DYNAMICS OF A Z-FTNCH DISCHARGE I N ARGON by CECIL CHARLES DAUGHNEY • B.Sc, U n i v e r s i t y o f New B r u n s w i c k , 1961 M . S c , U n i v e r s i t y o f B r i t i s h Columbia, I963 A THESIS SUBLETTED I N PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e Department of PHYSICS J He a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e required standard. THE UNIVERSITY OF BRITISH COLUMBIA May, 1966 In presenting this thesis in p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, 1 agree that the Library shall make it freely available for reference and study. I further agree that permission for ex- tensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financ i a l gain shall not be allowed without my written permission. Department of ^k<jUg 3 The University of B r i t i s h Columbia Vancouver 8, Canada Date Wei,, /), THE UNIVERSITY OF BRITISH COLUMBIA FACULTY OF GRADUATE STUDIES PROGRAMME OF THEt. t FINAL ORAL EXAMINATION FOR THE DEGREE OF • DOCTOR OF PHILOSOPHY of CECIL CHARLES DAUGHNEY B.Sc, .Sc. The University of New Brunswick, 1961 The University of B r i t i s h Columbia, 1963 IN ROOM 303, HENNINGS BUILDING WEDNESDAY,, - MAY 11, 1966, AT 9:30 A. M, COMMITTEE IN CHARGE Chairman: B. Ahlborn A. J. Barnard F. L. Curzon F. Noakes R„ Howard M. M. Z. Kharadly R. A. Nodwell External Examiner: A. F o l k i e r s k i Imperial College University of London Research Supervisor: F. L. Curzon DYNAMICS O F A Z-PINCH DISCHARGE IN ARGON ABSTRACT A d i s c u s s i o n of probe measurement of the magnetic f i e l d i n a plasma i s presented with p a r t i c u l a r reference to the p e r t u r b a t i o n of the magnetic f i e l d caused by the probe. A c o r r e c t i o n procedure i s developed to compensate f o r t h i s p e r t u r b a t i o n . Using magnetic probes, r a d i a l v a r i a t i o n of the current density d i s t r i b u t i o n s are obtained f o r an argon plasma i n a z-pinch discharge. I n i t i a l argon pressures of 100, 250, and 500 yUHg are i n v e s t i g a t e d . The current density d i s t r i b u t i o n s are determined f o r Ijj^sec intervals between the i n i t i a t i o n of the discharge and the occurrence of the f i r s t pinch. These current density d i s t r i butions are compared with photographic observations. The experimental r e s u l t s are discussed i n terms of the snowplow model and the shock wave model. Mathemat i c a l l y , the non-linear snowplow equation i s solved using an approximation technique which r e s u l t s i n a n a l y t i c s o l u t i o n s . The shock wave equation i s solved by a graphical technique. An extension of the shock wave model i s proposed f o r a b e t t e r understanding of the experimental results. GRADUATE STUDIES F i e l d of Study: Physics Non Linear Systems Electronic Instrumentation S t a t i s t i c a l Mechanics Advanced Plasma Physics A. C. Soudack F. Bowers R. Barrie A. J. Barnard PUBLICATIONS 1. Simple Method of Overcoming the Effects of Noise Signals Produced by Spark Gap Switches FcL. Curzon and C C Daughney; Rev. S c i , Insti 34, 430, 1963, 2. Triggering System for Pulsed. Discharge C i r c u i t S = S Medley, F.L, Curzon and C C , Daughney; Rev.ScioInstr. 36, 713, 1964. e 3. Investigation of Noise Signals i n Pulsed Discb' Devices - S.S. Medley, F.L Curzon, and C C Daughney; Can.J.Phys. 43, 1882, 1965. e ABSTRACT A d i s c u s s i o n o f probe measurement o f t h e magnetic field i n a plasma i s p r e s e n t e d w i t h p a r t i c u l a r r e f e r e n c e t o t h e p e r t u r b a t i o n of t h e magnetic f i e l d caused b y t h e p r o b e . c o r r e c t i o n procedure i s developed A t o compensate f o r t h i s perturbation. U s i n g magnetic p r o b e s , r a d i a l c u r r e n t d e n s i t y d i s t r i b u t i o n s a r e o b t a i n e d f o r an a r g o n plasma i n a z - p i n c h d i s c h a r g e . I n i t i a l a r g o n p r e s s u r e s o f 100, 2^0, and fJOO^Hg a r e i n v e s t i g a t e d . The c u r r e n t d e n s i t y d i s t r i b u t i o n s a r e determined between t h e i n i t i a t i o n f i r s t pinch. f o r 1/tsec intervals o f t h e d i s c h a r g e and t h e o c c u r r e n c e of the These c u r r e n t d e n s i t y d i s t r i b u t i o n s a r e compared w i t h photographic observations. The e x p e r i m e n t a l r e s u l t s a r e d i s c u s s e d i n terms of t h e snowplow model and t h e shock wave model. Mathematically, the n o n - l i n e a r snowplow e q u a t i o n i s s o l v e d u s i n g an a p p r o x i m a t i o n technique which r e s u l t s i n a n a l y t i c s o l u t i o n s . equation i s solved by a g r a p h i c a l technique. The shock wave An e x t e n s i o n o f t h e shock wave model i s proposed f o r a b e t t e r u n d e r s t a n d i n g o f the experimental r e s u l t s . - i - THE DYNAMICS OF A Z-PINCH DISCHARGE I N ARGON PAGE ABSTRACT TABLE OF CONTENTS LIST OF ILLUSTRATIONS LIST OF TABLES ACKNOWLEDGEMENT 1.0 i i i iv v i v i i INTRODUCTION 1 MAGNETIC PROBE MEASUREMENT 8 2.0 INTRODUCTION 9 3.0 3.1 3.2 MAGNETIC PROBES MEASURING CIRCUIT MAGNETIC PROBE ERROR 9 11 15 U.O U.l U.2 U.3 U.U ANALOGUE EXPERIMENT APPARATUS EXPERIMENTAL RESULTS THEORETICAL MODEL CONCLUSIONS 18 20 22 30 33' 5.0 5.1 MAGNETIC FIELD CORRECTION NUMERICAL COMPUTATION 36 38 5.2 TEST FOR NUMERICAL PROGRAM hi EXPERIMENTAL RESULTS 51 6.0 INTRODUCTION 52 7.0 7.1 8.0 8.1 8.2 8.3 APPARATUS DISCHARGE PARAMETERS MAGNETIC PROBE MEASUREMENT MAGNETIC PROBE CALIBRATION COMPUTED RESULTS ANALYSIS OF ERROR 53 60 67 75 78 92 9.0 9.1 9.2 9.3 PHOTOGRAPHIC MEASUREMENT MAGNETIC PROBE PERTURBATION COLLAPSE CURVES SHOCK VELOCITY MEASUREMENT PART I PART I I - ii - . 103 10U 108 109 THE DYNAMICS OF A Z-PINCH DISCHARGE I N ARGON (CONT'D.) PAGE 10.0 10.1 10.2 PART I I I COMPARISON OF MAGNETIC PROBE AND FRAMING CAMERA RESULTS DETERMINATION OF SHOCK AND FLOW VELOCITIES DETERMINATION OF ELECTRIC FIELD 115 120 123 COMPARISON OF THEORETICAL MODELS WITH EXPERIMENTAL RESULTS 128 11.0 INTRODUCTION 129 12.0 THEORETICAL MODELS 130 13.0 13.1 13.2 DYNAMIC EQUATIONS FROM SNOWPLOW MODEL ANALYTIC SOLUTION OF EQUATIONS COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS DYNAMIC EQUATIONS FROM SHOCK WAVE MODEL GRAPHICAL SOLUTION OF EQUATIONS COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS 132 135 lli.O 1U.1 Iii.2 15.0 PART TV PROPOSED MODEL OF THE Z-PINCH DISCHARGE CONCLUSIONS liiii l£6 162 168 171 l8k 16.0 DISCUSSION OF RESULTS 185 17.0 FINAL CONCLUSIONS 190 18.0 PROPOSALS FOR FUTURE WORK 193 CONSTRUCTION OF MAGNETIC PROBE NUMERICAL PROGRAMS I N FORTRAN LANGUAGE 196 198 REFERENCES 203 APPENDIX I APPENDIX I I - iii - LIST OF ILLUSTRATIONS FIGURE 1 2 3 k 5 6 7 8 9 10 11 12 13 Ik 15 16 17 18 19 20 21 22 23 2U 25 26 27 28 29 30 31-33 3k 35 36 37-12 U3-U5 l|6 hi U& h9 50 51 52 53 PAGE EXPERIMENTAL CONFIGURATION MAGNETIC PROBE MAGNETIC PROBE MEASURING CIRCUIT FREQUENCY RESPONSE CURVES ANALOGUE AND Z-PINCH GEOMETRY PHOTOGRAPH OF ANALOGUE EXPERIMENT EQUIVALENT CIRCUIT OF ANALOGUE EXPERIMENT MAGNETIC PROBE ROTATION ANALOGUE EXPERIMENTAL RESULTS UNPERTURBED MAGNETIC FIELD C ( z ) | VS I z | ( p = .35) C ( z ) l VS I z I (/D = .ii5) C ( z ) l VS | z I (THEORETICAL CURVES) C(z)|VS | z | (THEORY AND EXPERIMENT/O « .35) C ( z ) | VS I z | (THEORY AND EXPERIMENT & » .h$) SHEET APPROXIMATION TO CONTINUOUS CURRENT DENSITY DISTRIBUTION ERROR I N NUMERICAL APPROXIMATION TO CORRECTION MODIFICATION OF CORRECTION NEAR AXIS ANALOGUE TEST CURVE TOTAL FIELD CORRECTION PHOTOGRAPH OF APPARATUS SCHEMATIC DIAGRAM OF APPARATUS FRAMING CAMERA AND CONTROL EQUIPMENT OSCILLOSCOPE TRACE OF DISCHARGE CURRENT OSCILLOSCOPE TRACE OF DISCHARGE VOLTAGE OSCILLOSCOPE TRACE OF DISCHARGE LUMINOSITY OSCILLOSCOPE TRACE OF MAGNETIC FIELD COMPENSATED ATTENUATOR UPPER AND LOWER OSCILLOSCOPE TRACES SUPERPOSED FOUR SUPERPOSED OSCILLOSCOPE TRACES MEASURED MAGNETIC FIELD DISTRIBUTION CALIBRATION FOR MAGNETIC PROBE MAGNETIC FIELD CORRECTION AT WALL OF VESSEL RESULTS FROM TWO METHODS OF SOLUTION CORRECTED CURRENT DENSITY DISTRIBUTIONS CORRECTED MAGNETIC FIELD .DISTRIBUTIONS MEASURED AND COMPUTED DISTRIBUTIONS SIMULATED HIGH CONDUCTIVITY INSIDE CURRENT SHEET PLASMA INSTABILITY ESTIMATED ERROR I N J DUE TO ERROR I N B DEPENDENCE OF CURRENT DENSITY UPON MAGNETIC FIELD GRADIENT. SIDE-ON PHOTOGRAPH OF PROBE TOP-ON PHOTOGRAPH OF PROBE END-ON PHOTOGRAPH OF PROBE 2 10 12 lU 19 21 22 2k 26 27 28 0 29 0 32 3h 35 0 0 - Iv - 39 \& h6 hQ 50 5U 55 57 62 6h 65 68 69 71 71 72 76 79 80 82 89 ' 93 95 98 100 102 105 105 109 LTST OF ILLUSTRATIONS (CONT'D.) FIGURE 5U 55 56 57 58 59 60 61-63 6k 65 66 67 '68 69 70 COLLAPSE CURVES RADIAL POSITION OF SHOCK FRONT MEASUREMENT OF FORMATIVE TIME DELAY EFFECT OF AMPLIFIER DELAY RADIAL POSITION OF SHOCK FRONT VS TIME RADIAL POSITION OF CURRENT DENSITY MAX.VS TIME GRAPHICAL SOLUTION FOR wt COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS MODIFIED DISCHARGE CIRCUIT SHOCK WAVE PARAMETERS GRAPHICAL SOLUTION OF SAHA EQUATION GRAPHICAL SOLUTIONS OF RANKINE-HUGONIOT RELATIONS DISCONTINUITY I N PLASMA COLUMN ELECTROMAGNETIC FIELDS I N THE JOULE REGION PLASMA PARAMETERS I N JOULE REGION - v - LIST OF TABLES TABLE PAGE I APPARATUS 58 II DETERMINATION OF PINCH TIME WITH PHOTOMULTIPLIER 63 III RADIAL POSITIONS OF CURRENT DENSITY FEATURES 88 IV COMPARISON OF COMPUTED CURRENT DENSITY FROM TWO MEASUREMENTS OF MAGNETIC FIELD 101 V RADIAL POSITION OF SHOCK FRONT 115 VI FORMATIVE TIME DELAY 118 VII SHOCK FRONT AND PARTICLE FLOW VELOCITIES 123 VIII DETERMINATION OF ELECTRIC FIELD 126 IX DETERMINATION OF k ^ C b ) Hll X DETERMINATION OF A l5h XI MODIFIED SNOWPLOW CALCULATION 155 XII SHOCK WAVE PARAMETERS 157 XIII VALUES OF STATISTICAL WEIGHTS 161 XIV FLOW VELOCITY BEHIND SHOCK 168 XV THICKNESS OF THE SHOCK HEATED PLASMA 169 XVI APPARENT CONDUCTIVITY I N JOULE REGION 173 XVII "ELECTRIC FIELD PRESSURE'' 176 " vi " ACKNOWLEDGEMENT It i s a pleasure to acknowledge the stimulating supervision provided by Dr. F . L . Curzon throughout these i n v e s t i g a t i o n s . It i s a pleasure also to acknowledge the assistance of the f a c u l t y members and students of the plasma physics group. In p a r t i c u l a r , I wish to thank D r . R. A . Nodwell and D r . A. J . Barnard f o r t h e i r improvements i n the presentation of the t h e s i s . I wish to thank Mr. W. R a t z l a f f , Mr. J . Lees, Mr. P. Haas, and Mr. A . Fraser for t h e i r part i n the construction of the •apparatus. I g r a t e f u l l y acknowledge the f i n a n c i a l assistance awarded by the National Research Council. - vii - REFERENCES In the t e x t , references are denoted by, F i r s t Author (Year) i.e. - A l l e n (1957) An alphabetical l i s t of the complete references are given on page 2 0 3 . 1.0 INTRODUCTION The t h e s i s i s o r g a n i z e d i n t o f o u r p a r t s as o u t l i n e d i n t h e T a b l e of C o n t e n t s . the However, a r e a d i n g o f t h i s i n t r o d u c t i o n and c o n c l u s i o n ( P a r t I V ) w i l l be s u f f i c i e n t f o r an u n d e r s t a n d i n g of t h e p r i n c i p a l r e s u l t s . The i n v e s t i g a t i o n w h i c h we d e s c r i b e b e l o n g s t o the b r a n c h o f p h y s i c s known as p l a s m a p h y s i c s , i . e . t h e s t u d y of phenomena a s s o c i a t e d w i t h i o n i z e d gases. I n p a r t i c u l a r , we s t u d y t h e b e h a v i o u r of t h e plasma i n a z - p i n c h d i s c h a r g e . T h i s i s one o f s e v e r a l e x p e r i m e n t a l c o n f i g u r a t i o n s where a magnetic f i e l d i s used to c o n f i n e the plasma. I n t h i s c a s e , "the plasma i s s u s t a i n e d b y an a x i a l c u r r e n t , I (see f i g u r e 1.), w h i c h r e s u l t s f r o m t h e d i s c h a r g e of energy f r o m a c a p a c i t o r bank. The s e l f - i n d u c e d magnetic f i e l d , B , i n t e r a c t s w i t h the a x i a l current producing a r a d i a l f o r c e , e F | r and t h i s f o r c e causes t h e plasma t o c o n s t r i c t t o a s m a l l volume along the a x i s of the discharge v e s s e l . T h i s c o n s t r i c t i o n of t h e plasma i s known as t h e " p i n c h e f f e c t " . The z - p i n c h d i s c h a r g e was f i r s t i n t r o d u c e d i n t h e hope o f c r e a t i n g a t h e r m o n u c l e a r plasma (eg. Anderson (1957)), b u t r e c e n t work has r e v e a l e d t h e i m p r a c t i c a l i t y o f t h e s e p l a n s . The d e v i c e i s now used t o produce a l a r g e volume of h i g h t e m p e r a t u r e plasma w h i c h has a w e l l d e f i n e d geometry, and such a plasma i s s u i t a b l e f o r t h e p r e s e n t s t u d y o f "plasma dynamics". "Plasma dynamics" r e f e r s t o t h e m o t i o n o f t h e plasma as a whole as opposed t o t h e m o t i o n of t h e i n d i v i d u a l i o n s and e l e c t r o n s . - 1 - CAPACITOR BANK ( 5 3 / t F ) FIGURE 1. J B© F R r I z r 0 = = = = EXPERIMENTAL CONFIGURATION A x i a l Current Density A z i m u t h a l Magnetic F i e l d R a d i a l ( J x B ) Force Radius of D i s c h a r g e V e s s e l R a d i u s o f Plasma Column Discharge Current z e 1.0 INTRODUCTION (CONT'D.) In t h i s investigation we are interested i n the dynamics of the pinch e f f e c t . We wish to obtain a more complete physical model of the c o l l a p s i n g plasma than i s now a v a i l a b l e , and we wish to the temperature, estimate density, and degree of i o n i z a t i o n of the plasma i n the z-pinch discharge. In this study we must know the forces which arise from the flow of e l e c t r i c current and thus we must know the current density d i s t r i b u t i o n as a function of time i n the discharge vessel. This i s the chief objective of the present i n v e s t i g a t i o n . The current density d i s t r i b u t i o n can be computed from the associated magnetic f i e l d d i s t r i b u t i o n , and t h i s magnetic f i e l d can be determined experimentally with the use of "magnetic probes". A magnetic probe i s simply a small c o i l of wire which has an output voltage proportional to the change of magnetic f l u x through the coil. As these probes are the p r i n c i p a l experimental t o o l , Part I of the thesis i s a discussion of t h e i r l i m i t a t i o n s and the precautions required i n t h e i r use. The most serious problem associated with magnetic probe measurement i s that the probe must be immersed i n the plasma, perturbing i t , and therefore modifying the magnetic f i e l d which i t i s supposed to measure. An appreciable modification occurs because no e l e c t r i c current can flow i n the space occupied by the probe. The probe presents a non-conducting "hole" to the current flowing through the highly conducting plasma. This effect i s investigated i n d e t a i l i n the analogue experiment described i n section 3» - 3 - 1.0 INTRODUCTION (CONT'D.) In the analogue experiment we set up a w e l l defined current density d i s t r i b u t i o n with two concentric c y l i n d r i c a l sheets which carry equal and opposite e l e c t r i c a l currents. The true magnetic f i e l d , i . e . that which would e x i s t i n the absence of a probe, can be calculated. The perturbed magnetic f i e l d can be measured. "punctures" the current sheets). (The probe Hence the difference of these two magnetic f i e l d d i s t r i b u t i o n s gives the correction for the magnetic f i e l d of a given current sheet. Knowledge of this correction allows us to correct the magnetic f i e l d measured by the probe i n the z-pinch discharge. In t h i s t o t a l magnetic f i e l d correction i s obtained by integrating the corrections from each element of current (chosen to be a c y l i n d r i c a l sheet) over the whole current density distribution. This i n t e g r a t i o n i s performed by a numerical procedure and i t i s discussed i n section Using one of the analogue experimental curves i n the numerical correction .procedure we can t e s t the accuracy of the c o r r e c t i o n . The computed current density d i s t r i b u t i o n i s compared with the known current d i s t r i b u t i o n of the single current sheet of a given r a d i u s 0 Thus we conclude Part I knowing that we can correct the measured magnetic f i e l d which we obtain with a given magnetic probe. In Part I I we present a l l r e s u l t s of experimental measurement on the z-pinch discharge. The probe measures the magnetic f i e l d as a function of time at the r a d i a l p o s i t i o n of the probe c o i l . However, as the z-pinch discharge i s reproducible from discharge to discharge — ll <o 1.0 INTRODUCTION (CONT'D.) we can construct a p l o t of magnetic f i e l d as a function of radius f o r a given time. The magnetic f i e l d d i s t r i b u t i o n i s obtained at Ijjsec i n t e r v a l s up to the pinch time (the time required f o r the f i r s t c o n s t r i c t i o n of the plasma) f o r each of three i n i t i a l pressures of argon (100,250 and 500juHg.) We correct these measured results and we compute the r a d i a l current density d i s t r i b u t i o n from the corrected magnetic f i e l d . These current density d i s t r i b u t i o n s have two ing properties. First,there interest- i s a d i s t i n c t current density maximum, and second, there i s a region of almost constant current density - a "plateau" region. (See figures 37—14.2)• We supplement the magnetic probe technique with photographic techniques which have been used previously on the z-pinch discharge. F i r s t , a very high speed framing camera allows us to photograph the radius of the luminous plasma as a function of time. (Curzon (1962)). Secondly, an extension of t h i s technique has been developed by Folkierski (1963) which allows us to observe the propagation of the shock front i n the discharge plasma. We expect to observe such a shock front i n the z-pinch discharge because of the rapid r a d i a l motion of the collapse ( t y p i c a l l y 10^ meter per s e c ) . Part II with a correlation of the prominent features We conclude of the current density d i s t r i b u t i o n with those of the photographic measurements. In Part III we use two theoretical models to predict the experimentally measured plasma dynamics. These predictions compare favourably with the observed r e s u l t s and thus we are j u s t i f i e d i n . 5 - 1.0 INTRODUCTION (CONT'D.) i n using these t h e o r e t i c a l models to obtain a d d i t i o n a l information about the plasma (for example, temperature and d e n s i t y ) . The f i r s t t h e o r e t i c a l model i s the snowplow model of Rosenbluth (19!?1|). The current i s assumed to form a very t h i n sheet near the v e s s e l w a l l which then collapses toward the axis of the v e s s e l gathering a l l the gas as i t does so. With this model, we can develop both an equation of motion f o r the c o l l a p s i n g current sheet and an equation f o r the current i n the discharge c i r c u i t . These two equations must be solved simultaneously f o r the radius of the current sheet and the discharge current as functions of time. An approximation technique i s used to solve these second order d i f f e r e n t i a l equations - one of which i s non l i n e a r . The second t h e o r e t i c a l model i s known as the shock wave model and i t has been discussed i n connection with shock tube flow by Reynolds (I96l). In simple shock theory, the Rankine-Hugoniot r e l a t i o n s are derived from the conservation of mass, momentum, and energy across the shock d i s c o n t i n u i t y . In a plasma, these r e l a t i o n s must be supplemented by Saha's equation for the determination of the microscopic constituents of the plasma. We solve these equations with a graphical technique. Both of these models provide u s e f u l concepts f o r the understanding of the z-pinch discharge. However, neither of the models predicts the r e l a t i v e l y large thickness of the luminous plasma, the "plateau" region of the current density d i s t r i b u t i o n or the unusual dependence - 6 - 1.0 INTRODUCTION (CONT'D.) upon i n i t i a l pressure and time of the e l e c t r i c f i e l d in the plasma. These f a c t s lead us to propose a new model of the plasma i n the collapse stage of the z-pinch discharge. by two d i s t i n c t l a y e r s . The model i s characterized The inner layer has a higher p a r t i c l e density and a lower temperature than the outer l a y e r . The model i s discussed i n section 15.0 with p a r t i c u l a r reference to p e c u l i a r i t i e s mentioned above. F i n a l l y , i n Part 17 we present the conclusions of our investigation. - 7 - PART I MAGNETIC PROBE MEASUREMENT - 8- 2.0 INTRODUCTION It i s recognized that the introduction of a probe into a plasma must cause some perturbation, and indeed, Ecker (1962) has shown t h i s to be the case i n a t h e o r e t i c a l treatment of probe e r r o r . However, he has used a numerical technique to solve the problem and t h i s i s not e a s i l y adaptable to a given experimental configuration. In t h i s Part of the t h e s i s , we f i r s t describe our magnetic probe (section 3) and then examine the probe perturbation by means of an analogue experiment (section k ) » The results of the analogue experiment compare extremely well with the simple case of a plane current sheet punctured by a probe as examined t h e o r e t i c a l l y by Malmberg (196I4) (section I1.3). F i n a l l y we develop the numerical program f o r the a p p l i c a t i o n of these results discharge 3.0 to the z-pinch (section MAGNETIC PROBES A magnetic probe consists of a small wire c o i l which must be supported and shielded inside the plasma i n a manner, f o r example, as described by Segre (I960) or as i l l u s t r a t e d i n figure 2. The d e t a i l s of construction are given i n Appendix I, but a few of the most important ideas are presented here. The probe i s shielded mechanically from the hot plasma by a quartz guide tube (see f i g u r e 5>) and i t i s shielded e l e c t r i c a l l y by a very t h i n s i l v e r f i l m which i s at ground p o t e n t i a l . -3 the s i l v e r f i l m (~10 The thickness of „1 m.m.) i s much l e s s than the skin depth ( i / l O m.m^ - 9 - FIGURE 2. MAGNETIC PROBE MAGNETIC PROBES (CONT'D.) and thus no currents are induced i n the shield which would perturb the magnetic f i e l d at the probe c o i l . I t i s possible to obtain undesireable signals which are caused by magnetic f l u x threading through the probe leads. These signals may be eliminated by t i g h t l y winding together the leads to the probe c o i l . In t h i s way the f l u x through one loop i n the leads approximately cancel's the f l u x through the adjacent loop, and the net noise s i g n a l i s zero. The magnetic probe produces an output voltage which i s proportional to the rate of change of magnetic f l u x passing through the plane of the probe c o i l . Thus the i n t e g r a l of the probe s i g n a l - 10 - 3.0 MAGNETIC PROBES (CONT'D.) i s proportional to the magnetic f i e l d . The i n t e g r a t i o n i s performed by a simple R C - c i r c u i t with a time constant of 112/Lsec figure 3.1 (see 3). MEASURING CIRCUIT In the presence of pulsed discharge c i r c u i t s , such as the z-pinch discharge, there are two sources of noise signals which d i s t o r t the experimental s i g n a l and must, therefore, be eliminated. The f i r s t source of noise i s "capacitative pickup". There w i l l be a stray capacitance (C) between the discharge c i r c u i t and the measuring c i r c u i t and as the voltage (V) on the discharge c i r c u i t and the discharge frequency (w) are very high (approximately 10^ v o l t s and 10^ c/sec r e s p e c t i v e l y ) , the r e s u l t i n g noise signals (ocwCV) on the measuring c i r c u i t may be appreciable. source of noise i s " r a d i a t i v e pickup". High frequency noise signals are radiated when the spark gap switch i n the c i r c u i t (see figure l ) i s "closed" i n i t i a l l y . Medley ( 1 9 6 5 ) The second discharge See, f o r example, . Capacitative pickup can be eliminated by using the balanced d i f f e r e n t i a l c i r c u i t which i s shown i n figure 3. An " A - B " , D-type Tektronix preamplifier i s used i n the o s c i l l o s c o p e , and f u r t h e r , the integrators are provided with a rheostat which i s adjusted to give an optimum common mode r e j e c t i o n of approximately h o £ 10 :1 f o r frequencies ranging from approximately 10^ to 10-^ cps. - 11 - MAGNETIC PROBE DELAY LINE OSCILLOSCOPE - INTEGRATOR R> — y ^ A — H -AAAAA Sfflffl*- X 2-o »A-B» AWAV R. PREAMP R m FIGURE 3 . MAGNETIC PROBE MEASURING CIRCUIT L,C = A . C . equivalent c i r c u i t of magnetic probe L = ,U5 XtH C » 15 pF Rm = Matching resistance = 220_n_ L^ = . 5 / J i s e c delay l i n e R^ = Rj_'= Rb = Cj_ = Integrating resistance Integrating resistance Balancing resistance Integrating capacitor - 12 - = 2.2 k_ru = 1.5 k-0_ = 0 - 2 . 5 kXL. = .05 MEASURING CIRCUIT (CONT'D.) Radiated noise signals are minimized by c o a x i a l l i z i n g the main spark gap switch and the triggering c i r c u i t r y (see figure 27). The measuring c i r c u i t i s also c o a x i a l l i z e d and shielded. These p r e - cautions reduce the noise signal but i t i s s t i l l s i g n i f i c a n t f o r 0.£>usec. Thus the measuring c i r c u i t of figure 3 i s provided with a 0.5^/sec delay l i n e which allows the desired signal to be delayed u n t i l the radiated noise s i g n a l has decayed. A complete discussion of noise signals and elimination of noise signals i s given by Medley (1965b). The frequency response of the measuring c i r c u i t must also be examined. I f we consider f i r s t the probe and delay l i n e , then we wish the output to be proportional to the input for a l l frequencies. The input signal w i l l be proportional to dB/dt. However, th? has an inductance and capacitance, and therefore i t ' h a s a resonance at a particular frequency. probe Following the procedure outlined by Segre (i960), we can c r i t i c a l l y damp this resonance by terminating the probe with i t s c h a r a c t e r i s t i c impedance of 120JL. Similarly, however, the delay l i n e has a resonant frequency and a c h a r a c t e r i s t i c impedance of 220JI. In our experiment we terminate the delay l i n e properly and we measure the frequency response of the probe and delay line to check that i t remains " f l a t " up to frequencies of approximately lO^cps. This i s s u f f i c i e n t f o r a study of magnetic f i e l d signals with the z-pinch discharge. associated It can be seen i n figure k that the • 1 1 c 1 ? ' > <i ; > r — 90° 45" PROBE RESPOl fSE » \ » nil, o > £.0 5.0 /o.o Ml 1-0 3 <,9o° BTEGRATOR RESPONSE 45" 4 •2 2.0 1.0" 5.0 IYnc/ )-*~ s FIGURE U. FREQUENCY RESPONSE CURVES Phase s h i f t Gain f = Frequency IAI - Hi - 3.1 MEASURING CIRCUIT (CONT'D.) frequency response i s " f l a t " up to approximately 5 x lO^cps and that the measuring c i r c u i t introduces no phase change below this frequency. (Note: If i t were necessary, the frequency response could be improved by terminating both probe and delay l i n e with t h e i r characteristic respective impedances, but this procedure further attenuates the signal.) We have shown now that the output s i g n a l from the probe and delay l i n e i s a f a i t h f u l representation of the input s i g n a l . However, the input s i g n a l i s proportional to dB/dt and we are interested i n measuring B. Thus we integrate the signal with the RC network shown i n figure 3. The condition for f a i t h f u l integration i s that the product of gain and frequency should be constant f o r a l l frequencies and furthermore the integrator should introduce a 90° phase change i n signals of a l l frequencies. Both of these conditions are f u l f i l l e d as may be seen i n figure Ij.. We are j u s t i f i e d i n considering these two portions of the measuring c i r c u i t separately as the integrator presents a large impedance to ths delay l i n e and hence does not " l o a d " i t when the two c i r c u i t s are used together as shown i n f i g u r e 3» 3.2 MAGNETIC PROBE ERROR The most serious problem i n probe measurement of the current density d i s t r i b u t i o n i s the perturbation of the current caused by the presence of the probe i n the plasma* - 15 - In the l i t e r a t u r e , the MAGNETIC PROBE ERROR (CONT'D.) problem has received only preliminary i n v e s t i g a t i o n . Lovberg (1959) suggested that only a small error i n the magnetic f i e l d would occur, because the f i e l d at any point i s l a r g e l y determined by contributions from the more remote portions of the current density. Ecker (1962) considered the problem t h e o r e t i c a l l y , However, when he found the probe error to be appreciable, p a r t i c u l a r l y f o r the region at the boundary of the current d i s t r i b u t i o n . Ecker considered a time independent current d i s t r i b u t i o n and he calculated the magnetic f i e l d r e s u l t i n g from the flow of current around the magnetic probe. He also calculated the magnetic f i e l d for the unperturbed current d i s t r i b u t i o n and the difference of these two f i e l d s he called the probe e r r o r . The calculations were performed numerically on several r o t a t i o n a l l y symmetric current distributions. Although Ecker's r e s u l t s i l l u s t r a t e the importance of considering probe e r r o r , they are impractical f o r use i n the correction experimentally measured magnetic f i e l d s . of The chief advantage of the error considerations i n the following two sections of t h i s thesis is that they may be r e a d i l y applied to any l i n e a r or inverse pinch device. Ecker introduced terminology which w i l l be u s e f u l i n the f o l l o w ing d e s c r i p t i o n s . "Boundary error" refers to the error introduced i n the magnetic f i e l d when a non-conducting hole i s formed i n a conducting medium. "Conductivity error" refers to the error i n the magnetic f i e l d caused by the cooling e f f e c t of impurities from the probe which - 16 » MAGNETIC PROBE ERROR (CONT'D.) are "boiled o f f " i n t o the plasma. Note that the impurities e f f e c t i v e l y increase the radius of the low conductivity h o l e . Hence we can combine the treatment of conductivity error with that of boundary error by considering the boundary error caused by an e f f e c t i v e probe size which i s greater than the actual probe s i z e . Experimentally, the problem has received l i t t l e attention. Harding ( 1 9 5 8 ) examined the influence of two probes, one upon the other, f o r probe separations of approximately 70 cm. He observed no change i n the general f i e l d configuration at one probe when the other probe was introduced into the plasma. Dippel ( 1 9 5 9 ) performed an analogue experiment to investigate the e f f e c t of the probe on measurement of the magnetic f i e l d . He investigated the f i e l d associated with a c y l i n d r i c a l conductor of known c o n d u c t i v i t y . The f i e l d was measured f o r several diameters of the probe hole and these r e s u l t s were then compared with the t h e o r e t i c a l magnetic f i e l d . The t h e o r e t i c a l magnetic f i e l d i s e a s i l y calculated as the skin depth of the current d i s t r i b u t i o n i n the conductor i s known. He repeated the experiment f o r conductors of various conductivity and he concluded that the probe error becomes n e g l i g i b l e only for large skin depths, i . e . low c o n d u c t i v i t y . From his data we observe that the measured magnetic f i e l d i s approximately independent of the conductivity over the range of 2 x lO^-mho/m to U x 10^mho/m which he has examined. This implies that i n a l l cases the s k i n depth i s small compared to the c h a r a c t e r i s t i c length of the f i e l d correction (see section i j . 2 , f i g u r e 9 ) and thus we can consider his current d i s t r i b u t i o n to be a = 17- 3.2 MAGNETIC PROBE ERROR (CONT'D.) sheet l i k e our proposed analogue experiment. ing It i s then encourag- to observe i n his data that the probe error i s i n s e n s i t i v e to the exact radius of the probe h o l e . The probe e r r o r varies by approximately $0% when the probe radius varies by approximately 80$, Thus i t appears that the effective probe radius i s not c r i t i c a l and a rough estimate of conductivity error w i l l be s u f f i c i e n t . We concentrate upon an examination of boundary error with the intention of compensating f o r the e r r o r rather than t r y i n g to reduce the error to a negligible effect. In compensating for the e r r o r i t i s advantageous to make the probe perturbation as geometrically simple as p o s s i b l e . In the actual discharge v e s s e l , a quartz tube i s placed d i a m e t r i c a l l y through the vessel and the magnetic probe i s moved within the tube (see f i g u r e £ ) » In t h i s way the perturbation of the plasma i s independent of the probe p o s i t i o n . A similar arrangement i s used i n the analogue experiment and this i s discussed i n the next s e c t i o n , U,0 ANALOGUE EXPERIMENT The purpose of the analogue experiment i s to investigate the effect of the above described boundary error on a w e l l defined current d i s t r i b u t i o n and hence on a known "true" magnetic f i e l d . As the experiment i s designed to correct measurement on the z-pinch discharge which i s characterized by a r o t a t i o n a l l y symmetric current sheet, the analogue magnetic f i e l d i s defined by two concentric cylinders of aluminum f o i l (see figure £ ) . - 18 - The outer cylinder QUARTZ GUIDE TUBE MAGNETIC PROBE ANALOGUE EXPERIMENT r = R a d i u s o f o u t s i d e sheet = 9 cm. r-j_ = R a d i u s o f i n s i d e s h e e t I *» C u r r e n t 0 QUARTZ GUIDE TUBE GNETIC PROBES Z-PINCH EXPERIMENT R r. R a d i u s o f r e t u r n conductor Radius o f discharge v e s s e l Discharge current 8.5 7.5 cm. cm. ANALOGUE AND Z-PINCH GEOMETRY FIGURE 5- - 19 - ANALOGUE EXPERIMENT (CONT'D.) represents the return conductor of the actual discharge c i r c u i t and the inner conductor, which can be varied i n diameter, corresponds to the plasma current. A s i n u s o i d a l current supply i s available and i t i s desireable to use the magnetic probes which w i l l be used i n the l a t e r experiments. 7 —1 The frequency (w) of the supply i s approximately h x 10'sec. output voltage The (V) from the probe i s given byt Ve^ = Kwle^ -10 where K, a geometrical f a c t o r , i s 2.5 x 10 h. I i s the current i n the sheet, and, w i s the frequency of o s c i l l a t i o n . -2 For an output voltage of approximately 10 use of an o s c i l l o s c o p e , the required current 1 amp. (I) v o l t s which allows i s of the order of Such a current at t h i s frequency suggests that the concentric conductors of the analogue experiment be included as the inductance of a p a r a l l e l resonant circuit. APPARATUS The analogue experiment i s scaled to approximate the a c t u a l experiment. The outer aluminum conductor has a radius of 9.1 cm. and the inner conductor radius can be selected from any of l . l i , 2.6, I 4 . I , 5«3> and 6.8 cm. These conductors are 60 cm i n length and approximately .02 mm thickness. A photograph of the experiment i s shown i n f i g u r e 6 and an equivalent e l e c t r i c a l c i r c u i t i s shown i n f i g u r e 7. - 20 - FIGURE 6. PHOTOGRAPH OF ANALOGUE EXPERIMENT APPARATUS (CONT'D.) The coaxial conductors form part of a p a r a l l e l resonant c i r c u i t and the power supply has a variable frequency range which allows tuning to the resonant frequency for each of the various inner conductors. To increase further the output signal voltage, a series resonant c i r c u i t i s used as shown i n figure 7. This c i r c u i t can also be tuned to the resonant frequency of the circuit. analogue A Rogowski c o i l i s used to monitor the current i n the analogue c i r c u i t while measurement of the magnetic f i e l d i s taken. As i n the actual discharge, a glass tube i s placed d i a m e t r i c a l l y through the concentric aluminum sheets and the probe can be moved - 21 - PROBE CIRCUIT UPPER o 4-8 Mc/s BEAM Eg MONITOR TANK GENERATOR OSCILLOSCOPE CIRCUIT FIGURE 7. U.1 EQUIVALENT CIRCUIT OF ANALOGUE EXPERIMENT APPARATUS (CONT'D.) within i t . The experiment consists of measuring the probe output as a function of r a d i a l p o s i t i o n both inside and outside the current carrying conductors. A t r a v e l l i n g microscope i s used to measure probe positions and a dual beam oscilloscope i s used t o observe the probe and monitor s i g n a l s . k*2 EXPERIMENTAL RESULTS In the analogue experiment we are t r y i n g to determine the correction f o r the magnetic f i e l d ( A B ( r ) ) , which i s defined as the difference between the unperturbed or true magnetic f i e l d - 22 - (Bu(r)) EXPERIMENTAL RESULTS (CONT•D.) and the perturbed or measured magnetic f i e l d (Bp(r)). For the region between the concentric c y l i n d e r s , see f i g u r e the unperturbed magnetic f i e l d i s known from Amperes law to f a l l o f f as V r . The perturbed f i e l d , so called because of the c i r c u l a r hole punched i n the aluminum conductor f o r the probe, i s measured as described i n section U . l . Hence, the correction i s e a s i l y . obtained f o r a t h i n c y l i n d r i c a l current sheet as the difference between these two f i e l d s . Before we proceed with the experiment, the output of the magnetic probe should be checked to ensure that the only coupling between the probe and the analogue c i r c u i t i s provided by magnetic f l u x through the probe c o i l . In section 3 * 0 we have discussed two further p o s s i b i l i t i e s f o r coupling. We have attempted to avoid capacitative coupling by using a grounded shield around the probe and we have attempted t o avoid voltages induced i n the probe leads by t i g h t l y winding the leads together. It i s the effectiveness of these precautions which we wish to check. If we rotate the magnetic probe about i t s own axis when the probe c o i l i s held at a given r a d i a l p o s i t i o n between the conductors of the analogue experiment, then we expect the s i g n a l to vary as shown i n f i g u r e 8A. Note that we observe only the amplitude of the probe s i g n a l by choosing a s u f f i c i e n t l y - s l o w sweep rate of the oscilloscope. - 2 3 - B. FIGURE 8 . C. DEPENDENCE OF PROBE OUTPUT UPON ANGLE OF PROBE ROTATION 0 - Angle between axis of probe c o i l and the direction of current flow. 8A 8B 8C - Desired output. Output indicates spurious e l e c t r o s t a t i c s i g n a l . Output indicates spurious magnetic s i g n a l . EXPERIMENTAL RESULTS (CONT'D.) If there are any capacitative s i g n a l s , these w i l l not vary appreciably as the probe i s rotated about i t s a x i s , and the base l i n e w i l l appear s h i f t e d as i n f i g u r e 8 B by an amount proportional to the capacitative s i g n a l . EXPERIMENTAL RESULTS (CONT'D.) If the probe leads are not t i g h t l y wound, then they w i l l present a net area to the magnetic f l u x which i s at some angle to the plane of the probe c o i l . Thus the resultant signal w i l l be the sum of the voltages induced i n the leads and i n the c o i l . As the probe i s rotated about i t s a x i s , t h i s resultant s i g n a l w i l l , i n general, have maxima and minima which are shifted away from 90° and 270° as shown i n f i g u r e 8C. The magnetic probe was rejected unless the output varied i n the manner shown i n figure 8A. A t y p i c a l r e s u l t from the analogue experiment i s shown i n figure 9. This r e s u l t corresponds to a probe radius (p^ an inner conductor radius (r.j) of 2.6 radius of 9.1 of .1x5 cm, cm, and an outer conductor cm (see also f i g u r e 5)» The unperturbed magnetic f i e l d d i s t r i b u t i o n i s also included i n figure 9. This curve i s obtained by p l o t t i n g the measured points of figure 9 on a ^/r scale as shown i n f i g u r e 10. extrapolated to r^ and r 0 The straight l i n e section i s then and thus we obtain values of Bu(r) which are then transferred back to f i g u r e 9. The i n s e r t i n f i g u r e 9 represents a plot of A B ( r ) - the difference of Bu(r) and B p ( r ) . Note that the s p a t i a l r e s o l u t i o n of the probe i s limited by the diameter of the c o i l which i s approximately 1 mm. However, AB(r) extends over a distance of approximately 10 mm., and i t i s , therefore, a r e a l correction and not a property of the probe. - 25 - £ 4 6 8 / 0 r (cm) — « — FIGURE 9 . ANALOGUE EXPERIMENTAL RESULTS i*i = Radius of inner current sheet r = Radius of outer current sheet z = Distance from inner current sheet * Measured magnetic f i e l d ° True magnetic f i e l d (figure 10.) True magnetic f i e l d minus measured magnetic f i e l d . 0 x - 26 - \ \ \ \ -+-> in *> CQ ri= Z.60cm. •5 lo • o 5 (SCALE PROPORTIONAL TO FIGURE 1 0 . o I$0 6 y ii RCcm) S> i< UNDERTURBED MAGNETIC FIELD Experimental points from f i g u r e 9 . Extrapolated magnetic f i e l d Iu2 9.15 cm distribution. EXPERIMENTAL RESULTS (CONT'D.) The experiment i s repeated f o r each of the inner conductors mentioned i n section l i . l and then the entire procedure i s repeated f o r a probe radius of . 3 5 cm. In order to examine the r e s u l t s of the analogue experiment more e a s i l y we define the f i e l d correction A B ( z ) , where z i s the distance of the f i e l d point from the p o s i t i o n of the current sheet, i . e . , z = r - r-j_. - 27 - > II = 5»6 cm. ]1 = 2.8 cm. • ]* = 2.2 cm. a ]} = 1.6 cm. O A • • A * j1 0 I c • p > 1 A 1i A. ( • oo • • : < .6 •2 FIGURE 11/ O AOQ 0 B h.2 % Izl C c m ) — C(Z)| vs .8 Izl correspond to positive values. correspond to negative values. EXPERIMENTAL RESULTS (CONT'D.) Furthermore we l e t B(z) - KC(z) Where K i s a constant of p r o p o r t i o n a l i t y and C(z) i s a dimensionless correction with the normalization C(0) = 1 - 2 8 - uo ( /Oo = .35 cm.) IJ3> 0 A ^ • .8 R = 6 . 8 cm. R 5.3 cm. R = I4..O cm. R *» l.lj. cm. a .6 • ^ A • • • .2 • 6 • o .4 • .6 .8 IZI tern) - FIGURE 12. C ( z ) l vs I z l O A O D correspond to p o s i t i v e values. correspond to negative values. • 1.0 • U5 cm.) EXPERIMENTAL RESULTS (CONT'D.) In f i g u r e 11 we plot the correction | C(z) | versus | z | f o r a l l measurements taken with a probe of radius . 3 5 cm. be noted. Two points are to F i r s t , the points corresponding to p o s i t i v e and negative Z do not d i f f e r f o r a given sheet radius i . e . the correction appears to be anti-symmetric. C(z) Second, the points corresponding to current sheets of d i f f e r i n g radius do not d i f f e r appreciably, - 29 - EXPERIMENTAL RESULTS (CONT'D.) i . e . i t i s reasonable to assume that the correction C(:z) i s independent of the radius of the current sheet. The second point i s even more obvious i n figure 12 where we have made the same p l o t , |C(z) | v s . | z | , f o r the measurements taken with a .1*5 cm. radius probe. THEORETICAL MODEL The analogue experiment i s designed to measure the magnetic f i e l d as a function of p o s i t i o n along the axis of a non-conducting cylinder (the probe) placed diametrically through two c o a x i a l , c y l i n d r i c , current carrying sheets. It i s a complicated mathematical problem to predict such a r a d i a l l y dependent magnetic f i e l d . In the present treatment we consider a s i m p l i f i e d model which, i t i s hoped, w i l l aid i n the i n t e r p r e t a t i o n of the experimental In the l i m i t that r 0 results. and r^_, the r a d i i of the current carrying sheets, are l a r g e , the sheet i n the v i c i n i t y of the probe hole i s approximately plane. A solution f o r t h i s problem has been obtained by Malmberg ( l ° 6 h ) . where B B i s the perturbed magnetic f i e l d 0 i s the unperturbed magnetic f i e l d yq, i s the probe r a d i u s . - 30 ~ THEORETICAL MODEL (CONT'D.) In our previous notation (see page 25) we have w r i t t e n : Bu(p) - Bp(r) where *AB(z) Bu(r) i s the unperturbed magnetic f i e l d Bp(r) i s the perturbed magnetic f i e l d ^B(z) i s the f i e l d correction 2= r - x i s the distance between the f i e l d point r and the current sheet at x. Furthermore f AB("Z) « KC(Z) Thus we now have: where JJio i s the permeability r i s the thickness of the current sheet Jo i s the current density i n the sheet and, cfc) - 2 / %fo> o if V (i +z p ) 2 / 2 e " ?r cot A, As the current d i s t r i b u t i o n i n the sheet i s perturbed appreciably over a distance of only a few times the probe r a d i u s , then there i s a contribution to the correction from a correspondi n g l y small area of the current d i s t r i b u t i o n . In our analogue experiment the current sheet i s approximately planar over t h i s small area and hence even t h i s simple model gives reasonable - 31 - z| FIGURE 1 3 . |c(z)| vs |Z| (cm.) (THEORETICAL CURVES) THEORETICAL MODEL (CONT'D.) predictions. The correction Cfe) i s plotted against z> i n f i g u r e 13 f o r the two cases of i n t e r e s t - /O = .35 e P" 0 ,U5 cm. cm. and, CONCLUSIONS Consider the procedure which we w i l l be following i n the next sections. We w i l l have a measured magnetic f i e l d which we w i l l wish to correct i n order to calculate the associated true current density distribution. However, we must correct the magnetic f i e l d by an amount which i s i t s e l f a function of the "true" current density distribution. Mathematically, t h i s problem i s more e a s i l y set up i f we have an analytic expression f o r the c o r r e c t i o n . Thus we compare our experimental points of section 1|.2 with the t h e o r e t i c a l l y predicted curves of section k»3» The agreement i s excellent f o r both the case of y0 = .35 cm. (figure l U ) and p > = ,U5 cm. (figure 15X o o For comparison, we have also included i n f i g u r e Iii the . experimental points which are obtained from a measurement of the magnetic f i e l d outside the return conductor of the actual z-pinch discharge v e s s e l . This apparatus i s described i n a l a t e r section of the thesis (section 7.0), but i t i s s u f f i c i e n t to state here that the return conductor i s a t h i n , fine-meshed, brass screen with a hole of approximately .35 cm. f o r the probe. These points are obtained when the plasma i s away from the w a l l of the discharge v e s s e l (500/LHg argon j 11.5,/U.sec) and hence we expect the error i n the magnetic f i e l d to be associated with the return conductor alone 0 These points also agree with the t h e o r e t i c a l p r e d i c t i o n . It i s observed that the correction i s a function of the radius of the probe hole. In section 3.2 we have included conductivity error i n our discussion with the introduction of an e f f e c t i v e probe FIGURE 1U. |C(z) | v s . |Z | {jQ= .35 cm.) CONCLUSIONS (CONT'D.) radius. Work of Dippel (1959) suggested that the exact value of the e f f e c t i v e radius was not c r i t i c a l . S i m i l a r l y , i n f i g u r e 13, where we have plotted the curves of / O «= .35 cm., and, f>0 » .U5 cm., we observe 0 that a 25$ change i n p0 r e s u l t s i n less than 10$ change i n f i e l d correction over that region where the correction i s appreciable. We conclude that an i n v e s t i g a t i o n of boundary error allows f o r a correction of the greatest perturbations caused by the immersion of - 3h - \ x \ X 1HEORETICAL C URVE MLOGUE POIN TS \x x\ t 1 \ y \* 0 ^ .2 X : * .2 .4 FIGURE 1 5 . U.U 8 X .6 |c(z)| vs. |ZJ Izl >: ) .8 (cm) 1.0 — ( p~ cm.) CONCLUSIONS (CONT'D.) the probe i n the plasma. derived by Malmberg (196U) We conclude that the analytic expression i s a reasonable approximation t o the correction. - 35 - 5.0 MAGNETIC FIELD CORRECTION In t h i s section we consider the correction of that magnetic f i e l d which i s obtained from measurement on the z-pinch discharge. The apparatus and the experimental procedure w i l l be explained i n l a t e r sections (sections 7 and 8, r e s p e c t i v e l y ) , but i n t h i s section we simply assume the measured, or perturbed, magnetic f i e l d (Bp) i s known. Under experimental conditions the current density d i s t r i b u t i o n i s unknown and yet the f i e l d correction (AB) i s a function of current density (see section U . 3 ) . The correction procedure i s performed i n the following way. Assume that we know the unperturbed f i e l d Bu, then we can obtain the "true" current density from a solution of Maxwell's equation: V * = Moi B Using the c y l i n d r i c a l geometry of the z-pinch discharge and the f a c t that we have only an azimuthal magnetic f i e l d , we have: J(r) « _1_ A r 6_ <^r (rB(r)) where r i s measured from the axis of the discharge v e s s e l . We must now determine the f i e l d correction A B ( r ) . For t h i s discussion, we use the variable " r " f o r the f i e l d point and the variable "x" f o r the source point, i . e . the position of an element of current density. The current density i s thus J ( x ) . In t h i s notation, the f i e l d correction A B ( z ) , of the analogue experiment becomes A B ( r - x ) since Z , ' t h e distance from f i e l d to source p o i n t , i s given by ( r - x ) . - 3 6 - MAGNETIC FIELD CORRECTION (CONT'D.) I f we have a single current sheet of radius x , 0 section k»h' then from ' B(r-x ) - J(x ) A t c 0 I T ( I (r-xo)//V (1+ c o t " /O,o then the t o t a l ( A B ( r ) ) must be calculated as the i n t e g r a l over a l l elements of the current d i s t r i b u t i o n . AB(r) where = JAB(r-x)dx C(r-x) = _2_ / TTV » \ J(x) (r-x)/y0 ( l + (r-x)*/fa*) 0 C(r-x)dx. - cot" (r-x) ^ Po J 1 and where R i s the inner radius of the discharge v e s s e l . F i n a l l y we must add t h i s f i e l d correction to the measured magnetic f i e l d to get the unperturbed f i e l d . Bu(r) » Bp(r) + A B ( r ) . Thus the equation which must be solved f o r Bu(r) i s : Bu(r) • Bp(r) + 1 2 \ J o C(r-x) x d \_ x Bu(x)j dx dx where Bp(r) and C(r-x) are known f u n c t i o n s . We may also set up the problem to obtain an equation f o r the current density d i r e c t l y . AB(r) = Us 2 and, ^ 1 (r*o)Vp>*) However, i f we have a current d i s t r i b u t i o n J(x), correction factor - Bu(r) = [ J(x) C(r=x)dx Jo J&p j x J(x) dx o r - 37 5.0 MAGNETIC FIELD CORRECTION (CONT'D.) Thus, r x J(x) JU T R dx = Bp(r) + JU *2 J We must solve t h i s equation f o r j(x) J(x) C(r-x) dx J i n terms of the known functions Bp(r) and C ( r - x ) . As we do not, i n f a c t , have an analytic expression f o r the measured f i e l d Bp, but rather a set of measurements taken at given r a d i a l p o s i t i o n s , then a numerical solution becomes necessary. 5.1 NUMERICAL COMPUTATION Tn the previous section we have developed two equations which must now be solved. r 1. * JU f x J(x) dx - Bp(r) + JU Jo 2J J(x) C(r-x) dx r 2. Bu(r) = Bp(r) + 1 2 C(r--x) x d_ dx 0 (xBu(x)) dx In order to solve the f i r s t equation f o r J(x) following approximation. we consider the We construct a set of concentric carrying sheets^, each of thickness Y current = 1 mm., to replace the continuous current d i s t r i b u t i o n of the z - p i n c h . The magnitude of the current density i n each sheet i s determined by i t s r a d i a l p o s i t i o n as i s shown i n f i g u r e 16. At the same time, we replace the mathematical integration over the current d i s t r i b u t i o n by a summation over the constructed - 38 - FIGURE 16. SHEET APPROXIMATION TO CONTINUOUS CURRENT DENSITY DISTRIBUTION NUMERICAL COMPUTATION (CONT'D.) sheets. If we have one current sheet at x = x , 0 then from the analogue experiment (section U«3) we haves dB(r-Xo) - Jl/Y J ( x ) C(r-xo) 0 But i f we have a number of current sheets, we haves B(r) - 2- JAaX & ) C(r-oc) J x=o Similarly, Bu(r) - JL> Bu(r) - JA£ / r Y> 2 x J(x) dx becomes, r 2L. x=»o x j ( x ) NUMERICAL COMPUTATION (CONT'D.) Thus, we may write equation 1. a s : Now the i n t e r v a l (T) i s 1 mm. and we have values of Bp(r) from r=0 to r=80 mm. section h*3* C(r-x) i s also known f o r 1 mm. i n t e r v a l s from Hence there are 80 equations (one f o r each r ) of the form just shown which must be solved simultaneously f o r the eighty values of J(x). To s i m p l i f y the mathematical procedure these equations are written i n matrix notation and the problem i s then solved on a computer. In matrix notation, the equation becomes: JJ. y c Srx Jx = (Bp) r + JU*7 Crx Jx 2~ where Jx Srx i s an 80 - component vector i s an 80 x 80 matrix of the form: with i n d i v i d u a l elements (Srx) given by: (Srx) = x r if x^r (Srx) = 0 if x>r - 10 - NUMERICAL COMPUTATION (CONT'D.) (Crx) - 2 / (r-x)/Po TT I (1 + (r,-x)V^b 2 ) - cot" (r-x) 1 /O 0 Since the magnitude of (Crx) f a l l s to 1% at ( r - x ) «= 9 mm., we consider only 9 sub-diagonals above and below the main diagonal i n the matrix representation of Crx. The remainder of the matrix i s zero. The solution of equation 1. i s s J xa JL < '* 1 ) Srx Crx MoT i.e. 2 -1 (Bp)r we must invert an eighty by eighty matrix. S i m i l a r l y , we can solve equation 2. (page 3 8 ) . notation t h i s equation becomes: (Bu)r » (Bp)r + 1 2 - Ul - Crx Dxx' (Bu)x' In matrix NUMERICAL COMPUTATION (CONT'D.) where Dxx' i s an eighty by eighty matrix of the forms 12.3 ... with the i n d i v i d u a l elements (Dxx ) given byt 1 (Dxx») ° ( x ' + 1) 8 (x-oc'+l) - ( x ' - l ) & ( x - x ' - l ) 2x' 2x« where & (x-x«+l) = 1 if (x-x'-l) = 0 fS(x-x'+l) = 0 if (x-x'-l) H 0 The solution of t h i s equation i s given formally by: -1 (Bu) t = ( I r x « - 1 Crx Dxx) (Bp)r x where Irx 1 i s the i d e n t i t y matrix. From equation 1. we solve d i r e c t l y f o r the current density Jx and from t h i s we solve f o r the unperturbed magnetic f i e l d (Bu)r from: (Bu)r * ^ o T S r x Jx However, i n equation 2 . we solve d i r e c t l y f o r the unperturbed magnetic f i e l d (Bu) and then solve f o r the current density Jx r from: Jx = _1_ ATT Dxx' (Bu)x' NUMERICAL COMPUTATION (CONT'D.) We program the computer f o r both of these solutions and then compare the r e s u l t s (see section 8.3). In the remaining portion of t h i s section we discuss several points which arise i n programming the construction of the matrix Crx. In order to calculate any element of the matrix Crx we must have a value f o r the e f f e c t i v e probe radius pa . We had introduced the concept of e f f e c t i v e probe radius i n section 3.2 i n order to consider so c a l l e d "conductivity e r r o r " . Later, i n section li.lj. we found that the correction factor was i n s e n s i t i v e to the precise value of pQ . The plasma i n the v i c i n i t y of the probe was photographed (section 9.3) but there was no observed change i n luminosity i n t h i s region which could be used as an estimate of p0 . Thus f o r our " f i r s t order" correction we use the radius of the quartz guide tube which i s .37 cm. Tn order to minimize the error introduced by replacing the continuous functions by discrete numerical values, we arrange the x - coordinates (the coordinates of the current density values) to coincide with the h a l f - i n t e g e r s of the r - coordinates (the coordinates of the corresponding magnetic f i e l d v a l u e s ) . can best be seen i n a diagram. The effect In f i g u r e 17 we consider the correction factor due to a current sheet of thickness 1 mm. which i s centered at x . In f i g u r e 17B where we have x and r coincident, i . e . obtain f i e l d values at the mid point of the current sheet, then we 5.1 NUMERICAL COMPUTATION (CONT'D.) have a much larger error i n the numerical approximation to the correction factor than i n the case with x and r values staggered. One further consideration must be examined c l o s e l y and t h i s i s the choice of Crx f o r the region of x-*-0» The correction f a c t o r must be modified for those current sheets which are within the c h a r a c t e r i s t i c distance of the correction factor the axis of the discharge v e s s e l . ( 9 mm.) from The modification i s obtained by assuming the current d i s t r i b u t i o n i s r o t a t i o n a l l y symmetric and then c a l c u l a t i n g the correction factor as the algebraic sum of the two over lapping correction factors as i s shown i n figure 18. To obtain the complete correction factor we assume that the area under the correction as a function of radius i s the same both inside and outside the current sheet. The r e s u l t s of the analogue experiment (section I ; . 2 ) indicate that t h i s assumption i s reasonable. Thus the correction factor outside the current sheet i s normalized by the area under the curve, rather than the normalization C(0) = 1. (See figure 18). Once the i n d i v i d u a l matrices Crx and, either Srx or Dxx' (depending upon the method of s o l u t i o n ) , have been constructed, the problem i s solved by the inversion of a matrix which i s a combination of these matrices. D i f f i c u l t i e s are encountered i n t h i s procedure because the determinant i s very s m a l l . This determinant has 80 x 80 elements and the magnitude of each element i s less than one. — UU — 7 FIGURE 1 7 . 17 A 17 B ERROR IN NUMERICAL APPROXIMATION TO CORRECTION r r coincides with x coincides with x + 1 / 2 - - OVERLAPPING CORRECTIONS FIGURE 18. 5,1 MODIFICATION OF CORRECTION NEAR AXIS NUMERICAL COMPUTATION (CONT'D.) The problem i s solved simply by multiplying a l l matrix elements •»l!i by 2 . The resultant determinant i s then of the order of 1 0 t h i s l i e s w e l l within the range of the computer ( 1 0 " * ^ ) . and The f i n a l r e s u l t s must then a l l be multiplied by 2 (since i t i s the i n v e r s e ) . A second d i f f i c u l t y associated with the large size of the matrices i s the problem of accumulation of computer "round-off" error* This accumulation arises from the many operations involved i n the inversion of an 8 0 x 8 0 matrix. The results of the program are greatly improved when the inversion i s performed with variables defined i n double p r e c i s i o n . (For d e t a i l s of the Fortran computer program see Appendix II.) «=» I4.6 «= NUMERICAL COMPUTATION (CONT'D.) It should be pointed out that matrix inversion i s not the only method of s o l u t i o n . Indeed, the form of equation 1. and 2., suggests the use of an i t e r a t i v e type procedure. Two i t e r a t i v e methods of solution were attempted, but i n both cases spurious, negative "spikes" occurred i n the current density d i s t r i b u t i o n which prevented the i t e r a t i o n s from converging. These negative spikes could be eliminated through a modification of the computer program. However, as negative current densities have been observed at the walls of the discharge vessel by Komelkov (1958 )> this somewhat a r t i f i c i a l elimination i n the program i s most undesireable. There appeared to be no other way to get the program to converge to a solution and these methods were abandoned. TEST FOR NUMERICAL PROGRAM We w i l l postpone a detailed discussion of the possible errors i n applying the results of the analogue experiment to the z-pinch measurements u n t i l section 8.U. By then, we w i l l have calculated the true magnetic f i e l d and current density distributions o f the z-pinch discharge. However, we do wish to test the numerical program of the preceding s e c t i o n . Fortunately, the magnetic f i e l d measurements of the analogue experiment (see f i g u r e 9> ) provide an excellent test for the program. We can calculate the current d i s t r i b u t i o n from the measured magnetic f i e l d using our program and then compare t h i s r e s u l t with the known distribution (essentially a § - function). U7 - 5.2 TEST FOR NUMERICAL PROGRAM (CONT'D,.) The r e s u l t of the correction procedure may be observed i n figure 19., where we have plotted the uncorrected, corrected, and true current d e n s i t i e s . It i s apparent that the corrected current density i s an improvement over the uncorrected current density, but i t s t i l l f a l l s w e l l below the true current density which i s known from the analogue experiment. A discussion of these r e s u l t s follows. The correction applied to the measured curve is greatest at a point of d i s c o n t i n u i t y i n the current d i s t r i b u t i o n . seen from the diagram of figure 20. This can be At a point of approximately constant current density, the i n d i v i d u a l corrections cancel when the t o t a l f i e l d correction i s c a l c u l a t e d . However, at a point of d i s c o n t i n u i t y t h i s c a n c e l l a t i o n does not occur and A B ( r ) i s considerable. Although the c o r r e c t i o n i s therefore greatest at a point of d i s c o n t i n u i t y , the numerical c a l c u l a t i o n i s poorest at such a p o s i t i o n because of the f i n i t e i n t e r v a l (here 1 mm.) which i s used i n the numerical approximation. been examined i n f i g u r e 17. This effect has already As a r e s u l t , the corrected current density curve has s t i l l only one quarter of the true magnitude, but the improvement over the uncorrected current density curve increases peak current by $0% and reduces h a l f width by k0% (see f i g u r e 19.) - i s s u f f i c i e n t to give an appreciable change i n the experimentally measured current density d i s t r i b u t i o n . w i l l be discussed i n a l a t e r section (section - k9 - 8.U). This change FIGURE 20. TOTAL FIELD CORRECTION TEST FOR NUMERICAL PROGRAM (CONT'D.) The important conclusion i s the f o l l o w i n g . From these r e s u l t s i t i s apparent that any i n d i c a t i o n of a "spike" i n the current density d i s t r i b u t i o n represents a lower l i m i t to the peak i n the true current density. - 50 - PART II EXPERIMENTAL RESULTS - 51 - 6.0 INTRODUCTION This i n v e s t i g a t i o n has been designed to study the plasma dynamics of the z-pinch discharge. In this Part we present the observations which w i l l be compared with predictions based upon c e r t a i n physical models of the plasma i n Part III. The most important r e s u l t i s the measurement of the current density d i s t r i b u t i o n which has been corrected for the probe error as described i n Part I . However, i t i s desireable to have an independent method of i n v e s t i g a t i o n which we can use 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 our magnetic probe r e s u l t s . As the plasma i s highly luminous, and as photographic techniques have been described previously i n the literature, f o r example Curzon(l962) and F o l k i e r s k i (1963), we use these techniques to obtain r e s u l t s which can be compared with our current density measurements. The photographic measurements are discussed i n section 9 and the comparison of photographic and magnetic probe r e s u l t s i s given i n section 10. Thus Part I I of the thesis contains a description of the experimental apparatus and continues with a presentation and discussion of a l l experimental r e s u l t s . The presentation of r e s u l t s i s divided according to the diagnostic technique which has been used to obtain the results - magnetic probes i n section 8 and the framing camera i n section 9. - 52 - APPARATUS A photograph of the apparatus is shown i n figure 21. and a schematic diagram of the discharge c i r c u i t and t r i g g e r i n g c i r c u i t r y i s shown i n f i g u r e 22. The e s s e n t i a l features of a l l components of t h i s apparatus are given i n tabular form (see Table I at end of t h i s section). B r i e f descriptions are given f o r the three main components - the discharge v e s s e l , the capacitor bank, and the t r i g g e r i n g c i r c u i t . We also include a description of the framing camera which i s used i n the photographic studies. Discharge V e s s e l : The discharge vessel i s a large Pyrex tube (17 cm. O . D . , 15 cm. I . D . , and 70 cm. i n length) which i s sealed with plane brass electrodes at each end. The vessel may be evacuated and then f i l l e d to the required pressure with the desired gas.(argon for these investigations). An e l e c t r i c current i s discharged through this gas which thus forms the plasma. The discharge vessel i s designed f o r use with both photographic and magnetic probe diagnostics. To allow for photographic observation perpendicular to the axis of the discharge, a f i n e mesh brass screen i s used as the return conductor of the discharge c i r c u i t (refer to figure 21.and f i g u r e 2 2 . ) . Furthermore, one of the end electrodes i s perforated and i t i s provided with a large glass window which allows "end-on" photography, i . e . photography along the axis of the discharge v e s s e l . (The discharge vessel i s evacuated and f i l l e d through the second electrode.) - 53 - To allow access to the plasma FIGURE 21. PHOTOGRAPH OF APPARATUS APPARATUS (CONT'D.) D i s c h a r g e V e s s e l : (Cont'd.) f o r t h e magnetic probes a q u a r t z t u b e o f ,7k cm. o u t e r d i a m e t e r i s p l a c e d d i a m e t r i c a l l y a c r o s s t h e Pyrex d i s c h a r g e v e s s e l . T h i s tube i s s e a l e d a t t h e d i s c h a r g e v e s s e l w i t h A r a l d i t e epoxy r e s i n . As a magnetic probe c a n be i n s e r t e d t h r o u g h e i t h e r end o f t h e t u b e , we can o b t a i n two r a d i a l o b s e r v a t i o n s o f t h e magnetic f i e l d f r o m a single discharge. •& - V(+ve) ISOLATED TRIGGER PUISE Le .--j-\ * > _C2_ TRIGGERING CIRCUIT DISCHARGE CIRCUIT SCHEMATIC DIAGRAM OF APPARATUS FIGURE 22. Ld Le Cl C2 51 52 Rl R2 V T 7.0 = = = = = = = «= = Inductance of discharge Inductance of external c i r c u i t (,12^/UH) Main capacitor bank (53/IF) Trigger capacitor (,06/lF) Main spark gap switch Trigger spark gap switch Variable resistance (0 - 70 M i l | 10 M U steps) Voltage d i v i d e r (150 M U ) Charging voltage Trigger Pulse APPARATUS (CONT'D.) Capacitor Banks The capacitor bank ( C l of the schematic diagram of f i g u r e 22.) i s formed by f i v e N . R . G . , low inductance, storage capacitors. These capacitors are charged to 10 kV f o r a l l experimental investigations which means that we have 2.6 k j . of stored energy. - 55 - APPARATUS (CONT'D.) Capacitor Bank: (Cont'd.) In the e a r l y part of the investigation the capacitors were often destroyed after repeated use ( A/one hundred discharges), but since small (<\>166 p F . ) high frequency, capacitors have been placed across each of the storage capacitors, there has been no such damage. It i s thought that these capacitors may by-pass high frequency, high voltage transients on the discharge l e a d s . Triggering C i r c u i t : The t r i g g e r i n g c i r c u i t f o r the present system i s shown i n figure 2 2 . This c i r c u i t has two important features. F i r s t , the . switch S 2 of figure 2 2 . i s o l a t e s the high voltage discharge c i r c u i t from the i n i t i a l t r i g g e r generator. Secondly, this c i r c u i t i s constructed c o a x i a l l y i n order to minimize the creation of radiated noise s i g n a l s . The triggering c i r c u i t i s described i n d e t a i l by Medley (1965a). Framing Camera: A Barr and Stroud framing camera i s used to obtain photographic measurements. In this camera the f i l m remains stationary and a r o t a t i n g mirror i s used to pass l i g h t from the discharge through a series of f i x e d lenses as shown i n the schematic of figure 23. In order to photograph the desired t i m e - i n t e r v a l of the discharge, the position of the rotating mirror i s used to t r i g g e r the discharge. - 56 - COUNTER FILM TRACK FRAMING CAMERA AND CONTROL EQUIPMENT FIGURE 2 3 , 7.0 P.M.2. APPARATUS (CONT'D.) Framing Camera: (Cont'd.) A block diagram of the required t r i g g e r i n g c i r c u i t i s also included i n figure 23., but we refer to Hodgson (196k) f o r a d e t a i l e d explanation of the t r i g g e r i n g operation. We w i l l , however, explain the operation of the frequency meter designated by counter i n figure 235 ^ o r this meter determines the exposure time for the photographs. A small l i g h t source i s provided i n the framing camera which i s focused on the rotating mirror and, at a certain mirror p o s i t i o n , the r e f l e c t i o n i s recorded by a photo m u l t i p l i e r . - 57 - Thus the photo APPARATUS (CONT'D.) Framing Camera: (Cont'd.) m u l t i p l i e r records a pulse corresponding t o each complete rotation of the mirror and a C . M . C . Model 201C counter i s used to record the period-(T) between these pulses at the time of f i r i n g the discharge. The manufacturers state that a mirror r o t a t i o n of 5«5 kc/sec corresponds to a f i l m exposure time of .125 XL sec. Thus we can determine the exposure time (e) from the period (T) as: e = .125 x 5.5 x l O ^ x T where e and T are measured i n micro seconds. T y p i c a l l y T i s of the order of 350 JJL sec and thus e i s approx imately ,2h / t s e c . TABLE I CAPACITOR BANK AND LEADS (N.R.G. L O W Inductance Storage Capacitors) T o t a l Capacity Total Inductance Width of Leads Length of Leads Separation of Leads (Polyethylene) 53 > F .12 + .01 ^ H 15 cm. 1 m« 2 mm. DISCHARGE TUBE Pyrex Brass 59 cm. 15 cm. 17 cm. Discharge Tube Electrodes Electrode Separation Inside Tube Diameter Outside Tube Diameter - 58 - APPARATUS (CONT'D.) TABLE I (CONT'D.) VACUUM SYSTEM (Type 17 Balzers O i l D i f f u s i o n Pump) (Hyvac 1U Cenco Backing Pump) Base Pressure Leak Rate 1 /U-Hg 7 /LHg/hr. PRESSURE MEASUREMENT (Type GP-110 P i r a n i Vacuum Guage) (Vacustat) max. 1000 JiEg VOLTAGE MEASUREMENT (Conway Microammeter) (in series w i t h A . V . O . M u l t i p l i e r ) Charging Voltage 25 kV d . c , 500 M 10»0 + .3 kV CURRENT MEASUREMENT (Rogowski C o i l ) Integration Time Constant Sensitivity Maximum Discharge Current Discharge Current Frequency 180 jxsec 15.0 kamp v o l t . " l 150 kamp. 50 kc/s MAGNETIC FIELD MEASUREMENT (Type 551 Tektronix Dual Beam Oscilloscope) (Type D "Plug i n " Units) (Type 2620 Dumont Polaroid Oscilloscope Camera) Magnetic Probe S e n s i t i v i t y U.9 wbm"^ v o l t PHOTOGRAPHIC MEASUREMENT (Model CPS Barr & Stroud Framing Camera) (H.P. 3 I l f o r d Photographic Film) Number of Frames/Discharge Exposure Time Time Between Frames - 59 - 60 .25 /i .25 /Isec sec DISCHARGE PARAMETERS In t h i s section we present four separate experimental measurements. Three of these are observations of phenomena associated with the plasma discharge; namely, the discharge current, voltage, and emission of l i g h t . The fourth i s a measurement of the" inductance of the discharge c i r c u i t - excluding the inductance of the discharge v e s s e l . calculations i n Part This inductance i s required f o r t h e o r e t i c a l III. Discharge Current: The discharge current i s measured with a Rogowski c o i l which i s e f f e c t i v e l y a large magnetic probe. It i s constructed from a 15 cm. length of the inner conductor of RG6£ A / U delay l i n e . The c o i l i s placed between the f l a t copper leads to the discharge vessel w i t h the axis of the c o i l perpendicular to the d i r e c t i o n of current flow i n the leads. Like the magnetic probe the output signal of the Rogowski c o i l must be integrated to obtain a s i g n a l proportional to the magnetic f i e l d or the discharge current. The same R . C . - type integrator is used as has been described f o r the magnetic probe i n section 3 . 1 , but now the integration time constant i s 180 ji, sec« Again, as with the probe, we must check the frequency response of the Rogowski c o i l and measuring c i r c u i t (including the The s i g n a l cable i s terminated with - 60 - lj.7-0- integrator). as i s , then, the Rogowski c o i l . DISCHARGE PARAMETERS (CONT'D.) Discharge Current: (Cont'd,) This impedance i s much less than the characteristic impedance of the Rogowski c o i l (1000-D-) and we expect, therefore, that signals at high frequencies w i l l be attenuated (see Segre ( l ° 6 0 ) ) » Still, the conditions f o r r e l i a b l e integration of the input s i g n a l (see section 3 « l ) are s a t i s f i e d f o r frequencies (f) less than 500 kc/s (experimentally v e r i f i e d ) and as the frequency of the discharge current i s approximately 50 kc/s (see, for example, figure 29.), then more accurate matching is unnecessary. T y p i c a l current traces corresponding to the three initial pressures of 100, 250, and 500 >tHg argon are i l l u s t r a t e d i n f i g u r e 2U« These p a r t i c u l a r traces are obtained by magnifying the polaroid f i l m from the oscilloscope camera with a photographic enlarger. The traces are normalized i n the following way. An enlargement of a graticule area of 100 jx sec-volts i s also included i n figure 2U. This area i s used to c a l i b r a t e a planimeter which i s then used to determine the area under the current traces i n JUL s e c - v o l t s . It i s known that t h i s area must be equivalent to the t o t a l charge on the capacitor bank: J l d t » Q - CV where I i s the discharge current Q i s the charge on the capacitor C i s the capacitance V i s the charging voltage. - 61 - DISCHARGE PARAMETERS (CONT'D,) Discharge Current: (Cont d.) 1 We conclude that 1 volt of d e f l e c t i o n on the oscilloscope i s equivalent to a discharge current o f 15,000 amperes. The error associated with the c a l i b r a t i o n procedure i s r e l a t i v e l y large (10$). The largest source of error i s i n the planimeter measurement of the area under the current t r a c e . Discharge Voltage: Voltage measurement is obtained by using a 1000:1 Tektronix high voltage probe Type P6013. The measurement i s taken across the discharge end of the current leads. The results are shown i n figure 25*3 and i t i s obvious that considerable noise signals are associated with t h i s measuring technique (observe the accompanying current wave farms i n figure 25. as compared with those of figure 2I4..) However, f o r the present purpose an estimate i s s u f f i c i e n t , and i t i s noted that any d e t a i l i n f i g u r e 25. is unreliable. At higher pressures the voltage across the discharge tube i s approximately constant at 2,5 - 3.0 kV f o r the first k INITIAL PRESSURE 100 jiEg JUsec, PEAK LUMINOSITY Argon 5«5 + .5 M sec 7.7 + .5 /(sec 250 yUHg Argon 500 ^ H g Argon 10.0 + .5 ftsec TIME OF MAXIMUM LUMINOSITY TABLE I I - 63 - 100 //.Hg Argon 250 >lHg Argon 500 jlBg FIGURE 25. Upper T r a c e : Lower T r a c e : Argon OSCILLOSCOPE TRACES OF DISCHARGE VOLTAGE V o l t a g e a c r o s s d i s c h a r g e l e a d s (2,000 v o l t s / c m . ) C u r r e n t f r o m Rogowski c o i l (75,000 amps/cm.) 5 Ji sec/cm. Time Base: DISCHARGE PARAMETERS (CONT'D.) Discharge Luminosity: The i n t e n s i t y o f t h e l i g h t e m i t t e d b y t h e d i s c h a r g e i s m o n i t o r e d w i t h a P h i l l i p s photo m u l t i p l i e r Type CVP-l^O. The photo m u l t i p l i e r i s p l a c e d a t t h e s i d e o f t h e d i s c h a r g e v e s s e l a t the h e i g h t o f t h e - 6l» - 100 pEg Argon 250 jXEg Argon 500 jiEg FIGURE 26. Upper Trace: Lower Trace: OSCILLOSCOPE TRACES OF DISCHARGE LUMINOSITY Rogowski c o i l (75,000 amps/cm.) Photo m u l t i p l i e r (10 volts/cm.) Time Base: 7.1 Argon 2 ^sec/cm. DISCHARGE PARAMETERS (CONT'D.) Discharge Luminosity: (Cont'd.) discharge a x i s , and i t samples the l i g h t emitted from a s e c t i o n of the discharge midway between the magnetic probe guide and the discharge electrode. The photo m u l t i p l i e r has a -65 - 1000load DISCHARGE PARAMETERS (CONT'D.) Discharge Luminosity: (Cont'd*) resistance and i t has 8 £ 0 volts d » c . anode p o t e n t i a l . The l i g h t input to the photo m u l t i p l i e r i s attenuated by a f a c t o r of 1000 by using a neutral density 3*00 f i l t e r and s i g n a l output i s displayed on an oscilloscope and photographed on polaroid f i l m , , The r e s u l t i n g photograph for the three i n i t i a l pressures of argon - 100, 25>0, and 500 yU.Hg, i s shown i n figure 26. and the time corresponding to the peak l i g h t emission i s given i n Table I I f o r the three i n i t i a l pressures. We assume that peak l i g h t emission coincides with the time of the p i n c h , i . e . the plasma has been constricted to i t s minimum volume* External C i r c u i t -Qiductance: The external c i r c u i t inductance, i . e . the inductance of discharge capacitors and leads, i s determined by the following procedure. The electrodes of the main spark gap switch (see f i g u r e - 2 2 . ) are placed i n contact thus shorting out the switch. The capacitor bank i s charged to a few hundred v o l t s with the discharge c i r c u i t open-circuited at the discharge tubeo The discharge leads are then connected ~ shorting out the discharge v e s s e l inductance and completing the c i r c u i t (see f i g u r e 2 2 . ) - and the r i n g i n g frequency of the c i r c u i t current i s monitored with the Rogowski coil. Assuming the capacitance of the c i r c u i t i s known (C]_ of f i g u r e 2 2 . ) then the inductance may be c a l c u l a t e d . inductance i s calculated t o be .12 + . 0 1 ^tiH. - 66 - The external MAGNETIC PROBE MEASUREMENT In t h i s section we describe the experimental procedure which i s used to obtain measurement of the magnetic f i e l d i n the z-pinch discharge. Probes of the type described' i n section 3» are inserted into each end of the quartz guide tube which runs d i a m e t r i c a l l y across the discharge v e s s e l midway between the electrodes. The outputs from the probes are displayed simultaneously on a dual beam o s c i l l o s c o p e . We are assuming that the discharge plasma i s r o t a t i o n a l l y symmetric and thus we can meaningfully compare the two signals. The r a d i a l p o s i t i o n of the probe c o i l i s determined by f i r s t centering the probe c o i l on the axis of the discharge vessel (using a t r a v e l l i n g microscope). With the probe i n this p o s i t i o n , the probe stem i s marked at a point coinciding with the end of the quartz guide tube. Then the probe stem i s marked off i n 2.5 mm. i n t e r v a l s r e l a t i v e to this "zero" mark. Thus we place the probes at the desired r a d i a l p o s i t i o n s , trigger the discharge, and measure the magnetic f i e l d as a function of time at these given r a d i a l positions (see, f o r example, f i g u r e 27.). However, i n order to calculate the current density d i s t r i b u t i o n we have seen i n section 5 that we require the r a d i a l d i s t r i b u t i o n of the magnetic f i e l d at a given time and not the magnetic f i e l d as a function of time at a given radius (as we measure experimentally). If we assume that the discharge process i s reproducible from discharge to discharge, and i f we assume that the oscilloscope i s - 67 - IB t FIGURE 2 7 . Upper Traces Lower T r a c e : OSCILLOSCOPE TRACES OF MAGNETIC FIELD r = 2.2? cm. (.21*5 wb/m /cm.) r • 3.0 cm. (.21*5 wb/m^/cm.) Time Base: 1 Time D e l a y : 0.5 I n i t i a l Pressure: 100 JJ.sec/cm. jbLsec /LEg. MAGNETIC PROBE MEASUREMENT (CONT'D.) t r i g g e r e d a t t h e same p o i n t i n t h e development o f t h e d i s c h a r g e f o r each d i s c h a r g e , t h e n from each o f our observed t r a c e s o f the magnetic f i e l d a t g i v e n r a d i u s we c a n p i c k o u t t h e v a l u e o f the f i e l d a t a s e l e c t e d t i m e and t h u s c o n s t r u c t t h e d e s i r e d r a d i a l d i s t r i b u t i o n of t h e magnetic f i e l d c o r r e s p o n d i n g t o the s e l e c t e d t i m e . - 68 - ) lOpF AAAMAAr INPUT UYlL: (RCGOWSKI COIL) ' 330-TL' FIGURE 28. 8.0 .OOluF OUTPUT (OSCILLOSCOPE TRIGGERING) o COMPENSATED ATTENUATOR MAGNETIC PROBE MEASUREMENT (CONT'D.) This measured magnetic f i e l d d i s t r i b u t i o n i s then corrected by the procedure outlined i n section 5„ In the above procedure we have made three assumptions which we must now v e r i f y . But f i r s t we must describe the method of t r i g g e r i n g the oscilloscope t r a c e s . Throughout the experimental measurements, we have triggered the time base of the oscilloscope by using the external t r i g g e r i n g mode with a s i g n a l supplied from the Rogowski c o i l . In this case, the s i g n a l i s not integrated as we are interested i n the f a s t time (.2 jl s e c ) of the large d l / d t s i g n a l (500 v o l t s ) . This s i g n a l must be attenuated and the compensated attenuator figure 28. i s used which thus retains the f a s t r i s e time. - 69 - of rise MAGNETIC PROBE MEASUREMENT (CONT'D.) Now, to experimentally check the three previous assumptions - r o t a t i o n a l symmetry, r e p r o d u c i b i l i t y of the plasma discharge, and r e p r o d u c i b i l i t y of t r i g g e r i n g - we perform the following two t e s t s . In figure 29., we superimpose the two traces from the probes which are placed at corresponding r a d i a l positions on opposite sides of the discharge. The r e s u l t shows that the discharge i s symmetrical within our measuring accuracies and i t also i l l u s t r a t e s equivalence of the two probe measuring c i r c u i t s . the In figure 30., we superimpose the traces of several discharge shots f o r f i x e d probe positions and again within experimental accuracies the t r i g g e r i n g and discharge r e p r o d u c i b i l i t i e s are apparent. A previous i n v e s t i g a t i o n by Andrianov (1959) has also i l l u s t r a t e d that the z-pinch plasma i s reproducible up to times corresponding to the f i r s t c o n s t r i c t i o n of the discharge plasma. The experimental procedure i s to record magnetic f i e l d v s . time (t) 85 mm. (B) at 2.5 mm. i n t e r v a l s from radius (r) equal to 0 to Then from this set of r e s u l t s i t i s possible to determine magnetic f i e l d (B) VS. radius ( r ) . We choose to examine B(r) at 1 ji sec intervals with t=0 corresponding to the i n i t i a t i o n of discharge current. f i g u r e s 31 - 33* The experimental results are shown i n The c a l i b r a t i o n of the magnetic probe i s given i n the following s e c t i o n , and i n section 8.2 we compute the corrected magnetic f i e l d and current - 70 - density d i s t r i b u t i o n s . —6- t FIGURE 29, Upper Traces Lower Traces r » 6.25 cm. (.196 wb/m /cm.) r • 6.25 cm. (.196 wb/m^/cm.) 2 Time Bases Time Delay: Pressure: Upper Trace: Lower Traces TWO TRACES OF OSCILLOSCOPE SUPERIMPOSED 1 /(sec/cm. 0.5 jLsec 500 jjflg Argon r • 5«0 cm. (.2U5 wb/nrVcm.) r • 5»0 cm. (.2ii5 wb/m2/cm.) Time Base: Pressures Time Delays 1 JJLsec/cm. 250 p$g Argon 0.5 j^pec. FIGURE 3 1 . MEASURED MAGNETIC FIELD ( 1 0 0 / L H g Argon) o t = 1 jmsec • t = 2 /tsec X t = 3/tsec • t = U ^sec + t = 5/tsec - 72 - 8 6 4 2 -*-r(cm.) FIGURE 32. Xt o t A t • t = = = = MEASURED MAGNETIC FIELD (250/lHg Argon) 1/Lsec 2/Lsec 3/tsec U^tsec +t * t • t - 73 - = = = 5/dsec 6/tsec 7/tsec 8 6 4 2 0 r(cm.) FIGURE 33. o t = l/tsec t = 2 jxsec n t = 3 /isec xt A MEASURED MAGNETIC FIELD (500juHg Argon) o t = 5/isec + t = 6/(.sec • t » 7As e c = 8 - 7U - jLiseo 8.1 MAGNETIC PROBE CALIBRATION The magnetic probe can be calibrated i f a well known magnetic f i e l d is available. A simple technique i s now described which makes use of the experimental apparatus and the r e s u l t s of the analogue experiment of section 3 « I t has been shown i n section k»3 that i n the plane of the current sheet, i . e . z = 0 , AB(0) - 1 2 J Q Now i f we consider the magnetic f i e l d at a point i n the plane of the return conductor of the discharge vessel (where the probe punctures the return conductor), then we expect the measured f i e l d to be due to the f i e l d error alone as there would be no magnetic f i e l d outside the return conductor i f there were no probe hole through i t . Thus, B (r , t ) = Q 0 AB(o) = 1 jj^y J ( t ) 0 0 2 where r 0 t D i s the radius of the return conductor i s the time of observation J (*o) - K t ) / 2TT r T 0 lo(to) * 0 s t n e 0 current i n t h e return conductor Y i s the thickness of the return conductor. However, as the magnetic f i e l d gradient i s large about the point r = r , and as i t i s d i f f i c u l t to position the magnetic 0 probe c o i l accurately, the c a l i b r a t i o n procedure i s improved by - 75 - 6 4 o z a FIGURE 3U. 4 zCcm)—*~ 6 DETERMINATION OF B(o) Analogue correction for Correction 1 Correction 2 Correction 3 yO 0 = .3? cm. B(o) = .038 v o l t s B(o) = .OijO volts B(o) = .0U2 volts O Experimental points measured near return conductor of discharge c i r c u i t . 8.1 MAGNETIC PROBE CALIBRATION (CONT'D.) measuring the magnetic f i e l d as a function of r a d i u s , for points both inside and outside the current sheet. These points are then f i t t e d to t h e o r e t i c a l curves with varying normalization constants and from the curves of figure 3h» i t appears that B ( r , t o ) = .Oij.0 v o l t s . 0 - 76 - MAGNETIC PROBE CALIBRATION (CONT'D.) These t h e o r e t i c a l curves are plotted from the analogue r e s u l t s of section k»3» The time ( t ) 0 i s chosen as the time when l ( t ) choice has several advantages. largest at t = t ; 0 = I max. 0 This F i r s t , the measured s i g n a l i s second,, the current magnitude i s i n s e n s i t i v e to measuring error i n t 0 at l ( t ) 0 = I max.5 and t h i r d , the discharge current has moved w e l l away from the wall of the discharge v e s s e l at t h i s time and the f i e l d i n the v i c i n i t y of the return conductor i s due to the return current alone. 5 0 0 jjjlHg of argon, t 0 With an i n i t i a l pressure of i s approximately U jj.sec* A measurement of the discharge current by the Rogowski c o i l i s required t o complete the c a l c u l a t i o n , s i n c e : 4B(0) = JX I 0 (t ) 0 0 HTT^O However, the Rogowski c o i l has already been c a l i b r a t e d (section 7 . 1 ) , and from figure 2 1 ; . , the current at t « It ^ s e c f o r an i n i t i a l pressure of 5 0 0 //.Hg argon i s known to be 166 kamp. Thus, AB(0) - . 0 U 0 or, x CAL. = jjj UTT x I 0 0 (t ) x 1.66 x 1 0 ^ Lffl X 8 . 5 x l O " 1 0 7 CAL. «= U.9 wb/m2 volt - 77 Q - 2 COMPUTED RESULTS The corrected current density distributions as functions of time i n the z-pinch discharge are obtained by using the computer program developed i n section 5>.l with the measured magnetic f i e l d distributions of section 8.0. These measured results must, however, be adjusted i n two ways. F i r s t , the measured f i e l d values are converted t o M.K.S. units by using the calibration factor obtained in section 8.1. Secondly, the measured f i e l d values i n the vicinity of the discharge wall (i.e. for large values of r) must be corrected for perturbation of the field caused by the penetration of the probe through the return conductor. As the current density in the return conductor i s much greater than the current density i n the hypothesized current sheets of the plasma (see figure 16.), the f i e l d correction associated with the return conductor is appreciable over a larger range (about 20 mm.) compared to the usual correction (about 10 mm.). The effect i s shown in the diagram of figure 35>. The correction is easily applied as we know the current i n the return conductor from the Rogowski c o i l measurement of section 7.1 and the farm of the correction from section U«3» It is interesting to note that this correction must be added to the already considerable correction which i s applied to the magnetic field i n the vicinity of the wall of the discharge vessel (see figure 35«)» In particular, i f the thickness of the wall of the vessel i s comparable to,or less than,the radius of the hole punctured in the return conductor, then this correction would be very large. If the correction were large and i f i t were not applied - 78 - FIGURE 35. MAGNETIC FIELD CORRECTION AT WALL OF VESSEL COMPUTED RESULTS (CONT'D.) to the measured magnetic f i e l d , then the c a l c u l a t i o n would r e s u l t i n an apparent negative current at the wall of the v e s s e l . In section 5.1 we have developed two methods of s o l u t i o n . In one case we solve f i r s t f o r the current density (Equation 1.) and - 79 70 60 50 40 —s- FIGURE 36. Time: 30 20 X* ( m m . ) RESULTS FROM TWO METHODS OF SOLUTION 3/isec I n i t i a l Pressure: 2^0//Eg - 80 - Argon 8.2 COMPUTED RESULTS (CONT'D.) i n the other case we solve f i r s t f o r the magnetic f i e l d (Equation 2.). In f i g u r e 36* we have examined the results from one experimental measurement (2^0 JiKg, 3 ^sec) and we have determined both the magnetic f i e l d and the current density f o r these two methods of s o l u t i o n . The comparison shows that the two solutions are equivalent. Using Equation 1. (section £.1) we determine the current density d i s t r i b u t i o n s corresponding to each of the measured magnetic f i e l d d i s t r i b u t i o n of figures 31. - 33« These current density d i s t r i b u t i o n s are plotted i n figures 37* - U2. One characteristic feature of these graphs i s the presence of a d i s t i n c t current density spike at the outer edge of the distribution. which carries This corresponds to a r e l a t i v e l y t h i n current sheet a considerable f r a c t i o n of the discharge current. We have seen i n section £.2 that our numerical program tends to give a lower l i m i t to the a c t u a l current density maximum. Thus i t i s most probable that the actual current density maximum i s as much as four times greater than that indicated i n the graphs of figures 37. - 1+2., and that the current sheet i s correspondingly t h i n n e r . The knowledge of the existence of a t h i n current sheet i s important since a l l t h e o r e t i c a l models f o r z-pinch dynamics assume the presence of such a sheet although t h i s has never been established experimentally. These t h e o r e t i c a l models w i l l be discussed i n Part I I I of the thesis. - 81 - FIGURE 37. COMPUTED CURRENT DENSITY DISTRIBUTION I n i t i a l Pressure: Time: IjJLsec 2 //sec 3/isec - 82 100 /<»Hg Argon • o x - FIGURE 38. COMPUTED CURRENT DENSITY DISTRIBUTION I n i t i a l Pressure: Time: kjtH 5/1 sec sec -83 - 100juMg Argon ' x FIGURE 39. COMPUTED CURRENT DENSITY DISTRIBUTION I n i t i a l Pressure: Time: Jlsec 2£0//Hg • 1 2 /(.see 3 p. sec - 8U x o - Argon ii//sec Zjisec + * 00 a 8 tt *^ r ( c m . ) FIGURE U0. COMPUTED CURRENT DENSITY I n i t i a l Pressure: 2^0/UHg Argon Time: 6^Usec • 7/lsec x 1 - 85 - DISTRIBUTION FIGURE U l . COMPUTED CURRENT DENSITY DISTRIBUTION I n i t i a l Pressure: Time: 1 //.sec 2 //sec 7 psec 500JlEg • X o - 86 - Argon 8 4 6 2 — r FIGURE U2. 0 (cm.) COMPUTED CURRENT DENSITY DISTRIBUTION I n i t i a l Pressure: Time: 3/isec U^sec 5 /Jlsec 6 /(sec 500/HEg Argon • ^ X o «• 87 ** 8*2 COMPUTED RESULTS (CONT'D.) A second interesting feature of the current d i s t r i b u t i o n i n the z-pinch discharge i s the region of approximately constant current density which l i e s within the current sheet already mentioned. region has appeared as only a s l i g h t i r r e g u l a r i t y i n the r e s u l t s This of previous investigators who have not corrected t h e i r r e s u l t s for the e f f e c t of the probe, (for example, see results quoted by F o l k i e r s k i (1963)). In Table III we summarize some of the data from figures 37. U2. - This information i s required at a l a t e r stage of the i n v e s t i - gation where we wish to know the v e l o c i t y of the moving current density maximum and also the v e l o o i t y of the leading edge of the plateau r e g i o n . These r e s u l t s are plotted i n figures 58. and 59., of section 10.0, TABLE I I I RADIAL POSITIONS OF ESSENTIAL CURRENT DENSITY FEATURES R «* Inner radius of discharge v e s s e l Rc » R a d i a l p o s i t i o n of current maximum Rp « R a d i a l p o s i t i o n of leading edge c f i current 0 Pressure ( 100 250 ULER) Time( //.sec) Rc (mm.) Rp(mm.) Yc^/Ro YpJfe/Ro 2 + .3 62+2 U9 + 2 .83 + .03 .65 + .03 2 65 58 U9 3 ~ U 3 500 plateau. 51 ~ 30 U 5 U9 3U 3 U 5 65 60 6 U9 39 - 88 «* 33 " .68 " .UO .87 .77 36 •65 .U5 56 U5 37 .87 .80 20 21 .65 .52 .UU " .65 .U8 .27 .75 .60 .U9 .28 * tr i 72- * * *• + * + T ) * + o° CM X* 3 CQ X X .36 + o * H ¥ ******** * * ** o O <* o * < > < X ,24 0 X f + o X 0 X • o X e X X v|2 + V /• • X o • « « 8 1 > < o X • 6 *X A + ° o o *Kx 0 a *— r ( c m . ) ' FIGURE U3« TRUE MAGNETIC FIELD DISTRIBUTION I n i t i a l Pressure:' 100/tHg Argon • U/xsec + Time: 1/4 sec 2/t sec x 5 yttsec * 3/isec - 89 - * Y • ) .60 * •rt 48 o - +( i V v +J _Q * 0 ° r 36 \ ' QQ V *• V 0 + * *+ ' V 3 V * o + V cvl V * * V V 0 * y • 24 ** * + + 12 4 * 0 t y • ) 0 4- X • T < X Ah o + x • 0 0 ¥ * f *- 4. J T X;" ; + ( X +- x * 0 +• O x 7 ••../xxx. 2f.X X + + r (cm) 8 4 FIGURE hh. + + X TRUE MAGNETIC FIELD DISTRIBUTION I n i t i a l Pressure: 2j?0/xHg Argon Time: l/tsec A 5/tsec 2 ^csec • 6/fsec V 3/<,sec x 7/4 sec * ii /4,sec + 0 90 - o c f .60 ocoo .48 o f .36 3 CD .24 .12 6 FIGURE k5. r Ccm.) o TRUE MAGNETIC FIELD DISTRIBUTION I n i t i a l Pressure: 500^Hg Time: 1 jisec A 5 /tsec o 2 /isec • 6 //sec V 3 /isec x 7 /isec * U /tsec + - 91 8.2 COMPUTED RESULTS (CONT'D.) Using Equation 2 . (section 5 . 1 ) we determine the corrected magnetic f i e l d d i s t r i b u t i o n s corresponding to each of the measured magnetic f i e l d d i s t r i b u t i o n s of figures 3 1 » - 33* magnetic f i e l d d i s t r i b u t i o n s are given i n figures These corrected li3» - hS» F i n a l l y i n f i g u r e U 6 . , we compare the measured magnetic f i e l d and corresponding current density with the corrected values. It is apparent that our correction procedure modifies the r e s u l t s appreciably. We must, therefore, examine a l l possible sources of error i n order to determine that the calculated phenomena are r e a l (eg. the appearance of a plateau region i n the current density distribution). An analysis of error i s given i n the following section. 8.3 ANALYSIS OF ERROR There are three, main, possible sources of error i n our calculated values of the current density d i s t r i b u t i o n . First, there are assumptions which are i m p l i c i t when we apply our analogue r e s u l t s to the c a l c u l a t i o n of the current density and magnetic f i e l d i n the z-pinch discharge. Secondly, i t i s possible that we make errors i n measuring and i n p l o t t i n g the magnetic f i e l d d i s t r i b u t i o n (section 8 . 0 ) . And t h i r d l y , i t i s possible that there are errors i n our numerical c a l c u l a t i o n s . Analogue E r r o r : In using our analogue results f o r the z-pinch (see section - 9 2 - 5>ol) CURRENT DEMilTY + Measured 1falue o Corrected Value 1 MAGNETIC FU : L D o Measured falue x Corrected Value 1 1 j 3 f° 1 xs—o—* \ r IT X. X • <«• 4 1r,^ « ) V * \° X _L ° w R \ • t IP .Id 0 eo 10 * \ X X +1 s 60 60 ** AO ** •* r (mm) • 20 MEASURED AND CORRECTED CURRENT DENSITY AND MAGNETIC FIELD DISTRIBUTIONS FIGURE U 6 . Times 3/ttsec I n i t i a l Pressure; - 93 - 250/lHg Argon ANALYSIS OF ERROR (CONT'D.) Analogue Errors (Cont*d.) the most obvious assumption i s that we can apply the r e s u l t s obtained from a single current sheet to the i n t e r p r e t a t i o n of measurement upon, e f f e c t i v e l y , many current sheets. Actually,.we are assuming here that the correction i s not affected by the conductivity of the surrounding media. In the analogue experiment a given current sheet i s immersed i n a i r of low condui&tivity,, while i n the a c t u a l experiment a given current sheet i s immersed i n plasma of high c o n d u c t i v i t y . However, we do have evidence from one experiment which shows that t h i s assumption i s reasonable. In t h i s experiment we have used the analogue experiment exactly as i t has been described i n section U . l . We measure the perturbed magnetic f i e l d ; fasra f u n c t i o n of radius and then we repeat the experiment with the probe guide tube wrapped with aluminum f o i l as shown i n f i g u r e U7. As the analogue experiment i s performed with frequencies of the order of l O ^ c / s , t h i s aluminum f o i l simulates a region of high conductivity i n s i d e the current sheet and i t ' r e s t r i c t s any magnetic f l u x which "drips i n " through the probe hole i n the current sheet to the region of the probe guide tube. The comparison of the measured magnetic f i e l d s i n the v i c i n i t y of the current sheet i s given i n figure U7» and i t i s apparent that the conductivity of the surrounding media has a n e g l i g i b l e e f f e c t . Another major approximation or assumption which i s implied i n our using analogue r e s u l t s f o r plasma measurement i s that the 00 £ 1 OQ X o 8 O o FIGURE U7. SIMULATED REGION OF HIGH CONDUCTIVITY X Measured with aluminum f o i l around probe o Measured without aluminum f o i l around probe. i t t i - 95 - 8.3 ANALYSIS OF ERROR (CONT'D.) Analogue E r r o r : (Cont'd.) r e d i s t r i b u t i o n of current i n the plasma around the probe remains i n the o r i g i n a l plane o f the sheet. This must occur i n the analogue experiment but there i s no reason why there might not be a t h r e e dimensional r e d i s t r i b u t i o n of current i n the plasma. Note that this e f f e c t would only be important where there i s a d i s c o n t i n u i t y i n the current d i s t r i b u t i o n , as f o r example, at the p o s i t i o n of a current density " s p i k e " (refer to section 5>.l). This e f f e c t introduces an error which can not be accounted f o r i n our c o r r e c t i o n . Note, however, that t h i s effect would tend to decrease the height of the measured current density spike, with some of the current appearing to flow i n neighbouring current sheets. Hence t h i s consideration reconfirms our knowledge that the height of a current spike observed by our measuring technique i s a lower l i m i t of the true height (see section 5 . 2 ) . I f we do have a "three dimensional" perturbation of the current d i s t r i b u t i o n , then i t i s very l i k e l y that appreciable plasma i n s t a b i l i t i e s w i l l occur. It i s known that the c o l l a p s i n g plasma of the z-pinch discharge i s unstable and a perturbation w i l l grow i f excited (see Hodgson (196U)). Again we can not account, in detail, f o r t h i s e f f e c t i n our correction procedure, but we can obtain a rough estimate of the l i m i t a t i o n s which i t imposes on our r e s u l t s . From a p l o t of the r a d i a l p o s i t i o n of the current density spike versus time, we obtain a "collapse curve". - 96 - A t y p i c a l collapse curve ANALYSIS OF ERROR (CONT'D.) Analogue E r r o r : (Cont'd.) i s shown i n figure U8B. Now suppose the plasma i s not a x i a l l y symmetric but at a time t = t 0 current sheet of magnitude A r we have a perturbation i n the Q as shown i n figure h8A. What happens to the magnitude of this perturbation as a function of time? In f i g u r e U8B. we define a time A t which i t would take for the current sheet to c o n s t r i c t to the radius already reached by the perturbation. If A t i s f a i r l y small, then we can assume that the r a d i a l force acting on the current sheet does not change appreciably i n a time A t . Thus the radius of the current sheet at the perturbation w i l l follow a second t r a j e c t o r y which i s i d e n t i c a l to the t r a j e c t o r y of the normal collapse curve but displaced i n time by an amount A t . i n figure h8B. These trajectorie s are shown That the perturbation r e s u l t s i n an i n s t a b i l i t y i s apparent from an examination of A r ( t ) , the distance between the two collapse curves. It i s observed that A r ( t ) increases with time, i . e . the perturbation grows. Note, however, that the perturbation grows very slowly during the e a r l i e r stages,of the c o l l a p s e , but the growth i s rapid at times approaching the pinch time tp (see figure lj8B.). Thus the most r e l i a b l e measurements of the current density are made during the e a r l y stages of the collapse where any perturbations have not grown appreciably. However, at times close to the pinch - 97 - FIGURE u 8 . PLASMA INSTABILITY — — — Normal plasma boundary - — — Position of perturbation 8.3 ANALYSIS OF ERROR (CONT'D.) Analogue Errors (Cont'd.) time these i n s t a b i l i t i e s could a f f e c t appreciably the measured current d i s t r i b u t i o n and our information f o r these times i s unreliable. Measuring Errorss The second source of error which has been mentioned at the beginning of t h i s section i s the error associated with the ment of the magnetic f i e l d . The oscilloscope traces have been recorded on polaroid f i l m (see f i g u r e 27.) the aid of a g r i d . measure- and then measured with The g r i d , which has a spacing equal to the smallest gradicule spacing, i s ruled on perspex and can be placed - 98 - ANALYSIS OF ERROR (CONT'D.) Measuring E r r o r s : (Cont'd.) over the polaroid f i l m . The reading error i s approximately + .2 of the smallest d i v i s i o n . There i s also an error i n the measurement of the magnetic f i e l d from two different discharge shots which i s due to the v a r i a t i o n of the charging voltage on the capacitors. i n the magnetic f i e l d . This causes a 2 - 3% error "Error f l a g s " corresponding to these errors are shown on the magnetic f i e l d d i s t r i b u t i o n s of figures 3 1 . - 33o What i s the e f f e c t of t h i s measuring error i n the magnetic f i e l d upon the computed current density? Since, r &r r B ( r ) = B(r) + d B ( r ) r 5~r we see that the current density i s affected by both error i n the magnetic f i e l d and error i n the gradient of the magnetic f i e l d . A c t u a l l y , the gradient term i s usually dominant and the error associated with this term may be very large. For example, near the peak of the magnetic f i e l d d i s t r i b u t i o n curves where there are few experimental points to determine the r a p i d l y changing gradient, the error may be as large as 50$. We conclude t h i s discussion of measuring error with f i g u r e U9. which i l l u s t r a t e s the c o r r e l a t i o n of the c h a r a c t e r i s t i c features of t y p i c a l magnetic f i e l d and current density d i s t r i b u t i o n s . At the peak of the magnetic f i e l d d i s t r i b u t i o n we expect large error i n the corresponding current density. However, f o r r a d i a l positions inside - 99 - FIGURE 19* ESTIMATED ERROR IN " J " DUE TO ERROR IN "B» ANALYSIS OF ERROR (CONT'D.) Measuring Errorss (Cont'd.) and outside t h i s peak we have a w e l l established f i e l d gradient and the corresponding current densities have errors of less than 10$. Fortunately, the i n t e r e s t i n g regions of the current density d i s t r i b u t i o n coincide with the regions of greater accuracy. It i s apparent from f i g u r e U9» that we can not make meaningful measurements of the current density i n the region which is outside the current density peak. - 100 - just ANALYSIS OF ERROR (CONT'D.) TABLE I? COMPARISON OF COMPUTED CURRENT DENSITY FROM TWO MEASUREMENTS OF MAGNETIC FIELD (500 jXEg argon - 7 Asec) RADIUS 15 20 MAGNETIC FIELD 1 2.80 30 1 1 - 2 0 2 6 12 .21 .20 .20 .17 .15 20 15 30 23 7 15.0 17.2 lit. 9 17.6 llt.2 17.8 l l t . l 15.9 13.U llt.9 lit 18 2lt 12 10 .13 .10 .09 .09 . 0 9 8 9 18 10 0 13.2 11.5 11.3 11.5 11.1 12.1 11.5 10.2 10.6 11.2 6 0 io 9 1 -> = 10.7 10.lt 3 .20 .19 .20 .20 3.69 1 0 0 2 2 .17 .17 .lit .lit .lit U.60 U.71 U.81 U.90 k.93 U.99 2 2 2 1 1 .12 .11 .11 .10 .09 5.02 1 3.7U 3.92 3.90 It.io li.30 U.U5 U.50 lt.6l U.72 It. 8 3 5.08 — 2 0 5 10 15 .20 .20 .18 .17 3.08 3.28 3.U8 1 16.1 16.2 I6.lt 15.2 16.5 h h h 3 3 3.00 3.21 3.U0 3.58 CURRENT DENSITY 17.lt 17.2 16,1 16.1 llt.6 2.67 2.89 U.08 U.21 u.37 25 2 FIELD GRADIENT m. — Numerical E r r o r : In order to check that the numerical r e s u l t s do not include more error than we have already discussed, we perform the following t e s t . We p l o t two possible curves through the measured values of magnetic field (B) plotted against radius (r) for a given d i s t r i b u t i o n (actually 500 yUHg at 7 ^//Csec) and we calculate the corresponding values of the current density (J), - 101 - 0 K O V a r i a t i o n i n magnetic f i e l d V a r i a t i o n i n magnetic f i e l d gradient V a r i a t i o n i n computed current density. ANALYSIS OF ERROR (CONT'D.) Numerical E r r o r : (Gont'd.) The comparison of the two calculations may be observed i n Table IV or from the plot of percentage change i n the magnetic f i e l d , the magnetic f i e l d gradient, and the current density versus r a d i a l p o s i t i o n as given i n f i g u r e 50. From t h i s f i g u r e , i t i s obvious that the v a r i a t i o n i n the computed current density i s proportional to the v a r i a t i o n i n the gradient of the magnetic f i e l d and not to the v a r i a t i o n i n the f i e l d i t s e l f . Since the v a r i a t i o n i n the current density i s of the order of or less than the v a r i a t i o n i n the f i e l d gradient, we can assume that the numerical computation does not introduce any further e r r o r . = 102 - Note, though, that we have ANALYSIS OF ERROR (CONT'D.) Numerical E r r o r : (Cont'd.) seen i n section £.2 that considerable errors are introduced by the numerical approximation to the measured magnetic f i e l d and correction (see f i g u r e 17. (pge.U5)) curves. This l a t t e r e r r o r , however, i s only- appreciable at regions where the current density d i s t r i b u t i o n i s discontinuous. PHOTOGRAPHIC MEASUREMENT The reason f o r introducing photographic measurement i n t h i s i n v e s t i g a t i o n i s two f o l d . F i r s t we wish to examine o p t i c a l l y the perturbation of plasma by the magnetic probe. In p a r t i c u l a r , we wish to obtain an estimate of the " e f f e c t i v e " probe radius which has already been discussed i n section 3.2. Secondly, we wish to have some independent method of examining the plasma i n order to aid i n the interpretation of our magnetic probe measurements. Curzon (1962) and F o l k i e r s k i (1963) have used the same type of high speed framing camera which we have described i n section 7«0« Measurement with a camera obviously does not perturb the plasma as the magnetic probe does, but we s t i l l must recognize l i m i t a t i o n s when we interpret the photographic r e s u l t s . For example, when we plot the collapse curves of section 9.2 with points obtained from the photographs, we are assuming that the luminous region of the discharge coincides with the region of high current density. Curzon (1963) had reason to question t h i s assumption. The r a d i a l position of the radiating ions and neutrals does not necessarily » 103 - 9,0 PHOTOGRAPHIC MEASUREMENT (CONT'D.) coincide with the r a d i a l p o s i t i o n of the electron current f l o w . We w i l l examine this effect* A second source of error i s apparent when determining the radius of the luminous boundary of the films (see figure 5 3 « ) « The boundary i s d i f f u s e and may even be a function of exposure or developing time. A sharp boundary may also appear d i f f u s e because of so c a l l e d "smearing e r r o r " . I f the plasma v e l o c i t y i s very high, and the exposure time i s not correspondingly small then the image on the f i l m w i l l be "blurred" or "smeared" and the boundary w i l l appear d i f f u s e . This reading error i s the source of the error which i s estimated on the collapse curves of f i g u r e £U. A l l photographs are obtained on I l f o r d H . P . 3 f i l m . The f i l m i s developed i n I l f o r d I D - 2 developer f o r approximately twelve minutes and fixed i n Amfix f o r approximately s i x minutes. 9.1 MAGNETIC PROBE PERTURBATION We can use the framing camera to investigate the perturbation of the discharge plasma which i s caused by the magnetic probe. p a r t i c u l a r aspects of this phenomenon are examined. Two F i r s t , we l i n e the camera up with the axis of the magnetic probe guide tube and we obtain so called " s i d e - o n " photographs (see f i g u r e £ 1 . ) . These photographs are examined f o r any v a r i a t i o n i n l i g h t i n t e n s i t y from the region surrounding the probe. Such a v a r i a t i o n would be explained by the "conductivity error" which has been discussed i n - 10U - Photograph on the l e f t hand side of following page FIGURE 5 1 . SIDE-ON PHOTOGRAPH OF DISCHARGE F i l m s t r i p corresponding to f i r s t Probe guide tube appears as l i g h t Time sequence zigzags - .23 / i s e c Time increases from bottom to top Cathode i s at r i g h t . 3 JX ° f 10° A ^ S discharge c i r c l e i n centre square between frames of page. sec Photograph on the r i g h t hand side of following page FIGURE 52. TOP-ON PHOTOGRAPH OF DISCHARGE F i l m s t r i p corresponding to f i r s t 3 JLsec of 1 0 0 Probe guide tube appears as horizontal dark bar Time sequence zigzags - .23 ,6ksec between frames Time increases from bottom to top of page Cathode i s at top. - 1 0 5 - //.Hg discharge MAGNETIC PROBE PERTURBATION (CONT'D.) section 3«2 and thus we could obtain an estimate of the e f f e c t i v e probe r a d i u s . An examination of figure 51* reveals a strong perturbation of the i n i t i a l a x i a l current flow. Characteristic "Mach l i n e s " are observed and these reverse when the p o l a r i t y of the discharge electrodes i s reversed.', However, once the discharge plasma i s f u l l y developed and i t begins to c o n s t r i c t , luminosity about the probe. there i s no v a r i a t i o n i n the For t h i s reason, we have used the actual radius of the quartz guide tube i n the c a l c u l a t i o n s f o r magnetic f i e l d correction (see section 8.2) where an estimate of the e f f e c t i v e probe radius i s required. The second examination of the magnetic probe perturbation i s accomplished with "top-on" photography of the discharge. In t h i s case the camera i s i n l i n e with the axis of the discharge v e s s e l , but i t i s directed toward a mirror above the discharge which gives an image of the probe (see f i g u r e 52.) The purpose of t h i s photograph i s t o investigate any three dimensional perturbation of the discharge current around the probe. We have discussed this p o s s i b i l i t y i n section 8,3 and we expect that i f t h i s perturbation does e x i s t , i t s growth rate w i l l be small i n the early stages of the collapse. Figures' 51, and 52., are both obtained during the initial stages of a discharge formed i n an i n i t i a l gas pressure of 100//B-g argon. Again the Mach l i n e s are obvious, but observe that they - 107 - 9.1 MAGNETIC PROBE PERTURBATION (CONT'D.) occur at s p e c i f i c r a d i a l points - not a l l along the length of the probe. The anticipated e f f e c t s of three dimensional perturbation by the probe (see section 8.3) are a l s o observed i n f i g u r e $2* The c o n s t r i c t i o n at the probe i s p a r t i c u l a r l y obvious i n the l a s t frames. The perturbation e f f e c t i s largest at low pressures because of the higher plasma acceleration. For 100//Hg the e f f e c t i s appreciable when the plasma column i s as large as k cm. r a d i u s . In general, our probe measurement w i l l not be r e l i a b l e when the radius of the plasma column i s small (say, 9.2 less than 2 cm.). COLLAPSE CURVES A collapse curve i s a plot of the diameter of the luminous plasma as a function of time. In figure 5U. we have plotted the collapse curve for each of the three i n i t i a l pressures of the discharge gas which have been used throughout these i n v e s t i g a t i o n s . To obtain these measurements, the f i l m images, which are approximately 8 mm. square, are magnified ten times with the o p t i c a l system of a J a r r e l l - A s h microdensitometer. The magnified image i s viewed on a ground glass screen and the diameter of the plasma column can be measured with an ordinary r u l e . A t y p i c a l image i s reproduced i n figure 53• Scaling factors are avoided by taking a l l measurements to the inner diameter (d ) 0 of the discharge v e s s e l . relative By measuring the diameter (d) of the luminous plasma, we obtain the dimensionless diameter (or r a d i u s ) y defined as d / d » 0 - 108 - FIGURE 5 3 . END-ON PHOTOGRAPH OF DISCHARGE I n i t i a l pressure 5 0 0 / l H g argon Time of photograph i s k ,Usec after i n i t i a t i o n of discharge current Probe guide tube i s horizontal Black c i r c l e i s on brass electrode. SHOCK VELOCITY MEASUREMENT When the plasma current collapses toward the a x i s , i t forms a piston which sweeps up the gas before i t . If the piston moves much more r a p i d l y than the speed of sound i n the medium then we have the formation of a strong shock f r o n t . In our experiment, the piston v e l o c i t y i s approximately 1 0 ^ m/sec, or about UO times the speed of sound. A strong shock front i s characterized by a sharp d i s - continuity i n density and temperature and hence i t should be r e a d i l y observed experimentally. 109 Graph on page FIGURE 5hA» 111. COLLAPSE CURVES O Outer luminous boundary f o r 100 jX-Hg argon « Inner luminous boundary f o r 100 j&Rg argon A Outer luminous boundary f o r 500 ^t/.Hg argon A Inner luminous boundary for 500 jxKg argon Graph on page 112. FIGURE 5iiB. COLLAPSE CURVE X Outer luminous boundary for 250 /^Hg argon -fr Inner luminous boundary f o r 250 JbLEg argon. - 110 - - 112'- SHOCK VELOCITY MEASUREMENT (CONT'D.) A technique for observing shock fronts i n the z-pinch discharge and for measuring t h e i r v e l o c i t y of propagation has been reported by F o l k i e r s k i (1963). It i s possible t h a t the plateau region of the current density d i s t r i b u t i o n s may represent a shock heated plasma and we wish to f i n d the position of the shock front r e l a t i v e to the plateau region* F o l k i e r s k i ' s technique i s based upon the f a c t that a r e f l e c t e d shock wave i s u s u a l l y luminous due to the increase i n enthalpy of the gas at the point of r e f l e c t i o n . C y l i n d r i c a l b a f f l e s are placed i n the discharge v e s s e l and photographs r e v e a l the times at which the shock front s t r i k e s the b a f f l e s . A t y p i c a l example as given i n f i g u r e 55« shows the frames from the photograph which i l l u s t r a t e a r r i v a l of the shock f r o n t . the Four c y l i n d r i c a l b a f f l e s of approxi- mately 2-3 cm. i n length and of diameters 6.0, 7.8, 8.8, and 9.8, are used, i n t u r n , i n the discharge. The cylinders are mounted i n d i v i d u a l l y on a glass rods which l i e s along the discharge a x i s . The glass rod i s supported by a f i t t i n g i n the brass electrode which places the cylinders approximately 1$ - 20 cm. from the electrode. The glass cylinder obviously perturbs the plasma once the current sheet reaches the cylinder and measurements (such as pinch time) are unreliable at times following t h i s i n t e r a c t i o n . We assume that the flow i s undisturbed f o r the time preceding the i n t e r a c t i o n , but even t h i s assumption appears doubtful. - 113 - A careful FIGURE 55. RADIAL POSITION OF SHOCK FRONT Top-on p h o t o g r a p h s . Time sequence z i g z a g s . Time i n c r e a s e s f r o m l e f t t o r i g h t . I n i t i a l pressure • 500 jlEg a r g o n B a f f l e diameter • 8.8 cm. Time between frames • »2k JXseo, Surface o f c y l i n d r i c a l b a f f l e corresponds t o b r i g h t e s t r e g i o n . B a f f l e i s a p p r o x i m a t e l y 2 cm. l o n g , w i t h a x i s h o r i z o n t a l . SHOCK VELOCITY MEASUREMENT (CONT'D.) e x a m i n a t i o n o f f i g u r e 55. shows t h a t t h e c u r r e n t s h e e t appears t o c o n s t r i c t more r a p i d l y i n t h e r e g i o n o f t h e c y l i n d r i c a l b a f f l e . A sequence o f photographs i s t a k e n f o r each c y l i n d e r w i t h e a c h of the three i n i t i a l pressures. The r e s u l t s a r e p r e s e n t e d i n Table V. and p l o t t e d l a t e r i n f i g u r e 58. - 1H| - 9.3 SHOCK VELOCITY MEASUREMENT (CONT'D.) PRESSURE I ( U&K) 100 D (cm.) 500 N + 1 . -9.8 8.8 7.8 6.0 6.0 250 (/isee) .65 .59 .52 .UO .Uo 9.8 8.8 7.8 6.0 6.0 .65 .59 8.8 7.8 6.0 .59 .52 .Uo .ho .52 .Uo .26 .26 .275 .29 .275 .295 .27 .27 .26 .25 .26 .28 .26 T ( i^secT + .3 9 9 12 13 2.3 2.3 3.3 3.8 1U.5 U.o 11 12.5 1U.5 3.2 3.U 3.9 17.5 17.5 U.6 U*9 20 20 2U 5.2 5.0 6.2 TABLE V. POSITION OF SHOCK FRONT D = Diameter of b a f f l e Y = Dimension less diameter corresponding to D Tg Time between frames on photographic f i l m N «= Number of frames between i n i t i a t i o n of discharge and i l l u m i n a t i o n of b a f f l e T = Time f o r shock front to reach given p o s i t i o n Y e 10.0 COMPARISON OF MAGNETIC PROBE AND FRAMING CAMERA RESULTS One of the purposes of the photographic investigation i s t o correlate the v i s i b l e features of the discharge with the current density d i s t r i b u t i o n . But to do t h i s , we must f i r s t know how times measured on the oscilloscope traces correlate with times measured on the photographic f i l m s . which i s common to both. We have no reference point I f , however, we could show that the i n i t i a t i o n of luminosity and the i n i t i a t i o n of the discharge = 115 - 10.0 COMPARISON OF MAGNETIC PROBE AND FRAMING CAMERA RESULTS (CONT'D.) current are simultaneous, then we would have the required reference point. To v e r i f y t h i s simultaneity we perform the following experiment. In this experiment, we must t r i g g e r the oscilloscope from the Rogowski c o i l , i . e . from the break down of the discharge gas, as t h i s i s the mode of t r i g g e r i n g which has been used f o r a l l previous experiments. A l s o , we must trigger the discharge i t s e l f from the photo m u l t i p l i e r of the framing camera i f we wish to obtain photographs of the discharge. With these t r i g g e r i n g arrangements we wish to obtain some reference signal on the oscilloscope trace which coincides with a known time on the f i l m . The most obvious reference s i g n a l i s the pulse from the photo m u l t i p l i e r which immediately precedes f i r s t lens of the framing camera (see figure 23.)• the However, t h i s pulse i s also used to t r i g g e r the discharge and as there i s a delay between t r i g g e r i n g and break down of the discharge (called the formative time delay (tp)i t h i s s i g n a l occurs before the o s c i l l o scope i s t r i g g e r e d . This problem i s solved by placing a known time delay (tjj) i n the s i g n a l lead between the photo m u l t i p l i e r of the framing camera and the o s c i l l o s c o p e . If the time delay (tj) formative time delay (tp), i s greater than the then the photo m u l t i p l i e r pulse w i l l appear on the oscilloscope trace at a time t - 116 - 0 as shown i n figure 56. MEASUREMENT OF FORMATIVE TIME DELAY FIGURE 56 Upper t r a c e : Lower t r a c e : T r i g g e r p u l s e d e l a y e d by 8.9 p^sec (20 v o l t s / c m . ) D i s c h a r g e c u r r e n t (75 kamp/cm.) Time b a s e : Top: Middle: Bottom: .95 IXsec/cm. 100 / x H g argon 250 MJ&g argon 500 / L H g argon. COMPARISON OF MAGNETIC PROBE AND FRAMING CAMERA RESULTS (CONT'D.) F u r t h e r m o r e , we measure t 0 f r o m t h e o s c i l l o s c o p e t r a c e and thus determine t h e f o r m a t i v e t i m e d e l a y , tp • t n - t 0 On t h e c o r r e s p o n d i n g p h o t o g r a p h i c f i l m s we count t h e number - 117 - 10.0 COMPARISON OF MAGNETIC PROBE AND FRAMING CAMERA RESULTS (CONT'D.) of frames which have passed between the t r i g g e r i n g of the discharge and the i n i t i a t i o n of luminosity. Since we know the time between photographic frames from the monitored period of the mirror r o t a t i o n (see section 7.0), lag. then we have another value f o r the formative time I f these two values are the same, then we know that the i n i t i a t i o n of discharge current and luminosity c o i n c i d e . In figure 56, we present oscilloscope traces which show the p o s i t i o n of the delayed pulse (upper beam) r e l a t i v e to the discharge current (lower beam) f o r each of the three i n i t i a l gas pressures. formative time delay obtained from t h i s photograph i s entered i n Table V I . PRESSURE (>Hg) NO. FRAMES T ( Xisec) 100 250 11 500 8 TFL ( Wee) (Msec) 3.6U 3.U8 .16 .26 2.86 2.6U .22 .315 2.52 2.20 .32 COINCIDENCE OF INITIATION OF DISCHARGE CURRENT AND LUMINOSITY T = Time between f i l m frames T F L *= Formative time delay as measured to i n i t i a t i o n of discharge luminosity Tpc Formative time delay as measured to i n i t i a t i o n of discharge current ATp = TF L - TFC - 118 - pisec .26 TABLE VI = ( The TRIGGER PULSE FROM CAMERA TRIGGER PULSE FROM AMPLIFIER INITIATION OF CURRENT .1 sea* APPARENT DELAY ACTUAL DELAY FIGURE 57. 10.0 EFFECT OF AMPLIFIER DELAY COMPARISON OF MAGNETIC PROBE AND FRAMING CAMERA RESULTS (CONT'D.) Table VI also includes the value obtained from the photographic films. The comparison shows that the i n i t i a t i o n of discharge current and luminosity are coincident within our experimental error of _+ .25 ji. sec. There i s a measuring error i n determining the number of frames preceding luminescence. This error i s approximately + 1 frame which corresponds to approximately + .25 ^ s e c . A l s o , there i s a s l i g h t delay (^.1 ^LLsec) i n the amplifier which i s between the photo m u l t i p l i e r and the t r i g g e r p u l s e . This w i l l give an apparent formative delay from the current measurement which i s approximately .1/isec too short (see figure 57.) - 119 - 10.1 DETERMINATION OF SHOCK AND FLOW VELOCITIES Having obtained the desired reference time f o r the f i l m and oscilloscope t r a c e , we wish to make two comparisons between the features of the luminous plasma and those of the current distribution. density F i r s t , we wish to compare the p o s i t i o n of the shock front with the p o s i t i o n of the inner edge of the current plateau as functions of time. density Second, we wish to compare the p o s i t i o n of the outer edge of the luminous plasma with the p o s i t i o n of the peak current density - again as functions of time. In f i g u r e 58. we have plotted the p o s i t i o n of the shock front as a function of time from the data of Table V (section 9.3)• The p o s i t i o n of the inner edge of the current density plateau i s obtained from Table III (section 8.2). For a l l three i n i t i a l gas pressures it i s apparent that the p o s i t i o n of the shock f r o n t and the inner edge of the current density plateau are coincident w i t h i n our experimental error of + .25 cm. In f i g u r e 58. we have also included the p o s i t i o n of the inner edge of the luminous plasma as determined from the collapse of figure 5U« Within our experimental error, curves t h i s inner edge coincides with the p o s i t i o n of the shock f r o n t . In f i g u r e 59. we have plotted the positions of the current peak (again obtained from Table III) with the positions of the outer edge of the luminous plasma (as given i n figure 5U«) as functions of time. Within our experimental error of + .25 cm., i t i s apparent that the outer edge of the luminous plasma does coincide with the p o s i t i o n of - 120 - 5 FIGURE 58. 6 SHOCK FRONT (cjs) VS. TIME Magnetic-"Probe "Results '(section*. 8; 2) ID 100 /iHg Argon A . 250 yUHg Argon <t> 500 pJAg Argon Photographic Results (section 9.2) O 100 MHg Argon ^ 250/cHg Argon O 500 >LHg Argon Photographic • lOOjLtHg A 250/ttHg • 500 /IHg - Results (section Argon Argon Argon 121'- 9.3) 7 5 6 t (jAsec) FIGURE 59. CURRENT PEAK Photographic • 100 /iHg A 250 /iHg o 500 pHg (L]C) - VS TIME Results (figure 5U») Argon Argon Argon Magnetic Probe Results (figures 37. - U2.) • 100 jmEg Argon A 250//Hg Argon • 500 /iHg Argon - 122 7 10.1 DETERMINATION OF SHOCK AND FLOW VELOCITIES (CONT'D.) maximum current density. Thus the luminous plasma of the photo- graphs corresponds to the plateau region of the current density distributions. From figures 58. and 59., we see that the shock v e l o c i t y and the p a r t i c l e flow v e l o c i t y behind the shock front are approximately constant at t h i s stage of the c o l l a p s e . In Table VII we present the shock and flow v e l o c i t i e s f o r the three i n i t i a l gas pressures as determined from the slopes of these graphs. PRESSURE (uJlg) TIME .(/isec) Vs (cm.///sec) Vp 100 2.5 1.27 1.15 250 U 1.12 1.0U 500 5 .91 .85 (cm.7iZsec) TABLE VII SHOCK AND FLOW VELOCITIES Vg = Vp = 10.2 V e l o c i t y of shock front V e l o c i t y of p a r t i c l e flow behind shock f r o n t . DETERMINATION OF ELECTRIC FIELD From the r e s u l t s of the previous section, we have seen that the region of high luminosity and the plateau region of the current density d i s t r i b u t i o n s c o i n c i d e . From section 8.2 we have seen that the r e l a t i v e l y small area of the discharge characterized by the - 123 - 10.2 DETERMINATION OF ELECTRIC FIELD (CONT'D.) plateau region contains almost a l l of the discharge current. The large current and high luminosity i n d i c a t e that t h i s i s a region of high temperature and high c o n d u c t i v i t y . To determine the conductivity we must know the e l e c t r i c f i e l d i n the plasma and t h i s may be calculated from the experimental data of sections 8.2 and 10.1. From Maxwell's equation, and, we have for the p a r t i c u l a r geometry of our experiment, where we have used c y l i n d r i c a l coordinates with the z-axis coinciding with the axis of the discharge vessel (see f i g u r e 1.). Integration of t h i s expression y i e l d s : The second term on the r i g h t hand side of the expression represents an e l e c t r i c f i e l d the c u r l of which i s equal to z e r o . This f i e l d could be a space charge e l e c t r i c f i e l d i n the plasma. However, we assume that we may neglect t h i s term compared to the e l e c t r i c f i e l d induced by the change of the magnetic f i e l d . The e l e c t r i c f i e l d ( E £ ) seen by the plasma i s modified by the plasma motion. Hence, - 12U - 10.2 DETERMINATION OF ELECTRIC FIELD (CONT'D.) where v r i s the r a d i a l flew v e l o c i t y of the plasma as determined i n section 10.1. The i n t e g r a l of the rate of change of magnetic f i e l d i s proportional to the area between the magnetic f i e l d curves of - f i g u r e s k3» - U5» as f o l l o w s . The procedure f o r evaluating t h i s i n t e g r a l i s Suppose we wish to determine the i n t e g r a l f o r the case of 5>00/j.Hg i n i t i a l pressure, 5JLlsec from i n i t i a t i o n of the discharge, and at a r a d i a l p o s i t i o n of 37 mm. The area (A) between the magnetic f i e l d curves corresponding to Ii^sec and 6^usec i s measured from the axis of the discharge out t o a radius of 37 mm. This measurement performed w i t h a planimeter and the area i s evaluated i n wb/m. is The i n t e g r a l i s f i n a l l y determined as: <^Be dr J o = <5t~ The e l e c t r i c f i e l d A "V/m. 2 x 10" 6 (E ) of the plasma i s calculated as a 1 function of radius through the plateau region of the current density for the various i n i t i a l gas pressures at 1 jj. sec i n t e r v a l s . r e s u l t s are presented i n Table V I I I . - 1 2 5 - These 10.2 DETERMINATION OF ELECTRIC FIELD (CONT'D.) PRESSURE AND TIME 100 jLRg Argon 2 JJsec 100 pEg Argon 3 usee 250 /xHg Argon 3 usee 250 >i,Hg Argon 500 ^ H g Argon 3 jusec RADIUS (mm) AREA E (kv/m.) B (wb/m2) B (kv/m.) ii8 51 5U 57 60 62 75 5.19 6.09 7.37 8.U0 9.67 10.39 13.87 3.20 3.75 U.55 5.18 5.95 6.U0 8.5k .17 .21 .25 .29 .327 .35U 1.95 2.U0 2.90 3.35 3.75 U.05 1.25 1.35 1.65 I.83 2.20 2.35 32 35 38 Ul UU U7 51 75 6.81 8.28 9.7U 11.28 12.70 13.91 15.U8 18.5U U.20 5.10 6.00 6.95 7.80 8.55 9.55 11. Uo .189 .23U .282 .327 .375 .U23 .U90 2.18 2.69 3.2U 3.76 U.32 U.87 5.65 2.02 2.U1 2.76 3.19 3.U8 3.7U 3.90 U9 52 55 57 59 75 8.81 11.61 13.31 1U.U7 15.58 19.75 2.72 3.58 U.10 U.U6 U.80 6.08 .220 .263 .308 .3U5 .38U 2.20 2.63 3.08 3.U5 3.8U .52 .95 1.02 1.01 .96 37 39 Ul U3 U5 U7 U9 75 12.19 13.55 15.12 3.70 U.13 in60 5.00 5.U0 5.7U 6.08 7.17 .2U0 .282 .318 .35U .390 •U23 .U63 2.U0 2.82 3.18 3.5U 3.90 U.23 U.63 5U 57 60 63 67 6.05 7.66 9.15 10.82 11.U8 1.86 2.36 . 1 8 1.53 1.95 2.3U 2.96 3.26 16.U2 17.73 18.81 19.9U 23.5U ' 2.82 3.33 3.5U 1. Cfl .23 .285 *3U8 .38U TABLE VIII DETERMINATION OF ELECTRIC FIELD - 126 - E-YB (kv/m.) T . - 1.30 1.31 1.U2 1.U6 . 1.50 l.5l ' 1.U5 .33 .Ul .U8 .37 .28 10.2 DETERMINATION OF ELECTRIC FIELD (CONT'D.) PRESSURE AND TIME 500 jitRg Argon U jisec RADIUS (mm) Argon 5 jisec 5oo jing Argon 6 juisec B E (kv/m.) (wb/m2) 6.87 8.72 10.92 12.67 13.56 15.58 2.12 2.69 3.36 3.90 U.17 U.80 .213 .282 .3U8 37 39 Ul U3 U5 U7 U9 75 7.28 8.31 9.39 10.51 11.58 12.3U 13.18 2.22 2.52 2.85 3.20 3.53 3.75 U.00 U.2U 21 25 29 3U 39 U.OU 1.2U 1.77 2.39 3.03 3.5U U6 50 5U 57 60 75 500 ^ H g AREA 13.92 5.85 7.77 9.83 11.50 1 .31 .29 .228 .26U .300 .336 .372 .U08 1.9U 2.2U 2.55 2.86 3.16 3.U7 3.77 .28 .156 .216 1.32 1.83 2.33 2.96 3.60 -.08 -.06 .06 .07 -.06 .U68 .uuu " .27U .3U8 .U23 T ft (CONT'D.) DETERMINATION OF ELECTRIC FIELD - 127 - E-VB (kv/m.) 1.81 2.U0 2.96 3.U0 3.98 .uoo TABLE VIII TB (kv/m.) .UO .50 .19 .28 .30 .3U .37 .28 .23 PART III COMPARISON OF THEORETICAL MODELS WITH EXPERIMENTAL RESULTS - 128 - 11.0 INTRODUCTION I n P a r t I I I o f t h e t h e s i s we w i s h t o compare our experimental r e s u l t s w i t h v a r i o u s t h e o r e t i c a l models o f t h e z - p i n c h d i s c h a r g e . There are p r e s e n t l y t h r e e b a s i c models - t h e "snowplow" model, t h e "shock wave" model and the "independent p a r t i c l e " model - and we d e s c r i b e t h e p h y s i c a l s i g n i f i c a n c e of each model i n t h e f o l l o w i n g section. The models a p p l y t o d i f f e r e n t ranges of e x p e r i m e n t a l c o n d i t i o n s , but o f t e n t h e range o f v a l i d i t y a l l o w s more t h a n t h e o r y t o be a p p l i e d t o a g i v e n e x p e r i m e n t . s e c t i o n 12.0 one I t w i l l be shown i n t h a t b o t h t h e snowplow model and t h e shock wave model p r o v i d e u s e f u l concepts f o r t h e p r e s e n t experiment. t h r e e models are d i s c u s s e d i n s e c t i o n 12.0 However, a l l f o r t h e purpose o f comparison. S e c t i o n 13 d e a l s w i t h t h e snowplow a p p r o x i m a t i o n . e q u a t i o n i s developed The dynamic and t h e n s o l v e d u s i n g an a p p r o x i m a t i o n technique. The p r o b l e m has been s o l v e d p r e v i o u s l y by n u m e r i c a l t e c h n i q u e s (see Curzon (l°63)), but i t i s f e l t t h a t an a n a l y t i c s o l u t i o n i s advantageous because o f t h e e x p l i c i t appearance o f the discharge parameters i n t h e f i n a l s o l u t i o n . analytic The v a l i d i t y of t h e a p p r o x i m a t i o n must be t e s t e d by a c o m p a r i s o n of t h e o r e t i c a l and experimental r e s u l t s . This comparison i s given i n s e c t i o n In s e c t i o n l l ; , the Rankine Hugoniot r e l a t i o n s are f r o m t h e shock wave model. 13.2. developed A system o f s i x e q u a t i o n s r e s u l t s t h e s e a r e s o l v e d by a g r a p h i c a l t e c h n i q u e . and Again the t h e o r e t i c a l r e s u l t s are compared w i t h e x p e r i m e n t a l r e s u l t s (see s e c t i o n 1^.2). - 129 - 11.0 INTRODUCTION (CONT'D.) In section II4. we f i n d that the shock wave model i s inadequate f o r understanding of the experimental r e s u l t s . In section 1$ we discuss a proposed model which i s an extension of the shock wave model. 12,0 This model does describe the experimental r e s u l t s . THEORETICAL MODELS The physical concepts associated with the three theoretical models - the "snowplow", "shock wave", and "independent p a r t i c l e " models - w i l l now be discussed. In a l l three models i t i s assumed that the plasma conductivity i s i n f i n i t e . As the " s k i n depth" ( c h a r a c t e r i s t i c thickness of current d i s t r i b u t i o n ) i n the plasma i s i n v e r s e l y proportional to the square root of the conductivity, we see that t h i s assumption implies that the plasma current restricted plasma. is to a very t h i n sheet on the surface of the discharge The assumption of i n f i n i t e conductivity also implies that the magnetic f i e l d does not penetrate the current sheet and i t will be shown that a magnetic pressure i s formed outside the current sheet which forces the sheet to c o n s t r i c t . F i n a l l y , the assumption implies that we have neglected the energy d i s s i p a t i o n due to heating. Of course, Joule energy i s s t i l l transmitted to the plasma through the motion of the current sheet. The independent p a r t i c l e model i s discussed by Gartenhaus (I96I4). It i s assumed that the plasma p a r t i c l e s are r e f l e c t e d by the collapsing current sheet and they return to the confined plasma with a v e l o c i t y equal to twice the v e l o c i t y of the current - 130 - sheet. 12.0 THEORETICAL MODELS (CONT'D.) In t h i s model the confined plasma i s approximately homogeneous, it i s bounded by a very t h i n current sheet, and there i s a vacuum between the current sheet and the discharge w a l l . The dynamic equation i s derived from the r a t e of change of momentum of the i n d i v i d u a l plasma p a r t i c l e s due to the increasing magnetic pressure. I f , however, the current sheet i s strongly accelerated such that the r e f l e c t e d p a r t i c l e s are swept up by the c o n s t r i c t i n g sheet, then we have the "snowplow" model of the l i n e a r pinch. was f i r s t described by Rosenbluth (195U). The model In t h i s case the central region of the discharge i s l e f t undisturbed u n t i l i t i s encountered by the collapsing sheet i t s e l f and the p a r t i c l e s which are then swept up are assumed to form a very t h i n mass l a y e r . Thus we have a cold gas which i s enclosed by a t h i n mass sheet and a t h i n current sheet and the remainder of the discharge vessel w i l l have been evacuated by the "snowplow". The dynamic equation i n t h i s case i s derived from the rate of change of momentum of the mass layer due to the increasing magnetic pressure. The derivation of the dynamic equation f o r both the independent p a r t i c l e model and the snowplow model does not take into account the i n t e r n a l pressure due to the confined plasma or gas and thus these analyses are i n v a l i d f o r small r a d i i of the current sheet. In the t h i r d model, i t i s again assumed that we have a t h i n current sheet moving under the influence of the magnetic pressure, but i t i s now assumed that the r e f l e c t e d p a r t i c l e s - 131 - experience 12.0 THEORETICAL MODELS (CONT'D.) c o l l i s i o n s and a shock front i s produced which precedes the collapsing current sheet. Thus we have a basic difference from the independent p a r t i c l e model where a c o l l i s i o n l e s s plasma i s considered. The equations governing the motion of the shock front are the standard Rankine-Hugoniot relations since we have assumed that the magnetic f i e l d i s completly external to the current Several authors have discussed these equations. sheet. A l l e n (1957) has investigated the equations f o r plane geometry while Kuwabara (1963) has done the same for c y l i n d r i c a l geometry. 13.0 DYNAMIC EQUATION FROM SNCWPLCW APPROXIMATION In t h i s section we set up the mathematical equation which results from the snowplow model of the z-pinch discharge. It is observed that the magnetic pressure which causes the plasma to c o n s t r i c t i s a function of the discharge current, but the current i s affected by the c o n s t r i c t i n g plasma through the change of inductance of the discharge. The result of t h i s interdependence i s that the problem requires the solution of two simultaneous, second order equations one of which i s n o n - l i n e a r . The f i r s t equation i s derived from the snowplow model. It is assumed that the discharge forms a t h i n current sheet and t h i s sheet sweeps up a l l the gas as the current sheet c o n s t r i c t s . Internal pressure i s neglected and the external magnetic pressure i s determined by the discharge current. - 132 - 13.0 DYNAMIC EQUATION FROM SNOWPLOW APPROXIMATION (CONT'D.) The equation i s , t h e r e f o r e , determined from Newton's law: d_ (TL0b(r - r ) dr) - - 2TTr. B £ dt where *o 3t B7 >c 2 is is is is dt the the the the 13.1 2 o 2JJQ radius of the current sheet radius of the discharge v e s s e l , i n i t i a l mass density of the gas magnetic pressure. To s i m p l i f y the following algebra, we l e t Ij = r / r 0 and we use the r e l a t i o n s h i p : B where I « Mol 2Tfr 13.2 i s the discharge current i n the sheet. Equation 13.1 becomes: d_ (1 - c j ) dc|« - Jlo dt L u r ^ ^ r ^ 2 dt I 2 . 13.3 A second equation i s obtained f o r the current i n the discharge circuit. The discharge c i r c u i t i s shown i n figure 1. (page 2.) and the c i r c u i t equation i s : , d2(Le dt + Ld)I+I«0 c 2 where Le Ld C i s the external c i r c u i t inductance i s the inductance of the discharge i s the t o t a l capacitance. - 133 - 13.U DYNAMO EQUATION FROM SNOWPLOW APPROXIMATION (CONT'D.) However, the discharge inductance i s a function of Ld - - JlollnL - - MoH Inu +/r)ia) ' 2 TT where J R R i s the length of the discharge i s the radius of the return conductor. The t o t a l inductance of the discharge c i r c u i t i s : Le + Ld « Le (H - GiViCj) where and, H =1 - G l n ( r / R ) 0 I f we l e t , I'- (Le + Ld) I and, W = (LeC)" 0 2 1 equation 13.U becones, d !' 2 dt 2 + W Q 2 H-Glncj with the i n i t i a l conditions: I'(0) = 0 dl'(O) = V dt - 13U - I' = 0 DYNAMIC EQUATION FROM SNOWPLOW APPROXIMATION (CONT'D.) 13.0 S i m i l a r l y equation 1 3 . 3 becomes, 1 3 . 1 3 where F 2 - jU 1 3 - 1 U 0 Fff /0 r 2 and the i n i t i a l conditions 0 Q J are: H(0) = 1 1 3 . 1 5 ANALYTIC SOLUTION OF EQUATIONS 13.1 Equations 1 3 . 1 1 and 1 3 * 1 3 roust be solved f o r I ' functions of time. and Equation 1 3 . 1 3 i s a non-linear equation and there i s no standard method of s o l u t i o n . solved numerically (Curzon This equation has been but i f we could obtain even an ( I 9 6 3 B ) , approximate analytic s o l u t i o n , we could determine the dependence of I and C| upon the variables we have defined above ( i . e . / 0 , 0 as r , Q R, 1 , Le, C, and V ) . In order to find an approximate analytic s o l u t i o n we assume I ' to be of the form: I' = I Then and, 0 siiiC^i(t) = di' = I dt 0 CX , cosCKi I' = d I ' = - I CX 2 - 0 1 3 5 - 1 3 . 1 6 2 sinO(, + P C X , cosO( 13.1 ANALYTIC SOLUTION OF EQUATIONS (CONT'D.) The I n i t i a l conditions give CX,(0) = 0 , I Q - V^,(0) S i m i l a r l y , we assume <-| = t j Then, cos(X (t) 0 Cj =_£j Cx. 0 and 13.17 2 2 sinO^ =-C| a cos6X 0 2 2 -C| c\ 0 s i °^ n 2 The i n i t i a l conditions giveC<2,(0) = 0 , l | = 1 0 Substituting these expressions i n the current equation (l3.ll), we f i n d , CX, cosCX, - e x s i n ^ + W s i n C ^ 2 • 0 2 H - Glr}cos(X or, CX. (H - G l n c o s c O » tan<X (o< (H - GlncosO< )- W ) ' 2 i t 2 2 2 0 S i m i l a r l y , the snowplow equation (H 2 - GlncosC^) (coso( 2 2 ( I 3 . I 3 ) s i n 3cxzc<2 + 3 13.18 yields, sin o< cos o(2ex )=F i sin 0(; 2 2 2 2 2 o 2 or, (H - Glncos0fe) (cos<X 2 2 sin CK 0< 3 2 2 + I sin 2CX C< )=F I^ sirfo(, 2 2 2 2 2 13.19 Equations 13.18 and 1 3 . 1 9 must be solved simultaneously. do t h i s expand 0< F andOCg ^ "terms of t . To We use the facts that Of, (0) andO< (0) are zero and thatC*., and OC> are, i n general functions, of 2 odd powers of t only. - 136 - 2 13.1 ANALYTIC SOLUTION OF EQUATION (CONT'D.) That i s , CX, - Wit + at + " ' 13.20 - W t + bt + " ' 13.21 3 0< 3 2 We also make use of the following expansions: tanCX, -CX, +CX? +"• 3 (valid forCX.,^2) s±nO( -CX| +••• (valid f o r allCXg) cosCXg <= 1 - ( X + ••• (valid f o r all0(g) cosO( (valid f orCK < "^2) 2 2 In 2 -CX , - #| B 2 2 Using these expansions, equation 13*18 now becomes, t o order t , 3 6at (H + G w , t ) 2 - 2 (wit +(a+w )t )((wi+3at ) (H+Gw2 t ) - w ) 3 2 3 3 L 2 2 2 13,22 2 S i m i l a r l y equation 13.19 becomes, to order t^, (H+G w § t ) ( ( w t ) 3 6bt + 3 (2w t + (2b - 8 w3)t ) (w? + 3 b t ) ) 2 2 2 3 9 = F I ( w i t + (a - w i ) t ) 6 2 0 2 3 - 137 - 3 2 2 2 2 13.23 13.1 ANALYTIC SOLUTION OF EQUATION (CONT'D.) For these series expansions to be true f o r a r b i t r a r y time, the c o e f f i c i e n t s of t must each vanish. n Thus from equation 13.22, the c o e f f i c i e n t of t gives, 6aH = wi (Hwi - w ) I3.2U 2 2 and the c o e f f i c i e n t of \ ? gives, 3aGw£ = 6aw?H + G w?wjj + (Hw - W o ) ( a + w?) 2" 3 13.25 2 S i m i l a r l y , from equation 13,23, the c o e f f i c i e n t of t 3H w^ = F I 2 2 0 2 2 gives, w? 13.26 and the c o e f f i c i e n t of t^ gives, 6bH w^ + l8w|bH + 6bH w3 2 2 2 + 3HGW2 » F Ii, 2w,(a - wj.) 2 2 or. (3G * h) v& + 30bw2 « F I « 2 w . (a -• w?) H ^ "H ^ Z 2 2 2 13.27 2 The four parameters a, b , Wj^and w , may now be determined. 2 Equation 13.21; gives, a - w, (Hw? - w ) 2 6H 13.28 Equation 13.26 gives, w 2 "J Y3H I2 - 138 - 1 3 '29 13.1 ANALYTIC SOLUTION OF EQUATION (CONT'D.) or by s u b s t i t u t i n g 13.12 and 13.16 i n equation 13.29, wo =/FV\^ 13.30 lf3B/ 2 Combining 13.25 and 13.26, we obtain F I 2 Q 2w! (a - w j ) = wjw£ 2 H ' H 6~ 2 13.31 and thus equation 13.27 becomes, b = (UH - 30) w2 - w w„ 13.32 2 3 0 30IT H The f o u r t h parameter, wt, i s obtained from a combination of equations 13.2k and 13.25. (9Hw? - w ) 2 (Hw? - w ,) =-3Gw w = -VT 2 2 2 FGV w H 13.33 2 In order to solve equation 13.33 f o r w t , we write x 2 = Hw /* 2 2 and write equation 13.33 as, (9x where 2 - 1) (x 2 - 1) - - k 13.3U k = f j FGV If we assume that x = 1 - £ where £ « 1 , 6 ( 1 - 26) - k Iff - 139 - 13.35 then, 13.36 13.1 ANALYTIC SOLUTION OF EQUATION (CONT'D.) Therefore, £= 1 + (1 - k k)= 13.37 U 9 and finally, wi - 3+ 1 WQ I (1 - Uk) \ 13.38 2 T Considering the case ofyOo~*"°° ( i . e . when the plasma can not c o n s t r i c t ) we f i n d that the p o s i t i v e square root must be taken. t h i s case, F ^ ) -*-0 13.35) and thus w, (see equation I3.H4.) and k(^\ )—^0 =) In (see equation w /^~H. 0 The following argument shows that t h i s i s c o r r e c t . w w t 0 We know =/ Le \g V Ld + Le j 13.39 and a l s o , when the plasma can not c o n s t r i c t , Ld + Le = Le (H - Inejp = Le (H - In 1) 13.U0 = LeH Thus f o r the case of^0 -*»cO, equations 13.39 and 13.U0 0 that W| —^^WQ/^H" show which i s the desired r e s u l t . Equation 13.3H can also be solved graphically as shown i n f i g u r e 60. We plot f ( x ) = (9x - l ) (x 2 2 - l ) against x and the s o l u t i o n i s obtained at the i n t e r s e c t i o n of t h i s function with the constant k(^->). For our experimental configuration k ^ o ) = 0 .13 x K f F^O ). 6 Q - mo- 13.1 ANALYTIC SOLUTION OF EQUATION (CONT'D.) Values of k ( ^ ) corresponding to i n i t i a l pressures of 1 0 0 , 2 5 0 , and 5 0 0 JA,Eg of argon are given i n Table IX. PRESSURE 100 18.2 x 1 0 6 2 . 3 7 250 1 1 . 5 x 1 0 6 1 . 5 0 8,15 500 x 1 0 1 . 0 6 6 TABLE IX DETERMINATION OF k(p ) 0 F C A J ) given by equation 1 3 . 1 U k(^>) given by equation 1 3 . 3 5 Figure 6 0 . shows that r e a l roots are obtained f o r wt only when the i n i t i a l argon pressure i s greater than 200//lHg. The s i g n i f i c a n c e of t h i s r e s u l t can be seen i n the following d i s c u s s i o n . The equation f o r the discharge current i s (13.9)s 1=1' I = _ V _ s i n (w,t + wtL » at ) 3 V z(wT7 where Z(w,) i s an apparent impedance. Thus f o r i n i t i a l pressures less than 200 WHg (where we have a complex s o l u t i o n f o r w i ) , we have / a c i r c u i t which i s represented by a complex impedance. Our c i r c u i t shows r e s i s t i v e effects which are due to the rapid rate of change of - HA - -£ too I I -M I FIGURE 60. 13.1 I 1 I GRAPHICAL SOLUTION FOR wi ANALYTIC SOLUTION OF EQUATION (CONT'D.) the inductance of the discharge. d_ (LI) - L d i + I dL dt dt dt The second term represents the r e s i s t i v e term. 13.U1 As the collapse i s more rapid i n low pressure gases, we f i n d that t h i s term becomes appreciable at argon pressures less than 200 y&Hg. - 1U2 - 13.1 ANALYTIC SOLUTION OF EQUATION (CONT'D.) To summarize t h i s s e c t i o n , we have obtained an analytic approximation f o r the radius of the collapsing plasma discharge. LJ*> cos (w t + b t ) 1 3 . U 2 3 2 W wo =/ FV \ ^ where 2 \{m ) b «= (UH - 3G) v| - wjwj 30H 30H F and, 2 = , M-o H = 1 w where } r R 0 Le C V and, is is is is is is is the the the the the the the 2 Gln(r /R) 0 - 1/LeC length of the discharge vessel inner radius of the discharge vessel radius of the return conductor i n i t i a l gas density external c i r c u i t inductance external capacitance i n i t i a l voltage on the capacitor. We have a l s o obtained an analytic approximation f o r the discharge current. 1 where = V s i n ( ,t wiLe(H - Glnty) w w k = Q FGV win - X U 3 - + at ) 3 13.U3 13.1 ANALYTIC SOLUTION OF EQUATION (CONT'D.) a » wi (Hw? - w ) 2 m 0 ' and' the remaining variables are as defined f o r equation 13.1*2. 13.2 COMPARISON OF THEORETICAL AND EXPERIEMENTAL RESULTS In t h i s section we calculate the discharge radius (cj) and current (I) using equations 13.1;2 and 13«1|3 with the discharge parameters of the experimental apparatus as given i n Table I (section 7.0). We then compare these r e s u l t s with the observed values. From Table I we obtain the f o l l o w i n g . Discharge C i r c u i t : C « $3JJj Le - .12 pE V - 10 kV Discharge V e s s e l : i = .75 m r - .075 m R = .085 m Q P = 2.13 x K T x (Pressure i n /(Hg) kg/m3 6 Q In the following three examples, M.K.S.Q. units are used. Example 1. 500 jlUg Argon: The following variables are defined i n terms of the discharge parameters by equations 13.^2 and 13«U3» F = 8.15 x 10 G - 1.3 H • 1.1U w - .396 x 1.0 sec"?" wi = .339 x 10j? secTjW2 • .20h x 10° sec** 6 0 6 x - IliU - 13.1 COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS Example 1. $00J)Kg Argon: (Cont'd.) sec" a = - .OOIOU x 1 0 b .OOO7I4 x 1018 sec~3 k = 1.06 18 H = cos t (Vlsec) 0 1 2 3 k 5 6 7 8 9 10 11 (CONT'D.) (w t + b t ) 3 2 bt 3 -y- 0 .001 .006 .020 .0U7 .093 .160 .251i .379 0 • 20U .1*08 .612 .816 1.020 1.22U 1.U28 1.632 I.836 2.01*0 2.2UU 3 1.00 .98 .92 .83 .72 .60 .U85 .385 .31 .26 .27 .305 0 .203 •U02 .592 .769 .927 1.06U 1.17U 1.253 1.296 1.300 1.260 .5U0 .7U0 .98U In 14 0 -.022 .083 .186 .3U8 .510 .72U .955 1.17 1.3U5 1.77 1.19 I - V s i n (wit + at3) wiLe(H - GiIncj) t (Msec) 0 1 2 3 h 5 6 7 8 9 10 sinfl [ wit at .339 .678 1.017 1.356 1.695 2.03U 2.373 2.712 3.051 3.39 -.001 -.008 -.028 -.067 -.130 -.225 -.360 -.533 -.758 -1.01+0 3 .338 .670 .989 1.289 1.565 1.809 2.013 2.179 2.293 2.35 - llt5 - 0 .33 .62 .8U. .96 1.00 .95 .91 .82 .75 .71 -Glnn 1 .029 .108 .2U2 .U5U .66U .9U2 1.2li 1.52 1.75 2.30 I(kA) 0 69 122 150 1U9 137 112 9h 76 6U 51 13.1 COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS (CONT'D.) Example 2. 250 >Hg Argon? F - 11.5 x 10 G - 1.3 H = l.lU k » w » .396 x 10J? sec"*' w, - .325 x 10° sec"jw = «2U3 x 10° s e c " 6 0 a - -.0012U x 1 0 b » .00081 x 1 0 1.5 = cos (//sec) 0 1 2 3 k 5 6 7 8 9 1 2 wgt .i|86 .729 .972 1.215 1.U58 1.701 1.9UU 2.187 sec'3 sec"3 2 <X A- ? 0 .001 .006 .022 .052 .101 .175 .278 .iii5 .590 .2h3 1 8 (w t + bt3) bt3 0 l 8 1 .97 .89 .76 .60 0 c2k2 .U80 .707 .920 l.llli 1.283 . 1.U23 .10* .28 .15 •ou 1.529 1.597 0 -.035 .117 .27U .510 .820 1.27 1.90 3.22 -.02 I = V si (w,t + at3) w,Le(H - GlOLj) yUsec) 0 1 2 3 h 5 6 7 8 9 at .325 .650 .975 1.300 1.625 1.950 2.275 2.600 2.925 cx, 3 .321; -.001 -.009 -.033 -.050 -.155 -.268 -.U25 -.635 -.905 - .6U1 • 9U2 1.250 1.1; 70 1.682 1.850 1.965 2.020 li;6 - (kamp) sma, .32 .60 .81 .95 .99 .99 .96 .92 .90 .0U6 .152 .357 .661; 1.066 1.65 2.U7 0 70 120 1U9 136 115 91 68 13.1 COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS (CONT'D.) Example 3. 100 J/Eg Argon: F G H k » 18.2 x 10 = 1.3 = 1.1U = 2.37 w » .396 x 10 sec „ -1 w = .305 x 10° sec b = -.00109 x 1 0 sec 6 0 w 2 i a w. » (.271 + . 0 2 l i ) x 10 s e c " « -(.00U3 - .00032) x 10lo sec 6 -j < = cos (w2t + bt 3 1 ) t (/(.sec) 0 1' 2 3 k 5 6 bt w?t 0 .305 .610 .915 1.220 1.525 1.830 c*2 -4- 0 .301* .601 .886 1.150 1.389 1.595 1 .95 .82 .63 .ill .18 -.02 3 0 .001 .009 .029 .070 .136 .235 1 » V sin (wit + a t w,Le(H - G IOC| ) 3 0 -.051 .198 .ll62 .891 1.71 ) t ( JJ, sec) 1 2 3' U 5 Wit .271 .51*2 .813 1.08U 1.355 + + + + + at sin 3 - .001* -.031* + .002 i =.116 + .008 » -.275 + .019i - .538 + .037* .02li .01*21 .O631 .081*1 .1051 (wit + at3) .27 .19 .61* .72 .73 + .019 L + .038i + .05hi + .069 i + .095 i (continued) t ( yUsec) 1 2 3 k 5 -Gh .066 .258 .600 l a 16 2.22 V wtLe(H-Glnt|) x I03 ' 257 222 178 1 3 U 92 201 17t iui 10 i 7 i - 11*7 - I 3 x 10 3 69.2 109 113 96 67 + 10i + 16i + 21i +16*+13i I (kamp) 70 110 115 98 68 / \ X / i i /*/ \ \ i 1 1 / 1 0 \ \ \ i \ t \ ^ \\ \ / / y \ \ \ ] H i \\ V i il ^ ^ ^ ^ ^ J{4 o \\ 1 H > • \ \\ * \\ 1 1 1 If 1 \ \ \ \ \ n i i \ \ \ \ V \ \ * \ v •° \. i f il i i 1-4 \ \ *\ \ \\ i ii \ \ \ \\ \ . \ ii i \ \. i\ FIGURE 61. —*-Experimental Current —Experimental Collapse SNOWPLOW MODEL AND EXPERIMENTAL RESULTS (500/Mg Argon) — Current from Snowplow Model — Collapse from Snowplow Model O Corrected Snowplow Model - 1U8 - Z FIGURE 62. 6 A 8 SNOWPLOW MODEL AND EXPERIMENTAL RESULTS ( 2 5 0 / i H g Argon) -X-Experimental Current — E x p e r i m e n t a l Collapse —Current from Snowplow Model —Collapse from Snowplow Model OCorrected Snowplow Collapse - 1U9 - FIGURE 63. —X-Experimental Current —•-Experimental Collapse SNOWPLOW MODEL AND EXPERIMENTAL RESULTS (100 jx Hg Argon) — Current from Snowplow Model —Collapse from Snowplow Model - 150 - 13.1 COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS (CONT'D.) In figures 61 - 63 we have plotted experimental and t h e o r e t i c a l curves of both I and L| for each of the three i n i t i a l argon pressures used i n the present experiment. The experimental value of the current and plasma radius are obtained from figure 29 (section and figure %k (section 7.1) 9.2),respectively. The experimental curves of figures 61 - 63 show the trend of the t h e o r e t i c a l l y predicted curves, but the t h e o r e t i c a l current less than the observed current. is Even with a current which i s too small, the theoretical c a l c u l a t i o n predicts a collapse which is too fast. This i s a surprising r e s u l t and i t implies that not a l l the discharge current i s used to accelerate the plasma toward the axis of the discharge v e s s e l . Indeed the equivalent c i r c u i t of figure 6U i s probably a more r e a l i s t i c approximation of the discharge c i r c u i t , than the c i r c u i t assumed i n section 13.1. What i s the p h y s i c a l significance of the inductance L ? 0 current density d i s t r i b u t i o n s obtained i n section 8.2, In the we have observed that there i s a considerable current which l i e s i n s i d e the "knee" of the current density curve. We have also observed that the shock f r o n t , i . e . the concentration of mass, coincides with the p o s i t i o n of the "knee" of the current density curve. Therefore, the current l y i n g i n s i d e t h i s "knee" does not accelerate the mass of gas. It i s the current which we assume flows through the inductance L . D - 1 5 1 - FIGURE 6U. 13.1 MODIFIED DISCHARGE CIRCUIT COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS (CONT'D.) For s i m p l i c i t y we assume that Lo i s proportional to L ^ . Lo = Ld 13.kk X Thus the t o t a l inductance of the c i r c u i t i s now, L = Le + (1 +A)" 1 Id 13.U5 and the c i r c u i t equation becomes, d !' + dt2~ 2 where I ' - IL wp I' = 0 H' - G ' l n ^ 2 13.U6 and I i s the t o t a l discharge current. G' = (1 +A)"*1 G H' = 1 - G' l n ( r / R ) 0 - 152 - 13.kl 13.1*8 13.1 COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS (CONT'D.) The snowplow equation i s , d_ (1 - o ) dt ' 2 where du= - F ' ! ' dt' cf (H« - G'/ncj) 2 13.U9 2 F ' » ( l +/\ ) F . The advantage of an analytic solution to the problem now becomes obvious. The new t o t a l current I ' can be determined from the calculated values of I above as; I ' = H - G|nq x I H' - G'lnuj 13.50 and to a good approximation, Lj'^cos (w i 1+ A t 2 + b l/l+At3) 13.51 3 i . e . the new tj' i s determined from the previously by a simple expansion of the time scale, t' = 13.52 1+At We now require an estimate f o r X• X bas been determined e x p e r i - mentally and i t i s most e a s i l y obtained from the data of Table V I I I . Since I ( r ) , the current within a radius r i s proportional to r » B ( r ) , then rk-.B(r ) i s proportional to the current within k where r-^- i s the r a d i a l p o s i t i o n of the knee of the current density d i s t r i b u t i o n . This value i s thus a measure of ILO* Similarly rp«B(rp) i s proportional t o the current within the r a d i a l p o s i t i o n of the density peak. Thus r « B ( r p ) - r^'BCr^) i s a measure of I j ^ . p - 153 - current 13.1 COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS Now, (CONT'D.) ILQ = Ld * A lid Lo The r e s u l t s are given i n Table X. PRESSURE TIME .*(IM + ^ klLp kI I*1 X 100 2 3 220 250 80 60 lUO 190 .6 .3 2^0 3 U 227 108 89 120 ll+o .9 .6 500 3 U 5 6 257 281 218 165 97 98 160 180 135 130 .6 .6 .6 .3 227 8U 83 TABLE X DETERMINATION OF A As an example of t h i s procedure we correct the r e s u l t s of Example 1. f o r the effects of the inductance Lo. From equation 13.52 we see that the collapse curve i s corrected through a 2$% expansion of the time base. The corrected discharge current must be calculated from equation 13.50. This c a l c u l a t i o n i s given i n Table X I . The corrected results compare very favourably with the measured collapse curve and discharge current f o r the case of 500 /cHg argon. - 1514 - 13.1 COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS (CONT'D.) TIME ( /JUsec) H - Glow 1 1 ' 2 3 ii 5 6 7 8 9 10 I' (kamp) H' - G'|*t| 1.17 1.25 1.38 1.59 1.80 2.08 2.37 2.66 2.89 3*hh 1.11 1.16 1.2U 1.37 1.51 1.67 1.87 2.ou 2.19 2.53 73 132 167 173 163 1U0 119 99 85 69 TABLE XI MODIFICATION OF SNOWPLOW CALCULATION H » 1.1U G *= 1.3 H' » 1.09 G' -= .81 I ' determined from eqtiation 13.50 The rather large error on the measured current i s due to the 1% uncertainty i n c a l i b r a t i n g the Rogowski c o i l . In figure 62. we have also plotted the corrected collapse curve although the assumption of constant A i s questionable f o r the case of 250JUEg argon. It should be observed that the collapse curve for 100//Eg argon i s i n poor agreement with theory. The current discrepancy i s corrected by the inductance e f f e c t discussed above, but the discrepancy of the collapse curve i s accentuated by t h i s correction. It i s probable that the collapsing plasma does not sweep up a l l of the i n i t i a l gas and t h i s would account for the more rapid collapse than has been predicted by theory. - 155 - 13.1 COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS (CONT'D.) The very good agreement f o r the cases of 250 and ^OO^Hg indicates an e f f i c i e n t sweeping up of the gas. is This observation important f o r the discussion of the shock wave model i n the next s e c t i o n . lli.O DYNAMIC EQUATION FROM SHOCK WAVE MODEL The snowplow model predicts the p o s i t i o n of the c o n s t r i c t i n g plasma i n the z-pinch as a f u n c t i o n of time, but i t y i e l d s no information on the plasma temperature, the i o n i z a t i o n r a t i o , plasma pressure and the mass density of the plasma. the To investigate these macroscopic parameters we make use of the "shock wave" model. As our magnetic probe measurements are r e s t r i c t e d to regions of large LJ (see section 8.3) due to possible errors i n the regions of small we can obtain estimates f o r the above parameters only at early times i n the discharge process, i . e . 2-1*/Isec after initiation. It can be seen from the collapse curves of figure 51*. that the r a d i a l v e l o c i t y of the collapsing plasma i s approximately constant i n t h i s region and t h i s allows an accurate measurement of the v e l o c i t y of the shock f r o n t . A l s o , f o r large Cf (0.1* - 0.6), we can use the plane shock wave model because effects of curvature of the c y l i n d r i c a l plasma sheet are small. We include the e f f e c t of i o n i z a t i o n i n the Rankine - Hugoniot relations Resler (1952)). - 156 - (see, f o r example v =o V=V s S SHOCK FRAME /Oi,pi,ht V=0 .LAB FRAME DISCONTINUITY FIGURE 65. SHOCK WAVE PARAMETERS Parameters d e f i n e d i n T a b l e X I I CX. Q h m n IIQ hi t\± p Qi Qii qii R T Vp Vs 0 First ionization ratio = pjf/n Second i o n i z a t i o n r a t i o = /n E n t h a l p y / u n i t mass Atomic mass o f a r g o n = 6.68 x 10"* °kg I n i t i a l d e n s i t y o f poles/m3 F i n a l d e n s i t y o f neutrals/rn3 F i n a l d e n s i t y o f s i n g l y i o n i z e d atoms/m3 F i n a l d e n s i t y o f d o u b l y i o n i z e d atoms/m3 P r e s s u r e (nt/m?) _ F i r s t i o n i z a t i o n p o t e n t i a l ~ 2.51 x 10~:ri:j Second i o n i z a t i o n p o t e n t i a l = lie 39 x 1 0 ~ l " j F i r s t i o n p o t e n t i a l / u n i t mass = .376 x H P j / k g Second i o n p o t e n t i a l / u n i t mass - .656 x 1 0 j / k g Mass d e n s i t y (kg/m3) Gas c o n s t a n t / u n i t mass « 208 j / k g Temperature (°k) F l o w v e l o c i t y o f f l u i d b e h i n d shock f r o n t V e l o c i t y o f shock wave. 1 : L 2 8 TABLE X I I SHOCK WAVE PARAMETERS - 157 - lli.O DYNAMIC EQUATION FROM SHOCK WAVE MODEL , (CONT'D.) The variables i n the Rankine - Hugoniot relations are defined i n Table XII and i l l u s t r a t e d i n f i g u r e 65. The variables subscripted "0" correspond to the i n i t i a l conditions and the variables subscripted "1" correspond to the shock heated conditions. From the conservation of mass across the shock d i s c o n t i n u i t y , we obtain, A V S =/>t (V - V ) F S m.i » VS S~ V A i.e. V Po F From the conservation of momentum across the shock f r o n t , we obtain: Po + /OQVS - P> + pt 2 -v ) 2 F m.a- In w r i t i n g t h i s equation we have assumed that we can neglect the d i s c o n t i n u i t y of the magnetic pressure at the shock front - a term which has been introduced into the Rankine - Hugoniot relations by de Hoffman (l<?50). From our t h e o r e t i c a l model t h i s i s j u s t i f i e d as we have assumed no magnetic f i e l d i n the shock heated region. A l s o , the measured magnetic f i e l d (see section 8.2) shows no discontinuity,although there i s a magnetic f i e l d within the main current sheet. We further simplify the above relationship by neglecting the pressure ahead of the shock wave compared with the pressure behind the shock wave. - 158 - 1U.0 DYNAMIC EQUATION FROM SHOCK WAVE MODEL (CONT'D.) Then, upon substitution of p V with the assumptions: from equation ll+.l, we obtains x - F U..3 Pi B,<\>B P »Po lk»h 0 t From the conservation of energy across the shock d i s c o n t i n u i t y , we obtain: h where 1 V 2 + 0 s 2 - h , + 1 (V - V ) 2 h - £ (Ha+3S) R T <X-<li 0 2 q + + v 2 lli.J / £ (<li qii) + Assuming t h a t h « h ] _ and upon substitution of equation llj.,3, we 0 have, T g / E i _f 2P V 0 S 2 - P , \ -<X.q± -yS(qi+qii)\ U*.6 Assuming that we have an i d e a l gas, we can write the equation of state ass. 2J_ - (l-HX+2£) R T G and upon substitution of T i " 1U.7 f equation l l i . l and equation lI|o3« Jpi ? ' (/CbVS W S ^ - P-Pi) , ] (1-KX+^)R 2 C - 159 - U|o8 1U.0 DYNAMIC EQUATION FROM SHOCK WAVE MODEL (CONT'D.) F i n a l l y we obtain from Sana's equation, two equations f o r the i o n i z a t i o n r a t i o s (X a n d ^ Sana's equation a c t u a l l y gives the r a t i o of the number densities o f the various constituents the plasma as a f u n c t i o n of temperature. of They are cf the form; n-iHe - f i (Ti) lk.9 HiiH - fli(T,) where where 3/2 - Qi 3/2 - Q ii / VJX1 1)4*10 - 1 \ g i s the s t a t i s t i c a l weight defined as, -5- g = (2J +1) e ± - £i Ep, 1U.13 The values of Jj_ and £j_ - the angular momentum quantum number and the corresponding energy r e l a t i v e to the ground state i v e l y - can be found i n spectroscopic tables (see Moore respect(I9h9)% The s t a t i s t i c a l weights are presented i n Table XIII f o r a temperature T f of 1.5 ev. or approximately 1.8 x 1 0 ^ ° k . - 160 lil.O DYNAMIC EQUATION FROM SHOCK WAVE MODEL (CONT'D.) 2 1 6.0 So gi Sii IO.I4 TABLE XIII VALUES OF STATISTICAL WEIGHTS The l a s t term i n the f i n a l bracket of equations l i i . l l and lh.12 i s a correction term which arises from the Stark effect i n the plasma, i . e . there i s a d i s t o r t i o n of the atomic energylevels and a resulting change i n the i o n i z a t i o n potential due to the f r e e e l e c t r i c charges i n the plasma. In the calculations we w i l l assume the correction term i s small, but we must cheque t h i s assumption with the r e s u l t i n g value o f n . e A basic assumption implied by the use of Saha's equation i s that of thermal equilibrium i n the region behind the shock f r o n t . We do not have s u f f i c i e n t information to check t h i s assumption and thus the accuracy of t h i s solution can not be determined without a d d i t i o n a l information about i o n i z a t i o n relaxation times i n argon. In terms of OC" andy&, the Saha equations (Hj.9 and i U . l O ) may be written, oc n 2 TT^o<) » f±(T,) - 161 - H1.U4 lll.O DYNAMIC EQUATION FROM SHOCK WAVE MODEL (CONT'D.) Since, - IH P. Iit.l6 (l+0<+2 S) kTi / we have, C< (l^F « kT, Pi fi*(Ti) tMl+oW-2^) « kT, ^(T,) 2 (1-KX) lit.17 . _ AQX+2&) and, f l Pi Thus we have determined s i x equations - l i t . l , lit«3, Dt.6, lit.8, lit,17, and lii»l8 - f o r the s i x unknown variables -fa, Vp, p , , 0 < , ^ 3 , Tt. lit.l GRAPHICAL SOLUTION OF SHOCK WAVE EQUATIONS In the previous section we have arrived at a system cf equations f o r six unknowns. six In general, a s o l u t i o n of these equations i s d i f f i c u l t , but i n the present case a technique i s possible which s i m p l i f i e s the s o l u t i o n . an i n i t i a l estimate of p t The technique depends upon suggested from the experimental fact that the illuminated layer of the plasma does not change appreciably i n thickness as the layer collapses, i . e . Vg ^ Vp. Equation 1U„3 becomes, Pi £ A s v 2 l h ' 1 9 With t h i s value, graphical techniques are used to solve equations lit.6, lit.17, and lit.18, f o r C X , ^ , and T. A more accurate value of pi i s then obtained from equation lit.8 and f i n a l l y Vp and - 162 - Ui.l GRAPHICAL SOLUTION OF SHOCK WAVE EQUATIONS (CONT'D.) are determined from equations 1U.3 and II4..I, respectively. In t h i s section we consider three examples f o r the measured values of i n i t i a l pressure and shock v e l o c i t y . is The f i r s t example calculated i n d e t a i l and the r e s u l t s of a l l three examples are given i n tabular form. At the conclusion of this section we examine the approximations considered i n section II4..O. Example 1. 100 /(.Hg Argon: The discharge v e s s e l has an i n i t i a l pressure of lOO/CHg argon and the shock v e l o c i t y at 2.5 /^»sec after i n i t i a t i o n of the discharge i s 1.27 x 10^ m/sec. = 2.13 x 10-k kg/m3 p0 V s = 1.27 x 10"^ of p, We obtain an estimate from equation l U . l , Pt ^ 3 . U U x 1& We l e t p, ~ 3.2 x lcA n t / m 2 m/sec 0 t/m2 n as our f i r s t estimate. We assume that there are no neutral atoms i n the plasma. That i s , 1U.20 - 163 - 11;. 1 GRAPHICAL SOLUTION OF SHOCK WAVE EQUATIONS (CONT'D.) Substituting equation 11;.20 i n equation 11;. 18 we e l i m i n a t e ^ , Equations 1U.18 and figure 6 7 . ) . llj.6 are solved graphically f o r CH and T , Equation 11;.18 i s , i t s e l f , (see a complicated equation which can be solved graphically i n the following manner, Equation l l ; . l 8 gives» or, e - fl l-<x{ + J \ k / g -,g ymek \ 3/2 p, ^ ^27frf ) n g o T 5/2 i Upon substitution of numerical values, we have, 31.8 x 10** ^ 1 = J * 1 ( 1 1-CXV + 1 \ * ^ 2±^ 0 1 2 ^' 21 p, X2^J The R . H . S . and the L . H . S . of equation 11;.21 are plotted as functions of T, f o r given values of CXand p figure 6 6 . t i n the graph of The i n t e r s e c t i o n of these curves gives (Xvs. T, f o r a given value of p i . These points are transferred to f i g u r e 6 7 . , and the intersection of this curve with the curve of equation 11;.6 determines the solution for (X and T t . nt/m 2 With the value pi = 3»2 x and the use of equation 11;.20, Uj.6 becomes, T t * 10 8 * VP*) 520 ( .8 - (l.03~.66CX)\ V ) - 16U - 1U.22 f FIGURE 66. GRAPHICAL SOLUTION OF SAHA EQUATION - 165 - ^ a y /100 * i i i s Areon ) TT /.Or 250 /H3 g Argon .9 !W .8 2.05*1 TT- .8 500 jjgl .6 Argon \ 3.0*10 FIGURE 6 7 . GRAPHICAL SOLUTIONS OF RANKINEHUGONIOT EQUATIONS - 166 - lll.l GRAPHICAL SOLUTION OF SHOCK WAVE EQUATIONS (CONT'D.) The s o l u t i o n is T , = 2.25 x 10 Iii. 8 i s plotted i n f i g u r e 67., u °k and CX = .75- and the r e s u l t i s p, = 3.20 x 10^- nt/m which confirms our i n i t i a l estimate. 2 we obtain Vp = 1.18 fa/ft) ~ ^* Example 1. Given: ^ Solution: 3 S pt Pressure behind shock= 3.2 1 CX Fraction of s i n g l y ionized/atoms = .75 Fraction of doubly ionized/atoms = .25 Plasma temperature - 2.25 x 10^ °k P o l e . v e l o c i t y i n flow behind shock =• 1.18 Density r a t i o across shock = Iii Pressure behind shock = 3*20 x lCH-nb/m 250/tHg Argon (U.O^sec) Ci = I n i t i a l mass density = 5«2 ' V = Shock v e l o c i t y = 1.12 S Assume: x 10^m/sec 2 Pt Given: results are presented below. I n i t i a l mass density = 2.13 x 10"*^kg/m Shock v e l o c i t y = 1.27 x 10^ m/sec T, Vp Example 2. x 10^ m/sec and from equation l l u l we obtain n e s e /CL CX + From equation Iii.3 100/IHg Argon (2.5/t.sec) V Assume: Equation x 10-^kg/m x lO^m/sec 3 pi = Pressure behind shock - 6.0 x 10^ nt/m2 CX + ft • 1 Solution: CX = = Tt = = = Pt a Fraction of single ionized atoms = .96 Fraction of doubly ionized atoms - .oil Plasma temperature = 2.05 x 10^°k P o l e . v e l o c i t y i n flow behind shock = l.Oii x 10^m/sec Density r a t i o across shock = Hi Pressure behind shock = 6.02 x 1 c W m 2 - 167 - lli.l GRAPHICAL SOLUTION OF SHOCK WAVE EQUATIONS (CONT'D.) Example 3. Given: 500/XRg Argon (5.0/Asec) PQ *» I n i t i a l mass density = 10.65 x l O ^ k g / m V = Shock v e l o c i t y = .91 x 10%i/sec 3 s Assume: p, = Pressure behind shock = 8.2 x 10^nt/m A Solution: CX l-OC T, Vp A/o 'pt 0 LU.2 2 = o = = = = = = Fraction of s i n g l y ionized atoms = .72 Fraction of neutral atoms = ,.28 Plasma temperature = 1.57 x 10^°k P c l e . v e l o c i t y i n flow behind shock = .81^5 x lCwn/sec Density r a t i o across shock = l U . Pressure behind shock = 8.18 x lO^nt/m 2 COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS An examination of the predicted flow v e l o c i t y of p a r t i c l e s behind the shock front shows very good agreement with the measured values obtained i n section 10.1, The flow v e l o c i t i e s are presented i n Table XIV and the agreement indicates that the shock model i s a reasonable approximation of our plasma. PRESSURE (/lEg) THEORY (m/sec) EXPERIMENT (m/sec) loo 1.18 x ioh 1.15 x ich 250 1.0U x l£h 1.0U x lCr* 500 .85 x icA TABLE XIV FLOW VELOCITY BEHIND SHOCK - 168 - .85 x id* 1U.2 COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS,(CONT'D.) From the density r a t i o which has been calculated from the shock wave model, together with the assumption that a l l the i n i t i a l gas i s swept up by the collapsing plasma (see section 13.2), we c a n ' calculate the thickness of the shock heated layer of plasma. This value may then be compared with the observed plasma thickness. PRESSURE TIME C URg) THEORY (mm) (jXsec) (mm) EXPERIMENT (mm) 100 100 2 3 U8 32 2.5 5.0 Iii 19 250 250 3 U U6 h9 2.U 2.7 10 Hi 500 500 500 500 3 k 5 $h 1.8 2.7 13 I46 37 21 6 U.O 8.0 TABLE XV THICKNESS OF SHOCK HEATED REGION r i s the radius of the shock front Theoretical thickness i f obtained from density r a t i o , g THEORY - ^ x (r Q 2 2 - r r 3 2 ) s Experimental thickness i s obtained from figures 37. - U2., as thickness of plateau region. -169- Iii 12 18 Hi.2 COMPARISON OF THEORETICAL AND EXPERIMENTAL RESULTS.(CONT'D.) The large discrepancy shows that the plateau region (also the luminous region) i s not shock heated. The shock heated region i s very small and we f i n d that we have a t h i n sheet of high mass density as was assumed i n the snowplow model. However, we must explain the presence of the thick luminous layer which we have also shown to contain the majority of the discharge current. In the following section we propose a new model f o r the collapse stage of the z-pinch discharge. ASSUMPTIONS In t h i s section we have made two assumptions which we now wish to j u s t i f y . F i r s t , we assumed that the v a r i a t i o n of the magnetic f i e l d i s n e g l i g i b l e at the shock f r o n t (equation l i i . 3 ) . The t h i c k - ness of the shock heated plasma i s approximately k mm. as may be seen from Table XV. Over t h i s distance the magnetic f i e l d varies by approximately . 0 £ wb/m and t h i s corresponds to a pressure of 2 about l O n t / m . 3 2 This pressure i s n e g l i g i b l e compared to the p a r t i c l e pressure of 5 x 10^ n t / m . 2 The second assumption involves the neglect of the stark effect i n Sana's equations i . e . we assume, I •3/2 Using the calculated values of the n e and T i , this term i s approx- imately .08 f o r a l l examples considered here. That is,we have an error of approximately 8% i n our t h e o r e t i c a l values. This i s the range of accuracy of our experimental observations. - 170 a]so 15.0 PROPOSED MODEL OF THE Z-PINCH DISCHARGE In the previous section we have considered the simple shock theory with the i n c l u s i o n of i o n i z a t i o n e f f e c t s . Experimentally we have observed the presence of the shock f r o n t , (see figure 55.)» The simple shock theory does predict the flow v e l o c i t y successfully (see Table XIV). In t h i s s e c t i o n we propose an extension of the shock wave model. The key to this model i s an observation by F o l k i e r s k i (1963). He reports the presence of an unexplained discontinuity between the shock f r o n t and the outer- edge of the plasma column. We now suggest that the f i r s t region (between the shock front and the d i s c o n t i n u i t y ) i s a region of shock heated gas. This region has a high density (approximately llpc the i n i t i a l density) and a r e l a t i v e l y low temperature (approximately 2 e v . ) . We suggest that the second region (the remainder of the luminous plasma layer) i s a region of low density (approximately equal to the i n i t i a l density) and high temperature (approximately 6 e v . ) . These suggestions are i n agree- ment with the luminosities observed by F o l k i e r s k i ; the f i r s t region i s weakly luminous while the second region i s strongly luminous (see f i g u r e 68.). Later i n t h i s s e c t i o n we w i l l pursue the implications of this model and we w i l l show that i t explains the observed effects i n a consistent manner. However, we would f i r s t l i k e to give a plausible explanation f o r the o r i g i n of t h i s double layered plasma column. We suggest that i n the early stage of the discharge, the discharge current occurs i n a region near the walls of the - 171 - discharge REGION OF WEAK LUMINOSITY -DISCONTINUITY •REGION OF STRONG LUMINOSITY DISCONTINUITY I N PLASMA COLUMN FIGURE 68. 15.0 PROPOSED MODEL OF THE Z-PINCH DISCHARGE vessel. (CONT'D.) As t h e c o n d u c t i v i t y o f t h e plasma i s low a t t h i s time, there i s c o n s i d e r a b l e J o u l e h e a t i n g o f t h e gas. T h i s hot r e g i o n at t h e w a l l expands, forming a s h o c k heated r e g i o n i n f r o n t o f i t , and t h e n b o t h t h e J o u l e heated and shock heated r e g i o n s a r e f o r c e d t o c o n s t r i c t toward t h e a x i s o f t h e d i s c h a r g e v e s s e l under t h e i n f l u e n c e o f t h e magnetic p r e s s u r e . Thus, i n terms o f t h e snowplow model, we have i n s e r t e d a b u f f e r r e g i o n o f low d e n s i t y , h i g h temperature plasma between t h e mass l a y e r and t h e magnetic pressure. I f t h e t h i c k n e s s o f t h e b u f f e r r e g i o n i s s m a l l compared t o t h e r a d i u s o f t h e d i s c h a r g e v e s s e l , then t h e r e w i l l be n e g l i g i b l e upon t h e p r e d i c t e d snowplow r e s u l t s . effect I f the thickness of t h e b u f f e r region i s appreciable, then f o r a given p o s i t i o n of the c u r r e n t d e n s i t y maximum, t h e c o l l a p s i n g plasma i s more h e a v i l y mass loaded than the t h e o r y p r e d i c t s . Hence i t c o l l a p s e s more s l o w l y than p r e d i c t e d . - 172 - 15.0 PROPOSED MODEL OP THE Z-PINCH DISCHARGE (CONT'D.) In the remainder of t h i s section we wish to examine the experimental results in terms of t h i s p a r t i c u l a r model. We assume that the shock theory of section lit. i s applicable to the shock heated region of our model, and the results computed there are assumed correct for the f i r s t region of our double layered plasma. We wish to examine the Joule heated region i n t h i s s e c t i o n , and we begin with a discussion of the measured f i e l d i n the Joule r e g i o n . electric This e l e c t r i c f i e l d has been calculated from our magnetic probe r e s u l t s (see Table V I I I ) , and we repeat the essential r e s u l t s i n Table XVI below. features which should be observed. There are several unusual The apparent conductivity ( J / E £ ) increases with increasing pressure and, furthermore, f o r z low pressures the apparent conductivity decreases with increasing time. Both of these effects PRESSURE ( /JB$) TIME (jUsec) are p h y s i c a l l y improbable. Ik (amp/m.2) xlOY C7 R (mTVoT™. 100 100 2 3 1750 3100 i.U 1.8 0.8 0.6 250 250 3 h 1000 1U50 1.5 2.0 1.5 1.U 500 500 500 500 3 ii 5 6 U20 1.7 1.8 2.0 1.9 Il.O IiOO 320 20 U.5 6.3 10.5 • TABLE XVI APPARENT CONDUCTIVITY IN JOULE REGION E^ = E l e c t r i c f i e l d i n plasma (Table VIII) J = Current density i n plateau region (figure 37.-U2.) CTfl = J z / E | i s the apparent conductivity. z - 173 - 15.0 PROPOSED MODEL OF THE Z-PINCH DISCHARGE (CONT'D.) In an attempt to understand these unusual r e s u l t s l e t us consider the processes which occur i n the Joule heated region of the plasma. As we have seen that the plasma i s moving with a constant r a d i a l v e l o c i t y at the times of i n t e r e s t (refer to figures £8. and 59.), we choose to view the plasma from the frame of reference moving with t h i s r a d i a l v e l o c i t y . We have seen i n Table XVI that the a x i a l e l e c t r i c f i e l d E^ i s not zero i n this frame of reference, and with crossed E and B f i e l d s we can expect a H a l l e f f e c t . The E x B force acts upon both electrons and ions and tends to move them both r a d i a l l y inward. Because o f these d r i f t s a pressure gradient i s established which exerts an outward force on both electrons and i o n s . However, the mobility of the electrons i s much greater than the mobility of the ions and thus we obtain a net force on the electrons i n an inward d i r e c t i o n and a net force on the ions i n an outward direction. The plasma i s at rest i n t h i s p a r t i c u l a r frame of reference and these opposing forces must be balanced by a space charge e l e c t r i c f i e l d which keeps the ions and electrons from separating. Thus we have a r a d i a l e l e c t r i c field. Now i n exactly the same way, this r a d i a l e l e c t r i c f i e l d , crossed with the magnetic f i e l d , results i n a d r i f t of p a r t i c l e s i n the a x i a l d i r e c t i o n . The difference of p a r t i c l e m o b i l i t i e s r e s u l t s i n the creation of an a x i a l space charge e l e c t r i c field. A diagram i l l u s t r a t i n g the d i r e c t i o n s of these f i e l d s and forces i s given i n figure 69. - 17U - CATHODE FIGURE 69. B J E E E Vp e z z z r ELECTROMAGNETIC FIELDS IN THE JOULE REGION Magnetic F i e l d Current Density Induced A x i a l E l e c t r i c F i e l d Space Charge A x i a l E l e c t r i c F i e l d Space Char.ge Radial E l e c t r i c F i e l d Pressure Gradient - 17U - 15.0 PROPOSED MODEL OF THE Z-PINCH DISCHARGE (CONT'D.) I f we r e t u r n t o s e c t i o n 10.2 where we o r i g i n a l l y c a l c u l a t e d E^, we see t h a t we made t h e a s s u m p t i o n t h a t t h e space charge e l e c t r i c f i e l d s were n e g l i g i b l e . I f t h i s assumption i s i n c o r r e c t , t h e n we must r e c a l c u l a t e E^. and t h i s may account f o r t h e d i s c r e p a n c i e s observed i n t h e apparent c o n d u c t i v i t y w h i c h have been discussed. We can o b t a i n an e s t i m a t e o f t h e r a d i a l e l e c t r i c field from our e x p e r i m e n t a l measurements, f o r we know t h a t t h e plasma i s moving w i t h c o n s t a n t v e l o c i t y (see f i g u r e s 58« and 59.) We c o n s i d e r a L o r e n t z a p p r o x i m a t i o n o f t h e plasma ( S p i t z e r (1956) pg.83) i . e . , t h e i o n s a r e a t r e s t i n t h i s frame o f r e f e r ence and t h e e l e c t r o n s do not i n t e r a c t w i t h one a n o t h e r . The e l e c t r o n s , however, have no n e t r a d i a l m o t i o n i n t h i s frame and hence t h e n e t f o r c e on t h e e l e c t r o n s must be z e r o . Jz © - V r P - n e E B e r = 0 That i s , 15.1 I f we u s e t h e f a c t t h a t , J = _1_ V x B 15.2 A and i f we i n t e g r a t e t h e above e x p r e s s i o n t h e plasma, we have through the t h i c k n e s s o f 15.0 . PROPOSED MODEL OF THE Z-PINCH DISCHARGE (CONT'D.) PRESSURE TIME r i n JS (Msec) 100 3 250 k .ouo .ouo 500 5 .037 HpSURE (y/Hg) pjr-J W m rout. TmT Bfcln) (wb/nr) B(r ut, (wb/rn^) .051 .30 .U9 .052 .30 .50 .052 .23 .50 B (r )/2«L (nt/m^j 2 1 n 2 ) xlO^ 0 B 2 0 p* G£/m?J ^ ) / ^ (rt/ng) xlO^ . xipU xlOU 9.5 3 100 3 3.5 250 6 3.5 10 .5 500 8 2 10 .0 TABLE XVII ELECTRIC FIELD PRESSURE where r , ^ and r . a r e t h e r a d i a l p o s i t i o n o f t h e i n n e r and o u t e r m out edge o f t h e luminous plasma as determined f r o m f i g u r e 5U. That i s , t h e d i f f e r e n c e between t h e sum o f k i n e t i c and magnetic p r e s s u r e s e v a l u a t e d a t t h e o u t e r and i n n e r plasma b o u n d a r i e s g i v e s a measure of t h e " e l e c t r i c f i e l d < PE > •* < H e (r e pressure". t - r i n t )_B ( r ) - p ( r JUo 2 Uo ) E r > ••= B 2 ( r 2 2 o u i n i n ) I5.U S i n c e we assume t h a t t h e plasma sweeps up a l l o f the i n i t i a l gas t h e r e c a n be no p a r t i c l e s o u t s i d e t h e plasma column,, and t h u s t h e k i n e t i c p r e s s u r e a t t h e o u t e r plasma boundary must be z e r o . - 176 - 15.0 PROPOSED MODEL OF THE Z-PINCH DISCHARGE (CONT'D.) The v a l u e o f t h e k i n e t i c p r e s s u r e a t t h e i n n e r boundary i s d e t e r m i n e d f r o m t h e shock wave model o f s e c t i o n 1U.2. of B ( r i ) and B ( r n o u The v a l u e s ^ ) may be determined from f i g u r e s h3* - U5. The c a l c u l a t i o n i s p r e s e n t e d i n T a b l e X V I I and i t may be seen t h a t the e f f e c t o f t h e r a d i a l e l e c t r i c f i e l d i s n e g l i g i b l e f o r t h e case of 500/xHg, b u t t h e e f f e c t i n c r e a s e s w i t h d e c r e a s i n g p r e s s u r e u n t i l the radial field has a n e f f e c t upon t h e plasma w h i c h i s comparable t o t h e k i n e t i c p r e s s u r e f o r t h e c a s e o f lOO^iUHg. Certainly i n this l a t t e r case we a r e n o t j u s t i f i e d i n n e g l e c t i n g space charge fields electric effect. S i n c e we c a n n e g l e c t space charge e f f e c t s i n t h e i n v e s t i g a t i o n of plasma dynamics f o r t h e plasma c r e a t e d i n 5 0 0 / L H g i n i t i a l p r e s s u r e o f a r g o n , we choose t h i s case f o r our s t u d y o f t h e J o u l e heated r e g i o n . I n p a r t i c u l a r , we w i s h t o show t h a t t h e J o u l e heated r e g i o n i s one of h i g h t e m p e r a t u r e and l o w d e n s i t y compared t o the High shock heated r e g i o n . Temperature: I n the $00jlXBg c a s e , t h e a p p a r e n t c o n d u c t i v i t y as determined from Table XVI i s : Op = J ^ = 5 x 10^ mho/m E z and t h i s c o r r e s p o n d s t o t h e c o n d u c t i v i t y c o n s i d e r e d b y S p i t z e r ((1956) pg. 8U): ° 177 - 15.5 .0 PROPOSED MODEL OF THE Z-PINCH DISCHARGE (CONT'D.) C - T / mho/m 2^0" 3 2 15.6 Therefore, we obtain an estimate of the temperature i n the Joule heated region of approximately 5.3 x ick°k. The experimentally observed conductivity i s constant over the whole Joule r e g i o n , (Refer to c a l c u l a t i o n of E | as a f u n c t i o n of radius (Table V I I I ) ) , and graphs of J z as a function of radius (figures 37. - 1|2.). This constant conductivity implies then^that there i s a constant temperature i n the Joule region. The temperature i s high compared with the shock heated gas which has a temperature of 1.6 x 10^°k (see Example 3., section lii.2). Low Density; If we can neglect space charge e f f e c t s , we can also determine the r a d i a l dependence of the k i n e t i c pressure (p), and hence the r a d i a l d i s t r i b u t i o n of the p a r t i c l e density (Hj) i n the Joule heated r e g i o n . Hi(r) where k T Oi, ^tS - p(r) (1+^+2/2,+'") kT i s Boltzmann*s constant i s the p a r t i c l e temperature are the degrees of single and double ionization respectively. - 178 - 15.7 .0 PROPOSED MODEL OF THE Z-PINCH DISCHARGE (CONT'D.) Low Density: (Cont'd.) As the r a d i a l e l e c t r i c f i e l d can be neglected, we have, = 0 or, B (r) 2 + p(r) = constant l£.8 ¥ e can determine the constant by evaluating the l e f t hand side of the equation at the discontinuity between the Joule and shock heated regions. The k i n e t i c pressure p(r^) i s equal to the shock measured pressure of section lit.2. This l a s t statement follows from the f a c t that the plasma i s unaccelerated must be constant across the d i s c o n t i n u i t y . and hence the pressure Since we w i l l show that the shock heated region i s very t h i n , we make a n e g l i g i b l e error i n the magnetic f i e l d i f we evaluate the f i e l d at the r a d i a l p o s i t i o n of the "knee" of the current density curves (see figures 37. - 1*2.). We c a l l this value B ( r g ) . Thus at 500 ^ H g : B2_(rD) + p ( r o ) - constant « (.23) _ + 8.2 x lcA » 10,3 x l O ^ t / m ZJUO 8TTxlO-7 2 2 In figure 70., we have sketched the r a d i a l dependence of the various parameters which we are considering i n t h i s s e c t i o n . We know that the magnetic f i e l d increases l i n e a r l y with radius through the Joule region from f i g u r e s 1*3* - U5« - 179 - *d SHOCK HEATED REGION JOULE HEATED REGION PLASMA. PARAMETERS IN JOULE HEATED REGION FIGURE 70. 15.0 PROPOSED MODEL OF THE Z-PINCH DISCHARGE (CONT'D.) Low Density; (Cont'd.) For our special case of 500/lHg at 5/J.sec after i n i t i a t i o n of the discharge, we have, B(r) - (18 r - ,UU) wb/m 15.10 2 where r i s the r a d i a l p o s i t i o n i n meters. Thus the k i n e t i c pressure, i s given by equations 15.8, 15.9, and 15.10, ass p(p) » 10.3 x 10U - (I8r 8TTx - 180 - .hh) 10-Y 2 15.11 15.0 PROPOSED MODEL OF THE Z-PINCH DISCHARGE (CONT'D.) Low Density: (Cont'd.) I f we use the value of temperature estimated above, and i f we assume that the plasma i s 100$ doubly ionized argon ( i . e . 0( = 0, /5 s 1) then we can evaluate the p a r t i c l e density as a function of radius from equations 15. 1 and 15.11. 9 nj(r) = _1_ 3kT 15.12 .hhf~) (10.3 x Hr* - (I8r - ~&n7W 7 A c t u a l l y we wish t o evaluate an average p a r t i c l e density f o r the Joule heated r e g i o n . (10.3 x 1 0 - (I8r U N where and 3AkT J M ) rdr 15.13 2 arrx io* 7 A i s the annular area bounded by T-Q - the r a d i a l position of the discontinuity r = the r a d i a l p o s i t i o n of the current maximum r 01*9 <^> r ,21 - .91 x 10 (10.3 x 1 0 - (I8r - .l*l*) )rdr = 2.1*6 x 10 m~3 ~f U 2 8TTX 10- J .037 The i n i t i a l p a r t i c l e density n 0 i s known and thus we can determine the density r a t i o f o r the Joule heated region ass <n,i> = 2.1*6 x I O n 0 2 2 2.0 x 1023 - 181 - '.] 22 15.0 PROPOSED MODEL OF THE Z-PINCH DISCHARGE (CONT'D.) Low Density: (Cont'd.) Hence the density r a t i o i s indeed low compared to the shock heated density r a t i o of l l i . (See Example 3 « , section lli.2). The previous discrepancy i n the p a r t i c l e density has arisen from the f a c t that the shock model calculations have predicted a density r a t i o of lU while the observations give a value of approximately U. This observed value can be determined from a measurement of the thickness of the plasma column. (If we assume that a l l of the i n i t i a l gas has been swept up by the plasma.) That i s , <Q>= ( r where and, r TQ rg 0 2 p -r 2 q 1 5 . 1 U ) i s the inner radius of the discharge v e s s e l i s the radius of the current maximum i s the radius of the shock f r o n t . For our p a r t i c u l a r case of 5 0 0 / L E g <Q>a ( . Q 7 5 2 - ( . 0 U 9 2 - . Q 3 7 2 . 0 3 7 at 5 / t s e c , we have, ) 2 - U ) Now we can resolve t h i s discrepancy i f we assume that we have a r e l a t i v e l y t h i n , high density shock heated region and a t h i c k , low density Joule heated r e g i o n . In order to obtain the observed average density of the plasma, we estimate that the shock heated region forms - 1 8 2 - .0 PROPOSED MODEL OF THE Z-PINCH DISCHARGE (CONT'D.) Low Density: (Cont'd.) approximately 20% of the t o t a l plasma volume and the Joule heated region forms the remaining Q0%, We expect a s i m i l a r Joule heated region w i l l explain the apparent density r a t i o discrepancy f o r the cases of 100>lHg and 2^0/iHg i n i t i a l pressure. The shock theory predicts a density r a t i o of lit f o r both cases (see section 11*.2) while the observed value i s 5 for 100/tHg and 1* for 2£0/(Hg. - 183 - PART IV CONCLUSIONS - 18U - 16.0 DISCUSSION OF RESULTS In Part I we have investigated the perturbation of the magnetic f i e l d by the probe which performs the measurement. perturbation can not be neglected. This probe An analogue experiment has been used to obtain a c o r r e c t i o n which compensates far t h i s perturbation. This correction agrees extremely well with the t h e o r e t i c a l l y determined correction given by Malmberg (X96I4.) f o r the case of a plane current sheet. A numerical procedure has been developed f o r the a p p l i c a t i o n of t h i s correction to the measured magnetic f i e l d i n the z-pinch d i s charge. This i d e n t i c a l procedure can be used f o r any z-pinch or inverse pinch device. A feature which greatly s i m p l i f i e s t h i s procedure i s our use of a guide tube f o r the probe which i s placed d i a m e t r i c a l l y across the discharge v e s s e l . The probe moves i n s i d e t h i s guide tube and hence the perturbation i s independent of the probe p o s i t i o n . The numerical computation i s extended to determine the unperturbed current density d i s t r i b u t i o n . We have tested the accuracy of t h i s numerical program by using one of the measured magnetic f i e l d s from the analogue experiment. The r e s u l t i n g current density d i s t r i b u t i o n should then be a § - type current d i s t r i b u t i o n at the radius corresponding to the current sheet. This t e s t shews that the numerical approximation i s quite poor. The computed current density maximum i s only one quarter of the true current density, and there i s a corresponding increase i n the width - 185 - 16.0 DISCUSSION OF RESULTS (CONT'D.) of the current d i s t r i b u t i o n . Even so, t h i s correction procedure increases the current maximum and decreases the current width" by approximately $0% compared w i t h the uncorrected calculations. This c o r r e c t i o n procedure gives a s i g n i f i c a n t improvement i n the d e t a i l of the current density d i s t r i b u t i o n . Compare our r e s u l t s w i t h , f o r example, those of Burkhardt (1958). This improvement i n measuring technique allows us to observe two features of the current density d i s t r i b u t i o n which have not been observed by previous investigators of the z-pinch discharge. We observe a d i s t i n c t maximum or peak i n the current density at the outer edge of the plasma column. region i n the current d e n s i t y . We also observe a "plateau" This i s a region of constant current density which extends over the entire thickness of the luminous plasma. We w i l l now discuss these two features. A c a r e f u l examination of possible error has been given i n section 8.3 where we have shown that the measured magnitude of the current density peak i s a lower l i m i t of the true current density peak. Indeed, t h i s measured peak may represent a very high, very t h i n spike i n the true current density d i s t r i b u t i o n . Since the mean free path f o r electrons i n the Joule heated region (high temperature, low density) i s large (several centimeters), and since the p a r t i c l e density i s p a r t i c u l a r l y low at the outer edge of the plasma (see f i g u r e 70.), t h i s high current density may represent a beam of runaway electrons. - 186 ° (See Kruskal (196LJ). 16.0 DISCUSSION OF RESULTS (CONT'D.) Note that t h i s current peak i s not due to a skin depth phenomenon as has been assumed i n the snowplow and shock wave models. We have seen that the concept of skin depth requires c a r e f u l examination for the case of a plasma which can create space charge electric f i e l d s . These f i e l d s can cancel the applied e l e c t r i c f i e l d i n such a way that the plasma "sees" a constant field. electric Thus w - * - 0 i n the skin depth formula and the s k i n depth It i s t h i s rearrangement of e l e c t r i c f i e l d s i n the plasma which allows the creation of the plateau region i n the current density distribution. The magnitude of the current density over the plateau region i s l i m i t e d by the external c i r c u i t parameters of the discharge c i r c u i t rather than by any property of the plasma i t s e l f . The current density i n this region i s approximately 2 x 107 amps/m2 for a l l i n i t i a l gas pressures which have been used i n this i n v e s t i g a t i o n . An important r e s u l t follows from the f a c t that the e l e c t r i c as seen by the plasma i s approximately constant with time. field In t h i s s i t u a t i o n i t i s easy f o r the current to d i f f u s e into the centre of the c y l i n d r i c a l current sheet. This i s a disadvantage as f a r as heating the plasma i s concerned f o r t h i s means that a large f r a c t i o n of the t o t a l current i s inside the p o s i t i o n of the shock front and thus has only a small influence upon the motion of the plasma. Experimentally, we have observed that approximately one t h i r d of the t o t a l current l i e s inside the radius of the mass sheet. - 18? - This discharge effect 16.D DISCUSSION OF RESULTS (CONT'D.) explains the discrepancy between observed and predicted c o l l a p s e curves as reported by Curzon (1963a.). Their c a l c u l a t i o n s p r e d i c t a plasma c o l l a p s e which i s too r a p i d because they have assumed that the t o t a l discharge current causes the c o n s t r i c t i o n . The same e f f e c t s have been observed i n the present i n v e s t i g a t i o n and they are discussed i n d e t a i l i n s e c t i o n 13.2. I n Part III of the t h e s i s we have examined t h e o r e t i c a l models of the z-pinch discharge. The snowplow gives a r e l i a b l e p r e d i c t i o n of the discharge current and the plasma radius as functions of time f o r the two cases of higher i n i t i a l (250 and gOO^uuHg largon). pressure At lOO/UHg argon the plasma c o n s t r i c t s more r a p i d l y than expected and t h i s may be due t o i n e f f i c i e n t trapping of the gas p a r t i c l e s at low pressures. Since p a r t i c l e trapping depends upon numerous atom-ion c o l l i s i o n s , i t i s reasonable to expect that these c o l l i s i o n s w i l l be l e s s frequent i n a low density plasma. I n the shock wave model, i f we can assume that we have thermal e q u i l i b r i u m i n the region behind the shock f r o n t (and t h i s i s questionable), then we need only determine the shock f r o n t v e l o c i t y and the i n i t i a l gas density (or pressure) t o c a l c u l a t e the pressure, temperature, density, degree of i o n i z a t i o n , and the f l o w v e l o c i t y of the plasma behind the shock f r o n t . The plane shock f r o n t theory which we have used r e s t r i c t s our p r e d i c t i o n s t o those regions of large r a d i u s where the converging shock f r o n t i s unaffected by the - 188 - 16.0 DISCUSSION OF RESULTS (CONT'D.) c y l i n d r i c a l geometry. The predicted flow T e l o c i t y i s i n excellent agreement with the measured values (see Table XIV"). At 500/iHg argon, where we obtain our most r e l i a b l e r e s u l t s , we have a predicted temperature of 1 6 x 10^°k ( ^ l . i t ev.) and a k i n e t i c B pressure of 8.2 x 10^ n t / m . 2 30% neutral argon. The plasma i s 10% s i n g l y ionized and At 100/iHg argon the plasma temperature 2.3 x 10k°k (A>2 e v . ) , is the k i n e t i c pressure i s 3.2 x ick nt/m and 2 the plasma i s 1$% s i n g l y ionized and 2$% doubly ionized argon. In a l l cases, the estimates are made when the radius of the shock front i s approximately k cm. For the three i n i t i a l pressures which have been considered, the density r a t i o across the shock front i s Hj.. Assuming that a l l the i n i t i a l gas i s swept up by the c o l l a p s i n g plasma, we can estimate the density r a t i o since we have measured the plasma thickness (see f i g u r e 5>3»)« Me obtain a density r a t i o of approximately k f o r the three gas presures. To explain this discrepancy i n the plasma density and to explain the high temperature (*>£ ev.) which can be determined from the measured plasma conductivity, we have proposed an extension to the shock wave model. consists of two l a y e r s . We suggest that the plasma The inner layer corresponds to the shock heated plasma and the outer l a y e r , which i s much larger i n volume (approximately four times), corresponds to a Joule heated region. This outer layer has a lower density r a t i o - 189 - ( « l ) and a 16.0 DISCUSSION OF RESULTS (CONT'D.) higher temperature ( £ ev,) than the shock heated region. The term "Joule heated" i s used because we believe that t h i s region i s created during the i n i t i a l stages of the discharge when the conductivity i s low and Joule heating i s , therefore, appreciable (see section 15»0). The p a r t i c l e density and temperature i n the Joule heated region could only be calculated f o r the investigations performed i n an i n i t i a l gas pressure of %00jJLti.g argon. In this case we could neglect the effects of the r a d i a l e l e c t r i c f i e l d compared with the e f f e c t of the pressure gradient. It has been shown, however, that the r a d i a l e l e c t r i c f i e l d can not be neglected i n investigations at lower i n i t i a l pressures. In these cases the dynamics of the z-pinch i s very complex and much more e x p e r i mental i n v e s t i g a t i o n i s r e q u i r e d . 17.0 FINAL CONCLUSIONS We divide our f i n a l conclusions into two classes - those which present an improvement i n diagnostic technique and those which present new physical understanding of the dynamics i n the z-pinch discharge. DIAGNOSTIC TECHNIQUES 1. We have developed a high frequency, low noise magnetic probe and measuring c i r c u i t (section 3)« - 190 - 17.0 FINAL CONCLUSIONS (CONT'D.) DIAGNOSTIC TECHNIQUES (CONT'D.) 2. We have developed a numerical program which corrects the measured magnetic f i e l d for the error due to the nonconducting hole i n the plasma caused by the magnetic probe (section 3. $), We have obtained an analytic solution to the snowplow equation which f a c i l i t a t e s ment (section comparison of theory with e x p e r i - 13.1). NEW PHYSICAL CONCEPTS 1. The region of high luminosity observed i n photographs of the pinch coincides w i t h the region of high current density within our experimental error of + .25 cm. (section 2. 10.l). There i s a sharp maximum i n the current density d i s t r i b u t i o n at the outer edge of the plasma s h e l l (section 8.2). This current peak i s not due to skin depth e f f e c t s , a current of runaway electrons 3. (section but i s probably 16). There i s a region of constant current density and constant axial electric f i e l d (as functions of r a d i u s ) i n the collapsing plasma s h e l l . This i s approximately 1 cm. t h i c k f o r i n i t i a l gas pressures ranging from 100 to 5>00^lHg argon (section 8.2). - 191 - 17.0 FINAL CONCLUSIONS (CONT'D.) NEW PHYSICAL CONCEPTS (CONT'D.) U» A large portion of the discharge current flows through the region inside the plasma s h e l l and thus has l i t t l e upon the plasma dynamics. effect For i n i t i a l gas pressures of 1 0 0 - 500jXKg argon, t h i s f r a c t i o n i s approximately one-third of the t o t a l current 5. (section 1 3 . 2 ) . At the higher i n i t i a l gas pressures ( 2 5 0 and 500/jHg argon) i t i s reasonable to assume that the plasma s h e l l sweeps up a l l of the i n i t i a l gas as i t collapses. This i s not a good assumption for lower i n i t i a l gas pressures (section 6. (100/AMg argon) 1 3 . 2 ) . The c o l l a p s i n g plasma s h e l l a c t u a l l y consists of two separate regions. The inner region occupies approximately 2 0 $ of the t o t a l plasma volume and i t has a r e l a t i v e l y high p a r t i c l e density and low temperature. The remaining plasma volume forms the outer region which has a high temperature and a low p a r t i c l e density (section 7. 15). For the p a r t i c u l a r case of an i n i t i a l gas pressure of 5 0 0 p E g argon, the temperature i n the inner region has reached l.o x reached and the temperature i n the outer region has 5 » 3 x 1 0 ^ ° k at 5 A s e c a f t e r i n i t i a t i o n of the discharge. At the same time the p a r t i c l e density r a t i o s are l U a n d . l , r e s p e c t i v e l y , as compared with the i n i t i a l density (section - 192 - 15). 18.0 PROPOSALS FOR FUTURE WORK Perhaps the most remarkable result of the present i n v e s t i g a t i o n i s the awareness of the complex nature of the z - p i n c h . However, the z*»pinoh i s s t i l l simple compared t o other experimental plasmas because of i t s w e l l defined geometry and i t s high degree of reproducibility. These features make i t an i d e a l plasma source to further consider the e f f e c t s which we have observed i n t h i s i n v e s t i gation. For example, there has been no experimental i n v e s t i g a t i o n of the i n i t i a l stages of the z - p i n c h . In the very early stages of the pinch we have observed t y p i c a l "Mach l i n e s " which r e s u l t from, most probably, a beam of electrons (see figures 51. and 52.)• It has been shown that a r e v e r s a l of p o l a r i t y of the discharge electrodes s t i l l r e s u l t s i n a flow from cathode to anode. From the photographs i t appears that this flow of p a r t i c l e s occurs at a d i s t i n c t r a d i a l p o s i t i o n - not uniformly. The effect might be caused by a r a d i a l d i s t r i b u t i o n of electrons which f a l l s o f f with increasing radius while the e l e c t r i c f i e l d r i s e s w i t h increasing' radius because of the s k i n e f f e c t . The electron flow would occur at the p o s i t i o n where the product of electron density and e l e c t r i c f i e l d were a maximum. It would be i n t e r e s t i n g t o observe the r a d i a l p o s i t i o n of the electron beam as a function of i n i t i a l gas pressure and charging voltage. There has been no experimental i n v e s t i g a t i o n of the formation of the discharge current at the vessel w a l l s . - 193 - initial This 18.0 PROPOSALS FOR FUTURE WORK (CONT'D.) could e a s i l y be done with a more sensitive magnetic probe than we have used here, and there are several i n t e r e s t i n g to i n v e s t i g a t e . features The phenomena of the delayed collapse has been observed by Curzon ( 1 9 6 3 a . ) . The plasma s t a r t s to collapse toward the a x i s , i t reforms at the vessel w a l l , and f i n a l l y i t does c o l l a p s e . We have also observed an "accelerated collapse". For an i n i t i a l pressure of 1 0 0 / t H g argon, the plasma appears to be at the v e s s e l w a l l i n the f i r s t photographic frame and then i n the next frame ( ^ 2 5 / ( . s e c ) i t has moved approximately 3 ram* away t from the w a l l . It remains here f o r approximately 0.5>M.sec and then continues to c o l l a p s e . We have suggested i n the present i n v e s t i g a t i o n that the Joule heated region of the plasma i s created at t h i s time, and i t would be i n t e r e s t i n g to study the above phenomena i n terms of the various magnetic, e l e c t r i c , and k i n e t i c pressure f o r c e s . For example, the J x B force i s strongly dependent upon the experimental configuration (in p a r t i c u l a r , the inductance), but the plasma dynamics r e s u l t i n g from the i n i t i a l acceleration of the plasma mass by the f o r c e i s strongly dependent upon the mean free path of the p a r t i c l e s i n the plasma, i . e . the plasma dynamics are strongly dependent upon the i n i t i a l pressure. If the pressure i s high, the J x B force and the i n e r t i a l force w i l l be balanced mainly by a pressure gradient. If the pressure i s low the two forces w i l l be balanced mainly by an electric f i e l d . The importance of these mechanisms can vary as the mean free path varies - e s p e c i a l l y i n the i n i t i a l stages. - 19h - 18.0 PROPOSALS FOR FUTURE WORK (CONT'D.) There are many i n t e r e s t i n g problems which remain to be solved before the collapse stage of the z-pinch i s understood. 1^.0 we have demonstrated the presence of space charge In section electric f i e l d s i n the plasma but we have been unable to include these f i e l d s i n a dynamic model of the z - p i n c h . Indeed, the plasma i s even more complex, because the a x i a l d r i f t v e l o c i t y w i l l r e s u l t i n an a x i a l pressure gradient. Thus the mass density w i l l increase i n the plasma column as we look from cathode to anode. Thus the plasma column w i l l collapse more r a p i d l y at the cathode than at the anode. This "taper e f f e c t " has been observed i n framing camera photographs of a nitrogen plasma by Curzon (1962). This means that there may be a r a d i a l component of the current density and thus the complete generalized Ohm's law should be considered. This i s given by Rose (1961), page 126, as: m* , £i = E rrig H e E V J B <\7p C e and, + (V-J_)xB ne IV where A e is is is is is is is is is the the the the the the the the the + V£-J %e cr electron mass electron density electron charge e l e c t r i c f i e l d applied to the average plasma v e l o c i t y current density magnetic f i e l d pressure gradient conductivity. - 195 - electrodes APPENDIX I CONSTRUCTION OF MAGNETIC PROBE In t h i s appendix we give a d e t a i l e d d e s c r i p t i o n of the c o n s t r u c t i o n of a magnetic probe. These techniques are t h e r e s u l t of s e v e r a l d i f f e r e n t attempts and the f i n a l procedure i s f a s t and simple. 1. Using #I(.0 B. & S. copperwire, (.003 i n diameter), wind a 20 t u r n s i n g l e layer solenoid on a 1mm. diameter wire form. (The c o i l can be set w i t h n a i l p o l i s h ) . 2. Bend the wire form such that the probe c o i l i s held i n the chuck of a d r i l l press, but perpendicular t o the axis of r o t a t i o n . Using the d r i l l press t w i s t the wire leads t o the probe c o i l u n t i l the i n d i v i d u a l t w i s t s are approximately 1mm. i n length. (The t w i s t can be made very uniform by gluing t h e ends of t h e probe leads t o the ends of a c a r d board s t r i p approximately 10 cm. i n length.) 3. Obtain a piece o f glass tubing of /v/lmm.I.D. and ^2 mm. O.D., and <v 20 cm. i n l e n g t h . S i l v e r t h i s tubing using the Rochel3e S a l t Process as described, f o r example, by Strong i n "Procedures of Experimental Physics". (Several tubes can be s i l v e r e d at once.) h* Check that t h e s i l v e r layer i s much t h i n n e r than the s k i n depth. For s i l v e r : p - 1.6 x 10"^ ohm - m or, CJ = 6 x 10? mho/m. The s k i n depth i s approximately , 0 k mm. at 30 Mc/s. The thickness of the s i l v e r l a y e r may be estimated by measuring the r e s i s t a n c e of the l a y e r . - 196 « APPENDIX I (CONT'D.) I CONSTRUCTION OF MAGNETIC PROBE (CONT'D.) R = fi x 3- gfrh area i en where, area = TTx O.D. x thickness. (Our experimentally measured resistance i s approximately 2$fL 9 and t h i s corresponds to a thickness of approximately .0002 mm.) 5. Thread the probe leads through the s i l v e r e d tube. Cut the perspex end piece and b o l t i t to the "twinax" connector as shown i n f i g u r e 2. Then solder the leads from the c o i l to the connector - including a connection from the s i l v e r e d tube t o the base of the connector. 6. TEST ALL CONNECTIONS. Using tape to form a mold, pour molding epoxy r e s i n i n t o the region between the perspex and the connector. epoxy over the s i l v e r e d glass t u b e . when heated f o r several minutes. - 197 - Form a coating of (The epoxy sets more r a p i d l y APPENDIX I I NUMERICAL PROGRAMS IN FORTRAN LANGUAGE In t h i s appendix we give the numerical programs f o r the two equations of seetion 5.0. In the "J-Method" solution we solve directly f o r the current density d i s t r i b u t i o n and we then obtain the magnetic f i e l d from the current density. In the "B-Method" solution we solve d i r e c t l y for the magnetic f i e l d and i n d i r e c t l y f o r the current density. The "J-Method" i s perferable as the solution requires a numerical integration rather than a numerical d i f f e r e n t i a t i o n as i n the "B-Method". Both programs require the inversion of a large (80 x 80) matrix. Two " t r i c k s " are r e q u i r e d . to be inverted by two. F i r s t , we multiply a l l elements of the matrix This i s necessary as a l l of the i n i t i a l elements are less than or equal to one and hence the i n i t i a l determinant i s too small to be handled by the computer. Of course, we must then multiply a l l the elements of the inverted matrix by two to obtain the correct inversion. Second, we perform a l l inversion operations i n double precision format to minimize round-off error. given below. J - METHOD SOLUTION $ C 10 15 DOUBLE PRECISION D, DET, R, S DIMENSION D(80,80), C(80,8o) C-MATRIX INCLUDES - l/2 CALL ZERO (C,80,80) DO 10 J « 2,8 K = J + 7 READ (5,15) ( C ( I , J ) , I = 1,K) CONTINUE FORMAT (12F6.3) DO 20 I - 9,80 - 198 - The f o r t r a n programs are APPENDIX II (CONT'D.) NUMERICAL PROGRAMS IN FORTRAN LANGUAGE (CONT'D.) J - METHOD SOLUTION (CONT'D.) 26 C(1-8,1) - .01 C(1-7,1) » .02 C (1-7,1) = .01; C(1-5,1) = .07 C ( I - U , D » .13 C(I-3,I) - .205 C(1-2,1) = .305 C ( I - l . I ) - .U3 C(I,I) - - U3 IF (I+1.LT.81) C(1+1,1) « -.305 IF (I+2.LT.81) C(I+2,I) « -.205 IF (I+3.LT.81) C(1+3,1) » -.13 IF (I+U.LT.81) C(l+U,I) = -.07 IF (I+5.LT.81) C(I+5,I) » -.01; IF (I+6.LT.81) C(1+6,1) « -.02 IF (I+7.IT.81) C(1+7,1) « -.01 CONTINUE DO 30 I = 1,80 DO 30 J o 1,1 AI » I AJ = J C ( I , J ) - 2.* (C(I,J) + (AJ-.5)/Al) CONTINUE INVERSION DO 1;0 I = 1,80 DO kO J = 1,80 D ( I , J ) - DBLE ( C ( I , J ) ) CONTINUE DET « 1.0DO DO k2 K = 1,80 # 20 30 C UO DET » DET-5€)(K,K) R = l ODO/D(K,K) D(K,K) - l.ODO DO 1;!; J a 1,80 D(K,J) » R*D(K,J) DO U2 I = 1,80 IF (I-K) U6,Uft,U6 S = D(I,K) D(I,K) « 0.0D0 DO U8 J » 1,80 D ( I , J ) = D(I,J) - S*D(K,J) CONTINUE DET - DET-!® (80,80) DO 50 I - 1,80 DO 50 J = 1,80 C ( I , J ) = 2*D(I,J) e UU U6 U8 U2 - 199 - APPENDIX I I (CONT'D.) NUMERICAL PROGRAMS I N FORTRAN LANGUAGE (CONT'D.) J - METHOD SOLUTION (CONT'D*.) 50 $ CONTINUE WRITE (I*) C CALL EXIT END G RESULTS I N M.K.S. UNITS DIMENSION D(80,80), C(80.80), B(80,80), 1 BU(80) BP(80), DENS(80), DENSP(80), CN(26) READ Ox) C READ (5,20) (CN(I) I = 1,26) 15 READ (5,25) PRESS, TIME, ANORM, CURR READ (5,30) (BP(I) I = 1,80) 20 FORMAT (6F7.5) 25 FORMAT (UF12.lt) 30 FORMAT (12F6.2) DO 35 I = l 51i B U ( I ) B B P ( I ) * ANORM 35 CONTINUE DO 50 I « U5,8o BU(I) » BF(l)-*ANORM + CURR * CN(l-5U)/850. Ii5 CONTINUE BP(1) = BU(1) BP(80)= BU(80) DO 55 I « 2,79 B P ( I ) = (BU(I-l) + BU(I) + B U ( l + l ) ) / 3 . 55 CONTINUE CALL MATYEC (C,BP,DENS,80,80) DO 60 I = 1,80 DO 60 J « 1,1 AI = I AJ = J B ( I , J ) - 2.*(AJ - . 5 ) A D 60 CONTINUE CALL MATVEC (B,DENS,.BU,80,80) DO 65 I - 1,80 DENS(I) « .796* ( l O . * * 9 ) * DENS(I) 65 CONTINUE WRITE (6,80) PRESS,TIME WRITE (6,90) WRITE (6,100) ( I , B P ( I ) , B U ( I ) , D E N S ( I ) , I » 1,80 I F (PRESS.LT.500.) GO TO 15 I F (TIME.LT.8.) GO TO 15 80 FORMAT(iffL,20H I N I T I A L PRESSURE OF F6.0, 6 X l 6 H RESULTS FOR T » F 6 . 0 ) ? ? ; - 200 - APPENDIX I I (CONT'D.) NUMERICAL PROGRAMS IN FORTRAN LANGUAGE (CONT'D.) J - METHOD SOLUTION (CONT'D.) 90 1 0 0 FORMAT (IX, 67H RADIUS MEASURED FIELD TRUE FIELD CURRENT DENSITY) FORMAT ( I 6 , 3 E 2 0 . U ) CALL EXIT END B - METHOD C 20 DIMENSION C ( 8 0 , 8 0 ) , D ( 8 0 , 8 0 ) , C-MATRIX INCLUDES -l/2 CALL ZERO ( C , 8 0 , 8 0 ) SAME AS J-METHOD CONTINUE CALL ZERO ( D , 8 0 , 8 0 ) D ( l , l ) = 2. D(2,l) * - . 6 6 7 D(2,2) » 1 . 3 3 3 D(80.79) = - . 9 9 U D(80.80) « 1 . 0 0 6 DO 3 0 I AI = I Bl « I CI = I DI = I + EI » U*I D(l,I-2) D(1,1-1) D(I,I) D(1,1+1) 30 E ( 8 0 , 8 0 ) = 3 , 7 9 2 1 1 « = » - 2 AI/EI BI/EI CI/EI DI/EI CONTINUE CALL MULT ( C , D , E , 8 0 , 8 0 ) CALL UNIT ( D , 8 0 , 8 0 ) CALL ADD ( D , E , E , 8 0 , 8 0 ) WRITE (U) E CALL EXIT END - 2 0 1 - APPENDIX I I (CONT'D.) NUMERICAL PROGRAMS IN FORTRAN LANGUAGE (CONT'D.) B - METHOD (CONT'D.) I C 1 C 1*0 RESULTS IN M . K . S . UNITS DOUBLE PRECISION D , D E T , R , S DIMENSION C(80,80), D(80,80) B U ( 8 0 ) , B P ( 8 0 , . DENS(80), DENSP(80), CN(26) READ (k) C MULTIPLY MATRIX ELEMENTS BY TWO DO UO I = 1,80 DO HO J » 1,80 C(I,J) - 2.*C(I;J) D ( I , J ) » DBLE ( C ( l , j ) ) CONTINUE DET = l . O D O SAME AS J-METHOD 50 CONTINUE READ (5,20) (CN(I) I = 1,26) SAME AS J-METHOD CONTINUE CALL MATVEC (C,BP,BU,80,80) DO 6 0 I = 2,78,2 • DI - I DENS (I) » ((DI+1.)*BU(I+1)*(DI-1*)*BU(I-1))/(8^I*3»1U* 10.**(-7)) 60 CONTINUE DO 70 I = 2,78,2 DI a I 55 DENSP(I) = ((DI+1.)*BP(I+1)-(DI-1.)*BP(I-1))/(8.^I*3.1U* 10***(-7)) CONTINUE WRITE (6,80) PRESS, TIME WRITE (6,90) WRITE (6,95) ( I , B P ( I ) , B U ( I ) , D E N S P ( I ) , D E N S ( I ) , I » 2,78,2) 80 FORMAT ( 1 H 1 2 0 H INITIAL PRESSURE OF F6.0,6X,l6H RESULTS ' FOR T = F6 0) 90 FORMAT (LX,86H RADIUS MEASURED FIELD 1 TRUE FIELD PERTURB DENSITY CURRENT DENSITY) 95 FORMAT ( I 6 . U E 2 0 . M IF (PRESS.LT.500.) GO TO 15 IF (TIME.LT.8*) GO TO 15 CALL EXIT END 70 S e - 202 - REFERENCES A l l e n , J.E. (1957) Proc.Phys.Soc. B70, 2l* Anderson O.A.et.al.,(1957) Proc.3rd.International Conf.on Ionization Phenomenon i n Gases(Venice),62. Andrianov,A.M.et.al.,(l958) U.N. Peaceful Uses of Atomic Energy 3J.,3l*8 Andrianov,A.M. Bazelevskaia, O.A. & Prokhorov, V.G.(1959) Plasma Physics 2,271* Burkhardt, L.C. & Lovberg,R.H. & P h i l l i p s , J.A. (1958) Nature 181,221* Curzon,F.L. & Churchill,R.J.(l962) Can. J.Phys.1*0,1191 Curzon,F.L.,Hodgson,R.T., & C h u r c h i l l , R.J.(1963a.) Can.J.Phys.1*1,151*7 Curzon,F.L.,Churchill,R.J., & Howard,R. (1963b.)Proc.Phys.Soc.8l,3l*9 de Hoffman,F. & T e l l e r , E.(1950)Phys.Rev.80,692 Dippel,K.H., & Teckneberg W.(I959)l*th.Ionization Conference 1,533 Ecker Von G.,Kroll, W., & Z o l l e r ,(l962)Annalen der Physik 7,220 F o l k i e r s k i , A., Frayne, P.G., & Latham, R.(1962) Nuclear Fusion (Supplement),627 F o l k i e r s k i , A. & Latham, r.(l963)Phys.Fluids 6,1780 Gartenhaus, S. (l961;)Elements o f Plasma Physics (Holt Rinehart & Winston) Harding,G.N.et.al.,(1958) U.N.Peaceful Uses of Atomic Energy 32,365 Hodgson,R.T.(l961*)Ph.D.Thesis,University of B r i t i s h Columbia Komelkov,V.S.,Skvortsov,U.V., Uses of Atomic Energy 31,371* & Tserevitinov,S.S.(1958) U.N. Peaceful Kruskal,M.D.,& Bernstein,I.B. (1961*)Phys.Fluids 7,1*07 Kuwabara,S.(1963)Journal Phys.Soc. Japan 18,713 Lovberg,R.H.(l959) Ann.Phys.8,311 Malmberg, J.H. (l96i*)Rev.Sci.Instr .35,1622 - 203 - REFERENCES (CONT'D.) Medley,S.S., C u r z o n , F . L . , & Daughney,C.C., (1965a. ) R e v . S c i . I n s t r . 3 ^ , 7 1 3 M e d l e y , S . S . , C u r z o n , F . L . , & Daughney, C.C.,(1965b.)Can. Phys.li3_,l882 Journal Moore, C . ( l 9 U 9 ) Atomic Energy Levels(U.S.Govt.Printing O f f i c e ) R e s l e r , L i n & Kantrowitz, A . (1952) Journal Applied Phys.23_,1398 R o s e , D . J . , & Clark,M»(1961) Plasma and Controlled Fusion(John Wiley & Son) Rosenbluth,M.N.(195u) Report LAV1850 Los Alamos S c i e n t i f i c Laboratory, New Mexico, U . S . A . S p i t z e r , L.(1956) Physics of F u l l y Ionized - 20l| - Gases(interscience).
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Dynamics of a Z-pinch discharge in Argon.
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Dynamics of a Z-pinch discharge in Argon. Daughney, Cecil Charles 1966
pdf
Page Metadata
Item Metadata
Title | Dynamics of a Z-pinch discharge in Argon. |
Creator |
Daughney, Cecil Charles |
Publisher | University of British Columbia |
Date Issued | 1966 |
Description | A discussion of probe measurement of the magnetic field in a plasma is presented with particular reference to the perturbation of the magnetic field caused by the probe. A correction procedure is developed to compensate for this perturbation. Using magnetic probes, radial variation of the current density distributions are obtained for an argon plasma in a z-pinch discharge. Initial argon pressures of 100, 250, and 500 μHg are investigated. The current density distributions are determined for 1 μsec intervals between the initiation of the discharge and the occurrence of the first pinch. These current density distributions are compared with photographic observations. The experimental results are discussed in terms of the snowplow model and the shock wave model. Mathematically, the non-linear snowplow equation is solved using an approximation technique which results in analytic solutions. The shock wave equation is solved by a graphical technique. An extension of the shock wave model is proposed for a better understanding of the experimental results. |
Subject |
Plasma (Ionized gases) Magnetohydrodynamics |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-08-17 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085336 |
URI | http://hdl.handle.net/2429/36750 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-UBC_1966_A1 D2.pdf [ 11.97MB ]
- Metadata
- JSON: 831-1.0085336.json
- JSON-LD: 831-1.0085336-ld.json
- RDF/XML (Pretty): 831-1.0085336-rdf.xml
- RDF/JSON: 831-1.0085336-rdf.json
- Turtle: 831-1.0085336-turtle.txt
- N-Triples: 831-1.0085336-rdf-ntriples.txt
- Original Record: 831-1.0085336-source.json
- Full Text
- 831-1.0085336-fulltext.txt
- Citation
- 831-1.0085336.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085336/manifest