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Studies of the Z-pinch discharge in high pressure helium Preston, Jonathan Melvin 1974

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STUDIES OF THE Z-PINCH DISCHARGE IN HIGH PRESSURE HELIUM by JONATHAN MELVIN PRESTON B . S c , McMaster U n i v e r s i t y , 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of PHYSICS We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1972* In presenting th is thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary shal l make it f ree ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l ica t ion of th is thes is for f inanc ia l gain sha l l not be allowed without my wri t ten permission. Department of P h y s i c s The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada Date 4 Oct 74 ABSTRACT Improvements to the l a s e r e x c i t e d i n t e r f e r o m e t e r p r e v i o u s l y constructed i n t h i s l a b o r a t o r y are d e s c r i b e d . The use of a r o t a t i n g r e t r o - r e f l e c t o r i n the i n t e r f e r o m e t e r c a v i t y , and e l e c t r o n i c c i r c u i t r y , permit d i r e c t r ecording of e l e c t r o n d e n s i t i e s . The s e n s i t i v i t y of the i n t e r f e r o m e t e r i s 5 x 1 0 1 5 cm - 2, and the temporal r e s o l u t i o n i s 0.1 usee. Pyrex tubes, terminated by windows, en c l o s i n g the l a s e r beam, have been used to improve s p a t i a l r e s o l u t i o n . The e f f e c t s of the tubes, and of a l l other sources of e r r o r associated with i n t e r f e r o m e t r i c measurements of e l e c t r o n d e n s i t i e s , have been c a r e f u l l y assessed. A Z-pinch discharge i n 4 t o r r helium has been studied with t h i s instrument, and with s p e c t r o s c o p i c deter-minations of e l e c t r o n temperature. Under the c o n d i t i o n s chosen, the pinching plasma does not reach the axis of the discharge tube, but stops at a radius of 2.7 cm. The shock f r o n t caused by the pinching a c t i o n has been shown to be weak. L o n g i t u d i n a l s t r u c t u r e , due to heat f l u x , c h a r a c t e r i s t i c cathode behaviour, and v a r i a t i o n s i n timing of the pinching a c t i o n , has been found. i i The discharge has been assessed as a spectroscopic source and as a medium for laser scattering experiments. The characteristics of two suitable regions are presented. These are the axial region and the hollow cylinder of plasma formed by the arrest of the pinching plasma. Both are free of i n s t a b i l i t i e s . At the appropriate time the latter is also current free, and shows longitudinal electron density varia-tions of less than 2%. The length of time that the current spends near the wall of the vessel is determined by the balance between kinetic and magnetic pressures. 1 1 1 TABLE OF CONTENTS Page Abstract i i Lis t of Tables . v i i i Lis t of I l lustrations ix Acknowledgements . x i i Chapter 1 INTRODUCTION 1 PART A - DIAGNOSTICS 2 DIRECT READING HIGH FREQUENCY LASER EXCITED INTERFEROMETER -4 2.1 Introduction 4 2.2 Principles of Laser Excited Interferometery . . . 7 2.2.1 Refractive Index of a Plasma . . . . 7 2.2.2 Early Fabry-Perot Interferometers. . 10 2.3 Fractional Fringe Techniques. . . . . . . . 13 2.3.1 Introduction 13 2.3.2 Rotating Retro-Reflector . . . . . . 18 i v Chapter Page 2.3.3 Integrating Frequency Modulation Detector 21 2.3.4 Calibrating the Integrating Frequency Modulation Detector. . . . 26 2.4 Defining the Plasma Length 28 2.4.1 Introduction 28 2.4.2 Interaction between the Plasma and the Tubes 29 2.4.3 Refractive Index Gradients . . . . . 34 2.5 Extraneous Contributions to the Optical Path 35 2.5.1 Introduction 35 2.5.2 Other Terms in the Refractive Index 36 2.5.3 Geometrical Contributions 41 2.6 Spurious Fringes . . 44 2.7 Experimental Application 47 2.7.1 Construction . 47 2.7.2 Alignment 51 2.7.3 Triggering 53 2.8 Sensit ivity and Accuracy. 55 2.8.1 Introduction 55 2.8.2 Sensit ivity 55 2.8.3 Accuracy 56 2.9 Comparison with Electron Density as Determined by Stark Broadening Measurements 59 v Chapter Page 3 ELECTRON TEMPERATURE MEASUREMENTS 65 3.1 Introduction 65 3.2 Theory - Local Termodynamic Equilibrium? 66 3.3 Apparatus 74 PART B - PINCH DYNAMICS 4 RADIAL DYNAMICS OF A Z-PINCH IN 4 TORR HELIUM 79 4.1 Introduction 79 4.2 Apparatus 81 4.3 Radial Dynamics 86 4.3.1 Introduction . 86 4.3.2 The Precursor Shock 93 4.3.3 The Pinch Phase. . . . . . 102 4.3.4 The Post-Pinch Phase 106 4.3.5 Effects of Polarity Reversal . . . . 113 5 LONGITUDINAL STRUCTURE . 122 5.1 Introduction 122 5.2 Longitudinal Heat Transport 123 5.3 Longitudinal Electron Temperature and Density Distributions 134 6 CONCLUSIONS 141 6.1 Contributions to Plasma Diagnostics . 141 vi Chapter Page 6.2 Contributions to Pinch Dynamics 142 6.3 The Z-Pinch as a Spectroscopic Source or a Scattering Medium 143 6.4 Proposals for Future Work 150 REFERENCES • 152 APPENDIX: WALL-HANG-UP TIME IN THE Z-PINCH. . . 156 vi i LIST OF TABLES Tab! e Page 2-1 L i m i t a t i o n s of Conventional Interferometers 5 2-2 Angular Frequencies Associated with a Plasma 9 2-3 Anomalous R e f r a c t i v i t y of a Helium Plasma at 632.8 nm and at 441 .6 nm. . . . 37 2-4 C r i t e r i a f o r V a l i d i t y of PLTE 38 2-5 Maximum C o n t r i b u t i o n s of Various Species to the R e f r a c t i v e Index 40 2-6 Interferometer Components 49 2- 7 Accuracy of the Interferometer 58 3- 1 Populations of the He II Ground S t a t e . 72 4- 1 Z-Pinch Components 82 A-l Wall Hang-up Times . . 161 v i i i LIST OF ILLUSTRATIONS Figure Page 2-1 Fractional Fringe Interferometer -Schematic 6 2-2 Unmodulated Interferometer 11 2-3 Airy Function 14 2-4 Ray Diagram at the Retro-Reflecting Prism 19 2-5 Operation of the IFMD 24 2-6 Typical Electron Density Prof i le . . . . 24 2-7 Plasma-Tube Interaction - Electron Density 32 2-8 Geometrical Contributions to the Optical Path 42 2-9 Spurious Fringes 46 2-10 Effect of Spurious Fringes 46 2-11 Fractional Fringe Interferometer - Scale 48 2-12 Triggering and Recording 54 2-13 Recorded Profile of He I X 389 nm . . 61 i x Figure Page 2- 14 Comparison between Electron Densities Measured Interferometrically and by Spectral Line Widths 63 3- 1 Plasma-Tube Interaction - Line Emission 75 3-2 Optical System 76 3- 3 Typical Record of Total Line Intensities 78 4- 1 Circui t Diagram of Z-Pinch 84 4-2 Breakdown in a Z-Pinch 87 4-3 Pinch Dynamics (r , t ) Diagram 91 4-4 Current Density and Magnetic Field Densi ty 92 4-5 Electron Temperature and Density at 6 usee. . . . 104 4-6 Electron Temperature and Density at 9 usee. . . . 105 4-7 Electron Temperature and Density of Pinching Plasma 107 4-8 Electron Density Profiles at Radii of 2.6 and 2.8 cm 109 4-9 Electron Temperature and Density at 11.5 usee . . I l l ,4-10 Breakdown Mechanisms in both Polarit ies 114 4-11 Electron Temperature and Density of Pinching Plasma - Anode as In i t ia l High Field Electrode 117 4-12 Pinch Dynamics (r , t ) Diagram - Anode as In i t ia l High Field Electrode 119 x Figure Page 5-1 Contours of Electron Density during Pinch 131 5-2 Longitudinal Structure on Locus of Electron Density Maxima 132 5-3 Longitudinal Structure at r = 4.5 cm, t = 7.0 usee 136 5-4 Longitudinal Structure at r = 3.0 cm, t = 10.0 ysec 137 5-5 Longitudinal Structure on Axis at t = 13.0 ysec 139 A-l Wall Hang-up Times as a Function of F i l l i n g Pressure 163 A-2 Dynamics of Current Density Shells at t = t 166 we xi ACKNOWLEDGEMENTS I would l ike to thank Professor F . L . Curzon for his supervision and encouragement at a l l stages of this work. Jack Bosma and Dick Haines provided valuable technical support and instruction. The assistance of Doug Sieberg and Jim Zanganeh in maintaining the apparatus was most important. The current density measurements which were so useful in understanding the results presented here were obtained by Jaroslav Pachner in this laboratory in 1971. Without his work this thesis would have been much less complete. One of the most important advantages of working in a group such as the UBC Plasma Physics group is that the collective knowledge available is considerable. Many dis-cussions with Gary Albach, Dave Camm, and Mark Churchland are gratefully acknowledged, as are dealings with Bruce Armstrong. Lome Gettel digit ized much of the data presented here. Many of the figures were drafted by B i l l Basaraba, Esther Finlay, or Dale Stevenson. A f i r s t rate typing job has been done by Shari Haller. Al l of them deserve warm thanks. xi i The financial assistance of the National Research Council and the Atomic Energy Control Board has been gratefully received. x i i i Chapter 1 INTRODUCTION The work described in this thesis involves three areas of plasma physics; plasma diagnostics, pinch dynamics, and plasma spectroscopy. Interesting and useful new con-tributions have been made in a l l these f ie lds . An intro-duction to each topic is presented at the beginning of the relevant part of the thesis. Chapters 2 and 3 constitute Part A. The f i r s t of these describes the fractional fringe, direct reading, laser interferometer used to measure electron densities in a pinch discharge. Improvements in this interferometer, resulting in an instrument of unparalleled convenience, high sens i t iv i ty , and good accuracy, have been major con-tributions of this thesis. Chapter 3 relates the accepted techniques which were used for the measurement of electron temperature. Part B of the thesis consists of Chapter 4 and 5. Therein are described the radial and longitudinal dynamics, 1 2 respectively, of the high pressure z-pinch. Emphasis has been placed on the contributions made during this work. These include a new understanding of the shock front caused by the pinching plasma, and the f i r s t investigation of longitudinal electron density and electron temperature variations in the high pressure z-pinch. The high pressure regime is partiicularljf interesting for the last mentioned study, since in this regime longitudinal variations are not dominated by i n s t a b i l i t i e s . This is true because the pinching plasma does not reach the axis, but stops at a radius of several centimetres. Instabi l i t ies are important neither in the pinching plasma nor in the plasma which eventually forms on axis. In the concluding chapter, the data described in part B, which were measured by the techniques of Part A, are analyzed to assess the su i tab i l i ty of the high pressure z-pinch as a spectroscopic source, or as a medium for laser l ight scattering experiments. It is concluded that there are two regions of the discharge which are potentially valuable for these purposes. An additional experiment, concerning the length of time that the plasma spends near the walls of the discharge tube, before commencing its pinching motion, has been rele-gated to the appendix. This was done not because no conclusion was reached, nor because the results were not 3 interesting, but rather because the experiment raised more questions than i t answered. Hopefully this experiment wil l be expanded into some definit ive work on the formation of the pinching electron density and current density shel ls . Each chapter is terminated by a paragraph summariz-ing the new contributions to plasma physics described in the chapter. Throughout, an effort has been made to place experimental details in tables so that the thesis remains uncluttered. A l i s t of these tables can be found in the introductory pages. In the diagrams, least squares f i t s to experimental data are shown as solid l ines , smooth curves drawn through data points to show trends are interrupted at the points. Chapter 2 DIRECT READING HIGH FREQUENCY LASER EXCITED INTERFEROMETER 2.1 Introducti on One of the major contributions of this investigation was the improvement of the interferometer previously developed in this laboratory (Medley, 1970; Curzon and Funk, 1970; Funk et al. , 1972). It is the function of sections 2 through 4 of this chapter to explain the principles and past short-comings of the interferometer, and to show how these have been corrected or reduced. Figure 2-1 and Table 2-1, showing respectively a schematic of the present interferometer and a l i s t of limitations of past interferometers and their present solutions, are included here for ready reference. The present interferometer has been shown to be an accurate and convenient measurement tool with high sensit ivity and resolution. Another contribution has been the careful study of a l l possible sources of error in interferometric 4 5 Table 2 - 1 Limitations of Conventional Interferometers Limi tation Present Solution Secti on 1 . Light Source Lasers 2 i Monochromatic i i Intense i i i Coherent 2 . Lack of Sensit ivity Fractional Fringe Techniques 3 3 . Turn-over Points •I H 3 4 . Temporal Resolution H II 3 5 . Refraction due to Density Gradients causes: i Loss of overlap i i Enlarges optical path a. Two lenses focus beam at centre of plasma b. Glass tubes l imit path 4 5 6 . Spatial Resolution i Transverse to beam i i Parallel to beam Beam is focussed Moveable Glass tubes 4 5 7 . Other changes in optical path i Mechanical Vibration i i Imperfect components Short time scale 8 CW laser M, z pinch Plasma M3 M 4 c Rotating prism retroreflector 10m 0^ 7 investigations of this nature. These are described in sections 5, 6, and 8. Section 7 contains details of the interferometer as here constructed. These details have been withheld until section 7, so as to emphasize the general appl icabi l i ty of this interferometric technique. Also, details of the plasma which served as the test model have been withheld until Chapter 4 to avoid confusion. Suffice i t to say that this plasma is a typical laboratory plasma with electron density between 10 1 5 and 5 x 10 1 7 cm"3 and an electron temperature of 2 or 3 ev. in a magnetic f ie ld of up to 1 weber m - 2 . F inal ly , section 9 describes a comparison between this interferometer and measurement of widths of broadened spectral lines as methods for determining electron densities. The good agreement obtained provides a firm foundation for further work with this interferometer. 2.2 Principles of Laser Interferometry 2.2.1 Refractive Index of a Plasma The measurement of the refractive index of a plasma as a means of determining the electron density has been in use for many years and in many different applications. Its popularity is due to the simple relationship between the refractive index and the electron density. . 8 The refractive index of a Vlasov plasma is easily calculated. One uses Maxwell's equations and the Vlasov equation to calculate the response of the plasma to incident plane electromagnetic waves. The resulting dispersion equation yields the velocity of these waves in the plasma and thus the refractive index. The plasma may contain bound electrons also; these contribute to the refract iv i ty due to anomalous dispersion near their spectral l ines . Thus the refractive index, n, is given by w2 1 - n 2 = e • 1 oo f . N . + Y , J J . 1 . i i - Mw* - (Of) j = s p e c t r a l 1 i nes e j (2-Al l these frequencies and their definitions are l i s ted in Table 2-2. The co l l i s ion frequencies are estimates by Spitzer (1956). This table also l i s t s values of these frequencies for a typical laboratory plasma (N g = 10 1 6 cm"*3, T f i = 3 ev, B = 1 weber m~ 2), and for the He-Ne laser (X = 633 nm). Most of these frequencies may clearly be ignored. Throughout this experiment, then, we can write Table 2-2 Angular Frequencies Associated with a Plasma Symbol Defi ni tio i Value rad s ec - 1 (He piasma) Typical Value rad s e c - 1 Plasma electron Plasma ion Cyclotron electron Cyclotron ion Col 1isions: electron-electron ion-ion ion-electron Incident Radiation Spectral "Be uBi ee v i i v i e a. e 2 N. N z 2 e 2 N. m. e, eB/m, ezB/nu e^ N ir.h 25.8 /if e0 /~rr. k3 T ; " e e e-z" N. «n,V 25.8 /? e„ /n.T ks T 3 i t v e e / 2 / 2 2irc/n> 2ifc/nX ^ 5.646 x 10" /N~ e 30.7z /FTj" 1 .76 x 10 1 1 B 9.58 x 107 zB ! 3 x IO"6 JlnA N /T 3/2 e' e 2 x 102 v Q ee ee 1 .883 x 101 8 / X 5.6 x 10 1 2 3.1 x 10 s 1,8 x 10" 9.6 x 107 4 x 10 1 0 8 x 108 1 x 10 1 0 3.0 x 10 1 5 spectral lines of species where: A 12TT (eo kT)" e 3 /N~ e 3/2 = 2 X 10 l o e = charge of electron me,m-j = mass of electron, ion N e > i = electron, ion density (cm - 3) B = magnetic f i e ld (W m - 2 ) T e -j = electron, ion temperature (ev) z = charge of ion c = speed of l ight e0 = permittivity of free space k = Boltzmann's constant X = wavelength of incident radiation f. = osc i l lator strength of jth l ine N. = lower state population of jth l ine J u. = frequency of jth l ine J 10 2 2 co , co f. N. s p e c t r a l e J 1 i nes Also, in calculating x , a negligible error is produced by setting n = 1. For the moment we wil l drop the spectral term. It wil l be considered again in section 2 . 5 . 2 . The difference between the refractive index of a plasma and a vacuum is thus co- 2 / 2 w 2 , which is of the order of 10' 6 . This is advantageous, since the probing radiation does not perturb the plasma, but also challenging since sensitive measurement techniques are required to detect such a difference. Interferometric techniques are well nigh mandatory since a change of optical path length which exceeds but one-half wavelength of the probing radiation can be detected in that way. 2 . 2 . 2 Early Fabry-Perot Interferometers If the plasma has an axis of symmetry one approach is to surround the plasma with a Fabry-Perot cavity along that axis and excite that cavity with a laser, as depicted in Figure 2 - 2 . This was f i r s t done, almost simultaneously, by Ashby and Jephcott (1963) and by Gerardo and Verdeyen (1963). 11 LASER TUBE PLASMA -SO-UGHT DETECTOR Mi Figure 2-2. Unmodulated Interferometer M 2 , M3 semi-reflective mirrors. The phase, $, of l ight completing a double transit of such a cavity wil l change at the rate 2nL (2-3) where L is the total cavity length and L/2 - _ 1 n = r n(x) dx (2-4) -L/2 12 is the average refractive index. This assumes that the coherence length of the laser exceeds 2L. From eq. 2-2 and 2-3 Thus dt * ^ Z 1 - A m e e 0 8 P ^ T F 3_le 3t 2irC2m e. e 1 *Ve2A $(t) = -2.809 x 101 6 cm-^sec-1 »(t) (2-5) (£ in cm) Note that the total cavity length, L , has been replaced by the length of plasma in the cavity, l, since outside the plasma 8n/8t vanishes. N g is defined analogously to rf 1/2 1 (x) dx (2-6) -%I2 Therefore 13 111 Ne(x)dx = 2.809 x 1 01 6 A<f>(t) (2-•1/2 N g in c m - 3 , $ in s e c - 1 Thus by deducing the phase of the cavity from the detector signal one can record the integral of electron density along the l ine of sight. Equation 2-7 is the basis of a l l laser excited interferometric techniques, of which this simple system is only the beginning. 2.3 Fractional Fringe Techniques 2.3.1 Introduction The simple interferometer described in the pre-ceding section suffers from several drawbacks. They are l is ted in Table 2-1. Here the reasons for the lack of sens i t iv i ty , ambiguities, and poor temporal resolution are considered. These flaws are greatly relieved by fractional fringe techniques, which are described in the remainder of this section. Equation 2-7 does not predict any l imit to the sensi t iv i ty . Nevertheless the l imitation arises from physical 14 rather than technical, considerations. Assuming a linear detector, the recorded intensity from a Fabry-Perot cavity varies with phase in accordance with the Airy function V(t) - ] ^ _ 1 + F s i n 2 where F is the finesse of the cavity. This function is sketched in Figure 2-3. Figure 2-3. The Airy Function for two Values of Cavity Finesse. Now the finesse is not generally known. Even i f i t were measured shortly before the plasma was introduced into the cavity i t would then change, because any refractive index gradients in the plasma deflect l ight rays traversing the cavity and thus alter the degree of interference. Other 1 5 possible causes are absorption or, more l i k e l y , incoherent plasma emission at the laser wavelength, and possible mechanical vibrations. The situation becomes further con-fused i f the cavity is modulated over-rapidly (Williamson and Medley, 1969; Dangor and Fielding, 1969). Thus the finesse is unknown. A glance at Figure 2-3 wil l show that, without knowledge of F, a value of, say, V = 0.5 cannot be interpreted in terms of phase. Indeed only maxima and minima of V correspond to unique values of phase, namely $ = 2m7r or 2(m + l/2)ir. The sensit ivity is therefore limited by the fact that only extrema represent points of well defined phase, and the phase must change by ir between these points. The electron density can be accurately measured only at these extrema. Note that this is true of a l l interferometers, not only of Fabry-Perot cavi t ies , although i t is most severe in the latter case. In other systems the degree of inter-ference s t i l l depends on the amount of overlap of the sample and reference beams on the detector, which changes for the same reasons the Fabry-Perot finesse changes. This is a serious drawback to simple interferometric technique since the minimum detectable electron density is thus 16 1/2 Ne(x)dx = 2.809TT x 10 1 6 = 8.8 x 10 1 6 cm"2 (2-9) -%I2 (e.g. N g = 9 x 10 1 5 cm - 3 i f a 10 cm ray through the plasma is used.) Further, since the i n i t i a l phase at the commence-ment of data collection is s imilarly known only to within IT , an error of this magnitude is attached to each value obtai ned. The reason for low temporal resolution also follows from this discussion; data is obtained only at these points of well defined phase. A different problem, one that is just as serious, wil l also be discussed here since its solution is the same as the solution to the poor sensit ivi ty and poor temporal response. This problem is the ambiguities arising from 'turn-over' points. With reference to Figure 2-3 consider the effect on the emitted intensity of a change of sign of d<f>/dt. This change of sign is called a turnover point. Occasionally such points are noticeable as discontinuities in the recorded intensity V(t) . However at certain values of phase, such as 2(m + l/2)ir, they would pass unnoticed. Such ambiguities are rarely acceptable, since grossly inaccurate conclusions may be reached. 1 7 The solution to a l l these problems l ies in the application of fractional fringe techniques. A fringe is a maximum in intensity which occurs when the phase has one well defined value, namely a multiple of 2i\. Fractional fringe techniques increase the number of fringes, and thus the number of points at which the phase is well-defined. This is achieved by altering the optical length of the cavity hy some external mechanism at a rapid rate, while, simul-taneously, the changing electron density also affects the optical length. Since a fringe is produced for each change of 2IT in phase, the changes due to electron density modulate the frequency of fringes produced by the external mechanism. The minimum detectable electron density is thus reduced by the number of externally produced fringes which occur in the time period in which one plasma-produced fringe would have occurred. Clearly the temporal resolution is increased by the same factor as the sens i t iv i ty , namely the increase in the number of data points. Furthermore the ambiguities disappear since changes in phase due to the plasma alone result in increases or decreases in the modulated frequency. This principle has been applied to the measurement of electron densities in several forms (Baker et al., 1964; Gibson and Reid, 1964; Kricker and Smith, 1965; Sawyer et al., 1967; Herold and Jahoda, 1969; Curzon and Funk, 1970), 18 differing primarily in the choice of the external mechanism. The use of the laser as its own detector, as favoured by several of these authors, imposes a restraint on the temporal response, and raises d i f f i cu l t i e s concerning cavity response to rapid fringing rates (Williamson and Medley, 1969; Dangor and Fielding, 1969). These are avoided in the present work, as well as in other work in this laboratory (Curzon and Funk, 1970). 2.3.2 Rotating Retro-Reflector The external mechanism used in this investigation to vary the cavity length consists of a right-angle prism (counter-balanced by another) on a 28" aluminium bar mounted on a diameter of a gramaphone turntable. The path of the laser beam through the prism is shown in Figure 2-4 while the location of the turntable in the cavity is shown schematically in Figure 2-1. In this figure the Fabry-Perot cavity extends between semi-ref 1 ecting mirrors Mi and Mi», mirror M 3 is included solely for convenience. The principal advantage of such a retro-ref lector , apart from its simple and inexpensive nature, is the large fraction of each rotation during which interference is maintained and during which the resulting fringe frequency is nearly constant. It is thus a significant improvement over work done earl ier in 1 9 Figure 2-4. Ray Diagram at the Retro-Ref1ecti ng Prism. 20 t h i s l a b o r a t o r y by Curzon and Funk (1970) i n which the external mechanism s e l e c t e d did not y i e l d a constant f r i n g e frequency. To c a l c u l a t e the f r i n g i n g frequency an expression f o r the o p t i c a l path from A to B (Figure 2-4) must be found. I t i s P = 2r AB a - a 3"! /2 a 1 + 6 Un - V) - a — - — J { N J where a and 6 (both assumed small) are defined i n Figure 2-4 and n (n =1.6) i s the r e f r a c t i v e index of the prism The frequency produced at small a i s (2-10) f o ( - t ) = f 2 d P A B X dt 2 do dP. = 2Q_ X dt da X 2r 1 -a /2d a + 6 - a n - 1 (2-11) which, i f the t u r n t a b l e r o t a t e s at fi = 45 rpm i s f 0 - 10 MHz. 21 The advantage of the retro-reflector is that an f 0 ( t ) of several MHz can be made constant to any desired accuracy during a period T. Since ft, the angular speed of the turntable, does not vary during the observation period x (T << 1/ft), the only variation is from changes in a. If a = 0 at the centre of this observation time, then a < ftx/2 and o(0) - f 0 fTToT ftx d ft x 2/2 r n - 1 (2 which has the value, for ft = 45 rpm and x = 100 ysec, of 10~ 6. Variations in fo(t) during an observation period wil l be entirely negligible in comparison with the modula-tions in frequency produced by the plasma i f the dimensions and speed of rotation are chosen correctly. Thus henceforth fo(t) = f 0 , a constant. 2.3.3 Integrating Frequency Modulation Detector When recording the electron density, the rotating retro-ref lector, or other external means, extends the optical length of the Fabry-Perot cavity at a constant rate, producing a fringe whenever the change in length amounts to 22 one-half wavelength. In addition the changing electron density in the plasma in the cavity also changes the optical length, but at a slower rate. The change in the phase $(t) is then found by counting the number of fringes that have occurred up to time t. After subtracting the change of phase due solely to the rotating retro-ref lector , the balance is due to the change in electron density. Then eq. 2-7 can be applied to determine the path integral of electron density in the cavity. That i s , i f f ( t ) is the recorded frequency of fringes (f(t) = $(t)/2ir) 111 N (x)dx e -111 The calculation of the integral on the R.H.S. is tedious, and a c i rcu i t exists to accomplish this task, on l ine , in analogue form (Funk et al.3 1972). Brief ly the operation of the c i rcu i t is as follows: Each time the intensity from the cavity, as recorded by the photomultiplier, changes sign (any DC component having been removed) a bistable multivibrator is tr iggered. 1 This occurs at frequency 2f(t) . ' i n s t e a d of the c i r c u i t counting the z e r o - c r o s s i n g s of the p h o t o m u l t i p l i e r output, one might argue that i t should count the extrema, s i n c e they are the point s of well d e f i n e d phase. T h i s i s c o r r e c t , and could be accomplished by f i r s t d i f f e r e n t i a t i n g the d e t e c t o r s i g n a l . However, f o r approximately s i n u s o i d a l f r i n g e s , at high frequency, as used here, the d i f f e r e n c e s w i l l be minimal. = 1.765 x 10 1 7 f ( t ' ) - f 0j dt' (2-13) 23 The resultant pulses, containing equal quantities of positive charge, are summed on a capacitor. The charge on this capacitor is t Q(t) f (t 1 )dt' Simultaneously the charge on the capacitor is reduced by a constant current source. Before the measurement the constant current source is adjusted such that, i f the input to the c i rcu i t is at frequency f 0 the total charge on the capacitor remains constant. Therefore the constant current source alone would discharge the capacitor during time t by Q'(t) fo dt The output of the c i r c u i t , V(t ) , is proportional to the total charge on the capacitor. Positive current is supplied by the multivibrators, and negative current by the constant current source. Thus F i c'ire 2-5. Operation of the IFMD Upper trace IFMD output 100 mv/div Lower trace Interferometer fringes 10 mv/div Time scale 500 nsec/div (from right to left) Figure 2-6. Typical Electron Density Profile Vertical sensit ivity 2.53 x 10 1 6 electrons cnrVdiv Note timing spike at 3 usee 25 which is proportional to the R.H.S. of eq. 2-13. The c i rcu i t is an Integrating Frequency Modulation Detector (IFMD). An example of the operation of the IFMD is shown in Figure 2-5. Th bottom trace shows the photomultiplier signal at frequency f ( t ) , and the upper trace is the IFMD output, a voltage proportional to the electron density. One of the major contributions of this investiga-tion has been to apply this detector to an interferometer which involves a rotating retro-ref lector . As a resul t , the electron density in a plasma can be found by simply measuring the height of an oscilloscope trace above a straight baseline. There is no other technique in which this can be done with such accuracy and resolution. In interpreting this signal there are two points to be made here (and one further in section 2-6). The f i r s t is that, although ft, the rotation speed of the turntable, is constant during a 100 ysec observation time, i t wi l l fluctuate from one revolution to another by about 0.1%. This wi l l give a slope to the straight baseline, since f 0 a ft. It is thus desirable to record the slope during each observa-tion time. This may be accomplished by simply triggering the IFMD and oscilloscope twice during the observation time of 100 ysec and triggering the plasma on only one of these occasions. Figure 2-6 is a typical electron density temporal prof i le; the baseline is also recorded. 26 Secondly the IFMD involves a short delay due to the f in i te speed of the multivibrators. This delay is 0.35 ± 0.01 ysec. Thus instead of eq. 2-14 we should write where t^ = 0.35 ysec. The IFMD also has a f in i te bandwidth, namely up to 22 MHz. It is this frequency which determines the temporal resolution of the interferometer. The ultimate temporal resolution, with an ideal IFMD, would be equal to the double pass transit time of the cavity. 2.3.4 Calibrating the Integrating Frequency  Modulation Detector From eqs. 2-13 and 2-15 we have t-t dt (2-15) 0 1/2 r N e ( x , t - td )dx = k 1.765 x 1 0 1 7 V(t) (2-16) -1/2 and i t is necessary to determine the constant of propor-t ional i ty , k. 2 7 By setting f(t) = 0 in eq. 2 - 1 5 , and integrating for a period l / f 0 , we see that k is the voltage produced by the IFMD during one cycle of the input, which is a change of phase of 2TT in the Fabry-Perot cavity. Thus we define The differentiation of eq. 2 - 1 5 shows how to determine V 2 Tr. It yields If a tunable osc i l la tor is used in place of the interferometer, then and then V 2 T T can be determined graphically. Such calibrations were frequently carried out during the course of this investigation. After least squares f i t t ing the slope of dV/dt vs. f Q S C was typical ly V 2 T T = 0 . 1 1 volts . V = k. ( 2 - 1 7 ) ( 2 - 1 8 ) 28 2.4 Defining the Plasma Length 2.4.1 Introduction The preceding section described the measurement of the path integral of the electron density 111 N e (x, t) dx -£/2 in a fractional fringe, direct reading interferometer. Clearly the electron density can only be deduced from the integral i f the distribution of electron density over the interval ( - £ / 2 , £ / 2 ) is known. Tradit ional ly this impasse is c i r -cumvented by assuming the distribution to be rectangular, i . e . sharp plasma boundaries surrounding uniform electron density. Occasionally more sophisticated distributions are assumed. Regardless of the assumption the fact is the spatial resolution of the interferometer along the beam is no shorter than the vessel containing the plasma, unless the distribution is unfolded. In this and other investigations (Medley, 1970; Curzon and Funk, 1970) the spatial resolution in this direction is much improved. This is accomplished by the insertion of glass tubes^ into the plasma, which define the interval Whe ends o f t h e g l a s s t u b e s w h i c h a r e embedded i n t h e p l a s m a a r e s e a l e d w i t h w i n d o w s , s i n c e o t h e r w i s e , i f t h e p l a s m a were u n d e r p r e s s u r e , i t w o u l d t r a v e l up t h e t u b e s . T h i s w o u l d r e n d e r t h e p a t h i l l - d e f i n e d o n c e more, and d e f e a t t h e e n t i r e p u r p o s e o f t h e t u b e s . 29 (-&/2, 1/2). By moving the tubes one can record the path integral in short steps across this interval , using a reduced spacing between the tubes, and thus elucidate the d i s tr ibu-tion of electron density. The need for assumptions is much reduced. The limitation to this technique l i e s , of course, in the interaction between the plasma and the tubes. Another contribution made in this investigation was the careful study of this interaction. Perhaps surpris ingly, the interaction is quite limited on a short time scale, and so the use of glass tubes has been established as a valid and useful technique. This investigation is described in the next section. Finally section 2.4.3 explains an additional bonus of this technique. 2.4.2 The Interaction between Plasma and Tubes One would expect that the effect of inserting glass tubes into a plasma would i n i t i a l l y be to cool the plasma in a region near the glass, and then s l ight ly later to con-taminate the plasma in that region with material evaporated from the glass. The size of this region would increase at roughly the thermal diffusion speed. Values of electron density measured under these conditions would i n i t i a l l y be too low, and then too high as electron-rich s i l icon and oxygen entered the plasma. 30 The usefulness of the glass tubes depends completely on the percentage changes in electron density that they induce, and on the size of the region over which these changes occur. These values were measured experimentally. If a region of the plasma is observed in which the unperturbed electron density is homogeneous, then the » electron density contributed by the tubes (2 N.(x')dx' cm - 2 where N t is the change in density induced by the tubes and x the length of the region of interaction) wil l not vary with tube spacing, I, provided I > 2x. The total density measured by the interferometer would then be 111 111 N ' (x) dx = N I + 2 • i l l N t(x) dx (2-19) Hence a plot of 111 •111 N e '(x) dx against % should give a straight line of slope N , the unperturbed electron density, 111 and intercept 2 N t(x) dx, the electron density change caused by the tubes 31 After finding a homogeneous region of the discharge (described in Chapter 4) the path integral of electron density was recorded for several different tube spacings, £ . The results are plotted in Figure 2-7. The data for l ine (a) were recorded approximately 1 ysec after the plasma reached the tubes; l ine (b) represents the situation 3 ysec later . Each point is the mean of ten to twelve shots and the error bars are the standard deviations. (The additional significance of the points plotted as c irc les wi l l be mentioned in section 2.5.2. Fewer shots were recorded at these points.) Encouragingly the results are entirely in agreement with the simple model presented above. At f i r s t N t is negative, later i t changes sign. The influence of the tubes is indeed seen to be small. Line (a) has the equation N ' (x) dx = (4.89 ± 0.08) x 10 1 6 £ - (1.3 ± 0.3) x 10 1 6 while l ine (b) is N' (x) dx = (2.67 ± 0.08) x 10 1 6 £ + (1.1 ± 0.3) x 10 1 6  e in cm" 3, £ in cm Using a tube spacing, £ , of 10 cm, for example, the contr i -bution from the interaction with the tubes constitutes at 32 Figure 2-7. Plasma-Tube Interaction - Electron Density. 33 most 2.6% of the total path integral . For many applications this may be acceptable as experimental error. If i t is not acceptable then a scaling graph such as Figure 2-7 should be prepared for the precise situation under study. The correction to be applied to the result of a single measurement can then be deduced from this graph. It is also interesting to note that the length of the region of contamination, x, is small, since l i t t l e deviation from a straight line in Figure 2-7 is noticeable at small tube separation. This is discussed further in Chapter 5, and is in agreement with the results found there. The use of glass tubes to define the length of a transient plasma for interferometric purposes is thus a useful technique for deducing the electron density from the path integral . Therefore we can f ina l ly write an expression for the electron density measured by this i nterferometer « . < * • " t d ) - 1 - 7 ! 5 S " " *M ( The fractional fringe, laser excited, direct reading, inter-ferometer does indeed measure electron densities, as a function of time, at a given point! 34 2.4.3 Refractive Index Gradients As the probing laser beam traverses the plasma i t may encounter a gradient of electron density, which is also a gradient of refractive index. If this gradient is transverse to the resonator axis the beam will be bent through an angle 1/2 8 = \f- dx (2-21) J -£/2 where n, the refractive index, is a function of r , the distance from the resonator axis. This deflection has several consequences. One which has already been mentioned brief ly in section 2.3.1 is also the most serious, namely that a l l interference may be lost i f the returning beam fa i l s to overlap the source after a double transi t . The solution introduced by Curzon and Funk (1970), namely to make the cavity concentric, with the plasma mid-point and cavity centre coincident, was used here. This preserves beam over-lap to f i r s t order in 3n/3r. Since this is often insuff ic ient , 8 was further reduced simply by l imiting £ by use of the glass tubes, pioneered by Medley (1970). Thus the glass'tubes have a dual benefit; to increase spatial resolution along the line of sight and to reduce problems associated with radial gradients. Note also 35 (eq. 2-20) that the sensit ivity is proportional to the tube separation. Thus for a particular experiment the choice of tube separation is usually a compromise. 2.5 Extraneous Contributions to the Optical Path 2.5.1 Introduction The last two sections described the interferometer developed and used in this investigation. The purpose of this section and the one following is to outline the experi-mental errors and occasional p i t f a l l s associated with such an instrument. As before, details of the application to a specific plasma are postponed, in order to emphasize the general appl icabi l i ty of the interferometer. Essentially an interferometer is a device which measures the distance between two surfaces with an accuracy of the order of the wavelength of the l ight employed. Thus i t behooves one to consider changes in optical path due to other than the two principal effects, the presence of electrons and the rotating retro-ref lector , since very small changes may be s ignif icant . These contributions f a l l under two headings: extraneous additions due to the plasma refractive index and those due to geometric effects. 36 2.5.2 Extraneous C o n t r i b u t i o n s due to Other Terms  i n the R e f r a c t i v e Index The plasma r e f r a c t i v e index does not depend s o l e l y on the e l e c t r o n d e n s i t y . The approximations undertaken to w r i t e the s i m p l i f i e d r e f r a c t i v e index n = 1 - V 2w2 s p e c t ra1 1 i nes to 2 f . 3_ A*2 - to J 7J (2-are amply j u s t i f i e d i n most l a b o r a t o r y plasmas, but the neglect of the s p e c t r a l term i s perhaps not. In t h i s s e c t i o n t h a t term, which represents the anomalous d i s p e r s i o n of s p ecies i n the plasma, w i l l be considered. The s i g n i f i c a n c e of the s p e c t r a l term can only be assessed i n each f a i r l y s p e c i f i c case. In t h i s i n v e s t i -g a t i o n a (predominantly) helium plasma was probed at two l a s e r wavelengths, 632.8 nm and 441.6 nm. A l l He I and He II t r a n s i t i o n s w i t h i n 50 nm of e i t h e r l i n e were considered, and those which c o n t r i b u t e d s i g n i f i c a n t l y to the r e f r a c t i v e index were t a b u l a t e d i n Table 2-3. A l l oxygen, s i l i c o n , and n i t r o g e n l i n e s (three) w i t h i n 0.1 nm were also i n c l u d e d . The source f o r these wavelengths, o s c i l l a t o r strengths f . , degeneracies g., and energies of the lower s t a t e s was the National Bureau of Standards ta b l e s compiled by Weise et al. (1966). 37 Table 2-3 Terms Contributing to the Anomalous Refractivity of a Helium Plasma at 632.8 nm and at 441 .6 nm He I Wave- Osc. O J 2 N . 2 f- N. ( i, w J J lines length strength O J 2 - 0J f mi N e 2 NEZ ( O J 2 - OJ A . nm J fi 10- 2 2 cm 3 IO" 2 0 cm3 23P - 33D 587.6' 0.62 -6.26 44.4 -1 .72 2*P - 3XD 667.9 0.73 9.77 13.6 0.97 23P - 33S 706.5 0.07 5.05 44.4 0.16 23P - 53D 402.6 0.05 -4.92 44.4 -0.10 2-P - 5XD 438.8 0.04 -78.11 13.6 -0.46 23P - 43D 447.1 0.12 105.89 13.6 2.18 2*P - 4XD 492.2 0.12 5.13 - 13.6 0.08 He II N.xlO1* " 2 f j N J Ne N e(.oj* - O J J ) 4 - 6 656.4 . 0.18 14.16 1 .26 3 x IO" 4 5 - 16 641 .3 0.003 37.98 1 .79 2 x IO" 5 4 - 10 434.1 0.012 -28.69 1 .26 -4 x 10"5 : 4 - 9 454.3 0.019 18.14 1 .26 4 x IO" 5 3 - 4 468.8 0.84 8.88 0.89 7 x IO"1* N II « N • J cm-3 »2 fi N j O J 2 - O J 2 J cm'3 : 3D0 - 3P 632.84 0.028 -2.0x10 3 2 x 10 1 1 1 x 10 1 3 3P - 3D 441.79 0.009 2.0xl0 3 2 x 10 1 1 4 x 10 1 2 0 II 2P - 3D0 441 .70 0.55 1 .1x10- 4 x 109 2 x 10 1 3 38 The populations (N.) of lower states used in these calculations were those of partial local thermodynamic equilibrium. Anticipating the discussion (and nomenclature) of section 3.2.2, the conditions required for the PLTE assumption to be valid down to the lowest lying state l is ted in Table 2-3 of each species is (Griem, 1964) Table 2-4 Cri ter ia for Validity of PLTE Speci es Lowest Energy State Electron Density Must Exceed cm- 3 Plasma Lifetime Must Exceed ysec He I n = 2 7.4 x 10 1 5 3 x 10-"* He II n = 3 1.5 x 10 1 6 4 x IO" 3 N II n = 3 1.5 x 10 1 6 2 x 10-" 0 II n = 2 4.7 x 10 1 7 3 x IO" 3 The density requirements (except that for He II which requires a high temperature and thus a high density for its existence) were not always satisfied during this investigation. However, to anticipate another result of section 3.2.2, unless the level in question is a ground state or a metastable level its population (N.) wil l be less than its LTE population. This will be true provided 39 states of that species of somewhat higher energy are in PLTE, which is true here. Thus the calculated contributions to the refractive index from these states may be too large, but are assuredly not too small. For He I, then, the N. were calculated from the J Saha equation Ne NHe+ 2g- 2 IT m k T e e 3/2 \ E + " E i l exp k T J L e J (2-22) where T g is the temperature describing the electron d i s t r i -bution, g + the degeneracy of the free ion, h Planck's constant, and E + the f i r s t ionization potential of helium. N ^ e + , the population of the He II ground state, was taken as equal to the electron density, which is again an overestimate. For He II i t was adequate to assume that almost al l the He II density (- N g) was in the He II ground state with an additional (but very small) amount distributed as N. - g. N exp J j e E. /k T J e (2-23) 40 where E. i s the energy of the j L " s t a t e above the He II ground s t a t e . For 0 II and N II the procedure was s i m i l a r . The maximum p a r t i a l pressures of these species were estimated at 1 y. based on the leak rate and estimates of wal l e r o s i o n . The l a r g e s t c o n t r i b u t i o n s occur f o r helium at high e l e c t r o n d e n s i t i e s and f o r i m p u r i t i e s at low e l e c t r o n d e n s i t i e s . Table 2-5 l i s t s the r a t i o of these c o n t r i b u t i o n s to the r e f r a c t i v e index to the c o n t r i b u t i o n of f r e e e l e c t r o n s to the r e f r a c t i v e index. Thus the values l i s t e d appear experimentally as e r r o r s i n the e l e c t r o n d e n s i t y determination. Table 2-5 Maximum C o n t r i b u t i o n s of Various Species to the R e f r a c t i v e Index as a F r a c t i o n of the E l e c t r o n C o n t r i b u t i o n Species at X = 633 nm at X = 442 nm at N e (cm- 3) He I He II N 0 3 x 10- 3 3 x 10- 6 1 x 10" 2 8 x IO" 3 7 x 10"1* 4 x IO" 3 2 x 10" 2 5 x 1 0 1 7 5 x 1 0 1 7 1 x 1 0 1 5 1 x 1 0 1 5 41 Thus for a He-Ne laser (A = 632.8 nm) at least 99% of the refract iv i ty is due to free electrons while for a He-Cd laser (A = 441.6 nm) at least 98% is so due. This result was experimentally ver i f i ed . In Figure 2-7 the data plotted as c ircles were obtained using a He-Cd laser while those marked as crosses were measured with a He-Ne laser exciting the cavity. No corrections for neutral species have been applied. Thus the refractive index is indeed dominated by free electrons. 2.5.3 Other Extraneous Contributions There are two other sources of increments to the optical path. One is simply mechanical vibrations of the structure supporting the cavity. These must be suppressed, or alternatively their period must greatly exceed the observa-tion time (100 usee, during which data is collected for , at most, 20 ysec). In this investigation the latter was true. The structure supporting the interferometer had a resonant frequency of 1 kHz with amptitude of only a few wavelengths. The other source is the gradient in refractive index transverse to the cavity axis. Since the beam is deflected i t travels further. To f i r s t order the fact that the cavity is concentric will null this additional path. However four possible contributions s t i l l remain. Each of 42 these i s b r i e f l y described below, and the a d d i t i o n a l path lengths t a b u l a t e d . a) R e f r a c t i o n at windows. The gradient sweeps the beam across the windows of the plasma v e s s e l . Figure 2-8(a) ^m^m AP = AC-AB l i i i l i = d n e 2 ( l / n - 1) MUM WINDOW LENS The c i r c l e on which B and C l i e i s a wave f r o n t which w i l l be plane a f t e r passing through the l e n s . Due to the l e n s , OX and OB w i l l be paths of equal l e n g t h , but the d i f f e r e n c e between AC and AB w i l l be an a d d i t i o n a l c o n t r i b u t i o n . b) Focussing e r r o r s . Figure 2-8(b) AP = AC - AB = AB x d 2 9 2 MIRROR 43 If FX = f, the focal length of the lens, then no additional path will be caused i f there is no gradient or i f F and 0 are coincident. If they are separated by a distance d, however, an increment results . c) Additional path in the plasma Figure 2-8(c) HYPERBOLIC PATH The ray follows a hyperbolic path through the plasma while the lens corrects for the difference between OX and OA. Using eq. 2-21 the hyperbolic path length can be calculated. d) Non-flat optical components Figure 2-8(d) AP = d(l - n) COMPONENT 44 Clearly distortions in optical surfaces wil l cause additional optical path length i f the beam is swept across those surfaces. The significance of these contributions relative to those contributions arising from the electron density must be calculated for each interferometer, as is done in section 2.8. 2.6 Spurious Fringes If a minimum or maximum is missing from the fringe ' t r a i n , ' or i f one has been added, such a signal is referred to as a spurious fringe. If not understood they can be a p i t f a l l in the application of this interferometer. There are several causes. a) E l e c t r i c a l noise. Locating the IFMD and photomultiplier several metres from the plasma, or preferably in a screened room, was most helpful in suppression of noise. b) R e f r a c t i v e index g r a d i e n t . This may prevent interference (or reduce the fringe amplitude below that required by the IFMD). Alternatively i t may establish a phase change of 2IT or more across a diameter of the wave front of a beam traversing the plasma, which wil l cause several fringes to be 'missing.' Decreasing the tube separation is the remedy. 45 c) Modulation exoeeding 100%. If (in eq. 2-15) f(t) exceeds f 0 , i . e . the fringing rate of the plasma exceeds that due to the retro-ref lector , a turn-over point wi l l occur. Subsequent fringes which really represent a negative frequency w i l l , of course, be counted as posit ive. Decreasing the tube separation and thus the length of plasma in the cavity is the solution. d) The cavity finesse is too high. The additional fringes caused during rapid modulation of a Fabry-Perot cavity have been analyzed by Williamson and Medley (1969) and by Dangor and Fielding (1969). They find that unless cavity losses exceed 90% per double transit there are, in general, more than two extrema per 2TT change of phase. Also, the number of extrema is not constant but varies with frequency. Figure 2-9 shows several examples encountered in this investigation. The remedy employed here was simply to use a lossy cavity, the output from which changes sign only twice per 2TT change of phase. Since the IFMD counts zero-crossings a l l wi l l be well . Although not caused by spurious'fringes , the opposite problem to case c can occur i f the frequency response of the IFMD is exceeded. The result wi l l be analogous to the presence of spurious fringes. Again, decreasing the tube separation is the remedy. 46 Figure 2-9. Typical fringe trains from resonators with (a) low to (c) high finesse showing in -creasing spurious fringes. Time scale is 100 nsec/div so (a) is 19 MHz, (b) and (c) 10 MHz Figure 2-10. Effect of Spurious Fringes Vertical sensitivity 1.05 x 10 1 6 electrons cm" /d Fringes were lost 6 ysec after the timing spike, so the IFMD output does not return to its original value. 47 The experimenter is alerted to any of these problems by the fai lure of the IFMD output to return to the baseline, as shown by the example in Figure 2-10. In general, such measurements are discarded. In some situations, though, i f the cause of the spurious fringes is known to be b or c, the data obtained can s t i l l be used except during the period in which the spurious fringes existed. Thus spurious fringes, whilst demanding some care in their interpretation, are not a cause of additional experimental error. On occasion they may prevent the record-ing of electron density at isolated moments. 2.7 Experimental Application 2.7.1 Construction The interferometer described in this chapter was used in two forms. The f i r s t , as used for the measurements mentioned thus far, is sketched in Figure 2-1. For later work i t was decided to record the spectrum emitted by the plasma. This required an additional fold in the cavity axis. The result is shown to scale in Figure 2-11. Al l components used are l i s ted in Table 2-6. The dimensions of the prism mount and the speed of rotation were chosen such that the fringes produced by the TO CAPACITOR BANK TO VACUUM PUMPS LASER ft TO 'PHOTO-MULTIPLIER TURNTABLE F Stop H Tube Mount L Lense M Mirror P Pentaprism PD Photodiode S Scale T Pyrex Tube I METRE Figure 2-11. Scale Drawing of Interferometer co 49 Table 2-6 Interferometer Components Laser 1 . University Laboratories He-Ne Model L261 Power - 4 mw 2. Spectra-Physics He-Ne Model 235 Power - 4 mw 3. Spectra Physics He-Cd Model 185 Power - 40 mw (excessive) Lenses Edmunds. Simple, uncoated Focal lengths 80.0, 103.1, 126.9 cm as required to focus in centre of plasma. Mirrors M i Front mirror of laser M 2 Valpey Corp P/N 782-2202 99.9% R at 633 nm and 45° incidence T > 70% for X < 500 nm M3 Valpey Corp P/N 782-2203 98% R at 633 nm and 45° incidence or Aluminium-coated glass about 70% R broad band Mit Perkin-Elmer 582-1073 about 50% R at 633 nm 70% R at 442 nm M 5 Aluminium-coated glass Pri sm 90° - 45° - 45° Hypotenuse 2.54 cm Broad band anti-ref1ection coating on hypotenuse. (Total internal reflection from sides) Turntable Dual 1010 Wow < 0.3% at 45 rpm Interference fi1ter Baird-Atomic 4.3 nm bandpass (max 60%) around 633 nm (used a monochromator at 442 nm since no f i l t e r was readily available.) CONTINUED • 5 0 Table 2 - 6 (Continued) Photo-mul t i p i i er P h i l l i p s 1 5 0 CVP Risetime 8 nsec Fluke Model 4 1 2 B power supply Resonator tubes Pyrex 0 D = 8 mm Opaque coating on i n s i d e . Pyrex windows, (diam = 8 mm), glued to one end > I n t e g r a t i ng FM Detector Frequency Response 0 - 2 2 MHz S e n s i t i v i t y - Requires 2 0 mv Noise suppression Double copper walled screened room / 51 retro-reflector had a frequency near 10 MHz, the centre of the IFMD frequency response. This helped reduce the occurrence of spurious fringes. The laser, lenses, and mirrors M i and M2 were mounted on an aluminium beam which could be translated horizontally. This allowed measurements at any desired radius of the discharge tube. Each resonator tube was supported by two mounts connected by cables to outside the discharge vessel so that the tubes could be moved without releasing the vacuum. Drawings of such mounts may be found in Funk's thesis (1971). 2.7.2 Alignment Before obtaining a measurement the following must be true. 1. The laser beam ( i . e . the cavity axis) must be parallel to the axis of the discharge vessel and a known (but variable) distance from i t ; 2. The glass tubes must enclose the laser beam except over the distance to be probed, where i t traverses the plasma; 3. The laser beam must return on i t s e l f so that interference occurs. 52 The procedure was as follows. 1. To provide access to the plasma at any desired radius slots were milled in the electrodes (Figure 4-1). Inserts were made which f i t snugly into the centre of each electrode. These inserts had a 1.2 mm hole in their centres, and these holes defined the axis of the electrodes. (During experiments the inserts were removed.) The laser beam was made to pass centrally through the holes in the insert by adjusting the vertical supports of the laser head and translating the aluminium beam on which i t was mounted. Once this had been achieved the readings on scales S x and S 2 (Figure 2-11) were noted. Henceforth these scales could be set to these readings plus the desired radius. 2. The tube mounts were then inserted such that the tube axes and the laser beam could be made co-linear at any radius by use only of the cables. This was sure to meet with frustration i f the slots in the electrodes were not horizontal and para l l e l . 3. With the retro-reflector in motion the return-ing beam was noticeable on, for example, a window of the discharge vessel. Mirror M^  was provided with perpendicular micrometer adjustments and so an adequate fringe amplitude could be obtained. 53 2.7.3 Tri ggeri ng As explained in section 2.3.2 there is a period about 100 ysec long during each revolution of the turntable in which the retro-reflecting prism is producing fringes of essentially constant frequency. During this period, f i r s t the detection system alone was triggered to record the base-l ine , and then i t and the discharge were triggered together to record the electron density. Figure 2-12 shows how this was done. The photo-diode (FPT-100) was struck by a reflection from the rotating prism just before the fringes were produced, and the result i pulse in i t iated the trigger sequence. The delay unit produced the second trigger pulse, about 80 ysec after the f i r s t . The IFMD must be triggered because i f i t were on always i t would be driven to saturation; i t was triggered 3 ysec before the oscilloscopes to ensure i t was fu l ly turned on when recording. To avoid using another o sc i l l o -scope for simultaneous recording of the discharge current (which never showed any measurable fluctuations) the second time base of the 7704 oscilloscope was triggered by the current waveform when i t reached a certain value. The gate pulse from this second time base (50 nsec long), thus occurred at a known time after current in i t ia t ion and was added to the IFMD record (at a time of l i t t l e interest) 545A ' SCOPE o (MODIFIED) TRIG 9 PHOTO DIODE IN S. S. R E S E T + GATE A j - C O - . SET TO A D'LYD BY B SINGLE SWEEP 1 TO 3 CHANNEL DELAY UNIT TRIG. SPECTRAL-U N E INTENSITIES-515 DUAL-BEAM SCOPE TRIG, i -<&-IN 3^ SEC.DELAY"| 1 Figure 2-12. Triggering and Recordi ng. SEC. DELAY 545A SCOPE -3-FMD RC INTEGRAT. PHOTO-MULTIPLIER t FROM INTERFEROMETER 7704 SCOPE AMPS BASES SIGNALjPELAY FROM ROGOWSKI COIL SCREENED ROOM 55 to provide a sy n c h r o n i z a t i o n 'spike.' For an example, see the t y p i c a l IFMD output (Figure 2-6), i n which the added 'spike' has been l a b e l l e d . 2 .8 S e n s i t i v i t y and Accuracy 2.8.1 I n t r o d u c t i o n Now that the d e t a i l s of the in t e r f e r o m e t e r have been presented i n the l a s t s e c t i o n , the conclusions of sect i o n s 3 through 6 can be a p p l i e d to c a l c u l a t e the sen-s i t i v i t y and accuracy of t h i s p a r t i c u l a r i n t e r f e r o m e t e r . 2.8.2 S e n s i t i v i t y Equation 2-20 summarized the performance of the d i r e c t r e a d i n g , f r a c t i o n a l f r i n g e i n t e r f e r o m e t e r . V * - * • „ ) • u f » ; ; ° " v ( t ) ( 2 . V(t) represents the t o t a l change of phase between two points of well defined phase, and thus there i s no t h e o r e t i c a l l i m i t to the accuracy with which i t could be measured. Such a l i m i t was provided i n p r a c t i c e by noise sources such as the f r i n g e s themselves, the p h o t o m u l t i p l i e r , the 56 laser, and the discharge. In this investigation the minimum detectable signal was about 3 mv. Since V 2TT » the calibration constant of the detector, was typical ly 0.11 volts , eq. 2-20 gives N A I H I M I I M = 5 x 10 1 5 cm"2 e MINIMUM This is better than an order of magnitude improvement over the sensit ivi ty of the unmodulated interferometer (eq. 2-9). If the entire discharge length {Z = 61.3 cm) were used the minimum measurable electron density would be Ne MINIMUM = 8 X 1 0 1 3 C m ~ 3 2.8.3 Accuracy In applying the equations of section 5 to calculate the experimental errors a value for the largest possible refract iv i ty gradient is needed. A l imit does exist because, i f a phase change of 2n existed across the diameter of a wave front of the laser beam, there would be no inter-ference and the fringes would become spurious. This would occur i f (2-24) 57 where b is the diameter of the laser beam, (assume b = 0.5 mm) and r is the dimension perpendicular to the beam From eq. 2-2 a n 8 N | £ = 3.6 x 10" 2 2 r in cm, N g in cm"3 Thus the largest possible product of gradient and path is 3 N £ * 3 x 10 1 8 cm"2 which, using eq. 2-21, yields a maximum possible deflection 9 M A X * 1 X 1 0 ' 3 1/2 A v = 1 x I O - 3 radians n£ £ / 2 This value was used in calculating the errors as summarized in Table 2-7. This table l i s t s the calculated percentage error in electron density for the smallest and largest electron densities encountered in our discharge (Chapter 4 ) . In this discharge i t was also true that significant electron density gradients only occurred coincident with large densities In other applications this should be carefully ver i f i ed . 58 Table 2-7 Accuracy of the Interferometer Source X = 633 nm X = 442 nm large H Q small N e large N g small N^ e Variations in f 0 - - - -Calibration of IFMD 1% 1% 1% 1% Extraneous contributions C o l l i s i o n , cyclotron frequenci es - - - -Spectral lines 0.3% 1% 0.8% 2% Refractivity gradients Refraction at windows 0.05% - 0.08% -Focussi ng errors 2 x 10-6 - 3 x 10- 6 -Addition path in the plasma 0.1% - 0.2% -Non-flat optical components 0.1% - 0.1% -TOTAL 1.1% 1 .4% 1 .3% 2.2% The flatness of the optical components was examined in a Twyman-Green interferometer. Provided they were clean no surface irregulari t ies were v i s ib l e . Nevertheless a pessimistic value of X/5 per cm of surface was assumed. 59 By l a r g e e l e c t r o n d e n s i t y i s meant about 5 x 1 0 1 7 cm" 3, small i m p l i e s about 1 x 1 0 1 5 cm - 3. Two f u r t h e r sources of e r r o r must yet be added. One i s the e r r o r associated with reading the o s c i l l o s c o p e t r a c e , about 3% u s u a l l y , but l a r g e r near the l i m i t of sen-s i t i v i t y . Secondly i f one e l e c t s not to perform a s c a l i n g check ( s e c t i o n 2.4.2) the use of the resonator tubes produces an u n c e r t a i n t y . Spurious f r i n g e s are not a source of experimental e r r o r at a l l , j u s t a source of occasional confusion. Thus the o v e r a l l accuracy of the i n t e r f e r o m e t e r i s about 2%, and that of the associated o s c i l l o s c o p e i s about 3%. The t o t a l i s often smaller than the standard d e v i a t i o n of measured e l e c t r o n d e n s i t i e s . 2.9 Comparison with E l e c t r o n D e n s i t i e s Determined by  Stark Widths E l e c t r o n d e n s i t i e s can also be determined by measuring the widths of Stark broadened s p e c t r a l l i n e s . However t h i s i n t e r f e r o m e t e r i s p r e f e r a b l e because of i t s higher accuracy, b e t t e r s p a t i a l and temporal r e s o l u t i o n , s i n g l e shot r e c o r d i n g , and d i r e c t reading. Nevertheless agreement between the two methods would be a c o n c l u s i v e v e r i f i c a t i o n of the r e s u l t s derived i n t h i s chapter. In p a r t i c u l a r the e f f e c t of the tubes on the plasma would be 60 further scrutinized. For non-uniform density distributions the Stark broadening results would be weighted to regions of large electron density and higher temperature (higher emission) while the interferometer would give equal weight to a l l regions. Further, i f the electron density between the tubes was non-uniform the spectral line profiles would not be similar to theoretical prof i les . Therefore a comparison between the two methods was made. The optical system used for the collection of spectra is shown in Figure 3-2. In that chapter i t is shown that the emitted l ight is collected from a region not much larger than the focussed laser beam. Using a Spex monochromator (dispersion 0.33 nm/mm in third order) and an EMI 9558B photomultiplier of verified l inear i ty , the He I l ine at 388.9 nm was scanned shot to shot. The monochromator pass-band was trapezoidal with a width of 0.041 nm. A typical prof i l e , recorded on axis 6 ysec after discharge i n i t i a t i o n , is shown in Figure 2-13. The theo-ret ical l ine shape was calculated from the line shape data of Griem (1964) using the broadening parameters of Cooper and Oertel (1969). Since the l ine shape is very weakly dependent on temperature, an electron temperature of 30,000 °K was assumed. The calculated profi le was f i t ted to the experimental data by varying the half width. The best f i t INTENSITY i — « — i — • — i — 1 — i — 1 — F I A ^ — i — 1 I 1 I 1 I " 1 f WAVELENGTH Figure 2-13. Recorded P r o f i l e of He I X 389 nm. 62 h a l f width was co r r e c t e d f o r instrument width and the corresponding e l e c t r o n density deduced. This was repeated f o r f i v e times during the plasma l i f e t i m e . In a d d i t i o n the e l e c t r o n d e n s i t i e s as measured by the i n t e r f e r o m e t e r on the same shots were averaged. Both sets of data are p l o t t e d i n Figure 2-14. For the i n t e r f e r -ometric data the e r r o r bars are the standard d e v i a t i o n of the mean while f o r the Stark data they represent the un-c e r t a i n t y i n f i t t i n g . The recorded p r o f i l e s were s i m i l a r to the t h e o r e t i c a l p r o f i l e s . A l s o , the e l e c t r o n d e n s i t i e s measured by the two methods were i n agreement. This was concrete evidence that the i n t e r f e r o m e t e r was performing as expected. There were two points of p a r e n t h e t i c a l i n t e r e s t connected with t h i s experiment. One was that the l a r g e shot-to-shot v a r i a t i o n s i n the plasma were l a t e r found to be due to accumulated deposits on the p i t t e d inner w a l l s of the discharge v e s s e l . L a t e r , replacement of the vessel produced a marked improvement i n r e p r o d u c i b i l i t y . Secondly, the curve which produced the best f i t to the e l e c t r o n d e n s i t y data was a decaying e x p o n e n t i a l , as found by a semi-log p l o t (not shown). These d e n s i t i e s were measured on the a x i s , near the c e n t r e , of a low pressure z-pinch. (Chapter 4 contains a l l d e t a i l s of the INTERFEROMETER STARK WIDTH HE I \ 3 8 9 Figure '2-14. Comparison between E l e c t r o n D e n s i t i e s measured I n t e r f e r -o m e t r i c a l l y and by Spec t r a l Line Widths 64 discharge.) The decay ra t e i s too large f o r j u s t r a d i a t i v e recombination, so more information i s needed f o r an e x p l a n a t i o n . The c o n t r i b u t i o n s made during t h i s i n v e s t i g a t i o n to the development of the d i r e c t reading i n t e r f e r o m e t e r have a l l been discussed i n t h i s chapter. For convenience they are summarized here. 1. The marriage between the r o t a t i n g r e t r o -r e f l e c t o r and the IFMD has produced an instrument of unsurpassed convenience, accuracy, and s e n s i t i v i t y f o r the measurement of e l e c t r o n d e n s i t i e s . 2. A d e t a i l e d study of the u n c e r t a i n t i e s a s s o c i a t e d with such an instrument has shown them to be u s u a l l y n e g l i g i b l e . Included were the tube plasma i n t e r -a c t i o n , and a l l other c o n t r i b u t i o n s to the o p t i c a l path. Chapter 3 ELECTRON TEMPERATURE MEASUREMENTS 3.1 Introducti on The last chapter described the techniques employed for the measurement of electron densities. To carry out the studies to be described in subsequent chapters i t was also necessary to measure the electron temperature of the plasma. That measurement is the subject of this chapter. The electron temperature was measured in two ways. Most commonly used was the ratio of the total inten-s i t ies of the spectral lines He II 468.6 nm and He I 388.9 nm. Occasionally the intensity ratio of the latter l ine to a 5 nm wide band of continuum was recorded. Potential ly, the ratio of the intensities of lines of two species is a very accurate measurement of electron temperature. Often, however, d i f f i cu l t i e s in interpretation cloud the results . These d i f f i cu l t i e s are considered in the next section. Section 3 is a brief description of the optical system used for simultaneous recording of the two intensit ies . 65 66 3 . 2 Theory - Local Termodynamic E q u i l i b r i u m ? The i n t e n s i t y of a s p e c t r a l l i n e i s d i r e c t l y p r o p o r t i o n a l to the population i n i t s upper s t a t e . I f the r e l a t i v e populations of two or more upper s t a t e s are measured, and i f some type of e q u i l i b r i u m i s i n existence which includes these upper s t a t e s , a temperature can be a s c r i b e d to that e q u i l i b r i u m . In plasmas of moderate e l e c t r o n density (N e > 10 l l f cm - 3 or so) t h i s e q u i l i b r i u m , i f i t e x i s t s , i s c o n t r o l l e d by electron-atom c o l l i s i o n s , s i n c e e l e c t r o n s have by f a r the l a r g e s t c o l l i s i o n c r o s s - s e c t i o n s of any species in the plasma. The r e s u l t i n g population d i s t r i b u t i o n i s determined by a Boltzmann f a c t o r (exp {W/kTe>) where T g i s the e l e c t r o n temperature and W the energy d i f f e r e n c e between the s t a t e s . These energy l e v e l s are then s a i d to be i n l o c a l termodynamic e q u i l i b r i u m , or LTE. (Local because the temperature may vary i n space.) To produce LTE, then, the populations of s t a t e s of a species must be c o n t r o l l e d by e l e c t r o n c o l l i s i o n s with that s p e c i e s . The r a d i a t i v e rate from a l e v e l must be n e g l i g i b l e i n comparison with the e l e c t r o n c o l l i s i o n depopulation of that l e v e l . The f a c t that t h i s i s o f t e n true i s the basis of a l l the spectroscopic methods (but one) f o r the measurement of e l e c t r o n temperature. The dominance of the c o l l i s i o n a l depopulation rate over the r a d i a t i v e r a t e holds more e a s i l y f o r h i g h l y 67 ex c i t e d s t a t e s because t h e i r r a d i a t i v e l i f t i m e s tend to be longer, and t h e i r c o l l i s i o n c r o s s - s e c t i o n s l a r g e r , than those of low l y i n g s t a t e s . Often LTE i s not a v a l i d d e s c r i p t i o n of the populations of a l l s t a t e s of a species but does hold f o r some hi g h l y e x c i t e d s t a t e s . Such a s i t u a t i o n i s c a l l e d p a r t i a l LTE, or LTE down to a s p e c i f i c quantum number.1 Griem (1964), using the c r i t e r i o n that the c o l -l i s i o n a l depopulation rate be ten times the r a d i a t i v e r a t e , has estimated the c o n d i t i o n that a s t a t e with p r i n c i p a l quantum number n have a population w i t h i n 10% of i t s LTE popula t i o n . I t i s N > 2 x 1 0 1 8 -^rpz T * (3-1) n where N g i s i n cm - 3, T e i s ev, and z - 1 i s the charge of the species The a s s u m p t i o n made i n c a l c u l a t i n g a n o m a l o u s c o n -t r i b u t i o n s t o t h e p l a s m a r e f r a c t i v e i n d e x ( s e c . 2.5.2) i s now see n t o be g e n e r a l l y t r u e . In t h a t s e c t i o n e s t i m a t e s o f t h e maximum p o p u l a t i o n o f t h e l o w e r s t a t e s o f v a r i o u s t r a n s i t i o n s w ere r e q u i r e d . T h e s e l o w e r s t a t e s w e r e a l l e x c i t e d s t a t e s a n d , a l t h o u g h LTE down t o t h e s e s t a t e s c o u l d n o t be a s s u m e d , some o f t h e h i g h e r e n e r g y s t a t e s o f t h a t s p e c i e s were i n LTE. S i n c e most c o l l i s i o n s i n v o l v i n g s u c h a l o w e r s t a t e a r e e x c i t a t i o n s o r d e - e x c i t a t i o n s t o o r f r o m t h e s e h i g h e r s t a t e s t h e r a t e s o f c o l l i s i o n a l p o p u l a t i o n and d e p o p u l a t i o n a p p r o x i m a t e t h o s e o f LTE. However t he r a d i a t i v e d e p o p u l a t i o n r a t e o f t h e l o w e r s t a t e i s s t i l l s i g n i f i c a n t . Thus t h e maximum p o p u l a t i o n o f t h e s e l o w e r s t a t e s u s u a l l y does n o t e x c e e d t h e LTE p o p u l a t i o n . 68 This c o n d i t i o n i s s u f f i c i e n t i n a s t a t i o n a r y , homogeneous, o p t i c a l l y t h i n plasma. He has also c a l c u l a t e d the time required f o r the establishment of such an e q u i l i b r i u m , which i s simply the r e c i p r o c a l of the c o l l i s i o n a l e x c i t a t i o n r a t e . In temperature measurements i n subsequent chapters of t h i s t h e s i s two l i n e s were used. The c o n d i t i o n s required f o r them to be i n LTE with higher energy s t a t e s of t h e i r species were: N > 2 x IO 1* cm"3 e x > 2 x 10~ 3 ysec N > 1 .3 x 1 0 1 5 cm-3 e x > 4 x IO - 1* ysec He I X = 388.9 nm 2 3S - 3 3P He II X = 468.6 nm 3 - 4 where x i s the plasma l i f e t i m e . These were c a l c u l a t e d f o r T g = 2 ev, the lowest temperatures encountered i n t h i s i n v e s t i g a t i o n . During the experiments described i n subsequent chapters these c o n d i t i o n s were always f u l f i l l e d . Thus the upper s t a t e of the He I l i n e at 388.9 nm was i n LTE with higher s t a t e s of He I and thus also with the ground s t a t e of He I I . The upper s t a t e of the He II l i n e at 468.6 nm was s i m i l a r l y i n LTE with higher states of He II and with the He I I I p o p u l a t i o n . 69 Total l o c a l thermodynamic e q u i l i b r i u m did not e x i s t . Thus there were two d i f f e r e n t population s e t s , the upper s t a t e s of He I i n c l u d i n g the ground s t a t e of He I I , and the upper s t a t e s of He II i n c l u d i n g He I I I , which were both i n t e r n a l l y i n p a r t i a l LTE, but not i n LTE with each other. In order to measure the e l e c t r o n temperature by recording the i n t e n s i t y r a t i o of two l i n e s , one from each s e t , i t was e s s e n t i a l to know the population r a t i o between these two s e t s . This r a t i o i s determined, of course, by the rates of c o l l i s i o n a l and r a d i a t i v e e x c i t a t i o n and d e - e x c i t a t i o n between the two s e t s . Three of these four rates can be c a l -culated as before. However since a ground ( i . e . h e a v i l y populated) s t a t e i s involved the f o u r t h r a t e , photon a b s o r p t i o n , cannot be neglected. The most e n e r g e t i c , and thus the slowest, t r a n s i t i o n i n e x c i t a t i o n s to the upper sta t e s of He II i s between the ground s t a t e and the N = 2 s t a t e . This i s the Lyman a l i n e of He I I , at 30.4 nm. Reabsorption of these photons would g r e a t l y increase the r a t e of growth of the population of the He II upper s t a t e s . Griem (1964) assumed, i n his c a l c u l a t i o n of t h i s population r a t i o , that each of the Lyman a photons was reabsorbed, i . e . that the plasma was o p t i c a l l y t h i c k to t h i s l i n e . For low de n s i t y plasmas, or those i n which one dimension i s s m a l l , t h i s assumption i s que s t i o n a b l e . Mewe 70 (1966) has c a l c u l a t e d the escape p r o b a b i l i t y of Lyman a photons, and ap p l i e d the r e s u l t s to c a l c u l a t e the f a c t o r s b n ^ , which are the r a t i o of the actual population of the n**1 l e v e l to the LTE population of that l e v e l . As an example, he f i n d s that the He II ground s t a t e , i n a c y l i n d r i c a l plasma of radius 1 cm with N = 4 x 1 0 1 5 cm"3 and T = 2.75 ev r e e (2 ) has a population 170 times the LTE population ( i . e . b[ - 170) Using these f a c t o r s , an expression f o r the i n t e n s i t y r a t i o of a He II l i n e to a He I l i n e can f i n a l l y be w r i t t e n . He I I He I b 2 -P-(O gf/A gf/x 2 N (•») N i exp k T (3-2) where E^ i s the i o n i z a t i o n p o t e n t i a l of hydrogen Here the He II l i n e has a wavelength X 2 and an o s c i l l a t o r strength f 2 , while i t s upper s t a t e has a p r i n c i p a l quantum number p and a degeneracy g 2 . The s u b s c r i p t 1 r e f e r s to the He I l i n e whose upper s t a t e has p r i n c i p a l quantum number q. The density r a t i o N ^ V N J 2 ^ , the He I I I d e n s i t y to the He II ground s t a t e d e n s i t y , i s given by a Boltzmann equation again modified by the b f a c t o r s 71 ( 3 ) 2 ^ b i 2 } g i 2 ) N( 2 7T m k T e e 3/2 exp k T (3-3) ( 2 ) where Ej 7 i s the reduced i o n i z a t i o n p o t e n t i a l of the He II ground s t a t e . Thus Y ^ i = 7.1 x 1 0 2 1 , r ( , b [ 2 ) T,-« 389 exp 609 T • i n 10 3 °K e (3-4) where the values f o r the two l i n e s , He I 388.9 nm and He II 468.6 nm, used to measure temperature i n t h i s t h e s i s have * been i n s e r t e d . To make these measurements the i n t e n s i t y r a t i o and the e l e c t r o n d e n s i t y were measured. Then eq. 3-4 was solved i t e r a t i v e l y f o r T g by computer, using Mewe's t a b l e of b v a l u e s , part of which i s reproduced here. *For plasma dimensions of 10 cm, the o p t i c a l t h icknesses are about M and 5% r e s p e c t i v e l y . 72 Table 3-1 Populations of the He II Ground State log b j 2 ) N e cm - 3 T • e = 2.75 ev T „ = 5.5 ev d (cm) d (cm) 1 10 100 1 10 100 IO 1" 4.74 4.54 4.25 4.96 4.84 4.76 10 1 S 2.97 2.21 1 .51 3.51 3.37 2.82 10 1 6 0.60 0.36 0.32 1 .82 1 .46 1.15 10 1 7 0.04 0.04 0.04 0.31 0.20 0.20 10 1 8 0 0 0 0 0 0 where d i s a c h a r a c t e r i s t i c plasma dimension Since, as discussed e a r l i e r , the upper states of the l i n e s were i n LTE, b^ 1^ and b[2^ were very nearly u n i t y . The e r r o r s i n the r e s u l t i n g values of T are 3 e mainly due to the t h e o r e t i c a l u n c e r t a i n t i e s i n the b values. These produce an e r r o r i n temperature of up to 10% at low d e n s i t i e s (Mewe 1966). Errors caused by experimental un-c e r t a i n t i e s i n N g and the l i n e i n t e n s i t i e s are n e g l i g i b l e by comparison. 73 U n f o r t u n a t e l y , i n order that t h i s theory apply there i s one c o n d i t i o n yet to be s a t i s f i e d . I t s demands reduce the a p p l i c a b i l i t y of the method. I t concerns the time sc a l e necessary f o r the establishment of t h i s e q u i l i b r i u m . The slowest step i s the c o l l i s i o n a l e x c i t a t i o n of the He II resonance l i n e . At the density required f o r p a r t i a l LTE (N g - 1 0 1 5 cm - 3) and a temperature of 2 ev the inverse of t h i s r a t e exceeds 100 ysec (Griem, 1964), compared to a plasma l i f e t i m e of 10 ysec. The pinch phase of the d i s c h a r g e , however (to a n t i c i p a t e some of the d i s c u s s i o n of the next c h a p t e r ) , i s c h a r a c t e r i z e d by a d e n s i t y of about 1 0 1 7 cm - 3 and a temperature of 3 ev, in which case the e q u i l i b r a t i o n time i s about 3 ysec. Thus 3 ysec a f t e r these c o n d i t i o n s become e s t a b l i s h e d the temperatures w i l l be r e l i a b l e . However, during the 3 ysec e x c i t a t i o n time of the He II resonance l i n e the pinching plasma sweeps up a con-s i d e r a b l e q u a n t i t y of cold gas. Thus t h i s plasma probably always has an excess of population in the He II ground s t a t e and below. On the other hand the presence of a la r g e current density and a high e l e c t r i c f i e l d w i l l enhance the c o l l i s i o n a l e x c i t a t i o n r a t e s . Thus the t o t a l e r r o r i s d i f f i c u l t to assess. However even an upper l i m i t of an order of magnitude in the i n t e n s i t y r a t i o r e s u l t s i n only a 20% e r r o r i n the temperature. This f i g u r e was taken to be a reasonable maximum e r r o r i n the temperature measurements. 74 Another sp e c t r o s c o p i c method was used i n an attempt to avoid these la r g e e r r o r s . The i n t e n s i t y r a t i o of the He I 389 nm l i n e to the continuum s u f f e r s from none of these problems since LTE between f r e e e l e c t r o n s and the 389 nm l i n e upper s t a t e i s amply j u s t i f i e d and very q u i c k l y e s t a b l i s h e d . Unfortunately (there's always a catch) i t cannot be used above 30 x 10 3 °K because recombination and bremsstrahlung on He I I I become important. Attempts to c a l c u l a t e these c o n t r i b u t i o n s would i n v o l v e p r e c i s e l y the same d i f f i c u l t i e s discussed above. The i n t e r a c t i o n of the glass tubes with the plasma might be expected to a f f e c t the measured r a t i o of l i n e i n t e n s i t i e s . In view of the time s c a l e s i n v o l v e d t h i s was thought to be u n l i k e l y , but nonetheless was v e r i f i e d e x p e r i m e n t a l l y . Figure 3-1 i s a p l o t of the l i n e i n t e n s i t i e s of He II 469 nm and He I 389 nm against tube s e p a r a t i o n . The e r r o r bars are the standard d e v i a t i o n of the mean. C l e a r l y the tubes do not a f f e c t the emission of e i t h e r l i n e . (Data were recorded at mid-point of tube at r = 5 cm.) 3.3 Apparatus The o p t i c a l system used to record the r a t i o of l i n e i n t e n s i t i e s , or the line-continuum i n t e n s i t y r a t i o , was s t r a i g h t f o r w a r d . Lens L 2 (see Figure 3-2), which was 75 TOTAL LINE INTENSITY mv TUBE SEPARATION cm Figure 3-1. E f f e c t of Tubes on Line Emission. also part of the i n t e r f e r o m e t e r , c o l l i m a t e d l i g h t emitted by the plasma which had t r a v e l l e d through the resonator tube. This was focussed by lens L 3 onto the entrance s l i t s of the two monochromators. The l i m i t i n g stop of the system ISPEX Stop Tube Mount Lense Mi r r o r Pentaprism Scale Pyrex Tube I RCA 17265 F H L M P S T I METRE Figure 3-2. Scale Drawing of O p t i c a l System. 77 was F i which had a diameter of 5 mm. I t was shown that only l i g h t emitted from between the tubes from a c y l i n d e r 1 mm in diameter, centred on the resonator a x i s , was c o l l e c t e d . Thus the i n t e r f e r o m e t e r and the o p t i c a l system were sampling e s s e n t i a l l y the same volume of plasma. When sampling from mid-way along the plasma the f number of the system was approximately 100. The monochrom-ators had i d e n t i c a l t r a p e z o i d a l pass bands, with a measured width of 0.25 nm, which was s u f f i c i e n t to pass most of the Stark broadened and s h i f t e d l i n e s . C o r r e c t i o n s , using Grien' (1964) p r o f i l e s , were ap p l i e d f o r the'missing wings and also f o r any underlying continuum. For recording continuum i n t e n s i t i e s the width of the pass band was enlarged to 4.3 nm The Spex monochromator had a f i r s t order d i s p e r s i o n of 1 nm/mm, the J a r r e l l - A s h one of 1.5 nm/mm. Other d e t a i l s of the o p t i c a l and recording systems are contained i n Figures 3-2 and 2-12. The s p e c t r a l s e n s i t i v i t y of the e n t i r e system was measured by pl a c i n g a tungsten ribbon lamp at stop F x. E m i s s i v i t y data compiled by DeVos (1954) was used. For recording the continuum i n t e n s i t y the region chosen was 460 ± 2 nm. This region was scanned at 0.2 nm per shot to confirm the absence of s p e c t r a l l i n e s , which could have a r i s e n from a d i r t y plasma. 78 The f i r s t step i n recording was to set both mono-chromators to the same l i n e to record t h e i r r e l a t i v e s e n s i -t i v i t y w ith that p a r t i c u l a r alignment and supply v o l t a g e s . Then one was set to the other l i n e , or to 460 nm, and the desi r e d i n t e n s i t y r a t i o recorded at a s i n g l e shot, s i m u l -taneously with the i n t e r f e r o m e t r i c measurement of e l e c t r o n d e n s i t y . Figure 3-3 i s a t y p i c a l r e s u l t . p r a t Figure 3-3. T y p i c a l Record of Total Line I n t e n s i t i e s Upper - He I I 469 nm 20 mv/div Lower - He I 389 nm 10 mv/div Time Scale - 2 ysec/div Tube Separation - 5.1 cm This chapter has been a d i s c u s s i o n of the tempera-ture measurement technique, i t s l i m i t a t i o n s and e r r o r s . This and the previous chapter, d e s c r i b i n g the i n t e r f e r o m e t e r , c o n s t i t u t e Part A of t h i s t h e s i s , in which the d i a g n o s t i c s have been described. The next chapter begins Part B, the Physics of the discharge. Chapter 4 RADIAL DYNAMICS OF A Z-PINCH IN 4 TORR HELIUM 4.1 I n t r o d u c t i o n A z-pinch i s the discharge formed by the a p p l i c a t i o n of a s u f f i c i e n t voltage d i f f e r e n c e between the ends of a hollow c y l i n d e r c o n t a i n i n g a gas. A f t e r breakdown a c y l i n d r i c a l current s h e l l flows i n the plasma. This produces an azimuthal magnetic i n d u c t i o n which pinches the plasma toward the a x i s of the c y l i n d e r . The study of the pinch e f f e c t began i n 1934 when Bennett p r e d i c t e d the c o n s t r i c t i o n of a plasma under the ac t i o n of i t s own cu r r e n t . The f i r s t pinches were b u i l t i n 1950, with b r i g h t hopes that they would provide c o n t r o l l e d f u s i o n r e a c t i o n s . By 1958, however, Tuck had reported t h e i r s u s c e p t i b i l i t y to i n s t a b i l i t i e s . Fusion plans f o r pinches of simple geometry d i e d , although self-confinement i s s t i l l a major f e a t u r e of many proposed r e a c t o r s . Present i n t e r e s t i n simple pinches i s centred i n two areas. The pinched plasma can be, and o c c a s i o n a l l y has been, used as a spec t r o s c o p i c source or as a medium f o r 79 80 l a s e r l i g h t s c a t t e r i n g or wave mixing experiments. The plasma l i f e t i m e i s s u f f i c i e n t , and the range of e l e c t r o n density and temperature a v a i l a b l e to the experimenter i s l a r g e . Secondly simple pinches are a useful study i n plasma dynamics. In both these areas the superior pinch i s a z-pinch of l a r g e dimensions, such as that used i n t h i s work. The c y l i n d r i c a l plasma permits a x i a l access to a long path length of uniform plasma f o r s p e c t r o s c o p i c or l a s e r s c a t t e r -ing a p p l i c a t i o n s . The dynamics are not perturbed by curva-ture e f f e c t s as i n a 0-pinch; n e i t h e r are they d i s t u r b e d by e l e c t r o d e e f f e c t s i f the i n t e r - e l e c t r o d e spacing i s l a r g e . The aim of t h i s t h e s i s has been to study a long l i n e a r pinch and make c o n t r i b u t i o n s to both these areas of current i n t e r e s t . That aim has been f u l f i l l e d . This and the next chapter, and the appendix, discuss the knowledge gained about the pinch dynamics. In the concluding chapter the most s u i t a b l e region f o r s c a t t e r i n g and s p e c t r o s c o p i c a p p l i c a t i o n s i s i d e n t i f i e d and analyzed. The subject of the present chapter i s the r a d i a l dynamics of the discharge. Thus a t t e n t i o n i s confined to a plane p a r a l l e l t o , and midway between, the e l e c t r o d e s . The aspects i n which t h i s work has made c o n t r i b u t i o n s i n c l u d e measuring the t r a j e c t o r y of the precursor shock and the temperature of the gas i n f r o n t of and behind i t , a n a l y z i n g 81 the r e f l e c t i o n of t h i s shock from the a x i s , and recording the d i f f e r e n c e s produced by r e v e r s i n g the p o l a r i t y of the discharge. Experimental data i s presented in each of these areas, and w i l l a l so be used i n analyzing the region s e l e c t e d as a s p e c t r o s c o p i c source. For convenience, the f o l l o w i n g convention i s mentioned here. In a l l f u r t h e r d i s c u s s i o n s the o r i g i n of time i s the i n s t a n t of discharge current i n i t i a t i o n . 4.2 Apparatus The z-pinch used throughout t h i s experiment was s i m i l a r to that described by Medley et al. (1965). The dimensions and operating c o n d i t i o n s are l i s t e d i n Table 4-1, and Figure 4-1 contains the c i r c u i t diagram. The double spark can used f o r t r i g g e r i n g y i e l d e d both higher repro-d u c i b i l i t y and reduced noise (Medley et al.3 1965). The discharge current was measured with a Rogowski c o i l c o n s i s t i n g of a 12 cm length of RG65-A/U delay l i n e with the outer conductor removed. This c o i l was i n s e r t e d between the leads c a r r y i n g the discharge c u r r e n t with the c o i l a x i s normal to the d i r e c t i o n of current flow. The c o i l output was i n t e g r a t e d using a passive RC i n t e g r a t o r of time constant 1.1 msec to give a s i g n a l p r o p o r t i o n a l to the discharge c u r r e n t . This s i g n a l was c a l i b r a t e d by manually 82 Table 4-1 Z-Pinch Components Discharge Tube Materi al Pyrex Length 76.2 cm Electrode separation 61.3 cm Inner diameter 15.2 cm Outer diameter 17.1 cm Vacuum System Mechanical pump Cenco-Hyvac 14 D i f f u s i o n pump Balzers o i l DIFF 170 Connections 2" ID Vacuum gauge CVC P i r a n i GP-110 Base pressure < 0.5 u Hg Leak r a t e < 1 u Hg/hr Gas Helium 99.995% pure Canadian L i q u i d A i r Manometer con t a i n i n g d i b u t y l p h t h a l a t e , p = 1.047 gm cm-3. D i f f e r e n c e height measured with t r a v e l l i n g mi croscope Valve Edwards High Vacuum s o l e n o i d a l i s o l a t i o n valve E l e c t r i c a l Power supply Sorenson 1020-30 (0-20 kV, 0-30 ma) Capaci t o r s 5 x 10.3 yF NRG type 203 Leads Copper 1.6 mm x 10 cm x 1.3 m Inductance of bank and 1eads 0.12 ± 0.01 yH CONTINUED 83 Table 4-1 (Continued) Electrodes 1/4" Brass Return conductor Brass gauze Voltage measurement Simpson microammeter i n s e r i e s with p r e c i s i o n 50 Mft r e s i s t o r Charging voltage 12.0 ± 0.1 kV Swi t c h i ng 1. Thyratron, voltage doubled by Theophanis u n i t , to photon-triggered spark gap which f i r e s 2 . Mai n spark gap. Electrodes 1" diam. 84 PLASMA, • V B I A • c DISCHARGE VESSEL 200 k 51-5 F ^ 12 kV D 10 M 200 k | - V V V I 0 M - A A A o l O M j-AAAHIOM - W V i IOM - A / W M O M - A A A q l O M pAAAr IOM •I20M •*vw 30 k A ^ = 3 Figure 4-1 C i r c u i t Diagram of Z-Pinch A. Brass E l e c t r o d e s B. Pyrex Discharge Tube Wall C. High Voltage Power Supply D. Main C a p a c i t o r Bank E. f'.ain Spark Gap F. Quartz Envelope of U l t r a v i o l e t Source G. Return Conductor (Brass Gauze) H. Resonator Tubes (Pyrex) 85 i n t e g r a t i n g i t since the r e s u l t i n g i n t e g r a l had to be equal to the t o t a l charge stored on the c a p a c i t o r s . The current waveform was under-damped, with a period of 22.4 ysec and a peak value of 175 kA. A s o l e n o i d a l valve i n s e r i e s with a leak valve permitted p r e c i s e f i l l i n g of the discharge tube. The pressure was monitored by a manometer and a t r a v e l l i n g microscope. For a l l the experiments described i n t h i s chapter the d i s -charge tube was f i l l e d to 4.00 ± 0.01 t o r r . I mpurities i n the discharge gas can play a s i g -n i f i c a n t r o l e i n determining the behaviour of the plasma. Throughout a l l parts of t h i s experiment the tube was evacuated and then f i l l e d with f r e s h gas before each shot. F u r t h e r , before each run the pinch was f i r e d at l e a s t three times, with r e f i l l i n g , before data was c o l l e c t e d . Thus j u s t before the plasma was t r i g g e r e d at most one i n 10 5 atoms i n the tube were not helium. A f t e r the plasma has formed many species are released from the w a l l s . One of the advantages of the z-pinch i s that the m a j o r i t y of these p a r t i c l e s remain near the w a l l s . To study the r a d i a l dynamics the glass tubes of the i n t e r f e r o m e t e r were a l i g n e d symmetrically about the plasma mid-plane. Using various tube separations the e l e c t r o n density and e l e c t r o n temperature were measured, by the techniques discussed i n the l a s t two chapters, at r a d i i 86 from the ax i s to 6.5 cm i n 5 mm steps. At l e a s t four shots were recorded i n each case, and the e r r o r bars on a l l the data represent the standard d e v i a t i o n of the mean of those shots. In a d d i t i o n , a systematic e r r o r of up to 20% may be associated with the temperature data before the upper s t a t e s of He II reach t h e i r LTE po p u l a t i o n s , and 10% a f t e r they so do, as discussed i n the previous chapter. 4.3 Radial Dynamics 4.3.1 I n t r o d u c t i o n A f u l l explanation of the dynamics of a high pressure l i n e a r pinch would be a lengthy document. The. purpose of t h i s s e c t i o n i s to present the c o n t r i b u t i o n s made i n t h i s area by t h i s t h e s i s . To put these points i n context the s a l i e n t points of the dynamics are f i r s t presented. The sequence of events i n a z-pinch begins when the main spark gap i s broken down and the f u l l bank voltage i s a p p l i e d across the tube. The e l e c t r i c f i e l d d i s t r i b u t i o n at t h i s i n s t a n t has been studied by MacLatchy (1972). By f a r the highest f i e l d s are at the circumference of the cathode, as shown by his diagram (Figure 4-2). E l e c t r o n s are a c c e l e r a t e d by the e l e c t r i c f i e l d out of the plasma i n t o the co l d gas, and i o n i z e the gas by c o l l i s i o n s . Thus the plasma advances i n a breakdown wave along the vessel w a l l . 87 Figure 4-2. Breakdown i n a Z-Pinch (from MacLatchy, 1972). 88 Once the breakdown wave has reached the anode a conductive path has been e s t a b l i s h e d and current s t a r t s to flow. ( I t i s t h i s i n s t a n t which has been s e l e c t e d as the time o r i g i n . ) The current I ( t ) i s thus f l o w i n g in a large c o - a x i a l 'cable' composed of the plasma (radius r , con-d u c t i v i t y a, and c r o s s - s e c t i o n a l area A) and the re t u r n conductor (radius r , a = °°) and terminated at the anode. Km* The impedance per u n i t l e n g t h , Z ( t ) , i s then Z ( t ) I ( t ) I ( t ) In FTtT + HO Aa (4-1) Thus the current continues to flow at the maximum p o s s i b l e r , along the vessel w a l l . The azimuthal magnetic i n d u c t i o n produced by t h i s current i s , at any radius r , B ( r ) = r J ( r ) dr (4-2) where now the f i n i t e width of the current s h e l l i s recognized by i n t r o d u c i n g the current d e n s i t y J ( r ) . Since t h i s c u r r e n t density i s a x i a l , perpendicular to the magnetic f i e l d there i s a self-imposed Lorentz f o r c e on the current of 89 F(r) = 2 TT J ( r ) B(r) rdr (4-Within 2 ysec of current i n i t i a t i o n t h i s f o r c e i s s u f f i c i e n t to a c c e l e r a t e the current s h e l l toward the a x i s . (The time of departure i s considered f u l l y i n the appendix.) The motion of t h i s high temperature plasma produces a shock wave which i s dri v e n by the Joule heated plasma at a nearly constant v e l o c i t y of 1 cm/ysec toward the a x i s . This shock wave i s discussed i n s e c t i o n 4.3.2. The r e s t r a i n i n g f o r c e on the pinching plasma i s the c o l l i s i o n s i t s u f f e r s with the gas through which i t i s sweeping. Most of t h i s gas i s entrained with i t . As the current decreases i n the second quarter period of the discharge current c y c l e , the for c e d r i v i n g the current sheet i s reduced. At a radius of 2.7 cm the current sheet stops. I t s motion, and the motion of the s h e l l of plasma which i s 'dragged' behind i t , are described i n s e c t i o n 4.3.3. Almost simultaneously with the a r r e s t of the current peak, the shock wave reaches the a x i s , and the gas in that region i o n i z e s r a p i d l y . Much of the current d e n s i t y switches to the a x i s . The growth of e l e c t r o n d e n s i t y on a x i s , and the passage of the r e f l e c t e d shock away from the a x i s , are considered i n s e c t i o n 4.3.4. 90 A l l of these features are p l o t t e d i n the 'streak' diagram, an ( r , t ) p l o t , Figure 4-3. F i n a l l y s e c t i o n 4.3.5 discusses the pronounced changes produced by rev e r s i n g the p o l a r i t y of the ap p l i e d e l e c t r i c f i eld . The c u r r e n t density measurements made by Pachner (1971) on an i d e n t i c a l z-pinch were i n v a l u a b l e i n under-standing these r e s u l t s . His data f o r 4 t o r r helium, pre-sented i n Figure 4-4, were stated to have an accuracy of 4%. One f e a t u r e common to a l l phases of the discharge i s the absence of i o n i z a t i o n a l LTE. This i s to be expected from the c a l cul ated' r a t e of f i r s t i o n i z a t i o n . The c a l c u l a -t i o n of t h i s r a t e proceeds s i m i l a r l y to that f o r the r a t e of second i o n i z a t i o n i n s e c t i o n 3.2. Very few e l e c t r o n s , on c o l l i d i n g with a helium atom, give up enough energy to i o n i z e t h a t atom. A l a r g e r number are capable of donating enough energy to e x c i t e the atom to the upper s t a t e of the resonance l i n e . From here subsequent e x c i t a t i o n to an i o n i z e d s t a t e proceeds e a s i l y s i n c e comparatively l i t t l e a d d i t i o n a l energy i s i n v o l v e d . Thus the rat e of c o l l i s i o n a l e x c i t a t i o n of the He I resonance l i n e (58.4 nm, - 2*P) i s the r a t e which determines the production of He 11 and e l e c t r o n s . The i n v e r s e of t h i s r a t e , f o l l o w i n g Griem (1964), i s 8 x 10 2, 13, 2 ysec f o r T g = 20, 30, 40 x 10 3 °K. These values were c a l c u l a t e d f o r N e = 1 x 1 0 1 6 cm - 3, and sca l e with the *This ts- an oversTmpl'Ifixra-t-ton-, since other rates have been neglected, but the i o n i z a t i o n rates are roughly c o r r e c t Figure 4-3. Pinch Dynamics ( r , t ) Diagram. 92 48 35 "E 24 12 He 4 Torr Ml f! ' ! 1 V \ \ • \ \ I J I ' 1 / 1 \\ / G 7 \ \ I . S I 1 1 1 1 t •' h'l ' /''*' I f V' /' / / \ \ Vs. 4S 36 < 24 1 2 3r(cm) 4 5 b " "7 V 'fl He 4 Torr \ \ i / I l\ i \ . i i !'! \ \ 1! ! i \ /.' i / / 9 • ! \ J w \ v \ \ \ \ \ 10 l ? \ \ \ \ \ \ \ \> \ ure 4-4. Current Density and Magnetic F i e l d Density from Pachner (1971). Numbers i n d i c a t e time i n ysec. 93 r e c i p r o c a l of the e l e c t r o n d e n s i t y . The e f f e c t i v e i o n i z a -t i o n a l r e l a x a t i o n times may be longer, since these values hold only a f t e r the degree of i o n i z a t i o n approaches 1%, which may take a considerable time i n pure helium. Thus f u l l i o n i z a t i o n a l LTE w i l l not normally be found since these times are comparable with the period spent by a helium atom i n the hot portions of the dynamic pinch. 4.3.2 The Precursor Shock The motion of the pinching current sheet occurs at a speed of about 1 cm/ysec, ten times the speed of sound i n room temperature helium. Thus shock wave phenomena are i n v o l v e d . I f the time to pinch and the discharge c u r r e n t h a l f period are comparable then the Lorentz f o r c e on the current s h e l l decreases during p i n c h i n g . In such cases the shock wave may precede the c u r r e n t d e n s i t y peak which d r i v e s i t . In t h i s d i s c h a r g e , mean f r e e paths are of the order of microns. Thus these shocks are fundamentally d i f f e r e n t from the col 1 i s i o n l e s s shocks found i n f a s t pinches (Paul et al. 3 1 965). C o l l i s i o n dominated precursor shocks were f i r s t observed by F o l k i e r s k i and Latham (1963) by i n s e r t i n g glass c y l i n d e r s of various diameters i n t o , and c o a x i a l w i t h , the 94 discharge v e s s e l . These b a f f l e s probably perturbed the pinch (Daughney, 1966), and gave information only about the shock v e l o c i t y . These experiments were performed i n argon, in which the sound speed i s lower. In t h i s experiment i t was p o s s i b l e to observe the t r a j e c t o r y of the shock f r o n t from 1 cm from the vessel wall to the plasma on a x i s , w i t h i n an accuracy of ± 0.2 ysec. Further, the temperature behind the shock was measured. Valuable new i n s i g h t i n t o the nature of the precursor shock in helium has been obtained. The recording of the shock f r o n t t r a j e c t o r y was accomplished p r i m a r i l y by use of the i n t e r f e r o m e t e r with a large tube s e p a r a t i o n , which resolved the e l e c t r o n d e n s i t y created behind the shock. I t was also found that the He I and continuum emission were f i r s t observed d i r e c t l y behind the shock. This was also used to l o c a t e the shock f r o n t . P a r e n t h e t i c a l l y , Funk (1970) did not r e q u i r e as high s e n s i t i v i t y to l o c a t e the shock f r o n t , since i f the helium i s l e s s pure i o n i z a t i o n proceeds more r a p i d l y , and so the e l e c t r o n d e n s i t y produced i s l a r g e r . This i s an a l t e r -n a t i v e , although p e r t u r b i n g , d i a g n o s t i c . The recorded t r a j e c t o r y (Figure 4-3) c l e a r l y shows that i n i t i a l l y the current density i s la r g e at the shock f r o n t , but that l a t e r the shock precedes the regions of peak current d e n s i t y . The shock f r o n t v e l o c i t y i s constant 95 at 1.03 ± .03 cm/ysec i n t o a radius of 1.5 cm. By the time the shock f r o n t reaches 1.5 cm i t i s no longer m a g n e t i c a l l y driven by the current s h e l l which i s coming to r e s t . The shock wave continues to the a x i s , d e c e l e r a t i n g s l i g h t l y . On reaching the plasma on axis i t i s r e f l e c t e d outwards, as described i n s e c t i o n 4.3.4. In c o n s i d e r i n g t h i s shock wave, i n t e r e s t centres on the temperature r i s e which i t causes. At r = 4.5 cm, f o r example, the measured c o n d i t i o n s behind the shock were an e l e c t r o n d e n s i t y of 3 ± 0.5 x 1 0 1 5 cm - 3 and an e l e c t r o n temperature of 28 ± 5 x 10 3 °K. The temperature, which was recorded by the r a t i o of the He I 389 nm l i n e to the continuum, has a l a r g e e r r o r a s s o c i a t e d with i t due to the small emission i n t e n s i t i e s . No emission or e l e c t r o n d e n s i t y could be recorded i n f r o n t of the shock. The s i g n i f i c a n c e of these values i s emphasized by c a l c u l a t i n g the temperature which occurs behind a shock f r o n t of t h i s speed propagating through room temperature helium. A n t i c i p a t i n g the equations which f o l l o w s h o r t l y , an expression f o r the temperature r a t i o across a strong shock i s Ll = 2 M 2 " i l l 11 ( Y + 1 ) 2 (4-4) 96 where M i s the Mach number of the shock (shock speed d i v i d e d by sound speed). Since i o n i z a t i o n i s not e x p l i c i t l y included i n t h i s e x p r e s s i o n , but a s i g n i f i c a n t e l e c t r o n d e n s i t y i s found behind the shock, the value of y , the r a t i o of s p e c i f i c heats, i s set at 1.2. With T x = 300 °K and M = 10 we f i n d T 2 = 3 x 10 3 °K. Thus shock heating alone cannot account -for the temperature measured behind the shock. Two mechanisms preheat the gas i n t o which the shock wave i s t r a v e l l i n g . The more important i s the con-s i d e r a b l e current density which flows at r a d i i l e s s than the radius of the shock. Another c o n t r i b u t i o n i s due to r a d i a -t i v e t r a n s f e r from the hot plasma behind the current peak (Lederman and Wilson, 1967; Grieg and Palumbo, 1969). In helium, due to i t s energy l e v e l s t r u c t u r e , t h i s t r a n s f e r would be l a r g e l y to impurity atoms. Thus one might say that the high pressure z-pinch plasma "preheats" i t s e l f . The c a l c u l a t i o n , from data measured behind the shock, of the amount of preheating was one of the more i n t e r e s t i n g c o n t r i b u t i o n s of t h i s t h e s i s . This c a l c u l a t i o n s t a r t s with the conservation equations f o r mass, momentum, and energy across a shock f r o n t i n a monatomic gas, a l l o w i n g e x p l i c i t l y f o r f i r s t i o n i z a t i o n : 97 P2 u 2 (4-5) Pi + p i Ui P 2 + P2 U, (4-6) Y P i + a i 1 + " J _ (1 + u\ - Y £2. . a 2 I . U 2 2 ( . ?> The equation of s t a t e i s Pn- = Pi R T. (1 + a.) i = 1 ,2 (4-8) where p = mass density u = v e l o c i ty P = pressure T = k i n e t i c temperature a = degree of i o n i z a t i o n W = dimensionless term representing energy d e p o s i t i o n i n the shock f r o n t R = gas constant per u n i t mass I = f i r s t i o n i z a t i o n p o t e n t i a l (24.6 ev) m = atomic mass Y = r a t i o of s p e c i f i c heats Su b s c r i p t 1 -> i n f r o n t of shock Subscript,2 ->• behind shock 98 Second i o n i z a t i o n i s not important i n t h i s context. The r a t i o of the s p e c i f i c heats w i l l have an averaged value of close to 5/3 since i o n i z a t i o n has been e x p l i c i t l y taken i n t o account, although e l e c t r o n i c e x c i t a t i o n has not. To make a rough allowance f o r the l a t t e r a value of y = 1.5 was adopted. This matter i s discussed by Ahlborn and S a l v a t (1967). I f W i s l a r g e , i . e . i f there i s s i g n i f i c a n t energy d e p o s i t i o n i n the f r o n t , then t h i s could account f o r the high temperatures observed behind the f r o n t , even i f there were n e g l i g i b l e preheating. However, i n the c a l c u l a t i o n to f o l l o w , eqs. 4-5 to 4-8 w i l l be a p p l i e d across only the precursor shock, not across the whole pinching s t r u c t u r e . I t was j u s t behind the precursor shock that the temperature of 28,000 °K was measured. Since the width of t h i s region was only a few m i l l i m e t r e s , i t i s u n l i k e l y that there was s i g n i f i c a n t Joule heating i n t h i s f r o n t . Further there appears to be no s i g n i f i c a n t heat t r a n s f e r to t h i s f r o n t . Thermal conduction i s too slow ( s e c t i o n 5.2). R a d i a t i v e t r a n s f e r cannot be s i g n i f i c a n t i n helium. Thus there i s l i t t l e energy deposited at the precursor shock f r o n t , and so W w i l l be neglected. Another approach to t h i s question i s to consider the e n t i r e pinching s t r u c t u r e as a s i n g l e magnetohydrodynamic phenomenon. In t h i s case there would indeed be s i g n i f i c a n t Joule heating i n the pinching wave, since the current d e n s i t y 99 peak i s part of t h i s s t r u c t u r e . The pinch i s then well described by the formalism of a Chapman-Jourget detonation (Ahlborn, p r i v a t e communication), i f preheating i s neglected. Thus a v e r i f i c a t i o n of the amount of preheating would be d e s i r a b l e (see s e c t i o n 6.4). These f i v e equations contain ten unknowns (p^ , P.., T. , u., a,. , i = 1,2). Thus f i v e experimental values are needed to perform c a l u c l a t i o n s . S p i t z e r (1956) has c a l c u l a t e d the time required f o r the e l e c t r o n and ion k i n e t i c temperatures to approach the same value. At a d e n s i t y of 1 0 1 5 cm"3 and a temperature of 28,000 °K the r e l a x a t i o n time i s 5 x 10" 8 sec. F u r t h e r , the ion and neutral temperatures w i l l be c l o s e l y t i e d together by c o l l i s i o n s . Thus the use of the e l e c t r o n temperature as the gas k i n e t i c temperature i n the equation of s t a t e i s j u s t i f i e d . However, since the plasma i s r a r e l y i n LTE the use of the Saha equation to r e l a t e a and T g i s not reasonable. Instead, a i s estimated from the measured values of e l e c t r o n d e n s i t y . The r e s u l t i n g e r r o r s i n a are not very important. Equations 4-5, 4-7, and 4-8 y i e l d a useful form: 100 T i ( l + 0 l ) = T 2 ( l + a 2) RY 1 -P i 2^ 1 P 2 2J + ( a i - a 2 ) — v 'm (4-9) This equation can be used to estimate the temperature ahead of the precursor shock. In t h i s case Ui = 1.03 ± .03 x 101* m/sec T, = 28 ± 5 x IO 3 °K N « N = 3.0 ± 0.5 x 1 0 1 5 cm"3 e I e 2 and T i i s to be estimated. A c a l c u l a t i o n to obtain T i i s not p o s s i b l e s i n c e , of the f i v e values r e q u i r e d , only two are known and two more can be estimated. Nevertheless i t t r a n s p i r e s that a lower l i m i t can be placed on T i from the data a v a i l a b l e . Since the den s i t y of helium atoms to which the tube was f i l l e d i s N 0 = 1.33 x 1 0 1 7 cm" 3, a 2 w i l l be about 3 x 10- 2 and cn w i l l be n e g l i g i b l y s m a l l . ( I f the gas i n f r o n t of the shock has expanded these values w i l l be too s m a l l . The e r r o r thus produced i n T x w i l l not be important compared to the u n c e r t a i n t y i n T 2.) Thus, 101 T i « T 2 ( l + a 2) + - 1 I a 2 — -. z m (4-10) C l e a r l y the lower l i m i t of Ti i s obtained by n e g l e c t i n g i o n i z a t i o n ( i . e . a 2 - 0 i n s i d e the s.quare b r a c k e t s ) , and t a k i n g p i / p 2 = 0. I t i s T r = 2.0 x TO* °K. A f t e r c o n s i d e r i n g the experimental e r r o r s involved i t i s found that Thus the shock i s q u i t e weak. I t only increases the tempera-ture of the plasma from about 20,000 °K to about 28,000 "K. ment s i n c e n e i t h e r the He I 389 l i n e nor the continuum would u s u a l l y be de t e c t a b l e at t h i s temperature with our o p t i c a l system. (M = U i / c = Ui/m/yRT) at most 1.3 ± 0.2. Here c i s the speed of sound. Thus the shock i s very weak. This r e s u l t i s ra t h e r p a r a d o x i c a l , since the i n i t i a l o b servation was that the temperature behind the shock was anomalously high. We f i n d that the gas i n f r o n t of the shock has been s t r o n g l y preheated, and that the shock T i > 2.0 ± 0.3 x 10 K This temperature i s not i n c o n s i s t e n t with e x p e r i -The Mach number of the precursor shock i s then 102 wave i s weak. This i s i n c o n t r a d i s t i n c t i o n to the common model of the z-pinch which i n v o l v e s r a p i d heating by a strong shock. In f a c t most of the heating i s Joule heating by the current d e n s i t y , both before and a f t e r the passage of the shock wave. 4.3.3 The Pinch Phase During the pinch the current s h e l l e n t r a i n s a s i g n i f i c a n t f r a c t i o n of the gas ^ through which i t passes. This gas, already heated to about 28,000 °K by Joule heating and by the precursor shock, i s heated f u r t h e r to about 40,000 °K by the peak current d e n s i t i e s , and i o n i z e d to e l e c t r o n d e n s i t i e s of about 7 x 1 0 1 6 cm - 3. This plasma t r a i l s the current peak by about 1 cm, which i s curious s i n c e the Lorentz f o r c e (eq. 4-3) i s sharply peaked at the l o c a t i o n of the current density maximum. York and Jahn (1970) have shown that the region of current d e n s i t y i s l i n k e d to the region of high e l e c t r o n d e n s i t y by an e l e c t r i c f i e l d . This f i e l d a c c e l e r a t e s e l e c t r o n s produced i n the c o l l i s i o n s between the gas atoms and the p a r t i c l e s of the tenous current c a r r y i n g plasma to the pinch v e l o c i t y . The t r a j e c t o r y of the peak Lorentz f o r c e was derived by Pachner (1971) from his measurements of magnetic f i e l d i n t e n s i t y i n an i d e n t i c a l z-pinch. This t r a j e c t o r y and that of the peak e l e c t r o n d e n s i t y are p l o t t e d i n Figure 4 103 The temperature and density of the imploding features were measured i n t h i s experiment, and are presented i n Figure 4-5 and 4-6 f o r times representing the middle and l a t e stages of the p i n c h , t = 6 ysec and t = 9 ysec. Figures 4-5 and 4-6 provide good i n s i g h t i n t o the s t r u c t u r e of the imploding current sheet and d e n s i t y peak. The l o c a t i o n of the precursor and the peak Lorentz for c e density are marked on each. These temperatures were deduced from the measured i n t e n s i t y r a t i o of the He II 469 nm to He I 389 nm l i n e s . At a given r a d i u s , about 3 ysec a f t e r passage of the current d e n s i t y peak, the temperature i s seen to r i s e from about 40,000 to about 50,000 °K. (At a given time, as i n Figure 4-5, the temperature peak t r a i l s the current peak by 1.5 cm, s i n c e at t h i s time the pinch v e l o c i t y i s about 0.5 cm/ysec.) I t appears as i f the response of the r a t i o of l i n e i n t e n s i t i e s to a sudden change of temperature has been delayed, s i n c e , s u r e l y , the k i n e t i c temperature r i s e i s a s s o c i a t e d with the current peak. This appearance i s c o r r e c t . In s e c t i o n 3.2 i t was shown that the r e l a x a t i o n time f o r popu-l a t i o n of the upper st a t e s of He I I , under the c o n d i t i o n s of the p i n c h , was about 3 ysec. This i s p r e c i s e l y the period observed to elapse before the i n t e n s i t y of the He II 469 nm l i n e increases i n i n t e n s i t y a f t e r the temperature r i s e . 105 i 106 These observations are a good example of the importance of t h i s r e l a x a t i o n time. He II l i n e s , under thes c o n d i t i o n s , do not respond to changes i n k i n e t i c temperature u n t i l 3 ysec a f t e r the change. Fu r t h e r , i f the k i n e t i c temperature i s changing r a p i d l y the population of these e x c i t e d s t a t e s w i l l not reach t h e i r LTE values, i n t r o d u c i n g a d d i t i o n a l u n c e r t a i n t y , as mentioned i n s e c t i o n 3.2, i n t o the e l e c t r o n temperature determinations by t h i s method.^ Figure 4-7 shows e l e c t r o n temperature and de n s i t y of the imploding e l e c t r o n density peak at a l l times during the c o l l a p s e to the a x i s . To produce an improved value f o r the temperature, the i n t e n s i t y of the He II 469 nm l i n e was advanced 3 ysec before the r a t i o of l i n e i n t e n s i t i e s was c a l c u l a t e d . The r e s u l t s would be expected to have a systematic e r r o r of no more than 10%, as discussed i n se c t i o n 3.2. A f t e r the passage of the mass peak, at any r a d i u s , the temperature and density decay i n the wake r e g i o n . 4.3.4 The Post-pinch Phase Nine ysec a f t e r current i n i t i a t i o n the precursor reaches the a x i a l r e g i o n . The whole character of the ^The t e m p e r a t u r e d a t a o f t h e p r e c e d i n g s e c t i o n , 4.3.2, was m e a s u r e d f rom t h e r a t i o o f t h e He I 389 nm l i n e to the c o n t i n u u m i n t e n s i t y . P L T E p o p u l a t i o n s b e t w e e n t h e s e two s t a t e s a r e e s t a b l i s h e d v e r y r a p i d l y . T h u s no d e l a y in r e s p o n s e to a t e m p e r a t u r e c h a n g e w i l l be f o u n d . 1 0 7 108 discharge changes. E l e c t r o n d e n s i t i e s exceeding 2 x 1 0 1 7 cm"3 are produced on a x i s , and a considerable f r a c t i o n of the current density switches to t h i s conductive path. The f o r c e d r i v i n g the c o l l a p s i n g c u r r e n t s h e l l i s much reduced, and so i t s r a d i a l motion ceases abr u p t l y due to the k i n e t i c pressure of the swept-up plasma. The c o l l i s i o n of the precursor shock at the a x i s (a c o n c e n t r i c shock), as well as the considerable preheating which has occurred i n the a x i a l region produces a hot, dense plasma. There are then two paths of s i m i l a r c o n d u c t i v i t y a v a i l a b l e to the c u r r e n t , the a x i s and the pinching c u r r e n t s h e l l . The impedance per u n i t length of these paths can be c a l c u l a t e d from eq. 4-1. However, there i s a back emf associated with the path through the pinching c u r r e n t s h e l l . This a r i s e s since the s h e l l i s moving r a d i a l l y through an azimuthal magnetic f i e l d , and so a x i a l current i n the s h e l l i s acted on by a v x B f o r c e , where y_ i s the v e l o c i t y of the current sheet. Thus, even though the path on a x i s has a higher inductance, i t has a comparable impedance to the current s h e l l , and about h a l f the current d e n s i t y switches to the region near the a x i s (Figure 4-4). This hot a x i a l plasma, heated f u r t h e r by the current d e n s i t y , expands outward, as depicted as the shaded area i n Figure 4-3. The highest density i s always found on a x i s . 109 The r a p i d reduction of current i n the s h e l l brings the s h e l l to an abrupt h a l t . The mass peak stops a l s o . The a r r e s t of t h i s plasma occurs at a radius of 2.7 cm. This i s c l e a r l y shown i n Figure 4-8, which d e p i c t s the e l e c t r o n d e n s i t y recorded as a f u n c t i o n of time at 2.8 and 2.6 cm. Figure 4-8. E l e c t r o n Density P r o f i l e s . —* — Radius = 2.8 cm Radius = 2.6 cm V e r t i c a l Scale - 2.12 x 1 0 1 6 cm~ 3/div Time Base - 2 usec/div Synchronization pulse occurs at * 2 ysec This a r r e s t i s both i n t e r e s t i n g in the study of the dynamics and of considerable p o t e n t i a l b e n e f i t i n a p p l i c a t i o n s of the z-pinch as a spectroscopic source or s c a t t e r i n g medium. The advantages of t h i s region are the n o reduced gradients i n den s i t y and temperature, which are important s p e c t r o s c o p i c a l l y , and the existence of a current f r e e plasma, which i s important f o r several p o s s i b l e s c a t t e r i n g a p p l i c a t i o n s . These points are considered f u r t h e r i n Chapter 6. Meanwhile, back at the a x i s , the precursor has r e f l e c t e d from the a x i a l plasma. I t s t r a j e c t o r y back through the various f e a t u r e s of the pinch can e a s i l y be seen as an increase i n e l e c t r o n d e n s i t y (Figure 4-3 and 4-8). Thus i t i s a p o s s i b l e d i a g n o s t i c device f a r these regions (Ahlborn et al.} 1973). At 11.5 ysec, f o r example, s h o r t l y a f t e r the current zero, the temperature and density d i s t r i b u t i o n i s as shown i n Figure 4-9. The den s i t y r i s e i s the com-pression caused by the r e f l e c t e d shock. ( I t i s a weak shock, causing l i t t l e i o n i z a t i o n . Also since the ion den s i t y i s t i e d to the e l e c t r o n density by charge n e u t r a l i t y , and the neutral d e n s i t y connected to the ion de n s i t y by c o l l i s i o n s , t h i s increase i n e l e c t r o n density i s the same as the increase i n mass de n s i t y . ) A l s o , the temperature r i s e at r = 2 cm i s caused by the r e f l e c t e d shock. To see t h i s , and also to prove that the r e f l e c t i o n i s s t i l l a shock, eq. 4-9, a form of the Rankine-Hugoniot equations can be a p p l i e d . In t h i s case the known q u a n t i t i e s are Figure 4-9. E l e c t r o n Temperature and Density at 11.5 ysec. 112 Ti = 38 ± 4 x 10 3 °K ui = 1.14 ± 0.05 cm/ysec P 1 / P 2 = 0.85 ± 0.05 a i = a 2 = 0.30 ± 0.05 The values of a , which are not c r i t i c a l , have been estimated by assuming that the t o t a l gas d e n s i t y at these l a r g e r a d i i has returned to roughly i t s i n i t i a l value. With the luxury of the r e q u i s i t e f i v e values the c a l c u l a t i o n of T 2 proceeds smoothly, and y i e l d s T 2 = 41 ± 5 x 10 3 °K. The temperature as measured from the He II 469 to He I 389 l i n e i n t e n s i t y r a t i o agrees best with t h i s value about 1.5 ysec a f t e r the shock has passed (Figure 4-9). This i s again due to the delays i n r e l a x a t i o n to the upper l e v e l s of He I I , e x a c t l y as before. The good agreement shows that the r e f l e c t e d wave i s indeed a shock wave. Presumably i t i s now dri v e n by the rap i d expansion of the hot pinched gas back to l a r g e r r a d i i . Joule heating by the second h a l f - c y c l e of the current w i l l also be important. This r e f l e c t e d shock i s a l s o a weak shock; i t s Mach number i s 1.18 ± 0.07. 113 4.3.5 E f f e c t s of P o l a r i t y Reversal In the l a s t few s e c t i o n s a d i s c u s s i o n of the r a d i a l dynamics of the z-pinch has been given , with emphasis on the points which have been e l u c i d a t e d by t h i s work. This s e c t i o n contains a q u a l i t a t i v e d i s c u s s i o n of the e f f e c t s of p o l a r i t y r e v e r s a l , and presents some data. The i n i t i a l breakdown i n a high pressure z-pinch occurs along the wall of the c y l i n d r i c a l v e s s e l . The electrodes are not symmetrical, since one i s i n i t i a l l y i n a high e l e c t r i c f i e l d while the other i s not. The highest f i e l d s occur at the circumference of that e l e c t r o d e which i s not connected to the return conductor. I t i s here that the gas i s f i r s t i o n i z e d , and from here that a breakdown f r o n t proceeds along the vessel wall to the other e l e c t r o d e . This f r o n t propagates d i f f e r e n t l y f o r the two p o l a r i t i e s , as shown s c h e m a t i c a l l y i n Figure 4-10. In making the measurements discussed so f a r the high f i e l d e l e c t r o d e has always been the cathode. With t h i s p o l a r i t y the e l e c t r o n s i n the r e s u l t i n g breakdown plasma are ac c e l e r a t e d from the plasma i n t o the c o l d gas, i o n i z i n g i t by c o l l i s i o n s . Thus the breakdown wave moves along the wall at the d r i f t speed of the e l e c t r o n s in the a p p l i e d f i e l d . With the anode as the high f i e l d e l e c t r o d e there are two modes of propagation that might be considered. One can q u i c k l y be d i s c a r d e d , namely that the f i e l d a c c e l e r a t e s 114 + IF HIGH."'FIELD ELECTRODE IS / ANODE / / -< N \ ELECTRON ^ PATHS ^RETURN CONDUCTOR . 1 . i i • • • i • i i • • • i , , • , , . . . . . v . , . . . . . .•, , HIGH Z - P I N C H INITIAL ELECTRIC ^FIELD k ^ P Y R E X WALL I V / \ / ^ - _ < \ \ \ \IF HIGH\FIELD ELECTRODE IS CATHODE. ELECTRON PATHS + Figure 4-10. Breakdown Mechanism i n Both P o l a r i t i e s . i 1 1 5 ions from the plasma i n t o the cold gas. The slow ion d r i f t speed would make t h i s process much slower than the observed breakdown time. The other p o s s i b i l i t y i s e l e c t r o n production i n the gas. These e l e c t r o n s w i l l m u l t i p l y as they f a l l toward the plasma, thereby heating the gas and advancing the plasma boundaries. The i n i t i a l production of these e l e c t r o n s would be by p h o t o i o n i z a t i o n , presumably i n v o l v i n g i m p u r i t i e s . The propagation of the breakdown wave should be slower i n t h i s p o l a r i t y . 1 This model was confirmed by measuring the t r a n s i t times f o r the breakdown f r o n t along the tube, f o r the two p o l a r i t i e s . A c t u a l l y , the period between voltage a p p l i c a t i o n and current i n i t i a t i o n was measured. I t was 0.23 ± 0.02 ysec with the cathode as the high f i e l d e l e c t r o d e , and 0.34 ± 0.04 ysec i n the other p o l a r i t y . This period includes the time required to produce the f i r s t few e l e c t r o n s , which i s the same i n both p o l a r i t i e s . Thus the d i f f e r e n c e i n t r a n s i t time between the two p o l a r i t i e s exceeds 50%, but i s l e s s than a f a c t o r of two or three. The longer t r a n s i t time was, as expected, associated with the anode being i n the high f i e l d posi t i o n . ^ I t i s a l s o p o s s i b l e t h a t w i t h t h e anode as t h e h i g h f i e l d e l e c t r o d e b r e a k d o w n may o c c u r a l o n g t h e e n t i r e t u b e s i m u l t a n e o u s l y , by p h o t o i o n i z a t i o n , r a t h e r t h a n as a b r e a k d o w n wave. T h i s w o u l d n o t a f f e c t t h e o t h e r a r g u m e n t s o f t h i s s e c t i o n . 116 With the i n i t i a l high f i e l d s at the anode ( p o s i t i v e p o l a r i t y ) the plasma formed by breakdown w i l l have a greater r a d i a l depth than with negative p o l a r i t y . This i s because the e l e c t r o n s emitted by the plasma i n the negative case are s t r o n g l y a t t r a c t e d to the wall as they f a l l forward i n the breakdown wave. In the p o s i t i v e case, however, the plasma i s e l e c t r o n d e f i c i e n t . Not only are the e l e c t r o n s that propagate the plasma created from a l a r g e r volume of gas, but as they f a l l toward the plasma they are r e p e l l e d by the w a l l . Thus a s u b s t a n t i a l l y t h i c k e r plasma i s created. Both s i t u a t i o n s are shown s c h e m a t i c a l l y i n Figure 4 - 1 0 . Once the i n i t i a l conductive path has formed a l a r g e current flows along i t . P o s i t i v e p o l a r i t y produces a d i f f u s e c u r r e n t , so the current d e n s i t y i s everywhere (except l a t e r , on a x i s ) l e s s than with negative p o l a r i t y . Thus with the anode i n the high i n i t i a l e l e c t r i c f i e l d ( p o s i t i v e p o l a r i t y ) the temperature, and thus the e l e c t r o n d e n s i t y , would be reduced at a l l times during the pinch. Figure 4 -11 shows the d e n s i t y of the plasma with p o s i t i v e p o l a r i t y , and compares i t with negative p o l a r i t y . (Note that these values were obtained w h i l s t a l t e r n a t i n g the p o l a r i t y , to ensure that other c o n d i t i o n s were i n f a c t i d e n t i c a l . ) With p o s i t i v e p o l a r i t y there was never, during the pinch phase, enough He II 469 nm.line emission to record with any accuracy. This implies an upper l i m i t of about 117 Figure 4-11. E l e c t r o n Temperature and Density of Pinching Plasma - Anode as high f i e l d e l e c t r o d e . Data with cathode as high f i e l d e l e c t r o d e shown dashed. 117a 118 35,000 °K f o r the temperature. Within 6 ysec of discharge i n i t i a t i o n , however, the temperature recorded by the r a t i o of the He I 389 l i n e i n t e n s i t y to the continuum i n t e n s i t y exceeded 30,000 °K, p r o h i b i t i n g measurement by t h a t technique. Thus only an upper and a lower l i m i t could be deduced; 30,000 °K < T e < 35,000 °K. The reduced temperature slows the dynamics (although one would expect the current s h e l l to be l i t t l e a f f e c t e d ) . The two p o l a r i t i e s are compared i n the ( r , t ) p l o t i n Figure 4-12. The p o s i t i v e p o l a r i t y d ensity s h e l l moves more s l o w l y . I t s precursor seems to be l i t t l e d i f f e r e n t , although the d i f f u s e i n i t i a l plasma may have given i t a "head-start." The most s t r i k i n g d i f f e r e n c e s are the reduced temperature and d e n s i t y i n the p o s i t i v e p o l a r i t y case. The d i f f e r e n c e s i n the plasma on axis caused by r e v e r s i n g p o l a r i t y are much sm a l l e r . This i s not s u r p r i s i n g , s ince the precursor i s s i m i l a r i n both cases, and the a x i a l current d e n s i t i e s are probably i d e n t i c a l . The primary aim of t h i s chapter has been to describe the aspects of high pressure pinch dynamics which have been e l u c i d a t e d by t h i s work. These have i n c l u d e d : I . The h i g h p r e s s u r e z - p i n c h p r o v i d e s i t s own, s u b s t a n t i a l , p r e h e a t i n g t o t h e g a s b e f o r e i t i s r e a c h e d by t h e i m p l o d i n g s h o c k ; 119 RADIUS cm / i s Figure 4-12. Pinch.Dynamics ( r , t ) Diagram Data points and s o l i d l i n e s recorded with anode as high f i e l d e l e c t r o d e . Dashed l i n e s are cathode as high f i e l d e l e c t r o d e . 1 20 2. The m a j o r i t y CM t h e t e m p e r a t u r e r i s e i s due t o J o u l o h e a t i n g r a t h e r t h a n s h o c k h e a t i n g ; 3. The r e l a x a t i o n t i m e t o t h e He II u p p e r s t a t e s c a n n o t ho n e g l e c t e d i n i nter^-p r e t i n g t e m p e r a t u r e s m e a s u r e d by t h e r a t i o o f l i n e I n t e n s i t i e s . 4. The s h o c k wave a s s o c i a t e d w i t h t h e p i n c h i n g c u r r e n t p r e c e d e s t h e c u r r e n t a n d , on r e a c h i n g t h e a x i s , a i d s i n s w i t c h i n g much o f t h e c u r r e n t d e n s i t y t o t h e a x i s . A f t e r r e f l e c t i o n t h i s s h o c k c a n be u s ^ d t o m e a s u r e t h e p r o p e r t i e s o f t h e p l a s m a . 5. D r a m a t i c c h a n g e ^ i n t e m p e r a t u r e and e l e c t r o n d e n s i t y a r e c a u s e d by r e v e r s i n g t h e d i r e c t i o n o f t h e i n i t i a l e l e c t r i c f i e l d a p p l i e d a c r o s s t h e d i s c h a r g e t u b e . Another aim has been to c h a r a c t e r i z e a region of the discharge which i s highly s u i t a b l e as a spec t r o s c o p i c 121 source or as a s c a t t e r i n g medium. The data presented here w i l l y i e l d the gradients i n temperature and d e n s i t y a s s o c i a t e d with that source. Gradients i n another dimension, the a x i a l d i r e c t i o n , can be e q u a l l y as important, however. Thus the next chapter i s concerned with l o n g i t u d i n a l v a r i a t i o n s i n the z-pinch piasma. Chapter 5 LONGITUDINAL STRUCTURE 5.1 I n t r o d u c t i on This study of the l o n g i t u d i n a l s t r u c t u r e of the z-pinch was undertaken f o r the same two reasons as prompted the study of the r a d i a l s t r u c t u r e described i n the l a s t chapter. These are, the importance of gradients of den s i t y and temperature i n a p p l i c a t i o n s of the z-pinch as a spec-t r o s c o p i c source, and i n t e r e s t in the p h y s i c a l mechanisms r e s p o n s i b l e f o r l o n g i t u d i n a l v a r i a t i o n s . This work has enabled us to i d e n t i f y and assess the most s u i t a b l e regions f o r spectroscopic a p p l i c a t i o n s , and to reach a q u a l i t a t i v e understanding of the physics i n v o l v e d . Section 5.2 points out which heat t r a n s p o r t mechanisms govern the l o n g i t u d i n a l temperature v a r i a t i o n s . The experimental data r e l e v a n t to t h i s s e c t i o n was recorded along the length of the e l e c t r o n d e n s i t y peak at a p a r t i c u l a r time. This complication was necessary because the discharge does not pinch at the same time a l l along the tube, but 1 22 1 23 i s advanced near the e l e c t r o d e s . Section 5.3 discusses t h i s phenomena b r i e f l y , and presents data taken at both a constant radius and a constant time, as would be usual f o r spectro-scopic a p p l i c a t i o n s . For a l l measurements discussed i n t h i s chapter the cathode was chosen to be the e l e c t r o d e i n the i n i t i a l high e l e c t r i c f i e l d ( s e c t i o n 4.3.5). 5.2 L o n g i t u d i n a l Heat Transport To e x p l a i n the temperature d i s t r i b u t i o n produced by l o n g i t u d i n a l heat t r a n s p o r t i t i s f i r s t necessary to examine the energy losses from t h i s plasma, to determine which energy f l u x e s are important. The period during which the discharge i s of i n t e r e s t , e i t h e r dynamically or s p e c t r o s c o p i c a l l y , i s from about t = 2 to 15 ysec. By 2 ysec the plasma i s s t a r t i n g to p i n c h , and by 15 ysec i t i s expanding toward the w a l l s , where i t w i l l l o s e most of i t s energy. During t h i s time energy i s being deposited i n the plasma by Joule heating (and some by shock h e a t i n g ) , and i s being l o s t by r a d i a t i o n and by heat t r a n s p o r t . The three mechanisms by which energy i s r a d i a t e d are bremsstrahlung, l i n e r a d i a t i o n , and recombination r a d i a t i o n . The bremsstrahlung power loss i s (Cooper, 1966) 1 24 P D = 1 .4 x IO" 3 1* N 2 T ' B e e watts cm- 3 (5-1) f o r a predominantly s i n g l y - i o n i z e d plasma (T i n °K, H Q i n c m - 3 ) . An upper l i m i t to the l i n e r a d i a t i o n losses ( i m p u r i t y l i n e s are unimportant f o r a discharge of t h i s low temperature) i s , approximately, the r a d i a t i o n r a t e of the He I resonance l i n e , as i f i t were o p t i c a l l y t h i n . This i s P. = 2.6 x I O " 3 0 N. 2 exp L e 3.9 x 10" watts cm _ 3 (5-2) Free-bound r a d i a t i o n releases energy at the ra t e (Cooper, 1966) P D = 4.4 x I O - 2 9 N 2 T "* K e e watts cm - 3 (5-3) Under t y p i c a l c o n d i t i o n s (N g - 5 x 1 0 1 6 cm - 3, T g = 40,000 °K) the power l o s s per u n i t c r o s s - s e c t i o n a l area i s found by m u l t i p l y i n g the above expressions by the length of the plasma. The r e s u l t i s 4 x 10 3, 8 x 1 0 s , and 3 x 10 3 watts cm - 2 f o r bremsstrahlung, l i n e r a d i a t i o n , and recombina-t i o n r a d i a t i o n r e s p e c t i v e l y . 1 25 The maximum rate of Joule heating i n t h i s plasma i s about 4 x 10 6 watts cm - 2, where again a u n i t c r o s s -s e c t i o n a l area has been considered. Thus, of the energy deposited i n the plasma, about 20% i s r a d i a t e d , and the remainder heats the gas being swept up by the pinching plasma. Note that the p r i n c i p a l r a d i a t i o n power l o s s mechanism, l i n e r a d i a t i o n , increases e x p o n e n t i a l l y with i n c r e a s i n g temperature. Thus i f a d d i t i o n a l energy i s deposited i n some region of the plasma the r i s e i n tempera-ture w i l l be reduced, since the rate of r a d i a t i v e energy l o s s w i l l r i s e r a p i d l y . There are al s o two modes of heat t r a n s p o r t . One i s thermal d i f f u s i o n . The other i s the t h e r m o e l e c t r i c e f f e c t , that i s , the e l e c t r o n s which c a r r y the discharge current also t r a n s p o r t heat with them. D i f f u s i o n can occur both r a d i a l l y and l o n g i t u d i n a l l y , but s i g n i f i c a n t currents only flow l o n g i t u d i n a l l y . Radial d i f f u s i o n i s i n s i g n i f i c a n t once the plasma has l e f t the w a l l s . I t turns out t h a t , f o r most of t h i s d ischarge, l o n g i t u d i n a l thermal d i f f u s i o n i s al s o unimportant. To see t h i s the thermal d i f f u s i o n equation was solved £ 9 2T _ r 3T K 3 ^ " c p at (5-4) 126 where K i s the thermal c o n d u c t i v i t y , and c the heat c a p a c i t y at constant pressure, of the gas of density p. The boundary c o n d i t i o n s chosen involved one el e c t r o d e o n l y , since the r e s u l t shows that thermal d i f f u s i o n i s important only very near the e l e c t r o d e . In a pinch of these dimensions the other e l e c t r o d e i s not i n v o l v e d . Thus the model was a h a l f -i n f i n i t e c y l i n d e r of plasma at T = T 0 to which, at z = 0, an e l e c t r o d e , at T = 0, i s a p p l i e d at t = 0. The temperature in the plasma i s then T(z) = T 0 erf z_ 2 cp Kt (5-5) where the e r r o r f u n c t i o n i s er f (x) _2_ - S ds (5-6) I f the e f f e c t s of thermal d i f f u s i o n are to be unimportant then the value of the e r r o r f u n c t i o n must remain near i t s maximum value of u n i t y . To r e q u i r e that the value exceed 0.9 i s e q u i v i l a n t to demanding that i t s argument must exceed 1.17 (Abramowitz and Stegun, 1965). This c o n d i t i o n 127 determines how f a r i n t o the plasma the e f f e c t of thermal d i f f u s i o n w i l l penetrate. The thermal d i f f u s i o n c o e f f i c i e n t , K , has been c a l c u l a t e d by S p i t z e r (1956) and by Marshall (1960). For a non-magnetized plasma at 35,000 °K and a t o t a l d e n s i t y of 2 x 1 0 1 7 cm - 3 t h e i r r e s u l t s are 0.17 and 0.15 watts/cm °K, r e s p e c t i v e l y . I f the heat f l u x i s transverse to a magnetic f i e l d t h e i r r e s u l t s should be d i v i d e d by 1 + (w /2irv • •) where w c e i s the e l e c t r o n c y c l o t r o n frequency and v g e the e l e c t r o n - e l e c t r o n c o l l i s i o n frequency. Using the expressions i n Table 2-1 i t i s found that w c e / 2 i T V e e = 0.17 f o r a magnetic f i e l d of 0.5 weber m."2, the l a r g e s t found i n t h i s plasma (Pachner, 1971), and an e l e c t r o n d ensity of 6 x 1 0 1 6 cm" 3. Thus the magnetic f i e l d e f f e c t s are unimportant, and the thermal d i f f u s i o n c o e f f i c i e n t can be taken as K = 0.16 watts/cm °K. Thus the c o n d i t i o n that thermal d i f f u s i o n e f f e c t s be important, out to a distance z from an e l e c t r o d e , becomes, f o r p = 2 x 1 0 1 7 cm"3 and c - 12k (L i c k and Emmons, 1962) z > 0.2 / t cm where t , the time since the i n i t i a t i o n of the discharge, i s in microseconds, and k i s Boltzmann's constant. During 15 128 ysec thermal d i f f u s i o n only a f f e c t s the 8 mm of plasma nearest the e l e c t r o d e s . Between these r e g i o n s , which i s where a l l measure-ments were made, the heat f l u x vector i s q = <J> J z where f, the thermioeTeetrTe c o e f f i c i e n t , has, f o r t h i s plasma, a value of -10 v o l t s ( M a r s h a l l , 1960). I t i s negative because the heat f l u x - i s a n t i p a r a l l e l to the c u r r e n t . At the maximum current d e n s i t y found i n t h i s d i s c h a r g e , J z = 45 MA m~2 (Pachner, 1971), the heat f l u x i s 4.5 x IO1* watts cm - 2, d i r e c t e d toward the anode. Total' energy Tost to the anode i s about 20 j o u l e s . This heat f l u x i s not an i n s i g n i f i c a n t f r a c t i o n of the energy i n the plasma. Thus one expects to f i n d a c o n t i n u a l l y i n c r e a s i n g temperature as one proceeds from the mid-plane of the discharge toward the anode. The temperature r i s e i s i n h i b i t e d , though, by the f a c t that the l i n e r a d i a -t i o n increases e x p o n e n t i a l l y with i n c r e a s i n g temperature. Since the temperature dependence of a l l these energy g a i n , l o s s , and t r a n s p o r t mechanisms i s known, i t should be p o s s i b l e to proceed f u r t h e r and c a l c u l a t e the l o n g i t u d i n a l temperature d i s t r i b u t i o n . A c t u a l l y much more inf o r m a t i o n i s r e q u i r e d . One important question i s whether or not the energy d e n s i t y i s f a i r l y constant i n time, i . e . 1 29 whether or not the plasma i s i n a s t a t i o n a r y s t a t e . Several authors have assumed t h i s to be true and then have a l s o required other assumptions to c a l c u l a t e the temperature d i s t r i b u t i o n . De Borde and Haas (1961) also assume a s k i n c u r r e n t , uniform pressure, and no r a d i a l v a r i a t i o n i n tempera-t u r e . Haines (1961 and 1965) c a l c u l a t e s the current density (although his r e s u l t d i f f e r s c o nsiderably from the York-Jahn model a p p l i c a b l e to t h i s p i n c h ) , but otherwise r e q u i r e s the same assumptions. None of these holds i n t h i s discharge. There i s a complex r a d i a l s t r u c t u r e of current density and temperature, and pressure v a r i a t i o n s s i n c e , during a pinch, there i s inadequate time f o r a sound wave to t r a v e l the length of the discharge. Further, although the temperature of the pinching plasma i s f a i r l y constant (Figure 4-6) the energy d e n s i t y i s not, since the pressure i s c o n t i n u a l l y i n c r e a s i n g as gas i s swept up by the pinch. The plasma i s not i n a steady s t a t e . C a l c u l a t i o n of the temperature d i s t r i b u t i o n along the tube i s thus i m p r a c t i c a l , and recourse to experiment becomes necessary. The measurement of the l o n g i t u d i n a l v a r i a t i o n s of e l e c t r o n temperature and density was s t r a i g h t f o r w a r d . The i n t e r f e r o m e t e r tubes, separated by about 5 cm, were a l i g n e d at various l o c a t i o n s along the discharge tube. Using the methods of e a r l i e r chapters the density and temperature were recorded f o r each of at l e a s t four shots. Care was 1 30 taken to ensure that c o n d i t i o n s v a r i e d as l i t t l e as p o s s i b l e during the measurements. A l s o , the order i n which data was c o l l e c t e d was random, ra t h e r than progressing from one end of the tube to the other. Data were c o l l e c t e d at various i n t e r v a l s over a period of almost three months. The only complication arose from the f a c t that the timing of the pinching a c t i o n v a r i e d l o n g i t u d i n a l l y . Near the e l e c t r o d e s the c o l l a p s i n g d e n s i t y s h e l l was f u r t h e r advanced than i n the centre of the tube. Figure 5-1 i s a map of the contours of e l e c t r o n density at the time s e l e c t e d f o r these measurements (t = 6.1 ysec, near the middle of the pinch phase.) To study the e f f e c t s discussed above i t c l e a r l y was necessary to record temperatures and d e n s i t i e s along the locus of density maxima (path AA i n diagram), so as to study plasma with s i m i l a r h i s t o r i e s . (A locus p a r a l l e l to the maxima could have been chosen, but the maxima them-selves were more convenient.) Thus to record data the tubes were al i g n e d at the radius at which maximum e l e c t r o n density occurred at 6.1 ysec, f o r the value of a x i a l co-o r d i n a t e under study. The values of e l e c t r o n d e n s i t y and temperature (again corrected f o r the delay i n He II upper st a t e response) found at 6.1 ysec were recorded, and are p l o t t e d i n Figure 5-2. The measured temperatures agree q u a l i t a t i v e l y with the model discussed above. From the c e n t r a l region of RADIUS cm 131 CATHODE Figure 5-1 . 10 50 n m ANODE cm Contours of Electron Density during Pinch t = 6.1 ysec. Dots and error bars mark position of electron density peak at 6.1 ysec. N in units of I O 1 6 c u r 3 . e 132 Figure 5-2. L o n g i t u d i n a l S t r u c t u r e on Locus of E l e c t r o n Density Maxima at t = 6.1 ysec (along AA of Figure 5-1 ). 133 the tube toward the anode there i s a continual r i s e of temperature, due to the t h e r m o e l e c t r i c e f f e c t . A l s o , measure-ments i n c l u d i n g regions as near to the electrodes as 4 cm (since the tube spacing was about 5 cm) show no decrease that might be due to thermal d i f f u s i o n to the e l e c t r o d e s . Not p r e v i o u s l y explained i s the high temperature found near the cathode. Such a temperature p r o f i l e i s common i n a r c s . A considerable f r a c t i o n of the voltage drop i s found near the cathode, and the r e s u l t i n g high v e l o c i t y ions produce thermionic emission from the cathode surfa c e . There i s a tendency f o r the current to concentrate at a few small spots on the surfa c e . The c o n s t r i c t e d current d e n s i t y heats both the plasma and the surface f u r t h e r . O c c a s i o n a l l y the cathode surface w i l l be eroded at these "hot spots," i n c r e a s i n g the p r o b a b i l i t y that a hot spot w i l l form at the same l o c a t i o n during the next f i r i n g of the discharge. Then often the same hot spot w i l l occur during several successive shots. This simple model ex p l a i n s the increased temperature, the large standard d e v i a t i o n s (since an a d d i t i o n a l random f a c t o r has been i n t r o d u c e d ) , and the s c a t t e r . Away from the cathode region the e l e c t r o n d e n s i t y i s remarkably constant. Near the anode there i s some e l e c t r o n d e f i c i e n c y . 134 Thus the recorded values of d e n s i t y and temperature in the z-pinch at a given time can be understood on the f o l l o w i n g b a s i s . 1. Thermal d i f f u s i o n is only important w i t h i n a few c e n t i m e t r e s of each e I e c t r o d e . 2. The heat f l u x is governed by e l e c t r o n flow, so the temperature r i s e s toward the anode. 3. Hot spots on the cathod randomly cause c o n s t r i c t i o n s , and thus higher temperatures and increased v a r i a n c e s . 5.3 A x i a l E l e c t r o n Density and Temperature D i s t r i b u t i o n s U s u a l l y , when using the z-pinch as a s p e c t r o s c o p i c source, the o p t i c axis i s a l i g n e d c o i n c i d e n t w i t h , or p a r a l l e l t o , the axis of the discharge tube. (This i s discussed f u r t h e r i n s e c t i o n 6.3.) Thus, i f the e n t i r e tube length i s under o b s e r v a t i o n , the v a r i a t i o n s of d e n s i t y and tempera-ture due to the changes i n pinch timing along the tube, as well as the v a r i a t i o n s discussed i n the previous s e c t i o n , 135 w i l l i n f l u e n c e the r e s u l t s . P r e v i o u s l y i t has only been p o s s i b l e to measure these important v a r i a t i o n s of the spectroscopic source side-on. Unfolding of these r e s u l t s produce i n a c c u r a c i e s i n conclusions regarding the homogeneity of the source. However, with the d i a g n o s t i c s described i n the e a r l i e r chapters of t h i s t h e s i s , p a r t i c u l a r l y the resonator tubes, the l o n g i t u d i n a l v a r i a t i o n s i n e l e c t r o n d e n s i t y and temperature can now be measured with considerable c o n f i dence. The data presented i n t h i s s e c t i o n , then, p e r t a i n s d i r e c t l y to the use of the z-pinch as a s p e c t r o s c o p i c source. E l e c t r o n d e n s i t i e s and temperatures were recorded at points at most 5 cm apart from one end of the tube to the other, at a chosen time and a constant r a d i u s . Results found at r = 4.5 cm and t = 7.0 ysec, at r = 3.0 cm and t = 10.0 ysec, and on axis at t = 13.0 ysec are presented i n Figures 5-3, 5-4 and 5-5 r e s p e c t i v e l y . Note that since these data were obtained at a constant radius (path BB i n Figure 5-1), as in a s p ectroscopic source, there w i l l be a l a r g e r v a r i a t i o n than that found on path AA, the locus of e l e c t r o n d e n s i t y maxima at a p a r t i c u l a r time. Q u a l i t a t i v e l y , these f i g u r e s are most reasonable, i n view of the previous d i s c u s s i o n . From Figure 5-1, the shape of the pinching d e n s i t y peak, and Figure 4-3, the ( r , t ) p l o t i n the centre of the tube (z = 31 cm), i t i s found cm Figure 5-3. L o n g i t u d i n a l S t r u c t u r e at r = 4.5 cm, t = 7.0 usee. IO16 cm-3 T \ I I I I I CATHODE 10 20 30 40 50 ANODE cm Figure 5-4. L o n g i t u d i n a l S t r u c t u r e at r = 3.0 cm, t = 10.0 ysec, 1 38 that at t = 7.0 ysec the density peak i s at r = 4.5 cm i n the centre of the tube, and at smaller r a d i i elsewhere. Thus at t h i s radius and time, near z = 31 cm one f i n d s the density maximum, and at other values of z one f i n d s the outer regions of lower e l e c t r o n d e n s i t y . One would thus expect that along t h i s radius at t h i s time one would f i n d a density d i s t r i b u t i o n sharply peaked at the centre. The d e n s i t i e s measured at t h i s radius and time are presented i n Figure 5-3, and are as expected. The measured temperature i s a f f e c t e d s i m i l a r l y . At r = 3.0 cm and t = 10.0 ysec the d e n s i t y peak i s stopping i t s r a d i a l motion. The v a r i a t i o n i n pinch time along the tube w i l l thus be l e s s important, producing a l a r g e r length of uniform e l e c t r o n d e n s i t y , as i s found (Figure 5-4). For z < 10 cm and z > 50 cm i t appears t h a t the pinch progresses to r a d i i s m aller than 3.0 cm, and so lower e l e c t r o n d e n s i t i e s occur, at r = 3.0 cm, at distances l e s s than 10 cm from the e l e c t r o d e s . These two examples i l l u s t r a t e how both the a x i a l temperature s t r u c t u r e ( s e c t i o n 4.2) and the v a r i a t i o n s i n pinch time a f f e c t the e l e c t r o n temperature and d e n s i t y found at a constant radius at a c e r t a i n time. On axis a rather d i f f e r e n t s i t u a t i o n i s found, s i n c e , i n 4 t o r r helium, the c o l l a p s i n g current density does not reach the a x i s . However a high temperature, high 1 39 140 d e n s i t y plasma i s created on axis as the precursor shock a r r i v e s ( j u s t before the current zero - Figure 4-3). This plasma i s shock heated by the imploding precursor, and then Joule heated as a s i g n i f i c a n t f r a c t i o n of the discharge current s h i f t s to the a x i a l r e g i o n . The r e s u l t i n g plasma expands, r e l a t i v e l y unhindered, a f t e r the current reverses. Figure 5-5 shows the e l e c t r o n d e n s i t y and temperature on axis at t = 13.0 ysec, which i s during the expansion phase. Since the current has been fl o w i n g opposite to i t s o r i g i n a l d i r e c t i o n f o r 1.3 ysec, and i s s t i l l s m a l l , thermo-e l e c t r i c e f f e c t s on the temperature d i s t r i b u t i o n should be s m a l l . E x p e r i m e n t a l l y , only small v a r i a t i o n s i n temperature were found. The decrease i n e l e c t r o n density near the electrodes i s due p r i m a r i l y to plasma l o s s through the 1" diameter holes i n the e l e c t r o d e s , p a r t i c u l a r l y the anode. Behind the anode the i n t e g r a t e d e l e c t r o n d e n s i t y was measured as 1.3 ± 0.3 x 1 0 1 7 cm - 2, behind the cathode i t was 4.5 ± 1 x 1 0 1 6 cm"2. Some of the data presented i n t h i s s e c t i o n augur well f o r the use of the z-pinch as a s p e c t r o s c o p i c source or as a medium f o r s c a t t e r i n g experiments. In the concluding chapter t h i s data w i l l be combined with the data on r a d i a l s t r u c t u r e of t h i s plasma, to produce conclusions on these p o s s i b l e a p p l i c a t i o n s . Chapter 6 CONCLUSIONS 6 .1 C o n t r i b u t i o n s to Plasma Diagnostics The l a s e r e x c i t e d i n t e r f e r o m e t e r b u i l t f o r t h i s experiment i s unexcelled f o r the measurement of e l e c t r o n d e n s i t i e s i n an axi-symmetric plasma. F r a c t i o n a l f r i n g e techniques produced a s e n s i t i v i t y of 5 x 1 0 1 S cm - 2, more than an order of magnitude improvement over that of an un-modulated i n t e r f e r o m e t e r . Temporal r e s o l u t i o n i s b e t t e r than 100 nsec. There are two s i g n i f i c a n t c o n t r i b u t i o n s to the development of the i n t e r f e r o m e t e r described i n t h i s t h e s i s . The f i r s t i s the i n t r o d u c t i o n of the r o t a t i n g r e t r o - r e f l e c t o r f o r c a v i t y modulation. Since the r e s u l t i n g f r i n g e s have an e s s e n t i a l l y constant frequency, the detector c i r c u i t p r e v i o u s l y developed by Funk et al. (1972) can, f o r the f i r s t time, be a c c u r a t e l y used f o r d i r e c t r e c o r d i n g . .The e l e c t r o n density i n the plasma i s simply p r o p o r t i o n a l to the height of an o s c i l l o s c o p e trace above a s t r a i g h t b a s e l i n e . Without t h i s o n - l i n e data reduction i t would have been most i m p r a c t i c a l to obtain a l l the r e s u l t s quoted i n t h i s t h e s i s . 141 1 42 The second c o n t r i b u t i o n i s the c a r e f u l a n a l y s i s of a l l p o s s i b l e sources of e r r o r i n i n t e r f e r o m e t r i c i n v e s t i -gations of t h i s nature. I t was found t h a t , f o r most a p p l i c a -t i o n s , the accuracy of the i n t e r f e r o m e t e r would be about 5%. The use of glass tubes ( f i t t e d with windows) to define the plasma length has proven to be an i n v a l u a b l e a i d to the measuring techniques. As a r e s u l t , v a r i a t i o n s of the plasma along the resonator a x i s can now be studied with an accuracy not p r e v i o u s l y a v a i l a b l e . 6.2 C o n t r i b u t i o n s to Pinch Dynamics Even though z-pinch dynamics have been studied f o r decades, i t was p o s s i b l e to make several c o n t r i b u t i o n s i n t h i s f i e l d . Most important of these was the e l u c i d a t i o n of the precursor shock; the measurements show that the shock i s weak because i t propagates i n t o gas which has been pre-heated by the discharge c u r r e n t . I t was also shown t h a t , a f t e r r e f l e c t i o n , t h i s shock wave could be used to diagnose the plasma. Data of t h i s nature emphasized the e f f e c t of r e l a x a t i o n times on emission from the upper sta t e s of He I I . For some time i t has been r e a l i z e d that the i n i t i a l p o l a r i t y of the discharge electrodes a f f e c t s the breakdown and c h a r a c t e r i s t i c s of the discharge. Measurements described i n t h i s t h e s i s demonstrate how profound these changes are. 143 The l o n g i t d u i n a l s t r u c t u r e of the high pressure z-pinch has been experimentally i n v e s t i g a t e d f o r the f i r s t time. During the pinch phase i t was shown that the l o n g i -t u d i n a l temperature .structure was due to a t h e r m o e l e c t r i c heat f l u x to the anode and 'hot spots' on the cathode. The magnitude of both these e f f e c t s i s reduced by the strong temperature dependence of r a d i a t i o n c o o l i n g . The a x i a l plasma showed l i t t l e l o n g i t u d i n a l temperature v a r i a t i o n . I t s e l e c t r o n d e n s i t y v a r i a t i o n s were governed by l o s s of plasma through holes i n the e l e c t r o d e s . L o n g i t u d i n a l v a r i a t i o n s in the timing of the p i n c h i plasma have been observed. The i m p l i c a t i o n s of these v a r i a -t i o n s on the a p p l i c a t i o n of the z-pinch as a s p e c t r o s c o p i c source are discussed i n the next s e c t i o n . 6.3 The Z-Pinch Plasma as a Spectroscopic Source or a  S c a t t e r i n g Medium Spectroscopic s t u d i e s at temperatures i n excess of 2 or 3 e l e c t r o n v o l t s have been hampered by the s c a r c i t y of adequate sources. Shock tubes, f o r example, have been used (Griem, 1964). The r e s u l t i n g plasma i s s m a l l , h i g h l y t r a n s i e n t , and inhomogeneous, a l l of which are u n d e s i r a b l e . The i n t e r a c t i o n between plasma and intense l a s e r r a d i a t i o n i s a matter of considerable current i n t e r e s t . 144 Work, at present, i s d i r e c t e d toward plasma heating, super-compression, and plasma d i a g n o s t i c s . Since the s c a t t e r i n g c r o s s - s e c t i o n of a plasma i s determined by i t s d i s t r i b u t i o n f u n c t i o n , i t i s d e s i r a b l e to perform basic experiments on a plasma with a well c h a r a c t e r i z e d d i s t r i b u t i o n f u n c t i o n . This u s u a l l y r e quires a s t r i c t l y Maxwellian d i s t r i b u t i o n f u n c t i o n , f o r which the plasma must not be c a r r y i n g any c u r r e n t . Some e f f e c t s of the non-Maxwel1ian d i s t r i b u t i o n f u n c t i o n of a carbon a r c , f o r example, are described by Churchland and Nodwell (1974). One of the major aims of t h i s t h e s i s has been to assess the z-pinch plasma i n both these areas. For s p e c t r o -scopic a p p l i c a t i o n s , one important reason f o r the a t t r a c -t i v e n e s s of the z-pinch i s the p o s s i b i l i t y of end-on obser-v a t i o n , i n which case A b e l - u n f o l d i n g i s not r e q u i r e d . Thus the z-pinch has already received some a t t e n t i o n as a spectro-scopic source. The a n a l y s i s given by Roberts (1972) was hampered by his use of an unmodulated i n t e r f e r o m e t e r without improved l o n g i t u d i n a l r e s o l u t i o n . Jenkins and Burgess (1971), and Burgess and Cairns (1971) also considered t h i s question i n t h e i r s p e c troscopic work on helium. Although they analyzed r a d i a l v a r i a t i o n s c a r e f u l l y they did not consider the l o n g i t u d i n a l s t r u c t u r e . I t was concluded that the i d e a l source should show e s s e n t i a l l y no v a r i a t i o n s i n 145 e l e c t r o n d e n s i t y over the region from which l i g h t was c o l l e c t e d , u s u a l l y a c y l i n d e r about 1 mm i n r a d i u s . This c o n d i t i o n was necessary because of the e f f e c t s of Stark broadening on l i n e p r o f i l e s , p a r t i c u l a r l y near the forbidden components of allowed t r a n s i t i o n s . Temperature v a r i a t i o n s are not as c r i t i c a l , s i n c e they simply weight some regions of the discharge more h e a v i l y than others as c o n t r i b u t o r s to the l i n e p r o f i l e . A n a l y s i s of the data presented i n Chapters 4 and 5 showed th a t there are two regions i n the high pressure z-pinch which are s u i t a b l e as s p e c t r o s c o p i c sources. One of these i s also v a l u a b l e as a medium f o r s c a t t e r i n g e x p e r i -ments. Both of these are discussed below. 1. The axial plasma. In the high pressure z-pinch the pinching plasma does not reach the a x i s . The precursor shock does, however, and a plasma on a x i s i s created. This plasma i s thus f r e e of i n s t a b i l i t i e s . I n i t i a l l y i t i s a c y l i n d e r of such a small radius that e l e c t r o n d e n s i t y gradients would be excessive f o r spectroscopic work. How-ever i t d i f f u s e s outward. Three microseconds a f t e r f o r m a t i o n , f o r example, the r a d i a l e l e c t r o n d e n s i t y d i s t r i b u t i o n i s roughly Gaussian, with a h a l f - w i d t h at h a l f maximum exceed-ing 1 cm and a peak value of 5.7 x 1 0 1 6 cm - 3. I t s l o n g i t u d i n a l s t r u c t u r e (Figure 5-5) i s governed by e l e c t r o n l o s s through the anode. However between 10 and 30 cm from the cathode no 146 v a r i a t i o n was found. Only small temperature v a r i a t i o n s are present, both r a d i a l l y and l o n g i t u d i n a l l y . A v a r i e t y of e l e c t r o n d e n s i t i e s are a v a i l a b l e at d i f f e r e n t times (Figure 2-14). Thus t h i s region i s indeed a s u i t a b l e source, provided that tubes s i m i l a r to those described i n Chapter 2 are used, so that only l i g h t from the s u i t a b l e region near the cathode i s recorded. I t i s p o s s i b l e that the l o n g i t u d i n a l v a r i a t i o n of e l e c t r o n density could be reduced by the use of s o l i d e l ectrodes (with s u i t a b l e windows) to prevent plasma l o s s . These would probably lead to no more contamination of the plasma than the tubes do ( s e c t i o n 2.4.2). S o l i d e l e c t r o d e s were not i n v e s t i g a t e d i n t h i s experiment since these would not permit r a d i a l motion of the i n t e r f e r o m e t e r tubes. Thus the e f f e c t s of l o n g i t u d i n a l v a r i a t i o n s on the spectra obtained by Burgess and Cairns (1971) and Jenkins and Burgess (1971) could not be assessed. In both these papers spectra were measured on the a x i s of a z-pinch. In the former, quartz windows i n the elec t r o d e s were used; the l a t t e r paper does not describe f u l l y the elect r o d e c o n f i g u r a t i o n . F u r t h e r , t h e i r d i s -charges were run at lower pressures, so that the pinched plasma reached the a x i s . Thus, i n s t a b i l i t i e s might have a f f e c t e d the a x i a l plasma. Since they assumed that the mean e l e c t r o n d e n s i t y was present throughout the length of 147 the discharge tube, regions of higher or lower density could have a profound e f f e c t on the i n t e r p r e t a t i o n of t h e i r measured p r o f i l e s , p a r t i c u l a r l y the p r o f i l e s of forbidden components of s p e c t r a l l i n e s . Without d e t a i l e d data of l o n g i -t u d i n a l s t r u c t u r e , f o r t h e i r discharge, i t i s not p o s s i b l e to say whether these forbidden components would be reduced or enhanced. The r e s u l t s obtained by these authors have been v e r i f i e d at a density of 1 x 1 0 1 5 cm"3 (Stevenson, 1973), and agree q u i t e well with recent theory (Barnard and Cooper, to be p u b l i s h e d ) . Thus the e f f e c t s of l o n g i t u d i n a l d e n s i t y gradients cannot be l a r g e . I t has not been proven that they are i n s i g n i f i c a n t . 2. T h e p i n c h i n g p l a s m a a f t e r its m o t i o n h a s  been a r r e s t e d . This region has not p r e v i o u s l y been proposed, or used, as a spectroscopic source, but i t appears to be wel l s u i t e d . Between 10.5 and 11.5 ysec a f t e r discharge i n i t i a t i o n there e x i s t s a s t a t i o n a r y hollow c y l i n d e r of plasma, radius 2.8 cm (at 4 t o r r ) . This i s the pinching e l e c t r o n d e n s i t y s h e l l which i s no longer driven by the decreasing discharge current ( s e c t i o n 4.3.4 and Figure 4-3). I t s r a d i a l e l e c t r o n density d i s t r i b u t i o n i s peaked at 5 x 1 0 1 6 cm"3 at a radius of 2.8 cm, with a h a l f - w i d t h at h a l f maximum exceeding 5 mm (as i n Figure 4-8). L o n g i t u d i n a l l y , one should observe along the locus of d e n s i t y maxima (path AA 1 48 in Figure 5-1), i n which case the e l e c t r o n d e n s i t y would be as shown i n Figure 5-2. I f only the region between 5 and 40 cm from the cathode i s observed (the remainder being excluded by use of tubes) then l o n g i t u d i n a l d e n s i t y v a r i a -t i o n s w i l l be l e s s than 2%. This would be f a r p r e f e r a b l e to observing p a r a l l e l to the a x i s , i n which case a d i s t r i -bution such as Figure 5-4 would c o n t r i b u t e to the p r o f i l e . Radial and l o n g i t u d i n a l temperature v a r i a t i o n s are i n s i g n i f i -cant f o r spectroscopic purposes. To obtain a range of e l e c t r o n d e n s i t i e s i t i s necessary to change the energy supplied to the discharge. One cannot simply wait f o r d e n s i t y decay, since the r e f l e c t e d precursor shock, and other dynamic f e a t u r e s , e v e n t u a l l y d i s r u p t t h i s plasma. Changing the energy suppli e d to the discharge w i l l change the radius at which the pinching density s h e l l stops. Thus the use of t h i s s h e l l may be inconvenient compared to the a x i a l r e g i o n . However the improved l o n g i t u d i n a l density homogeneity may j u s t i f y the a d d i t i o n a l e f f o r t , f o r some a p p l i c a t i o n s . One question which has not been considered so f a r i s the degree of e q u i l i b r i u m present in these two regio n s . For some a p p l i c a t i o n s , such as Stark broadening measurements, t h i s may not be important. Following the d i s c u s s i o n of Chapter 3 we see that PLTE down to the n = 2 l e v e l i s q u i c k l y e s t a b l i s h e d i n both these regions of the plasma. However e q u i l i b r i u m between the upper states of the two s p e c i e s , 149 He I and He I I , r e q u i r e s a r e l a x a t i o n time of .3 ysec i n the a x i a l plasma and 3 ysec i n the c y l i n d r i c a l plasma. There are few a p p l i c a t i o n s f o r which these r e s t r i c t i o n s would be pro hi bi t i ve. This hollow plasma c y l i n d e r could also be most useful as a medium f o r s c a t t e r i n g experiments. For t h i s a p p l i c a t i o n , gradients are not important, since they need only be small over about 100 y, the s i z e of the focussed l a s e r beam. This i s true almost anywhere i n the discharge. The advantage of the a r r e s t e d plasma c y l i n d e r i s t h a t , at the moment of discharge current zero, i t i s f r e e of c u r r e n t , and so i t s d i s t r i b u t i o n f u n c t i o n w i l l be s t r i c t l y Maxwellian. The existence of a s t a t i o n a r y , q u i e s c e n t , and s u f f i c i e n t l y l o n g - l i v e d plasma i s of considerable p o t e n t i a l b e n e f i t f o r experiments i n l a s e r plasma i n t e r a c t i o n s . To summarize, both these regions have adequately small e l e c t r o n d e n s i t y and temperature gradients to be useful as s p e c t r o s c o p i c sources. The a r r e s t e d pinching plasma has two a d d i t i o n a l advantages, high l o n g i t u d i n a l u n i f o r m i t y ( f o r spectroscopy), and the absence of cur r e n t d e n s i t y ( f o r l a s e r s c a t t e r i n g ) . A simple change to the apparatus which would increase the usefulness of the discharge f o r these a p p l i c a -t i o n s i s to s h o r t - c i r c u i t ("crowbar") the c a p a c i t o r bank at about the time of current zero. L o n g i t u d i n a l v a r i a t i o n s 150 caused by the t h e r m o e l e c t r i c e f f e c t would be e l i m i n a t e d , i n c r e a s i n g the s u i t a b i l i t y of the discharge as a spectroscopi source. Further, with no current flow a l l regions of the plasma would have a Maxwellian d i s t r i b u t i o n f u n c t i o n . Thus the a x i a l plasma, i n p a r t i c u l a r , could then be used as a s c a t t e r i n g medium. 6.4 Proposals f o r Future Work As often happens, proposals f o r many f u t u r e e x p e r i -ments come to mind at t h i s p o i n t . A b r i e f d i s c u s s i o n of some of the p o s s i b i l i t i e s opened up by t h i s work i s app r o p r i a t e . There i s s t i l l disagreement between theory and experiment on the question of enhancement of forbidden components of allowed helium t r a n s i t i o n s by Stark f i e l d s i n a plasma (Barnard, p r i v a t e communication). Using the tech-niques discussed here a f u r t h e r experimental determination of these p r o f i l e s could be made, t h i s time with c e r t a i n t y as to the e f f e c t s of l o n g i t u d i n a l v a r i a t i o n s . A s u i t a b l e o p t i c a l system f o r such work might i n v o l v e tubes i n s e r t e d r a d i a l l y i n t o the plasma, with small m i r r o r s at t h e i r ends. Since these tubes would be s u b s t a n t i a l l y s h o r t e r than the resonator tubes used here, a lower f number could be obtained Their length should exceed the discharge tube r a d i u s , so that the p o r t i o n of plasma to be observed i s d r i v e n onto the tubes 1 51 Wave mixing i n a plasma, as f i r s t reported by S t a n s f i e l d et al. (1971), i s p r e s e n t l y a f i e l d of experimental i n t e r e s t . The quiescent plasma during current zero i n a high pressure z-pinch would be, f o r such experiments, a happy medium. A more accurate measurement of the extent of pre-heating of the gas, before i t encounters the pinching plasma, would be d e s i r a b l e . This could be accomplished by monitoring the sound speed i n the gas (Gautum et al., 1965). An audio pulse could be generated by a small spark, and recorded by two pressure probes. The pressure r a t i o across the precursor shock would also be recorded, as a v e r i f i c a t i o n of the r e s u l t s of s e c t i o n 4.3.2. I t i s not c l e a r why the timing of the pinching plasma v a r i e s l o n g i t u d i n a l l y . These shorter pinch times near the electrodes are probably a t t r i b u t a b l e to both higher temperatures, and increased i o n i z a t i o n due to the l a r g e e l e c t r i c f i e l d present there. This matter should be f u r t h e r i n v e s t i g a t e d . Another f i e l d i n which f u r t h e r work would prove f r u i t f u l i s the question of w a l l hang-up times, described i n the appendix. These times are important since they determine, to a large extent, the impurity concentration i n the plasma. REFERENCES Abramowitz, M. and Stegun, I.A. 1965. "Handbook of Mathematical Functions." Dover P u b l i c a t i o n s . Ahlborn, B. andc SaTvat, M. 1967. Z. Naturforschung , 22.,- 260. Ahlborn, B. S. Mikoshiba, P. 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APPENDIX WALL HANG-UP TIME IN THE Z-PINCH A.1 I n t r o d u c t i o n The concentration of i m p u r i t i e s i n a high tempera-ture plasma can be of c r u c i a l importance i n determining the r a d i a t i o n losses from that plasma. The source of the m a j o r i t y of these i m p u r i t i e s i s u s u a l l y the wall of the discharge tube. They are l i b e r a t e d from the wall surface by the plasma i f , as u s u a l l y happens, the plasma i s i n i t i a l l y formed at the w a l l s . Thus i t i s d e s i r a b l e to reduce the wall hang-up time, t , which i s the period between current i n i t i a t i o n and the removal of plasma from the w a l l s . To do so, i t i s c l e a r l y e s s e n t i a l to know the cause of w a l l hang-up time. At present there are four models under con-s i d e r a t i o n , and an experiment was performed to decide which of these was o p e r a t i v e f o r the high pressure z-pinch. The f i r s t of these models i s c a l l e d the neutral d e p l e t i o n theory. I t s t a t e s that i t i s e s s e n t i a l to i o n i z e a l l p a r t i c l e s i n the v i c i n i t y of the w a l l s before the 1 56 1 57 current can leave that region. I f t h i s were not done, the theory s t a t e s , the p a r t i c l e s l e f t behind by the current would be i o n i z e d r a p i d l y by r a d i a t i o n from the plasma, and would thus present a conducting path of lower impedance to the current d e n s i t y . The current s h e l l would then r e t u r n to the w a l l . Thus the w a l l hang-up time i s . the 1 en gin q.f time required to el^iHiiiifnia^e n e u t r a l s from the wall r e g i o n . O'sovets (1959) c a l c u l a t e d t h i s time to be t = 1 6 ir R 2 e ' , W 1 3 Q I where R = tube radius e = mean energy required to produce an i o n i z a t i o n (- 100 ev) M = mass of f i l l i n g gas per u n i t length of the tube I = discharge current at t = t (assumed 1inear with time) Q = c r o s s - s e c t i o n f o r i o n - n e u t r a l exchange (* 3 x 1 0 - 1 5 cm 2) The values l i s t e d above are from Osovets (1959). For helium i n a tube of radius R = 7.5 cm t h i s becomes t,. = 3.3 x 10 6/P I - 3 (A-2) 1 58 f o r t , i n ysec, P i n t o r r , and I i n kA. The second model i s that the current d e n s i t y i s a c c e l e r a t e d as soon as the magnetic pressure exceeds the k i n e t i c pressure of the gas i n t o which i t i s moving. Under a s i m p l i f i e d model of a s k i n current c o i n c i d e n t with the concen t r a t i o n of plasma, t h i s c o n d i t i o n i s ^rw = N' k r (A_3) Here the a d d i t i o n a l symbols are N' and T', the number de n s i t y and the temperature of the gas i n t o which the current sheet i s moving. (The primes are to d i s t i n g u i s h these' q u a n t i t i e s from those of the f i l l i n g gas, N 0 and T 0. However N' = N 0 at these e a r l y times.) A t h i r d model i s based on a ma g n e t i c a l l y reduced e l e c t r i c a l c o n d u c t i v i t y i n the plasma. The magnetic i n d u c t i o n produced by current flow decreases the c o n d u c t i v i t y by a f a c t o r ( M a r s h a l l , 1960). 1 + 0.55 x 1 + 6.73 x + 1.07 xz where x = w /2TTV . u i s the e l e c t r o n c y c l o t r o n frequency a n d v p P the e l e c t r o n - e l e c t r o n c o l l i s i o n frequency. The 1 59 c o n d u c t i v i t y of the plasma w i l l be l a r g e r at smaller r a d i i , where the magnetic f i e l d i s l e s s . I f the change i s s u f f i c i e n t the current w i l l move to smaller r a d i i (Wheeler, 1974a). A f o u r t h model might apply i f s u f f i c i e n t i m p u r i t i e s were released r a p i d l y from the w a l l . These would also decrease the c o n d u c t i v i t y at large r a d i i , and cause the current to move inwards (Wheeler, 1974a). A.2 Measurement of Wall Hang-up Time The d e f i n i t i o n of wall hang-up time adopted f o r measurement purposes was the period between current i n i t i a -t i o n and the moment at which the e l e c t r o n d e n s i t y at the wall decreased to 10% of i t s maximum value. This was measured with the in t e r f e r o m e t e r described e a r l i e r . Data were recorded at two r a d i i , 6.6 and 6.0 cm. The tube radius was 7.5 cm, but the c o n s t r u c t i o n of the electrodes prevented measurement at that r a d i u s . The time at which, at r = 6.6 cm, the e l e c t r o n d e n s i t y decreased to 10% of i t s maximum value, was noted. The v e l o c i t y of the pinching e l e c t r o n s h e l l was found by comparing these r e s u l t s with those obtained at r = 6.0 cm. Since (Figure 4-3) the v e l o c i t y of the pinching plasma i s i n i t i a l l y constant, i t 160 was p o s s i b l e to deduce the time at which the plasma l e f t the w a l l . This i s the wall hang-up time. The measurement was c a r r i e d out f o r 6 pressures i n the range 1 to 10 t o r r , without a l t e r i n g any other c o n d i t i o n s i n the discharge. A . 3 Results and Discussion The measured wall hang-up times, as a f u n c t i o n of pressure, are l i s t e d i n column a of Table A - l . Which of the four models p r e d i c t the data obtained i n t h i s experiment? F i r s t , both of those based on changes i n the plasma c o n d u c t i v i t y can q u i c k l y be discarded. In s e c t i o n 5.2 i t was shown that w /2TTV was, at most, 0.17 ce ee ' i n t h i s discharge. Thus the magnetic f i e l d hardly a f f e c t s the c o n d u c t i v i t y . Wheeler (1974b) has c a l c u l a t e d the temperature r i s e of the inner edge of a pyrex w a l l of a z-pinch. Although his r e s u l t s are not d i r e c t l y a p p l i c a b l e , they i n d i c a t e that i n t h i s case the upper l i m i t i s about 100°C. This temperature r i s e i s i n s u f f i c i e n t f o r r e l e a s e of s u f f i c i e n t i m p u r i t i e s to reduce the plasma c o n d u c t i v i t y appreci ably. The two remaining models are neutral d e p l e t i o n and pressure balance. Both of these w i l l be examined i n more d e t a i 1 . I f the pressure balance c o n d i t i o n , eq. A-3, i s r e w r i t t e n with the s u b s t i t u t i o n I ( t ) = I t , then Table A -l Wall Hang-up Times Pressure Balance* (t wc ^  Neutral Depletion (a) / (b) (c) (d) •(e) ( f ) t we Current Requi red Time at which t h i s Occurred Current D e n s i ty Msmnts Current at t = t w ( e x p t ) 3.3 x 10VP r s 0 t o r r ysec kA ysec ysec kA ysec 1 1.4 ± .3 49 0.6 0.6 ± .1 100 3.3 2 2.6 ± .4 69 1 .0 1 .1 ± .1 150 1 .4 4 3.3 ± .4 99 1 .4 1.4 ± .2 174 1.2 6 3.5 ± .4 1 20 1 .7 179 1 .5 8 4.4 ± .5 140 2.0 185 1 .5 10 5.2 ± .7 155 • 2.4 174 2.0 Assuming T' = 20,000 °K The data i n column d was measured by Pachner (1971). 162 t w = V2 TT .R* No k T 1 / I (A-4) * x i f I and T' are constant, then t,, <* N 0 2. The neutral deple-w t i o n theory p r e d i c t s that i f I ( t ) = I t then t <* N 0 (see eq. A-2). Thus i t should be p o s s i b l e to decide which of these models i s c o r r e c t , f o r t h i s discharge, on the basis of the dependence of the wall hang-up time on the f i l l i n g pressure. The upper l i n e of Figure A-l i s a l o g - l o g p l o t of t against P 0, the f i l l i n g pressure (which i s , of course, p r o p o r t i o n a l to N 0 ) . A computer f i t of i t s slope gave the value 0.504 ± 0.006. This was strong evidence i n support of the pressure balance model. The approach so f a r has been s u p e r f i c i a l i n two i ways. One i s the assumption that I , the rat e of current r i s e , was constant. Examination of the current waveform (Figure 4-3) shows that the current increases l i n e a r l y with • time, to w i t h i n 10%, ( I ( t ) = I t ) only u n t i l t = 2 ysec. A f t e r t h i s time one should not expect t « /F7, since I W i s varying (eq. A-4). Expe r i m e n t a l l y , however, t h i s propor-t i o n a l i t y held u n t i l at l e a s t 5.2 ysec. ^The a s s u m p t i o n t h a t T' i s c o n s t a n t w i l l c o n t i n u e t o be made. I t i s r e a s o n a b l e b e c a u s e r a d i a t i o n l o s s e s f r o m a h i g h t e m p e r a t u r e g a s i n c r e a s e r a p i d l y w i t h t e m p e r a t u r e ( s e c t i o n 5 .2) , s o a g r e a t l y i n c r e a s e d e n e r g y p e r p a r t i c l e i s r e q u i r e d t o r a i s e t h e t e m p e r a t u r e s i g n i f i c a n t l y . e 163 Figure A - l . Wall Hang-up Times as a Function of F i l l i n g Pressure. 1 64 The explanation of t h i s p r o p o r t i o n a l i t y i s found by a c l o s e r examination of the second s i m p l i f i c a t i o n made so f a r , namely neglect of the r a d i a l s t r u c t u r e . In Chapter 3 i t was shown that the leading f e a t u r e of the imploding discharge i s the precursor shock, followed by the current d e n s i t y peak, followed by the e l e c t r o n d e n s i t y peak (shown s c h e m a t i c a l l y i n Figure A-2). The l a s t two are connected by a r a d i a l e l e c t r i c f i e l d (York and Jahn, 1970). A c t u a l l y , at t h i s e a r l y stage, the shock f r o n t and the current d e n s i t y peak w i l l not have separated. Thus we are lead to consider two wall hang-up times, one f o r the current d e n s i t y , t, . WC and one f o r the e l e c t r o n d e n s i t y , t . I t w i l l always be J ' we J true that t,,„ < t , _ since the force which removes the wc we e l e c t r o n d e n s i t y peak from the w a l l i s the r a d i a l e l e c t r i c f i e l d a p p l i e d to i t by the pinching current d e n s i t y peak. C l e a r l y the pressure balance c r i t e r i o n (eq. A-3) a c t u a l l y p r e d i c t s t,,_, since i t deals with the Lorentz WC f o r c e , which acts on the current d e n s i t y . To f i n d the values of t p r e d i c t e d by the pressure balance model the current required f o r pressure balance was c a l c u l a t e d from eq. A-3 f o r each of the pressures s t u d i e d . For t h i s the value of T', the temperature of the gas i n t o which the current sheet i s propagating, was r e q u i r e d . I t was not large enough f o r a s p e c t r o s c o p i c determination. In accordance with the d i s c u s s i o n of s e c t i o n 4.3.2 a temperature 165 of about 20,000 °K, due to the preheating of the gas by a p o r t i o n of the current d e n s i t y , was i n d i c a t e d . The values of current c a l c u l a t e d from eq. A-3, with T' = 20,000 °K, are l i s t e d i n colum b of Table A - l , and the times at which these currents occurred (from the discharge current waveform) are l i s t e d i n column c. Column d of that t a b l e contains the times, e x t r a c t e d from the current density measurements of Pachner (1971), at which the current density peak moved through a radius of 7.3 cm, 3 mm from the discharge tube w a l l . These e x p e r i -mental current density wall hang-up times (also p l o t t e d i n Figure A - l ) are q u i t e c l o s e to those p r e d i c t e d by the pressure balance model. This i s f u r t h e r evidence s t r o n g l y f a v o u r i n g the pressure balance model. Turning now to the w a l l hang-up time f o r the e l e c t r o n d e n s i t y , t , i t i s found that f u r t h e r a n a l y s i s of the current density and e l e c t r o n density s h e l l s i s r e q u i r e d . The dynamic r e l a t i o n between the two s h e l l s i s sketched i n Figure A-2. The Lorentz f o r c e acts everywhere that there i s current d e n s i t y , but p r i n c i p a l l y i n the region of peak current d e n s i t y , J . A r a d i a l e l e c t r i c f i e l d , E, i s e s t a b l i s h e d by charge separation as the current d e n s i t y pinches toward the axis ( i n accord with the model of York and Jahn, 1970). The f o r c e density exerted on e l e c t r o n s near 166 Figure A-2. Dynamics of Current and E l e c t r o n Density S h e l l s at t = t . we 167 the w a l l i s then e 0E 2/2, where e 0 i s the p e r m i t t i v i t y of f r e e space. From the current density measurements of Pachner (1971), there i s l i t t l e current density at the w a l l , so a comparatively small Lorentz f o r c e acts on e l e c t r o n s at the wall at t - t . Thus the c o n d i t i o n f o r e l e c t r o n d e n s i t y to break away from the wal l i s > p (A-5) where P i s the pressure j u s t i n s i d e the e l e c t r o n d e n s i t y s h e l l . I f one assumes that P « P 0, the f i l l i n g pressure, and that the e l e c t r i c f i e l d increases l i n e a r l y i n time, then i t f o l l o w s that t <* /F7» as observed e x p e r i m e n t a l l y . Neither of these assumptions i s unreasonable, but f u r t h e r i n v e s t i g a t i o n i s required f o r a complete understanding. The pressure balance model has c o r r e c t l y pre-d i c t e d the values of the current d e n s i t y wall hang-up times, as a f u n c t i o n of f i l l i n g pressure. An adequate model f o r the e l e c t r o n d e n s i t y wall hang-up times r e q u i r e s more work, but the above d i s c u s s i o n i n d i c a t e s a path to f o l l o w . The neutral d e p l e t i o n model i s completely incon-s i s t e n t with the d e t a i l e d s t r u c t u r e of the pinching plasma. In t h i s plasma i t has been shown that the current density s t a r t s to pinch while there i s s t i l l a h i g h l y conductive, 1 68 lower inductance, l a y e r of plasma behind i t , j u s t i n s i d e the vessel w a l l . Even though agreement was not t h e r e f o r e , expected, the wall hang-up times according to t h i s model were c a l c u l a t e d by eq. A-2. They are l i s t e d i n column f of Table A-l . None of them f a l l s w i t h i n experimental e r r o r of the measured values. A. 4 Conclusions Even though the experiment i s f a i r l y i n d i r e c t , and even though the e f f e c t s of the complex r a d i a l s t r u c t u r e have only been s e m i - q u a n t i t a t i v e l y incorporated i n the d i s -c u s s i o n , the r e s u l t s unambiguously s e l e c t the pressure balance model as that which c o r r e c t l y p r e d i c t s the wal l hang-up time i n the high pressure z-pinch. In a recent paper, Wheeler (1974a) i n v e s t i g a t e d the same problem i n a z-pinch f i l l e d with argon at 0.20 t o r r . He concentrated on the dependence of wall hang-up time, t , on the rate of current i n c r e a s e , I. A considerable u n c e r t a i n t y was introduced i n t o his r e s u l t s by his method of determining the moment of wall departure. This was done by deducing, from the voltage waveform, the commencement of the back emf produced by current sheet motion in a magnetic f i e l d . This method proved to have considerable s c a t t e r . 1 6 9 For a l i n e a r l y i n c r e a s i n g c u r r e n t , the neutral d e p l e t i o n theory p r e d i c t s t « j-o-75 ( f r o m e q . A - l ) , while W the pressure balance model p r e d i c t s t « I " 1 (eq. A-4). He measured an exponent of -0.66 ± 0.1, which favours the former model. In comparing the actual values p r e d i c t e d by these models he was u n w i l l i n g to consider that the gas i n t o which the current s h e l l was propagating might have a temperature g r e a t l y exceeding room temperature, since he thought a high temperature would produce considerable i o n i z a t i o n . In t h i s t h e s i s i t has been shown that the current d e n s i t y does indeed preheat t h i s gas, but i o n i z a t i o n i s i n i t i a l l y small due to the i o n i z a t i o n a l r e l a x a t i o n time. I f a temperature more reasonable than room temperature (- 15,000 °K at his l a r g e s t c u r r e n t s ) had been i n s e r t e d i n t o the pressure balance equation (A-4), then the p r e d i c t e d w a l l hang-up times would have agreed much more c l o s e l y with his experimental values. Furthermore, the discrepancy i n the dependence of t on I could also have been explained by assuming that T', the temperature of the gas i n t o which the current i s propagating, depended weakly on I. (He v a r i e d I over a much l a r g e r range than was done i n t h i s experiment, thus t h i s v a r i a t i o n had no e f f e c t here.) Say T 1 «. I The value of n w i l l be p o s i t i v e but f a i r l y s m a l l . Eq. A-4 170 gives t « T'.^I - . Therefore t,, I T ]~ 1. I t would not be 3 w w s u r p r i s i n g i f n -1 turned out to have a value near -0.66, his experimental r e s u l t . In summary, t h i s present experiment, and Wheeler's paper, i f his r e s u l t s had been c o r r e c t l y i n t e r p r e t e d , unambiguously point to the c r i t e r i o n of pressure balance as c o r r e c t l y p r e d i c t i n g the wal l hang-up time i n the high pressure z-pinch. The present experiment also r a i s e s some i n t e r e s t i n g questions about the formation of the current and e l e c t r o n density sheets. I t would be well worth i n v e s t i -gating f u r t h e r the r a d i a l e l e c t r i c f i e l d , and the pressures and temperatures i n v o l v e d , i n order to obtain a c l e a r e r idea about the formation and i n i t i a l motion of these sheets. Once these questions are answered, and the high pressure case understood, i t w i l l be p o s s i b l e to proceed to s i t u a t i o n s c l o s e r to those of f u s i o n o r i e n t e d d i s c h a r g e s , with lower pressures and much more q u i c k l y r i s i n g c u r r e n t s . 

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