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Method to measure line shapes and relative transition probabilities Camm, David Malcolm 1971

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A METHOD TO MEASURE L I N E SHAPES AND R E L A T I V E TRANSITION P R O B A B I L I T I E S by DAVID MALCOLM CAMM A T H E S I S SUBMITTED IN P A R T I A L F U L F I L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY In the Department of PHYSICS 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 U N I V E R S I T Y OF B R I T I S H COLUMBIA A u g u s t , 1971 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e 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 r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f /7v> - < / / - c T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a D a t e J/j^j6 ABSTRACT An a c c u r a t e method t o m e a s u r e l i n e s h a p e s a n d r e l 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 b e t w e e n t r a n s i t i o n s w i t h a common l o w e r s t a t e has been d e v e l o p e d . A r g o n was u s e d as a t e s t g a s t o show t h a t t h i s m e t h o d i s more a c c u r a t e t h a n o t h e r c u r r e n t p r o c e d u r e s . The e x p e r i m e n t a l o b s e r v a t i o n s a g r e e w e l l w i t h t h e t h e o r y d e v e l o p e d i n t h i s t h e s i s . I t i s shown t h a t t h e r a t i o o f t h e L o r e n t z i a n t o D o p p l e r h a l f - w i d t h s o f t h e l i n e c a n be m e a s u r e d t o a p r e c i s i o n o f 10% when t h e D o p p l e r w i d t h i s as much as one h u n d r e d t i m e s l a r g e r t h a n t h e L o r e n t z i a n h a l f - w i d t h . The r e l 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 c a n be m e a s u r e d t o w i t h i n 10% f o r weak l i n e s a nd t o 1 % f o r s t r o n g l y a b s o r b i n g l i n e s . The l i n e s h a p e s m e a s u r e d a r e c o n s i s t e n t w i t h t h o s e p r e d i c t e d by G r i e m (2) and t h e r e l 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 a g r e e w e l l w i t h t h e v a l u e s g i v e n by W i e s e (5) . T h i s a b s o r p t i o n e x p e r i m e n t u s e s a g l o w d i s c h a r g e f o r t h e s o u r c e and a b s o r b e r . The a b s o r p t i o n t u b e i s p l a c e d b e t w e e n N i c o l p r i s m s and a l o n g i t u d i n a l m a g n e t i c f i e l d i s a p p l i e d t o Zeeman s p l i t t h e a b s o r p t i o n l i n e s . The t r a n s -m i s s i o n as a f u n c t i o n o f f i e l d d e p e n d s on b o t h t h e t r a n s i t i o n i i p r o b a b i l i t i e s and t h e l i n e s h a p e o f t h e a b s o r p t i o n l i n e s . T he h i g h r e s o l v i n g power r e q u i r e d t o m e a s u r e t h e l i n e s h a p e s i s o b t a i n e d f r o m t h e Zeeman s p l i t t i n g o f t h e a b s o r p t i o n l i n e s r a t h e r t h a n a s p e c t r o m e t e r . The m e t h o d i s an i n e x p e n s i v e , a c c u r a t e way t o m e a s u r e l i n e s h a p e s and r e l 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 e s p e c i a l l y s u i t e d f o r s t r o n g l y a b s o r b i n g l i n e s . TABLE OF CONTENTS Page ABSTRACT. i i TABLE OF CONTENTS . . i v INDEX OF TABLES v i i INDEX OF FIGURES v i i i ACKNOWLEDGEMENTS X C h a p t e r 1 INTRODUCTION 1 2 THEORY 5 2.1 I n t r o d u c t i o n 5 2.2 Waves i n a P l a s m a 7 2.3 P o l a r i z a t i o n Due t o A t o m i c P a r a m e t e r s . . 14 2.4 E f f e c t o f T h e r m a l M o t i o n on P o l a r i z a b i 1 i t y 17 2.5 A n o m a l o u s Zeeman E f f e c t 22 2.6 F r e e E l e c t r o n s 24 2.7 The E i n s t e i n T h e o r y o f R a d i a t i o n . . . . 25 2.8 E x p e r i m e n t a l M e t h o d s 30 i v C h a p t e r Page 2.8-1 S o u r c e P r o f i l e . 36 2.8- 2 N o n - c o n s t a n t M a g n e t i c F i e l d . . . . 38 2.9 L i n e Shape C o n s i d e r a t i o n s 39 2.9- 1 D o p p l e r L i n e S h a p e 42 2.9-2 L o r e n t z L i n e S h a p e - N a t u r a l . . . . 43 2.9-3 C o l l i s i o n a l B r o a d e n i n g - R e s o n a n c e . 44 2.9-4 C o l l i s i o n a l B r o a d e n i n g - Van d e r W a a l s . 45 2.10 Comments 49 3 EXPERIMENT AND PROCEDURE 51 3.0 I n t r o d u c t i o n . 51 3.1 E x p e r i m e n t a l A p p a r a t u s 51 3.1-1 T e s t G a s . . 52 3.1-2 S o u r c e and A b s o r b e r . 52 3.1-3 P o l a r i z e r s 53 3.1-4 M o n o c h r o m a t o r 54 3.1-5 D i v e r g e n c e and A p e r t u r e s . . . . . . 54 3.1-6 M a g n e t 55 3.1-7 D e t e c t i o n S y s t e m 56 3.2 M e a s u r e m e n t P r o c e d u r e 64 3.3 O p e r a t i n g C o n d i t i o n s 66 3.4 R e p r o d u c i b i l i t y 67 3.5 ' B e s t F i t ' 68 v C h a p t e r Page 3.6 U n i q u e n e s s o f F i t 70 3.7 E r r o r L i m i t s 71 3.8 S t i m u l a t e d E m i s s i o n 74 3.9 M a g n e t i c F i e l d E f f e c t s on the D i s c h a r g e 75 3.10 A d d i t i o n a l P a r a m e t e r s 75 4 RESULTS 78 4.1 R e s u l t s - R e l 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 78 4.2 R e s u l t s - L o r e n t z i a n L i n e Shape. . . . . 80 5 COMPARISON WITH OTHER METHODS 84 5.1 Wall S t a b i l i z e d A r c s and Shock Tubes . . 84 5.2 Hook Method 85 5.3 Zeeman S c a n n i n g . 86 6 CONCLUDING DISCUSSION. . 90 FIGURES . . . . . . . . . . . . . . 94 APPENDIX I Theorems 118 APPENDIX II E v a l u a t i o n o f the Complex E r r o r F u n c t i o n . . 124 APPENDIX I I I Computer Program. . . . . . . . 138 v i LIST OF TABLES Tab le Page 1 R e l 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 „ . 79 2 L o r e n t z i a n Ha l f -W id th s . . . . 81 3 R e l a t i v e L o r e n t z i a n Ha l f-Widths 82 v i i TABLE OF FIGURES Figure Page 1-1 Anomalous Dispersion . 94 1- 2 Par t ia l Term Diagram for Neutral Argon . . . . . 95 2- 1 Experimental Geometry. . 96 2-2 Po la r i za t ion E l l i p s e . 97 2-3 Experiment and Theory (8115) 98 2-4a Experiment and Theory (7067) 99 2-4b Experiment and Theory (8014, 7147) 100 2-4c Experiment and Theory (7635, 6965) 101 2-5 Zeeman S p l i t t i n g Term Diagram 102 2- 6 Line P ro f i l e s 103 3- 1 Apparatus 104 3-2 Discharge Tubes 105 3-3 Beam Width 105 3-4 Solenoid 106 3-5 Solenoid Power Supply. . 107 3-6 Waveforms of the D ig i ta l Phase Sens i t ive Detector 108 3-7 Block Diagram for the DPSD 109 3-8 Chopping Wheel . . . . . . . . . 110 3-9 Single Photon Pulse af ter the D isc r iminator . . .111 v i i i Figure Page 3-10 Photo Tube and Discriminator I l l 3-11 Changes in k 0 l 112 3-12 Changes in ' a ' 113 3-13 Uniformity of Magnetic Field 114 A-1 Rotation of Co-ordinate System 115 A-2 Integration Path. . . . . . . . . . 115 A-3 Integration Path. 116 A-4 Integration Path. 116 ix ACKNOWLEDGEMENTS I w i s h t o t h a n k D r . F . L . C u r z o n f o r s u g g e s t i n g a n d s u p e r v i s i n g t h e work d e s c r i b e d i n t h i s t h e s i s . The a s s i s t a n c e o f D r . J . H . W i l l i a m s o n i n t h e d e v e l o p m e n t o f t h e d i g i t a l p h a s e s e n s i t i v e d e t e c t o r a n d o f Mr. J . L e e s f o r g l a s s b l o w i n g s e r v i c e s i s a l s o a c k n o w l e d g e d . F i n a l l y I w o u l d l i k e t o t h a n k D r . F . L . C u r z o n , D r . R. N o d w e l l and E d i t h Camm f o r t h e i r w o r k i n h e l p i n g t o p r e p a r e t h i s t h e s i s . Ot-X CHAPTER 1 INTRODUCTION A c c u r a t e v a l u e s o f t r a n s i t i o n p r o b a b i l i t i e s a r e i m p o r t a n t i n o r d e r t o o b t a i n a c c u r a t e i n f o r m a t i o n f r o m q u a n t i t a t i v e s p e c t r o s c o p y . In a s t r o p h y s i c s 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 a r e n e e d e d i n o r d e r t o d e d u c e t h e c o n d i t i o n s r e s p o n s i b l e f o r t h e e m i s s i o n a n d / o r a b s o r p t i o n s p e c t r a o b s e r v e d . Q u a n t i t i e s s u c h as e l e m e n t a l a b u n d a n c e , d e g r e e o f i o n i z a t i o n , t e m p e r a t u r e s and e l e c t r o n d e n s i t i e s c a n be d e t e r m i n e d f r o m d e t a i l e d e x a m i n a t i o n o f t h e s p e c t r a l l i n e s h a p e s and i n t e n s i t i e s . In t h e l a b o r a t o r y , p l a s m a s p e c t r o -s c o p y u s e s t h e s e same p r o b a b i l i t i e s i n a s i m i l a r m anner t o d e t e r m i n e t h e p r o p e r t i e s o f t h e p l a s m a s b e i n g o b s e r v e d t h e r e . E v e r s i n c e t h e o r i g i n a l work o f E i n s t e i n (6) e x p e r i m e n t a l and t h e o r e t i c a l work has be e n c a r r i e d o u t t o d e t e r m i n e a c c u r a t e v a l u e s f o r t h e s e t r a n s i t i o n p r o b a b i 1 i t e s . U n f o r t u n a t e l y , i n g e n e r a l , t h e r e has n o t b e e n good a g r e e m e n t b e t w e e n v a r i o u s w o r k e r s who h a v e m e a s u r e d t h e same q u a n t i t i e s . 1 2 Due t o t h i s l a c k o f a g r e e m e n t i t was d e c i d e d t o d e v e l o p y e t a n o t h e r method t o m e a s u r e r e l 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 . T h i s new method i n v o l v e s t h e p r o p a g a t i o n o f l i n e a r l y p o l a r i z e d l i n e r a d i a t i o n a l o n g a m a g n e t i c f i e l d w h i c h i s embedded i n a p l a s m a . The d i s p e r s i o n i n t h e v i c i n i t y o f t h e o p t i c a l r e s o n a n c e a l o n g w i t h t h e e f f e c t o f t h e m a g n e t i c f i e l d i s shown i n F i g . 1-1. ( A l l f i g u r e s a r e a t t h e end o f t h e s i s . ) The f r e q u e n c y o f t h e u n s h i f t e d s o u r c e l i n e i s c e n t e r e d on f r e q u e n c y V0 and t h e r i g h t - and l e f t - h a n d c i r c u l a r l y p o l a r i z e d c o m p o n e n t s o f t h e a b s o r p t i o n l i n e ( s u b s c r i p t s + and - r e s p e c t i v e l y ) a r e s e p a r a t e d due t o t h e Zeeman e f f e c t . R e f e r r i n g t o F i g . 1-1 l e t us c o n s i d e r l i n e a r l y p o l a r i z e d l i g h t o f f r e q u e n c y V p r o p a g a t i n g t h r o u g h t h e p l a s m a w i t h a m a g n e t i c f i e l d s t r e n g h o f H a . S i n c e t h e l e f t - and r i g h t - h a n d c o m p o n e n t s o f t h e l i n e a r l y p o l a r i z e d s o u r c e h a v e d i f f e r e n t i n d i c e s o f r e f r a c t i o n t h e r e w i l l be a p h a s e s h i f t b e t w e e n t h e two p o l a r i z a t i o n s r e s u l t i n g i n a r o t a t i o n o f t h e p l a n e o f p o l a r i z a t i o n . The d i f f e r e n c e i n t h e a b s o r p t i o n c o e f f i c i e n t s ( k ) a t f r e q u e n c y V ' c h a n g e s t h e p o l a r i z a t i o n f r o m l i n e a r t o e l l i p t i c a l . As t h e m a g n e t i c f i e l d i n c r e a s e s t h e s p l i t t i n g o f t h e a b s o r p t i o n l i n e w i l l i n c r e a s e , r e s u l t i n g i n a c h a n g e i n t h e a b s o r p t i o n and r o t a t i o n a t f r e q u e n c y 2s'. 3 In t h e p r e s e n t work t h e t r a n s m i s s i o n o f t h e s o u r c e r a d i a t i o n i s m e a s u r e d as a f u n c t i o n o f m a g n e t i c f i e l d t o d e t e r m i n e t h e a b s o r p t i o n c o e f f i c i e n t k (cm~^) an d t h e i n d e x o f r e f r a c t i o n n as a f u n c t i o n o f f r e q u e n c y . I t i s shown t h a t ( n - 1 ) and k a r e p r o p o r t i o n a l t o t h e t r a n s i t i o n p r o b -a b i l i t y and t h a t t h e f r e q u e n c y d e p e n d e n c e g i v e s i n f o r m a t i o n a b o u t t h e s t a t e l i f e t i m e s and t h e r e f o r e a b s o l u t e t r a n s i t i o n p r o b a b i 1 i t i e s . The L o r e n t z i a n c o m p o n e n t o f t h e l i n e p r o f i l e i s p r o d u c e d by i n t e r r u p t i o n s o f t h e o s c i l l a t o r . Thus an e x t r a p o l a t i o n o f t h e L o r e n t z i a n c o m p o n e n t t o z e r o p r e s s u r e could g i v e an e s t i m a t e o f t h e l i f e t i m e s o f t h e s t a t e s i n -v o l v e d . T h e s e l e v e l s must h a v e a common l o w e r s t a t e t o g i v e t h e r e l 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 b u t t h e L o r e n t z i a n c o m p o n e n t o f t h e l i n e s h a p e c a n be d e t e r m i n e d f o r any l i n e . The p r e s e n t work m e a s u r e s t h e r e l 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 f o r a number o f a r g o n l i n e s and d i s c u s s e s t h e f e a s i b i l i t y o f m e a s u r i n g a b s o l u t e l i f e t i m e s by t h i s m e t h o d . The f i r s t a p p l i c a t i o n o f t h e m e t h o d was t o n e on by S e k a and C u r z o n ( 8 ) . T h e i r work i n d i c a t e d t h a t t h e a g r e e m e n t b e t w e e n e x p e r i m e n t and t h e o r y was s u f f i c i e n t t o g i v e r e l 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 t o 1 0 % - 2 0 % an d t h a t t h e r e was i n s u f f i c i e n t L o r e n t z i a n c o m p o n e n t i n t h e l i n e s h a p e t o be m e a s u r e d . The c r i t e r i o n t h e y u s e d t o i g n o r e t h e 4 L o r e n t z i a n c o m p o n e n t was t o i n t r o d u c e t h e L o r e n t z i a n i n t o t h e c a l c u l a t i o n o f t h e a b s o r p t i o n c o e f f i c i e n t and n o t e t h a t t h e r e was l i t t l e e f f e c t on t h e c a l c u l a t e d c u r v e s . The p r e s e n t work i n t r o d u c e s t h e L o r e n t z i a n l i n e s h a p e i n t o t h e c a l c u l a t i o n o f b o t h t h e a b s o r p t i o n c o e f f i c i e n t and t h e i n d e x o f r e f r a c t i o n a l o n g w i t h v a r i a t i o n s i n t h e p o p u l a t i o n o f t h e e n e r g y l e v e l s w i t h m a g n e t i c f i e l d . The e f f e c t o f t h e L o r e n t z i a n c o m p o n e n t i s f o u n d t o be g r e a t e s t i n t h e i n d e x o f r e f r a c t i o n w h i c h d e t e r m i n e s t h e r o t a t i o n o f t h e p l a n e o f p o l a r i z a t i o n . T h e s e a d d i t i o n s i m p r o v e t h e ' f i t ' b e t w e e n e x p e r i m e n t and t h e o r y t o t h e p o i n t w h e r e t h e y a g r e e w i t h i n e x p e r i m e n t a l e r r o r a n d as a r e s u l t t h e r e l 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 a r e d e t e r m i n e d t o an a c c u r a c y o f 1% f o r s t r o n g l y a b s o r b i n g l i n e s . In a d d i t i o n t h e new t r e a t m e n t d e t e r m i n e s t h e r a t i o o f t h e L o r e n t z i a n t o D o p p l e r c o m p o n e n t s w i t h i n 10%. T hus t h e m ethod as a p p l i e d i n t h i s t h e s i s g i v e s t h e r e l 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 t o an a c c u r a c y much b e t t e r t h a n t h e ' b e s t ' p r e v i o u s e s t i m a t e s by W i e s e ( 5 ) and i t a l s o g i v e s t h e s m a l l L o r e n t z i a n c o m p o n e n t o f a b a s i c a l l y D o p p l e r l i n e s h a p e ( L o r e n t z i a n = 7% D o p p l e r ) t o a p r e c i s i o n o f a b o u t 1 0 % . F u r t h e r i t must be s t r e s s e d t h a t t h e d e t a i l e d l i n e s h a p e ( L o r e n t z i a n h a l f - w i d t h ~ 1 0 " 4 c m - 1 ) i s m e a s u r e d by u s i n g a m o n o c h r o m a t o r w h i c h need o n l y i s o l a t e t h e l i n e w h i c h i s b e i n g m e a s u r e d (10 cm"^ i n s t r u m e n t w i d t h ) . CHAPTER 2 THEORY 2.1 INTRODUCTION v T h i s c h a p t e r d e a l s w i t h t h e p r o p a g a t i o n o f a m o n o c h r o m a t i c c i r c u l a r l y p o l a r i z e d wave a l o n g a c o n s t a n t m a g n e t i c f i e l d B embedded i n a p l a s m a ( F i g . 2 - 1 ) . The v e c t o r n o t a t i o n u s e d t h r o u g h o u t t h i s work w i l l be o f t h e f o r m Ej w h i c h i s d e f i n e d as t h e c o m p o n e n t o f t h e t h r e e d i m e n s i o n a l v e c t o r E i n t h e i - t h d i r e c t i o n . The s u b s c r i p t s t a k e on t h e v a l u e s 1, 2, and 3 w h i c h r e f e r t o t h e t h r e e o r t h o g o n a l a x e s o f a r i g h t h a n d e d c a r t e s i a n c o - o r d i n a t e s y s t e m . D e r i v a t i v e s w i t h r e s p e c t t o p o s i t i o n a r e w r i t t e n i n t h e f o r m Eli j - i - \7 - and w i t h t h e a l t e r n a t i n g u n i t t e n s o r s o C u r l c a n be w r i t t e n as e l e c t r o m a g n e t i c r a d i a t i o n i s t o d e t e r m i n e what t h e c h a r a c t e r i s t i c modes o f p r o p a g a t i o n a r e f o r o u r p a r t i c u l a r s i t u a t i o n . S e c t i o n 2 o f t h i s c h a p t e r d o e s t h i s a nd t h e n 1 ""Ft The f i r s t s t e p i n t r e a t i n g t h e p r o p a g a t i o n o f 5 6 p r o c e e d s t o e x p r e s s t h e p r o p a g a t i o n p a r a m e t e r s f o r t h e s e modes i n t e r m s o f t h e p o l a r i z a b i 1 i t y o f t h e p l a s m a . Once t h e p r o p a g a t i o n p a r a m e t e r s a r e known t h e n t h e amount o f l i g h t w h i c h w i l l be t r a n s m i t t e d t h r o u g h t h e e x p e r i m e n t a l p l a s m a c a n be c a l c u l a t e d . S i n c e we h a v e c a l c u l a t e d t h e s e p a r a m e t e r s i n t e r m s o f t h e p o l a r i z a b i 1 i t y we t h e n p r o c e e d t o c a l c u l a t e t h e p o l a r i z a b i 1 i t y i n t e r m s o f a t o m i c p a r a m e t e r s . In s e c t i o n 2-3 we c a l c u l a t e t h e c o n t r i b u t i o n made t o t h e p o l a r i z a b i 1 i t y by a c l a s s i c a l h a r m o n i c o s c i l l a t o r , and t h e n assume t h a t i n t h e quantum m e c h a n i c a l c a s e t h e p o l a r i z -a b i l i t y i s m u l t i p l i e d by t h e c o r r e s p o n d i n g o s c i l l a t o r s t r e n g t h f . The p o l a r i z a b i 1 i t y o f t h e c l a s s i c a l o s c i l l a t o r was c a l c u l a t e d f o r an i s o l a t e d o s c i l l a t o r a t r e s t ; t h e n e x t two s e c t i o n s g e n e r a l i s e t o t h e e x p e r i m e n t a l s i t u a t i o n . S e c t i o n 2.4 i s d e v o t e d t o t h e D o p p l e r s h i f t due t o t h e r m a l m o t i o n o f t h e a t o m s ; s e c t i o n 2.5 t a k e s i n t o a c c o u n t t h e Zeeman s p l i t t i n g o f t h e a b s o r p t i o n l i n e s p r o d u c e d by t h e a p p l i e d m a g n e t i c f i e l d . The p o l a r i z a b i 1 i t y due t o t h e f r e e c h a r g e s i n t h e p l a s m a i s c a l c u l a t e d i n s e c t i o n 2.6 and c o m p a r e d t o t h e e f f e c t s o f t h e bound c h a r g e s i n o r d e r t o show t h a t t h e f r e e c h a r g e s c a n be i g n o r e d . A t t h i s p o i n t we h a v e e x p r e s s e d t h e p r o p a g a t i o n p a r a m e t e r s f o r t h e c h a r a c t e r i s t i c modes o f p r o p a g a t i o n i n t e r m s o f t h e a t o m i c p a r a m e t e r s by means o f t h e p o l a r i z a b i 1 i t y . S e c t i o n 2.7 7 u s e s E i n s t e i n ' s t h e o r y o f r a d i a t i o n t o d e r i v e an e x p r e s s i o n f o r t h e r e l 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 i n t e r m s o f a t o m i c p a r a m e t e r s w h i c h were m e a s u r e d e x p e r i m e n t a l l y . The e x p r e s s i o n s f o r t h e p r o p a g a t i o n p a r a m e t e r s a r e t h e n u s e d i n s e c t i o n 2.8 t o c a l c u l a t e how t h e l i g h t t r a n s m i t t e d by t h e p l a s m a d e p e n d s on t h e a p p l i e d m a g n e t i c f i e l d a n d t h e a t o m i c p a r a m e t e r s . D u r i n g t h e d e r i v a t i o n o f t h e s e r e s u l t s i t i s f o r m a l l y a ssumed t h a t t h e f r e q u e n c y d e p e n d e n c e o f t h e a b s o r p t i o n c o e f f i c i e n t c a n be a p p r o x i m a t e d by a c o n v o l u t i o n o f a L o r e n t z i a n and D o p p l e r l i n e s h a p e , i . e . , a V o i g t p r o f i l e . S e c t i o n 2.9 d i s c u s s e s t h e f o r m o f t h e s e two l i n e s h a p e s and t h e n c o n t i n u e s t o g i v e some o f t h e phenomena w h i c h w o u l d p r o d u c e t h e s e s h a p e s i n t h e e x p e r i -m e n t a l s i t u a t i o n , t o g e t h e r w i t h f o r m u l a e t o c a l c u l a t e t h e amount o f e a c h t y p e o f b r o a d e n i n g . The c h a p t e r i s c o n c l u d e d w i t h a few comments i n s e c t i o n 2.10. 2.2 WAVES IN A PLASMA - Why u s e c i r c u l a r l y p o l a r i z e d l i g h t ? C o n s i d e r a m o n o c h r o m a t i c e l e c t r o m a g n e t i c wave o f f r e q u e n c y V p r o p a g a t i n g i n t h e d i r e c t i o n a l o n g a m a g n e t i c f i e l d embedded i n a p l a s m a ( F i g . 2 - 1 ) . The r a d i a t i o n w i l l i n t e r a c t w i t h t h e atoms t o a s i g n i f i c a n t e x t e n t o n l y when t h e f r e q u e n c y o f t h e r a d i a t i o n 8 i s c l o s e t o t h e f r e q u e n c y o f an a l l o w e d a t o m i c t r a n s i t i o n . U s i n g s t a n d a r d n o t a t i o n we w i l l d e f i n e t h e c o m p o n e n t o f t h e a n g u l a r momentum i n t h e d i r e c t i o n o f t h e f i e l d B t o be e q u a l t o m t l , where f l i s P l a n c k ' s c o n s t a n t d i v i d e d by 2TT . P h o t o n s h a v e two p o s s i b l e d i r e c t i o n s o f s p i n c o r r e s p o n d i n g t o m=+l a n d - 1 . The p h o t o n s w i t h m=+l a r e r i g h t h and c i r c u l a r l y p o l a r i z e d a nd t h e o t h e r s w i t h m=-l a r e l e f t - h a n d e d . The p o l a r i z a t i o n i s s a i d t o be r i g h t - o r l e f t -h a n d e d a c c o r d i n g t o w h e t h e r t h e e l e c t r i c v e c t o r a t a f i x e d p o i n t r o t a t e s i n a c l o c k w i s e o r c o u n t e r c l o c k w i s e d i r e c t i o n f o r an o b s e r v e r l o o k i n g t o w a r d t h e s o u r c e . The o n l y a t o m i c t r a n s i t i o n s w h i c h c a n a b s o r b t h e p h o t o n s m o v i n g a l o n g t h e f i e l d l i n e s a r e t h o s e w i t h Am(atom) e q u a l t o +1 o r - 1 . S i n c e a n g u l a r momentum must be c o n s e r v e d , Am(atom) + Am(photon)=0 and t h e r e f o r e t h e r i g h t - h a n d e d wave c a n o n l y i n t e r a c t w i t h t r a n s i t i o n s w i t h A m=+l and t h e l e f t -h a n d e d wave c a n o n l y i n t e r a c t w i t h t r a n s i t i o n w i t h Am=-1. T h u s t h e r i g h t - and l e f t - h a n d waves i n t e r a c t w i t h d i f f e r e n t t r a n s i t i o n s w i t h t h e r e s u l t t h a t t h e y a r e n o t c o u p l e d by t h e a t o m s . The m a g n e t i c f i e l d a f f e c t s t h e atoms i n s u c h a way t h a t t h e f r e q u e n c i e s o f t h e t r a n s i t i o n s w i t h Am=+1 a r e c h a n g e d i n an o p p o s i t e s e n s e t o t h o s e w i t h Am=-1. T h e r e f o r e t h e r i g h t - a nd l e f t - h a n d waves o f t h e same 9 f r e q u e n c y w i l l " s e e " atoms w i t h d i f f e r e n t p o l a r i z a b i l i t i e s b u t t h e two waves r e m a i n u n c o u p l e d . I t w i l l be shown ( E q . 2-19) t h a t t h e m o t i o n o f c h a r g e d p a r t i c l e s c a n n o t c o u p l e t h e w a v e s . The f a c t t h a t t h e s e two p o l a r i z a t i o n s a r e u n c o u p l e d w i l l be u s e d t o a l l o w us t o t r e a t i n d e p e n d e n t l y t h e two p o l a r i z a t i o n c o m p o n e n t s o f t h e i n c i d e n t wave. 2.2-1 PROPAGATION OF CIRCULARLY POLARIZED LIGHT The e l e c t r i c v e c t o r s a s s o c i a t e d w i t h t h e c i r c u l a r p o l a r i z a t i o n s a r e w r i t t e n i n c o m p l e x n o t a t i o n as h a n d e d w a v e s . T h a t i s , E + r o t a t e s a t a f i x e d p o i n t i n a r i g h t - h a n d s e n s e f o r an o b s e r v e r l o o k i n g a l o n g t h e p r o p a g a -t i o n v e c t o r t o w a r d t h e s o u r c e and E_ r o t a t e s i n a l e f t -h a n d s e n s e . and t h a t t h e v e l o c i t i e s a r e n o t r e l a t i v i s t i c t h e n M a x w e l l ' s e q u a t i o n s g i v e ( 2 - 1 ) w h e r e E+ and E a r e r e s p e c t i v e l y r i g h t - and l e f t -A s s u m i n g t h a t t h e s p a c e c h a r g e d e n s i t y i s z e r o (2-2) 10 w h e r e i s t h e e l e c t r i c f i e l d i n t e n s i t y , i s t h e c u r r e n t d e n s i t y , a n d Dj^  i s t h e e l e c t r i c d i s p l a c e m e n t . A s s u m i n g t h a t a l l n o n - l i n e a r e f f e c t s c a n be i g n o r e d , t h e p r o p a g a t i o n o f t h e wave a t f r e q u e n c y V w i l l i n d u c e r e s p o n s e s o f t h e medium a t t h e same f r e q u e n c y , t h a t i s , t h e s c a t t e r i n g i s c o h e r e n t . The r e s p o n s e s o f t h e medium a t f r e q u e n c y V c a n be f o u n d by t a k i n g t h e f o u r i e r c o m p o n e n t a t V . Thus t h e t i m e d e p e n d e n c e f o r E, J , D, X, ... w i l l be o f t h e f o r m e x p ( i ZTCV t ) . S i n c e t h e i o n s o r atoms h a v e a l a r g e mass c o m p a r e d t o t h e e l e c t r o n s ( 1 : 1 0 " ) i t i s as s u m e d t h a t t h e i r v e l o c i t i e s a r e u n a f f e c t e d by t h e i m p o s e d f i e l d and t h a t as a r e s u l t a l l t h e i n d u c e d c u r r e n t s a r e due t o t h e m o t i o n o f t h e e l e c t r o n s . T h e p l a s m a w i l l be t r e a t e d as a d i e l e c t r i c medium s o t h a t D k = f * E k + p k ( 2 - 3 ) J k = 0 w h e r e € c i s t h e p e r m i t t i v i t y o f f r e e s p a c e a nd i s t h e p o l a r i z a t i o n o r d i p o l e moment p e r u n i t v o l u m e . S u b s t i t u t i n g t h i s e x p r e s s i o n f o r D. i n t o ( 2 - 2 ) , we o b t a i n E K , j j ( 2 - 4 ) 11 w h i c h , i n t e r m s o f t h e c h a r a c t e r i s t i c modes d e f i n e d i n 2-1 c a n be w r i t t e n as E ± j j j =r 0E ± + P ± ( 2 - 5 ) where P± = Re( Rj + i R> ) ( 2 - 6 ) I t s h o u l d be n o t e d t h a t t h e t e r m P K i n 2-4 i s n o t p r o p o r t i o n a l t o E K s i n c e t h e p o l a r i z a t i o n i n t h e d i r e c t i o n d e p e n d s on t h e f i e l d and E^. However t h e t e r m P* i n 2-5 i s p r o p o r t i o n a l t o E+ s i n c e E + and E_ a r e u n c o u p l e d . T h i s w i l l be shown i n more d e t a i l when t h e m i c r o s c o p i c m o t i o n s a r e c o n s i d e r e d l a t e r . The r e s u l t o f t h i s u n c o u p l i n g i s t h a t when ( 2 - 5 ) i s r e w r i t t e n a s t h e n t h e p e r m i t t i v i t y i s a v e c t o r r a t h e r t h a n a t e n s o r and i s g i v e n by £±=e,+ -p- =e0 ( e'± • i € ^ > ( 2 - 8 ) C- ± w h i c h d e f i n e t^. and c + E q u a t i n g t h e r e a l and i m a g i n a r y p a r t s o f 2-8 g i v e s t h e r e l a t i o n b e t w e e n p e r m i t t i v i t y and t h e m i c r o s c o p i c R. v a r i a b l e ( c r- ) n a m e l y 12 e l = 1 + Re(-^-) (2-9) €± = I m C ^ r ) (2-10) 2.2-2 PERMITTIVITY AND PROPAGATION PARAMETERS OF  CHARACTERISTIC MODES S i n c e E+ a r e the c h a r a c t e r i s t i c modes t h e y can be e x p r e s s e d i n the form o f a s t a n d a r d e q u a t i o n f o r a mono-c h r o m a t i c a t t e n u a t e d wave, i . e . E.H=E„„ex p|-j27r^(n t-iK 4)-5L-t* 0\ (2-11) n* i s t h e i n d e x o f r e f r a c t i o n , c the speed o f l i g h t , K + i s . r e l a t e d to the a b s o r b t i o n c o e f f i c i e n t and AX- i s the phase f a c t o r ( r v and a r e r e a l ) . S u b s t i t u t i n g t h e s e e x p r e s -s i o n s f o r E ± i n t o the wave e q u a t i o n s (2-7) e n a b l e s us to e x p r e s s the p r o p a g a t i o n p a r a m e t e r s n and K i n terms o f the p e r m i t t i v i t y £ (rv K J 2 =MX(e; + ie: ) c 2 (2-12) N o t i n g that/C^^-4; and e q u a t i n g the r e a l and i m a g i n a r y p a r t s o f 2-12 g i v e s 13 n? - K ! = e; -2n + K ± = £ 'I The c o n n e c t i o n b e t w e e n t h e c o n s t a n t s i n t h e wave e q u a t i o n and t h e p e r m i t t i v i t y i s now com-p l e t e d by s o l v i n g t h i s p a i r o f e q u a t i o n s f o r n + and K + , l . e 2 2 ~ ± ( 2 - 1 3 ) 2 2 ( 2 - 1 4 ) K ± and t\± c a n be r e l a t e d t o £ + by t h e d i r e c t a p p l i c a t i o n o f 2-13, 14 b u t n o t i n g t h a t f o r t h e e x p e r i m e n t a l p l a s m a | ^  10~ 5 a n d | £ * | ^ 10~ 5 so t h a t £' = 1 and ( 2 - 1 3 and 2-14) c a n be s i m p l i f i e d t o r\ + The r e l a t i o n b e t w e e n £ ± and t h r o u g h 2-9 and 2-10 i . e . i s t h e n o b t a i n e d r u = 1 +1 Re 2 = Ilmr R 2 P± o— I _ ( 2 - 1 5 ) ( 2 - 1 6 ) 14 Thus we h a v e an e x p r e s s i o n f o r t h e p r o p a g a t i o n p a r a m e t e r s n and K i n t e r m s o f t h e p o l a r i z a b i 1 i t y " . The p r o b l e m now i s t o f i n d t h e f u n c t i o n a l d e p e n d e n c e o f t h e p o l a r i z a b i 1 i t y on w a v e l e n g t h . T h i s w i l l be a c c o m p l i s h e d by d e r i v i n g an e x p r e s s i o n f o r t h e p o l a r i z a b i 1 i t y i n t e r m s o f p r o p e r t i e s o f t h e atoms and t h e c o n d i t i o n s i n t h e d i s -c h a r g e . F i r s t we w i l l d e a l w i t h an i s o l a t e d h a r m o n i c o s c i l l a t o r . 2.3 POLARIZATION DUE TO ATOMIC PARAMETERS The p o l a r i z a b i 1 i t y c o n t r i b u t e d by e a c h s p e c t r a l l i n e i s p r o p o r t i o n a l t o t h a t w h i c h a c l a s s i c a l o s c i l l a t o r w o u l d c o n t r i b u t e and t h e p r o p o r t i o n a l i t y c o n s t a n t ' f i s c a l l e d t h e ' o s c i l l a t o r s t r e n g t h ' ( R e f . 3., Chp. 4 - 9 ) . F i r s t t h e p r o b l e m o f t h e c l a s s i c a l o s c i l l a t o r w i l l be t r e a t e d . C o n s i d e r an e l a s t i c a l l y b o u n d e l e c t r o n o f mass m, c h a r g e q , n a t u r a l f r e q u e n c y U , and h a l f - l i f e T i n a m a g n e t i c f i e l d B. The e q u a t i o n o f m o t i o n f o r s u c h a p a r t i c l e when p e r t u r b e d by an e l e c t r o m a g n e t i c f i e l d i s w h e r e £ i s t h e a l t e r n a t i n g u n i t t e n s o r . 15 The d i p o l e moment o f t h i s s i n g l e o s c i l l a t o r i s qX. so the p o l a r i z a t i o n from i t w i l l be p^. g i v e n by P t = q (Xj, *ixa) =qxt (2-18) Note t h a t p^. r e f e r s to one o s c i l l a t o r w h i l e P ± r e f e r s t o t o t a l medium. W r i t i n g 2-17, the e q u a t i o n o f m o t i o n , i n terms o f X + g i v e s the f o l l o w i n g e q u a t i o n x\ +lx + + (jfx,= £L( E + ?i B 3 x+) (2-19) T " m " J We have made the a s s u m p t i o n t h a t the r e s p o n s e o f the medium t o the e l e c t r i c f i e l d i s l i n e a r i n o r d e r t o d e r i v e (2-9,10) i . e . , the r e l a t i o n between p e r m i t t i v i t y and the m a c r o s c o p i c v a r i a b l e " . T h i s same a s s u m p t i o n i s to E+ imposed on the c l a s s i c a l o s c i l l a t o r by t a k i n g the f o u r i e r component o f e q u a t i o n (2-19) a t f r e q u e n c y /. J X . = X * (2-20) 27TV Thus we have assumed t h a t a l l n o n - l i n e a r o r se c o n d o r d e r e f f e c t s can be n e g l e c t e d compared t o the c o h e r e n t e f f e c t s . U s i n g (2-20) the e q u a t i o n o f moti o n a t f r e q u e n c y V can be w r i t t e n as 16 w h e r e LJa i s t h e e l e c t r o n c y c l o t r o n f r e q u e n c y q E>/fn , The s o l u t i o n o f t h i s p a i r o f d i f f e r e n t i a l e q u a t i o n s f o r X+ i s X + ( 2 - 2 2 ) " " m u \ - U { U ±uB) * i u/r E l i m i n a t i n g X ± f r o m ( 2 - 1 7 ) by u s e o f ( 2 - 2 2 ) e n a b l e s us t o e x p r e s s p ^ y / f , E * i n t e r m s o f t h e a t o m i c p a r a m e t e r s , n a m e l y P. q 2 1  ( 2 - 2 3 ) T h i s t h e n i s t h e p o l a r i z a t i o n due t o a c l a s s i c a l o s c i l l a t o r ; t o o b t a i n t h e c o r r e c t q u a n t um m e c h a n i c a l e x p r e s s i o n t h i s m ust be m u l t i p l i e d by t h e o s c i l l a t o r s t r e n g t h f t o g i v e — E a — - _ _ J ( 2 - 2 4 ) € C E ± e o m ut - U{ u ± U9 )* i u/r The t o t a l p o l a r i z a t i o n due t o t h e bound e l e c t r o n i s f o u n d by summing o v e r a l l p o s s i b l e s p e c t r u m l i n e s , b e i n g c a r e f u l t o a l l o w f o r t h e m o t i o n o f t h e atoms r e l a t i v e t o t h e s o u r c e . F i n a l l y i t s h o u l d be n o t e d t h a t e q u a t i o n ( 2 - 2 4 ) i s v a l i d o n l y f o r t r a n s i t i o n s w i t h a n o r m a l Zeeman e f f e c t . The a n o m a l o u s Zeeman e f f e c t i s t r e a t e d i n s e c t i o n 2.5. 17 2.4 E F F E C T OF THERMAL MOTION ON P O L A R I Z A B I L I T Y I t must be s t r e s s e d t h a t t h e f r e q u e n c y U u s e d i n ( 2 - 2 4 ) i s t h e f r e q u e n c y o f t h e r a d i a t i o n i n t h e r e s t f r a m e o f t h e atom. A l t h o u g h t h e m o t i o n o f t h e i o n s i n r e s p o n s e t o t h e f i e l d c a n be i g n o r e d ( d u e t o t h e a t o m i c m a s s ) t h e t h e r m a l m o t i o n o f t h e atoms must be c o n s i d e r e d . T h e o s c i l l a t o r s o f t h e p r e v i o u s s e c t i o n a r e as s u m e d t o be m o v i n g a t t h e t h e r m a l s p e e d s o f t h e atoms i n m o t i o n . We assume t h a t t h e atoms a r e i n t h e r m a l e q u i l i b r i u m s o t h e v e l o c i t y d i s t r i b u t i o n i s M a x w e l l i a n . H e n c e , t h e number o f atoms w i t h X - b e t w e e n V and V + d V i s d N w h e r e w h e r e N i s t h e number d e n s i t y o f atoms i n t h e g r o u n d s t a t e o f t h e t r a n s i t i o n b e i n g c o n s i d e r e d , M i s t h e mass p e r a t o m , 3 d N = J ± e x p ( - y 2 ) d y ( 2 - 2 5 ) ( 2 - 2 6 ) T i s t h e t e m p e r a t u r e i n *K, and k e i s B o l t z m a n n ' s c o n s t a n t . R e w r i t i n g e q u a t i o n ( 2 - 2 4 ) i n a s l i g h t l y d i f f e r e n t f o r m P. = f q2 1 ( 2 - 2 7 ) 18 b u t u n d e r e x p e r i m e n t a l c o n d i t i o n s we have t h u s £10 -5 f q 4 1 €0£+ €omU02(U0-U ? ) + J _ 2 r w h i c h c a n be w r i t t e n i n a d i m e n s i o n l e s s f o r m P , _ f ( f f n2 1 ~ €mDU ia+W+ ( 2 - 2 8 ) ( 2 - 2 9 ) w h e r e D T W+=-2VJn2 ('(»/-te± Ul/fe ) and D i s t h e f u l l w i d t h a t h a l f maxium o f t h e D o p p l e r b r o a d e n e d s p e c t r a l l i n e and i s g i v e n i n u n i t s o f a n g u l a r f r e q u e n c y by D = 2W>j]n2 ( 2 - 3 0 ) ( 2 - 3 1 ) ( 2 - 3 2 ) w h e r e U0 i s t h e f r e q u e n c y o f t h e c e n t e r o f t h e s p e c t r a l l i n e ; c i s t h e s p e e d o f l i g h t ; LJ i s t h e f r e q u e n c y a t w h i c h t h e p o l a r i z a b i 1 i t y i s e v a l u a t e d . T y p i c a l v a l u e s f o r t h e D o p p l e r h a l f - w i d t h f o r a r g o n l i n e s a t room t e m p e r a t u r e a r e ~ . 0 1 A = .02cm"^ = 20mk. 19 N o t e t h a t p o l a r i z a b i 1 i t y i s d e f i n e d i n t e r m s o f t h e a n g u l a r f r e q u e n c y OO w h i c h i s s e e n i n t h e f r a m e a t r e s t w i t h r e s p e c t t o t h e atom. The f r e q u e n c y s e e n i n t h e f r a m e o f r e f e r e n c e o f t h e s o u r c e i s g i v e n by iaJ . I f t h e atom i s m o v i n g w i t h X 3 - V t h e n t h e D o p p l e r s h i f t e d f r e q u e n c y i s U = u / | l - ^-j ( 2 - 3 3 ) s u b s t i t u t i n g 2-33 i n t o 2-31 we f i n d w; =-2(^'[l-^]-<4±l«y8 ) VJn2* w ± = w ; f y w h e r e W*' =-2Vjn2 (UJ'-Um ± ) D 2 y = Thus t h e p o l a r i z a b i 1 i t y p e r atom w i t h X g = V i s £ 0 E t ~ 6 o m D ( e / y + W ± ' + i a p The t o t a l p o l a r i z a b i 1 i t y '- i s f o u n d by i n t e g r a t i n g o v e r a l l atoms ( 2 - 3 4 ) 20 o r s u b s t i t u t i n g ( 2 - 2 5 ) and ( 2 - 2 9 ) i n t o t h i s i n t e g r a l we o b t a i n p ± OO k«> A J _ /exp(-y*) dy 7 7 , 2 7 r ; y + w ; + i a -00 ( 2 - 3 5 ) w h e r e k. _ Nf q 2 V T n 2 £ f lmc D o z E q u a t i o n ( 2 - 3 5 ) c a n be w r i t t e n i n t h e f o r m (j)(Z 2) PH. i ^  k„ € "E+ = 2 7T o -where (J)(Z ) i s t h e c o m p l e x e r r o r f u n c t i o n 0>(Z) =exp(-Z 2) 1+ 2L /exp( + t 2 ) d t by u s i n g A p p e n d i x I ( T h e o r e m 1.3) w h i c h shows _L / exp(-y zr J y * w ; + -y 2) dy = 1 (J)(zt) w h e r e - 0 0 Z<. = W * + i a ( 2 - 3 6 ) ( 2 - 3 7 ) ( 2 - 3 8 ) 21 S u b s t i t u t i n g the e x p r e s s i o n f o r p o l a r i z a b i 1 i t y from (2-37) i n t o (2-15, 16) g i v e s n + = 1 - Ak,lm(<l)(ZJ ) (2 ATT K+ = Ak,Re( <J)(Z.) ) (2 47T Thus we have an e x p r e s s i o n f o r the p r o p a g a t i o n p a r a m e t e r s i n terms o f the complex e r r o r f u n c t i o n (t) and k0 which i s g i v e n i n e q u a t i o n (2-36) as k = Nf qVln2 V i f I f k„ and a a r e s e t then the p r o p a g a t i o n p a r a m e t e r s a r e c o m p l e t e l y d e t e r m i n e d which g i v e us the e x p e c t e d t r a n s m i s s i o n f o r our e x p e r i m e n t a l c o n d i t i o n s . A l t e r n a t e l y we can t r e a t k0 and 3 as pa r a m e t e r s which a r e v a r i e d i n o r d e r t o g i v e the b e s t p o s s i b l e agreement between e x p e r i m e n t and t h e o r y . B e f o r e t h i s i s done t h e r e r e m a ins one f u r t h e r c o r r e c t i o n . When the e f f e c t o f the m a g n e t i c f i e l d was c a l c u l a t e d the f a c t t h a t a bound e l e c t r o n w i l l have i t s a n g u l a r momentum q u a n t i z e d was not t a k e n i n t o a c c o u n t c o r r e c t l y ( c l a s s i c a l Zeeman e f f e c t ) . T h i s q u a n t i z a t i o n w i l l be t r e a t e d i n the next s e c t i o n . The s e c t i o n f o l l o w i n g t h a t w i l l show t h a t the p o l a r i z a b i 1 i t y due to f r e e c h a r g e s can be i g n o r e d n e a r o p t i c a l r e s o n a n c e s because o f the much l a r g e r p o l a r i z a b i 1 i t y 22 f r o m t h e bound e l e c t r o n s . Hence e q u a t i o n s ( 2 - 3 9 ) , ( 2 - 4 0 ) 2.5 ANOMALOUS ZEEMAN E F F E C T In t h e t r e a t m e n t o f t h e s e m i - c l a s s i c a l o s c i l l a t o r t h e f a c t t h a t t h e a n g u l a r momentum o f an e l e c t r o n i n a bound s t a t e w i l l be q u a n t i z e d was i g n o r e d . I f we i m p o s e t h e c o n d i t i o n t h a t t h e a n g u l a r momentum i n t h e d i r e c t i o n o f t h e m a g n e t i c f i e l d i s q u a n t i z e d t h e n t h e s h i f t due t o t h e m a g n e t i c f i e l d i s n o t as i n d i c a t e d i n e q u a t i o n ( 2 - 3 1 ) b u t r a t h e r , as i n r e f e r e n c e ( 3 ) , g i v e t h e f i n a l f o r m f o r n ± and K + . 2 = 2TT/X9(M£g2 -M, g, ) B w h e r e M^ i s t h e m a g n e t i c q u a ntum number i n s t a t e i g. i s t h e L a n d e g - f a c t o r f o r t h e s t a t e i Jj, i s t h e B o h r M a g n e t o n = 1 . 3 9 9 6 x l 0 6 s e c " 4TT m q i s t h e c h a r g e o f t h e e l e c t r o n w i t h mass m and h i s P l a n c k ' s c o n s t a n t . ( s e e F i g . 2 - 5 ) . Thus t h e e x p r e s s i o n s d e r i v e d f o r t h e p o l a r i z a b i 1 i t y a r e v a l i d i f e q u a t i o n ( 2 - 3 1 ) i s r e p l a c e d by w; = 2V ln2 [W - U ±^27TB(M2g2-M, g,)] 23 r e w r i t i n g e q u a t i o n s ( 2 - 3 9 ) , ( 2 - 4 0 ) we h a v e n ± = 1 - Ak„ lmfd)(w; +ia)l A i r r 1 ' -i J K± = JLk ,Re[C^ (w :+ia)J Thus we hav e t h e v a l u e s f o r n ± and K+ i f t h e Zeeman s p l i t t i n g c o n s i s t s o f one c o m p o n e n t w i t h e a c h p o l a r i -z a t i o n . M o s t Zeeman l i n e s c o n s i s t o f many c o m p o n e n t s and e a c h one m u s t be t a k e n i n t o a c c o u n t . T h i s c a n be done by summing o v e r a l l t h e Zeeman c o m p o n e n t s . U n+-1 = I (ru. -1) )9i U H 1 ( 2 - 4 1 ) w h e r e U i s t h e t o t a l number o f c o m p o n e n t s w i t h e a c h p o l a r i -z a t i o n , 9^ j i s t h e r e l a t i v e i n t e n s i t y o f t h e i - t h c o m p o n e n t , K± , ru a r e t h e p r o p a g a t i o n p a r a m e t e r s f o r t h e i - t h com-p o n e n t as d e f i n e d i n e q u a t i o n ( 2 - 1 1 ) The r e l a t i v e i n t e n s i t y , / ^ i s d e r i v e d i n r e f e r e n c e ( 3 ) as /3j =C(J*M)(J±M+1) i f J - J ; M - M ± 1 fi. =C(J ± M)(J t M + 2)if M—M±1 fi. =CU ? M)(J * M-1)if J*J-1 ; M-M±1 where J i s t h e t o t a l a n g u l a r momentum and M i s t h e c o m p o n e n t 24 i n t h e d i r e c t i o n o f t h e f i e l d . T he c o n s t a n t C i s u s e d t o n o r m a l i s e t h e i n t e n s i t i e s so t h a t U i=1 1 2.6 FREE ELECTRONS S e c o n d l y t h e f r e e e l e c t r o n s m u s t be c o n s i d e r e d S i n c e t h e y a r e f r e e t h e r e s o n a n t f r e q u e n c y w i l l be z e r o . W r i t i n g ( 2 - 2 3 ) f o r t h i s c a s e we g e t P+ q* 1 u\ 1 w h e r e r^ N q y f n €0 i s t h e p l a s m a f r e q u e n c y , N_ i s t h e e l e c t r o n d e n s i t y a n d f = l ( 2 - 2 4 ) w i t h t h e a s s u m p t i o n t h a t t h e r a d i a t i o n i s n e a r r e s o n a n c e a n d t h a t iaJ » (J we h a v e 1 + i 1 7(j F o r t h e e x p e r i m e n t a l p l a s m a w i t h * ~ 6 0 0 0 A a n d a s s u m i n g t h a t *\/(jT& 1 ( c o l l i s i o n f r e q u e n c y s m a l l c o m p a r e d t o Ct/ ) t h e n ( 2 - 4 2 ) 25 10" ' * (2 T h i s i s c o m p a r e d 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 f o r _ 5 t h e b o u n d e l e c t r o n s o f 10 . Thus t h e e f f e c t o f t h e f r e e e l e c t r o n s on t h e p o l a r i z a b i 1 i t y i n t h e v i c i n i t y o f a s p e c t r a l a b s o r p t i o n l i n e c a n be i g n o r e d . T h e r e f o r e by i n v e s t i g a t i n g t h e v a l u e s o f t h e i n d e x o f r e f r a c t i o n and a b s o r p t i o n c o e f f i c i e n t c l o s e t o s p e c t r a l l i n e s we w i l l be a b l e t o d e t e r m i n e t h e a t o m i c p a r a m e t e r s ( f and a) s i n c e t h e y d o m i n a t e t h e p o l a r i z a b i 1 i t y . One o f t h e a t o m i c p a r a m e t e r s w h i c h c a n be d e t e r m i n e d i s t h e o s c i l l a t o r s t r e n g t h ( f ) o f s e c t i o n 2.3 b u t t h e p a r a m e t e r w h i c h i s o f i n t e r e s t i s t h e t r a n s i t i o n p r o b a b i l i t y . The r e l a t i o n b e t w e e n t h e r e l a t i v e o s c i l l a t o r s t r e n g t h s and r e l 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 f o r two t r a n s i t i o n s w i t h a common g r o u n d s t a t e i s d e v e l o p e d n e x t . 2.7 THE E I N S T E I N THEORY OF RADIATION C o n s i d e r t h e a t o m i c s y s t e m shown i n t h e t e r m d i a g r a m b e l o w . 1 F o l l o w i n g t h e t r e a t m e n t o f M i t c h e l l and Zemansky ( 1 3 , p. 93) we d e f i n e t h e p r o b a b i l i t y c o e f f i c i e n t s : 26 B^2^iy = t n e P r o b a b i l i t y p e r s e c o n d t h a t t h e atom i n s t a t e 1 e x p o s e d t o i s o t r o p i c r a d i a t i o n o f f r e q u e n c e b e t w e e n V and V + d V and i n t e n s i t y I v , w i l l a b s o r b a q u a n t u m hV and p a s s t o t h e s t a t e 2. A-2-^ 1 = t' i e P r o b a b i l i t y p e r s e c o n d t h a t t h e atom i n s t a t e 2 w i l l s p o n t a n e o u s l y e m i t , i n a random d i r e c t i o n , a quantum hV and p a s s t o t h e s t a t e 1. ^2 1 ^"V = P r o b a b i l i t y p e r s e c o n d t h a t t h e atom w i l l u n d e r g o t h e t r a n s i t i o n f r o m 2 t o 1 when i t i s e x p o s e d t o i s o t r o p i c r a d i a t i o n o f f r e q u e n c y b e t w e e n V and V + d V i n t e n s i t y I v , e m i t t i n g t h e r e b y a quantum i n t h e same d i r e c t i o n as t h e s t i m u l a t i n g q u a n t u m . k v = t h e a b s o r p t i o n c o e f f i c i e n t and i s t h e p r o b -a b i l i t y p e r u n i t l e n g t h o f a b s o r b e r t h a t a p h o t o n o f e n e r g y hi/ t o h(V +dV ) w i l l be a b s o r b e d . T = t h e mean l i f e o f an i s o l a t e d atom i n s t a t e 2. 9 ( } 9 2 - t h e s t a t i s t i c a l w e i g h t s o f s t a t e 1 and 2 r e s p e c t i v e l y . From t h e d e f i n i t i o n o f IT andA2_^ i t i s c l e a r t h a t 27 By c o n s i d e r i n g t h e t h e r m o d y n a m i c e q u i l i b r i u m b e t w e e n t h e r a d i a t i o n and t h e a t o m s , E i n s t e i n showed t h a t 3 A2-1 _ V ( 2 - 4 5 ) B 1~2 c * 9 2  B 2-1 9<i B 1-2 g 2 ( 2 - 4 6 ) w h e r e c i s t h e v e l o c i t y o f l i g h t . C o n s i d e r a p a r a l l e l beam o f l i g h t o f f r e q u e n c y b e t w e e n V and U+d V and i n t e n s i t y I v t r a v e l l i n g a l o n g t h r o u g h a l a y e r o f atoms b o u n d e d by X^ a n d X ( + d X j . I f i s t h e number d e n s i t y o f atoms i n s t a t e 1 c a p a b l e o f a b s o r b i n g I and d N 2 v i s t h e number d e n s i t y o f atoms i n s t a t e 2 c a p a b l e o f s t i m u l a t e d e m i s s i o n i n t h e f r e q u e n c y r a n g e V t o V +d V and n e g l e c t i n g t h e e f f e c t o f s p o n t a n e o u s e m i s s i o n s i n c e i t o c c u r s i n a random d i r e c t i o n we c a n w r i t e t h e d e c r e a s e i n e n e r g y o f t h e beam as d( I „dV )=j l t hI/dx(B 2 . 1 c lN| , - B , ^ d N l w ) 4 TT w h e r e I v /ATT 1 s t n e i n t e n s i t y o f t h e e q u i v a l e n t i s o -t r o p i c r a d i a t i o n f o r w h i c h and a r e d e f i n e d . T h i s c a n be r e w r i t t e n as 28 1 d U d l / . -hJ/(BT dN. -B dNL ) 4 8 ) The l e f t - h a n d s i d e o f t h i s e q u a t i o n i s t h e p r o b a b i l i t y p e r u n i t l e n g t h o f a b s o r p t i o n w h i c h i s j u s t - k - ^ d ^ I n t e g r a t i o n o v e r t h e e n t i r e l i n e , n e g l e c t i n g t h e s m a l l v a r i a t i o n i n V g i v e s k„ 6 2 / =_hi^ ( B^ 2 N1 - B 2 ^ N 2 ) ( 2 . 4 9 ) w h i c h by u s e o f ' e q u a t i o n ( 2 - 4 4 ) , (-45) and (-46) Qvrgj I g 2 N, ( 2 - 5 0 ) L e t us now p r o c e e d t o d e r i v e an e x p r e s s i o n f o r t h e i n t e g r a l i n ( 2 - 5 0 ) i n t e r m s o f t h e m e a s u r e d p a r a m e t e r k The i n t e n s i t y o f r a d i a t i o n due t o E+ i s p r o -p o r t i o n a l t o (ReE*)2 , w h i c h f r o m t h m I - 2 , A p p e n d i x I , i s e q u a l t o l R e ( E * E i ) S u b s t i t u t i n g t h e e x p r e s s i o n f o r E ± f r o m ( 2 - 1 1 ) g i v e s Ioc E^ ±exp(-47TK i x 3 / \ ) t h u s t h e i n t e n s i t y as a f u n c t i o n o f X ^ i s g i v e n by I = r oexp(-4irK yx 3 ) ( 2 _ 5 1 ) 29 w h e r e B i s s e t e q u a l t o z e r o so K+ =K_ = t h e r e f o r e t h e a b s o r p t i o n c o e f f i c i e n t i s g i v e n by k ^ = 4T7KV X ( 2 - 5 2 ) o r u s i n g ( 2 - 4 0 ) g i v e s k1/ = k.Re(|)(Zv) ( 2 - 5 3 ) s i n c e Z+ = Z_ = Z V for B = 0 i f B 4= 0 t h e n k - ^ k n . 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 s , t h e r e f o r e OO CO / k ± dv = k 0 jRe(j)(Z±) dV ( 2 - 5 4 ) o r u s i n g T h e o r e m 11-4, A p p e n d i x I I CO f K dV = k . D - x H T J0 " 2 v l n 2 c o m p a r i n g (2-50) and (2-55) shows t h a t k 0 . 2 A l g a N , a D 8Jg77 i i— 9eN ( J A/Io2. V f f = 2 A ^ N . - . r i n ^ A' Dg8TTg V TT d 1 ( 2 - 5 5 ) ( 2 - 5 6 ) 30 w h e r e k 0 ^ r e f e r s t o k 0 f o r t r a n s i t i o n f r o m s t a t e 1 t o s t a t e 2 1 -9, N, g N 2 i ( s e e s e c t i o n 2.8-5) I f t h e r e a r e two t r a n s i t i o n s w i t h a common l o w e r s t a t e t h e n i f we c a l l t h e s e c o n d u p p e r s t a t e 3 and t a k e t h e r a t i o o f t h e two a b s o r p t i o n c o e f f i c i e n t s ( E q . 2-56) 2 . j 3 ^ ^ . D a O a _ A a 9 2 ( 2 . 5 8 ) ( 2 - 5 7 ) °3 T h u s i f we a r e a b l e t o d e t e r m i n e t h e v a l u e s k „ and k,, f o r two a t o m i c t r a n s i t i o n s w i t h a common °Z 3 g r o u n d s t a t e t h e n i f g^  we know t h e r e l 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 f o r t h e s e l i n e s ( s e e s e c t i o n 3 . 2 - 7 ) . 2.8 EXPERIMENTAL METHODS The e x p e r i m e n t a l p l a s m a i s a g l o w d i s c h a r g e as shown i n F i g . 2-1 and 3-1. T h e l i g h t e n t e r s t h e p l a s m a a t Xg=0 l i n e a r l y p o l a r i z e d i n t h e X^ d i r e c t i o n and a f t e r t r a v e r s i n g a d i s t a n c e X i n t h e X^ d i r e c t i o n p a s s e s t h r o u g h an a n a l y s i n g n i c o l p r i s m w h i c h has b e e n r o t a t e d t h r o u g h an a n g l e .0. ( s e e F i g . 2-1) w i t h r e s p e c t t o t h e X^ d i r e c t i o n F o r t h i s c a s e ( 2 - 1 1 ) c a n be w r i t t e n as 31 = exp \2TTV(t -ru - i K * ( 2 - 5 9 ) \ " c / s i n c e f o r t h i s c a s e E<54.=EV From t h e o r e m 1-1 i n A p p e n d i x I t h e a m p l i t u d e o f t h e e l e c t r i c v e c t o r p a s s i n g t h r o u g h t h e a n a l y s i n g n i c o l a t t h e a n g l e Q, i s g i v e n i n t e r m s o f E + and E_ as EJI =4" (E + exp [ - i n ] + E_exp[if i ] ) ( 2 - 6 0 ) The o b s e r v e d i n t e n s i t y i s p r o p o r t i o n a l t o t h e t i m e a v e r a g e o f t h e a m p l i t u d e s q u a r e d o f t h e e l e c t i c v e c t o r , i . e . o r u s i n g t h e o r e m 1-2 i n A p p e n d i x I t h i s c a n be w r i t t e n as 1 _* I A =^Re(E_^Ej;) ( 2 - 6 1 ) w h e r e fc.^ i s t h e c o m p l e x c o n j u g a t e o f . E l i m i n a t i n g E ^ u s i n g e q u a t i o n ( 2 - 6 0 ) g i v e s as t h e o b s e r v e d i n t e n s i t y Ij^  I. - Re E E*+ E E* + E E*exp(-i2Q) + E_E*exp(i2J2)j ( 2 - 6 2 ) 32 w h i c h c a n be s i m p l i f i e d by u s i n g e q u a t i o n ( 2 - 5 9 ) f o r s i n c e cos( r ) = exp( i r ) +exp(-ir) 2 t o g i v e I„= E* |exp(-kj) • exp(-k J )+2cos(0 -2g)exp(-i[k.+ k_]) where k+ - 4JL K + A ~~2 ( 2 - 6 3 ) ( 2 - 6 4 ) ( 2 - 6 5 ) Now k+ and 0 c a n be e x p r e s s e d i n t e r m s o f k 0 and by u s i n g e q u a t i o n s ( 2 - 3 9 ) and ( 2 - 4 0 ) , i . e . , k £j?= k 0iRe[(|)(Z t)] e = k 0 ^ l l m <l><zj -Im -i \ ( 2 - 6 6 ) ( 2 - 6 7 ) The e x p e r i m e n t c o n s i s t s o f o b s e r v i n g t h e t r a n s -m i t t e d i n t e n s i t y ( E q . 2-63) as a f u n c t i o n o f m a g n e t i c f i e l d a nd t h e n c o m p a r i n g t h e o b s e r v e d v a r i a t i o n s w i t h t h o s e p r e -d i c t e d by t h e o r y . The v a l u e s o f k 0 ( E q . 2-36) and 'a' ( E q . 2-30) were u s e d as p a r a m e t e r s w h i c h were v a r i e d i n o r d e r t o o b t a i n a ' b e s t f i t . ' T h r o u g h e q u a t i o n ( 2 - 5 8 ) k 0 g i v e s us t h e r e l 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 a nd 'a' g i v e s 33 g i v e s us t h e l i n e s h a p e ( i n p a r t i c u l a r t h e L o r e n t z l i n e w i d t h p r o v i d e d t h e D o p p l e r w i d t h i s known). I t i s now e v i d e n t t h a t i f t h e p o l a r i z i n g n i c o l N 1 ( s e e F i g . 3-1) were o m i t t e d and t h e s o u r c e were n o t p o l a r i z e d f o r any o t h e r r e a s o n t h e n t h e t o t a l i n t e n s i t y o b s e r v e d I w o u l d be g i v e n by 2. & 2 exp(-k+|) + exp(-kj) ( 2 - 6 8 ) w h i c h i s i n d e p e n d e n t o f t h e p h a s e c h a n g e ( F i g . 2 - 2 ) . By u s i n g t h e p o l a r i z e r s we a r e a b l e t o o b t a i n i n f o r m a t i o n on b o t h t h e a b s o p t i o n c o e f f i c i e n t and t h e i n d e x o f r e f r a c t i o n s i n c e we now m e a s u r e t h e a m p l i t u d e s and t h e r e l a t i v e p h a s e o f t h e two t r a n s m i t t e d waves . L e t us now l o o k a t t h i s e q u a t i o n f o r t h e t r a n s -m i t t e d i n t e n s i t y more c l o s e l y . The maximum i n t e n s i t y i s o b t a i n e d when £2 = .©. n a m e l y 8 exp(-k+l) +exp(-k_l >+2exp( {k^ » _ k 4 I ) ( 2 - 6 9 ) and t h e minimum when .Q = Q±7f and i s g i v e n by t h e 2 e x p r e s s i o n fyriin) "-^  o exp(-kj)+exp(-kj)- 2 e x p ( - [ M k . ] j g ) | ( 2 - 7 0 ) 34 T h u s t h e p l a n e p o l a r i z e d l i g h t f r o m t h e s o u r c e i s t r a n s m i t t e d as e l l i p t i c a l l y p o l a r i z e d l i g h t w i t h t h e m a j o r a x i s r o t a t e d by 0 / 2 , w h i c h d e p e n d s o n l y on t h e d i f f e r e n c e i n t h e v a l u e s o f n + and n_. The r e l a t i v e a m p l i t u d e o f t h e m a j o r and m i n o r a x i s i s d e t e r m i n e d by t h e v a l u e s o f k, and k . I f k, = k t h e n I (max)= -5=^ , + + 2 I ( m i n ) = 0 and t h e t r a n s m i t t e d r a d i a t i o n i s p l a n e p o l a r i z e d . The r o t a t i o n o f t h e a x i s o f t h e p o l a r i z a t i o n o f t h e t r a n s -m i t t e d l i g h t i s r e f e r r e d t o as F a r a d a y r o t a t i o n ( F i g . 2 - 2 ) . I t must be s t r e s s e d t h a t t h e e x p r e s s i o n g i v e n i n ( 2 - 6 3 ) i s v a l i d o n l y i n t h e n e i g h b o u r h o o d o f a s p e c t r a l l i n e and i t assumes t h a t t h e s o u r c e i s m o n o c h r o m a t i c a t f r e q u e n c y V . In o u r e x p e r i m e n t a l s e t - u p two i d e n t i c a l p l a s m a s were u s e d f o r t h e s o u r c e and a b s o r b e r . The o n l y d i f f e r e n c e b e t w e e n t h e two i s t h a t t h e a b s o r b e r was i n a m a g n e t i c f i e l d w h i l e t h e s o u r c e was k e p t a t z e r o f i e l d . S i n c e t h e s o u r c e t h e n o n l y e m i t s l i g h t i n t h e n e i g h b o u r h o o d o f s p e c t r a l l i n e s ( 2 - 6 3 ) i s v a l i d a nd t h e o n l y r e q u i r e m e n t o f t h e m o n o c h r o m a t o r i s t h a t i t a l l o w t h e e n t i r e l i n e f r o m t h e s o u r c e t o p a s s b u t e x c l u d e t h e n e i g h b o u r i n g l i n e s . The t o t a l i n t e n s i t y o b s e r v e d i s t h e n g i v e n by i n t e g r a t i n g t h e i n t e n s i t y f r o m ( 2 - 6 3 ) o v e r t h e e n t i r e l i n e co T0' = J l^[exp(-kJO+exp(-kJ) ( 2 -+2cos(e-2&)exp(-l/k-^ k_l)j 61/ 35 The t e r m E 2 , i s t h e s o u r c e i n t e n s i t y a t f r e -q u e n c y V and must be d e t e r m i n e d e x p e r i m e n t a l l y . The f o r m o f t h i s t e r m w i l l be d e a l t w i t h i n s e c t i o n 2 .7-1. F i g . 2-3 i s a g r a p h o b t a i n e d f r o m t h e e x p e r i -m e n t a l a p p a r a t u s . T h i s shows t h e t r a n s m i s s i o n o b s e r v e d when t h e two n i c o l s a r e i n p o s i t i o n ( T ^ i n e q u a t i o n ( 2 - 7 1 ) ) and a l s o t h a t o b s e r v e d i f t h e p o l a r i z i n g n i c o l i s n o t i n p o s i t i o n , i . e 4" The r e l a t i v e amounts o f i n f o r m a t i o n i n t h e s e two c u r v e s w i l l be d i s c u s s e d i n g r e a t e r d e t a i l i n s e c t i o n 5.3. In t h e e x p e r i m e n t t h e p e r c e n t a g e t r a n s m i s s i o n i s a c t u a l l y m e a s u r e d so we must d i v i d e t h e t r a n s m i s s i o n w i t h t h e a b s o r b e r on by t h a t w i t h i t o f f . Thus we g e t w i t h t h e p o l a r i z i n g n i c o l N, i n T 0= jk[ekJ.e*-K2cos(Q-2n)6{K+w%v dV ( 2 - 7 2 ) - C O CO and w i t h t h e n i c o l o u t .00 4 JEI d v ( 2 - 7 3 ) TT = ^ E ' ( e M * e " k - i ) d v (2-74) 6V oo 36 2.8-1 SOURCE PROFIL E In o r d e r t o e v a l u a t e t h e r e l a t i v e t r a n s m i s s i o n , To and T T ( E q . (2-73) ( - 7 4 ) ) , t h e s o u r c e p r o f i l e Ev m u s t be known. S i n c e t h e s o u r c e and a b s o r b e r a r e i d e n t i c a l p l a s m a s t h e t r a n s m i s s i o n p a r a m e t e r s k ± w i l l be i d e n t i c a l f o r b o t h . The s o u r c e i s k e p t f r e e o f any m a g n e t i c f i l e d so we hav e k+ = k_ =ky . T h i s s e c t i o n d e r i v e s an e x p r e s s i o n f o r t h e s o u r c e i n t e n s i t y i n t e r m s o f k v . S i n c e t h e s o u r c e and a b s o r b e r h a v e t h e same p r o p a g a t i o n p a r a m e t e r s t h e r e i s o n l y one s e t o f c o n s t a n t s w h i c h c a n be v a r i e d i n o r d e r t o o b t a i n a t h e o r e t i c a l f i t t o t h e e x p e r i m e n t a l r e s u l t s . L e t us c o n s i d e r a c y l i n d e r o f p l a s m a as shown b e l o w . dx I ~ \ — E v The p r o b a b i l i t y t h a t a p h o t o n o f f r e q u e n c y V i s e m i t t e d f r o m a s l a b o f t h i c k n e s s dx i n a d i r e c t i o n p a r a l l e l t o t h e a x i s o f t h e s o u r c e ( P v ) i s e q u a l t o b N v A whe r e b i s a c o n s t a n t , N v i s t h e number d e n s i t y o f atoms c a p a b l e o f e m i t t i n g a p h o t o n o f f r e q u e n c y and A i s E i n s t e i n ' s A c o e f -f i c i e n t . The p r o b a b i l i t y t h a t a p h o t o n o f f r e q u e n c y V w i l l be a b s o r b e d p e r u n i t l e n g t h i s ky w h i c h i s e q u a l t o b Ny B where H'u i s t h e number d e n s i t y o f atoms c a p a b l e o f a b s o r b i n g t h e p h o t o n and B i s E i n s t e i n ' s B c o e f f i c i e n t . 37 The p r o b a b i l i t y t h a t a photon o f f r e q u e n c y V w i l l be e m i t t e d i n dx and w i l l l e a v e the plasma i n the Xg d i r e c t i o n i s _ k v P v e v (2-82) Thus the t o t a l i n t e n s i t y l e a v i n g the plasma i s L - k X P„ e " d X (2-83) which i s e q u a l t o where I i s a c o n s t a n t , c A & B a r e c o n s t a n t s t h a t a r e d e t e r m i n e d by the i n t e r a c t i o n o f the atom w i t h the r a d i a t i o n f i e l d and have the same v a l u e s whether t h e r e i s t h e r m a l e q u i l i b r i u m o r n o t . We c a n n o t c l a i m t h e r m a l e q u i l i b r i u m f o r the glow d i s c h a r g e but we can assume t h a t the number d e n s i t i e s o f the v a r i o u s s t a t e s have r e a c h e d a c o n s t a n t v a l u e so t h a t P v / k y i s a r a t i o which i s i n d e p e n d e n t o f f r e q u e n c y . S i n c e t h i s e x p e r i m e n t measures r e l a t i v e t r a n s -m i s s i o n o n l y , an a r b i t r a r y c o n s t a n t i n the s o u r c e s t r e n g t h w i l l c a n c e l so i t need not be c o n s i d e r e d . In the c a l c u l a -t i o n l c was chosen a r b i t r a r i l y so t h a t 38 ( 2 - 8 5 ) 2.8-2 NON-CONSTANT MAGNETIC F I E L D In t h e d e r i v a t i o n o f e q u a t i o n ( 2 - 3 1 ) i t was i m p l i c i t l y assumed t h a t n and K were c o n s t a n t o v e r t h e l e n g t h o f t h e a b s o r b e r . I f t h e y a r e n o t t h e n t h e e q u a t i o n f o r t h e t r a n s m i t t e d i n t e n s i t y g i v e n i n e q u a t i o n ( 2 - 6 3 ) c a n be c o r r e c t e d i f t h e f o l l o w i n g s u b s t i t u t i o n s a r e made f o r ( 2 - 6 8 ) and ( 2 - 6 7 ) . o ( 2 - 8 7 ) e ( 2 - 8 8 ) R e c a l l f r o m e q u a t i o n ( 2 - 3 8 ) t h a t •{Tr o where Z ± = W+ + i a . I f we a r e c o n s i d e r i n g t h e t r a n s m i s s i o n o f t h e r a d i a t i o n i n a m a g n e t i c f i e l d H ( X ) a t t h e norm-a l i z e d f r e q u e n c y U and t h e Zeeman s p l i t t i n g ( 0 = c o n -s t a n t ) t h e n we c a n w r i t e f o r t h e a b s o r b e r ( 2 - 8 9 ) 39 Thus t h e v a l u e o f <|>(Z±) as a f u n c t i o n o f f i e l d ( a n d h e n c e o f X ) i s known. In o r d e r t o c a r r y o u t t h e i n t e g r a t i o n s i n ( 2 - 8 7 ) and ( 2 - 8 8 ) t h e v a r i a t i o n o f t h e m a g n e t i c f i e l d a l o n g t h e o p t i c a l p a t h was d e t e r m i n e d e x p e r i m e n t a l l y and t h e c o r -r e c t e d v a l u e s g i v e n by ( 2 - 8 7 ) and ( 2 - 8 8 ) were u s e d t o c a l c u l a t e t h e t r a n s m i t t e d i n t e n s i t y . The maximum c h a n g e c a u s e d by t h i s p r o c e d u r e was a p p r o x i m a t e l y e q u a l t o t h e s t a n d a r d d e v i a t i o n o f t h e e x p e r i m e n t a l p o i n t s on t h e T v s . H c u r v e s . I t i s e s t i m a t e d t h a t t h e maximum e r r o r a f t e r t h e c o r r e c t i o n i s c a r r i e d o u t , i s o n e - t e n t h o f e x p e r i m e n t a l e r r o r f r o m o t h e r s o u r c e s and as s u c h t h i s p a r t i c u l a r e r r o r has bee n i g n o r e d i n t h e t r e a t m e n t o f e r r o r s . 2.9 L I N E SHAPE CONSIDERATIONS Now t h a t t h e f o r m a l d e r i v a t i o n s h a v e b e e n com-p l e t e d i t i s p r o f i t a b l e t o r e t u r n t o some key e q u a t i o n s and d i s c u s s t h e s i g n i f i c a n c e o f t h e d i s t r i b u t i o n o f s p e c t r a l i n t e n s i t y w h i c h was a s s u m e d . U s i n g e q u a t i o n s ( 2 - 1 6 ) and ( 2 - 6 4 ) we s e e t h a t T a k i n g t h e c a s e when t h e m a g n e t i c f i e l d i s e q u a l t o z e r o we kj oc Im _Jk ( 2 -c a n w r i t e k = k + v n+ - n_ E E n v 40 L e t us f i r s t c o n s i d e r t h e s p e c t r a l d i s t r i b u t i o n o f i n t e n s i t y o f t h e l i g h t e m i t t e d f r o m an o p t i c a l l y t h i n s o u r c e , i . e . t a k e t h e l i m i t o f e q u a t i o n ( 2 - 8 5 ) as t h e s o u r c e t h i c k n e s s g o e s t o z e r o lim j £ = l i rnd - e k v L ) = k^L L-o 2 L - 0 F o r t h e s p e c t r a l d i s t r i b u t i o n o f an a b s o r p t i o n l i n e we t a k e t h e l i m i t o f e q u a t i o n ( 2 - 6 3 ) t o g i v e I i m Iv oc 1 - kv L +. . . T h u s when we r e f e r t o l i n e s h a p e s i n t h i s work we a r e d e a l i n g w i t h t h e s p e c t r a l d i s t r i b u t i o n o f t h e a b s o r p t i o n c o e f f i c i e n t ky L e t us f i r s t assume t h a t t h e t h e r m a l m o t i o n o f t h e atoms c a n be i g n o r e d i n o r d e r t h a t we c a n u s e e q u a t i o n ( 2 - 2 4 ) f o r t h e p o l a r i z a b i 1 i t y t o g i v e us 1 ( 2 - 9 1 ) N o t i n g t h a t o v e r any g i v e n a b s o r p t i o n l i n e V~V9 we may w r i t e koc A U J } . , ( 2 - 9 2 ) w h e r e 2 \ ^ = l / 7 " i s r e f e r r e d t o as t h e L o r e n t z i a n h a l f -w i d t h ( s e e F i g . 2-6). 41 In t h e f i n a l e x p r e s s i o n f o r p o l a r i z a b i 1 i t y ( 2 - 3 7 ) t h e L o r e n t z i a n c o mponent i s u s e d t o d e f i n e t h e V o i g t ' a ' f a c t o r , i . e . , a - V l n 2 1 . V f n 2 A ^ D T 0 ( 2 - 9 3 ) Thus i f t h e e f f e c t o f t h e r m a l m o t i o n c a n be i g n o r e d t h e n D « (*JU and a — o o . T h i s t h e n l e a d s t o a L o r e n t z i a n l i n e p r o f i l e d e s c r i b e d by e q u a t i o n ( 2 - 9 2 ) . I f t h e o p p o s i t e c a s e i s t a k e n t h e n t h e t h e r m a l m o t i o n d o m i n a t e s t h e l i n e s h a p e and e q u a t i o n ( 2 - 6 6 ) c a n be s i m p l i f i e d f o r t h i s c a s e t o Z* = W*+ia W. =W = Wv = 2V ln2 ( U - (*>0) D so t h a t I i m a-*o Re (|)(ZV) = e ( 2 - 9 4 ) ( F r o m e q u a t i o n ( 2 - 3 8 ) ) T h u s we have f o r t h i s c a s e - w ( 2 - 9 5 ) 42 T h i s d e p e n d e n c e o f t h e s p e c t r a l d i s t r i b u t i o n i s c a l l e d D o p p l e r B r o a d e n i n g ( s e e F i g . 2 - 6 ) . N o t e t h a t t h e v a l u e o f k v w i l l d r o p t o h a l f i t s maximum v a l u e when Thus D i s c a l l e d t h e D o p p l e r h a l f - w i d t h o r more p r e c i s e l y t h e f u l l w i d t h a t h a l f maximum. Thus t h e d e r i v a t i o n o f t h e p o l a r i z a b i 1 i t y and t h e o b s e r v e d t r a n s m i s s i o n has a s s u m e d t h a t t h e s p e c t r a l d i s -t r i b u t i o n o f t h e a b s o r p t i o n c o e f f i c i e n t c a n be a p p r o x i m a t e d by a c o n v o l u t e d D o p p l e r and L o r e n t z i a n l i n e s h a p e . The t e m p e r a t u r e o f t h e gas t h a t w i l l d e t e r m i n e t h e v a l u e o f t h e D o p p l e r h a l f - w i d t h and t h e i r r a t i o ( V o i g t 'a') i s l e f t as a p a r a m e t e r t o be d e t e r m i n e d when t h e e x p e r i m e n t a l and t h e o r e t i c a l r e s u l t s a r e c o m p a r e d . The phenomena w h i c h g i v e r i s e t o t h e L o r e n t z i a n c o m p o n e n t w i l l be d i s c u s s e d i n s e c t i o n 2.9-2. 2.9-1 DOPPLER L I N E SHAPE The D o p p l e r l i n e s h a p e i s p r o d u c e d by an e n s e m b l e o f e m i t t e r s m o v i n g w i t h a M a x w e l l i a n v e l o c i t y d i s t r i b u t i o n ( s e c t i o n 2 . 4 ) . T h i s t r e a t m e n t s h o u l d be v a l i d s i n c e t h e r e i s n e g l i g i b l e h e a t i n g o f t h e atoms a b o v e room t e m p e r a t u r e and t h e t e m p e r a t u r e i s c o n s t a n t t h r o u g h o u t t h e e x p e r i m e n t . The mean f r e e p a t h o f a r g o n a t 3 2 0 ° K a t 2 mm Hg p r e s s u r e s _ 3 a p p r o x i m a t e l y 10 cm s o c o l l i s i o n a l n a r r o w i n g c a n be 43 i g n o r e d ( 1 ) . A t e m p e r a t u r e o f 3 0 0 ° K was u s e d ( s e c t i o n 3.2-7) t o c a l c u l a t e t h e D o p p l e r h a l f - w i d t h and t h e V o i g t 'a' was d e t e r m i n e d e x p e r i m e n t a l l y . From e q u a t i o n s ( 2 - 3 2 ) and ( 2 - 2 6 ) t h e D o p p l e r h a l f - w i d t h D i s g i v e n by 2.9-2 LORENTZIAN L I N E SHAPE - NATURAL A s t o c h a s t i c p r o c e s s w h i c h a f f e c t s t h e atom by s h o r t e n i n g t h e d u r a t i o n f o r w h i c h i t r a d i a t e s w i l l h a v e t h e e f f e c t o f g i v i n g t h e l i n e a L o r e n t z i a n c o m p o n e n t ( 1 ) . I f an atom e m i t t e d an i n f i n i t e l y l o n g s i n u s o i d a l wave t h e s p e c t r a l l i n e p r o d u c e d w o u l d be a D i r a c 6"-f u n c t i on . The m o s t o b v i o u s l i m i t t o t h e d u r a t i o n o f t h e wave t r a i n i s t h e f a c t t h a t an atom w i l l e m i t r a d i a t i o n i n a f i n i t e l e n g t h o f t i m e . I f t h e h a l f - l i f e ( T ) o f t h e e x c i t e d s t a t e i s t h e f i n i t e l e n g t h o f t h i s t r a i n t h e n a L o r e n t z i a n l i n e p r o f i l e o f h a l f - w i d t h A VN w i l l r e s u l t w h e r e sec (2 F o r e x p e r i m e n t a l c o n d i t i o n s A =8115 A ° and D ~ 24x10 c m 1 2 tr r 44 From e q u a t i o n ( 2 - 4 4 ) we s e e t h a t t h e l i f e t i m e o f s t a t e 2 ( T ) i s r e l a t e d t o t h e t r a n s i t i o n p r o b a b i l i t y f r o m s t a t e 2 t o s t a t e 1 (A 2_^-|) by A - - ! i f t h e t r a n s i t i o n f r o m s t a t e 2 t o s t a t e 1 i s t h e o n l y t r a n s i t i o n p o s s i b l e . In g e n e r a l t h e l i f e t i m e o f an atom i n s t a t e i n i s e q u a l t o -L. -v 5 A n - n ' w h e r e n' i s summed o v e r a l l p o s s i b l e s t a t e s . I f t h e u p p e r and l o w e r s t a t e s b o t h h a v e f i n i t e l i f e t i m e s t h e n t h e n a t u r a l w i d t h o f t h e l i n e A V*. i s g i v e n by rr rr (2 F o r t h e t r a n s i t i o n s t r e a t e d i n t h e p r e s e n t e x p e r i m e n t & V H ~ 2 x 1 0 ~ 4 c m J C O L L I S I O N A L BROADENING - RESONANCE BROADENING I f a p a r t i c l e c o l l i d e s w i t h t h e r a d i a t i n g atom t h i s w i l l a l s o m o d i f y i t s l i f e t i m e , p r o d u c i n g a L o r e n t z i a n l i n e p r o f i l e . I f t h e r a d i a t i n g and p e r t u r b i n g atoms a r e o f t h e same k i n d t h e n t h e e x p e c t e d b r o a d e n i n g i s g i v e n by G r i e m ( 2 ) and 45 A N 8 T 7 2 r m c where N i s t h e number d e n s i t y o f t h e p e r t u r b e r s , e i s t h e e l e c t r o n i c c h a r g e , m i s t h e e l e c t r o n i c m a s s , c i s t h e s p e e d o f l i g h t , gp and gr a r e t h e s t a t i s t i c a l w e i g h t s o f t h e s t a t e o f t h e p e r t u r b e r and r a d i a t o r , r e s p e c t i v e l y , A i s t h e w a v e l e n g t h o f t h e t r a n s i t i o n b e t w e e n t h e p e r t u r b e r and r a d i a t o r s t a t e s and f p r i s t h e 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 f o r t h a t same t r a n s i t i o n . T he glow d i s c h a r g e , w h i c h we u s e i n t h i s w o r k , i s a s l i g h t l y i o n i z e d , low d e n s i t y p l a s m a and as s u c h t h e m a j o r i t y o f t h e a r g o n atoms a r e i n t h e i r n e u t r a l g r o u n d s t a t e . Thus t h e m a j o r i t y o f t h e p e r t u r b e r s w h i c h w i l l c o l l i d e w i t h t h e r a d i a t i n g atom w i l l be t h e s e u n e x c i t e d n e u t r a l a t o m s . The t r a n s i t i o n s w h i c h a r e t r e a t e d i n t h i s work a r e b e t w e e n e n e r g y l e v e l s w h i c h a r e n o t c o n n e c t e d t o t h e g r o u n d s t a t e by an a l l o w e d t r a n s i t i o n w h i c h i m p l i e s t h a t f P r i s e q u a l t o z e r o f o r c o l l i s i o n s w i t h n e u t r a l s . T h e r e f o r e r e s o n a n c e b r o a d e n i n g c a n be i g n o r e d . C O L L I S I O N A L BROADENING - VAN DER WAALS F o l l o w i n g G r i e m ( 2 ) l e t us c a l c u l a t e t h e i m p a c t p a r a m e t e r ^m\n » w h i c h g i v e s r i s e t o p r a c t i c a l l y c o m p l e t e 46 d e s t r u c t i o n o f the c o r r e l a t i o n between the s t a t e s o f the system b e f o r e and a f t e r a Van Der Waals c o l l i s i o n . m , n herr fvE* R* = 3 ^ f X , - X , U f e 1 E M a! ^ E - E E.-E- . , (2-98) where N i s the number d e n s i t y o f the p e r t u r b i n g atoms, m i s t h e mass o f an e l e c t r o n , e i s the e l e c t r o n i c c h a r g e , V i s the r e l a t i v e v e l o c i t y o f the r a d i a t o r and p e r t u r b e r , E p i s e x c i t a t i o n e n e r g y o f the f i r s t e x c i t e d r e s o n a n c e s t a t e o f the p e r t u r b e r which i s c o n n e c t e d to the ground s t a t e by an a l l o w e d o p t i c a l t r a n s i t i o n and Roc i s the sq u a r e o f the c o - o r d i n a t e v e c t o r o f the r a d i a t i n g e l e c t r o n w i t h r e s p e c t t o i t s n u c l e u s i n a t o m i c u n i t s , QQ i s the f i r s t Bohr r a d i u s , E^ and E^, a r e the i o n i z a t i o n p o t e n t i a l s o f h y d r o g e n and the r a d i a t i n g atom r e s p e c t i v e l y , E ^ i s the e x c i t a t i o n p o t e n t i a l o f the upper s t a t e o f the l i n e , l ^ i s i t s o r b i t a l quantum and Z i s the c h a r g e o f the r a d i a t o r w i t h the r a d i a t i n g e l e c t r o n removed. Under e x p e r i -mental c o n d i t i o n s , N~7x10"m" 3 ;EH«13.53ev ; E *1.8xld' d joule =11.3ev v 4 0 0 m sec - i E -OO OC p 3 ev P mir) 6xld'°m F£~40 47 The c o l l i s i o n f r e q u e n c y c o r r e s p o n d i n g t o Q m m i s g i v e n b y 2 -3 _ j P v N - 10 cm ' m m w h i c h g i v e s r i s e t o a L o r e n t z i a n c o m p o n e n t e q u a l t o &Vv = P^v N * N v^9 f?RZ WQm E; ( 2 - 9 9 ) A l o n g w i t h t h e b r o a d e n i n g o f t h e l i n e t h e r e i s a s h i f t g i v e n by S H I F T = ( 2 - 1 0 0 ) In o r d e r t o o b t a i n t h e a b o v e e s t i m a t e s o f t h e l i n e w i d t h and s h i f t due t o Van d e r W a a l s ' i n t e r a c t i o n s t h e i m p a c t a p p r o x i m a t i o n was u s e d t o t r e a t a d i a b a t i c c o l l i s i o n s . The i m p a c t a p p r o x i m a t i o n a s s u m e s t h a t 22 -3 ~ Under e x p e r i m e n t a l c o n d i t i o n s N -10 m , j O m i ~ 1 0 m t h u s t h e p e r t u r b e r d e n s i t y i s low c o m p a r e d t o t h e i n t e r -a c t i o n v o l u m e . In o r d e r f o r t h e c o l l i s i o n t o be a d i a b a t i c t h e p e r t u r b e r must p a s s t h r o u g h t h e i n t e r a c t i o n v o l u m e w i t h o u t ^ a p p r e c i a b l e c h a n g e o f e n e r g y , i . e . , 48 "ft v « E p P 'mm - 2 J - 16 o r e x p e r i m e n t a l l y 10 « 10 In o r d e r t h a t t h e c o l l i s i o n p a r a m e t e r P m ( n c a n h a v e any p h y s i c a l s i g n i f i c a n c e i t must be l a r g e r t h a n t h e sum o f t h e " r a d i u s " o f t h e b r o a d e n e d s t a t e ( d0V R£ ) and t h e p e r t u r b i n g g r o u n d s t a t e . E x p e r i m e n t a l l y we h a v e a 0 V^~4xl6 ' °m < P „ , n The e x p r e s s i o n f o r R^ as g i v e n i n e q u a t i o n ( 2 - 8 4 ) i s d e r i v e d f o r a o n e - e l e c t r o n s y s t e m and i s a goo d a p p r o x i m a t i o n w h e n e v e r t h e Coulomb a p p r o x i m a t i o n i s v a l i d . (An e x c i t e d s t a t e i n a r g o n i s a p p r o x i m a t e d v e r y w e l l by a o n e - e l e c t r o n s y s t e m . ) N o t e t h a t t h e v a l u e Pm i B i s t h e v a l u e o f t h e i m p a c t p a r a m e t e r w h i c h w i l l d e s t r o y t h e c o r -r e l a t i o n b e t w e e n t h e s t a t e s o f t h e s y s t e m b e f o r e and a f t e r t h e c o l l i s i o n and t h a t c o l l i s i o n s w i t h i m p a c t p a r a m e t e r s g r e a t e r t h a n t h i s h a v e b e e n i g n o r e d . T h u s we c a n e x p e c t t h a t t h e b r o a d e n i n g p r e d i c t e d by t h i s t r e a t m e n t w i l l g i v e a minimum v a l u e . I f we c o n s i d e r two u p p e r s t a t e s oc and o c ' t h e n t h e r a t i o o f t h e b r o a d e n i n g o f t h e two s t a t e s i s g i v e n by e q u a t i o n ( 2 - 8 5 ) 49 A fry- ( 2 - 1 0 1 ) E x p e r i m e n t a l l y =.65 f o r I = 2 and 1^ = 1 Even i f t h e m a g n i t u d e o f t h e b r o a d e n i n g i s i n e r r o r t h e r a t i o o f t h e b r o a d e n i n g o f two d i f f e r e n t l e v e l s i s e x p e c t e d t o be g i v e n by t h e a b o v e f o r m u l a s i n c e t h e v a l i d i t y o f t h e r a t i o d e p e n d s s o l e l y on t h e f a c t s t h a t a r g o n c a n be t r e a t e d as a o n e - e l e c t r o n s y s t e m and t h e b r o a d e n i n g i s o f t h e Van d e r W a a l s t y p e . 2.10 COMMENTS The a s s u m p t i o n s made t o o b t a i n t h e e x p r e s s i o n f o r t h e o b s e r v e d t r a n s m i s s i o n ( e q u a t i o n s ( 2 - 7 3 ) and ( 2 - 7 4 ) ) must be s t r e s s e d . The atoms must h a v e a M a x w e l l i a n v e l o c i t y d i s t r i b u t i o n and t h e s p e c t r a l l i n e s h a p e must be a p p r o x i -m a t ed by a V o i g t p r o f i l e ; t h a t i s , i t must be a c o n v o l u t i o n o f a D o p p l e r and a L o r e n t z i a n l i n e s h a p e . The L o r e n t z i a n c o m p o n e n t r e l a t i v e t o t h e D o p p l e r c o m p o n e n t o f t h e l i n e s h a p e g i v e s t h e v a l u e o f t h e V o i g t ' a 1 w h i c h i s u s e d as a p a r a m e t e r t o p r o d u c e t h e ' b e s t f i t ' b e t w e e n e x p e r i m e n t and t h e o r y . 50 I t must a l s o be s t r e s s e d t h a t , s i n c e t h e two d i s c h a r g e s a r e made as s i m i l a r as p o s s i b l e , t h e same p r o p a g a t i o n p a r a m e t e r s c a n be u s e d t o c a l c u l a t e t h e s o u r c e f u n c t i o n as w e l l as t h e t r a n s m i t t e d i n t e n s i t y . Once t h e p r o p a g a t i o n p a r a m e t e r s h a v e b e e n d e t e r m i n e d f o r a g i v e n s o u r c e t h e n t h i s known s o u r c e c a n be u s e d t o i n v e s t i g a t e o t h e r a b s o r b e r s t o d e t e r m i n e t h e p r o p a g a -t i o n p a r a m e t e r s and a t o m i c p a r a m e t e r s i n them. S i n c e t h e r e l 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 a r e i n t r i n s i c a t o m i c c o n s t a n t s t h e s e s h o u l d n o t c h a n g e w i t h d i f f e r e n t d i s -c h a r g e s b u t t h e L o r e n t z i a n c o m p o n e n t w i l l c h a n g e s i n c e i t i s d e t e r m i n e d by t h e c o n d i t i o n s s u r r o u n d i n g t h e a t o m . CHAPTER 3 EXPERIMENTAL CONSIDERATIONS 3.0 INTRODUCTION C h a p t e r 2 has d e v e l o p e d t h e t h e o r y n e c e s s a r y t o a n a l y s e t h e r e s u l t s o f t h e e x p e r i m e n t . The f i r s t s e c t i o n i n t h i s c h a p t e r ( 3 . 1 ) o u t l i n e s t h e a p p a r a t u s u s e d w h i l e f o l l o w i n g s e c t i o n s g i v e t h e m e t h o d o f t a k i n g d a t a and t h e a c c e p t a n c e c r i t e r i o n . T h i s c h a p t e r t h e n d i s c u s s e s some o f t h e m o d i f i c a t i o n s w h i c h had t o be i n t r o d u c e d i n o r d e r t o t a k e a c c o u n t o f t h e n o n i d e a l n a t u r e o f t h e e x p e r i m e n t . 3.1 EXPERIMENTAL APPARATUS A b l o c k d i a g r a m o f t h e e x p e r i m e n t a l a r r a n g e m e n t i s shown i n F i g . 3.1. The s o u r c e and a b s o r b e r a r e i d e n t i c a l g l o w d i s c h a r g e t u b e s ( F i g . 3 . 2 ) . The l i g h t f r o m t h e s o u r c e i s l i n e a r l y p o l a r i z e d by N i c o l p r i s m N-j and t h e n t r a v e r s e s t h e a b s o r p t i o n t u b e b e f o r e p a s s i n g t h r o u g h t h e a n a l y s i n g N i c o l Ng t o t h e m o n o c h r o m a t o r . The l i g h t i n t e n -s i t y p a s s e d by t h e m o n o c h r o m a t o r i s d e t e c t e d by a p h o t o -m u l t i p l i e r and a d i g i t a l p h a s e s e n s i t i v e d e t e c t o r ( D P S D ) . 51 52 3.1-1 T EST GAS The c h o i c e o f a r g o n f o r t h e t e s t gas was made f o r a number o f r e a s o n s . S i n c e i t i s a n o b l e gas i t i s e a s i l y h a n d l e d and t h e p r o d u c t i o n o f t h e g l o w d i s c h a r g e s u s e d was r e l a t i v e l y s i m p l e . I t i s a gas w h i c h i s u s e d w i d e l y i n t h e d e v e l o p m e n t s t a g e s o f t h e p l a s m a d e v i c e s u s e d i n t h i s l a b o r a t o r y and i n d e e d t r a n s i t i o n p r o b a b i l i t i e s h a v e b e e n m e a s u r e d h e r e by J a c o b s o n ( 7 ) . In a d d i t i o n t h e r e h a v e b e e n p r e v i o u s r e p o r t s o f m e a s u r e d t r a n s i t i o n p r o b a b i l i t i e s ( R e f . 7 - 1 1 ) . 3.1-2 SOURCE AND ABSORBER The s o u r c e and a b s o r b e r ( F i g . 3-2) w ere made as d i m e n s i o n a l l y s i m i l a r as p o s s i b l e . In o r d e r t o r e d u c e t h e i m p u r i t i e s t o an a b s o l u t e minimum b o t h t u b e s w ere c o n n e c t e d t o t h e same h i g h vacuum s y s t e m a t t h e same t i m e . - 8 A p r e s s u r e o f l x 10" t o r r . was m a i n t a i n e d f o r s i x h o u r s w h i l e t h e t u b e s w ere h e l d a t 4 0 0 ° C b e f o r e b e i n g s i m u l -t a n e o u s l y f i l l e d t o 2 t o r r . w i t h r e s e a r c h g r a d e a r g o n ( A i r c o ) . When t h e t u b e s w ere f i r s t u s e d i t was f o u n d t h a t t h e l i g h t o u t p u t had a h i g h e r t h a n e x p e c t e d v a r i a n c e b u t a f t e r b e i n g l e f t on f o r a number o f h o u r s t h e l i g h t o u t p u t became more c o n s t a n t . To a v o i d t h e v a r y i n g s o u r c e s t r e n g t h t h e t u b e s w ere l e f t on f o r a t l e a s t 48 h o u r s b e f o r e t h e 53 e x p e r i m e n t was c a r r i e d o u t . I t was f o u n d t h a t t h e t r a n s -m i s s i o n c u r v e s v a r i e d w i t h t h e c u r r e n t i n t h e a b s o r b e r . In o r d e r t o e l i m i n a t e e r r o r s f r o m t h i s , t h e c u r r e n t was k e p t a t 1±.02 ma. The t r a n s m i s s i o n d i d n o t d e p e n d on t h e c u r r e n t i n t h e s o u r c e as l o n g as i t was k e p t b e t w e e n 1 and 10 ma. No e v i d e n c e o f s t r i a t i o n s was o b s e r v e d i n t h e s o u r c e o r t h e a b s o r b e r e v e n when v i e w e d s i d e on w i t h a p h o t o m u l t i p i i e r . The d i s c h a r g e t u b e s were o p e r a t e d f r o m a 1500 v o l t r e g u l a t e d power s u p p l y ( r e g u l a t i o n and r i p p l e ~ 5 % ) . The c u r r e n t was c o n t r o l l e d by a s e r i e s p e n t o d e r e g u l a t o r ( r e g u l a t i o n and r i p p l e ~ l % ) . 3.1-3 POLARIZERS N i c o l p r i s m s w e re u s e d as p o l a r i z e r and a n a l y s e r b e c a u s e t h e y g i v e e x c e l l e n t p o l a r i z a t i o n and have good t r a n s m i s s i o n q u a l i t i e s . The a n a l y s e r N 2 ( F i g . 3-1) was p e r m a n e n t l y a l i g n e d w i t h t h e m o n o c h r o m a t o r so t h a t t h e l i g h t o u t p u t was a maximum. T h i s was done b e c a u s e t h e m o n o c h r o m a t o r p o l a r i z e s t h e l i g h t i t s e l f . The p o l a r i z e r N-j was p o s i t i o n e d a c c u r a t e l y i n t h e c r o s s e d p o s i t i o n w i t h r e s p e c t t o Ng by o b s e r v i n g t h e minimum o f t h e t r a n s m i t t e d i n t e n s i t y . The m o u n t i n g o f N, had m a c h i n e d s l o t s 54 by w h i c h i t was r o t a t e d a c c u r a t e l y t o t h e p a r a l l e l p o s i t i o n . T h i s p r i s m i t s e l f c o u l d be r e m o v e d a n d r e p l a c e d w i t h o u t c h a n g i n g t h e p o s i t i o n o f t h e mount. As an a d d i t i o n a l c h e c k t h a t t h e p o l a r i z e r s were p a r a l l e l t h e m a g n e t i c f i e l d was r u n t h r o u g h z e r o t o v e r i f y t h a t t h e e x p e r i m e n t a l c u r v e s were s y m m e t r i c a b o u t z e r o f i e l d . From e q u a t i o n ( 2 - 6 3 ) i t c a n be s e e n t h a t t h e t r a n s m i t t e d i n t e n s i t y w i l l be s y m m e t r i c i f and o n l y i f ft = 0, t h a t i s , i f t h e two p o l a r i z e r s a r e p a r a l l e l . 3.1-4 MONOCHROMATOR A s p e x g r a t i n g m o n o c h r o m a t o r ( f / 1 0 , 10 A ° / m m i n f i r s t o r d e r ) was u s e d t o i s o l a t e i n d i v i d u a l s p e c t r a l l i n e s . The s l i t w i d t h s v a r i e d f r o m l i n e t o l i n e b u t t h e y w ere t y p i c a l l y a b o u t 400/{. T h i s maximum was i m p o s e d b e c a u s e t h e p o r t i o n o f t h e e n t r a n c e p l a n e , w h i c h i s i l l u m i n a t e d by t h e s o u r c e , i s l i m i t e d by t h e a p e r t u r e s t o p s ( F i g . 3 - 1 ) . 3.1-5 DIVERGENCE AND APERTURES The s i z e o f t h e a p e r t u r e s u s e d were d e t e r m i n e d e x p e r i m e n t a l l y by s t a r t i n g w i t h l a r g e d i a m e t e r s and r e d u c i n g t h e s i z e o f t h e a p e r t u r e s u n t i l t h e o b s e r v e d r e l a t i v e t r a n s m i s s i o n was i n d e p e n d e n t o f t h i s s i z e ( F i g . 3 - 3 ) . T h i s 55 p r o c e d u r e was c a r r i e d o u t t o e l i m i n a t e any r a d i a l v a r i a t i o n i n t r a n s m i s s i o n due t o v a r y i n g number d e n s i t y o r any o t h e r c a u s e . T h e a p e r t u r e s u s e d were 2mm i n d i a m e t e r w h i c h g i v e s a d i v e r g e n c e o f a b o u t . 7 ° . 3.1-6 MAGNET The magnet u s e d was c o n s t r u c t e d i n t h e l a b o r a t o r y a c c o r d i n g t o t h e s p e c i f i c a t i o n s shown i n F i g u r e 3-4. The maximum f i e l d p r o d u c e d was a b o u t 2400 g a u s s f r o m a c o i l c u r r e n t o f a p p r o x i m a t e l y 35 amps. The c u r r e n t was m e a s u r e d u s i n g a one m i l l i o h m s h u n t a c r o s s w h i c h a d i g i t a l v o l t m e t e r was c o n n e c t e d . I t was e x p e r i m e n t a l l y v e r i f i e d t h a t t h e s h u n t was t e m p e r a t u r e c o m p e n s a t e d t o 5 0 ° C , w h i c h i s w e l l a b o v e t h e o p e r a t i n g t e m p e r a t u r e s ( ^ 3 0 ° C ) . The c u r r e n t was o b t a i n e d f r o m two power s u p p l i e s i n p a r a l l e l as shown i n F i g . 3-5. The r i p p l e on t h e c u r r e n t p a s s i n g t h r o u g h t h e m agnet was t h u s k e p t b e l o w . 1 % . The v o l t m e t e r was c a l i b r a t e d i n t e r m s o f t h e m a g n e t i c f i e l d i n two wa y s . An a b s o l u t e c a l i b r a t i o n was e s t a b l i s h e d by t h e u s e o f an N.M.R. p r o b e w h i c h g a v e t h e f i e l d t o w i t h i n .05% and f o r r e l a t i v e f i e l d v a l u e s b e l o w t h e r a n g e o f t h e N.M.R. p r o b e ( 1 2 0 0 g a u s s ) a H a l l p r o b e g a u s s m e t e r ( B e l l "240" I n c r e m e n t a l G a u s s m e t e r #41228, p r o b e #4274) was u s e d . In t h e r a n g e b e t w e e n 1200 and 2400 g a u s s where b o t h p r o b e s 56 c o u l d be u s e d t h e y g a v e r e a d i n g s w h i c h were c o n s i s t e n t t o w i t h i n .5% and t h e f i e l d was l i n e a r w i t h r e s p e c t t o c u r r e n t t o w i t h i n t h e same a c c u r a c y , f r o m 0 t o 2400 g a u s s . The v a r i a t i o n o f f i e l d w i t h l e n g t h a l o n g t h e a x i s a n d t r a n s v e r s e t o t h e a x i s was m e a s u r e d u s i n g t h e H a l l p r o b e . The p e r c e n t a g e v a r i a t i o n a l o n g t h e a x i s was f o u n d t o be s y m m e t r i c a b o u t t h e c e n t e r o f t h e c o i l and i n d e p e n d e n t o f t h e c o i l c u r r e n t . The v a r i a t i o n a c r o s s t h e r a d i u s o f t h e t u b e was f o u n d t o be l e s s t h a n .5% o v e r t h e l e n g t h o c c u p i e d by t h e a b s o r b e r ( s e e F i g . 3 - 1 3 ) . 3.1-7 DETECTION SYSTEM In o r d e r t o m e a s u r e a d e s i r e d s i g n a l i n t h e p r e s e n c e o f u n w a n t e d b a c k g r o u n d n o i s e t h e d e t e c t i o n s y s t e m must be a b l e t o d i s t i n g u i s h b e t w e e n t h e s e two k i n d s o f i n p u t . In t h e e x p e r i m e n t a l s e t - u p t h e d e s i r e d s i g n a l i s t h e l i g h t w h i c h t h e s o u r c e e m i t s and w h i c h i s t r a n s m i t t e d t h r o u g h t h e a b s o r b e r ( F i g . 3 - 1 ) . The u n w a n t e d n o i s e comes f r o m two s e p a r a t e p l a c e s . The d e t e c t i o n s y s t e m w i l l c o n t r i b u t e some n o i s e ( p h o t o m u l t i p l i e r d a r k c u r r e n t ) b u t t h e m a j o r c o n t r i b u t i o n o f n o i s e i s f r o m l i g h t w h i c h t h e a b s o r b e r i t s e l f e m i t s . S i n c e t h e s o u r c e and a b s o r b e r w ere i d e n t i c a l p l a s m a s t h e y w i l l e m i t i d e n t i c a l s p e c t r a e x c e p t f o r t h e s m a l l Zeeman s p l i t t i n g i n t h e a b s o r b e r . The m o n o c h r o m a t o r was u s e d t o i s o l a t e i n d i v i d u a l s p e c t r a l l i n e s b u t due t o 57 t h e i r s i m i l a r i t y , t h e s p e c t r a o f t h e s o u r c e and a b s o r b e r c o u l d n o t be d i s t i n g u i s h e d f r o m e a c h o t h e r . In o r d e r t o s e p a r a t e t h e s o u r c e i n t e n s i t y f r o m t h e a b s o r b e r " n o i s e " i t was d e c i d e d t o i m p o s e a s q u a r e wave m o d u l a t i o n on t h e s o u r c e and u s e a DPSD ( D i g i t a l P h a s e S e n s i t i v e D e c t e c t o r ) t o o b t a i n a s i g n a l . The DPSD w i l l be t r e a t e d i n t h r e e s e p a r a t e s u b s e c t i o n s . The t h e o r y i s p r e s e n t e d f i r s t f o l l o w e d by a s e c t i o n w h i c h g i v e s a s t a t i s t i c a l t r e a t m e n t o f t h e e r r o r s o f a DPSD. The t r e a t -ment c o n c l u d e s w i t h o p e r a t i o n a l d e t a i l s on t h e s y s t e m u s e d i n t h i s e x p e r i m e n t . DPSD - THEORY The s i g n a l w h i c h i s o b s e r v e d f r o m t h e p h o t o m u l t i -p l i e r (PM) ( F i g . 3-10) w i l l be made up f r o m t h r e e s o u r c e s ; s i g n a l ( S ) , a b s o r b e r n o i s e (N) and p h o t o m u l t i p i i e r d a r k c u r r e n t ( D C ) . I f t h e s o u r c e i s c o n s t a n t i n t i m e t h e o u t p u t i s shown s c h e m a t i c a l l y i n t r a c e A o f F i g . 3-6. I f t h e s o u r c e i s now m o d u l a t e d w i t h a s q u a r e wave as i n t r a c e B t h e n t h e PM. o u t p u t w i l l be as shown i n t r a c e C. A g a t i n g p u l s e i s now g e n e r a t e d ( t r a c e D) s u c h t h a t a s e r i e s o f e q u a l l e n g t h p u l s e s a r e p r o d u c e d one f o r e a c h 'on' and ' o f f s e c t i o n o f t h e r e f e r e n c e ( t r a c e B ) . T h i s g a t i n g p u l s e ( t r a c e D) i s u s e d t o g a t e t h e PM o u t p u t t o p r o d u c e 58 a s e r i e s o f b u r s t s o f s i g n a l s e p a r a t e d by an o f f p e r i o d ( t r a c e E ) . N o t e t h a t t h i s s e r i e s o f p u l s e s a r e a l l o f e x a c t l y t h e same l e n g t h and t h a t e a c h one i s c o m p l e t e l y w i t h i n one ' o f f o r 'on' s e c t i o n o f t h e r e f e r e n c e . The l a s t s t e p i s t o d i r e c t t h e b u r s t s o f p u l s e s t o two s e p a r a t e o u t p u t s . One t a k e s a l l t h e b u r s t s p r o d u c e d w h i l e t h e r e f e r e n c e i s 'on' and t h e o t h e r t a k e s a l l t h o s e p r o d u c e d w h i l e t h e r e f e r e n c e i s ' o f f . ' The 'on' o u t p u t w i l l h a v e S+N+DC w h i l e t h e ' o f f o u t p u t w i l l h a v e o n l y N+DC. T h u s t h e s i g n a l s ( S ) i s j u s t t h e d i f f e r e n c e b e t w e e n t h e two o u t p u t s . L e t us now d e a l w i t h some s t a t i s t i c s . DPSD - S T A T I S T I C S T h i s s e c t i o n w i l l d e a l w i t h v a r i o u s s o u r c e s o f e r r o r i n t r o d u c e d by t h e DPSD u s e d i n t h i s e x p e r i m e n t . F i r s t l e t us d e a l w i t h g a t i n g t i m e s . F i g u r e 3-6 shows t h a t t h e g a t e #1 ( t r a c e D) p a s s e s a s e r i e s o f p u l s e s o f e q u a l l e n g t h , one f o r e a c h 'on' and ' o f f p e r i o d o f t h e r e f e r e n c e p u l s e ( t r a c e B ) . S i n c e t h e same p h y s i c a l e l e m e n t g a t e s b o t h t h e 'on' and t h e ' o f f c h a n n e l s t h e mean l e n g t h o f t h e p u l s e s w i l l be t h e same b u t t h e v a r i a n c e o r j i t t e r o f t h e g a t i n g t i m e s n e e d n o t be c o r r e l a t e d . I f t o n i s t h e g a t i n g 'on' t i m e and •^f-c 1 S t n e f o l l o w i n g ' o f f t i m e t h e n 59 d i s t r i b u t i o n a s s u m e d ) . The p e r c e n t a g e d i f f e r e n c e i n t h e g a t i n g t i m e f o r t h e s e s u c c e s s i v e p u l s e s i s t h u s g i v e n by I f now Ng s e t s o f s u c c e s s i v e 'on' and ' o f f t i m e s a r e c o n s i d e r e d t h e p e r c e n t a g e e r r o r i n t r o d u c e d b e c a u s e o f j i t t e r i n t h e g a t i n g t i m e s i s The minimum t i m e o f c o u n t i n g f o r any e x p e r i m e n t a l p o i n t s i s 100 s e c . T h e r e f o r e a t 1 KHZ } N Q ~10 5 T h u s i f t h e mean o f t h e g a t i n g t i m e i s c o n s t a n t t h e n t h e e r r o r i n t r o d u c e d due t o t h e j i t t e r i n t h e l e n g t h w i l l be l o w e r t h a n t h e v a r i a n c e o f one p u l s e by a f a c t o r o f t h e _ 3 o r d e r o f 10 . The e x p e r i m e n t a l t e s t t o p r o v e t h a t t h e g a t i n g t i m e s a r e e q u a l i s t o r e p l a c e t h e PM. w i t h a s i g n a l g e n e r a t o r and o b s e r v e t h e r e s u l t i n g v a l u e s on b o t h c o u n t e r s . The s i g n a l g e n e r a t o r p r o d u c e s a c o n s t a n t f r e q u e n c y f so t t 60 t h e c o u n t s r e c o r d e d i n t h e two c h a n n e l s w i l l be g i v e n by f NG t and f N r t „ Q O N G o f f and t h e d i f f e r e n c e i n t h e two c h a n n e l s i s F NG - F NG t o K ( e x p e r i m e n t a l l y I O - 8 ) The r e a s o n f o r t h i s e x t r e m e a c c u r a c y i s t h a t t h e mean t i m e t n e e d o n l y be c o n s t a n t o v e r s u c c e s s i v e 'on' and ' o f f t i m e s w h i c h e x p e r i m e n t a l l y r e q u i r e s o n l y a p p r o x i m a t e l y l m s e c . T h u s i n o r d e r t o p r o d u c e e q u a l 'on' a nd ' o f f t i m e s t h e g a t e #1 ( F i g . 3-6) need o n l y be c o n s t a n t o v e r t h i s same t i m e s c a l e . The e x p e r i m e n t a l p r o c e d u r e r e q u i r e s t h a t t h e r a t i o o f t h e t r a n s m i t t e d s i g n a l w i t h t h e a b s o r b e r on t o t h a t w i t h t h e a b s o r b e r o f f be t a k e n . The c o u n t s r e c o r d e d i n t h e two c h a n n e l s a r e g i v e n by C-j and r e s p e c t i v e l y , w h e r e X ' i s t h e t o t a l g a t i n g t i m e C, = (N + DC+S)TW(N+DC*S)7' C 2 = ( N + D C ) X ' i -\J(N*DC )T' t h u s t h e t r a n s m i t t e d s i g n a l i s S C, - C 8 = S T ' i V[2(N + DC)+S]T' I f t h e a b s o r b e r i s now t u r n e d o f f t h e t r a n s m i t t e d s i g n a l i s s ' and N ' i s t h e n o i s e c o u n t 61 < - C / r S ' T ' ± y [ 2 ( N ' + D C ) + S ' ] 7 t h u s t h e r a t i o o f t h e t r a n s m i t t e d s i g n a l s T i s g i v e n by T - C, - C ? = _S c ; - c: S' " " where AT T (2(N+DC)+ S f (2(N'+DC)+S'f 1 . Y r The p e r c e n t a g e e r r o r i n t h e t r a n s m i s s i o n was m a i n t a i n e d a t a p p r o x i m a t e l y 1% t h r o u g h o u t t h e e x p e r i m e n t by a d j u s t i n g t h e t o t a l g a t i n g t i m e (100 s e c .< jT '^.1000 s e c ) . DPSD - OPERATION A b l o c k d i a g r a m o f t h e s y s t e m u s e d i s g i v e n i n F i g . 3-7. The s q u a r e wave m o d u l a t i o n i s i m p o s e d on t h e s o u r c e by t h e c h o p p i n g w h e e l ( F i g . 3-8) and t h e r e f e r e n c e i s t a k e n o f f t h e w h e e l as shown i n t h a t f i g u r e . The w h e e l i s d r i v e n by a s y n c h r o n o u s m o t o r i n o r d e r t o m a i n t a i n a c o n s t a n t c h o p p i n g s p e e d o f 990 Hz. T h i s f r e q u e n c y was c h o s e n t o m i n i m i z e t h e e f f e c t o f t h e 60 c y c l e n o i s e i n t h e room. L e t us now go t h r o u g h t h e b l o c k d i a g r a m ( F i g . 3-7) 62 e x p l a i n i n g t h e d i f f e r e n t c o m p o n e n t s and g i v i n g t h e c r i t i c a l s p e c i f i c a t i o n s . In o r d e r t o i n t e g r a t e t h e two o u t p u t s f o r a l o n g t i m e (up t o 30 min.) i t was d e c i d e d t o u s e a d i g i t a l s y s t e m . In t h i s s y s t e m t h e i n d i v i d u a l p h o t o n p u l s e s w ere a m p l i f i e d t o g i v e p u l s e s w h i c h c o u l d be c o u n t e d i n s t a n d a r d c o u n t i n g c i r c u i t s . The p h o t o n p u l s e s w ere a m p l i f i e d a t t h e PM. t o g i v e an o u t p u t p u l s e as shown i n F i g . 3-9. The PM. u s e d (EMI 9 558B) had a p u l s e w i d t h o f 4 2 n s . so t h e p u l s e c i r c u i t r y u s e d t o t r e a t t h e p u l s e s had t o be made s u c h t h a t i t was c a p a b l e o f h a n d l i n g p u l s e s a t r a t e s up t o a o M H * . A l l t h e c i r c u i t r y u s e d was d e s i g n e d t o h a n d l e p u l s e s a t t h i s r a t e i n o r d e r t h a t t h e s y s t e m w o u l d be f r e e f r o m p i l e up e f f e c t s ( i . e . , two p u l s e s a r r i v e so c l o s e t o g e t h e r t h a t t h e c o u n t e r s o n l y s e e them as one p u l s e ) . A t u n n e l d i o d e c u r r e n t d i s c r i m i n a t o r ( F i g . 3-10) was u s e d t o p r o d u c e a s t a n d a r d s h a p e f o r a l l p h o t o n p u l s e s . The d i s c r i m a t i o n l e v e l was s e t by e x p e r i m e n t a l l y r e d u c i n g i t and n o t i n g t h e s i g n a l t o n o i s e p r o d u c e d f o r a c o n s t a n t i n p u t . The d i s c r i m i n a t i o n l e v e l was s e t t o g i v e t h e maxium s i g n a l t o n o i s e . The o p timum l e v e l was f o u n d t o be i n d e -p e n d e n t o f s i g n a l s t r e n g t h . 63 G a t e #1 ( F i g . 3-7) i s t h e g a t e w h i c h d e t e r m i n e s t h e l e n g t h o f t i m e t h a t t h e ' o n ' a n d ' o f f ' c h a n n e l s a r e o p e n . G a t e #2 i s d e s i g n e d t o s e n d t h e b u r s t s g a t e d by #1 t o t h e two o u t p u t s . The t i m i n g f o r g a t e #2 i s n o t c r i t i c a l p r o v i d e d t h a t t h i s g a t e a l l o w s t h e e n t i r e b u r s t t o go t o t h e c o r r e c t o u t p u t . The r e a s o n f o r t h i s p a r t i c u l a r a r r a n g e -ment i s s o t h e same p h y s i c a l e l e m e n t w i l l a l t e r n a t e l y g a t e b o t h t h e on and t h e o f f t i m e s . P r o v i d e d t h a t t h e g a t i n g t i m e o f #1 c h a n g e s s l o w l y c o m p a r e d t o t h e r e f e r e n c e f r e -q u e n c y ( l k c . ) t h e n t h e sum o f t h e 'on t i m e ' w i l l be e q u a l t o t h e sum o f t h e ' o f f t i m e . ' T h i s was c h e c k e d e x p e r i -m e n t a l l y by i n t r o d u c i n g a s m a l l l e a k t o t h e PM. l i g h t s e a l and v a r y i n g t h e i n t e n s i t y o f t h e l i g h t t o g i v e c o u n t i n g r a t e s o v e r t h e r a n g e f o u n d d u r i n g t h e e x p e r i m e n t . I t was f o u n d t h a t t h e two c h a n n e l s g a v e r e a d i n g s e q u a l t o e a c h o t h e r t o w i t h i n t h e e x p e c t e d v a r i a n c e . T h i s was c h e c k e d p e r i o d i c a l l y t h r o u g h o u t t h e e x p e r i m e n t and i n a l l c a s e s t h e r e s u l t s w e re c o n s i s t e n t w i t h t h e h y p o t h e s i s o f e q u a l g a t i n g t i m e s . In o r d e r t o c h e c k t h a t t h e r e was no p i l e u p e f f e c t s a s m a l l s i g n a l was f e d i n t o t h e s y s t e m and c o u n t e d a t minimum n o i s e l e v e l . A l i g h t l e a k was t h e n i n t r o d u c e d t o i n c r e a s e t h e b a c k g r o u n d c o u n t i n g r a t e t o w e l l a b o v e any r a t e s e n c o u n t e r e d d u r i n g t h e e x p e r i m e n t and t h e s i g n a l 64 was m e a s u r e d a g a i n . The r e s u l t i n g s i g n a l was f o u n d t o be i n d e p e n d e n t o f t h e s i z e o f t h e l i g h t l e a k . S i n c e t h e s i g n a l c o u n t s were t h e same none had be e n l o s t due t o p i l e u p and t h e r e f o r e we c a n i g n o r e t h e e f f e c t s o f p i l e u p f o r t h i s e x p e r i m e n t . 3.2 MEASUREMENT PROCEDURE A l l t h e e l e m e n t s o f t h e o p t i c a l p a t h shown on F i g . 3-1 e x c e p t t h e m o n o c h r o m a t o r and c h o p p i n g w h e e l were m o u n t e d on a h e a v y o p t i c a l b e n c h . A He-Ne CW l a s e r was p o s i t i o n e d as shown t o d e f i n e t h e o p t i c a l a x i s . The a p e r -t u r e s t o p s and t h e m o n o c h r o m a t o r were t h e n p l a c e d i n p o s i -t i o n w i t h r e s p e c t t o t h e l a s e r beam. I t was f o u n d t h a t t h e m o s t r e p r o d u c i b l e r e s u l t s w e re o b t a i n e d when t h e s o u r c e and t h e a b s o r b e r w ere p o s i t i o n e d so t h a t t h e o p t i c a l p a t h was down t h e c e n t e r o f t h e t u b e r a t h e r t h e n p o s i t i o n i n g them f o r maximum l i g h t i n t e n s i t y . , The maximum l i g h t f r o m t h e s o u r c e i s o b t a i n e d when t h e ' h o t a r e a ' ( F i g . 3-2) i s on t h e o p t i c a l p a t h b u t s i n c e t h i s s p o t moves w i t h t i m e t h e s o u r c e i s n o t as c o n s t a n t as when t h a t r e g i o n i s o f f t h e a x i s . The a b s o r b e r t e n d s t o t r a n s m i t l i g h t down t h e g l a s s w a l l s and u n l e s s t h e t u b e i s a l i g n e d t o keep t h e w a l l s o f f t h e a x i s u n r e l i a b l e r e s u l t s w i l l r e s u l t . 65 The s o u r c e was t u r n e d on and t h e a b s o r b e r was l e f t o f f i n o r d e r t o a d j u s t t h e p h a s e o f t h e r e f e r e n c e w i t h r e s p e c t t o t h e m o d u l a t i o n o f t h e s o u r c e . The p h a s e was a d j u s t e d s o t h a t t h e i n c r e a s e i n c o u n t i n g r a t e was i n one c o u n t e r o n l y ( F i g . 3 - 8 ) . T h a t i s , a l l t h e c o u n t s due t o t h e s o u r c e i n t e n s i t y were r o u t e d t o one c o u n t e r w h i l e t h e c o u n t i n g r a t e o f t h e o t h e r c o u n t e r r e m a i n e d u n c h a n g e d w h e t h e r t h e s o u r c e was on o r o f f . Once t h e p h a s e was a d j u s t e d t h e b i a s o f t h e two c h a n n e l s was c h e c k e d by t u r n i n g t h e a b s o r b e r on and p l a c i n g a b l a c k c l o t h b e t w e e n t h e a b s o r b e r and t h e c h o p p i n g w h e e l ( s e e F i g . 3-1) t o i n s u r e t h a t t h e r e was no s i g n a l p a s s i n g t h r o u g h t h e w h e e l . The c o u n t e r s t h e n r e a d t h e s a m e , f o r any c o u n t i n g p e r i o d ^ w i t h i n t h e e x p e c t e d v a r i a n c e b e t w e e n two P o i s s o n p r o c e s s e s w i t h t h e same mean. I t was f o u n d t h a t t h e two c h a n n e l s w e r e t h e same t o w i t h i n I : 1 0 8 o r b e t t e r . In o r d e r t o m i n i m i s e t h e e f f e c t o f t h e m a g n e t i c f i e l d on t h e e l e c t r o n i c s , s o u r c e , p h o t o m u l t i p l i e r and d i s c r i m -i n a t o r t h e y a l l w ere c o v e r e d w i t h ^ - m e t a l . W i t h t h e s o u r c e on and t h e a b s o r b e r o f f t h e m a g n e t i c f i e l d was i n c r e a s e d t o t h e maximum and t h e c o u n t i n g r a t e r e m a i n e d c o n s t a n t t h r o u g h o u t . 66 The a c t u a l m e a s u r e m e n t was c a r r i e d o u t by f i r s t r e m o v i n g t h e p o l a r i z i n g N i c o l N^ ( F i g . 3-1) and m e a s u r i n g t h e s i g n a l w i t h t h e s o u r c e on and t h e a b s o r b e r o f f , f o r z e r o f i e l d . The a b s o r b e r was t h e n t u r n e d on and t h e s i g n a l was r e m e a s u r e d . The r a t i o o f t h e s e two s i g n a l s was t h e n t h e d e s i r e d r e l a t i v e t r a n s m i s s i o n T T . T h i s p r o c e s s was t h e n r e p e a t e d f o r a s e r i e s o f e q u a l l y s p a c e m a g n e t i c f i e l d s . N-j was t h e n r e p l a c e d and t h e p r o c e d u r e was r e p e a t e d t o g i v e T 0 . The o n l y r u n s t h a t were c o n s i d e r e d as r e l i a b l e w e re t h e o n e s d u r i n g w h i c h t h e m a g n e t i c f i e l d c o u l d be i n c r e a s e d t o a maximum and b a c k t o z e r o i n i n c r e m e n t s and t h e t r a n s -m i s s i o n s m e a s u r e d were c o m p l e t e l y r e p r o d u c i b l e . I t was f o u n d t h a t t h e r e l i a b l e r u n s done on d i f -f e r e n t d a y s , e v e n w i t h new o p t i c a l a l i g n m e n t s b e t w e e n r u n s , a g r e e d w i t h i n e x p e r i m e n t a l e r r o r . The a c t u a l d a t a u s e d was t h e a v e r a g e o f a l l t h e r e l i a b l e r u n s t h a t were made on a g i v e n l i n e . 3.3 OPERATING CONDITIONS T y p i c a l d a r k c u r r e n t c o u n t s e n c o u n t e r e d i n t h i s e x p e r i m e n t were o f t h e o r d e r o f 500 Hz. w h i l e t h e a b s o r b e r n o i s e v a r i e d f r o m l i n e t o l i n e w i t h a t y p i c a l v a l u e t h a t v a r i e d f r o m 1 kHz. f o r t h e weak l i n e s t o as much as 10 kHz. f o r t h e s t r o n g e r l i n e s . The s o u r c e s t r e n g t h was u s u a l l y t h e same o r d e r as t h e a b s o r b e r s t r e n g t h s . The t i m e s f o r i n t e g r a t i o n were v a r i e d f r o m p o i n t t o p o i n t t o g i v e a 67 c o n s t a n t s t a n d a r d d e v i a t i o n o f t h e t r a n s m i s s i o n o f a b o u t 1% f o r a l l d a t a p o i n t s . The sum o f t h e s t a n d a r d d e v i a -t i o n s f o r t h e i n d i v i d u a l l i n e s a r e g i v e n i n t h e t a b l e w i t h t h e r e s u l t s . The DPSD was c h e c k e d o v e r a r a n g e o f 50 Hz. t o 10MHz. and o v e r i n t e g r a t i o n t i m e s o f 1 s e c . t o 120 m i n . P i l e - u p o f p u l s e s l i m i t e d t h e u p p e r f r e q u e n c y l i m i t and PM. d a r k c u r r e n t l i m i t e d t h e l o w e r e n d . The s i g n a l m e a s u r e d i n c r e a s e d l i n e a r l y w i t h t i m e f o r as l o n g as t h e s o u r c e r e m a i n e d c o n s t a n t and t h e r e a s o n t h e l i m i t o f two h o u r s was i m p o s e d was t h a t t h e l o n g e s t t i m e u s e d i n t h e e x p e r i m e n t was 30 m i n . T h e r e d o e s n o t a p p e a r t o be any u p p e r l i m i t on t h e i n t e g r a t i o n t i m e due t o t h e DPSD. 3.4 REPRODUCIBILITY A minimum o f two i n d e p e n d e n t r u n s t a k e n on d i f f e r e n t d a y s w i t h d i f f e r e n t o p t i c a l a l i g n m e n t s were a v e r a g e d f o r any g i v e n l i n e . I t was f o u n d t h a t i f t h e i n t e g r a t i o n t i m e s were i n c r e a s e d t h e r e p r o d u c i b i l i t y o f t h e r e s u l t s w o u l d i m p r o v e u n t i l t h e e r r o r p e r p o i n t was a b o u t 1% o r l e s s and i f t h e i n t e g r a t i o n t i m e were i n -c r e a s e d f u r t h e r t h e r e p r o d u c i b i l i t y d i d n o t i m p r o v e b e y o n d t h i s l e v e l . The i n t e g r a t i o n t i m e was a d j u s t e d t o g i v e a maximum a c c u r a c y ( ^ 1 % ) f o r e a c h i n d i v i d u a l l i n e . 68 The a c c u r a c y a t t a i n e d by i n c r e a s i n g t h e i n t e g r a -t i o n t i m e was l i m i t e d by v a r i a t i o n s i n s o u r c e s t r e n g t h . T h e s e v a r i a t i o n s had two e f f e c t s on t h e r e a d i n g t a k e n . F i r s t t h e v a r i a t i o n s w i t h p e r i o d s much l e s s t h e n t h e m e a s u r i n g p e r i o d d i d n o t a f f e c t t h e mean b u t d i d i n c r e a s e t h e v a r i a n c e o f t h e i n d i v i d u a l r e a d i n g s . S e c o n d l y t h e r e w e re v a r i a t i o n s i n t h e mean o f t h e s o u r c e s t r e n g t h w h i c h o c c u r r e d a t random w i t h a p e r i o d o f t h e o r d e r o f one h o u r o r more. The mean w o u l d s h i f t by a p p r o x i m a t e l y 1 t o 5% i n a s t e p f u n c t i o n and t h e n m a i n t a i n t h e new mean. Due t o t h e s e jumps i n l e v e l i t was n o t p o s s i b l e t o e x t e n d t h e i n t e g r a t i n g t i m e p e r r e a d i n g l o n g e r t h e n a b o u t 5 m i n . I t must be s t r e s s e d t h a t when t h e t r a n s m i s s i o n was m e a s u r e d b e f o r e and a f t e r t h e 'jump' t h e r e s u l t was c o n s t a n t . T h a t i s , t h e p e r c e n t a g e t r a n s m i s s i o n was c o n s t a n t w i t h r e s p e c t t o a c h a n g e i n t h e s o u r c e s t r e n g t h . 3.5 'BEST F I T ' The e x p e r i m e n t a l l y m e a s u r e d t r a n s m i s s i o n a r e n o t e x a c t v a l u e s b u t r a t h e r a r e e s t i m a t e s o f t h e t r u e v a l u e w i t h an e x p e r i m e n t a l v a r i a n c e O e . The m a g n i t u d e o f t h e v a r i a n c e i s d e t e r m i n e d f r o m t h e r e p r o d u c i b i l i t y o f t h e d a t a . A s e c o n d v a r i a n c e U c i s t h e mean s q u a r e d e v i a -t i o n b e t w e e n t h e e x p e r i m e n t a l l y m e a s u r e d mean and t h e 69 theoretically calculated value, i.e. Cfc =± £<T. -T. )' where T„. is the experimentally measured mean of the ' transmission at a magnetic field designated by i Tc. is the calculated transmission for the same ' field J is the total number of data points taken If the theoretical curve is the true transmission curve then TE. =<T„.> where <C\ ^  means expectation value, which means that Oc is an estimate of the true experimental variance. Thus if theory agrees with experiment the values of C£ and ^ are both estimates of the same variance. To test the equality of variances we use Snedecor's '3' distribu-tion with the degrees of freedom for both 6C andC^ choosen equal to J . 3* is defined by at or rewriting Ol we have 70 i. - _ f _ where J p F » I V T , / T h u s f o r any g i v e n c u r v e t h e v a l u e s o f t h e param-e t e r s ( k 0J? , a e t c . ) a r e a d j u s t e d t o g i v e a minimum v a l u e t o F ( l e a s t s q u a r e s ) w h i c h g i v e s a minimum v a l u e t o 9^ . I f *y 3% ( "3v i s t h e v a l u e o f *9T w h i c h one w o u l d e x p e c t t o e x c e e d by c h a n c e a l o n e 5% o f t h e t i m e ) t h e n i t i s a s s u m e d t h a t t h e d i f f e r e n c e b e t w e e n e x p e r i m e n t and t h e o r y i s due t o e x p e r i m e n t a l v a r i a n c e s and a ' f i t ' i s a s s u m e d . 3.6 UNIQUENESS OF F I T I t was f o u n d t h a t t h e e f f e c t s o f t h e v a r i o u s p a r a m e t e r s w ere n e a r l y i n d e p e n d e n t . As an e x a m p l e t h e e f f e c t s o f c h a n g i n g k0J? and a a r e shown i n F i g s . 3-11 and 3-12. Thus i t i s s e e n t h a t i n c r e a s i n g k>/ moves t h e maximas and m i n i -mas o f t h e c u r v e s T0 and T T t o h i g h e r f i e l d w i t h v e r y l i t t l e c h a n g e i n t h e d e p t h o f t h e minimum w h i l e c3 d e t e r m i n e s t h e d e p t h o f t h e m i n i m a s . D u r i n g t h e f i t t i n g p r o c e d u r e i t was f o u n d t h a t t h e e f f e c t s 71 o f a l l t h e p a r a m e t e r s c o u l d be s e p a r a t e d and a u n i q u e s e t o b t a i n e d w i t h i n t h e e r r o r s q u o t e d f o r them. In o r d e r t o remove any b i a s o f t h e a n a l y s t i n t h e c h o i c e o f t h e p a r a m e t e r s t h e f i t t i n g p r o c e d u r e was c o m p l e t e d b e f o r e t h e r e s u l t s f r o m t h e l i t e r a t u r e were c o n s i d e r e d . 3.7 ERROR L I M I T S The e r r o r l i m i t s f o r a p a r a m e t e r a r e d e t e r m i n e d by c h a n g i n g t h e s e t o f p a r a m e t e r s and n o t i n g t h e e f f e c t on F. The e r r o r l i m i t s f o r any g i v e n p a r a m e t e r a r e s e t a t t h e amount t h e p a r a m e t e r must be c h a n g e d t o g i v e a value o f F e q u a l t o t w i c e Fm,n . T h e s e l i m i t s must be c o n s i d e r e d maximum p o s s i b l e e r r o r s ( s e e s e c t i o n 3 . 2 - 4 ) . Why t h i s i s t h e c a s e c a n be s e e n f r o m t h e g r a p h s h o w i n g t h e e f f e c t o f c h a n g i n g <3 . Note t h a t t h e m a j o r e f f e c t i s t o c h a n g e t h e d e p t h o f t h e m i n i m a so p o i n t s n e a r t h e m i n i m a h a v e t h e s t r o n g e s t e f f e c t s on F . The c r i t e r i o n f o r a ' f i t ' i s t h e v a l u e o f F w h i c h w e i g h t s a l l p o i n t s e q u a l l y . Thus t h e d i f f e r e n c e n e a r t h e mini m a must become much l a r g e r t h a n e x p e r i m e n t a l e r r o r s f o r t h e s e p o i n t s i n o r d e r t o i n c r e a s e F t o t w i c e F^.n . 72 The p r e c i s i o n t o w h i c h 'a' and ' k o l ' c a n be m e a s u r e d i s a f u n c t i o n o f t h e m a g n i t u d e o f t h e s e p a r a m e t e r s . F o r e x a m p l e , i f t h i s e x p e r i m e n t were r e p e a t e d w i t h a l o w e r p r e s s u r e i n t h e a b s o r b e r t h a n t h e v a l u e o f 'a' w o u l d de-c r e a s e . T h e o r e t i c a l c a l c u l a t i o n s i n d i c a t e t h a t f o r t h e s e s m a l l e r v a l u e s o f 'a' t h e a c c u r a c y o b t a i n e d w o u l d i n c r e a s e . The t h e o r y a l s o p r e d i c t s t h a t t h e u n c e r t a i n t y i n t h e v a l u e o f ' a ' i s a minimum f o r l a r g e v a l u e s o f k 0 l . In g e n e r a l t h e m o st a c c u r a t e v a l u e s o f a l l t h e p a r a m e t e r s u s e d t o f i t t h e t h e o r e t i c a l c u r v e s t o t h e e x p e r i -m e n t a l r e s u l t s a r e o b t a i n e d f o r s t r o n g l y a b s o r b i n g l i n e s ( k 0 l > 1 0 ) . The r e a s o n f o r t h i s c a n be s e e n f r o m F i g . 2-3, 4 w h i c h shows s i x l i n e s w i t h d i f f e r e n t k Q l s . The l a r g e r t h e v a l u e o f k 0 l t h e more a b s o r p t i o n a t h i g h e r f i e l d s and t h e more maxima and m i n i m a o b s e r v e d i n T 0 . The p o r t i o n o f t h e s e c u r v e s w h i c h c h a n g e s most r a p i d l y w i t h c h a n g i n g p a r a m e t e r s i s i n t h e r e g i o n o f t h e e x t r e m e s , t h u s i n c r e a s e s i n k 0 l i n c r e a s e t h e s e n s i t i v i t y o f t h e m e t h o d . leaf 73 ommitted in page numbering 74 3.8 STIMULATED EMISSION The e f f e c t o f s t i m u l a t e d e m i s s i o n on t h i s e x p e r i -ment c a n be s e e n i n e q u a t i o n ( 2 - 5 7 ) . S t i m u l a t e d e m i s s i o n f r o m an u p p e r s t a t e 2 t o a l o w e r s t a t e 1 c a n be i g n o r e d c o m p a r e d t o t h e a b s o r p t i o n f r o m t h e l o w e r s t a t e 1. As d i s c u s s e d i n s e c t i o n 2.7 s t i m u l a t e d e m i s s i o n i n v o l v e s t h e d e - e x c i t a t i o n o f an atom f r o m an u p p e r s t a t e 2 t o a l o w e r s t a t e 1 i n r e s p o n s e t o a p a s s i n g p h o t o n w h i l e t h e t r a n s i t i o n p r o b a b i l i t y w h i c h t h i s e x p e r i m e n t i s a t t e m p t -i n g t o m e a s u r e i n v o l v e s t h e a b s o r p t i o n o f t h e p a s s i n g p h o t o n by an atom i n s t a t e 1 w h i c h r a i s e s t h e e n e r g y o f t h i s atom t o s t a t e 2. I f t h e p o p u l a t i o n o f t h e u p p e r s t a t e 2 i s much l e s s t h e n t h e p o p u l a t i o n o f t h e l o w e r s t a t e t h e n e f f e c t s o f t h e s t i m u l a t e d e m i s s i o n c a n be i g n o r e d . T h i s c a n be s e e n f r o m e q u a t i o n ( 2 - 5 7 ) s i n c e t h e n N, « N x and From t h e m e a s u r e d v a l u e s o f k j u s i n g e q u a t i o n ( 2 - 3 6 ) ) and 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 f r o m W i e s e ( 5 ) t h e p o p u l a -" -3 t i o n d e n s i t y o f t h e l o w e r s t a t e N, i s - 1 0 C m . From t h e o b s e r v e d i n t e n s i t y , p h o t o m u l t i p i i e r e f f i c i e n c y , e x p e r i -m e n t a l g e o m e t r y and a g a i n 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 o f 75 W i e s e t h e p o p u l a t i o n d e n s i t y o f t h e u p p e r s t a t e N 2 i s e s t i m a t e d as ~10 C m . Thus 1 - A , : ~ 1 0 3.9 MAGNETIC F I E L D E F F E C T S ON THE DISCHARGE In o r d e r t o o b t a i n a c o n s i s t e n t f i t b e t w e e n t h e e x p e r i m e n t a l and t h e o r e t i c a l r e s u l t s i f was n e c e s s a r y t o make a f u n c t i o n o f B ( m a g n e t i c f i e l d ) . I t was f o u n d t h a t t h e d e p e n d e n c e o f k 0 l on m a g n e t i c f i e l d was o f t h e f o r m = k 0 ^ ( 1 + ocjBl + o c g B + ° C 3 IBI ) ~1.5x10~* , B (Gauss) , o c 2 = 0 = oc 3 A s i m i l a r d e p e n d e n c e was i m p o s e d on & , b u t t h e ' b e s t ' v a l u e f o r o c a i s z e r o . The same d e p e n d e n c e o f k j was f o u n d f o r a l l l i n e s w i t h a common l o w e r s t a t e . A p o s s i b l e e x p l a n a t i o n i s t h a t t h e number d e n s i t y i n t h e g r o u n d s t a t e ( 3P£ ) i s a f u n c t i o n o f t h e m a g n e t i c f i e l d . The v a l u e o f o c ^ was d e t e r m i n e d e x p e r i m e n t a l l y u s i n g t h e o r e s u l t s o f t h e s t r o n g e s t 1 ine ( X s8115A ) The same v a l u e was t h e n u s e d f o r a l l t h e r e s t o f t h e l i n e s w i t h t h e same l o w e r l e v e l t o o b t a i n a c o n s i s t e n t s e t o f p a r a m e t e r s . 3.10 ADDITIONAL PARAMETERS The t e m p e r a t u r e o f t h e atoms i n t h e t u b e i s assumed t o be t h e same as t h e i n s i d e o f t h e d i s c h a r g e t u b e . In 76 o r d e r t o e s t i m a t e t h e t e m p e r a t u r e t h e t o t a l power d i s s i p a t e d i n t h e d i s c h a r g e was c a l c u l a t e d f r o m t h e v o l t a g e d r o p a c r o s s t h e t u b e and t h e c u r r e n t p a s s i n g t h r o u g h . A s s u m i n g t h a t t h i s power was c o n d u c t e d t h r o u g h t h e w a l l s o f t h e t u b e , we know t h e t e m p e r a t u r e d i f f e r e n t i a l a c r o s s t h e g l a s s t u b e . From t h i s d i f f e r e n t i a l ( ^ 2 . 0 ° K) and k n o w i n g t h e t e m p e r a t u r e o f t h e o u t s i d e o f t h e t u b e ( ^ 2 9 0 ° K) we a r e a b l e t o e s t i m a t e t h e maximum p o s s i b l e t e m p e r a t u r e o f t h e i n s i d e w a l l s ( ^ 3 1 0 ° K ) . T h i s e s t i m a t e i s an u p p e r l i m i t s i n c e we h a v e a s s u m e d t h a t a l l t h e power i s d i s s i p a t e d t h r o u g h t h e w a l l s and h a v e i g n o r e d t h e power d i s s i p a t e d by t h e e l e c t r o d e s . T he a c t u a l v a l u e o f t h e t e m p e r a t u r e was a l s o t r e a t e d as a v a r i a b l e i n f i t t i n g t h e t h e o r y t o t h e m e a s u r e d t r a n s m i s s i o n c u r v e s . V a r y i n g t h e t e m p e r a t u r e d i d i m p r o v e t h e v a l u e o f F m m b u t f o r v a r i a t i o n s o f t e m p e r a t u r e b e t w e e n room t e m p e r a t u r e and t h e maximum e s t i m a t e as g i v e n a b o v e t h e r e w e r e i n s i g n i f i c a n t e f f e c t s on t h e b e s t v a l u e s f o r & and t h e r e l a t i v e v a l u e s o f k e l. Thu s t h e t e m p e r a t u r e a ssumed ( 3 0 0 ° K ) , as l o n g as i t i s w i t h i n r e a s o n a b l e l i m i t s , w i l l h a v e l i t t l e e f f e c t on t h e a t o m i c p a r a m e t e r s m e a s u r e d . In d e r i v i n g t h e s o u r c e s t r e n g t h ( E q . ( 2 - 8 5 ) we h a v e a s s u m e d t h a t t h e r e i s a u n i f o r m p l a s m a i n t h e s o u r c e t u b e ( F i g . 3 - 2 ) . The a r g o n atoms w h i c h c o l l i d e w i t h end 77 windows w i l l t e n d t o be d e - e x c i t e d t h u s p r o d u c i n g a t h i n l a y e r o f atoms w h i c h a r e a l l i n t h e l o w e r s t a t e . S i n c e t h e y a r e i n t h e g r o u n d s t a t e t h e y w i l l n o t be a b l e t o e m i t any r a d i a t i o n b u t t h e y w i l l s t i l l be c a p a b l e o f a b s o r b i n g r a d i a t i o n . The e m i t t e d r a d i a t i o n w i l l t h e n n o t be as g i v e n i n e q u a t i o n ( 2 - 8 5 ) b u t r a t h e r E_V' =EI e k ^ = d - eKL) e k ^ ' 2 2 w h e r e J^' i s t h e t h i c k n e s s o f t h i s n o n r a d i a t i n g l a y e r o f g a s . I t i s a s s u m e d t h a t t h e t h e r m a l m o t i o n and number d e n s i t y o f t h i s t h i n l a y e r i s t h e same as i n t h e p l a s m a w h i c h i s e m i t t i n g E V , so t h a t t h e a b s o r p t i o n c o e f f i c i e n t w i l l be t h e same, i . e . , k^  = k v . The p a r a m e t e r | was v a r i e d i n o r d e r t o p r o d u c e a g r e e m e n t b e t w e e n t h e o r y and e x p e r i m e n t and i n a l l c a s e s t h e ' b e s t ' v a l u e was z e r o . T h e r e f o r e t h e l a y e r o f n o n e m i t t i n g gas c a n be i g n o r e d . CHAPTER 4 RESULTS 4.1 RESULTS - R E L A T I V E TRANSITION P R O B A B I L I T I E S The r e s u l t s o b t a i n e d f o r t h e r e l 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 a r e g i v e n i n t h e t a b l e b e l o w . The a r e com-p a r e d w i t h t h o s e o f W i e s e (5) b e c a u s e h i s v a l u e s were o b t a i n e d f r o m a s u r v e y o f t h e l i t e r a t u r e . W i e s e c l a i m s t h e u n c e r t a i n t y i n h i s a b s o l u t e v a l u e s d o e s n o t e x c e e d 2 0 % and t h e u n c e r t a i n t y i n h i s r e l a t i v e v a l u e s d o e s n o t e x c e e d 10%. N o t e t h a t f o r a l l l i n e s t h e r e l 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 a g r e e w i t h W i e s e t o w i t h i n t h e a c c u r a c y o f 10% w h i c h he c l a i m s . The v a l u e o f S n e d e c o r 1 s 9 ^ i s i n a l l c a s e s l e s s t h a n t h e v a l u e f o r a 9 5 % c o n f i d e n c e l i m i t . T h u s t h e o b s e r v e d v a r i a n c e s b e t w e e n e x p e r i m e n t and t h e o r y a r e w i t h i n t h e l i m i t s p r e d i c t e d by o b s e r v e d e x p e r i m e n t a l u n c e r t a i n t i e s . 78 T a b l e 1 RE L A T I V E TRANSITION P R O B A B I L I T I E S [ A ( r e l ) ] WAVELENGTH A ° k 0 l (EXP) A ( r e l ) (EXP) A ( r e l ) W i e s e * ± 1 0 % No. o f D a t a P o i n t s a E *s 8115 29.0 ± 1% 1 .00 1.00 66 .006 .006 .6 1.5 8014 5.25 ± 5% 0.262 ± 2% 0.264 42 .006 .007 1.0 1.7 7635 12.52 ± 2% .725 ± 1% .750 62 .002 .002 .5 1.5 7147 .160 ± 10% .0188 ± 10% .0179 10 .001 .001 .06 3.0 7067 1 .52 ± 5% .111 ± 3% .108 32 .001 .003 .13 1.8 6965 1 .52 ± 5% .193 ± 3% .184 32 .001 .002 .20 1 .8 * I S . R e f e r e n c e ( 5 ) W i e s e . See s e c t i o n 3.5. 80 4.2 RESULTS - LORENTZIAN L I N E WIDTH The r e s u l t s o b t a i n e d f r o m t h i s e x p e r i m e n t f o r t h e L o r e n t z i a n c o m p o n e n t o f t h e l i n e s h a p e a r e p r e s e n t e d i n T a b l e 2. From t h e v a l u e o f t h e V o i g t 'a' ( E q . ( 2 - 3 0 ) ) and t h e D o p p l e r h a l f - w i d t h D ( E q . ( 2 - 9 6 ) ) t h e e x p e r i m e n t a l L o r e n t z i a n h a l f - w i d t h i s c a l c u l a t e d and e n t e r e d i n c o l u m n t h r e e . The n a t u r a l h a l f - w i d t h i s t h e n c a l c u l a t e d u s i n g e q u a t i o n ( 2 - 9 7 ) and W i e s e ' s v a l u e s ( 5 ) f o r 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 o r l i f e t i m e s . S u b t r a c t i n g t h e n a t u r a l w i d t h f r o m t h e o b s e r v e d w i d t h we a r e l e f t w i t h t h e e x p e r i m e n t a l l y m e a s u r e d v a l u e o f t h e Van d e r Wa a l s b r o a d e n i n g . S i n c e t h e s o u r c e and t h e a b s o r b e r a r e a t t h e same p r e s s u r e and t e m p e r a t u r e t h e r e w i l l be no s h i f t o f t h e s o u r c e and a b s o r p -t i o n l i n e s w i t h r e s p e c t t o e a c h o t h e r . Column s i x g i v e s t h e e s t i m a t e o f a l o w e r l i m i t o f t h e Van d e r W a a l s b r o a d e n i n g as g i v e n i n e q u a t i o n ( 2 - 9 9 ) . The e x p e r i m e n t a l v a l u e s a r e a b o v e t h i s l o w e r l i m i t f o r a l l t h e l i n e s m e a s u r e d . I f t h e b r o a d e n i n g i s p r o p o r t i o n a l t o t h e p o l a r i z -a b i l i t y o f t h e u p p e r s t a t e ( 1 ) , as i s e x p e c t e d f o r Van d e r W a a l s b r o a d e n i n g , t h e n t h e r a t i o o f t h e b r o a d e n i n g f o r two l e v e l s w i t h a common g r o u n d s t a t e i s g i v e n by e q u a t i o n ( 2 - 1 0 1 ) ( T a b l e 3 ) . T a b l e 2 LORENTZIAN HALF-WIDTHS WAVELENGTH VOIGT a ( l O ^ c m '1 ) ( 1 0 ~ 3 c n f ] ) ± 2 0 % ( l O ^ c m '1 ) THEORY* ( l O ^ c m ' 1 ) UPPER LEVEL 8115 .072 ±.01 2.1 ± .3 .196 1.9 ± .3 .66 \ 8014 .065 ±.01 1 .9 ± .3 .184 1.8 ± .3 .67 \ 7635 .072 ±.01 2.2 ± .3 .202 2.0 ± .3 .70 \ 7147 .08 ±.05 2.6 ± 1 . 5 .191 2.5 ± 1 . 5 1.0 \ 7067 .10 ±.0 2 3.3 ± .7 .204 3.1 ± .7 1.0 \ 6965 .10 ±.02 3.4 ± .7 .212 3.2 ± .7 1.0 N a t u r a l l i n e w i d t h u s i n g Eq. 2-83 and W i e s e ' s v a l u e s o f A and IT + V a n d e r W a a l s B r o a d e n i n g ( 2 ) , Eq. 2-85. 82 T a b l e 3 R E L A T I V E LORENTZIAN HALF-WIDTHS AVERAGE LORENTZIAN d o " 3 ™ " 1 ) RATIO 1 = 2 1 =1 1 =2 EXP 3 . 0 ± . 6 1 .9±.2 .63+.20 THEORY 1 .0 .68 * .68 * E q u a t i o n 2-86. T h u s t h e o b s e r v e d L o r e n t z i a n c o m p o n e n t s a r e a b o v e t h e l o w e r l i m i t c a l c u l a t e d by G r i e m ( 2 ) and t h e r a t i o o f t h e L o r e n t z i a n h a l f - w i d t h s f o r l i n e s w i t h a common g r o u n d s t a t e a g r e e w i t h t h a t r a t i o e x p e c t e d i f t h e b r o a d e n i n g i s p r o p o r t i o n a l t o t h e p o l a r i z a b i 1 i t y o f t h e u p p e r s t a t e ( 1 ) , t h a t i s , Van d e r W a a l s b r o a d e n i n g . CHAPTER 5 COMPARISONS WITH OTHER METHODS 5.1 WALL S T A B I L I Z E D ARCS AND SHOCK TUBES B o t h t h e s e m e t h o d s i n v o l v e e m i s s i o n e x p e r i m e n t s ( 9 , 1 0 ) and d e p e n d on l o c a l t h e r m o d y n a m i c e q u i l i b r i u m ( L T E ) f o r t h e i r v a l u e s t o be v a l i d . A h i g h d i s p e r s i o n i n s t r u m e n t must be u s e d i n o r d e r t o f i n d t h e i n t e g r a t e d l i n e i n t e n s i t y and t h e l i n e s h a p e must be d e t e r m i n e d a c c u r a t e l y so t h e i n t e n s i t y i n t h e w i n g s c a n be c a l c u l a t e d . The p r e s e n t m e t h o d d o e s n o t d e p e n d on L T E and i n d e e d t h e g l o w d i s c h a r g e i s n o t i n L T E . A l l t h a t o u r meth o d r e q u i r e s i s t h a t t h e p o p u l a t i o n l e v e l s be i n d y n a m i c e q u i l i b r i u m . S i n c e s o u r c e and a b s o r b e r a r e i d e n t i c a l d i s c h a r g e s t h e y a r e b o t h d e t e r m i n e d by t h e same p a r a m e t e r s ( ke > a ) and s e l f - a b s o r p t i o n i s e a s i l y d e a l t w i t h ( s e c t i o n 2 . 8 - 2 ) . I f a s e r i e s o f p r e s s u r e s a r e u s e d i n t h e a b s o r b e r o n c e t h e s o u r c e c o n d i t i o n s have b e e n d e t e r -m i n e d t h e n t h e l i m i t i n t h e v a l u e o f t h e L o r e n t z i a n com-p o n e n t o f t h e l i n e as p r e s s u r e g o e s t o z e r o w i l l g i v e t h e a b s o l u t e v a l u e o f t h e l i f e t i m e s ( E q . 2-97) w i t h o u t h a v i n g 84 85 t o make any a b s o l u t e i n t e n s i t y m e a s u r e m e n t s . T he e x t r a p o -l a t i o n t o z e r o p r e s s u r e i s o n l y v a l i d i f d e - e x c i t a t i o n by c o l l i s i o n s w i t h t h e w a l l s i s n e g l i g i b l e c o m p a r e d t o t h e n a t u r a l l i f e t i m e . S i n c e n a t u r a l l i f e t i m e s a r e o f t h e - 8 o r d e r o f 10 s e c . and t h e atoms have an a v e r a g e v e l o c i t y 4 -1 o f 4 x 10 cm. s e c . w a l l c o l l i s i o n s c a n be i g n o r e d p r o -v i d e d t h e c h a r a c t e r i s t i c l e n g t h i n t h e t u b e s .>> 4 x 1 0 " ^ cm. 5.2 HOOK METHOD The R o s c h d e s t w e n s k y 'hook' method ( 1 1 ) u s e s t h e r a t e o f c h a n g e o f t h e i n d e x o f r e f r a c t i o n as a f u n c t i o n o f w a v e l e n g t h ("^"^ ) i ' n o r d e r t o m e a s u r e r e l 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 . T h e ' h o o k s ' o c c u r a t w a v e l e n g t h s when t h e d e r i v a t i v e i s e q u a l t o z e r o and t h u s t h e y a r e s e p a r a t e d by a few l i n e w i d t h s . In o r d e r t o o b t a i n a r e l a t i o n s h i p b e t w e e n t h e s e p a r a t i o n o f t h e h o o k s and t h e t r a n s i t i o n p r o b a b i l i t y i t i s u s u a l l y a ssumed t h a t t h e hooks o c c u r f a r e n o u g h f r o m t h e l i n e c e n t e r t h a t t h e l i n e s h a p e c a n be as s u m e d t o be c o m p l e t e l y L o r e n t z i a n . Thus t h i s m ethod r e q u i r e s t h e u s e o f a h i g h d i s p e r s i o n i n s t r u m e n t t o m e a s u r e t h e s e p a r a t i o n b e t w e e n two h o o k s s e p a r a t e d by a few l i n e w i d t h s . T h i s s i n g l e m e a s u r e m e n t i s t h e n u s e d t o d e t e r m i n e t h e t r a n s i t i o n p r o b a b i l i t y . T h e r e i s no i n d i c a t i o n f r o m t h i s m e t h o d what t h e l i n e s h a p e i s and an a s s u m p t i o n a b o u t 86 t h e s h a p e must be made t o d e t e r m i n e t h e t r a n s i t i o n p r o b -a b i l i t y . I f t h e a b s o r b e r i s n o t o f u n i f o r m o p t i c a l t h i c k -n e s s w i t h r e s p e c t t o r a d i u s t h e n t h e method w i l l g i v e un-r e l i a b l e r e s u l t s . The m e t h o d u s e d i n t h e p r e s e n t work u s e s t h e e n t i r e t r a n s m i s s i o n v s . m a g n e t i c f i e l d c u r v e t o d e t e r m i n e k D and ' a ' . The i n d e p e n d e n c e o f t h e s e p a r a m e t e r s d e p e n d s on t h e f a c t t h a t e a c h p a r a m e t e r a f f e c t s d i f f e r e n t p a r t s o f t h e t r a n s m i s s i o n c u r v e i n d i f f e r e n t w a ys. I f o n l y two p o i n t s on t h e i n d e x o f r e f r a c t i o n c u r v e a r e t r e a t e d t h e n k 0 and ' a ' c a n n o t be e v a l u a t e d i n d e p e n d e n t l y . 5.3 ZEEMAN SCANNING Zeeman s c a n n i n g ( 1 2 ) has b e e n u s e d t o d e t e r m i n e l i n e s h a p e s . N o r m a l l y when Zeeman s c a n n i n g , o n l y one c i r -c u l a r p o l a r i z a t i o n i s u s e d b u t t h e m e a s u r e d t r a n s m i s s i o n w i l l s t i l l be o f t h e f o r m g i v e n i n e q u a t i o n ( 2 - 6 8 ) i f t h e s o u r c e a n d a b s o r b e r a r e n o t s h i f t e d w i t h r e s p e c t t o e a c h o t h e r . I n o r d e r t o c o m p a r e Zeeman s c a n n i n g t o t h e p r e s e n t m e t h o d a l l we n e e d do i s c ompare t h e r e l a t i v e amount o f i n f o r m a t i o n , i n t h e two m e a s u r e d c u r v e s T Q and T^ ( e q u a t i o n s ( 2 - 7 3 ) , ( 2 - 7 4 ) ) s i n c e t h e p r e s e n t method u s e s b o t h c u r v e s t o d e t e r m i n e k 0 and ' a ' . The t r a n s m i s s i o n m e a s u r e d w i t h t h e N i c o l s i n p o s i t i o n ( T 0 ) i s much more s e n s i t i v e t o c h a n g e s o f k c and 87 'a' t h e n t h e n o r m a l Zeeman s c a n n i n g ( T ^ ) ( p o l a r i z i n g N i c o l r e m o v e d ) . T h i s c a n be s e e n by c a l c u l a t i n g t h e two q u a n -t i t i e s 5 = N (k . r k.,) £<T«,-T„ ): 1 = 1 M(k„ r Kx) w h e r e T, i s t h e t r a n s m i s s i o n m e a s u r e d w i t h k 0 = k 0 ^ T->. i s t h e t r a n s m i s s i o n m e a s u r e d w i t h k 0 = k 0 g F0 and F T a r e o b t a i n e d by summing o v e r t h e c u r v e d , T 0 and T^ r e s p e c t i v e l y . A t y p i c a l v a l u e f o r t h e r a t i o o f t h e s e two c h a n g e s f o r ( k o - j - k o g ) c o m p a r a b l e t o t h e e x p e r i m e n t a l u n c e r t a i n t y i s wi t h w h e r e r r i i s t h e s t a n d a r d d e v i a t i o n o f t h e e x p e r i m e n t a l p o i n t s . I f t h e a b s o r p t i o n l i n e c e n t e r i s s h i f t e d w i t h r e s p e c t t o t h e c e n t e r o f t h e e m i s s i o n l i n e f r o m t h e s o u r c e t h e n t h e t r a n s m i s s i o n c u r v e s w i l l be m o d i f i e d . I f t h e m a g n i t u d e o f t h e s h i f t i s u s e d as a p a r a m e t e r t o f i t t h e o r y 88 t o e x p e r i m e n t t h e p r e s e n t method w i l l g i v e an e s t i m a t e o f t h e s h i f t b u t f r o m t h e symmetry r e l a t i o n s o f e q u a t i o n ( 2 - 7 3 ) i t c a n be s e e n t h a t t h e t r a n s m i s s i o n c u r v e s a r e i n d e p e n d e n t o f t h e d i r e c t i o n o f t h e s h i f t . T h e s e symmetry r e l a t i o n s e s s e n t i a l l y show t h a t i f an e m i s s i o n l i n e i s moved f r o m one s i d e o f a s y m m e t r i c a b s o r p t i o n l i n e t o t h e o t h e r t h e r e i s no c h a n g e i n t h e t r a n s m i s s i o n . The d i r e c -t i o n o f t h e s h i f t c a n be d e t e r m i n e d by t h e Zeeman s c a n n i n g t e c h n i q u e i f o n l y one o f t h e c i r c u l a r l y p o l a r i z e d c o m p o n e n t s a r e u s e d . W i t h o n l y one p o l a r i z a t i o n t h e a b s o r p t i o n l i n e i s no l o n g e r s y m m e t r i c a b o u t i t s p o s i t i o n a t z e r o f i e l d ; t h u s t h e d i r e c t i o n i s g i v e n f o r t h e s h i f t . The Zeeman s c a n n i n g t e c h n i q u e s u s e d t o d a t e h a v e b e e n l i m i t e d t o n o r m a l Zeeman s p l i t t i n g i n o r d e r t o s i m p l i f y t h e c o m p u t a t i o n s . However t h e method c o u l d be e x t e n d e d t o ' a n o m a l o u s ' s p l i t t i n g s s o t h i s r e s t r i c t i o n t o n o r m a l s p l i t t i n g s i s n o t an i n t r i n s i c d i s a d v a n t a g e o f t h e Zeeman s c a n n i n g t e c h n i q u e . I t was f o u n d t h a t i t was n o t p o s s i b l e t o o b t a i n a m u t u a l l y i n d e p e n d e n t s e t o f p a r a m e t e r s i f t h e Zeeman s c a n n i n g c u r v e a l o n e w ere u s e d ( t r a n s m i s s i o n c u r v e w i t h p o l a r i z i n g N i c o l r e m o v e d ) . T h a t i s , we were a b l e t o o b t a i n more t h a n one s e t o f v a l u e s o f k 0 , a and oc ^ w h i c h w o u l d f i t Ty a l o n e b u t when b o t h T Q and T^ w e r e u s e d t h e r e was o n l y one s e t o f p a r a m e t e r s p o s s i b l e . 89 Thus t h e method we h a v e u s e d i s much more s e n i -t i v e t h a n t h e s t a n d a r d Zeeman s c a n n i n g t e c h n i q u e ; g i v e s b e t t e r , i n d e p e n d e n t v a l u e s f o r t h e r e l 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 and L o r e n t z i a n l i n e w i d t h s . In a d d i t i o n t h e p r e s e n t m e t h o d g i v e s a c c u r a t e r e s u l t s e v e n when t h e a b s o r p t i o n l i n e w i d t h i s c o m p a r a b l e t o e m i s s i o n l i n e w i d t h . CHAPTER 6 CONCLUDING DISCUSSION The u s e o f t h e d i s p e r s i o n i n t h e v i c i n i t y o f a b s o r p t i o n l i n e s has b e e n e s t a b l i s h e d as a p o w e r f u l s p e c -t r o s c o p i c t o o l . T h i s m e t h o d was u s e d t o d e t e r m i n e t h e r e l 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 b e t w e e n t r a n s i t i o n s w i t h a common l o w e r l e v e l . A l t h o u g h i t g i v e s t h e most a c c u r a t e v a l u e s f o r s t r o n g l y a b s o r b i n g l i n e s i t a l s o g i v e s v a l u e s as a c c u r a t e as any o t h e r m e t h o d f o r t h e w e a k e r l i n e s . In a d d i t i o n t o t h e r e l 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 t h i s m e t h o d g i v e s i n f o r m a t i o n a b o u t t h e l i n e s h a p e . U s i n g t h e p r e s e n t method we c a n d e t e r m i n e t h e l i n e s h a p e more a c c u r a t e l y and w i t h l e s s t e c h n i c a l d i f f i c u l t i e s t h a n w i t h any o t h e r m e t h o d . F o r e x a m p l e i n t h e p r e s e n t work t h e l i n e s h a p e o f t h e 8115 A 0 l i n e was m e a s u r e d ( D o p p l e r w i d t h .024 cm" 1; L o r e n t z i a n w i d t h .002 c m " 1 ) . The m e a s u r e -ment o f t h e D o p p l e r w i d t h r e q u i r e d a s i m p l e t e m p e r a t u r e m e a s u r e m e n t and t h e L o r e n t z i a n c o m p o n e n t was d e t e r m i n e d by t h e d e p t h s o f t h e maxima and m i n i m a i n t h e t r a n s m i s s i o n c u r v e ( F i g . 2 - 3 ) . The m e a s u r e m e n t o f t h i s L o r e n t z i a n 90 91 c o m p o n e n t w i t h a c o n v e n t i o n a l method w o u l d r e q u i r e t h e u s e o f a h i g h d i s p e r s i o n i n s t r u m e n t s u c h as a F a b r y - P e r o t i n t e r f e r o m e t e r , w i t h i t s accompanying l o s s o f s i g n a l s t r e n g t h ( 1 4 ) , and d i f f i c u l t i e s i n a l i g n m e n t w i t h i n f r a r e d r a d i a t i o n . A F a b r y - P e r o t i n t e r f e r o m e t e r w i t h a h i g h f i n e s s e w o u l d r e q u i r e a p l a t e s e p a r a t i o n o f a p p r o x i m a t e l y 60 cm t o g i v e e q u i v a l e n t r e s o l u t i o n . C a l c u l a t i o n s i n d i c a t e t h a t t h e a c c u r a c y o f t h e p r e s e n t m e t h o d i n c r e a s e s as t h e p e r c e n t a g e o f t h e L o r e n t z i a n c o m p o n e n t d e c r e a s e s p r o v i d e d t h e s t r e n g t h o f t h e l i n e ( k Ql) r e m a i n s c o n s t a n t . F o r low p r e s s u r e s i n t h e a b s o r b e r ( ~ . l t o r r ) t h e L o r e n t z i a n c o m p o n e n t i s d e t e r m i n e d by t h e n a t u r a l l i n e - 3 -1 w i d t h ("-.2x10 cm ) w h i c h c a n be m e a s u r e d a c c u r a t e l y ( ± 1 0 % ) p r o v i d e d t h e l i n e i s s t r o n g ( k o l £ 2 0 ) . T h e r e a r e o t h e r a d v a n t a g e s o f t h e p r e s e n t m e t h o d . No e x p e n s i v e h i g h d i s p e r s i o n d e v i c e i s r e q u i r e d b e c a u s e t h e t r a n s m i s s i o n o f t h e e n t i r e s o u r c e l i n e i s . m e a s u r e d . The m o n o c h r o m a t o r n e e d o n l y e x c l u d e a l l o t h e r l i n e s r a t h e r t h a n r e s o l v e t h e l i n e b e i n g i n v e s t i g a t e d . The a p p a r a t u s d o e s n o t r e q u i r e c r i t i c a l a l i g n m e n t and t h e d a t a i s e a s i l y o b t a i n e d . S i n c e t h e r e s o l v i n g power o f t h i s m ethod d e p e n d s on t h e Zeeman e f f e c t i n t h e a b s o r b e r any p a r t o f t h e e l e c t r o -m a g n e t i c s p e c t r u m c a n be i n v e s t i g a t e d p r o v i d e d t h e r e a r e p o l a r i z e r s and d e t e c t o r s a v a i l a b l e . As t h e Zeeman s p l i t t i n g 92 i n c r e a s e s t h e a b s o r p t i o n o f b o t h c i r c u l a r p o l a r i z a t i o n s w i l l d e c r e a s e l e a v i n g t h e r o t a t i o n o f t h e p l a n e o f p o l a r i -z a t i o n as t h e m a j o r e f f e c t o f t h e p l a s m a . Thus t h e w i n g s o f t h e i n d e x o f r e f r a c t i o n c u r v e a r e i n v e s t i g a t e d a t a m a g n e t i c f i e l d w h e r e t h e t r a n s m i t t e d i n t e n s i t y i s a maximum. The r e l 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 w h i c h w e r e m e a s u r e d by t h i s m ethod a g r e e w i t h t h e b e s t e s t i m a t e s a v a i l a b l e i n t h e l i t e r a t u r e ( 5 ) and t h e v a l u e s o f t h e L o r e n t z i a n components a r e c o n s i s t e n t w i t h b r o a d e n i n g o f t h e Van d e r W a a l s t y p e ( 2 ) . T h e v a r i a n c e b e t w e e n t h e e x p e r i m e n t a l a n d t h e o -r e t i c a l t r a n s m i s s i o n c u r v e s i s w i t h i n l i m i t s s e t by e x p e r i -m e n t a l u n c e r t a i n t i e s . F u r t h e r m o r e t h e s e t o f p a r a m e t e r s w h i c h a r e a d j u s t e d t o g i v e a ' f i t ' (k oj0,a, e t c . ) a r e d e t e r m i n e d i n d e p e n d e n t l y o f e a c h o t h e r . F o r e x a m p l e , when t h i s m e t h o d was a p p l i e d by S e k a (8) w i t h ' a ' = 0, t h e v a l u e s o f t h e r e l 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 o b t a i n e d w e r e t h e same as t h o s e o b t a i n e d by f i t t i n g t h e e x p e r i m e n t a l c u r v e w i t h t h e c o r r e c t v a l u e o f ' a ' . T h i s i s b e c a u s e t h e e x p e r i m e n t a l c u r v e s were f i t t e d s o t h e maxima and m i n i m a ( F i g . 2-3) o c c u r r e d a t t h e c o r r e c t m a g n e t i c f i e l d s . The o n l y e f f e c t o f u s i n g an i n c o r r e c t 'a' was t h a t t h e d e p t h o f t h e minimum p r e d i c t e d by t h e o r y d i d n o t a g r e e w i t h t h e o b s e r v e d v a l u e . 93 Now t h a t t h e a d v a n t a g e s o f t h e me t h o d h a v e b e e n e s t a b l i s h e d , i t i s p o s s i b l e t o u s e i t t o m e a s u r e t r a n s i t i o n p r o b a b i l i t i e s and l i n e s h a p e s t o a c c u r a c i e s w h i c h w o u l d be v e r y d i f f i c u l t , i f n o t i m p o s s i b l e , t o m e a s u r e i n any o t h e r m a n n e r . I f t h e p r e s s u r e i n t h e a b s o r b e r i s c h a n g e d w h i l e t h e s o u r c e i s l e f t c o n s t a n t t h e n e a c h o f t h e a b s o r b e r p r e s s u r e s w i l l y i e l d a new m e a s u r e m e n t o f t h e r e l 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 w h i c h s h o u l d be i n d e p e n d e n t o f t h e p r e s s u r e u s e d . P l o t t i n g t h e v a r i a t i o n o f t h e L o r e n t z i a n c o m p o n e n t w i t h p r e s s u r e w i l l g i v e two p a r a m e t e r s . The e x t r a p o l a t i o n o f t h e p l o t t o z e r o p r e s s u r e w i l l g i v e t h e a b s o l u t e v a l u e o f t h e s t a t e l i f e t i m e s and t h e s l o p e o f t h e p l o t w i l l g i v e i n f o r m a t i o n on t h e b r o a d e n i n g m e c h a n i s m s i n v o l v e d ( 1 2 ) . T h i s method c a n t h u s be a p p l i e d t o many p r o b l e m s w h e r e a c c u r a t e t r a n s i t i o n p r o b a b i l i t i e s o r l i n e s h a p e s a r e r e q u i r e d . I t g i v e s t h e m o s t a c c u r a t e r e s u l t s when t h e l i n e s b e i n g c o n s i d e r e d a r e s t r o n g l y a b s o r b e d and as s u c h i t i s w e l l s u i t e d t o s t u d y r e s o n a n c e t r a n s i t i o n s . V0 frequency ANOMALOUS DISPERSION F I G . 1-1 3 P 1 1^2 3 D 2" 3 D f -^ D 2 2 P 2 2P3 ^4 2P£ .2P "2P, 8 o c JZ u '1 o< P 2 -c CD > 3) c\i C\J 5 ro LO co CD o CO CO OJ LO CO O OJ OJ CO o 2R ro LO CD O) CD 10 1S« PARTIAL TERM DIAGRAM FOR NEUTRAL ARGON FIG.1-2 Plasma Slab EXPERIMENTAL GEOMETRY FIG.2-1 Locus of I 0 (Equ.2-63) POLARIZATION ELLIPSE FIG. 2-2 EXPERIMENT AND THEORY(8115) 8115 A 29.0 (without Nicol) + + Experiment — Theory 1 Magnetic Field ( KG ) FIG. 2-3 (with Nicols) EXPERIMENT AND THEORY (7067) A. - 7067 A M = 1.5 + ++ Experiment —— Theory Magnetic Field ( KG ) FIG. 2-4a 100 RESULTS (8015A,7147A) o Magnetic Field F IG.2-4b RESULTS (6965A_,7635A) ° 0 1.0 Kg Magnetic Field FIG.2-4c B = 0 J»1 J*1 B / 0 / / / / \ \ \ 102 g > § E 41 I O Z MeBg. tt^iBg, < 0 -1 « - A B a M.Bg, •1 M s B g M 8= Bohr Magneton B = Magnetic Field g = Lande' g-factor ZEEMAN SPLITTING TERM DIAGRAM FIG. 2-5 Abs. Coef f. U o Index of Refraction LINE PROFILES FIG. 2-6 APPARATUS -FIG. 3-1 105 Hot SpotS7 ^Windows -Kovar Electrodes I 2^-•10-26 4 0 3=t 3 = h IO Al l Dimensions in cm. Constructed of Pyrex Glass DISCHARGE TUBES FIG.3-2 O </) to z < . 5 | 0 o I I I A-8115A B = 0 gauss I 2 4 6 8 Aperture Diameter (mm) B E A M WIDTH FIG. 3-3 Co i l Run Immersed in Flowing Cold Water Const ructed of 3.2 mm Brass SOLENOID FIG.3-4 o CTt Digital Voltmeter 1 mft SHUNT Sorensen Nobatrons DCR150-15A DCR80-18 (DC Power Supplies) Solenoid SOLENOID POWER SUPPLY FIG. 3-5 o 1 08 A IJUL Signal T i r m T T r r n n i on B >ff-i D Reference Modulated Signal c t u n m T r i T T - M T i r i T m Gate 1 Gated Signal E i mrnnrc n " T " - n r n r T r r i \ ?n-|TTTTTT! Output (on) 'TTIITTT! Output (off) T T " T ' S+N + DC ! N+DC i S+N+DC WAVEFORMS FOR THE DIGITAL PHASE SENSIT IVE DETECTOR FIG. 3-6 -See Fig. 3-10 •Pulse Shape Given in Fig.3-9 PM Amplifier Disriminator B J c Gate #1 E Gate **2 B Let ters Refer to Waveforms in FIG.3-6 See F IG .3-8 A BLOCK DIAGRAM OF THE DPSD FIG.3-7 S 110 motor CHOPPING WHEEL FIG.3-8 n i Type 549 Storage Scope Type1S1 Sampling Unit R ise t ime^.35nsec . 5nsec./div. SINGLE PHOTON PULSE AFTER THE DISCRIMINATOR EMI 9 5 5 8 B - ^ -1500v. r r=56 C=.0 1/lf c 4= r± EC=f= r^ 1 Tunnel diode current discriminator with amplification (output is shown in Fig.3-9) current gain-10 input Z = 5Q output Z = 5tt ri set ime » 3 nsec PHOTO TUBE AND DISCRIMINATOR FIG.3-10 Magnetic Field ( KG ) FIG. 3-11 ro Magnetic Field ( KG ) FIG. 3-12 Used in calculations H—I- Observed Length along axis with respect to centre of coil (cm.) UNIFORMITY OF THE MAGNETIC F IELD FIG.3-13 FIG.A-2 1 1 6 BIBLIOGRAPHY 1. G. T r a v i n g , P l a s m a D i a g n o s t i c s ( N o r t h - H o l l a n d Pub-l i s h i n g Co. - A m s t e r d a m , 1 9 6 8 ) , pp. 6 6 - 1 3 1 . 2. H. G r i e m , PI asma S p e c t r o s c o p y ( M c G r a w - H i l 1 , New Y o r k , 1 9 6 4 ) , pp. 9 8 - 1 0 2 . 3. E. U. Condon and G. H. S h o r t l e y , The T h e o r y o f A t o m i c S p e c t r a ( C a m b r i d g e U n i v e r s i t y P r e s s , New Y o r k , 1 9 6 4 ) . 4. M. A b r a m o w i t z and I . A. S t e g u n , Handbook o f M a t h e m a t i c a l F u n c t i o n s ( D o v e r P u b l i c a t i o n , New Y o r k , 1 9 6 4 ) , pp. 2 9 7 - 3 3 0 . 5. W. L. W i e s e , P r o c . V I I I I n t . C o n f . I o n , G a s e s ( V i e n n a , 1968, p.UT. 6. A. E i n s t e i n V e r h a n d l . D e u t . P h y s . G e s . Ij8, 318 ( 1 9 1 6 ) . 7. T. V. J a c o b s o n , Ph.D. T h e s i s , UBC ( 1 9 6 9 ) . 8. W. S e k a and F. L. C u r z o n , JQSRT, 8, 1147-1162 ( 1 9 6 8 ) . 9. J . B. Shumaker and C. H. P o p e n o e , J . O p t . S o c . Am. 5 7 , 8 ( 1 9 6 7 ) and J . R e s . N a t l . B u r . S t d s . 69A, 495 ( 1 9 6 5 ) . 10. M. G. D o h e r t y , Can. J . P h y s . , 4 6 , 227 ( 1 9 6 8 ) . 11. A. P e r y - T h o r n e and J . E. C h a m b e r l a i n , P r o c . Phy. S o c . 8 2 , 133 ( 1 9 6 3 ) . 12. J . C. B u r n e t t , M.Sc. T h e s i s , UBC ( 1 9 6 9 ) , JQSRT, 10, 7 9 9 - 8 0 4 , 1970. 13. A.C.G. M i t c h e l l and M.W. Z e m a n s k y , R e s o n a n c e R a d i a t i o n and E x c i t e d Atoms ( C a m b r i d g e U n i v e r s i t y P r e s s , C a m b r i d g e , 1 9 6 1 ) . 14. W. C. K r e y e and F . L . R o e s l e r , J . O p t . S o c . Am., 60, 1100 ( 1 9 7 0 ) . 117 118 APPENDIX I THEOREMS THEOREM I - l — ROTATION OF CO-ORDINATES I f a c a r t e s i a n c o - o r d i n a t e s y s t e m i s r o t a t e d a b o u t t h e a x i s by an a n g l e Q. t h e n t h e new c i r c u l a r l y p o l a r i z e d c o m p o n e n t s E+ c a n be e x p r e s s e d i n t e r m s o f t h e o l d c i r c u l a r l y p o l a r i z e d c o m p o n e n t s E * by E* = E* e A - i PROOF: R e f e r r i n g t o F i g u r e A -1 t h e d e f i n i t i o n o f c i r -c u l a r p o l a r i z a t i o n g i v e s El = E\ t X Ea' A - 2 From t h e f i g u r e we s e e t h a t E,'= E, COSS1 - E 2 s m i2 A - 3 E 2 ~-E, sin_S£ + E £ cos sz A - 4 S u b s t i t u t i n g A - 3 and A - 4 i n t o A - 2 g i v e s 118 119 El = ( E, ±si E2) cosSl + C E, ± ^ Ea) s i nSl A - 5 b u t E ± — E, ± ^ E £ t h e r e f o r e / Ei = C C O S i 2 T > i S I nsi) El = E ± e THEOREM 1-2 The t i m e a v e r a g e o f t h e p r o d u c t o f t h e r e a l p a r t s o f two c o m p l e x f u n c t i o n s f and F w h i c h v a r y s i n u s o i d u a l l y i n t i m e w i t h f r e q u e n c y U) and Lz? r e s p e c t i v e l y , i s g i v e n by R e C O R e ( F ) =^ R e ( f F*) £( iv- &') Re(* V) PROOF: I t i s a s s u m e d t h a t f and F ha v e a s i n u s o i d a l t i m e d e p e n d e n c e so we may w r i t e _ _ CtVi * cc) •f = -f* e F = F0 e w h e r e o c a n d oc' a r e a r b i t r a r y p h a s e f a c t o r s . Thus we have R e C - O R e C F ) = -f 0 F 0 c o s f f c H . c o s CcV'-fc + oc ' ) 120 w h i c h c a n be r e w r i t t e n as + e 3J t a k i n g t h e t i m e a v e r a g e o v e r t h e l o w e s t common p e r i o d g i v e s ReC-0 freer} =1 R eC - F f = * ) SC<V-k/0 -+±Ree* r j $(6J-U>') N o t e i f ^ = &/'and f Q = F Q we have THEOREM 1-3 4.00 I H 2 - - i f ,2 </y' Cy = -y'J The r e a l p a r t o f t h i s e x p r e s s i o n g i v e s t h e ' V o i g t l i n e s h a p e ' o f a b s o r p t i o n c o e f f i c i e n t . T h e r e a r e many c o m p u t e r r o u t i n e s w h i c h h a v e b e e n w r i t t e n t o e v a l u a t e t h i s r e a l p a r t b u t f o r t h e p r e s e n t work i t r e q u i r e d b o t h t h e r e a l and i m a g i n a r y p a r t s o f t h e e x p r e s s i o n In o r d e r t o e v a l u a t e I i t i s d e r i v e d i n t e r m s o f t h e c o m p l e x e r r o r f u n c t i o n 0 w h i c h i s g i v e n by The r e l a t i o n b e t w e e n t h e s t a n d a r d V o i g t f u n c t i o n and t h e c o m p l e x e r r o r f u n c t i o n i s w h e r e (J) i f t h e c o m p l e x e r r o r f u n c t i o n PROOF: M u l t i p l y i n g t h e n u m e r a t o r and d e n o m i n a t o r o f I by C -V 1 = - L \ e dy w h e r e we d e f i n e £ + = w + + ^'4 From L a p l a c e t r a n s f o r m s we r e c a l l t h a t 1 (V 0 0 _<*X 0 0 122 S u b s t i t u t i n g t h i s i n t o X fl^es \ civ \ € rr c o l l e c t i n g t h e t e r m s and i n t e g r a t i n g o v e r y n o t i n g t h a t J e doc -J^f g i v e s J = _L ( ^ x e \ dye^ * J e IT i -2* c o m p l e t i n g t h e s q u a r e and i n t e g r a t i n g o v e r t h e l a s t v a r i a b l e 9 i v e s 0 0 ^ 1 Jiff .2. w h e r e ^ B U + /V R e f e r r i n g t o F i g . A-2 t h e i n t e g r a t i o n a l o n g p a t h C, must e q u a l t h e sum a l o n g Cj, » C3 and s i n c e t h e r e a r e no p o l e s w i t h i n t h e c o n t o u r . 123 S u b s t i t u t e ± = j,£ t j* = -JL t ~ J e~z = PjL + u ^ e + t o S u b s t i t u t e i n t o A-7 w h e r e ! . e- 5 5 2 { i * a* r V * } = <t>& c o m p l e x e r r o r f u n c t i o n . ^ - c o y'+ w + a APPENDIX I I EVALUATION OF THE, COMPLEX ERROR FUNCTION ^ fW  J* A-30 2 - w + -c" A The e v a l u a t i o n was c a r r i e d o u t n u m e r i c a l l y w i t h an a c c u r a c y o f a p p r o x i m a t e l y f i v e s i g n i f i c a n t f i g u r e s i n o r d e r t o i n s u r e t h a t t h e p o s s i b l e e r r o r i n t h e c a l c u l a t i o n be s m a l l c o m p a r e d t o t h e e x p e r i m e n t a l u n c e r t a i n t y . T h e o r e m E-l shows t h a t j e dt = ) e d x - e ) e s m s w y r f y A-31 0 O 0 W 2 _V2 i + e i e coswy dy o p u t t i n g t h i s i n t o t h e e x p r e s s i o n f o r d)(2)in A-30 we o b t a i n - e ^ i e " V Sin 2 W y c / y ] ] S e p a r a t i n g t h e r e a l a nd i m a g i n a r y p a r t s we g e t Re<uVB)> = A(w) c o s C 2 w « ) +- sc'^vJ s i o ( i w a ) A_ 3 3 A-32 124 125 Im{d>(Z)} = &Cvv) COS (• aw<0 - A f w O Sin ( a w«0 A-34 w h e r e a -w 0 e _y . A ( ^ ) j e - 2 e v e COS 2 w y c/y Z^fT- o A-35 B f w ) 5 2 e " J" F ( w ) ~ $ 6 V s»r> a w y c/y "1 t^r 1 3 A-36 p ( v y ) = e W j e e f t (Dawson's I n t e g r a l ) A-37 o T h u s t h e p r o b l e m has bee n r e d u c e d t o e v a l u a t i n g r - y r - y j e C ^ 5 C 2 w y ) « / y > e S * n f R w y ) </y, 4 n e / P ( vA-The r a n g e o f t h e v a r i a b l e s w h i c h must be c o v e r e d i s „ oo < W < - f -OO O £ 4 \. O F o r w < 3.<? and <a ^ 3 we e v a l u a t e A-38 d i r e c t l y . DAWSON'S INTEGRAL F(W) By n o t i n g t h a t F(-W) = -F(W) we need o n l y d e a l w i t h O ^ W ^ O O . The meth o d u s e d f o r W 5 i s g i v e n by Hummer (THE VOIGT FUNCTION: An e i g h t - s i g n i f i c a n t f i g u r e t a b l e and G e n e r a t i n g P r o c e d u r e , U n i v e r s i t y o f C o l o r a d o NBS. 38 126 J I LA R e p o r t 24 (Nov. 23, 1964) I t i n v o l v e s a C h e b y s c h e v e x p a n s i o n and t h e C l e n s h a w A l g o r i t h m . F o r 5 t h e s e r i e s e x p a n s i o n F ( w J . - J _ + _ l ^ I '3 » . 3 • S A-39 was u s e d as g i v e n by ERDELY A., OBERHETTINGER F., TRICOMI, I.G. "53 ( H i g h e r T r n a s c e n d e n t a l F u n c t i o n s , V o l . I I , McGraww-H i l l Book Co. I n c . , New Y o r k ) . The a b o v e c a l c u l a t i o n was done i n t h e s u b r o u t i n e DAWSON (W, Y, RL, R L N ) . S i n c e we a r e d e a l i n g w i t h s m a l l ' a ' i n t h e e x p e r i -- y 2 m e n t a l s i t u a t i o n , we c a n e x p a n d t h e t e r m e t o g i v e a c c o r d i n g t o T h e o r e m B-2.0. S e~v c o s u ^ W ^ y = <S T n A-40 w h e r e 2 w n.' L a w a w « j A-41 and j = S i n 2, w a ° 2 W A-42 N o t e t h a t t h i s s e r i e s w i l l c o n v e r g e q u i c k l y i f W i s l a r g e b u t f o r s m a l l W we must e x p a n d c o s ( 2 W Y ) t o g i v e a c c o r d i n g t o T h e o r e m U-2.Z. 127 ) e cos a w y ofy s £ T „ n » o w h e r e _w£ jr„«, -+• c-o a (aw) e > n 2 (2n;/ J and r v 2 N o t e t h a t t h i s s e r i e s c o n v e r g e s q u i c k l y f o r W s m a l l b u t A-40 i s b e t t e r f o r l a r g e r W. S i m i l a r e x p a n s i o n s c a n be o b t a i n e d f o r t h e l a s t i n t e g r a l T h e o r e m 13-2.1 I0 - | - c o s A-43 j n - -A Jn-i H - c-0 a (aw; e t A-44 A-45 _ , n f 2"-/ en ~[ •*-»!-< +• C-'J J r\a S m a w « - _ a c o s « w « > a w o r u s i n g T h e o r e m ( 2 w < J t £ l ) a 1 r, = w i - e 128 T h e r e f o r e A-38 i s e v a l u a t e d f o r W<3.9 and a < 1 . 0 . F o r W >3.9 o r a > 3 t h e n a s e r i e s e x p a n s i o n e x i s t s o f t h e f o r m ( H a n d b o o k o f M a t h e m a t i c a l F u n c t i o n s by A b r a m o w i t z and S t e g u n , p 328) d> (2) * ^ 2 £ _JZ_ A ] = .4613135 A 2 = .09999216 A „ = .002883894 - 6 B 1 = 0.1901635 B 2 = 1.7844927 B 3 = 5.5253437 F o r W > 6 o r a > 6 t h e n a n o t h e r s e r i e s o f t h e f o r m i*> h - 0 j . A ] = 0.5124242 A 2 = 0.05176536 - 6 0.2752551 B 2 = 2.724745 T h i s i s done i n s u b r o u t i n e a p p r o x (W, Y, RL, RLN) THEOREM H - l When e v a l u a t i n g t h e c o m p l e x e r r o r f u n c t i o n we a r e c o n f r o n t e d w i t h t h e i n t e g r a l i n t h e c o m p l e x p l a n e T h i s i n t e g r a l i s c a l c u l a t e d by b r e a k i n g i t up i n t o t h r e e i n t e g r a l s i n t h e r e a l p l a n e , i . e . 129 ) e ctt = ) 6 ax - e J Q Sin a w y c*y + e ) G cos R w y PROOF: S i n c e Q i s e n t i r e a b o v e t h e r e a l a x i s t h e sum o f t h e c o n t o u r i n t e g r a l shown i n F i g u r e A-3 must g i v e c, <\ Cj i . e . , - 7 -*CX , ( ( V f + V y ) ' o o r w r i t i n g t h e e x p o n e n t i a l i n i t s r e a l and i m a g i n a r y p a r t s g i v e s o o o r ~ ^ c -y* r -y* , Q - ) e - c J e> w * w y «rfy -*-^ c e ) c s^y o o ° The f o l l o w i n g s e r i e s o f t h e o r e m s number I I - 2 . 0 t o I I - 2 . 3 a r e t o p r o v e t h e a l g o r i t h m s u s e d t o c a l c u l a t e t h e c o m p l e x e r r o r f u n c t i o n o v e r d i f f e r e n t r a n g e s o f t h e a r g u -ment. THEOREM 11-2 .0 Used f o r 2Wa >1 130 t 00 where J « 2 w J A r s i n a w a 2V/V PROOF: _ v 2 AO „ e y = z c-vf 4 J = C-0 > y Cos 2 wy cjy ? X n Hse» f j ' O 1 1 : 6 I n t e g r a t i n g by p a r t s g i v e s us t h a t a a B i T _ r a n 8 » r an-/ a w y y - s i n ^ w g - a w V y si * a w y ^/y and by p a r t s a g a i n en a n - / 0 a£«-0 n 1.1 c 0 5 i n a w a 0 a a w 4 - n (aw-o ( y cos a w V ay c _ 0 h 2 ^ 2 w 2 ~ F w 2 ~ o r „ . a ' / A j j j , + S i 4 ± « : ; 5 y " M , TJ0 - $ cos a w y eiy TT S m 5 W a 2 ~ O E O Theore ry» I I - 2. (2 w «. > I ) In a way a n a l o g o u s t o o) i t c a n be shown t h a t T = ^ e V S i n a w y V y = ^ o w h e r e J c r i - c«»s aw<* 2wci T - 3 v>-» T_ + (-0 J ~ a g g j a w a r> a s i n * w « L THEOREM I I - 2 . 2 Used f o r 2Wa £ 1 I : ) e COS 2 wy o( y = £ Jf « .2 w h e r e J * S e y o/y o PROOF: We must f i r s t c o n s i d e r t h e s e r i e s e x p a n s i o n o f COS 2 w y " ( a w y j M-o Sim ! 132 r V e c o s 2 w y cty = £ (-0 ( 2 w J J y e T l L e t us now d e v e l o p an a l g o r i t h m t o e v a l u a t e t h e i n t e g r a l C o n s i d e r P r<*. *.) = X e _ < * y * ^ y ejmB , f 2 m - s t y " 1 <• ex"1 0 , _ | . < - . ) " * y * " e v ' - / y * ^ «* m | o s u b s t i t u t i n g t h i s i n t o T I g i v e s 133 -r C > - r - rvi a k M - i a t-vi - * ") 2 ( a w ) i j o THEOREM I I - 2 . 3 f o r 2Wa $-1 S . y e. Sm a w y o/ y r I*. 1 P R O O F oo m 2 m + • NOW 5 i n ( 2 W / J r: (-1) ( z ^ y j •••r-» '(-> ( g w ) C ^ 2. f S r v , + i ) « . J „ 134 I <r\*e$ r « + i n g by p a r t ' s + w » c J ' v e s a 2 THEOREM E-3 F o r v a r i o u s r e a s o n s i t i s d e s i r e d t h a t t h e l i m i t -i n g v a l u e s o f t h e i n t e g r a l s d e r i v e d i n T h e o r e m 1-3 s h o u l d be known f o r lv^(ji) za. «? . T h e s e l i m i t i n g v a l u e s a r e , i e c * 5 ( z w O *c~L - yfW e A - 2 0 + * F C * ~ ) A - 2 1 o w w h e r e F i s Dawson's i n t e g r a l F ( w ) = e~ ) £ e/y PROOF: In o r d e r t o c a r r y o u t t h e i n t e g r a l s g i v e n i n A-20 and A-21 n o t e t h a t t h e y f o r m t h e r e a l and i m a g i n a r y p a r t s o f t h e e n t i r e f u n c t i o n I w h e r e 135 0 0 *-C - t + 2 x w C X = 3 e «v-t i . e . ~ _ t R e f e r r i n g t o t h e c o n t o u r s i n F i g . A-4 we ha v e 2 C. ~ A-22 wh e r e Z-j = w ^ ' i s o dZ-j = i d t S i n c e I i s e n t i r e w i t h i n t h e c o n t o u r i n F i g . A-4 we ha v e c ' c a c3 *«. C o n s i d e r i n g c o n t o u r C 2 we s e e t h a t i f 2 •=:*-»Vythen - V 7 i m y— oo • c t h e r e f o r e u s i n g A-22 + A-23 we h a v e 136 I = e oo s i n c e 2 * ~ ^ <?/ i A . ~ oi " E 3 ^ -*'y o/n3 = - t ' ^ y t h e r e f o r e ~, _ w J - ^ - e w h i c h g i v e s us R e { T} ^ e ' w * v/7T THEOREM 11-4 C P l«2 PROOF: From t h e d e f i n i t i o n o f (Pfe*) g i v e n i n A p p e n d i x I I we s e e t h a t 137 CO ^ _ ^ ( y - w ^ + « 2 t h e r e f o r e m a k i n g t h e s u b s t i t u t i o n ; D g i v e s +«*> Cf p e W ^ P ( e ^ - O /T5T N o t e t h a t t h e l o w e r l i m i t o f i n t e g r a t i o n has been s e t t o - oo r a t h e r t h a n £ [u><, 7 - J • T h i s i s v a l i d s i n c e UJ0 U>a i . e . t h e r e i s n o t a b s o r p t i o n a t z e r o f r e q u e n c y , due t o t h e t r a n s i t i o n i n q u e s t i o n . APPENDIX I I I COMPUTER PROGRAM The c o m p l e x e r r o r f u n c t i o n was e v a l u a t e d as i n A p p e n d i x I I . The s u b r o u t i n e f o r c a l c u l a t i n g t h i s f u n c t i o n was shown t o be a c c u r a t e for-oo^W<° ° } ^1 by c o m p a r i n g t h e r e s u l t s i n t h i s r a n g e w i t h t h e v a l u e s g i v e n i n t h e t a b l e i n r e f e r e n c e 4. The r e q u i r e d i n t e g r a l s g i v e n i n e q u a t i o n s ( 2 - 7 3 ) and ( 2 - 7 4 ) were c a r r i e d o u t u s i n g a Simpson's r u l e n u m e r i c a l i n t e g r a t i o n r o u t i n e . The i n t e g r a t i o n was c a r r i e d o u t , t h e n t h e number o f i n t e g r a t i o n s t e p s was d o u b l e d and t h e sum was r e p e a t e d . T h i s d o u b l i n g o f t h e number o f i n t e g r a -t i o n s t e p s was c o n t i n u e d u n t i l two s u c c e s s i v e a n s w e r s d i f f e r e d by l e s s t h a n . 1 % . T h i s r o u t i n e i n t e g r a t e d t h e l i n e o u t t o a v a l u e o f W = 8.5 ( E q . 2 - 3 1 ) . B e y o n d t h i s l i m i t S i m p s o n ' s r u l e was u s e d w i t h i n c r e a s i n g w i d t h o f i n t e g r a t i n g i n t e r v a l o u t t o W = 4 2 . 0 . B e y o n d t h i s u p p e r l i m i t an a n a l y t i c a l c o n t i n u a t i o n was u s e d . 138 

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