DYNAMICS OP A THETA PINCH DISCHARGE I.N A TRANSVERSE MAGNETIC FIELD by WILLIAM LEUNG LEE B.Sc, University of Washington, i960 M.Sc, University of Purdue, 1962 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1966 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x -t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s m a y b e g r a n t e d b y t h e H e a d o f my D e p a r t m e n t o r b y h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n -c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a D e p a r t m e n t o f The Uni v e r s i t y of B r i t i s h Columbia FACULTY OF GRADUATE STUDIES c PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of WILLIAM LEUNG LEE B.Sc. (Eng.Phys.)s U n i v e r s i t y of Washington, 1960 M.Sc. (Aeronautical Eng.), Purdue University, 1962 IN ROOM 303, HENNINGS BUILDING THURSDAY, MAY 12, 1966 AT 10:30 P. M. COMMITTEE IN CHARGE Chairman: C = A. McDowell Bo Ahlborn R. Howard A. J. Barnard R. A. Nodwell F. L. Curzon T. Watanabe External Examiner: H. M. Skarsgard Department of Physics U n i v e r s i t y of Saskatchewan Research Supervisor: F. L. Curzon DYNAMIC PROPERTIES OF A THETA-PINCH DISCHARGE IN A TRANSVERSE MAGNETIC FIELD ABSTRACT The suppression of r a d i a l hydromagnetic o s c i l l a -tions of a theta pinch plasmoid produced i n a i r has been investigated with magnetic probes and a framing camera, In the presence of e s s e n t i a l l y s t a t i c and uniform magnetic bias f i e l d s (produced by Helmholtz c o i l s ) i t was found that the o s c i l l a t i o n s are suppressed by a f l i p i n s t a b i l i t y of the plasmoid and the d i r e c t i o n of f l i p i s c o n t r o l l e d by the transverse magnetic f i e l d s . Using the suppression of r a d i a l hydromagnetic o s c i l l a t i o n s as the c r i t e r i o n for f l i p s i t i s found experimentally that i f % i s greater than a c r i t i c a l value,, then the plasma r i n g f l i p s , DT i s the strength of the bias f i e l d perpendicular to the axis of the theta c o i l , 'a' the radius of the discharge vessel and 'p' i s the gas pressure. The experimental r e s u l t s are interpreted i n terms of a 'snowplough' model for the discharge, i n which the plasma current forms two concentric c y l i n d e r s . The pr e d i c t i o n of the theorgy agrees with the. experimental observations. GRADUATE STUDIES el d of Studyt Physics Quantum Mechanics Plasma Physics Magne t ohydrodynamic s Advanced Plasma Physics E l e c t r o n i c s W. Opechowski L„ de Sobrino Fo L. Curzon A. J. Barnard W. A. G. Voss ABSTRACT The suppression of r a d i a l hydromagnetic o s c i l l a t i o n s of a t h e t a p i n c h plasmoid produced i n a i r has been i n v e s t i g a t e d w i t h magnetic probes and a framing camera. In the presence of e s s e n t i a l l y s t a t i c and uniform magnetic b i a s f i e l d s (produced by Helmholtz c o i l s ) i t was found that the o s c i l l a t i o n s are suppressed by a f l i p i n s t a b i l i t y of the plasmoid and the d i r e c t i o n of f l i p i s c o n t r o l l e d by the transverse magnetic f i e l d . Using the suppression of r a d i a l hydromagnetic o s c i l l a t i o n as the c r i t e r i o n f o r f l i p , i t i s found experimentally that i f i s greater than a c r i t i c a l v a l u e , then the plasma r i n g f l i p s . °BT° i s the s t r e n g t h of the b i a s f i e l d p e r p e n d i c u l a r to the a x i s of the t h e t a c o i l , ? a ' the rad i u s of the discharge v e s s e l and 5 p 9 i s the gas pressure. The experimental r e s u l t s are i n t e r p r e t e d l n terms of a snowplough model f o r the discharge, i n which the plasma current forms two c o n c e n t r i c c y l i n d e r s . The p r e d i c t i o n s of the theory agree w i t h the experimental observations. i i i TABLE OF CONTENTS Abstract i i L i s t Of I l l u s t r a t i o n s v Acknowledgements i x CHAPTER I - In t r o d u c t i o n 1 CHAPTER I I - Experimental Apparatus 7 Section A - Theta Pinch Apparatus 7 Section B - Bias F i e l d Apparatus lk 1 „ Large F i e l d Bank 1 ^ 2 „ Small F i e l d Bank 1 7 3 o Helmholtz C o i l 1 9 Section C - Diagnostic Equipment 2 2 l o Magnetic Probes 2 2 a„ S i n g l e A x i a l Probe 2k b. D i r e c t i o n a l Probe 2 7 2 0 High Speed Framing Camera 3 0 CHAPTER I I I - Results 3 3 Section A - Discharge Of Theta Bank With No Bias F i e l d 3 3 Section B - Theta Pinch With A Bias F i e l d ^ 0 1 „ Photographic Measurements kk a. <b = 9 0 ° kk b 0 (j) ^ 0 ° kB iv 2. Probe Measurements 51 a* 0 ~ 0 ° 51 b„ 0 = 90° 56 3. Variation In The Discharge Radius 59 4. Directional Probe 6 l Section C - Theory 68 1. Radial Equation Of Motion 69 2. Axial Equation Of Motion 7^ 3. Discussion Of The Assumptions 79 Discussion Of The Interpre-t a t i o n Of The Suppression Of O s c i l l a t i o n s As The F l i p I n s t a b i l i t y 87 5„ Variations Of The Theore-t i c a l Model 90 Section D - Suggestions For Further Work 93 CHAPTER IV - Conclusions 95 APPENDIX A - Effect Of Preionization On The S t a b i l i t y Constant 97 B - Change In The S t a b i l i t y Constant With Varying Theta C o l l Length 99 C - Summary On The Argon Experiments 102 D - Pictures Of F l i p At 100 ^Hg And 250/zHHg, Air 105 E - Characteristic Impedance Of Magnetic Probe 108 REFERENCES 111 V LIST OP ILLUSTRATIONS PIG. I . 1 - Cross S e c t i o n Of Theta Pinch 2 2 - Top View Of Plasma F l i p k FIG. I I . 1 - Theta Capacitor Bank 8 2 - Cross S e c t i o n a l View Of Spark Gap Switch, S 1 9 3 - U l t r a V i o l e t T r i g g e r Generator 10 k - Rogowski C o i l C i r c u i t 13 5 - Helmholtz Spark Gap Switch 15 6 - Trigger Pulse Generator 15 7 - Helmholtz Trigger P i n 16 8 - Current Waveform For The Small Magnetic F i e l d Capacitor Bank 18 9 - Small Magnetic F i e l d Capacitor Bank C i r c u i t Diagram 18 10 - Geometric P o s i t i o n Of The Helmholtz C o i l 20 11 - Magnetic F i e l d D i s t r i b u t i o n Along Axis AAe And BB 9 Of Helmholtz C o i l s 21 12 - RC I n t e g r a t e r Network 22 13 - A x i a l Magnetic Probe 2k lk - Method Of Mounting Probe In The Theta Pinch Discharge Vessel 25 15 - C a p a c i t a t i v e Coupling S i g n a l Prom Magnetic Probe 27 v i PIG. 11.16 - Frequency Response Of A x i a l Magnetic Probe 28 17 - D i r e c t i o n a l Probe 29 18 - Frequency Response Of D i r e c t i o n a l Probe 31 19 - Framing Camera Schematic 32 FIG. I I I . 1 - Schematic Of Theta Pinch Discharge 3^ 2 - Theta Current Waveform 36 3 - Formation Of The Plasma S h e l l 36 4 - Integrated Magnetic Probe S i g n a l 38 5 - End-On Photograph (B H = 0 ) 39 6 - Top-On Photograph ( B H = 0 ) 39 7 - Bias F i e l d T r i g g er C i r c u i t For The Magnetic Probe 4 l 8 - Bias F i e l d T r i g g e r C i r c u i t For The Framing Camera ^3 9 - Sketch Of Plasma Photographs 46 10 - Top-On Photograph Of Plasma (tf) = 90°) ^7 11 - Side-On Photograph ((j) * 0°) ^9 12 - End-On Photograph (<f) » 0°) 4-9 13 - End-On Photograph Of Plasma With No Transverse Bias F i e l d ((j) = 0°) 50 14 - End-On Of Plasma With A Transverse Bias F i e l d 51 15 - A x i a l Probe S i g n a l s With Changes In *08 53 16 - Value Of B„sin0 For D i f f e r e n t Gas H r c Pressures ( A i r ) 5^ v i i FIG. I I I . 1 ? - V a r i a t i o n Of The C r i t i c a l Transverse F i e l d With Pressure ((f) ^  0°) 57 18 - V a r i a t i o n Of The C r i t i c a l Transverse F i e l d With V a r i a t i o n In Pressure (0=90°) • S9 19 - V a r i a t i o n In S t a b i l i t y Constant With Tube Diameter 60 20 - V]_ - V 2 Waveform 63 21 - V x Waveform 63 22 - Probe Waveforms At O ~ 10° 65 c 23 - The Angular R o t a t i o n 66 24 - A x i a l S e c tion Of The Theta Pinch Discharge 75 25 - S p e c i f i c a t i o n Of The Angle 76 26 - Probe S i g n a l s Showing B Q Is Constant For D i f f e r e n t Pressures At F l i p Conditions 80 27 - End-On Photographs Of The Plasma At F l i p Conditions 82 28 - Plasma F i e l d Strength At F l i p Conditions 85 29 - Magnetic F i e l d Lines Of Theta C o i l 89 APPENDIX A . l - E f f e c t Of P r e i o n i z a t i o n On The S t a b i l i t y 98 A.2 - S t a b i l i t y Constant V a r i a t i o n With Coil. Length 101 A. 3 - Value Of B j j S i n ^ For D i f f e r e n t Gas Pressures (Argon) 103 A.4 - Comparison Of The S t a b i l i t y Constant Of Argon And A i r 104 v i i i APPENDIX A.5 - Top-On Photograph Of Plasma At 2 0 ( J M H g Pressure ( A i r ) 106 A.6 - End-On Photograph At 200/AHg Pressure ( A i r ) 107 A.7 - Top-On Photograph At 100/tfIg Pressure ( A i r ) 107 A.8 - E q u i v a l e n t C i r c u i t Of Magnetic Probe 108 A.9 - Damped Probe C i r c u i t 109 ix-ACKNOWLEDGMENTS I wish to thank Dr, F.L. Curzon f o r his encouragement, helpful c r i t i c i s m s and supervision of the entire project and the Ph.D. Committee, Dr. R.A, Nodwell, Dr. A,J. Barnard and Dr. R. Howard, for t h e i r constructive c r i t i c i s m s i n the writing of t h i s t h e s i s . I would l i k e to express my appreciation to Dr. R.J. Churc h i l l f o r guidance i n the construction of the theta pinch apparatus. The help of the technical s t a f f , W, Ratzlaff and J.H, Turner i n the f i e l d of e l c t r o n i c s , A. Fraser and his s t a f f i n the machine shop work, J. Lees i n the glass work, are a l l g r a t e f u l l y acknowledged. To my colleagues go my sincere thanks f o r t h e i r helpful suggestions, p a r t i c u l a r l y to C.C, Daughney f o r pointing out the two current rings i n the photographs. Lastly, I wish to thank the Atomic Energy Control Board of Canada f o r f i n a n c i a l l y supporting the work. CHAPTER I INTRODUCTION The t h e t a p i n c h has been e x t e n s i v e l y i n v e s t i g a t e d because i t can generate r e l a t i v e l y pure, high temperature plasmas ( T ^ I O 5 °K) having high e l e c t r o n d e n s i t i e s (n ~ 1 0 l 6 cm" 3). To generate the plasma, a l a r g e current i s discharged through a c o l l surrounding a c y l i n d r i c a l discharge tube. The r e s u l t a n t azlmuthal e l e c t r i c f i e l d induced i n the discharge gas generates an e l e c t r i c current r e s u l t i n g i n the formation of a plasma c y l i n d e r . For short c o i l s the plasma tends to form a r i n g discharge. The discharge v e s s e l i s sometimes immersed i n a constant magnetic f i e l d ( i . e . b i a s magnetic f i e l d ) i n order t o improve the breakdown c h a r a c t e r i s t i c s of the discharge gas and to s t a b i l i z e the plasma. From the geometry of the e l e c t r i c and the magnetic f i e l d s It can be seen that the plasma i s compressed r a d i a l l y , that i s , a t h e t a p i n c h occurs. (The " t h e t a " s p e c i f i e s the azimuthal d i r e c t i o n of the current flow i n the plasma) <> A f t e r the plasma has compressed to a minimum r a d i u s , i t experiences r a d i a l o s c i l l a t i o n s due to the pressures exerted by the magnetic f i e l d s w i t h i n and e x t e r n a l to the plasma ( N i b l e t t and Green, 1 9 5 9 ) . - 2 -I I = Plasma current P B c = Theta c o i l magnetic f i e l d Bp = Plasma magnetic f i e l d © = Azimuthal angle P i g , I . l o Cross S e c t i o n Of The Theta Pinch - 3 -I t has been observed (W, C i l l i e r s et al,„ 19^3) that under c e r t a i n experimental c o n d i t i o n s , a f t e r the plasma has pinched, the magnetic f i e l d enclosed by i t suddenly disappears as may be detected by a sm a l l search c o i l I nserted along the a x i s of the discharge vessel„ When the i n t e r n a l magnetic f i e l d vanishes, the r a d i a l o s c i l l a t i o n s of the plasma a l s o disappear, The I n v e s t i g a t i o n of t h i s disappearance i s of considerable p r a c t i c a l i n t e r e s t i n thermonuclear research work because i t Indicates that the plasma confinement i s u n s t a b l e , that i s , the plasma has a short l i f e t i m e . In t h i s t h e s i s , i t i s shown that the disappearance of the o s c i l l a t i o n s can be a t t r i b u t e d to the r o t a t i o n of the plasma r i n g about a diameter,. The r o t a t i o n i s due to the I n t e r a c t i o n of the plasma current w i t h a superimposed transverse magnetic f i e l d which induces a x i a l motions. The transverse f i e l d need not be l a r g e and may be produced by an inhomogeneous t h e t a c o i l f i e l d . The combination of the r a d i a l and a x i a l motion of the plasma causes the plasma r i n g to r o t a t e about I t s diameter. As the plasma r o t a t e s , the magnetic f i e l d can no longer confine i t w i t h the r e s u l t that the plasma and the enclosed f l u x are d i s s i p a t e d . This r o t a t i o n was f i r s t observed by Clarke and Wuerker (1962}, and they r e f e r r e d to i t as the " f l i p i n s t a b i l i t y " They used an image converter camera to take successive p i c t u r e s of the r o t a t i o n . They d i d not use a b i a s magnetic f i e l d t o i n f l u e n c e the plasma. B a r t o l i and Green ( 1 9 6 4 ) gave a t h e o r e t i c a l model of the f l i p , t r e a t i n g the plasma as a r i g i d magnetic d i p o l e i n a reverse magnetic f i e l d , (see P i g o I o 2 0 ) o B c = Theta C o i l Magnetic F i e l d Bp « Plasma Current Magnetic F i e l d Brp = Transverse Magnetic F i e l d I = Plasma Current P F l g o I o 2 „ Top. View Of Plasma F l i p To the author's knowledge, no q u a n t i t a t i v e experimental s t u d i e s of the f l i p i n s t a b i l i t y have been madec This i s p r i m a r i l y because the plasma tends to r o t a t e about d i f f e r e n t diameters i n successive discharges making i t extremely d i f f i c u l t to measure the r a t e of r o t a t i o n of the plasma r l n g 0 -5-In the present work, the plasma Is given a p r e f e r e n t i a l a x i s of r o t a t i o n by a p p l y i n g a small t r a n s v e r s e magnetic f i e l d . An a x i a l motion of the plasma i s caused by the i n t e r a c t i o n of the plasma current w i t h t h i s t ransverse f i e l d and the plasma r o t a t e s about an a x i s perpendicular to the t h e t a and transverse f i e l d s . I f the a x i a l v e l o c i t y i s s m a l l , the r a d i a l v e l o c i t y can be l a r g e enough f o r r a d i a l o s c i l l a t i o n s t o develop. However, as the a x i a l v e l o c i t y i n c r e a s e s , the o s c i l l a t i o n s w i l l be suppressed because the plasma w i l l escape out through the ends of the c o i l before r a d i a l o s c i l l a t i o n s can develop. In t h i s t h e s i s the o s c i l l a t i o n s are detected by a magnetic probe and the suppression of o s c i l l a t i o n Is chosen as the c r i t e r i o n f o r the occurrence of the f l i p i n s t a b i l i t y . The occurrence of f l i p i s determined as a f u n c t i o n of the f o l l o w i n g parameters? the d e n s i t y of the discharge gas ( a i r ) , the s t r e n g t h and the d i r e c t i o n of the b i a s magnetic f i e l d 9 and the r a d i u s of the discharge tube. I t i s found that the f l i p i n s t a b i l i t y occurs whenever -J^k- > 0.02. ^t b en <*> « Where "Bm" i s the transverse magnetic f i e l d , "a" i s the r a d i u s of the discharge v e s s e l , and "p " i s the d e n s i t y of the gas. The a x i a l and r a d i a l equations of motion of the plasma are d e r i v e d and solved approximately and p r e d i c t that I f > i s J greater than a given constant, the f l i p i n s t a b i l i t y w i l l occur, i n agreement w i t h the experimental r e s u l t s . In the development of the a x i a l equation of motion, a two current model, of the plasma i s used (T.S 0 Green, 1962)0 One ourrent maintains the magnetic f l u x trapped by the plasma and the second current i s induced by the t h e t a c o i l magnetic f i e l d . The two currents flow i n opposite azimuthal directions,, The v a l i d i t y of t h i s model i s supported by framing camera photographs of the discharge i n a transverse magnetic f i e l d . In the absence of the transverse f i e l d , the magnetic f o r c e s a c t i n g on the current w i l l -be i n opposite r a d i a l d i r e c t -ions and the current sheets are t h e r e f o r e compressed together, rendering them i n d i s t i n g u i s h a b l e from each other. However, a transverse magnetic f i e l d , moves the current r i n g s In opposite a x i a l d i r e c t i o n s enabling the r i n g s to be d i s t i n g u i s h a b l e from each other. Although the t h e o r e t i c a l model i s only a crude r e p r e s e n t a t i o n of the a c t u a l plasma, the agreement w i t h the experimental r e s u l t s i n d i c a t e that the model i s a reasonable approximation,, A summary of the o r i g i n a l c o n t r i b u t i o n s of t h i s t h e s i s Is given i n the a b s t r a c t . The apparatus used i n t h i s work I s discussed i n the f i r s t p art of Chapter I I and the d i a g n o s t i c techniques are discussed i n the l a t t e r part of Chapter I I . The experimental r e s u l t s are presented i n Chapter I I I , together w i t h the t h e o r e t i c a l i n t e r p r e t a t i o n of the r e s u l t s . -7-CHAPTER I I EXPERIMENTAL APPARATUS This chapter i s d i v i d e d i n t o three s e c t i o n s which c o n t a i n d e s c r i p t i o n s of: Sect i o n A - Theta Pinch Apparatus, S e c t i o n B - Bias Magnetic F i e l d Apparatus, S e c t i o n C = Diagnostic Equipment 0 S e c t i o n A - Theta Pinch Apparatus A t h e t a p i n c h Is produced by r a p i d l y d i s c h a r g i n g a l a r g e c a p a c i t o r bank through a copper c o i l that i s wound around a glass tube. The r e s u l t a n t changing magnetic f l u x produces an azimuthal e l e c t r i c f i e l d i n the gl a s s tube. I f the gas.pressure i s a p p r o p r i a t e , the gas i s i o n i z e d and forms a plasma cylinder„ The plasma current flows In the azimuthal d i r e c t i o n and i n t e r -a c t s w i t h the a x i a l magnetic f i e l d of the t h e t a c o i l 0 Considering the d i r e c t i o n of the magnetic f o r c e s (see Fig„ I 0 1 0 ) 9 i t i s seen that the plasma s h e l l c o n t r a c t s r a d i a l l y g i v i n g a pin c h effect,, The plasma l o s e s energy r a p i d l y through r a d i a t i o n , thermal conduction, e t c To produce and maintain a hot plasma i t i s th e r e f o r e necessary to provide a l a r g e amount of energy i n a short time t o o f f s e t these energy losses„ To achieve the necessary high Input power, a l a r g e c a p a c i t y 9 high v o l t a g e c a p a c i t o r bank i s employed,, The input power i s optimized by minimizing the inductance i n the discharge circuit„ The c h a r a c t e r i s t i c s of the t h e t a p i n c h apparatus used i n t h i s work ares Charging P o t e n t i a l 1 0 . 4 kv 1 3 o 3 kv 1 5 . 3 kv T o t a l Capacitance - 2 5 jut T o t a l Inductance - 1 7 0 nH -M a t e r i a l of Discharge Vessel - Pyrex Glass Length of Discharge V e s s e l - 6 9 cm -Inner Diameter of Discharge Vessel 3 . 4 cm 5 . 1 cm 7 . 1 cm Discharge P e r i o d - 1 3 ynsec -Peak Magnetic F i e l d — 2 . 8 webers/m 2 The e s s e n t i a l components of the t h e t a c a p a c i t o r bank are depicted i n F i g , I I . l , Q Theta C o l l = Spark Gap Switch C^ =a C a p a c i t o r Bank F i g . I I . 1 . Theta Capacitor Bank - 9-ISlectrodes f1 h Brass Can T r i g g e r P i n Polythene I n a u l a t o r Bakeli.te I n s u l a t o r A n t i Tracking Grooves Perspex F l g o I I * 2 „ Cross S e c t i o n a l View O f Spark Gap Switch, Sj^ 10-The spark gap s w i t c h , 3^t (see F i g . I I o 2 0 ) i s t r i g g e r e d by a p p l y i n g a f a s t r i s e time, high v o l t a g e pulse t o a t r i g g e r p i n . The c a p a c i t o r bank, C^, i s then discharged through the spark gap s w i t c h , S^, i n t o the t h e t a c o i l . The command s i g n a l i s provided by a pulse generator (Theophanis, I960), which produces an 18 kv pulse w i t h a 40 nsec r i s e time. This generator can be activated:, manually or by a low volt a g e p u l s e . In p r i n c i p l e the high v o l t a g e pulse produced by the Theophanis u n i t could be used to t r i g g e r the t h e t a c o i l spark gap. However, t h i s p ulse i s too weak f o r r e l i a b l e o p e r a t i o n and a l s o the t r i g g e r u n i t i s not i s o l a t e d from the t h e t a bank c u r r e n t . To avoi d these d i f f i c u l t i e s an u l t r a v i o l e t (uv) t r i g g e r system i s used. The uv t r i g g e r c o n s i s t s of a c a p a c i t o r , C 2, a spark gap, S 2, and a r e s i s t o r , R 2 , a l l connected i n s e r i e s (see F i g . I I o 3 o ) 0 Tungsten Electrodes / Quartz Bulb B a k e l i t e S 9 = Spark Gap Switch OUT c 2 = 0 o 0 5 ^ F R 2 = 50 XL-F i g . I I . 3 . U l t r a V i o l e t ' T r i g g e r Generator, -11-The pulse from the Theophanls u n i t i s sparked across two tungsten e l e c t r o d e s contained i n a quartz bulb mounted near the elec t r o d e s of Sg (see F i g , I I , 3 « ) o The spark produces photons that pass through the quartz tube and cause a breakdown of the spark gap Sg, The ca p a c i t o r , Cg, discharges and a vo l t a g e pulse i s produced across Rg, This pulse i s used to t r i g g e r the t h e t a spark gap, S^, The uv t r i g g e r system e f f e c t i v e l y a m p l i f i e s the energy a v a i l a b l e from the Theophanls u n i t from a f r a c t i o n of a j o u l e up to 3 j o u l e s o Also the operating equipment i s e l e c t r i c a l l y I s o l a t e d from the main discharge c i r c u i t by the quartz bulb, thus improving the s a f e t y of the equipment and reducing the e l e c t r i c a l n o i s e coupled from the high discharge current i n t o the measuring apparatus. The c a p a c i t o r , Cg? i s charged w i t h a p o t e n t i a l d i v i d e r across the t h e t a bank C^ (see F i g , I I I , 1 0 ) , The j i t t e r of the complete t r i g g e r i n g c i r c u i t i s l e s s than l/2yt/sec. More d e t a i l s of the t r i g g e r i n g c i r c u i t can be found i n the note by Curzon and Smy (1963), The t h e t a c a p a c i t o r bank c o n s i s t s of 5 C o r n e l l D a b l l l e r Model NRG 201 c a p a c i t o r s connected i n p a r a l l e l . Each c a p a c i t o r has a c a p a c i t y of S/A F and a maximum operat i n g p o t e n t i a l ' d i f f e r e n c e of 20 kvdc, A low Inductance geometry i s obtained by connecting the c a p a c i t o r s together w i t h a p a r a l l e l p l a t e t r a n s m i s s i o n l i n e c o n s i s t i n g of copper sheets 2mm t h i c k . The spark gap (see F i g , 11,2,) Is l o c a t e d at one end of the bank and Is connected In s e r i e s w i t h the bank and the t h e t a c o l l ( F i g , 11,1,), •12-The tr a n s m i s s i o n l i n e from the spark gap t o the t h e t a c o l l i s a p a i r of p a r a l l e l copper p l a t e s 2 mm t h i c k , 13 cm wide, and 100 cm lo n g separated by a 3 mm l a y e r of polyethylene and clamped together w i t h wooden clamps, The spark gap el e c t r o d e s are I n s u l a t e d from each other by a d i s c of perspex 0 The d i s c has been grooved t o Increase the l e n g t h of the t r a c k i n g path between the e l e c t r o d e s , thus decreasing the p o s s i b i l i t i e s of spurious breakdown ( P i g . 11,2,),' The p h y s i c a l dimensions of the t h e t a c o i l s are as fo l l o w s ? M a t e r i a l of Theta C o i l Copper Gauze C o i l Diameter 4,1 cm 6,1 cm 7,6 cm C o i l Length ^,6 cm 6,4 cm 7»6 cm The c u r r e n t , I » i n the t h e t a c o l l i s monitored w i t h a y c * Rogowski c o i l c o n s i s t i n g of a s o l e n o i d made from an 11 cm l e n g t h of delay cable (RG 65 AU) w i t h the outer ground s h i e l d removed,, A matching r e s i s t a n c e i s placed i n s e r i e s w i t h the c o l l t o dampen the resonance response of the c o l l , (For more d e t a i l s 9 see Appendix E j , The c o i l i s placed i n between the high current leads to the t h e t a c o i l . The output v o l t a g e of the Rogowski c o i l i s p r o p o r t i o n a l t o the r a t e of change of c u r r e n t , d i / d t . To o b t a i n the current an RC I n t e g r a t o r i s used (see P i g . 11 , 4 , ) , 13 Coaxial Cable To Oscilloscope .2 = R l = C = Matching Impedance, 220£2_ Terminating Resistance, 5Q£)~ Integratlng Resistance, 100K Integrating Capacitor, 1/4 F Fig, IIAo Rogowski C o l l C i r c u i t The vacuum system consists of a Precision Model 150 roughing pump and a Balzer Model 120 d i f f u s i o n pump. The base pressure of the complete vacuum system i s < 1 ju Hg measured with a Macleod Gauge,, The leak rate of the discharge v e s s e l Is ~'20/uHg per hour. 14 Section B - Bias Magnetic Field Apparatus The source of the bias magnetip field used In the experiment is a pair of Helmholtz coils. The coils are powered by two different capacitor banks. One bank is used to produce a large field ( 0 , 6 weber/m ), and it is connected to the Helmholtz coils through a spark gap switch. At low fields (^  0 , 1 weber/m ), this spark gap switch cannot be reliably triggered and this necessitates using a second bank which has a solid state triggering device. This extends the operating range of the Helmholtz field to lower values ( ~ 0 , 0 0 9 weber/m ), Section B,l - Large Field Bank The bank consists of 3 Cornell Dubilier (Model NRG 212) capacitors, 2 2 0 ^ F each and rated at 5 kvdc, which are connected ln parallel by copper plates 2 mm thick. The capacitors are connected to the Helmholtz colls through a spark gap switch which is located at one end of the bank. The spark gap switch (see Fig, I I 0 5 o ) is triggered by a high voltage pulse produced by amplifying the output of a thyratron pulse generator (see Fig, 1 1 , 6 ,) with a pulse transformer with amplification of ~ 33« A 660 ohm resistor (1 watt, carbon) is connected in series with the pulse transformer and the trigger pin (see Fig, I I , 7 » ) . This reduces the electrical noise signals produced by the trigger spark. The characteristics of the bank are given on Page 1 ? . - 1 5 -' Brass Electrodes ZZ1 \ \ \ \ \ \ 5/ mi / \ \ \ z v N S=3 Perspex I n s u l a t i o n Scale? Half Size F i g . I I . 5 . Helmholtz Spark Gap Switch 2 D 2 1 1 : 3 3 n t 8 K V . , 3 OUT ^ u l s l Transformer F i g . I I . 6 . T r i g g e r Pulse Generator ( F i r e s . When 'A0 i s Shorted t o Ground) -16 F l g o I I 0 7 o Helmholtz Bank T r i g g e r P i n -17-Capacitance 660 JA F Charging P o t e n t i a l 500 v 3000 v T o t a l Inductance (with c o i l ) 69/4H Peak Magnetic F i e l d Discharge P e r i o d I c 35 msec 0.6 weber/m Helmholtz Capacitor Bank The electrodes of the spark gap sw i t c h tend t o burn away q u i c k l y because of the long time constant of the c i r c u i t 0 I t i s necessary t o readjust the polythene i n s u l a t i o n (see F i g 0 I I , 7.) on the t r i g g e r p i n every shots,otherwise the spark gap does not have r e l i a b l e t i m i n g c h a r a c t e r i s t i c s . S e c t i o n B . 2 - Small F i e l d Bank The low f i e l d bank c o n s i s t s of ten (DCM 539=2657-01) c a p a c i t o r s r a t e d at 10,000 ^  F each, at 75vdc, This bank Is connected to the Helmholtz c o i l s through a s i l i c o n c o n t r o l l e d r e c t i f i e r (C80 Type 2 N 2 5 4 2 ) , which serves as a t r i g g e r s w i t c h . The maximum current produced by t h i s bank i s ~ 400 amps and the time r e q u i r e d f o r the c o i l current t o reach i t s maximum value i s ^ 2 msec, A t r a c e of the discharge wave form i s given i n F i g . I I . 8 , -18-Time Scale: lmsec/cm V e r t i c a l Scale: l60amp/cm F i g . I I . 8 . Current Waveform For The Small Magnetic F i e l d Capacitor Bank A c i r c u i t diagram i s given i n F i g . I I . 9 . Diode — Zener Diode ^ (70V) 110 A.C.: ' • - ^ Transformer Helmholtz C o i l T r i g g e r F i g . I I . 9 . Small Magnetic F i e l d Capacitor Bank C i r c u i t Diagram -19-S e c t l o n B.3 - Helmholtz C o l l The Helmholtz c o i l i s constructed as a p a i r of c o i l s , each having 19 turns of No, 2 AWG copper w i r e . The inner and outer diameters of c o l l are 25 cm and 28 cm respectively,, The c o i l frames are made of b a k e l i t e and wood, and brass rods are used to hold the frames together,, The angle between the a x i s of the Helmholtz c o i l and the a x i s of the t h e t a c o l l i s measured w i t h a l a r g e p r o t r a c t o r of 25 cm r a d i u s (see F i g . 11.10.). The f i e l d d i s t r i b u t i o n s along the axes AA' and BB 9 ( F i g , 11.10.) were measured by a s o l i d s t a t e 9 B e l l 240 Incremental Gaussmeter 9 and the r e s u l t s are given i n F i g . I I . 11. To ensure proper measurements, the gaussmeter probe i s r o t a t e d about i t s a x i s to l o c a t e the p o s i t i o n of maximum s i g n a l before the measurements are taken. 20-End View P i g . 11.10. Geometric P o s i t i o n Of The Helmholtz C o l l F i g . 11.11, Magnetic F i e l d D i s t r i b u t i o n Along Axis AA' and BB' Of Helmholtz C o l l s ( C o l l Current: 2.5 amp/coll) -22-Seotlon C - Diagnostic Equipment Two main d i a g n o s t i c techniques are used i n t h i s experiment. In the f i r s t , magnetic probes are employed t o measure the magnetic f i e l d s a s s o c i a t e d w i t h the plasma. In the second, a framing camera i s employed to measure the shape of the luminous plasma. S e c t i o n C . l . - Magnetic Probes A magnetic probe i s e s s e n t i a l l y a small search c o i l , (the small s i z e ( 1 mm) means that measurements of the magnetic f i e l d s can be made w i t h good s p a t i a l r e s o l u t i o n ) . An emf, V, i s produced between the ends of the c o l l by a change i n the magnetic f l u x through the c o i l . Thus a c o i l p laced i n a time dependent magnetic f i e l d which remains f i x e d i n d i r e c t i o n w i l l generate a s i g n a l p r o p o r t i o n a l to the r a t e of change of that magnetic f i e l d . I f t h i s s i g n a l i s i n t e g r a t e d w i t h an RC c i r c u i t ( see P i g . 1 1 , 1 2 . ) the r e s u l t a n t s i g n a l w i l l be p r o p o r t i o n a l t o the t o t a l change of the magnetic f i e l d B. Integrated S i g n a l R = Resistance C = Capacitance P i g , 11.12. RC I n t e g r a t i n g Network 2 3 ' When a s i g n a l i s i n t e g r a t e d , i t Is a l s o attenuated. The s i z e of the output s i g n a l and the d i s t o r t i o n both vary i n v e r s e l y w i t h the time constant, RC, of the i n t e g r a t i n g c i r c u i t . Therefore, the time constant must be small enough to produce a measureable s i g n a l , but s u f f i c i e n t l y l a r g e to cause n e g l i g i b l e d i s t o r t i o n i n the i n t e g r a t e d waveform. In t h i s experiment, RC i s chosen to be ten times the c h a r a c t e r i s t i c time of the measured event, ( i , e , RC = 10 ^ sec f o r magnetic probes). In the theory of the probe presented above, the inductance and the s t r a y capacitance of the c o i l i t s e l f have been Ignored, These produce a resonant response i n the probe s i g n a l . Therefore the g a i n of the c i r c u i t w i l l depend on the frequency of the s i g n a l measured. This i s an undesireable e f f e c t which can be removed by damping the resonant response of the probe w i t h a s e r i e s r e s i s t a n c e (Segre and A l l e n , I960), This r e s i s t a n c e i s r e f e r r e d t o below as the matching Impedance and I t s value i s equal t o where L i s the s e l f inductance of the probe and C i s the s t r a y capacitance (see Appendix E )., The frequency response of the "damped" probe has been measured by mounting i t i n s i d e a s o l e n o i d that i s connected to a s i g n a l generator. The v o l t a g e across the leads of the s o l e n o i d i s measured w i t h an o s c i l l o s c o p e and from t h i s , the r a t e of change i n magnetic f i e l d can be c a l c u l a t e d . The output s i g n a l from the magnetic probe i s recorded on the o s c i l l o s c o p e w i t h the c i r c u i t r y normally employed i n the plasma i n v e s t i g a t i o n . -2& The plasma current Is monitored by two types of magnetic probes. The f i r s t i s a s i n g l e a x i a l probe t o detect the r a d i a l hydfomagnetlc o s c i l l a t i o n s i n the plasma ( N l b l e t t and Green 9 1959)5 f o r f u r t h e r d e t a i l s see Chapter I I I , The second, i s a d i r e c t i o n a l probe to measure the r o t a t i o n of the plasma r i n g about i t s diameter. Each probe i s described below i n more d e t a i l . S e c t i o n C l a - Single, A x i a l Probe The s i n g l e a x i a l probe c o n s i s t s of 35 turns of t h i n copper w i r e , No, 44 (AWG) wound on a g l a s s rod ~ 1 mm i n diameter. Much care i s used i n winding t h e c o i l u n i f o r m l y and the c o i l Is he l d together by epoxy glue. The leads are t i g h t l y t w i s t e d together so that the net magnetic f l u x enclosed by the c o i l leads ( 1 B o dA ) Is n e g l i g i b l e In comparison t o the magnetic f l u x enclosed by the c o l l (see F i g , II 0 13o) , Copper C o i l F i g , II,13o A x i a l Magnetic Probe -25-F i g o II . l 4'3 Method Of Mounting Probe In The Theta P i n c h Discharge Vessel The probe rod i s Joined t o another gla s s rod of l a r g e r diameter f o r r i g l d i t y 0 The whole probe system i s then i n s e r t e d i n t o a g l a s s tube 75 cm long and 5 nan dia0..to. protect the. probe, from the plasmao The g l a s s tube i s supported by a f l e x i b l e j o i n t at one end of the discharge tube as shown i n F l g o I I * 1 ^ and i s p o s i t i o n e d i n the center of the t h e t a c o i l by a set of a d j u s t i n g screws on the electrode,, The probe leads are connected t o the o s c i l l o s c o p e by RG 58 c o a x i a l cable which i s terminated i n i t s c h a r a c t e r i s t i c Impedance so as to reduce r e f l e c t i o n s of the s i g n a l In the c a b l e 0 -26 The matching Impedance of the probe I s c a l c u l a t e d (see Appendix rO t o be 4-3SL . The t e r m i n a t i n g r e s i s t a n c e f o r the cable Is 50_O_ and t h i s r e s i s t o r t h e r e f o r e a l s o serves to match the impedance of the probe. The s i g n a l s are d i s p l a y e d on a Te k t r o n i x 5 5 1 double beam o s c i l l o s c o p e w i t h a Type G d i f f e r e n t i a l p r e a m p l i f i e r , and are recorded by a P o l a r o i d Land camera. The probe s i g n a l i s p r o p o r t i o n a l to ^ r ( I c+ I p ) where I c i s the c o i l and I p i s the plasma current per l e n g t h . To ob t a i n a s u i t a b l y attenuated s i g n a l from the Rogowski c o i l ( i . e . ^ j r P 9 see Page 13) Is s ubtracted from the magnetic probe s i g n a l . This procedure w i l l be r e f e r r e d t o below as the 'double probe technique', (Green, 1962). A p o t e n t i a l d i v i d e r i s placed across the output leads of the Rogowski c o i l and i s adjusted to o b t a i n the s u i t a b l y attenuated s i g n a l . Thus, 1'f the t h e t a bank i s discharged w i t h no r e s u l t a n t plasma formation, then there should be no net s i g n a l produced. To o b t a i n the plasma current I p , the probe and the Rogowski c o i l s i g n a l s are i n t e g r a t e d by an RC c i r c u i t (see F i g . I I . 4 „ ) . The c h a r a c t e r i s t i c time of the plasma current o s c i l l a t i o n i s ~ l y M s e c , thus the i n t e g r a t i n g time constant, RC, chosen i s lOyMsec. I t i s found t h a t the c a p a c i t a t i v e c o u p l i n g between the plasma and the magnetic probe leads to no recordable s i g n a l . This i s determined by r e p l a c i n g the normal probe by a s t r a i g h t wire and comparing the s i g n a l obtained w i t h the normal probe s i g n a l (see F i g . 11.15.). -27-4 : 1 1 ! I 1 1 1 1 1 f >. t 1 0 jysec Amplitude Of Probe S i g n a l For The Peak Theta C o i l Current = 3V F i g . 1 1 . 1 5 . C a p a c i t a t i v e Coupling S i g n a l From Magnetic Probe The frequency response of the a x i a l probe i s given i n F i g . 1 1 . 1 6 . S e c t i o n C.lb - D i r e c t i o n a l Probe In general i f the magnetic f i e l d v a r i e s i n d i r e c t i o n , 3 probes are needed t o measure the f i e l d completely. However, i f the plane of r o t a t i o n i s known, then only two probes are needed. For the experiments described below, a "V shaped probe has been used f o r t h i s purpose. Using the geometric r e l a t i o n s and the s i g n a l s from the two probes, the d i r e c t i o n and s t r e n g t h of the magnetic f i e l d i n the plane of the probe can be c a l c u l a t e d . The d i r e c t i o n a l probe c o n s i s t s of two c o i l s ( 2 0 turns) of No. 4 4 AWG copper w i r e wound on a 'V8 shaped g l a s s rod ^  1 mm i n diameter a n d 2 mm i n l e n g t h (see F i g . I I . 1 7 o ) o 0 o 3 V o l t s ° ° 2 0 . 1 O 10 9 8 7 6 4 Output Input j 2 -—f 1 1 1 1 1—r-t-.3 .5 1 ^ 1 — — — i 1 1 — i — i —t— t — 5 10 Frequency (Megacycles) F l g o I I 0 l 6 „ Frequency Response Of A x i a l Magnetic Probe (Integrated Output) 29-B = Plasma Magnetic p F i e l d F i g o II.17 . D i r e c t i o n a l Probe For convenience the response of both probes should be i d e n t i c a l f o r a given f l u x change through each probe because the probe s i g n a l s are q u a n t i t a t i v e l y compared w i t h one another. To s a t i s f y t h i s requirement, both probes are constructed w i t h i d e n t i c a l geometry and the output s i g n a l s are f i n e l y balanced by use of a p o t e n t i a l d i v i d e r on one of the probes. The stem of the d i r e c t i o n a l probe i s i d e n t i c a l to the stem of the s i n g l e a x i a l probe. By i n t e g r a t i n g the s i g n a l s from the two probes and u s i n g the double probe technique (Page 2 6 ) , the net s i g n a l s w i l l then correspond t o the plasma magnetic f i e l d , B „ To c a l c u l a t e the d i r e c t i o n of the magnetic f i e l d from the output s i g n a l s of the »V° shaped probes, i t i s necessary that the magnetic f i e l d be approximately uniform over the volume occupied by the two probes and be coplanar w i t h probes 1 and 2 (see F i g . I I . l ? . ) . Then the s i g n a l s r e c e i v e d from probe 1 ( ) and - 3 0 -probe 2 ( V 2 ) are V± = V Q c o s ( | +& ) (1) = V0(cos^oos6 - s l n j s i n 6 ) and V 2 = V 0 c o s (f - S ) ( 2 ) = V Q ( C O S - | C O S ^ H- s i n ^ s i n & ) , where ft i s the angle between the axes of the two probes and o" i s the angle between B p and the t h e t a c o i l a x i s (see P i g . 11.17.). By combining these two s i g n a l s and l e t t i n g tf= 90°, then _VL — C 4 5 ° cos £ - sin ^".sln^i /o \ = c o s & - ,c.m& Therefore, knowing the values of and V 2 as a f u n c t i o n of time, the angle o can a l s o be determined as a f u n c t i o n of time. The frequency response of each probe i s given i n F i g . 11.18. (For the experimental procedure, see Ghgptee'.III, S e c t i o n B . 4 ) . S e c t i o n C.2 - High Speed Framing Camera The B a r r and Stroud Model C P 5 high speed r o t a t i n g m i r r o r framing camera can photograph 60 frames at a maximum r a t e of 8 X 10 frames/sec. I t c o n s i s t s e s s e n t i a l l y of an o b j e c t i v e l e n s , (see F i g . 11.19.) and a ma g n i f i e r l e n s , L 2 , which focuses the image on the surface of the r o t a t i n g m i r r o r . A le n s quadrant, L^, focuses the m i r r o r surface on the curved f i l m t r a c k . The p o l i s h e d s t a i n l e s s s t e e l m i r r o r runs i n f l e x i b l e mounted white metal bearings. I t i s d r i v e n at a maximum r a t e of 330,000 r.p.m. by an a i r t u r b i n e operated at 4 0 p s l g . - 3 1 -^ 0 2 . 0 1 l . o j 0 . 9 0 . 8 0 . 7 0 . 6 -0 . 5 -0 . 4 O . 3 l H • — i -2 . 0 l l . o -0 . 9 : 0.8-0.7-0.6-0.5 0.4 0 . 3 1 - I—I—t --I h - 1 — 1—H-0 . 5 1 . 0 Frequency Megacycles (a) Probe 1 5 . 0 1 0 . 0 H 1 H—I 1 I—h 0 . 5 l . o H 1 1 — 1 — e n -(b) Probe 2 5 . 0 10.0 F i g . 11.18. Frequency Response Of D i r e c t i o n a l Probe -32-PM2 £ i TRIGGER V V D E L A Y S Y N C H P U L S E = Objective Lens Lg = M a g n i f i e r Lj = Lens Quadrant F i g . I I 8 1 9 o Framing Camera Schematic The event t o be photographed must be synchronized w i t h the p o s i t i o n of the r o t a t i n g m i r r o r , since r e c o r d i n g can only take place during 1 5 ° of each h a l f r e v o l u t i o n of the m i r r o r . A pulse to i n i t i a t e the t h e t a p i n c h discharge i s provided by one of the two p h o t o m u l t i p l i e r s , PM 1 or PM 2. Light from a 6 v o l t lamp i s r e f l e c t e d from a part of the m i r r o r surface and d i r e c t e d by a prism to the p h o t o m u l t i p l i e r , so i t gives pulses synchronized w i t h the m i r r o r p o s i t i o n . The m i r r o r speed Is monitored by a frequency meter which r e c e i v e s a s i g n a l from a search c o i l adjacent t o a magnetized c o l l a r on the m i r r o r s h a f t . - 3 3 -CHAPTER I I I RESULTS In t h i s chapter, the experimental measurements of the dynamic behaviour of the plasma w i t h and without a b i a s magnetic f i e l d are presented and discussed. The c h a r a c t e r i s t i c s of the magnetic probe s i g n a l s and the framing camera p i c t u r e s d e s c r i b i n g the plasma w i t h no b i a s f i e l d are s t u d i e d . Then by comparing these c h a r a c t e r i s t i c s w i t h those f o r a discharge i n a b i a s f i e l d , the e f f e c t of the b i a s f i e l d on the plasma can be obtained, A t h e o r e t i c a l model i s then constructed to a s s i s t i n the i n t e r -p r e t a t i o n of the r e s u l t s . The p r e d i c t i o n s of t h i s model are compared w i t h the experimental r e s u l t s and the l i m i t a t i o n s of the model are discussed. S e c t i o n A - Discharge Of Theta Bank With No Bias F i e l d  Experimental Procedure The discharge tube i s evacuated t o < l / ^ H g pressure and then f i l l e d w i t h a i r to the r e q u i r e d pressure as measured by a .4. Macleod Gauge (- 5%)° The discharge tube i s then i s o l a t e d from the pumping system by a v a l v e and the t h e t a bank i s charged to the r e q u i r e d p o t e n t i a l . Subsequently the Theophanous pulse generator i s t r i g g e r e d (manually f o r the magnetic probe experiments, or by p h o t o m u l t i p l i e r pulses f o r the framing camera experiments), thus c l o s i n g the t h e t a spark gap s w i t c h (see Page 8 ) . 330k 20 kv supply 1» R2 ~ P o t e n t i a l D i v i d e r C l = 25/vF c 2 = 0 . 0 5 ^ S l = Theta Bank Spark Gap s 2 = U l t r a V i o l e t T r i g g e r F i g . I I I . l . Schematic Of Theta P i n c h Discharge - 3 5 -and d i s c h a r g i n g the c a p a c i t o r bank. Using the Integrated Maxwell's equation, the s t r e n g t h of the azimuthal e l e c t r i c f i e l d induced i n the discharge tube v a r i e s i n the f o l l o w i n g manners ( 1 ) E = ^ f f j e , where r = ra d i u s from the a x i s I = t h e t a c o i l current per u n i t l e n g t h c For c o l l currents I which vary s i n u s o i d a l l y w i t h time, c •ggc (and t h e r e f o r e E) i s maximum when the current goes through zero. Hence, w i t h appropriate c o n d i t i o n s , a plasma c y l i n d e r w i l l be formed when the t h e t a c o i l current I , goes through zero. c I f the plasma forms before the t h e t a current r e v e r s e s , (see F i g . I I I . 2 . P o s i t i o n B) the plasma current i s compressed t o the w a l l s of the discharge v e s s e l by the magnetic f o r c e s . Hence, before r a d i a l c o n t r a c t i o n begins, the plasma s h e l l has a d e f i n i t e mass (see F i g . I I I . 3 o ) . When no b i a s magnetic f i e l d I s used, the gas i n i t i a l l y breaks down at the end of the f i r s t h a l f c y c l e of the c o i l current (see F i g . I I I . 2 , P o s i t i o n B). As soon as the magnetic f i e l d e x t e r n a l t o the plasma becomes l a r g e r than the magnetic f i e l d w i t h i n i t , the plasma s h e l l c o n t r a c t s r a d i a l l y and then executes r a d i a l o s c i l l a t i o n s about an e q u i l i b r i u m p o s i t i o n (see F i g . I I I . 2 . P o s i t i o n D). Further d e t a i l s of these o s c i l l a t i o n s (which have been discussed by N i b l e t t and Green, 1 9 5 9 ) are given i n the next s e c t i o n (Page 6 8 ) . -36-F i g 0 I I I 0 3 „ Formation of The Plasma S h e l l 37 As the t h e t a current approaches i t s maximum value ( F i g . i l l . 2. P o s i t i o n E) the e l e c t r i c f i e l d goes t o zero and the plasma d i s s i p a t e s . When the t h e t a c o i l current again approaches zero ( F i g . I I I . 2 . P o s i t i o n F) the breakdown c y c l e i s r e i t e r a t e d . This phenomenon recurs at each h a l f c y c l e of the c o i l current u n t i l the peak induced e l e c t r i c f i e l d i s too small t o cause a break-down i n the gas. A magnetic probe placed on the a x i s i n the t h e t a c o l l records the r a t e of change i n the magnetic f i e l d s t r e n g t h w i t h i n the c o l l cross s e c t i o n . The i n t e g r a t e d s i g n a l (see Page SB ) corresponds t o the change l n the magnetic f i e l d . In the absence of a plasma, the probe records the change i n the th e t a c o i l magnetic f i e l d . However, when the e l e c t r i c c u r rents flow i n the plasma, these produce magnetic f i e l d s which are superposed on the t h e t a c o i l f i e l d s . From the f l u x conservation theorem 8 that i s , \B.dA = constant,the t o t a l magnetic f l u x enclosed i n a plasma r i n g i s constant w i t h respect to the r a d i a l motion. Consequently, when the radius of the plasma r i n g c o n t r a c t s , the s t r e n g t h of the enclosed magnetic f i e l d i n c r e a s e s . S i m i l a r l y , when the plasma expands r a d i a l l y , the enclosed magnetic f i e l d s t r e n g t h decreases. Thus, when plasma executes r a d i a l o s c i l l a t i o n s , the corresponding o s c i l l a t i o n s i n the magnetic f i e l d are recorded on the magnetic probe waveform. A t y p i c a l i n t e g r a t e d probe s i g n a l i s given i n F i g . I I I . 4 . - 3 3 -Time - 2 u sec/cm Pres. - 50 /^Hg, Gas-Air ' (upper waveform) P i g . I I I . 4 . Integrated Magnetic Probe S i g n a l The corresponding framing camera p i c t u r e s of the discharge show t h a t , at breakdown, a plasma sheet i s formed at the w a l l s of the discharge tube. This then c o l l a p s e s r a d i a l l y t o the e q u i l i b r i u m p o s i t i o n about which i t executes r a d i a l o s c i l l a t i o n s . The r a d i a l o s c i l l a t i o n s are not c l e a r l y seen because the time r e s o l u t i o n of the camera i s approximately 1/2 the p e r i o d of r a d i a l o s c i l l a t i o n . Thus, the o s c i l l a t o r y r a d i a l motions are smeared out. Framing camera p i c t u r e s have been taken from the ends of the c o i l (see F i g . I I I . 5 . ) and from the top of the c o i l ( F i g . I I I . 6 . ) . There i s no evidence of a x i a l motion. - 3 9 -Time Increases L e f t to Right, Frame P o s i t i o n s are Staggered, Pres,'50yuHg A i r , Exposure Time ^  0 . 3 5 ^ 8 6 0 . F i g . I I I . 5 . End-On Photograph (Bg=0) Pres: 75/1 Hg F i g . I I I . 6 . Top-On Photograph (Bg=0) -40-Since the c o i l dimensions and charging p o t e n t i a l of the t h e t a bank are not changed i n the experiments described between pages 39 and 68, they are s t a t e d below and are r e f e r r e d t o wherever the inform a t i o n i s r e q u i r e d , l . e . In f i g u r e s . Standard Discharge Conditions Theta c o i l diameter 6.1 cm Theta c o i l l e n g t h 6.4 cm Charging p o t e n t i a l of the t h e t a bank 13°3 kv Capacity of t h e t a bank 25 yuf Peak current i n the t h e t a bank 165 kamp Pe r i o d of the t h e t a c o i l current 13.0ynsec Sec t i o n B - Theta P i n c h With A Bias F i e l d In the presence of a transverse magnetic f i e l d (see F i g . I I I . 3 . ) , a x i a l Lorentz f o r c e s e x i s t and these tend t o d r i v e the plasma out through the ends of the c o l l . The combination of the a x i a l and r a d i a l motions causes the plasma to appear t o r o t a t e ( i . e . f l i p ) about a diameter s p e c i f i e d by the d i r e c t i o n B„ x B 4 , where B„ i s the b i a s f i e l d and B i s the t h e t a —H — c o i l —H — c o l l c o i l f i e l d . I f the a x i a l v e l o c i t y i s low, r a d i a l o s c i l l a t i o n s can develop. However, as the a x i a l v e l o c i t y increases the plasma escapes out through the ends of the c o l l before the o s c i l l a t i o n s can develop, i . e . o s c i l l a t i o n s are suppressed. The b i a s f i e l d B^ i s produced by d i s c h a r g i n g a c a p a c i t o r bank through the Helmholtz c o i l s . Although the b i a s magnetic f i e l d o s c i l l a t e s ( p e r i o d i s ^ 1 . 3 msec.), the p e r i o d i s l o n g - i H -compared to the l i f e time of the discharge plasma ( ^ 2 - 3 / ( S e c ) , Hence f o r the duration of the discharge, the bias magnetic f i e l d s are approximately constant. In practice, the experiments are performed by f i r i n g the discharge when Bg i s at i t s maximum value. To achieve the proper t r i g g e r i n g of the theta and Helmholtz banks, two di f f e r e n t methods are employed. The method used i n the magnetic probe experiments i s r e l a t i v e l y simple and i s discussed f i r s t . This w i l l be followed with the method employed i n the framing camera experiments. Magnetic Probe System AMPLIFIER SPARK 6AP HELMHOLTZ CAPACITOR BANK TRIGGER' DE LAN-UNIT PULSE AMPLIFY U V TRI6GER SPAR< SAP CAPACITOR, R-, , R c Potential Divider Fi g . III. 7 . Bias F i e l d Trigger C i r c u i t For The Magnetic Probe - 4 2 -For the magnetic probe measurements, the Helmholtz bank i s t r i g g e r e d manually and a p o t e n t i a l d i v i d e r across the leads of the Helmholtz c o i l generates a s i g n a l which a c t i v a t e s the delay u n i t ( F i g . I I I . 7 . ) . Then at a s p e c i f i e d delay time, l e , when the Helmholtz f i e l d i s maximum, the delay u n i t emits a pulse which t r i g g e r s the t h e t a bank. Framing Camera System In the framing camera experiments the method f o r t r i g g e r i n g i s more complicated because the t h e t a bank must be discharged when the r o t a t i n g m i r r o r i s at a given p o s i t i o n (see Page 3 2 ) and the Helmholtz f i e l d i s maximum. This i s accomplished i n the f o l l o w i n g way, (see F i g . I I I . 8 . ) . When the framing camera r o t o r i s running at the r e q u i r e d speed, the output pulse from one of the p h o t o m u l t i p l i e r s t r i g g e r s the Helmholtz bank. The same pulse a l s o t r i g g e r s a delay u n i t which i s adjusted t o give an output pulse when the Helmholtz f i e l d , B H, i s at i t s maximum value. The delayed pulse i s f e d i n t o a gate c i r c u i t together w i t h the p h o t o m u l t i p l i e r pulses from the framing camera. Hence the f i r s t p h o t o m u l t i p l i e r pulse a r r i v i n g a f t e r B^ . has reached i t s peak , valu e , t r i g g e r s the t h e t a p i n c h discharge. Consequently, the framing camera records photographs of a t h e t a p i n c h discharge which has been produced i n an e s s e n t i a l l y constant b i a s f i e l d , B H. In order to f i r e the t h e t a bank at peak values of B H, the tnis an intE^raV'. m u l t i p l e of m i r r o r speed I s s e l e c t e d so thatjjbhe time " i n t e r v a l Between the p h o t o m u l t i p l i e r pulses : ':/.r (where - 4 3 -PjHQTOMULTtPUER© , MANUAL HELMHOLTZ TRIGGER AMPLIFIER-:DELAY CIRCUIT • ® • 'AND' CIRCUIT © f THETA TRIGGER. © S i g n a l s At S p e c i f i e d P o i n t s •tM -1 Helmholtz F i e l d Waveform F i g . I I I . 8 . B i a s F i e l d T r i g g e r C i r c u i t Used For The Framing Camera - 4 4 -t M i s the time r e q u i r e d f o r the b i a s f i e l d to reach i t s maximum value. B i a s Magnetic F i e l d Strength The s t r e n g t h of the Helmholtz f i e l d at the time of the discharge can be c a l c u l a t e d by superposing a s i g n a l corresponding to the time of the discharge of, the t h e t a bank, on the Helmholtz vo l t a g e waveform. By c a l c u l a t i n g the current i n the Helmholtz c o i l s at t h i s time, assuming a simple LC discharge, the corresponding magnetic f i e l d at t h i s time can be c a l c u l a t e d . The Helmholtz v o l t a g e waveform i s monitored by a p o t e n t i a l d i v i d e r across the Helmholtz c o i l ; the s i g n a l corresponding t o the discharge of the t h e t a bank i s given by the Rogowski c o i l s i g n a l . The experimental r e s u l t s i n t h i s s e c t i o n are d i v i d e d i n t o the photographic r e s u l t s and the probe measurement r e s u l t s . Since the photographic i n v e s t i g a t i o n i s mainly f o r q u a l i t a t i v e purposes, we d i s c u s s t h i s f i r s t , t r e a t i n g the two cases, 0 = 9 0 ° and 0^0° s e p a r a t e l y , where  %0 % i s the angle between the axes of the discharge v e s s e l and the Helmholtz c o i l . S e c t i o n B . l - Photographic Measurements a) 0 = 9 0 ° At breakdown, the plasma current begins t o flow i n a r i n g whose a x i s i s approximately p a r a l l e l t o the t h e t a c o i l a x i s , (see F i g . I I I . 9 . ) . The current r i n g then separates i n t o two current r i n g s which r o t a t e about an a x i s p a r a l l e l t o B c x B_H - 4 3 -and l n opposite d i r e c t i o n s . The outer r i n g then c o l l a p s e s r a d i a l l y whereas the inner current r i n g appears to have a n e g l i g i b l e r a d i a l displacement d u r i n g t h i s time, (see P o s i t i o n s 1 - 3 , F i g . I I I . 9 & . ) A f t e r the outer r i n g has c o l l a p s e d t o an e l l i p t i c a l shape from the end on photographs ( P o s i t i o n 3» P i g . I I I . 9 b . ) , the inner r i n g begins t o c o l l a p s e r a d i a l l y . The outer r i n g i s i d e n t i f i e d by the f a c t that i t absorbs the l i g h t r a d i a t e d by the inner r i n g The two current r i n g s r o t a t e i n opposite d i r e c t i o n s i n d i c a t i n g that the current flows i n opposite d i r e c t i o n s i n the two r i n g s . In t h i s case one r i n g should c o l l a p s e r a d i a l l y whereas the other should be pressed i n t o the w a l l of the discharge v e s s e l , ( J x B f o r c e s are i n opposite d i r e c t i o n s ) . This i s observed t o be t r u e . We suggest t h a t the two r i n g s correspond to the current l a y e r s proposed by N l b l e t t and Green, (1959). The outer one i s induced by the t h e t a c o i l f i e l d (B_) whereas the inner one i s induced by the conservation of the f l u x trapped i n the plasma r i n g . In t h i s model the two c u r r e n t s flow i n opposite azimuthal d i r e c t i o n s . Therefore, the t r a n s v e r s e f i e l d w i l l 'tear* the, r i n g a x i a l l y producing the two r i n g s which appear i n P i g . I I I . 9 " ' ' . From the N l b l e t t and Green model, we expect t h a t the outer r i n g w i l l c o l l a p s e t o the a x i s before the inner one does. This i s confirmed by our observations. The reason t h a t the inner r i n g c o l l a p s e s e v e n t u a l l y i s that the B_ f i e l d induces more current i n t h i s r i n g causing a net inward f o r c e on i t . 46-. Outer Ring b 0 End On View Of Plasma P i g . I I I . 9 . Sketch Of Plasma Photographs I B T = -0.020 weber/m 2, Gas:Air, Pres=50/vHg, 0 = 90° a. 2 O B T = +0.016 weber/m , Pres = 50yuKg, 0 = 90 b. F i g . I I I . 1 0 . Top On Photograph Of Plasma - 4 8 -b) 0 ^ 0 ° In t h i s case the d i r e c t i o n and the st r e n g t h of the b i a s f i e l d r a d i c a l l y a f f e c t s the breakdown of the gas 0 Breakdown occurs when the t o t a l a x i a l magnetic f i e l d i s approximately zero. This i s v e r i f i e d by u s i n g the magnetic probe measure-ments to c a l c u l a t e the magnetic f i e l d of the t h e t a c o l l , B c o i l ' at the time of the gas breakdown. I t i s found that when the b i a s f i e l d , Bg, i s p a r a l l e l to the i n i t i a l d i r e c t i o n of B c o i l , ( I . e . p a r a l l e l b i a s f i e l d ) , the plasma i s comparatively unstable to the f l i p I n s t a b i l i t y ( i . e . the plasma escapes from the c o i l ) . I f the b i a s f i e l d B>H, i s a n t i - p a r a l l e l to the i n i t i a l d i r e c t i o n of B ( i . e . a n t i - p a r a l l e l b i a s f i e l d ) , the plasma i s comparatively s t a b l e against the f l i p i n s t a b i l i t y . The gas breaks down at the end of the f i r s t h a l f c y c l e of the t h e t a c o i l c u r r e n t , I c , when p a r a l l e l b i a s f i e l d i s used. However, i f an a n t i - p a r a l l e l b i a s f i e l d of s u f f i c i e n t s t r e n g t h i s employed, the gas breaks down at the beginning of the f i r s t h a l f c y c l e of I c . Therefore, the breakdown of the gas can be enhanced or suppressed by u s i n g an a n t i - p a r a l l e l or p a r a l l e l b i a s f i e l d r e s p e c t i v e l y . The plasma dynamics appear to be q u a l i t a t i v e l y s i m i l a r to those observed when 0 = 90° (see F i g . I I I . 1 1 . ) . ( 0 * 0 ° ) , B T = 0.068 weber/m 2 Pres. = 50/ARS F i g , I I I . 1 2 . End On Photograph - A i r -50-The r a t e at which the plasma r i n g f l i p s , depends c r i t i c a l l y on the alignment of the t h e t a c o i l and b i a s f i e l d . When 0 - 0°, no f l i p i s observed f o r the experimental c o n d i t i o n s reported above, (see F i g . I I I . 1 2 . ) . However, a misalignment of 0 2° w i l l 9 f l i p " the plasma r i n g . The most d i s t i n c t e f f e c t of the b i a s f i e l d i s observed f o r the f i r s t breakdown of the discharge gas. Sometimes, the plasma r i n g d i s i n t e g r a t e s i n l e s s than 5 x 10 v^j sec. (see F i g . 111.14.) 0 ^ 0 ° Pres : 50// Hg - A i r F i g . I I I . 1 3 . End On Photograph of Plasma With No Transverse Bias F i e l d -51-B T = 0.04 weber/m Pres. = 100yMHg, Gas-Air F i g . 111.14. End On Photograph Of Plasma With Transverse Bias F i e l d S ection B.2a. - Probe Measurements (0^0°) The i n i t i a l breakdown at the end of the f i r s t h a l f c y c l e of the the t a current i s chosen f o r the probe i n v e s t i g a t i o n . Under these c o n d i t i o n s the i n f l u e n c e of the bias f i e l d on the , plasma dynamics i s most pronounced and the discharge gas i s purer. Breakdown o f the discharge gas i s achieved i n the presence of a bias f i e l d , B^, p a r a l l e l to the I n i t i a l d i r e c t i o n of the magnetic f i e l d , B„, produced by the t h e t a c o l l . c From the double probe system, (see Page 2 6 ) , hydromagnetic o s c i l l a t i o n s of the plasma are observed and as 0 i s increased, these o s c i l l a t i o n s are suppressed. The corresponding framing camera p i c t u r e s show that when the o s c i l l a t i o n s are auppressed, 5 2 -a plasma r i n g forms and disappears i n ^ 1 ^ /sec, i n d i c a t i n g that the plasma has d i s i n t e g r a t e d . The f l i p of the plasma i s not r e a d i l y observed photo-g r a p h i c a l l y under t h i s c o n d i t i o n (see F i g . 1 1 1 . 1 3 . ) , because the plasma moves too r a p i d l y . The f l i p i s c l e a r l y observed i n framing camera photographs when there are a l a r g e r number of o s c i l l a t i o n s recorded by the probe. However, the suppression of o s c i l l a t i o n s was chosen as a c r i t e r i o n f o r f l i p because i t i s a r e a d i l y d i s t i n g u i s h a b l e c h a r a c t e r i s t i c i n the probe waveforms, (see F i g . 1 1 1 . 1 5 . )<> The gas pressure i n the discharge v e s s e l and the s t r e n g t h of the b i a s f i e l d at the time of the t h e t a bank discharge are f i r s t set at s e l e c t e d v a l u e s . A f t e r each discharge the angle 0 i s v a r i e d i n one degree i n t e r v a l s over the range - 18° and the corresponding magnetic probe waveforms are recorded. In the angular r e g i o n where f l i p occurs, 0 i s v a r i e d by 1 / 2 ° i n t e r v a l s t o a s c e r t a i n more a c c u r a t e l y the a c t u a l value of 0 at which f l i p occurs, 0 , and r e f e r r e d to as the c r i t i c a l v a lue. c Keeping the pressure constant, the Helmholtz f i e l d i s v a r i e d and the value of 0 ( B u ) i s determined f o r v a r i o u s values of the b i a s C xl f i e l d s t r e n g t h , B R . The bias f i e l d istrength was v a r i e d from 2 2 2 o . l webers/m to 0 . 6 webers/m i n 0 . 0 5 webers/m i n t e r v a l s . The value of B H s i n # C ( B H ) as a f u n c t i o n of the f i e l d s t r e n g t h , B H , i s determined and found t o be a constant f o r a given pressure ( F i g . I I I . l 6 0 ) 0 The above procedure i s repeated -53-DIp Dt 0=OC 0=3C 0=4C 0=5° B J J = 0.6 weber/m' O s c i l l a t i o n Peak / Disappears Gas A i r 10 ju{ sec Pressure = lOO^Hg F l i p Occurs At 0=5° A x i a l probe s i g n a l s f o r changes i n s0'. C o i l dimensions and discharge c o n d i t i o n s are s p e c i f i e d on Page 40„ F i g . 111.15. 5^ B H (weber/m ) -- 0,6 0.2 Pressure A 25 /A Hg A i r • 50 /A Hg A i r O100 ^ Hg A i r Other c o n d i t i o n s are given on Page 40 -15 -10 •5 0 5 a. 0 Degrees 10 15 0.070 -BjjSin 0Q weber/m' O.035 --0 0.2 0.3 0.4 0.5 0.6 b. B H weber/m' F i g . III.16. Value of Bj^sln 0c For D i f f e r e n t Gas Pressures c. 0 Degrees 0 . 0 5 0 -r BjjSin 0 cweber/m^ 0 . 0 2 5 0 . 2 GT O--0--o--Q--O--o--©-0 : 3 0 . 4 0 o 5 -o 0 . 6 d. B J J (weber/m ) F i g . 1 1 1 , 1 6 . Value of BgSin 0C For D i f f e r e n t Gas Pressures f o r the f o l l o w i n g gas pressures? l6ywHg 9 25ynHg9 3 5 y U H g 9 50^UHg, 75/wHg, lOO^Hg, 125/KHg, 150/wHg, 2 0 0 / MHg, and 2 5 0 ^ Hg. I t i s found that f o r pressures between l6yu Hg and 75yWHg, the value of T_/2 BgSin 0C v a r i e s as (pressure) ' , (see F i g . I I I . i 7 . J 9 however, above 75/AHg, the r e l a t i o n Is no longer v a l i d . Thus f o r a given c o i l , B ^ l n 0 C = constant, which w i l l be P* r e f e r r e d to as the s t a b i l i t y constant. The experimental r e s u l t s that the value of BgSin 0C i s a constant f o r a f l i p at a given pressure confirms the hypothesis that the component of the b i a s f i e l d t r a n s v e r s e to the discharge a x i s i s the important parameter i n the f l i p i n s t a b i l i t y . For t h i s reason a d d i t i o n a l experiments have been performed i n which only a transverse f i e l d was used, ( i . e . 0 = 90°). S e c t i o n B.2b - 0 « 9 0 ° In these experiments the b i a s f i e l d has only a transverse component. F o l l o w i n g the procedure adopted above, the value of B H when f l i p occurs i s denoted as the c r i t i c a l v a l u e , ( B H ) C . To o b t a i n (BH)„ at a given pressure, the b i a s f i e l d s t r e n g t h i s xi c 2 2 v a r i e d from 0.004 webers/m to 0.024 webers/m In.increments of 0,002 webers/m a f t e r each t h e t a bank discharge. The corresponding magnetic probe waveforms are recorded and i n the region cf Hip tte magnet Ic f i e l d increment i s changed to 0 . 0 0 0 5 webers/m „ The above procedure i s repeated f o r the following gas pressures; l 6 / U H g , 25/uRsi 35/wHg, 50/6<Hg, 65^,Hg, 75/<Hg9 -57-For Discharge Conditions see Page 40 —I H — : 1 1 1 1 1 1 1 1-20 30 40 50 60 70 80 90100 150 Log Pressure y Hg F i g . III.17. V a r i a t i o n Of The C r i t i c a l gransverse F i e l d With Pressure 0 ) F i g . III.18 V a r i a t i o n Of The C r i t i c a l Transverse F i e l d With V a r i a t i o n In Pressure (0 = 9 0 ° ) 80 y^Hg, 100 ^ 4 Hg, and 150 /i/iHg. I t i s found that the c r i t i c a l 1/2 value of Bg v a r i e d as (pressure) ' w i t h i n experimental e r r o r f o r pressures between 16 Hg, to 75/MHg, however t h i s r e l a t i o n -ship i s not v a l i d f o r pressures greater than 75 yt/iHg (see P i g . I I I . 1 8 ) . S e c t ion B .3 - V a r i a t i o n In The Discharge Radius The e f f e c t on the s t a b i l i t y constant BgSin0 c l n the l i n e a r pressure r e g i o n (16 ^x. Hg t o 7 5 ^ ^ ) d u e t o a v a r i a t i o n of the diameter of the discharge tube has a l s o been s t u d i e d . The i n t e r n a l diameters of the tubes employed were 3.^ cm., 5.1 cm., and 7.1 cm. Theta c o i l s of constant aspect r a t i o were used and the charging p o t e n t i a l of the t h e t a bank were v a r i e d to keep the peak magnetic energy d e n s i t y approximately constant. The procedure described i n the previous Section B.2a i s used t o determine the value of B g S i n ^ at v a r i o u s pressures f o r each of the three tubes. An approximately p a r a l l e l b i a s f i e l d was used f o r these experiments so as to have r e p r o d u c i b l e breakdown c o n d i t i o n s . A summary of the experimental c o n d i t i o n s i s given below. -60-0.020J-B H s l n ^ c  0.015+ weber m ^ H g f 0.0104-0 .0051 A ' s Q / / / / / • / 3 cm 4cm 5cm 6cm Tube Diameter Gas - A i r Pressure • Hg o 36/iHg A 25^/iHg 7cm P i g . III.19. V a r i a t i o n In S t a b i l i t y Constant With Tube Diameter (Other Experimental Conditions Appear on Page 40) -61-Experimental Conditions For Varying Discharge Radius Inner Tube Diameter C o i l Diameter C o i l Length Bank Voltage 7.1 c i 7.6 cm 7.6 cm 15.3 - .5 kv 5.1 cm 6.1 cm 6.4 cm 13.3 - .5 kv 3.4 cm 4.1 cm 4.6 cm 10.4 - .5 lev The dependence of the s t a b i l i t y constant, • BH s i n^c, on the p"* ~ : diameter of the discharge i s shown i n F i g . I I I . 1 9 . I t was found experimentally that the plasma showed l e s s tendency to f l i p as the diameter of the t h e t a tube was increased; f u r t h e r , the s t a b i l i t y constant v a r i e s roughly l i n e a r l y w i t h the tube diameter. However, there appears a minimum c o i l l e n g t h , below which there are no magnetic o s c i l l a t i o n s under the present experimental c o n d i t i o n s . S e c t i o n B .4 - D i r e c t i o n a l Probe The d i r e c t i o n a l probes (see Chapter I I , S e c t i o n C . l b . ) , are devised to measure the angular r o t a t i o n of the plasma r i n g . A T e k t r o n i x 551 double beam o s c i l l o s c o p e w i t h two d i f f e r e n t i a l (Type G) p r e a m p l i f i e r s i s employed. On the upper beam, V , the voltages from the two probes ( F i g . I I I . 2 0 , ) are subtracted. Therefore, -62-(2) V u = V l - V 2 » where = fjr (cos & - s i n S )» V 2 = y=-(cos 8 + s i n 8 ), 6 = angle between the a x i s of the plasma r i n g and the t h e t a c o i l a x i s (see F i g . 11.17.). V 0 corresponds t o the s t r e n g t h of the plasma f i e l d , B , ( F i g , 11.17.) hr (3) Therefore, V y « -fz V Q sin6. B = Plasma Magnetic F i e l d F i g . 11.17. D i r e c t i o n a l Probe - 6 3 -1 2 3 5 Time - Sec F i g . I I I . 2 0 . V , - V ? Waveform ^ ; — \ 1 1 1-1 2 3 ^ 5 Time - y^Sec P i g . I I I . 2 1 . V, Waveform Probe 1 measures the component of the t o t a l magnetic f i e l d along the a x i s of probe 1. On the lower beam, V^9 the th e t a c o i l component of the magnetic f i e l d i s subtracted from the output of probe 1, i . e . double probe technique (see Page 26). Thus, measures the component of the plasma f i e l d , B p, along the a x i s of probe 1, (4) i . e . V T = ^ r - (cosS - sin& ). (seeFig. III.21) By combining equations (3) and (4) the expression f o r '£ ' i s obtained i n terms of the measured s i g n a l s , V u and V^, (5) i . e . & = a r c t a n v ^ z v Equation (5) i s only a p p l i c a b l e when the a x i s of r o t a t i o n of the plasma r i n g i s perpendicular t o the plane of the 8V° probe. In order to l o c a t e t h i s plane experi m e n t a l l y , the probe i s r o t a t e d about i t s stem (see F i g . I I . 1 ? . ) u n t i l a p o s i t i o n Is found f o r which V ^ V g , at a l l times f o r successive discharges,, In p r a c t i c e t h i s p o s i t i o n , © = 90°, was determined c t o w i t h i n -10°. For t h i s c o n f i g u r a t i o n , the plane of the °V* i s c l e a r l y p erpendicular to the plane c o n t a i n i n g and the P discharge a x i s . A f u r t h e r r o t a t i o n of 90° about the stem, t h e r e f o r e , brings the plane of the 'V i n t o the d i r e c t i o n r e q u i r e d f o r equation (5) to be v a l i d . Using the two probe s i g n a l s - 2.V * t h e ^ S 1 1 1 3 1 " n o t a t i o n of r e s u l t s agree q u a l i t a t i v e l y w i t h (see F i g , I I I . 2 3 ) . and equation (5)» a r c t a n the plasma i s c a l c u l a t e d . The the framing camera photographs, Lower Trace , Upper Trace V 2 -Probe Waveforms At &c + 1 0 ° Pres.- 50//Hg Gas - A i r L&wpair Traoe , Uppers Trace V 2 - V-^  Probe Waveforms At 4 - 10° Pres. - 50 ^ Hg c Gas - A i r F i g . III.22. -66-Pig.III.23. The Angular R o t a t i o n ' 6 ' -67-In summary, the s i g n i f i c a n t e f f e c t s of a b i a s magnetic f i e l d on the t h e t a p i n c h are as f o l l o w s \ Photography 1 . A trans v e r s e b i a s magnetic f i e l d causes a r a p i d f l i p of the plasma r i n g i n a t h e t a p i n c h discharge. 2. The two current model of the t h e t a p i n c h i s confirmed. Magnetic Probes 1. For a given a i r pressure, c a p a c i t o r bank energy, and discharge c o i l , i f BgSinjtf i s greater than a c r i t i c a l constant, f l i p occurs. i s the Helmholtz f i e l d and 0 i s the angle between the axes of the Helmholtz f i e l d and the t h e t a c o i l . 2. For a given discharge c o i l and c a p a c i t o r bank energy, BgSlnjtf, v a r i e s as p^ " where "p" i s the a i r pressure. This r e l a t i o n i s v a l i d i n the r e g i o n of 1 6 ^ Hg t o 75 ^//Hg a i r . 3. With t h e t a c o l l s of constant aspect r a t i o , BH g* n^o v a r i e s pi-as the r a d i u s of the discharge tube, provided the magnetic energy d e n s i t y i n the c o i l i s kept constant. 68-Sectlon C - Theory In t h i s s e c t i o n , a t h e o r e t i c a l model i s proposed t o describe the r a d i a l and a x i a l motions of the plasma.. E x p e r i -mentally i t i s found that I f the plasma moves a x i a l l y a d i s t a n c e greater than ZN (Z_ i s the a x i a l displacement r e q u i r e d t o c c escape the containment of the c o i l ) , before i t c o l l a p s e s t o 0.8a (where *a s i s the i n i t i a l r a d i u s ) , then the plasma o s c i l l a t i o n s are suppressed, i . e . , c r i t e r i o n f o r f l i p . By u s i n g the experimental r e s u l t s given i n the preceding s e c t i o n , the v a l i d i t y of the t h e o r e t i c a l model can be checked i n the f o l l o w i n g way. F i r s t the a x i a l and r a d i a l equations of motion are solved simultaneously by a power s e r i e s approximation. Then from the s o l u t i o n of the r a d i a l equation, the time, t , f o r the plasma to c o l l a p s e to r ~ 0 . 8 a Is c a l c u l a t e d . S u b s t i t u t i n g t h i s value of t i n t o the s o l u t i o n to the a x i a l equation, the a x i a l displacement, Z Q, at t h i s time i s found. I f t h i s value of Z Q i s g r e ater than Z (see above), then the plasma o s c i l l a t i o n s c w i l l be suppressed, that i s , f l i p occurs. This value ZQ i s a f u n c t i o n of gas pressure, p, r a d i u s of the discharge tube, a, transverse magnetic f i e l d , B T, e t c . Thus, the dependence of the t h e o r e t i c a l f l i p c r i t e r i o n on the experimental parameters ( i . e . Z 0(p,B,p,a)> constant) can be compared w i t h that given i n the experimental r e s u l t s . The r a d i a l equation Is discussed f i r s t , f o l l o w e d by the d i s c u s s i o n of the a x i a l equation of motion. -69 Section C l - Ra d i a l Equation Of Motion The r a d i a l equation of motion i s d e r i v e d by equating the r a d i a l magnetic f o r c e s t o the r a t e of change of r a d i a l momentum. (see P i g . I I I . 3 . ) . At the beginning of the r a d i a l c o n t r a c t i o n , 2 we assume the plasma s h e l l has a f i n i t e mass m-fl a 6 because of the f i n i t e t h i c k n e s s of the induced current s h e l l . I f we assume that a l l the gas p a r t i c l e s are swept up by the plasma and concentrated i n a t h i n s h e l l (Rosenbluth, 1954), then the momentum of the s h e l l per u n i t l e n g t h I s , (6) m TT ( a 2 - r 2)§§ + m i r a 2 * | | , where m = mass d e n s i t y of the gas, a = i n t e r n a l r a d i u s of the discharge v e s s e l , r = rad i u s of the plasma r i n g , 2 mira £ = i n i t i a l mass of the plasma r i n g per l e n g t h . The f i r s t term i s the momentum r e s u l t i n g from the mass swept up by the c o l l a p s i n g s h e l l . The second term represents the momentum a r i s i n g from the i n i t i a l mass of the s h e l l . The I n e r t i a f o r c e of the s h e l l i s ; ( 7 ) d_ dt m Tt (& 2 \dr r } dT d_ dt 2 dr intra fe ^ -70. The r a d i a l m a g n e t i c f o r c e s a r e due t o t h e d i f f e r e n c e s I n m a g n e t i c f i e l d s on t h e i n s i d e a n d on t h e o u t s i d e o f t h e p l a s m a s h e e t o Two a p p r o x i m a t i o n s a r e made i n f o r m u l a t i n g t h e e x p r e s s i o n f o r t h e s e f o r c e s . The f i r s t i s t h a t t h e m a g n e t i c f i e l d s p r o d u c e d b y t h e p l a s m a c u r r e n t s a n d t h e t h e t a c o l l c u r r e n t s c a n be r e p r e s e n t e d b y t h e f i e l d o f a n i n f i n i t e s o l e n o i d , ( f o r d i s c u s s i o n s e e Page 81). (8) i „ e 0 B , where B = m a g n e t i c f i e l d , I - c u r r e n t p e r u n i t l e n g t h , yMc= p e r m e a b i l i t y . , The s e c o n d a p p r o x i m a t i o n i s t h a t t h e e l e c t r i c a l c o n d u c t i v i t y o f t h e p l a s m a i s i n f i n i t e so t h a t t h e m a g n e t i c f l u x e n c l o s e d by t h e p l a s m a i s c o n s e r v e d . T h e r e f o r e t h e i n t e r n a l m a g n e t i c f i e l d , B^, i 0 e , t r a p p e d f i e l d , c a n be r e p r e s e n t e d by, 2 _ o ' i B r ta (9) B, = r where B = i n i t i a l m a g n e t i c f i e l d e n c l o s e d by t h e p l a s m a , a = i n i t i a l r a d i u s o f t h e p l a s m a , r = r a d i u s o f t h e p l a s m a . -71-Thus, the r a d i a l magnetic f o r c e per u n i t l e n g t h can be represented by the f o l l o w i n g expression: ,2 (10) r a d i a l 2 TT r 2 4 K 4. B„a c + o L 2A The t h e t a c o i l f i e l d I n i t i a l l y increases approximately l i n e a r l y w i t h time, t h e r e f o r e , l e t (11) B c ^ B c ( t V ) f where B = r a t e of change of the t h e t a c o i l c f i e l d , A = a constant to be determined by the " I n i t i a l conditions,, S u b s t i t u t i n g t h i s value of B_ i n t o the expression f o r the c r a d i a l f o r c e , Eqn 0 (10), we have. (12) F r a d i a l " 2 1 1 r 2 / f ° 2 / ^ r ^ From the i n i t i a l boundary c o n d i t i o n t h a t at (13) t = 0 , F r a d l a l = 0 , r = a , then s u b s t i t u t i n g these values i n t o Eqn„ (12), i t i s found t h a t , -72-(14) B o The r a d i a l equation of motion can now be w r i t t e n u s i n g Eqn. ( 7 ) , (12) and ( I ' M : ^ dt' To w r i t e Eqn 0 (15) i n dimensionless form, t h e ' v a r i a b l e s y , l , and T are introduced where, (16) y T a Ama B. The dimensionless equation of motion i s , (17) dt S o l v i n g t h i s equation by a power s e r i e s s o l u t i o n and u s i n g the I n i t i a l boundary c o n d i t i o n , (18) at t = 0 ; y = i • = o dt u ' i t 3s found that -73-(19) VJ - I - 3 f f t 3 - T z t t * + S f ^ 5 t — - . Prom the framing camera photographs i t i s found that at the c r i t i c a l c o n d i t i o n s f o r suppression of o s c i l l a t i o n s , the plasma r a d i u s c o l l a p s e s to roughly 0 o8 of i t s o r i g i n a l v a l u e , i . e . y - 0 .8, before i t disappears. When y~0.8, the f o l l o w i n g c o n d i t i o n on t i s v a l i d ? (20) > 1 > ^ (see P.84for numerical v e r i f i c a t i o n and P.<?o f or t h e o r e t i c a l r e s u l t s i f t h i s approximation i s not v a l i d ) . Using t h i s c o n d i t i o n , Eqn. (20), i t can be shown that »y» can be represented by? (21) y ^ 1 - ~ ^ 0.8 The value of t when y 0.8 i s denoted by the v a r i a b l e /T C, It f o l l o w s then t h a t , (22) % * (2.4 £)* Note that t h i s c r i t i c a l time, T c, i s not dependent on^ or T. The a x i a l motion produced d u r i n g the c r i t i c a l t i m e , „ i s discussed below. -74-S e c t i o n C.2 - A x i a l Equation Of Motion In t h i s case, i t Is necessary t o take i n t o account the presence of the two current r i n g s i n d i c a t e d by the framing camera photographs, (see F i g . I I I . 9 .) . At breakdown, the changing f l u x from the t h e t a c o i l produces a current at the outer surface of the discharge gas which causes i n t e n s i v e ohmic heating. The heated l a y e r expands across the magnetic f i e l d trapped w i t h i n the plasma c y l i n d e r , thereby generating a current on the inner surface of the plasma c y l i n d e r . This current conserves the f l u x w i t h i n the c y l i n d e r . I t i s apparent from P i g . III.2 5 * that i n the snowplough approximation the a x i a l momentum of the elemental mass contained by the angle A 6 i s , (23) § *efoNr l j$ + § A e < f e ^ per u n i t l e n g t h , where ^ = a x i a l displacement, Ae = angle e n c l o s i n g a small p o r t i o n of the plasma, ( F i g . I I I . 2 5 . ) . -75-Plasma Ring I = Theta C o i l Current c I = Plasma Current P B T = Transverse Magnetic F i e l d B = Plasma Magnetic F i e l d P B = Theta C o i l Magnetic F i e l d c F i g . I I I . 2 4 . A x i a l S e c tion Of The Theta Pinch .Discharge - 7 6 -The other v a r i a b l e s have been p r e v i o u s l y s p e c i f i e d (Page 6 9 ) . The assumptions made i n d e r i v i n g ; the equation of r a d i a l motion (Page 69) are employed i n t h i s equation a l s o . In Eqn. (23), the f i r s t term i s the momentum of the mass swept up by the c o l l a p s i n g s h e l l - and the second term i s the momentum of the i n i t i a l mass s h e l l of the plasma. The t o t a l magnetic f o r c e a c t i n g on the plasma i s , r AG cos Q B R s i n 0 I 0 , where I = the plasma current which conserves P the i n i t i a l trapped f l u x , I c = the plasma current induced by the t h e t a c o i l , -77-0 = angle enclosed by the t h e t a f i e l d and Helmholtz f i e l d a x i s , Q = azimuthal angle of the plasma, (Pig. I I I . . 2 5 . ) . The probe s i g n a l produced by the double probe technique corresponds t o the net plasma f i e l d . However, i t i s assumed that when the i n n e r current r i n g escapes confinement, the plasma c o n d u c t i v i t y drops and the r a d i a l hydromagnetic o s c i l l a t i o n s are suppressed. I f v i s c o s i t y i s ne g l e c t e d , there i s no co u p l i n g between the a x i a l motions of the two current sheets and the motion of the in n e r current sheet can be considered s e p a r a t e l y . The motion of the inner current r i n g i s discussed below. The magnetic f i e l d produced by I p , u s i n g the l o n g s o l e n o i d approximation i s then given by the equation belowJ (25) IP * where B Q i s the magnetic f i e l d i n i t i a l l y enclosed by I p . I f we l e t AK be the f r a c t i o n a l mass i n i t i a l l y occupied by the inner current l a y e r , then the i n e r t i a f o r c e of the mass element i s , (26) K ^ L h r 1 f ? ( ^ - ^ + Q l A ) where 'K8 i s the f r a c t i o n of the mass through which the current s h e l l passes that i s r e t a i n e d i n t h i s s h e l l (K<^1). The equation of a x i a l motion f o r t h i s mass element becomes, -78-The maximum a x i a l motion occurs at points where Q = 0°* If the variables y,€ , and T previously s p e c i f i e d [Eqn0 (16)] are used, then the dimensionless equation of motion at point of maximum a x i a l motion becomes, (29) where 2 = ^ Substituting y = |— 3 ^ l T Vt*+ , ^ f 3 ? 5 , Eqn. (19),, and using a power series approximation, i t can be shown that, Thus, the a x i a l displacement i n the time f*. , w i l l be, j . However, T t = (2.4Oz from Eqn, (22), thus i f A ° K ^ i s a constant, then the a x i a l distance t r a v e l l e d by the elemental BjjSlntf mass i n the time "CT i s proportional to ^ — . If Z Q i s greater than some c r i t i c a l value, Z , then a f l i p occurs because the c B„sin0 plasma confinement vanishes. This Implies that ^ & — must exceed some c r i t i c a l value. However, m* i s proportional to p* , so that the experimental and computed c r i t e r i a f o r f l i p are the same, -79-(32) i . e . y constant Seotlon C .3 - Discussion Of The Assumptions S e c t i o n C.3a - The Trapped F i e l d . B Q. Is Constant With V a r i a t i o n In Gas Pressure From the Integrated magnetic probe waveforms, the amplitude of the s i g n a l s correspond to the s t r e n g t h of the magnetic f i e l d . Thus, by measuring the amplitude of the plasma s i g n a l when the o s c i l l a t i o n s are Just suppressed, one can estimate the correspond-i n g magnetic f i e l d . I t i s found that the plasma s i g n a l Is i d e n t i c a l ( w i t h i n accuracy of the measurements, ^ 10%)t f o r f l i p c o n d i t i o n s at a l l pressures between 1 6 H g t o 75 Hg. fhus i m p l y i n g that the trapped magnetic f l u x i s approximately the same f o r a l l pressures, (see F i g . I I I . 2 6 . ) . - 8 0 -a. Pres 1.15 /A. Hg Time = lyu sec/cm b. Pres.' 25 ^  Hg c. Pres 5 0 ^ Hg d. Pres lOOyMHg Top Of Double Beam Traoe Corresponds To The Plasma Magnetic F i e l d . P i g . I I I . 2 6 . Probe S i g n a l s Showing B Q Is Constant For D i f f e r e n t Pressures At F l i p C o n d i t i o n -81-Seotlon C . 3 b - € Is Constant With Respect To V a r i a t i o n In Gas Pressure I t i s very d i f f i c u l t to v e r i f y q u a n t i t a t i v e l y that the f r a c t i o n a l p o r t i o n of the mass, 6 , i n the i n i t i a l c o l l a p s i n g plasma s h e l l i s approximately constant w i t h v a r y i n g gas pressure. However, i t appears to be a reasonable assumption because the framing camera p i c t u r e s show that the t h i c k n e s s of the plasma s h e l l before i t c o l l a p s e s under the f l i p c o n d i t i o n s , i s constant w i t h v a r y i n g pressures, (see F i g . III.27. ). Thus, one would expect the f r a c t i o n a l mass, € , i n the s h e l l t o be approximately constant w i t h change i n gas pressure. S e c t i o n C . 3 c - The Magnetic F i e l d Can Be Approximated By The Magnetic F i e l d Of An I n f i n i t e Solenoid The i n f i n i t e s o l e n o i d a l approximation wduld be v a l i d f o r a c o i l of h i g h aspect r a t i o ( i . e . ^ ^ ^ ^ e r ^ b u t i n t h i s experiment, the aspect r a t i o i s ^ 1. However, i n the r e g i o n c l o s e t o the w a l l s of the discharge v e s s e l , the s o l e n o i d a l f i e l d r e p r e s e n t a t i o n becomes a reasonable approximation. Thus, i n the r a d i a l equation of motion, when y>0.8 ( i . e . , the r e g i o n of I n t e r e s t i n t h i s experiment), a s o l e n o i d a l f i e l d approximation i s reasonable. From t h e o r e t i c a l c a l c u l a t i o n s , the f i e l d changes by ^ 10# i n t h i s r e g i o n , (y = 1 -> y ^ 0.8). The plasma f i e l d i n the r a d i a l equation i s a second order e f f e c t when y > 0,8 and thus the s o l e n o i d a l approximation i s not Important. -82-GassAir c. 100 y u Hg F i g . I I I . 2 7. End On Photographs Of The plasma At F l i p Conditions -83-In the case of the a x i a l equation of motion, the plasma current i n t e r a c t i n g w i t h the transverse magnetic f i e l d i s the dominant f o r c e . However, t h i s f o r c e , F a x j _ a T _ s i s p r o p o r t i o n a l to IB™,and thus the geometry of the magnetic f i e l d produced by I i s not important. The only necessary requirement i s that the trapped f i e l d , B Q , be p r o p o r t i o n a l to I , which i s tr u e f o r a plasma of any given geometry, i . e . , B Q = I F ( r , z ) 0 -84-SeotIon Co 3d - CJ) ^ > ^ c > A p p r o x i m a t i o n — — iiijuiirriiiiinvwiwm^nmnirwalffrfi—i I I I IITM*-" 1111 " L—J main • • --r i In t h i s s e G t i o n 9 the approximation (^ -)36 > T t > ^  i s i n v e s t i g a t e d by s u b s t i t u t i n g t y p i c a l experimental values f o r the v a r i a b l e s . This approximation i s used i n d e r i v i n g the expression y = 1 - • N o w » (16) T « (14) w here ^ = p e r m e a b i l i t y = 4 TT x 10° 7 Henries/m m ~ mass d e n s i t y % ?,9 i 10"-* kgm/m3 f o r p = 50yu Hg 9 a i r , a - ra d i u s of the discharge v e s s e l = 0o025 meters, g = r a t e of change i n the t h e t a c o i l c 6 2 magnetic f i e l d AS 1,2 x 10 weber/m sec so that T 0,5^/1 sec IT" where B Q = i n i t i a l trapped magnetic f i e l d ~ 5 0 cmax° The v a l u e of B^ was deri v e d from the f a c t that under f l i p o c o n d i t i o n s , the amplitude of the i n t e g r a t e d probe s i g n a l due t o the plasma magnetic f i e l d i s ^  -j^ the amplitude of the s i g n a l corresponding t o the t h e t a c o i l f i e l d , (see P I g 0 I I I o 2 8 0 ) o - 8 5 -Upper Trace - Plasma Magnetic F i e l d : 0.05 V/cm, Lower Trace -Theta C o i l Magnetic Fie l d : l V / c m , Time ; 1 ^ sec /cm, Pressure 25/uHg, a i r , Other c o n d i t i o n s s p e c i f i e d on Page 4o„ F i g , HI.28. Plasma F i e l d Strength At F l i p Conditions p B = 2 . 8 weber/m max B^ » 0.056 weber/m 2 o • 6 9 B = 1.2 x 10 weber/m sec c then A = O.O^-o^sec. S u b s t i t u t i n g i n t o , (33) 86 Prom Eqn. (22), (22) t T = 0 ' ^ * From the end on photographs of the plasma, ( F i g . 111 . 2 7 .), i t i s seen that the t h i c k n e s s of the plasma sheet before the c o l l a p s e i s ^-a. Therefore I t i s reasonable t o assume that . Although the value of € i s d i f f i c u l t to determine a c c u r a t e l y , I t Is seen from Eqn. (22) that TJC i s a weak f u n c t i o n of <c . The p a r t i c u l a r value assumed f o r £ does not t h e r e f o r e c r i t i c a l l y a f f e c t the v a l i d i t y of the i n e q u a l i t y , ( j j e > l t > ^ . Using L * -jL-(34) 0.7 From the above values f o r T, fi , and 6 , i t f o l l o w s that (35) ( j j e ^ 100 The I n e q u a l i t y ( (y/fc >Tt >^ , Page 84 ), then becomes (36) 100 > 0.7 > 0.1-Hence the expression employed f o r y In the a x i a l equation ( i . e . y 1 ) Is v a l i d w i t h i n the l i m i t s of experimental accuracy. -67 S e c t i o n Co4 - A Discussion Of The I n t e r p r e t a t i o n Of The Suppression Of O s c i l l a t i o n s As The F l i p I n s t a b i l i t y Two p l a u s i b l e ways i n which the suppression of o s c i l l a t i o n s w i t h i n c r e a s i n g transverse magnetic f i e l d can be I n t e r p r e t e d are the f o l l o w i n g t f i r s t , the plasma c o n d u c t i v i t y becomes too small to support the plasma current and thus the plasma d i s s i p a t e s , or secondly, the plasma has escaped from the ends of the c o i l because of a x i a l motion 0 In the c o n d u c t i v i t y model, the suppression of the o s c i l l a t i o n s i s explained by assuming that as the trans v e r s e b i a s f i e l d i s increa s e d , the c o n d u c t i v i t y of the plasma i s decreased thus causing the plasma to disappear,, However, t h i s model has a number of inherent d i f f i c u l t i e s . lo One would expect p r e i o n i z a t i o n t o enhance the hydromagnetic o s c i l l a t i o n s and thus e f f e c t i v e l y increase the s t a b i l i t y constant. Yet, when p r e i o n i z a t i o n i s a p p l i e d , the s t a b i l i t y constant Is not a f f e c t e d (see Appendix, A ) D -88-2 , One would expect the s t a b i l i t y constant t o not increase I f the l e n g t h of the c o i l i s i n c r e a s e d and the peak induced e l e c t r i c f i e l d i n the discharge tube i s kept constanto Yet, on v a r y i n g the l e n g t h of the c o i l (see Appendix B) i t i s found that the s t a b i l i t y constant increases as the l e n g t h of the c o i l i s increased,, 3 . One would not expect the t r a n s v e r s e f i e l d to be Important when the t h e t a c o l l f i e l d becomes l a r g e . Yet i n the magnetic probe waveforms, the o s c i l l a t i o n s are slowly damped when B T ^ 1 ( 2 - 3 Msec a f t e r the breakdown of the plasma)„ On the other hand, i f the suppression of the o s c i l l a t i o n s i s due t o a x i a l motion of the plasma, then one would expect the s t a b i l i t y to increase w i t h the l e n g t h of the c o i l because the plasma must t r a v e l a longer distance t o escape the t h e t a c o i l . A l s o , a x i a l motion of the plasma w i l l e x i s t even when | j ~ ^ 3 0 * » c causing a gradual suppresion of o s c i l l a t i o n because the a x i a l f o r c e on the plasma i s a d i r e c t f u n c t i o n of B~ and not B , At the time of suppression of o s c i l l a t i o n s , from the end on framing camera photographs ( F i g . I I I , 1 3 o ) , there Is no c l e a r evidence that the plasma has a l a r g e a x i a l motion because the plasma forms c l o s e to the edge of the c o l l and thus escapes a f t e r only a small displacement. However, from framing camera p i c t u r e s where the a x i a l v e l o c i t y can be measured approximately, i t i s seen that f o r 1 0 0 ^ Hg gas pressure, the a x i a l v e l o c i t y of the plasma ls.~L cm / ^ sec (see F i g . I I I . l l ) . Thus, the - 8 9 -a x i a l motion can cause the disappearance of the plasma i n < 1 yM. sec, As the plasma moves a x i a l l y out of the c o i l i t i s probably f o r c e d i n t o the w a l l s because the magnetic f i e l d l i n e s outside the discharge tube, I n t e r s e c t the w a l l s of the tube. The contact w i t h the w a l l s r a p i d l y cools the plasma and i t d i s s i p a t e s , (see PIgo. IH.2'9.). Because of the experimental evidence of the e f f e c t of p r e l o n i z a t i o n and the l e n g t h of the discharge c o l l on the s t a b i l i t y constant, and a l s o the evidence from the framing \ camera p i c t u r e s , the pla^ na's axlaL.motion has been chosen to e x p l a i n the suppression of the hydromagnetic oscillations„ P i g , I I I . 2 9 ; Magnetic F i e l d Lines Of Theta C o i l - 9 0 -Sectton Co5 - V a r i a t i o n s Of The T h e o r e t i c a l Model In t h i s s e c t i o n v a r i a t i o n s i n the t h e o r e t i c a l model are made that could describe the a x i a l motion of the plasma 0 The corresponding c r i t e r i o n f o r the f l i p i n s t a b i l i t y i s then d e r i v e d . 1, Suppose that the T 3 term i n the expression •119) i s the dominant term, I.e. £ < Then the value of f c would be (37) 0,6 6 T l 3 where T S u b s t i t u t i n g t h i s value of "T i n t o the s o l u t i o n of the a x i a l equation of motion, Eqn. (31), then i t f o l l o w s that (38) Z ; _ TT B0 6h S i n ( h ^ ~< So that the c r i t e r i o n f o r f l i p becomes, (39) z o varies as B n f m f f > constant , a 1 m3 -91 Hence f o r t h i s model, the s t a b i l i t y constant would vary as (am) 3 o This i s not borne out by the experimental r e s u l t s . Suppose the a x i a l motion of the outer current sheet i s r e s p o n s i b l e f o r the disappearance of the o s c i l l a t i o n s 0 Then u s i n g the same assumptions as given i n the t e x t , Page 699 the power s e r i e s approximation t o the r a d i a l equation can be approximated by (21) Thus (22) <CC * (2.4-1)* . The a x i a l equation of motion of the outer sheet Is (4o) ( . i - K ) _ 2.TT 8„Bu Sl'n^ CO SG-where(|-K) fa^TT = the I n i t i a l mass of the outer current sheet. Using the boundary c o n d i t i o n s : (41) t - o; z = o 92 and u s i n g the u s u a l assumptions described p r e v i o u s l y , (Page 6 9 ) , the power s e r i e s approximation t o t h i s equation i s (42) so that Z o B„ c o s e * ' iw qi> / TT \+ ^ (l-K)|0* ft* W ™ ' ° where (43) BgSlnjeJ 2_ cX; ^ > constant , a.7- m^ Hence f o r t h i s model the s t a b i l i t y constant would v a r y as a m * , which again i s not borne out by the experimental r e s u l t s . Consider the case where the a x i a l motion of the outer current sheet i s r e s p o n s i b l e f o r the disappearance of the o s c i l l a t i o n s but the X3 term i n the r a d i a l s o l u t i o n (19) y Is the dominant term, i o e 0 (44) then (37) where B 93-S u b s t i t u t i n g t h i s value of ^ i n t o the s o l u t i o n t o the a x i a l equation f o r the outer current sheet 9 Eqn„ (42), the c r i t e r i o n f o r f l i p then becomes (45) 2 Q vanes as BjjSin0 c > constant Note that the f l i p c o n d i t i o n i n t h i s case i s Independent of the mass d e n s i t y of the discharge gas and the rad i u s of the discharge v e s s e l •= a p r e d i c t i o n which Is again not confirmed by the experimental data» Prom the above d i s c u s s i o n , i t i s seen that the model of the f l i p t h a t Is used In the t e x t of the t h e s i s Is the model which best f i t s the experimental r e s u l t s and a l s o the experimental conditions,. In t h i s model i t i s assumed that the plasma ceases to e x i s t as soon as the Inner current l a y e r i s d r i v e n a c r i t i c a l d i s t a n c e In the a x i a l d i r e c t i o n of the d i s c h a r g e 0 S e c t i o n D Suggestions For Further Work l o In the work reported above, the accuracy i n the d i r e c t measurement of the r o t a t i o n of the plasma has been l i m i t e d by the time r e s o l u t i o n of the framing camera 0 However, the r e s u l t s from the d i r e c t i o n a l probes I n d i c a t e that t h i s method of measuring r o t a t i o n , when p e r f e c t e d . Is f e a s i b l e to o b t a i n q u a n t i t a t i v e results» To improve the s p a t i a l r e s o l u t i o n of the probe waveforms, a discharge v e s s e l of l a r g e r diameter could be employed; a l s o the d i r e c t i o n a l response of the probe could be improved by -94-I n c r e a s i n g the l e n g t h of the search c o l l (see F i g 0 I I . l ? . ) . To Improve the time r e s o l u t i o n of the probe waveforms, a heavier gas could be employed to slow down the dynamics of the plasma. A second improvement on the time r e s o l u t i o n would be t o increase the frequency response of the probe. 2. Prom the theory, i t i s seen that the object of t h i s i n v e s t i g a t i o n i s the i n n e r plasma current which conserves the trapped magnetic f i e l d . To i n v e s t i g a t e the behaviour of the outer current sheet, i t i s necessary that the trapped f l u x i n the plasma be n e g l i g i b l e . Then u s i n g the double probe technique (see Page 26 ), the magnetic waveform would correspond to the magnetic f i e l d produced by the outer current sheet. For t h i s case, the t h e o r e t i c a l f l i p c r i t e r i o n I s , (46) ? varies as ^' S i n^ > constant In t h i s case, the s t a b i l i t y constant v a r i e s as a ^ m^ . To attempt t o o b t a i n the experimental c o n d i t i o n s when the trapped f i e l d i s n e g l i g i b l e , a heavy p r e l o n i z a t l o n of the discharge gas could be a p p l i e d by a Z p i n c h discharge. I f the c o n d u c t i v i t y of the plasma i s s u f f i c i e n t l y high at the beginning of the t h e t a bank discharge, then there w i l l be a n e g l i g i b l e amount of magnetic f l u x that can d i f f u s e Into the plasma r i n g and become trapped. In t h i s experiment the suppression of o s c i l l a t i o n would be I n v e s t i g a t e d at the beginning of the f i r s t h a l f c y c l e of the t h e t a c u r r e n t . - 9 5 -CHAPTER IV CONCLUSIONS The r e s u l t s reported show that the f l i p i n s t a b i l i t y can be produced i n a t h e t a p i n c h discharge and that the d i r e c t i o n of r o t a t i o n can be c o n t r o l l e d by a p p l y i n g a magnetic f i e l d t r ansverse to the t h e t a p i n c h f i e l d . Using the suppression of hydromagnetic o s c i l l a t i o n s t o detect the f l i p I n s t a b i l i t y , i t i s found that I f BgSinjZf i s greater than a given constant, then the plasma r i n g a p* f l i p s . The r e s u l t s a l s o c l e a r l y demonstrate that I f b i a s f i e l d s are used t o improve the discharge c h a r a c t e r i s t i c s , then these f i e l d s must be a c c u r a t e l y a l i g n e d w i t h the discharge a x i s . A misalignment of 0 ^ 2 w i t h a 0,3 weber/m b i a s f i e l d can produce c a t a s t r o p h i c f l i p i n s t a b i l i t i e s . Misalignment problems are best avoided by u s i n g the t h e t a c o i l I t s e l f t o produce any b i a s f i e l d r e q u i r e d . Another fea t u r e of the i n v e s t i g a t i o n i s that the v a l i d i t y of the two current models of a t h e t a p i n c h discharge i s confirmed. This i s d i f f i c u l t to do i f only the r a d i a l dynamics of the plasma are observed. However, the transverse b i a s f i e l d c l e a r l y separates at l e a s t two current r i n g s , A t h e o r e t i c a l model of the plasma f l i p Is constructed t o a s s i s t i n the i n t e r p r e t a t i o n of the experimental r e s u l t s . In 96-t h i s model 9 the plasma c o n s i s t s of two main current r i n g s 0 The outer current r i n g I s the r e s u l t of the change In the magnetic f l u x produced by the t h e t a c o l l 0 The Inner current r i n g Is a r e s u l t of the conservation of the magnetic f l u x trapped by the plasma. When the inner current r i n g escapes from the c o l l , the r a d i a l hydromagnetic o s c i l l a t i o n s are suppressed. The c r i t e r i o n f o r f l i p p r e d i c t e d by t h i s model agrees w i t h the experimental r e s u l t s given on Page 95 . - 9 7 -APPENDIX A EFFECT OF PREIONIZATION ON THE STABILITY CONSTANT A t e s l a c o i l Is used to p r e i o n i z e the discharge gas before the t h e t a bank i s discharged. The purpose of t h i s experiment i s to determine the e f f e c t of p r e i o n i z a t i o n on the f l i p i n s t a b i l i t y . The experimental procedure f o r determining the c r i t i c a l value of the s t a b i l i t y constant, B H s i n 0Qi as a f u n c t i o n of pressure i s i d e n t i c a l to the experimental procedure as given f o r the i n v e s t i g a t i o n of a i r (Page 52), except that the gas i s Ionized by a t e s l a c o i l d u r i n g the t h e t a bank discharge. I t i s found that there i s no appre c i a b l e d i f f e r e n c e i n the experimental r e s u l t s . A p l o t of B H s i n 0 as a f u n c t i o n of pressure i s given f o r both w i t h and without p r e i o n i z a t i o n i n F i g . A . l . - 9 8 -2,4 -10 20 30 40 50 60 70 80 90100 Pressure Hg a. Without P r e i o n i z a t i o n F i g , A . l . P l o t s Showing E f f e c t Of P r e i o n i z a t i o n On S t a b i l i t y -99-APPENDIX B CHANGE IN THE STABILITY CONSTANT WITH VARYING THETA COIL LENGTH The purpose of t h i s experiment i s t o determine how the B„sinjZf constant, — = = — r — v a r i e s w i t h a v a r i a t i o n i n the l e n g t h of the th e t a c o i l . In theory, a f l i p occurs i f Z > Z . where Z i s p r o p o r t i o n a l to the le n g t h of the c o i l . This i n e q u a l i t y can be expressed as x > constant which v a r i e s as the l e n g t h of the P " c o l l . Thus, the theory p r e d i c t s that the s t a b i l i t y constant should be p r o p o r t i o n a l to the l e n g t h of the c o i l , I.e., _c v a r i e s as the c o i l l e n g t h In t h i s experiment the value B H s i n 0Q was found f o r a given gas pressure (100 ^ Hg, a i r ) , f o r d i f f e r e n t c o i l l e n g t hs. Three c o l l lengths were used and the t h e t a bank v o l t a g e was I 2 v a r i e d to keep _ c o l l constant, see below, l e n g t h -100-Experlmental Conditions For D i f f e r e n t C o i l Lengths C o i l Length 3.^ cm 5.1 cm 6.8 cm Theta Bank Voltage 9.4 kv 11.5 kv 13.5 kv Induced E l e c t r i c F i e l d 1.^ E„ 1.2 E 0 K o ( I n f i n i t e s o l e n o i d approximation) I t i s seen that the s t a b i l i t y constant increases approximately l i n e a r l y w i t h increase i n the t h e t a c o l l l e n g t h . ( F i g . A.2.) However, the r e s u l t s are only q u a l i t a t i v e because of the crude approximations that are made. I f the s t a b i l i t y constant i s a c o n d u c t i v i t y phenomenon, then one would expect the constant to decrease w i t h decreasing induced e l e c t r i c f i e l d , but t h i s i s not supported by the experimental r e s u l t s . The i n v e s t i g a t i o n was not c a r r i e d on f o r longer c o i l lengths because of the l i m i t a t i o n s of the peak charging voltage of the t h e t a bank. -101-Gas s A i r C o i l Length A 3 A cm © 5°1 cm 4> 6.8 cm 0.0250-B g S i n ^ (weber/m2) 0.0125 G L A _ A - A -o 0.2 0.4 o B H (weber/m ) 0.6 0.0250 -B H s i n 0 c (webei/m2) 0.0125 --0 0 " A Gas s A i r Pres = lOOy* Hg 4 cm b. C o i l Length 8 cm F i g . A.2. S t a b i l i t y Constant V a r i a t i o n With C o i l Length -102-APPENDIX C SUMMARY ON THE ARGON EXPERIMENTS An experiment t o determine the v a r i a t i o n of the c r i t i c a l value of the s t a b i l i t y constant, B H s i n 0C (see Page 51 ), as a f u n c t i o n of pressure i s taken f o r argon 0 The experimental procedure i s i d e n t i c a l to the procedure given f o r the i n v e s t i g a t i o n of a i r (Page 51 )° The pressures i n v e s t i g a t e d are 8 ^ H g , 1? yMHg, 25 yuHg, and 41 /(/IHg, Argon at a given pressure was found t o be more s t a b l e to the f l i p i n s t a b i l i t y than a i r , (see F i g , A , 4 0 ) , The non-l i n e a r p o r t i o n of the curve was not determined because the b i a s magnetic f i e l d was not s u f f i c i e n t l y s trong to suppress the o s c i l l a t i o n s at pressures greater than 50 y u Hg, Within experimental e r r o r , the constant s i n 0 C, v a r i e s as (pressure (see F i g , A.4,)„ -103-B H (weber/m ) Pressure Q 8/iHg Ar A 17/A Hg Ar © 2 5 ^ Hg Ar O klyuRs Ar 5T 10 OwlO-B H s i n 0 c 2 (weber/m ) 0 o 05 0 o 2 a, 0 Degrees -A-o.'3 0 — C L Q JZL T2J-"oT4~ - 0 -o*5 b 0 B H (weber/m ) A F i g , A,3. Value of B g S i n j ^ For D i f f e r e n t Gas Pressures (Argon) 2„3 ' 1 1—i 1 1 1 1 1 1 1 — i — 7 8 9 10 20 30 40 50 60 70 80 90 Log Pressure -=• yuHg Figo A„4 0 Comparison Of The S t a b i l i t y Constant Of Argon and A i r -105-APPENDIX D PICTURES .OF FLIP AT 100 ^AHg AND 250 MEK« AIR Framing camera p i c t u r e s of the plasma were taken f o r d i f f e r e n t gas pressures. As the mass of the plasma i s increased, the dynamics of the plasma slows down and i t i s found that the a c t u a l r o t a t i o n of the plasma about i t s diameter becomes more c l e a r . At high pressures (i,e„ 250^ Hg, a i r , see F i g , A,5») a new phenomenon a l s o appears. As at lower pressures, two current r i n g s are observed. However, the outer current r i n g i s not the f i r s t to c o l l a p s e r a d i a l l y . This i n d i c a t e s that the two current r i n g s are not produced by a x i a l t e a r i n g of the N l b l e t t and Green model of the plasma. The t e a r i n g motion may be suppressed at high pressures because of the increased i n e r t i a of the discharge. We suggest that the outer current r i n g at high pressures i s produced In m a t e r i a l evaporated from the w a l l of the discharge v e s s e l . At lower pressures ( ~-50 y^Hg), the discharge spends l e s s time near the w a l l so that the impurity r i n g s do not develop so r e a d i l y . The s l u g g i s h movement of the Impurity current r i n g I n d i c a t e s that the current I s probably f a i r l y low, or that the r i n g i s f a i r l y massive. -107 B - 0 o042 weber/m 2, (0 » 0°) T Figo A.7. Top On Photograph At 100 M Hg Pressure - A i r -108-APPENDIX E CHARACTERISTIC IMPEDANCE OF A MAGNETIC PROBE One d e s i r a b l e c h a r a c t e r i s t i c of a magnetic probe when used to make measurements i s that the frequency response be f l a t In the frequency range of i n t e r e s t (see Page 23 ). In a magnetic probe, there Is a s e l f Inductance and s t r a y capacitance which combine to form a resonant c i r c u i t , , However, the resonance can be e f f e c t i v e l y damped (see Segre and A l l e n , I960) by i n t r o d u c i n g an appropriate r e s i s t o r , R, (see F i g . A. 6.). The equivalent c i r c u i t f o r the magnetic probe can be w r i t t e n as shown below. -/WW-r L = S e l f Inductance CQ= Stray Capacitance G = Generator r Resistance Of Probe R = A r b i t r a r y R e s i s t o r F i g . A.8. Equivalent C i r c u i t Of Magnetic Probe 109 S o l v i n g the c i r c u i t equation, i t can be shown that i f R >y^(2.--C^j'*' , then the a t t e n u a t i o n curve has a resonance (1) CO -In p r a c t i c e a L < 5*, (2) t h e r e f o r e 2 0,1?-2. Thus i f we l e t (3) R J 2C then the resonance frequency decreases t o z e r o 0 To o b t a i n the values of L and C Q, the f o l l o w i n g c i r c u i t i s used 0 L - S e l f Inductance r = Resistance Of Probe Stray Capacitance App l i e d Capacitance F i g , Ao9» C i r c u i t For Damped Probe C i r c u i t - 1 1 0 The c i r c u i t I s e x c i t e d by a pulse and the resonance frequency ^ Is measured. ( 4 ) I.e. a), = J L < C o + °1> Then the c a p a c i t o r C^, Is changed to C 2 and the corresponding resonance frequency OJX Is found, ( 5 ) = 1 From equations ( 4 ) and ( 5 ) , the values of C Q and L can be c a l c u l a t e d . Using these values, the matching Impedance Is found f o r each probe and given below. c o L Magnetic Probe 5 ^ 0 pf l o 9 / & H 4 - 3 _n_ Rogowski C o l l 1 ? 4 pf 2 5 ^ H 2 ? 0 SL. -111-REPERENCES B a r t o l i , Co and Green T.S. 1963, Nuclear Fusion _3_, Bodin, H.A.B., Green, T.S., N l b l e t t , G.B.F., Peacock, N.J., Qulnn, J o M o P o , Reynolds, J.A., T a y l o r , J.B. 1962, Nuclear Fusion Supplement 2, 511. Bogen, P „ , H l n t z , E 0 , S c h l u t e r , J 0 1964, Nuclear Fusion 4 a 1310 C l l l l e r , W.A., D r i v e r , H.S., I r v i n g , J 0 , Lewis, I . 1963 Nuclear Fusion 2» 78. Clarke, G 0L 0 and Wuerker, R.F. 1962 Phys. of F l u i d s £, 1503 Curzon, F.L. and C h u r c h i l l , R.J. 1962 Can.J.Phys0 40, 1191 Curzon, F . L o and Smy, P . R . 1961 R e v o S c i o I n s t 0 320 756 Eberhagen, A 0 and Glaser, H„ 1964 Nuclear Fusion 4, 296. Green, T.S. 1962. Nuclear Fusion 2, 92. N l b l e t t , G.B.F. and Green, T.S0 1959 Proco Phys.Soco (London) Ser. 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