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UBC Theses and Dissertations

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UBC Theses and Dissertations

Study of a perturbation in a glow discharge Baldis, Hector Alberto 1968

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A STUDY OF A PERTURBATION IN A GLOW DISCHARGE by H e c t o r A. B a l d i s L i e . , U n i v e r s i d a d N a c i o n a l de C o r d o b a , 1964 A T H E S I S SUBMITTED I N P A R T I A L F U L F I L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n t h e D e p a r t m e n t o f P H Y S I C S We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA D e c e m b e r , 1967 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the requirements f o r 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 t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and s t u d y , I f u r t h e r agree that p e rmission f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s represen-t a t i v e s . It i s understood that copying o r p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department nf The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada Date X><roe \M\OQ:V V\ <ol ABSTRACT The p e r t u r b a t i o n i n a g l o w d i s c h a r g e i n h y d r o g e n i s s t u d i e d t h e o r e t i c a l l y a n d e x p e r i m e n t a l l y . I n an o r d i n a r y g l o w d i s c h a r g e a p e r t u r b a t i o n h a s b e e n p r o d u c e d b y a p p l y i n g a s m a l l s i n u s o i d a l h i g h f r e q u e n c y e l e c t r i c f i e l d . T h i s m o d u l a t e s t h e e l e c t r o n e n e r g y a n d h e n c e t h e number d e n s i t y o f a t o m s i n e x c i t e d s t a t e s ; The r e l a t i v e p h a s e a n d a m p l i -t u d e o f l i g h t e m i t t e d i n t h e f i r s t f o u r l i n e s o f t h e B a l m e r s e r i e s h a v e b e e n m e a s u r e d . A t h e o r y b a s e d on t h e w o r k o f B a t e s e t a l , i s d e v e l o p e d i n t h i s t h e s i s a n d t h e r e s u l t s p r e d i c t e d g i v e b e t t e r a g r e e m e n t w i t h t h e e x p e r i m e n t a l v a l u e s t h a n do a l t e r n a t i v e t h e o r i e s w h i c h h a v e b e e n p u b -l i s h e d r e c e n t l y . The s e l f a b s o r p t i o n o f l i n e r a d i a t i o n b y t h e d i s c h a r g e i s shown t o h a v e a c r i t i c a l e f f e c t on t h e b e h a v i o u r o f t h e p l a s m a . The l i m i t a t i o n s a n d v a l i d i t y o f a s s u m p t i o n s a r e d i s c u s s e d a n d a m o d i f i e d e x p e r i m e n t i s s u g g e s t e d . - i i i -T A B L E OF CONTENTS P a g e ABSTRACT i i TABLE OF CONTENTS i i i L I S T OF I L L U S T R A T I O N S i v ACKNOWLEDGEMENTS v C h a p t e r I INTRODUCTION 1 C h a p t e r I I THEORY 3 S e e . l E l e m e n t a r y p r o c e s s e s i n t h e p l a s m a 3 S e c . 2 S t e a d y s t a t e s o l u t i o n . . . . 7 S e c . 3 S o l u t i o n w i t h a p e r t u r b a t i o n . . 8 C h a p t e r I I I D E S C R I P T I O N OF EXPERIMENT 12 S e c . l G e n e r a l 12 S e c . 2 D i s c h a r g e t u b e 15 S e c . 3 O p t i c a l s y s t e m . 15 S e c . 4 P h a s e s e n s i t i v e d e t e c t o r . . . 16 C h a p t e r IV RESULTS 18 S e c . l T h e o r e t i c a l s o l u t i o n . . . . . 18 S e c . 2 E x p e r i m e n t a l v a l u e s 20 C h a p t e r V CONCLUSIONS 24 S e c . l D i s c u s s i o n o f r e s u l t s . . . . 24 S e c . 2 C o m p a r i s o n w i t h o t h e r w o r k . . . 25 S e c . 3 S u g g e s t i o n s f o r f u t u r e w o r k . . 26 A p p e n d i x I ESTIMATION OF ELECTRON ENERGY AND DENSITY 28 A p p e n d i x I I SOLUTION OF ELECTRON ENERGY MODULATION . 30 A p p e n d i x I I I COMPUTER PROGRAM 34 BIBLIOGRAPHY 42 - i v -L I S T OF I L L U S T R A T I O N S F i g u r e Page 1 Modulation of Balwer l i g h t 12 2 Expe r i m e n t a l arrangement - 14 3 R e s u l t s at 4 MHz 21 4 R e s u l t s at 8 MHz 22 5 R e s u l t s at 15 MHz 23 6 E l e c t r o n energy at d i f f e r e n t degrees of modulation 33 -v-ACKNOWLEDGEMENT S I w o u l d l i k e t o s i n c e r e l y t h a n k my s u p e r v i s o r D r . J . H . Y / i l l i a m s o n , f o r h i s h e l p f u l g u i d a n c e a n d p a -t i e n c e t h r o u g h o u t t h i s w o r k a n d 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 am a l s o g r a t e f u l t o t h e members o f t h e t e c h n i c a l s t a f f , i n p a r t i c u l a r t o M r . J . D o o y w e r d , f o r t h e i r h e l p a n d a d v i c e . CHAPTER I - INTRODUCTION When a plasma i s not f u l l y i o n i z e d there e x i s t i n i t n e u t r a l atoms at d i f f e r e n t l e v e l s of e x c i t a t i o n . I f the degree of i o n i z a t i o n of the plasma i s low, or i f the plasma i s decaying from a giv e n degree of i o n i z a t i o n , these e x c i t e d atoms are an important c o n s t i t u e n t of the plasma. The p o p u l a t i o n of these atoms a t the d i f f e r e n t e x c i t e d l e v e l s are i n g e n e r a l o b t a i n e d from the balance c o n d i t i o n s f o r a l l the elementary processes t h a t l e a d to p o p u l a t i o n and d e p o p u l a t i o n of the d i f f e r e n t l e v e l s . A s t a t i s t i c a l approach has been used by Bates e t al.(1962a) to s o l v e t h i s problem and to d e s c r i b e the g e n e r a l decay mecha-nism i n a plasma. D e t a i l e d c a l c u l a t i o n s have been c a r r i e d out f o r a hydrogen plasma and r e s u l t s have been s u c c e s s f u l l y a p p l i e d to r e l a t e d problems (Bates and Kingston 1963,1964). Other approaches have been giv e n by D'Angelo(1961), by Hinnov and Hirshberg(1962) and by Byron et a l . ( 1 9 6 2 ) . These are u s e f u l approximate t h e o r i e s and s i m p l i f i c a t i o n s have been made i n the treatment of the process of decay. The theory of Bates et a l . i s i n reasonable agreement w i t h e xperimental v a l u e s , but i t can be a p p l i e d only to a decaying plasma. Furthermore, the e f f e c t s of boundaries have been ignored so r e s u l t s can be a p p l i e d only t o l a r g e plasmas. In order to t e s t the model f u r t h e r , a s m a l l p e r t u r b a t i o n can be a p p l i e d to the plasma and the response observed. In t h i s t h e s i s , the theory i s extended to a l l o w p r e d i c t i o n s to be made f o r such a plasma. Experimental data are then obtained by -1--2-a p p l y i n g a s m a l l s i n u s o i d a l v o l t a g e p e r t u r b a t i o n t o t h e p l a s m a i n a h y d r o g e n g l o w d i s c h a r g e . I n o r d e r t o u s e a f i r s t o r d e r a p p r o x i m a t i o n , t h e r e s u l t a n t e n e r g y p e r t u r b a t i o n h a s t o be s m a l l c o m p a r e d w i t h t h e a v e r a g e e l e c t r o n e n e r g y , t h a t i s t o s a y , we h a v e t o i m p r e s s m e r e l y a r i p p l e on t h e a v e r a g e e n e r g y . Due t o e l e c t r o n c o l l i s i o n s ( e l a s t i c , i n e l a s t i c a n d s u p e r -e l a s t i c ) w i t h a t o m s , t h i s m o d u l a t i o n i n t h e mean e l e c t r o n e n e r g y ' i s i m p r e s s e d u p o n t h e number d e n s i t y o f e x c i t e d a t o m s . The i n t e n s i t y o f l i g h t r a d i a t e d i n e a c h o f t h e s p e c t r a l l i n e i n t h e B a l m e r s e r i e s w i l l t h e n h a v e s i n u s o i d a l c o m p o n e n t s a t t h e p e r t u r b a t i o n f r e q u e n c y . The r e l a t i v e a m p l i t u d e s a n d p h a s e s o f t h e s e c o m p o n e n t s a r e m e a s u r e d and c o m p a r e d v / i t h t h e t h e o r e -t i c a l p r e d i c t i o n s . U n d e r t h e c o n d i t i o n s o f o u r d i s c h a r g e , t h e r e c o m b i n a t i o n due t o e l e c t r o n d i f f u s i o n t o t h e w a l l s i s i m p o r t a n t and h a s t o be t a k e n i n t o a c c o u n t i n t h e e q u a t i o n s r e l a t i n g t h e d i f f e r e n t e l e m e n t a r y p r o c e s s e s . I n a d d i t i o n t h e r e i s d e e x c i t a t i o n by a t o m i c d i f f u s i o n t o t h e w a l l s , a l t h o u g h t h i s i s n o t so i m p o r -t a n t . T h e s e d i f f u s i o n p r o c e s s e s a r e n o t c o n s i d e r e d i n t h e w o r k o f B a t e s e t a l . A c o m p a r i s o n i s made w i t h t h e w o r k o f H a m b e r g e r ( 1 9 6 3 ) and D a v y ( 1 9 6 7 b ) who s t u d i e d r e s p e c t i v e l y a h i g h f r e q u e n c y d i s -c h a r g e i n h y d r o g e n a n d a p e r t u r b e d g l o w d i s c h a r g e i n h e l i u m . ' CHAPTER I I - THEORY 1)- E l e m e n t a r y p r o c e s s e s i n t h e p l a s m a . The t o t a l r a t e o f c h a n g e o f p o p u l a t i o n i n t h e d i f f e r e n t l e v e l s o f e x c i t a t i o n a n d i o n i z a t i o n o f a d e c a y i n g h y d r o g e n p l a s m a c a n be o b t a i n e d c o n s i d e r i n g t r a n s i t i o n s c a u s e d by t h e f o l l o w i n g e l e m e n t a r y p r o c e s s e s a ) t o g ) . The s y m b o l s H ^ , H + a n d e d e n o t e a h y d r o g e n a tom w i t h p r i n c i p a l q u a n t u m number i , a h y d r o g e n i o n a n d a n e l e c t r o n r e s p e c t i v e l y . W r i t e n ^ a n d n e f o r t h e number d e n s i t i e s o f a toms i n s t a t e i a n d e l e c t r o n s r e s p e c t i v e l y . Due t o t h e m a c r o s c o p i c n e u t r a l i t y o f t h e p l a s m a , t h e number d e n s i t y o f i o n s i s t h e same a s f o r t h e e l e c t r o n s . The e l e m e n t a r y p r o c e s s e s t a k e n i n t o a c c o u n t a r e shown w i t h t h e i r p a r t i c u l a r d e c a y e q u a t i o n e x p r e s s e d i n t e r m s o f t h e r a t e o f c h a n g e f o r a toms i n l e v e l i : a ) downward r a d i a t i v e c a s c a d i n g : w i t h — t — The c o e f f i c i e n t A - y a r e t h e E i n s t e i n c o e f f i c i e n t f o r s p o n t a -n e o u s e m i s s i o n o f r a d i a t i o n o f f r e q u e n c y V>ij • The f i r s t t e r m g i v e s t h e d e p o p u l a t i o n o f l e v e l i due t o t r a n s i t i o n s f r o m l e v e l i . The s e c o n d t e r m g i v e s t h e p o p u l a t i o n due t o t r a n s i -t i o n t o w a r d s l e v e l i ; -3-_4-b ' i n e l a s t i c a n d s u p e r e l a s t i c c o l l i s i o n s w i t h e l e c t r o n s : H i t- e H i w i t h =-7 n ; n Q Gj -t* I r>j n c Cjt dJC J J A c o l l i s i o n i s i n e l a s t i c i f i < j a n d s u p e r e l a s t i c i f i > j . The c o e f f i c i e n t s C . . ( a n d a l s o t h o s e d e f i n e d b e l o w ) a r e f u n c -t i o n s o f t h e e l e c t r o n e n e r g y T . E l e c t r o n i c t r a n s i t i o n s due t o a t o m - a t o m , a t o m - i o n a n d i o n - i o n c o l l i s i o n s a r e n e g l e c t e d . T h e s e h a v e a s m a l l e f f e c t c o m p a r e d w i t h e l e c t r o n c o l l i s i o n s ( B a t e s e t a l . 1 9 6 2 a , page 2 9 8 ) . A u n i f o r m d i s t r i b u t i o n a m o n g s t t h e d e g e n e r a t e s t a t e s o f a l e v e l i s a s s u m e d . E l a s t i c c o l l i -s i o n s must be s u f f i c i e n t l y r a p i d t o e n s u r e s u c h a d i s t r i b u t i o n ; c) t h r e e b o d y r e c o m b i n a t i o n : ^ N J & J ^ ^ Wt. + e + e w i t h an; 3 r- C t ) d ) t h e i n v e r s e o f c ) , c o l l i s i o n a l i o n i z a t i o n : H ; + e H + + e + e \ w i t h cJ-t = — H e f l i C i c ; - 5 -e ) r a d i a t i v e r e c o m b i n a t i o n : I H4. -f- e *=~ H i J<- taP l } w i t h . a m = 0<? Act ; — f j a b s o r p t i o n o f r a d i a t i o n . I f t h e p l a s m a i s o p t i c a l l y t h i c k , u p w a r d t r a n s i t i o n s a n d p h o t o i o n i z a t i o n b y l i n e a b s o r p -t i o n w i l l o c c u r . The c o r r e s p o n d i n g downward t r a n s i t i o n s a r e t h e n b a l a n c e d by t h e r e v e r s e u p w a r d t r a n s i t i o n , a n d t h e t e r m s t h a t c o r r e s p o n d t o t h e s e t r a n s i t i o n s i n t h e s u m m a t i o n i n p a r t a ) a n d e) must be e x c l u d e d . I n o u r c a s e , t h e p l a s m a was f o u n d t o be p a r t i a l l y t h i c k t o w a r d s t h e m a i n s e r i e s o f t h e s p e c t r u m , so a c o m p l e t e b a l a n c e b e t w e e n u p w a r d s and downwards t r a n s i t i o n s d o e s n o t e x i s t . The e f f e c t i v e r a d i a t i v e d e c a y i s o b t a i n e d by d i v i d i n g e a c h A c o e f f i c i e n t by a t r a p p i n g f a c t o r ( H o l s t e i n , 1 9 4 7 ) . T h i s t r a p p i n g f a c t o r d e p e n d s on t h e b o u n d a r y c o n d i t i o n s o f t h e d i s c h a r g e a n d u p o n t h e l i n e s h a p e o f t h e r a d i a t i o n ; g) a n o t h e r i m p o r t a n t p r o c e s s t h a t h a s t o be t a k e n i n t o a c c o u n t i s t h e d e i o n i z a t i o n c a u s e d by a m b i p o l a r d i f f u s i o n o f e l e c t r o n s t o t h e w a l l s . T h i s r e c o m b i n a t i o n g i v e s a c o n t r i b u -t i o n t o t h e g r o u n d s t a t e a t o m s . I n a s i m i l a r way t h e r e i s a c o n t r i b u t i o n t o t h e g r o u n d s t a t e due t o d i f f u s i o n o f e x c i t e d a toms t o t h e w a l l s . T h i s a t o m i c d i f f u s i o n i s i n c l u d e d b u t i s n o t i m p o r t a n t c o m p a r e d w i t h t h e c o n t r i b u t i o n due t o a m b i p o l a r d i f f u s i o n o f e l e c t r o n s . The r a t e o f change due t o t h e s e p r o c e s s e s a r c g i v e n b y . -6-i n e . ^ tzAOS?^ n c (-Df- = |.26 X lO%Te(^ )scc-) dnc „ ^ 2 . 4 Q 5 ^D n . 3 . 7 ? x 10* sec") w h e r e D & and D a r e t h e a m b i p o l a r a n d a t o m i c d i f f u s i o n c o e f -f i c i e n t s r e s p e c t i v e l y a n d r i s t h e r a d i u s o f t h e c a p i l l a r y o f t h e d i s c h a r g e t u b e . T a k i n g i n t o a c c o u n t a l l t h e e l e m e n t a r y p r o c e s s e s c o n s i d -e r e d a b o v e , we c a n w r i t e f o r t h e t o t a l r a t e o f c h a n g e o f p o p u l a t i o n o f a t o m s i n l e v e l i : 4- JL OJ A l f - Z n c n c C « j + L n j n c Cji - n . n i C i c _(*^ esf Dni ^ H H W + n S C eJ + n o A c i + D a n e " w h e r e A ^ j a r e t h e r a d i a t i v e t r a n s i t i o n p r o b a b i l i t i e s v/hen t h e s e l f a b s o r p t i o n o f r a d i a t i o n i s t a k e n i n t o a c c o u n t . When t h e s t e a d y s t a t e p o p u l a t i o n s have b e e n c a l c u l a t e d , an i t e r a t i o n i s p e r f o r m e d t o a l l o w f o r c h a n g e s i n A ^ j , The l a s t e q u a t i o n c a n be c o m p a c t e d i n t o t h e f o r m : -7-f o r i a n d j r u n n i n g f r o m 1 t o N i f N l e v e l s a r e t a k e n i n t o a c c o u n t . T h i s i s a s y s t e m o f N e q u a t i o n s w i t h N unknowns t h a t c a n be s o l v e d t o g i v e t h e d e c a y o f t h e p l a s m a . The v e c t o r Rj_ i s : fK ^ ( ~ — ) 2 D o . 0* -j- A c i Oc - 1 - C a D e • CO E q u a t i o n (1) g i v e s t h e r a t e o f c h a n g e o f t h e number o f a toms i n e a c h one o f t h e l e v e l s . The r a t e o f c h a n g e o f t h e number o f i o n s ( o r number o f f r e e e l e c t r o n s ) i s g i v e n by t h e c o n s e r -v a t i o n o f t h e t o t a l number o f p a r t i c l e s Y_r>i -v- lie = O 4 o 4 a l ^ L w h i c h y i e l d s a r a t e o f c h a n g e cine _ „ y Mi 2 ) - S t e a d y s t a t e s o l u t i o n . I n o r d e r t o f i n d a s o l u t i o n o f e q u a t i o n (1) a f t e r a p p l y i n g a p e r t u r b a t i o n i n t h e e l e c t r o n e n e r g y , t h e s t e a d y s t a t e s o l u -t i o n must be f o u n d f i r s t . F o r t h i s p a r t i c u l a r c a s e , e q u a t i o n (1) r e d u c e s t o -s-w h e r e t h e m a t r i x Qj_j a n d t h e v e c t o r Rj_ a r e k n o w n . The s o l u t i o n o f t h i s s y s t e m o f e q u a t i o n s w i l l h a v e t o be n o r m a l i z e d i n o r d e r t o f u l f i l e q u a t i o n (3). 3 ) - S o l u t i o n w i t h a p e r t u r b a t , i o n . When a p e r t u r b a t i o n i s a p p l i e d t o t h e e l e c t r o n e n e r g y , e q u a t i o n (4j i s no l o n g e r v a l i d . The m a t r i x Qj_j a n d t h e v e c t o r R l c o m p u t e d a b o v e , a r e i m p l i c i t f u n c t i o n s o f T e t h r o u g h c o e f -f i c i e n t s C - • , e t c . a n d an e x p l i c i t f u n c t i o n o f n e . So n o t o n l y do we h a v e t o c o n s i d e r t h e c h a n g e o f Q j j a n d R^ f r o m t h e s t e a d y s t a t e s o l u t i o n due t o a Change i n T g , b u t we must a l l o w f o r t h e e f f e c t o f c h a n g e s i n n e . T h i s c a n be done by l i n e a r i s i n g QiA a n d R..- w i t h r e s p e c t t o s m a l l c h a n g e s i n n ^ . F o r a c h a n g e S T g i n t h e e l e c t r o n e n e r g y t h e s t e a d y s t a t e v a l u e s o f Q . . and R^ a r e r e p l a c e d by Q «j + 6 Q i j P.; + S Ri w i t h t h e c o r r e s p o n d i n g c h a n g e i n atom p o p u l a t i o n s nf + S n? We h a v e now - 9 -Using e x p r e s s i o n (3) and r e t a i n i n g o n l y the f i r s t order t e r c i t J j " ms • As R i ( ) e y H e ) we have SQij = 5nc + § C L 0 ' ST, SR; = ?_Ei §n e -;- a£i §T< then (5) The matrix of the f i r s t term i s of order N and has a l r e a d y been c a l c u l a t e d f o r the value T g i n the steady s t a t e s o l u t i o n , The p o p u l a t i o n of ions (or of f r e e e l e c t r o n s ) Sne c a n ^ e considered as the p o p u l a t i o n of a f i c t i t i o u s (N+l)th l e v e l to be i n c l u d e d a f t e r the N r e a l l e v e l s . Thus the c o e f f i c i e n t s of ^ n e ^ n equation (5) can be w r i t t e n as the (N+l) column elements of a new matrix Q j j of order (N+l) : -10-T h i s t e r m g i v e s t h e c o n t r i b u t i o n s t o t h e c h a n g e i n p o p u l a t i o n i n a n y l e v e l due t o t h e c h a n g e i n i o n p o p u l a t i o n . T h i s c h a n g e i n i o n p o p u l a t i o n i s o b t a i n e d f r o m e q u a t i o n (3) a n d c a n be a d d e d t o t h e N e q u a t i o n s i n (5) a s a N + 1 r o w i n t h e new m a t r i x E q u a t i o n (5) c a n be c o m p a c t e d a s : ^r(on;) = Z Q i j opj + ?; (fo) w h e r e P . i s t h e p e r t u r b a t i o n : T a k e a s m a l l p e r t u r b a t i o n ^ T e w i t h a t i m e d e p e n d e n c e o f t h e f o r m e x p ( j w t ) ( s e e a p p e n d i x I I ) . A s we c o n s i d e r e d o n l y f i r s t o r d e r t e r m s i n t h e d e r i v a t i o n o f e q u a t i o n (5), we c a n assume t h e s o l u t i o n o f £ n ^ t o h a v e t h e same t i m e d e p e n d a n c e . The n e g l e c t e d s e c o n d o r d e r t e r m s w o u l d have no c o m p o n e n t s a t t h e p e r t u r b a t i o n f r e q u e n c y . E q u a t i o n (6) c a n t h e n be w r i t t e n : j u o S o ; « ^ Q \ j & D j + P i ( 8 ) -11-Q! . a n d R. a r e known a n d r e a l . The s y s t e m o f e q u a t i o n s c a n t h e n be s o l v e d f o r t h e c o m p l e x q u a n t 5 . t i e s Sni« The c o m p u t e r p r o g r a m u s e d f o r t h i s i s g i v e n i n a p p e n d i x I I I and t h e r e s u l t s o b t a i n e d a r e l i s t e d i n c h a p t e r I V . The m o d u l a t i o n i n i n t e n s i t y o f l i n e V'IJ d e p e n d s o n t h e m o d u l a t i o n o f t h e p o p u l a t i o n s i n b o t h l e v e l s i a n d j . To f i r s t o r d e r Sly - W q A S U n i - T ^ n . - ^ ) where w i s t h e p r o d u c t o f t h e p e n e t r a t i o n d e p t h t i m e s t h e p o p u -l a t i o n o f l e v e l i . A s i m i l a r c o r r e c t i o n must be a p p l i e d t o t h e m a t r i x Q ! . . The e f f e c t o f t h i s i s t o add an e f f e c t i v e r e v e r s e r a d i a t i v e t r a n s i t i o n T h e s e c o r r e c t i o n s made v e r y l i t t l e d i f f e r e n c e t o t h e f i n a l r e s u l t s f o r o u r d i s c h a r g e . CHAPTER I I I - D E S C R I P T I O N OF EXPERIMENT 1)- General The p l a s m a s t u d i e d h e r e was p r o d u c e d by a p p l y i n g a D . C . e l e c t r i c f i e l d i n a h y d r o g e n d i s c h a r g e t u b e a n d t h e p e r t u r b a -t i o n i n t h e e l e c t r o n e n e r g y was p r o d u c e d by s u p e r i m p o s i n g a s m a l l A . C . e l e c t r i c f i e l d o n t h e d i s c h a r g e . The e m i t t e d l i g h t was o b s e r v e d , a n d t h e r e l a t i v e a m p l i t u d e o f m o d u l a t i o n and r e l a t i v e p h a s e among t h e d i f f e r e n t l i n e s o f t h e B a l r a e r s e r i e s were m e a s u r e d . T h e s e m e a s u r e m e n t s w e r e t a k e n f o r t h e f i r s t f o u r l i n e s H ( 6 5 6 2 . 7 9 A ) , H g ( 4 8 6 1 . 3 3 A ) , 11^(4340.47 A ) and He ( 4 1 0 1 . 7 4 A ) . A t y p i c a l o u t p u t f r o m t h e p h o t o m u l t i p l i e r i s shown i n F i g . l . D . C COM PON M H F i g . l - M o d u l a t i o n o f B a l m e r l i g h t The a m p l i t u d e s c a l e s a r e a r b i t r a r y , - 1 2 --13-Th3 r e l a t i v e m o d u l a t i o n were o b t a i n e d by a v e r a g i n g o v e r a t l e a s t t e n p i c t u r e s . The v e r t i c a l s c a l e s o f t h e d i f f e r e n t l i n e s i n t h e s e p i c t u r e s a r e a r b i t r a r y s i n c e t h e l i g h t i n t e n s i t y i s d i f f e r e n t f o r e a c h l i n e . F o r e a c h t r a c e , t h e z e r o o f i n t e n -s i t y was o b t a i n e d b y b l o c k i n g t h e e n t r a n c e s l i t o f t h e mono-c h r o m a t o r . To o b t a i n t h e p h a s e m e a s u r e m e n t , a p h a s e s e n s i t i v e d e v i c e was u s e d . Due t o s t a t i s t i c a l n o i s e f r o m t h e D . C . d i s c h a r g e , a n y d i r e c t m e a s u r e m e n t o f p h a s e w o u l d h a v e b e e n i n a c c u r a t e . A p h a s e s e n s i t i v e d e t e c t o r i s a n o n - l i n e a r d e v i c e , i n w h i c h t h e m o d u l a t e d s i g n a l i s i n some way m i x e d w i t h a r e f e r e n c e s i g n a l o f t h e same f r e q u e n c y . I n t h i s e x p e r i m e n t , t h e r e f e r e n c e s i g n a l c o u l d h a v e b e e n t a k e n f r o m t h e h i g h f r e q u e n c y o s c i l l a t o r t h a t f e e d s t h e A , C . v o l t a g e t o t h e d i s c h a r g e t u b e , b u t t h e r e was a l a r g e i n c r e a s e o f e l e c t r i c a l p i c k u p o f n o i s e when a s t r a i g h t c o n n e c t i o n was made f r o m t h e H . F . o s c i l l a t o r t o t h e p h a s e d e t e c t o r . I n s t e a d o f t h i s , t h e r e f e r e n c e s i g n a l was o b t a i n f r o m a s e c o n d p h o t o m u l t i p l i e r l o o k i n g a t a p a r t i c u l a r l i n e o f t h e d i s c h a r g e . T h i s a r r a g e m e n t h a d t h e d i s a d v a n t a g e t h a t o n l y r e l a t i v e p h a s e s c o u l d be m e a s u r e d . The o u t p u t f r o m t h e p h a s e d e t e c t o r was i n t e g r a t e d a n d o b s e r v e d w i t h an o s c i l l o s c o p e . The g e n e r a l s e t u p o f t h e e x p e r -i m e n t i s shown i n F i g . 2 . - 1 4 -F - o p t i c a l -f i\-l*c*" P . M . - phoTornoH'tpUer OB L A V U M I T PHASE sewsmve O S T E C T O R t N T £ 6 f * A T O f t o s c a u o s c o p e F i g . 2 - E x p e r i m e n t a l o v - r a n c j m e n T - 1 5 -2 ) - D i s c h a r g e t u b e . A s t a n d a r d h y d r o g e n G e i s s l e r t u b e was v ised a s a s o u r c e . Two e x t e r n a l e l e c t r o d e s were l o c a t e d o n t h e c a p i l l a r y r e g i o n o f t h e t u b e i n o r d e r t o a p p l y t h e H . F . e l e c t r i c f i e l d . The D . C . e l e c t r i c f i e l d was a p p l i e d t o t h e i n t e r n a l e l e c t r o d e s . I n t h i s way e l e c t r i c i s o l a t i o n was o b t a i n e d b e t w e e n H . F . o s c i -l l a t o r a n d D . C . power s u p p l y w i t h o u t u s e o f c a p a c i t o r s a n d c h o k e s . The s t e a d y d i s c h a r g e was s e t u p w i t h a c u r r e n t o f 10 mA, a n d u n d e r t h i s c o n d i t i o n t h e a x i a l r e d u c e d e l e c t r i c f i e l d i n t h e p o s i t i v e c o l u m n X / p was e s t i m a t e d t o be 35 v o l t s c m ~ ^ t o r r ~ ^ ( a p p e n d i x I ) . T h i s g i v e s e s t i m a t e s o f e l e c t r o n e n e r g y and d e n s i t y o f T e = 3 x l 0 4 °K a n d n g = 1 0 1 2 c m " 3 . The H . F . o s c i l -l a t o r was s e t u p t o g i v e a p e a k v a l u e o f a t e n t h o f t h e a p p l i e d D . C . v o l t a g e . T h i s g i v e s a p e r t u r b a t i o n s l i g h t l y l e s s t h a n 10 % i n t h e e l e c t r o n e n e r g y ( a p p e n d i x I I ) . 3 ) - O p t i c a l s y s t e m . The m o d u l a t e d l i g h t was o b s e r v e d w i t h a J a r r e l - A s h E b c r t m o n o c h r o m a t o r , m o d e l 8 2 - 0 1 0 , w i t h a l i n e a r d i s p e r s i o n o f 16 A/mm a n d a s p e e d o f f / 1 0 . The s i g n a l was o b t a i n e d w i t h a RCA 1P21 p h o t o m u l t i p l i e r t u b e . The r e f e r e n c e s i g n a l was o b t a i n e d w i t h a s i m i l a r p h o t o m u l t i p l i e r and t h e p a r t i c u l a r l i n e u s e d as r e f e r e n c e was i s o l a t e d w i t h f i l t e r s . O n l y H Q and H g were s t r o n g e n o u g h t o be u s e d a s r e f e r e n c e s i g n a l s . - 1 6 -In o r d e r t o m e a s u r e t h e d e p t h o f m o d u l a t i o n , i t was n e c -e s s a r y t o o b t a i n t h e D . C . s i g n a l f r o m t h e p h o t o m u l t i p l i e r . I n a d d i t i o n , a 50 XI l o a d was n e c e s s a r y t o m a t c h t h e i n p u t i m p e d -a n c e o f t h e p h a s e s e n s i t i v e d e v i c e . S o , a n e m i t t e r f o l l o w e r c o u l d n o t be u s e d a n d 50 Si had t o be c o n n e c t e d t o t h e anode o f t h e p h o t o t u b e . I n o r d e r t o a l l o w a r e a s o n a b l e v o l t a g e o u t p u t w i t h o u t s a t u r a t i o n , t h e c u r r e n t i n t h e d y n o d e b i a s i n g r e s i s -t o r s was made as l a r g e a s p o s s i b l e (10 roA). A f u r t h e r i n c r e a s e i n t h e o u t p u t c u r r e n t a t t h e anode was o b t a i n e d by u s i n g s p e e d up c a p a c i t o r s i n t h e l a s t f i v e s t a g e s . The amount o f e l e c t r i c p i c k up n o i s e i s shown i n F i g . l i n t h e z e r o r e f e r e n c e t r a c e f o r IIg , w h i c h was t a k e n w i t h a v e r t i c a l s e n s i t i v i t y o f 1 mV/cm. 4 ) - P h a s e s e n s i t i v e d e t e c t o r . A H e w l e t t - P a c k a r d m i x e r ( m o d e l 1051A) was u s e d as a p h a s e s e n s i t i v e d e v i c e . T h i s m i x e r c o n s i s t s o f a r i n g o f f o u r b a l -a n c e d d i o d e s ( W i l c o x , 1959) a n d o p e r a t e s b e t w e e n 0 . 2 MHz a n d 500 M H z . I t i s d e s i g n e d f o r u s e w i t h 50 11 s y s t e m s a n d t h i s l o w i m p e d a n c e i m p l i e s a s m a l l s i g n a l o p e r a t i o n . B u t t h i s was a w o r t h w i l e p r i c e t o pay f o r c l e a n n o i s e l e s s o p e r a t i o n a t t h e f r e q u e n c i e s o f i n t e r e s t b e t w e e n 4 MHz and 15 M H z . D i f f e r e n t p h a s e s e n s i t i v e d e t e c t o r c i r c u i t s wore t r i e d ( F a u l k n e r a n d H a r d i n g , 1 9 6 6 ; S c a n t l e b u r y , 1961) b u t i t was n o t - 1 7 -p o s s i b l e t o g e t a s a t i s f a c t o r y p e r f o m a n c e w i t h , s u f f i c i e n t l y s m a l l p i c k u p n o i s e a t t h a t f r e q u e n c i e s . A s B.C. l o a d a t t h e o u t p u t o f t h e m i x e r a h i g h i m p e d a n c e may be u s e d , b u t t h e A . C . l o a d s h o u l d be 5 0 1 L o r l e s s a t t h e i n p u t f r e q u e n c y . A l a r g e c a p a c i t o r was t h e n u s e d a s l o a d a n d t o i n t e g r a t e t h e o u t p u t s i g n a l . T h i s g a v e a s t e a d y D . C . s i g n a l t h a t was o b s e r v e d w i t h a T e k t r o n i x 585A o s c i l l o s c o p e a t a v e r t i c a l s e n s i t i v i t y o f 1 mV/cm. W i t h t h i s s e t u p i t was p o s s i b l e t o d e t e c t a p h a s e s h i f t c o r r e s p o n d i n g t o a d e l a y b e -t w e e n s i g n a l s o f a t e n t h o f a n a n o s e c o n d . The p h a s e s e n s i t i v e d e t e c t o r gave a maximun o u t p u t when t h e s i g n a l s a r e i n p h a s e o r 180° o u t o f p h a s e . F u r t h e r m o r e , a z e r o o u t p u t s i g n a l i s o b t a i n e d when s i g n a l s a r e i n q u a d r a -t u r e . So t h e m i x e r was u s e d a s a n u l l - d e t e c t o r w i t h o u t p u t i n d e p e n d e n t o f i n p u t a m p l i t u d e . I n o r d e r t o g e t a n u l l r e a d i n g , t h e r e f e r e n c e s i g n a l was d e l a y e d i n s t e p s o f 0 . 1 n s e c , u s i n g e x t r a p i e c e s o f 50 i l c a b l e a n d a c a l i b r a t e d b o x . CHAPTER IV - RESULTS 1) - The o r e t i c a 1 _ J s o 1 u t i o n . To f i n d t h e e x p e c t e d v a l u e s o f p o p u l a t i o n s , a n d t h e d e p a r -t u r e s o f t h e s e f r o m e q u i l i b r i u m when a p e r t u r b a t i o n i n e l e c t r o n e n e r g y i s a p p l i e d , t h e s y s t e m o f e q u a t i o n s ( 8 ) m u s t be s o l v e d . I n o r d e r t o do s o , t h e m a t r i x Q j • ( e q u a t i o n ( 7 ) ) must be c o m -p u t e f i r s t . The v a l u e s o f t h e E i n s t e i n A c o e f f i c i e n t s a r e w e l l known a n d a r e o b t a i n e d f r o m t h e same s o u r c e s a s i n B a t e s e t a l . ( 1 9 6 2 a ) . R e g a r d i n g t h e c o l l i s i o n a l p r o c e s s e s , p r e c i s e r e s u l t s a r e n o t a v a i l a b l e and s o t h e c o e f f i c i e n t s h a v e b e e n c o m p u t e d i n a s i m i l a r way t o B a t e s e t a l , u s i n g t h e f o r m u l a e o f G r y z i n s k i ( 1 9 5 9 ) . T h e s e c o l l i s i o n a l c o e f f i c i e n t s a r e f u n c t i o n s o f e l e c t r o n e n e r g y T e . An e s t i m a t i o n o f t h i s e n e r g y i s made i n a p p e n d i x I . Once t h e v a l u e s o f T e a n d n e a r e e s t i m a t e d , t h e p r o g r a m i s u s e d t o o b t a i n e d t h e e q u i l i b r i u m p o p u l a t i o n s and h e n c e t h e 17 —"3 t o t a l number o f p a r t i c l e s n t o t a l . T h i s s h o u l d be a r o u n d 10 'cm b u t i t s e x a c t v a l u e i s n o t v / e l l known s i n c e t h e d e g r e e o f d i s -s o c i a t i o n o f h y d r o g e n m o l e c u l e s i s n o t k n o w n . The c h a n g e i n p o p u l a t i o n s & n ± a r e c o m p l e x q u a n t i t i e s , s o we c a n w r i t e « XI + j vfi R e p l a c i n g t h i s e x p r e s s i o n s f o r 6 n^^ i n e q u a t i o n ( 8 ) : - 1 8 -- 1 9 -k S e p a r a t i n g r e a l p a r t and i m a g i n a r y p a r t we have the 2(N + 1) e q u a t i o n s : ( i ,W=| y N + l) (°0 w i t h 2(N + 1) unknowns x ^  and y^. E l i m i n a t i n g x^ g i v e s : T h e . v e c t o r i s c a l c u l a t e d by u s i n g the p e r t u r b a t i o n i n the e l e c t r o n energy ST e. T h i s i s g i v e n i n terms o f the m o d u l a t i o n i n the e l e c t r i c f i e l d i n appendix I I . When Q J ^ and a r e known, the system of e q u a t i o n s (9) can be s o l v e d . The s o l u t i o n f o r the f i r s t f o u r l i n e s of the Baimer s e r i e s i s shown i n f i g u r e s 3, 4 and 5. The r e l a t i v e a m p l i t u d e s a r e g i v e n w i t h r e s p e c t t o the e q u i l i b r i u m s t a t e t h u s t h e s e f i g u r e s r e p r e s e n t the depth of m o d u l a t i o n of each e m i t t e d l i n e i n the s e r i e s . The r e l a t i v e phases a r e a-lso g i v e n i n t h e s e f i g u r e s and are c o n s i d e r e d w i t h r e s p e c t to the phase of l e v e l n 3 and a r e expresed as an e q u i v a l e n t d e l a y i n nanoseconds. C a l c u l a t i o n s were performed f o r t h r e e d i f f e r e n t p e r t u r b a t i o n f r e q u e n c i e s , namely 4, 8 and 15 MHz. The c a l c u l a t i o n of s e l f a b s o r p t i o n i n the plasma and the -20-e v a l u a t i o n o f t h e t r a n s i t i o n p r o b a b i l i t i e s due t o e l e c t r o n c o l l i s i o n s were p e r f o r m e d n u m e r i c a l l y u s i n g s u b p r o g r a m s w r i t t e n b y D r . J . H . W i l l i a m s o n . T h e s e a r e i n c l u d e d i n a p p e n d i x I I I . A common s i m p l i f i e d m o d e l o f a h y d r o g e n p l a s m a assumes t h a t t h e Lyman l i n e s a r e c o m p l e t e l y t r a p p e d w h e r e a s a l l t h e h i g h e r s e r i e s a r e n o t a b s o r b e d a t a l l . The r e s u l t s f o r t h i s m o d e l , w h i c h were o b t a i n e d b y r e p l a c i n g A-y b y z e r o , a r e a l s o g i v e n f o r c o m p a r i s o n i n f i g u r e s 3, 4 a n d 5. 2)- E x p e r i m e n t a l v a l u e s . The m e a s u r e m e n t s h a v e b e e n t a k e n w i t h p e r t u r b a t i o n f r e -q u e n c i e s o f 4, 8 a n d 15 M H z , a n d t h e v a l u e s o b t a i n e d a r e g i v e n i n f i g u r e s 3, 4 a n d 5. The p h a s e s a r e e x p r e s s e d a s an e q u i v a -l e n t d e l a y i n n a n o s e c o n d s , a n d t h e v a l u e s o f a m p l i t u d e s h a v e b e e n n o r m a l i z e d t o t h e v a l u e s o f H a o b t a i n e d t h e o r e t i c a l l y . I t i s p o s s i b l e t o do t h i s s i n c e t h e s e a r e o n l y r e l a t i v e v a l u e s . The m e a s u r e m e n t s o f a m p l i t u d e m o d u l a t i o n were t a k e n f r o m p h o t o g r a p h s o f t h e o s c i l l o s c o p e t r a c e . Due t o s t a t i s t i c a l f l u c -t u a t i o n i n t h e d i s c h a r g e , t h e s e m e a s u r e m e n t s h a v e a n a c c u r a c y o f 5%. W i t h r e s p e c t t o t h e p h a s e m e a s u r e m e n t s , t h e s e n s i t i v i t y o f t h e p h a s e d e t e c t o r i s a f u n c t i o n o f t h e s i g n a l a m p l i t u d e . C o n s i d e r i n g f o r e x a m p l e t h e weak He s i g n a l , a d i f f e r e n c e o f 0.1 n s e c was e a s y d e t e c t a b l e a t a f r e q u e n c y o f 15 M H z , a n d o f 0.5 n s e c a t 4 M H z , T h i s b r i n g s a maximum e r r o r o f 10% f o r t h i s p a r t i c u l a r l i n e . The c o r r e s p o n d i n g e r r o r s f o r e a c h l i n e a r e a l s o g i v e n i n f i g u r e s 3, 4 a n d 5. F; 9 . 3 - Results at 4 MHz . O -Proper s e l f absorpt ion; A -Lyman t h i c k ; • -Davy's approach; J -Exper imental va lues . Ff 9. 4 - Re so! is S T 8 MHz. O -Proper s e l f sb s o r p t i o n ; A -Lyman t h i c k ; 0 -Davy's approach; J -Exper imental v a l u e s . F f 9 . 5 - R e s e l l s s f 15 M H z . . © -Proper s e l f a b s o r p t i o n ;A -Lyman t h i c k ; Q -Davy 's approach; £ -Exper imental v a l u e s . CHAPTER V - CONCLUSIONS 1)- D i s c u s s i o n o f r e s u l t s . F i g u r e s 3 , 4 a n d 5 shows r e a s o n a b l e a g r e e m e n t b e t w e e n t h e e x p e c t e d a n d o b s e r v e d v a l u e s o f t h e m o d u l a t i o n i n t h e e x c i t e d l e v e l s i n m o s t c a s e s . T h e d i s c r e p a n c i e s i n some c a s e s may be c a u s e d b y e r r o r s i n t h e c o l l i s i o n c o e f f i c i e n t s p r e d i c t e d b y G r y z i n s k i ' s t h e o r y . F u r t h e r e x p e r m e n t s , a s s u g g e s t e d i n s e c t i o n 3 o f t h i s c h a p t e r , c o u l d be c a r r i e d o u t t o c h e c k t h i s . A l t e r n a -t i v e l y , t h e a s s u m p t i o n s made i n t h e m o d e l s h o u l d be r e v i s e d a n d r e c o n s i d e r e d . T h e e l e c t r o n e n e r g y d i s t r i b u t i o n was assumed t o be M a x w e l l i a n , a n d t h i s i s a weak p o i n t i n t h e t h e o r y s i n c e an e l e c t r i c f i e l d i s a p p l i e d a n d s o t h e a x i a l e n e r g y d i s t r i b u t i o n i s n o t M a x w e l l i a n a n y l o n g e r . The same a p p l i e s t o t h e r a d i a l d i s t r i b u t i o n , s i n c e e l e c t r o n s d i f f u s e t o t h e w a l l s . E l e c t r o n i c t r a n s i t i o n s due t o a t o m - a t o m , a t o m - i o n o r i o n - i o n c o l l i s i o n s h a v e b e e n i g n o r e d , a u n i f o r m d i s t r i b u t i o n amongst t h e d e g e n e r -a t e s t a t e o f l e v e l s was a s s u m e d , a n d t h e e f f e c t s o f m o l e c u l e s h a v e b e e n n e g l e c t e d . The most i m p o r t a n t o f t h e s e a s s u m p t i o n s i s t h e l a s t o n e , a n d i t was n o t p o s s i b l e t o e s t i m a t e t h e e f f e c t o f m o l e c u l e s s i n c e t h e d e g r e e o f d i s s o c i a t i o n was u n k n o w n . The d i s c h a r g e showed s t r i a t i o n s a t t h e e n d s o f t h e c a p i l l a r y r e g i o n o f t h e d i s c h a r g e , w h i c h i n d i c a t e s t h a t m o l e c u l e s a r e a n i m p o r -t a n t c o n s t i t u e n t o f t h e p l a s m a . The i m p o r t a n t r o l e o f s e l f a b s o r p t i o n i n t h e d i s c h a r g e i s shown by t h e d a t a i n f i g u r e s 3 , 4 and 5 . The c o r r e c t e d - 2 4 --25-c a l c u l a t i o n of s e l f absorption showed that the plasma i n our experiment was o p t i c a l l y t h i c k towards Balmer l i n e s and, '".o a less e r degree, towards the remaining s e r i e s . This indicates that a correct c a l c u l a t i o n of s e l f absorption i s important, and inaccuracy i n the predicted values could also be due to inaccuracy i n the evaluation of the trapping f a c t o r since the r a d i a l d i s t r i b u t i o n of ground state and excited atoms i s not known. 2)- Comparison with other work. Hamberger(1963) using a s i m i l a r c a p i l l a r y discharge with a pure A.C. e l e c t r i c f i e l d , calculated the r e c i p r o c a l of the time constant f o r the decay of electron density, and even for pressures as low as 0.2 t o r r , he found i t to be of the order 5 —1 of 10 sec . On t h i s ground, he concluded that there w i l l be no change i n electron density when the frequency of applied 7 -1 e l e c t r i c f i e l d i s of the order of 10 sec , due to the long d i f f u s i o n decay constant. The t h e o r e t i c a l r e s u l t s obtained here disagree with h i s conclusions and show that n e varies j u s t as do the populations i n the excited l e v e l s . As the r e c i p r o c a l mean l i f e - of l e v e l 5 i s comparable with a discharge frequency of 10 MHz, Bamberger expected to f i n d appreciably less modulation of H^ .. In fact he found no notice-able decrease of modulation. In our experiment, a s l i g h t i n -crease of modulation of H i s found experimentally, i n agree-ment with the t h e o r e t i c a l predictions. - 2 6 -A m o d u l a t e d g l o w d i s c h a r g e i n h e l i u m h a s b e e n s t u d i e d by D a v y ( 1 9 6 7 b ) , a l t h o u g h h i s m e a s u i e m e n t s w e r e a l l made b e l o w 1 M H z . H i s t h e o r e t i c a l m o d e l ( 1 9 6 7 a ) a c c o u n t s o n l y f o r a m b i -p o l a r d i f f u s i o n t o t h e w a l l s a n d i o n i z a t i o n d i r e c t l y f r o m t h e g r o u n d s t a t e . He p r e d i c t s t h a t t h e m o d u l a t i o n o f e l e c t r o n d e n s i t y s h o u l d l a g -ir/2 b e h i n d t h e m o d u l a t i o n o f e l e c t r o n e n e r -g y , a n d t h a t t h e p h a s e o f t h e l i g h t r a d i a t e d f r o m e x c i t e d s t a t e s s h o u l d l i e b e t w e e n 0 a n d T T / 2 . When a p p l i e d t o t h e d i s c h a r g e s t u d i e d h e r e , h i s f o r m u l a e p r e d i c t s p h a s e s h i f t s f o r t h e Hg t o H g l i n e s o f 0 . 5 1 , 0 . 7 3 a n d 0 . 8 4 n s e c r e s p e c t i v e l y a t a f r e q u e n c y o f 4 M H z . Our m e a s u r e m e n t s show p h a s e s h i f t s 10 t i m e s l a r g e r t h a n t h e s e v a l u e s . I n a d d i t i o n , o u r t h e o r e t i c a l r e s u l t s show t h a t 8n e a n d $ T e a r e n o t n e c e s s a r i l y i n q u a d r a t u r e . 3 ) - S u g g e s t i o n s f o r f u t u r e w o r k . F o r f u t u r e w o r k , a d i s c h a r g e w i t h a l a r g e d i a m e t e r s h o u l d be u s e d , o p e r a t e d a t a l o w e r g a s p r e s s u r e . A l a r g e r d i a m e t e r w o u l d d e c r e a s e t h e d i f f u s i o n t o t h e w a l l s a n d s h o u l d i m p r o v e t h e c o n d i t i o n s t o w a r d s a M a x w e l l i z a t i o n o f e l e c t r o n e n e r g y . L o w e r g a s d e n s i t i e s w i l l d e c r e a s e t h e s e l f a b s o r p t i o n o f r a d i -a t i o n w h i c h was f o u n d t o p l a y a n i m p o r t a n t r o l e i n t h e d i s c h a r g e A l a r g e r t u b e w i l l a l s o a l l o w a d i r e c t m e a s u r e m e n t o f T e a n d n e . I n o r d e r t o g e t a b e t t e r i n s i g h t i n t o t h e m e c h a n i s m s g o v e r n i n g t h e d i s c h a r g e , a p e r t u r b a t i o n o f d i f f e r e n t k i n d - 2 7 -should be used. Instead of producing the perturbation through a modulation of the electron energy, an absorption experiment could be performed, using a modulated source of l i g h t ( s i m i l a r to that described by Hamberger) with the same gas as i n the discharge under study. The l i g h t from such a source could be modulated up to 100% and used to produce a d i r e c t perturbation to a p a r t i c u l a r atomic l e v e l , by pumping between two given l e v e l s . In our experiment, the e l e c t r o n i c pumping was done mainly from ground state to a l l the upper l e v e l s . -A P P E N D I X I - E S T I M A T I O N OF ELECTRON ENERGY AND DENSITY T h e s e q u a n t i t i e s h a v e n o t b e e n m e a s u r e d i n t h i s e x p e r i m e n t a n d e s t i m a t e d v a l u e s were u s e d . V a l u e s o f e l e c t r o n e n e r g y a r e g i v e n b y v o n E n g e l(1965) a s a f u n c t i o n o f t h e r e d u c e d d i s c h a r g e r a d i u s c p R , w h e r e c i s a c o n s t a n t d e p e n d e n t o n l y o n t h e n a t u r e o f t h e g a s , p i s t h e g a s p r e s s u r e a n d R i s t h e r a d i u s o f t h e p o s i t i v e c o l u m n . I n t h e d i s c h a r g e t u b e u s e d i n t h e e x p e r i m e n t , t h e r a d i u s o f t h e c a p i l l a r y r e g i o n i s 0.05 cm a n d t h e gas p r e s -s u r e i s a s s u m e d t o be 10 t o r r . U s i n g c - l . 0 5 x 10 a s g i v e n by v o n E n g e l f o r h y d r o g e n , we h a v e cpR = 0.525 x 10 , W i t h t h i s v a l u e , we o b t a i n f o r t h e e l e c t r o n e n e r g y ( v o n E n g e l , 1965 p a g e 243) Te = 3 x 10 4 °K. The e l e c t r o n number d e n s i t y c a n be e s t i m a t e d i f a v a l u e o f t h e a x i a l r e d u c e d e l e c t r i c f i e l d i s o b t a i n e d . I n o r d e r t o do s o , t h e i m p e d a n c e o f t h e p o s i t i v e c o l u m n was m e a s u r e d . T h i s was p e r f o r m e d b y m e a s u r i n g t h e i m p e d a n c e a s a f u n c t i o n o f f r e -q u e n c y a t t h e e x t e r n a l e l e c t r o d e s where t h e A . C . e l e c t r i c f i e l d was a p p l i e d . The c a p a c i t i v e r e a c t a n c e o f t h e g l a s s b e t w e e n e l e c t r o d e s a n d p o s i t i v e c o l u m n was n e g l i g i b l e a t f r e q u e n c i e s o v e r 5 M H z , c o m p a r e d w i t h t h e c o l u m n i m p e d a n c e . The i m p e d a n c e m e a s u r e d i n t h i s way was Z p c = 35 k i l / c m ( a s s u m e d t o be p u r e l y r e a l ) . The D . C . c o m p o n e n t o f the- t o t a l c u r r e n t i n t h e d i s c h a r g e was 10 mA, s o t h e a x i a l r e d u c e d e l e c t r i c f i e l d i s X / p = 35 v o l t c m " 1 t o r r " 1 . Now, t h e d r i f t v e l o c i t y o f e l e c t r o n s i n h y d r o g e n i s g i v e n b y t h e m o b i l i t y e q u a t i o n : -28--29-Ve = 3 J x |0 5 X ; (cm sec"') and the current density i s : j a e He Ve 2 So then, i f we write f o r the t o t a l current I = n R j , we have: n« «• - I = ( 0 . 5 4 *io'*) a 1 3.7 x 10= -rr Ra e X/p ^ R *X/p Using the value of X/p obtained above we get for the electron 12 —3 number density n e= 10 cm APPENDIX I I - SOLUTION FOR ELECTRON ENERGY I i T i s t h e mean e n e r g y o f an e l e c t r o n a t some i n s t a n t t , t h e c n a n g e d ( T e ) d u r i n g a t i m e d t i s g i v e n by t h e d i f f e r e n c e b e t w e e n t h e e n e r g y g a i n due t o a d i s p l a c e m e n t dx i n t h e e l e c t r i c f i e l d , a n d \,he l o s s due t o c o l l i s i o n s w i t h t h e g a s a toms ( H a r r i s a n d v o n E n g e l , page 4 9 6 ) : w h e r e e i s t h e e l e c t r o n c h a r g e , p t h e e l e c t r o n c o l l i s i o n f r e -q u e n c y , IC t h e a v e r a g e f r a c t i o n o f e n e r g y l o s t i n e a c h c o l l i -s i o n a n d E t h e ( e x t e r n a l ) e l e c t r i c f i e l d . I n o u r p a r t i c u l a r c a s e t h e e l e c t r i c f i e l d E h a s a D . C . c o m p o n e n t p l u s a s m a l l A . C . m o d u l a t i o n . So E c a n be e x p r e s e d a s : w h e r e X i s t h e D . C . c o m p o n e n t o f t h e e l e c t r i c f i e l d a n d a i s t h e f r a c t i o n o f m o d u l a t i o n a t f r e q u e n c y w ( a « 1 ) . A t h i g h g a s p r e s s u r e ( 1 0 t o r r i n o u r e x p e r i m e n t ) t h e e l e c t r o n d r i f t v e l o c i t y i s i n p h a s e w i t h t h e o s c i l l a t i n g e l e c -t r i c f i e l d , s o •^ 5- = M l ( | fQsinco-t) (A-2>) 4* 1 P -. - ' w h e r e z i s t h e a x i a l c o o r d i n a t e i n t h e d i s c h a r g e t u b e a n d \i i s t h e m o b i l i t y o f t h e e l e c t r o n s . T h i s e x p r e s s i o n i s j u s t i f i e d i n o u r c a s e s i n c e a t t h i s p r e s s u r e t h e e l e c t r o n c o l l i s i o n d l T a ) = e E d x - IC P T e c i t ( A - 0 gr - )( (i T- a. s\n u)Jc ) (A-2) f r e q u e n c y i s p ^ 10 11 s e c 1 a n d t h e h i g h e s t a n g u l a r f r e q u e n c y - 3 0 -- 3 1 -a p p l i e d i s <J^ 1 0 ° s e c - J \ T h i s a l s o j u s t i f i e d t h e u s e o f v a l u e s o f C ,P a n d u d e r i v e d f r o m e x p e r i m e n t a l d a t a o b t a i n e d i n D . C . d i s c h a r g e s . U n d e r t h i s c o n d i t i o n s a n d f o r v a l u e s o f r e d u c e d e l e c t r i c f i e l d X / p > 15 v o l t cm 1 t o r r \ H a m b e r g e r ( 1 9 6 3 ) u s e s t h e a p p r o x i m a t i o n f o r KV/p: = 3.23 x l 0 7 ( e , 3 , T a - ] ) U-4) P S i n c e i n o u r d i s c h a r g e X / p i s f o u n d t o be 35 v o l t cm 1 t o r r " 1 ( a p p e n d i x I ) , t h i s a p p r o x i m a t i o n i s j u s t i f i e d . U s i n g e x p r e s s i o n s ( A - 2 ) , ( A - 3 ) a n d ( A - 4 ) , e q u a t i o n ( A - l ) c a n be w r i t t e n : -dls..^ ep-^-O+a s ir?co-tI"-3.23 xio7(e°31 Te~l)pTe U s i n g n u m e r i c a l v a l u e s f o r u a n d X / p : eUf = A( l + a sinco-tf - B ( e 0 , 3 , T e - i ) T c CA-S) d -t w h e r e A = 4 . 5 3 x 1 0 9 eV s e c " 1 a n d B = 3 . 2 3 x 1 0 8 s e c " 1 . E q u a t i o n ( A - 5 ) i s s o l v e d n u m e r i c a l l y u s i n g R u n g a - K u t t a method f o r d i f f e r e n t d e g r e e s o f m o d u l a t i o n . S o l u t i o n s a r e shown i n F i g . 6 . A n e n l a r g e d p l o t h a s b e e n made f o r 10% m o d u l a t i o n a n d t h e v a r i a t i o n i n e l e c t r o n e n e r g y i s f o u n d t o be s i n u s o i d a l . I f t h e m o d u l a t i o n i s i n c r e a s e d t h e p a t t e r n becomes d i s t o r t e d . I f a p u r e A . C . f i e l d i s a p p l i e d , t h e v a r i a t i o n o f e l e c t r o n e n e r g y i s f o u n d t o h a v e a f r e q u e n c y 2w a n d i t s p a t t e r n i s a l s o p l o t t e d f o r c o m p a r i s o n i n -32-Fig.3. This p a r t i c u l a r r e s u l t has been obtained previously by Hamberger (1963). - 3 3 -Te F i q . 6 - e l e c t r o n energy a t d i f f e r e n t d e g r e e s o f m o d o I s i t o o . APPENDIX III - COMPUTER PROGRAM -34-s ^ F O R T R A N . 9 C C O M P U T E E Q U I L I B R I U M P O P U L A T I O N S A N D T H E I R C H A N G E W I T H R E S P E C T T O c _C_ A P E R T U R B A T I O N I N E L E C T R O N E N E R G Y ' 3 C U N I T S A R E D E G R E E S » " N A N O S E C , C U B I C M I C R O N S C . -D I M E N S I O N QM_( 2 0 »_20J_»_P0P ( 20 ) * RECOMBJ_20_) » R E C 0 P J 2 O J »_TR ( 5 0 _ _ c [ ~l 1 " v ( p o ) , X ( 7 0 > , A V 3 ( ? 0 > , A \ G ( 2 0 ) , A R ( 2 0 , 2 0 ! , Q v o ("Q , 70) , Q v O M v ( 2 0 , 2 0 ) , " Z 2 Q Q O M ( 2 0 , 2 0 ) , F ( 2 0 , 2 0 ) , A ( 2 0 , 2 0 ) , P O P S T ( 2 0 ) , W W H ( 2 0 > , P E R T ( 2 0 > j COMMON / Q A / A A ( 20 • 2 0J_ . J COMMON / T P / S H ( 2 0 > , C ( 2 0 , 2 0 ) , C C ( 2 0 ) , C I C ( 2 0 ) , A C ( 2 0 ) , A C E ( 2 0 ) N = 10 NP = N + 1__ , PI=3 . 1 A 1 5 9 2 6 5 D I F F = . 2 2 E - 3 READ I N E I N S T E I N A C O E F F I C I E N T S R E A D ( 5 , 1 0 1 " ) ( ( A ( I , J ) ,1 = 1 , 1 0 ) , J = l , 1 0 C A L L COPY ( A , A A , 1 0 , 2 0 ) DO 1 1 I = 1 » N A; = 1*1 „ DO 1 1 J - 1 » I 0 1 AJ = J * J _ . — — • — — — — — 1 1 "W = T . / A J - l . / A I F ( L , J ) = 2 . 3 3 8 6 * A U , J > * A I / ( A J * W*W*W) RE A D ( 5 , 1 0 1 ) ( TR ( K ) , K = 1 , 2 5 ) ; C C S O L V E FOR E Q U I L I B R I U M P O P U L A T I O N S AT E N E R G Y T E AND D E N S I T Y DE R E A D ( 5, 1 0 1 ) T E , D E DE= D E * 1 . E - 1 2 C A L L T R P R O B ( T E , N ) C I N I T I A L E S T I M A T E S OF E Q U I L I B R I U M P O P U L A T I O N S A R E F E D I N R E A D ( 5 , 1 0 1 ) ( P O P ( I ) , I = 1 , N ) 1 8 DO 19 1=1,N 19 P O P S T ( I ) = P O P ( I ) C R A D I A T I V E C O E F F I C I E N T S A R E M U L T I P L I E D B v T R A P P I N G "FACTOR'S" DO 4 1 I = 2 , N I A = I - l DO 3 3 J = ! , I A " ~ ~ - - - - - -W W = F ( I , J ) * P O P ( J ) * 0 . 0 0 2 9 I F ( J . E Q . 2 ) W W H ( I ) = WW_ WL = A'LOG (WW*160Vr/ArOGT2V) ' WL = AMAX 1 ( 2 . , A M I N 1 ( W L , 2 4 . ) ) KWL = I N T ( W L + 0 . 5 ) XWL= WL - F L O A T ( K W L ) A A < I * J ) = 2 . * ( T R ( K W L + l ) * X W L * ( l . + X W L ) / 2 . + T R ( K W D * ( 1 . - X W L * X W L ) 1 -__ TR_( KWL - 1_)*XWL*U. ._-XWU / 2 _ ) _ * A ( I • _ J . ) _ / _ L i ^ t _ _ 3 3 AB i I, J ) =' A A ( F, J ) * WW * POP ( I ) ? ( ( 1. + WW ) *P O P ( J ) )' 4 1 A A ( I » 1 ) = A A ( I , 1 ) + D I F F C EQ_UIL I B R I U M _ P O P U L A T I O N S A R E R E E V A L U A T E D C A L L QMATR ( QM » R E C O M B",T E"» D E V N ) C A L L C O P Y ( Q M » QMOMM » N P • 2 0 ) DO 20 J = 1 » N . 2 0 P O P ( J ) = - R E C O M B ( J ) . C A L L S O L T N ( Q M O M M , P O P , N » 2 0 » D E T ) SUM=DE 1 P O P ( N P ) = D E P O PCON = 0. DO 15 1=1,N I F ( A B S ( P O P ( I ) - P O P S T ( I ) ) . G T . . 0 1 * P O P S T ( I ) ) P O P C O N = 1 15 " S U M = S U M + P O P ( I ) I F ( P O P C O N . G T . 0 . 0 1 ) G O TO 18 C I T E R A T I O N I S C O M P L E T E D WHEN P O P . A R E U N C H A N G E D W I T H I N 1 P E R C E N T ^ C ~ f C E N E R G Y P E R T U R B A T I O N ( 1 0 P E R C E N T ? , , '_ T E P = T E * 1 . 1 C A L L T R P R O B ( T E P , N ) C A L L QMATR. ( Q M P ,_R E C O P , T E P , D E , N ) C C A L C U L A T E A P P L I E D P E R T U R B A T I O N TO P O P U L A T I O N S DO 6 0 I = 1 ,NP _ P E R T ( I ) = R E C O P ( I ) ; DO 6 0 J = 1 » N 6 0 P E R T ( I ) = P E R T ( I ) + Q M P ( I , J ) # P O P ( J ) C ALLOW FOR M O D U L A T I O N C A U S E D BY V A R Y I N G O P T I C A L T H I C K N E S S DO 6 1 I = 2,N I A = 1-1 DO 6 1 J = 1»IA _ QM( J , J ) = QM ( J , J ) - A B U , J ) 6 1 Q M ( I , J ) = Q M ( I » J ) + A B ( I , J ) WWH ( 1 ) __= 0 . WWH ! ?.) = 0 . WWH(NP) = 0. C C S O L V E FOR R E S P O N S E AT F R E Q U E N C Y OM 10 R E A D ( 5 , 1 0 1 ) OM W R I T E ( 6 » l l l ) T E » D E » O M W R I T E ( 6 » 1 3 5 ) S U M W R I T E ( 6 » 1 1 9 ) ( P O P ( I ) » 1 = 1 , N ) 0 M M = 0 M * l . . E - 9 DO 2 2 J = 1 , N P DO 2 2 1 = 1 , N P 2 2 Q M O M M J I , J ) = Q M ( I , J ) / O M M C A L L MULT'( QMOMM ,QMOMM , QQOM ,NP , 2 0 ) DO 7 0 1 = 1 , N P Y ( I ) =-PERT ( I ) /OMM 7 0 Q Q O M ( I , I ) = Q Q O M ( I , I ) + l . C S O L V E FOR I M A G I N A R Y P A R T OF R E S P O N S E C A L L S O L T N ( QQOM , Y , N P , 2 0 , _ D E J ) C S O L V E P O R R E A L P A R T OF R E S P O N S E C A L L M A T V E C ( Q M O M M , Y , X , N P , 2 0 ) DO 8 0 I = 1 , N P , " X ! I ) = X ( I ) * 1 0 0 . / P O P ( I ) Y ( I ) = Y ( I ) * 1 0 0 . / P O P ( I ) C O R R E C T FOR T R A P P I N G OF B A L M E R S E R I E S X ( I T= X ( I ) -X ( 2 r * W W H ( I ) / ( 1 . +WWH ( I ) ) Y ( I ) = Y ( I ) - Y ( 2 )*WWH( I )/(1.+WWH( I ) ) AMP( I ) =SQRT ( X ( I ) *X ( I ) + Y ( I )_*Y ( I )_) 8 0 A N G ( I ) = A T A N 2 ( Y ( I ) , X ( I ) !* 1 8 0 . / P I W R I T E ( 6 » 1 3 Q ) W R I T E ( 6 , 1 1 9 ) ( X ( I ) , I = 1 , N P J W R I T E ( 6 , 1 2 9 ) W R I T E ( 6 , 1 1 9 ) ( Y ( I ) , 1 = 1 , N P ) W R I T E ( 6 , 1 3 2 ) _ W R I T E ( 6 , 1 2 0 ) ( A N G ( I ) , I = 1 , N P ) W R I T E ( 6 » 1 3 3 ) W R I T E ( 6 , 1 2 0 !_( AM P U J , I =J_» N_PJ 6 0 TO 1 0 1 0 1 FORMAT ( 1 0 F 8 . 0 ) 1 1 1 FORMAT ( 5 H 1 T E , F 1 0 . 0 , 5 H N E , 1 P E 1 0 . 2 ,__5 H OM , E 1 0 . 2 ) 1 1 9 FORMAT ( IX»lP'll'ETl .2') "~ ~" 1 2 0 F O R M A T ( I X , 1 1 F 1 1 . 2 ) 1 2 9 FORMAT ( 3 2 H - I M A G I N A R Y _ P A R T _ O F R E S P O N S E (_Y)J 1 3 0 F O R M A T ( 3 2 H - R E A L P A R T O F R E S P O N S E ( X j " ) 1 3 2 F O R M A T ( 3 2 H - A N G ) 1 3 3 F O R M A T ( 3 2 H - R E L A T I V E C H A N G E _ _ I N P O P U L . ) 1 3 5 F O R M A T ( 3 2 H - P O P U L A T I O N S ~" "SUM =, T P E T O . 2 ) E N D S U B R O U T I N E Q M A T R ( Q M , R E C O M B > T E , D E , N ) • 3 COMPUTE D E C A Y M A T R I X Q ( I » J > FROM T R A N S I T I O N P R O B A B I L I T I E S L D I M E N S I O N _QM ( 2 0_» 2_0 )jRECOMB ( 2 0 ) . 8 COMMON / O A / A A ( 2 0 , 2 0 ) 6 COMMON / T P / S H ( 2 0 ) , C ( 2 0 , 2 0 > * -CC(20)» C I C ( 2 0 ) , AC ( 2 0 ) , A C E ( 2 0 ) , (H A M D I F F = 0 . 7 3 E - 6 * T E u NP = N+1 ' Z i DO 50 J = 1 » N t S I 6 = C I C ( J )*DE . — . . — h Q M ( N P * J ) = S I G DO 5 1 1=1,N QM ( I , J ) = AA ( J , I )_+DE_*C__J_,_IJ_ 5 1 S I G = S I G + Q M ( I , J ) 5 0 Q M ( J , J ) = - S I G QNN = AMD I F F R EN = AMD I F F # D E " DO 5 5 I = 1 , N QM ( I , NP )_= ( 2 v * A _ J J _ + 3 . _ * _ C C J J _ ) *pE_*_D_E RECOMB ( T ) = ( AC ( I ) + CC~7l ) *DE f*DE*DE~ ^ QNN = QNN + QM ( I ,NP ) °° 5 5 REN = REN+RECOMB (_!___ Q M ( 1 , N P ) = Q M ( 1 , N P ) + A M D I F F R E C O M B ( 1 ) = R E C O M B ( 1 ) + A M D I F F * D E Q M ( N P , N P ) = - Q N N R E C O M B ( N P ) = - R E N R E T U R N END _ _ S U B R O U T I N E ~ T R P R O B ( T E , N ) C O E F F I C I E N T S FOR S A H A E Q U I L I B R I U M P O P U L A T I O N S AND T R A N S I T I O N P R O B A B I L I T I E S B E T WEE N_B0UNJD_ _3T A T E_S_A N D _ T_0_A N D F R O M_C 0 NTIH_UU M U N I T S ARE" "DEGREES" >~ NAN S E C AND C U B I C M I C R O N S D I M E N S I O N X L G ( 6 ) , WLG ( 6 ) » E X ( Z O ) . COMMON_/TP/ SH(20_) , _C( 2 0 ,20 )_• CC ( 2 0 ) __ C I C ( 2 0 )_, AC___2_0 ) , _ A C E _2 0J X L G ( l ) = 0 . 2 2 2 8 4 6 6 0 4 2 ~ " X L G ( 2 ) = 1 . 1 8 8 9 3 2 1 0 1 7 X L G ( 3 ) _ = 2 . 9 9 2 7 3 6 3 2 6 " X L G ( 4 f "="5 . 7 7 5 143^5 6 9 X L G ( 5 ) = 9 . 8 3 7 4 6 7 4 1 8 X L G ( 6 ) _ = 1 5 . 9 8 2 8 7 3 9 8 W L G d T =' . 4 5 8 9 6 4 6 7 3 9 5 W L G ( 2 ) = . 4 1 7 0 0 0 8 3 0 7 7 W L G ( 3 ) = . 1 1 3 3 7 3 3 8 2 0 7 4 W L G ( 4 ) = " . 0 1 0 3 9 9 1 9 7 4 5 3 W L G ( 5 ) = . 2 6 1 0 1 7 2 0 2 8 E - 3 WLG ( 6 )___=_ . 8 9 8 5 4 7 9 0 6 4 3 E - 6 TR'"= 1 5 7 8 9 0 . / T E S Q T E = S Q R T ( T E ) S Q T R = _ S Q R T ( . T R ) DO 3 0 I = 1,N P P => 1*1 S H ! I ) = 0 . 0 E X ( I ) = E X P ( - T R / P P ) I F ( T R / P P . L E . 8 8 . ) S H ( I ) = P P * 4 . 1 3 6 E - 4 / ( T E * S Q T E * E X ( I ) DO 3 1 I = 1,N P = I P P = P*P TF = T R / P P SUM = 0.0 SUM1 = 0,0 S U M 2 = 0.0 DO 3 2 K = 1 , 6 U = X L G ( K ) / TF X = U + 1.0 , _ H = ( P * X ) * * ( - 2 . / 3 . 1 G = ( 1 . 0 + 0 . 1 7 2 8 * ( U - 1 . ) * H - 0 . 0 4 9 6 * ( U * U + 4 . * U / 3 . + l . > * H * H ) / X S U M 1 = S U M 1 + W L G ( K ) * G . : S U M 2 . = ' S U M 2 " " + ' V L G T K T * G ~ * X L G ( K ) """ I F ( X . L T . 2 . ) F = 2 . * ( U * 2 . 0 / ( X + l . ) 1 * * 1 . 5 I F ( X . G E . 2 . ) F = ( 5 . * X - 6 . ) * S O R T ( X / J X+_._)_#* 3 1 3 2 S U M = S U M + W L G ( K ) * F A C ( I ) = 5 . 1 9 7 E - 1 1 * S Q T R * S U M 1 / P A C E ( I__ _= 5 . 1 9 7 E - l l * S Q T R * S U M 2 * T E / P H =Vo 1 1 5 1 * S U M '* pp / S Q T E C I C ( I ) = H * E X ( I ) C C ( I ) _ = H * P P * 4 . 1 3 6 E - 4 / ( T E * S Q T E ) C ( T Y l ) = 0 . I F ( I . E Q . N ) G O T O 3 1 J A = I + l _ _ _ D O 3 3 J = J A , N Q = J Q Q = Q*Q _ A = Q Q / ( G Q - P P ) T F = T R / ( A * P P ) SUM„,= 0 . 0 D O 3 4 K = 1 , 6 U = XI_G( K> / T F Y_ = U + 1 . 0 I F ( U . L T . A ) E = ( 2 . - A + Y * Q . + 4 . * A ) ) * S Q R T ( U * ( 1 . + A ) 1 / ( ' ( Y + A ) * * 3 * A ) ) I F ( U . G E . _A )__ E__ = ( - 3 . * A + Y * ( 3 . + 4 ._* A )_)_* S QR T_(_Y / (_Y+AJ * * 3 3 4 S U M = S U M +" WLG ('< ) * E H = . 0 2 3 0 2 * S U M * P P * P P # A * A / ( Q Q * Q * S Q T E ) C (J » J ) = f i _*_ E X ( J J _ / _ E _ X U _ ) _ _____ 3 3 C ( J » I ) = H * P'P - / QQ 3 1 C O N T I N U E .. . .. R E T U R N E N D S E N T R Y 4 . 6 7 E - 1 5 . 5 ^ E - 7 1 . 7 7 E - 2 4 . 1 0 E - 3 1 . 6 4 E - 3 7 . 5 3 E - 4 3.8 5 E - 4 2 . 1 3 E - 4 1 . 2 6 F - 4 4 . 3 9 E - 2 8 . 3 7 E - 3 2 . 5 2 E - 3 9 . 6 8 E - 4 4 . 3 7 E - 4 2..20E-4 1 . 2 1 E - 4 7 . 0 8 F - 5 8. 9 4 E _ 3 _ 2 . 19E-3__7_._74E__4_3_.J4E_-4_1_. 64E_-4__8_. 8 5 E - 5 _ 5 _ n F -5 " *" " " 2 V 6 8 E - 3 7 . 6 7 E - 4 3 . 0 3 E - 4 1 . 4 2 E - 4 7 . 4 3 E - 5 4 . 1 2 E - 5 1 . 0 2 E - 3 3 . 2 4 E - 4 1 . 3 8 E - 4 6 . 8 7 E - 5 3 . 7 8 F - 5 4 . 5 0 E - 4 _ _ 1 . 5 5 E - 4 J 7 . 0 3 E - 5 _ 3 . 6 7 E - 5 - 2 . 2 6 E - 4 8 . 1 5 E - 5 3 . 8 8 E - 5 1 . 2 3 E - 4 A . _ 5 E - 5 7 . 1 2 E - 5 . 4 9 2 0 4 . 1 6 7 5 4 _ . 0 7 9 4 3 3 0 0 0 0 4 2 9 4 0 ^ 1 . 0 E 0 8 * 0 . 5 E 0 8 . 2 5 E 0 8 _ . 4 8 7 2 5 14JL00_ . 0 7 6 5 5 1 . E 1 2 4 7 . 7 . 4 8 0 4 7 1 2 5_14_ . 0 7 3 9 2 3 . 5 5 . 4 7 1 6 5 J _ 1 J J _ L _ . 0 7 1 3 0 . 7 5 8 . 4 6 1 1 7 1 0 6 0 9 _ . 0 6 9 2 ! . 3 0 6 . 4 4 8 5 9 . 0 9 9 6 5 . 4 7 8 5 5 ) 0 9 4 2 4 . 3 8 6 0 6 . 0 8 9 6 4 . 3 0 8 4 6 . 0 8 6 0 0 .7 7 1 4 1 . 0 8 7 5 7 . 1 8 8 . 1 5 8 . 1 6 1 . 1 8 2 . 7 1 3 1 •t B I B L I O G R A H Y B a t e s D . R . , K i n g s t o n A . E . a n d M c W h i r t e r R . W . P . ( 1 9 6 2 a ) P r o c . . R o y . S o c . A 2 6 7 , 297} ( 1 9 6 2 b ) P r o c . R o y . S o c . A 2 7 0 . 1 5 5 . B a t e s D . R . a n d K i n g s t o n A . E . ( 1 9 6 3 ) P l a n e t . S p a c e S c i . 1 1 , 1 ; ( 1 9 6 4 a ) P r o c . R o y . S o c . A 2 7 9 . 1 0 ; ( 1 9 6 4 b ) P r o c . R o y . S o c . A 2 7 9 . 3 2 . B y r o n S . , S a b l e r R . C . a n d B o r t z P . I . ( 1 9 6 2 ) P h y s . R e v . L e t t . _ , 3 7 6 . D ' A n g e l o N . ( 1961 ) P h y s . R e v 5 0 5 . D a v y P . ( 1 9 6 7 a ) R e v . P h y s . A p p l . _ , 6 5 ; ( 1 9 6 7 b ) E i g h t I n t e r n a t i o n a l C o n f e r e n c e o n Phenomena i n I o n i z e d G a s e s , V i e n n a . F a u l k n e r E . A . a n d H a r d i n g D . W . (1966 ) J . S c i . I n s t r . 4 3 , 9 7 . G r y z i n s k i M . ( 1 9 5 9 ) P h y s . R e v . 1 1 5 , 3 7 4 . H a m b e r g e r S . M . (1963 ) P l a s m a P h y s . 5., 7 3 . H a r r i e s W . L . a n d v o n E n g e l A . ( 1 9 5 4 ) P r o c . R o y . S o c . A 2 2 2 . 4 9 0 . H i n n o v E . a n d H i r s c h b e r g J . G . ( 1 9 6 2 ) P h y s . R e v . 1 2 5 , 7 9 5 . H o l s t e i n T . ( 1 9 4 7 ) P h y s . R e v . 7 2 , 1 2 1 2 . S c a n t l e b u r y G . S . P . ( 1 9 6 1 ) E l e c t r o n i c E n g i n e e r i n g , D e c . 6 1 , 8 0 3 . v o n E n g e l A . ( 1 9 6 5 ) I o n i z e d G a s e s . O x f o r d . W i l c o x R . H . ( 1 9 5 9 ) R e v . S c i . I n s t r . 3 0 , 1 0 0 9 . - 4 2 -

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