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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 p r e s e n t i n g t h i s  thesis  an advanced degree at the L i b r a r y I further for  freely  of  the  requirements  B r i t i s h Columbia, I agree  available  for  t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f  this  representatives. thesis for  It  financial  The  g a i n s h a l l not  PHYSICS  U n i v e r s i t y of B r i t i s h Columbia  2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5  Date  °3 OCTOBER 1977  that  this  thesis or  i s understood that copying or p u b l i c a t i o n  written permission.  Department of  for  r e f e r e n c e and study.  s c h o l a r l y purposes may be granted by the Head of my Department  by h i s of  fulfilment  the U n i v e r s i t y of  s h a l l make it  agree  in p a r t i a l  be allowed without my  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 A emission line.  The  1  measurements show that the plasma has parameters of 10 eV and 10 just before pinching and 37 eV and 8 x l 0 phase.  18  cm  3  18  cm  3  during the 400 ns pinch  The results show that the plasma is well suited to the requirements  of the future light mixing experiments.  ii  TABLE OF CONTENTS  Page  0  Abstract  i i  List of Tables  v  List of Illustrations  vi  Acknowledgements  viii  Chapter 1  INTRODUCTION  1  2  DETAILS OF THE Z-PINCH AND THE DISCHARGE CIRCUIT . . . .  3  2.1 The Z-Pinch 2.2  3  Impedance of the Discharge Circuit and the Pinch  2.3 Electric Circuit and Circuit Equations 3  9  ELECTRICAL MEASUREMENTS AND THE CURRENT SHELL RADIUS  12 i  3.1 Rogowski Coil  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  3.6 4  5  From the dl/dt Curve  21  I n i t i a l Conditions  24  SN0WPL0W MODEL 4.1  Snowplow Equation  26 iii  26  Chapter  5  Page 4.2  K i n e t i c Pressure  27  4.3  Snowplow Model With K i n e t i c P r e s s u r e Term . . . .  28  4.4  Taylor  29  4.5  Numerical I n t e g r a t i o n  32  4.6  Comparison With Experiment  32  4.7  E s t i m a t e s of E l e c t r o n D e n s i t y and Temperature  S e r i e s Expansions  . .  EXPERIMENTAL ARRANGEMENT USED FOR TAKING END-ON PHOTOGRAPHS  6  39  5.1  The E x p e r i m e n t a l Set-up  39  5.2  Sequence of Events  41  5.3  Alignment o f the O p t i c a l System  43  END-ON PHOTOGRAPHS AND PLASMA  RADIUS  AS FUNCTION OF TIME  7  45  6.1  End-on Photographs  45  6.2  Radius v s . Time Graph  53  EXPERIMENTAL ARRANGEMENT USED FOR SPECTROSCOPIC MEASUREMENTS  8  55  7.1  The E x p e r i m e n t a l Set-up  55  7.2  Sequence o f Events  57  THE HE I I 4686 & PROFILES AND  TIME-RESOLVED  ELECTRON DENSITY AND TEMPERATURE  9  37  59  8.1  Response of the 0. M. A.  59  8.2  C a l i b r a t i o n of t h e 0. M. A  60  8.3  The He I I 4686 £ P r o f i l e s  62  8.4  E l e c t r o n D e n s i t y and E l e c t r o n Temperature  DISCUSSION AND CONCLUSIONS REFERENCES  iv  . . . . .  64 71 73  LIST OF TABLES  Table  Page  2-1  Parameters  o f t h e Z-Pinch and D i s c h a r g e Bank  2-2  D e t e r m i n a t i o n o f the Inductance L  c  6  of t h e  Discharge C i r c u i t  8  4-1  Constants Used i n t h e I n t e g r a t i o n  8-1  D e t e r m i n a t i o n of E l e c t r o n D e n s i t y and Temperature  v  33 . . .  69  LIST OF  ILLUSTRATIONS  Figure  Page  2-1  The Z-Pinch  4  2-2  The D i s c h a r g e C i r c u i t  7  2- 3  Equivalent C i r c u i t After t = 0  9  3- 1  Rogowski C o i l  12  3-2  Rogowski C o i l S i g n a l  13  3-3  d l / d t , C u r r e n t I and C a p a c i t o r V o l t a g e V  3-4  Passive Integrator C i r c u i t  3-5  Discharge Current  3-6  C a p a c i t o r V o l t a g e and D i s c h a r g e C u r r e n t  3-7  Illustrating  3- 8  T o t a l Inductance, C u r r e n t S h e l l Radius C a l c u l a t e d From the d l / d t Curve  16  c  17 .  18 19  the E f f e c t of the Changing Inductance and  . . .  21  Velocity 23  4- 1  Snowplow Model Without K i n e t i c P r e s s u r e Term  34  4-2  Snowplow Model With K i n e t i c P r e s s u r e Term  35  4-3  C a p a c i t o r V o l t a g e and D i s c h a r g e C u r r e n t f o r Three F i l l i n g P r e s s u r e s  36  4- 4  Snowplow Model f o r V a r i o u s P r e s s u r e s  37  5- 1  O p t i c a l and E l e c t r i c a l Arrangement Used f o r Taking End-on Photographs  40  5-2  D i s c h a r g e C u r r e n t With Three Exposure P u l s e s Added  5-3  Image C o n v e r t o r  5-4  Alignment  Camera Exposure P u l s e s  of the P i n h o l e Camera vi  . . .  42 42 43  Figure 6-1  p  age  Frame S i z e 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  6-6  End-on Photographs  6-7  Radius v s . Time as Measured Photographs  6- 8  • • •  50 51  on the End-on 52  Radius v s . Time Graph Comparing the Photographs With the C u r r e n t S h e l l Radius and the Snowplow Model . . .  7- 1  650 V G a t i n g P u l s e  7-2  O p t i c a l and E l e c t r i c a l Arrangement  .  54 55  Used f o r Measuring  Time-Resolved L i n e P r o f i l e s of the He I I 4686 R L i n e  . .  56  7- 3  Discharge  . .  57  8- 1  Mask Used t o Mask the E n t r a n c e Monochromator  C u r r e n t With G a t i n g P u l s e Added Slit  of the 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  8-5  He I I 4686 X P r o f i l e s  66  8-6  He I I 4686 £" P r o f i l e s  67  8-7  E l e c t r o n D e n s i t y and E l e c t r o n Temperature  70  Oscillograms  . ,  65  vii  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, i t s 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 i s gratefully acknowledged.  viii  Chapter 1  INTRODUCTION  In t h i s t h e s i s the t h e o r e t i c a l and experimental i n v e s t i g a t i o n s of a f a s t Z-pinch are described.  The project was c a r r i e d out i n prepar-  a t i o n of various l i g h t mixing experiments which are to be conducted i n the near f u t u r e . D e t a i l e d d e s c r i p t i o n s 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 d l / d t has provided a wealth of d e t a i l e d information about the pinch.  The various parameters which have been determined as  functions of time using the d l / d t curve and the c i r c u i t equations are given i n Chapter 3.  The snowplow model, modified to include the e f f e c t of  k i n e t i c pressure i s discussed i n the f o l l o w i n g chapter.  With three  c i r c u i t equations included, the model c o n s i s t s of a closed set of equations which have been solved numerically. model are compared w i t h the experimental  The p r e d i c t i o n s of the  data.  In Chapter 5 the o p 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. presented i n the f o l l o w i n g chapter.  The photographs are  The plasma radius as f u n c t i o n of time  measured on the photographs i s compared to the current s h e l l radius c a l c u l a t e d from the d l / d t 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 f o r spectroscopic 1  2  measurements  i s d e s c r i b e d i n Chapter 7.  The measured l i n e p r o f i l e s  H e l l 4686 7A l i n e a r e p r e s e n t e d i n Chapter 8.  of the  The e l e c t r o n temperature and  e l e c t r o n d e n s i t y which were deduced from t h e measured l i n e p r o f i l e s a r e compared to p r e d i c t i o n s of t h e m o d i f i e d 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 v e s s e l i s a 45.7 cm pyrex c y l i n d e r . The i n s i d e and outside diameters are 10.2 cm and 11.4 cm r e s p e c t i v e l y . separation i s 35.6 cm.  The electrode  The electrodes were made of copper and brass as  shown i n Figure 2-1. The constituent parts of the electrodes were s o l d ered together using s i l v e r solder.  Holes of 4.2 cm diameter were made  i n t o the electrodes to allow i n future experiments convergent CO^ l a s e r r a d i a t i o n passage i n t o the pinch. Two holes of 1.9 cm diameter were d r i l l e d i n t o the v e s s e l w a l l allowing f o r ruby l a s e r s c a t t e r i n g experiments.  Neither the CO^ l a s e r  nor the ruby l a s e r were used i n the experiments described i n t h i s r e p o r t . The holes i n the v e s s e l w a l l were simply sealed o f f w i t h l u c i t e p l a t e s during these  experiments.  Helium gas was fed through continuously w i t h a very low flow r a t e while the v e s s e l was pumped on continuously.  The pressure was  measured at the v e s s e l as i n d i c a t e d i n the diagram, and was adjusted by varying the flow r a t e , 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 Electrodes © (D Cable Header © End Plates (2) Helium  _  1.9cm Dia. Holes Copper Brass Brass Lucite . _  ^ Bank 5  k  V  /  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 t h i s report was 1.22 Torr.  The i n i t i a l  number density of helium atoms i n the t a b l e was c a l c u l a t e d using the relation 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 c a p a c i t o r s , each w i t h i t s own four electrode spark gap, only one  capacitor and one spark gap has been i n d i c a t e d f o r s i m p l i c i t y .  spark gap i s connected to the pinch with f i v e 16 Q cables. s i m p l i c i t y , only one cable has been shown. to the cable header.  Each  Again, f o r  There are t h i r t y cables leading  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 c a l c u l a t e the inductances of a l l the components and adding them up.  The inductances of the 30 cables,  the 6 c o a x i a l spark gaps, the cable header, and the anode and cathode of the pinch, a l l of which were c a l c u l a t e d using these formulas, are l i s t e d i n Table 2-2.  The inductance of the energy storage capacitors i s a l s o  included i n the t a b l e .  The t o t a l inductance, not i n c l u d i n g the inductance  of the pinch i s thus L  c  =  33 nH ± 4 .  6  Table  2-1  Parameters o f the Z-Pinch and D i s c h a r g e Bank  P  n  1.22  0  Helium f i l l i n g  Torr  4.01xl0  1 6  cm ~3  pressure  I n i t i a l number d e n s i t y o f h e l i u m atoms a t T = 294 °K  0  0  0  p  2.68xl0"  o  lt  kg n f  3  I n i t i a l mass d e n s i t y  5.08 cm  Inner r a d i u s o f v e s s e l  R  5.72 cm  Outer r a d i u s o f v e s s e l  £  35.6 cm  Electrode  C  84 yF  Bank  capacitance  32 nH  Bank  inductance  11.5 kV  Charging  5.6 k J  Bank energy  r  L  0  c  - CV 2  L V  C  2  separation  voltage  F i g u r e 2-2  The D i s c h a r g e  Circuit.  8  Table 2-2 Determination of the Inductance L  (  of the Discharge C i r c u i t  Number  30  Unit  16 ft Coaxial cable, 3 m long  Inductance per u n i t (nH)  Parallel Combination (nH)  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 p r o p e r t i e s a r e concerned, t h e p i n c h may be c o n s i d e r e d  t o be a shorted c o a x i a l t r a n s m i s s i o n l i n e o f l e n g t h I,  c o n s i s t i n g o f two c o n c e n t r i c , c o n d u c t i n g ,  hollow  cylinders.  The o u t e r  conductor r e p r e s e n t s  t h e r e t u r n conductor o f r a d i u s R, and t h e i n n e r  conductor r e p r e s e n t s  t h e c u r r e n t f l o w i n g a t t h e s u r f a c e o f t h e plasma,  whose r a d i u s v a r i e s w i t h time.  The i n d u c t a n c e  u s i n g Ampere's law t o be ( H a l l i d a y and Resnick,  L  2.3  Electric  Circuit  p  ^ .  "  "aF  and C i r c u i t  £ n  o f the l i n e may be found 1967)  R  7 •  Equations  We assume t h a t t h e r e s i s t a n c e o f t h e d i s c h a r g e c i r c u i t and o f the p i n c h a r e b o t h n e g l i g i b l e .  Once t h e bank has been t r i g g e r e d , t h e  spark gaps become s h o r t c i r c u i t s . and  The c i r c u i t i s then simply a c a p a c i t o r  a v a r i a b l e i n d u c t o r i n s e r i e s as shown i n F i g u r e 2-3.  F i g u r e 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  L  "  L  =  L  C  V  +  c  +  R  V  "ir * 7 • n  <-> 2x  The capacitor voltage i s r e l a t e d to the current flowing out of the capacitor by  dV  C  I  The inductor voltage i s r e l a t e d to the magnetic f l u x <J> by Faraday's law  of induction  dt  L  The inductor voltage i s r e l a t e d to the capacitor voltage by  L  C '  so one may w r i t e  =  dt  v  C  (2-3)  The f l u x i s the product of the i n d u c t a n c e and c u r r e n t  •  so e q u a t i o n 2-3  These c i r c u i t chapters.  L I  »  (2-4)  becomes  d(LI)  two  =  _  V  (2-5)  e q u a t i o n s w i l l be made use of i n the f o l l o w i n g  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 c o i l .  This c o i l consists of 50 turns of enamelled  wire wrapped around a 0.5 cm diameter cylinder. The c o i l 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 c o i l  was placed in a location in the cable header where the magnetic field due to the discharge current was high, with the c o i l axis parallel to B as shown in Figure 3-1.  \ V  R  50U  Figure 3-1 Rogowski c o i l 12  5  13  The voltage V„ in the cable is proportional to the f i r s t time R derivative of the magnetic flux which passes through the c o i l , according to Faraday's law of induction: d_ dt  R  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  dt  where A is a constant.  =  AV  (3-1)  AV R '  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 f i r s t  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)  +  Kt)  =  KO)  +  A  rt  v dt. R  The i n i t i a l current i s zero, so one may w r i t e  Kt)  =  (3-2)  V dt.  A  R  The second i n t e g r a t i o n gives the capacitor voltage:  v  - V  c  ( t )  c  ( t )  0 )  " t  I d t  o trt  v  "  v  c  (0)  -  t  '  V., d t ' dt.  (3-3)  K  Solving f o r A and s e t t i n g t * » one obtains C{V_(0) - V_(~)} A  =  I To V  (3-4)  V R dt* dt  The charging voltage used throughout these experiments was V (0) = 11.5 kV and the voltage remaining on the c a p a c i t o r s a f t e r the bank has been f i r e d was V (°°) = 0.75 kV. C  value a f t e r about t = 25 ys.  The s i g n a l V  has n e g l i g i b l e K  The s i g n a l V  was d i g i t i z e d using the  o s c i l l o g r a m 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 f o r a n a l y s i s on the IBM 370/168 computer. The f i r s t and second time i n t e g r a l s of V  R  were computed using  the trapezoid r u l e and stored as functions of time i n computer memory. The f i n a l value of the second i n t e g r a l was found i n t h i s way to be t o  V,, d t ' dt = R  6.59 d i v i s i o n s y s , 2  so the constant A according to equation 3-4 i s  A  =  137 ( k A / y s ) / d i v i s i o n .  Each d i v i s i o n on the d l / d t o s c i l l o g r a m thus represents 137 kA/ys. Using t h i s value of A and the stored i n t e g r a l s , the d l / d t , current, and capacitor voltage curves were found from equations 3-1, 3-2, and 3-3. These are p l o t t e d i n Figure 3-3 from t = 0 to t = 5 ys.  The  high frequency of the d l / d t curve i s attenuated so much by the i n t e g r a t i o n 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 a t a l l on the voltage curve.  The current has i t s f i r s t maximum a t t = 1.1 ys.  The  value of the current at t h i s time i s  I  3.2  max  =  174 kA.  (3-5)  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 " i n t e g r a t e " 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 y s . 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 rt V  If t «  I  (  t  RC, then t ' «  )  =  fe  -  t-t' RC V (t«) d t ' . R  RC, since t ' < t , and the exponential f a c t o r i s  approximately equal to u n i t y , so one may w r i t e  v (t) I  1_ RC  V_(t) d t , i f t « R  RC.  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 i s 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 w i l l be amplified as well and the signalto-noise ratio ultimately decreases. The measured signal V  Figure 3-5  =  is shown in Figure 3-5.  Comparison  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 C a p a c i t o r  Voltage  Using a h i g h v o l t a g e 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 v o l t a g e of the uppermost e l e c t r o d e of one of t h e c a p a c i t o r s was the spark gap,  measured.  N e g l e c t i n g t h e v o l t a g e drop a c r o s s  t h i s v o l t a g e i s a good measure of the c a p a c i t o r v o l t a g e .  The v o l t a g e measured i n t h i s way the c u r r e n t s i g n a l . the d l / d t curve  i s shown i n F i g u r e 3-6a,  signal.  with  A comparison w i t h the v o l t a g e curve computed from  (see F i g u r e 3-3)  shows t h a t v a r i o u s h i g h f r e q u e n c i e s have  been added i n the measured s i g n a l , n o t a b l y a 1.5 10 MHz  together  MHz  s i g n a l and  a small  I f one a g a i n i g n o r e s t h e s e h i g h f r e q u e n c i e s , t h e o s c i l l o g r a m  agrees v e r y w e l l w i t h the computed c u r v e .  I n p a r t i c u l a r , the two  agree as  to the time when the v o l t a g e c r o s s e s z e r o , about 6 ys a f t e r i n i t i a t i o n  of  the c u r r e n t .  (a) F i g u r e 3-6  Using  (b)  C a p a c i t o r V o l t a g e and D i s c h a r g e C u r r e n t With (a) B = 0, and (b) B = 120 G; Upper T r a c e : 20 Lower T r a c e : 1 V. Time: 1 us.  two magnetic f i e l d  coils,  each c o n s i s t i n g of 1200  w i r e c a r r y i n g a c u r r e n t of 4 A, an a x i a l magnetic f i e l d  of 120 G  V,  t u r n s of was  20  applied.  Figure 3-6b shows the voltage and current measured w i t h the  magnetic f i e l d turned on.  The s i g n a l s w i t h the magnetic f i e l d turned  on show considerably l e s s 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 = T .  I f the current  Q  s h e l l were to remain a t r = r , the t o t a l inductance L would have the o  constant value  V  L  L  o  =  = L  =  o  32 nH  +  a  l  '  R  L + -Tr; An C 2TT r r  (Air«10-*H/m)(.36 m)  L  =  Q  £  ^057  n  40 nH.  (3-6)  o  This could be r e a l i z e d 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 C Q  and amplitude  I  q  =  "Co/(to L ) Q  O  frequency corresponds t o a quarter period of 2.9 ys, and w i t h V  . This = 11.5 kV,  the amplitude i s I = 525 kA. In Figure 3-7 t h i s current i s p l o t t e d together w i t h the measured current. The inductance L as f u n c t i o n of time v a r i e s approximately as  I sin w t 0  0  1-22 Torr  4 [ps] Figure 3-7  I l l u s t r a t i n g the Effect of the Changing Inductance.  the r a t i o of these two curves: L I T  •  •:~  "  where I i s the measured current.  s i n a) t  o. o  o_  I 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 c i r c u i t i s n e g l i g i b l e ,  at least during the time i n t e r v a l of interest 0 < t < 2.5 us, the  22  inductance may be found using only the d l / d t 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  v e l o c i t y may be found. The f l u x may be found by i n t e g r a t i n g  equation 2-3.  Since the  current s t a r t s a t zero, the f l u x s t a r t s at zero, so  c J  o  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 f o r r i n equation 2-1:  r  =  R e  .  0  (3-9)  We w i l l r e f e r to t h i s radius as the current s h e l l r a d i u s .  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 d i f f e r e n c e formula  ft •  <-> 3 0 1  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 a f t e r the current  23  55[Lc-L]  dr AT dt " A t  Figure 3-8  Total Inductance, Current Shell Radius and Velocity Calculated From the dl/dt Curve.  24  minimum a t t = 1.8 y s . The maximum speed reached by the current s h e l l i s 3.8 cm/ys, at t = 1.7 y s .  3.6  I n i t i a l Conditions The radius r should s t a r t a t the inner v e s s e l radius w i t h zero  velocity:  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 w i t h respect to the d l / d t curve.  The adjustment  has the e f f e c t of d i s p l a c i n g the f l u x curve v e r t i c a l l y , but the current and voltage curves are hardly a f f e c t e d .  The time o r i g i n was displaced  to the r i g h t by 80 ns. The f a c t that the d l / d t o s c i l l o g r a m s t a r t s 80 ns e a r l y , according to the new time s c a l e , i s a t t r i b u t e d to the i n t e r f e r e n c e caused by the switching of the high voltage.  The d l / d t curve was set to  zero before t = 0 as shown i n Figure 3-8. The minimum radius i s very i n s e n s i t i v e to the 80 ns time o r i g i n shift.  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 e f f e c t of s c a l i n g the e n t i r e radius curve.  The other i n i t i a l c o n d i t i o n ,  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 v e s s e l radius r . o The purpose of varying  i n t h 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 f o r the inductance L^.  The value  found as a r e s u l t of t h i s s c a l i n g procedure i s L  c  =  32 nH ± 3,  i n good agreement with the value c a l c u l a t e d 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 b e t t e r than 10 %.  Chapter 4  SNOWPLOW MODEL  4.1  Snowplow E q u a t i o n A dynamic model has been developed t o d e s c r i b e t h e r a d i a l  e v o l u t i o n o f t h e plasma i n a Z-Pinch d i s c h a r g e (Rosenbluth e t . a l . , 1954). D e s c r i p t i o n s a r e a l s o g i v e n , f o r example i n Uman (1964) and Jackson In t h i s model t h e assumption  i s made t h a t t h e plasma has z e r o r e s i s t a n c e ,  so i t cannot be p e n e t r a t e d by t h e magnetic  field  due t o the c u r r e n t I . The  c u r r e n t thus flows on t h e o u t s i d e o f t h e plasma, cylindrical  (1975).  s h e l l of i n f i n i t e s i m a l  i n t h e shape o f a  thickness.  The c u r r e n t s h e l l i s d r i v e n inward by t h e j x B f o r c e , but i s impeded by c o l l i s i o n s w i t h t h e m o l e c u l e s i n i t s p a t h .  According to the  model, t h e m o l e c u l e s a r e swept up and become p a r t o f t h e i n f i n i t e s i m a l s h e l l of radius r .  The momentum b a l a n c e e q u a t i o n f o r such a system i s  irp £ o  The r i g h t  4—  dt  . -  ( r  r  , £1  o  dt  .  - \  l  H  4iTr  s i d e r e p r e s e n t s t h e JVB f o r c e and t h e l e f t  s i d e t h e time  rate  of change o f momentum o f t h e current/mass s h e l l whose mass i n c r e a s e s according to m  =  up SL ( r o o  2  26  -r ) 2  27  4.2  Kinetic Pressure After colliding with the current/mass shell, each particle i s  given approximately an energy of sweeps up  -2Trr£  p  ^dr N dt particles.  increased to u  -  -ir£  dr .12 dt  m  In a time dt the shell  At time t the internal energy has •t  [drl dt  0  dt.  ^ J  Assuming total sweep-up, the volume i s approximately  v  =  trr-H,  and the corresponding average mass density i s r p  =  p O  r  2  o  ,  2  The kinetic pressure i s given by  P *K  so the force  =  ^ 3v '  F = 2irr£P i s K  F  =  4irp_o £r o_^ 3r  rt , 1 r^A dr r dt  dt.  28  4.3  Snowplow Model w i t h K i n e t i c P r e s s u r e Term The  snowplow e q u a t i o n i s thus, d i v i d i n g through by  4r  d_ dt  •  fc  o  3r  4ir p r 2  Q  In  2  J  0  1 r  dr dt  dt.  (4-1)  a d d i t i o n to t h i s m o d i f i e d snowplow e q u a t i o n we use the c i r c u i t  e q u a t i o n s i n t r o d u c e d i n Chapter  L  2:  L  =  c  +  ^  i  n  (4-2)  f  I  dt  (4-3)  C '  m i dt  =  v  C  .  (4-4)  E q u a t i o n s 4-1 through 4-4 form a s e t of f o u r e q u a t i o n s i n f o u r unknowns, so a s o l u t i o n e x i s t s . one  second  There a r e two f i r s t  order and  o r d e r e q u a t i o n s i n the s e t , so i n o r d e r to s o l v e we need  four i n i t i a l  conditions.  These a r e  r(0)  =  r  r(0)  =  0 ,  Q  ,  (4-5) 1(0)  V  0 )  -  0  "  V  Co  29  The n u m e r i c a l s o l u t i o n poses o n l y one o b s t a c l e . must t e l l us what the f i r s t increment for  increment  Equation  4-1  i n r a d i u s s h o u l d be f o r a g i v e n  i n time, but i f we s o l v e f o r dr as a means f o r o b t a i n i n g Ar  a g i v e n At we get  ft 4ir p 2  Ar  o>o  = r  T  — r  2  2  o  dt  r r j J 1  r  0  0  - r  2  4r  1  fdrl r [dtj  2  2  dt At.  2  o  o  dt'  (4-6) At time z e r o , r = r , so both terms i n the l a r g e b r a c k e t s a r e o f the Q  form zero d i v i d e d by z e r o . for  While e q u a t i o n 4-6 g i v e s a v a l i d  increment  a l l s u c c e e d i n g s t e p s , 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 c h a p t e r t h a t the second  term has a h i g h e r o r d e r time dependence around t = 0, so i t i s n e g l i g i b l e at  4.4  e a r l y times compared t o the magnetic p r e s s u r e term.  Taylor Series In  Expansions  o r d e r to f i n d the b e h a v i o u r o f I and r around t = 0, we  w r i t e a T a y l o r s e r i e s expansion  around t = 0:  Y  I(t)  =  1(0)  +  1(0) t  +  1(0)  r(t)  =  r(0) +  r(0) t  +  r ( 0 ) j-  t  Using the f i r s t  2  +  •••  + ...  t h r e e o f the i n i t i a l c o n d i t i o n s 4-5 these become  may  30  = 1(0) t  I(t)  +  1(0) (4-7)  r(t)  = r  +  r(0)  f  +  When the s e r i e s 4-7 a r e s u b s t i t u t e d i n t o the s e t of equations through  4-4, one can s o l v e f o r the c o n s t a n t s 1(0)  K0)  4-1  and r ( 0 ) :  = (4-8) e  r(0)  where L 3r6.  Q  =  n -\  -  Co 2irL r o o  i s the i n i t i a l v a l u e of the i n d u c t a n c e L , g i v e n by e q u a t i o n  The T a y l o r s e r i e s 4-7 become  Kt)  =  V  Co  t  + (4-9)  r(t)  The d e s i r e d i n i t i a l  =  increment  Ar  Co  r  =  [3p  JJ  *• o  4TTL r  t  +  z  oo  i n r a d i u s i s thus  -  Co 4ITL r  (At)  (4-10)  :  o o  The i n i t i a l of  increment  Ar i s q u a d r a t i c i n the time increment  At.  This  c o u r s e e x p l a i n s why e q u a t i o n 4-6, which i s l i n e a r i n At l e a d s t o a  r e s u l t of z e r o d i v i d e d by z e r o .  31  U s i n g the i n i t i a l c o n d i t i o n s 4-5, may  4-1  through  4-4  be w r i t t e n i n i n t e g r a l form, so the s e t becomes  rT. r  equations  =  r  4ir2p  r.  3  r  o '  r  1  - r  2  o  o  -  L c  dt'  dt dt,  0  R  2^r*  +  r J  f \ dr dt  2  V * L  p i  1 r  n  7' (4-11)  V C  =  V  V  *  1  For the f i r s t succeeding I(t)  1  I dt,  c  Co  ft  L  vc dt.  step we must use 4-10  s t e p s 4 - l l a may  be used.  f o r Ar, and  for a l l  F o r e a r l y times i t i s b e s t to use  and r ( t ) g i v e n by e q u a t i o n s 4-9  i n the e v a l u a t i o n of the  integrals  o c c u r r i n g i n e q u a t i o n 4 - l l a as f o l l o w s :  '  I  o Co  2  dt  4ir2p  4u p 2  L  oo  t 2  r  2  dt  o  =  Co  12ir p L 2  oo  o  2  r  o  t3, (4-12)  4r  2  4r  rt  3  if  rtft t  o  The  first  the second has a t  5  of these has a t dependence.  3  3  d t ' dt  =  3  3J-  dependence around t = 0,  t , 5  while  The k i n e t i c p r e s s u r e term i s t h e r e f o r e  n e g l i g i b l e compared to the magnetic p r e s s u r e term a t e a r l y  times.  32  4.5  Numerical I n t e g r a t i o n In order  written.  The  to perform the i n t e g r a t i o n a computer program  i n t e g r a l s were computed simply  step s i z e used was  At = 10 ns.  i n the i n t e g r a t i o n a r e l i s t e d  as sums.  The v a r i o u s c o n s t a n t s i n Table  f u n c t i o n s of time a r e p l o t t e d i n F i g u r e 4-1  •pressure  term.  The  The i n t e g r a t i o n which were used  4-1.  The r e s u l t s of the i n t e g r a t i o n g i v i n g I , V^,  comparison, the i n t e g r a t i o n was  L, and  and F i g u r e 4-2.  done both w i t h and without  d e r i v a t i v e s I and  was  r as For  the  kinetic  r have been computed u s i n g  the  f i n i t e d i f f e r e n c e formulas  i  = At '  r  =  r  By comparing F i g u r e 4-1 pressure  4.6  term has  and F i g u r e 4-2  At '  i t may  be seen t h a t the  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  kinetic ys.  Comparison With Experiment The  shape of the c u r r e n t curve g i v e n by  agrees q u i t e w e l l w i t h maximum c u r r e n t was measured v a l u e s The  167  the measured curve kA a t t = 1.07  (see e q u a t i o n  the model, F i g u r e  shown i n F i g u r e 3-3.  4-2  The  y s , i n good agreement w i t h  the  3-5).  r a d i u s of the model a l s o agrees c l o s e l y w i t h the  s h e l l r a d i u s shown i n F i g u r e 3-8.  current  These w i l l be p l o t t e d on the same  graph, t o g e t h e r w i t h the photographic  measurements i n Chapter  6.  33  Table 4-1 Constants Used in the Integration  r  0  R  L  c  5.08 cm  I n i t i a l radius of current/mass shell  5.72 cm  Radius of return conductor  32 nH  Inductance of external circuit  4irxl0~ H m"l  Permeability of free space  35.6 cm  Electrode separation  11.5 kV  Charging yoltage  c  84 uF  Capacitance of bank  p  1.22 Torr  Helium f i l l i n g pressure  T  294°K  Temperature of helium  n  4.01xl0  7  a  V  Co  0  0  p  0  o  16  cm  3  Number density from P  0  2.68xl0  _lt  kg m  3  = n kT 0  0  Mass density from p = m, n = 4m n o He o po T  Figure 4-1  Snowplow Model Without K i n e t i c Pressure Term.  35  Figure 4-2  Snowplow Model With K i n e t i c 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 i s good agreement between model and experiment.  (a)  (b)  (c)  Figure 4-3  Capacitor Voltage and Discharge Current for Three F i l l i n g 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  Figure 4-4  4.7  "  1.5 "  Snowplow Model for Various Pressures.  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 i s n  r 2n —  =  e  o  2  r  2  .  (4-13)  Using this density, the total density of ions and electrons is  n  =  n  i  +  n  e  =  2e ' n  so we may estimate the temperature using this density and the kinetic  38  pressure given by the model, as follows:  P„ K  =  nkT  +  o o  P  R  T =  n.kT. + 1 i  n kT , e e  = nkT,  K nk  (4-14)  The electron temperature and density have been computed using the values of r and P  K  given by the program, and w i l l be compared in Chapter 8 with  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 i t 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 camera package.  was included only because i t is part of the TRW  Its inclusion has the effect of decreasing the size  of the image on the photocathode. shown) was kept wide open. must be the pinhole.  The aperture behind lens  (not  In any pinhole camera the aperture stop  Any other apertures can easily ruin the image. 39  40  TRW IMAGE CONVERTER Photo anode  Photo cathode  Z-PINCH  3  He-Ne Laser  Rogowski " Coil DISCHARGE BANK  c  o  "1  3-  •TZZF  Thyratron Trigger Unit  2V TRW Delay Unit  Photomultiplier  Pulse Generator  i  0  10 x Atten  •8V  Add  Invert  2V  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.  open for .01 seconds, described below occur.  It remains  It is within this time interval that the events 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 i t s 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 gating pulses, to serve as a monitor.  to the three high voltage  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  Figure 5-3  Discharge Current With Three Exposure Pulses Added. Upper Trace: 1 V, Lower Trace: 2 V. Time: 500 ns.  Image Converter Camera Exposure Pulses, With Exposure Times of (a) 5 ns, and (b) 20 ns. I V , 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 Figure 5-4  Electrodes 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;  i f 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 i n 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 i s a pinhole camera "focussed at infinity". no perspective effects to be corrected for. as function of radius and angle directly.  There are  The photos show intensity  The z-dependence has been  integrated out by the optical set-up. The plasma i s optically thin at visible wavelengths, so the intensity on the photographs i s , assuming the pinhole i s small:  X(r,9)  oc  I(r,6,z) dz,  I -il  z  where I(r,6,z) i s the intensity of light emission in the plasma. The form of this integral shows that the signal-to-noise ratio i s 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 i s shown in each figure together with a scale representing radial 45  46  distance.  The p i n h o l e diameter d, the m a g n i f i c a t i o n f a c t o r M, and the  exposure time a r e s t a t e d f o r each. i n F i g u r e 6-2  F o r the s e r i e s of photographs shown  the f o c a l l e n g t h of l e n s  (see F i g u r e 5-1) was  F o r a l l the r e s t , the f o c a l l e n g t h of l e n s  was  12  23;cm.  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 photographs a r e not r e a l . the TRW  They a r e caused by the g a t i n g g r i d w i r e s ,  within  camera. A f i n i t e p i n h o l e s i z e has the e f f e c t o f b l u r r i n g t h e image by  an amount a p p r o x i m a t e l y equal  to the p i n h o l e d i a m e t e r .  a c c o r d i n g l y kept v e r y  F o r t u n a t e l y t h e r e was  small.  The p i n h o l e  enough plasma  was  light  to make t h i s p o s s i b l e . In F i g u r e 6-1 a t o t a l l y over-exposed photograph i s shown i n o r d e r to i l l u s t r a t e the frame s i z e of the photographs shown i n F i g u r e s 6-2  through  6-6.  F i g u r e 6-1  Frame S i z e of the Photographs Taken With the TRW Camera.  P l o t s of r a d i u s a g a i n s t time a r e shown i n F i g u r e 6-7. l e n g t h s of the v e r t i c a l l i n e s r e p r e s e n t  The  the t h i c k n e s s of t h e luminous  r e g i o n as measured on the end-on photographs of F i g u r e s 6-2  through  6-6.  0 2 A 6  R(cm)  Vessel  L60 165 1-70 175 No filter  1-80  B=0 d =1mm 5ns exp.  M = VZ83  Figure 6-2  1-85 1-90 1.95  End-on Photographs, Showing the Luminous Region of the Z-Pinch Plasma.  0 12 i  Figure 6-3  i  i  3 4 5 6 (cm) i  i  i ,i  »  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  Figure 6-4  i  i  i  i  i  End-on Photographs, as in Figure 6-3, But Using a He II 4686 A F i l t e r , 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  Figure 6-6  I  I  I  I  1  L_#»  End-on Photographs, as in Figure 6-5, But Using a He II 4686 A F i l t e r , and a Longer Exposure Time.  52  r [cm]  1  (a)  1  (b)  [ ]  2-0  15  us  h II 2-0  1-5  (c) 2-0  (d)  /A  (e)  IlIIL  1  1  1-5  2-0  1-5  2-0  -  1 r-  0 Figure 6-7  v/ Radius vs. Time as Photographs: (a) (b) (c) (d) (e)  Measured on the End-on From Fig. 6-2, From Fig. 6-3, From Fig. 6-4, From Fig. 6-5, 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 wavelengths.  The plasma emits primarily continuum radiation, which i s  proportional to the square of the electron density.  The photographs  which were taken without the He II f i l t e r thus show the regions of high electron density.  The photographs which were taken with the He II  f i l t e r show preferentially the hot (T  6.2  > 7 eV) region of the plasma.  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. uncertainties are indicated in Figures 6-2 through 6-6.  The  I  Current shell Snowplow with Snowplow with P. = 0 Fig. 6-7a, B = 0  I  Fig.6-7b,  B= 0  I  Fig. 6-7c,  6 = 0,4686?! filter  Figure 6-8 Radius v s . 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. gating pulse width of 50 ns.  It was used i n gated mode with a  An oscillogram of the gating pulse i s  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 i s shown in Figure 7-2. The pinch discharge i s imaged onto the entrance s l i t of a monochromator, and 55  56 Z-PINCH  He-Ne Laser  Rogowski Coil  1  DISCHARGE BANK  Thyratron Trigger Unit  P.B. Photo multiplier  50V Unit  Generator  JUUL •  OMA Sync. M 880x Atten  X-Y Plotter  T V Y  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 f i r e .  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 i s 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 i s 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 i s 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 i t 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 (crosstalk) 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 i s 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 i t s normal orientation.  The s l i t was masked along i t s length using the mask  shown in Figure 8-1.  The masked s l i t i s 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  The lens  Mask Used to Mask the Entrance S l i t of the Monochromator (True Size).  of Figure 7-2 was removed, and the mirror M  masked with white paper.  2  was  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 aperture A  2  The size of  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 f i r s t monochromator mirror.  If any light i s lost at  this mirror aperture, the image would again be non-uniform.  61  Intensity (Arb. units)|  D.C.  (a) 0  Channel number  499  200 300 400  Intensity f (Arb. units)  500 550 600 650 700  (b)  Q)  CT  a  4-*  800  "o  900  CT C *-»  1000  jh^A.ib.v.  1100 1200  Channel number Figure 8-2  499  Response of the 0. M. A. in (a) D.C. Mode and (b) Gated Mode.  > a * CD  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. the 0. M. A. response was not uniform even at 650 V. is essentially as shown in Figure 8-3a and 8-3b.  As stated earlier, The response curve  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 f l a t .  63  Intensity f (Arb. units)  1 5000A 2 5100& 3 5200%  (a) 0 +  Channel number  499  134 A  Intensity (Arb. units)  4 4900 & 5 4900 A  (b) 04-  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 f i 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 i n 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 i s the same time scale as was used in previous  chapters. Figure 8-4 shows typical oscillograms which were used while taking the data i n order to correlate the gating pulse with the pinch discharge.  8.4  The times correspond to those in Figure 8-5 and 8-6.  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_ _„, = 2 AX ) , and the relationship (Griem, 1964): FWHM HWHM T  TTT7T7Vt  3  N  e  = C(N , T ) e e  FWHM  AX  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  Figure 8-5  I  i  I  L  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  II 4686 A* P r o f i l e s : an " E y e b a l l F i t " to the Data of F i g u r e 8-5.  He  68  l i n e a f t e r s u b t r a c t i n g the continuum of F i g u r e 8-6. found  (Griem, The  were measured from the l i n e  U s i n g the r a t i o of these a r e a s the temperature may  be  1964). l i n e to continuum  intensity AAg^^  r a t i o , the h a l f - w i d t h s a t h a l f maximum  and the c o r r e s p o n d i n g e l e c t r o n d e n s i t y and  a r e l i s t e d as f u n c t i o n of time i n T a b l e 8-1. temperature  profiles  are p l o t t e d  i n F i g u r e 8-7.  The e l e c t r o n d e n s i t y  The  and  The e r r o r b a r s r e p r e s e n t the  s c a t t e r of the d a t a i n F i g u r e 8-5 which were used t o f i n d and temperature.  temperature  the d e n s i t y  e r r o r bars i n the temporal d i r e c t i o n i n d i c a t e the  d u r a t i o n of the g a t i n g p u l s e over which the s p e c t r a were i n t e g r a t e d the 0. M.  by  A. The s o l i d c u r v e s i n F i g u r e 8-7  found u s i n g e q u a t i o n s 4-13  and 4-14,  a r e the d e n s i t y and  i n S e c t i o n 4.7.  While  temperature these  e s t i m a t e s agree f a v o r a b l y w i t h the measurements b e f o r e t = 1.7  y s , which  a c c o r d i n g to the p h o t o g r a p h i c measurements i s about 50 ns a f t e r  the  plasma has reached the Z - a x i s , the measurements show t h a t these  parameters  a r e 3-4  times h i g h e r than the simple a n a l y s i s would i n d i c a t e d u r i n g the  p i n c h phase.  Table 8-1 Determination of Electron Density and Temperature  Time Cys)  Line to Continuum Ratio  HWHM  AX  (i)  N  T e (eV)  (10  18  e 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 y /  £  I  1.5 F i g u r e 8-7  I  I  L  J  I  I  2.0 E l e c t r o n D e n s i t y and E l e c t r o n  I  I  L  2.5 [us] 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 photographs and the measured current respectively. Had the resistance been signifigant, inconsistancies between each of these analyses and the photographs 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 f i r s t 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 f i r s t 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 S p e c t r o s c o p y , McGraw-Hill Book Co., New H. R. Griem, S p e c t r a l L i n e Broadening by Plasmas, Academic York, 1974.  York, Press,  D. H a l l i d a y and R. R e s n i c k , P h y s i c s , John W i l e y and Sons, I n c . , York, 1967.  1964. New  New  J . D. Jackson, C l a s s i c a l E l e c t r o d y n a m i c s , John W i l e y and Sons, I n c . , New York, 1975. J . Pachner, Ph.D.  T h e s i s , The U n i v e r s i t y of B r i t i s h Columbia,  1971.  M. N. R o s e n b l u t h , R. G a r v i n , and A. R o s e n b l u t h , Report LA-1850, Los Alamos S c i e n t i f i c L a b o r a t o r y , New Mexico, 1954. M. A. Uman, I n t r o d u c t i o n t o Plasma P h y s i c s , McGraw-Hill Book Co., York, 1964.  t  73  New  

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