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Time correlated study of the Z-pinch discharge in helium Dimoff, Kenneth 1968

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A TIME CORRELATED STUDY OF THE - Z^PINCH DISCHARGE IN HELIUM by KENNETH DIMOFF  B.A., University of Toronto, 1959 M.A., University of Toronto, 1960  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the department of PHYSICS  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA May, 1968  In p r e s e n t i n g  for  this  thesis  in partial  an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y  that  the L i b r a r y  Study.  thesis  shall  I further  make i t f r e e l y  agree that  f o r s c h o l a r l y p u r p o s e s may  publication  of t h i s  w i t h o u t my w r i t t e n  thesis  Columbia  1 agree  for reference  f o r extensive  and  copying of  this  be g r a n t e d b y t h e Head o f my  It i s understood  for financial  permission.  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  of B r i t i s h Columbia,  available  permission  D e p a r t m e n t o r b y hits r e p r e s e n t a t i v e s .  or  f u l f i l m e n t of the requirements  gain  shall  that  n o t be  copying  allowed  ii ABSTRACT The s t r u c t u r e i n the c o l l a p s e stage of a l i n e a r Z-pinch discharge i n helium has been s t u d i e d by o p t i c a l methods.  Observations w i t h a framing  camera, r o t a t i n g m i r r o r spectrograph, and monochromator have been c o r r e l a t e d w i t h magnetic f i e l d and current d i s t r i b u t i o n s determined by Tam The luminous regions i n a helium p i n c h are very f a i n t .  (1967).  Therefore, up  to twenty exposures have to be superimposed on the same framing camera or rotating mirror record.  This r e q u i r e s a high degree of r e p r o d u c i b i l i t y i n  the i n i t i a t i o n o f the discharge. At high i n i t i a l p r e s s u r e s , a non-luminous shock wave at the inner edge o f the c o l l a p s i n g current s h e l l precedes the luminous plasma l a y e r towards the centre o f the discharge v e s s e l .  This shock f r o n t i s followed by a region  o f predominantly Hel emission, while most of the H e l l r a d i a t i o n occurs i n the outer regions o f the c o l l a p s i n g plasma s h e l l .  The separation i n t o Hel and  H e l l r a d i a t i n g regions i s consistent w i t h spectroscopic measurements o f temperature: higher temperatures occur at l a r g e r r a d i i .  Pressure and density  i n the non-radiating shock wave r e g i o n are determined by c a l c u l a t i o n s based on a simple model. At low f i l l i n g pressures, the Hel and H e l l regions c o i n c i d e .  The  p o s i t i o n of maximum l u m i n o s i t y i s observed to correspond w i t h the p o s i t i o n o f maximum current d e n s i t y . The l u m i n o s i t y and current s h e l l s coincide w i t h no shock wave preceding the luminous f r o n t . Strong continuum r a d i a t i o n i s emitted from the centre of the discharge tube as soon as the l e a d i n g edge of the current s h e l l reaches the a x i s . l e a d i n g edge i s luminous at low i n i t i a l pressures, but becomes a nonr a d i a t i n g shock f r o n t at higher f i l l i n g pressures.  This  -iiiTABLE OF CONTENTS Page Abstract Table of Contents List of Tables L i s t of Figures Acknowledgements  -  c  Chapter 1.0 Introduction 2.0  The Superposition of Photographic Records 2.1 The Control Circuit 2.2 The Z-Pinch Discharge 2.2a Discharge Vessel and Capacitor Bank 2.2b Discharge Circuit and Trigger Generator 2.2c The Discharge Switch 2.3 Current Measurements  3.0 Pinch 3.1 3.2 3.3 3.4 4.0  and Post-Pinch Stages of the Discharge The Timer-Resolved Spectrograph Time-Resolved Spectra of the Pinch Pinch Times Behaviour of the Discharge at Pinch and Post-Pinch Times  Kinematics of the. Discharge 4.1 The High-Speed Framing Camera 4.2 Data Reduction 4.3 Corrections for Perspective 4.4 Peak Radiation as a Function of Time 4.5 Synchronization of Framing Camera Photographs with Other Discharge Parameters 4.6 Structure of the Z-Pinch Discharge  5.0 Determination of Densities and Temperatures within the Luminous Plasma 5.1 The Monochromator-Photomultiplier Method of Time Resolution 5.2 Absolute Line Intensity Measurements 5.3 Electron Temperatures from the Semi-Coronal Model 5.4 The Effect of Self Absorption on Line Profiles 5.5 Relaxation Times 6.0  Dynamics of the Pinch 6.1 Densities within the Shock Layer 6.2 Evidence of Guderley Flow 6.3 The Piston Model of Shock Formation  7.0 Summary and Proposals for Future Work  i i i i i v vi viii 1 7 10 12 14 14 17 22 26 27 32 37 40 49 50 52 57 59 65 74 83 85 89 95 97 102 107 108 115 122 127  -iv-  Page Bibliography  130  Appendix I. II. III. IV. V.  VI. VII.  The Control Circuit  132  Rogowski Coil and Integrator Circuit  137  Correlation of Monochromator Traces with Time-Resolved Spectroscopic Plates Monochromator-Photomultiplier Calibration  143 148  Retrieval of Information from "Spoiled" Monochromator Traces  153  Partition Functions for Helium  157  Self Absorption Effects  163  -vLIST OF TABLES N o  » .  Title  P a  2~i  Discharge system  3- i  Time-resolved spectrograph  3*-ii  Pinch times  4^i  Observational sequence and conditions for framing camera photographs  4- i i  g  e  25  , c  31 38  u  53  Transmission of the "neutral density" f i l t e r used in determining boundaries of the collapsing shell  54  5 r-i  Absolute intensity evaluation of n , T  93  5- i i  Absolute temperature measurements  e  e  5 - i i i Griem's semi-coronal model 5- iv  96  Influence of self absorption on line intensities and half-widths  6- i  95  Relative densities for shock fronts  6-ii Numerical relations for Guderley flow parameters 6-iii Comparison of pressures determined from velocity, spectroscopic and magnetic f i e l d measurements  100 113 117 124  -viLIST OF FIGURES Ho.  Title  Page  1.1  Z-pinch discharge  1  2.1  Photomultiplier trigger system for time-resolved photography  2.2  Control circuit and corresponding pulse trains  10  2.3  Discharge apparatus  13  2.4.  Triggering system and discharge circuit  15  2.5  Trigger generator  2.6  Electrode systems for spark gap switch  18  2.7  Oscillograms of discharge current  18  2.8a b  Electrode assembly Discharge switch S2  20 20  2.9  Output of Rogowski circuit vs. time  23  2.10  Comparison of discharge current i n helium Z-pinch  24  3.1  Experimental arrangement for the time-resolved spectrograph  28  3.2  Four consecutive discharges in 1,500 uHg helium  29  3.3  Time-resolved spectra in helium  33  3.4  Time-integrated spectra in helium  34  3.5  Determination of pinch time Tp  38  c  .  7  1  5  3.6-9 Composite trace of on~axis continuum for 500, 1,000, 2,000 and 4,000 uHg in helium  42-45  4.1  Framing camera  51  4.2  Colour sensitivity of Ilford HP3 film  53  4.3  Transmission curves for interference f i l t e r s  53  4.4  Segment of framing camera film  55  4.5  Analysis of framing camera images  55  4.6  Effect of perspective on film image  58  -viiNo. 4.7-10  Title Radius of radiation peaks vs. time for 500, 1,000, 2,000 and 4,000 uHg in helium  Page 61-64  Synchronization of framing camera images with the total discharge current  66  4.12-15 Luminosity curve for 500, 1,000, 2,000 and 4,000 ;aHg i n helium  70-73  4.16-19 Current shell collapse, magnetic probe method for 500, 1,000, 2,000 and 4,000 jiHg i n helium  76-78  4.11  4.20  Schematic diagram of Jukes's model for Z-pinch contraction  80  5.1  Experimental arrangement for monochromator  85  5.2  Hell 4686 X line profile  87  6.1  Model of shock formation  109  6.2  Model of the collapsing current shell  110  6.3  Discharge column and associated circuit  115  6.4  2,000 jiHg, "Hell peak" vs. time  120  6.5  Inner luminosity radius r ^ vs. time  121  6.6  Piston driven shock front  123  -viii-  ACKNOWLEDGEMENTS I wish to thank my supervisor, Dr. F.L. Curzon, for his advice and help during the course of research.  I am deeply indebted to Dr. B. Ahlborn  for discussions and his help in the preparation of the thesis.  Dr. J.H.  Williamson's invaluable suggestions and Dr. R.A. Nodwell's patient counsel helped greatly in the completion of this work. I am grateful to Messrs. W. Ratzlaff and J . Dooyeweerd for assembling the electronic systems. My thanks are also due to Mr.- A. Fraser and Mr. J . Lees and their staffs, who provided excellent service in the construction of the apparatus.  In particular, I owe much to Mr. A. Knop  whose patience and s k i l l made a viable switching device, and hence much of this research, possible. Consultation with Dr. W.L. Lee and fellow students H.A. Campbell, H.G. James, S.S. Medley, and R.N. Morris helped clarify difficulties in dealing with experimental results.  I am most deeply indebted to my wife  whose understanding and encouragement far outweigh the assistance she provided in reducing data, drawing diagrams, and typing the manuscript. Finally, I wish to acknowledge the generous financial assistance provided in the form of fellowships by the University of British Columbia and the National Research Council of Canada.  1.0  INTRODUCTION  c  Attempts to solve the problem of confining a very hot gas by means of a magnetic f i e l d have produced a variety of experimental configurations. The earliest and simplest arrangement is the Z-pinch discharge.  Aside  from i t s potential usefulness as a preionising device, the Z-pinch provides a simple arrangement in which the formation and behaviour of a plasma can be studied. In the Z-pinch discharge the plasma is sustained by an axial current I,  (Figure 1.1),  which is created by the discharge of energy from-a  capacitor bank. The axial current (I) produces an azimuthal magnetic f i e l d (B ), which i n turn interacts with (I) tc form a radial force ( F ) . &  r  This force causes the plasma to constrict along the axis of the discharge tube.  Such a constriction of the plasma is named the "pinch effect",  while the period of maximum constriction is known as the "pinch stage" of the discharge. An advantage of this device is that the high current  electrode  iin i  (brass  P.  7  s p a r k gap s w i t c h  11 11  i  1111  11  i ii ii  11  i  11  - JV;  r  I  /  7>  C a p a c i t o r bank ( 1 0 6 uF) F i g u r e 1.1  discharge v e s s e l (pyrex) r e t u r n conductor (brass screen)  Z-pinch discharge  Ro = r a d i u s of d i s c h a r g e v e s s e l r = r a d i u s of plasma column - plasma  -2simultaneously generates and confines the plasma. The early stage of constriction of the discharge is called the' "collapse" or "pre-pinch" phase. During this period the plasma forms a hollow cylindrical shell of excited gas which collapses onto the axis of the discharge vessel.  There is usually an abrupt, f i n i t e change i n  pressure and particle velocity between the inner surface of the hollow cylinder and the region of cold gas trapped within i t .  This zone of  discontinuity is known as the "shock front". The time interval between the initiation of discharge and the arrival on axis of the inner boundary belonging to the collapsing current shell is known as the "pinch" or collapse time. the particular discharge used.  Its magnitude i s characteristic of  It spans the time taken to form the pinch  and can be used to define the collapse and pinch stages of the discharge. Many investigations have been made of the Z-pinch discharge in helium because of i t s relatively simple spectrum. Detailed studies of the luminosity front i n the pre-pinch stage have been conducted by Botticher (1961 to 1965) using photoelectric and photographic techniques.  The action of  the pinch i n the axial region of the discharge has been investigated by H. Zwicker (l964a,b). Employing time-resolved spectroscopy, he has shown the occurrence of on-axis continua before and after the complete collapse of the luminosity front. Both authors demonstrate agreement with a theoretical model of the fast-pinched discharge proposed by Jukes (1958) and Allen (1957), which in turn is based on a theory f i r s t presented by Guderley (1942).  -3-  -  ..  Botticher (1963) equated the motion of the luminosity front with that of a cylindrical shock wave and refined the Guderley theory to describe i t s action,,  Zwicker (1964a) invoked Jukes' version of the same theory to  explain the appearance of on-axis continua  D  Magnetic probe studies of the Z-pinch discharge have been conducted at this laboratory under the direction of F.L, Curzon.  A "gradient probe"  technique perfected by C C . Daughney (1966) and S„ Tarn (1967) has yielded current and magnetic f i e l d distributions of the helium discharge i n the pre-pinch stage.  The experimental results agree with predictions based  on a modified snowplow equation.  However, this method does not yield  information about the particle densities and temperatures within the collapsing current shell. The research efforts mentioned above have concentrated on different aspects of the Z-pinch discharge because of limitations i n the diagnostic devices employed. They provide detailed, but essentially uncoordinated information about the action of the pinch~-from i t s inception to well after i t s formation i n time.  By extending the range of experimental observations  over the complete life-history of the discharge, and by correlating them with known results, the following question can be c l a r i f i e d . -=What relation i s there between the luminosity front and the collapsing current sheet which sustains the discharge? This thesis attempts to resolve the problem above through the use of three different optical diagnostic methods. The advantage of investigating light radiated by the discharge i s that the measuring devices leave the plasma undisturbed.  1)  Photographic time-resolved spectroscopy with a fast medium-quartz  spectrograph provides a survey of a large number of lines and their . behaviour in time.  Because of the calibration d i f f i c u l t i e s inherent  in photographic work, only qualitative results are considered. However, this method provides a guide to the identification, intensity, and time of formation for lines of interest.  A qualitative estimate  can be made of the purity of the discharge, as well as a choice of suitable interference f i l t e r s for use in high speed photography. 2)  A high speed framing camera can give information concerning the  stability and gross structure of the discharge.  By photographing  the  complete period of collapse and formation of the pinch with interference f i l t e r s , any interesting features requiring further investigation can be correlated with respect to the time of their appearance, 3)  The use of a grating monochromator serves to isolate lines  determined by the above photographic procedures.  A detailed study of  line profiles can then be carried out to yield electron densities and temperatures within the plasma. By using a Rogowski c o i l to monitor the discharge current, the above three methods can be correlated in time to give an overall picture of the Z-pinch discharge.  The behaviour of the discharge in the collapse and  pinch stages is then compared with results obtained from magnetic probe work. Most of the techniques for time-resolution are standard and highly developed.  However, the problem of adequate spatial resolution in photo-  graphic work for faintly luminous gases, such as helium, has been a serious obstacle until now.  Unless the event can be reproducibly recorded, only  diagnostic devices with a severely limited range of spatial or wavelength  resolution can be used. This i n turn complicates the interpretation of  J  observational results. Two of the recording devices used in this work employ a rotating mirror.  The discharge must be triggered by the motion of the mirror, since  a recording depends on i t s proper placement and rotational speed. Because the light level of a single discharge in helium is far too low to produce a well exposed negative, a superposition technique must be introduced. This requires a high degree of accuracy in triggering the discharge. Fluctuations in the breakdown time duration of the discharge must also be eliminated so that an accurate superposition of many recordings can be achieved.  Chapter 2 outlines, solutions to the technical problems of  accurate switching and timing of the discharge and diagnostic apparatus. The successful superposition of as many as 20 photographic records of the Z-pinch discharge in helium provides a valuable method for i t s study. Chapter 3 shows the application of the time-resolution technique in spectroscopy.  Low light levels in the collapse stage of the discharge  make i t impossible to obtain a viable record for even 20 superpositions off-axis.  Because of this, only on-axis observations are executed system-  atically.  These results are then correlated v)ith monochromator recordings  of the continuum and synchronized with the behaviour of the discharge current.  The procedure gives an accurate determination of pinch times.  It serves as a basis for comparison between the discharge system employed in this study and that used by Tarn (1967) in magnetic probe work.  Current  traces and the appearance of secondary continua reveal the action o f the discharge in the pinch and post-pinch regimes. contrasted with Zwicker's (l96/ a,b) findings. +  This is discussed and  -6Chapter 4 deals with framing camera results which are also based on a time-resolved photographic technique.  Records are obtained of the collapse  and pinch stages of the discharge. These are correlated with the observations of Chapter 3 to give a complete description of the discharge in the form of "collapse curves". By comparing them with Tarn's (1967) results, the development of the pinch from the collapse stage is explained and the relation of luminosity fronts to the current sheet becomes clear.  The  conclusions arrived at by experimental method agree well with a qualitative description provided by the Jukes-Allen (1958) theory of fast-pinched discharges. The structure of the' collapsing plasma shell, revealed in Chapter 4, is now subjected to a quantitative analysis.  Chapter 5 discusses in detail  the roonochromator-photomultiplier method of time-resolution.  Electron  temperatures and densities are determined in the luminous regions of the plasma.  Calculations by Griem (1964) for relative intensities and relaxation  times are compared with the experimental results.  It is shown that an L.T.E.  approximation can be assumed to be valid within the limits of experimental error. The results of Chapter 4 indicate the presence of a non-radiating front which precedes the luminous shell for high f i l l i n g pressures. Chapter 6 contains a proposed model for calculating the density in this non-luminous region, using electron densities and temperatures of Chapter 5. The results show that the non-luminous front can be regarded as a shock wave, in which very l i t t l e kinetic energy of the gas passing through i t is converted to ionization  0  Evidence in support of this conclusion, is pro-  vided by the application of Guderley flow theory and a simple piston model for shock wave formation.  -72.0  THE SUPERPOSITION OP PHOTOGRAPHIC RECORDS It i s often d i f f i c u l t to obtain good photographic records when the  i n t e n s i t y o f l i g h t emitted by a discharge i s low. Such a s i t u a t i o n occurs during the i n i t i a l collapse stage o f the Z-pinch.  I f the discharge has  reproducible o p t i c a l p r o p e r t i e s , i t i s possible to obtain usable records by superimposing This procedure  several exposures o f the same event on a s i n g l e piece of film. i s valuable i n extending the l i m i t s o f high speed photography  and time-resolved spectroscopy.  I t assures the recording o f s u f f i c i e n t  l i g h t to exploit the advantages o f s p a t i a l r e s o l u t i o n and a wide f i e l d o f view, which are c h a r a c t e r i s t i c o f high speed framing camera p i c t u r e s . I t also allows the use o f conventional spectrographs  i n place o f large aperture  o p t i c a l systems, which are c o s t l y and generally have a lower wavelength resolution.  t r i g g e r beam source-  discharge tube  objective lens  UJ dis charg 3 switch  rotating / \iJ> mirror 4  photomultiplier  recording film trigger pulse generator  Figure 2.1  control circuit  Photomultiplier t r i g g e r system f o r time-resolved photography Trigger beam, mirror i n p o s i t i o n A Recorded r a d i a t i o n , mirror i n p o s i t i o n B when discharge s t a r t s .  -8-  Figure 2.1 illustrates the time-resolution method for photographic records.  Light from the discharge tube is reflected from the surface of a  rotating mirror and imaged on the recording film.  As the mirror rotates,  the image of the discharge moves across the film, producing a record which has a time axis along the sweep direction of the mirror.  (For a time-  resolved spectrogram, the s l i t of the spectrograph replaces the f i l m Section 3.1. In the high speed framing camera, a series of lenses i s interposed between the rotating mirror and recording film to form a sequence of images of the discharge—Section 4.1). For faint discharges, several records must be superimposed to produce a useful photograph. This requires the discharge to be triggered at a certain mirror position so that each recorded event starts at the same point on the film.  The discharge must also be triggered each time at the same frequency  of rotation of the mirror, since this controls the scale of the time axis. Both position and frequency selection are performed by a control circuit through the action of a photomultiplier trigger system (Figure 2,1).  This  arrangement automatically fires the discharge when the mirror reaches the preset speed and recording position.  The trigger beam system is aligned i n  such a way that light from i t s source does not f a l l onto the film area swept out by the discharge image. The performance of the oontrol circuit i n conjunction with the trigger beam is outlined in Section 2.1. o  It is necessary to assure commencement of the discharge when the rotating mirror is properly placed.  This required the construction of a  special discharge switch and a change i n design of the conventional trigger pulse generator employed at this laboratory.  -9The rest of Chapter 2 may be omitted by a reader who i s not interested in technical details.  Apparatus and the solutions of associated technical  d i f f i c u l t i e s , implied i n the outline above, are dealt with in the following order; Section 2.1 gives a brief sketch of the control circuit governing the time of the discharge of the Z-pinch.  J i t t e r in the signal  emanating from this system is found to be less than 0.1 usee. Section 2.2 outlines the equipment and circuitry involved in producing the plasma. Before this thesis was undertaken, existing apparatus exhibited a considerable variation in the time of current breakdown over a long series of discharges.  This posed a considerable  problem in the execution of adequate time-resolved photography. A description is given of changes made to achieve an overall variation in breakdown of less than + 0,2 usee. Section 2.3 shows the method of measuring the discharge current. It is used for correlating the various tochniques of time-resolution and is referred to frequently throughout the thesis.  A comparison of  parameters with those characteristic of the discharge studied by Tarn (1967) is also given, since use w i l l be made of his experimental results. Descriptions of the time-resolved spectrograph, framing camera and monochromator arrangement are inserted in. the remainder of the text to avert the monotony inherent in a large body of technical detail.  2.1  THE CONTROL CIRCUIT The rotating mirror is the only moving part i n the system. A l l events  must be timed to happen when i t is in the proper position, as indicated in Section 2.0.  The timing action of the discharge is regulated by a control  circuit designed by F.L. Curzon.  It i s described in detail by Neufeld (1966)  and a synopsis of that account is provided in Appendix I for the sake of completeness. Since the discharge must occur when the mirror is rotating at a given speed, the control system must contain a frequency gate.  The trigger beam  system causes the monitoring photomultiplier to emit a palse each time the mirror swings through a certain position (mirror position A, Figure 2.1). The gate is tripped open only when the time interval (T) between pulses from the photomultiplier reaches a required value (waveforms e.and f, Figure 2.2)  •«3_ _{rf T  Uja.'Se&r-m <2) ~1| &om  |J &—  LP  Waveform  b)  c  ana/ Xn verte. r  'A  'fektron/x. ir  Vu./sZ. qo.te  UJ&t/e$>r/n e) does  r~L_r~L_ni '  •——<  9  (  ^  —  Figure 2.2 C o n t r o l c i r c u i t and c o r r e s p o n d i n g p u l s e trains  'T'hcjvxX.'tron  trigger switch  The output pulse from the gate is then delayed by a time T^, allowing the mirror to swing into the proper recording position (mirror position B, Figure 2.1).  The discharge is initiated by this delayed pulse.  The j i t t e r for the delay system is smaller than 0.0$ psec.  The only  other source of fluctuations would be from a change i n speed of the rotating mirror.  Section 3.1 deals with this problem i n connection with the time-  resolved spectrograph.  2.2 THE Z-PINCH DISCHARGE It i s essential that timing characteristics of the discharge be extremely reproducible to accomplish an accurate superposition of timeresolved records. Since pinch formation times are the' order of 10 |isec, the time between triggering the system and the breakdown of the discharge must be kept constant to within a fraction of a microsecond. •Such a condition imposes requirements not easily f u l f i l l e d by conventional equipment used until now i n this laboratory. It was found necessary to modify the trigger pulse generator (Medley, 1965) and the discharge switch (Curzon, 1961). This decreased the variation i n discharge breakdown to less than 0.2 usee, for as many as seventy consecutive firings of the discharge. A sketch of the Z-pinch apparatus i s provided in Figure 2.3. Functions of the component parts are discussed i n the following subsections: Subsection 2.2a i s a brief description of the gas vessel and adaptations made for observing radiation emitted by the discharge. Subsection 2.2b outlines the action of the trigger generator and the change made in coupling i t to the control c i r c u i t . Subsection 2.2c gives the design and operation of the discharge switch. Table 2 i at the end of this chapter summarizes the main 0  features of the equipment involved and compares them with the apparatus used by Tam (1967)..  'to McLeod gauges  main valve  O^to gas supply ,to Pirani gauge  pressure screws \f  pressure cross-struts sliding brass end plate (46 x 21 x 1.3 cm.)  -to backing pump  to diffusion pump cutaway section of "ground" electrode brass mesh on "ground"electrode  quartz window ^ held in place by brass screv-in holder pulse transformer (B) —, from thyratron pulse generator (A)  pressure struts to hold leads together main spark gap (C^) O R p o t e n t i a l c.  trigger spark gap switch (S-j) trigger generator (D)  high tension leads wrapped in polyethylene  Figure 2.3  Discharge apparatus  (letters in brackets refer to Fig. 2.4)  -u2.2a  DISCHARGE VESSEL AND  CAPACITOR BANK  The horizontally-mounted vessel is a pyrex glass cylinder (dimensions given in Table 2-i, Section 2.3) within which brass electrodes are located at both ends.  It is encased in a brass gauze (mesh interval 0.2 cm.) which  serves as a return conductor and an efficient electrical screen.  Provision  is made for evacuating the tube and introducing the test gas at a desired pressure through perforations at the high tension end electrode (Figure  2.3).  To allow .for photographic observations along the axis of the tube, the other electrode consists of a brass mesh beyond which an end plate is mounted, sealing the tube.  This end plate is a large glass window through  which the whole discharge cross-section can be photographed.  In spectro™  scopic work, where only a small region of the plasma cross-sectional area is to be investigated, a small, easily replaceable quartz window is used instead.  This i s mounted on a one-half inch thick (1.27  (Figure 2 . 3 ) ,  in which a quarter-inch ( 0 . 6 4 cm.)  cm.)  brass plate  hole is d r i l l e d .  The  brass plate is large enough so that the window can be s l i d to any position across the diameter of the tube. The capacitor bank (C2 of Figure 2.4)  is formed by ten N.R.G. low  inductance storage capacitors, giving an overall capacity of 106 ^uF. These capacitors are charged to 10.5 K.V. 2.2b  for a l l experimental investigations,  DISCHARGE CIRCUIT AND TRIGGER GENERATOR Figure 2.3 provides an idea of the placement and relative size of some  of the component parts of the apparatus used in generating the Z-pinch. schematic diagram of the whole system is given in Figure 2<,4o  A  -15  V(-ve) RF  400 k r u  J?2  33  k-o-  '  Figure  2.4  Triggering system mid discharge circuit A—thyratron pulser, triggered by pulse from control circuit B — p u l s e transformer D—trigger generator circuit E—discharge vessel --potential divider; RJ<(180MA) total resistance of potential divider G—high-voltage charging unit  Trigger generator components: S-j—three -elect rode trigger spark C-j—0.06 pF Ri  -50sA,  Main discharge components: Sp—.three-electrode spark gap switch C 2 — ( l 0 0 | i . F ) capacitor bsnk  charging lead 0 . 0 6 pj  Insulator: bakelite CD \ wrapped in mylar sheet  condenser  tungsten "t^wire  1  output lead 22 Ohm resistor (part of R )  trigger lead  1  Figure  AV  2.5  Allen 'screw holding ':insulator in place  Trigger generator Brass Perspex fiTTTO Polyethylene  The control circuit (Section 2.1) sends put a pulse that trips the 2D21  thyratron pulse generator, A of Figure 2.4.  Generator A forms a  negative pulse (-300 volts, rise time 2 psec.) across the primary winding of pulse transformer B.*  This in turn causes gap S^ to break dawn, discharging  capacitor Cj and generating a pulse across resister R-|.  This output pulse  initiates breakdown of the three-electrode spark gap switch S2 which discharges bank Cg through vessel E. The trigger circuit and generator D are modifications of those described by Medley (1965 ) .  That system used a pulsed light source, consisting of an  open a i r spark gap enclosed by a quartz bulb, in place of the present pulse transformer B in Figure 2.4. Such an arrangement proved awkward to operate, since fogging of the quartz surface, changing atmospheric conditions, and damage to the electrodes in gap S^ contribute to j i t t e r in the overall breakdown time of the system over a large number of shots.  Its replacement by the  pulse transformer s t i l l serves, in conjunction with switch S-^, to isolate the discharge circuit from the control electronics. Figure 2.5 shows the trigger generator D in detail.  It is constructed  coaxially to supress production of spurious electromagnetic signals.  The  50 i l output cable is terminated by the resistance R-j (Figure 2.4) which consists of five, chains of 22 S l resistors arranged as a sheath around capacitor C-j (Figure 2.5).  The trigger capacitor  with C^ through the potential divider F.  is charged in parallel  The whole system can then be primed  to discharge when switch S-j is shorted by command from the control circuit.  * Pulse transformer Type U.558, manufactured by Atkins, Robertson and Whiteford Ltd., Glasgow. Step-up i-atio 3:100.  -172.2c  THE DISCHARGE SWITCH The greatest source of j i t t e r in the breakdown time of the system is  caused by irregularities occurring i n the main trigger switch Sg (Figure 2.4).  At tho time this work was begun, a satisfactory discharge  switch for 1-2 kilojoule condenser banks had been designed (Curzon, 1961). However, such a switch proved unsatisfactory for the use of a 5.8 kilojoule condenser bank and the r i g i d control of breakdown time over ranges of at .'least thirty shots.  The salient features of design in the previous and  f i n a l electrode systems are sketched in Figures 2.6a and 2.6b respectively. The improvements depicted by Figure 2.6b are incorporated into the discharge switch drawn in Figure 2.8. In the electrode assembly of Figure 2,6a, i t was found that the c r i t i c a l design parameter was the distance labelled B.  This distance  controls the delay between the breakdown of tho trigger spark and the main spark gap channel: for B %  3,2 mm.,  the delay is about 2  ^Jisec.  When the system depicted by Figure 2.6a was used in conjunction with the 5.8 kilojoule bank, i t was found that such a minimum delay of 2^usec. was indeed observable.  However, when a series of ten shots was taken, a  random fluctuation in occurrence of breakdown was noticed that varied from 4 to 10 usee. By advancing the trigger pin T to a position indicated in  L  Figure 2.6b, the range of this fluctuation was cut down to approximately 4 j-tsec.  However, as the polyethylene insulator around tungsten tip T was  eaten back, the fluctuations in the breakdown delay increased again (see Figure 2,7).  Figure 2„6 Electrode systems for spark', gap switch a) Switch for 2 kilojoule capacitor banks b) Switch for 3 kilo joule- capacitor banks  L  ,  1  (  1  1  :  ,  t (psec)—t> A t ~- 2 yasec. '  Figure 2.7  Oscillograms of discharge current ' (measured by Rogowski coil) A r,-—breakdown time^2 microseconds if ~-fluctuations in breakdown time^6 microseconds  A  a  Since i t was inconvenient to replace the polyethylene insulator at frequent intervals, a conventional spark plug was tried instead.  The best  insulation material available for our purposes was. found to be the ceramic coating on the Lodge C-j type spark plug.  In order to eliminate problems  due to wearing away of the conventional spark plug electrode t i p , i t was replaced by a tungsten rod 2.7 ram. i n diameter. The rod projected 3,2 mm. beyond the surface of the ceramic insulator (Figure 2.8a).  It was cemented  into the ceramic with epoxy resin. The hollow "ground" electrode, in which the trigger pin is lodged, was enlarged so that the specially prepared spark plug could be inserted without altering the geometry of the ceramic insulator. In order to prevent short circuiting of the main discharge to the walls, the sharp edge D of Figure 2.6a was rounded off to the geometry of Figure 2.6b. A major problem i n the design of the trigger switch is the insulation of the side walls of the housing that contains the electrode assembly (Figure 2.8b). Unless the walls are far enough away from the ground electrode and insulated adequately, the discharge can occur from wall to trigger electrode rather than across the gap C in Figure 2.8b. A thick coating of glyptal i s applied to the inside of tho brass container. The perspex bottom is corrugated with circular indentations (Figure 2„8b) to prevent accumulated debris from serving as a conductor from wall to electrode.  Eventually, the glyptal coating on the 'walls w i l l give way after  about 200 shots.  Before this can occur, the container is replaced by a  spare one made up for this purpose. Both housings are used alternately on separate experimental runs after, a fresh coating of insulator has been applied.  -20. U  Hollow " g r o u n d " ~& electrode  in.-  J72  UP  ?  E l e ctrode separat i o n ^ G  (?1T  Tungsten rod  rora h i g h tension side of capacitor bank  Spark plug  From ground s i d e of,, d i s c h a r g e tube^  From t r i g g e r generator  ure 2.8a E l e c t r o d e assembly  Ceramic insulator  FJLgure 2.8b, D i s c h a r g e s w i t c h So  For optimum timing characteristics, the potential differences across gaps S^ and S^ (Figure 2.4) must be as close to the overvolting value as possible.  The overvoltage is the potential difference across the gap which  produces breakdown even though no trigger pulse is applied. The overvolting potential of gap  is f i r s t adjusted to 11.5 kilovolts by charging the  capacitor bank (Cg, Figure 2.4) to 11.5 kilovolts and by increasing the length of the gap S^ u n t i l i t stops breaking down. Since each spark in S^ produces a trigger spark i n S^ the l i d of S^ is removed (Figures 2.4 and S  2.8b). This prevents  from discharging while the adjustment is made.  The top of the main discharge switch S^ is then put in place and the electrode separation C (Figure 2.8b) is adjusted until the whole system overvolts at 10.75 kilovolts.  The system is then ready for experimental  work at an operational voltage of 10.5 kilovolts.  It was found that this is  the highest voltage at which the main discharge switch can be operated for an extended series of shots without serious damage to the electrodes. With this arrangement, the overall j i t t e r is less than 0.2jusec. for a series of 50 to 100 shots.  -222.3  CURRENT MEASUREMENTS The measurement of the total discharge current employing a Rogowski  c o i l i s a standard experimental procedure.  Current traces for i n i t i a l  pressures in helium of 500, 1,000, 2,000 and 4,000 uHg are illustrated i n Figure 2.9.  The period of the discharge current is approximately 34 to  37^J.sec. Because  a relatively short integration time was used, an  accurate calibration of the total current can prove d i f f i c u l t .  However,  this is compensated for in that a superimposed high frequency structure becomes apparent during the f i r s t half-cycle of the discharge current (Figure 2.9).  This effect would be smoothed out i f a longer "integration  time" were used. This problem is dealt with in Appendix: II, where i t is shown that the observed current trace can give an underestimate of the "true" reading by a 20$ error at worst.  A l l current traces in the following sections are  derived from the observed or "uncorrected" oscillograms. As outlined in Appendix II, a calibration constant (k) can be estimated which relates the discharge current to the observed voltage output of the Rogowski c o i l circuit: I = kv  ......(1)  where k = 9.4 x 1cA amps/volt + 20$ ......(2) I = the discharge current v = the output voltage of the integrator circuit Equations (1) and (2) can be considered to hold true for the times of interest (0 to 10 usee, after initiation of the discharge).  In order to better understand the action of the discharge, i t is necessary to correlate the results of optical investigations with magnetic f i e l d and current distributions observed by the magnetic probe method (Tarn, 1967)o  The discharge system used i n this study differed from Tarn's  only in the size of the condenser bank, which serves as the energy source for plasma production (Table 2 - i ) .  This in turn affects the discharge  parameters of the system. A comparison of total discharge currents, observed by Tarn and in the present Z-pinch system, i s given in Figure 2.10. Although the magnitude of the discharge current differs, i t w i l l be shown that the action of the pinch in the collapse stage is identical for the two systems (Sections 3.3 and 4 . 6 ) .  0  1  2  3  Time (jasec.) Figure 2.10  4  >  Comparison of discharge current in helium Z-pinch -Tarn's (1967) system -500 juHg f i l l i n g pressure •4,000 uHg f i l l i n g pressure  Present system  -25-  Table 2-i A.  Discharge system  COMPARISON OF DISCHARGE CIRCUIT PARAMETERS  Condenser Bank: Total capacity Total i n i t i a l inductance Charging voltage Current Measurement Maximum discharge current Discharge current frequency Rogowski c o i l sensitivity Integrator (RC passive, integration time constant)  B.  Prdsent system  Tarn (1967) 53 uF t 0.01 pli t 0.2 kV  0.12 10.0  200 kAmp 46 kHz (1.86  +  .11)  amp/volt  x  106  0.09 10.75  uF £ 10$ * 0.05  280 kAmp 30 kHz x 10 t amp/volt  10 0.95 5  47 jisec.  24 msec.  APPARATUS COMMON TO BOTH SYSTEMS  Discharge Tube (pyrex) Inner diameter Outer diameter Electrodes Electrode separation  cm. cm. Brass 59 cm. 15 17  Vacuum System Type 17 Balzers O i l Diffusion Pump Hyvac 14 Cenco Backing Pump Vacuum attainable 1 uHg Leak rate 7 uHg/ hr. MacLeod Gauges 0-1 mm Hg, 0-10 mm Hg Pyrani Gauge (Type GP-110 Pirani Vacuum Gauge) Voltage Measurement Convoy Micrbammeter in series with A.V.0. Multiplier  25 kV d.c,  pi kV  10J6  -263.0  PINCH AND POST-PINCH STAGES OF THE DISCHARGE It is important to establish a correspondence  between optical observ-  ations and the magnetic probe work conducted by Tam (1967).  Ths correlation  of results from these two different diagnostic methods can provide an otherwise unattainable insight into the action of the discharge. A test for such a relation can be carried out by comparing "pinch times" measured by optical methods with those obtained through magnetic probe investigations.  The pinch time-  is characteristic of the discharge circuit,  geometry, and i n i t i a l conditions of the Z-pinch apparatus.  Identical values  of this parameter would also prove the similarity of the discharge used by Tam with that employed here.  The most accurate optical determination of the  pinch time is achieved by synchronizing the discharge current with the output of a monochromator (Section 3.3).  In order to determine a suitable wave-  length for those measurements, several spectra were recorded by means of a time-resolved spectrograph.  The time-resolved spectrum provides a history  over a wide range of wavelength for visible radiation emanating from the discharge (Section 3.2). Discharge current traces, in conjunction with time-resolved spectrograph^ and monochromator results, also yield information about the postpinch stage of the discharge.  The interpretation of these observations is  contrasted with the work of Zv/icker (Section 3.4).  Section 3.1 outlines the  optical arrangement and experimental procedure for obtaining time-resolved spectra of the Z-pinch discharge.  -273.1  THE TIME-RESOLVED SPECTROGRAPH A schematic diagram of the apparatus is presented in Figure 3.1.  The  instrumental arrangement is based on a design f i r s t proposed by A. Gabriel (i960).  The rotating mirror assembly is a Barr and Stroud rotor type  CP 5911, which is driven by compressed air,  A brass housing with three  replaceable quartz windows was constructed, and a light-tight box containing the trigger beam arrangement was attached to i t .  Two of the windows, at  90° to one another, were used as entrance and exit ports for the plasma light source.  The third window was used for focussing the trigger beam  onto the rotating mirror as shown in Figure 3.1. The optical arrangement for the system is as follows. Light emanating from the discharge tube is focussed on s l i t S^.  The s l i t S^ is focussed via  the concave mirror Mj, the plane mirror Mg, and the rotating mirror M^ on to the s l i t Sg, which is the entrance s l i t of a Hilger medium quartz spectrograph.  S l i t s S^ and Sg restrict the scanned cross-sectional area of plasma  to less than 0.4 cm  2  at the centre of the tube.  0.03 cm. and 20 microns respectively.  They are set at widths of  The trigger beam system consists of  a direct current lamp which reflects from the rotating mirror onto the photomultiplier P^, in such a way that i t causes a sharp pulse output every half-revolution of the mirror. In an experimental run, the condenser bank is f i r s t charged and set ready for f i r i n g .  The rotating mirror is switched on and the pulses from  the photomultiplier are fed into the control unit (Figure 3.1).  When the  mirror reaches the preset speed, the control unit automatically fires the discharge so that an image is cast on s l i t Sg.  The operator then switches  off the mirror, re-charges the condenser bank, and repeats the process to superimpose a second image on the f i r s t .  Figure 3 . 1  Experimental arrangement for the time-resolved spectrograph  - 2* Oscilloscope #1 monitors the j i t t e r in the time interval between the triggering of the discharge and the breakdown of the discharge current. This is done by simultaneously displaying the Rogowski coil output and the pulse emitted by the control circuit (i.e., the pulse of the delay unit, Figure 3.1).  Oscilloscope #2 monitors j i t t e r due to the irregular rotation  of the mirror by exhibiting the f i r s t photomultiplier pulse which is produced after the discharge occurs.  Typical oscillograms are displayed in Figure 3.2.  Figure 3.2a shows a j i t t e r in current breakdown of about 0.2 p.sec.  a) Figure 3.2  b)  Four consecutive discharges in 1,500 ^iHg helium  a) Oscilloscope #1: upper trace—signal from delay unit, 20 v/cm. lower trace—Rogowski c o i l signal, 1 v/cm. time scale = 2 jisec./cm., reading from right to left b) Oscilloscope #2: output from amplifier, 5 v/cm. time scale = 4 jusec./cm., reading from right to left  -30The j i t t e r in the photomultiplier pulse (Figure 3.2b)  is of the order  0 1 p-sec for a half revolution of the mirror over a period of 1 msec o  This  gives rise to an uncertainty of 0.01 psec in registering time-resolved spectra, since the recording of an event does not start until after the f i r s t 100 usee of each mirror sweep (i.e., the delay time between tripping and recording positions of the mirror). The frequency gate can be preset to f i r e at the desired mirror speed by replacing the photomultiplier with a pulse generator and electronic counter.  Optical alignment was made with the aid of a Spectraphysics  Type 130C helium-neon laser. time-resolved  spectrograph.  Table 3-i l i s t s the component parts of the  -31-  Table 3-i Rotating Mirror Assembly Barr and Stroud Rotor  Time-resolved spectrograph  Type CP5911, Ser. No. 1  Optical System Quartz-water achromat Concave mirror S l i t (S-,, Figure 3.1) Spectrograph  Optical VIorks Ltd., London, U.K., T 543 Hilger F. 1497 Hilger, medium quartz  Spectrographic plates  Kodak, Type 1-0, I-F  Frequency Calibration Pulse generator Electronic counter Trigger Beam System Photomultiplier Power supply Control Circuit Frequency gate Delay unit  Hewlett Packard Hewlett Packard  focal length = 32 cm. diameter = 4»5 cm. focal length = 6 feet diameter = 8 inches s l i t width = 0.03 cm. f number ^ 6 s l i t widthsi20 microns  Model 212A Model 521C  R.C.A, Type IP28 Hewlett Packard, H.V. supply Model 6516A  0-3,000 volts 6 mA  Tektronix Type 162 waveform generator Tektronix Type 163 pulse generator same as for frequency gate  Monitoring Equipment Oscilloscope #1 (Figures 3.1, 3.2a) Oscilloscope #2 (Figures 3.1, 3.2b) Rogowski c o i l (Table 2-i)  Type 551, Dual-Beam, Tektronix Type 545A, Single Beam, Tektronix  -323.2: TIME-RESOLVED SPECTRA OF THE PINCH A wavelength must be chosen that is suitable for monochromator measurements of the pinch time.  Hence, a survey of radiation behaviour  i n time must f i r s t be conducted over the visible spectrum. These experimental, requirements are f u l f i l l e d by the time-resolved spectrograph. The plasma column contracts to i t s minimum width, or pinch stage, in the axial region of the Z-pinch discharge. intensity of plasma radiation occurs.  It i s here that the greatest  Even so, a superposition of at least  ten discharges is found necessary to form a satisfactory time-resolved spectrographic plate.  Therefore, only the "pinch" and "post-p5.nch" stages  of the discharge can be analyzed by this method. Figure 3.3 shows a series of time-resolved spectra taken end-on along the axis of the discharge vessel at different f i l l i n g pressures.  The be-  ginning of the time scale coincides with the i n i t i a l appearance of continuum at the bottom of each plate.  For low f i l l i n g pressures (250 to 1,000p.Eg)  s  a break (x) i n the f i r s t continuum (i) is noticeable. In this same pressure region secondary continua (II and III) appear, which gradually decrease in intensity and f i n a l l y disappear for the 2,000 jiHg f i l l i n g pressure.  The  persistence., self absorption, and broadening of the Hel lines (6678, 5876, 4470, and 3889 A) offer a striking contrast to the behaviour of the Hell 4686 A* line over the duration, of the discharge and for increasing f i l l i n g pressures.  A decrease in the intensity and number of impurity  lines i s also evident as the f i l l i n g pressure is increased.  - 33 -  b)  500  uHg  filling  pressure  d)  1,000  ^.Hg  filling  pressure  e)  !,500  uHg  filling  pressure  f)  2,000 uHg f i l l i n g  pressure  g) D i s c h a r g e w i t h m i r r o r i n s t a t i c F i g u r e 3.3  II—second  continuum  position  Time-resolved  s p e c t r a i n helium  - if •  Figure 3.4 a) b) c) d)  Time-integrated spectra i n helium  250 uHg f i l l i n g pressure i n helium (I-£ plate) 500 uHg f i l l i n g pressurt i n helium (1-0 plate) 750 ;iHg f i l l i n g pressure i n helium (1-0 plate) 1,000 jiHg f i l l i n g pressure i n helium (I-F plate) (Calibrated against iron arc spectrum)  6678  Hel 5876  -35-  An estimate of the pinch time can be made from the geometry, mirror speed, and discharge current characteristics employed in recording the spectra of Figure 3.3.  These measurements give values of 4 to 1 jxsec. for  the 500 and 2,000 jiRg cases respectively. A much more accurate method would be to use a photomultiplier with time resolution better than the limit of 0.25 psec. for the rotating mirror spectrograph (Figure 3.3g). Its output can be correlated easily with the discharge current, thus avoiding experimental errors in the determination of mirror speed and position (Section 3 . 3 ) .  -  It is desirable to measure the intensity and time of occurrence for secondary continua, as well as the pinch time which can be characterized by the f i r s t appearance of radiation continuum at the axis. For this purpose a monochromator, set at a chosen continuum wavelength, is used in conjunction with the photomultiplier. A careful examination of the spectra in Figure 3.3 shows that a wavelength in the 46OO A region is a good choice for monitoring the on-axis continua.  The photomultiplier sensitivity is  high and no impurity lines of consequence are encountered (Appendix III) Spectral lines appearing in the pinch column can be identified by comparing the helium discharge spectrum with that of an iron arc. Figure 3.4 displays records obtained by the spectrograph without time resolution.  They are referred to as "time-integrated" spectra.  The  value of time-resolved spectroscopy is amply demonstrated by comparing them with Figure 3.3.  In the time-integrated spectra, the secondary continua  are superimposed and there is no hint of self absorption in the Hel lines.. Figures 3.4a to 3.4c were recorded on Kodak type 1-0 spectrographie plates, while a type I-F was used in Figure 3.4d. Spectral lines of wavelength  -36-  higher than Hell 4686 appear on the last plate only because of the different sensitivity for the two plate types.  A l l spectra i n Figure 3.3 were re-  corded on Kodak type I-F plates. Figure 3.3g shows the discharge spectrum with the mirror in the static position.  This gives an estimate of 0.25/isec. for the limit i n time-  resolution by the rotating mirror method, for no j i t t e r in breakdown or recording.  -373.3  PINCH TIMES The "pinch time" is an easily measurable characteristic of the Z-pinch  discharge.  It is defined as the time interval between the initiation of the  discharge and the on-axis arrival of the inner current shell boundary. This parameter can be observed by different diagnostic techniques and is therefore valuable in relating magnetic probe results with optical measurements. The sudden appearance of continuum radiation at the axis of the discharge (Figure 3.3) indicates the presence of material which becomes highly excited in this region.  This phenomenon can be used to measure the  pinch time, since i t signifies the beginning of the "pinch" stage of the discharge.  It is found convenient to use the 4.64 0 A wavelength in monitoring  continuum radiation from the axial region of the discharge (Appendix III), This allows the•formation of subsequent continua to be timed as well. A monochromator is focussed along the axis of the vessel and i t s output is displayed simultaneously with the discharge current on a double beam oscilloscope (Figure 3.5).  The termination of spark gap noise from  the trigger switch is considered as the time at which initiation of the discharge current occurs, since a backwards extrapolation of the discharge current trace gives the current I = 0 at this point (Figure 3.5).  Such a  situation is expected because the onset of current changes the impedance of the system and thus, suppresses the spark gap noise.  Hence, the "formation"  time of the discharge, as well as the occurrence of on-axis continuum, may be read to give the pinch time as defined above (T  in Figure 3.5).  I  1  ,  1  1  1  j  H  0 12 3 4 5 6 7 t (pec.) Figure 3.5  >  Determination of pinch time T helium discharge at 2,000 jaHg f i l l i n g pressure.  Pinch times observed by Tarn (1967) are shown in column (2) of Table 3 - i i belovj.  They are taken from "current collapse shells" in Chapter 4 and  represent the time of arrival on axis for the inner boundary of the current shell.  The appearance times for on-axis continuum (Tp, Figure 3.5) are  shown i n column (3). Table 3 - i l Pinch times (3) (1) (2) I n i t i a l pressure "Collapse times" 0n-axis continuum Tarn "(1967) detectable at in helium (usee) (usee) (juHg) 500 1,000 2,000 4,000  3.5 4.5 5.5—6 6.5—7  Tarn's results are accurate to within 0.5jusec.  3.5 4.7 5.4 7 The optically determined  pinch times i n column (3) agree with Tarn's measurements for the same i n i t i a l f i l l i n g pressures. The experimental error between the two methods is approximately 10$.  This agreement in pinch times forms the basis for relating Tarn's 'magnetic probe measurements with a detailed, optical study of the collapsing shell.  A different diagnostic device must be used to study radiation  behaviour for this preliminary phase of the discharge.  However, spectro-  graph^ and monochromator records obtained thus far contain additional information about the pinch and post-pinch behaviour of the discharge column. For this reason, a further correlation of optical and magnetic probe results is deferred u n t i l Chapter 4.  -/ o~ +  3.4  BEHAVIOUR OF THE DISCHARGE AT PINCH AND POST-PINCH TIMES Synchronization of time-resolved radiation records with the discharge  current reveals the action of the discharge i n the pinch and post-pinch regimes.  In Figures 3.6 to 3.9, the discharge current is compared with  composite radiation traces which are built up from time-resolved monochromator and spectrographic records (Appendix III). A break i n the f i r s t continuum is observable for pinches i n the 250 to 1,000 uHg f i l l i n g pressure region (Figures 3.3, 3.6, and 3.7).  This  break, or "dark line", occurs because the discharge axis does not correspond exactly with that of the vessel.  A calculation verifying this conclusion is  made in Section 4.5 on the basis of information gained from the collapse stage of the discharge.  The appearance of the f i r s t continuum coincides  with the on-axis arrival of the inner current shell boundary (Section 3.3). For this reason, i t w i l l be referred to as the "pinch continuum". The occurrence of secondary continua in the 250 to 1,500 yuHg f i l l i n g pressure region (Figures 3.3a to 3.3e, and 3.6, 3.7) can be most readily explained by a second collapse of the current shell.  Framing camera pictures  in Chapter 4 show no evidence of additional collapse shells following the one responsible for the pinch continuum. The trapping probability of gas within the collapsing shell is very nearly 1 (Tarn, 1967). This means that very l i t t l e gas would be left free to form secondary collapse shells after the passage of the f i r s t current shell.  Furthermore, the plasma column  remains constricted in the axial region of the discharge for several microseconds after the f i r s t pinch (Chapter 4, Figures 4.12 to 4.15). during this stage that the secondary continua appear.  It is  Kenee, these continua  -41can arise only from an interaction within the expanding gas column after the pinch. The mechanism for the second collapse occurs in the following way.  The  kinetic energy of radial motion is thermalized during the f i r s t contraction responsible for the "pinch continuum". This causes an enormous pressure increase within the pinch column sufficient to overcome the confining magnetic pressure.  The column therefore, expands reducing the brightness  of the continuum. Because of the high density, the gas cools rapidly by radiation.  This is borne out by the onset of self-absorption in the HeI  lines and the formation of Hell lines in the latter half of the pinch continuum (Figure 3.3).  Hence, the kinetic pressure is greatly reduced,  but the electric current s t i l l exerts a large enough radial pressure to cause another collapse. The disappearance of secondary continua as the f i l l i n g pressure is increased can be understood by relating current discharge behaviour to the formation of the pinch.  In Figures 3.6 to 3.9, the current always goes  through a minimum when the pinch continuum appears and then rises again to a maximum. For low f i l l i n g pressures, this maximum is the highest possible value the discharge current can reach (Figures 3.6, 3.7).  At higher i n i t i a l  pressures, the pinch is not formed until the discharge current is already waning. Only a small, relative maximum can be attained afterwards by the discharge current (Figures 3.8, 3.9).  Hence, magnetic pressure generated  by the current is not sufficient to recompress the plasma column.  Composite trace of on-axis continuum, 1,000  •at  t (p.sec.)  > (relative time) Figure 3,8 Composite trace of on-axis continuum, 2,000 uI-L  -46Section 3.3 and supporting evidence in Chapter 4 imply that the f i r s t minimum in the discharge current appears very close to the time the contracting shell reaches i t s minimum diameter.  It i s well known that a sharp  discontinuity occurs i n the current waveform at the time of pinch formation (Zwicker, 1964a), brief.  The time interval for maximum contraction can be very  This would explain the sharpness i n the current minima.  A second minimum i n the discharge current trace occurs for the 500 and 1,000 uHg cases (Figures 3.6 and 3 . 7 ) .  The second minimum for the  1,000 uHg case i s more in the nature of an "inflexion point", but i t i s distinguishable nevertheless.  These second minima are reproducible from  shot to shot under the same i n i t i a l conditions.  Furthermore, they occur  just about the time of appearance for the second continuum i n the 500 >iHg case and for tho third continuum i n the 1,000 jiRg  i n i t i a l pressure case.  Figure 3.3 shows significant broadening in the helium line before and after the secondary continua (II, III). during the third continuum.  Indeed, the Hell 4686 line is widest  Since the line width increases with electron  density, the existence of secondary continua can be ascribed definitely to a second contraction of the plasma cylinder. Zwicker (1964 a,b) has conducted experiments similar to those described in this chapter. His observations were made at right angles to the axis of the Z-pinch discharge i n helium. The range of i n i t i a l pressures was comparable, although the discharge parameters were different from those used  -47-  here.  He also observed secondary on-axis continua and related the f i r s t  "pinch" continuum to the kink (or minimum) in his Rogowski c o i l trace. Zwicker links the appearance of secondary continua to the onset of instabilities in the on-axis plasma column. The argument is that local instabilities occur along the axis of the discharge and, therefore, carry continuum across the diameter of the pinch column. As evidence of this, he cites the irreproducibility of the secondary continua from shot to shot for the same i n i t i a l conditions. In this thesis, observations of the pinch were taken end-on, along the axis of the discharge. Figure 3.3 shows distinct separations between the secondary continua. In Appendix III, Figures 1 and 2 show a possible j i t t e r of 0.3 usee, in the monochromator traces where secondary continua appear. Such a jitter seems too small for the secondary continua to be generated solely by instabilities.  Although he does not state the magnitude of  variation in time for the occurrence of secondary continua, Zwicker is quite emphatic that i t is large from shot to shot for the same i n i t i a l conditions. A comparison of Zwicker's (1964a) and our discharge parameters shows that: i) Zwicker's current is much greater and its ringing period is much shorter than for this thesis: i.e., Maximum discharge current 380 ka vs. 280 IcA observed here; Discharge ringing period 14 jusec. vs. 36 fxsec. observed here. i i ) the change in amplitude of the discharge current, between its maximum value and the minimum attained at the f i r s t pinch, is greater in Zwicker's experiments than here: i.e., 190 Id for 330 uHg vs. 19 kA for 500 ;.iHg observed here; 95 kA for 1,000 uHg vs. 29 kA for 1,000/iHg observed here.  -48This suggests that changes in the conditions responsible for holding the plasma column together after the pinch vary much more violently in Zwicker's case than for the discharges studied in this thesis.  Figure 3.3  shows further evidence of the comparative stability of the discharge studied here in contrast to that used by Zwicker.  The highest f i l l i n g pressure at  which secondary continua occur is 1 , 5 0 0 uHg.  Moreover, for the 10 jasec.  interval covered by the plate, only one such additional continuum appears at this pressure.  Zwicker shows two secondary continua for 2,200 p.llg i n  helium, and at least one appears .as high as 3,000 jiBg (Zwicker, 1964a). The occurrence of a second minimum in the discharge current traces, the broadening of helium lines and the small jitter-time for the secondary continua, point to a second collapse of the pinch column. This j i t t e r of 0.3 usee, can be caused by Rayleigh-Taylor instabilities as Zwicker suggests.  Although instabilities are not primarily responsible for the  formation of secondary continua, they can affect the reproducibility in their appearance on axis. Zwicker's interpretation may be valid for his system, since i t is much faster and therefore, more unstable.  It is f e l t , however, that his secondary  continua disappear with increasing f i l l i n g pressure for the same reason as that given here—namely, the diminution of the discharge current at the pinch. Qualitative agreement has been established with Zwicker's observations of the post-pinch regime for the Z-pinch discharge.  On the other hand, the  similarity of discharge systems demonstrated i n Section 3.3 indicates that magnetic probe work done previously in this laboratory can be linked to our optical studies. Hence, further optical investigations .in the discharge stages observed by Tam can provide a general picture of how magnetic and current distributions act to produce the Z-pinch.  -49-  4.0  KINEMATICS OF THE Z-PINCH DISCHARGE "Kinematics" describes the motion of the plamsa as a whole, without  reference to the force or mass. Information can be obtained about the motion and macrostructure of the contracting plasma cylinder, from the time of i t s formation until i t s culmination in the pinch, by photographing successive stages of the discharge during the collapse regime.  The high  speed framing camera is preferable to a "smear" camera or image converter because of spatial resolution over a wider f i e l d of view, the larger number of film images available and the relative ease of interpreting the resulting photographs. The action of the framing camera is based on the same principle used in time-resolved spectroscopy (Section 4.1).  A similar superposition is  followed in obtaining useable records (Section 4.2).  Only a simple correction  for perspective is needed to determine the true position and thickness of the hollow collapsing cylinder of plasma (Section 4.3). Spatial distributions of Hel and Hell can be studied by taking framing camera photographs through suitable interference f i l t e r s (Section 4.4). By synchronizing these results with other discharge parameters, the collapse stage can then be related to the post-pinch regime discussed in Chapter 3 (Section 4.5).  A comparison with measurements performed by Tarn (1967) can  also be carried out to determine the connection between current distributions and radiation regions within the collapsing shell (Section 4.6),  This  provides an understanding of the action of the Z-pinch discharge which corresponds well with the Jukes-Allen theory for fast-pulsed discharges.  -504.1  THE HIGH-SPEED FRAMING CAMERA A Barr and Stroud Model CP5 framing camera is used, i n conjunction with.  interference f i l t e r s , ' t o photograph the collapsing discharge column of the Z-pinch.  In this camera the film remains stationary and a rotating mirror  is used to pass light from the discharge through a series of fixed lenses (Figure 4.1). The triggering process i s identical to that of the time-resolved spectrograph (Section 3.1).  A small light source (not shown in Figure 4.1)  is mounted inside the camera and focussed on the rotating mirror.  At a  certain position, the reflection is recorded by the photomultiplier every time the mirror completes a f u l l half-turn. For  a rotational speed of 5.5 kc/sec. the manufacturers give:  t-j = time interval for image beam to travel from f i r s t to last lens. =7.5 usee. tg = time period from trigger position of photomultiplier #1 to the f i r s t lens position.  = 3.4 ^J.sec.  For  the pinch times observed in helium, i t was found convenient to  operate the framing camera at a speed of 2 kc/sec. This makes t^ = 20.6 usee. t  For  60  2  = 9.4 ysecc  60 lenses at 2,000 c.p.s., the time between images on the film = r  A slight innovation was introduced to enable one person to operate the entire system.  Contact switches were placed on the lever controlling the  -51-  compressed a i r input to the rotating mirror assembly.  This allowed the  •operator to turn on the trigger beam, throw open the a i r valve and then open the camera shutter in one motion. By recording the discharge current and the appropriate output signals from the control system, events recorded by the framing camera and the timeresolved spectrograph can be correlated through the use of a monochromator. The oscillograms monitoring the mechanical and electronic j i t t e r are the same i n every respect as those of Figures 3.2 a,b for the rotating mirror spectrograph.  4.2  DATA REDUCTION The magnetic f i e l d and current distributions in a helium Z-pinch have  been established by Tam 4000 uHg.  ( 1 9 6 7 ) . f o r f i l l i n g pressures of 500, 1000, 2000, and  Since the system used in this thesis is identical to Tarn's in i t s  important features, i t was decided to investigate the discharge at the same pressures.  Spectroscopic observations in Chapter 3 indicate the plasma to  be almost pure helium before the f i r s t pinch occurs., particularly for higher i n i t i a l pressures.  The spatial distributions of Hel and Hell can therefore  be established by photographing the discharge end-on through suitable interference f i l t e r s .  This provides information which can be compared with  Tain's data.  A l l photographs are made on Ilford HP3 film developed for 10 minutes in Ilford developer, number ID2.  From i t s response curve (Figure 4.2) and an  examination of helium lines dominant in the plasma (Figure 3.3), only the Hell 4686 fi and the Hel 4470, 5876 fi 1 incs w i l l register successfully on the film.  Figure 4.3 shows the transmission curves for interference f i l t e r s  used in photographing the light emitted by these three helium lines. Table 4-i gives the experimental conditions used in obtaining the photographs.  The term "no f i l t e r " in the table means that no interference  f i l t e r s were employed. Such a record was made at tho beginning and end of each experimental run as an additional check that no error in timing occurred Figure 4.4 shows a typical sequence of framing camera photographs. In order to correlate the photographs with current distributions measured by Tam ( 1 9 6 7 ) , the dimensions of the luminous shell must bs measured.  To  determine the extent of the luminous regions in the plasma, half the f i e l d  -53-  3£oo  'taoo  6O90  Sooo  6590  Figure 4.2 Colour sensitivity of Ilford HP3 film  Table 4 - i  Order of observation  Type of filter  1 2 3  No f i l t e r Hell 4686 Hel 44.72 Hel 5876 No f i l t e r  4  5  Observational sequence and conditions for framing camera photographs  No, of shots per film strip in helium 500 pHg  1,000 jiHg  2,000 uHg  4,000 uHg  4  1 20 10 10 1  1 20 10 10 1  1 30 15 15 1  30 15 15  4  of view is masked by a "neutral density" f i l t e r of known transmission (Table 4 - i i ) .  For the purposes of this thesis, the boundaries of the  luminous shell are then determined by the radii of the shell at which the intensity f a l l s to about 3 5 $ of its peak value. Figures 4 . 5 a,b show how the dimensions of the luminous shell are determined by measuring densities of the photographic image with a J a r r e l Ash microphotometer.  The typical structure of the microphotometer records  appears in Figure 4 . 5 c. The most important features are a luminous shell which collapses onto the discharge axis, and a luminous spot appearing at the  centre of the discharge.  Table 4 - i i  Transmission of the "neutral density" f i l t e r used in determining boundaries of the collapsing shell  Wavelength  Transmitted radiation Incident radiation  . Hel 4472 Hell 4686 Hel 5876  0.34 0.37 0.48  -5-5"-  —56—  as seen through "neutral density"  c) Spatial].;/ resolved parameters r — r a d i u s o f axial spot a  ry - i n n e r radius o f luminous plasma shell  Tp-radius o f peak intensity r - o u t e r radius of luminous plasma shell RQ— radius of discharge vessel 0  Analysis of framing camera images  4.3  CORRECTIONS FOR PERSPECTIVE In order to ascertain the true dimensions of the plasma shell a  correction must be applied to the microphotometer traces (Figure 4.5 c). Images recorded on film are affected by the length of the plasma column, as demonstrated in Figure 4 . 6 . The procedure is to treat the inner radius (r^) of the film image as light coming from the back of the discharge tube, and the outer radius ( r ) 0  as radiation from i t s front end.  The radiation peak (rp, Figure 4.5 c) is  regarded as emanating from halfway down the tube.  The later frames of the  discharge in Figure 4.4 clearly show the high tension, or rear electrode of the discharge, tube.  Hence, the effect of perspective can be corrected by  measuring the image and true cross-sections of the electrodes at both ends of the discharge. If R  r  = the radius of the rear electrode image (Figure 4 . 6 ) ,  R^ - the radius of the front electrode image (Figure 4.6) which corresponds to R of Figure 4.5 c, Q  and Rj = the true dimension of the electrode radius, then the true r^ = image of r^ on film x ^T the true r = image of r on film x %• Q  Q  The optical system of the framing camera experiment is such that a linear interpolation between the two electrode image sizes (R^, R ) is sufficient to place the position of peak intensity: the true r -c^^image of r on film P  P  x  2  %  + R  r  -58-  The soundness of the above assumptions for perspective is shown for the 500 uHg case where there is excellent correspondence with the current collapse shell in Figure 4.16. The close correlation between luminous and current shells is discussed in Section 4.6.  Figure 4.6 Effect of perspective on film image.  -59-  4.4  PEAK RADIATION AS A FUNCTION OF TIME Before proceeding to a f i n a l synthesis of photographic data, the  accuracy in recording the discharge can be checked by examining the behaviour of the peak radiation front within the collapsing shell.  The radius of the  radiation peak (rp, Figure 4-.5c) was measured in each image and plotted against the number of the frame. such graphs exhibit r  Since the frame interval is 0.34 usee,  as a function of time (Figure 4.7 to 4.10).  A striking feature appears in these graphs as the f i l l i n g pressure is increased: i) At low pressures (500 and 1,000 uHg), the radiation peaks observed through Hel and Hell f i l t e r s coincide with the peak observed when no f i l t e r is used at a l l . i i ) At the higher pressures (2,000 and 4,000. uHg), the radiation peak observed through Hel f i l t e r s coincides with the peak observed when o no f i l t e r is used.  However, the peak of the Hell 4686 A radiation  consistently follows behind the Hel radiation peak. Only one discharge is required to obtain a usable photograph when no f i l t e r is used.  At least ten or fifteen shots are needed to produce a viable  photograph through the Hel f i l t e r s (Table 4 - i ) .  The high degree of overlap  for radiation peaks photographed with and without Hel f i l t e r s demonstrates the smallness of errors in timing. The recording process is therefore accurate to 0.34  ec, at least (i.e., the duration of the interval between  framing camera images).  -60Because timing errors are so small, the spatial separation of the Hel and Hell peaks at higher pressures must be regarded as significant. It can be explained by noting that gas in the outer boundaries of the luminous shell has been heated for a longer time than the gas on the inner edge. Furthermore, Joule heating i s more efficient in the neighbourhood of the Hell peak since i t has been established that the current density increases towards the outer edge of the collapsing shell (Tarn, 1967).  -61-  -63-  Figure 4.9 Radius o f r a d i a t i o n peaks corrected f o r perspective vs. time, He 2,000 juHg 0 — H e l l 4686 combination o f no f i l t e r , Hel 5876, Hel 4470  -654.5  SYNCHRONIZATION OF FRAMING CAMERA PHOTOGRAPHS WITH OTHER DISCHARGE PARAMETERS The reproducibility in creating and recording the Z-pinch discharge  is now established. In order to make maximum use of information provided by the framing camera records, i t is desirable to correlate them with other discharge parameters. By synchronizing each set of film images with the total discharge current, a connection can be made between the collapse phase and the pinch stage explored in Chapter 3. A direct relation between the discharge current and film image can be determined through the use of a monochromator. This instrument is placed behind a small window in the end electrode of the discharge tube. The distance of the window from the axis of the discharge is set at 1 inch or 2 inches.  (In reality these distances are 3.18 cm. and 5.08 cm. re-  spectively since the axis of the discharge does not coincide exactly with that of the vessel.)  The monochromator is set on the Hell 4686 §. line.  Its output is fed into a double beam oscilloscope and displayed simultaneously with the discharge current (Figure 4.11). Synchronization is achieved by noting that the framing camera image showing Hell peak radiation at a radius of 1 inch ( or 2 inches) must occur  when the maximum in the  f i r s t monochromator pulse registers on the oscillogram. The position of this point in time can then be read directly from the oscillogram with respect to the initiation of the discharge.  -66 spark gap no is G  •^discharge current f i r s t peak in monochromator t monochromator trace for "•"Hell 46S6 X at 1" (2») from discharge axis  Time ( u s e e ) — > o  / i  s  * s *e  i—1—«•.  i ? g~<? /° // a 30  35  ® ©0  Figure.4-,11  O  Hell 4686 X film strip  . Synchronization of framing camera image; Hell discharge radiationcurrent peak at radius of 1" with the total  The graphs in Figures 4.12  (2")  to 4.15 were plotted in this manner. They  w i l l be referred to henceforward as "luminosity graphs" to distinguish them from the current collapse graphs obtained by the magnetic probe method (Tarn, 1967). They are composite images built up from the film strips described in Table 4-i (Section 4.2). and "Hel" film strips.  The inner radius is that of the "no-filter"  The outer radius of the luminosity shells is that  of the "no-filter" and "Hell." film strips.  The peak radiation position is  that belonging to the peak radiation radii of the "Hell" film strips. Because of low light levels in the 500 ^iHg case, only the "no-filter" film strip was used.  This is the reason no bars representing the spread in  observation points are shown on the inner and outer boundaries of the luminosity curve of Figure  4.12.  -67The on-sxis continua discussed i n Section 3.4 are included in the luminosity graphs as well. against time.  The photomultiplier voltage is displayed  It can be seen that intense continuum radiation appears in  the axial region of the discharge only when the collapsing luminous shell approaches i t s minimum diameter.  Just one collapse shell is evident before  the appearance of on-axis continuum. This and the persistently small crosssection of the plasma column during the post-pinch regime support the conclusion that secondary continua occur because of an additional collapse within the pinch column (Section 3.4). The sequence of framing camera images in Figure 4.4 clearly shows an "axial spot" which appears i n association with appreciable radiation before the luminous shell is truly formed. The outer radius of this radiant spot i s drawn in the luminosity graphs (Figures 4.12 to 4,15).  The  luminosity curves indicate what seems to be a partial collapse before the main trigger switch is completely conducting.  These effects are not evident  on Tarn's collapse curves (Figures 4.16 to 4.19) since his observations are triggered by the rise i n discharge current after the i n i t i a l transient has died away. The monochromator response to "axial spot" radiation can be estimated from framing camera pictures.  Assuming i t s radiation to be continuum-like  for the 2,000 and 4*000 jiHg i n i t i a l pressure cases, i t s photomultiplier signal would be about 1/30 of the signal produced by the pinch continuum. Since the monitoring oscilloscopes were adjusted to measure pinch continuum intensities, there would be insufficient gain for the detection of the axial spot.  -68Further information is necessary to determine the nature of axial spot radiation.  Observations might be conducted in a direction perpendicular to  the discharge axis and close to the high-tension electrode. It may then be possible to decide whether the axial spot can be regarded simply as an "electrode burn", or i f i t is evidence of at least a partial column of luminous plasma appearing before the main pinch takes place. The accuracy in relating the collapse stage observations to the pinch and post-pinch regime can be demonstrated by considering the "dark line" in the f i r s t continuum for low i n i t i a l pressure discharges (Figures 3.3, and 3.7).  3.6,  This break in the pinch continuum can be explained by the fact  that the discharge axis is slightly eccentric with respect to the axial viewing hole.  For the 500 uHg f i l l i n g pressure case in Figure 4.12, the  velocity of the outer boundary of the luminous shell can be evaluated as i t hits the axis: = 3.33 cm/usec. The pinch continuum f i r s t appears at 7.43 usee, on the luminosity graph and the "dark line" occurs 0.85 usee, later.  If the break in the f i r s t  continuum is truly caused by eccentricity in the viewing hole, then a l l of the luminous shell w i l l have passed this observation point i n O.SSjisoc, i.e., the predicted thickness of the luminous shell at 7.43 ^usec.  _ ~ °"  85  x  j  3  j  c m  ° " ° 2  8 3  cra  '  Since continuum appears at 7.43 /asec. in. Figure 4.12, the inner radius of the total current shell is zero. 7.43 psec.  Therefore, the thickness of the shell at  can be read from the luminosity graph as the  outer radius of the  luminous region. The value obtained is 2.83 cm., v/hich is identical to that deduced above.  -69.  This calculation shows that the "axial" viewing port is not located on the axis of the discharge.  In fact, a burn on the electrode caused by the  pinch is found 0.25 inches away from the geometrical centre of the hightension electrode.  The agreement between predicted and observed values of  shell thickness attests the accuracy in timing for the observational methods used. The break in the f i r s t continuum caused by the eccentric viewing port shows that the diameter of the pinch must be less than 0.5 inches in the 500 jiHg case.  In higher i n i t i a l pressure plasmas (2,000 to 4*000 }iBg)  s  this "dark line" is not observed since more material is pushed onto the axis, making the pinch column larger in diameter.  discharge current  ,  r17 discharge current minimum at 16 .45 usee. 15  O •f & Q  outer radius of luminosity shell position of radiation peak inner radius outer radius of "axial spot"  ."dark line"  CD  o "on-axis" continuum 9 ~7 (intensity vs. time)  2 H* Ui  Time (usee.) Figure 4.12  Luminosity curve, 500 jxUg  F i g u r e 4.13  L u m i n o s i t y c u r v e , 1,000  pEg  •20  f i r s t discharge current >> maximum at 7./+ usee.  •19  ar -o—  •hs •17 •16 o  •15 at 11.65  a p.  discharge current minimum jxsec.  •  u  •13  outer radius of luminosity shell position of radiation peak A inner radius CE3 outer radius of "axial spot"  I 3  ©  Spark gap noise ends at t = 3.56  1  "on-axis" continuum (intensity vs. time)  0  /S  b 3  /V  a'oj a> 7 /  Frame Time (usee.)  p  G  G  *v 8 ^  i< akj 9  -11  -10  jisec.  P  ••12  z'r  AS  A? *0  Figure 4 . 1 4  ?/  5£ S 11  35  jsr 36  12  3r  Jig  39  13  Luminosity curve, 2 , 0 0 0 pRg  7o~p/ 14  «  J 4v  15  ^6 j  16  5  First maximum a t 8 . 1 5 pxsec  discharge current -•22  O •jr A •  S p a r k gap noise ends a t t =" 2.57 p e e .  outer radius of luminosity .shell position of radiation peak inner peak outer radius of "axial spot"  ••21 •-20  19-  •^17| f i r s t discharge current minimum at 14.18 jisec.  4-  14  3-  •-13 12 o -11  r * *. * ? * *  1 -  i  0  '?  V  -.1/  -frame ?? Tirce ( p s e c j '  21  -10  5 -'~ J  r  -9  i  -r-  13\ ZV  1 r 2ST Z6 37  ??|  so  lb  ^igure 4 . 1 5  5/  3 k  is  i—i 3S~  1;  i— 3 C S T  3 3  3 ?  3  L u m i n o s i t y c u r v e , 4,000 uHg  V a  .  4<S |  Y z V S  14  15  16  -74-  4.6  STRUCTURE OF THE Z-PINCH DISCHARGE The similarity in pinch times, measured by magnetic probe and optical  methods, forms the basis for relating current density distributions obtained by Tam (1967) to the luminosity regions observed in the collapse of the plasma shell.  Figures 4.16 to 4.19 display the current shells measured by  Tam for 500, 1,000, 2,000 and 4,000 uHg f i l l i n g pressures i n helium. The boundaries of the current shell are determined by the radii at which the current density f a l l s to 50$ of i t s maximum value. Tam shows that most of the mass i s contained within these boundariesby employing a modified snowplow equation: d at  1  oC (R - r ) _dr_ 2  2  dt J  A>  fo  f 'n'2 (r'B * - r i n  It i s assumed that the inner surface of the shell moves with the same velocity as that of the center of mass across the layer, where,  r Cr') = radius of the inner (outer) surface of the current shell R = radius of the inner wall of the discharge vessel B (B*)  ~ measured magnetic flux density at inner (outer) surface  f<> = i n i t i a l density of gas within the current.' shell at the beginning of the discharge <?( = trapping factor showing the fraction of gas retained by the collapsing shell. Numerical solutions of the above equation were obtained for the collapse curve of the inner current shell boundary and compared with those measured by the gradient probe. Tam determines the best f i t between theory and experiment to be:  -75oC= 0.9, for helium at 1, 2, 4 mmlig. i n i t i a l pressure cC - 0.75, for helium at 500 uHg. i n i t i a l pressure. Therefore, the boundaries of the current shell in Figures 4.16 to 4.19 contain almost a l l the material swept up by the collapsing current. The luminosity shells of Figures 4.12 to 4.15 are represented by the cross-hatched areas in the current shell collapse graphs of Figures 4.16 to 4.19.  An important result of correlating the luminosity and current  collapse shells is that the Hell peak radiation region coincides with Tarn's observation points for peak current density.  This holds true for a l l cases.  The luminosity shell can therefore be identified with the main currentbearing region of the collapsing plasma column. Although the boundaries of the luminosity shell are defined by a drop to 35$ of the peak intensity, its width corresponds well with the dimensions of the current shell in the 500 and 1,000 jiHg. i n i t i a l pressure cases (Figures 4.16 and 4.17), From the considerations of the previous paragraph this means that almost a l l the moving gas is visible in the framing camera pictures. For higher f i l l i n g pressures (2,000 and 4,000 uHg.) no longer holds true.  such a situation  Figures 4.18 and 4.19 show that a non-luminous,  conducting body of gas precedes the radiating region of the current shell. It is the on-axis arrival of this relatively cold gas which is responsible for the appearance of continuum in the axial region. In the 4*000 j.iHg. case, an examination of Tarn's current density profiles shows the formation of a subsidiary current shell in front of the radiating region.  Its peak value is i n i t i a l l y less than 50$ of the maximum current  density dn the luminous region of the collapsing plasma column.  Its inner'  Current- shell collapse magnetic prob  ethod  0 outer shell radius +- current peak position A inner shell radius position of minor current peak inner radius of minor current density profile  beginning of axial continuum  -79boundary is sketched in Figure 4.19 and represents the radius at which the current density in this "minor" current shell f a l l s to half i t s peak value. Hence, the apparent "splitting" of the inner current shell boundary i n the 4,000 juHg. f i l l i n g pressure case. In Figures 4.18 and 4.19 the luminosity shell containing most of the discharge current is slowed down.  It is brought to rest before i t reaches  the axis for the 4,000 /aHg. f i l l i n g pressure case.  This is due to pressure  exerted by material carried onto the discharge axis in front of the luminous region.  Tarn's (1967) current density profiles for 4,000 uHg f i l l i n g pressure  show the beginning of a reflection for the "minor" current shell after i t reaches the axis. This explains the relatively poor agreement in pinch times (tp) when he used the snowplow model (tp ^ served current collapse {t ~  8 yusec.) to f i t his ob-  7 usee.) for this case.  The model assumes  negligible kinetic pressure outside the main current shell. The slowing down of the current bearing layer can be explained qualitatively by the "Jukes-Allen" model for a fast-pinched discharge. The theory for this model (Jukes, 1958) is outlined in Section 6.2 in connection with Guderley flow.  The action of the collapsing plasma column can best be  understood by assuming that most of the current resides within a narrow shell around the radiation (current) peak. A cold shock front precedes the current layer.  The kinetic energy of this shock is thermaliaed upon hitting the  discharge axis. Material carried by i t is heated and expands, exerting pressure on the incoming current layer. This is known as the "throttling process": the reason the current layer can never reach the axis (Figure 4„20)„ This phenomenon is well illustrated in the correlated lurainosity and current shells of Figures 4.18 and 4.19.  -80Tube wall  r(t)  Tube axis, r = 0  t  Figure 4.20 Schematic diagram of Jukes's model for Z-pinch contraction The Jukes-Allen theory is valuable in that i t predicts the behaviour of the forerunning shock front independently of the current layer. That i t does not hold true for the low i n i t i a l pressure cases ( 5 0 0 and 1,000/uHg) w i l l be demonstrated more f u l l y in Section 6 . 2 .  However, Figures 4 . 1 6 and  4.18 show that the luminous region coincides almost exactly with the current shell and that, therefore, a firm demarcation between shock and current layers is not possible. For low i n i t i a l pressures ( 5 0 0 to 2 , 0 0 0 /aHg) the snowplow model seems adequate. For higher i n i t i a l pressures ( 2 , 0 0 0 pBg  and  above) the Jukes-Allen theory must be invoked to account for significant kinetic pressure outside the main collapsing shell. The above discussion of-discharge action helps to explain qualitatively the occurrence of the "dark line" in the pinch continuum for lew pressures (Figures 4 . 1 2 , , 4.13).  initial  If the viewing port is eccentric with respect  to the discharge axis, then the continuum appearing before the "dark line" can be identified with material rushing onto the axis. A l l continuum after  -81-  this break arises from excited material expanding outwards and away from the axis of the discharge. The following collapse stage characteristics of the Z-pinch discharge emerge from observations discussed in this and previous sections: i) Appreciable luminosity appears before the current shell is fully formed. This is caused by an i n i t i a l transient current from the triggering apparatus.  It is also possibly responsible for the form-  ation of an inner "axial spot", i i ) The time of appearance for continuum in the axial region of the discharge coincides with the arrival time on the discharge axis of the inner current shell boundary. These "pinch" or "collapse" times can be predicted by the Snowplow model of the Z-pinch discharge at low i n i t i a l f i l l i n g pressures (500 to 2,000/aHg). i i i ) The peak Hell radiation of the collapsing luminosity shell coincides with the peak current region of the corresponding current shell. The luminosity shell can therefore be identified with the main current-carrying region of the collapsing plasma column, iv) At lower f i l l i n g pressures (500 to 1,000/iHg in helium), both current and luminosity shells almost completely coincide. v) At higher f i l l i n g pressures (2,000 to 4,000/aHg in helium), the current shell is slowed down and, in the /+,000 ;.illg case, brought to rest before i t reaches the discharge axis. There is a discernable separation between current and luminosity fronts. In addition, a difference in spatial distributions of Hel and Hell becomes distinguishable within the luminosity shell.  This behaviour can be  described qualitatively by the Jukes-Allen model.  o -82Frora the observations above, the collapsing shell is composed of two regions: 1) an inner, relatively cold shock layer followed by 2) a hotter, luminous shell. In helium pinches generated with low i n i t i a l pressures (500 to 1,000  )iHg)  the "shock" and "luminous" layers almost coincide. Progressing through higher i n i t i a l pressures (2,000 uHg and above), the distinction between these two regions becomes more apparent.  -835.0 DETERMINATION OF DENSITIES AND TEMPERATURES WITHIN THE LUMINOUS PLASMA Measurements made until now have furnished insight only into the spatial distribution of helium and front velocities i n the collapsing current shell. In order to understand the plasma more completely i t i s necessary to know its temperature and density.  Having evaluated these parameters within the  axcited gas during several stages of i t s formation, the consistency of a given dynamical theory can then be checked.  This in turn would help to  explain the basic mechanisms responsible for the action of the Z-pinch. The most highly developed method of determining temperature and density is through the analysis of plasma radiation (Griem, 1964j Cooper, 1966). This chapter is therefore concerned with measuring these parameters only in the highly luminous regions of the plasma.  A model for calculating plasma  densities i n the faintly luminous shock zone of the collapse w i l l be outlined i n Chapter 6. The difficulties of radiation calibration inherent i n photographic work has been avoided by the use of a monochromator-photomultiplier arrangement (Section 5.1).  Electron densities and temperatures in the collapse  stage of the discharge are determined by analyzing helium line profiles. Absolute intensity measurements are used to evaluate these parameters (Section 5.2). (L.T.E.).  This entails the assumption of local thermal equilibrium  Plasma temperatures and particle densities are then computed at  two separate sta.ges of the discharge collapse.  -abusing relative intensity measurements, values of T are compared for e  L.T.E. and a semi-coronal model developed by Griem (Section 5.3). It i s shown that an error of 20$ in the estimate of T  Q  can occur because of the  different de-excitation mechanisms involved. The effect of self absorption on the shape of measured line profiles is examined (Section 5.4).  Calculations demonstrate that errors arising  from this phenomenon are less than, or of the order of, those due to corrections made to the monochromator traces.  The effect of self absorption  can also be used to estimate the accuracy of this correction procedure. An evaluation of electron-particle collisional relaxation times shows that an approximation to L.T.E. conditions may be assumed. This allows the equating of particle temperatures with deduced electron temperatures (Section 5 . 5 ) . Thus, estimates of particle densities can be made within the collapse stage of the discharge and later used to verify the shock structure of the plasma i n Chapter 6.  -855.1  THE MONOCHROMTOR-PHOTOMULTIPLIER METHOD OF TIME RESOLUTION A Spex Model #1700 II monochromator was used in conjunction with an  EMI photomultiplier tube Type 9558B to observe plasma radiation (Figure 5.1). A dual beam oscilloscope displayed the discharge current simultaneously with the photomultiplier output at the wavelength investigated.  This serves  as a useful t i e - i n with results from the time-resolved spectrograph and the framing camera.  Figure 5.1  Experimental arrangement for monochromator  By using a monochromator, the radiation intensity at only one given wavelength can be determined each time the discharge is fired.  For a study  of continuum radiation only one such oscillogram is necessary to determine the electron density or temperature, depending on the optical depth of the plasma (Section 5.3).  However, in the prepinch stage of the discharge, i t  is light emitted from the bound levels of the helium atoms and ions which predominates.  In this case, electron densities can be determined by  observing the broadening of these helium lines which is caused by the  -86-  perturbing electrons.  It i s therefore necessary to record line profiles;  i.e., plots of intensity versus wavelength for a spectral region occupied by a helium line at a specific instant of time during the discharge.  This  may be done by changing the wavelength monitored by the monochromator each time the discharge i s f i r e d .  Since the discharge characteristics are  reproducible, the resulting sequence of oscillograms can be used to construct the required line profile (Figure 5.2). Each profile required at least 30 shots.  They are therefore determined only when the peak intensity reaches  •points at 3.18 cm. or 5.08 cm. from the discharge axis. Only the collapse stage of the 2,000 and 4,000 ^iHg cases v/as thoroughly explored i n this manner because of considerable impurity radiation in.the low i n i t i a l pressure discharges (Figure 3 . 3 ) . Radiation profiles of the Hel 3889 A and Hel 5876 A lines v/ere used to evaluate electron densities. Electron temperatures for the collapse stage were determined by measuring the absolute intensity of the Hell 4686 % l i n e .  This was done by  evaluating the area of the profile (Figure 5.2) and then calculating the total power radiated by the line calibrated against black body emission. The monochromator was calibrated for this purpose by using a carbon arc. The reciprocal dispersion and instrumental broadening for the f i r s t order spectrum of the monochromator were found to be practically constant for the region of interest (3,800 to 6,000 A). For a s l i t width of 8 microns the instrumental broadening i s 0.2 X.  -88A description of the measurements for these instrumental functions and the absolute intensity calibrations is provided in Appendix IV. Optical alignment of the system vias made with the aid of a Spectraphysics Type 130C helium-neon laser.  The same quartz-water achromat was  used as for the time-resolved spectrograph (Section 3.1).  The f number for  the achromat = 7.0, and matches that of the monochromator with an f number = 6.8. It was decided to use the same control unit described in Section 2.2 in order to duplicate the timing of events as much as possible. The signals issued by the triggering photomultiplier (Figure 2.1) are replaced, by the output of a pulse generator in Figure 5.1.  -895.2  ABSOLUTE LINE INTENSITY MEASUREMENTS Electron temperatures (T ) and densities (n ) can be evaluated from Q  @  line profiles of the type displayed in Figure 5.2. The method is as follows. By choosing an appropriate electron temperature (T ) as a starting e  point, Griem's (1964) table of half-widths can be used to obtain the corresponding electron density (n ) from the observed half-width of a Hel e  line (Hel 5876ftor Hel 3889 ft). If one assumes that double ionization is negligible, ( n ^ then;  e + +  « n^,)  n, = n He+ e r  J  Knowing the total power radiated per; unit solid angle by the Hell 4686 line £i2j.j  e+  (-! - 4686)  from absolute intensity measurements, the following  formula can be used:  where  A^  = Einstein emission coefficient from upper state m to lower state n g = the s t a t i s t i c a l weight factor for state m He+( e) partition function (Appendix VI) £ = length of the discharge tube = 59 cm. m  z  T  =  Solving for T , using Hell 4686, equation (1) becomes: e  -  T  Q,  5.8972 x 1 0 ,  T  |i.i,i,i ,.,.ii, UM  mu^iM^^uu- .  2.3026 l o g  1 0  •nr-j  II  (?)  5  •  i  i t]  i.  i  0.11003 x "He Z (f )r e^86T H e +  with values J  u ... r •-. • i  e  H  ~ 4686 x 10"^ cm. ~ 1 .732 x 108 sec" (Griem, 1964) 1  Em E = 50.80 e.v. - 81.38 ergs. (Moore, 1959) =  m  3  2  • O O O O O V ^ - /  For a given T , the following parameters are known: n  H e +  (T )—from e  Griem (1964)  % +(T )--calculated by computer (Appendix VI) e  e  ^H +(4686)—determined by observation and remains fixed (Table 5-i) e  T  0  can be evaluated from equation (2). Starting with 35,000° K, the right  hand side of equation (2) is iterated until two successive values of T agree to the second decimal place. Usually not more than three iterations, using a desk calculator, are necessary.  This procedure also yields a final  value for the electron density n . e  For Hel 5876, equation (1) becomes: n for  H e 0  -\ A  = 59.7833^0(5876) Z  ( T ) exp( e  ^MBjU^L)  ( 3 )  = 5876 x 10-^cm. = 0.706 x 10  m n  E  H e 0  m  s  sec" , g 1  m  = 1 5 (Wiese,  1966)  = 22.97 e.v. = 36.78 ergs (Moore, 1959)  % o ( T ) = a complicated partition function given by Wiese (Appendix VI) e  e  The density of neutral atoms n^o  can be determined by equation (3) since  the following parameters are known: T  e  —from the iteration procedure of equation (2)  Z o(T )—since T He  e  g  is now fixed (Appendix VI)  o(5876)—determined by absolute intensity measurements (Table 5-i) It should be noted that equation ( 1 ) and consequently equations (2) and (3) depend on the existence of thermal equilibrium in the He+ excited states right down to the ground state.  (He ) 0  This i s implied by assuming  Such an assumption may not be valid. helium plasma, of n  ~10  1 7  Indeed, Mewe (1967) shows that for a  and 35,000° K < T  < AO,000° K, the excited  levels of the He+ species can be considered in thermal equilibrium with the continuum for m 5 2. s  The ground state population (m = 1) of He+ can depart  quite significantly from an L.T.E. distribution. If the plasma is optically thin with respect to Lyman radiation of He+,  the ground state population  can be 30 times greater than that predicted by L.T.E.  However, for a  plasma which is optically thick to radiation from the ground He+ state, the departure from equilibrium is greater by only a factor of 1.1 expected error in  to 2.0.  The  is approximately 20% i f n^e* is within a factor 10 of  the value predicted from thermal equilibrium. arithmic dependence of T  e  This is due to the log-  on n\] * in equation (2), where n -^. 1 o"*^cnT^ Q  e  and T -^4.0,000° K. e  For the neutral helium species in the same plasma, L.T.E. can be assumed valid only for m ^ 3.  Mo estimates are readily available for the  departure from L.T.E. values in cases of m < 3.  At most, a "departure  factor" of the same order quoted for the helium ions can exist, since the ground state to f i r s t level excitation energy is less for neutral helium (20 e.v. for He°, versus 40 e.v. for He ). +  Even with the ground state  population of neutral helium raised by a factor 30 above i t s expected L.T.E. value, neutral helium would s t i l l be negligible compared with ionized helium.  The range of T  determined here f a l l s in the 2 to 11  e.v.  e temperature region, where i t is known that rapid "burning out" of neutral helium takes place (Mewe, 1967). Table 5 - i contains the measured radiated power and half-widths'of lines necessary for the iterative procedure outlined above. The mono-  the  -92chromator response to radiation v/as calibrated by means of a carbon arc (Appendix IV). Plasma line intensities were calculated by measuring the area under line profiles,..such as the one .in Figure 5.2. These values were then used to determine the power radiated by the Hel 5876 A* and Hell 4686 A* lines.  The width of the observed plasma lines is assumed to be a combination  of Lorentzian broadening due to electron collisions and a Doppler type of instrumental broadening.  The instrumental broadening is quite small  (0.2 X In Appendix IV) compared with the observed f u l l line half-widths. Nevertheless, a correction was applied of the form u) where  =VkP - j - (0.2)2 2  ^ LO ^  - f u l l half-width used in calculations = observed f u l l half-width  to better approximate the natural line width. The Hel and Hell regions referred to in Table 5 ~ i are a consequence of the  different spatial distributions for these two species (Section 4.4).  The "Hel region" is more densely populated and at a slightly lower temperature than the "Hell region"-—hence, the terminology, since more Hel v/ould be in the f i r s t region. The temperature T v/as determined by equation (2) for the bracketed e  values of n  in Table 5 ~ i .  The choice was based on the*.quality and degree  of change in the profile from the "Hel" to the "Hell" region. n  Q  The other  for the corresponding region was calculated using the deduced T . e  served as a check on the accuracy of the results.  This  The apparent discrepancy  of 20$ is due to the variation in optical depth of the plasma for light of different frequencies.  This effect i s discussed in Section 5.4'.  Equation (2) shows that the calculated temperatures have a maximum "error of less than 10$, assuming L.T.E. holds true down to the ground state.  -93Table 5 - i Absolute intensity evaluation of n , T e  A,  e  I n i t i a l pressure 2,000 pBg in helium  i • Off-axis observ'ninvestpo'm t ifz ted. (cm. )  ffe/iu m / ine  a)  H e l l 4686 Hel ' Hel 5876 Hel 3889 3.18  <?(X) era/sec. sr.  1.295 x 10 1.466 x 10  9 9  lo  rteu.tra.1 fie/it/m  3,  17 X ID  in  (A)  (°K)  2.65 5.421 3.106  37.66  (1.486) 15 1.295  39.23  (0,891) 20 1.159  8,93  15,6  Hell  H e l l 4686 Hel 5876 Hel 3889  1.454 x 109 1,008 x 10  2.65 3.247 4.270  Hel  H e l l 4686 Hel 5876 Hel 3889  0,598 x108 2.958 x 10  0.925 1.952 1.3  33,40  0,532 (0,516)  3  6,24  H e l l 46S6 Hel 5876 Hel 3889  0.716 x 1 0 1.965 x 10  0,925 1,36 0,967  0,371 (0,384)  3  3.51  34,21  12  39,44  1.627 (1.863)  5.8 5.071 4.270  40,55  18 1,394 (1,710)  2,05 2,956  35.32  0,808 0.996  23  12.0  35.68  0,89  13  8.4  5.08  B.  - Fu./I "^Tx ha/f-us/ctih  9  s  8 8  I n i t i a l pressure 4*000 jiRg in helium  Hel  H e l l 4686 Hel 5876 Hel 3889  4,90 3,292 x 10 2.329 x 109 5.922 4,661  Hell  H e l l 4686 Hel 5876 Hel 3889  3.861 x 10 0.987 x109  Hel  H e l l 4686 Hel 5876 Hel 3889  2,795 x 10 7,934 x 10  H e l l 4686 Hel 5876  3,273 x 1G 6,023 x 10  3.18  5.08 Hell  n e i S'O&i  * sr. ~ unit steradian  9  8  8  9  2,505 8  8  2,15 3,256 2,266  0.902  20.7  7.86  -94Since the parameters n ^ ,  , <£  e+  H e +  ( 4 6 8 6 ) enter into the logarithmic part  of equation ( 2 ) , this small error appears even i f there i s : an error of 20$ in estimating n  e  an additional 20$ margin for determining the radiated power £ (a ), and a possible 20$ error for the oscillator strength (and hence, Einstein emission coefficient). Table 5 - i i summarizes electron temperature (T ) and density ( n j Q  determinations, together with the resulting kinetic pressures: P  kin  and densities  5°  K  =  =  n  +  i i m  i o>  n  +  +  n  n  o o m  +  VJ:  n  e e ra  It is assumed that n„ = n., and T ~ T. ~ T en;'' e I o vi here subscript e = electron mass and density i = helium ion mass and density o = neutral helium mass and density Q  n2  The table also contains estimates of the magnetic pressure P*., ~ — £ — .  2u  0  which were calculated from magnetic f i e l d distributions measured by Tam (1967).  These values are for the maximum pressure exerted on the outer  boundary of the current shells in Figures 4.18 and 4.19. The comparison of P ^ j ^ i n e  the determination of n  G  c  with  provides a check for consistency in  and T . At the 5.18 cm. radial position, the gas  pressure cannot exceed the magnetic pressure, since the shell is s t i l l accelerating.  Since the velocity changes slowly this would mean  ~ P» m  It should be noted that this condition is not f u l f i l l e d close to the axis for the higher i n i t i a l pressure (Table 5 ~ i i ) .  However, this agrees with  an earlier observation that the current layer never reaches t h e axis in t h e 4,000yj.Hg f i l l i n g pressure case (Section 4 . 6 ) .  -95-  Table 5 - i i Absolute Temperature Measurements I n i t i a l Off-ax is Region e pressure position invest- x igated 103K° T  X  7  1  p  3  P  Hel Hell  37.7 : 1.49 39.2 0.89  9.96  2.28  1.55  5.98  1.37  0.97  5.08  cm  Hel Hell  33.4 34.2  0.52 0.38  3.46 2.58  0.79 0.59  0.48 0.36  1.09  3.18  cm  39.4 40.6  1.86 1.71  12.49 11.46  1.43 1.31  2.03 1.91  0.29  5.08  cm  Hel Hell Hel Hell  35.3  0.90  6.05  0.69  0.88  35.7  0.90  6.01  0.69  0.88  2,000  5.3  2  cm  3.18  4,000  Densities Pressures g m / c r n ^ Relative to dyne/cm x 10° We x initial kin m 10 ycm 10 ' density 1.54  0.99  ELECTRON TEMPERATURE FROM THE SEMI-CORONAL MODEL The temperature determinations reported in the previous section are  based on the assumption that the plasma i s in local thermal equilibrium. For this to occur, collisional excitation and de-excitation rates of the atomic energy levels must greatly exceed population rates determined by radiative processes. This may not occur for low-lying energy levels, in which case collisional excitation can be balanced by radiative decay. is the coronal or Elwert steady-~state model.  This  In such a condition, the  population of the ground states of Hel and Hell can exceed the value expected from thermal, equilibrium calculations, as shown in Section 5.2 For this model, Griera (1964, pp. 273-4) shows that the electron temperature can be determined from the intensity ratio of the Hell 4-686 ?i and Hel 5876 i spectral lines.  -96Table 5 ~ i i i is constructed from Griem's calculations and the intensity ratios in Table 5-i. The spread in electron densities calculated from Hel 3889 and Hel 5876 half-widths ranges from 2$ to 30$ and i s included in the table below. The magnetic pressure calculations of Table 5 - i i have again been included to compare with the estimated kinetic pressures. It is seen that R^in'* T  in almost a l l cases, so that the assumption  = T(ions) = T(atoms) seems doubtful i f this model is to be employed. Table 5 - i i i Griem s semi-coronal model 8  T I n i t i a l Off-axis Region e © X press\ire position invest X igated 10 K° 10l7/ 3  %  N  3  2,000 jiHg  4,000 /iHg  3.18 cm 5.08 cm 3.18 cm 5.08 cm  Hel Hell Hel Hell  48.4 50.0 43.9 45.7  Hel Hell Hel Hell  50.3 57.5 45.5 46.9  cm  1.51 0.91 0.52 0.39 1.90 1.76 0.92 0,91  in n  e  20 31 2 5 21  24 23 2  Densities Pressures gm/cm Relative dyne/cm x 10 x 10~ i n i to tial p m density kin 3  3  6  7  r  10.09 6.06 3.51 2.62  2.31 1.38 0.80 0.60  2,01 1,25 0.63 0,49  12.73 11.76 6.13 6.10  1.45 1.34 0,70 0,70  2.64 2,64  0.29  1.15 1,18  0.99  1.54 1.09  Electron temperatures ranging from 35,000° K to 44,000° K are obtained when ths L.T.E. model is applied to the observed Hel 5876 to Hell 4686 intensity ratios (Griem, 1964; Mews, 1967). Although experimental error in determining line intensities may affect the actual, value of T , the application of the same intensity ratios to the L.T.E. and coronal models should, show the effect of departures from equilibrium.  As stated in Section 5.2 ,  a 20$ error iii T.e would be caused bv a factor of 10 in the Hell•ground state population.  Such an error is seen to occur on comparing the ranges of T : e  44,000° K ^  T  e  (coronal)^ 50,000° K  35,000° K <z T  e  (L.T.E.)^ 44,000° K  By assuming a lower starting temperature for the iteration procedure in Section 5.2 , we are assured of a better estimate of the total Hel population (equation ( 3 ) , Section 5.2 ). This, and the consistency criterion involving magnetic pressures, justifies the use of Table 5 - i i for further investigations of the Z-pinch in Chapter 6.  5.4  THE EFFECT OF SELF ABSORPTION ON LINE PROFILES For the same plasma conditions, a consistent difference of about 20$  appears between electron densities calculated alternatively from Hel 3S89 and Hel 5876 line profiles (Tables 5 - i and 5 - i i i ) .  The reason is that the  plasma is not uniformly transparent to radiation from a l l wavelengths; in the visible spectrum.  For an absorbing, homogeneous, L.T.E. plasma, the  ratio of observed lane intensity 1 ( 4 ) to that emitted in the ideal (or optically thin) case, E ( A ), is  ,  v  E(* )8  -  1-  A )  - e n>  h)  U )  where <7J' (A ) is the optical depth,at wavelength V  t  is the geometrical depth of plasma along the line of sight  E(~\ ) i.s the intensity per unit volume in the optically thin case The magnitude of f  ( A ) is greatest at the centre of the line profile  This causes a depression in the maximum intensity region of the profile, thus distorting i t s shape and causing observed half widths to appear great than their true value.  An expression for the effect of self absorption on  -98the observed line half-width can be derived for Lorentzian profiles assuming that emission and absorption occur as in a black body cavity. The ratio of the observed half-width (S")  of a spectral line to i t s true  half-width (to ) is found to be  T° In [  1  +  e  x  -1 ?  ( ^ j  U  (2)  J  where *7~ is the optical depth at the line centre 0  D  .  Self-absorption causes the observed total intensity (1^.) of the line to be smaller than i t s true value ( £ ^ ) . For Lorentzian profiles, the relation is  -jp-  ~  M  -  +  U  -JLS'O - ± ^o 16 38.4 2  ) ..(3)  3  Derivations of equations (2) and (3) are provided in Appendix VI along with an expression for f  0  and a graph showing i t s influence on observed half-  widths. Table 5-iv displays values of f  calculated from observed line half-  widths, particle densities and temperatures given in Table 5 - i .  The ex-  pected underestimate of total line radiation is calculated from equation (3) for the Hel 5876 and Hell 4686 lines.  The table shows that observed  intensities can be 20$ less than their true value.  However, a correction  procedure must be made to the monochromator traces before the helium line profile can be constructed (Appendix V).  This results in roughly a 20$  overestimate of the intensity at each monitored wavelength in the profile.  -99Hence, the correction procedure would tend to compensate errors in observed intensities due to self-absorption.  Therefore, an error in line intensities  of the order of 10$ can be expected. The consistency of the correction procedure in Appendix V can be demonstrated by comparing the effects of line broadening due to s e l f absorption with the observed line half-widths.  The spread in half-width  due to optical thickness is worked out for each line.  The percentage  difference in broadening due to self-absorption between Hel 5876 and Hel 3889 is then shown. Since the electron densities (n ) of Sections e  5.2;  and 5.3  were calculated using the half-widths of these lines, the  effect of self-absorption should be reflected in the percentage difference of h .  These values are recorded in Table 5-iv, for the absolute intensity  measurements and the Griem semi-coronal model. The last three columns of the table show a discrepancy between calculated and "observed" percentage difference of at most 10$.  This justifies the assumption of the 10$ error  in line intensities stated above. It should be noted that self-absorption effects for the Hel 3889 line are negligible. Therefore, electron densities calculated from the halfwidth of these line profiles would be closest to their true value. quantities were used whenever possible.  These  Since the slope of the line  profile in i t s half-height region varies slowly, a 20$ error in evaluating its shape would cause an error much smaller than this in determining i t s observed half-width.  However, a maximum error of 20$ is assigned to the  calculation of electron densities.  Table 5-iv  pressure  Region observ'n in t/estpoint  Influence of s e l f absorption on line intensities and half-widths  />rie  rfo  error Cz/cu^/ated in to ia/ Spreac/ cr>  hajf-widt/-, A A  intensify  Self&bsorpt/t>f  0.745 0.933 0.047  16 19  v. /a j 21 26.5 1  0.789 1.007 0.022  16 20  0.120 0.627 0.028  3 14  0.131 0.57 0.024  3 12  0.956 1.237 0.047  18 24  1,074 0.612 0.021  21 13  0.228 1.022 0,044  5 20  0.251 0.688 0.029  6 15  X/o~ $cm  (cm.)  Hel 3.13 Hell 2,000 Hel 5.03 Hell  Hel 3.18 Hell 4,000 Hel 5.08 Hell  Hell 4686 Hel "5876 Hel 3889.  1.325 2.711 1.553  37.66  Hell 4686 HeI 5876 HeI 3889  1.325 1.624 2.135  39.23  Hell 4686 Hel 5876 Hel 3889  0.463 0.976 0.650  33.40  Hell 4686 Hel 5876 Hel 3889  0.463 0.680 0.484  34.21  4686 5876 3889 4686 5876 3889  2.45 2.961 2.33 2.9 2.536 2.135  Hell 4686 Hel 5876 Hel 3889  1.025 1.478 1.252  35.32  Hell 4686 Hel 5376 Hel 3889  1.075 1.628 1.133  35.68  Hell Hel Hel Hell Hel Hel  39.-44  40.55  22 31 << 1  ' d/^f'ce. /or sdf-ai&arptiQr,  ^ difAe ir> ^rom observed /-/eT ha//- ujic/fhs  Corona./ //e J />nr,s 7&6/<z 5"- /Ta-He: S-'tn  25  15  20  31  20  31  3 17.5 < 1  7  3  2  3.5 16 •< 1  16  3  5  27 36.5 1  36  12  20  31 17 < 1  17  18  24  6 29 1  28  23  23  7 19 < 1  19  13  2  -101-  The relatively small values of f  indicate an average error of 20$  i n the observed intensities and calculated n absorption effects.  g  due to the neglect of self  This is of the order of the estimated experimental  error in constructing the line profiles.  The values in Tables 5 ~ i and 5 - i i  must therefore be regarded as estimates only.  However, as demonstrated in  Section 5.2 ••., they are adequate in determining electron temperatures and y i e l d kinetic pressures consistent with Tarn's magnetic probe results. It should be noted that self absorption can be quite significant for • higher particle densities occurring near the inner boundary of the collapsing s h e l l .  Indeed, Botticher (1964) has found that the optical  depth in this zone can be so great as to render the method of Section 5.2.-. useless.  However, knowing the extent of the current layer and the much  lower particle densities in i t s luminous regions, one can construct a model to give estimates of the density at the shock front. This w i l l be done in Chapter 6. Extreme cases of self absorption are noticeable in the axial radiation spectra, of Figure 3.3.  Calculations (Appendix VIIc) show that for expected  plasma values of n ^ ~ 10 cm~ , T l8  plasma is of the order  3  e  > 40,000° K, the optical depth of the  - 4640 A) $? 4.  For this reason, and because of  possible hon-homogeneity in the heating of the axial plasma column along its length, the monochromator voltages in Figures 3.6 to 3.9 (Section 3./+) have not been used to calculate electron densities or temperatures.  -102-  5.5  RELAXATION TIMES Two methods of measuring the electron temperature have been used in  this thesis. The f i r s t method, using absolute intensity measurements, assumes that excited states of the atom are brought into local thermal equilibrium (L.T.E.) with the free electrons.  The free electrons have a  Maxwellian velocity distribution characterized by a temperature T .  In the  second method, where Griem's (1964) calculations for the semi-coronal model are employed, i t is assumed that the upper excited states are in L.T.E, with the free electrons.  However, the relative populations of Hel and Hell are  controlled by a balance between collisional ionization and radiative recombination.  It is essential to establish whether population densities can  relax to their steady state values much faster than a time characterizing changes in macroscopic plasma properties. (~10  In most cases, the pinch time  usee.) determines the rate of change in the plasma temperature and.  is therefore used as a standard of comparison for relaxation times. The observed plasma conditions can be represented by: n  = 10 /cm , 17  3  T  = 40,000° K = 3.4  e.v.,  He+ ~^10  n  3  n  ......(1)  He°  These are averages of values found in Tables 5-i to i i i . Let us f i r s t consider electron-electron and electron-particle (Hel, Hell) relaxation times. Spltzer (1956) gives the electron-electron relaxation time as  3/?  r'ee ^ -0.266 e T  n  where T  e  In A  is in ° K and In A is a slowly varying function of n  usually of the order of 10.  ......(2) and T„,  -103-  From statement (1), i t i s seen that f k  ee  ^-r 5 x l O^1. s2e c .  ...o..(3)  and therefore, the electron velocity distribution is Maxwellian. Griem (1964; p. 155, equ'ns 6~~69a,b) derives electron-particle relaxation times i n a homogeneous transient plasma. and  _kT  v«*10 . Z. 8  is:  For a single ionization  cm/sec, the electron-neutral equilibriation tirae  To-f3x10-  EH  7  3  /  2  N 1 -  "KT  1  6  1  K  J  N  a 1 a  M  sec.  ...... (/J  n  and the electron-ion relaxation time i s ; 3 x 10" g % kT 7  where,  '  N  v Z M m z a  N  a  2  3 , / 2  N j"  1  Z _M_ m  = velocity of the electrons = 0 for neutrals, Z - 1,2.....for = atom or ion mass = electron mass = ion density  sec.  .......(5)  ions  = total density  Griem gives values of:  t-K° ~  s e c  °  (electron-neutral)  = 10-8 sec. (electron-ion) for a hydrogen plasma of N  e  10 6/cm , 1  kT = 1 e.v., and 10$ ionization.  3  Substituting values of statement (1) for a helium plasma in equations (4) and (5) gives: p r  ^K° ~ 2c,5 x 10"°° sec. (electron-neutral)  ^j^1  2.5 x 10'" sec. (electron-ion) 8  ....(6a)  ......(6b)  -104Statements (3) and (6a,b) show the electron-electron and electron-7 particle relaxation times are a l l less than 10  seconds»  Hence, i t can be  assumed that the velocity distributions of electrons, Hel and Hell particles are a l l controlled by the identical temperature T . However, for L.T.E. to hold, the relaxation time is determined by the rate of.collisional de-excitation of the f i r s t excited state.  For hydro-  genie systems this can be expressed as (Griem, 196/,.; p. 153, equ'n 6~«65):  T i ~ ^ Z  1.1 x , 1 0 ¥  1  >21 e N  V  <V * V  "  TT^Z-1 a  z  s  52  - 1  ?  2 2  E  H  .,  J "  KT M  ^ H'  exp  z-1 E2 kT  sec. (?)  o o . o e . V  '/  7 1s where, E2" " * ' = excitation energy of the resonance line for Hell (40e.v.) .^s. - accounts for the fact that not a l l ground-state atoms y z YI Z-1 or ions need be excited or ionized a a Eg = ionization energy of hydrogen = 13.6 e.v. 1  +  = absorption oscillator strength from ground to resonance state According to Griem, equation (2) yields a value of the collisional excitation time for ionized heliums T^'  ^v. 0.3 usee,  2  where his helium plasma has T = 4 e.v,,  n  - 10^/cm  3  e  Using the average values in statement (1) yields a relaxation time T-] * 1  2  9.0 jusec  ......(8)  This i s comparable to the pinch time of 10^isec. The relaxation times for the higher excited levels of the Hell ions are much less than the pinch time. For helium plasma conditions of statement (l) and principal quantum number m, an excited state of the Hell ions  -105(m, >1)  has the following equilibration time (Griem, 1964:  equ'n 6--67): f^ "" 2  so that, for  m = 2,  8.9 x l O "  1  ^2^-1  ^  x 10  1 0  p.  IAA, m 3  154, e o « c o « (9)  3/2 x 10-3 ^isec.  Thus, the relaxation times for states of m>1  (10)  are so short that partial  L.T.E. between excited states is established almost  instantaneously.  Although the relaxation time for the lowest state of Hell is comparable to the pinch time, i t s population is probably different by less than a factor of 10 from the value assumed'in L.T.E. Since the electron temperature depends logarithmically on the population density, the maximum error in T would s t i l l be only 15$ (Section 5.2,  equation (2)).  For the coronal model, Mclihirter (1965) gives a relaxation time of 19_2^ n  seconds. Hence, for the electron densities observed (n  10 ^cm" ), 1  3  e  the relaxation time is 10 ^asec, which is again comparable to the pinch time.  Temperatures calculated from the coronal model s t i l l depend log-  arithmically on the intensity ratios of the Hell 4686 end Hel 5876 lines. As Griem's results show, at 40,000° K, an error of a factor of ten in the intensity ratio causes an. error in T  e  of roughly 14$.  It is evident from the above discussion that a true estimate of T  e  can  only be obtained by solving in detail the rate equations for the excited levels in the two atomic species (Hel and Hell). that T  Nevertheless, the fact  depends logarithmically on the population densities lends confidence  to the estimate of T  as 40,000° K t 15$.  Since the temperatures deduced  from the L.T.E,, model and the semi-coronal model are practically the same  -106-  when relative intensity ratios are used (Section 5 . 3 ),-no conclusions can be drawn regarding which model best describes the plasma in the Z-pinch.  This question cannot be resolved by comparing relaxation time  which turn out to be almost identical.  -107-  6.0  DYNAMICS OF THE PINCH The motion of the collapsing luminous shell has been analyzed and  related to that of the current shell.  Time and spatially resolved measure-  ments of temperature and density" within the luminous plasma have been conducted at successive stages of the collapse. A synthesis of this information can now be made in an effort to understand the dynamics of the Z-pinch for high i n i t i a l pressure discharges (2-4- Torr). Framing camera pictures show that a non-luminous front precedes the luminosity shell.  Its existence is disclosed by the occurrence of axial  continuum before the luminous shell can reach the axis of the discharge. An attempt is made to determine the density of material in this leading (shock), edge of the collapsing current shell.  It is d i f f i c u l t to measure this  density directly because of the faintly luminous and optically thick region in which i t occurs.  However, a simple model can be constructed to relate i t  to densities observable in the luminous region of the current shell (Section 6.1).  From evaluations of this density, i t is found that shock heating occurs  in the non-luminous region, but that very l i t t l e kinetic energy is used in ionizing the gas as i t enters the collapsing current shell.  These findings  are confirmed by a more rigorous theory which takes the cylindrical geometry of the discharge and ionisation processes into account (Section 6.2). The pressure at the shock front can be calculated from particle velocities observed in the luminous region behind i t .  i i simple model is used of a shock  produced by t h e uniform motion of a piston through a gas at rest.  This yields  a gas pressure in t h e shock front which approxim&tely equals the spectroscopically determined pressure further back in t h e collapsing shell.  Both are  -108in turn, consistent with the driving magnetic pressure, so that the inward moving shell is a dynamically stable structure,  6,1  DENSITIES WITHIN THE SHOCK LAYER The determination of densities in the shock layer can elucidate the  processes responsible for heating the gas as i t is trapped by the collapsing current shell.  This can be demonstrated by considering the following  simple model. Since the current peak converges towards the discharge axis at supersonic speeds, i t w i l l drive a shock wave ahead of i t (Figure 6.1),  Also,  as the radius versus time curves of Sections 4,5 and 4,6 show, the collapse velocity can be regarded as approximately constant.  Provided the shock  front i s far enough from the discharge axis for curvature effects to be negligible, one would expect the condition of the shock heated gas to be f a i r l y well described by the standard Rankine-Hugoniot theory (Thompson, 1962).  In a'.frame of reference at rest with respect to the shock front  the following conservation equations hold: mass:  $> ^ u. = J°2 ?  ......(1)  U  momentum:  p-j + j^u-j = p  energy:  £ u-j +  2  2  p  1  .,,.,.(2)  ?2 2  +  n  2  2  , = i u ^ +  (TT^^Tfl  2  "  «P2__-—  +  W  ..(3)  (So - )f 2 1  with p, j> and u representing the gas pressure, density and velocity respectively.  Subscript 1 refers to the cold gas ahead of the shock front end  2 denotes the gas behind i t (Figure 6.1).  W is the energy used up per unit  mass in ionising the cold gas as i t enters the stock front.  The parameter g  is the ratio of enthalpy to internal energy and. is represented by y - 5/3  -109for a monatomic gas in which no ionization process occurs (Lunkin,  1959;  Abloom, 1967).  Motion of shock front  Relative density  current shell  -c3-  hot gas ^ region behind"// shock (2) I / / /  cold gas region ahead of shock (1)  -P*- radius of tube 5  Figure 6.1  Model of shock formation  If V/ is zero (i.e., no kinetic energy is used up in ionizing the particles) then i t can be easily shown that the density behind the shock front (j°2)  i s  f°  u r  j 1 9  "tines that of the cold gas entering the shock (f -|) s  u  /3 + 1  u2 •  g2  -  7T  1  If the kinetic energy of cold gas entering the shock, front is partially used up in ionizing i t , then u when ¥ = 0 in equation (3). increase correspondingly.  2  w i l l be reduced below the value observed  It follows from equation (1) that  2  will  Botticher (1963) has shown that for ionization  of helium by shock heating  For helium (T ^ 30,000° K and p ^ 1 atmospheres), g 1967).  2  is about 1.2  (Ahlborn,  Hence, a measurement of the particle densities behind the shock  front can provide decisive evidence on whether the incoming gas is ionized . by shock Densities have been measured within the "luminosity" shell at 5.OS ciru and 3.1S  cm. off-axis for the 2,000 and 4,000 uHg cases.  As stated in  -110Section 4.6, there is a definite separation of the collapsing current shell into shock and luminosity regions.  Since the cold shock front does not  register on framing camera film before i t hits the axis, Tarn's collapse curves must be used to f i x i t s approximate position (Figures 4.18 and 4.19). Furthermore, Tarn's work can provide an estimate for the overall average density of the total material within the collapsing shell. Since the densities shown in Table 5 - i i are local, and not average, i t is necessary to assume an overall structure for the collapsing shell before information can be extracted concerning the shock layer density. This structure is outlined in Figure 6.2  The outer boundaries of the  current shell have been established by current density measurements (Tam, 1967). It has been shown that at least 90$ of the gas is swept up into the current shell that passes through i t and resides in the region bounded by the positions at which the current density f a l l s to half its maximum value. Figure 6.2  relative density  Model of the collapsing current shell (1  a  ) P  I  A™ o I  I  irn  r  il  p1  r  II r  p2  om  R  axis of tube Let:  R = radius of vessel wall "ijij = inner ") , > radii of Tarn's "magnetic probe" shell ^om ~ outer j r-ji = inner radius of the luminosity shell r 1 = radius of the "Hel peak" r 2 = radius of the "Hell peak" Q  p  Particle densities at the Hel (^-j) and Hell ( p e a k  radiation positions  have been determined spectroscopically. It is assumed that the particle density (j° ) i n the region between the inner edges of the current and x  luminous shells varies slowly.  Since relatively l i t t l e light and current  is detectable in this zone, J °  represents, at worst, an average value  x  i f i t is treated as a constant. The evaluation of the particle density (j° ) in the non-luminous region x  is carried out in the following manner. The overall average density of the current shell  (_p ) T(n  is determined f i r s t .  Then observed local densities  j -] and j°2 are used to readjust the density distribution within the shell 0  according to Figure 6 . 2 . i) If ^TM  =  the average density within the total "magnetic shell"  as measured from Tarn's collapse curves (relative to the i n i t i a l density f> ) 0  then,  r  /  r• im so that,  R  o m  J^TM  r  dr  =  r. im  Q  r d r  ±  Ro - *Y "ML T* 2 — r• 2 om MI takes into account the fact that not a l l particles are swept up by 2  2  I  where ^  the collapsing current layer. Tam (1967) shows <=C = 0.9 for 2,000 and 4,000 jiHg.  i i ) Integrating over the various layers of the collapsing shell gives: ^<n  r  'f  Jpi  If"-  /' ° r  t V l  /  fr^rdr- .J  "i^i  '•••i  where:  f rdr  *• J f  x  r 4r +- J  x  T'e  r  f rdr  +J fm  %  rdr  ( 5 )  P'  P  = the unknown density of the shock layer (relative to the i n i t i a l density / ° ) j i t is assumed a constant and therefore, an average quantity; >o j = the density of the shell bounded by the inner radius of the ' luminosity layer and the radius of the "Hel peak", (relative to « / ° ) j /°TJ the density of the shell bounded by the "Hel peak" and the ' "Hell peak", (relative to 7° ); PlTL ~ the density of the shell bounded by the "Hell peak" and the outside radius of Tarn's "magnetic probe" shell, (relative toJ^ ). 0  0  =  0  0  For the three areas with relative density J°s>  s  I,II,HI, a linear  =  interpolation is used between known densities, so that:  /s v/here,  = K  r  (6a)  + b  s  s  Kg - pinner boundary ~ J°puter boundary inner boundary *" *~ outer boundary  (6b)  r  bs ~ /"inner Tputer - yputer >~ inner boundary ~  r  inner  (6c)  i  outer boundary  At rj_]_—e> the density is assumed s t i l l close to that of J ; i.e., that of the shock layer. At rp-j —e> the density is j° ^, the relative density measured at the "Hel peak"; from Table 5 - i i . At rp^—1> the density Is f 2, the relative density measured at the "Hell peak"; from Table 5 - i i . At r —> the density isj° 2/2. 3  x  Inserting equations (6a,b,c) into equation (5) and integrating gives: J°TM ( om ~ im ) r  2  r  2  =  fx ( * i l  2  - r  2 i n  > + b ( r 2 - r ^ ) + _2_ K (r 3 T  +  +  where bj sxvd. K-j- are functions ofj* .^. 3  b  H  ( r  p 1  p2  bni(r  2  o n  T  " p1 r  2 -  r  2  2 )  2)  * +  K  ?  ^ K 3  II  IIT  ( r  3 p 2  "  (r ,3 CJ  i ) 3  p1  r i  l  r P  r  3  )  3) 1  ......(7)  -113This i s solved for J ° , which is tabulated below for 2,000 and 4,000 uHg x  i n i t i a l pressures when the Hell peak passes the 5.08 and 3.18 cm. observation points. Table 6-i  Relative densities for shock fronts From Figs.  4.9, 4 J 0  Initial pressure  From Figs, 4.18  4,000  ^Hg R  Q  and  4 ° 15, 4«16  Table 5 - i i  4.19 r  2,000  Densities (relative)  im  r. T il  om  r  p1  r  r P  2  A A  0 3.5  4.03  2.5 2.93 4.0 4.78  3.18 5.08  2.28  5.75  0 3.0  4,6 5.6  2.4 2.75 4.5 4.83  3.18  1.43 0.69  5.1  C  = 7.62 cm.  Calculated shock average density (relative)  0.79  1.37 0.59  1.31 0.69  f J  X  5.51 4.73 5.26 3.76  off-axis observation point  The results in Table 6-i show that the relative density across the shock front is quite close to 4 .  This i s the value expected for plane  shock fronts in which shock ionization is unimportant.  It should, however,  be kept in mind that this calculated density depends on the model used. A magnitude of approximately 5 for the 3.18 cm. radius is not surprising when one considers the effect of a cylindrical geometry. In the luminous regions, the mass density can be less than that i n the cold gas (y^  or. j>  2  ^ 0.7 f )» Q  This result indicates that gas  passing through the shock heated region into the luminous shell is heated by otonic dissipation of the discharge current. the gas and hence, reduces i t s density.  This ohmic heating expands  -1-U-  The main Conclusion to be drawn from a study of the mass distribution in the collapsing shell is that the shell consists of two regions.  Within  the inner region the gas is shock heated, but very l i t t l e kinetic energy of the cold gas is used in ionizing i t .  In the outer, luminous region  the density i s lower than that of the cold gas and indicates that Joule heating predominates here.  i  -1156.2  EVIDENCE OF GUDERLEY FLOW The conclusions of the previous section can be confirmed by an appli-  cation of the Guderley theory for shock formation in a compressible gas (Guderley, 1942). The influence of cylindrical geometry and ionization processes is manifested in the time behaviour of observable fronts.  Limit-  ations of this theory render i t useful only i n the analysis of higher i n i t i a l pressure discharges (2,000 yEg and above).  Using Jukes's (1958) theoretical  development and Botticher's (1963) notation, the Guderley flow theory can be summarized as follows. Assume the discharge consists of a cylinder of ionized gas (Figure 6.3) in which,  i ) electrical conductivity is infinite; i i ) the current rises instantaneously from zero to a f i n i t e value on a bounding cylindrical sheet (the subsequent time variation is to be found)j  i i i ) the thickness of a shock wave is very small compared with the dimensions of the discharge.  Current J  Figure 6.3  z  Discharge column and associated circuit  -116Consider the equations of the gas motion between the current sheet and the ingoing, cylindrical Shockwave preceding i t . From Courant  (1948),  these equations with cylindrical symmetry are: v "*Lf t• i° 5. ( ) lit 7>f rf ( ^ + ^ ^ \ = -  continuity:  *  momentum:  - °  r v  0) ......(2)  and the equation of particle-isentropy:  (2. where "X.  =  + v L  ) ( f .f"*) =o  .  (3)  Y ~ 5/3, i f the gas ionization processes do not occur.  Bo'tticher  introduces "X , to account for the possibility that shock ionization can occur, v (r,t) is the radial velocity of the gas; p, the pressure; JJ, the density. One solution of these partial, differential equations is obtained by assuming a similarity, or "progressing wave" solution of the form: v  =  nr t  V (£ )  P = ( JE-,) where ^  2 2  f  ='il(^)  U)  P (? )  is a parameter defined as: (-t)n  in which n i s an eigenvalue (to be found), and the observed times (-t) are represented with the pinch time (tp) as origin (tp = 0). The v Jb_, P^-i-j > f appear constant to an observer moving on a trajectory of constant tZ . The velocity of the observer is nr . a moving shock front, £  If the flow contains  must remain constant on the trajectory of the front.  -117The relationship between the ratio of specific heats (denoted by g), 7 C and n is quite complicated, but can be summarized below: Table 6 - i i Numerical relations for Guderley flow parameters g  OC  1.667 1.4 1.31 1.195 1.195  1.667 1.4 1.235 1.285 1.25  source  n 0.817 0.834 0.8494 0.8568 0.8588  Jukes Guderley  Jukes develops a criterion for the behaviour of the current. If the total current flowing on the cylinder is I, when i t s radius i s r„, the pressure at the cylinder wall p  2u  D  is just the magnetic pressure outside,  ,;>  8?rr ' c  In a Guderley flow p , r , and therefore, l ( t ) are known as functions c  of time.  c  From Figure 5.1, the applied f i e l d E, causing the flux change  satisfies: E  =  _ d _ ( L x I ) - _ d _ ( 2 log j o x I ) dt dt c d  ......(7)  r  From equations (4) and (5), Jukes (1958),shows that for a Guderley flow: I must decay almost linearly with time up to the f i r s t pinch, Lj x I increases almost linearly from zero. This condition is satisfied by a constant applied f i e l d .  Botticher's (1961) experimental conditions are as follows (compare with Table 2 - i ) . Discharge vessel: 18 cm. diameter by 100 cm. long F i l l i n g pressure: 6 Torr Helium • Condenser bank: 100 kJoule; charging voltage 35 kv, i n i t i a l inductance 4.5 nHenries Maximum current: 500 kAmp. Pinch time~~ 5 usee, followed by additional contractions at 10 and 1 5 jxsec.  From the foregoing, i t can be seen that a necessary, but not sufficient, condition for Guderley flow is that the current I(t) decays almost linearly while the pinch is being formed. From the luminosity graphs, this occurs only for the 2,000 and 4,000 uHg cases (Figures 4.I4 and 4 . 1 5 ) .  For the  500 and 1,000 pHg graphs (Figures 4.12 and 4 . 1 3 ) . the discharge current increases while the pinch is being formed. This explains why, when (log r) vs.  (log t) values v/ere plotted for 500 and 1,000 yaHg, slopes v/ere obtained  which did not f a l l in the range, 8.1 <n <8.6, predicted by the Guderley theory. For 2,000 and 4,000 uHg i n i t i a l pressures, according to equation ( 5 ) , the slope of the line (log r) vs. (log t) should give n, the eigenvalue for the similarity solution. By using Table 6 - i i connecting n and g, an estimate can be obtained of the "specific heat ratio", g. tried for Tarn's inner radii ( F i g u r e 3 4 . 1 8 , 4.19). which f a l l s outside the .range predicted by theory.  This procedure was  His values give an n This is not too sur-  prising, since he used the half-height of his current peak as the radius for the inner boundary of the collapsing shell.  The real shock front is  probably a l i t t l e further ahead than he predicts. From Sections 4.6 and 6.1 though, i t seems reasonable that most of the mass within the shock layer is found behind this boundary. The (log r) vs. (log t) graphs for the inner radius of the luminosity shell (r^j) are given in Figure 6.5. case is provided in Figure 6.4.  A "Hell peak" graph for the 2,000 ^.Hg  The "Hel peak" front yields a graph with a  similar slope and is therefore not included.  Graphs for the Hel and Hell  peak radiation curves in the 4,000 uHg case yield an n outside the range predicted by theory.  This can. be explained by noting that the motion of  -119the Hel and Hell, peak radiation regions was assumed to be governed by Guderley flow.  However, as w i l l be demonstrated in the following section,  these highly luminous shells act as a current piston which is brought to rest, i.e., the luminous region does not form an intrinsic part of the Guderley flow.  The Guderley flow i t s e l f continues onward towards the  discharge axis, giving rise to the current peak and axial continuum preceding the luminous shell of Figure 4.19.  Essentially, in this case, the  shock front is driven to the axis by i t s own momentum, leaving the current piston of the Jukes model behind. According to Figure 6.5, a slope of n = 0.83 for the inner boundary of the luminosity shell indicates that: 1.4 < g S 1.667  (8)  which would be characteristic for relatively l i t t l e to no shock ionization. The slope for the "Hell peak" in Figure 6.4 for the 2,000 uHg case gives n = 0.86,  so that: 1.2<g<1.3  (9)  which is the region Botticher (1963) claims for ionization processes to occur in a helium plasma. Although the accuracy of the graphs is not precise, they at least show a trend in g. relatively cold gas.  Statement (8) shows a range of g characteristic of a This is in the region of the luminosity front. The  range of g in statement (9) indicates higher temperatures occurring deeper within the luminosity shell (g «s1.2 for helium at 30,000° K and 1 atmosphere pressure). and 6.1.  This agrees with the findings of Sections 5.3  -120The inner edge of the luminosity shell at 4,000 p.Hg conforms to Guderley flow.  This corresponds to no shock ionization.  Since the current  peak region does not exhibit Guderley flow i n this case, the boundary between the current "piston" and shock heated gas i s somewhere close to the edge of the luminous shell. The occurrence of Guderley flow in the 2,000 jiRg case is valuable from another point of view. This lies in the current decay criterion, and the constant applied voltage E required by equation (7). For i n i t i a l pressures  2,000 uHg, the current is s t i l l rising as the  pinch is formed. Equations (3) and (6) are violated as the current "diffuses" into the collapsing shell and excites helium throughout the whole shell by direct electron-atom  ——  log (-t)  1  1  collisions.  1  $. 4t = 0.34  1——j—  (.—j—  pec.  Figure 6.4 2,000 u, "Hell peak" vs. time  Figure 6.5  Inner luminositv radius r.  -1226.3  THE PISTON MODEL OF SHOCK FORMATION In the previous section, qualitative agreement was found to exist with  the Jukes-Allen theory of Guderley flow at the inner edge of the luminous shell for the 2,000 and 4,000 jiHg f i l l i n g pressure cases.  An implicit  assumption of this theory i s that the current-bearing layer can be regarded as distinct from the shock region upon which i t acts as a piston. From the logarithmic "radius versus time" graphs of Figure 6 . 5 , i t would appear that the value of parameter g  2  in the shock front is close to  the specific heat ratio of a perfect rnonatomic gas: Y= 5/3.  This is  corroborated in Section 6.1, where i t was found that £2-  =  X ± - L  %  4  Such a result is expected for a plane, non-ionizing shock propagating through an ideal gas.  The cylindrical geometry of the collapsing current  shell can therefore be relaxed to considerations of plane fronts at the 5.08 and 3.18 cm. off-axis observation points. The pressure behind a plane shock front can be calculated, from particle velocities observed in i t s wake. This serves as an independent check on spectroscopic and magnetic f i e l d measurements.  If these values are com-  parable, then the conclusions arrived at in the previous two sections with regard to shock flow are confirmed as well. Consider a serai-infinite cylindrical pipe, f i l l e d with gas, and terminated at one end by a piston (Figure 6 . 6 ) .  At an i n i t i a l instant of  time, the piston begins to move into the pipe with constant velocity U. The gas adjacent to the piston must move with the same velocity, while the  P from piston velocity  -123-  4  gas flow across shock front-  P spectrescopic  P magnetic .  U <(D  (2)  I  shock front Figure 6.6  piston  Piston driven shock front  gas farther ahead is compressed and accelerated. and moves along the pipe.  A shock wave is formed  At f i r s t , the shock and piston are close together.  Since the shock veloctiy is slightly larger than U, the shock front subsequently separates from the piston and a region of gas lies between them (region 2).  In front of the shock wave (region 1) the gas pressure is equal  to i t s i n i t i a l value p-j and i t s velocity relative to the pipe is zero. In region (2), the gas moves with constant velocity, equal to the velocity U of the piston.  The gas pressure P2 between the piston and the shock wave  can be expressed in the following manner (Landau and L i f s h i t z , 1959, p. 358):  where These equations can be applied to our experiment.  P-j andj -] are the 3  i n i t i a l f i l l i n g pressure and density of the gas in the discharge tube. The velocity U of the gas in front of the piston is assumed to be equal to the velocity of the Hel radiation peak (Figures 4.9 and 4.10). assumptions, pressures P  2  With these  can be calculated from the above equation and  -124compared with the pressures available from spectroscopic and magnetic probe data (Table 5 - i i ) . values of n  e  and T  The spectroscopic pressures (P ) are determined from s  e  at the Hel radiation peak. The magnetic pressure  is calculated from the maximum magnetic f i e l d which occurs at the outer edge of the current shell (Tam, 1967). A l l three measurements are independent and refer to different positions within the shell (see Figure 6.6).  Table 6 - i i i Comparison of pressures determined from velocity, spectroscopic and magnetic f i e l d measurements I n i t i a l Observation Velocity v ?s Pm pressure point of Hel velocity spectroscopically magnetic f i e l d off-axis peak determined determined determined (uHg.) (cm.) cm/usec pressure pressure pressure p  2,000  3.18 5.08  4,000  3.18 5.08  -  1.42 0.93  1.18 0.51  1.55 0.48  1.54  1.09  0.95 0.80  1.06 0.75  2.03 0.88  0.29 0.99  Pressures in units of dyne/cm^ x 10 ^ 1  In the light of the approximations and experimental errors involved, the agreement between velocity and spectrcscopically determined pressures at 5.08 cm. off-ax is is quite reasonable (Table 6-iii).  This supports the form  of density dependence across the shock front assumed in Section 6.1.  The  larger magnetic pressures indicate an acceleration of the shell towards the discharge axis.  Closer to the axis, a disparity in these values is apparent,  For the lower i n i t i a l pressure case, the agreement between spectroscopic and magnetic pressures shows that the shell is moving uniformly.  The  velocity determined pressure is lower by 20%, but this may be due to curvature effects not taken into consideration by the model. The lack of  -125agreement close to the axis for the higher i n i t i a l pressure case might be explained by increased Joule heating of the shock heated gas which would raise the pressure above the level predicted by ideal shock theory. The qualitative agreement between spectroscopically and velocity determined pressures, particularly far off the discharge axis, confirms the validity of the model proposed in Section 6.1.  Hence, the conclusion that  • = U has been reinforced. This result contrasts with Botticher's (1965) findings in which he concludes that -II  *=>10.  /1 The evidence i n the last three sections points to the following explanation for plasma formation i n the Z-pinch discharge.  The discharge  current rapidly energizes gas i n i t s immediate vicinity by Joule heating. The predominant species i n this region is ionic, so that the build up of thermal pressure is counteracted on the outside edge of the plasma by a large magnetic pressure.  The helium ions are therefore confined and are  only free to move Inwards, towards the axis of the discharge vessel. The gradient between magnetic and thermal pressures in this highly excited region is responsible for the inward motion of the plasma. Thus, the current layer acts as an inward moving piston which sets up a shock wave in front of i t s e l f .  This shock i n turn heats up the quiescent gas through which i t  passes, but does not posses sufficient energy or time to ionize i t . It should be noted that the model proposed above holds only for higher i n i t i a l density plasmas (2,000 |iHg and above).  In the previous section, i t  was found that the Guderley theory does not provide an adequate description of low i n i t i a l pressure discharges (500 to 1,000 uHg).  This failure  -126-  suggests that Joule heating pervades the whole plasma shell.  Since no  adequate theory has been devised to take this energy input into account, the question of shock ionization would remain unsolved.  Also,, the high  incidence of impurities observed in such low pressure discharges would render spectroscopic results doubtful in the best of circumstances.  -1277.0 SUMMARY AND PROPOSALS FOR FUTURE WORK A linear Z-pinch has been studied using a framing camera, timeresolved spectrograph, and monochromator. The results of these optical investigations have been correlated with magnetic probe work conducted by Tam (1967).  The pre-pinch stage of the discharge i n helium emits very  l i t t l e light.  Framing camera and time-resolved spectrographic observations  yield meaningful results only i f many discharges can be superimposed on one film record. This required the development of a switching and trigger system'accurate to within ± 0.2 ynsec. The structure of the collapsing plasma shell differs markedly with i n i t i a l f i l l i n g pressures. At low i n i t i a l pressures the luminosity and current shells overlap almost completely.  Peak Hel and Hell intensities  coin'c ide at the position of maximum current density. Intense continuum radiation appears when the leading edge of the luminous shell reaches the discharge axis. At high f i l l i n g pressures, the Hell region lags behind the Hel zone. This separation was f i r s t observed on framing camera records taken through interference f i l t e r s .  Confirmation was obtained by quantitative spectro-  scopic measurements which show higher temperatures at larger r a d i i .  Peak  Hell radiation s t i l l coincides with the position of maximum current density. However, the current shell extends inward beyond the leading edge of the Hel radiation zone. Strong continuum radiation occurs in the axial region as the current shell reaches the discharge axis, even though the luminous zone remains several centimeters off axis.  The early appearance of axial  -128continuum and the. extension of the current shell beyond the visible plasma region indicate a string perturbation of the gas ahead of the luminous plasma shell. Pressure and density determinations behind this discontinuity show that i t can be interpreted as a non-luminous shock front. The pressure was calculated from a simple model in which the maximum current density region acts as a piston. The velocity of this piston i s assumed to be that of the Hel radiation peak. The results compare well with two independent measurements: i) the magnetic pressure B 2 / 2 U  0  i i ) the pressure i n the luminous region obtained from spectroscopic temperature and density measurements. The density i n the shock front was found from a model of the density profile across the observed current shell.  The width of the current shell  and the total mass contained within i t are known. By determining the local density at two points within the current shell, the density i n the shock front can be calculated with the aid of the model. The resulting density ratio of 4 is characteristic of a non-ionizing shock. Future work might be directed at detecting the non-luminous shock wave using schlieren and interferometer techniques.  The nature of the intensely  radiating pinch continuum can be investigated a) at sufficiently short wavelengths so that electron densities can be obtained from an optically thin approximation; b) at a long wavelength for which the optically thick approximation w i l l yield electron temperature.  -129-  The faint axial spot appearing ahead of the pinch continuum (Figure 4.4) is a subject worth further inquiry.  Finally, information of the discharge  action would be more complete i f reliable measurements of the voltage waveform can be conducted simultaneously with the discharge current of the Z-pinch.  -130-  BIBLIOGRAPHY Ahlborn, B. (1964), Doctoral Dissertation, Technische Hochschule, Munich. Ahlborn, B. (1966), Physics of Fluids %  1873-4.  Ahlborn, B., and Salvat, M. (1967), Zeitschrift fur Naturforschung 22a, 260-3. Allen, J.E. (1957), Proc. Phys. Soc. (London) B70, 24-30. Botticher, W. (1961), Proceedings of V International Conference on Ionization Phenomena in Gases, 2, 2182-90. Botticher, W., 580-6.  and Dammann, H. (1963), Zeitschrift fur Naturforschung 18a,  Botticher, W., and Berge, O.E. (1964), Zeitschrift fur Naturforschung T9_a, 1460-5. Botticher, W. (1965), Zeitschrift fur Naturforschung 20a, 1262-8. Conrads, H. (1966), Report^No. 360, Institut fur Plasmaphysik, Kernforschunganlage Julich. Cooper, J.R. (1966), Reports on Progress in Physics, Volume XXIX, Part I, p. 35. Curzon, F.L., and Daughney, C.C. (1961), Rev. Sci. Instr. 3^, 430-1. Courant, and Friedrichs (1948),"Supersonic Flow and Shock Waves", (Interscience Publishers). Daughney, C.C. (1966), Doctoral Thesis, University of British Columbia. Drawin, H.J.7., and Felenbok (1965), "Data for Plasmas in Local Thermodynamic Equilibrium", (Gauthier-Villars, Paris). Finkelnburg, W., and Peters, Th. (1957), Handbuch der Physik, Bd. 38, Spectroscopy II, p. 97. Gabriel, A.H. (1960), J . S c i . Instr. 37, p. 50. Griem, H.R.  (1964), Plasma Spectroscopy, McGraw-Hill.  Guderley, G. (1942), Luftfahrtforschung 19^ 302-312. Jukes, J.D. (1958), A Theory of the Fast-pinched Discharge, A.E.R.E.,GP/R 2293. Lunkin, Y.P. (1959), Soviet Phys.-Techn. Phys. 4, 155.  -131-  McWhirter, R.VJ.P. (1965), "Plasma Diagnostic Techniques", edited by R.W. Huddlestone and S.L. Leonard, Academic Press, (New York, London). Medley, S.S., Curzon, F.L., and Daughney, CC. (1965), Review of Scientific Instr., 36, 713. Mewe, R. (1967), B r i t . J . Appl. Phys. 1£, 107-118. Moore, C.E. (1959), A Multiplet Table of Astrophysical Interest, Nat'l Bureau of Standards, Technical Note 36. •Null, M.R., and Lozier, W.W.  (1962), J . Opt. Soc. Am., j>2, 1156.  Spitzer, L. (1956), "Physics of Fully Ionized Gases", (New York: Interscience). Tam, S.Y.K. (1967), Doctoral Thesis, University of British Columbia. Thompson, W.B. (1962), "Introduction to Plasma Physics", Pergamon Press. Wiese, W.L., Smith, M.W., and Glennon, B.M. (1966), Atomic"Transition Probabilities, Nat'l Bureau of Standards, (N.S.R.D.S.-N.B.S. 4, v o l . 1). Zwicker, H. (1964a), Zeitschrift fur Physik 177, 54-67. Zwicker, H. (1964b), Zeitschrift fur Physik 178, 189-199..  & Neufeld, R.C. ( 1 9 6 6 ) ,  Doctoral Thesis, University of B r i t i s h  Columbia.  -132-  APPENDIX I THE CONTROL CIRCUIT  The following account is an enlargement on the discussion in Section 2.1  A photomultiplier monitors the speed of the rotating mirror.  Since the discharge must occur when the mirror rotates at a preset speed (i.e., the "tripping frequency"), the control system must contain a frequency gate.  The frequency gate is opened only when the time interval  between pulses of the photomultiplier (T) reaches a required value t^. (the "tripping" time interval).  As the mirror speed is increased from  zero, the interval, between pulses decreases, f i n a l l y reaching the value t^.. The frequency gate must then issue a signal that initiates the Z-pinch discharge.  However, this signal must be delayed a time T^ to allow the  mirror to swing into the proper recording position.  A block diagram of the  control circuit i s given in Figure 1, while the pulse sequences formed by i t are shown in Figure 2.  A l l figure numbers relate to this appendix unless  stated otherwise. a) The Frequency Gate The pulse sequence emanating from the trigger beam photomultiplier (Figure 2a) Is fed into a pulse inverter and shaper (Figure 1).  This unit  inverts, amplifies, and improves the rise time of the incoming pulses (Figure 2b).  Each of these pulses is fed in turn to a system composed of a  Tektronix type 162 wave form generator and a type 163 pulse generator (Figure 1).  These units combine to form a delayed pulse and generate the  sequences of Figures 2c and 2d respectively.  The trailing edge of each  pulse in 2d appears exactly t^ microseconds (the "tripping" frequency time interval) after the leading edge of each pulse in the sequence 2b.  -133The undelayed pulse of 2b and the delayed pulse of 2d emerging from the pulse generator are then fed into the coincidence unit (Figure 1). The coincidence circuit sends out a very small pulse (about 5 volts) when either input pulse appears separately at one of the two input points (Figure 2e). than t^..  This occurs when the time interval T between pulses is greater  As the mirror speed increases, the pulses in Figure 2a to 2e  crowd closer together.  When the "tripping" frequency is reached, T = t^,  the t r a i l i n g edge of the pulse in Figure 2d coincides with the leading edge of the next pulse in Figure 2b. Both input pulses to the coincidence unit now occur simultaneously, giving an output of 25 volts characterized by the shape in Figure 2f. The f i r s t pulse from the coincidence unit in the sequence 2f triggers a delay unit (Figure 1). As the mirror speed increases, the pulse in Figure. 2b "moves forward" along the pulse in Figure 2d causing the coincidence circuit to produce a pulse sequence of the form in Figure 2g. b) The Delay System The delay system setting is chosen so that the f i r s t visible light from the discharge hits the top of the spectrograph s l i t (Figure 3.1, Section 3.1).  This corresponds to the time T^ required by the mirror to  swing from the triggering to recording position (positions A and B, Figure 2.1, Section 2.0),  This system is made up of a second pair of  Tektronix type 162 and 163 units.  Their action is the same as that  described above in the frequency gate portion of this section. The pulse sequence from the coincidence circuit in Figure 2f takes on the role of the sequence in Figure 2b. This causes an output from the waveform  . ' -134generator of the type 2c which in turn causes the pulse generator to form a pulse of the type in Figure 2d.  In 2d the time delay T^ = x, and for  the rotating mirror i t is of the order of 120 microseconds. No observable j i t t e r is apparent ( i . e . , ^ 0.08 microseconds). The output pulse from the delay unit fires a bistable multivibrator (Figure 1). c) The Bistable Multivibrator The bistable multivibrator functions as a single shot device. When manually reset, i t is ready to accept pulses above a discrimination level greater than 5 volts.  This is necessary since smaller pulses emanating  from the coincidence circuit might be passed through the delay unit, causing a premature discharge.  The arrival of a 25 volt pulse from the  delay unit sets the unit into i t s other stable operating mode, simultaneously producing a positive pulse of about 20 volts to f i r e the 2D21 thyratron pulse circuit (Figure 2.4, Section 2.2b). Subsequent pulses fed into the multivibrator do not cause pulse output to the 2D21 thyratron circuit until the multivibrator is manually reset. This "single shot" feature increases the l i f e span of the thyratron pulse generator which would otherwise be required to f i r e every t^ microseconds as long as the mirror rotates above the tripping frequency. d) Operating Procedure The sequence of operations in using the rotating mirror is as follows. The single shot unit is set manually and the mirror speed is increased gradually from zero. When the correct speed is reached, a trigger pulse  -135produced by the control circuit fires the bank. The a i r supply to the turbine which drives the rotating mirror is cut off immediately the bank fires.  This reduces the mirror speed again to zero. The above sequence  is repeated to give the desired number of exposures on the film strip or spectrographic plate. It should be noted that the mirror sweeps past the film again at a time Tjj + tf., since by then i t has turned a half-revolution (t^ = 250 microseconds for framing camera,  1 millisecond for the spectrograph).  No complications arise from this because radiation from the plasma dies away i n a time of the order T^ + 150  microseconds.  -136 Pd/se  a  i  Tekirronin Ttype /&3 puke teener Ate>y  Jha- per Uric/  from gj, photo muJtip/tcr X n\>erter or pu/ae 2<Ln*<-<zf t  Coineic/snen gate  e, f, g 7)ef<xy  /nti/tt-  unit  Yit>ra.-tor  Trigger Unit"  /p dr/gger Soj/tct)  Appendix I, Figure 1 Block diagram of control circuit (Letters refer to pulse trains in Figure 2 , below) T a)  -1.5vT 25v.  \  i_r  Pulse from photomultiplier T is time interval between pulses  u  T  Same pulse after inversion and shaping  b)  Output of waveform generatortriggered by pulse from b)  c)  d)  -60v.l 25v.-  i  i  i  I  e)  5v.-r_  U  I  U  25v.-f)  Output of coincidence unit Mirror rotating at less than tripping frequency Output of coincidence unit -Mirror rotating at tripping frequency  5v.-_ 25v.-  g)  Output of pulse generator .Pulse initiated when sawvoltage reaches preset value  Output of coincidence vmit Mirror rotating at greater than tripping frequency  5v.--  Appendix I, Figure 2 Pulse sequences used in triggering the discharge  -137:APPENDIX II ROGOWSKI COIL AND INTEGRATOR CIRCUIT The total current (I) through the discharge is measured by a Rogowski coil.  An 11.4 cm. length of RG65 A/U delay line, with i t s outer ground  shield removed, is placed between the insulated flat high current leads which conduct current to the discharge tube (Figure 1, this appendix). If the total current (I) is uniformly distributed over the f l a t current leads, the magnetic flux (B) through the coil(L) is proportional to (I). The voltage (v^) induced on the c o i l (L-j) i s proportional to dB/dt and therefore, dl/dt.  The output signal of the c o i l is integrated by the  passive integration circuit producing an output signal (v) proportional to I. A schematic diagram of the circuit i s given in Figure 2 of this appendix. The usual, method of calibrating the current i s to put I = kv  (1) (2)  where v I Q v c  = = = = =  the output voltage.of the integrator circuit, the discharge current, the charge on the condenser bank, the charging voltage of the condenser bank, the capacitance of the condenser bank.  Integrating the area under the curves in Figure 2.9 (Section 2.3), a value of k can be obtained by applying equ'n (2) above. Using this method, i t was found that k =  .094 x 10° amp/volt + 10$  (3)  -138Because of the relatively short time constant of the integration circuit used, equ'n (l) must be replaced by:  1  I = k'(v +  , / vdt)  (1a)  R i C i ;  so that equ'n (2) becomes: Q = k'l/vdt + - J — / / vdt dt' i i ^ R  where, as before:  c  (2a)  J  Q = the charge on the condenser bank v = output voltage of integrator circuit  but RJL, C^= the resistance and capacitance of the integrating circuit k' = calibration constant for the current (equ'n (1)) now modified for the short integration time. To check what effect the additional term of equ'n (1a), (i.e., [ Y.dt) J RC would contribute to the measured oscilloscope voltage, both "uncorrected" and "corrected" Rogowski traces were plotted for the i n i t i a l pressure 4,000 pHg helium (Figure 3, this appendix). From Figure 3, the "uncorrected" trace gives an underestimate of a "true" current reading that varies: from 10$ at 6 usee, to 35$ at 10 usee. Since the times of interest for the lower i n i t i a l pressures (i.e., 500, 1,000 and 2,000 pHg helium) occur sooner than the 4,000 jiHg case, the percentage error of the "uncorrected" traces with respect to a proper current calibration would be even less (see Figure 2.9, Section 2.3). Using the calibration constant k of equ'n (3), the maximum discharge current for each i n i t i a l pressure can be calculated, i.e., I •s kv max * max - the maximum oscilloscope voltage i n the f i r s t half-cycle of the discharge. x  where  v  -139The ringing period of the discharge and the maximum discharge current in the f i r s t half-cycle for the range of i n i t i a l pressures is as follows (from Figure 2.9): Pressure  v  500 uHg 1,000 uHg 2,000 '/iHg 4,000 jiHg  ^nax  w a x  2.3 volts 2.5 volts 2.75 volts 3.0 volts  216 x 235 x 260 x 232 x  Ringing period  10 amps 103 amps 103 amps 10 amps  37 x 37 x 35 x 34 x  3  3  10-6 sec. 10~° sec. 10=° sec. 10-6 . s e c  For two discharge systems which differ markedly only i n the size of capacitance C of the charging condenser bank, i t can be shown that: c  12  \  ?2  1  *  C  • • • • o f t  (/)  2  where  I = discharge current T = ringing period of the discharge C = capacitance of the condenser bank and the indices 1,2 designate the two systems.  From Tarn's thesis: C = 53 jaF,  l2,max  2  =  200 x 10  3  amps, T  2  = 22  psec.  The value of the capacitance in the discharge used here is C-j = 106 yF. The substitution of these values in equ'n (4) yields: L  1,max  = IT x 200 x 1 CP = 283 x 103 amps. *  T-j = J2 x 22 x 10-  6  = 31 x 10" sec. 6  These results compare favourably with the 4,000 p.Hg case in the table above. A similar trend of increasing maximum discharge current and decreasing ringing period over a range of i n i t i a l pressures is observed in argon (Daughney 1966, p. 62).  In that case, the integration constant was  8 times that of the ringing period of discharge. Despite the relatively short time constant of the integration circuit, the recorded values of the oscilloscope voltage can s t i l l give a reasonable measure of the discharge current. From Figure 3, i t can be seen that for  -140the time of interest (0 usee to 10 usee), the recorded voltage closely follows the "true" integrated trace.  With the proper choice of calibration  constant k (equ'n 3), the discharge current can be estimated at worst by a 20$ error.  A l l current traces in the text are derived from the "uncorrected"  voltage readings. The frequency x gain response of the integrator circuit was found to be f l a t up to 1 megacycle (Figure 4* this appendix).  Since the discharge  current frequency is approximately 20 kc/sec, the integration of input signals is reliable to the accuracy discussed above. However, the high frequency minima superimposed on the discharge trace are not smoothed out by the integrator.  If these minima are sharply peaked, then their  occurrence in time can be regarded as accurate within the bounds of experimental error (i.e., less than 0.3^isec). From the theory of the Z-pinch (Daughney 1966), the discharge current behaves as:  Kt)  =  V WL  s i n (Wt + at )  (5)  3  t  where,  For time t' ^  2y W  W a V L.j.  = = = =  the ringing frequency' a constant the applied external voltage the total inductance of the system.  , L^^. L. = the total inductance of the system at the beginning of the discharge,  so that I(t') = _V_ Li  t'  ......(6)  By measuring the i n i t i a l slope of the traces in Figure 2.13, an estimate of the i n i t i a l total inductance can be made: = 0.09 uHenries ± 10$. .  R  i  A/l/VW  Appendix II, Figure 2 Circuit diagram of Rogowski c o i l and integrator L—Rogowski coil L = 15 uH C = 16 5 x 10  .1 f f  . specific ajnpedance - 950 JX  R-j— "attenuating" impedance ~ 1500/1 integrator resistorimpedance = 470 R-i = 4 70 A = 47 A integrator t R2—cable terminating c^—integrator capacitanice = 0.1 x 10*"^f j constant = 47j v'—induced voltageo< dl/dt v - output voltage  -142-  I  o.3S  1—  1  1  !  1  !  1  e>.$ o.b D.7 o.? a.f/o  WJSS-) * &  1  X  1  J  1—:  t  1  1  1  1  f  £  7  $  —>Appendix II, Figure 4 Frequency response o f integrator  —A—  I  (  i \  f tc Zo So  ^  -143-  APPENDLX III CORRELATION OF MONOCHROMATOR TRACES WITH TIME-RESOLVED SPECTROGRAPHS PLATES A relatively long time constant was discovered in the time constant of the circuit connecting the photomultiplier to the monitoring oscilloscope. Appendix V discusses the method of overcoming this d i f f i c u l t y . In correcting the photomultiplier traces, some information is lost.  However,  Figures 1, 2 , and 3 of this appendix show that gross characteristics of the discharge, such as the appearance of continua, can s t i l l be determined. The break i n the f i r s t continuum for the 500 and 1,000 juHg cases is lost i n the reduced traces. By correlating the time axis of the spectrographic plate with a monochromator trace, an estimate can be made of the time-duration of the "dark line" or break i n the f i r s t continuum. Figures 3 . 6 to 3 . 9 are derived from this data reduction procedure. The sharp outline of this break implies a reproducibility of better than 0.25 usee, in the formation of the f i r s t continuum. An estimate of the j i t t e r in reproducibility of secondary continua can be obtained by analyzing the monochromator traces.  In Figure 1 of  this appendix, the 4695 A and. 4642 X traces show the f i r s t secondary continuum to occur 0.286 usee, apart.  Since the 4695 2. trace is close  to the Hell 4686 line, i t i s affected by broadening from Hell.  Therefore,  a further inspection of the two traces for j i t t e r in the second secondary Figure 2 of this appendix shows the 1,000 uHg o monochromator traces for two separate shots at 4640 A. There is good continuum is unproductive.  -U4correspondence for the f i r s t (pinch) continuum and also for the f i r s t secondary continuum.  The second secondary continuum shows a j i t t e r of  0 . 2 9 Jisec. Hence, the f i r s t (pinch) continuum occurs within a time interval of less than 0 . 2 5 usee, from discharge to discharge.  The appearance of  secondary continua can be affected by a j i t t e r of about 0 . 3 usee.  first d ord continuum *  to  ? n  Monochromator trace for A s 4695 &  tfith neutral  density f i l t e r  Monochromator trace of 4695 8 in this region  4686 8 -(Oil 4649) •Hell  —  Monochromator trace of lM2 8 in this region  Photomultiplier i n overdriven region  h"U0  3 "120 100  Monochromator trace  11 J "60  for  80  5  \ m 4642 8  no neutral density f i l t e r  "40  ^••20  t (jisec.)-  0  1  2  3  4  Appendix III, Figure 1  5  T 500 jiHg f i l l i n g pressure  Hel 5876 1  Monochromator trace  for A = 4696 8  with neutral density f i l t e r  3 4  t (jisec)*£)  5 6  05 Monochromator trace  +>  of 4696 I in this region  $ 9  H  •Hell 4686 8  CD  - ( o n 4649) Monochromator trace of 4640 8 in this region  cn  -p  H  !  -100 -80  1  •60 •40  I  r1  .  Monochromator trace for A = 4640 ft without neutral density f i l t e r  ,  •20 1  1  t  h—  1 — 1 — 1 — 1 —  iec)-*6 1 2 3 4 5 6  Monochromator trace for \ 4640 & with neutral density f i l t e r s  tOtisec)*  r  1  2  3  4  5  Appendix III, Figure 2  1,000^iHg f i l l i n g pressure  -«7-  -Hel 5876 2  Monochromator trace for A = 4702 8 jith neutral density f i l t e :  Mcnochromator trace of 4702ftin •this region Hell 4686 1 ' Mono chro a ato r trace of 404O 2. in  this region  Monochromator trace for X = 4640 a without neutral density f i l t e r  t(/usec)  3  4  5  Appendix III, Figure 3 2,000 /uHg f i l l i n g pressure  .-US-  APPENDIX IV MONOCHROM&TOR-PHOTOMULTIPLIER CALIBRATION a) Photomultiplier Responses . The photomultiplier tube (EMI, Type 9558B) was connected to a high voltage power supply with an output of 1450 volts.  To check the linear  response range of the tube, the monochromator was set at the Hell 4686 peak, and a set of neutral density f i l t e r s was used to step down light from the discharge tube. The intensity or radiation transmitted by the f i l t e r s is calculated from the relation: IT x  where I j D  10-D  incident intensity n.d. f i l t e r "index" transmitted intensity  Figure 1 of this appendix shows that "saturation" sets in for o s c i l l o scope readings above 25 volts. the same photomultiplier-power  Work done by others at this laboratory, with supply combination, shows that this linear  response region extends well below 4 volts, down to about 10 millivolts. To avoid saturation effects, a neutral density f i l t e r was used for cases in which the photomultiplier response approached the 20 volt region. The same f i l t e r was used throughout.  Its relative transmission v/as  calibrated with a tungsten lamp for the wavelength regions of interest. Wavelength  Hel  5876 1  Hell 4686 1  Hel  3SS9 £  Incident radiation Transmitted radiation  5.98 6.17  5.14  -149-  b) Instrumental Function and Response. The reciprocal dispersion and instrumental broadening in the f i r s t order spectrum of the apparatus were measured by using Geissler tubes for several elements.  The monochromator was swept through accessible regions  nearest the helium lines of interest by a continuous drive mechanism.  The  photomultiplier response was traced out by chart-recorder. Arc  Region measured (2)  Iron Argon Mercury Neon  3865-3903 4259-4272 5770-5791 702A-7032  For  Reciprocal dispersion (2/mm.) 10.2 10.1 9.9 10.0  an estimate of instrumental broadening, the following lines were  investigated: Spectral line (°0 Hgl 3 1 2 5 . 6 6  Hgl 3131.55 Hgl 3131.83  Hgl 3 6 5 0 . 1 5 Hgl 3 6 6 2 . 8 8 Hgl 4 0 4 6 . 5 6  Hgl 4077.81 AI 4 1 9 8 . 3  Pull half-width (£) 0.19 (0.22) 0.19 0.24  0.20 0.23 0.20 0.19  Spectral line (A) AI Hgl Hgl Hgl Nel Nel Nel Nel  4200.67 4358.35 5769.59 5790.65 5852.49 5881.89 6328.17 6532.88  Full half-width (§) 0.19 0.22 0.20 0.29  0.16 0.16  0.110.19  •taken with He~Ne laser.  A helium Geissler tube proved unsatisfactory since considerable broadening of the lines takes place. 0.7 °u  Hel 5876 gave a f u l l half-width of  The variation in half-width i n the table above occurs because the  lines scanned are not infinitesimally narrow. A reasonable approximation for the instrumental broadening can be taken as 0.2 2, v/hich remains constat over the region of interest (3,800-6,000 2).  -1.50c) Absolute Intensity Calibration. A carbon arc was used for absolute intensity calibration.  The results  for the Hell 4686 and Hel 5876 regions are given in Figures 2 and 3 of this appendix.  The operation of the arc is that outlined by Null and Lozier  (1962) where:  ^Q/T) =i!c_U£) =1 W  B B  (1)  (A,T)  for W = power radiated by carbon arc, KBB power radiated by black body; and the arc temperature T = 3,800° K, for a direct current 10 amps, 150 volts. c  =  The arc was a standard commercial model, made by Leybold, with the projection lens removed. The electrodes were oriented at 90° as prescribed by Null and Lozier.  Ringsdorff spectroscopic carbons  anode and cathode respectively.  RW202  and RW4OI served as the  A 150 volt, 15 amp regulated direct current  power supply (Sorensen Nobatron DOR 150-15) was employed in series with a high current, variable carbon resistor.  The arc was operated just below  the "hissing point" as described in the literature. By calculating the black body power emitted by the carbon arc, V7  )  from the formula> w  c(-* ) ~ 2hc  2  (  h c e  /  A k T  - 1)-  1  erg/ secern?  str.(AA  ,cm.)  (2)  the monochromator response can be calibrated at the-wavelengths of interest. The experiment was designed originally without absolute intensity measurements i n mind. The radiation calibration was done after timeresolved line profiles had been made of the discharge. The monochromator was realigned upon an optic axis outside the discharge tube and parallel to its axis.  The carbon arc was placed at the focal plane of the quartz-water  achromat. A quartz window and screen, (corresponding to the viewport and low tension electrode of the discharge tube), were placed at appropriate positions along the optic axis to duplicate the light path followed by the radiation from the discharge.  It was found that the anode spot image of  the arc was smaller than the s l i t height of 2 mm. used when analyzing the discharge radiation.  The s l i t height was stepped down during the; calibration  process to ensure that only the bright inner region of the anode spot was monitored.  The difference in s l i t heights was taken into account when the  power output of the discharge radiation was calculated. Hence, the monochromator response, in volts, can be related to the power radiated from Figures 2 and 3, and equ'ns ( 1 ) , ( 2 ) . ( 5800-5876 1 = A.26 x 10 - erg/ sec. cm? s t r . volt ( ^ ,cm.) ) 5876-5900 A = 4.29 x 1 0 erg/ sec. cm? s t r . volt ,cm.) 14  Hel  5876  1/f  '4600-4686 A = 5.53 x l O ^ erg/ sec. cm? s t r . volt ( A 4 ,cm.) 1  Hell A686  4686^ 940 1 ~ 5.59 x 1 0 ^ erg/ sec. cm. s t r . volt ( JS«\ ,cm.) 1  +  -152-  Appendix IV, Figure 2 Instrument response 4686 2 region  —  > Zo.  w  V-&>© ¥$60 Woc>  o  x (R)  1—  bo o  iH  —— — •  tftoo  •  1• . .  V7ff» ¥-8ao  •  Shoo S/0&  SU  ©  o 01 (0  U U  O •H rH P. •H  p  e  /o  f  s•  7-  ST-  o  Ph  O  •I -Z .3  .V . r 6 .7  .?  /./  Neutral density f i l t e r index D Appendix IV, Figure 1 Photomultiplier response V (in volts) as a function of intensity  S"?oo S30o 59oo  &/co  -A (2) Appendix IV, Figure 3 Instrument response •. 5876 A region  4>Zoo  -153APPENDIX V RETRIEVAL OF INFORMATION FROM "SPOILED" MONOCHROMATOR TRACES The relation between photomultiplier current, l ( t ) , and the voltage displayed on the oscilloscope, V(t), can be shown to be I (t) = J / _ + C dV R dt =  1 R  fV  (1a)  (t) + T  L  d dt  V (t)  (1b)  J  where ^ = RC = the time constant of the "integrator circuit" formed by the f i n a l stage resistor and cable capacitance (see Figure 1, this appendix). For times of interest^"Tl , equ'n (1b) becomes the standard formula I(t) = V(t)/R.  However, i n measuring the rise time of the photomultiplier,  i t was found that t' = 5.2 usee. For times r  5.2 usee, the second term  of equ'n (1b) becomes significant. The circuit was properly terminated and i t s response compared with the arrangement used under experimental conditions (Figure 2, this appendix).  Figure 1 Circuit involving f i n a l stage resistor and coaxial cable Js>  Photomultiplier current l ( t )  .  1 |^ I  j  4""  £j>  Oscilloscope voltage V(t)  R = f i n a l stage resistor of photomultiplier C = capacitance of cable  This provides a test of the value ' f = 5.2 jusec. in Figure 2a,b  The oscilloscope traces  (this appendix) were provided by S.S. Medley. They show  monochromator responses taken perpendicular to the discharge axis in argon at 1,000-^Hg. Figure 3a shows an oscilloscope trace for a discharge in helium at 2,000 yuHg. The observation was made 1" parallel to the discharge axis. Figure 3b displays the proper response after corrections for the neutral density f i l t e r and equ'n (1) are applied.  With the aid of a ruled  magnifying glass, the traces can be read to 0.05^p.sec. for an oscilloscope display of 2 usee./cm. Figure 5.2 (Section 5.1) shows a typical line profile constructed from monochromator traces illustrated by Figure 3b.  The multiplicity of  observation points and the shape of the line near i t s half-height ensure a determination of "half-widths" by, at worst, 20% error.  Intensity and  half-width measurements were made from graphs approximately four times the size of Figure 5.2.  As mentioned in Section 5.1, only two points in time  were used from each trace; since a differentiation according to equ'n (1b) is involved at each of the 30 points used in constructing Figure 3b (this appendix).  Although the procedure is lengthy, the circuit construction  allowed an absolute intensity calibration to be made. It was noted in Appendix TVc that the experiment was not originally designed with absolute intensity measurements in mind. Had the circuit been terminated properly, the relatively weak carbon arc signal would not have registered above the photomultiplier noise signal.  -155-  a) Oscilloscope display of properly terminated photomultiplier circuit  CO  p  1  o  >  5  10 15  . t (jusec.)  M  20 25  30  35 40  45  50  —  40 b) Oscilloscope display of photomultiplier c i r c u i t as used in experimental conditions  30 20  /  $ 10 O  5  10  15  20  25 30 35 40  45 50  (usee.)—  c) Comparison of waveforms • — - - t r a c e a) enlarged by constant factor trace b) corrected The ordinate shows the true voltage response to the photomultiplier current l ( t )  Appendix V, Figure 2  Correction of monochromator trace?, and check on retrievability of information  -15.6-  a)  Hel 5876 1 for helium at 2,000 jiHg, 1" off-axis.  to rH O >  t  0 2 (jisec.)  4 6 8 012  Taken with neutral density f i l t e r .  >  Appendix V, Figure 3 Correction of monochromator trace  -157APPENDIX VI PARTITION FUNCTIONS FOR HELIUM The partition function (Z) is and expression giving the distribution of atoms occupying different energy states i n a system:  z=  Bn  e-  (En  ^ -  (  T)  D  where the summation is taken over a l l energy states of the system. g E T k n  n  = = = =  the s t a t i s t i c a l weight of the n** state the energy of the n^* state absolute temperature the Boltzmann constant 1  1  For neutral helium, the partition function i s as follows (energies i n electron volts): _ xo*7  X°S3  1 + 3e~  +  k r  a 3-62.  +  3e " "fr  +e  +  20e Ar  +  +  51 e  ^3 feV  e"  + Kr  3e~  +  i  + i9e  a n -*T 13VJ  + 3e~Tr  k  +  ~  l  23-fl *T  +  + e  *T  + e  ~W  13-V 3e~ * r  3  _ e  +  33 ?f  *r  9e~ *  T  23  ?v  + 96e kr  it-II  + 1446  *  r  + 196e *  T  + 256e ~ kr (2)  where the summation is taken up to n = 8. This estimate was given by Wiese, based on energy values from Moore*s tables (1959). It i s applicable for L.T.E., but does not take into account the lowering of ionization energy (^E) due to electric microfields within the plasma. These restrictions apply to the following formula for the partition function of singly ionized helium:  8 Z  - (Vn2)]  2  = 2 ^*2nV J  H e +  +  2 with E = 54.4 electron volts. +  () 3  -158-  Equation (3) i s based on the hydrogenic behaviour of Hell. Values for Z^ o (T) and Z e  H q +  (T) are tabulated on the follov/ing  pages for 10,000° <r T <" 60,000° K.  They are compared with more detailed  results published by Drawin (1965), where the effect of-a E i s taken into account.  The calculations based on equations (2) and (3) were  obtained from a computer program furnished by Mr. R.N. Morris. Depression of the ionization limit (A E), due to the Stark effect, can be ignored since for T  n  e e  = 40,000° K 17 -3 = 1.7 x 10 cm  j-  A  E = 0.19 e.v.  The tables show the calculated partition functions to be consistent with a A E in the region of 0.20 electron volts for an error of about 5$.  -159-  Table (i) Partition function (Z) for neutral helium Temperature  Z  Z (Drawin- 1960 AE =0.10 AE = 0.25 AE =0.50 AE = 1.0 AE= 2.0 (e.v.)  17500 18000 18500 19000 19500 20000 20500 21000 21500 22000 22500 23000 23500 2/000 24500 25000 25500 26000 26500 27000 27500 28000 28500 29000 29500 30000 30500 31000 31500 32000 32500 33000 33500 +  34000  34500 35000 35500 36000 36500  1.0001 1.0002 1.0003 1.0004 1.0006 1.0008 1.0011 1.0015 1.0021 1.0028 1.0037 1.0048 1.0062 1.0079 1 .0100 1.0125 1.0155 1.0190 1.0232 1.0282 1.0339 1.0406 1.0482 1.0570 1.0670 1.0782 1.0910 1.1053 1.1213 1.1391 1.1588 1.1807 1.2047 1.2311 1.2600 1.2915 1.3259 1.3631 1.4034  (e.v.)  (e.v.)  A E = 3.0  (e.v.)  (e.v.)  (e.v.)  1.000  1.000  1.000  1.000  1.000  1.000  1.001  1.000  1.000  1.000  1.000  1.000  1.001  1.000  1.000  1.000  1.000  1.000  1.002  1.001  1.000  1.000  1.000  1.000  1.004  1.002  1.001  1.000  1.000  1.000  1.008  1.003  1.001  1.001  1.000  1.000  1.014  1.005  1.003  1.001  1.001  1.001  1.025  1.009  1.004  1 .002  1.001  1.001  1.048  1.014  1.007  1.003  1.002  1.002  1.072  1.024  1.012  1.005  1.003  1.003  1.117  1.038  1.018  1.008  1.004  1.004  1.186  1.060  1.029  1.011  1.006  1.006  1.289  1.093  1.044  1.017  1.009  1.009  1.440  1.140  1.066  1.025  1.013  1.013  1.656  1.208  1.096  1.036  1.019  1.019  -160-  Table (i) (cont'd)  Z (Drawin, 1965) Temperature 37000 37500 38000 38500 39000 39500 40000 40500 41000 41500 42000 42500 43000 : 43500 44000 44500 45000 45500 46000 46500 47000 47500 48000 48500 49000 49500 50000  Z 1.4469 1.4937 1.5441 1.5980 1.6558 1.7174 1.7831 1.8530 1.9271 2.0057 2.0889 2.1767 2.2694 2.3669 2.4695 2.5773 2.6902 2.8086 2.9324 3.0617 3.1966 3.3373 3.4838 3.6362 3.7945 3.9589 4.1294  LE  =0.10 AE = 0.25 AE=0.50 AE= 1.0 (e.v.) (e.v.) (e.v.) (e.v.)  A E =2.0 AE=3.0 (e.v.) (e.v.)  1.961  1.302  1.138  1.051  1.026  1.026  2.332  1.431  1.196  1.071  1.036  1.035  2.954  1.605  1.273  1.097  1.049  1.047  3.717  1.837  1.375  1.130  1.065  1.063  4.719  2.140  1.507  1.173  1.085  1.063  6.017  2.529  1.676  1.228  1.110  1.107  7.671  3.024  1.888  1.295  1.141  1.137  -161-  Table ( i i ) Partition function (Z) for singly ionized helium Z (Drawin, 1965) Temperature  36000 36500 37000 37500 38000 38500 39000 39500 40000 40500 41000 41500 42000 42500 43000 43500 44000 44500 45000 45500 46000 46500 47000 47500 48000 48500 49000 49500 50000 50500 51000 51500 52000 52500 53000 53500  Z  2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.000 2.001 2.001 2.001 2.001 2.001 2.001 2.001 2.001 2,. 002 2.002 2.002 2.002 2.003 2.003 2.003 2.004 2.004 2.005 2.005 2.006  AE =0.10 AE =0.25 AE= 0.50 A E =1.0 AE =2.0 AE = (e.v.) (e.v.) (e.v.) (e.v.) (e.v.) (e. 2.000  2.000  T = 3&182 2.000 2.000  2.000  2.000  2.000  2.000  2.000  2.000  2.000  2.000  2.000  2.000  2.001  2.001  2.001  2.00-1  2.002  2.002  2.003  2.00.5  T = 37,991 2.001  r  2.000  2.000  2.000  T = 39,891 2.001  2.000  2.000  2.000  T =41,885 2.003  2.001  2.000  2.000  T = 43,980 2.006  2.002  2.001  2.001  T = 46,179 2.211  2.004  2.002  2.001  T = 48,488 2.021  2.007  2.003  2.002  T = 50,912 2.039  2.012  2.006  2.003  T = 53,458 2.069  2.022  2.011  2.006  -162-  Table ( i i )  (cont'd)  Z (Drawin, 1965) Temperatur 54000 54500 55000 55500 56000 56500 .57000 57500 58000 58500 59000 59500 60000 60500 61000 61500 62000 62500 63000 63500 64000 64500 65000  3  z  2.006 2.007 2.008 2.008 2.009 2.010 2.011 2.012 • 2.013 2.014 2.015 2.017 2.018 2.020 2.021 2.023 2.025 2.027 2.029 2.031 2.034 2.036 2.039  E = 0.10' E = 0.25 (e.v;) (e.v.)  '2.121  2.038  E = 0.50 E = 1 . 0 ' E = 2.0 (e.v.) (e.v.) (e.v.)  E =3.0 (e.v.)  T = i56,131 2.018 2.009  2.006  2.004^  2.009  2.007^  2.014  2.011'  2.022  2.017  T = i58,937 ^2.205  2.065  2.030  2.015  T = t>1,884 '2.340  2.106  2.049  2.024  N  T = t>4,978 '2.551  2.171  2.078  2.038  N  -163APPENDIX VII SELF ABSORPTION EFFECTS The following analysis pertains to an absorbing, homogeneous, L.T.E. plasma.  If absorption and emission of radiation are considered to occur  as in a black body cavity, the observed intensity 1(A) can be expressed in general as:  n-t \ \ -i  I U)  =B U)  where I ^ U ) =  f 1 -e-  C ( A )  J  E(4 )g  (1) (2)  BUT" with  ^(-\) B(~\ ) E(-\ ) £  = the optical depth at wavelength \ = black body radiation at wavelength A = intensity of radiation emitted per unit volume ='the geometrical depth of the plasma along the line of observation.  a) Considerations of Optical Depth • If  )  1, the plasma is said to be optically thin to radiation  of wavelength A . From equ'ns (1) and (2), the observed intensity in the optically thin case I f c h i n ^ ^ ^  1 f t )  s  or 1, partial absorption occurs for radiation at wavelengthJv .  From equ'ns (1), (2), and (3), the ratio of observed radiation intensity l(A ) to the intensity emitted in the optically thin case I-fchin(^ ) ^  J  thinUT  E(^)£  f l D  s  .(4)  For "£"(4 ) » 1, the plasma i s s a i d to be o p t i c a l l y t h i c k w i t h respect to r a d i a t i o n o f wavelength  .  In t h i s case, equ'n (1) becomes  I(A ) = B(>v ) = 2h£  [V^lcT „ -fl -1  and the plasma emits r a d i a t i o n as a b l a c k body.  ( 5 )  -164-  b) Self Absorption for Lorentzian Line Profiles The optical depth at line centreT"( 4 ) for a Lorentzian line profile A  is (Griem, 1964; Cooper, 1966):  T<J.)-- T = £ where  r  o  V  (1 - .-W*„M > j  n  8  ......  (6)  w  £ = length of the discharge tube - wavelength of the line investigated (cm.) W = half of the true line half-width (cm.) f*mn = absorption oscillator strength for the line W = number of atoms' i n the lower state n (cm ) r = 2.82 x 10~ cm. ^  0  -3  n  13  Q  For radiation emanating at wavelength^ ^- «o, i n the line profile, the V  optical thickness*^ ( A ) can be expressed as:  r'°  s  0  t  2  ( 7  )  1 *,U -jlp)' W  2  I f the observed intensity I(A ) is found to f a l l to half i t s maximum value at  + <£~, then from equ'ns (1) and (7) 1 - e" °  where  = a1 - exp  c  o  is half of the observed half-width of the line centred at 4 . o  The expression above, thez-efore, yields the ratio of the observed halfwidth (2&) to i t s true value (2W)  r  W  f  In  T  0  [ 1 + exTP^r]  (8)  The effect of self absorption on the observed total intensity of the line (1+) can be calculated by integrating equ'n (1) over a l l wavelengths  -165-  oo  (9)  since B(J^ ) varies slowly over the line profile.  Integrating the right  hand side of equ'n (9) term by term, and using equ'n (2) for ^ = X , Q  gives: E(>„ ) £  L  A  16  °  384  .  .  The evaluation of equ'ns (8), (10) and Figure 1 of this appendix were provided by Messrs. H.D. Campbell and H.G. James. Assuming an L.T.E. distribution within an ionic species, N of n  equ'n (6) can be computed by the standard formula N.  n  where H  Q  = Ne-E-VkT  g  n  TfrT  i s determined in Table 5 - i , Section 5.2.  The absorption  oscillator strengths ( f ) are taken from Griem (1964). jnn  Spectral line  Transition  f ^  g  Hell 4686  3D - 4 F  1.016  18  Hel Hel  5876 3889  2  2  2P - 3D 23s - 3 P 3  3  3  0.62 0.066  n  9 3  c) Absolute Continuum Intensities Cooper (1966) provides a complete discussion of contributions to the absolute continuum intensity from bremsstrahluag and recombination radiation.  For our purposes, we consider only the bremsstrahlung component  which, for a hydrogenic plasma (Conrads, 1966), exhibits an optical depth <  ^'g(j^ ) for wavelength A. s  2" (-| B  ) = 3.44 x 10-  26  H  Ml ~ e x p A -  e  /  fc n n gg ^3 / E V % x p A E ^ \kT / v kT 2  i  c  1  he ^] \ kT G  .0) where ^ ( n ) = the number of ions (electrons) per cm"  3  g  EJ-J  = the ionization energy for hydrogen  £ = the geometrical depth of the plasma along the line of observation ^03"  A  =  ?P f •y-e_ \ 4TT6,•-, 1[ 4ire 4ifenn, J) 2  >i  <:  0  2  P  e  = lowering of ionization energy due to electric microfields within the plasma  gB (T«,4 ) = J T ln(2.25 AKL'e) = the Gaunt factor for TT be bremsstrahlung at large wavelengths. Z  • • e, 6  Q)  k, h, c  = the customary constants in c.g.s. units.  For a plasma, optically thin to radiation of wavelength -I , the power radiated per unit volume Eg(j^ ) from bremsstrahlung radiation becomes: M A  ) = 6.36 x - l C P  4 7  n.n  Z  2  1  g ? 01  \ )  (2)  Hence, for an appropriate choice of wavelength ( A ) , the density of ions (or electrons) can be calculated from equ'n (2). For  T *«80,000 K, 0  e  which would be expected in the plasma column at the pinch stage, the wavelengths most suitable for this investigation would be in the near ultraviolet range of the spectrum, i.e., J, < 3,000 \.  -167-  Appendix VII, Figure 1 Ratio of self absorbed, halfwidth <§"" to optically thin halfwidth tO for Lorentzian line shapes *?o = optical depth at line centre  


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