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UBC Theses and Dissertations

Limitations of magnetic probe measurements in pulsed discharges Tam, Yun-Kwong Sebastian 1967

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THE UNIVERSITY OF BRITISH COLUMBIA FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of YUN-KWONG SEBASTIAN TAM. B.Sc. (Special Honours), University of Hong Kong, - 1962. M.Sc, University of B r i t i s h Columbia, 1965. TUESDAY, JULY 18, 1967 AT 3.30 P.M. IN ROOM 301,. PHYSICS (HENNINGS) BUILDING. COMMITTEE IN CHARGE Chairman: B. N. Moyls B. Ahlborn R. Nodwell A.J. Barnard L. de Sobrino M.M.Z. Kharadly J.H. Williamson Research Supervisor: F. L. Curzon. External Examiner: P. Savic, National, Research Council, Ottawa. L I M I T A T I O N S , O F M A G N E T I C P R O B E M E A S U R E M E N T S I N P U L S E D P L A S M A S A B S T R A C T A " g r a d i e n t p r o b e " c o n s i s t i n g o f t w o s e a r c h c o i l s h a s b e e n d e v e l o p e d t o m e a s u r e t h e c u r r e n t d e n s i t y . . i n a p u l s e d p l a s m a . T h i s p r o b e m e a s u r e s b o t h t h e m a g n i t u d e a n d t h e g r a d i e n t o f t h e m a g e n t i c f i e l d s i m u l t a n e o u s l y e n a b l i n g m o r e a c c u r a t e m e a s u r e m e n t s t h a n t h e c o n v e n t i o n a l m a g n e t i c p r o b e w h i c h h a s o n l y o n e c o i l . I t h a s b e e n u s e d t o m e a s u r e t h e c u r r e n t d e n s i t i e s a n d t h e m a g n e t i c f i e l d s i n z - p l h p h d i s c h a r g e s i n h e l i u m a t p r e s s u r e s b e t w e e n / 5 0 0 y U / a n d 4 mmHg.. T h e c o l l a p s e c u r v e s o b t a i n e d a g r e e d w i t h t h e p r e d i c t i o n s o f a m o d i f i e d s n o w -p l o w e q u a t i o n w h i c h a l l o w e d f o r t h e l o s s o f p a r t i c l e s f r o m t h e c o l l a p s i n g c u r r e n t s h e l l . T h e f l o w o f c u r r e n t i n t h e p l a s m a i s d i s t o r t e d b y t h e p r e s e n c e o f a p r o b e . S u c h a n e f f e c t s p o i l s t h e s p a t i a l r e s o l u t i o n s o t h a t t h e m e a s u r e d v a l u e s o f t h e c u r r e n t d e n s i t y J a r e T P a v e r a g e s o f t h e t r u e c u r r e n t d e n s i t y J q o v e r a f i n i t e region. To .investigate t h i s , a correction formula which relates J to P J q has been developed. Our error analyses showed that any scatter i n J due to P experimental errors was magnified twenty times i n J Q - For a pulsed plasma, there-fore, one should t r y to reduce the perturbation of the probe instead of r e l y i n g on the correction procedure. " GRADUATE STUDIES . F i e l d of Study: Plasma Physics Plasma Physics • L.- de Sobrino Advanced Plasma Physics A. J. Barnard Plasma Dynamics F. L. Curzon Elementary Quantum " ' Mechanics W. Opechowski Advanced Quantum Mechanics H. Schmidt ..Waves '• R. M. E l l i s Electromagnetic Theory G. M. Volkoff Special R e l a t i v i t y Theory P. R a s t a l l S t a t i s t i c a l Mechanics R. Barrie AWARDS 1957-60 Hong Kong Government Bursary .1961-62 Mrs. Eng Wong Yuk-Mui ' s Scholarship 1964-65 National Research Council Studentship PUBLICATIONS F.J. Jankulak, L.G. de Sobrino and Y.K.S. , Tarn,. "A Solution of the Fokker-Planck Equation for Homogeneous Plasmas", Can-.J.Phys. 42, 1743 (1964). LIMITATIONS OF MAGNETIC PROBE MEASUREMENTS IN PULSED DISCHARGES by YUN-KWONG SEBASTIAN TAM B.Sc. (Special Honours), U n i v e r s i t y of Hong Kong, 1962 M.Sc, Un i v e r s i t y of B r i t i s h Columbia, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS OF THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA July, 1967 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that copying or publication of t h i s thesis 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 permission. Department of PHYS/CS ^ The University of B r i t i s h Columbia Vancouver 8, Canada Date fL&t. /», I <?6 J ABSTRACT A " g r a d i e n t probe" c o n s i s t i n g of two s e a r c h c o i l s ; has been developed to measure the, c u r r e n t d e n s i t y i n a pulsed discharge. T h i s probe measures b o t h the magnitude and the g r a d i e n t of the magneitc f i e l d s i m u l t a n e o u s l y e n a b l i n g more a c c u r a t e measurements than the c o n v e n t i o n a l magnetic probe which has o n l y one c o i l . I t has been used to measure the c u r r e n t d e n s i t i e s and the magnetic f i e l d s i n z - p i n c h d i s c h a r g e s i n helium at p r e s s u r e s between 500 and 4 mmHg. The c o l l a p s e curves o b t a i n e d agreed w i t h the p r e d i c t i o n s of a m o d i f i e d snow-plow: :equation which al l o w e d f o r the l o s s of p a r t i c l e s from the c o l l a p s i n g c u r r e n t s h e l l . The flow of c u r r e n t i n the plasma 1 i s d i s t o r t e d by the presence of a probe. Such an e f f e c t s p o i l s the s p a t i a l r e s o l u t i o n so that the measured v a l u e s o f t h e " c u r r e n t d e n s i t y J are averages of the t r u e c u r r e n t d e n s i t y J p • o over a . f i n i t e r e g i o n . To i n v e s t i g a t e t h i s , a c o r r e c t i o n f o r m u l a which r e l a t e s J to J has been developed. p o Our e r r o r , a n a l y s e s showed t h a t any s c a t t e r i n J due P i • to e x p e r i m e n t a l e r r o r s was m a g n i f i e d twenty times i n ^ 0 ° ! F o r a p u l s e d plasma, t h e r e f o r e , one s h o u l d t r y to reduce the. p e r t u r b a t i o n of the probe i n s t e a d of r e l y i n g on the c o r r e c t i o n procedure. - i i i TABLE OF CONTENTS PAGE A b s t r a c t i i Ta b l e of Contents i i i L i s t of T a b l e s i v L i s t o f F i g u r e s v Acknowledgement v i i Chapter 1. I n t r o d u c t i o n 1 2. Measurements of the Current D e n s i t y o f a Plasma i n a Pulsed Discharge 2.1 I n t r o d u c t i o n 7 2.2.1 The Model by Ecker, K r O l l and Z O l l e r 18 2.2.2 Malmberg's Model 25 2.2.3 L i m i t a t i o n s of the E-K-Z Model 33 2.3 T r a n s f o r m a t i o n of the I n t e g r a l E q u a t i o n 43 i n t o a M a t r i x E q u a t i o n S u i t a b l e f o r S o l u t i o n w i t h a D i g i t a l Computer -2.4 E r r o r A n a l y s i s 48 2.4.1 Round-off E r r o r s due to Numerical 50 C a l c u l a t i o n s 2.4.2 The I n f l u e n c e of F l u c t u a t i o n s 52 2.4.3 S p a t i a l R e s o l u t i o n 58 2.4.4 C o n c l u s i o n s from E r r o r A n a l y s i s 61 3. E x p e r i m e n t a l R e s u l t s 3-1 I n t r o d u c t i o n 62 3.2 Apparatus 64 3»3 The G r a d i e n t Probe and the Delay L i n e 67 3.3.1 B a l a n c i n g of Probe and C a l i b r a t i o n C i r c u i t s 72 3-3»2 Frequency Response w i t h Loading C i r c u i t s 75 3.3.3 R e p r o d u c i b i l i t y o f Probe S i g n a l s 78 3.4.1 E x p e r i m e n t a l R e s t i l t s 80 3.4.2 Comparison of Dynamics w i t h Theory 99 4. C o n c l u s i o n s and P r o p o s a l s f o r F u t u r e Work 4.1 C o n c l u s i o n s 108 4.2 P r o p o s a l s f o r Fu t u r e Work 112 - i v PAGE Bibliography 1 1 3 Appendix I. C a l c u l a t i o n of B, f o r the E-K-Z Model -\ 1 k P I I . Fortran IV Programme f o r Solving the 121 Integral Equation LIST OF TABLES PAGE Table I 1 k Table II 51 Table I I I 53 Table IV 55 Table V 6k Table VI 70 Table VII 7^  - V -LIST OF FIGURES NO. PAGE 1 . A t y p i c a l arrangement f o r m a g n e t i c p r o b i n g 8 2(a). S k e t c h o f m i n i a t u r e Rogowski c o i l probe . 11 (b) . C r o s s - s e c t i o n of. Rogowski c o i l i n plasma 3(a). S k e t c h o f g r a d i e n t probe 12 (b ) . Connections i n g r a d i e n t probe and b a l a n c i n g u n i t 4 (a.). C y l i n d r i c a l c u r r e n t s h e e t w i t h a probe 16 p a s s i n g t h r o u g h i t d i a m e t r i c a l l y (b ) . A p p r o x i m a t i n g c y l i n d r i c a l s u r f a c e by p l a n e s ( c ) . C r o s s - s e c t i o n o f c u r r e n t s h e e t s (d) . R e p r e s e n t a t i o n o f plasma c u r r e n t by c y l i n -d r i c a l c u r r e n t s h e e t s 5. P e r t u r b a t i o n o f probe on plasma c u r r e n t 19 6. A p p r o x i m a t i n g a c u r v e d s u r f a c e by a p l a n e 21 7. P l o t s o f C ' ( r , s , a ) , f ( r , s , a ) d s and g ( r , s , a ) d s 24 v e r s u s r 8(a). D i s t o r t e d c u r r e n t f l o w i n c u r r e n t s h e e t due 28 t o c i r c u l a r h o l e (Malmberg's model) ( b ) . C r o s s - s e c t i o n i n ^ p l a n e o f the o b l a t e s p h e r o i d a l c o o r d i n a t e s (Malmberg's model) 9. Neighbourhood o f h o l e i n r e t u r n c o n d u c t o r 31 10. P l o t o f - J p ( r ) a t r e t u r n c o n d u c t o r o f z - p i n c h 32 11. D i s t o r t e d c u r r e n t f l o w i n p l a n e l a y e r w i t h }6 h o l e 12. Diagram showing form o f m a t r i x A 45 13. F i g i x r e showing magnitudes o f m a t r i x elements 46 o f A 14. Diagram showing c r o s s - s e c t i o n o f r i d g e 46 p a r a l l e l to r a x i s 15- Diagram showing c r o s s - s e c t i o n o f A p a r a l l e l 47 to r a x i s 16(a)• Computer r e s u l t f o r a t y p i c a l J ( r ) and i t s 49 c o r r e c t e d v a l u e J ( r ) P (b)'. P l o t s o f J ( r ) anS. J ( r ) v e r s u s r assuming . J ( r ) t o bis a 5 - f u n c t i o n o v ' 17• Computer r e s u l t o f J ( r ) showing s m o o t h i n g 52 e f f e c t o f probe on f E n e s t r u c t u r e o f J ( r ) 0 18. K r r o r s i n J ( r ) due'to e r r o r s i n J ( r ) 55 o v / P 19« E r r o r s i n J ( r ) due to e r r o r s i n J ( r ) 57 a s s i g n i n g J ^ r ) t o be a £>-function*5 - v i -NO. PAGE 20 . Comparison between ^ ^ P ^ r ^ and ^ g ^ r ^ 60 21 . C i r c u i t o f d i s c h a r g e system 66 22 . C r o s s - s e c t i o n o f d i s c h a r g e tube 66 23,. D e s i g n o f g r a d i e n t probe 70 24. C i r c u i t o f g r a d i e n t probe and m e a s u r i n g 71 system 2 5 . C a l i b r a t i o n ! c i r c u i t 72 26. C a l i b r a t i o n s i g n a l s 74 2 7 ( a ) . C i r c u i t f o r d e t e r m i n i n g f r e q u e n c y r e s p o n s e 75 of probe and l o a d i n g c i r c u i t ( b ) . S p e c i a l c o v e r t o r e p l a c e b o t t o m o f c a s i n g J6 o f b a l a n c i n g u n i t 28. F r e q t i e n c y r e s p o n s e o f i n t e g r a t o r 77 29« F r e q u e n c y r e s p o n s e o f probe 77 JO. R e p r o d u c i b i l i t y o f probe s i g n a l s 79 31 . Probe s i g n a l s showing absence o f common 79 modes 3 2 ( a ) . C r o s s - s e c t i o n t h r o u g h a x i s o f d i s c h a r g e <tube 81 (b) . C r o s s - s e c t i o n t h r o u g h g u i d e tube 33- Comparison between v a l u e s o f d i s c h a r g e 82 c u r r e n t measured by g r a d i e n t probe and by Rogowski c o i l 3 4 - 3 7 . E x p e r i m e n t a l c u r v e s f o r 1 J ( r ) and B ( r ) f o r 83-98 /.-pinch d i s c h a r g e s i n helSuru 1 a t 5 0 0 * \ i , 1 mm, 2 mm, and 4 mmHg r e s p e c t i v e l y 3 8 ( a ) , ( b ) . To o b t a i n e x p e r i m e n t a l v a l u e s o f r ' , B ' , 102 r,B a t a g i v e n time t a f t e r s t a r t o f ^ c o l ? a p s e 3 9 - 4 2 . C o l l a p s e c u r v e s f o r h e l i u m a t 500 p, 1mm, 104-2 mm, and 4 mmHg r e s p e c t i v e l y 107 4 3 ( a ) , ( b ) . G e o m e t r i c a l r e l a t i o n s between AB' and 116 4 J . U - v i i -ACKNOWLEDMENTS I w i s h t o thank my s u p e r v i s o r , Dr. F. L. Cu r z o n f o r h i s encouragement, h e l p and i n v a l u a b l e s u g g e s t i o n s t h r o u g h o u t t h i s work.. I am i n d e b t e d to Dr. C. C. Daughney f o r the p r e c i o u s i n f o r m a t i o n I o b t a i n e d from him p e r s o n a l l y and from h i s Ph. D. t h e s i s . I would a l s o l i k e t o thank Dr. B. A. Ahlborn Dr. L. de S o b r i n o and Dr. J . H. W i l l i a m s o n who have; h e l p e d me t o improve the p r e s e n t a t i o n o f t h i s t h e s i s C h a p t e r .1 INTRODUCTION I n t h i s t h e s i s , we a t t e m p t t o improve the method of m e a s u r i n g the c u r r e n t d e n s i t y i n a p u l s e d plasma by u s i n g a mag n e t i c p r o b e , and to i n v e s t i g a t e i n d e t a i l how the p e r t u r b a t i o n o f a probe on t h e plasma a f f e c t s t h e measured c u r r e n t d e n s i t y . The l o c a l c u r r e n t d e n s i t i e s and m a g n e t i c f i e l d s o f a plasma can, i n p r i n c i p l e , be d e r i v e d (a) from i n d u c e d v o l t a g e s i n s e a r c h c o i l s o r (b) from the d e f l e c t i o n o f i n j e c t e d beams o f c h a r g e d p a r t i c l e s . However, the i n t e r -p r e t a t i o n s o f the second t e c h n i q u e a r e u s u a l l y too c o m p l i c -a t e d t o be o f p r a c t i c a l i m p o r t a n c e ( H u d d l e s t o n e , 1965? p.69)« So f a r the o n l y t o o l f o r m e a s u r i n g s u c h q u a n t i t i e s i s s t i l l t he m a g n e t i c probe i n the form o f a s e a r c h c o i l . E v e r s i n c e i t was f i r s t r e p o r t e d by a group o f R u s s i a n s c i e n t i s t s ( A r t -s i m o v i c h e t a l . , 1 9 5 6 ) , the m a g n e t i c probe has been e x t e n -s i v e l y used f o r m e a s u r i n g the m a g n e t i c f i e l d . To d e t e r m i n e the c u r r e n t d e n s i t y , t h e g r a d i e n t o f the f i e l d has t o be computed n u m e r i c a l l y a f t e r a mapping o f the f i e l d i n t h e plasma has been o b t a i n e d . I t . i s o b v i o u s t h a t s u c h a method i s t e d i o u s and i n a c c u r a t e . We t h e r e f o r e a t t e m p t e d t o d e v e l o p a g r a d i e n t probe w h i c h d i r e c t l y measures b o t h the m a g n e t i c f i e l d and i t s g r a d i e n t , and g i v e s much more a c c u r a t e measure-ments o f c u r r e n t d e n s i t i e s i n a pla s m a . The g r a d i e n t probe we used c o n s i s t s o f two s e a r c h c o i l s c o n n e c t e d i n s u c h a way t h a t i t s o u t p u t can g i v e the d i f f e r e n c e AB between the m a g n e t i c f i e l d s measured by the two c o i l s . When the s e p a r a t i o n £ r between the c o i l s i s a c c u r a t e l y known, the g r a d i e n t - j — o f the m a g n e t i c f i e l d can be t a k e n as ^ . The g r a d i e n t probe we use i n t h i s t h e s i s i s 7 mm i n d i a m e t e r (see F i g . 2 3 ) . I t has a f r e q u e n c y r e s p o n s e w h i c h i s f l a t up to 1 MHz and a , s p a t i a l r e s o l u t i o n b e t t e r t h a n 3 mm (see ^ 2 . 4 . 3 ) Any probe d i s t u r b s the c u r r e n t f l o w by i t s mere p r e -sence i n t h e plasma. I f we know the n a t u r e o f t h i s d i s t u r b -a nce, t h e n i n pri.-nci.ple, we can c o r r e c t f o r i t , and o b t a i n t h e u n d i s t u r b e d c u r r e n t d i s t r i b u t i o n . A p a r t f rom the work o f E c k e r e t a l , ( 1 9 6 2 ) , Malmberg ( 1 9 6 4 ) and Daughney ( 1 9 6 6 ) , v e r y l i t t l e has been done to d e v e l o p s u c h a c o r r e c t i o n p r o -cedure f o r the probe p e r t u r b a t i o n . We f i n d i t i m p o r t a n t t o i n v e s t i g a t e t h i s p o i n t b e f o r e we can o b t a i n c o n c l u s i v e r e s u l t s f r o m probe measurements. We t h e r e f o r e s t u d y the p e r t u r b a t i o n c a u s ed by t h e p r o b e , i t s e f f e c t s on the measured v a l u e s o f t h e m a g n e t i c f i e l d s and t h e c u r r e n t d e n s i t i e s o f a. plasmai, and the l i m i t a t i o n s o f s u c h a .method o f p r o b i n g . Based on t h e s t e a d y s t a t e t h e o r y , E c k e r e t a l . ( 1 9 6 2 ) have done'some computer c a l c u l a t i o n s to see how a probe p e r -t u r b s t h e c u r r e n t f l o w i n l i n e a r d i s c h a r g e s . They s t u d i e d how the measured probe s i g n a l s could be used t o c a l c u l a t e t h e - 3 -t r u e c u r r e n t d e n s i t y . I n t h e i r c a l c u l a t i o n s , the d i s c h a r g e i s s p l i t up i n t o a s e t o f c u r r e n t l a y e r s . The p e r t u r b e d m a g n e t i c f i e l d due to t h e d i s t o r t e d c u r r e n t f l o w i n each c u r r e n t l a y e r i s f i r s t c a l c u l a t e d . By a d d i n g the f i e l d s o f th e d i f f e r e n t l a y e r s , the r e s u l t a n t m a g n e t i c f i e l d i s o b t a i n e d . T h i s w i l l t h e n g i v e t h e v a l u e o f t h e f i e l d t h a t would be mea-s u r e d by a m a g n e t i c p r o b e . By c o r r e c t i n g the measured probe r e s u l t , the t r u e c u r r e n t d e n s i t y and magnetic f i e l d i n the d i s c h a r g e can be computed. T h e i r t h e o r y i s based on time i n d e p e n d e n t c a l c u l a t i o n s and has not c o n s i d e r e d moving c u r r e n t s s u c h as t h o s e o b s e r v e d In z - p i n c h d i s c h a r g e s . N e v e r t h e l e s s , we s h a l l show t h a t t h e i r t h e o r y , a l s o a p p l i e s ; t o time dependent c a s e s and moving c u r r e n t s p r o v i d e d t h a t the c u r r e n t i s not moving too q u i c k l y . The c o n d i t i o n under w h i c h the t h e o r y a p p l i e s i s |~vj<S;<oo" where CrV and € a r e the f r e q u e n c y and the. s k i n d e p t h o f t h e c u r r e n t r e s p e c t i v e l y and v i s the v e l o c i t y a t w h i c h t h e c u r r e n t l a y e r c o l l a p s e s r a d i a l l y . Malmberg (1964) has o b t a i n e d an a n a l y t i c s o l u t i o n f o r the d i f f u s i o n o f mag n e t i c f i e l d t h r o u g h the h o l e o f a p l a n e c u r r e n t s h e e t o f i n f i n i t e e l e c t r i c a l c o n d u c t i v i t y cr. T h i s model i s d i s c u s s e d i n the.-, t h e s i s because i t h e l p s i n t h e u n d e r s t a n d i n g o f the pr o b l e m and i t i s r e l e v a n t t o h i g h f r e q u e n c y c u r r e n t flows i n h i g h l y c o n d u c t i n g m a t e r i a l such as the r e t u r n c o n d u c t o r o f t h e d i s c h a r g e v e s s e l . However, f o r p l a s m a s , i t i s g e n e r a l l y n o t a p p l i c a b l e because the con-d u c t i v i t y i s too low. The most e x t e n s i v e , e x p e r i m e n t a l s t u d y o f the p e r t u r b -a t i o n o f a probe on the plasmai i n a z - p i n c h was c a r r i e d out by - k -Daughney (1966). By m o d i f y i n g Malmberg's f o r m u l a and s p l i t t i n g the plasma c u r r e n t i n t o a number o f c u r r e n t s h e l l s , he has d e v e l o p e d an i n t e g r a l e q u a t i o n r e l a t i n g the measured m a g n e t i c f i e l d s t o the v a l u e s o f the f i e l d o c c u r r i n g i n the absence o f the p r o b e . However, the c u r v a t u r e o f the c y l i n d r i c a l c u r r e n t s h e e t s was n o t p r o p e r l y a c c o u n t e d f o r i n h i s t r e a t m e n t . As 4 a r e s u l t , h i s i n t e g r a l e q u a t i o n has a s i n g u l a r k e r n e l . I n the p r e s e n t t h e s i s t h i s p a r t i c u l a r e r r o r has been e l i m i n a t e d f r o m Daaighney' s. t r e a t m e n t o f the pr o b l e m . B e s i d e s , he has not made a q u a n t i t a t i v e a n a l y s i s o f the e f f e c t s o f e x p e r i m e n t a l e r r o r s . To i n v e s t i g a t e how e x p e r i m e n t a l e r r o r s a f f e c t the c o r -r e c t e d v a l u e s o f the c u r r e n t d e n s i t y , we employ the Monte Ca.rlo "method'in w h i c h random e r r o r s a r e added to the measured v a l u e s o f the mag n e t i c f i e l d (B) and i t ' s g r a d i e n t ( ^ ). The r e s u l t s c-f the i n v e s t i g a t i o n show t h a t the probe g r e a t l y smooths out the d e t a i l s o f t h e t r u e c u r r e n t d e n s i t y . Thus the probe measure-mehts have to be e x t r e m e l y a c c u r a t e ( w i t h i n -|$>) i n o r d e r t o o b t a i n t h e f i n e s t r u c t u r e o f the t r u e c u r r e n t s i n c e any s m a l l e r r o r s i n the probe measurements w i l l be m a g n i f i e d by a f a c t o r o f 20 i n the c o r r e c t i o n p r o c e d u r e . The p e r t u r b a t i o n on a t h i n c u r r e n t s h e e t i s s e v e r e and t h e a p p a r e n t c u r r e n t d e n s i t y d i s -t r i b u t i o n ( i . e . no c o r r e c t i o n made ) i s broadened and g r e a t l y r e d u c e d i n a m p l i t u d e . However, f o r smooth c u r r e n t d e n s i t y d i s t r i b u t i o n s , the c o r r e c t i o n p r o c e d u r e i s u n n e c e s s a r y . The m a g n e t i c probes d e s c r i b e d i n t h i s t h e s i s have been employed t o s t u d y the i n i t i a l c o l l a p s e o f c y l i d r i c a l c u r r e n t l a y e r s i n z - p i n c h d i s c h a r g e s . T h i s work has been c a r r i e d out i n h e l i u m o v e r the p r e s s u r e range from 500 y^S "to ^  mmHg. By e q u a t i n g the m a g n e t i c f o r c e to the r a t e o f change of momentum o f the c o l l a p s i n g c y l i n d r i c a l c u r r e n t s h e e t , the r a d i u s o f the s h e e t as a f u n c t i o n o f time can be p r e d i c t e d ( t h e s o - c a l l e d c o l l a p s e c u r v e ) . The t h e o r e t i c a l l y p r e d i c t e d , c o l l a p s e c u r v e s agree v e r y w e l l w i t h the e x p e r i m e n t a l o b s e r -v a t i o n s , p r o v i d e d t h a t t h e i n i t i a l mass o f the s h e l l i s t a k e n i n t o a c c o u n t . I n e f f i c i e n t t r a p p i n g o f gas by the s h e l l i s a l s o a c c o u n t e d f o r i n the t h e o r y . We s h a l l d i s c u s s the t h e o r e t i c a l a s p e c t s o f the mag-n e t i c probe measurement i n . C h a p t e r 2 and the e x p e r i m e n t a l r e s u l t s i n C h a p t e r 3» I n the t h e o r e t i c a l p a r t , we f i r s t i n t r o d u c e the c o r r e c t i o n f o r m u l a e u s i n g the models by E c k e r e t a l . ( 1 9 6 2 ) ( £ 2 . 2 . 1 ) , and Malmberg ( 1 9 6 4 ) (j> 2.2.2) r e s -p e c t i v e l y . I n each c a s e , we d e r i v e an i n t e g r a l e q i i a t i o n w h i c h r e l a t e s the g r a d i e n t probe measurements t o the t r u e c u r r e n t d e n s i t y o f a z - p i n c h a l l o w i n g f o r t h e p e r t u r b a t i o n of t he p r o b e . The l i m i t a t i o n s o f the model o f E c k e r e t a i . a r e t h e n g i v e n ( £ 2 . 2 . 3 ) -To c o n v e r t t h e e q u a t i o n t o a form s u i t a b l e f o r n u m e r i c a l c o m p u t a t i o n s , we t r a n s f o r m i t i n t o at m a t r i x equa-t i o n ( £>2.3) w h i c h can be used t o c a l c u l a t e the t r u e c u r r e n t d e n s i t y f r o m the probe s i g n a l s u s i n g t h e IBM JOkO d i g i t a l - 6 -computer. S u b s e q u e n t l y , we s t u d y t h e s t a b i l i t y o f t h e e q u a t i o n w i t h the h e l p o f the Monte C a r l o method (Fox, 1 9 6 2 , p. 4 2 5 ) by s i m u l a t i n g e x p e r i m e n t a l e r r o r s w i t h random e r r o r s ( ^ 2 . 4 ) . I n the e x p e r i m e n t a l p a r t , we g i v e d e t a i l s about the c o n s t r u c t i o n ( £ 3 • 3 ) » c a l i b r a t i o n ( ^ 3 . 3 . 1 ) , the f r e q u e n c y r e s p o n s e ( ^ 3 «3«2) and t h e r e p r o d u c i b i l i t y o f the g r a d i e n t probe s i g n a l s ( y 3 » 3 - ) ' The measurements o f the c u r r e n t d e n s i t y and m a g n e t i c f i e l d i n z - p i n c h d i s c h a r g e s i n h e l i u m a r e t h e n o b t a i n e d ( j> 3*4.1 ). The r e s u l t s a r e used to c a l c u l a t e t h e a c c e l e r a t i n g m a g n e t i c f o r c e on t h e c o l l a p s i n g c u r r e n t s h e l l i n the m o d i f i e d snow-plow model ( £ 3 * 4 . 2 ) . C h a p t e r 2 MEASUREMENTS OF THE CURRENT DENSITY OF A PLASMA IN A PULSED DISCHARGE 2.1 I n t r o d u c t i o n We a r e i n t e r e s t e d i n the dynamics o f a p l a s m a . S i n c e magnet ic.„ f o r c e s p l a y t h e dominant r o l e i n t h e m o t i o n o f s u c h s y s t e m s , i t i s i m p o r t a n t t o be a b l e t o measure the magnetic f i e l d a c c u r a t e l y . The s i m p l e s t d e v i c e t o a c c o m p l i s h s u c h measurements i s a m a g n e t i c p r o b e . The most c o n v e n i e n t probe i s t h e s e a r c h c o i l w h i c h i s i n t r o d u c e d i n t o the plasma i n an i n s u l a t i n g s h i e l d . A c o n v e n t i o n a l m a g n e t i c probe c o n s i s t s o f as s m a l l c o i l e n c l o s e d i n a c y l i n d r i c a l i n s u l a t i n g j a c k e t . When p l a c e d i n a^ time v a r y i n g plasma, an emf i s i n d u c e d w h i c h i s p r o p o r t ionasl to the r a t e o f change o f the t o t a l m a g n e t i c f l u x t h r e a d i n g t h r o u g h the s e n s i n g c o i l . To o b t a i n the m a g n e t i c f l u x , the o u t p u t s i g n a l s from the probe a r e i n t e g r a t e d by an RC c i r c u i t as shown i n F i g . 1. T h i s k i n d o f m a g n e t i c probe was f i r s t u s ed e x t e n s i v e l y by a group o f R u s s i a n s c i e n t i s t s ( A r t s i m o v i c h e t a l . , 1 9 5 6 ) . A p a r t from the i n v e s t i g a t i o n s o f Malmberg (1.964) and E c k e r e t a l . ( 1 9 6 2 ) , l i t t l e has been done to improve the t e c h n i q u e s and i n t e r p r e t a t i o n s o f s u c h probe measurements. F o r t h i s r e a s o n , we a t t e m p t t o i n v e s t i g a t e i n d e t a i l t h e v a r i o u s problems i n m a g n e t i c p r o b i n g and d e v e l o p t e c h n i q u e s f o r more a c c u r a t e mea? surements o f c u r r e n t d e n s i t i e s . 8 -SPARK GAP SWITCH. . A N O D E . T R A N S M I S S I O N L I N E x ; v-- a- R =±= c CONDEMsea SANK R O G - O W S K I .. C O I I TERMINATING-RESIST*? a R C P A S S I V E . . . ... I N T E G R A T O R .VERTICAL INPUT . E X T E R N A L . T R I G G E R : ._C.RO. . F I G . 1 A T Y P I C A L A R R A N G E M E N T . . . F O R M A G N E T I C . . ; P R O B I N G .. - 9 -( i ) I n a c c u r a c y i n the G r a d i e n t o f the Measured M a g n e t i c F i e l d One g r e a t d i f f i c u l t y i n m e a s u r i n g the c u r r e n t d e n s i t y w i t h a m a g n e t i c probe i s the i n a c c u r a c y i n o b t a i n i n g the g r a d i e n t o f the m a g n e t i c f i e l d . I n a p u l s e d plasma w i t h good r e p r o d u c i b i -l i t y , t he m a g n e t i c f i e l d o f the plasma, i s mapped by t a k i n g meat-surements f o r d i f f e r e n t d i s c h a r g e s w i t h the probe a t d i f f e r e n t p o s i t i o n s . F o r a l i n e a r p i n c h , the d i s c h a r g e i s assumed t o be a x i a l l y s y m m e t r i c . I n c y l i n d r i c a l c o o r d i n a t e s , the c u r r e n t d e n s i t y J " z ( r ) i n t h e a x i a l d i r e c t i o n a t a r a d i u s r i s r e l a t e d to the a z i m u t h a l m a g n e t i c f l u x d e n s i t y B ^ ( r ) k y t h e ( M a x w e l l ) e q u a t i o n ( D J ( r ) = — + \ ' z • ' y. L r d r Here ji i s t h e m a g n e t i c p e r m e a b i l i t y o f the plasma; and e q u a t i o n ( l ) i s i n mks u n i t s . T h e r e f o r e t o d e t e r m i n e t h e c u r r e n t d e n s i t y J ( r ) , we s h o u l d measure b o t h B / ( r ) and d.Bp' ( r ) y. ' 0 d r A c o n v e n t i o n a l probe measures t h e m a g n e t i c f l u x d e n s i t y B ^ ( r ) o f a d i s c h a r g e w i t h r e a s o n a b l e a c c u r a c y . However> the v a l u e s o f ( r ^ a r e u s u a l l y o b t a i n e d by d i f f e r e n t i a t i n g B , ( r ) d r 0 n u m e r i c a l l y . T h i s g i v e s u n r e l i a b l e v a l u e s o f c ^ ^ ( r ) e s p e c i a l l y d r when e x p e r i m e n t a l v a l u e s o f B ^ ( r ) always have a s c a t t e r about an unknown c u r v e . T h i s c a n be seen as f o l l o w s . A m a g n e t i c probe w h i c h measures B ^ ( r ) a t i n t e r v a l s A r g i v e s ^the measured c u r r e n t d e n s i t y - 10 -( 2 ) j , ( r ) ...1. ( - ^ 4 - , - ^ z B l ) } V > z v / Y r 2 + r 1 r 2 - r . r +r at r = 1 2 , where B and B a r e t h e r e s p e c t i v e f i e l d s a t 2 1 d r ^ and w h i c h a r e s e p a r a t e d by a. d i s t a n c e A r . I f terms i n v o l v i n g second and h i g h e r d e r i v a t i v e s o f B^ (see j 2 . 4 . 3) are n e g l i g i b l e , e q u a t i o n ( 2 ) g i v e s the t r u e v a l u e J ( r ) when • z • -B ^ ( B ^ j r ^ and r ^ a r e known e x a c t l y . We now denote t h e s t a n d a r d e r r o r s i n the measurements o f J , Bi/ and r by C T , a l and o" r e s p e c t i v e l y . Under z .0 ^ J B / r 1 • . r z p t y p i c a l e x p e r i m e n t a l c o n d i t i o n s , we have <r* ~ ^ r and r ^ > A r . S t a n d a r d c a l c u l a t i o n s w i t h the h e l p o f ( 2 ) g i v e s ( 3 ) [(g*)2^)2!* ~ ^ .  for ?a=ft.-J z - A r ' L Bp7 T V r ' J Ar Bp" \ , r.,-!-, c o r r e s p o n d i n g t o g r a d u a l changes i n B i ^ and ( 4 ) f ^ ~ 2 . A - J z ot,, 2 o" 2 B, ' v A r f o r B 2r£l ^ §4 r 2 " r 1 ^ r c o r r e s p o n d i n g , to s h a r p changes i n . Here the symbol " ~ " means " o f the same o r d e r o f magnitude as ". From ( 3 ) and ( 4 ) , we i m m e d i a t e l y see t h a t the f r a c t i o n a l e r r o r i n the computed v a l u e o f J z i s e i t h e r (~—) t i m e s the l a r g e r o f the f r a c t i o n a l e r r o r s i n the measurements o r comparable t o . u n i t y . I n o r d e r t o a v o i d t h i s d i f f i c u l t y , t he c u r r e n t d e n s i t y may be measured by a method r e p o r t e d by W r i g h t and J a h n ( 1 9 6 5 ) . They use a a m i n i a t u r e Rogowski c o i l e n c l o s e d i n a s m a l l t o r o i d a l i n s u l a t i n g tube (see F i g . 2 ( a ) and ( b ) ) v However, s u c h a,probe n o t o n l y has a poor s p a t i a l r e s o l u t i o n , b u t a l s o i t s t o r o i d a l geometry p e r t u r b s the plasma c u r r e n t s e v e r e l y . B e s i d e s , i t i s always d i f f i c u l t t o d e t e r m i n e what f r a c t i o n o f t h e t r u e c u r r e n t p a s s e s t h r o u g h the l o o p . F i g . 2(ai) . S k e t c h o f m i n i a t u r e F i g . 2 ( b ) . C r o s s - s e c t i o n of Rogowski C o i l P r o b e . , Rogowski c o i l p r o b e . I n an a l t e r n a t i v e a p p r o a c h , Ohkawa e t a l . (1963) use a probe c o n t a i n i n g f o u r t e e n c o i l s s e p a r a t e d a t i n t e r v a l s o f 1 cm. The probe i s p l a c e d a c r o s s the d i s c h a r g e tube a l o n g a i minor d i a m e t e r o f a s t a b i l i z e d t o r o i d a l p i n c h . The c o i l s a r e a r r a n g e d i n s e v e n p a i r s . Each p a i r has one c o i l c o u p l i n g t h e a x i a l s t a b i l i z i n g f i e l d and the o t h e r c o u p l i n g the t r a n s v e r s e f i e l d due to the p i n c h . Such an arrangement g i v e s c o n s i s t e n t r e s u l t s when t a k i n g measurements i n a system o f poor r e p r o -d u c i b i l i t y . However, the m a g n e t i c f i e l d i s d e t e r m i n e d o n l y a t s e v en p o s i t i o n s a c r o s s the d i s c h a r g e tube w h i c h has a minor d i a m e t e r o f 14 cm. The s p a t i a l r e s o l u t i o n o f the system i s t h e r e f o r e not v e r y s a t i s f a c t o r y . To improve t h e s p a t i a l r e s o l u t i o n o f Ohkawa's syste m , we have f o l l o w e d L o v b e r g ' s s u g g e s t i o n ( H u d d l e s t o n e et; a l . , 1965>P» 7 9 ) hy d e v e l o p i n g a m i n i a t u r e probe w h i c h measures b o t h the f l u x d e n s i t y and i t s r a d i a l g r a d i e n t i n one measurement,with comparable a c c u r a c i e s . I n f u t u r e , t o d i s t i n g u i s h i t from t h e c o n v e n t i o n a l m a g n e t i c p r o b e , we s h a l l c a l l i t the g r a d i e n t p r o b e . I t c o n s i s t s o f two s m a l l i d e n t i c a l s e a r c h c o i l s wound i n t he same d i r e c t i o n and c o n n e c t e d as shown i n F i g . 3 ( a ) , ( b ) (see F i g . 23,2k a l s o ) . A f t e r p a s s i n g t h r o u g h an e x t e r n a l b a l a n c i n g c i r c u i t , the o u t p u t s i g n a l s a r e f e d i n t o the d i f f e r -e n t i a l a m p l i f i e r o f an o s c i l l o s c o p e . The b a l a n c i n g c i r c u i t e n s u r e s a z e r o s i g n a l f o r a..uniform f i e l d w h i l e the d i f f e r -e n t i a l a m p l i f i e r h e l p s t o e l i m i n a t e common mode s i g n a l s . IusuLATiN^ JACKET --8 1 PRO&B COILS BAtAr/c/t/f CIRCUIT SYSTEM F i g . 3 ( a ) . S k e t c h o f g r a d i e n t . , F i g . 3 ( h ) . C o n n e c t i o n s i n g r a -probe.. d i e n t probe and b a l a n c i n g c i r c u i t The g r a d i e n t probe has s e v e r a l a d v a n t a g e s . I f a; d u a l beam scope i s u s e d , one beam can measure the d i f f e r e n c e o f t h e s i g n a l s f r o m t h e two c o i l s and a n o t h e r beam p i c k s up s i g n a l s f r om one o f them. T h i s e n a b l e s s i m u l t a n e o u s measurements o f dB^ b o t h BS^  and ^ r —• U s i n g e q u a t i o n ( l ) , t he l o c a l c u r r e n t d e n s i t y J ( r ) can be c a l c u l a t e d from the i n f o r m a t i o n o f one d i s c h a r g e o n l y . T h i s a v o i d s t h e t e d i o u s p r o c e d u r e o f mapping B i j ^ ( r ) , and i n a d d i t i o n , the v a l u e s o f ~ [ ~ ^ o b t a i n e d have com-p a r a b l e a c c u r a c y as t h a t o f B . ^ ( r ) . The s p a t i a l r e s o l u t i o n i s b e t t e r t h a n t h a t o f t h e Rogowski c o i l probe ( W r i g h t and J a h n , 1 9 6 5 ) . ( i i ) I n t e r a c t i o n o f Probe w i t h the D i s c h a r g e A n o t h e r d i f f i c u l t y w h i c h l i m i t s t he use o f a. probe i s t h a t i t p e r t u r b s the plasma a p p r e c i a b l y . A m a g n e t i c probe not o n l y c o o l s the plasma i n i t s n e i g h b o u r h o o d ( t h e r e b y con-t a m i n a t i n g the p l a s m a ) , i t a l s o d i s t o r t s the c u r r e n t f l o w and the m a g n e t i c f i e l d by i t s geometry and by t h e r e a c t i o n o f t h e s e n s i n g c o i l on the f i e l d . The c o n t a m i n a t i o n and c o o l i n g o f the plasma, e s s e n t i a l l y i n c r e a s e t h e e f f e c t i v e r a d i u s o f t h e p r o b e . F o r a plasma i n a l i n e a r p i n c h h a v i n g a... temperature: o f 10 eVL o r l e s s , the i n c r e a s e i n the e f f e c t i v e r a d i u s i s s m a l l ( H u d d l e s t o n e , 1965>P- 1 0 3 ) . A l s o s i n c e the c u r r e n t f l o w i n g i n the c o i l i s n e g l i g i b l e compared w i t h t h e d i s c h a r g e current,, the r e a c t i o n o f the c o i l can be i g n o r e d . The most i m p o r t a n t c o r r e c t i o n i n measurements u s i n g m a g n e t i c , p r o b e s i s t h e r e f o r e the d i s t o r t i o n o f the c u r r e n t f l o w . - ^k -I f we can o b t a i n the changes i n B caused by t h e probe;, we can i n p r i n c i p l e a l l o w f o r them, and deduce the v a l u e o f J w h i c h w o u l d o c c u r in-, the absence o f t h e prob e . E c k e r e t a l . ( 1 9 6 2 ) have made a computer s t u d y o f the probe p e r t u r b a t i o n on t h e s t e a d y c u r r e n t f l o w around a c y l i n d r i c a l p r o b e , and Malmberg ( T 9 6 4 ) has g i v e n an a n a l y t i c e x p r e s s i o n f o r t h e p e r t u r b e d mag-n e t i c f i e l d n e a r the h o l e i n a t h i n p l a n e i n f i n i t e c o n d u c t o r . I n M almberg ' s i c a l c u l a t i o n s , he; has assumed t h a t the ma g n e t i c f i e l d Bi i s t a n g e n t i a l t o the c o n d u c t i n g s u r f a c e . T h i s assump-t i o n i s v a l i d f o r h i g h f r e q u e n c y a l t e r n a t i n g c u r r e n t s . Ho.wev.er, f o r a low f r e q u e n c y f i e l d i n w h i c h the s k i n d e p t h 6 i s much g r e a t e r t h a n t h e t h i c k n e s s o f the c o n d u c t i n g s h e e t , s u c h an a s s u m p t i o n r e q u i r e s f u r t h e r r e f i n e m e n t s . F o r example, a t ai f r e q u e n c y o f 1 MHz, the s k i n d e p t h o f b r a s s i s about .1 mm whereas the s k i n d e p t h o f a plasma, i n a f a s t l i n e a r p i n c h can be l a r g e r t h a n 1 cm (see T a b l e i ) . I t i s t h e r e f o r e n ot c o r r e c t to assume t h a t B i s t a n g e n t i a l t o the m e t a l s u r f a c e w i t h o u t f u r t h e r j u s t i f i c a t i o n s . T a b l e I C o r i d u G K p i ^ ^ r e q ^ u e n c y M a g n e t i c S k i n t i v i t y ' p e r m e a b i l i t y d e p t h cr (mhos/m) to ( r a d / s e c ) u (H/m) S (mm) B r a s s 1 O 7 1 O 6 kr x 1 0 ~ 7 .5 Plasma i n 1 0 - t o 1 0 5 1O 6 klrx^O~^l 14 t o 5 a z - p i n c h - 15 -The p r e v i o u s models a p p l y r i g o r o u s l y o n l y t o t h e c u r r e n t f l o w around the h o l e o f an i n f i n i t e p l a n e c o n d u c t i n g s h e e t . However, the r e s u l t s c a n a l s o be used f o r c u r r e n t f l o w s around a c i r c u l a r h o l e i n a c y l i n d r i c a l s h e e t (see F i g . 4 ( a i ) , ( b ) , ( c ) , and ( d ) ) , p r o v i d e d t h a t t he r a d i u s o f the c y l i n d e r i s much l a r g e r t h a n t h a t o f the h o l e ( s e e a p p e n d i x i ) . I n t h e case o f a z - p i n c h d i s c h a r g e , we w i l l a p p r o x i m a t e the plasma c u r r e n t by a t ^ s u p e r p o s i t i o n o f c o a x i a l c y l i n d r i c a l c u r r e n t s h e e t s o f d i f -f e r e n t r a d i i . I n what f o l l o w s , we s h a l l b r i e f l y i n t r o d u c e the E-K-Z model ( E c k e r e t a l ' . , 1962) and Malmberg's model (Malmberg, 1 96k) . I n each c a s e , an i n t e g r a l e q u a t i o n f o r c a l c u l a t i n g t h e t r u e u n p e r t u r b e d c u r r e n t from the probe measurements i s o b t a i n e d ( ^  2.2.1 and ^  2.2.2..). We t h e n c o n s i d e r t he l i m i t a t i o n s o f the E-K-Z model f o r time dependent c u r r e n t s i n moving plasmas ( § 2.2.3). I n o r d e r t h a t t he i n t e g r a l e q u a t i o n can be s o l v e d n u m e r i c a l l y , i t i s t r a n s f o r m e d i n t o a m a t r i x e q u a t i o n s u i t a b l e f o r a d i g i t a l computer (£> 2.3). Monte C a r l o t e c h n i q u e s a r e th e n a p p l i e d t o i n v e s t i g a t e how the e x p e r i m e n t a l e r r o r s i n the probe measurements a f f e c t t he s o l u t i o n f o r t h e u n p e r t u r b e d c u r r e n t d e n s i t y ( ^  2.4). T h i s i s done by a d d i n g random e r r o r s t o known c u r r e n t d i s t r i b u t i o n s and s t u d y i n g t h e i r e f f e c t s on the s o l u t i o n o f t h e i n t e g r a l e q u a t i o n employed i n t h e c o r r e c -t i o n p r o c e d u r e . B e f o r e we i n t r o d u c e the v a r i o u s models, we d e f i n e t he f o l l o w i n g symbols and e x p l a i n t h e i r p h y s i c a l meanings: - 16 -R£J$/OAlS WHERE CURRENT FLOWS AR£ VlSTORTBP •b'STVRTFO CURRENT FLOW P i g . 4 ( a ) . C y l i n d r i c a l c u r r e n t sheet w i t h a probe p a s s i n g t h r o u g h i t d i a m e t r i c a l l y . F i g . 4(b) A p p r o x i m a t i n g c y l i n d r i c a l s u r f a c e by p l a n e s . CURRENT sneers F i g . 4 ( c ) . C r o s s - s e c t i o n , o f c u r r e n t s h e e t s p e r p e n d i c u l a r t o a x i s o f . c y l i n d e r . F i g . 4 ( d ) . R e p r e s e n t a t i o n o f plasma c u r r e n t by c y l i n d r i c a l c u r r e n t s h e e t s B q , the u n p e r t u r b e d m a g n e t i c f l u x d e n s i t y -- the a c t u a l f l u x d e n s i t y i n the absence o f p r o b e s , Bp, t h e p e r t u r b e d m a g n e t i c f l u x d e n s i t y -- the o b s e r v e d f l u x d e n s i t y p r o d u c e d by the plasma whose c u r r e n t f l o w p a t t e r n has been d i s t o r t e d by p r o b e , A B, t h e change i n m a g n e t i c f l u x - d e n s i t y due t o the p r e s e n c e o f t h e p r o b e , i . e . B; = B. - B , • p o J q , t h e t r u e c u r r e n t d e n s i t y -- the c u r r e n t d e n s i t y i n t h e absence o f a p r o b e , J p , t h e ^ « a « M ^ -- the computed v a l u e -of t h e c u r r e n t d e n s i t y t a k i n g B^ as the t r u e f l u x d e n s i t y , A J , the change i n c u r r e n t d e n s i t y due t o t h e p r e s e n c e o f the pr o b e . I n t h e f u t u r e , we s h a l l denote the v a l u e o f a q u a n t i t y f ait a c o o r d i n a t e r by f ( r ) . U n l e s s o t h e r w i s e s t a t e d , t h i s does not mean t h a t f ( r ) i s a f u n c t i o n o f r o n l y . I n t h e a p p l i c a t i o n o f the c o r r e c t i o n p r o c e d u r e o f t h i s t h e s i s , we a r e m a i n l y i n t e r e s t e d i n the a x i a l c u r r e n t d e n s i t y and the a z i m u t h a l mag-n e t i c f l u x d e n s i t y i n a l i n e a r d i s c h a r g e w i t h an a x i a l symmetr T h e r e f o r e , f o r c o n v e n i e n c e , we s h a l l use the symbols B, , B. o p and J q , J p to s t a n d f o r t h e s e components o f the c o r r e s p o n d i n g f i e l d s . - 18 -2.2.1 The Model by E c k e r , K r O l l and Z O l l e r (E-K-Z model) We now s t u d y t h e p e r t u r b a t i o n o f a c y l i n d r i c a l probe o f r a d i u s a on the p l a s m a a c u r r e n t f l o w i n a l i n e a r d i s c h a r g e i n s i d e a c y l i n d r i c a l e n c l o s u r e o f r a d i u s R. We s h a l l d e r i v e a f o r m u l a w h i c h r e l a t e s the p e r t u r b e d c u r r e n t d e n s i t y t o the t r u e c u r r e n t d e n s i t y . The a p p r o a c h we use h e r e d i f f e r s f r o m t h a t u s e d by E c k e r eifc al».(l962) . However, t h e r e s u l t i s e s s e n -t i a l l y the same. We s h a l l t h e r e f o r e c a l l i t the E-K-Z model. F o r c o n v e n i e n c e , we f i r s t c o n s i d e r t h e time i n d e p e n d e n t c a s e . The problem w i l l be e x t e n d e d t o time dependent cases i n ^ ' 2 . 2 . 3 . We now assume the f o l l o w i n g . (a) I n the absence o f the p r o b e , the system i s a x i a l l y sym-m e t r i c and the d i s c h a r g e c u r r e n t i s i n the a x i a l d i r e c t i o n , i . e . (1 ) J = J ( r ) k , • o o > ' k b e i n g a u n i t v e c t o r i n the a x i a l d i r e c t i o n (see F i g . 5 ) (b) There i s no r a d i a l f l o w o f c u r r e n t j i . e . ( 2 ) ( J 0 ) r = o . ( c ) The probe p a s s e s t h r o u g h the e n c l o s u r e d i a m e t r i c a l l y . ( T h i s i s the arrangement we use i n our e x p e r i m e n t ; see $3.1) - 19 -From M a x w e l l ' s e q u a t i o n s and t h e i r l i n e a r i t y , we r e a d i l y o b t a i n the change i n a z i m u t h a l f i e l d a l o n g the a x i s o f the probe as (?). . . A ^ ( r ) . . f e / d . . E ^ a i i i * ^ . Here "r and "s* r e f e r t o the f i e l d p o i n t and the s o u r c e p o i n t r e s p e c t i v e l y . The r e s t o f the symbols have t h e i r u s u a l meanings PROBB J/ICKET VISTCKTED PLASMA cumtrr F i g . j. P e r t u r b a t i o n o f probe on plasma c u r r e n t i '• S i n c e t h e r e i s no r a d i a l f l o w , we can s p l i t up t h e d i s c h a r g e c u r r e n t i n t o c y l i n d r i c a l l a y e r s and the c u r r e n t i n each l a y e r w i l l be c o n f i n e d w i t h i n the l a y e r i t s e l f . I f t h e r a d i u s o f the c y l i n d e r i s much l a r g e r t h a n b o t h the t h i c k n e s s o f the l a y e r and t h e r a d i u s o f the h o l e (see A p p e n d i x i ) , t he l a y e r may be r e g a r d e d as i n f i n i t e l y t h i n and the r e g i o n where - 20 -t h e d i s t o r t i o n o f c u r r e n t i s s i g n i f i c a n t w i l l be s m a l l compared w i t h the whole l a y e r . U s i n g the f a c t t h a t t h e a r c and t h e t a n g e n t s u b t e n d e d by a a s m a l l a n g l e d i f f e r by a > q u a n t i t y o f s e c ond o r d e r i n t h e a n g l e , the r e g i o n o f the c u r v e d l a y e r o v e r w h i c h A J i s s i g n i f i c a n t can be r e p l a c e d by t h e t a n g e n t p l a n e a t the c e n t r e o f t h e h o l e . Aj can now be s o l v e d w i t h the; h e l p o f Ohm's law, M a x w e l l ' s e q u a t i o n s and t h e i r l i n e a r i t y , a s suming the r e g i o n o f f l o w to be p l a n a r . F o r a c y l i n d e r o f j r a d i u s s and i n f i n i t e s i m a l t h i c k n e s s d s , ^ J s a t i s f i e s t h e e q u a t i o n s ( 4 ) d i v ( AJ) = 0 , ( 5 ) c u r l - - ( , A J ) = 0 , f o r } ds 4 ^ ^ a- ds and p a ( s e e Fig° 6). The v boundary c o n d i t i o n s f o r A J a r e (6) ^A ;6S f a r away from the probe and the n o r m al component (7) ( A J ) = - ( J ) v ' \ .. < n • • v o'n aft,, the boundary o f the h o l e p u n c t u r e d by the p r o b e . E q u a t i o n s ^ (6) and (7) do not s p e c i f y the p r o blem u n i q u e l y because we do not know the t a a a g e n t i a l component o f A J a t the boundary. We can r e s o l v e t h i s d i f f i c u l t y by u s i n g the symmetry o f the; pro b l e m and assumption! (b) w h i c h c o r r e s p o n d s t o z e r o f l o w a c r o s s t h e p l a n e (see ^ 2 . 2 . 3 ) • U s i n g the c y l i n d r i c a l c o o r d i n -a t e s ( p , 0, ) d e f i n e d i n F i g . 6, s t a n d a r d c a l c u l a t i o n s g i v e (8) A J = ( A y A J 0 , AJ^ ) = JQ(s) F ( jo, 9 , a) = J Q ( s ) ( - ( ^ ) 2 c o s 0 , + ( ^ ) 2 s i n 0 , 0 ) , a b e i n g t h e r a d i u s ^ o f t h e prob e . I n s i d e t h e h o l e , we have; •SuflFACF OF CYLINDBR. OF RADIUS S < a. -TANGENT PLAHB AT CENTRE OF HOLE PROBE JAO&T F i g . 6. A p p r o x i m a t i n g a c u r v e d s u r f a c e by a p l a n e From e q u a t i o n ( 8 ) , i t i s c l e a r t h a t (10) |**| ~ (p2 | j j , p > a T h e r e f o r e A J i s s i g n i f i c a n t f o r j> < a' , wherei a'^v a « s aaad the r e g i o n where A J i s s i g n i f i c a n t i s s m a l l compared w i t h t h e c y l i n d e r . S u b s t i t u t i n g ( 8 ) arid ( 9 ) i n t o ( 1 ) and i n t e g r a t i n g o v e r p and 0 (see A p p e n d i x I ) , we o b t a i n /•to (11) A B Ar) = i jx P J ds J Q ( s ) C ( r . s . a ) J ^ where (12) G - ( r . s . a ) = C ( ^ ) - C ( ^ ) , and C.(u) i s d e f i n e d by the f u n c t i o n a l r e l a t i o n (13) . 0 ( u ) = - js g n (u) - u ( 1 + u 2 . r The symbol d e n o t e s t h e p r i n c i p a l v a l u e o f the i n t e g r a l . We have two terms c o n t a i n i n g C ( r + a ) and C ( r ) r e s p e c t i v e l y i n the i n t e g r a l o f (1 .1 ) because t h e r e a r e two h o l e s i n each c y l i n d e r . I n d e r i v i n g , e q u a t i o n (11), we have assumed. t h a t a, c i r c u l a r h o l e d i s t o r t s t h e c u r r e n t f l o w i n i t s neighbourhood, i n a c y l i n d r i c a l l a y e r i n the same way as i t d i s t o r t s t h e f l o w i n a p l a n e c u r r e n t s h e e t . T h i s i s a.good a p p r o x i m a t i o n f o r c y l i n d e r s o f r a d i i l a r g e compared w i t h t h a t o f the h o l e , but. i t does not h o l d f o r c y l i n d e r s o f s m a l l r a d i i . However, i f we a r e i n t e r e s t e d i n A B ^ ( r ) f a r away from t h e a x i s s u c h t h a t (—)« 1, the predominant c o n t r i b u t i o n t o t h e i n t e g r a l o f equar t i o n (11) comes from c u r r e n t s h e l l s o f l a r g e r a d i i f o r w h i c h the a p p r o x i m a t i o n h o l d s . B e s i d e s , i n a z - p i n c h d i s c h a r g e s t h e c u r r e n t d e n s i t y near t h e a x i s i s n e g l i g i b l e b e f o r e t h e c u r r e n t s h e l l has c o l l a p s e d t o t h e a x i s . E q u a t i o n s h o u l d t h e r e -f o r e be v a l i d f o r a z - p i n c h d i s c h a r g e b e f o r e t h e f o r m a t i o n o f the f i r s t p i n c h . Biy the d e f i n i t i o n o f A B y ( r ) (see; ^2.1 ), the p e r t u r b * a z i m u t h a l m a g n e t i c f i e l d B» ( r ) a l o n g the a x i s o f t h e probe : P : g i v e n by ( 1 k) \ (^k) B p ( r ) = B Q ( r ) + B ^ ( r ) \ The measured a x i a l c u r r e n t d e n s i t y ( j ) i s t h e n g i v e n by > p' z. r ( J ) = ( c u r l B ) \ \ p' z v • P z \ o r x \ , r 3B (r') B ( r ) 7 (15) J ( r ) = 4- B p Y S u b s t i t u t i o n o f (14) i n t o (15) g i v e s '00 (16) J p ( r ) = J J Q ( s ) K ( r , s , a ) ds - 2k -where ( 1 7 ) K ( r , s , a ) = 2. PC ' ( r , s , a.) 3 r From e q u a t i o n ( i i ) and ( 1 6 ) , the v a l u e s o f A B ^ ( r ) and J p ( r ) due to a c y l i n d r i c a l l a y e r o f r a d i u s s and i n f i n i t e s i m a l t h i c k n e s s ds can be w r i t t e n as ( 1 8 ) ( 1 9 ) f ( r , s , a ) ds = \ y. J q ( S ) ds C ' ( r , s , a ) , g ( r , s , a ) ds = J ( s ) ds 3 c . ' ( r , s , a ) *. , o \ . » » / o x ' J r r r e s p e c t i v e l y . The p l o t s o f C ' ( r , s , a ) , f ( r , s , a ) d s and g ( r , s , a ) d s a r e g i v e n i n F i g . 7 ( a ) , ( b ) and ( c ) . ' J (a) I C'(r,Sya) i r : 1 -2a. ~°- 0 1 . i — // / - i f ( c ) E-k-z MODEL. F i g . 7. P l o t s o f C ' ( r , s , a) , f ( r , s , a )ds and g ( r , s , a ) d s v e r s u s r - 25 -2.2.2 Malmberg's Model A s e p a r a t e c a l c u l a t i o n o f the p e r t u r b a t i o n o f the mag-n e t i c f l u x d e n s i t y by a t i h o l e i n a p l a n e i n f i n i t e c o n d u c t o r i s g i v e n by Malmherg. I n h i s c a l c u l a t i o n , he. assumes t h a t the-mag n e t i c f l u x d e n s i t y n e a r the c o n d u c t o r i s t a n g e n t i a l . T h i s i s v a l i d i n the case o f a c o n d u c t o r i n a f i e l d o f s u c h a h i g h f r e q u e n c y t h a t the s k i n d e p t h i s n e g l i g i b l e compared w i t h the. t h i c k n e s s o f the s h e e t . However, i n the case o f a plasma s h e e t i n a f i e l d o f 1 MHz w h i c h i s t y p i c a l i n a f a s t l i n e a r p i n c h , such a c o n d i t i o n i s u s u a l l y v i o l a t e d . I n f a c t , f o r a t y p i c a l k plasma i n a z - p i n c h having a conduc t i v i t y o f the o r d e r o f 10 -10^ mhos/m (Tuck, 1 9 5 8 ) , t a e s k i n d e p t h S a t t h i s f r e q u e n c y i s o f the o r d e r .5 t o 1.4 cm (see t a b l e i ) . T h e r e f o r e , f o r au plasma s h e e t o f t h i c k n e s s 1 t o 2 mm, Malmberg's model i s no l o n g e r a p p l i c a b l e . H i s c a l c u l a t i o n i s e s s e n t i a l l y as f o l l o w s . C o n s i d e r a t h i n i n f i n i t e p l a n e c o n d u c t i n g s h e e t i n the ^ - p l a n e . A u n i f o r m c u r r e n t o f c u r r e n t d e n s i t y J q f l o w i n g i n the p o s i t i v e £ d i r e c t i o n i n the s h e e t p r o d u c i n g a ma g n e t i c f l u x d e n s i t y ) g i v e n by (1 ) = i fi J Q a^ , ( 2 ) B L o ( f ) = - i f J f l d f , where dC i s the t h i c k n e s s o f t h e c u r r e n t s h e e t and a a <o, £ > o, - 26 -u n i t v e c t o r i n the d i r e c t i o n . I f a c i r c u l a r h o l e o f r a d i u s a. i s now b o r e d a t t h e o r i g i n , the f l o w o f the c u r r e n t n e a r the h o l e w i l l be much d i s t o r t e d (see F i g . 8 ( a ) ) . The p e r t u r b e d f i e l d B; o u t s i d e • _ p / the s h e e t i s g i v e n by t h e M a x w e l l ' s e q u a t i o n s ( 3 ) d i v B = 0, • P (k) c u r l B p = 0, a ssuming t h a t B i s t a n g e n t i a l a t the boundary s u r f a c e S o f t h e c u r r e n t l a y e r , i . e . ( 5 ) B . n = 0, S where n i s the o u t e r normal o f S (see F i g . 8 (a) and ( b ) ) The boundary c o n d i t i o n a t p o s i t i o n s f a r away f r o m th e h o l e i (6) t p = ± i j i J o -T , <r S i n c e B i s i r r o t a t i o n a l as c a n be seen from (k), i t can b p r e g a r d e d as t h e g r a d i e n t o f a s c a l a r f u n c t i o n y, o r w r i t i n g e x p l i c i t l y (7) B = g r a d y . S u b s t i t u t i o n o f t h i s i n t o ( 3 ) g i v e s a L a p l a c e e q u a t i o n f o r - 27 -t ( 8 ) y 2 ^ = 0. Now i f we choose the o b l a t e s p h e r o i d a l c o o r d i n a t e s ( oC, p> , Q ) (Morse and F e s h b a c h , 1 Q53> P - 1292) t a k i n g the a x i s as the i, ' ' a x i s o f symmetry, the b o u n d a r y . s u r f a c e S c o i n c i d e s w i t h the c o o r d i n a t e s u r f a c e |3 = 0 (see F i g . 8 ( b ) ) and t h e L a p l a c e equar t i o n . ( 8 ) i s s e p a r a b l e . U s i n g t h e boundary c o n d i t i o n s ( 5 ) and ( 6 ) , the s o l u t i o n o f ( 8 ) i s (9 j ; .^(o^.p ,-e) = - ^ j Q d f a s m e [(cd 2 + 1 )j( 1 -/jj.^jf 1 ^ ) J | , f o r O ^ p ^ 1 , -0 ^ ^ < oo , 0 ^ 9 ^ . 2 7r , and ( 9 a ) y ( o C , A , e ) = - i u J Q d f a s i n 0 [ ( o 6 2 + l ) ( l - p 2 ) ] * j l ^ f o r 0 ^ ^ 4 1, ~<x>4°C4 0 , 04 9 427T. Here the i n v e r s e t a n -gent i s d e f i n e d f o r a p o s i t i v e q u a n t i t y ^ i n the b r a h c h (10) 0 ^ t a n " 1 ? 4 i T T The 71 component o f the p e r t u r b e d f i e l d B p a t the - a x i s i s t h e n g i v e n ( 1 1 > = - ¥ J o d ? [ ^ f f - ^ f p - t ^ " 1 <f )Jj • i >  0 a.' - 28 -F i g . 8 ( a ) . D i s t o r t e d c u r r e n t F i g . 8 ( b ) . G r o s s - s e c t i o n i n f l o w i n c u r r e n t s h e e t due to ^ - p l a n e o f t h e o b l a t e s p h e r -c i r c u l a r h o l e o i d a l c o o r d i n a t e s S u b s t i t u t i n g ( 1 ) and ( 2 ) i n t o ( 1 1 ) , we o b t a i n ( 1 2 ) A B . ^ ) = - i y u J q dc- C ( i ) where C(u) i s a . c o r r e c t i o n f a c t o r d e f i n e d by the f u n c t i o n a l r e l a t i o n 2 (1.3.) C(u) = u - sgn (u) tan"V-itrr) To o b t a i n a c o r r e c t i o n f o r m u l a s i m i l a r t o ( 1 1 ) o f ^ 2.2,1 (p. 22) f o r a l i n e a r d i s c h a r g e i n a c y l i n d r i c a l e n c l o -s u r e , we a g a i n s p l i t up t h e d i s c h a r g e c u r r e n t i n t o c y l i n d r i c a l c u r r e n t l a y e r s . The c o n t r i b u t i o n t o A B ^ due t o a c u r r e n t -c y l i n d e r o f an average r a d i u s s and an i n f i n i t e s i m a l t h i c k -ness, ds i s t h e n o f the f o r m ( 1 4 ) f ( r , s , a ) ds, = i u J q ds C ' ( r , s , a) J , where we have put ( 1 5 ) C - ' ( r . s . a ) = C - C ( ^ ) . We; have added a f a c t o r ^ i n ( l 4 ) (compare ( l 4 ) w i t h ( 1 2 ) ) t o a l l o w f o r the c u r v a t u r e o f t h e l a y e r ( s ee A p p e n d i x i ) . The two terms C ( r ~ S ) and C (r'*'s ) a r i s e f r om the f a c t t h a t t h e r e a r e two h o l e s . C o mbining t h e e f f e c t s o f a l l s u c h l a y e r s , the r e s u l t a n t change i n a z i m u t h a l f i e l d g i v e s (16> AB-^(r). = i jiP JQ (s )C..« ( r ,s ,a)~ , (compare ( 11 ) of £2.2.1). W i t h the same p r o c e d u r e as t h a t used i n £2.2.1 , we a g a i n o b t a i n . ( 1 7 ) B V / p ( r ) = ^ / j j o ( s ) d s + i y i ^ p / J o ( s ) C ' ( r , s , a i ) | d s , ( 1 8 ) J p ( r ) = J J Q ( s ) K ( r , s , a ) d s , 0 where ( 1 9 ) K ( r ) S > a ) , f 3 ^ ( r , s ? a ) ^ and (20) g ( r , s , a ) ds = J ( s ) ds 3 0 ' ( r > s 7 a ) * . The p l o t s o f C ' ( r , s , a")"7 f ( r , s , a)d s and - g ( r , s , a ) d s a r e g i v e n ins F i g . 7 ( a ) , (b) and ( c ) . \ The main d i f f e r e n c e between Malmberg's model and the \ E-K-Z model l i e s i n t h e f a c t t h a t t h e f o r m e r r e q u i r e s Bp t O ; be t a n g e n t i a l everywhere a t the s u r f a c e o f the c u r r e n t s h e e t w h i l e t h e l a t t e r r e q u i r e s the c u r r e n t t o be t a n g e n t i a l a t t h e boundary o f t h e h o l e . I n . our e x p e r i m e n t , we do not have a l l the i n f o r m a t i o n about the boundary c o n d i t i o n s o f the f i e l d s . To c o r r e c t f o r the.' p e r t u r b a t i o n o f a h o l e on the f l o w o f e l e c t r i c c u r r e n t i n a, c u r r e n t l a y e r , we t h e r e f o r e choose the s i mp l e s t model a p p r o -p r i a t e to the e x p e r i m e n t a l c o n d i t i o n s . To do s o , we f i r s t c o n s i d e r the f o l l o w i n g l i m i t i n g c a s e s . (a) Low f r e q u e n c y l i m i t . F o r a low f r e q u e n c y m a g n e t i c f i e l d s u c h t h a t the s k i n d e p t h 8 i s comparable w i t h the d i m e n s i o n s o f t h e l a y e r , the) f i e l d w i l l d i f f u s e i n t o t h e l a y e r g i v i n g r i s e t o a n o r m al component a t the s u r f a c e and i n c r e a s i n g the e f f e c t i v e r a d i u s o f t he h o l e . The boundary c o n d i t i o n i n Malmberg's model i s t h e r e f o r e not v a l i d . I n t h i s case the E-K-Z model i s a p p l i c a -b l e . (b) H i g h f r e q u e n c y l i m i t . I f the f r e q u e n c y o f the f i e l d i s so h i g h t h a t S i s n e g l i g i b l e compared w i t h t h e d i m e n s i o n s o f the l a y e r , the ma g n e t i c f i e l d , a t the s u r f a c e o f the l a y e r i s t a n g e n t i a l . Malmberg's model i s e x p e c t e d t o be more s u i t a b l e i f the l a y e r i s not too t h i c k . A c o n v e n i e n t t e s t o f the models i s t o measure: f o r t h e r e t u r n c o n d u c t o r ' o f the z - p i n c h d i s c h a r g e . I n our a p p a r a t u s (see F i g . 2 2 ) , the r e t u r n c o n d u c t o r i s 1 mm t h i c k and i s made o f a g r i d o f b r a s s w i r e s f o r c o n v e n i e n c e i n t a k i n g p i c t u r e s o f the d i s c h a r g e . The h o l e s i n t h e g r i d a r e 1 ..5 mm X 1 . 5 mm s q u a r e . T h e r e f o r e b o t h t h e ; : ; ^ h ^ i e g ^ # ^the holes, a r e s m a l l compared w i t h s i z e o f the c i r c u l a r h o l e (7 mm i n d i a m e t e r ) p u n c t u r e d by the pr o b e . The s q u a r e s s u r r o u n d i n g t h e iprobe a r e f i l l e d up w i t h s o l d e r so t h a t t h e n e i g h b o u r h o o d o f t he probe becomes a c u r r e n t s h e e t i n s t e a d o f a; g r i d (see F i g . 9)» Under s u c h c o n d i t i o n s , we may a p p r o x i m a t e the r e t u r n c o n d u c t o r by a c u r r e n t s h e e t . „ _ ' BRASS Wite GRID -biAkEf^-oP: wifi£ =,. ,5mm PiSTrBerfee/s Axes,. Z... ' ' OF mge = l>rAiAETE& OF-HOLE = im m SoLPER F i g . 9» Neighbourhood o f h o l e i n r e t u r n c o n d u c t o r We have measured J ( r ) Tor the r e t u r n c o n d u c t o r o f 6 t h e z - p i n c h d i s c h a r g e ( c o ~ i O r a d / s e c ) and f o u n d t h a t the most s i g n i f i c a n t p a r t o f the e x p e r i m e n t a l v a l u e s o f J ^ ( r ) l i e s between the v a l u e s p r e d i c t e d by Malmberg's model and the E-K-Z model r e s p e c t i v e l y (see F i g . 1 0 ) . From T a b l e I , we r e a l i z e tha;t f o r t h e r e t u r n c o n d u c t o r ( b r a s s ) , S • 5 mm w h i c h i s comparable w i t h the t h i c k n e s s (~1.5 mm) o f the c o n d u c t o r . The e x p e r i m e n t a l c o n d i t i o n s t h e r e f o r e l i e somewhere between the; low and the h i g h f r e q u e n c y l i m i t s r e s p e c t i v e l y . Under s u c h c o n d i t i o n s , the r e s t r i c t i o n t h a t the m a g n e t i c f l u x be t a n g e n -t i a l a t the h o l e i s u n r e a l i s t i c . We t h e r e f o r e choose t h e E-K-Z model w h i c h r e q u i r e s a more r e a l i s t i c boundary c o n d i t i o n i . e . J = O a t the hole;. 7^ «W g-3 S.S g~l c/,3 r (cm) F i g . 10. P l o t o f - J ( r ) a t r e t u r n c o n d u c t o r o f z - p i n c h - 33 -2.2.3 L i m i t a t i o n s o f the E-K-Z model I n the d e r i v a t i o n o f the c o r r e c t i o n f o r m u l a e ( l 4 ) and ( 1 6 ) o f ^ 2 . 2 . 1 (p. §23)> w e assumed t h a t the d i s c h a r g e was made up o f a s e r i e s o f c u r r e n t l a y e r s . F o r t h e time i n d e p e n d e n t s y s t e m , the E-K-Z model i s v a l i d . However, i f the c u r r e n t i s moving, the Ohm's law on w h i c h the c a l c u l a t i o n s a r e ba s e d con-t a i n s e x t r a terms w h i c h a r e not c o n s i d e r e d i n the s t e a d y s t a t e c a l c u l a t i o n s . I n a d d i t i o n , f o r time dependent c u r r e n t s , e l e c -t r i c f i e l d s d e p e n d i n g on the r a t e o f change o f t h e magnetic, f i e l d a l s o o c c u r . These a r e not i n c l u d e d i n t h e s t e a d y t h e o r y e i t h e r . The c a l c u l a t i o n p r e s e n t e d below i s i n t e n d e d t o e s t a b l i s h the l i m i t i n g speed and f r e q u e n c y o f the d i s c h a r g e c u r r e n t s u c h t h a t t h e s t e a d y s t a t e E-K-Z ! t h e o r y r e m a i n s v a l i d . ( i ) Moving s h e e t C o n s i d e r a c u r r e n t l a y e r c a r r y i n g at c u r r e n t d e n s i t y J and moving w i t h a v e l o c i t y v. Ohm's law f o r s u c h a system i s (1 ) J = <f (E + v x B) where a" i s the e l e c t r i c a l c o n d u c t i v i t y o f the c o n d u c t i n g medium, E and B a r e t h e e l e c t r i c and magnetic. f i e l d s r e s p e c t i v e l y . S i n c e the model p r o p o s e d by E c k e r e t a l . (see £ 2 . 2 . 1 ) uses the Ohm's law f o r a s t a t i o n a r y c u r r e n t , i . e . 34 -we would l i k e to investigate under what conditions equation ( 2 ) can replace ( l ) . To do so, we compare the magnitudes of E and v x B , ( 3 ) From Maxwell's equations, we have c u r l B = u J . r I f 1 i s the c h a r a c t e r i s t i c length over which the f i e l d s change appreciably, equation ( 3 ) gives (<0 B r1 Assuming that vX B i s at the most of the same order of magnitude as the other terms i n ( 1 ) , we have (5) E ~ * l J The various terms i n equation (1) therefore bear the approx-imate r a t i o s (6) V E : a" v x B. = 1 : 1 : 11 c v 1 Therefore i f (7) <y v 1 « 1 , the term <r v x B can be neglected, Condition (7) i s s a t i s f i e d at the i n i t i a l stage of a 4 3 f a s t l i n e a r pinch discharge where (f ^ 10 mhos/m, V<V10 m/sec, 1 = o ~ 10 to 10~ m, <^  1 0~ H/m, and ji c r ^ v 1 ~ v ( £ t o ) ~ .01 to . 1 , 1 being taken aa the skin dept h S, (6" = /.. 2 ) s i n c e t h e g r a d i e n t o f t h e f i e l d i n s i d e t h e c u r r e n t l a y e r i s g o v e r n e d b y t h e s k i n d e p t h ( s e e T a b l e I , p 14). A t t h e l a t e r s t a g e o f t h e d i s c h a r g e w h e r e t h e c o l l a p s i n g c u r r e n t s h e l l h a s b e e n a c c e l e r a t e d a n d h a s c o m e c l o s e r t o t h e d i s c h a r g e a x i s , t h i s c o n d i t i o n i s q u i t e m a r g i n a l s i n c e b o t h t h e v e l o c i t y v a n d t h e c o n d u c t i v i t y w i l l h a v e i n c r e a s e d . I n p r i n c i p l e , i t i s p o s s i b l e t o i n c l u d e t h e e f f e c t o f t h e t e r m v X Bi i n e q u a t i o n .(1 •) . T h e c h a n g e i n c u r r e n t A J i n a p l a n e c u r r e n t s h e e t d u e t o a h o l e w i l l t h e n b e d i f f e r e n t f r o m t h a t o b t a i n e d b y E c k e r e t a l . ( s e e e q u a t i o n (8) i n £2.2.1.) H o w e v e r , t h e c a l c u l a t i o n s w i l l b e v e r y c o m p l i c a t e d . I n f a c t i t i s o b v i o u s t h a t b o t h ~v a n d B d e p e n d o n J a n d e q u a t i o n ( i ) w i l l g i v e a n o n l i n e a r e q u a t i o n i n J . We m u s t r e a l i z e t h a t i n e q u a t i o n (11) w e h a v e n e g l e c t e d g r a v i t y , t h e k i n e t i c p r e s s u r e o f t h e i o n s a n d t h e t i m e v a r i a -t i o n s i n t h e G e n e r a l i z e d O h m ' s L a w w r i t t e n i n ;the f o r m ( s e e R o s e a n d C l a r k , 'T9-6.1 a n d S p i t z e r , 1962) w h e r e 7 0 < v g i s t h e g r a v i t a t i o n a l f i e l d , p t h e k i n e t i c p r e s s u r e d u e t o t h e i o n s . o n l y a n d t h e r e s t o f t h e s y m b o l s h a v e t h e i r u s u a l m e a n i n g s . 22 — 3 U n d e r o u r e x p e r i m e n t a l c o n d i t i o n s , w e h a v e n = n . v 1 0 rn , e x ' — 3 — 31 1 k T ^ . l e V , rn^lO V - v I O J k g , V 0 M O N k g T h e ; v a r i o u s t e r m s i n ( l a ) t h e r e f o r e b e a r t h e a p p r o x i m a t e r a t i o s 1 :1 :10~1 :10~5:10"2:10~13:10"3 s h o w i n g t h a t o u r a p p r o x i m a t i o n i s v a l i d . ( i i ) Time Varying F i e l d s We now consider time varying f i e l d s i n a plane current layer of i n f i n i t e s i m a l thickness ds having a c i r c u l a r hole of radius a (see F i g . 1 1 ) . I f the unperturbed current density i s (8) J ; ( ? . t ) = J Q ( * , t ) , a^ being a unit vector i n the ^ d i r e c t i o n , we w i l l show that, under r e s t r i c t i v e conditions to be given below, the hole changes the current density, by an amount ( 9 ) A J ( p , e , f , t ) = J Q ( f , t ) F(p,0,a), p > a-, = - (<c . * ) . p < a, where F(p,9,a) i s the c o r r e c t i o n f a c t o r already defined i n equation (9) of £ 2 „ 2 . 1 (p. 21 ) . F i g . 1 1 . Distorted current flow i n plane layer with hole. - 37 -To prove (9)> we assume the following: (a) A l l the f i e l d s involved are simple harmonic and are proportional to e^ W*, (j = • This does not r e s t r i c t . i w t -the problem to f i e l d s varying with e" " .only^ s ±irce a more general time varying f i e l d can be regarded as a superposition of such harmonics. (b) The hole does not a f f e c t the f i e l d at jo -* oo , re q u i r i n g (10) A J(f>-»o° , 0 , * , t ) = 0. (c) The conductivity C i s constant across the layer. (d) v i s small such that Ohm's law takes the simple form j " = o'Tif . (e) There i s no accumulation of charges insi d e the current layer, i . e . (1 1 ) div J = 0. (f) O J i s so small that the displacement current i s n e g l i g i b l e . i s ) There i s no current flow across the layer, (12 ) = 0. This i s a d i r e c t consequence of the assumption = 0 f o r a current c y l i n d e r i n the E-K-Z model (see assumption (b) of £2.2.1, p. 1 8 ) . With these assumptions, the unperturbed current density J'(£,t) - 38 -for a plane current layer i s given by ( 1 3 ) t o ( f , t ) = J Q Z ( f e j W t a^ , where z ( f ,<*>) i s a s o l u t i o n of the equation 2 ( 1 4 ) = j u V w Z , (see Landau and L i f s h i t z , 1°60, p. 1 9 0 ) . The e x p l i c i t form of z(£,w) i s given by ( 1 5 ) Z ( * , « « ) = c o a h f ( ^ i ) f ¥e may now set up a boundary value problem to solve AJ i n terms of J ^ . From Maxwell's equations, Ohm's law and t h e i r l i n e a r i t y i n the f i e l d s , the above assumptions require ( 1 6 ) d i v (AJ) = 0, ( 1 7 ) V 2 ( A J ) = A 2(Aj i) , 2 ( 1 8 ) X = j ji o-to For the boundary conditions, we know ( 1 9 ) (AJ) = -(J') v ' v vn v o' n at the hole and ( 2 0 ) J (p-*oo,9,^,t) = 0 at large distances from the hole. Equations ( 1 9 ) and (20) do - 39 -not completely specify the boundary conditions f o r a. unique s o l u t i o n of A j since we do not know the tangential component of A j at the surface of the layer. To overcome such a d i f f i c u l t y , we use assumption (g), i . e. = 0 , and the) symmetry of the problem. This w i l l greatly s i m p l i f y the c a l c u l a t i o n s . In c y l i n d r i c a l coordinates (p,9 , f - ) , equations ( 1 2 ) , ( 1 6 ) arid ( 1 7 ) give < 2 D A + 1 - h - * —2 * —2 —2 * >AJP = ^ J P • dp2 f 3 f jo 2 p 2 9 0 2 ?><r2 J> f ( 2 2 ) ( P * J . ) + " ° ' v • ' 2 p xy j>•'• d 9 ( 2 3 ) AJ^ = 0 . The? boundary conditions at p 00 and at the boundary of the hole give, r e s p e c t i v e l y , ( 2 4 ) A j ( p - ^ o , 9 , £ , t ) = 0 , (25) ( p=a,9 s ?:,t) = - ( J ^ . Biy symmetry about the ^  axis and about the c e n t r a l plane of the layer, we also have ( 2 6 ) A JQ( p ,9 ,£ , t) = A j g ( j>,-Q,{,t) , ( 2 7 ) ^ J ( p , 0 , f , t ) = A J ( p,9,-p,t) . m, - 40 -We now solve equation. ( 2 1 ) by separation of variables (Morse and Feshbach, 1 9 5 3 ) - Putting (28) A J (p,e,£,t) =j§(p,Q) Z'(£) e j W t , equation ( 2 1 ) gives ( 2 9 ) - M + i -T— + -h. + A - -3^2 ) £ ( p . e ) = -( 3 0 ) d " Z ' ^ ) = m + X2 . Z'd<^  Here m i s an undetermined constant and may be obtained by considering the low frequency l i m i t of the problem. At steady state, i ; e. ^ = j u o"6o = 0 (see ( 1 8 ) , p. 3 8 ) , A j must be; constant across the thickness of the layer and d Z.' (£ ) = 0, Z'drp2 Equation ( 3 0 ) therefore; requires m to be zero. Rewriting ( 2 9 ) and ( 3 0 ) , we have ( 3 2 ) d 2 Z ' ( r l _ x 2 Standard c a l c u l a t i o n s using (24 ) ,( 2 5 ) , ( 2 6 ) , ( 2 7 ) , ( 2 8 ) , (31 ) and ( 3 2 ) give ( 3 3 ^ AJ(p,e, f ,t) = J D Z ( f ,<*>) e j w t F(p,9 5a) , jo > a'. Here} we have put Ok) z.( £ , « ) = Z ' ( £ ) to stress on the cc? dependence. S u b s t i t u t i n g ( 8 ) and ( 1 3 ) into ( 3 3 ) » we f i n a l l y obtain ( 3 5 ) A J(j0,0,<r ,t) = J^(£,t) Ffy>,e,a) , jo ^  a. The s o l u t i o n of A J f o r p< a i s t r i v i a l and i s given by ( 3 6 ) Aj(p,«,£,t) = - J Q ( f , t ) , P < a, because the o r i g i n a l ctirrent J q ( £ , t ) i n the hole i s removed when the hole i s introduced. It i s important to r e a l i z e that f o r a current layer of f i n i t e thickness, the unperturbed current density J ^ defined i n ( 8 ) f o r a plane d i f f e r s from the unperturbed i. current density J i n a cylin d e r i n t h e i r s p a t i a l v a r i a t i o n s over the thickness even i f t h e i r average magnitudes are the same. In f a c t , f o r the c y l i n d r i c a l layer, J Q ( 5 « t ) should be a.l i n e a r combination of the Hankel functions of the f i r s t and the second kinds. However, at low frequencies such that the thickness of the layer i s small compared with the skin depth 6 , the v a r i a t i o n s of J ' and J over the thickness become u o o i n s i g n i f i c a n t and t h e i r d i f f e r e n c e vanishes. Under such a condition, equation ( 8 ) , ( 1 3 ) and ( 3 3 ) give ( 3 1 ) A ~j = A j ^ p . G . t ) = J q e> J W t F ( j 3 , 0 , a ) . Equation ( 3 1 ) can now be applied to time dependent - k2 -cases i n the E-K-Z model i f we replace J q ( S ) i n equation (8) o ^2.2.1 by J q ( S ) e^"* corresponding to the unperturbed time dependent current density i n a t h i n current cyl i n d e r of radius s . - k3 -2.3 Transformation of the Integral Equation into a.. Matrix Equation Suitable f o r S o l u t i o n with a D i g i t a l Computer In p r i n c i p l e , equation ( 1 6 ) of £2.2.1 (p. 23)».i. e-( 1 ) J p ( r ) =y ds J Q ( s ) K(r,s,a) where ( 2 ) K(r,s,a) . i f »C'(r,s,a) ? can be used to solve f o r J Q ( r ) a n a l y t i c a l l y i f J p ( r ) i s a* known function. However, since J p ( r ) i s only given as a set of measured values at d i f f e r e n t values of r, we need to solve the problem numerically. One standard technique i s to convert the i n t e g r a l to a summation and solve the i n t e g r a l equation as a matrix equation (Fox, 1^62, p. 159)= The range of i n t e g r a t i o n i s taken from 0 to a value R corresponding to a range over which J p ( r ) i s appreciable. We divide the i n t e r v a l [0,1*1, into . K equal i n t e r v a l s each of a width DX. The value s^ of s at the Nth i n t e r -v a l (t&< K) i s then taken as the value of s at the centre of the i n t e r v a l , i . e. ( 3 ) s N = (N - .5) DX. I f r i s the value of s at the midpoint of the Mth i n t e r v a l , we can e a s i l y see; that (k) r - s = (M - N) DX kk -and I f we put, (6) ( 7 ) and (8) + s = (M + N - 1 ) DX, u(M) = J (r) | r=(M-.5)DX v(M) = J (r) | r=(M-.5)DX A(M,N) = K(r,s,a) DX | , r=(M-.5)DX s=(N-.5)BX equation (1) can be approximated by a set of equations of the form (9) K X ! A(M,N) U(N.) = v ( M ) , N=1 (10) Here we have defined (11) In matrix notations, (9) takes the form A u = v . A( 1 , 1 ) A.( 1,2) . . A ( 1 , K ) | A . ( 2 , l ) A(2,2) A(2,K) A ( K , 1 ) A(K,2) A ( K , K ) A = and u and v are column vectors whose Mth components are U(M) and V(M) r e s p e c t i v e l y . - k5 -In order to eliminate unnecessary round o f f errors i n the computation, we reduce the matrix A to a sparse matrix by putting small matrix elements equal to zero. In the matrix A, the diagonal elements are of the same order of magnitude and the nondiagonal matrix elements decrease r a p i d l y as we go away from tjrie diagonal. In f a c t , with the matrix that we are using, A(M,N) i s reduced to a value smaller than . 1$> of A V(M,N) as the dif f e r e n c e of M and N i s larg e r than 10. We there-fore put ( 1 2 ) ( 1 3 ) A(M,N) = K(r,s,a) DX r=(M- .5)I>X s=(N-. 5)DX A(M,N) = 0 M-N 4 10, M-N >10, The shape of the matrix i s shown i n the.following f i g u r e F i g . 12, Diagram showing form of matrix A. Here the shaded area represents nonvanishing elements. The; dummy indices M- and N give the changes i n r and s re s p e c t i v e l y . A, 3-dimensional fig u r e of the magnitudes of the matrix - 46 -elements i s shown i n F i g . 13« It consists; of a ridge along the diagonal of the matrix. A vje,£:1t^  ridge p a r a l l e l to the r axis i s of the shape shown i n — - -F i g . 14. \ F i g , 13. Figure showing magnitudes of matrix elements of A, F i g . 14. Diagram showing cross-section of A parallel,, to r axis (a = 3^5 nim) . - 47 -The e x p l i c i t form of A ( M , N ) depends on the form of the correct i o n f a c t o r C ( r , s , a ) and hence the model we choose f o r the cor r e c t i o n procedure. The s o l u t i o n of ( 1 0 ) i s then given by = A. -1 u V . Therefore we need only invert the matrix A l once and the so l u t i o n f o r d i f f e r e n t v's obtained from experiments may then be calc u l a t e d . In our c a l c u l a t i o n s , the 5 0 X 5 0 matrix A. obtained using the E-K-Z model has a; condition number of 7°4 (Turing, 1 9 4 8 ) . The inverted matrix A ^ has the largest matrix elements near the diagonal. A s i m i l a r cross-section f o r A i s shown i n F i g . 15-haa. s+sa. r F i g . 15. Diagram showing cross-section of A ~ p a r a l l e l to r axis (a = 3•5 mm). - 48 -2.4 Error Analysis The purpose of t h i s analysis i s to show under what condition the c o r r e c t i o n procedure described above i s l i k e l y to be useful.- In order to do t h i s , we f i r s t e s t a b l i s h that the computer solutions of equation i ( 1 ) J p ( r ) = / d s J o < s ) K(r,s,a.) are not a f f e c t e d by round-off errors i n the computer program (see £2.4.1 below). Since the computer program i s s a t i s f a c t o r y , the next step i s to determine the dependence of J on J f o r the p o kind of current d i s t r i b u t i o n expected i n the z-pinch. This analysis (see F i g . 16(a), (b)) shows that i f J q i s a smooth function of r, the d i f f e r e n c e between J (r) and J (r) i s P 0 very small (see F i g . 1 6 ( a ) ) . However, i f J Q ( r ) has the " l i m i t i n g form of a ^-function, the corrections become very large ,.j(see F i g . 1 6 ( b ) ) . For p r a c t i c a l purposes, i f J Q ( r ) i s a 6 -function, we s h a l l assume i t to be the current density d i s t r i b u t i o n due to a current sheet of thickness equal to the mesh i n t e r v a l of the numerical c a l c u l a t i o n . The next stage i n the c a l c u l a t i o n ( §2.4.2) i s to investigate how f l u c t u a t i o n s i n J a f f e c t the calculated P values of J . There are two sources of fluctuations i n J ; one o p i s due to actual v a r i a t i o n s i n J ; the other can be ascribed o to the processes employed i n measuring B p. In general, i t i s not possible to d i s t i n g u i s h between these two possible sources - k9 -of f l u c t u a t i o r s i n J ^ . However, by considering i d e a l cases; i t i s shown that measuring errors are more s i g n i f i c a n t than errors which might be expected from f l u c t u a t i o n s of reasonable magnitude i n J . rCcm) F i g . 16(b). Plot of J (r) and J (r) versus r assuming J' (r) to f>.e" a ^-function, o - 50 -2.4.1 Round-off Errors due to Numerical Calculations Since we solve the i n t e g r a l equation (see ( l ) of ^ 2 . 4 , p. 48) by; transforming i t into a matrix equation ( 1 ) A u ==v , (see ( 1 0 ) of £ 2 . 3 , p. 4 4 ) where A... i s a 5 0 X 50 matrix, we have to make sure that round-off errors, are ins i g n i f leant i, To do so, we use a known current density d i s t r i b u t i o n J Q ( r ) and c a l c u l a t e J ^ ( r ) from ( 1 ) numerically. With the calcu-lated values of J ( r ) , ' the s o l u t i o n J * ( r ) of equation ( 1 ) i s obtained. The r e s u l t of J*(r) i s then compared with J ( r ) . I f the c a l c u l a t i o n i s accurate, the diffe r e n c e J * ( r ) - J Q ( r ) should be n e g l i g i b l e compared with J Q ( r ) . Such a procedure i s done numerically using equation ( 1 ) . I f J Q ( r ) i s represented by the vector u, and v i s calcula t e d from equation ( l ) , then J£(r) i s the vector A 1Au. We -1 therefore have to compare A, Au with u. Computer r e s u l t s for t y p i c a l trapezoidal current d i s t r i b u t i o n s with f l u c t u a t i o n s are obtained (see F i g . 17)° In a l l cases, we f i n d that the f r a c t i o n a l . errors (A.'^Au .T U ) / U £ 10 [showing that round-o f f errors i n the numerical c a l c u l a t i o n s are n e g l i g i b l e . We have t r i e d d i f f e r e n t mesh i n t e r v a l s from 1 mm to 4 mm, taking the range of i n t e g r a t i o n over r from 0 to 10 cm correspon-ding to the radius over which the measured values, of are not zero. The condition numbers (Turing, 1948) of the matrices corresponding to the various mesh i n t e r v a l s are given i n - 51 -Table II Table II Mesh In t e r v a l Dimension of Matrix Condition Number 4 mm 25X25 1.4 2 mm 50X50 7-4 1 mm 1 0 0 X 1 0 0 1 0 1 2 Since the s p a t i a l r e s o l u t i o n of the probe i s about 3 mm, we choose the 2 mm mesh i n t e r v a l . The 50X50 matrix thus obtained i s reasonably well-conditioned. - 52 -2.4.2 The Influence of Fluctuations From ee. measured set of values of J , i t i s not P possible to as c e r t a i n whether any scatte r about a mean value at a given point corresponds to r e a l f l u c t u a t i o n s i n J q , or whether the f l u c t u a t i o n s are merely due to inaccuracies i n measurements. I f we can be sure there are no i n s t a b i l i t i e s , then f l u c t u a t i o n s i n J (the true current density) are unlike\l o v ' However, i f this i s not the case, we need some method of de-termining the causes of f l u c t u a t i o n s i n the measured values J . e p Some assessment of the r e l a t i v e importance 1 of the two c o n t r i -butions can be obtained by considering i d e a l i z e d cases. ;V Suppose f o r example that f l u c t u a t i o n s i n can only a r i s e from f l u c t u a t i o n s i n J Q . A current spike i s greatly smoothed and reduced i n magnitude by the perturbation of a probe. We therefore expect that any fl u c t u a t i o n s i n J Q ( r ) w i l l be much smoothed out also (see F i g . 17)° F i g . 17» Computer r e s u l t of J (r.) show,ing smoothi ng ; e f f e c t of. probe" on': f?ne structure of ~ J ,('r").'. ' - 53 -To investigate the smoothing caused by the probe, we consider the e f f e c t s of random errors added to a trapezoidal d i s t r i b u t i o n f o r J Q ( r ) ( t y p i c a l of z-pinch discharges). The errors have a, normal d i s t r i b u t i o n with a, standard deviation cr of 3 0 % of J (r) , i . e . ' o • 7 max cr" = .3 J (r) o v 7 , (see Table III) max v 7 They are assigned to various points r, using tables of pseudo-random numbers generated by the computer (Shreider, 1 9 6 6 ) . These c a l c u l a t i o n s show that the standard deviation <T of the errdrs produced i n J (r) i s given by the p • *. 1 P equation o- ~ 3 cr P Here we use trie J (r) given i n F i g . 16(a) Table IIJL Fluctuations i n J ; (r) due to Fluctuations i n J (r) J (r) cr cr $ o.-V- 7 max - p a* ( 1 0 7 Amp m~2) (l0 7Amp m - 2) (l0 7Amp m~2) P 2 0 . 1.0 .35 2.9 2 0 . 2.8 1.1 2.6 20. 4.8 1.3 3 . 7 2 0 . 7-4 3.1 2.4 2 0 . 9-3 3-7 2.5 - 54 -The reduction i n the standard deviation arises because the probe "c o r r e l a t e s " random errors occurring at neighbouring points. I t might at f i r s t sight appear therefore that to determine J to an accuracy cr', we only need to evaluate o Jp to an accuracy of c//3« Unfortunately t h i s i s not the case, because the errors i n J ( a r i s i n g from random errors P i n J Jhave a p a r t i c u l a r s p a t i a l d i s t r i b u t i o n , i . e . the errors at neighbouring points are correlated (see F i g . 17)° However, J w i l l also have measuring errors which w i l l be t r u l y random i . e. no s p a t i a l c o r r e l a t i o n s . These errors should be removed from the values of J before using the c o r r e c t i o n P procedure. In p r i n c i p l e , t h i s could be done by measuring at whole set of J ^ ( r ) several times and using the mean. How-ever, t h i s procedure i s generally impracticable, and i n any case evidence f o r i n s t a b i l i t i e s might also be l o s t . The p r a c t i c a l s o l u t i o n i s therefore to examine what happens i f the measuring errors are fed through the c o r r e c t i o n program. We again employ a Monte Carlo method, s t a r t i n g with a Jp corresponding to a,u trapezoidal d i s t r i b u t i o n f o r J Q (Fig. 1 6 ( a ) ) . Normal errors with a standard deviation ° ^p are assigned at random to the values of J ^ , and the r e s u l t i n g deviations i n J (cr* ' ) , the standard deviation cr ', are o \ p computed from equation ( 1 ) of £ 2 . 4 , ( 1 ) J (r.) = / ds J (s) K(r , s,a) . A t y p i c a l computer r e s u l t i s shown i n F i g . 18. Pig. 18. Errors i n J (r) due to errors i n J ( r ) . o P • cr' o f J ( r ) ' v .005 x l j I; ( / o f J ( r W . 1 x | j p p v ' I p maxl o N ' 1 o maxi Repeating the c a l c u l a t i o n f o r d i f f e r e n t assignments of the errors leads to the following r e s u l t s . Table IV Fluctuations i n J (r) due to Fluctuations — ; ; : ' : O ' — - * — : —; • ;—• i n , J _ ( r ) pv ' T ^1 ,i <£' J cr cr —,. o max p Or' P (l0 7Amp m~2) (l0 7Amp rn"2) (107Amp m~2) 20. 2.0 .1 20. 20. 5.0 .28 18. • 20. 9.7 .48 20. 20. 14. .74 19. 20. 15. .93 16. - 5 6 -From Table IV, i t i s obvious that the standard deviation i n the measured r e s u l t s i s increased by an order of magnitude i n the c o r r e c t i o n process. In f a c t , we have CT • ^  20 cr* ' p This r e s u l t has very s e r i o u s r i m p l i c a t i o n s as can be seen by the following argument. Let J q be the true current which would be calculated i f J could be measured accurately, /s and l e t J be the value of the "true" current calculated o when J has a measuring error with a*standard deviation p to cK ' . We have from above f o r a trapezoidal current d i s t r i -P bution (2) | Jo ~ J P | ' ° 5 However, from the above c a l c u l a t i o n s (3) J - J h v 20 1 o P | P I f c/' i s less than 1$> of J ( a rather u n r e a l i s t i c P ' P assumption), then we have i J - J o p < .2; Equations(2) and (3) indicate that ( 5 ) o I f we compare the i n e q u a l i t i e s ( 4 ) and ( 5 ) , we see that th« measured current J (with errors) i s probably a better approx-P • A. imation to the true current J than the values J computed - 57 -from J p by means of eqtiation ( l ) . Equation (k) therefore indicates that unless one has reason to believe that measuring errors are less than j$>, the measured current i s normally the best approximation to the current which would flow i n the absence of the probe. However, f o r "spiky" current d i s t r i b u t i o n s , the c o r r e c t i o n technique f o r c a l c u l a t i n g J i s s t i l l of value since the c o r r e c t i o n i s o several times la r g e r than both J and the errors i t introduces P (see F i g . 19 f o r a t y p i c a l computer r e s u l t ) . .5 -/ + J 0 (r) + Sjjr) 1 — I :p 7-r + 4-8- + r(om) F i g . 19- Errors i n J Q ( r ) due to errors i n J ^ ( r ) assuming J (r) to be a g-function, o - 58 -r 2.4.3 S p a t i a l Resolution A f i n a l problem which must be considered i s the s p a t i a l r e s o l u t i o n of the probes. In our experiment section, we dB define the values of B and , p dr p— at r by the expres-sipns B p = | , ( B 1 + B 2 ) , - f g g - = B ^ B 2 , (see ^ 3 - 3 ) , where 2b i s the spacing between the sensing c o i l s i n the gradient probe If, we denote the defined quantities by a s t e r i s k s , then we have CD Pv (r) = t[B ( r + b ) + B (r-b)] , ( 2 ) dB^(r) = _L dr 2b •[B (r +b)-B ( r - b ) ] , where B p ( r ) i s the measured values of B p ( r ) at the points. Expanding these equations y i e l d s the following r e s u l t s , ( 3 ) B*"(r) = B (r) + p • p v ' 2 d Bg ( r ) ( D ^Bg(r) = dB p(r) b 2 d J B r ( r ) dr dr dr 0-neglecting higher order terms i n the Taylor expansions. Hence fo r the defined values to be equal to the values which would a c t u a l l y be measured at r , we must have (5) (6) B,(r) - 1 ( | ^ - 4 | P ) ; ; P dr* dB, -1 dr 2 3 /b_ _d^B p v ( 3 l ~ d ^ P ) « 1 » f o r B (r) = B ( r ) , •'/ F'v- '1 tf : << 1 t f o r ^ 1 =  d | ? ^ - 59 -For comparison, we compute the values of B ( r ) , dBp(r) d 2 B p ( r ) 2! dr* a n d * f d 3B ( r )  a n d 31 d?3 p" '' dr for the return conductor using the E-K-Z model (see ^2.2.1). The computed r e s u l t s show that for the gradient probe used i n th i s t h e s i s , we have (7) B (r) -1 ,h2 d 2 B ^ r ) ^ ( — v 2 : dr < .001 Therefore we may assume that B (r) = B p ( r ) . For the gradient, however, we obtain (8) d B r ( r ) ~1( b f d 3 B p ( r ) v . dr - V 3 i d r 3 '' < ..1, showing that dBt dB dr dr < 10%, Therefore we plot both d ^ d i - r ^  a n d d^P^ r ^— against r i n F i g . 20 These r e s u l t s show that the gradient probe we use should reproduce true values of B p ( r ) and dBp(r) dr Hence the major influnce; on the computed values of J ( i s the radius of the probe, and not the c o i l spacing. This r e s u l t together with the observed form of the c o r r e c t i o n f a c t o r indicates that the s p a t i a l r e s o l u t i o n of the probe i s approximately 3 m m ( ~ a ) • - 6o -- 6 1 -2.'h.h C6i^ 7IulsJi"6»S' -f-r"o.m' Error?; Analyst's From the error analysis presented i n previous several sections, we have shown that magnetic probe destroys information about the f i n e structure of the current i n a.plasma. The errors i n the unperturbed current density J Q ( r ) are about twenty times the errors i n the measured values of J ( r ) . Since i n most cases, the d i f f e r e n c e J - J (r) o p v ' i s about 5% of J (r) P ' except where J o ( r ) i s a sharp current spike (see equation ( 2 ) , p.- 5 6 ) 9 the c o r r e c t i o n i s not meaningful unless J (r) i s OP measured with an accuracy of better than - j % . Such an acciiarcy i s quite impossible i n most cases. In addition, highly unstable pulsed discharges present further d i f f i c u l t i e s due to a lack of r e p r o d u c i b i l i t y i n the probe s i g n a l s . However, i f we are c e r t a i n that the current d i s t r i b u t i o n has a sharp "spike", we must correct the measured value of J (r) to obtain J ( r ) . - - - • • • P o . For t h i s s p e c i a l case, the c o r r e c t i o n i s usually several times larger than both J p ( r ) a n c l the possible errors introduced by the c o r r e c t i o n process. :«f • 3 ..... EXPERIMENTAL" "RESULTS 3.1 Introduction In this chapter we describe the experimental technique used i n me.asurto-g the current density of a l i n e a r pinch d i s -charge i n helium. We choose helium because several workers i n our laboratory have been studying z-pinch d i s c h a r g e s i n helium using such techniques as time resolved spectroscopy. Stark broadening measurements, and l a s e r interferometry, and i t i s p r o f i t a b l e to compare our current density measurements with t h e i r r e s u l t s . This w i l l provide a better picture f o r the structure and the dynamics of the z-pinch. The apparatus for the z-pinch discharge used f o r t h i s experiment consists of a Pyrex tube of 15 cm i . d . , a vacuum system and a 5 KJ condenser bank having a natural o s c i l l a t i o n period of about 20 usee. De t a i l s of this equipment appear i n Table V. For current density measurements, we use a gradient probe which has already been b r i e f l y described i n ^ 2 . 1 . The> theory, properties and noise problems of the gradient probe are outlined i n £3° 3° Because the s e n s i t i v i t y of a magnetic probe i s usually low, i . e . i t can only detect strong f i e l d s such as those produced by pulsed discharges, experimental c a l i b r a t i o n requires strong pulsed f i e l d s and i s i t s e l f an important technique. This w i l l be explained i n £3.3.1. £3.3.2 describes the c a l i b r a t i o n and the frequency response of the - 62 -- 63 -probe and ^3°3°3 explains why r e p r o d u c i b i l i t y of probe signals i s necessary. In §3°4„1, we give the experimental r e s u l t s of the current density measurements f o r z.-pinch discharges i n helium at i n i t i a l pressures of 500 ji, 1 mm, 2 mm, and k mmHg respec-t i v e l y . To explain the r e s u l t s , a modified snow-plow equation which considers a loaded, thick current sheet i s developed. Using the magnetic pressure measured from the probe, the equa-t i o n i s solved numerically with the help of the IBM JOkO computer. The computed radius of the c o l l a p s i n g current s h e l l i n the discharge i s then compared with experimental f i n d i n g s . 3•2 Apparatus The apparatus used i n th i s experiment consists mainly of a l i n e a r pinch discharge system and a. gradient probe c i r c u i t . In this section, we only give d e t a i l s of the l i n e a r pinch device and s h a l l leave the d e t a i l e d d e s c r i p t i o n of the probe c i r c u i t i n the next sec t i o n ( ^ 3«3 )-The discharge system consists of a 5 KJ condenser bank power supply, a t r i g g e r i n g system, a vacuum system and a 15, cm i o i d . pyrex discharge tube. The condenser bank and the t r i g g e r -ing system are conventional i n design (Medley, 1965a)» The basic specifications are l i s t e d i n Table V, and the c i r c u i t diagram appears i n F i g . 2 1 . The d e t a i l e d design of the discharge tube i s shown i n F i g . 2 2 , because i t d i f f e r s s l i g h t l y from the usual discharge tube. The main difference i s that the probe moves i n a guide-tube mounted across the diameter of the discharge v e s s e l . In a, normal system, the probe i s introduced into the vessel through an o-ring seal i n the wall. Table V Apparatus (A) Energy Source f o r Discharge ( 1 ) Cond enser Bank To t a l capacity ( 5 * ( 1 0 ±-.1) uF N.R.G. Low Inductance Storage Capacitors i n p a r a l l e l ) 53 T o t a l Inductance .12^.01 p.H Charging Voltage 10 .01.2 kV Maximum Discharge Current 200 k Amp Ringing Frequency 100 kHz - 65 -( 2 ) T r i g g e r i n g System (Medley, 1 9 6 5 a ) Pulse generator (produces -9 kV spike voltage of a r i s e time of kO nsec and a duration of 6, usee ), Theophanis unit (doubles spike voltage from pulse generator) U l t r a v i o l e t t r i g g e r Main t r i g g e r i n g spark gap (3) Voltage Measurement Convoy Microameter A.V.O. M u l t i p l i e r (25 kV d.c.) ( 4 ) Current Measurement Rogowski C o i l ( S e n s i t i v i t y : ( 1 .86+ . 11 ) 105Amp V o l t - 1 ) Integrator (RC passive, i n t e g r a t i o n time constant: 24 msec) . (B) Discharge Syste m ( 1 ) Discharge tube (pyrex) Inner diameter 15 cm Outer diameter 17 cm Length 61 cm Electrodes Brass Electrode separation 59 cm ( 2 ) Vacuum System Type 17 Balzers O i l D i f f u s i o n Pump Hyvac 14 Cenco Backing Pump Vacuum attainable 1 P-Hg Leak Rate 7 uHg/hr Macleod Gauges 0-1 .mmHg 0-10 mmHg Pyrani Gauge (Type GP-110 P i r a n i Vacuum Gauge) - 66 -< «. M A I * 4 S P A R K <SAP J i / u i T C r t —'« C t MA'M CAPACITOR. &.AMK R.^  VARlA&l-E RESISTANCE ( 0-7© MA IO MA, STCPS) PlSCHARQE TUBE S - TCU&SeR. SPARK <SAP Su/l-rCrt ; ITSELF TRl<kGcft£.I> UCTftAVioLeT RATMA'TIOKI F R O M T C l - TRIGGER. CAPACITOR, ( . o&yiF) T = ULTRAVIOLET TRIGGER. WHICH EMITS ULTRAi / tOLET RAUIATIOM FR.0M SPARK INSIUE auAR.Tz. tuiuLATi^j TUBE GX ink A A / W -V\AA/ -J50 To POWER. SUPPLY 1/ 0*0 T F i g . 2 1 C i r c u i t oi' discharge system (from Daughney's. thesis, 1966) G-LASS PYREX..TU&E .15 CM I.D. CluARTZ. TUBE . . . R E T U R N . C O NDOcx iNOr T O V A C U U M V P U k t p CAPACITOR. LEADS F i g . 2 2 . Cross-section of discharge tube. - 67 -3 o 3 The Gradient Probe and the Delay Line A gradient probe consists of two small search c o i l s connected i n such a way that they not only measure the magnetic f i e l d j but also the difference of the f i e l d s at the respective positions of the c o i l s . For use i n current density measurement! of a pulsed plasma, the probe should have ( a x ) small size f o r minimum perturbations on the plasma, (b) small sensing c o i l s f o r good s p a t i a l r e s o l u t i o n (this requires small areas and small numbers of turns), (c) large bandwidth response so that i t gives true information from high frequency f i e l d s (this requires small induc-tances or small c o i l s with very few turns), (d) good s e n s i t i v i t y f o r accurate measurements and large s i g n a l to noise r a t i o s . Conditions ( a ) 9 (b) and (c) are compatible with each other since they a l l require c o i l s with small cross sections and a low number of turns. However, these a l l help to v i o l a t e condition (d) which requires exactly the opposite. To meet a l l these requirements, therefore, a compromise has to be made. -r ^ 4 . dBj-p (r) . . . ... In order to measure —'—, i t i s often necessary dr . • to record accurately the d i f f e r e n c e between two large signals which are almost equal to each other. This requires that the two probe c i r c u i t s be as c l o s e l y i d e n t i c a l to each other as; possible. In p a r t i c u l a r they must be balanced so that i n a - 68 -dB uniform magnetic f i e l d , the measured value of dr*^ i s zero ( i . e. the differ e n c e between t y p i c a l output signals from the two c o i l s i s 100 times smaller than the-signal from each c o i l ) . dBi Accurate values of -—:—**— also requires the elimination dr ' of common mode signals from the probe. This i s accomplished by the d i f f e r e n t i a l amplifier i n the recording o s c i l l o s c o p e , which i s c a r e f u l l y adjusted to give a common mode r e j e c t i o n 4 of 10 . A*'design which meets the above requirements i s shown i n F i g . 23. It. consists of two small c o i l s and L„ attached to the t i p of a small glass tube. Each.coil i s made of 100 turns of AWG NO. 4 2 enamelled copper wire wound on a P . V . C . i n s u l a t i n g tube; of 1 .8 mm o.d. The c o i l s are held i n place side by side.by means of Scotch tape. The leads from each c o i l are twisted t i g h t l y together so that the only emf induced is., produced by f l u x changes i n the c o i l s themselves. They are then connected to a t.resistor and a miniature potentiometer •*> R.^  9 which are enclosed i n a metal casing (see F i g . 2 4 ) . and L,, are connectefd i n such a way that they produce signals of the same sign when the probe i s placed i n a uniform f i e l d . To screen out noise s i g n a l s , the probe outputs are fed to an oscill o s c o p e through a screened twin-feeder. By delaying the sig n a l s , they can be displayed a f t e r "hash" produced i n the scope by.the discharge has disappeared from the oscillogram (Medley, 1 965t>) . The delay i s produced by two delay units (General Radio 3 l 4 s 8 6 , c h a r a c t e r i s t i c impedance 220 ohms, - 6 9 -maximum delay .5 p s e c --- see F i g . 2 4 ). Since we are interested i n values of the magnetic f i e l d B p , the probe outputs must be integrated with respect to time. This i s accomplished by a passive RC integrator (see F i g . 2 3 ) . dr*^ ° a n b e c a l c u x a t e d from the equation = — — , where B„ and B„ are the . dr AT , 1 2 measured f i e l d s obtained from the two integrated probe s i g n a l s s and Ar i s the distance between the c o i l s (see F i g . 2 3 ) . ^ r ^ is taken to be the value of dr^ midway between the two c o i l s . B^ - B^ i s measured d i r e c t l y by the d i f f e r e n t i a l ampli-fier, i n the os c i l l o s c o p e (Tektronix D i f f e r e n t i a l Amplifier Type ¥; Tektronix Scope 551 ) ° The output from one c o i l 'of the probe i s also amplified by another a m p l i f i e r (Tektronix Type G), aad displayed on the other beam of the o s c i l l o s c o p e . This s i g n a l i s proportional to B^„ The value of B p midway between the two c o i l s is-taken to be B p = B 1 + i ( B 2 - B,) where B^ and (B^-B^) are both recorded s i g n a l s . The integrator has a time constant of 100 usee and R^, R^, Rj. and R^ are terminating r e s i s t o r s (Fig. 24) of 220 ohms each. R_ i s a 2.2 k ohm r e s i s t o r and R„ i s composed of I o a 1 . 0 k ohm r e s i s t o r i n series with a 1 . 5 k ohm miniature potentiometer. The c i r c u i t i s made as symmetrical as possible i n order to obtain a good balance f o r the two probe c o i l s . The signals at the screen of the scope are recorded with a polaroid camera. The essentials of the gradient probe and the accompanying measuring devices are shown i n Table VI. - 70 -Table VI Gradient Probe unci Measuring Devices 1) Gradient Probe and Loading C i r c u i t s • S e n s i t i v i t y i n gradient measurements (seo Table VII) S e n s i t i v i t y i n flu x density measurements (see Table VII) Transmission l i n e (10 meters UG 88U twin-feeder) Delay Unit (two General Radio 314S86 delay units; c h a r a c t e r i s t i c resistance: 220 ohms; delay time:.5 psec) Integrator (passive RC network; in t e g r a t i o n time constant: 100 usee) 2 ) Oscilloscope and Accesories Oscilloscope (Tektronix Type 551 1 Dual Beam Scope) Plug-in Units: Tektronix Type W D i f f e r e n t i a l Amplifier Rejection r a t i o : 10 Pass band: dc to 20 MHz at 1 mV/cm dc to 8 MHz at 1 mV/cm Tektronix Type G D i f f e r e n t i a l Amplifier Rejection r a t i o : 10"' Pass band: dc to 18 MHz from .5 V/cm to 20 V/cm) SENS_IM<Ji COILS GLASS TU&6 43. 0 r> ~ P 0 T S f 4 T l O * \ G T S R . A f c t M u S « F C O I L . I w m L E M C T H o f c o i L 7.5 " " " i F i g . 23- Design of gradient probe (to be inserted into a guide tube across the discharge tube; see F i g . 3 2 ( a ) , and (b) ctlso). SENSING BALANClMCr TWIN FEEDER DELAY DUAL . . . . . TCKTRONIX TYPE 551 _.COILS . UNIT TRANSMISSION UNIT INTEGRATOR. DUAL &EAM OSCILLOSCOPE LINE C I R C U I T OP G R A D I E N T P R O B E ANIP ME/)SURlM<5 S Y S T E M 72 3.3-1 Balancing of Probe and Ca l i b r a t i o n > C i r c u i t s To balance and :calib"rrate**the gradient probe, a uniform magnetic f i e l d i s generated by discharging the condenser bank through two p a r a l l e l leads connected to a ..sres^is taiic^e. as shown-tee4ow . NONLINEAR. RESISTOR. (M0R$M6tiM im201/22) ROGOWSKI COIL SWITCH CONDENSER BANK y= 10 F i g . 25• C a l i b r a t i o n c i r c u i t (b = width of current leads = 20 cm; h = separation between leads = 1 cm ) With i t s complete loading c i r c u i t connected (see F i g . 2 k ) , the probe i s f i r s t balanced by adjusting the potentiometer R.j so that the probe gives zero output when the condenser bank i s discharged (see F i g 0 2 6 ) . The uniform f i e l d B^ between the copper s t r i p s i s u i/b , where b i s the width of the s t r i p s and I i s the current. Since the s i g n a l from each c o i l of the probe i s proportional to the f i e l d B 1 , we have V.^  = k f aB 1 where k b i s a, constant. I f I i s measured, B^  i s known. Therefore the above equation serves - 73 -to c a l i b r a t e the probes. To complete the c a l i b r a t i o n , we have to know I. This i s measured by a Rogowski c o i l . The s i g n a l from the Rogowski c o i l i s fed into the scope through an integrator having a time constant of 2.4 msec which i s much longer than the s i g n i f i c a n t time' constant of the discharge (20 usee). The output from th i s integrator (VT) i s proportional to the current i n the c i r c u i t , i . e. V = k.I, where k. i s a constant. Integrating t h i s eqaution with respect to time gives r o t /<*> ( 1 ) / I dt = Q = k T 1 / V dt Jo 14> r where Q i s the t o t a l charge discharged from the capacitor. This i s equal to'CV , the product of the capacity and the r charging p o t e n t i a l . Since / V rdt can be measured and C and are known, equation ( l ) enables k^ to be evaluated. The probe s e n s i t i v i t y k^ i s then given by the f i n a l equation (2) k - -2 ^ • \ ' , b ~ b k.v e i. i . 1 r ... ... In our e x p e r i m e n t J V rdt Ls measured on an enlarged o s c i l -logram of V r (see F i g . 2 6 ) with the help of a planimeter. Ve have made two gradient probes (Probe No.1 and No.2). .Their s e n s i t i v i t i e s are given i n Table VII. Table VII Probe b e n s i t i v x t i e s S e n s i t i v i t y l o r f l u x density rneasurements K b (Wb in" 2 V o i r - 1 ) Probo No . 1 Probo No . 2 87±.0? .tt!±.07 S e n s i t i v i t y 1 or -f i e l d gradient measurements K •' (Vb m--3 V o l t " 1 ) S p a t i a l Resolution sis; 3 4 2 0 1 3 0 3 mm 400+30 , 3 mm ..Rogowski c o i l s i g n a l (..05V/ cm)-Pro'be• -sxgnal Iroin one co11 \ (.05V cm; T xsn e (5 us e c / cin } (bl C i S t Time (5 isse.c^cm :j probo >5 J f n..l l rn»i ci<ij No • i .;. (:.05V/cru j .probe si/,n.il J row c o . l No. 2 ::(.05V / c i rO ••••-v. ' •y'.;g_Balanred pro DO t-x^nali-, x.e • . ; t f;difforcjice between ci t , i i . i l b j ro'm bo r n r o x J t» ^ . 05 V*/cu ) 'Sam U[<j>e^ tract . Time !, 5 usee / cm } F±tS. 2 o . C a l i b r a t i o n -si^naLs _ 75 -3o3 - 2 Frequency Response with Loading C i r c u i t s In p r i n c i p l e , the frequency response of a*magnetic probe could be obtained by i n s e r t i n g the probe into a time varying, strong magnetic f i e l d . However, i t i s d i f f i c u l t 3 to design such systems which give f i e l d s greater than 1.0 gauss at frequencies higher than 1 MHz. Instead of using a strong magnetic f i e l d to produce an emf i n the c o i l , we: therefore use a., s i g n a l generator connected i n series with the c o i l (see F i g . 2 7 ( a ) ) . In order to minimize stray signals s a l l connections are kept as short as possible and a s p e c i a l cover (see F i g . 27 (b)) i s made to replace the bottom cover of the balancing unit of the probe (see Fig. 23 p. 71)» The grounded lead of one of the c o i l s i s now d i s -connected from the casing and soldered on to the ce n t r a l pin of the connector T (see F i g . 27(a) and (b)). PEMY ItfTBK Dun TO (JFfER F i g . 2 7 ( a ) . C i r c u i t f o r determining 1 frequency response of probe and loading c i r c u i t (see F i g . 2k a l s o ) . F i g . 27(b)» Special cover to replace bottom of casing of balancing u n i t . To obtain a sine wave responses the s i g n a l from a sine wave s i g n a l generator (Tektronix Type 1 90B Constant-Amplitude Signal Generator) i s input at T.« The output from the probe through the loading c i r c u i t i s then observed. The r a t i o arid the phase s h i f t between the input and output signals are shown i n F i g . 2 9 . In the same fi g u r e , we also plot the frequency response of the probe and the induced emf i n one c o i l due to the other. The frequency response of the integrator i s given i n F i g . 28. - 77 -8 id .as-—t— .50 1 P^ASt SHIFT — r -.75 -45° F i g , 2'8, Frequency response of integrator (" <k'~ .--=•" amplication; ? f .£'frequen'c-y^$= .pba's'e s h i f t ) , " RESPONSE (tf) r- -5 F i g , 2 9 , Frequency response of probe (with complete loading c i r c u i t except i n t e g r a t o r ; | e Q / e q I = r a t i o between signals from c o i l s ) . - 78 -3°3°3 R e p r o d u c i b i l i t y of Probe Signals There are many factors that would a f f e c t the probe si g n a l s . I f t h e i r e f f e c t s are s i g n i f i c a n t , the measured signals might be d i f f e r e n t from one discharge to another even though experimental conditions are not varied i n successive discharges. Changes i n experimental conditions a r i s e from errors i n the i n i t i a l charging voltage- of the condenser bank and errors i n the f i l l i n g pressure of gas i n the discharge v e s s e l . These can be reduced by performing the experiments with greater care and patience. However, there i s one i n t r i n s i c d i f f i c u l t y i n pinch discharges which i s d i f f i c u l t to allow f o r , i f i t i s present. This i s the problem of i n s t a b i l i t i e s . They a r i s e spontaneously from small perturbations during the discharge (Rose and Clark, . 1961 ), and often cause v a r i a t i o n s i n the probe signals from successive discharges. It i s therefore most important i n magnetic probe measurements to ensure that the observed signals are reproducible. In our experiment, we found that the signals from the probe were reproducible u n t i l the f i r s t pinch occurred. F i g . 30 shows a t y p i c a l -.record-" of several-, super-posed signals from various discharges under the same experimental conditions. .79. -of'-.cxo . 0 2 '// F i g . '30.. R e p r q c m c i b i l i t y ^ o f pirO'be'r's'ignals ^'.?f or z-pinch': d i s c h a r g e i n He at h mm i n i t i a l pressure• (upper t r a c e : d.B, dr^— a t * 0 2 V/cm; lower . t r a c e : at.2 V/cra; sweep: 1 usec/cm ) . I . .J One o t h e r c h a r a c t e r i s t i c which.;.wasj-Jch-ecked.;.before embarking on raeasuremeiits was whether.. common ' m'ode"- s i'ghals were: appreciable'. T h i s was done by r e e o r a i n g . the probe.;,;.-'-^ t p i ; , r i A i and r o t a t i n g the probe through \}i0°- about i t s stern; ;r The s i g n a l s o b t a i n e d f o r the two o r i e n t a t i o n s were m i r r o r .-i.uaf es o f each other,, i n d i c a t i n g the absence o f common uio.de s i g n a l s ( i . e . no;.,spurious : "pick-up") (see Fig'. - 31 )• :'-u'v.. •'VERTICAL '• •VEFL££7I&/*' OR; CfiO "If" - • ;t> O W I/Cm . F i g . 3 1 . 'Prdbljy^ign^ modes; f o r z.-pinch_dis charge', in. .Ue:.._?it:.'4.. iiim- i n i t ial._p.res sure . - 80 -3'k, 1 Experimentai Results In this section we explain how the gradient probe i s used f o r f i n d i n g the current density of a l i n e a r pinch discharge. We f i r s t describe the arrangements of the probe and explain how measurements are taken and recorded. Then we give the measured current d e n s i t i e s of discharges i n helium at 500 u, 1 mm, 2 mm and k mmHg i n i t i a l pressures r e s p e c t i v e l y . F i n a l l y we present graphs of the discharge radius as functions of time. The l o c a t i o n of the probe i n the discharge, tube i s shown in. F i g . 32(a) and (b). To map the current density of a l i n e a r pinch discharge, probe signals f o r B and p dB. r^-'P" are obtained at d i f f e r e n t probe positions f o r d i f f e r e n t discharges under the same experimental conditions. In order to reduce contamination, each discharge i s produced i n fre s h ^gas (except f o r measurements i n helium at k mmHg i n i t i a l pressure when the e f f e c t of contamination i s apparently n e g l i g i b l e - the probe signals at a given p o s i t i o n are iden-t i c a l , i n several 'successive discharges without changing gas;)'. dB^ . The measured values of B and • . . are then p dr employed to calcu l a t e the perturbed current density J^(r)* By applying the r e s u l t s of Chapter 2~y•-'••the'1 unperturbed' current density, J Q ( r ) and magnetic f i e l d B Q ( r ) can be evaluated. (juiDB TUBE DISCHARGE F i g . 32(a ; ) o Cross-section through axis of discharge tube. F i g . 3 2 ( b ) . Cross-section through guide tube. - 82 -To test the accuracy of the probe measurements, we also compute the t o t a l current of the discharge and the t o t a l current through the return conductor. Since these should be equal, the diffe r e n c e of the computed values should give an idea of the accuracy of the probe measure-ments. An independent check i s obtained by comparing these with the current measured by a Rogowski c o i l (see F i g . 3 3 ) ' This check shows that i n most cases calculated values of J (.r)' are consistent With values measured by the Rogow-sk i c o i l to within 10-fo. The experimental r e s u l t s obtained! with the gradient probe are summarized i n figures i n the following pages,. For each pressure, we plot J ^ ( r ) and B (r) r e s p e c t i v e l y ( F i g . 3i+(a) to F i g . 3 7 ( d ) ) . TIME- C/*S-&c) F i g . 3 3 - Comparison between-values of discharge current measured by gradient probe and by Rogowski c o i l (He)k mmHg). - 83 -- 85 -- 86 -- 88 --,&S> -Fig.35(c).. . H E L I U M HT"'. 1 G O 0 o CO 0 • 5 jisec A 1 .0 wsec + 1 .5 usee 2 . 0 ps ec 2 .5 us ec C--5G ' ,D30 '• • :JDD * 6?) - 90 -- 9'J -- 9h -- 95 -- y° -F i e . 3 7 ( c ) . H E L I U M RT .4000. 'MICR0N5 Y (m, - • 98 o F i f t . 3 7 ( d ) . H E L I U M R T 4 0 0 0 u M I C R O N S o UD i n . 0 A 6 u s e e 7 )isec 8 ^ s e c 9 fisec s 2 -a -.ODD ,020 ,040 060 ' ,080 M O D .120. _ 99 -3*^.2 Comparison of Dynamics- with Theory In t h i s section, we study the dynamics of the z-pinch with the information from the measured p r o f i l e s of the current density and the magnetic f i e l d . To Obtain the; physical picture of the c o l l a p s i n g current layer, we'develop a< ..modified snow-plow eqtiation taking into account the finite.* conductivity of the plasma and the thickness of the s h e l l . I f we assume that the z-pinch discharge consists of a t h i n c o l l a p s i n g c y l i n d e r of i n f i n i t e conductivity and i n f i n i t e s i m a l thickness, and that the sheet sweeps a l l the p a r t i c l e s on i t s way, then the snow-plow equation can be written as <') 4r[/.* <?* - *2*.-5?-J - -where r ( t ) i s the radius of the c o l l a p s i n g c y l i n d e r , r o i t s i n i t i a l value, O q the i n i t i a l density of the gas and - F ( t ) the net magnetic force acting on a unit length of the cylinder (see Rose and Clark, I96l,p. 335) • Equation (l ) i s too simple to account f o r a l l the e f f e c t s that govern the dynamics of the discharge. Nevertheless, despite i t s s i m p l i c i t y , i t predicts the collapse of the current layer with s u r p r i s i n g accuracy under favourable conditions (Curzon et a h , 1962) . The discrepancy of the snow-plow equation given i n ( l ) usually a r i s e s from the - 100 -f a c t that the c y l i n d r i c a l conducting layer i n the discharge does not have an i n f i n i t e conductivity and i t i s by no means " t h i n " . Consequently magnetic fluxes can di f f t i s e through i t and the layer may expand during the collapse. To take into account these e f f e c t s , a modified equation of ( l ^ i s nojw used to obtain the rate of collapse of the current layer i n the discharge. For s i m p l i c i t y j we now make the following assump-tions % -(a) The current layer has an i n i t i a l thickness d, and hence an i n i t i a l mass m „ as confirmed by the measured o probe s i g n a l s . (b) The inner surface of the layer moves with the same v e l o c i t y as that of the centre of mass across the layer This i s introduced to allow f o r the expansion of the current layer due to Joule heating. (c) The k i n e t i c pressure outside the current layer i s n e g l i g i b l e . The theory therefore applies only before the current layer has collapsed so close to the d i s -charge axis that the ^kinetic pressure becomes s i g n i -f i c a n t o (d) The amount of gas escaped from the current layer due to i n e f f i c i e n t trapping i s always a constant f r a c t i o n 1> of the t o t a l mass i n the layer. - 101 -With these assumptions 9 the i n i t i a l mass i s given by (2) mo = / o 7 r ( R 2 - r j ) , « where R i s the radius of the inner wall of the discharge v e s s e l , r the i n i t i a l radius of the inner surface of o the layer (r = R - d) . Since t h i s mass w i l l be c a r r i e d by the current layer as the l a t t e r collapses, i t should be added to equation,(i) such that where the constant J> i s introduced to allow f o r the i n e f f i c i e n t trapping of p a r t i c l e s . The a c c e l e r a t i n g force - F ( t ) i s mainly due to the magnetic force acting on the current layer and i s given by (see F i g . 38)= Here and B^ are the magnetic f l u x d e n s i t i e s at the inner and outer surfaces r e s p e c t i v e l y , and r ' i s the radius of the outer surface. S u b s t i t u t i n g (2), (k) into ( 3 ) 9 the modified equation i s written as - 102 -or (6) 4- |>(R 2-r 2) = - -4-^ ( r ' B ' 2 - r B 2 ) . v ' dt L v , ' d t J jx pQ p P From our probe measurements, we have obtained the current density p r o f i l e s and the magnetic f l u x density at d i f f e r e n t stages of the collapse. We can therefore obtain the experimental values of r ' , Bp and r, Bp corresponding to the r a d i i and flux densities at the inner and outer surfaces of the c o l l a p s i n g sheet (see F i g . 38(a) and (b)). The> inner and outer edges of the s h e l l are taken as the r a d i i at which the current density J p ( r ) drops to half of i t s peak values. T r ' y F i g . 38. To obtain experimental values of r',Bp,r,Bpat a given time t a f t e r s t a r t of collapse. - 1 0 3 -We have solved numerically the modified snow-plow equation i n the form of ( 6 ) f o r the collapse curve of the current layer and compared the solutions with those measured by the gradient probe. By varying the parameter JJ , we obtain an idea of the e f f i c i e n c y of trapping of p a r t i c l e s by the magnetic f l u x . The best f i t between experiment and theory i s obtained with the following values f o r 1> ; JJ .sa 0 . 9 > f o r helium at 1 , 2 , k mmHg i n i t i a l pressures, }J = 0 . 7 5 » f o r helium at 5 0 0 ulig i n i t i a l pressure. There are two possible explanations f o r the above calculated r e s u l t s : i n e f f i c i e n t trapping of p a r t i c l e s by the current s h e l l and experimental b i a s . A, consistent bias of 5°/o i n the current measurements would change•the a c c e l e r a t i n g force by 10% This margin of error i s reasonable f o r the experimental set-up,and could explain the discrepancy between theory and experiment at 1, 2 and k mmHg i n helium. The fact that u j does not vary with pressure over t h i s range strongly suggests that the discrepancy i s due to errors i n measurements rather than imperfect trapping. However at 5 0 0 yaHg, the value f o r U i s 0 . 7 5 s which suggests that at this pressure not a l l the gas i s swept up by the c o l l a p s i n g current s h e l l s , i„ e. trapping i s i n e f f i c i e n t (Liebing, 1 9 6 3 ) ° . ijfie th-eo-re-t^^j^^^e^qp^^p^^. _jnen1;a-I^ po.lb£iEp.s*e :^ i^ y-esb_.ajpte;- siiown i n F i g . 39-42. - 104 -8 c t (/usee) F i g . 39. Collapse curves (He 500 uHg), ® Outer radius of s h e l l P o s i t i o n of current peak ^ Inner radius of s h e l l Theory (jf = .75) - 105 -8x" F i g . 4 0 o Collapse curves (lie 1 oimHg). © Outer radius of s h e l l •+• P o s i t i o n of current peak A Inner radius of s h e l l X Theory ( If = „ 9) - 106 -F i g . 41 o Collapse curves (.He 2 iniiiHg).: Outer radius of s h e l l P o s i t i o n of current peak Inner radius of s h e l l Theory ( J/ = . 9) + X - 107 -F i g . 42= Collapse curves (lie 4 mmHg) © Outer radius of s h e l l T P o s i t i o n of current peak ^ Inner radius of s h e l l X Theory ( I) = .9) Chapter 4 CONCLUSIONS AND PROPOSALS FOR FUTURE WORK 4. 1 Conclusions. The main contributions of this thesis to the measurements of the current densities using a magnetic probe are the following:- , (1) We have examined the l i m i t a t i o n s of the E-K-Z model f f o r the current flow round a hole i n a c y l i n d r i c a l current sheet. Our investi g a t i o n s show that the model i s valid? (a.) f o r time dependent cases provided that the conductivity cr' i s independent of time, (b) f o r a moving current layer with a v e l o c i t y v, i f v«.w6', where co i s the c h a r a c t e r i s t i c frequency of the current and & i s i t s skin depth f o r the layer, and (c) the skin depth £ i s much larger than the thickness of the sheet. This model i s used to obtain a c o r r e c t i o n formula r e l a t i n g the true current density of a plasma to the apparent current density ( i . e. without c o r r e c t i o n f o r the perturbation of the probe ) measured by a magnetic probe. (2) In the formula which corrects f o r the perturbation of a magnetic probe on the true ciirrent d i s t r i b u t i o n , _ - 10.9 -dB we have removed the d i s c o n t i n u i t y i n (appearing i n Daughney 1s thesis) by taking into account the curvature of the current sheet'*. This i s an important- , improvement because i t eliminates the s i n g u l a r i t y f r o m the i n t e g r a l equation from which the current i s computed. We,have obtained quantitative r e s u l t s of the corrections necessary f o r the measured current density J to give? P • the true current density J . In general f o r smooth current d i s t r i b u t i o n s J '', • J - J / J \4.5%> However, o ' p o / p l N i f J i s due to a i t h i n current sheet, a probe w i l l o smear out J and measure aamuch thicker current. • • p In such cases, |^ 0 - v^p| c a n ^ e several times) larger " f than I J I. -I P I We have assessed the accuracy of the c o r r e c t i o n proce-dure for. p r a c t i c a l a p p l i c a t i o n s . Employing the Monte Carlo method, we have simulated experimental errors with random numbers and have found that f o r smooth current d i s t r i b u t i o n s the errors i n the corrected current density J were usually twenty times the ; o ; experimental errors i n the measured current density J 2Bti^lg©fori»-e , %y^%ffieoexpl§rimenWi--eka^r:«as^v.J-: are larger than -|%, the errors i n the calcul a t e d values of J w i l l be larger than 5 % of J . Thus f o r a,, smooth o • ' ' • p current density d i s t r i b u t i o n J , such a c o r r e c t i o n procedure i s unnecessary. However, i f J q i s known to be a current,"spike" ( i . e. J i s due to a t h i n * V o current sheet), a p robe measures a much more broadened current several times reduced i n magnitude. In such cases, c o r r e c t i o n i s important. Due to the d i f f i c u l t y i n r e p r o d u c i b i l i t y , ' the c o r r e c t i o n of the measured i n a pulsed plasma cannot be achieved since the s c a t t e r i n experimental parameters i s usually larger than 2%.. We have developed a gradient probe using two search c o i l s . It has a l l the advantages of a magnetic probe i n the form of one single c o i l and i t measures both the magnetic f i e l d and the gradient of the f i e l d d i r e c t l y . This enables much more accurate measurements of.., plasma current d e n s i t i e s to be made. Measurements of the current densities , the magnetic fl u x d e n s i t i e s i n z-pinch discharges i n helium at pressures between 500 u and k. mmHg have been obtained. A modified snow-plow equation has been used to study the dynamics of the z-pinch discharges i n helium at i n i t i a l pressures ranging from 500 u to k mmHg. This equation has allowed f o r the thickness of the c o l l a p -sing current s h e l l and the escape of gas from i t . The a c c e l e r a t i n g forces on the s h e l l due to magnetic pressures have been taken from the probe measurements. The comparison between experiment and theory sugge'sts that at pressures between k mmHg and 1mmHg i n helium, trapping i s better than and that the observed discrepancy i s due to a systematic bias i n the c a l i -brations of the measuring equipment. In helium at 500 uHg, there i s evidence that some of the, discharge gas escapes through the c o l l a p s i n g current s h e l l . - 112 -k.2 Proposals for Future Work The probe was o r i g i n a l l y designed to give enough signals f o r a Tektronix Type G d i f f e r e n t i a l a m p l i f i e r which has a much smaller a m p l i f i c a t i o n than the Tektronix Type W d i f f e r e n t i a l a m p l i f i e r now being used. It i s therefore possible to reduce the si z e of the probe to about 3 mm i n diameter and about 10 turns i n each of the sensing c o i l s (the gradient probe used i n this thesis i s 7 mm i n diameter and has 100 turns i n each of the c o i l s ) . This w i l l probably reduce the probe signals to about a few m i l l i v o l t s , but the frequency response w i l l probably be f l a t up to 5 MHz. Using such a smaller probe, the perturbation on the plasma current w i l l be reduced and better frequency response and s p a t i a l r e s o l u t i o n should be achieved. In the study of the perturbation of a magnetic probe on the current flow of a plasma, we have not considered other possible "dynamical e f f e c t s " on the current layer. Apart from the d i s t o r t i o n of current flow round the probe as discussed i n the present t h e s i s , the probe might s p l i t up the current layer or induce i n s t a b i l i t i e s . Boundary layer *~ e f f e c t s w i l l also be appreciable. Further experimental and t h e o r e t i c a l work should be done to investigate such e f f e c t s . For the z-pinch discharges, our r e s u l t s show that the trapping of p a r t i c l e s by the c o l l a p s i n g s h e l l depends on the.' f i l l i n g pressures. Experiments may be done to investigate the amount and the nature of gas l e f t behind by the c o l l a p -sing current s h e l l . BIBLIOGRAPHY Artsimovich, L. A. et a l . ( 1 9 5 6 ) , At. Energy, 1_, 76; r e p r i n t i n J . Nucl. Energy 4> 203 ( 1 9 5 7 ) . Curzon, F. L. , C h u r c h i l l , R. J., and Howard, R. ( 1 9 6 3 ) . Proc. Phys. Soc. 81 , 34,9. Daughney, C. C. ( 1 9 6 6 ) Ph.D. Thesis,• U n i v e r s i t y of B r i t i s h Columbia. 't 1 • Ecker, G., K r O l l , W. , and ZOller, 0. ( 1 9 6 2 ) Ann. Physik 10, 2 2 0 . Fox, L. ( 1 9 6 2 ) Numerical Solution of Ordinary and P a r t i a l D i f f e r e n t i a l Equations (Addison-Wesley )•• v Huddlestone, R. H. , and Leonard,, S. L. (1965 )>• Plasma Diagnostic Techniques (Academic Press). Landau, L. D., and L i f s h i t z , E. M. ( i 9 6 0 ) , Electrodynamics i of C6ntinuous Media-;* (Addison-Wesley). , . , ' Liebing, L. , (,1963)> Motion and Structure of Plasma in.'a R a i l Spark-gap, Phys. Flu i d s 6, 1036 and ,IPP 3/1.6 ( 1 9 6 4 j , Garching. ^ Malmberg, J. H. ( 1 9 6 4 ) , Rev. S c i . Instr. 3 5 , • 1 6 2 2 . Medley, S. S., Curzon, F. L., and Daughney, C. C. ( 1 9 6 5 a ) , Rev. S c i . Instr. 36.» 7 1 3 . Medley, S. S., Curzon, F. L., and Daughney, C. C, ( 1 9 6 5 b ) , , Can. J . Phys. 4 3 , 1882.. Morse, P. M. , and Feshbach, H. (1953) , Methods of Theoretical. Physics,, Part I and I I , (McGraw-Hill). Ohkawa, T., Forsen, If. K. , Scllupp, A. A. J r , and Kerst, D. W. ( 1 9 6 3 ) , Phys. Fluids 6, 846. Rose, D. J . , and Clark, M. (19 61 ) , Plasma and Controlled Fusion (Wiley). Shreider, Yu. A. ( 1 9 6 6 ) , The Monte Carlo Methods (Pergamen). Smythe,W. R . ( 1 9 5 6 ) , S t a t i c and Dynamic E l e c t r i c i t y (McGraw-Hill). Spi t z e r , L . J r . ( 1 9 6 2 ) , P h y s i c s of F u l l y Ionized Gases(Interscience) Tuck, J..L. ( 1 9 5 8 ) , Proc. 2nd Univ. Nat. Int. Conf. on Peaceful Uses of Atomic Energy, 3j2, 3 .• Turing, A. M. (1948) , Rounding of Errors i n Matrix Processes , J. of Mech. and App. Maths.,' J_, 2 8 7 . Wright, E. S., and Jahn, R. G. ( 1 9 6 5 ) , Rev. S c i . In s t r . 3 6 , 1 8 9 1 . - 113 -Appendix i " SACCULATION OF FOR THE E-K-Z MODEL In t h i s appendix, we derive equation (1.1 ) of ^ 2.2 a (p. 2 2 ) . We w i l l f i r s t show that to f i r s t order i n —, A J s i s the same f o r both the plane current and the current cyli n d e r of radius s and i n f i n i t e s i m a l thickness ds. Then we w i l l prove that the corresponding values of AB SL ' d i f f e r by a quantity of the order —. /. s From equation ( 9 ) of ^ 2 . 2 . 1 (p. 2 1 ) , the change i n current density AJ i n a plane current layer due to a c i r c u l a r hole of radius a i s given by (1) A J = (-J o (p 2co S9, +J Q ( j|) 2sine, 0), j o ^ a , = - J , p < a. o " f This immediately shows that ( 2 ) | A J I ~ (f)2 J o | , j>>a. r A J i s therefore s i g n i f i c a n t over a f i n i t e region with < a', a' being comparable to a. In fac t we have £ .1 f o r j o > a ' ~ 3 a A J Consider a current c y l i n d e r of radius s and i n f i n i t e s i m a l thickness ds and the tangent plane at the centre of the hole. Using the fac t that the arc and the tangent subtended by a small angle d i f f e r by a.quantity of second order i n the angle, the region of the current cylinder over which AJ i s s i g n i f i c a n t can be replaced by - 115 -the tangent plane i f we r e t a i n quantities up to the f i r s t order i n —- It i s therefore obvious that the s o l u t i o n of s AJ given i n . (1) f o r a plane can be used as the s o l u t i o n a 1 2 i n the current c y l i n d e r ( (---—) .1 f o r s ^ 3 a ' ^ 9 a ) . To show that A and AB^ f o r the plane and the c y l i n d r i c a l current layers r e s p e c t i v e l y d i f f e r by a f i r s t order quantity i n we consider equation ( 3 ) of £2.2.1 (p. 1 9 ) , i . e. ( 3 ) **j In the same coordinates systems (see F i g . ^3(a) and (b)) as those of F i g . 6 (p. 21), the required component -^ B^ , due to the current c y l i n d e r i s given by (k) AB^(r) =-AB (^) where ( 5 ) I , = (6) I 2 = I 1 oos28(|) 2 , and (7) e = a 2s F i g . 43(b). Geometrical r e l a t i o n s between AB (f) and Aj(<) 0). 7 - 117 -In ( 4 ) , we have taken a 1 as the upper limit of p. Since the contribution to the i n t e g r a l fo'r current elements l y i n g beyond a' i s n e g l i g i b l e , we can replace the upper l i m i t by p —»oo. It i s d i f f i c u l t to evaluate the i n t e g r a l s of ( k ) d i r e c t l y because sinO appears i n the complicated expres- , sion i n the denominators of the integrands. However, we can evaluate the i n t e g r a l s approximately by expanding the i n t e g r a l s i n power series of e such that and 2 < S u b s t i t u t i n g (5a) and (6a) into { k ) and i n t e g r a t i n g over and 9 , we obtain  ^ 2 (6a) 1 = ^ " p T/2 + e a s i n 2 0 [l -3 2 2 1 t 0U2)1 2 ( ^ 2 + p 2 ) 3 / 2 ( f L ( f 2 + p 2 ) J J ( 8 ) AB^(r) =-AB^K) i u J o ( s ) d s . | ( s g n ( ^ ) - ? 2 l ) ( l - f ) ' (•£ +a ) 2 aT 2 a 2 x• • + j a ! , / 2 ] + 0(€ 2)} t ( r 2 + a 2 ) * ( C 2 + a 2 ) 3 / 2 J J \<   ) 2 +a )• For values of |£ |^2a» the expansion procedure gives accurate values f o r -AB^  whereas for | £ . | ^ 2 a , the approxim-ations are not accurate but the absolute value of which decreases r a p i d l y with |£ | and A B^ becomes AB - 118 -i n s i g n i f i c a n t . We are therefore interested i n values of A B y (r ) with 1| = r " s I £ . s The corresponding quantity AB^(r) f o r a plane current layer i s obtained from ( 8 ) by l e t t i n g s-*.oa , i.e ( 9 ) AB>($) = - i u J o ( s ; d s | [ s g n ( f / - — 0 ( 6 2 ) j . Comparing ( 8 ) and (9)=» we therefore' have d o ) A y p ^ - ^ J o ( s ) d s | [ s e n ( ^ ) - ^ ^ i j u - ^ M ' : ' ; | ' o 2 or neglecting 0(fe ) ( 1 1 ) AB^(.r) = i u J o ( s ) d s C ( ^ ) a.. r where C ( ~ ^ ) has been defined i n (.13) of ^ 2 . 2 . 1 . We have neglected the second term inside the brackets | j of ( 8 ) . This means that equation ( 1 1 j i s only correct provided,, terms of 0(g"jr) are n e g l i g i b l e . Equation ( 1 1 ) now gives us the value of AB^( r)- due to a current cyli n d e r of radius s. and i n f i n i t e s i m a l thickness ds, having an - •a'x-i-a;l'41^#rr^h:t' density J (s).- It also enables us to obatin AB^(r) from the corresponding value of'a plane current. For a r a d i a l d i s t r i b u t i o n of current, equation ( 1 1 ) gives (,1?) A B ^ ( r ) = i u f / d s J o ( s ) C ^ J - 1 1 9 -00 r-s s_ r 0 We have shown i n equation ( 1 1 ) that AB^ and ^B^ fo r a curved and a plane 1ayers,respectively d i f f e r by a m u l t i p l i c a t i o n f a c t o r I f |r-s|«s., the diff e r e n c e i s s small and we might be tempted to replace by unity. However, we now show that t h i s f a c t o r i s e s s e n t i a l i n the c a l c u l a t i o n of the gradient of B^(rJ, and hence J ^ i r ) , f o r a curved layer. Consider the same current cylinder of radius s and infinite'simal * thicknes'ss^ ,:. -Whe perturbed azimuthal f i e l d along the axis of the probe i s ( 1 3 ) B (r) = B ( r) + AB ,(r; p O 0 r > s , = i n J o ( s ) d s C ( ^ ) f , , r < s. D i f f e r e n t i a t i n g B (r) with respect to r, we obtain which i s continuous at r=s . However, i f the fa c t o r f - 120 -i n AB^(r) i s n e g l e c t e d t h e gradient of B^(r) becomes r > s ^ J o U ) d s 2dr" ' r < S ' which gives i i 6 ) w j w 7 ; = U J (s}d S - pf <) r .. • j o * L s 2flr v T I \a 3c ( 0 ~ ) + r=s . r=s C ( ^ ) Since , i s continuous, i t i s easy to see that k — — M i s discontinuous at r=s . Such a... d i s c o n t i n u i t y cannot occur i n a region where there i s no current flow. Further, a, d i s c o n t i n u i t y i n ( g ) creates a s i n g u l a r i t y i n the i n t e g r a l equation (see equation (16) of j>2.2.1 , p. 23) used f o r c a l c u l a t i n g the current density. I t i s therefore important to include the f a c t o r fr . Appendix IX FORTRAN IV PROGRAMME FOR SOLVING. THE INTEGRAL EQUATION In th i s appendix, we give the complete Fortran IV. computer programme f o r solv i n g the i n t e g r a l equation .(11) / J o ( s ) K ( r , s , a ) d s = J p ( r ) 0 which was already transformed into a set of equations K ( 2 ) A(M,N) u(N) = v(M), M = 1,2,...,K, (see equation ( l ) and ( 9 ) of ^2.3)° '^e u s e the f o l -lowing notation i n the computer programme :-DX = width of mesh i n t e r v a l , K = number i n t e r v a l s used i n the c a l c u l a t i o n , RHO = probe radius, AMU = magnetic permeability, GPSENS =s s e n s i t i v i t y of probe f o r measuring the gradient of the magnetic f i e l d , PBSENS = s e n s i t i v i t y of probe f o r measuring the magnetic f i e l d , x(M) = r = r a d i a l coordinate of the centre of the Mth i n t e r v a l , AJ(M) = u(M) = J o ( r ) , PJ(M) ='v(M) = J p ( r ) , - 122 -B(M) = B ( r ) , B P ( M ) = B p ( r ) , D B ( M ; = BP^ r) , C(M) = C l ^ 1 ^ ) , f o r r-s = M.DX and a = RHO , SL DC(M) = -|-DX. BP ^ r ^  , f o r r-s = M.DX and a = RHO. In the programme, the matrix i s f i r s t inverted, and then m u l t i p l i e d by d i f f e r e n t sets of input data to give the output values of AJ.(M). $$ $ J O B S P A G E 7 9 0 4 6 Y . K . S . TAM 3 0 MA.fi 6 0 $ T I ME 5 $ I B F T C TAMM C P R O G R A M M E N O . 5 C . E - K - Z MODEL C C C NO PLOT O U T P U T . TO C O M P U T E J ( X ) AND B ( X > FROM M E A S U R E D V A L U E S OF B P ( X ) AND DB ( X ) . P J ( I ) I S SMOOTHED IN THE P R O G R A M M E . • c . c c K = N O . OF M E S H I N T E R V A L S B ( I )=i ) N P E R T U R B E D M A G N E T I C F L U X D E N S I T Y c c c B P ( I ) = P E R T U R B E D B. D B ( I ) = G R A D I E N T O F B P I I ) C ( I ) = C O R R E C T I ON F A C T O R c c c DC ( I )='. 5 * D X * ( G R A D I E N T O F C l l l l A J ( I ) = U N P E R T U R B E D J P J ( I ) ^ P E R T U R B E D J c c c X ( I ) = R A D I A L C O O R D I N A T E OF C E N T R E OF I T H I N T E R V A L A ( M » N ) = M A T R I X E L E M E N T • AMU = MAGNET I C P E R M E A B I L I T Y • • " ' " ' c c c D X = W I D T H OF M E S H I N T E R V A L S C = S C A L I N G F A C T O R FOR M A T R I X E L E M T N T S RHO= P R O B E R A D I U S ro c c c P B S E N S = P R O B E S E N S I T I V I T Y FOR M E A S U R I N G B G B S E N S = P R O B E S E N S I T I V I T Y FOR M E A S U R I N G G R A D I E N T OF B i c D I M E N S I O N A ( 1 0 0 » 1 0 0 ) » B D ( T O O ) » D B ( 1 0 0 ) * C ( 2 0 0 ) > D C ( 2 0 0 ) , A J ( 1 0 0 ) D I M E N S I O N B ( 1 0 0 ) i P J ( 1 0 0 ) » X ( 1 0 0 ) TO E V A L U A T E C < I ) AND DC ( I ) 2 0 1 K D = 1 0 0 R E A D ( 5 » 2 0 1 ) A M U » D X , K FORMAT ( 2 E 2 0 . 8 » I 3 ) 2 0 2 R E A D ( 5 * 2 0 2 ) RHO FORMAT ( E 2 0 . 8 ) W R I T E ( 6 » 3 0 1 ) R H O » D X » A M U » K 3 0 1 FORMAT ( 1 H 1 » 5 H R H O = » E 2 0 . 8 » 5 X » 3 H D X = » E 2 0 . 8 » 5 X » & H A M U = » E 2 0 » 8 t 5 X » 2 H K = » I 13 ) R=DX/RHO KK=K+K P I J = . 5 * R 1 DO 11 M = l» 10 , : AM = M ( . AMR= AM*R - S S=1./ + AMR-*AMR ) , S R = S Q R T ( S S ) t ' ' C ( M ) = . 5 * < - 1 . + A M R * S R ) i 12 11 DC(M)=.5*R*SS*SR DO 9 9 7 1=11,KK C ( I ) = 0 . 9 9 7 D C ( I ) = 0 . K1=K-1 C TO EVALUATE A ( M »N ) DO 122 M = 1» K MM=M+M . AM = M AM = l . / ' ( AM-.5) DO 121 N=1»K IF(N.EQ.M) GO TO 121. A N = N AN=AN-.5 MN =I ABS(M-N) NM=M+N A(M,N)=AM*AN*(DC(MN)-DC(NM)) 121 CONTINUE • 122 A(M,M)=PIJ-DC(MM) l C 10 SCALE A(M,N) SO THAT NO NUMBER CAN BE.LARGE THAN 1.E38 SC=1 . / P I J is: DO 15 M=1,K DO 15 N = l » K 1 15 A(M,N)=SC*A(M,N) C ' TO INVERT A CALL INVERI(A»K»KD»DETiCOND) WRITE ( 6 , 302 ) DET,COND 302 FORMAT (6X»4HDET=,E20.8,5HC0ND=»E20.8) l h ( DL'I • L 1 • ( l . t - 2 0 ) ) S IOP IF (DET.GT.(1 .E38) ) STOP READ (5,203) PBSENS,GPSENS .; 203 FORMAT (2F1 0 . 5 ) DO 1111 M=1.K DO 1111 N=1»K 1 i l l A(M»N)=SC*A(M»N) C TO READ DATA . 1 READ (5,207) GAS,P,TIME 5 \ 4 3 207 FORMAT (A6,2F10.5) ' IF ( P . E Q . O . ) STOP READ (5» 204) (DB(I ) , I = 1,K) READ (5,204) (BP( I ),I = 1,K) 2 04 FORMAT (8F10.5) KM1=K-1 DX1=.5*DX DO 998 1=1,K 998 BP(I) = B P (I) + D X 1 * D B (I) DDX=1./DX AAU=1./AMU DO 205 1=1,K C I - I CI I = C I - • 5 X(I)=CII*DX BP(I)=BP(I)*PBSENS DB(I)=DB(I)*GPSENS 205 PJ(I)= AAU*(BP(I) *DDX/CII+DB(I)) C TO CALCULATE A J ( I ) CALL MATVEC(A»PJ,AJ,K,KD) . C TO CALCULATE B(X) AXU=.5*AMU*DX DO 17 M=1»K 1 SUM=0. 1 - CM = M MM 1 = M-1 MP1=M+1 DO 18 N=1»M,M1 MN=M-N 18 SUM=SUM+AJ(N)*C(MN) DO 19 N=MP1,K • N M = N - M 19 SUM= SUM-AJ(N)#C(NM) . 17. B(M)=3P(M)-AXU*SUM WRITE (6,500) GAS, P,TIME 500 FORMAT (1H1 , IX,A6 ,3HAT , F 1 0 . 5 , 11HMICRONS AND,F10.5,6HU S E C . ) WRITE (6,501) PBSENS,GPSENS 501 FORMAT (1X,21HM. PROBE SENS ITI V ITY=,E20.8>21HWEBERS/CU.METER/VOLT. 1,2X,21HG. PROBE SENSITI V ITY=,E20.8 ) • WRITE (6,502) ( X ( I ) , B P ( I ) , D 5 ( I ) , P J ( I ) , B ( I ) , A J ( I ) , I = 1 , K ) 502 FORMAT (74X,6KX (M.) ,9X , 11 HBP (W/SO•M),9X,11HD3 (W/CU.M) ,8X,12HJP( 1AMP/SQ.M1,2X,18HC0MPUTED B(W/SQ.M),2X,2OHCOMPUTED J ( A M P / S Q . M ) / 3 5 I -4 2 ( F 1 0 . 5 , 4 E 2 0 . 8 » E 2 2 . 8 ) ) ("., C TO CHECK EXPERIMENTAL RESULTS. C TO CALCULATE PI SCHARGE CURRENT SUM1 AND CURRENT IN THE RETURN C CONDUCTOR SUM 2 - - - - - ... * ( , - . ' . . SUM 1 = 0. SUM2 = 0 . ; . [ ' •  DO 50 5 1 = 1 »K ~ IF ( A J ( I ) . L T . O . ) GO TO 507 SUM1=SUM1+AJ( I )*X ( I ) - . ____ GO TO 505 . *  5 07 SUM2 =SUM2+AJ(I)*X(I) 505 CONTINUE ._ . : ; •  DI=3.14159265*2.*DX , SUM1=SUM1*DI SUM2 = SUM2*.DI ._ ' . WRITE ( 6 , 508) SUM1,SUM2 ~ ' ' . ;'. 508 FORMAT (//24H TOTAL POSITIVE CURRENT=, E20.8»6H AMPS./ 124H TOTAL NEGATIVE CURRENT=,E20.8?6H AMPS.) • WTTTTE ( 7,601) GAS,P»TIME T ~' 601 FORMAT ( A 6 , 2 X , F 1 0 . 5 , 2 X , F10.5) ' WRITE (7,600 ) SUM1,SUM2  : 6015 FORMAT (2E2U781 — WRITE (7,602) (DB( I ) • BP( I ) , B(I ) , P J ( I ) , A J ( I ) , I = 1,K) ( ' 602 FORMAT (5E16.8) ; ' \ . j _ GO TO 1 {• • . ' END N SENTRY Y 1 12 M 2 !0 9 3 8 '• 1 i 5 

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