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UBC Theses and Dissertations

The microwave rotational spectrum of methane in the ground vibronic state Holt, Craig Ward 1976

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THE MICROWAVE ROTATIONAL SPECTRUM OF METHANE IN THE GROUND VIBRONIC STATE by CRAIG WARD HOLT B.A., N o r t h w e s t e r n U n i v e r s i t y , 1963 M . S c , U n i v e r s i t y o f C a l i f o r n i a at B e r k e l e y , 1966 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department o f C h e m i s t r y We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1976 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the 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 that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree 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 the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that 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 not 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 . Department o f The U n i v e r s i t y o f B r i t i s h Co lumbia 2 0 7 5 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V 6 T 1W5 Date I 3 ABSTRACT The microwave r o t a t i o n a l spectrum o f ground s t a t e methane has been o b s e r v e d f o r the f i r s t t i m e . Seven AJ=0 Q-branch t r a n s i t i o n s have been o b s e r v e d between 7.8 and 20 GHz w i t h peak a b s o r p t i o n c o e f f i c i e n t s < 6 X 1 0 " 1 1 cm - 1: t h e s e t r a n s i t i o n s are the weakest e v e r o b s e r v e d w i t h a S t a r k - m o d u l a t e d waveguide a b s o r p t i o n microwave s p e c t r o m e t e r . The s p e c t r o m e t e r employed b o l o m e t e r d e t e c t o r s i n a s o u r c e -n o i s e c a n c e l l a t i o n scheme. The s p e c t r o m e t e r s i g n a l was i n t e g r a t e d by a s i g n a l a v e r a g e r w i t h i n t e g r a t i o n t i m e s r a n g i n g up to one week. S p e c i a l e x p e r i m e n t a l t e c h n i q u e s a re d i s c u s s e d i n d e t a i l . The b o l o m e t e r time c o n s t a n t i s c a l c u l a t e d from h o t - w i r e anemometer h e a t l o s s d a t a . The t h e o r y r e q u i r e d t o p e r f o r m t h e experiment i s p r e s e n t e d u s i n g the o c t a h e d r a l group 0^ and i t s r o t a t i o n a l subgroup 0 a l o n g w i t h the t e t r a h e d r a l group T^. An u n d e r s t a n d i n g o f the groups employed by v a r i o u s workers i n t h e f i e l d removes many o f t h e c o n t r a d i c t i o n s between them. The symmetric top r o t a t i o n a l w a v e f u n c t i o n p a r i t y i n t r o d u c e d by Wang i s used t o o b t a i n t h e c o r r e c t t o t a l w a v e f u n c t i o n p a r i t y from t h e e l e c t r o n i c , n u c l e a r s p i n , v i b r a t i o n a l and r o t a t i o n a l w a v e f u n c t i o n s . The one q u a r t i c t e n s o r d i s t o r t i o n c o n s t a n t D,p = 132943.41 + .71 Hz, t h e two s e x t i c c o n s t a n t s H 4 T =-16.9839 +.0076 Hz and H 6 T = 11.0342 +.0086 Hz, and f o r t h e f i r s t t i me t h e t h r e e o c t i c d i s t o r t i o n c o n s t a n t s L ^ T = (20.07 +.24) X 10"" Hz, L . T = (-26.77 +.35) X lO" 1* Hz and L R T = (-30.0 +1.8) X I0~h Hz i i i have been d e t e r m i n e d u s i n g the seven t r a n s i t i o n f r e q u e n c i e s measured h e r e , t h e J=2 o r t h o - p a r a s p l i t t i n g known f o r methane and the two J=7 Q-branch t r a n s i t i o n s measured e a r l i e r i n an i n f r a - r e d l a s e r - microwave double r e s o n a n c e e x p e r i m e n t . The above e r r o r s a r e s t a n d a r d d e v i a t i o n s g i v e n by w e i g h t e d l i n e a r l e a s t squares a n a l y s i s . The e s t i m a t e d a b s o l u t e e r r o r s a r e a l s o g i v e n . Term v a l u e s are p r e s e n t e d w h i c h a l l o w t h e a c c u r a t e c a l c u l a t i o n o f a l l ground s t a t e s p l i t t i n g s o f methane up t o J=21. i v TABLE OF CONTENTS CHAPTER Page 1. INTRODUCTION 1 2. THEORY OF THE MICROWAVE ROTATIONAL SPECTRUM OF METHANE • 4 2.1 I n t r o d u c t i o n and H i s t o r i c a l Development . . . 4 2.2 Symmetry o f t h e R o t a t i o n a l W a v e f u n c t i o n s o f Methane 8 2.3 The R o t a t i o n a l H a m i l t o n i a n o f Methane . . . . 17 2.3.1 F o u r t h Degree Tensor H a m i l t o n i a n . . . 22 2.3.2 S i x t h Degree Tensor H a m i l t o n i a n . . . 23 2.3.3 E i g h t h Degree Tensor H a m i l t o n i a n . . . 24 2.3.4 E x a c t E v a l u a t i o n o f the Tensor H a m i l t o n i a n E i g e n v a l u e s 26 2.4 D i p o l e Moment 27 2.5 S e l e c t i o n R u l e s 29 2.6 S t a r k E f f e c t 30 2.7 P h y s i c a l I n t e r p r e t a t i o n 31 2.8 I n t e n s i t i e s o f T r a n s i t i o n s 33 3. EXPERIMENTAL APPARATUS 35 3.1 I n t r o d u c t i o n 35 3.2 S t a r k M o d u l a t i o n 36 3.3 S t a r k C e l l 40 3.3.1 X-band 41 3.3.2 C e l l above X-band 41 3.4 Frequency S t a b i l i z a t i o n and C o n t r o l 43 3.4.1 Microwave R e f e r e n c e Frequency . . . . 44 3.4.2 S t a b i l i z e d VHF Re f e r e n c e O s c i l l a t o r . 44 V CHAPTER Page 3.4.3 S t a b i l i z e d Microwave O s c i l l a t o r s . . . 46 3.4.4 Frequency D e t e r m i n a t i o n 47 3.5 Time A v e r a g i n g 48 3.6 N o i s e M i n i m i z a t i o n 55 3.6.1 S t r a y P i c k - u p 55 3.6.2 D e t e c t o r t o P r e a m p l i f i e r Impedence Match 56 3.6.3 M i n i m i z a t i o n o f Microwave O s c i l l a t o r N o i s e 57 3.6.4 N o i s e C a n c e l l a t i o n 59 3.6.5 C e l l N o i s e 61 3.7 Bolom e t e r D e t e c t o r s 64 3.7.1 Bolometer N o i s e 64 3.7.2 Power S a t u r a t i o n 66 3.7.3 C o n v e r s i o n G a i n 70 3.7.4 Bolom e t e r Time Co n s t a n t 74 3.7.5 Bolom e t e r M a t c h i n g Impedence 76 4. EXPERIMENTAL CONDITIONS AND RESULTS 78 4.1 I n t r o d u c t i o n 78 4.2 I n i t i a l S e a r c h 79 4.3 Second E x p e r i m e n t a l Phase 80 4.4 F i n a l E x p e r i m e n t a l Phase 88 4.5 E x p e r i m e n t a l C o n d i t i o n s 90 4.5.1 L i n e S t r e n g t h s 96 4.5.2 Sweep Times 96 4.5.3 S t a r k F i e l d s 97 4.6 L i n e I d e n t i f i c a t i o n . 98 v i CHAPTER Page 4.7 D e t e r m i n a t i o n o f D i s t o r t i o n C o n s t a n t s . . . . 100 4.8 I n f l u e n c e o f D e c t i c and H i g h e r Order D i s t o r t i o n C o n s t a n t s . 105 5. CONCLUSIONS 115 APPENDIX A BOLOMETER TIME CONSTANT 119 APPENDIX B THE SYMMETRIC TOP ROTATIONAL WAVEFUNCTION PARITIES 126 BIBLIOGRAPHY 129 v i i L I ST OF TABLES TABLE Page I C h a r a c t e r T a b l e f o r the F u l l M o l e c u l a r Symmetry Group o f Methane 11 I I R e d u c t i o n o f t h e I r r e d u c i b l e R e p r e s e n t a t i o n s o f 0^ i n t o the Subgroup T^ and the R o t a t i o n a l Subgroup 0 : 12 I I I R e d u c t i o n o f the R e p r e s e n t a t i o n s and o f r g u the F u l l R o t a t i o n - I n v e r s i o n Group onto T^ . . . . 14 IV R e d u c t i o n o f the R e p r e s e n t a t i o n s and o f r g u the F u l l R o t a t i o n - I n v e r s i o n Group onto 0^ . . . . 15 V The A l l o w e d Symmetry S p e c i e s o f the W a v e f u n c t i o n s o f Methane i n the Ground V i b r o n i c S t a t e 18 VI N610B/38B7 Bolom e t e r C o n v e r s i o n G a i n 71 V I I The P r o d u c t G-P r f 72 V I I I R e l a t i v e B o l o m e t e r S i g n a l Output Power G i v e n by the P r o d u c t G « P r f 73 I X P r e d i c t e d B o l o m e t e r B e h a v i o r w i t h I n c r e a s e d B i a s C u r r e n t 74 X Bolom e t e r O p e r a t i n g V a l u e s n e a r 200 ohms 77 XI E x p e r i m e n t a l F r e q u e n c i e s o f X- and P-band AJ=0 T r a n s i t i o n s o f 1 2 C H i t i n the Ground V i b r o n i c S t a t e t o g e t h e r w i t h P r e d i c t i o n s based on t h e D i s t o r t i o n C o n s t a n t s D T, H ^ T , and H g T o f C u r l and Tarrago . . 82 X I I E x p e r i m e n t a l C o n d i t i o n s 95 X I I I C e n t r i f u g a l D i s t o r t i o n C o n s t a n t s o f 1 2 C I U i n the Ground S t a t e 101 XIV The AJ=0 T r a n s i t i o n s o f Ground S t a t e ^CH* . . . . 103 v i i i TABLE Page XV Frequency C o n t r i b u t i o n s o f the Terms A s s o c i a t e d w i t h the D i s t o r t i o n C o n s t a n t s L i s t e d i n T a b l e X I I I to the T r a n s i t i o n Frequency 106 XVI P r e d i c t e d T r a n s i t i o n F r e q u e n c i e s based on the D i s t o r t i o n C o n s t a n t s o f T a r r a g o e t a l and t h e D i s t o r t i o n C o n s t a n t s d e t e r m i n e d h e r e 108 X V I I Tensor D i s t o r t i o n Energy E T f o r 1 2C1U i n the Ground S t a t e I l l X V I I I B o l o m e t e r Heat D i s s i p a t i o n 121 i x L I ST OF FIGURES FIGURE Page 3.1 S p e c t r o m e t e r used between 7.8 and 18 GHz w i t h BWO as Source and S i n g l e Ended D e t e c t i o n Scheme . 37 3.2 S p e c t r o m e t e r used between 18 and 24 GHz w i t h K l y s t r o n Source and B a l a n c e d N o i s e C a n c e l l i n g D e t e c t i o n Scheme 38 3.3 HP-K15-8400B Microwave S p e c t r o s c o p y Source C o n f i g u r a t i o n used a t X- and P-bands 45 3.4 Improvement i n S i g n a l to N o i s e R a t i o R e s u l t i n g from the Summation o f Two S e p a r a t e Three Day Runs t o g i v e a S i x Day Run 50 3.5 Frequency A n a l o g V o l t a g e C o n d i t i o n i n g A m p l i f i e r . 53 3.6 B o l o m e t e r E q u i v a l e n t C i r c u i t g i v e n by Cohn . . . 76 4.1 512 Sample minus 512 B a s e l i n e 51.2 second Scans to g i v e t h e J=12 E 1 -»• J=12 E 2 T r a n s i t i o n o f Ground S t a t e 12CR* 83 4.2 The J=12 R o t a t i o n a l Energy L e v e l s f o r 1 2 C H H . . . 84 4.3 512 Sample minus lk 1024 B a s e l i n e 51.2 second Scans t o g i v e t h e J=13 E 1 -> J=13 E 2 T r a n s i t i o n o f Ground S t a t e 1 2CH-t 85 4.4 1024 Sample minus 1024 B a s e l i n e 51.2 second Scans to g i v e t h e J=14 E 2 -»• J=14 E 3 T r a n s i t i o n o f Ground S t a t e 12CH-» 86 4.5 512 Sample minus 512 B a s e l i n e 51.2 second Scans t o g i v e t h e J=15 E 1 + J=15 E 2 T r a n s i t i o n o f Ground S t a t e 1 2 GHi» ." 87 X FIGURE Page 4 . 6 3 8 4 Sample minus 3 8 4 B a s e l i n e 5 1 . 2 second Scans t o g i v e the J = 1 4 E 1 J = 1 4 E 2 T r a n s i t i o n o f Ground S t a t e . ^ C H i , 9 1 4 . 7 The J = 1 4 R o t a t i o n a l Energy L e v e l s f o r . 1 2 CHi» . . . 9 2 4 . 8 3 3 2 8 Sample minus 3 3 2 8 B a s e l i n e 5 1 . 2 second Scans t o g i v e t h e J = 1 6 E 2 -»- J = 1 6 E 3 T r a n s i t i o n o f Ground S t a t e LZCHK 9 3 4 . 9 3 0 7 2 Sample minus % 4 0 9 6 B a s e l i n e 5 1 . 2 second Scans to g i v e t h e J = 1 8 E 2 •> J = 1 8 E 3 T r a n s i t i o n o f Ground S t a t e 1 2 C 1 U 9 4 ACKNOWLEDGEMENTS x i The work r e p o r t e d i n t h i s t h e s i s was completed i n t h e Department o f C h e m i s t r y a t the U n i v e r s i t y o f B r i t i s h Columbia under the p a t i e n t and a b l e d i r e c t i o n o f Dr.M.C.L. Gerry . I would l i k e to thank Dr. G e r r y f o r h i s emphasis on f a i r - p l a y , h i s p l e a s a n t manner, h i s o p t i m i s m ( e s p e c i a l l y t h r o u g h o u t the l o n g " i n i t i a l s e a r c h p e r i o d " ) , h i s r e a d i n e s s t o l i s t e n t o new i d e a s , and t h e l a r g e degree o f independence a l l o w e d a l l my work. Dr. I r v i n g O z i e r i s g r a t e f u l l y acknowledged f o r p r e s e n t i n g the p r o b l e m t r e a t e d i n t h i s t h e s i s as w e l l as f o r the i n e x h a u s t i b l e e n t h u s i a s m w h i c h he gave t o the e xperiment. The a u t h o r s o f v a r i o u s d i s c i p l i n e s c i t e d i n t h e B i b l i o g r a p h y are t o be c r e d i t e d f o r t h e i n d i s p e n s a b l e work th e y p e r f o r m e d l a y i n g the f o u n d a t i o n upon w h i c h t h i s e x p e r i m e n t r e s t s . F o r the use o f c r i t i c a l equipment d u r i n g v a r i o u s s t a g e s o f the experiment Dr. M. Bloom, Dr. A.V. B r e e , Dr. J.E. E l d r i d g e , Dr. J.B. Farmer, Dr. D.C. F r o s t , Dr. W.N. Hardy, and Dr. C.A. McDowell are g i v e n thanks. The many d i s c u s s i o n s and c o n s t a n t encouragement g i v e n by Dr. W a l t e r N. Hardy have been q u i t e u s e f u l . I would l i k e t o e x p r e s s my g r a t i t u d e t o the U n i v e r s i t y o f B r i t i s h Columbia f o r e x t e n d i n g a Graduate F e l l o w s h i p t o me f o r s e v e r a l y e a r s and the N a t i o n a l R esearch C o u n c i l o f Canada f o r a b u r s a r y . My w i f e and t h r e e c h i l d r e n r e c e i v e s p e c i a l a p p r e c i a t i o n x i i f o r a c c e p t i n g a S p a r t a n e x i s t e n c e , f o r g o i n g many conveni e n c e s and p l e a s u r e s f o r a c o n s i d e r a b l e p e r i o d o f t i m e . Dr. and Mrs. F. C l a r k White are thanked f o r t h e i r l i b e r a l f i n a n c i a l s u p p o r t o v e r the y e a r s w h i c h became i n d i s p e n s a b l e d u r i n g t h e l a s t . Thanks are g i v e n Dr. Robert F. C u r l f o r the communication o f h i s e x p e r i m e n t a l r e s u l t s p r i o r t o p u b l i c a t i o n , a p o s t - d o c t o r a l f e l l o w h i p and much p a t i e n c e d u r i n g the c o m p l e t i o n o f t h i s t h e s i s . 1 CHAPTER 1  INTRODUCTION Microwave s p e c t r o s c o p y i s a w e l l e s t a b l i s h e d f i e l d o f r e s e a r c h [25,26,91,93,97,105] w h i c h a l l o w s t h e d e t e r m i n a t i o n o f , amongst o t h e r t h i n g s , bond d i s t a n c e s and a n g l e s , m o l e c u l a r d i p o l e moments, n u c l e a r quadrupole c o u p l i n g s , b a r r i e r s t o i n t e r n a l r o t a t i o n , n u c l e a r magnetic moments and n u c l e a r q u a d r u p o l e moments [25,26]. The f i e l d i s c h a r a c t e r i z e d by v e r y p r e c i s e f r e q u e n c y measurements. The f r e q u e n c y o f a m o l e c u l a r t r a n s i t i o n i s t y p i c a l l y d e t e r m i n e d w i t h a p r e c i s i o n o f one p a r t i n a m i l l i o n . T h i s p r e c i s i o n a l l o w s the above d e t e r m i n a t i o n s t o be done a c c u r a t e l y . The t r a n s i t i o n s o b s e r v e d i n microwave s p e c t r o s c o p y o c c u r between r o t a t i o n a l l e v e l s o f a m o l e c u l e . The p a t t e r n o f the l e v e l s i s d i c t a t e d by the m o l e c u l a r shape. These a l l o w c l a s s i f i c a t i o n o f the m o l e c u l e s i n t o v a r i o u s t y p e s : asymmetric r o t o r s , i n w h i c h a l l t h r e e p r i n c i p a l moments are u n e q u a l ; symmetric r o t o r s , w h i c h have two e q u a l p r i n c i p a l moments; l i n e a r m o l e c u l e s , w h i c h a re a c t u a l l y a su b s e t o f symmetric r o t o r s ; and s p h e r i c a l t o p s , i n w h i c h a l l t h r e e p r i n c i p a l moments are e q u a l . The o b s e r v a t i o n o f a m o l e c u l a r pure r o t a t i o n a l spectrum r e q u i r e s the m o l e c u l e t o have a permanent e l e c t r i c o r magnetic d i p o l e moment [ 2 6 ] , Most m o l e c u l e s p o s s e s s a permanent e l e c t r i c d i p o l e moment; t h e s e i n c l u d e l i n e a r m o l e c u l e s w i t h o u t 2 a c e n t e r o f symmetry, n o n - p l a n a r symmetric t o p s , and v i r t u a l l y a l l asymmetric tops ( w i t h o u t a c e n t e r o f symmetry). A few m o l e c u l e s , such as 0 2 , NO, N0 2, C10 2 and NF 2, p o s s e s s l a r g e m agnetic moments w h i c h p e r m i t r o t a t i o n a l t r a n s i t i o n s t o be ob s e r v e d . A l l d i a m a g n e t i c s p h e r i c a l t o p s , on t h e o t h e r hand, p o s s e s s z e r o e l e c t r i c d i p o l e moment i n t h e i r e q u i l i b r i u m c o n f i g u r a t i o n s , and u n t i l r e c e n t l y were b e l i e v e d t o p o s s e s s no pure r o t a t i o n a l spectrum. T h i s t h e s i s w i l l d e a l w i t h the r o t a t i o n a l spectrum o f the s p h e r i c a l top methane, CHi*. I t was shown r e c e n t l y by Watson [100] t h a t , c o n t r a r y t o p o p u l a r b e l i e f , s p h e r i c a l tops w i t h o u t a c e n t e r o f symmetry, and i n p a r t i c u l a r t h o s e h a v i n g t e t r a h e d r a l symmetry l i k e CHi* o r CCT i t , can develop a v e r y s m a l l d i p o l e moment i n i t s ground s t a t e . Such a moment i s i n d u c e d by c e n t r i f u g a l d i s t o r t i o n , and may be thought o f i n s i m p l e terms as a r i s i n g by r o t a t i o n about one o f the c h e m i c a l bonds, such as the C - H bond i n methane. On the o t h e r hand m o l e c u l e s l i k e SF6, w h i c h do have a c e n t e r o f symmetry, cannot develop such a d i p o l e moment. The e x i s t e n c e o f a s m a l l d i p o l e moment i n the t e t r a h e d r a l m o l e c u l e s l e d t o the p o s s i b i l i t y o f o b s e r v i n g pure r o t a t i o n a l t r a n s i t i o n s . Watson [100] c o n s i d e r e d i n p a r t i c u l a r t h e case o f methane, and p r e d i c t e d t h a t c e r t a i n t r a n s i t i o n s w h i c h o c c u r a t microwave f r e q u e n c i e s s h o u l d have peak a b s o r p t i o n c o e f f i c i e n t s o f up t o 4 . 4 X 1 0 " 1 1 cm"1 ( i e . t h a t 4.4 X 1 0 " 1 1 o f i n c i d e n t r a d i a t i o n s h o u l d be absorbed p e r c e n t i m e t e r ) . Sugden and Kenney [93] g i v e t h e p r a c t i c a l minimum a b s o r p t i o n c o e f f i c i e n t w h i c h can be d e t e c t e d w i t h a 3 c o n v e n t i o n a l S t a r k modulated microwave s p e c t r o m e t e r t o be i n the r e g i o n o f 1 0 ~ 9 cm - 1. Townes and Schawlow [97] s t a t e t h e l i m i t o f s e n s i t i v i t y f o r such a s p e c t r o m e t e r t o be from 10" 9 t o 10~ 8 cm - 1 i n t y p i c a l c a s e s . Long [ 5 4 ] , however, had r e p o r t e d t h e o b s e r v a t i o n o f a l i n e w i t h a peak a b s o r p t i o n c o e f f i c i e n t o f 1 . 3 3 X 1 0 " 1 0 cm" 1 a t d r y - i c e t e m p e r a t u r e . T h i s was the weakest l i n e e v e r r e p o r t e d p r i o r t o t h i s experiment u s i n g t e c h n i q u e s s i m i l a r t o t h o s e used h e r e . S i n c e t h e i n i t i a l methane l i n e had a c a l c u l a t e d i n t e n s i t y o n l y t h r e e times weaker than t h a t o f t h e l i n e o b s e r v e d by Long e t a l [ 5 4 ] , the methane experiment seemed t o have a r e a s o n a b l e chance o f s u c c e s s and was thus u n d e r t a k e n . The r e s u l t was the d e t e c t i o n o f l i n e s w i t h a peak a b s o r p t i o n c o e f f i c i e n t , ym , o f r r max 2 X 1 0 " 1 1 cm" 1 a t a s i g n a l t o n o i s e r a t i o o f t h r e e . These were c l e a r l y the weakest e v e r o b s e r v e d u s i n g a S t a r k modulated s p e c t r o m e t e r . I n the work t o f o l l o w , c h a p t e r 2 c o n t a i n s a d i s c u s s i o n o f the t h e o r y needed t o p e r f o r m t h e e x p e r i m e n t . Chapter 3 c o n t a i n s a d e t a i l e d account o f t h e a p p a r a t u s used i n t h i s e xperiment. Chapter 4 l i s t s the e x p e r i m e n t a l c o n d i t i o n s used and d i s c u s s e s the r e s u l t s o b t a i n e d . The c o n c l u d i n g c h a p t e r 5 l i s t s p o s s i b l e f u t u r e uses o f t h e r e s u l t s o b t a i n e d h e r e . Appendix A c o n t a i n s i n t e r e s t i n g and u s e f u l b o l o m e t e r d e t e c t o r c a l c u l a t i o n s . Appendix B d e a l s w i t h some a s p e c t s o f p a r i t y and i t s a p p l i c a t i o n t o methane. 4 CHAPTER 2 THEORY OF THE MICROWAVE ROTATIONAL SPECTRUM OF METHANE 2.1 I n t r o d u c t i o n and H i s t o r i c a l Development A r o t a t i o n a l energy l e v e l o f a t e t r a h e d r a l s p h e r i c a l top such as methane, c h a r a c t e r i z e d by t h e t o t a l a n g u l a r momentum quantum number J i n t h e ground s t a t e , has a f i e l d f r e e degeneracy o f ( 2 J + 1 ) 2 i n the r i g i d r o t o r a p p r o x i m a t i o n i f n u c l e a r s p i n s t a t i s t i c s are i g n o r e d . T h i s degeneracy i s the p r o d u c t o f a (2J+1) K-degeneracy c o r r e s p o n d i n g t o t h e number o f ways the t o t a l a n g u l a r momentum, J , can be o r i e n t e d i n the m o l e c u l a r framework and a (2J+1) m-degeneracy c o r r e s p o n d i n g t o the number o f ways J i s o r i e n t e d i n space upon a p p l i c a t i o n o f an e x t e r n a l f i e l d . The r o t a t i o n a l energy o f a r i g i d methane m o l e c u l e i s g i v e n by [ 3 1 ] : E j = BoJQJ+1) (2.1) where Bo = h 2 / 8 i T 2 I i s the r o t a t i o n a l c o n s t a n t i n H e r t z , h i s P l a n c k ' s c o n s t a n t and I = ? m i r 2 i s the moment o f i n e r t i a [26] where mi i s the mass o f t h e i - t h p a r t i c l e and r i i s the p e r p e n d i c u l a r d i s t a n c e from the a x i s o f c o n c e r n t o t h e i - t h p a r t i c l e . The r i g i d r o t o r degeneracy i s p a r t i a l l y l i f t e d by c e n t r i f u g a l d i s t o r t i o n . F o r a g i v e n J l e v e l o f methane i n t h e ground s t a t e , the components o f d i f f e r e n t symmetry s p e c i e s ( s e c 2.2), K, under the m o l e c u l a r p o i n t group are a f f e c t e d d i f f e r e n t l y by c e n t r i f u g a l d i s t o r t i o n , r e s u l t i n g i n a s p l i t t i n g 5 o f the r o t a t i o n a l l e v e l s . T h i s ground s t a t e r o t a t i o n a l s p l i t t i n g f o r a t e t r a h e d r a l m o l e c u l e i s due t o s p h e r i c a l t e n s o r o p e r a t o r s , tt^, i n the H a m i l t o n i a n where Z g i v e s the t e n s o r rank [ 8 4 ] . T h i s ground s t a t e s p l i t t i n g was f i r s t c o n s i d e r e d i n d e t a i l by Hecht [30] i n 1960. The t e n s o r s p l i t t i n g o p e r a t o r , Qn, c o n s i d e r e d by Hecht was f o u r t h degree i n J . W i t h the i n c l u s i o n o f a s c a l a r o p e r a t o r f o u r t h degree i n J he showed t h a t t h e energy l e v e l s become [ 3 0 ] : Ej k = B o J ( J + l ) - D S J 2 ( J + 1 ) 2 + D T f ( J , K ) (2.2) where D g and D ^ are q u a r t i c s c a l a r and t e n s o r d i s t o r t i o n c o n s t a n t s r e s p e c t i v e l y i n Hz and t h e e i g e n v a l u e s o f S l i t , f ( J , < ) , a r e s p l i t t i n g f u n c t i o n s w h i c h v a r y w i t h J and the symmetry s p e c i e s K. The term i n D g c o l l e c t i v e l y s h i f t s a l l the components o f a g i v e n J t h e same amount. The term i n Drj, s p l i t s a r o t a t i o n a l l e v e l c h a r a c t e r i z e d by J i n t o components w i t h a t o t a l s t a t i s t i c a l w e i g h t o f (2J+1) i f the components o f symmetry s p e c i e s ( s e c 2.2) A, E and F a r e a s s i g n e d t h e w e i g h t s 1, 2 and 3 r e s p e c t i v e l y . T h i s s p l i t t i n g c o r r e s p o n d s to a removal o f t h e K-degeneracy d e s c r i b e d above. The s p l i t t i n g f u n c t i o n , f ( J , < ) , w h i c h cannot be g i v e n i n c l o s e d form, was f i r s t t a b u l a t e d by Hecht [30] who a l s o gave t h e v a l u e Drr, = .12 MHz. I t was assumed methane d i s p l a y e d no pure r o t a t i o n a l s p e c t r u m i n t h e ground v i b r o n i c s t a t e because i t s h i g h 1 degree o f symmetry i n the e q u i l i b r i u m c o n f i g u r a t i o n r u l e d out a permanent d i p o l e moment. I n 1971, Watson [ 1 0 0 ] , Fox [21] and A l i e v [1,2] proposed a mechanism by w h i c h a m o l e c u l e w i t h no permanent d i p o l e moment, but w i t h o u t a c e n t e r o f 6 symmetry, a c q u i r e d one i n the ground s t a t e by c e n t r i f u g a l d i s t o r t i o n . T h i s d i p o l e moment, r e p r e s e n t e d by the c o n s t a n t Q xy, t o g e t h e r w i t h the t e n s o r s p l i t t i n g s o f Hecht, paved the z way f o r t h e o b s e r v a t i o n o f pure r o t a t i o n a l s p e c t r a o f t e t r a h e d r a l m o l e c u l e s d e s c r i b e d i n t h i s t h e s i s . A l s o i n 1971, O z i e r [71] det e r m i n e d the v a l u e o f Watson's d i p o l e moment f a c t o r 0 X^ t o be 2 . 4 1 X 1 0 " 5 Debye by means o f a m o l e c u l a r beam expe r i m e n t . T r a n s i t i o n s due t o t h e ground s t a t e c e n t r i f u g a l d i s t o r t i o n d i p o l e moment o f methane appear i n two r e g i o n s : t h e AJ=1, R-branch t r a n s i t i o n s appear i n the f a r i n f r a r e d and t h e AJ=0, Q-branch t r a n s i t i o n s appear i n the r a d i o frequency-microwave r e g i o n . F a r IR, R-branch pure r o t a t i o n a l t r a n s i t i o n s o f methane have been o b s e r v e d by Rosenberg e t a l [ 8 9 ] , Rosenberg and O z i e r [88] and C o l e and Honey [ 1 4 ] . The term v a l u e s o f e q u a t i o n (2.2) l e a d t o t r a n s i t i o n f r e q u e n c i e s : Vj = 2B 0(J+1) - 4 D g ( J + l ) 3 (2.3) T h i s spectrum [88] has g i v e n t h e r o t a t i o n a l c o n s t a n t Bp = (5.245±.004) cm" 1 and the s c a l a r d i s t o r t i o n c o n s t a n t Dg = (1.19±.09) X10~" c m - 1 f o r methane. S i m i l a r R-branch t r a n s i t i o n s have been o b s e r v e d f o r t h e t e t r a h e d r a l m o l e c u l e s s i l a n e [85,87] and germane [75,86] by Rosenberg and O z i e r . One AJ=0, Q-branch J=2 s p l i t t i n g was det e r m i n e d by O z i e r i n 1971 [ 7 0 ] . I n 1972 Dorney and Watson [17] extended Hecht's s p l i t t i n g f u n c t i o n f ( J , < ) t o J=20 and c a l c u l a t e d t h e l i n e s t r e n g t h s and f i r s t - o r d e r S t a r k c o e f f i c i e n t s f o r t h e E r o t a t i o n a l components 7 o f methane. W i t h a c c u r a t e s p l i t t i n g f u n c t i o n s , l i n e s t r e n g t h s and S t a r k c o e f f i c i e n t s [17] and a good v a l u e o f D^ , d e r i v e d from the J=2 s p l i t t i n g o f methane by O z i e r [ 3 5 , 7 0 ] , t h e exper i m e n t w h i c h i s r e p o r t e d h e r e was begun. The a c c u r a t e measurement o f two J=7 Q-branch t r a n s i t i o n s o f ground s t a t e methane i n a m i c r o w a v e - i n f r a r e d l a s e r double resonance ex p e r i m e n t by C u r l e t a l [16] and C u r l [15] demonstrated the need f o r f u r t h e r r e f i n e m e n t o f t h e ground s t a t e t h e o r y o f t e t r a h e d r a l m o l e c u l e s . T h i s was done t o s i x t h degree i n J by K i r s c h n e r and Watson [45] and M o r e t - B a i l l y [63,65]. U s i n g t h e n o t a t i o n o f K i r s c h n e r and Watson [ 4 5 ] , the r o t a t i o n a l e n e r g i e s t o s i x t h degree i n J a r e : E J K = B ° J ( J + 1 ) - D SJ 2(J+1.). 2. + H g J 3 ( J + l ) 3 + [ D T + H 4 T J ( J + l ) ] f ( J , K ) + H 6 T g ( J , K ) (2.4) where Bo, Dg, D,j, and f ( J , < ) are as p r e v i o u s l y d e f i n e d , Hg i s a s e x t i c s c a l a r d i s t o r t i o n c o n s t a n t i n Hz, i s a s e x t i c t e n s o r d i s t o r t i o n c o n s t a n t i n Hz a s s o c i a t e d w i t h the f o u r t h rank s p h e r i c a l t e n s o r o p e r a t o r oli» w h i c h g i v e s the e i g e n v a l u e f ( J , K ) , and H ^ i s a s e x t i c t e n s o r d i s t o r t i o n c o n s t a n t i n Hz a s s o c i a t e d w i t h the s i x t h r ank s p h e r i c a l t e n s o r o p e r a t o r fi6, w h i c h g i v e s the e i g e n v a l u e g(J,<) by f i r s t - o r d e r p e r t u r b a t i o n t h e o r y [ 4 5 ] . V a l u e s o f t h e s e x t i c t e n s o r d i s t o r t i o n c o n s t a n t s i n t r o d u c e d by K i r s c h n e r and Watson, H^ T and Hg^,, a l o n g w i t h a new v a l u e o f D,p, were d e t e r m i n e d by C u r l [15] ; t h e s e were o f c o n s i d e r a b l e v a l u e i n d e f i n i n g t h e s e a r c h r e g i o n s employed i n the e x p e r i m e n t r e p o r t e d h e r e . The f i r s t f o u r AJ=0, Q-branch microwave t r a n s i t i o n s o f 8 methane o b s e r v e d i n t h i s work showed the need t o c a r r y the ground s t a t e t h e o r y o f t e t r a h e d r a l m o l e c u l e s t o y e t h i g h e r o r d e r . A c c o r d i n g l y the t h e o r y was extended t o e i g h t h degree i n J by O z i e r [ 7 2 ] , M i c h e l o t , M o r e t - B a i l l y and Fox [ 5 9 ] , H i l i c o and Dang-Nhu [33] and Watson [102]. The e i g h t h degree t h e o r y combined w i t h t h e seven microwave t r a n s i t i o n s [35] r e p o r t e d i n t h i s t h e s i s , the J=2 s p l i t t i n g d e t e r m i n e d by O z i e r [71] and the two J=7 t r a n s i t i o n s o b s e r v e d by C u r l [15] p e r m i t t e d the d e t e r m i n a t i o n o f t h e s i x d i s t o r t i o n c o n s t a n t s Drj,, H ^ , Hgrp, L ^ , Lgrj, and L g T r e p o r t e d h e r e ( t a b l e X I I I ) and a l l o w e d an a c c u r a t e p r e d i c t i o n o f a l l AJ=0 ground s t a t e methane r o t a t i o n a l t r a n s i t i o n s t o J=21 ( t a b l e X V I I ) . The d i s c u s s i o n thus f a r has concerned a c e n t r i f u g a l l y i n d u c e d d i p o l e moment whi c h i s the p r i n c i p a l c o n t r i b u t i o n t o the d i p o l e moment i n t h e ground s t a t e o f a t e t r a h e d r a l m o l e c u l e . A s i m i l a r mechanism d e s c r i b e d by Watson [100] g i v e s r i s e t o a d i p o l e moment p e r p e n d i c u l a r t o the f i g u r e a x i s o f symmetric t o p s such as PH 3 and AsH 3 i n a d d i t i o n t o the s t r o n g d i p o l e moment a l o n g the f i g u r e a x i s . T r a n s i t i o n s due t o t h i s c e n t r i f u g a l l y i n d u c e d d i p o l e moment p e r p e n d i c u l a r t o the f i g u r e a x i s have been o b s e r v e d by Chu and Oka [ 1 2 ] . 2.2 Symmetry o f the R o t a t i o n a l W a v e f u n c t i o n s o f Methane The p a r i t y a s s i g n e d t o a r o t a t i o n a l w a v e f u n c t i o n t o g e t h e r w i t h the m o l e c u l a r p o i n t group used governs the symmetry s p e c i e s d e s i g n a t i o n s used to c l a s s i f y t he r o t a t i o n a l l e v e l s . I n a d d i t i o n t o b e i n g a b a s i s o f t h e nomenc l a t u r e used, p a r i t y i s i m p o r t a n t h e r e because i t h e l p s t o e x p l a i n t h e 9 s e l e c t i o n r u l e s and the f i r s t - o r d e r S t a r k s h i f t o f the E l e v e l s upon w h i c h the s u c c e s s o f t h i s experiment l a r g e l y depended. The p a r i t y o f a s t a t e i s g i v e n by the quantum number ±1 w h i c h r e s u l t s from the a p p l i c a t i o n o f t h e i n v e r s i o n o p e r a t o r [ 1 0 4 ] , i , t o t h e t o t a l w a v e f u n c t i o n : ^ t o t a l - ^ t o t a l <2'5> i i i The i n v e r s i o n o p e r a t o r i s g i v e n by X •+ —X, Y -»• —Y, Z •*• — Z. P a r i t y i s i m p o r t a n t because the H a m i l t o n i a n o f an i s o l a t e d system commutes w i t h the i n v e r s i o n o p e r a t o r so t h a t p a r i t y i s c o n s e r v e d . Moreover, s i n c e p a r i t y i s a p r o p e r t y o f f i e l d - f r e e space, i t i s always a good quantum number under t h i s c o n d i t i o n [58]. I t i s u n i v e r s a l l y a c c e p t e d t h a t methane has t e t r a h e d r a l symmetry, and thus b e l o n g s to the p o i n t group T^ [ 9 6 ] . The o p e r a t i o n s o f t h i s group, b e s i d e s the i d e n t i t y , c o n s i s t o f e i g h t t h r e e - f o l d r o t a t i o n s , t h r e e t w o - f o l d r o t a t i o n s , s i x p l a n e r e f l e c t i o n s and s i x r o t a t i o n - r e f l e c t i o n s [ 9 6 ] . V a r i o u s groups have been used t o c l a s s i f y the r o t a t i o n a l w a v e f u n c t i o n s o f methane. F o r a time [31,103] the r o t a t i o n a l w a v e f u n c t i o n s o f a m o l e c u l e o f p o i n t group T^ were c l a s s i f i e d by o n l y th e s p e c i e s A, E and F o f t h e r o t a t i o n a l subgroup T o f T^. Anderson and Ramsey [4] used the f u l l T^ p o i n t group and gave the r o t a t i o n a l w a v e f u n c t i o n s o f methane the s p e c i e s A 2 , E and F i . H e r z b e r g [31] p o i n t e d out t h a t i f i n v e r s i o n o f a t e t r a h e d r a l m o l e c u l e were c o n s i d e r e d , the p r o p e r group t o use was 0, w i t h the r o t a t i o n a l subgroup 0. Under 0 t h e r o t a t i o n a l 10 w a v e f u n c t i o n s assume the s p e c i e s A i , A 2 , E, F i , and F 2 . Hougen [36] has i n t r o d u c e d a f u l l p e r m u t a t i o n - i n v e r s i o n group f o r methane based on the i d e a s o f L o n g u e t - H i g g i n s [ 5 5 ] . The c h a r a c t e r t a b l e f o r t h e s e groups i s g i v e n i n t a b l e I . The o p e r a t i o n s t o the l e f t o f t h e d i v i d i n g l i n e a l l c o n s i s t o f r o t a t i o n s and are " f e a s i b l e " , whereas t h o s e t o t h e r i g h t o f the d i v i d i n g l i n e i n v e r t the m o l e c u l e and a r e t h u s " n o n - f e a s i b l e " . I t i s assumed t h r o u g h o u t the t h e s i s t h a t a t e q u i l i b r i u m the f u l l p e r m u t a t i o n - i n v e r s i o n group [36] i s i d e n t i c a l t o the p o i n t group 0^ and t h a t the subgroup o f a l l " f e a s i b l e " o p e r a t i o n s i s i d e n t i c a l t o t h e r o t a t i o n a l subgroup 0 o f 0^ [ 3 1 ] . Hougen [37] has made the i d e n t i f i c a t i o n between the " f e a s i b l e " o p e r a t i o n s and r o t a t i o n s w i t h o u t i n v e r s i o n : t h i s s u p p o r t s the use h e r e o f the r o t a t i o n a l subgroup 0 o f 0^ as an a l t e r n a t e t o the somewhat more c o n f u s i n g d e s i g n a t i o n " i s o m o r p h i c T^". The i d e n t i f i c a t i o n o f the group 0 w i t h p e r m u t a t i o n - i n v e r s i o n T^ r e s o l v e s the " c o n t r a d i c t i o n s " between th e work o f Hougen [38] and t h o s e o f Anderson and Ramsey [4] and M o r e t - B a i l l y [ 6 4 ] . T h i s r e s o l u t i o n i s a i d e d by use o f the c o r r e l a t i o n t a b l e , t a b l e I I . Two d i f f e r e n t systems o f p a r i t y have been a s s i g n e d t o the r o t a t i o n a l w a v e f u n c t i o n s o f methane thus f a r . Jahn [40] a s s i g n e d t h e s p h e r i c a l s u r f a c e harmonic p a r i t y (—1)^. Hougen [36] c o n c l u d e d by means o f t r i g o n o m e t r i c i d e n t i t i e s among v a r i o u s r o t a t i o n a l w a v e f u n c t i o n s t h a t the p a r i t y was always even. Because o f the p a r i t y assumed, Jahn a s s i g n e d t h e J = l r o t a t i o n a l w a v e f u n c t i o n t h e symmetry s p e c i e s F 2 , b u t Hougen a s s i g n e d t h e symmetry s p e c i e s F i ( b o t h symmetries T a b l e I C h a r a c t e r T a b l e f o r the F u l l M o l e c u l a r Symmetry Group o f M e t h a n e a T d E 8C3 3C 2 6S-, 6a d 0 E 8C3 3C 2 6C- 6C 2 °h E 8C 3 3C2 6C-* 6C 2 i 8S 6 3 a h 6S-» 6 a d P-I E 8(123) 3(12)(34) 6(1423)* 6(L2)* * E 8(123)* 3(12) ( 3 4 * 6(1423) 6(12) A i A i A i e A i + i 1 1 1 1 1 1 1 1 1 A 2 A 2 O A 2 n - A 2 + 1 1 1 -1 -1 1 1 1 -1 -1 E E o E g E + 2 -1 2 0 0 2 -1 2 0 0 F i F i o F i + 3 0 -1 1 -1 3 0 -1 1 -1 F 2 F 2 6 F * g F 2+ 3 0 -1 -1 - 1 3 0 -1 -1 1 A 2 A i A i u A f 1 1 1 1 1 -1 -1 -1 -1 -1 A x A 2 A 2 u A 2" 1 1 1 -1 -1 -1 -1 -1 1 1 E E E u E ~ 2 -1 2 0 0 -2 1 -2 0 0 F 2 F i F i u F i " 3 0 -1 1 -1 -3 0 1 -1 1 F i F 2 F 2 u F 2~ 3 0 -1 -1 1 -3 0 1 1 -1 Columns a r e l a b e l l e d w i t h the symmetry o p e r a t i o n s o f T^, 0 and 0^ t o g e t h e r w i t h t h e p e r m u t a t i o n - i n v e r s i o n ( P-I) o p e r a t i o n s o f L o n g u e t - H i g g i n s [37,55]. A l l o p e r a t i o n s i n a column are i d e n t i c a l when a p p l i e d t o a m o l e c u l e o f t e t r a h e d r a l symmetry i n an e q u i l i b r i u m c o n f i g u r a t i o n . 12 T a b l e I I R e d u c t i o n o f the I r r e d u c i b l e R e p r e s e n t a t i o n s o f 0^ i n t o the Subgroup T, and t h e R o t a t i o n a l Subgroup 0 [104]. 13 i n T d" t a b l e I I I ) . The t o t a l w a v e f u n c t i o n o f methane, t o t a l ' has been shown by v a r i o u s a u t h o r s [22,68,74,106] t o be o f e i t h e r o r b o t h p a r i t i e s f o r a g i v e n J v a l u e . Assuming t h e r o t a t i o n a l w a v e f u n c t i o n p a r i t y o f e i t h e r Jahn [40] o r Hougen [ 3 6 ] , when w a v e f u n c t i o n i s w r i t t e n i n the Born-Oppenheimer [6] a p p r o x i m a t i o n : the l e f t s i d e o f t h e e q u a t i o n y i e l d s o n l y one p a r i t y f o r a g i v e n J , i n c o n t r a s t t o the two p a r i t i e s g i v e n on the r i g h t s i d e . A t t e m p t s have been made by Y i , O z i e r and Anderson [106] and O z i e r and Fox [74] t o add a p s e u d o s c a l a r o f u n i t modulus, c o , o r an i n v e r s i o n f u n c t i o n , u (p) , t o t h e l e f t s i d e o f t h e above e q u a t i o n (2.6) t o se c u r e the r e q u i r e d two p a r i t i e s . These a t t e m p t s have been h e l p f u l and have l e d t h e a u t h o r t o co n c l u d e t h a t the symmetric top b a s i s f u n c t i o n p a r i t y g i v e n by Wang [98] i s a p p r o p r i a t e t o t h e methane problem. The ground s t a t e e l e c t r o n i c w a v e f u n c t i o n o f methane has even p a r i t y and i s A X g i n 0^, c o r r e s p o n d i n g t o A i i n 0 and T^. The n u c l e a r s p i n w a v e f u n c t i o n s o f methane are o f even p a r i t y [37,58]. These w a v e f u n c t i o n s a r e n o t a l l o f symmetry A j , so t h e y must be c o n s i d e r e d f u r t h e r . The f o u r s y m m e t r i c a l l y p l a c e d p r o t o n s o f s p i n 1/z combine t o g i v e 2'*=16 w a v e f u n c t i o n s . The c o r r e c t l i n e a r c o m b i n a t i o n s o f t h e s e w a v e f u n c t i o n s w h i c h t r a n s f o r m as i r r e d u c i b l e r e p r e s e n t a t i o n s o f T [103] and T d [4] have been de t e r m i n e d : t h e f i v e 1=2 (M I=2,1,0,-1,-2) s p i n f u n c t i o n s o f meta-methane t r a n s f o r m as A i , the one ^ e l e c ' W V L b ^ r o t = * t o t a l (2.6) Table I I I R e d u c t i o n o f the R e p r e s e n t a t i o n s D and D o f r g u the F u l l R o t a t i o n - I n v e r s i o n Group onto T, T d DJ g DJ u J , I = 0 Ax A 2 1 F i F 2 2 E + F 2 E + F i 3 A 2 + F i + F 2 A i + F i + F 2 4 A i + E + F i + F 2 A 2 + E + F i + F 2 5 E + 2 F i + F 2 E + F i + 2 F 2 6 A i + A 2 + E + F i + 2 F 2 A i + A 2 + E + 2Fj + F 2 7 A 2 + E + 2 F i + 2 F 2 A i + E + 2 F i + 2 F 2 8 Ax + 2E + 2 F i + 2 F 2 A 2 + 2E + 2 F i + 2 F 2 9 A i + A 2 + E + 3 F i + 2 F 2 A i + A 2 + E + 2 F j + 3 F 2 10 A i + A 2 + 2E + 2 F i + 3 F 2 A i + A 2 + 2E + 3 F i + 2 F 2 11 A 2 + 2E + 3 F i + 3 F 2 A i + 2E + 3 F i + 3 F 2 12 2Ai + A 2 + 2E + 3F! + 3 F 2 A i + 2A 2 + 2E + 3 F i + 3 F 2 Table IV R e d u c t i o n o f t h e R e p r e s e n t a t i o n s D and D o f the F u l l R o t a t i o n - I n v e r s i o n Group onto 0, °h DJ g D J u J,£ = 0 1 2 A l g E g F>g + F ^ g A i u E u F i u + F 2 u 3 A 2 g + F>g + F ^ g A 2 u + F i u + F 2 u 4 A l g + £ g + F l g + F 2 g A i u + E u + F i u + F 2 u 5 E g + 2 F l g + F ^ g E u + 2 F i u + F 2 u 6 A l g + A 2 g + E g + F>g + 2 F 2 g A i u + A 2 U + E u + F i u + 2 F 2 U 7 A 2 g + E g + 2 F l g + 2 F 2 g A 2 U + E u + 2 F 1 U + 2 F 2 u 8 A l g + 2 E g + 2 F l g + 2 F 2 g A i u + 2 E U + 2 F i u + 2 F 2 U 9 A i g + A 2 g + Eg + 3 F l g + 2 F 2 g A i u + A 2 U + E u + 3 F a u + 2 F 2 U 10 A l g + A 2 g + 2 E g + 2 F l g + 3 F 2 g A i u + A 2 U + 2 E U + 2 F m + 3 F 2 U 11 A 2 g + 2 E g + 3 F l g + 3 F 2 g A 2 U + 2 E U + 3 F i U + 3 F 2 u 12 2 A l g + A 2 g + 2 E g + 3 F l g + 3 F 2 g 2 A i U + A 2 U + 2 E U + 3 F i u + 3 F 2 U 16 do u b l y degenerate 1=0 (Mj=0) s p i n f u n c t i o n o f para-methane t r a n s f o r m s as E, and the t h r e e t r i p l y degenerate 1=1 (Mj=l,0,-1) s p i n f u n c t i o n s o f ortho-methane t r a n s f o r m as F 2 . S i n c e t h e s p i n f u n c t i o n s a re o f even p a r i t y , t h e s p e c i e s under [4] g i v e the s p e c i e s under 0 and 0^, where a g i s t o be added t o the s p e c i e s d e s i g n a t i o n i n 0^. These s p i n f u n c t i o n s a r e c o n v e n i e n t l y l i s t e d by Y i , O z i e r and Anderson [106]. The ground s t a t e v i b r a t i o n a l w a v e f u n c t i o n o f methane has even p a r i t y [27] and i s A i g i n 0^, c o r r e s p o n d i n g t o A i i n 0 and T d. I f t h e r o t a t i o n a l w a v e f u n c t i o n s o f methane are c o n s t r u c t e d u s i n g a b a s i s s e t o f Wang f u n c t i o n s [ 9 8 ] : i J m K * ) - ( 2 ) " 1 / 2 (| JmK) ± |Jm-K)), (2.7) w a v e f u n c t i o n s o f e i t h e r p a r i t y can be c o n s t r u c t e d f o r a g i v e n J . F o r J even, | JmK+^> has even and |JmK~^ > has odd p a r i t y . F o r J odd, |jmK+^> has odd and | JmK~^> has even p a r i t y . The r o t a t i o n a l w a v e f u n c t i o n | Jm0^ > has the p a r i t y (—1)^. The Wang f u n c t i o n s a r e 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 App e n d i x B. The symmetry o f t h e t o t a l w a v e f u n c t i o n must now be c o n s i d e r e d [ 5 5 ] ; i t can be deduced i n the f o l l o w i n g way. F e r m i - D i r a c s t a t i s t i c s d i c t a t e t h a t t h e t o t a l w a v e f u n c t i o n must be a n t i - s y m m e t r i c t o an odd exchange o f p r o t o n s . Under T^, the o p e r a t i o n s E, C 3 and C 2 g i v e an even number o f exchanges r e q u i r i n g the c h a r a c t e r +1, w h i l e Si* and g i v e an odd number o f exchanges r e q u i r i n g t h e c h a r a c t e r - 1 ; thus the t o t a l w a v e f u n c t i o n must be o f s p e c i e s A 2 under T^. Under 0,, the o p e r a t i o n s n o t i n T-, more than m e r e l y i n t e r c h a n g e i d e n t i c a l p a r t i c l e s [ 5 5 ] : t h e y t a k e p r o t o n s from even o c t a n t s i n t o odd o c t a n t s ( t h e m o l e c u l e f i x e d frame d e f i n e d by the Si* axes) and thus the P a u l i p r i n c i p l e i s n o n - c o m m i t t a l w i t h r e s p e c t t o t h e s e o p e r a t i o n s . Under 0^, t h e r e f o r e , the t o t a l w a v e f u n c t i o n can be e i t h e r A i o r A 2 , and under 0 e i t h e r o A i o r A 2 . The o n l y p o s s i b l e symmetry s p e c i e s o f the methane w a v e f u n c t i o n s are g i v e n i n t a b l e V c o n s t r u c t e d u s i n g the i r r e d u c i b l e r e p r e s e n t a t i o n s o f 0^. The A and F l e v e l s have a n u c l e a r s p i n degeneracy o f 5 and 3 r e s p e c t i v e l y . The E l e v e l s on the o t h e r hand e x h i b i t a t w o - f o l d degeneracy where the two l e v e l s have o p p o s i t e p a r i t i e s . I f t h e r o t a t i o n a l subgroup 0 o f 0^ i s used t o c l a s s i f y the r o t a t i o n a l w a v e f u n c t i o n s , the s u b s c r i p t s g and u are t o be dropped. I n t h i s case the symmetry s p e c i e s o f the r o t a t i o n a l w a v e f u n c t i o n s o b t a i n e d by r e d u c i n g and i n t o 0 are the same as o b t a i n e d from the r e d u c t i o n o f i n t o T, [17,32,37] g i v i n g a c o m p a t i b l e system. 2.3 The R o t a t i o n a l H a m i l t o n i a n o f Methane The r o t a t i o n a l H a m i l t o n i a n o f methane can be w r i t t e n i n terms o f v a r i o u s s p h e r i c a l t e n s o r o p e r a t o r s , fi^, where I g i v e s the rank o f the t e n s o r [ 8 4 ] . The number o f independent t e n s o r s [30,40,62,102] o f rank £ i s g i v e n by the number o f times the i r r e d u c i b l e r e p r e s e n t a t i o n A ( t h e H a m i l t o n i a n i s t o t a l l y symmetric and has even p a r i t y ) appears upon the r e d u c t i o n o f t h e r e p r e s e n t a t i o n s o f the f u l l r o t a t i o n -Table V The A l l o w e d Symmetry S p e c i e s o f t h e W a v e f u n c t i o n s o f Methane i n the Ground V i b r o n i c S t a t e ^ e l e c S p i n Deg. * v i b * r o t * t o t a l Degeneracy A l g A i g 5 A l g A 2 g A * g 5 A l g A i g 5 A i g A i u A i u 5 A l g 1 A i g Eg A 2 g 2 A l g E 8 1 A l g E u A i u * A l g F 2 g 3 A i g F>g A 2 g 3 A l g F * g 3 A l g F 2 u A i u 3 19 i n v e r s i o n group [96] i n t o t h e i r r e d u c i b l e r e p r e s e n t a t i o n s o f 0^ w i t h J assuming the r o l e o f £. T h i s r e d u c t i o n i s g i v e n i n t a b l e IV. The i r r e d u c i b l e r e p r e s e n t a t i o n A i g appears once f o r each o f J = £ = 0,4,6,8,9,10 and t w i c e f o r J = £ = 12. The s p h e r i c a l t e n s o r o p e r a t o r o f r a n k £ = 0, &Q , i s a s c a l a r a s s o c i a t e d w i t h the s c a l a r d i s t o r t i o n c o n s t a n t s , ft2 does n o t appear and fig has t o be e x c l u d e d because o f time r e v e r s a l symmetry [ 3 7 ] . The r o t a t i o n a l H a m i l t o n i a n , W t > can thus be w r i t t e n [30,40,45,72] i n terms o f v a r i o u s even [37] degrees o f t h e a n g u l a r momentum o p e r a t o r , w h i c h when ta k e n t o e i g h t h degree i s : W r £ J t = [ B 0 J 2 - D S ( J 2 ) 2 + H S ( J 2 ) 3 + L s ( J 2 ) - ] ^ o (2.8) + [ D T + H 4 T J 2 + L 4 T ( J 2 ) 2 ] f i l t + [ H 6 T + L 6 T J 2 ] f i 6 + L g T f i 8 The q u a r t i c , s e x t i c and o c t i c d i s t o r t i o n c o n s t a n t s D, H and L r e s p e c t i v e l y c o r r e s p o n d to terms o f f o u r t h , s i x t h and e i g h t h degree i n J . The s u b s c r i p t s S and T d i f f e r e n t i a t e s c a l a r and t e n s o r c o n s t a n t s . The n u m e r i c a l s u b s c r i p t s 4, 6 and 8 o f the d i s t o r t i o n c o n s t a n t s r e f e r t o t h e rank o f the a s s o c i a t e d s p h e r i c a l t e n s o r o p e r a t o r . I n t h i s e x p r e s s i o n t h e s c a l a r fio has been shown e x p l i c i t l y , b u t n o r m a l l y i t i s a s s i m i l a t e d i n t o the s c a l a r d i s t o r t i o n c o n s t a n t s Bo, Dg, Hg and Lg. The e f f e c t s o f t e n t h and h i g h e r degree terms a r e a s s i m i l a t e d i n t o a l l t h e c o n s t a n t s g i v e n above and w i l l change them s l i g h t l y from t h e i r t r u e v a l u e s when a f i t t o e x p e r i m e n t a l d a t a i s made. The H a m i l t o n i a n can be d i v i d e d i n t o a s c a l a r p a r t ( a l l terms a s s o c i a t e d w i t h fio) and a t e n s o r p a r t ( a l l o t h e r t e r m s ) : 20 W r o t (2.9) The s c a l a r H a m i l t o n i a n Wg a f f e c t s a l l components o f a g i v e n v a l u e o f J e q u a l l y . To e i g h t h degree i n J i t s e i g e n v a l u e s are g i v e n by: Eg = B 0 J ( J + 1 ) - D g J 2 ( j + l ) 2 + H g J 3 ( J + l ) 3 + Lg J k ( J + l ) * (2.10) The energy Eg would g i v e t h a t o f a r o t a t i o n a l l e v e l i f a l l t h e t e n s o r d i s t o r t i o n terms D^ ,, H ^ , . . , Lg^, assumed a v a l u e o f z e r o . I n t h e experiment d e s c r i b e d i n t h i s t h e s i s , o n l y Q-branch (AJ=0) t r a n s i t i o n s were o b s e r v e d ; thus t h e s c a l a r terms need n o t be c o n s i d e r e d f u r t h e r . The t e n s o r H a m i l t o n i a n t o e i g h t h degree i n J i s t h u s : WT = [ D T + H 4 T J 2 + L 4 T ( J 2 ) 2 ] S U + [ H 6 T + L 6 T J 2 ] f i 6 + L g T f i e (2.11) where t h e terms a r e as d e f i n e d e a r l i e r . I t s p l i t s t h e l e v e l s o f a g i v e n J v a l u e i n t o v a r i o u s A,E and F components w h i c h are c o n v e n i e n t l y l i s t e d i n t a b l e X V I I t o J=21. I t i s t h e s e s p l i t t i n g s w h i c h were o b s e r v e d i n t h i s work. The degree o f s p l i t t i n g i s g i v e n by t h e v a l u e s o f the t e n s o r d i s t o r t i o n c o n s t a n t s , and t h e s p h e r i c a l t e n s o r o p e r a t o r s , and They can be c o n s t r u c t e d from g e n e r a l symmetry arguments [30,72] and a r e : r (2.12) (2.13) C 8 Ve/53 T 8 ,o + ^ / ^ " ( T e , , + TB,_k ) + V s / ^ T e . s + T 8 j - e ) (2.14) where the C's are c o n s t a n t s and t h e T, a r e o b t a i n e d from J ,m the s p h e r i c a l harmonics by r e p l a c i n g x ± i y and z w i t h J ± and J o : a n d s y m m e t r i z i n g any r e s u l t i n g p r o d u c t s . I n terms o f t h e C a r t e s i a n components o f t h e t o t a l r o t a t i o n a l a n g u l a r momentum, J , o r i e n t e d a l o n g t h e m o l e c u l e f i x e d Sit axes o f methane, J , J , and J , tin [17,106] and X y Z ft6 [45] assume t h e form: f - 7 2 (35J z+25j;) + (15J;+3)J' - 7 2 ( J 2 ) 2 - 5A(j;+Jl) ( ftG = 2 V i e ( 1 U Z + 3 5 J Z + 1 4 J Z ) - 1 5 / i e ( 2 1 J Z + 3 5 J Z + 4) J 2 + 7 i 6 ( 2 1 J Z + 8 ) ( J 2 ) 2 - 5 / i e ( J ) - 2 1 / 3 2 [ J + ( H J Z - J +6 ) J + ] - 2 1 / 3 2 [ J ! ( H J Z - J +6 )J 2 ] where J ± - J x T i J y (2.15) (2.16) (2.17) The o p e r a t o r ft8 e x p r e s s e d i n C a r t e s i a n components i s q u i t e l e n g t h y . The e x p r e s s i o n s f o r T 8 i 0, T ^ i * , and T 8 j± 8 have been g i v e n by O z i e r [72], Watson [102] has g i v e n an e x p r e s s i o n f o r the t e n s o r o p e r a t o r ft8 i n terms o f s p h e r i c a l t e n s o r o p e r a t o r s o f lo w e r r a n k : 3465/1430 ft8 = 3003 ft* - 2080(4J2 -183) ft< + 252(12j't -489J 2 + 2645)ft-- 572J 2(J 2-2) (4J 2-3)(4J 2-15) (2.18) E i g e n v a l u e s f o r t h e t e n s o r H a m i l t o n i a n can be o b t a i n e d by d i a g o n a l i z i n g i t . The pr o c e d u r e used i n the p r e s e n t work w i l l be d e s c r i b e d below. However, the method i s q u i t e complex 22 and depends on the v a l u e s o f the d i s t o r t i o n c o n s t a n t s . Because o f t h i s , and because, f u r t h e r m o r e , t h i s degree o f a c c u r a c y i s n o t r e q u i r e d i n some c i r c u m s t a n c e s , v a r i o u s approximate methods have been dev e l o p e d . They were o f use a t v a r i o u s s t a g e s o f t h e p r e s e n t work, and i t i s a p p r o p r i a t e t o d e s c r i b e them here. We d e s c r i b e f i r s t a l l the approximate methods and then the p r o c e d u r e f o r e x a c t d i a g o n a l i z a t i o n . 2.3.1 F o u r t h Degree Tensor H a m i l t o n i a n T h i s a p p r o x i m a t i o n was used i n the i n i t i a l s e a r c h p e r i o d o f the work. I t c o n s i d e r e d W^, t o c o n t a i n o n l y terms f o u r t h degree i n J , making i t : The e i g e n f u n c t i o n s and e i g e n v a l u e s have been d e t e r m i n e d t o J=13 by Hecht [30] and extended t o J=20 by Dorney and Watson [ 1 7 ] , u s i n g C a r t e s i a n axes x,y, and z o r i e n t e d i n the m o l e c u l e i n two d i f f e r e n t ways: 1) a l o n g t h e S•+ axes and 2) w i t h t h e e i g e n f u n c t i o n s w h i c h d i a g o n a l i z e n-* a r e t h e z e r o t h o r d e r w a v e f u n c t i o n s used i n t h e f o l l o w i n g s e c t i o n s . The e i g e n v a l u e s a r e : where D^ , i s an e x p e r i m e n t a l l y d e t e r m i n e d d i s t o r t i o n c o n s t a n t and Wm = Drpftlt (2.19) z - a x i s a l o n g a C 3 a x i s and t h e x - a x i s i n a p l a n e . The (2.20) (2.21) i s a t a b u l a t e d s p h e r i c a l t e n s o r s p l i t t i n g f u n c t i o n w h i c h 23 depends on J , t h e symmetry s p e c i e s K, and a r u n n i n g i n d e x , t , w h i c h l a b e l s components o f a g i v e n J and K i n o r d e r o f i n c r e a s i n g energy. 2.3.2 S i x t h Degree Tensor H a m i l t o n i a n A s i x t h degree p e r t u r b a t i o n t r e a t m e n t was used t o c a l c u l a t e energy l e v e l s d u r i n g the p e r i o d i n w h i c h t h e f i r s t f o u r l i n e s o f methane were o b s e r v e d . The H a m i l t o n i a n was assumed t o be: WT = [ D T + H ^ J 2 ] ^ + H 6 T f t 6 (2.22) The f o u r t h degree t e n s o r H a m i l t o n i a n D^fiif was d i a g o n a l i z e d y i e l d i n g the z e r o t h o r d e r w a v e f u n c t i o n s . Upon g o i n g t o s i x t h As degree, the added terms i n t h e t e n s o r H a m i l t o n i a n H ^ J 2 ^ and Hg,pft6 can be t r e a t e d u s i n g f i r s t - o r d e r p e r t u r b a t i o n t h e o r y w i t h the z e r o t h o r d e r w a v e f u n c t i o n s . As A l t h o u g h t h e t e r m H ^ J 2 ^ c o u l d be t r e a t e d u s i n g f i r s t o r d e r p e r t u r b a t i o n t h e o r y , t h i s i s n o t n e c e s s a r y because i t As can be t r e a t e d e x a c t l y . The o p e r a t o r J 2 i s d i a g o n a l i n t h e z e r o t h o r d e r b a s i s s e t ; thus H ^ j J 2 ^ ! ^ ^ = H ^ ^ J 2 ^ = H 4 T J ( J + 1 ) ^ Q = H 4 T J ( J + l ) f ( J , K , t ) (2.23) F i r s t o r d e r p e r t u r b a t i o n t r e a t m e n t o f Hg^,ft6 r e q u i r e s e v a l u a t i o n o f the m a t r i x elements fi6) = g ( J , K , t ) (2.24) o 24 These have been t a b u l a t e d by K i r s c h n e r and Watson [ 4 5 ] . The e i g e n v a l u e s o f t h e t e n s o r s p l i t t i n g H a m i l t o n i a n t a k e n t o s i x t h degree i n J a r e t h u s d e t e r m i n e d u s i n g f i r s t o r d e r p e r t u r b a t i o n t h e o r y : E ( J , K , t ) = [ D T + H 4 T J ( J + l ) ] f ( J , K , t ) + H 6 T g ( J , K , t ) (2.25) where Drj,, H ^ and Hg^ a r e e x p e r i m e n t a l l y d e t e r m i n e d c e n t r i f u g a l d i s t o r t i o n c o n s t a n t s and f ( J , K , t ) and g ( J , K , t ) have been t a b u l a t e d by K i r s c h n e r and Watson [ 4 5 ] . 2.3.3 E i g h t h Degree Tensor H a m i l t o n i a n T h i s o f c o u r s e i s the f u l l t e n s o r H a m i l t o n i a n r e q u i r e d i n the p r e s e n t work. I t has the form: (2.26) WT = [ D T + H 4 T J 2 + L 4 T ( J Z ) 2 ] S A + [ H 6 T + L 6 T J 2 ] f i 6 + L g T f i 8 where a l l t h e parameters are as d e s c r i b e d e a r l i e r . Two methods have been used t o o b t a i n i t s e i g e n v a l u e s . One, an approximate method u s i n g second o r d e r p e r t u r b a t i o n t h e o r y was d e v e l o p e d by O z i e r [ 7 2 ] . I n t h i s case, f o u r t h degree terms a r e t r e a t e d e x a c t l y , s i x t h degree terms r e q u i r e a second o r d e r p e r t u r b a t i o n t r e a t m e n t , b u t e i g h t h degree terms r e q u i r e o n l y a f i r s t o r d e r p e r t u r b a t i o n t r e a t m e n t . As b e f o r e , s i n c e J 2 i s d i a g o n a l i n the z e r o t h o r d e r w a v e f u n c t i o n s , a second o r d e r t r e a t m e n t i s n o t r e q u i r e d f o r H 4 ^ , J 2 f i i , s i n c e i t can be t r e a t e d e x a c t l y . L i k e w i s e a f i r s t o r d e r t r e a t m e n t i s n o t r e q u i r e d f o r L 4 ^ , ( J 2 ) 2 f i i t because i t , t o o , can be t r e a t e d e x a c t l y . The e x a c t s o l u t i o n o f t h e term a s s o c i a t e d w i t h fl*. i s : [ D T + H 4 T J ( J + 1 ) + L 4 T J 2 ( J + l ) 2 ] f ( J , K , t ) (2.27) 25 U s i n g t h e above argument, t h e f i r s t o r d e r t r e a t m e n t o f T2 6r L / - r r J 2 f i 6 becomes L 6 T J 2 f i e ) = L 6 T ( j 2 f t f = L 6 TJ.(J+ D ( n . / o = L 6 T J ( J + l ) g ( J , K , t ) (2.28) The second o r d e r p e r t u r b a t i o n t r e a t m e n t o f H g T f i 6 g i v e s [ 7 2 ] : ( H 6 T f i 6 ^ = H 6 T g ( J , K , t ) + [ H 6 T 2 / D T ] g ( J , K , t ) (2.29) where ( j , K , t ) = y i<j , < > t i f i 6 i J,K,t ' > i 2 ( 2 3 0 ) Z _ J [ f ( J , K , t ) - f ( J , K , t ' ) ] t ^ t ' The f i r s t o r d e r p e r t u r b a t i o n t r e a t m e n t o f Lg^fts r e q u i r e s the m a t r i x elements ^fi8^)o = h ( J , K , t ) (2.31) These have been t a b u l a t e d by O z i e r [72] a l o n g w i t h t h e above g ( J , K , t ) . The e i g e n v a l u e s o f t h e t e n s o r s p l i t t i n g H a m i l t o n i a n t a k e n t o e i g h t h degree u s i n g a p p r o p r i a t e p e r t u r b a t i o n t h e o r y a r e t h u s : E T = [ D T + H 4 T J ( J + 1 ) + L 4 T J 2 ( J + l ) 2 ] f ( J , K , t ) + [ H 6 T + L 6 T J ( J + l ) ] g ( J , K , t ) + L 8 T h ( J , K , t ) + [H 6 T 2/D T]£(J,K,t) (2.32) where D T, H 4 T , Hg T, L 4 T , L g T and L g T a r e e x p e r i m e n t a l l y d e t e r m i n e d c o n s t a n t s , f ( J , K , t ) and g ( J , K , t ) have been t a b u l a t e d by K i r s c h n e r and Watson [ 4 5 ] , and g ( J , K , t ) and h ( J , K , t ) have been t a b u l a t e d by O z i e r [ 7 2 ] . 2.3.4 E x a c t E v a l u a t i o n o f t h e Tensor H a m i l t o n i a n  E i g e n v a l u e s T h i s method was the one u l t i m a t e l y used i n e v a l u a t i n g t h e d i s t o r t i o n c o n s t a n t s p r e s e n t e d i n t h i s t h e s i s . Because the w a v e f u n c t i o n s r e s u l t i n g from the d i a g o n a l i z a t i o n depend on t h e d i s t o r t i o n c o n s t a n t s [ 7 2 ] , the method was an i t e r a t i v e one. I t was d e v e l o p e d by Fox and O z i e r [22] and i s d e s c r i b e d h e r e . The p r o c e d u r e f i r s t assumes t h a t a l l d i s t o r t i o n c o n s t a n t s a r e z e r o e x c e p t f o r D T > The m a t r i x <^ JK' K | W T | JKK^> i s c o n s t r u c t e d i n the s o - c a l l e d " p r i m i t i v e " b a s i s |JKK^>, w h i c h i s a s u i t a b l y d e f i n e d s e t o f Wang f u n c t i o n s [22,98], I t i s t h e n t r a n s f o r m e d t o a new, s o - c a l l e d " p r e l i m i n a r y " , b a s i s iJKt) [75] by the method o f Fox and O z i e r [ 2 2 ] . T h i s b a s i s has no p a r t i c u l a r p h y s i c a l s i g n i f i c a n c e [72] and has m a t r i x elements ^ J i c t 1 |WT| J<t) . The symbols J , K, K, t and WT a r e as p r e v i o u s l y d e f i n e d . * The m a t r i x i n t h e " p r e l i m i n a r y " b a s i s i s d i a g o n a l i z e d to g i v e the e i g e n f u n c t i o n s w h i c h a r e c a l l e d t h e "second-o r d e r " b a s i s . F o r t h e f i r s t i t e r a t i o n , s i n c e o n l y Drj, was assumed n o n - z e r o , th e e i g e n f u n c t i o n s a r e t h o s e d e t e r m i n e d by Jahn [ 4 0 ] , Hecht [30] and Dorney and Watson [ 1 7 ] . The d i a g o n a l elements o f the s p h e r i c a l t e n s o r o p e r a t o r s P ^ , fi6 * Fox and O z i e r [22] use yp i n s t e a d o f K. The s p e c i e s o f ^ r o t i n T i s g i v e n by y . The v a l u e {1,2} g i v e n p i n d i c a t e s t h e s p e c i e s { A i , A 2 ) i n T^ o f the p s e u d o s c a l a r o r i n v e r s i o n f u n c t i o n [22,74,106] r e q u i r e d t o g i v e t h e c o r r e c t < I V o t a i p a r i t y [68] assuming t h e ^ r o t p a r i t y o f Jahn [ 4 0 ] : (-1)^ and even p a r i t y f o r ^ e l e c * ^ N S * ^ v i b . and fie a r e n e x t c a l c u l a t e d i n the " s e c o n d - o r d e r " b a s i s w h i c h f o r t h e f i r s t i t e r a t i o n a r e f ( J , K , t ) [ 1 7 ] , g ( J , K , t ) [45] and h ( J , K , t ) [72] r e s p e c t i v e l y so t h a t the e n e r g i e s can be e x p r e s s e d as a l i n e a r f u n c t i o n o f t h e s i x d i s t o r t i o n c o n s t a n t s : D T, H 4 T , Hg T, L ^ T , L g T and L g T . A new s e t o f d i s t o r t i o n c o n s t a n t s i s o b t a i n e d by a l i n e a r l e a s t squares f i t t o the e x p e r i m e n t a l f r e q u e n c i e s . T h i s new s e t o f c o n s t a n t s i s used t o s e t up a new m a t r i x i n the " p r i m i t i v e " b a s i s a t wh i c h p o i n t the second i t e r a t i o n i s begun. The i t e r a t i o n s a r e c o n t i n u e d u n t i l t he d i s t o r t i o n c o n s t a n t s converge to the d e s i r e d a c c u r a c y . 2.4 D i p o l e Moment U s i n g p e r m u t a t i o n - i n v e r s i o n arguments Hougen [37] has shown t h a t each o f the s p a c e - f i x e d components o f the e l e c t r i c d i p o l e moment o p e r a t o r i n i s o m o r p h i c T^ (our 0) i s o f s p e c i e s A 2. The t r a n s f o r m a t i o n between m o l e c u l e - f i x e d axes x,y,z and s p a c e - f i x e d axes X,Y,Z i s g i v e n by e q u a t i o n (B.2) i n a p p e n d i x B. Hougen [38] has shown t h a t the m o l e c u l e f i x e d components ux>Vy,Vz o f the d i p o l e moment are r e l a t e d t o the s p a c e - f i x e d components by a s i m i l a r t r a n s f o r m a t i o n , and has d e t e r m i n e d as a r e s u l t t h a t the s p e c i e s o f t h e s e components a r e F 2 F 2 ,F 2 r e s p e c t i v e l y . X y WW I n o r d e r f o r a m o l e c u l e t o p o s s e s s a permanent d i p o l e moment a t l e a s t one t r a n s l a t i o n must be t o t a l l y symmetric. Because t h i s i s n o t so [ 9 6 ] , as i s w e l l known, methane has no permanent d i p o l e moment i n i t s e q u i l i b r i u m c o n f i g u r a t i o n . However, r o t a t i o n about one o f t h e C 3 axes o f methane can d i s t o r t the m o l e c u l e i n such a manner t h a t a d i p o l e moment 28 i s c r e a t e d [ 1 7 ] . The d i s t o r t i o n o f t h e m o l e c u l e can be d e s c r i b e d i n terms o f the normal c o o r d i n a t e s , Q. Of the 3N — 6 = 9 normal c o o r d i n a t e s , o n l y the s i x c o r r e s p o n d i n g t o the i n f r a - r e d a c t i v e v i b r a t i o n s v 3 and Vi» ( o f symmetry s p e c i e s F 2 ) need be c o n s i d e r e d . The c e n t r i f u g a l l y i n d u c e d d i p o l e moment i s g i v e n by [ 1 0 0 ] : 6 ^ a - £ ( 3V 9V f iQk (2.33) where t h e d i s t o r t i o n o f t h e i n d i v i d u a l n o r m a l c o o r d i n a t e s , Q^, produced by c e n t r i f u g a l d i s t o r t i o n i s g i v e n by [100] : 60, = 8TT 2 X k By 3 Y / 9 Q k > J 6 J Y (2.34) .-1 w i t h an element o f the I " t e n s o r and t h e f o r c e c o n s t a n t o f t h e a s s o c i a t e d normal v i b r a t i o n [104]. The c e n t r i f u g a l l y i n d u c e d d i p o l e moment can be w r i t t e n i n the e q u i v a l e n t a l t e r n a t e form [100]: 6y a / i a 3 y BY (2.35) where the J a are t h e components o f J i n u n i t s o f h/2iT and 03Y = 0 Y6 = 2 a a 31 3Y 3Ql f a y a 3Qi (2.36) i s a c o n s t a n t t h i r d rank t e n s o r o f symmetry A i w i t h B^ the r o t a t i o n a l c o n s t a n t about the a a x i s , 1^ t h e p r o d u c t o f i n e r t i a about the B , Y axes and t h e f r e q u e n c y o f the normal v i b r a t i o n Q^. The d i p o l e moment i s an e v e n . f u n c t i o n 29 of J because o f t i m e - r e v e r s a l symmetry. There i s o n l y one independent d i p o l e moment c o n s t a n t 0 ^ f o r methane because t h e i r r e d u c i b l e r e p r e s e n t a t i o n A j appears o n l y once i n the d i r e c t p r o d u c t [32,100]: T(T) X [ r ( R ) ] 2 = F 2 X [ F i ] 2 \ / sym L 1sym = F 2 X [ A i + E + F 2 ] = A i + E + 2 F i + 3 F 2 (2.37) Watson [100] f i n d s f o r a t e t r a h e d r a l m o l e c u l e such as CEU s i x n on-zero v a l u e s o f b u t o n l y one i s independent: 0 y z = 0 z y = 0 x z = 0 z x = 9 x y = 0 y x ( 2 3 g ) x x y y z z N / where t h e x,y,z axes a r e c o i n c i d e n t w i t h the Si* axes o f the m o l e c u l e . The v a l u e o f © x^ has been c a l c u l a t e d by Watson [100] to be 2.6 X 1 0 " 5 Debye, c a l c u l a t e d by Fox [20] t o be 1.8 X 10" 5 D, and measured by O z i e r [71] as 2.41 X 10" 5 D. 2•5 S e l e c t i o n R u l e s S e l e c t i o n r u l e s a r e d e t e r m i n e d by m a t r i x elements o f the form: ^a|y|b^ where the d i p o l e moment o p e r a t o r , u, i s i n t he space f i x e d frame. A l l o w e d t r a n s i t i o n s , a«-+-b, a r e th o s e w i t h non-zero m a t r i x elements. These a re non-ze r o o n l y i f t h e d i r e c t p r o d u c t r & X X c o n t a i n s the t o t a l l y symmetric r e p r e s e n t a t i o n A x [ 9 6 ] . The p o i n t group 0^ w i l l be used t o d e r i v e t h e s e l e c t i o n r u l e s because p a r i t y i s an i m p o r t a n t c o n s i d e r a t i o n h e r e . I n the p o i n t group 0^, = A 2 u i n the space f i x e d frame. The above d i r e c t p r o d u c t w i l l c o n t a i n A i g o n l y i f r & X c o n t a i n s A 2 u . The o n l y 30 a l l o w e d t r a n s i t i o n s i n 0^ a r e : A l u •«-*• A 2 g , Eg «-»• E u , and F i g •<-»• F 2 u . The s e l e c t i o n r u l e s i n t h e subgroup 0 are t h e same as thos e i n 0^ ex c e p t t h a t the s u b s c r i p t s g and u ar e dropped. 2.6 S t a r k E f f e c t I f an e l e c t r i c f i e l d i s a p p l i e d t o a m o l e c u l a r system, a p e r t u r b a t i o n term Wg t ^ ^ s added t o the H a m i l t o n i a n . E v e r y term i n the H a m i l t o n i a n o f methane i s t o t a l l y symmetric, A i , under the o p e r a t i o n s o f the m o l e c u l a r p o i n t group T^ because t h e s e o p e r a t i o n s m e r e l y i n t e r c h a n g e i d e n t i c a l p a r t i c l e s and thus have no e f f e c t on the energy w i t h o r w i t h o u t an a p p l i e d e x t e r n a l f i e l d . The f i e l d f r e e terms o f the H a m i l t o n i a n are o f even p a r i t y because p a r i t y i s c o n s e r v e d f o r an i s o l a t e d system [ 5 8 ] . W i t h the a p p l i c a t i o n o f an e x t e r n a l f i e l d , the system i s no l o n g e r i s o l a t e d . The term W g t a r^ mixes s t a t e s o f o p p o s i t e p a r i t y [58] and thus Wgtark i s o f odd p a r i t y . The S t a r k H a m i l t o n i a n o f methane i s A i under T^, but the odd p a r i t y l e a d s t o the s p e c i e s A 2 u under 0^ and A 2 under 0 (see t a b l e I I ) . The f i r s t - o r d e r p e r t u r b a t i o n S t a r k H a m i l t o n i a n i s : W S t a r k - - ^ Z ( 2 " 3 9 ) w i t h the e l e c t r i c f i e l d , e , d e f i n e d a l o n g t h e space f i x e d Z a x i s . The S t a r k H a m i l t o n i a n i s e v a l u a t e d by t a k i n g t h e average over t h e u n p e r t u r b e d w a v e f u n c t i o n s [ 2 6 ] , thus E S t a r k = - £ ( a l ^ z l a ) <2'40> S i n c e i - s °f o c*d p a r i t y ( A 2 u under 0^) , t h e f i r s t - o r d e r 31 S t a r k energy, E g t a r ^ , i s z e r o u n l e s s a g i v e n l e v e l c o n t a i n s b o t h p a r i t i e s . As shown i n t a b l e V, o n l y t h e E r o t a t i o n a l components have w a v e f u n c t i o n s o f b o t h p a r i t i e s s i m u l t a n e o u s l y and thus o n l y the E r o t a t i o n a l components e x h i b i t a f i r s t - o r d e r S t a r k s h i f t . An a c c i d e n t a l degeneracy o f c e r t a i n l e v e l s can g i v e r i s e t o a f i r s t - o r d e r S t a r k s h i f t . The f i r s t - o r d e r S t a r k m a t r i c e s a r e seen t o be v e r y s i m i l a r t o thos e g o v e r n i n g t h e s e l e c t i o n r u l e s : a f i r s t - o r d e r S t a r k s h i f t w i l l a r i s e o n l y w i t h an a c c i d e n t a l degeneracy o f A i u and A 2 g l e v e l s o r F i g and F 2 u l e v e l s . N e a r l y degenerate l e v e l s o f the p r o p e r symmetry may become p s e u d o l i n e a r a t h i g h f i e l d s [ 1 7 ] . The s i g n e d S t a r k c o e f f i c i e n t s C ( J , E , t ) r e q u i r e d t o c a l c u l a t e the S t a r k energy s h i f t s : A E ( J , E , t ) = C ( J , E , t ) m 0 x y e (2.41) z where A E ( J , E , t ) i s the S t a r k energy s h i f t i n Debye«kV/cm, m i s t h e magnetic quantum number, © x y i s t h e - d i p o l e moment f a c t o r i n Debye [100] and e i s the e l e c t r i c f i e l d s t r e n g t h i n kV/cm, have been c a l c u l a t e d by Dorney and Watson and are l i s t e d i n t a b l e I I I o f r e f e r e n c e [ 1 7 ] . The S t a r k s h i f t f o r a g i v e n t r a n s i t i o n i s c a l c u l a t e d u s i n g the d i f f e r e n c e between t h e two a p p r o p r i a t e s i g n e d S t a r k c o e f f i c i e n t s . 2.7 P h y s i c a l I n t e r p r e t a t i o n Dorney and Watson [17] p r e s e n t a p h y s i c a l model t o e x p l a i n the c e n t r i f u g a l l y i n d u c e d d i p o l e moment and S t a r k e f f e c t w h i c h i s v a l i d f o r h i g h J and maximum p o s i t i v e and n e g a t i v e v a l u e s o f t h e t e n s o r s p l i t t i n g f u n c t i o n f ( J , < , t ) . 32 They n o t e t h a t l e v e l s pushed h i g h e s t i n energy by the t e n s o r o p e r a t o r ft* o c c u r i n n e a r l y degenerate s e t s w i t h a s p i n - f r e e degeneracy o f 8, whi c h i s a t t r i b u t e d t o two d i r e c t i o n s o f r o t a t i o n about each o f the f o u r C 3 axes o f methane. R o t a t i o n about a C 3 a x i s tends t o p i v o t t h e o f f - a x i s hydrogens away from the C 3 a x i s c r e a t i n g a d i p o l e moment a l o n g t h a t a x i s and l o w e r i n g t h e m o l e c u l a r symmetry from T^ t o C^v-The l a r g e s t d i s t o r t i o n d i p o l e moment g i v e s r i s e t o the l a r g e s t f i r s t - o r d e r E l e v e l S t a r k s h i f t s . On the o t h e r hand, t h e energy l e v e l s pushed l o w e s t i n energy by the t e n s o r o p e r a t o r o c c u r i n n e a r l y degenerate s e t s w i t h a s p i n - f r e e degeneracy o f 6, a t t r i b u t e d t o the two d i r e c t i o n s o f r o t a t i o n about each o f the t h r e e S^ axes. R o t a t i o n about an Sn a x i s tends t o lower t h e m o l e c u l a r symmetry from T^ t o T>2^ but n o t to ind u c e a d i p o l e moment. S i n c e a f i r s t - o r d e r S t a r k s h i f t i s p r o p o r t i o n a l t o t h e m o l e c u l e f i x e d d i p o l e moment t h r o u g h the d i r e c t i o n c o s i n e s , r o t a t i o n about an S i , a x i s w i l l g i v e no ( o r a q u i t e s m a l l ) f i r s t - o r d e r S t a r k s h i f t . The d i r e c t i o n c o s i n e s a re the n i n e c o s i n e s r e l a t i n g the t h r e e m o l e c u l e f i x e d axes t o t h e t h r e e space f i x e d axes [91] and are the t r a n s f o r m a t i o n (B.2) i n appendix B. I n the p a r t i c u l a r l i m i t i n g cases d i s c u s s e d above i t i s to be remembered t h a t the d i p o l e moment a l o n g t h e C 3 a x i s : y,. [17] o r a l o n g t h e S 4 a x i s : y = 0 X ^ J J i s n o t the same £ 0 z z x y as the d i p o l e moment f a c t o r 0 * ^ = 2.41 X 1 0 " 5 D, w h i c h i s a c o n s t a n t . 33 2.8 I n t e n s i t i e s o f T r a n s i t i o n s The l i n e s t r e n g t h s f o r i s o t r o p i c r a d i a t i o n have been c a l c u l a t e d by Dorney and Watson [17] u s i n g the t r a n s i t i o n moment: \ 1 summed ov e r the t h r e e d i r e c t i o n s o f space, f , and a l l components a and 3 o f a and b r e s p e c t i v e l y . The c a l c u l a t i o n can be performed f o r t h e s p i n - f r e e problem; n u c l e a r s p i n and i n v e r s i o n d o u b l i n g i s then t a k e n i n t o account i n t h e f o l l o w i n g manner: t h e e f f e c t o f i n t e r n a l degeneracy i s removed by d i v i d i n g the i n t e n s i t i e s o f A, E, and F t r a n s i t i o n s by 1, 2, and 3 r e s p e c t i v e l y , then the i n t e n s i t i e s o f the A and F t r a n s i t i o n s a r e i n c r e a s e d by t h e f a c t o r s 5 and 3 r e s p e c t i v e l y t o t a k e i n t o a ccount n u c l e a r s p i n degeneracy and the E l e v e l i n t e n s i t i e s a r e m u l t i p l i e d by 2 t o t a k e i n t o a c count the i n v e r s i o n d o u b l i n g . Dorney and Watson have t a b u l a t e d t h e re d u c e d l i n e s t r e n g t h [ S ^ B / ( 0 X Y ) 2 ] f o r a d i s t o r t i o n d i p o l e f a c t o r 0 X Y = 1 Debye i n t a b l e IV o f r e f e r e n c e [ 1 7 ] , where S'-^ i s the s p i n f r e e l i n e s t r e n g t h d i v i d e d by the A,E,F i n t e r n a l degeneracy o f 1,2,3 r e s p e c t i v e l y . I n microwave s p e c t r o s c o p y t h e peak a b s o r p t i o n c o e f f i c i e n t (2.42) fa3 Y. 8TT 2 P v 0 S a b e »b -E, /kT (2.43) max 3cQ(kT) 2Av i s o f i n t e r e s t [ 1 7 ] . I n t h i s e x p r e s s i o n Q i s the p a r t i t i o n f u n c t i o n , P/Av i s t h e r e c i p r o c a l o f the p r e s s u r e b r o a d e n i n g parameter w h i c h i s n e a r l y independent o f p r e s s u r e , Vo i s the 34 a b s o r p t i o n c e n t e r f r e q u e n c y , S ^ i s the t r a n s i t i o n moment -E /kT eq. ( 2 . 4 2 ) , and e b' i s the Boltzmann f a c t o r . Dorney and Watson [17] have t a b u l a t e d y ( c m - 1 ) assuming a p r e s s u r e 111 3.X b r o a d e n i n g parameter o f 1 M H z / t o r r , T = 300°K, B-= 5.241 c m - 1 , D T = .12 MHz and 0 * y = 2.41 X 1 0 " 5 Debye. The peak a b s o r p t i o n c o e f f i c i e n t s , Y , f o r the t r a n s i t i o n s o b s e r v e d r max' i n t h i s e x p e r i m e n t a r e l i s t e d i n t a b l e X I I c a l c u l a t e d on t h e b a s i s o f a p r e s s u r e b r o a d e n i n g parameter o f 2.7 MHz/torr [76] and D T = .133 MHz. C a l c u l a t e d v a l u e s o f Y M A X range from 2 X 1 0 " 1 1 t o 6 X 1 0 " 1 1 cm" 1. C o n s i d e r a t i o n o f h i g h e r o r d e r terms i n the d i p o l e moment e x p a n s i o n [78] su g g e s t s t h a t the a b s o r p t i o n c o e f f i c i e n t s o f methane may be s m a l l e r by a f a c t o r o f two. The p r e s s u r e b r o a d e n i n g parameter o b s e r v e d i n t h i s e x p e r i m e n t [73] was a p p r o x i m a t e l y 5 MHz/torr w h i c h f u r t h e r reduces the v a l u e o f Y_ 1 max 35 CHAPTER 3  EXPERIMENTAL APPARATUS 3.1 I n t r o d u c t i o n T h i s e x p e r i m e n t i n v o l v e d o b s e r v a t i o n o f e x t r e m e l y weak a b s o r p t i o n s r e a d i l y masked by n o i s e . To improve the s i g n a l t o n o i s e r a t i o , e f f o r t s were t a k e n t o enhance the s i g n a l as much as p o s s i b l e w h i l e m i n i m i z i n g t h e n o i s e . The c o n v e n t i o n a l method o f S t a r k m o d u l a t i o n combined w i t h phase s e n s i t i v e d e t e c t i o n d e s c r i b e d i n s e c t i o n 3.2 re d u c e d n o i s e i n t h r e e ways: 1) t h e d e t e c t i o n f r e q u e n c y was i n c r e a s e d from n e a r z e r o t o 1 kHz., d r a s t i c a l l y r e d u c i n g the e f f e c t i v e n e s s o f 1/f n o i s e s o u r c e s ; 2) t h e n o i s e bandwidth was r e d u c e d by means o f a s i m p l e p o s t d e t e c t i o n time c o n s t a n t ; and 3) t h e phase d e t e c t o r d i d n o t r e s p o n d t o n o i s e i n q u a d r a t u r e w i t h the s i g n a l . The s i g n a l was i n c r e a s e d a p p r o x i m a t e l y f o u r - f o l d by c o n s t r u c t i n g a 13 meter c e l l t o r e p l a c e t h e 3 meter c e l l s n o r m a l l y used i n the l a b o r a t o r y . The d i f f i c u l t i e s c r e a t e d by h i g h e r - o r d e r modes [82] i n t h e c e l l a r e d i s c u s s e d i n s e c t i o n 3.3. The use o f a s i g n a l a v e r a g i n g computer (see s e c t i o n 3.5) to accumulate many scans a l s o enhanced t h e s i g n a l . The l o n g scan t i m e s used r e q u i r e d p r e c i s e f r e q u e n c y c o n t r o l w h i c h i s d e s c r i b e d i n s e c t i o n 3.4. I n a d d i t i o n t o the S t a r k m o d u l a t i o n mentioned above, n o i s e was r e d u c e d by a) m i n i m i z i n g s t r a y p i c k u p , b) matching the 36 d e t e c t o r s t o the p r e a m p l i f i e r , c) m i n i m i z i n g microwave o s c i l l a t o r n o i s e , d) s u b t r a c t i n g microwave o s c i l l a t o r n o i s e from the c e l l d e t e c t o r o u t p u t , e) m i n i m i z i n g n o i s e g e n e r a t i o n i n the c e l l , and f ) u s i n g low n o i s e d e t e c t o r s . Steps a) t h r o u g h e) a r e d i s c u s s e d i n s e c t i o n 3.6. The s u b j e c t o f low n o i s e b o l o m e t e r d e t e c t o r s i s t r e a t e d i n s e c t i o n 3.7 and a p p e ndix A. I n X-band (8+12.4 GHz.) and P-band (12.4+18 GHz.) the microwave g e n e r a t o r s were backward wave o s c i l l a t o r s c o n t a i n e d i n a H e w l e t t - P a c k a r d 8690A microwave sweeper o f a HP-K15-8400B microwave s p e c t r o s c o p y s o u r c e ( f i g u r e s 3.1 and 3.3). Between 18 and 24 GHz. ( r e f e r r e d t o h e r e as K-band) OKI 20V10 and 24V10 a i r c o o l e d k l y s t r o n s were used ( f i g u r e 3.2). Schematic diagrams o f the c o n f i g u r a t i o n s o f the s p e c t r o -meters used between 8 and 18 GHz. and between 18 and 24 GHz. a r e g i v e n i n f i g u r e s 3.1 and 3.2 r e s p e c t i v e l y . 3.2 S t a r k M o d u l a t i o n Microwave s p e c t r o s c o p y i n i t s s i m p l e s t form c o n s i s t s o f p a s s i n g the t u n a b l e , v i r t u a l l y monochromatic power from a microwave g e n e r a t o r t h r o u g h a gas f i l l e d c e l l and d e t e r m i n i n g the amount o f power absorbed as a f u n c t i o n o f f r e q u e n c y by means o f some d e t e c t o r . S i n c e t h e c r o s s - s e c t i o n a l dimensions o f t h e c e l l s n o r m a l l y used a r e q u i t e s m a l l i n r e l a t i o n t o the w a v e l e n g t h o f t h e microwave r a d i a t i o n w h i c h p a s s e s t h r o u g h them, s m a l l i r r e g u l a r i t i e s i n the t r a n s m i s s i o n p r o p e r t i e s o f t h e c e l l r e s u l t . These t r a n s m i s s i o n i r r e g u l a r i t i e s , a l t h o u g h s m a l l , can be v e r y much l a r g e r t h a n t h e l i n e s under o b s e r v a t i o n because o f t h e c h a r a c t e r i s t i c weakness o f microwave a b s o r p t i o n s . Microwave Spectroscopy Source H P - K I 5 - 8 4 0 0 B Synchronizer HP 8 7 0 8 A | DC-I3m. CELL STARK 1kHz Reference Oscillator FM MOD RAMP SEPTUM tt PHASE DET PAR 128 5mfd "He-f I L N6IOB/38B7 Bolometer /77 ""TRIAD rh G-IO SIGNAL AVERAGER NICOLET 1072 2K WW — 12V. -OSCILLOSCOPE T -X-Y RECORDER F i g u r e 3.1 Spe c t r o m e t e r used between 7.8 and 18 GHz w i t h BWO as Source and S i n g l e Ended D e t e c t i o n Scheme. OKI 20VI0 IkHzJ~L GENERATOR 20db Xlodb SEPTUM Power Supply Narda 62 Al l3m.CELL Synchronizer ] Sage 244 UJUU MA460DR 25MHz IF Counter HP5246L 5252A Prescaler N X o i n ro RF^ 500mv HP8467B SLIDE SCREW TUNER 2KWW 20db RF ERROR N6I0B/38B7 Bolometer reference N6I0B/38B7 sample PHASE DEI PAR 128 Synchronizer HP8708A DC-FM <—<DCf MOD^ 1kHz Reference Oscillator jf RAMP SIGNAL AVERAGER NICOLET 1072 OSCILLOSCOPE X-Y RECORDER F i g u r e 3.2 Spec t r o m e t e r used between 18 and 24 GHz w i t h K l y s t r o n Source and B a l a n c e d N o i s e C a n c e l l i n g D e t e c t i o n Scheme. 39 I f t h e t r a n s m i s s i o n c h a r a c t e r i s t i c o f t h e waveguide c e l l has a s m a l l 1% d i p , f o r i n s t a n c e , i t w i l l make t h e o b s e r v a t i o n o f a .001% m o l e c u l a r a b s o r p t i o n d i f f i c u l t because t h e d i p i s one-thousand t i m e s l a r g e r than t h e a b s o r p t i o n due t o the m o l e c u l e . The i m p o r t a n c e o f t r a n s m i s s i o n i r r e g u l a r i t i e s can be s i g n i f i c a n t l y r e d u c e d by em p l o y i n g S t a r k m o d u l a t i o n combined w i t h phase s e n s i t i v e d e t e c t i o n [ 3 9 ] . Here t h e microwave a b s o r p t i o n i s t u r n e d on and o f f a t t h e m o d u l a t i o n f r e q u e n c y whereas t h e t r a n s m i s s i o n i r r e g u l a r i t i e s a r e i n p r i n c i p l e u n a f f e c t e d ; s i n c e the phase s e n s i t i v e d e t e c t o r responds o n l y t o t h e m o d u l a t i o n f r e q u e n c y , t h e e f f e c t o f ! t h e t r a n s m i s s i o n i r r e g u l a r i t i e s i s g r e a t l y r e d u c e d ( i n the f o r e g o i n g example, from 1000X t o .01). I t i s assumed the S t a r k c e l l i s p e r f e c t l y r i g i d so t h a t the S t a r k m o d u l a t i o n does n o t a f f e c t t r a n s m i s s i o n i r r e g u l a r i t i e s . I f a microwave s p e c t r o m e t e r i s o p e r a t e d i n i t s s i m p l e s t form, t h e d e t e c t e d b a n dwidth can be r e s t r i c t e d t o a s m a l l range about the r e p e t i t i v e sweep f r e q u e n c y t o m i n i m i z e n o i s e . I f t h e same r e p e t i t i v e sweep f r e q u e n c y i s used w i t h a S t a r k modulated s p e c t r o m e t e r , t h e same bandwidth i s r e q u i r e d , b u t i t i s c e n t e r e d a t the m o d u l a t i o n f r e q u e n c y , n o t t h e sweep f r e q u e n c y . T h i s i s o f g r e a t advantage because most power s u p p l i e s , microwave g e n e r a t o r s and d e t e c t o r s e x h i b i t much l e s s n o i s e p e r u n i t bandwidth as the c e n t e r f r e q u e n c y i s r a i s e d . I n t h i s e x p e r i m e n t , t h e m o d u l a t i o n f r e q u e n c y was s e t as h i g h as p o s s i b l e c o n s i s t e n t w i t h the b o l o m e t e r d e t e c t o r r e s p o n s e and was chosen t o be 1 kHz. T h i s low m o d u l a t i o n f r e q u e n c y 40 e n a b l e d the 13 meter c e l l employed t o be d r i v e n t o 2 kV/cm w i t h ease s i n c e the power the S t a r k g e n e r a t o r [9] must s u p p l y t o the septum v a r i e s d i r e c t l y as the f r e q u e n c y [ 9 1 ] . The S t a r k m o d u l a t i o n was s u p p l i e d as a z e r o based square wave w i t h a peak v o l t a g e j u s t s u f f i c i e n t t o g i v e complete (99+7o) m o d u l a t i o n o f the l i n e . An u n n e c e s s a r i l y h i g h S t a r k f i e l d was n o t used because t h i s would a g g r a v a t e a n o n - z e r o b a s e l i n e s l o p e w h i c h r e s u l t e d i n some cases from S t a r k f i e l d i n d u c e d v i b r a t i o n s o f t h e n o n - r i g i d c e l l . The S t a r k m o d u l a t i o n was o b t a i n e d from a s o l i d - s t a t e square wave g e n e r a t o r w h i c h has been d e s c r i b e d i n d e t a i l by B r i t t [ 9 ] . The square wave g e n e r a t o r w i t h seven s e r i e s c o n n e c t e d t r a n s i s t o r s t o charge the c e l l and seven a d d i t i o n a l t r a n s i s t o r s t o d i s c h a r g e the c e l l was d e s i g n e d t o o p e r a t e a t 25 kHz; o p e r a t i o n a t 1 kHz was a c c o m p l i s h e d w i t h o u t m o d i f i c a t i o n . An e x t e r n a l a u d i o o s c i l l a t o r o p e r a t i n g a t a p p r o x i m a t e l y 1 kHz (992 Hz) drove the square wave g e n e r a t o r and s i m u l t a n e o u s l y s u p p l i e d a r e f e r e n c e s i g n a l t o the PAR-128 phase s e n s i t i v e d e t e c t o r . 3.3 S t a r k C e l l The c e l l c o n s i s t e d o f one 12 f t . and t h r e e 10 f t . l e n g t h s o f X-band copper waveguide co n n e c t e d i n s e r i e s t o g i v e a t o t a l l e n g t h o f 12.8 meters; each l e n g t h c o n t a i n e d a S t a r k septum h e l d midway between and p a r a l l e l t o the b r o a d f a c e s o f the waveguide by T e f l o n s t r i p s , t o g i v e a septum s p a c i n g o f 0.47 cm. I n o r d e r t o f i t the c e l l i n t o a v a i l a b l e l a b o r a t o r y space and t o a l l o w t h e c e l l ends t o be a d j a c e n t f o r a n o i s e c a n c e l l a t i o n scheme, f i v e 90° elbows were used t o f o l d t h e 41 c e l l onto i t s e l f i n the shape o f a paper c l i p opened 90°. M i c a windows s e a l e d t h e ends o f t h e c e l l and a l o n g i t u d i n a l s l i t i n the c e n t e r o f t h e b r o a d f a c e o f the waveguide s e r v e d as t h e gas e n t r y p o r t . 3.3.1 X-band A t X-band (8-<->12.4 GHz) the c e l l a t t e n u a t i o n c o e f f i c i e n t was « c = 1.4 X 10~ 3/cm so t h a t 60 mw o f i n p u t power y i e l d e d 10 mw o f d e t e c t o r power. I n t h i s case the optimum c e l l l e n g t h [26] ( L Q p t = 2 a " 1 ) was 14.3 meters w h i c h was c l o s e t o the 12.8 meters used. A t h i g h e r f r e q u e n c i e s , i t was found t h a t microwave g e n e r a t o r powers o f g r e a t e r than 100 mw gave l e s s t h a n 10 mw o f power a t the d e t e c t o r as a r e s u l t o f i n c r e a s e d c e l l a t t e n u a t i o n . The c e l l had g r e a t e r than optimum l e n g t h a t h i g h e r f r e q u e n c i e s , b u t i t was c o n s i d e r e d i m p r a c t i c a l t o v a r y the l e n g t h o f t h e c e l l w i t h the o p e r a t i n g f r e q u e n c y . 3.3.2 C e l l above X-band F o r a g i v e n f r e q u e n c y o f o p e r a t i o n , the waveguide s i z e i s n o r m a l l y chosen so t h a t i t can t r a n s m i t energy i n o n l y one mode: the dominant mode. I f i t i s d e s i r e d t o double t h e f r e q u e n c y o f o p e r a t i o n , f o r i n s t a n c e , a waveguide i s chosen t w i c e as s m a l l t o i n s u r e t h a t o n l y t h e dominant mode i s pr o p a g a t e d . The dominant mode i s advantageous because i t i s t r a n s m i t t e d w i t h the l e a s t l o s s and because i t g i v e s the b e s t performance w i t h most microwave components. S i n c e t h i s e x p e r i m e n t was o r i g i n a l l y c o n c e i v e d as an X-band e x p e r i m e n t , t h e X-band c e l l was c o n s t r u c t e d f o r low a t t e n u a t i o n and p r e d i c t a b l e performance. I f the dominant mode c o u l d have been m a i n t a i n e d as the s o l e means o f energy 42 t r a n s m i s s i o n as the f r e q u e n c y o f o p e r a t i o n was i n c r e a s e d , t h e c e l l would have e x h i b i t e d no more a t t e n u a t i o n a t 40 GHz than a t 8 GHz [ 8 2 ] . But above 12 GHz, X-band waveguide t r a n s m i t s h i g h e r modes c h a r a c t e r i z e d by h i g h e r l o s s e s , w h i c h a t 16 GHz are a t l e a s t 2X g r e a t e r t h a n t h a t o f the dominant mode. Above 12 GHz, any d i s c o n t i n u i t y i n the c e l l w h i c h d i s t o r t s the microwave f i e l d o f the dominant mode (a mis-a l i g n e d f l a n g e o r a S t a r k septum end, f o r i n s t a n c e ) c r e a t e s h i g h e r modes. The S t a r k s e p t a w i l l have no e f f e c t on the p r o p a g a t i o n o f a mode p r o v i d e d i t i s p a r a l l e l t o the microwave magnetic f i e l d and p e r p e n d i c u l a r t o the microwave e l e c t r i c f i e l d a t e v e r y p o i n t [ 8 1 ] . The s e p t a thus do n o t i n t e r f e r e w i t h the dominant TEio mode or any o f t h e r e l a t e d h i g h e r T E n Q modes. But t h e h i g h e r T E a i and TM X a modes [81] a r e t o t a l l y r e f l e c t e d by the s e p t a . The S t a r k s e p t a pass a l l TE modes but r e f l e c t o t h e r s . r r no The t a p e r e d waveguide t r a n s i t i o n s u sed a t each end o f the c e l l i n f r e q u e n c i e s above X-band, however, r e f l e c t a l l h i g h e r modes. A l l T E n Q modes n o t r e c o n v e r t e d t o the dominant T E i 0 mode are r e f l e c t e d back and f o r t h o v e r the S t a r k s e p t a i n the c e l l u n t i l d i s s i p a t e d . O ther h i g h e r o r d e r modes n o t r e c o n v e r t e d t o the dominant TEio mode are r e s t r i c t e d t o S t a r k septum f r e e waveguide r e g i o n s and are a l s o r e f l e c t e d back and f o r t h u n t i l d i s s i p a t e d . I n an e f f o r t t o cope w i t h h i g h e r modes, t h e waveguide was n o t t a p e r e d a t the d e t e c t o r end. I n s t e a d an X-band s l i d e screw t u n e r and X-band d e t e c t o r mount w i t h t u n a b l e stub were used. I n t h i s way the h i g h e r modes w h i c h d i d r e a c h t h e d e t e c t o r end t o g e t h e r w i t h t h e dominant mode c o u l d be c o u p l e d to the d e t e c t o r element i n the b e s t p o s s i b l e manner. The a p p l i c a t i o n o f t h e 1 kHz square wave S t a r k v o l t a g e t o the s e p t a produced an a u d i b l e v i b r a t i o n i n t h e S t a r k s e p t a -waveguide system [ 9 1 ] . The v i b r a t i n g s e p t a gave r i s e t o r e f l e c t i o n s and h i g h e r modes whi c h v a r i e d a t t h e m o d u l a t i o n f r e q u e n c y i n phase w i t h the m o l e c u l a r a b s o r p t i o n and c o u l d n o t be d i s c r i m i n a t e d a g a i n s t . F o r t u n a t e l y the " s i g n a l " due to the v i b r a t i n g s e p t a was b r o a d i n c h a r a c t e r i n comparison w i t h a methane l i n e and thus appeared as a s l o p i n g b a s e l i n e w h i c h was e a s i l y compensated f o r . 3.4 Frequency S t a b i l i z a t i o n and C o n t r o l The sharpness o f the microwave t r a n s i t i o n s o b s e r v e d and the l o n g p e r i o d s o f time r e q u i r e d t o o b t a i n them r e q u i r e d c l o s e f r e q u e n c y c o n t r o l . T h i s c o u l d n o t have been o b t a i n e d w i t h f r e e r u n n i n g ( u n s t a b i l i z e d ) X- and P-band backward wave o s c i l l a t o r s c o n t a i n e d i n the HP-8690A sweeper o r w i t h f r e e r u n n i n g OKI 20V10 o r 24V10 k l y s t r o n s . T h i s c o n t r o l was a c h i e v e d by means o f c o m m e r c i a l l y a v a i l a b l e s y n c h r o n i z e r s w h i c h n o t o n l y s t a b i l i z e d t h e f r e q u e n c y , b u t were r e s p o n s i v e enough t o keep t h e c o n t r o l l e d o s c i l l a t o r i n phase s t e p w i t h a w e l l c o n t r o l l e d microwave r e f e r e n c e f r e q u e n c y . T h i s was i m p o r t a n t because t h e FM n o i s e [67] ( o r s p e c t r a l l i n e w i d t h ) o f t h e microwave o s c i l l a t o r s was r e d u c e d t o t h a t o f the p u r e r r e f e r e n c e o s c i l l a t o r . The microwave r e f e r e n c e f r e q u e n c y was o b t a i n e d by harmonic m u l t i p l i c a t i o n o f a s i g n a l from a s t a b i l i z e d v e r y 44 h i g h f r e q u e n c y (VHF) r e f e r e n c e o s c i l l a t o r . T h i s had the b e n e f i t o f a l l o w i n g t h e microwave f r e q u e n c y t o be d e t e r m i n e d and swept by p e r f o r m i n g t h e s e same o p e r a t i o n s on a VHF o s c i l l a t o r much lo w e r i n f r e q u e n c y . 3.4.1 Microwave R e f e r e n c e Frequency The microwave r e f e r e n c e f r e q u e n c y was a harmonic o f t h e o u t p u t o f an HP-8466A (200^-450 MHz) VHF s o l i d s t a t e v a r i a b l e f r e q u e n c y o s c i l l a t o r (VFO), see f i g u r e s 3.2 and 3.3. A t X-and P-bands t h e o u t p u t o f t h i s VHF r e f e r e n c e o s c i l l a t o r was f e d d i r e c t l y t o a harmonic m i x e r . A t K-band th e o u t p u t o f the VHF r e f e r e n c e o s c i l l a t o r was f e d t h r o u g h a HP-8767B b r o a d band power a m p l i f i e r (which gave a maximum o f 500 mw power a t 350 MHz) t o a MA-460DR v a r a c t o r harmonic m u l t i p l i e r i n an X-band n o n - t u n a b l e d e t e c t o r mount. 3.4.2 S t a b i l i z e d VHF R e f e r e n c e O s c i l l a t o r The HP-8466A (200++450 MHz) VHF r e f e r e n c e o s c i l l a t o r was n o t p a r t i c u l a r l y s t a b l e , so i t had t o be s t a b i l i z e d w i t h a HP-8708A s y n c h r o n i z e r , see f i g u r e s 3.2 and 3.3. T h i s VHF s y n c h r o n i z e r was employed i n the u s u a l manner ex c e p t t h a t t h e s a m p l i n g r a t e (a f r o n t p a n e l c o n t r o l ) was s e t a t 50 kHz. T h i s spaced the microwave l o c k p o i n t s a p p r o x i m a t e l y 3 MHz a p a r t and s i m p l i f i e d i n s t r u m e n t o p e r a t i o n when s e a r c h i n g r e g i o n s 3 MHz a t a time; The HP-8708A VHF s y n c h r o n i z e r had a d-c c o u p l e d FM i n p u t , a d e s i r a b l e f e a t u r e because i t p e r m i t t e d v e r y slow, l i n e a r f r e q u e n c y sweeps. The d e v i a t i o n from l i n e a r i t y o f t h e e n t i r e system was checked u s i n g f r e q u e n c y markers and was found t o be w i t h i n 1/2%. 45 f r e q u e n c y a n a l o g v o l t a g e HP-5246L Counter w i t h 5252A P r e s c a l e r HP-8466A VHF R e f e r e n c e (-L O s c i l l a t o r v T f r e q 0 m o n i t o r out e r r o r fm HP-8708A i n S y n c h r o n i z e r v h f sample HP-8690A Microwave Sweeper ou t * * h e l i x v W 20 db HP-8709A S y n c h r o n i z e r I r r o r l n °" d i r e c t i o n a l c o u p l e r O m + 4-o o CM 0) > Cti CD O P J u o • H to S Harmonic M i x e r HP-934A X-band HP-932A P-band x 20 db to c e l l F i g u r e 3.3 HP-K15-8400B Microwave S p e c t r o s c o p y Source C o n f i g u r a t i o n u s e d a t X- and P-bands 46 The use o f the FM i n p u t t o sweep t h e f r e q u e n c y produced a d i s a d v a n t a g e . I n t h i s mode, the o v e r a l l f r e q u e n c y s t a b i l i t y and p u r i t y was u l t i m a t e l y d e t e r m i n e d by an L-C o s c i l l a t o r . T h i s was so because t h e FM i n p u t v a r i e d the f r e q u e n c y o f a .3 t o .4 MHz VFO i n t h e HP-8708A s y n c h r o n i z e r . By means o f a phase l o c k l o o p t h i s v a r i a t i o n was t r a n s f e r r e d i n an a d d i t i v e manner t o a 19.95 t o 20.05 MHz VFO i n the same s y n c h r o n i z e r . T h i s f r e q u e n c y t r a n s f e r r e d u c e d the s t a b i l i t y demands on t h e .3 t o .4 MHz VFO by a f a c t o r o f 50, but the o v e r a l l s t a b i l i t y was s t i l l worse t h a n t h a t o b t a i n e d from a c r y s t a l o s c i l l a t o r . A l o n g term f r e q u e n c y s t a b i l i t y o f 1 p a r t i n 10 6 (.02 MHz a t 20 GHz) was m a i n t a i n e d by " t r i m m i n g " t h e f r e q u e n c y o f t h e s p e c t r o m e t e r e v e r y few hours t o compensate f o r t h e e f f e c t s of s l o w temperature d r i f t s . The f r e q u e n c y o f t h e 19.95 t o 20.05 MHz VFO i n t h e HP-8708A s y n c h r o n i z e r was t h e n d i v i d e d by 400 t o d r i v e a s a m p l i n g gate a t 50 kHz. The o u t p u t f r e q u e n c y o f t h e HP-8466A VHF r e f e r e n c e o s c i l l a t o r was sampled a t a 50 kHz r a t e (which meant o n l y a f r a c t i o n o f a c y c l e was sampled e v e r y 7000 c y c l e s a t a c o n t r o l l e d f r e q u e n c y o f 350 MHz) t o g i v e a c o r r e c t i v e -e r r o r s i g n a l back t o the sampled VHF o s c i l l a t o r . The s t a b i l i t y o f the 19.95 t o 20.05 VFO was thus t r a n s f e r r e d t o the 200-<~*-450 MHz VFO, b u t i n a m u l t i p l i c a t i v e manner so t h a t t h e r e was no g a i n i n r e l a t i v e s t a b i l i t y . 3.4.3 S t a b i l i z e d Microwave O s c i l l a t o r s The f r e q u e n c y o f t h e microwave o s c i l l a t o r s was s t a b i l i z e d by means o f s e r v o l o o p s ( f i g u r e s 3.2 and 3.3). I n a l l c a s e s , one p e r c e n t o f the o u t p u t power was mixed w i t h the microwave 47 r e f e r e n c e f r e q u e n c y t o g i v e a b e a t f r e q u e n c y . T h i s b eat f r e q u e n c y was i n t u r n compared t o a c r y s t a l c o n t r o l l e d o f f s e t f r e q u e n c y by means o f a phase s e n s i t i v e d e t e c t o r , g i v i n g a c o r r e c t i v e e r r o r v o l t a g e t o the microwave o s c i l l a t o r , c l o s i n g the l o o p . A t X- or P-band ( f i g u r e 3.3) the m i x e r was e i t h e r a HP-934A o r HP-932A harmonic m i x e r w h i c h a l s o s e r v e d t o g e n e r a t e t h e microwave r e f e r e n c e f r e q u e n c y . The s y n c h r o n i z e r was a HP-8709A w i t h an o f f s e t f r e q u e n c y o f 20 MHz. A t K-band ( f i g u r e 3.2) t h e m i x e r was a 1N26 c r y s t a l mounted i n a t u n a b l e K-band mount. The microwave r e f e r e n c e f r e q u e n c y was s u p p l i e d by a v a r a c t o r i n an X-band mount combined w i t h an X- t o K-band t r a n s i t i o n . B o t h the m i x e r and t h e v a r a c t o r harmonic m u l t i p l i e r were mounted on o p p o s i t e arms o f a 20-db c r o s s g u i d e d u a l d i r e c t i o n a l c o u p l e r , w h i c h was p l a c e d between th e k l y s t r o n and t h e i s o l a t o r t o m i n i m i z e i n t e r f e r e n c e from c e l l r e f l e c t i o n s . The s y n c h r o n i z e r was an LFE 244 ( l e s s the harmonic g e n e r a t o r ) w i t h an o f f s e t o f 25 MHz. Phase l o c k was a s c e r t a i n e d by o b s e r v i n g t h e c o r r e c t i v e e r r o r s i g n a l on an o s c i l l o s c o p e . 3.4.4 Frequency D e t e r m i n a t i o n The f r e q u e n c y o f the microwave o s c i l l a t o r was always m a i n t a i n e d ±20 o r ±25 MHz w i t h r e s p e c t t o t h e microwave r e f e r e n c e f r e q u e n c y a t X- and P-bands or K-band, r e s p e c t i v e l y . I f t h e f o r e g o i n g s i g n was known, a l o n g w i t h the VHF r e f e r e n c e o s c i l l a t o r f r e q u e n c y and t h e c o r r e c t harmonic number, the microwave f r e q u e n c y was e a s i l y d e t e r m i n e d . A t X- and P-bands ( f i g u r e 313) t h i s d e t e r m i n a t i o n was 48 e x c e p t i o n a l l y easy. Here th e f r e q u e n c y was c o r r e c t l y d e t e r m i n e d by a f a c t o r y m o d i f i e d HP-5246L c o u n t e r w i t h a 5252A p r e s c a l e r i f the system was i n l o c k and the HP-8690A sweeper d i a l s e t t i n g was i n approximate agreement w i t h t h e f r e q u e n c y d i s p l a y e d on the c o u n t e r . The l i n e a t 7861 MHz was below the n o r m al range o f the HP-8690A sweeper w i t h an X-band p l u g - i n , but c o u l d be r e a c h e d i n f r e q u e n c y by the a p p l i c a t i o n o f an a p p r o p r i a t e d-c v o l t a g e from the f r o n t p a n e l FM i n p u t . S i n c e i n t h i s case t h e f r o n t p a n e l f r e q u e n c y c a l i b r a t i o n was no l o n g e r v a l i d , the f r e q u e n c y had t o be d e t e r m i n e d by e s t a b l i s h i n g phase l o c k w i t h no FM i n p u t a t 8 GHz and t h e n s i m u l t a n e o u s l y i n c r e a s i n g t h e FM i n p u t and d e c r e a s i n g the r e f e r e n c e f r e q u e n c y w h i l e m a i n t a i n i n g phase l o c k u n t i l 7861 MHz was r e a c h e d . T h i s f r e q u e n c y was i n agreement w i t h an independent d e t e r m i n a t i o n made w i t h a HP-5245L c o u n t e r u t i l i z i n g a HP-5255A h e t e r o d y n e c o n v e r t e r p l u g - i n . A t K-band ( f i g u r e 3.2) t h e k l y s t r o n f r e q u e n c y was d e termined t o w i t h i n 10 MHz w i t h a Narda 12K1 c a v i t y wavemeter. T h i s was s u f f i c i e n t t o d etermine unambiguously the harmonic number and the c o r r e c t o f f s e t f r e q u e n c y s i g n , w h i c h when combined w i t h t h e p r e c i s e l y d e t e r m i n e d r e f e r e n c e o s c i l l a t o r f r e q u e n c y gave an a c c u r a t e k l y s t r o n f r e q u e n c y . 3.5 Time A v e r a g i n g The o b s e r v e d s i g n a l t o n o i s e r a t i o i n an e xperiment can be improved by r e s t r i c t i n g t h e n o i s e bandwidth o f t h e i n s t r u m e n t employed. T h i s can be done s i m p l y by i n c r e a s i n g t h e o u t p u t time c o n s t a n t . I n o r d e r f o r t h e o u t p u t t o f o l l o w 49 a s p e c t r a l f e a t u r e r e a s o n a b l y , however, t h e sweep r a t e must be slowed s i m u l t a n e o u s l y . But as the sweep r a t e i s l o w e r e d , t h e t i m e r a t e o f change o f any s p e c t r a l f e a t u r e becomes l e s s , i n c r e a s i n g , t h e r e l a t i v e i m p o r t a n c e o f any slow v a r i a t i o n i n e x p e r i m e n t a l c o n d i t i o n s w h i c h might be due, f o r example, t o temperature or power s u p p l y v o l t a g e f l u c t u a t i o n s . On t h e o t h e r hand, the s i g n a l t o n o i s e r a t i o can be improved by a d d i t i v e l y s u p e r i m p o s i n g many scans o f t h e i d e n t i c a l f r e q u e n c y r e g i o n , w h i l e r e t a i n i n g t h e o r i g i n a l sweep r a t e . I n t h i s case t h e s i g n a l i n c r e a s e s l i n e a r l y w i t h t i m e , but t h e n o i s e i n c r e a s e s o n l y as t h e square r o o t . The i n f l u e n c e o f low f r e q u e n c y n o i s e i s s i g n i f i c a n t l y r e d u c e d because i t i s d i v i d e d i n t o many time segments and summed. T h i s i s the f a m i l i a r p r o c e s s o f " s i g n a l a v e r a g i n g " [29,47]. F i g u r e 3.4 shows t h e improvement i n the s i g n a l t o n o i s e r a t i o r e s u l t i n g from the summation o f two s e p a r a t e t h r e e day runs t o g i v e a s i x day run. A s i g n a l a v e r a g i n g computer a l l o w s the sweep time t o be v a r i e d a lmost a t w i l l , w i t h the number o f sweeps a d j u s t e d t o g i v e the same t o t a l sweep tim e . I n t h i s e x p e r i m e n t , the a b i l i t y t o det e r m i n e t h e f r e q u e n c y w i t h p r e c i s i o n s e t t h e upper l i m i t t o t h e sweep r a t e . I f , f o r i n s t a n c e , t h e c o u n t i n g r a t e o f an e l e c t r o n i c c o u n t e r i s 1 MHz, the c o u n t e r must have a d w e l l time o f 1 second t o determine a f r e q u e n c y w i t h a p r e c i s i o n o f 1 p a r t i n 1 0 6 . I f t h e p r e c i s i o n need be o n l y 1 p a r t i n 1 0 5 , the r e q u i r e d d w e l l time i n t h i s example can be re d u c e d t o .1 second and t h e sweep r a t e may be c o r r e s p o n d i n g l y i n c r e a s e d . 18562 18563 FREQUENCY (MHz) F i g u r e 3.4 Improvement i n S i g n a l t o N o i s e R a t i o R e s u l t i n g from the Summation o f Two S e p a r a t e Three Day Runs (top two t r a c e s ) t o g i v e a S i x Day Run (bottom t r a c e ) . o The s i g n a l a v e r a g e r was a N i c o l e t #1072 w i t h a memory o f 1024 c h a n n e l s d i v i d e d i n t o q u a r t e r s w i t h p r o v i s i o n t o t r a n s f e r d a t a i n v a r i o u s ways between q u a r t e r s . T h i s f e a t u r e a l l o w e d b a s e l i n e s u b t r a c t i o n . The b a s e l i n e s l o p e i n some i n s t a n c e s exceeded t h e l i n e h e i g h t by a f a c t o r o f from 16 t o 32 w i t h i n a 3 MHz range. The s l o p i n g b a s e l i n e was compensated f o r t o a c o n s i d e r a b l e e x t e n t by a l t e r n a t e l y a c c u m u l a t i n g sample and b a s e l i n e runs o f s e v e r a l hours each i n s e p a r a t e q u a r t e r s and t a k i n g the d i f f e r e n c e t o g i v e a methane a b s o r p t i o n on a r e l a t i v e l y f l a t b a s e l i n e . The sample and b a s e l i n e runs were i d e n t i c a l i n a l l r e s p e c t s e x c e p t f o r the p r e s e n c e o f methane i n the c e l l . The b a s e l i n e s u b t r a c t i o n method worked w e l l because t h e b a s e l i n e s l o p e remained r e a s o n a b l y c o n s t a n t w i t h time. The s i g n a l t o n o i s e r a t i o i n p r i n c i p l e was degraded by 1.414 u s i n g t h i s method, b u t i f t h e b a s e l i n e s l o p e was c o n s i d e r e d n o i s e i n s t e a d o f s i g n a l , t h e r e was a g r e a t improvement. On o c c a s i o n , t h e i n s t r u m e n t b r o k e down d u r i n g a r u n d e s t r o y i n g t h e i n f o r m a t i o n b e i n g accumulated. The s e r i o u s n e s s o f any breakdown was d i m i n i s h e d by p e r i o d i c a l l y t r a n s f e r r i n g t h e c o n t e n t s o f t h e w o r k i n g q u a r t e r t o an a p p r o p r i a t e s t o r a g e q u a r t e r ; i n t h i s manner o n l y a few hours work a t most wou l d be r u i n e d p e r i n t e r r u p t i o n . Because some runs were un u s a b l e on o c c a s i o n , t h e b a s e l i n e s u b t r a c t i o n c o u l d n o t always be performed d i r e c t l y . I n t h e s e c a s e s , t h e d a t a was m u l t i p l i e d by a b i n a r y c o n s t a n t d u r i n g t r a n s f e r , w h i c h when done t h e c o r r e c t number o f times gave e q u a l w e i g h t t o sample and b a s e l i n e . 52 The N i c o l e t #1072 o f f e r e d two modes o f o p e r a t i o n : " f l y b a c k " i n w h i c h the memory ch a n n e l s were swept always i n one d i r e c t i o n , or "sweepback" i n w h i c h t h e memory cha n n e l s were f i r s t swept up and then down i n number. The sweepback mode was chosen because t h e " t i m e c o n s t a n t d i s t o r t i o n " [91] s u f f e r e d by a l i n e was s y m m e t r i c a l , b r o a d e n i n g the l i n e b u t n o t a f f e c t i n g the peak a b s o r p t i o n f r e q u e n c y . The sweepback mode was chosen a l s o because i t l a c k e d t h e l a r g e d i s c o n t i n u i t y a t f l y b a c k o f t h e o t h e r mode. T h i s d i s c o n t i n u i t y i n t h e f l y b a c k f r e q u e n c y a n a l o g v o l t a g e caused t h e s y n c h r o n i z e r s e r v o l o o p s t o "drop out o f l o c k " f r e q u e n t l y . I n a d d i t i o n , a s l o p i n g b a s e l i n e y i e l d e d a d i s c o n t i n u o u s b a s e l i n e on f l y b a c k , w h i c h when a c t e d upon by th e phase s e n s i t i v e d e t e c t o r o u t p u t f i l t e r caused a v e r y c u r v e d b a s e l i n e a t t h e b e g i n n i n g o f each scan r e s u l t i n g i n an a p p r e c i a b l e l o s s o f i n f o r m a t i o n . B o t h t h e s e d i s a d v a n t a g e s were a v o i d e d w i t h the sweepback mode. The s i g n a l a v e r a g e r p r o v i d e d an o u t p u t v o l t a g e p r o p o r t i o n a l t o t h e memory c h a n n e l b e i n g swept. T h i s was a 0 t o 4 v o l t s t a i r c a s e ramp and was used as a f r e q u e n c y a n a l o g v o l t a g e t o d r i v e b o t h t h e HP-8708A s y n c h r o n i z e r ( f i g u r e s 3.2 and 3.3) and a n o i s e t r a c k i n g e l e c t r o m a g n e t ( f i g u r e 3.2). S i n c e n e i t h e r t h e r e q u i r e d s y n c h r o n i z e r FM i n p u t v o l t a g e n o r the r e q u i r e d e l e c t r o m a g n e t v o l t a g e was 0 t o 4 v o l t s , two v a r i a b l e g a i n , v a r i a b l e o f f s e t p r e c i s i o n d-c a m p l i f i e r s ( f i g u r e 3.5) were c o n s t r u c t e d . The " s t a r t i n g p o i n t s " were s e t w i t h the o f f s e t c o n t r o l s . The g a i n c o n t r o l s had no e f f e c t on the " s t a r t i n g p o i n t " s e t t i n g s because h e r e t h e ramp v o l t a g e lOKft A / W 5KQ< 10KQ uA777 + u n i t y g a i n i n v e r t i n g a m p l i f i e r +3Vo lOKft A A A / 30Kfi - w v uA741 and yA777 o p e r a t i o n a l a m p l i f i e r s and uA7805 - 5 v o l t v o l t a g e r e g u l a t o r m a n u f a c t u r e d by F a i r c h i l d S emiconductor © SYNCHRONIZER A / W lOKfi OFFSET 3X g a i n i n v e r t i n g a m p l i f i e r w i t h v a r i a b l e o f f s e t lOKfi W V v o l t a g e f o l l o w e r +4V o V V V lOKfi OFFSET 30Kft A A A / m out RELAY MAGNET common 1 amp 3X g a i n i n v e r t i n g a m p l i f i e r w i t h v a r i a b l e o f f s e t F i g u r e 3.5 Frequency A n a l o g V o l t a g e C o n d i t i o n i n g A m p l i f i e r 54 was 0.0 v o l t s . The s e t t i n g s c o r r e s p o n d i n g t o a ramp v o l t a g e o f 4.0 v o l t s were th e n made "on the r i m " w i t h the g a i n c o n t r o l s . The c u r r e n t g o i n g t o the n o i s e t r a c k i n g e l e c t r o -magnet had t o be a d j u s t e d on o c c a s i o n midway t h r o u g h a r u n , f o r t h i s r e a s o n a v o l t a g e f o l l o w e r was p u t i n t o the a m p l i f i e r to p r e v e n t t h e g a i n adjustment o f t h e e l e c t r o m a g n e t a m p l i f i e r f r o m u p s e t t i n g the e n d - p o i n t f r e q u e n c i e s . The s t a i r c a s e ramp o f the N i c o l e t 1072 i n t r o d u c e d a n o i s e s p i k e i n t o t h e system each time the f r e q u e n c y was d i s c o n t i n u o u s l y i n c r e a s e d from one v a l u e t o t h e n e x t . These n o i s e s p i k e s were o b s e r v e d i n the s i g n a l c h a n n e l o f the PAR 128 phase s e n s i t i v e d e t e c t o r . To smooth out t h e s e d i s c o n t i n u i t i e s the ramp was f e d t h r o u g h an R-C f i l t e r w i t h a time c o n s t a n t o f 1/2 second. T h i s f i l t e r a f f e c t e d t h e t r i a n g u l a r ramp v o l t a g e i n two ways: i t rounded t h e sharp peaks and t r o u g h s and i t r e t a r d e d t h e wave i n time by an amount e q u a l t o t h e time c o n s t a n t [61]. The e f f e c t was t o add h y s t e r e s i s t o t h e f r e q u e n c y sweep system. I n t h e c e n t e r o f the sweep, th e f r e q u e n c y was s h i f t e d a l t e r n a t e l y 1% low and 17o h i g h as the f r e q u e n c y a n a l o g v o l t a g e was swept up and down. Sharp f r e q u e n c y markers appeared d o u b l e d due t o t h i s h y s t e r e s i s . W i t h a 3 MHz sweep range, each marker was a l t e r n a t e l y s h i f t e d down and up i n f r e q u e n c y 30 kHz. T h i s was e s t i m a t e d t o have broadened t h e methane a b s o r p t i o n s a p p r o x i m a t e l y 45 kHz, w h i c h was s m a l l compared w i t h t h e u s u a l t o t a l w i d t h a t h a l f - h e i g h t o f 450 kHz. To t e s t the o v e r a l l f r e q u e n c y l i n e a r i t y o f t h e system and the a c c u r a c y o f t h e method o f f r e q u e n c y d e t e r m i n a t i o n , s e v e r a l 55 r u n s were made u s i n g marker d o u b l e t s spaced 1 MHz a p a r t t o g i v e 3 d o u b l e t s w i t h i n a 3 MHz sweep range. An e x t r a p o l a t i o n employing d o u b l e t c e n t e r s gave e n d - p o i n t f r e q u e n c i e s t o w i t h i n .01 MHz ( c o u n t e r a c c u r a c y ) o f t h o s e d e t e r m i n e d by o b s e r v i n g t h e c o u n t e r "on the r u n " . The t o t a l system d e v i a t i o n from l i n e a r i t y was e x p e r i m e n t a l l y d e t e r m i n e d t o be l e s s than l/27o w h i c h was c l o s e t o the minimum w h i c h c o u l d be e x p e c t e d : a 1 c h a n n e l e r r o r i n 256 g i v e s a .4% d e v i a t i o n . 3.6 N o i s e M i n i m i z a t i o n I n a d d i t i o n t o t h e n o i s e m i n i m i z a t i o n a f f e c t e d by u s i n g S t a r k m o d u l a t i o n combined w i t h phase s e n s i t i v e d e t e c t i o n , o t h e r s t e p s were t a k e n t o m i n i m i z e n o i s e . These a r e d i s c u s s e d i n d e t a i l below. 3.6.1 S t r a y P i c k - u p Many components c r i t i c a l t o t h i s e xperiment e x h i b i t e d microphonism: t h e s e were the microwave and s t a b i l i z i n g o s c i l l a t o r s , the c e l l , and t h e d e t e c t o r s . The l a b o r a t o r y was s i t u a t e d i n a basement below grade w h i c h p r o b a b l y h e l p e d m i n i m i z e t h e p i c k - u p o f b u i l d i n g v i b r a t i o n s . I n a d d i t i o n , c a r e was e x e r c i s e d t o i n s u r e t h a t t h e v i b r a t i o n o f vacuum pumps d i d n o t c o u p l e i n t o the system. Troublesome c o o l i n g f a n s were s u p p o r t e d i n such a manner t h a t they d i d n ' t a f f e c t the system. To m i n i m i z e e l e c t r i c a l p i c k - u p , much a t t e n t i o n was g i v e n t o t h e d e t e c t o r i n p u t system w h i c h o p e r a t e d a t the l o w e s t s i g n a l l e v e l . The ground l o o p formed by t h e c e l l , d e t e c t o r mounts and i n p u t c i r c u i t r y , was b r o k e n by c o v e r i n g t h e d e t e c t o r mount f l a n g e s w i t h p o l y e s t e r tape and s e c u r i n g them 56 w i t h p l a s t i c screws. The T r i a d G-10 i n p u t t r a n s f o r m e r was v e r y w e l l s h i e l d e d : t h r e e u-metal and two copper s h i e l d s t o g e t h e r w i t h humbucking c o i l s were s p e c i f i e d t o g i v e up t o a 135 db (6 X 1 0 6 v o l t a g e r a t i o ) r e d u c t i o n i n p i c k - u p . I n a d d i t i o n , t h e l e a d s f rom t h e d e t e c t o r s and t o the phase s e n s i t i v e d e t e c t o r were double s h i e l d e d and s h o r t . The b o l o m e t e r b i a s b a t t e r y (Eveready 732) was s h i e l d e d and a l l c o n n e c t i o n s t o i t and t h e i n p u t t r a n s f o r m e r were made w i t h i n a s t e e l box. The 2000 ohm l o a d r e s i s t o r s i n t h e b o l o m e t e r b i a s c i r c u i t were n o n - i n d u c t i v e l y wound f o r low p i c k - u p . I n s p i t e o f w e l l r e g u l a t e d power s u p p l i e s , l a r g e n o i s e s p i k e s were i n t r o d u c e d i n t o t h e system when th e room l i g h t s were s w i t c h e d on o r o f f . T h i s was remedied by l e a v i n g the l i g h t s on i n t h e l a b o r a t o r y around t h e c l o c k . C l e a r l y , however, s i n c e the l u x u r y o f an e l e c t r o m a g n e t i c a l l y s h i e l d e d room [66, 69] was n o t to be had, t r a n s i e n t s o r i g i n a t i n g i n o t h e r p a r t s o f the b u i l d i n g had t o be t o l e r a t e d (the o v e r a l l e f f e c t o f t h e s e t r a n s i e n t s c o u l d n o t be a s s e s s e d ) . 3.6.2 D e t e c t o r t o P r e a m p l i f i e r Impedence Match The b o l o m e t e r d e t e c t o r s used were low impedence d e v i c e s o p e r a t e d a t a d-c r e s i s t a n c e o f about 200 ohms. The o p e r a t o r ' s manual [79] f o r a PAR 128 l o c k - i n a m p l i f i e r showed t h a t f o r t h i s r e s i s t a n c e w i t h a m o d u l a t i o n f r e q u e n c y o f 1 kHz, the n o i s e f i g u r e was a p p r o x i m a t e l y : 11 db; t h a t i s , t h e s i g n a l t o n o i s e r a t i o o b t a i n e d from the d e t e c t o r would have been degraded by a f a c t o r o f 3.V2 upon p a s s i n g t h r o u g h the f i r s t few s t a g e s o f the phase s e n s i t i v e d e t e c t o r . However, the use o f an i n p u t t r a n s f o r m e r i n c r e a s e d t h e impedence seen by 57 t h e l o c k - i n by the t r a n s f o r m e r t u r n s r a t i o squared. I f t h e sou r c e impedence seen by t h e PAR 128 was between 250 k i l o - o h m s and 7 megohms a t 1000 Hz, a n o i s e f i g u r e o f b e t t e r t h a n .05 db was a c h i e v e d , t h a t i s the s i g n a l t o n o i s e r a t i o i s o n l y degraded 1%. Assuming a b o l o m e t e r d e t e c t o r dynamic r e s i s t a n c e o f 275 ohms ( s e c t i o n 3.7.5) and a t u r n s r a t i o o f 1:37.7,: t h e impedence seen by the l o c k - i n a m p l i f i e r i s 400 k i l o - o h m s , g i v i n g a n o i s e f i g u r e o f b e t t e r t h a n .05 db. A 5 m i c r o f a r a d p o l y e s t e r c a p a c i t o r was p l a c e d i n s e r i e s w i t h the p r i m a r y o f the t r a n s f o r m e r t o p r e v e n t a d e l e t e r i o u s d-c m a g n e t i z i n g c u r r e n t from f l o w i n g . 3.6.3 M i n i m i z a t i o n o f Microwave O s c i l l a t o r N o i s e The microwave o s c i l l a t o r s produced b o t h AM and FM n o i s e . B o t h t y p e s o f n o i s e were i m p o r t a n t f o r t h i s e xperiment. AM n o i s e was i m p o r t a n t because the d e t e c t o r s responded d i r e c t l y to i t . FM n o i s e was i m p o r t a n t because i t gave the microwave o s c i l l a t o r s s p e c t r a l w i d t h ( s i m i l a r t o t h e e f f e c t o f o p t i c a l s l i t w i d t h ) w h i c h i n p r i n c i p l e broadened t h e a b s o r p t i o n l i n e s . FM n o i s e was a l s o i m p o r t a n t because i t c o u l d be c o n v e r t e d t o AM n o i s e i n v a r i o u s ways and thus a l s o be d e t e c t e d by the b o l o m e t e r s . The phase s t a b i l i z a t i o n o f the microwave o s c i l l a t o r s r e d u c e d t h e FM n o i s e t o the p o i n t t h a t i t approached t h a t o f the microwave r e f e r e n c e f r e q u e n c y . The AM n o i s e was u n a f f e c t e d by the s y n c h r o n i z a t i o n . A t X- and P-bands the o n l y n o i s e r e d u c t i o n o f the backward wave o s c i l l a t o r s was br o u g h t about by t h e s y n c h r o n i -z a t i o n p r o c e s s . The power l e v e l l i n g o p t i o n i n c r e a s e d the n o i s e 58 seven f o l d a t 1 kHz and was n o t used. A t K-band, the n o i s e o f the k l y s t r o n s , a l t h o u g h perhaps n o t improved, was k e p t f rom g e t t i n g worse. The o u t p u t power o f a k l y s t r o n appears as a "mode"; the p l o t o f o u t p u t power v e r s u s r e f l e c t o r v o l t a g e ( f r e q u e n c y ) i s p a r a b o l i c . A k l y s t r o n w h i c h i s s t a t i o n a r y i n f r e q u e n c y can be p l a c e d on the t op o f the mode, w h i c h has z e r o power v e r s u s f r e q u e n c y s l o p e , by s u i t a b l y a d j u s t i n g the r e f l e c t o r v o l t a g e and t h e m e c h a n i c a l s i z e o f t h e c a v i t y . A t t h i s p o i n t , t h e minimum n o i s e i s o b t a i n e d because a f r e q u e n c y v a r i a t i o n g i v e s the s m a l l e s t power v a r i a t i o n , t h a t i s , FM t o AM n o i s e c o n v e r s i o n i s l e a s t e f f i c i e n t . FM t o AM n o i s e c o n v e r s i o n [69, 92] was i m p o r t a n t because t h e FM n o i s e o f a microwave o s c i l l a t o r i s u s u a l l y much g r e a t e r t h a n the AM n o i s e [7, 4 1 ] . Any mechanism w h i c h a f f e c t e d FM t o AM n o i s e c o n v e r s i o n d i d n o t need t o be v e r y e f f i c i e n t t o produce a l a r g e i n c r e a s e i n AM n o i s e . The amount o f e x t r a AM n o i s e produced by a k l y s t r o n upon t u n i n g o f f the top o f the mode depends on the FM n o i s e p r e s e n t and t h e p o w e r - v e r s u s - f r e q u e n c y s l o p e . M u e l l e r [66] found t h e AM n o i s e (as an average .3 t o 5 kHz o f f c a r r i e r ) o f a 2K39 i n c r e a s e d a mere I V 2 db upon g o i n g f rom t h e top o f t h e mode t o the h a l f power p o i n t , b u t t h a t o f an X-13 k l y s t r o n i n c r e a s e d 19 db ( n e a r l y 100X). I n t h e case o f the OKI k l y s t r o n s used h e r e , t h e AM n o i s e a t a m o d u l a t i o n f r e q u e n c y o f 1 kHz was found t o i n c r e a s e t y p i c a l l y 8X w i t h i n 3 MHz a t 19 GHz. The AM n o i s e g e n e r a t e d by a k l y s t r o n can be m i n i m i z e d a c r o s s a f r e q u e n c y range by v a r y i n g t h e s i z e o f t h e k l y s t r o n 59 c a v i t y i n s y n c h r o n i s m w i t h the f r e q u e n c y sweep. The k l y s t r o n c a v i t y s i z e a f f e c t s o n l y t h e n o i s e i n t h i s case because t h e f r e q u e n c y i s d e t e r m i n e d by t h e microwave r e f e r e n c e f r e q u e n c y t h r o u g h t h e s y n c h r o n i z e r s e r v o l o o p and t h e r e f l e c t o r v o l t a g e . T h i s n o i s e r e d u c t i o n a c r o s s a f r e q u e n c y range was a c h i e v e d w i t h an OKI 20V10 k l y s t r o n by u s i n g an e l e c t r o m a g n e t t o v a r y t h e s i z e o f the c a v i t y . The e l e c t r o m a g n e t was s e c u r e d t o the t u n i n g knob b r a c k e t ; t h i s a c t e d upon a h i n g e d i r o n p l a t e w h i c h was l i n k e d by a r o d t o the l a r g e h o l l o w l e v e r adjustment screw on top o f the k l y s t r o n . I n t h e r e g i o n o f ma g n e t i c s a t u r a t i o n , i t was found the r e q u i r e d e l e c t r o m a g n e t c u r r e n t was a l i n e a r f u n c t i o n o f the f r e q u e n c y and t o be w i t h o u t s i g n i f i c a n t h y s t e r e s i s . A v a r i a b l e g a i n , v a r i a b l e o f f s e t d-c a m p l i f i e r ( f i g u r e 3.5) was c o n s t r u c t e d . I t a l l o w e d the f r e q u e n c y a n a l o g v o l t a g e o f the s i g n a l a v e r a g e r t o d r i v e the e l e c t r o m a g n e t so t h a t the n o i s e minimum p r e c i s e l y t r a c k e d the f r e q u e n c y . The AM n o i s e o f t h e k l y s t r o n was thus m i n i m i z e d a c r o s s the sweep range. 3.6.4 N o i s e C a n c e l l a t i o n Even w i t h the k l y s t r o n m a i n t a i n e d on t h e top o f t h e mode f o r minimum AM n o i s e , t h e n o i s e was s t i l l s e v e r a l t i m e s g r e a t e r t h a n t h a t o f the backward wave o s c i l l a t o r s used a t X- and P-band; f o r t h i s r e a s o n a n o i s e c a n c e l l a t i o n scheme was t r i e d and s u c c e s s f u l l y employed w i t h t h e K-band k l y s t r o n s . I n t h i s scheme t h e " k l y s t r o n n o i s e " s i g n a l d e t e c t e d b e f o r e t h e c e l l was s u b t r a c t e d from t h e "methane a b s o r p t i o n p l u s k l y s t r o n n o i s e " s i g n a l d e t e c t e d a f t e r t h e c e l l l e a v i n g o n l y a "methane a b s o r p t i o n " s i g n a l . I f the d e t e c t o r s c o n t r i b u t e d no n o i s e o f 60 t h e i r own and i n a d d i t i o n , the c e l l c o n t r i b u t e d no n o i s e , then i n p r i n c i p l e t h e system s h o u l d have worked v e r y w e l l . I n p r a c t i c e , i t d i d work v e r y w e l l , t o o . Each b o l o m e t e r d e t e c t o r r e q u i r e d a p p r o x i m a t e l y 10 mw o f microwave power f o r optimum o p e r a t i o n . T h i s was s u p p l i e d t o t h e " n o i s e r e f e r e n c e " b o l o m e t e r by s u i t a b l y a t t e n u a t i n g a t e n p e r c e n t f r a c t i o n o f t h e microwave power g o i n g t o the c e l l . The r e m a i n i n g 90% o f the power, w h i c h was s e v e r e l y a t t e n u a t e d by the f o u r s e c t i o n - 13 meter c e l l , gave a p p r o x i m a t e l y 10% o f the t o t a l power t o the "sample" b o l o m e t e r w i t h a s s o c i a t e d a t t e n u a t o r . The s u b t r a c t i o n was pe r f o r m e d by b r i d g i n g the ou t p u t o f the two b o l o m e t e r s a c r o s s the p r i m a r y o f the T r i a d G-10 i n p u t t r a n s f o r m e r ( f i g u r e 3.2) where t h e n o i s e common t o b o t h d e t e c t o r s was r e j e c t e d as a common mode v o l t a g e . The power l e v e l s g o i n g t o t h e b o l o m e t e r s were a d j u s t e d w i t h a t t e n u a t o r s f o r b e s t n o i s e c a n c e l l a t i o n c o n s i s t e n t w i t h optimum b o l o m e t e r b i a s power. The n o i s e c a n c e l l a t i o n scheme worked so w e l l t h a t some-times t h e microwave power c o u l d be t u r n e d c o m p l e t e l y o f f w i t h no d e t e c t a b l e d i f f e r e n c e i n t h e n o i s e . Y e t w i t h the microwave power back on, when t h e microwave power g o i n g t o e i t h e r d e t e c t o r i n d i v i d u a l l y was removed, t h e n o i s e v o l t a g e would i n c r e a s e by a f a c t o r o f twenty o r more. The n o i s e c a n c e l l a t i o n scheme was so e f f e c t i v e t h a t the e l a b o r a t e n o i s e t r a c k i n g scheme d e s c r i b e d i n t h e p r e c e d i n g s e c t i o n made o n l y a m a r g i n a l d i f f e r e n c e . The d e t e c t o r n o i s e , o f c o u r s e , was 1.4 times g r e a t e r t h a n 61 i f o n l y one b o l o m e t e r were used, b u t t h i s was no p e n a l t y compared w i t h the b e n e f i t s o f n o i s e c a n c e l l a t i o n . T h i s s i m p l e n o i s e c a n c e l l a t i o n scheme worked e x t r e m e l y w e l l because i t worked a t a u d i o f r e q u e n c i e s and n o t a t microwave f r e q u e n c i e s where the e x a c t e q u a l i t y o f sample and r e f e r e n c e arms n e c e s s a r y f o r n o i s e c a n c e l l a t i o n over a f r e q u e n c y span i s many t i m e s more d i f f i c u l t t o a c h i e v e . A phase change o f 180° i n one arm o f a microwave b r i d g e t u r n s n o i s e c a n c e l l a t i o n i n t o n o i s e r e i n f o r c e m e n t . The same microwave f r e q u e n c y phase change w i t h a u d i o f r e q u e n c y b a l a n c i n g w o u l d have caused o n l y a n e g l i g i b l e change (1 kHz/10 GHz = 10" 7) because the d e t e c t o r s were envelope d e t e c t o r s and d i d n ' t r e s p o n d t o i n d i v i d u a l microwave c y c l e s . 3.6.5 C e l l N o i s e The optimum n o i s e c a n c e l l a t i o n d i s c u s s e d i n t h e p r e c e d i n g s e c t i o n was n o t o b t a i n a b l e a t e v e r y f r e q u e n c y w i t h the equipment d e s c r i b e d . T h i s was due t o AM n o i s e g e n e r a t e d w i t h i n the c e l l , w h i c h made t h e microwave n o i s e b e f o r e and a f t e r the c e l l o f d i f f e r e n t c h a r a c t e r . Complete n o i s e c a n c e l l a t i o n under t h i s c o n d i t i o n was n o t p o s s i b l e . F o r i n s t a n c e , the n o i s e would i n c r e a s e t y p i c a l l y 3X from optimum w i t h i n 3 MHz; a t any p o i n t w i t h g r e a t e r t h a n optimum n o i s e , c o n d i t i o n s c o u l d n o t be improved by m a n i p u l a t i o n o f e i t h e r a t t e n u a t o r a l o n e . The c e l l g e n e r a t e d AM n o i s e from FM n o i s e . I f t h e r e was FM n o i s e , some o f t h i s was c o n v e r t e d t o AM n o i s e i f the c e l l had a t r a n s m i s s i o n c h a r a c t e r i s t i c w h i c h was n o t f l a t o r e x h i b i t e d a n o n - l i n e a r phase s h i f t [ 9 2 ] . The f r e q u e n c y r e g i o n 62 o f i n t e r e s t was 1 kHz each s i d e o f the microwave f r e q u e n c y a t any p a r t i c u l a r t i m e . S i n c e the f r e q u e n c y was swept 3 MHz, i t was d e s i r a b l e t o have f l a t a t t e n u a t i o n and a l i n e a r phase s h i f t o v e r t h i s e n t i r e r e g i o n . To a c o n s i d e r a b l e e x t e n t the c e l l was " t u n e d " t o m i n i m i z e FM t o AM n o i s e c o n v e r s i o n . T h i s was done by i n s e r t i n g an X-band s l i d e - s c r e w t u n e r i m m e d i a t e l y b e f o r e the t u n a b l e X-band sample d e t e c t o r mount and c a r e f u l l y t u n i n g t h e t h r e e v a r i a b l e s f o r minimum n o i s e a c r o s s the e n t i r e 3 MHz sweep range. W i t h good t u n i n g t h e b a l a n c e d n o i s e c o u l d be k e p t t o w i t h i n a f a c t o r o f two o f the d e t e c t o r n o i s e over a 3 MHz sweep range. 3.7 Bolometer D e t e c t o r s B o l o meters were chosen as d e t e c t o r s i n t h i s e xperiment because o f t h e i r l o n g s t a n d i n g r e p u t a t i o n as n e a r l y pure Johnson ( t h e r m a l ) n o i s e s o u r c e s o p e r a t i n g a t an e l e v a t e d t e mperature [5,24,91,92,93,97]. However, s e v e r a l t h e o r e t i c a l p a pers [11,42,60] i n d i c a t e t h a t i n a d d i t i o n t o Johnson n o i s e , n o i s e due t o s t a t i s t i c a l t e m p e r a t u r e f l u c t u a t i o n i s g e n e r a t e d . An a t t empt has been made by L i and Chen [51] t o measure t h i s , b u t a comparison w i t h the work o f Feher [19] w i l l i n d i c a t e more work needs t o be done on t h i s s u b j e c t . Bolometer d e t e c t o r s are n o t n o r m a l l y used i n microwave s p e c t r o s c o p y because o f t h e i r low c o n v e r s i o n g a i n , G, a t the m i c r o w a t t power l e v e l s used. The power l e v e l s a r e u s u a l l y r e s t r i c t e d because o f s a t u r a t i o n e f f e c t s [ 9 7 ] . But b o l o m e t e r s c o u l d be used i n t h i s e xperiment because th e power l e v e l s r e q u i r e d f o r a good c o n v e r s i o n g a i n were p e r m i s s i b l e due t o 63 the s m a l l d i p o l e m a t r i x element o f methane and t h e h i g h e r than u s u a l c e l l p r e s s u r e m a i n t a i n e d . I t w i l l be shown t h a t the same degree o f power s a t u r a t i o n g i v e n by one m i c r o w a t t t o the J = 0 -> 1 t r a n s i t i o n o f OCS i s g i v e n by 340 m i l l i w a t t s t o the most e a s i l y s a t u r a t e d component o f the seven microwave methane t r a n s i t i o n s o b s e r v e d h e r e . The b o l o m e t e r s p u r c h a s e d f o r use i n t h i s experiment were Narda type N610B/38B7 b o l o m e t e r s i n a 1N23 s t y l e c a r t r i d g e . B o l o m e t e r s are microwave t e r m i n a t i n g r e s i s t o r s t h e r e s i s t a n c e o f w h i c h v a r i e s i n p r o p o r t i o n t o the heat energy s u p p l i e d by t h e microwaves. I n o r d e r t h a t the r e s i s t a n c e change r a p i d l y w i t h v a r i a t i o n s i n microwave power, th e b o l o m e t e r s were c o n s t r u c t e d w i t h a v e r y s m a l l t h e r m a l c a p a c i t y : t h e r e s i s t i v e element was a f i n e p l a t i n u m w i r e t y p i c a l l y 3 / L > t o l 1 /2 m i c r o n s (30 t o 60 m i c r o i n c h e s ) i n d i a m e t e r . Wire o f such a s m a l l d i a m e t e r i s v e r y d i f f i c u l t t o h a n d l e , l e t a l o n e draw [ 2 8 ] ; t h e r e f o r e t h e p l a t i n u m w i r e was encased i n a s i l v e r j a c k e t and drawn to t h e r e q u i r e d d i a m e t e r , mounted i n the b o l o m e t e r case and t h e n enough o f t h e s i l v e r was e t c h e d away w i t h d i l u t e n i t r i c a c i d t o g i v e t h e p r o p e r r e s i s t a n c e . The l e n g t h o f exposed p l a t i n u m t o g i v e a room temperature r e s i s t a n c e o f 116 ohms v a r i e d from 1 / 2 mm f o r a 3/^ m i c r o n d i a m e t e r w i r e t o 2 mm f o r a I V 2 m i c r o n d i a m e t e r p l a t i n u m w i r e . A l t h o u g h t h e r e s i s t a n c e o f a b o l o m e t e r changes i n accordance w i t h t h e a p p l i e d m o d u l a t i o n , no s i g n a l v o l t a g e i s g e n e r a t e d . To r e c o v e r a s i g n a l a d-c c u r r e n t was p a s s e d t h r o u g h the b o l o m e t e r . I n t h i s experiment an a p p r o x i m a t e l y c o n s t a n t c u r r e n t was o b t a i n e d by u s i n g a w e l l s h i e l d e d 12 v o l t 64 z i n c - c a r b o n b a t t e r y and a 2000 ohm w i r e wound l o a d r e s i s t a n c e chosen f o r v e r y low 1/f n o i s e . The b o l o m e t e r s employed were d e s i g n e d t o o p e r a t e a t a t o t a l r e s i s t a n c e o f 200 ohms. T h i s gave t h e b e s t impedence match f o r l e a s t r e f l e c t i o n s and optimum power t r a n s f e r w i t h i n t h e d e s i g n f r e q u e n c y range o f the t u n a b l e waveguide d e t e c t o r mounts used. A t 200 ohms, the p l a t i n u m w i r e i s h e a t e d t o 516°K o r 1.7 times room temperature assuming a room temperature (300°K) r e s i s t a n c e o f 116 ohms. T h i s f i g u r e i s o b t a i n e d from the approximate r e l a t i o n AT = AR(l+aT)/aR (3.1) where the 0°C temperature c o e f f i c i e n t o f r e s i s t i v i t y , a = .0037 f o r p l a t i n u m [ 4 4 ] , R = 116 ohms and T = 27°C. 3.7.1 Bolometer N o i s e I f t h e b o l o m e t e r n o i s e were p u r e l y t h e r m a l o r Johnson n o i s e , t h e n o i s e v o l t a g e would be g i v e n by the e x p r e s s i o n E . c a = /4kTRAv (3.2) n o i s e ' where k i s Boltzmann's c o n s t a n t , T i s the a b s o l u t e temperature i n °K, R i s the r e s i s t a n c e i n ohms and Av i s t h e bandwidth i n H e r t z . The n o i s e v o l t a g e was e x p e c t e d t o i n c r e a s e by a f a c t o r o f 1.7 upon i n c r e a s i n g the b o l o m e t e r r e s i s t a n c e from 116 ohms a t 300°K t o 200 ohms a t 516°K, because b o t h the bo l o m e t e r r e s i s t a n c e and temp e r a t u r e were i n c r e a s e d by an approximate f a c t o r o f 1.7. I n p r a c t i c e t h i s was r o u g h l y t h e i n c r e a s e i n n o i s e v o l t a g e found upon the a p p l i c a t i o n o f 6 ma b i a s c u r r e n t i n the absence o f microwaves. The n o i s e found was c o n s i s t e n t w i t h t h a t g i v e n by a r e s i s t a n c e o f 200 ohms o p e r a t e d a t a p p r o x i m a t e l y t w i c e room temp e r a t u r e . 65 Feher [19] found u s i n g a d-c b i a s c u r r e n t o f 8 ma, t h a t the n o i s e temperature o f 20 b o l o m e t e r s o b t a i n e d from f o u r d i f f e r e n t m a n u f a c t u r e r s v a r i e d from 4 t o 40 (one apparently-d e f e c t i v e u n i t had a n o i s e temperature o f 1000). Some o f t h e s e n o i s e t e m p e r a t u r e s were s i g n i f i c a n t l y h i g h e r t h a n the e x p e c t e d v a l u e o f 2. Feher removed the a i r from one b o l o m e t e r w i t h a n o i s e temperature o f 10 and o b s e r v e d the n o i s e temperature drop t o t h e e x p e c t e d v a l u e o f 2. Feher s u g g e s t e d t h a t the u n e x p e c t e d l y h i g h n o i s e t e m p e r a t u r e s found f o r s e v e r a l b o l o m e t e r s might be due t o a i r c u r r e n t s w i t h i n t h e b o l o m e t e r case. L i and Chen [51] have p u b l i s h e d a paper e n t i t l e d " N o i s e i n M e t a l B o l o m e t e r s " i n w h i c h t h e y a t t e m p t e d t o measure t h e s t a t i s t i c a l t e m p e r a t u r e f l u c t u a t i o n n o i s e w h i c h has been shown t h e o r e t i c a l l y [11,42,60] t o be g e n e r a t e d by a v e r y s m a l l body i n a d d i t i o n t o Johnson n o i s e . At a f r e q u e n c y o f 1 kHz, t h e y measured v e r y h i g h n o i s e t e m p e r a t u r e s o f 250 and 3500 a t d-c b i a s c u r r e n t s o f 6 and 8.7 ma, r e s p e c t i v e l y . The b o l o m e t e r s used were t h r e e Narda N821B b o l o m e t e r s w h i c h a r e e l e c t r i c a l l y e q u i v a l e n t t o the N610B/38B7 b o l o m e t e r s used i n t h i s e x p eriment. L i and Chen [51] comment t h a t two o f t h e t h r e e b o l o m e t e r s burned o u t b e f o r e complete measurements were o b t a i n e d . The work o f Feher [19] suggests t h a t the one b o l o m e t e r on w h i c h they d i d complete a l l measurements was s i m p l y d e f e c t i v e . L i and Chen [51] have s t a t e d " t h e h e a t l o s s o f t h e (N821B bo l o m e t e r ) w i r e i s m a i n l y due t o r a d i a t i o n " . On the c o n t r a r y , the h e a t l o s s was e x p e r i m e n t a l l y d e t e r m i n e d i n our l a b o r a t o r y 66 • t o be a p p r o x i m a t e l y 99% due t o f r e e a i r c o n v e c t i o n . A s m a l l h o l e was b o r e d t h r o u g h the b o l o m e t e r case and the b o l o m e t e r was p l a c e d i n a vacuum chamber. Upon e v a c u a t i o n o n l y .166 mw o f d-c power were r e q u i r e d t o heat the p l a t i n u m w i r e t o 200 ohms, b u t upon r e t u r n i n g t h e b o l o m e t e r t o a t m o s p h e r i c p r e s s u r e , i t took 15.3 mw o f power t o b r i n g t h e b o l o m e t e r up t o 200 ohms. Thus the c o m b i n a t i o n o f r a d i a t i o n and c o n d u c t i o n down t h e l e a d s a c c o u n t s f o r o n l y 1% o f the h e a t l o s s . T h i s i s i n agreement w i t h the low e m i s s i v i t y o f p l a t i n u m a t 492°K (219°C), w h i c h i s o n l y 3.9% [ 5 6 ] . I r o n i c a l l y , i t had been shown p r e v i o u s l y by Langmuir [50] i n 1912 and K i n g [44] i n 1914 t h a t r a d i a t i v e l o s s e s a c c o u n t f o r o n l y a p p r o x i m a t e l y 1% o f the h e a t l o s s o f a l o n g p l a t i n u m w i r e h e a t e d t o a p p r o x i m a t e l y t w i c e room temperature i n s t i l l a i r . 3.7.2 Power S a t u r a t i o n B o l o m e t e r s c o u l d be used i n t h i s e x periment because the power l e v e l s r e q u i r e d f o r a good d e t e c t o r c o n v e r s i o n g a i n c o u l d be t o l e r a t e d by methane w i t h o u t s i g n i f i c a n t power s a t u r a t i o n . The r e d u c t i o n i n t h e peak a b s o r p t i o n c o e f f i c i e n t due t o power s a t u r a t i o n i s g i v e n by [ 9 7 ] : V a x _ 1 Yo,max f l 6 i T 2 S a b v I t 2 T T T ) (3.3) 1 + 3ch where i s the peak a b s o r p t i o n c o e f f i c i e n t i n a r a d i a t i o n f i e l d o f i n t e n s i t y I ( i n u n i t s o f qu a n t a p e r second p e r u n i t c r o s s s e c t i o n a l a r e a ) , y i s t h e peak a b s o r p t i o n o, max r r c o e f f i c i e n t as t h e r a d i a t i o n i n t e n s i t y approaches z e r o , S & b i s 67 the l i n e s t r e n g t h [17,26,97], v i s the f r e q u e n c y i n H e r t z , c i s the speed o f l i g h t , h i s P l a n c k ' s c o n s t a n t , x i s the mean time between c o l l i s i o n s , and t i s the i n v e r s e r a t e o f approach to e q u i l i b r i u m by c o l l i s i o n ( p r o p o r t i o n a l t o T ) . Good v a l u e s f o r t and T are n o t r e a d i l y a v a i l a b l e t o c a l c u l a t e a b s o l u t e degrees o f power s a t u r a t i o n , b u t they a r e p r o p o r t i o n a l t o t h e i n v e r s e o f t h e l i n e w i d t h , A v - 1 , w h i c h a l l o w s a r e l a t i v e c a l c u l a t i o n . The p r o d u c t o f i n t e n s i t y and f r e q u e n c y , I v , i s p r o p o r t i o n a l t o (power/area). The r e d u c t i o n i n t he peak a b s o r p t i o n c o e f f i c i e n t due t o power s a t u r a t i o n can be w r i t t e n : Ymax _ . 1 ( 3 4 ) Yo,max { S ^ • p o w e r 1 + K [ a r e a : p r e s s u r e 2 • ( A v / P ) 2 J where K i s a c o n s t a n t dependent on t h e gas and (Av/P) i s the p r e s s u r e b r o a d e n i n g parameter i n MHz/torr. F o r a g i v e n m o l e c u l e , p r e s s u r e , c e l l and power l e v e l , the t r a n s i t i o n w i t h the g r e a t e s t v a l u e f o r S ^ w i l l be t h e most e a s i l y s a t u r a t e d . Of the seven methane t r a n s i t i o n s o b s e r v e d h e r e , t h e J = 18 E 2 J = 18 E 3 t r a n s i t i o n was the most e a s i l y s a t u r a t e d as seen by r e f e r e n c e t o the reduced l i n e s t r e n g t h s t a b u l a t e d i n t a b l e IV o f Dorney and Watson [ 1 7 ] : S a b / ( 0 z y ) 2 = 6 - 7 3 x l ° 5 : U s i n g the v a l u e o f 0 x y g i v e n by O z i e r [ 7 1 ] : 0 x y = 2.41 X 1 0 " 5 D, S a b = (6-73 X 1 0 5 ) ( 2 . 4 1 X 1 0 - 5 ) 2 = S ^ I X I O - 1 * The l i n e s t r e n g t h S ^ i s g i v e n i n terms o f t h e reduced l i n e s t r e n g t h S'^ by t h e r e l a t i o n [ 1 7 ] : S a b - Sa,b Sab ( 3 - 5 ) 68 where g , *= 2 i s the degeneracy common t o t h e E l e v e l s . I n c l , D t h i s c a l c u l a t i o n o n l y one. t r a n s i t i o n a r i s i n g . f r o m t h e do u b l y d e g e n e r a t e E l e v e l i s o f i n t e r e s t , so g_ , i s ta k e n as 1 cl t D and S a b = S a b = 3.91.X 10"•* i n t h i s case. The l i n e s t r e n g t h u s ed by Dorney and Watson [ 1 7 ] : S a b = ^ J ( a | v f | b > | 2 (3.6) faS i s summed o v e r a l l components a o f a, and 3 o f b and o v e r t h e t h r e e space f i x e d d i r e c t i o n s , f ; thus S & b c o r r e s p o n d s t o i s o t r o p i c r a d i a t i o n . But the J = 18 t r a n s i t i o n under c o n s i d e r a t i o n has 2J' + 1 = 37 S t a r k components. The m = 18 components s a t u r a t e most e a s i l y because t h e i n t e n s i t y o f the i n d i v i d u a l S t a r k components v a r i e s as m2 [17,97]. A " w o r s t c a s e " c a l c u l a t i o n w i l l be p e r f o r m e d u s i n g an m = 18 S t a r k component. The l i n e s t r e n g t h S a b (eq 3.6) i s summed o v e r a l l 37 S t a r k components f o r J = 18. J S QK - S^^Tm2 (3.7) 'ab u a b m=-J f ( 1 / s ) J ( J + D ( 2 J + D S ^ 1 where S & b i s the l i n e s t r e n g t h f o r t h e m = 1 S t a r k component. The l i n e s t r e n g t h f o r t h e m = 18 S t a r k component o f t h e J = 18 E 2 -»• J = 18 E 3 t r a n s i t i o n o f methane i s : Qm=J _ T 2 cm=l b a b " J b a b = 3 J [ ( J + l ) ( 2 J + l ) ] - 1 S a b (3.8) = 3. 0 1 X 1 0 " 5 The l i n e s t r e n g t h f o r i s o t r o p i c r a d i a t i o n o f t h e 69 J = 0 •*• 1 t r a n s i t i o n o f OCS i s [ 2 6 ] : S a b S = ^ < J + 1 > = - 5 0 4 where t h e d i p o l e moment, u, has been t a k e n as .71 D [ 2 6 ] . For the same degree o f power s a t u r a t i o n , t h e e x p r e s s i o n : S a b-power [ a r e a * p r e s s u r e 2 • ( A v / P ) 2 J must be t h e same f o r t h e two gases, assuming (3.9) A vOGS / t :OCS = A vCH, / t :CH,- ( 3 ' 1 0 ) U s i n g a p r e s s u r e b r o a d e n i n g parameter (Av/P) = 6 MHz / t o r r f o r OCS [26] and 2.7 MHz/torr f o r methane [ 7 6 ] , the r e l a t i o n : S a b " ' p o w e r S a b S * P o w e r p r e s s u r e 2 • ( A v / P ) 2 p r e s s u r e 2 • ( A v / P ) 2 3. 0 1 X 1 0 - 5 -power = .504(.001) 6 0 2 - ( 2 . 7 ) 2 6 2 ( 6 ) 2 g i v e s a power o f 340 mw r e q u i r e d t o g i v e the same degree o f power s a t u r a t i o n i n the m = 18 S t a r k component o f the J = 18 E 2 -»- J = 18 E 3 t r a n s i t i o n o f methane a t 60 micro n s p r e s s u r e as one m i c r o w a t t produces i n t h e J = 0 1 t r a n s i t i o n o f OCS a t 6 m i c r o n s p r e s s u r e . The o b s e r v e d J=18 E 2 J=18 E 3 methane a b s o r p t i o n l i n e was the s u p e r p o s i t i o n o f ( 2 J + 1) - m v a l u e s . The lower v a l u e s o f m were more d i f f i c u l t t o s a t u r a t e and thus the 340 mw f i g u r e i s c o n s e r v a t i v e : at th e power l e v e l s employed (^60 mw f o r t h e BWO's and 'vlOO mw f o r t h e k l y s t r o n s ) no s i g n i f i c a n t amount o f s a t u r a t i o n f o r any o f th e o b s e r v e d microwave methane a b s o r p t i o n s was ex p e c t e d . 70 3.7.3 C o n v e r s i o n G a i n The p r e v i o u s c a l c u l a t i o n has shown t h a t the microwave energy d e n s i t y i n t h e c e l l c o u l d be m a i n t a i n e d a t a r e l a t i v e l y h i g h l e v e l w i t h o u t s i g n i f i c a n t power s a t u r a t i o n . T h i s microwave power l e v e l p e r m i t t e d a good b o l o m e t e r c o n v e r s i o n g a i n , G, w h i c h has been g i v e n by Long [53] and Feher [19] a s : 4P I 2 (dR/dP^ 2 G = b o l o m e t : : e r s i g n a l power o u t p u t _ r f v ' ' bol o m e t e r s i g n a l power i n p u t R ( l - I 2 d R / d P ) 2 P P G = 4 ( d R / d P ) 2 — (3.11) *1 _ J * c dR R dP, The c o n v e r s i o n g a i n o f the N610B/38B7 b o l o m e t e r s used can be shown t o have a maximum i n t h e v i c i n i t y o f 10 mw microwave power. These b o l o m e t e r s have a t y p i c a l room temperature r e s i s t a n c e , R c, o f 116 ohms and t y p i c a l l y r e q u i r e 15.3 mw o f microwave p l u s d-c power t o b r i n g them up t o a r e s i s t a n c e o f 200 ohms. The r e s i s t a n c e o f microwave b o l o m e t e r s i n g e n e r a l f o l l o w s t he e m p i r i c a l r e l a t i o n [ 2 8 ] : f r om w h i c h R = R c + K P - 9 (3.12) dR/dP « .9(R - R )/P = 5 ohms/mw (3.13) a t 200 ohms. The c o n v e r s i o n g a i n w i l l be c a l c u l a t e d as a f u n c t i o n o f microwave power, P r£. w h i l e m a i n t a i n i n g the r e s i s t a n c e a t 200 ohms. S u b s t i t u t i n g the v a l u e s dR/dP = 5 ohms/mw and R = 200 ohms i n t o the above e q u a t i o n , 4P J P J 6 - r f d c (3.14) (40 - P d c ) 2 71 where P r£ and P^ c are e x p r e s s e d i n m i l l i w a t t s and P^ c=15.3-P r£ U s i n g e q u a t i o n (3.14) t h e c o n v e r s i o n g a i n , G, o f the N610B/38B7 b o l o m e t e r s was c a l c u l a t e d as a f u n c t i o n o f t h e microwave power f o r a t o t a l power o f 15.3 mw. The r e s u l t s a r e t a b u l a t e d i n t a b l e VI w h i c h shows a maximum c o n v e r s i o n g a i n i n the v i c i n i t y o f 10 mw microwave power. T a b l e VI N610B/38B7 Bolometer C o n v e r s i o n G a i n P r f (mw) G P d c (mw) .001 .01 .1 1.0 10.0 15.0 .0001 .001 .0099 .087 .18 .014 15.3 15.3 15.2 14.3 5.3 .3 The b o l o m e t e r s i g n a l i s p r o p o r t i o n a l t o the p r o d u c t G s P r £ w h i c h i s t a b u l a t e d i n t a b l e V I I . T h i s g i v e s a maximum s i g n a l s t r e n g t h i n the v i c i n i t y o f 9 mw microwave power and 6.3 mw d-c power a t a b o l o m e t e r r e s i s t a n c e o f 200 ohms. E x p e r i m e n t a l c o n d i t i o n s were a r r a n g e d such t h a t 6 t o 7 mw o f d-c power were d i s s i p a t e d by the b o l o m e t e r s depending on the s t a t e o f charge o f the b i a s b a t t e r y and 8 t o 9 mw o f microwave power were s u p p l i e d t o b r i n g t h e bo l o m e t e r r e s i s t a n c e t o 200 ohms. T h i s gave near optimum c o n d i t i o n s f o r a t o t a l b o l o m e t e r power o f 15.3 mw. T a b l e V I I The P r o d u c t G-P 72 P r f (mw) G ' P r f P Q C (mw) 1 .09 14.3 2 .30 13.3 3 .57 12.3 4 .89 11.3 5 1.17 10.3 6 1.42 9.3 7 1.62 8.3 8 1.74 7.3 9 1.80 6.3 10 1.76 5.3 11 1.63 4.3 12 1.41 3.3 13 1.09 2.3 14 .68 1.3 15 .17 .3 T a b l e V I I I shows t h e manner i n w h i c h the r e l a t i v e b o l o m e t e r s i g n a l o u t p u t power i n c r e a s e s as t h e microwave power i s i n c r e a s e d from 1 m i c r o w a t t t o 10 m i l l i w a t t s by s t e p s o f t e n . I n t h i s example, t h e microwave power i n c r e a s e s by IO1*, b u t the b o l o m e t e r s i g n a l o u t p u t power i n c r e a s e s by 1 . 8 X 1 0 7 . On the o t h e r hand, t h e t h e r m a l n o i s e remains c o n s t a n t because t h e t o t a l power i s f i x e d a t 15.3 mw. T h i s u n d e r s c o r e s the i m p o r t a n c e o f m a i n t a i n i n g a h i g h microwave 73 power l e v e l a t the d e t e c t o r f o r optimum b o l o m e t e r performance and demonstrates why b o l o m e t e r s cannot be used i n microwave s p e c t r o s c o p y i f power l e v e l s a r e r e s t r i c t e d t o m i c r o w a t t s because o f power s a t u r a t i o n . T a b l e V I I I R e l a t i v e Bolometer S i g n a l Output Power G i v e n by t h e P r o d u c t ( G - P r f ) a Microwave r p Power b ' * r f 1 yw .0000001 10 yw .00001 100 uw .001 1 mw .09 10 mw 1.8 a F i x e d t o t a l power = 15.3 These c a l c u l a t i o n s can be extended t o show t h a t t h e b o l o m e t e r s i g n a l o u t p u t can be i n c r e a s e d , w i t h o u t an i n c r e a s e o f microwave power, by i n c r e a s i n g the d-c b i a s c u r r e n t . T h i s has been done i n t a b l e IX. The c o n v e r s i o n g a i n i n c r e a s e s 2.8 f o l d w i t h a 20% i n c r e a s e i n b o l o m e t e r r e s i s t a n c e . S i n c e t h e p l a t i n u m w i r e t e m p e r a t u r e i s p r o p o r t i o n a l t o the r e s i s t a n c e , the t h e r m a l n o i s e v o l t a g e v a r i e s as the r e s i s t a n c e u s i n g e q u a t i o n ( 3 . 2 ) . W i t h c o n s t a n t microwave power, the b o l o m e t e r s i g n a l power i n c r e a s e s as the c o n v e r s i o n g a i n . The b o l o m e t e r s i g n a l v o l t a g e i n c r e a s e s as /R-P '. 7 s i n c e P . = E 2 . /R. ° s i g n a l s i g s i g The e x p e c t e d i n c r e a s e i n s i g n a l t o n o i s e w i t h i n c r e a s i n g b i a s 74 c u r r e n t i s t a b u l a t e d i n t a b l e IX: a 53% i n c r e a s e i n the s i g n a l t o n o i s e r a t i o i s e x p e c t e d by s i m p l y i n c r e a s i n g the b i a s c u r r e n t 46%,. T a b l e I X P r e d i c t e d Bolometer B e h a v i o r w i t h I n c r e a s e d B i a s C u r r e n t R (ohms) . 200 220 240 P t o t a l ( m w ) 15.3 20.3 25.0 P r f (mw) 9.0 9.0 9.0 I (ma) 5.6 7.2 8.2 P d c ( m w ) 6.3 11.3 16.0 G .20 .38 .56 R e l a t i v e t h e r m a l n o i s e v o l t a g e 1.0 1.1 1.2 R e l a t i v e s i g n a l power . . . . 1.0 1.9 2.8 R e l a t i v e s i g n a l v o l t a g e 1.0 1.44 1.84 R e l a t i v e s i g n a l / t h e r m a l n o i s e 1.0 1.31 1.53 W i t h the methane l i n e s now f i r m l y e s t a b l i s h e d and an adequate s u p p l y o f b o l o m e t e r s the above method o f i m p r o v i n g the s i g n a l t o n o i s e r a t i o c o u l d be t e s t e d . T h i s p r o c e s s cannot be c a r r i e d too f a r because o f t h e p o s s i b i l i t y o f b o l o m e t e r b u r n out o r i n c r e a s e d n o i s e due t o damaged s o l d e r j o i n t s [19,51]. 3.7.4 Bolometer Time Co n s t a n t The b o l o m e t e r time c o n s t a n t was i m p o r t a n t i n t h a t i t gave an upper l i m i t t o the S t a r k m o d u l a t i o n f r e q u e n c y . I n a d d i t i o n i t c o u l d have been i m p o r t a n t i f i t v a r i e d s i g n i f i c a n t l y w i t h t h e microwave power l e v e l because t h e optimum phase adjustment 75 o f t h e l o c k - i n a m p l i f i e r would have v a r i e d : i f , f o r i n s t a n c e , the phase were a d j u s t e d w i t h OCS a t a bo l o m e t e r r e s i s t a n c e o f 200 ohms, t h a t same phase might n o t have been optimum w h i l e s e a r c h i n g f o r a methane l i n e w i t h l i m i t e d power. Cohn [13] s t a t e t h e " t h e o r e t i c a l c o m p u t a t i o n ( o f t h e time c o n s t a n t o f a b o l o m e t e r ) i s e x c e e d i n g l y d i f f i c u l t " . U s i n g the t h e o r y o f the time c o n s t a n t o f a h o t - w i r e anemometer o f B u r g e r s [10] and Dryden & Kuethe [18] and t h e d a t a o f K i n g [ 4 4 ] , t h e t i m e c o n s t a n t o f a b o l o m e t e r i s c a l c u l a t e d i n appendix A. T h i s c a l c u l a t i o n shows t h a t whereas the b o l o m e t e r t i m e c o n s t a n t i s 360 ysec when a l l the power s u p p l i e d t o the bolo m e t e r i s d-c power, t h e time c o n s t a n t f a l l s t o 240 ysec under normal c o n d i t i o n s o f o p e r a t i o n i n t h i s e x p e r i m e n t , and r i s e s s l i g h t l y t o 250 yse c i n the case o f l i m i t e d microwave power w h i c h g i v e s a b o l o m e t e r r e s i s t a n c e o f o n l y 180 ohms.' The phase r e t a r d a t i o n , <{>, due t o the bo l o m e t e r time c o n s t a n t , T ( i n s e c o n d s ) , i s g i v e n by [ 6 1 ] : <j> = - a r c t a n 2-rrfx (3.15) where the f r e q u e n c y , f , i s i n H e r t z . A t the 1 kHz S t a r k m o d u l a t i o n f r e q u e n c y used, t h e d i f f e r e n c e i n phase r e t a r d a t i o n due t o the 240 and 250 m i c r o s e c o n d t i m e c o n s t a n t s o f d i f f e r i n g power l e v e l s i s l e s s t h a n 2 degrees r e s u l t i n g i n an a m p l i t u d e r e d u c t i o n due t o a m i s - a d j u s t e d l o c k - i n a m p l i f i e r [79] o f l e s s than 0.1%, whi c h i s n e g l i g i b l e . The b o l o m e t e r time c o n s t a n t , T, i s r e s p o n s i b l e f o r a r e d u c t i o n i n d e t e c t o r s i g n a l v o l t a g e g i v e n by [ 6 1 ] : A = l / v / l + ( 2 7 r f x ) i ! = .56 (3.16) The b o l o m e t e r time c o n s t a n t o f 240 ysec combined w i t h a m o d u l a t i o n f r e q u e n c y o f 1 kHz g i v e s a s i g n a l w h i c h i s r e d u c e d i n a m p l i t u d e t o 56% o f what i t would be i f e i t h e r the time c o n s t a n t or the m o d u l a t i o n f r e q u e n c y were z e r o . The s i g n a l t o n o i s e r a t i o c o u l d be improved s i g n i f i c a n t l y by e mploying a b o l o m e t e r w i t h a s m a l l e r time c o n s t a n t o r o p e r a t i n g a t a l o w e r m o d u l a t i o n f r e q u e n c y ( p r o v i d e d 1/f n o i s e o f t h e microwave o s c i l l a t o r c o u l d be removed). 3.7.5 Bolometer M a t c h i n g Impedence When m a t c h i n g the b o l o m e t e r t o the p r e a m p l i f i e r o f t h e phase s e n s i t i v e d e t e c t o r , the dynamic impedence o f the b o l o m e t e r s h o u l d be used f o r the b e s t match. T h i s can be found u s i n g t h e e q u i v a l e n t c i r c u i t s w h i c h have been d e v e l o p e d f o r b o l o m e t e r d e t e c t o r s by Jones [ 4 3 ] , S o r g e r [ 9 0 ] , and Cohn [ 1 3 ] . The e q u i v a l e n t c i r c u i t o f Cohn [13] i s g i v e n i n f i g u r e 3.6 where R i and C a r e o f dynamic o r i g i n and R i s t h e d-c o p e r a t i n g r e s i s t a n c e . V i s the open c i r c u i t s i g n a l v o l t a g e as the m o d u l a t i o n f r e q u e n c y approaches z e r o and i s the l o a d impedence. AAA/V 3 C F i g u r e 3.6 Bolometer E q u i v a l e n t C i r c u i t G i v e n by Cohn [13] 77 Cohn [13] has g i v e n an e x p r e s s i o n f o r R^: 2R(R-R ) R i = — (3.17) [R c+(R/9>] w h i c h y i e l d s the v a l u e R i = 244 ohms u s i n g a " c o l d " r e s i s t a n c e , R c, o f 116 ohms and a d-c o p e r a t i n g r e s i s t a n c e , R, o f 200 ohms. The sum o f t h e d-c and dynamic r e s i s t a n c e s : R + R i = 444 ohms i s i n good agreement w i t h t h e d-c dynamic impedence [ 4 3 ] , Z = 430 ohms, det e r m i n e d a t an o p e r a t i n g p o i n t o f 200 ohms. The d-c dynamic impedence, Z, has been d e f i n e d by Jones [ 4 3 ] : r, _ dE AE .099 / oo t, / o i ON HT = AT = TTJWI = 4 3 0 O H M S ( 3 ' 1 8 ) The v a l u e o f Z was de t e r m i n e d u s i n g d a t a o b t a i n e d i n our l a b o r a t o r y g i v e n i n t a b l e X. Tab l e X Bolom e t e r O p e r a t i n g V a l u e s n e a r 200 ohms R (ohms) E ( v o l t s ) AE I (amps)' AI 197 1.711 .00868 .099 .00023 203 1.810 .00891 The r e s i s t a n c e seen a t the t e r m i n a l s o f the b o l o m e t e r a t t h e m o d u l a t i o n f r e q u e n c y , R^, i s g i v e n by [ 1 3 ] : R = R + R i = 200+^ 241 = 275 ohms (3.19) 1 1 1 l + ( 2 T T f T ) 2 where f = 1 kHz, x = 240 y s e c , and R and R i a r e as p r e v i o u s l y d e f i n e d . The v a l u e R^ = 275 ohms was used t o match t h e b o l o m e t e r s to t h e phase s e n s i t i v e d e t e c t o r . . 78 CHAPTER 4 EXPERIMENTAL CONDITIONS AND RESULTS 4.1 I n t r o d u c t i o n T h i s d i f f i c u l t , c h a l l e n g i n g e xperiment can be b r o k e n i n t o t h r e e p a r t s . The f i r s t p a r t was based on the q u a r t i c d i s t o r t i o n c o n s t a n t , D T = 132,841 ±9 Hz d e t e r m i n e d by O z i e r [70] and the ( a t t h a t time) u n t r i e d c a l c u l a t i o n s o f Dorney and Watson [ 1 7 ] . The d i f f i c u l t y o f t h i s e x periment a r o s e from the extreme weakness o f the l i n e s , and r e s u l t e d i n n e c e s s a r i l y l o n g s e a r c h t i m e s . The i n i t i a l s e a r c h was conducted t h i r t y megahertz each s i d e o f 10,570.2 MHz. A l t h o u g h t h i s phase o f the e xperiment d i d n o t g i v e a methane a b s o r p t i o n , i t d i d produce a v e r y s e n s i t i v e , s t a b l e i n s t r u m e n t . The second p a r t o f t h e e x p e r i m e n t was based on the a d d i t i o n a l s i x t h degree t h e o r y o f K i r s c h n e r and Watson [45] and the q u a r t i c d i s t o r t i o n c o n s t a n t D^ and t h e s e x t i c d i s t o r t i o n c o n s t a n t s H ^ and d e t e r m i n e d by C u r l [ 1 5 ] . W i t h t h i s d a t a , the f i r s t f o u r l i n e s were o b s e r v e d w i t h i n a p e r i o d o f a few months r e s u l t i n g i n the p u b l i c a t i o n by H o l t , G e r r y and O z i e r [ 3 4 ] . The t h i r d p a r t b r o u g h t the experiment t o a s u c c e s s f u l c o n c l u s i o n . Here t h e i n s t r u m e n t was extended i n f r e q u e n c y coverage and the n o i s e c a n c e l l a t i o n scheme was employed. The t h e o r y was extended t o e i g h t h degree by O z i e r [72] and t h r e e more l i n e s were measured e n a b l i n g an e x c e l l e n t f i t o f the 79 e x p e r i m e n t a l l i n e s t o the t h e o r y [ 3 5 ] . 4.2 I n i t i a l S e a r c h P r i o r t o t h i s e x p e r i m e n t , O z i e r [70] had de t e r m i n e d the J=2 E 1 J=2 F i s p l i t t i n g o f methane [ 3 5 ] . From the s p l i t t i n g o f 7.97046 ±.00054 MHz [35] and t h e f ( J , K , t ) a v a l u e o f D T = 132,841 ±9 Hz was determined. At t h a t t i m e , D^ ,f ( J , K , t ) was c o n s i d e r e d t o be the o n l y t e n s o r s p l i t t i n g term so the i n i t i a l s e a r c h was c e n t e r e d a t 10,570.2 MHz. The v a l u e s o f t h e h i g h e r degree terms a f f e c t i n g the i n i t i a l s e a r c h f r e q u e n c y were unknown, b u t c o n s i d e r i n g t h a t a t J=12, the r o t a t i o n a l energy E = B 0 J ( J + 1 ) o f methane i s a p p r o x i m a t e l y 2 . 5 X 1 0 7 MHz, t h e y were n o t c o n s i d e r e d t o be i n s i g n i f i c a n t . The s e a r c h was a r b i t r a r i l y r e s t r i c t e d t o t h i r t y megahertz each s i d e o f 10,570.2 MHz because o f the g r e a t amount o f time r e q u i r e d t o s e a r c h a g i v e n f r e q u e n c y r e g i o n . To s e a r c h 3 MHz r e q u i r e d a t l e a s t one day's t i m e ; i f some f e a t u r e i n the 3 MHz wide sweep r e g i o n l o o k e d i n t e r e s t i n g , i t took on the average two more days o f sweep time t o r e v e a l i t s t r u e c h a r a c t e r . To s e a r c h t h o r o u g h l y the above r e g i o n r e q u i r e d the o r d e r o f one month's ti m e . D u r i n g t h i s p e r i o d , many o f the improvements d e s c r i b e d e a r l i e r were made t o the i n s t r u m e n t w h i c h w a r r a n t e d s e v e r a l more s e a r c h e s o f t h i s r e g i o n . The J=12 l i n e was l a t e r found t o be a t 10,321.91 MHz, a p p r o x i m a t e l y 250 MHz from t h e o r i g i n a l s e a r c h s i t e . As seen i n t a b l e XV, i t was n o t a poor v a l u e o f Drr, w h i c h h e l d back i n i t i a l s u c c e s s : use o f o n l y the w e l l d e t e r m i n e d v a l u e o f D T = 132,943.41 Hz gave a t r a n s i t i o n p r e d i c t i o n o f 10,578.34 MHz w h i c h i s even f u r t h e r from the e x p e r i m e n t a l f r e q u e n c y b u t 80 w i t h i n t h e o r i g i n a l s e a r c h range. C l e a r l y i g n o r a n c e o f t h e v a l u e s o f the s i x t h degree c o n s t a n t s , and Hg,p, was a s e v e r e h a n d i c a p . Moreover, even i f b o t h D T and had been known t o c u r r e n t a c c u r a c y , i n i t i a l s u c c e s s would n o t have been a c h i e v e d . These d i s t o r t i o n c o n s t a n t s g i v e a p r e d i c t e d f r e q u e n c y o f 10,367.5 MHz, over 45 MHz away from the e x p e r i m e n t a l f r e q u e n c y and o u t s i d e t h e o r i g i n a l 30 MHz sweep range. T h i s r e v i e w o f t h e f i r s t p a r t o f the experiment i s g i v e n t o p o i n t out t h e e x p e r i m e n t a l d i f f i c u l t y and the need t o know where t o l o o k i f s u c c e s s i s t o be a c h i e v e d w i t h i n a r e a s o n a b l e l e n g t h o f t i m e . A t l e a s t one y e a r o f p e r s i s t e n t work would have been r e q u i r e d t o f i n d t h e f i r s t l i n e s t a r t i n g from 10,570.2 MHz u s i n g t h e 3 MHz/day s e a r c h r a t e c o n s i d e r e d n e c e s s a r y . 4.3 Second E x p e r i m e n t a l Phase The s u c c e s s o f t h e e x p e r i m e n t s o f C u r l e t a l [15,16] gave an improved v a l u e o f the q u a r t i c d i s t o r t i o n c o n s t a n t : Drj, = 132,933 ±10 Hz and t h e two s e x t i c t e n s o r d i s t o r t i o n c o n s t a n t s : H 4 T = -16.65 ±0.2 Hz and H & T = 10.2 ±1 Hz o f methane. U s i n g O z i e r ' s m o l e c u l a r beam r e s u l t s [70] t h e s e d i s t o r t i o n c o n s t a n t s were c o r r e c t e d s l i g h t l y ( t h e c o r r e c t i o n s were i n s i d e C u r l ' s e r r o r l i m i t s ) . W i t h t h e s e d i s t o r t i o n c o n s t a n t s the f r e q u e n c y u n c e r t a i n t y o f the microwave methane t r a n s i t i o n s between 7 and 15 GHz were reduced t o l e s s than 10 MHz each (see t a b l e X I ) . The J=12 E 1 -»• J=12 E 2 ground s t a t e Q-branch r o t a t i o n a l t r a n s i t i o n o f methane was found 81 w i t h i n a week. The t h r e e r e m a i n i n g t r a n s i t i o n s l i s t e d i n t a b l e XI were found w i t h i n a few months. The f o u r l i n e s o b s e r v e d a r e r e p r o d u c e d i n f i g u r e s 4.1, 4.3, 4.4, and 4.5. The l i n e s t r e n g t h s and e x p e r i m e n t a l c o n d i t i o n s a re g i v e n i n s e c t i o n 4.5. An energy l e v e l d iagram based on [17] f o r t h e J=12 l e v e l o f methane i s g i v e n i n f i g u r e 4.2, where t h e o b s e r v e d t r a n s i t i o n i s i n d i c a t e d by the arrow, the i r r e d u c i b l e r e p r e s e n t a t i o n s , K, o f t h e r o t a t i o n a l w a v e f u n c t i o n s a r e g i v e n i n the p o i n t group 0, t h e p a r i t y i s i n d i c a t e d by ±1, the l e v e l s w i t h i n an i r r e d u c i b l e r e p r e s e n t a t i o n a r e l a b e l l e d i n o r d e r o f i n c r e a s i n g energy by t , and t h e " z e r o " r e f e r e n c e energy i s g i v e n by the s c a l a r terms i n t h e H a m i l t o n i a n ( a p p r o x i m a t e l y 25,000 GHz). A l l t h e t r a n s i t i o n s o b s e r v e d h e r e were E t y p e e l e c t r i c d i p o l e t r a n s i t i o n s , w h i c h were t h e o n l y ones t o e x h i b i t a f i r s t - o r d e r S t a r k s h i f t . T a b l e X I g i v e s the o b s e r v e d t r a n s i t i o n f r e q u e n c i e s w i t h e s t i m a t e d e x p e r i m e n t a l e r r o r s o f t h e X- and P-band l i n e s . A l s o l i s t e d i n t a b l e XI a r e t h e p r e d i c t e d f r e q u e n c i e s w i t h s t a n d a r d d e v i a t i o n s and d i f f e r e n c e s based on C u r l ' s d i s t o r t i o n c o n s t a n t s (as c o r r e c t e d ) : D T = 132,942 ±10 Hz, H 4 T = -16,8 ±.19-Hz, and H 6 T = 10.2 ±.9 Hz and the e q u a t i o n : v = [ D T + H 4 T J ( J + l ) ] A f ( J , K , t ) + H 6 T A g ( J , K , t ) , (4.1) where the f r e q u e n c y e x p r e s s i o n , Af and Ag were o b t a i n e d from K i r s c h n e r and Watson [ 4 5 ] . A l s o l i s t e d i n t a b l e XI are the p r e d i c t e d f r e q u e n c i e s w i t h s t a n d a r d d e v i a t i o n s [52] and d i f f e r e n c e s g i v e n by Tarrago e t a l [ 9 5 ] . T a r r a g o , Dang-Nhu and P o u s s i g u e [95] r e d e t e r m i n e d D^, H/rp and H c r p based on the then t o t a l o f seven r a d i o f r e q u e n c y T a b l e XI E x p e r i m e n t a l F r e q u e n c i e s o f X- and P-band AJ=0 T r a n s i t i o n s o f 1 2 C H 4 i n the Ground V i b r o n i c S t a t e t o g e t h e r w i t h P r e d i c t i o n s based on the D i s t o r t i o n C o n s t a n t s D„, H, T, and H f i T o f C u r l [15] and Ta r r a g o [95] t t i E x p e r i m e n t a l P r e d i c t e d F r e q P r e d i c t e d F r e q J K L + K L D i f f a D i f f a Frequency (MHz) C u r l [15] Tarr a g o [95] 12 E ^ E 2 10,321. 91 ±.1 10,322, .0 ±4. 9 -.09 10,322. 20 ±1. 30 -.29 13 E ^ E 2 11,261. 37 ±.1 11,256. ,0 ±6. 7 5.37 11,261. 03 ±1. 52 .34 14 E 2+E 3 7,861. 67 ±.1 7,855. .8 ±8. 3 5.87 7,862. 01 ±1. 14 -.34 15 E ^ E 2 14,151. 81 ±.1 14,145. 3 ±8. 3 -6.51 14,151. 73 ±2. 19 .08 D i f f e r e n c e = E x p e r i m e n t a l mean - P r e d i c t e d mean oo N3 J I i : i : L 10321.0 10322.0 10323.0 F R E Q U E N C Y ( M H z ) F i g u r e 4.1 512 Sample minus 512 B a s e l i n e 51.2 second Scans t o g i v e t h e J=12 E 1 -*• J=12 E 2 T r a n s i t i o n o f Ground S t a t e l2CEtt 8 4 +10 J L M X >-c r LU LU o -< o ° in k^ > A F i g u r e 4 . 2 The J = 1 2 R o t a t i o n a l Energy L e v e l s f o r 1 2 C H i » . Wg g i v e s the z e r o r e f e r e n c e energy. F i g u r e 4 . 1 i l l u s t r a t e s the t r a n s i t i o n . 11261 11262 11263 FREQUENCY (MHz) -Figure 4.3 512 Sample minus lk 1024 Baseline 51.2 second Scans to give the J=13 E 1 •*• J=13 E 2 T r a n s i t i o n of Ground State 1 2CHi, CO F i g u r e 4 . 4 1 0 2 4 Sample minus 1 0 2 4 B a s e l i n e 5 1 . 2 second Scans t o g i v e t h e J = 1 4 E 2 -> J = 1 4 E 3 T r a n s i t i o n o f Ground S t a t e 1 2 C H i , 14151 14152 14153 FREQUENCY(MHz) *-F i g u r e 4 . 5 5 1 2 Sample minus 5 1 2 B a s e l i n e 5 1 . 2 second Scans t o g i v e the J = 1 5 E 1 * J = 1 5 E 2 T r a n s i t i o n o f Ground S t a t e l 2 C H i , 00 88 and microwave l i n e s . They o b t a i n e d the v a l u e s : D T = 132,858 ±13 Hz, H 4 T = -16.151 ±.062 Hz, and Hgrr, = 10.468 ±.027 Hz under t h e handicap o f the time c o n s t a n t d i s t o r t i o n o f O z i e r ' s l i n e [34,35,71] and t h e l a c k o f an e i g h t h degree t h e o r y ( a l t h o u g h the s i x t h degree t h e o r y used d i d t a k e i n t o account t h e o f f d i a g o n a l elements o f ft6). The d i f f e r e n c e s between t h e e x p e r i m e n t a l f r e q u e n c i e s and the p r e d i c t i o n s o f T a r r a g o e t a l a r e s i g n i f i c a n t l y l e s s than t h o s e between the e x p e r i m e n t a l f r e q u e n c i e s and the p r e d i c t i o n s based on C u r l ' s c o r r e c t e d c o n s t a n t s as would be e x p e c t e d because f o u r a d d i t i o n a l e x p e r i m e n t a l f r e q u e n c i e s have been used. However, the d i f f e r e n c e s between the e x p e r i m e n t a l f r e q u e n c i e s and t h e p r e d i c t i o n s o f Ta r r a g o e t a l [95] a r e s t i l l up t o t h r e e t i m e s g r e a t e r t h a n 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 i s i s due t o undetermined h i g h e r o r d e r d i s t o r t i o n c o n s t a n t s w h i c h have been a s s i m i l a t e d i n t o Drj, H ^ and Hg T. The s i z e o f t h e undetermined e i g h t h degree terms t o g e t h e r w i t h the a b i l i t y o f e f f e c t i v e l ower o r d e r c o n s t a n t s t o p r e d i c t f r e q u e n c i e s i n s p i t e o f the s e h i g h e r terms w i l l be used i n s e c t i o n 4.8 i n e v a l u a t i n g the goodness o f the s i x d i s t o r t i o n c o n s t a n t s e v a l u a t e d t h e r e . 4.4 F i n a l E x p e r i m e n t a l Phase The experiment was o r i g i n a l l y c o n c e i v e d f o r X-band (8-*--*-l2.4 GHz) and t h e s p e c t r o m e t e r was c o n s t r u c t e d such t h a t optimum performance was o b t a i n e d w h i l e s e a r c h i n g f o r . t h e J=12 and J=13 t r a n s i t i o n s . The J=15 t r a n s i t i o n , a l t h o u g h o u t s i d e o f X-band a t 14.15 GHz was found w i t h o u t d i f f i c u l t y u s i n g t h e o r i g i n a l s p e c t r o m e t e r w i t h o u t m o d i f i c a t i o n except 89 f o r the s u b s t i t u t i o n o f a P-band backward wave o s c i l l a t o r . The J=14 t r a n s i t i o n a t 7.86 GHz a p p a r e n t l y was beyond t h e range o f the i n s t r u m e n t w i t h the BWO p l u g - i n s a v a i l a b l e i n our l a b o r a t o r y . The range o f the i n s t r u m e n t was l o w e r e d below 8 GHz i n the manner d e s c r i b e d i n c h a p t e r 3 t o g i v e the l i n e a t 7862 MHz. Phase l o c k s t a b i l i t y was poor a t 7862 MHz r e q u i r i n g the s t o r a g e o f each sweep i n a b u f f e r memory so t h a t one mishap would n o t s p o i l hours (days) o f work. T h i s l i n e was t h e most demanding because a p p r o x i m a t e l y 1000 scans were put i n t o t h e s i g n a l a v e r a g e r one a t a time. The l i n e s a t 18 and 19 GHz were d e f i n i t e l y beyond the range o f the o r i g i n a l i n s t r u m e n t ; no d-c v o l t a g e c o u l d be a p p l i e d t o t h e microwave sou r c e t o i n c r e a s e t h e f r e q u e n c y t o the e x p e c t e d l i n e f r e q u e n c i e s . K l y s t r o n s had t o be employed, and t h e y p r o v e d t o be much n o i s i e r t h a n t h e BWO's used a t lower f r e q u e n c i e s , n e c e s s i t a t i n g t h e development o f a n o i s e c a n c e l l a t i o n scheme. Phase s t a b i l i t y had t o be i m p a r t e d t o t h e k l y s t r o n s . A s y n c h r o n i z e r d e s i g n e d f o r use a t lower f r e q u e n c i e s on stand-by s t a t u s i n the department was m o d i f i e d t o p e r f o r m the t a s k . The a u t h o r w i s h e s t o thank P r o f e s s o r s J.B. Farmer and C.A. McDowell f o r t h e use o f t h e i r s y n c h r o n i z e r . D u r i n g m o d i f i c a t i o n o f the s p e c t r o m e t e r , O z i e r [72] completed c a l c u l a t i o n s o f the t e n s o r d i s t o r t i o n energy o f t e t r a h e d r a l m o l e c u l e s c a r r i e d t o e i g h t h degree i n J and f i t D,p, H ^ , Hgrj, and t h e o c t i c c o n s t a n t s L ^ , L^^, and Lg^- t o t h e seven t r a n s i t i o n s then a v a i l a b l e . These were the m o l e c u l a r beam J=2 s p l i t t i n g [35] , t h e two 3=1 l i n e s o f C u r l [15] and the f o u r r e c e n t l y found microwave l i n e s [ 3 4 ] . A l l 90 c o n s t a n t s e x c e p t Lgrp f i t w e l l , e n a b l i n g the J=14 l i n e a t 19,288 MHz t o be p r e d i c t e d t o w i t h i n a s t a n d a r d d e v i a t i o n (68% p r o b a b i l i t y [52]) o f ±1.5 MHz. A l i n e was found w i t h i n t h i s 1.5 MHz range. No o t h e r t r a n s i t i o n was found i n s e a r c h e s 20 MHz each s i d e o f t h e p r e d i c t i o n . F i t t i n g t h e above c o n s t a n t s t o e i g h t l i n e s a l l o w e d t h e 18 GHz J=16 and J=18 methane l i n e s t o be p r e d i c t e d t o w i t h i n a s t a n d a r d d e v i a t i o n , a, o f ±.5 MHz. The J=16 l i n e was found w i t h i n ±.2 MHz. F i n a l l y 9 l i n e s were used t o p r e d i c t the t e n t h , J=18, l i n e t o w i t h i n a = ±.2 MHz. The f i n a l f i t g i v e n i n t a b l e XIV l e a v e s l i t t l e doubt as t o the v a l i d i t y o f the assignment and the t h e o r y . The t h r e e t r a n s i t i o n s o b s e r v e d i n t h i s phase o f the e x p e r i m e n t are r e p r o d u c e d i n f i g u r e s 4.6, 4.8 and 4.9. An energy l e v e l d iagram f o r t h e J=14 l e v e l o f methane i n t h e ground s t a t e i s g i v e n i n f i g u r e 4.7. 4.5 E x p e r i m e n t a l C o n d i t i o n s E x p e r i m e n t a l c o n d i t i o n s w h i c h v a r i e d from l i n e t o l i n e a re l i s t e d i n t a b l e X I I . A l l the microwave t r a n s i t i o n s o f methane o b s e r v e d h e r e were a t a sample p r e s s u r e of a p p r o x i m a t e l y 60 mTorr and a t e m p e r a t u r e o f a p p r o x i m a t e l y 300°K. The microwave power l e v e l g o i n g i n t o t h e c e l l was the f u l l power l e v e l o f t h e backward wave o s c i l l a t o r or k l y s t r o n employed, t y p i c a l l y from 60 t o 120 m i l l i w a t t s . The phase o f t h e synchronous phase d e t e c t o r was s e t u s i n g an OCS sample, the g a i n was s e t a t 1 m i c r o v o l t and t h e o u t p u t f i l t e r c o n s i s t e d o f two s i m p l e R-C f i l t e r s w i t h a time c o n s t a n t o f 1 second each i n cascade. Figure 4.6 384 Sample minus 384 Baseline 51.2 second Scans to give the J=14 E 1 J=14 E 2 Transition of Ground State 12CKk 92 A, A 2 E F, £ Figure 4.7 The J=14 Rotational Energy Levels for 12CH.». W£ gives the "zero" energy. Fig. 4.4 illustrates the upper, and f i g . 4.6 the lower transition. 18562 ' 18563 FREQUENCY (MHz) — Figure 4.8 3328 Sample minus 3328 Baseline 51.2 second Scans to give the J=16 E 2 -»• J=16 E 3 T r a n s i t i o n of Ground State 1 2CH^ VO U> Figure 4.9 3072 Sample minus 3A 4096 Baseline 51.2 second Scans to give the J=18 E 2 •*• J=18 E 3 Transition of Ground State 12Cm vO -P> Table X I I E x p e r i m e n t a l C o n d i t i o n s J F r e q (MHz) Y X 1 0 1 1 'max (cm" 1) T o t a l Sweep Time (hr) A C ( J ,K , t ) [17] S t a r k F i e l d (kV/cm) A v S t a r k f o r m=l (kHz) 12 E x+E 2 10,321.91 5.09 77 4.025 1.06 104 13 11,261.37 3.54 73 12.857 .43 133 14 E 2+E 3 7,861.67 2.32 29 17.496 .34 145 14 E ^ E 2 19,288.63 5.88 28 5.544 .85 115 15 E ^ E 2 14,151.81 5.53 66 6.457 .81 127 16 E2-*-E3 18,562.40 2.06 47 17.494 .28 117 18 E 2+E 3 18,528.94 2.06 44 10.144 .47 116 VO 96 4.5.1 L i n e S t r e n g t h s The l i n e s t r e n g t h s i n t a b l e X I I a r e c a l c u l a t e d i n t h e manner o f Dorney and Itfatson [17] w i t h the same p a r a m e t e r s , e x c e p t f o r the l i n e b r o a d e n i n g parameter w h i c h has been g i v e n a v a l u e o f 2.7 M H z / t o r r based on R b r a n c h t r a n s i t i o n s o f CH 3D [76] and t h e v a l u e o f Drr,, w h i c h has been r e v i s e d from .12 t o .133 MHz. The l i n e s o b s e r v e d i n t h i s e x p e r i m e n t a r e t h e weakest e v e r o b s e r v e d w i t h a c o n v e n t i o n a l S t a r k modulated microwave s p e c t r o m e t e r . 4.5.2 Sweep Times Each f r e q u e n c y sweep c y c l e c o n s i s t e d o f two phases: t h e f i r s t i n w h i c h the f r e q u e n c y was i n c r e a s e d a t a c o n s t a n t r a t e f o r 51.2 seconds, f o l l o w e d by a second phase i n w h i c h the f r e q u e n c y was d e c r e a s e d a t the same c o n s t a n t r a t e f o r a n o t h e r 51.2 seconds. The advantages o f t h i s "sweepback" mode were t h a t t ime c o n s t a n t d i s t o r t i o n d i d n o t e f f e c t the l i n e c e n t e r f r e q u e n c y and t h e s y n c h r o n i z e r d i d n o t f a l l out o f l o c k because o f t h e a p p l i c a t i o n o f a d i s c o n t i n u o u s f r e q u e n c y a n a l o g d r i v i n g v o l t a g e . The t o t a l t i me spent sweeping a l i n e v a r i e d because o f l i n e i n t e n s i t y , t e s t s o r e x p e r i m e n t a l c o n d i t i o n s . The l i n e s a t 10, 11, and 14 GHz r e c e i v e d more sweep t i m e , n o t out o f n e c e s s i t y , b u t because i t was h e r e t h a t t e s t s were performed t o t e s t power s a t u r a t i o n , adequate S t a r k m o d u l a t i o n , and sample p u r i t y . Of the l i n e s a t 18 and 19 GHz, the 19 GHz l i n e r e q u i r e d l e s s sweep t i m e because o f i t s g r e a t e r l i n e s t r e n g t h . The l i n e s a t 18 GHz r e q u i r e d 6 days sweep time each t a k i n g i n t o account t h e r e q u i r e d b a s e l i n e sweep t i m e ! The l i n e a t 7861.67 MHz, a l t h o u g h o f o n l y 29 hours sweep d u r a t i o n , had t o be done one sweep a t a t i m e . A l t h o u g h v i r t u a l l y the same s t r e n g t h as t h e 18 GHz l i n e s , as good a s i g n a l t o n o i s e r a t i o was a c h i e v e d i n l e s s time because the i n s t r u m e n t was i n i t s prime o p e r a t i n g range ( d i s c o u n t i n g the phase l o c k p r o b l e m ) . 4.5.3 S t a r k F i e l d s S t a r k m o d u l a t i o n was s u p p l i e d as a z e r o - b a s e d , 1 kHz square wave th r o u g h o u t the exp e r i m e n t . The S t a r k s h i f t c o e f f i c i e n t s , C ( J , K , t ) , f o r the E l e v e l s o f methane have been c a l c u l a t e d by Dorney and Watson [ 1 7 ] . S i n c e the microwave e l e c t r i c v e c t o r was p a r a l l e l t o the S t a r k f i e l d , Am = 0 t r a n s i t i o n s were o b s e r v e d [97] w h i c h gave a S t a r k s h i f t : A v s t a r k = A A C ( J , K , t ) m G x y e (4.2) where A v S t a r k i s i n kHz, A = 5.035 X 1 0 5 k H z • D - 1 ( k V / c m ) - 1 i s the S t a r k e f f e c t c o n s t a n t [ 9 7 ] , A C ( J , K , t ) i s the a b s o l u t e d i f f e r e n c e between the S t a r k s h i f t c o e f f i c i e n t s f o r the l e v e l s c oncerned as t a b u l a t e d by Dorney and Watson [ 1 7 ] , m i s the magnetic quantum number, 9 x y = 2 . 4 1 X 1 0 " 5 D i s the d i p o l e z moment f a c t o r [17,71,100] and e i s the e l e c t r i c f i e l d s t r e n g t h i n kV/cm. Thus the components w i t h b i g g e s t m show t h e l a r g e s t s h i f t . S i n c e the S t a r k f i e l d had a d i r e c t b e a r i n g on t h e s e v e r i t y o f the b a s e l i n e s l o p e , i t was d e s i r a b l e t o keep t h e S t a r k f i e l d t o a minimum. S i n c e t h e S t a r k component i n t e n s i t y v a r i e s as m2 f o r AJ = Am = 0 t r a n s i t i o n s [ 9 7 ] , i t was p o s s i b l e t o l e a v e a few o f the S t a r k components under t h e l i n e and y e t 98 a c h i e v e s i g n i f i c a n t m o d u l a t i o n . U s i n g the v a l u e s o f A C ( J , K , t ) from Dorney and Watson [17] l i s t e d i n t a b l e X I I , the f r e q u e n c y s h i f t s o f the m = 1 components a r e c a l c u l a t e d from the r e l a t i o n : A v S t a r k = 2 4 . 2 7 [ A C ( J , K , t ) ] e (4.3) The a s s u m p t i o n w i l l be made t h a t i f a S t a r k component was s h i f t e d more than 300 kHz ( I V 2 t i m e s a t y p i c a l l i n e h a l f - w i d t h a t h a l f - h e i g h t ) i t modulated t h e l i n e , b u t a s h i f t o f l e s s than 300 kHz r e s u l t e d i n no m o d u l a t i o n . From t a b l e X I I t h e S t a r k components s h i f t e d l e a s t were those o f J=12. I n t h i s case, t h e r e l a t i v e t o t a l S t a r k i n t e n s i t y was g i v e n by the sum o f the squares o f t h e i n t e g e r s up t o and i n c l u d i n g 12, w h i c h i s 650. S i n c e two S t a r k components were moved l e s s t h a n 300 kHz, the r e l a t i v e S t a r k m o d u l a t i o n w h i c h was i n e f f e c t i v e i s l 2 + 2 2 = 5 . The p e r c e n t S t a r k m o d u l a t i o n was t h u s : 100(650-5)/650 = 99+% f o r J=12. The S t a r k m o d u l a t i o n was more th a n 997o e f f e c t i v e f o r the s i x o t h e r microwave t r a n s i t i o n s o b s e r v e d . 4.6 L i n e I d e n t i f i c a t i o n A g r e a t d e a l o f e f f o r t was expended t o prove t h a t the t r a n s i t i o n s were due to methane and n o t t o a s m a l l amount o f i m p u r i t y w i t h a r e l a t i v e l y l a r g e permanent d i p o l e moment. I m p u r i t i e s c o u l d be o u t g a s s i n g from v a r i o u s p a r t s o f the c e l l (perhaps remnants o f samples used i n p r i o r e x p e r i m e n t s ) o r t hey c o u l d be l e a k i n g i n t o the c e l l f r o m t h e room. These concerns were h e i g h t e n e d because t h e c e l l d i d show a d e f i n i t e , s l o w p r e s s u r e r i s e d u r i n g l o n g scan t i m e s . These p o s s i b i l i t i e s 99 were e l i m i n a t e d by e v a c u a t i n g t h e c e l l , i s o l a t i n g i t and a l l o w i n g the p r e s s u r e t o i n c r e a s e s p o n t a n e o u s l y o v e r a l o n g p e r i o d o f time t o 60 mTorr a t w h i c h time a f t e r a s u i t a b l e s e a r c h p e r i o d , o b s e r v i n g t h a t no spectrum appeared. The i m p u r i t i e s c o u l d have been p u r c h a s e d from t h e Matheson Co. w i t h the sample. The i m p u r i t y l e v e l i n the U l t r a - H i g h P u r i t y methane p u r c h a s e d was g i v e n as 3 p a r t s p e r 10,000 maximum. The sample was f r e q u e n t l y s o l i d i f i e d w i t h l i q u i d n i t r o g e n so t h a t accumulated a i r ( a l o n g w i t h methane vap o r ) c o u l d be pumped o f f , b u t t h i s would n o t remove t h e g r e a t preponderance o f i m p u r i t i e s w h i c h have a n e g l i g i b l e v apor p r e s s u r e a t the tem p e r a t u r e o f l i q u i d n i t r o g e n . I n s t e a d o f removing t h e i m p u r i t i e s from the s o l i d methane, t h e y were k e p t t h e r e and the u s a b l e methane vapor p r e s s u r e a t l i q u i d n i t r o g e n t emperature were used t o g i v e an a b s o r p t i o n spectrum. T h i s t e s t e l i m i n a t e d t h e p o s s i b i l i t y o f v i r t u a l l y e v e r y e f f e c t i v e i m p u r i t y e x c e p t p o l a r permanent gases such as carbon monoxide. The p o s s i b i l i t y t h a t the l i n e s were due t o i m p u r i t i e s were f u r t h e r r e d u c e d by t h e f o l l o w i n g c o n s i d e r a t i o n s . The l i n e s t r e n g t h s agreed w i t h t h e c a l c u l a t e d l i n e s t r e n g t h s t o w i t h i n a f a c t o r o f two, based on a comparison w i t h known OCS l i n e s t r e n g t h s [ 4 6 ] , and t a k i n g i n t o account the power s a t u r a t i o n o f OCS. The l i n e w i d t h s agreed t o w i t h i n a f a c t o r o f 2 o f those e x p e c t e d from a p r e s s u r e o f 60 mTorr and a CH 3D b r o a d e n i n g parameter o f 2.7 MHz/Torr [ 7 6 ] ; the l i n e w i d t h s were a p p r o x i m a t e l y e i g h t times l e s s than t h o s e o f OCS under i d e n t i c a l c o n d i t i o n s i n d i c a t i n g a v e r y s m a l l t r a n s i t i o n 1 0 0 moment. The S t a r k f i e l d s r e q u i r e d f o r complete m o d u l a t i o n o f the l i n e s f o l l o w e d those t a b u l a t e d by Dorney and Watson [ 1 7 ] . More p r o o f t h a t the l i n e s were due t o the c e n t r i f u g a l d i s t o r t i o n d i p o l e moment o f methane i s g i v e n by the e x c e l l e n t f i t o f a l l the microwave l i n e s t o t h e t h e o r y c a r r i e d t o e i g h t h degree as shown i n the n e x t s e c t i o n . Subsequent to t h i s e x p e r i m e n t , O z i e r , Lees and G e r r y [ 7 7 ] have had s u c c e s s o b t a i n i n g the 1 3 C H i | spectrum s l i g h t l y s h i f t e d i n f r e q u e n c y from the above, g i v i n g a d d i t i o n a l p r o o f t h a t the t r a n s i t i o n s o b s e r v e d a re due t o methane and n o t an i m p u r i t y . 4 . 7 D e t e r m i n a t i o n o f D i s t o r t i o n C o n s t a n t s I n a d d i t i o n t o the seven t r a n s i t i o n f r e q u e n c i e s d e t e r m i n e d h e r e , the two J = 7 double resonance t r a n s i t i o n s d e t e r m i n e d by C u r l [ 1 5 ] and the J = 2 o r t h o - p a r a s p l i t t i n g d e t e r m i n e d by O z i e r [ 3 5 , 7 0 ] were used i n a w e i g h t e d l i n e a r l e a s t squares f i t t o d e t e r m i n e t h e s i x d i s t o r t i o n c o n s t a n t s l i s t e d i n t a b l e X I I I : D T, H ^ , H ^ T , L ^ T , L ^ T and L g T . The s e x t i c d i s t o r t i o n c o n s t a n t s , a n d ' H ^ , a r e s m a l l e r t h a n t h e q u a r t i c d i s t o r t i o n c o n s t a n t , D^ ., by an approximate f a c t o r o f 10"*. The o c t i c d i s t o r t i o n c o n s t a n t s , L ^ , L g T , and Lgrp, i n t u r n a r e a p p r o x i m a t e l y 1 0 K t i m e s s m a l l e r t h a n the s e x t i c d i s t o r t i o n c o n s t a n t s . T h i s r a t i o between d i s t o r t i o n c o n s t a n t s w h i c h i s c l o s e t o t h a t o f Born-Oppenheimer [ 6 ] i n d i c a t e s t h a t t h e d i s t o r t i o n c o n s t a n t s d e r i v e d have a p h y s i c a l s i g n i f i c a n c e and a r e n o t m e r e l y f i t t i n g parameters [ 9 9 ] . The e x p e r i m e n t a l f r e q u e n c i e s t o g e t h e r w i t h e x p e r i m e n t a l e r r o r s and d i f f e r e n c e s between e x p e r i m e n t a l f r e q u e n c i e s and T a b l e X I I I C e n t r i f u g a l D i s t o r t i o n C o n s t a n t s o f . 1 2 C H i , i n t h e Ground S t a t e C o n s t a n t V a l u e (Hz) °LS 3 D T 132 943. 41 .71 3.7 H 4 T -16. 983 9 .007 6 .023 H 6 T 11. 034 2 .008 6 .019 L 4 T 002 027 .000 024 .000 085 L 6 T —. 002 677 .000 035 .000 12 L 8 T —. 003 00 .000 18 .000 27 S t a n d a r d d e v i a t i o n i n Hz o b t a i n e d by f i t t i n g f r e q u e n c i e s t o e i g h t h degree t h e o r y by w e i g h t e d l i n e a r l e a s t s q u a r e s . E s t i m a t e d s t a n d a r d d e v i a t i o n o f a b s o l u t e e r r o r i n Hz. 102 f r e q u e n c i e s p r e d i c t e d on t h e b a s i s o f the above d i s t o r t i o n c o n s t a n t s are l i s t e d i n t a b l e XIV. The f r e q u e n c i e s used i n the l i n e a r l e a s t squares f i t were w e i g h t e d a c c o r d i n g t o the square o f t h e r e c i p r o c a l o f the e x p e r i m e n t a l u n c e r t a i n t y to a r r i v e a t the s t a n d a r d d e v i a t i o n s , cr^g, l i s t e d i n t a b l e X I I I . A comparison o f 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 w i l l r e v e a l t h a t the J=2 s p l i t t i n g was g i v e n a p p r o x i m a t e l y 300,000 ti m e s the weight of an X-band microwave t r a n s i t i o n i n s p i t e o f the f a c t t h a t the J=2 e x p e r i m e n t a l e r r o r e x p r e s s e d as a p e r c e n t a g e i s a p p r o x i m a t e l y 7 times t h a t o f an X-band microwave t r a n s i t i o n . T h i s heavy w e i g h t on the J-2 s p l i t t i n g i s j u s t i f i e d by c o n s i d e r a t i o n t h a t the energy o f the J=2 l e v e l s i s a f f e c t e d o n l y by t h r e e o f t h e s i x d i s t o r t i o n c o n s t a n t s : f o r J=2, E T = [ D T + 6H A r p o f r a n k 6, 8, and h i g h e r do n o t a f f e c t the energy o f t h e J=2 r o t a t i o n a l l e v e l f o r the f o l l o w i n g r e a s o n . The J=2 r o t a t i o n a l w a v e f u n c t i o n s a r e s p h e r i c a l t e n s o r s o f r a n k 2. The energy i s a s c a l a r q u a n t i t y ( s p h e r i c a l t e n s o r o f r a n k 0 ) . A s p h e r i c a l t e n s o r o p e r a t o r o f rank I i n t h e H a m i l t o n i a n can a f f e c t the energy o f the m o l e c u l e o n l y i f a v e c t o r o f l e n g t h I can combine w i t h two v e c t o r s o f l e n g t h J c o r r e s p o n d i n g t o t h e r o t a t i o n a l w a v e f u n c t i o n s t o g i v e a t r i a n g l e e q u a l i t y [ 8 4 ] . T h i s i s because t h e energy i s g i v e n by E = <^a|P^|b^ w h i c h r e q u i r e s : rank 0 - rank J + r a n k I + rank J : where t h e a d d i t i o n i s v e c t o r a d d i t i o n . I f I > 2 J , the t r i a n g l e cannot be c l o s e d and the energy c o r r e s p o n d i n g t o t h e t e n s o r o p e r a t o r ft. i s z e r o f o r the J v a l u e under c o n s i d e r a t i o n . For J=2, The s p h e r i c a l t e n s o r o p e r a t o r s i n the H a m i l t o n i a n T a b l e XIV The AJ=0 T r a n s i t i o n s o f Ground S t a t e 1 2 J Exp. F r e q . (MHz) D i f f ? 2 7.97046 ±.00054 a .00003 7 F 2 + F 2 2 1 423.02 b ±.02 .008 7 F J + F 2 1 2 1,246.55 b ±.02 -.003 14 E 2+E 3 7,861.67 ±.1 -.003 12 E*+E 2 10,321.91 ±.1 -.003 13 E X-*E 2 11,261.37 ±.1 .000 15 E ^ E 2 14,151.81 ±.1 .003 18 E 2+E 3 18,528.94 ±.2 -.004 16 E 2+E 3 18,562.40 ±.2 .003 14 E ^ E 2 19,288,63 ±.2 .000 a O z i e r [70,35] b C u r l [15] c D i f f e r e n c e (MHz) = P r e d i c t e d Frequency - E x p e r i m e n t a l Frequency based on t h e d i s t o r t i o n c o n s t a n t s g i v e n i n t a b l e X I I I 104 o n l y f t i » a f f e c t s the energy. R e g a r d l e s s o f whether the J=2 s p l i t t i n g i s l e f t o u t o f c o n s i d e r a t i o n , t h e s i x d i s t o r t i o n c o n s t a n t s so o b t a i n e d f a l l w i t h i n the range o f t h e l e a s t square d e v i a t i o n s , a L g , l i s t e d i n t a b l e X I I I . Moreover, a f i t t o 8 l i n e s o m i t t i n g the h i g h J (J=16 and 18) l i n e s , o r a f i t to o n l y the 7 microwave l i n e s d e t e r m i n e d h e r e , o r a f i t t o a l l 10 l i n e s w e i g h t e d e q u a l l y gave, i n a l l c a s e s , d i s t o r t i o n c o n s t a n t s w i t h i n the l e a s t square d e v i a t i o n s l i s t e d i n t a b l e X I I I . The f a c t t h a t the c o n s t a n t s d i d n o t change a p p r e c i a b l y when one, two, o r t h r e e t r a n s i t i o n s were l e f t out o f c o n s i d e r a t i o n shows t h a t no one l i n e (J=2 i n p a r t i c u l a r ) was d i s t o r t i n g t h e f i t . The f i t was p e r f o r m e d i n the manner d e s c r i b e d by O z i e r [ 7 2 ] . D u r i n g t h e f i r s t i t e r a t i o n , a l l d i s t o r t i o n c o n s t a n t s except D^ , were s e t e q u a l t o z e r o . The H a m i l t o n i a n then assumed the form: WT = D TJU (4.4) w h i c h was d i a g o n a l i z e d t o g i v e the " z e r o t h - o r d e r " w a v e f u n c t i o n s . The o p e r a t o r s ftij, fi6 and ft8 then gave t h e d i a g o n a l m a t r i x elements: ^u) 0o = f , (^&)0 . - g and - = h. For each t r a n s i t i o n f r e q u e n c y a l i n e a r e q u a t i o n was w r i t t e n . F or i n s t a n c e , f o r J=12, v = [ D T + H 4 T J ( J + 1 ) + L 4 T J 2 ( J + l ) 2 ] A f (4.5) + [ H 6 T + L 6 T J ( J + l ) ] A g + L g T A h 3*6322 X 1 0 6 = 79570 D T + 12413 X 1 0 3 H 4 T + 19364 X 10 s L 4 T - 4676 X 1 0 3 H 6 T - 72946 X10* L 6 T - 20853 X 1 0 3 L g T where the d i s t o r t i o n c o n s t a n t s are i n H e r t z . 105 U s i n g from 7 t o 10 l i n e a r e q u a t i o n s o f t h i s type i n 6 unknowns ( o n l y 3 unknowns f o r J=2), the v a l u e s o f the unknown d i s t o r t i o n c o n s t a n t s were det e r m i n e d by w e i g h t e d l i n e a r l e a s t squares f i t . The " z e r o t h - o r d e r " d i s t o r t i o n c o n s t a n t s thus d e t e r m i n e d were used as the b a s i s f o r the second i t e r a t i o n . The H a m i l t o n i a n was s e t up u s i n g a l l s i x d i s t o r t i o n c o n s t a n t s and d i a g o n a l i z e d t o g i v e a new s e t o f w a v e f u n c t i o n s . M a t r i x elements were th e n c a l c u l a t e d i n terms o f the new w a v e f u n c t i o n s , new l i n e a r e q u a t i o n s were s e t up and a n o t h e r improved s e t o f d i s t o r t i o n c o n s t a n t s d e t e r m i n e d by w e i g h t e d l i n e a r l e a s t s q u a r e s . The i t e r a t i o n was c o n t i n u e d u n t i l t h e i n d i v i d u a l c o n s t a n t s converged. For a l l t h e s u b s e t s o f l i n e s c o n s i d e r e d , t h e t h i r d i t e r a t i o n gave c o n s t a n t s w h i c h agreed w i t h t h o s e o f the second i t e r a t i o n t o w i t h i n 10% o f t h e s t a n d a r d d e v i a t i o n s o f the second i t e r a t i o n . 4.8 I n f l u e n c e o f D e c t i c and H i g h e r Order D i s t o r t i o n C o n s t a n t s I t i s i n t e r e s t i n g t o c o n s i d e r t h e f r e q u e n c y c o n t r i b u t i o n s o f the terms a s s o c i a t e d w i t h the v a r i o u s d i s t o r t i o n c o n s t a n t s as a f u n c t i o n o f J . These a r e g i v e n i n t a b l e XV. T a b l e XV i l l u s t r a t e s t h e e x p e c t e d : as J i n c r e a s e s , t h e i m p o r t a n c e o f h i g h e r degree terms i n c r e a s e s [D i s f o u r t h , H i s s i x t h and L i s e i g h t h d e g r e e ] . W i t h i n a g i v e n degree, t h e terms a s s o c i a t e d w i t h the l o w e r rank t e n s o r ( g i v e n by the s u b s c r i p t 4, 6, o r 8) are g e n e r a l l y more i m p o r t a n t . But the t r e n d s a r e n o t always smooth i n t h i s i r r e g u l a r spectrum. Note th e a b n o r m a l l y l a r g e v a l u e s f o r the J=16 terms a s s o c i a t e d w i t h . T a b l e XV Frequency C o n t r i b u t i o n s ( i n MHz) o f the Terms A s s o c i a t e d w i t h the D i s t o r t i o n C o n s t a n t s L i s t e d i n Ta b l e X I I I t o the T r a n s i t i o n Frequency Exp F r e q J t t ' K +K D T H 4 T H 6 T L 4 T L 6 T L 8 T 7.970495 2 E 1 7.976605 -.000611 - 0 - .000044 - 0 - - 0 -423.02 7 F22->F2 425.76 -3.05 .30 .02 -.004 -.002 1246.55 7 1255.32 - 8 . 98 .15 .06 -.002 .002 10321.91 12 E ^ E 2 10578.34 -210.82 -51.55 3.92 '. 1.95 .06 11261.37 13 E*+E 2 11452.13 -266.27 73.18 5.78 -3.23 -.22 7861.67 14 E 2-*E 3 7977.08 -214.01 98.60 5.36 -5.02 -.34 19288.63 14 E*+E 2 19886.97 -533.53 -82.79 13.37 4.22 .39 14151.81 15 E ^ E 2 14663.55 -449.59 -78.97 12.88 4.60 -.65 18562.40 16 E2->E3 18884.62 -656.22 335.68 21.30 -22.15 -.83 18528.94 18 E 2+E 3 19411.73 -848.13 -72.63 34.62 6.03 -2.68 107 The f o u r t h and s i x t h degree d i s t o r t i o n c o n s t a n t s d e t e r m i n e d h e r e can be compared w i t h t h o s e d e t e r m i n e d by Tar r a g o e t a l [95] ( i n d i c a t e d by a prime ' f o r c l a r i t y h e r e ) . The d i s t o r t i o n c o n s t a n t s d e t e r m i n e d by Tarr a g o e t a l u s i n g t h e 3=2 t r a n s i t i o n o f O z i e r [ 3 5 , 7 0 ] , t h e two 3=1 t r a n s i t i o n s o f C u r l [15] and the J=12,13,14,15 t r a n s i t i o n s r e p o r t e d h e r e [34] a r e : Dj, = 132858 ±13 Hz, H 4 T = -16.151 ±.062 Hz and H £ T = 10.468 ±.027 Hz. I n t h i s work, D T = 132943.41 ±.71 Hz (removed 6.54 a' from D^), H 4 T = -16.9839 ±.0076 Hz (removed 13.43 a' from H 4 T ) and H 6 T = 11.0342 ±.0086 Hz (removed 20.96 a' from H £ T ) . The r e a s o n the d i s t o r t i o n c o n s t a n t s d e t e r m i n e d h e r e a r e so f a r from those d e t e r m i n e d by T a r r a g o e t a l [95] i s because t h e y i n c o r p o r a t e d t h e e i g h t h and h i g h e r degree c o n s t a n t s i n t o D^ ,, H 4 T and H ^ whereas h e r e t h e e i g h t h degree c o n s t a n t s have been s e p a r a t e d out (but t e n t h and h i g h e r degree terms a r e s t i l l i n c o r p o r a t e d ) . T a b l e XVI g i v e s p r e d i c t e d f r e q u e n c i e s w i t h s t a n d a r d d e v i a t i o n s based on the c o n s t a n t s g i v e n h e r e and thos e g i v e n by T a r r a g o e t a l [ 9 5 ] , The d i f f e r e n c e s between the p r e d i c t e d and e x p e r i m e n t a l f r e q u e n c i e s a r e a l s o l i s t e d , as a r e t h e t o t a l e i g h t h degree t e r m c o n t r i b u t i o n s t o t h e f r e q u e n c i e s ( o b t a i n e d from t a b l e XV). The e f f e c t i v e d i s t o r t i o n c o n s t a n t s d e t e r m i n e d by T a r r a g o e t a l p r e d i c t the f r e q u e n c i e s r e a s o n a b l y w e l l up t h r o u g h t h e maximum J v a l u e used i n the f i t (J=15) i n s p i t e o f some l a r g e e i g h t h degree c o n t r i b u t i o n s . Above J=15, t h e a b i l i t y o f T a r r a g o ' s c o n s t a n t s t o p r e d i c t f r e q u e n c i e s d i s i n t e g r a t e s r a p i d l y . The s i x d i s t o r t i o n c o n s t a n t s d e t e r m i n e d h e r e a re a g r e a t T a b l e XVI P r e d i c t e d T r a n s i t i o n F r e q u e n c i e s based on t h e D i s t o r t i o n C o n s t a n t s o f Ta r r a g o e [95] and the D i s t o r t i o n C o n s t a n t s determined h e r e . , E x p e r i m e n t a l P r e d i c t i o n (MHz) D i f f a T o t a l P r e d i c t i o n (MHz) D i f f a J Kt->Kt 8 t h , F r e q . (MHz) based on t h i s work degree based on Tarr a g o [95] 12 E ^ E 2 10321. 91 1 10321. 91 + .014 -.003 5. 93 10322. 20 ±1. 30 .29 13 E J+E 2 11261. 37 1 11261. 37 + .014 .000 9. 23 11261. 03 ±1. 52 — .34 14 E2->E3 7861. 67 + _ 1 7861. 67 + .02 -.003 10. 72 7862. 01 ±1. 14 .34 14 E ^ E 2 19288. 63 2 19288. 63 + .035 .000 17. 98 19288. 54 ±2. 76 — .09 15 E ^ E 2 14151. 81 1 14151. 81 + .03 .003 18. 13 14151. 73 ±2. 19 — .08 16 E2->E3 18562. 40 + # 2 18562. 40 + .06 .003 44. 28 18565. 42 ±3. 14 3 .02 18 E 2+E 3 18528. 94 2 18528. 94 + .22 -.004 43. 33 18520. 37 ±3. 64 -8 .57 D i f f e r e n c e (MHz) = P r e d i c t e d f r e q u e n c y - E x p e r i m e n t a l f r e q u e n c y T o t a l e i g h t h degree c o n t r i b u t i o n e q u a l t o the sum o f the a b s o l u t e v a l u e s o f the terms a s s o c i a t e d w i t h L ^ , L ^ T and L g T g i v e n i n t a b l e XV. 109 improvement o v e r t h e t h r e e d e t e r m i n e d by Tarr a g o e t a l [ 9 5 ] . The p r e d i c t e d f r e q u e n c y i s e q u a l t o the e x p e r i m e n t a l f r e q u e n c y t o a l l the s i g n i f i c a n t f i g u r e s used i n t h i s e xperiment. The s t a n d a r d d e v i a t i o n s a r e two o r d e r s o f magnitude b e t t e r f o r a l l the J v a l u e s used by Ta r r a g o e t a l . But j u s t as the comparison o f t h i s work w i t h t h a t o f Tarrago e t a l has shown ( s e c t i o n 4.3), the a b s o l u t e v a l u e s o f t h e s i x d i s t o r t i o n c o n s t a n t s t a b u l a t e d i n t a b l e X I I I cannot be e x p e c t e d t o be w i t h i n the l e a s t square s t a n d a r d d e v i a t i o n s , o^g, l i s t e d i n t a b l e X I I I because o f the i n c l u s i o n o f t e n t h and h i g h e r o r d e r terms. U s i n g t r e n d s shown i n t a b l e XV, t h e magnitudes o f the f o u r d e c t i c terms can be e s t i m a t e d as a f u n c t i o n o f J . F o r i n s t a n c e , t h e r a t i o s D T/H 4 T and E^/L^rj, can be used t o g i v e an approximate r a t i o f o r L^/M^y. w h i c h can be used t o approximate The c o n t r i b u t i o n o f terms i n M ^ and Mgy. are e s t i m a t e d t o be o f t h e o r d e r o f 0.1 MHz f o r J=12 and 1 MHz f o r J=18; w h i l e the c o n t r i b u t i o n s o f Mg^ and a r e e s t i m a t e d t o be n e g l i g i b l e . Based on t h e s e o r d e r o f magnitude e s t i m a t e s , the a b s o l u t e e r r o r s a r e g i v e n i n t a b l e X I I I . A l t h o u g h i n some cases i s s e v e r a l t i m e s a^g, the l e a s t w e l l d e t e r m i n e d c o n s t a n t , Lgy., i s n o t s i g n i f i c a n t l y p o o r e r i n terms o f (known t o w i t h i n 9%) t h a n i n terms o f a^g (known to w i t h i n 6%). Based on t h i s , no attempt t o dete r m i n e any o f the d e c t i c c o n s t a n t s seemed w a r r a n t e d . Even though the c o n t r i b u t i o n s o f t h e d e c t i c terms up t o J=17 a r e e s t i m a t e d t o be up t o 1 MHz, t h e e f f e c t i v e c o n s t a n t s d e t e r m i n e d h e r e s h o u l d be a b l e t o p r e d i c t f r e q u e n c i e s f o r 110 t r a n s i t i o n s up t o J=17 t o w i t h i n 0.1 MHz. As J goes above the h i g h e s t v a l u e used i n t h i s f i t (J=18), the a b i l i t y o f the s i x c o n s t a n t s d e t e r m i n e d here t o p r e d i c t a c c u r a t e l y i s e x p e c t e d t o get p o o r e r r a p i d l y . T a b l e X V I I l i s t s term v a l u e s i n MHz f o r a l l t h e r o t a t i o n a l l e v e l s o f methane up t o J=21 w i t h r e s p e c t t o t h e energy g i v e n by the s c a l a r terms i n the H a m i l t o n i a n . Here t a k e s a maximum v a l u e o f 2.4 MHz. Use o f t h i s t a b l e s h o u l d g i v e t r a n s i t i o n s t o w i t h i n 0.1 MHz f o r J < 16, t o w i t h i n 0.4 MHz f o r J < 18, t o w i t h i n 1.3 MHz f o r J < 20, and t o w i t h i n 3.4 MHz f o r J < 22. I l l T a b l e X V I I Tensor D i s t o r t i o n Energy E^ f o r 1 2CH^ i n t h e Ground S t a t e a J t K Err. (MHZ) a E (MHz) J K* Err. (MHz) 0 fE 0 Ax -0- F i 1 -855.411 0. 003 1 F i -0- 8 F 2 2 1,323.223 0. 003 E 2 1,173.921 0. 003 2 F 2 3.18820 0.00002 F i 2 678.946 0. 002 E -4.78230 0.00002 F 2 1 112.200 0. 002 3 A 2 47.78042 0.00023 E 1 -1,393.578 0. 004 F 2 7.96885 0.00004 F i1 -1,451.344 0. 004 F i -23.89566 0.00011 A i -1,549.761 0. 004 4 F 2 103.41425 0.00045 9 A 2 2,202.189 0. 005 E -15.85721 0.00008 F 22 1,966.574 0. 004 Fa -55.69044 0.00024 F i3 1,665.052 0. 003 A i - 111.45700 0.00049 A i 944.442 0. 004 F i 2 174.479 0. 002 5 Fx 2 218.4807 0.0009 E -14.596 0. 003 E 166.7862 0.0007 F 2 1 -2,371.901 0. 005 F 2 -111.0389 0.0005 F i 1 -2,473.351 0. 005 Fa 1 -218.6326 0.0009 10 F 2 3 3,157.929 0. 006 6 A i 499.9253 0.0017 E 2 2,753.177 0. 005 F i 380.8742 0.0013 F i 2 2,389.525 0. 004 F 2 2 246.0790 0.0009 A i 1,284.364 0. 007 A 2 -260.9141 0.0011 F i 1 126.668 0. 002 F 2» -404.9655 0.0014 F 2 2 -299.909 0. 004 E -452.4870 0.0016 A 2 -3,729.734 0. 007 F 2 1 -3,820.052 0. 007 7 F i 2 814.164 0.002 E 1 -3,861.732 0. 007 E 558.015 0.002 F 2 2 391.136 0.001 11 F i 3 4,535.927 0. 007 A 2 80.229 0.001 E 2 4,246.546 0. 006 F 2 1 -748.642 0.002 F 2 3 3,583.832 0. 007 112 •(MHz)-(MHz)+ a. F i 2 1, 761. 548 0. 009 E 1 353. 813 0. 003 F 2 2 634. 969 0. 004 A 2 - 1 , 072. 371 0. 007 F 2 1 - 5 , 706. 826 0. 008 F i 1 - 5 , 777. 211 0. 008 A i 2 6, 553. 297 0. 009 F i 3 6, 190. 678 0. 008 F 2 3 5, 722. 754 0. 008 A 2 4, 916. 127 0. 013 F 2 2 2, 490. 198 0. 009 E 2 2. 043. 089 0. 010 F i 2 - 1 , 219. 810 0. 005 F 2 1 - 1 , 754. 304 0. 007 E 1 - 8 , 278. 818 0. 010 F i 1 - 8 , 307. 660 0. 010 A i 1 - 8 , 363. 535 0. 010 F i k 8, 746. 728 0. O i l E 2 8, 228. 119 0. 012 F 2 3 7, 697. 822 0. 013 A 2 4, 634. 261 0. 013 F 2 2 3, 188. 315 0. 007 F i 3 2, 227. 735 0. 009 A i - 2 , 300. 808 0. 013 F i 2 - 2 , 821. 418 0. 008 E 1 - 3 , 033. 251 0. 007 F 2 1 - 1 1 , 617. 690 0. 014 F i 1 - 1 1 , 662. 556 0. 015 F 2 11, 678. 606 0. 014 E 3 11, 258. 425 0. 015 F i 3 10, 539. 840 0. 021 F 2 3 5, 996. 395 0. 012 E 2 3 396. 758 0. 013 F i 2 2 631. 683 0. O i l A i 1 417. 734 0. 013 F i 1 - 4 , 478. 110 0. 014 F 2 2 -4, 861. 682 0. 010 A 2 -15 840. 180 0. 031 F 2 1 -15 874. 791 0. 032 E 1 -15 891. 872 0. 032 A 2 2 15 566. 36 0. 02 F 2 «» 15 110. 80 0. 02 F i <t 14 551. 75 0. 03 A i 13 780. 48 0. 04 F i 3 7 ,941. 17 0. 02 E 2 7 119. 81 0. 02 F 2 3 3, 130. 48 0. 03 F i 2 1 ,446. 81 0. 02 E 1 -7 032. 00 0. 02 F 2 2 -7 205. 62 0. 02 A 2 1 -7 522. 97 0. 01 F 2 1 - 2 1 141. 36 0. 07 F i 1 - 2 1 ,167. 20 0. 07 F 2 19 830. 81 0. 04 E 3 19 271. 74 0. 05 F i it 18 683. 31 0. 05 A i 2 11 671. 59 0. 03 F i 3 9 936. 68 0. 04 F 2 3 8, 309. 86 0. 04 A 2 3 ,100. 74 0. 07 F 2 2 1 ,316. 30 0. 05 E 2 709. 34 0. 04 F i 2 -10,389. 76 0. 02 113 (MHz)-(MHz)->-o-T F21 -10,671. 70 0. 02 E 1 -27,620. 56 0. 15 Fx 1 -27,630. 19 0. 15 A i 1 -27,649. 34 0. 15 17 F i 5 25,212. 81 0. 08 E 3 24,730. 41 0. 09 F 2 * 24,064. 01 0. 10 F i * 14,508. 30 0. 08 E 2 11,576. 04 0. 09 F z 3 10,083. 40 0. 09 A 2 7,268. 66 0. 09 F 2 2 592. 50 0. 09 F i 3 -569. 15 0. 08 Ai -14,569. 26 0. 02 F i 2 -14,799. 92 0. 02 E 1 -14,910. 06 0. 03 F . 1 -35,454. 28 0. 29 F i 1 -35,468. 39 0. 29 18 A i 2 31,896. 50 0. 16 F i * 31,417. 67 0. 17 F 2 5 30,866. 96 0. 18 A 2 2 30,206. 78 0. 19 F 2 * 18,366. 30 0. 15 E 3 17,080. 93 0. 16 F i 3 12,678. 37 0. 18 F 2 3 8,543. 50 0. 18 E 2 -1 ,448 . 01 0. 15 F i 2 -2 ,048 . 51 0. 14 A i 1 -3 ,067 . 85 0. 14 F i 1 -20,060. 82 0. 06 F 2 2 -20,236. 33 0. 06 A 2 1 -44,803. 52 0. 53 F 2 l -44 , 813. 78 0. 53 E l -44 818. 90 0. 53 F i 5 39 289. 32 0. 30 E 3 38 768. 65 0. 31 F 2 5 38, 224. 91 0. 33 A 2 2 24 388. 52 0. 27 F 2 4 22 369. 99 0. 28 F i it 20 088. 76 0. 28 Ai 14, 797. 03 0. 30 F i 3 10 126. 98 0. 30 E 2 8 858. 77 0. 31 F 2 3 - 4 347. 94 0. 22 F i 2 - 5 374. 93 0. 23 E 1 -26 666. 72 0. 15 F 2 2 -26 735. 82 0. 15 A 2 1 -26 , 871. 25 0. 15 F 2 1 -55 856. 23 0. 92 F i 1 -55 , 863. 60 0. 92 20 F 2 5 48, 207. 28 0. 53 E * 47, 747. 63 0. 54 F i 5 47, 195. 85 0. 56 F 2 * 29 474. 97 0. 46 E 3 26 378. 01 0. 46 F x * 24 115. 55 0. 47 A i * 18, 097. 89 0. 56 F i 3 11 180. 89 0. 48 F 2 3 8 690. 79 0. 50 A 2 - 7 780. 74 0. 31 F 2 2 - 8 676. 10 0. 32 E 2 - 9 076. 79 0. 33 F i 2 -34 ,666. 31 0. 31 F 2 1 -34 771. 79 0. 32 J KFC Err, + (MHz) + a E E 1 - 6 8 , F i 1 - 6 8 , A i 1 - 6 8 , 2 1 A 2 2 5 8 , F 2 5 5 8 , F i 6 5 7 , A i 2 5 7 , F i 5 3 6 , E 3 3 4 , F 2 * 2 9 , F i * 2 1 , E 2 1 0 , F 2 3 8 , A 2 1 6 , F 2 2 - 1 3 , F i 3 - 1 3 , A i 1 - 4 4 , F i 2 - 4 4 , E 1 - 4 4 , F 2 1 - 8 3 , F i 1 - 8 3 , 7 7 5 . 4 0 1 . 5 2 7 7 8 . 0 3 1 . 5 2 7 8 3 . 3 0 1 . 5 2 7 9 5 . 4 0 0 . 8 7 3 6 3 . 1 9 0 . 8 9 8 9 2 . 5 0 0 . 9 1 3 7 3 . 1 5 0 . 9 3 1 1 9 . 3 5 0 . 7 4 3 6 1 . 9 6 0 . 7 5 9 6 6 . 2 1 0 . 7 3 2 1 7 . 5 2 0 . 8 9 3 9 9 . 5 4 0 . 7 2 8 8 3 . 8 9 0 . 7 6 4 4 2 . 0 4 0 . 8 2 2 2 2 . 6 0 0 . 4 3 8 9 4 . 8 0 0 . 4 6 1 8 6 . 5 2 0 . 6 0 2 6 6 . 6 6 0 . 6 0 3 0 6 . 3 6 0 . 6 1 7 4 9 . 8 2 2 . 4 3 7 5 3 . 5 6 2 . 4 3 a O g i s t h e s t a n d a r d d e v i a t i o n c a l c u l a t e d by l e a s t squares a n a l y s i s . 115 CHAPTER 5  CONCLUSIONS T h i s e x p e r i m e n t has produced the f i r s t microwave spectrum o f a s p h e r i c a l top m o l e c u l e u s i n g a g e n e r a l purpose i n s t r u m e n t . The method does n o t r e l y on f o r t u i t o u s c o i n c i d e n c e s and thus can be used t o i n v e s t i g a t e t h e e n t i r e c l a s s o f s p h e r i c a l t o p s p r o v i d e d the microwave t r a n s i t i o n s o f i n t e r e s t are o f s u f f i c i e n t i n t e n s i t y and the s e a r c h r e g i o n i s r e a s o n a b l y w e l l d e f i n e d . The need t o know where t o l o o k has been amply demonstrated by t h i s experiment. The t h r e e o c t i c d i s t o r t i o n c o n s t a n t s L ^ , Lgy. and L g T o f 1 2 C H i t have been d e t e r m i n e d f o r the f i r s t time i n t h i s e xperiment. The l e a s t w e l l d e t e r m i n e d d i s t o r t i o n c o n s t a n t , Lgy, i s known 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 1 p a r t i n 10 w h i c h i s the same as Dy. was f o u r y e a r s ago [ 1 7 ] . The s i x d i s t o r t i o n c o n s t a n t s d e t e r m i n e d here ( t a b l e X I I I ) have p e r m i t t e d the a c c u r a t e p r e d i c t i o n o f a l l the Q-branch t r a n s i t i o n s o f methane t o J=21 ( t a b l e X V I I ) . T h i s a c c u r a t e p r e d i c t i o n c o u l d p o s s i b l y r e s u l t i n the d e t e c t i o n o f methane i n i n t e r s t e l l a r space u s i n g t h e t e c h n i q u e s o f r a d i o astronomy. I n t e r e s t has even been e x p r e s s e d i n u s i n g the microwave a b s o r p t i o n s o f methane t o i d e n t i f y t h i s gas i n o i l w e l l b o r e s [ 2 3 ] . The s i x t e n s o r d i s t o r t i o n c o n s t a n t s d e t e r m i n e d here ( t a b l e X I I I ) w i l l a l s o be u s e f u l i n the d e t e r m i n a t i o n o f t h e s c a l a r d i s t o r t i o n c o n s t a n t s o f methane when the t r a n s i t i o n s w h i c h depend on b o t h the t e n s o r and s c a l a r d i s t o r t i o n c o n s t a n t s 116 are measured w i t h s u f f i c i e n t p r e c i s i o n ( u s i n g l a s e r t e c h n i q u e s p o s s i b l y ) . The t r a n s i t i o n s o b s e r v e d h e r e s h o u l d a i d i n the s e t t i n g - u p o f new microwave s p e c t r o m e t e r s d e s i g n e d t o d e t e c t a b s o r p t i o n s due t o v e r y weak t r a n s i t i o n moments because o f the d i f f i c u l t y o f power s a t u r a t i n g t h e methane l i n e s . I t i s hoped t h i s e x p e r i m e n t , w h i c h s u c c e s s f u l l y employed a s i m p l e microwave s o u r c e n o i s e c a n c e l l a t i o n scheme u s i n g t h e r m a l d e t e c t o r s combined w i t h a modern system o f d a t a c o l l e c t i o n has s e r v e d t o advance th e a r t o f the d e t e c t i o n o f weak s i g n a l s b u r i e d i n n o i s e . A l t h o u g h t h e microwave t r a n s i t i o n s r e p o r t e d h e r e a r e the weakest e v e r o b s e r v e d w i t h a c o n v e n t i o n a l S t a r k modulated microwave s p e c t r o m e t e r , i t i s n o t f e l t the u l t i m a t e l i m i t has been reached. Longer p a t h l e n g t h s , q u i e t e r d e t e c t o r s , and equipment o f improved d e s i g n s h o u l d y i e l d t r a n s i t i o n s even weaker t h a n t h o s e o b s e r v e d h e r e . The t h e o r y o f the u l t i m a t e s e n s i t i v i t y o f a microwave s p e c t r o m e t e r has been summarized by Gordy [ 2 6 ] , Townes and Schawlow [97] and S t r a n d b e r g [ 9 1 ] . I t i s i n t e r e s t i n g t o compare the s e n s i t i v i t y a c h i e v e d w i t h the s p e c t r o m e t e r used i n t h i s e xperiment w i t h an " u l t i m a t e " s e n s i t i v i t y c a l c u l a t e d by S t r a n d b e r g [ 9 1 ] . S t r a n d b e r g ' s " u l t i m a t e " s e n s i t i v i t y i s 1 0 " 1 1 cm - 1 based on the f o l l o w i n g a s sumptions: a) a bandwidth o f .01 Hz e q u i v a l e n t t o a time c o n s t a n t o f 314 seconds (RC = T T / B, r e f . [ 9 1 ] ) , b) a c e l l p a t h l e n g t h o f 60 m, c) a s i g n a l t o n o i s e r a t i o o f 1, and d) a c e l l a t t e n u a t i o n c o e f f i c i e n t o f 1 0 " 3 cm - 1. I n o r d e r t o compare s e n s i t i v i t i e s , t h e amount o f time r e q u i r e d t o scan a t y p i c a l methane l i n e 117 w i t h S t r a n d b e r g ' s h y p o t h e t i c a l s p e c t r o m e t e r must be determined. From f i g u r e 15 o f r e f e r e n c e [91] i t appears t h e f a s t e s t t o l e r a b l e sweep r a t e i s t h a t i n w h i c h h a l f t h e l i n e w i d t h a t h a l f - h e i g h t i s swept i n one t i m e c o n s t a n t . S i n c e a t y p i c a l methane l i n e h a l f - w i d t h a t h a l f - h e i g h t was 1 / I* MHz, t h e f a s t e s t sweep time would be 12 time c o n s t a n t s o r a p p r o x i m a t e l y 1 hour f o r maximum t o l e r a b l e d i s t o r t i o n o f a methane l i n e i n a 3 MHz scan. I f S t r a n d b e r g ' s c e l l l e n g t h were r e d u c e d a p p r o x i m a t e l y f i v e f o l d f r om 60 meters t o t h e 13 meters used i n t h i s e x p e r i m e n t , a p p r o x i m a t e l y 5 2 = 25 hours o f sweep time o r 1 day would have been r e q u i r e d t o a c h i e v e t h e same " u l t i m a t e " s e n s i t i v i t y o f 1 0 " 1 1 cm - 1. Furthermore i f a s i g n a l t o n o i s e r a t i o o f 3 i n s t e a d o f 1 were d e s i r e d , the sweep time would have t o be i n c r e a s e d t o 3 2 = 9 days f o r a s e n s i t i v i t y o f 1 0 " 1 1 cm - 1. I f S t r a n d b e r g ' s c e l l a t t e n u a t i o n c o e f f i c i e n t ( I O - 3 cm - 1) were i n c r e a s e d t o c o r r e s p o n d w i t h t h e b e s t 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 (1.4 X 1 0 - 3 c n r t 1 ) , the " u l t i m a t e " s e n s i t i v i t y w o u l d have been degraded t o 1.4 X 1 0 " 1 1 cm - 1 ( e q u a t i o n X.8, r e f e r e n c e [ 9 1 ] ) . Two days o f c o n t i n u o u s sample sweep t i m e were r e q u i r e d t o o b s e r v e t h e weakest methane l i n e s (2 X 1 0 " 1 1 cm - 1) a t a s i g n a l t o n o i s e r a t i o o f 3. Four days o f sweep time would have been r e q u i r e d t o r e a c h the above c a l c u l a t e d s e n s i t i v i t y o f 1.4 X 1 0 " 1 1 cm - 1. These c a l c u l a t i o n s show t h a t S t r a n d b e r g ' s " u l t i m a t e " s p e c t r o m e t e r m o d i f i e d w i t h a s h o r t e r , more l o s s y c e l l would r e q u i r e 9 days t o produce the same s i g n a l to n o i s e r a t i o w h i c h the s p e c t r o m e t e r used i n t h i s e x p e r i m e n t c o u l d 118 produce i n 4. I n o t h e r words, the e x p e r i m e n t a l s p e c t r o m e t e r used here was i n a sense " b e t t e r t h a n u l t i m a t e " . T h i s can be e x p l a i n e d i n terms o f a n o t h e r assumption o f S t r a n d b e r g : t h a t the performance i s l i m i t e d by the s o u r c e n o i s e o f a v e r y low n o i s e k l y s t r o n , whose n o i s e power i s 160 db/Hz below the c a r r i e r (an e x c e p t i o n a l l y good k l y s t r o n ) . The s p e c t r o m e t e r used h e r e employed n o i s e - c a n c e l l a t i o n w h i c h r e d u c ed t h e e f f e c t o f k l y s t r o n n o i s e s i g n i f i c a n t l y . The e f f e c t o f k l y s t r o n n o i s e may have been reduced t o t h e p o i n t t h a t i t was no l o n g e r the l i m i t i n g f a c t o r . 119 APPENDIX A BOLOMETER TIME CONSTANT The b o l o m e t e r time c o n s t a n t can be c a l c u l a t e d u s i n g t h e h o t - w i r e anemometer time c o n s t a n t t h e o r y o f B u r g e r s [10] and Dryden & Kuethe [18] t o g e t h e r w i t h t h e d a t a o f K i n g [ 4 4 ] . The o p e r a t i o n o f a h o t - w i r e anemometer i s s i m p l e : a i r f l o w i n g over a h o t w i r e removes heat energy c o o l i n g the w i r e , as the w i r e i s c o o l e d , the r e s i s t a n c e f a l l s w h i c h can be used t o measure the a i r v e l o c i t y . The e f f e c t i v e n e s s o f f o r c e d c o n v e c t i o n c o o l i n g o f a h o t w i r e can be seen by b l o w i n g on t h e e l e c t r i c h e a t i n g element o f a d o m e s t i c t o a s t e r . H o t - w i r e anemometers and b o l o m e t e r s a r e q u i t e s i m i l a r i n t h a t t h e y b o t h c o n s i s t o f f r a g i l e p l a t i n u m w i r e s o f comparable d i m e n s i o n [57] h e a t e d a p p r o x i m a t e l y the same degree above ambient w i t h a i r c o n v e c t i o n the dominant h e a t l o s s mechanism. The b o l o m e t e r h e a t l o s s i s due m a i n l y t o f r e e - c o n v e c t i o n , whereas the h o t - w i r e anemometer heat l o s s i s due m a i n l y to f o r c e d -c o n v e c t i o n . H o t - w i r e anemometer t h e o r y can be a p p l i e d t o b o l o m e t e r s by s e t t i n g t h e f o r c e d a i r v e l o c i t y to z e r o w h i c h l e a v e s o n l y f r e e - c o n v e c t i o n terms. The h e a t l o s s o f a b o l o m e t e r due t o f r e e - c o n v e c t i o n has been shown i n s e c t i o n 3.7.1 t o be a p p r o x i m a t e l y 99% o f the t o t a l h e a t l o s s . The r a t e o f change o f h e a t energy o f a h o t - w i r e anemometer has been g i v e n by Dryden & Kuethe [ 1 8 ] : dH/dt = i 2 R - (K + C v ^ K T - T 0) ( A . l ) where C and K a r e c o n s t a n t s , V i s t h e a i r v e l o c i t y , T i s the 120 temperature o f the w i r e , and To i s the tem p e r a t u r e o f the a i r i n °K. I n o r d e r t o a p p l y t h i s e q u a t i o n t o a b o l o m e t e r , t h e a i r v e l o c i t y must be s e t e q u a l t o z e r o and an a d d i t i o n a l term must be added t o acc o u n t f o r the he a t s u p p l i e d to t h e w i r e by the microwave power. When t h i s i s done, t h e f o r e g o i n g e q u a t i o n becomes: dH/dt = i 2 R + P r f - K(T - T 0) (A.2) I n the s t e a d y s t a t e , dH/dt = 0; t h i s y i e l d s : i 2 R + P r f = K(T - To) P = K(T - To) (A.3) where K has been g i v e n by K i n g [44] f o r a w i r e o f d i m i n i s h i n g l y s m a l l d i a m e t e r : K = 2.5 X 1 0 - l t [ l + .00114(T — To)] (A.4) i n u n i t s o f w a t t s p e r °K per u n i t l e n g t h . B e f o r e p r o c e e d i n g t o t h e time c o n s t a n t c a l c u l a t i o n , t h e s e e q u a t i o n s can be used t o d e r i v e t h e " e m p i r i c a l b o l o m e t e r r e s i s t a n c e law " , R = R 0 + LP- 9 (A. 5) g i v e n by G r i e s h e i m e r [ 2 8 ] , where L i s a c o n s t a n t w h i c h v a r i e s f r om b o l o m e t e r t o b o l o m e t e r . The c o n s t a n t K i s e v a l u a t e d a t two p o i n t s 100° and 200°C above ambient. The power d i s s i p a t i o n p e r u n i t l e n g t h o f w i r e i s then c a l c u l a t e d from e q u a t i o n (A.3). These v a l u e s are t a b u l a t e d i n t a b l e X V I I I . L e t t h e r e s i s t a n c e o f a b o l o m e t e r be g i v e n as an e x p o n e n t i a l f u n c t i o n o f P: R = Ro + L P X ( A , 6 ) where t h e e x p o n e n t i a l x i s t o be determined. The v a l u e o f R T a b l e X V I I I Bolometer Heat D i s s i p a t i o n 121 T - To (°C) K w a t t s 'C-unit l e n g t h J p [ w a t t s [ u n i t l e n g t h 100 200 2.785 X 10-* 3.07 X 10"* 2.785 X 1 0 - 2 6.14 X 1 0 " 2 can be o b t a i n e d from the r e l a t i o n g i v i n g the temp e r a t u r e dependence o f r e s i s t a n c e : R = R 0 [ 1 + a(T - T 0 ) ] (A.7) where a i s the temp e r a t u r e c o e f f i c i e n t o f r e s i s t i v i t y ( ° C ) _ 1 . Combining the two p r e c e d i n g e q u a t i o n s g i v e s : R 0 a ( T - T 0 ) = L P X Roa = L P X / ( T - T 0 ) (A. 8) The l e f t hand s i d e o f the above e q u a t i o n i s a c o n s t a n t f o r t h e two p o i n t s 100° and 200°C above ambient. U s i n g v a l u e s from t a b l e X V I I I , t h e f o l l o w i n g e q u a t i o n i s o b t a i n e d : LP loo _ L P 2 o o ToU 2 W - 2 \ X = tc 1 / . v i n - 2 \ X (A. 9) 2(2. 785 X 1 0 " 2 ) A = ( 6 . 1 4 X 1 0 - 2 ) ( 2 . 2 0 4 6 ) x = 2 w h i c h g i v e s the v a l u e : x = .877 Thus the r e s i s t a n c e o f a b o l o m e t e r i s g i v e n by: R = Ro + LP- 8 7 7 (A. 10) i n e x c e l l e n t agreement w i t h e q u a t i o n ( A . 5 ) . The above d e r i v a t i o n i s based on e x p r e s s i o n s and c o n s t a n t s f o r t h e f r e e c o n v e c t i v e h e a t l o s s o f p l a t i n u m w i r e s g i v e n i n the e x c e l l e n t paper by K i n g [ 4 4 ] . 122 The b o l o m e t e r time c o n s t a n t i n seconds, x, can be c a l c u l a t e d from an e q u a t i o n g i v e n by Dryden & Kuethe [18] m o d i f i e d t o t a k e i n t o account a d d i t i o n a l i n c i d e n t microwave power: x = m s < T ~ T°> (A. 11) P r f + i 2 R o where m = mass o f the w i r e (gm), s = s p e c i f i c h e a t o f the w i r e (joules/gm°C) and the r e m a i n i n g q u a n t i t i e s a r e as p r e v i o u s l y d e f i n e d . F o r a b o l o m e t e r w i r e o f l e n g t h , I, e q u a t i o n s (A.3) and (A.7) g i v e t h e r e l a t i o n : P = K' (T — To) = K ' ( R - R 0 ) / a R 0 (A.12) where K' = KA. U s i n g (A.12) and t h e r e l a t i o n P f = P - i 2 R , t h e time c o n s t a n t becomes: msP/K' = msP/K'  P + i 2 ( R 0 - R) (P/K')K' + i 2 ( - a R 0 ) P / K ' T = — (A:13) K' - i 2 a R 0 B e f o r e t h e b o l o m e t e r time c o n s t a n t can be c a l c u l a t e d , the mass o f t h e b o l o m e t e r w i r e must be determined. The l e n g t h may be d e t e r m i n e d from t h e h e a t l o s s because f o r v e r y s m a l l w i r e d i a m e t e r s , t h e heat l o s s i s independent o f t h e d i a m e t e r [ 4 4 ] . Once t h e l e n g t h i s known, the d i a m e t e r can be c a l c u l a t e d from t h e volume r e s i s t i v i t y o f p l a t i n u m and t h e t o t a l r e s i s t a n c e . W i t h the l e n g t h and d i a m e t e r , i t ' s a s i m p l e m a t t e r t o c a l c u l a t e t h e w i r e mass. A t 200 ohms the p l a t i n u m b o l o m e t e r w i r e i s h e a t e d 216°C above an ambient o f 27°C assuming a c o l d (27°C) r e s i s t a n c e o f 116 ohms; t h i s i s o b t a i n e d from t h e approximate r e l a t i o n 123 AT = AR(1 + aT)/aR (A. 14) where the temperature c o e f f i c i e n t o f r e s i s t i v i t y o f p l a t i n u m , a C C ) " 1 , i s g i v e n t h e v a l u e .0037 [ 4 4 ] , R = 116 ohms and T = 27°C. I t was e x p e r i m e n t a l l y d e t e r m i n e d i n l a b o r a t o r y t h a t 15.3 mw o f power were r e q u i r e d t o i n c r e a s e t h e r e s i s t a n c e o f t h e b o l o m e t e r from 116 t o 200 ohms. From the r e l a t i o n P = K' (T — To) .0153 w a t t s = K'(216°C) K* = 7.1 X 1 0 " 5 watts/°C (A.15) f o r the Narda N610B/38B7 b o l o m e t e r s used The f r e e c o n v e c t i v e h e a t l o s s p e r u n i t l e n g t h , K, o f p l a t i n u m w i r e 216°C above ambient can be c a l c u l a t e d from the work o f K i n g [ 4 4 ] : e q u a t i o n ( A . 4 ) : K = 2 . 5 X 1 0 - * [ 1 +.00114(216)] = 3.1 X10"* w a t t s (A.16) cm- °C The l e n g t h o f the p l a t i n u m w i r e i n t h e b o l o m e t e r i s t h u s : l e n g t h = — = 7 - 1 X 1 Q " 5 = .229 cm = 2.29 mm (A. 17) K 3.1X10-* from e q u a t i o n s (A.15) and (A.16). The r e s i s t a n c e o f t h e b o l o m e t e r i s g i v e n by: R = p a / A = .1(2290)/A (A.18) where p i s the volume r e s i s t i v i t y : p = .1 ohm-micron; I i s the l e n g t h i n m i c r o n s ; and A i s the a r e a i n m i c r o n s 2 . S o l v i n g f o r A: A = 1.975 * 2 m i c r o n s 2 = 2 X 1 0 " 8 c m 2 (A.19) The r e s u l t i n g w i r e s i z e : 2.3 mm X 1.6 m i c r o n s d i a m e t e r i s r e a s o n a b l e . The mass o f t h e w i r e i s g i v e n by: m = l-A-density (A.20) 124 where t h e d e n s i t y i s 21.4 gm/cm3. m = .229 cm(2 X 1 0 " 8 c m 2 ) 2 1 . 4 gm/cm3 = 9.8X10- 8gm (A.21) The time c o n s t a n t o f t h e bolom e t e r i s now c a l c u l a t e d f o r the case i n w h i c h i t i s b r o u g h t up t o 200 ohms s o l e l y w i t h d-c power, t h a t i s , i = 8.75 ma: T = — — — = 360 mi c r o s e c o n d s (A. 22) K' - i 2 a R 0 where m = 9 . 8 X 1 0 " 8 gm; s = s p e c i f i c h e a t o f p l a t i n u m = .139 joules/gm-°C; K' = 7 . 1 X 1 0 " 5 watts/°C; R 0 = 116 ohms; and a = .0037 C C ) " 1 from K i n g [ 4 4 ] . The time c o n s t a n t o b t a i n e d o f 360 mic r o s e c o n d s i s i n e x c e l l e n t agreement w i t h the c a t a l o g v a l u e o f 350 mic r o s e c o n d s f o r a Narda N610B/38B7 b o l o m e t e r . Cohn [13] f i n d s t he c a t a l o g v a l u e o f the time c o n s t a n t o f the e l e c t r i c a l l y e q u i v a l e n t S p e r r y 821 t o be r e l i a b l e . I f t h e b o l o m e t e r i s broug h t up t o 200 ohms w i t h 6.3 mw d-c b i a s power and 9 mw microwave power, t h e time c o n s t a n t T = m s = 240 mi c r o s e c o n d s (A.23) K* - i 2 a R 0 where i = 5.6 ma. T h i s i s t h e time c o n s t a n t used i n t h e t e x t . I t i s u s e f u l t o know t h e time c o n s t a n t when the b i a s c u r r e n t i s m a i n t a i n e d a t 5.6 ma, b u t microwave power i s l i m i t e d such t h a t the bo l o m e t e r can be brought up o n l y t o , say, 180 ohms. A t 180 ohms, t h e temperature r i s e i s l e s s : (T - To) = AT = A R ( 1 + aTo) = 64[1 +.0037(27)] aR 0 .0037(116) AT = 164°C (A.24) A s m a l l e r t e m p e r a t u r e r i s e g i v e s a s m a l l e r v a l u e f o r K i n g ' s [44] c o n s t a n t K ( A . 4 ) : 125 K = 2 . 5 X 1 0 " * [ 1 +.00114(164)] = 2.96X10"* U-25) w h i c h i n t u r n g i v e s a s m a l l e r v a l u e o f K' = K£ K' = .229(2.96 X10"*) = 6.8 X 1 0 " 5 watts/°C (A.26) The time c o n s t a n t i s : m s = 250 microseconds (A.26) K' - i 2 a R 0 where m, s, a, and R 0 a r e as p r e v i o u s l y g i v e n K' = 6.8 X I O " 5 watts/°C i = 5.6 ma I n t h e case o f l i m i t e d microwave power (b o l o m e t e r r e s i s t a n c e down to 180 ohms from 200 ohms), t h e time c o n s t a n t r i s e s s l i g h t l y f r o m 240 t o 250 m i c r o s e c o n d s . 126 APPENDIX B THE SYMMETRIC TOP ROTATIONAL WAVEFUNCTION PARITIES K r o n i g [49] found t h e wave e q u a t i o n f o r a d i a t o m i c m o l e c u l e t o be i n v a r i a n t under the t r a n s f o r m a t i o n : x -»• x y - y z z tj) -+• IT + tj) x - x T h i s t r a n s f o r m a t i o n i s g i v e n by e q u a t i o n (15) r e f e r e n c e [ 4 9 ] . The symbols mip and a<J> i n [49] have been changed t o m<(> and Kx t o c o r r e s p o n d w i t h c u r r e n t usage. The x,y,z are the m o l e c u l e f i x e d C a r t e s i a n c o o r d i n a t e s . The E u l e r a n g l e s tj>, 6 , and x used h e r e have been d e f i n e d by K r o n i g [49] as f o l l o w s : 6 i s t h e a n g l e between t h e p o s i t i v e space f i x e d Z - a x i s and the p o s i t i v e m o l e c u l e f i x e d z - a x i s , cj> i s t h e a n g l e between the p o s i t i v e space f i x e d X - a x i s and t h e p o s i t i v e l i n e of nodes, and x i s t n e a n g l e between the p o s i t i v e l i n e o f nodes and t h e p o s i t i v e m o l e c u l e f i x e d x - a x i s . The r o t a t i o n s a r e c a r r i e d out a c c o r d i n g t o the r i g h t hand r u l e i n t h e f o l l o w i n g manner: f i r s t t h e m o l e c u l a r c o o r d i n a t e system i s r o t a t e d by tj) about the space f i x e d Z - a x i s , t h e n by 6 about t h e x - a x i s , and f i n a l l y by x about t h e z - a x i s . The E u l e r a n g l e s used h e r e are i l l u s t r a t e d i n Symon [94] where \> r e p l a c e s x used h e r e . Wang [98] found t h e wave e q u a t i o n f o r an asymmetric t op to be i n v a r i a n t under the same t r a n s f o r m a t i o n ( B . l ) . The 127 t r a n s f o r m a t i o n c o n s i s t s o f a r e f l e c t i o n i n t h e m o l e c u l a r x , z - p l a n e ( t h e r e f l e c t i o n i s p a r a l l e l t o the y - a x i s ) f o l l o w e d by a t w o - f o l d r o t a t i o n about t h e "m o l e c u l e f i x e d " y - a x i s . T h i s t r a n s f o r m a t i o n i s e q u i v a l e n t t o a m o l e c u l e f i x e d i n v e r s i o n . The t r a n s f o r m a t i o n g i v e s a space f i x e d i n v e r s i o n a l s o . T h i s i s demonstrated by a p p l y i n g the above t r a n s f o r m a t i o n t o K r o n i g ' s e q u a t i o n s r e l a t i n g t h e space f i x e d C a r t e s i a n c o o r d i n a t e s t o the E u l e r a n g l e s [48] and t h e m o l e c u l e f i x e d C a r t e s i a n c o o r d i n a t e s ( e q u a t i o n (1) r e f e r e n c e [ 4 8 ] ) : X = x(cosxcoscj) — cos9sinxsin<j)) + y ( — sinxcos(|> — cos6cosxsin<|>) + z ( s i n e sine))) Y = x(cosxsintj) + cos6sinxcos(f>) + y (— sinxsin<|> + cos6cosxcosc|>) + z ( — sin6cosc|)) Z = x ( s i n 6 s i n x ) + y ( s i n 8 c o s x ) (B.2) + z ( c o s B ) When s u b j e c t e d t o K r o n i g ' s t r a n s f o r m a t i o n ( B . l ) , t h e above e q u a t i o n s y i e l d : X + -X Y + -Y (B.3) Z + -Z Thus K r o n i g ' s t r a n s f o r m a t i o n g i v e s b o t h a space f i x e d and a m o l e c u l e f i x e d i n v e r s i o n o f a l l p a r t i c l e s t h r o u g h t h e o r i g i n . The " K r o n i g ' s symmetry" found by Wang i s p a r i t y . I t w i l l be shown t h a t the Wang f u n c t i o n s s a t i s f y p a r i t y . T h i s w i l l be done u s i n g u n n o r m a l i z e d symmetric top w a v e f u n c t i o n s 128 [N =/(2J+l)/8T T z ]. The symmetric top w a v e f u n c t i o n s a r e : D m ± K < * 6 X > - e - ^ ^ d i . C e ) um±K" (B.4) Under th e i n v e r s i o n o p e r a t i o n the E u l e r a n g l e s become: e + T T - 0 , <{>-><J> + TT, X ^ " - X (B.5) U s i n g Wang f u n c t i o n s d e f i n e d i n e q u a t i o n ( 2 . 1 2 ) : iJmK1) = = [ ^ ( ^ . ^ - e . - x ) ± D^_ K ( 7 r+c} ) , 7 r - e , - x ) ] (B.6a) (B.6b) (B.6c) ± e-i [ t n ( 7 r+*)-K ( - x)] dJ ( 7 T_ e ) m-K = ( - l ) m f e - i ( m * - K ^ d ^ K ( T r - e ) ± e - ^ ^ ^ d l ^ C T t - e ) ] m-lT (B.6d) = ( - l ) J [ e - i ( m < f , - K x ) d ^ _ K ( e ) ± e - ^ + ^ d ^ e ) ) (B.6e) 1 P m K ± D m - K = (-1)' ,J + D J m-K - mK (B.6f) where e ~ l m i T = ( - l ) m has been used t o o b t a i n (B.6d) and d ^ C e ) = ( - l ) J " m d J v ( T T - 6 ) has been used [8] t o o b t a i n ( B . 6 e ) . mix m—ix E q u a t i o n (B.6f) can be r e - w r i t t e n t o show Wang's p a r i t y c l e a r l y : i | J m K ' > = ( - 1 ) " | J m K ' > (B.7) J m K + ) = - l ) J J m K + )i | JmK") = - (-1) J | JwK~y where Wang's f u n c t i o n s a r e : . A */£ JntfO = (2)" / 2[|JmK^> + | Jm-K) (B.8) 129 B i b l i o g r a p h y [1] A l i e v , M.R., Zh. Eksp. Teor. F i z . Pis'ma. Red., 14, 600 (1971), Sov. Phys. JETP L e t t . , 14, 417 (lF7l) [2] A l i e v , M.R., and V.M. M i k h a y l o v , O p t i k a i S p e k t r o s k o p i y a , 35, 251 (1973) [3] A l i e v , M.R. , and V.M. M i k h a y l o v , J . M o l . S p e c t r o s c , 49, 18 (1974) [4] Anderson, C.H., and N.F. Ramsey, Phys. Rev., 149, 14 (1966) [5] B e r i n g e r , R., and J.G. C a s t l e , J r . , Phys. Rev., 78, 581 (1950) ~ [6] Born, M., and J.R. Oppenheimer, Ann. d. Phys., 84, 457 (1927) [7] Bosch, B.G., and W.A. Gambling, J . B r i t . I.R.E., 24, 389 (1962) ~~ [8] B r i n k , D.M., and G.R. S a t c h l e r , A n g u l a r Momentum, 2nd ed., C l a r e n d o n P r e s s , O x f o r d , 1963 [9] B r i t t , CO., Rev. S c i . I n s t r . , 38, 1496 (1967) [10] B u r g e r s , J.M., K o n i n k l i j k e Akademie van Wetenschappen, P r o c . Roy. N e t h e r l . A c a d . . S c i e n c e s (Amsterdam), 29, 547 (1926) [11] Chasmar, R.P., W.H. M i t c h e l l and A. R e n n i e , J . Opt. Soc. Amer., 46, 469 (1956) [12] Chu, F.Y. and T.Oka, J . Chem. Phys., 60, 4612 (1974) [13] Cohn, S.B., IEEE T r a n s . , MTT-16, 536 (1968) [14] C o l e , A.R.H. an d F . R . Honey, J . M o l . S p e c t r o s c , 55, 492 (1975) ~~ [15] C u r l , R.F., J . Mol. S p e c t r o s c , 48, 165 (1973) [16] C u r l , R.F., T. Oka, and D.S. S m i t h , J . Mol. S p e c t r o s c , 46, 518 (1973) [17] Dorney, A . J . , and J.K.G. Watson, J . M o l . S p e c t r o s c , 42, 135 (1972) [18] Dryden, H.L., and A.M. Kuethe, N a t i o n a l A d v i s o r y Committee f o r A e r o n a u t i c s A n n u a l R e p o r t s , 15, r e p o r t no. 320, p. 359 (1929) [19] F e her, G., B e l l Sys. Tech. J . , 36, 449 (1957) 130 [20] Fox, K., Phys. Rev. L e t t . , 2 7 , 233 (1971) [21] Fox, K., Phys. Rev., A6, 907 (1972) [22] Fox, K., and I . O z i e r , J . Chem. Phys., 52, 5044 (1970) [23] G a r b u t t , H., p r i v a t e communication [24] Gordy, W., Revs. Mod. Phys., 20, 668 (1948) [25] Gordy, W., and R.L. Cook, Microwave M o l e c u l a r S p e c t r a , I n t e r s c i e n c e , New Yo r k , 1970 [26] Gordy, W., W.V. Smith and R.F. Trambarulo, Microwave  S p e c t r o s c o p y , Dover, New Y o r k , 1966 [27] Green, R., p r i v a t e communication [28] G r i e s h e i m e r , R.N., i n Technique o f Microwave Measurements, C.G. Montgomery, ed., MIT R a d i a t i o n Lab. S e r . , v o l . 11, M c G r a w - H i l l , New Y o r k , 1947 [29] Gwinn, W.D., A.C.Luntz, C.H. Sederholm, and R. M i l l i k a n , J . Comput. Phys., 2, 439 (1968) [30] Hecht, K.T., J . Mol. S p e c t r o s c . , 5, 355 (1960) [31] H e r z b e r g , G., I n f r a r e d and Raman S p e c t r a o f P o l y a t o m i c M o l e c u l e s , Van No s t r a n d , New York, T9"45— [32] H e r z b e r g , G., E l e c t r o n i c S p e c t r a and E l e c t r o n i c S t r u c t u r e o f P o l y a t o m i c M o l e c u l e s , Van N o s t r a n d , Princeton,~T966 [33] H i l i c o , J.C., and M. Dang-Nhu, J . P h y s i q u e , 35, 527 (1974) ~~ [34] H o l t , C.W., M.C.L. G e r r y , and I . O z i e r , Phys. Rev. L e t t . , 31, 1033 (1973) [35] H o l t , C.W., M.C.L. G e r r y , and I . O z i e r , Can. J . Phys., 53, 1791 (1975) [36] Hougen, J.T., J . Chem. Phys., 39, 358 (1963) [37] Hougen, J.T., J . Chem. Phys., 55, 1122 (1971) [38] Hougen, J.T., i n MTP I n t e r n a t i o n a l Review o f S c i e n c e  P h y s i c a l C h e m i s t r y , S e r i e s Two, S p e c t r o s c o p y , D.A. Ramsay, ed., t o be p u b l i s h e d [39] Hughes, R.H., and E.B. W i l s o n , Phys. Rev., 71, 562 (1947) ~~ [40] Jahn, H.M., P r o c . Roy. Soc. (London), A168, 469 (1938) 131 [41] Johnson, S.L., B.H. S m i t h , and D.A. C a l d e r , P r o c . IEEE, 54, 258 (1966) [42] J o n e s , R.C., Advances i n E l e c t r o n i c s , 5_, 1 (1953) [43] J o n e s , R.C., J . Opt. Soc. Amer., 43, 1 (1953) [44] K i n g , L.V., P h i l . T r a n s . Roy. Soc. London, A214, 373 (1914) [45] K i r s c h n e r , S.M. , and J.K.G. Watson, J . M o l . S p e c t r o s c . , 47, 347 (1973) [46] K i s l i u k , P., and C. H. Townes, M o l e c u l a r Microwave  S p e c t r a T a b l e s , NBS C i r c . 518, Washington, D.C. 'T95IT [47] K l e i n , M.P., and G.W. B a r t o n , J r . , Rev. S c i . I n s t r . , 34, 754 (1963) [48] K r o n i g , R. de L., Z e i t s . f. P h y s i k , 46, 814 (1928) [49] K r o n i g , R. de L., Z e i t s . f . P h y s i k , 50, 347 (1928) [50] Langmuir, I . , Phys. Rev., 34, 401 (1912) [51] L i , S.C., and T.M. Chen, J . A p p l i e d P h y s i c s , 38, 3449 (1967) ~ [52] L i v i n g s t o n , R., P h y s i c o C h e m i c a l E x p e r i m e n t s , M a c m i l l a n , New Y o r k , 1957 [53] Long, M.W., Rev. S c i . I n s t r . , 31, 1286 (1960) [54] Long, M.W., Q. W i l l i a m s , and T.L. W e a t h e r l y , J . Chem. Phys., 33, 508 (1960) [55] L o n g u e t - H i g g i n s , H.C., M o l . Phys., 6, 445 (1963) [56] Lummer and Kurlbaum, V e r h . Phys. Ges., B e r l i n , 17, 106,(1898) — [57] M e l n i k , W.L., and J.R. Weske, ed., Advances i n Hot-Wire  Anemometry, P r o c . I n t ' l Symposium on Hot-Wire Anemometry, U.S. A i r F o r c e O f f i c e o f S c i e n t i f i c R e s e a r c h f i n a l r e p o r t 68-1492 [58] M e rzbacher, E., Quantum M e c h a n i c s , W i l e y , New Y o r k , 1961 [59] M i c h o l e t , F., J . M o r e t - B a i l l y , and K. Fox, J . Chem. Phys., 60, 2606 (1974) [60] M i l a t z , J.M.W., and H.A. van der V e l d e n , P h y s i c a , 10, 369 (1943) 132 [61] M i l l m a n , J . , and H. Taub, P u l s e , D i g i t a l and S w i t c h i n g Waveforms, M c G r a w - H i l l , New Yo r k , 1965 [62] M o r e t - B a i l l y , J . , C a h i e r s Phys., 15, 237 (1961) [63] M o r e t - B a i l l y , J . , J . Mol. S p e c t r o s c . , 15, 344 (1965) [64] M o r e t - B a i l l y , J . , J . Mol. S p e c t r o s c , 50, 483 (1974) [65] M o r e t - B a i l l y , J . , L. G a u t i e r , and J . M o n t a g u t e l l i , J . M o l . S p e c t r o s c . , 15, 355 (1965) [66] M u e l l e r , R., T r a n s . IRE, ED-1, 42 (1954) [67] N a r a t h , A., and W.D. Gwinn, Rev. S c i . I n s t r . , 33, 79 (1962) — [68] Oka, T. , J . Mol. S p e c t r o s c , 48, 503 (1973) [69] O n d r i a , J.G., IEEE T r a n s . , MTT-16, 767 (1968) [70] O z i e r , I . , u n p u b l i s h e d d a t a (1971) [71] O z i e r , I . , Phys. Rev. L e t t . , 27, 1329 (1971) [72] O z i e r , I . , J . Mol. S p e c t r o s c , 53, 336 (1974) [73] O z i e r , I . , p r i v a t e communication [74] O z i e r , I . , and K. Fox, J . Chem. Phys., 52, 1416 (1970) [75] O z i e r , I . , and A. Rosenberg, Can. J . Phys., 51, 1882 (1973) ~ [76] O z i e r , I . , W. Ho, and G. Birnbaum, J . Chem. Phys., 51, 4872 (1969) [77] O z i e r , I . , R. L e e s , and M.C.L. G e r r y , t o be p u b l i s h e d [78] O z i e r , I . , A. Rosenberg, and D.B. L i t v i n , J . Mol. S p e c t r o s c . , 58, 39 (1975) [79] P r i n c e t o n A p p l i e d R e s e a r c h C o r p o r a t i o n , I n s t r u c t i o n  Manual, L o c k - I n A m p l i f i e r , Model 128, P r i n c e t o n , New J e r s e y , 1T72 [80] R a d u z i n e r , D.M., andN.R. G i l l e s p i e , i n P r o c . IEEE -NASA Symp. Short-Term Frequency S t a b i l i t y , NASA-SP80, USGPO, 1964 [81] R e i c h , H.J., P.F. Ordung, H.L. K r a u s s , and J.G. S k a l n i k , Microwave Theory and T e c h n i q u e s , Van N o s t r a n d , P r i n c e t o n , 1953 133 [82] R e i c h , H.J., J.G. S k a l n i k , P.F. Ordung, and H.L. K r a u s s , Microwave P r i n c i p l e s , Van N o s t r a n d , P r i n c e t o n , 1957 [83] R e i c h e , F., and H. Rademacher, Z e i t s . f . P h y s i k , 39, 444 (1926); 41, 453 (1927) ~~ [84] Rose, M.E., E l e m e n t a r y Theory o f A n g u l a r Momentum, W i l e y , New York, 1957 [85] Rosenberg, A., and I . O z i e r , Chem. Phys. L e t t . , 19, 400 (1973) [86] Rosenberg, A., and I . O z i e r , J . Chem. Phys., 5_8, 5168 (1973) [87] Rosenberg, A., and I . O z i e r , Can. J . Phys., 52, 575 (1974) [88] Rosenberg, A., and I . O z i e r , J . M o l . S p e c t r o s c , 56_, 124,(1975) [89] Rosenberg, A., I . O z i e r , and A.K. K u d i a n , J . Chem. Phys., 57, 568 (1972) [90] S o r g e r , G.U., IRE T r a n s . , 1-4, 165 (1955) [91] S t r a n d b e r g , M.W.P., Microwave S p e c t r o s c o p y , Methuen, London, 1954 [92] S t r a n d b e r g , M.W.P., Rev. S c i . I n s t r . , 43, 307 (1972) [93] Sugden, T.M. , and C.N. Kenney, Microwave S p e c t r o s c o p y  o f Gases, Van N o s t r a n d , London, 1965 [94] Symon, K.R., M e c h a n i c s , 2nd ed., Addison-Wesley, Reading, M a s s a c h u s e t t s , 1960 [95] T a r r a g o , G. , M. Dang-Nhu, and G. P o u s s i g u e , C R . Acad. S c i . , B278, 207 (1974) [96] Tinkham, M., Group Theory and Quantum M e c h a n i c s , M c G r a w - H i l l , New York, TMU [97] Townes, C.H., and A.L. Schawlow, Microwave S p e c t r o s c o p y , M c G r a w - H i l l , New York, 1955 [98] Wang, S . C , Phys. Rev., 34, 243 (1929) [99] Watson, J.K.G., J . Chem. Phys., 46, 1935 (1967) [100] Watson, J.K.G. , J . M o l . S p e c t r o s c , 40, 536 (1971) [101] Watson, J.K.G., J . Mol. S p e c t r o s c , 50, 281 (1974) [102] Watson, J.K.G., J . Mol. S p e c t r o s c , 55, 498 (1975) [103] Wilson, E.B., J. Chem. Phys., 3, 276 (1935) [104] Wilson, E.B., J.C. Decius, andP.C. Cross, Molecular Vibrations, McGraw-Hill, New York, 1955 [105] Wollrab, J.E., Rotational Spectra and Molecular  Structure, Academic Press, New York, 1967 [106] Y i , P., I. Ozier and C.H. Anderson, Phys. Rev., 165, 92 (1968) 

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