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An analogue computer study of the Townsend and linear Z-pinch gas discharges Gallaher, Donald Frederick McGeer 1963

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AN ANALOGUE COMPUTER STUDY OF THE TOWNSEND AND LINEAR Z-PINCH GAS DISCHARGES ,  DONALD FREDERICK McGEER GALLAHER B.A.Sc,, University of B r i t i s h Columbia, 1959 > M.A., University of Toronto, I960.  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in the Department of Physics We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1963.  In presenting this thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t freely available for reference and study.  I further agree that permission  for extensive copying of this thesis for scholarly purposes may  be  granted by the Head of my Department or by his representatives. It i s understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of  Physics;  The University of B r i t i s h Columbia, Vancouver 8, Canada. Date  A p r i l , 1963  ABSTRACT  This t h e s i s describes a study o f gas discharges employing analogue computing techniques.  Two discharges are examinedi  The e x t e n s i v e l y  documented Townsend discharge and the r e c e n t l y i n v e s t i g a t e d l i n e a r z-pinch discharge o f plasma p h y s i c s . The Townsend discharge theory i s presented, w i t h t h e o r e t i c a l and experimental r e s u l t s given f o r the gas i o n i z a t i o n c o e f f i c i e n t s «< , Y and the attachment c o e f f i c i e n t ^ ,» Simulation o f the general Townsend equation i n c l u d i n g «< , Y and ^ i s described and an e r r o r a n a l y s i s i s given.  Computer i o n i z a t i o n curves are generated from experimental  values o f °( , Y  and *\ i n three common gases and compared w i t h  corresponding experimental i o n i z a t i o n curves. equation f o r «<  Townsend s e m p i r i c a l 1  i s used t o extend i o n i z a t i o n curves t o r e l a t i v e l y l a r g e  values o f E"/p » T h e o r e t i c a l formulae are presented f o r the primary ionization coefficient  i n neon-argon gas mixtures.  The z-pinch discharge i n the case o f i n f i n i t e plasma c o n d u c t i v i t y i s discussed and the snowplow equation f o r the time v a r i a t i o n o f the plasma tube radius i s d e r i v e d . An analogue simulator f o r the snowplow equation i s presented and an e r r o r a n a l y s i s given.  Analogue computer  r a d i a l c o l l a p s e curves a r e generated f o r argon and n i t r o g e n and comparisons are made between analogue, d i g i t a l computer and experimental c o l l a p s e curves. I t i s concluded that the analogue computer i s w e l l s u i t e d t o the s o l u t i o n o f problems i n v o l v i n g the Townsend and z-pinch discharges.. The possibility  o f t r e a t i n g magnetohydrodynamic power generation by analogue  methods i s mentioned.  iii  ACKNCMLEDGEMENT  The author would l i k e t o express h i s gratitude f o r the invaluable d i r e c t i o n and encouragement o f h i s research  supervisor,  Dr. R.J. C h u r c h i l l , i n both the research and w r i t i n g f o r t h i s thesis.  He would also l i k e t o thank Dr. A^M. Crooker f o r  p r o v i s i o n o f the analogue computing f a c i l i t i e s , the E l e c t r i c a l ^Engineering Department f o r the use o f an e l e c t r o n i c d i v i d e r and Mr. J . Turner f o r . t e c h n i c a l assistance w i t h the analogue computer. The work was p a r t l y supported by the Atomic Energy Control Board of Canada.  iv  7A3LE OF G Da TENTS  TITLE PAGE  Page i  ABSTRACT  i i  ACKNOWLEDGEMENT  i i i  LIST OF TABLES  vi  LIST OF FIGURES  INTRODUCTION  vii  ,  1  CHAPTER 1. THE TOWNSEND DISCHARGE .  ,. 8  1.1.  Townsend Gas Discharge Theory  1.2,  Computer Simulation of Townsend Discharge Equations  20  Preliminary simulation problemes leading up to simulation of modified (general) Townsend Equation  20  A u x i l i a r y Computing apparatus and error analysis  32  1.3.  Comparison of Computer Results with Experiment  36  1.3.1.  Possible applications of the computer programme  36  1.3.2,  Preliminary computer solutions  39  1.3.3,  Extension of ionization curves to large values of Ey'p  44  Computer ionization curves compared to experimental ionization curves  56  Townsend's Primary Ionization Coefficient i n Gas Mixtures  6l  1.2.1.  1.2.2.  1.3.4.  8  -  1.4,  V  TABLE OF CONTENTS  CHAPTER 2.  THE LINEAR Z-PIKCH DISCHARGE  Page  69  2.1.  Linear Z-pinch Discharge Theory  69  2.2.  Computer Simulation of the Snowplow Equation  74  2.3»  Comparison of Computer Results with Z-pinch Experimental Data and Associated D i g i t a l Computer Predictions  84  CONCLUSION  •. ,  REFERENCES APPENDIX A.  APPENDIX B.  100 103  An Analogue Computer Study of the Townsend Gas Discharge (a paper submitted i n support of this thesis).  106  An Analogue Computer Study of the Collapse Stage of the Linear Z-pinch (a paper submitted i n support of this thesis).  127  vi  LIST OF TABLES  TABLE I  Page Values of A, B and  p v a l i d i t y range  for Townsend relation (1.5) ( ^ / j , = 4 e ~ ^ 6  E  )  i n various gases,  II  III  45  Calculated values of << at j> = 1 mm Hg and various ^/j> i n certain gases.  48  Values of various parameters for snowplov equation.  87  vii  LIST OF FIGURES  FIGU.RE  Pag©  1, Schematic diagram of experimental apparatus for Townsend Discharge2.  13  Analogue simulation of restricted Townsend equation ( 0 . 2 ) , I - - r e * .  22  3. Analogue simulation of restricted Townsend equation C u i ) , X - I o [ j f ^ -^~] •  24  t J  o  4.  Simulation of modified Townsend equation ( 0 . 4 ) including ionization and attachment coefficients ^ , y and i .  31  5. C i r c u i t to detect possible amplifier output saturation. 6.  Ramp function generation test of amplifier output saturation.  7.  Graphic recorder response curve.  8.  Computer and calculated ionization characteristics from equation (0.2),~X = I e * ^ = 2 , 4 > 6 , 8 , 1 0 ionizations/cm)  34 35  0  9.  0  ionizations/cm; \ - 1 , 2 , 3> 4 > 5 attachments/cm),  12.  47  Computer ionization curves for A from equation H-Toe.^ , using results of TABLE I I .  49  (0.2),  Computer ionization curves for Xe from equation X ^ r o C ^ * , using results of TABLE I I .  (0.2),  50  1  13.  Computer ionization curves f o rfarfrom equation f using results of TABLE I I .  (0.2),  T-To^^ 14.  15.  43  Plots of ^/j> against !*/£• from Townsend relation ( 1 . 5 ) (*</|, = 4c" ^/C ) for various gases. 8  11.  42  Computer and calculated ionization characteristics from equation ( l . l ) , X = X & ~^JJ - j ^ J ( <K= ,*< = 4, 0  10.  34  Computer ionization curves for Ne from equation r » r . e.*^ > using results of TABLE I I . Computer ionization curves f o r H 2 from equation 1 X ^Tce.*- * using results of TABLE I I .  51 (0.2),  52 (0.2), 53  viii  16.  Computer i o n i z a t i o n curves f o r N~ from equation 1 =l e' ' > u s i n g r e s u l t s of TABLE I I . c  (0.2),  54  L  0  17. 18.  19.  20.  21.  22.  Computer i o n i z a t i o n curves f o r dry a i r from equation I - I o e ^ J u s i n g r e s u l t s o f TABLE II.  (0.2), 55  Comparison o f computed and experimental i o n i z a t i o n curves i n dry a i r .  58  Comparison of computed and experimental i o n i z a t i o n curves i n COg.  59  Comparison of computed and experimental i o n i z a t i o n curves i n H2.  60  Schematic diagram o f experimental apparatus f o r l i n e a r z-pinch discharge.  70  Analogue s i m u l a t i o n o f equation  (2.15),  79 23.  Analogue s i m u l a t i o n of snowplow equation  (0.1),  24. .Function generator c i r c u i t diagram.  80 82  25,  Function generator approximation t c parabola .  82  26.  Collapse curves f o r A a t 2 5 ^ Hg pressure.  90  27.  Collapse curves f o r A a t 5 0 ^ Hg pressure.  91  28.  Collapse curves f o r A a t 1 0 0 ^  Hg pressure.  92  29.  Collapse curves f o r A a t  Hg pressure.  93  30.  Collapse curves f o r A a t 500m Hg pressure.  94  31.  Collapse curves f o r  37.5m  Hg pressure.  95  32.  Collapse curves f o r N2 a t 7 5 ^  Hg pressure.  96  33.  Collapse curves f o r N2 a t 1 5 0 ^  Hg pressure.  97  34.  Collapse curves f o r N2 a t 3 7 5 ^  Hg pressure.  98  35.  Collapse curves f o r  Hg pressure.  99  250^<  N2 a t  N2 a t  750/<  1  INTRODUCTION  Plasma physics, the study.of the motions and properties of ionized gases, i s a rapidly expanding f i e l d of physical science which has come into prominence only during the l a s t few years.. The main objective of this f i e l d of physics i s to . learn how to control the fusion process which takes place i n highly ionized gases at very high temperatures..  Fusion i s a  process In which atomic particles of given mass combine to form single particles of. larger mass, the mass excess being released as an equivalent amount of energy according to Einstein's well known relation between energy and mass, E=  mc . l  Thus, a means of tapping an inexhaustible source of energy i s available, i f one can maintain a plasma at high enough temperatures for- fusion to occur, A crucial problem i n the discipline of plasma physics i s that of confinement of the plasma, which i s an extremely hot ionized gas, at temperatures frequently above those of s t e l l a r i n t e r i o r s .  It  i s evident that since no known physical material could enclose a gas at such temperatures, other less obvious means of confinement must be devised.  Such means are found i n the  electromagnetic forces derived from large magnets or current carrying coils of wire.. Many specialized machines employing different configurations of magnets and current carrying coils have been devised to produce these necessarily large plasmaconfining magnetic f i e l d s .  2  One type o f plasma discharge which i s considered here, >  i s the l i n e a r z-pinch discharge»  I n t h i s discharge a t u b u l a r  shaped plasma i s confined w i t h i n a narrow c y l i n d r i c a l conducting wall,.  The c y l i n d e r ends are maintained a t a d i f f e r e n c e i n  p o t e n t i a l causing a current to flow i n a t h i n s h e l l a t the plasma surface. Applying Newton's second law t o t h i s  system,  and assuming that the plasma has i n f i n i t e c o n d u c t i v i t y , one obtains a simple d e s c r i p t i o n of t h i s z-pinch i n the form of a d i f f e r e n t i a l equation c a l l e d the "snow-plow" equation.  The  equation i s s i named because o f the p h y s i c a l a c t i o n o f the l i n e a r z-pinch  t  Plasma i s swept up i n f r o n t o f the discharge  and pinched i n t o a narrow l a y e r or sheath immedietely behind the advancing shock wave accompanying the discharge, w i t h an a c t i o n l i k e a snowplow* The snowplow equation i s the f o l l o w i n g p a r t i a l d i f f e r e n t i a l equation which may, i n a c e r t a i n approximation, be w r i t t e n i n ordinary d i f f e r e n t i a l form t  A  ((.  ,)h,l  - J ^ £ l ! _ -  (o i). #  In equation (O.l) t i s the time, measured from the beginning o f the current pulse of the discharge, X ( t ) i s the current pulse induced i n the plasma sheath by the l a r g e e l e c t r i c f i e l d along the c y l i n d e r ; i t may, w i t h considerable accuracy, be approximated to by a s i n u s o i d ,  ^  i s the gas d e n s i t y i n the undisturbed  0  s t a t e before the p o t e n t i a l d i f f e r e n c e i s a p p l i e d t o the c y l i n d e r , - ^* ^ / R O  *  s a  dimensionless v a r i a b l e where R  the plasma radius a t any time t" and  s  R  0  (tj  is  i s the c y l i n d e r r a d i u s .  3  <^.(t=oJ = | since, at t ~O  i  R, = R, and the gas completely f i l l s s  0  the tube; after the e l e c t r i c f i e l d i s applied to the cylinder the gas i s constricted by the pinching effect of the magnetic f i e l d caused by X > and ^. decreases from unity. performed on the gases A and  Experiments  by Curzon and Churchill''" and •2 '  by Curzon, Churchill and Howard i l l u s t r a t e this pinch effect; their results s h a l l be u t i l i z e d i n CHAPTER 2. The second gas discharge problem to be examined i s the well known Townsend discharge.  Unlike the more recently studied  linear z-pinch, the Townsend discharge has a long h i s t o r i c a l background.  The f i r s t experiments on this discharge were 3  performed about the year 1900 by Townsend , and many workers have subsequently repeated his experiments with greater precision on many different gases.  The essential idea of the  Townsend discharge i s to cause a pre-breakdown ionization current to flow i n a gas between two p a r a l l e l plate electrodes. The electrodes are maintained at a difference i n potential with a uniform electric f i e l d produced between the plates,  Current  flow i s i n i t i a t e d by illuminating the cathode with u l t r a - v i o l e t l i g h t from an external mercury vapour source.  A usual modem  experimental arrangement consists of the following.  The  experimental gas i s contained i n a highly evacuated ionization chamber and i s carefully guarded from contamination with other external gases which may be present.  Two plane p a r a l l e l  electrodes whose surfaces are machined to a Rogowski p r o f i l e to f a c i l i t a t e the production of a uniform e l e c t r i c f i e l d are fixed i n the ionization chamber, and their relative potential i s  established with great accuracy by an external voltage source. The pre-breakdown ionization current 1 i s i n i t i a t e d by shining u l t r a - v i o l e t l i g h t through a window i n the w a l l of the ionization chamber onto the cathode.  The illumination i s provided by an •  external mercury vapour lamp. This illumination gives rise to an i n i t i a l photoelectric current I  0  at the cathode, and the  pre-breakdown current builds up between the electrodes i n a cascade fashion, with ionization of the gas molecules by the photoelectrons.  The ionization current X i s measured by  discharging i t through a very high resistance and measuring the voltage drop produced; of the order of 10"  12  the currents are always very small,  amp.  With this experimental arrangement, Townsend deduced the expression  I  = Xe e  .  (0.2)  for the t o t a l current X flowing at distance d from the cathode.  I f the current at the anode i s desired, as i s usual,  d w i l l be the electrode, separation. X  c  i s the i n i t i a l photo-  e l e c t r i c current produced at the cathode, as described above, and *C i s a parameter called the Townsend primary ionization coefficient, oi . i s the number of ionizations produced by an electron d r i f t i n g one cm i n the gas i n the direction of the e l e c t r i c ' f i e l d E ; that i s , •< describes the efficiency of ionization of the particular gas by incident: electrons. I t i s found that this coefficient «C depends on the ratio of the e l e c t r i c f i e l d E to the pressure  p  of the gas i n the ion-  ization chamber;. for a given gas and a given value of E /_ ,  5  oC w i l l have a definite value. The experiments are performed with the ionization chamber kept at a fixed sras pressure p , Both the voltage V  across  the electrodes, and their separation J , are varied so as to keep the electric f i e l d |£ = ^/A. between the p a r a l l e l plates constant.  Thus, the experiments are carried out at various  fixed values of the parameter ^/p  which i s found to play an  important role i n the Townsend theory. The theoretical expression (0.2) neglects other possible atomic processes occurring i n the gas.  Townsend made a  correction to equation (0.2) when he considered the p o s s i b i l i t y of secondary ionization wherein other factors beside that of the primary photoelectrons may produce additional gas ionization'. Llewellyn-Jones"''''' distinguishes several p o s s i b i l i t i e s for secondary ionization;  by far the most important of these  processes i s secondary ejection of electrons from the cathode, and this was considered by Townsend who described i t with a parameter X • The parameter  Y , analogously to «<. , i s  the number of secondary ionizations per cm of travel i n the direction of the e l e c t r i c f i e l d E  produced by one primary  electron. With this correction the Townsend equatiim becomes i  I  £ ^ ^  Now the p o s s i b i l i t y of breakdown i s present, when the denominator of the right hand expression of equation (0.3) becomes zero*  In this event current w i l l flow without an  6 a p p l i e d f i e l d , , and sparking occurs j the  the equation expressing  Townsend gap breakdown c r i t e r i o n i s then More r e c e n t l y , f u r t h e r refinements have been included  . i n the Townsend i o n i z a t i o n mechanism,.  I t i s found t h a t  appreciable e r r o r a r i s e s i n the c a l c u l a t i o n o f the c o e f f i c i e n t s «< and Yunless considered.  the process of attachment i n the gas i s This attachment i s d i s t i n g u i s h e d by the parameter  *\ which i s the number o f attachments produced i n one cm i n the  d i r e c t i o n o f the e l e c t r i c f i e l d E by a s i n g l e d r i f t i n g  electron. becomes:  The Townsend equation Including t h i s m o d i f i c a t i o n ~r _  ° "~ ^ L  :  n  ^  and t h i s i s the form i n which i t i s u s u a l l y employed.  A  5  f u r t h e r m o d i f i c a t i o n has been proposed by Howard  who a l s o  takes i n t o account i o n p a i r formation denoted lay a parameter , s i m i l a r i n i t s i n t e r p r e t a t i o n to *< , \ above.  and K  described  The Townsend equation then assumes the forma T*l*±£  Equation (0,5)  ,  i s almost never used, however, since  i s only  s i g n i f i c a n t f o r very high ^/p values not u s u a l l y encountered experimentally, and i t w i l l not be used i n subsequent d i s c u s s i o n of the Townsend equation. The modified (general) Townsend equation (0,4)  i n c l u d i n g •£ > ^ and y  employed w i l l be used here.  which i s almost always  n f i  ,  7  These two problems:  the z-pinch discharge, and the  Townsend discharge; w i l l be analyzed by analogue computation methods. The literature and existing experimental data, i n particular, on the Townsend discharge, i s very large; and lends i t s e l f ideally to computer treatment.  A programme to  simulate the equations describing the physical process may be conveniently set up on the computer with the system parameters readily variable i n the computer. By varying the system parameters a whole spectrum of solutions may be generated with relative ease and much data may be analyzed. This thesis describes the analogue computer simulation of the Townsend equation (O.4) and the "snowplow equation 11  (O.l)  describing the linear z-pinch discharge, • The computer  results are then compared with the results of experiment,.  It  i s shown that analogue methods do indeed provide a feasible means i n which to study and correlate the existing data i n these two types of gas discharge, and that these methods may well be used successfully i n the study of other types of discharge i n the gas discharge and plasma physics f i e l d .  8  CHAPTER 1  THE TOhTNSEN'D DISCHARGE  1.1  Townsend Gas Discharge Theory. Townsend envisioned a simple charge accumulation process  responsible f o r the production of i o n i z a t i o n currents.  His  experiments were performed to i n v e s t i g a t e i o n i z a t i o n currents produced i n a gas between two p a r a l l e l p l a t e electrodes enclosed i n an-evacuated, i o n i z a t i o n chamber, the lathode being i n i t i a l l y i r r a d i a t e d w i t h u l t r a - v i o l e t . l i g h t from a mercury vapour source through a window i n the chamber.  The f o l l o w i n g  d i s c u s s i o n outlines, the d e r i v a t i o n given hy Townsend.for the r e l a t i o n between pre-breakdown i o n i z a t i o n current i n the. gas and the  r e l a t e d p h y s i c a l parameters of the discharge. Consider a lamina of gas of thickness dx and u n i t area..at  a distance x  from the cathode.  There are n electrons per  u n i t volume of gas a t t h i s distance x from the  outhodo  These electrons have been produced by bombardment and subsequent i o n i z a t i o n of gas molecules by the i n i t i a l e l e c t r o n shower ejected from the cathode by p h o t o e l e c t r i c a c t i o n w i t h u l t r a v i o l e t cathode i r r a d i a t i o n .  I f one e l e c t r o n d r i f t i n g a t the  distance X causes oi i o n i z a t i o n s of gas molecules and so produces oC more electrons i n a u n i t distance measured i n the d i r e c t i o n of the e l e c t r i c f i e l d E ; i n a distance dx  at  X*  then i t w i l l produce  electrons  Hence, the increase A a i n number  of electrons from v\ to  M+JH  i n the distance Aye i s just  ov\ -e(v\6.K> nd so by integrationft= A e. a  a  number of electrons per unit volume at  where yi  0  i s the  X — O . Ho iff  proportional to the i n i t i a l photoelectric current density X  0  at the cathode.  Thus,I = Xo£  > since the pre-breakdown  current X i s proportional to the number of electrons V\ , and I„ i s the ini t i a l photoelectron current density at the cathode.  I f the electrodes are a distanced apart the  ionization current registered at the anode w i l l be I = I € ' ^ equation (0.2).  (0,2) i s Townsend's f i r s t gas  <  D  ionization equation i n which *C i s called Townsend's primary ionization coefficient and i s found to be a function of  #  The ions produced, i n this cascade have a positive charge, and so are attracted to the cathode where they may dislodge electrons from the cathode surface into the gas to produce a secondary electron emission.  The t o t a l number of ions  produced i n the electrode gap i s obtained by integrating dn , the number of electrons i n a thin slab of gas of unit area and thickness d * "t distance X , over a l l X from 0 to d . a  This  value i s je( n</x =to„(e^- l)> ^ w i l l represent the ion current • a n  at the cathode.  I f each ion which bombards the cathode releases  Y electrons then there w i l l be an electron number density at the cathode given by  i n addition to the i n i t i a l  photoelectric number density Y\ ^ 0  Thus, the t o t a l electron  density at the cathode i s ^ . ( e ^ - l ) -Ielectron density must equal  >  a n d  |rt , Then M„ £l 0  ^hia t o t a l Y(e*^-  l)j=-Ho  10  and one hasto= n „ c  •=  . Therefore the  ionization current at a distance x between the electrodes including the process of secondary emission i s X = XoC° /i-K(d' ^ ~ ' ) tX  i  > ^ *he current registered anc  at the anode where X=d i s found now by: r  -  i  p  (0.3).  £  (0.3) i s Townsend's discharge equation including primary and secondary ionization.  The coefficient Y i s the num.ber of  secondary electrons produced, in this case, by one primary electron d r i f t i n g a unit distance i n the f i e l d direction. For many years equation (0.3) was thought to describe the process with sufficient refinement, but discrepancies between theoretical and actual ionization currents were found. Geballe and Reeves considered the process of dissociative attachment whereby a molecule AB, say, could be separated into nonionized and ionized atoms A and B,  respectively,  by an incident electron according to the reaction AB + e = (AB ) = A 4 B,  where the intermediate state (AB ) represents  an "excited state of the molecule„ Although other attachment processes such as direct attachment represented by AB+e = AB, or dissociation into ions represented by AB+e = A +  B+e  could be spectroscopically observed to exist i n certain gases, the most prevalent reaction appeared to be the one of dissociative attachment, and so the modification to the Townsend discharge produced by this effect was investigated. In considering the process only small electrode separations ,  11 cl were allowed, to prevent the p o s s i b i l i t y of secondary ionization; so that the simultaneous processes of direct ionization represented by  and dissociative attachment which  was represented by a parameter vy were .considered.  The  parameter i\ was called the attachment coefficient and gave, analogously to o(_ , the mean number of attachments produced by an electron d r i f t i n g unit distance i n the direction of the electric f i e l d .  Continuity equations for the number of  positive ions H , the number of negative ions v\_ and the +  number of electrons V) with e  IT}. , |r_ and  representing  their respective d r i f t speeds i n the gas could then be written down. These equations are:  \ lf ^ t  =. «iVeH  e  - *|V~e V\e >  and their steady state solution with, appropriate boundary conditions, was found to be:  (i.D. . Equation ( l . l ) , however, neglects the secondary .ionization procesE represented by )f.  Including this,  secondary.process leads to the p o s s i b i l i t y of gas breakdown. The Townsend equation including both primary and secondary ionization and attachment i s then found to be:  r  as used by Prasad  __£L  ( w - i ; J _ ,) e  and others. The secondary process  (0.4)  12 represented by :  allows the p o s s i b i l i t y of gas breakdown at  which current w i l l flow without the external stimulus of a photoelectron cathode current. In this case current increases very rapidly as the denominator becomes smaller and sparking  w i l l occur.  The relations Y(e^-\\ = '  a n d  (<^"^ - f) = I J  express this breakdown c r i t e r i o n ; respectively, with and without including the attachment process.  The modified  Townsend equation ( 0 . 4 ) including primary and secondary ionization and attachment i s the most general one usually employed, but there i s another discharge equation due to 5  Howard including the process of ion pair formation.  The ion  pair formation reaction proceeds according to the mechanism AB4 e=A + B + e and i s represented by a parameter </\  the  mean number of ion pairs produced by one electron t r a v e l l i n g a unit distance i n the direction of the e l e c t r i c f i e l d .  The  discharge equation with this modification becomes:  The Howard equation ( 0 , 5 ) including 0 i s almost never employed, since the ion pair coefficient i s only found to be appreciable for very large values of ^/p , These E/p values are not usually encountered experimentally, so Howard's modification to the general Townsend discharge equation (O.4) i s very seldom used.. While performing Townsend'sexperiment i n an ionization chamber the ionization currents I are measured, keeping the pressure p  constant and varying the voltage / and gap  separation d so that a constant f i e l d E =  , and hence a  13 constant value of ^/p  , results for each ionization curve.  The ionization curves are plotted as I against J at various constant values of ^/y  > ^ from measurements of their a n  curvature the coefficients << , ^ and Y are obtained. »( affects the overall slope of the curve, being the most dominant parameter; the c o e f f i c i e n t . ^  contributes'to an i n i t i a l  concavity of the curve i n regions of small <J , while  Y,  the parameter which causes gas breakdown, i s manifested by a sharp upcurving to higher currents at larger values of A . Thus, the f u l l ionization curve for appreciable values of J i n gases with moderate attachment w i l l have an approximate S-shape; i n i t i a l l y concave down i n the region of dominance of  , then straightening out under the influence of o( and  f i n a l l y concave up when the Y  process takes over.  Although  the coefficient oC i s f a i r l y easily obtained by curve analysis, • the other two parameters'^ and/' are exhibited by usually slight variations i n curvature and are considerably more d i f f i c u l t to evaluate by this technique. \ and Yare dependent on ^ p  Since the three coefficients ^ , , the usual method i s to derive  from the ionization curves their values as functions of ^/j> , *^/p  i n practice the reduced parameters  and ^p  plotted against ^~j p . Sometimes the coefficient i s plotted as a function of ionizations per volt.  p. where  ;  are = *^/p  fy^-  f= represents  This i s an experimentally more apparent  coefficient than the coefficient °^/p  which i s measured i n  ionizations per unit distance per unit pressure.  I t i s usual  i n these investigations to measure the gas pressure i n mm Hg  and the electrode separation d i n cm. and Penning  favour the plotting of  The workers Kruithof rather than °S/p .  Figure 1 shows a schematic diagram for the Townsend discharge experiment. .Figure _1.  . Schematic diagram of experimental apparatus' for Townsend discharge. U I t r a v/t ole"t rua^tt lens  "V-awode •w cafcl> ode "to \onviia.ticti\  System  detector Although the coefficient ^ r e p r e s e n t i n g secondary ionization appears to be predominantly due to secondary ejection of electrons from the cathode," other processes contributing to secondary ionization exist i n different proportions i n various gases, and careful studies by Llewellyn '  .  4  .Jones  have disclosed at least four other different secondary  ionizing processes. These processes producing secondary • ionization'are:  ionization by impact of excited atoms at  the cathode represented by a coefficient $ , ionization by c o l l i s i o n of positive ions i n the gas represented by a coefficient ^ , ionization by impact of positive ions at the cathode represented by the coefficient Y and  L  5  ionization produced by bombardment of the cathode with photons generated i n the gas ionization current, represented by a. coefficient  F . Taking a l l of these processes into account  Llewellyn-Jones writes a generalized secondary ionization coefficient w as the sum of these processes;  where o( i s the Townsend primary ionization coefficient.  It  i s the generalized coefficient CJ of equation (1.2) representing a l l the secondary ionizing processes which i s responsible for gap breakdown.. Theoretical evaluation of the.process  which represents  photoelectric action at the cathode due to photons produced i n  9 the gas was calculated by Davies et a l , who showed that the generalized secondary ionization, coefficient account only the f  fo( taking into  process was given by: (1.3)  where r  /^ ) /  ]  0  i s the constant} _ ! _  iLjLg  H  C  H  ^  H (1.4).  In expressions (1.3) and (1.4) © i s the number of photons produced by one electron moving unit distance In the f i e l d direction, \( i s the photoelectric efficiency of the photons, ^. i s a geometrical factor dependent upon the angle subtended at the cathode by the head of the electron avalanche,^, i s the mean absorption coefficient of the photons i n the gas, u.  16 i s the most probable velocity with which electrons leave the cathode, \*/_ i s their d r i f t velocity, T  i s the lifetime of  the excited state of the gas molecule which gives r i s e to ' emission of photons, ?(<^) i s the probability of destruction of this excited state by collisions of the second kind, Irf  = ^/y  wnere  J i s "the c o l l i s i o n frequency of the gas  molecules and p  i s the gas pressure.  Thus, since  )  0  i s a constant for given values of ^/p and p«/ , where ^/p and ^ ~/*.  P  are functions of ^/p ; i t i s seen that ^/ad  '  depends l i n e a r l y on the pressure p . This dependence was 9 experimentally confirmed by Davies et a l i n Hg gas where the  process was the dominant secondary process, thus  validating this theory. It has also been found that the secondary ionization coefficient i s dependent on the nature of the cathode surface and of i t s material.  This effect i s shown to be due to a  reduction i n the effective work function of the cathode because of the deposition of atoms i n a thin metallic layer on i t s surface.  The effect was studied by Llewellyn-Jones^  who noticed that the curves of the generalized secondary coefficient plotted against  p were dependent on the state  of the cathode surface and i t s material; by outgassing the ionization chamber to remove the thin layer on the sathode, or by allowing deposition to build up on the cathode, marked changes could be noted i n the curves of  against  p .  I t was Townsend who f i r s t wrote down a simple empirical expression for ^ j^  as a function of E/p •  H e  found that for  17  many gases i n moderate ranges of ^/p  less than magnitudes p  of 200, the expression:  /P " Z  (1.5)  1e  with A and B empirical constants gave a good f i t to the curves.  Plotting *^/p against P/g: on a semilogarithmic  chart resulted i n f a i r l y straight l i n e s , a factor from which A and B could be determined.  Later theoretical 11  studies by .Von Engel and Steenback  postulated a more  complex equation for °< fp . Loeb has shown how the Von Engel-Steenback  equation reduces to Townsend's  equation (1.5) for values of ^/p i n the range of 200 for 13 the case of a i r .  Sen and Ghosh  confirmed this using  experimental results of Sanders^f and showed that the results for *^/p calculated from Townsend's relation (1.5) and "(/j> obtained experimentally by Sanders, are i n good agreement f o r ^ / p between 100 and 200 i n the case of a i r . The results for the gases, A, Ne, Kr, Xe, H , N2, and a i r 11 2  as well as for other gases were analyzed by Von Engel  ,  and the constants A and B as well as ranges of v a l i d i t y of Ey/p were compiled according to the Townsend relation (1.5). Attempts have also been made to f i n d an empirical 13  y  t . Sen and Ghosh  equation f o r the secondary coefficient  studying electrodeless discharges i n a i r have found that the variation of )f with ^/p could be represented by the empirical equation: '  r  =  W r )  +  4)  (  + ' c  (1  -  6)  18  where A ' , B» and C are empirical experimentally determined constants.  They experimented over large ranges of  from zero to approximately 3200 and found that Y depended both on ^yp and p . As Davies et a l showed, the dependence on p i s due to photoionization at the cathode, the other dependence on E/p  being the normal dependence of If.  Sen  and Ghosh assumed the four processes-studied by LlewellynJones to-be taking place:  positive ions striking the glass  surface of the tube i n which the electrodeless discharge was taking place, electrons s t r i k i n g the glass surface, ionization by positive ions and photoionization. how  Y  Ttiey then demonstrated  could be represented by their empirical equation (1.6)-.-  As far as the author i s aware no theoretical or empirical equations are known for the attachment coefficient f . A further theoretical evaluation of the primary ionization  ^  15  coefficient e( has been made by Emeleus, Lunt and Meek  with  16 later emendations by Blevin and Haydon  . Emeleus et a l made  assumptions about the electron velocity distribution i n the gas and related the coefficient  to the distribution; from  this they were able to determine the theoretical variation of *^/p  with ^/p  i n various gases. They assumed both a  Maxwellian distribution of electron velocities and a Townsend distribution and were able to show, f o r the case of a Maxwellian velocity distribution,, that:  ^  v.-  j7 ' "  (1.7);  where V i s the mean electronic agitation energy, W i s the  19 average electron d r i f t velocity i n the direction of the electric f i e l d E , V i s the mean energy, \f- i s the ionization potential of the atoms of the gas and P(/J i s the efficiency of ionization of the electrons i n the gas.  For the case of  Hg gas at values of E/"p between 0 and I4O i t was found that agreement between the theoretical value of "^p given by Emeleus et a l i n (1.7) and experiment was good for ^ p > 30, but for 0 < ^p"^ 30 there was a considerable discrepancy. If one writes, following Blevin and Haydon}  H*)'p(v-vc)  ( 1  the integral i n the expression (1.7) for ^  . . 8 )  becomes tractable  and i t i s then found that:  <l  = r-4g x 10"^ w~'V " ' ( y . ' + i - s i V)e~,'f  where ^ i s a constant.  7  (1.9)  From equation (1.9) one deduces that: u  =  K  ^  (1.10)  '-5- U ^ i » - f ? " ' ' " J  ( 1  -  u )  i s larger the lower E, E^p i s , Blevin and Haydon show that a percentage change of 5% in Y gives rise to a percentage change i n l ^ / p i ^ ^ . °f about 50$ at ^/p = 30 and about 15$ at ^  = 100.  I t was suggested  by Blevin and Haydon that the results of Emeleus et a l for 1/ may show a systematic error causing the large discrepancy between their (Emeleus et al) values of ( °^pJ | of (1.9) ca  c  •ao and the experimental values (^/y ) p "t low values of ^/p a  et  whereas for higher E/p values the agreement was observed to be better. I f this effect i s not operative, and the values' of V used are correct, then they suggest that the assumed electron distribution of velocities may not be correct, or that the relation (1.8) for the efficiency of ionization i s not v a l i d .  1.2 1.2.1  Computer Simulation of Townsend Discharge Equations. Preliminary simulation problems leading up to simulation of modified (general) Townsend equation. The method chosen to analyze the experimentally  determined ionization curves was the device of analogue computation. This type of analysis of physical problems i s p a r t i c u l a r l y adapted to cases where there are large amounts of data for which e x p l i c i t mathematical evaluation would prove too tedious and laborious. I f the equations describing the physical variables of the problem are reducible to d i f f e r e n t i a l form (either ordinary or p a r t i a l d i f f e r e n t i a l ) the analogue computer becomes an ideal tool since i t i s especially suited to simulate d i f f e r e n t i a l equations. Ordinary d i f f e r e n t i a l equations may be simulated on the computer most easily.  P a r t i a l d i f f e r e n t i a l equations may also be simulated,  though with considerably less f a c i l i t y , since a single p a r t i a l d i f f e r e n t i a l equation must f i r s t be written as a number of algebraic f i n i t e difference equations, and the set of these difference equations then programmed for the computer. The accuracy of computer solution of the original p a r t i a l d i f f e r e n t i a l  21 equation increases with the number of f i n i t e difference intervals "chosen for i t . . Since there exists a very great deal of experimental data on the Townsend gab discharge i t seemed appropriate to describe feasible methods for a systematic reduction and analysis of the data by analogue means; especially since the gas~ ionization equations are readily reducible to ordinary d i f f e r e n t i a l form. The analogue; 'Computer used for. simulating the Townsend gaa discharge equation (O4) and larter also for the simulation of the linear z-pinch discharge equation (O.l) (snowplow equation) of a plasma was a small Heath model.  I t incorporated  15 d c . d r i f t - s t a b i l i z e d high gain computing amplifiers with a n c i l l a r y apparatus on the computer cabinet such as coefficient set potentiometers (both high precision and less accurate v a r i e t i e s ) , toggle switches and i n i t i a l condition set control switches. A l l of this specialized computing apparatus used for the simulation of the Townsend equation (0.4) plus auxiliary computing equipment and parts, w i l l be described i n d e t a i l i n Section 1.2.2,  The aim here i s to describe the steps by which  an eventual simulation of the modified (general) Townsend equation (0„4) was achieved.' The f i r s t Townsend equation to be analyzed was that one including only the parameter °( ; namely equation (0.2) given by T=loC ^° a(  % differentiation of (0.2) with respect to d  one obtains the physical d i f f e r e n t i a l equation; J  J  which may be easily simulated since (1.12) i s a f i r s t  (1.12)  22 order ordinary d i f f e r e n t i a l equation.  No time integrals  appear anywhere i n equation (1.12), and since the coefficient i s usually constant i n time, i t follows that the physical to analogue correspondence may be made so that A = K | t where K, i s a constant scaling factor for time scaling of this equation.  The voltage or amplitude scaling may be  accomplished by choosing 1= K e where Y i s another constant 2  z  scaling factor and 6. i s analogue voltage; with these correspondences and the i n i t i a l condition = X ( t °) - l o equation (0.2) could be simulated as shown i n =  the c i r c u i t of Figure 2. Figure 2.  Analogue simulation of restricted Townsend equation (0.2), T - X « * .o( i s Townsend's primary ionization coefficient i n ionizations/**/! is electrode separation i n cm, I i s pre-breakdown current and T „ i s i n i t i a l photoelectric current. J  Q  In the c i r c u i t of Figure 2 use i s made of a voltage divider which i s incorporated as an accurate (0.25$ l i n e a r i t y ) coefficient set potentiometer on the computer to precisely vary the parameter ^ By choosing R = ^ = 1 megohm andC = j/*r" one has a time scale 1  1  23 such that 1 second computer time =1 unit of A and a voltage -12 scale -which may be chosen a r b i t r a r i l y , 1 volt - 10 amp i s a convenient figure to simuls-te the. minute ionization currents.  The amplifier appearing i n Figure 2 with the 1  megohm input resistance and 1 megohm feedback resistance reverses the sign of the input voltage.  The amplifier which has  a feedback capacitor instead of a feedback resistance integrates the input voltage e,^ i n the followiog way.  I f the input  resistance i s R and the feedback capacitor i s C then the amplifier output  In the case  of a multiplying amplifier with input resistor R, and feedback o  resistor R  2  the resulting relation w i l l be C »t ~ ^ S . These =  0  w  relations may be derived by applying Kirchoff's current law to the amplifier input node, assuming the amplifier input impedance to be i n f i n i t e . . Thus the net current at the node i s equated to zero.  These and other general analogue computing principles with 17  various technical applications may be found i n Johnson  .  The equation (1.12) i s correctly simulated using this c i r c u i t , with the l i m i t s of < from 0 to 1. I f one wishes to extend the l i m i t s of oi from 0 to 10 the desired multiplicative factor may be introduced without the use of extra or different equipment by reading the time scale differently,.  One reads 10 seconds on the  unit of distance. The of i n idistance, tial time scale as one unit instead of 1 second for one i s introduced across the integrating capacitor as shown i n Figure 2.  24 The next ionization equation to be simulated was the equation  T - T P*— x  - j . o -  e.^"' n  -—1  yU  z  *  which includes the effect of attachment i n the coefficient ^ . By differentiation, again with respect to d , one finds, holding I  0  and ^ constant;  once more a f i r s t order ordinary d i f f e r e n t i a l equation.  The  i n i t i a l conditions on equation (1.13) are also X (d = © ) = X  0  as i n the case with the previous equation ( 0 . 2 ) , X ~ X o C  •  By making the same correspondence as before where <j i s simulated by the time "t and I  i s simulated by the voltage V , the correct  c i r c u i t becomes the one shown i n Figure 3. Fjy^re_J3. Analogue simulation of restricted Townsend equation (1.1); f ~ T F * - ^- O * - ~r --1 (  t  <  L  tf  <\ i s Townsend's coefficient i n attachments/cm, d i s electrode separation i n cm, X i s prebreakdown current and X i s i n i t i a l photoelectric current. 0  s + 1X0  •WW—1  -Ic  in  id  25  The sole essential difference between the c i r c u i t of Figure 3 and the c i r c u i t used to simulate equation (0.2) given i n Figure 2, i s the introduction of a constant external voltage source proportional to i\ Xo « The parameter *^ i s varied by means of a precise coefficient set potentiometer and X„ i s a fixed,  arbitrary  voltage which i s derived from, i n i t i a l condition voltage terminals on the computer cabinet. In the c i r c u i t of Figure 3 the second potentiometer simulates the quantity o(-i^ instead o£s< as before. that this c i r c u i t reduces to the previous one I f  I t i s evident  = 0,, just as equation  ( l . l ) , ! * ! . ] ^ - ^ ' ! ^ - ^ " ] reduces to equation (0.2), I = X c ' 1  ( i  0  for /j = 0, so that the problem including  i s correctly simulated.  A further generalization of the Townsend equation must include the secondary ionization coefficient  . This immediately changes  the character of the equation markedly because now a breakdown mechanism i s present which i s incorporated into the denominator of the equation for I . Starting with equation (0.3) given by X =• T0e*^/\-Y(e *tC  including only primary  \)  and secondary (^j ionization the problem  i s to simulate (0.3) on the computer, a more d i f f i c u l t task than the l a t t e r two simulations since equation (0,3) has a denominator which contains the variable J . Two ideas regarding the simulation of (0,3) were explored and f i n a l l y rejected as unsuitable i n favour of a t h i r d method. F i r s t , by straightforward differentiation with respect to A one has»  26  Equation (I.I4) is not impossible to simulate, the essential extra mechanism required i n this case being a squaring device which, given an input voltage X > w i l l produce an output voltage KX^ where K i s some constant*  Thusj a workable c i r c u i t could  be incorporated consisting basically of one integrator for the derivative ^ 5 . The integrator output voltage - J could then be AA  squared i n some variety of electronic squaring device and multiplied and X i n this X. c i r c u i t one could then carry this sum back to the integrator as-  by the constant  To  . Forming the sum of  At  Hi  and the c i r c u i t i s complete since equation (I.I4) i s  satisfied. The second idea on simulating equation (0.3) including < and Ywas to take the logarithmic derivative with respect to A of equation (0.3). In this way one finds: I o<j ~$-f= -  rU^-O)**!  Therefore  -\o}(\-  -dft&l = <+ — T~Ja r T  e  Y(e^  sothat  U  loj &  ^  - \ ))\  =«/ /t I O j  This procedure gives the same equation (I.I4) as before so  V  27 that no new method of simulation emerges. The f i n a l successful method chosen f o r simulation' of equation (0.3) including a^-and /'was to use an electronic divider and simulate numerator and denominator i n separate c i r c u i t s , then to feed the separate numerator and denominator signals into the divider and so produce'the solution X .  If  the modified Townsend equation i n the form:  i s differentiated with respect to cJ one finds the rather complicated relation:  u  , _  -, J  which cannot be put into the simple form C -/TJ + C^^/^Jwhere (  C, and C  0  are constants depending on «<, ^ and Y as was the :  x  case with the simpler equation (0.3) where the relation  -^It^uM  i^rJ with C , = o{ and C  was found.  x  (  ,  U)  For this reason, and  predominantly because the other Townsend discharge equationsare a l l special cases of the modified (general) equation (O.4) • including U., ^  and Y ; the solution to the problem  incorporating a divider was chosen. If the physical gas discharge equation i s written i n the  r =V x=  i.[t^^-^  Y  (1  - ^  ]  .  15)  (1.1*)  28  i s the numerator and Y«  | _ J J L ( *-%U C  _ , J  (1.17)  i s the denominator, one finds by differentiation with respect to d  the two equations  X(A = oJ r£ X  The i n i t i a l conditions on X and Y are Y(J«  ,  0  o j = , (1.20). The electronic divider operated i n the following way.  If  € , and (f are the two voltage inputs to the divider, c being (  the numerator and £ is C j ^ u t  =  L  K  < had the value 100.  the denominator; the output of the divider where, i n the particular divider used here, The divider would only function i f the  divisor C was a negative voltage, and i t s output was limited to 2  the range from 0 to 50 volts.  For this reason the denominator  generated i n the computer was -100 \  and the numerator - X >  the output from the divider then being Coutput = k (_ y ) = I  0 0  ldo  - */y  =  (-n^>f)  I j 'the ionization current as desired. Thus the  c i r c u i t of Figure 4 was set up to simulate the modified (general) Townsend equation (0.4) where the analogue parameters, voltage and time, are indicated i n the c i r c u i t . The two analogue equations are obtained from the numerator and denominator c i r c u i t s by equating the net current to zero at  29  the input of each integrating amplifier i n the same way as has been done before: e C  "  it  "  dt  °  =  m  ( 1  0  ftp ,  KP  -21)'  (1.22). Z  Choosing the correspondence i n the analogue equations (1.21 and (1.22) to the physical equation (0.4) so that - €.^j , cf , Co p  J  (  > <f > 4>x. > $j  a r e  t  ^  n e  e x a c  ' t analogues of X >  , D( , (j , Y ; one has, by setting =  " ^ k l " R> t  t h e  = C « C and p  following equations:  RC  df  X>  RC  RC  R C  (1.24).  Assigning to ftC any fixed value, a time scale may be chosew, and by setting e« = K T o an amplitude or voltage scale may also be chosen. Setting, say, (^C  =  1 with C - 1>*F and  =• 1 megohm the time scale i s such that 1 second computer time — 1 second r e a l time = 1 unit of electrode separation d . With • 2, one has 2 seconds real time = 1 unit of electrode separation. An arbitrary voltage can be chosen to represent a given current; i t seemed convenient to l e t 1 volt correspond  -12 to a current of 10  amp. Thus the time and amplitude  scaling factors can be chosen for this c i r c u i t as indicated.  30 This completes the correct simulation of the Townsend equation (0.4.) including e( , \ and  Y.  The other three gas discharge  equations discussed above (equations (0.2), (0.3) and ( l . l ) are a l l special cases of the more general equation (0.4) and hence their c i r c u i t s are special cases of this general c i r c u i t given i n Figure 4 obtained by setting ^ = ^*0,  1 = 0  or Y* °.  31  Figure 4. Simulation of modified Townsend equation (0.4) including ionization and attachment coefficients < ^ ^ a n d parameters- ^ , ^ ^ , e > £ *> t^'/e (  r  T e  o  V  T e s e n  ^  Analogue  respectively ^  ^,  ,.T>.X »• Y> I p  •->1ELEC TRONIC  -Ilo<vtL  -  100 V  ^  e  o  WV—i  ,  |  1  0  ^  H H ±±poe  0  32  1.2.2.  Auxiliary computing apparatus'and error analysis.  The special components necessary for the simulation of the Townsend equation include accurate coefficient set potentiometers, an electronic divider mechanism and a graphic recorder; The most general c i r c u i t given above i n Figure J+ includes four accurate ten turn h e l i c a l l y wound wire potentiometers with a mean scale l i n e a r i t y of 0.25% and a resistance of 10K ohms (10.1%).  These coefficient set potentiometers simulate the  parameters o( , >\ and <T as indicated i n Figure 5; to simulatevalues of ^ , ^ and JT greater than unity a change i n time scale suffices.  Each of these 10K ohm potentiometers was fed  into a 1 megohm resistance.  The potentiometer loading error  incurred with a potentiometer of t o t a l resistance R, feeding into a resistance of value R, at a potentiometer decimal A  fractional scale setting of 4, i s :  & ~^ ~  -  ~ ~~~  where 6 i s the value which must be  added to each potentiometer setting value of (X. between 0 and 1 to obtain the true potentiometer setting. i n the reading of O. i s  The percentage error  * 100%«. This loading error arises  because the resistance into which the current i s feeding i s not infinite.  With the c i r c u i t used here, R , « .lOKyt, ft « 1000K-A., v  and with a setting of A. - 0.5 at which maximum percentage error i n the loading error occurs, the actual percentage error incurred  33 is ( l  [££f  ) 100 = (1 - ° °  i s completely negligible.  |0  ) 100 ^ 0.249$ which  Grid current error arises when  amplifier voltage d r i f t s gradually from a fixed value with time.  I t may be minimized by reducing the time of the  computer runs and was so done here; typical computer runs lasting 7 seconds or less. Amplifier output saturation induces another type of error which may occur with an amplifier feeding into a low resistance potentiometer winding; here the potentiometer may draw large enough currents from the amplifier to saturate i t s output and distort the upper voltage portions of the amplifier output. The c i r c u i t chosen to check on this error p o s s i b i l i t y i s shown i n Figure 5 where an integrating amplifier integrates a constant input voltage of 100 v o l t s .  This would theoretically  give r i s e to a precisely linear output curve given by 100 "t. The current drawn from the amplifier output by the 10K./1. potentiometer may saturate the amplifier output and f l a t t e n off the upper portion of the output curve. As can be seen i n Figure 6 the upper flattening of the curve i s almost unnoticeoble, thus amplifier output saturation error i s v i r t u a l l y negligible i n the general c i r c u i t of Figure 4.  34  FajQjre 5_« C i r c u i t to detect possible amplifier output saturation. Testing function used i s ramp function ^ = lOOt where t i s time. Amplifier output feeds into l M-a resistor v i a a 10K.A. r e s i s t o r .  H23-2I£L6»  Ramp function generation test of amplifier output saturation. Ramp function used i s y = lOOt where "t i s time. Slight bending of curve near 100 volts indicates very s l i g h t output saturation.  Time  (SCCOIAJS  35 The electronic divider i s a commercial Philbrick model employing a non-quarter square c i r c u i t .  As mentioned before,  the output of the divider i s given by <?oot^4 "100 ^'/<?*. where and €  x  are the voltage inputs; ^  negative i n sign.  necessarily being  The output of the divider varies i n absolute  value from 0 to 50 v o l t s . Special adjustment of the divider wasnecessary due to d i f f i c u l t i e s i n i t s operation with very small denominators when the quotient of C, and C becomes very large. z  I t i s thought that the main source of computing error i n the c i r c u i t arises from this factor, one which would be d i f f i c u l t to calculate and i s larger than a l l other errors. The output signal from the divider was brought to a recording milliammeter for readout. The milliammeter had a theoretical frequency response curve 3 db down at 5 cps according to the manufacturer.  This response was checked by means of an  o s c i l l a t o r and found to be quite accurate as may be seen i n Figure 7. Figure 7. Graphic recorder response curve. Response i s 3 db down at 5 cps and linear u n t i l about 2 cps. •Experimental check of recorder response curve agrees well with quoted manufacturers response curve.  4  »  IL  -frequency ( c J>5 J  /6  to  36  The recorder had a nominal internal resistance of 6 K J L and several chart speeds which could be conveniently switched on by a finger t i p control, the speeds ranging from 0 . 7 5 inches per hour through settings of 1 . 5 , 3 . 0 , 6 . 0 , 1 2 . 0 inches per hour and 0 , 7 5 inches per minute to 1 2 inches per minute. This current recorder was converted into a voltmeter by inserting an accurate (0.1%) external resistance of 34-K ohms between i t s terminals and the divider output. Since the recorder gave a f u l l scale deflection for a current of 2 . 5 mA this converted i t into a voltmeter with a f u l l scale deflection at 1 0 0 volts, a conveniently chosen figure.  The gas discharge problem was most often run at the 1 2  inches per minute speed on the recorder since this gave adeq&tte resolution of the ionization curve and minimized grid current error. F i n a l l y , a l l passive computer components such as resistances and capacitances were of ± 1% quality and would not measurably contribute to error.  Taking into consideration a l l of the above  sources of error i t i s thought that the largest error contribution arises from the divider mechanism, and that for reasonable experimental values of the coefficients <( , ^ and Y(the values usual i n experiment) the combined error i s estimated to be a maximum of + 5%.  1.3.  Comparison of Computer Results with Experiment.  1.3.1.  Possible applications of the computer programme.  With the computer set up as indicated i n Section 1 . 2 to  37 solve the modified (general) Townsend equation ( 0 . 4 ) , a number of applications become possible. F i r s t , experimental data might be used directly to obtain values of the three characteristic gas coefficients c( , ^ Y  (at given values of  versus £/p  ) from plots of * ' ,  and  and ^/p  ; the ionization curves could then be generated  on the computer and a direct comparison of them made with the experimental ionization curves from which <K , *\ and Y were o r i g i n a l l y taken.  This programme would provide a v e r i f i c a t i o n  of the r e l i a b i l i t y of the c i r c u i t . . In connection with this procedure i t might be possible to extrapolate the experimental ' ^f ?  ^(P  v e r s u s  ^(p  c u r v e s  0  higher values of ^/p >  at these higher ^p values  thence derive values of c( , *\ and f and from these extrapolated a( , ^  f  and Rvalues generate  ionization curves i n the computer. This procedure would be a method of extension of experimental ionization curves to higher Ey/p values by extrapolation of the •^p versus  ,^  and typ  curves.  Secondly, one might generate on the computer a whole spectrum of ionization curves for various «( , 1^ and  values, and then  visually or graphically compare these curves with actual experimental ionization curves.  I f enough computer curves are  generated, a one to one correspondence between certain computer ionization curves and certain experimental ionization curves could be obtained, and gas coefficient curves might then be predicted from the computer results.  Thus, i f no experimentally  deduced gas coefficient curves are available, this method of  f  38  electronic curve f i t t i n g could be employed to predict the coefficient curves.  In practice the method would prove quite  d i f f i c u l t due to the very s l i g h t variations i n slope of many of the ionization curves.  For the more prominent coefficients  o{ and *^ the programme might be useful; with Y i t i s unlikely that the method would be very successful since  usually  very much smaller than either ^ or ^ . Thirdly, the computer programme might u t i l i z e gas coefficients deduced theoretically or empirically, and from these coefficients predict theoretical gas ionization curvesi In extremely high ranges of the parameter  usually  inaccessible  to experiment i t may be possible to exclude the less prominent coefficients grounds.  or f  or both on physical and experimental  One could then predict ionization curves from  theoretical values of the most significant primary ionization coefficient ^ « Since theoretical expressions are available for  this method would be useful where, i n particular, one might  use Townsend's simple empirical relation (1,5) f or "(/p of E  / p  i n terms  ,  F i n a l l y , before any one of these programmes i s implemented, for each of which abundant data i s available, the  simulating  c i r c u i t must be tested to ensure that the resulting curves are within reasonable agreement with mathematically calculated  curvoo.  When this i s so, any or a l l of the above three programmes may be followed with impunity.  As a systematic procedure for checking  f.ha computer one can generate ionization curves for given  39  combinations of the  gas coefficients e( , ^  and Y and then compare  these with mathematically calculated curves, the discrepancy indicating the amount and tendoncy of the computer error.  1.3.2.  Preliminary solutions.  The following coefficient combinations were used i n testing the computer:  (1) *| = r  =  0;  (2) 1 = 0,^= (3)  =  2, 4, 6, 8, 10.  6; ^=.02, .04, .06, .08, .10.  0, ci = 6; <j = 1, 2, 3, 4, 5.  (4) *j = 3,*=  6j/<= .02, .04, .06, .08, .10.  Thus, according to the above c l a s s i f i c a t i o n by gas coefficients the following forms of the modified Townsend equation are being used: (0.2)  1=  I  € ^  e  (varying ^ ) ,  (0.3) X =  (1.1) X = Xo  :  rT^—  (0.4)1 =  (fixed«(-, varying 0 ).  e^" ^ 1  17  - T ] 3 -  ( f i x e d * ; varying 1  ),  (fixedvarying^).  The computer solutions for ( l ) , (2), (3) and (4) were plotted on semilogarithmic and  paper withX^/j-^as the logarithmic  ordinate  <J as the linear abscissa. With case (l) the errors were smallest; at U. - 2 there i s no  40  error and as <C increases from 2 to 10 the computer curve ordinates exceed the calculated ordinates i n a l i n e a r l y increasing manner. On the logarithmic scale the error at ol — 10 appears to be about 3%.  Thus for small values of o(  up to about 4> the error w i l l be insignificant i n the simplest Townsend equation (0.2). Since e( values are frequently i n this range the computer may be used with confidence here. In case (3) the errors are greater than i n case ( l ) but are not excessive. They also increase l i n e a r l y with an increase i n ^ , the computer curves again having ordinates uniformly greater than the calculated curves. the largest value of *J = 5 i s about 5%.  The error at  Error i s nearly zero  for 1 - 1. Since attachment coefficients are usually very small and rarely exceed values of 2, except i n highly attaching gases such as CO^j the Townsend equation ( l . l ) with ^ and ^ may be likewise simulated with assurance. Errors with the coefficients e( and Y only are the largest of the four cases.  The coefficient Y triggers the breakdown  mechanism and, i n regions of small denominators gives r i s e to errors which are impossible to compensate. However, the values of Y  u s e d  f o r  from Y =  '  t h ef o r m  (°«3) of the Townsend equation ranged  .02 to Y •=• .10. Since secondary ionization i s  usually very small compared to primary ionization and attachment (normal values of )f being from about 10" to 10 ^ ) , the error 8  w i l l be much smaller for these very small Y values occurring experimentally.  In this case the computer'ordinates are  4.1  uniformly less than the calculated ordinates, the error again increasing l i n e a r l y with  .  F i n a l l y , i n the general Townsend equation (0,4.) with fixed o( and ^ and varying l v a l u e s , the error i s smaller than i n case (2), being intermediate i n magnitude between cases (2) and ( 3 ) . The computer' ordinates are again uniformly less than the calculated ordinates, the discrepancy being approximately constant. Again, as i n case (2), since the experimental )f values are so much smaller than these values used here, the error w i l l not be significant i n actual practice. These four checks indicate that although error exists i n "the c i r c u i t , most of i t attributable to the divider;  i t will  not materially, affect the generation, of experimental..and' predicted results for reasonable experimental values of the gas coefficients,  ^hus, the computer may be used with confidence  in the actual applications;. The curves for the f i r s t two cases are given i n Figures 8 and 9*  42  Figure 8.  G'omptiter and calculated ionization characteristics;  from equation (0.2),, X = X e ^ ^ 0  ELECTRODE  ( d — 2,4>6,8,10 ionizations/cm):..  SEPARATION  IN  CM  43  Figure 9.. Computer and calculated ionization characteristicsfrom equation (l.lX, I = I ( Y-  -  0  0, << = & ionizations/cm; *| = 1,2,3,4>5' attachments/cm).. !  ELECTRODE  SEPARATION  :  IK]  CM  44  1.3.3.  Extension of ionization curves to largo values of ^/p .  Two programmes for the application of the computer were chosen from the three indicated.  The f i r s t programme was an  extension of gas ionization curves to large  values using  values of o( derived from Townsend's empirical equation (1.5), *^/p / \ d ~ ^ ' / ^ =  5 and, at the same time, setting f{ =.  >  O .  In this case the curves X=X!oC^^of (0.2) are generated. *^ = o for the inert gases since by definition the inert gases do not chemically react with others, Y  i s also very small,  even for higher H^p values i n the noble gases, so i t i s expected that the approximation vy —V= gases.  § w i l l he quite good for these  In other common gases such as dry a i r and R^, *y i s  very small and Y i s small as w e l l , so the approximation 13 • may also be j u s t i f i e d there. A recent study by Sen and Ghosh already alluded to shows that, i n electrodeless discharges i n dry a i r  Y  i s less than .01 at as. high a value of  1000, and Y '- also small for high H/p values i n :  s  Writing Townsend's relation (1.5) for ^yp with Y - 1 -  r m  %  p as and Ng.  as c( = A C  the usual units i n which the equation  variables are given) one finds 1°^^  =  I  A ~ ^/f£  where  A and B are the empirically determined constants. The values of these constants are chosen so that the relation (1.5) f i t s most closely the experimental >(^p versus ^ jp  curve i n the  v a l i d i t y range of this Townsend relation and they were taken 18 from a table given by Brown Von E n g e l . 11  which had been compiled by  Part of this table i s given below as TABLE I  IS  where, for each gas, the  range of v a l i d i t y of equation (l,5)  and the values of A and B i n the appropriate units are given,  TABLE I A > B  Values of  ail(  for Townsend relation (1,5), (^/p i n various gases. ^/p Gas  E/p  3 v a l i d i t y range o-f ^  (°( i n ionizations/cm,  i n volts/cm/mm Hg). Range of v a l i d i t y of E  /p  . A (ionizations/cm/mm Hg)  ] (volts/cm/mm  A  100  -  600  H  180  Kr  100  -  1000  17  240  Ne  100  - 400  4  100  Xe  200  - 800  26  350  2  150  -  600  5  130  Air  100  ~ 800  15  365  2  100  -  12  342  H  N  600  Setting p - 1 mm Hg one has, as above, lo^  -( •=  4 - ^ (\$  ('^/£  j  so one obtains a straight l i n e plot on- a cemilogarithmic graph of  ^  against  a(.  intercept i s A and the ^ ^ / J C intercept i s ^  lo  £  for given constant values of A and B,  The  (o^ A . These  straight l i n e semilogarithmic plots are given i n Figure 10 for the gases i n TABLE I. From the straight l i n e plots one may, f o r given values of ( /p e  « ^ E  - £ when p -1 mm Hg), f i n d ^ j and so for each gas  ^/p  46  at particular values of '•yp generate ionization curves. TABLE I I cites calculated values of o( for selected values of E/"p above gases.  i n the  Ionization curves may be generated i n each gas  within the particular v a l i d i t y range of  using TABLE I I .  On a  semilogarithmic plot these curves w i l l be straight lines since they are the solutions of equation.(0.2), Figures 11 to 17 exhibit these ionization curves for the gases A, Xe, Rr, Ne, H , N and dry a i r . 2  2  For a l l of these curves  J» = 1 mm Hg and they are a l l plotted with^y'jrj^ as ordinate and  d  a s  abscissa..  L i t t l e experimental data i s available i n these large ranges of ^/f> i "  o r  the gases used here so comparison curves  are d i f f i c u l t to f i n d , and the computer curves must serve as f i r s t order predictions to future experimental curves. They are f i r s t order predictions only since the refining parameters  \  and Y have been neglected. I t would be expected that these curves w i l l give good approximations to experiment taken at thes-e high  values. The computer i s a useful device  enabling one to generate many of these curves with maximum ease and f a c i l i t y .  43  TABLE I I  Calculated values of oi at j> = 1 mm Hg and various (^/p  7r  =  100  i  n  p i n certain gases.  volts/cm/mm Hgj ^  i n ionizations/cm)  150  200  250  300  350  400  450  500  Gas A  2.33  4.23  5.70  6.83  7.70  8.37  8.92  9.38  9.77  Kr  1.55  3.44  5.12  6.50  7.61  8.55  9.32  9.95  10.50  Ne  I.48  2.06  2.69  2.87  3.00  3.11  4.51  6.40  8.05  9.51  10.82  11.85  12.90  Xe H N  2  2  Air =  '  2.42  .40  2.10  2.61  2.97  3.24  3.43  3.60  3.73  3.84  .40  1.23  2.18  3.05  3.83  4.52  5.10  5.60  6.05  .40  1.32  2.42  3.4-9  4. . 4 4  5.30  6.04  6.68  7.23  600  650  700  750  800  850  900  950  12.83  12.99  13.20  550  .  Gas A  10.10  10.42  Kr  IO.96  11.42  11.70  12.03  12.35  12.56  Xe  13.68  14.47  15.07'  15.70  16.20  16.68  3.93  4..01  2  6.43  6.78  Air  7.75  8.20.  8.90  9.23  9.52  H  2  N  8.55  49  Figure 11.- Computer ionization curves for A., from equation (0.2),. I = I o C ^ , using results of TABLE I I . . ( ^/p i n volts/cm/mm Hg>, pressure p = 1 mm Hg).  ELECTROPE  SEPARATION  IN  CM  50  Figure 12.- Computer ionization curves for Xe from equation (0.2 X =I e '> using results of TABLE II.. e<e  0  C^/p  i n volts/cm/mm Hg, pressure j? = 1 mm Hg).  ELECTRODE  SEPARA\\Ohi  IM C l * |  51  Figure 13»  Computer ionization curves for Kr from equation (0.2).,. I =I e 0  (^/f  r f d  , using results of TABLE. II..  i n volts/cm/mm Eg> pressure  ELECTRODE  SEPARATION  p - 1 mm Hg).'.  IN  CM  52  Figure 14. Computer ionization curves: for Me from equation (0,2),. I = r«e » rfJ  C^/p  u s i n  g results, of TABLE II..  i n volts/cm/mm Eg', pressure p = 1 mm HgO».  E L E C T R O D E  ;  SEPARATION)  INJ  CM  53  Figure 15.- Computer ionization curves f o r  ELECTRODE  from equation (0,2),,,  S E PAR4TIOK)  IM C t i  54  Eigure 16.. Computer ionization curves f o r  from equation (0.2),  I = I e ^ . using results of TABLE II.. 0  c (  i n volts/cm/mm Hg,, pressure  ELE.CTROPE  p> = 1 mm Hg)".  SEPARATION  55  Figure 17.- Computer ionization curves for dry a i r from equation (0.2), X =X e 0  y using results of TABLE I I .  ELEC T R O P E  SEPARATION  I (v) C M  56  1.3.4.  Computer ionization curves compared to experimental ionization curves. o(  One may obtain values- of the coefficients the experimental gas coefficient curves H/p  >1/p  , ^/^  and )f v s  from  ^/p  ^>  feed these coefficient values into the computer and thereby generate resulting ionization curves.  The ionization curves generated on the  computer may then be compared with the experimental ionization curves. By taking intermediate E/"p values from the gas coefficient curves one may generate computer ionization curves for E/"p  values different from  those already employed experimentally. Extrapolation of the gas coefficient curves to higher E/p  values and generation of the  ionization curves for these larger Ey'p  values may also be done,  although i t was not attempted here. Results for the gases Hg, CO^ and dry a i r were used. Harrison and Geballe"^ give gas coefficient curves of ^/p  and 1 ^ vs  ^/p  for moderate E/'-p ranges from about 2 5 to 6 0 volts/cm/mm Hg. <^ and *| values were obtained from these curves and from them the ionization curves generated for a i r . These curves are shown i n Figure 18 where one notices the reasonable agreement between experimental and computer curves. 20  The results i n COg of Bhalla and Craggs  with numerical  values from 3 0 to 4 6 were next generated on the computer. These ionization curves appear i n Figure 19.  The agreement i s not so close  as i n the case of dry a i r . The computer ordinates uniformly exceed the experimental ordinates, the error not increasing with increased values of E ^  .  57  With  the data of Hopwood, Peacock and Wilkes  for low values of narrow range.  21  was used  from 19 to 22.5 volts/cm/mm Hg, a very  Figure 20 shows that the agreement here, as i n the case  of dry a i r , i s quite good} the computer curves showing, however, a marked tendency to upcurving with increasing A  values.  The  computer ordinates again exceed experimental ordinates, the discrepancy increasing s l i g h t l y with increasing Since, i n the case of CO2,  ^ •  the coefficient ^  i s appreciable,  i t i s seen why the variance between computer and experiment i s largest here.  In both H  2  and dry a i r ^  i s very small. In H^,  Y  i s taken into account but i t i s nevertheless very small.. The error thus does not appear so large i n H  2  and dry a i r .  Many other gases might have been treated but these representative ones give reasonably close agreement between computer and  experiment.  58  Figure 18., Comparison of computed and experimental ionization curves' i n dry a i r . Experimental results- due to Harrison  ELECTRODE  SEPAR ATlO/v  IM  CM  -  59 Figure 19. Comparison of computed and experimental ionization 20 curves i n COg. C-^/p  i n  Experimental results due to Bhalla and Craggs .-  volts/cm/mm Hg, pressure p = 100 mm Hg,• T = 1 0 " 0  ELECTROPE  SEPARATION)  (M  Ctt  11  A-).  60  Figure 20. Comparison of computed and experimental ionization curves i n H^.  ( ^ Jy  21 Experimental results due to Hopwood; et a l ,  i n volts/cm/mm Hg, pressure? p = 700 mm Hgy I = l O * A). -  0  ELECTRODE  5EF4 RAT \DS)  I k)  CM  11  61  1.4.  Townsend's Primary Ionization Coefficient i n Gas Mixture; One important possible application of this analogue programmme  i s to the prediction of ionization curves i n mixtures of gases. The procedure required here would be to derive theoretical values of the ionization coefficients o< ,  for a gas mixture; and from  these values to generate ionization curves for the mixture considered. This i s a predominantly theoretical problem;  once the  gas mixture coefficients are derived i t i s a very much simpler matter to introduce the coefficients into the computer and generate the resulting ionization curves. For single gases theory can to a certcdn extent predict the values of the primary and secondary ionization coefficients as was 15 seen i n the work of Emeleus, Lunt and Meek ,. Von Engel and Steenback' 3 and Townsend on the primary coefficient o( , and i n the work of Sen and Ghosh"^ with the secondary coefficient For gas mixtures the theory i s no longer simple and hasonly been attempted for certain mixtures and then only for the Townsend primary ionization coefficient *C * The gas treated i n this way was a Ne-A mixture i n which minute percentages of A gas were added to the pure gas Ne and the resultant coefficient <^ of the Ne-A mixture was compared to the value of <?( for the original pure Ne. The process supposed to take place was one of ionization of the argon atoms by metastable neon atoms whose excitation potential i s higher than the ionization potential of A. Theoretical and experimental work hasbeen done on this problem by Kruithof and a 22 23 Penning , Glotov and Moralev . In particular'the work of  62  Moralev  describes c a l c u l a t i o n s to evaluate the c o e f f i c i e n t t>{ i n  a pure gas and i n gas mixtures.  Glotov made use of these  calmilations  o f Moralev and applied -them to the ease of the Ne-A mixture, f i n d i n g f a i r agreement with h i s own experimental work.. Glotov s work w i l l 1  bo o u t l i n e d here i n d e t a i l since i t might p o s s i b l y be generalized t o other gas mixtures from which comparisons could be made w i t h experiment.  Although. Glotov d i d an a n a l y s i s of the i o n i z a t i o n  - c o e f f i c i e n t o( f o r the case of pure Ne as w e l l as f o r the Ne-A mixture only hid work on the Ne-A mixture w i l l be considered sinne other te-chniques f o r f i n d i n g  i n . a pure gas have been p r e v i o u s l y  mentioned, and the determination f o r t h i s mixture i n v o l v e s , i n i t s d e r i v a t i o n , the vaiie of ^  f o r the.pure gas*  Certain approximations have been made by Glotov..  He assumes  plane p a r a l l e l electrodes and homogeneous f i e l d s j he only takes i n t o 3  account the e x c i t a t i o n of the ?^ atomic l e v e l i n . A i n c a l c u l a t i n g i t s e x c i t a t i o n p r o b a b i l i t y ; and he f u r t h e r assumes that metastable Ne atoms are only destroyed by c o l l i s i o n s with. A_ atoms ...with a p r o b a b i l i t y of u n i t y  Despite these s i m p l i f i c a t i o n s  a c l e a r dependence of  Glotov-observes  on the admixture o f A i n the gas., and  demonstrates how the r e s u l t s of h i s c a l c u l a t i o n s i n d i c a t e the elementary processes occurring i n the gas discharge. Accordingly, Glotov sets the i o n i z a t i o n c o e f f i c i e n t *>( o f the Ne-A mixture as <'= <^+ v  + * j ( 1 . 2 5 ) where °(, and V  A  are,  r e s p e c t i v e l y , the i o n i z a t i o n c o e f f i c i e n t s of pure Ne and pure A, and ^  i s a conventional i o n i z a t i o n c o e f f i c i e n t determined by the  c o l l i s i o n s of metastable neon atoms w i t h argon atoms. Adding t o  63  the original pure Ne gas a quantity of A leads to an increase i n the number of i n e l a s t i c collisions of the electrons i n the discharge with the gas atoms, and so alters the most probable velocity of the electron.  The dependence of the most probable electron velocity  (X-fc- with the A admixture i s found from a formula due to Moralevs r k L  where  and ^  0  are the respective mean free paths of an electron  in Ne and A at a pressure of 1 mm Hg; j) and p (  2  are the respective  pressures of the original gas (pure Ne) and of the admixture (A) in mm of Hg; \f  and U/„'  corresponding to the n  are, respectively, the energy  c r i t i c a l potential of the A gas i n volts  and the mean excitation probability of this l e v e l ; v* and  are  these same quantities for the main component Ne; ^/p i s the ratio of e l e c t r i c f i e l d to pressure of the primary gas Ne, and H and VH. are the masses of the atom and electron respectively. With a value of ^ p  of 10 volts/«m/mm Hg and a value of p of  2 mm Hg; knowing the mean free paths \ hi ,  and y„  0  , A  0  and the quantities M  , the dependence of Ht on the concentration of A  admixture expressed by p c o u l d be found i f W„ and W„ , the  64  respective excitation probabilities of the f i r s t metastable  3  levels  of argon and neon, were known. For t h i s , another formula due to Moralev was used, giving the probability of exciting the f i r s t  3  P^  metastable l e v e l of ari atom i n the form: 3 fT ^  f\kr \  €  3*-  I—(—*  —'7  (1.27).  W t  In formula (1.27) <*. and b are constants, Up i s the excitation -rpotential of the metastable 3 l e v e l of the atom, £_ i s the error integral or error function, U  t  i s the most probable velocity of  the electron and o( i s Townsend's primary ionization coefficient.  d. and t are found by curve f i t t i n g from experimental data with the empirical excitation function formula: H  VU J*<l€  (1.28).  r  Knowing the values of <a. , t , UV and «^ (the l a t t e r from experiment) P r can be calculated as a function of (X-t i  With  £/p  = 10 volts/cm/mm Hg Glotov found the excitation probability 3 -3 for the f i r s t metastable P^ l e v e l i n neon as 1.25 X 10 ' and i n -2 argon as 1.3 X l0> . With these respective values f o r the excitation probabilities W 'and , the variation of Mt vlfch J  K  argon concentration can be found and wasplotted by Glotov. The theoretical relation given by Moralev for p( i s the following:  ,. 19  =  xip,  ' P<-'  (1-29)  65  where U t  is the energy corresponding  to the most probable electron  velocity in volts, P ' is the probability of ionization of an atom L  in a oollision, E/p l and X  9  the ratio of f i e l d strength to pressure,  s  is the electron mean free path for a pressure of 1 mm Hg.  The probability of ionization is calculated from the formula  (1.30) where A = 1.5 + 4, is a constant obtained experimentally, « J is the ionization potential of the .gas and c" is another constant obtained by curve f i t t i n g another empirical formula for the ionization probability,  .  * I  -  to experimental data,  c  n  u  /  ({.3i)  e t  Formulae (1.30) and (l,3l) are also due to  Moralev. With these formulae one may calculate the dependence of ol \ for neon on the argon admixture by using Moralev's result, (1.29) °^/p  ^ ,  =  cm/mm Hg,  ^  o  •--  •  ?i .  One has ^/o =• 10 volts/  is known for Ne as .08.cm, the dependence of ?i on  (equation 1.30) and hence on the admixture of A is known since i t s e l f can be determined as a function of the admixture (equation 1.26), using a formula for the excitation probability  3 of the f i r s t metastable 1,27),  level in its determination (equation  Glotov finds a decrease in * , as the A admixture is increased,  beginning with an admixture of ,01/6. The coefficient  <( f o r argon i s found in exactly the same  66  way as above, i n s e r t i n g the appropriate constants and parameters pert a i n i n g to argon,  Glotov f i n d s i n t h i s way t h a t the d i r e c t  i o n i z a t i o n of argon atoms by electrons i s n e g l i g i b l e . The l a s t c o e f f i c i e n t o(^ i n the c o n t r i b u t i o n t o the t o t a l ^ i s found from the r e s u l t 4  3  where ^  = n r, s  L  (1.32)  s  b  i s the number o f metastable atoms i n a l a y e r o f gas. 1 cm '  t h i c k produced by one e l e c t r o n t r a v e l l i n g across t h i s l a y e r , £ , i s the p r o b a b i l i t y o f c o l l i s i o n o f a metastable, neon atom w i t h an atom of the a d m i x t u r e , ^ is. the p r o b a b i l i t y of i o n i z a t i o n i n 3uch a c o l l i s i o n and  the p r o b a b i l i t y f o r the atom to t r a v e l unaffected  by c o l l i s i o n s w i t h .atoms o f the primary gas a distance d by the electrode separation. ^ Moralev,  determined  i s found, f r o n another formula due t o  •/  3  where a l l variables have been p r e v i o u s l y defined. The p r o b a b i l i t y o f d e s t r u c t i o n o f metastable states by c o l l i s i o n s w i t h electrons and cumulative ioniza--'<">n has been neglected as being very s m a l l , +  C, i s found from  .Where  ..  ;  _ 1/,  L< "  j , , ..; 7  ;  r- I ( L 3 5 )  i s the mean f r e e path of a metastable atom r e l a t i v e t o atoms o f the admixture. L  0  characterizes the mean f r e e path o f a metastable Ne  atom r e l a t i v e to atoms o f the ..admixture and can be found from k i n e t i c theory-.  £^ i s assumed t o be. equal t o unity}, that i s , i t i s assumed  •that a metastable neon atom i s destroyed i n every c o l l i s i o n w i t h an  67  argon atom and hence ionizes the argon atom. 5^ i s found from  where ^  i s a quantity characterizing the probability of the  metastable state being destroyed i n c o l l i s i o n s with atoms of the mean gas and L i s the mean free path of the metastable atoms relative to the primary gas atoms at a given pressure.. From these expressions «^ i s found i n the following form:  In this expression £^  1 hasbeen assumed since y3 «. 1,2 X 10 f o r  pure Ne. Knowing now the values of ^, j <^ and x  calculated i n  this way for the mixture, the value of <?{ as a function of argon admixture at an ^  of 10 volts/cm/mm Hg i s determined,  ^lotov  finds a good qualitative agreement with his experimental results f o r the variation of o{ with A admixture; he finds that minute quantities of A from ~L0~^% to .3% influence very highly the value of °t f o r the .mixture, --increasing; i t ..several times, the .maximum ofoC, corresponding -.to ,.a concentration of 0.3% to,0.4$ A.., The variation-.of,,the- function , \..,viith. experimental conditions can. be., determined, f^om the ...relevant formula,  (equation .1.33)  .-- • .  rln.-this. way.,Glotov finds that  . for; ,.the, Ne-A. mixture-.ris.es to ;  a maximum at a 0.3% admixture of A and then decreases.  He interprets  the decrease as due to electron energy loss with consequent decrease i n the most probable electron velocity C( and hence i n ei. . The t  68  i n i t i a l trend of <*f f o r the mixture with a small admixture of A i s confirmed by experimental data on Ne-A mixtures. Glotov's work has been mentioned hero i n d e t a i l since i t affords a general method of calculating the variation of <^ with a mixture component given mixture percentage and various values of ^/p the theoretical dependence of '(Ip on E/p experimental results* component gas mixtures.  could be determined and compared with  The formulae of Moralev apply to two They were not employed i n the Townsend  computer work for any gas mixture since the computer programming of the ionization curve i s a very small part once the coefficient * has been obtained i n t h i s way,  I t i s f e l t that Glotov's work i s  eminently useful, and calculations might well be performed i n the same way for other gas mixtures of the requisite type.  69  CHAPTER 2  THE LINEAR Z-PINCH DISCHARGE .  2.1  L i n e a r Z-pinch Discharge Theory; The l i n e a r z-pinch i s a type o f discharge i n which gas i s  enclosed i n a s t r a i g h t c y l i n d r i c a l conducting tube; a l o n g i t u d i n a l e l e c t r i c f i e l d i s suddenly a p p l i e d to the c y l i n d e r causing an i n c r e a s i n g l y l a r g e l o n g i t u d i n a l current pulse t o flow, mainly i n a t h i n s h e l l a t the surface o f the now i o n i z e d gas or plasma.  When t h i s current has become l a r g e enough f o r  the magnetic f i e l d pressure on the plasma to exceed the gaspressure, the plasma column i s c o n s t r i c t e d or pinched and contracts r a d i a l l y , hence the name "pinch" discharge.  This  e f f e c t has been, and i s now, under i n t e n s i v e i n v e s t i g a t i o n because of i t s more general connection w i t h the i n t r i g u i n g problem o f plasma physics,- c o n t r o l l e d thermonuclear power. Although other plasma-confining configurations- such as m i r r o r - l i k e devices and the so c a l l e d s t e l l a r a t o r geometry devices are thought t o be more promising f o r c o n t r o l l e d thermonuclear power, as i n d i c a t e d i n a recent plasma physics t e x t by Rose and C l a r k ^ ; the z-pinch discharge i s u s e f u l f o r experimental and t h e o r e t i c a l studies o f b a s i c plasma physics.  The major disadvantage o f  t h i s p a r t i c u l a r discharge type f o r p r a c t i c a l use i s Its s u s c e p t i b i l i t y t o various i n s t a b i l i t i e s of the sausage, kink or f l u t e types, S"0 named from t h e i r p h y s i c a l appearance.  70  The schematic diagram o f the -usual experimental arrangement f o r the l i n e a r z-pinch i s shown below i n Figure 21, Figure 21.  Schematic diagram o f experimental apparatus f o r l i n e a r z-pinch discharge.  iwsv/laTotr  'r~ piwtcli "tote, r e t u r n cekiiuctop Je^ter-iow (or other  I'HCJ \j  gasj  etanee  I t may be seen from the diagram i n Figure 21 t h a t the basic l i n e a r z-pinch mechanism i s a l a r g e c a p a c i t o r bank connected i n s e r i e s t o a spark gap v i a an unavoidable, f a i r l y large c i r c u i t inductance.  The main inductance i s t h a t o f the  discharge tube w i t h a value o f about 0,2/A. H; the gap inductance i s about .05/<H and the inductance o f the leads about ,01>< H. Current increases i n a t h i n l a y e r a t the plasma surface a f t e r the onset o f the discharge, and the plasma tube begins a r a p i d r a d i a l c o n t r a c t i o n , w i t h a shock wave developed which advances ahead of the plasma sheath.  This shock wave i s  r e f l e c t e d a t the axis o f the tube a t a time c a l l e d the "pinch" time o r "bounce" time and then proceeds outward, causing a r e v e r s a l o f the current d i r e c t i o n i n the sheath.  The con-  t r a c t i o n and expansion o f the sheath continues u n t i l i n s t a b i l i t i e s destroy the r e g u l a r o s c i l l a t o r y motion.  71  The original theoretical treatment of this r a d i a l motion of the linear z-pinch discharge was carried through by 25 Rosenbluth, Rosenbluth and Garwin  who made the simplifying  assumption of i n f i n i t e plasma conductivity.  They used a snow-  plow model i n which the mass of gas i s swept up by the advancing shock front and compressed into an i n f i n i t e s i m a l l y thin layer just behind the front so that the current sheath and the shock wave are effectively the same interface. Anderson et a l remark that this snowplow model i s probably quite r e a l i s t i c since the fraction of neutral gas atoms inside the contracting current sheath i s quite large; these atoms are ionized and pinched i n the current layer. The speed of radial contraction of the sheath characteri s t i c a l l y corresponds to a shock of an intensity from Mach 10 to Mach 60 so that hydrodynamical treatment of the plasma i s confined to the "strong shock" l i m i t .  Anderson et a l remark that  the l i m i t i n g compression ratio behind a plane p a r a l l e l strong shock i s given by Yf > Y- 1  where  i s the ratio of the  gas specific heats. This ratio i s 5/3 for a simple gas with three degrees of freedom, resulting i n a value for the compression ratio of 4 , a poor approximation to the snowplow or strong shock l i m i t .  With compression ratios of 8 to 10  obtainable i n monatomic argon or deuterium, the snowplow l i m i t i s achieved, and the theory here i s v a l i d , A simple derivation of the snowplow equation describing the time evolution of plasma radius up to the pinch time w i l l now be  given  72  Let R  Q  be the radius of the metal tube, ^  the i n i t i a l  9  gas density,X (+) the current i n the contracting sheath at time ,  "t  -9  and ft the associated f i e l d of magnetic induction.  concept of magnetic pressure must f i r s t be defined.  The  Newton'a  second law for a hydromagnetic system of charged particles' 2A (see Rose and Clark , page 125) i s : "H* *t* « ~ ? - f ^ p (2tl) Y  where  V i s the average particle v e l o c i t y , j  density, £  the e l e c t r i c f i e l d ,  the mass density, p density and  8  the, current  the magnetic induction,  the plasma pressure, ^  the charge  the gravitational potential energy.  The l e f t  hand side of equation (2.1) i s the kinetic reaction or reversed effective force (mass X acceleration) of d'Alembert, while on the right hand side are the forces on the- plasma: electromagnetic force ^ t +-J X 8 force  -Vj?  the Lorentz  , the pressure, gradient  and the gravitational force  y> V^« M  Many of these  terms may be neglected for practical applications: within a plasma the net charge density ^  i s zero; gravitational forces  are always negligible except i n very specialized applications i n astronomy, and i f only the steady state i s considered one: finds ^ f  ~ J *  ^  (2.2).  In the steady state' the Maxwellian displacement current V vanishes so that Ampere's law becomes  Vx8  j  ;  ( 2 , 3 )  '  73  Introducing the value (2.3) hydro-magnetic equation  Vf  -  j^e  =^  for  (2.2)  0  j  i n t o the s i m p l i f i e d  one f i n d s  (VXB3X8  = -jjr  i f the l i n e s of magnetic induction 8 '"Vjb  so that  e  y  B  "  (  ——  )  are s t r a i g h t and p a r a l l e l  = 0 (as i n the l i n e a r zrpinch disdharge),  thus obtains V (v +  2 < 4  Che  \= 0 so that p 4-  For t h i s reason " / i ^ i s  =  const.  (2.5).  c a l l e d the magnetic pressure.  It is  the pressure exerted on a plasma by the magnetic i n d u c t i o n ^ie"l<J  E  < ¥ith t h i s r e s u l t we may apply' Newton's"'Second law.; to., the ;  imploding plasma;  This law states in-this,,case that the force,  on a s e c t i o n of u n i t height and circumference i-Tl^j i s equal to  the time r a t e of change of momentum of the m a t e r i a l swept up  by the imploding plasma.  One obtains the  equation  =  the negative s i g n a r i s i n g since a c c e l e r a t i o n increases.-  R, 5 decreases as the. m a t e r i a l  In equation  (2-6)  R.  0  i s the  initial  plasma tube radius equal to the radius.of the c y l i n d r i c a l discharge tube, K$-(tJ i s the radius of the plasma tube at time .f  and ^>  0  i s the i n i t i a l d e n s i t y of the c o l d gasj  measured a t the surface of the plasma tube. system  ft  i s r e l a t e d to  X Q  B  is  For t h i s c o a x i a l ,  by Ampere's- law so t h a t  _  X  (2.7).  74  Thus one derives the equations  i  v  -  U  ^  J  ^ F T , (2.8),  and d e f i n i n g the ditnensionless v a r i a b l e ^ - ^ / R ^ 0  where  £ ^ £ 1. equation (2.8) becomes  » t i  l  - ^  ,  J  t  l -  ( 2  . ) 9  in MKS units. Equation (2.9) i a the snowplow equation* relation^  «=4TT  One employs t h e  x 10 ^ J/amp^m t o change t o the u n i t s used by  Curson and Churchill"*" which are c h a r a c t e r i s t i c o f the magnitudes of tke experimental parameters. eoulombe^. sec, ^  0  i n gm/cm^,  They use the unite..-JJ i n  (R* i n  cm and  t iny«, sec,  o b t a i n i n g the form o f the snowplow equation t o be employed i n the analogue-treatment given i n Section 2,2 f o l l o w i n g :  Rose and C l a r k ^ remark t h a t t h i s snowplow theory *»ould be 2  made more r e a l i s t i c by accounting f o r f i n i t e plasma c o n d u c t i v i t y , r e v e r b e r a t i o n of-shock waves from the-center o f the.discharge tube-..and the f i n i t e Larmor o r b i t s o f the i o n i z e d atoms.  2.2.  Computer S i m u l a t i o n of the Snowplow Equation. A d i s c u s s i o n of analogue computer elements f o r summing,  m u l t i p l y i n g and. i n t e g r a t i n g voltages as w e l l as the method o f  75  operation o f c o e f f i c i e n t set potentiometers has been given i n . CHAPTER I on the s i m u l a t i o n o f the Townsend equation and w i l l be omitted here.  This S e c t i o n presents the analogue  simulation o f the l i n e a r z-pinch discharge i n i t s simplest form c a l l e d the snowplow equation. The snowplow equation deduced i n the l a s t s e c t i o n , i s the f o l l o w i n g p a r t i a l d i f f e r e n t i a l equation*  itt^hf) Here ^ =  -  Z^JjTy  .  (0tl  >-  i s a dimensionless v a r i a b l e between 0 and 1  where R i s the instantaneous plasma radius a t time '"fc and R f  0  the i n i t i a l plasma radius a t time t - 0 , ^> (measured i n gm/cm^) 0  i s the i n i t i a l gas d e n s i t y and  i s the l o n g i t u d i n a l current  pulse developed in the i n i t i a l l y cold gas*  coulombs/^ sec,) pulse waveform X (t)  (Measured i n  The time "t i s measured in^a sec.  The current  may be closely approximated by a p r e c i s e l y  s i n u s o i d a l waveform w i t h angular frequency u> (measured i n radian?///* sec) and an amplitude X 1 (t)  0  o f the order o f 200 kiloampaxjes.  i s thus w r i t t e n as - I  0  ^  u t  (2.10).  The snowplow Equation (0,1) i s u s u a l l y w r i t t e n i n ordinary d i f f e r e n t i a l form and was so w r i t t e n here f o r purposes o f analogue simulation*  The e r r o r i n c u r r e d w i l l not be l a r g e s i n c e discharge. ,  times i n v o l v e d are very s m a l l being o f the order o f ^ < s e c . Accordingly then, we w r i t e ( 0 , l ) a s :  76  The form (2.11) o f Equation (0,1) i s time i n t e g r a t e d w i t h t h e result  J f"  — (£  j an expression  which i s beginning t o appear s u i t a b l e f o r s i m u l a t i o n . We now make the transformation Y = the i n i t i a l c o n d i t i o n , on Y ^  Y  and  JY  =  (2.12) so that  ( + =0)  «  50 ^ (t=0) «• r o j  both being pure numbers.  Thus we have:  or  becomes Y  50  ^foo  JX  2 f =  °°  f  I C t p l f  T^itTil  (2.13).  The snowplow equation (.0.1.) has now been reduced t o a form s u i t a b l e -  f o r analogue s i m u l a t i o n .  One notes the occurrence o f Y  i n the i n t e . ^ r o - d i f f e r e n t i a l equation (2,13).  and 1  S p e c i a l i z e d means  w i l l have t o be u t i l i z e d t o produce-these squares i n the f i n a l circuit. Consider f i r s t the f u n c t i o n 1 multiple of  , and define a constant  given by:  lorf^.RJ  i . . ^ , R,+  By a s i n g l e time d i f f e r e n t i a t i o n o f X  (2.U).  one obtains  — Sin z u>\  •jf-=  and a f u r t h e r time d i f f e r e n t i a t i o n r e s u l t s i n  = -f-Lo (  -  +•  employing trigonometric r e l a t i o n s .  =  -460  X +. —  77  Thus the ordinary d i f f e r e n t i a l equation  i s obtained for the function X • As i n the simulation of the Townsend equation of CHAPTER l  one chooses here the sane manner  r  of analbjrTue to physical correspondence so that the quantities €, , xj> and €  0  are the exact analogues of the respective physical  variables X > **> and  1^—  —  . hen by equating the net current T  to zero at the input node to the f i r s t integrating amplifier (amplifier l ) as required by Kirchoff's current law for e l e c t r i c a l networks, one obtains the following analogue equations R  3  "  By choosing R , = R ,  R,  (\  y  =  K  *  = R, and  becomes  ,  c  *  at-  C , =  C\  =  C  (2.16)  J*-  «CRC/"~L  e o ^ - 4 f e ,  (2.17)  and a time scale may be fixed by giving to  any convenient value,  as noted before i n the Townsend equation simulation.  The i n i t i a l  condtions on X  and-^pp  a n d  X (0) = 0 and A*  "J^*  c o r r e s  P  o n  ^ S ^° those on e, n  (0) ~ 0 (2.18) sinceX varies as rf*i  i s seen that for any value of Rc  A  are  wt . It  which fixes the time scale, and  any amplitude scale defined by the voltage chosen for (f the above 0  analogue expiation (2 17) corresponds precisely to the physical f  equation  / i ^ - - ^  H-S'w ^ —: 1  X  X  +  and so equation (2.15) has been correctly simulated. The simulating  78 c i r c u i t for (2.15) i s shown i n Figure 22. The reduced snowplow equation  has been simulated i n a precisely similar way to %hat of (2.15) The additional analogue to physical correspondence of €± to represent Y i s necessary here.  One also uses the i n i t i a l condition  Y (0) = 50 (0) = 50 i n absolute value.  By examining Figure 23 on  which the f i n a l complete analogue c i r c u i t of the snowplow equation i s shown one obtains, by again equating to zero the net current at the input to amplifier 5, the analogue equation:  Setting R,-R. =I\ =R =Rs-=Rand C, = d i  3  +  =  - C«r * C  i n (2.20) one finds the f i n a l analogue equation  and for any time scale given by K C and amplitude scale defined by 3ACJ  <e. the analogue equation (2.21) corresponds exactly to the 0  physical reduced snowplow equation  Jv _ *• 10  fx Jt  The two sets of i n i t i a l conditions "Y (0)= 50 and  =  = 0 are  set on amplifiers 4 and 5 as shown. The snowplow equation i s thus correctly simulated, the simulating c i r c u i t being given i n Figure 23.  79  Figure 22.  Analogue simulation of Equation.(2.15), Analogue variables ^ , €• a n d g , represent (  (>o ,  and X .  80  Figure 23. Analogue simulation of snovplow equation (0.1)., 12-fie Analogue parameters <j> ,e ,€ , 6 represent respectively U) , 0  t  f  Z  OK-*GRAPHIC ft H A A r  ±  ±  ELECTRONIC DIVIDE*  SQUARES  •SoV  "->{ELEC:TROKIK  DlV IDER.  81  Time and amplitude scale factors were chosen as i n the Townsend analysis to reduce amplifier d r i f t or grid current error due to long computing times., and to ensure that the amplifiers would not be greatly overloaded during the computer runs.. The time scale was chosen with R, C = 1, resulting i n 1 second computer time — l _ ^ c sec i n the physical equation (O.l) (snowplow equation) =• 1 second real time =- l/5 inch on the recording paper (which was run at a maximum speed of 12 inches/min). This produced maximum computing times of about 7 seconds so that g r i d current error was minimized. The resolution of the solution on the recorder was quite satisfactory with this time scale.  The amplitude scale was a r b i t r a r i l y but  conveniently chosen so that 50 volts corresponded to a numerical w_ value of unity. The voltage outputs of the commercial electronic dividers incorporated i n the c i r c u i t were limited to + 50 volts. A specialized item i n the f i n a l c i r c u i t i s the squaring mechanism.  For this operation a Heath function generator was used,  a device which approximates to a continuous function by means of ten equally spaced straight l i n e segments.  I f the continuous  straight l i n e segments symmetrically disposed about the ^. axis, 5 segments on either side with the segment ends joined together at equally spaced )C intervals. The function generator was used i n the c i r c u i t as a squarer to generate  I eo  from the output  of ."amplifier 5. The schematic  c i r c u i t of the function generator i s shown i n Figure 24.  I t employs  82  ''double diodes i n a modified bridge" c i r c u i t tt* generate each l i n e segment. Figure 24.  Function generator c i r c u i t diagram. Each line. segment i s generated by a modified bridge c i r c u i t as shown. Biassed diodes are used to achieve end point voltages of the segments. replace C wrt/i iti  0  X  ootJ>ot  -f>J—AY.-tf-t The segments chosen were at the points c? = 0, 10., 20, 30, 40 2  and 50 volts so that (^fj y ^ 9, 16 and 25.  eo  had the respective voltages 0, 4,  A comparison between the segmented approximation  and the mathematical parabola for the function C^) y^ 1  e 00  i s shown i n  Figure 25. Figure 25-  Function generator approximation to parabola. The close agreement to the continuous parabola with only five'straight segments w i l l be noted.  83  A systematic c i r c u i t error analysis reveals the following separate vcontributions to error:  component errorj grid current error; potentio-  meter loading error; divider induced error; error due to approximation of the actual current wave form by a sinusoid; error due to approximation of the continuous function  V loo  by 5 straight l i n e segments i n the  function generator and amplifier output saturation error. Most of these errors are of small consequence; the main contributing cause to error being, as i n the case of the simulation of the Townsend equation, divider induced error. t 1% tolerance as before.  Passive components were a l l rated  at  Computing times were of the order of 10  seconds or less to minimize grid current error i n the amplifiers*  The  output currents from the amplifiers vere limited by resistors to the range within which amplifier output saturation i s not reached.,  '^he  two 10 K-rt_potentiometers i n the c i r c u i t fed into lM-rt- resistors causing, as i n the Townsend error analysis, 0.24.9$ loading error.  Error due to  the 5 segment approximation of the continuous function ^2 /(  00  estimated to be not larger than 2%.  is  ^he error caused by approximating  to the actual current wave form by a sinusoid i s estimated to have a maximum value of 5%.> most often being le3s*  F i n a l l y , commercial divider  induced error due to divider non-linearity i n regions of small denominators overshadows the rest of the contributing error causes, and i s estimated to add i 10% error,, each divider contributing an uncertainty of-5%» Taking a l l of the above errors into consideration the resulting accuracy of the reading doubt by 1 10%.  i n the f i n a l c i r c u i t i s estimated to be i n  84  2,3.  Comparison of Computer Results with Z-pinch Experimental Data  and Associated D i g i t a l Computer Predictions. With the snowplow equation (Q.l) correctly simulated as given In the l a s t section, actual experimental values of the parameters i n equation (O.l) could now be inserted via the coefficient potentiometers of the general c i r c u i t and solutions obtained for cases of physical interest.  The experimental curves with which the analogue results are 1 compared are due to Curzon and Churchill and Cturzon et a l » Cureon and 2  Churchill investigated the linear z-pinch discharge i n the gases A, Ee, I^, although only their results for A and  a r e  used.  Their apparatus consisted basically of a 7.5 cm radius pyrex glass c y l i n d r i c a l tube enclosing the gas to be investigated.  3y subjecting  the ends of the glass tube to a sudden large difference i n potential a pulsed current i s made to flow,, according to snowplow theory, i n a thin shell on the surface of the heated gas which i s now hot enough to be a plasma. In the f i r s t approximation this pulsed current has a sinusoidal wave form with an amplitude In these experiments of about 2001CA and a quarter period varying from about 2 to 4^u. sec. The discharge i s photographed at right angles to the axis of the cylinder with a synchronized framing camera which takes sixty pictures of each discharge, the average exposure time and framing interval being 0.2 yA. sec.  From these photographs one obtains the collapse curve of the  plasma radius with time.  The collapse curve i s plotted on a graph of  the dimensionless variable ^ = ^ /f{ against time t liiy*. sec where s  0  s  •= R  0  at t > 0. ^s(t)  i s the instantaneous radius of the tubular  plasma and R i s the glass tube radius, 7.5 cm. I n i t i a l l y one has a 0  85  cold gas entirely f i l l i n g the tube to i t s v a i l s *  The current i s measured  with a Hogowski c o i l and i s plotted simultaneously on the graph of ^ against fj , I t builds up from an i n i t i a l zero value with time. As a theoretical comparison Curzon and Churchill  adopted the  simple snowplow theory and numerically integrated the snowplow equation by the Runge-Kutta method, setting y -  1 and  =  0 at t -  0 as  i n i t i a l conditions. The numerical integration of the equation was programmed for the University of B r i t i s h Columbia d i g i t a l computer. The resulting theoretical curves are plotted on the,same graph as are the current curves and experimental collapse curves.  For this  numerical integration the actual experimental current curves X(t) were used rather than \their close sinusoidal approximation. The present analogue programme supplements these curves of the l i n e a r z-pirich discharge r a d i a l collapse., A sinusoidal current curve has been used instead of the actual current curve.  Analogue computers  are much more subject to error than d i g i t a l computers. Since this analogue employs two dividers which induce error because of their nonl i n e a r i t y i n regions Of small denominators, i t would not be expected that analogue results w i l l agree as closely with the experimental curves as^ wduid' tfre more" accurate ones obtained on the d i g i t a l computer. Nevertheless, reasonable results have been obtained, although i t was'necessary to use an analogue* •circuit; of'moderate complexity and, i n particular, no way was seen to avoid the use of the two dividers, either one or both of which would have been discarded i f possible. Analogue results have been taken for ten different discharges., five in N  9  and five i n A,  Respective pressures of the i n i t i a l l y cold gas were  86  25/H > 50A* »• 100,6* * 250^ and 500** for A;  ran Hg) and 37i5y^ > 75^ , 150^* the universal gas law of ^  c  and 750^< for N ;  375/A.  p> ~ f°  -3  (1^ pressure i s 10 2  From  one may then find the values  > the i n i t i a l gas density occurring i n the snowplow equation  (0;l) knowing that the cold gas was at room temperature or about 20° C;  The amplitude  current pulse T(tJ  J  and period T =  0  27 (fj  the constant multiples ^  ^  and  t*J  X  and  k)*  ^  ?  —  -  which are to be set up  TABLE I I I below gives the  (coulombs/^** sec); T"  0  of the  » This enables one to calculate  on the coefficient potentiometers. X  /w  have been obtained for each of the ten  different current curves  values of  lfl  (/(  sec); ^  0  (gm/cm^);  for the ten different pinch discharges  of the experiment in both A and Hg.  87  TABLE I I I  Values of various parameters for snowplow equation. Argon Gas Pressure  i -  (rad*//* s e c ) ^ Hg) 2  0  ( c o u l / ^ sec)  | sec) (gm/cm^)  (coul /gm/cm yu sec2)  .806  25  .123  7,00  1.37(10" )  139  ,394.  50  .154-  10.00  2.74(10" )  109  .315  100  .170  11..20  5„47(10- )  66.6  .274.  250  ..200  12,00  13.,68 d o ' )  36.8  .171  500  ,207  15.20  27.35(10- )  19.7  9  9  9  9  9  "itrogen Gas .586  37.5  ,132  8.20  2,05(10-9)  .315  75  .160  11.20  4..10(10~ )  78.6  .315  150  .180  11.20  8,21(l0~ )  49.7  .180  375  .172  14,80 20.50(10" )  18.2  .162  750  .206  15.60 41.00(10 )  13.0  9  9  9  Each of the ten discharges i s plotted on linear axes.. ordinateills  The l e f t  ranging from 0 to 1,0 and the right ordinate i s I (t)  ranging from zero to about 200 K'A . The time "p the abscissa.  107  sec i s plotted on  These curves are given i n Figures 26 to 35, On each f^gur,e  has been plotted the experimental collapse curve as a s o l i d l i n e j the d i g i t a l computer collapse curve as a dot-d&s-h l£c&; the analogue computer  88  collapse curve as a dotted l i n e ; the experimental current curve as a s o l i d l i n e and the sinusoidal approximating current curve as a dotted line.  This l a s t curve was obtained by f i t t i n g a sinusoidal curve onto  the actual current curve giving the sinusoidal curve the same amplitude and frequency as the experimental current curve. By examining Figures 26 to 35 one notes that the sinusoid approximates within better than 5% to seven of the tfiO. current curves, the approximation f a l l i n g to 10% i n N*2, 150/x  and 750^  and i n A, 5 0 0 ^  .  The actual current pulse i s thus nearly sinusoidal, and i t i s often taken in this form for theoretical treatments.  Sometimes a further approximation  i s taken by only retaining the f i r s t term of the sine series.  As a general  rule, i t would seem here that for small pressures (less than 100^u. Hg) the current pulse i s very nearly sinusoidal, above a pressure of 100/* Hg to 7 5 0 ^ Hg i n N2 the approximation becomes less accurate although the discrepancy does not appear to increase l i n e a r l y with increasing gas • pressure. The correlation of the three collapse curves may be noted.  It w i l l  be seen from an examination of Figures 2o to 35 that the d i g i t a l solution ordinates are less than the experimental collapse curve ordinates i n every case, the underestimation being not quite so evident for 2 5 ^ , 50p. and loox i n A, while the discrepancy becomes quite marked for 7^< and 7 5 0 ^  in N , 2  This means that the actual collapse of the plasma  occurs less rapidly than that calculated from simplified snowplow theory as worked out on the d i g i t a l computer.  I t seems reasonable to assume  that this discrepancy i s due to the simplified nature of the snowplow theory, assuming i n f i n i t e plasma conductivity and neglecting the gas  89  specific heats, as well as to the fact that the experimental conditions are not those required by snowplow theoryi In the experiments the capacitor bank voltage do&3 not remain constant and the gas i s not completely trapped by the Collapsing current s h e l l as snowplow theory requires. The analogue computer results a l l appear to have just the opposite effect from, the d i g i t a l results; i n every case the analogue comruter ordinates exceed the experimental collapse curve ordinates, i n most cases this excess being greater than the underestimate of the d i g i t a l co^mter.  The analogue curves a l l appear  to exhibit a peak or maximum, discrepancy from the actual collapse curves i n approximately the same general region of each curve, at a ^- value of about 0.8.  This systematic peaking effect i s  thought to be due to divider non-linearity and i s a spurious effect, not characteristic of the analogue solution. Although the peaking effect i n the analogue collapse curves i s marked, the pinch times or the  t  values at which y: = 0 closely  agree i n the analogue and experimental results.  The pinch time  parameter i s important i n more detailed considerations of the dynamic properties of the actual collapse mechanism, and since reasonable agreement between analogue and experimental curves.is obtained here, i t i s thought that the analogue treatment has been a successful, useful one? supplementing the d i g i t a l method of solution, and of generally greater ease of working out.  A study  of the analogue curves also shows that greater agreement between them and the experimental curves occurs for N and  5 0 0 .  2  and also i n A at 250^  90  Eigure 26.  Collapse curves for 1 at 25/i Hg pressure.  Current pulsel6t)in kiloaraperes (kA.). y. =^ /R where R •= instantaneous' plasma radius and Ro = 7.5 cm = glass tube radius. Collapse curves: experimental Current curves: experimental . analogue sinusoidal digital S  0  s  91  Figure 27. Collapse curves for A.at  50/cKg  pressure.  Current p u l s e l (t)in kiloamperes. (kA.),y.= VR where R. = instantaneous plasma radius and = 7,5 cm - glass tube radius. Collapse curves: experimental Current curves; experimental analogue sinusoidal digital 0  T I M E  -  M I C R.OS E C O N J 0 5  y  92  Figure 28. Collapse curves for A.at 100/t Hg pressure. Current pulseI^t)in kiloamperes (kA). y. = ^/f^where R = instantaneous plasma radius and R = 7.5 cm - giass tube radius. Collapse curves: experimental Current curves: experimental analogue sinusoidal digital 5  0  TIME  -  MlcROSECOfO P 5  93  Figure 29. Collapse curves for A. at 250/iHg pressure. Current pulseI(t)in kiloamperes- (kA). ^ /% where R s = instantaneousplasma radius and R 0 = 7.5 cm - glass tube radius. Collapse curves: experimental Current curves: experimental analogue sinusoidal digital s  T I M E  -  M I C R O S E C O N D S  94  Figure 30. Collapse curves for A"., at 500/x Hg pressure. Current pulsel(jb)in kiloamperes (kA). = / R where R = instantaneous plasma radius and R =7.5 cm = g/lass tube radius. Collapse curves: experimental Current curves: experimental • analogue sinusoidal • digital 5  0  S  0  T I M E  -  tiicRose^owps  95  Figure 31. Collapse curves f o r  at 37„5/- Hg' pressure.  Current pulsel(fc)in kiloamperes? (kit), if = ^Vfc where R, = instantaneous' plasma radius and £ = 7.5 cm - glass tube radius. Collapse curves: experimental Current curves: experimental -analogue sinusoidal digital 0  0  TIME-  -  MICROSECONDS  s  96  Figure 32.. Collapse curves f o r  at 75/tHg pressure.  Current p u l s e l ( t ) i n kiloamperes (kA). ^ = * J / R 0 where R s = instantaneous' plasma radius and R 0 = 7.5 cm = glass tube radius. Collapse curves: experimental Current curves: experimental analogue sinusoidal digital  T I M E  -  H IC R.O S E C O t v i D S  97  Figure 33. Collapse curves for  at 150/c Hg pressure.  Current pulseI^tJ In kiloamperes (kA'). = ^V^owhere 8 5 = instantaneous plasma radius and R = 7.5 em = glass tube radius. Collapse curves:experimental Current curves:experimental analogue sinusoidal digital 0  TIME  -  MlCROSECOMPS  \  98  Figure 31.  Collapse curves for F  2  at  Hg pressure.  Current p u l s e l f t j i n kiloamperes (kA), y. = /R where R = instantaneous plasma radius' and R = 7.5 cm = glass tube radius'. Collapse curves: experimental Current curves:: experimental analogue sinusoidal digital RJ  0  0  TIME  -  Mtc  KOSECOMDS  S  99  Figure 35. Collapse curves for  at 750/*. Hg pressure.  Current pulsel(.tjin kiloamperes (k&.). iy = */R<,where R = instantaneous plasma radius and R„ =• 7.5 cm •=. glass tube radius. Collapse curves: experimental Current curves: experimental analogue sinusoidal digital 5  s  COLLAPSE  Z o o  X  y  n  \  o-e  N \  \  \  0-& -  \ / CURREN)  l  oo  •  \  0-4  \ 0-2 -  M  2  -  ISOju  \  8  T I ME  -  M 1 C R O S E CONJPS  100  CONCLUSION  The aim of t h i s study has been t o demonstrate the s p e c i a l usefulness of the analogue computer technique In dealing w i t h these two gas discharge problems? the l i n e a r z-pinch and the Townsend discharge.  The reason f o r  the success of the analogue technique i n t h i s i n v e s t i g a t i o n i s because both p h y s i c a l discharge problems can be simply formulated as d i f f e r e n t i a l equations, and the analogue computer i s most v e r s a t i l e i n the s o l u t i o n o f d i f f e r e n t i a l equations.  As p r e v i o u s l y mentioned, the computer deals much  more e a s i l y w i t h ordinary d i f f e r e n t i a l equations than w i t h p a r t i a l d i f f e r e n t i a l equations which i t solves by converting the s i n g l e p a r t i a l d i f f e r e n t i a l equation i n t o a number of f i n i t e d i f f e r e n c e a l g e b r a i c equations. The analogue s o l u t i o n of the general Townsend discharge was noted as e n t a i l i n g a maximum e r r o r of 5% which was predominantly a t t r i b u t a b l e to the n o n - l i n e a r i t y of the e l e c t r o n i c d i v i d e r f o r small values of the denominator f u n c t i o n f e d i n t o the d i v i d e r .  I t i s f e l t that analogue s i m u l a t i o n i f  p o s s i b l e without the use of the d i v i d e r i s d e s i r a b l e to avoid t h i s dominant e r r o r producing f a c t o r , and t h a t the r e s u l t a n t e r r o r without a d i v i d e r mechanism i n the c i r c u i t would be considerably s m a l l e r .  The  Townsond computer proved to have q u i t e v a r i e d p o s s i b i l i t i e s f o r use once the simulator had been c o r r e c t l y constructed, and only a few of these p o s s i b l e programmes were e x p l o i t e d here.  One p o s s i b l e programme f o r the  Townsend computer not attempted i s as an " e l e c t r o n i c curve f i t t i n g device" whereby l a r g e numbers of i o n i z a t i o n curves might be e a s i l y and conveniently c l a s s i f i e d according to t h e i r i o n i z a t i o n and attachment c o e f f i c i e n t s o( , and h  . Another moit important use would be i n the problem of gas  101  mixtures,  I f theory can predict the primary ionization coefficient V f o r  a gas mixture then the computer w i l l easily generate theoretical ionization 22 curves.  I t was seen i n Glotov's  mixture was very complicated,  work that the theory for the Ne-A  The generation of the ionization curves  on the simulator from the theoretical values of c( was f e l t to be a r e l a t i v e l y minor operation compared with the theoretical deviation of the expression for <K , Attempts were made to construct an empirical expression for the primary ionization coefficient £>( i n the gas mixture of dry a i r . A i r was approximated to by a two component gas mixture composed of 78$ nitrogen gas and 22% oxygen gas by volume. Various forms for the resultant coe f f i c i e n t oC^ f. were t r i e d but a l l of them proved to be unsatisfactory, some more so than others. A form given where K. i s an empirical curves 19 of Harrison and Geballe for dry a i r f a i r l y w e l l , but not with any great degree of accuracy..  I t i s f e l t that an application of Glotov's  complicated analysis to dry a i r treated as a two component gas mixture "as above might prove f r u i t f u l and, i f successful, the same type of analysis might be applied to gas mixtures for which no experimental data has been taken.  One essential point of Glotov's analysis was that he  treated a two component mixture only; not a mixture with more than two components. Another point i s that he 'stated one gas component of the mixture was i n a metastable excited state and could thereby ionize atoms of the other gas component. These points would need to be observed i n any extension of his analysis to other gas mixtures.  The simulation of  the snowplow equation provided an interesting and useful correspondence  102  with experimental and d i g i t a l computer results.  I t was noted that the  peaking effect i n the analogue collapse curves was due to divider nonl i n e a r i t y , and that i f the curves were smoothed out as are the d i g i t a l curves, agreement between analogue, d i g i t a l and experimental collapse curves would be quite close, The good agreement of the analogue pinch time with d i g i t a l and experimental pinch time was also noted.  Since the  pinch time parameter i s most important i n z-pinch studies the analogue solution again proves useful.  I t would be instructive to see i f the  more general linear z-pinch theory not assuming i n f i n i t e plasma conductivity could be simulated; i f so more refined  predictions and  comparisons of collapse curves might be made. F i n a l l y , a further application of the analogue technique i n the general f i e l d of plasma physics deserves mention.  The problems  associated with magnetohydrodynamic power generation could possibly be solved by analogue techniques i f the very complicated magnetohydrodynamic equations could be simulated.  These equations are p a r t i a l d i f f e r e n t i a l  equations v a l i d i n three dimensions and their solutions w i l l depend on the variables x,^ ,2- and £ .  I f the equations were linearized completely,  then written i n ordinary d i f f e r e n t i a l form and one or two of the space co-ordinates neglected some success might be had i n their solution. programme would be complementary to the simulation of the snowplow equation of plasma physics.  This  103  REFERENCES 1.  Curzon, F.L. and C h u r c h i l l , R,J, 1962. Framing Camera Studies o f the z-pinch i n Nitrogen, Can. J . Phys. 4Q> 1191.  2.  Curzon, F.L,,, C h u r c h i l l , R.J. and Howard, R, 1962,, St i n the Nitrogen Pinch. Can. J . Phys. L e t t e r s 8, 301.  3.  Townsend, J.S., 1915 (see Loeb, L,B, and Meek, J,M. 1941. The Mechanism o f the E l e c t r i c Spark (Stanford Universit?/- P r e s s } ) .  4.  Llewellyn-Jono3, F. 1957. I o n i z a t i o n and Breakdown i n Gases (Methuen).  5.  Howard, P,R» 1958, C o r r e l a t i o n Between I o n i z i n g (cxf ) and Attachment ( t ) C o e f f i c i e n t s and Breakdown C h a r a c t e r i s t i c s f o r Carbon T e t r a f l u o r i d e , Nature (London), 181, 645,  6. Geballe, R* and Reeves, M.L,, 1953 (see Geballe, R, and Harrison, M.A, 1953. Simultaneous Measurement of I o n i z a t i o n and Attachment C o e f f i c i e n t s . Hiys, Kev, 9JL, 1,1), 7.  8.  Prasad, A.N., 1959. Measurement o f Ioniaation an&JRfctachment C o e f f i c i e n t s i n Dry A i r i n Uniform F i e l d s and the Mechanism o f Breakdown, Proc, Phys, Soc, 24, 33. fruithof, A.A, and Penning, F.M, 1937, Townsend I o n i z a t i o n C o e f f i c i e n t * f o r Neon -Argon Mixtures. Physica. 4, 430.  9. Davies, D.K., Dutton, J , and Llewllyn-Jfones, F, 1958. Secondary I o n i z a t i o n Processes i n Hydrogen a t High Gas Pressures, Proc., Phys. Soc. 7_2_, 1061, 10.  Llewellyn-Jones, F, and Davies, D.E, 1951. Mechanism o f Secondary I o n i z a t i o n i n Low-Pressure Breakdown i n Hydrogen, Proc. Phys. Soc, 64, 519.  11.  Von Engel, A>, 1955 (see Sen, S.N. and Ghosh, A.K, 1 9 6 1 , V a r i a t i o n of Townsend's Second C o e f f i c i e n t i n k l e c t r o d e l e s s Discharge. Proc. Phys. Soc. 22, 1> 507.,).  12.  Loeb, L.B., 1947 (see Sen, S.N, and Ghosh, A,K. 1961, V a r i a t i o n o f Yownoend'e Second Coefficient i n Eloctrodeless Discharge Proc, Phys, Soc 72, 1 , 507). t  13.  Sen, S.N and Ghosh, A.K, 1961, Variation of Townsend's Second C o e f f i c i e n t i n ELectrodeless Discharge, Proc, Phys, Soc, 22, 1 . 507.  14.  Sanders, F.H„ 1933 (see Loeb, L.B, and Meek, J.M, 1941. The Mechanism o f the E l e c t r i c Spark (Stanford U n i v e r s i t y P r e s s ) ) .  t  10  4  15*  Emeleus, K,G,, Lunt, R»W, and Meek, C,A,» 1936* I o n i z a t i o n , E x c i t a t i o n , and Chemical Reaction i n Uniform' E l e c t r i c F i e l d s . Proc. Roy, Soa. A 1£6, 394.  16.  B l e v i n , H.A. and Haydon, S.C, 1957, 10, 4> 590.  The T h e o r e t i c a l Evaluation  1  17.  Johnson, C.L* 1958  Analog Computer Techniques (McGraw - H i l l ) .  18.  Brown, S^C, 1959* Basic Data of Plasma Physics (Technology Press and W i l e y , N.Y,).  19.  Harrison, M.A, and Geballe, 1953.. Simultaneous Measurement of I o n i z a t i o n and Attachment C o e f f i c i e n t s . Phys* Rev, 9_1> 1.  20.  B h a l l a , M.S., and Craggs.> J^D» I960., Measurement of I o n i z a t i o n and Attachment C o e f f i c i e n t s i n Carbon Dioxide i n Uniform i e l d s . Proc. Phys, Soc. 26, 369. F  21»  Hopwood, W,, Peacock, N.J, and Wilkes, A, 1956, A ^tudy of I o n i z a t i o n C o e f f i c i e n t s and E l e c t r i c a l Breakdown i n Hydrogen. Proc. Roy. Soc, A2J1, 334.  22,  Glotov, I . I . 1938, C a l c u l a t i o n o f the C o e f f i c i e n t of "Volume I o n i z a t i o n f o r Pure Neon and Neon-Argon Mixtures. Phys, Z e i t , Sowjetunion 13_> 84.  23.  Moralev, S.K., 1937 (Russian) (See Glotov, I . I , 1938 C a l c u l a t i o n of the C o e f f i c i e n t of Volume I o n i z a t i o n f o r Pure Neon and Neon-Argon Mixtures. Phys, Z e i t . Sowjetunion 13_, 8 4 ) .  24*  Rose, D.J. and Clark J.., M e l v i l l e . 1961. P l a s m a s and C o n t r o l l e d Fusion (M.I.T. Press and John Wiley, N.Y.),  25.  Rosenbluth, M.N., Rosenbluth, A.W. and Garwin, R.L., 1954> (see Anderson, 0,A», Baker, W,R,, Colgate, S.A., Ise J r . , John and Pyle,, R,V. 1958 Neutron Production i n Linear Deuterium Pinches. Phys. Rev, 110, 6, 1375.)  26,  Anderson, O.A., Baker, W.R., Colgate, S.A., Ise J r . , John and P y l e , R*V, 1958, Neutron Production in. L i n e a r Deuterium Pinches. Phys. Rev, 110. 6, 1375.  105 APPENDICES  Papers Presented i n Support of t h i s Thesis  APPENDIX A.  R.J. C h u r c h i l l and D.F. Gallaher.  An Analogue  computer Study o f the Townsend Gas Discharge. Journal o f E l e c t r o n i c s and C o n t r o l .  1963.  (In the p r e s s ) .  APPENDIX B.  R*J. C h u r c h i l l and D.F, G a l l a h e r .  An Analogue  Computer Study o f the Collapse Stage o f the Linear Z-pinch,  Journal o f E l e c t r o n i c s and  Control.  ( i n the p r e s s ) .  1963*  106  AN ANALOGUE COMPUTER STUDY OF THE TOWNSEND GAS DISCHARGE . Ri J. Churchill and 1. Fi Gallaher^  Department of Physics University of B r i t i s h Columbia Vancouver, CANADA  ^*  Now at: Department of Physics Acadia University W o l f v i l l e , N. S., CANADA  io?  ABSTRACT  The paper describes an analogue computer Which simulates the modified Townsend gun discharge e'quaticn including primary and secondary ionization and attachment.  Pre-breakdown ion-  ization current characteristics for various gases have been extended to r e l a t i v e l y large values of the parameter ^ / p Computer solutions of the .modified Townsend equation are i n good agreement with experimental results.  •  108  I.  INTRODUCTION  . Analogue computers have been employed i n many-studies in physics and engineering (Korn and Korn 1956; Churchill and Cluley I962), and are especially useful where the equations describing the physical processes are expressible i n d i f f e r e n t i a l form.  Only a few applications of analogue techniques to gas  discharges and plasma physics have been reported to date (Braffort and Chaigne 1958; Hart I 9 6 2 ) . One particular gas discharge problem amenable to analogue treatment i s the growth of pre-breakdown ionization current between p a r a l l e l plate electrodes i n a uniform e l e c t r i c f i e l d . Following Townsend's investigations (Llewellyn-Jones 1957) many later workers have produced a wealth of experimental data relating the pre-breakdown current to the electrode separation for a wide range of operating conditions i n many gases. From the experimental results i t i s usually necessary to evaluate the Townsend ionization coefficients by a process of graphical curve f i t t i n g (Prasad and Craggs I960).  An  alternative method makes use of an analogue computer as an 'electronic curve f i t t i n g ' device.  In this case the computer  simulates the Townsend gas discharge and the ionization and attachment coefficients are found from potentiometer settings when the cojputer generates the transfer function for the physical system.  Conversely, from a knowledge of the physical  coefficients i t i s possible to generate the growth of pre-  109  breakdown ionization current f o r various experimental c o n d i t i o n s . This paper describes an analogue computer which simulates the general Townsend equation i n c l u d i n g the e f f e c t of e l e c t r o n attachment.  The simulator has extensive uses i n the p r e d i c t i o n  and evaluation of experimental r e s u l t s and has a l s o proven to be a valuable teaching a i d ;  110 II.  •  THE TOUNS^TD DISCHARGE  In the o r i g i n a l a n a l y s i s of current growth between p a r a l l e l plates i n uniform e l e c t r i c f i e l d s Townsend 1957)  (Llewellyn-Jones  deduced the r e l a t i o n s h i p  (1) X  where  X  =. t o t a l pre-breakdown current e  = fke i n i t i a l p h o t o e l e c t r i c  current  from, cathode i r r a d i a t i o n d  =  electrode  separation  o( — Townsend primary i o n i z a t i o n coefficient. The primary i o n i z a t i o n c o e f f i c i e n t , oC y represents  the number  of electrons produced i n the path of a s i n g l e electron t r a v e l l i n g a distance of 1 cm i n the d i r e c t i o n of the e l e c t r i c f i e l d . In f u r t h e r introducing the p o s s i b i l i t y of secondary i o n i z a t i o n processes operative i n the gas, Townsend a r r i v e d at Xo  the equation  X  e  =  Here, (f i s Townsend's secondary i o n i z a t i o n c o e f f i c i e n t representing the number of secondary electrons produced by a p o s i t i v e i o n t r a v e l l i n g 1 cm i n the d i r e c t i o n of the Llewellyn-Jones  (1957) d i s t i n g u i s h e s  field.  between several p o s s i b l e  secondary mechanisms the most important of which i s the e j e c t i o n of secondary electrons from the cathode by i n c i d e n t p o s i t i v e i o n s .  Ill  It i s usual to consider Y[^^  ~ \) — I ' the as  Townsend breakdown c r i t e r i o n i n discharges of this type; Penning (1938), Harrison and Geballe (i953) and Prasad (1959) have included an attachment coefficient, ^  ,  analogous to *(  representing electron capture i n electron  attaching gases.  Inclusion of ^  leads to the general gas  discharge eauation  1  I  =  ° L< -1  -i  Z  J  ~  (3),  In the usual experimental conditions (Keek and Cragg's 1953)  the pre-breakdown current (  10  A) i s determined ^/p  as a function of electrode separation for fixed  , E  being the e l e c t r i c f i e l d intensity (V/cm), and  p  pressure (mm Hg).  parameters,  I t i s found that the reduced  the gas  112  III.  SIMULATION OF Tu'JJjSEv'J RATION  Preliminary considerations led to an analogue computer where X  which solved Equation (3) as the quotient X/ y Y  are the numerator and denominator respectively.  expressions for X  Y  and Y  the parameters X©?  In'the  << , >\  and  are assumed to be constant for any particular discbarge  condition for a fixed value of X  and  and Y  . Thus, differentiating  with respect to electrode separation gives  =0  JJ  - n/X  +t  Xo  (4),  Eoth Equation ( 4 ) and ( 5 ) are similar i n form and have the following i n i t i a l conditions,  X(<i=oJ = I  Y  C  = o) - I  The physical variables are related as I the electronic analogue comuter variables are \] i t i s convenient to set the computer time "t electrode separation d X  and  and  Y  Y  •  T n e  (6). = f (J,) / (*tj  and so  i n terms of  and to l e t computer voltages simulate  voltages <f  (  and  £  t  representing X  respectively, are fed into an electronic divider which  produces a voltage proportional to X / y of the problem.  , and thus the solution  113  THE PRACTICAL CIRCUIT (U)  The analogue c i r c u i t for the solution of Equation reauires a feedback capacitor to simulate the term input proportional to X represent ^ I  an<  ^  a  an  constant current input to  . Equation (5) i s treated similarly.  0  A  c i r c u i t embodying these requirements i s shown i n Fig, 1 where the voltages at the outputs of amplifiers 1 and 3 represent X  Y > respectively.  and  With the usual assumption of high amplifier gain and negligible input current the c i r c u i t s connected to amplifiers 1 and 3 obey the relations  h° c  i£A  c  =  +  (7)  and  C  ^  (  o  o  ^  e  =  C^.' TV)«oog*  loo^.-^J +  <ftj  (8).  With appropriate scale factors Equations (7) and (8) become the analogues of the modified Townsend Equations (/.) and (5). Choosing a scale factor of 1 sec computing time equal to an electrode separation of 1 cm, and grouping some of the coefficients leads to the practical c i r c u i t i n Fig. 1. correspondence i s indicated i n the Figure. pre-breakdown current X  The analogue to physical The relation between  and computer voltage was such that  1 volt represents lO*"-^ A and two decades of. ionization current could be obtained i n general. Y  I n i t i a l conditions for X  and  are represented by charging the integrating capacitors  C, and C" to the voltages - C*„ and - 100 V , respectively. z  114  It i s necessary to introduce the factor of 100 since the electronic divider output i s |oo ' e  representing  ; The output signal ^ / / f  the pre-breakdown current,  X  > i s recorded on  a graphic recording milliammeter. SIMULATOH COMPONENTS AND ACCURACY The sirmlator i s provided v;ith standard unstabilized d.c. amplifiers having an open loop gain of 50,000 and a long term d r i f t of 0.5 mV.  The electronic divider i s a commercial  time-division multiplier/divider having a maximum ovitput of 50 volts i n the division mode. The graphic recorder i s of the. r e c t i l i n e a r recording type having variable chart speeds and a frequency response 3 db down at 5 c/s.  High quality + 1%  passive components are used throughout the  simulator.  Potentiometer loading errors are well below 1% and the amplifiers operate always in the region of l i n e a r i t y . In order to reduce errors due to amplifier d r i f t , computing tim.es are kept below 1 minute and are t y p i c a l l y 20 seconds. The commercial electronic divider has an accuracy dependent on denominator magnitude, but i t i s estimated computing errors should not exceed 5% for the results given i n the following section.  115  IV.  RESULTS AMD' DISCUSSION  PRELIMINARY COMPUTER SOLUTIuNS To determine the accuracy of the analogue computations, solutions of the Townsend equation (Equation (3)) were generated f o r comparison with t h e o r e t i c a l and experimental results.  In general, the greatest discrepancy occurred f o r  the l a r g e s t values of oC , Y exceed 5% f o r  e<  -  4,  max  and ^ Y*  but the e r r o r s do not — 0.05 and ^  I  max  V  —  4.  m a x  GENERATION OF PRE-BREAKDOWN IONIZATION CURRENTS The computer was employed t o generate pre-breakdown i o n i z a t i o n currents f o r r e l a t i v e l y large values of E/p i n various gases using the f o l l o w i n g t h e o r e t i c a l r e l a t i o n s h i p given by Townsend (llewellyn-J ones 1957)  "</  f/e  r  i n which the constants A  and %  Table I gives the values of A  (9)  are e m p i r i c a l l y determined.  and 8 and the ^/p range of  v a l i d i t y f o r various gases. I t Is evident that Equation (9) neglects the e f f e c t of secondary i o n i z a t i o n and of attachment ( i . e . Y=  ^ -  0)  and that the t o t a l I o n i z a t i o r current i s given by Equation (l). since  There i s some j u s t i f i c a t i o n f o r t h i s simple approach :  Y  and ^  are often very much l e s s than °t  i n the rare gases where  Y  i s very small and  (Llewellyn-Jones 1957; Sen and Ghosh 1961).  especially  non-existent  Even i n e l e c t r o n -  116  attaching gases ^ <V <^ Craggs  for high values of E/'p  (Bhalla and  1962).  The values of <?(  for a particular value of E/p  are  obtained from the theoretical curves (Fig. 2 ) which result from the Townsend relationship given in Equation ( 9 ) .  The resulting  primary ionization coefficients are fed into the simulator i n order to generate the ionization growth curves shown i n Fig. 3. Experimental results for nitrogen at ^ p  = 1 0 0 (Posin. 1 9 3 6 )  are i n reasonable agreement with the computer data. Ionization currents have been predicted by employing the analogue of the modified Townsend equation (Equation ( 3 ) ) including e£  ,Y  and ^  for low values of £/"p  with experimental work i s possible. relationships exist for  /p  a  where direct comparison  Theoretical and empirical  nd Yf p  but i n a l l cases  experimental values of t^p were employed in the computer. Figure 4- shows some t y p i c a l computer results for hvdrogen generated for comparison with the work of Hopwood, Peacock and Wilkes  (1956).  The rapid up-curving of the computed results  f o r l / j ^ near 5 0 i s due to divider non-linearity near i t s maximum output of 5 0 v o l t s . dry a i r with E/j>  Computed ionization currents for  ranging from 37.5 to 52.5 V/cm mm Hg are  ' i n good agreement with the measured values of Harrison and Geballe  (1953).  Only a few examples have been given here, but any other applications of the Townsend discharge simulator are possible. One particular problem concerns ionization current growth i n  117  mixtures of gages following the early theoretical and experimental work of Glotov (1938) and Kruithof and Penning (1937)* For neon-argon mixtures these workers associated the increase i n o/ v  over mixture  &( ^ neon  with ionization of the argon atoms  by metastable neon atoms; I t i s desirable to extend this treatment, to other gas mixtures and work i s i n progress for this purpose *  lis V.  CONCLUSION  Ah analogue computer, designed to solve the modified Townsend gas discharge equation, has been emploved to extend ionization current growth characteristics for various g-ses to r e l a t i v e l y large values of E/p  %  Ionization currents, predicted for conditions i n  which ionization and attachment coefficients are known are i n good agreement with experimental-, results; The Townsend simulator performs a useful auxiliary function as a teaching aid since the effects of varying, system, parameters.may be readily examined. . The analogue computer technique may be extended to other important problems connected with the- Townsend gas discharge, particularlv the growth of ionization currents i n gas mixtures.  ACKNOWLEDGEMENTSThe work was partly supported by the Atomic Energy Control Board of Canada. The idea for a Townsend discharge computer arose from discussions by one of us (R. J . C.) with Dr. M. S. Bhalla of the Electronic Engineering Department, University of Liverpool.  ii.9  REFERENCES Bhalla, Mi S. and Craggs; J . D 1962. ProCi Physi Soc; 80^ 151. s  Braffort, Pi and Chaigne, M. 1958* Proceedings of the Second United Nations International Conference on the Peaceful Uses o f Atonic Energy, Geneva, 3_lj. 247; Brown, Si C; 1959; Basic Data of Plasma Physics (Technology Press and Wiley, N. Y . ) . Churchill, R. J . and Cluloy, J . C; 1962; Electronic Technology 3JL. 292. Glotov, I. I., 1938. Phys. Zeit. Sowjetunion 13» 84. Harrison, M . A. and Geballe, R. 1953. Phys. Rev. 9_lj_ 1. Hart, P. J . 1962. Phys. Fluids, i * . 38. Hopwood, W., Peacock, N. J . and Wilkes, A. 1956. Proc. Roy. Soc. A235, 334. Korn, G. A. and Korn, T, 1956. Electronic Analog Computers (McGraw-Hill, N. Y . ) . Kruitbof, A. A. and Penning, F. M. 1937. Physica, 4j. 430. Llewellyn-Jones, F., 1957. Ionization and Breakdown i n Gases (Methuen). Meek, J . M . and Craggs, J . D., 1953. E l e c t r i c a l Breakdown of Oases, (Oxford University Press). Penning, F. M. 1938. Ned. Tijdschr. Natuurkde, 5_i. 33. Posin, D. Q. 1936. Phys. Rev., 50.-650. Prasad, A. N. 1959. Proc. Phys. Soc. 74, 33. Prasad, A. N. and Craggs, J . D. I960. Proc. Phys. Soc. 76,. 223. Sen, S. N., and Ghosh, A. K. 196I. Proc. Phys. Soc. 22a.  1 6 0  •  120 TABLE I  Coefficients A and % i n Equation (9) for Various Oases  Gas  Range of v a l i d i t y of  ^(p  A ionizatiohs/cTn/irun Hg  B volts/cn/nip Hg  A  100 - 600  H  180  Kr  100 - 1000  17  240  Ne  100 - 400  4  100  Xe  200 - 800  26  350  H  2  150 - 600  5  130  Air  10T- - 800  15  365  N  100 - 600  12  . 342  2  121  LIST OF FIGURE CAPTIONS Figure 1.  Simulation of modified Townsend Equation; ^  represents  (  v  <t>x  *l  .  r i. X  e e  u  >/e  l  Figure 2.  i,  Y  »  r  against p /g-  Plots of  from Equation (8) for  various gases. Figure 3.  Computer ionization characteristics using some of the results from. Figure 2. Parameter  = H^p  (V/cm mm Hg)  Pressure  = 1 mm Hg  lo  = 10" A 11  Experimental curve from Posin (1936). Figure 4 .  Comparison of computed and experimental results i n hydrogen. Experimental results (Hopwood et a l , 1956) Parameter  = E/p (V/cm mm Hg)  Pressure  = 700 mm  T  = ID"  0  11  1  A  122  Figure 5;  Growth of ionization currents i n dry a i r ; Experimental results (Harrison and Geballe 1 9 5 3 ) Parameter =• E/p Pressure l o  (V/cm mm  - 6 0 mm Hg = ID"  11  A  Hg)  i23  Figures i n this paper (An Analogue Corvput Study of the Townsend Gas Discharge) duplicated by Thesis Figures* .  Figure 4 (see Figure 20 i n thesis). Figure 5 (see Figure 18 i n thesis);  124  125  Eigure 2.  126  Figure 3.  E L E C T R O D E  SEPAR/\T\Ohj  IN/ C M  127 AN ANALOGUE COMPUTER STUDY OF TEE COLLAPSE STAGE OF THE LINEAR Z-PINCH Rj J . Churchill and D. F. Gallaher^"  Physics Department University of B r i t i s h Columbia Vancouver, Canada.  Now at: Physics Department, Acadia University W o l f v i l l e , Nova Scotia, Canada.  128  ABSTRACT  The paper describes a special-purpose analogue computer which solves the so-called snow-plough equation describing the i n i t i a l collapse stage of the linear Z-Pinch discharge.  Collapse  curves, generated by the computer for a low energy (5.A k j ) Z-Pinch i n argon and nitrogen, are presented•.  These compare  favourably with d i g i t a l computer calculations and with photographic measurements of the discharge developmentj  129  I.  ',  INTRODUCTION  In recent investigations of the collapse stage of the linear Z-pinch discharge (Curzon et a l 1963) experimental results were compared with theoretical collapse characteristics computed from, the so called snow-plough equation (Rosenbluth et a l 1954) by means of a d i g i t a l computer. For subsequent theoretical studiesah analogue computer was adopted to simulate the dynamic behaviour of the discharge.  This technique provides considerable f l e x i b i l i t y  in the variation of the discharge parameters as well as a means of readily examining the effects of the various approximations usually made i n the theoretical analysis. The design of the required special-purpose computer i s described i n this naper and computer results are compared with those obtained from a photographic study of a low-energy Z-pinch discharge in argon and nitrogen.  130  II.  COLLAPSE STAGE OF THE Z-PINCH  In the present work the experimental arrangement for the linear Z-pinch consists of a c y l i n d r i c a l glass tube (15 cm internal diameter) with plane brass electrodes mounted at either end.  A  brass gauze surrounding the tube serves as a coaxial return lead for the discharge current and allows side-on photographic observations of the pinch dynamics. A 12 kV capacitor bank i s connected across the electrodes by means of a triggered spark gap switch. Upon breakdown of the inter-electrode gas the discharge current flows f i r s t near the walls of the tube, but as the current builds up, the interaction of the rras current with i t s self-magnetic f i e l d causes a l a t e r a l constriction of the discharge known as the pinch effect.  The period from the i n i t i a t i o n of the discharge up to the  f i r s t maximum contraction i s called the collapse stage and the time at which this maximum, constriction occurs i s designated the pinch time.  Subsequent to the f i r s t pinch, i n s t a b i l i t i e s of various types  develop on the discharge column but these are not considered here. The simplest theoretical description of the collapse stage of the pinch discharge Is that given by Rosenbluth et a l (1954). Their analysis i s based on the assumption that the discharge current flows i n a thin c y l i n d r i c a l shell which contracts r a d i a l l y due to the Lorentz force acting upon i t .  The rapid contraction  drives a shock wave into the gas contained within, the current s h e l l . By equating the time rate of change of momentum of this shock to the Lorentz force the so-called snow-plough equation i s derived as  131  Kt)  >t  ;  (1)  '(' " 1  loo |(  r  +  where; | -  H  i  ,  / r .  h(t) = radius of contracting channel as a function of time r,, — radius of discbarge tube (7.5  cm)  yd = i n i t i a l gas density 0  X("t) = discbarge current as a function of time "t  = time.  It i s found experimentally that the snow-plough equation applies very well to the collapse stage of pinch discharges.  132  III.  SIMULATION OF THE SNOW-PLOUGH EQUATION  To obtain solutions of the snow-plough equation describing the collapse stage of the pinch discharge, measured values of discharge current for various i n i t i a l conditions are substituted i r t o Equation 1. Examination of the current measurements reveals that the waveforms may oe closely represented by sine functions.  This considerably  simplifies the generation of X CtJ i n the analogue computer since  X (t) = I  e  5 in u)t  () 2  where, Io  =  P ^ current measured from the waveforms ea  U) — angular frequency derived from the time to peak current. With Equation 1 written i n ordinary d i f f e r e n t i a l form, a single time integration and substitution of Equation 2 gives  By introducing the transformation Y=.T©<4. Equation 3 nay be written as  it  133  At this point i t i s convenient to define a new variable X  X -  — 7  as  (5)  —  \ bo i r jo o so that the snow-plough equation i n reduced form becomes  (6).  After consideration of several alternatives the computer c i r c u i t of Fig. 1 was chosen to simulate the collapse stage of the Z-pinch by solving the reduced snow-plough equation. The required d i f f e r e n t i a l equations for the varia ble X are found by successive time differentiation of Equation 5 as /IHICJL  =  and  TP  =  _  4  ^ +  The e x p l i c i t appearance of X  (8).  i n the simulator i s effected by the  c i r c u i t s connected to amplifiers 1, 2 and 3. With the assumption of zero input current to the computing amplifiers, the summation of currents at the input of amplifier 1 gives c  C  1 (c  i l l ! - -Jt£±L  ^ i t ^ ' i t )  if—  4. fJ  < * '  r~-  e  «  t  (9)  a  \  134  which reduces to  when R, = f N i ~ R 3 = K  ana  ^ ^- i -C.-t.~C. The proper choice  of scale factors makes Equation 10 the exact analogue of Equation 8 with -e. , representing X • The factor (jvx)  determines the  time scaling of the analogue, and i s chosen such that 1 second of computing time corresponds to 1 microsecond i n the physical problem.  In addition to the overall time scaling, the potentiometer  setting <f> accounts for the variation i n angular frequency of the discharge current for varying experimental conditions. Amplitude or voltage scaling i s dependent upon the value of f „ which represents the quantity * ^ 2  conditions.  }  f  or  different physical  Within the computer the following i n i t i a l conditions  must be s a t i s f i e d : 4$ (t» o) ^ 0 and x(t aT  = o)  = 0.  In a similar manner the complete solution of Equation 6 i s found by summation of currents at the input of amplifier 5. R^ •= R $- « R  By choosing <JU)  1  and  £--^i-  - C$- = £7 , i t results that Jt  (u).  With appropriate scale factors Equation 11 i s the analogue of Equation 6 since -~ represents Y • The same tim.e constant, appears i n Equation 11 so that the whole computer operates on the same time scale.  Amplitude scaling i s governed both by ~ tapped {  135  from the output of amplifier 2 and by the i n i t i a l condition of 50 volts on integrator 5 required by the relationship, Y ( t = oj -  50 ^.(t = 0) =  50, for the physical problem.  The solution of  Equation 6 available at the output of integrator 5 i s fed into a graphic recording milliammeter: This method of simulation requires two divisions and one function squaring operation to be performed*  The electronic  dividers generate an output voltage of 100 i/g>^ where €, and €  are the input voltages.  In addition the denominator must be  negative and the maximum, output Is 50 v o l t s .  To generate the  function € *~ , a commercial electronic squaring unit was employed x  to produce  /100  from -£f inout. r  the continuous function for C  2  The generator represents  by five straight-line segments  evenly spaced between 0 and 50 volts.  Maximum computing errors  introduced by the squaring unit and dividers are 2 and 5% respectivelv. Potentiometer loading errors are less than 0,5$ and precision passive components (t. 1$) have been used throughout the simulator.  Commercial d,c. computing amplifiers having an  open loop gain of 50,000 are employed and errors due to amplifier d r i f t are minimized by the use of short ( •< 10 sec) computing times.  136  IV;  RESULTS AND DISCUSSION  The linear Z-pinch apparatus used i n the present work i s described i n d e t a i l by Curzon and Churchill (1962).  The i n i t i a l  collapse stage i s photographed by means of a rotating mirror framing camera at an average exposure time of 0,2y. sec; Typical experimental results are shown i n Figures 2 and 3 where the reduced r a d i i of successive images are plotted as the measured p r o f i l e s . Table I l i s t s the pinch times i n argon and nitrogen for the common range of i n i t i a l gas densities investigated,  For comparison  with the experimental results, the collapse curves were calculated both by d i g i t a l and analogue techniques. Measured values of the discharge current  were  substituted into Equation 1 which was integrated on an IBM 1620 d i g i t a l computer. The resultant values of y (t) plotted i n Figures 2 and 3 and the pinch times included i n Table I show good agreement with the experimentally determined dynamic characteristics of the Z-pinch, Typical collapse curves generated with the analogue c i r c u i t i n Figure 1 are shown i n Figures 2 and 3. these solutions, the computer parameters <j> and calculated from the physical quantities <o and respectively.  The values of u> and X  measured current profiles and  0  0  To obtain €„  are  a -r  lo*" —-  are derived from, the  i s the i n i t i a l gas density  calculated from the universal gas law for the particular experimental conditions. Agreement between analogue and d i g i t a l  137  results i s satisfactory In view of the simplifying assumptions i n the analogue treatment*  The approximation contained i n Equation  2 i s shown to be reasonable up to about peak current, while the representation ofvftj as five straight l i n e segments i n the electronic squaring unit i s found to have i t s greatest deviation i n the region 0*8 £ yt{) < 1*  The largest errors i n the electronic dividers  w i l l also appear for yl\)> 0,8 and w i l l approach 5% as u ( f } - * l . Nevertheless the analogue solutions follow the measured profiles f a i r l y well and provide good predictions of the pinch time defined in section I I . Since the pinch time i s usually an important parameter for comparison, this i s a desirable feature of the analogue solutions. Pinch times from the analogue results are also included i n Table I. Over the common range of i n i t i a l gas densities investigated, the calculated and measured pinch times i n Table I are i n good agreement. However, at the highest pressures studied, namely 50C/*Hg i n argon and 750^Hg i n nitrogen, the measured pinch times are much larger than the calculated values.  This discrepancy i s  due to an observed delay i n the formation of the pinch at high pressures and not due to computing inaccuracy. At 750/1 Hg the delay has been photographically measured as 2.3/1 sec.  After this  i n i t i a l delav, the discharge column collapses i n the usual manner, and i s closely described by the solutions of the snow-plough equation obtained with the analofltie and d i g i t a l computers.  138  V; '  CONCLUSION  The relative simplicity of the special-purpose analogue computer described makes i t a useful tool i n the laboratory study of the linear Z-pinch and related plasma dynamics;  lis  decreased accuracy as compared with the d i g i t a l computer i s offset by the ease with which various experimental problems are programmed on the simulator. This l a t t e r feature i s of particular importance with respect to several magnetohydrodynamic problems now being investigated i n this laboratory.  ACKNOWLEDGEMENTS  The experimental measurements and d i g i t a l calculations were performed i n conjunction witl: Dr. F. L. Curzon to whom the authors are grateful for many helpful discussions.  The work i s  supported i n part by a research grant from the Atomic Energy Control Board of Canada.  139  REFERENCE^  Curzon, F. L. and Churchill, R. J i ; 1962, Canadian Ui Physicsj 42i 1191;  Curzohi F. L., Churchill; R; J . and Howard) R i j 1963} Proc; Physi Soc;, 82, 394. Rosenbluth, M. N., Rosenbluth, A. W. and Garwin) R, L,; 1954 > Los Alamos S c i e n t i f i c Report LA-1850 (New Mexico, U; Si A;)'i  HO  TABLE I COMPARISON OF CALCULATED AND MEASURED PINCH TIMES  Gas  Pressure Eg  Pinch Time Digital  Microseci Analogue  Measured  Argon  25  2.60  2.67  2.75  Nitrogen  37.5  2.63  3;07  2.90  Argon  50  3.11  3.19  3.44  Nitrogen  75  3.22  3.80  3.89  Argon  100  3.85  4.00  4.00  Nitrogen  150  3.75  4.24  4.25  Argon  250  4.70  4.90  5.40  Nitrogen  375  5.75  6.15  6.30  Argon  500  6.30  6.07  7.00  Nitrogen  750  6.75  6.72  8.75  ui  LIST OF FIGURE CAPTIONS  Figure 1.  Analogue Computer Circuit e. i represents X  i n Equation 6  '-x. represents  Y  i n Equation 6  € „ represents  - • • - — i n Equation 8 ioey r ^ <f> represents LO i n Equation 8. 0  Figure 2.  0  Collapse Curves for Argon. = i^(t:)/ro  I ( t ) i s discharge current measured with a Rogowski c o i l . D i g i t a l and analogue curves calculated from snow-plough equation.  Measured curve from  photograph. Figure 3.  Collapse Curves for Nitrogen. See legend of Figure 2.  U2  Figures i n this paper (An Analogue Computer Study of the Collapse Stage of the Linear Z-pinch) duplicated by Thesis Figures.  Figure 1 (see Figure 23 i n thesis). Figure 2 (see Figure 29 i n thesis). Figure 3 (see Figure 34 i n thesis).  

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