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Theoretical and experimental studies of a high density Z-pinch Houtman, Hubert 1977

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THEORETICAL AND EXPERIMENTAL STUDIES OF A HIGH DENSITY Z-PINCH by Hubert Houtman B.A.Sc., University of British Columbia, 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1977 © Hubert Houtman, 1977 In present ing th is thes is in p a r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f r ee 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 representa t ives . It is understood that copying or pub l i ca t ion of th is thes is for f i n a n c i a l gain sha l l not be allowed without my wr i t ten permiss ion. Department of PHYSICS The Un ivers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date °3 OCTOBER 1977 ABSTRACT A fast Z-Pinch in 1.22 Torr helium has been investigated electrically, photographically, and spectroscopically in order to determine the important parameters of the discharge. The dl/dt oscillogram has been used in conjunction with the circuit equations to find the current shell radius and velocity as functions of time. The dynamics have been modelled numerically using a modified snowplow model. The luminous zone of the plasma has been photographed end-on using a TRW image convertor camera. The radial distribution of the plasma measured on the end-on photographs is found to agree with both the electrical determination of the current shell radius and the radius given by the model. The time-resolved electron temperature and electron density were measured spectroscopically using the He II 4686 A1 emission line. The measurements show that the plasma has parameters of 10 eV and 10 1 8 cm 3 just before pinching and 37 eV and 8xl0 1 8 cm 3 during the 400 ns pinch phase. The results show that the plasma is well suited to the requirements of the future light mixing experiments. i i TABLE OF CONTENTS 0 Page Abstract i i List of Tables v List of Illustrations vi Acknowledgements v i i i Chapter 1 INTRODUCTION 1 2 DETAILS OF THE Z-PINCH AND THE DISCHARGE CIRCUIT . . . . 3 2.1 The Z-Pinch 3 2.2 Impedance of the Discharge Circuit and the Pinch 5 2.3 Electric Circuit and Circuit Equations 9 3 ELECTRICAL MEASUREMENTS AND THE CURRENT SHELL RADIUS 12 3.1 Rogowski Coil i 12 3.2 Passive Integrator Circuit 15 3.3 Measurement of the Capacitor Voltage 19 3.4 The Changing Inductance 20 3.5 Calculation of the Current Shell Radius From the dl/dt Curve 21 3.6 Initial Conditions 24 4 SN0WPL0W MODEL 26 4.1 Snowplow Equation 26 i i i Chapter Page 4.2 K i n e t i c Pressure 27 4.3 Snowplow Model With K i n e t i c Pressure Term . . . . 28 4.4 Taylor Series Expansions 29 4.5 Numerical Integration 32 4.6 Comparison With Experiment 32 4.7 Estimates of Electron Density and Temperature . . 37 5 EXPERIMENTAL ARRANGEMENT USED FOR TAKING END-ON PHOTOGRAPHS 39 5.1 The Experimental Set-up 39 5.2 Sequence of Events 41 5.3 Alignment of the O p t i c a l System 43 6 END-ON PHOTOGRAPHS AND PLASMA RADIUS AS FUNCTION OF TIME 45 6.1 End-on Photographs 45 6.2 Radius vs. Time Graph 53 7 EXPERIMENTAL ARRANGEMENT USED FOR SPECTROSCOPIC MEASUREMENTS 55 7.1 The Experimental Set-up 55 7.2 Sequence of Events 57 8 THE HE II 4686 & PROFILES AND TIME-RESOLVED ELECTRON DENSITY AND TEMPERATURE 59 8.1 Response of the 0. M. A. 59 8.2 C a l i b r a t i o n of the 0. M. A 60 8.3 The He II 4686 £ P r o f i l e s 62 8.4 Electron Density and Electron Temperature . . . . 64 9 DISCUSSION AND CONCLUSIONS . 71 REFERENCES 73 i v LIST OF TABLES Table Page 2-1 Parameters of the Z-Pinch and Discharge Bank 6 2-2 Determination of the Inductance L c of the Discharge C i r c u i t 8 4-1 Constants Used i n the Integration 33 8-1 Determination of Electron Density and Temperature . . . 69 v LIST OF ILLUSTRATIONS Figure Page 2-1 The Z-Pinch 4 2-2 The Discharge C i r c u i t 7 2- 3 Equivalent C i r c u i t A f t e r t = 0 9 3- 1 Rogowski C o i l 12 3-2 Rogowski C o i l Signal 13 3-3 d l / d t , Current I and Capacitor Voltage V c 16 3-4 Passive Integrator C i r c u i t 17 3-5 Discharge Current . 18 3-6 Capacitor Voltage and Discharge Current 19 3-7 I l l u s t r a t i n g the E f f e c t of the Changing Inductance . . . 21 3- 8 T o t a l Inductance, Current S h e l l Radius and V e l o c i t y Calculated From the d l / d t Curve 23 4- 1 Snowplow Model Without K i n e t i c Pressure Term 34 4-2 Snowplow Model With K i n e t i c Pressure Term 35 4-3 Capacitor Voltage and Discharge Current for Three F i l l i n g Pressures 36 4- 4 Snowplow Model f o r Various Pressures 37 5- 1 O p t i c a l and E l e c t r i c a l Arrangement Used for Taking End-on Photographs 40 5-2 Discharge Current With Three Exposure Pulses Added . . . 42 5-3 Image Convertor Camera Exposure Pulses 42 5-4 Alignment of the Pinhole Camera 43 v i Figure page 6-1 Frame Size of the Photographs Taken With the TRW Camera • • 46 6-2 End-on Photographs 47 6-3 End-on Photographs 48 6-4 End-on Photographs 49 6-5 End-on Photographs • • • 50 6-6 End-on Photographs 51 6-7 Radius vs. Time as Measured on the End-on Photographs 52 6- 8 Radius vs. Time Graph Comparing the Photographs With the Current S h e l l Radius and the Snowplow Model . . . . 54 7- 1 650 V Gating Pulse 55 7-2 O p t i c a l and E l e c t r i c a l Arrangement Used for Measuring Time-Resolved Line P r o f i l e s of the He I I 4686 R Line . . 56 7- 3 Discharge Current With Gating Pulse Added . . 57 8- 1 Mask Used to Mask the Entrance S l i t of the Monochromator 60 8-2 Response of the 0. M. A. i n D.C. Mode and Gated Mode . . 61 8-3 0, M. A. Response 63 8-4 Monitor Oscillograms . , 65 8-5 He II 4686 X P r o f i l e s 66 8-6 He II 4686 £" P r o f i l e s 67 8-7 Electron Density and El e c t r o n Temperature 70 v i i ACKNOWLEDGEMENTS I would like to thank Dr. J. Meyer for his excellent supervision throughout a l l stages of this work. I am indebted to many members of the Plasma Physics group, especially G. Albrecht, for valuable discussions and assistance. For help in interpreting the line profiles I would like to thank Dr. E. Kallne. Discussions with Dr. C. Tai, especially those concerning the electrical properties of the pinch are gratefully acknowledged. While a detailed description of the low-inductance discharge bank has been included in this report, its construction was not part of the work described here. It was built and used for earlier experiments with a plasma focus by J. Burnett, under the supervision of Dr. J. Meyer. I would like to express my thanks to glassblowers E. Williams and J. Lees for their excellent work. Financial assistance of the National Research Council is gratefully acknowledged. v i i i Chapter 1 INTRODUCTION In t h i s thesis the theoretical and experimental investigations of a fast Z-pinch are described. The project was carried out i n prepar-ation of various l i g h t mixing experiments which are to be conducted i n the near future. Detailed descriptions of the pinch, the energy storage bank, the c i r c u i t equations and the inductance of the c i r c u i t are given i n Chapter 2. The measurement of dl/dt has provided a wealth of detailed information about the pinch. The various parameters which have been determined as functions of time using the dl/dt curve and the c i r c u i t equations are given i n Chapter 3. The snowplow model, modified to include the effect of k i n e t i c pressure i s discussed i n the following chapter. With three c i r c u i t equations included, the model consists of a closed set of equations which have been solved numerically. The predictions of the model are compared with the experimental data. In Chapter 5 the op t i c a l and e l e c t r i c a l arrangement which was used to photograph the plasma end-on i s described. The photographs are presented i n the following chapter. The plasma radius as function of time measured on the photographs i s compared to the current s h e l l radius calculated from the dl/dt curve and the radius given by the snowplow model. The o p t i c a l and e l e c t r i c a l arrangement for spectroscopic 1 2 measurements i s described i n Chapter 7. The measured l i n e p r o f i l e s of the H e l l 4686 7A l i n e are presented i n Chapter 8. The electron temperature and electron density which were deduced from the measured l i n e p r o f i l e s are compared to predictions of the modified snowplow model. Chapter 2 DETAILS OF THE Z-PINCH AND THE DISCHARGE CIRCUIT 2.1 The Z-plnch A diagram of the Z-pinch used throughout t h i s experiment i s shown i n Figure 2-1. The vessel i s a 45.7 cm pyrex cylinder. The inside and outside diameters are 10.2 cm and 11.4 cm respectively. The electrode separation i s 35.6 cm. The electrodes were made of copper and brass as shown i n Figure 2-1. The constituent parts of the electrodes were sold-ered together using s i l v e r solder. Holes of 4.2 cm diameter were made into the electrodes to allow i n future experiments convergent CO^ laser radiation passage into the pinch. Two holes of 1.9 cm diameter were d r i l l e d into the vessel wall allowing for ruby laser scattering experiments. Neither the CO^ laser nor the ruby laser were used i n the experiments described i n t h i s report. The holes i n the vessel wall were simply sealed off with l u c i t e plates during these experiments. Helium gas was fed through continuously with a very low flow rate while the vessel was pumped on continuously. The pressure was measured at the vessel as indicated i n the diagram, and was adjusted by varying the flow rate, using a needle valve on the gas i n l e t l i n e . 3 4 Vessel © Pyrex Return Conductor (D Brass Mesh <D 1.9cm Dia. Holes Electrodes © Copper (D Brass Cable Header © Brass End Plates (2) Lucite ^ 5 k V Bank Helium _ . _ / 11.5 kV Bank Figure 2-1 The Z-Pinch. 5 In Table 2-1 the parameters of the pinch and the discharge c i r c u i t are l i s t e d . Unless stated otherwise, the f i l l i n g pressure used for the experiments described i n th i s report was 1.22 Torr. The i n i t i a l number density of helium atoms i n the table was calculated using the r e l a t i o n P = n kT . 0 0 0 The discharge c i r c u i t i s shown i n Figure 2-2. Although there are s i x 14 uF capacitors, each with i t s own four electrode spark gap, only one capacitor and one spark gap has been indicated for s i m p l i c i t y . Each spark gap i s connected to the pinch with f i v e 16 Q cables. Again, for s i m p l i c i t y , only one cable has been shown. There are t h i r t y cables leading to the cable header. The spark gaps are triggered by pulses produced by f i r i n g one three electrode spark gap shown i n the diagram. 2.2 Impedance of the Discharge C i r c u i t and the Pinch The discharge c i r c u i t inductance was determined to an accuracy of 12 % using various standard formulas to calculate the inductances of a l l the components and adding them up. The inductances of the 30 cables, the 6 coaxial spark gaps, the cable header, and the anode and cathode of the pinch, a l l of which were calculated using these formulas, are l i s t e d i n Table 2-2. The inductance of the energy storage capacitors i s also included i n the table. The t o t a l inductance, not including the inductance of the pinch i s thus L c = 33 nH ± 4 . 6 Table 2-1 Parameters of the Z-Pinch and Discharge Bank P 0 1.22 Torr Helium f i l l i n g pressure n 0 4.01xl0 1 6 ~3 cm 0 I n i t i a l number density of helium atoms at T = 294 °K 0 p o 2.68xl0"l t kg n f 3 I n i t i a l mass density r 0 5.08 cm Inner radius of vessel R 5.72 cm Outer radius of ve s s e l £ 35.6 cm Electrode separation C 84 yF Bank capacitance Lc 32 nH Bank inductance 11.5 kV Charging voltage - CV 2 2 L V C 5.6 kJ Bank energy Figure 2-2 The Discharge C i r c u i t . 8 Table 2-2 Determination of the Inductance L( of the Discharge C i r c u i t Number Unit Inductance per unit (nH) P a r a l l e l Combination (nH) 30 16 ft Coaxial cable, 3 m long 500 ± 75 16.6 ± 2.5 6 Coaxial spark gap 34 ± 6 5.7 ± 1 6 14 uF Capacitor 15 2.5 1 Cable header 5.4 ± 0.4 5.4 ± 0.4 1 Anode 1.0 ± 0.1 1.0 + 0.1 1 Cathode 1.7 ± 0.1 1.7 ± 0.1 Total: L = 33 ± 4 nH 9 As f a r as i t s e l e c t r i c a l properties are concerned, the pinch may be considered to be a shorted c o a x i a l transmission l i n e of length I, c o n s i s t i n g of two concentric, conducting, hollow c y l i n d e r s . The outer conductor represents the return conductor of radius R, and the inner conductor represents the current flowing at the surface of the plasma, whose radius v a r i e s with time. The inductance of the l i n e may be found using Ampere's law to be (Halliday and Resnick, 1967) ^ . R L p " "aF £ n 7 • 2.3 E l e c t r i c C i r c u i t and C i r c u i t Equations We assume that the resistance of the discharge c i r c u i t and of the pinch are both n e g l i g i b l e . Once the bank has been triggered, the spark gaps become short c i r c u i t s . The c i r c u i t i s then simply a capacitor and a v a r i a b l e inductor i n seri e s as shown i n Figure 2-3. Figure 2-3 Equivalent C i r c u i t A f t e r t=0. 10 In the diagram, C i s the t o t a l bank capacitance of 84 uF and L i s the t o t a l inductance i n the c i r c u i t L " LC + V V R L = L c + "ir *n 7 • <2-x> The capacitor voltage i s related to the current flowing out of the capacitor by d VC I The inductor voltage i s related to the magnetic f l u x <J> by Faraday's law of induction dt L The inductor voltage i s related to the capacitor voltage by L C ' so one may write = v dt C (2-3) The f l u x i s the product of the inductance and current • = L I » (2-4) so equation 2-3 becomes d(LI) _ V (2-5) These c i r c u i t equations w i l l be made use of i n the following two chapters. Chapter 3 ELECTRICAL MEASUREMENTS AND THE CURRENT SHELL RADIUS 3.1 Rogowski Coil The rate of change dl/dt of the discharge current was measured using a small Rogowski coil. This coil consists of 50 turns of enamelled wire wrapped around a 0.5 cm diameter cylinder. The coil was fastened to the end of a 50 Q coaxial cable, with one end of the wire soldered to the center conductor, and the other end to the outer conductor. The coil was placed in a location in the cable header where the magnetic field due to the discharge current was high, with the coil axis parallel to B as shown in Figure 3-1. V R 50U \ 5 Figure 3-1 Rogowski coil 12 13 The voltage V„ in the cable is proportional to the first time R derivative of the magnetic flux which passes through the coil, according to Faraday's law of induction: R d_ dt Coil Since 5 at every point is proportional to the discharge current I according to Ampere's law, the surface integral is also proportional to I. It follows that = AV dt A VR ' (3-1) where A is a constant. Figure 3-2 shows the signal V R measured when the pinch is fired. Figure 3-2 Rogowski Coil Signal; 100 V, 1 ys. The simplest method which may be used to find the constant of proportionality A involves integrating this curve twice. The first integration with respect to time gives the current, s t i l l in terms of A, 14 flowing out of the capacitor: I ( t ) = 1(0) + r t Kt) = KO) + A v R dt . The i n i t i a l current i s zero, so one may write Kt) = A V R d t . (3-2) The second integration gives the capacitor voltage: v c ( t ) - V 0 ) " t I d t ' o v c ( t ) " v c ( 0 ) - t t r t V., dt' dt. K (3-3) Solving for A and setting t * » one obtains C{V_(0) - V_(~)} A = I T V V o R dt* dt (3-4) The charging voltage used throughout these experiments was V (0) = 11.5 kV and the voltage remaining on the capacitors a f t e r the bank has been f i r e d was V (°°) = 0.75 kV. The signal V has n e g l i g i b l e C K value after about t = 25 ys. The signal V was d i g i t i z e d using the oscillogram shown i n Figure 3-2 and others having d i f f e r e n t time scales from t = 0 u n t i l t = 25 ys, and the numbers were typed onto computer 15 cards for analysis on the IBM 370/168 computer. The f i r s t and second time integrals of V R were computed using the trapezoid rul e and stored as functions of time i n computer memory. The f i n a l value of the second integral was found i n t h i s way to be t V,, dt' dt = 6.59 div i s i o n s y s 2 , R o so the constant A according to equation 3-4 i s A = 137 (kA/ys)/division. Each d i v i s i o n on the dl/dt oscillogram thus represents 137 kA/ys. Using t h i s value of A and the stored integrals, the dl/ d t , current, and capacitor voltage curves were found from equations 3-1, 3-2, and 3-3. These are plotted i n Figure 3-3 from t = 0 to t = 5 ys. The high frequency of the dl/dt curve i s attenuated so much by the integration that i t i s barely v i s i b l e on the current curve, and not v i s i b l e at a l l on the voltage curve. The current has i t s f i r s t maximum at t = 1.1 ys. The value of the current at t h i s time i s I = 174 kA. (3-5) max 3.2 Passive Integrator C i r c u i t A good approximation to the current curve can be obtained by using a passive R-C c i r c u i t to "integrate" the Rogowski c o i l s i g n a l . 16 17 The c i r c u i t i s shown i n Figure 3-4. A 1 kil r e s i s t o r and a 0.1 uF capacitor were used, so the time constant RC should be 100 ys. The time constant was measured to be 88 ys. Figure 3-4 Passive Integrator C i r c u i t The impulse response of the c i r c u i t i s a decaying exponential; t RC The c i r c u i t output V_ i s a convolution between f ( t ) and the input V (t) I R t - t ' r t -V I ( t ) = fe RC VR(t«) dt'. If t « RC, then t ' « RC, since t ' < t , and the exponential factor i s approximately equal to unity, so one may write v I ( t ) 1_ RC V_(t) dt, i f t « RC. R 18 If we are concerned with times much smaller than RC, the signal is a good approximation to the integral. The choice of R and C depends on various considerations. The value of R should be much larger than 50 ft in order that the cable sees a 50 ft resistive impedance. The time constant RC should be much larger than the time over which the integral is sought. However, i t should not be too large because the resulting signal decreases with RC. Since the signal has to be amplified by the scope, the danger arises that any R.F. noise present at the scope input will be amplified as well and the signal-to-noise ratio ultimately decreases. The measured signal V = is shown in Figure 3-5. Comparison Figure 3-5 Discharge Current; 1 V, 500 ns. with the computed current curve of Figure 3-3 reveals that a high frequency component has been added to the integrated signal. If one ignores this 10 MHz frequency, the oscillogram agrees very well with the computed integral. On the oscillogram, one division represents 120 kA. 19 3.3 Measurement of the Capacitor Voltage Using a high voltage p o t e n t i a l probe which was constructed with 50 Q impedance c h a r a c t e r i s t i c s , the voltage of the uppermost electrode of one of the capacitors was measured. Neglecting the voltage drop across the spark gap, t h i s voltage i s a good measure of the capacitor voltage. The voltage measured i n t h i s way i s shown i n Figure 3-6a, together with the current s i g n a l . A comparison with the voltage curve computed from the dl/dt curve (see Figure 3-3) shows that various high frequencies have been added i n the measured s i g n a l , notably a 1.5 MHz signa l and a small 10 MHz s i g n a l . I f one again ignores these high frequencies, the oscillogram agrees very w e l l with the computed curve. In p a r t i c u l a r , the two agree as to the time when the voltage crosses zero, about 6 ys a f t e r i n i t i a t i o n of the current. (a) (b) Figure 3-6 Capacitor Voltage and Discharge Current With (a) B = 0, and (b) B = 120 G; Upper Trace: 20 V, Lower Trace: 1 V. Time: 1 us. Using two magnetic f i e l d c o i l s , each c o n s i s t i n g of 1200 turns of wire carrying a current of 4 A, an a x i a l magnetic f i e l d of 120 G was 20 applied. Figure 3-6b shows the voltage and current measured with the magnetic f i e l d turned on. The signals with the magnetic f i e l d turned on show considerably less 10 MHz noise. 3.4 The Changing Inductance In f i r s t approximation, the current i n a Z-Pinch may be considered to flow i n an i n f i n i t e l y t h i n c y l i n d r i c a l s h e l l of radius r (Uman, 1964). At time t = 0, the radius i s r = TQ. I f the current s h e l l were to remain at r = r , the t o t a l inductance L would have the o constant value V a l ' R L = L = L r + -Tr; An -o C 2TT r Q L o = 32 nH + (Air«10-*H/m)(.36 m) £ n ^057 L = 40 nH. (3-6) o This could be realized experimentally by f i l l i n g to a very high pressure. The c i r c u i t would be an ordinary L-C c i r c u i t whose current would be a sinusoid of frequency u>o = 1//L QC and amplitude I q = " C o / ( t o Q L O ) . This frequency corresponds to a quarter period of 2.9 ys, and with V = 11.5 kV, the amplitude i s I = 525 kA. In Figure 3-7 t h i s current i s plotted together with the measured current. The inductance L as function of time varies approximately as I 0 sin w0t 1-22 Torr 4 [ps] Figure 3-7 Illustrating the Effect of the Changing Inductance. the ratio of these two curves: L I sin a) t T •:~ o. o o_ • " I where I i s the measured current. From the diagram, i t i s evident that the measured current shows a strong dependence on the inductance. Consequently the measured current i s a very sensitive measure of the inductance. 3.5 Calculation of the Current Shell Radius from the dl/dt Curve Assuming that the resistance of the cir c u i t i s negligible, at least during the time interval of interest 0 < t < 2.5 us, the 22 inductance may be found using only the dl/dt curve and the c i r c u i t equations. Once the inductance i s known the current s h e l l radius and vel o c i t y may be found. The f l u x may be found by integrating equation 2-3. Since the current starts at zero, the f l u x starts at zero, so J o c dt. (3-7) Using equation 2-4 the inductance i s L = f • (3-8) The inductance depends only on the r a d i a l d i s t r i b u t i o n of the current density, the average radius of which may be found by solving for r i n equation 2-1: r = R e 0 . (3-9) We w i l l refer to t h i s radius as the current s h e l l radius. The v e l o c i t y of the current s h e l l i s defined using the f i n i t e difference formula ft • <3-10> The f l u x , inductance, and current s h e l l radius and v e l o c i t y found using equations 3-7 through 3-10 are plotted i n Figure 3-8. The radius has a minimum value of r . =1.1 cm, 80 ns after the current 23 55 [Lc -L ] dr AT dt "At Figure 3-8 Total Inductance, Current Shell Radius and Velocity Calculated From the dl/dt Curve. 24 minimum at t = 1.8 ys. The maximum speed reached by the current s h e l l i s 3.8 cm/ys, at t = 1.7 ys. 3.6 I n i t i a l Conditions The radius r should start at the inner vessel radius with zero v e l o c i t y : r(0) = r Q , (3-11) r(0) = 0. In order to meet the second of these requirements i t was necessary to adjust the time o r i g i n with respect to the dl/dt curve. The adjustment has the effect of displacing the f l u x curve v e r t i c a l l y , but the current and voltage curves are hardly affected. The time o r i g i n was displaced to the righ t by 80 ns. The fact that the dl/dt oscillogram starts 80 ns early, according to the new time scale, i s attributed to the interference caused by the switching of the high voltage. The dl/dt curve was set to zero before t = 0 as shown i n Figure 3-8. The minimum radius i s very insensitive to the 80 ns time o r i g i n s h i f t . The s h i f t causes a change i n * m^ n of only 10 %. The changes i n the radius and v e l o c i t y curves around t = 0 are much more pronounced. I t may be seen from equation 3-9 that the radius curve depends i n an exponential way on the external inductance L^. Varying L^ , thus has the effect of scaling the entire radius curve. The other i n i t i a l condition, 25 equation 3 - l l a was s a t i s f i e d by varying L^ , u n t i l the i n i t i a l radius coincided with the inner vessel radius r . o The purpose of varying i n th i s way i s two f o l d . F i r s t we obtain the properly scaled radius and v e l o c i t y curves shown i n Figure 3-8 and second we obtain another value for the inductance L^. The value found as a result of t h i s scaling procedure i s L c = 32 nH ± 3, i n good agreement with the value calculated i n the previous chapter using the dimensions of the components (see Table 2-2). The accuracy of the present method i n determining L r i s estimated to be better than 10 %. Chapter 4 SNOWPLOW MODEL 4.1 Snowplow Equation A dynamic model has been developed to describe the r a d i a l evolution of the plasma i n a Z-Pinch discharge (Rosenbluth et. a l . , 1954). Descriptions are also given, f o r example i n Uman (1964) and Jackson (1975). In t h i s model the assumption i s made that the plasma has zero resistance, so i t cannot be penetrated by the magnetic f i e l d due to the current I. The current thus flows on the outside of the plasma, i n the shape of a c y l i n d r i c a l s h e l l of i n f i n i t e s i m a l thickness. The current s h e l l i s driven inward by the jxB force, but i s impeded by c o l l i s i o n s with the molecules i n i t s path. According to the model, the molecules are swept up and become part of the i n f i n i t e s i m a l s h e l l of radius r . The momentum balance equation f o r such a system i s irp £ 4— o dt ( r . - r , £ 1 . - \ l H o dt 4iTr The r i g h t side represents the JVB force and the l e f t side the time rate of change of momentum of the current/mass s h e l l whose mass increases according to m = up SL (r 2 - r 2 ) o o 26 27 4.2 Kinetic Pressure After colliding with the current/mass shell, each particle is given approximately an energy of mp dr dr .12 dt In a time dt the shell sweeps up -2Trr£ ^ N dt particles. At time t the internal energy has increased to u - -ir£ •t [drl 0 dt ^ J dt. Assuming total sweep-up, the volume is approximately v = trr-H, and the corresponding average mass density is r 2 o p = p , O r2 The kinetic pressure is given by P = ^ *K 3v ' so the force F = 2irr£PK is 4irp £r ^ rt , r^A F = _o o_ 3r 1 r dr dt dt. 28 4.3 Snowplow Model with K i n e t i c Pressure Term The snowplow equation i s thus, d i v i d i n g through by d_ dt 4 i r 2 p Q r 4r 2 o 3r •fc 1 dr r J 0 dt dt. (4-1) In a d d i t i o n to t h i s modified snowplow equation we use the c i r c u i t equations introduced i n Chapter 2: L = L c + ^ i n f (4-2) dt I C ' (4-3) m i = v . dt C (4-4) Equations 4-1 through 4-4 form a set of four equations i n four unknowns, so a s o l u t i o n e x i s t s . There are two f i r s t order and one second order equations i n the set, so i n order to solve we need four i n i t i a l conditions. These are r(0) = r Q , r(0) = 0 , (4-5) 1(0) - 0 V 0 ) " VCo 2 9 The numerical s o l u t i o n poses only one obstacle. Equation 4-1 must t e l l us what the f i r s t increment i n radius should be f o r a given increment i n time, but i f we solve f o r dr as a means for obtaining Ar for a given At we get Ar = 4ir2p o > ft T 2 — dt r o 4r 2 o r 1 r 1 fdrl j 0 r J r 0 [dtj dt' dt r 2 - r 2 o 2 2 o At. (4-6) At time zero, r = r Q , so both terms i n the large brackets are of the form zero divided by zero. While equation 4-6 gives a v a l i d increment f o r a l l succeeding steps, we c l e a r l y must use a d i f f e r e n t approach f o r the f i r s t step. I t w i l l be shown l a t e r i n t h i s chapter that the second term has a higher order time dependence around t = 0, so i t i s n e g l i g i b l e at e a r l y times compared to the magnetic pressure term. 4.4 Taylor Series Expansions In order to f i n d the behaviour of I and r around t = 0, we may write a Taylor ser i e s expansion around t = 0: I( t ) = 1(0) + 1(0) t + 1(0) Y  + ••• t 2 r ( t ) = r(0) + r(0) t + r(0) j- + ... Using the f i r s t three of the i n i t i a l conditions 4-5 these become 30 I(t) = 1(0) t + 1(0) (4-7) r(t) = r + r(0) f + When the serie s 4-7 are substituted into the set of equations 4-1 through 4-4, one can solve f o r the constants 1(0) and r ( 0 ) : K 0 ) = r(0) = -(4-8) e n -\ Co 2irL r o o where L Q i s the i n i t i a l value of the inductance L , given by equation 3r6. The Taylor s e r i e s 4-7 become V K t ) = Co t + r ( t ) = r Co [3p J 4TTL r *• oJ o o t z + (4-9) The desired i n i t i a l increment i n radius i s thus Ar = - Co 4ITL r o o (At) : (4-10) The i n i t i a l increment Ar i s quadratic i n the time increment At. This of course explains why equation 4-6, which i s l i n e a r i n At leads to a r e s u l t of zero divided by zero. 31 Using the i n i t i a l conditions 4-5, equations 4-1 through 4-4 may be written i n i n t e g r a l form, so the set becomes rT. r r = r. 4ir2p o ' r 3 1 p i f \ dr r 1 o J r 0 dt dt' dt r 2 - r 2 o dt, V * R L - L c + 2^ r * n 7 ' V = V VC Co 1 c I dt, (4-11) ft 1 * L vc dt. For the f i r s t step we must use 4-10 for Ar, and for a l l succeeding steps 4 - l l a may be used. For early times i t i s best to use I ( t ) and r ( t ) given by equations 4-9 i n the evaluation of the i n t e g r a l s occurring i n equation 4 - l l a as follows: 4ir2p ' I 2 dt o Co 4 u 2 p L 2 r o o o t 2 dt = o Co 4r 2 rt o 4r 3 r t f t 12ir 2 p L 2 r o o o if 3 t3, (4-12) t 3 dt' dt = 3 J - t 5 , The f i r s t of these has a t 3 dependence around t = 0, while the second has a t 5 dependence. The k i n e t i c pressure term i s therefore n e g l i g i b l e compared to the magnetic pressure term at e a r l y times. 32 4.5 Numerical Integration In order to perform the i n t e g r a t i o n a computer program was written. The i n t e g r a l s were computed simply as sums. The i n t e g r a t i o n step si z e used was At = 10 ns. The various constants which were used i n the i n t e g r a t i o n are l i s t e d i n Table 4-1. The r e s u l t s of the i n t e g r a t i o n giving I, V^, L, and r as functions of time are plotted i n Figure 4-1 and Figure 4-2. For comparison, the i n t e g r a t i o n was done both with and without the k i n e t i c •pressure term. The d e r i v a t i v e s I and r have been computed using the f i n i t e d i f f e r e n c e formulas i = At ' r = r At ' By comparing Figure 4-1 and Figure 4-2 i t may be seen that the k i n e t i c pressure term has n e g l i g i b l e e f f e c t u n t i l a f t e r about t = 1.3 ys. 4.6 Comparison With Experiment The shape of the current curve given by the model, Figure 4-2 agrees quite well with the measured curve shown i n Figure 3-3. The maximum current was 167 kA at t = 1.07 ys, i n good agreement with the measured values (see equation 3-5). The radius of the model also agrees c l o s e l y with the current s h e l l radius shown i n Figure 3-8. These w i l l be plotted on the same graph, together with the photographic measurements i n Chapter 6. 33 Table 4-1 Constants Used in the Integration r 0 5.08 cm Initial radius of current/mass shell R 5.72 cm Radius of return conductor L c 32 nH Inductance of external circuit 4irxl0~7 H m"l Permeability of free space a 35.6 cm Electrode separation VCo 11.5 kV Charging yoltage c 84 uF Capacitance of bank p 0 1.22 Torr Helium f i l l i n g pressure T 0 294°K Temperature of helium n 0 4.01xl016 cm 3 Number density from P = n kT 0 0 0 po 2.68xl0_lt kg m 3 Mass density from p = m,T n = 4m n o He o p o Figure 4-1 Snowplow Model Without Kinetic Pressure Term. 35 Figure 4-2 Snowplow Model With Kinetic Pressure Term. Figure 4-3 shows three oscillograms which were obtained by firing the pinch at three different i n i t i a l f i l l i n g pressures. The program was run for these pressures and the results are plotted in Figure 4-4. There is good agreement between model and experiment. (a) (b) (c) Figure 4-3 Capacitor Voltage and Discharge Current for Three Filling Pressures: (a) 0.1 Torr, (b) 1.0 Torr, (c) 1.5 Torr. Upper trace: 20 V, Lower trace: 1 V. Time: 1 ys. 1 .1 Torr 2 1.0 " 3 1.5 " Figure 4-4 Snowplow Model for Various Pressures. 4.7 Estimates of Electron Density and Temperature Assuming total sweep-up, a l l the helium atoms are within a cylinder of radius r. The average density of electrons assuming total ionization is r 2 n = 2n — . (4-13) e o r2 Using this density, the total density of ions and electrons is n = n i + ne = 2 ne ' so we may estimate the temperature using this density and the kinetic 38 pressure given by the model, as follows: P„ = nkT + n.kT. + n kT , K o o 1 i e e P R = nkT, T = K nk (4-14) The electron temperature and density have been computed using the values of r and P given by the program, and will be compared in Chapter 8 with K the spectroscopic measurements, Chapter 5 EXPERIMENTAL ARRANGEMENT USED FOR TAKING END-ON PHOTOGRAPHS 5.1 The Experimental Set-up In order to photograph the plasma a TRW image converter camera with a fast framing plug-in unit was employed. The set-up used to photograph the plasma end-on is shown in Figure 5-1. The object to be photographed is a long luminous cylinder. It was possible to photograph the luminous cylinder only after about t= 1.5 ys since before this time it is hidden behind the electrodes. The radius of the holes in the electrodes through which the photographs were taken is 2.1 cm. The optical set-up is a modified pinhole camera. It accepts only that light which is emitted parallel to the Z-axis, to within a tolerance defined by the diameter of the pinhole. The luminous cylinder projects onto a circle on the photocathode, provided that the pinhole is small. The lens was included only because i t is part of the TRW camera package. Its inclusion has the effect of decreasing the size of the image on the photocathode. The aperture behind lens (not shown) was kept wide open. In any pinhole camera the aperture stop must be the pinhole. Any other apertures can easily ruin the image. 39 40 TRW IMAGE CONVERTER Photo anode Photo -cathode Z-PINCH c o 3 He-Ne Laser Rogowski " Coil DISCHARGE BANK "1 3-•TZZF Thyratron Trigger Unit 10 x Atten 2V TRW Pulse Delay Unit Generator •8V Add Invert 2V Photo-multiplier i 0 R-C Integrator 1V SHIELDED ROOM Figure 5-1 Optical and Electrical arrangement Used for Taking End-on Photographs (Schematic). 41 5.2 Sequence of events In order to take a picture the shutter is activated. It remains open for .01 seconds, It is within this time interval that the events described below occur. The Polaroid photographs containing three images is subsequently removed from the rear of the TRW camera. There exist a pair of electrical contacts in the TRW camera which close at the time the shutter is opened. The trigger unit for the capacitor bank is activated by this switch. When the bank fires, current begins to flow in the vessel. Light produced by this current flow is transported into the shielded room via the light fiber, where i t is converted into a negative electrical signal by means of a photomultiplier. A positive pulse is created at the start of this signal, which is fed into the TRW delay unit. The delay unit waits for the time t^ e ^ which has been dialed in, and then produces a 300 V pulse which is fed into the trigger input of the camera. The camera then opens and closes its gating grid three times and simultaneously deflects the electron beam so that i t is focussed at three positions on the photoanode. By means of a resistor divider network within the TRW camera, three 8 V pulses are produced, corresponding to the three high voltage gating pulses, to serve as a monitor. These pulses are fed back into the shielded room, attenuated using a 10x attenuator, and added to the discharge current signal using the "Add" feature of the oscilloscope. This composite signal constitutes the upper oscilloscope trace. A typical oscillogram is shown in Figure 5-2. The lower trace is the photomultiplier signal, which has been made to appear positive using the "Invert" feature 42 Figure 5-2 Discharge Current With Three Exposure Pulses Added. Upper Trace: 1 V, Lower Trace: 2 V. Time: 500 ns. Figure 5-3 Image Converter Camera Exposure Pulses, With Exposure Times of (a) 5 ns, and (b) 20 ns. IV, 20 ns. 43 5.3 Alignment of the Optical System In order to align the pinhole of the pinhole camera, the TRW camera was put aside, leaving only the pinhole, the lens L^, and the pinch. First the lens was aligned using the Helium-Neon laser beam which had previously been adjusted to go through the center of each electrode. The lens was fixed in the position where i t was found that the laser beam was not deflected by the lens. The laser was then turned off and the pinhole was introduced and aligned in the x,y, and z directions using the method illustrated in Figure 5-4. l i n L-3 Pinhole Electrodes Figure 5-4 Alignment of the Pinhole Camera. The eye is placed very close to the pinhole, and the pinhole is moved in the three directions until the electrode apertures appear superimposed. If the pinhole is at the wrong x or y position, the electrodes appear displaced with respect to eachother; if i t is at the wrong z position the electrodes appear to have different sizes. 44 Using this method the pinhole was aligned to an accuracy of ±0.5 mm in each of the three directions. Once the pinhole was aligned, the TRW camera was put into place as near as possible to the pinhole as shown in Figure 5-1. Chapter 6 END-ON PHOTOGRAPHS AND PLASMA RADIUS AS FUNCTION OF TIME 6.1 End-on Photographs The luminous zone of the plasma was photographed using the arrangement which was described in detail in the previous chapter. The optical set-up is a pinhole camera "focussed at infinity". There are no perspective effects to be corrected for. The photos show intensity as function of radius and angle directly. The z-dependence has been integrated out by the optical set-up. The plasma is optically thin at visible wavelengths, so the intensity on the photographs is, assuming the pinhole is small: X(r,9) oc I I(r,6,z) dz, - i l z where I(r,6,z) is the intensity of light emission in the plasma. The form of this integral shows that the signal-to-noise ratio is very high. It represents an averaging process in the z-direction, averaging over a length I = 36 cm of plasma. The photographs are shown in Figures 6-2 through 6-6. The vessel is shown in each figure together with a scale representing radial 45 46 distance. The pinhole diameter d, the magnification factor M, and the exposure time are stated f or each. For the s e r i e s of photographs shown i n Figure 6-2 the f o c a l length of lens (see Figure 5-1) was 23;cm. For a l l the r e s t , the f o c a l length of lens was 12 cm. The h o r i z o n t a l s t r i a t i o n s which appear on some of the photo-graphs are not r e a l . They are caused by the gating g r i d wires, within the TRW camera. A f i n i t e pinhole s i z e has the e f f e c t of b l u r r i n g the image by an amount approximately equal to the pinhole diameter. The pinhole was accordingly kept very small. Fortunately there was enough plasma l i g h t to make t h i s p o s s i b l e . In Figure 6-1 a t o t a l l y over-exposed photograph i s shown i n order to i l l u s t r a t e the frame s i z e of the photographs shown i n Figures 6-2 through 6-6. Figure 6-1 Frame Size of the Photographs Taken With the TRW Camera. Pl o t s of radius against time are shown i n Figure 6-7. The lengths of the v e r t i c a l l i n e s represent the thickness of the luminous region as measured on the end-on photographs of Figures 6-2 through 6-6. Vessel 0 2 A 6 R ( c m ) No filter B = 0 d =1mm 5ns exp. M = VZ83 L60 165 1-70 175 1-80 1-85 1-90 1.95 Figure 6-2 End-on Photographs, Showing the Luminous Region of the Z-Pinch Plasma. 0 1 2 3 4 5 6 (cm) i i i i i i ,i » Figure 6-3 End-on Photographs, as in Figure 6-2, But With a Higher Magnification, and a Smaller Pinhole. 0 1 2 3 4 5 6 (cm) j i i i i i Figure 6-4 End-on Photographs, as in Figure 6-3, But Using a He II 4686 A Filter, and a Longer Exposure Time. Figure 6-5 End-on Photographs, as in Figure 6-3, But With an Applied Axial Magnetic Field of 120 Gauss. 0 1 2 3 4 5 6 (cm) I I I I I 1 L_#» Figure 6-6 End-on Photographs, as in Figure 6-5, But Using a He II 4686 A Filter, and a Longer Exposure Time. r [cm] 1 (a) (b) (c) (d) (e) 1 h 15 I I 1-5 /A 1 r-0 v/ 1-5 IlIIL 1-5 2-0 2-0 1 1 -2-0 2-0 52 2-0 [ u s] Figure 6-7 Radius vs. Time as Measured on the End-on Photographs: (a) From Fig. 6-2, (b) From Fig. 6-3, (c) From Fig. 6-4, (d) From Fig. 6-5, (e) From Fig. 6-6. 53 The plasma appears very different when photographed using only its emitted 4686 X light than i t does when photographed using a l l wave-lengths. The plasma emits primarily continuum radiation, which is proportional to the square of the electron density. The photographs which were taken without the He II filter thus show the regions of high electron density. The photographs which were taken with the He II filter show preferentially the hot (T > 7 eV) region of the plasma. 6.2 Radius vs. Time Graph In Figure 6-8 the measurements of a l l the end-on photographs which were taken with B = 0 are plotted, as in Figures 6-7a, 6-7b, and 6-7c. For comparison, the current shell radius from Chapter 3 (see Figure 3-8), and the radius given by the snowplow model (see Figures 4-1 and 4-2) are also plotted in Figure 6-8. In order that the diagram remain as uncluttered as possible, the uncertainties of the measurements in the temporal direction have not been shown in Figure 6-8. The uncertainties are indicated in Figures 6-2 through 6-6. Current shell Snowplow with Snowplow with P. = 0 I Fig. 6-7a, B = 0 I Fig.6-7b, B = 0 I Fig. 6-7c, 6 = 0,4686?! filter Figure 6-8 Radius vs. Time Graph Comparing the Photographs With the Current Shell Radius From Chapter 3 and the Snowplow Model From Chapter 4, Both With and Without Kinetic Pressure P . Chapter 7 EXPERIMENTAL ARRANGEMENT USED FOR SPECTROSCOPIC MEASUREMENTS 7.1 The Experimental Set-up In order to obtain time-resolved line profiles of the singly ionized helium line He II 4686 X a commercial optical multichannel analyzer (0. M. A.) was employed. It was used in gated mode with a gating pulse width of 50 ns. An oscillogram of the gating pulse is shown in Figure 7-1. This pulse is produced by a circuit which discharges a length of high voltage cable charged to 1300 V through a 50 n terminating resistor. Figure 7-1 650 V Gating Pulse; 1 V, 20 ns. The arrangement used is shown in Figure 7-2. The pinch discharge is imaged onto the entrance s l i t of a monochromator, and 55 56 Z-PINCH He-Ne Laser Rogowski Coil DISCHARGE BANK 1 Thyratron Trigger Unit P.B. Photo -multiplier M 880x Atten Unit Generator Y JUUL • 50V OMA Sync. X-Y Plotter T V R-C Integrator 1V 100 V SHIELDED ROOM Figure 7-2 Optical and Electrical Arrangement Used for Measuring Time-Resolved Line Profiles of the He II 4686 % Line (Schematic). 57 the 0. M. A. head is mounted in place of the exit s l i t . 7.2 Sequence of Events The activation of the push button alerts the "0. M. A. Sync" unit to output a pulse at the time i t receives the next clock pulse from the 0. M. A. console. The pulse is fed into the trigger unit of the capacitor bank, causing the pinch to fire. Light from the pinch is sent into the shielded room via the light fiber, and converted into a negative electrical signal by the photomultiplier. An inverted pulse generator creates a 2 V pulse at the start of the light signal, and this pulse is fed into the TRW delay unit. After a time t, .. has passed this unit • delay puts out a 25 V pulse which triggers the square pulse generator. The 650 V square pulse is sent to the 0, M. A. head gate input, and by means of a "T" connector is returned into the shielded room. After attenuating the pulse by a factor of 1/880, the 0.7 V signal is added to the current signal using the "Add" feature of the oscilloscope. This constitutes the upper oscilloscope trace. A typical oscillogram is shown in Figure 7-3. Figure 7-3 Discharge Current With Gating Pulse Added; Upper Trace: 1 V, Lower Trace: 2 V. Time: 500 ns. The lower trace is the photomultiplier signal, which has been made to appear positive using the "Invert" feature of the oscilloscope. The light which f e l l onto the 500 channels of the 0. M. A. head during the 50 ns square pulse is displayed on a second oscilloscope and plotted by the console on the X-Y plotter. Chapter 8 THE HE II 4686 A" PROFILES AND TIME-RESOLVED ELECTRON DENSITY AND TEMPERATURE 8.1 Response of the 0. M. A . The line profiles of the He II 4686 R line were measured using the arrangement which was described in detail in the previous chapter. While the profiles were being collected during the course of the experiment it became apparent that the response of the 0. M. A. was not at a l l uniform across the 500 channels, and that there was considerable blurring (cross-talk) occurring between the various channels. The electron optics within the 0. M. A. head was producing a very poor image. The gating pulse height which had been used until that time was 1000 V. In order to improve the situation, the response of the 0. M. A. to an input consisting of a series of "spikes" was monitored while varying, from shot to shot the gating voltage. In this way i t was possible to monitor both the overall shape of the response curve and the cross-talk while varying the gating voltage. To calibrate the 0. M. A. i t would have been undesirable for a number of reasons to dismount the head from the monochromator. It was accomplished without dismounting the head using the following method. 59 60 8.2 Calibration of the 0. M. A. First the monochromator was turned to the A = 0 setting. This setting places the zero-order light which is reflected from the grating onto the 0. M. A. head. Next the entrance s l i t of the monochromator was dismounted and remounted with the s l i t horizontal, 90° relative to its normal orientation. The s l i t was masked along its length using the mask shown in Figure 8-1. The masked s l i t is thus imaged onto the 0. M. A. head via two curved mirrors and the grating acting as a plane mirror, in a way which is not wavelength dependent. The s l i t could s t i l l be adjusted in order to vary the amount of light. •IIIIIIIIIIIIIH Figure 8-1 Mask Used to Mask the Entrance Slit of the Monochromator (True Size). The lens of Figure 7-2 was removed, and the mirror M2 was masked with white paper. The purpose of removing the lens and covering the mirror with white paper was to ensure that whatever fraction of pinch light which entered the monochromator would pass through each opening of the masked entrance s l i t with equal intensity. The size of the aperture A 2 was reduced somewhat in order to ensure that light from any part of the white paper which passed through the masked entrance s l i t is accepted by the first monochromator mirror. If any light is lost at this mirror aperture, the image would again be non-uniform. 61 Intensity (Arb. units)| (a) 0 D.C. Channel number 499 Intensity f (Arb. units) (b) jh^A.ib.v. 200 300 400 500 550 600 650 700 800 900 1000 1100 Q) CT a 4-* "o > CT C *-» a * CD 1200 Channel number 499 Figure 8-2 Response of the 0. M. A. in (a) D.C. Mode and (b) Gated Mode. 62 The resulting 0. M. A. signal using D.C. mode, illuminating the white paper with a 100 Watt lamp, is shown in Figure 8-2a. In Figure 8-2b are shown the resulting signals using gated mode, with various values of the gating voltage. Fortunately, the pinch at t = 1.8 ys gave enough light within 50 ns to make this possible. On the basis of this plot, i t was decided that the He II profiles which had already been collected at a gating voltage of 1000 V should be discarded, and that a new set of data should be taken at a gating voltage of 650 V, While 650 V is clearly the best voltage to use, there s t i l l remains some cross-talk between the channels, primarily near the edges. Also, the response at 650 V is s t i l l not uniform. The overall shape of the response curve is quite similar for voltages of 500 V through 1200 V. A gating voltage of 650 V is a signifigant improvement over 1000 V, but the improvement is mainly a reduction of cross-talk. 8.3 The He II 4686 i Profiles After returning to the configuration shown in Figure 7-2, many shots were made at various monochromator settings, at various times relative to the pinch current initiation, using a gating voltage of 650 V and a pulse duration of 50 ns as shown in Figure 7-1. As stated earlier, the 0. M. A. response was not uniform even at 650 V. The response curve is essentially as shown in Figure 8-3a and 8-3b. Figure 8-3a shows the 0. M. A. signal at three wavelength settings far enough removed from the 4686 &* line to show only continuum. If the 0. M. A. response had been uniform a l l these curves would have been essentially flat. 63 Intensity f (Arb. units) (a) 0 + 1 5000A 2 5100& 3 5200% Channel number 499 134 A Intensity (Arb. units) (b) 04-4 4900 & 5 4900 A Channel number Figure 8-3 0. M. A. Response. These Shots Were Made at t = 1.8 ys at Wavelengths Far Enough Removed From the He II 4686 7A. Line to Show Primarily Continuum. 64 Figure 8-3b shows the 0. M. A. signal at 4900 R with and without a neutral density fi l t e r of .3 density. The curve labelled "4" was used as the response curve and a l l the data were divided by this curve. The line profiles which were found in this way are presented in Figure 8-5. These profiles are the result of piecing together a mosaic of curves, after dividing each one by the curve labelled "4" in Figure 8-3b. The times are in microseconds relative to the initiation of the pinch current. This is the same time scale as was used in previous chapters. Figure 8-4 shows typical oscillograms which were used while taking the data in order to correlate the gating pulse with the pinch discharge. The times correspond to those in Figure 8-5 and 8-6. 8.4 Electron Density and Electron Temperature Figure 8-6 shows an "eyeball f i t " to the data of Figure 8-5. These profiles were used to determine the temperature and density of the electrons. The electron density was found using the line full-widths at half maximum (AX_T_„, = 2 AXTTT7T7Vt) , and the relationship (Griem, 1964): FWHM HWHM N = C(N , T ) e e e AXFWHM 3 The values of C(N , T ) were calculated using tabulated Stark profiles e e (Griem, 1974). The electron temperature was found using the line to continuum ratio. To do this the area of 100 & of continuum and the area under the T(us) 2.5 2.4 2.3 2.2 2.1 2.0 1.9 1.8 1.7 1.6 1.5 Figure 8-4 Monitor Oscillograms, as in Figure 7-3, Showing the 50 ns Gating Pulse Added to the Current Signal, and the Photomultiplier Signal. The Times Are Those Used for the He II Line Profiles of Figures 8-5 and 8-6. Upper Trace: 1 V, Lower Trace: 2 V. Time: 500 ns. 66 J I i I L Figure 8-5 He II 4686 A Profiles. These are the 0. M. A. Results, After Having Been Divided by the Instrument Sensitivity Curve Shown in Figure 8-3b. The Curves Have Been Normalized to Maximum Intensity. 67 Figure 8-6 He II 4686 A* P r o f i l e s : an "Eyeball F i t " to the Data of Figure 8-5. 68 l i n e a f t e r subtracting the continuum were measured from the l i n e p r o f i l e s of Figure 8-6. Using the r a t i o of these areas the temperature may be found (Griem, 1964). The l i n e to continuum r a t i o , the half-widths at ha l f maximum in t e n s i t y AAg^^ and the corresponding electron density and temperature are l i s t e d as function of time i n Table 8-1. The electron density and temperature are p l o t t e d i n Figure 8-7. The error bars represent the scatter of the data i n Figure 8-5 which were used to f i n d the density and temperature. The error bars i n the temporal d i r e c t i o n i n d i c a t e the duration of the gating pulse over which the spectra were integrated by the 0. M. A. The s o l i d curves i n Figure 8-7 are the density and temperature found using equations 4-13 and 4-14, i n Section 4.7. While these estimates agree favorably with the measurements before t = 1.7 ys, which according to the photographic measurements i s about 50 ns a f t e r the plasma has reached the Z-axis, the measurements show that these parameters are 3-4 times higher than the simple analysis would i n d i c a t e during the pinch phase. Table 8-1 Determination of Electron Density and Temperature Time Cys) Line to Continuum Ratio AXHWHM (i) T e (eV) N e (10 1 8 cm"3) 1.5 >9 5 <10 .3 1.6 >9 14 <10 .8 1.7 9.2 24 11 1.7 1.7 1.5 45 37 3.6 1.8 1.6 64 36 5.2 1.9 1.5 73 37 6.7 2.0 1.5 85 37 8.0 2.1 1.4 53 38 4.4 2.2 1.8 22 33 1.5 2.3 >8 16 <13 1.0 2.4 >8 13 <13 .8 2.5 >8 10 <13 .6 40 [eV]30 20 10 £ I I I L J I I I I L y / 1.5 2.0 2.5 [us] Figure 8-7 Electron Density and Electron Temperature. DISCUSSION AND CONCLUSIONS The equivalent circuit (see Figure 2-4) and the circuit equations are based on the assumption that the circuit resistance is negligible. That this is a good assumption at least during the time interval of interest 0 < t < 2.5 ys has been verified indirectly in two ways. First, the current shell radius found from the dl/dt curve agrees with the end-on photographs. Second, the radius and current given by the modified snowplow model in Chapter 4 are in good agreement with the photo-graphs and the measured current respectively. Had the resistance been signifigant, inconsistancies between each of these analyses and the photo-graphs and the measured current would have resulted. The current shell radius found from the dl/dt curve shows clearly the expected piston-like behaviour (see Figure 6-8). The outer radius of the luminous zone stays approximately 0.5 cm ahead of the current shell throughout the pinch. That i t should stay ahead by this distance makes sense since the current shell has a finite thickness. A detailed analysis of the current density distribution which was done using a Z-Pinch of somewhat different parameters (Pachner, 1971) shows that while there is some current flowing at a l l radii at a l l times, most of the current flows within about .5 cm of the current density maximum, The parameters resulting from the numerical integration of the modified snowplow model (see Figure 4-1) are in good agreement with those 71 72 found from the dl/dt oscillogram. In particular, the agreement between the current of the model and the measured current during the first two microseconds shows that the kinetic pressure term gives a reasonable estimate of the pressure. Using the pressure and radius of the model the electron density and temperature have been predicted. The end-on photographs show that the plasma shell first reaches the Z-axis at t = 1,65 ys. Up until this time the temperature and density found from the model agree favourably with the spectroscopic observations (see Figure 8-7). After the shell has reached axis, however the measured parameters are 3-4 times higher than those found from the model. The discrepancy is attributed to shock heating and shock compression which were not included in the model. REFERENCES H. R. Griem, Plasma Spectroscopy, McGraw-Hill Book Co., New York, 1964. H. R. Griem, Spectral Line Broadening by Plasmas, Academic Press, New York, 1974. D. H a l l i d a y and R. Resnick, Physics, John Wiley and Sons, Inc., New York, 1967. J . D. Jackson, C l a s s i c a l Electrodynamics, John Wiley and Sons, Inc., New York, 1975. J . Pachner, Ph.D. Thesis, The Un i v e r s i t y of B r i t i s h Columbia, 1971. M. N. Rosenbluth, R. Garvin, and A. Rosenbluth, Report LA-1850, Los Alamos S c i e n t i f i c Laboratory, New Mexico, 1954. M. A. Uman, Introduction to Plasma Physics, McGraw-Hill Book Co., New York, 1964. t 73 

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