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Confinement of a Z-pinch plasma with cold gas end plugs Milne, Andrew F. 1984

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C o n f i n e m e n t o f a Z - P i n c h Plasma W i t h C o l d Gas End P l u g s by Andrew F. M i l n e B.A.Sc, U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1980 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE 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 a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A u g u s t 1984 © Andrew F. M i l n e , 1984 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . ^ ^ Physics Department of The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 22 August 1984 i i ABSTRACT Plasma e s c a p i n g f r o m t h e ends o f a Z - p i n c h c a n be s l o w e d by a p l u g o f c o l d g a s . Model c a l c u l a t i o n s p r e d i c t t h a t i n c r e a s i n g t h e p l u g d e n s i t y w i l l r e d u c e t h e p l a s m a m o t i o n , a l t h o u g h e x p e r i m e n t s show t h a t t h i s e f f e c t i s g a i n e d a t t h e c o s t o f g r e a t e r h e a t l o s s . The h e l i u m p l a s m a u s e d i n t h i s work has a t e m p e r a t u r e o f 35000K and a d e n s i t y of 8 X 1 0 2 2 m"3. The measured h e a t f l u x i n t o a h e l i u m end p l u g a t 300K and 525 p a s c a l s i s 130 MW.rn"2. Outward m o t i o n of t h e p l a s m a adds a m e c h a n i c a l f l u x of 830 MW.rn"2. I n c r e a s i n g t h e p l u g d e n s i t y by a f a c t o r o f f o u r r e d u c e s t h e m e c h a n i c a l f l u x t o 200 MW.rn-2, but i n c r e a s e s t h e h e a t f l u x t o 420 MW.rn"2. The c o r r e s p o n d i n g h e a t l o s s t o a s o l i d e l e c t r o d e i s a p p r o x i m a t e l y 400 MW.rn"2. In te r m s o f t h e t o t a l e n e r g y l o s t , t h e " w a l l o f g a s " i s n e a r l y a s e f f e c t i v e a s a s o l i d b a r r i e r . L i n e a r s y s t e m s a t h i g h e r t e m p e r a t u r e s c o u l d a v o i d t h e i m p u r i t i e s c o n t r i b u t e d by p l a s m a / s o l i d i n t e r a c t i o n by r e p l a c i n g t h e s o l i d w i t h a c o l d gas end p l u g . i i i TABLE OF CONTENTS ABSTRACT i i L I S T OF TABLES v L I S T OF FIGURES v i ACKNOWLEDGEMENTS v i i i I . INTRODUCTION 1 1.1 The L i n e a r M a g n e t i c F u s i o n C o n c e p t 3 1.2 End C o n f i n e m e n t As An I n i t i a l V a l u e P r o b l e m 5 1 . 3 O u t l i n e Of The E x p e r i m e n t a l Program 7 I I . THEORETICAL DISCUSSION 9 I I . 1 The R o l e Of F o r m a l M o d e l s 10 1 1 . 2 D e f i n i t i o n Of The Plasma S t a t e 12 1 1 . 3 O b s e r v a t i o n Of S p e c t r a l L i n e s 19 1 1 . 4 C a l c u l a t i o n Of The L i n e Shapes 24 1 1 . 5 T e s t i n g The A d i a b a t i c L i m i t 28 1 1 . 6 T e m p e r a t u r e And D e n s i t y Measurement 32 1 1 . 7 F o r m a l i z i n g The I n i t i a l V a l u e P r o b l e m 35 1 1 . 8 The Shock Tube Model 43 1 1 . 9 D e r i v a t i o n Of The U n i f o r m Flow M o d e l 46 1 1 . 1 0 P r e d i c t i o n Of Improved End C o n f i n e m e n t 50 1 1 . 1 1 Summary 51 I I I . EXPERIMENTAL SETUP 54 1 1 1 . 1 The C o m p l e t e System 55 1 1 1 . 2 The Z - P i n c h D i s c h a r g e 56 1 1 1 . 3 The Gas P u f f V a l v e 60 i v 111.4 The S t r e a k Camera 63 111.5 The 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 65 111.6 Other E x p e r i m e n t a l A p p l i a n c e s 67 111.7 T i m i n g And S y n c h r o n i z a t i o n 71 I I I .8 Summary 74 IV. DATA AND OBSERVATIONS 79 IV. 1 Comparison W i t h P r e v i o u s Work 80 IV.2 S t r e a k Photographs Of The E s c a p i n g Plasma 82 IV.3 H e l i u m L i n e S p e c t r a 85 IV.4 I n t e r a c t i o n W i t h A H i g h e r D e n s i t y Gas P l u g 90 IV. 5 P r e p a r a t i o n F o r Computer A n a l y s i s .94 V. ANALYSIS 96 V. 1 Plasma Temperature 97 V.2 Plasma D e n s i t y 103 V.3 Comparison W i t h Model P r e d i c t i o n s 110 V.4 C a l c u l a t i o n Of Heat T r a n s f e r 111 V. 5 Summary 115 V I . CONCLUSIONS 117 REFERENCES 120 V L I S T OF TABLES I I - 1 Model N o m e n c l a t u r e 41 I I - 2 Summary Of N o m i n a l P a r a m e t e r s 42 111 — 1 E x p e r i m e n t a l E q u i p m e n t 75 I I I - 2 E x p e r i m e n t a l C o n d i t i o n s 78 V-1 Summary Of E x p e r i m e n t a l R e s u l t s 116 v i L I S T OF FIGURES I - 1 P l a s m a And Gas I n i t i a l S t a t e 5 II -1 U n i f o r m Flow S t r u c t u r e 38 I I - 2 The S i m p l e Shock Wave 43 I I - 3 E n e r g y B a l a n c e In The Shock R e g i o n 48 I I - 4 The Shock Wave W i t h Heat F l u x P r e s e n t 50 I I I -1 Z - P i n c h , V a l v e And Shock Tube 54 I I I - 2 P h o t o g r a p h Of The A p p a r a t u s 55 111-3 D i s c h a r g e C i r c u i t 56 I I I - 4 P u f f V a l v e 61 I I I - 5 Gas V a l v e F i r i n g C i r c u i t 62 I I I - 6 S t r e a k Camera And O p t i c s 63 111-7 R o t a t i n g The S l i t Image 64 I I I - 8 OMA And C o l l e c t i o n O p t i c s 66 111-9 Vacuum And HV Systems 69 111-10 T i m i n g C i r c u i t I 72 111 — 11 T i m i n g C i r c u i t I I 73 I I I - 12 Common T i m i n g Sequence 74 IV- 1 Z - p i n c h B e i n g F i r e d 79 IV-2 R a d i a l C o l l a p s e Of Plasma 81 IV-3 D i s c h a r g e C u r r e n t 82 IV-4 25mm Tube - 120mm S e c t i o n 84 IV-5 25mm Tube - 40mm S e c t i o n 84 IV-7 OMA G a i n P r o f i l e 86 IV-6 OMA Dark C u r r e n t 86 v i i IV-8 Raw And P r o c e s s e d D a t a V e c t o r s 88 IV-9 E v o l u t i o n Of S p e c t r a l L i n e s 89 IV-10 Gas P u f f P r e s s u r e P r o f i l e 91 IV-11 I n t e r a c t i o n W i t h The D e n s i t y S t e p 92 IV- 12 A S t e p C l o s e r I n 93 V- 1 S o l i d End L i n e I n t e n s i t i e s 97 V-2 I n t e n s i t y C r o s s - C o r r e l a t i o n s 99 V-3 Open-End M i d p o i n t L i n e R a t i o s 100 V-4 Open-End Q u a r t e r p o i n t L i n e R a t i o s ..101 V-5 S o l i d - E n d M i d p o i n t L i n e R a t i o s 102 V-6 T e m p e r a t u r e s Compared 102 V-7 M i d p o i n t D e n s i t y F o r B o t h E l e c t r o d e T y p e s 105 V-8 Open-End M i d p o i n t And Q u a r t e r p o i n t D e n s i t i e s 105 V-9 Plasma P r e s s u r e s 106 V-10 C r o s s - C o r r e l a t i o n Of Open-End P r e s s u r e s 107 V-11 Combined T e m p e r a t u r e , D e n s i t y And C u r r e n t 108 V-12 Heat F l u x V e r s u s P l u g D e n s i t y 113 v i i i ACKNOWLEDGEMENTS My i n t e r e s t i n c o n t e m p o r a r y p h y s i c s was f i r s t s t i m u l a t e d by P e t e r M a r t i n , whose l e c t u r e s on quantum m e c h a n i c s b r i g h t e n e d up a d r e a r y f o u r t h y e a r i n e l e c t r i c a l e n g i n e e r i n g . My s u b s e q u e n t work on t h e Z - p i n c h was s u g g e s t e d and s u p e r v i s e d by Boye A h l b o r n . Mark D a v i s b u i l t t h e gas p u f f v a l v e , R o s s Nogami d i d t h e s p e c t r o s c o p y , and Ron F l e t c h e r t o o k many o f t h e s t r e a k p h o t o g r a p h s . Susan M a i r o f t h e UBC Computing C e n t r e h e l p e d w i t h APL and g r a p h i c s s o f t w a r e . U s e f u l comments on t h e m a n u s c r i p t were made by Ross W a l k e r and F r a n k C u r z o n , t h e l a t t e r i n p a r t i c u l a r g i v i n g me t h e b e n e f i t of h i s many y e a r s e x p e r i e n c e i n Z - p i n c h r e s e a r c h . My u n d e r s t a n d i n g o f p h y s i c s w o u l d n o t be c o m p l e t e were i t not f o r Tony A r r o t and L u i s de S o b r i n o , a c t i v e members o f S c i e n c e f o r Peace who s e t a f i n e example f o r t h e new p h y s i c s s t u d e n t . I am i n d e b t e d t o them f o r t h e i r c r i t i c i s m a nd e n c o u r a g e m e n t . Most o f a l l , I am g r a t e f u l t o my w i f e , B r e n d a , who a l w a y s e n c o u r a g e d me t o f o l l o w t h e s t a n d a r d s I had c h o s e n . In t h e words o f T.S. E l i o t : We s h a l l n o t c e a s e f r o m e x p l o r a t i o n And t h e end o f a l l o u r e x p l o r i n g S h a l l be t o r e t u r n t o where we s t a r t e d And know t h e p l a c e f o r t h e f i r s t t i m e . A u g u s t , 1984 I.INTRODUCTION 1 I . INTRODUCTION The g o a l of t h i s work was to t e s t the plasma/gas i n t e r a c t i o n model proposed by Ahlborn and Sinnot [ A H L B 7 7 , S I N N 7 7 ] , which i n t u r n arose from the end confinement problem of l i n e a r magnetic f u s i o n . The i s s u e at hand i s the behavior of plasma and c o l d gas when they are brought suddenly i n t o c o n t a c t . Ahlborn and Sinnot b e l i e v e that under c e r t a i n c o n d i t i o n s the plasma w i l l t r a n s f e r l e s s heat to the gas than i t would to a s o l i d w a l l . The end l o s s e s of a l i n e a r f u s i o n r e a c t o r c o u l d i n p r i n c i p l e be reduced by the presence of a gas end p l u g . The work d e s c r i b e d i n t h i s t h e s i s covers four main areas: an examination of the s p e c t r o s c o p i c theory used to measure plasma temperature and d e n s i t y , an e m p i r i c a l comparison of s o l i d and gas end plugs, d e r i v a t i o n of a uniform flow model s i m i l a r to that of Ahlborn and S i n n o t , and experimental t e s t s of the model p r e d i c t i o n s . The s p e c t r o s c o p i c t h e o r i e s were found to r e l y on the " a d i a b a t i c assumption", which was o f t e n b u r i e d i n the mathematics i f not s t a t e d e x p l i c i t l y . As a r e s u l t , problems a r i s e i f the same s p e c t r a l l i n e i s used to c a l c u l a t e both temperature and d e n s i t y , and i n Chapter I I i t w i l l be shown that the " r u l e s of thumb" used to j u s t i f y simultaneous measurement cannot be a p p l i e d c o n s i s t e n t l y . Accepted p r a c t i c e ( i n t h i s l a b o r a t o r y ) has been to c a l c u l a t e d e n s i t y from the broadening of the He I I 588nm l i n e , and temperature from the i n t e n s i t y r a t i o of He II 469nm to He I 588nm. The circumstances under which t h i s can be done without c o n t r a d i c t i o n are a l s o examined i n I.INTRODUCTION 2 C h a p t e r I I . The p r o b l e m does n o t seem t o be a p p r e c i a t e d i n t h e c u r r e n t l i t e r a t u r e , and needs t o be e xamined f u r t h e r . S i m i l a r l y , t h e A h l b o r n - S i n n o t p r o p o s a l r e s t s on a s s u m p t i o n s w h i c h need t o be t e s t e d f o r c o n s i s t e n c y . A l t h o u g h t h e scheme i s e s s e n t i a l l y an e x t e n s i o n o f t h e w e l l - k n o w n shock t u b e m o del, t h e d i f f e r e n t i a l e q u a t i o n s u s e d t o d e r i v e t h e l a t t e r w i l l n o t r e m a i n v a l i d i n a s y s t e m where t h e f l o w r e g i o n s a r e u n i f o r m and w i t h o u t g r a d i e n t s . In C h a p t e r I I t h e model i s r e d e r i v e d i n i t s p r o p e r a l g e b r a i c c o n t e x t . I t i s u s e d l a t e r i n C h a p t e r V t o c a l c u l a t e t h e h e a t f l u x f r o m t h e p l a s m a t o t h e g a s . The f i r s t i n v e s t i g a t i o n of gas end p l u g g i n g was c a r r i e d o u t a t t e m p e r a t u r e s and d e n s i t i e s much low e r t h a n t h o s e needed t o a t t a i n f u s i o n . I n a s t r a i g h t e m p i r i c a l a s s e s s m e n t , s o l i d and o pen-ended e l e c t r o d e s were mounted on a Z - p i n c h and d i r e c t l y compared by l o o k i n g f o r d i f f e r e n c e s i n t h e h e l i u m l i n e s p e c t r a m e a sured a t t h e p i n c h c e n t e r . No s u c h d i f f e r e n c e s were f o u n d , and s u b s e q u e n t a t t e m p t s t o c o n t i n u e t h e i n v e s t i g a t i o n c l o s e r t o t h e ends o f t h e p i n c h were hampered by t h e r a d i a l e x p a n s i o n o f t h e p l a s m a . N o n e t h e l e s s , s t r e a k p h o t o g r a p h s t a k e n o f t h e p l a s m a / g a s i n t e r a c t i o n showed c l e a r e v i d e n c e o f c o n f i n e m e n t . The l a c k o f m e a s u r e a b l e d i f f e r e n c e s i n t h e l i n e s p e c t r a a l s o t u r n e d o u t t o be c o n s i s t e n t w i t h t h e e n e r g y b a l a n c e a t t h e p l a s m a / g a s i n t e r f a c e . I t w i l l be shown t h a t t h e A h l b o r n - S i n n o t model i s v a l i d f o r c e r t a i n t y p e s o f g a s f l o w s , a l t h o u g h s e v e r a l q u e s t i o n s r e m a i n t o be a n s w e r e d by f u t u r e work. I.INTRODUCTION 3 The A h l b o r n - S i n n o t model r e p r e s e n t s a f r e s h a p p r o a c h t o m a t e r i a l end c o n f i n e m e n t , and t h e Z - p i n c h t o o o f f e r s p r a c t i c a l a d v a n t a g e s n o t s e e n by p r e v i o u s w o r k e r s i n t h i s f i e l d . The i m p l i c a t i o n s f o r m o d e l l i n g and measurement a r e examined i n C h a p t e r I I . The e q u i p m e n t i s d e s c r i b e d i n C h a p t e r I I I , and C h a p t e r s IV and V d e a l w i t h t h e d a t a and i t s a n a l y s i s . C o n c l u s i o n s a r e a d v a n c e d i n C h a p t e r V I . The r e m a i n i n g p a r t s o f t h i s c h a p t e r i n t r o d u c e t h e c o n t e x t i n w h i c h t h e work s h o u l d be i n t e r p r e t e d , p r e s e n t i n g t h e end c o n f i n e m e n t p r o b l e m f i r s t a s a p a r t of f u s i o n r e s e a r c h , t h e n as an i n i t i a l v a l u e p r o b l e m f o r w h i c h m o dels c a n be f o r m u l a t e d . The e x p e r i m e n t s u s e d t o t e s t t h e s e m o d els a r e b r i e f l y o u t l i n e d i n t h e f i n a l s e c t i o n . I.1 The L i n e a r M a g n e t i c F u s i o n C o n c e p t A v a r i e t y of e l e m e n t s a r e s u i t a b l e f o r f u s i o n , b ut t h e t e m p e r a t u r e s needed t o c o n v e r t them t o p l a s m a a r e g r e a t e r t h a n a m a t e r i a l v e s s e l c a n s u s t a i n . I t i s o n l y b e c a u s e t h e atoms i n t h e p l a s m a a r e i o n i z e d t h a t t h e i r m o t i o n c a n be i n f l u e n c e d by t h e L o r e n t z f o r c e , where t h e p r e s e n c e o f a m a g n e t i c f i e l d w i l l d i r e c t t h e i r t r a n s v e r s e m o t i o n i n t o s p i r a l o r b i t s a r o u n d t h e f i e l d l i n e s . P r o d u c i n g a f i e l d a r r a n g e m e n t c a p a b l e o f h o l d i n g t h e s e p a r t i c l e s l o n g enough f o r h e a t i n g and r e a c t i o n i s t h e main g o a l of m a g n e t i c f u s i o n r e s e a r c h . E x p e r i e n c e up t i l l now [BAKE81] s u g g e s t s t h a t t h e b e s t c a n d i d a t e s f o r a r e a c t o r a r e t h o s e b a s e d on e i t h e r a doughnut s h a p e d d e v i c e c a l l e d a tokamak, or a l i n e a r s y s t e m known a s t h e m i r r o r m a c h i n e . I.INTRODUCTION 4 The l i n e a r s y s t e m i s s i m p l e r t h a n t h e tokamak, and was a c t u a l l y t h e f i r s t t y p e t o be c o n s i d e r e d . The Z - p i n c h i t s e l f was an e a r l y f a v o r i t e t h a t even now ( d e s p i t e i t s p o o r s t a b i l i t y ) o f f e r s hope of becoming a r e a c t o r [HAIN78]. O t h e r w i s e i t s main use i s as a s o u r c e o f p l a s m a f o r p u l s e d l i n e a r e x p e r i m e n t s . From t h i s t y p e o f work a s i m p l e machine l i k e t h e Z - p i n c h g i v e s i n s i g h t i n t o more complex d e s i g n s . A p r a c t i c a l l i n e a r s y s t e m has i m p o r t a n t a d v a n t a g e s o v e r t h e tokamak: The p l a s m a i s more s t a b l e , m a i n t e n a n c e i s e a s i e r , and m o d u l a r c o n s t r u c t i o n r e d u c e s t h e c o s t . The main d i s a d v a n t a g e of t h e l i n e a r g e o m e t r y i s t h a t e n e r g y c o n f i n e m e n t i s l i m i t e d by t h e f r e e s t r e a m i n g e s c a p e o f p a r t i c l e s f r o m t h e open e n d s . The c o n f i n e m e n t t i m e s c a l e s a s L / a , where L i s t h e h a l f l e n g t h of t h e r e a c t o r and a i s t h e i o n t h e r m a l v e l o c i t y . C a l c u l a t i o n s show t h a t l e n g t h s between one and t e n k i l o m e t e r s a r e needed i n an open-ended s y s t e m . One p o s s i b l e s o l u t i o n i s t o p l a c e m a t e r i a l b a r r i e r s a t t h e ends o f t h e r e a c t o r v e s s e l . N u m e r i c a l s t u d i e s by Malone and M orse [MAL077] show t h a t c e r t a i n e l e m e n t s c a n w i t h s t a n d t h e p l a s m a l o n g enough t o o b t a i n r e a c t i o n s . E x p e r i m e n t s c a r r i e d o u t a t l o w e r t e m p e r a t u r e s by Commisso [COMM77] have shown t h e c o n c e p t t o be v a l i d . In t h i s t h e s i s , t h e s t u d y i s e x t e n d e d t o m a t e r i a l b a r r i e r s f o r m e d d y n a m i c a l l y by a " w a l l o f g a s " . In c o n t r a s t t o t h e s o l i d end p l u g , w h i c h r e m a i n s s t a t i o n a r y as t h e p l a s m a e r o d e s i t , t h e gas end p l u g c a n be t h o u g h t o f a s a shock t u b e a t t a c h e d t o t h e end o f t h e c o n f i n e m e n t v e s s e l . T h i s model o f f e r s p o s s i b i l i t i e s n o t o b v i o u s i n i t s s o l i d c o u n t e r p a r t . I.INTRODUCTION 5 1.2 End Conf inement as an I n i t i a l V a l u e P r o b l e m F i g u r e 1-1 i s a s i m p l i f i e d p i c t u r e o f t h e h o t p l a s m a i n s i d e t h e r e a c t i o n v e s s e l , j u s t a f t e r i t has come i n c o n t a c t w i t h t h e u n d i s t u r b e d gas i n t h e shock t u b e . I f t h e r a d i a l m o t i o n o f t h e p l a s m a i s c o n s t r a i n e d by a m a g n e t i c f i e l d , o r i f t h e r a d i a l m o t i o n c a n a t l e a s t be t a k e n a s i n d e p e n d e n t o f t h e s m a l l a m p l i t u d e a x i a l m o t i o n , t h e n t h e s y s t e m shown i n F i g u r e 1-1 forms an i n i t i a l s t a t e f o r a one d i m e n s i o n a l f l o w p r o b l e m . V V V V IP 1 BRASS ELECTRODE • • • T £ HOT PLASMA^ 3 COLD GAS A A A A 9 A A A A MAGNETIC CONFINEMENT (ACTUAL SIZE) 1-1 Plasma and Gas I n i t i a l S t a t e The f i r s t s t e p i n t h e a n a l y s i s o f C h a p t e r I I i s t o assume t h a t t h e p l a s m a and gas do n o t mix, so t h a t t h e " c o n t a c t s u r f a c e " between them c a n be m o d e l l e d a s a m a s s l e s s b a r r i e r . A f u r t h e r a s s u m p t i o n t h a t t h e r e i s no h e a t f l o w a c r o s s t h e b a r r i e r I . I N T R O D U C T I O N 6 w i l l s e p a r a t e t h e p r o b l e m i n t o two h a l v e s : The s y s t e m on t h e r i g h t i s a p i s t o n d r i v e n s h o c k ; t h e s y s t e m on t h e l e f t i s a n e x p a n d i n g c y l i n d e r . The c o n t a c t s u r f a c e a p p e a r s t o b o t h a s a s o l i d p i s t o n c a p a b l e o f e x e r t i n g f o r c e , w i t h t h e o n l y l i n k b e t w e e n t h e t w o s y s t e m s b e i n g a p r e s s u r e m a t c h i n g c o n d i t i o n a t t h e i n t e r f a c e . T h i s i d e a l i z e d " d u a l s y s t e m " i s t h e s t a r t i n g p o i n t f o r many a n a l y s e s ( e . g . WHIT74). I n t h i s w o r k i t w i l l be r e f e r r e d t o a s t h e s h o c k t u b e m o d e l . The A h l b o r n - S i n n o t m o d e l i s a s t r a i g h t f o r w a r d e x t e n s i o n o f t h e s h o c k t u b e m o d e l . The t w o f l u i d s a r e s t i l l a s s u m e d n o t t o m i x , b u t h e a t i s a l l o w e d t o f l o w a c r o s s t h e i n t e r f a c e , r a i s i n g t h e g a s p r e s s u r e a n d i n c r e a s i n g t h e s h o c k v e l o c i t y . The h e a t f l u x i s a s s u m e d t o be s m a l l ( s o t h a t t h e p l a s m a e x p a n s i o n r e m a i n s a p p r o x i m a t e l y a d i a b a t i c ) b e c a u s e a l a r g e f l u x w o u l d i n t r o d u c e t r a n s p o r t a n d d i f f u s i o n e q u a t i o n s n o t c o n s i s t e n t w i t h t h e s h o c k t u b e m o d e l a n d i t s " m a s s l e s s b a r r i e r " . I n t h i s c a s e i t w o u l d be n e c e s s a r y t o r e - e x p r e s s t h e u n d e r l y i n g c o n s e r v a t i o n e q u a t i o n s i n c h a r a c t e r i s t i c f o r m , a n d t h e n d e r i v e a h i e r a r c h y o f wave v e l o c i t i e s . S u c h w o r k i s b e y o n d t h e s c o p e o f t h i s t h e s i s . The p r e d i c t i o n s o f b o t h t h e s h o c k t u b e m o d e l a n d t h e A h l b o r n - S i n n o t m o d e l w e r e t e s t e d by o b s e r v i n g t h e p l a s m a / g a s i n t e r a c t i o n a t t h e e n d o f a Z - p i n c h d i s c h a r g e ( h e n c e t h e b r a s s e l e c t r o d e i n F i g u r e 1-1). The d a t a s h o w e d t h a t t h e A h l b o r n -S i n n o t m o d e l g a v e a b e t t e r d e s c r i p t i o n . A g e n e r a l o u t l i n e o f t h e e x p e r i m e n t s i s g i v e n i n t h e n e x t s e c t i o n . F u l l d e t a i l s a r e p r e s e n t e d i n C h a p t e r s I I I a n d I V . I.INTRODUCTION 7 I.3 O u t l i n e o f t h e E x p e r i m e n t a l P rogram The main r e q u i r e m e n t s o f an e x p e r i m e n t a l t e s t a r e a s o u r c e o f p l a s m a and a t u b e o f gas t o a c t a s t h e end p l u g . The p l a s m a was p r o d u c e d i n a s m a l l , h i g h p r e s s u r e Z - p i n c h , 600mm i n l e n g t h by 150mm i n d i a m e t e r . F i l l e d w i t h h e l i u m t o a p r e s s u r e o f 525 p a s c a l s , and f i r e d a t 12kV, t h e p i n c h c r e a t e s a p l a s m a w i t h a peak t e m p e r a t u r e o f 40000K and a peak e l e c t r o n d e n s i t y o f 2x1 0 2 3 n r 3 . The d e v i c e i s s i m i l a r t o t h a t u s e d by P r e s t o n [PRES74] and P a c h n e r [PACH71], b u t m o d i f i e d so t h a t vacuum and h i g h v o l t a g e c o n n e c t i o n s a r e made a t t h e same e l e c t r o d e . The o t h e r end was l e f t f r e e t o s i m p l i f y measurement of t h e p l a s m a / g a s i n t e r a c t i o n . The p i n c h e l e c t r o d e s had open c e n t e r s t o p e r m i t t h e a t t a c h m e n t of m e t a l p l u g s o r g l a s s s h o c k t u b e s . The t u b e s were e i t h e r l e f t c l e a r , i n w h i c h c a s e t h e p l u g i s i d e n t i c a l t o t h e Z - p i n c h f i l l g a s , o r " p r e - f i l l e d " by i n j e c t i n g gas w i t h a f a s t p u f f v a l v e . T h i s a r r a n g e m e n t made i t p o s s i b l e t o t e s t t h e A h l b o r n - S i n n o t model o v e r a r a n g e of p l u g d e n s i t i e s . In t h e e m p i r i c a l c o m p a r i s o n o f s o l i d and open end s , t e m p e r a t u r e and d e n s i t y were m e a s u r e d s p e c t r o s c o p i c a l l y a t t h e p i n c h m i d p o i n t . T e m p e r a t u r e and d e n s i t y i n t h e open end p i n c h were a l s o m e a s u r e d a t t h e q u a r t e r p o i n t ( h a l f way between t h e g r o u n d e l e c t r o d e and t h e m i d p o i n t ) , b e c a u s e t h e sh o c k t u b e model p r e d i c t s an i n w a r d moving e x p a n s i o n wave. Shock waves moving o u t w a r d i n t o t h e gas were p h o t o g r a p h e d w i t h a s t r e a k camera, and t h e h e a t f l u x c a l c u l a t e d f r o m t h e sh o c k v e l o c i t i e s . The h i g h e r d e n s i t y p l u g s were p h o t o g r a p h e d b u t n o t measured s p e c t r o s c o p i c a l l y . . I.INTRODUCTION 8 A number of i n n o v a t i o n s were i n t r o d u c e d by t h i s work. In C h a p t e r I I i t i s shown how t h e m o d e l s o f p h o t o n e m i s s i o n u s e d i n p l a s m a s p e c t r o s c o p y c a n be t e s t e d w i t h a s i n g l e a s y m p t o t i c p a r a m e t e r . The s p e c t r o s c o p i c d a t a i t s e l f ( i n t h e form of l i n e p r o f i l e s ) was r e c o r d e d d i g i t a l l y and a s s e m b l e d i n t o an i n t e r a c t i v e d a t a b a s e . Waves a l o n g t h e a x i s o f t h e Z - p i n c h c o u l d t h e n be s t u d i e d by c r o s s - c o r r e l a t i n g t i m e s e r i e s t a k e n a t d i f f e r e n t p o s i t i o n s , a s r a d i a l o s c i l l a t i o n s i n t h e pl a s m a made i t d i f f i c u l t t o o b s e r v e t h e s e d i r e c t l y . L i n e i n t e n s i t i e s u s e d t o measure t e m p e r a t u r e were c h e c k e d f o r r e l a x a t i o n e f f e c t s by c r o s s - c o r r e l a t i n g t h e He I 588nm and He I I 469nm t i m e s e r i e s . An i m p o r t a n t r e s u l t o f t h e s e l a t t e r c a l c u l a t i o n s was t o r u l e o u t t h e t h r e e m i c r o s e c o n d t i m e s h i f t a d v o c a t e d i n PRES74, s u g g e s t i n g t h a t t h e t e m p e r a t u r e s i n t h a t work s h o u l d n o t be t a k e n as c o r r e c t . The d e t a i l s of t h i s a n a l y s i s a r e g i v e n i n C h a p t e r s IV and V. The f u l l r e s u l t s and t h e i r i m p l i c a t i o n s a r e d i s c u s s e d i n C h a p t e r V I . II.THEORY 9 I I . THEORETICAL DISCUSSION T h i s c h a p t e r c o v e r s two o f t h e f o u r main a r e a s i n t h i s work: c a l c u l a t i o n o f t h e p l a s m a t e m p e r a t u r e and d e n s i t y , and d e r i v a t i o n o f t h e u n i f o r m f l o w m o d e l . The f o r m e r w i l l a l s o be n e e d ed f o r t h e e m p i r i c a l c o m p a r i s o n of s o l i d and open-ended Z - p i n c h e s . In g e n e r a l , t h e p u r p o s e o f t h i s c h a p t e r i s t o s e t up t h e e p i s t e m o l o g i c a l framework, so t h a t measurements i n t h e l a t e r c h a p t e r s c a n be f i t t e d i n t o a c o h e r e n t w o r l d v i e w . The o v e r a l l o b j e c t i v e i s t o a s s e s s t h e A h l b o r n - S i n n o t model a s a p p l i e d t o t h e c o n f i n e m e n t o f a Z - p i n c h p l a s m a w i t h c o l d gas end p l u g s . In s e c t i o n II.1, t h e p r o b l e m i s e x a m i n e d f i r s t i n t h e c o n t e x t o f m o d e l l i n g i n g e n e r a l , s p e c i f i c a l l y t h e need t o r e l a t e f o r m a l m a n i p u l a t i o n t o t h e w o r l d o f d i r e c t e x p e r i e n c e . The n e c e s s a r y d e f i n i t i o n o f t h e p l a s m a and gas s t a t e v a r i a b l e s i s t h e n e x p l o r e d i n s e c t i o n II.2. T h e r e i t w i l l a l s o be shown t h a t t h e c o n f l i c t between an a d i a b a t i c t h e o r y o f p h o t o n e m i s s i o n and c o l l i s i o n d o m i n a t e d l o c a l e q u i l i b r i u m c a n n o t be r e s o l v e d i n any s i m p l e way. D i s c u s s i o n of t h e d e n s i t y measurements i n s e c t i o n s II.3 t o II.5 w i l l f u r t h e r i l l u s t r a t e t h e d e e p r o o t s o f t h e a d i a b a t i c t h e o r y . S e c t i o n II.6 w i l l a l s o use t h i s t h e o r y i n e x p l a i n i n g how t e m p e r a t u r e c a n be c a l c u l a t e d f r o m t h e i n t e n s i t y r a t i o o f two s p e c t r a l l i n e s . S e c t i o n I I . 7 w i l l i n t r o d u c e t h e r e m a i n i n g a s s u m p t i o n s n e e ded t o d e r i v e t h e m o d els o f s e c t i o n s I I . 8 and I I . 9 , and t h e s e w i l l s e t t h e s t a g e f o r t h e d e r i v a t i o n o f s e c t i o n 1 1 . 1 0 : a s c a l i n g law t h a t s u g g e s t s i m p r o v e d e n e r g y c o n f i n e m e n t w i t h h i g h e r d e n s i t y end p l u g s . I t i s shown h e r e a n d i n t h e p r e v i o u s II.THEORY 1 0 s e c t i o n how e x p e r i m e n t a l measurements o f t h e p l a s m a e s c a p e v e l o c i t y c a n be u s e d t o c a l c u l a t e t h e h e a t f l u x i n t o t h e c o l d gas end p l u g . A summary o f a l l t h e a s s u m p t i o n s needed f o r t h e model and i t s measurement i s g i v e n i n s e c t i o n 1 1 . 1 1 . D i s c u s s i o n of t h e e x p e r i m e n t a l s e t u p f o l l o w s i m m e d i a t e l y i n C h a p t e r I I I . I I . 1 The R o l e o f F o r m a l M o d e l s The s c a l i n g law t o be d e r i v e d i n s e c t i o n 1 1 . 1 0 c a n be t h o u g h t o f a s an i n f e r e n c e , t h e p r o d u c t o f a f o r m a l m o d e l . T h i s s e c t i o n w i l l e x p l a i n how t h e c o n c e p t s of f o r m a l l o g i c can be u s e d t o s t r e n g t h e n e x i s t i n g c o n c e p t s o f p h y s i c s a s an e x e r c i s e i n model b u i l d i n g . I t w i l l a l s o be shown t h a t t h e need f o r l o g i c a l c o n s i s t e n c y c a n a f f e c t t h e c h o i c e o f e x p e r i m e n t a l t e c h n i q u e . A c o m p l e t e f o r m a l model may be b r o k e n i n t o t h r e e p a r t s : a s e t of i n i t i a l a s s u m p t i o n s , a s e t o f r u l e s f o r making i n f e r e n c e s , and a s e t o f p r o c e d u r e s f o r t e s t i n g t h o s e i n f e r e n c e s i n t h e e x t e r n a l w o r l d . The f i r s t p a r t may be d i v i d e d f u r t h e r i n t o f a c t s and c o n j e c t u r e s . The f o r m e r may be t a k e n a s e s t a b l i s h e d . The l a t t e r w i l l be s u b j e c t t o e x p e r i m e n t a l t e s t s . The frame o f r e f e r e n c e e s t a b l i s h e d i n t h i s work w i l l e n a b l e t h e A h l b o r n - S i n n o t model t o be t e s t e d a s an e x p e r i m e n t a l c o n j e c t u r e . The s p e c i f i c n a t u r e o f t h e p l a s m a and gas w i l l a l l o w some of t h e model s t a t e m e n t s t o be f o r m a l i z e d i n a r a t h e r s i m p l e way. As f l u i d s , b o t h c a n be d e s c r i b e d w i t h a s m a l l number o f m a c r o s c o p i c s t a t e v a r i a b l e s , and p r o v i d e d t h i s i s done c o n s i s t e n t l y ( e . g . m a t e r i a l s a t t h e same f o r m a l t e m p e r a t u r e II.THEORY s h o u l d have no n e t e n e r g y f l o w between them), t h e f o r m a l m a n i p u l a t i o n c a n be p a r e d down t o a s m a l l number o f a l g e b r a i c t r a n s f o r m a t i o n s . The r e s u l t i n g s i m p l i c i t y i s a g r e a t a i d t o b o t h i n t u i t i v e judgement and e x p e r i m e n t a l d e s i g n . F u r t h e r m o r e , r u l e s o f m a n i p u l a t i o n s e t down i n d i f f e r e n t i a l form may n o t a l w a y s meet t h e need f o r c o n s i s t e n c y . The t h e o r e m s of c a l c u l u s , f o r example, o f t e n embody d e e p a s s u m p t i o n s a b o u t t h e d e n s e n e s s o f p o i n t s on t h e r e a l number s p a c e , s o m e t h i n g w h i c h c a n n o t be g u a r a n t e e d f o r t h e l a r g e b ut f i n i t e number o f atoms i n a p h y s i c a l s y s t e m . An a l g e b r a i c model i s l e s s l i k e l y t o have p r o b l e m s i n t r o d u c e d by h i d d e n a s s u m p t i o n s i n t h e m a t h e m a t i c s . H a v i n g s e t f o r t h t h e a s s u m p t i o n s and t h e i r i n f e r e n c e r u l e s , t h e model c a n be t e s t e d by t h e p r e d i c t i o n s i t makes a b o u t t h e w o r l d o f d i r e c t e x p e r i e n c e . E x p e r i m e n t s a r e d e s i g n e d t o r e l a t e t h e model i n f e r e n c e s t o t h e numbers a p p e a r i n g on t h e d i g i t a l r e a d o u t s o f t h e m e a s u r i n g i n s t r u m e n t s . The t e s t i n g p r o c e d u r e s a r e t h e m s e l v e s m o d e l s , whose i n t e r n a l l o g i c s h o u l d be c h e c k e d f o r c o n s i s t e n c y w i t h t h e model d e s c r i b i n g t h e p h y s i c a l p r o c e s s . T h e r e i s a l o n g c h a i n of i n f e r e n c e s f r o m t h e atoms and p h o t o n s i n t h e p l a s m a t o t h e s i x t e e n b i t i n t e g e r s a t t h e o u t p u t p o r t o f t h e 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 . The e x p e r i m e n t a l d a t a w i l l -be m e a n i n g l e s s i f t h e c h a i n r e q u i r e s a s s u m p t i o n s i n c o n s i s t e n t w i t h t h e model under t e s t . The r e m a i n d e r o f t h i s c h a p t e r d e a l s w i t h t h e p r a c t i c a l a s p e c t s o f b u i l d i n g a model, a l w a y s w i t h an eye t o i t s u l t i m a t e v e r i f i c a t i o n by e x p e r i m e n t . S t a t e m e n t s a b o u t t h e Z - p i n c h p l a s m a and i t s s p e c t r o s c o p i c measurement a r e a d v a n c e d i n s e c t i o n s I I . 2 II.THEORY 12 t o I I . 6 . A d d i t i o n a l s t a t e m e n t s needed t o c o n s t r u c t t h e one d i m e n s i o n a l i n i t i a l v a l u e p r o b l e m ( i . e . a r e s t r i c t i o n of t h e a l l o w e d i n f e r e n c e r u l e s t o t h o s e d e p e n d i n g on a s i n g l e p o s i t i o n c o - o r d i n a t e ) a r e a d v a n c e d i n s e c t i o n I I . 7 . The s y m b o l s and t h e i r f o r m a l r e l a t i o n ( t h r o u g h c o n s e r v a t i o n e q u a t i o n s e t c . ) a r e p r e s e n t e d i n s e c t i o n s I I . 8 t o 11.10. The f i n a l s e c t i o n II.11 s ummarizes t h e model i n a n t i c i p a t i o n of t h e e x p e r i m e n t a l t e s t s . I I . 2 D e f i n i t i o n of t h e Plasma S t a t e A t a f i r s t g l a n c e , i t would a p p e a r t h a t e x i s t i n g knowledge i n f l u i d d y n a m i c s makes a n y t h i n g t h a t c o u l d be s a i d i n t h i s s e c t i o n r a t h e r e l e m e n t a r y . In f a c t i t w i l l t u r n o u t t h a t t h e o p p o s i t e i s t r u e . The p a r a d i g m imposed by model c o n s t r u c t i o n w i l l b r i n g o u t an i m p o r t a n t p r o b l e m : i n c o n s i s t e n t a s s u m p t i o n s i n t h e common methods of t e m p e r a t u r e and d e n s i t y measurement. T h i s s e c t i o n w i l l d e f i n e t h e meaning of p l a s m a s t a t e v a r i a b l e s i n s u c h a way t h a t no i n c o n s i s t e n c y w i l l a r i s e . The s t a t e of a c o l d g a s end p l u g i s d e f i n e d e a s i l y i n terms of t h e i d e a l gas t h e o r y . D e n s i t y i s t h e amount of mass p r e s e n t i n a g i v e n volume, p r e s s u r e i s t h e f o r c e p e r u n i t a r e a e x e r t e d by t h e g a s on i t s c o n t a i n e r , and t e m p e r a t u r e i s a measure o f t h e m i c r o s c o p i c e n e r g y d i s t r i b u t i o n . I n h e l i u m a t 525 p a s c a l s and 300 k e l v i n , a l m o s t a l l o f t h i s e n e r g y w i l l be f o u n d i n t h e f r e e m o t i o n k i n e t i c e n e r g y o f n e u t r a l h e l i u m atoms. An e q u i v a l e n t d e f i n i t i o n o f t h e p l a s m a s t a t e has no d i f f i c u l t y w i t h t h e d e n s i t y , b u t t h e p r e s s u r e and t e m p e r a t u r e need somewhat more c a r e . An o p e r a t i o n a l d e f i n i t i o n t h a t II.THEORY 13 r e q u i r e s t h e p l a s m a t o e x e r t f o r c e begs t h e q u e s t i o n o f what t h a t f o r c e c a n be e x e r t e d on. T h e r e would h a r d l y be a need f o r m a g n e t i c c o n f i n e m e n t i f t h e p l a s m a c o u l d be p u t i n a s o l i d chamber! F o r t u n a t e l y , i n t e r a c t i o n w i t h t h e c o l d gas end p l u g makes i t p o s s i b l e t o compare t h e p l a s m a t e m p e r a t u r e t o t h e common s t a n d a r d o f an i d e a l gas t h e r m o m e t e r . The mechanism f o r t r a n s f e r r i n g e n e r g y from t h e p l a s m a t o t h e gas c a n be u s e d t o e q u a t e t h e i r p r e s s u r e s . A r e l a t i o n between t h e p r e s s u r e and t h e o b s e r v a b l e t e m p e r a t u r e and d e n s i t y w i l l d e f i n e t h e p l a s m a s t a t e . In a g e n e r a l a n a l y s i s , t h e p l a s m a would c o n s i s t of a l a r g e number o f e l e c t r o n s and h e l i u m n u c l e i i i n t e r a c t i n g t h r o u g h an e l e c t r o m a g n e t i c f i e l d . The p h y s i c s of t h e s y s t e m would t h e n be d e s c r i b e d by w r i t i n g down t h e H a m i l t o n i a n . F r e e s t a t e s o f t h e s y s t e m would c o r r e s p o n d t o a l l o f t h e p a r t i c l e s moving i n d e p e n d e n t l y , and t h e bound s t a t e s would a p p e a r whenever any o f th e e l e c t r o n s and n u c l e i i c ombined t o f o r m atoms. In t h e Z - p i n c h p l a s m a however, most o f t h e p a r t i c l e s a r e e i t h e r s i n g l y i o n i z e d atoms o r f r e e e l e c t r o n s , a l l o f w h i c h move s l o w l y enough t o i n t e r a c t t h r o u g h an e l e c t r o s t a t i c p o t e n t i a l . I t w i l l t h e r e f o r e be c o n v e n i e n t t o t h i n k o f t h e s y s t e m a s a c o l l e c t i o n o f p a r t i c l e s w i t h f r e e m o t i o n and i n t e r n a l bound s t a t e s . I t w i l l be a r g u e d l a t e r i n s e c t i o n I I . 3 t h a t t h e s u b s p a c e s c o r r e s p o n d i n g t o t h e f r e e and bound a t o m i c s t a t e s a r e s e p a r a t e and w e a k l y c o u p l e d , so t h a t e n e r g y i r i t h e s y s t e m a s a whole can be made up f r o m t h e sum o f i n d i v i d u a l k i n e t i c a nd p o t e n t i a l t e r m s . In t h i s s e c t i o n i t w i l l s i m p l y be assumed t h a t t h i s c a n be done, b e c a u s e t h e i s s u e of i n t e r e s t h e r e i s t h e d i s t r i b u t i o n II.THEORY of e n e r g y w i t h i n e a c h s u b s p a c e , s p e c i f i c a l l y t h e number o f p l a s m a p a r t i c l e s t o be f o u n d a t e a c h e n e r g y l e v e l t h e r e i n . The p r a c t i c a l a p p l i c a t i o n comes from t h e way i n w h i c h e n e r g y i s t r a n s f e r r e d between t h e p l a s m a and t h e g a s . The d o m i n a n t mechanism a f f e c t s o n l y t h e f r e e s t a t e s , whereas measurement by s p e c t r o s c o p y w i l l depend on p h o t o n s e m i t t e d f r o m t h e bound s t a t e s . The r e l a t i o n s h i p between t h e s e two s u b s y s t e m s i s o f c r i t i c a l i m p o r t a n c e i n d e f i n i n g and m e a s u r i n g t h e p l a s m a t e m p e r a t u r e . A n a l y s i s of t h e t e m p e r a t u r e s t a r t s w i t h t h e e n e r g y t r a n s f e r mechanism a t t h e p l a s m a / g a s i n t e r f a c e . Here i t i s n e c e s s a r y t o r e s t a t e t h e a s s u m p t i o n t h a t t h e two f l u i d s do n o t mix, so t h a t a p r e s s u r e b a l a n c e i s a c h i e v e d t h r o u g h p a r t i c l e c o l l i s i o n . A s t a t i o n a r y i n t e r f a c e means t h a t t h e p l a s m a p r e s s u r e i s t h e same as t h a t o f t h e g a s . F u r t h e r m o r e , p l a s m a t e m p e r a t u r e can a l s o be d e f i n e d by t h e same k i n e t i c t h e o r y t h a t a p p l i e s t o t h e c o l d g a s . When t h e number and v e l o c i t y o f e l e c t r o n s and i o n s i s s u c h t h a t no n e t momentum i s t r a n s f e r r e d a c r o s s t h e i n t e r f a c e , knowledge of t h e i r number d e n s i t y i s s u f f i c i e n t t o d e t e r m i n e t h e i r v e l o c i t y d i s t r i b u t i o n . The v e l o c i t y d i s t r i b u t i o n i s d i r e c t l y r e l a t e d t o t h e i d e a l gas t e m p e r a t u r e . I n some p l a s m a s , d i s t r i b u t i o n s f o r t h e i o n s and e l e c t r o n s c a n be c h a r a c t e r i z e d by d i f f e r e n t t e m p e r a t u r e s ( e . g . an " i o n t e m p e r a t u r e " and an " e l e c t r o n t e m p e r a t u r e " ) , but c o n d i t i o n s i n t h e Z - p i n c h w i l l t e n d t o make them t h e same. II.THEORY 1 5 A l t h o u g h a p r e s s u r e b a l a n c e e n s u r e s t h a t no n e t momentum i s t r a n s f e r r e d , c o l l i s i o n s between t h e f a s t p l a s m a p a r t i c l e s and t h e s l o w e r gas atoms w i l l t e n d t o a c c e l e r a t e t h e l a t t e r a t t h e e x p e n s e o f t h e f o r m e r . The n e t e f f e c t i s t o t r a n s f e r e n e r g y ( i n t h e f o r m of h e a t ) and r a i s e t h e t e m p e r a t u r e o f t h e c o l d gas end p l u g . In t h e Z - p i n c h p l a s m a , t h e s e c o l l i s i o n s w i l l be t h e o n l y means of e n e r g y t r a n s f e r , a s r a d i a t i o n i s n e i t h e r e m i t t e d n o r a b s o r b e d i n amounts s u f f i c i e n t t o c a u s e a m e a n i n g f u l c h a n g e . ( I n s e c t i o n I I . 3 i t w i l l be shown t h a t a b s o r p t i o n c o u l d s e r i o u s l y c o m p l i c a t e t h e measurement p r o c e s s . ) The o n l y r o l e o f r a d i a t i o n i n t h i s e x p e r i m e n t i s t o s e r v e a s a m e s s e n g e r f r o m t h e bound s t a t e s u b s y s t e m t o t h e o u t s i d e w o r l d . The i n f o r m a t i o n c a r r i e d by t h e e s c a p i n g r a d i a t i o n w i l l d e pend on t h e c o u p l i n g between t h e bound s t a t e and f r e e s t a t e s u b s y s t e m s . A l t h o u g h t h i s c o u p l i n g i s assumed t o be weak, i t i s o n l y b e c a u s e i t e x i s t s a t a l l t h a t atoms w i l l be e x c i t e d t o e m i t p h o t o n s . The q u e s t i o n now i s one o f e q u i l i b r i u m : I f t h e e n e r g y d i s t r i b u t i o n i n t h e bound s t a t e s c o r r e s p o n d s t o t h e same t e m p e r a t u r e a s f o u n d i n t h e f r e e s t a t e s , t h e e n e r g y d i s t r i b u t i o n o f t h e p h o t o n s c a n be c a l c u l a t e d f r o m t h e f r e e s t a t e t e m p e r a t u r e . The d i f f i c u l t y comes i n knowing whether t h e e q u i l i b r i u m h a s been e s t a b l i s h e d . A u s u a l c r i t e r i o n i s t h a t no n e t e n e r g y f l o w s between t h e two s u b s y s t e m s , b u t i f t h e p l a s m a i s o p t i c a l l y t h i n , so t h a t p h o t o n s e s c a p e w i t h o u t r e a b s o r p t i o n , t h e n e x t r a e n e r g y must be c o n t i n u a l l y s u p p l i e d t o t h e bound s t a t e s o f t h e e m i t t e r s . The measurement o f t e m p e r a t u r e and d e n s i t y by p l a s m a s p e c t r o s c o p y r e s t s on t h e a s s u m p t i o n t h a t II.THEORY 16 e n e r g y g i v e n t o t h e e m i t t i n g atoms by c o l l i s i o n a l e x c i t a t i o n w i l l be t r a n s f e r r e d o u t by r a d i a t i o n . The v a l i d i t y o f t h e measurements w i l l depend on t h e r e b e i n g l o c a l t h e r modynamic e q u i l i b r i u m between t h e f r e e m o t i o n k i n e t i c s t a t e s and t h e p h o t o n - e m i t t i n g bound s t a t e s . I t i s i n t e r e s t i n g t o n o t e t h a t when t h e t e m p e r a t u r e i s measured f r o m s p e c t r a l l i n e s a s s o c i a t e d w i t h bound s t a t e t r a n s i t i o n s , some models r e q u i r e t h a t most o f t h e e n e r g y p u t i n t o an atom by c o l l i s i o n a l e x c i t a t i o n be r e l e a s e d t h e same way, so t h a t e q u i l i b r i u m between t h e f r e e m o t i o n and a t o m i c s t a t e s i s c l e a r l y e s t a b l i s h e d . In o t h e r words, t r a n s i t i o n s between t h e f r e e and bound s t a t e s s h o u l d be much more f r e q u e n t t h a n t r a n s i t i o n s between t h e bound s t a t e s t h e m s e l v e s . G r i e m , f o r i n s t a n c e , h as a r g u e d t h a t t h e c o l l i s i o n a l d e p o p u l a t i o n r a t e s h o u l d be t e n t i m e s t h e r a d i a t i o n r a t e [ G R I E 6 4 ] . T h i s i s i n d i r e c t c o n f l i c t w i t h t h e a d i a b a t i c a s s u m p t i o n u s e d i n d e s c r i b i n g t h e e m i s s i o n o f p h o t o n s (and hence t h e s p e c t r a l l i n e s h a p e s ) , an a s s u m p t i o n t h a t c a n be so d e e p l y b u r i e d i n t h e quantum m e c h a n i c a l t e c h n i q u e t h a t c e r t a i n a u t h o r s have d e n i e d making i t even when t h e i r m a t h e m a t i c s shows o t h e r w i s e ( e . g . SMIT69). The n e t e f f e c t o f t h i s c o n t r a d i c t i o n i s t o r u l e o u t any a p r i o r i t e s t o f e q u i l i b r i u m i f t h e l i n e p r o f i l e s a r e t o be o b s e r v e d w i t h an a d i a b a t i c e m i s s i o n m o d e l . A p o s s i b l e a p o s t e r i o r i t e s t i s d i s c u s s e d i n C h a p t e r V. The c o n f l i c t between, t h e s e t h e o r i e s h as n o t been s t u d i e d i n t h e p a s t , m a i n l y b e c a u s e t h e p o i n t o f v i e w h e l d by most r e s e a r c h e r s was t h a t p l a s m a t e m p e r a t u r e e x i s t e d i n d e p e n d e n t l y o f 11.THEORY 17 t h e means c h o s e n t o o b s e r v e i t . In t h i s view t h e r e i s no p r o b l e m when t h e means of o b s e r v a t i o n r e l y on i n c o n s i s t e n t a s s u m p t i o n s , s i n c e what m a t t e r s i n t h e end i s t h e number t h e y a s s i g n t o t h e i n d e p e n d e n t t e m p e r a t u r e . The d i f f i c u l t y comes when t h e m i c r o s c o p i c e n e r g y d i s t r i b u t i o n c a n i m p l y t h e e x i s t e n c e of s e v e r a l " t e m p e r a t u r e s " : one f o r e a c h t y p e o f p a r t i c l e , one f o r e a c h s e t of bound s t a t e s w i t h i n a c e r t a i n e n e r g y d i s t a n c e of e a c h o t h e r , e t c . e t c . T h e o r i e s w h i c h a l l o w t h i s t o happen make i t d i f f i c u l t t o s e t up a thermodynamic model w i t h a s m a l l s e t of c o n s i s t e n t p a r a m e t e r s . The v i e w a d v a n c e d i n t h i s t h e s i s i s t h a t t e m p e r a t u r e i n a p l a s m a must be d e f i n e d p r e c i s e l y i n t e r m s o f what c a n be m e a s u r e d , and t h a t t h e o r i e s s h o u l d be e x p r e s s e d i n s u c h a way t h a t o b s e r v a t i o n s can be p r e d i c t e d and t e s t e d . In t h i s frame of r e f e r e n c e t h e p r i m e s t a t e m e n t i s t h a t "under c e r t a i n c o n d i t i o n s f o l l o w i n g a s p e c i f i c e x p e r i m e n t a l p r e p a r a t i o n , p l a s m a t e m p e r a t u r e T w i l l l e a d t o s p e c t r o s c o p i c o b s e r v a t i o n S." The a l g o r i t h m u s e d t o c a l c u l a t e T i s t h e r e f o r e an a t t e m p t t o i n v e r t t h e l o g i c . One assumes t h a t t h i s c a n be done, but t h e i m p o s s i b i l i t y of knowing T d i r e c t l y means t h a t no t e s t c a n be a p p l i e d t o p r o v e t h e a l g o r i t h m v a l i d . The b e s t t h a t c a n be done i s t o use S a s a t e s t o f t h e p r o c e s s model t h a t p r e d i c t e d T. I f enough t e s t s show a p a r t i c u l a r model t o be v a l i d , t h e f o r m a l t e m p e r a t u r e i t c o n t a i n s c a n be u s e d a s a b a s i s f o r o t h e r c a l c u l a t i o n s . T h e r e w i l l be no need t o make s t a t e m e n t s a b o u t a " p l a s m a t e m p e r a t u r e " whose e x i s t e n c e c a n n o t be p r o v e d . II.THEORY 18 The model of p l a s m a t e m p e r a t u r e t o be u s e d w i t h t h e Z - p i n c h i s b a s e d on t h e a s s u m p t i o n t h a t most o f t h e e n e r g y coming i n t o t h e p l a s m a w i l l a p p e a r f i r s t i n t h e f r e e m o t i o n s t a t e s o f t h e c h a r g e d p a r t i c l e s , as t h e s e r e s p o n d t o t h e a p p l i e d e l e c t r i c f i e l d . The " t h e r m o m e t e r " i s an o u t p u t s u b s y s t e m formed by t h e bound s t a t e s i n v o l v e d i n p h o t o n e m i s s i o n . The e n e r g y d i s t r i b u t i o n o f t h e e m i t t e d p h o t o n s i s f o r m a l l y r e l a t e d t o t h e number d i s t r i b u t i o n of p a r t i c l e s i n t h e bound s t a t e s . T h i s i n t u r n i s r e l a t e d t o t h e v e l o c i t y d i s t r i b u t i o n i n t h e f r e e s t a t e s , w h i c h i s t h e n assumed t o be t h e same as i t w o u l d be i n t h e p r e s e n c e o f an i d e a l gas r e s e r v o i r . A n o t h e r way o f e x p r e s s i n g t h i s r e l a t i o n i s t o p o s t u l a t e t h e e x i s t e n c e o f an i d e a l gas r e s e r v o i r whose m a c r o s c o p i c p r o p e r t i e s would c o r r e s p o n d t o t h e o b s e r v e d d i s t r i b u t i o n of p h o t o n e n e r g i e s . I t s h y p o t h e t i c a l t e m p e r a t u r e c a n t h e n be a s s i g n e d t o t h e p l a s m a . In summary, t h e p l a s m a s t a t e may be d e f i n e d a s an e x t e n s i o n o f t h e i d e a l gas s t a t e , g e n e r a l l y on t h e e p i s t e m o l o g i c a l b a s i s o f an i d e a l gas t h e r m o m e t e r , and p r a c t i c a l l y i n t h i s e x p e r i m e n t b e c a u s e t h e main o b j e c t i v e i s t o measure t h e e n e r g y t r a n s f e r between a p l a s m a and an i d e a l g a s . The p r o b l e m f o r e x p e r i m e n t a l m o d e l l i n g i s t o f i n d a p h o t o n e m i t t i n g s u b s y s t e m w h i c h a l l o w s t h e c a l c u l a t i o n of p o p u l a t i o n and t r a n s i t i o n r a t e s i n t e rms of k i n e t i c t e m p e r a t u r e and p l a s m a d e n s i t y , and w h i c h i s known ( o r c a n be t e s t e d ) t o be i n l o c a l thermodynamic e q u i l i b r i u m w i t h t h e f r e e m o t i o n s t a t e s . A b a l a n c e must be s t r u c k between t h e a d i a b a t i c r e q u i r e m e n t t h a t t h e t r a n s i t i o n r a t e s be s m a l l ( o t h e r w i s e t h e quantum m e c h a n i c s g e t s t o o d i f f i c u l t ) and t h e II.THEORY 19 e q u i l i b r i u m r e q u i r e m e n t t h a t t h e t r a n s i t i o n r a t e s be f a s t ( o t h e r w i s e t h e bound s t a t e p o p u l a t i o n w i l l n o t change q u i c k l y enough t o f o l l o w t h e c h a n g e s i n k i n e t i c t e m p e r a t u r e ) . A d e t a i l e d a n a l y s i s o f t h i s l a t t e r p r o b l e m has y e t t o a p p e a r i n t h e l i t e r a t u r e . I I . 3 O b s e r v a t i o n o f S p e c t r a l L i n e s S p e c t r o s c o p i c measurement o f t e m p e r a t u r e and d e n s i t y c a n be s u b d i v i d e d i n t o two m a j o r t o p i c s : e m i s s i o n o f l i g h t by t h e i n d i v i d u a l atoms, and p r o p a g a t i o n o f t h a t l i g h t f r o m e m i t t e r t o o b s e r v e r . The two p a r t s a r e l i n k e d when t h e e m i t t e r s a b s o r b i n a d d i t i o n t o e m i t t i n g , b u t i n many c a s e s a s t r i c t s e p a r a t i o n c a n be made. The c o n d i t i o n s u nder w h i c h t h i s may be v a l i d a r e e x p l o r e d i n t h e p r e s e n t s e c t i o n . The s i n g l e most i m p o r t a n t c o n d i t i o n i s t h a t c o u p l i n g between t h e p l a s m a d y n a m i c a l s t a t e s and t h e f r e e s p a c e r a d i a t i o n s t a t e s n o t i n d u c e c o u p l i n g between e m i t t i n g atoms. In o t h e r words, t h e p r o b a b i l i t y o f one atom e m i t t i n g a p h o t o n must n o t be a f f e c t e d by t h e i n t e r n a l s t a t e o f a n o t h e r atom. A s l i g h t l y s t r o n g e r c o n d i t i o n i s t o s t a t e t h a t p h o t o n e m i s s i o n must i n v o l v e o n l y a s m a l l number o f atoms, i n s u c h a way t h a t t h e e m i t t i n g atoms a r e s t a t i s t i c a l l y i n d e p e n d e n t . T h i s s t a t e m e n t has e x t r e m e l y d e e p i m p l i c a t i o n s . S t a t i s t i c a l i n d e p e n d e n c e j u s t i f i e s t h e s e m i c l a s s i c a l t h e o r y o f p h o t o n e m i s s i o n , w h i c h depends on t h e c l a s s i c a l w a v e l e n g t h b e i n g much l a r g e r t h a n t h e e m i t t e r s c a l e l e n g t h , and h e n c e v e r y l i k e l y e n c o m p a s s i n g more t h a n one e m i t t e r . I n c o n j u n c t i o n w i t h t h e II.THEORY 20 e r g o d i c h y p o t h e s i s i n s t a t i s t i c a l m e c h a n i c s , . i t a l l o w s t h e e m i t t e r s t o be t r e a t e d as members o f an ensemble whose p r o p e r t i e s c a n be c a l c u l a t e d f r o m t h e b e h a v i o u r of a s i n g l e atom. W i t h o u t t h i s a s s u m p t i o n a c o m p l e t e f o r m a l t h e o r y would be n e x t t o i m p o s s i b l e . The i n d e p e n d e n c e o f e m i t t i n g atoms i s t r a d i t i o n a l l y a n a l y z e d by c a l c u l a t i n g t h e " o p t i c a l d e p t h " , a d i m e n s i o n l e s s number d e f i n e d as t h e i n t e g r a l o f t h e a b s o r p t i o n c o e f f i c i e n t o v e r t h e p a t h t a k e n by t h e l i g h t . An a b s o r p t i o n c o e f f i c i e n t o f z e r o i n d i c a t e s t r a n s p a r e n c y , so i n d e p e n d e n c e w i l l r e q u i r e t h a t t h e o p t i c a l d e p t h be s m a l l . Upper bounds on t h e Z - p i n c h o p t i c a l d e p t h have been c a l c u l a t e d by P r e s t o n [P R E S 7 4] and B e r n a r d [ B E R N 7 9 ] , i n b o t h c a s e s o v e r p a t h l e n g t h s l o n g e r t h a n t h o s e o f t h e p r e s e n t work, and i n b o t h c a s e s s howing t h e p l a s m a t o be o p t i c a l l y " t h i n " . I t i s assumed t h a t i f t h e e m i t t e r s do n o t i n t e r a c t t h r o u g h t h e p h o t o n f i e l d t h a t t h e y do n o t i n t e r a c t a t a l l . E a c h w i l l e x p e r i e n c e t h e r e s t o f t h e p l a s m a as i f t h e r e were no o t h e r e m i t t i n g atoms p r e s e n t . In a d d i t i o n t o d i r e c t a b s o r p t i o n , t h e r e i s a l s o t h e p o s s i b i l i t y t h a t e l e c t r o m a g n e t i c waves may n o t be a b l e t o p r o p a g a t e , an e f f e c t p r e d i c t e d by t h e d i s p e r s i o n r e l a t i o n f o r a s y s t e m o f c h a r g e d p a r t i c l e s i n t e r a c t i n g t h r o u g h a p o t e n t i a l . The r e s u l t i n g i n d e x o f r e f r a c t i o n f o r a wave o f f r e q u e n c y CJ i s r " i 1 / 2 n = 1 - CJ§AJ2 , where CJ0 i s t h e p l a s m a f r e q u e n c y II.THEORY 21 r i 1 / 2 OJ0 = Ne_^ + N Z e 2 L e 0m e0M -I and m and M a r e t h e e l e c t r o n and i o n masses, e i s t h e u n i t e l e c t r i c c h a r g e , Z i s t h e a t o m i c number of t h e i o n , and N i s t h e number d e n s i t y o f e l e c t r o n s a n d i o n s . In g e n e r a l t h e f i r s t t e r m w i l l be much l a r g e r t h a n t h e s e c o n d . The a b s o r p t i o n e f f e c t w i l l be n e g l i g i b l e when a l a r g e v a l u e of co makes n c l o s e t o u n i t y . I t i s n o t p o s s i b l e t o h a v e co t o o l a r g e however, b e c a u s e t h e s e m i c l a s s i c a l model o f p h o t o n e m i s s i o n r e l i e s on t h e w a v e l e n g t h b e i n g g r e a t e r t h a n t h e a t o m i c r a d i u s . I n t h i s model t h e H a m i l t o n i a n f o r an e l e c t r o n i n a Coulomb p o t e n t i a l i s m o d i f i e d t o i n c l u d e t h e momentum of a c l a s s i c a l e l e c t r o m a g n e t i c f i e l d w i t h v e c t o r p o t e n t i a l A ( r , t ) , H = p 2 - Z e 2 + eA'p + e 2 A 2 - e<t>, 2m r mc 2mc 2 and weak c o u p l i n g means t h a t A must be s m a l l ( w h i c h i t i s f o r t h e Z - p i n c h - s t r o n g c o u p l i n g o n l y o c c u r s i n l a s e r e x p e r i m e n t s where v e r y h i g h f i e l d s a r e p r e s e n t ) . The w a v e l e n g t h r e s t r i c t i o n comes f r o m a s s u m i n g t h a t t h e f i e l d i s s i n u s o i d a l , j (k • r_-o;t) - j (k • r-<jt) A = A 0 e + Aoe , where |kj = 2ir/\ - co/c. L a r g e co i m p l i e s t h a t k«jr w i l l change s i g n i f i c a n t l y o v e r t h e r a n g e o f v_ i m p o r t a n t t o t h e a t o m i c wave f u n c t i o n . I t would n o t t h e n be p o s s i b l e t o expand t h e e x p o n e n t i a l a s demanded by t h e m o d e l . II.THEORY 22 The Z - p i n c h p l a s m a f a l l s n i c e l y between t h e s e two e x t r e m e l i m i t s . The Bohr r a d i u s a 0 o f s i n g l y i o n i z e d h e l i u m i s a p p r o x i m a t e l y 5x10" 1 1m, so X=469nm i m p l i e s k«r_^0.00l. The p a r a m e t e r i n t h e c l a s s i c a l i n d e x of r e f r a c t i o n i s a l s o s m a l l : o>0/o=*0.01 . The s e m i c l a s s i c a l model i s v a l i d , and f r e q u e n c i e s i n t h e o p t i c a l band w i l l p r o p a g a t e c l a s s i c a l l y w i t h o u t a b s o r p t i o n . A f u r t h e r s i m p l i f i c a t i o n may be made i f t h e e m i t t i n g atoms have o n l y weak c o u p l i n g between t h e i r t r a n s l a t i o n a l and i n t e r n a l s t a t e s . I t w i l l t h e n be p o s s i b l e t o f a c t o r t h e e m i t t e r d e n s i t y o p e r a t o r i n t o t r a n s l a t i o n a l and i n t e r n a l p a r t s ( i . e . by f a c t o r i n g t h e H i l b e r t s p a c e i n t o a t e n s o r p r o d u c t ) , i n t h e same way t h a t v e l o c i t y components a r e f a c t o r e d i n an i d e a l gas v e l o c i t y d i s t r i b u t i o n . The p r o b a b i l i t y d i s t r i b u t i o n f o r t h e e n e r g y of e m i t t e d p h o t o n s w i l l t h e n depend on b o t h i n t e r n a l and t r a n s l a t i o n a l e f f e c t s . A d i f f e r e n t way o f l o o k i n g a t t h i s i s t o t h i n k of e a c h i n d e p e n d e n t e m i t t e r a s a f i x e d o b j e c t s u r r o u n d e d by a moving p l a s m a ( a l w a y s p o s s i b l e i n a n o n - r e l a t i v i s t i c s y s t e m ) . Summing l i g h t f r o m d i f f e r e n t e m i t t e r s means t h a t c o r r e c t i o n must be made f o r t h e i r d i f f e r e n t v e l o c i t i e s r e l a t i v e t o t h e o b s e r v e r . The p h o t o n f r e q u e n c i e s w i l l be s h i f t e d by t h e D o p p l e r e f f e c t . An e m i t t e r moving a l o n g t h e l i n e o f s i g h t a t v e l o c i t y v has a f r e q u e n c y s h i f t g i v e n by ACJ/CO = v / c . I f t h e e m i t t e r s move i n d e p e n d e n t l y and have a n o r m a l i z e d v e l o c i t y d i s t r i b u t i o n F ( v ) , t h e f r e q u e n c y s h i f t d i s t r i b u t i o n w i l l be L(ACJ)dAw = F (C A C O / C J ) C / O ) dAo>. II.THEORY 23 I f t h e l i n e p r o f i l e of t h e i n d e p e n d e n t e m i t t e r s i s P(co), t h e n t h e D o p p l e r b r o a d e n e d p r o f i l e w i l l be g i v e n by t h e c o n v o l u t i o n o f L(Aco) and Pico). The c o n n e c t i o n w i t h t h e H i l b e r t s p a c e s e p a r a t i o n comes from a s s u m i n g t h a t t h e d e n s i t y o p e r a t o r f o r a p l a s m a i n t h e r m a l e q u i l i b r i u m w i l l i n c l u d e F ( v ) a s one o f i t s f a c t o r s . M u l t i p l i c a t i o n i n t h e t i m e domain goes o v e r t o c o n v o l u t i o n i n t h e e n e r g y ( i . e . f r e q u e n c y ) domain. The e x t r a l i n e w i d t h i n t r o d u c e d by t h i s a c t i o n i s n o r m a l l y c a l c u l a t e d by a s s u m i n g t h a t F ( v ) t a k e s on i t s maximum e n t r o p y f o r m , a G a u s s i a n d i s t r i b u t i o n whose s t a n d a r d d e v i a t i o n A C J = C J . I kT c L M i s a good measure of t h e D o p p l e r b r o a d e n i n g . The weak c o u p l i n g a s s u m p t i o n d o e s n o t a c c o u n t f o r t h e f a c t t h a t e x c i t e d atoms have a l a r g e r Bohr r a d i u s , and t h a t f a s t e r v e l o c i t i e s l e a d t o more c o l l i s i o n s . T h e o r i e s w h i c h i n c l u d e t h e s e e f f e c t s a r e r e v i e w e d by Berman [BERM75]. F o r t h e Z - p i n c h p l a s m a however, D o p p l e r b r o a d e n i n g adds o n l y 0.015nm t o t h e t o t a l e x p e r i m e n t a l l i n e w i d t h o f n e a r l y 4nm, s u g g e s t i n g t h a t w h a t e v e r c o u p l i n g e x i s t s makes so l i t t l e c o n t r i b u t i o n t h a t p r a c t i c a l o b s e r v a t i o n s c a n n e g l e c t i t . The l i n e p r o f i l e i n t h i s e x p e r i m e n t i s c o m p l e t e l y d o m i n a t e d by t h e i n t e r a c t i o n o f e m i t t e r s and p e r t u r b e r s . The t h e o r y , t h e l i n e s p e c t r u m o f a s i n g l e e m i t t e r may be c a l c u l a t e d f r o m an ensemble o f i n d e p e n d e n t e m i t t i n g atoms. In p r a c t i c e however, t h e l i n e s p e c t r u m i s o b t a i n e d by s a m p l i n g II.THEORY 24 p h o t o n s from a f i n i t e i n t e r v a l , p o s s i b l y even s e v e r a l p h o t o n s f r o m t h e same e m i t t i n g atom. The s t a t i s t i c a l i n d e p e n d e n c e o f t h e s e atoms i s e s s e n t i a l i f t h e ensemble model i s t o be u s e d . E a r l i e r i n t h i s s e c t i o n i t was shown t h a t an o p t i c a l l y t h i n p l a s m a would g u a r a n t e e t h i s i n d e p e n d e n c e , a l t h o u g h c a r e must be t a k e n i f t h e p l a s m a has been s p e c i a l l y p r e p a r e d ( e . g . where m e t a s t a b l e s t a t e s have been a t t a i n e d t h r o u g h o p t i c a l p u m p i n g ) . In t h i s work t h e r e a r e no s u c h c o n d i t i o n s , so t h e D o p p l e r e f f e c t may be " f o l d e d i n " a f t e r t h e i n d e p e n d e n t s p e c t r u m has been c a l c u l a t e d . The p r o b l e m i s r e d u c e d t o c a l c u l a t i n g t h a t p r o f i l e a s a f u n c t i o n o f t h e p l a s m a t e m p e r a t u r e and d e n s i t y . 11. 4 C a l c u l a t i o n o f t h e L i n e Shapes T h e o r i e s of s p e c t r a l l i n e b r o a d e n i n g d a t e back as f a r as 1895, when A l b e r t M i c h e l s o n p o i n t e d o u t t h a t a t o m i c v i b r a t i o n s w ould be a l t e r e d by c o l l i s i o n s w i t h o t h e r p a r t i c l e s [MICH95]. S i n c e t h a t t i m e t h e f i e l d h as a d v a n c e d s u b s t a n t i a l l y . The modern t e c h n i q u e f o r a quantum s y s t e m w e a k l y c o u p l e d t o t h e f r e e s p a c e r a d i a t i o n s t a t e s i s t o w r i t e t h e e m i s s i o n s p e c t r u m a s t h e power s p e c t r a l d e n s i t y Pico) = 4CJ* Z 8( oi-cj ) <a\p\a> 3 c 3 a/3 a/3 </3|d|a> where t h e sum i s o v e r a l l s t a t e s |a> and |0> o f a l l t h e e m i t t e r s and p e r t u r b e r s . The d e n s i t y o p e r a t o r p and d i p o l e o p e r a t o r d i n p r i n c i p l e c o n t a i n e v e r y t h i n g t h a t may be known a b o u t t h e s y s t e m . II.THEORY 25 S i m i l a r e x p r e s s i o n s have been u s e d as a s t a r t i n g p o i n t by P e a c h [PEAC81], S m i t h [SMIT69], and G r i e m [ G R I E 6 4 ] . The power s p e c t r a l d e n s i t y P(co) i s i n t u r n g i v e n by t h e F o u r i e r t r a n s f o r m Pico) = J _ 2TT-3cos C ( s ) e d s , of t h e a u t o c o r r e l a t i o n f u n c t i o n C ( s ) = I <a|d|0>><0|U* ( s , 0 ) d u ( s , 0 ) | a> <a|p|a>, aj3 where <a|d|/3> i s t h e e x p e c t a t i o n v a l u e of t h e c l a s s i c a l d i p o l e moment. The e q u i v a l e n t c l a s s i c a l s t a t e m e n t l e a d s t o t h e L i n d h o l m - F o l e y impact p h a s e s h i f t m odel. S i n c e \a> and |/3> a r e s t a t i o n a r y s t a t e s o f t h e f u l l s y s t e m , t h e p r o b l e m becomes one o f f i n d i n g U and p. T h i s i n t u r n r e q u i r e s c h o i c e of a H i l b e r t s p a c e and i t s a p p r o p r i a t e r e p r e s e n t a t i o n . The e a r l i e r a s s u m p t i o n s o f s t a t i s t i c a l i n d e p e n d e n c e and e r g o d i c i t y mean t h a t a l l of t h e p r o p e r t i e s e x p e c t e d f o r an ensemble o f e m i t t e r s c a n be c a l c u l a t e d f r o m t h e t i m e a v e r a g e o f a s i n g l e e m i t t e r . The p r o b l e m may t h e r e f o r e be r e - e x p r e s s e d as t h a t of a s i n g l e e m i t t e r a f f e c t e d by a l a r g e number o f p e r t u r b e r s , w i t h a c o r r e s p o n d i n g H i l b e r t s p a c e formed by t h e t e n s o r p r o d u c t o f a s i n g l e atom s p a c e and w h a t e v e r e l s e i s n eeded t o d e s c r i b e t h e p l a s m a . The l i n e p r o f i l e s n e e d ed f o r s p e c t r o s c o p y a r e c a l c u l a t e d from t h e t i m e e v o l u t i o n o f t h e a t o m i c s t a t e s . II.THEORY 26 The t i m e d e p e n d e n t e v o l u t i o n o p e r a t o r i s d e t e r m i n e d by t h e t i m e d e p e n d e n t i n t e r a c t i o n p o t e n t i a l , and c a n be c a l c u l a t e d f o r m a l l y w i t h a Dyson s e r i e s i n t h e i n t e r a c t i o n r e p r e s e n t a t i o n . The end r e s u l t i s a s e r i e s (a c l u s t e r e x p a n s i o n ) w h i c h d e s c r i b e s t h e p e r t u r b a t i o n due t o one p e r t u r b e r , two p e r t u r b e r s , t h r e e , and so on. The d i f f i c u l t y o f w o r k i n g w i t h t h i s s e r i e s m o t i v a t e s t h e two a p p r o x i m a t e l i m i t s : t h e i mpact model where p e r t u r b e r s i n t e r a c t w i t h t h e e m i t t e r one a t a t i m e , and t h e q u a s i s t a t i c model where t h e y a l l a c t a t o n c e . The i m p a c t model g e t s i t s name from i t s assumed s e q u e n c e o f c o l l i s i o n s between p e r t u r b e r s and e m i t t e r s . A d i s t r i b u t i o n o f v e l o c i t i e s a n d i mpact p a r a m e t e r s g e n e r a t e s a t i m e s e r i e s o f c o l l i s i o n p e r t u r b e d e n e r g y l e v e l s , w h i c h a r e t h e n s a m p l e d u n i f o r m l y . I t i s sometimes assumed t h a t t h e i m p a c t s do n o t a l t e r t h e t r a n s i t i o n p r o b a b i l i t i e s ( i . e . t h e i n t e r a c t i o n i s a d i a b a t i c ) , b u t o t h e r m o d e l s c a n be d e r i v e d f r o m s c a t t e r i n g t h e o r y . T h e s e however a l m o s t a l w a y s have t o assume t h a t t h e s c a t t e r i n g e v e n t s a r e i s o l a t e d , so t h a t e a c h goes t o c o m p l e t i o n b e f o r e t h e n e x t e v e n t o c c u r s . T h i s a s s u m p t i o n w i l l a l s o s e t a l i m i t on t h e d e n s i t y and i n t e r a c t i o n r a t e . The q u a s i s t a t i c model e l i m i n a t e s t i m e d e p e n d e n c e a l t o g e t h e r . A l l o f t h e p e r t u r b e r s a c t a t o n c e , c o l l e c t i v e l y p r o d u c i n g an " e l e c t r i c m i c r o f i e l d " w h i c h a l t e r s t h e s t a t i o n a r y s t a t e e n e r g i e s o f t h e e m i t t e r . T r a n s i t i o n s t a k e p l a c e u n i f o r m l y a t a r a t e d e t e r m i n e d by t h e d i p o l e o p e r a t o r . The d i s t r i b u t i o n o f t h e m i c r o f i e l d i s c a l c u l a t e d f r o m t h e d e n s i t y o p e r a t o r on t h e p e r t u r b e r s u b s p a c e , and power s p e c t r a l d e n s i t y P(CJ) comes from a II.THEORY 27 u n i f o r m s a m p l i n g o f p e r t u r b e d a t o m i c e n e r g i e s . T h i s s e p a r a t e a r e a o f p l a s m a t h e o r y has been s t u d i e d e x t e n s i v e l y by B a r a n g e r , Mozer [BARA59, MOZE60], and Hooper [HOOP66, HOOP68]. The q u a s i s t a t i c model i s a d i a b a t i c by n a t u r e , s i n c e i t d e a l s w i t h s t a t i o n a r y s t a t e s . The same i s t r u e o f i m p a c t m o d e l s , a l t h o u g h sometimes t h i s c a n be h i d d e n i n t h e f o r m a l i s m , e i t h e r a s t h e " e l a s t i c c o l l i s i o n " a s s u m p t i o n , where t h e p r i n c i p a l quantum number o f t h e e m i t t e r i s l e f t u n changed by c o l l i s i o n , o r (more s u b t l y ) when t h e t i m e o r d e r i n g o p e r a t o r i s d e l e t e d f r o m t h e Dyson s e r i e s . ( S u c h a d e l e t i o n i s made i n SMIT69, a l t h o u g h t h e a u t h o r s do n o t seem t o be aware o f i t s i m p l i c a t i o n s . ) Time o r d e r i n g i s n e e ded i f t h e p o t e n t i a l o p e r a t o r a t t i m e t 0 does n o t commute w i t h t h e o p e r a t o r a t t > t 0 . E l i m i n a t i n g t h i s o p e r a t o r i s e q u i v a l e n t t o c h o o s i n g a p o t e n t i a l w h i c h v a n i s h e s o u t s i d e some f i n i t e t i m e i n t e r v a l , and w h i c h commutes a t a l l t i m e s w i t h i n t h a t i n t e r v a l . S u c h a p o t e n t i a l must commute w i t h t h e u n p e r t u r b e d H a m i l t o n i a n , p r e c i s e l y t h e r e q u i r e m e n t o f t h e a d i a b a t i c t h e o r y . R e l a x i n g t h i s a s s u m p t i o n i s by no means e a s y , and t h e i n t e r m e d i a t e s i t u a t i o n between t h e two l i m i t s s t i l l l i e s b e yond t h e r a n g e o f c u r r e n t l i n e b r o a d e n i n g t h e o r i e s . When t h e s y s t e m r e a l l y i s a d i a b a t i c however, t h e l i n e s p e c t r u m P(a>) c a n be c a l c u l a t e d f r o m t h e power s p e c t r a l d e n s i t y o f t h e e l e c t r i c f i e l d i n s i d e t h e p l a s m a , even when t h e f i e l d i s n o t assumed t o be s t a t i c . An o u t l i n e o f t h i s t e c h n i q u e and a t e s t o f i t s v a l i d i t y a r e g i v e n i n t h e n e x t s e c t i o n . II.THEORY 2 8 1 1 . 5 T e s t i n g t h e A d i a b a t i c L i m i t The a d i a b a t i c t h e o r y i s de p e n d e n t on a q u i t e s i m p l e a s s u m p t i o n , w h i c h c a n be s t a t e d i n f o r m a l l y a s a r e q u i r e m e n t t h a t t h e p r o b a b i l i t y o f p h o t o n e m i s s i o n n o t depend on t h e p e r t u r b a t i o n of t h e e n e r g y l e v e l s . (A f o r m a l s e t o f r e q u i r e m e n t s c a n be f o u n d i n a l m o s t any s t a n d a r d t e x t b o o k , e . g . MESS 5 8 , MERZ70) The d i p o l e o p e r a t o r i n t h e e x p r e s s i o n f o r P ( C J ) i s r e d u c e d t o a c o n s t a n t f u n c t i o n of t h e s t a t e s i n v o l v e d i n t h e t r a n s i t i o n . The l i n e b r o a d e n i n g p r o b l e m i s r e d u c e d t o a c a l c u l a t i o n o f t h e l o c a l a t o m i c s t a t e s and t h e i r e n e r g y l e v e l s . Suppose t h a t t h e a t o m i c H a m i l t o n i a n i s o f t h e form where H 0 r e p r e s e n t s t h e t i m e i n d e p e n d e n t H a m i l t o n i a n o f t h e u n p e r t u r b e d e m i t t e r , and V ( t ) i s t h e p o t e n t i a l e n e r g y due t o t h e p r e s e n c e o f o t h e r p a r t i c l e s . The s m a l l p a r a m e t e r X i s b o t h a measure of t h e i n t e r a c t i o n and a means f o r d e v e l o p i n g t h e a s y m p t o t i c s t r u c t u r e o f t h e s y s t e m . The S c h r o d i n g e r e q u a t i o n i s t h e n u s e d t o d e f i n e t h e t i m e e v o l u t i o n o p e r a t o r U. S i n c e H 0 i s known, U may be f a c t o r e d i n t o 11=11011, , and t h e e q u a t i o n f o r U, H ( t ) = H 0 + X V ( t ) , j h d U ( t , t 0 ) = H ( t ) U ( t , t 0 ) , d t U ( t 0 , t 0 ) = 1, jfcdU, = X U 0 V ( t ) U 0 U d t U, ( t 0 ) = 1 , II.THEORY 29 may be r e w r i t t e n f o r t o = 0 a s t h e i n t e g r a l e q u a t i o n , U , ( t ) = 1 - 0 ds U o ( s ) V ( s ) U 0 ( s ) U , ( s ) , w h i c h may t h e n be s o l v e d by i t e r a t i o n f o r a power s e r i e s i n X. H i g h e r o r d e r t e r m s i n X a r e d i f f i c u l t t o c a l c u l a t e , u n l e s s V ( t ) has t h e same e i g e n s t a t e s a s H 0 , and can t h e r e f o r e commute w i t h U 0 and U,. T h i s i s t h e m a t h e m a t i c a l e f f e c t o f a s s u m i n g t h a t no t r a n s i t i o n s o c c u r , a l t h o u g h i n s t r i c t t e r m s t h e o n l y r e q u i r e m e n t i s t h a t p r o b a b i l i t i e s o f t r a n s i t i o n due t o X V ( t ) be a s y m p t o t i c a l l y s m a l l . The power s e r i e s i s t h e n g i v e n by I f V ( s ) commutes w i t h H 0 f o r a l l s, t h e n V ( s ) w i l l commute w i t h V ( s ' ) , a l l o w i n g t h e s i m p l e r e x p r e s s i o n w h i c h i s f o r m a l l y i d e n t i c a l t o t h e s e r i e s o b t a i n e d when t i m e o r d e r i n g i s removed f r o m t h e t h e f u l l e x p a n s i o n . W i t h o u t g o i n g i n t o a l o n g d i s c u s s i o n o f t h e method, t h e l i n e b r o a d e n i n g mechanism c a n be i l l u s t r a t e d by t h e c h a n g i n g bound s t a t e e n e r g y i n t h e d i a g o n a l e l e m e n t s o f U, U 0 U = e x p - j h " 2 e t i i L i II.THEORY 30 The f i r s t e x p o n e n t i a l i s s i m p l y t h e t r a n s f o r m of a d e l t a f u n c t i o n , b u t t h e s e c o n d depends on t h e t i m e h i s t o r y o f t h e p e r t u r b a t i o n . F l u c t u a t i o n s i n V ( t ) w i l l " s p r e a d o u t " t h e F o u r i e r t r a n s f o r m o f t h e p r o d u c t . The c h a n g i n g e n e r g i e s o f two s t a t e s i n a l i g h t e m i t t i n g t r a n s i t i o n w i l l d e t e r m i n e t h e e n e r g y d i s t r i b u t i o n o f t h e e m i t t e d p h o t o n s . The s q u a r e m a g n i t u d e o f t h e t r a n s f o r m l e a d s d i r e c t l y t o t h e power s p e c t r a l d e n s i t y P ( C J ) . The f i n a l t a s k i s t o c h e c k t h e v a l i d i t y o f t h e a s y m p t o t i c e x p a n s i o n , w h i c h i m p l i e s t h a t s o m e t h i n g must be s a i d a b o u t t h e p r e c i s e f o r m o f V ( t ) . A g a i n i t i s b e s t t o a v o i d d e t a i l s , and i n s t e a d r e l y on a c r u d e e s t i m a t e made by a s s u m i n g t h a t t h e i n t e r a c t i o n i s g o v e r n e d by e l e c t r i c f i e l d s . The s m a l l p a r a m e t e r X c a n t h e n be e s t i m a t e d from t h e r a t i o o f t h e Coulomb f i e l d t o t h e e l e c t r i c f i e l d imposed by t h e p l a s m a . The p r o b a b i l i t y d i s t r i b u t i o n o f t h e l a t t e r was f i r s t c a l c u l a t e d i n 1919 by H o l t s m a r k [HOLT19], and t h e s t a n d a r d s c a l e p a r a m e t e r i s u s u a l l y r e f e r r e d t o a s t h e H o l t s m a r k m i c r o f i e l d , The Coulomb f i e l d i s e s t i m a t e d c l a s s i c a l l y u s i n g t h e Bohr r a d i u s o f t h e s t a t e w i t h p r i n c i p a l quantum number n, F o r t h e He I I n=4 t o n=3 t r a n s i t i o n a t 469nm, t h e Bohr r a d i u s a 0 i s a p p r o x i m a t e l y 5.3.10" 1'm. The n o m i n a l t e m p e r a t u r e and d e n s i t y o f t h e Z - p i n c h g i v e E 1 . Ze 47re 0 a j C n z ( n + 1 / 2 ) ( n + 1 ) II.THEORY 31 X = E / E * 0.01 . H C A l t h o u g h t h e H o l t s m a r k d i s t r i b u t i o n o f f i e l d s t r e n g t h s e x t e n d s f r o m z e r o t o i n f i n i t y , most o f t h e d i s t r i b u t i o n l i e s c l o s e enough t o z e r o t o l e a v e X a s a r e a s o n a b l y s m a l l number. A f u l l d e s c r i p t i o n o f t h e l i n e b r o a d e n i n g mechanism i s s t i l l q u i t e c o m p l i c a t e d , and n o t h i n g has been s a i d a t a l l a b o u t t h e m o t i o n o f t h e p e r t u r b e r s . T h e r e a r e a l s o q u e s t i o n s a b o u t t h e a c c u r a c y (and c o n v e r g e n c e ! ) o f t h e a s y m p t o t i c s e r i e s . The v a l u e of X g i v e n i n t h i s s e c t i o n i s r e a l l y n o t s m a l l enough t o j u s t i f y s t o p p i n g w i t h j u s t one t e r m . N o n e t h e l e s s , t h e s e t o p i c s a r e w e l l b e y o n d t h e s c o p e o f t h i s t h e s i s . E v e r y t h i n g beyond t h i s s e c t i o n w i l l be done w i t h t h e a s s u m p t i o n t h a t t h e a d i a b a t i c l i m i t i s c o r r e c t . II.THEORY 32 I I . 6 T e m p e r a t u r e and D e n s i t y Measurement G i v e n t h a t t h e p l a s m a w i l l e m i t l i g h t a c c o r d i n g t o t h e a d i a b a t i c t h e o r y , s e v e r a l methods o f s p e c t r o s c o p i c measurement a r e a v a i l a b l e f o r e x p e r i m e n t a l u s e . In t h e c a s e o f d e n s i t y , c u r r e n t b e s t p r a c t i c e t e c h n i q u e i s t o use t h e s e m i - e m p i r i c a l f o r m u l a s g i v e n by G r i e m [ G R I E 7 4 ] . T h e s e r e p r e s e n t a l i m i t i n g c a s e where D o p p l e r , i m p a c t , and q u a s i s t a t i c b r o a d e n i n g a r e assumed t o be i n d e p e n d e n t i n s u c h a way t h a t t h e i r r e s p e c t i v e e f f e c t s c a n be a d d e d t h r o u g h c o n v o l u t i o n . In t h e c a s e of t e m p e r a t u r e , p r e v i o u s o b s e r v a t i o n s of s i m i l a r Z - p i n c h p l a s m a s have r e l i e d on Mewe's c a l c u l a t i o n o f t h e i n t e n s i t y r a t i o between He I I 469nm and He I 588nm [MEWE67]. A p p r o x i m a t e forms of MEWE67 and GRIE74 were u s e d i n t h i s e x p e r i m e n t . A g e n e r a l o u t l i n e o f t h e d e n s i t y measurement was g i v e n i n s e c t i o n s I I . 3 t o I I . 5 , so f u r t h e r r e i t e r a t i o n w i l l n o t be needed h e r e . The r e m a i n i n g p a r t o f t h i s s e c t i o n i s d e v o t e d e n t i r e l y t o t h e c a l c u l a t i o n o f p l a s m a t e m p e r a t u r e f r o m s p e c t r a l l i n e i n t e n s i t y r a t i o s . F o l l o w i n g MEWE67, t h e t o t a l i n t e n s i t y I o f a s p e c t r a l l i n e o f w a v e l e n g t h X c a n be r e l a t e d t o t h e p o p u l a t i o n d e n s i t y n + ( r ) i n i t s u p p e r l e v e l ( w i t h p r i n c i p a l quantum number r ) and s t a t i s t i c a l w e i g h t g + ( r ) by I = C . n « ( r ) . gf , ? t T H X* p r o v i d e d t h a t t h e p l a s m a i s o p t i c a l l y t h i n ( i . e . no s e l f a b s o r p t i o n ) , and t h a t t h e l o w e r s t a t e o f t h e t r a n s i t i o n i s n o t II.THEORY 33 t h e g r o u n d s t a t e . The c o n s t a n t s g and f a r e r e s p e c t i v e l y t h e s t a t i s t i c a l w e i g h t o f t h e l o w e r l e v e l and t h e " a b s o r p t i o n o s c i l l a t o r s t r e n g t h " d e t e r m i n e d by a v e r a g i n g o v e r a l l t h e f i n e s t r u c t u r e components. ' The c o n s t a n t C a r i s e s f r o m t h e o p t i c a l g e ometry and i s t h e r e f o r e i d e n t i c a l f o r a l l s p e c t r a l l i n e s . The s t a t i s t i c a l w e i g h t g + ( r ) i s known f r o m t h e quantum m e c h a n i c s o f t h e atom. R e l a t i n g t h i s l i n e i n t e n s i t y t o t h e p l a s m a t e m p e r a t u r e i n v o l v e s c a l c u l a t i n g n + ( r ) f r o m t h e s y s t e m d e n s i t y o p e r a t o r , w h i c h i n t h e r m a l e q u i l i b r i u m w i l l be g i v e n by t h e maximum e n t r o p y p r i n c i p l e . The v a l u e o f n + ( r ) so o b t a i n e d i s u s u a l l y r e f e r r e d by t h e name of i t s c l a s s i c a l e q u i v a l e n t : t h e S a h a - B o l t z m a n n p o p u l a t i o n . The t e m p e r a t u r e and d e n s i t y e x p e c t e d f o r t h e Z - p i n c h w i l l a l l o w n + ( r ) t o be c a l c u l a t e d f r o m t h e c l a s s i c a l e q u a t i o n of t h e same name, a l t h o u g h i n g e n e r a l t h e r e w i l l be c o r r e c t i o n f a c t o r s b ( r , Z ) t o a c c o u n t f o r p o p u l a t i o n d e n s i t i e s w h i c h a r e f a r f r o m t h e i r e q u i l i b r i u m v a l u e s . T h i s m o d i f i e d S a h a - B o l t z m a n n e q u a t i o n i s g i v e n i n MEWE67 as where n + ( r , Z ) i s t h e p o p u l a t i o n d e n s i t y o f t h e l e v e l w i t h p r i n c i p a l quantum number r , and Z r e f e r s t o t h e n e t c h a r g e on t h e e m i t t i n g atom (Z=0, 1 o r 2 f o r He I , I I , o r I I I r e s p e c t i v e l y ) . The o t h e r d e n s i t i e s N and n,(r,Z+1) r e f e r t o t h e e l e c t r o n s and t h e n e x t i o n i z e d s p e c i e s . The s t a t i s t i c a l n . ( r , Z ) = b ( r , Z ) g + ( r , Z ) II.THEORY 34 w e i g h t s g + and g, have t h e same meaning a t t a c h e d t o t h e i r s u b s c r i p t s and arguments a s N and n t . F i n a l l y , t h e e n e r g y t e r m E ( r , Z ) i s t h e e n e r g y o f i o n i z a t i o n f r o m l e v e l r t o t h e c o n t i n u u m . F o r t h e h y d r o g e n i c i o n He I I a n d t h e q u a s i -h y d r o g e n i c u p p e r l e v e l s o f He I , t h i s i s g i v e n by E ( r , Z ) = 1 3 . 6 ' ( Z + 1 ) 2 . r " 2 . The r e m a i n i n g symbols have t h e i r u s u a l meaning. S u b s t i t u t i n g t h i s e x p r e s s i o n f o r t h e p o p u l a t i o n d e n s i t i e s i n t o t h e e a r l i e r e q u a t i o n f o r I , and t a k i n g a r a t i o of two l i n e i n t e n s i t i e s t o e l i m i n a t e t h e c o n s t a n t C, t h e He I I 469nm t o He I 588nm l i n e r a t i o c a n be p u t i n t h e form Iq.s<> = K 0- [kTJ / • exp J" E_p_ 1 . I 5 8 8 Kl" The c o n s t a n t s K 0 and E 0 a r e d e t e r m i n e d by t h e p r e v i o u s e q u a t i o n f o r E ( r , Z ) and t a b l e s o f b and g g i v e n i n MEWE67. An i n v e r s e f o r m o f t h i s e x p r e s s i o n w i l l be u s e d i n C h a p t e r V. The i n v e r s e i s o n l y a p p r o x i m a t e however, b e c a u s e u l t i m a t e l y t h e r e a r e o n l y two r e a s o n s i n t h i s e x p e r i m e n t f o r w a n t i n g t o know t h e t e m p e r a t u r e and d e n s i t y i n s i d e t h e Z - p i n c h : f i r s t t o compare t h e s o l i d a nd open ended p i n c h v e s s e l s , a n d s e c o n d t o d e t e r m i n e t h e e x i s t e n c e o f a p l a s m a e x p a n s i o n wave. I t i s t h e r e f o r e p o s s i b l e t o s i m p l i f y t h e e x a c t c a l c u l a t i o n s i n GRIE74 and MEWE67 by r e p l a c i n g them w i t h T a y l o r s e r i e s c l o s e t o t h e t e m p e r a t u r e and d e n s i t y a l r e a d y known t o o c c u r i n t h e s o l i d end s y s t e m . Any l a r g e d e v i a t i o n w i l l g i v e n o t i c e t h a t a f r e s h II.THEORY 35 c a l c u l a t i o n s h o u l d be done, b u t i n p r a c t i c e t h i s t u r n s o u t t o be u n n e c e s s a r y . C h a p t e r V w i l l examine t h i s t o p i c i n more d e t a i l . A f i n a l p o i n t t h a t s h o u l d be made i s t h a t MEWE67 r e l i e s on t h e a d i a b a t i c t h e o r y of p h o t o n e m i s s i o n , j u s t a s much a s t h e d e n s i t y t h e o r i e s o u t l i n e d e a r l i e r o r i n GRIE74. Any a t t e m p t t o j u s t i f y t h e n e c e s s a r y l o c a l thermodynamic e q u i l i b r i u m w i t h c o l l i s i o n r a t e c a l c u l a t i o n s w i l l be s u b j e c t t o t h e same i n c o n s i s t e n c y . T e s t s a p p l i e d i n s t e a d t o t h e s p e c t r o s c o p i c d a t a i t s e l f w i l l be d i s c u s s e d a l o n g w i t h t h e o t h e r c a l c u l a t i o n s i n C h a p t e r V. I I . 7 F o r m a l i z i n g t h e I n i t i a l V a l u e P r o b l e m Now t h a t t h e t e m p e r a t u r e and d e n s i t y a r e r e d u c e d t o m a t t e r s o f p r a c t i c a l c a l c u l a t i o n , t h e same must be done w i t h t h e models t h a t a r e u s e d t o p r e d i c t them. The t h r e e m a j o r a s s u m p t i o n s n e e ded t o do t h i s w i l l be c o v e r e d i n t h e p r e s e n t s e c t i o n . The f i r s t m a j o r a s s u m p t i o n i s t h a t p l a s m a i n t h e Z - p i n c h c a n be t r e a t e d a s a one d i m e n s i o n a l s y s t e m . T h i s r e q u i r e s f i r s t t h a t t h e r a d i a l d e p e n d e n c e o f t h e p l a s m a s t a t e v a r i a b l e s be u n i f o r m a l o n g t h e e n t i r e a x i s , and s e c o n d t h a t t h e f o r c e s e x e r t e d on t h e p l a s m a p a r t i c l e s be s e p a r a b l e i n t o a x i a l and r a d i a l c o mponents. The f o r m e r i s shown c o n c l u s i v e l y i n PRES74 by t h e s m a l l c u r v a t u r e o f t h e p i n c h i n g s hock wave ( l e s s t h a n 3mm o v e r a 600mm l e n g t h ) . The l a t t e r f o l l o w s i n t h a t c u r r e n t w i t h an e q u a l l y u n i f o r m d i s t r i b u t i o n w i l l p r o d u c e a u n i f o r m c i r c u l a r m a g n e t i c f i e l d . F o r c e s e x e r t e d on t h e p l a s m a p a r t i c l e s w i l l t h e r e f o r e be s e p a r a b l e i n t o p a r a l l e l and p e r p e n d i c u l a r II.THEORY 36 components, w i t h t h e a x i a l and r a d i a l d i r e c t i o n s b o t h b e i n g p e r p e n d i c u l a r . The a s s u m p t i o n o f i n d e p e n d e n t a x i a l d y n a m i c s i s t h e r e f o r e l i k e l y t o be j u s t i f i e d . The s e c o n d major a s s u m p t i o n i s t h a t p l a s m a t e m p e r a t u r e and d e n s i t y a r e r o u g h l y c o n s t a n t o v e r t h e p e r i o d o f gas d y n a m i c a l a c t i o n . T h i s a s s u m p t i o n i s somewhat more d i f f i c u l t t o j u s t i f y , m a i n l y b e c a u s e o f t h e l a r g e i n i t i a l d e n s i t y peak c r e a t e d by t h e p i n c h i n g s h o c k wave. The i n i t i a l i m p e t u s g i v e n t o t h e c o l d gas end p l u g may v e r y w e l l d i s t o r t t h e o v e r a l l e n e r g y b a l a n c e . I n C h a p t e r V, f o r example, i t w i l l be shown t h a t t h e c o n t a c t s u r f a c e i s s l o w e r , and t h e shock p r e s s u r e h i g h e r , t h a n t h o s e p r e d i c t e d by a c o n s t a n t p r e s s u r e s hock m o d e l . The argument i n s e c t i o n I I . 9 e x p l a i n s t h i s d i s c r e p a n c y by p o s t u l a t i n g a h e a t f l u x , b u t an e q u a l l y v a l i d h y p o t h e s i s i s t h a t t h e s hock r e g i o n i s e x p a n d i n g a f t e r an i n i t i a l s t r o n g c o m p r e s s i o n by t h e peak d e n s i t y p l a s m a . The argument a g a i n s t t h i s t h e o r y i s t h a t shock v e l o c i t i e s a r e o b s e r v e d t o be c o n s t a n t o v e r a p e r i o d o f a t l e a s t t e n m i c r o s e c o n d s , l o n g e r t h a n t h e f o u r m i c r o s e c o n d d u r a t i o n o f t h e d e n s i t y p e a k . T h e r e i s a l s o a f u r t h e r two m i c r o s e c o n d l a g between t h e f i r s t p l a s m a t o g a s c o n t a c t and t h e f i r s t a p p e a r a n c e o f t h e shock wave i n t h e f i e l d o f v i e w o f t h e s t r e a k c a m e r a . T h i s l o n g p e r i o d s t a b i l i t y i s c o n s i s t e n t w i t h a u n i f o r m p r e s s u r e i n t h e d r i v i n g p l a s m a , s u g g e s t i n g t h a t any c o n t r i b u t i o n f r o m t h e s t r o n g i n i t i a l t h r u s t c a n be a v e r a g e d o v e r t h e e n t i r e i n t e r a c t i o n t i m e . II.THEORY 37 G i v e n t h e s e two major a s s u m p t i o n s , b a s i c e q u a t i o n s f o r t h e c o n s e r v a t i o n of mass, e n e r g y , and momentum may be w r i t t e n f o r a one d i m e n s i o n a l f l o w . The t h i r d major a s s u m p t i o n i s t h a t t h e f l o w can. be b r o k e n up i n t o a number o f u n i f o r m r e g i o n s , so t h a t c o n s e r v a t i o n e q u a t i o n s a r e f u r t h e r r e d u c e d t o jump c o n d i t i o n s a t t h e r e g i o n i n t e r f a c e s . E a c h e l e m e n t a l r e g i o n i s a volume o f m a t e r i a l e x i s t i n g a t some d e f i n i t e s t a t e . Changes i n i t s s i z e and p o s i t i o n a r e r e l a t e d ( t h r o u g h t h e c o n s e r v a t i o n e q u a t i o n s ) t o t h e i n t e r n a l s t a t e s of n e i g h b o r i n g r e g i o n s and t h e f l o w o f e n e r g y a c r o s s t h e c o n t a c t s u r f a c e . The n e t r e s u l t i s a s e t o f a l g e b r a i c e q u a t i o n s c o n n e c t i n g t h e p r o p e r t i e s o f a l l t h e r e g i o n s i n t h e s y s t e m . Knowledge o f t h e i r v olumes and i n t e r n a l s t a t e s a t any one t i m e i s s u f f i c i e n t t o p r e d i c t t h e i r e n t i r e l a t e r d e v e l o p m e n t . The d i f f e r e n c e between a u n i f o r m f l o w model and one d e r i v e d from a f u l l s e t o f d i f f e r e n t i a l e q u a t i o n s i s t h a t u n i f o r m r e g i o n s have no i n t e r n a l g r a d i e n t s ( i . e . f u r t h e r s u b d i v i s i o n w i l l not improve t h e a c c u r a c y ) . I n a u n i f o r m f l o w model t h e r e i s n o t h i n g t o be g a i n e d by h a v i n g more r e g i o n s t h a n a r e n e e ded t o d e s c r i b e t h e p r o c e s s e s c u r r e n t l y u n d e r s t u d y . The m o d els i n s e c t i o n s I I . 8 and I I . 9 w i l l a l s o t a k e a d v a n t a g e o f e x i s t i n g knowledge a b o u t f l u i d s y s t e m s w i t h l a r g e i n i t i a l p r e s s u r e jumps. I t i s a l r e a d y known, f o r example, t h a t t h e i n i t i a l s t a t e o f F i g u r e 1-1 w i l l e v o l v e i n t o some s o r t o f wave s t r u c t u r e , and r e g a r d l e s s o f t h e d y n a m i c s t h e r e w i l l a l w a y s be two r e g i o n s f a r enough away from t h e i n t e r f a c e t o be c o n s i d e r e d u n d i s t u r b e d . P a s t s t u d i e s o f s i m i l a r s y s t e m s have shown t h a t t h e edges o f t h e s e u n d i s t u r b e d r e g i o n s w i l l be formed II.THEORY 38 by a shock wave i n t h e gas and an e x p a n s i o n wave i n t h e p l a s m a . A model o f t h e s y s t e m between t h e s e two waves c a n t h e n be d e v e l o p e d t o w h a t e v e r l e v e l o f d e t a i l seems n e c e s s a r y . I n t h i s t h e s i s , i n v e s t i g a t i o n o f t h e A h l b o r n - S i n n o t end c o n f i n e m e n t scheme makes i t a d e q u a t e t o p r e d i c t t h e m o t i o n o f t h e p l a s m a / g a s i n t e r f a c e a n d t h e h e a t f l u x a c r o s s i t . The r e l e v a n t s e t o f p a r a m e t e r s i s t h e minimum needed t o d e s c r i b e t h e s e e f f e c t s . 4 EXPANSION FAN CONTACT SURFACE SHOCK FRONT 11 — 1 U n i f o r m Flow S t r u c t u r e The f o r m a l v e r s i o n o f t h e A h l b o r n - S i n n o t model s t a r t s w i t h a f o u r r e g i o n f l o w s y s t e m : a s t r u c t u r e o f t h e f o r m shown i n F i g u r e I I - l . R e g i o n s one and f o u r a r e t h e u n d i s t u r b e d g a s and p l a s m a . R e g i o n two i s t h e shock c o m p r e s s e d gas a h e a d o f t h e r i g h t w a r d m oving c o n t a c t s u r f a c e . B e h i n d t h e c o n t a c t s u r f a c e t h e expanded p l a s m a forms r e g i o n t h r e e , and t h e p r e s s u r e II.THEORY 39 g r a d i e n t between t h i s r e g i o n and t h e u n d i s t u r b e d p l a s m a o f r e g i o n f o u r i s n o r m a l l y c a l l e d t h e e x p a n s i o n f a n ( f r o m t h e shape of i t s c h a r a c t e r i s t i c t r a c e s ) . The l e a d i n g edge of t h e e x p a n s i o n f a n t r a v e l s a t t h e p l a s m a sound v e l o c i t y . F i n a l l y , h e a t t r a n s f e r f r o m t h e p l a s m a t o t h e gas i s a c c o u n t e d f o r by a h e a t f l u x q, a l t h o u g h any d e r i v a t i o n t h a t i n c l u d e s q w i l l a l s o r e q u i r e t h a t h e a t f l o w w i t h i n t h e r e g i o n s be i n s t a n t a n e o u s , so t h a t p r e s s u r e and t e m p e r a t u r e s t a y u n i f o r m t h r o u g h o u t . The assumed e x i s t e n c e of u n i f o r m f l o w r e g i o n s w i l l r e s t r i c t q t o c o m p a r a t i v e l y low v a l u e s . S e t t i n g q t o z e r o makes t h e A h l b o r n - S i n n o t model e q u i v a l e n t t o t h e c l a s s i c a l shock t u b e . C a l c u l a t i o n o f t h e h e a t f l u x q w i l l depend on two c o n d i t i o n s : f i r s t t h a t i t s c o r r e s p o n d i n g s e t o f model v a r i a b l e s i s a d e q u a t e t o d e s c r i b e t h e s y s t e m , and s e c o n d t h a t t h e p r e d i c t e d model s t a t e s a r e o b s e r v a b l e t h r o u g h means c o n s i s t e n t w i t h t h e o t h e r model a s s u m p t i o n s . The g e n e r a l o b j e c t i v e w i t h i n t h i s frame o f r e f e r e n c e i s t o u n i f y p r o c e s s and measurement i n t o a s i n g l e s t a t e m e n t of t h e f o r m : "Plasma P i s b r o u g h t i n c o n t a c t w i t h gas G. In t h e c o u r s e of t h e i r i n t e r a c t i o n S s h o u l d be o b s e r v e d . " F a i l u r e t o o b s e r v e S i n p r a c t i c e means t h a t t h e r e i s a f a l s e a s s u m p t i o n i n t h e f o r m a l s y s t e m . The b a s i c p r e m i s e of t h i s e x p e r i m e n t i s t h a t s u c h a s s u m p t i o n s a r e c o n f i n e d t o t h e p r o c e s s m o d e l s , and i n f a c t a l l o w a c l e a r c h o i c e t o be made between them. F i n a l l y , much o f t h e i n f o r m a t i o n u s e d i n t h i s t h e s i s was o b t a i n e d f r o m p r i o r work i n r e l a t e d f i e l d s . The shock t u b e model was t a k e n f r o m an e x i s t i n g v e r s i o n p r e s e n t e d by Whitham II.THEORY 40 [WHIT74], s u b j e c t t o t h e m i n o r change t h a t t h e shock s t r e n g t h z a p p e a r s i n t h i s t h e s i s under t h e new name of $. Whitham's a n a l y s i s was t h e n j o i n e d w i t h t h e a s s u m p t i o n s of A h l b o r n and S i n n o t t o d e r i v e t h e u n i f o r m f l o w m o d e l . D i m e n s i o n a l c h e c k i n g was s i m p l i f i e d by u s i n g Systeme I n t e r n a t i o n a l u n i t s t h r o u g h o u t . F u r t h e r m o r e , t h e Z - p i n c h p l a s m a i s a l r e a d y known i n p a r t f r o m PRES74, and t h i s h e l p e d i n s e l e c t i n g measurement p r o c e d u r e s v a l i d f o r t h e e x p e c t e d r a n g e o f d a t a . O t h e r u s e f u l m a t e r i a l on t h e e q u i l i b r i u m h e l i u m p l a s m a was a v a i l a b l e f r o m t h e p u b l i s h e d t a b l e s o f L i c k and Emmons [ L I C K 6 2 ] . The model n o m e n c l a t u r e i s summarized i n T a b l e I I - l , and t h e n o m i n a l v a l u e s o f t h e p l a s m a and gas s t a t e v a r i a b l e s i n T a b l e I I - 2 . The m o d e l s t h e m s e l v e s a r e p r e s e n t e d i n t h e f o l l o w i n g two s e c t i o n s . .THEORY V a r i a b l e s A a c E h M N P q R T t U V z 7 P S u b s c r i p t s 1 2 3 4 11 — 1 Model N o m e n c l a t u r e c r o s s s e c t i o n a l a r e a o f shock t u b e (m 2) sound v e l o c i t y ( m . s - 1 ) s p e c i f i c h e a t a t c o n s t a n t volume ( J . k g _ 1 . K " 1 e n e r g y ( j o u l e s ) s p e c i f i c e n t h a l p y ( J . k g ~ 1 ) mass p r e s e n t i n r e g i o n 2 (kg) number d e n s i t y (m~ 3) p r e s s u r e ( p a s c a l s = N . n r 2 ) h e a t f l u x from r e g i o n 3 t o 2 (W.irr 2) h e l i u m gas c o n s t a n t ( J . k g " • .K~ 1) t e m p e r a t u r e ( k e l v i n ) t i m e ( s e c o n d s ) shock f r o n t v e l o c i t y (m.s~ 1) c o n t a c t s u r f a c e v e l o c i t y (m.s~ 1) a x i a l p o s i t i o n c o - o r d i n a t e (m) e n t h a l p y c o e f f i c i e n t ( d i m e n s i o n l e s s ) mass d e n s i t y ( k g . n r 3 ) m e c h a n i c a l e n e r g y f l u x (W.nr 2) shock s t r e n g t h $ = ( p 2 ~ P i ) / p i ( d i m e n s i o n l e s s ) u n d i s t u r b e d c o l d gas shock c o m p r e s s e d gas expanded p l a s m a u n d i s t u r b e d p l a s m a II.THEORY I I - 2 Summary o f N o m i n a l P a r a m e t e r s C o l d Gas Ahead o f Shock C o m p o s i t i o n E n t h a l p y C o e f f i c i e n t T e m p e r a t u r e P r e s s u r e Mass D e n s i t y Sound Speed Plasma D r i v e r ( f r o m t=15/us t o t=48us) C o m p o s i t i o n : e q u i l i b r i u m mole f r a c t i o n s [ L I C K 6 2 ] E n t h a l p y C o e f f i c i e n t T e m p e r a t u r e P r e s s u r e Mass D e n s i t y E l e c t r o n Number D e n s i t y Sound Speed V a r i a t i o n i n N„ and T a H e l i u m (99.995% p u r e ) 7 i = 1.667 T, = 300K p, = 525 p a s c a l s p, = 8.44x10-" kg.m" 3 a, = 1018 m.s" 1 He 0.28% He + 49.68% H e + + 0.12% e- 49.92% 7a = 1.18 T, = 35000K p„ = 101300 p a s c a l s p« = 6.98x10-" kg.m" 3 N« = 8 X 1 0 2 2 m'3 a„ = 13090 m.s" 1 +10%, -30% S c a l e d V a r i a b l e s T e m p e r a t u r e Number D e n s i t y T = T/T, n = N/N, G e n e r a l I d e a l Gas C o n s t a n t i n p=pRT R = 2073 J . k g - 1 . K " 1 s p e c i f i c t o h e l i u m II.THEORY 43 I I . 8 The Shock Tube Model I f t h e r e i s no h e a t t r a n s f e r a c r o s s t h e c o n t a c t s u r f a c e , t h e s y s t e m shown i n F i g u r e 11 — 1 w i l l e v o l v e a s shown below. i * EXPANSION FAN I HOT PLASMA !§fm / CONTACT SURFACE / SHOCK FRONT J&m COLD GAS Z = o I I - 2 The S i m p l e Shock Wave Once a g a i n , r e g i o n one i s t h e u n d i s t u r b e d gas p l u g ; r e g i o n f o u r i s t h e u n d i s t u r b e d p l a s m a . The v e l o c i t y o f t h e c o n t a c t s u r f a c e i s g i v e n by V, and t h e c o r r e s p o n d i n g s hock v e l o c i t y i s g i v e n by U. E a c h r e g i o n i s i n e q u i l i b r i u m w i t h i t s s t a t e d e f i n e d by a p r e s s u r e p and d e n s i t y p. The i n t e r n a l e n e r g y o f e a c h r e g i o n i s g i v e n by i t s s p e c i f i c e n t h a l p y , h = _x_ £, 7-1 p where t h e c o e f f i c i e n t y may a l s o depend on p and p, but i n a l l II.THEORY 44 c a s e s w i l l l i e between 1.1 and 1.67. The v a l u e o f 1.67 a p p l i e s t o t h e i d e a l gas where 7 i s t h e r a t i o o f s p e c i f i c h e a t s , and where p r e s s u r 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 t h e e n e r g y s t o r e d i n e x t r a e n e r g y s t o r e d i n t h e bound s t a t e s o f e x c i t e d atoms. The f a c t o r o f 7/(7-1) means t h a t a s m a l l e r 7 w i l l i n c r e a s e t h e s p e c i f i c e n t h a l p y a t any g i v e n p r e s s u r e . The shock t u b e model i s w e l l known, and i t s u f f i c e s t o q u o t e t h e g o v e r n i n g e q u a t i o n f r o m WHIT74, where t h e shock s t r e n g t h S=(p2~Pi )/P i c a n be f o u n d by i t e r a t i o n . The e n t h a l p y c o e f f i c i e n t s 7, and 7, a r e t a k e n from t h e t a b l e s o f L i c k and Emmons [ L I C K 6 2 ] . The sound v e l o c i t i e s a, and a f t a r e c a l c u l a t e d f r o m t h e r e l a t i o n a 2=7p/p. The n o m i n a l i n i t i a l v a l u e s g i v e an e s t i m a t e d p l a s m a p r e s s u r e ( a g a i n from LICK62) o f a p p r o x i m a t e l y one a t m o s p h e r e : 101300 p a s c a l s . I f a, p, and 7 a r e g i v e n t h e i r n o m i n a l v a l u e s , t h e n $=108. I t s s e n s i t i v i t y t o e r r o r c a n be c h e c k e d by v a r y i n g T „ , a r g u a b l y t h e l e a s t - k n o w n p a r a m e t e r . A t h i r t y p e r c e n t change i n T a c h a n g e s 7, and $ by o n l y two p e r c e n t , and a l t h o u g h t h i s change i n t e m p e r a t u r e a l s o c h a n g e s p,, t h e p r e s s u r e f a c t o r i n $ i s r a i s e d t o an i n f l u e n c e s t i f l i n g power o f 0.08. A shock s t r e n g t h o f 108 c a n t h e r e f o r e be u s e d w i t h c o n f i d e n c e . A shock o f t h i s s t r e n g t h c a n be a n a l y z e d w i t h t h e a s y m p t o t i c r e l a t i o n s f o r l a r g e $ ( a l s o f r o m WHIT74). I f U i s t h e f r e e m o t i o n k i n e t i c s t a t e s . S m a l l e r v a l u e s a c c o u n t f o r t h e 2 • a n 7«-1 a . II.THEORY 4 5 t h e o b s e r v e d shock v e l o c i t y , t h e n t h e c o n t a c t s u r f a c e v e l o c i t y V and shock p r e s s u r e p 2 a r e g i v e n by V = 2 U + o(rM, 7+1 p 2 = 2 P l u 2 + o ( r 1 ) . 7+1 When t h e n o m i n a l v a l u e s f r o m T a b l e I I - 2 a r e i n s e r t e d i n t o t h e shock t u b e model, i t s c h i e f p r e d i c t i o n i s an e x p a n s i o n wave w h i c h r e d u c e s t h e p l a s m a p r e s s u r e by f i f t y - s i x p e r c e n t ( t o t h e shock p r e s s u r e p 2 ) . T h i s r e d u c t i o n may a p p e a r i n e i t h e r t h e t e m p e r a t u r e o r d e n s i t y ( o r p o s s i b l y b o t h ) , but i n any e v e n t a f i f t y - s i x p e r c e n t change s h o u l d n o t be d i f f i c u l t t o t e s t e x p e r i m e n t a l l y . A n o t h e r p r e d i c t i o n i s t h a t t e m p e r a t u r e i n t h e shock w i l l be r o u g h l y 14500K, h o t enough t o e m i t l i g h t d e t e c t a b l e w i t h a s t r e a k c a m e r a . A p l a s m a t e m p e r a t u r e o f 35000K means t h a t c l e a r d i f f e r e n c e s s h o u l d be seen between t h e v a r i o u s r e g i o n s . I f no e x p a n s i o n wave i s f o u n d , o r i f t h e shock c o n d i t i o n s a r e n o t t h o s e p r e d i c t e d by t h e model, t h e n one of t h e u n d e r l y i n g p r e m i s e s must be f a l s e . The most l i k e l y c a n d i d a t e i s t h e f i r s t a s s u m p t i o n o f no h e a t t r a n s f e r a c r o s s t h e c o n t a c t s u r f a c e . The f o l l o w i n g s e c t i o n w i l l p r e s e n t an a l t e r n a t e model where t h i s a s s u m p t i o n i s r e l a x e d and h e a t f l u x i s a l l o w e d . II.THEORY 46 11.9 D e r i v a t i o n o f t h e U n i form Flow M o d e l An i n t u i t i v e argument s u g g e s t s t h a t h e a t f l o w i n g from t h e p l a s m a t o t h e gas w i l l i n c r e a s e t h e gas p r e s s u r e and hence t h e v e l o c i t y o f t h e shock wave. In t h i s s e c t i o n i t i s shown t h a t a h e a t f l u x q w i l l a f f e c t t h e v e l o c i t i e s o f b o t h t h e shock wave and t h e c o n t a c t s u r f a c e , a s w e l l as i n c r e a s i n g t h e shock p r e s s u r e and r e d u c i n g t h e e x p a n s i o n o f t h e p l a s m a . The f o l l o w i n g d e r i v a t i o n shows how e x p e r i m e n t a l measurements of U, V and c a n be u s e d t o c a l c u l a t e q . I f t h e p r e s s u r e i n t h e gas i s u n i f o r m and c o n s t a n t , and i f U and V do n o t change w i t h t i m e , t h e n t h e u p s t r e a m d e n s i t y w i l l d e t e r m i n e t h e d e n s i t y o f t h e s h o c k . The shock r e g i o n sweeps up mass a t a r a t e p,UA. The shock volume expands a t a r a t e (U-V)A. I n i t i a l mass and volume a r e b o t h z e r o , so t h e r a t i o p 2 o f shock mass t o s h o c k volume i s j u s t t h e r a t i o o f t h e r a t e s , namely p 2 = p,U, U-V w h i c h a l s o i m p l i e s t h a t mass i s c o n s e r v e d a t t h e i n t e r f a c e . N o te t h a t i f t h e p r e s s u r e p 2 i s f i x e d by t h e d r i v i n g p i s t o n , and i f d e n s i t y p 2 i s f i x e d as a bove, t h e n t e m p e r a t u r e i n t h e s h o c k must a l s o be c o n s t a n t . The e x p e r i m e n t a l i m p l i c a t i o n i s t h a t e n e r g y f r o m q must show up i n t h e v e l o c i t i e s . The shock gas c a n n o t be h e a t e d w i t h o u t c h a n g i n g U and V. The p r e s s u r e p 2 i s r e l a t e d t o p, by t h e c o n s e r v a t i o n of momentum. In an i n f i n i t e s i m a l t i m e 6 t , t h e shock gas r e c e i v e s II.THEORY 47 new momentum p 2 A 6 t f r o m t h e p l a s m a p i s t o n . In t h e same 6 t , new mass p,UA5t i s add e d t o t h e shock and a c c e l e r a t e d t o v e l o c i t y V . I t t h e r e f o r e g a i n s momentum p,UA6tV, e x a c t l y t h a t s u p p l i e d by t h e d r i v i n g p i s t o n . The A5t c a n c e l s , l e a v i n g p 2 = P T U V . E n e r g y c o n s e r v a t i o n f i n i s h e s t h e p i c t u r e . The e n e r g y i n p u t comes from t h e h e a t f l u x q and t h e f o r c e o f t h e p i s t o n a c t i n g o v e r d i s t a n c e V 5 t . In t i m e 5t t h e s e c o n t r i b u t e e n e r g y 6 E + = p 2 A V 5 t + q A S t . B e c a u s e t h e shock gas t e m p e r a t u r e i s c o n s t a n t , i t s i n t e r n a l e n e r g y s t a y s u n c h a n g e d . A l l o f 8 E + goes i n t o t h e new mass P l U A 5 t , p a r t l y a s k i n e t i c e n e r g y ( p ! U A 6 t ) V 2 / 2 , b ut a l s o i n t h e h e a t i n g and c o m p r e s s i o n n e e d e d t o b r i n g t h e new mass t o t h e t e m p e r a t u r e and d e n s i t y o f t h e s h o c k . The p r o c e s s i s shown i n F i g u r e I I - 3 . The s t a t e change e n e r g y SE o f an i d e a l gas depends s o l e l y on t h e amount o f mass M and i t s change i n t e m p e r a t u r e 6T, 6E = Mc6T = _ M _ 6 ( p / p ) . 7-1 The s p e c i f i c h e a t c i s t a k e n a t c o n s t a n t volume, b e c a u s e t h e mass o f g a s i s e x t e r n a l l y d r i v e n f r o m t h e P 1 / P 1 s t a t e t o t h e p 2 / p 2 s t a t e . I t do e s n o t do work on i t s s u r r o u n d i n g s . II.THEORY 48 3[ HEAT 2 1 | NEW MASS' : at time t N x x v \ V v \ X \ V p v ^ 1 ! 2 WORK J i mmm mmm at time t+5t I I - 3 E n e r g y B a l a n c e i n t h e Shock R e g i o n The 8(p/p) t e r m can t h e n be expanded t o g i v e 6E = _ 1 _ p_2_ - p_L p,UA5t 7 - 1 L p2 p, J = _JL_ £JLP2 " P i UA6t. 7-1 L p 2 -I The A 8 t f a c t o r c a n be c a n c e l l e d and 6E e q u a t e d t o t h e i n p u t 6 E + : P1UV2 + 1 j £ i P 2 - p, |U = p 2 V + q. 2 7-1L P 2 J The c o n s e r v a t i o n o f momentum e q u a t i o n s i m p l i f i e s t h e f i r s t t e r m : p 2 V + _1_| £ J J ? 2 " P i |U » P 2 V + q, 2 7-1 L P 2 J II.THEORY 49 w h i c h g i v e s q a s a f u n c t i o n o f e x p e r i m e n t a l q u a n t i t i e s : q = 1 I £_LP 2 " P 7-1 L 92 F o r q=0, t h i s e q u a t i o n r e d u c e s t o t h e a s y m p t o t i c s t r o n g shock r e l a t i o n s i n t r o d u c e d e a r l i e r . The e v o l u t i o n of t h e s y s t e m i s as shown i n F i g u r e I I - 2 . When q i s s m a l l , i n c r e a s e d p r e s s u r e i n t h e shock r e g i o n w i l l " s p r e a d " t h e c h a r a c t e r i s t i c t r a c e s of U and V, c h a n g i n g t h e e v o l u t i o n d i a g r a m o f F i g u r e I I - 2 i n t o s o m e t h i n g l i k e t h a t o f F i g u r e I I - 4 . N o n e t h e l e s s , d e r i v a t i o n s i n t h i s s e c t i o n and i n s e c t i o n 11.10 i m p l i c i t l y assume t h a t U and V d e s c r i b e t h e m o t i o n o f r e a l i n t e r f a c e s s e p a r a t i n g r e g i o n s w i t h r e a l d i f f e r e n c e s i n t h e i r p r o p e r t i e s . Too l a r g e a h e a t f l u x w ould smooth t h e s y s t e m i n t o one l o n g r e g i o n w i t h a s t e a d y change i n t e m p e r a t u r e , t h u s n e g a t i n g t h e a s s u m p t i o n t h a t t h e f l o w i s g r a d i e n t - f r e e . L a t e r i n C h a p t e r IV i t w i l l be shown t h a t i n t e n s i t y c h a n g e s on t h e s t r e a k p h o t o g r a p h s mark t h e b o u n d a r i e s between u n i f o r m f l o w r e g i o n s . The h e a t f l u x c a l c u l a t i o n s i n t h i s s e c t i o n and t h e n e x t c a n be j u s t i f i e d by t h i s e x p e r i m e n t a l t e s t . II.THEORY 50 1 SLOWER CONTACT SURFACE SMALLER EXPANSION FAN / i  / / W HIGHER PRESSURE / y SHOCK REGION ^FASTER SHOCK FRONT Z= II - 4 The Shock Wave w i t h Heat F l u x P r e s e n t 11.10 P r e d i c t i o n of Improved End C o n f i n e m e n t The t o t a l e n e r g y f l u x a t t h e open e n d . i n c l u d e s b o t h t h e h e a t f l u x q and t h e p r e s s u r e - v e l o c i t y f l u x w h i c h a c c o u n t s f o r t h e m e c h a n i c a l e n e r g y u s e d t o "push" t h e c o n t a c t s u r f a c e : $ = p2V = p,UV2. The t o t a l l o s s i s t h e n g i v e n by t h e sum q+*. I t i s c l e a r f r o m t h e c o n s e r v a t i o n o f mass t h a t i n c r e a s i n g p, w i l l r e d u c e * by r e d u c i n g U and V. I t i s a l s o c l e a r t h a t t h e h e a t f l u x q w i l l t e n d t o r e d u c e V a n d i n c r e a s e U. The q u e s t i o n i s w h ether p, c a n be c h o s e n t o m i n i m i z e q+$, and whether t h i s minimum w i l l be l e s s t h a n t h e h e a t l o s s t o a s o l i d e l e c t r o d e . II.THEORY 51 I f q i s s m a l l compared t o i t i s c o n v e n i e n t t o i g n o r e h e a t c o n d u c t i o n and use t h e s t r o n g shock e q u a t i o n s f o r p 2 and V. The s c a l i n g r e l a t i o n f o r # i s t h e n g i v e n by r -.1/2 3/2 -1/2 * ~ Ll_ P 2 Pi • L 7 + 1 J I f p 2 i s f i x e d by t h e p l a s m a , and i s r o u g h l y c o n s t a n t d u r i n g t h e t i m e o f g a s d y n a m i c a l a c t i o n , t h e n r a i s i n g p, by a f a c t o r of f o u r w i l l c u t $ i n h a l f . The c o r r e s p o n d i n g e f f e c t on t h e p l a s m a e s c a p e v e l o c i t y i s f o u n d by s o l v i n g t h e e n e r g y b a l a n c e e q u a t i o n f o r V. I f q i s s m a l l , t h e s q u a r e r o o t t e r m c a n be expanded a s y m p t o t i c a l l y , so t h a t V i s g i v e n by V ~ 2 u| 1 - (xM ) R T i U " 2 - (j±±) (7-1 )U" 3 g 1. 7+1 L 2 2 p, J I f U s c a l e s a s t h e i n v e r s e s q u a r e r o o t o f p,, t h e n V (and $) w i l l be r e d u c e d when q i s p r e s e n t . The e x p e r i m e n t a l t e s t w i l l be t o f i n d q and $ as f u n c t i o n s o f p,. 11.11 Summary In s e c t i o n II.1 i t was s t a t e d t h a t a f o r m a l model c o u l d be b r o k e n i n t o t h r e e main p a r t s : b a s i c s t a t e m e n t s o f f a c t and c o n j e c t u r e , f o r m a l r u l e s f o r w o r k i n g o u t t h e i r i n f e r e n c e s , and e x t e r n a l p r o c e d u r e s f o r t e s t i n g t h e model i n e x p e r i m e n t a l r e a l i t y . S e c t i o n s I I . 2 t o I I . 7 have l a r g e l y d e a l t w i t h t h e b a s i c s t a t e m e n t s , whereas s e c t i o n s I I . 8 t o 11.10 c o v e r e d t h e II.THEORY 52 f o r m a l r u l e s . T h i s s e c t i o n p r o v i d e s a summary i n a n t i c i p a t i o n o f t h e t e s t i n g p r o c e d u r e s t o be i n t r o d u c e d i n C h a p t e r I I I . The f i r s t p a r t o f t h e model can be summarized i n t o t h e f o l l o w i n g l i s t o f b a s i c p r e m i s e s : 1. The p l a s m a c o n d i t i o n s a r e c o n s t a n t a f t e r t h e i n i t i a l p r e s s u r e peak. 2. The number d i s t r i b u t i o n o f v e l o c i t i e s i n t h e p l a s m a may be c h a r a c t e r i z e d by t h e same t e m p e r a t u r e as t h e number d i s t r i b u t i o n i n t h e bound s t a t e s c o r r e s p o n d i n g t o He I 588nm and He I I 469nm ( i . e l o c a l thermodynamic e q u i l i b r i u m e x i s t s ) . 3. The number d i s t r i b u t i o n i n t h e bound s t a t e s i s a t t a i n e d t h r o u g h a c o l l i s i o n r a t e low enough f o r t h e p h o t o n e m i s s i o n t o be a d i a b a t i c . 4. The p l a s m a i s o p t i c a l l y t h i n . 5. The p l a s m a and gas do n o t mix. 6. The s o l e e n e r g y t r a n s f e r between p l a s m a and gas i s t h r o u g h p a r t i c l e c o l l i s i o n a t t h e i n t e r f a c e . 7. The f l o w i s one d i m e n s i o n a l , w i t h n e g l i g i b l e t r a n s f e r o f e n e r g y between r a d i a l and a x i a l m o t i o n . 8. The f l o w may be d i v i d e d i n t o u n i f o r m r e g i o n s w h i c h a r e i n t e r n a l l y i n e q u i l i b r i u m . 9. I n t e r f a c e s between t h e r e g i o n s may be o b s e r v e d e x t e r n a l l y a s c h a n g e s i n l u m i n o s i t y . 10 The h e a t f l u x a c r o s s t h e p l a s m a / g a s i n t e r f a c e i s s m a l l enough t o p r e s e r v e b o t h t h e s h a r p b o u n d a r i e s of t h e u n i f o r m f l o w r e g i o n s and t h e a d i a b a t i c model o f p l a s m a e x p a n s i o n . The s e c o n d p a r t o f t h e model i s embodied i n t h e f o r m a l d e s c r i p t i o n s o f s e c t i o n s I I . 8 and I I . 9 . The majo r p r e d i c t i o n t h a t h i g h e r d e n s i t y gas p l u g s w i l l c o n f i n e t h e p l a s m a more e f f e c t i v e l y i s d e r i v e d i n s e c t i o n 11.10. II.THEORY 53 F i n a l l y , the t h i r d p a r t of the model may be s t a t e d c o n c i s e l y i n the f o l l o w i n g way: "Helium plasma a t 35000K and e l e c t r o n d e n s i t y 8 X 1 0 2 2 m~3 i s brought i n t o c o n t a c t w i t h c o l d h e l i u m a t 300K and 525 p a s c a l s . The shock tube model w i t h no heat t r a n s f e r p r e d i c t s a shock v e l o c i t y U=9500m.s - 3, a c o n t a c t s u r f a c e v e l o c i t y V=7000m.s" 3, and a plasma e x p a n s i o n wave which reduces the plasma p r e s s u r e p„ by f i f t y - s i x p e r c e n t . The A h l b o r n - S i n n o t c o n j e c t u r e i s t h a t heat f l u x q w i l l r a i s e U and lower V, w h i l e k e e p i n g p a c l o s e r t o i t s nominal v a l u e . I t i s f u r t h e r e x p e c t e d t h a t r a i s i n g p, w i l l reduce the net energy l o s s q+<i>." I f U and V a r e not as p r e d i c t e d by the shock tube model, the u n i f o r m f l o w model w i t h heat t r a n s f e r can be brought i n t o r e p l a c e i t . The c a l c u l a t e d heat f l u x f o r the open end Z-pinch can then be compared w i t h the s o l i d end v a l u e r e p o r t e d i n PRES74. Such a comparison w i l l be more s e n s i t i v e than the crude e m p i r i c a l comparison of m i d p o i n t t e m p e r a t u r e and d e n s i t y . The r e m a i n i n g p a r t s of t h i s t h e s i s d e a l w i t h the a p p a r a t u s and s p e c i f i c measurements used t o c a r r y out t h i s work. Ill.EXPERIMENT 54 I I I . EXPERIMENTAL SETUP The experiment c o n s i s t s of thr e e p a r t s : a g l a s s shock tube, a Z-pinch to supply the d r i v i n g plasma, and a gas puff v a l v e to p r e - f i l l the shock tube with a high e r d e n s i t y gas p l u g : vac 111-1 Z-Pinch, Valve and Shock Tube The above diagram shows the general arrangement. The dis c h a r g e takes p l a c e between the negative e l e c t r o d e on the l e f t and the p o s i t i v e e l e c t r o d e on the r i g h t . The p o s i t i v e e l e c t r o d e i s i n tu r n connected to the brass mesh r e t u r n conductor on the ou t s i d e of the pin c h v e s s e l . On the f a r r i g h t i s the gas puff v a l v e , j o i n e d to the pi n c h by a 150mm l e n g t h of 25mm ID g l a s s t u b i n g . The i n i t i a l s t a t e shown i n F i g u r e 1-1 i s c r e a t e d where the g l a s s shock tube j o i n s the p o s i t i v e e l e c t r o d e . I l l . E X P E R I M E N T 55 I I I . 1 The C o m p l e t e S y s t e m The s c h e m a t i c r e p r e s e n t a t i o n o f F i g u r e I I I - 1 may be s e e n i n c o n t e x t i n t h e p h o t o g r a p h b e l o w : 111-2 P h o t o g r a p h o f t h e A p p a r a t u s The p i n c h i t s e l f r u n s d i a g o n a l l y a c r o s s t h e c e n t e r o f t h e p i c t u r e . The s h o c k t u b e a n d g a s v a l v e a r e m o u n t e d a t t h e u p p e r l e f t h a n d e n d . A t t h e u p p e r r i g h t i s t h e 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 (OMA) u s e d t o r e c o r d t h e s p e c t r a l l i n e s . O u t o f s i g h t b e t w e e n t h e OMA a n d t h e p i n c h i s a n o p t i c a l t a b l e w i t h t a p p e d h o l e s f o r a n c h o r i n g c o m p o n e n t s . U n d e r n e a t h t h e l a r g e r t a b l e i s t h e Z - p i n c h c a p a c i t o r b a n k . The s e c o n d b a n k on t h e . l e f t i s u s e d t o d r i v e t h e g a s p u f f v a l v e . The v a c u u m a n d g a s s u p p l y a r e j u s t v i s i b l e a t t h e f a r r i g h t h a n d e d g e . Ill.EXPERIMENT 56 I I I . 2 The Z-Pinch Discharge The Z-pinch i s dr i v e n by the energy stored i n a f i f t y m icrofarad c a p a c i t o r . The discharge c i r c u i t i s as shown below. Main c a p a c i t o r C, i s charged i n i t i a l l y to twelve k i l o v o l t s . The 0.05MF t r i g g e r c a p a c i t o r i s charged i n p a r a l l e l through the 200k°, r e s i s t o r . The bank energy of four k i l o j o u l e s i s about enough to heat a cup of cof f e e three degrees C e l s i u s . pinch HV + o 200 k V — V v V H V W 10M 190M 50 uF 1 C I 42 fi AAA/—" 0.05MF tr-1 UV flash I I I - 3 Discharge C i r c u i t The discharge begins w i t h a 9kV pulse from a t h y r a t r o n , which f l a s h e s a small spark gap placed i n s i d e a quartz tube between the e l e c t r o d e s of gap S 3. . The r e s u l t i n g - u l t r a v i o l e t photons t r a v e l through the tube and s t r i k e the ele c t r o d e s u r f a c e s , d r i v i n g o f f e l e c t r o n s . The gap breaks down, the III.EXPERIMENT 57 t r i g g e r c a p a c i t o r i s d i s c h a r g e d , and t h e v o l t a g e a c r o s s i t s 42J2 l o a d a p p e a r s a c r o s s t h e t r i g g e r p i n o f main gap S , . The c i r c u i t i s a r r a n g e d so t h a t v o l t a g e on t h e S T t r i g g e r p i n i s o p p o s i t e i n p o l a r i t y t o t h e v o l t a g e a c r o s s t h e gap. The i n s t a n t a n e o u s v o l t a g e i s t h e r e f o r e a l m o s t t w i c e as g r e a t as t h e s t a t i c gap v o l t a g e . Breakdown w i l l o c c u r even when t h e e l e c t r o d e s have been b a d l y e r o d e d . C u r r e n t f l o w s t o t h e Z - p i n c h t h r o u g h f l a t c o p p e r l e a d s 100mm w i d e . C o n n e c t i o n t o t h e p i n c h i s made c o a x i a l l y , t h e g r o u n d e l e c t r o d e b e i n g a t t a c h e d t o t h e b r a s s mesh r e t u r n c o n d u c t o r . The two c u r r e n t l e a d s a r e wrapped i n p o l y e t h y l e n e and c l a m p e d t o g e t h e r t o f o r m a p a r a l l e l p l a t e w a v e g u i d e . F o l l o w i n g a common p r a c t i c e i n microwave e n g i n e e r i n g , s h a r p c o r n e r s were e l i m i n a t e d t o m i n i m i z e c o u p l i n g between w a v e g u i d e modes and r a d i a t i v e f r e e s p a c e modes. E l e c t r i c a l n o i s e s h o u l d i n t h e o r y be r e d u c e d . I n p r a c t i c e t h e r e was t o o much n o i s e f r o m o t h e r s o u r c e s t o t e s t t h i s c o n c e p t p r o p e r l y . The c u r r e n t was m easured w i t h a Rogowski c o i l , i n s e r t e d between t h e l e a d s and o r i e n t e d t o m a x i m i z e t h e s i g n a l / n o i s e r a t i o . The c o i l o u t p u t , w h i c h i s p r o p o r t i o n a l t o t h e r a t e o f change o f t h e c u r r e n t , was i n t e g r a t e d by a p a s s i v e RC network w i t h a t i m e c o n s t a n t of 100MS. ( F o r c o m p a r i s o n , t h e , p i n c h r i n g i n g p e r i o d i s a p p r o x i m a t e l y 26/zs.) The i n t e g r a t e d s i g n a l was d i s p l a y e d on an o s c i l l o s c o p e and p h o t o g r a p h e d e v e r y t i m e t h e p i n c h was f i r e d . C o m p a r i s o n o f t h e s e v e r a l h u n d r e d p h o t o g r a p h s e v e n t u a l l y c o l l e c t e d showed t h e p i n c h t o be h i g h l y r e p r o d u c i b l e . I l l . E X P E R I M E N T 58 T h e r e a r e two ways t o c a l i b r a t e a Rogowski c o i l . One i s t o i n t e g r a t e t h e c u r r e n t s i g n a l ( i t s e l f t h e i n t e g r a l of t h e c o i l v o l t a g e ) o v e r t h e i n t e r v a l f r o m t h e t i m e t h e c u r r e n t b e g i n s t o t h e t i m e o f z e r o c a p a c i t o r v o l t a g e . T h i s w i l l n o r m a l l y c o r r e s p o n d t o t h e f i r s t q u a r t e r c y c l e ( i . e . 90°) o f t h e c u r r e n t waveform. The i n t e g r a t e d c u r r e n t c a n t h e n be e q u a t e d t o t h e t o t a l c h a r g e on t h e c a p a c i t o r s . A s e c o n d method i s t o t r e a t t h e Z - p i n c h a s an RLC c i r c u i t , so t h a t an e x p o n e n t i a l l y d e c a y i n g s i n e wave c a n be f i t t e d t o t h e m easured s i g n a l . The i n i t i a l e n e r g y s t o r e d i n t h e c a p a c i t o r s c a n t h e n be e q u a t e d t o t h e s t a n d a r d i n t e g r a l f r o m z e r o t o i n f i n i t y o f c u r r e n t s q u a r e d t i m e s r e s i s t a n c e . The two methods a r e e q u i v a l e n t when t h e Z - p i n c h c u r r e n t i s c l o s e t o i t s model v a l u e . In t h i s e x p e r i m e n t , t h e f i r s t t e c h n i q u e i n d i c a t e d a peak c u r r e n t o f 163kA. The s e c o n d g i v e s a much l o w e r v a l u e o f 131kA. Be c a u s e t h e Z - p i n c h c h a n g e s i t s i n t e r n a l c u r r e n t d i s t r i b u t i o n a f t e r t h e f i r s t c y c l e , t h e r e b y c h a n g i n g i t s RLC p a r a m e t e r s , t h e f i r s t c a l c u l a t i o n i s e x p e c t e d t o be t h e more a c c u r a t e . A peak c u r r e n t o f l 6 3 k A w i l l be u s e d t h r o u g h o u t t h i s work. C a l i b r a t i o n o f t h e Rogowski c o i l d o e s n o t a l w a y s p r o c e e d t h i s s m o o t h l y . In BERN79 and DAUG66, f o r example, t h e c u r r e n t i n t e g r a l s o f t h e f i r s t method were m i s t a k e n l y p e r f o r m e d o v e r 180° i n s t e a d o f 9 0 ° . The e r r o r r e m a i n e d u n n o t i c e d b e c a u s e t h e d i s c h a r g e c u r r e n t r a r e l y a p p e a r s i n l a t e r c a l c u l a t i o n s . I t was o n l y d i s c o v e r e d i n t h i s work b e c a u s e t h e Z - p i n c h had t o be compared w i t h e a r l i e r m a c h i n e s o f t h e same t y p e . A p p a r e n t d i s c r e p a n c i e s i n t h e c u r r e n t s h o u l d be seen w i t h t h i s i n mind. I l l . E X P E R I M E N T 59 P r i o r t o f i r i n g t h e d i s c h a r g e , t h e p i n c h v e s s e l i s e v a c u a t e d and b a c k f i l l e d w i t h l a b o r a t o r y g r a d e h e l i u m . The d i s c h a r g e b e g i n s when t h e breakdown a t s p a r k gap S, s w i t c h e s t h e v o l t a g e on c a p a c i t o r C, o n t o t h e e l e c t r o d e s . Gas a l o n g t h e i n s i d e w a l l o f t h e p i n c h v e s s e l i s q u i c k l y i o n i z e d , and when a c o n d u c t i o n p a t h i s e s t a b l i s h e d , t h e " s h e l l " o f c u r r e n t e x p e r i e n c e s a n e t JxB f o r c e i n t o w a r d s t h e a x i s . The i m p l o d i n g c u r r e n t s h e l l d r i v e s a shock w h i c h h e a t s and c o m p r e s s e s t h e c e n t r a l g a s , a r r i v i n g a t t h e a x i s t e n m i c r o s e c o n d s a f t e r t h e i n i t i a l breakdown. Gas on t h e a x i s i s i o n i z e d and t h e p i n c h p h a s e i s c o m p l e t e . S h o r t l y t h e r e a f t e r , c u r r e n t i n t h e s y s t e m p a s s e s t h r o u g h t h e f i r s t o f i t s s e v e r a l z e r o s , and t h e p l a s m a b e g i n s t o d r i f t t o w a r d s t h e w a l l . By t w e n t y m i c r o s e c o n d s t h e e n t i r e p i n c h v e s s e l w i l l be f u l l . From t h i s p o i n t on, t h e p l a s m a seen by t h e sho c k t u b e i s e s s e n t i a l l y c o n s t a n t . T h e r e i s nowhere e l s e f o r t h e p l a s m a t o expand ( t h e n o m i n a l mass d e n s i t y i s v e r y c l o s e t o th e i n i t i a l f i l l d e n s i t y ) , and t h e e n e r g y l e f t i n C, i s s u f f i c i e n t t o m a i n t a i n t h e t e m p e r a t u r e . O n l y a b o u t a q u a r t e r o f t h e i n i t i a l f o u r k i l o j o u l e s was n e e d e d t o d r i v e t h e p i n c h . F u r t h e r m o r e , t h e s u p e r s o n i c r a d i a l s h ock ( i . e . t h e p i n c h ) b r i n g s t h e p l a s m a " i n s t a n t l y " i n t o c o n t a c t w i t h gas i n t h e shock t u b e . The i n i t i a l p l a s m a / g a s i n t e r f a c e i s t h e r e f o r e v e r y s h a r p . I f t h e j o l t f r o m t h e i n i t i a l d e n s i t y peak i s n o t t o o s t r o n g , t h e n t h e one d i m e n s i o n a l f l o w s y s t e m o f F i g u r e 1-1 i s r e p r o d u c e d i n a c t u a l l i f e . I l l . E X P E R I M E N T 60 I I I . 3 The Gas P u f f V a l v e The s i m p l e s t gas end p l u g i s a shock t u b e f i l l e d t o t h e same p r e s s u r e a s t h e Z - p i n c h , b u t h i g h e r d e n s i t y p l u g s a r e a l s o n e e d e d t o t e s t t h e p r e d i c t i o n s o f t h e m o d e l . F o r t h i s p a r t of t h e e x p e r i m e n t t h e shock t u b e was " p r e - f i l l e d " w i t h a d d i t i o n a l gas f r o m a f a s t m a g n e t i c p u f f v a l v e . A l t h o u g h t h e r e s u l t i n g d e n s i t y s t e p i s n o t a s s h a r p a s t h a t c r e a t e d by s e p a r a t i n g two r e g i o n s o f gas w i t h a t h i n p l a s t i c membrane, f i r i n g t h e v a l v e i s much more c o n v e n i e n t . The gas p u f f w i l l a p p e a r e q u a l l y s t a t i c when t h e oncoming shock wave t r a v e l s much f a s t e r t h a n t h e s p e e d o f sound. The v a l v e was t a k e n f r o m a p u b l i s h e d d e s i g n by Kuswa, Stamm and S t a l l i n g s [KUSW70]. The body was made f r o m D e l r i n p l a s t i c , and t h e d i a p h r a g m from 7075 a i r c r a f t aluminum. T h i s l a t t e r m a t e r i a l i s s t r o n g e r t h a n t h e 2024 a l l o y i n d i c a t e d i n t h e p l a n s . The s i n g l e t u r n d r i v i n g c o i l was m a c h i n e d from 6mm b r a s s p l a t e , and f i t t e d i n t o g r o o v e s c u t i n t h e D e l r i n p l a s t i c . The c o i l a nd i t s a t t a c h e d l e a d s were s e a l e d i n p l a c e w i t h epoxy r e s i n . F i g u r e I I I - 4 shows a c r o s s s e c t i o n o f t h e f i n i s h e d v a l v e . The i n l e t p a s s a g e and s p r i n g - p u s h r o d a r r a n g e m e n t have been s l i g h t l y e n l a r g e d t o show d e t a i l . I l l . E X P E R I M E N T 6 1 HV gas ::::::::::: * * * * — 1 i 1 " 1 1 1 1 I ' 1 -i • s. i 3 r 1 — ^ 80 mm i n 1 -4 P u f f V a l v e The v a l v e i s opened by a 3 5 k A p u l s e o f c u r r e n t f l o w i n g t h r o u g h i t s s i n g l e t u r n c o i l . The r i s i n g m a g n e t i c f i e l d i n d u c e s eddy c u r r e n t s i n t h e aluminum d i a p h r a g m , and t h e r e s u l t i n g f o r c e d r i v e s t h e d i a p h r a g m away f r o m i t s O - r i n g s e a t . Gas f l o w s a r o u n d t h e d i a p h r a g m and o u t o f t h e v a l v e t h r o u g h g r o o v e s c u t i n t h e s i d e s o f t h e plenum chamber. P u f f d u r a t i o n i s a p p r o x i m a t e l y e i g h t h u n d r e d m i c r o s e c o n d s ; o p e n i n g t i m e i s l e s s t h a n t e n . When t h e plenum has been e m p t i e d , t h e d i a p h r a g m i s r e s t o r e d t o i t s o r i g i n a l p o s i t i o n by t h e p r e s s u r e of t h e s p r i n g and p u s h r o d , a i d e d by t h e f l o w o f f r e s h gas c o m i n g t h r o u g h t h e i n l e t p a s s a g e . A n o t h e r s h o t c a n be f i r e d w i t h v e r y l i t t l e d e l a y . I l l . E X P E R I M E N T 62 I I I - 5 Gas V a l v e F i r i n g C i r c u i t F i g u r e I I I - 5 shows t h e s p e c i a l " l o w - n o i s e " c a p a c i t o r bank t h a t was b u i l t f o r use w i t h t h e v a l v e . Two NRG t y p e 203 ( l 0 . 2 y F ) c a p a c i t o r s were mounted b e n e a t h a c o p p e r g r o u n d p l a n e , and t h e s w i t c h i n g c i r c u i t s were p l a c e d i n s i d e c o - a x i a l s h e a t h s . C a p a c i t o r C 2 i s c h a r g e d t o -5kV by t h e h i g h v o l t a g e s u p p l y , and c a p a c i t o r C, i s c h a r g e d t o -9kV by t h e t h y r a t r o n u n i t . When t h e t h y r a t r o n i s f i r e d , C, i s d i s c h a r g e d t h r o u g h R,. The v o l t a g e d r o p i s c a p a c i t i v e l y c o u p l e d t o t h e S 2 t r i g g e r p i n . The gap b r e a k s down and c u r r e n t i s c o n d u c t e d t o t h e v a l v e by a s h o r t l e n g t h o f RGU-8 c a b l e . The v a l v e worked r e l i a b l y t h r o u g h o u t t h e e n t i r e e x p e r i m e n t , m i s f i r i n g o n l y on v e r y r a r e o c c a s i o n s . F u r t h e r d e t a i l s of i t s o p e r a t i o n w i l l be g i v e n i n C h a p t e r V. I l l . E X P E R I M E N T 63 I I I . 4 The S t r e a k Camera The o u t w a r d m o t i o n o f t h e u n i f o r m f l o w s y s t e m i s r e c o r d e d w i t h a s t r e a k camera of t h e t y p e shown s c h e m a t i c a l l y i n F i g u r e I I I - 6 . L i g h t f r o m t h e p l a s m a and shock h e a t e d gas i s c o l l e c t e d by a c o n v e x l e n s and f o c u s s e d o n t o t h e camera e n t r a n c e s l i t . The r o t a t i n g c o n c a v e m i r r o r " s t r e a k s " an image o f t h e e n t r a n c e s l i t a c r o s s t h e f i l m . By s e l e c t i n g l i g h t f r o m a h o r i z o n t a l l i n e I I I - 6 S t r e a k Camera and O p t i c s segment a l o n g t h e a x i s o f t h e s h o c k t u b e , m o t i o n o f t h e l u m i n o u s m a t e r i a l c a n be "smeared o u t " i n t o a t r i a n g u l a r image. W i t h a known s t r e a k s p e e d , and a d i s t a n c e s c a l e m e asured f r o m t i m i n g marks, t h e s p e e d o f t h e l u m i n o u s f r o n t c a n be c a l c u l a t e d from t h e a n g l e o f t h e image b o u n d a r y . I l l . E X P E R I M E N T 64 I I I - 7 R o t a t i n g t h e S l i t Image The a r r a n g e m e n t o f m i r r o r s shown i n F i g u r e I I I - 7 r o t a t e s t h e s l i t t o o b s e r v e m o t i o n i n t h e v e r t i c a l d i r e c t i o n , and a l t h o u g h t h e r e i s some l o s s o f d e f i n i t i o n i n v i e w i n g a c y l i n d r i c a l o b j e c t f r o m t h e s i d e , t h i s t e c h n i q u e was u s e d t o f i n d t h e r a d i a l s h ock v e l o c i t i e s i n s i d e t h e Z - p i n c h . S t r e a k p h o t o g r a p h s o f b o t h t y p e s a r e p r e s e n t e d i n C h a p t e r IV. The s t r e a k camera was c a l i b r a t e d by p l a c i n g a h e l i u m - n e o n l a s e r i n f r o n t o f t h e e n t r a n c e s l i t , so t h a t t h e l a s e r beam t r a v e l l e d t h e same p a t h a s t h e p r i n c i p a l r a y from o b j e c t t o image. The r o t a t i n g m i r r o r i n t h e s t r e a k camera w o u l d t h e n sweep t h e beam a c r o s s t h e f i l m p l a n e a t t h e same s p e e d a s t h e image o f t h e s l i t . The s t r e a k v e l o c i t y was measured w i t h a s p e c i a l c a l i b r a t i o n d e v i c e (two p h o t o d i o d e s mounted on a P o l a r o i d f i l m c a s s e t t e ) , a s c h a r t s a t t a c h e d t o t h e machin e by I l l . E X P E R I M E N T 65 p r e v i o u s r e s e a r c h e r s were f o u n d t o be i n e r r o r by o v e r f i f t y p e r c e n t . Measurements t a k e n a t d i f f e r e n t t i m e s showed a f i v e p e r c e n t d r i f t was t h e b e s t t h a t c o u l d be e x p e c t e d . I I I . 5 The 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 I n s i d e t h e p i n c h , t e m p e r a t u r e and d e n s i t y were m e a s u r e d by p l a s m a s p e c t r o s c o p y . L i g h t f r o m t h e p i n c h a x i s was c o l l e c t e d by a c o n v e x l e n s and f o c u s s e d o n t o t h e e n t r a n c e s l i t o f a monochromator. The h e l i u m s p e c t r a l l i n e s were r e c o r d e d w i t h 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 (OMA) mounted a t t h e e x i t . T h i s d e v i c e i s s i m i l a r t o t h e p i c k u p t u b e u s e d i n a v i d e o camera, but w i t h a f i v e h u n d r e d segment anode t o " d i g i t i z e " t h e p i c t u r e l i n e . L i g h t coming i n t o t h e monochromator i s d i s p e r s e d by t h e d i f f r a c t i o n g r a t i n g . The f i v e h u n d r e d OMA c h a n n e l s a t t h e e x i t c o v e r a r a n g e o f a p p r o x i m a t e l y t e n n a n o m e t e r s . The d i s c r i m i n a t i n g power o f t h e o p t i c s was c h e c k e d b e f o r e t h e Z - p i n c h was a s s e m b l e d . A G e i s s l e r t u b e was p l a c e d a t t h e c e n t e r o f t h e p i n c h (where t h e p l a s m a would b e ) , and t h e l e n s e s were a l i g n e d t o m a x i m i z e t h e OMA s i g n a l . The G e i s s l e r t u b e was t h e n moved away from t h e a x i s i n a slow s p i r a l . O n l y l i g h t f r o m w i t h i n 10mm of t h e a x i s r e a c h e d t h e d e t e c t o r . A f t e r t h i s t h e s i g n a l went t o z e r o . I f t h e p l a s m a i s o p t i c a l l y t h i n , so t h a t p h o t o n s c a n e s c a p e w i t h o u t a b s o r p t i o n , t h a n a l l t h e p h o t o n s e m i t t e d f r o m w i t h i n 10mm o f t h e a x i s w i l l be p i c k e d up by t h e OMA. I t s o u t p u t s i g n a l t h e r e f o r e r e p r e s e n t s an a v e r a g e o v e r t h i s s m a l l c y l i n d r i c a l r e g i o n . Ill.EXPERIMENT 66 I I I - 8 OMA and C o l l e c t i o n O p t i c s Time r e s o l u t i o n was p r o v i d e d by. an image i n t e n s i f i e r mounted i n f r o n t of the main d e t e c t o r . Normally opaque, the i n t e n s i f i e r c o u l d be "switched on" by a thousand v o l t p u l s e , i n t h i s case of one microsecond i n d u r a t i o n . A l l of the l i g h t f a l l i n g onto the d e t e c t o r d u r i n g t h i s "exposure time" would be i n t e g r a t e d i n t o the f i n a l p r o f i l e . ( F l u c t u a t i o n s l a s t i n g l e s s than one microsecond were t h e r e f o r e f i l t e r e d out.) I t was p o s s i b l e to r e c o r d an e n t i r e l i n e p r o f i l e i n a s i n g l e shot. For each of these shots, a r e c o r d of the f i v e hundred OMA channels was t r a n s f e r r e d to a d i s k f i l e on the Amdahl 470 system operated by the U n i v e r s i t y of B r i t i s h Columbia Computing Centre. Records were taken at the p i n c h midpoint f o r two s p e c t r a l l i n e s (He I 588nm and He II 469nm), and two types of e l e c t r o d e ( s o l i d and open). An a d d i t i o n a l sequence of l i n e s was taken at the I l l . E X P E R I M E N T 67 q u a r t e r p o i n t o f t h e open-end p i n c h . The s e q u e n c e s r a n from z e r o t o f o r t y - e i g h t m i c r o s e c o n d s w i t h a r e c o r d i n t e r v a l o f one m i c r o s e c o n d f r o m s i x t o t w e n t y m i c r o s e c o n d s (where t h e d a t a c h a n g e s most r a p i d l y ) , and two m i c r o s e c o n d s b e f o r e and a f t e r . A d d i t i o n a l r e c o r d s were t a k e n f o r c a l i b r a t i o n and r e f e r e n c e . The w a v e l e n g t h s c a l e o f t h e monochromator was m a tched a g a i n s t t h e s p e c t r u m of a G e i s s l e r t u b e , and t h e e f f e c t s of n e u t r a l d e n s i t y f i l t e r s and d i f f e r e n t s l i t w i d t h s were m e a s u r e d w i t h b o t h t h e G e i s s l e r t u b e and r e p e a t e d f i r i n g s o f t h e Z - p i n c h . The c o m p l e t e s e t o f d a t a e v e n t u a l l y o c c u p i e d some two h u n d r e d f i l e s . I I I . 6 O t h e r E x p e r i m e n t a l A p p l i a n c e s As shown i n F i g u r e I I I - 2 , t h e Z - p i n c h works i n c o n c e r t w i t h a number o f o t h e r m a c h i n e s . T h e s e a r e n e e d e d t o c h a r g e t h e bank, f i l l t h e p i n c h v e s s e l , measure t h e p a r a m e t e r s , and so f o r t h . A l l o f t h e s e d e v i c e s must be a c c u r a t e , r e l i a b l e , and i n t h e c a s e o f h i g h v o l t a g e , s a f e . T h e r e were two vacuum pumps i n t h e s y s t e m : a W e l c h D u o - S e a l r o t a r y pump, c a p a b l e o f r o u g h i n g t h e s y s t e m t o a p p r o x i m a t e l y f i v e p a s c a l s , and a 100mm d i a m e t e r o i l d i f f u s i o n pump. The two i n s e r i e s c o u l d r e d u c e t h e p r e s s u r e t o l e s s t h a n a h u n d r e d m i l l i p a s c a l s . The f i l l g a s , l a b g r a d e h e l i u m , was p u r e t o one p a r t i n t w e n t y t h o u s a n d . C o n t a m i n a t i o n f r o m t h e gas l e f t a f t e r pumping was l e s s t h a n one p a r t i n f i v e t h o u s a n d . The f i l l p r e s s u r e was m e a s u r e d w i t h an MKS T r u - T o r r e l e c t r o n i c guage w h i c h had been c a l i b r a t e d e a r l i e r w i t h a manometer c o n t a i n i n g I l l . E X P E R I M E N T 68 d i b u t y l e s t e r p t h a l i c a c i d ( C 1 6 H 2 2 0 « ) . A c c u r a c y h e r e was b e t t e r t h a n h a l f o f one p e r c e n t . The f i l l gas i t s e l f was a d m i t t e d t o t h e p i n c h by a h i g h vacuum s o l e n o i d v a l v e c o n t r o l l e d by t h e e l e c t r o n i c guage. Pumping and f i l l i n g were done t h r o u g h a s i n g l e l i n e c o n n e c t e d t o t h e same e l e c t r o d e a s t h e h i g h v o l t a g e , and t h e f i r s t f i r i n g o f t h e " Z - p i n c h " a c t u a l l y t o o k p l a c e i n s i d e t h e vacuum s y s t e m ! The o r i g i n a l s h o r t c o n n e c t i o n t h e r e f o r e had t o be l e n g t h e n e d t o i n c l u d e some e x t r a i n s u l a t i o n . G l a s s t u b e s were t r i e d f i r s t , b u t t h e j o i n t s c o u l d n o t w i t h s t a n d t h e m e c h a n i c a l s t r e s s of r e p e a t e d f i r i n g . The 25mm ID r u b b e r hose t h a t f i n a l l y s o l v e d t h e p r o b l e m i n t r o d u c e d some new p r o b l e m s o f i t s own: o s c i l l a t i o n s i n t h e f i l l s e q u ence as t h e v a l v e opened and c l o s e d i n r e s p o n s e t o gas s u r g i n g back and f o r t h t h r o u g h t h e h o s e . F o r t u n a t e l y , t h e s e o s c i l l a t i o n s were s u f f i c i e n t l y c o n s i s t e n t t h a t a d d i t i o n o f a few n e e d l e v a l v e s s u c c e s s f u l l y r e d u c e d t h e f i n a l p r e s s u r e e r r o r t o l e s s t h a n two p e r c e n t . The power s u p p l y was a r r a n g e d so t h a t b o t h c a p a c i t o r banks c o u l d be c h a r g e d f r om a s i n g l e h i g h v o l t a g e b u s . E a c h bank was e q u i p p e d w i t h t h e same c i r c u i t : two 120VAC s o l e n o i d s w i t c h e s o p e r a t e d by remote c o n t r o l . One s o l e n o i d s w i t c h , t h e "dump" s w i t c h , s h o r t c i r c u i t e d t h e bank c a p a c i t o r s when t h e y were n o t i n use ( p r e v e n t i n g t h e r e l a x a t i o n v o l t a g e f r o m b u i l d i n g up t o d a n g e r o u s l e v e l s ) . The o t h e r s o l e n o i d s w i t c h c o n n e c t e d t h e bank t o t h e b u s . T h i s s w i t c h was c l o s e d f o r c h a r g i n g and opened d u r i n g u s e . I II.EXPERIMENT 69 HV SUPPLY + DUMP jj ^ HV BUS ISOLATE V\Ar VALVE BANK X I DUMP _L-1^ ISOLATE MAIN BANK H X H OIL 0 @ ( T ^ i ^ HELIUM GAS ROTARY PUMP I I I - 9 Vacuum and HV Systems Ill.EXPERIMENT 70 B o t h s e t s o f s w i t c h e s were o p e r a t e d f r o m an i n s u l a t e d box mounted on t o p o f t h e h i g h v o l t a g e s u p p l y c a b i n e t . T h e r e were t h r e e 120VAC l i n e s f o r e a c h bank: two h o t w i r e s f o r t h e s o l e n o i d s and a s i n g l e r e t u r n . A m a s t e r s w i t c h t u r n e d on w a r n i n g lamps whenever t h e s o l e n o i d s were a c t i v e . S t r o n g e f f o r t s were made t o e n c o u r a g e s a f e t y . The f i r i n g p r o c e d u r e was r i g i d l y p r e s c r i b e d by c h e c k l i s t , b a r r i e r s were s e t up a r o u n d t h e e q u i p m e n t , w a r n i n g s i g n s were p o s t e d , and e a r d e f e n d e r s were s u p p l i e d t o o t h e r l a b o r a t o r y w o r k e r s . I n h e r e n t d a n g e r s were d e m o n s t r a t e d t o any new worker t h a t m i g h t come i n c o n t a c t w i t h t h e m a c h i n e . T h e r e were no a c c i d e n t s d u r i n g t h e e n t i r e p e r i o d o f t h e work. V o l t m e t e r s f o r t h e c a p a c i t o r banks were f a s h i o n e d f r o m m i c r o a m m e t e r s , n o m i n a l l y a c c u r a t e t o w i t h i n one p e r c e n t . E a c h ammeter was c o n n e c t e d i n s e r i e s w i t h a l a r g e r e s i s t o r , and t h i s a r r a n g e m e n t p l a c e d i n p a r a l l e l w i t h t h e s m a l l r e s i s t o r o f a v o l t a g e d i v i d e r ( t o p r e v e n t h i g h v o l t a g e f r o m a p p e a r i n g a t t h e m e t er t e r m i n a l s i f t h e meter c o i l b u r n s o u t ) . The r e s i s t a n c e o f t h e d i v i d e r was m e asured w i t h a p r e c i s i o n ohmmeter. The p r e d i c t e d c u r r e n t s f o r f i v e and t w e l v e k i l o v o l t s were marked on t h e m e ter f a c e s f o r use d u r i n g o p e r a t i o n . The e q u i p m e n t t i m e b a s e s were c a l i b r a t e d by o b s e r v i n g t h e i r s i g n a l s on a T e k t r o n i x 555 d u a l beam o s c i l l o s c o p e . The o s c i l l o s c o p e i t s e l f was c a l i b r a t e d w i t h a f r e q u e n c y g e n e r a t o r and d i g i t a l c o u n t e r . O n l y t h e u p p e r beam had a f u n c t i o n i n g t i m e b a s e , and t h i s was f o u n d t o be slow by a p p r o x i m a t e l y two p e r c e n t . The x ± 3 % a c c u r a c y s p e c i f i e d i n t h e m a n u f a c t u r e r ' s Ill.EXPERIMENT 71 c a l i b r a t i o n p r o c e d u r e showed t h a t r e c a l i b r a t i o n w ould n o t be u s e f u l . Time measurements g i v e n i n t h i s t h e s i s s h o u l d t h e r e f o r e be t a k e n a s h a v i n g t h i s t h r e e p e r c e n t a l l o w a n c e . Some o f t h e c a l i b r a t i o n d e v i c e s t h e m s e l v e s had s e r i o u s e r r o r s , m a i n l y due t o o l d age and l a c k of p r o p e r c a r e . S e v e r a l i n s t r u m e n t s had t o be c h e c k e d f o r c o n s i s t e n c y b e f o r e f i n d i n g a u s e a b l e s t a n d a r d . I I I . 7 T i m i n g and S y n c h r o n i z a t i o n A l t h o u g h t h e Z - p i n c h i s t h e o b j e c t o f i n t e r e s t , t i m i n g i s c o n t r o l l e d by t h e OMA and s t r e a k camera. E a c h has an i n t e r n a l c y c l e w h i c h a l l o w s d a t a t o be t a k e n o n l y f o r a s h o r t i n t e r v a l . B e c a u s e t h e s e c y c l e s a r e d i f f e r e n t , t h e two d e v i c e s c a n n o t be ru n a t t h e same t i m e , o r a t l e a s t n o t v e r y e a s i l y . One o r t h e o t h e r must s u p p l y t h e m a s t e r t i m i n g s i g n a l . N o n e t h e l e s s , t h e s e d e v i c e s were n o t u s e d f o r t h e u l t i m a t e s y s t e m r e f e r e n c e . The t i m e o r i g i n was c h o s e n i n s t e a d t o be t h e moment a t w h i c h t h e Z - p i n c h t h y r a t r o n r e c e i v e s i t s command t o f i r e . T h i s i s t h e l a t e s t p o i n t a t w h i c h e l e c t r i c a l p u l s e s s t i l l work r e l i a b l y , a s n o i s e f r o m t h e t h y t a t r o n d i s r u p t s t h e s y s t e m i m m e d i a t e l y a f t e r w a r d s . Time measurements a r e g i v e n i n t h i s t h e s i s r e l a t i v e t o t h e s y s t e m o r i g i n u n l e s s o t h e r w i s e s p e c i f i e d . T i m i n g f o r t h e d i s c h a r g e w i t h o u t gas i n j e c t i o n i s shown i n F i g u r e I I I - 1 0 . The OMA " r e a d y " s i g n a l t r i g g e r s t h e o s c i l l o s c o p e . The p i n c h t h y r a t r o n i s f i r e d two m i c r o s e c o n d s l a t e r , s o t h a t t h e z e r o r e f e r e n c e p u l s e c a n be v e r i f i e d on i t s s c r e e n . The OMA g a t e s i g n a l , w h i c h s e t s o f f t h e h i g h v o l t a g e p u l s e f o r t h e image i n t e n s i f i e r , c a n be s e t t o any d e l a y between I l l . E X P E R I M E N T 72 SCOPE TRIGGER 3-CHANNEL DELAY UNIT u O O O O MONITOR SYSTEM ZERO OMA READY HV PULSER FIRE SYSTEM ZERO THYRATRON ROGOWSKI COIL INT HV TO DETECTOR I FIRE PINCH I I I - 1 0 T i m i n g C i r c u i t I z e r o and f o r t y - e i g h t m i c r o s e c o n d s . A m o n i t o r . s i g n a l from t h e h i g h v o l t a g e p u l s e r t h e n r e t u r n s t o t h e o s c i l l o s c o p e , where i t ca n be p h o t o g r a p h e d a l o n g s i d e t h e d i s c h a r g e c u r r e n t . S u c c e s s f u l r e c o r d i n g o f a s p e c t r a l l i n e p r o f i l e t e r m i n a t e s t h e s e q u e n c e . The s t r e a k camera works w i t h a s i m i l a r a r r a n g e m e n t . I t s r o t a t i n g m i r r o r i s b r o u g h t up t o s p e e d , a " s y n c " p u l s e t r i g g e r s t h e o s c i l l o s c o p e , and s h o r t l y t h e r e a f t e r t h e p i n c h i s f i r e d . The d e l a y i s c h o s e n t o make t h e image o f t h e e s c a p i n g p l a s m a a p p e a r a t t h e b e g i n n i n g o f t h e f i l m . F o r t h e s e t u p shown i n F i g u r e 111-11 , a t i m e d e l a y o f e i g h t h u n d r e d m i c r o s e c o n d s was f o u n d t o be a p p r o p r i a t e . I l l . E X P E R I M E N T 73 SCOPE TRIGGER o son 3-CHANNEL DELAY UNIT o ROGOWSKI COIL STREAK CAMERA PRESSURE SYSTEM ZERO SYNC THYRATRON 1 I FIRE VALVE I THYRATRON 2 I FIRE PINCH 111 — 11 T i m i n g C i r c u i t I I S t u d i e s o f t h e h i g h e r d e n s i t y gas p l u g s f o r m e d by p u f f i n j e c t i o n made use o f t h i s e i g h t h u n d r e d m i c r o s e c o n d gap t o l o c a t e t h e p u f f "edge" a t d i f f e r e n t p o s i t i o n s i n s i d e t h e shock t u b e . F o r t h i s p a r t o f t h e e x p e r i m e n t t h e r e was no s p e c t r o s c o p y , so t h e g a i n i n c o m p l e x i t y c a u s e d by h a v i n g two c a p a c i t o r banks was p a r t i a l l y o f f s e t by h a v i n g o n l y t h e s t r e a k camera t o c o n t e n d w i t h . T i m i n g s e q u e n c e s f o r b o t h s e t s o f e x p e r i m e n t s a r e shown i n F i g u r e 111 — 12 on t h e f o l l o w i n g p age. I l l . E X P E R I M E N T 74 OMA READY SYSTEM ZERO 111-12 Common T i m i n g Sequence I I I . 8 Summary The e x p e r i m e n t a l s e t u p d e s c r i b e d i n t h i s c h a p t e r i s d e s i g n e d t o p r o d u c e t h e one d i m e n s i o n a l f l o w s y s t e m f o u n d i n t h e m o d e l s o f C h a p t e r I I . P l a s m a i s c r e a t e d by t h e Z - p i n c h , and d e n s i t y i n t h e c o l d gas end p l u g i s c o n t r o l l e d by t h e f i r i n g c o n d i t i o n s o f t h e gas p u f f v a l v e . O b s e r v a t i o n s of t h e p l a s m a and g a s a r e made w i t h t h e s t r e a k camera and 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 . A f u l l l i s t o f t h e e x p e r i m e n t a l e q uipment i s g i v e n i n T a b l e 111 — 1. C o m p a r i s o n w i t h t h e e a r l i e r work o f P r e s t o n a p p e a r s i n T a b l e I I I - 2 , and s i m i l a r t a b l e s may be f o u n d i n BERN79, PRES74, PACH71, and DAUG66. The f i r s t s e c t i o n of t h e f o l l o w i n g c h a p t e r w i l l c o n f i r m t h a t t h e p l a s m a o f PRES74 i s r e p r o d u c e d i n t h i s e x p e r i m e n t a l work. I l l . E X P E R I M E N T 75 111 — 1 E x p e r i m e n t a l E q uipment D i s c h a r g e V e s s e l M a t e r i a l D i m e n s i o n s E l e c t r o d e S e p a r a t i o n P y r e x L e n g t h 760mm, ID 150mm, OD 170mm 600mm Vacuum System R o t a r y Pump D i f f u s i o n Pump Gauges F i l l i n g Gas Base P r e s s u r e Leak R a t e W e l c h D u o - S e a l Model 1402 C o n s o l i d a t e d Vacuum, 100mm Model MC275-01 V a r i a n 801, t h e r m o c o u p l e MKS " T r u - T o r r " t y p e VT-8 H e l i u m ( l a b g r a d e ) 99.995% p u r e l e s s t h a n 1OOmPa (200 m i c r o t o r r ) a p p r o x 25 Pa/h (50 m i l l i t o r r / h o u r ) D i s c h a r g e C i r c u i t H i g h V o l t a g e S u p p l y C a p a c i t o r s V o l t a g e Measurement U n i v e r s a l V o l t r o n i c s model BAL-22-35 NRG 203, 5X10.3MF Conway micro-ammeter i n s e r i e s w i t h TRW HV " T i g e r " r e s i s t o r s I n d u c t a n c e 340 nH, L = ( w 2 C ) - 1 I l l . E X P E R I M E N T ( T a b l e I I I - l Gas P u f f V a l v e M a t e r i a l HV S u p p l y C a p a c i t o r s Plenum Guage D i a g n o s t i c Equipment S t r e a k Camera G a t h e r i n g O p t i c s Imaging O p t i c s W r i t i n g Speed S l i t D i m e n s i o n O p t i c a l M u l t i c h a n n e l A n a l y G a t h e r i n g O p t i c s Monochromator H i g h V o l t a g e S u p p l y L o c a l P l o t t e r D i s p l a y O s c i l l o s c o p e O s c i l l o s c o p e s Gas P r e s s u r e T r a n s d u c e r 76 c o n t i n u e d . ) D e l r i n p l a s t i c , B r a s s 7075 Aluminum same a s d i s c h a r g e NRG 203, 2X10.2J U F Edwards S p e e d i v a c C.G.3 UBC P l a sma P h y s i c s [HUNI68] S i n g l e Convex Lens f / 5 Conc a v e M i r r o r f / 5 1.6±0.1mm/MS 70mm by 0.2mm ( n o m i n a l ) P r i n c e t o n A p p l i e d R e s e a r c h model 1205, 500 c h a n n e l s S i n g l e Convex Lens f/4 SPEX 1704 • b l a z e d a t 500nm • a p e r t u r e f / 9 • d i s p e r s i o n 0.8nm/mm F l u k e 412B H e w l e t t - P a c k a r d 7035B T e l e q u i p m e n t D61 T e k t r o n i x 555 d u a l beam (1A1 and B t y p e p l u g - i n s ) T e k t r o n i x 454 d u a l c h a n n e l C e l e s c o LD-25 Ill.EXPERIMENT 77 ( T a b l e I I I - 1 c o n t i n u e d . ) C a l i b r a t i o n E q u i p m e n t F r e q u e n c y G e n e r a t o r K r o h n - H i t e model 1000 (0.2Hz - 3MHz) D i g i t a l C o u n t e r F l u k e model 1900A (6 d i g i t s ) Ohmmeter K i e t h l e y model 168 (4 d i g i t s ) Manometer d i b u t y l e s t e r p t h a l i c a c i d ( C 1 6 H 2 2 O f l , 1047 kg/m 3) I l l . E X P E R I M E N T 78 I I I - 2 E x p e r i m e n t a l C o n d i t i o n s P a r a m e t e r PRES74 T h i s Work C h a r g i n g V o l t a g e F i l l P r e s s u r e D u r a t i o n o f f i r s t h a l f c y c l e Time o f breakdown a f t e r s y s t e m z e r o Time o f s h o c k wave a r r i v a l on a x i s 1/e R e l a x a t i o n Time B e s t F i t F r e q u e n c y Maximum C u r r e n t P e r i o d o f O b s e r v a t i o n Gas V a l v e C a p a c i t o r V o l t a g e Gas V a l v e Plenum P r e s s u r e N o m i n a l E l e c t r o n D e n s i t y N o m i n a l Mass D e n s i t y N o m i n a l P l a s m a T e m p e r a t u r e Peak E l e c t r o n D e n s i t y Peak Plasma T e m p e r a t u r e I n i t i a l Gas P l u g D e n s i t y I n i t i a l Gas P l u g P r e s s u r e Gas P u f f P r e s s u r e -12kV 533 p a s c a l s (4 t o r r ) 10.5MS 0MS (assumed) 9MS n o t g i v e n 44kHz l75kA 0 t o 13MS 20X1 0 2 2 u r 3 48000K -12kV 525 pascals (3.94 t o r r ) 1 3MS 1 MS 1 1 MS 50MS 38kHz 1 63kA 0 to 48MS 5kV l O l k P a Oatm) 8 X 1 0 2 2 n r 3 5x10-° k g . n r 3 35000K (3.0 eV) 30X1 0 2 2 n r 3 (3.75 x nominal) 41000K (3.5 eV) (1.167 x nominal) 8.4x10-* k g . n r 3 525 pascals 2100 pascals (4 x i n i t i a l ) IV.DATA 79 IV. DATA AND OBSERVATIONS In c o n t r a s t t o t h e p r e v i o u s c h a p t e r s , w h i c h d e s c r i b e d t h e t h e o r y and means of p r o d u c t i o n , t h i s c h a p t e r d e a l s w i t h t h e t h i n g s t h a t a r e p r o d u c e d : s t r e a k p h o t o g r a p h s an d OMA r e c o r d s . The use o f d a t a f r o m p r e v i o u s work is a l s o assessed by c o m p a r i n g t h i s Z - p i n c h t o t h e s i m i l a r machine i n P R E S 7 4 . F i g u r e IV-1 shows t h e Z - p i n c h b e i n g f i r e d . To t h e l e f t o f t h e p i c t u r e a r e t h e monochromator and OMA d e t e c t o r head. ( F o r c o m p a r i s o n , t h e p h o t o g r a p h i n F i g u r e I I I - 2 was t a k e n f r o m a p o i n t n e a r t h e d a r k o p e n i n g v i s i b l e on t h e f a r w a l l . ) The gas p u f f v a l v e i s n o t mounted, and t h e m e t a l s c r e e n shown i n F i g u r e I I I - 8 h a s been removed f o r b e t t e r v i s i b i l i t y . IV.DATA 80 IV.1 C o m p a r i s o n w i t h P r e v i o u s Work The Z - p i n c h h a s been s t u d i e d f o r a number of y e a r s i n t h i s l a b o r a t o r y . I n v e s t i g a t i o n s have been made o f d i f f e r e n t f i l l g a s e s and p r e s s u r e s , and some o f t h i s p a s t work i s d i r e c t l y a p p l i c a b l e . The Z - p i n c h most c l o s e l y r e l a t e d t o t h i s e x p e r i m e n t i s t h a t u s e d by P r e s t o n [PRES74], t h e main d i f f e r e n c e s b e i n g t h e l e n g t h (613mm v e r s u s 600mm), and t h e f i l l p r e s s u r e (533 v e r s u s 525 p a s c a l s ) . T h e r e were a l s o d i f f e r e n c e s i n e l e c t r o d e shape and l e a d a t t a c h m e n t t h a t a r e n o t so e a s y t o q u a n t i f y , a l t h o u g h a g e n e r a l c o m p a r i s o n of t h e two d e v i c e s was g i v e n i n T a b l e I I I - 2 . P r e s t o n made a v e r y t h o r o u g h s t u d y o f t h e f i r s t t h i r t e e n m i c r o s e c o n d s o f t h e d i s c h a r g e , and t h e p l a s m a he d e s c r i b e s s h o u l d f o r m t h e i n i t i a l s t a t e f o r t h i s e x p e r i m e n t . The f i r s t t a s k i n c o m p a r i n g t h e two m a c h i n e s was t o measure t h e v e l o c i t i e s o f t h e p i n c h i n g s hock waves. To do t h i s , t h e r a d i a l c o l l a p s e was v i e w e d f r o m t h e s i d e w i t h t h e s t r e a k camera s l i t r o t a t e d a s i n F i g u r e I I I - 7 . A t y p i c a l p h o t o g r a p h i s shown i n F i g u r e I V - 2 . C a l c u l a t i o n s show t h a t t h e shock wave h i t s t h e a x i s e l e v e n m i c r o s e c o n d s a f t e r s y s t e m z e r o , s h a r p l y i n c r e a s i n g t h e a x i a l c u r r e n t and c a u s i n g t h e b r i g h t f l a s h v i s i b l e on t h e f i l m . The i n w a r d shock v e l o c i t y was f o u n d t o be 8mm/MS. The r e f l e c t e d s h o c k t r a v e l l e d a t lOmm/MS. P r e s t o n ' s v a l u e s a r e 1 0mm//xs and 1 2 . 5mm//zs r e s p e c t i v e l y . The t w e n t y p e r c e n t d i f f e r e n c e i s n o t a s s e r i o u s as i t l o o k s . S t r e a k p h o t o g r a p h s i n t h i s e x p e r i m e n t have a ± 1 0 % e r r o r i n i n t e r p r e t a t i o n ( i . e . t h e r a d i a l shock i s n o t w e l l d e f i n e d ) , and a p o s s i b l e f i v e p e r c e n t d r i f t i n t h e s t r e a k c a m e r a . I t i s IV.DATA 81 i i i i i i i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 5 10 15 20 25 30 35 40 45 50 TIME IN MICROSECONDS IV-2 R a d i a l C o l l a p s e o f Pl a s m a a l s o p o s s i b l e t h a t t h e l u m i n o u s f r o n t d o e s n o t c o r r e s p o n d t o t h e d e n s i t y f r o n t m easured by P r e s t o n . D a t a i n PRES74 shows t h e r a d i a l shock s t r u c t u r e t o be f a i r l y w i d e , g i v i n g a r a n g e o f v e l o c i t i e s f r o m 10mm//us i n t h e shock t o 5mm/MS i n t h e t r a i l i n g e l e c t r o n d e n s i t y peak. Such a r a n g e w o u l d i n c l u d e t h e 8mm/MS o b s e r v e d on t h e s t r e a k p h o t o g r a p h s i n t h i s e x p e r i m e n t . O t h e r v a r i a b l e s a l s o matched. The d i s c h a r g e c u r r e n t i n t h i s e x p e r i m e n t had a b e s t f i t complex e x p o n e n t i a l f r e q u e n c y o f 38kHz. In PRES74 t h e f r e q u e n c y i s a p p r o x i m a t e l y 44kHz. The d i s c h a r g e c u r r e n t s and shock v e l o c i t i e s a r e compared i n F i g u r e I V - 3 . Measurements o f t h e plas m a d e n s i t y showed a t h i r t y p e r c e n t d i f f e r e n c e a t t h e peak, b u t r e l a t i v e c l o s e n e s s e v e r y w h e r e e l s e . An a d d i t i o n a l c h e c k i s made b e c a u s e t h e v a l u e s i n PRES74 were c a l c u l a t e d u s i n g i n t e r f e r o m e t r y r a t h e r t h a n l i n e IV.DATA 82 TIME IN MICROSECONDS IV-3 D i s c h a r g e C u r r e n t b r o a d e n i n g . B o t h s t u d i e s u s e d MEWE67 f o r t h e t e m p e r a t u r e however, and t h e s e measurements t e n d e d t o be c l o s e r . I t would be s a f e t o c a l l t h e two d e v i c e s s i m i l a r but n o t i d e n t i c a l . IV.2 S t r e a k P h o t o g r a p h s o f t h e E s c a p i n g Plasma In C h a p t e r I I i t was shown t h a t t h e p l a s m a / g a s i n t e r a c t i o n c o u l d be m o d e l l e d w i t h a s e t o f u n i f o r m f l o w r e g i o n s . In s e c t i o n s I I . 8 and I I . 9 i t was f u r t h e r shown t h a t e x i s t e n c e o f t h e s e r e g i o n s c o u l d be c o n f i r m e d by t h e d e t e c t i o n o f s t r o n g d i f f e r e n c e s i n t h e l u m i n o s i t y o f t h e f l o w i n g m a t e r i a l . T h i s s e c t i o n d e s c r i b e s how t h e s t r e a k camera of s e c t i o n I I I . 4 i s u s e d t o o b s e r v e t h e moving r e g i o n s o f t h e u n i f o r m f l o w m o d e l . IV.DATA 83 A l t h o u g h t h e r e were a q u i t e a l a r g e number o f s t r e a k p h o t o g r a p h s t a k e n , most a r e o f t h e two t y p e s shown i n F i g u r e s IV-4 and I V - 5 . The shock t u b e b e i n g s t u d i e d i s 150mm l o n g . F i g u r e IV-4 c o v e r s t h e l e f t m o s t 120mm ( c l o s e s t t o t h e Z - p i n c h ) , and F i g u r e IV-5 shows t h e l e f t m o s t p a r t o f t h i s r e g i o n m a g n i f i e d by a f a c t o r o f t h r e e . C l o s e t o t h e e l e c t r o d e t h e r e i s not enough t i m e f o r t h e shock wave t o d e v e l o p , s o p h o t o g r a p h s o f t h e F i g u r e IV-5 t y p e show t h e e s c a p i n g p l a s m a d i r e c t l y . The s e p a r a t i o n o f t h e shock wave from t h e c o n t a c t s u r f a c e c a n be seen b e t t e r n e a r t h e t o p o f F i g u r e I V - 4 . A v e r a g i n g t h e d a t a from s e v e r a l 120mm segments g i v e s a shock v e l o c i t y U = 1 2 7 0 0 ± 6 0 0 m / s . A s i m i l a r a n a l y s i s o f t h e m a g n i f i e d segment has t h e p l a s m a moving a t 8 8 0 0 ± 4 0 0 m / s , and t h i s i s t a k e n t o be t h e c o n t a c t s u r f a c e v e l o c i t y V. An e x p e r i m e n t was a l s o done w i t h a 50mm ID t u b e i n w h i c h two aluminum b l o c k s were mounted. The o u t g o i n g shock wave was t o o weak t o be l u m i n o u s , b u t i t s t i m e o f f l i g h t c o u l d be measured f r o m t h e l u m i n o u s bow s h o c k s f o r m e d a t t h e m e t a l o b s t a c l e s . T h e s e t u r n e d o u t t o be q u i t e b a d l y s c a t t e r e d . S t i l l , knowing t h a t t h e 50mm t u b e w i l l s u p p o r t a shock h e l p s t o j u s t i f y u s i n g a one d i m e n s i o n a l model f o r t h e s m a l l e r 25mm t u b e . A 50mm ID t u b e w i t h o u t t h e b l o c k s mounted i s shown i n F i g u r e IV-1 a t t h e b e g i n n i n g o f t h i s c h a p t e r . IV.DATA 84 1 1 40 mm 0 IV-5 25mm Tube - 40mm S e c t i o n IV.DATA 85 IV.3 H e l i u m L i n e S p e c t r a The s t r e a k camera r e g i s t e r s m o t i o n on a l a r g e s c a l e , and i t s f i l m i s s e n s i t i v e t o a l l v i s i b l e w a v e l e n g t h s . W i t h t h e 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 t h i n g s a r e n o t so s i m p l e . To b e g i n w i t h , p h o t o n s have t o r e a c h t h e d e t e c t o r : f i r s t f r o m t h e p l a s m a i t s e l f , s e c o n d t h r o u g h t h e w a l l s o f t h e v e s s e l , t h i r d t h r o u g h t h e b r a s s mesh c o n d u c t o r , i n t o t h e monochromator and so f o r t h . Then comes e l e c t r i c a l ' d i s t o r t i o n : The OMA does n o t a m p l i f y a l l i t s c h a n n e l s e q u a l l y , nor i s t h e r a d i o n o i s e f r o m t h e p i n c h c u r r e n t e q u a l l y d i s t r i b u t e d . The raw s p e c t r u m must be p r o c e s s e d f o r i t s w i d t h o r m a g n i t u d e t o have any meaning. I n s i d e t h e OMA d e t e c t o r t u b e , l e a k a g e c u r r e n t i s so s e v e r e t h a t t h e r a n g e o f o b s e r v a t i o n b a r e l y spans a d e c a d e . (An i r o n i c c o n s e q u e n c e i s t h a t a l l t h e c h a n n e l s l i e w i t h i n t h e i r l i n e a r r a n g e . ) T h i s l e a k a g e c u r r e n t , o r " dark c u r r e n t " , i s measured by p u l s i n g t h e OMA's e l e c t r o n i c s h u t t e r w i t h o u t f i r i n g t h e p i n c h . A t y p i c a l example i s shown i n F i g u r e I V - 6 . Dark c u r r e n t s t e n d t o d r i f t w i t h t i m e , but m a i n l y on t h e l o w e r c h a n n e l s w h i c h a r e r a t h e r u n r e l i a b l e anyway. (The OMA g a i n i s s i m i l a r . ) The smoothed a v e r a g e o f f o u r r a n d o m l y s e l e c t e d r e c o r d s was u s e d as a s t a n d a r d i n t h e c a l c u l a t i o n s . The f i r s t s t e p i n c o r r e c t i n g a raw s p e c t r u m i s t o s u b t r a c t t h i s a v e r a g e s i g n a l . E v en t h o u g h t h e c h a n n e l s were l i n e a r , e a c h had a d i f f e r e n t g a i n . T h e i r r e l a t i v e a m p l i f i c a t i o n was m e asured by t u n i n g t h e monochromator t o t h e c o n t i n u u m r e g i o n a t 480nm, where t h e h e l i u m s p e c t r u m i s e s s e n t i a l l y f l a t a c r o s s t h e lOOnm s y s t e m p a s s b a n d . The r e s u l t i n g OMA s i g n a l i s shown i n F i g u r e I V - 7 . A smoothed IV.DATA 86 o m m CN i I i i i i I i i i i I i i i i I i i i i I i i i i I i i i i I i i i i I i i i i I i i i 50 100 150 200 250 300 350 400 450 500 OMA Channel IV-6 OMA Dark Current c 3 O U o o « o o o i n I i i i i | i i i r | i i i i | 50 I i i i i I i i i i I i i i i I i i i i '[• 100 150 200 250 300 350 400 450 500 OMA Channel IV-7 OMA Gain P r o f i l e IV.DATA 87 v e r s i o n was n o r m a l i z e d t o make t h e sum o v e r a l l t h e c h a n n e l s e q u a l t o u n i t y , and e a c h c h a n n e l o f t h e raw s p e c t r u m ( l e s s t h e d a r k c u r r e n t ) was d i v i d e d by i t s own s t a n d a r d g a i n . S m o o t h i n g was done by t r e a t i n g t h e f i v e h u n d r e d OMA c h a n n e l s as e l e m e n t s o f a " d a t a v e c t o r " , and a p p l y i n g a s m o o t h i n g o p e r a t o r d e f i n e d on t h e a p p r o p r i a t e d i s c r e t e s p a c e . The f a v o r e d t e c h n i q u e i s t o s e t up a s m o o t h i n g m a t r i x from t h e f i n i t e d i f f e r e n c e r e p r e s e n t a t i o n of t h e d i f f u s i o n e q u a t i o n , s i n c e t h i s p r e s e r v e s t h e s i g n a l e n e r g y w h i l e f i l t e r i n g o u t h i g h f r e q u e n c y " s p i k e s " . A s i m i l a r method ( l e s s e l e g a n t l y d e r i v e d ) i s d i s c u s s e d by B e v i n g t o n [ B E V I 6 9 ] . The argument can be made t h a t t h e OMA s i g n a l a l r e a d y r e p r e s e n t s an e n e r g y , b e i n g p r o p o r t i o n a l t o t h e i n t e n s i t y of l i g h t f a l l i n g on t h e d e t e c t o r . In t h i s c a s e t h e d i f f u s i o n o p e r a t o r s h o u l d be r e p l a c e d by an a r e a - p r e s e r v i n g t r a n s f o r m a t i o n . On t h e o t h e r hand, t h e g a i n i s n o r m a l i z e d a f t e r s m o o t h i n g , and t h e d a r k c u r r e n t s a r e a v e r a g e d f i r s t . In t h e end i t was d e c i d e d t o a v o i d t h e i s s u e by n o t s m o o t h i n g t h e l i n e p r o f i l e s a t a l l . I t was a l s o d e c i d e d n o t t o compensate f o r i n s t r u m e n t b r o a d e n i n g ( w h i c h i n any e v e n t was v e r y s m a l l ) , s i n c e t h i s would have s i m p l y "unsmoothed" t h e d a t a even more. A f t e r c o r r e c t i n g f o r d a r k c u r r e n t and g a i n , t h e d a t a v e c t o r s were r e d u c e d t o t h e two h u n d r e d and f i f t y - s i x e l e m e n t s c e n t e r e d a r o u n d t h e s i g n a l peak, and t h e n j u s t l e f t a s t h e y were. F i g u r e IV-8 shows a raw p r o f i l e and i t s p r o c e s s e d c o u n t e r p a r t . A l t h o u g h t h e r e were no more r e l a t i v e c h a n g e s , a f i n a l u n i f o r m s c a l i n g was a p p l i e d t o compensate f o r c h a n g i n g s l i t IV.DATA 88 o OMA Channel IV-8 Raw and P r o c e s s e d D a t a V e c t o r s w i d t h s and n e u t r a l d e n s i t y f i l t e r s . The r e f e r e n c e i n t e n s i t y was e s t a b l i s h e d w i t h no f i l t e r s and a + 120/um r e a d i n g on t h e s l i t m i c r o m e t e r b a r r e l . L i n e s r e c o r d e d a t a s m a l l e r s l i t w i d t h were s c a l e d up so t h a t t h e i r i n t e g r a t e d a r e a would be t h e same as i f t h e y had been t a k e n a t t h e r e f e r e n c e s l i t w i d t h . S i m i l a r l y , l i n e s t a k e n t h r o u g h a n e u t r a l d e n s i t y f i l t e r were s c a l e d up by t h e e n e r g y t r a n s m i s s i o n f a c t o r . E x t r a s h o t s were sometimes t a k e n when c o n d i t i o n s c h a n g e d , and a v e r a g e s were u s e d when more t h a n one r e c o r d was a v a i l a b l e f o r a g i v e n l i n e . C o m p a r i s o n o f t h e d a t a showed t h e Z - p i n c h t o be h i g h l y r e p r o d u c i b l e . F i n a l l y , t h e p r o c e s s e d l i n e s were c o l l e c t e d i n t o m a t r i c e s and p l o t t e d as shown i n F i g u r e I V - 9 . T h e s e m a t r i c e s c o m p l e t e l y summarize t h e s p e c t r o s c o p i c work. IV.DATA 89 4) o-He I 588nm Open End Quarterpoint ?5Q ^ j i t l ffllC 10 2 0 ^ s He I I 469nm Open End Quarterpoint 4 0 in *5Q rfjtfle 30 ; r f 0 S e 10 20 co**5 IV-9 E v o l u t i o n of S p e c t r a l Lines IV.DATA 90 IV.4 I n t e r a c t i o n w i t h a H i g h e r D e n s i t y Gas P l u g The h i g h e r d e n s i t y gas p l u g s were p r o d u c e d by f i r i n g t h e p u f f v a l v e j u s t a h e a d o f t h e Z - p i n c h d i s c h a r g e . The c o n n e c t i n g t u b e ( i n c l u d i n g t h e f i t t i n g s ) was 190mm l o n g , and t h e p u f f v e l o c i t y was a p p r o x i m a t e l y 460m/s (0 . 46mm//is) . The s t a n d a r d v a l v e - t o - p i n c h d e l a y of 250MS p l a c e d t h e main d e n s i t y s t e p a b o u t 70mm out from t h e e l e c t r o d e edge. A s m a l l amount of gas f l o w i n g a h e a d o f t h e p u f f a t t h e h e l i u m sound v e l o c i t y o f I0l8m/s s l o w e d t h e shock by a b o u t t w e n t y p e r c e n t b e f o r e t h e main i n t e r a c t i o n t o o k p l a c e . The i n t e r a c t i o n i t s e l f i s shown i n t h e s t r e a k p h o t o g r a p h s o f F i g u r e s IV-11 and IV-12. C o n n e c t i n g t u b e s s h o r t e r t h a n 150mm c o u l d n o t be u s e d , b e c a u s e s t r e a k p h o t o g r a p h s showed t h a t t h e " f r o n t s u r f a c e " o f t h e p u f f d i d n o t become smooth u n t i l t h e p u f f had moved a good d i s t a n c e down t h e t u b e . T h i s i s t o be e x p e c t e d , s i n c e gas e x i t i n g p a s t t h e v a l v e d i a p h r a g m must e n t e r t h e tub e a s an a n n u l a r f l o w . The f r o n t s u r f a c e r e a c h e s u n i f o r m i t y o v e r a f l o w l e n g t h s e v e r a l t i m e s t h e t u b e d i a m e t e r , w h i c h p o s e s a r e s t r i c t i o n , s i n c e gas d i f f u s i n g t r a n s v e r s e l y i s a l s o d i f f u s i n g l o n g i t u d i n a l l y , s m o o t h i n g o u t t h e d e n s i t y g r a d i e n t . In t o o l o n g a t u b e t h e d e n s i t y c hange w i l l be n o t be s h a r p enough t o "bend" t h e x - t t r a c e on t h e s t r e a k p h o t o g r a p h s . A s i m i l a r e f f e c t o c c u r s a t t h e e l e c t r o d e end, where t h e plas m a d r i v e n shock wave must a l s o t r a v e l some d i s t a n c e b e f o r e i t becomes u n i f o r m . A l l o f t h e s h o t s were made w i t h t h e v a l v e plenum f i l l e d t o l O l k P a , and t h e bank c h a r g e d t o 5kV. T h e s e s t a n d a r d f i r i n g c o n d i t i o n s c r e a t e d a p u f f o f a p p r o x i m a t e l y 2100 Pa i n a IV.DATA 91 O O O co CO o < o _ IT) U CN / \ CO O / \ PA O _ o CN / \ z O O - / \ m / \ LLi o / \ Q£ o _ / V o / CO / ^ — — ^ . CO o J LU o -Q£ m Q. o 1 1 1 1 1 1 1 1 i 1 I I I I I I I I I I 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 T I M E IN M I C R O S E C O N D S IV-10 Gas P u f f P r e s s u r e P r o f i l e b a c k g r o u n d p r e s s u r e o f 525 Pa, w h i c h i n model c a l c u l a t i o n s i s t a k e n as a d e n s i t y s t e p w i t h an e x a c t f a c t o r of f o u r . The p u f f p r e s s u r e shown i n F i g u r e IV-10 was measured w i t h a p i e z o e l e c t r i c t r a n s d u c e r mounted a t t h e v a l v e o u t l e t . The u n i f o r m f l o w model p r e d i c t s t h a t h i g h e r d e n s i t y p l u g s w i l l s l o w t h e shock by a f a c t o r p r o p o r t i o n a l t o t h e s q u a r e r o o t o f t h e d e n s i t y i n c r e a s e , and t h a t h e a t t r a n s f e r i n t o t h e gas w i l i s l o w t h e c o n t a c t s u r f a c e even more. The p h o t o g r a p h s i n F i g u r e s IV-11 and IV-12 c l e a r l y show a "bend" i n t h e shock t r a j e c t o r y , and a l t h o u g h t h e c o n t a c t s u r f a c e i s somewhat i l l - d e f i n e d , i t a p p e a r s t o change more t h a n t h e shock i t s e l f . The e f f e c t i s p a r t i c u l a r l y s t r o n g on t h e s e c o n d and t h i r d p i n c h i n g c y c l e s . C o m p a r i s o n w i t h F i g u r e IV-4 shows t h a t c o l d gas end p l u g s a r e s u c c e s s f u l a t b l o c k i n g t h e p l a s m a . IV.DATA 92 Puff Edge shock contact surface 20 /*s pinch 160 mm 40 mm 0 IV-11 I n t e r a c t i o n w i t h t h e D e n s i t y S t e p U n f o r t u n a t e l y , i t was not p o s s i b l e t o d e t e c t t h e bend i n t h e s hock t r a j e c t o r y once t h e s t e p had been moved c l o s e r t o t h e e l e c t r o d e , n o r was i t p o s s i b l e t o s h a r p e n t h e s t e p by moving t h e p u f f v a l v e c l o s e r . I n b o t h c a s e s t h e p l a s m a / g a s i n t e r a c t i o n t o o k p l a c e b e f o r e a u n i f o r m f l o w s y s t e m was e s t a b l i s h e d . The i n i t i a l s h o c k v e l o c i t y i n F i g u r e IV-11 i s 9 6 0 0 ± 3 6 0 m / s , down from t h e "no g a s " v e l o c i t y o f 1 2 7 0 0 ± 6 0 0 m / s shown i n F i g u r e I V - 4 . The l i k e l y c a u s e o f t h e d e c l i n e i s l o n g i t u d i n a l d i f f u s i o n i n t h e p u f f . T h i s c a n p r o b a b l y be i g n o r e d ; what m a t t e r s most i s t h e f i n a l s h o c k v e l o c i t y o f U ' = 5 4 4 0 ± 1 9 0 m / s , a n d t h e f i n a l c o n t a c t s u r f a c e v e l o c i t y o f V = 2 0 0 0 ± 1 O O m / s . F i g u r e IV-11 has a v a l v e - t o - p i n c h d e l a y o f -250MS, w h i c h p l a c e s t h e p u f f edge some 35mm i n f r o m t h e e l e c t r o d e end. F i g u r e IV-12 shows t h e i n t e r a c t i o n w i t h a p u f f i n j e c t e d f i f t y IV.DATA 93 II] Puff Edge z-<—I 1 1 160 mm 40 mm 0 IV-12 A S t e p C l o s e r In m i c r o s e c o n d s e a r l i e r . Compared t o F i g u r e I V - 1 1 , t h e p u f f edge w i l l have a d v a n c e d a n o t h e r 23mm t o w a r d s t h e e l e c t r o d e , p l a c i n g i t a mere 12mm i n t o t h e t u b e . The f l o w h e r e i s e v e n more p o o r l y d e f i n e d t h a n i n t h e p r e v i o u s p h o t o g r a p h . A t t h i s p o i n t t h e r e i s l i t t l e more t o be g a i n e d from s t r e a k p h o t o g r a p h y , as enough d a t a now e x i s t s t o c a l c u l a t e t h e h e a t f l u x i n t o t h e g a s . The f o l l o w i n g s e c t i o n o u t l i n e s some of t h e work needed t o p r e p a r e t h i s d a t a f o r machine c o m p u t a t i o n . IV.DATA 94 IV.5 P r e p a r a t i o n f o r Computer A n a l y s i s The e x p e r i m e n t a l p r o c e d u r e c a l l e d f o r e a c h f i r i n g of t h e Z - p i n c h t o be f o l l o w e d by a d a t a t r a n s f e r f r o m t h e l o c a l s y s t e m t o t h e main U n i v e r s i t y c o m p u t e r . E a c h s h o t was a s s i g n e d t o a d i s k f i l e of i t s own. S u p p l e m e n t i n g t h e s e f i l e s were t h r e e v olumes of OMA p l o t s and a l a r g e c o l l e c t i o n of s t r e a k p h o t o g r a p h s . E a c h OMA p l o t a l s o had a p h o t o g r a p h showing t h e h i g h v o l t a g e m o n i t o r p u l s e i n r e l a t i o n t o t h e d i s c h a r g e c u r r e n t , and a w r i t t e n r e c o r d of t h e f i l l p r e s s u r e and o p t i c a l a r r a n g e m e n t . A s s o c i a t i n g e a c h d i s k f i l e w i t h a p a p e r p l o t meant t h a t p r o c e s s i n g a t t h e end o f a l i n e s e q u e n c e c o u l d be c r o s s - c h e c k e d a g a i n s t r e c o r d s made a t t h e t i m e o f f i r i n g . C l e r i c a l e r r o r s were weeded o u t and t h e r e s u l t i n g d a t a base was v i r t u a l l y w i t h o u t f a u l t . The end r e s u l t of t h e p r o c e s s i n g was a s e t of s p e c t r a l m a t r i c e s r e s i d i n g i n an APL w o r k s p a c e . (Two of t h e s e were p l o t t e d i n F i g u r e I V - 9 . ) Backup c o p i e s were k e p t on m a g n e t i c t a p e . When t h e d a t a g a t h e r i n g was o v e r , t h e w o r k s p a c e o c c u p i e d some one h u n d r e d and n i n e t y t h r e e d i s k p a g e s : s e v e n h u n d r e d and s e v e n t y - t w o k i l o b y t e s o f m a g n e t i c s t o r a g e . Use of t h e APL a r r a y p r o c e s s i n g l a n g u a g e s i m p l i f i e d t h e a n a l y s i s , w h i c h was i n t e r a c t i v e and o r i e n t e d t o w a r d s p r o d u c i n g g r a p h s . A r r a y f u n c t i o n s f o r i n t e g r a t i o n and w i d t h measurement o p e r a t e d on t h e s p e c t r a l m a t r i c e s t o p r o d u c e t h e v e c t o r t i m e s e r i e s . O t h e r f u n c t i o n s were t h e n d e f i n e d f o r t i m e s h i f t i n g , c o r r e l a t i o n , i n t e r p o l a t i o n and t h e l i k e . The end p r o d u c t was a s e t o f v e c t o r s w h i c h c o u l d be p l o t t e d on a p p r o p r i a t e d i s p l a y IV.DATA 95 d e v i c e s . I n t e r a c t i v e a n a l y s i s makes i t e a s y t o t e s t t h e model p r e d i c t i o n s a g a i n s t t h e a c t u a l d a t a , and h e r e a g a i n i t was p o s s i b l e t o c o n s t r u c t f u n c t i o n s w h i c h would g i v e t h e d e s i r e d r e s u l t s . T h i s s t r o n g l y c o m p u t a t i o n a l a p p r o a c h w i l l be e v i d e n t i n t h e f o l l o w i n g c h a p t e r . V.ANALYSIS 96 V. ANALYSIS The p r e v i o u s c h a p t e r d e a l t w i t h t h e d a t a p r o d u c e d by t h e d i a g n o s t i c e q u i p m e n t , s u g g e s t i n g t h a t i t c o u l d be t r a n s f o r m e d i n t o c h a r t s and g r a p h s w h i c h b e t t e r r e p r e s e n t t h e i n f o r m a t i o n g a t h e r e d . T h i s c h a p t e r makes use o f t h o s e c h a r t s and g r a p h s t o a n a l y z e t h e two a r e a s o f t h i s t h e s i s n ot c o v e r e d i n C h a p t e r I I : e m p i r i c a l c o m p a r i s o n of t h e s o l i d and open-ended Z - p i n c h , and e x p e r i m e n t a l t e s t i n g o f t h e models i n t r o d u c e d i n s e c t i o n s I I . 8 and I I . 9 . B r o a d e r i m p l i c a t i o n s and s u g g e s t i o n s f o r f u t u r e work a r e d i s c u s s e d i n C h a p t e r V I . C o m p a r i s o n of t h e s o l i d and open ended Z - p i n c h r e l i e d on t h e t e m p e r a t u r e and d e n s i t y measurements made a t t h e p i n c h m i d p o i n t . T h e s e i n t u r n were c a l c u l a t e d f r o m T a y l o r s e r i e s a p p r o x i m a t i o n s d e r i v e d f r o m t h e a c c e p t e d models of GRIE74 and MEWE67. T h e r e a r e no h i d d e n t r i c k s i n t h e c a l c u l a t i o n s . A d i r e c t c o m p a r i s o n o f t h e end p l u g s c an s t i l l be made r e g a r d l e s s o f w hether t h e measurement t h e o r y i s c o r r e c t . T e s t i n g t h e models o f C h a p t e r I I was i n i t i a l l y a c a s e o f l o o k i n g f o r t h e e x p a n s i o n wave p r e d i c t e d by t h e shock t u b e m o d e l . When no s u c h waves c o u l d be f o u n d , t h e shock t u b e model was c h e c k e d a g a i n s t t h e v e l o c i t i e s m e asured f r o m t h e s t r e a k p h o t o g r a p h s . The u n i f o r m f l o w model w i t h h e a t t r a n s f e r matched t h e d a t a more s u c c e s s f u l l y . The r e s u l t s a r e compared w i t h t h e PRES74 c a l c u l a t i o n o f h e a t l o s s t o a s o l i d e l e c t r o d e , and w i t h t h e f r e e s t r e a m i n g e n e r g y f l u x e x p e c t e d f o r a t r u e open end w i t h o n l y a vacuum p r e s e n t . The h i g h e r d e n s i t y gas p l u g i s shown t o c o n f i n e t h e p l a s m a more e f f e c t i v e l y . V. ANALYSIS 97 V. 1 Plasma T e m p e r a t u r e The k i n e t i c t e m p e r a t u r e i s c a l c u l a t e d f r o m t h e i n t e n s i t y r a t i o o f He I I 469nm t o He I 588nm, f o l l o w i n g t h e t h e o r y d i s c u s s e d e a r l i e r i n s e c t i o n I I . 6 . The l i n e p r o f i l e s o f F i g u r e IV-9 a r e i n t e g r a t e d o v e r w a v e l e n g t h t o p r o d u c e t h e t i m e s e r i e s o f i n t e n s i t i e s shown i n F i g u r e V - 1. The upper l i n e i s I 5 B B . The l o w e r l i n e i s I A 6 9 s c a l e d up by a f a c t o r of t e n . o DE o -DE o SOLID ELECTRODE in -z o • '• O • \ < ! ; o LU o \ > CN - Al He 1 588nm / \ i < O i * LU II 469nm (xlO) i i i i i i i i i i i 1 1 1 1 i i 1 1 O d * 1 ' T ' f 1 » I'T 1 1 1 1 1 1 I 1 1 1 1 He 1 1 1 1 1 ! 1 1 0 5 10 15 20 25 30 35 40 45 50 TIME IN MICROSECONDS V-1 S o l i d End L i n e I n t e n s i t i e s An o b v i o u s p r o b l e m w i t h t h e l i n e r a t i o c a l c u l a t i o n s i s t h e i r s e n s i t i v i t y t o e r r o r when t h e l i n e i n t e n s i t i e s a r e s m a l l . F o r t h i s r e a s o n i t i s unwise t o c a l c u l a t e t e m p e r a t u r e s u n t i l a b o u t n i n e m i c r o s e c o n d s i n t o t h e d a t a , w h i c h i s why t h e g r a p h s shown l a t e r a r e b l a n k b e f o r e t h i s t i m e . V.ANALYSIS 98 The d i s c u s s i o n i n s e c t i o n I I . 6 showed t h a t l i n e i n t e n s i t y a s a f u n c t i o n of t e m p e r a t u r e c o u l d be s i m p l i f i e d t o a s i n g l e e x p r e s s i o n of t h e form r i 3 / 2 I a 6 9 = K 0 • |^kTj -exp 5 8 8 kT J D a t a f r o m F i g u r e 4 o f MEWE67 was t h e n u s e d t o c o n s t r u c t t h e a p p r o x i m a t e r e l a t i o n l o g i o Iu e 9 = 1 + 1.47(kT - 3 . 0 ) , 1 5 8 8 w h i c h i s a c c u r a t e i n t h e r e g i o n c l o s e t o n o m i n a l t e m p e r a t u r e T« when t h e p l a s m a d e n s i t y i s c l o s e t o N „ . I f T i s t h e n d e f i n e d a s t h e r a t i o o f a c t u a l t o n o m i n a l t e m p e r a t u r e s , t h e a p p r o x i m a t i o n - 23 J 1 + l o g 1 0 l > f i 9 1, T = T = 1 + 0 Ti( - J- 5 8 8 may be u s e d t o c a l c u l a t e t h e " r e l a t i v e t o n o m i n a l " t e m p e r a t u r e s shown l a t e r i n t h i s s e c t i o n . P r e s t o n c a l c u l a t e d t h a t t h e r e l a x a t i o n t i m e of t h e He I I u p p e r l e v e l s was a p p r o x i m a t e l y t h r e e m i c r o s e c o n d s , and i n PRES74 t h e t e m p e r a t u r e s were c a l c u l a t e d w i t h t h e I « 6 9 v a l u e s s h i f t e d by t h a t amount. In t h i s e x p e r i m e n t t h e s p e c t r a l l i n e s were t e s t e d f o r r e l a x a t i o n e f f e c t s by c o m p a r i n g t h e a u t o c o r r e l a t i o n o f I 5 8 8 ( w h i c h i s known t o have a v e r y s m a l l r e l a x a t i o n t i m e ) w i t h t h e c r o s s - c o r r e l a t i o n o f I588 and I«6 9 • D e l a y s i n t r a n s f e r r i n g e n e r g y f r o m t h e f r e e m o t i o n s t a t e s t o t h e bound s t a t e s p e r t i n e n t t o I f l 6 j w i l l show up a s a s p r e a d i n g and s h i f t i n g o f t h e V.ANALYSIS 99 CO UJ -Q ZD A He 1 588nm • / \ AUTOCORRELATION z o < — — _ UJ . He 1 588nm. He II 469nm > / \ CROSSCORRELATION -< _ l 111 UJ• 11111111 j 1111111111111111111 -50 -40 -30 -20 -10 0 10 20 30 40 50 TIME SHIFT IN MICROSECONDS V-2 I n t e n s i t y C r o s s - C o r r e l a t i o n s c o r r e l a t i o n peak, and t h i s o f f e r s a s e n s i t i v e t e s t o f t h e l o c a l t h e rmodynamic e q u i l i b r i u m n e e d e d f o r m e a s u r i n g t h e t e m p e r a t u r e . A s l i g h t s h i f t i s e v i d e n t i n F i g u r e V-2, but i t i s nowhere n e a r t h e t h r e e m i c r o s e c o n d s a d v o c a t e d i n PRES74. The t e m p e r a t u r e s i n t h i s t h e s i s w i l l t h e r e f o r e be c a l c u l a t e d w i t h t h e l i n e r a t i o s l e f t u n s h i f t e d . T h e r e i s a t l e a s t one use f o r t i m e s h i f t i n g however, a s i t g i v e s an e s t i m a t e of t h e f l u c t u a t i o n s i n t h e d a t a . The r e l a t i v e p r e c i s i o n o f t h e measurements i s t h e r e f o r e c h e c k e d by c a l c u l a t i n g t h e l i n e r a t i o s w i t h a r a n g e o f d e l a y s f r o m z e r o t o f i v e m i c r o s e c o n d s . Sample c a l c u l a t i o n s a r e shown i n F i g u r e s V - 3 , V-4 a n d V - 5 . ( I n t e r p o l a t e d l i n e i n t e n s i t i e s have been u s e d where t h e d a t a was n o t t a k e n a t e v e r y p o i n t . ) The z e r o - o f f s e t t e m p e r a t u r e s ( r e l a t i v e t o T „ ) a r e shown t o g e t h e r i n F i g u r e V- 6 . V.ANALYSIS 100 o o i n OO CN o ° I— o < o m co 2 Z 6 UJ 469nm DELAY 0 (HIGHLIGHTED)' i i i i I i i i i 10 15 20 25 30 35 40 45 50 TIME IN MICROSECONDS V -3 Open-End M i d p o i n t L i n e R a t i o s The s o l i d - e n d Z - p i n c h i s much more s t a b l e t h a n i t s o p e n - e n d t w i n , w i t h a s l o w r i s e i n t e m p e r a t u r e most l i k e l y due t o t h e t h r e e k i l o j o u l e s l e f t i n t h e c a p a c i t o r bank a f t e r t h e f i r s t c u r r e n t h a l f - c y c l e . The s e c o n d and t h i r d c y c l e s have l i t t l e e f f e c t , b u t t h e c o r r e s p o n d i n g open-end t e m p e r a t u r e f l u c t u a t e s s t r o n g l y . A p o s s i b l e e x p l a n a t i o n i s t h a t c u r r e n t i n t h e open-end p i n c h c a n n o t f l o w f r o m t h e e l e c t r o d e c e n t e r s , and t h e r e f o r e must f o r m a t u b e c l o s e t o t h e e n d s . R a d i a l a c c e l e r a t i o n o f t h e c h a r g e c a r r i e r s p r o b a b l y i n d u c e s some t y p e o f f u r t h e r o s c i l l a t i o n . I t would be i n t e r e s t i n g t o examine t h e s e f l u c t u a t i o n s i n a p i n c h w i t h s m a l l e r e l e c t r o d e h o l e s t o l e s s e n t h e r a d i a l e f f e c t s . S o m e t h i n g e l s e w o r t h n o t i n g i s t h e a b s e n c e o f s e c o n d and t h i r d p i n c h p e a k s . S e c t i o n V-2 w i l l e x p l a i n why t h i s i s s o . V.ANALYSIS 101 O co m CO CN o ° I— o <. o m t o 2 Z d LU ? 6 469nm DELAY 0 (HIGHLIGHTED) i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 10 15 20 25 30 35 40 TIME IN MICROSECONDS 45 50 V-4 Open-End Q u a r t e r p o i n t L i n e R a t i o s The f l u c t u a t i o n s c a n p r o v i d e an e s t i m a t e o f t h e e r r o r made i n c a l c u l a t i n g t e m p e r a t u r e , b ut f i r s t t h e y must be c h e c k e d f o r t h e p r e s e n c e of t h e p l a s m a e x p a n s i o n wave. The s e p a r a t i o n of t h e q u a r t e r p o i n t and m i d p o i n t i s I50±2.5mm, and from T a b l e I I - 2 th e n o m i n a l p l a s m a sound v e l o c i t y i s 13090m/s. The e x p a n s i o n wave t r a n s i t t i m e w i l l t h e r e f o r e be a b o u t e l e v e n m i c r o s e c o n d s . C o m p a r i s o n o f t h e open-end m i d p o i n t and q u a r t e r p o i n t t e m p e r a t u r e s shows t h a t any common f e a t u r e s t r a v e l l i n g a t t h i s s p e e d must be v e r y s m a l l . The f a c t t h a t f l u c t u a t i o n s i n t h e t e m p e r a t u r e s t a y w i t h i n f i f t e e n p e r c e n t o f t h e mean i m p l i e s t h a t t h e f i f t y - s i x p e r c e n t p r e s s u r e r e d u c t i o n p r e d i c t e d by t h e shock t u b e model must show up i n t h e d e n s i t y i f i t i s t o show up a t a l l . F u r t h e r i m p l i c a t i o n s o f t h i s c o n s t a n t t e m p e r a t u r e w i l l be d e a l t w i t h i n t h e f o l l o w i n g s e c t i o n . V.ANALYSIS 1 02 V - 5 Solid-End Midpoint Line R a t i . CN < z z O ~ z O H O oo P O H u o-i d CONDITIONS SOLID MIDPOINT OPEN MIDPOINT .0?"EN„_QUARTERy6TNT" 10 15 20 25 30 35 40 TIME IN MICROSECONDS 45 50 V . A N A L Y S I S 103 V . 2 Plasma D e n s i t y D e n s i t y , l i k e t e m p e r a t u r e , c a n be c a l c u l a t e d r e l a t i v e t o a c o n v e n i e n t n o m i n a l v a l u e . S e c t i o n s I I . 3 t o I I . 6 of C h a p t e r I I showed t h a t t h e d e n s i t y was r e l a t e d t o t h e w i d t h of s p e c t r a l l i n e s . T h i s s e c t i o n u s e s t h e s t a n d a r d a n a l y s i s of GRIE74 t o d e v e l o p an a p p r o x i m a t e f o r m u l a v a l i d i n t h e n e i g h b o r h o o d of n o m i n a l d e n s i t y N „ = 8 X 1 0 2 2 n r 3 . In GRIE74, t h e h a l f w i d t h a t h a l f maximum W i s g i v e n by where N 0 , W0 and A 0 a r e a l s o g i v e n i n GRIE74, and R 0 i s d e f i n e d ( i n e l e c t r o s t a t i c CGS u n i t s ) t o be From t h e t a b u l a t e d v a l u e s N o = l 0 2 2 n r 3 , T o=20000K, W o=0.0l76nm, A o=0.059, and R o=0.2950, t h e n o m i n a l t e m p e r a t u r e T „ = 3 5 0 0 0 K , and n o m i n a l d e n s i t y N „ = 8 X 1 0 2 2 n r 3 i m p l y t h a t He I 588nm has a G r i e m w i d t h o f W=0.159nm. I f n i s t h e r a t i o o f a c t u a l t o n o m i n a l d e n s i t y , and r i s t h e r a t i o o f a c t u a l t o n o m i n a l t e m p e r a t u r e , t h e n t h e G r i e m f o r m u l a c a n be s h i f t e d t o N A and T „ , where n=r=1: = 6 1/3 1/6 1/4 5/12 W(n) = 0.141n 1 + 0.173n - 0.0401n T V.ANALYSIS 104 T h i s e x p r e s s i o n c a n i n p r i n c i p l e be i n v e r t e d , and expanded as t h e T a y l o r s e r i e s n = 1 + dn(W,)•(W-W,) + 0( (W-W,) 2 ), dw where t h e f i r s t d e r i v a t i v e i s f o u n d by i m p l i c i t d i f f e r e n t i a t i o n and W,=0.159 i s now d e f i n e d a s t h e w i d t h a t n=1. I f t h e s m a l l t e m p e r a t u r e d ependence i s n e g l e c t e d , t h e n n(W) = 1 + 6.13( W - 0.159). C l o s e t o n=1, t h i s e x p r e s s i o n w i l l be more a c c u r a t e t h a n t h e s i m p l e l i n e a r r e l a t i o n between N and W. In t h e ran g e of n from 0.25 t o 4.00, t h e e r r o r w i l l be l e s s t h a n t e n p e r c e n t : a c c u r a t e enough t o l o o k f o r t h e e x p a n s i o n wave. The e x p e r i m e n t a l l i n e w i d t h W i s c a l c u l a t e d f r o m t h e s p e c t r a l l i n e p r o f i l e s o f F i g u r e I V - 9 . F i g u r e V-7 shows t h e d e n s i t y a t t h e c e n t e r o f t h e Z - p i n c h . A p a r t f r o m t h e s l i g h t t i m i n g s h i f t t h e r e a r e no s i g n i f i c a n t d i f f e r e n c e s between t h e s o l i d and open e l e c t r o d e s , and c e r t a i n l y none i n t h e h y p o t h e t i c a l e x p a n s i o n r e g i o n t w e n t y m i c r o s e c o n d s a f t e r t h e p i n c h . A c l o s e r c h e c k i s g i v e n by t h e open-end m i d p o i n t and q u a r t e r p o i n t d e n s i t i e s shown i n F i g u r e V-8. Waves t r a v e l l i n g a t th e s p e e d o f sound s h o u l d l e a d t o common f e a t u r e s s e p a r a t e d by t h e t i m e o f p a s s a g e . E x a m i n a t i o n o f t h e g r a p h shows t h a t any s u c h f e a t u r e s w i l l be a l s o be v e r y s m a l l . V. ANALYSIS 105 V-7 Midpoint Density f o r Both Electrode Types o «© * 3 o Z 9 CO z O o u < 9 o d I H 3 I i 1 I L O C A T I O N MIDPOINT QUARTERPOINT >> i | i i i l | i i i i | i i i i | i i l l \ i i i i | i i i i | i i i i | i i i 10 15 20 25 30 35 40 45 50 T I M E IN M I C R O S E C O N D S V-8 Open-End Midpoint and Quarterpoint D e n s i t i e s V.ANALYSIS 106 in - J o < 9 < 1 0 o 0 Z o m -y CN < o OC —' I I m CN C O N D I T I O N S SOLID MIDPOINT .°„p.?N..M. lP.p.P!!siT. OPEN QUARTERPOINT ' ' I ' ' 10 15 20 25 30 35 AO 45 T I M E I N M I C R O S E C O N D S 50 V-9 Plasma P r e s s u r e s A f i n a l t e s t was made by c a l c u l a t i n g t h e m i d p o i n t and q u a r t e r p o i n t p l a s m a p r e s s u r e s from t h e p r o d u c t o f n o m i n a l t e m p e r a t u r e and d e n s i t y , and t h e s e a r e shown i n F i g u r e I V - 9 . The c r o s s - c o r r e l a t i o n o f t h e two s e r i e s was t h e n t a k e n i n s u c h a way t h a t a p o s i t i v e t i m e s h i f t w o u l d r e p r e s e n t t h e d e l a y i n wave m o t i o n between t h e q u a r t e r p o i n t and m i d p o i n t . A wave moving i n t o t h e p i n c h f r o m t h e end e l e c t r o d e s h o u l d p r o d u c e a peak a t p l u s e l e v e n m i c r o s e c o n d s . A l t h o u g h t h e u p p e r c u r v e i n F i g u r e V-10 shows a s m a l l r i s e a t t e n m i c r o s e c o n d s , t h e r e a l c a u s e i s p r o b a b l y n o t an e x p a n s i o n wave. The f e a t u r e i s i n s t e a d due t o t h e weak s e c o n d p i n c h f o l l o w i n g t h e c u r r e n t r e v e r s a l , where t h e s e c o n d p r e s s u r e peak i s c o r r e l a t e d w i t h t h e v e r y s t r o n g f i r s t p eak. The c u r r e n t h a l f c y c l e t i m e o f t h i r t e e n m i c r o s e c o n d s i s s i m p l y t o o c l o s e t o t h e V.ANALYSIS 107 CO LU Q 3 < LU > LU OPEN END PRESSURE: MID- AND QUARTERPOINT CORRELATION WITH FIRST PRESSURE PEAKS REMOVED 11 n 111111'11111 II II 1111111111111111111111111111111111111111 [ 11111 n 111111 II 111111 II 1111111111 u 1111 •50 -40 -30 -20 -10 0 10 20 30 40 50 TIME SHIFT IN MICROSECONDS V-10 C r o s s - C o r r e l a t i o n of Open-End P r e s s u r e s wave t r a n s i t t i m e o f e l e v e n m i c r o s e c o n d s . A p r o b e s e p a r a t i o n o f 150mm i s a l m o s t t h e w o r s t p o s s i b l e c h o i c e . When t h e f i r s t f i f t e e n m i c r o s e c o n d s o f t h e p r e s s u r e s e r i e s a r e removed, c o r r e l a t i o n w i t h t h e f i r s t peak a l s o goes away. The l o w e r c u r v e i n F i g u r e 11-10 shows no c o r r e l a t i o n between t h e m i d p o i n t and q u a r t e r p o i n t p r e s s u r e s . A b s e n c e o f an e x p a n s i o n wave means, among o t h e r t h i n g s , t h a t p r e s s u r e i n t h e o u t g o i n g shock wave must be c l o s e t o t h a t o f t h e p l a s m a . V.ANALYSIS 108 TEMPERATURE I* R A D 1 A L S HP C K \  y\jy^y ''jr-"" ~^" ' ""' — — DENSITY — i k A SOLID MIDPOINT I / \\ \ OPEN MIDPOINT ; //Ai \ OPEN QUARTERPOINT i i M I 1 1 M I 1 1 I 1 1 1 I 1 1 1 1 1 1 1 1 1 1 1 I I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 M 0 . 5 10 15 20 25 30 35 40 45 50 TIME IN MICROSECONDS V-11 Combined T e m p e r a t u r e , D e n s i t y and C u r r e n t F o r e a s e o f c o m p a r i s o n , t h e e x p e r i m e n t a l t i m e s e r i e s a r e shown c o l l e c t i v e l y i n F i g u r e V-11. The most s t r i k i n g f e a t u r e i s a c o m p l e t e l a c k o f o s c i l l a t i o n i n e i t h e r t h e d e n s i t y o r t e m p e r a t u r e , e v e n t h o u g h t h e p i n c h c u r r e n t i s r i n g i n g c o n t i n u o u s l y . T h i s s u g g e s t s t h a t p l a s m a i n t h e Z - p i n c h e xpands w i t h t h e o u t g o i n g s h o c k t o f i l l t h e e n t i r e v e s s e l , and t h a t c u r r e n t i n t h e p l a s m a i s u n i f o r m t h e r e a f t e r . S u c h a c u r r e n t w ould t e n d t o m a i n t a i n t h e p l a s m a t e m p e r a t u r e w h i l e h a v i n g l i t t l e e f f e c t on t h e d e n s i t y . F u r t h e r e v i d e n c e o f t h i s r a d i a l e x p a n s i o n c a n be s e e n i n t h e s t r e a k p h o t o g r a p h o f F i g u r e I V - 2 . The r a p i d d i f f u s i o n o f t h e p l a s m a p r e v e n t s any a x i a l waves f r o m b e i n g s e e n , a t l e a s t o v e r t h e t i m e s and d i s t a n c e s u s e d i n t h i s e x p e r i m e n t . The e x p a n s i o n wave c o u l d o n l y have been d e t e c t e d much c l o s e r t o t h e end e l e c t r o d e s . V.ANALYSIS 109 L o o k i n g s p e c i f i c a l l y a t t h e p e r i o d between t e n and f i f t e e n m i c r o s e c o n d s , d i f f e r e n c e s i n t h e d e n s i t y a r e more t h o s e of t i m i n g t h a n a n y t h i n g e l s e . V a r i a t i o n s i n t h e t e m p e r a t u r e s h o u l d be v i e w e d a s a p r o d u c t o f t h e f l u c t u a t i o n s v i s i b l e i n F i g u r e s V-3 t o V-5. Of a l l t h e s e r i e s however, o n l y t h o s e t a k e n f o r t h e open-end m i d p o i n t were d i r e c t l y t e s t e d f o r a c c u r a c y i n s p a c e as w e l l as t i m e ( s e e s e c t i o n I I I . 5 ) . I t i s p o s s i b l e t h a t movement of t h e p i n c h v e s s e l ( e . g . i n c h a n g i n g t h e e l e c t r o d e s ) c o u l d s h i f t t h e OMA c o l l e c t i o n o p t i c s s l i g h t l y o f f - a x i s . The r a d i a l shock would t h e n be p i c k e d up b e f o r e t h e a x i s had been r e a c h e d , a d v a n c i n g t h e peak t o an e a r l i e r p o i n t i n t h e t i m e s e r i e s . When t h e d e n s i t y p e a k s a r e a l i g n e d t h e t h r e e c u r v e s a r e v e r y much t h e same. C o n d i t i o n s a t t h e p i n c h e n d p o i n t s have no e f f e c t on t h e p l a s m a f o u n d n e a r t h e c e n t e r . N o n e t h e l e s s , t h e v o l u m e - f i l l i n g p l a s m a has a u n i f o r m t e m p e r a t u r e and d e n s i t y n o t a t t a i n a b l e i n i t s n a r r o w l y c o n f i n e d e q u i v a l e n t , w h i c h makes i t an a l m o s t i d e a l " d r i v e r " f o r t h e gas i n t h e a t t a c h e d shock t u b e . The a n a l y s i s o f t h i s s y s t e m w i t h t h e m o d els o f s e c t i o n s I I . 8 and I I . 9 i s d e v e l o p e d i n t h e n e x t two s e c t i o n s . V.ANALYSIS 110 V.3 C o m p a r i s o n w i t h M o d e l P r e d i c t i o n s S i n c e t h e e m p i r i c a l s t u d y o f s o l i d v e r s u s open ends showed no e v i d e n c e o f c o n f i n e m e n t , t h e q u e s t i o n n a t u r a l l y a r i s e s a s t o whe t h e r t h e p r o c e s s c o u l d o c c u r under d i f f e r e n t c o n d i t i o n s . In o t h e r words, d o e s t h e model p r o p o s e d by A h l b o r n and S i n n o t make t h e c o r r e c t p r e d i c t i o n s even i f t h e i n t e r i o r measurements seem t o show o t h e r w i s e ? The answer c a n be f o u n d i n t h e f l o w s t r u c t u r e o b s e r v e d a t t h e open end. Of a l l t h e s t r e a k p h o t o g r a p h s t a k e n , t h e 120mm s e c t i o n gave t h e b e s t p i c t u r e o f t h e shock wave. The a v e r a g e d v e l o c i t y of t h e c o n t a c t s u r f a c e was V = 8 8 0 0 ± 4 0 0 m / s , and t h a t o f t h e shock wave U = 1 2 7 0 0 ± 6 0 0 m / s . The b e s t f i t t o t h e s i m p l e shock model was f o r 5=178, whereby U=12200 and V=9080, b o t h w i t h i n e x p e r i m e n t a l e r r o r o f t h e i r m e asured v a l u e s . The b e s t f i t , however, i s n o t good enough i n t h e f a c e of s y s t e m a t i c e r r o r . The c o n t a c t s u r f a c e was s l o w , t h e shock wave was f a s t , a n d t h e p r e s s u r e d r o p was n o t even d e t e c t a b l e . I n f a c t , a shock s t r e n g t h of 5=178 i m p l i e s a d r i v i n g p r e s s u r e of 195000 p a s c a l s , a l m o s t t w i c e t h a t e x p e c t e d f o r an e q u i l i b r i u m p l a s m a a t t h e measured t e m p e r a t u r e and d e n s i t y [ L I C K 6 2 ] . The e v i d e n c e p o i n t s t o a h i g h e r shock p r e s s u r e c a u s e d by h e a t c o n d u c t i o n a t t h e c o n t a c t s u r f a c e . V.ANALYSIS 1 1 1 V.4 C a l c u l a t i o n o f Heat T r a n s f e r The f i r s t s t e p i n c a l c u l a t i n g q i s t o f i n d t h e v a l u e o f 7 t h a t a p p l i e s t o t h e shock wave, b e c a u s e t h e f a c t o r of ( I - 7 ) - 1 w i l l be s e n s i t i v e t o e r r o r i f 7 i s c l o s e t o u n i t y . F o r t u n a t e l y , t h e p r e s s u r e and d e n s i t y i n t h e shock a r e c o n s i s t e n t w i t h an e q u i l i b r i u m t e m p e r a t u r e o f 14300K [ L I C K 6 2 ] , where 7 has c h a n g e d a mere two p e r c e n t f r o m i t s i d e a l v a l u e of 5/3. C a l c u l a t i o n s w i l l be s i m p l i f i e d by u s i n g t h i s v a l u e t h r o u g h o u t . The h e a t f l u x i n t o t h e gas c a n be c a l c u l a t e d f r o m t h e e n e r g y b a l a n c e i n t h e u n i f o r m f l o w model, w h i c h i n s e c t i o n I I . 9 was f o u n d t o be S u b s t i t u t i n g t h e v a l u e s of V, U, p 1 f and P! f r o m T a b l e I I - 2 and C h a p t e r IV, t h e h e a t f l u x i s g i v e n by The ± 1 2 0 % e r r o r comes from t a k i n g t h e d i f f e r e n c e o f two l a r g e numbers w h i c h a r e r e a l l y w i t h i n e x p e r i m e n t a l e r r o r o f e a c h o t h e r . The v a l u e o f 1.3x10 8 W/m2 c a n b e s t be i n t e r p r e t e d as an u p p e r l i m i t s u i t a b l e f o r c o m p a r i s o n w i t h t h e e n e r g y l o s t t o a s o l i d e l e c t r o d e . In a d d i t i o n t o t h e h e a t f l u x however, gas p l u g s w i l l a b s o r b m e c h a n i c a l e n e r g y a s t h e p l a s m a p u s h e s them away. The p r o d u c t o f p r e s s u r e p 2 and v e l o c i t y V w i l l d e f i n e t h e m e c h a n i c a l e n e r g y f l u x B e c a u s e t h e shock s t r e n g t h i s l a r g e , p 2 and V c a n be q = ( 1 . 3 ± 1 2 0 % ) x 1 0 8 W/m2. V.ANALYSIS c a l c u l a t e d f r o m t h e a s y m p t o t i c r e l a t i o n s o f s e c t i o n I I . 8 , V = 2 U + 0 ( 5" 1) , 7+1 p 2 = _2_p,U 2 + 0 ( r '), 7+1 from w h i c h 7 may be e l i m i n a t e d , so t h a t p 2 V becomes $ = P T U V 2 . The v a l u e s of p 1 f U and V measured i n s e c t i o n s IV.2 t h e n g i v e * = ( 8 . 3 ± 1 . 3 ) X 1 0 8 W/m2, and t h e t o t a l f l u x i s t h e n g i v e n by 4>+q = ( 9 . 6 ± 2 . 6 ) X 1 0 8 W/m2. In c o n t r a s t , t h e h e a t c o n d u c t i o n t o a s o l i d e l e c t r o d e was e s t i m a t e d i n PRES74 t o be o n l y 4 x 1 0 8 W/m2. T h i s i n t u r n was b a s e d on c u r r e n t measurements g i v e n i n PACH71, and i n t h e o r y r e p r e s e n t s a maximum l o s s a t maximum c u r r e n t . C o m p a r i s o n w i t h t h e s o - c a l l e d f r e e s t r e a m i n g f l u x i s more i m p r e s s i v e . I f t h e r e were no b a r r i e r s a t a l l i n t h e s y s t e m , p l a s m a w o u l d e s c a p e f r o m t h e ends a t t h e t h e r m a l v e l o c i t y a f t . The f r e e s t r e a m i n g f l u x i s c a l c u l a t e d by a s s u m i n g t h a t t h e p l a s m a d o e s m e c h a n i c a l work, and t h e i n d i c a t e d p r o d u c t of p« and a, g i v e s a v a l u e o f 13X10 8 W/m2. Such a c o m p a r i s o n makes t h e gas end p l u g l o o k r a t h e r good. V.ANALYSIS U J oc 1— o o U J IT) !:.RJI.....!?IR.^AMJ.^.G„....f:.Lux, , , U J OC • < ZD o o _ TOTAL o o oo MECH ^ oc 11 1 -Q . o O O o -(— ID SOLID END HEAT FLUX 1— < < HEAT Jt o O — U J 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 < KILOGRAMS PER CUBIC METRE xlO 3 V-12 Heat Flu-x v e r s u s P l u g D e n s i t y The h e a t f l u x i n t o t h e h i g h e r d e n s i t y gas p l u g c a n a l s o be c a l c u l a t e d f r o m t h e u n i f o r m f l o w m o d e l . S u b s t i t u t i n g t h e v a l u e s of U' and V measured i n s e c t i o n IV.4 i n t o t h e same e q u a t i o n u s e d f o r q g i v e s q' = ( 4 . 2 ± 0 . 2 ) x 1 0 8 W/m2, $' = ( 2 . 0 ± 0 . 1 ) X 1 0 8 W/m2, $'+q' = ( 6 . 2 ± 0 . 3 ) X 1 0 8 W/m2. The g e n e r a l t r e n d o f t h e measurements i s shown i n F i g u r e V-12. V.ANALYSIS A f i n a l c h e c k must be made on one o f t h e a s s u m p t i o n s u s e d i n s e c t i o n I I . 9: t h a t q be s m a l l enough f o r t h e p l a s m a t o expand " a d i a b a t i c a l l y " . I t i s a l s o i m p o r t a n t i n t h e s e n s e t h a t a l a r g e r e s e r v e o f e n e r g y i n t h e pla s m a w i l l a l l o w t h e shock t o a b s o r b as much h e a t a s needed t o a t t a i n u n i f o r m i t y . A c o n s t a n t h e a t f l u x f l o w i n g i n t o a 25mm ID shock t u b e s u g g e s t s t h a t t h e p l a s m a c a n be t a k e n a s a column 0.3m i n l e n g t h ( i . e . h a l f t h e p i n c h l e n g t h ) by 0.0005m 2 i n a r e a . I t s n o m i n a l t e m p e r a t u r e and d e n s i t y t h e n g i v e a t o t a l e n e r g y c o n t e n t o f a p p r o x i m a t e l y one h u n d r e d j o u l e s . A f l u x o f 4 x 1 0 s W/m2 a c t i n g o v e r 22us ( i . e . t h e e n d - t o - m i d p o i n t wave t r a n s i t t i m e ) w i l l d e p l e t e t h i s r e g i o n by about f o u r j o u l e s . The a s s u m p t i o n t h a t q i s s m a l l i s t h e r e f o r e l i k e l y t o be j u s t i f i e d . V.ANALYSIS 1 1 5 V. 5 Summary In t h e i n v e s t i g a t i o n of gas dynamic end c o n f i n e m e n t , t h e s o l i d and open e l e c t r o d e s were d i r e c t l y compared by m e a s u r i n g t h e m i d p o i n t t e m p e r a t u r e and d e n s i t y of t h e p l a s m a . The f o r m e r shows a s l o w r i s e i n t e m p e r a t u r e . The l a t t e r f l u c t u a t e s , b ut i n g e n e r a l shows a l o w e r t e m p e r a t u r e . D e n s i t y i n t h e two c a s e s i s much t h e same. The r a d i a l d i f f u s i o n o f t h e p l a s m a i s so f a s t t h a t c h a n g i n g t h e e l e c t r o d e s has no s i g n i f i c a n t e f f e c t a t t h e p i n c h m i d p o i n t . B e h a v i o u r a t t h e open-end e l e c t r o d e was b e s t p r e d i c t e d by t h e u n i f o r m f l o w model w i t h h e a t t r a n s f e r . The c a l c u l a t e d h e a t f l u x was l e s s t h a n t h a t r e p o r t e d e a r l i e r f o r a s o l i d - e n d s y s t e m . A d d i t i o n a l l o s s from p r e s s u r e a c t i n g o v e r d i s t a n c e was r e d u c e d by a h i g h e r d e n s i t y gas p l u g (as p r e d i c t e d by A h l b o r n and S i n n o t ) , a l t h o u g h t h e h e a t f l u x t e n d e d t o i n c r e a s e as m e c h a n i c a l l o s s went down. A s i m p l e e x t r a p o l a t i o n shows t h a t a gas p l u g dense enough t o s t o p t h e c o n t a c t s u r f a c e w i l l a b s o r b as much h e a t as a s o l i d e l e c t r o d e . The main r e s u l t o f t h i s work has been t o show t h a t c o l d gas end p l u g s c a n i n d e e d b l o c k t h e e s c a p e of p l a s m a f r o m t h e ends of a Z - p i n c h . T h i s and o t h e r more g e n e r a l c o n c l u s i o n s w i l l be summarized i n t h e f o l l o w i n g and f i n a l c h a p t e r . V.ANALYSIS 1 16 V-1 Summary o f E x p e r i m e n t a l R e s u l t s 525 Pa P l u g (p,=8.4x10"" kg/m 3) D i r e c t l y M e a s u r e d Shock V e l o c i t y C o n t a c t S u r f a c e V e l o c i t y C a l c u l a t e d f r o m M o d e l and D a t a H e a t F l u x i n t o Gas M e c h a n i c a l E n e r g y F l u x ( p r e s s u r e x v e l o c i t y ) T o t a l E n e r g y F l u x U = 12700±600 m/s V = 88001400 m/s q = (1 .3±1 . 6 ) x l 0 ' 8 W/m2 $ = ( 8 . 3 ± 1 . 3 ) X 1 0 b W/m2 q+$ = ( 1 0 ± 3 ) X 1 0 8 W/m2 2100 Pa P l u g (p,=33.6x10"• kg/m 3) D i r e c t l y M e a s u r e d Shock V e l o c i t y C o n t a c t S u r f a c e V e l o c i t y C a l c u l a t e d f r o m M o d e l and D a t a H e a t F l u x i n t o Gas M e c h a n i c a l E n e r g y F l u x ( p r e s s u r e x v e l o c i t y ) T o t a l E n e r g y F l u x U' = 5 4 4 0 ± 1 9 0 m/s V = 20001100 m/s q' = ( 4 . 2 i 0 . 2 ) x l 0 8 W/m2 *>' = (2.010. 1 ) x l 0 8 W/m2 q'+4>' = ( 6 . 2 l 0 . 3 ) x l 0 8 W/m2 C o m p a r i s o n w i t h S o l i d B r a s s E l e c t r o d e H e a t F l u x [PRES74] 4 x 1 0 8 W/m2 C o m p a r i s o n w i t h Vacuum End F r e e S t r e a m i n g F l u x 13x10 s W/m2 VI .CONCLUSIONS V I . CONCLUSIONS As s u g g e s t e d i n t h e i n t r o d u c t i o n , most of t h e r e s u l t s o b t a i n e d i n t h i s t h e s i s c a n be g a t h e r e d i n t o f o u r main a r e a s . F i r s t of a l l , t h e commonly u s e d s p e c t r o s c o p i c t h e o r i e s were shown t o r e l y on t h e a d i a b a t i c a p p r o x i m a t i o n . T h i s was shown t o c r e a t e a c o n f l i c t between t h e a d i a b a t i c model o f p h o t o n e m i s s i o n and t h e c o l l i s i o n r a t e model of l o c a l t hermodynamic e q u i l i b r i u m . Two new t e s t s were p r o p o s e d : an e s t i m a t e of t h e a s y m p t o t i c e x p a n s i o n p a r a m e t e r i n t h e a d i a b a t i c m o d el, and a new e q u i l i b r i u m t e s t b a s e d on t h e a u t o - and c r o s s - c o r r e l a t i o n o f s p e c t r a l l i n e i n t e n s i t i e s . The l a t t e r c a l c u l a t i o n showed t h a t t e m p e r a t u r e c o u l d be measured w i t h o u t t h e t h r e e m i c r o s e c o n d r e l a x a t i o n s h i f t a d v o c a t e d i n PRES74. Sec o n d , t h e s o l i d a nd gas end p l u g s were d i r e c t l y compared by m e a s u r i n g t h e t e m p e r a t u r e and d e n s i t y a t t h e m i d p o i n t o f t h e Z - p i n c h . A n a l y s i s i n C h a p t e r V showed t h a t r a d i a l d i f f u s i o n of t h e p l a s m a p r e v e n t e d any d i f f e r e n c e s f r o m b e i n g o b s e r v e d . T h i r d , t h e A h l b o r n - S i n n o t end c o n f i n e m e n t scheme was a s s e s s e d i n terms of two ad hoc m o d e l s : a s i m p l e s h o c k t u b e w i t h no h e a t or p a r t i c l e t r a n s f e r , and a newly d e r i v e d u n i f o r m f l o w model where t h e h e a t t r a n s f e r was a l l o w e d . The new model can i n p r i n c i p l e be a p p l i e d t o any o t h e r s y s t e m f o r w h i c h i t s a s s u m p t i o n s c a n be s a t i s f i e d . F o u r t h , and f i n a l l y , e x p e r i m e n t a l t e s t s showed t h a t t h e u n i f o r m f l o w model w i t h h e a t t r a n s f e r matched t h e d a t a more c l o s e l y , a nd c o u l d t h e r e f o r e be u s e d t o c a l c u l a t e t h e h e a t f l u x i n t o t h e c o l d gas end p l u g . E s t i m a t e s i n PRES74 o f t h e h e a t VI .CONCLUSIONS 118 l o s t t o a s o l i d e l e c t r o d e showed t h a t t h e gas a b s o r b e d c o n s i d e r a b l y l e s s h e a t , a l t h o u g h t h e p r e v i o u s c h a p t e r a l s o showed t h a t p r o b l e m s w o u l d a r i s e i n s c a l i n g up t h e d e n s i t y , n ot t o m e n t i o n t h e f a c t t h a t h e a t l o s s a t t h e ends i s o n l y a s m a l l p a r t o f t h e t o t a l Z - p i n c h e n e r g y b a l a n c e . A number of a r e a s r e m a i n open f o r f u r t h e r work. The g e n e r a l c o n t e x t o f f u r t h e r r e s e a r c h c a n be f o u n d i n t h e f i r s t a r e a c o v e r e d i n t h i s work: t h e need t o c o n s t r u c t m o d els w i t h c o n s i s t e n t t h e o r i e s of measurement. T h i s i n t u r n c an be seen as a l o g i c a l c o n s e q u e n c e o f quantum p h y s i c s i t s e l f . U n l i k e i t s c l a s s i c a l p r e d e c e s s o r s , w h i c h s o u g h t t o make s t a t e m e n t s a b o u t e x t e r n a l and i n d e p e n d e n t r e a l i t y , quantum m e c h a n i c s makes s t a t e m e n t s s o l e l y a b o u t o b s e r v a b l e s and t h e i r e v o l u t i o n . I t s u l t i m a t e r e f e r e n c e i s e s s e n t i a l l y a s y s t e m f o r making i n f e r e n c e s a b o u t t h e w o r l d of d i r e c t e x p e r i e n c e . M o d e l s of p h y s i c a l p r o c e s s e s w h i c h r e l y on quantum t h e o r y f o r t h e i r measurement must ( t o be c o n s i s t e n t ) impose t h e quantum metaphor on e v e r y t h i n g e l s e t h e y c o n t a i n . I t i s t h e n t h e p u r p o s e o f n a t u r a l s c i e n c e t o expand t h e s e m o d e l s , t e s t t h e i r p r e d i c t i o n s , and m o d i f y t h e i r axioms so t h a t i n f e r e n c e s s t a y c o n s i s t e n t w i t h human e x p e r i e n c e . The p r a c t i c a l i m p l i c a t i o n s o f t h i s v i e w p o i n t w i l l be f e l t a s c o m p u t e r s and s y m b o l i c l o g i c p l a y a l a r g e r r o l e i n m o d e l l i n g p h y s i c a l p r o c e s s e s [ A L L E 8 3 ] . G i v e n t h a t any c o n c l u s i o n w i l l f o l l o w f r o m an i n c o n s i s t e n t s e t o f ax i o m s , a p p l y i n g t h e f u l l power o f " l o g i c a l i n f e r e n c e e n g i n e s " ( s o o n t o a p p e a r i n f i f t h g e n e r a t i o n c o m p u t e r s ) w i l l r e q u i r e an e f f o r t t o remove VI .CONCLUSIONS 119 c o n t r a d i c t i o n s , e s p e c i a l l y f r o m t h e t h e o r i e s of measurement w h i c h r e l a t e t h e f o r m a l i s m t o t h e m a c r o s c o p i c human w o r l d . The c o n s i s t e n t l i n k a g e of p r o c e s s and measurement a t t e m p t e d i n t h i s t h e s i s i s a s t e p t o w a r d s t h i s u l t i m a t e g o a l . A f u r t h e r s t e p would be t o r e d e r i v e t h e A h l b o r n - S i n n o t model i n a more g e n e r a l way. The a s s u m p t i o n s s h o u l d be s t a t e d more c l e a r l y , and t h e m a t h e m a t i c s s h o u l d be s e t up i n a s y m p t o t i c f o r m , f o r example by a p p l y i n g t h e method of m u l t i p l e s c a l e s t o a f u l l s y s t e m o f p a r t i a l d i f f e r e n t i a l e q u a t i o n s . The r e s u l t w ould be a model w i t h a c l e a r s e t of a x i oms and an e s t i m a t e of i t s own v a l i d i t y . A c o n s i s t e n t s t a t e m e n t of t h e end c o n f i n e m e n t c o n c e p t w o u l d do much t o s i m p l i f y i t s e x t e n s i o n t o c o n t i n u o u s f l o w s y s t e m s a t h i g h e r t e m p e r a t u r e s and d e n s i t i e s . F i n a l l y , a few r emarks s h o u l d be made a b o u t t h e f u t u r e r o l e of t h e Z - p i n c h , i n g e n e r a l f o r o b s e r v i n g t h e p l a s m a / g a s i n t e r a c t i o n , and s p e c i f i c a l l y f o r t e s t i n g t h e m o d els a l r e a d y m e n t i o n e d . The Z - p i n c h i s a w e l l known d e v i c e t h a t i s e a s y t o b u i l d and r e l i a b l e t o u s e , but i t s main a d v a n t a g e f o r t h i s work comes i n t h e c r e a t i o n o f i t s a x i a l p l a s m a by a s u p e r s o n i c r a d i a l s h o c k . The gas on t h e a x i s r e m a i n s unchanged u n t i l t h e s h o c k a r r i v e s , and t h e r e s u l t i n g i n i t i a l s t a t e can i n p r i n c i p l e be made v e r y c l e a n ( s t e e p g r a d i e n t s , p u r e m a t e r i a l s e t c . ) . The p r o c e s s a f t e r w a r d s i s e a s y t o o b s e r v e . M o d e l s o f p l a s m a - t o - g a s h e a t t r a n s f e r w i l l a l w a y s have i m p o r t a n t a p p l i c a t i o n s . A p r o p e r l y d e s i g n e d Z - p i n c h c o u l d be v e r y u s e f u l i n f u r t h e r s t u d i e s o f t h i s i n t e r e s t i n g f l u i d s y s t e m . 120 REFERENCES A h l b o r n , B. Confinement of a F u s i o n Plasma by a C o l d Gas B l a n k e t Canadian J o u r n a l of P h y s i c s v o l 55, pl 0 4 7 (1977) A h l b o r n , B. and L i e s e , W. Heat F l u x Induced Wave F r o n t s , P h y s i c s of F l u i d s , v o l 24 no 11 (1981) A l l e n , J . M a i n t a i n i n g Knowledge about Temporal I n t e r v a l s Communications of the ACM v o l 26, no 11 (1983) Baker, C , C a r l s o n , G., and K r a k o w s k i , R. Trends and 'Developments i n Magnetic Confinement R e a c t o r Concepts N u c l e a r T e c h n o l o g y / F u s i o n v o l 1, p5 (1981) Baranger, M. and Mozer, B. E l e c t r i c F i e l d D i s t r i b u t i o n i n an I o n i z e d Gas P h y s i c a l Review v o l 115, no 3, pp52l-525 (1959) Berman, P. A p p l i e d P h y s i c s v o l 6, p 283 (1975) B e r n a r d , J . MSc T h e s i s , U n i v e r s i t y of B r i t i s h Columbia 1979 UBC Plasma P h y s i c s Lab Report #69 B e v i n g t o n , P. Data R e d u c t i o n and E r r o r A n a l y s i s  f o r the P h y s i c a l S c i e n c e s M c G r a w - H i l l , New York, 1969 BEZZ69.1 B e z z e r i d e s , Bandel Theory of L i n e Shapes P h y s i c a l Review v o l 181, no 1, pp 379-399 (1969) BEZZ69.2 B e z z e r i d e s , Bandel T r a n s i t i o n Between the Q u a s i s t a t i c and Impact L i m i t s i n S p e c t r a l L i n e B r o a dening P h y s i c a l Review v o l 186, no 1, pp239-244 (1969) COMM77 Commisso, R., E k d a h l , C , F r e e s e , K. , e t a l S o l i d - E n d P l u g E x p e r i m e n t s on a 0-Pinch P h y s i c a l Review L e t t e r s v o l 39, no 3 (1977) AHLB77 AHLB81 ALLE83 BAKE81 BARA59 BERM75 BERN79 BEVI69 121 COOP66 C o o p e r , J . Plasma S p e c t r o s c o p y R e p o r t s o f P r o g r e s s i n P h y s i c s v o l 29, p35 (1966) COOP69 Co o p e r , J . L i n e B r o a d e n i n g L e c t u r e s i n T h e o r e t i c a l P h y s i c s v o l X I c Gordon and B r e a c h , New Y o r k , 1969 DAUG66 Daughney, C C . PhD D i s s e r t a t i o n , U n i v e r s i t y of B r i t i s h C o l u m b i a , 1966 GRIE64 G r i e m , 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 , New Y o r k , 1964 GRIE74 G r i e m , H.R. S p e c t r a l L i n e B r o a d e n i n g by P l a s m a s Academic P r e s s , New Y o r k , 1974 HAIN78 H a i n e s , M.G. P a r t i c l e o r b i t s , d i a m a g n e t i s m , and e n e r g y b a l a n c e i n a Z - P i n c h s a t i s f y i n g t h e Lawson c r i t e r i o n J o u r n a l o f P h y s i c s D v o l 11 p l 7 0 9 (1978) H0LT19 H o l t s m a r k , J . Ann Phys Lpz 58, 577 (1919) HOOP66 Hooper, C.F. E l e c t r i c M i c r o f i e l d D i s t r i b u t i o n s i n P l a s m a s P h y s i c a l Review v o l 149 no 1, pp77-91 (1966) HOOP68 Hooper, C.F. Low F r e q u e n c y Component E l e c t r i c M i c r o f i e l d D i s t r i b u t i o n s i n P l a s m a s P h y s i c a l Review v o l 165, no 1 (1968) HUAN63 Huang, K. S t a t i s t i c a l M e c h a n i c s J o h n W i l e y and Sons, New Y o r k , 1963 HUNI68 H u n i , J . P . PhD D i s s e r t a t i o n , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1968 ICHI73 I c h i m a r u , S. B a s i c P r i n c i p l e s of Plasma P h y s i c s A S t a t i s t i c a l A p p r o a c h W.A. B e n j a m i n , R e a d i n g Mass., 1973 1 22 KUSW70 Kuswa, G., S t a l l i n g s , C. and Stamm, A. Improved F a s t O p e n i n g Gas P u f f V a l v e , Review o f S c i e n t i f i c I n s t r u m e n t s , v o l 41 p l 3 6 2 (1970) LICK62 L i c k , W. J . and Emmons, H. W. Thermodynamic P r o p e r t i e s o f H e l i u m t o 50000 K H a r v a r d U n i v e r s i t y P r e s s , C a m b r i d g e , Mass., 1962 MAJK68 M a j k o w s k i , R.F. and Donohue, R . J . M e a s u r e d S h i f t s of C e s i u m A t o m i c L i n e s -C o r r e l a t i o n w i t h E l e c t r o n D e n s i t y D e r i v e d from W i d t h s P h y s i c a l Review v o l 173, 1968, p p 1 7 7 - l 8 2 MAL077 M a l o n e , R.C. and M o r s e , R.L., M a t e r i a l End P l u g g i n g o f S t r a i g h t T h e t a P i n c h e s , P h y s i c a l Review L e t t e r s v o l 39, no 3 (1977) MERZ70 M e r z b a c h e r , E. Quantum M e c h a n i c s 2nd e d . , c h a p t e r 22 Jo h n W i l e y and Sons, New Y o r k , 1970 MESS58 M e s s i a h , A. Quantum M e c h a n i c s v o l I , I I , c h a p XVI,XVII,XXI John W i l e y and Sons, New Y o r k , 1958 MEWE67 Mewe, R. R e l a t i v e I n t e n s i t y o f H e l i u m S p e c t r a l L i n e s a s a F u n c t i o n o f E l e c t r o n T e m p e r a t u r e and D e n s i t y B r i t i s h J o u r n a l of A p p l i e d P h y s i c s v o l 18, p l 0 7 (1967) MICH95 M i c h e l s o n , A. A s t r o p h y s i c a l J o u r n a l 2, 251 (1895) MOZE60 Mozer, B. and B a r a n g e r , M. E l e c t r i c F i e l d D i s t r i b u t i o n i n an I o n i z e d Gas I I P h y s i c a l Review v o l 118, no 3, pp6 2 6 - 6 3 l ( i 9 6 0 ) PACH71 P a c h n e r , J . PhD D i s s e r t a t i o n , U n i v e r s i t y o f B r i t i s h C o l u m b i a 1971 PEAC81 P e a c h , G. T h e o r y of P r e s s u r e B r o a d e n i n g and S h i f t s o f S p e c t r a l L i n e s A d v a n c e s i n P h y s i c s v o l 30, no 3, pp367-474 (1981) PRES74 P r e s t o n , J . PhD D i s s e r t a t i o n , U n i v e r s i t y of B r i t i s h C o l u m b i a 1974 UBC Plasma P h y s i c s L a b R e p o r t #40 1 23 ROYE80 R o y e r , A. S h i f t , W i d t h and Asymmetry o f P r e s s u r e B r o a d e n e d S p e c t r a l L i n e s a t I n t e r m e d i a t e D e n s i t i e s P h y s i c a l Review A v o l 22, no 4, pp1625-1654 (1980) SINN77 S i n n o t , T. and A h l b o r n , B. R e d u c t i o n o f P a r t i c l e End L o s s e s from L i n e a r M a g n e t i c F u s i o n D e v i c e s , P h y s i c s o f F l u i d s , v o l 20 no 11 (1977) SMIT69 S m i t h , E.W., C o o p e r , J . and V i d a l , C.R. U n i f i e d C l a s s i c a l P a t h T r e a t m e n t of S t a r k B r o a d e n i n g i n P l a s m a s P h y s i c a l Review v o l 185, no 1, pp140-151 (1969) TRAV68 T r a v i n g , G. I n t e r p r e t a t i o n of L i n e B r o a d e n i n g and L i n e S h i f t P lasma D i a g n o s t i c s c h a p t e r 2 L o c h t e - H o l t g r e v e n , W., e d i t o r N o r t h - H o l l a n d , 1968 WHIT74 Whitham, G. B., L i n e a r and N o n l i n e a r Waves Jo h n W i l e y and Sons, New Y o r k , 1974 

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