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Absolute intensity measurements in a helium plasma MacLatchy, Cyrus Shantz 1966

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ABSOLUTE INTENSITY MEASUREMENTS IN A HELIUM PLASMA by CYRUS SHANTZ MACLATCHY B.Sc,  Acadia U n i v e r s i t y ,  1964  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of PHYSICS  We accept t h i s t h e s i s as conforming required  t o the  standard  THE UNIVERSITY OF BRITISH COLUMBIA August,  1966  In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y available f o r reference and study,  1 further agree that permission-for extensive copying of t h i s  thesis f o r scholarly purposes may be granted by the Head of my Department or by his r e p r e s e n t a t i v e s „ or p u b l i c a t i o n of xhis  It  i s understood that copying  thesis f o r f i n a n c i a l gain s h a l l not be allowed  without my written permission.  Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada  -iiABSTRACT The temperature o f a Helium plasma produced i n a shock tube has been determined from a b s o l u t e i n t e n s i t y  measurements.  The plasma was considered t o be homogeneous, t r a n s p a r e n t and i n l o c a l thermodynamic  equilibrium.  The e m i s s i v i t y o f the  plasma has been measured by comparing the plasma r a d i a t i o n t o the r a d i a t i o n from a carbon a r c through a simple o p t i c a l  system.  The i n f l u e n c e o f e r r o r s i n measured parameters and the presence o f i m p u r i t i e s i s d i s c u s s e d .  Temperatures which depend  on l a r g e e x p o n e n t i a l terms are r e l a t i v e l y i n s e n s i t i v e t o both e r r o r s i n measurement and the i m p u r i t y content o f -the plasma. The accuracy o f a b s o l u t e i n t e n s i t y measurements i s a t best comparable t o r e l a t i v e i n t e n s i t y  measurements.  -iiiTABLE OF CONTENTS Page ABSTRACT  i i  LIST OF FIGURES  v  LIST OF TABLES  v  ACKNOWLEDGEMENTS  vi  CHAPTER I  INTRODUCTION  CHAPTER I I  EQUATIONS OF THE PLASMA  (a) I n t r o d u c t i o n  5  (b) L o c a l Thermodynamic E q u i l i b r i u m  5  (c) E m i s s i v i t y and E x c i t e d State  7  Populations  (d) L i n e Broadening and E l e c t r o n D e n s i t i e s  10  (e) E q u i l i b r i u m Equations o f t h e Plasma  12  ( f ) S e n s i t i v i t y o f the Temperature Measuring Techniques  21  CHAPTER I I I  THE ABSOLUTE INTENSITY CALIBRATION'  (a) I n t r o d u c t i o n  24  (b) L i g h t Gathering o f the Monochromator  24  (c) Standard Sources  28"  (d) Shock Tube Absorption  33  (e) Comparison o f L i g h t Sources  36  CHAPTER IV  APPARATUS AND TECHNIQUE  (a) I n t r o d u c t i o n  44  (b) Shock Tube  44  (c) O p t i c a l Equiment and Measuring Techniques  46  (d) Carbon Arc  46*  -ivPage CHAPTER V  RESULTS AND CONCLUDING REMARKS  (a) I n t r o d u c t i o n  50  (b) R e s u l t s  51  (c) D i s c u s s i o n  52  BIBLIOGRAPHY  58"  -vLIST OF FIGURES Figure  Page  1.  qkT as a f u n c t i o n o f L  15  2.  L i g h t g a t h e r i n g o f a monochromator w i t h a l a r g e condensing l e n s  25  3.  L i g h t g a t h e r i n g o f a monochromator with a small condensing l e n s  27  4.  Measuring the shock tube a b s o r p t i o n  35  5.  Radiance and l i g h t f l u x  37  6.  Comparison o f l i g h t  3$  7.  Schematic diagram o f apparatus  45  S.  L i n e sampling techniques  47  9.  Impurity e l e c t r o n s as a f u n c t i o n o f kT  53  H e l l population  54  10.  sources  as a f u n c t i o n o f kT  LIST OF TABLES Table  Page  1.  Sensitivity factors  23  2.  Final results  55  -viACKNOWLEDGEMENTS I am g r a t e f u l t o Dr. A.J. Barnard  f o r h i s guidance  and p a t i e n c e d u r i n g t h e p r e p a r a t i o n o f t h i s t h e s i s .  I  a l s o wish t o express my a p p r e c i a t i o n t o f e l l o w students and members o f the t e a c h i n g and t e c h n i c a l s t a f f f o r h e l p f u l d i s c u s s i o n s d u r i n g the course o f t h e work.  CHAPTER I INTRODUCTION T h i s t h e s i s i s concerned p r i m a r i l y with the o f the measurement o f absolute as a d i a g n o s t i c  i n t e n s i t i e s of s p e c t r a l l i n e s  technique i n the  measuring the absolute  evaluation  study of plasmas.  Although  value of the e m i s s i v i t y o f the  i s b a s i c a l l y a simple procedure, the  inference  plasma  of p h y s i c a l  p r o p e r t i e s o f the plasma from the measurement o f a s p e c t r a l l i n e depends c r i t i c a l l y on the degree o f e q u i l i b r i u m which i s a t t a i n e d i n the  plasma.  As a source o f plasma, a shock tube was has been c o n s i d e r a b l e  used.  d i s c u s s i o n i n the l i t e r a t u r e  There  concerning  the a c t u a l s t a t e of e q u i l i b r i u m i n shock tube plasmas g e n e r a l l y . For example, McLean et a l . (I960) c l a i m to have measured the temperature with e r r o r s o f 2% i n a shock heated plasma. r e c e n t l y , I s l e r and  Kerr (1965) and  Eckerle  and  (1966) have suggested d e v i a t i o n s from thermal Assuming t h a t one equilibrium,  McWhirter  equilibrium.  can produce a plasma which i s i n  a knowledge o f the absolute  l i n e can y i e l d u s e f u l i n f o r m a t i o n p r i n c i p l e , i t should be the  i n t e n s i t y of a s p e c t r a l  concerning the plasma.  simplest  spectroscopic  line profile.  In a d d i t i o n , i f one  can  a r e l i a b l e temperature by t h i s or other means, an v a l u e o f the a l l o w one  In  method of  measuring the k i n e t i c temperature since i t r e q u i r e s the o f only one  More  study  evaluate absolute  e m i s s i v i t y o f the plasma f o r a s p e c t r a l l i n e  to c a l c u l a t e the absolute  will  value o f the t r a n s i t i o n  -2probability. In recent years there have-appeared o n l y a few measurements o f a b s o l u t e i n t e n s i t i e s ,  f o r the most p a r t ,  experimenters  have r e l i e d on r e l a t i v e i n t e n s i t y measurements f o r e v a l u a t i n g the temperature.  T h i s technique i s g e n e r a l l y assumed  fairly  r e l i a b l e because o f the c o n s i s t e n t r e s u l t s one a c h i e v e s by using i t .  However, as w i l l be p o i n t e d out, l a r g e v a r i a t i o n s  i n measured parameters are r e q u i r e d b e f o r e n o t i c e a b l e changes i n the c a l c u l a t e d v a l u e s o f temperature  occur.  This property  tends t o n u l l i f y the e f f e c t o f r e l a t i v e l y poor measurements o f the e l e c t r o n d e n s i t y sometimes a c h i e v e d .  The a b s o l u t e  intensity  technique has been s t u d i e d i n the hope o f removing t h i s c u l t y and o f c a l c u l a t i n g a more r e l i a b l e temperature.  diffiIn the  chapters which f o l l o w , the absolute i n t e n s i t y technique i s d e s c r i b e d and compared t o r e l a t i v e i n t e n s i t y measurements i n terms o f both accuracy and s e n s i t i v i t y t o e r r o r s i n measurement. Chapter  I l c o n t a i n s a review o f the e q u i l i b r i u m  o f the plasma and i n d i c a t e s how  equations  these equations might be  used  to c a l c u l a t e e l e c t r o n d e n s i t i e s , e x c i t e d s t a t e p o p u l a t i o n s , and k i n e t i c temperatures  from s p e c t r a l o b s e r v a t i o n s .  s e c t i o n s on l o c a l thermodynamic e q u i l i b r i u m s e n s i t i v i t y and Stark broadening  The  (LTB"), plasma  (  are b r i e f d e s c r i p t i o n s o f  w i d e l y used t e c h n i q u e s . The f o u r t h s e c t i o n i s based  on the  Saha and Boltzmann equations which d e s c r i b e the plasma and i s e s s e n t i a l l y a m a n i p u l a t i o n o f these equations to f i n d temperature  o f a pure plasma and an i m p u r i t y  plasma. F i n a l l y , an attempt  the  contaminated  has been made t o compare the  -3s e n s i t i v i t y o f a b s o l u t e and r e l a t i v e i n t e n s i t y measurements t o e r r o r s i n measuring The  techniques.  t h i r d chapter i s mainly  geometrical optics"of"making tion.  concerned with  the  the a b s o l u t e i n t e n s i t y  calibra-  A g e n e r a l r u l e f o r s i m p l i f y i n g the c a l c u l a t i o n o f the  l i g h t g a t h e r i n g e f f i c i e n c y i s set f o r t h i n the f i r s t s e c t i o n , w h i l e a review of standard The  sources i s contained i n the  second.  a b s o r p t i o n o f r a d i a t i o n by the shock tube i s d i s c u s s e d  i n the t h i r d s e c t i o n with  s p e c i a l emphasis t o problems  p e c u l i a r t o the p a r t i c u l a r apparatus used.  The  f i n a l section  o f Chapter I I I d e s c r i b e s i n c o n s i d e r a b l e d e t a i l the o f the plasma to the standard  comparison  source.  Chapter IV I s a short d e s c r i p t i o n o f the apparatus and  experimental  technique.  A standard  shock tube with a  "coplanar d r i v e r " has been used as a plasma l i g h t  source.  H e l 5S76, 667S and H e l l 1+686 l i n e p r o f i l e s have been time r e s o l v e d u s i n g a J a r r e l l - A s h monochromator.  Absolute i n t e n -  s i t y c a l i b r a t i o n s were c a r r i e d out u s i n g a carbon a r c as a standard  source. In Chapter ¥, an i n d i c a t i o n o f the i m p u r i t y  content  o f the plasma i s i n c l u d e d along with a comparison o f tempera t u r e s c a l c u l a t e d from a b s o l u t e and r e l a t i v e  intensities.  I t i s p o i n t e d out t h a t c o r r e c t i o n s f o r i m p u r i t i e s have  little  e f f e c t f o r r e l a t i v e and a b s o l u t e i n t e n s i t i e s , as l o n g as the e x p o n e n t i a l terms i n the equations have l a r g e arguments. Temperatures c a l c u l a t e d from a b s o l u t e i n t e n s i t i e s are at best no more r e l i a b l e ' than those  c a l c u l a t e d from r e l a t i v e  intensities.  -4-  and i n some i n s t a n c e s are much worse.  Only under c e r t a i n  r e s t r i c t i v e c o n d i t i o n s i s the use o f t h e more t e d i o u s - a b s o l u t e i n t e n s i t y technique  warranted.  CHAPTER I I EQUATIONS OF THE PLASMA (a)  Introduction By measuring the e l e c t r o n d e n s i t y o f a plasma, some  conclusions  can be drawn concerning i t s s t a t e o f e q u i l i b r i u m .  Assuming t h a t LTE e x i s t s , one can deduce both the e l e c t r o n d e n s i t y and e x c i t e d s t a t e p o p u l a t i o n s  from the measurement  o f l i n e p r o f i l e s and i n t e g r a t e d absolute chapter, equations are i n t r o d u c e d  intensities.  In t h i s  from which t h e e q u i l i b r i u m  temperature o f the plasma can be c a l c u l a t e d from both and  absolute  intensities.  relative  F i n a l l y , a t h e o r e t i c a l comparison  o f the s e n s i t i v i t i e s o f these measuring t e c h n i q u e s i s made.  (bD  L o c a l Thermodynamic E q u i l i b r i u m In p r a c t i c e i t i s extremely d i f f i c u l t to'produce a  plasma whose e n t i r e volume i s i n thermodynamic Since the absolute  i n t e n s i t y measurement depends h e a v i l y on  the preponderance o f e q u i l i b r i u m ,  i t i s necessary t o be able  t o l o c a t e a r e g i o n i n the plasma which simulates l i b r i u m or LTE.  equilibrium.  In the most r e s t r i c t i v e  strict  equi-  sense, an LTE r e g i o n  i s one i n which the quantum s t a t e s o f the i o n i z e d gas are populated i d e n t i c a l l y t o those o f an equal volume o f gas which i s i n thermodynamic e q u i l i b r i u m and has the same t o t a l temperature and chemical composition as the system. the  l a b o r a t o r y plasma seldom approaches t h i s i d e a l ,  density,  However, so that  a l e s s s t r i n g e n t c o n d i t i o n , t h a t o f p a r t i a l LTE, i s g e n e r a l l y  -6-  sought.  In p a r t i a l LTE, only a r e s t r i c t e d number o f quantum  s t a t e s need be populated  according to e q u i l i b r i u m conditions.  The v a l i d i t y o f the r e s u l t s o f any set o f measurements depending on the assumption o f e q u i l i b r i u m i n the plasma  cannot  be e f f e c t i v e l y d i s c u s s e d u n l e s s c e r t a i n c r i t e r i a f o r LTE are e s t a b l i s h e d and evaluated f o r the p a r t i c u l a r plasma i n q u e s t i o n . One assumes a p r i o r i t h a t both the temperature and p a r t i c l e g r a d i e n t s are n e g l i g i b l e i n the r e g i o n o f o b s e r v a t i o n . (1964)  which  Griem  has been able t o suggest order-of-magnitude c o n d i t i o n s should h o l d i n order t h a t LTE c a l c u l a t i o n s be j u s t i f i e d .  Because o f the importance d f the LTE assumption, a b r i e f description follows. I f the plasma i s t o be considered i n LTE, i t i s e s s e n t i a l t h a t c o l l i s i o n a l processes dominate. with p o p u l a t i o n n  m  For a p a r t i c u l a r l e v e l m  one w r i t e s :  ott  XctL  cL± t r a c t .  The r a d i a t i v e r a t e o f decay can be found from the E i n s t e i n r e l a t i o n t o be i n t r o d u c e d i n the next s e c t i o n .  The c o l l i s i o n a l  term i s found from quantum mechanical c o n s i d e r a t i o n s . plasma i s c o n s i d e r e d t o be o p t i c a l l y t h i n .  The  For a l i n e whose  upper l e v e l i s 1m the e l e c t r o n p o p u l a t i o n i n a homogeneous, time independent plasma  should be such t h a t  -7E^ i s the i o n i z a t i o n energy f o r Hydrogen and  ^  i s the  frequency o f the r a d i a t i o n f o r a t r a n s i t i o n from l e v e l m t o l e v e l n. The  shock tube plasma i s a t r a n s i e n t plasma.  This  s i t u a t i o n puts l i m i t a t i o n s on the d u r a t i o n o f the p e r i o d o f o b s e r v a t i o n i n such a way t h a t a measurement must be made over a p e r i o d o f time much g r e a t e r than the e q u i l i b r a t i o n time o f t h e l e v e l s and much l e s s than t h e p e r i o d i n which g r o s s changes i n the plasma o c c u r . The second o f these cond i t i o n s i s i n g e n e r a l e a s i l y accomodated, but the f i r s t c o n d i t i o n must be d e a l t with c a u t i o u s l y s i n c e i t depends on the e q u i l i b r a t i o n time o f the l e v e l s .  'Essentially, i t i s  n e c e s s a r y that the quantum l e v e l s o f the plasma a r e i n a s t a t e o f "near e q u i l i b r i u m !  1  a t a l l times o f change o f the plasma.  Of the e q u i l i b r a t i o n times w i t h i n the plasma, bound s t a t e s i s the g r e a t e s t .  t h a t o f the  For t h i s reason, i t i s wise t o  check the e q u i l i b r a t i o n times o f these l e v e l s i n l i g h t o f the p h y s i c a l parameters  o f the experimental plasma.  Griem(1964)  has suggested order o f magnitude c a l c u l a t i o n s which a l l o w one t o form some c o n c l u s i o n concerning t h i s time. o f t e n remains  However, t h e r e  some doubt as t o the LTE nature o f the plasma.  (c) E m i s s i v i t y .and E x c i t e d S t a t e P o p u l a t i o n s The p o p u l a t i o n s o f e x c i t e d s t a t e s m a y be r e l a t e d t o the i n t e g r a t e d absolute- i n t e n s i t y o f s p e i r t r a l l i n e s .  To  i l l u s t r a t e the-assumptions-made i n r e l a t i n g t h e s e p r o p e r t i e s , the problem-of" l i g h t t r a n s m i s s i o n through the plasma w i l l be examined i n the g e n e r a l case and then s i m p l i f i e d t o meet the  c o n d i t i o n s presumed present plasma.  i n the author's l a b o r a t o r y  F i r s t l y , n o t i c e t h a t the 2  by  where v i s the frequency o f the r a d i a t i o n , x^ represent  s p a t i a l co-ordinates  the l i n e o f s i g h t , and £ ( V due  of the  X 3 ) along the l i n e o f s i g h t  r a d i a n t i n t e n s i t y I(V, xi> x , can be r e p r e s e n t e d  s p a t i a l gradient  > *!>  x-^, x  and  2  with X]_ o r i e n t e d x , 2  X3)  along  i s the e m i s s i v i t y  to spontaneous t r a n s i t i o n s o f e l e c t r o n s between s t a t e s  i n the plasma. represents  k(V,  x-p  X g , x^)  both a b s o r p t i o n and  i s a c o e f f i c i e n t which  induced emission  processes.  I f i t i s assumed t h a t the plasma i s homogeneous, the s p a t i a l dependence o f dl/dx^ may i f only non  be dispensed  with.  resonance r a d i a t i o n i n the plasma i s  the r e l a t i v e l y low p o p u l a t i o n  considered,  o f the e x c i t e d s t a t e s u s u a l l y  warrants the n e g l e c t i o n o f a b s o r p t i o n m i s s i o n processes,  Also,  and  induced• t r a n s -  both o f which are p r o p o r t i o n a l t o  the  number d e n s i t y o f the e x c i t e d s t a t e s which g i v e r i s e to radiation involved.  The  emissivityE(V ) mn  Under these c o n d i t i o n s ,  i s defined  by  the  -9where m and n r e p r e s e n t the upper and lower e x c i t e d s t a t e s Q£ a t r a n s i t i o n , n and  m  i s the p o p u l a t i o n o f the upper l e v e l ,  i s the E i n s t e i n c o e f f i c i e n t g i v i n g the  probability  o f an e l e c t r o n making a spontaneous t r a n s i t i o n from l e v e l m t o l e v e l no  The E i n s t e i n c o e f f i c i e n t must be e v a l u a t e d  quantum mechanical c o n s i d e r a t i o n s .  where r  Q  I t i s d e f i n e d by  i s the c l a s s i c a l e l e c t r o n r a d i u s , c i s the  of l i g h t , g  n  involved,  f  determination  and g  m  from  velocity  are s t a t i s t i c a l weights of the two  states  i s the a b s o r p t i o n o s c i l l a t o r s t r e n g t h whose r e q u i r e s the e v a l u a t i o n of matrix  d e s c r i b i n g the exchange between s t a t e s m and  elements  n.  where x^ r e p r e s e n t s the c o - o r d i n a t e s o f the e l e c t r o n making the t r a n s i t i o n . The  evaluation of f nm  a n a l y t i c a l l y and source  e x p e r i m e n t a l l y and may  o f e r r o r i n experimental  the predominant one  complete l i n e  be a s u b s t a n t i a l  mechanisms i n the plasma,  i n t h i s case being Stark broadening,  to i n t e g r a t e the e m i s s i v i t y £(V) shape.  v a r i a t i o n i n V-™  both  procedure o f t h i s t h e s i s .  Because o f v a r i o u s l i n e broadening  i s necessary  i s imprecise ^  F u r t h e r , one  over  it  the  assumes t h a t the percentage  i s small throughout the t o t a l l i n e .  Thus,  -10-  £ < 0  =• j^COWv)  2:8-  i.e.  Vw W  '  1  kv\  VA  <S *  7  Hence the excited state population may be calculated from  where  etfu  must be evaluated from considerations of  the o p t i c a l system and comparison with a standard source. (d) Line Broadening and Electron densities The shape of a spectral l i n e i s normally influenced by a number of broadening mechanisms.  To the plasma p h y s i c i s t ,  the two most important of these mechanisms are the Doppler broadening and Stark broadening.  The f i r s t of these i s caused  by the thermal motion of the p a r t i c l e s and the second by the e l e c t r i c f i e l d of the electrons and ions i n the neighbourhood of the emitting p a r t i c l e .  In low density plasmas, Doppler  broadening dominates and i s a function of the temperature of the emitting species Stark  bToadeniirg  only.  --dominates.  At high p a r t i c l e densities, the Stark broadening- i s found to be  only s l i g h t l y dependent on the temperature and strongly  -In-  dependent  on the e l e c t r o n d e n s i t y .  There are i n essence t h r e e approaches t o u s i n g the Stark e f f e c t to measure the e l e c t r o n d e n s i t y .  These a r e , i n  order o f i n c r e a s i n g accuracy, the l i n e s h i f t , the l i n e width and the l i n e p r o f i l e t e c h n i q u e .  half  L i n e s h i f t s cannot  g e n e r a l l y be measured a c c u r a t e l y i n dense plasmas because o f the e x c e s s i v e l i n e broadening. study o f the l i n e  shape o f  Of the o t h e r t e c h n i q u e s , a w i l l y i e l d about 5% accuracy  i n the c a l c u l a t e d e l e c t r o n d e n s i t y ; w h i l e h a l f widths o f o t h e r H l i n e s and o f He l i n e s are known t o about 10$.  (Griem,  1964)  The h a l f width, the width o f the l i n e at h a l f the peak i n t e n s i t y , i s a f u n c t i o n o f the e l e c t r o n d e n s i t y , and the e l e c t r o n temperature, TL . q  it  n , e  For non-hydrogenic atoms  i s g i v e n by  ^Wx  =C(T )-^ e  «»  where Wi. i s the measured h a l f width and W  Q  h a l f width at a s p e c i f i e d e l e c t r o n d e n s i t y ,  2 = 11  i s o n e - h a l f the n 6 >  o*  Since  the h a l f width i s a slowly v a r y i n g f u n c t i o n o f temperature, one can choose an approximate temperature and assume  He  Griem  =  C  t"0  ^°  2 : 1 2  (1964) has l i s t e d t a b l e s o f v a l u e s f o r C ( T ) from which e  -12n  i s easily calculated.  Once n  i s found, a f i r s t  t i o n t o the temperature i s c a l c u l a t e d . of n  and  T  are found by u s i n g i t e r a t i v e  (e) E q u i l i b r i u m Equations o f the The gas.  Successive  techniques.  Plasma  approaches t o the treatment  the plasma w i l l be d i s c u s s e d . t h a t there are two  Initially,  and  elements o f the plasma w i l l be respectively.  designated  The  notation for i n d i c a t i n g ionic  standard  These  by the  can be  subscripts  s p e c i e s w i l l be used such  component o f the plasma.  electron density n  two  spectroscopic!  (n* ^ )„_.•_ i s the number d e n s i t y of the i prxn.  the p r i n c i p a l  derived  t h a t d e r i v e d from i m p u r i t i e s  which i n a d v e r t e n t l y contaminate the plasma.  imp.  of  i t w i l l be assumed  components o f the plasma; t h a t  from the p r i n c i p a l gas,  that  values  l a b o r a t o r y plasma i s seldom d e r i v e d from a pure  For t h i s reason, two  p r i n . and  approxima-  t  n  Thus, the  ions  of  total  written  2:13  where a i s the atomic number o f the component gas.  Each  s p e c i e s o f i o n i n the plasma i s i n e q u i l i b r i u m with  the  electrons. i  n  . 7  n  L-M  -he  2:14  -13For t h i s reason, the Saha equation can be w r i t t e n i n the g e n e r a l form  2 :15  where i t i s understood  t h a t t h i s equation can be used t o  d e s c r i b e the r e l a t i v e p o p u l a t i o n o f a l l adjacent i o n i c i n the plasma.  In the Saha equation, k r e p r e s e n t s the  Boltzmann constant, T, the temperature o f the plasma, I s the i o n i z a t i o n  p o t e n t i a l o f the i  t  n  stage i o n .  and Y* are the p a r t i t i o n f u n c t i o n s of the i o n i c and  and Y ^" i+  s p e c i e s i+1  i r e s p e c t i v e l y and are d e f i n e d by  T where g and  specie  EJJJ  1  1 m  = Z  5^  — j  i s the s t a t i s t i c a l weight o f the m  2:16  th  i s the e x c i t a t i o n energy o f t h a t l e v e l .  excited l e v e l The  second  term i n the square b r a c k e t s i s u s u a l l y much l e s s than u n i t y and w i l l be n e g l e c t e d f o r the most p a r t i n t h i s The  treatment.  equilibrium populations of excited states i n  the p l a s m a , n ^ , are g i v e n by  -14In subsequent p a r t s o f t h i s s e c t i o n , i t w i l l be assumed t h a t both n  e  and n *  f o r any  m  s p e c i e s can be measured and  the  temperature,T, w i l l be t r e a t e d as an unknown parameter. two  approaches t o c a l c u l a t i n g the temperature w i t h i n  l i m i t s o f the preceding  information  The  the  are;  (a) Assume t h a t the i m p u r i t i e s i n the plasma are n e g l i g i b l e and  calculate a s e l f consistent  2:13,  2:15  and  2:17,  set o f values  n e g l e c t i n g the  from equations  second term i n  2:13.  (b) Find an approximate value of temperature u s i n g method (a). Using t h i s value o f T, f i n d a c o n s i s t e n t  2:12,2:15  equations  and  2:17  set of v a l u e s  assuming that the  from  impurities  i n the plasma are not n e g l i g i b l e . Turning to the f i r s t s i n g l y and density,  s o l u t i o n , and  doubly i o n i z e d s p e c i e s  one  has  For the time being, component gas  neglect  the  c o n t r i b u t e t o the  only  electron  2:13  from equation H e ^ ^ l A  assuming t h a t  3  3  2:15  1  s u b s c r i p t s f o r the p r i n c i p a l  s i n c e no ambiguity w i l l a r i s e from doing  so.  Keeping i n mind t h a t the Saha equation can be used to f i n d r a t i o of n  to n  n  , one  has  -  2:19  The  e v a l u a t i o n o f the temperature i s s l i g h t l y d i f f e r e n t f o r  n *  and  m  first.  n * * values. m  S i n g l y i o n i z e d s p e c i e s w i l l be  S o l v i n g equations 2:15  logarithms,  the  an i m p l i c i t  and -2:19  expression  examined  and -taking n a t u r a l  f o r the temperature r e s u l t s .  -15-  kT  2:20  i—  *  i/S  m  4  ft** Y  JLA  U s i n g a s c a l i n g f a c t o r s i m i l a r t o t h a t used by N e u f e l d ( 1 9 6 4 ) , the equation may  be transformed  a.  Q.KT  into  ' -  where  L  -f-  A  2:21  L  has been chosen  such t h a t  2-22  and  ^ L ^ i s understood t o be the c o r r e c t i o n term In [ 1 + 2 ( 1 ? / ^ ) ] .  E q u a t i o n 2:21 may Figure(1).  When n  t o the temperature  be p l o t t e d f o r L=f(o^kT) as has been done i n x m  and n  may  e  are known, a f i r s t  approximation  be found by n e g l e c t i n g In  Subsequent i t e r a t i o n s may  Ql+2(n'^'V "''^0 • n  then be c a r r i e d out u s i n g the value  o f n ^ * / n * * c a l c u l a t e d from the Saha e q u a t i o n .  Thus, the  measurement o f the p o p u l a t i o n o f an e x c i t e d s t a t e o f a i neutral species, n l e a d s t o an estimate o f the temperature. ffl  -17-  Equation  2:19 may be s o l v e d with equation 2:17, w r i t t e n f o r  a s i n g l y i o n i z e d s p e c i e s , as w e l l .  T h i s equation  In t h i s case one has  i s a l s o solved u s i n g an i t e r a t i v e  i n c o n j u n c t i o n with the Saha  technique  equation.  Suppose t h a t t h e r e are i m p u r i t i e s present in' the plasma and t h a t the i m p u r i t i e s are dominated by a p a r t i c u l a r ionic  species n  a  +  \  Then '  \  L—  i-.o  /  l  2:24  \  J. /  \  m  P  As b e f o r e , assume t h a t the p r i n c i p a l gas i s predominantly composed o f o n l y the f i r s t two i o n i c  f\ e  („ ) D  stages.  + (z* ) . 01  Then  2:25  With the h e l p o f the Saha equation, t h i s e x p r e s s i o n can be rearranged  t o become  -Itimp.  ^  a+1 \ (n )  ulation  m  o  i  m  u  D  n  &y measuring the e x c i t e d s t a t e  d  .  pop-  Then,  2:27  From equation  \  2:17,  V *1  Equations 2:26 prin  a  s  a  \  cx"  K~  /  and 2:2^ are most e a s i l y s o l v e d by p l o t t i n g ^  u n c - t  i°  n  °f T i  nt  n  e  region of solution  f o r the pure plasma and f i n d i n g the p o i n t o f i n t e r s e c t i o n o f the two curves d e f i n e d by 2:26  and 2:28 . >  For the e x c i t e d s t a t e s o f n e u t r a l s , to write  i t i s possible  an equation o f the form  2:29 x  n.  / -+-Z(-)  -19-  Th e i n t r o d u c t i o n o f a s c a l i n g parameter q s i m p l i f i e s the solution  =  o f the e q u a t i o n .  L  -f- * L ,  -  A  F i g u r e 1 can be used t o f i n d s o l u t i o n s o f equation 2 : 3 0 and i t e r a t i v e t e c h n i q u e s must be used t o get a which i s c o n s i s t e n t with a l l o f the measured  temperature  parameters.  In equation 2:30 AL^ i s d e f i n e d by  2:31  In e v a l u a t i n g the a b s o l u t e i n t e n s i t y procedure,  measurement  i t w i l l be advantageous t o compare t h e r e s u l t s  w i t h those o f r e l a t i v e i n t e n s i t y measurements made on the same apparatus.  So t h a t t h i s might be e a s i l y  accomplished,  the d e r i v a t i o n o f the a p p r o p r i a t e equation w i l l be o u t l i n e d and c i t e d f o r f u t u r e r e f e r e n c e .  I f t h e i n t e g r a l i n equation  2:5 i s r e p l a c e d by 1 ^ , then  iL  -  <~\»L  2  :  3  2  - 20Th e r a t i o o f the t o t a l i n t e n s i t y o f two s p e c i e s i s then given  The  l i n e s o f two  adjacent  by  e x c i t e d s t a t e p o p u l a t i o n s can be e l i m i n a t e d by u s i n g the  2:17  Holtzmann equation  and e l i m i n a t i n g the r a t i o  by u s i n g the Saha equation  2:15.  The  resulting  nVn^ + i  equation  c o n t a i n s o n l y constants o f known value, and the measurable ratio I ^ / l m  m n  ^ " as a f u n c t i o n o f  T h i s equation may  P  +  kT.  be w r i t t e n i n the form  - E 2.  2:35  The  s o l u t i o n o f equation 2:35  i s best c a r r i e d out by u s i n g  a s c a l i n g parameter and a g r a p h i c a l technique t h a t a l r e a d y demonstrated i n t h i s d i s c u s s i o n .  s i m i l a r to  -21-  ( f ) S e n s i t i v i t y o f the Temperature  Measuring Techniques  Both the a b s o l u t e and r e l a t i v e i n t e n s i t y measurements r e q u i r e the measurement o f two parameters o f the plasma.  I t i s i n s t r u c t i v e t o c a l c u l a t e the d i f f e r e n t i a l  v a r i a t i o n i n the c a l c u l a t e d temperature as a f u n c t i o n o f percentage v a r i a t i o n s i n the measured parameters. example, c o n s i d e r equation 2:20  For  f o r a neutral species of  a pure plasma r e w r i t t e n i n e x p o n e n t i a l  form.  2:36  I f the dependence o f the Saha t e n t  m  / n  on temperature  i s c o n s i d e r e d n e g l i g i b l e f o r s m a l l changes i n the a t u r e , then a r e l a t i v e l y simple e x p r e s s i o n A(kT)/kT to  A(n  I m  )/n  I m  temper-  relating  can be - f otrath by "d-i"f*erentiating  equation 2:36 w i t h r e s p e c t t o kT, assuming n ^ v a r i a b l e which depends on the temperature.  i s the o n l y  -22-  2:37  D i v i d i n g equation 2:37 one  by equation 2:36  and r e a r r a n g i n g ,  has  1*1) kT  2:3S  J 2.  where the d i f f e r e n t i a l s have been w r i t t e n as A(kT) A(n^)to  i n d i c a t e small macroscopic  and  changes i n the v a r i a b l e s .  S i m i l a r e x p r e s s i o n s can be found f o r other s p e c i e s i n the plasma w r i t t e n f o r changes i n other measurable parameters. The g e n e r a l e x p r e s s i o n may  be w r i t t e n as  ^ ^k k  / Percentage J=  V  T  change\  J  i n the measurable 1  [  p  a r  a  o e  t  e r  J  where X r e p r e s e n t s the f a c t o r which g i v e s the dependence of  (kT)/kT on -the percentage  parameter.  change o f t h e measured  Table 1 l i s t s v a l u e s f o r X d e r i v e d from the  v a r i o u s equations i n t h i s t h e s i s .  2:39  A c  + V  V 1J  o o  c H  •J  S 0  -0 <0 L  +  to  ! I  »-  i M  -J  m s  +  a) +>  X H  UJ I  +  is  0  c  •iJ +  CHAPTER I I I THE  ABSOLUTE INTENSITY CALIBRATION  (a) I n t r o d u c t i o n The measurement o f a b s o l u t e i n t e n s i t i e s r e q u i r e s the comparison o f the unknown r a d i a t o r t o a source.  In the simplest experimental  standard  arrangement, both  sources have the same g e o m e t r i c a l c o n f i g u r a t i o n and i s observed  through  an i d e n t i c a l o p t i c a l system.  each  However,  i n measuring the a b s o l u t e value o f plasma e m i s s i v i t y , t h e experimenter  i s o f t e n c o n f r o n t e d with comparing a plane,  homogeneous standard source to a vdumetric, source.  For simple geometries,  inhomogeneous  i t i s p o s s i b l e t o make  c e r t a i n assumptions which g r e a t l y reduce the  complexity  o f measuring the a b s o l u t e value o f the e m i s s i v i t y .  This  chapter d e a l s with o n l y one aspect o f t h i s problem, t h a t o f measuring the e m i s s i v i t y o f a homogeneousj, c y l i n d r i c a l l y shaped plasma.  C o r r e c t i o n s f o r the shock tube a b s o r p t i o n  have a l s o been i n c l u d e d * (b) L i g h t Gathering o f the Monochromator In t h i s d i s c u s s i o n , i t w i l l be demonstrated t h a t the proper c h o i c e o f condensing s i m p l i f y the g e o m e t r i c a l o p t i c s .  lens w i l l considerably An examination  of Figure  (2) r e v e a l s t h r e e d i s t i n c t r e g i o n s i n the image space o f the s l i t * These r e g i o n s are d i f f e r e n t from one  another  because o f the s o l i d a n g l e s i n t o which l i g h t from each i s radiated.  In r e g i o n 1, the s o l i d angle i s subtended by the  SiDg  VIEW t  ^ono/ecii/'n^ 2 encode.  FIGURE  Lews,  2.1 LL<jk* of^tkerc  or C o w o(e  CM*?  (.ins.  -fck  -26p r o j e c t i o n o f the c o l l i m a t o r on the condensing l e n s ; i n r e g i o n 2, by the s l i t the  image; and i n r e g i o n 3, by the s l i t  image and  p r o j e c t i o n o f the c o l l i m a t o r on the condensing l e n s . A  v o l u m e t r i c source occupying the r e g i o n o f t h e s l i t  image  would t h u s have p a r t s which are not e q u a l l y e f f i c i e n t i n producing a f i n a l  image i n the monochroraater.  For t h i s reason,  i n t e g r a t i o n must be c a r r i e d out i n each o f t h e e m i t t i n g r e g i o n s s e p a r a t e l y , t a k i n g i n t o account the t r a n s m i s s i o n p r o p e r t i e s p e c u l i a r t o each r e g i o n .  (1930)  Niel«swi  has a n a l y z e d the  p r o b l e m - g e o m e t r i c a l l y f o r both plane and volumetri-c sources, both with and without a condensing l e n s .  A rule-of-thumb  which a l l o w s one t o know i f a thorough approach i s n e c e s s a r y can -be -formulated. is  I f the f - v a l u e o f the condensing l e n s  g r e a t e r than the f - v a l u e o f t h e c o l l i m a t o r , then a  simpli-  f i c a t i o n o f the c a l c u l a t i o n o f t h e u s e f u l f l u x p a s s i n g through the  o p t i c a l system can be made. Figure 3 shows the s i t u a t i o n i n which the condensing  l e n s , having the l a r g e r f - v a l u e , l i m i t s the l i g h t f l u x i n the  optical  system.  are  contained i n the l i g h t  l e n s and the s l i t  A l l points emitting useful  cone subtended by the condensing  image, and the e f f e c t i v e  d e f i n e d by t h e condensing l e n s .  s o l i d angle i s  A l l r a d i a t i o n from p o i n t s  f u l f i l l i n g these c o n d i t i o n s , which the  radiation  i s not s c a t t e r e d out o f  system and which passes through t h e condensing  focused on t h e f i n a l image o f the system.  lens,is  -26I t should be noted t h a t t h i s technique i s p a r t i c u l a r l y i n e f f i c i e n t and cannot be used where low sources are t o be s t u d i e d .  intensity  On the o t h e r hand, the  sub-  s t a n t i a l g a i n i n s i m p l i f i c a t i o n by u s i n g a s m a l l e r condenser l e n s o f t e n warrants the use o f t h i s technique i n o b s e r v i n g b r i g h t plasma sources. (c) Standard  Sources  Standard sources are e s s e n t i a l t o the a b s o l u t e i n t e n s i t y measurement and i t i s the accuracy o f these standards which u l t i m a t e l y l i m i t s the v a l i d i t y o f data i n v o l v i n g absolute i n t e n s i t i e s .  Standard sources may  be  d i v i d e d i n t o two main c a t e g o r i e s : primary sources which are used i n standard l a b o r a t o r i e s , and  secondary  which are c a l i b r a t e d i n the standard l a b and f i n d t i o n i n other l a b o r a t o r y experiments.  sources applica-  The primary standard  i s i d e a l l y a b l a c k body whose r a d i a n t i n t e n s i t y obeys Plank's law, namely  where W(^.,T) i s the t o t a l power per u n i t area r a d i a t e d into- a s o l i d angle 27f o f wavelength  .  7L arrd T have the u s u a l meaning  and temperature,  and c-^ and c  2  depend on  atomic c o n s t a n t s o n l y . The major d i f f i c u l t y with a primary s o u r c e - i s d e f i n i n g i t s temperature.  Griem  (1964) has d e s c r i b e d  -29some o f the problems a s s o c i a t e d with accurate ature  determinations.  Molten g o l d or platinum  used as a primary standard. standard  which has  i s often  An a d d i t i o n a l primary  been used i n the  tungsten lamps i s the  temper-  standardization  cavity radiator.  Larrabee  f o r example, compared the r a d i a t i o n from a  (1959)  cylindrical  f i l a m e n t o f tungsten t o the r a d i a t i o n from a small i n the the  of  hole  c y l i n d e r i n order t o measure the e m i s s i v i t y o f  surface. The  most commonly used secondary  source i s the tungsten r i b b o n lamp. l a b o r a t o r y quotes the b r i g h t n e s s the lamp when a s p e c i f i e d c u r r e n t filament.  T, i s d e f i n e d b  standard  The  standard  temperature, T , D  for  i s f l o w i n g through  such t h a t a b l a c k body with  temperature T^ would have the  same r a d i a n t i n t e n s i t y as  the tungsten f i l a m e n t . o  at a given wavelength, u s u a l l y  6500 A. I f Wien's law  i s considered  imation  where  the  to be a good approx-  o f the r a d i a t i o n , then  "feC^O i s the t r a n s m i s s i o n o f the g l a s s envelope  and  £(3.<jr) i s the e m i s s i v i t y o f the tungsten s u r f a c e  the  c a l i b r a t e d wavelength X  C  .  B[7.T)  *  s  defined  by  at  -30-  =  where W  W  t  P  j  T  )  ,  3:3  and WB.B. ^ e the power radiated f r o * the a  T  tungsten ribbon and a black body respectively f o r the same temperature and wavelength.  A solution of equation  3:2 gives the relationship between the true temperature and the brightness temperature.  JL =_!..+. i X [ { ( ^ U  C J  T)l  3:4  Once T i s found, the radiant i n t e n s i t y can be found at other wavelengths.  W G T ) =-|; T  ™p(~)i  (2)  fiUj)  3=5  DeVos (1954) has elaborated on the relationship between true temperature and brightness temperature, and has also plotted values for^(^.,T) as a function of Ti f o r various values of T.  There i s some degree of controversy  concerning the precise values of £(/?.,T} and the exper-  -31-  imenter should be aware o f t h i s before u s i n g a p a r t i c u l a r set o f v a l u e s . be c o n s i d e r e d  The t r a n s m i s s i o n o f the envelope must f o r i n d i v i d u a l cases.  o f e r r o r are c o n s i d e r e d ,  When a l l sources  the tungsten  r i b b o n lamp  should g i v e a r e s p e c t a b l e degree o f accuracy. tungsten  I f the  r i b b o n lamp i s c a l i b r a t e d a t o n l y one p o i n t ,  measurements f o r wavelengths w i d e l y d i v e r g e n t from the c a l i b r a t i o n wavelength should be h i g h l y suspect.  Errors  may vary from about 3% i n the r e d end o f the spectrum t o about 12% i n the b l u e .  More c a l i b r a t i o n p o i n t s w i l l  reduce t h i s high l e v e l o f e r r o r .  The power requirements  f o r the lamp are small and the l i g h t i s both very and r e p r o d u c i b l e .  steady  However, the i n t e n s i t y o f the lamp  may be too low f o r some a p p l i c a t i o n s , and the range i s l i m i t e d t o the v i s i b l e p o r t i o n o f the spectrum. A much more i n t e n s e source  o f standard r a d i a t i o n  i s the carbon a r c . The c o l o u r temperature i s g e n e r a l l y quoted f o r the o p e r a t i o n o f the a r c . The c o l o u r temperature o f a r a d i a t o r i s t h a t temperature o f a b l a c k body which has the same s p e c t r a l d i s t r i b u t i o n o f energy i n the v i s i b l e r e g i o n as the secondary source.  According  to de Vos, the e m i s s i v i t y must be o f the form  3:6  where K i s some constant l e s s than or equal t o one, and k i s the Boltzmann c o n s t a n t .  Using Wien's Law,  3:7  Thus, the t r u e temperature i s r e l a t e d t o the c o l o u r temperature by  T There i s some d i s c u s s i o n i n the l i t e r a t u r e as t o the a c t u a l value o f T.  Euler  (1954) has suggested a temperature o f  3'995°K with an e m i s s i v i t y o f about 0.75, w h i l e l a t e r authors, N u l l and L o z i e r with  (3.,T)£0.95  (1962) have suggested a value o f 36*00°K depending on the value o f "X. In the v i s i b l e  r e g i o n i t appears t h a t one can make the assumption T =T=3$00 K c  with |S> U-,T)=1. f a l l s below orre.  In r e g i o n s beyond 6000 A, the e m i s s i v i t y In t h e r e g i o n below 2500 A, the a r c plasma  r a d i a t i o n comprises the major p o r t i o n o f t h e spectrum.  In  a d d i t i o n , there are -various r a d i a t i o n bands which can cause trouble" i n the v i s i b l e r e g i o n .  The-molecular r a d i a t i o n  bands i n t h e p y r o m e t r i c a r c spectrum, as t h e y appear i n N u l l and L o z i e r  (1962) a r e :  -33Source CN CN CN  4150,  35*5 3350, 3390 4225, 4500, 45$0 4700, 4730 5163 5633  These bands o f r a d i a t i o n should o f course be The  avoided.  o p e r a t i o n o f the a r c r e q u i r e s some c o n s i s t e n c y o f  technique.  In theory, one observes the anode spot whose  temperature  i s the s u b l i m a t i o n temperature  o f carbon.  However,  i n p r a c t i c e , o n e f i n d s t h a t the anode spot may  not be u n i f o r m l y  b r i g h t , nor does i t remain f i x e d and  The method o f  stable.  o p e r a t i o n used i n the course of t h i s experiment  follows closely  t h a t p r e s c r i b e d by N u l l and L o z i e r and w i l l be d e s c r i b e d l a t e r i n connection with the experimental  techniques.  (d) Shock Tube A b s o r p t i o n Because o f the method used tube  i n c o n s t r u c t i o n , the shock  changes i t s t r a n s m i s s i o n c h a r a c t e r i s t i c s d u r i n g a s e r i e s  o f shots.  I t i s suspected t h a t i m p u r i t i e s are evaporated  from  the m a t e r i a l i n the spark gap r e g i o n , and i t i s these i m p u r i t i e s condensing  on the i n s i d e o f the tube which cause the change  in transmission.  Suppose t h i s t o be the case and  consider  t h a t a l a y e r o f constant t h i c k n e s s i s d e p o s i t e d a f t e r each shot.  Let the l a y e r t h i c k n e s s be AX. and l e t the number o f shots  be n.  I f k i s a constant which depends on the d e p o s i t e d m a t e r i a l  and L i s the a b s o r p t i o n o f the g l a s s w a l l ,  I  then 3:9  -34-  yvhere I  Q  i s the i n t e n s i t y o f r a d i a t i o n  f a l l i n g on the  tube and I i s the t r a n s m i t t e d i n t e n s i t y .  Setting  k'^^c =k , one has  I = I . L e ^ p (-  3:10  The t o t a l l i g h t energy, A, t r a n s m i t t e d i n a s e r i e s o f shots n^ w i l l be  -JA  3:11  r I - e jo (- k  H o  )  Define a mean t r a n s m i s s i o n c o e f f i c i e n t by  0  =  I  L  G  3:12  and w r i t e the f i n a l i n t e n s i t y as  U  lo  =  LI  0  e.*p(-Wv\+)  3:13  Then  3:14  Combining e q u a t i o n s 3 : 1 1 , 3 : 1 2 and 3:13 one has  -35-  F i g u r e 4:  Measuring the shock tube  absorption  In f i g u r e 4 the shock tube i s shown p l a c e d i n f r o n t o f the monochromator so t h a t L can be measured f o r a p a r t i c u l a r wavelength.  The P.M.  v o l t a g e with the tube i n p l a c e i s Vy{/L)  and without the tube i t i s  .  I f the i n c i d e n t 2 i s I„ n d the t r a n s m i t t e d i n t e n s i t y i s L I . then o * o  intensity  a  3 s 16  -36S i m i l a r l y , a t the end o f a run V (2)  3:17  r  In order t o make t h i s measurement, i t i s e s s e n t i a l t o keep t h e d i s t a n c e d great  so t h a t most o f t h e l i g h t  s t r i k e s the tube a t n e a r l y normal i n c i d e n c e ; the  otherwise,  s c a t t e r i n g p r o p e r t i e s w i l l be much d i f f e r e n t  those  encountered when l i g h t  o f the tube.  i s emitted from the i n t e r i o r  A shot t o shot value o f the e f f e c t i v e  t r a n s m i s s i o n can be c a l c u l a t e d by f i n d i n g k from 3:9 and estimates  o f L and L~. i  A plot of i / l  course  equation  versus n o  y i e l d s the e f f e c t i v e t r a n s m i s s i o n f o r each shot. the  from  During  o f a run o f 40 shots, the e f f e c t i v e t r a n s -  m i s s i o n has changed by as much as 15% depending on t h e s p e c t r a l r e g i o n under  observation.  (e) Comparison o f L i g h t Sources In order t o c a l c u l a t e t h e p o p u l a t i o n o f an e x c i t e d s t a t e i n the plasma, i t i s necessary  ZfSX) The  t o measure the e m i s s i v i t y  o f the plasma a t a c a r e f u l l y d e f i n e d wavelength ?.,  technique  i n v o l v e d r e q u i r e s the comparison o f the energy  r a d i a t e d from a w e l l d e f i n e d volume o f the i o n i z e d gas t o the energy r a d i a t e d from a s p e c i f i e d area o f a standard  source.  There a r e two r e l a t e d q u a n t i t i e s a s s o c i a t e d with r a d i a t i n g s u r f a c e s which are i n d i s p e n s a b l e t o t h i s t e c h n i q u e . o f these  The f i r s t  i s t h e radiance B U) tfhich i s the amount o f energy 0  - 37radiated  normally from t h e s u r f a c e per second per u n i t  per s t e r i a d i a n . . radiated figure  area  The second q u a n t i t y i s the t o t a l power  i n t o the h a l f sphere and i s denoted  5, B(©-)=3  Q  cose  by W(2.).  From  and  3:13  where i n t e g r a t i o n Integration  i s c a r r i e d out over the h a l f  o f equation 3:13 y i e l d s W(*)-1TB  t o t a l power r a d i a t e d  sphere.  W.  The  i n t o the h a l f sphere f o r a b l a c k body  i s g i v e n by Plank's Law, equation 3:1»  One o f the p r i n c i p a l  equations o f photometry r e l a t e s the power, dP(/0, i n t e r cepted by an area, dA , which i s r a d i a t e d 2  of b r i g h t n e s s B (/0. Q  See f i g u r e  from a  5 for details.  surface,dA^, The area  elements are separated by a d i s t a n c e R, and the normals o f dA^ and dA the area  g  make angles o f  and #  2  with the l i n e  joining  elements.  3:19  (a) Figure 5:  (b) Radiance and l i g h t  flux  -39Suppose now t h a t an o p t i c a l system composed o f a condenser l e n s and a monochromator can be arranged i n such a way t h a t the condenser, f o c u s e s the s l i t  o f the monochromator  e i t h e r on the standard source or i n the i n t e r i o r o f the plasma. It  i s assumed t h a t the e f f e c t i v e f-number o f the condenser  is  much l a r g e r than the f-number o f the monochromator. Figure  6 shows the schematic diagram o f the apparatus.  Since the  f-number o f the condenser i s much l a r g e r than t h a t o f the monochromator, any l i g h t emitted from the image o f the s l i t and p a s s i n g through the condenser l e n s , which i s not absorbed o r s c a t t e r e d by the system, w i l l e v e n t u a l l y reach the e x i t of  the monochromator.  {/L)&X  Let P  slit  be the power r a d i a t e d  s from the standard source at wavelength 7- i n the wavelength interval  where  i s l i m i t e d by the s l i t  d i s p e r s i o n o f the monochromator.  width and  From e q u a t i o n 3;19  where t i s the transmission" c o e f f i c i e n t o f the-system. Since the areas i r e p a r a l l e l , c o s ^ =cos area o f the image o f the s l i t  -cos©.  The  i s very small and i s r e p r e -  -40sented by dAj.  An area element on the  r a d i u s a i s g i v e n by 2Tfrdr.  where -& 0  c i r c u l a r lens of  Integration y i e l d s  i s the angle subtended by the  condenser l e n s .  Assume t h a t the response o f the p h o t o m u l t i p l i e r i s l i n e a r . Then the v o l t a g e  output o f the p h o t o m u l t i p l i e r c i r c u i t i s  g i v e n by V_(3-)^2_  »  -SP [Xi  o f response f o r the  B  where S i s the  system. Equation 3:20  Z-tJ(P< 7r %U)  \ sM±l= /  becomes  S u i ' ^ X  T  In performing the  coefficient  3:21  same c a l c u l a t i o n f o r the  plasma  source, i t w i l l be n e c e s s a r y to i n t e g r a t e over the volume o f the e m i t t i n g plasma. considerably  i f one  The  calculation i s simplified  notes t h a t the e f f e c t i v e r a d i a t i o n  o r i g i n a t e s o n l y from plasma i n the r e g i o n bounded by cone, whose base i s d e f i n e d by the whose apex i s the image o f the  the  c o l l i m a t o r lens, and  slit.  Note t h a t the  mirror  image o f t h i s cone, r e f l e c t e d i n the plane which passes through the apex normal to the a x i s o f symmetry, a l s o emits l i g h t light  i n t o the o p t i c a l system.  r a y s which pass through the  Further,  slit  only  image or which,  when p r o j e c t e d backwards, would pass through the image w i l l enter the monochromator.  Equation  must be r e v i s e d t o f u l f i l l these c o n d i t i o n s . the e m i s s i v i t y o f the plasma i n the wavelength 4 7L to be  EpCl)^  The  slit  3:19 Consider interval  t o t a l power r a d i a t e d from a  small element o f volume 27T/odpdz i s given by 27pdpdz  £p{3.)  -41where  (^o, z,&)  a co-ordinate  are the c y l i n d r i c a l c o - o r d i n a t e s  of  system o r i e n t e d with the z - a x i s c o i n c i d e n t  with the o p t i c a l a x i s of the apparatus. demonstrates t h i s s i t u a t i o n .  Figure  (6)  I t i s f u r t h e r assumed  t h a t the e m i s s i v i t y i s not a f u n c t i o n o f p o s i t i o n i n the r e g i o n o f o b s e r v a t i o n .  As w e l l , there i s no  a b s o r p t i o n i n the p a r t o f the spectrum under c o n s i d e r ation.  Under these  circumstances,  the amount o f u s e f u l  f l u x r a d i a t e d by t h i s volume element depends on s o l i d angle which the area o f the s l i t  image dAj sub-  tends at a d i s t a n c e r from the volume element. the r a d i a t e d power which passes through the e x i t will  Thus, slit  be  P U W = i f T^Hf ^ W ^ — ^ f  The  the  3:22  47)'appears i n the denominator s i n c e £p(/L) i s the  energy r a d i a t e d i n a l l d i r e c t i o n s and t i s , as b e f o r e , the t r a n s m i s s i o n c o e f f i c i e n t o f the system. i n t e g r a t i o n o f equation 3:22 o f v a r i a b l e P=%fcan-&-.  £U)*7.where  0  Q  On  change  reduction  fsiVJefcl*  3:23  subtended by the l e n s and d i s  the depth o f the plasma.  P^CO** =  r e q u i r e s the simple  ^ ' ^ ^ f l i  i s the angle  The  -fcf.U)  I n t e g r a t i o n yields,  ^(/-cosg,)^  3:24  -42-  The output v o l t a g e o f the p h o t o m u l t i p l i e r ,  \Z [/L), i s p  then g i v e n by  VpU)A,lr  (\-c-o$%)<Jlfi*  3:25  Combining equations 3 : 2 1 and 3 : 2 5 y i e l d s t h e e m i s s i v i t y o f the plasma ZpiX)  a t a p a r t i c u l a r wavelength i n  terms o f known and measurable  S  P  U )  quantities.  '  3  W(X)  3:26  In p r a c t i c e , the plasma i s c o n f i n e d i n a c y l i n d r i c a l tube, so t h a t side-on o b s e r v a t i o n s do not s t r i c t l y adhere t o t h e c o n d i t i o n s o f o b s e r v i n g an i n f i n i t e o f depth d as assumed i n t h e above d i s c u s s i o n . if &  Q  slab  However,  i s r e l a t i v e l y small, the e r r o r i n assuming a plane  s u r f a c e i s not g r e a t .  Secondly, t h e g l a s s or q u a r t z  c y l i n d e r w a l l s absorb and r e f l e c t r a d i a t i o n .  For t h i s  reason, Vp(T-) must be c o r r e c t e d f o r l o s s o f r a d i a t i o n . As has been suggested, n over the l i n e shape.  1  depends on the i n t e g r a l o f  Thus, i t i s e s s e n t i a l t o c o r r e c t  a l l p o i n t s i n a p l o t o f Vp(T-) v e r s u s X .  When a c o r r e c t e d  l i n e p r o f i l e has been drawn, e q u a t i o n 3 : 2 6 i s used t o f i n d the a b s o l u t e value o f t h e peak r a d i a t i o n . methods may then be used t o f i n d  oJt>  Graphical  found i n  -43quation 2:10. alculated  from  The e x c i t e d s t a t e p o p u l a t i o n  can then  CHAPTER IV APPARATUS AND TECHNIQUE (a) I n t r o d u c t i o n The experiment was arranged t o measure a b s o l u t e intensities.  R e l a t i v e i n t e n s i t i e s were then c a l c u l a t e d from  the same set o f d a t a .  Temperatures  c a l c u l a t e d from each s e t  o f measurements were compared. Helium was used as a working gas i n a shock tube w i t h a coplanar d r i v e r .  Side-on o b s e r v a t i o n s were taken through  a simple o p t i c a l system designed t o f u l f i l l the p r e c e d i n g c h a p t e r .  the c o n d i t i o n s o f  A l l measurements were time r e s o l v e d  by u s i n g e l e c t r o n i c d e v i c e s , and a carbon a r c was used f o r the absolute i n t e n s i t y c a l i b r a t i o n .  The shock tube, o p t i c a l  arrangement, d e t e c t i o n d e v i c e s , and the carbon a r c are d e s c r i b e d i n t h i s chapter. (b) Shock Tube The shock tube used i n t h i s experiment has been amply d e s c r i b e d i n o t h e r r e p o r t s o r i g i n a t i n g from t h i s  laboratory.  For t h i s reason, only a short d e s c r i p t i o n i s g i v e n here and the i n t e r e s t e d r e a d e r i s r e f e r r e d t o Sirapkinson (196^) o r Neufeld  (1963).  The shock tube c o n s i s t s o f a one i n c h diameter  quartz o r g l a s s tube w i t h a two c e n t i m e t e r spark gap a t one end. Three 5/<-f c a p a c i t o r s a r e connected i n p a r a l l e l a c r o s s the spark gap w i t h a maximum c h a r g i n g v o l t a g e o f 20 KV.  The d i s c h a r g e  i s i n i t i a t e d through an open a i r spark gap s w i t c h .  In t h i s  p a r t i c u l a r a p p l i c a t i o n , o n l y one o f the t h r e e c a p a c i t o r s was  -46used a t a charging v o l t a g e o f 12 KV. an i n i t i a l pressure o f 300  Hg.  The gas used was He a t  Under these c o n d i t i o n s ,  James (1965) has suggested t h a t He plasmas produced shock tube e x h i b i t e q u i l i b r i u m  in this  characteristics.  (c) O p t i c a l Eqipment and Measuring  Techniques  The plasma was observed ,at r i g h t angles t o the tube at a p o i n t 17  c e n t i m e t e r s from t h e spark gap.  A Jarrell-Ash  (62-010) monochromator with a 50 centimeter f o c a l l e n g t h was arranged as shown i n f i g u r e 7.  The r o t a t i n g m i r r o r a c t s  both t o give a 90° d e f l e c t i o n o f the l i g h t when o b s e r v i n g the He plasma, and t o f u n c t i o n as a s h u t t e r when o b s e r v i n g the standard source. carbon a r c l i g h t signals.  In t h i s way, both the plasma and the  s i g n a l s were r e c e i v e d a t the PM as a.c.  The s t a t i o n a r y 90° d e f l e c t i o n m i r r o r a l l o w s e i t h e r  the plasma or the standard source t o be observed i d e n t i c a l lenses.  through  Note t h a t d i s t a n c e s d-^ and dg are equal  so t h a t the o p t i c a l paths a r e e q u i v a l e n t .  The l e n s e s L are  focused on the anode spot and the c e n t e r o f the shock tube. The p h o t o m u l t i p l i e r s used were an RCA 1P26 range 3000 A t o 6000  5000 1 t o 7000 A. on a cathode  and a P h i l i p s CVP 150 i n the r e g i o n  The p h o t o m u l t i p l i e r s i g n a l s were a m p l i f i e d  f o l l o w e r a m p l i f i e r and d i s p l a y e d on a T e k t r o n i x  545A o s c i l l o s c o p e . averages  I  i n the  The d i s p l a y s were photographed, and the  o f t h r e e or more r e a d i n g s were taken i n order t o  smooth out the shot t o shot f l u c t u a t i o n s i n l i g h t Before p l o t t i n g on graph paper,  intensity.  these r e a d i n g s were a d j u s t e d  f o r tube a b s o r p t i o n a c c o r d i n g t o the technique o u t l i n e d i n  -47earlier  sections.  s  Figure S:  (c)  L i n e Sampling  Techniques  There a r e t h r e e techniques f o r measuring line intensity.  These a r e demonstrated  the i n t e g r a t e d  i n f i g u r e 3 and  e x p l a i n e d beldw. (a) Choose a s l i t  width which covers the e n t i r e l i n e and the  output s i g n a l w i l l be p r o p o r t i o n a l t o the t o t a l l i n e s t r e n g t h . Griem (1964) d i s c u s s e s t h i s technique i n d e t a i l . it  Note t h a t  i s necessary t o c o r r e c t f o r both the continuum r a d i a t i o n  and the wings o f the l i n e . (b) Choose a s l i t  width which covers a known f r a c t i o n o f the  t o t a l l i n e width and then take v o l t a g e r e a d i n g s a t wavelengths in  such a way t h a t the i n t e r v a l s j u s t touch.  The sum o f the  v o l t a g e s w i l l be p r o p o r t i o n a l t o the t o t a l l i n e i n t e n s i t y .  It  -43i s again necessary  t o s u b t r a c t the continuum r a d i a t i o n but  the wings o f the l i n e can be accounted f o r .  There i s a l s o  some u n c e r t a i n t y as t o the a c t u a l f r a c t i o n o f the l i n e (c) Choose a narrow s l i t and measure the l i n e p r o f i l e .  sampled. In  t h i s case the t o t a l i n t e n s i t y w i l l be p r o p o r t i o n a l t o t h e area under the curve.  In a d d i t i o n , the e l e c t r o n d e n s i t y i s  e a s i l y c a l c u l a t e d from l i n e shape o r l i n e h a l f - w i d t h s .  As  b e f o r e , t h e continuum r a d i a t i o n must be s u b t r a c t e d . In t h i s experiment a l l three techniques s i d e r e d but s i n c e n  e  were con-  was t o be c a l c u l a t e d , the l i n e  method was chosen because t h i s technique  profile  allows the c a l c u l a t i o n  o f both t h e e x c i t e d s t a t e p o p u l a t i o n and the e l e c t r o n d e n s i t y . (d) Carbon Arc Both the tungsten  ribbon lamp and the carbon a r c were  c o n s i d e r e d as p o s s i b l e standard imentation tungsten  sources.  Considerable  exper-  i n d i c a t e d t h a t t h e r e l a t i v e l y low i n t e n s i t y o f t h e  lamp made i t unacceptable  with t h e e l e c t r o n i c t e c h -  n i q u e s which were r e a d i l y a v a i l a b l e .  The carbon a r c , on t h e  other hand, proved very s u i t a b l e because o f i t s r e l a t i v e l y h i g h temperature o f 3300°K.  A 150 v o l t , 15 amp, r e g u l a t e d  direct  c u r r e n t power supply  (Sorensen Nobatron DCR 150-15) was  used.  A h i g h c u r r e n t , v a r i a b l e carbon r e s i s t o r was p l a c e d i n  s e r i e s so t h a t t h e a r c c o u l d be operated r e s i s t a n c e i n the c i r c u i t . operation o f the a r c .  with an o v e r a l l p o s i t i v e  T h i s o f course  facilitated  stable  The c u r r e n t was a d j u s t e d t o about 10  amps depending on the e l e c t r o d e s e p a r a t i o n .  As p r e s c r i b e d i n  N u l l and L o z i e r (1962), the e l e c t r o d e s were o r i e n t e d a t 90*.  - 49Th e a r c was operated j u s t below the " h i s s i n g p o i n t " as d e s c r i b e d i n the l i t e r a t u r e .  The a r c i t s e l f was a standard  commercial model, made by Leybold, with the p r o j e c t i o n l e n s removed.  R i n g s d o r f f s p e c t r o s c o p i c carbons RW 202 and RW 401  were used f o r the anode and cathode diameter  respectively.  The  o f the anode spot image on the monochromator was  a d j u s t e d t o be c o n s i d e r a b l y l a r g e r than the s l i t  height.  T h i s ensured t h a t the image area was small as r e q u i r e d by the t h e o r y and reduced  f l u c t u a t i o n s i n the standard  i n t e n s i t y over the s l i t .  source  CHAPTER V RESULTS AND CONCLUDING REMARKS (a) I n t r o d u c t i o n E l e c t r o n d e n s i t i e s were measured u s i n g t h e h a l f - w i d t h s o f HEI l i n e s , and e x c i t e d s t a t e p o p u l a t i o n s were c a l c u l a t e d u s i n g techniques course  In the  o f the work, an attempt was made t o check on v a r i o u s  important and  d e s c r i b e d i n Chapters I I and I I I .  parameters such as the e l e c t r o n d e n s i t y ,  i m p u r i t y consent o f t h e plasma.  were checked a g a i n s t Griem's ( 1 9 6 4 )  absorption,  The e l e c t r o n d e n s i t i e s criteria  f o r LTE and t h e  absorption c a l c u l a t e d using t h e o r e t i c a l expressions  found i n  1  many t e x t s on spectroscopy.  The plasma appeared t o f u l f i l l  the c o n d i t i o n s o f LTE and transparency The  i m p u r i t y content was estimated  of impurity l i n e s .  r e q u i r e d by the t h e o r y .  u s i n g the a b s o l u t e  The e x c i t e d s t a t e p o p u l a t i o n s  intensity  could thus  be c a l c u l a t e d and t h e number o f e l e c t r o n s s u p p l i e d by the i o n i z e d s p e c i e s o f the i m p u r i t i e s e s t i m a t e d . In order t o a r r i v e a t a f i n a l estimate a t u r e , an i t e r a t i v e technique  was employed.  o f t h e temper--  Initially i t  was assumed t h a t the H e l l s u p p l i e d a l l the e l e c t r o n s .  A few  c a l c u l a t i o n s and comparison with t a b l e s i n d i c a t e d t h a t t h e number o f doubly i o n i z e d s p e c i e s present was s u b s t a n t i a l a t the temperature c a l c u l a t e d . both the i m p u r i t y content  Graphical representations of  and t h e degree o f i o n i z a t i o n o f  He as a f u n c t i o n o f temperature were used t o c a l c u l a t e a f i n a l temperature which would be c o n s i s t e n t with a l l o f the a v a i l a b l e data.  -51(b) Results  determined to be 7X10  with an error of 20%.  cm  The 20$  error has been assumed from research into accumulated data from the  same apparatus f o r i d e n t i c a l conditions collected both by  Simkinson (1964) and James (196$) and i s attributable largely to  the r e p r o d u c i b i l i t y of the plasma.  Shot to shot fluctuations  of  the plasma were found to be about 20%.  averages of several measurements were used.  For t h i s reason, Lines were time-  resolved, and a l l readings were taken at the time of maximum i n t e n s i t y of Hel l i n e s .  Although measurements are very  i n s e n s i t i v e to v a r i a t i o n s i n measured parameters, wide deviations from LTE are required before they would appear here. Two approaches have been used i n determining the temperature of the He plasma. to  The purpose of each method was  determine with some degree of certainty the actual number  of H e l l p a r t i c l e s i n the plasma.  The f i r s t method assumes a  pure plasma as presumed i n equation 2:19, while the second considers that there are a large percentage of impurity electrons assumed i n equation 2:26.  In order to estimate the actual  impurity content, CII 3919 and S i l l $056 were measured. Intensity Impurity l i n e  watts cm  CII  3 X 10  3919  S i l l 5056  1.6X  -1 1.6 X 10  6  10  ster  6  3.4 X10  10  10  The number of electrons supplied by these species i s  -52-  13 2*10  -3 cm  .  However, a c a l c u l a t i o n with the Saha equation  i n d i c a t e d t h a t the S i l l l p o p u l a t i o n i s comparable t o n temperatures' i n the 3 ev. range.  for  Q  Hence, u s i n g the S i l l l  5056  upper s t a t e p o p u l a t i o n value, a l o n g with the Saha equation, a plot of S i l l l The  as a f u n c t i o n o f kT was  r a t i o o f H e l l / H e l was  t h a t equations 2:19  drawn as i n f i g u r e  p l o t t e d from the Saha e q u a t i o n so  and a r e v i s e d form o f 2:26  p l o t t e d as i n f i g u r e 10.  9.  c o u l d be  The a b s o l u t e i n t e n s i t y o f H e l 6676,  5675 and H e l l 4666 were measured and e x c i t e d s t a t e p o p u l a t i o n s calculated.  The r e s u l t s o f these c a l c u l a t i o n s along with  the r e s p e c t i v e temperatures  are t o be found i n Table  2.  The agreement between H e l l a b s o l u t e i n t e n s i t i e s r e l a t i v e i n t e n s i t i e s appears  t o be w i t h i n experimental  and error,  the d i f f e r e n c e i n the r e s u l t s o f each method b e i n g about  6%,  The a b s o l u t e i n t e n s i t y o f n e u t r a l helium l i n e s on the o t h e r hand, y i e l d a s u b s t a n t i a l l y h i g h e r temperature about 15$ from the o t h e r s .  The e f f e c t o f i n c l u d i n g the  i m p u r i t i e s does not appear t o be or (c)  no e f f e c t on the c a l c u l a t e d  s i g n i f i c a n t , having  little  -temperatures.  Discussion The  i m p u r i t y content o f the plasma has been estimated  from the measurements of CII and S i l l p o i n t e d out above. in  which v a r i e s  l i n e i n t e n s i t i e s as  T h i s technique has c o n s i d e r a b l e weakness  t h a t i t r e q u i r e d the use o f a second e q u i l i b r i u m e q u a t i o n ,  the Saha equation, as w e l l as the Boitmnann equation . However-, -  t h i s estimate o f the i m p u r i t y e l e c t r o n content i n d i c a t e s t h a t  Figure 10:  H e l l p o p u l a t i o n as a f u n c t i o n o f  kT  TABLE 2 FINAL RESULTS OF RELATIVE AND ABSOLUTE INTENSITY MEASUREMENTS  Line  Hel  667#  n*(-25$)  Relative Intensities  11 3.2x 10  3.2 ev.  5376  3X 10  H e l l 4636  1.3 X10  11.  3.2 ev.  11  n = 7 X 1 0 — 20$  Pure Plasma  3.7 ev. 3.^ ev. 3.4 ev.  Impurities Considered  3.5 ev. 3.7 ev. 3.4 ev.  -56-  17% o f t h e t o t a l e l e c t r o n s s u p p l i e d by S i l l l  and t h a t  approximately 9% o f the t o t a l i o n s are S i l l l  ions.  Using the r e s u l t s o f Table 1, the s e n s i t i v i t y of  the temperature c a l c u l a t i o n t o changes i n measured  parameters i s almost i d e n t i c a l f o r both H e l l a b s o l u t e i n t e n s i t i e s and H e l l - H e l r e l a t i v e i n t e n s i t i e s .  In each  case, the percentage change i n the temperature i s about l/20  the percentage change i n the measured parameters.  Thus, the temperatures c a l c u l a t e d from these t e c h n i q u e s are  o f about equal r e l i a b i l i t y .  The H e l c a l c u l a t i o n ,  on the o t h e r hand, i s extremely s e n s i t i v e t o v a r i a t i o n s n  in  these measurements,.alone,  e  and n * .  Since e r r o r s o f about 20% are expected  in  m  no s i g n i f i c a n c e can be p l a c e d  on the h i g h e r temperature c a l c u l a t e d from the H e l l i n e s . Although the plasma has been found t o more than Griem's  fulfill  (1964) c r i t e r i a f o r e q u i l i b r i u m , f o r t h e above  reasons, the r e s u l t s o f t h i s experiment cannot be used to  support the LTE assumption. The n e c e s s i t y o f i n c l u d i n g the i m p u r i t y content  of the  the plasma i n the c a l c u l a t i o n s appears t o depend on l i n e under study.  The i n s e n s i t i v i t y o f the H e l l  technique t o changes i n the measured n  e  d e n s i t y would  i n d i c a t e t h a t i m p u r i t i e s can be n e g l e c t e d i n t h i s approach. On the o t h e r hand, i f H e l l i n e s are under study, i m p u r i t i e s must be c o n s i d e r e d . a g e n e r a l statement.  These c o n c l u s i o n s can be c a r r i e d For l e v e l s whose p o p u l a t i o n s  into  depend  -57on a l a r g e  exponential  The a b s o l u t e of  producing a direct  Also, two  in  line  intensity  some  term,  estimate  intensities.  no i n c r e a s e  i n accuracy  absolute  the  intensity  calculations  i f  has  excited be  the  does not  difficulty warrant  intensities  its  neglected. advantage  populations. to  compare  absolute reliability  there  by u s i n g a b s o l u t e  increased  state  w i t h good  However,  the  possible  instance,  used  chosen.  relative  of  may n o t  can be  lines  so t h a t  it  In t h i s  appropriate  ments,  are  i m p u r i t i e s may b e  i n t e n s i t y technique  situations  measurement  the  seems t o  intensity  can be  in  be  measure-  i n measuring use  i f  the  temperature  measured.  -53-  BIBLIOGRAPHY  de Vos, J.C., (1954) Physica 20, 690. de Vos, J.C.and Rutgers, G.A.W., (1954) Physica 20, 715. Eckerle, K.L. and McWhirter, R.W.P., (1966) Phys. Fluids 2, 31. Euler, J . , (1954) Ann. Physik 14., 145. Griem, H.G., (1964) Plasma Spectroscopy, McGraw-Hill. i B l e r , Ralph C. and Kerr, Donald E., (1965) Phys. Fluids 3, 1176. James, H.G., (1965) M.Sc. Thesis, University of B r i t i s h Columbia Johnson, F.S., (1956) J-OptSoc. Am. £6, 101. Larrabee, R.D., (1959)J.Opt. Soc. Am. /£, 619. McLean, E.A., Faneuff, C.E., Kolb, A.C., and Griem, H.R., (I960) Phys. Fluids 1, 343. Neufeld, C.R. (1963) M.Sc. Thesis, University of B r i t i s h Columbia Nielson, J . Rud, (1930) J . Opt. Soc. Am. 20, 701. Null, M.R. and Lozier, W.W., (1962) J . Opt. Soc. Am. £2, 1156. Simpkinson, W.V., (1964) Ph.D. Thesis, University of B r i t i s h Columbia.  

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