by Z-pinches. S i m u l t a n e o u s w i t h a d i p i n d l / d t , the plasma e m i t s a b u r s t of r a d i a t i o n i n the v i s i b l e , 63 TJ » I TJ ".0 .2 .4 .6 .8 10 12 1.4 16 18 20 t ime (ps ) F i g . 4-3(a) d l / d t t r a c e of vacuum spark .0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 t i m e (JJS ) J  F i g . 4-3(b) C u r r e n t t r a c e of vacuum spark .0 .2 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 t i m e (us ) F i g . 4-3(c) C a p a c i t o r and e l e c t r o d e gap v o l t a g e of a vacuum .'spark 64 u l t r a v i o l e t and the X-ray r e g i o n s . The v o l t a g e a c r o s s the gap s t a y s a t a c o n s t a n t v a l u e of 2.8kV, but r i s e s up i n t e r m i t t e n t l y when p i n c h e s o c c u r . D u r i n g the f i r s t q u a r t e r c y c l e , the p i n c h e s can be c l a s s i f i e d i n t o t h r e e t y p e s a c c o r d i n g t o t h e i r p i n c h d u r a t i o n s : the slow p i n c h , the f a s t p i n c h and the s u p e r f a s t p i n c h a c c o r d i n g t o t h e i r p i n c h d u r a t i o n s as measured from t h e i r d l / d t t r a c e s . The slow p i n c h has a p i n c h d u r a t i o n of 40ns or l o n g e r , and o c c u r s e i t h e r between 500ns t o 800ns a f t e r the breakdown or a f t e r 1.4ps a f t e r the breakdown. They a r e v e r y weak, and the c o r r e s p o n d i n g d i p i n the d l / d t t r a c e i s s h a l l o w . There i s no X-ray e m i s s i o n , i n d i c a t i n g t h a t the plasma formed from t h i s p i n c h i s r e l a t i v e l y c o l d . F i g . 4 - 4 ( a ) shows the d l / d t t r a c e of a slow p i n c h . In the same f i g u r e t h e r e i s a l s o an example of a f a s t p i n c h . A f a s t p i n c h has a p i n c h d u r a t i o n of t y p i c a l l y 20ns and o c c u r s around 700ns a f t e r the breakdown. The o c c u r r e n c e of the p i n c h i s i d e n t i f i e d i n the d l / d t t r a c e by a sharp d i p which u s u a l l y goes n e g a t i v e , i n d i c a t i n g an a c t u a l d e c r e a s e i n c u r r e n t i n t h i s i n t e r v a l . Accompanying such a d i p are s i m u l t a n e o u s e m i s s i o n s of X - r a y s , and subsequent e m i s s i o n s of v i s i b l e and UV l i g h t . Framing photographs show t h a t the v i s i b l e e m i s s i o n s a r e c o n f i n e d t o v e r y s m a l l r e g i o n s , t y p i c a l l y 80-120um i n d i a m e t e r ( s e e F i g . 4 - 5 ) . I t s h o u l d be noted t h a t the d i s t i n c t i o n between a slow p i n c h and a f a s t p i n c h i s an a r t i f i c a l one. There i s no sharp boundary i n the p i n c h d u r a t i o n s observed w i t h 65 F i g . 4~4(a) d l / d t and X-ray t r a c e of a spark (The PM tube s i g n a l s a r e n e g a t i v e - g o i n g p u l s e s ) p i n c h e s o c c u r r i n g w i t h v a r i o u s d u r a t i o n s . The f i r s t p i n c h t h a t o c c u r s i n a d i s c h a r g e i s always a slow p i n c h . I t o c c u r s f a i r l y r e p r o d u c i b l y a t 500ns a f t e r the breakdown w i t h a j i t t e r of 50ns. I t s i m p l o s i o n time f o l l o w s c l o s e l y t o t h a t p r e d i c t e d by Eq.2.12. The v i s i b l e 66 500mV , !dl dt X- ray F i g . 4-4(b) d l / d t and X-ray t r a c e of a s u p e r f a s t p i n c h (the d l / d t t r a c e i s d e l a y e d by 20ns w i t h r e s p e c t t o the X-ray t r a c e ) e m i s s i o n from t h i s p i n c h i s low, as the plasma i s s t i l l q u i t e tenuous d u r i n g t h i s t i m e . For the l a t e r p i n c h e s , n e i t h e r the t i m i n g nor the numbers are r e p r o d u c i b l e . F a s t p i n c h e s b e g i n t o appear f o r a c h a r g i n g v o l t a g e above 3.5kV f o r copper and above 6kV f o r carbon anode. However, r a i s i n g the c h a r g i n g v o l t a g e above lOkV does not i n c r e a s e a p p r e c i a b l y the o c c u r r e n c e of the f a s t p i n c h e s , but the e l e c t r o d e wear i s i n c r e a s e d s e v e r e l y . For t h i s reason a l l our subsequent measurements were done w i t h a c h a r g i n g v o l t a g e of lOkV. In i t s f i n a l c o l l a p s e , the f a s t p i n c h may t r a n s f o r m 67 i n t o a s u p e r f a s t p i n c h , w i t h p i n c h d u r a t i o n of 2ns or l e s s . F i g . 4.4(b) shows such a case w i t h the c o r r e s p o n d i n g X-ray e m i s s i o n . The o c c u r r e n c e i s r a r e , about once i n every f i v e f a s t p i n c h e s . T i m e - c o i n c i d e n t w i t h such a s u p e r f a s t p i n c h i s an e x t r a b u r s t of X-ray e m i s s i o n on t o p of the e m i s s i o n from the f a s t p i n c h i t s e l f . (See 6pm t r a c e of F i g . 4 - 1 3 ( b ) . ) The o c c u r r e n c e of a s u p e r f a s t p i n c h does not cause any o b s e r v a b l e i n c r e a s e i n the e m i s s i o n i n the v i s i b l e or UV r e g i o n of the spectrum. These o b s e r v a t i o n s a r e summarized i n T a b l e 4.1. 68 Slow p i n c h F a s t p i n c h S u p e r f a s t p i n c h most p r o b a b l e time of o c c u r r e n c e e i t h e r from 500ns t o 800ns or >1. 4us 700ns 700ns nominal p i n c h i n t e r v a l >40ns 20ns <2ns X-ray e m i s s i o n no yes yes v i s i b l e - U V e m i s s i o n weak i f <800ns s t r o n g i f >1.4us s t r o n g but d e l a y e d s t r o n g but d e l a y e d T a b l e 4.1 C l a s s i f i c a t i o n of t h r e e t y p e s of p i n c h e s . (The v i s i b l e e m i s s i o n s a re d i s c u s s e d i n S e c t i o n 4.4) 6 9 4.3 REPRODUCIBILITY The Z-pinch of the vacuum spark has a l o n g h i s t o r y of b e i n g v e r y i r r e p r o d u c i b l e . However, by o p t i m i z i n g the t r i g g e r i n g method and energy, some improvements have been made i n our e x p e r i m e n t . We f i n d t h a t the t r i g g e r must d e l i v e r s u f f f i c i e n t charge c a r r i e r s i n the s h o r t e s t p o s s i b l e time i n t o the i n t e r - e l e c t r o d e gap t o cause a r e l i a b l e breakdown of the main gap. T h i s i s a c h i e v e d by u s i n g d i r e c t d i s c h a r g e t r i g g e r i n g , i n which the f i r s t p i n c h o c c u r s r e p r o d u c i b l y and w i t h much reduced time j i t t e r . However, the appearance of t h e f a s t and the s u p e r f a s t p i n c h a r e s t i l l e r r a t i c . R e p r o d u c i b i l i t y and the f o r m a t i o n of a good p i n c h a r e s e n s i t i v e t o both the macro- and the micro-geometry of the e l e c t r o d e . R a d i a l m i s a l i g n m e n t of the e l e c t r o d e s by as l i t t l e as 0.5mm w i l l reduce s i g n i f i c a n t l y the p r o b a b i l i t y of a f a s t p i n c h . The d i r e c t d i s c h a r g e t r i g g e r i n g , which g i v e s the most " r e p r o d u c i b l e " slow p i n c h e s , the t i m i n g s of the f a s t p i n c h e s a r e c e n t e r e d around 700ns, w i t h a s t a n d a r d d e v i a t i o n of 100ns. A t y p i c a l s e t of me t a l e l e c t r o d e s w i l l l a s t f o r about one hundred s h o t s . The type of e l e c t r o d e m a t e r i a l s can s e r i o u s l y a f f e c t the r e p r o d u c i b i l i t y of the p i n c h e s . The i n t e n s e c u r r e n t d u r i n g the d i s c h a r g e s c r e a t e s c r a t e r s i n the e l e c t r o d e s as the molten m e t a l s a re bei n g pushed a s i d e by the magnetic f i e l d a s s o c i a t e d w i t h the c u r r e n t . G r a p h i t e 70 e l e c t r o d e s l a s t l o n g e r s i n c e they do not m e l t , but r a t h e r e v a p o r a t e . A l s o some m e t a l s such as i r o n form n e e d l e - l i k e c r y s t a l s " w h i s k e r s " when they s o l i d i f y . These a f f e c t the micro-geometry and cause l o c a l i n t e n s i f i c a t i o n of e l e c t r i c f i e l d s near the s h a r p p o i n t s . S i n c e e v e r y shot has d i f f e r e n t micro-geometry and hence d i f f e r e n t i n i t i a l c o n d i t i o n s , i t r e n d e r s the d i s c h a r g e u n p r e d i c t a b l e . E l e c t r o d e m a t e r i a l s w i t h low m e l t i n g p o i n t s such as aluminum and z i n c a r e c o m p l e t e l y deformed a f t e r a few d i s c h a r g e s . E l e c t r o d e shape a l s o a f f e c t the r e p r o d u c i b i l i t y . I t i s found t h a t t a p e r e d e l e c t r o d e s g i v e the optimum r e s u l t . Tapered e l e c t r o d e s cause the e l e c t r i c f i e l d t o c o n c e n t r a t e a t t h e i r p o i n t s , and hence f o r c e the plasma t o stem from a w e l l d e f i n e d p o i n t on the e l e c t r o d e s u r f a c e . However, s h a r p l y p o i n t e d e l e c t r o d e s s h o u l d be a v o i d e d s i n c e they w i l l wear out v e r y q u i c k l y . 71 4.4 EMISSIONS IN THE VISIBLE AND ULTRAVIOLET REGION 4.4.1 FRAMING CAMERA PICTURES F i g . 4 - 5 shows a s e r i e s of f r a m i n g camera p i c t u r e s of the vacuum spark e m i s s i o n i n the v i s i b l e r e g i o n a t d i f f e r e n t t i m e s . Due t o l i m i t a t i o n s of the i n s t r u m e n t t h e s e p i c t u r e s a r e not taken from one s i n g l e d i s c h a r g e , but r a t h e r from s e v e r a l s p a r k s . There i s l i t t l e or no e m i s s i o n b e f o r e the f i r s t p i n c h of the d i s c h a r g e . The f i r s t frame shows the luminuous plasma a t the end of a slow p i n c h . Note the b r i g h t s p o t s near the two e l e c t r o d e s , and a f a i n t r e g i o n i n the m i d d l e . The d i a m e t e r of the plasma column i n the middl e i s e s t i m a t e d t o be 250um. The second frame shows the plasma 100ns a f t e r a slow p i n c h . The plasma column has been d i s r u p t e d and has not y e t been reformed. The s p i r a l l i n g s t r u c t u r e c o u l d be the r e s u l t of m=1 mode i n s t a b i l i t y . The t h i r d frame shows the plasma column d u r i n g a f a s t p i n c h . The d i a m e t e r of the column i s e s t i m a t e d t o be 80um i n d i a m e t e r i n the h i g h e s t p i n c h e d r e g i o n . V i s i b l e l i g h t e m i s s i o n from the e n t i r e column i s a l s o reduced. S e v e r a l l a t e r frames show the plasma column a f t e r the p i n c h e s . The whole plasma column appears t o be bending and t w i s t i n g , i n d i c a t i n g the r e s u l t s of i n s t a b i l i t y . / 72 i t slow pinch ' 4 5 0 hs 6 0 0 ns • fast pinch m 8 2 0 n s 1060 ns 9 5 0 ns cathode _ V///A 5mm \—\ 1mm anode F i g . 4-5 Framing camera photographs of vacuum spark d i s c h a r g e w i t h copper-copper e l e c t r o d e s 73 4.4.2 VISIBLE AND UV SPECTROSCOPY The e m i s s i o n c h a r a c t e r i s t i c s a r e v e r y d i f f e r e n t f o r copper-copper e l e c t r o d e s and f o r carbon-copper e l e c t r o d e s . A copper-copper d i s c h a r g e s have m u l t i p l e (more than f o u r ) p i n c h e s , w h i l e a carbon-copper d i s c h a r g e o n l y has two or t h r e e p i n c h e s . Continuum r a d i a t i o n i n the v i s i b l e and UV r e g i o n i s v e r y i n t e n s e d u r i n g a copper-copper d i s c h a r g e and sometime i t even smears out the l i n e r a d i a t i o n s , making l i n e w i d t h measurement d i f f i c u l t . On the o t h e r hand d u r i n g a carbon-copper d i s c h a r g e the continuum r a d i a t i o n i s weak and the l i n e r a d i a t i o n i s s t r o n g and s t a n d s c o n s p i c i o u s l y above the continuum background. Our r e s u l t s showed t h a t the e m i s s i o n i s p r i m a r i l y c h a r a c t e r i z e d by the anode m a t e r i a l . Comparison between carbon-copper and copper-copper d i s c h a r g e shows t h a t b e s i d e s h a v i n g a l a r g e r e d u c t i o n i n the i n t e n s i t y of the continuum, the carbon-copper d i s c h a r g e c o n s i s t s m a i n l y of l i n e r a d i a t i o n of carbon l i n e s , w i t h the copper l i n e s v e r y much weakened or even s u p p r e s s e d . F i n a l l y , the p u l s e shape of the X-ray e m i s s i o n i s a l s o d i f f e r e n t between the two d i s c h a r g e s . The copper-copper e l e c t r o d e s g i v e a two temperature e m i s s i o n , w h i l e the carbon-copper e l e c t r o d e s g i v e s o n l y a one temperature e m i s s i o n . R e s u l t s of the measurements show t h a t the vacuum spark e m i t s a spectrum c o n s i s t i n g of the s u p e r p o s i t i o n of l i n e r a d i a t i o n s from many d i f f e r e n t i o n i z a t i o n s t a g e s on t o p of a continuum background. Except f o r the case of c a r b o n , the 74 l a c k of e x p e r i m e n t a l and t h e o r e t i c a l d a t a on the l i n e r a d i a t i o n of the h i g h i o n i z a t i o n s t a g e s t o g e t h e r w i t h the br o a d e n i n g of the l i n e s due t o the h i g h e l e c t r o n d e n s i t y make l i n e i d e n t i f i c a t i o n and i s o l a t i o n d i f f i c u l t . T h i s i s compounded w i t h poor s i g n a l - t o - n o i s e r a t i o from a h i g h continuum background and poor r e p r o d u c i b i l i t y from the spark i t s e l f t o make q u a n t i t a t i v e s p e c t r o s c o p y i m p o s s i b l e . The t i m i n g of the e m i s s i o n , however, can g i v e an i n s i g h t i n t o the p r o c e s s e s d u r i n g the p i n c h . Cu I , Cu I I , Cu I I I and Cu IV l i n e s a r e observed i n a copper-copper p i n c h , and t h e i r t i m i n g i s t a b u l a t e d i n Tab l e 4-2. C I I , C I I I and C IV and a t r a c e of are o b s e r v e d i n a carbon-copper p i n c h , g i v e s the t i m i n g of these l i n e r a d i a t i o n s . Cu I , Cu I I l i n e s T a b le 4.2 a l s o 75 F a s t / S u p e r f a s t p i n c h Slow p i n c h emi s s i o n t i m i n g from onset of p i n c h e m i s s i o n d u r a t i o n (FWHM) e m i s s i o n t i m i n g from onset of p i n c h e m i s s i o n d u r a t i o n (FWHM) Cu I 1 00ns 1 00ns f o l l o w s p i n c h p r o f i l e >160ns Cu I I 50ns 1 00ns f o l l o w s p i n c h p r o f i l e 1 30ns C u l l I 30ns 1 00ns f o l l o w s p i n c h p r o f i l e 1 00ns • C I I 200ns 1 00ns f o l l o w s p i n c h p r o f i l e >200ns C I I I 200ns 1 00ns f o l l o w s p i n c h p r o f i l e 1 00ns C IV 50ns 50ns f o l l o w s p i n c h p r o f i l e 1 00ns T a b l e 4.2 E m i s s i o n t i m i n g s of copper-copper p i n c h and carbon-copper p i n c h 76 The e m i s s i o n t i m i n g s t o g e t h e r w i t h the f r a m i n g camera p i c t u r e s i n d i c a t e t h a t l i n e r a d i a t i o n s d u r i n g a slow p i n c h a r e e m i t t e d from the compressed plasma volume i n the i m p l o d i n g s h e l l , so t h a t the e m i s s i o n p r o f i l e f o l l o w s the p i n c h p r o f i l e as more plasma a r e swept up by the i m p l o d i n g shock f r o n t . However, d u r i n g a f a s t or a s u p e r f a s t p i n c h t h e r e i s l i t t l e or no l i n e e m i s s i o n d u r i n g the p i n c h d u r a t i o n . Then a f t e r the p i n c h i s o v e r , l i n e s of h i g h e r i o n i z a t i o n s t a g e s a r e o b s e r v e d f i r s t . T h i s s u g g e s t s t h a t the plasma i s heated t o a h i g h temperature v e r y q u i c k l y d u r i n g the p i n c h . A f t e r the p i n c h i s d i s r u p t e d and the plasma expands, the h i g h l y s t r i p p e d i o n s recombine w i t h e l e c t r o n s t o form the lower i o n i z a t i o n s t a g e s , g i v i n g the e m i s s i o n c h a r a c t e r i s t i c s o b s e r v e d . F i g . 4 - 6 ( a ) shows a t i m e - i n t e g r a t e d i n t e n s i t y of a Cu I l i n e (4651A) and the continuum e m i s s i o n of the plasma over an e n t i r e copper-copper d i s c h a r g e , w h i l e F i g . 4 - 6 ( b ) shows the e m i s s i o n d u r i n g the f i r s t one m i c r o s e c o n d . These two p r o f i l e s a r e q u i t e d i f f e r e n t , but bear i n mind t h a t the plasma e m i s s i o n a t a l a t e r time (>2ps) c o n s i s t s m a i n l y of l i n e r a d i a t i o n as the plasma at t h a t time i s r e l a t i v e l y c o l d . S i n c e the OMA-obtained s p e c t r a are not t i m e - r e s o l v e d and a spectrum may c o n t a i n e m i s s i o n s from a number of p i n c h e s , r e l a t i v e i n t e n s i t y measurements w i l l be u s e l e s s . I n s t e a d , i n F i g . 4 - 7 , the i n t e n s i t i e s of l i n e r a d i a t i o n s 77 • i i i r 4 6 0 0 4700 wavdenqth ( A ) F i g . 4 - 6 ( a ) T i m e - i n t e g r a t e d i n t e n s i t y of e n t i r e Cu-Cu d i scharge n o r m a l i z e d by i t s background continuum a t v a r i o u s p o s i t i o n i n the i n t e r - e l e c t r o d e gap are p l o t t e d f o r some of the copper l i n e s . T h i s i s q u a l i t a t i v e l y u s e f u l as i t g i v e s an i n d i c a t i o n of the r e l a t i v e i o n d e n s i t y d i s t r i b u t i o n of the plasma. The graph shows t h a t Cu I and Cu I I l i n e s a r e c o n c e n t r a t e d near the two e l e c t r o d e s , w i t h h i g h e r c o n c e n t r a t i o n s near the cathode. T h i s i s i n agreement w i t h the f r a m i n g camera r e s u l t s . The i n t e n s i t y of the Cu I l i n e f a l l s o f f more r a p i d l y than t h a t of the Cu I I l i n e s as one moves away from the e l e c t r o d e s . T h i s i s r e a s o n a b l e s i n c e the average temperature i s h i g h e r i n the m i d d l e r e g i o n of the gap. 78 4600 wavelength ( A ) 4700 F i g . 4-6(b) Time-integrated in t e n s i t y of the f i r s t microsecond of Cu-Cu discharge The better signal-to-noise r a t i o and the r e l a t i v e s i m p l i c i t y of the carbon spectrum makes i t more amenable to analysis than that of copper. Fig.4-8(a) shows an OMA output of the C IV(5801.3A) spectrum and Fig.4-8(b) shows the coressponding dl/dt trace and the emission timing. Although the OMA is gated by a 1us pulse, i t s t i l l results in a time integrated spectrum of several pinches and their subsequent relaxations. The t h e o r e t i c a l widths of i s o l a t e d ion l i n e s can be estimated from a semi-empirical formula derived by Griem[20]. They are proportional to density with only a 79 i I I r distance? from cathode (mm ) F i g . 4-7 R a t i o of l i n e t o continuum i n t e n s i t y i n a copper-copper d i s c h a r g e weak dependence on t e m p e r a t u r e . These l i n e w i d t h s can be used t o determine the d e n s i t y of the plasma at d i f f e r e n t t e m p e r a t u r e s , as the measured l i n e w i d t h s , a l t h o u g h i r r e p r o d u c i b l e , have FWHM t h a t v a r y at most by h a l f an o r d e r of magnitude. I t s h o u l d be s t r e s s e d t h a t the a c t u a l d e n s i t y a c h i e v e d d u r i n g the p i n c h v a r i e s and the c a l c u l a t e d d e n s i t y s h o u l d o n l y be t r e a t e d as a rough e s t i m a t e of the average d e n s i t y . The p o p u l a t i o n of a p a r t i c u l a r i o n i z a t i o n s t a g e , however, depends s t r o n g l y on the temperature and the i n t e n s i t i e s of t h e l i n e s v a r y from shot t o shot over an o r d e r of magnitude. F i g . 4 - 9 shows the e l e c t r o n d e n s i t y a l o n g the 8 0 5760 5840 wavelength ( A ) F i g . 4-8(a) Time i n t e g r a t e d p r o f i l e of C IV(5801.3A) i JL I" -i I ! — -001 di c i v : c m i s s o n F i g . 4-8(b) d l / d t t r a c e and i n t e n s i t y of C IV(5801.3A) i n t e r - e l e c t r o d e gap p o s i t i o n d e r i v e d from l i n e w i d t h measurements of C I I ( 4 2 6 7 A ) , C I I I ( 5 6 9 6 A ) and C IV(5801A) l i n e s . The d e n s i t y i s h i g h e s t c l o s e t o the anode, v a r y i n g from 2 . 0 X 1 0 1 8 t o 3 . 0 x l 0 1 8 c m - 3 and d e c r e a s i n g g r a d u a l l y 81 towards the ca t h o d e , a t which p o i n t , the d e n s i t y i s from 0.5X10 1 8 t o 1.5x10 1 8cm" 3. The e l e c t r o n d e n s i t y i s a l s o h i g h e r a t a h i g h e r e l e c t r o n t e m p e r a t u r e , as seen from the o v e r a l l t r e n d s of the p o i n t s f o r d i f f e r e n t i o n i z a t i o n s t a g e s . Furthermore the l i n e w i d t h measured i s c o r r e l a t e d x CE? a cnr • cn v u 00 X I ** f * t * fr * * X a • • t * typical tP ? tP o error bars t 1 2 3 4 distance from cathode (mm.) F i g . 4-9 D e n s i t y d i s t r i b u t i o n a l o n g the i n t e r e l e c t r o d e gap (The e r r o r bars i n d i c a t e ' ' i n t r u m e n t a l u n c e r t a i n t i e s , w h i l e the s c a t t e r of the d a t a p o i n t s a re due t o s t a t i s t i c s ) t o the e m i s s i o n of X - r a y s . Space i n t e g r a t e d s p e c t r a from C II ( 4 2 6 7 A ) l i n e show t h a t f o r d i s c h a r g e s w i t h X-ray 82 e m i s s i o n s , the average e l e c t r o n d e n s i t y i s (2.5±0.7)x10 1 8cm~ 3. For d i s c h a r g e s w i t h o u t measurable X-ray e m i s s i o n , the average e l e c t r o n d e n s i t y i s (1.8±0.5)xl0 1 8cm" 3. I t i s more d i f f i c u l t t o e s t i m a t e the e l e c t r o n d e n s i t y i n a copper-copper d i s c h a r g e , due t o i t s enhanced background continuum which o b s c u r e s the l i n e r a d i a t i o n , and due t o the l a c k of p u b l i s h e d S t a r k w i d t h s of copper i o n s f o r compar i s o n . A rough e s t i m a t e of the e l e c t r o n d e n s i t y g i v e s a v a l u e of the o r d e r of 1 u 1 8 c r r r 3 , but we were unable t o determine the v a r i a t i o n i n d e n s i t y a l o n g the i n t e r e l e c t r o d e gap. I t s h o u l d be noted t h a t the above measurements a r e t i m e - i n t e g r a t e d r e s u l t s . Due t o the c o n s t a n t l y c hanging plasma r a d i u s , t e m p e r a t u r e , d e n s i t y and i o n i z a t i o n s t a g e s , i t i s d i f f i c u l t t o o b t a i n the e l e c t r o n d e n s i t y as a f u n c t i o n of r a d i u s . We e s t i m a t e , from the e m i s s i o n t i m i n g of the l i n e s and the f r a m i n g camera p i c t u r e s , t h a t a t a plasma r a d i u s of 300um, the e l e c t r o n d e n s i t y would be a p p r o x i m a t e l y I 0 1 8 c m ~ 3 . However t h i s r e s u l t s h o u l d be t r e a t e d w i t h c a u t i o n , b e a r i n g i n mind t h a t i t s h o u l d o n l y be used as an or d e r of magnitude e s t i m a t e . U s i n g these r e s u l t s , the e - f o l d i n g t i m e s f o r c o l l i s i o n a l i o n i z a t i o n and e l e c t r o n - i o n r a d i a t i v e r e c o m b i n a t i o n from ground s t a t e i o n s f o r d e n s i t i e s and te m p e r a t u r e s of f a s t and s u p e r f a s t p i n c h e s a re c a l c u l a t e d u s i n g f o r m u l a g i v e n by Book[8] and G r i e m [ 2 l ] ( r e f e r t o S e c t i o n 4.5.1 f o r measurement of t e m p e r a t u r e ) . The 83 i o n i z a t i o n e n e r g i e s f o r v a r i o u s copper i o n s are o b t a i n e d from K e l l y [ 2 3 ] . The r e s u l t s a r e p l o t t e d i n Fig.4-10 w h i l e the r e l a t i v e i o n d e n s i t i e s a t v a r i o u s t e m p e r a t u r e s f o r s t e a d y s t a t e corona e q u i l i b r i a a r e p l o t t e d i n F i g . 4 - 1 1 . F i g . 4 - 1 0 shows t h a t the plasma i s i n a t r a n s i e n t s t a t e d u r i n g the f a s t and the s u p e r f a s t p i n c h , w i t h the i o n i z a t i o n t i m e s l o n g e r than the p i n c h d u r a t i o n a t the e l e c t r o n t e mperature of 140-600eV a c h i e v e d by the f a s t p i n c h and the 2keV t o 4keV by the s u p e r f a s t p i n c h . I t i s p o s s i b l e t h a t the a c t u a l i o n i z a t i o n t i m e s are f a s t e r due t o e x c i t a t i o n of i o n s from t h e i r ground s t a t e s i n t o t h e i r e x c i t e d s t a t e s b e f o r e i o n i z a t i o n ( c o l l i s i o n - r a d i a t i v e model). R a d i a t i v e r e c o m b i n a t i o n times a r e two t o t h r e e o r d e r s of magnitudes l a r g e r than the i m p l o s i o n time i n a f a s t p i n c h . T h i s a c c o u n t s f o r the r e l a t i v e t i m i n g of the d i f f e r e n t l i n e e m i s s i o n s . Three-body r e c o m b i n a t i o n times a r e of the o r d e r of hundreds of nanoseconds i n our plasma c o n d i t i o n s due t o the h i g h temperature and the r e l a t i v e l y low d e n s i t y of the plasma and t h e r e f o r e a r e not p l o t t e d . From Fi g . 4 - 1 0 one can c o n c l u d e t h a t d u r i n g a f a s t p i n c h most of the copper i o n s are i n the i o n i z a t i o n s tage Cu XI or h i g h e r . C o n s e q u e n t l y v e r y l i t t l e v i s i b l e l i n e r a d i a t i o n i s e m i t t e d . 84 charge stats of copper ion F i g . 4 - 1 0 _ e - f o l d i n g time f o r c o l l i s i o n a l i o n i z a t i o n and e l e c t r o n - i o n r a d i a t i v e r e c o m b i n a t i o n f o r copper p l a s m a . 85 •0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 charge state Z of copper F i g . 4-11(a) R e l a t i v e i o n i c d e n s i t i e s f o r copper plasma i n a st e a d y s t a t e c o r o na e q u i l i b r i u m . 1.0 if) d > '•+-» D .2 •Q *1eV 3eV l ft30eVr 3 0 c i 10eV/ j \\ 100eV/ ' / -J _L_. \ j/ 0 2 3 4 5 charge state Z of carbon 6 E.i.g. .4-1 1 (b) ^ R e l a t i v e i o n i c d e n s i t i e s f o r carbon plasma i n ' a . ..steady . s t a t e ..corona e q u i l i b r i u m . 86 4.5 EMISSION IN THE X-RAY REGION For the h i g h t e m p e r a t u r e s e x p e c t e d t o be a c h i e v e d by the s p a r k , the e m i s s i o n i s s t r o n g e s t i n the X-ray r e g i o n . The temperature of the p i n c h i n g plasma can be measured from i t s X-ray e m i s s i o n . By p u t t i n g d i f f e r e n t t h i c k n e s s of aluminum f o i l i n f r o n t of the X-ray d e t e c t o r assembly, one can change the w i d t h of the passband of the aluminum f i l t e r and o b t a i n an i n t e n s i t y p r o f i l e of the X-ray b e i n g e m i t t e d . A d e t a i l e d p r o f i l e of the i n t e n s i t y vs f o i l t h i c k n e s s then emerges which can be compared w i t h e m i s s i o n s from t a b u l a t e d d a t a t o g i v e a temperature d i s t r i b u t i o n of the e m i t t i n g plasma. A more d e t a i l e d d e s c r i p t i o n of t h i s t e c h n i q u e i s g i v e n i n Appendix A. 4.5.1 TIME RESOLVED ABSORBER FOIL MEASUREMENTS X-ra y s a r e f i r s t produced d u r i n g the breakdown phase. However, th e s e X - r a y s , b e i n g g e n e r a t e d by b r e m s s t r a h l u n g and i n n e r s h e l l t r a n s i t i o n of the e l e c t r o d e m a t e r i a l [ 2 7 ] , a r e non-thermal and not of i n t e r e s t t o our p u r p o s e s . The X-ray e m i s s i o n p r o f i l e f o r a copper-copper d i s c h a r g e i s q u i t e d i f f e r e n t from t h a t of a carbon-copper d i s c h a r g e , and we w i l l d i s c u s s them s e p a r a t e l y . 87 X-RAY EMISSION FROM COPPER-COPPER PINCH An o s c i l l o g r a m showing s e v e r a l p i n c h e s f o r copper-copper d i s c h a r g e i s shown i n F i g . 4 - 1 2 . The o s c i l l o g r a m shows the t r a n s m i s s i o n t h rough a 6pm and a 150pm aluminum f o i l s w i t h a time s c a l e of 2 0 0 n s / d i v . The X-ray b u r s t s a r e t i m e - c o i n c i d e n t w i t h the p i n c h e s . F i n e r d e t a i l s of the e m i s s i o n shows up under a f a s t e r time s c a l e . F i g . 4 - 1 3 shows the e m i s s i o n p u l s e f o r v a r i o u s k i n d s of p i n c h e s a t a f a s t e r time s c a l e . The r i s e time of the X-ray p u l s e d i f f e r s from p i n c h t o p i n c h , depending on the r a t e of i m p l o s i o n . The d u r a t i o n of the b u r s t , u s u a l l y from 20-50ns, a l s o depends on the d u r a t i o n of the p i n c h . F i g . 4 - 1 3 ( a ) shows a f a s t p i n c h which t r a n s f o r m s i n t o a s u p e r f a s t p i n c h . The e x t r a sharp d i p a t the end of the u s u a l d l / d t d i p i s v e r y e v i d e n t . The c o r r e s p o n d i n g X-ray e m i s s i o n p u l s e a l s o shows an e x t r a b u r s t on t o p of a p e d e s t a l . F i g . 4 - 1 3 ( b ) shows the X-ray e m i s s i o n p u l s e t r a n s m i t t e d through a 6pm f o i l and a 150pm f o i l . The 6pm f o i l has an energy c u t o f f of 1.3keV, a l l o w i n g the t r a n s m i s s i o n from the e m i s s i o n of lower t e m p e r a t u r e plasma (See Appendix A ) . The 150pm f o i l , w i t h an energy c u t o f f of 15keV, o n l y t r a n s m i t s e m i s s i o n from h i g h e r temperature plasma. The o s c i l l o g r a m t r a c e of the s i g n a l from the 6pm f o i l shows a f a i r l y l o n g p e d e s t a l , superimposed on top of which i s a s h o r t p u l s e . The s i g n a l from the 150pm f o i l shows o n l y a s h o r t X-ray p u l s e , w i t h both the r i s i n g and f a l l i n g edge l i m i t e d by i n s t r u m e n t response. 88 F i g . 4-12 . d l / d t t r a c e of copper-copper d i s c h a r g e and accompanying X-ray e m i s s i o n I t i s w o r t h w h i l e t o note t h a t the t h i n f o i l s i g n a l s a r e c l o s e l y c o r r e l a t e d t o b oth the f a s t and the s u p e r f a s t p i n c h e s , w h i l e the t h i c k f o i l s i g n a l s a r e o n l y p r e s e n t d u r i n g / a s u p e r f a s t p i n c h . 89 200*V jfy F i g . 4-13(a) S u p e r f a s t p i n c h and X-ray e m i s s i o n (The d l / d t t r a c e i s d e l a y e d by 20ns w i t h r e s p e c t t o the X-ray t r a c e ) F i g . 4-13(b) E m i s s i o n through d i f f e r e n t f o i l s 9 0 Fig.4-14 shows a p l o t of the peak i n t e n s i t y vs a b s o r b e r f o i l t h i c k n e s s . Because the i n t e n s i t i e s of the X-ray b u r s t s a r e i r r e p r o d u c i b l e , the measured i n t e n s i t i e s a r e n o r m a l i z e d t o the i n t e n s i t y of the t h i n n e s t f o i l , which i s 6yxm i n t h i c k n e s s . From the s l o p e of the i n t e n s i t i e s we can 1.0 0.5h 0.2 Ui c > 0.1 o v .03-.02 .01 l I l I > o _i i « i i i i i i 1 1 1 s i > : * ¥ S » JL JL 1 • • • » T — r 5 6 10 20 5 0 100 2 0 0 A l a b s o r b e r foil t h i c k n e s s ( u m ) 500 F i g . 4-14 X-ray e m i s s i o n t h r o u g h d i f f e r e n t f o i l t h i c k n e s s e s t i m a t e the e l e c t r o n t emperature of the e m i t t i n g plasma, (see Appendix A) S i n c e the c o l d e r plasma i s much more abundant than the h o t t e r plasma, the measured i n t e n s i t i e s t h r o u g h the t h i n f o i l s a r e m a i n l y due t o the s o f t photons 91 e m i t t e d from the c o l d e r plasma. A s l o p e from the t h i n f o i l s then g i v e s the temperature of the c o l d e r plasma. On the o t h e r hand, u s i n g the t h i c k f o i l s the d e t e c t o r s o n l y see the h a r d e r photons from the h o t t e r plasma, and hence the s l o p e from the t h i c k f o i l s g i v e the temperature of the h o t t e r plasma. Because the e m i s s i o n s are d i f f e r e n t f o r each p i n c h , the s c a t t e r i n the d a t a i s m a i n l y due t o the a c t u a l v a r i a t i o n i n plasma c o n d i t i o n s . A range of temperature a c h i e v e d by the p i n c h i n g plasma can be i n f e r r e d from t h i s graph. B e s i d e s h a v i n g a range of t e m p e r a t u r e , more i m p o r t a n t l y , the graph a l s o i n d i c a t e s the e x i s t e n c e of a two t e m p e r a t u r e component plasma. The t h i n f o i l s g i v e an e l e c t r o n temperature of 140eV t o 600eV, w h i l e the t h i c k f o i l s g i v e an e l e c t r o n t emperature of 2keV t o 4keV. Our r e s u l t s show t h a t the h i g h e r t e m p e r a t u r e s are due t o the e m i s s i o n s from the s u p e r f a s t p i n c h e s . T i m e - r e s o l v e d m u l t i - f o i l measurements show t h a t b e f o r e the s u p e r f a s t p i n c h , the temperature i s c o m p a r a t i v e l y low, about 200eV as i n f e r r e d from the p e d e s t a l measurements. However, s i m u l t a n e o u s l y w i t h the e x t r a d i p i n the d l / d t t r a c e which i n d i c a t e s the presence of a s u p e r f a s t p i n c h , the temperature i n c r e a s e s r a p i d l y t o a maximum of 4keV i n about 4ns. The maximum r a t e of i n c r e a s e i s a t the b e g i n n i n g of the s u p e r f a s t p i n c h , i n which the measured r a t e i s d e t e c t o r l i m i t e d . In view of t h i s one q u e s t i o n s whether the two t e m p e r a t u r e s a r i s e from the same plasma a t d i f f e r e n t t i m e , or whether they are due t o two d i f f e r e n t plasma r e g i o n s . To answer t h i s q u e s t i o n , one has t o l o o k at the d a t a more 92 c a r e f u l l y . F i r s t l y , i t i s not p o s s i b l e t o a t t a i n such a ten f o l d i n c r e a s e i n temperature (e.g. 140eV t o 2keV) t h r o u g h t h e r m a l i z a t i o n of the c o n v e r g i n g f l o w by the r e f l e c t i o n shock a l o n e , as c a l c u l a t i o n s and measurements!27] show t h a t the plasma would a c q u i r e an i m p l o s i o n v e l o c i t y of 8 x l 0 6 c m / s at the end of an i m p l o s i o n . Complete t h e r m a l i z a t i o n of t h i s m e c h a n i c a l energy r a i s e s the temperature of the plasma by 30eV. Compression due t o the r e f l e c t i o n shock a p p r o x i m a t e l y d o u bles the t e m p e r a t u r e . T h i s i m p l i e s t h a t a d d i t i o n a l compression and h e a t i n g schemes a r e a t work d u r i n g a s u p e r f a s t p i n c h which produce the o b s e r v e d h i g h t e m p e r a t u r e and d e n s i t y . From the • 6pm f o i l measurements the i n t e n s i t y r a t i o between the h i g h temperature e m i s s i o n and the low temperature e m i s s i o n i s o n l y r o u g h l y 0.5:1, whereas c a l c u l a t i o n s i n Appendix A show t h a t i t s h o u l d be a t l e a s t 100:1 i f the e n t i r e plasma i s r a i s e d from 200eV t o 1keV. Thus we come t o the c o n c l u s i o n t h a t o n l y a s m a l l amount of the plasma i s r a p i d l y compressed and heated t o a keV t e m p e r a t u r e , w h i l e the m a j o r i t y of the plasma remains a t sub-keV t e m p e r a t u r e s . T h i s r e s u l t s i n the f a s t X-ray b u r s t a t the t o p of the slow p e d e s t a l i n the 6pm f o i l measurements and the f a s t p u l s e observed i n the 150pm f o i l a r e both t i m e - c o i n c i d e n t w i t h the e x t r a d i p i n the d l / d t t r a c e . 93 X-RAY EMISSION FROM CARBON-COPPER PINCH In a carbon-copper d i s c h a r g e , no o b s e r v a b l e X-ray e m i s s i o n i s e m i t t e d d u r i n g a f a s t p i n c h , but i t i s observed d u r i n g a s u p e r f a s t p i n c h . The e m i s s i o n p u l s e t h rough a t h i n a b s o r b e r f o i l does not show a p e d e s t a l as i n a copper-copper p i n c h , and the p u l s e s from d i f f e r e n t t h i c k n e s s e s of f o i l s a r e s i m i l a r . An o s c i l l o g r a m of t h i s i s shown i n F i g . 4 - 1 5 . I t c o n s i s t s of o n l y one f a s t p u l s e 150Lim foil 6jum foil F i g . 4-15 X-ray e m i s s i o n s t h rough d i f f e r e n t f o i l s i n carbon-copper p i n c h w i t h a d u r a t i o n of a p p r o x i m a t e l y 7ns FWHM. Absorber f o i l 94 measurements of the e m i s s i o n g i v e an e l e c t r o n temperature of 1 t o 3keV as shown i n F i g . 4 - 1 6 . S i n c e the lower 1.0 L 1.1 I I I i — r Q5l Q2l § 0.11 > .021 .01 I I I I I I 1 I ' • ' « ' « » 5 6 10 2 0 5 0 XX) 2 0 0 AL a b s o r b e r foil t h i ckness ( u m ) F i g . 4-16 X-ray e m i s s i o n s through d i f f e r e n t f o i l t h i c k n e s s t e m p e r a t u r e components a r e not p r e s e n t i n t h i s plasma, i t i n d i c a t e s t h a t the h i g h temperature plasma i s o n l y produced by the s u p e r f a s t p i n c h . 95 4.5.2 X-RAY PINHOLE PHOTOGRAPHS A s p a t i a l l y r e s o l v e d image of the X-ray e m i t t i n g plasma can be o b t a i n e d by the p i n h o l e camera. As mentioned p r e v i o u s l y , i n u s i n g the 25um p i n h o l e , o n l y the h i g h l y p i n c h e d plasma i s exposed i n the f i l m . T h i s means t h a t o n l y the e m i s s i o n s of the s u p e r f a s t p i n c h e d plasma a r e r e c o r d e d on the f i l m . Thus the measurements show t h a t the X-ray e m i t t i n g plasma i s c o n f i n e d t o a v e r y minute r e g i o n , t y p i c a l l y 30um i n d i a m e t e r and a p p r o x i m a t e l y c y l i n d r i c a l . I t i s not l o c a t e d a t the m i d - p o i n t of the i n t e r e l e c t r o d e gap, but a t a s m a l l d i s t a n c e above the anode. I f we assume t o t a l sweep-up of the plasma d u r i n g c o m p r e s s i o n , we can then i n f e r the e l e c t r o n d e n s i t y of t h i s compressed plasma t o be of the o r d e r of l 0 2 O c m ~ 3 . Some v e r t i c a l s t r u c t u r e s are a l s o p r e s e n t i n the plasma, as the plasma column i s seen t o be broken up i n t o one or two segments, each t y p i c a l l y lOOum i n l e n g t h . Fig.4-17 shows some of the p i n - h o l e photographs o b t a i n e d t h a t bear these s t r u c t u r e s . 96 :*J * 40/ jm F i g . 4-17 X-ray p i n - h o l e photographs of s u p e r f a s t p i n c h e s 4.6 COMPARISON OF EXPERIMENTAL RESULTS AND SHOCK WAVE MODEL Be f o r e comparing our e x p e r i m e n t a l r e s u l t s w i t h the shock model, we s h a l l examine the v a l i d i t y of the asumptions made i n the t h e o r y . The t h e o r y p r e d i c t s a shock i m p l o s i o n speed of da/dT=0.2 t o 0.6, which under our e x p e r i m e n t a l c o n d i t i o n s t r a n s l a t e s t o 2.5 t o 7.5xl0 6cm/s, whereas, by s e t t i n g T e = l 0 e V f o r the approximate plasma c o n d i t i o n s d u r i n g the l a t e r t imes of a d i s c h a r g e , the i o n a c o u s t i c speed i s 4.0xl0 5cm/s. T h i s v a l i d a t e s our assumption of the e x i s t e n c e of a shock even a t the l a t e r s t a g e s of the 97 d i s c h a r g e . But the shock e q u a t i o n s assume t h a t an e q u i l i b r i u m e x i s t s b e h i n d the shock f r o n t , which i n c l u d e s the c o m p r e s s i o n l a y e r and the r e l a x a t i o n l a y e r . However, as the i o n i z a t i o n r e l a x a t i o n time of the plasma i s l o n g e r than the p i n c h d u r a t i o n , the g v a l u e s h o u l d be a d j u s t e d t o compensate f o r the l o n g r e l a x a t i o n t i m e . We next c o n s i d e r the e f f e c t s of f i n i t e c o n d u c t i v i t y f o r the measured e l e c t r o n t e m p e r a t u r e s . (40eV f o r a slow p i n c h i n g plasma and 400eV f o r a f a s t p i n c h i n g p l a s m a ) . Then the e x p r e s s i o n of the r i g h t hand s i d e of Eq.2.28 g i v e s v a l u e s of the o r d e r of u n i t y and one hundred f o r the slow p i n c h and the f a s t p i n c h , r e s p e c t i v e l y . T h i s shows t h a t most of the e l e c t r i c a l energy i s used i n the form of magnetic compression i n a f a s t p i n c h but a s i g n i f i c a n t p o r t i o n of the e l e c t r i c a l energy i s used up i n Ohmic h e a t i n g i n a slow p i n c h . I f o t h e r energy terms are not c o n s i d e r e d , then t h i s would r e s u l t i n a h o t t e r plasma but w i t h a l a r g e r f i n a l p i n c h e d r a d i u s . 98 L i n e r a d i a t i o n i s by f a r the dominant heat l o s s mechanism. The c o o l i n g r a t e , which i s d e f i n e d as ( r a d i a t i v e p o w e r ) / ( t h e r m a l k i n e t i c energy) of the plasma, i s g i v e n by G r i e m [ 2 l ] 3 R l i n e = 3 x l 0 - 8 ( ! * i f N Z e x p | ^ | ( c m ^ e c " 1 ) , (4.1) VkT/ Z L kT_ 7 where E i s the i o n i z a t i o n energy of hydrogen, E^ i s the H 2 energy between the ground s t a t e and the f i r s t e x c i t e d s t a t e of an i o n i n charge s t a t e Z, and N i s the number d e n s i t y of the i o n . With an average v a l u e of 8 f o r Z and I 0 1 7 c m ~ 3 f o r N f o r the slow p i n c h , a v a l u e of 19 f o r Z and I 0 1 8 c m ~ 3 f o r N Z f o r the f a s t p i n c h , and E2 e q u a l t o h a l f the i o n i z a t i o n energy of the i o n , then the c o o l i n g r a t e f o r slow and f a s t p i n c h a r e 4 . 8 8 X 1 0 7 S ~ 1 and 2 . 5 4 X 1 0 7 S ~ 1 , r e s p e c t i v e l y . Thus i n a slow p i n c h the e l e c t r o n s a re s i g n i f i c a n t l y c o o l e d but not i n a f a s t p i n c h . 99 The c o o l i n g r a t e f o r b r e m s s t r a h l u n g and r e c o m b i n a t i o n l o s s e s a r e [ 2 l ] R c o n t * 1-75X10- 1 4 (^Hj Ejgf*gfb!!!!H|22NZ ( c ^ s e c - 1 ) # ( 4 > 2 ) where g ^ and g ^ are the Gaunt f a c t o r s f o r b r e m s s t r a h l u n g and r e c o m b i n a t i o n , r e s p e c t i v e l y . However, they a r e s e v e r a l o r d e r s of magnitudes lower than l i n e r a d i a t i o n l o s s e s . I t i s w o r t h w h i l e comparing the r e l a t i v e magnitude between Ohmic power g a i n and r a d i a t i o n power l o s s of the plasma i n a slow p i n c h . C a l c u l a t i o n s u s i n g I=50kA and the above f i g u r e s show t h a t the Ohmic h e a t i n g power i s of the ord e r of 107W, w h i l e r a d i a t i o n l o s s i s of the o r d e r of 108W. These r e s u l t s show t h a t - t h e r e s h o u l d not be a s i g n i f i c a n t change i n e l e c t r o n t e m p e r a t u r e i n the shock compressed plasma as a r e s u l t of the c o m p e t i t i o n between these two p r o c e s s e s . S i n c e most of the i n i t i a l parameters a r e unknown i n a vacuum s p a r k , i t i s not p o s s i b l e t o compare r i g o r o u s l y the t h e o r e t i c a l r e s u l t s d i r e c t l y w i t h e x p e r i m e n t a l measurements. However some o r d e r of magnitude comparisons a r e p o s s i b l e . We l e t the c a p a c i t o r v o l t a g e be 8kV a c c o r d i n g t o the r e s u l t i n F i g . 4 - l ( c ) . Supposing the temperature of the plasma a f t e r the f i r s t p i n c h t o be between 1eV and lOeV, then the i o n i z a t i o n s tage i s l e s s than 5 (see F i g . 4 ~ 1 2 ( a ) ). S i n c e 100 the average e l e c t r o n d e n s i t y i s of the o r d e r of I 0 1 8 c m " 3 d u r i n g t h i s t i m e , the d e n s i t y of the copper plasma i s of the o r d e r of 10""gm/cnr 3. F u r t h e r m o r e , g i s between 1.1 and 1.2 f o r the temperature range e x p e c t e d b e h i n d the shoc k ( s e e Appendix B ) . But as the plasma r e l a x a t i o n time i s l o n g e r than the p i n c h t i m e , we s h o u l d l e t g i=1.3 f o r our e s t i m a t e . U s i n g t h e s e v a l u e s , f o r a f a s t p i n c h of 20ns p i n c h d u r a t i o n one g e t s A Q=0 . 1 1 3cm, a n d r m i n = 4 5 ^ m -For a slow p i n c h of 40ns p i n c h d u r a t i o n one g e t s A o=0.223cm, a n d r m i n = 8 5 J j m ' Compare t h i s w i t h the nominal e x p e r i m e n t a l v a l u e s r m i n=40um f o r f a s t p i n c h and r m i n=120pm f o r slow p i n c h . The lower t h e o r e t i c a l v a l u e f o r a slow p i n c h i s ex p e c t e d as the model has not taken i n t o c o n s i d e r a t i o n the e f f e c t s of f l o w convergence d u r i n g the f i n a l s t a g e s of the i m p l o s i o n and the f r a c t i o n of energy used as Ohmic h e a t i n g . The c a l c u l a t e d r e s u l t f o r the f a s t p i n c h i s s u s p i c i o u s l y c l o s e t o the e x p e r i m e n t a l v a l u e . However, the c a l c u l a t i o n can be e r r o n e o u s . An i n c r e a s e i n the i n i t i a l d e n s i t y due t o i t s s m a l l e r i n i t i a l r a d i u s and a h i g h e r g due t o the s h o r t e r p i n c h d u r a t i o n can r e s u l t i n a s m a l l e r f i n a l r a d i u s . R a d i a t i o n l o s s e s d u r i n g the f a s t p i n c h can a l s o cause the f i n a l r a d i u s t o be much s m a l l e r . 101 The r e p r o d u c i b i l i t y of the f i r s t slow p i n c h a t 500ns i n d i c a t e s t h a t the i n i t i a l plasma c o n d i t i o n s are q u i t e c o n s i s t e n t i n any d i s c h a r g e s when u s i n g the h i g h - v o l t a g e d i r e c t d i s c h a r g e t r i g g e r i n g method. However, i n s t a b i l i t y s e t s i n a f t e r t h i s slow p i n c h phase and causes the subsequent p i n c h e s t o be i r r e p r o d u c i b l e due t o v a r y i n g plasma c o n d i t i o n s . The d i f f e r e n c e between the slow and the f a s t p i n c h can be e x p l a i n e d as f o l l o w s : i f a subsequent p i n c h s t a r t s b e f o r e the plasma column expands s u f f i c i e n t l y a f t e r t h e f i r s t p i n c h , then t h i s w i l l r e s u l t i n a f a s t p i n c h due t o i t s i n i t i a l l y s m a l l e r r a d i u s and h i g h e r t e m p e r a t u r e . I f the plasma column has expanded s u f f i c i e n t l y b e f o r e the next p i n c h s t a r t s , i t w i l l have a l a r g e r r a d i u s , r e s u l t i n g i n a lower magnetic p r e s s u r e and a l o n g e r p i n c h d u r a t i o n . F u r t h e r m o r e , the i n t i a l d e n s i t y and temperature w i l l a l s o be lower than t h a t of the f a s t p i n c h , which s u b s e q u e n t l y cause the f i n a l d e n s i t y and t e m p e r a t u r e t o be of lower v a l u e s . A f t e r the d i s r u p t i o n of a p i n c h , the over-compressed plasma w i l l expand w i t h the a c o u s t i c speed. T h e r e f o r e i n two c o n s e c u t i v e i m p l o s i o n s , the p i n c h d u r a t i o n of the second i m p l o s i o n w i l l be p r o p o r t i o n a l t o the time i n t e r v a l between the two i m p l o s i o n s . T h i s l i n e a r b e h a v i o r of the e x p e r i m e n t a l r e s u l t s as shown i n F i g . 4 - 1 8 g i v e good agreement w i t h the above e x p l a n a t i o n . The e x p e r i m e n t a l r e s u l t s d e v i a t e from the l i n e a r b e h a v i o r f o r l o n g e r time i n t e r v a l s . T h i s i s e x p e c t e d s i n c e we have not taken i n t o c o n s i d e r a t i o n the e f f e c t of the magnetic p r e s s u r e . 1 02 0 100 200 300 400 500 600 700 time interval between 2 pinches ( ns ) F i g . 4-18 P i n c h i n t e r v a l vs time i n t e r v a l between c o n s e c u t i v e p i n c h e s The shock model, however, f a i l s t o e x p l a i n the e x i s t e n c e of the s u p e r f a s t p i n c h e s . I t i s p l a u s i b l e t h a t the s u p e r f a s t p i n c h e s a re due t o i n s t a b i l i t i e s i n the p i n c h dynamics and we s h a l l examine t h i s more c l o s e l y i n the next c h a p t e r . 103 CHAPTER 5  INSTABILITY OF Z-PINCH The b e h a v i o u r of the slow and the f a s t p i n c h can be p r e d i c t e d by the dynamic models of Z-pinch o u t l i n e d i n Chapter 2. However, the s e models f a i l t o e x p l a i n the f o r m a t i o n of a s u p e r f a s t p i n c h , i t s h i g h d e n s i t y and te m p e r a t u r e . S i n c e the s u p e r f a s t p i n c h e s a r e obser v e d t o o n l y occur a t the v e r y f i n a l s t age of a f a s t p i n c h , i t i s c o n c e i v a b l e t h a t t h e i r o r i g i n s a r e r e l a t e d t o the dynamics of the p i n c h i n g mechanism. In t h i s c h a p t e r we s h a l l l o o k a t the s t a b i l i t y f o r each stage of a p i n c h i n o r d e r t o u n d e r s t a n d the development of a s u p e r f a s t p i n c h . As b e f o r e , we s h a l l make the assumption t h a t the plasma c o l l i s i o n s between p a r t i c l e s a r e s u f f i c i e n t l y f r e q u e n t t h a t the plasma p r e s s u r e can be t r e a t e d as a s c a l a r q u a n t i t y . A l s o the heat l o s s by r a d i a t i o n and heat g a i n by Ohmic h e a t i n g a r e assumed t o be n e g l i g i b l e . We s h a l l r e v i e w two well-known t y p e s of i n s t a b i l i t i e s t h a t occur i n a plasma column. The f i r s t t y p e a r i s e s when the plasma i s b e i n g a c c e l e r a t e d by the magnetic f i e l d i n the vacuum. We c a l l t h i s type the g r a v i t a t i o n a l i n s t a b i l i t y . The second type a r i s e s when the a z i m u t h a l magnetic f i e l d which c o n f i n e s the Z-pinch plasma i s i n t u r n caused by the e l e c t r i c c u r r e n t on the plasma s u r f a c e . We c a l l t h i s the 1 04 a z i m u t h a l i n s t a b i l i t y . These two ty p e s of i n s t a b i l i t y a r e v e r y s i m i l a r i n n a t u r e , and are g e n e r a l l y known as R a y l e i g h - T a y l o r i n s t a b i l i t y . However, d u r i n g a i m p l o s i o n the dynamics of the plasma column i s more c o m p l i c a t e d s i n c e the p r e s ence of a shock wave and the a s s o c i a t e d d i s c o n t i n u i t i e s can have s e r i o u s e f f e c t s on the two ty p e s of i n s t a b i l i t i e s . In Chapter 2 we showed t h a t d i f f e r e n t models are r e q u i r e d t o d e s c r i b e a p i n c h d u r i n g d i f f e r e n t s t a g e s of an i m p l o s i o n . T h e r e f o r e the plasma column w i l l a l s o e x h i b i t d i f f e r e n t s t a b i l i t i e s a t the s e d i f f e r e n t s t a g e s . As b e f o r e , the f r e e p a r t i c l e model i s a p p l i c a b l e d u r i n g the e a r l y stage of a p i n c h , and the i n s t a b i l i t y of the f r e e p a r t i c l e model i s d e s c r i b e d i n S e c t i o n 5.1. Because of a n a l y t i c a l d i f f i c u l t i e s i n the shock model, a s i m p l i f i e d approach i s taken and the s t a b i l i t y problem d u r i n g the i n t e r m e d i a t e stage i s i n v e s t i g a t e d i n S e c t i o n 5.2 u s i n g the snowplow model, which i s of c o u r s e the l i m i t i n g case of the shock model. Because the growth r a t e of the a c c e l e r a t i o n i n s t a b i l i t y i s r e l a t i v e l y s m a l l [ 3 7 ] we s h a l l o n l y c o n s i d e r the a z i m u t h a l i n s t a b i l i t y i n the above models. In the convergence s t a g e , s i n c e the dynamics of the shock f r o n t and t h e plasma-vacuum i n t e r f a c e can be t r e a t e d s e p a r a t e l y , t h e s t a b i l i t y of the shock f r o n t i s i n v e s t i g a t e d from the g e o m e t r i c a l approach i n S e c t i o n 5.3, w h i l e the two ty p e s of R a y l e i g h - T a y l o r i n s t a b i l i t i e s ( a c c e l e r a t i o n and a z i m u t h a l ) i n the plasma-vacuum i n t e r f a c e a r e i n v e s t i g a t e d i n 105 S e c t i o n 5.4. The n o n l i n e a r growths of these p e r t u r b a t i o n s a r e d i s c u s s e d i n S e c t i o n 5.5. The p o s s i b l e mechanisms f o r the g e n e r a t i o n of the s u p e r f a s t p i n c h i s then d i s c u s s e d i n S e c t i o n 5.6. For any q u a n t i t y F under c o n s i d e r a t i o n we can d e f i n e F u as i t s v a l u e i n the u n p e r t u r b e d s t a t e and F' the d i f f e r e n c e between the u n p e r t u r b e d s t a t e and the p e r t u r b e d s t a t e . In p a r t i c u l a r , the p r e s s u r e , magnetic f i e l d and the r a d i u s of the plasma column i n the p e r t u r b e d s t a t e a r e g i v e n by P • P u • P ' I = "5 • £ t (5.1) u » A = A u • A ' . We s h a l l a n a l y z e the p e r t u r b a t i o n i n normal mode u s i n g c y l i n d r i c a l c o o r d i n a t e ( F i g . 5 - 1 ) and assume a l l p e r t u r b a t i o n s have the form F ' ( r , * , Z , t ) = f ( r . t ) e i ( k x * " * ) , (5.2) where m i s the a z i m u t h a l number and k i s the a x i a l wave number of the p e r t u r b a t i o n . The g e n e r a l p e r t u r b a t i o n c o n s i s t s of a sum over a l l m and an i n t e g r a l over k. I f one c o n s i d e r s o n l y the l i n e a r c a s e , then each pert u r b e d , mode can be t r e a t e d s e p a r a t e l y . The view of a p e r t u r b e d column at d i f f e r e n t v a l u e s of m i s g i v e n i n F i g . 5 - 2 . 1 06 z r — A — M F i g . 5-1 C y l i n d r i c a l c o o r d i n a t e system f o r p e r t u r b a t i o n a n a l y s i s L e t the plasma column t o be i n s i d e an i n f i n i t e l y l o n g c y l i n d e r w i t h z e r o r e s i s t i v i t y . S i n c e no c u r r e n t f l o w s i n the vacuum, the p e r t u r b a t i o n i n the magnetic f i e l d i s g i v e n as V- B' = 0 , (5.3) where the r a d i a t i o n term due t o the c h a n g i n g f l u x i s n e g l e c t e d . 1 07 F i g . 5-2 A p e r t u r b e d column at d i f f e r e n t v a l u e s of m U s i n g t h e boundary c o n d i t i o n s t h a t the normal magnetic f i e l d v a n i s h e s a t r=R 0 and r=A one g e t s the s o l u t i o n a f t e r some t e d i o u s m a n i p u l a t i o n s [ 2 4 ] im A' u K' (kR ) I ( k r ) I'(kR )K ( k r ) I'(kA )K'(kR )-K»(kA ) I ' ( k R ) m uJ mv oJ mv uJ m oJ i f £i B Q ( k r ) k A uxm^ J u (5.4) where I , K are the h y p e r b o l i c B e s s e l ' s f u n c t i o n s . 1 The p e r t u r b e d magnetic f i e l d i s then = ( k * ' . ^7* . i k * ) . (5.5) and the p e r t u r b e d magnetic p r e s s u r e a t the plasma s u r f a c e p. . J i L i + • Q (kA ) . (5.6) Pmag 4 u A |_ k A u ^ u J 5.1 STABILITY OF FREE PARTICLE MODEL The p e r t u r b e d momentum e q u a t i o n of Eq.2.7 i s /3A X/SA'X L e t a p e r t u r b a t i o n be a p p l i e d a t t=0. 109 A p p l y i n g the r e s u l t of Eq.5.6 t o Eq.5.7 and a f t e r some m a n i p u l a t i o n s , Wyld[52] o b t a i n e d the p e r t u r b a t i o n of the plasma s u r f a c e as A ' ( t ) 2o A (0) u v ' exp I ii J 1 A y t t ) dr lm — * * * — * Q (kr) r kr Hm v J (5.8) which has the a s y m p t o t i c forms A ( t ) u ' A ' ( t ) A ^ T o T -m-1 f o r kR <Jm , o v * R << 1 A T O E X P u f o r kR > m , o * (5.9) R « 1 Thus the h i g h e r m modes ten d t o be s t a b l e . T h i s i s p h y s i c a l l y p l a u s i b l e s i n c e the h i g h e r m modes d i s t o r t the magnetic f i e l d s i n such a way t h a t i t tends t o push the plasma back t o the u n p e r t u r b e d p o s i t i o n . However, f o r the m=0 mode the column i s u n s t a b l e t o p e r t u r b a t i o n s of a l l wavelengths and the p e r t u r b a t i o n grows as the r e c i p r o c a l of the d e c r e a s i n g p i n c h r a d i u s . The growth of t h i s p e r t u r b a t i o n can be u n d e r s t o o d as f o l l o w s : a t the c r e s t of 110 the p e r t u r b a t i o n where the r a d i u s i s l a r g e s t , the magnetic f i e l d i s p r o p o r t i o n a l l y s m a l l e r , r e s u l t i n g i n a weaker magnetic p r e s s u r e . T h i s p a r t of the column t h e r e f o r e e x p e r i e n c e s a s m a l l e r i m p l o d i n g a c c e l e r a t i o n and causes the s i z e of the p e r t u r b a t i o n t o i n c r e a s e f u r t h e r . As the growth r a t e i s q u i t e s m a l l d u r i n g the i n i t i a l s t age of the i m p l o s i o n , we do not expect the s i z e of the p e r t u r b a t i o n t o i n c r e a s e a p p r e c i a b l y d u r i n g the time i n t e r v a l i n which the f r e e p a r t i c l e model i s v a l i d . 5.2 INSTABILITY OF SNOWPLOW PINCH The i n s t a b i l i t y of the snowplow p i n c h has a l s o been i n v e s t i g a t e d t h e o r e t i c a l l y by W y l d [ 5 2 ] . The u n p e r t u r b e d e q u a t i o n i s the same as Eq.2.18, w i t h gj=1 and t h e r e f o r e d = <r. C o n s i d e r a c y l i n d r i c a l l y symmetric but o t h e r w i s e a r b i t r a r y p e r t u r b a t i o n on the p i n c h . The p e r t u r b a t i o n e q u a t i o n s a r e 2 1 2 i re p 3 t2 ( A ; • A 2 ) A ' 1 d T~ I t u 7 _ d A ( A 2 - A 2 ) " A ' o u-'dt (5.10a)and d _ d t A I n j i I A u u •* d z A " = 0 (5.10b) 1 1 1 s i n c e I ' i s a f u n c t i o n o n l y of t i m e , we c o n c l u d e t h a t A» = f 1 ( z ) h 1 ( t ) + f 2 ( z ) h 2 ( t ) + h 3 ( t ) , ( 5 . 1 1 ) where f j ( z ) , f 2 ( z ) a r e d e t e r m i n e d by the i n i t i a l c o n d i t i o n of the p e r t u r b a t i o n . T h i s has the s o l u t i o n h 1 . h 2 - e ±T . / ( n / l)h TT2 k l e T / n ' e - T / n ' h_ = . + + k_- ( 5 . 1 2 ) 3 VCT 2 l - o 2 3 l - o 2 ' where T J ' = ^5 1— S ( l - i ) * I t can be shown t h a t a l t h o u g h t h e s e s o l u t i o n s p r e d i c t i n s t a b i l i t y , the i n c r e a s e i n p e r t u r b a t i o n a m p l i t u d e i s q u i t e s m a l l d u r i n g the time i n t e r v a l i n which the snowplow model i s v a l i d . 112 5.3 STABILITY OF CYLINDRICAL CONVERGING SHOCK FRONT When an i m p l o d i n g shock i s near i t s c e n t e r of c o l l a p s e , the shock f r o n t and the p i s t o n f r o n t can be c o n s i d e r e d s e p a r a t e l y . T h i s i s a v a l i d d e s c r i p t i o n when the compressed s h e l l t h i c k n e s s i s much l a r g e r than the r a d i u s of the i m p l o d i n g shock f r o n t [ 5 3 ] . A l t h o u g h s i m p l e c o n s i d e r a t i o n s show t h a t a c y l i n d r i c a l c o n v e r g i n g shock wave i s u n s t a b l e t o a l l f i n i t e s i z e p e r t u r b a t i o n s , t h e r e a r e c o n f i g u r a t i o n s i n which the shock f r o n t i s s t a b l e when a x i a l f l o w b e h i n d the f r o n t i s taken i n t o c o n s i d e r a t i o n . We l i k e t o know the s t a b i l i t y of the shock f r o n t t o p e r t u r b a t i o n s of d i f f e r e n t w a v e l e n g t h s , by t a k i n g i n t o c o n s i d e r a t i o n the e f f e c t s of a r e a v a r i a t i o n s i n the r a d i a l and the a x i a l d i r e c t i o n s . To a c h i e v e t h i s , we s h a l l c o n s i d e r the developments of i n f i n i t e s i m a l l y s m a l l p e r t u r b a t i o n s . We s h a l l look a t the s i m p l e case of the s t a b i l i t y of the c y l i n d r i c a l shock f r o n t t o m=0 mode p e r t u r b a t i o n . S t a b i l i t y r e q u i r e s t h a t the a r e a convergence a t the t r o u g h s h o u l d be l e s s than t h a t a t the c r e s t . I f such i s the ca s e , then the shock wave a t the c r e s t w i l l s t r e n g t h e n f a s t e r than t h a t a t t h e t r o u g h , t h e r e b y r e d u c i n g the s i z e of the p e r t u r b a t i o n . 1 1 3 Without l o s s of g e n e r a l i t y , we can assume i n i t i a l l y u n i f o r m shock f r o n t v e l o c i t y . The above argument l e a d s t o the f o l l o w i n g c r i t e r i o n f o r the s t a b i l i t y of the shock f r o n t t o i n f i n i t e s i m a l l y s m a l l p e r t u r b a t i o n s , kS > 1 . (5.13) The s t a b i l i t y a r i s e s from the f a c t t h a t the d i s t o r t i o n produced by the p e r t u r b a t i o n c auses a c o r r e s p o n d i n g change i n the f l o w b e h i n d the i m p l o d i n g shock f r o n t i n such a way t o m i n i m i z e the d i s t o r t i o n . ( T h i s i s the same reason why a p l a n e shock f r o n t i s s t a b l e . ) The c y l i n d r i c a l shock f r o n t i s , however, u n s t a b l e t o a p e r t u r b a t i o n of l o n g w a velength, where the e f f e c t of d i v e r g i n g a x i a l f l o w i s not s u f f i c i e n t t o compensate f o r e f f e c t of the r a d i a l c o n v e r g i n g f l o w caused by the p e r t u r b a t i o n . We have done a more g e n e r a l t r e a t m e n t of the growth of a f i n i t e p e r t u r b a t i o n by n u m e r i c a l l y c a l c u l a t i n g the l o c a t i o n of the i m p l o d i n g shock f r o n t a t a s e r i e s of time s t e p s u s i n g Whitham's ar e a r u l e [ 4 9 ] . R e s u l t s of these c a l c u l a t i o n s a r e shown i n F i g . 5 - 3 . F i g . 5 - 3 ( a ) shows the time e v o l u t i o n of an i n i t i a l l y s i n u s o i d a l p e r t u r b a t i o n of kA=lO on the shock f r o n t . I t can be seen t h a t the s i z e of the p e r t u r b a t i o n d e c r e a s e s i n time so the shock f r o n t i s s t a b l e t o t h i s p e r t u r b a t i o n . F i g . 5 ~ 3 ( b ) shows the i n s t a b i l i t y growth of an i n i t i a l l y s i n u s o i d a l p e r t u r b a t i o n of kA=0.1. In g e n e r a l , our r e s u l t s agree w i t h t h a t of 114 c / V Z 3 T3 Pol C O 00 u O J Z cn 1 1 1 - t, -— ________ tl -t 3  't, 1 I i 00 ft kz 37T 2 2 7 T F i g . 5-3(a) S t a b i l i t y of wavelength) p e r t u r b a t i o n an i n i t i a l l y kA=lO ( s h o r t 2 7 T F i g . 5-3(b) I n s t a b i l i t y wavelength) p e r t u r b a t i o n of an i n i t i a l l y kA=0.1 ( l o n g Eq.5.13. F u r t h e r m o r e , i t can a l s o be shown t h a t the growth r a t e of the p e r t u r b a t i o n i n c r e a s e s m o n o t o n i c a l l y w i t h d e c r e a s i n g k. In p a r t i c u l a r , the most dangerous 1 15 p e r t u r b a t i o n s i n a c o n v e r g i n g shock f r o n t a r e those w i t h l o n g w a v e l e n g t h s . 5.4 RAYLEIGH~TAYLOR INSTABILITY OF A PINCH COLUMN Having l o o k e d a t the i n s t a b i l i t y of the shock, we now t u r n t o the i n s t a b i l i t y of the plasma-vacuum i n t e r f a c e . K r u s k a l [ 2 4 ] has i d e n t i f i e d two t y p e s of i n s t a b i l i t i e s , namely the a c c e l e r a t i o n i n s t a b i l i t y and the a z i m u t h a l i n s t a b i l i t y d i s c u s s e d above. I f the p e r t u r b a t i o n a m p l i t u d e of the r a d i u s i s g i v e n by A = A + A 1 u i ( k z + m<j> + wt) then the ima g i n a r y p a r t of the complex f r e q u e n c y OJ i s the growth r a t e v of the p e r t u r b a t i o n . In the a c c e l e r a t e d frame of the plasma-vacuum i n t e r f a c e , we assume a q u a s i - s t a t i c p i n c h column i n which the magnetic p r e s s u r e a c t i n g on the plasma i s b a l a n c e d by the plasma k i n e t i c p r e s s u r e and the p r e s s u r e due t o " g r a v i t y " . With t h i s approach, the e q u a t i o n s i n the vacuum are the same as t h a t of the f r e e p a r t i c l e model(Eq.5.6). 1 1 6 We d e f i n e £ as the d i s p l a c e m e n t of a f l u i d element i n s i d e the plasma column from i t s u n p e r t u r b e d p o s i t i o n , w i t h the boundary c o n d i t i o n l ( A u ) = A U 6 ( r ) e i f k z + m * + u t ) ^ ( 5 1 5 ) B e a r i n g i n mind the dependence of the p e r t u r b a t i o n on the c o o r d i n a t e s and t i m e , we can w r i t e the e q u a t i o n of motion as . P l « 2 1 - - V p i + .PL**T . (5.16) where p'^  i s the plasma p e r t u r b a t i o n p r e s s u r e and g i s the " g r a v i t a t i o n a l " a c c e l e r a t i o n i n the a c c e l e r a t e d frame. When g i s assumed t o be u n i f o r m w i t h i n the p e r t u r b a t i o n , Eg.5.16 has the s o l u t i o n . _ pu) 26 I m ( l c r ) , i ( k z * m » - m » t ) , . P i ~ ~ 11(KXJ ' ( 5 ' 1 7 ) 2 p l w where K2 = k + m * u 2 1 17 We r e q u i r e t h a t the p r e s s u r e i s c o n t i n u o u s a c r o s s the i n t e r f a c e , and a r r i v e a t the e x p r e s s i o n f o r the complex f r e q u e n c y „2 -, u mv u , . 2 T - kA„ + m Q (kA ) - kg (5.18) In p a r t i c u l a r , the column i s u n s t a b l e t o p e r t u r b a t i o n s of a l l w avelengths i n the m=0 mode, and the growth r a t e v i s of the o r d e r of 4rrp where the f i r s t term on the r i g h t hand s i d e i s due t o a z i m u t h a l i n s t a b i l i t y and the second term i s due t o g r a v i t a t i o n a l i n s t a b i l i t y . For an i n c o m p r e s s i b l e f l u i d of kA—"1, Eq.5.19 can be s i m p l i e d t o (5.20) ie I . (K A ) 1 u A I (K A ) u o v u . + kg (5.19) where V = a i s the A l f v e n wave speed. C a l c u l a t i o n s by J u k e s [ 2 6 ] show t h a t the i n c l u s i o n of a l a r g e but f i n i t e c o n d u c t i v i t y i n t o the plasma column i n t r o d u c e s a d e s t a b i l i z i n g i n f l u e n c e i n t o 1 18 an o t h e r w i s e s t a b l e column by a l l o w i n g the s l i p p a g e of the magnetic f i e l d l i n e s which would o t h e r w i s e have s t a b i l i z e d the column. However the growth r a t e of t h i s new i n s t a b i l i t y i s s m a l l when compared t o the growth r a t e of the a l r e a d y u n s t a b l e modes. 5.5 NONLINEAR EVOLUTION OF m=0 RAYLEIGH-TAYLOR INSTABILITY When the p e r t u r b a t i o n grows t o a f i n i t e s i z e , the l i n e a r models are no l o n g e r a p p l i c a b l e and second o r d e r e f f e c t s have t o be taken i n t o c o n s i d e r a t i o n . Book[8] c o n s i d e r e d the m=0 a z i m u t h a l i n s t a b i l i t y f o r an i n c o m p r e s s i b l e f l u i d . H i s r e s u l t s show t h a t under the a s y m t o t i c c o n d i t i o n s when e i t h e r kA<<1 or kA>>1, the c y l i n d r i c a l l y symmetric t w o - d i m e n s i o n a l p e r t u r b a t i o n problem can be reduced t o a o n e - d i m e n s i o n a l problem, w i t h the g o v e r n i n g e q u a t i o n s s i m i l a r t o the g r a v i t a t i o n a l i n s t a b i l i t y . N u m e r i c a l methods a r e then used t o f o l l o w the e v o l u t i o n of i n i t i a l l y s m a l l p e r t u r b a t i o n s i n t o the n o n - l i n e a r s t a g e s . An i n i t i a l l y s i n u s o i d a l p e r t u r b a t i o n tends t o d e v e l o p i n t o a s p i n d l e s t r u c t u r e w i t h broad m i n i n a and s h a r p maxima i n plasma r a d i u s f o r both cases of kA<<1 and kA>>1. A l t h o u g h the l i n e a r growth r a t e s of the two ty p e s of i n s t a b i l i t i e s i n c r e a s e m o n o t o n i c a l l y w i t h i n c r e a s i n g k, s a t u r a t i o n i s e x p e c t e d t o occur when the a m p l i t u d e s of the p e r t u r b a t i o n s exceed the wavelengths. The r e s u l t i s a " f u z z y " plasma s u r f a c e due t o s h o r t 1 19 wavelength p e r t u r b a t i o n s , but the g r o s s sausage d i s t o r t i o n i s dominated by p e r t u r b a t i o n s w i t h the wavelength s c a l e of kA~1 . 5.6 THE GENERATION OF A SUPERFAST PINCH Based upon the above r e s u l t s , we propose the f o l l o w i n g mechanism f o r the g e n e r a t i o n of the s u p e r f a s t p i n c h : d u r i n g t h e i n i t i a l s t a g e s of a p i n c h the plasma column i s u n s t a b l e t o p e r t u r b a t i o n a c c o r d i n g t o the f r e e p a r t i c l e model. However, the growth r a t e i s r e l a t i v e l y s m a l l at. t h i s s t a g e . Then as the magnetic p i s t o n a c q u i r e s s u p e r s o n i c speed the plasma column i s s t a b i l i z e d by the accumulated mass i n f r o n t of the p i s t o n . The column a g a i n becomes u n s t a b l e a t the f i n a l moments of the c o l l a p s e when the shock has d e t a c h e d from the magnetic pusher. The shock f r o n t i s u n s t a b l e t o l o n g wavelength p e r t u r b a t i o n s w h i l e the plasma-vacuum i n t e r f a c e i s u n s t a b l e t o s h o r t w a v e l e n g t h p e r t u r b a t i o n s . Because of t h e s e c o n t r a d i c t i n g r e q u i r e m e n t s and the p o s s i b l e s a t u r a t i o n s of s h o r t wavelength p e r t u r b a t i o n s , the p e r t u r b a t i o n s t h a t a r e most l i k e l y t o d e v e l o p w i l l have wavelength s c a l e s kA~1 . These p e r t u r b a t i o n s grow i n t o s t r u c t u r e s w i t h broad minima and s h a r p maxima i n plasma r a d i u s , l e a d i n g t o the o b s e r v e d s u p e r f a s t p i n c h . R a p i d c o m p r e s s i o n a l h e a t i n g d u r i n g the e v o l u t i o n of t h i s sausage 1 20 i n s t a b i l i t y r a i s e s the compressed plasma t o keV t e m p e r a t u r e s . S e v e r a l e x p e r i m e n t a l r e s u l t s can be used t o t e s t the v a l i d i t y of the proposed mechanism. These t e s t s a r e as f o l l o w s : 1. The time between the f i n a l convergence stage of the f a s t p i n c h and t h e o b s e r v a t i o n of the s u p e r f a s t p i n c h s h o u l d be l o n g e r than the e - f o l d i n g time of the p e r t u r b a t i o n g i v e n by Eq.5.20. 2. The t emperature a c h i e v e d by the s u p e r f a s t p i n c h s h o u l d be of the o r d e r of t h a t e s t i m a t e d by a d i a b a t i c c o m p r e s s i o n a l h e a t i n g , i f t h i s i s the dominant h e a t i n g mechanism d u r i n g the growth of the i n s t a b i l i t y . 3. The r a t i o of the l e n g t h t o the r a d i u s of the s u p e r f a s t p i n c h e d plasma column s h o u l d be a p p r o x i m a t e l y e q u a l t o 2 T , as t h i s i s the dominant mode p r e d i c t e d by the t h e o r y . 4. The s i z e of the s u p e r f a s t p i n c h e d column s h o u l d be r e l a t e d t o the e x t r a d i p o b s e r v e d i n the d l / d t t r a c e . We now l o o k a t t h e s e c o n d i t i o n s i n more d e t a i l . U s i n g the r e s u l t s i n Chapter 4, the e - f o l d i n g time i s a p p r o x i m a t e l y 2ns w h i l e the e l a p s e d time between the convergence s t a g e and the s u p e r f a s t p i n c h i s a p p r o x i m a t e l y 10ns, t h e r e f o r e the f i r s t c o n d i t i o n i s s a t i s f i e d . The growth of the p e r t u r b a t i o n compresses the r a d i u s o'f the plasma column from 40pm t o 15pm. For the s u p e r f a s t p i n c h e d plasma as we a r e c o n s i d e r i n g the e l e c t r o n t emperature and s i n c e c o l l i s i o n a l i o n i z a t i o n i s r e l a t i v e l y slow compared t o 121 140 Y emitting radius in pm F i g . 5-4 R a t i o of r a d i u s vs l e n g t h of a s u p e r f a s t p i n c h e d plasma the growth of the p e r t u r b a t i o n , we can assume g 1=5/3 i n the c a l c u l a t i o n of a d i a b a t i c h e a t i n g . E x p e r i m e n t a l l y , the e l e c t r o n t emperature i n a f a s t p i n c h i s between 140 and 600eV. A f t e r the passage of the r e f l e c t i o n shock, i t s temperature a p p r o x i m a t e l y d o u b l e s . A f u r t h e r 4 - f o l d i n c r e a s e w i l l r e s u l t i n a temperature between 1keV and 4keV, which i s i n agreement w i t h the sausage compression scheme. The r a d i u s vs l e n g t h a re p l o t t e d i n F i g . 5 - 4 , and i t can be seen t h a t the dominant mode i s kA=1. S i n c e the i m p l o d i n g v e l o c i t y of the plasma column does not change a p p r e c i a b l y d u r i n g the the f i n a l s t a g e s of the p i n c h , the magnitude of the d l / d t d i p can be c o n s i d e r e d t o be i n v e r s e l y p r o p o r t i o n a l 122 t o the plasma r a d i u s i n t e g r a t e d over the p i n c h e d l e n g t h . From the f r a m i n g camera measurements the l e n g t h of a f a s t p i n c h i s 3±0.5mm, w h i l e from X-ray p i n - h o l e photographs i t i s I00±20um f o r a s u p e r f a s t p i n c h . Hence the e x p e c t e d d l / d t d i p due t o the p e r t u r b a t i o n i s 0.08 of the u n p e r t u r b e d d l / d t d i p . E x p e r i m e n t a l l y t h i s r a t i o i s l e s s than or e q u a l t o 0.1. Thus the above comparisons g i v e c o n f i d e n c e i n our proposed mechanism f o r the g e n e r a t i o n of a s u p e r f a s t p i n c h . 123 CHAPTER 6  CONCLUSIONS AND SUGGESTIONS The o p e r a t i o n of a vacuum spark as a s p e c t r o s c o p i c source has been i n v e s t i g a t e d w i t h r e g a r d t o i t s r e p r o d u c i b i l i t y and e m i s s i o n i n d i f f e r e n t r e g i o n s of the e l e c t r o m a g n e t i c spectrum. C u r r e n t and v o l t a g e measurements ar e made w i t h a Rogowski c o i l and a v o l t a g e probe. The v i s i b l e and the UV e m i s s i o n s a r e measured w i t h a f r a m i n g camera and a s p e c t r o m e t e r . The X-ray e m i s s i o n s a r e measured w i t h a PIN d i o d e , m u l t i - f o i l a b s o r p t i o n d e t e c t o r s and a p i n h o l e camera. L a s e r t r i g g e r i n g of the d i s c h a r g e has been found t o o f f e r no advantage over the more s i m p l e e l e c t r i c a l d i s c h a r g e t r i g g e r i n g , and the r e p r o d u c i b i l i t y of the Z-pinches has been improved by u s i n g d i r e c t d i s c h a r g e t r i g g e r i n g w i t h the t r i g g e r gap r e c e s s e d i n s i d e t h e c a t h o d e . By o p t i m i z a t i o n on the t r i g g e r i n g energy, the f i r s t p i n c h of our vacuum spark i s r e p r o d u c i b l e t o w i t h i n 50ns. L a t e r p i n c h e s a r e s t i l l e r r a t i c , but t h i s b e h a v i o r i s p r i m a r i l y due t o the r e s u l t of i n s t a b i l i t i e s which a l s o causes the d i s r u p t i o n of the p i n c h e s . In a vacuum s p a r k , the Z-pinches can be c l a s s i f i e d i n t o s low, f a s t and s u p e r f a s t a c c o r d i n g t o t h e i r p i n c h d u r a t i o n . The slow p i n c h has a nominal p i n c h d u r a t i o n of 40ns or l o n g e r . I t i s r e l a t i v e l y c o l d , w i t h e l e c t r o n t e m p e r a t u r e 1 24 l e s s than lOOeV and an i n f e r r e d e l e c t r o n d e n s i t y of the o r d e r of I 0 1 8 c m " 3 . V i s i b l e and u l t r a v i o l e t r a d i a t i o n are e m i t t e d d u r i n g the i m p l o s i o n , but no X-ray i s observed. The f a s t p i n c h has a nominal p i n c h d u r a t i o n of 20ns, w i t h an e l e c t r o n temperature between 140eV and 600eV and an e l e c t r o n d e n s i t y of the o r d e r of I 0 1 9 c m ~ 3 . X - r a y s are e m i t t e d d u r i n g the i m p l o s i o n , but s i n c e the p i n c h i n g plasma has such a h i g h t e m p e r a t u r e , i t s e m i s s i o n s are p r e d o m i n a n t l y i n the vacuum UV and the s o f t X-ray wavelengths. The subsequent v i s i b l e and u l t r a v i o l e t e m i s s i o n s are e s s e n t i a l l y due t o r e c o m b i n a t i o n s i n the plasma a f t e r the d i s r u p t i o n of the p i n c h . The d i f f e r e n t c h a r a c t e r i s t i c s of the slow and the f a s t p i n c h are due t o t h e i r d i f f e r e n t i n i t i a l r a d i i a t the onset of the i m p l o s i o n . A l a r g e r i n i t i a l r a d i u s r e s u l t s i n a slow p i n c h and a s m a l l e r i n i t i a l r a d i u s i n a f a s t p i n c h . T h i s has been found t o be i n agreement w i t h our t h e o r e t i c a l shock wave model. The s u p e r f a s t p i n c h , however, d e v i a t e s from the t h e o r e t i c a l p r e d i c t i o n . I t i s observed o c c a s i o n a l l y a t the f i n a l s t age of a f a s t p i n c h . I t has a p i n c h d u r a t i o n of l e s s than 2ns, an e l e c t r o n t emperature between 2keV and 4keV, and an e l e c t r o n d e n s i t y of the o r d e r of l 0 2 O c m " 3 . Hard X-ray b u r s t s of 10ns FWHM are e m i t t e d by the plasma d u r i n g t h i s type of p i n c h . The p i n c h e d plasma i s a l s o v e r y s m a l l , t y p i c a l l y 40pm i n d i a m e t e r and 100pm i n l e n g t h . The f o r m a t i o n of the s u p e r f a s t p i n c h i s r e l a t e d t o the R a y l e i g h - T a y l o r i n s t a b i l i t y of the Z-pinch. I t i s proposed 125 t h a t the s u p e r f a s t p i n c h i s g e n e r a t e d as a r e s u l t of the n o n - l i n e a r growth of the R a y l e i g h - T a y l o r i n s t a b i l i t y . T h i s i n s t a b i l i t y i s examined i n d e t a i l and comparisons a r e made w i t h e x p e r i m e n t a l d a t a , i n p a r t i c u l a r the d i m e n s i o n s and the t i m i n g s of the s u p e r f a s t p i n c h , and good agreement o b t a i n e d . N u m e r i c a l c a l c u l a t i o n s , u s i n g the e x p e r i m e n t a l l y o b t a i n e d p i n c h d u r a t i o n s and t e m p e r a t u r e s , show t h a t the p i n c h i n g plasma i s i n a t r a n s i e n t s t a t e . T h i s , t o g e t h e r w i t h the i r r e p r o d u c i b l e m u l t i p l e p i n c h e s , render a- vacuum spark t o be of l i m i t e d use when a s p e c t r o s c o p i c measurement r e q u i r e s a c c u r a t e l y d e t e r m i n e d d e n s i t y or t e m p e r a t u r e . For example, a vacuum spark cannot be used as a s p e c t r o s c o p i c s o u r c e f o r the d e t e r m i n a t i o n of the o s c i l l a t o r s t r e n g t h or the S t a r k w i d t h of a p a r t i c u l a r i o n i c s p e c i e s . However, due t o i t s s i m p l i c i t y and r e l a t i v e ease of o p e r a t i o n , a vacuum spark i s s t i l l a u s e f u l t o o l f o r the p r o d u c t i o n and o b s e r v a t i o n of h i g h l y i o n i z e d plasma. ORIGINAL CONTRIBUTIONS My own c o n t r i b u t i o n s t o the vacuum spark a r e as f o l l o w s : I have dev e l o p e d a c y l i n d r i c a l shock wave model f o r the dynamics of a Z-pinch. T h i s model g i v e s a r e l i a b l e p r e d i c t i o n of the i m p l o s i o n time and p i n c h d u r a t i o n . I t a l s o g i v e s an o r d e r of magnitude r e s u l t f o r the f i n a l d e n s i t y and p r e s s u r e of the p i n c h e d plasma column. With 1 26 t h e s e r e s u l t s , the temperature reached by the Z-pinch can a l s o be c a l c u l a t e d . I have made c o n s i d e r a b l e improvements on the r e p r o d u c i b i l i t y of an e l e c t r i c a l l y t r i g g e r e d vacuum s p a r k . I have e x p l a i n e d t h a t the fundamental d i f f e r e n c e between a slow p i n c h and a f a s t p i n c h are i n t h e i r d i f f e r e n t i n i t i a l r a d i i . I have obse r v e d the s u p e r f a s t p i n c h u s i n g a f a s t response Rogowski c o i l . An e l e c t r o n t emperature measurement u s i n g m u l t i - f o i l t e c h n i q u e g i v e s a two tem p e r a t u r e component plasma, w i t h the f a s t p i n c h g i v i n g the lower temperature and the s u p e r f a s t p i n c h g i v i n g the h i g h e r t e m p e r a t u r e . I have proposed a s e r i e s of mechanisms f o r the g e n e r a t i o n of the s u p e r f a s t p i n c h , and have compared the proposed mechanisms w i t h e x p e r i m e n t a l measurements on the s u p e r f a s t p i n c h . 127 SUGGESTION FOR FURTHER WORK S i n c e the plasma r e s u l t i n g from the s u p e r f a s t p i n c h ' i s of c o n s i d e r a b l e i n t e r e s t because of i t s h i g h d e n s i t y and t e m p e r a t u r e , one would l i k e the s u p e r f a s t p i n c h t o be more r e p r o d u c i b l e . But as one cannot hope f o r r e p r o d u c i b l e r e s u l t s a f t e r the d i s r u p t i o n of the f i r s t p i n c h , the s u p e r f a s t p i n c h has t o be produced from the f i r s t p i n c h . One method t h a t can be t r i e d i s t o use a programmable c u r r e n t s o u r c e , which can g e n e r a t e e i t h e r an a d i a b a t i c c o m p r e s s i o n or a m u l t i p l e shock i m p l o s i o n , whose aim i s t o r a i s e the p r e s s u r e , d e n s i t y and temperature of the f i r s t p i n c h s u f f i c i e n t l y h i g h so t h a t any i n s t a b i l i t y i n the plasma column can d e v e l o p i n t o a s u p e r f a s t p i n c h . F u r t h e r d i a g n o s t i c s can then be a p p l i e d t o the r e p r o d u c i b l e f a s t p i n c h i n o r d e r t o study the c o n d i t i o n s f o r a s u p e r f a s t p i n c h t o d e v e l o p . F u r t h e r improvements can a l s o be made t o the shock model i n o r d e r t o t a k e i n t o c o n s i d e r a t i o n the e f f e c t s of n o n - u n i f o r m f l o w v e l o c i t y and volume convergence, a l t h o u g h t h i s would r e q u i r e d d e t a i l e d c a l c u l a t i o n s of the shock compressed r e g i o n . One method i s t o d i v i d e the compressed r e g i o n i n t o s h e l l s . Each s h e l l i s formed by the newly shock compressed plasma d u r i n g an increment i n time s t e p . C o n s e r v a t i o n e q u a t i o n s i n L a g r a n g i a n form can be used t o f o l l o w the time e v o l u t i o n s of these s h e l l s . V a l u a b l e computing time s h o u l d be saved by t h i s method when compared to a o n e - d i m e n s i o n a l s t a t i o n a r y g r i d method. 128 BIBLIOGRAPHY 1. A h l b o r n , B, Can.J.Phys. 53, 10, p976, 1975. 2. A h l b o r n , B. & S t r a c h a n , J.D., Can.J.Phys. 5J_, 13, p 1 4 1 6 , 1 973. 3. Ahmed, N. & Key, M.H., J.Phys. B5, p866, 1972. 4. A l f v e n , H. and Sanner, V.H., Nature 135, p l 8 0 , 1935. 5. A l l e n , J.E., Proc.Roy.Soc.B, V o l 70, J_, p24, 1957. 6. B e k e f i , G., Deutsch, C. & Y a a k o b i , B., " P r i n c i p l e s of L a s e r Plasma", J o h n - W i l e y , N.Y., 1976. 7. Bogen, P. i n "Plasma D i a g n o s t i c s " , ed. W.Lochte-Holtgreven, N o r t h . H o l l a n d , Amsterdam, 1 968. 8. Book, D.L., NRL Report 3332. 9. Book, D.L., O t t , E. & Lampe, M. , Phys F l u i d j_9, 12, P1982, 1976. 10. Buneman, 0., Phys.Rev. 115 3, p503, August 1959. 11. C h i l e s , J.A., J . A p p l . P h y s . 8, p622, Sept 1937. 12. C i l l i e r , W.A., D a t l a , R.U. & Griem, H.R., Phys.Rev. A12, 4, p1408, Oct 1975. 13. Cohen, L., Feldman, U., S w a r t z , M. & Underwood, J.H., J.Opt.Soc.Am. 58, 6, p843, June 1968. 14. C o u r a n t s , R. & F r i e d r i c h , K.O., " S u p e r s o n i c Flow and Shock Waves", I n t e r s c i e n c e P u b l i s h e r , N.Y., 1948. 15. Donaldson, T.P., Plasma Phys. 20, p1279, 1978. 16. E l t o n , R.C, NRL Report 6738, 1968. 17. E p s t e i n , H.M., G a l l a g h e r , W.J., M a l l o z z i , P . J. & S t r a t t o n , T.F., Phys.Rev. A2, 1, p.146, 1970. 129 18. Feldman, U., S w a r t z , M. & Cohen, L., A s t r o p h y s . J . 201, p225, Oct 1975. 19. F o w l e s , J.W., Phys.Rev. 64,7-8, p225, Oct 1943. 20. G o l a n t , V.E., Z h i l i n s k y , A.P. & Sakharov, I.E., "Fundamental of Plasma P h y s i c s " , J . W i l e y , N.Y., 1977. 21. Griem, H.R., "Plasma S p e c t r o s c o p y " , M c G r a w - H i l l , N.Y., 1 964. 22. Henke, B.L. & E l g i n , R.L., i n "Advances i n X-ray A n a l y s i s " , v o l 13, p638, Plenum N.Y., 1970. 23. K e l l y , R.L. & Palumbo, L . J . , NRL Report 7599, 1973. 24. K r u s k a l , M. & S c h w a r z c h i I d , M., Proc.Roy.Soc. (London) A223, p348, 1954. 25. Jahoda, F.C., L i t t l e , E.M., Quinn, W.E., Sawyer, G.A. & S t r a t t o n , T.F. Phys.Rev. 119, p843, 1960. 26. J u k e s , J.D., Phys F l u i d 4, 12, p1527, 1961. 27. Lee, T.N., A s t r o p h y s . J . J_90, p467, June 1 974. 28. L i e , T.N. & E l t o n , R . C , Phys.Rev. A3, 3, p865, March 1971. 29. M c W h i r t e r , R.W.P. i n "Plasma D i a g n o s t i c s " , Chapter 5, ed. R.H.Huddleston & S.L.Leonard, Academic P r e s s , N.Y., 1965. 30. M e y e r o t t , A . J . , F i s h e r , P.C. & R o e t i n g , D.T., R e v . S c i . I n s t u m . 3 5 ( 6 ) , p669, 1964. 31. M i l l i k a n , R.A. & Bowen, J.A., Phys.Rev. X I I 2, p167, 1918. 32. Negus, C R . & Peacock, N.J., J.Phys. D j_2, p 9 l , 1979. 33. P e l l i n e n , D.C, D i Capua, M.S., Sampayan, S.E., G e r b r a c h t , H. & Wang, M., R e v . S c i . I n s t r u m . 5 1 ( 1 1 ) , P1535, 1980. 34. P o p i l , R. & Meyer, J . , J . A p p l . P h y s . 5 2 ( 9 ) , p5846, 1981. 35. R o s e n b l u t h , M., Report #LA-1850, Los Alamos S c i e n t i f i c L a b o r a t o r y , (New M e x i c o ) , 1955. 36. R o s t o k e r , N . , N u c l e a r F u s i o n J_, p 1 0 l , 1961. 37. Samson, J.A.R., "Techniques of Vacuum U l t r a v i o l e t S p e c t r o s c o p y " , J . W i l e y , N.Y., 1967. • 130 38. Schwob, J.L. & F r a e n k e l , B.S., P h y s . L e t t 40A, 1, p81, 1972. 39. S h e a r e r , J.W., P h y s . F l u i d s J_9, p1426, Sept 1976. 40. Sh.iloh, J.H., Ph. D. T h e s i s , U n i v e r s i t y of C a l i f o r n i a , I r v i n e , 1978. 41. S i n h a , B.K & G o p i , N., P h y s . F l u i d s 2 3 ( 8 ) , p l 7 0 4 , 1980. 42. S l i v k o v , I.N. & M i k h a y l o v , V . I . , " E l e c t r i c a l Breakdown and D i s c h a r g e i n a Vacuum", t r a n s l a t e d by USAF F o r e i g n Technology D i v i s i o n , 1966. 43. S p i t z e r , L. & Harm, R. , Phys.Rev. I39, 5, p977, 1953. 44. S t a n y u k o v i c h , K.P., "Unsteady Motion of Co n t i n u o u s Media", Pergamon P r e s s , N.Y., 1960. 45. S t r i g a n o v , A.R. & S v e n t i t s k i i , N.S., "Tab l e s of S p e c t r a l L i n e s of N e u t r a l and I o n i z e d Atoms", Plenum, N.Y., 1 968. 46. Suprunenko, V.A., F a i n b e r g , Y.B., T o l o k , V.T., Sukhomlin, E.A., Reva, N.I., Burchenko, P.Y., Rudev, N.I. & V o l k o v , E.D., Plasma P h y s i c s , J . N u c l . E n e r g y P a r t C, 7, p297, 1965. 47. Turechek, J . , T e c h n i c a l r e p o r t #73-083, U n i v e r s i t y of M a r y l a n d , Department of P h y s i c s and Astronomy, 1972. 48. Welch, T.J. & C l o t h i a u x , E . J . , J . A p p l . P h y s . 4_5, 9, p3825, 1974. 49. Whitham, G.B., " L i n e a r and N o n l i n e a r Waves", John W i l e y and sons, N.Y., 1973. 50. W i n t e r b e r g , F., Z.Physik A 284, p43, 1978. 51. Wood, R.W., Phys.Rev. V 1, p.1, 1897. 52. Wyld, H.W., J . A p p l . P h y s . 29, 10, p1460, 1958. 53. Z e l ' d o v i c h , Y.B. and R a z i e r , Y.P., " P h y s i c s of Shock Wave and Hig h - t e m p e r a t u r e Hydrodynamic Phenomena", Vo l . 1 & I I , Acedemic P r e s s , N.Y., 1960. 131 APPENDIX A DETERMINATION OF ELECTRON TEMPERATURE Because of the h i g h temperature i n a vacuum spark plasma, most of the e m i s s i o n i s i n the vacuum u l t r a v i o l e t and the X-ray r e g i o n s . The f o i l a b s o r p t i o n t e c h n i q u e (Jahoda e l a t [ 2 5 ] ) i s used i n our experiment t o determine the plasma e l e c t r o n temperature from i t s continuum e m i s s i o n . T h i s method i s used because of the s m a l l s i z e of the plasma and i t s i r r e p r o d u c i b i l i t y e x c l u d e s the use of more a c c u r a t e methods such as Thomson s c a t t e r i n g . I t g i v e s a more r e l i a b l e r e s u l t than l i n e r a d i a t i o n measurements, because of the much s h o r t e r t h e r m a l i z a t i o n time of the f r e e e l e c t r o n s . F u r t h e r m o r e , the M a x w e l l i a n d i s t r i b u t i o n of the f r e e e l e c t r o n s are not as e a s i l y d i s t u r b e d by r a d i a t i o n l o s s e s as the Saha and Boltzmann d i s t r i b u t i o n s of the bound s t a t e s . The measured e m i s s i o n s can be due t o b r e m s s t r a h l u n g ( f r e e - f r e e t r a n s i t i o n s ) or r e c o m b i n a t i o n s ( f r e e - b o u n d t r a n s i t i o n s ) , p r o v i d e d t h a t the e l e c t r o n s have a M a x w e l l i a n d i s t r i b u t i o n . The c o n t r i b u t i o n from r e c o m b i n a t i o n i s e s t i m a t e d by c o n s i d e r i n g o n l y the r e c o m b i n a t i o n s w i t h ground s t a t e i o n s . 1 32 These e m i s s i o n c o e f f i c i e n t s f o r b r e m s s t r a h l u n g and r e c o m b i n a t i o n a re g i v e n by G r i e m [ 2 l ] where k i s Boltzmann's c o n s t a n t , n e i s the e l e c t r o n number Z d e n s i t y ( c m " 3 ) , T i s the e l e c t r o n t e m p e r a t u r e ( e V ) , N i s the i o n number d e n s i t y ( c m " 3 ) of charge Z, A i s the photon wav e l e n g t h , E_ i s the i o n i z a t i o n p o t e n t i a l i n eV of an Z, n e l e c t r o n i n the n - t h s h e l l of an i o n of charge s t a t e Z, n i s the l o w e s t l e v e l t h a t has a vacancy f o r r a d i a t i v e r e c o m b i n a t i o n and g_„, g„ a r e the Gaunt f a c t o r s f o r f f fb b r e m s s t r a h l u n g and r e c o m b i n a t i o n , r e s p e c t i v e l y . Hydrogenic a p p r o x i m a t i o n s a r e a l s o used. To c a l c u l a t e the ground s t a t e i o n d e n s i t i e s , the stea d y s t a t e corona e q u i l i b r i u m i s assumed. I f the e m i t t e d i n t e n s i t y i s I. , then the i n t e n s i t y I ' A A t h a t i s t r a n s m i t t e d through an a b s o r b e r f o i l of t h i c k n e s s D and a b s o r p t i o n c o e f f i c i e n t K i s then hc/X (Acm e r g / s t r ) )(A.1) I ' = I e X X - K D (A.2) 1 33 The a n a l y t i c a l e x p r e s s i o n f o r t h e mass a b s o r p t i o n c o e f f i c i e n t s f o r the c a l c u l a t i o n s a re ( X i n Angstrom) K = 4. 776-7. 32X + 3. 84X 2 + 13. 84X 3-0. 755X1* cm*/g , X <7.9 z (A.3) = 42-35X+6X 2 + 0.3838Xl* cm /g , X>7.9 . These p o l y n o m i a l s a r e o b t a i n e d from l e a s t square m i n i m i z a t i o n t o the t a b u l a t e d c o e f f i c i e n t s between lOOeV and 12keV o b t a i n e d from Henke[22]. T h i s e x p r e s s i o n has reduced the v a l u e of • x 2 by over 50% from the p r e v i o u s a n a l y t i c a l e x p r e s s i o n s [ 3 , 3 5 ] as they a r e found t o g i v e e r r o n e o u s l y h i g h temperature a t t h i n n e r a b s o r b e r f o i l t h i c k n e s s e s . The i n t e g r a t e d i n t e n s i t y over the wavelength between 0.1A and 50A have been computed f o r aluminum f o i l s of d i f f e r e n t t h i c k n e s s e s i n microns and f o r copper and carbon plasma a t d i f f e r e n t e l e c t r o n t e m p e r a t u r e s i n eV, i n which s t e a d y s t a t e c o r o n a e q u i l i b r i a a r e assumed. The r e s u l t s a r e g i v e n i n Fig.A-1 and Fig.A-2 w h i l e t h e i r r e l a t i v e , e m i s s i v i t y ( n o r m a l i z e d t o carbon plasma a t 80eV) i s g i v e n i n F i g . A - 3 . The a c t u a l e m i s s i o n can be o b t a i n e d by m u l t i p l y i n g the e m i s s i v i t y from Fig.A-3 w i t h the t r a n s m i t t e d i n t e n s i t y from Fig.A-1 or A-2 and w i t h the average charge s t a t e of the plasma t o account f o r the e x t r a e l e c t r o n s from h i g h e r i o n i z a t i o n s t a g e s . The e x p e r i m e n t a l plasma e l e c t r o n t emperature can then be d e t e r m i n e d by measuring the t r a n s m i s s i o n of X-ray 1 34 continuum through d i f f e r e n t f o i l t h i c k n e s s e s and comparing them w i t h the s l o p e s of the c a l c u l a t e d t r a n s m i s s i o n s a t d i f f e r e n t t e m p e r a t u r e s . In t h e s e c a l c u l a t i o n s the c o n t r i b u t i o n s due t o l i n e e m i s s i o n s a re i g n o r e d . S i n h a [ 3 6 ] has e s t i m a t e d t h e s e c o n t r i b u t i o n s a t an e l e c t r o n t emperature of lOOeV f o r copper. He found them t o be a t l e a s t 10 ti m e s l e s s than the r e c o m b i n a t i o n i n t e n s i t i e s and are l o c a t e d o u t s i d e the passband of the aluminum a b s o r b e r used. T h e r e f o r e f o r low temperatures(<400eV) the l i n e r a d i a t i o n can be n e g l e c t e d . However, a t tempe r a t u r e s i n e x c e s s of 1keV, the average i o n i z a t i o n i s a p p r o x i m a t e l y 22, and s t r o n g l i n e e m i s s i o n of these copper i o n s o c c u r s a t around 10A[23], which i s the maximum t r a n s m i s s i o n r e g i o n of the aluminum a b s o r b e r . Due t o the l a c k of e x p e r i m e n t a l d a t a t h e i r c o n t r i b u t i o n s cannot be e a s i l y c a l c u l a t e d and a r e i g n o r e d i n the p r e s e n t c a l c u l a t i o n s . One s h o u l d t h e r e f o r e bear i n mind t h a t t h i s w i l l r e s u l t i n the i n f e r r e d t e mperature b e i n g h i g h e r than the t r u e t e m p e r a t u r e . Carbon plasma temperature measurement on the o t h e r hand does not have t h i s d i f f i c u l t y as t h e r e a r e no carbon l i n e s i n the passband of the aluminum f i I t e r . The computer program used t o ge n e r a t e t h e s e r e s u l t s i s g i v e n i n the end of t h i s a p p e n d i x . The X-ray e m i t t i n g plasma a r e assumed t o be i n a corona e q u i l i b r i u m , and the i o n i z a t i o n e n e r g i e s of the plasma a re read i n t o the program i n f r e e format from L o g i c a l U n i t Number 4. 135 F i g . A - 1 ; R e l a t i v e t r a n s m i s s i o n of .copper plasma e m i s s i o n .through aluminum f o i l s . 1 36 E i g . v A - 2 " R e l a t i v e t r a n s m i s s i o n of carbon plasma r e m i s s i o n t h r o u g h .-aluminum f o i l s . 137 F i g . A-3 R e l a t i v e e m i s s i v i t y of carbon and copper plasma as a f u n c t i o n of t e m p e r a t u r e . ( n o r m a l i z e d t o e m i s s i v i t y of carbon a t 80eV) 138 2 C 3 C Program to c a l c u l a t e the r a t i o of transmitted 4 C i n t e n s i t y to emitted i n t e n s i t y from a hot 5 C plasma in corona e q u i l i b r i u m . 6 C 7 C A p r i l , 1982 8 >C R. Fong 9 C Plasma Physics 10 C UBC 1 1 C 12 C Absorber f o i l used i s Aluminum, thickness 13 C from 6 to 400um. 14 C i n t e n s i t y i n t e g r a t e d from 0.1A to 50A i n 15 C - steps of 0.1A. 16 C 17 C Plasma i o n i z a t i o n energy i s read in from LUN4 18 C data in free format. i:9iIIII|M 20 C r e f : 1. Griem, H.R."Plasma Spectroscopy", 21 C McGraw H i l l , N.Y.,1964. 22 C 2. Henke, B.L. & R.L. E l g i n , in 23 C "Advances in X-ray A n a l y s i s " 24 C Plenum, N.T.,1970, vol.13,p.639 25 C 3. Book, D., NRL Report 3332 28 C 29 REAL ENERGY(50), DEN(50), WAVE(500), ABSOR(500), TEN(10) 30 REAL RAY(500), TENNOR(10) 31 REAL MICRO, MU 32 INTEGER Z(50) 33 INTEGER FMT(1) /'*'/ 34 C 35 C ********* 36 C *********Change the following l i n e s for d i f f e r e n t f o i l thickness 37 C ********* NOFOIL = number of f o i l s used 3B Q ********* 40 REAL T(20) / 12.5, 18.5, 25., 37.5, 50./ 41 NOFOIL = 5 42 C 43 C input temperature ;4:4*IKlltt 45 CALL FTNCMD('DEFAULT 8=*SINK*1, 16) 46 WRITE (8,10) 47 10 FORMAT (' Temperature=?') 48 READ (5,FMT) TEMP 49 CALL ABSORB(ABSOR, WAVE) 50 CALL CORONA(TEMP, Z, ENERGY, DEN, N) 51 RNM = 0.0 52 C 53 C c a l c u l a t e t o t a l emission 54 C I=integration over wavelength 55 C J=summation over i o n i c species I56IIIIIM 57 DO 30 I « 1, 500 58 EMM =0.0 59 DO 20 J = 2, N 60 CALL EMITON(Z(J), N, TEMP, WAVE(I), ENERGY, CONT) 139 61 EMM = EMM + CONT * DEN(J) 62 20 CONTINUE 63 RAY(I) = EMM 64 RNM = RNM + EMM 65 30 CONTINUE 66 WRITE (6,40) TEMP, RNM 4/ 67 40 FORMAT ('-Relative emission at ', F8.2, 'eV i s , E l l . 68 1 '0 Thickness I n t e n s i t y ' / ' ') 69 DO 70 L = 1, NOFOIL MICRO = T(L) 70 71 CALL THICK(MICRO, MU) 72 C 73 C c a l c u l a t e transmitted i n t e n s i t y 74 C 75 ETOT = 0.0 76 DO 50 I = 1, 500 77 ETOT = ETOT + RAY(I) * EXP(-MU*ABSOR(I)) 78 50 CONTINUE 79 TEN(L) = ETOT 80 TENNOR(L) = TEN(L) / RNM 81 WRITE (6,60) MICRO, TENNOR(L) 82 60 FORMAT (' ', F10.2, 3X, E11.4) 83 70 CONTINUE 84 STOP 85 86 END 87 C 88 c C a l c u l a t e s the absorption c o e f f i c i e n t s 89 c using an a n a l y t i c a l expression[2]. 90 c I n i t i a l i z e d the wavelength i n t e g r a t i o n 91 c 92 c steps. 93 c 94 c . Parameters: entry-none 95 c e x i t -ABSOR absorption coef. 96 c -WAVE wavelengths 97 c 98 c 99 SUBROUTINE ABSORB(ABSOR, WAVE) 100 REAL ABSOR(500), WAVE(500) 101 DO 30 I = 1, 500 102 WAVE(I) = I * 0.1 103 IF (WAVE(I) .LE. 7.9) GO TO 10 104 IF (WAVE(I) .GT. 7.9) GO TO 20 105 10 ABSOR(I) = 4.77 - 7.32 * WAVE(I) + 3.94 * WAVE(I) ** 2 106 ABSOR(I) = ABSOR(I) + 13.841 * WAVE(I) ** 3 107 ' 1 - 0.75539 * WAVE(I) ** 4 108 GO TO 30 * WAVE(I) 109 20 ABSOR(I) = 42.11 - 35.03 * WAVE(l) + 6.00 ** 110 ABSOR(I) - ABSOR(I) + 0.3838 * WAVE(I) ** 3 1 1 1 30 CONTINUE 1 12 RETURN 113 END 1 1 4 115 C Converts thickness from um, to gm/cm2. 116 C 117 C 118 c Parameters: entry-MICRO micron 119 c e x i t -MU gm/cm2 120 c 1 40 121 C SUBROUTINE THICK{MICRO, MU) 122 123 REAL MICRO, MU 124 MU = 2.702 * MICRO * 1E-4 125 RETURN 126 END 127 128 C 129 c Converts energy u n i t s from Angstrom to 1 30 c eV 131 c 132 c Parametes: entry-ANG angstrom 1 33 c e x i t -TEV eV 134 c 135 c 136 SUBROUTINE WAVTOV(ANG, TEV) 137 TEV = 1.2399E4 / ANG 138 RETURN 139 END 140 141 c C a l c u l a t e s ion d i s t r i b u t i o n using 142 c 143 c corona e q u i l i b r i u m 144 c I o n i z a t i o n energy of ion i s read from 145 c LUN4 in free format 146 c 147 c Parameters: entry-TEMP temperature 1 48 c e x i t -Z charge state of ion 1 49 c -ENERGY i o n i z a t i o n energy 1 50 c -DEN ionix density 151 c -N no. of species 152 c (neutral+ionic) 153 c 1 54 c SUBROUTINE CORONA(TEMP, Z, ENERGY, DEN, N) 155 1 56 REAL ENERGY(50), DEN(50) 157 REAL COL(50), REC(50) 158 INTEGER Z(50) 159 INTEGER FMT(1) /'*'/ 160 N = 0 161 10 READ (4,FMT,END=20) ENERGY(N + 1) 162 N = N + 1 163 GO TO 10 164 c 165 c i n i t i a l i z e n e u t r a l atom 166 c 167 20 DEN(1) = 1E-50 168 c get c o l l i s i o n i o n i z a t i o n ^ r a t e 169 c 170 c get r a d i a t i v e recombination rate[3] 171 c 172 DO 30 I = 1 , N COL(I) = RION(ENERGY(I),TEMP) 173 174 REC(I) = RAD(ENERGY(I),TEMP,I) 175 30 CONTINUE 176 RNORM = 0.0 177 c c a l c u l a t e r a t i o of i o n i c d e n s i t i e s 178 c 179 c 180 DO 40 I = 1, N 141 181 1 8 2 1 8 3 4 0 184 C 1 8 5 C 1 8 6 C 1 8 7 1 8 8 1 8 9 1 9 0 191 1 9 2 5 0 1 9 3 194 1 9 5 1 9 6 C 1 9 7 c 1 9 8 c 1 9 9 c 2 0 0 c 2 0 1 c 2 0 2 c 2 0 3 c 2 0 4 2 0 5 2 0 6 2 0 7 2 0 8 2 0 9 2 1 0 211 c 2 1 2 c 2 1 3 c 2 1 4 c 2 1 5 c 2 1 6 c 2 1 7 c 2 1 8 c 2 1 9 2 2 0 221 2 2 2 2 2 3 2 2 4 2 2 5 2 2 6 , c 2 2 7 c 2 2 8 c 2 2 9 c 2 3 0 c 231 c 2 3 2 c 2 3 3 c 2 3 4 c 2 3 5 c 2 3 6 c 2 3 7 c 2 3 8 c 2 3 9 2 4 0 D E N ( I + 1 ) = C O L ( I ) / R E C ( I ) * D E N ( 1 ) RNORM = R N O R M + D E N ( I + 1 ) C O N T I N U E n o r m a l i z e i o n i c d e n s i t i e s R N O R M = R N O R M + D E N ( 1 ) : N = N + 1 DO 5 0 I = 1 , N D E N ( I ) = D E N ( I ) / R N O R M Z ( I ) = 1 - 1 C O N T I N U E R E T U R N E N D C a l c u l a t e s c o l l i s i o n a l i o n i z a t i o n r a t e [ 3 ] P a r a m e t e r : e n t r y - T t e m p e r a t u r e - E i o n i z a t i o n e n e r g y e x i t - R I O N r a t e R E A L F U N C T I O N R I O N ( E , T ) X = T / E R I O N = ( 1 E - 5 ) * S Q R T ( X ) * E X P ( - 1 . / X ) R I O N = R I O N / ( E * * 1 . 5 * ( 6 + X ) ) R E T U R N E N D C a l c u l a t e r a d i a t i v e r e c o m b i n a t i o n r a t e [ 3 ] P a r a m e t e r s : e n t r y - T t e m p e r a t u r e - E i o n i z a t i o n e n e r g y e x i t - R A D r a t e R E A L F U N C T I O N R A D ( E r T , I ) X = E / T R A D = 5 . 2 E - 1 4 * I * S Q R T ( X ) R A D = R A D * ( 0 . 4 3 + 0 . 5 * A L O G ( X ) + 0 . 4 6 9 * X * * ( - 1 . / 3 . ) ) R E T U R N ..END..... C a l c u l a t e r e l a t i v e e m i s s i v i t y f r o m b r e m s t r a h l u n g a n d r e c o m b i n a t i o n [ 1 ] o f c o p p e r p l a s m a u s i n g h y d r o g e n i c a p p r o x i m a t i o n P a r a m e t e r s : e n t r y - Z S c h a r g e s t a t e T E M P t e m p e r a t u r e E N E R G Y i o n i z a t i o n e n e r g y A N G w a v e l e n g t h o f e m i s s i o n e x i t - C O N T i n t e n s i t y o f e m i s s i o n S U B R O U T I N E E M I T O N ( Z S , I O N , T E M P , A N G , E N E R G Y , C O N T ) R E A L E N E R G Y ( 5 0 ) 1 42 241 I N T E G E R Z S , I O N 242 C A L L W A V T O V ( A N G , T E V ) 243 C 244 C c a l c u l a t e p r i n c i p l e q u a n t u m n u m b e r 245 c 246 N D I F F = I O N - ZS - 1 247 248 I F ( N D I F F . G T . 2) N = 2 249 I F ( N D I F F . G T . 10) N = 3 250 I F ( N D I F F . G T . 28) N = 4 251 F B = 0.0 252 DO 10 I = N , 8 253 E N G Y = E N E R G Y ( Z S ) / I * * 2 * N **2 254 I F ( T E V , L T . E N G Y ) GO T O 10 255 F B = F B + 2 * E N G Y / T E M P / I * E X P ( E N G Y / T E M P ) 256 10 C O N T I N U E 257 20 F F = 1 . 258 CONT = F F + F B 259 C O N T = C O N T / T E M P * * 0.5 * Z S * * 2 * E X P ( - T E V / T E M P ) 260 C O N T = C O N T / A N G * * 2 261 R E T U R N 262 E N D E n d o f F i l e 1 43 APPENDIX B SHOCK DYNAMICS The i n t e g r a l form of the c o n s e r v a t i o n e q u a t i o n s f o r mass, momentum and energy a c r o s s a shock f r o n t a r e [ l ] P v K o o • p l v l P o + p o V o = P l + p l V l (B.1a) (B.1b) (B.1c) where P, u, p and h a r e the d e n s i t y , the p a r t i c l e v e l o c i t y , the p r e s s u r e and the e n t h a l p y rn the s t a t i o n a r y f r o n t frame. The s u b s c r i p t s 0 and 1 r e f e r t o the c o n d i t i o n s ahead of and behi n d the f r o n t r e s p e c t i v e l y . 144 The above e q u a t i o n s can be r e a r r a n g e d ! 1 ] as P i -=. = • (B.2a) Pn I1 = r-= 1 + 'T*r ^  9 (B-2b) .,2 g = 1L____ (B.2c) 2.(g_*.l.) (g0-8I)M2.g0 ( « o - 1 H 8 l - g o M ^ Z (B.2d) (B.2e) where e i s always s m a l l . M i s the Mach number of the shock and g's a r e the e n t h a l p y c o e f f i c i e n t s . 1 45 In the frame of r e f e r e n c e i n which the gas ahead of the f r o n t i s s t a t i o n a r y , the p a r t i c l e v e l o c i t y i s u = v -v - (B.3) o 1 g M * The thermodynamics of the medium i s c o n t a i n e d i n the e n t h a l p y c o e f f i c i e n t . I t i s d e f i n e d as the r a t i o of the e n t h a l p y h t o the i n t e r n a l energy u. g u -r»™ .3, (B.4) £ N . ( E . + f k T ) • where N i are the number d e n s i t y of the i - t h s p e c i e s of i o n s , E| i s the i o n i z a t i o n energy of the i - t h i o n i c s t a t e s . g v a r i e s whenever the r a t i o of the t h e r m a l k i n e t i c energy t o the i o n i z a t i o n energy changes. The v a r i a t i o n of g as a f u n c t i o n of temperature f o r copper plasma i s p l o t t e d i n F i g . B - 1 . The v a r i o u s d i p s i n the c u r v e a r e due t o l a r g e d i f f e r e n c e s i n i o n i z a t i o n e n e r g i e s between d i f f e r e n t p r i n c i p a l quantum numbers. Note t h a t i t behaves o n l y as a monatomic i d e a l gas a t temperature above 1MeV. The e f f e c t s of a c o n v e r g i n g shock wave can be c o n s i d e r e d by u s i n g the d i f f e r e n t i a l form of the c o n s e r v a t i o n e q u a t i o n s i n r e g i o n s w i t h o u t d i s c o n t i n u i t y . 146 The q u a s i - o n e d i m e n s i o n a l forms w i t h no heat s o u r c e or s i n k a r e Ft 9r P3T ^ TT ~ ' ( B ' 5 A ) 3u 3t 3r p 3r (B.5b) If * -If - 4 H * -HI - ° • ( B- 5 c ) Here A i s an e l e m e n t a l a r e a . These e q u a t i o n s can be m a n i p u l a t e d i n t o the c h a r a c t e r i s t i c form dp du ^ 2pu 3A n at i E at + a h 37 " 0 • { B - 6 ) a l o n g the C+ and C- c h a r a c t e r i s t i c s . 147 Whitham[49] showed t h a t f o r s u f f i c i e n t l y s t r o n g shock, t h e C+ c h a r a c t e r i s t i c s can be a p p l i e d t o the shock f r o n t t o o b t a i n the shock s t r e n g t h e n i n g due t o a r e a convergence XMdM _ dA _ n — * — ~ 0 ' " (33.7) 2 g l A 8 2 where x = ( 2 o + 1 + — - i - ) ( 1 + - 2 — r ^ r - ) g g l ° (g - l ) M 2 + 2 a 2 = — 1 2 g l M 2 - ( g l - l ) For v e r y s t r o n g shock t h i s becomes K - 0 , (B.8) S M A 148 F i g . B-1 V a r i a t i o n of e n t h a l p y c o e f f i c i e n t g as a f u n c t i o n of t e m p e r a t u r e f o r copper plasma. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085775/manifest

Comment

Related Items