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Production of a highly-ionized gas by an electromagnetically-driven shock wave Cormack, George Douglas 1960

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PRODUCTION OF A HIGHU-IONIZED GAS BY AN ELECTR0M&.GNETICAILY-DRI7EN SHOCK WAVE  by GEORGE DOUGLAS CORMCK B.A.Sc, University of British Columbia, 1955  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M.Sc. i n the Department of PHYSICS We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA October, I960  In presenting the  this  r e q u i r e m e n t s f o r an  thesis in partial advanced degree a t  of B r i t i s h Columbia, I agree that it  freely  available  agree that for  the  f o r r e f e r e n c e and  permission for extensive  s c h o l a r l y p u r p o s e s may  D e p a r t m e n t o r by  be  gain  shall  not  Department  of  be  a l l o w e d w i t h o u t my  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r #, C a n a d a . Date  /O^JAt  /  shall  study.  I  Columbia,  the  of  University  copying of  his representatives.  copying or p u b l i c a t i o n of t h i s  the  Library  g r a n t e d by  that  fulfilment  make  further this  Head o f  thesis my  I t i s understood  thesis for written  financial  permission.  ABSTRACT  A study has been made of the properties of an electromagnetically generated shock wave travelling through both argon and helium. An i n i t i a l capacitive energy of kSO joules has been rapidiy discharged into the gas and has resulted i n shock wave velocities of up to five centimeters per microsecond. The high velocities obtained can be attributed to careful design of the main discharge circuit. Preliminary results of measurements of the properties of the shock wave are given.  Of primary interest has been the results of measure-  ments of the properties of the plasma associated with the shock wave. A magnetic f i e l d deflection method has been developed for the measurement of the electrical conductivity of this plasma.  (ii)  TABLE OF CONTENTS  Chapter  Page  I  INTRODUCTION  1  II  THE THEORY OF SHOCK WAVES  6  1. 2. 3.  6 8 9  HI  IV  V  Ho Ionization Low Degree of Ionization High Degree of Ionization  13  APPARATUS 1. General 2. Main Discharge Circuit 3. Conductors U. High Current Switch 5. Trigger Unit 6. Condensers 7. Electrode System 8. Luminosity Detector 9. Current Detector 10. Voltage Measurements 11. Electric Probes 12. Spectrophotometer 13. Time Integrated Spectroscopy H i . Apparatus to Measure Conductivity MEASUREMENT OF THE ELECTRICAL CONDUCTIVITY OF THE PLASMA ASSOCIATED WITH THE SHOCK WAVE  13 lU 18 22 23 2k 25 28 29 33 33 3k 3k 35  1. 2. 3.  36 k3 U9  Theory Apparatus Calibration  56  RESULTS 1. 2. 3. U.  36  The Shock Driving Mechanism The Shock Wave Conductivity Conclusions  r  56 59 6k 67 68  BIBLIOGRAPHY (iii)  ILLUSTRATIONS 1. 2. 3. k* 5. 6. 7. 8. 9. 10. 11. 12. 13. lh» 15. 16. 17. 18. 19. 20. 21. 22.  Following Page  Electromagnetic Drivers Characteristics of a Strong Shock Wave Shock Tube Discharge Circuits High Current Switch Synchronizing Circuit Trigger Circuits T-Tube Driver Current Crowbar Coplanar Driver Current Detector Electric Probes Coordinate System for Conductivity Measurements Signals from Conductivity Apparatus Pickup Coils Apparatus for Measuring Conductivity Experimental Evidence of Field Distortion During Conductivity Measurements on Shock Wave Distortion of Magnetic Field Daring Conductivity Measurements Luminosity of Shock Wave Velocity of Luminosity Front as a Function of Discharge Voltage Propagation Characteristics of Luminosity Front as a Function of Pressure, Gas and Electrode Configuration Propagation Characteristics of Luminosity Front as a Function of Discharge Voltage  TABLES  2 5 12 13 21 22 22 2k 25 26 29 32 35 36 U2 kk U5 5l 55 56 56 60  Page  I Calibration Data for Conductivity Apparatus II Measured Values of Maximum Conductivity of Plasma III Conductivity Versus Magnetic Field Intensity  (iv)  51* 65 66  ACKNOWLEDGEMENTS  The a u t h o r wishes t o express h i s g r a t i t u d e t o Dr. A . J . Barnard f o r h i s generous  a s s i s t a n c e and g u i d a n c e .  A warm thank y o u i s a l s o g i v e n t o Dr. K. Sodomsky f o r many h e l p f u l d i s c u s s i o n s and t o D r . G.V. B u l l , Canadian Armament Research and Development E s t a b l i s h m e n t , who has been r e s p o n s i b l e f o r i n t r o d u c i n g t h e a u t h o r t o t h i s extremely i n t e r e s t i n g a n d i m p o r t a n t f i e l d o f Plasma P h y s i c s .  I  INTRODUCTION  The possibility of harnessing the energy released by a fusion reaction i s one of the most tantalizing aims of present-day scientific research.  Scientists i n this f i e l d of study have realized that they are  tackling a far more d i f f i c u l t problem than that posed by the harnessing of a fission reaction.  Their attention i s now centered not primarily on the  fusion reaction i t s e l f , but rather on the conditions for obtaining an efficient reaction rate.  Because the most promising reaction w i l l only o  proceed efficiently at temperatures i n excess of 1 0 °K. according to Bishop ( 1 9 5 8 ) , the constituents must be highly ionized and of course i n gaseous form.  Such highly ionized gases are called plasmas and the study  of their properties, plasma physics. There i s much to be known of the properties of plasmas.  In his  efforts to harness this energy source of the stars, man has had to compromise and study not the hot, long-lived plasmas that he would wish to study, but rather the attainable alternatives of either a cool, long-lived plasma or a hot, short-lived plasma.  In either case he i s hampered by his lack of com-  plete knowledge of the properties of a plasma. One method of generating a plasma under conditions that are f a i r l y amenable to analysis i s to ionize a gas with a shock wave. It i s this method that has been developed i n the work upon which this thesis i s based. Gas atoms are abruptly heated by the shock front and then slowly impart their heat energy into excitation and ionization energy u n t i l temperature -  1  -  - 2 equilibrium between the constituents of the gas i s attained.  Subsequent  cooling then decreases the temperature of a l l constituents and the degree of ionization.  The time interval during which the gas i s ionized i s of  the order of one to ten microseconds, quite sufficient for spectrographic, photometric and electronic observations to be made. Design considerations are given i n this thesis for both the high energy electrical discharge shock driving equipment and the equipment used for measuring the properties of the shock wave, including the plasma following the shock front.  Preliminary  results of measurements obtained with this equipment are also presented. In a conventional shock tube a low pressure section i s separated from a high pressure section by a thin diaphragm.  The diaphragm i s ruptured  by slowly increasing the pressure differential u n t i l the diaphragm breaks. The shock wave thus generated then travels down the low pressure section of the shock tube. The characteristics of a conventional shock tube are reviewed by Penner, Harshburger and V a l i (1957). A conventional shock tube was used by Resler, Lin and Kantrowitz (19^2) to study the properties of argon at thermal equilibrium temperatures of up to 18,000°K. Their studies reached fruition with the publication of the papers by Petschek, Rose et a l (1955) and Lin, Resler and Kantrowitz (1955).  They were interested i n spectral line broadening and line shifts,  continuum radiation and the conductivity of the plasma associated with the shock wave. The method that they employed to measure the electrical conductivity has been adopted i n the present work. The method consists of allowing the shock wave to pass through a radial magnetic f i e l d . The force on the electrons results i n a circumferential current which can be  - to follow page  la  T - Tube  I b  Conical  Ic '•  I d  Figure  I  :  :  Rail  2 -  Driver  Driver  Driver  Co-planar  Driver  Electromagnetic  Drivers  - 3 measured by transformer action into a pick-up c o i l .  The output voltage  from the c o i l can then be related to the conductivity of the plasma. Petschek (1957) presented both experimental data and a theoretical analysis on the ionization processes that occur i n argon after the passage of a strong shock wave which has been generated i n a conventional shock tube. The temperature that can be obtained i n a conventional shock tube i s limited by the thermodynamic properties of the driving gases. A higher temperature can be obtained by adding an explosive to the high pressure section.  The maximum temperature w i l l again be limited by such factors  as the burning rate of the explosive and the thermodynamic properties of each of the gases i n the system. The highest temperature behind a shock wave can be obtained by generating the shock wave by electromagnetic means. Fowler et a l (1952) f i r s t proposed that strong shock waves could be generated by an electrical discharge.  This method consists of very quickly depositing the energy of  a charged condenser bank into the gas that i s between two electrodes at one end of a shock tube. The electrical discharge that occurs between the electrodes generates a shock wave. Kolb (1957) introduced the T-Tube driver (Figure l a ) that u t i l i z e s the Lorentz force caused by close coupling between the current-return conductor and the discharge i n the gas to give additional driving energy to the shock front.  Kolb studied the properties of very high  energy shock waves passing through deuterium. He was unable, however, to measure the temperature of the high temperature gas associated with the shock wave. In a subsequent publication Kolb (August 1957)  discusses a  successful method of increasing the velocity of the shock wave by applying a strong longitudinal magnetic f i e l d to the shock wave while i t i s leaving the  - h -  main discharge chamber. Kolb (1958) has also ec^erimented with colliding shock waves and has by this means obtained temperatures of about 2 x 10^ °K., as deduced from the shock velocities.  Josephson (1958) studied the pro-  perties of the conical driver shown i n Figure 1(b).  The advantage of this  system over that of Kolb was that this driver was symmetrical thus more l i k e l y ensuring i n the formation of a plane shock front.  A plane shock  front w i l l travel down a shock tube with less attenuation than a non-plane shock front, because the radially-travelling energy i n a non-plane shock front w i l l be attenuated upon reflection from the walls.  Josephson also  studied the shock driving properties of an electrodeless discharge which he obtained by discharging the energy of a charged condenser bank into a c o i l wrapped around one end of the shock tube.  The intense f i e l d i n the  gas caused by the c o i l resulted i n breakdown of the gas, absorption of energy and then the generation of a shook wave that had the characteristic of being free of electrode contamination. Hart (i960) experimentally optimized the angle of the cone that was used i n a driving assembly quite similar to that of Josephson and also experimented with various cone materials.  The r a i l  drive method (Figure 1(c)) i s discussed by Bostick (1958). The driver that has been developed for the present work i s shown i n Figure 1 (d). This driver i s similar to that used by Kolb (1957) but i s more efficient because a closer coupling has been employed between the current-return conductor and the discharge i n the gas. A l l experiments have been of an exploratory nature i n order that familiarily with techniques may be gained and a firm foundation l a i d for more elaborate experiments. The work to date has been somewhat hampered by lack of equipment.  In particular, the condenser bank that has been employed i s  - 5quite slow and has a very low energy storage capacity - 450 joules. Nevertheless, very careful design of the main discharge circuit has resulted i n the achievement of comparable temperatures, as deduced from the velocity of the shock front, to those obtained by Kash et a l (1958) who employed a f a r faster condenser bank. The experimental measurements on the shock wave have been concerned with obtaining time-integrated emission spectra transverse to the shock tube, distance-time relations for the luminosity associated with the shock wave and conductivity-position relations for the plasma associated with the shock wave. The investigation has been concerned with shock waves i n argon and helium possessing sufficient energy to heat the unexcited, un-ionized gas atoms immediately behind the shock front to a maximum temperature of about UOO,000°K. One of the objects of the investigation has been to make a detailed study of the properties of shock waves i n argon which produce a higher temperature than that considered  (27jOOO°K.  immediately behind the  shock front) by Petschek, Rose et a l (1955) and Petschek (1957). The present apparatus can be readily modified to yield far higher temperatures by replacement of the present condenser bank with a bank possessing less inductance and higher energy storage capacity.  These superior  condensers w i l l be installed as soon as they become available.  - to follow page  5 -  | Temperature  |  Luminosity  Ion  Density  Electrical Conductivity  Figure 2  :  Characteristics  of a Strong  Shock Wave  - 7U  T?Z  -  U  C  (5)  T  V  -  R,T  (.6)  r-1 where ^  = molecular weight of gas and  R,  = C  _ P  C  » R = gas con-  V  stant per mole « 8.315 x 10^ ergs per °K. mole (Perry, 19Ul), when P i s expressed i n dynes/sq. cm. and j> gas  K « 5/3.  Representative of the values quoted by Din (1956) are,  » 1.6U0 for P  1 atmosphere and 600°F.  3  For argon there i s considerable departure from this value  at high, pressures. y  i n grams/cm. . For an ideal monatondc  = 1 atmosphere and  T • 600°F.,  The value of  T • 90°F.,  * « 1.793 for P  K - 1.669 for P  = 200 atmospheres and  • T=  X • 5/3 for both argon and helium (Perry, 19U1) over  the pressure temperature range of interest appears to be quite justified. Equations 1 to 6 can be solved for T^,  P^, jo, , and u, i n terms  of the velocity of the shock front and the variables shock waves P i » PQ and  » %  0  and J> . For strong 0  and the equations reduce to  2(r-i)  T,  P  (7) (8)  P,  if r T,  •  £r  -  —  j>.  '«»  R,  %  (9)  *  (10)  •  4  5  •  (11) (12)  - 8 P,  -  f A  (13)  J>,  -  4  J>  Oh)  0  Alternatively these equations may be expressed as functions of Mach number i n the undisturbed gas at temperature TQ,  M  K« ~JI  // y (?, T  U  where  Rj_  cms./sec. 2.  83  ^*J^ fi Q  1 0  ergs/°K.gm.  for v  (15)  5  s o u n c  i  to be measured i n  i n argon.  Low Degree of Ionization Resler, Lin and Kantrowitz (1952) consider this case and state  that the internal energy of the gas when radiation i s neglected i s given by,  U Here  -  - ^ T T - R , T («••<) Y-  +  volts*  (16)  i s the fraction of the original monatomic gas which i s  singly ionized at a given temperature and pressure, stant  - x - ^  8.6l6 x 10~^ ev./°K.  k i s Boltzraann's con-  and x i s the ionization energy i n electron  The f i r s t term on the right hand side of this equation represents  the internal energy of the atom, ion and electron gas, there being 1 + °c times as many particles i n the ionized gas having three degrees of freedom as i n the un-ionized gas. The second term represents the internal energy that i s due to ionization.  Excitation and dissociation contributions to the  internal energy are neglected i n this equation. For a monatomic gas such as argon or helium, there i s of course no dissociation energy. The variable oc introduced i n equation 16 must be determined from an additional equation of state.  One approach to this problem i s to assume  - 9that theraodynamic equilibiriam holds for the ionization and recombination reactions that determine the density of ions and electrons i n the gas. Saha's equation then yields the relation (Lin, Resler and Kantrowitz, 1955), ~  Q  ~  (17)  e  where G i s a constant which for argon equals  1.73 x 10"  c  when T i s  i n °K. and p i n centimeters of mercury. Guman (1958) considers i n more detail the equations for a shock wave than do Lin, Resler and Kantrowitz (1955)* but unfortunately also assumes that Saha's equation i s valid.  There  i s considerable doubt that Saha's equation i s valid for the conditions i n the high temperature gas following a shock wave because the gas i s not i n an isolated system. The gas i s continually losing photons and the reactions that are believed to proceed do not proceed to equilibrium concentrations of the constituents,  X  +  Kv>  x + e where x 3.  ^  X*  +  (18)  e  ^ r x * ^ 2 e  (  1  9  )  represents an atom of the gas considered.  High Degree of Ionization When the gas behind the shock front i s highly ionized, the shock  wave w i l l have the characteristics shown i n Figure 2. It i s very l i t t l e trouble to extend the analysis of Petschek and Byron (1957) to be applicable for the description of a shock wave which i s sufficiently strong to result i n multiple-ionization and any degree of ionization. 2L p  ;  y  ;  Excitation energy could be included as an extra term of the form i n the internal energy equation.  However, since excitation  - 10 energies would be quite d i f f i c u l t to measure i n the present apparatus, the present analysis w i l l neglect excitation energy.  ^  would be the exci-  tation energy per gram of the gas atoms which have the excitation energy }j;  and p  would be the fraction of the original gas atoms that have the  x  excitation energy if; • Let i* -ionized,  X. = ionization potential of the  h  P  l»  p 0  >  » fraction of the i n i t i a l gas which i s  %  i ^ - i o n i z e d ion.  Then for  »  P,  =  m ( U, *  ^Lof)  P,  (21)  rr, CC, =  P , cu,  (22)  = ( i+ r^)joR,T,  (23)  From equations 21 and 22,  U,  = j  The temperature of  (2$)  T  behind the shock front i s desired as a function  2£ and «^ , so a solution of the above five equations for CO and t  yields, From equations 20 and 21,  P.  - J>.  (26)  ~ >) < u  u  From equations 23 and 26, (l+ I x V )  R T, t  =  (27)  From equations 24 and 25, l  2  '  '  r-l  r,  - r  R,t  ^  •  \  *>«•'*•)  .  +  ^-  *.  £  (28)  T^  - 11 U, as a function of V and  i s found by combining equations  %  27 and 28, U.  1- 2 Z  LL.^-U,)  « 3  2  (  2  0  )  Y = 5/3,  In particular, i f <*,  -r>  ZA  - 5 ^,  S  2  2 Z *  1-  X.  (30)  ~  Therefore, or  *  (3D  Combining equations 27 and 31 yields the desired equation for T^,  T, =  i ^ S * - ^ £ Y j 2^ +  fs 2^ 44 R. ( i + r **..) \  kA  V  5  v  Z «.y(-  *  5  <</  (32)  It i s of interest to calculate the value of /u  t  from equation 31  for singly-ionized argon (x^ - l5«75 ev.), 3 ZT  • S  U,  + J<f  z/s  1  -  i2. IF *  io oc, l%  (33)  For helium (x - 2U.58 ev.), x  ' "  ?  Once more an additional relation between oc and T i s needed. K  Petschek and Byron (1957) have obtained a relation by balancing the rate of change of the thermal energy of the electrons with the net rate of energy input to the electron gas.  The net rate of energy input to the electron gas  - 12 i s controlled by elastic collisions between the electrons and the atoms and ions i n the gas.  The rate of energy loss by the electrons i s controlled  by electron-atom collisions.  Khorr (19J?8) has derived a general expression  that i s applicable for multiple-ionization and that relates oc^ to  T  by  considering equilibrium between the photo-recombination and ionization by collision reactions included i n equations 18 and 19*  It i s beyond the scope  of this thesis to consider these theories i n detail or to either expand or adapt them to be applicable to the high temperature conditions obtained i n the present apparatus. One point that should now be mentioned i s that at the high temperatures now obtained, there w i l l be appreciable radiation loss. An adequate theory yielding a relation between T  and «.  (or  T  and o c for multiple;  ionization) should include consideration of the loss of de-excitation and de-ionization photons and continuum radiation.  Figure 3  :  Shock  Tube  Ill  1.  APPARATUS  General A schematic showing the dimensions and layout of the shock tube  i s given i n Figure 3.  Not shown i s a grounded shield enclosing the main  discharge circuit and consisting of a box made of 1/32 inch thick aluminum sheeting. The shock tube was evacuated with a Cenco Hi Vac Ih mechanical pump rated at 0.1 microns.  The lowest system pressure that could actually  be obtained was 0.1 microns.  The working evacuation pressure before admis-  sion of argon (99.98$ pure) or helium (unknown purity) was about 1 micron. The minimum system pressure during f i r i n g was $00 microns and the pressure rise due to f i r i n g was then about 90 microns.  The concentration of impur-  i t i e s i n the ambient argon gas i n front of the shock wave was, therefore, equal to or less than 6 x 10"^.  Pressures were read from a Pirani Gauge  which was periodically standardized against a McLeod Gauge. The spectrum of the luminosity emitted by the shock wave contained no lines that could be attributed to gaseous impurities.  A more elaborate pumping system was,  therefore, not justified. The high voltage power supply was a N.J.E. Corporation 0-30 Model H-51  KV.  unit. Electrical signals were observed with two Tektronix oscilloscopes,  one type 5>35 and one "type f&l.  The pre-amplifiers that have been used i n - 13 -  -  to fol|ow  page  13 -  L  4a  Single  Components  -HRT  W  1  Y  condensers per switch  1  -r x 4 b  Multiple  ;  Components  ^oTP L  c  C:  4 c  :  • Shared  Inductance  M  c 4d  Figure  Mutual  4  :  Inductance  Discharge  Circuits  S  switches  - lit these oscilloscopes have included types G, K and 53/51* D. Permanent records of signals have been obtained with a Dumont polaroid oscilloscope camera. 2. Main Discharge Circuit The shock wave i s driven by an electrical discharge.  The para-  meters of importance i n the discharge circuit are the rise-time of the current, the energy stored and the ratio of load to lead resistance.  A pre-  liminary observation of the current waveform revealed that i t was essentially a damped sinusoid.  The simplest equivalent circuit that w i l l result i n this  current waveform i s a circuit having non-time-dependent components and consisting of a series resistance, capacitance and inductance. The following analysis applies to such a circuit (See Figure l*a),  A (t = o) = 0 ±L dt  (  =  (±,o)  (36) (37)  L  The general solution for equations 35 to 37 for an underdamped discharge i s , f±\  (tj  V  -  °  Q  ~  S  +•  •  t  (38)  tot;  Sin  co L  where u> = cyclic frequency ficient =  = ^J-~-  - $  1  and  S = damping coef-  R  2L The power dissipated i n the resistance has the form, p  W '(sr)*  R e  _  1  "sin*«t  Both the power and the current have maxima when ^ •tan  u>t -  (39)  • 0 or when, (1^0)  -15 The extension of this analysis to the case of multiple condensers and spark gaps (Figure lib) i s readily made and yields the results, s  (ia)  I  _ ait  Vo  (42)  where  (43)  and  (U4)  Note that the damped period of the waveform i s ,  •  r  =  (45)  and the undamped period,  (46) The maximum current and maximum power dissipated i n the load occur when  —  = 0  or when tan wt • ^  or when wt = —  - arc tan —  .  A first-order approximation yields,  - ri.  Vo  (47)  2 co  R  e  (48)  e  A second-order approximation, letting wt » — - — 2  yields,  - 16 -i/i-  _L)  i!0  ,/JL - A.)  Vo R ,  (50)  e  Equations k9 and 50 can be further simplified for small damping by letting nC » C  T  and  (51) Then,  Cr  7T e  U  ^ " 4  . Lc ^ I „  s  n  s  f t r +r x= 7Tr 3  (52) *  L  '  2. / ^ U ., L, "5" • vr +  (53)  Usually the value of the exponential factor i s very close to unity. Making this approximation yields, Cr  Pj J  *  v  /  .  (5U)  (55)  The time integral of the power dissipated i n the gas i s the energy that i s absorbed by the gas.  This energy w i l l generate a shock wave i f i t  i s deposited i n the gas sufficiently fast.  The gas w i l l expand shortly after  i t absorbs energy; subsequent energy that i s put into the gas behind the shock wave does not affect the propagation characteristics of the shock  - 17 wave. The actual energy i n the shock wave would be extremely d i f f i c u l t to calculate but can be expressed as, DO  E,h.cU where  -  j Pj(±)  (56)  g(t) i s an unknown function that includes the effects of the expanding  plasma i n the main discharge. The effect of shared inductance i s to increase the period of the ringing of the current.  A simplified case i s shown i n Figure Uc and the  following analysis applies (The impedance Z i s the output impedance plus load impedance):  Z  =  +  J L o L .  ;  2 = O Uen  -wcfL^Lj +  (57)  l-^UCJHl-^Xcyi-^cfL^u))^  85  or when,  2.  2  0 3  Lj * 2 U  i- L, - ^  2 C ( ( L , +UyL«  =  +L^) *  4  L / + L,  x  Lj L )  Consider the special case when L, =• Lj - o.i L  0 9  r  " o.4 3, U C =  (  e  c  = 2 TT^O.^OO  )  (60) LC C  (61)  T  The corresponding period for L,.-0 and L^= o.» L T  9  and C,.- 2 C ,  T  2TT^O.i>3\  *  LC t  T  c  is, (62)  Therefore, the period of the main discharge i s increased slightly by shared inductances.  - 18 Mutual inductance i n the main discharge circuit can decrease the period of the ringing of the current. The following equations are the result of an analysis of the simplified circuit shown i n Figure Ud:  T  =  2 7 T  V  / C  T  (  L  +  S  U=  The coupling coefficient  (6k)  w i l l be negative for the current  flow directions given i n Figure Ld. Therefore, for the usual side-by-side placement of condensers (resulting i n a negative value for k ) the total circuit inductance w i l l be decreased by the mutual inductance between condensers.  The resonant frequency i s correspondingly increased and the period  decreased. 3.  Conductors The design for the conductors that carry the high current i n the  main discharge circuit i s based on the consideration of four problems: i ) the inductance must be a minimum  i i ) the resistance must be a minimum  i i i ) the conductors must have sufficient mechanical strength to prevent either separation or rupture caused by forces exerted by the current pulse. iv) skin depth effects must be considered. Both coaxial and coplanar type conductors are i n common use i n high current discharge circuitry.  A third, less commonly used method i s  to use many parallelled lengths of commercially available coaxial cable (Smith, I960).  In the driving mechanism used i n the present work both coaxial and coplanar conductors have been used.  The inductance per unit length for  a coaxial conductor when a l l current i s assumed to flow at the facing surfaces of the conductors i s , L  ~  =  henries/meter  where M = permeability of the dielectric, • k  x 10  (65) h./m. for free  space, /i- = inner radius of outer conductor, and A.^ = outer radius of 0  inner conductor. When edge effects are neglected the corresponding equation for the inductance of a coplanar conductor i s , L where  =  M — w  henries/meter  (66)  d • conductor separation, and w" » conductor width. It i s to be noted that the inductance of two closely spaced con-  ductors could be calculated exactly from a knowledge of the current d i s t r i bution i n the conductors. However, the major component of the total inductance i s caused by the current that flows on the facing surfaces of the conductor.  Inductances caused by interaction of currents elsewhere i n the  conductor are of higher value than that caused by the current flowing on the facing surfaces of the conductor.  The important point i s that a l l these  inductances are i n parallel, so the total inductance depends primarily on the one of lowest value - that caused by the close spacing between the inner faces of the electrodes.  Also, of course, a more c r i t i c a l treatment would  have to include both the skin effect and the fact that less than half the current flows on the inner surfaces.  - 20 For low inductance i t i s apparent that the insulating material between the conductors should be very thin - the separation being determined by the dielectric strength of the insulator. After experimentation with celluloid, polyethylene and mylar for use as an insulating material, i t has been found that mylar possesses the best properties.  It has less surface  leakage than either of the other materials tried, excellent dielectric strength - 6 x 10^ volts per centimeter as measured by Inuishi et a l (1957), and possesses the highly desirable property that i t i s adhered to by epoxy resin.  It has been found that epoxy resin i s an excellent high voltage  insulator so i t has been used extensively i n the construction of the high voltage equipment. The resistance of a conductor to high frequency current i s a function of the skin depth (Gray, 1957),  i  .  f j ~ y  where ^ - permeability,  (67)  cr uJ  or » conductivity, w  = angular frequency.  At a frequency of 200 KG the skin depth i n copper ( M^ 1.00, m  cr=  L _ x 10^ mhos/cm. (Hodgman,195U)) i s l.M>5 x 10"^ meters or 5.77 mils. 1.72U  The surface resistivity i s given by Gray (1957), R  s  ' ^^  ~  ohms/square  (68)  Two times the surface resistivity i s the total body resistivity when the skin depth i s appreciably less than the thickness of the conductor. For example, the resistance of a copper conductor 1/32 inch thick, 6 inches long and 1 inch wide to a 200 KG sinusoidal signal i s due to conduction on  - 21 both surfaces and equals  (£) -  x  x  10  S)  = 0.3U7 x 10~3 ohms. Equations 67 and 68 apply when the current i s sinusoidal.  For a  non-sinusoidal current such as that i n a damped oscillatory discharge, the solution to the f i e l d equations for the fields inside the current-carrying conductor becomes more involved. Gray (1957) analyzes the step and ramp current cases.  Levine et a l (1958) applies Gray's results to the design of  high-current high-frequency conductors.  One can use the curves given by Gray  to estimate, for example, the f i e l d strength at a depth of x « 10"^ meters after the f i r s t quarter wave of a sinusoidal current of period four microseconds.  One can approximate the sinusoidal current by a ramp current that  builds up to peak value i n one microsecond. *J^f  The value of the parameter  i s then 0.86 and from the curve given by Gray, the desired value of  the f i e l d i s H(io  w.,10  sees.) =  0.33 H(o»"-,Jo  sees.).  The conventional  definition of skin depth i s that depth at which the fields are attenuated by a factor of l/e.  The same definition applied to the case of a ramp on  for one microsecond yields the result that the effective skin depth i s about 10""^ meters, a value somewhat less than the skin depth for sinusoidal current having a frequency the same as the fundamental frequency of the ramp. At the present frequency of 200 KC, not much evidence of the non-sinusoidal nature of the discharge on the current distribution i n the conductors should be noticed.  However, at higher frequencies consideration of the pulse nature  of the current w i l l be necessary.  Another factor of importance i n the design of the conductors i s the mechanical force exerted by the currents. Between wires (from Gray,1957),  -  to follow page  1—  21  Driver  - I 2 |nel<  InStde  Diameter Tubing in  (  Brass  2 inches  Length  Sheet Conductor To  Tte  r m i n a  Is  o f  ^  ^rl \s s s  Condensers — < ^ .  ^  Mylar  s  S  S  s  S  y  y  S  S  S  S<  To  Terminals of Condensers Cross-sectional  View  Figure 5  :  High  Current  Switch  - 22 2 I, I F  -  •  -7  a  d  *  (69)  1 0  while between coplanar conductors,  F =  4  T  - *  -7  T  fa  2 o(  In equations 69 and 70  x  h '  y io  (70)  F i s the force i n newtons per unit length  and i s repulsive when the currents I'i and I2 flow i n opposite directions, d  i s the separation between the wires, b  coplanar conductors, a  i s the separation between the  i s the width of the coplanar conductors. For  typical values of a = .0254 m.,  b « 2.03 x 10'^ m.,  amperes we find from equation 70 that  Ii-  I » 65,000 2  F «• 102,000 newtons/meter = 230 lbs/cm.  This force only exists while the current i s large, a time of a few microseconds. The physical displacement of the conductors w i l l be governed by the mass and by the method of clamping.  A force of 230 lbs/cm. on for a few micro-  seconds can be handled by an extremely weak clamping arrangement. From equations 42 and 5>6 i t i s evident that to obtain maximum current and power i n the load the important object i n the design of the conductors i s to minimize their inductance and resistance. U. High Current Switch The main discharge must be turned on with a minimum of switching effects.  Both three-electrode spark gaps and ignitrons are popular for this  purpose.  A three-electrode spark gap has been selected for use on the shock  tube driver.  The construction i s shown i n Figure $ and the inductance can  be calculated by considering two coaxial conductors i n series of dimensions  ro = .0195 m., 1 = .01 m.  ^  31  .009 m.,  1 » .038 m., and TQ = .0195 m., r^ = .00079 m.,  The latter figures apply to the discharge channel, the radius  -  to follow  page  22Trtyratron  To  Tr ' g g e, r  " Trigger  Circuit  Electrode  Synclnr oni 2 ing  amera  Circuit  Other Instrumentation  RG/<2U Q Coble  X  to  Trigger Electrode (<73 ohm t e r m i n a t i o n at  trigger electroo/ej  7 b  Thyrotron  :  Trigger  Release J/  To  Fire  -fl To T r i g g e r Electrode  O.I y u f .  SparU  5 7  Figure  Spark  C  7  :  Trigger  Coil  Trigger  Circuits  - 23 being estimated by visual observation during f i r i n g .  The total inductance  of the switch from equation 65 i s , therefore, 6.1 milli-microhenries.  The  resistance of the switch during conduction i s unknown. 5.  Trigger Unit Instrumentation such as a framing camera or a rotating mirror  spectrograph w i l l probably necessitate synchronization of the main discharge with some event such as a certain angular position of the mirror i n the camera or spectrograph.  When such instrumentation becomes available, a  suitable synchronizing trigger circuit w i l l be required. The circuit shown i n Figure 6 has been designed and built for this purpose. The synchronizing circuit operates on an input pulse of at least +10 volts from the camera. Within 1.0  - 0.1 microseconds i t sends out a  200 volt pulse suitable for f i r i n g the large thyratron i n the high voltage trigger circuit (Figures 7a and 7b).  The 5 K7 pulse from the large thyratron  i s applied to the trigger electrode of the three-electrode high current switch i n the main discharge circuit.  The expected time Lag between the input  pulse and the breakdown of the main switch i s 1.5  - 0.2 microseconds.  There has been an alternate output built into the synchronizing circuit that supplies a 200 volt pulse i n a controlled time interval after the input pulse i s applied of from 3 to 1000 microseconds. The large thyratron can be t r i g gered from either the delayed or undelayed outputs.  The unused output of  the synchronizing circuit can, of course, be used to trigger other instrumentation. Push-button f i r i n g i s an alternate mode of operation that has been built into the synchronizing circuit should i t not be required to synchronize the firing to the instrumentation.  - 2k The interesting electronic features of the synchronizing circuit are:  i ) The use of a thyratron i n the input stage i n order to trigger the  monostable multivibrator with only one pulse.  A l l of the manual switches  that the author has tested for positive closure have been unsuitable because they possessed contact bounce. The effect of such multiple pulses when fed into a monostable circuit makes the operation of the unit unpredictable. The alternative of an electronic switch - a thyratron - was chosen as a suitable driver for the monostable circuit.  A second possible circuit which  could be substituted for the input thyratron should less time-lag be desired i s a binary,  i i ) The use of a diode clamp and a high pass f i l t e r on the  delay circuit i n the monostable unit. The diode clamp circuit decreases the j i t t e r on the delay and i s a commonly used circuit.  However, i t i s be-  lieved that the high pass f i l t e r i n this circuit i s an original contribution.  As the need has not yet arisen to use the synchronizing unit, a l l firings have been triggered with the spark-coil circuit shown i n Figure 7c. 6.  Condensers The energy stored i n the condensers must be transferred to the  load as quickly as possible. From equations 42 and $6 i t follows that desirable properties of the condensers i n order of importance are:  high  voltage, high capacity, low internal inductance, low internal resistance. The internal inductance and resistance are the only two properties that are d i f f i c u l t to measure. The inductance can be obtained approximately by connecting a large cross-section conductor between the terminals and leaving a small gap i n the circuit,  when the condenser i s discharged at  gap breakdown voltage, an approximate value for the inductance can be obtained  Figure 8  :  T-Tube  Driver  - 25 from the ringing frequency.  The internal resistance can be obtained by a  similar method by comparing the circuit resistance as determined from the logarithmic decrement of the discharge current waveform with a similarly obtained value when a known low resistance i s connected i n series with the gap. The condensers that have been used i n the present work are four of General Electric 1 microfarad, 15 K7 each having an internal inductance of about 250 milli-microhenries. 7.  Electrode System The object of the design of the electrodes i s to obtain a maximum  shock velocity for a given condenser energy. The type of electrodes that have been developed i n the present work efficiently use the Lorentz force between the currents i n the return conductor and the discharge plasma to obtain a high shock velocity. I n i t i a l experiments were performed with the driver shown i n Figure 8, which w i l l henceforth be called the T-Tube driver. The electrodes consisted of 3/h inch diameter brass rods and the return conductor one-half of a split length of ;L12 5V' inch diameter brass tubing.  The distance between  the electrodes and the return conductor was about 1/8 inch and was determined by the glass tubing of which the T-Tube was constructed.  The inductance of  this driver was quite high because of the large spacing between the electrodes and the return conductor.  The inductance as determined from the ringing  frequency of the current was 330 milli-microhenries.  A position-time curve  for the luminosity of the shock wave generated by this driver i s given i n Figure 2.1  .  iConde risers.  to  follow page  25  -  - 26 The driving effect of the return conductor was determined by moving i t two inches directly back from the discharge chamber. The velocity of the luminosity front decreased from 0.5>1 cms./microsecond to 0.39 microsecond at a distance of 25.5 were discharged from 15 KV.  cms./  cms. from the electrodes when the condensers  The period as measured from the ringing frequency 2  increased from 7«2 to 8.0 microseconds. The i n i t i a l energy (fcv ) was 450 joules. The effect on the shock velocity of the spacing between the electrodes was also determined. The result was that maximum velocity was obtained for an electrode spacing of 3*5 cms. between nearest points of the electrodes. This spacing roughly corresponded to that which would offer the smoothestwalled discharge chamber. The curve given i n Figure 2) was obtained with this electrode spacing and with the current return conductor tightly clamped to the T-Tube. The electrodes i n the series gap shown i n Figure 8 were spaced at such a distance that conduction occurred at a voltage of about 15 K7.  The  actual f i r i n g time of the T-Tube driver was thus somewhat undependable. An attempt was made to increase the i n i t i a l rate of use of the current i n the main discharge by shunting the base of the electrodes with a "current crowbar". The principle of this method i s shown i n Figure 9. The current crowbar consisted of a fuse mounted between electrodes spaced further apart than the 15  spark breakdown distance i n a i r .  The fuse  materials considered were, #32 copper wire, 6 ampere strip fuses and 10 ampere strip fuses.  The results indicated that the method did not increase  the shock velocity because too much power was dissipated i n igniting the fuse.  -  to follow  page  2 6 -  Epoxy  -Teflon(.03l3"j  Lucite  C o p p e r (.OI5fc") NVylar ( . 0 0 5 " )  Cross - section  Through  Electrodes  Figure  10  :  Co-planar  Driver  }  -27-  The results, however, were promising because they suggest that the method should be of value when the shock velocity i s dependent upon the rise-time of the condenser bank. The problem that was not solved but that probably could with considerable more experimentation would be to optimize the fuse ignition time and thermal properties.  These properties could probably be  optimized by varying the ambient pressure and temperature and the fuse material.  Subsequent to the completion of the experimentation just described,  a paper was found i n which James and Patrick (1958) describe a successful application of this principle. After some experimentation with the T-Tube driver, i t became obvious that a higher shock velocity should be obtained i f the inductance of the driver could be decreased.  The driver shown i n Figure 10 was then  developed and subsequently proven to produce a higher velocity shock wave than the T-Tube driver.  The development of the coplanar driver was directed  toward solving the following problems: i ) The main discharge plasma should not contact any material that i s easily ablated or changed by high temperatures.  This requirement i s impossible to meet because the temperature i n  the main discharge w i l l be at least UOO,000°K. However, an approach to a solution to this problem i s to use a ceramic material.  For ease of con-  struction i n the present apparatus, a compromise has been made and teflon used for the high temperature electrical insulator where required. Other methods that have been tried unsuccessfully are, f i r i n g aluminum oxide, anodizing and using Sauereisen Insalute cement, i i ) The conductors bringing the current to the discharge should have minimum inductance and resistance. The design adopted i s to use 10 mil mylar insulation around and between l/6k inch copper conductors.  The dimensions of each of the conductors i s about  - 28 1 inch by 6 inches, so the inductance and resistance at 200 KC as calculated from equations 66 and 68 i s 1.1  milli-microhenries and 0.0028 ohms, i i i ) The  problem of a i r leaks into the shock tube has been solved by potting the electrodes with the exception of the facing surfaces with epoxy resin, iv) Mechanical forces on the conductors caused by current interaction has been handled by clamping the electrodes together between sheets of Incite. The period of the main discharge current with the coplanar driver connected i n the main discharge circuit was I N 5 microseconds. The total circuit inductance was, therefore, 125 milli-microhenries.  The maximum current as  measured by the method described i n Chapter III , Section 9,  was 65,000  amperes. The low circuit inductance and high peak current can be attributed to both the design of the coplanar electrodes and the other circuit changes that were added simultaneously.  The other changes included the switch  described i n Section U of this chapter and low inductance closely-spaced sheet copper conductors that connected the condensers to the switch.  These  additions are shown i n Figure 10, and Figure 5. 8.  Luminosity Detector The luminosity associated with the shock wave i s viewed by two  photomultipliers spaced 5 cms. apart along the axis of the tube. Standard photoraultiplier circuits  are used i n this unit. The important character-  i s t i c s are, 0.2 microseconds rise-time and 3,000°A to about 6,0O0°A spectral response peaking at U,O00°A (photoraultiplier type 931-A). The f i e l d of view i s determined by adjustable s l i t s inside the box of the detector. of the firings the s l i t widths have been, outer - 0.2 mm., The s l i t height i s fixed at 1.88 s l i t s i s 15.3  cms.  For a l l  inner - .02  mm.  The spacing between the outer and inner  cms. and between the inner s l i t and the corresponding photo-  multiplier i s 2.5 cms.  The f i e l d of view i s thus 0.3 mm. by 3.6 cm. at the  - 29 axis of the shock tube when the outer s l i t s are 8.0 cms. from the axis of the tube.  Each photomultiplier i n the luminosity detector thus essentially  samples the luminosity from a cylindrical volume i n the shock tube of dimensions  rr x tube radius^ x 0.3 mm.  Nb attempt as yet has been made  to make absolute measurements of the intensity of the source emitting the light. The photomultipliers w i l l overload on the light from strong shock waves. Two methods of readily controlling the sensitivity and thus avoiding overload problems have been adopted: i ) neutral density f i l t e r s (fogged film) have been inserted between the photomultipliers and the inner s l i t s and i i ) the dyhode voltage has been controlled. The same power supply i s used for the photomultiplier i n the spectophotometer so that i t i s often necessary to control the sensitivity of the luminosity detector by the more laborious f i r s t method. 9.  Current Detector The current i n the main discharge can be determined by measuring  the voltage drop across a low known-value resistance that i s connected i n series with the main discharge.  Olsen and Huxford (1952) measured the  current i n a k KV discharge with a 0.053 ohm low inductance resistor. In the present apparatus the current can reach a peak value of about 65,000 amperes thus creating very large magnetic f i e l d s .  Because the dis-  charge i s oscillatory there also i s generated a strong electric f i e l d that i s 90° out of phase with the magnetic f i e l d .  The magnetic f i e l d generated  by the main discharge has been u t i l i z e d to measure the current thus bypassing  -  Figure  to follow  II = Current  page  2 9 -  Detector  - 30 the problems that would have been encountered should a resistor method have been adopted: i ) frequency dependency of the resistance of the resistor (a skin depth problem) i i ) spurious signals generated i n both the shunt resistance and the voltage measuring cable by the very large fields present and i i i ) loss of power i n the resistor. The magnetic f i e l d generated by the main discharge i s measured with two 100-turn coils jumble-wound on a l / l 6 inch diameter form with size B & S 36 wire and connected as shown i n Figure 11.  Connections between the  coils and the differential inputs of an oscilloscope were made so that the in-phase voltages (generated by pickup of the electric field) cancelled and out-of-phase voltages (due to magnetic f i e l d pickup) added. The capacitance of the connecting cables and the inductance of the coils was such that the 120 ohm resistors gave c r i t i c a l damping to the coils which then had a frequency response f l a t up to the resonant frequency of U mcs.  The coils  were placed inside the loop formed by the main discharge circuit and oriented so that the axis of the coils coincided with the axis of the loop formed by the main discharge circuit. The energy stored i n the condensers i s completely dissipated by a firing.  The total charge stored i n the condensers i s , therefore, equal to  the integral of the current flow,  dQ  hut  f  =  J^clQ  »o  I Jit  VC 0  (71) (72)  cO  therefore  V„ C  =  f  I  dt  (73)  - 31 The observed differential voltage induced across the coils f ( t ) i s given by Faraday's law,  fM «  -gr dt  (7U)  d  at where <J> i s the magnetic flux which links with the coils.  Since the magnetic  flux i s linearly proportional to the current generating the flux, we have,  = k  m  (75) at  where k i s a constant depending on the c o i l geometry and position with respect to the current-carrying conductors. Integrating equation 75 and noting that  >  - i  X(t  f  I(o) = 0 yields,  fMJf  ( 7 6 )  Q  Equation equations  73 can now be used to find the value of k. Combining  73 and 76 , * v  - ire  J  J  t H  v  )  ^  A  t  (7»  Equation 76 w i l l yield the value of I(t) when the value of k as determined from equation 77 i s substituted. The value of k w i l l change only when the geometry of either the c o i l or discharge circuit i s altered. Equations 76 and 77 are extremely general, applicable for determination of the current as a function of time regardless of the shape of the waveform. The c o i l method for measuring the current i s thus applicable to any circuit i n which energy of known i n i t i a l and f i n a l value i s dissipated sufficiently fast to produce a measurable magnetic f i e l d . When the current waveform i s known to be a damped sinusoid and the capacitance and energy change involved i s known, then the value of the current can be readily found.  The value that i s of most interest i n the present  - 32 apparatus i s the peak current - which occurs at the f i r s t maximum of the damped sinusoid. For the damped current, T(t) =  ¥L  e  "  J  s i ^ t  t  (78)  CO l _  the observed waveform w i l l be, -f(t)  k ^ I  =  ^  cotct)  (79)  Measurement of the ratio of the amplitudes at two successive maxima of f ( t ) , f ^ and f , w i l l give 2  J , (80)  e  or  In practice t - t ^ can be obtained more accurately by measurement 2  of the time interval between successive zero values of f ( t ) rather than maxima. The value of co  can also be readily obtained from LO  t  2  - t]_ ,  7T  t -t, 2  We f i n a l l y get, 3T»n«x  ^  =  Sl  *>  6c>t*  60 1_  (81)  where <3  (83)  n =  ^ - t,  L  =  t *  =  '• —  arc  (8U)  t a n  (85)  Equations 81 to 85 have been used i n the present work to determine the value of the peak current i n the main discharge.  -  to follow page  3 2 -  To  Oscilloscope  lo  Figure 12  :  Electric  Probes  - 33 10.  Voltage Measurements Very l i t t l e work has been done by the author to perfect a method  for measuring the voltages i n the main discharge.  The usual method of  tapping off the voltage with a resistive attenuator has been tried. The problems of skin effect, and anomalous voltage pickup due to flux linkage with the measuring cables and connections were not tackled. The only problem that was considered was the faithful reproduction of the voltage across a resistive attenuator.  A voltage fed into a high resistance ( R ) i n series  with a low resistance  (r) has a transfer function of the form  § L  =  I +  Em where R,  •  R  EfJL^.  (86)  c i s the total capacity across r , C i s the total capacity across r  E Q i s the output voltage across  r,  i s the input voltage across  R + r, and p i s the differential operator ...^ . To obtain a frequency independent transfer function the following condition must be met: CR  (87)  - cr  In actual circuit values,  R • 100,000 ohms, r » 100 ohms,  G » stray capacitance, and c required to give faithful response of a step input voltage was 11.  .005 microfarads.  Electric Probes Under special conditions, electric probe studies can yield electron  energy distribution, electron temperature (when the velocity distribution i s Maxwellian) and a rough estimate of the electron density. Basic probe theory i s discussed by Loeb (1955) and Francis (1956).  - 31* Electric probes have been used i n the present apparatus to measure the conductivity by the same method as used by Lin, Resler and Kantrowitz (1955). 12.  Figure 12 gives the circuit details and probe dimensions.  Spectrophotometer* A small Hilger spectroscope has been fitted with an exit s l i t , a  photomultiplier and a cathode-follower.  The electronic circuitry i s quite  standard. The spectral range covered by this unit i s 3000°A to 6000°A. The sensitivity of a spectrophotometer i s far greater than that of a spectroscope employing photographic plates, so this unit i s proving to be a valuable adjunct to the other spectroscopic equipment presently i n use.  The dis-  advantage that only a limited spectral range may be viewed at one time i s greatly compensated for by the sensitivity and the time resolution obtained. The spectral resolution of this instrument i s quite poor, so construction u t i l i z i n g either a spectroscope or a monochromator having higher dispersion is planned. 13.  Time Integrated Spectroscopy A considerable number of time-integrated spectra have been taken  with the spectrograph positioned transverse to the shock tube.  Progressively  more sensitive and higher dispersion spectrographs have been used.  The  instrument that has been f i n a l l y adopted i s a Hilger double automatic large quartz and glass spectrograph. The spectral range that has been investigated is  2800°A to 7500°A and both Kodak IIF plates and Ilford HP3 plates have been  •M- W.V. Sinrpkinson has been responsible for the construction and operation of this unit.  - 35 employed over their respective sensitive spectral ranges.  It was found  that multiple shots (up to 12) were required i n order to obtain sufficient exposure, even vrhen a collimator lens (f/2.0) was used to focus an image of the shock front onto the entrance s l i t . lU.  Apparatus to Measure Conductivity Considerable effort has been expended to develop a technique to  measure the conductivity of the plasma associated with the shock wave. A description of this apparatus would involve a large number of design considerations and i s , therefore, postponed to the next chapter.  -  to follow  page  3 5 -  Figure 13'•• Coordinate System For Conductivity Measurements  TV  MEASUREMENT OF THE ELECTRICAL CONDUCTIVITY OF THE PLASMA ASSOCIATED WITH THE SHOCK WAVE  1.  Theory An excellent method of measuring the conductivity of the plasma  associated with the shock wave was originally proposed and tested by Lin, Resler and Kantrowitz (1955).  The method consists of allowing the shock  wave to pass through a radial magnetic f i e l d .  The V x B force on the  electrons results i n a circumferential current which can be measured by transformer action into a pickup c o i l .  The output voltage from the c o i l  can then be related to the conductivity of the plasma. The apparatus can be calibrated by one of two methods, either by consideration of a l l geometric factors or by passing a slug of metal of known conductivity through the c o i l at a known velocity. has been employed by the author.  The output voltage obtained during c a l i -  bration as a function of distance waveform  V (t)  definition of  V (s) c  can be found from the observed  and the velocity of the slug  c  s  The latter method  v  c  and other important distances).  (see Figure 13 for a The function  V (s) c  approximates the Gaussian distribution,  Vc(s) where b  -~ V  cp  evp-CI)  2  (88)  i s a length that i s characteristic of the resolving power of the  apparatus. - 36 -  - t o  follow  page  3 6 -  3  4  V(»)  - 2 - 1  O  14a  Typical  :  2  I  Observed  5  £•  7  S in C m s .  7  S In c m s .  Waveform  to b  0.8  0.6  0.4-  a  2  -0.2 -Q4 14b  Figure 14  :  :  Theoretical  Waveforms  ( C - 0** e x p -  -^-^  Signals From Conductivity Apparatus  - 37 The output voltage obtained during a measurement on the shock wave V(s) can be found from the observed waveform  V(t) and the velocity  of the ionization front, U, which i s very nearly identical to the velocity of the luminosity associated with the shock wave. Because a l l of the conductivity measurements on the high temperature gas associated with the shock wave have resulted i n V(s) curves that were essentially Gaussian followed by somewhat periodic oscillations (see Figure lha), i t has been concluded that the conductivity very quickly reaches a maximum value and then decays.  The rate of decay does not yield  much useful information because of the presence of the multiple shocks and f i e l d distortion effects to be discussed i n Section 2 of this chapter. However, the conductivity as a function of distance over the length of the shock wave must be approximately known i n order that the maximum conductivity may be determined. Lin, Resler and Kantrowitz (1955) derive a relation between the output voltage V ( s ) and the axial conductivity distribution G"(f) behind the ionization front through V[s) =  "!?2l^L{o)V (s) c  i£T<r \  - tr(l)V (s-l)+f(r'(s-x)V (x)otA c  c  Jj  c  where U , i s the flow velocity given by equation 31,  J  (89)  I i s the current  i n the f i e l d c o i l during a measurement on the shock wave,  I  c  that during  calibration and cr the conductivity of the slug. The method suggested by c  Lin, Resler and Kantrowitz for determining  from equation 89 i s to  guess at an appropriate function o-[^) and then calculate V(s) from equation 89.  I f the correct function 0~(f) has been chosen, then the  calculated  V"(s) w i l l be identical with the experimental V(s).  A typical experimental curve f o r V(s) i s given i n Figure l U a . The shape of this curve suggests that an appropriate 0"(f) function may be,  <r(0 = o-* where  e  ~  f  /  f  (90)  !  f = s-x Substituting the following five equations: 0"( o) =  Cr*  (91)  ''cr (Jt) = O , -that is  = V  Vc(s)  (92) (93)  e  c p  <r'{.-«)  M~er(Jt)-*o  •  e  1  (  )  '  '  (94)  p  V,60 = V 1  P  e  (95)  into equation 89 yields, s_ b  V  ( 5  )  «  ff  \/  r  e  -(f)' ^1  (96)  or  V(s) =  y(s)  (97)  where V ( ) i s the reciprocal of the bracketed function i n equation 96. s  The maximum conductivity of the plasma i s related to the reciprocal of the maximum of the bracketed function i n equation 96.  Designating the  reciprocal of this value as V, , then, (98)  v., where  l / i s the maximum value of p  V(s).  - 39 Numerical integration of the bracketed function i n equation 96 has yielded the curves given i n Figure lUb. close to being identical to the observed  These curves are sufficiently  V(s) curves (after smoothing out  the high frequency oscillations) to warrant their use i n data reduction. The value of Y,  i n equation 98 that has been used during the analysis of  the data has been chosen as follows: Each experimental curve compared with the curves i n Figure llib. then yielded a value for V, of the conductivity.  V(s) was  The closest-fitting plotted curve  and a value for (3 , the logarithmic decrement  Equation 98 could then be solved for or* . The f i t t i n g  has been done with the object of matching the shape of the i n i t i a l maximum of the experimental  V(s) to that of a calculated V( s)  }  noting that the  amplitude of the negative portion of the experimental V(s) curve i s not dependable. The amplitude of this negative portion w i l l be decreased by the decrease i n  (see equations 31 and 98) and w i l l be increased by the  f i e l d distortion effect to be discussed i n Section 3 of this chapter. It would be desirable to perfect a technique to measure the temperature of the gas behind the shock front as a function of the distance from the shock front. a temperature.  In theory, the conductivity function G~[%) can be related to In the present work no simple theory i s very applicable for  reasons which w i l l now be discussed. The electrical conductivity of a partially-ionized gas i s dependent upon the temperature of the constituents and far less so on the electron density or degree of ionization (at least i n the range of temperature and density that i s being considered). Therefore, i t Is believed that maximum conductivity w i l l occur earlier i n time than does maximum ionization.  Figure 2 i s a diagram of the expected relation between  the conductivity and the atom temperature behind the shock front.  The atom  temperature i s neither the same as the electron temperature nor the ion temperature.  That i s , the plasma being studied i s not i n thermal equilibrium.  The electrons from the ionization by collision reaction (equation 19) are at a lower temperature than the gas atoms so the temperature as calculated from the conductivity measurements (yielding an electron temperature) i s expected to be lower than the atom temperature as calculated from the shock velocity. One point to note, however, i s that the maximum conductivity w i l l occur at an electron temperature that i s considerably higher than the electron temperature that i s attained later when equilibrium between the reactions of equations 18 atures.  and I 7 has resulted i n equal ion, atom and electron temper-  An adequate theory that would relate the conductivity to an electron  temperature and then to an atom temperature (providing a check on the atom temperature as calculated from the shock wave conservation equations) i s beyond the scope of this thesis. An approximate value for the electron temperature during maximum conductivity can be calculated from a relation applicable for the conductivity of a f u l l y ionized gas measured transverse to a strong magnetic f i e l d and for T > 8,000°K., as quoted by Spitzer (1956),  where  ,.3 1\ z I I v  y\ _  3  Ze?  V  7T  f  Me /  o  r a  singly-ionized gas.  The applicability of equation 99 i s i n doubt because the plasma being considered i s neither i n thermal equilibrium nor i s i t fully-ionized. The value of the electron density N  e  i n the conductivity-temperature  - Ui calculations made i n this thesis has been taken to be identical to 1 0 ^ electrons/cm.  This value corresponds to <* = 10  since the atom  density i n front of the shock front i s of the order of l O ^ atoms/cm. 1  at the pressures being used. suitable value for N . e  3  There are two reasons for this choice of a  The f i r s t reason only imposes logical bounds on the  value - which would be 10° < 5L» <  The upper bound has been chosen  on the basis of the work done by Lin, Resler and Kantrowitz (1955) who on theoretical grounds conclude that a plasma w i l l have a conductivity that i s nearly independent of the electron density when <*• > 10"^ and when the plasma i s i n thermal equilibrium (which i t i s not i n the present case; however this point w i l l not be considered here). Finally, soine value for N must be chosen i n order to use equation 99 . The true value of N e  g  w i l l be, of course, a function of the temperature of each constituent of the gas, or from another point of view,  should be stated as % ( £ )•  It i s also to be noted that the choice of a value of N does not greatly e  alter the value for the electron temperature that results from equation 99 because the value of the logarithmic term i n equation 99 does not change much even for an order of magnitude change i n N . e  Equation 99 i s only valid when the conductivity i s measured transverse to a strong magnetic f i e l d .  One of the criteria of applicability i s  that the cyclotron frequency (^o ) of the electrons must be greater than the electron-electron collision frequency ( i ^ ) . cd  =  —  rv\ C  For our case, B„, . -, \ aj(  therefore,  at  a  = 750 gauss  cO = 13.2 x 10^ rad/sec.  From Spitzer (1956), (100)  - k2 -  ^  (101)  3  where M • electron density i n number per cubic centimeter and ir~ average Q  electron velocity. Equating thermal energy to kinetic energy, §kT » |mv  2  ,  neglecting r e l a t i v i s t i c corrections and substituting i n values to obtain the highest possible collision frequency: (T = 6,000°K., N = 1 0 ^ per cubic e  centimeter), \? • 7.k x 10^ collisions per second. Spitzer states that the use of equation 101 i n calculating the collision frequency i n a partially-ionized gas results i n a value for v> that i s about an order of magnitude too high because this equation has been derived for close encounter collisions only.  Electrostatic forces f a l l off much more  slowly with distance than do the forces between neutral particles - upon which equation 101 i s based.  Therefore, collisions governed by electro-  static forces occur at greater distances and at a lower collision rate than given by equation 101 . In the present investigation the cyclotron frequency has, therefore, always been greater than the collision frequency during the conductivity measurements and equation 99 i s more applicable than the conductivity-temperature relation quoted by Spitzer (19$6), for the conductivity of a plasma when a strong transverse magnetic f i e l d i s not present,  a  =  : 4. S3  x io  (102)  A  -  15a  :  Winding  t o follow  For Best Caused  15 b  Figure 15  Rejection  By Electric  Winding  :  :  page 4 2 -  Pickup  of  Signals  Which  Fields  For Lowest  Coils  Capacity  Are  - U3 It i s of interest to calculate typical values for the electron cyclotron radius ( ^ y ) and the mean free path (L). For C  (  C  2^=» .3 x 10^ cm/sec) and a> = 13.2 x 10  9  T = 6,000°K.  rad/sec, then,  As the temperature increases both the cyclotron radius and the mean free path w i l l increase.  Because the electron cyclotron radius i s so  much smaller than the dimensions of the shock tube i t i s strongly suggested that the electrons w i l l follow the f i e l d lines and be essentially confined by the f i e l d .  However, the motion of the ions w i l l be the controlling  factor during confinement because neutrality i n each increment of volume of the gas w i l l tend to be maintained. Assuming again that  T = 6,0O0°K.,  then, *  002  ^  «*» 0.5U cms. This radius i s of the same order of magnitude as the tube radius (1.2 cm.).  There thus should be a small confinement effect on the high  temperature gas by the magnetic f i e l d of the f i e l d coils. 2.  Apparatus Lin, Resler and Kantrowitz (1955) employed an 800 turn f i e l d c o i l  situated slightly downstream from a pickup coil of 25 turns wound doubly (50 times total) as shown i n Figure 15a.  It has been found that their  technique could be improved by using a pickup c o i l wound as shown i n Figure 15b.  This c o i l has a lower capacitance (due to a lower turn-to-turn voltage)  and, therefore, a wider frequency response than the c o i l shown In Figure l 5 a . The effect of the method of winding on the capacity was determined by  - bh -  comparing two coils, one of 39 turns per winding wound as i n Figure l£a, and having a capacitance of 2k micro-microfarads, and one of U8 turns per winding wound as i n Figure l$b and having a capacitance of 5 micro-microfarads.  Low capacitance i s needed i n order that the frequency response of  the c o i l does not affect the faithful reproduction by the output voltage of the rate of change of flux linking with the c o i l . The problem of electric f i e l d pickup has been minimized by keeping the length of the pickup c o i l as small as possible. wire has been used.  Number !|0 enamelled  The in-phase voltages generated i n the pickup c o i l  by electric fields w i l l be filtered out by the differential amplifier. An electric f i e l d can also cause generation of a differential voltage that i s indistinguishable from the differential voltage generated by magnetic fields.  An order of magnitude test of the signal with and without the  magnetic f i e l d on has shown that differential voltages generated by electric f i e l d pickup are always between 10 and 100 times smaller than the voltage generated by the magnetic fields.  In any case, the signal due to magnetic  fields can be separated from that due to electric fields by making observations with and without a magnetic f i e l d on. Two pickup coils have been used.  The f i r s t i s a high sensitivity  c o i l having somewhat poor electrically-generated-differential-voltage rejection characteristics. as shown i n Figure l£b.  This c o i l has 96 turns wound as a single layer The length of the winding i s 0.8 cm. and the dia-  meter 1.17 inches. The resonant frequency i s 7.H mcs. and the resistance for c r i t i c a l damping 2.2 K. The second pickup c o i l has low sensitivity but excellent electric f i e l d signal rejection characteristics.  This c o i l has a  , — to follow page 4 4 -  Figure  16  :  Apparatus For Measuring Conductivity  - 45 t o t a l o f 6 t u r n s wound as a s i n g l e l a y e r as shown i n F i g u r e 15a. o f t h e w i n d i n g i s 0.5  mm.  and the d i a m e t e r 1.17  inches.  The  The l e n g t h  same damping  r e s i s t o r s have b e e n u s e d w i t h t h i s c o i l as f o r t h e 96 t u r n c o i l , so t h e f r e q u e n c y response up t o about 5 mcs.  The 6 t u r n c o i l  i s identical.  b e e n u s e d f o r a l l measurements t h a t have r e s u l t e d i n a good s i g n a l  has  amplitude.  In order t o r e a l i z e the f u l l p o t e n t i a l i t i e s o f the e x c e l l e n t f r e q u e n c y response o f t h e p i c k u p c o i l s , i t was  n e c e s s a r y t o couple  each  c o i l w i t h v e r y s h o r t l e a d s t o a c a t h o d e - f o l l o w e r h a v i n g extremely low i n p u t c a p a c i t y ( c i r c u i t i s shown i n F i g u r e 16). o f t h e c o i l s o f 7.4  mcs.  The  r e s u l t i n g resonant frequency  i s s u f f i c i e n t t o g i v e 96$  r e p r o d u c t i o n o f a ramp  s i g n a l r i s i n g t o maximum v a l u e i n one microsecond as c a l c u l a t e d by a modif i c a t i o n o f an e q u a t i o n g i v e n by Millraan and Taub  (1955),  (The minimum r i s e  time t o maximum v a l u e observed f o r any shock wave s i g n a l was  1  microsecond  so t h i s time f i x e d the r e q u i r e d h i g h f r e q u e n c y r e s p o n s e ) ,  A E  '  o u t  E.* f o r two  '  rrfT  cascade l a g networks and f o r  T = d u r a t i o n o f ramp,  ZJ E  0 u  ( 1 G 3 )  t « E i , where n  i s the i n p u t v o l t a g e and  f = break  A  frequency,  i s the e r r o r i n  the output v o l t a g e .  The v a l u e o f the damping r e s i s t a n c e has been c a l c u l a t e d the pickup c o i l equivalent c i r c u i t :  a s e r i e s i n d u c t a n c e (L),  and v o l t a g e g e n e r a t o r ( E i n ) shunted by s t r a y c a p l t a n c e C resistance  p  resistance ( r ) ,  and a damping  R,  E f  where  from  "  I  +  P[~  i s the d i f f e r e n t i a l operator  * - £ A ) t p LC 2  A—  (104)  -  t o follow  Field  Coils  page  4 5 -  DC.  17a  Before  a n d After  Pickup  Coijs  D.C.  17b  Figure  17  :  :  Field  Coils  After  Experimental Distortion  Measurements  Pickup  Evidence of  During  Coil  Field  Conductivity  on Shock Wave  - U6 For both the 6 turn and the 96 turn coils the h damping term R predominates and c r i t i c a l damping occurs when the roots of the denominator i n the preceding equation are equal, giving L was measured with a General Radio inductance bridge and  C  was determined from the undamped natural resonant frequency with the c o i l connected to the cathode followers. equation 105 yielded the value  These values when substituted into  R = 2.2 K. for the 96 turn c o i l .  To maintain  a good frequency response, the capacitance of this load resistance was decreased by employing for each load resistance, two 1 K. resistors connected i n series. Oscillations are observed following the main signal from the pickup c o i l as can be seen i n Figure 17.  It i s believed by Lin, Resler and  Kantrowitz (1955) that these oscillations were caused by the fundamental mode of radial oscillation of the high temperature gas.  The radial o s c i l -  lation of the gas w i l l cause a voltage to be generated across the pickup coils due to the interaction of the radially-moving charges with the axial field.  A theoretical check of this acoustical explanation has been made by  consulting Morse (19U8) for the radial oscillation equation of a gas i n a pipe, (106) where sure:y  CD  = ^  "~'  ac  ^*° "*  and <* = 0 , 00  u  f  a  ^ the radius of the pipe, p i s the press  = 0.586,  <*,. = 1.22,  etc.  - 47 It i s believed that the fundamental mode of radial oscillation would be independent of the angle ()> when the shock front i s planar. For the 10 mode, an i n i t i a l pressure of 500 microns i n argon, a shock velocity of 1.7 cms. per microsecond, a temperature of 260,000°K. immediately behind the shock front ( v  s c m n c  i = .95 x 10^ cms./sec.) and a tube radius of 1.225  centimeters, the radial oscillation frequency from equation 106 i s 475 kilocycles per second.  The frequency of the fundamental mode of radial  oscillation i n the region of the shock wave that contains partially-ionized gas would be even lower than 475 kilocycles per second because the velocity of sound i n this region would be considerably less than that i n the high temperature region immediately following the shock front.  The observed  oscillations had major frequency components of between 100 and 500 kilocycles per second and, therefore, could be of acoustical origin. An experimental check on this acoustical explanation for the observed oscillations was made by increasing the ratio of radial to axial magnetic f i e l d with the geometry shown i n Figure 17a.  Should the oscilla-  tions following the peak have been due to interaction of the radially moving plasma with the axial f i e l d , then the amplitude of the oscillations would depend upon the strength of the axial f i e l d .  The amplitude of the  main peak i s dependent upon the strength of the radial f i e l d i n the vicinity of the pickup c o i l .  Therefore, i f the oscillations are of acoustical origin,  a change i n the ratio of radial to axial magnetic f i e l d strength should result i n a change i n the relative amplitudes of the main peak and the oscillations following the main peak. The experimental waveforms shown i n  - U8 Figure 17a and 17b indicate that this change was not observed. The observed oscillations were, therefore, not of acoustical origin. The oscillation that occurs i n the signal before the maximum arrives i n Figure 17a  can be attributed to distortions of the applied magnetic  f i e l d caused by induced currents i n the high temperature gas.  Because of  this large f i e l d distortion effect the double f i e l d c o i l method shown i n Figure 17a was abandoned i n favour of the single f i e l d c o i l method as used by Lin, Resler and Kantrowitz (1955)•  It i s of interest to note that the  double f i e l d coils shown i n Figure 17a generate a cusped magnetic f i e l d . Should the f i e l d be sufficiently strong then trapping of the plasma inside the cusped f i e l d lines can occur.  Scott and Wenzel (i960) discuss the  successful trapping of a shock-wave-generated plasma by such a f i e l d . The magnetic f i e l d geometry was drastically changed i n an attempt to pin down the origin of the oscillations following the maximum. A single turn f i e l d c o i l carrying a current of about U50 amperes was employed.  The  current was obtained by discharging a condenser bank (2 of General Electric Type U+F97 rated 0.25 microfarads at 50 KV) into the single turn c o i l at the same time as the main discharge was fired.  A suitable inductance (spool of  high-voltage cable) was connected i n series i n order to increase the period of the discharge sufficiently to allow the current through the c o i l to reach a maximum at about the same time as the shock wave passed through the coils. The current i n the f i e l d c o i l was measured by the method discussed i n Section 9 of this chapter.  A comparison of this signal with that obtained  when the 656 turn f i e l d c o i l was employed revealed that the major component of the oscillations was independent of the geometry of the magnetic f i e l d .  -h9  -  The conclusion i s that the oscillations are primarily caused by a property of the plasma rather than by the magnetic f i e l d .  It then follows that the  plasma associated with the shock wave must have either a non-monotonicallydecreasing conductivity function following the i n i t i a l maximum or that radial oscillations of the plasma exist.  It was subsequently verified from  observations with the luminosity detector that multiple peaks of luminosity occur at the same times as the conductivity oscillation maxima. Kash et a l  (1958) have observed similar peaks and have attributed them to the oscillatory nature of the shock driving mechanism. An identical conclusion has been reached i n the present work but this point w i l l be discussed later i n Chapter V, Section 1.  A small component of the oscillations accounting for  about l/lO of the amplitude of the maximum would vary with the geometry of the f i e l d . effects.  It i s believed that this component i s caused by f i e l d distortion No method i s suggested for decreasing this component. The complete equipment for measuring the conductivity i s shown i n  Figure 16.  The f i e l d c o i l finally adopted contained 656 turns of No. 18  enamelled wire having a mean diameter of 6.1+6 3.15  cms. and a length of 2.83 cms.  cms., a minimum diameter of  The axial f i e l d strength at the c o i l  center was 125 gauss to 750 gauss for a current input of 1 ampere to 6 amperes. The 6 turn pickup c o i l has been used for a l l measurements except when the signal has had an inobservably small amplitude due to a low shock velocity. 3.  Calibration To calibrate the conductivity apparatus a slug of copper, 6 cms.  long, was dropped through the f i e l d c o i l , i t s f a l l being guided by a glass tube.  - 50 This method had the advantage that the velocity could be readily controlled and measured. In the present investigation the skin effect must be taken into account i n the interpretation of the conductivity measurements. Lin, Resler and Kantrowitz did not encounter this problem because they were concerned with lower velocities and conductivities.  A rigid solution of  the f i e l d equations to determine the magnitude of the skin effect would be extremely d i f f i c u l t .  However, when  (107)  = 0- VI  o- * U  the skin effect would be almost identical during calibration and during measurements on the shock wave.  (This i s true insofar as we can consider  the plasma as a slug of constant conductivity  0" moving with velocity  U ).  In order to see that equation 107 i s valid, consider the simpler one-dimensional problem of a slug moving into a f i e l d increasing linearly with distance. For a copper slug moving at velocity  into a f i e l d ,  (108)  H ( x = o, t ) = C X * e  i  where x  • 0 when t » 0, c  x  i s the moving coordinate that i s measured  into the slug from the front face, and x* = Z ^ ^ t , the f i e l d inside the copper i s given by Gray (1957) as  HJx, t )-c x V ( x i f ) ft  (109)  where yu i s the permeability of the copper.  For a slug of plasma moving at velocity noting that  x* » U t  p  U into the same f i e l d ,  , the f i e l d inside the plasma i s  - 51 The relation has been used that,  V  c  t  c  =  Ut  (111)  P  Substituting equation 107 and 111 into equation 110 yields,  =CxVfyR )  r\ (* t,) P  t  (112)  or, fx, t ) c  -  H  p  (113)  (x,t ) p  The f i e l d pattern inside the copper slug i s , therefore, identical to the f i e l d pattern inside the slug of plasma when the slugs have penetrated equal distances into the f i e l d .  Because the skin effect i s governed by  these f i e l d patterns, we can conclude that the skin effects are identical in this simplified case.  We cannot conclude that skin effects are identical  when equation 107 i s satisfied i f we consider the real case of three- dimensional diffusion of fields into a slug. We can only conclude that skin effects should be almost identical when equation 107 i s satisfied. The problem of skin effect can be considered from the point of view of the magnetic Reynolds number (RM)« The magnetic Reynolds number i s a measure of the domination of the transport of f i e l d lines by a conductor moving with respect to a magnetic f i e l d over the leakage of f i e l d lines into the conductor.  where  Cowling (1957) defines  %  as follows:  L i s a characteristic length of the flow, V" i s the velocity of  the flow and  =• ( 4 rrJA cr )~*~ x 10? with  = 1 for a plasma and  K  yU - relative permeability H  G~ = conductivity of the plasma i n mhos per centimeter.  -  t o follow  page  51 -  V (*) c  18a  :  Typical  18b  Figure  18  :  :  V (t)  Field  t  Waveform-.  Distortion  Distortion of Magnetic Conductivity  Field  During  Measurements  - 52 Thus for a typical value of L1.2  centimeters, we find  per second,  R  M  L » 1.2 centimeters,  » 0.6. % °  crv~  m  U2 x 10^ mhos per second and  Again for 3»8.  0~i)~<* 250 x 10^ mhos  Since  RJJ i s of the order of  unity we can conclude that a considerable amount of the magnetic f i e l d due to the f i e l d c o i l i s transported inside the moving specimen. The skin effect can also be considered from the point of view of the distortion of the magnetic f i e l d of the f i e l d coils which i s caused by induced currents i n the specimen. The second peak observed during c a l i bration has been consistently about 10$ larger than the f i r s t peak (Figure 18a). Also, a small signal i s always generated after step changes i n conductivity (Figure 18a).  Both of these observations can be attributed to distortion  of the main f i e l d by the f i e l d produced by circumferential currents generated i n the specimen when i t reaches the downstream side of the f i e l d c o i l . This f i e l d distortion effect i s shown i n Figure 18b.  The small signal that  i s generated after step changes i n conductivity i s also shown i n the c a l i bration signal given by Lin, Resler and Kantrowitz (1955), but they do not discuss this.  It was f i r s t believed by the present author that the increase  in the height of the second calibration peak over the height of the f i r s t peak was due to a higher velocity of the specimen when i t was leaving the pickup c o i l than when i t was entering. However, as a free f a l l drop of the specimen was being used during calibration, the equations of motion for the specimen could be readily solved to give the velocity of the specimen when i t entered the pickup c o i l and when i t l e f t the c o i l .  The increase i n  velocity that would explain the increase i n signal amplitude corresponded to an acceleration of at least 1800 cms./sec , an impossibly high value. 2  - 53 The alternative explanation for the signal amplitude increase - that the f i e l d was distorted by the slug - i s more plausible.  Resler, Lin and  Kantrowitz (1955) could have neglected this distortion effect because the method that they employed to drive the slug through their apparatus, a rubber-band drive, could easily result i n sufficient deceleration over the length of the specimen (a lower velocity at the trailing end would result i n a smaller signal) to compensate for the f i e l d distortion effect (which would result i n a larger signal when the trailing end passed through the pickup c o i l ) .  Another effect noted that supports the suggestion that  there i s distortion of the main f i e l d i s that the amplitude of the signal caused by the trailing edge was consistently larger than that caused by the leading edge when the slug was allowed to slide through the coils at an angle. When the sliding distance i s kept constant while the angle i s varied, the acceleration of the slug while passing through the c o i l w i l l decrease. This decrease i n acceleration should result i n a decrease i n the ratio of the peak amplitudes of the signal i f the peak amplitudes depended only on the velocity.  The experimental result was that the trailing edge peak was  s t i l l about 10% higher than the leading edge peak. Therefore, f i e l d distortion due to currents i n the specimen can cause about 10% error i n the signal from the pickup c o i l . An experimental approach to the problem of skin effect has revealed that  V ( s ) approximates a Gaussian distribution function when 6~ i s a c  c  step function (that i s , during calibration).  The value for b, as closely  as could be determined, i s independent of the skin-effect parameter over the range of 1;2 x 10^ mhos per second < (T l)~ < 250 x 10^ mhos per second. c  c  - $h -  This range of o"Oj_ corresponds to the range expected of 0** U  . Also,  c  V p i s , as closely as could be determined, independent of the skin effect C  over the same range of G" i^ • The skin depth that corresponds to a value c  of  O'c  = 2^0 x 10^ mhos per second can be calculated from equation 67 Tr  noting that o) i s approximately  iti  -^-g" •  T h i s  relation i s based on the  assumption that the Gaussian distribution may be approximated to by one peak of a sine wave. The resulting skin depth i s then 0.6 centimeters. The conductivity measurement apparatus was calibrated by passing a cylindrical slug of copper 6 centimeters i n length through the coils at a known velocity.  The average velocity was determined by measuring the  time lapse between the signals due to the leading edge of the slug passing through the c o i l and the trailing edge passing through the c o i l .  A first  order correction determined from the ratio of the maximum amplitudes (after a correction factor had been included to compensate for the f i e l d distortion effect discussed i n the preceding paragraphs) was applied to this average velocity to obtain the velocity when the slug entered the c o i l . of this correction to the velocity was always less than b  and  ±i—^  The amount  The values of  for the signal generated by the leading edge of the slug  and obtained from an average of many runs taken over the range of U2 x 10° mhos per second, <0 t) <250 x 10^ mhos per second was: t  Pickup Coil  t  b i n ens.  0 /  I  0  - i n amperes cm^/sec^ volt Vcp  6 turn  1.03  225 x 10  6  96 turn  1.02  12.7 x 10  6  TABLE I1  CALIBRATION DATA FOR CONDUCTIVITY APPARATUS  - 55 Only the signal generated by the leading edge of the slug was considered i n the determination of b and I±^L i n order to approximately match the skin effect errors that occur during calibration and during measurement of the conductivity of the shock wave. The value of (J^ was found by measuring the voltage drop between two axially-separated cross-sections of the slug when a direct current was passed through the slug.  The result was  0~ - 0.581 ± .006 x 10^ mhos/cm. t  To determine the effect of a possible separation of the high temperature gas from the walls of the shock tube (caused by either cooling of the high temperature gas by the walls or partial confinement by the magnetic f i e l d ) , a comparison was made of the signals that resulted when three slugs of different r a d i i were passed through the apparatus. The slug that was employed to obtain the data i n Table I had a diameter of 2.45 mm, approximately .005 mm less than the inner diameter of the shock tube.  The smaller  diameter slugs were increased i n diameter to 2.45 mm with cellulose tape i n order to ensure that the slug would pass through the coils symmetrically. For equal velocities for three slugs having different diameters, the signal amplitudes were: i ) diameter = 2.45 M m , V p = 1.00 volt, i i ) diameter » C  2.40 » j V p » 0.93 volt, i i i ) diameter = 2.35 mm, Vqp *» 0.83 volt. The C  conclusion i s that should the plasma associated with the shock wave be separated from the walls of the shock tube, then the observed signal w i l l be anomalously small.  -  to follow  page  5 5 -  Luminosity 8.2 c m s . From Discharge  <b  Luminosity 3.2 cms. From Discharge  Curre nt in Main Discharge  O  1  Argon,  Figure  19  2  3 Time  5 0 0 microns  :  4 5 6 7 microseconds  initial  Luminosity  pressure,  of Shock  8  co-planar  Wave  9 driver  V RESULTS  1.  The Shock Driving Mechanism No author has yet definitely stated what properties are most  desirable i n the driving mechanism. KLein and Brueckner (i960) come the closest to this problem - they consider the kinetics and driving requirements for a plasma ionized and driven by the magnetic f i e l d of a pulsed current i n a c o i l surrounding the plasma. Measurements with the luminosity detector close to the discharge have shown that there are multiple shock waves generated by the coplanar driver.  The i n i t i a l shock front can be attributed to the energy that i s  transferred into the gas over approximately the f i r s t half cycle of the power dissipated i n the gas.  The energy that i s transferred into the gas  during the second half cycle of the power generates a second shock wave. It i s probable that an explanation for this double shock characteristic i s that there i s a space and time dependent current path i n the discharge chamber that at f i r s t transfers energy into the i n i t i a l shock front and then, as the i n i t i a l shock front moves away from the electrodes, this energy is dissipated i n a lower impedance path closer to the electrodes that results i n the generation of the second shock front.  Subsequent shock fronts  are generated by each maximum of the power. The luminosity of the entire shock wave that i s generated i s shown i n Figure 19 along with the corresponding current waveform. These curves can be interpreted to yield a rough - 56 -  -  t o follow  page 5 6 -  Velocity  cms.  fisec.  OA  1  8  Discharge  Figure 2 0 as  :  a  Velocity of  10 Voltage  |2  kv.  Luminosity  Function of Discharge  Front Voltage  14  - 57 g(t) i n equation 56.  estimate of the form of the function  approximation can be made by assuming that  A first  g(t) i s a constant over the  f i r s t half cycle of the power. Then, rr  Vo -  front  C . ., T  t  ,Soot  s  cit  L  arid assuming  (Uii)  If we assume that the shock front energy can be expressed as kinetic energy, then, E  oc  initial shock front  VA  s  Z,,H-\  (115)  Combining equations l l U and 115 yields, Vs  (116)  «• V.  An experimental check of the velocity versus voltage relation resulted i n the two curves shown i n Figure 20.  The velocities were determined by  taking the slope of the distance-time luminosity curves at distances of 8 cms and 18 cms from the main discharge. Typical distance-time curves are  given i n Figure 21.  The 8 cm. data l i e very close to a straight line  so i t i s strongly suggested that the shock energy i s proportional to the square of the i n i t i a l voltage on the condensers.  The discontinuity i n  the 18 cm. curve can be attributed to decay of the main shock front and then reinforcement by shocks generated by subsequent current pulses. This conclusion was reached from interpretation of the multiple peaks observed with the luminosity detector.  - 58 Observation of the decay of the shock waves that follow the i n i t i a l shock wave has disclosed the interesting property that i n helium the  subsequent shocks tend to catch up with the i n i t i a l shock, whereas  i n argon the subsequent shocks tend to separate both from each other and from the i n i t i a l shock. The period of the main discharge i s U»5 microseconds and the peak current 65,000 amperes. The h microfarad condenser bank discharges U50 joules.  This energy was deposited i n the gas sufficiently quickly  to result i n a shock wave velocity at 10 cms. from the discharge of 5 cms. per microsecond i n helium at 300 microns pressure or 2 cms. per microsecond i n argon at 500 microns pressure. The voltage across the base of the coaxial driver built from l/6U  inch copper and measured by the method described i n Chapter III,  Section 10, revealed that the voltage was almost exactly 90 degrees out of phase with respect to the current after the f i r s t quarter of a current cycle - a result that indicated that the impedance of the driver i s mostly inductive after the f i r s t quarter of a current cycle.  From the known value  of peak current and the voltage observed across the driver at peak current (that i s , at the end of the f i r s t quarter cycle) the resistance of the driver could be calculated and was .05 ohms. The inductance could be determined from the ratio of v/ ^ '  90 degrees out of phase.  later when the voltage and current are  dt  The calculated value was 110 milliraicrohenries.  This value i s supposedly the inductance of the driver.  However, considering  that the total circuit inductance as calculated for the ringing frequency, i s only 125 millimicrohenries, i t i s extremely unlikely that there i s only 15 millimicrohenries of inductance i n the remainder of the circuit  - 59 (condensers, switch and conductors). The more likely explanation i s that the observed voltage waveform contains considerable electric f i e l d signal because the resistor chain used to observe the voltage was not shielded. 2.  The Shock Wave It has been presupposed throughout a l l of the experimental work  that the shock front velocity (% of the luminosity front.  ) i s approximately equal to the velocity  Petschek and Byron (1957) present data and theory  which confirm that this approximation i s valid when V~  i s greater than  s  0.1J5 cms./microsecond for a shock wave travelling i n argon. data has been found by the author for helium.  No similar  It i s thus apparent that  i t would be very desirable to measure the velocity of the shock front. The shock front i s characterized by an abrupt change i n pressure, temperature and density oecu ring over a distance of a few mean free paths of the gas atoms. The two major methods that have been used by other experimenters to observe the position of the shock front for temperature and density conditions similar to those i n the present apparatus are described i n detail by Petschek and Byron (1957).  Both methods require the intro-  duction of apparatus into the shock tube.  It would be a great advantage  to be able to detect the shock front with external apparatus so considerable effort has been directed i n this direction.  Lin, Resler and Kantrowitz  (1955) have noted that a pulse of extremely high conductivity (causing a closely spaced plus then minus voltage signal) precedes the main signal caused by ionization of the gas.  None of the firings with the present  equipment have shown this i n i t i a l pulse of conductivity.  This result favours  - 60 the alternative explanation proposed by Lin, Resler and Kantrowitz for their i n i t i a l signal as being due to electric f i e l d pickup.  In the present  work the 6 turn pickup c o i l very strongly rejects such signals. Another method that has been used to detect the shock front at low velocities i s to observe the luminosity emitted by the front. Petschek, Rose et a l (1955) and Lin, Resler and Kantrowitz observed this low-level luminosity for shock velocities of the order of Mach 8.  It has also been  observed i n the present investigation at velocities of this order with the luminosity detector.  The intensity appears to increase with the ambient  pressure i n front of the shock wave. This luminosity, however, disappears at higher velocities.  It would be extremely interesting to obtain a time-  resolved spectrum of this radiation - the spectrophotometer i s the only piece of equipment that has sufficient sensitivity for such a study. Petschek (1957) suggests but has not conclusively proven that impurities, notably carbon, are responsible for emission of this radiation. A novel method for detecting the shock front has been proposed and tested by the author.  It consists of weakly-ionizing the gas i n the  shock tube inside the pickup c o i l used for the conductivity measurements and detecting the signal that results from the motion imparted to the ions by the shock front. No shock front signal has been detected by this method because of limitations i n the equipment. The source of the ionizing energy (a 2,U50 MC. magnetron-powered radiotherapy unit, model GMD-10 supplied by Raytheon Mfg. Co.) emits a very strong U00 KC signal which conflicts with the desired signal. It i s possible that a U00 KC f i l t e r could be added to the radiotherapy unit, but this has not been attempted.  One result  to  follow page  6 0 -  50,  30  20 T i m e of A r r i v a l of Luminosity  Front  (microseconds) 10  5  10  2 0 Distance  Argon,  Figure  5 0 0 m'icrons  2 2  3 0  50  (cms.)  initial p r e s s u r e , c o - p l a n a r  driver  Propagation Characteristics of Luminosity Front as a Function of Discharge Voltage :  - 61 that has come out of these pre-ionization studies i s that the rate of ionization of the argon behind the shock front i s increased when the gas i s pre-ionized. The shock front can also be detected by interferometry techniques. For a typical case of  J>,/f>  B  a  k  i n an argon shock and over a path length  of 2.5 cms. at a wavelength of 5,000°A, the fringe shift i s about .02. Calculations have been made to ascertain the worth of a light refraction system to detect the shock front using a photomultiplier detector. However, the displacement for typical geometry i s of the order of .001 mm far too l i t t l e to detect even with a photomultiplier and excellent optics. Graphs showing the propagation characteristics for the luminosity front for various driving conditions are given i n Figures 21 and 22. It i s to be noted that the curves a l l approximately obey the relation x = at  b  (117)  where a and b are functions of the choice of gas, the ambient pressure and the driving energy. The 15 KV data with the exception of that for the T-Tube system, was taken with the coplanar driving electrodes built from electrode material l / l 6 inch thick. The data taken at voltages other than 15 KV was taken from a shock wave driven by a coplanar driver b u i l t with 1/6U inch thick electrodes.  It i s believed that the minor discontinuities  i n the lines are due to replacement of the leading shock front by subsequent shocks generated by the oscillatory main discharge driver (See Section 1 of the present chapter).  - 62 No successful attempt has been made to substantiate equation 117 by a theoretical treatment. It i s believed that the discontinuity i n the curves of Figure 21 at a distance of about 10 cms. from the driver i s due to the detachment of the shock wave from further driving energy. The major use of these curves has been to yield the velocity of the luminosity front for any desired condition of operation of the shock tube and at any desired distance from the driver. Typical waveforms as observed with the luminosity detector are shown i n Figure 19. The electric probes and the conductivity measuring equipment yielded data that was correlated i n time with the luminosity data. Thus there was a correspondence between the luminosity and the electron density i n the high temperature gas.  The electric probe data was analyzed for  conductivity for one case. The result was 0.06 mho/cm. for a luminosity front velocity of 0.27 cms./microsecond and an ambient pressure of 500 microns i n argon. The temperature immediately behind the shock front that corresponds to this velocity i s from equation 11: degree of ionization from equation 17:  10"^.  6,500°K, and the  The expected conductivity  as calculated by Lin, Resler and Kantrowitz (1955) for this temperature i s about 7 mhos/cms. Thus the measured conductivity with the electric probes was greatly i n error with respect to the expected conductivity.  Lin, Resler  and Kantrowitz (1955) obtained similar values and attributed the low values obtained to a low temperature for the gas which i s i n contact with the probe.  - 63 Time-integrated spectra that have been taken have shown that neutral argon atom excitation lines appear at low velocities and that de-excitation lines from singly-ionized argon atoms appear at higher velocities.  No lines have been thus far observed that could be attributed to  de-excitation from multiple-ionization levels.  It i s believed that the  observed absence of these transitions i s due to an extremely rapid theiroalization process that very quickly equalizes the atom, electron and ion temperatures once ionization has begun. This equalization process would greatly decrease the atom temperature, typical values being: Tmax  88  T qu • 17,U00°K. at e  « • 0.5 (from Knorr, 1958,  u  versus  87,000°K,  T relation).  The temperature of the ions would thus not be high enough for a sufficient time to result i n emission of an observable intensity of transitions from multiply-ionized levels. Although the majority of the experimental work has been on argon, there has been some work done with helium.  The velocity of the shock wave  i s higher i n helium than i n argon. The time-integrated spectra have a l l contained many impurity lines, notably  Hw, H  at U128.11°A  p  , H  r  , Hj , Na I at 5889.95°A  and la30.96°A  and  and 5895.92°A, Si I  C II at l;266.53°A. The spectra obtained  with helium i n the shock tube are quite similar to title spectra obtained when argon i s the working gas.  The major difference i s that  are observed from the shock wave i n helium gas.  F II  lines  The fluorine could be  either an impurity supplied i n the helium or i t could be due to ablation of the teflon i n the driving mechanism. Most of the impurity lines have been noticeably broadened.  In particular, analysis of the  H  and  H ^ line  profiles should result i n extremely useful values for the presently unknown  - 6k maximum electron density.  The analysis by Griem, Kolb and Shen  (19$9)  can be used to interpret the line profiles. 3.  Conductivity The maximum conductivity, conductivity decay constant and  electron temperature during maximum conductivity have been determined for various shock conditions.  Typical results are given i n Table II and  have been determined with the aid of equations 31, 32 and 99 and considering <x = 10""^ (See Chapter IV, Section 1). terras that contain  It has been found that a l l  oc i n equations 31 and 32 are negligible when <*.= 10"^.  We can conclude that maximum conductivity occurs under temperature, pressure, flow velocity and density conditions that are virtually identical to the conditions that exist immediately behind the shock front. Equations 11 and 12 are, therefore, applicable to the conductivity measurements because they are identical to equations 31 and 32 when  ~ 10"^.  - 65 -  Gas  Ionization Front Velocity  - U« v  3  cms.per second  Argon  500±20  166  119,000 1  6  537  152,000  6  3UU  98,000 1  556  76,000  I.l5xl0  6  Argon 500±2O microns  1.3xl0  Helium 280-10 microns  3.3xl0  microns  Helium  6oo±5o  2.9xl0  Maximum Axial Maximum Atom Magnetic Temp. f Field °K. Intensity gauss  6  .92  .92  microns TABLE II:  1 .92  1 .92  Maximum Electrical Conductivity cms. mhos/cm.  Calculated Maximum Electron Temp. °K;  16  135  60,000  16  113  53,000  16  180  76,000  16  115  53,000  MEASURED VALUES OF MAXIMUM ELECTRICAL CONDUCTIVITY OF PLASMA  The value of the electron temperature i n row one of Table I I i s too high, due to the use of an insufficiently large magnetic f i e l d . The magnitude of the magnetic f i e l d w i l l determine the choice between equation 99 or 102 as the applicable conductivity-temperature relation. A l l of the electron temperatures i n Table II are probably also i n error due to confinement of the plasma by the magnetic f i e l d .  Confirmation that  confinement probably does exist i s given by the following conductivity measurements made on argon at an i n i t i a l pressure of 500 microns and a shock speed of 1.3 x 10^ cms./second:  - 66 -  Field Coil Current amperes  0.352 0.960 4.30  V  W  P  volts  i n mhos/cm.  169 152 113  .53 1.354 4.20  axial  i n gauss  35.5 120 538  TABLE I I I : CONDUCTIVITY VERSUS MAGNETIC FIELD INTENSITY  Because the maximum conductivity &" i s a function of the magnetic f i e l d intensity, we can conclude that there i s either or both of a confinement effect and that the conductivity of the plasma i s a function of the magnetic f i e l d .  High speed photography of the luminosity associated  with the plasma as i t passes through the magnetic f i e l d would resolve which i s the maijor effect. Another source of error i n the determination of 0" has been the neglect of both wall cooling and the decrease i n velocity due to the wall. Investigation of these errors could be made by studying the conductivity of the plasma when i t i s propagated down various diameters of shock tube. Again, high speed photographic equipment would be of great help i n such an investigation. A check of any decrease i n velocity of the luminosity front caused by the conductivity measurement apparatus has revealed that there i s no detectable change i n velocity.  We can conclude that the magnetic f i e l d  of the f i e l d coils neither appreciably deflects nor changes the energy of the plasma following the shock front.  - 67 h*  Conclusions A high velocity shock tube has been assembled and operated.  Careful design of the electromagnetic shock driving mechanism has resulted i n the attainment of high shock speeds from a comparatively slow and small condenser bank. The methods that have been employed to measure the properties of the shock wave have proven to be reliable insofar as the results agree with the simple theory that has been presented.  Most of the methods have  been of a quite general nature and are, therefore, applicable to the;, study of plasmas generated by other means. It i s suggested that further experimental work be concentrated on obtaining a single shock wave instead of the multiple shock waves generated by the present driver.  One promising method consists of shorting  the driver with an ignitron at the end of the f i r s t current pulse. A second possibility that warrants investigation i s the use of a conical driver.  Kash et a l (1958) have obtained single shock waves with this type  of driver. It would be of great aid i n the interpretation of the experimental results i f a theoretical solution for two problems could be obtained. Applicable relations are needed for the conductivity, electron density, ion densities, electron temperature, ion temperatures, atom temperature and flow velocity as a function of the distance behind the shock front for d i f ferent shock velocities.  The second problem i s to explain the observed  shock wave propagation equation i n the present tube, x « at  b  BIBLIOGRAPHY  Bishop, A.S. (1958) Project Sherwood. Bostick, W.H. (1958) Boston Conference on Extremely High Temperatures. J. Wiley and Sons, Inc. Cowling, T.M. (1957) Magnetohydrodynamics. Din, F. (1956) Themodynamic Functions of Gases. Fowler, R.G. et a l (1952) Physical Review 88, 137, 1952. Francis, G. (1956) Handbuch der Physik, Volume 22, 1956. Gray, D.E. (1957) American Institute of Physics Handbook, 1957. Griem, H.R., Kolb, A.G. and Shen, K.Y. (1959) Physical Review 116, No.l,  U-16, 1959. German, W.J. (1958) Journal of Applied Physics 29* 109, 1958. Harris, E.G., et a l (1957) Physical Review 105, No. 1, U6-50, 1957. Hbdgman, CD. (1953-195U) Handbook of Chemistry and Physics. Inuishi, Y., et a l (1957) Journal of Applied Physics 28, 9, 1017, Sept. 1957. Janes, G.S. and Patrick, R.M. (1958) Boston Conference on Extremely High Temperatures, J . Wiley and Sons, Inc. Kash, S.W. et a l (1958) Velocity Measurements i n Magnetically Driven Shock Tubes, a Chapter i n the book "The Plasma In a Magnetic Field" by R.K.M. Landshoff. KLein, M.M. and Bruechner, K.A. (i960) Journal of Applied Physics 31, U+37. Knorr, G. (1958) Zeitschrift Fur Naturforschung 13a, 9U1-950, 1958. Kolb, A.C (1957) Physical Review 107, 3*i5, 1957. Kolb, A.C. (August 1957) Physical Review 107, 1197, 1957. Kolb, A.C per Landshoff, R.K.M. (1957) Experiments at U.S. Naval Research Laboratory, a Chapter i n the book "Magnetohydrodynamics" by R.K.M. Landshoff. - 68 -  - 69 Kolb, A . C .  (1958) G e n e v a C o n f e r e n c e P a p e r (1958) P/3U5. P r o g r e s s i n N u c l e a r E n e r g y S e r i e s X I , 1959.  Reprinted i n  L e v i n e , M . A . , S a m p s o n , J . L . , W a n i e k , R . W . (1958) B o s t o n C o n f e r e n c e o n Extremely High Temperatures, J .Wiley and Sons, I n c . Lin,  S - C , Resler, E . L . , and Kantrowitz, A.  (1955)  Journal of Applied  Physics  26,95, 1955. Loeb, L . B . Morse,  (1955)  P.M.  B a s i c P r o c e s s e s o f Gaseous E l e c t r o n i c s ,  (19U8^ V i b r a t i o n  Olsen, H.N. and Huxford, Penner,  and Sound.  W.S.  S . S . , Harshbarger,  (1952) P h y s i c a l  F . , and V a l i ,  V.  Review  (1957)  87, 922, 1952.  Combustion  and Reviews, Perry,  J.H.  (I9hl)  1955.  Chemical Engineer's  Petschek, H . E . , Rose, P . H . , e t a l  Researches  1957.  Handbook.  (1955)  Journal o f Applied Physics  26,  83, 1955. Petschek, H . E . and Byron, Resler,  S.  (1957)  Annals o f Physics  1, 270-315, 1957.  (1952) J o u r n a l o f A p p l i e d P h y s i c s 23, 1390, 1952. (I960) P h y s i c a l R e v i e w 119, N o . U, 1187-1188,  E . L . , L i n , S - C , and Kantrowitz, A .  S c o t t , F . R . and Wenzel,  R.F.  I960.  Smith, V . L .  (i960) I . R . E .  Spitzer,  (1956) P h y s i c s  L.  Transactions of Fully  Nuclear Science  Ionized  Gases.  NS-7, 13, I960.  

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